Handbook of Magnetic Materials, Volume 5 Elsevier, 1990 Edited by: E.P Wohlfarth and K.H.J. Buschow ISBN: 978-0-444-87477-1
by kmno4
PREFACE TO VOLUME 5 This Handbook on the Properties of Magnetically Ordered Substances, Ferromagnetic Materials, has a dual purpose. As a textbook, it is intended to help 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, the volumes of the Handbook are 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 materials science. The original aim of Peter Wohlfarth was to combine new developments in magnetism with the achievements of earlier compilations of monographs, to produce a worthy successor to Bozorth's classical and monumental book Ferromagnetism. It is mainly for this reason that Ferromagnetic Materials was chosen as title for the Handbook series, although the latter aims at giving a more complete cross-section of magnetism than Bozorth's book. Here one has to realize that many of the present specialized areas of magnetism were non-existent when Bozorth's book was first published. Furthermore, a comprehensible description of the properties of many magnetically ordered materials can hardly be given without considering, for instance, narrow-band phenomena, crystal-field effects or the results of band-structure calculations. For this reason, Peter Wohlfarth and I considered it desirable that the Handbook series be composed of articles that would allow the readers to orient themselves more broadly in the field of magnetism, taking the risk that the title of the Handbook series might be slightly misleading. Publication of the Volumes of the Handbook cannot be considered as having proceeded by a process of continuous growth. Volumes 1 and 2 first appeared in 1980, and were followed by Volume 3 in 1982. All three Volumes have been reprinted in the meantime and many of the articles contained in them have frequently been quoted in the scientific literature as providing an authoritative description of the achievements made in a given subfield of magnetism. In the
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PREFACE TO VOLUME 5
early 1980's the interest of the scientific community in magnetism declined considerably. Many a scientist had moved to greener pastures and, in a way, this had hampered completion of Volume 4, published in 1988. However, at the time that I joined Peter Wohlfarth as co-editor of the Handbook series early in 1986, we both had the impression that there was a pronounced increase in interest from the scientific as well as from the technological side, to the extent that one could speak of a revival of magnetism. Topics like the Kondo effect, spin glasses and valence fluctuations had kept the fires burning, and a considerable proliferation in research effort occurred in other areas like amorphous magnetism, permanent magnets and magneto-optics. The magnetism of thin films and multi-layers and heavy-electron systems became novel topics. Improvements in band-structure calculations and the possibility of applying these calculations to more complicated structures started to play an important role in the understanding of many features of magnetism. All these facts made it necessary that we again pose the question 'what is magnetism and where does it go?' The outcome of such considerations was that we eventually decided to include chapters in the Handbook series that were able to provide the readership with an insight into modern trends in magnetism and new achievements in this area. Several such topics have already been mentioned above. Other topics of this kind dealt with the increased activity and investigations of the magnetism of magnetic superconductors and with investigations of the magnetic properties of hydrides. We also felt that there should be an account of the progress that had been made in the understanding of first-order magnetic processes and of quadrupolar interactions in 4f systems and their role in magnetic ordering and in magneto-elastic effects. The magnetism of strongly enhanced itinerant alloys and compounds and the magnetism of Invar alloys already had a long-standing tradition. However new achievements were made in this area, and these were often obtained with the more sophisticated experimental techniques available nowadays. This also made it necessary to include chapters covering the progress made in these latter fields, and many of these latter topics are contained in this volume. It will be clear that we had to abandon the ideal of restricting the Handbook series to four volumes. The large number of topics will require several more volumes, the more so since the revival of magnetism is still going on. This is, for instance, manifest from the extraordinarily large number of contributions submitted for presentation at the International Conference on Magnetism, held in Paris in July 1988. The number of contributions (about 1300) exceeds by far the number of contributions (about 900) presented at the preceding ICM Conference held in San Francisco in 1985. This conference has also focused the attention on several new fields of interest such as the magnetism of high-temperature superconducting compounds and dilute magnetic semiconductors while it stressed the intimate connection existing between fundamental magnetism and high-density magnetic recording. Also these topics will be accommodated in future volumes of this Handbook series. The task to provide the readership with novel trends and achievements in magnetism would have been extremely difficult without the professionalism of the
PREFACE 'TO VOLUME 5
vii
North-Holland Physics Division of Elsevier Science Publishers and I would like to thank Peter de Ch~tel, Anita de Waard, Bas van der Hoek and Willem jan Maas for the great help and expertise. K.H.J. Buschow Philips Research Laboratories
CONTENTS Preface to V o l u m e 5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
- . .
v ix
C o n t e n t s of V o l u m e s 1 - 4 . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
List of c o n t r i b u t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
1. Q u a d r u p o l a r I n t e r a c t i o n s a n d M a g n e t o - e l a s t i c Effects in R a r e - e a r t h IntermetaUic C o m p o u n d s P. M O R I N a n d D. S C H M I T T . . . . . . . . . . . . . . . . . . . . . . 2. M a g n e t o - o p t i c a l Spectroscopy of f - e l e c t r o n Systems W. R E I M a n d J. S C H O E N E S . . . . . . . . . . . . . . . . . . . . . . 3. INVAR: M o m e n t - v o l u m e Instabilities in T r a n s i t i o n Metals a n d Alloys E.F. WASSERMAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Strongly E n h a n c e d I t i n e r a n t I n t e r m e t a l l i c s a n d Alloys R E . B R O M M E R and J.J.M. F R A N S E . . . . . . . . . . . . . . . . . 5. F i r s t - o r d e r M a g n e t i c Processes G. A S T I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. M a g n e t i c S u p e r c o n d u c t o r s O. F I S C H E R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 133 237 323 397 465
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
551
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . .
577
........
Materials I n d e x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
585
CONTENTS OF VOLUMES 1-4 Volume 1 1. 2. 3. 4. 5. 6. 7.
Iron, Cobalt and Nickel, b y E . P . Wohlfarth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Transition Metal Alloys: Spin Glasses, b y J . A . M y d o s h a n d G . J . N i e u w e n h u y s . . . Rare Earth Metals and Alloys, b y S. L e g v o l d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rare Earth Compounds, b y K . H . J . B u s c h o w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Actinide Elements and Compounds, b y W. T r z e b i a t o w s k i . . . . . . . . . . . . . . . . . . . . . . . . . . Amorphous Ferromagnets, by F . E . L u b o r s k y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetostrictive Rare Earth-Fe 2 Compounds, by A . E . Clark . . . . . . . . . . . . . . . . . . . . . . .
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, b y G . Y . Chin a n d J . H . W e r n i c k . . . . . . . . . . . . . . . . Ferrites for Non-Microwave Applications, b y P.I. Slick . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microwave Ferrites, b y J. Nicolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystalline Films for Bubbles, b y A . H . E s c h e n f e l d e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amorphous Films for Bubbles, b y A . H . E s c h e n f e l d e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recording Materia!s, b y G. Bate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferromagnetic Liquids, b y S.W. Charles a n d J. P o p p l e w e l l . . . . . . . . . . . . . . . . . . . . . . . . .
1 ~. .
55 189 243 297 345 381 509
Volume 3 1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry and Technology, b y U. E n z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Permanent Magnets; Theory, b y H. Zijlstra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R . A . M c C u r r i e . . . 4. Oxide Spinels, b y S. K r u p i 6 k a a n d P. N o v 6 k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite Structure, by H. K o j i m a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. 7. 8. 9.
Properties of Ferroxplana-Type Hexagonal Ferrites, b y M . S u g i m o t o . . . . . . . . . . . . . . . . . Hard Ferrites and Plastoferrites, b y H. Stiiblein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulphospinels, by R . P . van Stapele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transport Properties of Ferromagnets, by I . A . C a m p b e l l a n d A . Fert . . . . . . . . . . . . . . . . xi
1
37 107 189 305 393 441 603 747
xii
CONTENTS OF VOLUMES 1-4
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 lntermetallic Compounds, by K . H . J . Buschow.
1
131 211 309 493
chapter 1 QUADRUPOLAR INTERACTIONS AND MAGNETO-ELASTIC EFFECTS IN RARE EARTH INTERMETALLIC COMPOUNDS
P. MORIN and D. SCHMITT Laboratoire Louis N@el* C.N.R.S. B.P. 166 X 38042 Grenoble Cedex France
*Associated with Universit6 Joseph Fourier, Grenoble
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. I~ t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Fc rl~ alL, m . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. T h e c t b i c H a n n i l t c n i a n . . . . . . . . . . . . . . . . . . . . . 2.1.1. G e n e r a l e ~ p l e s s i o n . . . . . . . . . . . . . . . . . . . . 2.1.2. H a r m o n i c d e s c r i p t i o n of m a g n e t o s t r i c t i o n . . . . . . . . . . . . . 2.1.3. T h e f e t a l q ~ a & u p o l a r H a m i l ( o n i a n . . . . . . . . . . . . . . . 2.2. T r e a t m e n t of the c u b i c H a m i l t c n i a n . . . . . . . . . . . . . . . . . 2.2.1. F e r t m b a t f o n t h e o r y . . . . . . . . . . . . . . . . . . . . 2.2.2. C e n e r a l i z a t i o n of l h e s u s c e p t i b i l i t y f o r m a l i s m . . . . . . . . . . . . 2.2.3. B e h a v i o u r of l h e ~ a r i c u s s u s c e p t i b i l i t i e s . . . . . . . . . . . . . 2.2.4. M a g n e t i c a n d q u a & u p o l a r t r a n s i l i o n s . . . . . . . . . . . . . . 2.3. E x t e n s i o n to l o w e r s y m m e t r i e s ( h e x a g o n a l a n d t e t r a g o n a l ) . . . . . . . . . 2.3.1. T h e H ~ m i l t o n i ~ n . . . . . . . . . . . . . . . . . . . . . . 2.3.2. H a r m o n : ' c d e s c r i p t i o n of 1Tagnetostr c t i c n . . . . . . . . . . . . . 2.3.3. P e l t u r b a t i o n tl;eo~y . . . . . . . . . . . . . . . . . . . . 2.3.4. C o m p a r i s o n w i t h c u b i c s y m m e t r y . . . . . . . . . . . . . . . . 2.4. A p e c u l i a r c a . e : the semi-classical t r e a l m e n t . . . . . . . . . . . . . . 2.4.1. T h e p a r a m ~ g n e l i c lzhase . . . . . . . . . . . . . . . . . . . 2.4.2. T h e c r d e r e d p h a s e . . . . . . . . . . . . . . . . . . . . . 3. E x p e r i m e n t a l e v i d e n c e . . . . . . . . . . . . . . . . . . . . . . . 3.1. Q u a d l u p o l a r o r d e r i n g s . . . . . . . . . . . . . . . . . . . . . 3.1.1. F e r i c q u a d r ~ p o l a r older:'ng . . . . . . . . . . . . . . . . . . 3.1.1.1. T m Z n . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2. T m C d . . . . . . . . . . . . . . . . . . . . . . 3.1.1.3. C e A g . . . . . . . . . . . . . . . . . . . . . . 3.1.2. A n t : f e r l c q u a d ~ p o / ~ r o l d e r i n g . . . . . . . . . . . . . . . . 3.1.2.1. C e B 6 . . . . . . . . . . . . . . . . . . . . . . 3.1.2.2. T m G a 3 . . . . . . . . . . . . . . . . . . . . . 3.1.2.3. P I P b 3 . . . . . . . . . . . . . . . . . . . . . . 3.1.3. S t r u c t u r a l l r a n s i t i o n s p o s s i b l y of q u a d r u p o l a r o r i g i n . . . . . . . . . 3.1.3.1. P r C u z . . . . . . . . . . . . . . . . . . . . . . 3.1.3.2. U P d 3 . . . . . . . . . . . . . . . . . . . . . . 3.2. N a t u r e of q u a d r u p o l a r a n d m a g n e t i c t r a n s i t i o n s . . . . . . . . . . . . . 3.2.1. Q u a d r u p o l a r t l ~ n s t i o n s . . . . . . . . . . . . . . . . . . . 3.2.2. M a g n e t i c o r d e r i n g in t h e p r e s e n c e of q u a d r u p o l a r i n t e r a c t i o n s . . . . . . . . . . . . . . . . . 3.2.2.1. F i r s t - o r d e r t r a n s i t i o n s a n d tricriti.cality
5 6 7 7 11 12 12 13 16 17 20 24 24 27 28 30 30 30 32 34 34 34 35 36 36 38 38 39 40 41 41 42 43 43 43 44
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
3.2.2.2. Second-order transition in PrMg 2 due to negative qua drupol a r interactions . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Magnetic ordering in the q u a d r u p o l a r phase . . . . . . . . . . . . 3.2.3.1. C e A g , CeB 6 . . . . . . . . . . . . . . . . . . . 3.2.3.2. T m Z n . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Presence of additional couplings . . . . . . . . . . . . . . . . 3.2.4.1. DySb and rare earth monopnictides . . . . . . . . . . . . 3.2.4.2. The RA12 series . . . . . . . . . . . . . . . . . . 3.3. D e t e r m i n a t i o n of the q u a d r u p o l a r p a r a m e t e r s from susceptibility techniques . 3.3.1, Elastic constants . . . . . . . . . . . . . . . . . . . . . 3.3.2, Parastriction . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Third-order p a r a m a g n e t i c susceptibility . . . . . . . . . . . . . 3.4. Effects on the magnetostriction . . . . . . . . . . . . . . . . . . 3.4.1. Magnetostriction of dilute compounds . . . . . . . . . . . . . . 3.4.2. Magnetostriction in the ordered phase . . . . . . . . . . . . . . 3.5. Effects on the magnetization processes . . . . . . . . . . . . . . . . 3.5.1. P a r a m a g n e t i c phase . . . . . . . . . . . . . . . . . . . . 3.5.2. O r d e r e d phase . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Effect on the direction of easy magnetization . . . . . . . . . . . 3.6. Effects on the magnetic structures . . . . . . . . . . . . . . . . . 3.6.1. Multi-axial structures . . . . . . . . . . . . . . . . . . . 3.6.2. I n c o m m e n s u r a t e magnetic structures . . . . . . . . . . . . . . 3.7. Effects on magnetic excitations . . . . . . . . . . . . . . . . . . 3.8. Two-ion anisotropic magneto-elasticity . . . . . . . . . . . . . . . . 3.9. Isotropic magneto-elasticity and pressure effects . . . . . . . . . . . . . 3.9.1. Two-ion contribution . . . . . . . . . . . . . . . . . . . 3.9.2. Single-ion contribution . . . . . . . . . . . . . . . . . . . 3.9.3. Pressure effects . . . . . . . . . . . . . . . . . . . . . 3.10. Magneto-elasticity in the presence of lattice instability . . . . . . . . . . . 4. Magneto-elastic and pair interaction coefficients in rare earth intermetallic series . . . . 4A. CsCl-type structure compounds . . . . . . . . . . . . . . . . . . . 4.1.1. R Z n series . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. R C d series . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. R A g series . . . . . . . . . . . . . . . . . . . . . . . 4.1.4, R C u series . . . . . . . . . . . . . . . . . . . . . . . 4.1.5, R M g series . . . . . . . . . . . . . . . . . . . . . . . 4.2. NaCl-type structure compouods . . . . . . . . . . . . . . . . . . . 4.3. AuCu3-type structure compounds . . . . . . . . . . . . . . . . . . 4.4. Cubic Laves phase compounds . . . . . . . . . . . . . . . . . . . 4.4.1. RA12 compounds . . . . . . . . . . . . . . . . . . . . . 4.4.2. R M 2 compounds (M = Mg, Ni, Co) . . . . . . . . . . . . . . . 4.4.3. R F e 2 compounds . . . . . . . . . . . . . . . . . . . . . 4.5. Hexagonal CaCus-type structure compounds . . . . . . . . . . . . . . 4.5.1. RNi 5 compounds . . . . . . . . . . . . . . . . . . . . . 4.5.2. RCo 5 compounds . . . . . . . . . . . . . . . . . . . . . 4.6. Dilute rare earth systems . . . . . . . . . . . . . . . . . . . . . 4.6.1. R a r e earths diluted in cubic noble-metal hosts . . . . . . . . . . . 4.6.2. R a r e earths diluted in cubic pnictides . . . . . . . . . . . . . . 4.6.3. Other dilute systems . . . . . . . . . . . . . . . . . . . . 4.7. Miscellaneous compounds . . . . . . . . . . . . . . . . . . . . 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. G e n e r a l analysis . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Coherency of the various determinations . . . . . . . . . . . . .
3
49 49 50 50 50 50 52 53 54 56 57 61 61 64 67 67 69 70 72 72 75 77 79 79 80 80 81 82 83 83 84 85 85 90 92 93 95 95 95 98 100 102 102 102 106 106 107 110 111 113 113 113
4
P. M O R I N and D. SCHMITI"
5.1.2. Comparison of the magneto-elastic coefficients . . . . . . . . . . . 5.1.3. Comparison of the quadrupolar pair-interaction coefficients . . . . . . . 5.2. Origin of the quadrupolar interactions . . . . . . . . . . . . . . . . 5.2.1. The one-ion magneto-elastic coupling . . . . . . . . . . . . . . 5.2.2. Quadrupolar pair interactions . . . . . . . . . . . . . . . . . 5.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Symmetrized Stevens operators for the cubic symmetry . . . . . . . . . . A.2. Fourth- and sixth-rank one-ion magneto-elastic Hamiltonian ~ME~ . . . . . . . A.3. Perturbation theory: the tetragonal mode 3' . . . . . . . . . . . . . . A.4. Perturbation theory: the trigonal mode e . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114 116 117 117 119 120 122 122 123 124 125 126
1. Introduction
The study of multipolar interactions concerning the unfilled 4f shell has known a great development in rare earth (R) intermetallic compounds. It involves the complex problems of both the crystalline electric field (CEF) acting on a 4f ion and the pair interactions between different ions. The electron-lattice coupling can drive very characteristic magneto-elastic features, in particular in elastic constants and in magnetostriction, through symmetry lowering modes which depend on the 4f electronic states. According to the Jahn-Teller theorem, the magneto-elastic modulation of the CEF corresponds to a gain of energy larger than the elastic energy lost at the thermodynamic equilibrium. These properties have been extensively studied in the past in many systems exhibiting an orbital degeneracy as reported for cubic spinels (Englman and Halperin 1970, Englman 1972) in particular mixed chromites (Kataoka and Kanamori 1972). Gehring and Gehring (1975) and Melcher (1976) have provided the literature with very complete reviews of the cooperative Jahn-TeUer transitions occurring in rare earth insulators within the tetragonal zircon structure. All these reviews show the fundamental importance of studying elastic constants. Indeed they couple, as thermodynamical variables, the strain to the ultrasonic stress in the presence of a quadrupolar moment, describing the orbital character of the 4f shell. Another important feature is the large success met by the mean-field approximation for describing thoroughly the structural transitions. More recently, very large symmetry lowerings, which are mainly of one-ion origin have been observed to occur at the magnetic ordering in rare earth intermetallics (L6vy 1969, Morin et al. 1979). In the example of RFe 2 compounds which are ordered at room temperature, they are large enough to motivate industrial applications as magnetostrictive transducers which are able to compete with ceramics (Clark 1980). In all the series, they are associated with large softenings of elastic constants in the non-ordered phase (Mullen et al. 1974, Moran et al. 1973). From a theoretical point of view, they have required the development of a quantum treatment which takes into account all the characteristics of the 4f wave functions, as will be discussed in the present chapter. However, in R intermetaUics an additional coupling has been revealed, mediated from one site to the others by the conduction electrons. This quadrupolar pair interaction may dominate the magneto-elastic coupling and drive a
6
P. MORIN and D. SCHMITI"
quadrupolar ordering in the paramagnetic phase, as occurs in TmZn and TmCd (Morin et al. 1978b). According to the sign of the quadrupolar pair interactions, ferro- or antiferroquadrupolar orderings have been observed. The symmetry lowering is then, when it exists, a consequence of the quadrupolar symmetry through the magneto-elastic coupling. Such compounds are not stricto sensu Jahn-Teller compounds, because then the magneto-elastic coupling is the driving mechanism. In the presence of dominating bilinear interactions, which is the majority of cases, quadrupolar interactions act on the minimization of the total free energy, and magnetic properties cannot be understood without them in several series (Giraud et al. 1985). For different experimental techniques (Ltithi 1980a), the magneto-elasticity remains the most appropriate probe for determining the quadrupolar pair interactions. This chapter is organized as follows. First, the quantum Hamiltonian is built, which describes both the one- and two-ion, spin and quadrupolar couplings in systems of cubic, tetragonal and hexagonal symmetries. The power of the susceptibility formalism, which allows independent determinations of the various coefficients, is emphasized (section 2). The main experimental evidence of quadrupolar orderings is presented in section 3, before giving the quadrupolar consequences on the magnetic properties. The experimental techniques which are based on the susceptibility formalism of section 2 are illustrated. Section 4 gives a review of the magneto-elastic and pair interaction coefficients which have been determined in the rare earth intermetaUics. Due to the anisotropic character of the magneto-elasticity, studies on single crystals are of fundamental importance and these are the only ones considered here. Section 5 gives an analysis of all these results from both a macroscopic and a microscopic point of view.
2. Formalism
The existence of non-quenched orbital moments in rare earth ions is the source of their anisotropic magnetic properties. A classical description of these properties is no longer valid. In a quantum formalism, the 4f shell is represented by wavefunctions expanded in the IJ, M:) basis. The appropriate Hamiltonian, acting on these wave functions, depends on the point symmetry of the rare earth site under consideration. In addition to the crystalline electric field (CEF) Hamiltonian, any kind of interactions acting on the rare earths or between them (Heisenberg coupling, magneto-elasticity, quadrupolar pair interactions . . . . ) must be considered. In the literature, most of the experimental results reported on the magnetoelastic properties concerns rare earth compounds with cubic symmetry, because for these compounds the number of parameters required for analyzing these properties is the smallest. In the following two sections, we extensively develop the formalism and the treatment of magneto-elasticity and quadrupolar interactions for cubic symmetry. The case of lower symmetries (hexagonal and tetragon-
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
7
al) will be discussed more briefly in section 2.3, while the connection with the classical formalism will be made in section 2.4. 2.1. The cubic Hamiltonian 2.1. i. General expression
The Hamiltonian appropriate for cubic symmetry includes several terms. By using the operator-equivalent method (Stevens 1952) they are written as: ffb°= ~CEF + ~°Z + ~B + ~OQ+ '~ME1 + ~ME2 "l- (Eel + E B + EQ q- EME2) .
(I)
The different terms represent successively the crystal field term, the Zeeman coupling, the Heisenberg-type bilinear interaction, the two-ion quadrupolar term, the magneto-elastic coupling, the elastic energy, and bilinear and quadrupolar corrective energy terms. Each term is described in detail below. The crystalline electric field Hamiltonian: Y(CEF According to different authors, this Hamiltonian takes one of the following forms. The x, y and z axes have been chosen as the four-fold axes in the cubic crystal: ~ C E F = B4(O°4 + 5 0 4 )
-]- B6(0°6 - 2104)
= A4 ( r 4 ) ~ 1 ( 0 40+ 5 0 4 ) + A 6 ( r 6 ) y 1 ( O ~ - 2 1 0 4 = __
Wx (004 -~- 5 0 ~ ) +
F4
W(1 - - I X l )
F6
0
)
(2)
4
( 0 6 -- 2 1 0 6 ) .
The O~'s are the Stevens operators in the formulation of Hutchings (1964). 13I and 7i are the 4th- and 6th-order Stevens coefficients, the value of which depends on the rare earth. The CEF parameters A4
and A6(r 6> originate in the electric field acting on the 4f shell, which arises from the surrounding point charges (Hutchings 1964) or from the conduction electrons (Schmitt 1979). In the last formulation, W represents a scale factor, and x the relative proportion of 4th and 6th order CEF terms (Lea et al. 1962). The Zeeman Hamiltonian: Y(z ~ z = --gJtzB H" J ,
(3)
represents the Zeeman coupling between the 4f magnetic moment M = gj/x B( J ) and the internal field H (applied magnetic field corrected for demagnetization effects). The bilinear Heisenberg-type Hamiltonian: ~ ~fB = -- g J IxBnM " J ,
(4)
8
P. MORIN and D. SCHMITT
represents the Heisenberg-type bilinear interaction, written within the molecular field approximation (MFA), n being the bilinear exchange parameter. An alternative definition for this parameter is 0", with: O* n
=
-C
3kBO* =
(5)
g :2t z B2 J ( J + 1)
where C is the Curie constant. The two-ion quadrupolar Hamiltonian: ~ o In a form analogous to the Heisenberg-type bilinear coupling between magnetic dipoles, a two-ion coupling between 4f quadrupoles is possible (see Schmitt and Levy 1985 and references therein), for which evidence has been found in several rare earth intermetallic compounds (Levy 1973, Levy et al. 1979). According to symmetry considerations, the corresponding Hamiltonian, again in the MFA, may be written as: ~Q = -KY((O°)
O0 + 3 < 0 2 ) O 2 ) - Ke(Pxy + < P y z > G z + ( P z x ) P z x ) •
(6) In this expression, 0~, 022 and the Pi:'s are the second-order (quadrupolar) Stevens operators (Hutchings 1964, Sivardi6re 1975) (see table 1), and K ~ and K ~ are two-ion quadrupolar parameters corresponding to the two linear combinations of quadrupolar operators, which are invariant to all the cubic symmetry operations. As will be discussed below, they are associated with the two anisotropic normal strain modes in cases with cubic symmetry, namely the tetragonal (3/) and trigonal (e) modes. TABLE 1 Expression of the quadrupolar operators. Operator
Expression
0°2
3JZz - J(J + 1)
o~
J~ - s,2 = ~(s~++ :_)
exy
l(JxJy + JyJx) = ~- (J+ - j2_)
--i
Pyz
e zx ~_
1
0 2
2
~(Y~L + L :,) ½(LL+LL)
e
The magneto-elastic Hamiltonian: ggME The magneto-elastic properties of rare earth compounds have been extensively studied since the 1960s (Callen and Callen 1965, Du Tr6molet de Lacheisserie 1970, Du Tr6molet de Lacheisserie et al. 1978). The analysis of these properties employs one-ion and/or two-ion magneto-elastic Hamiltonians, it is usually
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
9
limited to the first order in the strain (harmonic approximation) and contains terms up to the sixth rank for the Stevens operators. Few studies have been made in connection with second-order terms in the strain (anharmonic coupling) (Du Tr6molet de Lacheisserie et al. 1978, Rouchy and Du Tr6molet de Lacheisserie 1979, Wang and Lfithi 1977b) and this second-order magneto-elasticity will be not considered in this chapter. The one-ion magneto-elastic Hamiltonian corresponds to the direct coupling between the deformations of the lattice and the 4f shell. It can be considered as the strain derivative of the crystal field Hamiltonian. The two-ion magnetoelasticity is related to the modification of the two-ion magnetic interactions by the strains. The one-ion magneto-elastic H a m i l t o n i a n : ~fME1
Group theory gives us the correct procedure for constructing this Hamiltonian. First, one writes linear combinations of the first-order strain components eq which transform according to the three irreducible representations F1, F3 and F5 or a, y, e (see table 2) (Du Tr6molet de Lacheisserie 1970). Second, one considers the linear combinations of the Stevens operators of rank I (l = 2, 4 or 6) which transform also according to the irreducible representations of the cubic group (see appendix 1). Finally, one obtains all the possible invariants by summing the direct products of the different combinations which belong to the same representation. A magneto-elastic coefficient then is associated with each invariant. For l = 2 , two magneto-elastic terms are obtained, coming from the two representations /"3(7) and Fs(e), WMEa(I = 2) = Y ~ m ( l = 2) + aq~°hEl(l 2 ) , -----
(7)
with ~IEI(/
= 2)
e
=
--
~, 2 X/3e202),
B~(e~O ° +
Y(MEI(I = 2) = - B
8
e
e
(e,Pxy + e2Py ~ + e3Pzx ) .
(8)
(9)
Thus there are only two normal strain modes associated with symmetry lowerings of the cubic cell, one for a tetragonal and the other for a trigonal strain. TABLE 2 Normalized, syrnmetrizedstrains for the cubic symmetry. Representation
Straincomponents
r~
E° = ~
1"5
E1 = %¢/2 x y '
1
(Ex~+ ~, + ~zz)
E2 = " ~ Eyz'
10
P. MORIN and D. SCHMITT
The l = 4 and l = 6 one-ion magneto-elastic Hamiltonians have also been given (Creuzet and Campbell 1981, Niksch et al. 1982). Their full expressions are given in appendix 2.
The two-ion magneto-elastic Hamiltonian: ~ME2 Although any kind of two-ion interaction can a priori depend on the strains, only the bilinear coupling will be considered here. The same theoretical considerations as above allow us to write the corresponding Hamiltonian as the sum of an isotropic term and two anisotropic ones: ~ME2
=
o~ ~ME2
+
ffb'O~vlE2 "[- ~ M E 2
(10)
"
The first term, (11)
~ME2 = -D~e~ ( J ) " J ,
is related to the volume dependence of the isotropic bilinear exchange interactions. On the contrary, the two other terms,
Yg~E2 =
-Dr[e~'(2(Jz)J~
)~ME2
-De[eel( ( Jx)Jy + ( Jy) Jx) + 1=:2(( J,)Jz + ( Jz)Jy)
-
(L)L
- (4)4)
+ v3e~((L)L
- (4)4)],
(12) =
+ e3((Jz)Jx + (Jx)Jz)],
(13)
are associated with the appearance of anisotropic bilinear interactions under a strain. As for the bilinear coupling itself [eq. (4)] the MFA has been used for this two-ion magneto-elasticity.
The elastic energy: Fez The elastic contribution to the free energy may be expressed as a function of strains and symmetrized elastic constants. In cubic symmetry, the three normal strain modes a, y, e appear in the expression of the elastic energy 1 .,-,al a x 2 1 3' Y 2 Eo, = ~,~ot~ ) + ~c0[(~,) + ( ~ D 21+ 1 Co[(e~) ~ ~ 2 + (e~) 2 + (e~)2].
(14)
The C~ are background elastic constants without magnetic interactions; they are related to the conventional constants by a 0 C O ~-- C 1 1 +
0
.
2C12 ,
C~
:
0 C11 -
0 C12
. ,
s
0
C O-= 2C44
.
(15)
The corrective energy terms E~, EQ, EMe 2 These terms arise from the fact that each rare earth pair is counted twice in the MFA treatment of two-ion magnetic couplings. This occurs for the bilinear [eq. (4)], the quadrupolar [eq. (6)] interactions, and for the two-ion magneto-elastic
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
11
terms as well [eqs. (11)-(13)]: E B = ½nM 2 ,
(16)
Eo=½K'((o°)Z+3(OZ2>z)+½K~(
(17)
EME2 = ½D"e~<J>2 + ½DV[e~(2<jz>2_ <Jx>2_ <jy>2)
+ V~E~(<jx>2__ <jy>2)]_~_De[E~<Jx><jy>q_E;<Jy><Jz> ~- e;<Jz) <J~>] .
(18)
2.1.2. Harmonic description of magnetostriction Starting from the above Hamiltonian, the actual magnetostriction, i.e. the equilibrium values for the various strains e~'s, may be obtained by minimizing the free energy F with respect to each of them: OF/Oe ~= O. That leads to the following general expressions for the 6 symmetrized strains for cubic symmetry ~, 0 + 5 0 4 > + B(6) Ea = ~1 [B(4)
./
1
+ ½V,~<j>2]
y,1
+ ½D~(2(jz)2_ <jr>2_ (jr>2)], =
(19)
(20)
'Y T,2 1 [~/-~B.V+ B~4)+ B(6)(O 6 >
- -
+ ½V~D~((j~>2_ <jy)2)],
(21)
~. l~l [()el,l\ ~2 81 = ~1 [Be+ B(4) + u(6)\~ 6 / -}-B(6)
+ m~(Jx)(Jy>], 1
E2 : ~ 0
(22)
[Be(PYz>+ B (~4 ) < O 4 ' 2 >
l:?a:l / /-'}el,2 ~ e2 062,2> + u(6)\"6 / -~- B(6)<
+D~(J,)(Jz>],
(23)
1 ~ el O;1,3 ]~,52 [ /~e2,3 k e 3 = ~ g [ B ~ ( P ~ > + B(4) -4- B(6)< > + u(6)\,j6 / Co
+ D~(L)
(24)
The macroscopic change of length of a magnetic lattice with cubic symmetry may then be written as a-
8l 1 1 l - V~ e~ + ~
v
2
2
e1(2/3z -/3x - / 3 2 ) + ~
+ V~(e~/3~/3y + e~/3,/3 z + 23/3~/3x).
1
~
2
e2(/3" - / 3 2 ) (25)
12
P. MORIN and D. SCHMITF
To obtain a reliable description of the various magnetic and elastic properties, the expectation values of the Stevens operators must be treated in a self-consistent manner. In particular, this yields the actual temperature dependence of the various strains [eqs. (19)-(25)] in the paramagnetic as well as ordered phases.
2.1.3. The total quadrupolar Hamiltonian By including the preceding equilibrium values of the strains in the starting Hamiltonian, one can write it in a form which does not explicitly depend on the e~'s, but only on the magneto-elastic coefficients and on the expectation values of the various Stevens operators. In most of the cases reported in the literature, one-ion magneto-elasticity involving the quadrupolar operators appears as the dominant contribution. This justifies focusing our attention on it in the following. The relative importance of other terms from an experimental point of view will be discussed later, in particular in sections 3.8 and 3.9. Keeping only the B " contribution leads to an expression for NMEI(1 = 2) which appears to be formally undistinguishable from the two-ion quadrupolar Hamiltonian No; both terms can be grouped together in the total quadrupolar Hamiltonian NOT fft°QT= -G'Y[(O~)O°2 + 3 ( 0 2 2 ) 0 2] - G [( p xy)Pxy ~- (Pyz)Pyz -I- (P~x)P~x] (26) where the new total quadrupolar parameters G ~ are defined as G ~ - (B~) 2 -
-
cg
- - -
C;
~IE ~- K ~ '
,
(27)
+ K ~ = GME + K ~ .
(28)
+
K "~ =
G
E
Equation (26) provides an alternative expression for the quadrupolar couplings, which may be used instead of eqs. (6), (8) and (9), every time that the e "'s are not explicitly needed.
2.2. Treatment of the cubic Hamiltonian The treatment of the Hamiltonian described above can be carried out in two ways: i) First, by diagonalizing the full Hamiltonian in a self-consistent manner. This allows the description of all the magnetization and magnetostriction processes, in particular in the ordered phase or under large external stresses (magnetic field or uniaxial stresses) as shown in section 3. ii) The second possibility is to apply perturbation theory to the above Hamiltonian in order to describe the magnetic and magneto-elastic properties in the
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
13
non-ordered phase and under small external stress. There are two advantages working in the cubic paramagnetic phase. First, analytical expressions can be obtained for the various susceptibilities connecting the various variables. Second, any strain mode can be systematically studied by applying external stress along the appropriate direction, whilst in the ordered phase only the spontaneous state may be usually achieved because of the strong anisotropy effects. In this section, the susceptibility formalism is developed by applying perturbation theory to the above Hamiltonian which is limited to second-rank quadrupolar terms. An analytical expression will be derived for the free energy, defining the various susceptibilities which appear in the analysis of the magnetization, parastriction and elastic constants measurements. Finally, the behaviour of these susceptibilities and their role in describing phase transitions will be described. 2.2.1. Perturbation theory
The zeroth-order Hamiltonian Y(0 consists of the CEF term [eq. (2)], the diagonalization of which provides the zeroth-order energies and CEF wave functions. According to the direction of external stress, the two normal cubic strain modes may be fully decoupled. Therefore they are considered separately. Tetragonal m o d e y
For a magnetic field applied along a four-fold axis, e.g. z, the only non-zero terms are M = g H a a 3 ( J z ) , Q = ( 0 2°) and e~. The perturbating Hamiltonian is then written as: YU1 = - g j l - % ( H + n M ) J z - (B3,e~ + K3,Q)O°2 + ~-~'0~,,'~3,to3,,= )~ + ½ n M : + -½K3,Q2 .
(29) Perturbation theory has to be carried out up to the second order for e~ and Q and up to the fourth order for H and M in order to derive analytical expression of the generalized Gibbs free energy F = - k B T l n Z. Here Z = T r e x p ( - / 3 Y ( ) is the partition function, k B the Boltzman constant and/3 = 1/(k B T ) . The details of the calculation may be found in appendix 3. The free energy expansion takes the form F3,= FVo - ½Xo(H + n M ) : -
~X3,tt~ 1 ,,,3, ea3, + K3,Q) 2
1. (3)(H + n M ) 4 - (2)tB3,e3,, + K 3 , Q ) ( H + n M ) 2 - zx3, \ •
- - , , I [ 3,
+ ~a"-'o,,ele~'t^3,'2)+ l n M 2 + 1K3,Q2 + . . . .
(30)
This expression includes four CEF single-ion susceptibilities which are defined without any magnetic or quadrupolar interactions, and depend only on the non-perturbed CEF state, i.e. on the cubic CEF level spacings and on the matrix elements of Jz and O ° between cubic CEF wavefunctions (see appendix 3). X0 is the usual (first-order) magnetic susceptibility, X3, is a quadrupolar strain susceptibility, X3, - (2) is a quadrupolar field-susceptibility which couples the quadrupolar
14
P. M O R I N
a n d D. S C H M I T T
operator O ° to the magnetic field, xz, - (3) is the third-order magnetic susceptibility which is related to the initial curvature of the magnetization curves and, finally, F 0 is the zeroth-order free energy corresponding to ~fcEF alone. The equilibrium values for M, e~ and Q are found by minimizing the generalized free energy F3, OF v
.
.
.
OM
OF3,
.
Oe~
OF3,
.
.
0.
oO
(31)
This yields a system of three coupled equations, the solution of which leads to the following relations M=
(3)11 u3 , XMH + XM
(32)
.
B ~'
~=Q ~ C~
(33)
'
Q = xoH 2 ,
(34)
XO
(35)
with XM -- (1 -- nXo) '
["~
1
X(M 3)=
(1
- nx0) 4
+ 203,
xv
(x~) ~ ] 1 - GrX3,J '
(36)
X(2) 3'
Xo
9
(1 - ~ X o ) 2 ( 1 - 6 % )
(37)
Note that we find for the equilibrium value of e~ the same expression as that found without recourse to perturbation theory [eq. (20)]. In the limit of zero field, OF3,/OQ = 0 leads to the strain dependence of the quadrupolar operator: B3,X3, -
Q = 1-
-
?
(38)
K:'X3, e l '
and the expression for the corresponding elastic constant can be written as: d2F 3, C3,= d(e~)2 = C~
( B y) 2Xv (1 - K~'X~,) '
(39)
or
Cy
C--~ = (1 - Grxr)/(1 - K~Xr).
(40)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
15
Trigonal m o d e e
When the magnetic field is applied along a three-fold axis, e.g. [111], the new symmetry (trigonal symmetry) implies that the only non-zero values involved are 1 M ( J x ) = <Jy> : <Jz> - V ~ -gJ~£B
,
(41)
( P x y ) = (Pyz) = ( P z x ) = P ,
(42)
and e 2~ =
E 1 ~
e 3~ ~
e"
(43)
The perturbating Hamiltonian takes the following form 1
~1 =
V ~ gJld"B(H "[- lq'm)(Jx -[- Jy -Jr Jz) 3 ~-~e[
ex2
- (B~e ~ + K P)(Pxy + Pyz + Pzx) + ~ C o t e ) + ½nM2 + 3 K ' p 2
.
(44)
For the calculations it is more convenient to rotate the coordinate axes so that [111] is the new z-axis (see appendix 4). In this new axes system, the same perturbation theory may be carried out to obtain the trigonal free energy F ". Then applying an inverse rotation allows us to express the latter in the initial coordinate system with four-fold symmetry F ~ = Fo _ ½Xo( H + n M ) 2 - ~ x9~ ( B
~e + K ~ P ) 2
- 3X~2)(Bee" + K ~ P ) ( H + r t M ) 2 - 1X(e3)(n + n M ) 4 3~e," + ~t~ol, e 8 \)2 + ½ n M 2 + 3K~p2
(45)
Here X0 is the same as in F r [eq. (30)] because of the isotropy of the (first-order) magnetic susceptibility, but the other CEF susceptibilities X~, )~z) and x~" (3) differ from the corresponding X~i) 's (see appendix 4). This yields the anisotropy of the properties related to the quadrupolar variables. The equilibrium conditions for M, e ~ and P are OF ~
OF ~
OF t
OM
Oe ~
3P
-0,
(46)
whence the following expressions .
(3) Lr3
M = XMH + X M I,
e = ~BP ,E Co
p
=
XpH 2
,
(47) (48) (49)
16
P. MORIN and D. SCHMITT
with XM as in eq. (35), F
1
X(M3 ) -
[)(~3)
4
+6
(1 - n X o )
1 - 3G~x, ] '
X(2) Xe = (1
-
nx0)2(1
(51)
3G~x~)
-
(50)
In the limit of zero field, OF~/OP leads to the strain dependence of the trigonal quadrupolar operators, 3B~x~ e ~ P = 1 - 3K~x, '
(52)
and the corresponding elastic constant, 1 d2F"
c . . 3 . d(e~) . .e
Co
C~ CO
•
3(B~)ZX~ 1 - 3K~x, '
(53)
or
1 - 3G~x~ ~ 1 - 3K X,
(54)
2.2.2. Generalization of the susceptibility formalism Symmetry considerations allow us to easily extend the preceding expressions for a magnetic field in any direction. For the third-order magnetic susceptibility, symmetry considerations show that there are two independent parameters in the expansion of the magnetization to order H 3 (Birss 1964), the general expressions are
Mx
x (3))I-I (I-Iy 2 + I-I z) ,
.~ (3) /At3 + . (3) Llr3
(3)
2
(3)
2
/ I ( M y I I Y ._[._ 1
(3) - XM~,)H,(H~ + H~Z), (3XM~
(3)/_/3 m z = X M , , / I I z .Jr- i
. (3) -- XMr)Hz(Hx + H y2) . (3XM~
.
.
(55)
. (3) and XM~ - (3) are the total third-order magnetic susceptibilities associated Here xa~:, with the four-fold and three-fold axes, respectively [eqs. (36) and (50)]. This allows us to predict the H 3 dependence of the magnetization for any direction; e.g., applying the magnetic field along any two-fold direction leads to the following relation
X~)l = ZXM~I-(3)+ ~,~,,(3) M~3 .
(56)
In a similar way, symmetry considerations lead to only two independent
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
17
parameters in the expansion of the quadrupolar susceptibilities to order H 2 ( 0 2 0 ) ~ ~XQ(2H~ 1 2 (O~)__
- H~2 - H y2 ) ,
1 2 ~XQ(Hx-H2y),
(57)
= 3XeI-I~t-IJ" (ij = xy, yz, zx) , where X~ and Xe are given by eqs. (37) and (51). These latter relations are useful in the expression for the parastriction, i.e. for the relative change of length A [eq. (25)] for a paramagnetic crystal in an applied magnetic field (see section 3.3.3). 2.2.3. Behaviour o f the various susceptibilities From the expressions of the CEF susceptibilities defined above, it is obvious that the crystal field plays an important role in their temperature dependence, in particular at low temperature where only a few CEF levels are populated. First-order magnetic susceptibility: Xo It is well-known that X0 varies as 1 / T at high temperatures, and that deviations from this law may occur at low temperatures, according to two typical behaviours: (i) with a magnetic CEF ground state, X0 diverges at 0 K; (ii) with a non-magnetic CEF level as ground state, a van Vleck-type behaviour is observed (see inset of fig. la). Third-order magnetic susceptibilities: X~ " (3) Because of the numerous terms in the expression of x (3)~ (/x = y, e), a more complex behaviour may be expected. While the high-temperature variation always follows a 1 / T 3 law, the behaviour of X- ~(3) at low temperature depends on the nature of the ground state. With a magnetic CEF ground state, X- ~(3) diverges at 0 K (fig. la). In the case of a non-magnetic singlet ground state, a van Vleck-type behaviour is observed (fig. lb). A very particular situation arises with a non-magnetic doublet ground state (fig. lc): if the van Vleck-type behaviour is observed for the trigonal susceptibility (H II[111]), a positive divergence occurs for the tetragonal one (HII[001]), due to the presence of mixed Curie-van Vleck terms in the expression of A~y - (3) • Thus, in addition to their anisotropy, the different (3) . (3) low-temperature behaviours oi Xr and x~ give valuable information about the CEF, in particular on the nature of the ground state. Strain susceptibilities: X~ The quadrupolar strain susceptibilities X~ give information about the quadrupolar character of the lowest CEF levels in the same way as X0 gives information about the magnetic character. Thus, for a singlet (fig. 2b) or a Kramers doublet (fig. 2d) ground state, there is no intrinsic quadrupolar moment and the low-temperature behaviour is of van Vleck-type. On the other hand, for a magnetic level as a
18
P. M O R I N and D. SCHMITI"
10
c° "~ I0
0
."
¢ .. /" --
0
20
40
0
q
20
TEMPERATURE
2ooi
-10
[.;t,)
--
40
(K)
Fig. 1. Typical temperature behaviour of the third-order magnetic susceptibility x~ - (3) for the two modes 3' (tctragonal) and e (trigonal) of cubic symmetry, and with a magnetic triplet (a)~ a singlet (b) and a non magnetic doublet (c) as ground state (CEF levels spacings are indicated). Inset of (a): corresponding temperature behaviour of the reciprocal susceptibility 1/Xo (dashed line: behaviour without CEF). (a)
'
--
G
- -
F,
'
r;'~ 1
80
J=6
~0
10
v
i
!
(e)
-
-
r,,
i
(d)
~ --
~5
J=6
20C
"
\
j =~
r;' r;" 100
10(
i
100
TEMPERATURE
'
lb0
(K)
Fig. 2. Typical temperature behaviour of the strain susceptibility X~ for both modes y and e, with a magnetic triplet (a), a singlet (b), a nonmagnetic, quadrupolar doublet (c) and a magnetic, nonquadrupolar Kramers doublet (d) as ground state. C E F levels spacings are indicated.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
19
triplet/,~t) (fig. 2a) or a quartet Fs, there is an intrinsic quadrupolar moment and the X,'S diverge at 0 K. Finally, the non-magnetic doublet F 3 (fig. 2c) is quadrupolarly active for the 3' mode but inactive for the e one, leading to a Curie and a van Vleck behaviour for X~ and X~, respectively. In all cases, the high-temperature variation of X, follows a 1/T law, as does X0-
Quadrupolar field-susceptibilities: X ~) The low-temperature variation of quadrupolar field susceptibilities depends on both the magnetic and quadrupolar character of the ground state. Thus a pure van Vleck-type behaviour is expected only for a non-magnetic and non-quadrupolar level, i.e. a singlet (fig. 3b). For a Kramers doublet, the presence of an intrinsic magnetic moment leads to the divergence of x~, - (2) (fig- 3d). For the non-magnetic doublet F 3 (fig. 3c), the van Vleck-type behaviour is observed only for the e mode for which there is no corresponding intrinsic quadrupolar moment. For the other cases, a divergence occurs at low temperatures. A very special case arises for the triplet F~ t) (fig. 3a), where the competition between the various components of the level leads to a change of sign of x- r(2) as a function of temperature: this leads to the so-called "reverse magnetostriction effect" (see section 3.4.1). Also, at high temperature, a 1/T 2 law is expected.
(a) 20
- -
r+
_ _
F'~ 1
8
100
\
--->
I --
r,,+++
--r]
10
1
----[-i" J=B
--Q J= 6
P,i
\ (b)
0
T
0 v
i
-1(
i
(c)
- -
r,
--15.
~>~
- -
F~2)
J=6
2°°~T
F~
40
t/ I\
\\+
100
TEMPERATURE
'++q
,.,
L = ,-+,,
20
100
( K)
Fig. 3. Typical t e m p e r a t u r e behaviour of the quadrupolar field-susceptibility X~ ) for both m o d e s 3' and e, with the same ground states as in fig. 2.
20
P. MORIN and D. SCHMITT
2.2.4. Magnetic and q u a d r u p o l a r transitions
In the presence of quadrupolar interactions, rare earth systems are characterized by up to five quadrupolar order parameters (Sivardirre et al. 1973) in addition to the usual magnetic order parameter M = g J P ~ ( J z ) . For cubic symmetry, there are only two independent order parameters, namely Q = (O °) and P = ( P i i ) , associated with the two quadrupolar parameters G r and G ~. In most cases, the bilinear interactions are dominant and drive one single magnetic phase transition, since the quadrupoles necessarily follow the dipoles as they magnetically order. Nevertheless, the quadrupolar interactions may influence the nature of this transition (see section 3.2). In a few cases, the quadrupolar interactions are dominant and drive a quadrupolar ordering without any magnetic dipolar ordering (see section 3.1). Several authors have investigated the phase diagrams associated with the presence of magnetic and quadrupolar interactions, with the frequent restriction of considering effective spins, i.e. by taking into account only the low lying levels. The cases S = 1 (Chen and Levy 1973), S = 3 (Sivardirre and Blume 1972) and S = ~ (Chen and Levy 1971) have been treated. Also the case of the F3-F5 magnetic system (S = 2) has been considered, explicitly including the related crystal field effects (Ray and Sivardi~re 1978). These calculations were all performed within the MFA. Numerous phase transitions have been found to occur, depending on the strength and the sign of the various parameters. Uniaxial ferro- (FQ) and biaxial antiferroquadrupolar (AFQ) orderings have been described (Sivardirre et al. 1973). More complex situations have been calculated by simultaneously considering the quadrupolar interactions belonging to both modes y and e, e.g., a ferriquadrupolar phase associated with the Q component followed by a FQ phase related to the P component (Chen and Levy 1973). In all the cases, the nature of the phase transitions was discussed and triple or tricritical points were found. While the existence of the various magnetic and quadrupolar phase transitions can be qualitatively described within the effective spin formalism, accounting for the real situations found experimentally requires one to consider the actual CEF level schemes, since the precise nature of the ground state as well as possible effects of excited levels via van Vleck-type matrix elements may noticeably change the physical properties. This can be done by using the susceptibility formalism developed above (Morin and Schmitt 1983a). Indeed, the expression for the free energy derived in section 2.2.1 can be used to investigate the possible phase transitions by taking H = 0. Equations (30) and (45) are then written as F v = F~ + ½n(1 - n X o ) M 2 - in4-,~,(3),~ar4 ~,~ + ½G~'(1- G r x r ) Q 2 - n 2r,'y t 1 X r(2) Q M 2 ,
F ~ = F 0 + ½n(1 - n x o ) M 2 - zin4"a t(3)hAr4 1,1 + 3G~(1-3G~x~)P 2 - 3n2G~2)pM
z"
(58)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
21
According to Landau theory, magnetic (tD) and quadrupolar (to_ and te) critical temperatures are defined by canceling the coefficient of the M 2, Qz or p2 terms in the expansion of F 1 - nXo(to) = 0,
1 -- G~Xr(tQ) = O,
1 - 3G~x~(tp) = 0.
(59)
The actual phase diagram as well as the nature of the transitions is determined by the relative magnitudes of tD, tQ and t e and by the signs and relative magnitudes of the other coefficients. The occurrence of a quadrupolar phase transition is a particularly interesting situation in cases with cubic symmetry where no quadrupolar component can spontaneously exist. In such a case (to_> t D for example) the transition is generally first order because the next term in the Q expansion of the free energy is of odd parity (Q3). The ordered value of Q is positive or negative depending on the sign of the corresponding coefficient. This coefficient is a second-order strain susceptibility [X(s2) in the notation of Morin and Schmitt (1983a)] and involves quadrupolar matrix elements between the various CEF levels. Note that the quadrupolar transition may be second order if Xs" (2) = 0 and the next term in the Q expansion of the free energy is positive. This occurs when the ground state is a quadrupolar doublet (F3 doublet, /"4, F5 triplets or F8 quartet) (Kanamori 1960). Examples of pure quadrupolar phases are shown in fig. 4 for both modes 3' and e (phases a 0 and ae, respectively). In the presence of dominant bilinear interactions (t D > tQ and tp), the magnetic transition is first- or second order, according to the sign of the next term in the M expansion of the free energy, namely x~ -(3) (/x = y, e). As seen in a previous section, xg -(3) is usually negative, whence a second-order magnetic transition. However, with a F3 doublet as ground state, xz, -(3) becomes positive at low temperature (see fig. lc), according to the characteristics of the low-lying CEF levels, leading to a first-order transition (see fig. 5). The limit between the two situations, i.e. xv - ( 3 ) = 0 , corresponds to the tricritical point C in the phase diagram. In the presence of competing bilinear and quadrupolar interactions, the situa-tion is more complex. First, various magnetic and Q or P quadrupolar phases may exist according to the relative strength of the associated coupling parameters n, G ~, G ~ (see fig. 4). Note in particular the possibility for two types of magnetic phases: (i) the phases b 0 or b e correspond to magnetic order where the ground state, coming from the F~ 1) CEF triplet, has mainly a Mj = 5 component; (ii) the phase co_ may be considered as a magnetic phase induced on the non-magnetic level, which is the ground state in the quadrupolar phase ao, and which has a dominant M : = 6 component. Another interesting feature of competing bilinear and quadrupolar couplings is the modification of the nature of the magnetic transition by the quadrupolar interactions. Indeed, the character of the phase transition is now determined by the sign o f -X M,'O (3) = X~)In=0 (see Morin and Schmitt 1983a). According to eqs. (36)
22
P. M O R I N and D. S C H M I T T
a) 5, / ~ ~ '
b) (]*=0 ~ aKp ~ 8'=2K
5 •
O'=4K/
W 0..
5"
,
)-
0'~=6K
o'=gK
bQ ,fQ 0
10
G I (mY,)
bp 0
100
G2(m K)
Fig. 4. Magnetic and quadrupolar phase diagrams for various bilinear coefficients 0* as a function of the quadrupolar parameters G ~ (=-G 1, left part) or G ~ (-=G 2, right part) for a T m 3+ ion (J = 6). a o, ap = pure quadrupolar phases; bo, bp = magnetic phases; Co: induced magnetic phase (from Morin et al. 1987c).
and (50), a positive quadrupolar parameter may drive the transition to be . (3) is larger than ]X~)[ (see fig. 6). Also first-order if its contribution to xM,0 tricritical points may be observed. The opposite situation may also occur, where a .(3) leads to a first-order transition, but a negative quadrupolar positive value of x~ . (3) changes it into a second-order one. Several examples of this contribution to XM,O will be discussed below (see section 3.2.2). Note that most of the experimental first-order transitions may be explained through the present model and do not require us to invoke critical fluctuations (see K6tzler 1984). In conclusion, the existence of bilinear and 3'- and e-type quadrupolar interactions gives rise to a variety of situations. Analyzing the complex related phase diagrams as well as the nature of the magnetic and quadrupolar transitions is qualitatively achieved within effective spin models, but requires a more realistic formalism in order to take the exact nature of the CEF levels involved into account correctly, and to better understand the experimental situations. This can easily be done through the present susceptibility formalism.
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
J=4 W=-5.SK x=.6B
(El 10
23
I-1 ~, ~ ? [ _ _ F 3
[001]
0 4c; 3O -
~
" -TD=tD
2C 1C I
0
/
I
20 TEMPERATURE(K)
40
Fig. 5. Temperature variation of xr - (a) (___X~3)) calculated for a Pr 3+ ion for the C E F level scheme indicated (upper part). Magnetic phase diagram (lower part); the dot-dashed line is the second-order magnetic transition line if neglecting X~3); the dashed line corresponds to a second-order magnetic transition, the full line (TD) to a first-order one; C is a tricritical point (from Morin and Schmitt 1983b).
~"
10 - -
:~
r 2
f2)
5
G1=13mK
--
r3
I 0 .-.
/
_'
10~-
/ I," 0
.
." T.-t IT.,
/
.~.-~_" . -
,
/
J=6 vv = u . ~ 3
n
x=- .34!
, , [oo ] 5 10 15 TEMPERATURE ( K )
20
Fig. 6. Temperature variation of X~)0 calculated for a Tm 3+ ion for the CEF level scheme indicated, for two G r (~G1) values (upper part). Lower part: magnetic phase diagram for G r = 13 inK; the solid (dashed) line represents a first- (second-) order transition; the tricritical point C occurs when XM,O - (3) vanishes; the dot-dashed line corresponds to a second-order transition which would occur if G ~= 0; T o is the quadrupolar ordering temperature, and t r the magnetic ordering temperature within the quadrupolar phase; T o and t o correspond to the first- and second-order magnetic transition, respectively (from Morin and Schmitt 1983a).
24
P. M O R I N
and D. SCHMITT
2.3. Extension to lower symmetries (hexagonal and tetragonal) The magneto-elastic and, more generally, quadrupolar properties have been studied less thoroughly in rare earth intermetallic compounds with a symmetry lower than cubic, e.g. in hexagonal and tetragonal compounds, because of the greater number of parameters involved in the CEF Hamiltonian as well as in the quadrupolar couplings. Only a few systems have been investigated such as metallic Pr (Hendy et al. 1979), dilute yttrium based rare earth compounds (Pureur et al. 1985), and more recently the compound PrNi 5 (Barthem et al. 1988). In this section, the formalism describing the magneto-elastic and quadrupolar properties of the rare earth compounds with hexagonal and tetragonal symmetries is developed. The full Hamiltonian is given, including all the relevant terms for both the one-ion magneto-elastic coupling and the two-ion quadrupolar interactions. Then analogies and differences with respect to the cubic symmetry are emphasized. 2.3.1. The Hamiltonian The general Hamiltonian appropriate for hexagonal and tetragonal symmetries takes the same form as eq. (1). However, the detailed expressions for most terms are different from those of cubic symmetry and are given below (see Morin et al. 1988). The CEF Hamiltonian The CEF Hamiltonian may be written, in terms of the Stevens operators (Hutchings 1964) and in the axes system where the z-axis is parallel to the c-axis of the hexagonal or tetragonal unit cell, as, ~Hex CEF =
~a'et
CEF =
0
0
0
0
0
0
6
(60)
6
B2 0 2 + B404 + B606 + B606 ,
0 0
0 0
4 4
0 0
4 4
B2 02 + B 404 + B404 + B606 q- B606 •
(61)
The bilinear Hamiltonian For hexagonal and tetragonal symmetries, the existence of one privileged direction, namely [001], dictates that there are two invariants in the full bilinear exchange coupling. Their expression is the same in both symmetries and takes the following form within the molecular field approximation ~ 1 = _ ( gjlJ.B)2nC~l ( j ) j , 2
~ B 2= --(g, lxB) n
~2
( 2 ( J z ) J z - (Jx)Jx - ( J y ) J y ) .
(62) (63)
The first term corresponds to the usual isotropic Heisenberg-type bilinear coupling, while the second term is an anisotropic bilinear coupling, forbidden in cubic symmetry, which leads to effective n coefficients which are different parallel and perpendicular to the c-axis in hexagonal and tetragonal symmetry. An experimental way to estimate the magnitude of this a2-term, would be to investigate the
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
25
anisotropy of the magnetic susceptibility in the gadolinium compounds where there is, in principle, no other source of anisotropy. The two-ion quadrupolar Hamiltonian In hexagonal and tetragonal compounds, the O ° quadrupolar component is already ordered (i.e. (O2°) ¢ 0) by the crystal field, but that does not exclude the possible existence of quadrupolar pair interactions, as in cubic symmetry. However, due to the lower symmetry, more than two quadrupolar parameters are involved, and the corresponding Hamiltonian may be written, in the MFA, as a function of the linear combinations of products of second-order Stevens operators, which are invariant under the symmetry operations of the hexagonal or tetragonal point groups: ~Hex
o
=-K
a
0
0
2
(O2)O 2-KÈ[(O2)O2+4(Pxy)Pxy ]
- K~[(Pyz)Py~ + ( P z ~ ) P ~ ] , ~o Tet
~o
0
=
-
K~(02)02
0
7 -
-- K ~ [ ( P y z ) P y z
K
+
2
(64)
2
( 0 2 ) 0 z - K (Pxy)Pxy
(P~)P~].
(65)
The magneto-elastic Hamiltonian As in cubic symmetry, the one-ion magneto-elastic couplings may be considered as the strain derivatives of the crystal field term itself. Within the same harmonic approximation, and limiting ourselves to the second-rank (quadrupolar) terms, the full one-ion magneto-elastic Hamiltonian is written as:
~Hex ME1 = __(B,~le~l
~-
B,~2e,~2)O0 _ B ~(elO ~ 2z + 2e2Pxy ~ ) _ B¢(e~pyz + eIpzx) ,
(66) ~ TME1 et =_(B,~lal
+ B.aa2)
0 2 o --
B,s,O~_Baeapxy
--
B~(s~pyz+e2Pzx).
(67) In these expressions the e~'s are the symmetrized strains (Du Tr6molet de Lacheisserie 1970) (see table 3) and the B"'s the magneto-elastic coefficients associated with the corresponding normal strain modes. Note that there are two independent fully symmetric a-terms, related to the two deformations which maintain the initial symmetry, namely the volume al-strain and the axial a2-strain, the latter one is associated only with a change of the ratio c/a. For the other terms, a symmetry lowering towards orthorhombic or even lower symmetry Occurs.
The two-ion magneto-elasticity is related to the modification of the two-ion magnetic interactions by the strains. In this section, we restrict ourselves to the two a-type bilinear couplings. For each one, two strain modes must be considered; they are associated with volume and axial strain dependences of bilinear interactions. For hexagonal and tetragonal symmetries, the corresponding two-ion
26
P. M O R I N and D. SCHMITI" TABLE 3
Normalized, symmetrized strains for the hexagonal and tetragonal symmetry. Hexagonal
Tetragonal
Expression
Representation
Strain
Representation
Strain
5
~°~
/'1
~°~
-~v~(~ + ~. + ~z~)
5
~°~
5
~°~
~ ( ~ - ½(% + ~x))
5
4
r4
d
v~ %
F6
eg
F5
s;
V ~ G~
magneto-elastic Hamiltonians take an identical form, and may be written in the M F A as o~1
~ME2
~--- t'r~otl otl D ~ 2 e ~2 --~LIalE At- al
gME2a2 = --IZDaXa2 E ~
+
)(j)j
(68)
D~2 J e~2)(2( z a 2L )
-
(L)L
- (Jy)Jy).
(69)
The elastic energy The elastic contribution to the free energy may be expressed as a function of the strains s" and of the appropriate symmetrized elastic constants C~" ( D u Tr6molet de Lacheisserie 1970). For hexagonal and tetragonal symmetries, they are given by:
EHeX : 1 C 0 1 ( G a l ) 2 q_ CO12Eo~IEa2 + 1 C 0 2 ( E a 2 ) 2 el
+ ~Co[(~1)2 + (~)21 + ½Coq(~)2 + (~)21, ETet e, = 1
Co,(eal)2 +
(70)
C012EotlEo~2 + ~ 1t ~..~ao 2 ,ks -ot2\2 ) l z~/
6\2
1
e
e 2
+ ½G(~) 2+ : ~ o ~ ) + ~Co[(<) + ( 4 ) q -
(71)
The background elastic constants without magnetic interactions Co~ are given in table 4.
TABLE 4
Symmetrized elastic constants in hexagonal and tetragonal symmetry.
Hexagonal
Tetragonal
Expression
C al
C ~1
½(2Cn +2C~2 +4Ca3 + C33)
C~2 C~2
C~12 Co2
C~ C~
C~ C~
_
C ~
-~V~(G~ + c,~ - c . - G~) 3'(C n + C ~ 2 - 4 C ~ + 2 G 3 ) C n - C12 2G~ 2(2"66
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
27
The two-ion corrective energy terms As in cubic symmetry, the molecular field treatment of the two-ion interactions requires the usage of corrective terms in the deduced single-ion free energy. According to the two-ion couplings considered above, three corrective terms have to be taken into account, corresponding to the bilinear, the quadrupolar and the two-ion magneto-elastic Hamiltonian, respectively.
Hexagonal symmetry EB'= ln~lM2,
(72)
]IA2 2 = ~1 n,~2[,-) t .... z - Mx2 - My) ,
E~2
. 1KO: < o O > 2
EHeX
o
2 2
+ ½K"[
(73) 1 ' ~K [<eyz> 2 "~ 2 ],
+4
(74)
j~?Hex l,~cd ~1 _}_'rja2Ea2)<j>2_}_ l z r ~ l ~1 a2Ea2 ) I,/.)28 + O,~ 2 ~ME2 = 2[,Ual 8 ~ctl x
[2(L) 2- (L) 2 -
<4>2]
(75)
.
Tetragonal symmetry E~ 1 and E~ ~ are the same as for hexagonal symmetry (eqs. 72 and 73), and 1 y 2\2 rTset=½K~(Oz°)Z+~K (O2/ + ~1K 8 (Pxy) 2 + ½ K * [ ( P y z ) 2 + ( P z x ) 2] (76)
ESet ME2 is the same as for hexagonal symmetry [eq. (75)].
2.3.2. Harmonic description of magnetostriction The equilibrium values of the strains eU's are obtained by minimizing the total free energy with respect to the strains. This leads to relations analogous to those for cubic symmetry, but they involve appropriately symmetrized coefficients.
Hexagonal symmetry E~I
=
[(BalCo2
+
- B
l g FI ot l['~ae2
X [ColCo
2 --
a2
~12
0
l_/rlalt,'~2
oe2 ,~12
C O ) ( O 2 ) + 2\~.-al~0 -- D~Co ) ( J ) a2
:Co~12 )(22 - < L
-D
>2
2
- <4)2)]
(Co'2)21-1 ,
(77)
e ~2 =[(B~2Co I _ B~1C012)( O ° ) + !gD,~2C,~1 2, =1 o - D ~ul I C ou12) ( J ) 2 1 //-,i ~ 2 / - ~ 1 + 2~,"~2~o -X [ColCo
B~
ea = ~
B ~
(022),
e~ = C'-~o(Pyz>,
al 412 Da2Co )(2(L)2 - (L) 2 -
2 - (Co12)2] -1 ,
~
2B ~
e 2= Co" (P~y),
(jy)2)] (78)
(79)
B ~
e~2= C---~o(Vzx)"
(80)
28
P. M O R I N and D. SCHMITT
Tetragonal symmetry e a l and e oe2 are the same as for hexagonal symmetry [eqs. (77) and (78)], and BT
e ~=
-
C~ (O~)
(81)
-
B ~
8 = ~g ( P x y ) , Co
(82)
B *
B ~
e~ = -~o (PYz)'
e2 = ~
(Pzx).
(83)
In the same way as for cubic symmetry, including these equilibrium values in the starting Hamiltonian allows us to express this Hamiltonian in a form which depends only on magneto-elastic coefficients. For example, considering only one-ion magneto-elasticity leads to t h e following total quadrupolar Hamiltonian ff(Hex QT = _ Goe(O2°) O ° - G~[(O~) 0 2 + 4(Pxy)Pxy]
-- G¢[(Pyz)Pyz + (Pzx)Pzx], ~fTet a 0 ~ Q T : -- G < 0 2 ) 0 2
0
2 2 __ G T ( 0 2 ) 0 2
(84) a a
-
-
(Pxy)exy
-- Ge[(Pyz)Pyz + (Pzx)Pzx],
(85)
where the total quadrupolar coefficients G ~ have contributions from both the one-ion magneto-elasticity and the quadrupolar pair interaction:
Hexagonal a2 (B a l ) 2 C O - 2B'~IB~2Co 12 + (B~2)2C01
G~ =
G ~, _
C01C02
(B~)
2
C----~"- +
K ~,
_
~
(C012) 2
: GME +
K~
;
+ R °e = G M E
(/*
=
e, ~'),
"{- K a ,
(86)
(87)
Tetragonal G oe is the same as above (eq. 86), and G~-
(B'~) 2
+ K ~ = C M~E + K ~ ;
=
(88)
2.3.3. Perturbation theory An analysis of the magnetic and magneto-elastic properties of hexagonal and tetragonal compounds within the paramagnetic phases may be done by using perturbation theory. The full development of the susceptibility formalism is treated elsewhere (Morin et al. 1988) and only the main results are reported below.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
29
Among the different modes, only the a- and e-modes in hexagonal symmetry (a-, y- and 6-modes in tetragonal symmetry) can easily be investigated by applying a magnetic field along the main symmetry directions, namely the [001], [100] and possibly the [110] directions. The quadrupolar components involved are ( O ° ) , ( 0 2) and possibly (Pxy)" Various associated susceptibilities may then be defined, namely strain susceptibilities X, and quadrupolar field-susceptibilities X~ - (2) (/z = a, e . . . . ). The only difference with respect to cubic symmetry is the explicit presence of a spontaneous quadrupolar moment (O°)0 in the expressions for X~, and Xtz -(2) • Also, the field dependence of the magnetization and quadrupole moments may be determined. For the quadrupolar components, relations analogous to those for cubic symmetry are found for hexagonal symmetry =
-
(89)
(9o)
2 ,
with ((2)
X~ = (1 - nx0)2(1 - G " X g ) '
(91)
and similar relations for tetragonal symmetry. For the magnetization, the following expressions are found M=
XMH--
4- . (3) L/3 X M 11
,
(92)
with X0 XM
--
,
(93)
1 - nXo
and X~ ) -
G a (x~(2),~2 ) . (2)-~2 q] [ (/l(e I |X~ 3) + 2 +2G ~ . (1 -- rtX0) 4 t_ 1 ~ G-~Xa 1 --- G- ~ 1
(94)
In these relations, all the susceptibilities assume different numerical values according to the direction of the applied magnetic field, although their expressions are formally equivalent. This is due to the different CEF wave functions determined by the field direction. This behaviour is well-known for the first-order magnetic susceptibility which is different for fields parallel and perpendicular to the c-axis in hexagonal and tetragonal compounds. For the third-order magnetic susceptibility, it turns out that only the G" contribution is present when the field is applied along the c-axis, while both G ~ and G ~ terms must be considered when the field is within the basal plane. Note that the y- and 6-modes in cases with
30
P. MORIN and D. SCHMITT
tetragonal symmetry may be separately investigated by applying a magnetic field along the [100] and [110] directions. Finally, only the isotropic bilinear exchange coefficient n ~1 has been considered here and is denoted n. 2.3.4. Comparison with cubic symmetry The main difference between hexagonal and tetragonal symmetries compared to cubic ones comes from the fact that the quadrupolar component (O~) is already ordered (i.e. ( 0 2°) # 0 ) by the crystal field. This leads to several consequences for the magnetic and magneto-elastic properties. First of all, from eqs. (77) and (78), there exists a spontaneous quadrupolar contribution to the volume (through e ~1) and to the ratio c/a (through e ~2) of the hexagonal and tetragonal unit cell. This feature manifests itself through an anomalous thermal expansion with regard to that of the corresponding compound with non-magnetic lanthanum or yttrium. This has been observed, e.g., in Pr (L/ithi and Ott 1980) and PrNi 5 (Barthem et al. 1988). Second, the a-term in YgOT[eqs. (84) and (85)] gives an additional quadrupolar contribution to the pure CEF second-order Hamiltonian as soon as either B ~1, B ~2 or K ~ is present. As for (O2°), this contribution is temperature dependent; this is a possible explanation for a thermal variation for the total effective second-order crystal field parameter B°2. As a consequence, the CEF level spacing will be temperature dependent, and it is worth taking this contribution explicitly into account besides the pure CEF one, at least in the case of strong quadrupolar coupling. The presence of a quadrupolar moment (020) does not exclude the possible existence of the corresponding quadrupolar pair coupling K ~. However, a quadrupolar phase transition involving this coupling is no longer allowed. This behaviour can be compared to the effect of high rank (fourth- and sixth-rank) pair interactions in cubic compounds which lead to apparent temperature-dependent fourth- and sixth-order CEF parameters (Morin and Williamson 1984). On the other hand, quadrupolar transitions involving the other couplings as K ~, K ~ (K r, K s, K" in tetragonal compounds) are quite possible and have been observed in several rare earth insulators (Gehring and Gehring 1975). 2.4. A peculiar case: the semi-classical treatment
In the case when bilinear interactions overwhelm the CEF coupling, the magnetic ordering temperature is large in comparison with the total CEF spacing. In the paramagnetic phase, the CEF levels may be considered as equally populated, while in the ordered phase, each of the CEF wave functions is "purified" by the exchange field to its dominant IJz) component, at least in the limit of low temperatures. Both situations may then be discussed within a semi-classical analysis. 2.4.1. The paramagnetic phase For vanishing CEF effect, the 4f multiplet is (2J + 1)-fold degenerate and the expressions of the various one-ion susceptibilities previously introduced become
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS C Xo = T
(95)
CQ T'
X~X(2)
(96)
2
2
CQ
(97)
= gjtx B 6 T 2 , C (3)
X(3) =
X(21 s
31
~--
r 3 ,
(98)
C~ ) T2 ,
(99)
where C is the usual Curie constant [see eq. (5)], and CQ, C ~ ) are quadrupolar constants and C (3) is the third-order Curie constant
CQ =
J(J + 1)(2J - 1)(2J + 3) 5k B ,
(lOO)
C(o2) = J(J + 1)(2J - 1)(2J + 3 ) ( 2 J - 3 ) ( 2 J + 5) 70k 2 C(3)
4 4 J(J+l)(2Jz+2J+1) = -gjtx B 90k 3
(101)
(102)
In the same way as C and C (3) are the coefficients of the first two terms in the H~ T expansion of the Brillouin function, CQ and C ~ ) are the coefficients of the first two terms in the s / T expansion of the response function of Q in the Brillouin-Boltzmann statistics 2
e C~ ) e Q ~ Ca -~ + ~5 +'".
(103)
Note that all the C's are isotropic. After renormalization by the bilinear and quadrupolar interactions, the actual susceptibilities . are
C XM -- T - O *
'
2 2 Co.T XQ = gllXB 6 ( T - 0 * ) 2 ( T - 0~) ' C(3)T
(i-U)
[1-
4
L
(lO4)
(2J-1)(ZJ+3)
?-2
75;-T
-T - -' O ~ J '
32
P. M O R I N and D. S C H M I T F
and the actual elastic constant is given by
C
T-O~
C0
T - 0~: "
(lo5)
This equation is very close to that given by Kataoka and Kanamori (1972). O~ = CQG and 0~ = CQK are defined by analogy with 0", as functions of the total and two-ion isotropic quadrupolar parameters G and K. The temperature dependences in eq. (104) have been extensively described by Morin and Schmitt (1981b) and Morin et al. (1980b) at high temperature as well as close to the magnetic ordering temperature. Less attention has been paid to the nature of the quadrupolar transition. It is easy to verify that for J = ½there is no quadrupole moment. For J = ~, C~2) is zero and the next term in the e expansion of Q is negative, the transition is then second-order. In all the other cases, C~ ) is positive and the transition is first order.
2.4.2. The ordered phase Several discussions of the energy in the ordered phase have been given in the past in particular by Clark in volume 1 of these handbooks (Clark 1980). Only those features that are necessary for describing properties encountered in the following sections will be derived here from our basic Hamiltonian. Following Callen and Callen (1963, 1965), the temperature dependences of the average value of the different operators O~' present in eq. (1) are expressed as functions of the normalized 4f magnetic moment, m, through modified hyperbolic Bessel functions. Their angular dependences are written as functions of the 4f magnetic moment cosines, ai, by using the harmonic polynomials. ( 0 O) = j ( j _
l ~
~)Is/2(~
-1
2
(m))(3a 3 - 1),
( 02> = J(J - 1)Is/2(~/~-1(m))(ot
2 -
ce2),
(Pij) = J ( J - ~) 1 ~1s / 2 f(~( -mi ) ) ( a i % ) (O04+504)=_40J(j -- ½ ) ( J _ l ) ( j _ 3
(i,j=1,2,3) 2 ~ 22 + ) I"9 / 2 ( ~ ' - 1 ( m ) ) ( O l l O
{O~ + 2 1 0 6) = 1 8 4 8 J ( J - ½ ) ( J - 1 ) ( J ^
-1
(106)
2
2
3)(j_2)(j_ 2
2
2
cycl. - ½)
~) 2
2
2
2
x I13,2(5g (m))[c~,c~2a 3 _ 1(12ct3 + c~3a a + a,t~2) +
~1
.
Du Tr6molet de Lacheisserie et al. (1978) have studied in detail the classical free energy of a cubic ferromagnet. The classical magnetostriction constants and anisotropy constants are found to be 0 --- 0 J ( J - ½)I5/2(5¢-1(rn)),
A100(T) = 2
Cll
-- C12
B ~
AHa(T ) - 3v~cO4 J ( J - ½)_15/2(~ ~(m)),
(107)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
33
K I ( T ) = - 4 J ( J - ½) ( J - 1)(J - 23-)[10B419/2(Gt?-'(m))
+ 42(J - 2)(J -
G(T) =
5 )B6~13/2(~ '
-'(m))],
~)B6?~l~/Z(iLf-l(m)).
1848Y(Y- 1 ) ( j _ 1 ) ( J - k ) ( J - 2 ) ( J -
(108)
The K 1 anisotropy constant receives two contributions from 4th and 6th rank polynomials. Owing to their different temperature dependences (fig. 7) they may be the causes of a change in the easy magnetization direction in the ordered phase. It is also well-known that the magnetostriction contributes to the magnetocrystalline energy, e.g. AK 1
=
9
-- ~(Cll
0
0 2 _ C,2)Aioo(T )
-I-
0 2 79 C44~,,11(T)
(109)
.
This magnetostriction contribution may be an appreciable part of the intrinsic K 1 constant: it reaches ~ of K 1 in TbFe 2 at room temperature [see table 8 in Clark (1980)]. In addition, due to different temperature variations, K 1 may change of
I
I
I
I
I
I
I
I
Is/2 10"1 19/2
10-2
113/2 10"3
1.0
.8
.6
.4
.2
0
m
Fig. 7. Normalized hyperbolic Bessel functions J5~2, 19/2 and I13/2 versus the normalized magnetization (from Clark 1980).
34
P. MORIN and D. SCHMITI"
sign. This situation occurs, e.g. in hexagonal RC% compounds as revealed by the variation of the lattice parameters and elastic constants (se e section 4). This classical description of magnetostriction and magnetocrystalline anisotropy gives a phenomenological connection between their temperature variation and that of the 4f magnetic moment. The details of the magnetic couplings leading to this observed magnetization itself are not of too great importance. It is useful in complex situations, as e.g., for rare earths dissolved in a magnetic metal (Fe, Co, M n , . . . ) . Indeed, at least at low temperatures, the magneto-elastic coefficient, in particular for the symmetry associated with the spontaneous state, may be determined by using the assumption that the 3d magneto-elastic contributions are negligible. However, both the necessary separation of the 4f and 3d magnetic moment and the existence of many sublattices limit the insight in the 4f magnetism. Obviously it would be unrealistic to search for quadrupolar pair interactions, when they are overwhelmed by bilinear ones. 3. Experimental evidence
3.1. Quadrupolar orderings To observe quadrupolar orderings it is necessary to find rare earth systems where the bilinear interactions are weak compared to the quadrupolar interactions. Favourable situations exist at each end of the lanthanides series. Indeed, first, the bilinear interactions are at their minimum for the series of compounds under consideration and, second, both the orbital moment L (Tm a+ : J = 6, L = 5; Ce 3+ : J = ~, L = 3; pr3+: J = 4 , L = 5 ) and the Stevens coefficient as, which deeply control the quadrupolar coefficients (Morin and Schmitt 1981a, Schmitt and Levy 1985), are large. In other cases, magnetic and structural transitions coincide (section 3.2.2). Quadrupolar orderings have been reported to occur in TmZn (Morin et al. 1978b), TmCd (Al6onard and Morin 1979), CeAg (Ushizaka et al. 1984, Morin 1988), PrPb a (Bucher et al. 1972a), TmGa 3 (Czopnik et al. 1985), CeB 6 (Effantin et al. 1985) and PrCu2 (Ott et al. 1977a). The first three of them are clearly ferroquadrupolar (FQ), the next three antiferroquadrupolar (AFQ) and the actual situation in the last one is not yet fully elucidated. UPd a which exhibits a structural transition may be relevant to the present review owing to its localized 5f electrons (3H 4 multiplet) (Andres et al. 1978). Structural transitions have been also observed in the RB 4 series, however, their origin has not been fully understood (Will et al. 1986).
3.1.1. Ferroquadrupolar ordering As for the ferromagnetic ordering (FM), the FQ ordering is characterized by a q = 0 propagation vector and is expected to be closely described by the singlesublattice formalism (see section 2.2).
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
35
3.1.1.1. TmZn. Cubic compounds within the CsCl-type structure are characterized by large quadrupolar interactions, TmZn being the archetype compound (Giraud et al. 1985). The temperature variation of the specific heat (fig. 8) reveals two well-defined transitions, close to each other (Morin et al. 1978b). TmZn undergoes a FQ ordering at T e = 8.55K, where ( O °) is the order-parameter. The transition is clearly first-order. An associated tetragonal strain takes place simultaneously due to the B y magneto-elastic coefficient. A FM ordering occurs in the quadrupolar phase at T c = 8.12 K, with the spins lying along the four-fold axis. The spontaneous tetragonal strain is reinforced by the bilinear interactions and reaches c/a - 1 = -8%0 at 1.5 K. From the jumps in the electrical resistivity at the two transitions, the electronic scatterings of quadrupolar and magnetic origins are of similar magnitude. Numerous properties investigated in the paramagnetic phase of TmZn have been consistently analyzed with a total quadrupolar coefficient G v= 25 mK; the dominant contribution to G v comes from the pair interactions (Givord et al. 1983). This G ~ value describes the temperature T e and no additional coupling (anharmonicity, higher order terms . . . . ) is needed. The FQ ordering lifts the degeneracy of the F~ 1) cubic ground state and splits it into a doublet and a singlet; the non-magnetic singlet is the ground state in the quadrupolar phase (structure a o in fig. 4 of section 2.2.3). Bilinear interactions are strong enough to induce a magnetic moment in this phase. Substituting L u 3+ for Tm 3+ in Tml_xLuxZn or Cu for Zn in TmCuxZnl_ x decreases the bilinear interactions quicker than the quadrupolar ones and the spin system remains paramagnetic: only the FQ ordering is observed (fig. 9) (Morin and Schmitt 1986). Note that in this case the splitting of the F~ 1) cubic triplet in the tetragonal paramagnetic phase has been observed by inelastic neutron scattering (Morin et al. 1981). 60
!
1
i
50 Tm Zn
o
E 40
30
20
l
T
r,, sss
T 8.12K= .
K , , ~ .
I
1
I
7
8
9
TEMPERATURE
( K )
Fig. 8. Specific heat of TmZn; T c is the magnetic ordering temperature, T O the quadrupolar ordering temperature (from Morin et al. 1978b).
36
P. MORIN and D. SCHMITT
10
% TmCu
v
Zn 1-c c
p
LU mr
2 ¢rUJ O.
;* 2 5 ~
:E
--.
hi
-_
.
,
/
f--
-5
20
/
oG 1 //
0.5
,,
15
)
C
Fig. 9. Phase diagram for TmCUl_cZnc; P: paramagnetic, M: modulated, AF: antiferromagnetic, F: ferromagnetic, Q: quadrupolar phases; lines are guides for the eye. Variation of the quadrupolar parameter G r (=G~) and bilinear exchange parameter 0* with c (from Morin and Schmitt 1986).
3.1.1.2. TmCd. Historically, the first FQ ordering in a rare earth intermetallic was observed in this compound at T o = 3.16 K. It was initially analyzed with a F3 cubic ground state, and the first-order character of the transition was assumed to be driven by anharmonic elastic effects (L/ithi et al. 1973b). However, RZn and RCd isomorphous compounds exhibit very similar properties, in particular the CEF (Alronard and Morin 1985). As a matter of fact, there is a close parallel between TmCd and TmZn and all the TmCd properties are fully explained with a G ~= 12 rnK total quadrupolar coefficient without the need of any other coupling (Alronard and Morin 1979). As in TmZn, pair interactions are the dominant contribution to the quadrupolar coupling and the ground state is also the non-magnetic singlet coming from the F~ 1) cubic triplet. Here, however, the spin system remains paramagnetic owing to the weakness of the bilinear interactions. As in TmZn, electrical resistivity measurements give clear evidence for the vanishing of the quadrupolar scattering below T o (fig. 10). 3.1.1.3. CeAg. CeAg is the only example of a FQ ordering for a Kramers ion. It
also belongs to the family of CsCl-type compounds. The cubic ground state is the F8 quartet, well-isolated from the F7 excited level (A = 266 K) (Schmitt et al. 1978). It exhibits a cubic-tetragonal transition at T o = 15.85 K with (O2°) as the order parameter (fig. 11). A FM transition occurs at T c = 5.2 K, the c axis of the tetragonal cell is the easy magnetization direction (Morin 1988). The c / a - 1 strain is very large (1.7% at 1.5 K). A strong dependence of the low-temperature properties on the metallurgical conditions is observed, which is related to the very large magneto-elastic coupling.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
37
5.6 I
,
,p..."
E 5.4
de_' dT
1
°
I.-te
E
W °
5.0
•"°
0
°
°¢,°°
""
I
i
2
½
t
~
n
I
4 i
4
3
TEM PE RATURE(K) Fig. 10. Low-temperature variation of the resistivity and its derivative (inset) in TmCd (from A16onard and Morin 1985).
u
30 CeAg
_~2o E
0
5
10
1~i
20
T (K) Fig. 11. Specific heat of CeAg. Anomalies at T c = 5.5 K and TQ = 15.5 K correspond to the ferromagnetic and the quadrupolar orderings, respectively (from Ushizaka et al. 1984).
The study of the quadrupolar properties in the cubic phase leads to G r = 150 inK. However, this value would drive a FQ ordering at only 10 K instead of T o = 15.85 K (Morin 1988). This difference is consistent with the weak softening of the Cll - C12 elastic mode (Takke et al. 1981) and reveals the existence of an additional term, i.e. the anharmonic elastic coupling of the Cll - C12 mode with a zone-boundary phonon, which softens at the M point of the Brillouin zone (Knorr et al. 1980). This last mechanism is associated with the incipient instability of the CsCl-type lattice for light rare earth-silver compounds which is of electronic origin (see section 3.10).
38
E MORIN and D. SCHMITI"
3.1.2. Antiferroquadrupolar ordering Several types of AFQ structures may exist. First, the lifting of the degeneracy of the cubic ground state may lead to wave functions different from one site to the other (Sivardi~re and Blume 1972). If the absolute values of the quadrupolar moments are different for the sublattices, that corresponds to a ferriquadrupolar ordering. Second, the ground state can be the same on each site, but the local z-axis may change, although belonging to a given star of crystallographic directions: the corresponding quadrupolar arrangement may be called multi-axial (Sivardi6re 1975). Lastly, more than one propagation vector may be necessary to describe the AFQ structure. In all the cases, magnetic structures with many sublattices may be favoured (see section 3.6). Whatever the actual structure may be, a multi-sublattice formalism is needed. The previous single-ion treatment may be used only for describing the occurrence of a magnetic or quadrupolar moment within a given sublattice. To describe the AFQ as well as AFM properties, no reliable information can be expected from static measurements in these intermetallics. In particular, the macroscopic symmetry may remain cubic. Only microscopic probes, such as neutron diffraction, are effective in revealing the existence of more than one magnetic sublattice.
3.1.2.1.
C e B 6. This cubic compound exhibits very complex properties due to the interplay of localized and delocalized 4f behaviours. The temperature variation of the specific heat reveals two anomalies at T o = 3.3 K and TN = 2.4 K (Peysson et al. 1986). Neutron diffraction experiments have shown that the compound remains paramagnetic down to 2.4 K (Horn et al. 1981). Magneto-elastic effects are present above TQ, although the C44 and Cll - C1~ elastic modes softens by only 2% (L/ithi et al. 1984). As the F8 ground state is well isolated (A = 540 K) (Zirngiebl et al. 1984), a quasi-full softening is expected in the case of a FQ ordering. On the other hand, the temperature variation of C44 and Cll - C12 may be analyzed with negative quadrupolar pair interactions. An AFQ ordering is then probable below TQ. This is confirmed by a neutron diffraction experiment on a monocrystalline sample in a magnetic field (Effantin et al. 1985). Indeed, two kinds of Ce ions have been observed which differ in zero field only by their magnetic susceptibility. The corresponding superstructure lines are described by a [½ ½ ½] propagation vector. The AFQ ordering then consists of an alternating sequence of (111) planes with different quadrupolar moments (fig. 12). Below TN, the spontaneous biaxial spin structure reflects the existence of the two sublattices driven by AFQ ordering. The magnetic field phase diagram reveals the interplay between localized and delocalized 4f magnetism as demonstrated by the anomalous increase of the AFQ phase under a magnetic field (fig. 13). In zero field, T O and the AFQ phase are reduced by the Kondo coupling but with an increasing magnetic field, the Kondo state is progressively suppressed and the AFQ ordering can manifest itself at a higher temperature.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
Magnetic ph<,.,,,,,
modulated
Anti ferro
~:[~/+ ,/, ,/2]
,
pho,.,,
:
39
quadrupolar
"~:[~]
z
(b)
(a)
Fig. 12. (a) Biaxial magnetic structure in CeB6 described by the propagation vectors indicated. (b) Antiferroquadrupolar structure described by the k o propagation vector; open and full circles correspond to the two different Ce-quadrupole sublattices (from Effantin et al. 1985).
' A- ,~ ' 'o ~,;too~j "-= ~6 *-I~//[111:]
20
t,....~ It ",
140
/
"iT
160
J/
120 100 80
0
+oIl,.,,,..=,,,,,,.+o,. q o!
60
pl ,+,,°,.o., V ,oI+,+..,.+.°,.o/,.
II
+~"'<""'i
0 1 " " @ 0 2
4
.
17 I~L_.
i
TernperaLure(K)
I 8
l
,o
=
0 10
Fig. 13. Magnetic phase diagram of CeB 6. Phases III and II are drawn in fig. 12 (from Effantin et al. 1985).
T m G a 3. This AuCu3-type cubic compound also exhibits two transitions in the temperature dependence of the specific heat, but very close to each other (TQ = 4.29 and Tr~ = 4.26 K) (fig. 14) (Czopnik et al. 1985). Here again, studies in the paramagnetic phase lead us to introduce quadrupolar interactions, but they 3.1.2.2.
40
P. MORIN and D. S C H M I T r
rN 60
50
I 40 30 20 10 o 4.1
4.2
4.3 T(K)
4.4 -
Fig. 14. Specific heat anomalies in TmGa 3. Arrows indicate the antiferromagnetic (TN) and antiferroquadrupolar (TQ) transitions (from Czopnik et al. 1985).
are too weak to induce a FQ ordering at T e (Morin et al. 1987c). In addition the spontaneous spin structure determined by neutron diffraction on a single crystal is multi-axial (see fig. 53 in section 3.6). This indicates there are different (111) directions as local z-axes at the 4f sites stabilized by AFQ interactions of trigonal symmetry (Morin et al. 1987d). Thus the quadrupolar ordering is of AFQ-type. As for the ferromagnetism in TmZn, the conditions of induced antiferromagnetism on the singlet ground state cause the temperature range of the paramagnetic AFQ phase to be very narrow. P r P b 3 . Another AuCu3-type compound, PrPb3, exhibits a non-magnetic transition at TQ = 0.35 K corresponding to the lifting of the F3 ground state degeneracy (fig. 15) (Bucher et al. 1972a). No spontaneous macroscopic strain has been detected in the quadrupolar phase. This may be due either to its weakness in the case of a FQ ordering or to its nullity in the case of an AFQ ordering. The softening of the C44 and Cll - C12 ultrasonic is far from complete and influenced by high-order magneto-elastic terms (Niksch et al. 1982). They may be analyzed by positive as well as negative quadrupolar interactions. However, both the parastriction and the third-order magnetic susceptibility need negative quadrupolar coefficients to be described (Morin et al. 1982). An AFQ ordering is then highly probable, and ( O °) is the order parameter. In conclusion, it appears that the case of AFQ ordering is not as well understood as the FQ ordering. This is due to experimental difficulties as well as theoretical complications. Indeed, none of the previously mentioned three compounds is a completely favourable candidate for investigating AFQ properties. In CeB 6 the quantitative analysis is complicated by the Kondo behaviour of the 3.1.2.3.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
41
0.55"K 0.7
0.6
PrPb 3
0.5
~
0.4
0.~
0.2
0.1
0
I
I
0.5
1.0
T(*K)
Fig. 15. Molar specific heat CJ R of PrPb 3 between 0.04 and 1.0 K (from Bucher et al. 1972a).
cerium ions. In TmGa 3 the temperature range of the AFQ phase is much too narrow. PrPb 3 orders at 0.35 K, a temperature characteristic of very weak AFQ interactions, which may easily be broken by the application of a magnetic field. From a theoretical point of view, the existence of many inter- and intralattice pair coefficients complicates the analysis of the data.
3.1.3. Structural transitions possibly of quadrupolar origin A few other structural transitions have been reported in the literature, which may be evidence for quadrupolar interactions. However, the actual situation is not yet fully determined, often because the crystal electric field is not known.
3.1.3.1. PrCu 2. This orthorhombic compound exhibits a second order structural transition at 7.5 K which has been observed in many different experiments, in particular by specific heat (Wun and Phillips 1974). From elastic constant measurements, the primary order parameter was found to be the ez, strain (Ott et al. 1977a). This agrees with neutron diffraction data which have shown that below 7.5 K there is a sizeable splitting of the (101) nuclear reflections corresponding to an angle/3 between the a- and c-axes of the orthorhombic cell that is no longer equal to 90° (Kjems et al. 1978) (fig. 16). A FQ ordering involving the (Pz~) quadrupolar component is likely. Unfortunately, there does not exist a determination of the CEF; thus a quantitative analysis of the observed properties is not possible.
42
P. MORIN and D. SCHMITI"
PrCu z
T=B.OK
T=4.75 g 1.0
0.5 ,
,
-0.01
0 i
~
~-I
0.01-0.01
0
0.01
k" ="
~ ~OR'¢
0.0
i-
4.0
L
~ I.v
=
A
6.0 8.0 TEMPERATURE (K)
10.0
Fig. 16. The upper part shows high resolution scans through the (101) Bragg peak below and above the structural transition temperature in PrCu2 at T = 7.5 K. In the lower part the observed splitting (A) is plotted versus the temperature. The full line represents a mean field calculation (from Kjems et al. 1978). 3.1.3.2. U P d 3. Contrarily to most of the metallic actinides, the hexagonal compound U P d 3 (P63/mmc) exhibits properties characteristic of localized magnetism associated with a 5f 2 configuration (3H 4 ground-state multiplet) (Andres et al. 1978). In a way reminiscent of Pr metal, two U sites exist with hexagonal and nearly cubic symmetries. T w o C E F transitions have been observed by inelastic neutron scattering on a single crystal. They have been attributed to an excitation process from a singlet ground state for both sites (Murray and Buyers 1980). In addition, U P d 3 exhibits a well-defined transition at 7.5 K, strongly depending on the metallurgical status of the sample (fig. 17). N o change of the hexagonal
4
t2
3 B~
-
ou -~-
,
0
2
4
6
4
8
10
12
14
o~_I
0
16
T(K)
Fig. 17. Specific heat of a UPd 3 monocrystalline sample minus that of ThPd 3 used as background
(open circles). The solid line (right-hand scale) is a plot of dp/dT. The slight difference in peak position of the two curves is caused by inaccuracies in the temperature scale of the resistivity measurement (from Andres et al. 1978).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
43
symmetry was observed from X-ray diffraction experiments on polycrystalline samples (Andres et al. 1978, Zaplinski et al. 1980). However, the increase of the c/a ratio observed at low temperatures might be associated with U atoms becoming more prolate ( ( O ° 2 ) < 0 ) at the quasi-cubic sites, according to an internal ferroquadrupolar process. This assumption might be checked by a full determination of the crystallographic structure on both sides of the transition by using a single crystal.
3.2. Nature of quadrupolar and magnetic transitions 3.2.1. Quadrupolar transitions As discussed in section 2.2.4, in cubic symmetry the transition at the quadrupolar ordering is usually first-order due to the existence of van Vleck matrix elements from the ground state. Indeed for the 6 cases (TmZn, TmCd, CeAg, CeB6, TmGa3, PrPb3) presented in section 3.1, the so called Xs(2~ susceptibility (Morin and Schmitt 1983a), which determines the order of the quadrupolar transition is calculated to be non zero. Thus the observed first-order quadrupolar transitions in the Tm compounds are well predicted by the present formalism. However a quantitative description of the specific heat anomalies is difficult in TmGa 3 and TmZn due to the vicinity of the magnetic and quadrupolar transitions. In TmCd, the measurements are not accurate enough, in particular in the low temperature phase. In PrPb3, the quadrupolar ordering temperature, T~ = 0 . 3 5 K , is small in comparison with the F3 - F4 spacing. The corresponding van Vleck terms do not affect the ordering process too drastically and the ( O °) jump is calculated to be weak in agreement with the temperature dependence of the specific heat (fig. 15). The same feature is also observed in CeB 6 and CeAg where the F8 - F7 splitting reaches 540 and 260 K respectively. For instance in CeAg the jump of (O2°) is calculated to reach only about 10% of its maximum value. The existence of anharmonic elastic terms in this compound makes it impossible to obtain a quantitative analysis of the T~ transition within the present formalism.
3.2.2. Magnetic ordering in the presence of quadrupolar interactions The nature of the magnetic transition in ferro- and antiferromagnetic cubic compounds has motivated a great number of studies, which can be classified according to whether they are based on an MFA analysis or renormalization group (RG) theory (K6tzler 1984). In both cases, the (isotropic) Heisenberg exchange is the starting point. The two approaches differ in the additional couplings which also have to be considered. The MFA model presented above, is in the line of models previously developed for transition metals as MnO, MnS (Lines and Jones 1965). However, instead of exchange striction, it considers for rare earth compounds quadrupolar pair and magneto-elastic interactions. The resulting coefficients of the free energy expansion are expressed in terms of single-ion susceptibilities and determining the phenomenological parameters for the different couplings then leads to the description of the magnetic transition (see section 2.2.4).
44
P. MORIN and D. SCHMITI"
On the other hand, the RG approach is based on symmetry considerations and includes critical fluctuations to predict the order of the magnetic transition. Within this hypothesis, continuous magnetic transitions become discontinuous ones according to the sign of the M4-term in the Landau expansion of the free energy. Note that the RG theory usually considers an incompressible lattice and nonquantized magnetic moments. The hypothesis of an incompressible lattice is in strong opposition with the evidences of large symmetry lowerings in cubic rare earth compounds. Due to the assumption of classical moments, the RG theory cannot predict the occurrence of a first-order transition associated with the positive X:, - (3) value in the case of a F3 non-magnetic ground state (see section 2.2.4 and Blume 1966). The aim of this section is to show through examples from the literature that the nature as well as the temperature of the magnetic transition may be predicted as soon as the quadrupolar magneto-elastic and pair interactions are reliably known (sections 3.2.2.1 and 3.2.2.2). Of course more complex situations may occur due to additional contributions from spin (and quadrupolar) fluctuations (section 3.2.3).
3.2.2.1. First-order transitions and tricriticality TmCu and TmAg A good example of the effect of the quadrupolar interactions on the magnetic transition is given by TmCu. This compound orders at TN = 7 . 7 K in a modulated antiferromagnetic structure, with all the moments parallel to the [001] axis. The transition is first-order (fig. 18). Various experiments performed in the paramagnetic state reliably provide the tetragonal quadrupolar parameter G r = 11 mK
1oo TmCu
-~ so
5 i
+I
i
10
,~-'"'"
.......
i
" "
.-
S
5
i
(b)
f" .,.." ,,, ~ ." " I 6
I
I
+
7
8
g
TEMPERATURE
(K)
Fig. 18. Temperature variation of the specific heat Cp (a) and of the entropy S (b) in TmCu. A n incommensurate antiferromagnetic structure is established at T N = 7.7 K and transforms at 6.7 K into a commensurate one (from Morin and Schmitt 1980).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
45
(Jaussaud et al. 1980). Just above TN the CEF third-order paramagnetic susceptibility Xz, - (3) is negative; this would lead to a second-order magnetic transition. However, the quadrupolar contribution to the total third-order magnetic susceptibility is positive and large enough to change the sign of X~)0, in the vicinity of TN, as seen experimentally. Consequently, the magnetic dipolar transition is expected to have a first-order character, as effectively observed (fig. 19a).
2
'
I'0.5
'
I
'
i -.
i
I
_m°!
,/ 3.5,,"
-~
T
ii..................
W =1~2K x -=-.42
'
1~o
'
f J" j/J
W=i.05K x=-.4.7
:.:" I
0
"
/ (~
.-"'' ( /"" T~
0/
0
I
I
10
TEMPERATURE(K) Fig. 19. (Top): temperature dependence of X~ ) observed for the tetragonal symmetry in TmCu (a) and TmAg (b). Full and dotted lines describe, for the CEF parameters W and x indicated, the cases with and without quadrupolar interactions, as indicated by the numerical values of the G v coefficient. (Bottom): the deduced magnetic phase diagram for both compounds, n is the bilinear coefficient (n N for an antiferromagnet). Dotted lines: second-order magnetic transition without quadrupolar interactions; dashed lines and full lines denote respectively: second- and first-order magnetic transition in the presence of quadrupolar transitions. C is a tricritical point (from Giraud and Morin 1986).
The isomorphous compound TmAg exhibits very similar CEF and bilinear interactions. It orders antiferromagnetically at T N =- 9.5 K, the transition is second order (Morin and Schmitt 1982a). Quadrupolar interactions within the tetragonal symmetry are weaker than in TmCu. They are characterized by G r = 4 m K (Giraud and Morin 1986). X~ ) is observed to be negative above 10 K, but is calculated to change sign at 6.2 K (fig. 19b). This defines a tricritical point between second-order transition (T > 6.2 K) and first-order ones (T < 6.2 K): the magnetic phase diagram predicts a second-order transition for a temperature of 9.5 K in agreement with the observation. An interesting feature is the large dependence of the tricritical temperature on quadrupolar interactions. Indeed, if G ~ reaches 5 mK it would shift the tricritical point up to 12.5 K, and the magnetic transition would be first-order for the same bilinear coefficient. Owing to the quadrupolar interactions, TmAg appears to be very close to tricriticality.
46
P. MOR1N and D. S C H M I T r
CeZn and CeMg Among other CsCl-type compounds, the CeX present a large set of different properties. Indeed whereas CeAg undergoes two well-separated magnetic and quadrupolar transitions (section 3.1.1.3), CeZn and CeMg order antiferromagnetically at T N = 30 and 19.5 K, respectively, the transition being first-order (Pierre et al. 1981, 1984). Contrary to the case of CeAg, the lattice stability is large; as in CeAg the Kondo-type behaviour for the Ce ion is not important. The tetragonal symmetry lowering below T N reaches record values ( c / a - 1 = 1.7 × 10 -2 and 1.3 × 10 -2 in CeZn and CeMg, respectively). Fitting magnetic properties in the ordered phase leads to G r = 220 and 50 mK in CeZn and CeMg, respectively. Note that these values have not been confirmed by measurements in the cubic paramagnetic phase. In CeZn the corresponding third-order magnetic susceptibility has been calculated to be positive in the vicinity of the N6el temperature, in agreement with the first-order character observed (fig. 20). In CeMg, the value G r = 50mK is a little bit too small: a value of about 75 mK would be large enough and more relevant.
/~-~=264 K
i
I'-~- ~=190KI
~
G.~=5OrnKI
CeZn
'75mK
1, ~ OOmK
\
0
I
/
20 TEM
I
PERATURE(K)
I
40
Fig. 20. Temperature dependence of the third-order magnetic susceptibility /~M,O - (3~ for CeZn and CeMg in the presence of quadrupolar interactions characterized by the G ~ values indicated.
erB 6 The same features are found again in P r B 6 , a compound isomorphous t o C e B 6. Here bilinear interactions are larger than the quadrupolar ones and a first-order antiferromagnetic transition is observed at 6.9 K (McCarthy et al. 1980). This first-order character seems to be stabilized by the same type of AFQ interactions as in the quadrupolar phase of C e B 6 ; indeed at low temperature the magnetic structure is described by the same propagation vectors (Burlet et al. 1988) and at high temperature large softenings of elastic constants are observed (Tamaki et al. 1985). By using quadrupolar coefficients deduced from f e B 6 (L/ithi et al. 1984) and NdB 6 (Tamaki et al. 1985) calculations within the present formalism show that the third-order magnetic susceptibility associated with the trigonal symmetry is positive. Measurements would allow to confirm this analysis.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
47
Tb pnictides Equi-atomic compounds within the NaCl-type structure have been extensively studied (Hulliger 1978). They are the earliest intermetallics observed to exhibit large one-ion magneto-elastic couplings (L6vy 1969, Mullen et al. 1974). In addition, from numerous studies, they have been shown to exhibit a more complex behaviour, they are at the meeting point of different couplings (section 3.2.4). However, a careful analysis of the antiferromagnetic TbX has shown that quadrupolar interactions unambiguously play a dominant role (K6tzler 1984). Indeed the AF transition is first order in TbP (T N = 7.1 K) and TbAs (T N = 12.5K) and second-order in TbSb (TN= 15.5K), TbBi (TN=17.5K), TbS (TN = 45 K), TbSe (TN = 49 K) and TbTe (T N = 51 K). All the compounds exhibit trigonal spontaneous distortions (Hulliger and Stucki 1978b). Figure 21 shows the resulting jumps of the magnetization and the susceptibility at the first-order transition in TbP compared to the behaviours observed in TbBi. In the t
__°_,--°_
i
TbP
MS(r) Ms(O)
i
i
°~Ooo~_-~ "~
(TN=7.1K )
.8 ! f
1.0
°~,
,
TbBi
\
IT, =17.5K)
~,
\
.6 .4
0.0
\
.2
o
0.4
Xo(T) Xo(T~)
0.6
0.8 T/TN
1.0
1.0
"'~ TbBi (TN :l?.SK)
~.0
0.5
.6
°
.8
a
1.0
3.5
..
0 (TN =7.1 K)
b o,
I
z
~
I
z,
~
I
6
~
T/TN
I
8
Fig. 21. Comparison between: (a) the spontaneous sublattice moments, and (b) the magnetic susceptibility for TbP and TbBi. Full lines are calculated within a formalism similar to that of section 2
(from Koetzler 1984).
48
P. MORIN and D. SCHMIT]"
paramagnetic phase, there are large magneto-elastic contributions to the elastic constants in TbP (Bucher et al. 1976) and TbSb (K6tzler 1984). The quadrupolar interactions are characterized by G" = 61, 32, 22 and 22 mK in TbP, TbAs, TbSb and TbBi, respectively (K6tzler 1984). These G ~ values lead to the temperature variations of X~u,o -(3) shown in fig. 22. Obviously the antiferromagnetic ordering occurs in TbP and TbAs at temperatures TN, for which xM,0 - (3) is positive; this explains the first-order transitions. From an experimental point of view, positive values of X}~) have been observed in TbP for the trigonal symmetry (fig. 23). For TbSb and TbBi, a second-order transition is predicted, as is experimentally
-/4
i
~32mK
I m
,
,~N
22m½
-10-4O
=
10 TEMPERATURE(K)
20
Fig. 22. Temperature dependence of the third-order magnetic susceptibility XM,O - (3) calculated in the presence of quadrupolar interactions in TbP (G ~= 61 mK), TbAs (G " = 32 mK) and TbSb, TbBi (G ~= 22 mK). W and x define the level scheme common to the four compounds. The first-order (second-order) transition at T N in TbP and TbAs (TbSb and TbBi) are driven by quadrupolar interactions.
0.00 B
,
,
X2)
~X =6inK
,
,
TbP ~ l~[I.]
o.oo/-, i i ~ 0.002 0.0
8
10
12
lt,,
T[K]
16
Fig. 23. Temperature dependence of the third-order magnetic susceptibility in TbP. Full lines are calculated with the indicated values for the trigonal quadrupolar coefficient Y( = 12G" (from Raffius and Koetzler 1983).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
49
observed. Note, however, that the critical point C strongly depends on G ~ and that TbSb and TbBi are not far from tricriticality of a quadrupolar origin.
3.2.2.2. Second-order transition in PrMge due to negative quadrupolar interactions. This compound orders ferromagnetically at T c = 10 K through a secondorder transition (Loidl et al. 1981). The ground state is a F 3 doublet. As mentioned in section 2.2.4, this CEF configuration gives rise to a positive value for Xz, - (3) in the vicinity of T c. This should produce by itself a first-order magnetic transition. On the other hand, the G ~ value suggested by Loidl et al. (1981) for explaining the temperature dependence of the spontaneous magnetic moment, i.e., G ~= -13.3 mK, leads to a negative total third-order magnetic susceptibility )¢(3) M,0" This now, accounts for the second-order character of the transition (fig. 24). I
.J.lo..
3
I
I
i
l
I
%%0 "
#2 PrMg21 0
5
T[K]
t~t 10
Fig. 24. Spontaneous magnetic moment, versus temperature in PrMg z. The dotted line is the Brillouin curve for J = 4. The full and broken lines give the result of a mean-field calculation with and without the quadrupolar interactions G ~= -13.3 mK (from Loidl et al. 1981).
Note that in the ferromagnetic phase, the ( O ° ) quadrupolar components are ferroquadrupolarly aligned by the dominant ferromagnetic interactions, whereas the negative quadrupolar interactions would prefer an AFQ arrangement and are thus frustrated.
3.2.3. Magnetic ordering in the quadrupolar phase The last case corresponds to a dipolar phase transition inside the quadrupolar phase (T < To). Below T c the quadrupoles are ordered and the crystal field is no longer cubic, e.g. it may have tetragonal symmetry. Therefore the various susceptibilities (X0, x~ " (3) , • • .) have new values (X~, x"~ T(3) . . . . ) according to the new tetragonal CEF level scheme. Note that this level scheme is now temperature dependent, like the ordered quadrupolar moment. The discussion of the nature of the dipolar transition is the same as in section 2.2.4, where /1-C(3) M,0 is replaced by )(T(3) 7 . Depending on the sign of X~ (3), the dipolar phase transition has a first- or second-order character.
50
P. MORIN and D. SCHMIqT
3.2.3.1. CeAg, f e B 6. In the case of CeAg, the ferromagnetic transition is second-order in the tetragonal phase (fig. 11) in accordance with theory. Indeed, . T(3) the x~ susceptibility is calculated to be negative as expected for a Kramers ion through the divergence of negative Curie-type terms associated with the magnetic doublet ground state. The same conclusion is also valid for CeB 6. 3.2.3.2. TmZn. In TmCd, bilinear interactions in the tetragonal phase are insufficient to induce a magnetic moment on the singlet ground state. On the contrary, in TmZn bilinear interactions generate a ferromagnetic ordering (T c = 8.12 K) slightly below the quadrupolar ordering (T o = 8.55 K). Calculations lead to a critical situation for Tc: the nature of the magnetic transition seems to be very sensitive to the values of the various parameters. For instance, T c is calculated to be of first order for G ~ = 25 inK, but would be of second-order with G v = 28 mK, as it experimentally appears. Consequently, the situation of TmZn with regard to the ferromagnetic transition seems to be not far from a tricritical point. 3.2.4. Presence of additional couplings In the compounds presented in section 3.2.2.1, the quadrupolar and Heisenberg bilinear interactions are dominant; any other interaction remains undetected; nonetheless, the competition between various couplings may be more balanced in other series such as rare earth pnictides and dialuminides. Two-ion magnetoelasticity and/or fluctuations may play a larger role in the minimization of the free energy. 3.2.4.1. DySb and rare earth monopnictides. This compound appeared in the past as an archetype for a first-order magnetic transition driven by quadrupolar interactions (Bucher et al. 1972b). It antiferromagnetically orders at TN = 9.5 K with a four-fold axis of easy magnetization. Associated with the magnetic structure, the lattice undergoes a structural symmetry lowering, mainly tetragonal, but with a small monoclinic component (L6vy 1969, Felcher et al. 1973) (fig. 25). This implies a magnetic moment not strictly parallel to the [001] axis. The existence of quadrupolar interactions has been unambiguously proved by the strong softening of C l l - C12 (Moran et al. 1973, Levy 1973); the value of G ~= 1 mK deduced is large enough to explain the first-order character of the joint spins and quadrupoles orderings, the third-order magnetic susceptibility is calculated to be positive (fig. 26). However, the analysis of the magnetization processes (Kouvel and Brun 1980, Everett and Streit 1979) and of the third-order magnetic susceptibility (A16onard et al. 1984b) in terms of only quadrupolar interactions fails (fig. 26). Indeed data are positive only in a very short range above TN and cannot be described by a unique G ~ value. Thus, additional spin interactions are present especially for the tetragonal symmetry. The most commonly proposed coupling is based on anisotropic bilinear interactions (Trammell 1963, Kim and Levy 1982, Jensen et al. 1980). It was in
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
.~;
,.y-e-V--L~c~-
Oa
h
....+, + , .
-I
l
~
-2-c
(t~tragonal)
-3-E
51
-4-__~n
n
,',
(cubique)
l° c/a
Oy P,
0.997 0.996
= Dy S b
0.995
-5--
Temperature
T° K
Fig. 25. Temperature dependence of the relative change of the tetragonal cell parameters in the ordered phase of DyP (two samples), DyAs and DySb; a 0 is the cubic cell parameter (from L6vy 1969).
a =
i
i
"
\
i
W=.2BK
i
~
x =-.82
GI~0"lmK 0
\.,~e3 1 ,,p o
=b
I
t
~
'
DI(kO~PB/ 3 0 ~
10
I
~
'
[0 0 1]
20 IEMPERATURE(K)
' W=.28K x =-.82 1~=-6.5K GI=1mK
30
Fig. 26. Temperature variation of the third-order magnetic susceptibility for H[I[001 ] in DySb. (a) tentative fits are drawn with the quadrupolar model alone, G 1 = G r = I mK being the value obtained from the softening of the Cu - C12 normal mode. (b) Fits here include, in addition to G: = i mK, the two-ion magneto-elastic coupling characterized by D1 = D r (from A16onard et al. 1984b).
52
P. MORIN and D. SCHMITT
particular postulated that rare earth pnictides, where the nearest-neighbour exchange is dominated by direct exchange and superexchange, are excellent candidates for large two-ion magneto-elastic effects (Lacueva and Levy 1985). From an experimental point of view, anisotropic bilinear couplings have been observed in the magnetic excitation spectrum of TbP (Loidl et al. 1979) and PrSb (Vettier et al. 1977). They have been also used for explaining the different antiferromagnetic structures found according to the temperature in GdX pnictides. Associated with these structures, different two-ion magneto-elastic strains have been also detected (Hulliger and Stucki 1978a, Hulliger and Siegrist 1979). For the experimental conditions which are used to study the third-order magnetic susceptibility , anisotropic bilinear interactions occur through their strain derivative, i.e., the two-ion magneto-elastic coupling, which may be described within the MFA formalism (Aldonard et al. 1984b). For DySb this coupling, characterized by a D 1 coefficient in fig. 26, is opposite to the quadrupolar interactions and improves the fit down to 12 K. However, a second-order transition would be expected. The same short-comings are also observed on the temperature dependence of the reciprocal susceptibility under an uniaxial [001] stress (Morin et al. 1985a). Even if present, two-ion magneto-elasticity, considered in the MFA, is not able to describe the features observed close to T~. On the other hand, a large number of studies has shown that NaCl-type compounds exhibit fluctuations through their critical behaviours close to the N6el temperature (Taub and Parente 1975, Hfilg et al. 1985). These critical fluctuations appear to be favoured in a deformable lattice (Taub and Williamson 1973). In DySb, as in HoSb, neutron diffraction measurements have shown an anomalously large magnetic scattering above T N. In the same type of experiments, the paramagnetic correlations in ErSb are characterized by a strong two-ion anisotropy (Knorr et al. 1983). In conclusion, the nature of the magnetic transition in numerous monopnictides seems to result from a well-balanced competition between different couplings. For many cases, it is difficult to determine the driving mechanism as e.g., in HoSb, which is very close to tricriticality (Jensen et al. 1980). The determination of quadrupolar interactions, coherent in CsCl-type compounds, cannot be achieved in these series. In addition the large dependence of anisotropic magnetic interactions on metallurgical stresses may change the nature of the transition in a given sample as shown in Er compounds (Hulliger and Natterer 1973). As in DySb the difficulty in properly considering fluctuations and anisotropic bilinear couplings may prevent one to fully understand the first-order transitions in DyBi (Hulliger 1980), DyP, DyAs (L6vy 1969), ErBi and ErSb (Hulliger and Natterer 1973) even if quadrupolar interactions are obviously present as proved by the temperature dependence of elastic modes. The R A l 2 series. RA12 compounds (C14 structure) appear to pose a similar problem as was observed in pnictides (del Moral et al. 1987). DyA12 orders ferromagnetically at T c --61.4K through a second-order process. The third-order magnetic susceptibility along the [001] easy magnetization direction is 3.2.4.2.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
-3 x 10"5~ /oB/KO¢3
I Gt=OrnK
I
70
80
53
mK i
-2 <3 -1
0
I 60
90
T(K]
Fig. 27. Temperature dependence of the third-order magnetic susceptibility A.... = X~ ) for HI[[001 ] in
DyAI2. Full lines are calculated with a bilinear coefficient n = 17.5 kOe//xB and the values indicated for the quadrupolar parameter G1 = G ~ (from Del Moral et al. 1987). negative in the entire paramagnetic range and cannot be described with a unique G ~ coefficient (fig. 27). However, the data close to T c do not indicate critical behaviour. The present range (8-10 mK) of G ~ is different from values deduced from magnetization measurements in the ordered phase, i.e. - 2 m K (Rossignol 1980). The same failure for describing the third-order susceptibility is found again in ErA12 (del Moral et al. 1987). Data go from positive values at high temperature to negatiye ones close to T c which countermands a first-order transition. The 2 possible G ~ values deduced are larger than in DyA12 in violation of the o~j variation law across the rare earth series (Schmitt and Levy 1985) (see section 5). All these features may be the signature of additional couplings and an improved knowledge would need systematic studies in particular of ultrasonic properties.
3.3. Determination of the quadrupolar parameters from susceptibility techniques Different ways of determining quadrupolar parameters are directly related to the various susceptibilities described in section 2. Indeed, these susceptibilities connect the magnetic variables on one hand and external stresses on the other hand. Thus, (i) the strain susceptibility represents the response of the quadrupolar moments to the corresponding stress; (ii) the quadrupolar field-susceptibility connects the quadrupolar moments to the magnetic field; (iii) the third-order magnetic susceptibility couples the magnetic moment, more exactly the initial curvature of the magnetization curves, and the magnetic field. Each of these three
54
P. M O R I N and D. S C H M I T T
susceptibilities may be acquired by a different experimental technique, namely measurement of the elastic constants, the parastriction and the low-field magnetization, respectively; these techniques are described below. 3.3.1. Elastic constants
As seen in section 2.2, the existence of a magneto-elastic coupling leads to the softening of the elastic constants of appropriate symmetry when the temperature is lowered, and this softening is enhanced by positive two-ion quadrupolar interactions. The experimental technique generally used is the ultrasonic velocity measurement, and several geometrical configurations are considered in order to observe the various ultrasonic propagation modes. In general, the normal (symmetrized) elastic constants are not directly accessible; they have to be obtained from a linear combination of those which are measured. The observation of a softening of elastic constants due to magneto-elastic coupling in the intermetallic rare earth compounds was realized in the early 1970s, and extensively studied in particular in cubic rare-earth antimonides (see figs. 28 and 29). Several behaviours are observed, according to the strength of the quadrupolar (magneto-elastic) coupling as well as the nature of the CEF ground state. The temperature variation of the elastic constants in LaSb is taken as the background for the other compounds. Large anomalies are often detected at the magnetic ordering temperatures and below; they are due to domain-wall stress effects. However, the most spectacular effect remains the softening above the ordering
I.C
0.8
0.6
O~
°il I
'
8b
'
I
J~o
'
z~,o
TEMPERATURE(°K)
'
3~o
Fig. 28. Change in C o = C ~ = (Cal - C12) as a function of temperature in DySb. A fit of the data leads to (g0) 2 = GEE = 1 . 4 m K and I o = K v = - 0 . 4 m K and thus G ~ = 1 mK (from Moran et al. 1973).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
14.0~t3.9~
55
TmSb
13.8 ~:
13.6 r ~
13.5 Ci1-C12 • I
1
5
.
2
el
~
~2.70 ee o
- • • ee~ e
~:__.z6.sz6.6z6.4z6.~
o
°
C44 ee
"
I
I
Ioo
200
300
T (°K)
Fig. 29. Temperature dependence of the symmetrized modes Clt - C12 = C 7 and C44 = ½ C e for TmSb. Full lines are calculated with only magneto-elastic couplings in the presence of CEF (from Mullen et al. 1974).
temperature, e.g., this softening reaches about 60% of the room-temperature value of C r at 10 K in DySb and occurs over a wide range of temperature (see fig. 28). A good description of the softening of the C ~ elastic constant requires the introduction of a two-ion quadrupolar interaction in addition to the one-ion magneto-elastic coupling (Levy 1973, Moran et al. 1973). The cubic TmX series (X = Zn, Cd, Cu, Ag, Mg) is another series which is particularly spectacular with regard to the softening of the C v elastic constant. For example in TmZn and TmCu (fig. 30) the softening reaches more than 50% of the room temperature value just above the ordering temperature, indicating very strong quadrupolar interactions (Liithi et al. 1979, Jaussaud et al. 1980). In the opposite limit, the softening is much less pronounced in TmAg and TmMg (Giraud et al. 1986, Giraud and Morin 1986). It is worth noting that the analysis of the temperature variation of the elastic constants may give information about the crystal field in addition to the magnitude of the quadrupolar parameters. Indeed, according to the nature and spacing of the low-lying CEF levels, anomalies may occur at low temperature in the strain susceptibilities, thus in the elastic constants (see section 2.2.3). This is particularly emphasized in singlet ground-state systems as in PrSb (Liithi et al. 1973a) and TmSb (see fig. 29) for both C ~ and C ~ modes.
56
P. M O R I N and D. S C H M I T T
I
I
I
l
l
C~(YCu) •/ ' 1 / " .~.I
/
,t~
/.
E
c" :½(c,-c,2
?
o
I
O,l 0
T
t..j
i
- -
experiment fit with G1 = l l . 5 m K R1 = 7,0 mK
J
I
l
100
I
I
2OO
300
T(K)
Fig. 30. T e m p e r a t u r e d e p e n d e n c e of the C' = ½C r = ½(C n - Ct2 ) m o d e in T m C u ; the full line is a theoretical fit with G 1 = G ~ and K 1 = K ~ ( f r o m J a u s s a u d et al. 1980).
3.3.2. Parastriction Parastriction constitutes a very suitable technique for investigating magnetoelastic coupling. It consists of measuring changes in the length of a sample in an applied magnetic field in the paramagnetic phase (Morin et al. 1978c, 1980b). It differs from the usual magnetostriction by the fact that only the low-field variation of the latter is analyzed, namely its H2-dependence. Indeed, as shown above (section 2.2.1), within the limits of validity of the perturbation theory, the coefficient of this HZ-variation is related to the quadrupolar field-susceptibility Xo (or Xp) which depends only on the crystal field and on the magnetic and magneto-elastic parameters. In addition, in agreement with the classical hightemperature approximation (Callen and Callen 1965), a linearization of this susceptibility may be achieved by plotting ) ( Q 1 / 2 a s a function of the temperature. Deviations from this linear behaviour may occur at low temperature due to CEF effects in the same way it does for the magnetic susceptibility. Experimentally the measurement of the relative change of length A = 8l/l is performed successively along and perpendicular to the direction of the applied magnetic fields; one eliminates the volume effects e ~ by considering the difference All - A~. As a consequence, for a magnetic field along a four-fold or three-fold direction, and according to the expression for 81/l [eq. (25)], one obtains respectively
(All- Al)v =
~93_B~ 2 C---~oXQH2'
3 B+ (All- A±)+ = V'2 C O x e H 2 '
(110)
for the two tetragonal and trigonal strain modes. It is worth noting that the same
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
57
relations are found, by considering a fixed measurement direction and by applying successively the magnetic field parallel and perpendicular to that direction. This latter geometry is habitually used by experimentalists for the sake of simplicity. As mentioned above it is useful to plot HI(I All - A± I) 1/2 in order to observe the linear high-temperature b~haviour. A good example is provided by the compound TmZn where the quadrupolar effects are particularly strong (fig. 31). According to eq. 110, the high-temperature slopes give the magneto-elastic coefficients B v and B', as the background elastic constants are known from other measurements. ~I~
~
I
0
,
v
o, o / / '
I
O•
."
I
I
1
Fig. 31. Parastriction of TmZn, measured for the tetragonal and trigonal symmetry lowering modes. Curves are calculated within the susceptibility formalism (G ~= Ga) (from Morin et al. 1980a).
In addition, large deviations from this linear behaviour are observed for the tetragonal mode below 100 K due to CEF effects and to a strong G v parameter. On the other hand the anisotropy of the parastriction is particularly obvious in this compound, showing that measuring the magnetostriction on polycrystals is quite meaningless in these anisotropic systems. Note also that the parastriction gives the sign of the magneto-elastic coefficients, while only its absolute value may be obtained from elastic constant measurements. This technique has been successfully used in several series of rare earth compounds, in particular in RZn compounds where the coefficients associated with both the tetragonal and trigonal strain modes have been obtained for the heavy rare earths (figs. 32 and 33).
3.3.3. Third-order paramagnetic susceptibility The third-order paramagnetic susceptibility has been introduced as a new method for studying quadrupolar interactions in rare-earth compounds (Morin and Schmitt 1979, 1981b). It consists of a detailed analysis of the magnetization induced by an external field in the paramagnetic phase. More precisely it
58
P. MORIN and D. SCHMITF =
i
1S
Er Zn/y
/.
H . [ oo]
.
..~
.
•
,/ TmZn
%s DyZn
"r
20o
too
300
(K)
TEMPERATURE
Fig. 32. Temperature variations of g/I;,, All 1/2 for tetragonal strain in RZn. The full lines are calculated dependencies (from Morin et al. 1980b). -
H //
8 ~o
1o
v
[111]
TmZn DyZn
5
TbZn
"I-
i
100
TEMPERATURE
200
300
(K)
Fig. 33. Temperature variation of H/l All- A±I 1/2 (rhombohedral strain) in RZn. The full lines are calculated with the susceptibility formalism (from Morin et al. 1980b).
corresponds to the second term (in H 3) of th e development of the magnetization as a function of the magnetic field, this term depends on the total quadrupolar interactions in addition to the crystal field (see section 2.2.1). An experimental way to investigate the magnetization curves is to plot M / H versus H2; this provides both the first-order magnetic susceptibility XM, i.e. the
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
59
isotropic null field value, and the third-order one X~ ) , i.e. the anisotropic slope of the linear low-field range. This is well-illustrated by the compound TmCd, where the anisotropic character of X~ ) is particularly obvious (see fig. 34). It is worth noting that the linear part of these curves diminishes as the temperature is decreased, since the higher-order terms (in H s , . . . ) become relatively more and more important. An alternative experimental way, although less direct, is to consider the Arrott's plots M 2 versus H/M. In low field, the variation is linear; the intersection with the x-axis yields (X~)) --1 while the third-order magnetic susceptibility is related to the slope dM2/d(H/M) through the relation
X(~ )
-(XM)4/(dM2/d(H/M)) .
=
The variation of the third-order magnetic susceptibility may then be drawn as a function of the temperature and analyzed with the above formalism (see fig. 35). As an other example, in fig. 36 we show the temperature dependence of X~ ) for the two main symmetry directions in PrPb 3. The anisotropic character of X~ ) is obvious from these curves again, as well as its dependence on the bilinear and quadrupolar interactions. The third-order magnetic susceptibility also explains the paramagnetic anisotropy of the magnetization observed in TbxLal_xA12 for a constant magnetic field within a given plane (Hoenig 1980). The technique of third-order magnetic susceptibility has also been applied to the tetragonal compound TmPO 4 which presents noticeable non-linear terms in the low-field magnetization within the basal plane (Andronenko et al. 1983). Another experimental way to extract X~ ) is to perform torque anisotropy 120
i
"
~
"- "
115
[001]
6 11o ~,
"
"
"
"
,
[111]
TmCd
' ~ 105 EI 77
"
~ - - - - ' -
"" ~ ~
""
76
K
[OO]l
75 74 .....
0
i 100
2
Hi
I ~ 200
I 300
400
(k(De2 )
Fig. 34. Experimentalvalues of M / H versus H 2 in TmCd (from Morin and Schmitt 1982c).
60
P, MORIN and D. SCHMITT
//// ///"
&"
!// II
-4
:o,5 K
,:o
/
×
~
5
,
=-o.34
i
10 15 20 TEMPERATURE ( K )
i
25
30
Fig. 35. Experimental and calculated variations with temperature of X~ ) along the [001] axis in TmCd system. Note the dependence of the coefficient G1 = G ~ on dilution (from Morin and Schmitt 1981b).
1 e*=-9K GI=O '. -
tI
i
l
H//[001]
~o O ~.~:E ~<
I
I
H//[111]
'
]
\", i L
i
PrPb3 W=-.65K x=.6
\
1¢---9K JG .3K
i . . . . . . . . . ;_:.=_._.....
K ............2 : 2 : ~ _-'r='- .....
'G =, --.9. -15mK -/ , //--GI="0' ' - I . IG2= ~ O*---09K \
,e*-gK ;.2K O,=G2__O// //
//
/
~_ /
// /
-1
I
0
5
I
I
10 5 TEMPERATURE (K)
/I
10
15
Fig. 36. Temperature variation of the third-order susceptibility along [001] and [111] in PrPb 3. Dots are experimental points. The dashed lines correspond to behaviours without quadrupolar interactions, but show the effects of the bilinear interactions (0" = - 9 K). The full lines show quadrupolar effects (G v= G1, G ~ = G2) (from Morin et al. 1982). m e a s u r e m e n t s which, in p r i n c i p l e , allow us to d e t e r m i n e p a r a m a g n e t i c susceptibility t e n s o r s o f f o u r t h (~- X ~ )) a n d e v e n sixth r a n k s a n d has b e e n a p p l i e d to R A I 2 c o m p o u n d s ( D e l M o r a l et al. 1987). A n a l t e r n a t e a p p r o a c h to a s c e r t a i n i n g t h e q u a d r u p o l a r effects o n t h e m a g n e t i z a t i o n is b a s e d o n t h e n o n - l i n e a r i t y o f t h e e x c h a n g e field Hexch as a f u n c t i o n o f M
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
61
(Kouvel and Brun 1980) Hex~h = AM
+ A'M 3 .
(111)
However, this procedure provides a )t' value which is temperature dependent, since it is related to the true quadrupolar parameter G" through the relation )t, =
2G u'(X~)) 2 (X0)4( 1 _ G~X# )
(112)
This ,V parameter is therefore connected with the quadrupolar part of X}~); its temperature variation has indeed been observed (Abu-Aljarayesh et al. 1986a)
3.4. Effects on the magnetostriction While parastriction allows us to describe the initial magnetostriction in the paramagnetic phase, i.e. the first term in its field development, a full diagonalization of the Hamiltonian describes the whole magnetostriction curve, in the ordered or non-ordered state, provided the magneto-elastic couplings and all the two-ion interactions are included.
3.4.1. Magnetostriction of dilute compounds The main advantage in investigating dilute magnetic compounds is to be free, at least to a large degree, of two-ion interactions. Indeed it is these interactions that make the analysis difficult. If the effect of the dilution on the macroscopic measured strain is correctly taken into account, one can determine the microscopic one-ion magneto-elastic coupling. Usually authors simply assume that the macroscopic deformation is proportional to the magnetic impurities concentration (Nieuwenhuys et al. 1982b, Pureur et al. 1985). A systematic study of the forced magnetostriction has been carried out for the cubic noble metals compounds (silver, gold) containing magnetic rare earth impurities (Creuzet and Campbell 1981, Campbell and Creuzet 1985). The magnetostriction follows the second-order Stevens coefficient a I (see figs. 37 and 38), indicating that a single-ion model is pertinent, and that the second-rank magneto-elasticity is preponderant in these systems. Nevertheless, a fourth-rank magneto-elastic coupling has also been determined, its contribution is about ten times smaller than the second-rank one. In addition to the magneto-elastic coefficients, the magnetostriction curves are strongly affected by the nature of the low-lying CEF levels. This allows one to choose between possible levels schemes through the shape as well as the temperature variation of the magnetostriction curves (fig. 39). Another spectacular effect related to the crystal field concerns the so-called "reverse" magnetostriction, i.e. the sign reversal of the magnetostriction between the low and high temperatures. This occurs in CexLal_xSb, SmxLaa_xSb (Nieuwenhuys et al. 1982a), and CexLal_~Bi (H~ifner et al. 1983) compounds.
62
P. MORIN and D. SCHMITF 14,
y
lu-,,-~nt
12
10' 8 6
~
14. •106x6L "~Ag Tb
Ag Tb
t
12.
10, 6
A
4 2 0
2
~
2 -4
--6 -8
kG
(a)
0
AgEr
-2
X,~Tm \
-t, -6 -e
~
o
H.kG 0AgEr AgTm (b)
Fig. 37. Parallel magnetostriction data for a field applied along the [001] axis of silver-0.5% rare earth single crystals at 1.4 K: (a) experimental data, (b) calculated neglecting the e~ contribution (from Campbell et al. 1979).
6, ' 106 x"~t
.
/
6-
/AgTb
4.
AgDy
4 / - -
A_gHo
4u , 4u , 6,0 ,H.kG 0[ ~
,
H,kG
-2 -4. •
(a)
-q
fo)
Fig. 38. Parallel magnetostriction data for a field applied along the [111] axis of silver-0.5% rare earth single crystals at 1.4 K: (a) experimental, (b) calculated neglecting the e~ contribution (from Campbell et al. 1979).
For example, the change of sign of the measured strain takes place at 2 K in LaBiCe and at 8 K in LaSbCe (see fig. 40). This can be explained by a simple crystal field theory, which, in some particular cases, leads to an expectation value of the quadrupolar operator O ° which is not always positive. For example, in Ce0.15La0.85Sb, the quadrupolar m o m e n t induced by an external magnetic field on the F 7 doublet ground state is negative at low temperatures. It changes its sign either for very large magnetic fields (about 40 T at 1.5 K) or above 8 K for any field, namely when the influence of the excited F 8 quartet becomes important. This change of sign of the initial magnetostriction curvature is accounted for by the formalism developed in section 2 and corresponds to the change of sign of the quadrupolar field susceptibility x~ - (2) (see fig. 41). Similar reverse magnetostriction
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
~-110-6) 20
40
60
t
t
i
63
H(kG)
_1
_2 ~2.8 t
_3
Fig. 39. [001] parallel magnetostriction in ErA g (concentration 0.5%) corrected for the volume contribution; circles are experimental data, solid lines calculated. Note the anomalous variation with temperature which is characteristic of the CEF level scheme (from Creuzet and Campbell 1982b).
LaBi:Ce (lO°/o)
6 4
-5T
2
/I 3T
O -2
_1 ~-
-4
o
~ ~\
3 2/
O
LaSb:Ce (15%)
4.8T k,,..~ 4 T
5
10 15 T , (K)
20
Fig. 40. Comparison of the magnetostriction of (a) LaBi : Ce (upper panel) and (b) LaSb : Ce (lower panel) as a function of temperature at various fields. As clearly seen the "reverse magnetostriction" occurs at a lower temperature for (a) than for (b). Solid lines represent a fit to theory (from H/ifner et al. 1983).
64
P. M O R I N and D. SCHMITT r
50
-50
o
LaSb:Ce
LoBi:Ce
/ /
40o
•>• 0
-400
lo
T (K)
20
Fig. 41. Temperature dependences of Xy(~) , the quadrupolar field-susceptibility, calculated in the presence of only the CEF for LaBi : Ce (a), LaSb : Ce (b), LuCd : Tm (c) and LuZn : Tm (d). In these four cases the CEF level scheme drives the so-called reverse magnetostriction.
effects may be expected in other situations, e.g., with a F5 triplet as ground state, as in TmCd or TmZn, for an appropriate dilution.
3.4.2. Magnetostriction in the ordered phase The magnetostriction observed in the ordered phase of magnetic compounds is generally much larger than that in the paramagnetic phase. This is due to the large influence of the exchange field which tends to saturate both the magnetic and quadrupolar moments. The symmetry of the cell distortion is directly related to the direction of the magnetic moments, e.g., in ferromagnetic compounds. In cubic antiferromagnets, the observation of a spontaneous strain may be very helpful in determining the actual magnetic structure (see section 3.6). A quantitative analysis remains more difficult than for a non-ordered state, because of two-ion effects. Depending on the strength of the bilinear exchange coupling, it has been done using either a semi-classical (see section 2.4) or a quantum treatment (see section 2.1.2). In the case where the bilinear interactions are big, large magnetostrictions have been measured at room temP6erature, as e.g. in TbFe 2 where a record magnetostriction of Am = 2400 × 10- was reported (see fig. 42). This motivated numerous investigations because of the interest of technological applications. Still higher strains have been measured at low temperature in the RFe 2 series (see section 4). Huge magnetostriction has also been observed at low temperature in numerous series where the alloyed metal is nonmagnetic, in particular within the NaC1- and CsCl-type structures (L6vy 1969, Morin et al. 1977). Large tetragonal strains of the order of 10 - 2 have been measured; e.g., c / a - 1 =--1.4% in PrZn at 1.5K (Morin and Pierre 1975). In TbZn a change of easy direction within the ordered
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
I
I
I Tb Fe2
'
L
]
65
|
2,000I
~.poly [hi
1,000 t Y
0
V
I
I
I
I
I
5
10
15 H [kOel
20
25
F
Fig. 42. Magnetostriction of TbFe z at room temperature for a single crystal (Aiq curve) and a polycrystal taken for two different orientations (a and b) (from Clark et al. 1974).
10 -2
10- 3
IO-~
Lu
TbZn
1
io-s
10 -6
u~o.s 10" ;' 0 10 - 8
.....
i
100 i
100
200 )
I
200 TEMPERATURE OQ
300
Fig. 43. Thermal variations of the spontaneous strain e 3 = e~ above the temperature at which there is a change in direction of the easy axis and the e 3 strain induced by a 1 kOe field in the paramagnetic state for TbZn ((3, strain-gauge data; Q, capacitance dilatometer data; full curves are calculated variations). Inset: experimental variation of the volume anomaly e v = X/3 e" (from Morin et al. 1977).
66
P. MORIN and D. S C H M I T r -2 10
10.3
10-4
10~ 5 U3
TbZn
10-6
TC
H: lkCe 10- 7
\ ,
10- 8
,
~"
q.
100 200 TEMPERATURE (K)
300
Fig. 44. V a r i a t i o n s w i t h t e m p e r a t u r e o f the spontaneous strain sxz = e~/~/2 b e l o w the temperature where the easy axis changes, and of the exz strain induced by a 1 kOe field in the paramagnetic range for ThZn ( 0 , strain-gauge data; @, capacitance dilatometer data; full lines are calculated variations) (from Morin et al. 1977). ,
Tm Zn
2
~
,
T - 107 K 1
0
-- '?1 _..
01i°, E
9 10 11 TEMPERATURE (K)
-4
o
2
,
6
;
APPLIED FIELD
;o
12
(kOe)
Fig. 45. Experimental variations of the strains parallel (All) and perpendicular ( A l ) to the four-fold direction of the magnetic field at 10.7 K in TmZn. Full and dashed lines are relative to fast and slow rates of variation of the field. The inset shows the temperature dependence of the critical field H c (from Morin et al. 1978b).
QUADRUPOLAR
EFFECTS IN RARE EARTH INTERMETALLICS
67
range occurs; this allows one to follow the temperature variation of the strain associated with the [001] and [101] moment axes (figs. 43 and 44). Finally, in TmZn one transition appears on the magnetostriction curves at temperatures above the Curie temperature (see fig. 45). This behaviour, also observed on the magnetization curves (see section 3.5), can be explained by the appearance of a strained ferromagnetic state above a critical field within the paramagnetic phase, arising from strong quadrupolar interactions.
3.5. Effects on the magnetization processes 3.5.1. Paramagnetic phase Perturbation theory has shown that quadrupolar interactions strongly influence the initial curvature of the magnetization as a function of field in the paramagnetic phase, i.e., the third-order paramagnetic susceptibility (section 2.2). In the same way, quadrupolar couplings influence the higher-order terms, i.e. they have an effect on the whole magnetization process. A diagonalization of the full Hamiltonian allows us to take these couplings into account particularly at high fields. As an example, the magnetization is predicted to be largest along a three-fold axis in [111]
(ooi] I
6
G,=9mK
.... +.:;,~ I
.0 ~+~+- +.+.+-16K
,
/ . +] ~o+- ~o+2OK I
4
-/'/+/4,/*/+°I o/'/+ 4,/ /0 I / /4,/ / ] 0 I/I /4" 4,0 /
d'i
2
o~ +~
•..'+25K,
/n
-
W=1.4 K x =-.42
•
• 4,o011/ +4,0 • 1+4,0 • o+~Oll u
+.,.
I
~'."
jEP=_ 3K
J
"'"
x =-42
~ ,
4 I-
W=I.4K E)w:I 3K ,
" 9K
+ ~...
..-'.~"
G.=gr~--I "11 -----
•
• +~/@/4,/0/
..,+0o., •
--
4~~
e~ e 0 i / @ .:+...-..
04"@0
0
.-1;:;. / 4+
|
e/o/../•
i+tOll t ~+ ll . .
~.-
I
G2=-80 mK
//+."
/."
/
.di.".
-
®// m II
,g .'"
0
I;::"
,
0
SO
/ 100
0 H(k~)
W=I.4K
? ." .e • •
x =-.42
!
50
it~=. 3 K 100
Fig. 46. Magnetization curves along the directions [001] and [111] for TmCu at various temperatures; lines are theoretical fits with the p a r a m e t e r s indicated ( G 1 = G ~, G 2 = G ~) ( f r o m J a u s s a u d et al. 1980).
68
P. MORIN and D. SCHMITF
TmCu in the absence of quadrupolar interactions; by using appropriate values of the parameters G ~ and G ~ one can restore the four-fold axis as the direction where the magnetization is maximum, as observed experimentally (see fig. 46). The magnetization curves show a more spectacular behaviour in case of huge quadrupolar couplings, as in TmZn (Morin et al. 1978b). In this compound, the quadrupolar coupling is strong enough to induce a transition from the paramagnetic to a ferromagnetic state in a wide range of temperatures above the quadrupolar ordering (see fig. 47). This transition occurs for a critical field which depends on the strength of the quadrupolar interactions and is well-described within the MFA. The critical field is also present along the directions of hard
.~
1K 111
0
20
(b
40
20
60
INTERNAL FIELD
80
100
120
(kO~)
Fig. 47. Magnetization curves of TmZn along the [001] axis at different temperatures; the insets show the temperature dependence of the critical field He; (a) experimental, (b) calculated (from Morin et al. 1978b).
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
69
magnetization, although it is larger than along the easy direction, i.e. [001]. For example, along the [101] direction, the magnetic moment is first aligned with the external field, then rotates towards the [001] axis when the component of the field along the [001] direction reaches the associated critical value; finally it comes back toward the initial [101] axis when the field is large enough.
3.5.2. Ordered phase In the ordered state, quadrupolar interactions play an important role in the free energy balance between the various crystallographic axes. In particular, when the field is applied along a direction of hard magnetization in a ferromagnetic compound, the magnetic moment rotates from the easy axis towards the field. The area between the corresponding curve and that extrapolated from the high field behaviour is related to the anisotropy of energy between both directions. This energy difference is strongly influenced by the quadrupolar couplings, as for example in ErZn (see fig. 48). This behaviour is enhanced when the free energies corresponding to two moment directions cross each other at a given temperature T R within the ordered phase. This leads to a first-order phase transition in which the direction of easy magnetization changes. This phenomenon is driven by entropy contributions induced by the anisotropic splitting of the CEF level scheme under the exchange field (see fig. 49). There are many examples in rare earth compounds, especially
6
5F//
b >
, 1mK-
~_o4~.4
* [101]
~I Z
=(I 1 I)
3 I
'
'
'
l
O N FLU Z L~ :E
,
Z/
~ 10
i:I \ 1 TEMIPERATUREI(K) .
4 0
;o
~o
INTERNAL
30 FIELD
40
50
(kOe)
Fig. 48. Magnetization curves in E r Z n along the three principal cubic directions at (a) 4.2 K and (b) 10 K; the inset shows the temperature dependence of the critical field; the points are the experimental data along [101] (O), [111] (zX) and [001] ( 0 ) ; the lines are the theoretical fits (broken curves: without quadrupolar terms; full curves: with G ~ = G 1 = 1.5 m K and G ~ = G 2 = 11 m K terms (from Morin and Schmitt 1978).
70
P. MORIN and D. SCHMITT - 250
-50P
///
HoZn 12
-
// /- ~" " i l l
v
>.-
L9
"Tl
-15OF
-26G
0~ W
Z W
/
W W
_.__~[i01]
//
----
// . . . .
G2:0
/
rr
b_
.... -270 0
G1 : - 4 mK G2= 10 mK
/ / ~[101] i 20
GI=0
i 40
TEMPERATURE
-2sot-
0
(K)
80
[lOl]
[111]
Fig. 49. (Left-hand part) calculated temperature variation of the free energy along [101] and [ l l l ] for HoZn, without quadrupolar terms (broken curves) and with G ~ and G ~ contributions (full curves); (right-hand part) arrangement of the lowest CEF levels in HoZn at T R = 25 K according to whether the moment lies along the [101] or [1ll] direction and taking into account the quadrupolar coefficients G ~= - 4 inK, G ~= 10 mK (from Morin and Schmitt 1978).
with Nd, Tb and Ho (see, e.g., Gignoux et al. 1975, Sankar et al. 1977, A16onard and Morin 1985). In each compound, the two-fold axis is the direction of easy magnetization at low temperature. In HoZn for example, the critical temperature T R is calculated to be 48K with tile CEF alone (Morin and Schmitt 1978). Introducing appropriate quadrupolar parameters G r and G ~ shifts the free energy curves and allows to reduce T R to 25 K in agreement with the experimental data (fig. 49). The effects of quadrupolar interactions on the magnetization processes in antiferromagnets are more difficult to analyze due to the existence of several sublattices (Jensen et al. 1980). On the other hand, they are more relevant for the stability of the actual magnetic structures. This will be discussed in the next section (3.6).
3.5.3. Effect on the direction of easy magnetization In the previous section, the dominant CEF determined the direction of easy magnetization and the quadrupolar interactions determined the exact temperature TR, for the change of the easy axis. In this section, we discuss some cases where the quadrupolar energies are larger than the CEF anisotropy. Good examples are provided by the ferromagnets DyCd and DyZn which have the CsCl-type structure (T c - - 8 0 and 140K, respectively). In both cases, a four-fold axis is observed as the direction of easy magnetization and a large tetragonal spontaneous strain occurs in the ordered phase (0.8% at 4.2K in DyZn) (Morin et al. 1977). Their cubic level schemes are given in fig. 50a together with those of the antiferromagnets DyCu and DyAg. Their anisotropic splitting by bilinear interactions always favours the three-fold axis: the CEF
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
zOO
71
100
I
r(3) DyAgi~--~. <..8 ~
!~-Y~.-~-~.~ /
200
c11"~i
®
.....
o
I
@easyaxis [I11]~k~O011
-200
2s~
W=-.57K \ 6
~
(-.}7)
-400
¢_-60K
',
(.9
\
T:4.2K J
(-5)
0 0
-.5
X
Fig. 50. (a) CEF level scheme in dysprosium compounds having the CsCl-type structure. E is the energy in Kelvin, W is a scale factor, the value of which is given in parentheses for each compound, x is the ratio between fourth- and sixth-order CEF terms; (b) x-dependence of the strain susceptibilities, X2 (=Xr) and X~ (=X~), for a given W at low temperature; (c) easy axis at low temperature according to x and G1 ( = G ~) for given values of W and 0". The full line is mainly related to the case of DyAg and DyCu; for DyZn and DyCd, the critical line would lie lower (from Al6onard et al. 1984a).
anisotropy between three-fold and four-fold axes is too large in DyAg and DyCu to be reversed by the tetragonal quadrupolar energy; the three-fold axes remain the easy ones and the (negative) trigonal quadrupolar interactions stabilize the muti-axial spin arrangement (section 3.6.1). In DyZn and DyCd, tetragonal quadrupolar interactions have the same order
8
-J - ~ m ] ~ ~ ~ _ _ .1 t~o~. . . . . . . . . . . -o-
7 "
i
~
" o
.
o
n
o
o[~OO1~
-
- -
2
GI=ImK ,G2=-22inK
N
ErNi 2
Z
. . . . . GI=O ,G2=0 T=1.5 K
:E'
o
2'o
go
go
8'0
'60
1~,o
INTERNAL MAGNETIC FIELD (kOe)
Fig. 51. ErNi2: magnetization variation at 1.5 K as a function of internal field. ©, • and [] are experimental values. Full lines are the variations calculated with 0* = 7 K, W = - 1.0 K, x = - 0.35, G1 (=G ~) = 1 mK and G 2 ( = G ~) = - 2 2 m K . Dashed lines are the variations calculated without quadrupolar terms (from Gignoux and Givord 1983).
72
E MORIN and D. SCHMITr
of magnitude as in DyCu and DyAg, but are more efficient due to the groundstate wave functions (see fig. 50b). They are then able to make the four-fold axis the easy direction above a G r critical value depending on the exact level scheme. This is realized for D y Z n as well as for DyCd (A16onard and Morin 1985). These two compounds represent the border-line case of quadrupolar effects on the temperature T R for the change of the axis of easy magnetization in the ordered phase (section 3.5.2); also in the paramagnetic state the quadrupolar effects are noticed through anisotropic magnetization processes. Similar features may be found also in other series specially when the C E F anisotropy is weak. This is, e.g., the case in ferromagnetic ErNi 2 which exhibits a [001] easy magnetization axis (fig. 51). The magnetization processes along the three main cubic directions are characterized by a crossing between the [001] magnetization curve and the [101] and [111] ones. Their description, as in D y Z n and DyCd, is impossible without quadrupolar interactions stabilizing the [001] axis as the easy one, instead of the [111] axis. 3.6. Effects on the magnetic structures 3. 6.1. Multi-axial structures In addition to their effects on the magnetic transition, the quadrupolar interactions also contribute to the determination of the magnetic structure through the minimization of the free energy. It is well-known that in cubic symmetry the spin structures that are described by one or more propagation vectors, q, which belong to the same star have the same bilinear energy and lead to identical powder neutron diffraction spectra (Wintemberger and Chamard-Bois 1972). The presence of several propagation vectors determine a multi-axial arrangement for spins (see, e.g., fig. 52). The knowledge of the C E F allows one to partly remove this ambiguity.
(17 FI O )
(a)
(b)
@=90 °
(c)
(d) Fig. 52. Possible configurationsfor the ( ½, 1,0)-type antiferromagneticstructure with 0 = 90°; (a) and (d) collinear; (b), (c) and (e) bi-axis; (f) multi-axis. All these structures lead to the same powder neutron diffraction spectrum as that observed in PrAg (from Morin and Schmitt 1982b).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
73
The magneto-elasticity can determine the actual structure. For instance with a four-fold easy axis, a large ( c / a - 1) magnetostriction may be a clear signature for a collinear arrangement as observed in many CsCl-type compounds (Morin et al. 1978a, Schmitt et al. 1978). On the contrary, the existence of collinear structures with three-fold easy axes has not been demonstrated for CsCl-type rare earth intermetallics. However, there is plentiful evidence for this in the NaCl-type series, in particular in Tb compounds (L6vy 1969, Hulliger and Stucki 1978b). In the presence of negative quadrupolar interactions, multi-axial spin structures are stabilized. This occurs within the trigonal symmetry in CsCl-type compounds as DyAg, DyCu (A16onard et al. 1984a) and NdZn (Morin and de Combarieu 1975) and in the AuCu3-type compound TmGa 3. In the latter compound for instance, powder neutron diffraction spectra have been indexed by using a [ ½ ½0] propagation vector and a [001] spin direction and the hypothesis of a collinear spin arrangement (Morin et al. 1987a). In spite of the sizeable magneto-elastic coupling found in the non-ordered phase (Morin et al. 1987c), no spontaneous strain was observed, making this collinear structure unlikely. The actual spin structure was determined by neutron diffraction under a magnetic field applied along the three main axes of a single crystal (Morin et al. 1987d). No change of magnetic intensities was detected when cooling the sample in different applied fields. This indicates the sample is spontaneously single domain; consequently, the actual structure is a triple-q structure which preserves the cubic symmetry (see fig. 53). This solution was confirmed by determining the different structures revealed by magnetization processes. In particular, along the Jill] direction, the quadrupolar structure and energy are unchanged at the lower critical field, He1, only the spins being reversed (fig. 54), the Zeeman energy corresponds mainly to this change of bilinear energy. At the high critical field, He2, the Zeeman energy has
•
• HC2
-,-10
I
I
2
3 4 5 TEMPERATURE (K)
I
TI
I
I
6
Fig. 53. Magnetic phase diagram for a magnetic field along a three-fold axis in TmGa 3 (from Morin et al. 1987d).
74
P. MORIN and D. SCHMITT
oooo0O0,°~° o°°° O
T=
1,5°4, " o •
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to compensate both the bilinear and quadrupolar energies as the AFQ arrangement is broken at this field. The existence of multi-axial spin structures in rare earth antiferromagnets with high symmetry appears to be a common consequence of AFQ interactions. In series other than the CsC1- and AuCu3-type ones, multi-axial structures also exist, e.g., as in rare earth hexaborides. In the AFQ phase of CeB6, the local anisotropy drives the spins to order in a multi-axial arrangement described by two propagation vectors [¼ 1 ½] and [1 _ ] ½] whereas the AFQ structure itself is described by [½ ½ ½] (fig. 12). The magnetic moments of the two cerium AFQ sublattices order along two mutually perpendicular directions (Effantin et al. 1985). The low-temperature magnetic structure of PrB 6 was initially described by a single [I ~ ½] propagation vector from powder neutron diffraction spectra (McCarthy et al. 1980). Recent experiments on a monocrystalline sample have shown that the structure is in fact double q, described, as in CeB6, by [¼ ¼ ½] and [ ~ - ¼ ½] and thus probably results from the effects of AFQ interactions (Burlet et al. 1988). Another complex situation is observed in the NaCl-type compound, HoP (Fischer et al. 1985). At Tc = 5.4 K it undergoes a second-order transition to a ferromagnetic state. At TF = 4.8 K there is a first-order transition to a flopside state with two Ho sublattices consisting of alternate ferromagnetic (111) planes. From one plane to the next one, the 4f moments change from one four-fold axis to another. Due to the antiferro- and ferromagnetic components, this lowtemperature structure is biaxial (fig. 55). The spontaneous and field-dependent properties were analyzed using the cubic model with two sublattices (Kim and Levy 1982). This is justified by the presence of six low-lying CEF levels nearly degenerate and well-separated from the excited ones. Bilinear and quadrupolar pair interactions were limited to nearest and next-nearest neighbours. The
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
75
lO-3Jn
10-
5-
10
5.OK I
I
I
I
~1 i
i
5
i
o
71 ""-------
3'0
eo
'
9'0
i
1~o
~'ec*~
Fig. 55. Temperature dependence of neutron diffraction pattern of a HoP powder sample. At 4.2 K the indexing corresponds to the magnetic unit cell with lattice constant a m = 2a. Insets indicate corresponding magnetic structures (from Fischer et al. 1985).
corresponding parameters were deduced from the magnetic excitations measured in the flopside phase by inelastic neutron scattering. Bilinear interactions are positive, in accordance with the ferromagnetic tendency, tetragonal quadrupolar ones are negative in accordance with the change of the z-axis from one plane to another. Thus the negative quadrupolar interactions stabilize the flopside structure. 3.6.2. Incommensurate magnetic structures The nature of bilinear interactions in rare earth intermetallic systems very often leads to modulated magnetic structures (Koehler 1972, Rossat-Mignod 1979). This corresponds to a maximum value of the Fourier transform of the bilinear interactions for an incommensurate wave vector. These structures are either
76
P. MORIN and D. SCHMITT
phase modulated or, when the magnetic moment is confined along a given direction by the anisotropy, amplitude modulated. When considering only bilinear interactions, other couplings being equal, the modulated structures can change when decreasing the temperature below the N6el transition in three ways: (a) For a non-magnetic ground-state (non-Kramers ions, Kondo systems) the modulated structure can remain stable down to 0 K as observed in HoNi0.sCu0. 5 (Gignoux et al. 1977) and in CeA12 (Barbara et al. 1980). (b) For a magnetic ground state (Kramers ions) entropy effects drive the structure to move progressively towards an antiphase structure with constant magnetic moments as in metallic Tm (Koehler et al. 1962). (c) A first-order transition to a commensurate structure (constant magnetic moments) occurs at a relatively high reduced temperature T]TN; e.g., T I T N = 0.87 in TmCu (Morin and Schmitt 1980), 0.82 in DyAg (Kaneko et al. 1987). In the presence of quadrupolar interactions, some modifications are expected. In particular for the first case (a), the nonmagnetic ground state can be quadrupolar (F3 doublet) and the same entropy effects exist for quadrupoles as for spins: consequently, the magnetic structure also has to change as in cases (b) or (c). For case (b), the tendency to a commensurate quadrupolar structure facilitates the squaring-up of a modulated spin arrangement. Unfortunately there is no clear evidence of these two situations up to now. In case (c) the temperature of the first-order commensurate transition Tt appears to be strongly dependent on the large quadrupolar interactions. In TmCu, Tt = 6.7K is close to the temperature To_= 5 K, at which it would ferroquadrupolarly order in the absence of bilinear interactions. Below Tt, both the magnetic and quadrupolar structures are collinear with tetragonal symmetry (Morin and Schmitt 1980). In DyAg, a modulated structure is also observed close to TN (Kaneko et al. 1987). However, the commensurate triple-q structure with spins along the three-fold axes is rapidly stabilized by the trigonal AFQ interactions through a first-order transition at Tt = 46.5 K. The influence of quadrupolar interactions on the commensurate-incommensurate transition appears to be a rather general feature and can be observed in other series such as RB 6 (Effantin 1985, McCarthy et al. 1980) and in RGa 3 (Morin et al. 1987a,d). From a purely magneto-elastic point of view, it is worth noting that a static strain wave is associated with the incommensurate spin wave. Its modulation is one half that of the spins because of the time-reversibility of the quadrupoles. It manifests itself through lattice satellite reflections at wave vectors ~'m~= 2~'n~, where r m is the magnetic propagation vector. They have been observed in metallic Cr (Tsunoda et al. 1974, Pynn et al. 1976), and in rare earth metals such as Ho (Keating 1969, Bohr et al. 1986) and Er (Gibbs et al. 1986). In the incommensurate helicoidal phase of these metals, 2%-satellites have been observed with X-ray diffraction and their intensity was related to the aspherical 4f charge density changing its orientation from plane to plane (fig. 56). In addition, % varies with temperature, the spin structure locks-in at commensurate values of % because it gains some magneto-elastic energy. These lock-ins are also observed in Dy
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS I
I
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Fig. 56. Temperature variation of the nuclear scattering at 2Tm in Er near the % = ~ lock-in transition (% is the magnetic propagation vector) (from Gibbs et al. 1986). ( G r e e n o u g h et al. 1981), they appear to be m o r e frequent in phase m o d u l a t e d structfires (existence of a quasi-isotropic plane) than in amplitude m o d u l a t e d structures (easy axis).
3.7. Effects on magnetic excitations T h e effects of quadrupolar interactions o n the e l e m e n t a r y excitations of m a g n e t i c systems w e r e first studied, by using effective spin m o d e l s , at the s a m e time as the
78
P. MORIN and D. SCHMITT
quadrupolar Hamiltonian itself. Cases with S = 1 (Chiu-Tsao et al. 1975, Sivardi~re 1975), S = 3 (Chiu-Tsao and Levy 1976), and S => 3 systems (Sivardibre 1976) were successively investigated. Two types of effects were considered in these calculations, namely: (i) the modification of the usual (spin waves) magnetic excitations by the quadrupolar couplings, and (ii) the occurrence of pure quadrupolar excitations corresponding to two-spin (AM = 2) deviations. The first effect consists simply of a variation of the size of the spin waves dispersion, at least in the ferromagnetic state. In the quadrupolar phase, however, an additional change of shape occurs, the dispersion curves becoming linear in q when q--~ 0. More exciting is the possibility of two-spin (AM = 2) excitations (quadrons) with the same type of dispersion as the AM = 1 magnetic excitations, but for different energies. Such quadrupolar excitations can be measured by inelastic neutron scattering experiments (Chiu-Tsao and Levy 1976, Levy and Trammell 1977). However, to our knowledge, they have never been observed experimentally. In more realistic calculations, the influence of the CEF must be taken into account by considering the whole level scheme, i.e., the actual CEF wave functions. Elementary excitations have then been obtained through a Green function (Sablik and Wang 1978, 1979) or a generalized susceptibility formalism (Morin et al. 1980c) with similar results. Attention was mainly focused on the influence of quadrupolar couplings on the magnetic (spin wave) excitations, which led to the same general conclusions as the effective spin models. In the magnetic phase, quadrupolar interactions modify the average energy of the dispersion curves, through the single-ion Hamiltonian, as well as their shape, through the Fourier transform of the two-ion quadrupolar couplings. The excitations are then dipolar and quadrupolar mixed excitations. The first effect has been widely used, as described in the literature, for adjusting the energy of the excited CEF levels with the average value of the dispersion curves (Furrer et al. 1977, Pierre et al. 1984). The second effect is difficult to observe; it has been invoked in NdSb in order to explain the shape of the experimentally observed excitation spectra (Sablik and Wang 1985). Surprisingly in the quadrupolar phase, the quadrupolar coupling which induces the static quadrupolar ordering does not produce any specific dispersion in addition to tha(arising from the bilinear coupling; it only modifies the overall amplitude of the dispersion curves. This has been clearly demonstrated in TmxLul_xZn, in which a 10% dilution of Tm by Lu suppresses the ferromagnetic ordering existing in TmZn and provides a pure quadrupolar phase below TQ = 5.8 K (Morin et al. 1981). In that phase, the total amplitude of the longitudinal excitation reaches 16 K, while the corresponding value is only 10K in the ferromagnetic phase of TmZn. The only expected effect in the quadrupolar phase is the modification of the transverse excitations by the second (trigonal) quadrupolar interaction. Unfortunately the latter is too weak in Tm0.9Lu0.1Zn to give rise to a noticeable effect.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
79
3.8. Two-ion anisotropic magneto-elasticity The magneto-elastic modulation of the pair interactions has been neglected in all the preceding sections. This may be justified by analyzing Gd compounds in the absence of any single-ion terms. Usually two-ion magneto-elasticity is only associated with the bilinear interactions. Its modification occurs through either the isotropic exchange striction (see section 3.9) or symmetry lowering modes which introduce anisotropic interactions. The corresponding equilibrium strains have been given in section 2 [eqs. (19)-(24)]. In the CsCl-type compound GdZn, a spontaneous tetragonal strain has been observed below T c = 270 K (Morin et al. 1977, Rouchy et al. 1981). Its zeroKelvin value is e~ = -4.5 x 10 -4, a value one order of magnitude smaller than observed in other RZn (see section 4). The two-ion strain is expected to vary roughly as the De Gennes factor across a series and to take its maximum value for the Gd compound. Thus its contribution to the tetragonal strain rapidly decreases towards each end of the rare earth sequence; e.g., it is only half a percent of the observed value in TmZn. The e-strain has been estimated to be two orders of magnitude smaller than the y-strain. In the Laves phase compound GdAI2, the opposite situation is encountered: e~ is negligible and the e-strain reaches e ~= 1.5 x 10 -5 at 0 K (Du Tr6molet de Lacheisserie 1988). As for RZn, the spontaneous strain is one order of magnitude smaller in GdZn than in other compounds (section 4). The ratio reaches 40 between~strains in TbFe 2 and GdFe 2 polycrystalline samples (Abbundi and Clark 1978). In hexagonal RCo 5 also anisotropic strains are noticeable in the presence of a rare earth orbital moment, but weak in YCo s (Andreev et al. 1985a). In NaCl-type Gd compounds, a spontaneous trigonal strain is observed (Hulliger and Stucki 1978a). The e ~ values observed at low temperatures are 8 x 10 -4, 12 x 10 -4 and 17 x 10 -4 in GdAs, GdSb and GdBi, respectively. For the same trigonal symmetry they are -37, -28 and -33 x 10 -4 in TbAs, TbSb and TbBi (Hulliger and Stucki 1978b). The ratio is then not as large as in the preceding examples, in particular for antimonides and bismuthides. This noticeable two-ion anisotropic magneto-elasticity seems to be a general feature of NaCl-type compounds as observed experimentally and expected theoretically (section 3.2.4). Although the two-ion symmetry lowering modes are usually negligible in comparison with the one-ion modes, the latter example shows that it is necessary to study the Gd compound in a given series systematically.
3.9. lsotropic magneto-elasticity and pressure effects The e ~ volume strain may be obtained from a combination of parallel and perpendicular changes-of length Ix/3 e ~ = AlL+ 2h. from eq. (25)]. Equation (19) shows that one- and two-ion magneto-elastic terms coexist in a given compound. Only the two-ion term is present in Gd compounds. As they are expected to
80
P. MORIN and D. SCHMI'IT
decrease from the Gd value according to the De Gennes law, this allows to evaluate the one-ion contribution in other compounds. 3.9.1. Two-ion contribution In addition to the change of length method, the volume strain may be deduced from measurements of the dependence of the magnetic transition on hydrostatic pressure. For vanishing CEF effects, a MFA treatment proposes an equivalent expression for e ~ as a function of the reduced magnetization o- (Bean and Rodbell 1962): e~-
3 J 2 J+l
ATe 2 NkB -Aff ~r '
where o" is the normalized magnetization. Both methods usually give compatible determinations in Gd compounds. In GdAI2, e.g., the isotropic strain at low temperatures is found to reach - 1 . 7 5 x 10 -3 from change of length measurements (Du Tr6molet de Lacheisserie 1988) and -1.85 × 10 -3 from pressure data (Jaakkola and Hanninen 1980). In GdZn, at T = 0 K a value of +5 x 10 -4 is obtained (Morin et al. 1977, Hiraoka 1974). In GdAg, at T = 0 K a large value, e ~ = -3.8 × 10 -3 may be deduced from ATN/AP = 1.17 K/kbar (Yoshida et al. 1987). This is characteristic of a strong dependence of the magnetic properties on the actual electronic structure, which is close to instability (see section 3.10). 3.9.2. Single-ion contribution For rare earths other than Gd, the isotropic strain also has a single-ion term describing the bulk modulation of the CEF. As the 4th and 6th order CEF terms are of comparable magnitude in rare earth (R) intermetallics, the two magnetoelastic coefficients B(4 ) and B(6 ) have to be considered [eq. (19)]. By comparing results on GdZn with those on TbZn and DyZn, e ~ has been found to be negative in the latter two ( - 3 × 10 -4 and -13 × 10 -4, respectively). This indicates that the CEF part is opposite to that of the exchange and at least twice as large in magnitude (Morin et al. 1977). With regard to the single-ion contribution, thermal expansion appears as the best experimental probe. It may be discussed in the paramagnetic phase within a susceptibility formalism (Morin and Williamson 1984) as well as in terms of Gr/ineisen parameters (Ott and Lfithi 1977). Thus, thanks to thermal expansion results, the pressure dependence of the magnetic susceptibility has been explained in TmSb by a modification of the single-ion susceptibility rather than by a change of the bilinear interactions (Ott and L/ithi 1977). In another singlet ground state system, PrSb, inelastic neutron scattering under pressure (Vettier et al. 1977) and thermal expansion results (Ott 1977) have consistently shown the CEF spacing to decrease when the volume decreases.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
81
3.9.3. Pressure effects Except for the previous examples of the magnetic susceptibility or the ordering temperature, the literature is rather scarce about pressure effects in rare earth intermetallics. Indeed the energies associated with pressure remain weak in comparison with the intrinsic couplings. Hydrostatic or uniaxial stress seems to be able to have dramatic effects only in the case of magnetic systems which are subcritical. For instance, hydrostatic pressure has been shown to drive an antiferromagnetic ordering in PrSb through a softening of the X point magnetic exciton, the zero-pressure value of which is determined by the intrinsic anisotropic bilinear interactions (McWhan et al. 1979). More relevant to the present review is the case of another single ground state system, Pr (Jensen 1979). The bilinear interactions between rare earths at hexagonal sites are just below the critical value for an antiferromagnetic ordering. This makes the spin system very sensitive to various influences, in particular to magneto-elastic effects with e-symmetry. They have been show to influence deeply the magnetic excitons ( H o u m a n n et al. 1979). As the B ~ magneto-elastic coefficient was deduced from ultrasonic measurements (Palmer and Jensen 1978), it was possible to calculate the e" critical value that is needed to modify the level spacing enough so as to drive the spin system to be critical; an uniaxial stress of 770 bar applied along the a-axis of the hexagonal cell was found to be necessary. Experimentally antiferromagnetic ordering was observed below 7.5 K just under about 800 bar (see fig. 57), thereby verifying the above prediction.
1.0 -
/---r-X
N
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Fig. 57. Dispersion relations for magnetic excitons propagating on the hexagonal sites in Pr, at 5.2 K, at room pressure (circles) and under an uniaxial pressure of 800 bar (squares). TO and LO branches correspond to transverse and longitudinal optical modes; the TA and LA transverse and longitudinal acoustic modes were not examined under stress. An asterisk (,) denotes the magnetic satellite position in the ordered phase (from McEwen et al. 1978).
82
P. MORIN and D. SCHMITT
3.10. Magneto-elasticity in the presence of lattice instability In all of the preceding sections, magneto-elasticity was acting on an otherwise stable lattice: As for the lattice parameter, the background elastic constants were assumed to vary weakly across a series as well as to be weakly dependent on the temperature. This situation is the norm, but there are some exceptions. For instance, the CsCl-type structure appears less stable than other crystallographic arrangements such as the MgCu 2- or NaCl-type ones. Indeed some series are incomplete (Iandelli and Palenzona 1979) with compounds exhibiting martensitic transformations (Gefen and Rosen 1981, Al6onard and Morin 1984, Sekizawa et al. 1981). For example, RAg compounds are cubic at all temperatures, and an incipient lattice instability is easily evidenced by hydrostatic pressures at temperatures which drastically depend on the pressure (Kurisu 1987). Chemical pressure obtained by substituting Ag by In produces analogous effects (Ihrig et al. 1975, Fujiwara et al. 1983). The martensitic transformation has been shown to occur mainly through a phonon softening at the M point of the Brillouin zone in LaAgxIn~_ x (Knorr et al. 1980, Abu-Aljarayesh et al. 1986b) rather than through a band Jahn-Teller coupling developing at the F point. Through an anharmonic coupling with the F-point acoustic phonon, this M-point phonon softening drives a partial softening of the lattice elastic constant. A full understanding of this lattice instability requires, in the presence of a strong electron-phonon coupling, dynamical models considering the full dielectric response function of the electronic system. The starting point must be the determination of the actual band structure as has been done for transition metals (Dacorogna et al. 1982). The coexistence of magneto-elasticity and lattice instability makes it difficult to determine the predominant coupling from the temperature dependence of the elastic constants. For instance, in light rare earth-silver compounds, the tempera-
1,4
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o
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Fig. 58. Temperature variation of ½Cv= ½(Cll - C1~) for LaAg, CeAg and PrAg. (PrAg)o is the lattice behaviour after correction for the magneto-elastic contribution (from Giraud et al. 1983).
Q U A D R U P O E A R EFFECTS IN RARE EARTH INTERMETALLICS
83
~.5
t t
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ture variation of Cll - C12 in the LaAg compounds is obviously anomalous and cannot be used directly as background in a magnetic compound (fig. 58). Across the series, the instability tendency progressively vanishes when proceeding to heavier rare earth compounds (fig. 59). As discussed from a theoretical point of view for the orthochromites (Kataoka and Kanamori 1972), the anharmonicity involved with the lattice instability manifests itself in the occurrence of the quadrupolar ordering. In CeAg, where the quadrupolar interactions have been determined by means of parastriction and third-order magnetic susceptibility, both the temperature and the discontinuity of the order parameter are increased, e.g., the quadrupolar ordering would occur around 10 K with only quadrupolar interactions, but this temperature is increased to 15.85 K due to the presence of the incipient lattice instability (Morin 1988). 4. Magneto-elastic and pair interaction coefficients in rare earth intermetallic series
4.1. CsCl-type structure compounds CsCl-type rare earth intermetallics RX have been extensively studied due to the availability of their single crystals. Their very simple structure allows a deep insight in the 4f magnetism; analyses are possible according to the alloyed non-magnetic metal (X = Mg, Zn, Cd, Cu, A g , . . . ) . They have revealed large quadrupolar interactions which usually prevent any analysis which would only consider CEF and bilinear interactions of the magnetic data. In particular large spontaneous strains exist with tetragonal symmetry; they reach record values ( c / a - 1 = 1.4% in PrZn and -0.8% in TmZn). Important elastic constant softenings are observed in the paramagnetic range. FQ orderings were discovered in TmZn and TmCd (section 3). The following subsections outline the numerous effects of quadrupolar interactions on the magnetic properties in each series. Quadrupolar interactions are largest in tetragonal symmetry. They are of a ferroquadrupolar type. In trigonal symmetry, they are negative and very often stabilize multi-axial spin structures when the three-fold axis is the direction of easy magnetization. Systematic studies of the TmX have shown how very relevant the (spin and) quadrupolar Hamiltonian is within the MFA. In particular the
84
E MORIN and D. SCHMITT
absence (or the weak value) of fluctuations leads to determinations of G r and G" from third-order magnetic susceptibility measurements in close agreement with those obtained from magneto-elasticity data. However, there are particular situations where the quadrupolar coupling may exist with additional terms (as in the case of the lattice instability for CeAg).
4.1.1. RZn series Light rare earth compounds are antiferromagnetic with a (½,0, 0) propagation vector. From SmZn to TmZn, a ferromagnetic behaviour is observed (table 5). The CEF parameters A4(r4), A6(r 6) are found to be negative by inelastic neutron spectroscopy, without exhibiting an erratic variation (Giraud et al. 1986). When the four-fold axis is the easy magnetization direction, a large spontaneous tetragonal strain is observed. The various magneto-elastic coefficients were determined by different experimental techniques. Note that when the temperature range investigated by parastriction is reduced (high Tc values), only a relationship between G ~ and B ~ is obtained. Some differences between B ~ values deduced from parastriction and spontaneous strain measurements may be noticed, in particular in TbZn, DyZn and HoZn. They are partly explained by an overestimation of the two-ion contribution to the spontaneous strain. Across the series, the coefficient B ~, normalized by the Stevens coefficient exhibits an average value of about -2050 K with the ErZn value slightly remote (see section 5.1). In trigonal symmetry, reliable values of B ~ are achieved in TbZn and HoZn from the spontaneous trigonal strain. In other compounds, only the parastriction may be efficient. However, as the tetragonal symmetry lowering mode is highly favoured in CsCl-type compounds, a very accurate orientation of the magnetic field along [111] is necessary in order to avoid the presence of any tetragonal contribution to the change of length measured. Here again the parastriction leads only to a relation between values of B e and G ~. B e values obtained with both techniques are in good agreement in TbZn and HoZn. The ratio B ~/aj obviously varies across the heavy rare earth compounds; in TbZn, it even Changes sign. The magneto-elastic contribution to G ~ is less than 1 mK for the trigonal symmetry, but reaches 5.2, 9 and 120 mK for tetragonal symmetry in TmZn, PrZn and CeZn, respectively. However, in RZn, the quadrupolar interactions originate mainly from the pair coupling, as proved by the ratio K~/G~jE of 1.6 and 4 which is found in ErZn and TmZn, respectively. They are systematically positive in tetragonal symmetry, negative in trigonal symmetry. In TmZn, they dominate the bilinear interactions, and induce a quadrupolar ordering in the paramagnetic phase. Depending on their sign, they may stabilize either collinear antiferromagnetic structures as in CeZn and PrZn or multi-axial ones as in NdZn. In other compounds, none of the magnetic properties can be described without the quadrupolar interactions as shown for the four-fold axis of easy magnetization in DyZn, the magnetization processes in HoZn and ErZn, the actual temperature for the change of easy axis in NdZn and HoZn and the first-order transition in CeZn (see section 3).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
85
4.1.2. RCd series This series has been less extensively studied than the RZn one, due to the large neutron absorption cross section of cadmium, which in particular forbids a direct determination of the CEF level scheme by neutron spectroscopy. However, the properties appear to be very close to those of the RZn series from a magnetic point of view. One of its peculiarities rests in the properties of the lattice: La, Ce and Pr compounds exhibit a well-defined martensitic-type transition, which is revealed by electrical resistivity or susceptibility measurements, at 61, 107 and 125 K, respectively (A16onard and Morin 1984). At 216 K CeCd undergoes an additional transition which is as well-defined as the previous one (Fujii et al. 1985). Compounds from Nd to Tm are ferromagnetic, CeCd and PrCd being antiferromagnetic (A16onard and Morin 1985). CeCd was also found to be ferromagnetic (Fujii et al. 1985). The easy magnetization axes and the magnetic moments observed in ferromagnets are closely reminiscent of the corresponding results in RZn. For instance, all of the properties studied in TmCd, in particular the quadrupolar properties, can be described with a CEF level scheme very close to that in TmZn. Assuming the CEF parameters as constant, quadrupolar coefficients may be obtained through the full description of the magnetization properties. This is true in particular for the four-fold easy direction in DyCd as well as in DyZn, and for the temperature of change of easy direction in NdCd as well as in NdZn. On the contrary, observing [111] axes as easy direction in CeCd and PrCd clearly reveals a deep modification of all the one- and two-ion terms driven by the lattice instability. In RCd ferromagnets, quadrupolar interactions are present having large analogies with the RZn case as proved by the comparison for a given rare earth (tables 5 and 6). However, from the thorough study of TmCd, both the pair-interaction coupling and the magneto-elasticity are smaller than in TmZn in tetragonal symmetry, the ratio Kr/G~E, however, is twice as large. This is emphasized by the T o value in both Tm compounds. In trigonal symmetry, the coefficients G ~ are negative as in RZn. 4.1.3. RAg series This series exhibits some specific behaviours. Bilinear interactions favour incommensurate magnetic structures. One notes, however, a tendency to ferromagnetism at the beginning of the series through a ferromagnetic component in PrAg and a ferromagnetic moment in CeAg. The observed properties strongly depend on the preparation. In an unannealed sample, the magnetic structure is systematically commensurate with the lattice periodicity. Removing the metallurgical stresses by annealing the sample restores the intrinsic nature of the bilinear interactions as it was demonstrated in HoAg, ErAg and TmAg (Nereson 1973a,b, Morin and Schmitt 1982a). Another important peculiarity rests in the incipient tendency to lattice instability, although all the RAg are stable within the CsCl-type structure in the absence of external stress (section 3.10). Under hydrostatic or chemical pressure, the instability towards a tetragonal-symmetry based structure (Maetz et al. 1980) is
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90
P. MORIN and D. SCHMITF
revealed by a martensitic transition at a strongly pressure-dependent temperature (Kurisu 1987). Its driving mechanism consists of a softening of the TA phonon at the M point on the Brillouin zone (Knorr et al. 1980) and induces anharmonic contributions to the elastic constant. In the absence of an external stress, the incipient lattice instability takes the opportunity of the quadrupolar ordering in CeAg to manifest itself through both an increase of the quadrupolar ordering temperature (see section 3.1.1) and a spontaneous tetragonal strain ( c / a - 1 = 1.9%) which is larger than expected from the one-ion magneto-elasticity only. In RAg, the tendency to instability is particularly clear in LaAg, CeAg and, to a smaller extent, in PrAg (see section 3.10). The tetragonal magneto-elastic parameters, normalized by the second-order Stevens coefficient, are of the same order of magnitude as in RZn. Their average value is around -2000 K/at., the CeAg value being somewhat smaller (table 7). The trigonal symmetry coefficient observed in PrAg and TmAg leads to positive BE/aj ratios (4400 and 2200 K/at., respectively). As in RZn and RCd, the pair interactions are positive in tetragonal symmetry, negative in trigonal symmetry. For example, in TmAg, they seem to be smaller than in RZn. Their effects are, however, numerous on the magnetic properties. The commensurate-incommensurate transition at 9.5 K in ErAg may be influenced by quadrupolar interactions as discussed for DyAg (section 3.6.2). The N6el temperature in TbAg is the highest one known with a first-order character, which agrees with the G v value determined in the paramagnetic phase. In PrAg, a neutron diffraction experiment with polarization analysis on a single crystal has, at least partly, removed the ambiguities among the possible structures of fig. 52 and has shown that a double-q structure (structure b or c) is very likely in agreement with the existence of rather negative quadrupolar interactions (Givord et al. 1985). 4.1.4. RCu series CsCl-type structure compounds range from GdCu to LuCu; equiatomic alloys with light rare earth crystallize within the FeB structure (Buschow 1980). Lattice instability develops itself in boundary compounds: GdCu undergoes a martensitictype transition with a very large hysteresis (Gefen and Rosen 1981) (table 8). In TbCu the martensitic transition seems to be triggered by the antiferromagnetic ordering (Pierre and Hennion 1982). All the other heavy rare earth compounds are stable down to 0 K. Antiferromagnetic interactions are observed across the series, they are characterized by a (1, ½,0) propagation vector. However, ( ½ - r, ½,0) modulated structures have been observed in TmCu and ErCu close to the N6el temperature in a way reminiscent of RAg compounds. Magneto-elastic coefficients are available for TmCu and DyCu in tetragonal symmetry. In TbCu, a B ~ value may be achieved from the spontaneous strain by assuming a maximum value for ( O °) due to the large exchange field at 4.2K. The resulting ratios Br/aj are -2300, -2450 and -2800 K/at. in Tb-, Dy- and TmCu, respectively: the same sign and order of magnitude are observed as in other CsCl-type series. In trigonal symmetry, B ~ is found to be positive in TmCu as in other TmX.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
91
"O
~Av
8
PI
o
o
~AV~ O
o
u~ II ~]~
0 ..~ N ..~ o
~4 tt~
8~
,,6
~"
V A A
O
,,6 II A
m M'~
~.~
V
w
I
~e +
E
E ~
rq ,,6
,--Z e., o
Z
~D
92
P. MORIN and D. SCHMITT
Quadrupolar pair interactions are also present. They are positive in tetragonal symmetry although they are a little weaker than in RZn; they dominate the magneto-elastic coupling. The total quadrupolar coefficient G r would drive a quadrupolar ordering at only T o = 5 K in TmCu. Its value in ErCu, deduced from the magnetic excitation spectrum, is lower than in ErZn. The tendency to negative quadrupolar pair interactions in trigonal symmetry is also established in RCu. As discussed in section 3.6.1, they determine the multi-axial structure of DyCu, the CEF being responsible of the three-fold direction of easy magnetization as in DyAg and TmGa 3 (section 3.6). 4.1.5. R M g series
Light rare earth compounds are antiferromagnets. In heavy rare earth compounds, the spin structure is either canted, surprisingly also in GdMg (Kim and Levy 1982), or more complex, because they mix ferro- and antiferromagnetic tendencies (A16onard et al. 1976). From a quadrupolar point of view, spontaneous strains are observed in CeMg and PrMg (table 9). From the extensive study of TmMg (Giraud et al. 1986), it was found that the magneto-elastic and pair quadrupolar interactions are weak in heavy RMg compounds (section 5.1). Then an analogy appears between quadrupolar and antiferromagnetic interactions: they are strong and well-established at the beginning of the series and they rapidly decrease in heavy rare earth compounds.
TABLE 9 Quadrupolar coefficients in the RMg series. Also indicated are the N6el temperature, the antiferromagnetic propagation vector Q, and the spontaneous strain at low temperature. RMg compound
Tr~ Q Easy axis c/a - 1 (%) B" (K) G ~ (mK) B ~ (K) a " (mK)
CeMg
PrMg
TmMg [5]
19.5 [11 1st order (~,0,0) [001] [2] 1.3 -+ 0.2 [2] 98 [6] (50, 150) [11
45 [3]
<1.3
(~,0,0) [001] [4] 1.4 -+ 0.2 [4] 30 [61
-5.4 (0, 3) (14, 21) (-100,-200)
References: [1] Pierre et al. (1984), from magnetic excitation studies. [2] Schmitt et al. (1978). [3] A16onard et al. (1976). [4] Morin et al. (1978a). [5] Giraud et al. (1986) from measurements of parastriction, elastic constants and X~ )[6] From c / a - 1 data, using (O °) ='8 (CeMg) and 28 (PrMg).
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
93
4.2. NaCl-type structure compounds Rare earth pnictides (rare earth compounds with group V-A elements) crystallize within the cubic NaCl-type structure. They have been extensively studied from a magnetic point of view (Hulliger 1978). Their semi-metallic behaviour leads to RKKY-type bilinear interactions driving rather low magnetic transition temperatures (T < 60 K). The low-temperature phase is antiferromagnetic of type-I or -II with some exceptions for Ce compounds. The small size of nitrogen ions is probably the origin of ferromagnetism in some nitrides. Associated with the semi-metallic behaviour, the strength of the magnetic interactions strongly depends on the stoichiometric conditions as may be deduced from the spread in the T s values listed in tables 10 and 11. The CEF parameters are very similar for compounds of a given rare earth ion and the dependence across the rare earth series follows, more or less fortuitously, point-charge estimates. However, a clear exception is the case of Ce compounds (Birgeneau et al. 1973). One of the consequences of the positive A4(r 4) parameter and of the weak A6(r 6) parameters, is the great number of nonmagnetic singlet ground states for Pr, Tb and Tm ions; over- (sub-) critical conditions are usually observed in Tb (Pr and Tm) compounds. The magnetoelastic coupling provides clear evidence for the actual magnetic structure, through the tetragonal or trigonal spontaneous strain. The number of multi-axial structures appears to be very small contrarily to the situation in CsCl-type compounds. Numerous compounds order at a first-order transition, in particular those with Tb, Dy and Er ions. In connection with the low-temperature range, this is explained by positive values for the third-order susceptibility driven by quadrupolar interactions in both tetragonal symmetry (Levy 1973) and trigonal symmetry (Koetzler 1984). In tetragonal symmetry, the magneto-elastic quadrupolar term has been observed to be dominant (Mullen et al. 1974, L6vy 1969). The B ~ values have signs opposite to those in the CsCl-type series and they roughly follow the aj variation with a magnitude slightly larger than in the CsCl-type structure (Liithi 1980b). This seems also to be valid for the coefficient in trigonal symmetry. As discussed in section 3.2.4, the anisotropic character of the bilinear interactions as well as the existence of an associated two-ion magneto-elastic coupling has been shown theoretically and experimentally. The occurrence of a sizeable trigonal spontaneous strain in Gd compounds, in particular in GdBi (HuUiger and Stucki 1978a), complicates the analysis of the magnetic properties, such as, e.g., the magnetic transition. Anisotropic bilinear interactions seem to have a dominant role in Ce pnictides due to hybridization effects (Cooper 1982, Takahashi et al. 1982). According to the De Gennes law, two-ion magneto-elastic effects are expected to be noticeable in the middle of the rare earth sequence, e.g., in Tb and Dy compounds. In DySb below TN, the actual direction for the magnetic moment is tilted from the c-axis of the tetragonal cell; the resulting monoclinic cell underlines the presence of the trigonal two-ion magneto-elasticity (Felcher et al. 1973). Above TN, in tetragonal symmetry, the parastriction also reveals the competition between one- and two-ion magneto-elastic terms (Al6onard et al.
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~
ir~
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~
~
÷
~
or ~I~
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~
F~
z
p~ t~
p~
c~ Tm~
p~ Z
94
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QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
95
1984b). As for Ho compounds, HoBi exhibits a positive tetragonal strain (Hulliger et al. 1983) which is opposite to the negative one expected on the basis of a dominant one-ion contribution as effectively observed in HoAs; this may be also a signature of this competition.
4.3. AuCu3-type structure compounds Numerous RX 3 intermetallics crystallize with the cubic AuCu3-type structure, in particular for X = Pb, In, T1, Pd, Sn and Ga. Only a small number of studies has been devoted to magneto-elastic and quadrupolar interactions. The non-magnetic transition observed at 0.37 K in PrPb 3 has been analyzed to be driven by an AFQ ordering (see section 3.1.2). This behaviour agrees with a study of TmGa 3 which shows that AFQ interactions drive quadrupoles to order slightly above TN and stabilize the multi-axial spin structure below T N (sections 3.1.2 and 3.6.1). A quadrupolar ordering was also claimed to occur in the paramagnetic phase of Stain 3 and SmSn3; a noticeable softening of the C44elastic constant is observed at high temperatures (Kasaya et al. 1985). The tendency to negative quadrupolar interactions seems to be general in AuCu3-type compounds and responsible for the multi-axial magnetic structures observed. From a purely magneto-elastic point of view, PrPb3, TmGa 3 and CePb 3 (Nikl et al. 1987, Morin et al. 1987b) are the only AuCu3-type compounds studied. The spontaneous strain is null in TmGa3, as the magnetic structure is cubic. It is unobserved in CePb3, owing to the low N6el temperature (TN = 1.1 K) and the non-quadrupolar F7 ground state, and in PrPb3, due to the low value of TQ and the AFQ structure. The magneto-elastic coefficients and their ratios to aj are given in table 12. In both symmetries, the ratios B"/aj are positive; a good coherency is achieved for the tetragonal mode with an average magnitude which is coherent with values in preceding series. Some scattering is observed for the trigonal mode; however, values for PrPb 3 and TmGa 3 are noticeably stronger than in CsC1- and NaCl-type compounds.
4.4. Cubic Laves phase compounds 4.4.1. RAI 2 compounds The RAI 2 compounds have been thoroughly studied for many years with regard to, first, the crystal field and the magnetic properties and, second, the magnetoelastic coupling. All the available experimental results are reported in table 13. Most of them are based on magnetostriction measurements in the ferromagnetic range. That explains why some values seem out of range, as e.g., B" for PrA12 or B r for TbAI2, because they have been obtained from measurements along an axis of hard magnetization in compounds with a large anisotropy. The high value of B ~ for CeA12 has been related to a coupling with optical phonons at the F-point (Liithi and Lingner 1979). All the other experimental values are rather consistent with each other, in particular in NdA12 where the B r value has been obtained by elastic constants, X-ray and magnetostriction measurements. Recently, some
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0
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o~ ~ ~ <
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0
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x
~
°
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96
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oo
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97
98
P. MORIN and D. SCHMITF TABLE 12 Magnetic and quadrupolar ordering temperatures and magnetoelastic coefficients from elastic constant, parastriction and g~ ) experiments in AuCu3-type compounds. AuCu3-type compound CePb3 [1] TN (K) To (K) B r (K) B~/oq (K) K" (inK) G"MZ(mK) a" (inK) B" (K) B~/a~ (K) K" (mK) GME (mK) G ~ (mK)
PrPb3 [2]
TmGa3 [31
0.35 [5] -41 1950 -13 8 -5 -190 [11 9050 -200 7 -200
4.26 [6] 4.29 17 1700 (8,14) 0.5 (8,14) 71 7000 20 10 30
1.1 [4] -80 1400
-190 3350
References: [1] Morin et al. (1987b). [2] Morin et al. (1982). [3] Morin et al. (1987c). [4] Lin et al. (1985). [5] Bucher et al. (1972a). [6] Czopnik et al. (1985). quadrupolar p a r a m e t e r s G v have been obtained by the torque method, which allows to extract the third-order paramagnetic susceptibility x- e(3) in a less direct way than from magnetization m e a s u r e m e n t s (Del Moral et al. 1987). As for the sign of magneto-elastic coefficients, they appear to follow well that of the Stevens factor aj. The ratios BY/aj and BE/aj range around 1000 K and - 3 0 0 0 K, respectively, with a few exceptions (see section 5.1). F r o m values of G r and G rME, the existence of two-ion quadrupolar coupling is suggested. This is in disagreement with elastic constants fits, which did not require considering this coupling.
4.4.2. R M 2 compounds (M = Mg, Ni, Co) For the R M 2 cubic Laves phase compounds where M is not A1 (section 4.4.1) or Fe (section 4.4.3), quadrupolar data are rather scarce. The PrM 2 compounds (M = Pt, Rh, Ru, It) have been investigated by magnetization, neutron diffraction, and specific heat m e a s u r e m e n t s (Greidanus et al. 1983a,b). A systematic overestimation of the ferromagnetic m o m e n t by a calculation including C E F and bilinear exchange interactions, with regard to the observed spontaneous magnetization, was explained by the presence of negative quadrupolar interactions. The same is true in PrMg2, where a quantitative analysis of the t e m p e r a t u r e variation of the ferromagnetic m o m e n t provided the quadrupolar p a r a m e t e r G r = - 1 3 . 3 m K (see table 14). F u r t h e r m o r e , this negative quadrupolar coupling leads
.i
+I
+I
e~
=
Z c~
~
i ~
I
¢J
8
c~c~ I i
~
~°~'~
Z ~
~
"~.~o~ °°
~ ~ ~oo 8
< •
e.
o Z
I
I
I
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S
~
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0
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4~ ~ +I
~:~o~
4
0
i ~°
+I
.o e.
-g X
X
~
99
~
100
P. M O R I N and D. SCHMITT T A B L E 14 Quadrupolar coefficients in some R M 2 compounds. R M 2 compound PrMg 2 [1]
ErNi 2 [2]
B ~ (K) B ~ (K) G ~' (inK) G ~ (mK)
TbCo 2 [3]
HoCo 2 [4]
~0 79 -13.3
1 -22
-2.7 -11.7
References: [1] Loidl et al. (1981). [2] Gignoux and Givord (1983), from magnetization measurements. [3] Gignoux et al. (1979), from X-ray measurements. [4] Castets et al. (1982), from magnetic excitation studies.
to the disappearance of the first-order transition which is expected without it (see section 3.2.2). Some magneto-elastic results are also available in the RNi 2 series, where nickel is not magnetic. The compounds order ferromagnetically at temperatures roughly three times smaller than the corresponding temperatures in the RAI 2 series. At low temperatures, the directions of easy magnetization are the same in RNi 2 and RAI 2 compounds (R = Tb, Dy, Ho); the spontaneous distortions are comparable in both series, except for HoNi 2 where the strain is noticeably larger than in HoA12 (e ~ = -1.1 x 10 -3 in TbNi2, c/a - 1 = - 2 x 10 - 3 and -1.5 x 10 -3 in DyNi 2 and HoNi2, respectively) (Markosyan 1980, 1981). The behaviour of ErNi 2 differs considerably from that of ERA12, since its direction of easy magnetization is [100] instead of [111]. This has been well accounted for by introducing the parameters G ~ and G ~ (see table 14). In the RCo 2 series, cobalt is magnetic and contributes mainly to the volume magnetostriction. A positive trigonal magneto-elastic coefficient B ~ associated with the rare earth has been determined in TbCo 2 (see table 14). This leads to a positive trigonal magnetostriction e ~= 3.1 x 10 .3 at 4.2K which is opposite in sign to that measured in TbNi 2 and TbAI 2. This feature has been related to environmental effects in this type of cubic structure (Gignoux et al. 1979). In HoCo 2, a consistent description of the magnetic excitations as well as the spin rotation temperature requires introducing negative quadrupolar terms in the Hamiltonian (table 14), indicating the probable existence of a contribution of two-ion origin.
4.4.3.
RFe 2
compounds
The magneto-elastic properties of cubic ferromagnetic Laves phase compounds RFe 2 are relevant for the classical analysis, owing to their high magnetic transition temperature (around 700 K in TbFe2). Indeed, the Fe magnetization induces an exchange field large enough to purify fully the 4f magnetic moment towards its maximum value at 0 K. Their magnetostriction has been studied, in particular at
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
101
room temperature, for industrial applications. The large set of available data has been collected in the first volume of this Handbook by Clark (1980). Here, only characteristic features and analyses are considered again. Comparing the strength of the tetragonal and trigonal symmetry-lowering modes leads to a conclusion opposite to that in CsCl-compounds. Indeed, in RF% the trigonal mode is many times larger than the tetragonal mode (fig. 60). TbFe2, the basic compound for applications, is the paradigm of the series. The temperature variation of the Aln trigonal spontaneous strain has been described using eq. (107). As pedagogically explained by Clark (1980), an important step is to determine the 4f magnetization at any temperature by means of neutron diffraction or magnetization measurements. In this latter case, the Fe magnetization (about 1.64/~) is deduced from M6ssbauer spectroscopic results. Noticeable uncertainties may be present at high temperatures; however, the saturation of the 4f moment observed at low temperatures allows one to derive reliable values of B ~ through the relation Be=
3v~ o%,,(o)
J(2J-
1)
5000
4000
-
'
'
'
'
I
.
.
.
.
~ j I ~ Tb Fez
3000
I
'
^
'
'
'
I
(£l(m))
-
~'111
(a)
2000
40"
7
0 -40 -80 -120 -160 -200
L
100
i
i
]
200
n
i
u
I
300
T (*K) Fig. 60. Temperature dependence of the spontaneous strain in TbF% (trigonal symmetry) and in DyF% (tetragonal symmetry). The An1 variation is classically described according to the Tb magnetic moment. For the DyF% data, the full line is only a guide for the eye (from Clark et al. 1977).
102
P. M O R I N and D. SCHMITI" T A B L E 15 Curie temperature, easy magnetization axis, trigonal spontaneous strain at 300 and OK, trigonal magneto-elastic coefficient B e and its ratio to the Stevens coefficient c~j in RFe 2 compounds [data are from tables 5 and 6 of Clark (1980)]. RFe z compound
T c (K) Easy axis hal 1 (300 K) × 1 0 6 )tll 1 (0 K) x 106 B" (K) B'/a: (K)
TbFe 2
DyFe 2
HoFez
ErFe 2
TmFe 2
SmFe 2
697 [111] 2460 4400 94 -9400
635 [001] 1260 3320 a 44 -7070
606 [001] 200 770 a 9 -4100
590 ' [1111 -300 -1760" -24 -9300
560 [111] -210 -3500 -75 -7500
676 -2100
Note: "estimated from the room-temperature value.
Across the series, B" is observed to vary as the second-order Stevens coefficient a: in sign and, with some erratic fluctuations as in other series, also in order of magnitude (table 15). The same technique, based on eq. (107), fails to describe the tetragonal spontaneous strain in DyFe 2 (fig. 60) and HoFe 2 (Abbundi et al. 1979). Indeed, the change of sign of hl00(T) in DyFe 2 and the general temperature variation in HoFe 2 are both unexpected within this model. Whereas the hi00 sign at 4.2 K follows the sign of aj in both compounds, its absolute value is about ten times smaller in DyFe 2 than in HoFe 2, in full disagreement with the ratio of the aj's. Cullen and Clark (1977) have developed an atomic model that shows that internal distortions drive the large external trigonal strain mode whereas the hi00 strain may be sorted out because of the high tetrahedral symmetry of the rare earth site.
4.5. Hexagonal CaCus-type structure compounds Among the series of hexagonal 3d-4f intermetallic compounds, only few RNi 5 and RCo 5 compounds have been investigated from a magneto-elastic point of view; nickel is not magnetic in the first series, but cobalt is in the second one.
4.5.1. RNi 5 compounds Whereas the RNi 5 compounds have been studied for many years with regard to their magnetic properties and CEF effects, only few results are available about their magneto-elastic properties. A first study concerned the thermal expansion of PrNi 5 (L/ithi and Ott 1980) which evidenced crystal field effects, and showed the existence of the two a-mode magneto-elastic coefficients. However, the latter were not determined because the elastic constants were not known at that time. Using2the values of C" from Barthem et al. (1988), we are able to calculate B ~1 and B from these early measurements (see table 16). These values have the same sign, but are 2-3 times smaller than more recent values determined by magnetostriction measurements. In addition, the latter provide an estimation of
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
103
TABLE 16 Quadrupolar coefficientsin RNi5 compounds. RNis compounds B'~1 (K) B ~z (K) B ~ (K) G ~ (inK) G ~ (mK)
PrNi5 [11
PrNi5 [21
TmNi5 [3]
-7.1 28.6
-22 68.6 <15 -10 -20
(-500,-100) (-300,-150) (-1500,-1000)
References: [1] L/ithi and Ott (1980), from thermal expansion measurements. [2] Barthem et al. (1988), from magnetostriction measurements. [3] Barthem et al. (1989), from magnetostriction measurements. the total quadrupolar parameters G ~ and GL By contrast, Barthem et al. (1988) deduce antiferroquadrupolar (negative) interactions between Pr ions in PrNi 5 (K ~ - - 1 5 mK, K ~ - - 2 0 i n K ) . Note that a magneto-elastic coupling was also deduced from inelastic neutron scattering in PrNi 5 (Aksenov et al. 1983), where it should contribute through the hybridization of magnetic excitons and phonons. However, no accurate determination could be obtained. A recent magnetostriction study of TmNi 5 also provided one- and two-ion magneto-elastic coefficients (Barthem et al. 1989). Particularly due to the CEF configuration of this compound, the one-ion and two-ion magneto-elasticities could be separated. More precisely, the axial strain dependence of the isotropic exchange interaction ( a 2 - m o d e ) leads to a strongly anisotropic anomaly in the thermal expansion along a and c-axes below the ferromagnetic ordering temperature T c = 4.5 K, while the magnetostriction is more sensitive to the one-ion magneto-elasticity. However, the deduced B " values are quite large with regard to those in PrNi 5 and do not scale with aj. This behaviour is still to be explained. 4.5.2. R C o 5 c o m p o u n d s
Just as the cubic RFe 2 compounds, RCo 5 compounds are typical cases for applying the classical formalism, as the Co-sublattice induces very large bilinear interactions: the Curie temperatures are around 1000 K (e.g., 940 K in YCos). It is therefore difficult to study paramagnetic properties or to determine the CEF for the rare earth. A main feature is the huge magnetocrystalline anisotropy even in the absence of a magnetic rare earth as observed in YCo 5 (Alameda et al. 1983). From a magneto-elastic point of view, large anomalies have been observed in the temperature variation of the lattice parameters (Andreev et al. 1983, 1985b). It is difficult to observe magneto-elastic properties other than the spontaneous ones except in several compounds which exhibit a change in the direction of easy magnetization, and thus a temperature range of reduced anisotropy. Indeed, as classically explained in hexagonal Co (Barnier et al. 1962) and Gd (Klimker and Rosen 1973), this p h e n o m e n o n is driven by the change of sign at a given temperature of the K 1 anisotropy constant, which phenomenologically includes
104
P. MORIN and D. SCHMITI"
magneto-elastic contributions. This occurs for compounds with a negative Stevens coefficient aj for the rare earth (PrCo 5, NdCo 5, TbCo 5, DyCo 5 and HoCos). The corresponding temperature regions range from 13 K (in TbCos) to 130 K (in HoCos); the rotation of the magnetization induces clear anomalies in both the lattice parameters (Andreev et al. 1983) (fig. 61) and the ultrasonic modes (Deryagin et al. 1985) (fig. 62). It has been shown by X-ray studies that the magneto-elastic coefficient B ~ is small. On the other hand, the relative changes Aa/a and Ac/c of the lattice
v.gs~ --
1*~t~f,
,, -4 v, A3
.!'""I
q.g#Z~
~
85 1
"
Oq. g
q.960
~ I ~ J'l'~+"~
390
350
i
l-I
qJo T K
Fig. 61. Temperature dependence of the lattice parameters a and c, and of the volume V of the unit-cell in the temperature range of the change of easy axis in TbCo 5 (from Andreev et al. 1983).
z+
r,~r~r2
"E i
t
;'¢.2 I0.8
5.2 ~ 0
r
i
200
t r ~"w
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Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
105
parameters at the average rotation temperature between the high-temperature (magnetization along the c-axis) and low-temperature (magnetization in the basal plane) phases indicate contributions of the two a-modes (see section 2.3). Temperature dependent magneto-elastic coefficients have been introduced (Deryagin et al. 1985),
V~
2(Aa 11 + 3 a
AC'~c~,
B12-
.
2 ( Aa
AC~c .
B~2=
AC'~c ~ 2 ( A a 2 (Aa X/3 2 - - + a c ! t 2 + 3 a
AC'}c ~
2 --a + c /
c /
c/
12 ,
22"
Experimentally, the values of (2Aa/a + Ac/c) have been observed to be very small. The 0 K values of B12 and B22 have been deduced assuming a temperature variation according to the Callen and Callen function is/z[sg-l(m)]. Note that some uncertainties may arise from inaccurate knowledge of the rare earth's magnetic moment, especially for the light rare earths (Andreev et al. 1983). The B~2(0 ) and B~2(0 ) are related to the B ~ and B "2 magneto-elastic coefficients of the present formalism through, BI:(O ) = 2B '~1
B~:(O) = 2V'-6 B "e
It is then possible to follow their variations through the entire series of rare earths. Table 17 collects the values for the different parameters in the series. The T A B L E 17 Magnetostriction constants at OK, h~2(0), h~2(0), associated to the relative change of lattice parameters at the rotation temperature, aa/a and Ac/c, elastic constants C~2 = 2X/2-7-gC~12, C~z = 2C ~2, B ~ and B ~2 magneto-elastic coefficients and their ratio to the Stevens coefficient a~ in RCo 5 compounds. RCo s compounds
h~2(0) x 103 [11 A~2(0) x 103 C~2 (101° Pa) C22 (1010Pa) B "1 (K) B "2 (K)
B~l/aj (K)
B"Z/a: (K)
PrCo s
NdCo s
SmCo 5
TbCo 5
DyCo 5
HoCo5
ErCo 5
TmCo 5
-1.1 2,2 1,6 a 20 a -6.6 -39 314 1860
-0.8 1.7 1.6 [2] 21.2 [2] -5.1 -25 793 3890
1.3 -2.6 1.6 = 22 a 24.6 143 597 3460
-1.8 3.5 1.7 [3] 23.3 [3] -5.5 -31.2 544 3090
-1.7 3.4 1.9 [31 25 [3] -3.7 -20.5 583 3230
-0.7 1.2 2a 27 ~ -1.2 -6.9 540 3100
0.6 -1.2 2" 27 a 1.4 7.6 551 3000
1.5 -3 2a 27" 5.5 30.8 544 3050
Note: aExtrapolated values. References: [1] Andreev et al. (1983). [2] Deryagin et al. (1984), [3] Deryagin et al. (1985).
106
R MORIN and D. SCHMITr
a f d e p e n d e n c e of B ~1 and B ~2 is very closely observed. The result indicates that the a - m o d e s depend on the rare earth and not on the cobalt. This conclusion is confirmed by the study of Y C % by X-ray diffraction. In this c o m p o u n d , the cobalt sublattice appears as contributing only through a two-ion magneto-elastic coupling, modifying the c/a ratio (Andreev et al. 1985a). The partial softening observed on the different elastic constants m a y be expected to be described using the coefficients B ~1, B ~a previously determined. Note that the softening observed on C44 during the rotation also indicates a sizeable magneto-elastic coefficient B ;. Some significant results might be provided by magnetostriction m e a s u r e m e n t s in high magnetic fields for the various symmetries in the rotation t e m p e r a t u r e range.
4.6. Dilute rare earth systems The magnetization of concentrated rare earth compounds generally includes oneand two-ion terms and m a y be strongly influenced by magnetic interactions. Dilute systems should, in principle, be easier to understand by limiting the magneto-elastic effects to one-ion contributions only. A dilution in the 1% range seems to be a satisfactory compromise between experimental sensitivity and residual two-ion effects. Most of the available studies concern rare earths diluted in noble-metal hosts, as Ag, A u and Cu.
4.6.1. Rare earths diluted in cubic noble-metal hosts The noble metals crystallize within the face centered cubic structure, and, because of the low solubility of rare earths in them, concentrations of rare earth impurities smaller than 1% were usually taken for the experiments. Silver-rare earth dilute compounds. A systematic study of heavy rare earths diluted in silver was carried out by magnetostriction m e a s u r e m e n t s on single crystals (Creuzet and Campbell 1981) (see table 18). Varying the cubic C E F TABLE 18 Magneto-elastic coefficients in the AgR series. All results were taken from Creuzet and Campbell (1982a) and from Creuzet (1982) and refer to magnetostriction measurements (rare earth concentration c = 0.5 at.%), except where indicated otherwise. AgR (R =) B r (K) B ~ (K) B(r4) (mK) B~4) (raN)
Tb
Dy
Ho
Er
Tm
2.72 3.9 [21 31.1 -6.3 -67
1.71 0.9 [21 12.1 1.9 45
0.60 3.5 [21 1.18 5.3 9
-1.24 <-0.39 [1] -7 -3.5 -29
-2.47 -3.3 [21 -16.1 -6.0 -200
References: [1] Garifullin et al. (1985) by EPR bulk, c = 0.03%. [2] Nieuwenhuys et al. (1982b) [alternate analysis of results of Creuzet and Campbell (1982a) with another set of CEF parameters].
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
107
parameters may affect the result of the magneto-elastic coefficients. However, the best agreement was achieved by using a CEF level scheme which also describes other results such as susceptibility measurements. They correspond to the same CEF parameters Aa(r 4) =--18 K and A6(r 6) = 7.6K across the series (except for Tm where A6(r 6) -= 6.5 K). These results have also been compared with estimates of the magneto-elastic coefficients obtained by EPR measurements (Pela et al. 1981). The EPR results appear to be systematically underestimated in experiments performed on strained thin films and, to a lesser extent, on strained bulk samples (Campbell and Creuzet 1985). Magnetostriction measurements were analyzed by considering second- and fourth-order one-ion magneto-elastic terms, the latter always contributing for about 10% to the total magnetostriction. The experimental values appear consistent along the series, and follow roughly the appropriate ~j or /3j Stevens coefficients. Typical values are, By oq
_
U~4)_
V~
3
V~
3
B ~
2V2 V2 ~ -270 K ,
U~4)
/3j - 16----~ V4 ~ - 4 0 K ,
3
5
O/j - W~ V 2 - -2100 K ~
5
/3j - 4V2 V4 ~ - 7 5 0 K ,
for all the dilute AgR compounds, the V~'s are defined by Campbell and Creuzet (1985). This coherency is reminiscent of the constant CEF parameters across the series. The isotropic one-ion magneto-elasticity was also investigated from these magnetostriction measurements (Creuzet 1982). The uncertainty in the coefficients is largely due to the weakness of the volume effects. Typical values for the AgR series are,
n(4)_ /3j
Weft
1
16V2 V4 - -120 K ,
n(6)
- -yj -
V3
1
64 V6~ -t- 135 K
Gold- and copper-rare earth dilute compounds. Compared to the AgR-series, rather less results are available on the A..__~uR and CuR dilute compounds. Systematic comparisons are therefore difficult. Nevertheless, the main result is the change of sign of B e and/or B y in AuR and C__uuRcompounds (see table 19), compared to AgR compounds. On the other hand, the volume magnetostriction appears to be still weaker in these compounds than in A__ggRones (Creuzet 1982). 4.6.2. Rare earths diluted in cubic pnictides A systematic study of the tetragonal magneto-elasticity was performed in the cubic NaCl-type LaSbR series by [001] magnetostriction measurements (Nieuwenhuys et al. 1982a). As in the A___ggRsystem, B y follows the sign of the Stevens factor % well (table 20). In comparison to corresponding concentrate
108
P. M O R I N and D. SCHMITT
T A B L E 19 Magneto-elastic coefficients in the A__._quRand C__._~uRseries. All the results are from Creuzet and Campbell (1982b,c) and Creuzet (1982); they were obtained by magnetostriction measurements (rare earth concentration e = 0.5 at.%), except when indicated otherwise. The C E F parameters used are A 4 ( r 4) = - 3 0 K, A 6 ( r 6) = 6 K for the A._.._uuRseries, A 4 ( r 4) = - 8 6 K, A 6 ( r 6) = 12 K for CuEr compound. Compound A..._~uTb B y (K)
1.7
B" (K)
-20
B~4~ (mK) B~4 ~ (mK)
12 -33
A..._~uEr [1]
AuYb
-0.8
-8.3 - 8 . 7 [11 89 44 [1] -17 1200
4.5
C__._~uEr[2] 0.44 -6.5
References: [I] Pela et al. (1981), from E P R measurements on films (c = 1%). [2] Garifullin et al. (1985), from E P R measurements on bulk samples (c = 0.03%).
compounds (section 4.2), these parameters appear to be quite comparable in magnitude, except in LaSbSm. The value obtained by magnetostriction in LaSbDy is consistent with that deduced from magnetization measurements. However, the latter may be considered to be more reliable than the former, because the magnetization reflects the lattice distortion in the immediate vicinity of the rare earth ions, while the magnetostriction cumulates the distortion effects far away from the rare earths. Indeed, it appears closer to the value in DySb than that obtained by magnetostriction. The most original feature in the series is the observation of a "reverse" magnetostriction in LaSbCe and LaSbSm, i.e., a change of sign of the distortion as a function of temperature, which is explained by the sign reversal of the expectation value ( O ° ) (see section 3.4.1). Magnetostriction was also observed for S-state Gd ions diluted in LaSb and LaBi (Hfifner et al. 1983). This has been explained within a phenomenological model which includes admixture effects of the first-excited 6p multiplet. However, corresponding magnetostriction values reach ~ l / l - - 8 x 10 -6 at 1.5 K in a 5 T field for 10%-dilute compounds. These values, scaled for the concentration, are only 1/10 of that observed in GdZn, where the distortion was interpreted as the manifestation of a two-ion magneto-elasticity (Morin et al. 1977). According to section 3.2.4, it could then be appropriate to consider the latter term explicitly. The magnetostriction measurements on LaSbR compounds were extended to the LaBiR series in order to check, in particular, the reverse magnetostriction in Ce and Sm based compounds (H/ifner et al. 1983). Although the CEF itself is smaller in bismuthides than in antimonides, the magneto-elastic coupling seems to be larger (see table 21).
QUADRUPOLAR
EFFECTS IN RARE EARTH INTERMETALLICS
i ~ ,...,
< II
~
~
~9
.~ o=
o C, t"q
Z
I
~
.I
cq
O
r.)
109
110
P. MORIN and D. SCHMITT TABLE 21 Crystal field parameters W, x and magneto-elastic coefficient B y in the diluted LaBiR series (c is the rare earth atomic concentration), from H~fner et al. (1983), from magnetostriction measurements. Rare-earth
W (K) x B y (K) c (%)
Ce
Tb
Er
1.91 1 -65.3 10
-0.45 -0.96 -43.7 3
0.20 0.79 10.0 3
4.6.3. Other dilute systems Few other cubic dilute rare earth systems have been studied with regard to magneto-elasticity. Only on Tb based compounds has there been a determination of magneto-elastic coupling parameters (see table 22). Creuzet et al. (1982) determined both coefficients B y and B e by magnetostriction measurements in cubic CsCl-type Tb0.01La0.99Ag compound. The B y value is roughly eight times smaller than in the concentrated TbAg compound (see table 7), which could be related to the incipient instability of the LaAg matrix (section 3.10). The same is valid for the value of G ~. On the other hand, the B e value seems to be doubtful, since it would lead to an unreasonable value of G~E, i.e., 1 K. Some studies have also been carried out on dilute systems having a lower symmetry, namely the hexagonal YR and MgR compounds. The three secondorder magneto-elastic coefficients B or1, Ba2 and B ~ were determined in YTb, __YDy and __YErcompounds (see table 23). A change of sign of all coefficients occurs for TABLE 22 Crystal field parameters W, x and magneto-elastic coefficients in some diluted rare-earth systems. Compound LaAgTb
w (K) x B ~ (K) B E (K) G y (mK) G" (mK)
0.5 -1 3.3 [1] - - 3 2 0 [1] - 1 [2]
LaA12Tb
0.27 0.9
0 [31 - 3 0 [31
References: [1] Creuzet et al. (1982), from magnetostriction measurements (c = 1%). [2] Hoenig et al. (1980), from magnetization (SQUID) measurements (c = 2%). [3] Hoenig (1980), from magnetization (SQUID) measurements (c = 1 to 6%).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
111
TABLE 23 Magneto-elastic coefficientsin hexagonal YR system "(fromPureur et al. 1985), using magnetostriction measurements (c = 1 to 2%). Compound B "' (K)
B~2 (K) Be (K)
YTb
YDy
YEr
6.1 18.9 14.8
9.4 8.5 20.4
-1.9
-4.4 -8.4
the compound YEr, due to the Stevens factor aj. In addition, the ratio B ~ / a j is roughly constant throughout the series, B ' ~ / a j -~ V'3 B'~2/a, ~ ~
V2(/'a,1)/Olj
~
--850 K ,
V2(F~,2)/a s ~ -1600 K ,
B~/aj ==-~X/-2 Vz(F~,) / a j ~ -3200 K .
These values lead to distortions which are quite consistent, after correction for the concentration, with those observed in pure rare earth metals (Pureur et al. 1985). In a few cases, the introduction of a fourth-order magneto-elastic coefficient slightly improved the fits. In spite of the low concentration, spin-glass effects have been observed at low temperatures, in particular with Tb. Lastly, magnetostriction for YGd compound (c = 1%) reaches ~c/c ~ 10 -6 for a magnetic field of 6 T along the c axis, which is again of the same order of magnitude as in LaSbGd and LaBiGd (see above). Magnetostriction and thermal expansion data are also available for the heavy rare earths dilute in a single-crystal Mg host (Bijvoet et al. 1980, De Jong et al. 1980, 1982) (concentration about 0.1%). Here, also the magnetostriction agrees with the sign of aj. Unfortunately, no complete analysis was performed in order to determine the actual magneto-elastic coefficients. 4.7. Miscellaneous c o m p o u n d s
The universal character of the magneto-elasticity is emphasized in the literature through accurately conceived data on individual compounds, however, without thorough and systematic analyses across a given series. One can find, e.g., magneto-elastic effects in RCo 3 (Pourarian and Tajabor 1980) for which magnetostriction data on single crystals are available. In other series, like rare earth metals and rare earth borides few experimental results have been quantitatively analyzed. Rare earth metals. Among the rare earth metals, dhcp Pr has been widely studied over the last decade, because of the magneto-elastic effects on the criticality of antiferromagnetic interactions; the strain mode involved in these effects is the e mode (section 3.9). A magneto-elastic coefficient B ~ = 72 K acting
112
P. MORIN and D. SCHMITr
on the hexagonal Pr sites, i.e., G ~ = 76 mK, leads to a consistent description of various experiments, such as elastic constant measurements, magnetostriction, magnetic excitations and the occurrence of an antiferromagnetic ordering under an uniaxial stress (Houmann et al. 1979). This e-mode magneto-elastic coupling was also suggested to explain the field dependence of the thermal expansion (Liithi and Ott 1980). Without any magnetic field, these thermal expansion experiments have been described with the a-mode magneto-elastic coefficients of the hexagonal symmetry B al (-----~V/-3Bv) = - 4 6 K and B a2 (--=V~ B3) -~ 35.5 K. On the other hand, a detailed study of the magneto-elasticity was carried out by magnetostriction measurements (Hendy et al. 1979). A complete analysis was made in terms of one-ion and two-ion magneto-elastic coupling. It was found that one-ion a2-strain coupling is the dominant mechanism between the lattice and the 4f ion. Unfortunately no value for B ~ or D ~ was extracted from the experiments; only the h ~ coefficients relative to both cubic and hexagonal sites of Pr have been obtained. When taking the values of elastic constants from Greiner et al. (1973), the following evaluations can be made for the Pr ions on hexagonal sites: B "2 = 2 2 . 6 K , B ~ = 4 2 K , nal = o a~1 l = D~I a2 ~ 0. These values have the same order of magnitude as those from Houmann et al. (1979) in which the contribution arising from the cubic sites was included. Rare earth borides. Among the six families of binary rare earth borides (Etourneau 1985), RB 4 and R B 6 have been the ones most studied. Their crystallographic and magnetic properties have been shown to depend drastically on the nature and rate of impurities and on the metallurgical conditions. R B 6 compounds. RB 6 compounds all crystallize within a CsCl-type cubic arrangement of B octahedra and R ions. The main part of the literature on this series is devoted to light R compounds. Magneto-elastic and quadrupolar properties have been studied mainly in CeB 6 and negative quadrupolar interactions appear to be the origin of the properties observed (section 3.1.2). The tetragonal magneto-elastic coefficient [gr3[ = 50 K leads to [B~/,~j[ -- 875 K and the trigonal one = 135K to IB'/a,I = 3300K (L/ithi et al. 1984). PrB 6 exhibits properties reminiscent of those of CeB 6 in particular regarding its magnetic structure (section 3.6). The first-order transition (section 3.2.2) may result from bilinear and quadrupolar couplings of equivalent strength, as the bilinear exchange is larger than in CeB 6. Sizeable effects on the elastic constants are observed in PrB 6 and in NdB 6 (Tamaki et al. 1985). Unfortunately no quantitative analysis is available at the present time. In addition, the first-order Nrel transition in GdB 6 sets a puzzling problem which cannot be analyzed in terms of pure quadrupolar interactions. RB 4 compounds. The rare earth tetraborides have tetragonal crystal symmetry (Will et al. 1981). The rare earth atoms are confined in boron cages with an mm local symmetry (4 g). The compounds are antiferromagnetic with the exception of PrB 4 (Buschow and Creighton 1972). They exhibit some magneto-elastic features, especially ErB 4 which undergoes a tetragonal-orthorhombic transition at the Nrel temperature. At low temperature, the lattice symmetry (Pbam) agrees with the
[grsI
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
113
magnetic one (Pb'am). However, the most interesting compound might be TbB 4 for which the structural transition occurs in the paramagnetic range at 80 K, far above the N6el temperature (T N = 43 K). Here again, lattice (Pbam) and magnetic (Pbam') symmetries are closely related. In both compounds the structural symmetry lowerings have been observed by X-ray diffraction on polycrystalline samples and are rather weak (Will et al. 1986). To the author's knowledge there are, at present, neither experimental determinations for the CEF nor for the magneto-elastic coupling. It is worth noting that the mm rare earth symmetry induces non-zero (O~) and (022) quadrupolar values. In addition, the easy magnetization directions do not vary across the series in the same way as the second-order Stevens coefficient, revealing that the anisotropy is not only of single-ion origin (Etourneau 1985). 5. Discussion
5.1. General analysis 5.1.1. Coherency of the various determinations To prove the relevancy of the basic Hamiltonian, coherent determinations of the quadrupolar coefficients in independent experimental ways are needed. For example, table 24 collects the various values obtained for the quadrupolar parameters in tetragonal symmetry for TmX compounds within the CsCl-type structure. A very good agreement is observed. In particular, the parastriction removes the ambiguity about the sign of the magneto-elastic coefficient which remains after the elastic constant analysis. The consistency between determinations obtained by direct (elastic constants) and indirect (parastriction and third-order susceptibility) ways, i.e., without and with the presence of an applied magnetic field, demonstrates the relevancy of the quadrupolar Hamiltonian and the weakness of the two-ion magneto-elasticity in this family of TmX compounds. However, we have to keep in mind that the latter contribution may be more significant for large Land6 factors (gj). Higher order one-ion magneto-elasticity as well as anharmonic elastic terms appear to be negligible here. From table 24, the one-ion magneto-elasticity of compounds with 4d elements (Ag, Cd) is observed to be weaker than that of compounds with 3d elements (Cu, Zn). The eccentric case of TmMg is consistent with the absence of d electrons on the Mg-ions (see section 5.2). The same behaviour is observed for the pairinteraction coefficient, which is at its maximum for II-B elements, in particular for zinc. In most cases, the pair interaction contribution to the total coefficient G ~ dominates the magneto-elastic one. TmMg is the only TmX compound within the CsCl-type structure not to be strongly influenced by tetragonal quadrupolar interactions. The microscopic origin of these different coefficients is analyzed in section 5.2. In the presence of cubic level schemes close to each other, this series provides us with clear experimental evidence of quadrupolar effects. Large softenings of
P. MORIN and D. SCHMITT
114
TABLE 24 Quadrupolar coefficients in TmX compounds within the CsCl-type structure. Compound
Coefficient
TmZn
TQ [1] Cll - C12 [2] Parastriction [2] X~ ~ [31
-+29 -31
20
r e [41 c n - c12 [4] Parastriction [5] x~ ~ [6]
-+12.7 -14
11.2
TmCd
B ~ (K)
K s (mK)
G ~ (mK)
Kr/~ ~ ~ M E
25 25 27 25 12.5 12.5 12 13
TmCu
Cn - C12 [7] Parastriction [7] X~ ) [31
-+30 -27
7
11.5 11 10.5
TmAg(8~
Cll - CI2
---20 - 20.3
2
4.4 4 3.5
-+5.4 -5.4
(0, 3) (0, 3) (0, 3)
(0, 3) (0, 3) (0, 3)
1.75 Parastriction X~ ) TmMg(9)
CI1 C12 Parastriction X~ ) --
References: [1] Morin et al. (1978b). [2] Givord et al. (1983). [3] Morin and Schmitt (1981b). [4] A16onard and Morin (1979). [5] Morin et al. (1978c). [6] Morin and Schmitt (1982c). [7] Jaussaud et al. (1980). [8] Giraud and Morin (1986). [9] Giraud et al. (1986). elastic constants exist, with t h e r m a l variations strongly d e p e n d i n g on the competition b e t w e e n the two q u a d r u p o l a r couplings, as shown in T m Z n and T m C d [fig. 1 in Liithi et al. (1979)]. Similar selective characteristics are o b s e r v e d in parastriction ( M o r i n et al. 1978c) and third-order m a g n e t i c susceptibility ( M o r i n and Schmitt 1981b) experiments.
5.1.2. Comparison o f the magneto-elastic coefficients T h e various magneto-elastic coefficients p r e s e n t e d in section 4 for cubic c o m p o u n d s are collected in figs. 63 and 64, after normalization to aj. T h e first c o m m e n t c o n c e r n s the Order of m a g n i t u d e which is relatively i n d e p e n d e n t o f the rare earth across a given series. This is reminiscent o f the b e h a v i o u r of 4th and 6th o r d e r C E F p a r a m e t e r s within a series. A relatively g o o d a g r e e m e n t is o b s e r v e d for the B~/aj ratio b e t w e e n concentrated R S b and dilute L a S b R c o m p o u n d s . This indicates, in particular, that the
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS i
I
I
5
I
i
i
i
I
RLoSb/ '~
\ /~-~Sb
,,
o. /..o
0
ePb3,,.," • C-3
"
o0
#0%
115
+
~
~
~
.0.-'%o. TmGo3I
÷
RA[2~
+
F
RMg
l'\
.
.
.
el
-5
RZn__
.
RAg
....... •
-
-
TmCd
RC-u". . . . . . . . ">'~,~~.>,...-:~~"
i
I
I
I
i
t
i
i
Ce
Pr
Nd
Tb
Dy
Ho
Er
Tm
Fig. 63. Magneto-elastic coefficient B y for the tetragonal symmetry normalized by the second-order Stevens coefficient in various series of rare earth cubic intermetallic compounds.
I
]
:PrPb 3 ,
L
TmGg3
.~ • PrAg lCePb3
TmMg.
,,t
0
Ce
o/-
Pr
Nd
Tb
Oy
Ho
Er
Tm
Fig. 64. Magneto-elastic coefficient B ~ for the trigonal symmetry normalized by the second-order Stevens coefficient in various series of rare earth cubic intermetallic compounds.
macroscopic strain measured in dilute samples is, in a first approximation, linearly related to the local deformation through the concentration of active sites. For a given series, the two magneto-elastic coefficients are often of opposite sign, as expected from point charge estimates as will be discussed in section 5.2. The only exceptions are the cases of AuCu3-type compounds and RAg dilute alloys. For a given type of structure, values have been observed to be similar in a first approximation. This is in particular true for the CsCl-type structure. However, the positive sign of B ~/as for TbZn is in opposition with the other determinations in the RZn series. This might be the signature of a two-ion contribution although it has been observed nowhere else for this symmetry. Note in addition the small average value of B~/a: in RZn.
116
P. M O R I N and D. SCHMITT
Among the different series, the values observed for the B~/aj's are of the same order of magnitude. This is reminiscent of the CEF parameters which usually correspond to total level spacings of a few hundred Kelvin. In the same way as peculiar surroundings or densities of electronic states drive small or large CEF spacings, as in RBe13 (Vigneron et al. 1980) and RB 6 (Zirngiebl et al. 1984), respectively, the magneto-elastic coefficients may also vary from one series to another. The actual spontaneous strain depends not only on the B" values, but also on other parameters. The CEF usually determines the axis of easy magnetization, then the symmetry lowering mode. The pair interactions state the purification of the CEF wave functions, then the various expectation values. For this reason, RFe 2 compounds develop "giant" strains at room temperature although their low-temperature values are in the same range as for other series. On the other hand, the spontaneous strain, which would be large in the case of a collinear spin structure, fully vanishes in the case of a multi-axial spin structure, preserving cubic symmetry as in DyCu. The magneto-elastic effects are of similar magnitude in rare earth intermetallics and insulators. For instance, in TbVO4, DyVO 4 and TmVO4, the vanadates which undergo a Jahn-Teller tetragonal-orthorhombic transition (Gehring and Gehring 1975), the low-temperature spontaneous strains are e ~= 12.7 × 10 -3, e ~ ' = 5.9 × 10 -3 and e ~= 3 × 10 -3, respectively. These strains are reminiscent of those in intermetallics. Unfortunately, whereas determinations of magneto-elastic coefficients in metals are numerous in cubic symmetry and scarce in tetragonal symmetry, the situation is the opposite in insulators; thus comparing the magneto-elastic coefficients is difficult. However, from the magneto-elastic contributions G~aE = 6 /[~1 and GME = /X2 determined from the ultrasonic velocity softenings (Melcher 1976), the following values IB~I--S7K, In l--80 and 22K are deduced in DyWO4, TbVO4, TmWO4, respectively. These magnitudes are similar to those observed in intermetallics.
5.1.3. Comparison of the quadrupolar pair-interaction coefficients There are fewer systematic studies for the pair interactions than there are for the magneto-elastic coupling. However, from table 24, their existence in TmX is well established and a very good coherency is observed in tetragonal symmetry. Within the CsCl-type structure, the tetragonal pair interactions dominate the magnetoelastic coupling and are the main origin of the ferroquadrupolar ordering. They are always positive and particularly important as soon as the aj coefficient is large. On the contrary, from the determinations in the paramagnetic phase, the trigonal pair interactions are negative and overwhelm the weak magneto-elasticity. For NaCl-type compounds, the tendency is less clear (fig. 65): K ~ is negative in DySb, but seems to be positive in HoSb and null in TmSb, in which no pair interactions seem to be present (Wang and Liithi 1977b). Tetragonal pair interactions are weaker than the magneto-elastic coupling in RSb pnictides. For the trigonal symmetry, the study of the TbX system indicates K ~ to be positive
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
20
-t
80
-Ag
117
.Ag -Go 3.~. '-Cd
-Cd
II
-'Cu
-Cu°
-Sb E 0 X)
- cd "zo ;Ag
"-P%
I
Ce
I
Pr
-Sb
-sbD
I
I
I
Tb Dy Ho Er
I
Tm
Fig. 65. Quadrupolar pair coefficient K ~ for the tetragonal symmetry in some rare earth cubic intermetallics (0: CsCl-type, +: AuCu3-type, [~: NaCl-type compounds).
and larger than G ~E. However, the total quadrupolar interactions remain smaller than the bilinear coupling and only modify the nature of the N6el transition. In insulators, such as the vanadates, the pair-interaction coefficients are fundamentally different. From the analysis of elastic constants (Melcher 1976) the coefficients K ~= h 1 and K ~= A2 are found to be negative and their absolute values are smaller than the magneto-elastic contribution (K ~= -0.35 mK in D y g O 4 , K ~ = -0.77 and - 1 1 mK in TmVO 4 and TbVO4, respectively). In these Jahn-Teller compounds, the structural transition is thus driven by the magnetoelastic coupling, whereas in the intermetallics the pair interactions drive the quadrupolar ordering and then, through the magneto-elastic coupling, the structural transition (Levy et al. 1979). In insulators the pair interactions are mediated by phonons (Gehring and Gehring 1975), and the h coefficient is a so-called "phonon exchange" parameter. It integrates the effective coupling of all the phonons to the Jahn-Teller ions for a given symmetry. The K/GME ratios are very close to --1/3, the theoretical expectation in the absence of any optic phonon mode contribution (Harley et al. 1972). On the other hand, the positive K values observed in intermetallics can originate neither from a k = 0 optical phonon in the NaC1- and CsCl-type structures, nor from the negative phonon self-energy term (Levy et al. 1979). In section 5.2 it will be shown that pair interactions in intermetallics are mainly mediated by conduction electrons.
5.2. Origin of the quadrupolar interactions 5.2.1. The one-ion magneto-elastic coupling From the comparison between the expressions for the cubic crystal field Hamiltonian [eq. (2)] and those for the one-ion magneto-elastic coupling [eq. (7)], it becomes obvious that the magneto-elastic coefficients are the strain derivatives of
118
P. MORIN and D. SCHMITT
the CEF parameters. For example, the second-order CEF parameter B ° vanishes in cubic symmetry, but its derivative with regard to a tetragonal distortion does not, B ~ , =-=
OB °
0~
"
The same is valid for a trigonal deformation, as well as for the higher-rank magneto-elastic coefficients, or if we consider crystals with lower symmetry, e.g., hexagonal or tetragonal ones. Therefore, the question of the origin of the magneto-elastic coupling is identical to the problem of the origin of the crystal field itself. In rare earth intermetallic compounds, it has been shown that the CEF parameters originate mainly from two contributions, namely the point charges and the contributions of the conduction electrons (Schmitt 1979). In cubic systems, these latter have been found to be preponderant, due to the strong anisotropic orbital character of the conduction electrons, as shown, e.g., by augmented-plane-wave band calculations (Belakhovsky and Ray 1975). In systems with hexagonal or tetragonal symmetries, the point-charge contribution seems to play a role which is relatively more important than in cubic systems, at least for the second-order CEF parameters. Then, calculating the magneto-elastic coefficients requires a precise knowledge of how these two CEF contributions depend on a given strain. First, in the presence of a weak strain, the localized charges are displaced from their previous non-strained arrangement, generating a new potential having the new symmetry of the lattice and proportional to the strain. This potential gives the point-charge contribution to the magneto-elastic coefficients. On the other hand, and to the first order of perturbation, this new potential modifies the conduction band. In particular, it lifts the orbital degeneracy of the conduction electron states in the vicinity of the Fermi level. According to Fermi statistics, this leads to a redistribution of the conduction electrons in the band which gives rise to a new CEF contribution, i.e., the band contribution to the magneto-elastic coefficients. Both contributions have been thoroughly investigated in cubic rare earth intermetallic compounds (Morin and Schmitt 1981a). In particular, the expressions for the direct and exchange Coulomb contributions of each type of conduction electrons (p, d, or f character) have been derived. It has been found that the magneto-elastic modifications of the cubic fourth- and sixth-order CEF parameters are negligible and that the second-order magneto-elastic coefficients result from the competition between numerous antagonistic contributions, which are of the same order of magnitude as the experimental values. Within the hypothesis of constant effective charges and a rigid-band model, all contributions are expected to vary as aj, ~j, yj for the second-, fourth- and sixth-order magneto-elastic coefficients, respectively. Furthermore, the pointcharge estimation leads to the constant ratio B ~ / B ~ = -4/V'-3 between the trigonal
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
119
and tetragonal second-order coefficients (see Creuzet 1982); this is not true for the band contribution. As shown in the previous section, this relation is far from being satisfied experimentally for a given compound, and this failure clearly indicates the importance of the conduction electrons. Nevertheless, the difference in sign between both parameters in most of the series as well as their overall magnitude (figs. 63 and 64) suggests a driving role for the point-charge contribution, contrarily to the case of the cubic CEF parameters. In realistic systems, because of the various contributions involved, the magnetoelastic coefficients are thus liable to vary significantly from one compound to another, especially if they belong to different series. This variation may occur via a shift of the values of the effective point charges. For the conduction-band contribution, the variation may be related to the modification of the partial densities of states at the Fermi level, which can be important when the Fermi level lies near a peak of the density of states. The importance of the d electrons on the rare earth site has also been suggested in dilute rare earth alloys (Campbell and Creuzet 1985). In these systems, the magneto-elastic coupling constants were evaluated within an indirect electronlattice interaction model, and using the nearly free electron approximation (Kozarzewski et al. 1985). The general features of the results obtained in this calculation are similar to those described above. The effect of screening of the magneto-elastic coupling by the conduction electrons has been found to significantly reduce the point-charge estimate of the parameters in Laves-phase rare earth intermetallics (Del Moral 1983). As a conclusion, ab initio evaluations of the one-ion magneto-elastic coefficients appear to be difficult to perform. As for the CEF parameters, they originate from numerous antagonistic contributions, and obtaining realistic values seems rather difficult. Qualitatively, their order of magnitude is well accounted for by point-charge and/or conduction electron contributions, and they follow the Stevens factor throughout a given series.
5.2.2. Quadrupolar pair interactions In metallic systems, conduction electrons have been shown to play an essential role in the coupling between magnetic ions, by propagating the local interactions between them and the localized magnetic electrons. This indirect exchange mechanism has been widely invoked in magnetic materials and is known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Most of the studies has been focused on bilinear-type indirect coupling, which, in general, is the dominant term. Much less attention was paid to the origin of quadrupolar-type interactions. In rare earth and transition metal insulators, direct coupling between electric multipole moments of the two atoms and indirect coupling through the lattice-ion interaction have both been investigated (Birgeneau et al. 1969). However, in rare earth intermetallic compounds with the NaC1- and CsCl-type structures, both couplings have been proved to be negligible and therefore cannot be the dominant mechanism for the quadrupolar pair coupling (Levy et al. 1979): the
120
P. MORIN and D. SCHMITF
primary source of such a coupling is the indirect Coulomb and exchange interaction via the conduction electrons. The coupling of quadrupoles via the conduction band is analogous to the RKKY interaction quoted above, which involves the spins at different sites through the local s - f exchange interaction. For quadrupoles, it is the direct and exchange part of the full 4f-conduction electron (k-f) Coulomb interaction that contributes to the couplings K ". The corresponding effective couplings were calculated in second-order perturbation theory and by using plane-wave states for the conduction electrons (Teitelbaum and Levy 1976). A more realistic ab initio calculation of bilinear and quadrupolar pair coupling was performed later in the cubic rare earth intermetallic compound DyZn, by using conduction-electron wave-functions and energy bands obtained by a selfconsistent augmented-plane-wave method (Schmitt and Levy 1984, 1985). Spin and orbital effects in the complete k - f interaction were taken into account. The results underline the dominant role played by the d conduction electrons in the bilinear and quadrupolar couplings. In this calculation, the Fourier transform of the bilinear 0*(q) and quadrupolar Kr(q) and K~(q) interactions were found to be of the correct order of magnitude. Their different q-dependence was explained by the behaviour of specific matrix elements of the k - f interaction: the magnitude of these matrix elements is directly related to the symmetry of the two k-states involved. In particular, the ferromagnetism and the ferroquadrupolar tendencies (y-mode) of DyZn are associated with the strong eg-character of the conduction electrons near the Fermi level. On the contrary, the antiferroquadrupolar tendency of the trigonal coupling K~(q) is related to the small amount of t2g-type conduction electrons near the Fermi energy. The dependence of the bilinear and quadrupolar couplings on the Fermi energy was also investigated. Only part of the coupling, i.e. the intraband terms at q = 0, follows the partial densities of states at the Fermi level, which are associated with the type of electrons (% o r t2g ) from which they arise. The other terms, in particular the interband ones, are not directly correlated with the Fermi energy but depend on details of the entire energy band structure, including energy bands well above the Fermi level, which contribute for a noticeable amount. At last, the orbital interactions are the main contributions to the quadrupolar couplings K" so that the latter are expected to vary as otj. 2 That explains the importance of quadrupolar effects at the beginning and end of the rare earth series, where weak spin contributions (small De Gennes factor) allow the quadrupolar coupling to be better revealed. As for the one-ion magneto-elastic coefficients, ab initio evaluations of the two-ion quadrupolar parameters are very difficult to perform and would be necessary for each compound under consideration. 5.3. Conclusion
The aim of this chapter has been to present an extensive overview of the effect of the magneto-elastic coupling and of the quadrupolar interactions in rare earth intermetallic compounds. A complete Hamiltonian has been detailed, which
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
121
includes all the relevant terms describing the one-ion magneto-elasticity as well as the two-ion quadrupolar coupling, for all the possible normal strain modes in cubic, hexagonal and tetragonal symmetries. The formalism for using this Hamiltonian has been developed, in particular that related to a perturbation theory. The various effects of the magneto-elastic coupling and of the two-ion quadrupolar interactions on the magnetic and elastic properties have been discussed, with a particular emphasis on the properties related to the perturbation methods: elastic constants, parastriction and third-order magnetic susceptibility. All these experimental manifestations appear as reliable ways to determine the quadrupolar parameters. Due to the strong anisotropic character of the magneto-elastic properties, using single crystals for experiments is quite essential. The magneto-elastic and quadrupolar data available in the literature have been summarized for a great number of rare earth series, within the present formalism, in order to make a comparison between them easier. Some series, especially those having the cubic CsC1- and NaCl-type structures, have been extensively studied in the last decades. Several others have been investigated much less, and some of them would be worth to be studied more thoroughly, in particular when single crystals become available. In most of the experimental results, the second-order one-ion magneto-elastic coefficients are concerned. The ratios B~'/%, in general, present a regular variation throughout a given series, but their magnitude and their sign can drastically change from one series to another. Nevertheless, their absolute value always remains in the 0-4000 K range. These differences show that, even if the point-charge contribution to the magneto-elastic coefficient is important, the role of the conduction electron is far from being negligible in these metallic compounds. An accurate ab initio estimate appears, however, difficult to obtain, due to the great number of different antagonistic contributions. The two-ion quadrupolar interactions have been studied less thoroughly in the rare earth intermetallic compounds. A consistent determination of the corresponding parameters has been successfully carried out in those series where the effects are strong, in particular those where a quadrupolar ordering occurs. Through their contribution to the free energy, the quadrupolar pair-interactions may strongly influence the character of the magnetic transitions, as well as, when they are negative, the stability of multi-axial spin structures. At last the dominant role of the conduction electrons has been emphasized in these systems, contrarily to the case of the rare earth insulators where the coupling between 4f quadrupoles is mainly due to the phonons. The existence of magneto-elastic and quadrupolar couplings then appear quite general in rare earth interrnetallic compounds. They need to be included in order to quantitatively understand the magnetic and elastic properties of these systems.
Acknowledgement It is a great pleasure to thank Professor P.M. L~vy from New York University for very fruitful discussions and a critical reading of the manuscript.
122
P. MORIN and D. SCHMITI"
Appendix I. Symmetrized Stevens operators for the cubic symmetry To completely express the magneto-elasticity of rank l (l = 2, 4, 6), all 21 + 1 Stevens operators are needed. In the present chapter we use the operators defined by Hutchings (1964), namely the quadrupolar operators of table 1 (l = 2) and the O / (m = 0 to l) and O'S(s) (m = 1 to l) for l = 4 and 6 (see table VIII of Hutchings 1964). The linear combinations of these operators, which transform according to the irreducible representations of the cubic symmetry group, read as, Second-rank 5"
O~ ' 1 = 002'
o l '2 = v ~ o i .
5"
Oi '1 = P~y,
° 2 '2 = Pr= ,
0 2 '3 = Pz=.
Fourth-rank ot 0 4 = 0 0 "~- 5 0 ~
FI:
r3:0~4
.
0~4'2 = -4X/-3 0 ] .
'1 = 004 -- 7 0 4 '
6,2
1
3
5:
064 '1 = O ~ ( S ) ,
0 4 = 0 4 -- 0 4 ,
5:
o i '1= o~4(s),
01,2 = --lOi(s
8,3
0 4 =
__ 0 1 ( S ) __ O I ( S )
) q- 7: O 43( s ) ,
0 4 , 3 = - - ~1 1O 4 -- 1 0 43.
Sixth-rank 5:O6
a
= O1--210~"
r~: 06~-- o l - 06 5:
O~ '1~- O 1 71-3 0 ~ 01,1 _-- 0 4 ( S ) ,
5:
01,2=
=
~5 r ~ 0 6 _2[ _
- - ~10 6I
~/'~
06 6 .
-- 5
~ 0 6 "31 - T11/',15 v6,
Y11~.5/ IJ6(S)", .
011,2 = ~ 01 16 ( s ) - 9
3 )-{- ~'~ O I ( s ) , ~06(S
1 9 3 33 [-15 0 6 1 3 = ~1 O 6-[~O 6 q- T t / 6 •
0621 = OI(S),
5:
O~,2
0~6,3 = 1o~6(s)_ ~5 O 63( s ) 06~, 1 = O I ( S ) ,
5:
,
3 1 0 6 ~ 3 = -~0 6-
5 3 ) + IOI(s), O12,2 = I O ~ ( s ) ~'- ~06(S 5 3 3 5 ~0 6 + ~0 6 •
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
123
Appendix 2. Fourth- and sixth-rank one-ion magneto-elastic Hamiltonian ~Mm Starting from the expressions in table 2 and appendix 1, three terms may be derived for l = 4: one isotropic term which keeps the cubic symmetry, and two anisotropic terms as for l = 2, O~MEI(I = 4) = O~MEI(I = 4) + Yg~m(l = 4) + Y(MEI(I = 4 ) , with ~x ~MEI(I
m --
4) =
~ -B(4)8
a
0 4 ,
--~(4)\e, ltJ4
~/~MEI(I e = 4)
At- /:,2tJ4
) ,
= - - /nJ ( 4. .). . ['Ell . . J 4 1 ..}_ E 2e,,~e I J 4 , 2 + E 3. .[.J. 4 3,) .
The case l = 6 is identical to l = 4, except that, for the e mode, there exist two independent sets of linear combinations of the Stevens operators. Therefore, ~MEI(I = 6) = ffffMEl(l=6) + ffff~ciEl(l=6)+ ffffMEX(/ el e2 a = 6) + ~MEI(/ = 6), with oz
o~
~1;
~MEI(I = 6)
= -B(6)8
F3:
Y(~E,(I
:
F5:
fft~MEI(/ el = 6):
F5:
g Me2E I ( I = 6)
:
6)
ot
o~
0 6 ,
- - D, (~6r) ~, E Ir~r,1 I J 6 + ~_r,-~r,2x 2~6 J , ... 1,1 ..[_ 8 2 0 ; 1 , 2 _]_ E 3e,-~el,3x, - - Dh e( 61 ) .I'EIIA'6 IJ 6 ) ,
= - - / ~ ( 6 ) .j E . .l.l J 6
2,1 + 406
,2
+ E;O;2,3)
,
All the above expressions are referred to axes with four-fold symmetry, i.e., with z-axis parallel to the [001] direction. In some cases, however, it may be useful to consider the z-axis along a [111] direction, e.g., when a magnetic field is applied along that direction. In that case, symmetry considerations imply that the tetragonal strain e r should vanish and that the trigonal strains e" are all identical, e 7 = e 2 = e; = e'. Therefore, the corresponding trigonal magneto-elastic Hamiltonians each involve a single linear combination of symmetrized operators which may be rewritten in the new axes system (according to the convention of Hutchings 1964) as, I=2:
t'xy+ e,z+ ezx-, o ,
1=4:
0 4,1+ 0~,2 + 0 4,3~ ~(O 1 4o +7V'2 O ] ) ,
l=6:
O ; 1 , 1 + O ; 1 , 2 _[_ O;1,3..._> _ _ ~ ( 0 6 1 0 __ ~ / ~ 0 6 3
O62,1 + 062,2 + O6Z,3__> -- T~ 1 ( O 60 --
_ 110~)
2 5 X / 2 O 63 +
130~)
124
P. MORIN and D. SCHMITr
Appendix 3. Perturbation theory: the tetragonal mode Y First we have to define the zeroth-order energies E~ and the eigenvectors [ik) corresponding to the CEF Hamiltonian ~fCEF alone, ~CEF [ik) = E i lik)
(k
= 1,...,
m,).
(Cl)
In each degenerate subspace i, the m~ independent eigenfunctions lik) have to be adapted to the perturbation Y('~ [eq. (29)]. This can be achieved easily by choosing the following expansion in the IJ, M) basis [for the complete expressions, see Lea et al. (1962)], [ik) = aiklJ,1
M ) + a~2lJ,
M
- 4) +
a~klJ, M
- 8) + " "
.
(C2)
A perturbation theory carried out to second-order in e~ and fourth order in H allows one to obtain analytic expressions of the perturbed energies Ezk,
4 Eik = E i + 2
n=l
~ikl~(n)+ ' ' "
(C3)
"
Then, the partition function Z, Z = Z e -t~E'k ,
(C4)
i,k
can be calculated, where/3 = 1 / ( k a T ) , where k B is the Boltzmann constant and T the temperature. One obtains, Z=
ZcEFZ
1 r~ ,BVeV1 + KVQ): .... [I + ½/3Xo(H+ nM)2 + :PXvt
+ PXv'~(2)'B~ert1 + K r Q ) ( H + n M ) 2
-1-1/3(X(3) -t- 1/3(X0)2)(n--}-/,/M) 4-1-.. "l,
(C5)
where ZCE v is relative to the Ei's and Z .... to the elastic energy and to the corrective energy terms [see eqs. (14), (16), (17)]. Finally, this leads to the expression of the total free energy F ~ = - k BT In Z given in eq. (30). . (2) and x~ - (3) are given The expressions of the four CEF susceptibilities X0, X~, x~ by,
x0 = g . . B
i z.12
Il Ok,j, 2 i,k
l
+ 1 ~B T
I/ k,,kl 2 , \ I ;~.i~l )
'
(c6)
(C7)
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS t,(2)
,
2 2
= gylxB
125
Jik,jlQjl,J'l'JY'l',ik q- 2Qik,jlJjl,j'l'Jy'l',ik fi jei,l £ (U i -- E i ) ( E i -- Ej,) j'~i,l' (
Ei,k
I4~,~,l O,~.,~ + 2Oik,jl41,ikJik,ik (
- ]¢i,I 2
7--V
1
1
)
(c8)
+ 2(kBT)2 [J~k,ikl Q,k,ik , __ ,y ,•(3)
1 4 4 ( Jik,jlJjl,j'l'Jj'l',j'T'Jj'T',ik 2k~r (x°)= + g'~"£i,k f' - 4 /#i,, £ (E,--~---E-7')--~i----Ef) j'#i,l' j"¢i,l" 2
2
IJ,k,j,I IJ,k,/','~ + 2J,'k,/,J/t,/','J/',',ikJik,ik(
+2 ~
/~,,,
(E,- g)(E,- E/,)
2
E,
j' #i,l'
2
IJ....~12l.l,,.,jll 2
-2 ,,i,,E
((E i - Ej)2
~ii:-~])
i
+ 6k3T 3 IJik,ik
i4)
2 q- ( E i _
E/)kuT
,
+©I (C9)
where
Jik,/, =
(C10)
a.~,/, = ,
(Cll)
and
are the matrix elements of Jz and O ~ in the CEF levels. For each CEF sublevel, 1 exp(-/3Ei) f / - ZCEF exp(-/3E~) = ~ exp(-/3E~)
(C12)
i,k
is the Boltzmann population factor. Appendix 4. Perturbation theory: the trigonal mode e
When the magnetic field H is applied along a three-fold axis, e.g., [111], it is more convenient to first rotate the coordinate axes so that [111] is the new z-axis. Then the two combinations of operators in eq. (44) may be rewritten as:
L+~+L--.V~L,
126
P. MORIN and D. SCHMITT
and P x y + P y z q-
1
0
P~x --> ~02 "
The perturbation Y(] takes the new form:
Y({'= --g,p.B(H + n M ) ] z - ½(B~e ~ + K~P)O ° 3 ,.-~e; e.,2 + ~t..oke ) + ½nM2 + ~K*P 2.
(D1)
In each subspace i, the new zeroth-order eigenfunctions Ilk }' (corresponding to ! the appropriate YgCEF) have to be adapted to the above Hamiltonian. This condition is always fulfilled if these basis functions are chosen so that AM = 3 in the IY, M) expansion: 2 [ik )' = bill J, M ) + b~klJ, M - 3) + b~klj, M _ 6 )
+ ..- .
(D2)
The same perturbation theory as in appendix 3 may then be carried out, leading to the following expression for the total free energy F ~, 1 ! F ~ = Foe -- ~Xo( H + nM)2 -- ~XvkD 1 ..... e +K~P) 2 1 (2)' [n~ -~X~, ~D e e + K ~ P ) ( H + n M ) 2 - - ~i a, . y (3)'(H + nM)4
+ ~3 o,,.~e,, te
e,,2
) + l n M 2 + ~K~P 2,
(D3)
¢
where the new CEF susceptibilities X~ are related to the operators Jz and O ° in the new system of axes. Their expressions are identical to those given in appendix 3, but their values differ from the corresponding unprimed susceptibilities, due to the new CEF eigenfunctions lik)'. Returning to the initial system of four-fold axes and defining new trigonal susceptibilities we are led to the expression of eq. (45) for the free energy, where: Xo
:
,
Xo
X~
=
1
,
~X~,
. (2)
X~
1.
(2)'
= ga~,
/~3)
. (3)'
= ,'(v
"
(D4)
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chapter 2 MAGNETO-OPTICAL SPECTROSCOPY OF f-ELECTRON SYSTEMS
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J. SCHOENES ETH ZEirich Laboratorium FEtr Festk&rperphysik 8093 Z(Jrich Switzerland
Ferromagnetic Materials, Vol, 5 Edited by K.H.J. Buschow and E.P. Wohlfarth1" © Elsevier Science Publishers B.V., 1990
CONTENTS 1. l ~ t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. ~[LeoJy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Ph e n o r L e n e l o g y . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. De flnition c f t h e o p t i c a l f u n c t i o n s . . . . . . . . . . . . . . . 2.1.2. F a r a d a y a n d K e r r effects . . . . . . . . . . . . . . . . . . 2.2. Microscol:ic Ir o d e l s to c a l c u l a t e ~ . . . . . . . . . . . . . . . . . . 2.2.1. Inlerle a n d t r a n s i t i o n s . . . . . . . . . . . . . . . . . . . . 2.2.2. Ce ndu c t i o n - e l e c l r o n e ffecls . . . . . . . . . . . . . . . . . . 2.3. T h e H a l l effect a n d the K r a m e r s - K r o n i g e q u a t i o n s . . . . . . . . . . . . 3. E x p e r i m e n t a l c o n s i & r a t i o n s . . . . . . . . . . . . . . . . . . . . . 3.1. T e c h n i q u e s . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. S a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . 4. L f g h t l ~ t e e a r t h s . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. N a C l - t y p e c e r i u m , n e o d y m i u m a n d s a m a r i u m c o m p o u n d s . . . . . . . . . . 4.2. ~IhsP4-type c e m p c u n d s . . . . . . . . . . . . . . . . . . . . . 4.3. I n t e r m e t a l l i c c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . 5. Half-filled .,hell i are e a l t h s . . . . . . . . . . . . . . . . . . . . . . 5.1. N a C l - t y p e e u r e p i u m c e m p c u n d s . . . . . . . . . . . . . . . . . . 5.2. D o p e d N a C l - t y F e c o m p e u n d s . . . . . . . . . . . . . . . . . . . 5.3. M iscell~ n e e u s e u r o p i u m c o m p e u n d s . . . . . . . . . . . . . . . . . 5.4. G a d o l i n : ' u m . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. l n t e r m e t a l l i c gadoliniu m c omF oun ds . . . . . . . . . . . . . . . . . 6. H e a ~ y r a t e e a r l h s . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. N a C l - t y p e c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . 6.2. H e a ~ y r a l e e a r t h m e t a l s . . . . . . . . . . . . . . . . . . . . . 6.3. M i s c e l l a n e o u s c o m p o u n d s c o n t a i n i n g h e a v y r a r e e a r t h s . . . . . . . . . . . 7. U r a n i u m ~omF o u ~ d s . . . . . . . . . . . . . . . . . . . . . . . . 7.1. U r a n i u m d i o x i d e . . . . . . . . . . . . . . . . . . . . . . . 7.2. B i n a r y N a C l - t y p e c o m p o u n d s . . . . . . . . . . . . . . . . . . . 7.3. P s e u d o - b : n a l y N a C l - t y p e c omit o u r ds . . . . . . . . . . . . . . . . . 7.4. qTh3P4 s~Iucture cc mlcounc's . . . . . . . . . . . . . . . . . . . . 7.5. T e r n a r y c o m p o u n d s a n d i n t e r m e t a l l i c s . . . . . . . . . . . . . . . . 8. C o n c l u s i c n ~ n d outloc k . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
135 137 139 139 141 147 150 153 156 157 157 160 162 162 169 172 175 176 186 190 192 194 196 196 202 203 206 206 208 215 222 225 230 231
1. Introduction
Magneto-optical phenomena have been known for more than a century. It was M. Faraday who reported in 1846 (Faraday 1846) the first genuine magneto-optical effect, namely the rotation of the polarization plane of linearly polarized light on transmission through a lead-borosilicate glass to which a magnetic field was applied parallel to the light propagation direction (fig. la). Since then, magnetooptics has gone through several highs. The discovery of what today we call Faraday rotation was a clear proof of the interaction of electromagnetic radiation with magnetized matter. With the experiments of Zeeman (1896, 1897) and the classical electron theory of Lorentz, magneto-optics allowed the determination of the sign of the electron charge and of the ratio e/rn. The developments of quantum mechanics prompted a large number of magneto-optical experiments on gases as well as on diamagnetic and paramagnetic solids to test the implements of the new atomic theory (see, e.g., Schiitz 1936). Microwave absorption on magnetized semiconductors opened the field of magneto-optical determination of band parameters in classical semiconductors like Ge, Si or InSb (Dresselhaus et al. 1955). The sixties brought a first wave of magneto-optical investigations of magnetic semiconductors which, in the following two decades, led to a wider recognition that magneto-optics used as spectroscopy is a very powerful tool to study the electronic structure of magnetic materials. The actual increasing research efforts in magneto-optics rely on one hand, on this basic physical interest and, on the other hand, on the potential for using magneto-optics for erasable optical memories. The present review focuses on the spectroscopic aspect of magneto-optics with the goal to relate the experimentally determined magneto-optical spectra to electronic transitions and, thus, to provide directions for searching for improved magneto-optical materials. We shall not discuss results of Raman scattering, microwave absorption or other resonance effects, which, in a wider sense of the word, are also magneto-optical effects. Instead, we shall mostly review Faraday spectra and polar magneto-optical Kerr effect spectra, and a few times, Voigt effect and equatorial and longitudinal Kerr effect spectra. This selection reflects the different interest which the various magneto-optical effects have received in the past 25 years. As will become clear in the following, the qualitative difference between these various effects is the different proportionality of the magneto135
136
W. REIM and J. SCHOENES
a) FARADAY ROTATION
8
o;I
BE--Re (~)= ~-~ (n+- n-) DF=-Im ( { ) = ~cl (k+- k_) h) MO KERR EFFECT
Polar
Longitudinal
Equatorial
N
n_-n+ 8pK= Im ~÷~__~----~ N N_-N.
PK = R e
~+~'-"1
Fig. 1. Scheme of the Faraday effect (a) and the various magneto-optical Kerr effects (h). O~Kand ~PK are the polar Kerr rotation and ellipticity, respectively, which in the text of this chapter are named OK and eK, for simplicity,if+ and ~ are the complex refractive indices for right and left hand circularly
polarized light, respectively (after Schoenes 1981). optical signal to the magnetic field. In the Faraday configuration, for which the field direction is parallel to the light propagation direction, the linearly polarized wave can be decomposed into a right- and a left-hand circularly polarized wave (fig. la). The rotation of the plane of polarization OF indicates a phase shift R e ( l ~ ) between the two partial waves. The Faraday rotation is, therefore, a magnetic circular birefringence. If the material absorbs light, the difference of the absorption indices for the two circularly polarized waves, k+ - k , gives rise to magnetic circular dichroism DF, which manifests itself as elliptically polarized light. Both effects are, to lowest order, linear functions of the magnetic field. In the Voigt (or Cotton-Mouton) configuration the external magnetic field is perpendicular to the light propagation direction. The difference in the refractive indices for fight polarized parallel and perpendicular to the magnetic field nil - n± gives rise to transverse or linear magnetic birefringence. The difference in the absorption indices k f f - k± leads to transverse or linear dichroism. Both effects are, to lowest order, quadratic functions of the magnetic field. As a consequence, these effects are generally smaller than the circular effects. In addition, they
MAGNETO-OPTICAL SPECTROSCOPY
137
cannot be used to determine the magnetization direction of domains and, therefore, they have attracted much less attention than the circular effects. If the polarized light is reflected from a magnetized solid, one speaks of the magneto-optical Kerr effect (Kerr 1877). The various orientations of the magnetic field relative to the plane of incidence and the direction of propagation allows to distinguish the polar, the longitudinal and the equatorial magneto-optical Kerr effect (fig. lb). In the polar Kerr effect, the light propagation direction is parallel (or antiparallel) to the field direction. This is similar to the situation of the Faraday effect in transmission and, indeed, the polar Kerr effect is, to lowest order, a linear function of the applied field. Yet, the polar Kerr rotation 0K and the polar Kerr ellipticity eK are not linear functions of the complex refractive indices rT+ and tT_ and the usual Kramers-Kronig relation does not hold between 0K and eK (Kahn et al. 1969). Up to this point we have always assumed the magnetic field as to be the driving force for the magneto-optical effects. This is correct, as long as the temperature is not too low and the field is not too high, for diamagnetic materials to come in the region of Landau quantization and for paramagnetic materials to reach paramagnetic saturation. Also, for magnetically ordered materials the field is to be replaced by the magnetization. The last remark which we would like to make in this introduction, regards the choice of the data to be included in this chapter. We have tried to be as complete as possible for the magneto-optical spectra of magnetically ordered systems containing as magnetic ions f-elements only. Problems arise for those materials which contain also transition elements other than the actinide or lanthanide elements. Examples are transition metal-rare earth intermetallics, many garnets and rare earth-ferrites. Magneto-optics of transition metal-rare earth intermetallics are widely discussed in chapter 5 of vol. 4 of this handbook and here we have only included data in direct connection with pure f-electron systems or very recent results. Garnets and ferrites have not been included to keep the article to a tractable size and because the signals related to the rare earth are in most cases only marginal. Albeit we did a computer search, we may have overlooked valuable data for this review. We hope that the authors of these overlooked publications, should there be any, will accept our apologies beforehand, and will understand our difficulties the more they themselves search in the labyrinth of terms for magneto-optical effects and the infinite number of compounds that all f-elements can possibly form. 2. Theory The analysis of the magneto-optical activity of a substance is generally carried out in two stages; first, the measured quantities like the Kerr rotation 0~: and ellipticity eK are related to the dielectric tensor g: or the conductivity tensor ~. In a second step, these functions are then interpreted in terms of band-structure features or may even be calculated directly from relativistic and spin-polarized band-structure calculations.
138
W. REIM and J. SCHOENES
The object of this theoretical section is to provide a self-consistent description of important interrelations between the different optical and magneto-optical functions. In section 2.1.1 and fig. 2 we give a sign convention on, which all graphs in this chapter will be based. For Fe in the visible spectral region, a negative Kerr rotation and a positive Faraday rotation are in agreement with this definition of the sign. The Faraday rotation of quartz glass in the visible region has a positive sign, as it is usually found in literature (Gmelin 1959, Landolt-B6rnstein 1950, Robinson 1964). In section 2.1, we derive the equations connecting the measured quantities Kerr effect or Faraday effect with the elements of the conductivity tensor & or the dielectric tensor g:, which describe the optical and the magnetooptical material properties. Section 2.2 deals with the interpretation of the frequency dependence of & in terms of the spin-polarized band structure of the material, and frequently used concepts will be reviewed. In the same section, we discuss one possible way of large Kerr effect engineering, namely by a splitting of the plasma edge of a metal for different light polarizations. The importance of the
a)
~
Z
Kerr-ellipticity X
6F
c)
Y
Faraday-ellipticity Y Fig. 2. Scheme to illustrate the sign convention for the Faraday and Kerr effects. (a) Electric field vector E at a fixed time t o for right ( + ) hand circularly polarized light (rcp). The incoming beam travels along the + z direction parallel to the applied field B. Looking always along the propagation direction of the light, E describes a clockwise rotation at fixed time t o for the incoming and the transmitted beams but a counter-clockwise rotation for the reflected beam. (b) Time dependence of E+ and E at fixed position z 0 after reflection from the sample for a material with dominant absorption of rcp-radiation, i.e., R+ > R _ and consequently [E+[ > [E I. The Kerr ellipticity is positive [eq. (2.20)]. (c) Time dependence of E+ and E_ at fixed position z 0 after transmission through the sample with dominant absorption of rcp-radiation, i.e., k+ > k and, consequently, IE÷ [ < [E 1. The Faraday ellipticity is positive [eq. (2.40)]. Note the different sense of rotation of the E vector for rcp-light in transmission and reflection.
M A G N E T O - O P T I C A L SPECTROSCOPY
139
normal optical constants for Kerr-effect enhancement will be critically discussed. Finally, section 2.3 shows the relation of non-optical measurements like the Hall effect to magneto-optics and how these additional data may be used efficiently. Also, the Kramers-Kronig integral equations and their application to magnetooptics will be discussed.
2.1. Phenomenology 2.1.1. Definition of the optical functions A circularly polarized wave propagating along the z-axis is described by the electrical field vector E+ = Re[E 0 exp{i(wt - 2rrff+Z/Ao)}(e x +-ie,)],
(2.1)
where upper ( + ) and lower ( - ) signs refer to right- and left hand circular polarization, A0 is the wavelength in vacuum, rT± is the complex index of refraction for the two states of polarization and e~ and ey are unit vectors in x- and y-direction, respectively. Looking in + z direction of a right-handed cartesian coordinate system (direction of propagation), at a fixed place z0, this definition of right (+) circular polarization describes a counterclockwise rotation of the electric field vector with time which is equivalent to a clockwise rotation of E along the direction of propagation at a fixed time to (see fig. 2). To obtain positive values for the absorptive quantities the choice of sign for the time factor in the exponent directly implies the definition ~ , = n± - i k ± ,
(2.2)
for the complex refractive index where n and k are the real indices of refraction and absorption, respectively. The magneto-optical effects are associated with the influence of a magnetic perturbation (either applied field H or intrinsic magnetization M) on the conductivity tensor & or the dielectric tensor g: of a medium. These two tensors are not independent but describe the same phenomena and are related by i47r ~ e]j = 6u - - - - o'u, ¢O
(2.3)
where 6u = 1 for i = j and 6u = 0 for the off-diagonal elements i # j. With the magnetization along the z-direction and applying general symmetry arguments (Roth 1964, Agranovich and Ginzburg 1966), we define the conductivity tensor as
(
O'xy
(2.4)
o and the dielectric tensor g: according to (2.3). All elements of ~ are complex
140
W. R E I M and J. S C H O E N E S
functions O-ij = O'lij "~-
i°'2i/•
(2.5)
By. decomposition of & into a Hermitian and an anti-Hermitian contribution (LeGall and Jamet 1971) it can be shown that the absorptive parts of the diagonal and the off-diagonal elements are O-ixx and %xy, respectively. Because ~ / / ( M ) = ~ j i ( - M ) the diagonal and the off-diagonal tensor elements are even and odd functions of the magnetization, respectively (Roth 1964). The given conductivity tensor applies to isotropic media, to cubic structures with crystallographic axes along x, y, z and to hexagonal and tetragonal structures with the c-axis along z. For crystals with lower symmetry, both the tensor and its relation to the optical properties become more complex (Emslic and Aronson 1983) and natural birefringence often inhibits magneto-optical measurements. The dielectric response of the medium is described by the Maxwell equations: V × E -
1 dB c dt'
(2.6)
V × H = - c 1 [ 47rJ+ d-~tl ,
(2.7)
and V - B = 0 and V - D = 0 in a conducting medium. Neglecting higher orders in the moment densities of the medium in the presence of applied fields, we write D = E + 4~-P and H = B - 4~-M for the new field quantities, electric displacement D and magnetic field H (compared to the averaged electric field E and magnetic induction B). Further, we neglect nonlinear effects in the material response and use P=oLIE , M = X 1 H and the Ohm law J=CrlE with the real tensors polarizability % and Xa and conductivity cr 1. Finally, we define the real dielectric function as e 1 = ~ + 4~-%, and the real permeability P.1 = 8 + 47rxi, with 8 defined as in eq. (2.3). Using these definitions in eqs. (2.6) and (2.7) and with the time dependence + itot as defined in eq. (2.1), we obtain, 1.
V x E = - - l~ttH, C
(2.8)
V×H=
(2.9)
1 [4¢rtr 1 ] dE c k iw + el dt "
Now, we define the complex dielectric function • 4~r g: = e 1 - 1 - - 0"1 ,
(2.10)
(O
which is equivalent to the definition of a complex conductivity (Tr=
{1¢ 1
-[- i o ~ a l
= Cr~ + i OJ(el -- 1 )
4rr
(2.11)
MAGNETO-OPTICAL
SPECTROSCOPY
141
Equation (2.11) together with eq. (2.7) finally allows a generalization of the Ohm law,
io) ( i o )
lq--~-~el E :
./tot =
~+~--~8 E,
(2.12)
with the total current Jtot = (1/4~-)D + J. In general, besides on frequency, the optical functions will also depend on the wavevector g which points along the propagation direction and whose length is complex and equal to 2~rrT/A0. However, as long as the wavelength of the radiation is large compared to atomic distances, which is certainly the case for photon energies of several eV as used for standard magneto-optical measurements, the wavevector dependence can be neglected. Applying the Maxwell equations eqs. (2.6) and (2.7) on an electromagnetic field with E and H defined according to (2.1) one obtains n~z =
~gx~,
(2.13)
which connects the complex index of refraction with the diagonal element of gz. It has sometimes been argued that in ferromagnets the linear relation M = R1H does not hold and/x ~ 1. We follow the argumentation of Landau and Lifshitz (1960) and Pershan (1967) that at optical frequencies one should take /x = 1 even for ferromagnets. With/x = 1, we obtain 6,1x x
=
n 2 -- k 2 ,
(2.14) (2.15)
e2x x = 2 n k ,
and ~xx is defined according to eq. (2.3). The optical reflectivity is given by the Fresnel equations (Born 1964). At normal incidence, the amplitudes r+ and r_ for right and left circular polarizations are r_+ =
Ir_+ I e ia÷ -
n_+
-
ik+
-
1
n_+ - ik_+ + 1 '
(2.16)
which leads to the well-known formula for the reflected intensity R+=r~= 2.1.2.
(n+ - 1) 2 + k 2 ( n + + l ) 2+kz+
(2.17)
F a r a d a y a n d K e r r effects
In reflection, the interaction of light with a magnetized medium is traditionally decomposed into three fundamental effects, namely the transverse (=equatorial), the longitudinal (=meridional) and the polar Kerr effect. These effects differ only in the orientation of the magnetization relative to the plane of incidence and relative to the surface of the medium (fig. lb). All effects are described by the
142
W. REIM and J. SCHOENES
same optical tensor & or ~:, however, only the polar effect and a specific equatorial configuration give linear and the other configurations can give terms linear as well as quadratic in magnetization (Pershan 1967, Hunt 1967). A theoretical formulation of these three effects extended to multi-layer thin films has been given by Smith (1965) and a general treatment of the Kerr effect for arbitrary orientation but restricting to effects linear in the magnetization was published by Hunt (1967). In this review, however, we will restrict to the polar Kerr effect, cbK = OK-- ie K ,
(2.18)
because most experimental work to be reported has been done in this configuration. In this geometry, the magnetization is perpendicular to the surface and parallel to the direction of light propagation, i.e., in + z direction (fig. 2). The eigenmodes are right and left circular polarizations and ~bK is expressed in terms of the Fresnel coefficients (2.16), without any approximations as OK = -
eK -
½(a+ - a _ ) ,
(2.19)
Ir+l-lr-I
ir+l + ir_l •
(2.20)
It should be noted that after reflection, the direction of light propagation is reversed, traveling now along - z . According to eq. (2.1) this implies for right ( + ) circular polarized light a counterclockwise rotation both at fixed time and at fixed position looking along the light propagation direction. The sign of the Kerr rotation has been chosen such, that the ellipticity ei~ will be positive at a photon energy dominated by absorption of right circular polarization (selection rule (Am = +1, see section 2.2) as shown in fig. 2 and accordingly the rotation will be positive at lower energies and negative at higher energies. One should note at this point that there will be no Kerr rotation for a non-absorbing material while there may still be some Kerr ellipticity. This is easily seen rewriting eq. (2.16) as phase and amplitude zl±=arctan
(
2+~-~+ n__+
)
1 ' --
~/(n 2 + kZ+ - 1) 2 + (2k+) 2 Ir±l = (n_+ + 1) 2 + k~
221, (2.22)
As we will see later, this behavior is opposite to the Faraday effect just by the fact that the phase shift, which always leads to magneto-optical rotation, in reflection is proportional to the absorption index k, eq. (2.21), while in transmission it is induced by the refractive index n, eq. (2.39). In order to relate ~bi~and & one makes the ansatz 4~K = 0 K - - i e K
= +i
r + -- r
r++r_
(2.23)
MAGNETO-OPTICAL SPECTROSCOPY
143
After some algebra one finds, that compared to the exact formulas given by eqs. (2.19) and (2.20), the following approximations have been made in eq. (2.23): sin(A+ - A _ ) = ( A + - A_), cos(A+ -- A_)~--I and (r+ - r _ ) 2 ~ 2 r + r _ . Even for Kerr rotations of several degrees, these approximations lead only to minor errors in the determination of ~. Substituting the Fresnel equation, eq. (2.16), into eq. (2.23) we find the following relation between the complex indices of refraction and the Kerr effect, OK = - I m
---~1
eK = - R e ( ~+ - ff
) )
(2.24)
'
(2.25)
Now we use eqs. (2.14), (2.15) and (2.3) and write ~+ . (n_+ . . ik_+) . 2.
1
i 4 7 r o-+ _ ,
(2.26)
o9
where c7_+can be related to ~,x and ~ y using eq. (2.12), °'+- = ~ x x
-T- i O ' x y
(2.27)
.
It is straightforward to relate rT_+ and the conductivity tensor by inserting eqs. (2.27) and (2.26) into eqs. (2.24) and (2.25), --2
-2
87J" ~
n + -- n _ =
n+--n-
-- --
o9
_
rT+rT_ - 1
Oxy
(2.28)
,
~xy
E~xy
i6.~r7 = i(~'xx - 1 ) g - ~ x
"
(2.29)
In here we have used the approximation r7 = ½(if+ + rT_). Taking the real and imaginary part according to eqs. (2.24) and (2.25) yields,
EK ~
4 ~ [ Ao'lxy - BOr2xy ] o9 [ A2-+ B 2 ] '
--
oK=-,;- L ~ + B 2
J'
(2.30) (2.31)
with the coefficients A = (O'2xxn - O'lxxk ) 4"rr = (n 3 - 3 n k 2 - n), O9
B = (o'l~n + o'z~k ) 4"rr = ( - k 3 + 3 n 2 k - k ) . O9
(2.32) (2.33)
144
W. R E I M and J. S C H O E N E S
It is obvious, that the coefficients A and B in eqs. (2.30) and (2.31) are complicated and non-trivial functions of the indices of refraction n and absorption k. Figure 3 gives the signed logarithm of A and B as function of k for different values of n. It becomes clear, that only at singular points and for k < 1 and n >> 1 one term dominates but, in general, it is not possible to neglect one of the prefactors in eqs. (2.30) and (2.31). This makes clear that it is absolutely necessary in the Kerr configuration to measure all four quantities OK, eK, n and k to obtain reliable information on CT~yand hence to be able to relate magnetooptics to spin-polarized band-structure features what certainly is the ultimate goal. The final formulas to calculate ~Txyfrom these measurements are obtained from eqs. (2.30) and (2.31) to be o) O'lxy =
4~r [BOK + AeK] '
(2.34)
O'2xy
60 4--~ [ A O K
(2.35)
=
-- B e K ] "
The equations for the complex Faraday effect (i.e., ~bF = 0v - i e v ) can be derived quite a bit easier and more direct. Linear polarized light incident on a medium at z = 0 may be written as a superposition of two right and left circular polarized waves of equal amplitude according to eq. (2.1) E = ½E0[(e x + i e y ) exp{i(wt - 27rff+Z/Ao) } + (e x - iCy) exp{i(wt - 27rr7 z/A0)}].
(2.36)
This equation can be written as E = ½ E 0 exp{i(o~t- 2~ffZ/Ao))[e x cos½ c~+ ey sin½ c5"],
(2.37)
x Ioglxl 2
0
-2 -3 0
5
10
k
15
Fig. 3. Coefficients of 0~ and e K in the calculation of ~ y [eqs. (2.32) and (2.33)] as a function of n and k (after Reim 1986). Solid lines: x = 3 n 2 k - k 3 - k ; dashed lines: x = n 3 - 3 n k 2 - n. Note the logarithmic scale of the vertical axis and the inclusion of the sign of x.
MAGNETO-OPTICAL SPECTROSCOPY
145
where we have defined the complex phase = (iT+ - ff_)21rz A0
(2.38)
Now it becomes obvious (see fig. 2) that a wave which is linearly polarized at z = 0 will be elliptically polarized after propagating a distance l o with the major axis of the ellipse rotated through the angle tol° n ( i f + - iT_))=-~-c ( + - n _ ) ,
0F = Re(½ c~)= R e ( - ~
(2.39)
and the ellipticity, defined as the ratio of the minor to the major axis of the ellipse, will be / / ~'l 0 / tol 0 e F = - t a n h ( I m ( l c ~ ) ) = -tanh~Im~-~--(if+ - if_))) = t a n h ~ (k+- k)) = wl° (k+ - k ). 2c
(2.40)
In eq. (2.40) we have used the approximation tanh(x) = x, neglecting terms of the order of x 3 and higher. Within this approximation the Faraday ellipticity equals the magnetic circular dichroisrn. Equations (2.39) and (2.40) clarify the interchanged roles of n and k in the Faraday effect as compared to the Kerr effect [eqs. (2.19)-(2.22)]. While the Kerr rotation OK is proportional to the absorption index k, the Faraday rotation 0v is determined by the index of refraction n. To derive the expressions between Faraday effect and the conductivity tensor, we start with eq. (2.28), insert eq. (2.2) and calculate the real and imaginary part 8~r [(n2+ - n2) - (k 2 - k2_)] - 2i[k+n+ - k _ n _ ] = - ~ (O'lxy "Jr-iO-2xy ) .
(2.41)
With the average values n ½(n+ + n ) and k = l(k+ + k_) and the differences An=n+-n andAk=k+-k we w r i t e e q . (2.41) as 8~2[n An - k Ak] = - - - O-lxy ,
(2.42)
8~2[k An + n Ak] = - - trzxy,
(2.43)
to
tO
The combination of eqs. (2.42) and (2.43) leads to expressions of An and Ak as function of ~Ty and finally using eqs. (2.39) and (2.40) the Faraday effect as function of G~y is given by 2~'1o [ rt°'lxy OF
C
[
2 to
e F - - -c
~+
-
-
ktr2,y ] k2
,
+ n 2xy ] k
n2 + k2
J "
(2.44) (2.45)
t46
W. REIM and J. SCHOENES
The final equations used to calculate ~xy from the measured functions n, k, OF and e F are: C
o'lxy --
2,0.io [nOv - keF],
(2.46)
¢
o'2xy = q - ~ l o
[koF + n e F ] .
(2.47)
Equations (2.44), (2.45) and (2.30), (2.31) express the measured magnetooptical effects in transmission and reflection in terms of the conductivity fir, which is the quantity describing the material property for the interaction of radiation with a magnetized medium. It is now possible to compare the size of the magneto-optical effects in reflection and transmission in general for a given medium characterized by the complex tensor components ~'xx and cT~y. In view of applications, it is most interesting to compare the size of the Faraday effect, 27rd °'e + o'2~y IthFI = V~F + e 2 - , - c -- n + k -~ '
(2.48)
of a thin film with thickness d with the bulk Kerr effect 4~
t / O.lxy 2 _1_ IT2xy 2
o~ ~
A 2 + B -~
(2.49)
'
of the same material, .I
FI .
I~KI
. ood
2c
.A/-~-~ .B2 ~ n2 + k2
2 " r r d Vo- x
2 _ -27rd
+ O'2xx -
c
I xxl .
(2.50)
c
Now, the thickness d of the film has to be chosen properly, to be able to compare the two effects. We adjust d such that the transmitted intensity, including reflection losses but neglecting interference effects (Kessler 1963),
.:.0Ii+
., exp(
with the absorption coefficient K = 2tok/c, equals the bulk reflectivity R of eq. (2.17). This leads to the final result
[~bld -
wk log [1 + (k/n)2](1 - R) z
I~xl
1 ( R ) k 2 1) 2 + (2nk) 2 4k log [1 + (k/n)2](1- R) 2 ~ / ( n 2 -
(2.52)
which shows that the ratio of Faraday and Kerr effect is independent of the
MAGNETO-OPTICAL SPECTROSCOPY I
I
147
I
a) "5 tu
3
Index of Refraction n
o
1
2
3
1
2
3
4
5
I
I
6
7
I n d e x of A b s o r p t i o n k
Fig. 4. (a) Ratio of the total Faraday and Kerr signals according to eq. (2.52) for a material with optical constants n and k. Compared is the Faraday rotation of a thin film with a thickness such that the transmittance equals the reflectivity of the bulk material. For large absorption the transmitted intensity is always lower than the reflected intensity and I~FI/I~K[ = 0. (b) As fig. 4a, but excluding reflectivitylosses in the Faraday configuration,i.e., the transmitted intensity has been calculated using ! = I 0 exp (-Kd) instead of eq. (2.51). off-diagonal element O'xy and only depends on the optical constants n and k. In fig. 4a, I~bFl/l~l is plotted versus k for different values of n. For materials with low absorption k < 1, the Faraday effect exceeds the Kerr effect by a large factor because the film can be chosen quite thick to obtain equal intensities in transmission and reflection. For larger k, on the other hand, the absorption is high which leads to very thin films in transmission and causes large reflectivity losses in the Faraday geometry. Consequently, the magnitudes of Faraday and Kerr effects become similar and the Kerr effect finally dominates. However, if the reflectivity losses in the Faraday geometry are neglected, the Faraday effect is the better choice for almost all combinations of the optical constants, as shown in fig. 4b. This fact is used in optical storage applications via quadrilayer structures (Bell and Spong 1978) or by directly chosing the double Faraday rotation instead of the Kerr rotation (Martens and Voermans 1984).
2.2. Microscopic models to calculate The basis of magneto-optical effects in solids is, of course, the occurrence of transitions between the magnetically quantized electronic states. For not magnetically ordered materials, Schlitz (1936) has reviewed the atomistic interpretation of the Faraday rotation using the Z e e m a n effect. The applied magnetic field splits the atomic levels of the initial and final states characterized by the m o m e n t u m
148
W. REIM and J. SCHOENES
quantum numbers S-L-J into sublevels characterized by the now non-degenerate magnetic quantum numbers m. The selection rules for right and left circular polarized light require Am = + 1, respectively. Thus the resonance frequency will, in general, be different for the two circular polarizations leading to the so-called diamagnetic Faraday effect, which is characterized by a line shape as shown in fig. 5b for the most simple magneto-optically active transition between the atomic 2S1/2(S 1, L = 0, J = ½) and the 2p1/2(S = ½, L = 1, J = ½) states. This contribution will be independent of magnetization and temperature, in a first approximation. A strongly different, so-called paramagnetic, line shape occurs if the occupation of the Zeeman-split ground states is different. This leads to a temperature dependent term, the so-called paramagnetic Faraday rotation line shape (fig. 5c) which is proportional to the magnetization. An extensive review of the magneto-optics of nonmagnetic materials has been given by Buckingham and Stephens (Buckingham and Stephens 1966, Stephens 1970). It should be noted that from the sign of the Faraday-rotation it is not possible to decide whether the material is diamagnetic or paramagnetic because, in contrast to the magnetic properties which are determined by the occupied states only, for magneto-optics the quantum numbers of the final states are of equal importance. It has been shown by several authors (Hulme 1932, Argyres 1955, Cooper 1965, Voloshinskii and Bolotin 1964, Bennett and Stern 1965, Erskine and Stern 1973, Roth 1964) that the several orders of magnitude larger magneto-optical effects observed for magnetically ordered materials can not be explained by the Zeeman effect but need the introduction of spin-orbit coupling. The influence of spinorbit coupling has been considered as a perturbation of either the initial (Argyres =
n ,/'i
m = + 1/2
A 1'
I
LCP
m=-
i'--~ n_-n+
1/2
/1
RCP
2s~,2J
/~n+
n /'i
m =+ I / 2
/
.... -l".:n_ .
.
.
.
i
i
.
OJ m =-
LOP
/,/ f~l--
RCP
I, .1 -
a)
I/2
-
~
ta,;
~°
Fig. 5. Dispersion of the Faraday rotation for a 2SI/2----~2p1/2 transition in an applied field which removes the m = -4-½degeneracy. (a) Energy level scheme. (b) Equal intensity transitions leading to a diamagnetic line shape. (c) Unequal intensity transitions (i.e., different occupation of the m = + ½ ground states) with paramagnetic line shape.
MAGNETO-OPTICAL SPECTROSCOPY
149
1955) or the final (Hulme 1932) state energies of an interband transition, or as a modification of the initial state wavefunctions (Kittel 1951), which is of importance if the ground-state orbital momentum is quenched. It has been pointed out that, in general, magneto-optical effects will be proportional to ($i X r E ) " Pi ,
(2.53)
i
which is the spin-orbit energy of an electron with spin s and momentum p moving through the electrical field - r V inside the medium and finally summed over all electrons of the medium. This vector sum indicates that the magneto-optical effects are proportional to the product of magnetization Es i and spin-orbit energy. In nonmagnets or at temperatures above the magnetic ordering temperature, there will be as many electrons with up-spin as there are with down-spin if no external field is applied and Es i will be zero. With an applied field, on the other hand, the electrons in the vicinity of the Fermi surface have a nonzero average value of the spin due to Pauli paramagnetism and we have Es i proportional to the applied field. It is obvious now that, in general, the magneto-optical effects will follow M(T, H , . . . ) and the interchanging roles of M and H are clear. To illustrate these proportionalities, fig. 6 shows the determination of the magnetic phase diagram M(T, H) of UAs (Reim et al. 1984a). In fig. 7, the field dependence of the Faraday rotation of a EuSe thin film is displayed (Schoenes and Wachter 1977). In this case, of a material with no crystalline anisotropy (S ground state), shape anisotropy plays the dominant role because of the large
UAs ( 100 ) hoJ = 1,6eV -2
-a
J
. 1.5
J J J
B IT]
~__ ~_ ~ J _J ~ J _ ~
n." -1 I J
~
-0.5
~ ~ --
0
i 0
t 50
I
J 100
I
I0.0 9.75 9.5 9.25 9.0 8.5 8.0 7.5 7.0 6.5 5.5 4.5 3.5 3.0 2.0 I
150
Temperature [K] Fig. 6. Temperature dependence of the magneto-optical Kerr rotation of a cleaved UAs single crystal for applied fields up to 10 T (after Reim et al. 1984b). Different ferro and antiferromagnetic phases are observed in agreement with the magnetic phase-diagram determined by neutron scattering (RossatMignod et al. 1982).
150
W. REIM and J. SCHOENES
22 r~ T
g
"~o~ = 2 . 3 5 e V
r4
T =2.5K
,o t?_
/ Applied Field Ha E T 3 i
i
i
i
i
=
i
Fig. 7. Field dependence of the specific Faraday rotation Ov/d of a EuSe thin film (after Schoenes and Wachter 1977). The Faraday rotation is proportional to the applied field H~, until H a exceeds the demagnetizing field 47rMs.
demagnetizing factor of 47r for a thin film. For applied fields H < 4~-M s, OF will be proportional to H and will saturate around 47rM s. Based on the concept of proportionality to spin-polarization and spin-orbit coupling, Bennet and Stern (1965) and Erskine and Stern (1973) have developed expressions for ~xy which allow quantitative calculations for interband and intraband transitions.
2.2.1. Interband transitions The expression for the absorptive part of the off-diagonal conductivity o-2xycan be written as the sum of separate contributions from spin-up and spin-down electronic states
O'2xy(to) = O'2xy i, (tO) -Jr-O'2xy~,(tO).
(2.54)
The contribution for transitions between spin-up states a and /3 is given by (Bennet and Stern 1965)
2"n'2e2 ~] tr2xy ~ (tO) = 4hm2VtO ~p
[1(/3 t 1~+ t 1(3/ t )[2__ 1(/3 t 1~'- t [a t )12]
× 6(tO~t3 t - tO),
(2.55)
where the operators ~'-+ = Wx --- i~y are linear combinations of the kinetic momentum operator defined by
(h),
¢r = p + ~
× IFV(r),
(2.56)
M A G N E T O - O P T I C A L SPECTROSCOPY
151
where p is the momentum operator and s × VV represents the spin-orbit contribution. Within this model, spin-flip transitions are neglected although the spinorbit contribution to ~- could couple spin-up to spin-down states. In fact, spin-flip transitions seem to give the dominant contribution to the magneto-optical effects of Fe, Co and Ni, as has been shown by direct determination of & from spin-polarized band-structure calculations (Tawil and Callaway 1973, Callaway and Wang 1973). For a given interband transition a---~b, Erskine and Stern (1973) introduced the concept of the total weight, (2.57)
< 2 y>a- b = fa b 1 2xyl dw
of that transition as the total magneto-optic absorption due to transitions between all sublevels a and/3 of the states a and b at all energies. With the approximation that the only effect of spin-orbit coupling is the splitting of degenerate energy levels, eq. (2.56) can be written as ~r ± = itom(x +_iy) and using eq. (2.55), the total weight is given by
27r2ne2
( O'2xy) a--~b = n-----h--- ~
[to + l(x + iy) l 2 - oJ2nl(x
-
iy),~t~12l%.n,~n¢,
(2.58)
where a and/3 now may be viewed as the spin-orbit split sublevels of a and b, respectively. The joint spin-polarization o) is defined as
n~ 1' nt3 t -n~ $ n~ $ o-j= n~ 1' nt~ 1' +n~ $ nt3 $ '
(2.59)
where n~ and nt~ are the occupied initial and the unoccupied final states, respectively. If the atomic wavefunctions of all states a and /3 are known, (tr2xy) a~b may be calculated. However, up to this stage the concept applies only to atomic states. In solids, where in general the initial and final state are energy bands, some further assumptions have to be included. First, we assume that due to strong exchange forces the spin-orbit coupling lifts the degeneracy of the orbital quantum n u m b e r and the spin-orbit split atomic energy levels will be labeled by rnt. This assumption, of course, holds only in the limit of strong magnetization, i.e., T < T c. Second, we assume that the oscillator strength of the atomic transition a---~ b is unchanged in the solid but is spread out in energy uniformly over the bandwidth E B. With these restrictions in mind, we may use spherical harmonics YT' for the orbital part multiplied by a radial contribution u(r) for the initial and final state wavefunctior/s, and write
Z ([( YT'~ ~]x + iy] YT' )]2 _ ]( y7,~1 ]x - iy] YT' ) ]2)Rib, a¢¢
(2.60)
152
w. REIM and J. SCHOENES
where
R ab
is the radial overlap integral
Rab = fa,b UaUbr2 dr .
(2.61)
Figure 8a,b summarizes the final treatment of a transition between two band states. According to the assumption of spreading the spin-orbit levels over the band, one expects the Am = +1 and Am = - 1 transitions being shifted by the spin-orbit energy Eso. The total weight will show to be proportional to (Eso/EB)O) (Schoenes 1975). It is often helpful to compare the off-diagonal weight (o'2xy)a_,b with the corresponding diagonal weight (osx x )a---~bwhich is defined in an analogous way by ( °rlxx) a---~b--
2 ~2ne 2 h %b](blrla) [2nanb ,
(2.62)
where r is the dipole operator. While all optical transitions contribute to eq. (2.62), only those states with a net spin-polarization add to eq. (2.58) which is an elegant way to separate, e.g., magnetic f-states from non-magnetic p-states. It should be noted that, although unpolarized, exchange split states may contribute to the magneto-optical response (Erskine and Stern 1973, Schoenes 1975). This fact demonstrates the limits of the total weight concept and is only a manifestation
Im(Txy /
,
~\,j"
x
,XiU -"
t ~J
c)
a) Im~xy
Im°x I ~i, /""""~ b)
d) Fig. 8. Schemes of signal versus frequency w to calculate ~2~yfor band to band transitions (after Erskine and Stern 1973, Schoenes 1975). (a) Absorption of right- and left-hand circularly polarized light shifted relative to each other by the spin-orbit energy, shown for one spin-direction only. EBand Eso indicate the bandwidth and the spin-orbit energy, respectively. (b) Resulting off-diagonal conductivitytr2xydisplayed separately for both spin directions. The magneto-opticaleffect will vanish, unless the contributing electronic states are spin-polarized. (c) Situation for unpolarized but exchange split states. (d) Resulting magneto-optical signal ~2xy.
MAGNETO-OPTICAL SPECTROSCOPY
153
of eqs. (2.53) and (2.55) showing that the spin polarization of states at a given binding energy is of importance, and not the integrated value. Figure 8c,d suggests how this type of transition may be integrated into the present concept. The S-shaped off-diagonal absorption bands in o-2~r for up-spins and down-spins are shifted relative to each other by the exchange energy. The difference between these two spectra is the quantity which is to be compared to experimental results. If the initial state of, e.g., a fn___>fn-1 d transition is localized, atomic coupling effects of the fn-~d final state configuration have been shown to be important (Schoenes 1975, Reim and Schoenes 1981). Then, atomic coupling schemes have to be used to describe the structures of the magneto-optical spectra. Depending on the relative magnitude of the Coulomb and the spin-orbit interactions, either L - S or ]-j coupling, or an intermediate scheme, is appropriate.
2.2.2. Conduction-electron effects It has been shown (Erskine and Stern 1973, Reim et al. 1984a, Doniach 1966) that intraband transitions may contribute to o-2xy with two different frequency dependences, namely o-2xy~ 0) -1 and O-2xy~ w -3. The theory gives a proportionality of cTxyto the conduction-electron concentration N via the plasma frequency
2
~Op =
4~rNe2 m*
'
(2.63)
where e is the electron charge and m* is the effective mass of the electron, and to the spin polarization n~ - n ~ O-¢o.d- - , n t + n~
(2.64)
where n t and n+ are the numbers of conduction electrons parallel and antiparallel to the total moment. In addition, proportional to the strength of spin-orbit coupling via the macroscopic dipole moment P0 and the product of electron velocity ev o (Reim et al. 1984a)
O'xy(0"))~-- ~
O"cond -- ~-~2 ~_ ( y q_ io9)2 +
with spin moment a second term is ratio of maximum charge and Fermi
--eVo 1 - •2 + (,y + i o ) ) 2 j ]
.
(2:65)
Here, y( = 1/r) and J2 are the relaxation and the skew-scattering frequency, respectively. The absorptive part of the free-electron contribution then calculates to 2 Or2xy((O) = " ~ O'eond (~-~2 ..~ , 2 -- 032)2 "Jr 4"yaw 2 r 0 [
(.o'y(a 2 --~ V 2 -t- 60 2)
11
eVo (0£ +--~--;-5-)~ +-~y2oo2 j j .
(2.66)
154
W. R E I M a n d J. S C H O E N E S
It is important to note that the sign of the two terms is opposite. The decisive number for the dominance of the first or the second term in eq. (2.66) at a frequency to is the ratio ( le~0[) / ( ~ 2 2 ) .
(2.67)
This ratio can be estimated using the relation Po/eV o = (ma2tOso)/(htOa~b) where a is the lattice constant and htOsn and htOa~ b a r e typical spin-orbit and interband energies, respectively (Stern 1965). The first term in brackets in eq. (2.66) with a OVzxy~ tO-3 frequency dependence in the high-frequency limit, gives the same frequency dependence as the classical Drude treatment of free electrons. This term dominates if the damping is small enough to allow a coupled plasma frequency tOp well below the lowest energy interband transition. It may be viewed as a splitting of the plasma edge for the two circular polarizations (different tOp for right and left circular polarization) and may lead to very large magneto-optical effects if the plasma edge in the optical reflectivity is steep. In the classical Drude treatment, one starts with the equation of motion d2r dr m*~-~ + m*y dt
e dr c dt × H = e E .
(2.68)
Using the time dependence e i¢ot, the dielectric constant for (-+) circular polarizations 47rNer+ to2 = 1+ ~ - 1 + to(_to + tO~+ iv) '
(2.69)
is obtained with the cyclotron frequency to~ = eH/m*c. Using eqs. (2.3), (2.26) and (2.27), we derive ~
Orxy--
tO
~
(~+ -- E_)--
2 to P
47r (ito
toc +
,}/)2 _[_0)2
"
(2.70)
Formally, this equation is equal to the first term of eq. (2.65) except for the conduction-electron spin-polarization prefactor O'cond in eq. (2.65) if toc is identified with the skew-scattering frequency J2. Figure 9 displays the experimental Kerr effect of paramagnetic silver for an applied field of 1 T (Schnatterly 1969), which is a typical example of a Kerr signal due to plasma-edge splitting. All details of the energy dependence of 0K and eK can be successfully modelled using eq. (2.69) and eqs. (2.19)-(2.22) with the input parameters tOp = 9.2 eV, y = 0.04 eV, to~ = 0.11 meV at 1 T, and an unpolarized contribution e b from interband transitions as derived experimentally (Ehrenreich and Phillipp 1962). The reason for the relatively large Kerr effect of several
MAGNETO-OPTICAL SPECTROSCOPY T -o
4
155
r
Silver B=IT
_o
-E K
'~-2
5.5
4
4.5
Photon Energy [eV]
Fig. 9. Polar Kerr rotation OKand ellipticitye~ for paramagnetic silver in an applied field of 1 T (in part after Schnatterly 1969). The screened plasma frequency of silver is 3.8 eV. The Kerr effect originates from the plasma-edge splitting for (±)-circularly polarized light. 10 -3 degrees for a paramagnet is the fact that both A and B in eqs. (2.30) and (2.31) are small and B changes sign where eK peaks. As expected, O'xy is three to four orders of magnitude smaller than in typical ferromagnets. Obviously, large Kerr effects may be generated if toc is enhanced relative to top either by reducing tOp or by increasing tOe. The former concept leads to a Kerr effect of up to 60 ° at room temperature, e.g., for n-type InSb with a plasma frequency of tOp * = 52 meV (Palik et al. 1962). The second way towards large rotations by an increase of the magnitude of the plasma-edge splitting using a ferromagnetic material has only recently been realized (Reim et al. 1984a) and will be discussed in detail in section 5.3. Recently, it has been claimed (Feil and Haas 1987), that a resonance-like enhancement of the Kerr effect may be expected at photon energies where e l x x equals one, as suggested by eq. (2.29). A detailed investigation of the influence of the normal optical constants on the Kerr effect (Schoenes and Reim 1988) has shown, however, that the magnitude of A and B in eqs. (2.30) and (2.31) is of major importance. For a typical ferromagnet, A and B vary by up to two orders of magnitude between 0.5 and 5 eV which indicates that both A, B and tT~y strongly affect the Kerr effect and have both to be treated in a proper way to allow any predictions. Multi-layered metallic thin films consisting of one material with small optical constants and a second material with the desired magnetic properties are one way to modify the coefficients A and B defined in eqs. (2.32) and (2.33) and also to increase OK via eq. (2.31). First results, which are promising, show an increase in Kerr rotation by as much as a factor of 4 without substantial decrease of reflectivity (Katayama et al. 1988, Reim and Weller 1988, 1989).
156
W. REIM and J. SCHOENES
2.3. The Hall effect and the Kramers-Kronig equations The energy range for magneto-optical measurements is limited both on the lowand on the high-energy side. However, very often a low-energy extrapolation to zero energy is necessary to be able to separate interband from intraband transitions and to deduce a model band structure near the Fermi energy. This extrapolation can be done quite accurately for O'lxy by using the Hall effect (Doniach 1966). At to = 0, both o-2,~ and o-2,yvanish as can be shown by symmetry arguments (S.D. Smith 1969). The shape of o-2~yat low energies may be obtained using the Kramers-Kronig integral equations between o-l~y and o-2xy. In general, the electrical resisitivity is defined as the inverse of the conductivity, I~" = ~ - 1 ,
(2.71)
where both & and ~ are tensors. By calculating the inverse tensor of ~ and restricting to terms up to first order in magnetization, one finds for the diagonal element the well-known equation 1
O-lxx(to __ O) = --,
(2.72)
P
and for the off-diagonal element
O'lxy(to
= 0) --
0H2 ,
(2.73)
P where p and PH are the DC resistivity and the Hall resistivity, respectively. If the resistivity is given in units o f / 2 cm, the conversion factor to %xy i n units of s-1 is 9 x 1011. The real and the imaginary part of 6"xy obey the well-known Kramers-Kronig equations (Bennet and Stern 1965) ____2 p f0 ~ to,~--_ tot 2 °'2xy(to') d w ' ,
Cr2xy(to) = 2to P
f0
1
tot2 __ ('02 fflxy(to') do)',
(2.74) (2.75)
where P denotes the principal value of the integral. Although this integral extends from zero to infinity, contributions at to' far off a given frequency to are small due to the local character of the integrand and to changes in sign of tTxy. This fact makes eqs. (2.74) and (2.75) very useful for extrapolating O'xy below the lowenergy limit E 1 of the measurement. For this purpose, an extrapolation of Orlxy below E 1 and towards O-lxy(W= 0) given by eq. (2.73) is chosen in such a way, that eq. (2.75) reproduces the measured trZxy in the energy range above El, and vice versa. We have found that for most of the materials reviewed in this work,
MAGNETO-OPTICAL SPECTROSCOPY
157
the calculated ~xy in the energy range from E 1 to E 1 + 2 eV depends quite strongly on the chosen extrapolation, thus, giving reliable information o n O'xy between E 1 and zero energy. It should be noted that also the Faraday rotation and ellipticity are connected by a Kramers-Kronig integral similar to eqs. (2.74) and (2.75) (Smith 1976), which allows a fairly accurate determination of the complex Faraday effect if only one part has been measured (Schoenes 1975). However, there is no such equation for the Kerr effect (Kahn et al. 1969). A way to gain additional information in spectroscopy is the use of sum rules for the optical constants (Altarelli and Smith 1973). Here we only want to mention the relation ~ 47r (9
2 0"2ij((9)d(9 = -27r o-lq((9 = 0),
(2.76)
between the zero-frequency magnitude of O-lq (i.e., resistivity or Hall effect) and the frequency dependence of o-2q. This is an additional way to check an extrapolation of ~r and to obtain information on optical transitions not covered by the experimental energy range. Recently, an extra contribution to the Faraday rotation besides intraband and interband terms has been proposed arising from the Hall effect (Hartmann and McGuire 1983). This result was deduced from an experimentally observed proportionality of OF to the Hall angle (OH = arctan(pH/p ) = pH/p) for a selected series of similar amorphous alloys. However, in the light of the basic theory given above, the Faraday rotation at a given photon energy will always be proportional to OH for different materials if the conduction-electron contribution dominates. This holds as long as the energy dependence of the conduction-electron contribution t o ~xy is similar for the compared materials and their optical properties are not too different. Faraday rotation and Hall effect are manifestations of the same material property but at different frequencies and the proportionality may be deduced from eqs. (2.46) and (2.73) introducing a factor f((9, ~-) to take care of the frequency dependence, OH/p = 5 × 109f((9, ~') (nO F -- k e F ) .
(2.77)
3. Experimental considerations
3.1. Techniques To determine the four quantities ~xx a n d O'xy at a photon energy h(9, in general four independent measurements are necessary. In the special case when the value of~r at only a few discrete wavelengths of laser lines is of interest, the complex tensor may be determined by the measurement of the reflectance change, AI = I+ - I_ I+ + I _ '
(3.1)
of p-polarized light over a wide range of angles of incidence (Corke et al. 1982).
158
w. REIM and J. SCHOENES
Here, I+ and I_ are the reflected intensities for the two directions of in-plane magnetization, respectively. The advantage is the need of only one relatively simple experimental set-up which, in addition, is easily automated. However, if the investigation is devoted to examine ~r over a large energy range, say 0.5-6 eV, this intensity method is limited by signal to noise ratio and one is restricted to methods specialized to either optics or magneto-optics. To determine ~Txxthe simplest way, in general, is an absorption measurement. However, the reduction of the data to obtain the absorption coefficient K from the ratio of the transmitted to incident intensity can be much harder than eq. (2.51) suggests. The problems arise from reflectivity losses and interference effects (Kessler 1963, Heavens 1965). Furthermore, intraband transitions and fundamental interband transitions lead to values for K in the order of 105106 cm -1. According to eq. (2.51), the thickness of the samples therefore should be less than 200 nm. Thus, thin films are needed. For bulk samples like single crystals, reflection measurements have to be used to examine ~xx. Two procedures exist: in the first category, two measurements are performed at each energy, either two reflectivities different by polarization or angle of incidence or the direct measurement of amplitude r and phase A in eq. (2.16) (Muller 1969, Kolb 1972). In the second category, only one normal incidence reflectivity measurement is performed at each photon energy and the Kramers-Kronig integral R(o~') A(w) = _tOTrp f o l n~--'~ -~-~- dto',
(3.2)
is used to calculate the phase A from the reflected intensity R (Smith and Manogue 1981). The real and imaginary part of the dielectric function are then obtained with the relations, e~x~
( 1 - - R) 2 - 4 R sin 2 A (1 + R - 2V'-R cos A)2 '
(3.3)
'E2xx
4(1 - R)X/-R sin A (1 + R - 2X/-R cos/1) 2
(3.4)
The advantage of this second type of reflectivity measurement is the use of unpolarized light and normal incidence. The main disadvantage comes from the need to extend the measurement over an energy range as wide as possible. It has turned out that the photon energy range needed to make safe use of eq. (3.2), extends roughly from 10 meV to 25 eV, thus covering phonons, plasmons, interband and intraband transitions. For most of the rare earth and actinide compounds to be reported on in sections 4-7, the normal optical constants have been determined by this method and will be used to calculate ~ y from the magnetooptical measurements. To the problem of generating, focussing and detecting monochromatic radiation of various energies, magneto-optics adds the task of polarizing it and detecting very small changes in the degree o f polarization. Figure 10 illustrates an ex-
MAGNETO-OPTICAL SPECTROSCOPY
Polarizing Monochromator Prism ~
He- Cryostat
so,,,
~_-~
o-;;:o,
Superconducting Phase Faraday Magnet Shifter Cell Amplifier
A
....... ~
159
Light
~
Source
Oetec,or Analyzing Prism
Preamp.
m i_ t Audio Osc.
Lock - in Amplifier
Fig. 10. Experimental set-up for polar Kerr effect measurements in the 0.5-5.7 eV photon energy range (after Reim 1986). A typical measurement of a 0.5-5 eV spectrum with less than 0.01 ° unaccuracy takes about 60 min. with a fully automated device (Weller and Reim 1988).
perimental set-up (Reim 1986) which is, among others (Krinchik and Artem'ev 1968, Erskine and Stern 1973, Badoz et al. 1977, van Engen 1983), very well-suited to high sensitivity measurements of Kerr rotation OK and ellipticity e K in the energy range 0.5-5.7 eV at variable temperature and high magnetic fields. The light from halogen, xenon or deuterium sources is polarized with an air-gap Glan Thomson prism. Several samples and one reference mirror are mounted in a gas-flow cryostat allowing temperatures in the range 1.5-320 K and magnetic fields up to 10 T generated by a split coil superconducting magnet. The angle of incidence is kept below 5° to assure complete detection of the polar Kerr component (Metzger et al. 1965). The reflected light is polarization modulated with a Faraday modulator with either a heavy glass or a quartz rod for energies below and above 2.5eV, respectively. Using the Jones matrix formalism (Theocaris and Gdoutos 1979) it can be shown, that by setting the analyzer to 90°-6 relative to the polarizer one obtains at the detector an intensity I = ~I0[(1 - J0(2~b0) cos 2(0K - 6)) + 2Ja(2~b0) sin 2(0K - 6) × sin cot . . . .
cos oJt. • - ] ,
(3.5)
for a Kerr rotation OK and a modulation angle ~b = ~b0 sin tot, where Jn(X) is the nth order Bessel function. Thus, the difference of the analyzer settings for vanishing oJ-signal for light reflected from the reference mirror and from the sample, respectively, gives the Kerr rotation OK of the sample corrected for all other rotations due to windows and optical elements in magnetic stray fields. Both gold and platinum are suitable as a reference mirror, showing a Kerr effect of less than 0.002 over the whole spectral region in a field of 1 T. The Kerr ellipticity e K
160
W. REIM and J. SCHOENES
is measured in the same way as the rotation, but with a Soleil-Babinet compensator at 41-Aretardation mounted between sample and modulator. An orientation with the fast axis of the compensator parallel to the major axis of the light ellipse results in linearly polarized light (Senarmont principle) rotated by e K (Theocaris and Gdoutos 1979). It should be noted that for accurate measurements, the Kerr rotation of the sample has to be compensated for by setting the polarizer to -Oi~(hoo). The absolute error of this method depends on the amount of birefringence in the optical components in the light path and the accuracy in the parallel adjustment of reference mirror and sample. Values of less than 0.001 ° are achievable in the 1-4 eV energy range and complete automatization is possible with a typical measuring time of 60 min for a 0.5-5 eV energy scan (Weller and Reim 1988). The same experimental techniques can also be used for magneto-optical measurements in the Faraday geometry. However, due to the much lower transmitted intensity in the spectral region of fundamental absorption compared to measurements in reflection, the accessible energy range will, in general, be reduced.
3.2. Samples As discussed in section 3.1, absorption and Faraday rotation measurements most often require samples in the form of thin films. With the use of modern sputtering techniques or feedback-loop controlled multielectron-gun evaporation systems, thin film preparation of multi-component compounds is relatively easy (Gambino and McGuire 1986). Most magneto-optical data of rare earth (RE) transition metal (TM) compounds and of the RE chalcogenides, both in reflection and in transmission, have therefore been obtained on thin films using the advantages of high composition flexibility and perfectly flat surfaces. One serious problem related with Faraday rotation measurements on thin films is birefringence in the substrate induced by different coefficients of thermal expansion of film and substrate or by a large magnetostriction of the magnetooptical film (McGuire et al. 1987). A second major problem stems from corrosion of the magneto-optical film if the preparation chamber is not built for in situ measurements, which, of course, is the most elegant way to obtain a well defined surface (Erskine and Stern 1973). If some portion of the material is corroded, the film acts like a multi-layer system both for Kerr and Faraday measurements, but with unknown values of the optical constants and thicknesses. In general, this leads to faulty results. Performing the Kerr rotation measurement through the substrate, as it is frequently done to avoid this difficulty, generates a new problem, because the magnitude of the Kerr effect depends on the optical constants of the surrounding material (Choe 1989) as can easily be seen using the appropriate Fresnel equation in eq. (2.16) and calculating OK and e K via eqs. (2.19) and (2.20). For R E - T M compounds on quartz substrates e.g., the measured Kerr rotation through the substrate is, in general, enhanced by about a factor 1.3 in the visible frequency region. If a laser with high correlation length
MAGNETO-OPTICAL SPECTROSCOPY
161
like the HeNe laser is used as the light source, interference effects within the substrate cause additional alteration of the measured values. Hence, in situ measurements as has been done for Gd (Erskine and Stern 1973), or a careful transfer of the sample into the measuring equipment under inert gas atmosphere are necessary for corrosion sensitive materials. Similar corrosion problems arise if single crystals are used, which was the case for the optical and magneto-optical measurements on most actinide compounds and some RE-chalcogenides. However, single crystals have the advantage that very clean and optically flat surfaces can be prepared by cleaving. Therefore, crystals of these materials have been cleaved in the ultrahigh vacuum of the reflectivity equipment (Schoenes 1980b) or liquid helium in the cryostat for magneto-optical measurements (Reim 1986), whenever it turned out to be necessary. If bulk samples have to be polished to obtain an optically flat surface, besides the mentioned corrosion problems, the uppermost 100nm are mechanically altered by the polishing grains. In reflectivity measurements, this leads to strong scattering losses at wavelengths <400 nm, and consequently to a reduced reflectivity in the UV (Kfing and Schoenes 1980). In magneto-optical measurements, the energy dependence of OK and eK has been found to be, in general, only a little affected (Reim 1985). However, for materials where the electronic or the magnetic structure are determined by a delicate balance between different energy terms, like in intermediate-valence compounds or materials with a complex magnetic phase diagram like UAs, polishing leads to dramatic changes of the critical properties in the uppermost 100 nm. As a typical example, fig. 11 displays the temperature dependence of the Kerr rotation for cleaved and polished UAs single crystals (Reim et al. 1984b) showing a complete wash-out of the magnetic phases for the polished surface.
UAs (loo) • cleaved A polished B = 100 kG E=1.6
o=
eV
~-.5 t hID
Y
I
I
I
t
I
50
100
150
200
250
Ternperoture (K) Fig. 11. The influence of mechanical polishing on the magnetic phases of single crystalline U A s in the uppermost 100 nm, as probed by the magneto-optical Kerr effect (after Reim et al. 1984b).
162
W. REIM and J. SCHOENES
4. Light rare earths
The study of the rare earth or lanthanide (Ln) series is not obstructed by radioactivity, scarcity or short life times as it is the case for most of its 5f homologues. Nevertheless, many white spots in our knowledge of the magnetooptical properties of lanthanides still exist. Disregarding here data for single lanthanide ions diluted in large-gap insulators like CaF 2 (for examples see: Shen and Bloembergen 1964, Weakliem et al. 1970), which are nonmagnetic and therefore not of central interest for this chapter, the reasons for the limited number of studied rare earths can be summarized in the two following points: (i) In materials with crystal-field splitting effects, lanthanide ions with an even number of f-electrons will generally have a nonmagnetic ground state. As a consequence, many high symmetry structures with Pr 3+, Tb 3÷, Ho 3+ and Tm 3÷ are nonmagnetic down to very low temperatures. (ii) The binding energy of the 4fn states in the lanthanide series varies over many electronvolts (Lang et al. 1981). Since in most cases only dipole-allowed transitions will lead to large magneto-optical signals that can be resolved in a metallic sample, the 4fn---~4fn-15d transition energy should preferably be in the easily accessible photon energy range, say 0.5 to 5 eV. This excludes, for most magneto-optical measurements, Sm 3+ and Gd 3+ and puts materials on the basis of Nd 3÷, Dy 3÷ and Er 3+ near the high-energy limit. From these arguments, it becomes clear that the majority of magneto-optical investigations is dealing with compounds and alloys of (a) the light rare earth ions Ce 3+ and Nd 3+, (b) the half filled shell ion Eu 2+ and (c) the heavy rare earths Tm 2+ and Yb 3+. Exceptions are Kerr effect measurements using synchrotron radiation (Erskine 1976, 1977) which allows probing of deep lying 4f states as in Gd 3+. Other exceptions are the thulium monochalcogenides with trivalent thulium. In these compounds one observes large magneto-optical effects due to intraband transitions (Reim et al. 1984a). After the recent discovery of similar contributions in some UTe samples (Reim 1986), in NdS (Br/indle et al. 1988) and in Nd3S 4 (Schoenes et al. 1988), we anticipate that this origin of magneto-optical signals will occur in other magnetic and metallic lanthanide compounds, as well.
4.1. NaCl-type cerium, neodymium and samarium compounds As most lanthanides, cerium, neodymium and samarium form face centered cubic binary chalcogenides with the V I A elements of the periodic table of the elements (S, Se, Te) and fcc pnictides with the V A elements (N, P, As, Sb, Bi). Table 1 collects the lattice constant a, the type of magnetic order, the ordering temperature, Tc or TN, the ordered magnetic moment/Zora and the crystal-field splitting A between the ground state and the excited states. For the Ce compounds the 4fI(:Fs/2) free-ion state splits into a F7 doublet and a F8 quartet, the F7 state in
MAGNETO-OPTICAL SPECTROSCOPY
163
TABLE 1 Crystallographic and magnetic data for cerium and neodymium monochalcogenides and monopnictides, a is the lattice constant of the NaCl-type compounds. The ordered moment /~ordis taken from neutron scattering, except for neodymium chalcogenides where the data are from magnetization measurements in fields of 200 kOe. The crystal-field excitation energies ~ [K] are also preferentially taken from inelastic neutron scattering. For the Ce compounds the energy is between the ground state F~ and the excited state F8. For the Nd compound the ground state is F~2~and the excited states are F6 and F~1~. a (/k)
Magnetic Structure
CeS CeSe CeTe
5.778 [1] 5.992 [1] 6.361
AF-II [21 AF-II [31 A F - I I [3]
CeN [6] CeP [7] CeAs [7] CeSb [7]
5.018 5.932 6.078 6.412
CeBi [7]
6.487
no order AF-I AF-I 6AF phases ( r < TN/2 : AF-IA) AF-I (T < TN/2 : AF-IA)
NdS [81 NdSe [8] NdTe [8]
5.695 5.911 6.284
NdN [9] NdP [9] NdAs [9] NdSb [9] NdBi [9]
5.131 5.826 5.958 6.31 6.41
Tc,N (K) 8 5.4 2.2
/Xord (p~)
A (K)
0.57 [21 0.58 -+ 0.05 [31 0.3 -+ 0.1 [31
142 -+ 6 [41 100 [11 32 [5]
8 8 16
0.83 0.85 2.06
172 159 37
25
2.1
AF-II AF-II AF-II
9.3 10.5 10.6
2.24 (at 200 kOe) 2.34 (at 200 kOe) 2.58 (at 200 kOe)
232 208 150
F AF-I AF-I AF-I AF-I
27 11 13 16 25
3.1 1.83 2.18 2.98 3.14
56,467 34, 170 30,157 18, 103 37, 105
8
References: [1] Hulliger et al. (1978). [2] Schobinger-Papamantellos et al. (1974). [3] Ott ct al. (1979). [4] Furrer (1987). [5] Rossat-Mignod et al. (1985). [6] Kaldis et al. (1982). [7] H~ilgand Furrer (1986). [8] Hulliger et al. (1975). [9] Hulliger (1979).
m o s t cases b e i n g t h e g r o u n d state. T h e e x c e p t i o n is C e N , w h i c h a p p e a r s to h a v e an u n s t a b l e v a l e n c e ( B a e r a n d Z f i r c h e r 1977) a n d for which, to o u r k n o w l e d g e , n o crystal-field e x c i t a t i o n s h a v e b e e n r e p o r t e d . T h e 4f3(419/2) f r e e - i o n g r o u n d state of N d 3+ splits in an o c t a h e d r a l crystal field into states with s y m m e t r y Fs(1), F 6 a n d F8(z). T h e two crystal-field v a l u e s given for t h e n e o d y m i u m p n i c t i d e s c o r r e s p o n d to t h e s e p a r a t i o n o f F 6 a n d F8(2), r e s p e c t i v e l y , f r o m t h e g r o u n d s t a t e F8(1). W i t h t h e e x c e p t i o n o f t h e n i t r i d e s , all c e r i u m a n d n e o d y m i u m chalc o g e n i d e s a n d p n i c t i d e s o r d e r a n t i f e r r o m a g n e t i c a l l y in z e r o field. C e N s h o w s n o o r d e r d o w n to 1.5 K , w h i l e N d N o r d e r s f e r r o m a g n e t i c a l l y . T h e p a r t i c u l a r situat i o n of t h e n i t r i d e s follows f r o m t h e i r small l a t t i c e c o n s t a n t , giving a L n - L n s e p a r a t i o n v e r y s i m i l a r to t h a t in t h e p u r e L n - m e t a l s . T h e small b i n d i n g e n e r g y o f t h e 4f 1 s t a t e in C e 3+ results in a s t r o n g h y b r i d i z a t i o n with o t h e r states, a r e d u c t i o n of t h e m a g n e t i c m o m e n t a n d t h e a b s e n c e o f m a g n e t i c o r d e r . O n t h e o t h e r h a n d , in n e o d y m i u m t h e 4f 3 b i n d i n g e n e r g y is of t h e o r d e r 4 - 5 eV ( L a n g et al. 1981)
164
W. R E I M and J. S C H O E N E S
leaving also in NdN the 4f3 state localized, and allowing the occurrence of ferromagnetic order, due to the dominance of direct exchange to the nearest neighbors over the indirect exchange via the anions to the next-nearest neighbors. The antiferromagnetic ordering in the large majority of these compounds is not very appealing for magneto-optical studies. Yet, in large enough magnetic fields substantial magnetic moments can be induced and the puzzling question of the nature and the binding energy of the 4f state in cerium compounds has triggered an investigation of CeSbxTel_ x with x = 0, 0.75 and 1 (Reim et al. 1986). Figure 12 shows the Kerr rotation OK and the Kerr ellipticity eK for these three compounds in the energy range from 0.5 to 5 eV. At a measurement temperature of 2 K and an applied field of 5 T, CeSb is saturated with a moment of 2.06 p~/Ce (Rossat-Mignod et al. 1985) while CeSb0.ysTe0.25 and CeTe are in an intermediate spin state with, e.g., 0.71/xB/Ce in the latter material (Hulliger et al. 1978). The most striking feature is the value of the maximum Kerr rotation in CeSb. 0~ reaches 14° at the low-energy limit of the measurement which was 0.5 eV. This is the largest ever reported rotation angle for a single reflection in any material. From a Kramers-Kronig transformation of O'2xy and vice versa (see section 2.3), one can anticipate that the maximum rotation will exceed 15° near 0.4 eV. A second interesting feature is the shift in energy of the observed structures from 2eV in CeTe to 1 eV in CeSb0.vsTe0.25, t o ~ 0 . 5 eV in CeSb. Although, as a general rule, assignments to electronic transitions should not be made on the basis of OK and EK spectra, but on the basis of Orlxy and Or2xy spectra, this shift points to a decrease in energy of the responsible transition. This idea also gains support from the fact that if one normalizes the maximum rotation of CeTe and CeSb0.75Te0.25 to the magnetic moment in CeSb, the former two materials would display a maximum rotation of ca. 10°, which is not too far from the 15° occurring in CeSb. Figure 13 displays the real and imaginary parts of the off-diagonal conductivity of CeSb and CeTe as computed from the above Kerr spectra and the optical constants. Below 0.5 eV the spectra have been extrapolated in such a way that the Kramers-Kronig-transformed spectra fit the experimental data above 0.5 eV as 14
O K
CeSb B=5T T = 2K
OK
3 2
"510
CeSboz5Teo.zs B=6T T=2K
CeTe B= 5 T T=2K
OK
1
~n ,
k
8
0
6
-1
#_4
', -~K
-2
~x 2 E
',~EK ~T-7"-
,
.
,
,
ir ii d ii ii g
80
1
2
3
4
5
6
1
2
3
4
Photon Energy
5
6
I
I
I
5
4
5
6
[eV]
Fig. 12. Polar magneto-optical Kerr rotation OK and ellipticity e~ for [100]-oriented cleaved single crystals of CeSb, CeSbo.75Te0.25 and CeTe (after Reim et al. 1986).
MAGNETO-OPTICAL SPECTROSCOPY
165
4
o
CeTe
;
T=2K B=5T
*** z.× ,
02xy
o
2
* o a!
l
0 '(0
7D - 2
!
~n
', ~ 1 - ,
o
,
~t I
3
a
+~o 1 ~ y
~-4
CeSb
+ ÷
T=2K B=5T
-6 -8 i
i
i
2
3
4
6
5 Photon
0 Energy
1
2
(eV)
Fig. 13. Absorptive part ~rzxyand dispersive part O'lxy of the off-diagonal conductivity of CeSb and CeTe (in part after Schoenes 1987). well as the extrapolation of the c o m p l e m e n t a r y spectrum. O n e recognizes that O'lxy of CeSb peaks at 0.4 eV w h e r e also the diagonal conductivity (fig. 14) shows a p r o n o u n c e d shoulder. A f t e r substracting the free-electron contribution or elxx from the total o-ixx the remaining bound-electron-transition part exhibits a maximum at 0.4eV, which has the theoretically expected weight for an f~---~d ~
CeSb
~xx
o v
b'-I
0
0.5
10
1.5
2_0
Photon Energy (eV)
Fig. 14. Absorptive part o'1=~ of the diagonal conductivity of CeSb(+ + + +) at room temperature and f b a decomposition into intra- (crlxx) and interband (~rl~x) contributions (after Schoenes and Reim 1985).
166
W. REIM and J. SCHOENES
transition with localized f-electrons (Schoenes and Reim 1985). It is thus concluded that the fl ~ d ~ transition occurs in CeSb at 0.4 eV and that it gives raise to a so-called diamagnetic line shape (see also section 2.2), i.e., a dispersive behavior in the absorptive part of O"xy(=O'2xy)and an absorptive behavior in the dispersive part of tTxy (=o-lxy). As we will see throughout this chapter, the diamagnetic line shape is the common energy dependence for the f---~d transition in lanthanide and actinide compounds. It is the result of at least two subtransitions with inverse dominances of absorption for right- and left-hand circularly polarized light. At 1.4eV, both o-ix~ and O'e~y show a maximum indicating a transition with a so-called paramagnetic line shape. Presumably, this is the onset of transitions from the valence band, which at its top has strong admixture of d-character and also some admixture of f-character. Coming to CeTe, fig. 13 shows that the large negative peak in Orlxy is shifted to 2 eV. At the same energy, o-1~~ displays a barely resolved structure which is, however, of the correct size for an f---~d transition. A similar result is found in CeS with an excitation energy of 2.6 eV (Schoenes 1984). Photoemission gives binding energies of 2.4 eV in CeS and 2.5 eV in CeTe (Gudat et al. 1981). Thus for Ce-Chalcogenides optics, magneto-optics and photoemission come to the corresponding conclusion that the f-state is localized with a binding energy in the order of 2 eV. The situation is definitely more complicated in the cerium monopnictides. In CeP, CeAs, and CeBi, valence-band photoemission displays two maxima, one near 0.5 and one near 3 eV binding energy, which have been attributed to 4f electron excitation on the basis of their intensity dependence of the incidentphoton energy (Franciosi et al. 1981). Many controversial interpretations have been put forward to explain this double structure and the historically interested reader may consult the proceedings of the various conferences on valence fluctuations held in Rochester (1976), Santa Barbara (1981), Z/irich (1982), K61n (1984) and Sendai (1985). We will not repeat here the various arguments favoring one or the other proposal, but only mention the two lines along which the double structure is interpreted today. The first model, which we may call the localized or many-body model, starts from a localized 4f level interacting with the electrons of a conduction band. A Kondo resonance is formed at the Fermi energy in addition to the bare 4f level 2-3 eV below E F. The relative intensities of the peaks in the photo-emission spectra are determined by the ratio of the hybridization width and the energy of the level (Gunnarson and Sch6nhammer 1983, Sch6nhammer and Gunnarson 1987). In the second model, which we may call the itinerant model, one starts from a band-structure calculation, and the f-states lie at the Fermi energy. This is the ground state of the hybridized system and it is taken to correspond to a well-screened f hole, i.e., a final fl state (H/ifner and Steiner 1982). The second peak is obtained by a calculation in which an f hole is created at a central atom in a larger unit cell and only conduction electrons are allowed to relax and screen the 4f hole (Norman et al. 1985). Thus, for large enough hybridization both models come to the conclusion that the peak near the Fermi energy corresponds to the ground state and the high binding-energy peak is merely a result of the photoemission process. It is then interesting to note that the
MAGNETO-OPTICAL SPECTROSCOPY
167
magneto-optical spectra of CeSb exhibit only one 4f peak and that its energy of 0.4 eV is an excellent measure of the ground-state energy of the 4f state. The second trivalent rare earth element in the series of NaCl-structured lanthanide compounds which is suitable for magneto-optical studies is neodymium. Compounds and alloys of this element have received more attention recently, when it was more widely recognized that the addition of terbium or dysprosium to iron or cobalt reduces the magneto-optical signals (see section 6). To help understanding the magneto-optical response of Nd 3+, the polar Kerr effect has been determined in a concentrated, but simple, magnetic system like NdS (Br/indle et al. 1989). Figure 15 displays the Kerr rotation and ellipticity spectra of a [100] cleaved NdS single crystal at 15 K in a field of 100 kOe. The rotation spectrum can be decomposed into two parts. In the neighborhood of 3 eV one observes a diamagnetic line shape very similar to that found for the first time in TmS and TmSe by Reim et al. (1984a), and which will be discussed in section 6. The energy of this structure corresponds to the plasma energy in NdS and the size of the rotation is similar to that found in TmS. Therefore, the interpretation of the structure follows the same lines as for TINS, for which quantitative fits of OK, eK, trlxy and Or2xy have been performed. These are based on the classical description of the optical and magneto-optical properties, as the sum of Lorentz oscillators for the interband transitions and a Drude term for the intraband transitions. It results in structures with the observed line shapes for OK and eK near the coupled plasma energy. However, to fit the large rotation and ellipticity quantitatively, the cyclotron resonance frequency entering the classical formula has to be enhanced by two orders of magnitude. This can be accounted
4
NdS (100)
3 "v o
T = 15K H = 100kOe
2
-OK
,..I.-
HJ
1 0 - - ~ . ~
O 13_ X
f
-1
JI
-2
E o q)
I
;J -3 -4
,
0
i
,
i
i
i
,
i
i
i
1 2 3 4 5 Photon Energy (eV)
Fig. 15. Polar magneto-optical Kerr rotation OK and ellipticity eK for a [100]-oriented cleaved single crystal of NdS at 15 K and in an applied field of 100 kOe (after Brfindle et al. 1989).
168
W. R E I M and J. S C H O E N E S
for by a microscopic theory for the scattering of conduction electrons in a magnetic material (Reim et al. 1984a). The second structure in fig. 15 is a broader peak near 4 eV which, as a preliminary computation of trxy shows, persists in the latter spectra and presents all the characteristics of an f--->d transition. The last light rare earth to be discussed in this section is Sm 2÷. Suryanarayanan et al. (1972) have co-evaporated Sm and Te onto heated substrates of pyrex, vycor or CaF: to produce thin films of SmTe. The compound formed is nonmagnetic, as is expected from the 7F0 ground state of Sm 2÷. It is also nonconducting, in agreement with the simplest ionic model. Samarium monochalcogenides attracted large interest in the beginning of the seventies, due to the discovery of pressure-induced semiconductor-metal transitions (Jayaraman et al. 1970). In the metallic phase, samarium was shown to be in a valance state intermediate between Sm 2÷ and Sm 3+ (see also section 6.1). The magnetic circular dichroism spectrum reproduced in fig. 16 (Suryanarayanan et al. 1972) is at normal pressure, i.e., for samarium in an integer divalent state. In the studied photon energy range from 2.5 to 4 eV, one observes two S-shaped structures which have corresponding shoulders in the absorption spectra. From a detailed analysis of the temperature dependence of the circular dichroism peaks, the authors derive that the responsible transitions are 4f6(7F0)-->4fS(6H)5deg and 4f6(TF0)--->4fS(6F)5deg. The corresponding two transitions into the 5d t2g subbands are assigned to two absorption peaks at 1.2 and 2.0 eV. These assignments have been corroborated by an investigation of the variation of the absorptionpeak energies as a function of pressure and temperature in the related compound SmS (Batlogg et al. 1974). Of course, an interesting question is whether Sm 2+ is a suitable ion for obtaining large magneto-optical signals. Although the data shown in fig. 16 are not normalized, a comparison with similar data for EuSe (Ferr6 1974) allows the estimation that the magneto-optical response per Gauss applied field is a factor 1000 smaller than in EuSe, which, as we will see in Section 5.1, is an excellent
t
~'~ 2
t
.....~oo K
~--~0 .......
- -
1.5~
.............
~0
111 a -2
/
f_J
~
/- - ~
11
-J
).57-o_ 0
PHOTON ENERGY (eV)
Fig. 16. Optical density K d [eq. (2.51)] and magnetic circular dichroism AD of a SmTe thin film vapor-deposited onto C a F 2. T h e M C D was m e a s u r e d for various temperatures in a field of 35 k O e (after Suryanarayanan et al. 1972).
MAGNETO-OPTICAL SPECTROSCOPY
169
magneto-optical material. In the spirit of what we said at the beginning of this section, the weak magneto-optical response for a nonmagnetic ground state is expected and, therefore, compounds with Sm 2÷ have not been investigated much further.
4.2. Th3P4-type compounds Many lanthanides also form compounds of the Th3P4 structure and its anti-type, the Gd4Bi 3 structure with the V A and V I A elements. It is also well-known that the normal Th3P4 structure is stable with holes in the 8-coordinated sites, leading to compositions between L n a S 4 and L n 2 S 3 (Hulliger 1979). Thus, neodymium forms semiconducting Nd2S 3 (Henderson et al. 1970) as well as the metallic N d 3 S 4 (Schobinger-Papamantellos et al. 1977) with the same cubic T h 3 P 4 s t r u c t u r e . N d a S 4 is especially suited for magneto-optical studies of Nd 3+, because it orders ferromagnetically at (47 --- 1) K (Schobinger-Papamantellos et al. 1977). Its ordered moment at 4.2K has been determined to be (2.08_+ 0.06)/zB. This reduced value, as compared to the free-ion value of gJ = 3.27/z B is ascribed to crystal-field splitting. Figure 17 shows the polar Kerr rotation and ellipticity measured on a single crystal of N d 3 S 4 held at 10K and exposed to an external field of 40kOe (Schoenes et al. 1988). One observes two maxima in the rotation 0K(tO), one very narrow peak reaching a rotation of 2.3 ° is located at 0.8 eV, and a second broader peak with 0.5 ° rotation is centered at 4 eV. To discuss the origin of these peaks and those in the ellipticity eK(tO) we need to know the off-diagonal conductivity.
Nd~S 4 -eK
B =4 T T =1OK
80 E
0
-1
I
I
I
I
I
1 2 3 4 5 Photon Energy [eV] Fig, 17. Polar magneto-optical Kerr rotation OK and ellipticity %: for a polished single crystal of Nd3S 4 at 10 K and in an applied field of 40 kOe (after Schoenes et al. 1988).
W. REIM and J. SCHOENES
170 100 r
80
Nd3$4
60
20
I
0
I
I
2
I
I
I
I
I
I
I
4 6 8 Photon Energy [eV]
I
I
10
I
12
Fig. 18. Near-normal incidence reflectivity of Nd3S 4 at room temperature. The surface is a polished [100] plane (after Studer and Schoenes 1977).
Figure 18 displays the near-normal incidence reflectivity of the same material (Studer and Schoenes 1977) which is needed to compute the diagonal and, together with 0K and eK, also the off-diagonal conductivity. The reflectivity indicates that Nd3S 4 is a metal with a plasma edge of about 1 eV. The dielectric function and the energy loss which have been calculated by Kramers-Kronig transformation of R(to) give for the coupled plasmon energy htop a value of 0.75 eV. This agrees with the low-energy structure in OK and eK, suggesting that the latter are to be associated with the coupled plasmon. A close comparison with TmS, TmSe and Tm0.sEu0.sSe (section 6.1), for which a similar effect has been observed for the first time (Reim et al. 1984a), shows that -O K has a positive maximum at tOp * in N d 3 S 4 and Tmo.sEuo.sSe and a negative extremum (minimum) at tOp in TInS and TmSe. This difference can be explained by the expected difference in the spin polarization of the conduction electrons. For Nd 3÷, L = 6, S = ~ and J = 9, thus the spin moment of the f-electrons is antiparallel to the total moment. The spin moment of the conduction electrons couples parallel with the 4f-spin moment leading to a negative spin-polarization of the conduction electrons relative to the total moment. In Tm 3+, with a more than half-filled 4f shell, L, S and J are parallel, so that the conduction electrons will have a positive spinpolarization. The case of Tm0.sEu0.sSe is complicated by two antiferromagnetically coupled sublattices, of which the Eu 2+ sublattice has the larger moment, and therefore determines the net magnetization direction. Figure 19 exhibits the off-diagonal conductivity spectra of N d 3 S 4. The dominant structures now occur in the 4 eV range, where OK showed a second pronounced peak. These structures are assigned to an f3___~f~5d transition. From fig. 19 one derives that for this transition Orlxy has an absorptive and Or2xy has a dispersive line shape. The transition energy is then ~4 eV, and is determined by the extremum in
MAGNETO-OPTICAL SPECTROSCOPY
171
t\
5
t
o ',
Nd3 $4 B= 4 T T =1OK
,/ / I °-zxY
0
._>
/
e0
!
e-
o -5
121 I
o -10
1
2
3
4
5
Photon Energy [eV] Fig. 19. Real part o-lxyand imaginary part ~r2xyof the off-diagonal conductivity of Nd3S" at 10 K and in an applied field of 40 kOe (after Schoenes et al. 1988).
o-lxy. This agrees reasonably well with a binding energy of the 4f3 level in Nd metal of 4.6 eV (Lang et al. 1981) and with the assignment in NdS (section 4.1). Below 3 eV two further, but much smaller, peaks appear in O-2xy. The peak at 0.8 eV falls together with oJ~ and might be an artifact, produced by the multiplication of strongly varying functions in this energy range. The peak near 2 eV, on the other hand, has its counterpart in O'lx~ and should, therefore, be considered real. Its assignment to an electronic transition is not as straightforward as for the p e a k at 4 eV. However, two explanations can be offered: (i) a transition of d-electrons into empty f-states. Such transitions occur in uranium compounds and they have the same positive sign. Also, the energy of 2.2eV is compatible with the energy of 1.8 eV for the 4f 4 peak observed by bremsstrahlung isochromat spectroscopy in Nd metal (Lang et al. 1981). Against this interpretation speaks the observation of a peak at similar energy in o-1~~ of La3S 4 (Studer and Schoenes 1977), for which the 4f 1 state is expected at only 5.5 eV. Also, the intensity is much larger than expected for a transition into localized f-states considering the results for uranium monochalcogenides (section 7.2). (ii) a transition from the top of the p-band into empty d-states. This interpretation is compatible with the Orlxx(¢-O) data for La3S 4 and may also give some basis for the observed sample-, i.e., stoichiometry, dependence of the intensity of the peak in O-lx~(w). The relative importance of the signal in O'2xycompared to 0-1~~ necessitates f-character to be admixed to the involved p-states in Nd3S 4.
172
W. REIM and J. SCHOENES
4.3. Intermetallic compounds The potential of thermomagnetic writing and magneto-optical read-out has prompted quite early investigations of intermetallic compounds and alloys of rare earths and 3d transition elements. Thus, after Nesbitt et al. (1962) reported a Kerr rotation of 2 ° in SmCos, several RCo s intermetallic compound films with R either Y, Sm, Gd or Nd, were studied by Stoffel and Strnat (1965) and Stoffel (1968). They found that the off-diagonal component of the complex permittivity tensor generally decreases with increasing rare earth content and they did not confirm the large Kerr rotation of SmCo 5. More recent work has been performed by Gambino and McGuire (1985, 1986), Suzuki et al. (1987), Choe et al. (1987), Choe (1989) and Weller and Reim (1989a,b). While the two first authors, as many others, quote only values for a single laser wavelength and find a maximum Kerr rotation of 0.59 ° at 1.96 eV in Pr0.2Fe0.4Co0. 4 (Gambino and McGuire 1986), the latter have investigated the spectral dependence. Figure 20 reproduces the Kerr rotation spectra for polycrystalline Co and various compositions of amorphous
5 i 0.5
Energy 5 I
4 I ~-Co
Thin
(eV)
2 I
Film
/
°~o~
~
~
_
o
0.4
/
0,5
FY 19C0.81
. .....~,~o
---m:T
0.2 o?..ol.dSo o
-~
0
"(29C°.71 I = I
,i
i,..
0.4 0..5 0.2
I
I
I
I
I
I
Nd.t9Co.81
o 0.5
"5 n,-
= I
I
"~-:~---o . . . ... . . .. . . . ~-~
Nd.24C° 76 J
~.......~-~" o- - ~ 2 " - - . __ Ce.Zl C°-79 N d.29C0.71 -"~--=---o --_._~__ .
.
.
.
.
.
" .
.
.
.
0.1 0
I 300
I
I I I = I 400 500 600 Wavelength (nm)
I 700
Fig. 20. Absolute value of the polar Kerr rotation at room temperature and an unspecified field for various compositions of amorphous Y-Co, Ce-Co, Nd-Co and polycrystalline Co films (after Choe et al. 1987).
MAGNETO-OPTICAL SPECTROSCOPY
173
L n - C o films, with Ln being Ce and Nd (Choe et al. 1987). For comparison, also shown, is the effect of alloying Y instead of Ce or Nd. One recognizes, that except for lower Nd-concentrations, the general trend is a decrease of the Kerr rotation at room temperature. The main reason for this behavior is presumably the decrease of the magnetic moment at room temperature. Choe et al. (1987) find that for T = 300 K the magnetization decreases by a factor of approximately 2 when the Ce, Nd of Y content increases from 15 to 30 at%. Nearly the same reduction is found for the Kerr rotation at 633 nm (=1.96 eV). Therefore, if one normalizes for every sample the Kerr rotation by its value at long wavelengths, one can obtain an idea of the spectral contributions of the rare earth atoms to the total Kerr effect. Figure 21 (Choe et al. 1987) displays such normalized spectra, which should now also be compared with the pure Co spectrum (fig. 20). Apparently, Nd leads to an increase of the Kerr rotation near 4 eV. This is the same energy at which Nd3S 4 shows a maximum (fig. 17), which, from an analysis of the diagonal and off-diagonal conductivity, could be assigned to an f3___~f2d transition. Because such an intra-cation transition is not expected to be very much dependent on the compound formed, we suggest that also in NdxCol_ x the peak near 4 eV is due to an f3__, f2d transition (Choe 1989, Weller and Reim 1989a,b). The Kerr effect of alloys of two light rare earths with iron is shown in fig. 22 (Suzuki et al. 1987). As in NdxCOl_ x an increase of the rotation with decreasing wavelength is observed between 800 and approximately 400 nm. For still shorter wavelengths, Suzuki et al. find a zero crossing, which, however, has not been observed by others (Weller and Reim 1989a,b). Very recently, the wavelength dependence of the Faraday rotation has been studied for a variety of amorphous Nd-Co, Nd-Fe and N d - A u thin films (McGuire et al. 1987). Figure 23 shows the results for the Nd-Co system Energy 5 1.5 - I
4 I
/ ~ . ~
__
_
E
5 I
(eV)
~
2 I
Nd.29 Co.71
".
Nd,24Co,76
~.~'~'- - ~--"~'~--~...-..~ "" -. j Nd 19 Co 81 " ' ~ . ' ~- Ce.28Co.72
•
0
'~
I
~00
.
I
/
I
400
I
I
I
500
Wavelength
I
600
I
I
TO0
(nm)
Fig. 21. Kerr rotation spectra for various amorphous Y-Co, Ce-Co and Nd-Co films, normalized by their Kerr rotation values at 700 nm (after Choe et al. 1987). The temperature is 300 K, the field has not been indicated.
174
W. R E I M and J. S C H O E N E S I
~ 0.1 v
I
I
I
I
I
I
o
~ -o.i ,~ -0.2
I \
-0.3 _~ -0.4
g
Nd38 Fe62
\~
~
I
t
Pr4oFe60 I
I
I
I
I
400 600 800 Wovelength X (nm) Fig. 22. Polar Kerr rotation at room temperature and fields of 7 kOe for amorphous films of NdsaFe62 and Pr40Fe60 (after Suzuki et al. 1987).
I
I
/ 0 0~,0 0
-I-++~
~
I
I+
I I NdloC090 0 Nd15C085 • Nd25C075 • NdsoC%0 0 Nd75C025 &&A&
¢ ÷,÷o/ \,-
4
",,.....
% ~ 3
L,t j~'/ •
,,~
/
÷,+ 'o OOooo+++o +*'...... +,***
O ++
~ 2 o Z
\
l d
\
T=20K B=32kG 0 0.0
I 0.5
I 1.0
I 1.5
I 2.0
I 2.5
I 3.0
I 3.5
4.0
Photon Energy {eV)
Fig. 23. Faraday rotation for amorphous Nd-Co films at 20 K and a field of 32 kOe (after McGuire al. 1987, and Reim and Gambino 1986)•
et
(McGuire et al. 1987, Reim and Gambino 1986)• With increasing Nd content, one observes a strong decrease of the rotation at low energies and the disappearence of the peak at 1 eV, which is present for 10 and 15 at.% Nd. On the high-energy side of the spectrum, one finds with increasing Nd concentration first the development of a peak, which then for higher Nd concentrations, shifts from about 3 eV for 25 at.% Nd to 1.5 eV for 75 at.% Nd. McGuire et al. (1987) assign this peak to either a 4f--->5d transition or a transition from below E F to an empty
MAGNETO-OPTICAL SPECTROSCOPY
175
4f state above E v. The energy shift of the peak with Nd concentration is ascribed to a shift of some 4f levels relative to E v. In the light of the Kerr data on N d - C o , NdS and Nd3S 4 and these off-diagonal conductivity spectra of Nd3S4, we would rather expect to find signs of 4f 3~ 5d transitions near 4 eV, which, unfortunately, is outside the range of the Faraday rotation measurements. A peak at 2 eV appears to be compatible with a 5d ~ 4f 4 transition in Nd. When alloying with Co, conduction electrons are transferred from Nd to Co, lowering E v and increasing the 5d---~4f4 transition energy. A shift of 1 eV is conceivable, considering the width of the occupied conduction-band part of approximately 2 eV in Nd (Lang et al. 1981) and observed shifts of the same order of magnitude in the systems C e - P d , T h - P d and U - P d (Schoenes and Andres 1982). This assumption, however, remains speculative as long as the off-diagonal conductivity spectra are unknown. The same is also true for the Faraday rotation data on N d - F e and N d - A u . Figure 24 displays the spectra for the latter system which is the simpler one, because it only contains one type of magnetic ion. One observes one peak near 2 eV, which roughly scales with the Nd content and which slowly shifts to higher energies when the A u content increases from 40 to 60 at%. Both effects are compatible with the assumption of a transition of conduction electrons into an empty 4f 4 state, but a definitive assignment has to await a computation of ~r y(O~) or ~:xy(o~). i
i
i
i
i
i
e,leeee
\ % 3
,"%
i
•
Nd4oAU6o
•
Nd60Au40
\
%
1 T=IOK B=32kG
0 0.0
I
I
I
I
I
I
I
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
P h o t o n Energy (eV)
Fig. 24. Faraday rotation for amorphous Nd-Au films at 10 K and a field of 32 kOe (after McGuire et al. 1987). 5. Half-filled shell rare earths
Although this section deals only with the two ions Eu 2+ and Gd 3+, it is certainly the richest in physics of the three sections on lanthanides. This results merely from the unique role that the europium monochalcogenides play for our basic understanding of magnetism of localized electron systems, as well as the many
176
W. REIM and J. SCHOENES
different magneto-optical effects which these materials display in a unprecedented clarity and magnitude.
5.1. NaCl-type europium compounds Table 2 collects some of the crystallographic, optical and magnetic data of the europium monochalcogenides (EuX, X = O, S, Se, Te). The Eu-monochalcogenides are model systems for isotropic Heisenberg exchange. The ground state is the spherically symmetric spin-only state 8S7/2. The nearest-neighbor exchange integral J1 is positive for all EuX-compounds, but decreases with increasing lattice constant. The next-nearest exchange integral -/2 is positive in EuO and negative in EuS, EuSe and EuTe. The two positive exchange terms in EuO and the net dominance of J1 over J2 in EuS lead to ferromagnetic ordering in these two materials. In EuTe, the negative J2 clearly dominates the positive J1 and antiferromagnetic order of the MnO type occurs below 9.6 K. In EuSe the absolute values of J1 and J2 are similar and the different sign of both terms results in a complex magnetic phase diagram. In zero field, EuSe enters an antiferromagnetic phase below 4.6 K. This phase consists of ferromagnetic (111) planes which are stacked in a + + - - sequence (Griessen et al. 1971). We call this phase AF-IIA in analogy to the AF-IA phase in which ferromagnetic (100) planes are stacked in a + + - - sequence and to the AF-II phase in which (111) planes are stacked in a + - + sequence. Below 2.8 K, EuSe consists of a mixture of an antiferromagnetic phase with order +-+and a ferrimagnetic phase with a stacking + + - + + (Fischer et al. 1969). The ratio of ~3 between the two phases determines the observed spontaneous magnetic moment of about 1.7/.~/Eu. Finally below 1.8 K, an MnO type, AF-II ordering occurs in zero field. Already, for fields of a few hundred Oe, ferrimagnetic phases are induced. Fields above 2 kOe lead to a ferromagnetic-like alignment of the moments (Griessen et al. 1971). In EuTe, the critical field for spin alignment is approximately 7 0 k O e (Oliveira et al. 1971, Schoenes and Wachter 1977). TABLE 2 Lattice constant a, magnetic structure, ordering temperature and exchangeparameters for europiumchalcogenides. The last column lists the gap energies at 300 K
EuO EuS EuSe
a (•) 5.141 [1] 5.968 [1] 6.195 [1]
Magnetic Structure F F AF-IIA ferri AF-II AF-II
T~,c (K) 69.33 [2] 16.57 [3] 4.60 [1] 2.8 [4] 1.8 [4] 9.58 [I]
J1/kB(K)
J2/kB(K)
0.55 [5] 0.22 [6] 0.11 [5]
0.15 [5] -0.10 [6] -0.09 [5]
Eo (eV) at 300 K 1.12 [7] 1.65 [7] 1.80 [7]
EuTe 6.598 [1] 0.06 [5] -0.20 [5] 2.00 [7] References: [1] Schwob (1969). [2] Kornblit et al. (1973). [3] Berkner (1975). [4] Griessenet al. (1971). [5] Zinn (1976). [6] Bohn et al. (1981). [7] Wachter (1972).
MAGNETO-OPTICAL SPECTROSCOPY
177
The energy gaPs as determined from re-emission and reflectivity measurements (Wachter 1972) increase with increasing lattice constant. This is contrary to the expectation for a p (anion)--~ d (cation) transition with increasing covalency, and points to an f--~d transition. As we will see, magneto-optical spectroscopy corroborates this suggestion. The first magneto-optical spectra have been published by scientists from IBM (Greiner and Fan 1966, Suits et al. 1966, A h n and Suits 1967). Although they found very large signals, the quality of some of the samples has been improved since then, and even larger signals and more structured spectra have been obtained. In addition the spectral range has been considerably extended and we will concentrate our discussion on these more recent investigations. Systematic studies of the Faraday rotation and the magnetic circular dichroism of EuS, EuSe and EuTe films have been published by Schoenes (1975) and Ferr6 (1974), respectively. Figure 25 shows the absorption and the Faraday rotation spectra as determined on polycrystalline EuS films in a field of 11.5 kOe for temperatures below as well as above T c (=16 K) (Schoenes 1975). At 8 K and 11.5 kOe the film is not magnetically saturated. Nevertheless, the Faraday rotation reaches a maximum value at 4.3 eV as high as 1.5 x 106 deg/cm. The presentation of the absorption and the Faraday rotation on the same energy scale demonstrates very nicely the larger resolution of magneto-optical spectroscopy and allows a rather straightforward assignment of the structures to electronic transitions. In fact, as
i
i
I
i
EuS I ~4 V 0
.-~' 500K 6K
---
/y z//_
~
_ H = 11,5 kOe 5
:~ 2 /
"\\~ ,
2
5
4
~
- - 22K
5
15
gin0
-.-
H
~o o ~
8K
= 11,5
- -
2
kOe
-
5 4 Photon Energy (eV)
~.~,:-~-r,.---
5
Fig. 25. Absorption (top) and Faraday rotation (bottom) of a polycrystallineEuS film at various temperatures and an applied field of 11.5 kOe (after Schoenes 1975).
178
W. R E I M and J. S C H O E N E S
we will argue quantitatively shortly below, magneto-optics discriminates p--~d against f--~ d transitions, while in absorption p--~ d transitions are giving the larger signals. Thus, fig. 25 indicates that f-~ d transitions occur near 2.1 and 4.3 eV, while p--* d transitions start around 3.5 eV. For substrate temperatures exceeding 500°C, EuS, EuSe and EuTe form (100) textured films on fused quartz (Schoenes and Wachter 1977). The magnetization and the Faraday rotation of these films saturates at 3 K for fields of about 30 kOe [EuS, EuSe (see fig. 7)] and 80 kOe (EuTe). Figure 26 shows the Faraday rotation of a (100) textured EuS film for magnetic saturation (Schoenes 1979, 1987). Also shown is the Faraday ellipticity e F or magnetic circular dichroism as has been computed from a Kramers-Kronig inversion of the Faraday rotation spectrum. Note that the sign of this eF spectrum has been reversed relative to the original to conform with the sign convention used in this chapter (see section 2.1.2). One recognizes a maximum rotation of 2.5 × 106 deg/cm which is, together with similar values for EuSe and EuTe (Schoenes 1979), the largest Faraday rotation of any material. The spectral dependence of the magnetic circular dichroism computed from the Faraday rotation can be compared with that measured by Ferr6 (1974) for films deposited on CaF 2 (fig. 27). The agreement is quite good, and is even better if Faraday measurements at similar small fields as for the circular dichroism are used (Schoenes 1975). Also shown in fig. 27 is the linear magnetic dichroism and the optical density of the used film. Faraday rotation spectra for EuSe and EuTe films are displayed in fig. 28 (Schoenes 1975). Note again that these spectra are not for magnetic saturation and that, at magnetic saturation, the rotation near 2.4 eV reaches 2.2 × 106 deg/ cm in both compounds (Schoenes and Wachter 1977). The similarities between
~2
EuS T = 2.5K H = 100 kOe
"u
%
,,',, /',,
1
/, I
I
I
~
',j
e
L
0
I
V
- - a . _ _
2 3 Phofon Energy (eV)
J
4
5
Fig. 26. Specific Faraday rotation OF and magnetic circular dichroism e F of a (100)-textured E u S film at magnetic saturation, e F is c o m p u t e d from 0F using the K r a m e r s - K r o n i g relation (after Schoenes 1979, 1987).
MAGNETO-OPTICAL SPECTROSCOPY x 10 -2 _
,..~ 2_ t_ '
x 10:
t~o~
o
112KK
.-~ a 3,5
i~,,,' - - ' ~
-,
-1
7K
179
I
c~
A
H~O,72T
H~O,O44T -
s,4 K
xlO -2-
0~5
-2 2 ¢'~
~
~10"3(~ ',
1 --
~ __._17,5K ~ ..... 2 6 K
--6
K
2oK
3
--
2
cn Q
o
i.............. ~
o
=~
H=0,0171 !IB J / --1 _ ~L,'e"C ~
.1
H : 0,14T
0,8 .
6 K
-.
_._2o~
-0,6
"~
(:3 b a;
I "
c i
- 0,4 _ 0,2
;
0 "Z-.
o 2
3
4
i 5
Photon Energy (eV) Fig. 27. Linear magnetic dichroism (top), circular magnetic dichroism (middle) and absorption (bottom) of an EuS film vapor-deposited onto CaF z. Note that the spectra are not normalized by the field nor by the film thickness and that the sign of the magnetic circular dichroism is opposite to the definition in this chapter (after Ferr6 1974).
the three compounds are quite obvious. The Faraday rotation spectra are dominated by two sets of structures which come closer to each other in energy when going from EuS via EuSe to EuTe. This reflects the decrease of the crystal-field splitting of the 5d final states with increasing lattice constant. Each set of structures consists of three or four single structures, which can be interpreted in the framework of atomic coupling schemes between the excited 5d-electron and the remaining 4f6 state (Ferr6 1974, Schoenes 1975). Before we do so, let us discuss Kerr effect measurements on single crystals which undoubtedly show the best spectral resolution. Figure 29 displays the longitudinal double Kerr rotation of EuO and EuS single crystals in an applied field of 2.5 kOe and temperatures of approximately 12 and 8 K, respectively (Greiner and Fan 1966). The photon energy range extends over the low-energy group of structures in EuS, but lacks one of the structures in EuO. The saturation field for the longitudinal effect is smaller than for the polar effect, due to the much smaller demagnetizing factor, and the authors claim that fields less than 2 kOe were sufficient to saturate the samples. The maximum single rotation is 1.3 ° in EuO and 0.8 ° in EuS. Considerably larger rotation angles have been reported for polar Kerr measurements.
180
W. REIM and J. SCHOENES 15
J
I
I
I
EuSe
A
10.
-
-
o
_
4,5K
-
,2 K
-
If",\
i
ii_,
i_.).,_;p_,,,
_
u'~-5 ,..2 "7 E
i
i
I
I
o v ~.
2
5
4
5
i
I
I
I
2
6
---9 K
EuTe i".
_
--15
u. 1
K
i H=ll,5kOe
i
o
__f! /
-t
~
l
i
i
~'~kq I'
-2
I
I
2
i "
VI 5
I
3 4 Photon Energy (eV)
6
Fig. 28. Faraday rotation of polycrystalline films of EuSe (top) and EuTe (bottom) at various temperatures and an applied field of 11.5 kOe (after Schoenes 1975). 3.0
2.0 ILl ILl n-" £.9 IJ.I
o
st
t
/
~l~EuO
1.0
4i
o
£
'
EuS
I J ~'X'I ~
~ i
~ l
on,.- -I.0 n~ n,, lull
v -2.0
-3.0
\
I
I
400
i
I
t
500
I
600
i
I
700
i
I
800
.,,'
i i i
900
I000
WAVELENGTH (m/.~) Fig. 29. Longitudinal double Kerr rotation of EuO and EuS single crystals in an applied field of 2.5 kOe and temperatures of 12 K (EuO) and 8 K (EuS) (after Greiner and Fan 1966).
MAGNETO-OPTICAL SPECTROSCOPY 8
I
i
i
/~°K
~6
L~
i
4
i
EuO
t] a}""--'~"~'K /i/ '~i
EKI'
6
H=40kOe T=|OK
I
I
o.
-8~
EuSe
A
4
H = 40 kOe
i] 1
2
,I
T1.~i = K A
y--",.-.,,"
-~0 E g-4
181
",./ V 1
2
5
4
5
I
2
3
4
5
Pholon Energy (eV)
Fig. 30. Polar Kerr rotation Ox and ellipticity eK of EuO (left) and EuSe (right) for [100]-cleaved single crystals at 10 K and 40 kOe (after Wang et al. 1986, Hiisser 1983, Schoenes and Kaldis 1987). Figure 30 shows results of such measurements extending also over large enough energy ranges to include both sets of structures for E u O (Wang et al. 1986) and EuSe (Hfisser 1983, Schoenes and Kaldis 1987). While a temperature of 10 K is sufficiently low to saturate the magnetization of E u O in a field of 40 kOe, this is not the case for EuSe. Figure 31 shows the effect of varying temperature on the polar Kerr rotation of EuSe (Hfisser 1983). Besides an increase of the maximum Kerr rotation up to +9.7 ° at 2.4 eV if the temperature is decreased to 2 K, one observes shifts and splittings of the structures. The most prominent shift occurs on the low-energy side, known as the magnetic red shift of the absorption edge (Wachter 1972). From fig. 31 one derives a shift of 0.19 eV for the lowest energy peak compared to a shift of 0.13 eV which has been derived for the absorption I0
r /~
5
/ /t~ ~ 4 - 0
$-5
T= 2K
T 1"-'EuSe kOe
tO
~0 ~8 hi
i~'l~ '
I
--- T='% l / -I0
1.75 2.0 2,25 2.5 Photon Energy (eV) Fig. 31. Polar Kerr rotation of EuSe for various temperatures and a field of 40 kOe (after H/isser 1983, Reim 1986, and Schoenes and Kaldis 1987).
182
W. REIM and J. SCHOENES
edge by re-emission measurements on powders (Wachter 1972). From either the Faraday rotation and the circular dichroism spectra (Schoenes 1975) or the Kerr rotation and ellipticity spectra (Schoenes 1987, Schoenes and Reim 1985), the off-diagonal conductivity spectra have been computed. Figure 32 shows the results of the latter for EuO and EuSe. One ascertains that the Faraday rotation of EuSe is very similar to -O-xxy, as expected for insulating materials. The similarities between 0K and O'2xy , o n one hand, and eK and Orlxy , o n the other hand, are less pronounced, due to the fact that the reflectivity is a function of both n and k (see section 2.1.2). The off-diagonal conductivity spectra have been used to make detailed assignments of electronic transitions to the various structures (Schoenes 1975, 1987). Figure 33 shows the atomic coupling schemes drawn to scale with the experimental data of fig. 32. Following a proposal of Kasuya (1972), the various interaction terms of the coupled 4f65d I final state are considered in order of decreasing magnitude. The largest term is the crystal-field splitting of the 5d states, which is found to be 3.7, 2.2, 1.9 and 1.7 eV in the series oxyde, sulfide, selenide and telluride. The next term is the exchange term between the excited 5d electron with SO-- i and the remaining 4f 6 state with Sgp = 3. This term amounts to 0.59 eV in EuO, and 0.43 eV in EuSe, the variation going parallel with the decreasing red shift of the absorption edge. The third term included in fig. 33 is the spin-orbit splitting between the total spin of the 4f65d 1 states and the angular momentum of the 4f 6 state. The resulting J terms have been drawn to obey the rule that two consecutive levels, J and J + 1, are separated by ifJ, where ~ is a spin-orbit interaction constant. The vertical arrows indicate the allowed transitions and the energies given are those of the peaks of O'2xy in fig. 32. For electric dipole-allowed transitions, the selection rules are AS = 0 and AJ = 0, - 1 . For spin-flip transitions, the selection rules are AS = - 1 and AJ = 0. The latter transitions become partly allowed, due to the admixture of states with ST = 7. The fit gives 55 meV in EuO and 46 meV in EuSe for the spin-orbit coupling constant ff (Schoenes 1987).
4f
2I-
0
~
o
~', I-
~t
~:)-1
-2~
-2 i! 1/
H= 40kOe T= 10 K
I! il
-ff
--'"
.:Xo~o~
l
,
I ~ il,
17, !
E,,Se
i"
,
/~,,
, h ,,,,
!i II ,t,"i
',J
"
/ q' "1
I/1ti '
'"J
.... V!i
ii
T: 1OK
',I ~,
-5 I
2
5
I
4
Pholon Energy
2
3
4
5
(eV)
Fig. 32. Real part O'lxyand imaginary part o'2xyof the off-diagonal conductivity of EuO (left) and EuSe (right), as computed from the data in fig. 30 and the optical constants (after Wang et al. 1986, and Schoenes and Reim 1985).
MAGNETO-OPTICAL SPECTROSCOPY 10Dq
183
ST=Ssd+S4f6d=S'T+I~4f6 (Fa} .
.
.
.
11/2
.
9/2
~
ST=Ssd+S4f6
J=ST+L4f6
(Fj)
9/2
7/2 5/2 5/2 I/2
5d "~
IIII
5d ~ 4f 6
10 Dq
5/2 1112 312 I12
. . . . ~(~--------T
/--
ls/~
5d 1 4f 6
1/2:
\ J .....
9/2 7/2
~ ST: 5
TI/2
S :
.
.
.
.
~,~
,,=
11/~ 5/z
------~3/2 9/2
~
---
-.,~
~;~
I/2
9/2 I/2 7/2
EuSe 1/2
4f 7
(8S7/2)
4 f ? (8S7/2)
Fig. 33. Atomic coupling schemes for EuO and EuSe and observed 4fv---~4f65d transitions and energies (after Wang et al. 1986, and Schoenes and Reim 1985).
From the earlier Faraday rotation measurements on films (Schoenes 1975), ff is derived to be 51 meV in EuS, 4 1 meV in EuSe and 32meV in EuTe. The single-ion value in t h e 4 f 6 ( 7 F ] ) configuration of Eu 3+ diluted in LaC13 is ~'io, = 30 meV. (Dieke and Crosswhite 1963). Thus, the europium chalcogenides constitute a quite particular class of materials. The initial state has no spin-orbit splitting and is unfavorable for obtaining large magneto-optical signals. On the other hand, it is 100% spin polarized with the largest possible pure spin moment. The necessary spin-orbit splitting is contributed by the excited state, making these compounds nice examples to show that magneto-optics is also an excited state spectroscopy. Another related final-state effect has been reported for the antiferromagnetic compounds EuSe and EuTe. Figure 34 shows the Faraday rotation of EuTe for various fields and temperatures in the energy range of the 4f7--->4f65d t2g transition (Schoenes 1979). For the lower field of 20kOe one observes with decreasing temperature the growth of a-peak P3 at 2.75 eV. This growth is opposite to the decrease of magnetization of EuTe for T < TN = 9.6 K and for H smaller than the critical field for spin alignment Her = 70 kOe. The same peak P3 disappears again if the field exceeds the critical field, as can be seen from the spectra at 100kOe. Hence, the size of this peak is intimately related to the antiferromagnetic order, and it has been shown that for T < TN the magnetic circular dichroism spectrum can be decomposed into two similar shaped parts,
184
W. R E I M and J. S C H O E N E S
2o
2.6 5
- :\
EuTe 9K~.~
b) H = IOOkOe
I0
o
L~ o
.......~.J// ". 7\ ~ "
-IC
P,
Pz
a) H=2OkOe
\
P~
/
K
&
-2
P~
2Ill
2J2 213
2.4
2!5
2!6
2!7
2:8
Photon Energy [eVl Fig. 34. Faraday rotation of a [100]-textured film of EuTe for fields of 20 and 100 kOe and various temperatures above and below T N = 9.6 K (after Schoenes 1979).
indicating the splitting of the 5d t2g sub-band into two further sub-bands. The latter splitting is attributed to the formation of the antiferromagnetic superlattice. Since 2.75 eV is also the energy of the structure which has been assigned to a spin flip transition in the atomic model, it is interesting to search for a relation between the atomic and band models. Schoenes (1975) has pointed out that the superlattice splitting of the conduction band corresponds to the exchange splitting between the two possible spin configurations of the final 4f65d state. In contrast to the exchange splitting in a ferromagnet, the superlattice splitting in the antiferromagnet occurs only at the superlattice zone-boundaries and both sub-bands contain spin-up and spin-down states. A last result from the Faraday rotation measurements of EuX compounds which we would like to mention here is the decomposition of o-2xy(~O) into single components (Schoenes 1975). Figure 8 gives a scheme of the expected single components in a band-like model. For an f t --~ d t transition from a completely spin-polarized f-state, a o~2xy(W) spectrum in the form of a lying S is expected (fig. 8b). For p--~ d transitions from unpolarized p-states into exchange split d-states a double-peak structure as shown in fig. 8d is expected. Figure 35 shows a decomposition of the experimentally determined o-2xy(~o) spectrum of EuS into two f--~ d like and two p--~ d like components. The integration of the positive part of a single component over the photon energies is half of the weight (o-2xy> as defined in eq. (2.57). The obtained experimental values have been compared with a computation using eq. (2.58) and atomic f and d wave functions. Good
MAGNETO-OPTICAL SPECTROSCOPY
185
1 1.80x I029sec-2
I 2.2 ~ 1029sec'2 ~
,~ 0.65~i029sec-a
0.9 x I0 29 sec-2
.~-./~///A ~
o2, Io",ee'
o
C I -4
3
4
5
eV
6
Photon Energy (eV) Fig. 35. Decomposition of the absorptive part tr2~yof the off-diagonal conductivity of an EuS film in components as expected for f-->d and p-->d transitions (see fig. 8). The numbers are the integral values of the positive parts of -~r:xe.
agreement is obtained for the f--> d t2g and the f--->d eg transition, corroborating also from a quantitative point of view the assignment of the dominant magnetooptical signal near 2 and 4.2eV to transitions of the f electrons into the crystal-field split d t2g and d eg sub-bands. Besides these classical magneto-optical measurements, various other magnetooptical effects have been studied in pure EuX-compounds. We mention here magneto-reflectance (Feinleib et al. 1969, Pidgeon et al. 1969, Scouler et al. 1969, G/intherodt 1974) thermo-reflectance (Mitani and Koda 1973), electro-reflectance (L6fgren et al. 1974) and magneto-absorption (Busch et al. 1964, Wachter 1972, Freiser et al. 1968, Schoenes 1975, 1979, Schoenes and Nolting 1978, Mitani et al. 1975, Llinares et al. 1973, Ferr6 1974). All these experiments can be interpreted within the same model of the electronic structure of the europium chalcogenides, but they often give less complete information. So we abstain from presenting them here in more detail, except for a beautiful example of magneto-reflectance. Figure 36 shows the data obtained by Scouler et al. (1969) for single crystals of EuS and EuSe. If we take the difference of the reflectance spectra at 1.5 K for right- and left-hand circularly polarized light of EuSe we obtain a spectrum which is quite similar to the Kerr-ellipticity spectrum at 2 K and a same field of 40 kOe (fig. 30). However, one notes also that the shoulder at 2.3 eV in the Kerr spectrum is not reproduced and that at energies around 4 eV the resolution of the differential Kerr effect is much higher. The last measurements which we would like to mention, although only shortly, because they are single-wavelength data, are Faraday rotation measurements to study critical exponents near the Curie temperature of EuO (Huang et al. 1974, Huang and Ho 1975) and EuS (Berkner 1975).
186
W. REIM and J. SCHOENES '
' I EuS 1.5*K H, 4 0 kOe
1.4
I
'
I
'
E2
(-~ 1.2
1.0' A
.= 0.8 ,
~'0.6 .Q
~
CrL
/
O"R
0.4
bJ U Z I~ 1.4 uJ
,
~'x
~--~
,
I
I
]
I
I
I
'
I
'
I
i
I
i
EuSe
1.5° K
H = 40 kOe
EI
1.2
E2
1.0 0.8 0.6
L
0.4 1.0
I
I
2.0
I
i
3.0
I 4D
, 5.0
PHOTON ENERGY (eV)
Fig. 36. Reflectance spectra on cleaved single crystals of EuS (top) and EuSe (bottom) for right- and left-hand circularly polarized light. The field is 40 kOe and the temperature is 1.5 K (after Scouler et al. 1969). Right- and left-hand circularly polarized light is interchanged compared to our definition.
5.2. Doped NaCl-type compounds The very large magneto-optical effects of the EuX-compounds (GdX compounds have not been studied so far and should have smaller effects due to their antiferromagnetic order and the large f---~d transition energies) has lead to considerable efforts to try to increase the rather low ordering temperatures to above room temperature. Unfortunately, these efforts were not successful to that measure, but interesting findings regarding the dependence of the magnetic exchange and the magneto-optical signal on the dopants have been made. Since europium is divalent in the EuX-compounds, the substitution of europium by trivalent rare earths will create "free" 5d conduction electrons which can enhance the ferromagnetic coupling via an indirect exchange mechanism of the Ruderman-Kittel type. This possibility has been realized as early as 1964 (Holtzberg et al. 1964) and, at the beginning, has prompted mostly magnetic and transport studies (von Molnar and Methfessel 1967, von Molnar and Shafer 1970, Methfessel and Mattis 1968). The trivalent rare earth commonly used is Gd 3÷, due to its 4fT-configuration being identical to Eu e÷. Tc of up to 135 K have been reported for EuO doped with 3.4 at.% Gd 3+ (Sharer and McGuire 1968). Yet, it has been realized that the local 4f spin of the dopant contributes little to an
MAGNETO-OPTICAL SPECTROSCOPY
187
enhancement of the ordering temperature, but on the contrary, may couple antiparallel to the Eu 2+ spin (McGuire and Holtzberg 1971). Therefore, efforts have been made to increase the free-carrier concentration by replacing the divalent anion by a monovalent ion like I + (Vitins and Wachter 1973). A third class of dopants and composite layers are transitions metals and particularly iron (Lee and Suits 1971, Almasi and Ahn 1970). The effect of Gd 3+ doping on the magneto-optical properties of EuO single crystals has been investigated by Kaldis et al. (1971) and Schoenes and Wachter (1974). A remarkable result of their absorption measurements was the discovery that the magnetic red shift of the absorption edge is substantially reduced upon doping. As fig. 37 shows, the total red shift of pure EuO amounts to AE = 0.26 eV and it follows a spin correlation function (dashed curve). For a free-carrier concentration at room temperature of 6 × 1020 cm -3 the red shift has decreased to AE = 0.12 eV, although the ordering temperature has increased from 69 to 115 K. Various interpretations have been put forward to explain this effect. Schoenes and Wachter (1974) have argued on empirical grounds that the simultaneous increase of T c and decrease of AE indicates a sub-linear decrease with increasing free-carrier concentration of the exchange Hex = - 2 ~'n Jn( r - R n ) s " S . where s is the spin of a conduction electron, S~ are the spins of neighboring Eu 2÷ ions, and J . ( r - R ~ ) is the distance-dependent exchange constant between electron and ion 0.9
~" "~'~• --°~o~o~L
I S0mpe
~ \ >~1.0 - - ' ~ t - ' ~
--'---•--
Nfree c0rrier [i0~9cr~3J
0 2
< 0.1 6.6
g_ 1.1
..... -
/am(T)d-r'\ " ~ . . ~ . , ° ~ o ~ - ~ - - - Y-
Urn(O)[~m(T)dT%,~I.~,=.~,___i. -
_
~
1.2 -- am= mogn.pertofthe exponsioncoeffieie~ 0 50
'
100
~
•
~
_
',,
_
150 200 Temperoture (K)
Fig. 37. Temperature dependence of the absorption edge of pure EuO and EuO doped with various amounts of gadolinium. The given carrier concentration N~r~ carrier is the conduction-electron concentration determined at room temperature by a combination of optical and magneto-optical data. The photon-energy values given here are for a constant absorption coefficient (~0), which otherwise may vary from sample to sample. The dashed line shows a fit of the shift of the absorption edge with the spin-correlation function Um (after Schoenes and Wachter 1974).
188
w. REIM and J. SCHOENES
spins. Nolting and O16s (1981), using a spectral density approach to compute the temperature dependence of the quasi-particle density of states for the conduction band, argued that the red shift depends on the band width and that the temperature-dependent filling of the conduction band causes the red shift to decrease. Schoenes and Wachter (1974) have also used Faraday rotation, absorption and reflectivity measurements to derive the three transport parameters of the free carriers, i.e., their concentration, mobility and effective mass, as a function of temperature. With this electrode-free method they could show that, for not too large doping, the activation energy of the Gd donors decreases on magnetic ordering while the mobility increases. Similar measurements have also been performed for EuTe doped with iodine (Vitins and Wachter 1973). The doping is found to introduce ferromagnetic clusters which give rise to a large wavelength-independent Faraday rotation. Figure 38 shows the field dependence of this wavelength-independent rotation and
I 3000
--~.
y
EUTeoI: T=8°K
]
2000
g o
o I000
06
o.u.
0
I
2
4
6
'.
4
,;
8
Applied Field
H
I0 I>-
WOe
/~
Applied
Field H
I
i:>
,e
Fig. 38. (top) Field dependence of the wavelength-independent Faraday rotation for two EuTe crystals doped with iodine at 8 K. (bottom) Field dependence of the magnetization at the same temperature of 8 K for the same doped samples and for pure EuTc (after Vitins and Wachter 1973).
MAGNETO-OPTICAL SPECTROSCOPY
189
compares it with the magnetization of the same two doped samples and a pure EuTe single crystal. One recognizes the rapid saturation of the frequencyindependent rotation, while the magnetization should only saturate above 70 kOe. An analogous violation of the proportionality to the magnetization is also observed for the temperature dependence of the wavelength-independent Faraday rotation. The authors suggest that the wavelength-independent Faraday rotation is due to magnetic resonance at lower frequency. The effect of doping with iron has been studied on Euo films by Ahn (1970) and McGuire et al. (1971) and on EuS films by Ahn and Sharer (1971). The Curie temperature was found to increase to 200 K for EuO with 8% Fe by weight and to 45 K for EuS with 16% Fe by weight. The magnetization decreases somewhat more than expected from the dilution of the Eu 2+ lattice. The f---~d t2g transition energy is shifted rather little as compared to pure films, while the absorption coefficient is enhanced by a broad background, originating from iron. Figure 39 displays the Faraday rotation measurements of McGuire et al. (1971) which were performed at 6 K in a magnetic field of approximately 20 kOe. The maximum rotation value is depressed from 8 × 105 deg/cm for pure EuO to 6 × 105 deg/cm for the iron-doped sample. The magneto-optical hysteresis loops of Fe-doped EuO are reported to be square-like for temperatures extending almost to the Curie points (Ahn 1970). The physical origin of the important increase of T c on doping with Fe is not completely clear, because it considerably exceeds the effect of adding free electrons. Suggestions are that either the presence of Fe ° around grain boundaries causes a kind of exchange coupling, as has been observed between films of permalloy and EuO (Almasi and Ahn 1970), or that some of the Fe 2+ ions present form (EUO)l_x(FeO)x, where FeO has a N~el temperature of 190 K. The chemical stability of EuO films doped with iron and europium has been investigated by Street (1972). The effect of Eu, Ag and Cu on the ferromagnetic exchange in EuO films has also been studied by Lee and Suits (1971). For the latter two nonmagnetic dopants, they find Curie temperatures of approximately
tO X 10 5
i
,6K,
-5'
4,000
6,ooo
I
i
~ooo
i~ooo
WAVELENGTH
Fig. 39. Faraday rotation for pure E u O a n d doped E u O f i l m a t 6 K a n d 20 kOe (a~erMcGuire et al. 1971).
190
W. R E I M and J. S C H O E N E S
150 K. Yet, the Faraday rotation as function of temperature increases only slowly down to approximately 80 K and accelerates its growth below this temperature. The last type of composite films containing Eu-chalcogenides which we would like to mention are those with enhanced Kerr effect. Ahn (1968) and Suits and Lee (1971) have reported data for EuO films deposited on mirror substrates and find a large increase of the Kerr rotation by closely matching the antireflection condition. Thus, Suits and Lee (1971) find a polar Kerr rotation of 70 ° for an EuO film deposited on a silver film (fig. 40). Yet, the reflectivity at the wavelength of maximum Kerr rotation is nearly zero, reducing again the effective figure of merit for the composite film. Another approach has also been taken by Ahn (1969) who optimized the thickness of the protective Eu20 3 layer on top of the magneto-optically active EuO film so as to enhance the apparent Kerr rotation. Of course, again the reflectivity of the composite film is depressed. Q)
6O OK
~6 "~ 40
/
• Reflectivity
~ 2O
\\ .
//----\/
,
~,. i I I
2O
o O_
-~o
-a •" ,
E o
0.6
/
,
",.
,
~ ""?-"i
0.8 1.0 1.2 WQvelength (/zrn)
o
,.
OS
I4
Fig. 40. Polar Kerr rotation, polar Kerr ellipticity, and reflectivity for an EuO film on top of a silver film on glass. The field is 20 kOe and the temperature is 5 K (after Suits and Lee 1971).
5.3. Miscellaneous europium compounds The large rotatory power of E u 2+ diluted in C a F 2 (Shen and Bloembergen 1964) has led many investigators to search for europium-containing compounds with specific properties. Although not all these compounds do show magnetic order in the temperature range studied, some of them are strongly paramagnetic and the moments might be paramagnetically aligned in a large enough magnetic field. There may then result magneto-optical signals per europium ion which are comparable to those of the europium chalcogenides and we will include a discussion of these materials in the following. The room-temperature Faraday rotation of EuF 2 has been determined by Suits et al. (1966). EuF 2 crystallizes in the cubic C a F 2 s t r u c t u r e and appears to be paramagnetic down to 0 K (Lee et al. 1965). The use of single crystals did limit the spectral range of the Faraday rotation measurement to the transparent region of E u F 2. This limit is given by the onset of 4f--->5d eg transitions. Figure 41 shows the absorption and the Faraday rotation of EuF 2 close to this absorption edge (Suits et al. 1966). From a plot of the reciprocal Verdet constant versus A2, the
MAGNETO-OPTICAL SPECTROSCOPY
-6
191
3O
"r=
tll 20
,'1"
8 _g
¢9
o]
400
5;0
xc.~l
~o
700
o
Fig. 41. Room-temperature Verdet constant (solid line) and absorption (dotted line) of a EuFz single crystal. The absorption is corrected for reflection losses (after Suits et al. 1966).
effective wavelength of the atomic transition causing the rotation and a wavelength-independent proportionality factor are obtained. Both values confirm that the observed Faraday rotation is due to an 4f7--~4f65d eg transition. Similar investigations have also been performed for glasses containing large amounts of Eu 2+ (Suits et al. 1966, Schoenes et al. 1979b). While the former measurements were restricted to room temperature, Schoenes et al. performed measurements down to 1.5 K in fields up to 100 kOe. Under these conditions paramagnetic saturation is reached and rotations of nearly 6000 deg/cm are reached at 0.8 p~m for a sample containing 12 at.% Eu 2+ (fig. 42). At room temperature, . the same sample at 0.8 txm has a Verdet constant of 3.5 x 10 -3 deg cm - 1 0 e -1 (Schoenes et al. 1979b). If we normalize this value to 33 at.% to simulate the Eu 2+ concentration in EuF2, we obtain V-- 0.58 min cm -I Oe -1, which is in very nice agreement with the data shown in fig. 41 for EuF 2. Besides amorphous silicate glasses, Eu 2+ forms the crystalline silicates EuSiO3, Eu2SiO 4 and Eu3SiO 5 (Sharer et al. 1963). The latter two order ferromagnetically near 9 K (Kaldis et al. 1974). Kaldis et al. reported for Eu3SiO 5 a magnetic red shift of 20 meV. Faraday rotation measurements appear to have only been performed for two wavelengths in Eu2SiO 4 (Shafer et al. 1963). At room temperature, the Verdet constant is similar to that in EuF 2. Europium has also been incorporated in various ferric oxide compounds (Kahn et al. 1969). In their comprehensive study, Kahn et al. (1969) show that the effect of europium and many other rare earths on the magneto-optical properties of these materials are only minor at room temperature. For europium, the main reason is that in orthoferrites and garnets europium should be trivalent. The negligible effects of some other rare earths can be understood by their small sublattice magnetization at room temperature. It has been demonstrated, that Eu enters divalent also in amorphous E u - F e C o alloys (Weller and Reim 1989a,b). However, no major contribution to the Kerr effect has been detected at room temperature, which is understood by the large optical constants of the intermetallic alloy and by weak exchange interaction.
192
W. REIM and J. SCHOENES
r~
"o t.a
._g
-o E
y_
o~
20
40 Magnetic
60
80
IO0
Field r'kOe]
2+ 3+ • Fig. 42. Field dependence of the Faraday rotation at 0.8 p~m of the glass Eu0.i2Eu0.02S1o.3100.55 for various temperatures between 1.5 and 300 K (after Schoenes et al. 1979b).
5.4. Gadolinium Regarding its magneto-optical properties, gadolinium is undoubtedly the best investigated rare earth metal. Gadolinium orders ferromagnetically at 293 K. As early as 1963, Lambeck et al. (1963) reported Faraday rotation measurements on vapour-deposited thin films of Gd. They found a negative rotation increasing by a factor of approximately 2 between A = 4000 and 6500 ~ . A t magnetic saturation, which Lambeck et al. reached by applying fields up to 30 kOe to samples cooled down to 93 K, the authors observed a rotation of - 3 . 2 5 × 105 deg/cm at A = 5890 A. Kerr effect measurements on thin films of Gd were first reported by Erskine and Stern (1973), who also presented a thorough analysis of the data. Although these authors used a set-up in which the magnetization direction is in the plane of the sample (longitudinal Kerr effect), Erksine and Stern separated that part of the Kerr effect, which is a linear function of the magnetization, by using a 30 ° angle of incidence and reversing the field. The obtained spectra are displayed in fig. 43. We note the small values compared with the spectra of the europium monochalcogenides. This indicates that the structures are hardly due to f---~d transitions and, in fact, Erskine and Stern assign the two prominent peaks near 2 and 4 eV to p--~ d transitions, the double structure being a consequence of a covalency splitting of the 5d band into two major sub-bands. To detect f---~d
MAGNETO-OPTICAL SPECTROSCOPY I0
I
I
i
I
193
I0 TATION
7 ,q
~5 I-
~ 4
q, Y
o,
T
T
I
T
~'
T
3
4
ti
5
ENERGY (eV)
Fig. 43. Magneto-optical Kerr rotation and ellipticity of polycrystalline Gd films at 105 K and a field of 2 kOe (after Erskine and Stern 1973).
transitions in Gd, the spectral range of the measurements has to be extended beyond the quartz transmission limit of 6 eV. Erskine (1976; 1977) performed such pioneering work at the Synchrotron Radiation Center at Stoughton, Wisconsin. Figure 44 shows the resulting too-2ry(W) spectrum including, as dots, the data from the former measurements in the quartz transmission range. Also shown is the absorptive part of the diagonal conductivity, tr~x. One recognizes that the two peaks at 2 and 4 eV, which we have already discussed before, lie in an energy 5.0
I
4.0
~
,~ 3.0
!
~_o z.o 'B
1.0
I
I
I
t
3.0~
.
_(2
-~' -'~U
2.5~ mm~
~L~ -2.C
/&5
(2] WCrxy
"
(i) / ~ ' - * " - ' " " " °-xx
"~
,~
~
~
,~
~-.0~ c* vff [.5
,,.o
ENERGY (eV)
Fig. 44. The product of the absorptive part of the off-diagonal conductivity tr2~yand the frequency to for polycrystalline Gd films. The dashed horizontal line is an estimate of the free-electron contribution. At the bottom the absorptive part of the diagonal conductivity tr~ is shown (after Erskine 1976, 1977).
194
W. REIM and J. SCHOENES
range which is dominated by free electrons. These free electrons give for o) >> 7 a o9-1 contribution to o-2xy. In a o)O-2xyplot, this flee-electron contribution is energy independent. The broken horizontal line in fig. 44 is the flee-electron contribution in Gd, as estimated by Erskine and Stern (1973). Near 6 and 8 eV, two additional and prominent structures appear, which are assigned to f--~ d transitions. Arguments regarding the sign and the weight of the peaks are presented favoring such an assignment (Erskine 1976). Indeed, a comparison of the line shape and the sign of the peaks with those found in the europium chalcogenides for f---~d transitions shows strong similarities. The more subtle question is why there are also two peaks for the f--~ d transitions in Gd. In the europium chalcogenides the two peaks reflected the crystal-field splitting of the final 5d state. Erskine and Stern (1973) have used a similar final-state splitting to explain the two peaks at 2 and 4 eV in terms of p---~d transitions. Yet, for the two peaks near 6 and 8 eV, Erskine (1976, 1977) discards such a final state splitting in favor of a model assuming two different screening mechanisms for the hole created in the optical excitation process. Erskine compares the energies of the two peaks with XPS data (Hedrn et al. 1972, Lang et al. 1981) where one finds 4f emission approximately 8 eV below the Fermi energy. It is then tempting to assign the 8 eV peaks in both spectroscopies to a same type of partial screening of the core hole, while the 6 eV peak in magneto-optics is ascribed to a fully screened 4f hole. As can be seen, the problem is related to what we have already discussed for cerium pnictides (section 4.1). Also, these different screening mechanisms were assumed to explain a peak separation of +2 eV. Yet, contrary to the cerium pnictides in which magnetooptics displayed one peak and photoemission two peaks, in Gd we encounter one peak in photoemission and two peaks in magneto-optics. On the other hand, we know that for the cerium chalcogenides, in which the 4f binding energy is larger than in the cerium pnictides, both magneto-optics and photoemission show only one peak. Thus, we favor an interpretation of the two peaks near 6 and 8 eV in o-2xy(W) of Gd in terms of a 5d band splitting, the difference in threshold energies for the optical and photoemission process being due to complete and only partial screening, respectively.
5.5. Intermetallic gadolinium compounds The magneto-optical properties of rare earth 3d-transition element alloys are presented in great detail in the chapter by Buschow (chapter 5 of vol. 4 of this handbook). Nevertheless, we would like to mention here, in connection with our discussion of Gd, Kerr rotation measurements for various Gd-containing intermetallic compounds. Figure 45 shows the double Kerr rotation at room temperature for bulk polycrystalline samples of Co, Gd2Co17, GdCos, GdCo 3 and GdCo 2 (Buschow a n d van Engen 1984). From the magnetic properties of Gd-Co, it follows that the moments of the Gd and Co sub-lattices couple antiparallel. For Gd2Co17 and GdCos, this results in a dominance of the Co-sublattice magnetization and for GdCo 3 and GdCo 2 in a dominance of the Gd-sublattice magnetization. To facilitate a comparison, Buschow and van Engen (1984) have, therefore,
MAGNETO-OPTICAL SPECTROSCOPY
195
-0.40 -0.20 f -0"10 t
o.o;l
GdCo3
.~ 0.o~,
~,~ 0.06
i
i
i
i
i
i
r
i
I - 0.20 -0.15
-0.25 -0.26 -0.46 -0.6£ i
i
i
i
-0.40 -0. 60
-0.80
~'0
q
2'0 '
i
~0
i
i
40
i
5.0
energy [eV)
Fig. 45. Double polar Kerr rotation of various polycrystaUine bulk Gd-Co compounds at room temperature and a field of 11.5 kOe. Note the reversing of the positive axis for GdCo2 and GdCo3. The vertical bars correspond to the peak position in Gd (after Buschow and van Engen 1984). reversed the direction of positive Kerr rotation for GdCo 3 and G d C o 2. The two vertical bars indicate the energy of the Kerr rotation peaks observed by Erskine and Stern (1973) in Gd (fig. 43). The rather good agreement of the structures in GdCo 2 with these two energies suggests an interpretation of these two peaks as arising from gadolinium. Yet, one should realize that Kerr rotation spectra of sublattices are, in general, not directly additive, this being contrary to the absorptive parts of the off-diagonal conductivity. Also, because the two structures near 2 and 4 eV in Gd are assigned to transitions between p- and d-band states, band-structures effects will play an important role in contrast, e.g., to the N d - C o system (section 4.3), in which the Nd derived structure at 4 eV is related to a transition of localized atomic-like 4f-states. The photon-energy dependence of the Kerr rotation and ellipticity of amorphous G d - C o and G d - F e films has been studied by Sato and Togami (1983). In their measurements through the glass substrate, they find maximum rotations near 0.9 eV of 0.4 ° in G d - C o and 0.58 ° in G d - F e . From a qualitative comparison with exy(OJ) of Co, Fe and Gd, Sato and Togami (1983) conclude that the magnetooptical effect of amorphous Gd-transition metal films in the visible and near infrared region essentially stems from the magneto-optical polarizability of the transition metal atom. Hansen and Urner-Wille (1979) have investigated the
196
W. REIM and J. SCHOENES
effect of alloying Bi to Gd-Fe. For the wavelength of the H e - N e laser (i.e. 633 nm), they found an increase of the Kerr rotation from 0.29 ° for a Gd0.26Feo.74 sample to 0.41 ° for a sample of (Gd0.26Fe0.74)0.89Bio.11 . Van Engelen and Buschow (1987) have investigated Gd/Fe14B and Gd2Co14B. Maximum Kerr rotations of -0.34 ° and -0.29 ° are reported for the Fe and Co compound, respectively. The Kerr rotation spectra of the Gd compounds are found to be quite similar to the corresponding La compounds, while for Nd2Fe14B a net increase of the Kerr rgtation is observed near 4 eV. 6. Heavy rare earths
Except for thulium chalcogenides, rather few magneto-optical spectra have been collected for the rare earth metals beyond gadolinium and their simple compounds. This is rather unfortunate, since alloys of these heavy rare earths with 3d-transition elements have received much attention and the knowledge from the simple systems could provide important hints for the interpretation of these data. As before, we will start with the discussion of the "simple" NaCl-type compounds, to move then to the pure metals, which are already complicated by the presence of free electrons and strong band-structure effects, and to end with results for a few selected miscellaneous compounds.
6.1. NaCl-type compounds The thulium monochalcogenides have raised much interest in the past due to the occurrence of intermediate valence in some of them. That is: TmS is an integral-valent metal with Tm 3÷, TmTe is an integral-valent semiconductor with 2+ Tm , and TmSe is an intermediate-valent metal with a valence between two and three. In TmSe the energies of the 4f 13 and 4f125d1 states are degenerate, leading to a ground state which is a linear combination of these two states. The unique feature of thulium compounds compared with intermediate-valent materials containing cerium, samarium or europium is that both ground-state configurations of thulium have a non-zero magnetic moment. This leads to magnetic ordering in the thulium monochalcogenides at low temperatures and promises that magnetooptics can probe these two groundstates. The Kerr rotation and ellipticity of pseudobinary TmSe0.32Te0.68 and TmSe is shown in fig. 46 (Schoenes et al. 1985). TmSe0.32Te0.68 has a lattice constant of 6.204 A and is semiconducting while TmSe is an intermediate-valent metal with a lattice constant that is 0.5 .~ smaller. To elucidate the physical property causing the very different magneto-optical spectra of these two materials, fig. 46 also exhibits the Kerr rotation and eUipticity for two more thulium monochalcogenides. We recognize that integral-valent metallic TmS (Reim et al. 1984a) and intermediate-valent TmSe0.87Te0.13 (H/isser 1983) have very similar spectra to intermediate-valent TmSe. It has therefore been concluded (Reim et al. 1984a, Schoenes et al. 1985) that the differentiating feature in these compounds is whether the material is semiconducting or metallic and not whether it is integral
MAGNETO-OPTICAL SPECTROSCOPY
0.5
b)
2t
197
'\
TmS~
-OK. 11"
0.25
"
\
0
-0.25
.
B =4T , =
-0.5
~o -£K
E
8
V
-0,75 I
2
4
3
5
Photon Energy (eV)
c)
d)
+'+ +;
-B K
:\
TmS
++ ÷ i
°"
v "5 a_
T1 2K B=4T
,,"~÷ t +o TmSe87Te13 o
° ~ °~++ -E K * *
'l
~o -E K
T=IOK B=4T
o
;
i 1
i 2
'i~l 3
h 4
i 5
Photon Energy (eV) Fig. 46. Polar magneto-optical Kerr rotation Or: and ellipticity e K of cleaved single crystals of TmSe0.32Te0.68(a), TmSe (b), TmSe0.87Te0.i3 (c) and TInS (d) (after Schoenes et al. 1985, Reim et al.
1984b, Hiisser 1983).
or intermediate-valent, nor whether it is a pseudobinary or a true binary compound. Figure 47 shows the real part of the diagonal conductivity and the complex off-diagonal conductivity of TmSe0.3zTe0.68 as a function of photon energy. If one compares the structures of o'2xy with those of crlxx one finds an agreement in their respective energies, if the peak at 0.5 eV is of paramagnetic line shape and the peaks in O-ixx at 1.7 and 2.5eV give rise to diamagnetic line shapes, i.e., correspond to extrema in o-lxy. Figure 48 displays the energy-level scheme for the 4f13---~ 4f~25d transition, which has been proposed by Schoenes et al. (1985) to account for the magneto-optical data of TmSe0.3zTe0.68. In contrast to the europium chalcogenides, the spin-orbit splitting of the final states (3H and 3F) of thulium is comparable in size to the electrostatic repulsion in the f shell. Thus, a clear hierarchy of the various interaction terms is not present and the 3H and SF terms strongly mix. The last column couples the angular momentum of the fa2 state with J = 3 for the 5d electron. Dipole-allowed transitions require Aj = 0 or - 1 . At low temperatures, one expects for J increasing by one a positive peak in O"2xy and for J decreasing by one a negative peak. These transitions and their
198
W. REIM and J. SCHOENES
TmSe32Te68/
o
to
O
d
O
,'' ,"f' ,o Q O I
2
3
4
TmS%2T%8
O-2*y
I I I
5
B =4T T =,0K
I
b) I
2
4
5
Photon Energy (eV)
Fig. 47. (a) Real part of the diagonal conductivity Glx. of TmSe0.3=Te0.58 and TmSe. (b) Real part Glxy and imaginary part cr2xyof the off-diagonaJ conductivity of TmSe0.32Te0.68 at 10 K and a field of 40 kOe (after Schoenes et al. 1985).
eg4f 12
5/2 7/2 9/2 11/2 7/2
1Dip
~Fs/~
5/2 5/2 1/2 9/2 7/2 5 / 2 9/2 5 / 2
7/2
11/2 9/2
11/2 5/2 15/2 11/2
7/2 7/2 5/2
9/2
1512 11/2 15/2
912
4f13 (
2F7/ 2
+
)
++
i
,n,.q ~ ~t..Qo O- ~ eJcJ c, b+ o
Fig. 48. Atomic coupling scheme for TmSeo.32Te0.68 and observed 4f13--o4f125d transitions and energies (after Schoenes et al. 1985).
MAGNETO-OPTICAL SPECTROSCOPY
199
expected signs and energies are also given in the last column. Transitions with •J = 0 have similar transition probabilities for right and left circularly polarized light and are omitted for clarity. The agreement with experiment is satisfying and indicates once more that the coupling of the excited 5d electron to the 4f"-1 state can not be neglected in the optical absorption process in rare earth chalcogenides and pnictides. Figure 49 shows the real (open circles) and the imaginary (solid circles) part of the off-diagonal conductivity for TInS (Reim et al. 1984a) and TmSe (Schoenes 1987). The large peaks in OK and ei~ at 3 and 2.5eV in TInS and TmSe, respectively, have transformed to small structures at the same energies in OSxyand o-2xy. On the other hand, the off-diagonal conductivity shows very large values at low energies with a marked extremum in o-2xynear 0.5 eV. Reim et al. (1984a) have given quantitative evidence that this behavior, observed for the first time in the metallic thulium monochalcogenides, is associated with the free carriers. In particular, they could show that the structures in OK and e K correspond to a splitting of the reflectivity curves near the plasma edge for right and left circularly polarized light of the order of 30 to 60 meV compared to the 0.4 meV that one expects from the cyclotron resonance frequency at 4 T. This is a strong indication that the dominant term is not the direct action of the field on the free carriers, but that the local moments have to interact with the free carriers. Consequently, skew scattering has been inferred and the full lines in fig. 49 for TmS correspond to fits with the first term of eq. (2.65). Another interesting question is, of course, what has happened with the f13__~fla5d transitions in intermediate-valent TmSe? Unfortunately, the presently
b
8
it
TmS
~xy
T 24KT °11 I B:= ~o[I/
~'2xy
~
1
2
3
~
~
4
5
I
T=2K
B=4T
-30 1
2
3
4
5
Photon Energy [eV] Fig. 49. Real part ~rlxy and imaginary part Or2xy of the off-diagonal conductivity of TmS and TmSe at 2 K and a field of 40 kOe. The full lines are fits with a skew scattering model (after Reim et al. 1984a, Schoenes et al. 1985).
200
W. R E I M a n d J. S C H O E N E S
available data cannot give a definite answer. Figure 49 shows some structures in the 2-5 eV energy range which are not explained by the skew scattering model for free electrons• If we assume a valence of 2.5 for Tm in TmSe, as may be derived from the Curie constant (Boppart and Wachter 1984), and we normalize the data obtained for TmSe0.32Te0.68 (fig• 47) by the factor ½, one anticipates peaks in O'xy of the order of 1013 S-1. This corresponds exactly to the order of magnitude of the deviation of the measured spectra from the free-electron fit in the 2-5 eV energy range (fig. 49). However, similar structures are also present in TmS which has no occupied 4f 13 state. In order to possibly find an answer to the above question, H/isser (1983) and Reim and Wachter (1985) have also investigated Tm0 sEu0.sSe. This material is so close in lattice constant to the semiconductor-metal transition that the metallic state can be induced at the surface by polishing the crystal (Batlogg 1981). Thus, a cleaved single crystal will be semiconducting and a polished one will have a transformed intermediate-valent layer which is thick enough that the crystal, in an optical reflectivity measurement, behaves like a bulk metallic sample. Figure 50 illustrates this difference in a plot of the real part of the optical conductivity for a cleaved and a polished crystal. Figure 51 shows the complex polar Kerr effect for the same two samples. Near 2 and 4 eV we find in both samples the 4f7--~ 4f65d t2g and - e g transitions of europium (see for comparison fig. 30). Outside these energy ranges the two spectra are quite different. For the cleaved Tm0 sEu0.sSe crystal, one recognizes below 2 eV the same structures as for TmSe0 32Te0 68 (fig. • • • " " • 46), Le., 4f 1 3 ~ 4f 1 2 5d transmons. The spectrum below 2 eV of the pohshed Tm0.sEu0.sSe sample, on the other hand, resembles the spectra for TmS and TmSe (fig. 46) with a reversed sign and a shift to 1 eV. This part of the spectrum is 3
I
I
I
I
I
Tm.5Eu.sSe T = 293 K polished --cleaved u' )
"~'N
2
0
i
v x
l
b* 0/
J
0 0
. ,f--I /
I
I
I
J
1
2
5
4
5
Photon Energy (eV) Fig. 50. R e a l p a r t of the optical conductivity Olx x at r o o m t e m p e r a t u r e of a cleaved a n d a polished Tm0.sEu0.sSe single crystal (after H/isser 1983).
MAGNETO-OPTICAL SPECTROSCOPY
2
Tm.sEUsSe
°° b," +t o
:o:o
t8
~. LU i
o
T=IOK,
:°°
o
o
.
~
Tm.5EUsSe
+
cleaved
* ~
polished
+o~
T=IOK, B=4T
+ ~÷
B=4T
201
~ o
?
~ Y :
-6 O_
Eo -I
~o
+~*-o
0
i
s
og
4
; Photon
Energy
(eV)
Fig. 51. Polar magneto-optical Kerr rotation OK and ellipticity e~ of a cleaved (left) and a polished (right) Tm0.sEu0,sSe single crystal at 10 K and a field of 40 kOe (after Hiisser 1983).
therefore assigned to the free carriers, which have a coupled plasma frequency of I eV and which are polarized antiparallel to the net moment of the sample. This antiparallel polarization is a result of an antiferromagnetic coupling of the moments of the Tm and Eu sublattices. The Tm-derived conduction electrons are polarized parallel to the Tm 4f electrons and antiparallel to the dominant Eu 4f electrons. Thus, again free electrons hinder an unambiguous identification of 4f13---* 4f125d transitions in an intermediate-valent thulium compound, and we do not know, at present, how much the f - d and/or f - p hybridization reduces the spin-orbit coupling and how much this affects the magneto-optical signals from f---> d transitions. The second class of heavy rare earth chalcogenides which have been studied by magneto-optical methods are the ytterbium-monochalcogenides (Suryanarayanan et al. 1970, 1974). Suryanarayanan et al. prepared thin films of YbSe (1970) and YbTe (1974) and measured the optical absorption as well as the magnetic circular dichroism (MCD). Figure 52 shows the results of these measurements for a 1000 A thick film of YbTe deposited onto CaF 2 by co-evaporation of the elements Yb and Te (Suryanarayanan et al. 1974). The magnetic field for the MCD measurements was 37.5 kOe. Absorption measurements down to wavelengths of 10 p.m do not show additional absorption bands, indicating that the peak near 2 eV corresponds to the lowest interband transition. Thus, YbTe is a semiconductor (or insulator) with divalent Yb and a completely filled 4f shell. Consequently, YbTe and YbSe are diamagnetic. The temperature dependence of the MCD spectra of the two compounds has been analyzed in terms of diamagnetic and paramagnetic line shapes, and these authors conclude that the relatively weak increase of the MCD peak intensity with decreasing temperature is a narrowing effect of the line and not a population effect of the ground-state. This indicates that the ground state is in fact diamagnetic. The numerous lines have, therefore, to be associated with splittings of the excited 4f135d state. Figure 52 shows also on
202
W. REIM and J. SCHOENES Photon Energy (eV)
2
5
I
4 5 l I
I
I 2.5
5/z ] 3
o
ci
%
~
- - 3&
I
-10
+ •
/52
I /--(
,-
---- 300 K
YbTe
4f155dJ
/ I /
/
/ I
Hcf(d) I
I
I
I
6
5
4
5
Wovelength
--Z 4v4
C
t
rl~ t2gil 7/2 /--
I
-
v3eg L , 7 , ~ - -
i
~.
--IOK
,>
Hso(f) Hso(d)
2
(10 3 ~ )
Fig. 52. (left) Magnetic circular dichroism AD and optical density D of a 1000~ thick film of YbTe at 300 and 10 K. Note that for AD at 300 K the scale on the right is valid and for 10 K the scale on the left is valid. The field is 37.5 kOe. (right) Atomic coupling scheme for YbTe (after Suryanarayanan et al. 1974). the right-hand side the coupling scheme proposed by Suryanarayanan et al. (1974) for YbTe. The crystal-field splitting of the 5d states is estimated in a point-charge model to be approximately 1 eV. The spin-orbit splitting of the 4 ) 3 state into 2F7/2 and 2F5/2 is taken from free-ion spectra to be approximately 1.2 eV. It results in four levels A, B, C and D, of which B and C are nearly degenerate. The peak at approximately 2 e V is assigned to the transition 4f14---~4f13(ZF-.~5de ", //z/ g denoted A, the ~roup of peaks near 3eV is assigned to 14 13 2 13 2 transition s from 4f into the final states 4f ( FT/2)5d t2g (C) and 4f ( Fs/z)5d t2g (B) and the peak at 4.24 eV is assigned to a transition into the 4f13(=Fs/2)5d eg final state. Besides this crude weak-coupling model between 4f 13 and 5d 1 states, these authors also give on the right-hand side of fig. 52 the excited levels in a more sophisticated coupling scheme, but the agreement with the experimental data is rather worse. 6.2. Heavy rare earth metals
Magneto-optical data for the heavy rare earth metals are extremely scarce. To our knowledge, only terbium and dysprosium have been studied, but, even for these two materials, little information is available. Thus, Kerr spectra have not been published and we have to content with a plot of the absorptive component of the off-diagonal conductivity multiplied with the photon energy (fig. 53) (Erskine 1975). As has been pointed out already in the discussion of Gd (section 5.4) this kind of plot eliminates the 1/to energy dependence of the free-carrier contribution to o-2xy. Thus, the spectra should reflect mostly interband transitions. Very little
MAGNETO-OPTICAL SPECTROSCOPY i
I
I
I
203
I
2.0 Gd t'M i U
1.0 t%l
o
)/
0
'
b~
3
Dy
-I.0 0
I
I
[
[
I
1
2
3
4
5
Energy (eV) Fig. 53. The product of the absorptive part of the off-diagonalconductivityo-z~yand the frequencyco for terbium, dysprosiumand gadolinium.The terfiperature and field have not been specified (after Erskine 1975). has been said about how these spectra are to be interpreted, except that the interpretation of the Tb and Dy spectra is complicated by the expected presence of transitions involving 4f-electrons. Judging from the very fiat diagonal conductivity spectrum of Dy (Erskine 1975), it appears, in fact, that s o p and d<--->p transitions dominate the diagonal and f <-->d transitions dominate the off-diagonal conductivity. This then, would indicate that the peak near 4 eV is possibly an f--> d transition in fair agreement with a binding energy of 3.8 eV determined by X-ray photo-electron spectroscopy (Lange et al. 1981).
6.3. Miscellaneous compounds containing heavy rare earths Due to the potential applicability for thermomagnetic writing, alloys based on (Tb,Gd)x (Fe,CO)l_ x have received much attention in the past ten years and the chapter by Buschow (chapter 5 of vol. 4 of this handbook) is reviewing these results in full. Here, we limit ourselves to the presentation of a few selected data, which might be explained on the basis of the previous discussion of the magnetooptical properties of elemental heavy rare-earths or simple heavy rare earths compounds. Choe et al. (1987) have measured the Kerr rotation of G d - C o and Tb-Co films at room temperature for wavelengths ranging from 250 to 700 nm. Guided by the observation that the Kerr rotation at long wavelength is almost the same for a same concentration of different rare earths, Choe et al. (1987) have normalized the Kerr rotation spectra by their respective values at 700 nm and compared it with Y-Co. Figure 54 shows then, that with increasing Tb-content the normalized
204
W. REIM and J. SCHOENES Energy
(eV)
5
4
3
2
I
I
I
I
,
E ¢::
r--°°,,1.0 ~
Gd35Co.65
Gd.26C°.74 ~-Gd19 Co 81
\/
.
.
~
3E
~- 0.5 '~
/ /" zY .25 .75 tL--~---~t Tb19 C081 i ii • •
Cb
i
/ OL,-,
s'~ff~Tb.29 C° 71 ii ,;~'
!! ~
500
Tb35C°65 I
400
I
I
500 Wovelength
I
I
600
,
I
I
700
(nm)
Fig. 54. Kerr rotation spectra for various amorphous Gd-Co and Tb-Co films, normalized by their Kerr rotation values at 700 nm. The temperatures is room temperature and the field is 13 kOe (after Choe et al. 1987). Kerr rotation becomes smaller at shorter wavelengths, while with increasing Gd content the normalized Kerr rotation increases at the shortest wavelengths. From the occurrence of magnetic compensation in the G d - C o and T b - C o systems one concludes that the two sublattices couple antiparallel in both systems. Looking then at fig. 53, which shows positive totr2xy values for Gd as well as for Tb, the different influence of these two rare earths in Co is not obvious, and this indicates once more the difficulty to discuss Kerr rotation spectra directly. As speculation we offer the following explanation. The o)Or2xy spectra are superpositions of free-carrier and interband contributions. An estimate of the former contribution in Gd (see fig. 44) gives a value of 1029 s -2. Because the diagonal optical conductivities of Tb and Gd are quite similar (Erskine 1975), one may also anticipate a similar free-electron contribution to the off-diagonal conductivity of the two metals, which would then lead to interband transitions of opposite sign for Gd and Tb. The system T b - F e has been investigated by Connell (1986), who also offers a microscopic model. In this model, the diagonal and off-diagonal conductivity below 3 eV are dominated by Fe d-states with, however, some negative contributions to Or2xy between 2 and 3 eV that are attributed to d--~ f transitions from Tb. A partial substitution of Tb by Gd, Nd or Sm is found to enhance the Kerr signal in agreement with the offered explanation. From a spectroscopic investigation of various R E - F e C o amorphous films Weller and Reim (1989a,b) conclude that f--~d transitions may contribute at maximum some tenths of a degree to the Kerr effect due to the large values of the optical constants in intermetallic alloys compared to NaC1 structure compounds. The polar Kerr effect of some polycrystalline R E - F e 2 compounds has been measured by Katayama and Hasegawa (1981). As fig. 55 shows the compounds where R E is either R E = Er, H o , Dy or Tb have a sign change in the rotation
MAGNETO-OPTICAL SPECTROSCOPY I
"-
I0
I
i
I
205
I
- ,f'\\./~
tO
•4= l:l "5
0
tr
x,, - I 0
/Y
',
ErFep_
\"'~ "HOFe2 I '--k,,.,//L---.-_ ~DyFez
H : I 2 kOe T= :500 K
0
a_ - 2 0 I
300
I
I
500 Wovelengih
I
i
700 (nm)
Fig. 55. Polar Kerr rotation spectra for polycrystalline bulk samples of GdFe 2, TbFe2, DyFe z, H o F e and ErFe2 at room temperature and a field of 12 kOe (after Katayama and Hasegawa 1981).
2
spectrum between 350 and 500nm, while GdFe 2 always stays positive. The rotation is largest in T b F e 2 , but does not exceed 0.4 °. The equatorial Kerr effect of nearly all R E - F e z Laves phases with heavy rare earths has been reported by Mukimov et al. (1985) and Shapirov et al. (1986). In contrast to the preceding works, these authors have reduced their data to obtain the off-diagonal conductivity for their polycrystalline samples. Figure 56 displays the absorptive and dispersive parts of the off-diagonal conductivity for GdFez, TbFe 2 and ErFe z (Shapirov et al. 1986). The differences are noted to be larger than in F e l T R E 2 compounds, which contain less rare earth, and they are attributed to the hybridization of the f electrons with the conduction electrons. Most of these interpretations are vague or very qualitative. The occurrence of contributions of equal importance from itinerant and localized states at the same energy renders assignments on the basis of simple energy-level schemes very difficult if not impossible. What would be needed here, are band-structure calculations that can handle itinerant and localized states on an equal footing, to then allow a computation of the diagonal and off-diagonal conductivity spectra. There is a long way to go to understand or even predict the magneto-optical properties of rare earths-transition metal intermetallic compounds. In order not to close this paragraph on a pessimistic note, we would like to mention that some rare earths have also been considered for devices such as optical isolators. Besides Eu 2+, Tb 3+ is very promising. Its Faraday rotation in a glass has been investigated, e.g., by Davis and Bunch (1984) at the H e - N e laser wavelength of 6328 ,~. For an unspecified concentration of Tb 3+ in the glass, a specific room-temperature Faraday rotation of 0.004°cm - 1 0 e -1 is quoted. At liquid-helium temperature the Verdet constant is approximately 0.3 ° cm - 1 0 e -1. This is nearly the same value as we have discussed for the E u 2+ glass (section 5.3) at 8000 A and allows a rotation of 45 ° for a i cm sample with application of only 150 Oz. As origin of this large rotation we anticipate a 4fs---~4fY5d transition because the 4f 8 state is about 2 eV below E v.
206
W. REIM and J. SCHOENES I
i
I
•
" f ' ~ ~
E z 0
! / / l//
/"-- !JiI
I I I
I
-72
~~
i
I -70
f"-.,~
•
/il
I
I
/ "~)
--
-
-
-
-
-
GdFe2 TbFea
ErFe2
JJ 7
E(or) Fig. 56. Real part trlxyand imaginary part tr2xr of the off-diagonal conductivity of GdFe 2, TbFe 2 and ErFe 2 polycrystalline bulk samples at room temperature. The field is cited to be 40e! probably 4 kOe (after Sharipov et al. 1986).
7. Uranium compounds Actinide compounds a p p e a r to be a promising field for magneto-optical research due to the fact that the 5f" states m a y be located near the Fermi energy. Regarding the magnitude of the magneto-optical effects c o m p a r e d to rare earth materials, an e n h a n c e m e n t due to the larger s p i n - o r b i t energy can be expected and was in part experimentally verified. A m a j o r drawback is the high radioactivity of most actinide elements and only compounds using depleted uranium have found their way into the laboratories on a b r o a d scale. The configuration of the outer electrons in uranium is 5f36d17s 2. In compounds, different ground-state configurations have been found, e.g., the idealized ground states 5f 2, 5f 3 and 5f36d I in UO2, U P and US, respectively. H o w e v e r , hybridization effects in the latter two metallic samples cause a m o r e general description in terms of 5f3-X6d x and 5f3-X6d 1÷~ states to be appropriate. With the chalcogens X = S, Se, Te and pnictogens Z = P, As, Sb compositions with atomic ratio 1 : 1, 1 : 2 and 3 : 4 are known. Most of the magneto-optical efforts up to now have b e e n focussed onto these materials.
7.1. Uranium dioxide U r a n i u m dioxide, UO2, was one of the first uranium compounds to be investigated intensively because of its use as nuclear fuel. As early as 1959 ( A c k e r m a n n
MAGNETO-OPTICAL SPECTROSCOPY
207
et al. 1959), stoichiometric thin films have been prepared, and the absorption coefficient K was determined showing that UO 2 is semiconducting with an energy gap of about 2 eV. However, only in the early eighties, the electronic structure was completely resolved by optical (Schoenes 1978, 1980b), magneto-optical (Reim and Schoenes 1981) and photoemission (Baer and Schoenes 1980) spectroscopies. Figure 57 displays the derived energy-level scheme consisting of an oxygen 2p valence band, an uranium 6d conduction band with a crystal-field splitting of 2.8 eV between the 5d t2g and 5d eg sub-bands and a localized 5f2 state. Thus, the energy-level scheme looks very similar to the one of EuO (section 5.1), and large magneto-optical effects can be expected. However, UO 2 orders antiferromagnetically at TN = 33 K (Leask et al. 1963), and with the knowledge of the sublattice moment of 1.73/x B from neutron scattering (Faber and Lander, 1976) it was estimated that an applied field of about 190T would be necessary for magnetic saturation. Figure 58 shows the specific Faraday effect at a field of 4 T and a temperature of 33 K as measured on thin films prepared by Knudsen-cell evaporation in ultra high vacuum (Reim and Schoenes 1981). The magnitude of the rotation is of the same order of magnitude as for the Eu-chalcogenides if it is scaled with the magnetization. For the interpretation of the energy dependence of the magneto-optical data, final-state coupling of the 5f16d I state has to be taken into account (Reim and Schoenes 1981). Both L - S and j - ] coupling for the final E EeV3
U02
5,60,201 5f16deg
I 5/25/2 J=3~ 7/2 3/2 J=31~-.,, ,'5/25/2 J= 4 - - - - - - 7/2 3/2 J=4 5/2 3/2 d=3 .lr--.-.--5/2.3/2 J=4 -'-T1 ]
o -'or2
m
5 / 2 512 d= 4
7/2 ~/2 J=4
2p ("F15")
localized states
Fig. 57. Density of states scheme for
UO 2 as
band states, D (E)
derived from optical and magneto-optical spectroscopy
(after Schoenes 1984b).
208
W. REIM and J. SCHOENES
/
U02 T = 55K B = 40kG
1
5
j/
] \o 1
@J ¢
"\
o/
J .o.o.O,,i .¢.
i
~
\ °%%
/
/ ,
"
/
i I
~
.2:
..,~'."
s!p ° ' °"'°'~-°'° ~*
Do/
j'°
/
....
!"\,.,
~
o
/
z
~j
XX~/i
Photon Energy [eV]
Fig. 58. Faraday rotation 0 and magnetic circular dichroism D for Reim and Schoenes 1981).
UO 2
1
at T = 33 K and B = 4 T (after
state of the 5fz---~5f16d 1 transition have been considered. Figure 59 reproduces the calculated energy-level schemes for the excited state in the two coupling approximations, the energy parameters entering into this computation being taken from independent spectroscopic data. The calculation of the transition probabilities shows that for the absorptive part of the off-diagonal conductivity O'2xy o n e expects merely one strong line in L - S coupling but two strong transitions separated by approximately 1 eV in the j-j coupling limit (fig. 59). A similar characteristic difference is found for the calculated absorptive part of the diagonal conductivity O-xxx [or approximately K, which is the absorption coefficient as defined in eq. (2.51)]. Figure 60 shows o-2xy and K of the same UO 2 thin film together with the calculated 5f2---~5f16d 1 transitions and their relative oscillator strengths for j-j coupling. From the good agreement of both the oscillator strengths and the energy separations, using no adjustable parameters, it has been concluded that the initial state of UO 2 is a localized 5f2 state and that the excited 5f16d 1 final state should be described in an intermediate coupling scheme close to the j-j limit. These results are included in fig. 57 on the left side showing schematically the most important localized states.
7.2. Binary NaCl-type compounas The face centered cubic uranium-monochalcogenides and monopnictides are the most investigated class of uranium compounds. This fact reflects, on one hand, the relatively easy handling of these materials from both an experimental and a theoretical point of view, and, on the other hand, the large variety of physical
MAGNETO-OPTICAL SPECTROSCOPY
209
LO
Ls
jj
Fig. 59. Coupling scheme for the 5f16d1 state in L - S approximation (left) and in j - j approximation (right). Dipole-allowedtransitions and their oscillator strengths normalized to the strongest transition are indicated by the vertical bars (after Reim and Schoenes 1981).
effects encountered in these systems. Similar to the Eu-chalcogenides in section 5.1, this class of materials can be thought of as a model system for the understanding of uranium compounds. The uranium-chalcogenides U X with X = S, Se, Te are ferromagnets (Lam and Aldred 1974). The magnetic moments are strictly confined to the ( l l l ) - e a s y axes and the total moment increases going from US to UTe. The moment obtained from magnetization measurements is found to be smaller than the neutron moment, which has been interpreted by antiparallel coupling of f and d moments (Busch et al. 1979). The uranium-pnictides U Z with Z = P, As, Sb, on the other hand, order antiferromagnetically (Lam and Aldred 1974). In general, they exhibit a very complex magnetization-field-temperature phase diagram showing various ferro- and antiferromagnetic phases (for an example see fig. 6). The magnetic and crystallographic parameters of U X and U Z compounds are summarized in table 3. The complex magnetic behavior of these materials goes along with
210
W. REIM and J. SCHOENES
~5
U02
% .'-.4
#
UO2 T =33K
T = 300 K
/
B : 40kOe
"E
t~ / i ',., 1
/
[ |,.
/
z.~-~-I
3
/
0
.Q
/
/
/
/
/
i
4
g
Photon Energy [eV]
Photon Energy [eV]
Fig. 60. Absorption coefficient K (left) and absorptive part of the off-diagonal conductivity o'z~y (right) for UO 2. Calculated transitions from the 5f2 (3H4)ground state into the 5fX6dIeg final state in j - j coupling approximation are shown by the vertical bars where the length is proportional to the oscillator strength. Transitions into the 5f16d1 t2g multiplet starting at about 5.7 eV are added in the case of the absorption coefficient (after Reim and Schoenes 1981). TABLE 3 Crystallographic and magnetic data for the uranium-monochalcogenides and monopnictides (RossatMignod et al. 1982, Vogt and Spirlet 1984, Schoenes 1980a). Tc.r~: Curie and N6el temperatures; a: lattice constant in ,~ngstr6m; N: number of molecules in units of 1022cm-3; /ztot: magnetic moment per uranium at the conditions of the magneto-optical measurement (fig. 63); /zB: magnetic moment measured in neutron scattering. Tc,~ (K) a (/~) N (1022 cm -3) tZ,o, (/zB) /zn (/.~)
US
USe
UTe
UP
UAs
177 5.488 2.42 1.55 1.7
160 5.74 2.11 1.8 2.0
102 6.155 1.71 1.9 2.25
125 5.589 2.29 0.16 1.7
124 5.767 2.08 0.42 2.25
USb0.85Te0.15 214 6.197 1.68 2.8 2.8
an u n u s u a l e l e c t r o n i c s t r u c t u r e . F i g u r e s 61 a n d 62 s h o w a s c h e m a t i c view o f t h e b a n d s t r u c t u r e s e v e r a l eV a r o u n d t h e F e r m i e n e r g y E F as d e r i v e d f r o m o p t i c a l reflectivity m e a s u r e m e n t s ( S c h o e n e s 1980a). O n e c h a r a c t e r i s t i c f e a t u r e is t h e s p a t i a l l y e x t e n d e d n a t u r e o f t h e f w a v e f u n c t i o n s f o r m i n g b a n d s of t y p i c a l l y 1 eV w i d t h in t h e vicinity o f t h e F e r m i level. T h e u r a n i u m 6d b a n d is p a r t i a l l y filled, w h i c h causes t h e m e t a l l i c c o n d u c t i v i t y o f all U X a n d U Z c o m p o u n d s . T h e s t r o n g f - d h y b r i d i z a t i o n c r e a t e s a d i p in t h e d d e n s i t y o f states n e a r E F w h i c h has b e e n s u b s t a n t i a t e d b y a c l u s t e r c a l c u l a t i o n f o r U S ( S c h o e n e s et al. 1979a). F o r t h e u r a n i u m p n i c t i d e s , also a s t r o n g f - p h y b r i d i z a t i o n has b e e n p o s t u l a t e d ( S u z u k i et al. 1982) w h i c h is s u g g e s t e d b y t h e o v e r l a p p i n g p a n d f s t a t e s in fig. 62. T h e d o m i n a n t o p t i c a l t r a n s i t i o n s for p h o t o n e n e r g i e s u p to 6 eV a r e o f f---~ d,
MAGNETO-OPTICAL SPECTROSCOPY US
USe
eg
_
UTe
"eg"
d,
~O . E F ~
211
d
....
-~~5fF~_~----'~5f
" ~ ~ 5 f
~-2 -6
Fig. 61. Density of states schemes for the uranium-monochalcogenidesas derived from optical spectroscopy (after Schoenes 1980a). UAs
USb
6
...... 6~..... eg. ~'~
4
:" 0
EF
Keg'
~i 'to' ~ - ~ =
5f
u_l -2 -4 -6 Fig. 62. Density of states schemes for the uranium-monopnictides UAs and USb as derived from optical spectroscopy (after Schoenes 1980a).
d---> f and p---> d type. While the latter excitation has been found to contribute for the pnictides at energies above 3 eV but only above 5 eV for the chalcogenides, f--->d and d--->f transitions are always present in the 0.5-5 eV energy range of the experiment. Table 4 shows the parameters used for a fit of the f--->d and d--->f transitions in the diagonal conductivity o-ixx with classical Lorentz oscillators (Reim 1986). The f--->d transition has equal oscillator strength and line width in all six compounds, pointing to very similar ground states. The d--->f transition is much weaker in the pnictides compared to the chalcogenides which reflects the lower d-state occupation. Figure 63 displays the complex polar Kerr-effect spectra of (100)-cleaved single crystals of the uranium-monochalcogenides and monopnictides. For US, USe and UTe the measurements have been performed at saturation magnetization which corresponds to a moment per uranium atom of 1.55, 1.8
212
W. REIM and J. SCHOENES
TABLE 4 Interband transition parameters for the f---~d and d---~f transitions derived from a fit of the absorptive diagonal conductivity ~r~ and transition energy and magneto-optical strength of the f---~d transition from ~ r (Reim 1986). hwa and h~%: f---~d and d----~f transition energies; hyA and hy~: line widths; fA and re: diagonal oscillator strengths; (Oxy)A: total weight for the f--+ d transition; f~r: normalized total weight ( O'xy) A/N //'tot"
~x
USe
UTe
UP
UAs
USb(USb0.85Te0ns) 0.9 1.2 0.9
f---~d
h~oA (eV) hy A (eV) fA
1.15 1.1 0.85
1.0 1.2 0.9
0.6 1.15 0.9
1.05 1.1 0.9
1.0 1.1 0.9
d--of
hwn (eV) hy~ (eV)
2.85 2.8 1.55
2.45 2.5 0.7
1.75 2.1 0.25
2.8 2.3 0.5
2.3 1.9 0.4
1.13 10 2.7
0.94 9.4 2.5
0.77 5.4 1.5
0.78 1.2 2.9
0.82 2.3 2.6
f~ ~xy
US
hwa (eV)
f---~d
(O'~y)a ( 1 0 2 9 S- 2 ) f~y(lO7 S-2 cm -3 ~ )
4
US
3
#~
eK
USe
~2
UTe
-8 K
/ }
-e K
0.65 16 3.3
/
~S-~
i
2
# #
b
:
;
£
Wo
**
%
t
×
-~-I
+
f
+
o
, \
V ÷÷
i
f
J
i
¸
t
t 2 3 4 5
.~
l
l
l
i
1 2 3 4 5 Photon Energy [eV]
UP
lo
'
÷/
/~,
UAs
B=IOT T-25K
B:IO T
i
5
", •
3
2 i °-EK
i
i
USb85Te.15 B=6T T=IOK
A
w .25 i
~'
°~CK
.
~, n
i
1 2 3 4 5
0
/
°\
21
o
!!l ,'
t
g ~ -.25 o
0
0
,
i
:
g
-1 i
1
2
3
4
5
1
i
i
i
2
3
4
i
5
°
r 1
i 2
i 3
i 4
r 5
Photon Energy [eV]
Fig. 63. Complex polar Kerr effect of the uranium-monochalcogenides at B = 4 T and T = 10 K and of the uranium-monopnictides at the indicated temperature and field conditions (after Reim 1986).
MAGNETO-OPTICAL SPECTROSCOPY
213
and 1.9/x B, respectively (Vogt 1980). In an applied field of 10T, ferrimagnetic phases exist for UP and UAs with magnetization values of 0.16 and 0.42/~, respectively (Vogt et al. 1980). USb is antiferromagnetic with very low magnetization even at an applied field of 10 T. Therefore, the Kerr measurements have been performed o n USb0.asTe0.15 at a field of 6 T (Reim 1986), where a ferromagnetic phase exists (Rossat-Mignod et al. 1982). The energy dependence of the Kerr effect shows, as expected from the band structure, similarities in the low-energy regime. The magnitude extends up to 6.5 degrees, a remarkable size with respect to the relatively low magnetic moment, e.g., compared to 7ixB in the Eu-monochalcogenides discussed in section 5.1. For a quantitative discussion of these effects fig. 64 displays the off-diagonal element cT,y (Reim 1986). In order to
~. :
6 4
:'++~
US
+ +
~y-axy
++&
USe
UTe
++ ~.
% ~.12r2 × y
.
\
2 o
O /
~ .... //~f
o
/i
/
-4
/
o
g
f
~ s
-6
[
1
2
t
I
3
4
5
Glxy
o o
o o oo
I
g
-2 ,, ;
o o o
%
o
-8
o
o
-4-
~f
~# =9
I
i
I
I
L
I
I
I
I
1
2
5
4
5
2
5
4
5
Photon Energy [eV]
2
5F
~%
~xy
/[ /
-1
UP
-1 //'
/ A +++ -2 " , ++
B =lOT
~A: ,+.
~
T=25K I
I
I
....
•
USb85Te.15
5 gas B=IOT T =20 K
O'lxy I
/:)j
~A* ;++ ;+
O-lx y I
I
B=6T T=IOK I
I
I
I
Photon Energy [eV]
Fig. 64. Off-diagonal conductivity ~ y of the uranium-mon0chalcogenides at magnetic saturation and for the monopnictides at the indicated field and temperature conditions. The dashed line displays the extrapolation of o-lxy for hoJ < 0.5 eV obtained by Kramers-Kronig inversion (section 2.3). The solid and dash-dotted curves show the estimated free-electron contribution to crz~y and crlxy, respectively. Note the different ordinate scales (after Reim 1986).
214
w. REIM and J. SCHOENES
separate the contributions from conduction electrons and from interband transitions, the peak positions in the absorptive part of the diagonal and the offdiagonal conductivity have been compared (Reim et al. 1983, 1984b, c). It was concluded, that the lowest energy interband transition, which was assigned to a f---~ d excitation, has a diamagnetic line shape (see fig. 5) similar to various rare earth compounds (sections 4-6), i.e., it shows a bell shaped form with a maximum at hto A in O'lxy and O'2xy(htOA)=0. This directly implies, that the conduction-electron contribution to Or2xy has to be positive as indicated in fig. 64 by the solid lines. From eq. (2.66) it then follows, that the conduction electrons are spin-polarized antiparallel to the f moment using the fact that the 1/w term in eq. (2.66) dominates (Reim et al. 1984b). This is equivalent to a negative O-lxy(~O= 0) for all compounds as indicated in fig. 64 and predicts the sign for the Hall effect according to eq. (2.73) which, in fact, has been verified (Schoenes et al. 1984). In a second step, the magnitude of the conduction-electron contribution was estimated (Reim 1986) using eq. (2.66). The numbers -40, - 3 5 and - 2 0 % have been derived for the spin-polarization o-d of the conduction electrons in US, USe and UTe, respectively, in excellent agreement with spin-polarized photoemission results (Erbudak and Meier 1980). In particular, the reduced size of trd for UTe was confirmed. For the uranium monopnictides at the temperature and field conditions indicated in fig. 64, o-d was found to amount to - 1 5 % , - 3 0 % and - 1 0 0 % for UP, UAs and USb0.85Te0.15, respectively. Taking into account the ferrimagnetic stucture of the former two materials, a - 1 0 0 % local spin-polarization (within a ferromagnetic sublattice) has been concluded, i.e., only the spin-down d-sub-band is occupied (Reim 1986). Subtracting the conduction-electron contribution from the m e a s u r e d ~xy gives the pure interband signal. The dominant feature around 1 eV for all six materials has been attributed to a f---~d transition. In table 4 the energy, the total weight according to eq. (2.57) and the normalized total weight fxy = (°-xy)/N/xtot are summarized, where/tZtot is the magnetization at the conditions for the magnetooptical measurement and N is the density of uranium atoms. The transition energy hw A decreases slightly along the chalcogenides and pnictides series thus showing a similar trend as in o-ix~ (table 4). However, the absolute numbers, in particular for the pnictides, disagree by up to 0.25 eV, which points to a shift of the transition energy between the paramagnetic (O-lx~) and the magnetically ordered (trxy) states. This phenomenon, which is well known for the Euchalcogenides (see section 5.1), will be discussed in detail in section 7.3. The quantity f~y shows that, except for UTe, the microscopic strength of the f---~d transition in the uranium-chalcogenides and pnictides is identical and may be summarized as (O-xx) = 3 . 3 × 103°N [s-2],
(7.1)
(O'xy)
(7.2)
-----'2.8 X 10 29 N/zto t [ s - 2 ] ,
where the units of N and /xtot are 1022 cm -3 and / ~ , respectively. It has been
MAGNETO-OPTICAL SPECTROSCOPY
215
demonstrated, that these numbers both for (o-x~) and for (Oxy) can be deduced in the framework of the microscopic model presented in section 2.2. However, it was found that the use of atomic f and d wavefunctions in the derivation of the radial overlap integral of eq. (2.61) leads to values too small by a factor of six while the wavefunctions from a LMTO band-structure calculation for US (Brooks 1984) gave much better results. This fact corroborates the assumed delocalized character of the f states (Schoenes 1984b, and Schoenes and Reim 1986). At this point of the discussion, some comments should be made about the magnitude of the conductivity O'xy from the f--~ d transition in a uranium compound compared to the maximum possible signal. With realistic values of the spin-orbit energy for f and d states in uranium it was shown (Reim 1986) that (Cry)/(cr~) =0.3 o'j where o) is the joint spin polarization of f and d states according to eq. (2.59). On the other hand, the combination of eqs. (7.1) and (7.2) gives (O-~y)/( ~rxx) = 0.085/Ztot. Hence, with the values of t~tot in table 3 it turns out that US reaches about 50% of the maximum signal, and USb0.ssTe0.15 shows the maximum reachable magneto-optical response for the f---~d transition in uranium. The width of that transition is similar for the two materials. Therefore, the Kerr effect is proportional to jL~tot, tOO, and thus, for these two materials, differs by a factor of two. Consequently, the only way to increase the Kerr effect of a material like USb0.85Te0.15 is to narrow the width of the transition but keeping (o-xr) constant. We think that this is not feasible because narrowing the transition would reduce the overlap integral and would in turn reduce (o-~r). In both the series of uranium-chalcogenides and pnictides, one further dominant peak, apart from the f---~d transition, is observed in CTxy.It was pointed out, however, that the microscopic origin is different (Reim et al. 1984b), being a d---~ f transition for the chalcogenides, and a p,f---~ d transition for the pnictides. For the former materials, a p---~d excitation can be excluded because of the high binding energy of the p states according to fig. 61. An interpretation in terms of a second f---~d transition is also improbable because of the different behavior of the two structures along the chalcogenide series in (o'~) and (O-xy) (Reim et al. 1984c). The observed reduction of the strength of this d-+ f transition along the series was explained by an increasing localization of the final state for the compounds with heavier anions. The d---~f transition is present in the uraniumpnictides, too. However, due to the low d-electron concentration per formula unit of about 0.4 in UAs and less than 0.15 in USb0.85Te0.15 compared to about 1.3 in US, this transition is accordingly weak and shows up only as the shoulder marked B for the light uranium-pnictides in fig. 64. Structure C in ~xy has about the same energy and weight for the three compounds. Its microscopic explanation is most probably the excitation of high orbital-momentum states at the top of the valence band, which then would indicate a similar pf-hybridization in UP, UAs and USb.
7.3. Pseudo-binary NaCl-type compounds Pseudo-binary rocksalt-structure uranium compounds have been investigated for different purposes: UAs~Sel_ x in order to verify the assignments of optical
216
W. REIM and J. SCHOENES
transitions given in section 7.2 (Reim et al. 1984b); USbxTel_~, to optimize the magneto-optical signal and to investigate the magnetization induced shift of the f--+d transition energy (Reim et al. 1984d, e, Reim 1986); UxYI_~Sb, to study the effect of dilution of uranium on the nature of the f-state and the magnetooptical response (Reim 1986). With the study of UA%SeI_ ~, the similarities and differences in the optical and magneto-optical properties for uranium-monochalcogenides and pnictides have been investigated (Reim et al. 1984b). Figure 65 shows ~7~yfor U A s , UAs0.7Se0. 3 and USe. The energy and strength of f---~d, d----~p and p,f---~ d transitions, labelled A, B and C in this figure, clearly support the arguments for these assignments given in section 7.2. The conduction-electron spin-polarization was shown to be negative with absolute values of the local polarization of -100, -75 and - 3 5 % for UAs, UAs0.7Se0. 3 and USe, respectively. These values have been interpreted in the framework of a rigid band model explaining both the decreasing spinpolarization and the increasing number of conduction electrons in going from UAs to USe (Reim 1986). A comparison of the f---~d transition energy hoJn in o-a~x and in ~ y (table 4) suggests a decrease of hw A with magnetization (red shift) in the order of 0.2 eV for USb0.85Te0.a5 but a blue shift of 0.1 eV for UTe. To study these aspects of the f---~d transition, different compositions of the USb~Tea_~ system have been investigated. In zero field, compounds with x > 0.82 are ferromagnetic while for x <0.82 ferri- or antiferromagnetic ordering occurs (Burlet et al. 1980). The lattice constant is nearly independent of x, which excludes lattice pressure effects for the explanation of the different behavior of UTe and USb. Figure 66 displays the Kerr rotation in the f---~d transition energy range for ferromagnetic USb0.85Te0.15 and ferrimagnetic USb0.9Te0. x at different temperatures. Obviously, the temperature dependence of the f---~d transition is similar in both compounds,
t" z
\ ~xy f "~
,,¢ o
B
,: ~
C1 0 %
+.+~3.E~--~*~'~
.
// ? //
:
.. A : -2 ', ~
~+ o-,,
UAs B ~10 T T ~20 K
° ~ C~2xY
6 o
%
4 2 0
/:
u~
:%
8
-2,
/;
/
UAso.,Se0.~
/, A + + + + ~lxy
B 5T T=123K
/
/
USe
,/,",,,,:
B: 4 T T=15 K
¥*-%xy
Photon Energy [eV]
Fig. 65. Off-diagonal conductivity ~xy of UAs, UAs0.7Se0.3 and USe at the indicated field and temperature conditions. The meaning of the dashed and solid lines is given in fig. 64 (after Reim et al. 1984b).
MAGNETO-OPTICAL SPECTROSCOPY
15 "••,
•.
15 - " %
USb85Te.15 1.5 B=6T
t
217
T(K)
114 o~".o~. 152 o ~
USb9Te 1. . %
B = 4T
I
c-
.£
%\
•
170 " ' ~ "
'5
1D
"
171 • ++++~++++",, o 180 ++."" 195 •
v
Ill t
•
"÷~ i
•.+,,o
I 0.5
2
225 ~
2601 "
f---. ,
0
I
• .'."
..,...,
•
~'.+~
~ .~,~ ••• ~ "+A
• ~.~
a~"~'~..
• ~! ~ * i "
T(K)
io.o,~OoO ]~:!]
"-,,,.... ".."."".'~".. 2
0
1
2
Photon Energy [eV]
Fig. 66. Polar Kerr effect of cleaved USbo.asTeo.15and USbo.gTeo1 at different temperatures. At the mentioned applied fields, the type of magnetic ordering is ferro and ferrimagnetic, respectively. Note the different scales (after Reim 1986). and the different magnetic order and the, by a factor of 4 higher, magnetization of USb0.asTe0.15 is only reflected by the different magnitudes of the Kerr rotation, which is quite astonishing. With decreasing temperature the main effect is a transfer of oscillator strength from the high-energy side to the low-energy side of the main structure. This effect can be seen even more pronounced in ~r2xyshown in fig. 67 for USb0.85Te0.as. According to the theory, the shift of the transition energy with temperature should be proportional to the magnetization in high applied fields. Figure 68 shows a plot of hWA, as determined from the peak energy in O-lxy, versus temperature for USb0.9Te0.1 in a field of 4 T. In the same figure the normalized magnetization measured in the same applied field of 4 T is displayed. The fit of ho)A is excellent and predicts a transition energy at zero magnetization of 0.85 eV in excellent agreement with o-~xx (table 4). The total magnetic red shift is deduced to be 0.2 eV. In the USb~Te~_ x system, it was observed that the double-structure behavior in o-2~yof the f---~d transition shown in fig. 67 abruptly changes into a single structure around x = 0 . 8 . This is visualized in fig. 69, comparing USb0.85Te0.~s with USb0.sTe0. 2 both for magnetic saturation. Together with this change in shape, the sign of the magnetic shift of the transition energy alters from red for x > 0.8 to blue for x ~ 0.8. Both compounds shown in fig. 69 have an f--~ d energy of 0.86 eV in the nonmagnetic state, but at magnetic saturation the resonance energy is 0.65eV and 0.95 eV for x = 0.85 and x = 0 . 8 , respectively. The composition dependence of the magnetic shift of the £--* d energy, the change from double to single structure at x = 0.8 and the negative d spin-polarization in USbxT%_ x have been explained in a complex band model (Reim 1986). The USb~TeI_ x system has also been used to optimize the magnitude of the
218
W. REIM and J. SCHOENES
~\ 1.5
USb85Te15
if~**~
!J/z o
T [K]
\ 2oo
#/} i//
i5 .5 i
z
',290
/
4 1
2
5
Photon Energy [eV] Fig. 67. Absorptive off-diagonal conductivity ~r2xyof USb0.ssTe0.15 at B = 6 T for different temperatures. Each curve is normalized to its maximum value. For clarity, the spectra for 200, 170 and 10 K are vertically shifted by 0.25, 0.5 and 0.75, respectively (after Reim 1986).
1
M/Mo
>.. ¢, t-
0.7
cO
0.5
.m
(,0
0.8 +.., "O I
0
r
100
i
,
i
,
i
0
200
Temperature (K) Fig. 68. f---~d transition energy in ~Txy versus temperature for USb09Te0.1 at B = 4 T (dots) and normalized bulk magnetization M / M o (solid line) for the same applied field (after Reim 1986).
Kerr effect (Reim et al. 1984e). Figure 70 shows the energy dependence of 0K for different values of x. With increasing x, both the saturation moment/Xtot increases and the transition narrows which results in a Kerr rotation enhancement by a factor of 2. Due to the shift of the f---~d transition energy around x = 0.8, large Kerr rotations, exceeding 6 degrees, can be generated at most energies in the range between 0.5 and 1.5 eV by variation of x. Using the fact, that the easy axis is along the (111) direction, rotation angles up to 9 degrees have been observed on a (lll)-oriented surface (fig. 71).
MAGNETO-OPTICAL SPECTROSCOPY
219
/ \ US%Ze., (.3 °.5
USb8Te.2
i
0 -e 0
i
I
I
I
E Z I
2
5
Photon Energy [eV] Fig. 69. Absorptive off-diagonal conductivity trZ~y normalized to its maximum values for the two compounds USb0.sTe0. 2 and USb0.ssTe0.15 at magnetic saturation (B = 4 T , T = 10K and B = 6T, T = 10 K, respectively). For the latter compound, the zero line has been shifted verticaUy by 0.4 (after Reim 1986).
10
~~~~÷
X=O
A
03 8 v"0 ,m
,&
tr •"(3) 4 v I
00
I
I
1
2
3
Photon Energy [eV] Fig. 70. Polar Kerr rotation of four different (100)-oriented cleaved USb x Tel_ x single crystals at magnetic saturation. For clarity the curves with x = 0.8, 0.5 and 0 are vertically shifted by 2, 4 and 7 degrees (after Reim et al. 1984e).
220
W. REIM and J. SCHOENES
f%
8
USbo..Teo.
g °
o
o
'::',,'~, ° : \o
6
+ 100 cleaved ~ ~071 polished
~s~
+20
+ +
C
o
+ +
"~4
×0
*
+ × []
+
£E
+ +
+ +
g2
+
v
~
+ +
! X
0
'
I
'~
I
'
I ×x×xx×
-20
I
1
o~oooo I
2
I
3
4
Photon Energy [eV] Fig. 71. Magneto-optical Kerr rotation of (100)- and (lll)-polished, compared to (lll)-cleaved crystals of USb0.sTe0. 2 at magnetic saturation (H = 4 T, T = 10 K) (after Reim 1986).
Although the spatial overlap of the 5f wavefunctions decreases with increasing size of the anion, even in USb the magneto-optical properties could be explained assuming a narrow f-band and no final-state effects common for localized states, like in UO 2 (section 7.1) or the Eu-chalcogenides (section 5.1), have been observed. Diluting the uranium sublattice in USb, increases the mean spatial separation of the magnetic ions and provides a way to study a possible localization of the 5f states and the effects on magneto-optics. In this experiment, trivalent yttrium with the atomic configuration 3d14s 2 has been used which decreases the number of f states whereas the d occupation remains almost unchanged. The compounds in the UxYl_xSb system order antiferromagnetically for x > 0.55 and ferromagnetically down to x = 0.2. For x < 0.2 paramagnetic behavior has been observed (Rossat-Mignod et al. 1982). All compounds exhibit metallic conductivity (Frick et al. 1984). Figures 72 and 73 show the Kerr effect and the off-diagonal conductivity ~xy the compositions U o . 7 Y o . 3 S b , U o . 4 Y o . 6 S b and U0.15Y0.ssSb, being antiferromagnetic, ferromagnetic and paramagnetic, respectively. The Kerr spectra are dominated by a sharp f---~d peak around 1 eV similar to all other uranium-monochalcogenides and pnictides. With increasing substitution of U by Y this f---~d feature narrows subtantially. Similar to the behavior of USb0.85Te0.15 displayed in fig. 64, a double structure in ~2xy is observed for the f--~ d transition. The resolution of these two peaks becomes better with increasing dilution due to the narrowing of the two lines. It was shown that with increasing dilution of uranium the f--+d transition energy does not decrease to zero as has been proposed from optical data (Frick et al. 1984), but approaches a value of 0.5 eV for the lowest uranium concentration. The double structured f--~ d transition has been tentatively explained in the framework of the band model proposed for the f o r
MAGNETO-OPTICAL SPECTROSCOPY
--
221
3 I
0
/. . . . I[ o
~1
I
0
1
+ t~!
B=4T
ili
T = 1OK
T= 181K
2
; I ,
0.5
f~
U,,,YH, Sb
+4r~ ;~ ~i~It,!~< J+
B =7T T = 15K
, t+~eK
I
I
I
3
4
5
a
0
1
2
3
4
5
0
I
i
1
2
Photon Energy [eV]
Fig. 72. Energy dependence of the polar Kerr rotation and ellipticity of (100)-cleaved single crystals of U~Y~_~Sb with x = 0.7, 0.4 and 0.15. Note the different ordinate scales and the expanded energy scale for x = 0.15 (after Reim 1986).
[~ -~ 0 10
~%
U0,Y, ~Sb g2xy
'~
[I 8I
"
o ~
B=7T T = 181 K
20
._> 0
-o t-
Sb
•
U ,Y0~,Sb
%/
1
,/
O
i c'2
1°j~ I i
0
0
U o ,Y,, "
O: xy
\ T= 15K
I
/<'
0
+
E
o
-1
-20
-10
.--
I
I~1
I
+ 0-
~
lxy
-2
I
-40 ~, II,
1
2
3
4
5
0
%y
I I
I
I I
0
T V i
1
I
I
I
I
2
3
4
5
0
1
2
Photon Energy [eV]
Fig. 73. Off-diagonal conductivity for three UxYl_xSb compounds. The dashed lines display the extrapolation of ~xy for h~o < 0.5 eV obtained by Kramers-Kronig inversion (section 2.3). Note the different ordinate scales and the expanded energy scale for x = 0.15 (after Reim 1986). U S b x T e l x s y s t e m ( R e i m 1986). T h e t o t a l w e i g h t o f t h e f----~d t r a n s i t i o n in U x Y 1 xSb was f o u n d to b e r o u g h l y p r o p o r t i o n a l to t h e p r o d u c t o f m a g n e t i c m o m e n t a n d u r a n i u m c o n c e n t r a t i o n , thus it shows a g a i n t h e s a m e b e h a v i o r as all o t h e r i n v e s t i g a t e d r o c k s a l t s t r u c t u r e u r a n i u m c o m p o u n d s . D o w n to a u r a n i u m c o n c e n t r a t i o n as low as x = 0.15 a l o c a l i z a t i o n o f t h e f states was e x c l u d e d a n d it has b e e n c o n c l u d e d t h a t t h e f - c o n f i g u r a t i o n is n e a r 5f 3 in t h e w h o l e s y s t e m . H o w e v e r , for x < 0 . 1 5 m a g n e t i z a t i o n a n d t r a n s p o r t d a t a p o i n t to an f 2 g r o u n d state.
222
W. REIM and J. SCHOENES
7.4. Th3P4 structure compounds
Two uranium compounds in the Th3P4-type bcc structure have been investigated by magneto-optics: U3P4 and U3As 4 (Reim et al. 1984e). They order ferromagnetically at 138 and 198 K with a saturation moment of 1.39 and 1.83tZB/U, respectively (Burlet et al. 1981). An unusually large anisotropy with respect to the (lll)-easy axis of the magnetic (Buhrer 1969, Trzebiatowski et al. 1971) magnetostrictive (Bielov et al. 1973) and electrical (Henkie 1980) properties has been reported. However, the magnetic moments on the uranium sites are not oriented along the (lll)-direction as it is the case for all ferromagnetic rocksaltstructure compounds mentioned in sections 7.2 and 7.3, but a noncollinear magnetic structure has been shown to exist (Burlet et al. 1981). Anisotropic pf-mixing was used to explain the observed magnetic structure (Takegahara et al. 1981, Suzuki et al. 1982) assuming a mixing energy in the order of 1 eV. Optical data on U3P4 and U3As 4 and on the corresponding nonmagnetic and semiconducting thorium compounds T h 3 P 4 and T h 3 A s 4 (Schoenes et al. 1983) strongly support this size of the mixing effect. From the optical data it was concluded that the main effect of the pf-mixing is shifting up the p-valence band by 0.85 eV, and pushing down the f-states by the same energy. The energy level schemes for the thorium and uranium compounds deduced from optical spectroscopy are given in fig. 74. The density of states around E F is dominated by the uranium 6dconduction band and the anion-derived p-valence band which both show several peaks due to crystal-field effects. Excitations from the 5f state were not identified. The polar Kerr effect of (ll2)-polished U3P4 and U3As 4 crystals at magnetic saturation is displayed in fig. 75. These spectra are characterized by an extremely large gradient of 0K at low energies with maximum values of up to 6°. Although these curves may show some similarities with the Kerr effect of the uraniummonopnictides (section 7.2), the off-diagonal conductivity displayed in fig. 76
Th3P4 (Th3As 4)
~Iii!~ ~6d u.i-i /
3p(4p)
U3P4 (U3As4 )
6d 3p(4p)
Fig. 74. Density of states schemes for Th3P4 and U3P4 as derived from optical spectroscopy. The same graphs apply to the corresponding As compounds (after Schoenes et al. 1983).
MAGNETO-OPTICAL SPECTROSCOPY
223
o
o
5
_13"
U3As 4
~8K
*% +04+
~4
U3P4
o
c
÷ LLI
'
o
3
++
o ~EK ++
~.
o
~K
~2 -6
° %%~,,~+~ o +
~_
K
,
(")
+
++
I
:
',
:
,
, I , 1 , 1 , 1 , 1 ~
-
q3
I
i
I
1
1
2
3
4
I
I
1 5 Photon Energy [eV]
I
I
2
3
~
I
5
Fig. 75. Polar Kerr effect of (ll2)-polished U3P 4 and U3As 4 at magnetic saturation (after Reim et al. 1984e).
o
10 r% / k
10 °-2xy
e
U3P4
o-
2xy
U3As 4
5 ill )
0
L_o °
+;++ ++
~" -5
-10
/
-5
+
+ (T1xy
C~xy :4T T:15K
B
B :4T T:15K
-10
-15
-15 0
+ + + +
i
i
I
I
I
1
2
3
4
5
0
I
i
i
I
I
1
2
3
4
5
Photon Energy [eV] Fig. 76. Off-diagonal conductivity of U3P4 and U3as 4 at magnetic saturation. The low energy extrapolation has been calculated by Kramers-Kronig inversion and verified using Hall-effect data (after Reim 1986).
224
W. R E I M and J. S C H O E N E S
clearly indicates the completely different origin of the large Kerr effect in these compounds. The absorptive part of ~xy displays for both materials a general to-1 dependence superimposed by some weak structures, which strongly points to a dominant conduction-electron part. This interpretation was corroborated by the absence of a strong interband transition in Olxx and by the extrapolation o f ~xy towards to = 0 shown in fig. 76, which was derived by Kramers-Kronig inversion of Olxy and vice versa (see section 2.3). The very large value of o-lxy(to = 0) given by the extrapolation agrees favorably with the Hall-effect data using eq. (2.73). While the transport measurements (Henkie 1972) give a ratio pH/p of 1/2.8 and 1/2.3 for U3P4 and U3As4, respectively, 1/2.9 and 1/1.8 are calculated from O'Ixy(O)/o'lxx(O ). Since only intraband transitions contribute to O-lxyat to = 0, the magnitude of O-lxy(0) verified by the comparison with transport data excludes any major interband contribution to the magneto-optical spectrum for these compounds. The intraband contribution has been analyzed using only the to-~ term in eq. (2.66). A conduction-electron spin-polarization of -100%, and about 0.65 charge carriers per uranium have been deduced. In order to enhance the weak interband structures in Or2xy compared to the intraband part, fig. 77 shows toOr2xy v e r s u s energy, where the solid line displays the pure intraband contribution as derived
,...>_.
4 3
15
xy /
+
*~'t
#
+
+
+
.
i
I0
+
+
b3cJ 5
I
c~0
o o ~i
UaP
:
4
÷ +
÷
0
I
I
I
I
2
3
4
5
Photon
Energy [eV]
Fig. 77. Experimental values of w~rzxyfor UaP 4 at magnetic saturation. The area below the solid line is the estimated conduction-electron contribution as derived from a fit of ~y. At the top of the figure the absorptive diagonal conductivity trlxx is displayed for comparison (after Reim 1986).
MAGNETO-OPTICAL SPECTROSCOPY
225
from a fit of O'2xy using eq. (2.66). In the (.O0"2xyrepresentation, the intraband term becomes energy independent for hw >>hy. To be able to compare interband contributions in the diagonal and off-diagonal element, o-lx~ is also shown. Each peak in O)Or2xycorresponds to a structure in Olxx and it was not possible to observe any additional transition with magneto-optical spectroscopy. From the comparison of the optical spectra of corresponding U and Th compounds, all structures in o-1~x have been assigned to p-excitations into the crystal-field split 6d band. Consequently, the structures in (DOr2xycorrespond to the same transitions. The relatively large total weight of these structures suggests that, due to strong pf-mixing, the magneto-optical response of the p---~ d excitations is enhanced by high orbital-momentum admixtures. We conclude that in these compounds pfhybridization prevents a typical f---~d response despite of the high magnetic ordering temperature and large saturation moment which both point to a high spin-polarization of the 5f states. 7.5. Ternary compounds and intermetallics
On the search for new high Kerr-rotation uranium compounds with Curie temperatures as high as possible, which then could be useful for optical memory applications, two different concepts have been used. First, to find new materials where uranium is the only magnetic constituent and the magnetic exchange is altered by the U - U separation and the type of neighboring atoms. Results are available for UAsSe (Reim et al. 1985), UCuP 2 (Fumagalli et al. 1988a) and UCuzP 2 and UCH2As 2 (Fumagalli et al. 1988b, Schoenes et al. 1989). Second, to alloy uranium with a transition metal. Here, magneto-optical results for UMnzSi 2 and UMnzGe 2 (van Engelen et al. 1988), UCos+ x (Br/indle et al. 1990) and (TbxFel_x)yUl_y (Dillon et al. 1987) have been published. The ground state of the metallic ternary compounds UXZ, with X = S, Se, Te and Z = P, As, Sb, has been reported to be 5f2 from magnetic data (Bazan and Zygmunt 1972), photoemission data (Brunner et al 1981) and crystallographic considerations (Hulliger 1968). Thus, a magneto-optical response similar to UO 2 (section 7.1), and different from the band-like 5f3-x in the uranium-monochalcogenides UX and monopnictides UZ, was expected. For UAsSe, magnetization and neutron scattering (Leciejewicz and Zygmunt 1972) indicate a Curie temperature of 109 K with a saturation moment ~tot 1"36/XB and a neutron moment/x n = 1.5/x B. The magnetic structure is collinear with ferromagnetic layers perpendicular to the c-axis in the PbFCl-type crystal structure. Figure 78 shows the complex Kerr effect at magnetic saturation for cleaved single crystalline UAsSe with maximum values reaching up to 1.5 ° at 2.8 eV (Reim et al. 1985). The energy dependence looks definitely different from that of the UX and U Z materials. However, O-xy displayed in fig. 79 shows that this different Kerr spectrum does not have its origin in a different electronic structure, but is rather due to different optical constants. The off-diagonal conductivity consists of two dominant structures, one with diamagnetic line shape at 0.74 eV and the second at 4.3 eV with paramagnetic line shape. The low-energy feature has been assigned to =
226
W. REIM and J. SCHOENES
~
13
K
4--
LLI I
t
+
0.5
o
4~
g
++
O 13..
o
%
o o
g
X
0
o_
~ I
E O
~
"9 "
',
:
I
Z
I :
UAsSe
B=5T
(D
T= 10K I
I
I
0 1 2 3 4 5 Photon Energy [eV] Fig. 78. Complex polar Kerr effect of cleaved UAsSe at magnetic saturation (after Reim et al. 1985).
%
÷ +
+
UAsSe
+
-4
+
B = 5T T = 1OK
+ + O-lxy
I
1 Photon
2
3 Energy
I
|
4
5 [eV]
Fig. 79. Complex off-diagonal conductivity for UAsSe at magnetic saturation. The extrapolation for h~o < 0.5 eV has been estimated using the Kramers-Kronig transformation (section 2.3) (after Reim et al. 1985).
MAGNETO-OPTICAL SPECTROSCOPY
227
an f---~d transition. The initial state was interpreted to be a narrow 5f band near E v which is corroborated by a high linear term in the specific heat (Blaise et al. 1980). For this transition, the zero crossing of o-2xy occurs at the same energy where o-w peaks which means that conduction-electron contributions are negligible for this material in agreement with optical spectroscopy (Reim 1986). This is the main reason for the difference in optical constants compared to UX and UZ materials and leads to the different shape of the Kerr spectrum. The broad structure at 4.3 eV occurs at the onset of p excitations in oixx, and has been interpreted similar as in UAs (section 7.2) as the representation of high orbitalmomentum admixtures to the top of the valence band. The uranium pnictogen ternary compounds with copper or nickel crystallize in a high-symmetry structure: UCuP:, UCuAs 2 and UNiAs e are tetragonal (Zolnierek et al. 1987b) and UCuzP a and UCu2As 2 are hexagonal (Zolnierek et al. 1987a). The U - C u ternaries order ferromagnetically in contrast to the U-Ni ternaries, which are all antiferromagnets. The magnetic ordering temperatues are among the highest known so far for uranium compounds reaching 216 K for UCu2P 2 (Zo{nierek et al. 1986). Figure 80 shows the energy dependence of the Kerr rotation 0~ and ellipticity eK of U C u P 2 a s measured on a natural grown surface perpendicular to the c-axis (Fumagalli et al. 1988@. The Kerr rotation reaches about 1.6 °, which is half the value of US (section 7.2). Since UCuP 2 has only half the uranium density of US, the magneto-optical strength of the corresponding transition is similar and the peak has been assigned to a f--~ d transition. Figure 81 shows the energy dependence of ~xy. The f--~ d transition energy is given by the peak in o'w, thus it has a diamagnetic line shape as in most other compounds. The second structure in o'2~yaround 2 eV has been attributed to excitations of high
~.~2 @ 'o v
.,~
.p
o
@ qqw L L @
t
UCuP~
~-8 K T = 113 K &
;
1
B = 4 T A
L O .-4
m
o 13_
x @
0
o. E 0 U
.,b
I
I
I
2
3
I
a
0
Photon E n e r g y
4
5
(eV)
Fig. 80. Polar Kerr rotation OK and ellipticity e K of UCuP 2 as measured on a natural grown surface perpendicular to the c-axis at magnetic saturation (after Fumagalli et al. 1988a).
228
W. R E I M and J. S C H O E N E S
~ i b ~ 2xy
i
/
@
,
O ,--i v
UC U P 2
A
-2
, t
-4
olx×
I0 K
T =
~'~
B = 4 T
O
I
I
I
I
1
2
3
4
Photon
Energy
J
(oV)
Fig. 81. Complex off-diagonal conductivity of UCuP 2 at magnetic saturation. The extrapolation (solid line) has been obtained using the Kramers-Kronig relation between ~rlxyand ~r2xy(after Fumagalli et al. 1988a).
orbital-momentum admixtures from the top of the valence band. Figure 82 displays the complex Kerr rotation of UCu2P 2 with even lower uranium concentration (Schoenes et al. 1989), as measured on a cleaved surface perpendicular to the c-axis. The width of the f---~ d peak is reduced compared to U C u P 2 similar to the U x Y I _ , S b system described in section 7.3, but the Kerr rotation is enhanced reaching up to 3.6 ° . ~4 @ -D
03
:
UCu2P2
. . -SK
cleaved
A
"
(4_ (4_ III I L
T=IOK B = 4T
A
2 L
aA
a_ 1
a
X
E 0
u
Oaal
-EK
mno o AA~ aaaooaooa % *A a% ooaoaouam°a°
0 0
A ~A A
o
I
I
I
I
1
2
3
4.
Photon
Enen9y
[eV]
Fig. 82. Polar Kerr rotation OKand ellipticity eK of UCuzP2 at magnetic saturation (after Schoenes et al. 1989).
MAGNETO-OPTICAL SPECTROSCOPY
229
For most of the alloys of uranium with transition metals, it has been found that the uranium atoms do not, or only very weakly, contribute to the magnetic properties. Consequently, it cannot be expected to find extraordinary magnetooptical properties for these materials. On the other hand, uranium intermetallics with the transition metals Fe, Co or Mn are one straightforward way to obtain uranium-containing materials with magnetic ordering temperatures well above room temperature, which is quite interesting for optical storage applications. (TbxFel_x)l_yUy has been extensively studied in the composition range 0.125 < x < 0.55 and y = 0, 0.04, 0.07 and 0.16 (Dillon et al. 1987). It was found that uranium does neither contribute to the magnetic properties nor to the magnetooptical properties. This finding agrees well with the fact that uranium in U F e 2 is almost nonmagnetic showing a magnetic moment of less that 0.05/~ per uranium (Hilscher 1982). A room-temperature Kerr rotation in excess of 2 ° has been reported for UCos. 3 with a Curie temperature of 360 K (Deryagin and Andreev 1976). However, a systematic analysis of the UCos+ ~ system has shown (Br/indle et al. 1990) that the rotation reaches only a few tenth of a degree and exhibits an energy dependence similar to T b C % , which makes contributions to the rotation from uranium very unlikely. Two of the few compounds where a significant uranium contribution to magnetism has been experimentally observed at room temperature are UMn2Si 2 and UMn2G % (van Engelen et al. 1988), although it should be noted that different authors find no uranium m o m e n t at room temperature (Szytula et al.
L
a
M
n
2
~
-0.10
/ -0.20
~ -0.30
/
/
IMn2Si2 h i
i
i
i
i
i
i
LaMn2Ge2
9-0.1
-0.2( lMn2Ge2 I
.o
2'.o
~:o
4:0
~.o
hv (eV)
Fig. 83. Energy dependence of the Kerr rotation (full line) and ellipticity (broken line) for UMn2Si2 and UMn2Ge2 at room temperature. The horizontal lines visualize the Kerr rotation of LaMn2Si2 and LaMn2G% which amounts to less than 0.01 degrees in the whole energy range (after van Engelen et al. 1988).
230
w. REIM and J. SCHOENES
1988). Both materials are ferromagnets with Curie temperatures of 375 and 380 K and a saturation magnetization of 3.9 and 4.2p~ per formula unit, respectively. Also the corresponding La compounds LaMn2Si2 and LaMn2Ge z are ferromagnets, but with reduced T c of 303 and 315 K and reduced magnetization of 3.1 and 3.0/x B per formula unit indicating a substantial contribution to magnetism from the uranium atoms in the former materials. All four compounds crystallize in the tetragonal ThCrzSi 2 structure. The polar Kerr effect of UMn2Si 2 and UMn2Ge2, measured at room temperature, is displayed in fig. 83. The Kerr effect of the corresponding La compounds was found to be less than 0.01 ° over the entire energy range and is indicated by the horizontal lines. The energy dependence of the Kerr effect of the uranium compounds thus may be attributed completely to uranium contributions. In fact, some similarities with the spectra of the Th3P4 structure compounds seem to exist. Extrapolating from the magnetic data, some increase of the magnitude of the Kerr effect towards lower temperatures can be expected. 8. Conclusion and outlook
This chapter should have made clear that magneto-optics of f-electron systems is a field in full development. Our understanding is rather good for localized f---~d transitions, where atomic models can be applied. However, still not all lanthanides and practically only one actinide have been thoroughly studied. Our understanding of the magneto-optical properties is rather poor for intermetallics and for itinerant f-electron systems, where band-dispersion effects become important. As has been pointed out by us already earlier (Schoenes and Reim 1986), a computation of ~xy(to) or ~'xy(to) from first principles would require a band-structure calculation including spin-orbit splitting, spin-polarization, correlation and screening for the f-electrons. Then the excitation energies and the energy-dependent matrix elements should be calculated. The computation is particularly demanding for moderately delocalized f-systems because localized and itinerant states should be handled on an equal footing. Band-structure calculations or magneto-optical spectra have been published for nickel by Cooper (1965) and Wang and Callaway (1974). Only very recently, a first calculation has been performed for an f-system. In a spin-polarized, semi-relativistic calculation Daalderop et al. (1988) predict a maximum Kerr rotation of 5° for the Heusler alloy NiUSn. It remains to be seen whether this prediction can be confirmed by experiment. In any case, more theoretical efforts in the field of magneto-optics are very welcome and should be encouraged, particularly for the most difficult cases of uranium compounds. In many of the latter systems, the f delocalization and hybridization is near the optimum size to give an enhanced f---~d oscillator strength in a moderately widened energy range, and thus a larger maximum magneto-optical signal than in the localized limit (Schoenes and Reim 1986, Schoenes 1987). The delocalization also favors higher ordering temperatures and in fact many uranium compounds have ordering temperatures which are one order of magnitude larger than those in similar lanthanide compounds.
MAGNETO-OPTICAL SPECTROSCOPY
10"
231
U3As4 - (112) polished (/-'~f USbo.sTeo.2-(111)polished i/ / ~ USbo,sTeo.2"(100) cleaved
10.2 "
!/
/ ~--
USbo sTeos - (100) cleaved
0:: * MnBi * PtMnSb
1(~~
lO
1
\ USe- (100)I "~leaved
2 3 4 Photon Energy [eV]
5
Fig. 84. Figure of merit R sin 2 OK as function of energy for different uranium compounds at magnetic saturation. For comparison, data for other good Kerr rotators are also given (after Reim et al. 1984e).
Figure 84 displays the figure of merit for Kerr rotators ~ = R sin 2 20K for the "best" uranium compounds, together with some data points for the "best" non-f materials. One recognizes that the uranium compounds generally have a one order of magnitude larger figure of merit. Unfortunately, the ordering temperatures of these materials are below room temperature and most applications for erasable optical disks call for operation temperatures at or above room temperature. Efforts are undertaken to increase the ordering temperature by alloying. A somewhat connected approach is the exchange coupling of thin films of good magneto-optical materials with thin films of ferromagnetic materials with highenough Curie temperatures. A third way consists in trying to enhance a weak magneto-optical signal of some known magnetic material with high-enough temperature of magnetic ordering. Such an enhancement may be achieved by increasing the off-diagonal conductivity or decreasing the optical coefficients A and B in eqs. (2.30)-(2.33) (Katayama et al. 1988 and Reim and WeUer 1988, 1989). The latter effect contributes, e.g., to the large Kerr signals occuring in NdS, Nd3S4, TmS and TmSe (sections 4 and 6). Besides the theoretical efforts, basic research activities in magneto-optics will also lead to the investigation of more exotic materials like heavy electron systems or unconventional superconductors. Finally, it is conceivable that magneto-optics will be applied in the future to study f states in trans-uranium compounds. References Ackermann, R.J., R.J. Thorn and G.H. Winslow, 1959, J. Opt. Soc. Am. 49, 1107. Agranovich, Y.M., and V.L. Ginzburg, 1966,
in: Spatial Dispersion in Crystal Optics and the Theory of Excitons (Interscience, London).
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chapter 3 I NVAR: MOMENT-VOLUME INSTABILITIES IN TRANSITION METALS AND ALLOYS
E. F. WASSERMAN Tieftemperaturphysik Universit&t Duisburg D-4100 Duisburg F.R.G.
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. Introduction and outline of the Invar problem . . . . . . . . . . . . . . . 2. Technical applications of Invar- and Elinvar-type alloys . . . . . . . . . . . . 3. Magnetic phase diagrams of Invar and Elinvar systems . . . . . . . . . . . . 3.1. Phase diagrams of binary systems . . . . . . . . . . . . . . . . . . 3.1.1. F e - N i system . . . . . . . . . . . . . . . . . . . . . . 3.1.2. F e - P t system . . . . . . . . . . . . . . . . . . . . . . 3.1.3. F e - P d system . . . . . . . . . . . . . . . . . . . . . . 3.1.4. F e - M n system . . . . . . . . . . . . . . . . . . . . . . 3.1.5. N i - M n system . . . . . . . . . . . . . . . . . . . . . . 3.1.6. C o - M n system . . . . . . . . . . . . . . . . . . . . . . 3.1.7. C o - F e and C o - N i systems . . . . . . . . . . . . . . . . . . 3.1.8. C r - F e system . . . . . . . . . . . . . . . . . . . . . . 3.1.9. C r - M n system . . . . . . . . . . . . . . . . . . . . . . 3.2. Phase diagrams of ternary systems . . . . . . . . . . . . . . . . . . 3.2.1. F e - N i - C r system . . . . . . . . . . . . . . . . . . . . . 3.2.2. F e - N i - M n system . . . . . . . . . . . . . . . . . . . . . 3.2.3. F e - N i - C o system . . . . . . . . . . . . . . . . . . . . . 3.2.4. F e - C o - M n system . . . . . . . . . . . . . . . . . . . . . 3.2.5. N i - C o - M n system . . . . . . . . . . . . . . . . . . . . . 3.2.6. C o - F e - C r system . . . . . . . . . . . . . . . . . . . . . 4. Fundamental properties of Invar and Elinvar systems . . . . . . . . . . . . . 4.1. Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . 4.2. Volume magnetostriction . . . . . . . . . . . . . . . . . . . . . 4.3. Forced volume magnetostriction and high-field susceptibility . . . . . . . . . 4.4. Pressure dependence of Curie (N6el) temperatures and magnetization . . . . . . 4.5. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . 4.5.1. Concentrations dependence of Curie and N6el temperatures . . . . . . . 4.5.2. Concentration dependence of magnetic m o m e n t . . . . . . . . . . . 4.5.3. Temperature dependence of the magnetization . . . . . . . . . . . 4.6. Spin waves and spin-wave stiffness . . . . . . . . . . . . . . . . . 4.7. Elastic properties . . . . . . . . . . . . . . . . . . . . . . . 4.8. Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . 5. Survey on theories of Invar . . . . . . . . . . . . . . . . . . . . . 5.1. Early Invar models . . . . . . . . . . . . . . . . . . . . . . . 5.2. M o m e n t - v o l u m e instabilities in 3d-elements . . . . . . . . . . . . . . 6. Towards a new understanding of the Invar effect . . . . . . . . . . . . . . 6.1. Theoretical evidence for high-spin-low-spin-state transitions in Invar alloys . . . . 238
240 245 248 249 249 249 250 251 253 254 256 256 257 258 258 259 260 261 262 263 264 264 268 271 273 276 276 277 278 280 285 292 299 299 301 305 305
INVAR 6.2. F i n i t e - t e m p e r a t u r e m o d e l s . . . . 6.3. E v i d e n c e for h i g h - s p i n - l o w - s p i n - s t a t e 7. C o n c l u s i o n s a n d o u t l o o k . . . . . . References . . . . . . . . . . . . .
. . . . . . . . . . . transitions from experiment . . . . . . . . . . . . . . . . . . . . . .
239 .
. . . . . . . . . . . . . . . . . . . .
310 310 315 317
1. Introduction and outline of the Invar problem
The 'Invar effect' originates from investigations by Ch.E. Guillaume who found, already in 1897, that ferromagnetic fcc FeNi alloys at concentrations around Fe65Ni35 show almost constant- 'invariant'-thermal expansion as a function of temperature in a wide range around room temperature (Guillaume 1897). According to his results the linear thermal expansion coefficient a = ( 1 / l ) / ( d l / d T ) of Fe65Ni35 Invar at 300 K is about 1.2 x 10 -6 K -a, thus an order of magnitude smaller than in the pure components Fe and Ni and even smaller than in a Pt-10% Ir alloy, the material used for the prototype meter defined since 1875. The practical importance of Guillaume's detection for the construction of precision instruments and seismographic devices as well as his finding of the temperature-independent elastic behavior of 3,-FeNiCr alloys (Guillaume 1920) named 'Elinvar' and used (with some precipitation-hardening additions) as hair spring material in watches- honoured him with the Nobel prize in 1920. Figure la schematically shows the relative volume (or length) change o~ = AV/V with temperature for an Fe65Ni35 Invar alloy. The full curve gives Wexp(T) as determined experimentally. It can be seen that, with respect to a nonmagnetic ('hypothetic') reference sample (tOnm(T), dashed curve), the Fe65Ni35 Invar alloy has a positive magnetovolume effect below the Curie temperature T c. The difference Wexp(T ) - ¢Onm(T) = ~os(T) is called the spontaneous volume magnetostriction (dashed-dotted curve in fig. la), with a maximum value at T = 0 of O~s0= 1.9 × 10 -2 for Fe65Ni35. Values of the same order of magnitude are found in other Invar systems, as we shall see later. Figure lb shows the temperature dependence of the thermal expansion coefficient a(T) as derived from fig. la. The full curve gives Otexp(T ) as determined from O~xp(T). The dashed curve shows the temperature dependence of the nonmagnetic reference sample, O~nm(T), as calculated from the Gr/ineisen relation, using data from specific-heat measurements. (The problem of finding a reference sample from first principles will be discussed later.) Assuming that Otexp(T ) = O~nm(T) + otto(T), where am(T ) denotes the magnetic contribution to the thermal expansion (dashed-dotted line in fig. lb), one can see that in this type of analysis, am(T ) is negative throughout the range from zero Kelvin to above the Curie temperature Tc. Ch.E. Guillaume (1897) also determined the concentration dependence of aexp
240
INVAR
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Fig. 2. Schematic representation of the concentration dependence of ~os0and %xp at room temperature
in the FeNi system. The vertical line marks the structural a - y transition in the system.
242
E.F. WASSERMAN
spontaneous volume magnetostriction tos0 reaches maximum values. However, nearness to a y - a transition is not a necessary condition for the occurrence of the Invar effect, as we shall see below. Returning to fig. lb and the temperature dependence of the expansion coefficient, two remarks should be added. First, at low temperatures O~exp(T) is negative and has a minimum. This means that the sample first shrinks when raising the temperature from T = 0. This minimum in aexp(T ) is not Invar typical as we have demonstrated recently (Wassermann 1989b). Second, concerning the high-temperature behavior, one can see that around and above the Curie temperature T c (arrow in fig. 1), the expansion anomaly does not vanish (nor has any sharp 'features'). This has to be explained in any type of model describing the Invar effect. A further example for the extraordinary properties of Invar at higher temperatures is given in fig. 3. where the heat capacity Cp(T) for three FeNi alloys as measured by Bendick et al. (1978) in the temperature range 300-1200 K is shown. Above the 'Gr/ineisen' background (broken line), which refers in the same way as
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Fig. 3. Specific heat Cp as a function of temperature as measured by Bendick et al. (1978) on three FeNi alloys. The respective Curie temperatures are marked by arrows. The temperature dependence of Cp for respective non-magnetic reference samples are shown by the dashed horizontal lines. The dashed-dotted lines give Cp(T) as calculated for FeNi by Kakehashi (1981) within the finitetemperature local-environment model (see section 6.2).
INVAR
243
TABLE 1 Invar- and Elinvar systems. fcc structure
3,-range at 20°C
Magnetic Order
Fel_xNi x Fel_xPt ~ Fe~_xPt x Fe~_xPd x Fe~_xMn x Nil_xMnx
0.3 ~<x ~<1.0 0.25 ~<x ~<0.5 0.23 ~<x ~<0.5 0.3 ~ x ~<0.6 0.2 ~ x ~<0.58 0 ~<x ~<0.45 0.6~<x ~<0.8 0.27 ~<x ~<0.5 0.05 ~<x ~<0.25 0.2 ~ x ~< 1 See resp. fig. See resp. fig. See resp. fig. See resp. fig. See resp. fig. See resp. fig. 0.29 ~<x ~<~0.4 0.1 ~ x ~<0.17
FM (SG) FM FM FM AF FM-SG AF FM-SG-AF FM FM FM-SG-(AF) FM-SG-AF FM FM-SG-AF FM-SG-AF FM FM FM
diso. ord. diso. diso.
Coa_xMn x Co~FG Col_~Nix (Fe1_~Ni~)yCr~_r
(Fel_xNix)yMnx_y (Fe~ ~Ni~)yCol_y (Co~ ~Mn~)yFex_r (Co~_~Mn~)rNil_y (Co~_~FG)yCr~_ r (Fe~_xNi~)92(Cu , V)8 FeTo(Pta_~Rex)30 bcc structure
a-range at 20°C
Magnetic order
Cr~_~FG Crl~xMn~
0 <~x ~<1.0 0 ~<x ~<0.6
FM-SG-AF AF
hex. structure
e-range at 20°C
Magnetic order
CO~_xCG
0 <~x ~<0.15
FM
amorphous structure
Components
Magnetic order
Fe~_~ETx Fel_~B~ Fe~_xP~ (Fel_~TMx)75_s0G25_2o
ET: Sc, Y, Zr, Hf, La
FM, SG FM FM
TM: Cr, Mn, Co, Ni G: Si~ B, P
FM, SG
Laves phases and Intermetallic compounds Components
Magnetic order
ETFe 2 RECo 2 (Zrl_xNbx)Fe 2 Zr(Fel_xCox) 2 REzFe14B RE2Fe17 Dy2(FexCol_~) 17 Dy2(FexAll_~)17 LaFela_xAl x
FM FM FM FM-SG FM AF, FM (Ferri) AF AF AF
ET: Sc, Ti, Zr, Hf RE: Gd, Tb; Dy, Ho, Er
RE: all except Eu RE: Y, Dy, Lu
Magnetic phase diagram Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. -
5 6 6 7 8 9 9 10 11 11 13 14 15 16 17 18
Magnetic phase diagram Fig. 12
244
E.F. W A S S E R M A N
in figs. la,b to a nonmagnetic reference alloy, there is in addition to the magnetic specific heat, which peaks at the respective T c values, a considerable excess specific heat-capacity due to the Invar effect. Characteristically, a broad 'bump' in Cp(T) occurs around 500K in all three alloys. Similar effects have been encountered by the same authors in other Invar alloys like, e.g., F%0(NixMnl_x)50 (as will be seen fig. 41) or Fe80_xNixCr20 (Bendick and Pepperhoff 1981). Note that for Fe65Ni35, where T c coincides with the 'bump', the A-type anomaly in C p ( T ) is washed out. The original work by Guillaume was the basis for widespread experimental and theoretical activities, in the 1950s and 1960s especially in Japan and Europe, and with the increasing understanding of solid-state magnetism, the number of publications about Invar and Invar-related topics increased drastically. The reason for this increase was two-fold. First, the observation of Invar anomalies remained by no means bound to ferromagnetic fcc FeNi alloys. As shown in table 1, Invar anomalies are, to date, observed in ferromagnetic (FM) as well as antiferromagnetic (AF) binary, ternary (and also multi-component) alloy systems. The lattice structure is of no influence, and Invar systems can have fcc, bcc, hexagonal and other structures (e.g., fct) or even be amorphous. Moreover, Invar anomalies are observed in rare earth (RE)-transition metal (TM) compounds with Lavesphase structure, e.g., RECo 2 or compounds like Fe14Nd2B. The key point is that the systems are rich in at least one (but a specific, as we shall see later) 3d-transition element. There are no purely 4f (nor insulating) Invar alloys or compounds. This shows that the Invar effect is obviously a problem of itinerant 3d-magnetism. Secondly, although the name Invar resulted from the anomaly in the thermal expansion, a broad variety of physical anomalies have, to date, to be considered as Invar typical. Table 2 lists the main physical properties in which Invar anomalies are observed, as a function of composition and variable external TABLE 2
List of physical properties in which Invar anomalies are observed. Property
Variable
Thermal expansion Lattice constant Spontaneous volume magnetostriction Spontaneous volume magnetostriction at T = 0 Heat capacity Derivative of the electrical resistivity Magnetization Hyperfinefield distribution Forced volume magnetostriction High-field susceptibility Pressure dependence of magnetization Pressure dependence of Curie temperature Pressure dependence of N6el temperature Young and bulk modulus Elastic constants Neutron scattering
(Al/l)(T), a(T) a(T) o,~ = (AV/V)(T)
~o~(T= O) = W~o cv( T)
dp/dT (M/Mo)(T) P(BHF) (O~o/OH)(T) XnF(T ) -(dM/dp)r,n -(drc/dp) -(dTN/dp) E(T), B(T) CL(T), C44(T), C'(T) D(T), oJ(k, T)
INVAR
245
parameters like temperature, magnetic field and pressure. The listings in tables 1 and 2 demonstrate the richness of the Invar effect and, at the same time, the complexity incorporated. Earlier reviews summarizing the field have been given in the literature (Nakamura 1976, Honda Memorial Series on Material Science, No. 3, 1978, Freeman and Shimizu 1979, Shimizu 1979, 1983). Recent updatings on the physical understanding of Invar have been given by Wassermann (1987, 1989 a,b) and in the proceedings of the International Symposium on Magnetoelasticity and Electronic Structure of Transition Metals, Alloys and Films (ISOMES), which took place in Spring 1989 in Duisburg, FRG (Wasserman e t al. 1989). Finally, we mention here that the name 'Invar' is somewhat misleading in describing all the observed anomalies. 'Invar' stands for alloys showing minimum thermal expansion coefficients (maximum spontaneous volume magnetostriction) in certain ranges of composition and temperatures (cf. Fe65Ni35 in fig. 2). A more general headline, summarizing all the features in a better way would be: 'moment-volume instabilities in 3d-element rich systems'. The relevance of this will be shown in section 6. 2. Technical applications of Invar- and Elinvar-type alloys Since their discovery, Invar- and Elinvar-type alloys have experienced increasing technical applications. Although the annual production of these materials has never been very high (as compared, e.g., to stainless steel) and the total amount produced world wide to date is difficult to figure (we estimate 104-105 tons per year), one can see from table 3 that the practical use of Invar and Elinvar is widespread. As shown in table 3, two major fields for technical applications can be distinguished. First, there is a group of materials, which meets the requirement of dimensional stability with temperature, i.e., low thermal expansion coefficients. To this group belong the classical FeNi Invar alloys (Guillaume 1897), the so-called 'Super-Invar' alloys, which are ternary FeNiCo alloys with 35 at% Ni and 4-6 at% Co (Masumoto 1931), and 'Stainless-Invar' (Masumoto 1934), which are ternary Co-rich alloys of CoFeCr with roughly 9 at% Cr and 52 at% Co. TABLE 3 Practical applications of Invar and Elinvar alloys.
Requirement: dimensional stability Bi-metals; wires for printers and x - y recorders; liquid natural gas (LNG) tanks and pipelines; precision machine tools; precision pendulums; precision capacitors; precision moulds; transistor bases; lead frames for integrated circuits; membranes; springs; glass (ceramic)-metal seals; pressure gauges; thermostats; bending meters; gravity meters; flow meters; astronomical telescopes; seismographic devices; microwave guides; resonant cavities; laser light sources; radar echo boxes. Requirement: elastic stability springs; segments, plates and wires for watches, chronometers, time recording instruments and various electronic and electromagnetic instruments; vibrators for standard frequencies; mechanical filters; reeds; reed relays; tuning forks; filters; delay lines.
246
E.F. WASSERMAN
The second class of materials is used for practical purposes, where stability of the elastic behavior (bulk- and shear modulus) with temperature is required. These materials often show low thermal expansivity too. To this group belong the classical Elinvar alloys of FeNi (40-45 at% Ni), ternary alloys of FeNiCr (512at% Cr) detected by Guillaume (1920), and the so-called 'Super-Elinvar' (39 at% Ni, 10 at% Co, 10 at% Cr, Fe balance). In 1952, Masumoto and Saito (1952) found that annealed ternary alloys with compositions in the neighborhood of Stainless-Invar (cf. table 4) show high elasticity but small temperature coefficients of the Young- and shear moduli. These alloys were named 'Co-Elinvar', since they are Elinvar-type alloys containing cobalt. With the addition of Ni, the concentration range of low shear- and Young-modulus coefficients can be widened, so that there are also Co-Elinvartype alloys in the quarternary system FeNiCoCr (Masumoto et al. 1954). In table 4 we have summarized the concentration ranges of the 3d-element constituents of the major Invar and Elinvar type alloys used technically and mentioned before. All these alloys have fcc y-type structure and they are ferromagnetic. Figure 4 shows, albeit in a qualitative way, where these alloys are positioned in the quarternary constitution diagram of FeNiCoCr. The diagram demonstrates the linear law of substitution of Co and Cr by Ni in both series, from Stainless-Invar to Super-Invar and Co-Elinvar to Super-Elinvar for the minimum temperature coefficients of thermal expansion and shear modulus, respectively (Masumoto et al. 1956). If the alloys listed in table 4 would consist of the 3d-elements only, they would not meet the necessary requirements for technical application, mainly because TABLE 4 Commercial Invar- and Elinvar-type alloys. Name
Composition (Fe balance) Ni(at%)
Co(at%)
Cr(at%)
Invar
35-39
Super-Invar
31-33
4-6
Stainless-Invar Elinvar
40-44 36-38 36-39
52-54 5-10
9-10
9-25
58-60 20-35
9-10 10-12
Super-Elinvar Co-Elinvar
Anti ferromagnetic Elinvar MnNiFe CrFeMn MnCu
Remarks Mn(at%) Additions for hardening and prevention of aging (in concentrations ~<0.5 at%): Be, Mn, Mo, Si, Se Cr replacable by Mo, W. Additions for hardening and reduction of AE-effect: Ti, Be, AI
5-12 5-10
20-28 85-95
Velinvar (V for Cr) Moelinvar (Mo for Cr) Thungelinvar (W for Cr) and/or additions for Ti, Mn, Si, AI
72-80 -1 50-55
Additions of Fe, Co, Mo, W
INVAR
247
CI"
Ni stointe~s~ Invar
\l,
Fe Fig. 4. Commercially used Invar compositions within the quarternary phase diagram of FeNiCoCr.
they show aging effects (caused by C impurities) and insufficient mechanical strength. Thus, their workability and machinability would be too limited. Therefore, in commercial Invar- and Elinvar-type alloys a broad spectrum of additional elements is used with the purpose of increasing the mechanical strength of the alloys (mainly through precipitation hardening after different heat treatments) and reducing the aging effects as well as the AE-effect. Moreover, the weldability and sometimes the corrosion resistance can be improved, if these factors are of importance. Some of the elements added to achieve commerciably usable Invarand Elinvar-type alloys are listed in table 4. The commercial alloys often have six to eight, sometimes even more, different constituents. The spectrum of brand names accompanying this variety is correspondingly broad. Detailed listening of compositions of commercial Invar and Elinvar would go far beyond the scope of this chapter. Although there are many data sheets describing Invar, sometimes the detailed composition of the alloys is kept a companies secret, while sometimes the concentrations given differ considerably in numerical values for the same type of material, or show other contradictory results. Nearly all the literature up to 1965 is listed by McCain and Maringer (1965). Excellent summaries on technical research on and application of Invar and Elinvar can be found in the Honda Memorial Series on Physics and Applications of lnvar (1978). Finally, we mention that sometimes the ferromagnetism of Invar and especially Elinvar alloys is an inconvenience for practical applications (e.g, hairsprings for watches, delay lines). Therefore, antiferromagnetic alloys have been developed. Although the spectrum of Invar and Elinvar systems showing antiferromagnetism is wide (cf., table 1), many of the alloys cannot be applied technically, again because of poor machinability, workability, etc.. In practical use are antiferromagnetic Cr-rich CrFe alloys (with small additions of Mn), MnCu alloys with additions of Fe, Co, Mo or W (in amounts of 1-10 at%) and ternary FeNiMn alloys. We are, however, convinced that with further understanding of the basic principles, more antiferromagnetic Invar and Elinvar systems suitable for practical purposes will be developed.
248
E.F. WASSERMAN
3. Magnetic phase diagrams of Invar and Elinvar systems In this section, we summarize the magnetic phase diagrams of the most important binary and ternary alloy systems showing m o m e n t - v o l u m e instabilities in general and Invar or Elinvar behavior in special composition ranges (cf., also table 1). Besides the magnetic-transition temperatures, Curie temperatures T c for ferromagnetic (FM) transitions, N6el temperatures T N for antiferromagnetic (AF) transitions and Tf for spin glass (SG) or re-entrant spin glass ( R S G ) like transitions, the diagrams will also contain information about the structural phases occurring in the systems• Conventional nomenclature is used (e.g., 3/for fcc, a for bcc and e for hexagonal structures)• In almost all systems presented, the magnetic order has been studied down to 4He-temperatures and the magnetic phasetransition lines are, in general, well-known. Concerning structural transitions, the knowledge of martensitic transformation temperatures M s is very important, since most of the experimental data are achieved on alloys quenched from high temperature (typical 1000-1200°C), where the solubility is correspondingly high. Ms-temperatures can be determined with satisfying accuracy from different types of studies, even at low temperatures• Equilibrium solubility lines, necessary to understand the decomposition of metastable alloys, are often less well-known (a famous example is the FeNi diagram, shown in fig. 5, which is still debated)• I
i
8OO l
/
T i//
i
.,,,... io ...__.,..,,,
i
/°-° T: o/ \;\
i
°'°\
'\.\\
(K)
°
I
\o
/
./"
/.
/'/
/J2i
\ ;
qi
//
L/ o"i
i
90%
~
/
/~/
/)~ 20
tXF~
/ /o i o~o- ,.~ ' . /~I
./
200
// ! //" ./- 3
r .~ / l ~
"% \
/\
[
,
,,~
/
• Fex N'l-x
i
40
60
I
c~ T'N~ ~
80-
100
ot % Fe
Fig. 5. Magnetic phase diagram of FexNi1 x based on experimental data by Crangle and Hallam (1963) and Stamm (1988). Curie temperatures Tc in the 3,-phase and the magnetic moment of the Fe-atoms in the a- and 3,-phase/%° as a function of concentration are given. The dashed-dotted lines show the structural equilibrium lines (Kubaschewski 1982) and the martensitic transformation lines (Jones and Pumphrey 1949), respectively. The hatched region indicates a low-temperature mixed magnetic phase as observed in ac susceptibility measurements (Miyazaki et al. 1985).
INVAR
249
These lines are known with the least accuracy at temperatures below about 500 K. This is due to the fact that the times needed for reaching structural equilibrium in solid-state reactions become extremely long, especially if broad miscibility gaps and extended eutectic (or peritectic) decomposition ranges occur. Therefore, the values for the stability ranges of the respective structural phases in the binary systems given in table 1 and the equilibrium stability lines in the ternary diagrams (section 3.2) have to be regarded with some care. It was believed for a long time that the Invar effect is bound to the occurrence of a y - a transition and a deviation of the magnetic moment tz from the Slater-Pauling curve, simultaneously observed on approach to the instability limit. Although the latter condition was put into doubt, because ordered FePt alloys around the composition F%Pt (Kussmann and von Rittberg 1950) did not show any weakening of the moment, we shall, in the following diagrams, give the values for the average magnetic moments ~ as a function of composition as well, if data are available from the literature.
3.1. Phase diagrams of binary systems 3.1.1. Fe-Ni system Figure 5 shows the magnetic phase diagram of the archetypical Invar system FexNil_ x. Data were originally published by Crangle and Hallam (1963) but they have repeatedly been verified since then (see e.g., Stamm 1988). One can see that on approach to the y - a phase-transition there is the well-known strong downward deviation of the moment/x from the Slater-Pauling curve, accompanied by a decrease in the Curie temperature T c. The dashed-dotted lines in fig. 5 give martensitic transformation temperatures M s as a function of concentration as determined by Jones and Pumphrey (1949) by quenching from high temperatures. The lines represent temperatures corresponding to 10 and 90% martensitic transformation, respectively. The equilibrium phase-transition lines in FeNi are still debated in this composition region (Kubaschewski 1982) and the question about the existence of an ordered Fe3Ni phase is still unanswered. Due to the structural mixing, the magnetic ground state of the alloys around the T - a transition region is also inhomogeneous. The presence of RSG-like properties has been proposed by Miyazaki et al. (1985), the RSG-region being indicated in fig. 5 by the hatched region. As shown in our own work (Zfihres et al. 1988) antiferromagnetic "y-Fe precipitations in a ferromagnetic Fe65Ni35 3/-type matrix probably describe the situation better. For the Invar effect in this system, these magnetic inhomogeneities are not of much relevance. 3.1.2. Fe-Pt system Figure 6 shows the magnetic phase diagram of FexPtl_ x in the concentration range 0.55 ~<x ~<0.95 around the concentration of the Fe3Pt-phase with Cu3Au (L12) superstructure. Values of the Curie temperatures in the ordered (Tc) and disordered state (T~) as well as the magnetic moment /xF e as a function of the Fe-concentration have been determined many times. We show very reliable data of Sumiyama et al. (1978). The structural equilibrium lines of Fe~Ptl_x are only
250
E.F. WASSERMAN
800 T
ff
i .,____FePt -.. L1 o ~
{K)
t i_ -ii
i
i [ : i I i
Fe3Pt L12
_
I ff 3
"" " " ' -
600
"Tc°\...
°-~'a j .
{laB}
Tcd>,S-;~,:" ~" 400
°',o, %,, . °'~ ~./
2
* ordered o disordered
°x
200
"d
o
rex Ptl x
I1' 60
o
,ii
70
, 80
'11
o
90
{at % Fe
Fig. 6. Magnetic phase diagram of FexPtl_ x as determined experimentally by Sumiyama et al. (1978) for ordered (full dots) and disordered (open dots) alloys in the range close to the y = a transition. Note that although the Curie temperatures (T c and TO for the ordered and disordered state, respectively) decrease on approach to the y - a phase boundary, the iron moment/~e does not deviate from the Slater-Pauling curve.
known inaccurately (Hansen and Anderko 1958). However, martensitic y - a transformations indicated by the dashed-dotted lines in fig. 6 occur on cooling, both in the disordered and in the ordered state. FePt alloys show Invar properties in the disordered as well as the ordered state, but no deviations of the Fe moment from the Slater-Pauling curve on approach to the y - a transition, and no mixed magnetic state at low T (cf., fig. 6). This fact, first realized by Kussman and von Rittberg (1950), for a long time gave this system a somewhat unique position in the discussion about the origin of the Invar effect, until other Invar systems with similar properties, like FexPdl_ x and amorphous Fe~BI_~, were discovered. We will discuss this in more detail below. 3.1.3. F e - P d system
Invar anomalies in the thermal expansion near 30 at% Pd in Fe were first investigated by Kussman and Jessen (1962). They reported that the thermal expansion coefficient of Fel_xPdx alloys quenched from the high-temperature y-phase shows a minimum around x = 3.0. Figure 7 gives (dashed-dotted lines) the high-temperature part of the structural equilibrium phase diagram as established by Raub et al. (1963). Obviously, an ordered Fe3Pd phase does not exist in this system, in contrast to FePt (cf., fig. 6). According to the data by Raub et al., FePd alloys in the Invar range consist of two phases (a + FePd) when in equilibrium. Thus, quenching is a necessary condition to prevent phase separation. On the other hand, quenching from the
INVAR I
1200 i T (K)
I
[
I
~"
251 I
I
I
P
Fel_ x P d x
(PB)
\
1000
3
800
2 600
'\ "~M~
400
200 ÷ .I
0
i
Fe
I
20
;I
I
I
I
I
40
I
60
I
0 80
ot% Pd
Fig. 7. Magnetic phase diagram of Fel_xPdx. Curie temperatures as determined by Matsui et al. (1980b) (open dots) as well as the average magnetic moment (crosses) and Curie temperatures measured by Fujimori and Saito (1965) (solid dots) are given. The high-temperature structuraltransition lines (dashed-dotted) are due to Raub et al. (1963), the low-temperature structuraltransition lines to Matsui et al. (1980a). y-range leads to martensite at low temperatures. This has been studied in detail by Matsui et al. (1980a), who determined the position of the Ms-line (see fig. 7) and showed that on lowering the Pd-concentration, other structural transitions occur in FePd, an fct y ' - p h a s e , a mixed region a - y ' , and a bcc a-phase below 28 at% Pd (for T = 0 K). The same group (Matsui et al. 1980b) also determined the concentration dependence of the Curie temperatures in the y-phase (open dots in fig. 7) as well as the concentration dependence of the average magnetic m o m e n t / 2 (crosses in fig. 7). While their data on T c agree well with earlier results by Fujimori and Saito (1965) (solid dots in fig. 7), there is a discrepancy in t h e / 2 ( x ) - d e p e n d e n c e for x ~< 0.3, where Fujimori and Saito (1965) found a deviation of the moment from the Slater-Pauling curve (data not shown in fig. 7). This was, however, obviously due to metallurgical problems. Later investigations by Matsui et al. (1983) confirmed that F e P d - like F e P t - does not show a decrease in magnetic moment near the y - a transition in the Invar range. This demonstrates the importance of FePd in the search for the basic understanding of the Invar effect. 3.1.4. F e - M n system An interesting alloy system, showing fcc structure in a wide range of composition but only antiferromagnetic order, is FeMn. The magnetic phase diagram is shown in fig. 8. Between 20 and about 60 at% Mn in the y-range the N6el temperatures
252
E.F. WASSERMAN I
I
I
I
I
800
]
I
I
Fe 1_× Mn x
{K)
,~ +D[Mn
i
600
O{Mn
~ i
i~ \
~oo
i~i
ll_n/~_
° II--o
°I°" x."="
AF XX~x~x.~TN x
2oo
/
'" AF
Ms
o
I
I
Fe
"
I
20
40
60
80
Mn
at % M n
Fig. 8. Magnetic phase diagram of Fel_xMnx as determined by Ishikawa and Endoh (1967) (solid squares) and Stamm (1988) (open and solid dots). The inset shows the non-collinear spin structure in the y-range as found by Endoh and Ishikawa (1971) from neutron-scattering experiments. N6el temperatures in the range with aM. structure are due to Nakai et al. (1986). rise from 350 to about 520 K as confirmed by different authors (Ishikawa and E n d o h 1967, Stamm 1988). The spin structure is noncollinear (Endoh and Ishikawa 1971) but isotropic, as indicated in the inset to fig. 8. On the Fe-rich side there is a steeply decreasing a - y transition line in the concentration range up to 10 at% Mn (Schumann 1967). Alloys with higher Mn concentrations, i.e., those in the y-range, undergo a martensitic transformation at lower temperatures into e-martensite with hexagonal lattice structure. This has been confirmed by M6ssbauer measurements (Trichter et al. 1978) and T E M investigations (Gartstein and Rabinkin 1979). The latter authors also showed that e-martensitic needles are present up to about 50 at% Mn, with the volume fraction of the needles in the rest-austenite being almost temperature and concentration independent. Their findings were supported in recent magnetization measurements by Stamm (1988) on Fei_xMnx, with 0.26 ~< x ~< 0.59, who observed SG like splittings between the field-cooled (FC) and zero-field cooled (ZFC) susceptibility curves at temperatures indicated by the full dots in fig. 8. The magnetic ground state of y-FeMn alloys is thus inhomogeneous due to the structural inhomogeneity. It remains an open question, whether the low-lying y - e transitions are accompanied by a transition from the noncollinear spin structure at high temperatures, into a collinear spin structure at low temperatures because of the tetragonal distortion of the lattice occurring simultaneously. In the range 0.53 ~< x ~< 0.69, the Fex_xMn x system shows a y--aMn two-phase range, and for x i> 0.69 up to 100% Mn there occurs an aM,-phase , which also
INVAR
253
orders antiferromagnetically. The N6el temperatures shown in fig. 8 (crosses) have been determined by Nakai et al. (1986). 3.1.5. N i - M n system In contrast to the FeMn system, the system NixMn I x has a wider 3~-range, and shows a broad spectrum of magnetic order, ferromagnetism on the Ni-rich side accompanied by RSG- and pure SG-behavior, ferromagnetism in the middle of the diagram around the ordered L10 phase of NiMn, and antiferromagnetism on the Mn-rich side. The magnetic phase diagram, which has been subject of many investigations, especially on the Ni-rich side, is shown in fig. 9. The Curie temperatures T~ of the disordered alloys decrease from 632 K for pure Ni to about 100 K at 24 at% Mn (Tange et al. 1978), where a pure SG phase starts to occur. There is also a reentrant SG like phase (hatched region in fig. 9). Data for the alloys in the 20-30 at% Mn range are results from magnetization measurements by Kouvel and co-workers (full dots in fig. 9) (see Abdul-Razzaq and Kouvel 1987 and references therein), and neutron scattering results by Hennion et al. (1984) (plus signs in fig. 9). The decrease of T~ with rising Mn concentration is accompanied by a deviation of the average magnetic moment /2 from the
\ \PMo \' 800 T
e--
iNiMn
",,-'~.4
I I
(K}
Ii
600
\. Tc
I
I
I
~- Mn1_x Nix
/
i i i
+/
i
o
t I,i Mn
20 Elinvar
I
/o"
,o
-
"D
~~ "
SG ~, • 0
60
40 at%
FM
I:
i i i i
ilili
200
,-I (laB~
?
i 400
P
X
Ni
80 ' Invar
Ni '
Fig. 9. Phase diagram of Mnl_xNi x. On the Ni-rich side, Curie temperatures (full squares) and the average magnetic moment (open squares) have been determined by Tange et al. (1978). Other transition temperatures, especially for the spin-glass phase, are from Hennion et al. (1984) (plus signs) and Abdul-Razzaq and Kouvel (1987) (solid dots). Curie temperatures in the ordered alloys T c (crosses) are from Kaya and Kussman (1931). Note that on the Mn-rich side the system orders antiferromagnetically. T N values (open dots) are due to Honda et al. (1976). Structural transition lines (dashed-dotted) are also due to these authors.
254
E.F. WASSERMAN
Slater-Pauling curve, shown by the dashed curve (open squares) in fig. 9 (Tange et al. 1978). This deviation and the occurrence of a metamagnetic behavior in NiMn around the composition Ni3Mn has also been widely studied theoretically in different types of models (see, e.g., Jo 1980, Kakehashi 1984 and references therein). The scattering of the data for T d and T~, as well as the discrepancies between experimental and theoretical results to our feeling originate from the fact that in the range around ordered Ni3Mn (the existence of that phase has not been proven yet), there are also high-temperature P M - F M transitions possible in annealed, ordered alloys. This has been known since a long time (Kaya and Kussmann 1931, K6ster and Rauscher 1948). Curie temperatures T c are shown by the crosses in fig. 9. Quenching of NiMn alloys with Mn concentrations exceeding 20 at% with different quenching rates therefore leads to different magnetic short-range order (SRO) in the disordered state, and to unreproducible magnetic data. Obviously, complete disorder can not be achieved experimentally. On the Mn-rich side of the NiMn system, the structural phase transformation lines are not very well established at high temperatures. There is probably a eutectic decomposition of/3Mn into aMn and a y-phase at 820 K, the y-phase being stable down to low temperatures above about 22 at% Ni. The y-type alloys have antiferromagnetic order. TN values shown in fig. 9 (open dots) are from Honda et al. (1976). Alloys with 22-27 at% Ni show Elinvar properties as confirmed by these authors. Alloys with Ni concentrations smaller than 22 at% (at T = 0 K) undergo complicated structural transitions. There are regions with fct structure with c/a > 1, fc orthorhombic structure, and fct structures with c/a ~< 1 as seen from the dashed-dotted lines in fig. 9 (Honda et al. 1976). 3.1.6. Co-Mn system This system is complicated by the existence of the two allotropic forms of Co, a close packed hexagonal e-form, stable at temperatures below 417 -+ 10°C, and a cubic y-form, stable at higher temperatures up to the melting point (1495°C), as well as the four allotropic forms of Mn. A reliable structural phase diagram has been published by Tsioplakis and G6decke (1971). We show in fig. 10 the magnetic phase diagram in the range from pure Co to the phase CoMn. The Curie temperatures in the y-range decrease sharply with increasing Mn-concentration [see fig. 10; full dots: H. Matsumoto et al. (1969); open dots: Matsui et al. (1970); open triangles: Bendick and Pepperhoff (1979)] and ferromagnetism vanishes around 35 at% Mn. The average magnetic moment /2 decreases linearly with increasing Mn content as reported by Matsui et al. (1970) (open dots) and Cable (1982) (solid squares), and vanishes also at 35 at% Mn. In the range 0.32 ~<x ~< 0.40 mixed magnetic behavior is observed (Rhiger et al. 1980; crosses in the figure) and the magnetic SRO is of complicated nature in that region, leading to contradictory results. A pure SG phase very likely does not exist. Pauli paramagnetic behavior was reported for x = 0.39 (Rhiger et al. 1980), antiferromagnetism for alloys with x = 0.36 and 0.37 (Matsui et al. 1970). Dorofeyev et al. (1988), who recently reinvestigated the COl_xMn x magnetic phase diagram also found
INVAR I
T
I
I
%
1200
255 I
COl_xMn x
(K)
\
1000
FM
800
~\"\
\
600
400
/
(tl B)
\\\ \\\ 'i
\, o\,
\. 2.°\,
200
/
\ V
;/'
\
\, // sp~o / / AF []?o\ /,/
[]
)°'"4
r 10
t 20 30 a t % Mn
x xo°
0 Co
SG?t ~0
i
0
50
Fig. 10. Magnetic diagram of CO 1 xMnx on the Co-rich side. Note the sharp decrease of the Curie temperatures [full dots: Masumoto et al. (1969), open dots: Matsui et al. (1970), triangles: Bendick and Pepperhoff (1979)] and of the average magnetic moment [open dots: Matsui et al. (1970), full squares: Cable (1982)] on approach of the possibly existing SG phase [crosses: Rhiger et al. (1980), open squares: Dorofeyev et al. (1988)] around 35 at% Mn. There might also be a superparamagnetic phase (SP) present in this range. On approach to CoMn the system shows antiferromagnetic order [plus signs: Adachi et al. (1973)].
some RSG like transition points in the FM alloys at temperatures around 100 K (see open squares in fig. 10). In our feeling, all these discrepancies stem from the fact that the y - e structural transition line is crossed, when quenching alloys from the high temperature y-range to lower temperatures, thus causing unreproducible SRO effects in the composition range in question. When approaching the composition CoMn (e/a = 8, like pure Fe) the system becomes antiferromagnetic. Although the L10 superstructure probably does not exist, Col_xMn x alloys with x = 0.48 and x = 0.5 show AF long-range order (see plus signs in fig. 10) as confirmed by neutron scattering investigations on powdered and single-crystalline samples by Adachi et al. (1973).
256
E.F. WASSERMAN
3.1.7. Co-Fe and C o - N i systems Figure 11 shows the magnetic phase diagrams for COl_xNi x (right-hand part) and the Co-rich side of the CoFe system (left-hand part), both diagrams have been taken from Bendick et al. (1979). The CoNi system forms a complete series of solid solutions with y-structure, and the Curie temperatures decrease continuously from Tc = 1390 K (Co) to T c = 632 K (Ni). In CoFe, the Curie temperatures decrease on addition of Fe, the high-temperature y-range, however, has a limit of about 25 at% Fe. In both systems, martensitic transformations into the e-phase are found close to pure Co (see dashed-dotted lines in fig. 11) on quenching as well as heating. i
T
i
i
i
I
i
i
Col_xNix
Col-x FOx
1400
(K) 1200
\. 1000
\. \
FM
\
80O '
/
-\\ \\ \ \,,
600 400
i~. \'~
2OO 0
I
30
0 at % Fe
10
Co
20
I
I
40 60 at% Ni
I
80
Ni
Fig. 11. Magnetic phase diagram of COl_xNix and part of the diagram of Col_,Fe x as determined by Bendick et al. (1979). Some structural-transition lines, especially into e-martensite are also given (dashed-dotted lines).
3.1.8. Cr-Fe system A well-studied bcc alloy system showing moment-volume instabilities and Invar properties in a wide range of composition is Fex_xCrx. The magnetic phase diagram is shown in fig. 12. It has been established from data by Burke et al. (1983) (solid dots for Tc, TN) and from Burke and Rainford (1978) (dashed curve for/~) in the range 0.65 ~<x ~< 1. Tc values on the Fe-rich side (triangles) are due to Fukusaka et al. (1986), and the open dots show the values of Aldred (1976) for the concentration dependence of the average moment/~ (open dots) in the range 0 ~<x ~<0:7. As one can see from the diagram, there is a continuous decrease of Tc and ~ with increasing Cr concentration, a transition region with RSG and SG order, and then AF-order on the Cr-rich side. Here, we neglect the complications and extra transition lines on the Cr-side caused by the occurrence of incommensurate (up to about 6 at% Fe) and commensurate spin-density waves (CSDW) at
INVAR 1200
T (K)
i
i
I
i
I
257 i
~ 820"c / . - f - TV--- •~ .
~
looo-
.
'~,,./
\,
I
. !
/./~,
8oo - \o, / . . . . . - ~
,00
k
I
I
\.
2.5
{5
570~ ~
i7
-.;----k
/-\-,, .... .// \\,, 7%
I
I=el-x Cr x
\~
(lae)
20
\
~__O'_C_-, 2o_~, .
\\
,5
400
1.0
2O0
0.5
I I IF 0
Fe
20
40
60
ot % Cr
I ~80 I Invar
I Cr
0
Fig. 12. Magnetic phase diagram of Fel_xCrx as determined by Burke et al. (1983) (solid dots for T c and TN) and Burke and Rainford (1978) (dashed curve for g). Tc values on the Fe-rich side (solid triangles) are due to Fukusaka et al. (1986). Open dots show Aldred's (1976) values for the concentration dependence of the average magnetic moment. Note that in structural equilibrium the system shows a decomposition of the high-temperature g-phase at 440-+20°C (Rivlin and Raynor 1980) or 570°C (Fukusaka et al. 1986). higher Fe-concentrations in Cr. The structural phase diagram of FeCr is also well-investigated. The exceptional feature is a decomposition of the high-temperature o--phase at 4 4 0 - 20°C (Rivlin and R a y n o r 1980) or 570°C (Fukusaka et al. 1986), so that in low-temperature equilibrium the alloys in the whole system would decompose into Fe- and Cr-rich components, respectively.
3.1.9. Cr-Mn system A n o t h e r bcc-type alloy system is Crl_xMnx, where the a - p h a s e is stable up to x = 0.6. The magnetic order of the alloys is solely antiferromagnetic, very small Mn-concentrations (x <~0.01) and problems due to incommensurate spin-density waves disregarded. The N6el temperatures rise sharply from T N = 312 K for pure Cr to T N = 750 K for Cr with addition of 2 at% Mn. A b o v e this concentration, T N is almost constant with rising Mn concentration up to the stability limit of the a - p h a s e (see, e.g., Maki and Adachi 1979).
258
E.F. WASSERMAN
3.2. Phase diagrams of ternary systems 3.2.1. Fe-Ni-Cr system Alloys in the y-range of FeNiCr are widely used as commercial Elinvar (see crosses in fig. 13) and are even m o r e important as austenitic stainless steels. T h e y also show Invar properties (Acet et al. 1987, 1988). Therefore, the system has been extensively studied in structural and magnetic investigations. The magnetic phase diagram is shown in fig. 13. D a t a for the T-range stability line (at r o o m temperature) and Curie temperatures projected into the zero-Kelvin plane are mainly taken from papers by Menshikov et al. (1979, 1982), who investigated the total T-area by neutron-scattering and magnetization measurements. While these data in the purely FM range agree well with other results in the literature, the antiferromagnetic, Fe-rich side close to the T - a transition is a problematic range, and contradictory results are reported. To date, it is almost certain that a h o m o g e n e o u s long-range ordered antiferromagnetic phase in the outside corner of the T-range never develops (Ishikawa et al. 1975, Takzey et al. 198i), although other authors claim A F order with low T N (30-60 K) to occur (Warnes and King 1976, M a j u m d a r and von Blanckenhagen 1984). The problem is analogous to that in binary s y s t e m s - like, e.g., FeNi. Close to the 7 - a phase boundaries, the exact positions of the martensitic transformation lines M s down to very low t e m p e r a tures are not known with sufficient accuracy. Moreover, M s depends critically on interstitial impurities like C or N. Very likely the M~-line at 0 K (see dashed line in fig. 13) lies inside the region with A F and FM S R O , while M s for 300 K definitely lies outside the stability range of the y-phase.
/
FeNiCr ~.~.~ ~:~~:~
j'~-range
.~'..--.-~. ~
i.'~.~!~
Ni
1(3
' " :.: : " ~ .
2(3
3(3
40
50 Fe { a t % )
60 ~,
70
/MsCoKI \
80
90
Fe
Fig. 13. Ternary magnetic phase diagram of FeNiCr in the y-range (Menshikov et al. 1979, 1982, Majumdar and von Blanckenhagen 1984). The full lines show contours of constant Curie temperatures projected into the zero-Kelvin plane. The RSG area is shown hatched, the pure SG area dotted. On the Fe corner within the y-range, a pure antiferromagnetic long-range order never develops (Ishikawa et al. 1975, Takzey et al. (1981) because of the occurrence of a martensitic transformation [M~-lines from Warnes and King (1976)]. The crosses mark commercial Invar and Elinvar compositions.
INVAR
259
Along the rim of the FM-range, FeNiCr alloys show R S G behavior at low temperatures (see hatched area in fig. 13) with a pure SG region adjacent (dotted area in fig. 13). This has been supported by our recent investigations of the frequency dependence of the A C susceptibility on Fes0_xNixCr20 alloys with 14 ~ x ~ 35 at% (Uffelmann 1987).
3.2.2. Fe-Ni-Mn system Ferromagnetic FeNi Invar alloys with small additions of Mn and A F Elinvar alloys of MnNi with small additions of Fe, both used commercially, can be found in the ternary system of FeNiMn. The magnetic phase diagram is shown in fig. 14. The system exhibits a wide range of y-stability, with exceptions at the corners of Fe (aFe and low-temperature e-martensite), Mn (aMn) and around the ordered phase of NiMn. M o m e n t - v o l u m e instabilities are found throughout the whole y-range. The magnetic transition lines, N6el temperatures T N for the A F F e - M n rich side, and Curie temperatures T c for the FM Fe-Ni-rich side projected into the 0 K-plane are given in the figure. The system has been intensively studied and the data shown are compiled from the literature (Shiga 1967, Menshikov et al. 1976, 1981, Ettwig and Pepperhoff 1974). Concerning the Invar behavior, alloys along the composition FesoNixMns0_ x have been recently reinvestigated by us in Mn
I
Fe
20
..
/.0
at% Ni
60
80
Ni
Fig. 14. Magnetic phase diagram of the ternary system FeNiMn in the fcc y-phase, compiled from different data in the literature (Shiga 1967, Menshikov et al. 1976, 1981, Ettwig and Pepperhoff 1974, Stamm 1988, Acet et al. 1987). Contour lines for constant N6el temperatures TN in the antiferromagnetic phase and Curie temperatures Tc in the ferromagnetic phase are shown. The dotted region marks a pure spin-glass phase separating the AF and FM regions. The hatched region gives the area of a re-entrant spin-glass phase, which occurs below the FM ordering. Crosses mark commercial Invar and Elinvar compositions.
260
E.F. WASSERMAN
detail (Acet et al. 1988). Transition temperatures from the A F and FM side fall into a deep valley, along which RSG behavior (hatched area in fig. 14) and a pure SG phase (dotted in fig. 14) are found. Compositions of the commercial alloys are denoted by crosses. 3.2.3. F e - N i - C o system The magnetic phase diagram of FeNiCo is shown in fig. 15. It is based on constitutional data taken from the Cobalt Monograph (1960) and early magnetic investigations by Masumoto (1931). The system shows a wide range of y-stability, a hexagonal e-phase near pure Co and a narrow o~-range along the F e - C o side. Between the a- and y-ranges lies a broad region of a + y mixtures. Some of the high to low temperatures M s transition lines (dashed-dotted) are given in fig. 15. The magnetic order is solely ferromagnetic throughout the whole diagram, with high Curie temperatures, as can be read from the respective projection lines in the figure. The projected Tc-lines end at the respective Ms-lines in the a + y range. Most of these M~-lines are omitted for clearness in fig. 15, but they, in principle, are running parallel to the 800 K line. The FeNiCo diagram is of practical importance. The commercial Invar compositions for 'Super-Invar' (around 0-18 at% Co) as well as certain FeNi Invars are indicated by the crosses. All these compositions are found close to the y - a instability limit. FeNiCo alloys of other compositions are also of practical importance, e.g., as high-damping materials, and, with the addition of Cr, as high-temperature-high-strength super alloys. Co
Fe
20
1.0
60
80
Ni
at°lo Ni
Fig. 15. Magnetic phase diagram of FeNiCo after Bozorth (1950) based on magnetic data by Masumoto et al. (1931) and structural data taken from the Cobalt Monograph (1960). Note the wide range of y-miscibilityin this system and the absence of any AF, RSG or SG order.
INVAR
261
3.2.4. F e - C o - M n system The structural equilibrium phase diagram of this system has been established by K6ster and Schmidt (1933). As can be seen from fig. 16, there is a relatively broad range of fcc, y-structure at room temperature (dashed-dotted lines), though none of the constituents is stable in the y-phase at this temperature. The magnetic data shown in fig. 16 originate from careful investigations by Adachi et al. (1971) and Matsui et al. (1973) and a recent study by Dorofeyev et al. (1988), the latter of the composition COl_x(FeMn)x. The system also shows a transition from ferromagnetic order with very high Curie temperatures on the Co-rich corner, through mixed magnetic RSG like order (hatched area in fig. 16), pure SG behavior (dotted area) to AF order on the FeMn side. In comparison to the binary CoMn system (cf. fig. 10), which probably does not show a SG phase, a broad SG region opens up on the addition of Fe into CoMn. In the FM alloys left hand to this SG range and close to the a -3' transition line (e/a = 8.4-8.5), Invar-like magnetovolume effects have been observed, while a binary Co70Mn30 alloy close to the y - e transition line (with the same ratio e/a = 8.4) did not show this effect (Matsui et al. 1973). This was not understood at the time it was detected, but will become clear on the basis of modern band-structure calculations, as we shall see later. We finally mention that the extension of the low-temperature e-martensite field (the e-martensite occurs on quenching of the y-phase alloys) is not well-investigated in the ternary FeCoMn Fe
S/,.gi/! A'
_. Nt1717," I I I ,i / i.I/
Co
20
A0
ot% Mn
60
2 °0
+
\
80
Mn
Fig. 16. Magnetic phase diagram of FeCoMn. The data originate from Adachi et al. (1971), Matsui et al. (1973) and Dorofeyev et al. (1988). Characteristic for this system is the deep 'valley' between the FM and the AF range, where a pure SG phase (dotted area) is observed. Structural-transition lines at room temperature (dashed-dotted) are due to K6ster and Schmidt (1933). Note the 'opening up' of the SG area on approach of the T - a transition region.
262
E.F. WASSERMAN
system. The phase-transition line drawn in fig. 16 (3' + (e)) is therefore tentative and roughly valid for M s = 300 K. The 3" + (e) field will stretch more into the direction of Mn, if alloys are quenched to lower temperatures. 3.2.5. N i - C o - M n system Figure 17 shows the magnetic phase diagram of NiCoMn. The system has a wide range of T-stability, and shows ferromagnetic order with high Tc-values on the CoNi side, which decrease drastically on addition of Mn. Along the line from Co36Mn64 t o Ni75Mn25there is a deep valley of SG order with low transition temperatures (Tf = 50-80 K). To the right-hand side of the valley antiferromagnetic order occurs, with TN values rising to more than 400 K with increasing Mn concentration. On either side of the SG valley there are regions of mixed magnetic behavior (hatched regions in fig. 17), which have RSG character on the FM side, and ferro-antiferromagnetism coexisting on the AF side. The structural part of the diagram is due to work by K6ster and Rittner (1954). In the Co corner there is e-order with structural transition temperatures lying below the respective Curie temperatures. At the Mn corner and along the Mn-Ni side the lattice structure is more complex due to the allotropy of Mn. According to K6ster and Rittner (1954), about 3 at% Co can be dissolved in ordered NiMn. However, the Ni
\
NiMn ? ,/
Co
20
RSG
40
60
ot% Mn
NiMn \
80
Mn
Fig. 17. Magnetic phase diagram of the ternary system NiCoMn (Cobalt Monograph 1960). Solid lines show constant Curie and N6el temperature contours projected into the zero-Kelvin plane. Note the deep lying, narrow spin-glass region (dotted area), which runs across the diagram, separating the FM from the AF region. Hatched areas indicate re-entrant spin-glass phases observed in the FM as well as the AF range. Structural-transition lines (at 20°C) (dashed-dotted) have been determined by K6ster and Ritmer (1954). Note the small solubility of Co in ordered NiMn, in contrast to the extended range of (3' + NiMn) phase mixtures.
INVAR
263
contour of the phase boundary (3' + NiMn)-~/boundary at room temperature (see dashed-dotted line in fig. 17) extends far into the T-region and phase mixtures occur even in alloys quenched from 1000°C.
3.2.6. Co-Fe-Cr system CoFeCr alloys are of importance for technical applications, since the FM fcc alloys 'Stainless-Invar' and 'Co-Elinvar' (cf., table 4) are found in this system. The structural phase diagram has a very simple form at high temperatures as seen in the inset to fig. 18 (Rideout et al. 1951). There are two wide ranges of solid solutions, with T-structure on the C o - F e side and a-structure on the Cr-rich corner. On lowering of the temperature, the diagram gets very complicated. Alloys of composition within the area labeled A in the inset to fig. 18 transform completely into bcc a-structure on quenching, while alloys of composition within area B decompose on quenching, mostly into the hexagonal e-phase (and e-omixtures), and only a small T-region is left. This T-region is shown by the dotted area in fig. 18. The respective composition lines at RT determined by K6ster and Hofmann (1959) are shown by the dashed-dotted lines. The structural part of the CoFeCr diagram reveals that the Stainless-Invar alloys with compositions (Fel_xCOx)90Crl0 and 0.52 ~ x ~<0.54 (Masumoto 1934), just lie within the narrow right-hand end of the T-stability range, very close to the boundaries of the a- and e-phase. This means that results concerning the magnetic properties of alloys in the Stainless-Invar range have to be regarded with care. Occurrence of phase mixtures is the immediate consequence of slight errors in the Cr concentration. This is less critical for Co-Elinvar, since for 58-
Co
20
z.O Fe(Gt %)
Fig. 18. Section of the magnetic and structural phase diagram on the Co-rich side of the ternary system CoFeCr [Curie temperatures from Masumoto (1934); structural data from K6ster and Hofmann (1959)]. 'Stainless-lnvar' and 'Co-Elinvar' compositions(see crosses) are found right at the l~-a phase-transition line. The inset shows the structural constitution at 1200°C [after Rideout et al. (1951)1.
264
E.F. WASSERMAN
60 at% Co and 9-10 at% Cr the y-phase is a little wider. We remark that on addition of Ni in the quarternary diagram CoFeCrNi (cf., fig. 4) the y-area opens up. Curie temperatures are hardly known for the y-range. Some Tc = const.-lines projected into the basal plane are shown in fig. 18 (Masumoto 1934). 4. Fundamental properties of Invar and Elinvar systems Following the listing in table 2, we will present in this section the principal behavior of the physical properties of Invar (and Elinvar) systems as a function of external parameters like temperature, magnetic field, or pressure. We will restrict ourselves to the presentation of selected data on specific systems, which are relevant for the general understanding of the Invar effect within modern band calculations presented below. This is also necessary for the sake of clearness, and in order not to confuse the reader with the overwhelming mass of data available for all the systems presented in section 3. These are well summarized in the Honda Memorial Series on the Invar problem (1978) up to the year 1978. Thus, we will mainly outline the new insights gained since then.
4.1. Thermal expansion Doubtlessly, the anomaly in the thermal expansion is the most characteristic feature of Invar. For ferromagnetic alloys, the principal behavior of the length change with temperature and the temperature dependence of its derivative - the expansion coefficient a( T ) - hav e already been shown in figs. la and lb. Less well-known is the dependence of a on the temperature in antiferromagnetic Invar. To exhibit the differences, we have measured the thermal expansion on systems, where a transition from FM to AF order through a SG phase is possible by change of the concentration of one of the constituents (Acet et al. 1988a). Together with data from the literature, this leads to a sizeable set of results that enables us to observe some general features of magnetovolume effects in FM, SG and AF systems (Acet et al. 1988a). Figure 19 shows, as an example, a(T) in Fe65NixMn35_x (Acet et al. 1988a, Hayase et al. 1971). In this system, a continuous transition from FM (Fe65Ni35) through a SG (Fe65Ni25Mn10) to AF (Fe65Mn35) is obtained by a variation of the Ni content (cf., magnetic phase diagram in the inset of fig. 19b and also fig. 14). As seen in fig. 19a, the FM alloys show positive magnetovolume effects with respect to the calculated Gr/ineisen curve. One might expect the SG alloy to represent the nonmagnetic Gr/ineisen behavior, but in fact it does not, since it is paramagnetic. It only follows the calculated Griineisen curve in the low-temperature range. Note that there are substantial contributions to a(T) above the respective Curie temperatures in the FM alloys. As seen from fig. 19b, the AF alloys show positive magnetovolume effects like the FM alloys to temperatures above TN, but then, in the PM range (T>> TN), their a(T) curves cross the
INVAR
265
T (K) 0 30
200 I
i
Fe65NixMn35_x
20
~00 i
i
FM
a.
x (at%) GrQneisen calc. ~
....
2sso
~
Z.;.."
........
..
...... ........ :'"
""
I
/ / ///
I
I
/
- 25 -17 - 13 .......... o
10
I
AF
b.~
x (at%) 20
.._...
~---.__'~--~.::~
- - - - - - 30 ~ ' - ' " ........... ~/ . / " -. ~ ' ~ " -
% "; o - ~"
600 i
/
~
o~
_ , , . , , -I~el' ~ ' : "
/ . ~ _ _ . _ f f ~ ..." ~ .. - ~ - - " ~ Z . ~ . ~ .'~'~. / .," f-'"
_.~-w- , . [T~ N
,
200
T
.4
..... ~2~ 3o I 600 l
.ill
z.00
27 ,
~o i
(K)
Fig. 19. Thermal expansion coefficient a as a function of temperature for different ferromagnetic (a), antiferromagnetie (b), pure spin-glass (SG) and re-entrant spin-glass (RSG) alloys of the system Fe65NixMn3s_x [after Hayase et al. (1971) and own data, Acet et al. (1988a)]. The Grfineisen curve is fitted to the data of the SG alloy (x = 25 at% Ni) at low temperatures. The inset shows the magnetic phase diagram, the arrows mark the respective concentrations, for which the a ( T ) curves are given.
Gr/ineisen curve, show an additional maximum and approach it from above for higher temperatures. This is better revealed in fig. 20, where a(T) for binary CoMn alloys is shown in the temperature range up to 1100 K (Pepperhoff 1989). Note the pronounced maxima and the decrease in a(T) in the high-temperature PM range. In relation to the discussion in section 3.1.6 we also mention here that in the FM CoMn sample with x = 32.5 at% Mn (Tc ~ 150 K) a positive magnetovolume effect is indeed absent around Tc. Howeyer, this alloy might not be a FM but superparamagnetic according to the phase diagram in fig. 10, so that the question for the absence of the Invar effect on approach to the 7 - e structural phase transition line has to be left open, until further data are available. The general a(T) behavior in ferromagnetic and antiferromagnetic Invar systems can be summarized in the following way: (i) a(T) curves for ferromagnetic alloys always lie below the respective
266
E.F. WASSERMAN 30
.,'T~" --"'"" 25
~ x: 20
_
~z.~.~"
x - - 15
.
COl_~Mn× o, O,o
//,-
//'1°-IT~ Igl~l'r. ,.~
-'-- . ........
"
3S ~3.2 &Z
10
5
100 F i g . 20. T h e r m a l
expansion
I
I
300
500
I
I
700 T (K)
c o e f f i c i e n t a as a f u n c t i o n
900
of temperature
1100 for different T-structured
Col_xMnxalloys[after Pepperhoff(1989)]. Note the maximaat high temperatures and the absenceof sizeable magnetovolumeeffects around the respective ordering temperatures. Gr/ineisen curve and show positive magnetovolume effects, regardless of the temperature. FM Alloys also have substantial magnetovolume effects above the respective Curie temperatures in the paramagnetic range which vanish, however, gradually for very high temperatures. (ii) Antiferromagnetic Invar alloys, in principle, show the same behavior, positive magnetovolume effects (although less in absolute values as compared to FM Invar) below and around TN. Yet, in the PM range for T > TN, their a(T) curves cross the respective Grfineisen curves and show distinct maxima, calling for the reappearance of a magnetovolume effect. This is shown schematically in fig. 21, where the relative volume change of ferromagnetic (FM) (full curve, cf., fig. la) and antiferromagnetic (AF) Invar (dashed curve) is given in comparison to a nonmagnetic (NM) Grfineisen-like reference. (iii) Pure SG alloys expand Gr/ineisen-like up to temperatures roughly five times their freezing temperature Tf, but then - like the AF alloys - show positive magnetovolume instabilities at high temperatures in the PM range and approach the Grfineisen curve from above for T going to infinity. The principal change of the expansion coefficient a with impurity concentration for FeNi at room temperature (RT) has been presented in fig. 2. In fig. 22 we have plotted aRT versus the electron per atom ratio e/a for a series of different Invar systems. Data are taken from the literature (Acet et al. 1988a, Honda Memorial Series 1978). A deep minimum with aRT = 0 (or even negative values in FePt) occurs in the range e/a = 8.5-8.7 for certain binary and ternary systems (Invar effect) which are all ferromagnetic. Note that in this e/a range the
INVAR
267 I
FM
0
Temperature
Fig. 21. Schematic representation of the volume change with temperature in ferromagnetic (FM) and antiferromagnetic (AF) Invar in comparison to a non-magnetic (NM) Or/ineisen alloy.
AF
20
I
SG ,j|,'-'~ =/o
I
=~
• /'®
FM~
cuR,l
I
/
15
/ ~10
\
-I
$clFe
•
./°
~o
o
m •
" °" v
o ?
v
t
o
÷+
x
#
0 -2
----
I 7s
--
•
MnCo
o
Fe Ni
[
•
Fe PI ord, FeP~ disord.
~. o
Fe65NixMn 35_x
=
Fe50Nix Mn 50_x
I V
•
Fe80_xNi x Cr20
+
Fe75_xCrx Nn25
•
( Fe0.5Mn0,5)lO0_ x Co x
V
(Fel-xNix}90C°10
"or xI
I 8.o Fe
=
vo
zx I |
e/a
o -- --
~ v
I tttt vt {M~)
~o
I g.o
Co
Fig. 22. Thermal expansion coefficient at room temperature aRT , as a function of the electron concentration e/a, for different Invar systems as compiled from the literature (Acet et al. 1988a, Honda Memorial Series 1978). Arrows on the abcissa mark the martensitic transition temperatures M~ at room temperature for the respective systems. Note the occurrence of two minima in ar~T(e/a) in the ferromagnetic range e / a = 8.5-8.7 and the antiferromagnetic range around e / a = 7.6. In the spin-glass range (e/a ~ 8.3), aRT for the respective Invar alloys is larger than aRT Of pure copper (see arrow).
268
E.F. WASSERMAN
respective systems are all close to the y - a structural transition. The respective martensitic transformation temperatures at RT, M RT are given by the arrows in fig. 22. For e / a > 9, aRT approaches a constant value of about 12 x 10 -6 K -1. Isothermes of a(e/a) for higher (lower) temperatures in principles look similar. The minimum stays in the range e / a = 8.5-8.7, the respective curves are just shifted upwards (downwards). A second salient feature of fig. 22 is the maximum in aRT at e / a ~ 8 . 3 - 8 . 4 , i.e., at the transition from FM to A F order in the SG range (exception: Fes0_,NixCrz0, SG phase for x =20, which corresponds to e / a -- 8.0). Note that here, aRT reaches values as high as 19x 10 -6 K -a, higher than the values of pure copper at RT. These alloys are thus used as active components in bi-metals together with low aRT Invar. Finally, we see in fig. 22 that a second minimum in aRT (e/a) is reached in the A F range at e / a = 7.5-7.6, which is, however, not as deep as on the FM side of the diagram. For bcc alloys ( C r - F e , Cr-Mn) minimum values of a = 0 are reached around e / a = 6 . 5 , yet, not for RT but at lower temperatures T~<200K, since the transition temperatures of these alloys are so low. Our presentation shows that for the understanding of the Invar-effect and magnetovolume effects of transition metal alloys in general, the 3d-electron concentration is of relevance. Critical values for low a are the ranges e / a = 8.5-8.7 for ferromagnetic alloys, and e / a = 7.5-7.6 for antiferromagnetic alloys. This will be supported by the theoretical results as we shall see below (section 6.1 and fig. 50e).
4.2. Volume magnetostriction The principal behavior of ~o = AV/V as a function of temperature has already been presented above (cf. figs. 1, 2 and 21). The spontaneous volume magnetostriction % is related to o% = aex p - a n r n , the magnetic contribution to the expansion coefficient, by, T
ogs(T) = 3 f a m d T , T1
(1)
where T I is a reference temperature, for which % is assumed to vanish. At this point again the problem of finding the 'correct' non-magnetic reference from first principles arises, since for the determination of the magnetic part O)s(T) the knowledge of tOnm(T ) o r anna(T ) is required. The usual approach (as indicated) is to use thermodynamics, i.e., a Grfineisen relation, KT
anm(T ) = ~
Co(T),
(2)
where K is the compressibility, y the Grfineisen parameter, V the atomic volume,
INVAR
269
and C o the lattice specific heat at constant volume. Integration of eq. (2) leads to, T
(3)
Cv(T ) dT.
Otnm(T ) = ~ o
This approach, however, just 'transfers' the problem, since now the knowledge of the lattice specific heat of the nm reference Co(T ) is required, which, in principle, is also unknown. Commonly, in the literature, Debye theory with 'suitable' Debye temperatures is used to evaluate Co, and the term Ky/V is determined by fitting eq. (3) to the slope of the experimentally determined volume (length) change curves in the 'linear' range above T c (or TN). This procedure completely neglects the sizeable contributions to o)exp(T) in the paramagnetic range of Invar systems (see figs. 19 and 20), and especially does not work for antiferr0magnetic Invar (cf., fig. 21), since a linear range in tOexp(T ) [or Otexp(T)] can hardly be defined for these alloys. Even pure SG alloys (see fig. 19) are not suitable as reference, because they show volume instabilities in the PM range too. All absolute values of tos for Invar are, therefore, burdened with some uncertainty. However, it is certainly correct to say that the spontaneous volume magnetostriction ~os0 in ferro- and antiferromagnetic Invar is large and positive as compared to other magnetic materials. This is seen from fig. 23, where to~0 as a function of concentration (e/a) is plotted for different fcc Invar systems (Data from Richter and Pepperhoff 1976, Wassermann 1987, 1989a,b). Maximum values I~AF
J
1.5
SG
I F M ~o FePIo+d. lJ~eNi
I
(FeC°)89Cr11~z~ t / .
I\
..-o 1
\I
\
~t~' + /
/=,/,d \
o.s /
I \i~'--
x/(P.eOr)6eMn~.0
/,;"
";
A\g
I
\i
J 7/ {FeCr}67Ni33"/
X
\\
~:!l/-/
Fe75xCrxt4n25 \ \ a \ ~
I
\\'~ \
\X
\o \ Fe65NIxMn35 x
, =, 1 1 "/ 0
P
I
75
8 Fe
~..~--/~
e/c]
I
r
8.5
9 Co
Fig. 23. Spontaneous volume magnetostriction at zero temperature tos0 versus electron concentration e/a for different Invar systems with fcc lattice structure [after Richter and Pepperhoff (1976) and Wassermann (1987, 1989a,b)]. Note that %o reaches maximum values in those concentration regions, where aRT (cf. fig. 22) has minimum values. On the other hand, tos0 vanishes in the SG range around e/a = 8.3, where aRT reaches maximum values.
270
E.F. WASSERMAN
of Ws0= 1.9 x 10 -2 are reached on the ferromagnetic side in the FeNi system, but the antiferromagnetic fcc alloys also experience a substantial volume magnetostriction, with maximum values of ws0 = 1.2 × 10 -z in FeMn. As expected, fig. 23 is a 'mirror image' of fig. 22. In the range where aRT in fig. 22 has minimum values ( e / a = 8.5-8.7 in the FM-range and e / a = 7 . 6 - 7 . 7 in the AF-range), maximum volume magnetostriction w~0 is reached in fig. 23. Vice versa, in the SG range around e/a = 8.3, the spontaneous volume magnetostriction vanishes, while agT reaches maximum values. In table 5 we summarize values for the spontaneous volume magnetostriction at zero temperature for other Invar systems as taken from the literature. Note that in the cubic Laves phases values of Ws0 are smaller- in RE2Fe17 and mixed compounds larger- than in the fcc transitionelement alloys. Very large spontaneous volume magnetostrictions are found in the RE2FeanB compounds. Due to the complex magnetic microstructure of these compounds this observation is far from being understood. The 'giant' value of ~os0~- 5% in AF YMn 2 is doubtful, since its occurrence is coupled to a first-order transition (Nakamura 1983). Amorphous Invar systems show w~0 values of the same order of magnitude as the crystalline 3d-alloy systems. Data have been collected by Xianyu et al. (1985). A maximum value of o~0 = 2.05 × 10 -2 is reached in a-Fe85B15.
TABLE 5 Spontaneous volume magnetostriction ws0 (×10 -2) for different Invar systems, Compound GdCo2 TbCo2 DyCoz HoCo2 ErCo2 YMn2 Y2Fe17 Lu2Fex7 Dy2Fe17
W~o×
Compound
10 -2
tos0 ×
0.72 [1] 0.68 [1] 0.55 [I] 0.55 [1] 0.47 [1] 5.0 [1] 1.7 [3] 1.5 [3] 1.7 [4]
ScFe 2 TiFe 2 ZrFe 2 HfFe YFe 2 Dyz(Fe0.8Coo.2)i7 Dy2(Fe0.82A10.18)~ 7 -
0.55 [2] 0.25 [2] 1.0 [2] 0.8 [2] small, negative [2] 2.25 [4] 0.75 [4]
2.52 3.21 2.79 2.79 2.73
Gd Tb Dy Ho Er
3.36 3.18 3.51 3.54 3.12
RE2FelTB [5], R E =
La Ce Pr Nd Sm
10 -2
References: [1] Nakamura (1983). [2] Shiga et al. (1979). [3] Gignoux et al. (1979). [4] Radwanski and Krop (1983). [5] Buschow and Gr6ssinger (1987).
INVAR
271
4.3. Forced volume magnetostriction and high-field susceptibility The anomalies in the thermal expansion and volume magnetostriction in Invar alloys originate- as mentioned a b o v e - from the instability of their magnetic moments. The size of the moment is sensitive not only to composition and temperature, but also to external magnetic fields. If the field is increased beyond values necessary to reach technical saturation, the volume of a ferromagnetic Invar alloy continues to increase with rising field strength. This phenomenon is called 'forced volume magnetostriction', expressed by Ow/OH, if oJ increases linearly with H. By a simple thermodynamic argument (Becker and D6ring 1939) aexp and Ow/OH can be related. In Invar, one expects a large positive forced volume magnetostriction since aex p is small (or negative). Figure 24 shows O)exp(H) at various temperature for Fe-36.4 at% Ni. Data have been determined by Ishio and Takahashi (1985). In the temperature range from 77 to 160 K, (o increases linearly with H, and the forced volume mangnetostriction is readily evaluated. However, if T is raised up to the Curie temperature T c (529 K), the slope of the w(H) curves decreases with increasing field, at least in 20^ 10
1C
% $
0~- .......... 0~ 0
738 t 10
.-,--~ 20 H (T)
~r
Fig. 24. Experimentally determined volume expansion OJexp as a function of magnetic field H for Fe-36.4 at% Ni at various temperatures (after Ishio and Takahashi 1985). The Curie temperature is T c = 529 K .
272
E.F. WASSERMAN
the low-field range 0 ~ H ~< 1 T. For T > T c, in the PM range, opposite behavior is observed, o~(H) has increasing slope with increasing field; Ishio and Takahashi (1985) even claim o~ ~ H 2 Thus values of Ow/OH cannot be defined at all for T > T c. We have recently found (Acet et al. 1988b) the same general behavior of w(H) as in fig. 24 on ferromagnetic FesoNiMns0_ x alloys in fields up to 6 T. Thus, in the temperature range around T c, it is not adequate to discuss magnetovolume effects in terms of Ow/OH in Invar. Unfortunately, the same statement is valid for low temperatures, where, e.g., for ordered and disordered FeyzPt28, values of Ooo/OH~lO x 10-1°Oe -1 have been reported for T = 50 K (Sumiyama et al. 1979), while in Fe65Ni35 at the same tempeature O~o/OH = 75 x 10 -l° Oe -1 (Ishio and Takahashi 1985). Moreover, for T ~<20 K, Oo~/OH in Fe65Ni35 suddenly drops and becomes time dependent in the low-temperature range. The same behavior of (0 ~o/ 0 H ) (T), with similar absolute values, has been observed by us on FesoNixMns0_x with x = 35 and 36.5 at% Ni. These alloys definitely show RSG states for T ~<50 K. We therefore conclude that the presence of mixed magnetic behavior or RSG phases inhibits a reliable determination of Ow/OH at low temperatures. For antiferromagnetic Invar alloys, which e x h i b i t - as we have s e e n - substantial spontaneous volume magnetostriction (cf. fig. 23), only a tiny forced volume magnetostriction occurs. This is seen from our data of w(H) for A F F%0.3Niz6.sMn23.2 (T N = 160 K) in fig. 25. The volume change in field is 102-103 times smaller than in ferromagnetic Invar (cf. fig. 24), and the o~(H)-curves are not linear either. In conclusion, we claim that the forced volume magnetostriction is not a good quantity to specify magnetovolume effects in Invar systems - at least from an experimental point of view. The same statement holds for the high-field susceptibility. In Stoner- or Heisenberg-type models, which h a v e - regardless of their validity- often been used to describe Invar properties, OoJ/OH is proportional to the slope of the magnetization curve in high fields (above technical saturation) (OM/OH)r = 24
i
i
i
i
i
AF Fe 50.3 Ni 26.5 Mn 23.2
/ A
/.
TN =160K 7°18~
~,
x
T (K)
o
//
o,,2
3 12
~,
201
a
3oo
250
,
~.
A/ /
.
-
/°
/ " / ' - -
/
o/
ii
3 6
~
0
1
2
•
3
4
5
a (T)
Fig. 25. Relative volume change at constant temperatures (Ato) as a function of applied field strength for antiferromagnetic Fes0 3Nizr.sMn23.2 Invar (after Acet et al. 1988b).
INVAR
273
(XHF)T, which
is called the 'high-field susceptibility'. Unfortunately, like the w(H)-curves, the M(H)-curves at T = const, are not linear either, especially, if T c is approached. The values of XHF determined experimentally depend on the maximum field-strength available. This has been demonstrated, e.g., on Fe36.4 at% Ni by Yamada and Du Tr6molet De Lacheisserie (1984), who showed that the absolute values of XHF(T) in H = 4 T in the 300~ < T~<650K range around T c (~537 K) are only half of the values taken in fields of H = 1 T. At low temperatures in alloys with mixed magnetic phases, the problem of defining reliable Xi~F-Values is even more severe. In a spin glass (or RSG) the magnetization never reaches technical saturation, at least in fields of laboratory scale, and XHF is therefore always ill-defined. Consequently, like 0 oJ/OH, the high-field susceptibility XI~Fis also not a suitable quantity for comparison of experimental results on Invar with what theoretical model so ever [see, e.g., discrepancies in the paper by Kakehashi (1982), comparing experimental data of XnF and Ow/OH of FeNi with his results within the local-environment model].
4.4. Pressure dependence of Curie (NOel) temperatures and magnetization From the large changes of the volume with magnetic field as well as concentration ('internal pressure'), one can expect large changes in the magnetic moment and the Curie or N6el temperatures with external pressure in Invar. This is known since a long time and experimentally well confirmed, at least for ferromagnetic Invar alloys. For not too high pressures (roughly up to 20 kbar), the Curie and N6el temperatures of Invar decrease linearly with pressure, so that values of dTc/dp (K/kbar) can be defined. Figure 26 shows a plot of (dTc/dp) (or dTN/dp) versus electron concentration (e/a) as taken from the literature for FeNi (open dots) (Leger et al. 1972, Hausch 1973), disordered FePt (open triangles) and FePd (solid triangles downwards) (Honda Memorial Series 1978), Fe65(NiMn)35 (open squares) (Nakamura et al. 1971), NiMn (full dots) (Arnold and Kamarad 1980), NiCr (full triangles upright) (Tange et al. 1981), for an Elinvar-alloy Fes2Ni36Crx2 (divided circle) (Hausch 1973), for antiferromagnetic Fe70Mn30 (cross) (Fujimori 1966) as well as for 7-Fe precipitations in Cu (plus sign) (Liu and Inga!ls 1979). One salient feature of fig. 26 is the huge increase of -dTc/d p in the ferromagnetic range from a positive value of dTc/dp = +0.32 K/ kbar for pure Ni to values of roughly - 8 K/kbar for Fe65Ni27.2Mn7.8. Less recognized by experimentalists (and in theory) as seen from the available amount of data, but to our feeling even more important, is the fact that in antiferromagnetic Invar appreciable values up to dTN/dp = - 2 . 5 K/kbar are reached too. Note that fig. 26 resembles very much the presentation of the spontaneous volume magnetostriction o)s0(e/a ) in fig. 23. A large volume magnetostriction occurs in the same e/a-ranges where large depressions of the Curie and N6el temperatures are observed. The large decrease of the ordering temperature with pressure is thus an important Invar characteristic feature, since it is present in both, FM and AF systems. Remember that plots like in fig. 26 or fig. 23 could be
274
E.F. WASSERMAN -8
°if\o o
-6 Z" v 5(
!riO
o z, • x
FeNi Fe Pt Fe Pd FeMn
• o
Ni Cr Fe65 (Ni Mn) 35
O
Fe52Ni36 Cr12 ,~Fe in Cu
+
~o
,% 00
x\ -2
.
°\ +
\
0
-AF
S6
* ~ ; ~
FM ~
I
I
I
I
8.0
8.5
9.0
9.5
10
e/(l
Fig. 26. Pressure dependence of Curie and N6el temperatures (dTcJdp) as a function of the electron concentration (e/a) for different Invar systems. Note that dTc/dp and dTN/dp reach maximum values in the same concentration regions, where tos0(e/a ) also peaks (cf. fig. 23) (for references see text).
obtained neither for the forced volume magnetostriction Oo)/OH nor for the high-field susceptibility XHF, because both quantities are negligibly small in antiferromagnetic Invar alloys. This supports the statement given at the end of the last chapter concerning the relevance of absolute values of O~o/OH and XHF for the understanding of the Invar effect. As mentioned above, from the theoretical side commonly Stoner-type 'ansiitze' have been used to explain ferromagnetic Invar behavior. For a homogeneous weak itinerant ferromagnetic (WIF) Wohlfarth (1983) claimed - ( d Tc/dp) to be proportional to 1 / T c. This is by no means correct for Invar as seen from fig. 27, where - ( d Tc/dp) versus the Curie and N6el temperatures are plotted [data from the same references as in fig. 26, for amorphous FeZr from Shirakawa et al. (1983)]. In all FM-Invar obviously - (d Tc/dp) decreases linearly with increasing Tc, at least if the Curie temperatures are not too low. This was proposed to be valid for inhomogeneous ferromagnetic alloys by Wagner and Wohlfarth (1981). It is doubtful at present to what extend FePt alloys are 'inhomogeneous' ferromagnets. Moreover, a comprehensive theoretical description of Invar has to explain the contrasting fact that in antiferromagnetic Invar alloys obviously - ( d T N / d p ) increases linearly with TN, as revealed in fig. 27. If the Curie temperature is dependent on pressure, the magnetization of a FM Invar alloys should be pressure dependent as well. From the thermodynamic relation (0 oo/ 0 H)p = - (OM/Op)n ,
(4)
we can see that (OM/Op), like (dTc/dp), is negative, since Oo)/OH- irrespective
INVAR I
[i
I
275 I
I
I
I
I
-8
a..
x,'
z
-2
,//AF t~/
0 0.5
Fev~dX
ge551NiNln135
I
100
I
200
400
600 Tc,T N (K)
800
Fig. 27. Pressure dependence of the Curie temperatures dTc/dp for ferromagnetic and of N6el temperatures d TN/dp for antiferromagnetic Invar alloys versus Tc and TN, respectively (for references see text). Note the increase in -dTN/d p with TN, in contrast to the decrease of dTc/dp with Tc in the systems with FM order.
of its absolute v a l u e - i s always positive. This is shown in fig. 28, where the pressure dependence of the spontaneous magnetization (relative to the magnetization at zero pressure) at 4.2 K and room temperature for Fe65Ni35 and disordered Fev2Pt28 is given (after Hayashi and Mori 1981). For Fe65Ni35 the magnetization decreases almost linearly with pressures up to about 20 kbar. The pressure coefficients, (dM/dp)/M(O), are - 4 . 7 x 10 -3 kbar -I at 4.2K and -1.1 x 10 -2 kbar -1 at RT, respectively. On the other hand, for Fe72Pt28 the pressure I
I
I
~-,,.~
~°~
~ o
F e 6 5 Ni35 ° ~
\
~-0.5
I Fe 72 Pt 28 d
1.0 ~ -'~-~z~'--~'-"---~--~
,~-
~.2K4.2K
\ ~ . ~ F e\ 7 2Pt 28 d.
~"
\
0
0
I
I
5
10
I
15 p (kbor)
~'~ I
20
~- 2 9 5 K
25
Fig. 28. Relative pressure dependence of the magnetization M(p)/M(O) versus pressure p for FeNi Invar and FePt Invar (d = disordered) at 4.2 K and room temperature (after Hayashi and Mori 1981).
276
E.F. WASSERMAN
coefficient of the magnetization at 4.2 K (dM/dp)/M(O) = - 7 x 10 - 4 kbar -1 is an order of magnitude smaller than in FeNi. At RT, for small pressures (p < 3 kbar), one finds the initial decreases of ( d M / d p ) / M ( O ) = - 2 . 4 x l O - 2 k b a r -1 for Fe72Pt28 to be of the same order of magnitude as in Fe65Ni35, but at higher pressures a rapid, nonlinear decrease of the magnetization of Fev2Pt28 is observed. This is due to the fact that the Curie temperature (T c = 380 K for p = 0) decreases to about Tc = 310 K at p = 20 kbar, thus lies only slightly above RT. Similar differences between the two systems have been observed in the pressure dependence of the spontaneous volume magnetostriction ws(p) (Oomi and Mori 1981). These authors also have shown that Fe65Ni26Mn9 becomes magnetically 'soft' in the RSG range (Oomi and Mori 1983) so that part of the magnetic 'softness' of Fe65Ni35 in comparison to the magnetic 'hardness' of FeTaPt2, at low temperatures is due to the presence of mixed magnetism in Fe65Ni35. Nevertheless, the magnetic instability, which causes the Invar anomalies, obviously starts from zero temperature for FeNi, but it appears only at higher temperatures for FePt. This fact has intrigued people for a long time. We think that it is explainable within a modern band picture of Invar, as we will show below. Principally, there is no difference in the origin of the Invar effect in FeNi and FePt and thus all Invar systems.
4.5. Magnetic properties 4.5.1. Concentration dependence of Curie and NOel temperatures Figure 29 shows Curie and N6el temperatures versus electron per atom ratio for different binary and ternary Invar systems in fcc y-structure. Data are mainly due to Richter and Pepperhoff (1976). Open circles give the respective ordering temperatures of the pure elements. Figure 29 reveals some luther systematics in the magnetism of 3d-elements and alloys and consequently the Invar effect. Comparison to figs. 22 and 23 shows that in the ranges e/a = 8.5-8.7 for the FM systems and e/a = 7.5-7.7 for the AF systems, where minimum thermal expansion (fig. 22) and maximum volume magnetostriction (fig. 23) occur, the Curie and N6el temperatures steeply decrease. In the range e/a = 8.1-8.3, where aRT reaches a maximum. (of., fig. 22) and the spontaneous volume magnetostriction vanishes (cf., fig. 23), the ordering temperatures in fig. 29 approach small values. As we have seen above, this is the e/a-range, where SG and RSG phases are observed. One might call these alloys 'itinerant spin glasses' since they differ appreciably from the common dilute spin glasses usually understood in a localmoment picture. Itinerant SG models have been proposed by Kakehashi (1988a) on the basis of the nonlinearity of the couplings between local moments, and by Wasserman (1989b) on the basis of the results of the 'fixed spin moment' method (see section 6). In this context, it is important to note that binary and ternary alloys composed of Fe with 3d-elements on the fight-hand side of Fe in the periodic table (e.g., FeNi, FeCo, FeNiCo) do not show the itinerant spin-glass state, but undergo a structural phase transition prior of reaching the SG phase. In contrast to that, all
INVAR 1#00
277
Fe- Co - ~ . / / ~ o - Ni
1200 1000 800 60(3 ~00
---
200 g~oo
-
F£~_~'~'Ni-Mn
z
600
Feso(Ni,Mn)5o
[Fe, Ni}8oCr2o
O
I
.... + (F%oMn.~o}xC°wo i1--11 ( Fe •C r)Goi'4n ~o ( Fe,Cr}zoMn3o
x
A--A-.& Fex(Co2oNi~o}wo_x o - - 0 (Fe, Ni}./oC% 0
I
.... ( Fe,Co)~gC rll 800 Cr Hn Ni Cu Fe e/a Co 7 8 9 6 10 11 Fig. 29. Curie and N6el temperatures as a function of the electron concentration for different fcc transition element alloys (after Richter and Pepperhoff 1976). Note that in comparison to figs. 22 and 23 in the FM (and AF) Invar range the transition temperatures decrease with falling (rising) electron concentration. SG phases occur around e/a = 8.3, where the transition temperatures reach minimum values.
binary and ternary alloy systems containingat least one 3d-element that is positioned left of Fe in the periodic table (e.g., CoMn, CoV, FeNiMn, FeNiCr), do show the itinerant SG phase. As will be demonstrated later (section 6) this difference can be understood on a theoretical basis. If the m o m e n t - v o l u m e instabilities (Invar effect) become overwhelming, the systems undergo a structural phase transition before reaching the itinerant SG phase. If the lattice is continuously enlarged by alloying elements with larger atomic volume (i.e., those 3d-elements left of Fe in the periodic table), the systems can reach the SG state and the antiferromagnetic e / a range as well. H e r e they show A F Invar behavior, since - as to be seen later - e.g., fcc Mn shows m o m e n t - v o l u m e instabilities too.
4.5.2. Concentration dependence of magnetic moment For a long time it was thought that the deviation of the average magnetic m o m e n t /2 from the Slater-Pauling curve, like it is observed in FeNi alloys on approach to the y - a - t r a n s i t i o n (see fig. 5) is one of the most prominent features of Invar, since it gives obvious indication for the instability of the magnetic moment. On the other hand, as seen in fig. 6 for FePt and fig. 7 for FePd, there are alloy systems which show the Invar effect on approach to the y - a instability limit, their Curie temperatures decrease in the respective concentration range, but the moment does not deviate from the Slater-Pauling curve. Consequently, deviation of the average magnetic m o m e n t from the Slater-Pauling curve is not an Invar relevant feature.
278
E.F. WASSERMAN
With respect to the discussion giv,~n in the preceding chapter, this can also be understood in the modern band picture. If the lattice volume is enlarged by alloying Fe with Pt and Pd, which both have a larger atomic volume than Fe, the internal pressure in the systems is reduced and there is 'room' for large moments and large volumes of the Fe atoms. No deviation of the moment from the Salter-Pauling curve is the consequence. If the Pt or Pd concentration is reduced, the internal pressure rises, and at a certain limit the moment-volume instabilities become too large for the lattice to 'bear'. A y - a transition is then the consequence. These features have been described in a finite-temperature model by Kakehashi (1985) (see also below, section 6.2). Decisive in this model is the amplitude of the local moment as defined in a spin-fluctuation picture (Kakehashi 1985). This amplitude might be experimentally accessible through neutron diffraction, but the amount of this type of data for Invar is scarce, and also their interpretation is problematic (see Kakehashi 1985). 4.5.3. Temperature dependence o f the magnetization Another characteristic property of Invar alloys is their unusual temperature dependence of the magnetization (or of the magnetic moment) in comparison to 'ordinary' ferromagnets. For the latter we assume, for simplicity, Brillouin behavior to be valid. To show the principal differences, we again choose the two Invar compositions Fe65Ni35 and disordered and ordered Fe72Pt28. In fig. 30, following data of Yamada et al. (1982), the reduced spontaneous magnetization Ms(T, H = O)/Ms(T = O, H = O) = Ms(T, O)~Ms(O, O) versus the reduced tempera-
1.0 ~
.
[ Fe72Pt21 disord. ~".
0.85 I o
, o.2
I
I 0.6
I
o.8
to
T/T=
Fig. 30. Relative spontaneous magnetizationMs(T,O)/M,(O,O) versusrelative temperature T/Tc for ordered and disordered Fe72Ptz8and Fe6sNi35Invar alloys. The full curve gives the behavior for pure Ni (Brillouin function) [all data after Yamada et al. (1982)]. The inset shows the low-temperature behavior on an enlarged scale.
INVAR
279
ture T / T c is plotted. The inset shows in enlarged scale the low-temperature behavior. Note that initially, with rising T the magnetization in all three Invar alloys decreases faster than in Ni. This means that assuming a Bloch dependence of the form, Ms(T, 0) = Ms(0 , 0) 1 - 2.61 Ms(0 ' 0------)
(5)
the spin wave stiffness D m in Invar alloys, as determined from the temperature dependence of the magnetization, is smaller than in, e.g., Ni, expressing the magnetic softness of the alloys (see section 4.7 below). However, if one analyzes the available low-temperature magnetization data of Invar more closely, a T 3/e dependence of M s ( T ) can never be fitted in a sizeable range. Attempts to fit Stoner expressions with M s ( T , O)/Ms(O , O) oc 1 - a T 2 also failed. Best fits resulted from a combination of two dependences in the form (Maruyama 1986), M s ( T , 0) = Ms(0 , 0) [1 - a T 3/2 - b T " ] ,
(6)
with n = 2 --- 0.3 and a, b = const. The validity of this equation is footed on the assumption that in Invar alloys, besides spin-wave excitations (which are definitely present as we will see below, section 4.6), a second type of excitation occurs. Since the physical nature of the second excitation is not fully understood at present, eq. (6) is more or less a working hypothesis. We shall show, however, in section 6 that on the basis of modern theories of Invar its presence can be made at least qualitatively plausible. One should also note that a determination of reliable values of Ms(0, 0) is problematic, since as discussed in section 4.3, the M ( H ) curves in Invar often do not saturate [XHF values depend on the maximum field available, see, e.g., Ono and Chikazumi (1974)], leading to errors in the extrapolation of the M(H)-curves to H = 0. Secondly, if pure ferromagnetic ground states do not exist because of the occurrence of SG-phases or for metallurgical reasons [e.g., Fe65Ni35 contains some antiferromagnetic precipitations (Zfihres et al. 1988) and ordered FeTePt28 shows some premartensitic transformation at roughly Mps = 60 K (Maruyama et al. 1986)], neither M,(O, 0) nor the low-temperature dependence of M s ( T ) regardless of what type of fit is attempted - can be determined satisfactorily. In turning back to fig. 30 we can see that with increasing temperature the magnetization of Fe65Ni35 always stays below the Ni- or BriUouin curve, whereas disordered and ordered FeTzPt28 both cross the Brillouin curve at some temperatures, and on approach to T c show a higher magnetization than a respective ordinary FM. This explains - s t a r t i n g from the 'crossing' t e m p e r a t u r s - t h e experimentally observed sharp decrease of the spontaneous volume magnetostriction (and consequently negative values in the a ( T ) curves) on both Fe72Pt~s alloys (Sumiyama et al, 1976). Simultaneously, the sharp drop of M s ( T ) on approach to T c is accompanied by a steep rise in a ( T ) to positive values for T > T c. As one can see in fig. 30, for T > T c the magnetization in all Invar alloys has a long 'tail'
280
E.F. W A S S E R M A N
up to T/T c = 1.2 or higher. This reflects the presence of m o m e n t - v o l u m e instabilities for T > T c in Invar, as we have already seen above, when discussing other physical properties. In other words, the splitting does not vanish at Tc, when we interpret these features in a band picture. The 'tail' in Ms(T) also makes a correct determination of the Curie temperatures from Ms(T ) curves almost impossible. This does not improve very much, when Arrot plots from magnetization measurements M(H, T) are used. The M 2 versus H/M curves for temperatures around T c are not linear (Shen et al. 1985).
4.6. Spin waves and spin-wave sttffness As mentioned in the preceding section, spin waves are exited with increasing temperature in Invar alloys. Figure 31 shows the spin-wave dispersion in the three major directions of the Brillouin zone as measured by inelastic neutron-scattering (INS) at 11 K for ordered FeTePt:8 (Ishikawa et al. 1980) and for Fe65Ni35 at 0
0.2
80-
?;L
0.4
i
0,2
0.2
,
Fe72PI28 ordered
-
T
11K
; /.0-
2I
z[
40-
/,
~%,'= J ,"
7-
.,/'
~:2o~ o .... o
'
0.5
I
u3
I
2 •/ 1.0
0.5
oz
0.2
~.],
0.5
0
110 q (,&-')
0.2
0.,-
80-
80
"--~.
,
[t, OO]
60 -
=~ OJ
0.4
i
Fe65Ni35 &2K
;
6O
4O
2O
0
~20,
,
1.0
0.5
Dsw = 142 meV/~2
/
%,0
.,/
0
,
0.5
,
b.
1.0 q (,&-l)
Fig. 31. Spin-wave dispersion in the three major directions [~'00], [sr~'0] and [~'(sr] of the Brillouin zone at 11 K for ordered Fe72Pt28 and Fe65Ni35 at 4.2 K (a) Ishikawa et al. (1980). (b) Kohgi et al. (1976). ~"= q/qzB ( Z B = zone boundary) is the relative wave vector in the respective direction of the Brillouin zone (~'zB ~ 0 . 5 in all directions).
INVAR
281
4.2 K (Kohgi et al. 1976). For Fe3Pt at low q-vectors a quadratic dispersion relation of the form E = Dswq2 holds, with a spin-wave stiffness D~w = 80--5 m e V A 2, independent of the lattice direction. For Fe65Ni35, D~w=142___ 5 meV ~2 is found. There are negligibly small energy gaps of the order of 0.05 meV at q = 0 in both systems. On approach to the zone boundary q = rr/a in Fe3Pt, only in the [if00] direction a negative magnon dispersion with a limiting value of EBz [if00] 40 meV is found. There is some uncertainty in the data (see dots and squares in fig. 31) because phonon peaks cross through the spin-wave maxima near the [ if00] zone boundary (Ishikawa et al. 1980). In both systems, however, the quadratic behavior of E(q) continues for both directions [ff~0] and [ffffff] up to energies above 80 meV, and no saturation of the magnon energies could be detected. Stoner-type exitations are not found either, in spite of intensive search at q = rr/a (and lower q). The magnon peaks in the INS spectra just become very broad at high energies as expressed by the error bars in fig. 31. The Stoner boundary in both systems thus should be located above 80 meV. Since Invar properties start to appear at very low temperatures, Stoner exitations cannot be, if they exist at all, the origin of the Invar effect. In most of the publications of neutron scattering on Invar alloys it is quoted that the spin-wave stiffness renormalizes with temperature (provided E = Dswq2 holds) according to, Dsw(T ) = Dsw(0)[1 - A(T/Tc)5/z].
(7)
This equation is based on the Heisenberg model (magnon-magnon damping), thus A is proportional to the exchange interaction (see Fernandez-Baca et al. 1987). As shown in fig. 32a, eq. (7) seems to be valid almost up to T c in ordered Fe72Pt28 (Ishikawa et al. 1979) and amorphous Fe86B14 (open triangles; Fernandez-Baca et al. 1987, closed triangles; Ishikawa et al. 1981), while it holds well for Fe65Ni35 up to (T/Tc)= 0.7 (Ishikawa et al. 1979) but not at higher temperatures, and is also not valid for Fe2(Zr0.7Nb0.3) at high temperatures [see crosses in fig. 32 (after Onodera et al. 1982)]. Surprisingly, eq. (7) is valid for FesoNis0, only for T / T c > 0.6. Since Invar systems are definitely not pure Heisenberg-like, the validity of eq. (7) to describe the temperature dependence of the spin-wave stiffness (equivalent to the renormalization of the spin-wave energy) with temperature is doubtful anyhow. Moreover, the experimental data are not sufficient in number and accuracy so t h a t - as revealed in fig. 32b - a plot of Dsw versus (T/Tc) z describes the results as well as, and at high temperatures close to T c even better than the (T/Tc) 5/2 dependence. However, note that both plots in figs. 32a,b are not very conclusive for (T/Tc) < 0.4, because of the 'compression' of the low-temperature range. Since Invar systems are neither purely Heisenberg nor purely Stoner-like, one can again use the working hypothesis that in Invar two different excitationsmagnons and 'Invar-type' spin fluctuations- are present simultaneously. A phenomenological approach (Izuyama and Kubo 1964) to the theory of magnetic
282
E.F. W A S S E R M A N
0.2 0.4 1 I
0.6 I
200
(T/To)
0.8 I
0.9 o Fe65 Ni35 • Fe72Pt28
[] +
N I ""~o, I ~0 0.2 0.2 0.4 '
'
+ Fe2(Zr0y Nb0.31
o~ I a @ ~,p.....~ F I 0.4 0.6 (T/To)S/2 1.0 0.6 0.8 (T/To) '
200
'
"*.. " A ~ .~
0 0
b.
\
~
100
ord.
zx,A o-FeB6 BlZ"
m
o,~.
e~ .~ >~ E -~
~.
I
~+.
"*--~'-~.,~o
",~
i
i
I
I
02
04
20.6
OS
10
(T/To)
Fig. 32. (a) Spin wave stiffness constant D,w versus (T/Tc//2 for F%oNis0, Fe65Ni35 and ordered Fe72Pt28 fall data after Ishikawa et al. (1979)], as well as amorphous Fes6B14 [open triangles: Fernandez-Baca et al. (1987), closed triangles: Ishikawa et al. (1981)] and the Laves phase compound Fe2(Zr0.7Nb0.3) (after Onodera et al. 1982). (b) Same data as in (a) plotted versus (T/Tc) 2.
materials with collective (itinerant) electrons, leads to the following temperature dependence of the magnon spectrum, (o(q, T) = ~o(q, 0)(1 -
AT 2 - BTS/2),
(8)
where the T 2 term is determined by Fermi excitations of the electrons and the T 5/z term by spin fluctuations. Theoretical approaches for evaluating the coefficients A and B have been discussed recently by Silin and Solontzov (1985, 1987). They showed that the anomalous frequency and temperature dependence of the magnon-spectrum rigidity is caused by the influence of the dynamics of the crystal lattice on the spin fluctuations in itinerant ferromagnets. Applied to Invar, their results could lead to an understanding of a well-known intriguing problem of Invar, i.e., the difference in spin-wave stiffness Dsw as determined from neutron scattering, and D m as determined from the temperature dependence of the magnetization according to eq. (5), i.e., from a T 3/2 law.
INVAR
283
In table 6 we list some values of Dsw and D m a s they have been taken from different references. Ratios of D s w / D m - - 1.3-2.5 are observed in Invar, while in non-Invar systems the ratio is always equal to unity. Note that in FesoNis0, although this compound shows a small invar effect (%0 = 0.5 x 10-2), D s w / D m is also unity. The discrepancies between Dsw and D m have led Ishikawa et al. (1986a) to the statement that in Invar some 'hidden excitations' exist which are not sensed by neutrons, i.e., on a short time scale. However, they do contribute to the magnetization, which means they are 'seen' in a long time-scale experiment. Although the problem is not yet fully solved at present, we think (see section 6) that the volume coupled high-spin-low-spin-state transitions, clearly revealed in modern band calculations, could give rise to these 'hidden excitations'. However, since band calculations are so far only done for T = 0, the dynamics and nature of the high-spin-low-spin-state transition and the temperature dependence involved remains an open question at present. Without going into the details of the dynamics we remark here that Ishikawa et al. (1979) and Onodera et al. (1981) have proposed that the rapid demagnetization of Invar-alloys with temperature is a consequence of an anomalous spin-wave damping. Indeed, in a neutron-scattering experiment longitudinal spin fluctuations (like possibly the high-spin-low-spin-state transition) could cause a considerable increase in the linewidth Fq. INS experiments on crystalline Fe3Pt and Fe65Ni35 (Ishikawa et al. 1979, Onodera et al. 1981) and amorphous Fe86B14 (Ishikawa et al. 1981) revealed that the spin-wave linewidths are not proportional TABLE 6 Spin-wave stiffness constants at 0 K. Invar systems
Fe65Ni35 FesoNis0 o.Fe72Pt2s Fe2(Zr0.7Nb0.3) a-Fes6B14 a-Fe82Bls a-FeTsSilsB10 a-FeToNi20Zrl0
T c (K)
Dsw (meV A 2)
D m (meV A 2)
503 785 504 380 556 617 710 455
142 --- 5 [1] 220 -4-10 [2] 80 +- 5 [2] 350 -+ 50 [3] 125-+7 [4] 165 [4] 220 -+ 10 [6] 112 [8]
59 [1] 225 [2] 60 [2] 125 + 10 [3] 65 [5] 71 [5] 127 --- 5 [7] 72 [9]
114 -+ 10 [10] 60-+ 2 [11] 310 -+ 10 420
115 4- 3 [10] 55 - 2 [11] 285 --- 10 415
~
Non-Invar systems
(Fe6sNi35)7sP16B6A13 FeToCr2oP13C7 Fe Ni References: [1] Hennion et al. (1975) [2] Ishikawa et al. (1979, 1981) [3] Onodera et al. (1982) [4] Fernandez-Baca et al. (1987) [5] Hasegawa and Ray (1979) [6] Minor et al. (1985)
[7] [8] [9] [10] [11]
Dmowski et al. (1984) Fernandez-Baca et al. (1986) Krishnan et al. (1984) Tarvin et al. (1978) Birgeneau et al. (1978).
284
E.F. WASSERMAN
to q4 like in ordinary ferromagnets, but rather follow the empirical relation
Fq ~ Fo(1 + CT~)q 2 ,
(9)
with a = 1 f o r F % P t and a ~< 1 for Fe6sNi35. Although there are some contradictions in the literature [e.g., Fernandez-Baca et al. (1987) report Fq oc q4[ T ln(k B T~ Eq)] 2 for a-Fe86B14 in the range 4 4 5 - 5 2 0 K < T c = 5 5 6 K ] , and the analysis depends on the spectral weight functions chosen and on resolution limits, there is no doubt that for Invar alloys the linewidths Fq are much larger than for ordinary Heisenberg or Stoner-type ferromagnets. This is seen from fig. 33 where we have collected all data Fq(q) known to us for FeaPt and Fe65Nia5 from the literature. FOr T < T c (data from Ishikawa et al. 1979, 1986b) Fq roughly follows the q2-dependence in both alloys (and FesoNi50 as well). Although there is some ambiguity in the exact value of' the exponent, a dependence Fq oc q4 like in Pd~MnSn or E u O is definitely not observed in Invar. I
I I
I
10 -
6.0 --
E O ~2.0
I
I I
[] • .
0.11 Tc 0.34 Tc 0.58 Tc
m o •
A V
0.68 Tc 0.78 Tc
T = Tc
o • +
1.20T c T=Tc 139Tc
+ q2.7 ~,/ + ~'
~[+
~0.6
+' , ,
-o,.
0.16 Tc Q8z, Tc 1.25 Tc
+,,,/;'
2 +
,(//
/
0.02 ["
{ "
/+F~
!
/q2
//
?
-
'
-
/~'"+'/
I
/
(0.59Tc)
/
,/i'
/-
,5( ',I' i ~ t/i
/"
!
_
i1~
r /// 0.0/.
I
/
,,/:+t
o.1
El/
,,
°
02
I
Fe 65 Ni 35
8 1.0
.-,
I
Fe 3 Pt
2o
4.0
I
I
0.04 Q06 Q1
'
/
/ {E ?Tc ) I
I
0.2
QL.
t
q ( k -I )
I I
0.1
I
0.2
I
I
t
0.4 0.6 1.0
Fig. 33. Double logarithmic plot of the spin-wave linewidth P(q, T = const.) versus wave vector q at constant temperatures f9r ordered Fe3Pt and Fe6sNi35. Some data for F%0Nis0 and the Heisenherg ferromagnets Pd2MnSn~and EuO are also shown for comparison [all data after Ishikawa et al. (1979, 1986b)].
INVAR
285
The insufficient accuracy of the data in fig. 33 also does not allow a closer examination of the critical behavior. At T ~ Tc (about 4 K above) in Fe3Pt one observes Fq ~ qn with n =2.7---0.1, which is larger than n = 2.48, as expected from critical-scaling theory and observed in pure a-Fe (Mezei 1982). For T > T c at all q-values investigated, there is a further nonmonotonic increase in linewidth above the values found at Tc in Fe3Pt (Brni et al. 1986) and Fe65Ni35 (Tajima et al. 1987). This might be considered an Invar-typical feature, since a fit to the data using a spin-fluctuation dependence of the form Fq oc q/(q2 + /¢2), where K is an inverse correlation length, did not give satisfying results either. More accurate measurements are obviously necessary to achieve a conclusive answer concerning the nature of the dynamical behavior in Invar alloys. Doubtlessly, however, the polarization of the 3d-bands persists to far above Tc, as already shown by Collins (1965) by means of paramagnetic neutron-scattering on Fe65Ni35. At twice the Curie temperature, he found an iron moment of (1.4---0.4)/x B.
4.7. Elastic properties As mentioned in the introduction, Guillaume found already in 1920 that the temperature coefficient of the elastic modulus of FeNi becomes very small around 45 at% Ni (Elinvar effect). Since then the elastic properties of FeNi and other Elinvar systems (see section 2) have been studied extensively, mainly because of their technical importance. Good summaries on the elastic properties of polycrystalline Invar and Elinvar alloys- expressed by the bulk, shear and elastic moduli as functions of alloy composition, temperature, and magnetic field - have been given in the Honda Memorial Series on the Physics and Application of Invar Alloys (1978) (chs.7, 23 and 24). Of more fundamental importance for the physical understanding of Invar, are studies of the elastic behavior of. singlecrystalline material, i.e., the determination of elastic constants. A cubic single crystal has three independent elastic constants c11, c12 and c44. There are also three different modes i of propagation for a sound wave with velocities vi. These velocities are related to the elastic constants (/9 = density) as indicated in table 7. One can see that a measurement of sound velocities in the [110] direction on one TABLE 7 P r o p a g a t i o n of s o u n d waves a n d respective elastic c o n s t a n t s for a crystal with cubic s y m m e t r y . Direction of propagation
Mode i
Direction of polarization
pv~ = cij
[100] [100]
L T
indifferent
cl 1 c44
[110] [110] [llOl
L T1 Tz
[1101 [110] [0011
1(Cll d- C12 d-2C44 ) ~ C L ½(cll - Q2) --= C' c,, =- C44
[111] [111]
L T
indifferent
1(Cll + 2c12 d- 4c4,t) ~ (c~1 - c n + c44 )
286
E.F. WASSERMAN
sample allows to determine the three independent constants Cll , c12, c44. Commonly one expresses the elastic properties of crystals with cubic symmetry through three elastic constants CL, C' and C44 ('Capital C') which are defined by linear combinations of the c u (see table 7) and related directly to the modes of propagation in the [110] direction by, 2 C ' = PUT1
C L = pv 2 ,
2 pvr2.
C44 =
,
(10)
The bulk modulus is given by B=CL
2c12).
-- ~1 .C,
(11)
-- C44 = 1 ( C l l _.}_
C L expresses a compression mode accompanied by a change in volume, C' corresponds to a pure shear mode with an expansion along one of the [110] axis (tetragonal distortion). C44 corresponds to a pure shear mode with an expansion occurring along the [111] axis (trigonal distortion). Both shear modes leave the volume unchanged. For ferromagnetic Invar alloys, a large softening in the shear behavior occurs when the temperature is lowered below the respective Curie temperatures. This is seen in figs. 34 and 35. In fig. 34 the elastic constant C' is plotted versus
/~/~,
Fe72Pt28 ord. I N £ ~ . ~
.{--+ /
~3
/
% z o o x ~2 "t..)
/
/
/
op
0
~ ~e Fe72 Pt28 disord.
+~.+"-~... "~,+~ E
•
/
ei"'
I
0
V /
~ .
°
./ j.,"
~.
Fe55Ni35 INS
/'(/~°~" -] / cl~ / Fe65Ni35 Fe59Ni/,1
/
~+.+~"
f
i
C = {c11-c12)/2 I 200
I
I 400
I
I 600
I
I 800
1000
T(K)
Fig. 34. Elastic shear constant C' as a function of temperature, as determined in ultrasonic measurements for pure Ni, FersNi35 and F%gNi41 by Renaud (1988) and ordered and disordered Fe72Pt28 by Hausch (1974). Data from inelastic neutron-scattering (INS) as determined by Endoh et al. (1977) for Fe65Nia5 and by Tajima et al. (1976) for ordered Fe72Pt28 are also given. The respective Curie temperatures are marked by arrows.
INVAR
287
temperature. Data for pure Ni and FeNi alloys are taken from a recent thesis by Renaud (1988), who measured the elastic constants in zero magnetic field in the temperature range from 77 to 1500 K by an ultrasonic method. The data for FevzPt28 (disordered and ordered) originate from the work of Hausch (1974), who measured C'(T) in zero field and a field of 0.6 T to suppress domain reorientation effects (AE effect). There is, however, no difference in the principal behavior of C'(T), although the AE effect in C' is large at low temperatures (in FePt -~ 30% for T---~0) but small at Tc. Figure 34 also shows the results of C'(T) from fits to phonon dispersion curves as measured on Fe65Ni35 with inelastic neutron scattering (INS) by Endoh et al. (1977) and by Tajima et al. (1976) on ordered Fe72Ptz8. Note that there is roughly equivalence between the neutron data and the ultrasonic measurements at low temperatures, w.hile for T around T c or T > Tc the sound velocities from INS measurements are somewhat higher than the velocities from ultrasonic (US) measurements. This discrepancy might be due to the difference between so-called 'zero' sound (neutrons) and first sound (ultrasonics) (Endoh et al. 1977). Grosso modo, the temperature behavior of the shear mode C'(T) is, however, similar in the INS and US investigations. Both methods show that substantial softening is present in the Invar systems on lowering the temperature, setting in almost at the respective Curie temperatures (see arrows in fig. 34). Note also that the softening is absent in Ni, where for T < T c even a small increase in slope of C'(T) with respet to the range T > T c is observed. Without going into details, we mention here that the softening in the shear constant C ' = ~(Caa- c12) was explained in a band picture within the Stoner model for F%Pt by Pettifor and Roy (1978). As can be seen from fig. 35, the shear constant C44 as a function of temperature shows almost the same overall behavior as C'(T). The data in fig. 35 are taken from the same references as the data in fig. 34. Note that in Fe65Ni35 below about 50 K there is a sudden increase in C44 on lowering the temperature, which also occurs in the shear constant C' as seen in fig. 34 (Hausch 1976). This increase is absent in F%Pt (Hausch 1974) and 9bviously related to the magnetic inhomogeneities present in Fe65Ni35 at low T (see magnetic phase diagram in fig. 5). This is supported by the observation of the low-temperature behavior of CL(T) on Fe65Ni30.sMn4.5 (Shiga et al. 1988) (cf., fig. 36) where below the SG-freezing temperature Tf a sudden increase in CL, i.e., a lattice hardening, sets in. This leads us to the discussion of the temperature behavior of the longitudinal elastic constant CL(T), which is shown in fig. 36. Data for Ni and FeNi alloys (in zero field) are due to Renaud (1988), for Fey2Ptz8 (ordered and disordered in B = 0.6 T) again to Hausch (1974) and for FeNiMn to Shiga et al. (1988). The INS results for Fe72Ptzs are calculated from data by Tajima et al. (1976), the results for Fe65Ni35 are due to Endoh (1979). In comparison to the shear behavior C'(T) and Ca4(T), the temperature dependence of the elastic compression CL(T) of Invar shows the following different features: (1) In the US measurements again a substantial lattice softening is observed, which, however, sets in at temperatures far above T¢.
288
E.F. WASSERMAN 14
I
I
I=
I
I
i
i
i
i
i
i
i
"~.
z 12
~
0
"~.
"~
l Ni
Fe72Pt28 ord.
/
--,-
Fe72Pt28 Otto,-,°~° •~ ~ .... ° Z."
o Q
~
i
~-
o0~. . . . . . . . . •
_
Fe5g Niz,1
• •
~'-'x'~ ~
"-,,.,.,
.
/ . Fe72Pt28 dlsord.
x~.
I I I/
tr-" 'J
8.~
I
I
0
i
i
200
I
z.00
I
i
600
i
i
800
I
I
1000
1200
T(K) Fig. 3,5. Elastic shear constant C44 as a function of temperature, as determined in ultrasonic
measurements, for pure Ni, Fe65Ni35 and Fe59Ni4t by Renaud (1988) and ordered and disordered Fe7zPt28 by Hausch (1974). Data from inelastic neutron-scattering (INS) for ordered Fe72Pt28 by Tajima et al. (1976) are also given. The respective Curie temperatures are marked by arrows.
I
.
34
" ~ ' ~ .
32
] ~
1
I
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I
"~'~
o_.._ u-....,u - . °
; 1" f " - . .
"...~Ooo ~..°~" • o a
..~ ~.
i
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I
{
INS Fe72Pt28 ord.
~
INS Fe65 Ni35
°°., f
,Feso Ni so
..&° 0o
'~.
|
t ~+",,--,"~+'7"~';~r~L.~:~m._~_
.-4
!
÷" +-:'- t. +..f.'.~- +. +.+ .+-.~""T'me Fe72 Pt28 ore.
II !I Tf
18 "~"
I
N
.
I
f
~•
I
I
200
.'1Fe.. ~ /z Pt.~ zo disord "
,/ ~"
I
400
I
//
I
I
600
I
800
I
I
1000
I
1
/
1200
T (K)
Fig. 36. Longitudinal elastic c o n s t a n t C L versus temperature, as determined in ultrasonic measurements, for pure Ni, Fe65Ni35, Fe59Ni41 and FesoNis0 by Renaud (1988), for ordered and disordered Fe72Pt28 by Hausch (1974) and for Fe65Ni30 5Mn4.5 by Shiga et al. (1988). Data from inelastic neutron scattering (INS) for ordered Fe72Pt28 by Tajima et al. (1976) and Fe6sNi35 by Endoh (1979) are also given. The arrows mark the respective Curie temperatures.
INVAR
289
(2) CL(T ) has a minimum below Tc, therefore irrespective of the increase through the spin-glass freezing (see FeNiMn and Fe65Ni35 in fig. 36), there is (3) increasing lattice stiffness for T---~0. (4) In inelastic neutron scattering, i.e., in zero sound, the softening in the longitudinal mode is not observed, neither in FeNi nor in FePt (cf. INS data in fig. 36). The latter observation (4) is reminiscent of the situation described in the preceding section, where we discussed the differences in spin-wave stiffness observed on short time scales with neutrons and long time scales in magnetization. Obviously, the Invar-typical volume changing high-spin-low-spin-state transitions (see section 6) are also not sensed in the phonon dispersion curves boo(q, T) determined by neutron spectroscopy on short time scales (zero sound). Remember that this is different when t h e - v o l u m e conserving-transversal (shear) modes are investigated. The elastic constants C'(T) and C44(T ) as derived from the phonon-dispersion curves ho~(q, T) in the [110] direction for q < 0.4A -1 in INS are almost equivalent (cf. figs. 34 and 35) to those determined from the transversal sound velocity OT(T ) in the US experiments at much smaller q-values (smaller frequencies). In this context we mention that an unexpected 'central peak' around ho~ = 0 was observed in Fe3Pt in addition to the soft-phonon side peaks (Tajima et al. 1976) in the [110] direction in the INS spectra, the intensity of this peak developing gradually on lowering of the temperature from 100 to 4.2 K. Tajima et al. (1976) suggested that the central peak was of static origin, possibly related to the premartensitic transformation in Fe3Pt , but a detailed explanation for its physical origin was never given. It remains an open question whether central peak is indeed only of static origin. In total, the frequency and temperature dependence of the longitudinal modes in Invar are not yet fully understood. It is necessary to investigate further these dependences, i.e., the dynamical and temperature behavior of the electronphonon coupling in the presence of longitudinal and/or transverse spin fluctuations. Since- as we have s e e n - phonon softening occurs only at small wave vectors q, i.e., at long wavelength, only distant forces are changed and the softening is not dominated by near-neighbour forces. This means that band-type calculations provide the right approach for the understanding of the problem. For this it is helpful to know absolute values of the purely magnetic contributions to the different modes. The determination is possible in a reliable way on the basis of the high-temperature ultrasonic investigations by Renaud (1988). Extrapolating the high-temperature (T >> Tc) linear behavior of CL(T ), C'(T) and C44(T ) to T = 0 provides the respective paramagnetic references for the temperature range T < T c. The magnetic contributions to the elastic constants are then given by the difference between the paramagnetic reference and the actually measured data (cf. figs. 34-36). Assuming similar linear dependences to hold for T >> T c in Fe72Pt2s, the magnetic contributions to the elastic constants (at ultrasonic frequencies) can be evaluated for these alloys (with larger error) too. Note again the difficulty discussed above that the reference is paramagnetic and not 'nonmagnetic' as would be actually required.
290
E.F. W A S S E R M A N
By means of eq. (11) the magnetic contribution to the bulk modulus ABm(T ) can then also be determined. The result is shown in fig. 37, where AB m versus the reduced temperature T~T c is plotted. The inset shows in the same type of plot the shear behavior AC'(T/Tc). A C 4 4 m ( T / T c ) ( n o t shown in the figure) has almost the same quantitative features as AC'(T/Tc). Note the onset of the softening in ABm, the longitudinal (volume changing) compression at T > Tc, where the pure shear modes AC" (and A C 4 4 m ) a r e zero, and the large drop in A B m in the range just below To where the softening in shear is also of minor influence. &BIn(T~ Tc) has for all Invar alloys a characteristic minimum, which is largest in disordered Fe72Pt28, still observable in FesoNis0 and not present in pure Ni, for which ABm(T/Tc)= 0 in the whole temperature range investigated (points omitted for clearness in fig. 37). A further characteristic feature is the increasing lattice hardness (rise in ABr~) on approach to zero temperature, which is more or less present in all Invar alloys in fig. 37. The differences between Fe65Ni35 and Fe72Pt28 are probably due to the differences in the forced volume magnetostriction Ow/OH, which is large in Fe6sNi3s at low temperatures but small in Fe72Pt28. Finally, we infer from fig. 37 that in the 0 ~< T/Tc<~0.3 range the magnetic contributions due to shear deformation A C ' , AC44m are larger or of the same order of magnitude (Fe65Ni35) as the magnetic contribution to the bulk modulus. Although many efforts (see, e.g., Hausch 1979) have been done to understand the salient features of the elastic behavior of Invar as presented in fig. 37, no
1 .~:"--'~"-~,i
'
-o.2-.~q~
-1
z -°o~ - 3
/
'
~o.6
='D- ~
\
FeSO"'SO
3
~\,,+~+
T/Tc
0.8
~ ~'
'
'
'
'
to
t2
~.~
t6
~"~"
.o//
yV _
x
= -~
<
I
~+-~
~++~.,~
~\ . ~
j'(
-5 -6
--7
~ +.....4-~ O.o.O~
I
I
+"7"
~
Fe69Ni31
~•
Id's°rd
o
.
I
,c~
o.~. ~T~ . ~'1° o .~
I ~
I
F •
+.i+i +_.+-+
I ~'~ " / *
,./+'÷] / / /
+.+r.~'/ . /
/*
.
/
Ez -~
"~
,~
.*~
Fig. 37. Magnetic contribution to the bulk m o d u l u s AB m versus reduced temperature T / T c as calculated by use of eq. (11) from the ultrasonic data in figs. 34-36 by extrapolating the results in the paramagnetic high-temperature range to T = 0 K, and taking the difference between extrapolated and m e a s u r e d values. Note the characteristic softening on approach to Tc, as well as its persistence to far above T c. The inset shows the magnetic contribution to the shear constant A C " versus T / T c determined in the same fashion for 3 alloys (symbols as in large scale figure). Note that the shear softening sets in at T c and increases continuously on lowering the temperature.
INVAR
291
satisfying answer has been given so far. Especially, the model of weak itinerant ferromagnetism due to Wohlfarth (1976) is not applicable at all. The situation is complicated by the fact that antiferromagnetic Invar alloys, in spite of having a substantial spontaneous volume magnetostriction (cf. fig. 23) exhibit a completely different temperature (and frequency) dependence of their elastic constants in comparison to ferromagnetic Invar. This is shown in fig. 38, where the temperature dependences of C L (fig. 38a), C' (fig. 38b) and C44 (fig. 38c) are given. Ultra sound measurements on a single crystal of Fe60Mn40 (open triangles) are due to Renaud (1988), on a crystal with the almost the same composition Fe61.sMn38.5 (solid dots) from Lenkkeri (1981) and for Mn85Ni15 (full curve) and Mnal.sNi18.5 (dashed curve) from Hausch et al. (1983). The INS results i
~nSl
i
~
S Nil85
-"-.j.~<. ~ . , . .
28
~"85 '~'15
~
I
i~
\
~
i
i
F,7OMn30tNS \.7
%A.
(31.
19~
. 'i,k
Z n4(
24
C
,h~,,,,,,%_.. ~
AF- Systems
b.
/ -+-+ re;o:°,o
{
"(_.1
17j
C J
5a
o
I 1z,
I
I
I
I
13
++4 0
I c
,~
10
I
f.+ ,.,
, 200
, T (K)
, ~'~ 400
Fe60Mn40
, 600
Fig. 38. Elastic constants CL, C' and C44 as a function of temperature for single-crystalline antiferromagnetic Invar alloys. Ultra-sound results on Fe60Mn40 (open triangles) are due to Renaud (1988), on a crystal with almost the same composition Fe61.sMn38 5 (solid dots) to Lenkkeri (1981), on MnasNils (full curves) and Mna~.sNi~s.s (dashed curves) to Hausch et al. (1983). The INS results on FeToMn30 have been determined by Endoh et al. (1981). Note that in contrast to ferromagnetic Invar (figs. 34-37), softening in antiferromagnetic lnvar is noticeable only in C'(T) right at the respective N6el temperatures (arrows).
292
E.F. WASSERMAN
on FeToMn30 (solid triangles) have been determined by Endoh et al. (1981). We infer from fig. 38 that: (1) the softening in the longitudinal mode CL(T ) is absent in AF Invar alloys, there is just a small dip at T = TN; (2) the discrepancy in the CL(T ) behavior between short-time neutron investigations (INS) and US measurements at lower frequencies does not exist, both methods show lattice hardening in the range T < TN; (3) Under trigonal distortion in C44(T), almost no softening effect is observable; (4) the only relaxation occurs in the tetragonal shear distortion in C'(T) right at TN, where a s u d d e n - first-order l i k e - drop takes place at temperatures just below the respective Nrel temperatures, while on further lowering of T there is again a linear increase in C'(T), with a slope similar to the one in the paramagnetic range. A determination of the magnetic contribution to the bulk modulus ABe(T) [eq. (11)] consequently gives a sudden hardening for AF Invar at T = T N. For T < TN, ABm(T ) is positive and almost constant to T = 0. If we assume - for good reasons, as demonstrated above - that the Invar effect in FM and AF systems is of the same physical origin, the results presented in fig. 38 are puzzling, and up to now not understood. In our opinion, the elastic behavior is strongly influenced by the atomistic type of magnetic ordering, which is quite different in the FM and AF systems. The FM single-crystalline systems presented above all have a collinear spin structure, while both AF systems in fig. 38 have a noncollinear spin structure. It is known from neutron-scattering experiments on FeMn (Endoh and Ishikawa 1971) and MnNi (Kawarazaki et al. 1988) that the spins in both alloys point in the [111] direction of the unit cell. This could lead to a 'compensation effect' between the spin components in different lattice directions under compression and trigonal shear (C L and (7.44) and explain the absence of softening in these modes. Moreover, the first-order like transition observed in C'(T) right at T N in the AF crystals can also be made plausible. It is known (Endoh and Ishikawa 1971) that fct FeMn has a collinear spin structure. Thus, under tetragonal distortion there is possibly a sudden transition from a non collinear to a collinear spin arrangement for T ~< T N.
4.8. Specific heat Since for the understanding of the Invar effect (and magnetoelastic properties in general) electronic and phonic contributions are of equal importance, measurements of the specific heat provide an important tool to study both contributions simultaneously. We first turn to an analysis of the electronic contribution, i.e., of the y-term in the electronic specific heat. Since in Invar alloys the instability of the moment is accompanied by a drastic change in density of states (DOS) of the d-electrons near the Fermi surface (see section 6), unusually high y-terms have been thought for a long time to be an Invar-typical feature. In fig. 39 we have plotted y-values as derived from the low-temperature specific heat as a function of e/a for
INVAR I
293
I
I
v
o •
V
~ ~ v
3O
+
L@
LO H
® c~v
~ a
I Fe Ni Ni Co
Ni M n
•
Ni V
x A
FeMnC4 FePt ord.
•
FePt
"
Fe50(NiMn)50 (Fe Ni)90 Cr i0 Fe65(Ni Cr)35
disord.
N
RSG 20
o e /
?
V e
ml" L~ H
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L
: o w n invest.
x
jo /
'°\3
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lo
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~'
i
. .~.o-,
+
0
75
L
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8.0 Fe
8.5 e/G
9.0 Co
9.5
10 Ni
Fig. 39. ,/-coefficients of the low-temperature electronic specific-heat as a function of the electron concentration for different Invar and non-Invar transition metal alloys (for references see text). Note that the enhanced values observed in the range e/a = 8.2-8.6 are caused by superposition of two linear terms in the specific heat at low temperatures, the electronic y-term and a spin-glass or re-entrant spin-glass (RSG) term originating from the low-energy excitations in the disordered state. For other details see text. different systems. Data for FeNi (open circles) and NiCo (solid circles) have been measured by Caudron et al. (1974), for NiMn (plus signs) FeMnC 4 (crosses) and NiV (solid triangles, downwards) by Gupta et al. (1964), for FePt (triangles upright) by Sumiyama et al. (1976), for Fe65(NiMn)35 (open squares) by Kawarazaki et al. (1972), for F%0(NiMn)50 (solid squares) and Fe65(NiCr)35 (open circles, divided) by Deryabin et al. (1983) and for (FeNi)90 Crl0 (open triangles downwards) by Deryabin et al. (1985). Several features are immediately obvious from fig. 39. First, it seems that in the FM Invar range for e / a > 8.5, where the spontaneous volume magnetostriction o)s0 is large (cf. fig. 23), the 3,-values increase to twice or three times the values observed in pure Ni or non-Invar systems like NiV or NiMn. Note, however, that, secondly, in ordered and disordered FePt ( 3 , = 9 . 6 m J / m o l K 2 and 3' = 7.6 m J / m o l K 2, respectively) 'normal' 3,-values like in Ni have been measured. Third, in the range e / a = 8.3-8.5, where o)s0 vanishes, the 3' values become very large, up to 30 m J / m o l K 2. This increase in 3, has nothing to do with the Invar effect. It originates from magnetic low-energy excitations in the SG or RSG phases, which are present in the respective alloys at low temperatures. It is well-known that spin glasses exhibit a linear magnetic specific heat as a function of temperature up to temperatures above the respective freezing temperatures, so
294
E.F. WASSERMAN
that the total specific heat in the alloy systems in question is given by,
Cp(T) = 3,T + asGT +/3 T3 .
(12)
In plots of (Cp/T) versus T 2, from which the electronic 3' terms (and the Debye-like/3 lattice term) are usually derived, both linear T terms are superimposed and cannot be distinguished. To separate the two terms, we recently have carefully reinvestigated the low-temperature magnetic properties [AC susceptibility x'(T) and X"(T)] and the specific heat in zero external field and in magnetic fields up to 6 T of FM RSG alloys Fe57Ni23Cr20 with T c = 23 K, Tf = 19K (Lecomte and Schubert 1988) and F%0Ni35Mn15 with T c = 2 0 8 K , Tf = 60 K (Lecomte et al. 1988a), as well as an AF alloy FesoNiesMne2 (Lecomte et al. 1988b) with T~ = 80 K, which is, however, not a good long-range ordered AF. Our 3/values as derived from the zero-field measurements are marked by an 'L' in fig. 39. They are in agreement with the earlier data. Yet, in FM + RSG samples SG-typical maxima are observed in the magnetic specific heat above the respective Tf. If one tries to 'freeze out' the SG contributions by measurement of Cp(T) in a magnetic field, appreciable reductions in the magnetic specific heat are observed. The 3,-values as determined in H = 6 T are marked by an additional 'H' in fig. 39. Yet, these values do not represent the 'true' 3,-values of the alloys, since the relative reduction Cp(H)/Cp(O) at constant T (e.g., 4.2K) shows a linear decrease with increasing H up to 6 T, and is therefore far from saturation. The previous discussion shows that an enhanced 3,-term of the electronic specific heat is not an Invar-typical feature. The magnitude of 3' is apparently not directly correlated to the magnetoelastic properties (compare figs. 23 and 39). There is no maximum in 3,(e/a) in fig. 39 in the AF-range, where ~Os0(fig. 23) reaches peak values too. The only alloy investigated there, i.e., FesnMn11C 5 [see cross with question mark in fig. 39; data from Gupta et al. (1964)], is very likely inhomogeneous (mixture of e-martensite and 3,-phase), and for a comprehensive discussion not reliable. The same holds for the carbon-stabilized sample Fe65Ni28C 7 [see circle with question mark in fig. 39 (Gupta et al. 1964)]. Also Fe65Ni35 (e/a = 8.7) exhibits mixed magnetic behavior at low temperatures (cf. magnetic phase diagram in fig. 5). In conclusion, reliable 3,-values for Invar are of the order of 10-11 mJ/mol K 2 and thus they are not 'anomalous' with respect to Ni or other magnetic non-Invar systems. We have indicated the range where 3, is enhanced through SG contributions by the dashed-dotted line in fig. 39. Concerning the lattice contribution to the specific heat, doubtlessly the softening of the lattice observed in Invar in the elastic properties should be directly reflected in the respective Debye temperatures 0D. In fig. 40 we have plotted the Debye temperature versus e/a as collected from the literature. Data symbols and references are identical to the references in fig. 39. Although most of the Debye temperatures in fig. 40 have been determined from the low-temperature specific heat and thus are ~ ( T = 0) = 0~, the analogy of the concentration dependence of 0~(e/a) to the spontaneous volume magnetostriction O~s0(e/a) (see fig. 23) is obvious. Low 01, i.e., lattice softening, occurs in the FM range around e/
INVAR
600
I
295
I
I
I
o + • A • • ?
• x
5OO
Fe Ni NiMn NiCo FePt F e 5 0 (NiMn) 50 FeMnC&
Fe 65[NiCr) 35 L
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°, B"
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400
HA
t-S6 I I
o o
o
+
300
~ord. I
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I
75
8.0 8.5 9.0 9.5 10 Fe e / ca Go Ni Fig. 40. Debye temperatures 0D versus electron concentration e/a for different Invar and non-Invar transition element alloys (for references see text). Note the 'softening' in the ferromagnetic Invar range around e/a = 8.7. With some uncertainty softening also occurs in the antiferromagnetic Invar range around e/a = 7.7. a = 8.5-8.7 and very likely also in the A F range around e / a - ~ 7.7, i.e., at those values of e / a where OJs0 reaches maxima. Not m a n y specific-heat m e a s u r e m e n t s have been reported in the literature for an intermediate t e m p e r a t u r e range between about 20 K and r o o m temperature. Measurements of Cp(T) in this range, however, provide important informations about Invar. First, recent studies on FesoNi35Mnx5 (Lecomte et al. 1988b) and F%TNi23Cr20 (Lecomte and Schubert 1988) reveal that Debye-like fits to the lattice contribution of the specific heat in the t e m p e r a t u r e range from 20 to 120 K lead to an appreciable reduction in the D e b y e temperature. 0D is - as expected - a function of t e m p e r a t u r e , OD(T). A r o u n d 50 K, 0o drops from the low-temperature values 0~ to values 0o at higher T and remains constant up to 120 K. The drop can be as large as 50 to 60 K, as shown by the data indicated by ' L ' and ' ~ ' in fig. 40. In principle, the behavior of OD(T) can also be derived from the phononfrequency distribution in inelastic neutron scattering and the t e m p e r a t u r e dependence of the elastic constants. Again, few data are available here. We mention the result on Fe72Pt2s, where Hausch (1974) derived 0~ = 320 K for the ordered alloy from the elastic constants (see value indicated by ' H ' in fig. 40), while the specific-heat m e a s u r e m e n t s by Sumiyama et al. (1976) gave 0~ = 525 + 20 K for the ordered alloy (see triangle with question m a r k in fig. 40) and 0~ = 450 + 10 K f o r the disordered alloy, respectively. The same group, on the other hand, reported 0~ = 460 K for ordered Fe3Pt earlier (see Kawarazaki et al. 1972). On
296
E.F. WASSERMAN
approach to Tc all elastic constants increase with increasing temperature, as has been demonstrated in the preceding section. This lattice hardening should manifest itself in an increase in the Debye temperature and Oo(T) should thus show a minimum. Yet, to our knowledge, no data are available so far to support this expectation. A second extraordinary feature found in the Cp(T) dependence of certain Invar alloys at elevated temperatures is the absence of the peak or 'smeared out' behavior at the respective Curie or N6el temperatures. This is seen in fig. 41,
T (K) 0
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/
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Fig. 41. Total specific heai as a function of temperature Cp(T) as measured on F%0NixMns0_x alloys with different Ni concentrations in the 0-300K range by Derayabin et al. (1983) and in the high-temperature range by Bendick et al. (1978). Because of mismatch in absolute values and differences in concentrations, the right-hand scale, valid for the high-temperature data, is shifted downwards by 4 J/mol K with respect to the left-hand scale, valid for the low-temperature data. The crossover range is indicated by the vertical dashed-dotted line. Note the absence of any ordering peaks at Tc, TN, if these ordering temperatures are small (~< 300 K) and the smeared out behavior if Tc, TN lies within the broad 'bump' in the 350-500 K range. All these features are caused by volume fluctuations associated with the moment instabilities. Remarkable are also the fluctuations in Cp(T) occurring in the paramagnetic high-temperature range. The inset shows the magnetic phase diagram (Acet et al. 1988a). Arrows mark the concentrations for which Cp(T) curves are given.
INVAR
297
where the total specific heat Cp versus T is plotted for alloys of the system FesoNixMns0_ x. Data in the 0-300 K range have been reported by Deryabin et al. (1983), the high-temperature data by Bendick et al. (1978). Unpublished data on two alloys with x = 10 and 40 at% Ni by Pepperhoff are also added. Unfortunately, there is a mismatch between the absolute values in Cpof the two groups, even when the Ni concentration is equal. We have indicated this by the vertical dashed-dotted line. The specific-heat values of Deryabin et al. (1983) are about 4 J / m o l K too small at RT, as we have recently confirmed (see fig. 42), but coincide with our data up to about 150 K. Nevertheless, we see from fig. 41 that there is almost no anomaly in Cp(T) accompanying the onset of magnetic long-range order, if the ordering temperatures lie in the range below RT. We have confirmed this on several alloys in the Fes0(NiMn)50 system by using a quasi-adiabatic heat-pulse technique (Schubert et al. 1989), in which the samples (~7 g) are in thermal equilibrium at each measuring temperature. Deryabin et al. (1983) also find no anomaly at Tc, T N in Fe65Ni35_xCr x with x = 6 ( T c = 270 K), x = 10 ( T c = 160 K) and x = 20 (SRO A F + FM alloy, T N = 100 K) at% Cr. If the Curie (N6el) temperatures increase to 4 0 0 - 5 0 0 K , maxima in Cp(T) accompanying the ordering are present, as can be seen in fig. 41 for the two alloys with x = 10 at% Ni (TN = 385 K) and x = 40 at% Ni ( T c = 475 K). Yet, both samples are far from showing any type of critical behavior at TN, i.e., a h-type anomaly as one would expect. Note also that the high-temperature specific-heat I
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Fig+ 42. Total specific heat as a function of temperature (left-hand scale) for FesoNi35Mn15 as m e a s u r e d in the 0-300 K range by Lecomte et al. (1988b) (full dots) and Schubert et al. (1989) (open triangles), in the 320-1200 K range by Bendick et al. (1978) (full dots at high T). T h e full curve is calculated from Cp(T)=3'T + / 3 T 3 with 3' = 4 m J / m o l K 2 and 0D = 3 6 0 K . T h e crosses show the temperature dependence of the thermal expansion coefficient a ( T ) (after Bendick et al. 1977) (right-hand scale). T h e dashed curve is a calculated Griineisen curve with 0D = 360 K.
298
E.F. WASSERMAN
data in fig. 3 show the same behavior for Fe65Ni35. Only if Tc lies beyond the respective temperature (energy) range, as, e.g., in Feg0Ni40 (T c = 600K) and FesoNis0 (T c = 785 K), sharp peaks in Cp(T) appear at the transition temperatures. Note that for all Invar alloys in fig. 41 and fig. 3 large excess specific heat is found in the paramagnetic range, with broad maxima between 400 and 500 K. We are certain that all these contributions are of magnetic origin and manifest the presence of the instability of the moments (spin fluctuations) for T ~> Tc, TN, since they are also observed in ordered Fe72Pt28 (see Wassermann 1989a) and AF Fel00_xMn x with x = 31.7, 39.7 and 50.2 at% Mn (see Stamm 1988). The moment instabilities are accompanied by lattice instabilities, as it is shown in fig. 42, where Cp(T) and a(T) for FesoNi35Mn15 as measured in the range from zero to 1200 K are plotted. Data for Cp(T) from 0 K to RT have been determined by Schubert et al. (1989) and from RT to 1200 K by Bendick et al. (1978). The full curve is calculated from Cp(T)= 3,T + fiT 3, with 3, = 4mJ/mol k2 and 0o = 360 K. Note again the absence of a peak in Cp(T) at the ordering temperature, and the excess specific heat in the paramagnetic range for T > Tc with a broad maximum around 500 K. The thermal expansion a(T) [data are a combination of results from Acet et al. (1988a) and Bendick et al. (1977)] simultaneously has no anomaly at T = Tc, but also shows a broad 'bump' around 500 K and lies above the Gr/ineisen curve (dashed line in fig. 42), which has been calculated with 0o = 360 K and fitted to the experimental results in the low-temperature range. We can summarize the results of this section in the following way: (1) Extraordinary high 3,-values determined from the specific heat in Invar alloys at low temperatures are not Invar typical, but result from contributions of mixed magnetic phases (SG, RSG or similar). 'True' 3' values of Invar (FM and AF) are of the order of 10-11 mJ/mol K 2, thus not much higher than in, e.g., pure Ni (3" = 7 mJ/mol K2). (2) A plot of the Debye temperatures determined at low (or elevated) temperatures versus the electron per atom ratio (e/a) shows analogy to the spontaneous volume magnetostriction of the lattice accompanying the Invar effect. (3) Ferromagnetic and antiferromagnetic Invar alloys with 100 K ~< Tc, TN ~< 400K, show no anomaly in the specific heat Cp(T) at the respective ordering temperatures. (4) Large excess specific-heat contributions are, however, observed in Cp(T) in all AF and FM systems, even in ordered Fe72Pt28 , in the range around 500 K. If the Curie (or N6el) temperatures reach values above this temperature range, the bump in the specific heat at about 500 K is still present, and a second h-type anomaly occurs at the respective Tc (or TN). If the Curie (N6el) temperatures lie in the range around 500 K, both anomalies overlap and lead to a 'smeared out' maximum of Cp(T) in the range around the ordering temperature. (5) The anomalies in the specific heat are accompanied by anomalies in the volume of the alloys. This fact reveals the principal, Invar-typical property because it means that the magnetic instabilities or moment fluctuations are coupled to lattice instabilities or phonon fluctuations. Invar is thus a probiem which calls for the theoretical understanding not only of
INVAR
299
the nature of the ground state but - more important to our feeling - also of the finite-temperature effects involved. 5. Survey of theories on Invar
5.1. Early Invar models In order to understand the Invar effect in 3d-transition metal alloys, there have been two different basic approaches. One is based on the localized electron picture (Heisenberg model), in which each atom has its own permanent and temperature independent moment. The other is based on the itinerant picture of magnetism (Stoner model) giving rise to the understanding of the composition dependence of the average moment (Slater-Pauling curve), with the drawback that the band splitting vanishes at Tc, which is, as we have seen, not the case in Invar. Within the years more than 20 different models for the understanding of the Invar effect have been published. The most important ones are listed in table 8. The early local models stressed the metallurgical and/or magnetic inhomogeneity as Invar relevant, since in the archetypical Invar system FeNi the magnetovolume effects reach a maximum near the "y-a transition at Fe65Ni35, where simultaneously a strong deviation of the average magnetic moment (or magnetization) from the Slater-Pauling curve was observed. This group includes the 'model of latent antiferromagnetism' (Kondorsky and Sedov 1960, Jo 1976), the 'local models with different short-range order' (Sidorov and Doroshenko 1966, Dubinin et al. 1971) the 'local environment models' (Schlosser 1971, Kanamori 1974), the 'inhomogeneity models' (Kachi and Asano 1969, Shimizu 1979) as well as a 'Zener-type model' (Coiling and Carr 1970). With the detection of the Invar effect on ordered Fe3Pt, a system which neither shows mixed magnetic behavior nor deviation of the average moment from the Slater-Pauling curve, all these models came principally into doubt. Important within the group of local models is the historically widely debated phenomenological '23'-state model' of Weiss (1963). Already in 1963 Weiss claimed that 3,-Fe can exist in two different magnetic states, the AF low-spin (LS) state 3'1, which has a small magnetic moment ( ~ =0.5txB) and small volume (a = 3.57 /k), and the FM high-spin (HS) state 3'2, with large moment ( / x = 2.87/-%) and large volume (a = 3 . 6 4 A ) . For FeNi alloys, Weiss assumed the energy difference between the two states to be a function of the Ni.concentration, so that in Fe65Ni35 the 3'2-state is the ground state, and with rising temperature the increasing population of the low-volume 3'l-state leads to the compensation of the lattice expansion. The Weiss model was extended and refined (Bendick et al. 1979, Chikazumi 1980) but a direct proof of its validity was never given. We shall see later that within modern band calculations, i.e., in a completely different type of approach, the 2~/-state model will experience a certain revival. Therefore, instead of stressing the inhomogeneity, homogeneous models on the basis of an itinerant picture seemed more promising. Mathon and Wohlfarth (1968) and Wohlfarth (1969) considered Fe65Ni35 to be a weak itinerant fer-
300
E.F. W A S S E R M A N
~
~_l
~
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o6
06
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=
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INVAR
301
romagnet (WIF) and successfully explained many Invar properties on the basis of a Stoner-type band model. However, ordered Fe3Pt is not a weak but a strong itinerant ferromagnet and the finding of Invar properties on this alloy thus puts the W l F approach also into doubt. Improvements of the itinerant model with distance-dependent band width proposed by Shiga and Nakamura (1969), and a CPA-type of approach by Hasegawa and Kanamori (1971) explained some of the Invar properties of FeNi and FePt, but failed to explain the finite temperature properties, especially in the paramagnetic range. Antiferromagnetic Invar was not discussed at all within all the models mentioned, and therefore, we, can regard them all as 'historical'.
5.2. Moment-volume instabilities in 3d-elements In the late 1970s band calculations within the local spin density approximation (LSDA) predicted the existence of two ferromagnetic states for fcc Fe, a high-spin state with large moment and large volume and a low-spin state with low m o m e n t and volume. The main literature is summarized in 8.1 of table 8. Other methods (8.2-8.4 in table 8) gave similar results too. In fig. 43 we show a plot of the magnetic m o m e n t / ~ e versus the radius of the Wigner-Seitz cell rws for fcc Fe as taken from different references. Note that all theoretical methods applied reveal more or less the instability of the Fe-moment and the coexistence of a HS, LS and
3.52
3.66
3
3.38
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ltFe for f,c.c. Fe
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N.M. I
rws (a.u.)
2,8
Fig. 43. Magnetic moment of fcc Fe as a function of the 'radius of the Wigner-Seitz cell rws as calculated by Bagayoko and Callaway (1983) (dashed-dotted lines), Wang et al. (1985) (dashed lines), Moruzzi et al. (1986) (full lines for the stability ranges in zero field), Krasko (1987) (plus signs and dashed-double dotted lines) and within a disordered local-moment (DLM) picture by Pinski et al. (1986) (crosses). The arrow marks the lattice constant as determined for AF coherent ~/-Fe precipitations in Cu (Bendick et al. 1977).
302
E.F. WASSERMAN
a non-magnetic (NM) state within an instability range of 2.640 ~< rws <~2.704 alu. or 3.570 ,~ ~< a ~< 3.657 _A of the lattice constant, respectively. This range coincides with the instability range in the phenomenological model of Weiss (1963), which thus experiences retrospectively a certain confirmation. This is better seen in fig. 44a, where the total energy (relative to a minimum energy) as determined by Moruzzi et al. (1986) is plotted versus the Fe moment with r,vs as a parameter. The HS, LS and NM states, characterized by energy minima in the curves for rws = const, can clearly be distinguished, and thus remind of the Weiss picture. Figure 44b shows E Emin as a function of rws as determined by Moruzzi et al. (1986) (full curves) and by Krasko (1987) (dashed curves). Note that both methods of calculation result in almost the same overall behavior, but differ in absolute values. While Moruzzi et al. (1986) give an energy difference of about 15 mRy between the HS and the NM state, Krasko (1987) finds A E = 7.5 toRy. The m o m e n t - v o l u m e instability is by no means bound to fcc Fe, but occurs in other fcc elements as well. This is seen from fig. 45, where the magnetic m o m e n t /x versus rw~ is plotted for fcc Fe and Co (Moruzzi et al. 1986) and fcc Mn (Brener et al. 1988). Note the instability in Co, which shows a transition from a FM HS state to a NM state, and Mn, where a transition from a ferromagnetic HS state to an antiferromagnetic ground state is observed. In this context it is most important to note that a computed m o m e n t of z e r o - like in the ground state of M n indicates only that the ground state is not ferromagnetic. The calculations cannot determine whether the ground state is non-magnetic or antiferromagnetic. For fcc -
r w s = 2 6 9 ~I Ul
b.
Fe{fcc)
20 -
-
----
M o r u z z i et c1L(1986) Krasko
(1987)
15
HS
t'~
f
E c 15 uJ I w
5
NM/ /I/ HS \/ i/N' /M
14 rws = 2.67 a.uj
1
i
I
F
2
(IJB)
2.6
rws
2.7 (a.u.)
Fig. 44. (a) Total energy (relative to a minimum energy) versus magnetic moment for fcc Fe at different radii of the Wigner-Seitz cell r,s within the instability region (cf. fig. 43). Note the occurrence of high-moment or high-spin (HS) and low-moment or low-spin (LS) states, as characterized by minima in the curves (after Moruzzi et al. 1986), (NM) marks the non-magnetic states. (b) Total energy (relative to a minimum energy) versus radius of the Wigner-Seitz cell rws as calculated by Moruzzi et al. (1986) (frill curves) and Krasko (1987) (dashed curves) for fcc Fe.
INVAR
303
3.4 3.6 3.8 ' Instability 'ranges '
4 {PPB3)
Co
Fe
,-//
4.2 a (]k) '
Mn S
Mn
~
Fe (FM)
Co ( F . . ~ . _ ( - - . - -
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z,.0 ~
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fcc. Elements
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(Mn:AF!)
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2.8
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Fig. 45. Magnetic moment/x (in/~B) versus radius of the Wigner-Seitz cell rw~ (in atomic units) for fcc Fe (full line), fcc Co (dashed-dotted line) [both after Moruzzi et al. (1986)], and for fcc Mn (after Brener et al. 1988). The dashed-dotted line shows the result for homogeneous pure amorphous Fe (Krompiewski et al. 1988).
i
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11b, fcc Fe on Cu3Au (001) hv = 60 eV ~-10 d •~- 8 c" (lJ
I / !
6 /
0
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~2 t
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4 3 2 1 0 :E F Binding Energy (eV) Fig. 46. Normal-emission, spin-resolved energy-distribution curves for an epitaxially grown 11/~ thick fcc Fe film on a single-crystalline Cu3Au substrate, using hv = 60 eV, s-polarized radiation (Carbone et al. 1988).
304
E.F. WASSERMAN
Mn it has been shown by additional wave-vector dependent susceptibility calculations (Fry et al. 1988) that the ground state has AF order. With respect to the occurrence of the Invar effect in amorphous alloys (a-FeB, a-FeZr) we show in fig. 45 that Krompiewski et al. (1988) have found an instability of the Fe-moment in amorphous, homogenously magnetized a-FeB (dashed line). An experimental proof that fcc Fe at large lattice constants is in the ferromagnetic HS state has been given by us (Carbone et al. 1988) recently through measurements of the spin-resolved energy distribution curves (SREDs) of a 11 y-Fe film grown epitaxially on a Cu3Au (001) single-crystalline substrate with a = 3.75 A. Figure 46 shows the SREDs for this film, for which a high spin polarization (i> 50%) was also observed. The peak positions in the SREDs in fig. 46 are in good agreement with the calculated critical-point energies at F and X points in the Brillouin zone of fcc Fe at a = 7.0 a.u. = 3.70 A (Bagayoko and Callaway 1983). The shoulder in the majority-spin EDC at E F can be attributed to Xst, which is predicted to be at 0.2 eV binding energy. Since Xst determines the top of the majority spin d-bands, we conclude that fcc Fe is a strong FM for a lattice constant of 3.75/~. This is in excellent agreement with the theoretical results (cf. fig. 43) and earlier magnetization measurements (Gradmann and 2.69 I
5 m
3.22 I
3.76 I
_Cr / / ~ M n _ .
bcc Elements
(~B)
7"
/
.- V '
~I
//
I
l...........
..": ............
i
Fe "'"
J
)/. /
~ . / f
//
.
/
/ i I 2.5
]'
I 3~)
I 3.5
rws (a.u.)
Fig. 47. Magnetic moment m versus radius of the Wigner-Seitz rw, for different 3d-elements in bcc lattice structure (after Moruzzi and Marcus 1988). The crosses mark the results for bcc Mn as calculated by Fry et al. (1987). Note the instabilities for bcc Mn, V and Cr. The arrows mark the respective equilibrium values of the lattice constants.
INVAR
305
Isbert 1980) for Fe on CuxAul_ x, where a decrease in the Fe m o m e n t with increasing Cu concentration (decreasing lattice constant) was observed. On the other hand, the situation for fcc Fe when grown epitaxially on Cu(001) with a = 3.61 ~ is still experimentally contradictory [see discussion in the paper by Carbone et al. (1988)]. FM and NM behavior has been observed, probably because at a = 3.61 A = rws = 2.669 a.u. one is right within the instability range of pure fcc Fe. On the other hand, coherent 3,-Fe precipitations in Cu are definitely antiferromagnetic (Tsunoda et al. 1987, Macedo and Keune 1988) with T N = 67 K at a = 3.570 ~ , but with a complicated canted spin structure of the Fe moments modulated in space (Tsunoda et al. 1988). Finally, we show'in fig. 47 that for 3d-elements in bcc lattice structure moment instabilities occur too. These results stem from very recent calculations by Moruzzi and Marcus (1988), who investigated carefully the complete 3d-series in bcc form. In fig. 47 we see that bcc Cr (full line) makes a H S - N M transition; bcc V (dashed-dotted line) and bcc Mn (dashed line and crosses from Fry et al. 1987) clearly make H S - L S - N M transitions around rws---3.5 and rw~ = 2.58, respectively, while the curves for a - F e and bcc Co are continuous, which means that the moments of these elements are not unstable with respect to changes in the lattice constant. 6. Towards a new understanding of the Invar-effect
6.1. Theoretical evidence for high-spin-low-spin-state transitions in Invar alloys Figure 48a shows constant energy contours projected into the m o m e n t - v o l u m e (or rws) plane as calculated for ordered F%Ni with the FSM method by Moruzzi (1988). Figure 48b shows an analogous presentation (Entel and Schr6ter 1989) calculated with parameters determined from band-structure calculations using a G i n z b u r g - L a n d a u like energy functional (Entel et al. 1989) for ordered Fe3Pt. The zero-field solutions ( H = d E / d M = 0) are given by the full curves labeled H = 0, the zero-pressure solutions ( p = - d E / d V = 0) are given by the dashed lines labeled p = 0 in both diagrams, respectively. Note that these zero-pressure and zero-field equilibrium curves in both systems cross twice, at a HS state marked by '0' and a LS state marked by 'X'. In Fe3Ni the I-IS state occurs at rws = 2.6 a.u. = 3 . 5 1 7 A with m = 1.5 p~/atorn, the LS state at r,~s = 2.55 a.u. = 3 . 4 4 9 A with rn =0.47/.~B/atom. The values for Fe3Pt are rws = 2 . 6 9 5 a . u . = 3.645 A with m = 1 . 7 5 / ~ / a t o m for the HS state and rw~ = 2.65 a.u. = 3.584 7k with m = 0 . 9 5 / ~ / a t o m for the LS state. We remark that the LS state in Fe3Pt in fig. 48 might not be stable, but the absolute values gained in these calculations should not be overestimated anyhow. Important is the principally analogous behavior for both systems which is evident from fig. 48. Both systems show HS and LS states, reminiscent of the Weiss 23~-states model (Weiss 1963). This is also revealed in fig. 49, where the volume dependence of the total energy along the zero-field lines is given (Moruzzi 1988, Entel and Schr6ter 1989). For Fe3Ni also the site-decomposed zero-field moments for Fe and Ni are
E.F. WASSERMAN
306
,
. . . - 4 ~ = G'
o.
Fe3N{(Cu3Au) O~ 2
0.5 mRy/ATOM CONTOURS
z~
~J = o
w
I=
2.5
2.6
217
rws (o.u.) b.
'
'
'
2"
~-4' - O
H=O
uJ o=
:£
//
i' w
Fe3P t (Cu3Au) 0
75
80
85
90
VOLUME {o.u 3)
Fig. 48. Total-energy contours projected into the moment-volume (or rws) plane for (a) ordered Fe3Ni calculated by Moruzzi (1988), and (b) ordered Fe3Pt calculated by Entel and Schrrter (1989) with the FSM method. The zero-field solutions H = d E / d M = 0 are given by the full curve labeled H = 0, the zero-pressure solutions p = - d E / d V = 0 by the dashed curve labeled p = 0 in both diagrams, respectively. Note that in both systems the zero-field and zero-pressure curves cross twice, corresponding to two energetically-stable states: a high-spin state (marked by '0') with large moment and large volume, and a low-spin state (marked by 'X') with small moment and small volume. The results are reminiscent of the phenomenological 2 y-states model of Weiss (1963).
shown. In particular, the low energy-difference of the order of 1 mRy between the HS and LS states in both systems is obvious. Although the present results are valid at T = 0 only, one can imagine that the smallness of the energy barrier allows the LS state to be accessible with temperature (or pressure), since 1 mRy corresponds to roughly 150 K (or about 10 kbar considering the volume difference between HS and LS states). From a band-theoretical point of view, thermal expansion is a direct consequence of the anharmonicity of the binding curves like in fig. 49. In non-Invar systems or pure 3d-metals usually the curves are skewed towards high volumes, reflecting positive anharmonicity and thus implying a tendency of the system to expand with increasing temperature (Moruzzi et al. 1988). In both Invar systems in fig. 49, on the other hand, the LS branch at low volumes effectively introduces a negative anharmonicity, which implies a tendency of the systems to contract with increasing temperature, which is indeed experimentally observed.
INVAR
A
i -- ~
Fe3Ni
307
, ii1~ II ~'re-
iI / m :::k ::L
o
~2 HA
2.5
26 rws (a.u.) 2.7
i
i
i
Fe3Pt HS
E ~2 E HA J HA
~Ry
0
i 2.6
2.7
rws
(a.u.)
I 2.8
Fig. 49. Zero-field total energy (relative to the minimum energy) versus radius of the Wigner-Seitz cell as calculated (a) for ordered Fe3Ni by Moruzzi (1988), and (b) for ordered Fe3Pt by Entel and Schr6ter (1989). For Fe3Ni also the site-decomposed magnetic moments versus rws are shown. Note the HS and LS states, characterized by the energy minima in both systems. For further details see text.
Moreover, we note a difference between Fe3Ni and Fe3Pt insofar as Fe3Ni exhibits a saddle point at rws = 2.55 a.u., while in Fe3Pt a real energy barrier is found with a height of about i mRy. The minima themselves show only an energy difference of about 0.3 mRy. Although again, for finite tempertures, the results in figs. 48 and 49 should not be taken too literally, this difference in the total energy curves makes some of the observed differences in the low-temperature behavior of these two Invar-systems plausible. Remember that, e.g., the magnetization at 4.2 K hardly depends on pressure up to p = 20 kbar in Fe3Pt , while a considerable decrease is observed in Fe65Ni3s (see fig. 28). Analogously, 'hard' magnetic behavior is found in Fe3Pt in the initial dependence of the magnetization on temperature (see inset in fig. 30) and in the magnetic contribution to the bulk modulus (see fig. 37), while in Fe65Ni3s we find magnetically 'soft' behavior in both properties. It is obviously more difficult to 'excite' Fe3Pt out of the narrow minimum at the FM HS state and over the energy barrier into the LS state, than it is to drive Fe3Ni up to the saddle point, i.e., out of the more shallow minimum at the HS state. Note also that, concerning experiments, we compare ordered Fe3Pt with a disordered Fe65Ni3s alloy. Disorder very likely will lead to a decrease of the energy difference between the HS and LS state in the total energy curves and favour magnetic softness of Fe65Ni35 in comparison to ordered Fe3Ni. Finally, a convincing result concerning the concentration (or e/a) dependence
308
E.F. WASSERMAN
of the total energy in the FeNi system can be derived from the FSM calculations. It has been demonstrated earlier (Williams et al. 1983) that in ferromagnetic FesoNis0 (e/a = 9) the HS state at high volume is the ground state and there exists an energy difference of AE = ENM -- Ens = 11.2 t o r y to the non-magnetic state at lower volume. We have also seen in fig. 44b that in comparison to FesoNis0 in fcc Fe the HS and NM states are energetically reversed, so that AE = ENM -- EHS = - 7 . 5 m R y (Krasko 1987) or A E = - 1 4 . 5 m R y (Moruzzi et al. 1986). In figs. 50a-d we have sketched schematically the total-energy curves as a function of the radius of the Wigner-Seitz cell rws for the FeNi series. Note the reversal in level position just mentioned. In fig. 50e the energy difference AE = ENM- EHS is plotted versus the e/a ratio. The straight full line results when we use the Krasko (1987) value for h E in fcc Fe, the dashed line when we use the results of Moruzzi et al. (1986) (cf., fig. 44b). The dashed-dotted line shows h E versus e/a as originally proposed by Weiss (1963) for the FeNi system. The plus sign gives the result for AE in pure fcc Co (Moruzzi et al. 1986). To our feeling, the similarity between the 'historic' phenomenological Weiss model and the modern band calculations is striking. Although Weiss assumed the antiferromagnetic yl-state to be the ground state in pure fcc Fe and we have used here the NM state within the band model, it is clearly seen from the figure that, with increasing Ni concentration, in both models the energy difference reverses sign in the range e/a = 8.4-8.6. Consequently, for the Invar alloys-especially Fe65Ni35 with e / a = 8 . 7 - the FM HS state (y2-state in the Weiss model) becomes the ground state. Moreover, in the Invar r a n g e - as e x p e c t e d - the energy difference between HS and NM state (or HS and LS state) is small as compared to fcc FE or FesoNis0, so that the LS state is then thermally accessible from the ground state. A special situation occurs when AE = 0. The corresponding energy curves are shown schematically in fig. 50c. In this case it 'costs' no energy for the system to change (or fluctuate) between the ferromagnetic HS state and the paramagnetic or possibly antiferromagnetic NM (or LS) state. We call an alloy of this composition an 'itinerant spin glass', since FM and AF order are energetically equally possible. Note that in FeNi in the respective composition range (e/a ~ 8.6) mixed magnetic behavior (but not a pure SG phase) is indeed observed at low temperatures (cf., magnetic phase diagram in fig. 5). We have demonstrated in fig. 29 that in the range e/a = 8.3-8.4 the ordering temperatures of Invar systems in general become small, SG phases occur and the spontaneous volume magnetostriction tos0 (cf. fig. 23) vanishes at the respective compositions. We do not doubt that future total-energy calculations for other 3d-Invar systems will reveal these general features too. In conclusion, the FSM calculations provide some new and comprehensive understandings of the ground state properties of Invar systems. In particular, the long debated difference between 'hard magneti c' Fe3Pt and 'soft magnetic' Fe65Ni35 is explainable. There is no principal difference between the two Invar systems. Furthermore, some of the general features concerning the e/a or concentration dependence of Invar properties in the broad variety of systems presented in section 4 of this chapter can at least be made plausible. However,
INVAR 2.5
rws
309
2.6
(a.u.) 2.7 l
~HS i
~
i
F%NNi~
/HS
T
" i
i
I
i
i
NM
~
i
LU
Fe(fcc)
f, HS
NM
~
3
+10
~ m
Q.
i
NM
/
-10
/ 4
--7"-
/
INV.-- FM~,~'+
.//"
~
f----"~ ,,,
d
L
--
Fe 8
w
~
!
-- AF--S8
c.
/ I ~
o/o
]u
g
)~
%o%o
~
/ e.
Fig. 50. Total energy curves versus radius of the Wigner-Seitz cell for (a) F%0Nis0 (Williams et al. 1983), (b) F%Ni (Moruzzi 1988), and (d) pure fcc Fe (Moruzzi et al. 1986). Note the reversal in the relative position of HS and NM (or LS) state with change of the composition from FesoNis0 to pure fcc Fe. In (c), when both levels are at equal energy, an 'itinerant spin-glass' state occurs. (e) shows the energy difference between the NM (or LS) and HS states, AE = ENM- EHS, versus the electron concentration e/a. The straight full line results, when the Krasko (1987) value of AE is used for pure fcc Fe, the straight dashed line, when using the AE value of Moruzzi et al. (1986). The dashed-dotted line shows AE(e/a) as given by Weiss (1963), Note the striking similarity between the band-calculation results and the phenomenological Weiss model. The results also show that for Fe65Ni35 Invar AE is small, of the order of 1-2 mRy, which means, the NM (or LS) state is thermally accessible from the HS ground state. The moment-volume instabilities, responsible for the Invar effect also drive the FeNi system into the martensitic 7 - a phase transition around e/a = 8.65. As a consequence, neither the itinerant SG state nor the ordered F%Ni phase can be reached experimentally in this system (cf. section 3.1.1 and fig. 5). Using the calculated value of AE for pure fcc Co [plus sign in (e) after Moruzzi et al. (1986)] an analogous discussion could be carried through for the FeCo system, which is, however, not stable in the -y-phase. c a u t i o n is r e q u e s t e d w i t h r e s p e c t to a n y i n t e r p r e t a t i o n at finite t e m p e r a t u r e s . T h e s t a t e s c o n n e c t e d w i t h p a r t i c u l a r p o i n t s o n t h e b i n d i n g s u r f a c e ( l i k e t h e o n e s i n fig. 48) r e p r e s e n t s t a t e s o f t h e w h o l e i t i n e r a n t m a g n e t i c s y s t e m . It is n o t k n o w n h o w l a t t i c e v i b r a t i o n s a n d s p i n d i s o r d e r will c h a n g e t h e e n e r g y s u r f a c e s w h e n t h e t e m p e r a t u r e is i n c r e a s e d . H o w e v e r , t h e r e a r e p r e s e n t l y w i d e s p r e a d t h e o r e t i c a l activities to u n d e r s t a n d f i n i t e - t e m p e r a t u r e I n v a r p r o p e r t i e s o n t h e basis o f t h e
310
E.F. WASSERMAN
FSM method in connection with spin and density fluctuations (see table 8, 9.5.1-9.5.3).
6.2. Finite-temperature models Early attempts to understand magnetic and magnetovolume effects of transition metals and alloys at finite temperatures have been scarce in the literature. A fundamentally new approach was first given by Moriya and Usami (1980) on the basis of the spin-fluctuation model for weak itinerant ferromagnets. Since Invar alloys are not WlF, this theory is not applicable to Invar. A rigid-band finite-T calculation by Hasegawa (1981) and an electron-phonon model by Kim (1982) could make some Invar properties at finite T plausible, but failed to explain the Invar features in detail. A promising way to describe finite-temperature properties of Invar has been shown by Kakehashi (see, e.g., Kakehashi 1981-1985, 1988a and references therein) within the so-called 'local-environment theory'. The theory self-consistently takes into account the number of nearest-neighbor atomic and magnetic configurations in different systems at finite temperatures. Calculation of the amplitude of the local moments with respect to the atomic environment as a function of temperature, pressure, magnetic field and/or composition enabled Kakehashi to qualitatively explain a broad spectrum of Invar anomalies and magnetic as well as magnetovolume effects in several binary systems; so far, however, not for Fe3Pt. Surprisingly, evidence for the existence of two distinct energy states in the high-temperature limit for, e.g., FeNi could not be given from these calculations. Moreover, the local-environment theory does not account for the typical features observed in Invar alloys in the temperature range T > T c. We show, e.g., in fig. 3 (dashed lines), the result of Kakehashi (1981) for the specific heat of FeNi in comparison to the experiments. The theoretical curves have been fitted at the respective experimentally determined Curie temperatures, since the absolute values computed theoretically are erroneous. Note that the second 'bump' in Cp(T) at temperatures below Tc is qualitatively correctly revealed by the calculations. On the other hand, the sudden drop in Cp(T) at T = T c in the theoretical curves is not observed experimentally. Refinement of the LE approach for the understanding of the Invar effect thus also seems necessary in the future.
6.3. Evidence for high-spin-low-spin-state transitions from experiment First experimental evidence for the existence of the HS-LS state transition has been given by us in investigations of the temperature dependence of the photoemission intensity on a single crystal of ordered Fe3Pt (Kisker et al. 1987a,b). In fig. 51a we show the spin-integrated energy distribution curves as measured below Tc = 450 K for F%Pt at T = 270 K (T/T c = 0.6) and above T c at T/T c = 1.22. As shown in fig. 51b, the data taken below Tc differ from the data above T c by a significant decrease in intensity within an energy range of about 0.4 eV below EF, and an increase in intensity within an energy range of about 0.5 eV above Ev,
INVAR
-~
10
i
i
T =270 K
-
311
i
-
i
i
i
~
a)
o= 8 o.
6
.
4 Fe3Pt
0
I Pt
I
-
b)
r II
r~
._.
I~
I
I~ i
~*
!
^
:
"
-10
~ -20
I Pi
31o c 0
I
Fe
20 ~
0
10011
t
"J
1
V
Data DOS-Difference for fcc Fe theor.-
• o, ° exp. .... I
I
i
I
i
I
2 0=E Energy below E F l e V )
/.
Fig. 51. (a) Spin-integrated energy-distribution curves from Fe3Pt(001 ) for normal emission and s-polarized light at hi, = 60 eV as determined at T = 270 K = 0 . 6 T c (full curve) and T = 500 K = 1 . 2 2 T c (dashed dotted curve). The intensities are normalized to the photon flux (Kisker e t al. 1987a). (b) Difference between the EDCs of (a) taken at 500 K (low-spin state) and 270 K (high-spin state) (dots and full line). The dashed line is the theoretically calculated DOS difference for fcc Fe at r ~ = 2.68 a.u. in the LS (cf. fig. 52c) and HS state (cf. fig. 52d), convoluted with a 0.4 e V C F W H M Gaussian-type resolution function (after Podgorny 1989).
while at E F the intensity does not change. We attribute these features to the flat band with A2 symmetry, which, in the HS state for T~< Tc, is found at about 0.4 eV below EF, but in the LS state about 0.2 eV above EF, because the increase in intensity above E F is an order of magnitude larger than expected from a variation of the Fermi function f(E, T) with temperature alone. As we have shown earlier (Carbone et al. 1987), because of symmetry reasons the band structure of fcc Fe in the X - W direction and the band structure of Fe3Pt in the F - X direction correspond to each other. Therefore, we can compare the DOS for pure fcc Fe in the HS and LS states with our results of the photoemission on Fe3Pt. Figures 52a and b gives the total (up and down spin) DOS for fcc Fe in the LS and HS states, respectively, as calculated very recently by Podgorny (1989) for a radius of the Wigner-Seitz cell rws = 2.68 a.u. Analogous features, i.e., the high DOS peak closely below E F in the HS state and slightly above E F in the LS state, are found in theoretical calculations for Fe3Ni and Fe3Pt (Podgorny 1988) as well.
312
E.F. WASSERMAN fcc Fe: ,
i
i
i
J
rws= 2.68 (Q.u.)
i_
i
L S - store
3.O
i
i
i
r
LS - s t a t e
3.0
c)-
"~ 2 . 0
~2.0
co
~- 1,0
0
i
-8
-6 t
-~ -2 E Energy (eV) i
i
-6
~ 2.o
-5
-4
-3 Energy
r
t
HS - s t e t e
3.0
0
2
i
i
-2
-1
EF
(eV)
t
i
HS - stote
3.0
d)
2.0
3. o
1.0
1.0
0 -8
-6
-4 -2 EF Energy (eV)
-6
-5
-4
-3 -2 -1 Energy (eV)
EF
Fig. 52. Total DOS (up and down spin) for fcc Fe at rws = 2.68 a.u. in (a) the LS and (b) the HS state, respectively, as calculated by Podgorny (1989). (c, d) the respective DOS-distribution curves after truncation of the Fermi function (DOS-Fermi statistic). If one subtracts from the total DOS in figs. 52a and b the effect of the Fermi function f(E, T) at the respective experimental temperatures ( T = 270 K for the HS state, T = 550 K for the LS state) this results in the DOS minus f(E, T) distributions curves shown in figs. 52 c and d. The difference of these two distribution curves convoluted with the experimental resolution function (0.4 eV F W H M ) , results in the dashed line shown in fig. 5lb. One can see that the agreement between the theoretically calculated DOS difference for fcc Fe and the experimentally (from fig. 51a) determined intensity difference between the LS and HS states of Fe3Pt is very good, suggesting that they are of the same physical origin. We therefore conclude that we have given clear experimental evidence for the occurrence of the high-spin-low-spin-state transition in Fe3Pt Invar with temperature. This volume-coupled transition, to our feeling, is the salient feature of Invar. The amplitude of the intensity above E v in photoemission curves should be a measure for the weight of the low-spin state if the interpretation given applies. As shown by Holden et al. (1984) the fractional change of the radius of the Wigner-Seitz cell with temperature, i.e., the fractional linear expansion AI/I = (I(T)-l(O))/l(O), is proportional to the change of the square of the local magnetic m o m e n t with temperature, m 2 ( T ) - rn2(0), where T is an elevated temperature T > T c. Therefore, the square of the intensity above E v in the EDCs, I:(T), should be proportional to ma(T) which is proportional to the volume change tos = (AV/V)(T). In fig. 53 we have plotted the experimentally
INVAR 'HI
I
1.o!
I
I
313 I
I
I
t
i
10A
Fe 3 Pt
.6 8b
////
-0.5
0.61
z~.j//
LL UJ
I2(T) .
.,0 0
3 -1.0
"~0"4 ~E
= /'/Us(T>'////~ "(3 D t3" 03
-1.5
O~/t
200
I
t
~
t
300 400 Temperature (K)
\\!
I
500
i
i
600
0
Fig. 53. Normalized spin-polarization as a function of temperature as determined experimentally for Fe3Pt (open triangles, full curve), and squared intensity above EF of the photoemission curves at different temperatures (open circles, dashed curve). The dashed-dotted curve shows (qualitatively) the negative fractional volume change ~%(T) for Fe3Pt as determined by Sumiyama et al. (1979).
determined squared intensity 12 (dashed line) as a function of temperature together with the behavior of the negative spontaneous volume magnetostriction ms(T ) as measured by Sumiyama et al. (1979) for ordered Fe3Pt. Also ploRed in fig. 53 is the spin polarization as determined by us experimentally on our Fe3Pt sample (actual concentration Fe72Pt28 ) as a function of temperature, leading to a T c value of 450 K, which corresponds to a degree of order of 60-70%. One can see that in the range where the polarization starts to decrease, both I2(T) and - w ~ ( T ) increase in the same way. This gives further support for the correctness of our analysis, and the presence of the high-spin-low-spin transitions in Fe3Pt on approach to T c as-well as on passing through T c. Our experimental XPS-results (fig. 51) on Fe3Pt have recently been rediscussed from a theoretical point of view concerning controversial aspects. On the one hand, it was quoted (Kakehashi 1988b) that the DOS-difference curve (fig. 51b) is in better agreement with curves obtained from the transition from a strong-band ferromagnetic state at T < T c to disordered local-moment states ( D L M ) with considerable large local moments at T > T c. On the other hand, this was put into doubt by calculations of Gollisch and Feder (1989), who showed that the magnetic structure of Fe3Pt Invar at 1.2 T c is characterized by large localmoments, however, with a substantial amount of magnetic SRO. This was also shown to be valid for pure bcc Fe, for which around T c a ferromagnetic SRO extending over 4 to 6 A was reported (Haines et al. 1985), ruling out the validity
314
E.F. WASSERMAN
of the D L M picture. Further work on photoemission, both theoretical and experimental, is obviously necessary. Very recently, a further convincing experimental p r o o f for the existence of H S - L S state transitions in Fe68Ni3z and ordered and disordered FeTzPt28 Invar has been given in m e a s u r e m e n t s of the pressure dependence of the M6ssbauer effect by A b d - E l m e g u i d et al. (1988) and A b d - E l m e g u i d and Micklitz (1989). Figure 54a shows their result of the pressure dependence of the normalized average effective hyperfine field [Beff(p)/Beff(O)] as measured at 4.2 K on these Invar alloys. Note that in accordance with the results of the FSM calculations in figs. 48 and 49, Beff of Fe68Ni32 initially decreases continuously on the application of pressure. The decrease in B~ff or magnetic m o m e n t sets in immediately with increasing p, driving the system out of the HS state through the reduction of lattice volume. A t p = 6 G P a , B~ff(p)/B~f~(O) starts to drop faster and at p c ~ 7 G P a the LS state is reached, supported by the observation that for p > P c
i
1.0
i
A~
,~\ 0.5 _
\...
.~
"~
\ \
4.2 K I
0
- -
~ "~
Fe?2Pt28o WFe=1.6p.B_.
~'~ Fe72Pt28dis. P'Fe=I"01~B ~)-0- Fe58.5Ni31.5
I
I
I
~Fe=06~B -
'
-~ 0.5 \ z ' ~ A\ ~ A ~ Fe72Pt28.d
/ ~" 0/ 0
\ , 2
, 4
&--&- Fe72Pt28 O~" O--o - F%8..5Ni31.5 6 8 P(GPa)
Fig. 54. (a) Average effective hyperfine field at pressure p, B~f~(p),normalized to Bo~fat p = 0 versus pressure p (in GPa) for disordered Fe68.sNi31.5 and ordered and disordered Fe72Pt28 Invar as determined from Mrssbauer experiments at 4.2 K (after Abd-Elmeguid et al. 1988 and Abd-Elmeguid and Micklitz 1989). Salient general feature is the experimental proof for the occurrence of HS-LS state transitions induced by pressure in all three alloys. Details can be quantitatively understood within the results of the FSM calculations (cf., figs. 48, 49). (b) Relative change of the Curie temperature Tc(p)/T(p = 0) versus pressure for the same samples as in (a).
INVAR
315
Beff(p)/Beff(O)~-const. The residual value of the effective moment in the LS state is ~ = 0.6/x B in accordance with the theoretical value (cf. fig. 49a). The behavior of the 'soft' magnetic material Fe68Ni32 is contrasted by the 'hard' magnetic behavior of ordered Fe72Pt28. We observe in fig. 54a that a critical pressure Pc = 4 GPa is necessary before this alloy can be driven out of the HS state with pressure, reflecting the deep energy-valley around the HS state revealed by the FSM calculations (cf., fig. 49b). For p >Pc the moment of ordered Fe3Pt then drops continuously until for p >t7 GPa the LS state is reached. As expected from a comparison with the results in figs. 48a and b, the residual moment of ordered Fe72Pt:8 in the LS state is much higher than in Fe65Ni35, although the absolute value of/x = 1.6p~ for the LS state of ordered Fe72Pt2s is only in qualitative agreement with the result given for ordered Fe3Pt by the calculation (cf., fig. 48b). As expected, disordered FeTzPt28 takes an intermediate position between the 'soft' magnetic Fe68Ni32 and the 'hard' magnetic ordered Fe7zPt2s. As can be seen from fig. 54a, at a critical pressure Pc = 2 GPa the reduction of the moment sets in for disordered FeTzPt28 , and at around p > 7 GPa the LS state is reached with a residual moment of/zF~ = 1.0/~ B. Unfortunately, no FSM calculations for disordered systems are presently available to allow further quantitative analysis. The reduction in Curie temperatures accompanying the HS-LS transitions are demonstrated in fig. 54b, where the normalized Curie temperatures T c ( p ) / T c ( 0 ) versus pressure are plotted. Although the overall behavior is similar for the three Invar systems, note again the magnetic hardness of Fe72Ptzs (cf., also fig. 28). In conclusion, the experimental results presented in this section to our feeling give clear evidence for the presence of HS-LS state transitions in Invar alloys, simultaneously supporting the results of the FSM calculations. 7. Conclusions and outlook
If we assume that in an itinerant ferromagnet, in general, there are short-ranged magnetic correlations - or longitudinal and/or transverse spin fluctuations - in the temperature range around and above Tc, we can ask the question, what is the reason for all the differences observed between the physical behavior of Invar and ordinary ferromagnets? To our feeling, the principal answer is as follows. Although there are moment-volume instabilities, characterized by high-spin (large volume), low-spin (small volume) or no-spin (small volume) states present in many pure 3d-elements, alloys or compounds, the essential feature is that in Invar the energy difference AE between the states is so small that the states become thermally accessible via low-energy excitations. Experiments (section 4) and band calculations (figs. 49 and 50) have shown that indeed Invar behaviori.e., maximum magnetovolume effects-occur, when AE is of the order of 1-2 mRy or 150-300 K. Clearly, the population of the states or their volume fraction is temperature dependent. Thus magnetovolume contributions- though small-initiate from zero temperature, but drastically increase in size, when k T ~ AE, as seen, e.g., in the dependence of w~(T) (cf., fig. 53) or the magnetic
316
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contribution to the bulk modulus ABm(T ) (cf., fig. 37). With increasing temperature, i.e., rising excitations of phonons, it also becomes possible that, vice versa, density fluctuations produce magnetic fluctuations in Invar through the strong magnetoelastic coupling (large magnetic Grfineisen parameter). This stems from the observation that the (equilibrium) volume difference between HS and LS (or NM) states is of the same order of magnitude (3-5%, cf., fig. 49) as thermal volume fluctuations (Renaud 1988). We are thus bound to assume that the unusual fluctuations observed in the high-temperature range of the specific heat (cf., figs. 3 and 41) find their origin in these density fluctuations, causing local magnetic fluctuations. This is supported by the fact that in non-Invar ferromagnets these high-temperature 'oscillations' in Cp(T) are not observed (Pepperhoff 1989), since the energy differences between the instability states- if they existare too large. Concerning the change of the energy difference between the instability levels AE with temperature, no reliable answer has been given so far on the basis of the theoretical FSM calculations. Again, we can only guess what happens. It is possible that the negative anharmonicity which is apparent only at low temperature in' Invar systems, eventually becomes a positive anharmonicity at high temperatures. This would imply that: (i) at a certain temperature HS, LS and NM states have the same energy (AE = 0), and it 'costs' no energy to go from one to the other. This could explain the absence of an ordering peak in the specific heat at T c (or TN) (cf. figs. 41, 42) if T c (or TN) is identified with this 'certain temperature'. If then (ii) at even higher temperatures the binding curves become skewed towards h!gh volumes (positive anharmonicity) then the LS state will be the ground state. Transitions from the LS (or NM) to the HS state then lead to the reappearance of magnetovolume effects in the paramagnetic range of an Invar alloy, which has indeed been observed experimentally (see figs. 19-21 and 42). As we have seen, the larger on many occasions in this chapter, large magnetovolume effects (Invar) are observed as a function of concentration (e/a) shortly before the systems undergo a structural first-order transition from the fcc ~/- into the bcc a- phase at some smaller e/a-values as compared to the Invar range. Note that the ~/-a transition is accompanied by a large increase in volume, from dense packed fcc to the much more open bcc structure. As we have seen, the larger the magnetovolume effects become, the smaller the energy difference AE between HS and LS states gets. Now AE decreases with decreasing e/a, as, e.g., shown for the FeNi system in fig. 50e. Therefore, one has the impression that at some (certainly system dependent) e/a value the positive volume magnetostriction would become so large that the system 'prefers' to undergo the 3'-a structural phase transition. Note that in FeNi at zero temperature this happens at e/a = 8.65 (cf. fig. 50e). If Mn, with a large atomic volume, is alloyed to FeNi, the structural phase transition does not occur. Instead, AE can get increasingly smaller so that around e/a = 8.4 for AE = 0 the itinerant spin-glass phase is observed [see inset of fig. 19 and also Wassermann (1989a)] and then for e/a < 8.4 in Fe65NixMn35_ x we find antiferromagnetic Invar.
INVAR
317
In total, in our opinion, the particular binding-energy surfaces computed at T = 0 for 3d-elements and alloys, and revealing the existence of certain local minima with corresponding fixed magnetic moments and atomic volumes, provide a new and general basis for the understanding of the large magnetovolume effects observed in these systems. To our feeling, the structural phase-transitions equally characteristic for 3d-elements (allotropy) and alloys are also ultimately connected to the peculiarities of these binding-energy surfaces. Certainly, more theoretical and experimental work is necessary to understand the properties of the systems at finite temperatures a n d - equally i m p o r t a n t - antiferromagnetic Invar.
Acknowledgement I am very much indebted to the late Peter Wohlfarth, who originally encouraged me to write this chapter. My thanks also go to W. Pepperhoff for many valuable discussions and sublet of unpublished data. Helpful discussions with Y. N a k a m u r a and V. Moruzzi and my own co-workers M. Acet, G. Lecomte and W. Stamm are also gratefully acknowledged. Finally, I thank M.M. Abd-Elmeguid for allowing me to use his pressure data prior to publication, and J. Kfistner for critically reading the manuscript. The work has been supported within Sonderforschungsbereich 166 Duisburg-Bochum.
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Tajima, K., Y. Endoh, Y. Ishikawa and W.G. Stifling, 1976, Phys. Rev. Lett. 37, 519. Tajima, K., P. B6ni, G. Shirane, Y. Ishikawa and M. Koghi, 1987, Phys, Rev. B 35, 274. Takzey, G.A., I.I. Sych, A.Z. Menshikov and A.Y. Teplykh, 1981, Phys. Met. & Metallogr. (USSR) 52, 49. Tange, H., T. Tokunaga and M. Goto, 1978, J. Phys. Soc. Jpn. 45, 105. Tange, H., T. Yonei and M. Goto, 1981, J. Phys. Soc. Jpn. 50, 454. Tarvin, J.A., G. Shirane, R.J. Birgeneau and H.S. Chen, 1978, Phys. Rev. B 17, 241. Trichter, F., A. Rabinkin, M. Ron and A. Sharfstein, 1978, Scr. Metall. 112, 431. Tsioplakis, K., and T. G6decke, 1971, Z. Metallkd. 62, 680. Tsunoda, Y., N. Kunitomi and R.M. Nicklow, 1987, J. Phys. F 17, 2447. Tsunoda, Y., S. Imada and N. Kunitomi, 1988, J. Phys. F 18, 1421. Uffelmann, D., 1987, Diplomarbeit (University of Duisburg), unpublished. Wagner, D., 1989, preprint. Wagner, D., and E.P. Wohlfarth, 1981, J. Phys. F 11, 2417. Wang, C.S., B.M. Klein and H. Krakauer, 1985, Phys. Rev. Lett. 54, 1852. Warnes, L.A.A., and H.W. King, 1976, Cryogenics 16, 659. Wassermann, E.F., 1987, Adv. Solid State Phys. 27, 85. Wassermann, E.F., 1989a, Phys. Scr. T 25, 209. Wassermann, E.F., 1989b, in: Cooperative Dynamics in Complex Physical Systems, ed. H. Takayama (Springer Verlag, Berlin) p. 157. Wassermann, E.F., K. Usadel and D. Wagner, eds, 1989, Proc. Int. Symp. on Magnetoelasticity and El. Str. of Trans. Met. Alloys and Films (Duisburg, Germany, 1989) [Reprint from: 1989, Physica B 161, 1-3]. Weiss, R.J., 1963, Proc. Phys. Soc. London 82, 281. Williams, A.R., V.L. Moruzzi, C.D. Gelatt Jr and J. K/ibler, 1983, J. Magn. & Magn. Mater. 31-34, 88. Wohlfarth, E.P., 1969, Phys. Lett. A 28, 569. Wohlfarth, E.P., 1976, J. Phys. F 6, L59. Wohlfarth, E.P., 1983, Physica B 119, 203. Xianyu, Z., Y. Ishikawa, S. Ishio and M. Takahashi, 1985, J. Phys. F 15, 1787. Yamada, O., and E. Du Tr6molet De Lacheisserie, 1984, J. Phys. Soc. Jpn. 53, 729.
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Yamada, O., F. Ono, I. Nakai, H. Maruyama, F. Arae and K. Ohta, 1982, Solid State Commun. 42, 473. Z~ihres, H., M. Acet, W. Stamm and E.F.
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chapter 4 STRONGLY ENHANCED ITINERANT INTERMETALLICS AND ALLOYS
P.E. BROMMER and J.J.M. FRANSE Natuurkundig Laboratorium Universiteit van Amsterdam Valckenierstraat 65 1018 XE Amsterdam The Netherlands
Ferromagnetic Materials, Vol. 5 Edited by K.H.J, Buschow and E.P. Wohlfarth-~ © Elsevier Science Publishers B.V., 1990
CONTENTS 1. Itinerant-electron magnetism . . . . . . . . . . . . . . . . . . . . 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Stoner-Edwards-Wohlfarth model . . . . . . . . . . . . . . . 1,3. Equations of state . . . . . . . . . . . . . . . . . . . . . . 1.4. Thermodynamic relations; Griineisen relations . . . . . . . . . . . . . 2. Arrott plots . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Characterization . . . . . . . . . . . . . . . . . . . . . . . 2.2. Arrott plots for homogeneous materials . . . . . . . . . . . . . . . 2.3. Arrott plots for inhomogeneous materials . . . . . . . . . . . . . . 2.4. Arrott plots in the critical region . . . . . . . . . . . . . . . . . 2.5. Arrott plots for local moments in a polarized matrix . . . . . . . . . . 3. Magnetovolume effects . . . . . . . . . . . . . . . . . . . . . . 3.1. The magnetic volume in homogeneous materials . . . . . . . . . . . . 3.2. Thermal expansion at low temperatures . . . . . . . . . . . . . . . 3.3. Magnetovolume effects in dilute PdMn alloys . . . . . . . . . . . . . 3.4. The combined analysis of specific heat and thermal expansion . . . . . . . 4. Perfectly ordered stoichiometric compounds . . . . . . . . . . . . . . . 4.1. Structural defects in off-stoichiometric compounds . . . . . . . . . . . 4.2. Magnetic properties of stoichiometric perfectly ordered Ni3A1 . . . . . . . 4.3. Magnetic properties of Ti(Fe,Co) compounds . . . . . . . . . . . . . Note on units . . . . . . . . . . . . . . . . . . . . . . . . . . List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
. . . . . . . . . . . . . . . . . . . . . . .
325 325 326 331 335 342 342 342 347 351 354 358 358 362 368 376 385 385 387 389 393 394 394
1. Itinerant-electron magnetism 1.1. Introduction
Itinerant-electron ferromagnetism was recognized in the 1930s by the work of Slater (e.g., 1936) and Stoner (e.g., 1938). In particular, the 3d transition metals Fe, Co and Ni were described as itinerant-electron ferromagnetic materials. Later on, the properties of many (mostly 3d-) alloys and compounds were discussed in this model. Topics were, e.g., the onset of ferromagnetism, the pressure and the concentration dependence of the Curie temperature. At present, an extensive literature exists on the subject. Wohlfarth (1980a) described the history and the future of the 'Stoner-Edwards-Wohlfarth' model (SEW-model). Shimizu (1981) reviewed very elaborately the theoretical and experimental situation. Although it was well-known that for strongly enhanced materials, i.e., weak and very weak itinerant ferromagnetics, fluctuations should be important near the transition point, the Stoner adepts were inclined to ignore them. Moriya advocated strongly the opposite point of view (see his book, 1985). More recently, the 'heavy fermion' systems attracted much attention. Here the f-electrons (of a rare earth, often Ce, or of an actinide, mostly U) are thought to become delocalized and thus to become part of the 'Fermi liquid'. These systems were reviewed, e.g., by Stewart (1984) and by Ott (1987), whereas Fulde et al. (1988) discuss the theory of the heavy-fermion systems. Since, as indicated above, various excellent reviews do exist, in this chapter we preferred to focus attention on the analysis of experimental results, from the point of view of the experimentalist. Most experimental data used as an example were naturally taken from the work of the Amsterdam group in which the authors have taken part. It proved to be necessary, however, to consider the results already published, together with some more recent or unpublished results, in the light of recent theoretical developments. Consequently, in many cases the experimental data are now interpreted in a new, more sophisticated way. Most of the theoretical framework is established in the remaining sections of this section. In section 1.2 the SEW model is described; here the (zero temperature) magnetic properties of the strongly enhanced materials under consideration are expressed in a few so-called fundamental parameters. More generally, however, a Landau-Ginzburg formalism is applied in which both the SEW-model 325
326
RE. BROMMER and J.J.M. FRANSE
and the fluctuations, as well as spatial variations, can be treated (section 1.3). Thermal expansion and specific heat data prove to be very important in the analysis. Thermodynamic relations between these quantities are treated in section 1.4. Magnetization data are analyzed by means of Arrott plots. In section 2 we discuss the four characteristic types: plots for homogeneous and for inhomogeneous materials, plots observed in the critical region, and plots for a system of local moments embedded in the strongly enhanced matrix. In section 3 magnetovolume effects are discussed. The magnetic volume at the transition temperature is a measure for the mean local magnetic moment, and thus for the influence of the fluctuations (section 3.1). The fluctuations may play an important role even at low temperatures (section 3.2). Local moments embedded in a strongly enhanced matrix (in section 3.3 Mn in Pd) give rise to a 'ferromagnetic' contribution and a 'paramagnetic' (or 'pair') contribution, which can be separated by a combined analysis of the thermal expansion and the specific heat. This procedure is discussed in more detail in section 3.4. Here, we also seized the opportunity to present a recent re-analysis of the 'heavy fermion' compound UPt 3. The magnetic properties of the strongly enhanced compounds appear to depend on the degree of order. In section 4 the properties of an ideally ordered stoichiometric compound are derived from the experimental data observed on a series of actual, off-stoichiometric compounds. A simple ordering model is explained in section 4.1. We apply this method to Ni3Ga and Ni3AI in section 4.2, and to Ti(Fe,Co) in section 4.3. 1.2, The S t o n e r - E d w a r d s - W o h l f a r t h model
The Stoner-Edwards-Wohlfarth model (SEW, see, e.g., Stoner 1983, Edwards and Wohlfarth 1968, Wohlfarth 1968, 1969, 1977) for weakly magnetic, itinerant electron systems can be formulated in terms of a Landau expansion of the molar free energy F = F(V, M, T ) of a homogeneous system (1)
F = F ° + ½ A M 2 + ~B M 4 + ' ' ' .
Here V is the molar volume and M the molar magnetic moment. Terms of order M 6 are omitted (see, however, Shimizu 1981). In this 'classical' model the influence of fluctuations is neglected (see section 1.3). The equations of state are, (2)
p~oH/M = A + B M ~ , -VP=
1
2
1
4
F ° + S A , o M + aBo~M .
(3)
Here A,o =-OA(V, T ) / O l n V , and so on. Equation (3) defines a 'magnetic pressure' given by: Vp m
= ~1 A ~ M
2
+IBM
4.
(4)
ITINERANT INTERMETALLICS AND ALLOYS
327
The magnetic volume is defined as the excess volume due to the magnetic pressure. For transition metals with atomic moments of the order of 1/% the magnetic pressure is of the order of -10 kbar, so non-linearities should be taken into account. Indeed, Janak and Williams (1976) apply the Bridgman quadratic relation between volume and pressure i n their estimates of the magnetic volume of the transition metals. For the enhanced itinerant systems under consideration, we take the relative 'magnetic volume' to be OJm = V m / V :
--Kern
,
where K is the appropriate compressibility [see also Brommer (1982a); and the discussion in section 3.1]. Experimentally, the magnetic volume appears to be proportional to M 2. We define a magnetovolume parameter C by, OV 1 KC = -~0 = - ~ K ( A , ~ + B~,MZ).
(5)
In most cases, the contribution proportional t o M 2 is neglected, so KC is taken equal to its 'paramagnetic' value -½ KA,o. In a band model we want to incorporate the exchange interaction explicitly. Its contribution to the free energy is given by -½12 M2 - ¼14M4, where the second-order term usually is written in terms of the 'Stoner interaction', I
2
= 2NA~BI2
,
or rather, in terms of the dimensionless parameter i = gI = I2/Aband(V, 0 ) .
The relevant magnetic free energy is, Fmag n = 1 A M
2 + 1
B M 4 = ~(Aband 1 1 _ I4)m 4 . -- I 2 ) M e + ~(Bband
(6)
The band part of the parameters A and B can be expressed in the density of states, g, and its (scaled) derivatives vj = g - 1 0 J g / O E J (all taken at the zerotemperature Fermi level). The electron-phonon interaction can be taken into account by assuming that for each subband the density of states is enhanced by a factor (1 + oe). Then we find, Aba,~(V , T) = Aband(V, 0)[1 + a2T2(v 2 - v2) + a4(a2T2)Z], Bband(V ,
T)
= g1 ~ B2A b3a n d ( V , 0 ) [ 3 P ~ -- P2 q-
bz(a2T2)l,
(7) (8)
where a 2 = ~-(1 + 00~r2k2 and Abana(V, 0) = (2NMz~g) -~. The coefficients a 4 and
328
RE. BROMMER and J.J.M. FRANSE
b 2 are given by 3 4 2 2( a4 = 2 v 1 -- 7 /J1/'2 + /"2 +
7 )
1+ 10(l+a)
7 /"1/33
10(l+a)
b2 = (15v 14__ 25v2v2 + 4v~ + 7v~ v3 - v4)/(3v 2I - p 2 )
1'4'
(9)
(10)
The corresponding expressions in the case a = 0 were given by De Chfitel en De Boer (1970). See also Shimizu's review (1981). For very weak itinerant ferromagnetism (VWIF) the temperature dependence of B is neglected, whereas in A only the quadratic terms are retained:
A(V, T) = Aband(V, 0){1 -- i + T2/T2}.
(11)
The temperature T v is a measure for the degeneracy temperature: TF 2 = az(V ~ --v2), which means that for a parabolic band without electron-phonon enhancement, TF equals (2/~-)V~ times the Fermi temperature. For a paramagnetic system (A > 0), the zero-temperature susceptibility, Ao 1, is -1 enhanced with respect to the band value Aband by a factor S = (1 - i) -1 . We use this definition of the Stoner enhancement factor for ferromagnetic systems as well (S < 0 ) . In this region the zero-field magnetic moment is given by M 2= M2t =-A/B. The critical temperature T s (later on referred to as the 'Stoner temperature') is given by the relation A(V, Ts) = 0, hence = v
/Isl •
The influence of a change of volume can be expressed by a small number of 'fundamental' parameters (Brommer 1982a), dlng =F do) '
dlnTF d ~ --
/'F,
dlni d----w--= FA
(12)
Under the so-called 'uniform scaling assumption' (Lang and Ehrenreich 1968) the parameters F and FF are equal to the proper Grfineisen parameter - d In W/do), where W is the band width. In section 1.4 Grfineisen relations are discussed in more detail. The volume dependence of the Stoner interaction I is given by d In 1 ~ d o ) = - F ( 1 - A). A physical meaning of the parameter A is provided by Kanamori's (1963) expression for the effective interaction I=Ibare/(l+ eIbare/W), where /bare is the 'bare interaction' and e a numerical constant. Provided that only the band width W depends on the volume, we have
ITINERANT INTERMETALLICSAND ALLOYS
329
Using eq. (12) we obtain expressions as dlnAba,d(V,O)/dw=
-F,
dlnBb~nd(V,O)/dw = 2FF- 3F.
The volume dependence of the 'inverse susceptibility' is enhanced, d In [A(V, O)l/d¢o = r)t(1 - S) - F . An analogous expression can be given for the parameter B, but we prefer to write simply d In B/doJ = 2F' - 3F. As indicated by Brommer (1982a) the relationships mentioned above can be found in many different, but analogous, forms in the literature. For paramagnetic materials one often expresses the ratio between the susceptibility and the specific-heat coefficient in terms of the so-called Wilson ratio. In the present model we may write R = ½ ( ~ k / p ~ ) 2 ( A e f ( y ) -1 ~ S, so dln R~ dto = d In S / d w . Hence, the relation din S
= F~,(S - 1),
do~
(13)
gives the volume dependence of the ratio R as well (see, e.g., Fawcett and Plushnikov 1983). Finally, we express the magnetovolume parameter in the fundamental quantities, 2
(14)
2
(lS)
4Na/ZBCp,ra = F M + r ( g -1 - I ) , 4NAI~BCf .... = F M + 2(F' - F ) ( g -~ - I ) .
Notice that the factor ( g - l _ I) can be determined from magnetization measurements, since we have, -
I)
=
(uS)
-1
=
2 _ 12) = 2NA/ZB(Aba.d
2NAI~ZA(V,O).
(16)
The magnetovolume parameter can be determined accurately from forced magnetostriction data in combination with magnetization data. Since ( g - 1 I ) = (gS) -a is small, a good estimate for the combination F M is obtained. Moreover, in an alloying system ( g S ) -~ varies as a function of the composition. By extrapolation down to ( g S ) -~ =0, a still more accurate value for F M can be determined at the critical composition. The fundamental parameter F, i.e., the electronic Gr/ineisen parameter, can be determined experimentally by measuring the thermal expansion coefficient and the specific heat as well [see eq. (39)]. The interaction energy I can be derived from an estimate of the density of states in combination with a determination of the inverse susceptibility A from magnetic measurements, because we have, I = g-1 _ 2 N A/.2A,
330
P.E. B R O M M E R and J.J.M. FRANSE
(see above). The density of states usually is estimated from the linear term in the specific heat, or by a band calculation. This procedure was carried for a series of Ni3AI compounds. From the magnetovolume experiments the product FAI was determined (fig. 1). At the critical composition (i.e., 74.6% Ni) we find FAI = 0.32 eV Ni spin. Franse (1977) derived a value for the Griineisen parameter, F = 3.6. In section 3.2, however, we show that the determination of the appropriate Griineisen parameter is not unambiguous at all. Buis et al. (1981a) adopted a value of the density of states g = 1.2 st/(eV Ni spin) for the perfectly ordered stoichiometric 'ideal' compound. From the analysis discussed in section 4.2 one finds ( g S ) -1 = 2 N A / ~ B2 A o = -0.0138 eV spin Ni, and hence A(=I/Ib) = 0.1 and, I = g-1 + 0.0138
[eV spin Ni].
Again, the determination of the density of states, (e.g., from the observed specific heat) is not straightforward, because the enhancement factor is not known. The same is true for the density of states values obtained by band calculations. We conclude that the fundamental parameters can be determined only approximately. In contradistinction, the magnetovolume parameter (i.e., FAI) can be determined rather accurately. For a number of strongly enhanced itinerant systems this product is given in table 1. The interaction energy I can be estimated to adopt
76 75.5 75 7h.8 I
0.5
I
I
\
':-
FXI
\
74.4 |
7/.,.2 7/, %Ni
1
I
I
t|
eV spin Ni
~eal'"
0.4
I
Ni3A[
+
+
0.3 I
I
I
-20
-10
0
I
I
10 20 (gS) -1 {10-3eVspinNi}
Fig. 1. The normalized magnetovolume parameter FAI as a function of the normalized inverse enhancement ( g S ) -1 for Ni3AI compounds. Data: ( + ) Kortekaas et al. (1974); (O) Kortekaas and Franse (1976). The arrow indicates the 'ideal' compound (see section 4.2).
ITINERANT INTERMETALLICS AND ALLOYS
331
TABLE 1 The magnetovolume parameter KCwt(=to/o"2) and FAI, the product of the fundamental parameters F(= 0 In g/0w), A(=1/Ib) and the effective interaction I for some typical itinerant-electron magnetic materials. Compound
106 KCwt (A m2/kg)-2
Pd
2.4 2.1
Ni3A1 0.64 0.79 3.0 3.3 0.33 1.9 1.7 1.4 5.0 8.9 1.0
Ni3Ga NiPt TiFe TiFe0.sCo0.5 TiCo ZrZn2 MnSi
Fhl (eV spin at) 0.46 (Pd) 0.40 (Pd) 0.32 (Ni) 0.39 (Ni) 0.31 (Ni) 0.21 (Ni) 0.23 (Ni) 0.16 (Fe) 0.91 (Fe) 0.799 (Ti) 0.648 (Co) 0.314 (f.u.) 0.554 (f.u.) 0.766 (Mn)
Remarks H61scher (1981) Keller et al. (1970) See text See table 4 From Oa/OP(0.2% Fe), Buis (1979) Ordered: Brommer and Franse (1988) Disordered (45.2% Ni): table 4 H61scher (1981) (experimental) H61scher (1981) (analysis) H6slcher (1981) H61scher (1981) Brommer and Franse (1984) Brommer and Franse (1984) Brommer and Franse (1984)
values between 0.5 and 1 eV at spin. The corresponding values for h range from say 0.1 to 0.5.
1.3. Equations of state In this section we derive the equations of state for both homogeneous and inhomogeneous materials in a phenomenological ' L a n d a u - G i n z b u r g ' formalism. The free energy is considered to be a functional of the magnetic m o m e n t distribution M(r). This distribution may be regarded as the result of an averaging process in which at least the very fast 'quantum-mechanical' fluctuations have been filtered out (see Lonzarich and Taillefer 1985). Moreover, variations over distances shorter than, say, the interatomic distance are supposed to be washed out b y an averaging procedure as well. The materials under consideration are • assumed to be isotropic, although one may choose n, the number of components of the magnetic moment, in accordance with a particular model (e.g., n = 1 for the one-dimensional Ising and n = 2 for the XY-model). The molar free energy is written as the functional
F = F(V, [M(~)], T) = F(V, O, r ) + Fro(V, [M(,')], T) = F0 + Fm, 1 Fm = Vo1
J" d~{-~A~oc(~)M~(~) + ~B(~)Mo(~)} - Fox(V, [~(~)], 7")
Fox(V, [~(")1, r) = ~
d~ d~'J(~, ~')M(~)M(~'),
(17)
(181 (19)
P.E. B R O M M E R and J.J.M. F R A N S E
332
where V is the molar volume and Vol is the integrated volume, e.g., determined by periodic boundary conditions. By expanding M(r') in a region around r the exchange free energy can be written in the Landau-Ginzburg form (see, e.g., Kronmfiller and F/ihnle 1980). The magnetic moment distribution in the presence of a magnetic field H(r) is then given by the equation of state (Euler equation)
A(r) M(r) + B(r) M2(r) M(r) - D AM(r) = i~on(r) .
(20)
We neglect the r dependence of the 'stiffness' parameter D. This parameter is directly related to the second moment of the exchange interaction J(r, r'), but is here considered as an independent quantity (to be determined experimentally). Further details can be found, e.g., in the work of Kronmiiller and Ffihnle (1980), where also an effective temperature dependence is introduced by incorporating a temperature-dependent correlation between the spins at r and r'. Moreover, these authors incorporate stray fields and magnetoelastic interactions in the equation of state. Since the influence appears to be small, we neglect the latter terms. The Landau-Ginzburg parameter A(r) in eq. (20) contains the corresponding local parameter [from eq. (18)], as well as a contribution caused by the exchange interaction. Denoting the exchange interaction, averaged in a region around r, by J(r) we have, (21)
A(r) = A,oc(r ) - J(r).
As a next step we decompose the magnetic moment distribution in its Fourier components:
M ( r ) = M + ~ m oexp(iq.r),
q¢O.
(22)
Both in homogeneous and in inhomogeneous materials thermally excited spin waves and spin fluctuations are taken into account by taking the thermal average of the (Fourier transformed) equation of state, eq. (20). The surviving secondorder longitudinal terms are denoted by,
An analogous relation defines the fluctuations of a transverse component, rn 2, and the total variance m 2(=mlf2 +
2m~)
In inhomogeneous materials the spatial variation of the Landau-Ginzburg parameters A(r) and B(r) leads to a corresponding variation of the magnetic moment distribution. Dropping the brackets { ) denoting the thermal average and
ITINERANT INTERMETALLICSAND ALLOYS
333
neglecting higher-order terms, we find, (24)
M~ = Xll( q){I.toH~ - M[Zq + Bq(M 2 + All)l} , M
± q =
x ± ( q)
± I'~on q ,
(25)
X~l(q) = A + Dq 2 + B[3M 2 + All],
(26)
X-~(q) = A + Dq 2 + B[M 2 + A±].
(27)
Here the corrections due to the fluctuations and the spatial variations are given by All = 3 ~ MIIM + ±_ q ± q -II -q + 2 ~ M q M q
3m~ + 2 m ~
(28)
q
a ± = E M l ~ M I l _ , + 4 E M ~ M S , + m l l +2 q
.
(29)
q ±
In a homogeneous field the transverse variations Mq vanish. Dropping superscripts and neglecting higher-order terms, we find the equation of state, tzoH = A M + B M ( M 2 + All) + ~] {A_,Mq + B_qMq(3M z + All)}.
(30)
q
For homogeneous materials the equation of state can be simplified. Written in the form of a so-called Arrott plot the result is I%H/M = A + B M 2 + B(3m~ + 2m2).
(31)
We remark here that the same result can be obtained by reducing the probability distribution e x p [ - ( V o l / V k T ) F ] , where F is given by eqs. (17)-(20), to a Gauss±an distribution in the variables M and Mq [see eq. (20)] by a Hartree-type pair-wise avaraging procedure (Murata and Doniach 1972, Shimizu 1981). The free-energy function F(M, Mq) is reduced to F(M, Mq) = ½(A + BrnZ)M 2 + B E (M~)Zrn2 + 1BM 4 v
+ ½ ~ ~, [m + B { ( M ) ( M ) + 2(M~) 2 q
v
....... + m 2 + 2 m 2}+ O q 2lJVlqM_q.
(32)
The inverse susceptibilities given in eqs. (26) and (27), can be read off immediately, as well as the equation of state, eq. (31) [the analogon of eq. (2)]. The magnetic pressure can be derived from the analogon of eq. (3), which is obtained by calculating F,o in the same approximation. Neglecting the volume dependence
334
P.E. B R O M M E R and J.J.M. FRANSE
of the stiffness parameter and omitting some terms of fourth order we find,
VP m = ½A,o(M 2 + m 2) + ¼Bo,M 4 + B,oM 2(3mll2 + 2 m ~ ) .
(33)
In the present model, the Curie temperature is much lower than the Stoner temperature (see section 1.2) because of the presence of the fluctuations. Since the Curie temperature is given by
A + B(3m~ + 2 m ~ ) = O ,
(34)
we have, with m~ = m2/n; rn 2 -
n Ms2t = 3 M s t2, n+2
(35)
where the last equality holds in the three-dimensional isotropic case. The 'Stoner' value of the squared magnetization is given by M2st = - A / B . The equation of state (eq. (31)) was derived by Lonzarich and Taillefer (loc. cit.) in a much more fundamental way. Following Ramakrishnan (1974), they calculated the fluctuations of wave vector q by integrating the imaginary part of the dynamical susceptibility weighted with the Bose-Einstein function. The result is that the 'classical' value (O/Vol)RTx(q), where O is the atomic volume and R the gas constant, should be multiplied with a function g(z)= (1 + 6z) -1, where z ~ q/x(q)T. The total fluctuations are obtained by performing the sum over the q-values. Introducing the correlation length Ell, ~ = D/A~re(O) (and an analogous transverse correlation length), the integrated 'classical' result is,
RTg2 (qll m ~ - 2,n.2D
arctan qll ~11) • ~1[
(36)
Here qu is a cut-off wave vector for the longitudinal fluctuations. An analogous expression is obtained for the transverse fluctuations. Notice that this expression, substituted in the equation of state, would lead to a first-order transition at T c (see Yamada 1975, Murata and Doniach 1972).* This first-order transition is signalled by a negative slope d M2/d(/~0H/M) of the calculated Arrott plots near T o which in turn is a consequence of the sharp decrease of m~ with increasing M2: at T c we find dm~/dM 2 = - 1 (this feature can be traced back to the contribution proportional to arctan ( q ~ ) / ~ ~-1 in eq. (36), and is generally considered to be an artefact of the model). Lonzarich and Taillefer argue that the correction to the 'classical' result effectively leads to a temperature-dependent cut-off. For instance, at T = T c we have x( q) ~ q-2, hence z oc q3/T and the integral f x( q) g(z) dq is proportional to T 1/3. This result can be reconstructed by assuming that in eq. (36) only the first *Lonzarich and Taillefer circumvented the unrealistic first-order transition by adopting a suitable approximate expression for the susceptibility.
ITINERANT INTERMETALLICS AND ALLOYS
335
term remains, with an 'effective' cut-off wave vector qc also being proportional to T 1/3. Consequently, the fluctuations m 2 appear to be proportional to q~T~ T 4/3, in accordance with the more detailed calculations. In fact, Lonzarich (1986) takes both spin waves and spin fluctuations into account, but we ignore this feature here. The work of Lonzarich and Taillefer extends the pioneering work of Moriya, Kawabata and many others (see, e.g., Moriya 1985), in particular by consistently taking both the longitudinal as well as the transverse fluctuations into account. Their main observation, however, is that the relevant parameters A, B, D and, say, the relaxation frequency of a spontaneous spin fluctuation, can be determined experimentally. Thus, essentially, no free parameters are available in the model. The good agreement obtained for Ni3A1 and MnSi - the only cases where all parameters have indeed been determined - gives strong support to this model. In general, however, not all parameters are available (mostly only A and B are known well from low temperature magnetic data), so the model cannot be severely tested for all materials. Shimizu (1981) found an equation of state analogous to eq. (31) including terms of higher order in the magnetic moment. His further derivations, however, were based on a temperature-independent cut-off wave vector. Consequently, his interpretation of experimental data is quite different from that of Lonzarich and Taillefer, and that of Moriya. Various model calculations are available at present (see, e.g., Turov and Grebbenikov 1988, Hirooka 1988). For inhomogeneous alloys in a homogeneous field, we have to combine eq. (30) with eqs. (24) and (26) in a self-consistent way. For small values of M we obtain a decrease of the first Landau coefficient,
A--~ A - ~, Xit(q)lAq + Bq a , I
2 .
(37)
q
We remark that such a shift is obtained also in the expression for Xil(q) , i.e., in eq. (27), when we take into account higher-order terms (in particular, m o d e mode coupling terms such a s Aq,Mq_q, and so on). The negative shift in the A-value points to an increase of the Curie temperature with respect to that of the 'averaged' homogeneous material. In section 2.3 we discuss the shift, and the Arrott plots of inhomogeneous materials, in more detail. The magnetic pressure, Pm, is determined by the equation of state eq. (3). Ignoring the volume dependence of B(r) we find the counterpart of eq. (33), em= 1
A~(M 2 + m 2) + ~q [ ½Zo,MqM q + c3Aq MM_q}
(38)
In most cases, however, the influence of the variations is neglected.
1.4. Thermodynamic relations; Griineisen relations In the study of magnetic systems the thermal expansion and the specific heat prove to be very important (see also section 3). In general, an arbitrary
336
P.E. BROMMER and J.J.M. FRANSE
contribution to the specific heat (cj) is related to a corresponding contribution to the thermal volume expansivity (/3j) by a so-called Gr/ineisen relation, V~j/Kcj=Fi .
(39)
A 'proper' Gr/ineisen relation of this kind arises when the volume dependence of the corresponding entropy contribution happens to occur in the scaling combination T / T j ( V ) only, where T j ( V ) represents the relevant volume-dependent energy parameter (e.g., Barron et al. 1980, Brommer 1982a). Then: Fj = - d In Tj(V)/d In V.
(40)
As an example, we mention the familiar Grfineisen parameter for the phonon or lattice contributions in the Debye model, flph = (K/V)FphCph(v)(OD/T) with Fph = - d In 0D/dto.
(41)
In principle, for each mode, with energy ej, an independent Gr/ineisen parameter, Fj = - 0 In ej/0to, may exist; in the Debye model the volume dependence of the mode energy is determined by the fact that there is only one single energy parameter (0D) for all modes. In eq. (39) the specific heat has to be determined at constant volume. In practice this means that the specific heat as determined at constant pressure has to be corrected (Cj(v) = cj(e) - V~fljT/K). In an anisotropic crystal, the energy parameter Tj may be a function of the volume and of a deformation (or of more deformations), say, Tj = T~(V, e) ,
with
e = 2ezz - exx - Eyy
=
3e= - to.
Defining a 'coefficient of thermal deformation' ~ax = d e / d T we can derive an analogous Griineisen relation between the contribution ~axj and the contribution to the specific heat cj(v, e), with an appropriate Gr/ineisen parameter, Fax j = - 0 In Tj(V, e)/Oe and the corresponding stiffness parameters instead of the compressibility K in eq. (39). Of course, the measured specific heat should be corrected correspondingly as well as the thermal expansion coefficients (see section 3.4). Griineisen relations also arise from the observation that, by virtue of the Maxwell relation 02V/O T 2 = O ( V ~ ) / O T = - T - 1 0 c / O P , an expansion of the specific heat c= TT + bT n + dTnln T,
(42)
corresponds to the concurrent expansion of the thermal-expansion coefficient: -V/3 = yaT + _1 bpTn + _1 dpTn In T n
n
i n
dpTn ,
(43)
ITINERANT INTERMETALLICS AND ALLOYS
337
where the subscript P denotes the pressure derivatives. For the terms TT, bT n and dT" In T Gr/ineisen relations exist, with F equal to a In .y/Oco, (1/n) 0 In b/Oco and ( l / n ) a In d/Oco, respectively. The spurious term (-1/nZ)de T" in -V/3 can be removed by rewriting eq. (42) in such a way that the logarithmic term has the form do(T/T*)" ln(T/T*), with of course doT*-" = d. Then a (proper) Gr/ineisen relation results, with F = - 0 In T*/Oco = (1/n) 0 In d/Ow as expected. It is well-known that the electron-phonon interaction enhances the electronic contribution to the specific heat. From the thermodynamic identity, eq. (43), it is clear that both the specific heat [yT = %(1 + a)T] and the coefficient of thermal expansion are enhanced*. The (improper) Gr/ineisen parameter for the linear term 7T is now, F = 0 In g ( E F ) / O c o + 0 ln(1 + a)lOoo,
(44)
where g(EF) is the unenhanced density of states. In fact, this relation holds irrespective of the nature of the enhancement. In general, we expect the Gr/ineisen parameter to increase with increasing enhancement. Thalmeier and Fulde (1986) and Fulde et al. (1988) proposed to scate the temperature as well as the magnetic field. They assumed that the free-energy contribution ~F(V, H, T) can be written in the form:
~F(V, H, T) = - Tf(x, y),
x = T/T*,
y = H/H*,
(45)
where f(x, y) does not explicitly depend on the volume V, only the parameters T* and H* do. If one thinks in terms of a 'magnetic mode', or of 'quasiparticles', with an effective moment /x, the partition function is governed by the combination (e - I~H)/T, hence H * ~ T*//z, where T* determines the energy scale (e ~ T*). From this point of view, the 'thermal parameter' T* is the more fundamental one. In case t~ does not depend on the volume or in case ~F is just a function of H / T (and T/T*), we have simply H* ~ T*. The extended Gr/ineisen relations can be obtained by comparing the expressions for the variations dS, dM and dP, as deduced from eq. (45), with the corresponding ones, expressed in the experimentally accessible quantities, i.e.,
- d P = (VK) -1 d V - hK-a d/z0H _/3K-1 d T ,
(46)
- d M = - h K - 1 d V - )~ dtz0H + ~ d T ,
(47)
- d S = --/3K-1 dV + ~d/%H - (~/T) d T ,
(48)
*Incorporating this enhancement in the usual derivation of the specific heat in the band model, we find under some natural assumptions the following expression for the paramagnetic specific heat:
c= Z(l + a)~rZk2g(Er)NT(l + ~
k T [su2-(l +
where (1 + a) is the electron-phonon enhancement factor, g(EF) is the density of states per spin per relevant atom, and N is the number of relevant atoms per mole. For a = 0 this expression reduces to an expression given by Shimizu (1981).
338
P.E. BROMMER and J.J.M. FRANSE
where ~ = X - Vh2K-1 ~ : ~ @ V h f l k - 1 and ~ / T = ( c / T ) - V~2K -1. X, ~ and c are the quantities determined experimentally at constant pressure. Notice also the definition of the susceptibility X = jIZo1 0 M / O H I p r , keeping in mind that we use molar quantities (so X is expressed in units A m2/mol T). In the following, the corrections to X, ~ and c are not indicated explicitly, and K-1 is replaced by the bulk modulus b. From eq. (45) we infer immediately that the entropy S contains a contribution, (49)
g(x, y) = f ( x , y) + xfx(x, y) .
Inspection of the variation - d S yields in an obvious notation gy = - t z o H * 8~ ,
gx = + T* 8 ( c / T ) .
(50)
Substituting this in the expression obtained for 8(bfl), and using the notation: F= -aln T*/Oto,
f'= -OlnH*/Oa,,
(51)
we find the extended Gr/ineisen relation, (52)
V 8(bfl) = F 8c -/~/x0H 8ft.
This relation was derived by Fulde et al. 1988 (their equation 4.10c). For vanishing field eq. (52) is reduced to the familiar Gr/ineisen relation of eq. (39). Again following Fulde et al. (1988), we use the (extra) magnetic moment 8M as another experimentally observable quantity. Consequently, we know the derivative fy = i x o H * T -1 8 M , and hence, with eqs. (49) and (50), also the second derivative fxy" T h e second d e r i v a t i v e fyy is equal to (/z0H*)2T -1 8X. Substituting these results in the expression obtained for 8(bh), we find, V 8(bh) = ( F - F ) 8 M - F T 8~ + I'tzo H 8 X .
(53)
This relation differs slightly from the analogous one obtained by Fulde et al. (1988, their equation 4.10b results if we adopt the linear approximation 6M = /*oH 8X). In order to proceed still further, we need another experimentally observable quantity. We take the 'extra pressure' 8P, which can be estimated by the determination of the excess volume: 8 P = - b 8 V / V = - b 8to. We define parameters F ' and/% by, 0 In ~v/&O = - ( F ' + 1),
and
0 In H~,lOto = -(/% + 1),
(54)
where T~ = O T * / a V and so on. Substituting these in the expression for 8b, we find V 8b = - T F 2 8c - ( V ' + 1)V 8/' +/,oH/~ 8M{Z(F - / ~ ) - ( r ' -/%)} + 2~oHTr?
8~" - (~oHP) 2 8X.
(55)
ITINERANT INTERMETALLICSAND ALLOYS
339
Of course, using the other Grfineisen relations one can eliminate, e.g., gff and/or 8X, and so on. Consequently, a variety of equivalent expressions can be derived.
Moreover, these expressions for the change of the bulk modulus can be simplified by further assumptions. For instance, taking i.e.
oZT*/ovZ=oZH*/OV2=O,
F'=/%=-l,
we find V 8b = - T F 2 8c + (IxoHF) 2 8 X - 2txoHFVb o 8 h ,
(56) ~
and so on. A more natural assumption would be to take T* ~ V - r and H * oc V -r, so F ' = F and/% =/~. Then we find bo I 8b = - T F ~
+ ( F + 1) ~oJ -/zoH/~ bh.
(57)
We consider eqs. (52), (53) and (57) to be the most useful extended Grfineisen relations. Notice that if, by some assumption, one of the quantities discussed is known as a function of V, H and T, we can find the corresponding part of the free energy and hence all other quantities. For instance, if we adopt (as in the example given by Fulde et al.) a linear approximation for 8 M = tzoH g x ( V , T ) with ~ x ( V , T ) = C / [ T + T*(V)],
the corresponding contribution to the free energy is - ½ { C / ( T + T*)}(/z0H) 2. This contribution can be written in the form given by eq. (49) with/x0H* = T* and f ( x , y) = ½ { C / x ( x + 1)}y 2. Hence, F and F ' equal /~ and /%, respectively, leading to further obvious simplifications. For nearly-ferromagnetic systems, on the one hand, and heavy-fermion systems on the other hand, the relevant contributions to the free energy can be written as, ~F(V, H , T ) = - ½ y T 2
1 2A~tf ( tx°H)2 "
(58)
The first contribution on the right-hand side is proportional to T ( T / T * ) with T * ~ W/Se, where W determines the energy scale of the unenhanced density of states (e.g., W~ band width) and where Se is an appropriate enhancement factor. There appears to be some ambiguity in the definition of the 'unenhanced' density of states, i.e., of the band-width parameter W. First of all, the periodic potential of the lattice causes, of course, a deviation from the free-electron value. Secondly, the electron-electron interactions may cause an enhancement, which may or may not have been incorporated in the band calculation. We shall adopt the usual point of view in assuming W to be the 'bare' band width, so that the enhancement of the density of states by the electron-electron interactions (and the periodic
340
P.E. B R O M M E R and J.J.M. F R A N S E
potential) are incorporated in the enhancement factor S e. In the case of an electron-phonon enhancement of (1 + a), we may write in terms of a Fermiliquid model, Se
=
(m*/m)(1 + ~ ) .
The density of states occurring in the susceptibility, A~f~, is not enhanced in the same way [see eqs. (7) and (8)]. The second contribution can be written in the form (1_+ T2/TZs)-IHz(W/S) -1, where S is the (positive) Stoner enhancement 1 / ( 1 - I). Here, again, we adopt the usual point of view that the Stoner enhancement is measured with respect to the 'bare' density of states, i.e., in a Fermi-liquid model we have, S = (m*/m)(a
+ F0) -1 ,
where Fg is the spin antisymmetric Landau parameter. Notice that in the ratio S/S e the mass enhancement (m*/m) cancels (see table 2). In the SEW model we have T~ = T2/S o~W2/(SSe) (see section 1.2). Obviously, the temperature scaling TABLE 2
The 'magnetic' Gr/ineisen parameter /~, the 'thermal' Griineisen parameter F, and the 'Wilson' ratio S / S e = (rr2k2/ /z~)(3yA) -1 of some nearly ferromagnetic alloys and some heavy-fermion and mixed-valence systems, as calculated by Kaiser and Fulde (1988) from volume magnetostriction data. Further data and references are given there too. F
S/So
Pd~_~Rhx x =0.01 0.03 0.05
4.8 9.5 11.1
2.9 3.4 3.9
7.3 8.3 9.5
14.2 44
4.1 6.0
35 124
x = 0.631 0.61 0.586
24 32 87
5.4 6.0 7.0
25 36
CePd 3
20
10
1.85
CeSn 3
15
10
2.9
CeA! 3
35
50
1.6
Ni~_~AIx x = 0.26 0.256
Nil_~Pt x
UPt 3 YbCuAI
I00
50-60
60
1.1
-90
-35
6.8
ITINERANT
INTERMETALLICS
AND
ALLOYS
341
of the two contributions is different in the SEW model and, hence, also in general. We have now two options: the first option is to consider the second contribution as a separate one indeed. The second option is to consider ~F as one entity, and to impose a temperature scaling according to the first contribution, neglecting the temperature dependence of the second contribution altogether. According to the first option the magnetic field scales with H* = ( W T s / S ) , in the SEW model H * ~ z T * ( S J S ) 1/4. According to the second option we find H* oc T*(Se/S) 1/2. In case S~ ~ S, we have in both options H* oc T*, or/~ = F. This situation arises, on the one hand, when the enhancement can be ignored ( S ¢ - S = I ) , like in the transition metals. On the other hand, according to Lonzarich (1986) one should expect Se (i.e., the mass enhancement m * / m in the specific heat) to be proportional to S also in the case of a large enhancement due to spin fluctuations• Quoting Lonzarich: 'In this case spin fluctuations may be described in terms of Fermi quasiparticles having high mass and vanishing residual (Landau antisymmetric) interactions in the low-T limit'. Kaiser and Fulde (1988) have applied eq. (53) in order to calculate/~ (O H in their notation) from magnetostriction data and the experimentally observed values for the susceptibility. The Gr/ineisen parameter F (OT in their notation) was calculated from the experimental values for ~, 3~and/3. F and F appear to be large and roughly equal in the heavy-fermion and the mixed-valence systems (see table 2, extracted from their work)• Returning to the strongly enhanced paramagnetic systems under consideration we reformulate eq. (53) in the form, I" = ½F + C/A o = ½F + ½ O In Xo/Oto ,
(59)
where C is the magnetovolume parameter [eq. (5)]. Alternatively, = F + ½d l n ( S / S ~ ) / d o .
(60)
Since d l n S / d w = F A ( S - 1 ) , as stated in eqs. (14) and (15), one expects to calculate very high F-values in case the Stoner enhancement is strong (i.e., when A 0 is small). The enlargement of the electronic (thermal) Griineisen parameter, governed by d In Se/doJ, may well be much smaller. In this section, we have used the Grfineisen parameter F as defined in eq. (52), 0 0 0 so given by F = 3Vaexp/KTexp. Here, aex p is the coefficient of the term proportional to T in the paramagnetic zero-field linear expansivity [see eq. (93)]. In • 0 0 section 3.2 we discuss the influence of the fluctuations on aex o and YCxpFo r a consistent discussion, e.g., on the basis of eq. (60), one should use the 'electronic' terms a ° and y ° in the calculation of the Griineisen parameter F~. This consideration shows once more that the 'magnetic' Grfineisen parameter is not directly related to (the volume derivative of) a physically relevant energy parameter. Nevertheless, the descriptive power of the presentation offered by Kaiser and Fulde is very appealing.
342
RE. BROMMER and J.J.M. FRANSE
2. Arrott plots
2.1. Characterization The so-called Arrott plots, i.e., plots of M e versus tzoH/M, are widely used in the analysis of magnetization measurements. In the simple Stoner model, discussed in section 1.2, the Arrott plots are expected to be straight lines, whereas it was shown in section 1.3 that extra contributions originate from the presence of fluctuations and of inhomogeneities [compare eqs. (2) and (31)]. These extra contributions cause a shift in the 'Landau'-coefficients A .and B, and also deviations from linearity. Experimentally, the Arrott plots are, indeed, often found to be curved. In the following sections we discuss four characteristic types: (a) no, or negligible, curvature, both at T < T c and at T > Tc; such a behavior is expected for homogeneous materials far from the critical region (section 2.2); (b) negative curvature at T < Tc, positive curvatures at T > Tc; inhomogeneous alloys (section 2.3); (c) negative curvatures both at T < T c and T > T c are expected in the critical region (section 2.4); (d) positive curvatures both at T < T c and T > T c are found in some systems consisting of local (foreign) moments embedded in a susceptible matrix (section 2.5).
2.2. Arrott plots for homogeneous materials For homogeneous materials, the Arrott plots are given by the equation of state, eq. (31). Away from the critical region we assume the following expansion for the mean fluctuations, 3m~ + 2m~ = 5m 2 - a2M 2 + oQM 4 .
(61)
A similar approach was applied by Wagner and Wohlfarth (1986), see also Wohlfarth and Mohn (1988). Adding a, presumably small, sixth-order contribution, ~ CM 6, to the free energy we find the 'Arrott plot relation',
txoH/M = (A + 5Bm 2) + B(1 - a2)M 2 + (C + Bot4)M 4 .
(62)
Because of stability requirements, we assume that C + B a 4 is positive; otherwise, higher-order terms should be taken into account. Consequently, the Arrott plots represented by eq. (62) may show a slight negative curvature which, near the critical region, grow into the pronounced curvatures to be discussed in section 2.4. At zero temperature, however, fluctuations are not present, so the Arrott plots should be quite straight (if C is small enough), yielding the low-temperature values of the Landau coefficients A and B. As an example we show in fig. 2 the magnetization data obtained on some Ni3AI compounds at two different pressures. The deviations from linearity at the lowest fields are typical for such
ITINERANT INTERMETALLICSAND ALLOYS I
I
I
I
J
Ni.~ AI o)
I
I
343
I
v 1 bar o 4.05 k b a r
150
76 % Ni E
b
. . v _ V - T ~ o - O "~
loo
w~V" ...O~O'v .V ~ - ~ 0 - '~-6 -o'°
75.5%
~v.v~ _ _ ..Vo-_%~
Ni
75 % Ni
vo -ydY~ "°'° -
50 ~ 0 ~
0
I
v.: .y6:VO~%'.VO'-%~'6" O'v
I
I
I
]
0.5 ~o H/ff
I
N
I
I
(Tkg/Am 2)
Fig. 2. Arrott plots for some Ni75+xAl:5_x alloys at 1 bar and at 4.05kbar. Data (at 4.2 K) from Buis et al. (1981a).
measurements. These deviations are usually ascribed to inhomogeneities (see also next section), probably in combination with demagnetizing effects. They are ignored in the determination of the parameters A and B. These experiments, however, were performed at relatively low fields (<5 T). Data obtained at higher fields (up to 20 T, De Boer 1969) yield a somewhat different slope for the ferromagnetic alloys. An analogous comparison of low-field (Kortekaas and Franse 1976) and high-field data (De Boer, 1969) for the paramagnetic alloy Ni74AI26, reveals much larger discrepancies. In the next section it is shown that these discrepancies are due to the combined effect of a few ppm Fe and a variation of the Ni concentration over the sample (say 0.1% Ni). In the analogous Ni3Ga-system even more intricate deviations were observed (fig. 3)' and (partly) ascribed to an anomalous magnetization dependence of the effective interaction (Schinkel et al. 1973), or even to many-body effects leading to a correction proportional to m 2 In M in the Arrott plot (Barnea and Edwards 1977). Corrections for the Fe-contamination or inhomogeneities, however, were not performed consistently. Barnea and Edwards recognized such corrections, but suggested that the deviations were mainly caused by many-body effects leading to a term proportional to M: In M in the Arrott plot relationship (see also Yamada 1975). Since the deviations from linearity are very small anyhow, we conclude that, especially at the higher fields, the magnetization data of these compounds can be described satisfactorily by Arrott plots for homogeneous materials. One should be
344
P.E. BROMMER and J.J.M. FRANSE I
I
I
/
% 50
I
//
i
Ni3Ga
o o
ff
I
75.0
75.3o 1oo
I
I
7/,.5 7/..0
o/
[i 5
o
~oH/0"
( T k g / h m 2}
Fig. 3. Arrott plots at 4.2 K for some Ni75+xGa2~_ x alloys. Data from Schinkel et al. (1973).
aware, however, t h a t - the other way a r o u n d - a straight Arrott plot cannot be considered as proof for the material to be homogeneous (see also section 2.5).
Further analysis of the experimental results The Arrott plots yield the pressure-dependent coefficients A(P, T) and /~(P, T) by fitting a straight line through the data points, IxoH/M = ffl(P, T) + B(P, T ) M 2 .
(63)
In order to find the volume-dependent Landau coefficient B(V, T), or rather the coefficient B(1 - a2) in eq. (62), one should perform the correction, B(1 - a2) =/~ + ~
1
(Offl/OP) 2 .
In general, however, this correction appears to be negligibly small. At zero temperature the coefficient A ( = / i ) immediately yields the value ( g - a - I ) [see eq. 16)], whereas its pressure derivative yields the main contribution to the magnetovolume parameter (see eq. (5): KC=--½KOA/OoJ= ½0A/OP). The further derivation of the fundamental parameters (F, h and I) was discussed in section 1.2. Again at zero temperature, according to the band model [see eq. (8)] the coefficient B yields 3 u 2 - v2 g3
3 4 48NA/zB/4 •
ITINERANT INTERMETALLICS A N D ALLOYS
345
In order to check the applicability of the SEW model to Ni3Ga , De Chfitel and De Boer (1970) combined this result with a value ( v ~ - u2) estimated from the temperature dependence of the coefficient A [using eq. (7)] and an estimate for the density of states g. They concluded that the contribution which is proportional to I 4 should be taken into account, indeed, in order to obtain a physically meaningful result (i.e., u 2 > 0). Hilscher and Gratz (1978) obtained an estimate for 3 v ~ - v2, by an analysis of the temperature dependence of the resistivity in Ti(Fe,Co) and Zr(Fe,Co). Neglecting a contribution proportional to I4, they calculated the density of states from the observed slopes of the Arrott plots (Hilscher et al. 1977), and compared these results with experimental values obtained from specific-heat measurements (Hilscher and Gmelin, 1978). Essentially using low-temperature data only, Buis (1979) showed that for the Fe-rich Ti(Fe,Co) alloys the strong concentration dependence of the slope is consistent with the observed concentration dependence of the density of states (see fig. 4). This alloying system is discussed from another point of view in section 4. Much time and effort have been put into the determination of the Curie temperature and its pressure derivative. Indeed, in the SEW model various expressions have been applied in order to demonstrate the fact that d In Ts/d~o diverges for T s $0. Conversely, such a divergence is taken as evidence for the itinerant character of the material. Wohlfarth's (1969) expression can be written as din Ts/dP= KCst(Ts)/A o (notice that this expression assumes very large negative values because A 0 is negative and tends to zero for Ts$0 ), where
I
I
I
I
Ti (Fel_x Co x ) ¢./
I
I
I
xe"
5/
/
/
~' \ " I \
E \
E 20~_
m
,,B
¢o i0
/ / /
~// +
/
--10
/ /
7/ i
TiFe
¢ 0
i
i
i
I 50
I
i x
(%)
I
TiCo I"~0 100
Fig. 4. The concentration dependence of B, the slope of the Arrott plots, and of % the coefficient of the linear term in the specific heat, for Ti(Fe,Co) alloys. ( + ) B - 1 / 3 , data from Buis (1979); ([~) B -1~3, see table 10; (O) % data from Buis (1979); (Q) y, H61scher (1981), see table 9.
346
P.E. B R O M M E R and J.J.M. FRANSE
Cst(rs)
is the value of the magnetovolume parameter at the Stoner temperature (i.e., at T¢ in the SEW model). In the notation introduced in section 1.2 we find (see Brommer 1982a), d In Ts/dw = ½rh(1 +
Isl)
- r~,
showing explicitly the proportionality to the enhancement factor S. In all these considerations, however, the influence of the fluctuations was neglected [compare eq. (64)]. Instead, the approach of Moriya (see section 1.3) was based on the assumption that the fluctuations would dominate the temperature dependence of the (inverse) susceptibility. Since in the neighbourhood of T c, the fluctuations are proportional to T 4/3 it was suggested (Moriya and Kawabata 1973, Wohlfarth 1980b) to use the relation T~= T~(/-1), where q = 2, 4 or 1 in the Stoner, Moriya or Doniach-Murata model, respectively. For a series of Ni3AI alloys a plot of In T c versus In [g-1 _ I1 (Buis et al. 1981a), as well as a plot of d i n T c / d P versus [a0[ -1 [or versus KC/[Ao[ (Brommer 1982a)], indicate a value q = 1.5 or 1.6. In order to give an interpretation of this value we follow, e.g., Lonzarich (1986) and combine eqs. (11) and (62) and Moriya's Ansatz m 2 oc (T/TF) 3/4 into, Aeff(Zc) =
A o+
A b a n d ( T c / T F ) 2 q- B ' ( T c / T F ) 4/3 = O.
(64)
Calculating q from the relation q d In T c / d P .~ d i n [Ao[/dP, we obtain the estimate q = ½(4+2p), where p is the fraction Aband(Tc/Tv)2/[Ao[ = ( T c / T s ) 2. Notice that p = 0 (q = 4/3), corresponds to the 'Moriya' case and p = 1 (q = 2) to the SEW model. The observed q-value between 1.5 and 1.6 corresponds to a decrease of [A(Tc) [ with respect to [A0[ of 0.25 < p <0.4. Another way of studying the temperature dependence is to investigate the concentration dependence of T c in an alloying system. From eq. (64) we infer that a small variation of Aband , B' or I will have little or no effect on the temperature-dependent terms (because T / T v ~ 1), whereas the same variation of Aband o r I has a profound effect on A 0 = Aband -- I. Indeed, a plot of A ( T , x) versus T for different values of the concentration x shows, in general, a series of curves which are shifted in the vertical direction only. Examples are Ni3A1 (De Boer 1969), Ni3Ga (Schinkel et al. 1973), and (Ni,Pd)3AI (Sato 1974, see fig. 5). The latter author demonstrated the linear concentration dependence of A ( T , x) at any fixed temperature, and subsequently observed that the relation T c o¢(x c x) 3/4 fitted the experimental data much better than the Stoner relation T c o¢(x c x) 1/2. A systematic analysis on the basis of eq. (64) was not performed, but the data as presented seem to exclude a sizeable reduction of A(Tc) for Ni3AI (p -0).
ITINERANT INTERMETALLICS AND ALLOYS
I
I
I
347
I 0
/,, {Nil_ x Pdx)75AI25
Ni60Pd15Ai2/
19.8
/o / ~3
o/
o
""6 0
/ °°
o/
o/
,.,/o
J
3 /
t 2.9h
otO"°oO -°" c~ 0
0
1-0/"
e~10 o~O
0
.0
0/0~0
/0.0,0"9/o,0/ c.O4"°
0
/o
.,~" .0 oo~" /o o.O" clO
25
50
00_
/o/Nie.At25 _/0 I 75
T (K) Fig. 5. The Landau coefficient A(T, x) as a function of the temperature T for some (Nil_xPdx)75Alz5 alloys. Data from Sato (1974).
2.3. Arrott plots for inhomogeneous materials The variation of the local chemical composition, as well as that of the local atomic arrangement can be the origin of the r dependence of the L a n d a u - G i n z b u r g coefficients in eq. (20). In a m o r p h o u s alloys the local atomic configuration is assumed to play an important role (see, e.g., Ffihnle and Kronmiiller 1980). For crystalline alloys, however, we follow Wagner and Wohlfarth (1981), in assuming that the spatial variation of both A(r) and B(r) is related to the r dependence of one and the same concentration p a r a m e t e r , c(r). We define c to be the concentration averaged over the sample, and introduce the Fourier expansion
(c(r) - c)/c = ~ 7q eiq'" •
(65)
q
F u r t h e r m o r e , we use the expansion
A(r) = Ao + A' -c(r)- - c + ½ A"( -c(r)- - c ) 2 , C
(66)
C
with B(r) expanded in an analogous way. Notice that the average of A(r) equals 1 ~tt 2 A = A 0 + ~,a 7 , where
7 =
( c(,') - c ) c
= ~ Yq%q" q
(66a)
348
P.E. B R O M M E R and J.J.M. FRANSE
Furthermore, we have Aq = A'yq and Bq = B'yq. Inserting these definitions into eq. (30) we obtain, I%H _ A + B ( M 2 + A) - y2fl [ A ' + B ' ( M 2 + a)](A' + 3 B ' M 2 + B ' a ) M A + B(3M 2 + A) 2r [A' + B ' ( M 2 + A)] 2 B M 2
+ 7
a)12
(67)
,
where A = 3m~ + 2m~ indicates the influence of the thermal fluctuations [see eq. (6)] and where,
=E
q 1 + q2~2 ,
(68)
q (1 + q2~2)2 •
(69)
Here ~ is the (longitudinal) coherence length (see section 1.3), i.e., 1~2 = D / [ A + B(3M 2 + A)].
Assuming a white-noise spectrum (Wagner and Wohlfarth 1979, Kronmfiller and Ffihnle 1980, Shapero et al. 1974) for yq up to a cut-off wave vector kc, we have: fl(z) = 4 (z - arctan z ) , z
3(
f2(z) = 2z 3 arctan z
(70)
z)
1 +z 2 '
(71)
where z = kc ~. Since away from the origin in the (tz0H/M, M 2)-plane, k c~ will be very small, it is customary to approximate f l ( z ) and f2(z) by their limiting value 1. In that approximation, we find the Arrott plot ~ o H / M = A + B ( M 2 + A) + ~ A ( B ' / B ) 2 - ~ A ' ( B ' / B ) - - - ~ A y2
- 272B(3M2 + a ) ( B ' / B ) 2 - }72(A + Ba) ( 3 A ' - A ( B ' / B ) + 2B' A ) 2 × \ -A + B(~-M-$ + A) "
(72)
These relations were derived by Brommer (1982b) in the case A = 0, thus under the assumption that the thermal fluctuations would have a negligible influence. Moreover, it was shown that eqs. (67) and (72) do contain various corresponding relations given in the literature (Brommer 1982b). Closely related equations were derived by Yamada and Wohlfarth (1973, 1974, 1975).
ITINERANT INTERMETALLICS AND ALLOYS
349
For so-called locally responding magnetic systems, the response of a magnetic site to the local effective field does not depend on the local atomic arrangement (e.g., Heisenberg type). Under a number of simplifying assumptions discussed by F/ihnle (1980), the resulting local Landau coefficients can be written as,
A(r)-
Aloc
c(r)
J'
B(r)-
Bloc
c(r) 3 ,
(73)
where A~oc, J and Bloc are now assumed to be constant [compare eq. (21)]. The relevant parameters become A o=(AloJc )-J, B o = Bloc/C 3 ,
A'=-AloJc
B' = - 3 B 0 and
,
A"=-2A',
B " = 12B 0 .
Inserting A = A o + ½A"7 2 and so on in eq. (72), and assuming A ~ J we find,
I % H / M = A o + B0(M 2 + Zl) - j y z _ yZ(A ° + B0zl ) A0 + B0( 3M2 + zl) (74)
Remarkable features of this model are that the high-field slope (Bo 1) does not depend on the concentration variations, and that the concentration variations do cause a shift of the high-field curves to the left ( - j 3 2 ) , consequently an apparent shift in the critical temperature of • T c = 3' 2J( OA e~f/OT) -1 with respect to that of the perfectly homogeneous alloy. In the Heisenberg model the shift is ATc/ T c = 32, (see Brommer 1982b). The general behavior of the curves is well expressed by i z o H / M = a + b M 2 - 3"2aj2(a + 3bM2) -2 ,
(75)
where a and b are effective values. The general behaviour is shown in fig. 6; the curvatures are negative for T < T c and positive for T > T c. Similar curves were calculated by Shtrikman and Wohlfarth (1972) and by Yamada (1975). In case the local response model is not appropriate, eq. (72) may be applied. The general behavior, however, is very much the same. A quantitative estimate can be obtained by neglecting A", B" and B', and using the experimentally observed quantity aA/Oc: 3"aA'2 is replaced by ~c:(OA/Oc) z. In the paramagnetic region, the deviation on the tzoH/M-axis is ( 1 / A ) ( O A / o c ) a S c 2. In the ferromag~. In this way, ne___tticregion, the decrease of M e at zero field is z~c 1 aMo(OA/Oc)Z/A a ~c 2 was estimated to be roughly (0.1% Ni) a in Ni3AI and NiPt alloys (Buis et al. 1981a, Kortekaas and Franse 1976, see fig. 7). Brommer (1982b) reported a thorough check performed on a Ni74Ala6 sample prepared carefully by Schalkwijk. After correcting for the presence of an internal field of 0.1 T due to some ppm Fe (as discussed in section 2.5), the resulting
350
P.E. B R O M M E R and J.J.M. F R A N S E I
I
I
o=
"E -~ 3~
-1 a=O
..Q i1
2
1
-
,4 0 0
I 2
1
3 x : htMl"l ( o r b . u n i t s )
Fig. 6. The influence of inhomogeneities on the Arrott plots. Equation (31) is writte.._n_nin the form x = a + y - (yJ)2a/(a + 3y) 2, for a equal to 1, 0 and - 1 , respectively. (3,J) 2 = 0.5 = 6c 2 ( a A / O c ) 2 .
I
I
I
'
Ni3AI 150 2£ I"M
76.0 % Ni
E
<
1.0
.0~0_0_0~0 -0-0-0"0 ~
b
1.2
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15
100 o o" ° ' ° - 7 5 " 5 _ 0_0~0
50
0
1.5
0
~,,,~ 7.~_..o~o_o_O~ o_o_O_o.OS6.o ~i~ 0.5 o
~0~0~0~0
0
I 0
~0~0
I 0.2
o ~°~°-
1.7~.
i
i
5
10
0.6 -
7~'8
~
u
I I 0.4 P.oH/O" (Tkg/Am 2)
Fig. 7. Arrott plots for Ni3AI. Deviations from linearity are ascribed to variations in the Ni concentration. Data (at 4.2 K) from Kortekaas and Franse (1976).
ITINERANT INTERMETALLICS AND ALLOYS Arrott plots were very well described by eq. (75), up to a field of 40 T. Again was found to be roughly (0.1% Ni) 2.
351 ~C 2
2.4. Arrott plots in the critical region In the critical region, the fluctuations grow to such magnitudes that the LandauGinzburg approach breaks down; scaling laws and renormalization group-techniques should be used, but lie outside the scope of this review. In the LandauGinzburg model itself, the range of validity can be estimated self-consistently, e.g., by the demand that the fourth-order fluctuations in the free energy are smaller than the second-order ones (Menyh~ird 1970). Hargitai et al. (1972) proposed to demand that the influence of the fluctuations in the magnetization is relatively small in the expression for the longitudinal susceptibility [see eq. (26)]: A + 3BM 2 > BA. The intersection with the Arrott plot (eq. (31)) yields,
tzoH/M + 2BM 2 > 2BA. Assuming a parabolic band, Hargitai et al. (1972) estimated the fluctuations to be given by B A = (8A/OT)(O.O7Tc) , and thus defined a 'critical triangle' in the (l~oH/M, M2)-plane. Another estimate would be to replace A by its zero-field value at T c, or BA = IA(Tc)[-- (1 - p ) B M ~ . Here p again indicates the fractional decrease of IA(T)I at the critical temperature (see preceding section). The straightness of the Arrott plots is consistent with the estimate by Hargitai et al. Fluctuations seem to have little influence on the Arrott plots, perhaps indicating that the decrease of [ A [ - i.e., the fraction p - is sizeable. Franse et al. (1981) considered the Laves phase compound UFe 2 as an example for an itinerant ferromagnet exhibiting critical phenomena (fig. 9). Since M 0 = 6.2Am2/mol (Aldred 1979), T c = 160K and O A / O T = O . O 6 T m o l / K A m 2, the data points fall well inside the critical region by either criterion. In fact, polarized neutron measurements indicate magnetic moments on the Fe and U sites of 0.59~B and 0.06/ZB, respectively (Lander et al. 1977), so the magnetic properties can be considered to be indeed, determined by the itinerant 3d electrons. The drawn curves in fig. 9 represent the approximation to the exact equation of state proposed by Kouvel and Comley (1968), here written in the form
IxoH/M = b
[T-Tc ( M )1/~] ~ T~c---+ - ~ e "
(76)
This equation yields the proper critical exponents in the limiting cases, X -1 -- /z°H M M=0 ~ ( T - T c ) ~
MIH=o o c ( T - T c ) t~,
for T > T c ;
T
whereas for T = Tc we have M o: H 1/~, with the allowed value 8 = 1 + 3,//3.
352
P.E. B R O M M E R and J.J.M. FRANSE I
I
I
I
I
I
I
i
I
'
4.2K •
10
/,10.2
N'42.9 Pt 57.1
/00/
.
16 5
/
,o/
/of/,o _ / o
/22.5 .y" / o /27.4
o°
<E b
o / o/
o/O"
/
o _o
/
/
/ /
/ . o /
o'/°/°/,
,
/
/.o
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{
°
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_o o/
__ /~uK
o/O" o,/O/ {
I:/,°://// L/ o /
o/
,
/
o
, ,
0
0.5
,
,
,
1.0 IJ,oH/O" (Tkg/Am 2)
Fig. 8. Arrott plots for Ni42.gPt57.1. Data from Kortekaas and Franse (1976). Both Hargitai's critical triangle (hatched left comer) and the line txoH/M= 2B(M20 - M 2) are indicated.
I
t~4
I
I
I
I
,
i
,
U Fe 2
~ E
,..+/ 154K z,,~ +.-+ ..+"
<E
+. ~57
,/+ +"
-
+ /.+/
,+/
5 /
+
÷/
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+/"
/
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,÷
+
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,,,+- 160 /+/
/'~
~,~+
/+" /+-+
+/ ,+
....K
/+/+
V
÷/ 0.0
I
0
I
I
+
~
I
I
~
0.5 I-I.oH/N1 (T tool U/Am 2)
Fig. 9. Arrott plots in the critical region for UFe 2. Data: Franse et al. (1981) or Frings (1984). The drawn lines represent the approximation of eq. (76), with y = 1.26 and fl = 0.46 (Aldred 1979). The other parameters are given in table 3.
ITINERANT INTERMETALLICS AND ALLOYS
353
Moreover, eq. (76) has, just as the fundamental equation of state, the property that the reduced plots of versus h / m = l % H / M [ t l ~, with t = ( T - T c ) / T c , show two separate branches for T < T c and T > Tc, respectively. The limiting values, m 0 and h / m o (in Kouvel's notation), obviously correspond to our parameters f M e and b, respectively. In fig. 9, the values for the critical exponents are 3' = 1.26 a n d / 3 = 0.46, so 6 = 3.7, and were taken from Aldred's work (1979). In the fundamental equation of state, M 0 is a factor f times the (fixed) local atomic moment. The constant f is model dependent, e.g., f = 1.732 for S = 0.5, the 'molecular field' model, f = 1.49 for S = 0.5, the 'Ising' model (in the Stoner model f = X / 2 = 1.41 and 3' = 1). In the itinerant systems under consideration, we identify M 2 with the zero-field mean-square local moment at the critical temperature. According to the considerations in section 1.3, we have M2e = m 2 = ri M2st = 71[- A ( Tc ) / B ], where we expect ~/= -~. Moreover, we define again M2t = (1 -p)M2o (see section 2.3). The pressure derivative OM2/OP is taken to be equal to - 2 K C / B I ~= 7~c= - 2 K C / B l r = o = aM2~ OP, where we assumed that the magnetovolume parameter C does not vary strongly with T. Experimentally, one determines f M e and its pressure derivative, as well as the zero-temperature magnetic moment M 0 and its pressure derivative. Subsequently, we calculate f and ~3(1- p) from the identity,
m2=M2/lt[2~
f 2=(fm'g'~20ln(fme)/OP \M 0/
OlnM0/0P
(77) '
and from 0 In M o / a P 77(1 - P ) = 0 l n ( f M e ) / O P
(78)
"
Notice that the calculated f-values increase in case the magnetovolume parameter is smaller at T c than at zero temperature. The experimentally determined values are given in table 3. The large values of the relative pressure dependence show the itinerant-electron character of the 3d-moments. From these values, we find from eq. (77), the very reasonable value f = 1.8---0.2 and, from eq. (78), ~ / ( 1 - p) = 0.55 --- 0.1. Hence, taking 7/= 3 ( p = 8%), we conclude that the reduction of the magnetic moment at T c would be only TABLE 3 Experimentally determined parameters for U F e 2 in the critical region. The parameter (X) and its pressure derivative are given. X
units
Mo Tc fM e b
A m2/mol K AmZ/mol Tmol/Am 2
0 In X/OP (Mbar-1) 6.22 + 0.02 160.0 +0.1 8.44-+ 0.08 19.2 +-1.5
-4.8 -+0.5 -3.2-+0.3 -8.5 + 1.3 -16 +5
354
RE. BROMMER and J.J.M. FRANSE
4% (compare, e.g., to MnSi in table 5). In conclusion, the experimental data appear to be described very well in this picture, in which the familiar equation of state derived for a system of local moments now is extended to describe an itinerant-electron system. We remark, however, that the Arrott plots in fig. 9 and the observed critical exponents (in particular 3 = 3.7) happen to be very similar to those shown by Herzer et al. (1980) for an amorphous alloy (Fea0Ni40PlaB6). Obviously, the micromagnetic theory used by these authors is able to reproduce the observed features in the critical region. Consequently, it would be interesting to study the assumptions made in our approach in the framework of the micromagnetic model. 2.5. Arrott plots for local moments in a polarized matrix
In this section, we discuss the Arrott plots for a binary system in which the atoms of the one kind form a paramagnetic polarizable matrix, whereas the atoms of the other kind (the 'foreign' atoms) carry local moments. The s-d-model (Takahashi and Shimizu 1965) describes such a system. Acker and Huguenin (1979) proposed an analogous model for PdNi alloys (where the local moments are formed by clusters of Ni atoms only) and showed in particular that rather straight Arrott plots may be expected in spite of the obvious inhomogeneity. Here we focus attention on a system in which relatively few foreign atoms form so-called 'giant moments' by polarizing the surrounding matrix. At low temperatures, these giant moments are already fully oriented in relatively low fields. The averaged molecular field exerted on the matrix is written as jfdMf~, where Jfd is the molecular field parameter, coupling the itinerant d-electron magnetic moment to M~, the (saturated) magnetic moment of the foreign atoms. Ignoring thermal fluctuations as well as the variation of the magnetization in the neighborhood of the foreign atoms, and denoting the total magnetic moment by Mr, we write the Arrott plot as, tZoH + jfdMf~ M t - Mrs
-
A
+ B(Mt
-
Mrs) 2 .
(79)
This relationship was applied to the analysis of the magnetization data observed in Ni75_xGa25Fe x in the pioneering work of Schalkwijk et al. (1971). The observed Arrott plots show a positive curvature, as mentioned in the introduction to this chapter (see fig. 10). In the Ni3Ga + Fe system it appeared to be possible to determine the foreign (Fe) moment and the molecular field parameter, Jfd, in the following way. For low fields and low Fe-concentrations, the higher-order terms in eq. (79) can be neglected, so the magnetization curve is linearized to, Mt = tZoXH + Mes(1 + xJfd),
(80)
where X = A-1. By extrapolation down to vanishing field, the effective moment
ITINERANT INTERMETALLICS AND ALLOYS I
I
I •
O
Iii
100 --
355
I
[] 0
Ni75_xGa25 Fe x
/
cM •
,-,,T E 80
•
0
• o 7 A Ni 75Ga25
<
•
% 60
0
"Z
--
"Z
40 x=0.1
/
20
[] 0.05
•
-
O/ /
•
0 0
I0.OAo 1
I 2
I I 3 4 IJ.o H/o- [T kg/Am 2}
Fig. 10. Reconstruction of the Arrott plot for 'pure' Ni~sGa25from the Arrott plots experimentally observed in some Ni75_xGa25Fexalloys, [see eq. (79)]. The internal field and the Fe-magnetization follow from J~d= 330 T mol Ni/A m2 and PTo= 1.9/zB- Solid symbols: experimental data from Schalkwijk et al. 1971. Open symbols: reconstructed pure Arrott plot. per foreign atom, ]'~eff, was determined for a variety of Ni concentrations, i.e., for various x-values. We have, /.t~ff = /.t,ze(1 +
XJfd)"
(81)
Consequently, one determines /'q:eJfd from the slope in fig. 11 (fairly accurately) and /XFe from the intercept of the /zeff-axis (rather roughly). Hence, also Jfd is known. From the results for the Ni3Ga + Fe alloys, given in the caption of fig. 11, we find that the effective field/%Her f acting on the matrix equals x(47 T) in an alloy with x % Fe-atoms. An analogous analysis was performed on some Ti(Fe,Co) alloys. In fig. 12, we show the reconstruction of the Arrott plot for the 'ideal' matrix from the high-field data observed on TiFe0.sCo0.5, at a temperature low enough for the foreign moments to be saturated. Here, the localized moments are thought to be connected to so-called anti-structure Fe-atoms. In section 4 this system is discussed in more detail. In particular, it is shown that the (paramagnetic!) susceptibility of the matrix can be determined for Ti(FeCo) by extrapolating the data observed on a series of off-stoichiometric compounds. This independent determination has been used in the present analysis, and, in fact, is indispensable, because the 'straightness' of the reconstructed Arrott plot is not a sharp criterion at all.
356
P.E. B R O M M E R and J.J.M. F R A N S E I
I
I
40
LL
=~30 O
:::L
j
20
+ Ni74 AI26
O
oJ
10 /
/
Co_
0
/ /
0
,/
,
,
I
I
I 0.05 ( A m 2 / T m o l Ni)
X=A-1
Fig. 11. The effective moment per Fe atom, /z, ef, as a function of X for various Ni3Ga alloys (©). Values for Fe in Pd (O) and in Ni3A1 ( + ) are given too, as well as the effective moment per Co atom in Ni3Ga (A). Data from Schalkwijk et al. (1971). From the slope (/zFejfa) and the intercept (/zFe) one finds: /zvo = 1.9/xB, Jfd = 330 T mol Ni/A m 2 [=27 T kg/A m2].
I
I
I
Ti Fe0 5 Co0. 5
" 200
jo /
o,
/o O4
/
b
/ o -°/ ° , , i d
eo I ,,
2
3
100 0
0 0
1
IJ.oH/O" [Tkg/Am 21 Fig. 12. The reconstruction of the Arrott plot for 'ideal' TiFe0.sCo0 5. Filled circles: experimental data at 4.2 K from Buis et al. (1981b). Open circles: reconstructed Arrott plot with extra internal field 10 T and trv~ = 0.5 A m2/kg (see section 4.3).
ITINERANT INTERMETALLICS AND ALLOYS
357
Finally, we want to comment on the meaning of the 'giant moments' defined in this section. In case the foreign moments are oriented, it doesn't really matter whether we define a low background polarization on which is superimposed a number of polarization clouds around each foreign atom (the 'giant moments'), or an averaged matrix polarization, caused by the effective field due tO the oriented foreign moments. The notion of 'giant moment' is only important at higher temperatures. Then, we imagine that each local foreign moment forms one entity with its induced, strongly correlated polarization cloud. This entity induces a 'long range' matrix polarization, and, thus, is coupled to the matrix and to the other giant moments. The important thing to realize is that the correlated polarization cloud does not influence the orientation of the giant moment itself. Hence, the effective 'long-range interactions' are much smaller than the molecular-field parameter, Jfa, would indicate. Furthermore, at high fields the response of the matrix is non-linear. All these features can be taken into account in the following way. Defining a uniform background matrix polarization, Mu, a saturated correlated polarization, Mc~, induced by the (oriented) foreign moments, i.e., by Mf~ = N ~ ( g S l ~ ) , we split the original Arrott plot relation in three parts, Mt = Mu + (Mcs + Mfs)Bs ( Z + Jlr -[- 3nM2)Mcs
( [.~OHeffgeffSlJ,B ~
+ nM3cs
( A + j~ + 3 B M c2s ) M u +
kT
/ '
= (Jfd -- Jlr)Mfs ,
BM3u=
~ 0 a e f f = /x0H + J l r M t ,
(82)
(83) (84)
where geff/g = (Mfs + Mcs)/Mfs , ]lr is a long-range molecular field parameter, for simplicity taken to be the same for the matrix-matrix and the matrix-foreign moment interaction and Bs(x ) is the Brillouin function for the spin quantum number S. At saturation, the matrix polarization is ( M t - Mrs ). The sum of eqs. (83) and (84) yields eq. (79), valid in case all foreign moments are oriented. Linearization of eqs. (82) and (84) yields the following expression for Tc, T c = Nf
g~ffS(S3k-}- 1)l£ 2
Jlr[ 1 nt- j~r(A
+ 3BM2~)-I l .
(85)
This expression has, of course, the salient feature that the critical temperature is governed by the reduced long-range coupling Jlr instead of by the full coupling jfa. Notice that in this expression, a direct long-range interaction exists between the foreign moments (i.e., Jlr). If we suppress this direct interaction the resulting T c is smaller [see eq. (118)]. We tested the applicability of this approach by reproducing successfully the Arrott plots at 4.2 K for three Ni75_xGa25Fex compounds with x = 0.05, 0.2 and 1.0, respectively (data from Schalkwijk, 1982, private communication). At this temperature, M~ is saturated, so, in fact, the analysis shown in figs. 10 and 11 was extended to very large Fe concentrations, but with one set of parameters A and B, together with the obtained J,d- and/XFe-Values (so S -- 0.95).
358
P.E. BROMMER and J.J.M. FRANSE
Moreover, the Arrott plots near the transition temperature could be reproduced for the alloys with x = 0.05 and x =0.02, adopting a quadratic temperature dependence of the parameters A and B. The test failed for the composition x = 1.0, probably because the extrapolation to the transition temperature (90 K) is not justified. As in this analysis the thermal fluctuations were ignored, we feel that a more searching reconsideration is necessary. Nevertheless, we conclude that the picture given, in particular the difference between a 'correlated giant moment' (with spin S) and the 'saturated induced moment per foreign atom' is very useful. In the analysis of magnetovolume effects in the next section an analogous picture will be applied. 3. Magnetovolume effects
3.1. The magnetic volume in homogeneous materials In this section, we discuss the analysis of various experiments in which the 'magnetic volume' is involved. In section 1 we derived expressions for the magnetic pressure, Pm (eqs. (33) or (38)). Denoting the magnetovolume parameter in the Stoner model by Cst = - ½ A o,, we find now, tOm(T, H ) . tOm .
KPm .
(KCst/V)( . . m2 + m2)
(K Cst/V)(Mloc). 2
(86)
The actual magnetovolume parameter is
Om2 C=K IOV/OM2 pT:Cst(l+~-~ PT)"
(87)
Since at finite temperature OmZ/OM2 is expected to be negative, the magnetovolume parameter is expected to decrease with increasing temperature, even in case Cst is taken to be equal to its zero temperature value Co = C(T = 0). At T = T c a rough estimate would be omZ/OMZ<=- 3 , corresponding to a z = 1 in eq. (61), so C(Tc)/C o_->0.4. In the Stoner model, however, a reduction of comparable magnitude can be expected (Brommer 1982a), in case the enhancement is not excessively strong. Consequently, we refrain from a further analysis of the temperature dependence of the magnetovolume parameter although the actually observed decrease is indeed of the order of magnitude estimated above (see table 4). Notice, however, that Suzuki and Masuda (1985) give an explicit expression for the temperature dependence of the magnetovolume parameter in the Moriya self-consistent renormalized spin-fluctuation model. They claim that in Ni3A1 the temperature dependence of C is described satisfactorily by their expression. The magnetovolume parameter can be determined by measuring the pressure dependence of the magnetic moment or, by virtue of the Maxwell relation OV/OlxoH= -OM/OP, very accurately by measuring the forced volume magnetostriction and the magnetic moment (preferably on the same samples). In fig. 13
ITINERANT INTERMETALLICS AND ALLOYS
359
TABLE 4 The magnetovolume parameter (K Cwt = tOm/O'0), 2 the specific magnetic moment (tro), and the Curie temperature (Tc) of some NiyAll00_y and NiyPtl00_y alloys. Data from Kortekaas and Franse (1976). KCwt(10 6(kg/A m2)2)
Tc
o"o (A m2/kg)
r =0
r = rc
(K)
0.63 0.56 0.54 0.63 0.64 0.67 0.76
0.42 0.36 0.39 0.40
30 43 59 72
4.5 4.0 3.92 3.47 3.32
3.37 2.18
14 55
Ni3A! 74 74.2 74.4 74.8 75 75.5 76
4,5 6.6 8.7 10,3
NiPt 36,9 39 41.4 42.9 45.2
_
8-
-
/
NiPt 4 ~2.90/o Ni Jo45.2 -
°
Ni3A I ]
/
]
75.5%Ni
/ 75'2°/°
-
*
1.32 4.2
76
.o-
g
'-l T I' / ? ? -~ O
-
,
0
6
/
o
g
O
_-¢ !:
O
-
O
o
_J .J
2
36.9 --
O
ij¢
0Z,, 0 0.5 0
10
20
/
I
o
60
I
I
80
100
I
120:
748 75%Ni
8 _
,,o
~ 74
°
~,,.,P,
30 0 2 0
o*
o
~ o°
20
: o
,
_
,o , , 40 60
o"2 {Am2/kg)2 Fig. 13. The forced magnetostriction in Ni3A1 and NiPt at 4.2 K. Data from Kortekaas and Franse (1976). The dashed line indicates the results of Suzuki and Masuda (1985) on Ni7~.aA124.s.
360
EE. BROMMER and J.J.M. FRANSE
we show some typical results, obtained on various Ni3AI and NiPt alloys (Kortekaas and Franse 1976). Notice that in these alloys the forced magnetostriction is isotropic within the experimental accuracy, so AV/V= 3 A L / L . In Pd and Pd based alloys, however, the forced magnetostriction appears to be anisotropic (Keller et al. 1970, Creuzet et al. 1981, H61scher 1981). The volume magnetostriction is then determined as the sum of the linear magnetostriction measured in three orthogonal directions, preferably on a monocrystalline sample. From the experimentally determined value of the magnetovolume parameter the magnetic volume at zero temperature is readily determined (see table 4), tOm(0, 0) = ( K C o / g ) m 2 0
= (KCwt)O-2 ,
(88)
where Moo is the molar magnetic moment at zero field and zero temperature. In fact, since M 2 can be determined separately, we can determine m 2 from eq. (86), by measuring the thermal expansion. Clearly, the experimental determination of tOm(T, 0) is of great interest for understanding the model. The value at T = T c can be expressed as, wm(Tc, O) : (K Cst/V)~TM2st(Tc) = (K C s t / V ) r l ( 1 - p)M2oo ,
(89)
where we used eq. (86) and the definition of p given in section 2.4. If eq. (11) holds, we have p = T2c/T 2, and, in our simple description, ~/ should have the value 3. For T ~ T c the magnetic volume is tom(T, 0) -- ( K C s t / V ) m 2 .
(90)
According to Moriya (see section 1.3), the fluctuations are expected to increase with temperature (oc T) until saturation occurs. According to Shimizu (1981), the temperature dependence is more likely governed by the temperature dependence of A. Since the inverse susceptibility A eff can be measured experimentally, Moriya and Usami (1980), or Usami and Moriya (1981), estimated for a paramagnetic material, tom(T, 0) = ~-~ ~
(Aeff - A ) .
(91)
and, in the same way, for a ferromagnetic material for T > Tc, tom(Tc, 0) - tom(Tc, 0) = ~
T
Aeff"
(92)
~m(T, 0) can be determined by measuring the thermal expansion, and subtracting the non-magnetic contribution. As we shall see in the next sections, a combined analysis of thermal expansion and specific heat data is most useful in this procedure. In most cases, the coefficient of (linear) thermal expansion is determined at low
ITINERANT INTERMETALLICS AND ALLOYS
361
temperatures only, in particular the contribution proportional to the temperature T. In that case we write the thermal volume expansivity in the form, [~m = 3 a ~ T + " " ,
i~ = 3 a e T " " ,
(93)
/3exp = 3aexpT + "'" •
Subsequently, we determine (~m / T)T+O = 3am = 3(aexp -- ae) from the experimentally observed (linear) expansivity and an appropriate value for the 'electronic' contribution a e. This electronic contribution can be determined by analyzing the data observed in an analogous paramagnetic alloy, or are estimated from the corresponding contribution to the specific heat (i.e., yeT) using a Griineisen relation. The 'magnetic' thermal expansion up to T = T c is then estimated to be 3 2c (negative!). Hence, equal to ~amT 3 2 t o m ( T c , 0 ) = t o m ( 0 , 0 ) + ~anaT c .
(94)
In a few cases, the thermal expansion has been measured in a wider temperature range. The electronic contribution is now determined by analyzing the data observed in the paramagnetic region (T --- Tc) at such high temperatures that the fluctuations are expected to give rise to a constant expansivity (see, e.g., Ogawa 1983) or even a vanishing contribution (see, e.g., Matsunaga et al. 1982). The lattice contribution is accounted for by introducing a term proportional to the corresponding specific heat in the Debye approximation. Finally, the change of the magnetic volume is determined by integrating the/3 m obtained in this way. In section 3.4, however, we show that the determination of the paramagnetic electronic contribution is not unambiguous at all. For that reason, we indicate in table 5 the results for several choices of a e, using the data reviewed by Brommer and Franse (1984). TABLE 5 Electronic and magnetic contributions to the coefficient of thermal expansion; aeT: electronic; a~T: magnetic. T h e relative magnetic volume tom at zero temperature and at T = T c. 77(1 - P ) = wm(Tc, 0)/ o~m(0, 0) (see section 2.4); p = 1 - M2(Tc)/M2oo = 1 - ~wm(Tc, 0)/OJm(0, 0). a~ (10 -8 K -2) ZrZn 2 T c = 18 K M20 = 0.38 (A m2/mol) 2 a~xv = - 4 . 8 × 10 -8 K -2 MnSi TN=30 K M020 = 4.5 (A mZ/mol Mn) 2 aexp : --8.0 X 10 8 K-2 NiTsAl2s T c = 43 K a~xp = --0.44 × 10 -8 K -z
M~0 = 1.8 (A m2/mol) 2
am (10 -s K -z)
Win(0, 0) (10 -6)
oJm(Tc, 0) (10 -6) 7.4 12 14
",7(1 - p)
p
0.19 0.31 0.36
0.68 0.49 0.40
1.7 0.7 0.2
-6.5 -5.5 -5.0
39
0.793
-8.8
670
370
0.55
0.08
0.00 0.20
-0.44 -0.64
28
16 10
0.57 0.36
0.05 0.40
362
P.E. BROMMER and J.J.M. FRANSE
From table 5 we conclude that in MnSi magnetism appears to be governed by fluctuations [p = 8%, i.e., M(Tc)/Moo = 0.96], while in ZrZn z the Stoner excitations may be important [e.g., p = 4 9 % , i.e., M(Tc)/Moo =0.72]. In Ni3AI, experimental evidence allows both models: p = 5%, M(Tc)/Moo = 0.98 in case a e happens to be small, whereas p = 100% [ M ( T c ) = 0] is obtained in case the high-temperature value, a e = 0.60, is the true value. Suzuki and Masuda (1985) performed an analysis of various Ni3A1 alloys in an analogous way, taking into account the spin fluctuations consistently, in particular, applying eqs. (91) and (92). The actual measurements are in good agreement with the results of Kortekaas and Franse quoted in table 5. The results of the analysis are discussed in the next section (see fig. 16). Finally, we mention also the work of Creuzet and Campbell (1983a,b) on TiBel.sCu0. 2.
3.2. Thermal expansion at low temperatures Spin-wave contributions (order of magnitude) As already stated in section 1, the general theory of spin fluctuations (Lonzarich, Moriya, loc. cit.) leads to spin-wave type excitations in the ferromagnetic region, whereas the phenomenological Landau-Ginzburg approach is less suited for a description of these excitations. Here, we restrict ourselves to an estimate of the order of magnitude of the spin-wave contribution to the thermal expansion, by a comparison with the corresponding contribution to the specific heat and the magnetization. These contributions can be written as (see, e.g., Shimizu 1981), Csw = O.113R(T/O~w)3/2 ,
(95)
AM~w/C~w = 0.70(g/2) A m 2 K / J ,
(96)
V[JswlKcsw = F~w = - d i n O~wldw .
(97)
In practice, we take the g-factor in eq. (96) equal to 2. In the Heisenberg model the spin-wave temperature, 0sw, is proportional to MooJ, where J is the exchange parameter. In the band model rather intricate expressions are found (Shimizu 1981), in which, again, the same combination does occur. As an example, we consider the analysis of magnetization data obtained on some Ni3AI alloys (De Chfitel and De Boer 1970). These authors estimated the possible spin-wave contributions in the magnetization (see table 6). In order to estimate the (possible) influence on the experimentally determined y-values, we calculated the corresponding contribution to the specific heat at 10K. This contribution appears to be less than 10% (table 6). The relative contribution to the coefficient of thermal expansion is negligibly small (unless Fsw assumes very large values). A similar analysis was performed by Beille and Towfiq (1978) on Ti(Fe,Co) compounds. See also H61scher's analysis of these compounds discussed in section
ITINERANT INTERMETALLICS AND ALLOYS
363
TABLE 6 Possible spin wave contributions in Ni3A1 from the analysis, M ( O , T ) / M ( O , O) = 1 - a ( T / T c ) 3/2 - b ( T / T ¢ ) 5/2 - c e ' T 2, as performed by De Chfitel and De Boer (1970). The coefficient a ' is not
reproduced here. 0swfrom eq. (95); y from De Dood and De ChUte1 (1973). flswfrom eq. (97), with Fsw= 2. units a b Tc
Ni75A125
Ni75.5A124.5 Ni76A124
K
0.133 0.280 41.5
Moo
A m2/molNi
0.42
0~w 0~w/Moo (cs,J T)110K y
K arbitrary mJ/mol at K2 mJ/mol at K2
215 512 0.97 6.5
280 483 0.5
334 477 0.4 6.9
lO-s K -2 10-8 K -2
0.02 -0.44
0.015 -0.50
0.01 -0.38
(/3sw/3T)lloK aexp
0.107 0.270 58.1
0.093 0.248 71.5
0.58
0.70
4.3. Spin-wave corrections have been carried out also in the interpretation of the thermal expansion and forced magnetostriction observed in MnSi (see preceding section).
The linear term in the coefficient of thermal expansion The thermal expansion usually is observed as a change of the length L of the sample with respect to the length L* at the reference temperature T* (say 4.2 K). Kortekaas (1975) (see also Kortekaas and Franse 1976) analyzed his data according to, L - L* 1 + B ( T 2 + T .2) A = L , ( T Z _ T , 2 ) = ~aex p
(98)
1 p by extrapolation to the (unphysical) A plot of A versus ( T 2 + T *z) yields ~aex zero o f ( T E + T'E). The slope yields B. Typical results observed on Ni3Al are shown in fig. 14. The upturns observed in the data on the paramagnetic alloys are ascribed to the presence of magnetic clusters, since the corresponding upturns in the specific-heat measurements (De D o o d and De Ch~tel 1973), in particular in Ni3Ga , were shown to be Schottky anomalies rather than 'upturns' (of T 3 In T type). In fields larger than 2 T, the upturns disappear (see fig. 14). Focussing attention on the linear term in the coefficient of thermal expansion, 3aexpT, we see a sharp decrease of aexp with increasing field. In our simple model, we find from eq. (86) a magnetic contribution (in the isotropic case), am = "~m
I T=0
2
=
( dM2 dm2'~ ~(KC/V) ~ + dTZ/r= °
(99)
364
P.E. BROMMER and J.J.M. FRANSE 0.8
o
I
I
I
%OOOOoo
~---0.6-
.~
o
Oo
+4-++
o
o
++
+
o
o
+
o
o
+
J
o7.~./~
+7L,.21
+
0T o °O ooooooooooooooO _
~,
J 2T 0.;
/
-,
7/..0
I
/
I
_
', .6T 0.0
I I
~ -
75
I -0.2
-
~ o o o o o o o o o o o o I
]
-0.~
oo__..~.~oo76--
4.6T
0
I 50
I
I
100
150
T2+L..22 (K 2)
Fig. 14. 'Thermal expansion plotted otherwise', zX = ( L - L * ) / L * ( T 2 T .2) versus T 2 + 4.2 K). Data for Ni3AI: Kortekaas (1975) and Kortekaas and Franse (1976).
T .2
(T* =
It is interesting to discuss at the same time the corresponding magnetic contribution to the linear t e r m in the specific heat, Ym" We assume that in the relevant t e m p e r a t u r e interval, the effective inverse susceptibility, A eff, depends quadratically on T, in accordance with the results of D e B o e r (1969) and the calculations of Lonzarich (1986). In consequence, the fluctuations m~, m .2 and m 2 are effectively proportional to T 2, too. The magnetic contribution to the specific heat is given by, Cm ~=7m
{ dAeff d m 2 -3
dA d T 2 -d-T 2 + 2 d~ T
d
cam2 Ir T 2 dT2J ,
(100)
with
Tm --
dAeff dm2 d T 2 M~ - A 0 d T 2 .
(101)
In eq. (101) the first t e r m on the right-hand side is the generalization of the well-known Wohlfarth correction, giving the correction due to an applied field in the paramagnetic region as well as the correction in the ferromagnetic region. The second t e r m changes sign with A0, thus yields a negative contribution in the paramagnetic region, and a positive contribution in the ferromagnetic region (hence a reduction of the Wohlfarth correction). Its m a x i m u m value can be
I T I N E R A N T I N T E R M E T A L L I C S A N D ALLOYS
365
estimated by neglecting the 'Stoner excitations' in Aeff and adopting, d m2 n OT 2 - ( n + 2 ) B
dAef f 3 dAef f OT 2 - 5B d T 2 '
(102)
with n = 3 in the isotropic case (see section 1.3). According to this estimate, the Wohlfarth correction at zero field in the ferromagnetic region is reduced to 2 of the first term (see fig. 15). Notice that the resulting (zero-field ferromagnetic) correction is equal to, 0 Tm = - A o
dM~t - 2 T----d~ dT2J •
(103)
The corresponding contributions [see eq. (52)] to the coefficient of thermal expansion are very small, unless the Griineisen parameter is large. Presumably, only the terms involving the pressure derivatives of A 0 or M02 are important, because the effective Grfineisen parameter, d i n IA01/do~, is very large indeed. Precisely these terms are given in eq. (99). Here, dM2/dT 2 can be obtained experimentally, whereas, again, dm2/dT 2 may be maximized by (3/5B)(dAeff/ dT2). In fig. 16 the influence of these corrections is shown. Analogous to eq.
i
9
_
I
I
Ni3AI
v
"6 -68 E
6
E
÷
*
7
1
6
5 73
I 7/~
I 75
I 76 I% Ni}
Fig. 15. The specific heat coefficient 3' for some Ni3A1 alloys. Data from Kortekaas (1975). Circles: raw experimental data (De Dood and De Chatel, 1973). (+): results of the 'Wohlfarth correction'. The arrows indicate the influence of the 'fluctuation correction'.
366
P.E. BROMMER and J.J.M. FRANSE
L<> j ° °e;'P
1.0
x ,, in /,.6T + "Wohlfarth" correction "fluctuotion" -
0
0
v
@,'r,
+ I i
e__ eS . M .
i
~0.5
0
I
I
I
',°;
,I-el ',/ /
l
o
0.0 •
X
V
-0.5 73
7/,
75
76
(%Ni)
Fig. 16. The linear term in the thermal expansivity as a function of the Ni concentration in Ni3A1. Data from Kortekaas and Franse (1977). The values extracted from the work of Suzuki and Masuda (1985) are indicated with S.M. (1~ and O).
(103), the zero-field ferromagnetic contribution can be written as, o flo am=" ~
/" d M ~ t
dm~\
T=O=2(KC/V)~ dT"2 - 2 - d ~ ) •
(104)
Again, as in fig. 15, the correction in the ferromagnetic region, which is proportional to dM2/dT 2, is reduced to 2/(n + 2) = 2 of its value. The corrections in the paramagnetic region are smaller than those in the ferromagnetic region, because the B-values happen to be larger. Moreover, we believe that the difference between the experimental zero-field a-values and the upper 'fieldcorrected' values in the paramagnetic region, may well be due to an underestimation of the influence of the 'clusters' in the extrapolation procedure applied by Kortekaas. Indeed, if we assume that the difference actually is caused by the 'clusters', we find, without performing a complete analysis, a value Fcl = 10 for the Gr/ineisen parameter connecting these upturns, to those found in the specific heat mentioned above. Obviously, we find in fig. 16 a broad band of possible ae-values [a e = aexp - am, see eq. (93)]. In the ferromagnetic region the lower values are obviously too low (even negative), so the applied correction for the fluctuations must be considered as too large. Indeed, if we assume that the a~-values well inside the paramagnetic region should not differ very much from those well inside the ferromagnetic region, the correction due to the fluctuations should not be large at all. In the preceding section (see table 5), we already saw that for a 75% Ni alloy the
ITINERANT INTERMETALLICS
AND ALLOYS
367
electronic contribution was determined at high temperatures ( T > 80K) to be 0.6 x 10 -8 K -2, in rather good agreement with the corrected value in fig. 16. There, the high-temperature value obtained on a 74% Ni alloy (0.7 x 10 -s K -2) is given too. Moreover, the results of the analysis performed by Suzuki and Masuda (1985) are indicated. In order to clarify the origin of the apparent discrepancy, we discuss the results obtained on the 74% Ni alloy in some detail. First of all, the values for a~xp obtained at low temperatures appear to coincide with the results of Kortekaas (for that reason they are not indicated in fig. 16). The high-temperature values are in satisfactory agreement, although the temperature range mentioned in Suzuki and Masuda's work (i.e., the assumed range of validity for the approximation a = aT + b T 3) is much lower (up to say 30K) than the range mentioned in Brommer and Franse's (1984) publication (T > 80 K). Consequently, Brommer and Franse interpreted their value as the (high-temperature) electronic contribution, whereas Suzuki and Masuda observed a non-saturated magnetic contribution in their temperature range and, consequently, by subtraction obtained a much lower electronic contribution. They considered the difference to be evidence for the presence of a (positive) fluctuation contribution. For the other alloys (74.6, 75 and 75.2% Ni, respectively), we extracted a similar presentation of the experimental data given in Suzuki and Masuda's paper (see fig. 16). Finally, in the paramagnetic region near the critical composition, an enhancement of both the %-value (fig. 15) and the ae-value (fig. 16) probably is present. We estimate the enhancement in ye to be of the order of, say, 20% above the background. This background value corresponds to an 'enhanced' density of i
i
I
Ni3AI
o
+
L 5 ~, + J
÷ +.
÷
i I I
1 I
0 73
I 7/,
I 75
I
I 76 (% Ni)
Fig. 17. T h e 'electronic' Griineisen p a r a m e t e r : circles: uncorrected values for a e = aex p (fig. 16) and for y (fig. 15). ( + ) : calculated using the 'field-corrected values'. T h e a r r o w s indicate the influence of the 'fluctuation corrections'.
368
P.E. BROMMER and J.J.M. FRANSE
states of 175 st/Ry cell or 2.14 st/eV Ni spin for Ni3A1, i.e., to an enhancement m * / m of 2.2-2.5, depending on the choice of the calculated density of states (see Buis et al. 1981a). Lonzarich (1986) derived a total enhancement of 2.8, in satisfactory agreement with the analysis presented above. The extra enhancement is more clearly visible in the coefficient of thermal expansion than in the specific heat. Consequently, the Grfineisen parameter Fe(=3Vae/K%) appears to be enhanced too. Franse (1977) reports an increase up to 6 for Ni3A1 and up to 7 for NiPt alloys, ignoring the corrections due to the fluctuations. These values were used by Kaiser and Fulde (1988) as discussed in section 1.2. If the 'fluctuation corrections' indicated in figs. 15 and 16 indeed have to be applied, much lower values for Fe will result (see fig. 17). Obviously, the determination of the true electronic Grtineisen parameter appears to be very difficult. 3.3. Magnetovolume effects in dilute P d M n alloys In this section we discuss the magnetovolume effects in a system consisting of well-defined local atomic moments, embedded in a polarizable matrix (like in section 2.5). In particular, we discuss dilute PdMn alloys as an example. Specific-heat and magnetization measurements (Star et al. 1975, Boerstoel et al. 1972, Smit et al. 1979) show a ferromagnetic interaction between the Pd matrix and the local Mn moments, and an antiferromagnetic interaction between Mn moments within the third-neighbour shell (Nieuwenhuys and Verbeek 1977). At concentrations above 3% Mn up to 10% Mn spin-glass formation occurs (Coles et al. 1975, Zweers and Van den Berg, 1975, Nieuwenhuys et al. 1979). In a combined analysis of the specific heat and the thermal expansion, Brommer et al. (1981) separated the two contributions (see below). H61scher (1981) incorporated also the forced magnetostriction in the analysis. Geerken and Nieuwenhuys (1982) studied the thermal expansion in Pd(H)Mn spin-glass alloys. They observed a scaling proportional to the effective Mn concentration as derived from specific heat measurements in the corresponding PdMn alloys (not containing hydrogen). We describe the Pd___Mnalloys in the following way. In the neighbourhood of a cluster (i) of Mn moments (at sites Rq) the matrix responds, in a strongly correlated way, to the total 'foreign' moment/~fi = Zj/.~q(Rq). According to the picture sketched in section 2.5 a part of the response is strongly correlated with /~fi, giving rise to an induced correlated magnetization distribution. The integrated correlated matrix moment is, Jfd
--
]lr
/Zci - A + J~r /*f~= (/)fd - - Vlr)(1 -{- XJfd)la'fi "
(105)
Here we applied eq. (83) ignoring the cubic terms and introducing the definitions,
xJfa ]~fd
-
-
1 + X]fd '
xJl, Vlr
1 + XJlr
(106)
ITINERANT INTERMETALLICSAND ALLOYS
369
with X = A-1. In an applied field these 'giant moments' are oriented. Together they induce a long-range uniform magnetization Mu,
Mu = IzoXH + ~, X],r( ( I-to,) + (/-ti)) , i
(107)
where we used eq. (84), again neglecting third-order terms. Notice, that for simplicity the long-range molecular-field coupling is taken to be independent of the kind of atom. The total induced molar magnetic moment, M, is given by,
M = M u + M¢ = M u + E (tx~i) = teoXH + xJfd E (~.Lfi) i
i
=
X{t~oH + jfdMf}. (108)
Consequently, the total magnetic moment is given by,
M, = M, + M = XtZoH + (1 + XJfa)Mf .
(109)
In fact, from the data of Star et al. (1975) we deduce gJ~d = 0.46. Hence, with X -1 = A = 140 T mol/A m 2 we find, Jfd ---~ 64.4
T mol/A m 2 and
l,'fd = 0.315.
Ignoring thermal fluctuations in t h e magnetic moments of the matrix we find that the magnetic volume is given by, tO m = ( K C / V ) ( M
2 (r)}
= ( K C / V ) { M 2 - M 2 + ~i• ~NA (/]fd- Plr)2( 1 + xj~a)2 (/z~)}.
(110)
Here, we used the assumption that the correlated parts of different clusters do not overlap. N i is the effective number of Pd atoms inside the cluster. The factor N A / N ~ represents the influence of the peaking of the magnetization near the cluster. Next, we have to consider the contributions of the various kinds of clusters. First of all, we remark that the contribution of the most simple 'clusters', i.e., the isolated single Mn moments, does not depend on field, temperature or the state of order (at low temperatures the temperature dependence of the susceptibility of the matrix can be ignored). The contributions of a pair (antiferromagnetically coupled) does depend on the field as well as on the temperature because the total 'foreign' moment assumes non-zero values. For a quantitative estimate we use
(iz 2) = (glxB)2(Sp(Sp + 1 ) ) ,
(111)
where Sp indicates the total spin quantum number of the pair. Taking (for Mn)
370
P.E. B R O M M E R and J.J.M. F R A N S E
g = 2 and S = ~ we have 0 =< Sv N 5. At high temperatures (and vanishing field) the pair is decoupled: (Sv(S p + 1))--* 2S(S + 1) = 17.5, again giving rise to a constant contribution. Hrlscher (1981) prefers to use (/XZp) ~ (S2p) ~ S(12S 2 + 10S + 1) / 3(2S + 1) = 14. The contribution of more intricate, isolated clusters can be written in an analogous way. For the dilute PdMn alloys under consideration, however, we ignore these higher-order dusters. For the higher concentrations, computer calculations are to be preferred (Smit et al. 1979). Using, M = Mt-
M e
(112)
= vfa(M t +j~a~lXoH),
and
Mc
= E
i
=
(/Zci)
=
-
(~'fa -- Vlr)(1 -
+
XJrd)Mf
(113)
x oH),
(114)
we can express the magnetic volume in a form suitable for the analysis of the experimental data,
(115)
(.0m ~---O)lr Jr- O.)p ,
O)lr: ~(b~C _ vfd(Mt2 + ]~l/x0H)2 _ (v,a _ b,lr)2(Mt_ KC
Xp
XI.t,oH)2},
Wp = --~ ~pp (gNA/.q3)z(Vfd - v,r)2(1 + xjfa)Z(Sp(Sp + 1 ) ) .
(116) (117)
Here, xp is the concentration of pairs (xpNA pairs per mole) and Np is the effective number of Pd-atoms inside the correlated 'cluster'. Notice that, in the present approximation, the formation of a spin-glass (at zero field) does not lead to a change in the volume, as far as each local moment or cluster retains its own correlated polarization cloud irrespective of being oriented in the spin-glass or not. Experimentally, the spin-glass formation indeed does not show up in the thermal expansion (Franse et al. 1980). In eqs. (115)-(117) the magnetic volume is split up in two distinct contributions, the 'long-range' contribution wlr and the 'cluster' or 'pair' contribution O)p, It is natural to assume that also the average energy of a cluster (i.e., a pair) is proportional to (/.~2p) or, with the estimate given in eq. (111), proportional to (Sp(Sp -~ 1)). Consequently, the cluster contribution to the coefficient of thermal expansion and the corresponding contribution to the specific heat are connected by a Grfineisen relation, albeit an improper one. Experimentally, the Griineisen parameter, Fp, can be determined in the paramagnetic (or spin-glass) region and indeed appears to be independent of the temperature. The values obtained are given in table 7.
ITINERANT INTERMETALLICS AND ALLOYS
371
TABLE 7 Magnetovolume effects in dilute PdMn alloys. /]1r is the Grfineisen parameter for the long-range contribution; tolr the long-range contribution to the relative magnetic volume at T= 0 K; S~r the entropy contained in the long-range contribution; xl the calculated concentration of isolated Mn atoms (see text); xp the calculated concentration of pairs: 2Xp = x - xi; Fp the Gr/ineisen parameter for the cluster (pair) contribution; Fp = 3VAo~/KAc in the paramagnetic region (V=8.87 x 10-6 m3/mol, K = 5.53 X 1 0 - 1 2 m3/j); top the 'pair' contribution to the relative magnetic volume up to 15 K; Sp the entropy contained in the cluster contribution up to 15 K; Cp(max) the maximum value of the 'pair' contribution to the specific heat; Tc(calc) the Tc calculated using eq. (119) with x~ as calculated (see fable); Tc(eff) as Tc(calc), now with x~ replaced by Slr/R In 6. Mn concentration 10% F~r 106t% SI,/R In 6 xi(calc)
2%
1.35%
0.96%
0.54%
-6.5 1.3 0.9 0.93
-6.5 0.95 0.8 0.81
-6.5 0.66 0.7 0.67
-6.5 0.2 0.4 0.44
6.5 3.4 1.0 1.07 2
5 0.9 0.3 0.55 0.75
5 0.4 0.2 0.29 0.5
5.5 0.3 0.1 0.10 0.25
4.8 5.0 5.8
4.3 4.3 5.0
3.8 3.6 3.2
2.1 2.4 1.8
(%) (%)
0.18
-13 1 1.5 0.71
2Cp(max)/R
(%) (%) (%)
5,5 12 3.6 9.82 12
7.2 3.2 4.29 6
Tc(calc) Tc(eff ) Tc(exp)t
(K) (K) (K)
1.0 -
/~P6 10 top Sp/R In 6 2xp
-
5% +0.35% Fe
-
3.8 8.0 10 Tsf = 2.5
Also in the f e r r o m a g n e t i c region (at zero field), the long-range contribution to the t h e r m a l expansion is e x p e c t e d to be related to a c o r r e s p o n d i n g contribution to the specific heat. H e r e , the Grfineisen p a r a m e t e r is estimated to be either equal to - d i n Tc/dto [ F f = - 1 3 for P d 5 % M n 0 . 3 5 % F e (Franse et al. 1980)] or to - d In x l d w [F~ = - 6 . 5 (Beille and C h o u t e a u 1975)]. T h e forced magnetostriction is given by eqs. ( 1 1 5 ) - ( 1 1 7 ) as well. T h e longrange terms are k n o w n , and thus can be subtracted f r o m tom. T h e resulting cluster contribution can t h e n be c o m p a r e d with a theoretical estimate for (Sp(Sp + 1)) in an applied field. In figs. 18 and 19 we show the excess coefficient of t h e r m a l expansion and the excess specific heat, respectively, for three P d M n alloys (data f r o m B r o m m e r et al. 1981). T h e excess t h e r m a l expansion, as d e t e r m i n e d by G e e r k e n and N i e u w e n h u y s (1982), appears to show a little m o r e structure (a dip at a b o u t 10 K). A s an example, their results on a Pd 3 % M n alloy are indicated in fig. 18. I n an a t t e m p t to establish the effect of M n pairs m o r e clearly, Oraltay p e r f o r m e d analogous experiments in o u r l a b o r a t o r y at still lower M n concentrations (see table 7). By c o m b i n i n g the t h e r m a l - e x p a n s i o n data and the specific-heat data in the way discussed in the next section, the 'pair' contribution can be separated f r o m the ' l o n g - r a n g e ' contribution. T h e results for Pd 2 % M n are indicated in fig. 19. F u r t h e r results are given in table 7 and are discussed below.
372
P.E. B R O M M E R and J.J.M. FRANSE I
I
I
Pd-Mn
x x lO%Mn XYx
5O XX Xx X
co
~x
i
0
xxx
X
xX2
5%Mn+ 0.35% Fe
~Tc 0 0
sf
x
~ o
X
o o°°
0
,"#~"~-+-+%"
.,.Tc~ +
Qx
~, :I.T
0
o
O0 0
O~
0
0o
,"
/A~ ~.- ~&
3%Mn "'-.
tL~ +
2%Mn
.__,,..,_fi ;,c . / ..~ , ++ +++++ ,~lr 12%)
-lo
"-. _[,"
0
5
I
I
10
15 T
(K)
Fig. 18. Excess linear thermal expansivity of some PdMn Alloys. The transition temperatures are indicated: T c ferromagnetic, T,f spin-glass. Data from Brommer et al. (1981): The dashed curve gives the long-range contribution for Pd2%Mn. The results of Geerken and Nieuwenhuys (1982) on a Pd3%Mn alloy are also indicated (A).
I
I
x
Pd -Mn
~0.4
x~
x x lO%Mn X
-5 E
X x XX
Xx ~Tc
O.3 o
5%Mn+
o o O c ~ ~ o ° Oooo0 o 0.35% Fe ~0 o
OOx
0.2
o
~Tsf O°°x×
2 ~>+÷ Tel% .x-
~
/ ~ . ~ " Oi
" "r'l"÷++++ +'1" 4"
'~ I
0
o
5
~ -+'+ 4 4 L + ~-
I
10
15 T(K)
Fig. 19. The excess specific heat of some PdMn alloys. Data from Brommer et al. (1981). For the Pd2%Mn alloy the 'long range' or 'ferromagnetic' contribution (broken curve) and the 'cluster' or 'pair' contribution (drawn curve) are indicated.
ITINERANT INTERMETALLICS AND ALLOYS
373
The pair contribution In our pair model the entropy under the cluster peak should be Sp=
2xpR ln(2S + 1) = 2xpR In 6. We integrated Ac/T up to 15 K (at higher temperatures these early data become too inaccurate, mainly because of empty-cell corrections). The corresponding number of Mn atoms bound in clusters (pairs) is given in the table as Sp/R In 6. As is obvious from fig. 19, for the 10%Mn alloy a large part of the entropy is still hidden above 15 K, in accordance with the expectation that at high concentrations more intricate, strongly coupled, clusters can be formed. Taking into account the inaccuracy of the integration procedure, we feel that for the other alloys the observed values for 2Xp = Sp/R In 6 are in satisfactory agreement with the calculated values. Here, we calculated 2Xp = x(1 - (1 - x)3S), assuming that an 'isolated' Mn atom has no Mn neighbour within 38 surrounding sites (see H61scher 1981, Star et al. 1975, Nieuwenhuys and Verbeek 1977). The maximum of the excess specific heat of one 'pair' is expected to be roughly R/N A [see, e.g., Oraltay et al. (1984) for the case S = 1; for S = the peak broadens, but the maximum value remains roughly the same, i.e., 1.07 R/NA]. So 2 ACmax/Rshould - again - be equal to the number of Mn atoms in the 'pairs' (clusters), i.e., equal to 2Xp. The values obtained are given in table 7, and appear to be too high by about a factor of two. This discrepancy casts some doubt on the interpretation of this peak as being caused by clusters (pairs) rather than by the (independent) Mn spins. The energy, integrated up to 15 K, under the cluster peak equals fp(Z)115 i~ = (V/KFp)Ogp= 0.25 x 106O)p , by virtue of the Grfineisen relation. Identifying Up(T) with xpRTp( Sp(Sp -k 1 ) ) r for pairs, we estimate Tp to be of the order of 1 K. Then, the maximum level of the pair has an energy 30 Tp, in accordance with estimates in the literature (Smit et al. 1979, Nieuwenhuys et al. 1979).
The 'long-range', or ferromagnetic contribution Since, in our picture, the ferromagnetic peak is due to the ordering of (mainly) the isolated Mn moments, the total entropy under the peak should be S~r = xiR ln6, where x i is the concentration of isolated Mn atoms. This prediction appears to be in nice agreement with the experimental results (table 7). According to eq. (116), the relative magnetic volume at zero field is given by, ¢-Olr(H = O)
=
K C Mt2{Vfd 2 _ (vf~
~
--
v~r) 2} .
(118)
At low concentrations, we ignore the pair contribution to the spontaneous magnetic moment. Moreover, at zero temperature the Mn magnetic moment is saturated (Mf =Mfs = XiNA/~n ). From eq. (109) we have,
Mt(H =
O) = (1 + Xjfd)Mfs = xi(1 +
Xjfd)NA~qVln.
374
P.E. B R O M M E R and J.J.M. FRANSE
So, at low concentrations O)lr appears to be proportional to x 2. In fig. 20 the integrated long-range thermal expansivity is plotted as a function of x~. From the slope we find utr=0.08 to 0.09, so with A = 1 4 O T m o l / A m 2 we have j~r = 13--14 T mol/A m z, much smaller than Jfd = 64.4 T mol/A m 2. From eq. (105) it follows that the correlated matrix moment is /Zci = 0.33 ~fl = 1.7/xB, hence, t h e total 'giant moment' equals 6.7/~. In section 2 we derived an expression for the Curie temperature T c [eq. (85)]. As remarked there, this relation contains a direct long-range interaction between the foreign moments. After correction for this spurious interaction we find, S ( S + 1)],r[geff( 1 ,2 Tc = xi (NA/~)2 3R
+
XJir) --
g2],
(119)
where geff/g = (Mrs + Mcs)/Mfs (see section 2). Inserting the numerical values we find T c = 5.5
x
(100xi) K .
The calculated T c values appear to be in reasonable agreement with those experimentally observed (see table 7). Notice that a simple indirect exchange model would lead to T c values larger by a factor of, say, jfd/]~,, i.e., by about a factor of five. We conclude that the model gives a very satisfactory description of the ferromagnetic contribution. I
I
I
I
I
I
I
i
I
• x i calculated o x i Sir/Rm6
Pd -Mn
I
/
• 0/2%
i O
/
5°IoMn] +0.35°•0 Fe.h+
10
/ /
O
9
1.35%
0o//0.96%
/ / /,~•
0.5/.%
/ o
t
,
,
l
I
l
,
i
0
,
I
,
I
x2
(%21
Fig. 20. The ferromagnetic contribution to the magnetic volume as a function of the square of the concentration of single Mn-moments. Measurements performed by G. Oraltay (private communication).
ITINERANT INTERMETALLICS AND ALLOYS
375
The forced magnetostriction H61scher (1981) measured the forced magnetostriction in some of the PdMn alloys mentioned above. Focussing attention on the P d 2 % M n alloy, we calculated 601rwith eq. (116) inserting the experimentally observed Mt(H ) values (at T = 4.2K). The calculated values are shown in fig. 21. Inserting a zero-field zero-temperature value for the magnetic moment of M t ( 0 , 0 ) = 0.4 A m2/mol, as deduced from Star's data (1975), we calculate ~0]r(0, 0) --- 1.4 x 10 -6, in reasonable agreement with the integrated value of 1.3 × 10 - 6 (see table 7). Furthermore, starting with the value of %~(0, 0) given in the table, and subtracting the thermal expansivity due to the long-range term integrated from zero temperature up to 4.2 K (see figs. 18 and 19), we find, 106¢91r(T = 4.2 K, H = 0) = 1.3 - 0.7 = 0.6, in good agreement with the value 0.65 calculated from eq. (118) (see fig. 21). In fig. 21 we show the observed forced magnetostriction (AV/V) as a function of the effective field acting on a 'pair', in the sense meant in eq. (82). Correcting again for the spurious long-range interaction, we have now, /xoH~ff
:
I%H + j,rMt - J,r(g/geff)Mf
u
u
i
(120)
•
i
I
I
i
Am_V+1.15x10 -6 V ~ A
Pd 2 % M n 4.2K
10
i
A_.V . V
i
A ~x
l-to H
A A
.
C°lr
• + + d •
+
0
+
OK /
•
+
+ A
o
o
Wp
O
'~ •
d
a m
+ 0
+ +
0 O
O A+O
o
~o 0
I
0
I
I
I
'~H:O
[
I
I
I
5 IJ.o Hef f
(T)
Fig. 21. The forced volume magnetostriction in Pd2%Mn. Points: A V / V as a function of the applied field (H61scher 1981). (A): A V / V plus the zero-field estimate, 1.15 x 10 -6, as a function of the effective field defined in eq. (120). ( + ) : the long-range contribution ca~ calculated with eq. (116). (O): the cluster contribution cap is obtained by subtraction, cap = A V / V + 1.15 x 1 0 - 6 - calf" The corresponding values in zero field at T = 0 K are shown also (filled symbols).
376
P.E. BROMMER and J.J.M. FRANSE
From the observed thermal expansion, the increase of the cluster contribution, top(T = 4.2 K) - wp(T = 0 K), is estimated to be 0.3 x 10 -6. Taking, more or less arbitarily, the zero-temperature cluster contribution wp(T = 0 K) to be equal to 0.25 z 10 -6 we have wp(T -= 4.2 K) = 0.55 x 10 -6 at zero field. Hence, we added, (~% + Wp)tn= 0 = 1.15 x 10 -6 to the observed AV/V values, in order to obtain a consistent presentation. At the maximum effective field, the cluster contribution is 4 x 10 -6, even larger than its high-temperature value at zero field, i.e., 3.1 x 10 -6. Obviously, the 'pairs' are opened up to, say, Sp in the order of 4 in an effective field of, say, 10 T. Since the Zeeman term, geffNal~SptxoHeff, must be of the same order of magnitude as the pair energy, RTpSp(Sp + 1), we now estimate Tp to be roughly 2 K, in reasonable agreement with the estimate above. We conclude that the model gives a satisfactory description of the volume magnetostriction too. In conclusion, we state that the simple 'pair' model reproduces most experimentally observed features qualitatively, and in most cases also quantitatively, the only obvious exception being the number of 'pairs', as deduced from the maximum in the cluster contribution.
3.4. The combined analysis of specific heat and thermal expansion In the preceding sections, we have stressed the importance of a combined analysis of thermal-expansion and specific-heat data. Here we discuss the procedure in more detail, focussing attention on the application to the heavy-fermion system UPt 3. Let us assume that the contributions cj to the specific heat at constant volume and the corresponding contributions/3j to the coefficient of thermal expansion are related by Grtineisen relations [see eq. (39), we consider only the case H = 0]. If the Fj values differ from each other and the temperature dependence of the contributions is not identical, the so-called effective Gr/ineisen parameter Feff(T) =
VI~/Kc = E I~jcl/c,
(121)
J
varies with the temperature T. The other way round, a temperature dependent Feff(T) indicates the presence of different contributions, unless a temperature scaling as assumed would be impossible altogether. We assume further that the necessary corrections have been carried out: c - - c ( e ) - V~ZT/K, or equivalently cj = c j ( e ) - V[3fljT/K, as well as the corresponding corrections in )6 and if, or the anisotropy corrections to be discussed presently. Since we have two independent sets of data, i.e., c(T) and/3(T), it is possible for any choice of F l and Fz, to split the functions c(T)= Q(T)+ c2(T ) and ~ ( T ) = ~ l ( T ) + ~ 2 ( T ) in such a way that apparently the Gr/ineisen relations [eq. (39)] are satisfied. The 'hat' (^) on a F indicates that this parameter is not
ITINERANT INTERMETALLICSAND ALLOYS
377
necessarily a true Grfineisen parameter. We find the equivalent expressions: cl(T)
= VIS(T)/K - l%c(r) = G f ( T ) -
Cl- f'2
K
(122)
1 - ~/Fj
ISa(T) = ~ ~ cI(T)= j~. i -- ~ c2(r) = c(r) - c
E P2 c(r) = E_ lr'-
(r) = 2
IS2(T ) = F 2 ~ c 2 ( T ) =
ISj'
1 - r/L
j 1 - ~/iff1 cj, ~Fj 11
Fj/~ tS,. Fz//~a
(123)
(124)
(i25)
In case F 1 equals the 'true' Grfineisen parameter F1, the true contribution c I does only contribute to ca(T ). In fact, a true contribution cj is divided over c~(T) and c2(T ) in the ratio (~ - F2): (Fa - ~). If, in particular, Pl ~ F1 >> [~1 (J ¢ 1), the function Cl(T ) represents the contribution c a quite accurately, with the exception of the region where Cl/C and ( q - F a ) / ( F 1 - / ~ 2 ) are of the same order of magnitude. Notice that in this situation the functions ISI(T) and IS2(T) are more difficult to interpret, because the ratio ~-/P2 may differ appreciably from unity. Moreover, a contribution to the specific heat not having a counterpart in the coefficient of thermal expansion (~ = 0) will show up not only in c2(T), but, perhaps misleadingly, by construction also in IS2(T). Nevertheless, if some contribution dominates the others in a certain temperature region, we can estimate the corresponding Grfineisen parameter from the F~fr(T) value (often a plateau) in that region. A typical example was treated in the preceding section, where we could estimate the Griineisen parameter for the 'Mn pairs' in PdMnalloys in the high temperature region. Some other aspects of this analysis are: (1) At low temperatures the terms linear in T survive. In general,"such a term may consist of different contributions. As an example, we recall that the familiar 'electronic' contribution (proportional to the density of states times T) has to be corrected with a 'magnetic contribution' (the so-called Wohlfarth correction), with a very different Grfineisen parameter. Consequently, linear terms are, in general, expected to show up in both ca(T ) and c2(T ). Notice, however, that for a choice /~a F1 would lead to a spurious contribution in c2(T ). (2) At higher temperatures, the phonon contributions dominate in most cases. It appears to be very useful to determine these contributions in an independent way, e.g., by a calculation using the phonon spectrum as determined by neutron scattering, or by performing experiments on an analogous, less enhanced, alloy. (3) The specific heat of a truly independent contribution is essentially positive, whereas a 'correction' may be negative as well. Examples are the magnetic
378
P.E. B R O M M E R and J.J.M, F R A N S E
Wohlfarth correction mentioned above and also the T 3 In T term in a heavyfermion system (see below). (4) A very important tool is to consider the physics behind an alleged contribution. Clues are the temperature dependence itself and last but certainly not least the total energy and the total entropy involved. The method just described has been applied to a variety of materials. First of all, we mention again the separation of the ferromagnetic contribution and the contribution ascribed to antiferromagnetic pairs in PdMn alloys (preceding section). An analogous reasoning was used in an analysis of YCo alloys (Oraltay et al. 1984). Also, most of the work discussed in section 3.1 can be considered as being analyzed along these lines (in particular Ogawa 1983, Matsunaga et al. 1982). Luong et al. (1985) applied the method extensively in the analysis of ErCu 2. Here, the crystal field acting on the rare earth gives a distinct contribution, with a Gr/ineisen parameter of the order of two. Recently, Franse et al. (1989) analyzed specific-heat and thermal-expansion data in the heavy-fermion compound UPt 3 in this way. Their analysis is discussed below. De Visser et al. (1988) applied the same method on the CeRu2Si 2 system and obtained analogous results. The thermal expansion of many materials (in particular UPt3) is anisotropic. For that reason, we now shortly discuss the anisotropy corrections. Taking into account the diagonal elements of the stress (strain) tensor only, we define the dilatation o9, the axial deformation eax and the deformation in the basal plane, epl , and the corresponding stresses as follows, (I.I =" Exx -1- Eyy + Ezz ,
eax =
2ezz
epl =
exx -- Eyy ,
P = l(exx
-- exx -- eyy ,
Pax =
Ppl = l ( P x x
-~- e y y -~- e z z ) , ½(Pzz
-
P)
,
(126)
(127) (128)
- eyy).
The compressibility and the other compliance constants can be expressed in the elastic stiffness constants cH, c12 , c~3 and c33 (the c-axis of the hexagonal cell is chosen as the third or z-axis): K -
do) d P - V ( 2 c 3 3 - 4c13 q" c l l "q- c 1 2 ) / D
'
(129)
Kax =
deax dPax - V(2c33 + 8c~3 + 4c H + 4c~2) / D ,
(130)
Kpl ~
dep---2 dPpl
(131)
V(c11
-
-
C12) - 1 ,
do) deax h = + dPax - + ~ = 2V(c33 + c13 - c H - c12) / D ,
(132)
ITINERANT INTERMETALLICSAND ALLOYS
379
with (133)
D = c33(Cll -]- c12 ) -- 2c213 .
The cross-terms depJdP and do)/dPpl vanish. Moreover, in UPt 3 no deformations occur in the basal plane (depJdT=O). Hence, we ignore the corresponding terms. The axial deformation eax (say, 2 8 In c/a) does change with temperature and the calculated value for h does not vanish. The scheme given in eqs. (46)-(48) can now be extended. For simplicity, we omit the magnetic terms. - d P =/~ do) + ( / ~ / K a x ) dea, - b-fl d T ,
(134)
-dPax = + (b/~/Kax) do) +/~ax deax - b~x/3~xd T ,
(135)
- - V -1 d S = - b ~
(136)
do) -- b a x ~ x deax- V---Td T ,
with
/~-1 = K(1 -- a2/KKax ) , /3--/3 1 + Kax~], = C -- V b ~
Z -
b~-lax= Kax(1 --
a2/KKax)
,
(137)
~ax=~ax"
Vbax~ax~ax
T .
(139)
These relations contain, of course, both the 'isotropic corrections', already indicated in section 1.4, and the 'anisotropy corrections'.
The heavy-fermion system UPt3 UPt 3 is a close-packed hexagonal (MgCd3-type) intermetallic compound, famous because it combines heavy-fermion behavior [y = 422mJ/K 2 m o l U (De Visser (1986)] and superconductivity (T c = 0.5K), as well as interesting (antiferro-) magnetic properties. Upon alloying with Pd, the pseudobinary compounds U(Pt,Pd)3 lose rapidly the superconducting properties (T c < 40 mK at 0.5% Pd), whereas around 5% Pd long-range antiferromagnetic order develops (T N _-<6 K) with uranium moments of about 0.6/xB (Franse et al. 1987, Frings et al. 1987). The compound UPt 3 itself is very close to an antiferromagnetic instability. Thermal and magnetic properties have been discussed widely based on a Fermi-liquid picture with strongly enhanced (antiferromagnetic) spin fluctuations, especially the T 3 ln T term in the specific heat and the thermal expansion, the T 2 term in the resistivity, and the maximum in the susceptibility as a function of temperature (see De Visser et al. 1985, De Visser 1986, and references therein). The presence of spin fluctuations
380
P.E. BROMMER and J.J.M. FRANSE
was verified by inelastic n e u t r o n scattering (Aeppli et al. 1985, J o h n s o n et al. 1986). U P t 3 is strongly anisotropic. T h e inverse susceptibility in the basal plane Aa, b equals 10.9 T m o l U / A m 2 at 4.2 K and passes a m i n i m u m of 9.31 T tool U / A m 2 at 20 K, w h e r e a s in the c direction at 4.2 K o n e finds A c = 2 2 . 0 T m o l U / A m 2, while no m i n i m u m is o b s e r v e d (Frings 1984). M o r e o v e r , at a field of 20 T in the basal plane, the i n d u c e d m o m e n t increases a n o m a l o u s l y , in contradistinction to the b e h a v i o r in a field along the c direction. A t a field of 30 T in the basal plane, the i n d u c e d m o m e n t is l p ~ / U , w h e r e a s the same field in the c direction induces only a m o m e n t of 0 . 3 6 p ~ / U . T h e t h e r m a l expansion is anisotropic too. A t low t e m p e r a t u r e s , the relative length c h a n g e is negative in the c direction but positive in the basal plane. T h e coefficient of t h e r m a l (volume) expansion is positive (see fig. 23). Since the elastic constants have b e e n d e t e r m i n e d f r o m the o b s e r v e d s o u n d velocities ( D e Visser et al. 1985, D e Visser 1986), we are in a position to calculate b o t h the 'isotropic' and the 'anisotropic' corrections. T a k i n g the values at T = 300 K we find f r o m eqs. (129)-(132), K = 0.479 M b a r -1
Kax = 5.258 M b a r -1
I
and
1
h = 0.027 M b a r -1
I
1.0 O
S °
r-~
0.5
o
° o°°
I 10
20
30 T (K)
Fig. 22. c/T as a function of Tfor U P t 3. Data from De Visser et al. (1984) and Franse et al. (1984). The drawn lines give the splitting in cl(T ) and c2(T ) with/~1 = 73 and ~ = 2.35, respectively. The dashed line gives the 'electronic contribution' obtained by subtracting the phonon contribution (see text).
ITINERANT INTERMETALLICS AND ALLOYS I
l
I
t
l
I
I
I
I
i
-
381
l
3
-101
i
,
,
,
o
[
~
,
,
,
50
I
,
I oo T(K)
Fig. 23. The linear thermal expansivity a as a function of T. Data from D e Visser et al. (1985). Notice the strong anisotropy.
Hence, the correction A21KGx is negligibly small and /~ = b = K-1 = 2.087 × 1011 J / m 3
(Vb =
/~ax = b ax = Kax--1 0.190 x 101~ J / m 3 =
8.85
(Vbax=
× ]0 6
0.81
J/mol), X 10 6
J/mol).
Furthermore, h/Kax 0.005 and h/K = 0.055, which indicates the order of magnitude of the possible relative corrections in /3 and /3ax, respectively. The corrections to the specific heat can be calculated straightforwardly and turn out to be rather large at higher temperatures, in particular because the subtracted phonon contribution is calculated as a specific heat at constant strains (see also fig. 26). The effective Grfineisen parameter, F~ff(T), decreases sharply with increasing T (fig. 24). Using the data up to T = 30 K Franse et al. (1989) calculated the function Cl(T ) according to eq. (121) with /~1 = 73 and ~ = 2.35. Van Sprang (1989) extended the thermal-expansion measurements up to room temperature. These data were combined with the high-temperature specific-heat results reported by Felten (1987). Since the calculated phonon contributions refer to the specific heat at constant volume and form, it has appeared to be necessary to correct the measured value c(e ). The corrections for constant volume are significant at the higher temperatures. The influence of these corrections is visible in fig. 26. Notice =
382
P.E. B R O M M E R and J.J.M. F R A N S E I
I
I
I 10
I 20
I 30
80
L_~
40
0 0
T(K)
Fig. 24. The effective Grfineisen parameter Fe~(T ) = Vbfl/c. Data from Franse et al. (1989).
i
I
2-
i
6d~C~o
I
i
U Pt 3
CI(T)
~
6
_
o
E
b
o
~o
o/
RIn2
~
s
'/ /\ ¢'
/
,
--
',',,,
',,',,,.
/
\ 0
~ 0
I 1
~
I 2
"" "--,
IogT
Fig. 25. The function cI(T) as a function of log T. Dashed line: from Van Sprang's measurements up to room temperature. Circles: from De Visser's measurements up to 35 K. Drawn peak: 'Bethe Ansatz' solution of the spin-½ Kondo problem. The entropy contained under the experimentally observed peak is given too. See text for details.
ITINERANT INTERMETALLICS A N D ALLOYS '
i
I
I
383
i
U Pt 3
/
-
/ /
-5
/ /
E ,/ / /
0
/ /
/
COl /
J
J
"
.." •
r
•
C21v
•" el
s
~
f
•
f
•s
f /
,,, /
,
j• i
S
~ C elelC /
i
~l
~'100
~
1 200
t
300 TIK)
Fig. 26. The function c~'(T) obtained from the 'electronic' contribution. ~ = 73; -P2 = 2. The influence of the correction for constant volume is shown. The function c~l(T) is also shown.
that a correction to the specific heat has only little effect on the function ca(T), because it is reduced by a factor F 2 / ( ~ - F2) [see eqs. (122) and (123)]. The relative correction for constant form [eq. (138)] amounts to about 1.5% (at 20 K). The corresponding correction t o C l ( T ) has about the same value, and thus can be neglected too. Consequently, the function cl(T ) is expected to represent a true contribution rather accurately. Felten (1987) calculated the phonon contribution to the (corrected) specific heat from the neutron-scattering results obtained by Renker et al. (1986), and subsequently obtained the presumably electronic contribution to the specific heat, ce~(T), by subtracting the calculated phonon contribution. The corresponding electronic contribution to the coefficient of thermal expansion, ae~(T), was obtained by subtracting a phonon contribution calculated with Fpu = 2.35 (actually, a Debye function with 0o = 200 K was used). Again applying eq. (121), now with F1 = 73 and ~ = 2, Franse et al. (loc. cit.) determined from these electronic contributions the functions c~l(T), and so on. In the low-temperature range, c11(T) coincides with the function cl(T ) obtained above, and appears to show a striking resemblance to a 'Kondo peak' (see fig. 25), thus confirming a conjecture of Renker et al. (1987). Notice, however, that Cl(T)/T decreases monotonously as a function of T (fig. 22), and can be described up to 5 K by,
cl(T ) = 7T + ~T 3 In T/T~f , with 7 = 433 (422) mJ/K 2 mol, ~ = 5.88 (1.38) mJ/K 4 mol and Tsf = 12.4 (27) K.
384
P.E. B R O M M E R and J.J.M. F R A N S E
The numbers between parentheses refer to De Visser's analysis (loc. cit.) of the total specific heat c(T) in the temperature interval 1.4-10 K. The total entropy in this contribution tends to R In 2, in accordance with the physical picture, i.e., a Kondo doublet. The fact that the peak is somewhat sharper than the 'Bethe Ansatz' solution of the spin-1 Kondo problem (Tsvelik and Wiegmann 1983), cannot be ascribed to the separation procedure, but must have a physical explanation (perhaps related to the formation of a coherent state). The integrated specific-heat peak contains an energy of 109 J/mol, i.e., R × 13K. This temperature scaling is remarkably close to the spin fluctuation temperature, Tse = 12.4 K. The thermal expansion amounts to a relative volume collapse of 9 × 10 -4. If this volume change is considered to be a magnetovolume effect, the high-temperature moment would be 1.24P.B/U (estimated using a magnetovolume parameter of -4 2 5.81 x 10 //XB, derived from forced-magnetostriction data with the field in the basal plane). The other way round, it is suggested that the formation of moments of that magnitude (either by a temperature of, say, 100 K or by a magnetic field of, say, 100 T) would lead to a loss of the heavy-fermion (Kondo) state. The situation, however, is far from clear, because UPt 3 is anomalous in the sense that the application of a magnetic field initially increases the heavy-fermion character (7 increases) in contradistinction to other heavy-fermion systems (Stewart 1984). Turning now to the function c2(T ) in fig. 22, or rather c~l(T) in fig. 26, we remark that a linear (~ T) contribution appears to be absent at low temperatures. Although this might be an artefact of the separation method (see remarks on the choice of ~ ) , the result that all electrons contribute to the enhanced linear term (with large Gr/ineisen parameter) is in accordance with De Haas-Van Alphen data (Taillefer et al. 1986). In the function c~l(T) we clearly see an extra peak. This peak cannot be explained by the corrections for constant form, which still are rather small. The corresponding contribution to the thermal-expansion coefficient is small in comparison with the value of the function a ll(T) in that region (fig. 27). Consequently, we cannot rule out the possibility that the specific-heat peak should not have a counterpart at all in the thermal volume expansion, e.g., because it is primarily connected to the anisotropic strains. On the other hand, Renker et al. (1986) suggested that this peak is caused by crystal-field effects. In that case the Gr/ineisen parameter should be low (say, lower than 5). The entropy in this contribution is estimated to be 3 J/mol K, which seems to be rather low, because one extra doublet above the Kondo doublet in the crystal-field scheme would correspond to an entropy of R In 2 = 5.8 J/mol K (note: with only one extra level the entropy would be R In 3 = 3.4 J/mol K). Finally, it is obvious that the different relative weights of the contributions 1 and 2 in the specific heat (fig. 25) and in the thermal expansivity (fig. 27), respectively, make the curves look very different. The other way round, it is now clear why the analysis mentioned above, which seemed to describe successfully the total specific heat up to 20 K, although Ts~ was as low as 29 K, apparently was
ITINERANT INTERMETALLICS AND ALLOYS 6
'
I
'
I
385
'
~5 Y ~D
~L~
m
-
~
__~e21 Ctel
-
--
0 100
200
300
T IKI Fig. 27. The functionsa~(T) and a~za(T)obtained from the 'electroniccontribution'. See caption to ~g. (26). not applicable to the thermal expansivity in the same temperature range. The good quality of the fitting result to the specific heat must now be considered to have been accidental. Although the present method yields, a priori, the same fitting to the specific-heat function C l ( T ) as to the thermal-expansivity function, al(T ) ~ cl(T), we regard this analysis as the more satisfactory one.
4. Perfectly ordered stoichiometric compounds
4.1. Structural defects in off-stoichiometric compounds The magnetic properties of ordered compounds appear to be strongly influenced by structural defects, i.e., vacancies and antistructure atoms. In the actual compounds these defects are always present. The properties of the perfectly ordered stoichiometric compound, however, can be derived from those of a series of off-stoichiometric compounds. The procedure is based on the assumption that the physical quantity under consideration (in particular, the Landau coefficient A in the Arrott plots) is a smooth function of the concentrations of the different defects. The interdependence of these concentrations is derived here from a simple ordering model, but extension to more intricate models is straightforward. In this section we discuss the ordering model. In the following sections we discuss applications to the Ni3A1 and Ni3Ga systems, and to the Ti(Fe,Co) system, respectively.
386
RE. BROMMER and J.J.M. FRANSE
Ordering model
We consider N unit cells of an ordered binary compound consisting of m equivalent A sublattices and n equivalent B sublattices. For a possibly offstoichiometric compound Am+xBn_ x we define: a the fraction of the B sites occupied by A atoms, b the fraction of the A sites occupied by B atoms, v~ the fraction of vacant A sites, v b the fraction of vacant B sites. The number of A atoms is [m(1 - b - G ) + n a ] N . The number of B atoms is [n(1 - a - 12b) -~ m b ] N . Hence, with v ( m v a + n v b ) / ( m + n), the total vacancy concentration, we have =
nm
x(1 - v) = na - m b + - -
n+m
(v b - / ) a ) "
(140)
The ordering model is based on a quasi-chemical approach, derived from a minimization of the free enthalpy G = G O - k T In W, under the assumption of a random distribution of the structural defects. The configurational entropy is determined by W=
(mN)!
[mN(1 -
b
(nN)!
- G)]!(mUb)!(mUva)!
[nU(1 - a - V b ) ] ! ( n U a ) ! ( n U V b ) ! " (141)
Introducing a Lagrange multiplier ~ for the subsidiary condition of eq. (140) and using (m + n)N(1 - v) = ( m + n ) N A , where N a is the total number of atoms per sublattice (here chosen equal to the Avogadro number), we find with some natural approximations, aG O -+ nh + nNAkT Oa
In a = 0
OG o -- mA + mNAkTln Ob OG °
m ( n - x)
Ov a
m + n
OG ° • n ( m -+ 00 b
(143)
b = 0,
2t + m N A k T
+ x)
(142)
~ + nNhkT
In v a = 0,
(144)
In v b = 0.
(145)
m ~- n
From these conditions we derive, [ ab= K=exp
( -
1 OG ° 10G°)/kT m~VA O----if- + n ~ Oa
]
(146)
ITINERANT INTERMETALLICS AND ALLOYS
ao b
(
387
1 OG ° 1 OG ° 1 m~VA O--b- + n---~A OV~-b'-- -m- N A Ov a
= M y a = v a exp -
(147) obl/m+Xoal/n-x : L 1 / m n
exp
1 °OG n ( m + x ) N A Ov b
1 6
m(n - x)N A (148)
From eqs. (140) and (146)-(148), the concentrations of the structural defects can be calculated, provided the configurational entropy G O is known. In practice we use the 'quasi-chemical' relations, ab = K ,
bv b =
Mv,,
VbUan ,, = L ,
(149)
where K, L and M are considered as 'reaction constants', which are to be determined experimentally. Obviously, the reaction constant K is related to the exchange of an A and a B atom. M corresponds to a process in which a vacancy on the A sublattice is filled by a B atom, leaving a vacancy on the B sublattice. L corresponds to the addition of an empty cell. In all cases, the added defects are distributed at random over the appropriate sublattices. Although the reaction constants are, in general, not very well known, and also the dependences of the quantity under consideration on the various structural defects is not clear, we do try a few reasonable choices and are thus able to derive a value for that quantity in the perfectly ordered stoichiometric compound. Where the extrapolated value appears to be rather independent of the choices mentioned above, we assume to have obtained a satisfactory value for the perfectly ordered compound. In the next section we apply this procedure to the ordered C % A u - t y p e compounds Ni3A1 and Ni3Ga. Equation (149) can then be applied with m = 3 and n = 1. In section 4.3 we apply the procedure to ordered CsCl-type Ti(Fe,Co) compounds. For these quasi-binary compounds we introduce separate parameters for both Fe and Co antistructure atoms and the quasi-chemical relationships are adapted accordingly. 4.2. Magnetic properties o f stoichiometric perfectly ordered Ni3Al The composition dependence of the Landau coefficient A [see A in eq. (63)] is shown in fig. 28. The procedure outlined in the preceding section is applied in the following form. First of all, we assume that the vacancy concentrations are negligible (Aoki and Izumi 1975). Denoting the fraction of Ni antistructure atoms by n and the fraction of Al (or Ga) antistructure atoms by g, we assume the 'inverse susceptibility' A to be a linear function of n and g (or of x -- 3g - n). Then we find, as
388
P.E. BROMMER and J.J.M. FRANSE
\~
~-8o ~
,
I x Ni3+ x Go1_x
5 E
4.2K
_
Ni3+ x All_ x ~ i~ o high field 4.2K \ ~ + + high temp. ,, ~''Q-~..~'~.~.........+ -t+l
73
I 74
x~O 75 (75+25x1
I 76 (°/o Nil
Fig. 28. The inverse susceptibility A as a function of the composition in a series of Ni3÷,AI 1 x compounds and a series of Ni3÷=Gal_ x compounds. Low-field data: Buis (1979) and Buis et al. (1980). High field (up to 30 T) and 'high temperature' (i.e., extrapolated down to T = 4.2 K, see text) data from De Boer (1969).
the result of a fitting procedure, n g = K = 3 x 10 -5 ,
A -- - 2 1 . 5 + 1000n - 1165x
(150) [T mol Ni/A m2].
(151)
Consequently, the concentrations of antistructure atoms at the stoichiometric composition would be n = 3g = 1%. The value for A o does not change very much in case we allow the vacancies to play a role in the ordering model a n d / o r in the dependence of A on the structural defects. The value given in eq. (151) corresponds to the high-field data obtained at 4.2 K. The error is estimated to be about 3 T m o l / A m z. In order to obtain the other magnetic parameters for the perfectly ordered compound, we assume that the concentration of defects is so small, that the unenhanced quantities are given by those of the actual compounds (see table 4). The temperature dependence of the inverse susceptibility a ( T ) = A ~ is given by De Boer (1969) (see also de Chfitel and D e Boer 1970, discussed in section 3.2), a ( T ) = a(O) + a l T 2 + a2 T4 ,
(152)
with a 1 = 3.13 x 1 0 - 4 T kg/A m 2 K 2 and a 2 = - 5 . 7 × 1 0 - 9 T kg/A m 2 K 4 for all Ni
ITINERANT INTERMETALLICS AND ALLOYS
389
TABLE 8 Magnetic and magnetovolume parameters of perfectly ordered stoichiometricNi3A1. a(T) is the inverse susceptibility; b the inverse slope of the Arrott plot; o"0 the spontaneous magnetic moment at zero temperature; Tc the Curie temperature. Quantity
Units
Value
a(T = 0 K) Oa/OP
Tkg/A m2 T kg/A m2Mbar
b %
T kg3/A 3 m 6
-1.46 15.4 0.02 8.49 0.077 0.108 -0.408 -5.29 63 -6.35
O%/OP 0 In Cro/OP Tc 0 In Tc/aP
A m2/kg /zB/at tz~/Ni /zB/atMbar Mbar-1 K Mbar-1
concentrations. As indicated in fig. 28 the a(0) values are shifted upwards with respect to the high-field values. The same shift is applied to the value for the perfectly ordered material, yielding a(0) = ( - 1 . 4 6 + 0.3) T K g / A m 2 or A(0) = ( - 2 1 . 5 + 4.4) T m o l N i / A m 2. With this temperature dependence the Curie temperature for the perfectly ordered stoichiometric compound is calculated to be 63 K. The Landau coefficient B is taken equal to the observed value for the actual stoichiometric compound. The resulting magnetic properties for the perfectly ordered material were used to construct table 8, taken from Buis et al. (1981a). A direct comparison between these experimental results and those calculated with the spin-density functional method (Hackenbracht and Kfibler 1980) is now possible. Buis et al. (1981a) remark that the calculated results cannot be described consistently in the Stoner model. Probably the deformation of the spin-polarized subbands should be taken into account. Nevertheless, the experimentally determined value for the magnetic moment per atom, i.e., 0.077p~, can be compared with the calculated values 0.023/ZB/at at the calculated lattice parameter, 0.059/.hJat obtained by linear extrapolation to the actual lattice parameter and 0.052/.q3/at obtained by extrapolation of o-2 rather than of o-, respectively. The calculated pressure dependence, Oo-/OP= - 0 . 5 0 2 / z B / a t Mbar agrees reasonably well with the experimental value --0.41/Xs/at Mbar. Finally, the calculated value 0 ln I/Ow = - 2 . 6 can be compared with the value F ( a - 1 ) = - 3 . 1 9 , derived from the fundamental parameters given in table 1.
4.3. Magnetic properties of Ti(Fe,Co) compounds In the Ti(Fel_xCOx) compounds, the Ti atoms occupy one of the simple cubic sublattices of the CsC1 structure. The Fe and Co atoms are distributed at random on the other sublattice. Ferromagnetism occurs roughly for 0.3 < x < 0.8 with, at x = 0.5, a maximum average magnetic moment of 0.18p~ per cell, i.e., 0.15/.q3 per
390
P.E. B R O M M E R and J.J.M. FRANSE
(Fe,Co) atom and 0.03/~ on a Ti atom. The density of states was calculated in the spin-density formalism by K/ibler (1980). Some magnetization data were discussed already in section 2.2 (see fig. 4). Giner and Gautier (1977) calculated the CPA density of states for an antistructure (a.s.) Fe or Co atom, and suggested that these defects, in particular the majority of Fe a.s. atoms, could play an important role in the occurrence of ferromagnetism in these compounds. Ikeda (1974) suggested that local moments would be formed on Fe a.s. atoms in off-stoichiometric Wil+yFel_y (if surrounded by 8 Fe atoms). Buis et al. (1978) demonstrated local moment behavior in the Fe-rich stoichiometic Ti(Fe,Co) compounds. Hilscher et al. (1981) showed by a careful analysis of M6ssbauer data that in the stoichiometric compounds Fe a.s. atoms do carry a local moment. Their analysis yielded up to 2% Fe a.s. atoms with a moment of up to 2/.q3. In this way the conjecture of Brommer et al. (1980) was confirmed (see section 2.5). H61scher (1981) found that specific heat peaks in the Fe-rich compounds were field dependent in accordance with this picture. Taking these peaks into account, H61scher determined accurate values for Yexp. For the ferromagnetic compound Ti(Fe0.5Co0.5) the 'paramagnetic' value was found by correcting for the field dependence in a direct way (see also section 3.2). Moreover, H61scher analyzed the thermal expansion data according to, O~ = a e x p T +
c T 3/2 ,
and observed rather large contributions proportional t o T 3/2 (table 9). Such contributions were not found in the analysis of the specific heat, although a combined analysis as discussed in section 3.4 was not carried out. Anyhow, the TABLE 9 The coefficient of the linear term in the specific heat, Yoxp (for TiFe0.65Co0.35 the corrected value is given too), and the coefficients of the expansion of the linear thermal expansivity according to a = a e x p T + c T 3/2 (see H61scher 1981). The entropy in the observed specific heat peak is given too (H61scher et al. 1980). aex p (10 SK-2)
C
%xp
x
(10 8K-5/2)
( m J / m o l a t K 2)
Speak (mJ/mol at K)
0.0 0.15 0.25 0.30 0.35 0.40 0.50 0.56 0.60 O.65 0.75 0.90 1.0
0.37 -1.0 -2.5 -1.6 -1.7 -4.0 -3.7 -2.7 3.2 3.9 1.8 1.3
0.0 0.19 -0.51 0.37 0.38 0.77 0.74 0.58 -4.4 -3.8 0.0 0.0
2.0 4.4
12 17
6.1--* 8.5
-0
13.55 12.0 10.3
12 7.5 0
ITINERANT INTERMETALLICS AND ALLOYS
391
Griineisen parameter for the T 3/2 term must be very large (>50), whereas the 'electronic' Gr/ineisen parameter F can be estimated to assume values between 5 and 15. In this section we apply the analysis discussed in section 4.1 to a series of off-stoichiometric Til+y(Fe0,sCo0.5)~_y compounds (see Buis et al. 1981b). In these compounds, the pressure dependence of T c diverges for vanishing Tc, thus indicating itinerant behavior. In contradistinction, 0 T c / O P remains finite in the Fe-rich stoichiometric compounds. For simplicity, here we assume, again, that the vacancy concentrations are negligibly low (see Ikeda 1974). Denoting by f and g the fractions of the Ti sites occupied by Fe and Co atoms, respectively, and by t the fraction of (Fe,Co) sites occupied by Ti atoms we find, t=f+
(153)
g-y,
tf=(1 -x)K1,
(154)
tg= xK 2 ,
t ( f + g) = g ( x ) = (1 - x ) g I + x g 2 .
(155)
Obviously, the formalism allows the reaction constant to be composition dependent. The (zero temperature) Landau coefficient A depends strongly on the deviation from stoichiometry (see fig. 29). Buis et al. (1981b) assumed A to depend linearly on the concentration of a.s. atoms and found, with K(0.5)= 1.07 x 10 -4, a = AM = 0.4 - 167(f + g) A = 3 . 8 0 - 1592(f + g)
[T kg/A mZ], or
[Tmolf.u./Am2].
(156) (157)
Hence, at the stoichiometric composition the antistructure atom concentrations are t = ( f + g)---1%. From the experimental data it is clear that contributions proportional to t or v should be small. Incorporating such contributions, we find only slightly lower values for A 0 (for vacancy concentrations up to 1%). The A 0 value found in fig. 29 was used in the analysis of the Arrott plots in section 2.5 (fig. 12). There it was assumed that (some) a.s. Fe atoms carry a local moment and thus trigger ferromagnetism, whereas the 'ideal' matrix is a strongly enhanced itinerant paramagnet. In such a picture it is perhaps more appropriate to approximate A by, A = A o-jtdMfJMt(H=
0),
see eq. (79) with Mrs ~ M t ,-~ M t ( H = 0). Since M , ( H = O) ,-~ ( j f d M ~ s / B ) 1/3, one expects A to have roughly the form, A = A o- A'(f
+ g)z/3 + . . . .
392
P.E. BROMMER and J.J.M. FRANSE I
E < o o
i
I
-6 E i--
/
o
/ _
I
'
'
I
/ 0
'
f
° ~
o
/
/
<
-20 Ti,i+y (Fe. 1.b Co5) . Y
/ y0 I
,
49
I
50
,
I
,
51 50 + 50 y
I
,
,
52 (%Ti)
Fig. 29. The zero temperature 'inverse susceptibility' A for a series of off-stoichiometric Til+r(Fe0.sCo0.s)l_y alloys. Data from Buis et al. (1981b). The drawn line is obtained by fitting according to eqs. (153)-(155) with K(0.5) = 1.07 × 10-a. The dashed line indicates the behavior of perfectly ordered compounds [see eq. (158)]. A n analysis along these lines yields, a = 0.57 - 5 0 0 ( f + g)2/3
[ T k g / A m2],
(158)
with K(0.5) = (0.75%) 2 This exercise again leads to the conclusion that the A 0 value can be determined rather well, irrespective of the precise physical or ordering model. The analysis of the A r r o t t plots of the stoichiometric Ti(Fel_xCox) compounds is h a m p e r e d by the lack of knowledge of the A 0 value, because for x # 0.5, no data on off-stoichiometric compounds are available. Nevertheless, from the experimental data it is clear that A 0 almost vanishes at x = 0.5, and obtains larger values in both the Fe-rich and the Co-rich compounds. B r o m m e r et al. (1980) assumed the molecular field acting on the matrix to be proportional to a function, f ( x ) = (1 - x)[1 - x s - 8x7(1 - x) - 28x6(1
-
x)2],
i.e., only Fe atoms with at least 3 Fe nearest neighbours do possess a local m o m e n t . Taking the reaction constant K ( x ) to be independent of the concentration x, the observed A r r o t t plots can now be corrected. The results are given in table 10, but should be regarded as an educated guess only. Nevertheless, it is gratifying that, e.g., the molecular field constant jfd(=290 T mol f . u . / A m 2) is of the same order of magnitude as the one found in the Ni3AI(Fe ) system (see caption to fig. 11: Jfd = 330 T mol N i / A mZ).
ITINERANT INTERMETALLICS AND ALLOYS
393
TABLE 10 The Landau coefficients A and B, the saturated antistructure-atom magnetic moment M~s, and the corresponding internal field,/x0Hof~= jf~Mfs, with J~d= 290 T tool f.u./A m z, for ideally ordered stoichiometric TiFel_xCox compounds. Constructed from data of: [1] Buis et al. (1981b), [2] Beille and Towfiq (1978) and [3] Waterman and Franse (1980). The value for Mrs at x = 0.50 is chosen to be 0.052 A m2/mol f.u. (0.50 A m2/kg) in accordance with a magnetic moment of 1.9tzB per (magnetic) Fe antistructure-atom. x
A (T (Am2/mol f.u.) -~)
B (T (Am2/m01 f.u.) -3)
Mr, (Am2/mol f.u.)
/xoHo~~ (T)
98.6 55.5 16.7 3.8 5.7 23.6 85.2
75 58 29 13 7 42 ?
0.18 0.09 0.084 0.070 0.052 0.033 0.007 0.0
35 26 24 20 15 10 2 0
0 [1] 0.25 [1] 0.30 [11 0.40 [1] 0.50 [1] 0.60 [1] 0.77 [2] 1.0 [3]
Note on units F o r h o m o g e n e o u s m a t e r i a l s at z e r o t e m p e r a t u r e t h e A r r o t t p l o t is given b y t h e r e l a t i o n [eq. (31)], txoH/M = A + BM 2 ,
w h e r e M is t h e m o l a r m a g n e t i c m o m e n t (units A m 2 / m o l ) . T h e i n v e r s e susceptibility A is e x p r e s s e d in units T m o l / A m 2, a n d B in units T ( m o l / A m2) 3. I n p r a c t i c e , h o w e v e r , t h e specific m a g n e t i c m o m e n t cr is u s e d m o r e often. W e w r i t e , I % H / o " = a + b o "2 .
Since o- = M / ~ a=A~
(~ : molar weight), we have, [ T k g / A m 2]
and
b=B~
3
[T/(kg/Am2)3].
T h e a c t u a l m a g n e t i z a t i o n ( m a g n e t i c m o m e n t p e r unit v o l u m e ) is po- ( p : d e n s i t y , units k g / m 3 ) , o r p M / ~ = M / V . T h e ( d i m e n s i o n l e s s ) SI s u s c e p t i b i l i t y is g i v e n b y , Xsi = M / V H
= ~ o ( V A ) -1 = t z o t ) / a .
Finally, we remark that, /x0H = 1 T o- = 1 A m e / k g M = 1 A m2/mol txoH/o- = 1 T k g / A m z ~oH/M = 1 T mol/A m 2
corresponds corresponds corresponds corresponds corresponds
to to to to to
B = 104 G ( H = 104 O e ) ; or = 1 e m u / g ; M = 103 e m u / m o l ; H/~r = 104 g / c m 3 ; a n d H / M = 10 e m u m o l .
394
P.E. BROMMER and J.J.M. FRANSE
List of symbols Some frequently occurring symbols are listed below. Unless explicitly stated otherwise, they have, in this chapter, the following meaning:
Symbol c
g 2 m P A Aband
Ao, B C, Cst F I, Ib
t2, I4 M, Mst P, (em)
S
Se
7"F, Zc, Ts V, W OL
/3 ~7 /(=b
-1
A
F
Quantity Unit molar specific heat (m)J/mol K (eV spin atom) -1 density of states (A mZ/mol) 2 fluctuations of the molar magnetic moment A(T¢)/A(OK) T mol/A m E inverse susceptibility, first Landau coefficient band part of A (and so on) OA/Oto = V OA/OV (and so on) T (mol/A m2) 3 second Landau coefficient T mol/A m 2 magnetovolume parameter, its Stoner value molar free energy J/mol Stoner interaction, its bare value eV spin atom coefficients describing the exchange interaction A m2/mol molar magnetic moment, its Stoner value molecular weight kg/mol (magnetic) pressure Pa, Mbar molar entropy J/mol K also: spin quantum number also: enhancement factor XlXband, also (1 - [)--1 'electronic' enhancement factor Y/Ybare Fermi, Curie, Stoner temperature K molar (magnetic) volume ma/mol band width coefficient term ~ T in linear thermal expansivity K -2 also: electron-phonon enhancement coefficient term ~ T in volume thermal expansivity K -2 2 2 ratio m/Mstl~c compressibility Mbar- 1 ratio I / I b (1/g) OJg/OE j relative (magnetic) volume Grfineisen parameter
References Acker, F., and R. Huguenin, 1979, J. Magn. & Magn. Mater. 12, 58. Aeppli, G., E. Bucher and G. Shirane, 1985, Phys. Rev. B 32, 7579. Aldred, A.T., 1979, J. Magn. & Magn. Mater. 10, 42. Aoki, K., and O. Izumi, 1975, Phys. Status Solidi A 32, 657.
Barnea, G., and D.M. Edwards, 1977, J. Phys. F 11, 2417. Barron, T.H.K., J.G. Collins and G.K. White, 1980, Adv. Phys. 29, 609. Beille, J., and G. Chouteau, 1975, J. Phys. F 5, 721. Beille, J., and F. Towfiq, 1978, J. Phys. F 8, 1999.
ITINERANT INTERMETALLICS AND ALLOYS Boerstoel, B.M., J.J. Zwart and J. Hensen, 1972, Physica 57, 397. Brommer, P.E., 1982a, Physica B 112, 343. Brommer, P.E., 1982b, Physica B 113, 391. Brommer, P.E., and J.J.M. Franse, 1984, J. Magn. & Magn. Mater. 45, 129. Brommer, P.E., and J.J.M. Franse, 1988, Physica B 149, 221. Brommer, P.E., J.J.M. Franse and H. H61scher, 1980, Inst. Phys. Conf. Ser. 55, 279. Brommer, P.E., J.J.M. Franse, B.M. Geerken, R. Griessen, H. H61scher, J.A.M. Kragtwijk, J.A. Mydosh and G.J. Nieuwenhuys, 1981, Inst. Phys. Conf. Ser. 55, 253. Buis, N., 1979, Thesis (Amsterdam). Buis, N., J.J.M. Franse and C.J. Schinkel, 1978, Inst. Phys. Conf. Set. 39, 389. Buis, N., P.E. Brommer, P. Disveld, M.S. Schalkwijk and J.J.M. Franse, 1980, J. Magn. & Magn. Mater. 15-18, 291. Buis, N., J.J.M. Franse and P.E. Brommer, 1981a, Physica B 106, 1. Buis, N., P. Disveld, P.E. Brommer and J.J.M. Franse, 1981b, J. Phys. F 11, 217. Coles, B.R., H. Jamieson, R.H. Taylor and A. Tari, 1975, J. Phys. F 5, 565. Creuzet, G., and I.A. Campbell, 1983a, J. Phys. Lett. 44, 547. Creuzet, G., and I.A. Campbell, 1983b, J. Magn. & Magn. Mater. 37, 173. Creuzet, G., A. Hanizi6 and I.A. Campbell, 1981, Solid State Commun. 39, 451. de Boer, F.R., 1969, Thesis (Amsterdam). de Chatel, P.F., and F.R. de Boer, 1970, Physica 48, 331. de Dood, W., and P.F. de Chatel, 1973, J. Phys. F 3, 1039. de Visser, A., 1986, Thesis (Amsterdam). de Visser, A., J.J.M. Franse, A. Menovsky and T.T.M. Palstra, 1984, J. Phys. F 14, L191. de Visser, A., J.J.M. Franse and A. Menovsky, 1985, J. Phys. F 15, L53. de Visser, A., P. Haen, E Lejay and J. Flouquet, 1988, J. Magn. & Magn. Mater., to be published. Edwards, D.M., and E.P. Wohlfarth, 1968, Proc. R. Soc. London 202, 127. F~hnle, M., 1980, Phys. Status Solidi B 99, 547. F~ihnle, M., and H. Kronmfiller, 1980, Phys. Status Solidi B 98, 219. Fawcett, E., and V. Plushnikov, 1983, Physica B 119, 161. Felten, R., 1987, Thesis (Darmstadt). Franse, J.J.M., 1977, Physica B 86-88, 283.
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Franse, J.J.M., H. H61scher and J.A. Mydosh, 1980, J. Magn. & Magn. Mater. 15-18, 179. Franse, J.J.M., P.H. Frings, F.R. de Boer and A. Menovsky, 1981, in: Physics of Solids under High Pressure. Franse, J.J.M., P.H. Frings, A. de Visser, A. Menovsky, T.T.M. Palstra, P.H. Kes and J.A. Mydosh, 1984, Physica B 126, 116. Franse, J.J.M., K. Kadowaki, A. Menovksy, M. van Sprang and A. de Visser, 1987, J. Appl. Phys. 61, 3380. Franse, J.J.M., M. van Sprang, A. de Visser and P.E. Brommer, 1989, Physica B 154, 379. Frings, P.H., 1984, Thesis (Amsterdam). Frings, P.H., B. Renker and C. Vettier, 1987, J. Magn. & Magn. Mater. 63-64, 202. Fulde, P., J. Keller and G. Zwicknagl, 1988, Solid State Phys. 41, 2. Geerken, B.M., and G.J. Nieuwenhuys, 1982, Physica B 115, 5. Giner, J., and F. Gautier, 1977, J. Phys. (France) Colloq. 7, 301. Hackenbracht, D., and J. Kiibler, 1980, J. Phys. F 10, 427. Hargitai, C., S. Shtrikman and E.P. Wohlfarth, 1972, Phys. Lett. A 39, 87. Herzer, G., M. F/ihnle, T. Egami and H. Kr6nmuller, 1980, Phys. Status Solidi B 101,713. Hilscher, G., and E. Gmelin, 1978, J. Phys. (France) Colloq., Suppl. 6, 774. Hilscher, G., and E. Gratz, 1978, Phys. Status Solidi A 48, 473. Hilscher, G., N. Buis and J.J.M. Franse, 1977, Physica B 91, 170. Hilscher, G., G. Wiesinger and R. Hempelmann, 1981, J. Phys. F 11, 2161. Hirooka, S., 1988, Physica 149, 156. H61scher, H., 1981, Thesis (Amsterdam). H61scher, H., J.C.P. Klaasse, J.J.M. Franse and P.E. Brommer, 1980, Inst. Phys. Conf. Ser. 55, 283. Ikeda, K., 1974, Phys. Status Solidi B 62, 655. Janak, J.F., and A.R. Williams, 1976, Phys. Rev. B 14, 4199. Johnson, S.M., A. Benoit, P.J. Brown, H. Capelmann, J. Flouquet, J.J.M. Franse, P.H. Frings and K.R.A. Zwiebeck, 1986, Z. Phys. 63, 335. Kaiser, A.B., and P. Fulde, 1988, Phys. Rev. B 37, 5357. Kanamori, J., 1963, Prog. Theor. Phys. 30, 275. Keller, R., J. Ortelli and M. Peter, 1970, Phys Rev. Lett. A 31, 376.
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Kortekaas, T.F.M., 1975, Thesis (Amsterdam). Kortekaas, T.F.M., and J.J.M. Franse, 1976, J. Phys. F 6, 1161. Kortekaas, T.F.M., and J.J.M. Franse, 1977, Phys. Status Solidi A 40, 479. Kortekaas, T.F.M., J.J.M. Franse and H. H61scher, 1974, Phys. Lett. A 50, 153. Kouvel, J.S., and J.B. Comley, 1968, Phys. Rev. Lett. 20, 1237. Kronm/iller, H., and M. F~ihnle, 1980, Phys. Status Solidi B 97, 513. Kiibler, J., 1980, J. Magn. & Magn. Mater. 15-18, 859. Lander, G.H., A.T. Aldred, B.D. Dunlap and G.K. Shener, 1977, Physica B 86-88, 152. Lang, N.D., and H. Ehrenreich, 1968, Phys. Rev. 168, 605. Lonzarich, G.G., 1986, J. Magn. & Magn. Mater. 54-57, 612. Lonzarich, G.G., and L. Taillefer, 1985, J. Phys. (USA) C 18, 4339. Luong, N.H., J.J.M. Franse and T.D. Hien, 1985, J. Magn. & Magn. Mater. 50, 153. Matsunaga, M., Y. Ishikawa and T. Nakajima, 1982, J. Phys. Soc. Jpn. 51, 1153. Menyhfird, N., 1970, Solid State Commun. 8, 1337. Moriya, T., 1985, Spin Fluctuations in Itinerant Electron Magnetism (Springer, Berlin). Moriya, T., and A. Kawabata, 1973, J. Phys. Soc. Jpn. 35, 669. Moriya, T., and K. Usami, 1980, Solid State Commun. 34, 95. Murata, K.K., and S. Doniach, 1972, Phys. Rev. Lett. 29, 285. Nieuwenhuys, G.J., and B.H. Verbeek, 1977, J. Phys. F 7, 1497. Nieuwenhuys, G.J., B.H. Verbeek and J.A. Mydosh, 1979, J. Appl. Phys. 50, 1685. Ogawa, S., 1983, Physica B 119, 68. Oraltay, G.G., J.J.M. Franse, P.E. Brommer and A. Menovsky, 1984, J. Phys. F 14, 737. Ott, H.R., 1987, Prog. Low Temp. Phys. 11. Ramakrishnan, T.V., 1974, Phys. Rev. B 10, 4014. Renker, B., F. Gompf, J.B. Suck, H. Rietschel and P.H. Frings, 1986, Physica B 136, 376. Renker, B., F. Gompf, E. Gering, P.H. Frings, H. Rietschel, R. Felten, F. Steglich and G. Weber, 1987, Physica B 148, 41. Sato, M., 1974, Solid State Commun. 15, 1863. Schalkwijk, M.S., P.E. Brommer, G.J. Cocq and C.J. Schinkel, 1971, J. Phys. (France) Colloq. 32, 997. Schinkel, C.J., F.R. de Boer and B. de Horn, 1973, J. Phys. F 3, 1463.
Shapero, D., J.R. Cullen and E.R. Callen, 1974, Phys. Lett. A 50, 303. Shimizu, M., 1981, Rep. Prog. Phys. 44, 329. Shtrikman, S., and E.P. Wohlfarth, 1972, Physica 60, 427. Slater, J.C., 1936, Phys. Rev. 49, 537, 931. Smit, J.J., G.J. Nieuwenhuys and L.J. de Jongh, 1979, Solid State Commun. 30, 243. Star, W.M., S. Foner and E.S. McNiff, 1975, Phys. Rev. B 12, 2690. Stewart, G.R., 1984, Rev. Mod. Phys. 56, 755. Stoner, E.C., 1938, Proc. R. Soc. London Ser. A 165, 372. Suzuki, K., and Y. Masuda, 1985, J. Phys. Soc. Jpn. 54, 326, 630. Taillefer, L., R. Newbury, G.G. Lonzarich, Z. Fisk and J.L. Smith, 1986, J. Magn. & Magn. Mater. 63-64, 372. Takahashi, T., and M. Shimizu, 1965, J. Phys. Soc. Jpn. 20, 26. Thalmeier, P., and P. Fulde, 1986, Europhys. Lett. 1, 367. Tsvelick, A.M., and P.B. Wiegmann, 1983, Adv. Phys. 32, 453. Turov, E.A., and V.I. Gebbenikov, 1988, Physica B 149, 150. Usami, K., and T. Moriya, 1981, in: Electron Correlation and Magnetism in Narrow Band Systems (Springer, Berlin) p. 84. van Sprang, M., 1989, Thesis (Amsterdam). Wagner, D., and E.P. Wohlfarth, 1979, J. Phys. F 9, 717. Wagner, D., and E.P. Wohlfarth, 1981, J. Phys. F 11, 2417. Wagner, D., and E.P. Wohlfarth, 1986, Phys. Lett. A 118, 29. Waterman, E.H., and J.J.M. Franse, 1980, J. Phys. F 10, 947. Wohlfarth, E.P., 1968, J. Appl. Phys. 39, 1061. Wohlfarth, E.P., 1969, J. Phys. (USA) C 2, 68. Wohlfarth, E.P., 1977, Physica B 91, 305. Wohlfarth, E.P., 1980a, Inst. Phys. Conf. Set. 55, 161. Wohlfarth, E.P., 1980b, Solid State Commun. 35, 797. Wohlfarth, E.P., and P. Mohn, 1988, Physica B 149, 145. Yamada, H., 1975, Phys. Lett. A 55, 235. Yamada, H., and E.P. Wohlfarth, 1973, Phys. Status Solidi B 58, K151. Yamada, H., and E.P. Wohlfarth, 1974, Phys. Status Solidi B 64, K71. Yamada, H., and E.P. Wohlfarth, 1975, Phys. Lett. A 51, 65. Zweers, H.A., and G.J. van den Berg, 1975, J. Phys. F 5, 555.
chapter 5 FIRST-ORDER MAGNETIC PROCESSES
G. ASTI Physics Department University of Parma 43100 Parma Italy
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Phenomenology of first-order magnetization processes in terms of anisotropy constants. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Uniaxial symmetry . . . . . . . . . . . . . . . . . . . . . . . 2.3. Cases of trigonal, hexagonal and tetragonal symmetry . . . . . . . . . . . 2.4. Cubic symmetry . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Mechanisms responsible for high-order anisotropy constants . . . . . . . . . 2.6. Domain wall processes at the transition point . . . . . . . . . . . . . . 3. Processes involving competition between exchange and anisotropy . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Role of exchange at transition points . . . . . . . . . . . . . . . . . 3.2.1. Multisublattice magnetic systems with uniaxial anisotropy . . . . . . . . 3.2.2. Field-induced spin reorientation transitions. Critical fields . . . . . . . . 3.2.3. Boundary conditions for first-order transitions . . . . . . . . . . . 3.2.4. The two-sublattice system . . . . . . . . . . . . . . . . . . 3.2.5. Linear regime and magnetic transitions in the canted phase . . . . . . . 3.3. The small-angle canting model . . . . . . . . . . . . . . . . . . . 3.3.1. General remarks . . . . . . . . . . . . . . . . . . . . . 3.3.2. The two-sublattice model . . . . . . . . . . . . . . . . . . 3.3.3. Some applications . . . . . . . . . . . . . . . . . . . . . 3.4. Hexagonal ferrites having block-angled and spiral structures. Role of antisymmetric exchange . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Considerations on the experimental methods . . . . . . . . . . . . . . . . 4.1. Measurements in high magnetic fields . . . . . . . . . . . . . . . . 4.2. The singular-point detection technique . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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399 402 402 402 414 419 421 423 427 427 428 428 430 432 433 440 447 447 448 453 454 455 455 457 462
1. General
The magnetization curve of a magnetically ordered crystal, ferro- or ferrimagnetic, can, under special circumstances, show anomalous discontinuities that appear as sudden increases of the magnetization with increasing magnetic field. In these conditions, the crystal actually undergoes a first-order magnetic transition, with characteristics that are, in general, strongly dependent on the relative orientation of the applied magnetic field direction and the crystal axes. The observed phenomenon is an irreversible process of a different nature than the ordinary magnetization reversal, which always occurs when the applied magnetic field intensity is equal to the coercive field; a fact that is reflected in the well-known symmetry of the hysteresis loop of all ferromagnets (by the general term 'ferromagnet', we shall refer hereafter to both types of material, properly called ferromagnetic or ferrimagnetic, when the magnetic order is considered to be irrelevant for the phenomenon under study). These anomalous magnetization jumps, being first-order transitions, also ought to be accompanied by hysteresis loops, which would essentially be localized in the first quadrant of the magnetization curve, but, in general, the hysteresis around the transition happens to be very small and can only be revealed with difficulty. The observed jump in the magnetization curve can be due to a rotation of the magnetization vector, to a change of the modulus of the magnetization vectors, or, in complex systems, to a change of the magnetic order. In the first case, we deal with changes in the magnetic state of the system that can effectively be ascribed to its magnetic anisotropy, while in the second case, we are, in most cases, faced with direct manifestations of microscopic effects, such as metamagnetic transitions that can be the consequences of particular crystal-field effects, and/or the effect of the magnetic field on the energy-band structure. The third case involves effects of the type of spin-flop transitions, with drastic changes in magnetic structures. In the present chapter we will not consider the second type of transitions, which has been treated in the reviews by Wohlfarth (1980, 1983), Buschow (1980) and Khan and Melville (1983). Some aspects of the third type of phenomena, i.e., the case of anisotropic ferrimagnets, will be examined in section 3.2. In section 2 we deal with the first type of field-induced first-order transition, hereafter referred to as FOMP [first-order magnetization process (Asti and Bolzoni 1980)]. In this case, 399
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G. ASTI
the magnetic anisotropy of the system by itself allows the existence of two inequivalent minima of the free energy that correspond to two distinct directions of the magnetization vector. The effect of the external field is to produce a relative shift between the energies of the two states involved in the transition. When the two states reach the same energy the transition can start, provided a mechanism exists that allows the nucleation and growth of the new phase. It is likely that this nucleation occurs inside the domain wall (see section 2.6). When the changing parameter is temperature, the system can make the transition in zero external field, making a sudden rotation of the easy-axis direction. This type of transition is usually referred to as a spin reorientation transition (SRT), and is essentially the same phenomenon as FOMP. In fact, in two-dimensional phase diagrams, energy versus H or T, the transition is represented by lines that in general can be crossed by changing either T or H. FOMPs in cubic crystals have been considered in the frame of the problem of phase transitions and critical phenomena (Mukamel et al. 1976, Cullen and Callen 1984) as examples of the Potts model (see section 2.4). However, the widest application of the concepts developed around the FOMP transition is indeed in studies of the anisotropic properties of multisublattice magnetic systems, especially rare earth (R) intermetallic compounds. In practice, it is very common to find this type of behaviour in these materials. The study of FOMP and the measurement of critical parameters, on one hand, provide a great deal of information about the magnetic anisotropy and, on the other hand, the interpretation of observed FOMPs represents a formidable test for a theory of magnetic anisotropy in these systems. Moreover, one has to consider the inherent precision and proper reliability of data about critical parameters, as compared to best fit approach to saturation and similar procedures. A phenomenological description of FOMP in terms of the anisotropy constants or the anisotropy coefficients shows that these transitions require high-order terms in the expansion of the free energy. Crystal-field theory applied to single ions does not in many cases justify the existence of such high-order terms. This fact by itself indicates how important it is to take the competing interactions typical of multisublattice systems into account, as, e.g., the competition between anisotropy of different sublattices (Cullen and Callen 1985). In this respect, the exchange interaction also plays an important role, especially for the case of ferrimagnetic order (Sarkis and Callen 1982). Indeed, the presence in R intermetallic compounds of exchange of the same order of magnitude as the anisotropy energy implies deviations from the collinear magnetic order. Even for small canting angles, the free energy of the system is strongly modified, and the macroscopic anisotropy properties are influenced in various ways. Hence, the classical phenomenological description is inadequate, sometimes leading to wrong conclusions. For instance, non-monotonic dependences of the anisotropy constant in pseudobinaries, such as (Y, Nd)Cos, (Y, Gd)Cos, and (Sm,Nd)Cos, have erroneously been considered as evidence of the non single-ion origin of the anisotropy energy (Ermolenko 1979). A convenient approach to the problem, which applies to the majority of cases, is to treat the
FIRST-ORDER MAGNETICPROCESSES
401
small-angle canting (SAC) between sublattice moments as a first-order perturbation (Rinaldi and Pareti 1979). The advantage of this model (hereafter referred as the SAC model) is that the system can again in many respects be treated as an ordinary ferromagnet having certain effective anisotropy constants K1, g2,.... It means that the effect of canting of the sublattice moments can be merely accounted for by using proper expressions for the effective anisotropy constants; these include the exchange-interaction parameters and reduce to the ordinary algebraic addition of the anisotropy constants of the individual sublattices when the anisotropy is negligible with respect to the exchange-energy density, a condition that corresponds to normal collinear magnetic order. Actually, the K's obtained are not rigorously constant because they depend on the intensity of the magnetic field, but the model in many cases allows to get a simple physical picture of the mechanisms responsible for the observed phenomena. As a matter of fact, it gives an explanation of the existence in these intermetallic compounds of effective anisotropy constants of high order. Certainly, canting effects play a role in the presence of FOMP and non-linear effects in the magnetization curves. As mentioned above, at high magnetic fields, pronounced transitions of the spin-flop type are possible for ferrimagnetically ordered RE intermetaUic compounds. They most frequently involve direct competition between magnetostatic energy and exchange. With increasing fields, magnetic configurations with large angles between the magnetic moments can be energetically favorable. A particular type of first-order transition of this category, named first-order moment reorientations (FOMR), was observed by Radwanski et al. (1985) and extensively studied in easy-plane ferrimagnetic compounds of the R2T17 type (here T is a transition metal). In this case, the applied field indeed lies in the basal plane and the six-fold symmetry plays a role in allowing non-collinear moment configurations to become energetically favorable. Computed curves for ferrimagnetic systems with magnetic anisotropy, presenting transitions between different magnetic configurations have also been described by other authors (Sannikov and Perekalina 1969, Kazakov and Litvinenko 1978, Rozenfeld 1978, Gr6ssinger and Liedl 1981, Ermolenko 1982, Radwanski 1986). A certain analogy exists with the phase diagrams of antiferromagnets that show spin-flop transitions. Classic treatments on angled spin structures but without anisotropy (Clark and Callen 1968), show continuous transitions between various structures. Among the peculiar effects found, are a temperature-independent susceptibility (Clark and Callen 1968) and an effective decoupling between the sublattices in the canted phase (Meyer 1964, Boucher et al. 1970, Acquarone and Asti 1975). A theoretical model (Acquarone 1981) of an antiferromagnetic chain including anisotropic and antisymmetric exchange, as well as anisotropy terms up to sixth order, explains the observed phenomena in the so-called 'angled block structures' in complex ferrimagnetic oxides of hexagonal and trigonal structure. Certain compositions show field-induced transitions and unusual effects, because the system behaves like an effective two-sublattice canted antiferromagnet, so as to give rise to various possible configurations: angled, conical helix or flat spiral.
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2. Phenomenology of first-order magnetization processes in terms of anisotropy constants 2.1. Introduction
The anisotropy energy of a ferromagnetic crystal can be expressed as a power series of the direction cosines of the magnetic moment with respect to the crystal axes. So the expansion has different forms depending on the symmetry of the crystal. The coefficients of the various terms are the anisotropy constants and, in general, a satisfactory description of the magnetization curve is obtained by limiting the expansion to the first few terms. Under normal conditions, this curve is continuous in the region preceding saturation, where the magnetization process takes place by reversible rotation of the magnetization vector M s . However, when the anisotropy constants fall within certain ranges corresponding to some specific conditions, irreversible rotations of M S are possible, implying first-order transitions between inequivalent magnetization states. In fact, depending on the values of the anisotropy constants, the system may possess two or more inequivalent minima of free energy E for different magnetization directions; so it is evident that an applied magnetic field H of suitable intensity and orientation can induce transitions between these minima. Actually, the addition of a magnetostatic term modifies the total energy surface in such a way that the two energy levels become equal. The existence of two minima of E in zero field is not a necessary condition: the effect of the magnetostatic energy component is in certain circumstances enough to produce an additional minimum besides the absolute one (easy direction). In principle, in the case of F O M P we expect to observe minor hysteresis loops around the critical field, for which the two states have equal energy. However, it is difficult to observe hysteresis in F O M P even at low temperatures (Melville et al. 1976, Ermolenko and Rozhda 1978). The reasons probably lie in the very low coercivity inherent in this type of magnetization process (see section 2.6). 2.2. Uniaxial s y m m e t r y
A very special but important case is that of uniaxial symmetry. In this case the free energy of the system can be expressed as E -- K 1 sin e 0 + K 2 sin 4 0 + K 3 sin 6 0 + . . . .
H M s cos(0 - q0,
(2.1)
where K1, Kz,/(3,... , are the anisotropy constants and 0 and ~p are the orientation angles of the magnetization vector M s and the magnetic field H with respect to the symmetry axis c, respectively, As a first step, it is important to know the phase diagram of easy and hard directions. For reasons of symmetry, the c axis (0 = 0) and the basal plane (0 = ½It) are always points of extremum. With the anisotropy energy expanded up to the sixth order, we can have, at most,
FIRST-ORDER MAGNETIC PROCESSES
403
two additional extrema along conical directions at angles given by sin 0c = ( [ - K 2 ++-( K 2 - 3 K1K3) 1/2]/3K 3 }1/2.
(2.2)
Let C ÷ and C - be the cones associated with the + and - sign in eq. (2.2). It turns out that C ÷ is always a minimum while C - is always a maximum, so that only C ÷ can be an easy direction. A convenient representation of the diagram of the easy directions and the other extrema is representation in terms of reduced anisotropy constants x = K 2 / K 1 and y -- K 3 / K 1. These two variables are sufficient to distinguish all the possible cases. In fig. 1 the phase diagram for the two cases K i > 0 (upper) and K 1 < 0 (lower) is shown. All the information concerning the easy directions and the other extrema are contained in a special symbol that marks every different region. It simulates a polar type of energy representation indicating existing extrema by concave (minimum) and convex tips (maxima). Vertical and horizontal stems refer to the symmetry axis and the basal plane, respectively. The left-hand and right-hand oblique stems indicate the C - and C ÷ cones, respectively. The absolute minimum (easy direction) is indicated by filling
I
4
¥
L, ×
-2 Fig. 1. Magnetic phase diagram of the uniaxial ferromagnet: (Top) K~ > 0; (Bottom) K 1 < 0. T h e coordinates are x = Kz/K ~ and y = K3/K~. For the explanation of the symbols see text (Asti and Bolzoni 1980).
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G. ASTI
of the tip. The boundary lines of the various regions in the phase diagram of fig. 1 are specified by the equations labelled by the same letters in table 1. The determination of the conditions for the existence of FOMP requires the analysis of the magnetization curve and its dependence of the values of the anisotropy constants, for different directions of the magnetic field. For most purposes it is enough to consider the cases of H parallel or perpendicular to the c axis, hereafter indicated as the A-case and P-case, respectively, where A denotes axial while P stands for planar. The general case is relevant when considering the effects of FOMP in polycrystalline materials (see section 4). Here, it suffices to mention that a peculiar phenomenon is only observed for H lying not in a symmetry direction; under certain conditions of anisotropy constants and field orientations, an additional transition is possible, so that one observes a double FOMP, consisting of an ordinary and extraordinary FOMP (Asti and Bolzoni 1985). The equilibrium equation is OE/O0 = 0, where E is given by eq. (2.1). For the P-case (~ = ½) it turns out to be, h = 2m(1 + 2 x m a + 3 y m 4 ) K 1 / l K l [ ,
(2.3)
where m = M / M s is the reduced magnetization, h = 2H/IHAII the reduced field and HA1 = 2K1/Ms, the c axis anisotropy field. Using eq. (2.3) and the condition that the total energy is equal in the two phases, we obtain the equation for the critical magnetization, (3m 2 + 2m + 1)x + (5m 4 + 4m 3 + 3 m 2 + 2m + 1)y + 1 = 0.
(2.4)
For the A-case, the equilibrium equations corresponding to eqs. (2.3) and (2.4) are h = - 2 m [ 1 + 2x(1 - m 2) + 3y(1 -
m2)2]KJ[KI(1+ 2x + 3y)[,
(1 + m)(1 - 3 m ) x + (1 + m)(1 - m2)(1 - 5 m ) y + 1 = 0,
and
(2.3') (2.4')
TABLE 1 Equations of the boundary lines of the various regions in the phase diagram (fig. 1) and in the F O M P diagram (figs. 3 and 4). The lines are labelled by a letter that appears in figs. 1 and 3. The couples of curves conjugated in the K ~ R transformation [eqs. (2.6)-(2.8)] appear on the same line in the table. Curve
Label
Conjugated curve
Label
3y+2x+l=0 x+y+l=0 15y + 6 x + 1 = 0
1 m n
K I = 0 , i.e., x+y+l=0 4x- 1 =0
1'
4y - x 2=0
o
3 y 2 + 2xy - x z + 4y =0
3y-
p
3y-x
q
3x z + 8xy + 12y z - 5y = 0
r
x4+
x 2 =0
5y - 3x z = 0
x,y---~
n' o'
2 =0 q'
Eq. (2.11) in the text
x3y - 8xay - 3 6 x y a -
27y 3 +
16y 2 = 0
r'
FIRST-ORDER MAGNETIC PROCESSES
405
where h = 2 H / I H A 2 ] with HA2 = -2(K~ + 2 K 2 + 3 K 3 ) / M ~ , the anisotropy field along the basal plane. There is a transformation, called the K*+ R transformation, of the anisotropy constants into conjugate quantities that allows to transfer immediately all the results obtained for the P-case to the A-case and vice versa, according to the following symmetrical dual correspondence,
basal plane
c axis
K1 K2 K3
(2.5)
R1 R2 R3
The K ~ R transformation is obtained by imposing on the energy expressions the same formal dependence on the reduced magnetization m, for the A and P cases. So one obtains the following linear transformation, RI=-Ka-2K2-3K
3,
R2=K2+3K
3,
3.
(2.6)
/£3 = - R 3 .
(2.7)
R3=-K
The inverse transformation obviously has the same form, K 1 = - R 1 - 2R 2 - 3R 3 ,
K2 = R 2 + 3R 3 ,
In terms of reduced variables it becomes, 2= (-x-
3y)/(1 + 2 x + 3 y ) ,
x = (-2-
337)/(1 + 22 + 337),
fi = y / ( 1 + 2 x +
3y), (2.8)
y = 37/(1 + 2£ + 337),
where 2 = R 2 / R 1 and 37= R 3 / R 1. One can easily recognize that the obtained correspondence is a projective transformation, i.e., a homology having the center in the point C ( - 1 , 0) and the axis being the line with equation 2x + 3y = 0, with an invariant equal to - 1 , i.e., the transformation is a harmonic or involutory homology. Accordingly, second-order curves are transformed into other secondorder curves. The correspondence for some important lines is reported in table 1. Two types of F O M P are distinguished, depending on the fact whether the final state after the transition is the saturation state (type-1 F O M P ) or not (type-2 FOMP). They are represented in fig. 2. From the degree and the symmetry of the equilibrium equations, it turns out that when the easy direction is coincident with a cone (cone C +) type-2 F O M P cannot occur. From the equality of the total energy in the two states and from the equilibrium equations, one obtains the conditions for the occurrence of the two types of FOMP. Figure 3 summarizes the results, giving the complete diagram of F O M P in the planes of reduced anisotropy constants x and y for the two cases K 1 > 0 and K 1 < 0. The various regions are
406
G. ASTI c-axis
'
(a)
.r--. ;
Ha2
H
(b)
(c)
Ib
Fig. ½. Examples of FOMP different types (right-hand side): (a) type 1. (b) type 1 with easy cone. (c) type 2. The diagrams on the left-hand side are cross-sections of the anisotropy-energy surface (Asti and Bolzoni 1980). The arrows on the vertical portion of the M(H) curve (full line) indicate that the transition occurs at the same critical field both with increasing and decreasing magnetic field, when magnetic states having equal free energy are involved. This is actually the most common case for experimentally observed FOMPs, because the hysteresis around the transition is either very narrow or even not detectable. Upward and downward arrows on the two sides indicate upper and lower limits for the critical field, located at the onset of instability. They define the maximum theoretical width of the hysteresis cycle connected with the FOMP. distinguished by the labels A 1 , A 2 , P1, P2, P1C, A 1 C which specify the m a g n e t i c field direction (A, axial; P, planar) and the type o f F O M P (1 and 2), and the easy-cone o f the regions with type-1 F O M P ( A 1 C , P1C). T h e b o u n d a r y lines are those specified in table 1. T h e areas that are c o n j u g a t e d in the K ~ R transformation are s h a d e d in the same way. T h e analysis of the equations also provides the values o f the critical field her and the critical m a g n e t i z a t i o n mcr as functions o f the c o o r d i n a t e s x, y a n d the sign of K 1. T h e critical magnetization is defined as the m a g n e t i z a t i o n at which the transition takes place. Figure 4 gives the same diagrams as in fig. 3, but n o w with the lines of constant h , and incr. T h e s e plots can be used in analyzing experimental observations o f F O M P , and allow precise
FIRST-ORDER MAGNETIC PROCESSES .....
~, :,.-':-,
4
~
-
.;:,(U:,-',-
407
"Y
-:','-'.- 2 . , ' - ' , , . " ~ - - ' ~
x
....,......,........... -"-'--',-" .......- , - , ~ , ~ - p ( ~ i ' .
.....
!iiii! ~
'
"
!~:!i!i!~:##::!!?!ii:~!ii?i!!!!: !!':::il;i!il}i ii!},
.
x
.......! :::::::::::::::::::::::
Fig. 3. Diagram of FOMP in the plane of reduced anisotropy constants x = K2/K1, y = K 3 / K x. Top and bottom figures refer to the cases Kx > 0 and K~ < 0, respectively. The different regions are distinguished by a letter, i.e., P for the P-case or A for the A-case (see text), followed by the numeral 1 or 2 indicating the type of FOME Type-1 FOMP in regions of easy cone are specified by the addition of the letter C. The boundary lines are those specified in table 1. The areas that are conjugated in the K ~ R transformation (see text) are shaded in the same way (Asti and Bolzoni 1980). e v a l u a t i o n s o f t h e r a t i o s o f t h e a n i s o t r o p y constan~ts. I t is w o r t h n o t i n g t h a t i n t h e case o f t y p e - 2 F O M P t h e s y s t e m o f e q u a t i o n s c a n b e a n a l y t i c a l l y s o l v e d g i v i n g e x p l i c i t e x p r e s s i o n s for her as well as mcr = m 1 a n d m2, t h e r e d u c e d m a g n e t i z a t i o n s for t h e i n i t i a l a n d final s t a t e s , r e s p e c t i v e l y . T h e s e a r e , for t h e P - c a s e , hc~ = ~ A { 2 - [17x 2 + x ( 6 0 y
-
llx2)1/2]/30y}, (2.9)
m x = ½(A-
D),
m 2 = ½(A + D ) ,
where A = {[-3x + (60y - llx2)~/2]/lOy} D = {[-5x - (60y - llx2)l/2]/6y}
~/2 , t/2
408
G. ASTI 6
-2
-4
¥
-6
- 4
-2
.3
Fig. 4. Lines of constant values for the reduced critical magnetization, mcr (dashed lines), and reduced critical magnetic field her (full lines), in the plane of reduced anisotropy constants x = K 2 / K 1 , y = K 3 / K ~. The various regions are denoted as in fig. 3. In the case of type-2 FOMP, mcr is the initial state, i.e., m 1. The spacing between the values relative to contiguous lines of mcr or her, changes only when crossing a line labelled by its value. (a) K1 > 0, x > 0, y < 0. (b) K 1 > 0, x < 0, y > 0. (c) K~ > 0, x<0, y<0.(d) Kl<0, x>0, y>0.(e) Kl<0, x>0, y<0.(f) K~<0, x<0, y>0(AstiandBolzoni 1980).
409
FIRST-ORDER MAGNETIC PROCESSES -~
X
-4
0
-2
(c)
e ___ ~--~-~_
--_
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . -
. . . . . . .
2
~
. . . .
4
(a) Fig. 4.
(Cont.)
. 3 s -
6
410
G. ASTI 2
4
6
~---<.~ ,+.<
-2
I
i
i
~
',
-..<)
-4, Y
(e)
~',
\~
\;
t
I /
,'/
/,,1¥
e""4 ~X" \ \ \
\\
\\
,
I
\\
i
., \\
I
,A2
\x
~\~~ \! \ '/'.
\\
,
A1
,
/I
//
I 111
'
II
~I
//
[ t /X,,'/ .J.-" 1.'/. ~..--"t=
- 7 \ / " -~
'~..~.'-,.~,~,7 ~7~ j ""-
~_.~_~
~~-~.~
J( -i
'
Fig.
(f) 4. (Cont.)
-~
'
0
FIRST-ORDER MAGNETIC PROCESSES
411
and, for the A-case, her = } A ' { 2 - (x + 3y)[17(x + 3y) + G l / [ 3 O y ( 1 + 2 x + 3y)]}, (2.10) m I = (A' - D')/2,
m 2 =
(A' + 0')/2,
where G = (60y - l l x 2 + 81y 2 + 5 4 x y ) 1/2 , D' = {[5(x + 3y) - G]/6y}
A' = ([3(x + 3y) + G l / l O y ) 1/2
1/~ .
If we impose the condition m 2 = 1, we find the boundary with the region of transitions of type 1 [line labelled 'r', see fig. 3 (top)] for the P case. The equation obtained is X 4 -- 5x3y + 61x2y 2 + 255xy 3 + 225y 4 -- 8x2y + 52xy 2 + 105y 3 + 16y 2 = 0 .
(2.11) After applying the R ~ K transformation it becomes the ' r " line [see fig. 3 (bottom) and table 1], which is relative to the A-case. Similar diagrams can also be obtained in terms of the anisotropy coefficients Y{Z,minstead of the constants K i. These coefficients apear in the expansion of the anisotropy energy in spherical harmonics. It is known, in fact, that there are advantages in using this representation when analyzing the t e m p e r a t u r e dependence of magnetocrystalline anisotropy, because of the congruence with the Callen and Callen (1966) theory. The relations between the two sets of p a r a m e ters are, K 1 = al ~'/2, 0 + bl ~'{4, 0 + cl if{6,0 + . . . , K2 =
(2.12)
b2•{4, 0 q- C2Y~'6, 0 "-[- . . . ,
K3 ~_~
c3Y{6,0 ql_ . . .
,
where aI
=
-V~
p ,
c2 = ~ X / ~ p ,
b a = -10p ,
cI = -7X/-~ p
c3 = - ~X/-~ p ,
b2 = ~p ,
,
p = 3/(4x/-~).
The inverse transformation is, ~/~'2,0 = a71K1 - b l ( a l b 2 ) - l K 2 ~/'4,0 = ffff6,0 =
b 21K2
+ ( b l C 2 - b2Cl)(alb2C3)-lK3 , - c 2(b 2c3)- 1K3,
c31K3 ,
(2.13)
412
G. ASTI
The FOMP diagrams in terms of the ratios u = Y{4,0/Y{2,0and v = ~{6,o/Y{2,o are reported in figs. 5 and 6, for the cases Y~2,0> 0 and Y{2,0< 0. The straight line labeled L' corresponds to K 1 = 0. Above this line, K 1 has a sign opposite to Y{2,0, and changes sign when crossing it. The same letters denoting the boundary lines in fig. 1 are used here in capitals to distinguish corresponding lines according to transformation (2.12). By the same transformation the FOMP diagrams of figs. 3 and 4 become those given in fig. 5 and 6, respectively. It is worth noting that the diagrams in the plane ff(2,o > 0 are symmetric to those in the plane ~Lr2,0< 0 in a projective sense, i.e., including the points at infinity and considering continuity through it; a property that is absent in the diagrams in terms of the K's. It is also of interest to consider a typical trajectory of the representative point at low temperatures as predicted by the ll(l + 1) power law according to the Callen and Callen (1966) theory. This yields u x m y and v ~ m is, so the line described by the
':!:!:i: i:!:i:i:!!!:i:i:!:i:i:ii!!!i!ii:!
:::!:!:::::::::: li~:}:ii ::~::i::~i!~:!~ P1C
ci!i!iiiii!iiiiiiiiii! ';'iiii!!ii;iiii iiiii?iii!i'oJ;:ii:iiiiii!!iii:!i:i:!i!ii::ii!i
:::::::::::::::::::::::::::::::::::::::::::::::::
;i;!S;~;{~ttt-~t::~:~':-:':-:-:-:-:':':-:': 'Ic !!ii!!iiiiii!!!iiii!iiii1 ~ ;-:-:-:-:-:-:-:':-:-:':':-:':~ v!ii i!i i!!i!i i i i i !i i ...F,2
i::i~ii!ii:!-ii!i!i~iii~Lo'::::::::::::::::::: | :::::::::::::::::::::::::::::: i.:;::P_t ,
o
~--..."::,.,:
Fig. 5. Diagram of F O M P in the plane of reduced anisotropy coefficients u = Y{n.01Y{z,0, v = Y{6,0/Sff2,0. Top and bottom figures refer to the cases Y{2,0> 0 and ~2,0 < 0, respectively. The different regions are distinguished as in fig. 3. The same letters denoting the boundary lines in fig. 1 are used here in capitals, to distinguish corresponding lines according to the transformation of eq. (2.12). The areas that correspond (according to the same transformation) to the conjugated areas represented in fig. 3 are shaded in the same way (Asti and Bolzoni 1980).
FIRST-ORDER
MAGNETIC
.... 06. . . . . . . .
A2
er-'O
\-,\
' ~
,
i
PROCESSES
I_~~--,
",
, ___2_.
-
413
f,,r.,-" PlC~
~
.
--l-
"-._
-..
~.'~
\
~-.%,
\
•o,,.-.~,~, \
A'iC ~ \ -08. `
~xx
~ / •
,
/
,, / , ,
\ , // ~
II
-
,,' .,~
.4//."" y/
//"" .-
,
,>..oo~,
(,IA2 ~,,oo4~._.-4 3
5
t'ig. 6. Lines of constant values for the reduced critical magnetization mot (dashed lines), and reduced critical magnetic field hot (full lines), in the planes of reduced anisotropy coefficients u = 5(4,0/5(2,0 and v = ~6,0/5(2,0 . T h e different regions and the values relative to the various lines of mcr and her are distinguished as in fig. 4. (a) and (b) figures refer- to the cases 5(2.0 > 0 and 5(2.0 < 0, respectively. T h e inset of (b) shows a part of the plane 5(2,0 < 0, with an expanded scale and with the origin (u = v = 0) in the same point. T h e curve . . . . . . . is an example of the ½1(l + 1) power law for the t e m p e r a t u r e dependence of the anisotropy. T h e equation of the curve is o = c u 1817 with c = 1 (Asti and Bolzoni 1980).
414
G. ASTI
representative point is a curve of the type v = CU18/7 or V = c u 2"57, with c constant. Figure 6a shows a typical curve for c = 1. It is worth noting that an analogy exists between the expression for the anisotropy energy [see eq. (2.1)] and the Landau free-energy expansion in the magnetization M, i.e., F = 1 A M z + ¼BM 4 + 1 C M 6 - H M .
This similarity was used by Khan and Melville (1983) to construct a complete set of phase diagrams for zero field and metamagnetic transitions from paramagnetic to ordered state. In particular, the critical magnetic field and specific magnetizations corresponding to first-order magnetic processes were determined. 2.3. Cases o f trigonal, hexagonal a n d tetragonal s y m m e t r y
Numerous ferrites of the hexagonal system may have either hexagonal or rhombohedral crystal structures, depending on whether there is or there is not a symmetry plane perpendicular to the three-fold axis. In this system of compounds, cobalt ions have been found to have strong effects on magnetic properties, introducing high-order anisotropy constants that favor, in general, easymagnetization directions in or near the basal plane, and spin-reorientation phase transitions (see section 2.5). This behaviour is attributed to the degeneracy, even in a crystal field having trigonal symmetry, of the ground state of the Co ions. Investigations (Bolzoni and Pareti 1984) on single crystals of the series of ferrimagnetic oxides Ba2(Zn1_xCOx)2Fea2022 (Y-type ferrites, abbreviated as (Zn,Co)2-Y) having trigonal symmetry, gave evidence of spin-reorientation phase transitions and FOMP. The transition is observed at 77 K with magneticfield direction along the symmetry c axis in all samples containing cobalt. For instance, in the sample with composition Znl.sCo0.5-Y, the critical field is l l . 4 k O e and the magnetization jumps to saturation. The critical parameters (magnetic field and magnetization) measured as functions of temperature and composition, together with the values of the anisotropy field when FOMPs are absent, allow to obtain complete information on the magnetic anisotropy of this system. For carrying out this analysis, Bolzoni and Pareti (1984) have studied the magnetic phase diagram of a trigonal system and determined the conditions for the observation of FOMP in terms of anisotropy-constant ratios. They use the following expression for the anisotropy energy of a trigonal crystal, E 1 = K 1 sin 2 0 + K z sin 4 0 + K t sin 3 0 cos 0 sin 3q~,
(2.14)
where 0 and q~ are the polar angles of magnetization vector M s. Proceeding in a way similar to that described in section 2.2, they obtained the magnetic phase diagram, reported in fig. 7, in terms of the anisotropy-constant ratios x = K 2 / K 1 and y = Kt/K1, giving easy-magnetization directions and other extrema of aniso-
FIRST-ORDER MAGNETIC PROCESSES
415
Y 3
t
X~ I -,
I
-9
0
I
2
I
4
Fig. 7. Magnetic phase diagram of a trigonal ferromagnet with K1> 0. The coordinates are x = K2/K ~ and y = K ] K 1. A symbol similar to that of fig. 1 has been utilized. It marks every region and gives informations about the easy direction of magnetization vector M~, as well as about the other extrema of the anisotropy energy surface with respect to variable 0 at q~= 90° (0 and ~0are the polar and the azimuthal angles of M~with respect to the tfigonal axis); vertical and oblique stems refer to the ternary axis and to 'cones', respectively. A concave tip denotes a minimum while a convex tip refers to a maximum of the energy. The absolute minimum (easy direction) is indicated by a filled tip. For K1< 0 no boundary line exists; the system is easy cone throughout: its representative symbol is "~ (Bolzoni and Pareti 1984).
tropy energy. The diagram of F O M P is reported in fig. 8 which gives the lines of constant reduced critical field her -- H c r M s / I K l l and critical reduced magnetization mcr = M , / M s. The shaded zone indicates the region where type-2 F O M P is present. Figure 9 instead reports the diagram of lines of constant values of the easy 'cone' angle together with the curves mcr--const. By using this approach, Bolzoni and Pareti (1984) have been able to obtain for the first time precise and consistent data about magnetic anisotropy in these trigonal ferrites. The value they obtained for K t (i.e., 3.5 x 106 erg/cm, extrapolated to 0 K) reconciles the dramatic discrepancies between theory and experiment. In fact, Bickford (1960) used a torque technique to measure anisotropy of C o 2 - Y and (K 1 and K2) data obtained by Casimir et al. (1959) in magnetic fields far below saturation. The experimental value thus obtained by Bickford for K t at llT K (i.e., 6 x 105 e r g / c m 3) was more than one order of magnitude lower than the one he calculated on the basis of atomic and structural considerations, i.e., 1 0 . 7 x 106 erg/cm. For rare earth transition metal compounds having hexagonal symmetry, a two-sublattice model has been developed by Sinnema et al. (1987). It applies to compounds of the type R2(Co,Fe)I 7. The attention was especially devoted to the study of easy-plane ferrimagnetic R2T~7 compounds (T stands for transition metal), which display first-order magnetic transitions at high magnetic fields. The
416
G. ASTI
,
Y I/
.
.
\
...
.
a,.
.
.
.
,
,, e
':iiii:iiii ,.....:.:..: ........ . . . . . Y." , O;,,,v
i::!$ii!:?'
- ::!~:::. . . . . . 5 : ~:!!i' ....... .7 - i!iiiii!:
. . . . . . . . .
3
F
64
0
-2
2 ,
\
t,
,
4
i - /
i
(b)
-3 ~
,,,;):
s
Y • --~
I
'
I I
i /~',,
,/
,~ ~
, ,
/ e°
'
Fig. 8. Lines of constant values for the reduced critical field h , = Hc,Ms/[KI[ (solid lines), and reduced critical magnetization rn, = M c r / M s (dashed lines), for type-1 FOMP, in the planes of reduced anisotropy constants x = K2/K~ and y = KJK~. (a)/(1 > 0. (b) K 1 < 0. The shaded zone indicates the region where type-2 FOMP is present (Bolzoni and Pareti 1984). f r e e - e n e r g y e x p r e s s i o n u s e d is,
E = E Kil sin 20i + Ki2 sin 40i + Ki3 sin 60i + Ki4 sin 60i
COS6 ~i
i
Jr nRTM R ' M T - M
R "H--M
T'H,
(2.15)
w h e r e i = T o r R , a n d 0i a n d ~i a r e t h e p o l a r a n d a z i m u t h a l a n g l e s o f m a g n e t i z a t i o n v e c t o r s M T o r M R w i t h r e s p e c t t o c axis a n d a axis, r e s p e c t i v e l y ; nRT is t h e i n t e r s u b l a t t i c e m o l e c u l a r field c o e f f i c i e n t a n d H is t h e e x t e r n a l m a g n e t i c field. T h e m a g n e t i z a t i o n c u r v e s f o r H a p p l i e d p a r a l l e l t o a, b a n d c a x e s h a v e b e e n o b t a i n e d
FIRST-ORDER MAGNETIC PROCESSES 4
,\
,
,\,
3-
,~
"
',
,
'
\ ",
,
, ;
~,
,
,
417
,
~, , ,
I
(a) --
1
e,
'4
.
""
.
"',
-...
o
64
-2
0
'
x'-L+ 2 X
"
4
4
-3
Y + -4
Fig. 9. Lines of constant values for the easy 'cone' angle 0c (solid lines), and reduced critical magnetization, mcr (dashed lines), in the planes of reduced anisotropy constants x = K J K 1 and y = K t / K ~. (a) Ka >0. (b) K~ <0 (Bolzoni and Pareti 1984).
by numerical methods. In easy-plane ferrimagnetic R2T17 compounds, the magnetization curves, with the field in the plane, show large first-order transitions (termed F O M R , from 'first-order moment reorientation processes') resulting mostly from competition between exchange and magnetostatic energy. The assumption is made that K~ is negligible as evident from investigations on Y2C017 and Y2Fe17 compounds. At zero field the resultant moment is directed along one of the six easy directions in the plane. As the magnetic field rises, successive transition points are reached where different moment configurations are energetically favorable. The critical fields correspond to the points where the energy of the two magnetic phases have the same value. An example of the obtained
418
G. ASTI
experimental and theoretical curves is shown in fig. 10 for Dy2Co17. The measurements have been performed at 4.2 K in a field up to 40 T at the high-field laboratory of the University of Amsterdam. A variety of compounds have been investigated using this approach, obtaining precise and consistent data about anisotropy constants for both sublattices. The obtained values of K i are, in most cases, one order of magnitude larger than those derived from other authors' low-field measurements and assuming rigid exchange bonding between the moments (see section 3.3.3). In tetragonal crystals, due to the presence of four-fold anisotropy in the basal plane, it is even more important to take the tetragonal term into account. An interesting case is that of the ternary intermetallic compound Nd2Fe14B , which represents a very important development in the field of hard magnetic materials [for a comprehensive review on this topic see the work of Buschow (1986)]. Working on this material, Asti et al. (1984) and Gr6ssinger et al. (1984) revealed, at temperatures below 210 K, the presence of FOMPs with critical fields above 14 T. For these experiments, the technique used was that of the singular point detection (SPD, see section 4.2). The existence of FOMP was confirmed by direct observations on single-crystal samples by Pareti et al. (1985a). However, the important role of the tetragonal symmetry appeared immediately (Bolzoni et al. 1987, Kajiwara et al. 1987), as the FOMP was not visible when the applied field was parallel to (110) axes. Subsequent measurements in other high-field laboratories [see references below in addition to Andreyev et al. (1985) and Sinnema (1988)] confirmed the existence of the transition in a variety of compositions in the system R2T~aB. For the understanding of magnetic anisotropy in this
Oy2[o1~ 6C
42
K
+ a axis
,
"--
x ~ X _
X
~
-
~
~
z
,
1
1
f
10 20 Mognefic Fietd B(T)
,
i
i
30 =
Fig. 10. Magnetization curves at 4.2K along different crystallographic directions of hexagonal Dy2Co17. The a axis is the easy magnetization axis within basal plane. The full lines are computed magnetization curves; after Sinnema et al. (1987).
FIRST-ORDER MAGNETIC PROCESSES
419
system a theoretical approach based on tetragonal symmetry is necessary. Theoretical models based on the crystalline electric field (CEF) Hamiltonian ~CZF = ~2 B~O~,
(2.16)
l,m
have been developed and applied to several compounds of this class. It is found that CEF terms up to fourth order are necessary to explain the observed FOMP in Nd2Fe14B and its dependence on crystal orientation (Kido et al. 1987). The case of Pr2FelgB is worth noting because the observed FOMP is of type 2, as the magnetization, after the abrupt increase, is well below saturation (Gr6ssinger et al. 1985, Pareti et al. 1985b, 1988, Huang Ying-Kai et al. 1987, Pareti et al. 1988, Pareti 1988). Moreover, the critical field changes from 13 to 16 T when the magnetic field is moved from [100] to [110] direction. In this case, the theoretical interpretation is complicated because it is found that CEF terms up to sixth order are necessary for reproduction of the observed phenomena (Hiroyoshi et al. 1987). On the other hand, the role of sixth-order terms in the case of type-2 FOMP was already clear from phenomenological arguments, even in uniaxial systems [Asti et al. (1980), see section 2.2]. The necessity of sixth-order terms has, in this context, also been evidenced by Zhao Tiesong and Jin Hanmin (1987), although the various authors adopt different criteria for simplifying the CEF Hamiltonian, because of the great number of adjustable parameters.
2.4. Cubic symmetry First-order magnetic processes of the same nature as those described in the preceding sections (FOMP) have also been observed in magnetic materials with cubic symmetry. Preliminary discussions of the effect were already present in a work by Bozorth (1936) and the theoretical description of the effect has been partially developed by Krause (1964) and Rebouillat (1971) on the basis of the usual phenomenological theory of magnetic anisotropy. Furthermore, Rebouillat (1971) gave some experimental evidence of a first-order transition of this kind in iron single crystals at low temperatures. Asti and Rinaldi (1974b) have examined the conditions for the existence of FOMP with the magnetic field H parallel to one of the crystallographic axes, taking into account only the first two anisotropy constants, K 1 and K2, both in the case of cubic and uniaxial crystals. Two different kinds of FOMP are possible: (1) When the irrreversible rotation of M~ occurs in the plane defined by the magnetic field direction and the easy direction, i.e., the initial direction of M~ (type-I transition): (2) When M~ starts to rotate in one plane and, at the critical field, jumps to another plane where it continues to rotate towards the magnetic field direction (type-II transition). The analysis of the various possible magnetization curves has been made assuming the direction of magnetic field H to be parallel to an extremal direction
420
G. ASTI
of the anisotropy energy surface, i.e., a stationary point but not an absolute minimum. In the presence of only K 1 and K 2 constants these directions coincide with crystallographic symmetry directions. The results are summarized in the graph displayed in fig. 11, which also gives the phase diagram of easy and hard directions in the (K 1, K2) plane. Depending
Fig. 11. Conditions for the existence of FOMP in cubic crystals, which depends on the ratio K 2 / K 1 of the first two anisotropy constants and the sign of K 1 (on the outside part). The letters A, B and C indicate magnetic field orientations parallel to (100), (110) and (111) directions, respectively. The superscript indicates whether the transition is of type I or II. The magnetic phase diagram of a cubic crystal is given by the inner shaded circular areas relative to the three crystallographic directions. The various ranges of K 2 / K ~ are distinguished by different symbols, indicating a minimum for the anisotropy energy (e), a maximum (h), a saddle point having a maximum for rotations of Ms in the {110) planes (sx) or in the {100} planes (sy). For K 2 / K ~ < - 2 (see the ranges indicated by 0 and ~oin the white circular zone) the anisotropy energy surface has saddle points along non-symmetry directions of M~ lying in the (110) planes and having two direction cosines equal to ( - K 1 / K 2 ) ~/2 For rotations of Ms within the (110) planes the anisotropy energy has a maximum or a minimum when K z / K ~ is in the range 0 or ~, respectively (Asti and Rinaldi 1974a,b). A continuous solid line, inside the shaded areas, joins all the ranges on the three crystallographic areas labelled 'e', where t h e crystal is easy axis (absolute minimum).
FIRST-ORDER MAGNETIC PROCESSES
421
on the direction of H, the following intervals for the anisotropy constants ratio k = K2/K 1 are indicated for the two types of FOMP: (1) H[I(100) - T y p e I for k > 4 with g I < 0 , and k < - 9 with K~ > 0 ; type II for-9 0. (3) H I I ( l l l ) - T y p e I for k < - 9 with K 1 < 0 and k > - 9 with K1 > 0 , i.e., the whole range where (111} directions are not easy axes; type II for k > 9 for K 1 > 0. In this range both types of transition coexist. A more detailed discussion of the different magnetic phases and modes of magnetization in the cubic ferromagnet has been given by Birss et al. (1977) including also the eight-order anisotropy constant K3, but again considering only magnetic field directions parallel to the symmetry axes. The results of Birss et al. (1977) are in agreement with those of Asti and Rinaldi (1974b) and differ in several respects from those of Rebouillat (1971). Thus, the behaviour of a cubic ferromagnet in an applied magnetic field is an area which is rich in examples of phase transitions. For this reason, it has been especially considered from the point of view of the study of the critical phenomena. Mukamel et al. (1976) showed that a cubic ferromagnet, in which (111) directions are hard axes, under the action of a magnetic field, is a realization of the three-state Potts model. Cullen and Callen (1984) extended this type of analysis, also including sixth-order terms of magnetic anisotropy, and calculating various aspects of the magnetic-field phase diagram depending on the ratio k = K2/K ~. That is, they display in a diagram the critical field-strengths f o r all field orientations, discussing the topological variations in the phase diagram in relation to the different k values. From the experimental point of view, besides the experiments on iron by Rebouillat (1971) and by Hathaway and Prinz (1981), it is important to mention that low-temperature measurements in cubic RFe 2 compounds (R is a heavy rare earth) reveal values of K 2 and K 1 which lie in all four quadrants of the (Ka, K2) plane (Atzmony et al. 1973). Good examples of anisotropic cubic ferromagnets are the RAI 2 compounds. For instance, DyA12 has been shown to have a large FOMP for fields applied in the (110) plane (Barbara et al. 1978). These works, besides the inherent importance for the study of critical phenomena, provide a reliable guide for the investigation of cubic ferromagnets and ferrimagnets which display FOMP type transitions. FOMPs also have remarkable effects of magnetostriction, giving rise to large discontinuities, as observed by Melville et al. (1981) in CeFe 2 as well as, more recently, in hexagonal systems, namely Er2(Col_xFex)17 (Kudrevatykh et al. 1986).
2.5. Mechanisms responsible for high-order anisotropy constants
From the above discussion, it is clear that the existence of FOMP in a ferromagnet implies high-order anisotropy terms. These circumstances are not easy to be
422
G. ASTI
included in the theory and, in general, require special mechanisms that take into account the nature of the magnetic ions and of their mutual interactions. For instance, in cubic crystals the f electrons of rare earth ions can support no term above sixth order in the CEF Hamiltonian (Cullen and Callen 1984). In mixed crystals, namely pseudobinary compounds, the competition between the individual anisotropies of the constituent rare earths is responsible for the onset of higher-order terms in the anisotropy energy. These terms are generated if we treat the crystalline electric field in higher than first-order perturbation (in terms of the ratio crystalline field/exchange field), either by admixing higher multiplets in the J quantum number, or by allowing a small canting between magnetic moments (Cnllen and Callen 1985). Indeed, section 3 is devoted to the effect of canting on the anisotropic behaviour of multisublattice systems on a phenomenological basis. As regards magnetic oxides of the transition metals, the mechanisms for generating high-order anisotropy constants must be different. In fact, we are in this case in the limit of large exchange and the situation for a 3d ion is very different as for the relative importance of the various terms in the Hamiitonian. As a matter of fact, there are examples of hexagonal ferrites that display large FOMPs within wide ranges of temperature and magnetic fields, both parallel and perpendicular to the symmetry c axis (Asti et al. 1978, Graetsch et al. 1984, Bolzoni and Pareti 1984, Paoluzi et al. 1988). All these examples have in common the presence of Co as a constituent. Lotgering et al. (1980) have given an explanation of the peculiar influence of Co 2+ ions on the anisotropic behaviour of all hexagonal ferrites. Similar effects have also been observed for Fe 2+ ions in L ~ 2+~ 3 + ~ are re11 u19 (Lotgering 1974). The interpretation is based on the Slonczewski theory (1961), which considers the incomplete quenching of the orbital momentum in the local crystal field, present at octahedral sites which have an uniaxial component, in addition to the cubic one. In fact the local uniaxial crystal field is not sufficient to completely remove the degeneracy and in one case we can have an orbital doublet as the lowest state. The orbital momentum is then either +h or - h along the local axis, so that the spin-orbit coupling gives a torque rTI = ALS sin 0 (where L = 1 and 0 is the angle of the spin with respect to the local axis). This fact implies a sign change in the orbital momentum at 0 = 90°, giving rise to a discontinuity in the torque curve and a very high anisotropy. Even if, in real cases, there are substantial deviations from such ideal conditions, due to various perturbations, the model gives an account of the strong distortions that can be induced in the pure sine function, representing the torque of the spin when only second-order anisotropy exists. As a consequence, high-order terms enter the anisotropy energy. If the orbital singlet is lowest, an anisotropy can again arise from admixing of the excited state with an unquenched orbital moment by spin-orbit coupling (Lotgering et al. 1980). The fact that Co 2+ ions in all hexagonal ferrites reside on lattice sites having octahedral coordination, allows to deduce relations between the anisotropy constants in the hexagonal structure. In fact the anisotropy term associated with Co 2+ in octahedral coordination has the form,
F I R S T - O R D E R M A G N E T I C PROCESSES E
2 2 2 2 2 2 = Kc(O/lO~ 2 + o~20t 3 + o t 3 0 t l )
423
(2.17)
where the a's are the direction cosines of the magnetization with respect to the (x, y, z) reference system centered at the cation site and with axes containing the coordination anions. The c axis of the hexagonal structure has direction cosines ( ½X/-3, IV~, )V~). Expressing eq. (2.17) in terms of polar coordinates (0, ~o) with the c axis as the polar axis, and the azimuthal axis having direction cosines (~X/2, ½V2, 0), we obtain, E=Kc( 1-2sin 20+7
sin 4 0 + } s i n 3 0 c o s 0 s i n 3 ~ ) .
(2.18)
The last term on the right-hand side has trigonal symmetry. However, for hexagonal structures (like W- and Z-type compounds) we must take into account that for each octahedron there is another one rotated by 180°. Averaging over the two, the trigonal term vanishes. Equation (2.18) indicates that a positive singleion cubic anisotropy contribution gives rise, in the hexagonal structure, to both a negative K 1 and a positive K 2 (Asti and Rinaldi 1977). There is another kind of transition in hexagonal ferrites that cannot be considered as ordinary FOMP, but is associated with a more complex magnetic behaviour (see section 3.4). These phenomena are characteristic of ferrites having block-angled magnetic structures and are caused by mechanisms which are different from those responsible for the above-mentioned FOMP. 2.6. Domain wall processes at the transition point
As we have seen, a FOMP is due to an irreversible rotation of the magnetization vector M s between two inequivalent minima of the free energy of the crystal under the action of the magnetic field. Normal hysteresis in a ferromagnetic material is caused by irreversible rotation processes between equivalent crystallographic directions and is therefore symmetrical. In the case of FOMP we expect to observe minor hysteresis loops displaced to an unsymmetrical position in the M versus H diagram. In reality, it is difficult to reveal the hysteresis associated with FOMP. The few known examples are experiments at liquid-helium temperatures, as reported by Barbara et al. (1978) and by Ermolenko and Rozhda (1978). The reasons are probably in the inherent low coercivity of the magnetization processes in the critical range of the phase transition. Mitsek et al. (1974) and Melville et al. (1987) have calculated the domain structure existing in this range. The analysis starts from the consideration of a 180° domain wall in a uniaxial ferromagnet with K 1 > 0 and x = K ~ / K 1 in the range ( - 1 , - ~1) , which corresponds to a P1 type FOMP (see section 2.2). The magnetic field H is oriented along the positive direction of the x axis while the plane of the wall is parallel to the xz plane and neighbouring domains are oriented in opposite directions of the z axis. Under the action of the magnetic field, both domain-magnetization vectors rotate by an angle 0 towards the field direction, so that the wall, polarized along the magnetic
424
G. ASTI
field direction, becomes a ( 1 8 0 - 20) degree wall. The free energy is
--de f E= k[ A \(d0) d y / 2 + K l s i n Z O + K 2 s i n g o - M s H s i n O ] dy,
(2.19)
where A is the exchange parameter. The Euler equation has the form, K1
d20
dy z
A sin 0 cos 0
2K2 MsH ~-- sin 3 0 cos 0 + ~ cos 0 = 0 ,
(2.20)
and the boundary condition are, d0 ~yy y=--+~= 0 ,
0 ( - ~ ) = OH,
0(~) = ~ - - OH,
where OH is the equilibrium angle corresponding to field H inside the domains. The solution of eq. (2.20) turns out to be
0
y =
fl
~ dO,
(2.21)
re/2 with U=
Kx K2 -)- (sin 2 0 - sin 2 OH) + ~ - (sin g 0 -- sin g OH)
]1/2 MsH (sin 0 - sin OH)] A
.
The slope dO/dy can be calculated from differentiation of both parts of eq. (2.21), obtaining, dO -- = U. dy
(2.22)
It is found that dO/dy vanishes at the center of the wall, i.e., for 0 = ½~-, when H is equal to the critical field of F O M P H , = ½HAahcr = Klhc]M ~. The reduced field h~ is given by eq. (2.3), where y is taken equal to zero and m is equal to the critical magnetization given by
mcr = Mc,./Ms = ½[-1 + ( - 2 - 3/x)1/2],
(2.23)
which is a solution of eq. (2.4). The fact that the derivative dO/dy vanishes at the center of the wall means that, at the critical field, the original 180 ° wall is split into two symmetrical walls of (90 - 0) degrees, giving rise to a nucleus of the new phase that appears inside the
FIRST-ORDER MAGNETICPROCESSES
425
domain wall. These new walls are in equilibrium in the presence of a magnetic field equal to Her , because the two adjacent domains belong to different phases, but have the same free energy. The same results can be obtained in a very direct way if one utilizes a simple energy criterion that is valid for domain structures in general, and for a single wall in particular. Let us consider a one-dimensional problem as before, so that we consider a system of plane domain walls parallel to the xz plane. Then we can rewrite eq. (2.19) as follows, cr2
KI(x + y2) + K2(x z + y2)2 _ Ms(xH~ + yHy + zH~) + 2~MZy 2 o"1
where x, y and z are the Cartesian coordinates of the unit vector Ms~Ms, Hx, H e and H z are the components of the magnetic field H and tr is an axis parallel to the y axis. In the case of static domains, if Hy = 0, we may exclude the y component because it gives rise to demagnetizing fields that would drive the wall motion. In this form, the integral of eq. (2.24) can be regarded as the action integral of a mechanical system consisting of a body of mass 2A, free to move on a sphere. The mechanical forces acting on it are opposite to the forces acting on the magnetic vector M s. They are a gravitation-like force directed opposite to H, and elastic forces repelling the body away from the z axis (when K 1 > 0) and away from the xy plane. Hence, cr represents the time, and the function in the integral is the Lagrangian function T - V of the mechanical system, where the kinetic energy T corresponds to the exchange term and the potential energy V is represented by the other terms with changed sign. This mechanical analogy enables derivation of an important property of magnetic domain structures directly from the energy conservation principle in the mechanical system. In fact, T + V must be a constant throughout the whole range of or. As a consequence, we can say that everywhere in the magnetic system the increase in exchange-energy density is balanced by an equal increment in the remaining part of the energy density, consisting of the addition of anisotropy and magnetostatic terms. This kind of equipartition principle has easy and immediate applications. For instance for the calculation of domain-wall thickness (see section 4.2) and to the above results concerning domain structure in the presence of FOMP. In fact, eq. (2.22) can be written directly from the balance of the energy densities, and, likewise, the vanishing of dO/dy at the center of the wall, when a nucleus of the new phase appears, is obvious, because the two phases have, by definition, the same energy density in terms of anisotropy plus magnetostatic contributions. As a consequence, there must be no variation in exchange energy density. The mechanical analogy also makes intuitive the effect on domain structure of the magnetization process of a uniaxial crystal with applied magnetic field parallel
426
G. ASTI
to the c axis, which is taken as easy magnetization direction. For instance, it is easy to understand why reverse domains are reduced with respect to those oriented parallel to magnetic field: If we consider the body on the sphere to rotate continuously, it means that we have a periodic system of parallel walls having alternate polarization, towards positive and negative x direction. Then, in the mechanical analogue the moving body spends less 'time' in the opposite domains because it has a higher kinetic energy there. Equilibrium is reached, because there is a repulsion between adjacent walls with opposite polarization. In the limit of high magnetic fields they join to form a 360° wall. Instead, solutions with magnetization vector pointing in the same direction must be unstable, because they present the opposite effect, i.e., the mean magnetization of the crystal will be opposite to H. All these conclusions are in agreement with detailed theoretical studies reported by Forlani and Minnaia (1969), Wasilewski (1973) and Odozynsky and Zieter (1977). The same approach can be used to study domain wall mobility and dynamic domain-wall structure, if one includes in eq. (2.24) the terms in y and effective dynamic forces deducible from the dynamic equation of magnetization, e.g., in the Gilbert form, dMJdt
= 7M x H - (M s x dM/dt)a/M
s.
(2.25)
If we admit that there are solutions characterized by a rigid displacement of the domain wall with a constant velocity v, differentiation with respect to time of a function L becomes d L / d t = v d L / d o - = v L , where we have indicated differentiaton with respect to o- by a point above the function. Then the dynamic equilibrium of the moving body can be written in the form ( 7 / M s ) M s × ~:
= O,
(2.26)
where ~ is the total dynamical force given by = (-grad V - 2 A ~ ' I I M s - v M s G I 7 - v a M l y ) ,
(2.27)
and G = M x ~ is a Lorentz-like force caused by a pseudo-magnetic field perpendicular to the surface of the sphere in outward direction and having a uniform intensity equal to 1/M s. The term containing the damping parameter a is responsible for the energy loss and acts as a typical viscous resistance. The dynamic equations are then obtained by projecting eq. (2.26) on the Cartesian axes, or from the condition that the component of ~r on the sphere must be equal to zero. This condition can be expressed in the form, o% = 0 ,
o~0 -- 0,
(2.28)
where ~ and ~0 are the components of ~: along the directions of the local spherical-coordinate lines ~p = const., and 0 = const., having taken the polar axis
FIRST-ORDER MAGNETIC PROCESSES
427
coincident with the z direction. If we suppose that the magnetic field H is applied parallel to the positive direction of the z axis and that we have a single domain wall with the boundary conditions z = 1 for cr---~ - ~ , z --- - 1 for or--->+ ~ , then we can see that a very simple solution exists, namely the one given by ~0 = const. [the solution given by Walker and reported by Dillon (1963)]. In fact, in this case eqs. (2.28) become,
o%~ = 4~yM~ cos q~ - vi(/lly = 27rM 2 sin 0 sin 2~o - v M / y = O, (2.29)
~o = MsH sin 0 - a v ~ l / y - 2Alf4/M s + 2 K 1 sin 0 cos 0 + 27rM 2 × sin 2 qo sin 20 = O. The solution corresponds to the condition that the viscous resistance is perfectly balanced on every 'instant' tr, by the term due to the applied field, i.e.,
MsH sin 0 = a v i f l / y .
(2.30)
This means that the magnetostatic energy - M . H is entirely dissipated inside the wall during its damped motion. This condition together with the first equation of (2.28) yields, sin 2~ = H / ( 2 7 r a M s ) .
(2.31)
Then h)/can be deduced from the principle of energy conservation, i.e.,
A(i(4/Ms) = K 1 sin 2 0 + 27rM~ sin 2 0 sin 2 q~,
(2.32)
so, for the wall velocity the following expression is obtained,
v = H ( y / a ) (A/K1)l/2[1 + 7rM2(1 - c o s 2~)/K1] - v 2 .
(2.33)
The case ~ = ¼7r corresponds to the maximum value for v, which is known as the Walker limit (Dillon 1963).
3. Processes involving competition between exchange and anisotropy 3.1. Introduction In the previous chapter we have seen that the simple consideration of high-order terms in the magnetic anisotropy energy expansion justifies the existence of F O M P in an 'effective' ferromagnet. By using this expression, we emphasize the fact that the internal magnetic structure of the system has no relevance, because during the magnetization process it was assumed to remain absolutely rigid. Examples are ordinary ferrimagnetic materials, such a s the oxides of transition
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G. ASTI
metals which, at ordinary field intensities, display a constant spontaneous magnetization as well as perfectly collinear magnetic moments, due to the extremely high exchange interactions. However, this is not true for all magnetically ordered substances. It is well-known indeed that in RE intermetallic compounds the situation can be such that the fundamental interactions have comparable intensities, so that we can have exchange and crystal-field terms in the Hamiltonian which are essentially of the same order of magnitude. The result is obviously that we can no longer consider the system as a rigid collinear magnetic structure, but we must allow for substantial deviations from the equilibrium configuration at zero field. Hence, the role of the exchange interaction in the magnetic process can in certain conditions be so critical, as to produce extraordinary effects. As a matter of fact, it can be responsible for: variations of the anisotropy fields by orders of magnitudes (even leading to change of sign); stabilization of easy-cone directions; or it can give rise to field-induced first-order transitions, even when only the first anisotropy constants of the individual sublattices are present. In treating effects of this kind in the next two sections, we shall use, once more, a phenomenological approach that is based on the classical model of exchange. Although simple in principle, it can be considered as a useful guide to the experimentalist, because it contains the essential features and allows to gain a better understanding of the fundamental processes which form the basis of the observed phenomena. In section 3.2, as a first application of this concept, a detailed analysis is made of the magnetic system in the vicinity of the transition points, so as to obtain exact relations between some fundamental parameters, such as the expressions for the anisotropy fields and the conditions for the existence of FOMP. In section 3.3, the same concept is developed within the approximation of small deviations from collinear magnetic order (small-angle canting). In this limit, the effect of canting can be accounted for, in a sufficiently accurate approximation, by the use of effective anisotropy constants that are slightly field dependent. So, the system can be conveniently treated as an effective ferromagnet having certain anisotropy constants. A typical application of this model is an immediate explanation of the easy-axis to easy-cone transition observed at 110 K in PrC%, thus, indicating in a very simple way the important role of canting (see section 3.3.3). In section 3.4, a brief review is given of other cases and treatments of the problem of canted systems and magnetic phase transitions.
3.2. Role of exchange at transition points 3.2.1. Multisublattice magnetic systems with uniaxial anisotropy We shall focus our attention on an ordered magnetic system with uniaxial magnetic anisotropy, consisting of L interacting sublattices, Ma, M b , . . . , ML, when it is near the transition point, in general, in the presence of an external magnetic field. In what follows, we will consider transitions involving a change in magnetic symmetry, in particular, transitions in which one of the phases is a collinear-type equilibrium configuration, i.e., a transition between a 'collinear' phase and an 'angular' or canted phase.
FIRST-ORDER MAGNETICPROCESSES
429
So, the equilibrium states we are considering in the collinear phase are essentially: the fundamental state in zero field, normally characterized by a collinear-type magnetic structure, and the various other collinear states that are stabilized by an external magnetic field H parallel to a crystallographic symmetry direction. The latter are intermediate saturated states and are stable in principle within certain regions on the temperature-magnetic field plane ( H - T plane). Among these states, the one at the lowest field, when possible, is that having the s a m e magnetic structure as the fundamental state at zero field and occurs at H = HA, the effective anisotropy field, or at H = Her, the first critical field, in case a FOMP is present. Other collinear states can be stabilized by higher fields. They are characterized by different magnetic structures, corresponding, in principle, to all possible combinations of the mutual orientations of vectors M i. Among these 'excited' states the one having the highest magnetization corresponds to the forced ferromagnetic order. The phase transitions associated with these magnetic states are of two types: s p o n t a n e o u s spin reorientation transitions (SRT), a n d f i e l d - i n d u c e d SRT. One can imagine the reverse situation, i.e., to keep the magnetic field H constant and to change the temperature, or even a combination of these two, which corresponds to crossing the transition line in the H - T plane along different directions. In this sense, the first type of SRT reduces to a particular case of the second type. As an example, let us consider again the phase diagram of figs. 1, 3 and 4. The line m represents first-order SRT. If the representative point, with changing temperature, crosses that line at a certain critical value T = Tcr, it means that the system undergoes a first-order SRT in which the easy-magnetization axis changes discontinuously from axis to plane or vice versa. Alternatively, the lines o and o' are those where discontinuous transitions occur of the type easy axis-easy cone or easy plane-easy cone. Obviously, the same lines have the meaning of lines where the value of the critical field of FOMP reduces to zero. Since in the present treatment we are confined to small deviations from collinear order, the orientation angles of the magnetic moments are treated as infinitesimal, in general, all of the same order. So in our considerations we are not limited to the classical 'large exchange' case, but we may consider systems in which the intensity of the anisotropy forces can be comparable to or even higher than, the exchange interactions. The analysis of the behaviour of the system in the neighbourhood of the transition point leads to exact relations between important magnetic parameters. Let us consider a system of L sublattices M a , M b , M ~ , . . . , M L having uniaxial anisotropy constants K l a , K l b , K l c , . . . , K1L, Kza , K2b , Kz~, . . . , K2L up to a certain order N, and intersublattice molecular field coefficients J~t3. The free energy of the system, in the presence of an external magnetic field H parallel to the symmetry axis c has the form, L
F=-
~ a,a >fl
J~M~M~
cos(0~ - 0~) + ~ a=a
N
L
~ Ki, ~ sin2i 0~ - ~ HM,~ cos 0~, i=1
a=a
(3.a)
430
G. ASTI
where Oa, 0b . . . . , 0 L are the orientation angles of the magnetization vectors M, with respect to the c axis. This is the typical situation we have when we magnetize a crystal along a crystallographic axis: when the field approaches a certain critical value, which in the ordinary cases is the anisotropy field H A relative to that direction (hard magnetization direction), all the O's are vanishing and at H = H A we have a metastable equilibrium. Above that field, the magnetic structure is collinear and does not change until H (in the case of ferrimagnetic order) eventually reaches a further critical value which starts another SRT, in general, towards a canted state having higher magnetization. In the limit of 'infinite' exchange, the effective anisotropy of the system is merely given by the sum of the sublattice anisotropies, so that the anisotropy field turns out to be, H A ---
2K13
M,
(3.2)
with M = E~L =, M . Moreover, when the exchange is of comparable magnitude as the anisotropy energy, we can have considerable modifications in that expression because there are important contributions to the total energy from relaxation processes due to substantial deviations from the coUinear order. The particular collinear configuration is specified by the sequence of the signs of Ms, and can be chosen independently from the sequence of the signs of J ~ , because, in general, the stability of a configuration cannot be decided a priori, being dependent on the values of the K's and H. Furthermore, we neglect here possible effects due to the splitting of one or more sublattices, a phenomenon of the Yafet-Kittel type, (Boucher et al. 1970) which is possible, in principle, if we consider competing intrasublattice-exchange interactions.
3.2.2. Field-induced spin reorientation transitions. Critical fields It is easy to write down the equations for the critical field Ht, at which a certain state having collinear order becomes unstable. The equilibrium equations of the system are, L
OF~dOe = ~
N
J~t~M~Mt~ sin(0~ - 0t~) + ~] Ki~2i cos 0~ sin 2i-1 0 a
/~=a
i=1
+ M ~ H sin O~ = O,
(3.3)
with a = a, b , . . . , L. Expanding up to first order in the O's, we obtain a system of homogeneous linear equations, L
/3=a
J.~M,~M~(O,~ - Or3) + (2K1~ + M~H)O~ = 0 ,
(a=a,b,...
,L). (3.4)
It is evident that in these equations the anisotropy constants of an order higher than two do not appear, because we have considered the transition with the field parallel to the c axis.
F I R S T - O R D E R M A G N E T I C PROCESSES
431
The condition at the transition point is that the determinant of the matrix of the coefficients is vanishing, i.e.,
a =o.
(3.5)
This is an equation of degree L, which, in principle, can be solved giving the expression for the critical fields H t as functions of the sublattice magnetizations, the anisotropy constants and the exchange coefficients. Note that, in principle, we have L distinct solutions for each collinear magnetic structure. So, already for a small number of sublattices the situation appears very complicated. Among the real solutions H = H 1, H 2 , . . . , Hp of eq. (3.5), with p ~< L, the solutions having physical meaning are those corresponding to transitions to (or from) a minimum of F. They define ranges of H for which a particular collinear magnetic structure is metastable. The other solutions of eq. (3.5) are, in general, only transitions between saddle points of different kinds or between a saddle point and a maximum, i.e., unstable states. An important case is when all the exchange coefficients are zero except those related to one particular sublattice, say M a. This can be a good approximation for several rare earth-transition metal compounds (RE-T), and especially for solid solutions containing various RE species or RE occupying different lattice sites. In this case, we can indeed neglect all interactions between RE and consider only those between RE and TM. Hence, the system of eq. (3.4) reduces to,
(Z
L J~M.M~
) L "}-2gla "]- M a l l Oa-- Z Jl3MaM,t3Ot3 -~-0,
-
¢~=b
(3.6)
- J y o M , Oa+(JyoM
+2K1 + M H)O =0,
(¢=b,c,... ,L),
where we have taken J~ = Ja.~. Then eq. (3.5) becomes,
-
JoMoM + 2Kla + Mon,
¢l=b
-
"y=b
(LMaM, + 2K,.,, + M ,Ht)
(JvM.M:, + 2K1. -I- M:,Ht) J ~: M a2M ~2/ ( J ~ M a M ~ + 2 K ~
+ M~Ht) = O, which yields, L
2K,,, + Mo, H t + ~, [(2Kit3 + Mt3Ht) -1
+ (J~MaMi3)-l1-1 = 0 .
(3.7)
In the case H t = 0 this equation, as well as eq. (3.5), gives the condition for the spontaneous SRT. All these results are easily extended to the case of a field perpendicular to the c axis, by the use of the K-~-->R transformations (see section
432
G. ASTI
2.2). In the most simple case, H t in eq. (3.7) can be identified with the anisotropy field H A. An important consequence of the effect of canting is that we have two different expressions for the anisotropy field, depending on whether we apply the magnetic field parallel or perpendicular to the symmetry axis of the crystal (see section 3.3.2). These directions of the applied magnetic field are typical experimental arrangements for measuring magnetization curves, depending on the fact that the symmetry direction is a hard or an easy direction, respectively. In the absence of canting effects the anisotropy field in both cases is given by eq. (3.2), if only second-order anisotropy constants contribute to the anisotropy energy. However, the anisotropy field given by eq. (3.7) could also have physical meaning when the symmetry direction is an easy direction, when we think of processes involved in phenomena like the nucleation and growth of a reverse domain, that have relevance for coercivity mechanisms. Equation (3.7) reduces to a very simple form in the case of two sublattices, i.e., (2Kla + m a n t ) -~ + (2Kab + MbHt) -~ + ( J m a m b ) -1 = O,
(3.8)
with J = Jb = Jab" Then, the condition for the spontaneous SRT becomes, (2Kaa) -a + (2Kab) -1 + ( J M a M b ) -1 = O,
(3.9)
which merely says that the sum of the compliances due to anisotropy forces is balanced by the exchange compliance. 3.2.3. B o u n d a r y conditions f o r first-order transitions
Another problem concerning these magnetic transitions is connected with their order. In fact, as we have seen in section 2.2, the approach to saturation in certain circumstances can occur with a first-order transition that is a discontinuous rotation of the magnetization vector (FOMP). The present approach, based on the analysis of an infinitesimal region around the collinear state, can be used to study the conditions for the existence of first-order magnetic processes in general, which, in particular cases, can be identified as ordinary FOMP. For instance, the transi:ion to the first intermediate saturation state can be considered to be an ordinary FOMP. Instead, first-order transitions involving higher magnetization states, such as the spin-flop transition, unavoidably bring drastic changes in the magnedc order. When necessary, it may be convenient to distinguish between these two types of FOMP, so, hereafter we will call them 'o.FOMP' and 'h.FOMP' (for 'ordinary' and 'high-magnetization state' FOMP, respectively). Let us consider the system of equilibrium equations (3.3) and expand up to terms of third order in the variables 0. The magnetic field will be expressed in terms of an infinitesimal reduced variable given by ~, = ( n -
H,)/I4 t ,
(3.10)
where H t is a real positive solution of eq. (3.7). The an~les 0 can be written as the
FIRST-ORDER MAGNETIC PROCESSES
433
sum of an unperturbed part, 0, plus an infinitesimal of an higher order, e, i.e., we write 0~ + e~ instead of 0~ for each a. We obtain a system of L linear equations in the variables e having the same coefficient matrix as the homogeneous system of eq. (3.4). The column of terms on the right-hand side are, instead, functions of the 0's and of y, so the system obtained is L
J ~ M Mt~(e~ - e¢) + (2K1,, + M~Ht)e ~ = 0 ( 0 3, TO), t3=a
(a = a, b , . . . ,
L),
(3.11)
Since A is equal to zero, the system has a solution only if the augmented matrix (i.e., the matrix including the column of the right-hand side terms) has the same rank as the matrix of the coefficients. This, in general, allows the writing down of a condition in the form A' = 0, where A' is one of the determinants obtained by replacing one of the columns of A with the column of the right-hand side terms. This existence condition allows one to find the relation between y and the angles 0, which means that we can obtain the slope of the magnetization curve M ( H ) at the transition: a negative slope indicates an unstable equilibrium, which implies that a F O M P is possible. That is, if M ( H ) connects stable states, the transition is of first order, otherwise it means that M ( H ) represents metastable states, and so the transition does not occur at all. In the particular case already considered, in which all the exchange interactions are zero except those with a single sublattice Ma, the coefficient matrix of system (3.11) reduces to that of system (3.6), while the column of the right-hand side terms turns out to be, L
-~ ~
Jt~Mt~Ma(Ot~ Oa)3 + ( ~ M a n t + 4 gla -4Kza)O 3 - MantOa,Y , -
[3=b
~Jt3Mt3M~(Ot3 - 0,) 3 + (~Mt3H t + 4K1~ - 4K2t~)013 - Mt3HtyOt~, (B=b,c,...
(3.12)
,L).
Note that anisotropy constants above fourth order give no contribution, thus indicating that the boundary between first- and second-order magnetic transitions depends, in general, only o n K 1 and K 2. This is consistent with the results obtained in section 2.2, where we have seen that the boundary line n', separating A1 and A I C from normal regions, is vertical, which means that K 3 has no effect. The equation of n' is, in fact, 4 K 2 = K 1.
3.2.4. The two-sublattice system In what follows, we shall limit ourselves to the simplest case of only two sublattices, M a and M b. As already mentioned, the smaller of the two magnetizations, Mb, can be either positive (ferromagnetic order) or negative (ferrimagnetic order), it is convenient to introduce the reduced variables
G. ASTI
434 x = 2Kla/JMM
a ,
y = 2Kab/JMM
v = 4 K 2a / J M M
b = 1 - a = Mb/M, h = H/JM,
a = M / Ma
b ,
a
,
w =4K2b/JMM
(3.13)
b ,
m : MII/M = a cos 0a + b cos 0b ,
h t = Ht/JM,
where M = M, + Mb, J = Jb = Jab and MII = M a cos O, + M b cos 0b is the component of the total magnetization parallel to H. Hence, we can write the existence condition A ' = 0 as follows, -a
[ _ l a(Oa _ 0b)3 + (~ht + 2 y _ w)O~ - htTOb]
A, = b+ht+x
[~b(O
a -
0b) 3 + ( l h t + 2 x - 0 ) 0 3 - htTOa]
=0.
(3.14)
Note that in the case of ferrimagnetic order we have J < 0 so that also both h and h t are negative. We have to take into account that h t is the value of the field at the transition. In the simplest case, it is the anisotropy field h A in the hard direction. In terms of the reduced variables it is related to the other p a r a m e t e r s by the condition that A = 0, i.e., -a
A=
a+ht+Y
b+h t+x
-b
=0,
(3.15)
which is identical to eq. (3.7) for L = b, i.e., to eq. (3.8). The direction angles 0 are determined but for a proportionality factor Oa/Ob = ( a + h t + y ) / a
(3.16)
.
A m o n g the p a r a m e t e r s a, b, x, y and h t only three are independent variables. By solving eq. (3.14) we can express (Y/0bz) in terms of y ' , h t , a, v and w, y/0~ = (h* - h t ) [ ( y ' + a) 4 + a3(1 - a ) ] / [ 2 a 2 h t ( Y '2 + 2 a y ' + a ) ] ,
(3.17)
where, h* = [-by'2(y'
+ 2a)2 _ 2 v ( y ' + a ) 4 - 2 w a 3 b l / [ ( y '
+ a) 4 + a 3 b ] ,
(3.18)
and x' = x + h t ,
y' =y + h t .
(3.19)
In case v = w = 0, our two-sublattice magnetic system is fixed by giving a, x and y. It can be represented by a point in the plane of the sublattice anisotropy fields (x, y). Transformation (3.19) allows reference to overall effective fields x', y ' which
FIRST-ORDER MAGNETIC PROCESSES
435
are a superposition of anisotropy and applied field. Equation (3.17) is related to the approach to saturation, because the component of the measured magnetization parallel to H is MII, or m = MII/M. In fact, by using eqs. (3.13), (3.15), (3.16) and (3.17) we obtain the slope of the magnetization curve at h = ht, S = d m / d ( h / h t ) l h = h t = a ( y '2 + 2 a y ' + a ) 2 h t / { [ ( y ' + a) 4 + a3b](ht - h*)}.
(3.20) The boundary between second- and first-order transitions is determined by the condition ht = h*.
(3.21)
In the limit of infinite exchange (J---~ oo), this condition gives (K2, + K2b )/(Kla + Klb ) = ~, which, as expected, is the equation of the boundary line n' of FOMP of type 1 (see section 2.2). The use of eq. (3.20) allows us to evaluate what is the role of the canting effects in determining FOMPs in different conditions. In particular, one is interested to ascertain whether FOMPs are possible when only K~ constants are present in the individual anisotropies of the sublattices. Hence, we shall examine the various situations under the hypothesis that g2a and K2b are vanishing, so that eq. (3.18) reduces to h* = - b y ' 2 ( y
' + 2 a ) 2 / [ ( y ' + a) 4 q- a3b] .
(3.22)
In the case J > 0 we have ferromagnetic order. It can immediately be seen that in this case S > 0: in fact, 0 < a < 1, h* < 0 and h t = h A > 0. So, in general, we ought not to expect a FOMP when the two sublattices are coupled ferromagnetically and the highest sublattice anisotropy terms are of second order. At least, we can exclude type-1 F O M P with S < 0. In the case of ferrimagnetic order (J < 0), the situation is more complex. In fact, we have three possible transitions with increasing field. The first one at H = Htl(=HA), where the system reaches the state of intermediate saturation with the two vectors antiparallel. The system does not change until another critical field H = Ht2 is reached that triggers the break of the collinear symmetry, and the two vectors start to open an angle. With further increase of H the system reaches the forced ferromagnetic state at a field H = Ht3, with the two vectors parallel. However, depending on the values of the parameters, it is possible that the intermediate saturation is completely absent, so that there is a continuous regime of the magnetization in the canted phase from H = 0 up to H = Ht3. This can happen because the intermediate saturation can be either unstable or metastable. The condition for the existence of FOMP of any kind, o.FOMP or h.FOMP, at each of the transition points H t l , Ht2 and Ht3 is deduced from eq. (3.20), where H t is always a solution of eq. (3.15). So, the negative value of S, relative to a certain transition point, together with the fact that the collinear state involved is
436
G. ASTI
stable, are sufficient conditions to assure that the transition occurs through a FOMP. In the case of ferrimagnetic order, a is larger than 1, the denominator of eq. (3.22) is less than zero for y' below a certain value Y0. So that for y' < y ~ we have S < 0 if h t < h*, i.e., Ihtl > Ih*l. In the case of the transition to forced ferromagnetic order (at H = H~3), we have 0 < a < 1 so that always h * < 0 and S < 0 if h , > h * , i.e., Ihtl < Ih*l. We have to remember that, in this case, the reduced variables (3.13) are referred to a different unit field, because for ferrimagnetic order it is J M = J ( M a - I M b l ) , while for ferromagnetic order it is J M = J(Ma + IMbl), which implies a factor (a - b) between the reduced expressions of the same quantities [see eq. (3.32)]. We can obtain the equation in parametric form for the boundary line, 'h*' between first- and second-order transitions. The definition equations are x0=x'-h*,
Y0=Y'-h*"
(3.23)
We must take into account that the effective fields x' and y', defined by eq. (3.19) obey eq. (3.15), that represents a hyperbola having an upper branch that crosses the origin, i.e., ax' + by' + x ' y ' = 0,
(3.24)
so that eq. (3.23) becomes, x o = -by'/(y'
+ a)- h*,
Yo = Y ' -
h*,
(3.25)
in which y' is the variable parameter. Its graph for the case M a = 21Mbl will be shown in fig. 17. For the aforementioned value y' = y;, the denominator of eq. (3.22) vanishes and one has h*---~ ~. In the graph the line denoted by 'Y0" is then the asymptote of line h*. From the above analysis, we can conclude that the canting effects can be responsible for FOMP transitions, even in the absence of anisotropy constants of higher than second order. However, this possibility is limited to ferrimagnetic order, certainly for h.FOMP, while it is doubtful that o.FOMP could exist. Obviously, by admitting K2a a n d / o r K2b # 0, there are a broad range of possibilities for a variety of FOMPs, and the equations obtained and critical parameters could be used for the analysis of the experimental curves. In the limit of J---~ 0o we find y' ~ 0, h* ~ 0 and S ~ 1, as expected. It may be useful to write down the explicit expressions for h t of the various critical fields. They can be deduced from eq. (3.15), which in explicit form is, ht2 + (1 + x + y ) h t + ax + by + x y = 0.
(3.26)
We take into account definitions (3.13) of the reduced variables, which are valid for all cases. As already mentioned, this implies that for the same set of values of the physical parameters, Ma, IMbl, gl,,, glb and IJI we obtain different sets of
FIRST-ORDER MAGNETIC PROCESSES
437
values for a, b, x and y in the various cases. For the case of ferromagnetic order (J > O) we have only one transition at (3.27)
htf = h A = ½(C + D ) , where, C= -1-x-y
,
D = [(1 + x - y ) 2 _ 4 a ( x - y)]1/2.
If htf is negative, it means that the c axis is an easy direction. The solution with negative sign has no physical meaning in this case, because it merely gives the value of the field at which the c axis changes from a maximum to a saddle point for the free energy F. A geometrical interpretation of htf is given in fig. 12. Given a representative point P(x, y) in the plane (x, y) of the reduced sublattice anisotropy fields, we draw the line g from P having angular coefficient equal to 1
~Y
X
-1
¢/:
Fig. 12. Magnetic phase diagram for the case of a two-sublattice system having ferromagnetic order ( J > 0 , M b > 0 ) with a=Ma/M= 3, in the plane of reduced sublattice anisotropy fields x = 2Kx,,/(JMM,~), y=2K1J(JMMb). Line o,~ is the upper branch of the hyperbola with equation ax + by + xy = 0. The region above o-[ corresponds to collinear magnetic states with the sublattice moments parallel to the c axis. When the representative point P(x, y) drops below ¢r[, the magnetic system again has the same magnetic order provided the magnetic field is stronger than the anisotropy field H A. A geometrical interpretation of this critical-field value is given by the components of vector PQ, oriented at 45° from the x axis, that turn out to be equal to the reduced anisotropy field h A = htf = HA/JM.
438
G. ASTI
and crossing the upper branch o-I of the hyperbola ax + by + x y = 0 ,
(3.28)
in a point Q: the components of the vector PQ are equal to htf. This procedure stems from the fact that the equation for the critical fields [(3.26)] is obtained from eq. (3.28) by the substitution x---~x + h, y---~y + h. The condition for the c axis to be a hard direction turns into the requirement that point P must not be above o-I . For the case of ferrimagnetic order (J < 0) we have, htl = h A = ½(C + D ) ,
(3.29)
htz= ½(C- D) .
Referring to fig. 13, we have that htl and ht2 are represented by the components of vectors PQ1 and P Q z , respectively, which, indeed, are both negative in the given example. Q1 and Q2 are the intersections with the upper branch o-a+ of the hyperbola given by eq. (3.28), which is different from o-f because now a > 1 (and
t,
(3.30)
In fact, in this case, a and b are both positive, as in the ordinary ferromagnet, but J < 0 so that F is a minimum below the lower branch o-~- of hyperbola (3.28) (see fig. 14). It is evident that when P drops below the straight line 'r', of equation y-x-p = 0 and tangent to O'a+, then the intermediate saturation state is unstable. The value of p comes from the condition D = 0 and turns out to be p = 1 - 2a + 2(a 2 - a) 1/2 .
(3.31)
If we want to compare ht3 with the other two critical fields related to the ferrimagnetic order, it may be convenient to express ht3 in the same units as used for htl and ht2. For this we have to refer to the same unit field J M - = J(M,, -[Mb]) instead o f J M + = J ( M a + IMbl), where we have denoted by M - and M + the modulus of the resultant magnetization in the cases of ferri- and ferromagnetic order, respectively. Then, assuming for all the reduced variables the meaning they have in the case of ferrimagnetic order, we must multiply ht3 by a factor, ~1 = M + / M - = (M,~ + [ M b l ) / ( M a - IMb[) = a -
b,
(3.32)
in order to obtain its value in the new units. Or we can give directly in terms of a, b, x and y (as defined in the ferrimagnetic order) the new expression h~3 , by using the substitution, x---~ x / ( a - b) ,
y---~ - y / ( a - b) ,
ht3--~ h~/rl .
(3.33)
F I R S T - O R D E R M A G N E T I C PROCESSES
439
::Y
-3
-2
)
~
X
u
+
.........................................
i:i ......
Fig. 13. Magnetic phase diagrams for the case of a two-sublattice system having ferrimagnetic order (J < 0, M b < 0 ) , with a = Ma/M = 2, in the plane of reduced sublattice anisotropy fields x = 2K1,,/ (JMMa), y = 2K1J(JMMb). Line o-~+ is the upper branch of the hyperbola with equation ax + by + xy = 0. The region above tra+ corresponds to collinear magnetic states with the sublattice moments parallel to the c axis but having opposite orientations. When the representative point P(x, y) drops between line O'a+ and the straight line r, which is the upper part of the tangent to (r~ in its vertex, the magnetic system again has the same magnetic order provided the magnetic field is stronger than the anisotropy field H A = Htx but lower than the second critical field, H,2, at which the collinear state becomes unstable (see section 3.2.2). A geometrical interpretation of these critical-field values is given by the components of vectors PQ1 and PQ2, oriented at 225 ° from the x axis, that turn out to be equal to the reduced critical fields, h A = ht~ = H~/JM and ht2 = H~2/JM , respectively. Line h* defines the boundary between second-order and first-order transitions [see eq. (3.22) and fig. 17]. In using this diagram one has to take i n t o a c c o u n t that, because of the fact that J < 0 and M b < 0, the reduced critical fields are negative, x has a sign opposite to K~a and y has the same sign as K~b.
So, we have, (3.34)
h~ = ½(a - b ) ( C - - D - ) ,
with, C- = -1-
x/(a-
b) + y / ( a -
b) ,
D - = {[1 + x / ( a -
b) + y / ( a -
b)] 2 - 4a(x + y ) / ( a - b)} 1/2 ,
(3.35)
which yields, h(3=X{b-a-x+y-[(a-b+x+y)2-4a(a-b)(x+y)]l/z}.
(3.36)
440
G. ASTI
Y
-2
-1 I
I
X
h* g
iP
-2
Fig. 14. Magnetic phase diagram for the case of a two-sublattice system having forced ferromagnetic order ( J < 0 , Mb>0), with a=Ma/M= z3 in the plane of reduced sublattice anisotropy fields x = 2Kxa/(JMMa), y = 2Kxb/(JMMb). Line try- is the lower branch of the hyperbola with equation ax + by + xy = 0. The region below trf corresponds to collinear magnetic states with the sublattice moments parallel to the c axis. When the representative point P(x, y) drops above line trf the magnetic system again has the same magnetic order provided the magnetic field is stronger than the third critical field Ht3. A geometrical interpretation of this critical field value is given by the components of vector PQ, oriented at 225° from the x axis, that turn out to be equal to the reduced critical field, hi3 Ht3/JM. Note that, in this case, h, x and y have signs opposite to H, Kla and K~b, respectively. Line h* defines the boundary between second-order and first-order transitions [see eq. (3.22) and fig. 17)]. =
A s a l r e a d y n o t i c e d , t h e a b o v e given e x p r e s s i o n s a n d r e p r e s e n t a t i o n s for t h e critical fields h t a r e g e n e r a l l y v a l i d , b e c a u s e t h e y d e p e n d o n l y On t h e s e c o n d o r d e r anisotropy constants. 3.2.5. L i n e a r regime a n d magnetic transitions in the canted p h a s e I t is k n o w n t h a t t h e m o l e c u l a r field t h e o r y a p p l i e d to a N r e l m u l t i s u b l a t t i c e s y s t e m , in t h e a b s e n c e o f m a g n e t i c a n i s o t r o p y , p r e d i c t s t h e p o s s i b i l i t y o f a
FIRST-ORDER MAGNETIC PROCESSES
441
non-collinear magnetic order that can be achieved via spontaneous SRT with changing temperature, or via field-induced SRT which are normally of second order. A peculiar feature of the magnetic behaviour of a canted system of this kind is that the resultant molecular field, acting on each one of the canted sublattice magnetization vectors, has the form,
lij = xj tj,
(3.37)
where Aj is a function of the set {J~K} of the molecular field coefficients only (Acquarone and Asti 1975). That is, the canted sublattices are effectively decoupled and evolve with temperature as an assembly of non-interacting ferromagnets, each with a different 'effective Curie constant' (°quasi-ferromagnetic' behaviour). In this respect, also the applied field H can be formally treated as one of the canted vectors to which a 'molecular field' x H = r. M i corresponds, i.e., the resultant magnetization. The consequence is that X, the susceptibility of the system, is also a constant, being only a function of the set {Jir}; this phenomenon was observed by Clark and Callen (1968) in rare earth-iron garnets. Indeed, they have observed temperature ranges over which the magnetization, at fixed field strength, is temperature independent. For a two sublattice N~el ferrimagnet having molecular fields, H a -= JxlMa + J 1 2 M b ,
l i b = JzlMa + J22Mb,
(3.38)
we have in the canted phase, (3.39)
X = - 1/.112.
It is worth noting that the two-sublattice ferrimagnet discussed in section 3.2.4, having only second-order anisotropy constants, does show a similar behaviour within certain conditions. In fact, it is found that if the representative point P ( x , y ) belongs to the upper branch '1' of the hyperbola (see fig. 17), b x + ay - x y
-= 0,
(3.40)
then it turns out that: (i) Either htl or ht2 is equal to --1; (ii) The same point P(x, y) in the reference frame of the forced ferromagnetic order (J < 0, M b > 0) belongs to the lower branch 1 of hyperbola (3.40) so that also ht3 = - 1 [or h~3 = - a + b, if expressed in ferrimagnetic reduced variables as in eq. (3.36)]; (iii) The magnetization curve is linear and takes the simple form m = - h , where h and m a r e the reduced field and magnetization defined by relations (3.13). This means that the magnetic susceptibility is a constant up to h = h~3 and is equal to, X = M H/H = m M / h J M
= - 1/J,
(3.39')
442
G. A S T I
an expression that coincides with eq. (3.39), which is related to a N6el ferrimagnet (without anisotropy). In order to demonstrate this property, we start from the equilibrium equations, b sin(0a - 0b) + h sin 0a + x sin 0~ cos 0, = 0 , (3.41) - a sin(0~ - 0b) + h sin 0b + y sin 0b cos 0b = 0. These equations can be obtained by means of relations (3.3) using the same definitions for x , y , a and b as given in connection with relations (3.13). Denoting s~ = sin 0~, c I = cos 0a, s 2 = sin 0~ and c 2 = cos 0b, and multiplying the first equation by c 2 and the second by Cl, we obtain after subtraction, ( m + h ) ( s l c 2 - seca)
+ (xs 1 -
YS2)CaC 2 = O.
(3.42)
Now, elimination of h from eqs. (3.41) yields, (b - Y)C2S~/S 2 - (a
2 -
X)ClS1/S
2
2 '[- a C 2 S 1 / S 2 - - b c 1 = O .
(3.43)
Taking into account condition (3.40), we can verify that XS 1 -- ys 2 = O,
(3.44)
is a solution of eq. (3.43), so that eq. (3.42) becomes, (m + h) sin(0a - 0b) = 0.
(3.45)
Since in the canted phase 0a ~ Ob, we conclude that m = - h . Note that hyperbola (3.40) is obtained by a translation of hyperbola (3.28) parallel to a vector v, i.e., (1, 1), having unitary components. In fact eq. (3.40) can be obtained from eq. (3.26) by taking h t = - 1 . The results are also consistent with eq. (3.20), where S is equal to the differential susceptibility at the transition point. In fact, substitution of h t = - 1 in eq. (3.20), where h* is given by eq. (3.22), yields, S = (y,2 + 2 a y ' + a ) 2 / ( y '2 + 2 a y ' + a) 2 = 1.
(3.46)
Moreover, we can show that if htl = - 1 (or ht2 = - 1 ) , then also ht3 = - 1 and vice versa. In fact, let x + and y+ be the coordinates of the representative point P(x +, y+) in the flame of forced ferromagnetic o r d e r (i.e., x + and y+ are expressed in units of J M + = J ( M , + [M b [). Then by the substitution (3.33) for x + and y+ in eq. (3.26), we obtain, ht23 + [1 + (x - y ) / r l ] h t 3 + (ax + b y - x y ) / r l 2 = O,
(3.47)
F I R S T - O R D E R M A G N E T I C PROCESSES
443
with 7/= a - b. If b x + a y - x y = 0, eq. (3.47) becomes ht2 + [1 + ( x - y ) / ' o ] h t 3 + ( x - y ) / r l = 0 ,
(3.48)
for which ht3 -- - 1 is a solution. Or, if we put ht3 = - 1, we obtain b x + a y - x y = 0, which implies that either htl or ht2 is equal to - 1 . Now we can observe that the variation of 01 and 02, according to the law of eq. (3.44), is such that, in general, the m i n i m u m value m 0 of the magnetization in the linear regime is above m = 0, and occurs at a certain critical field h 0 (when h t l = - 1 ) . Obviously, depending on the values of x and y , such a m i n i m u m can coincide with intermediate saturation (when ht2 = - 1 ) , or it can coincide with the origin (h = 0, rn = 0) when x = y [see the paragraph below eq. (3.49)]. That is, the range of the minimum m 0 is 0 ~< m 0 <~ 1. The transition to the linear regime is
Ill¸ 3
1 ...........
0
Ta,
T_'I~===.T 2
-I1.1
-ht2
I
2
;
-h
J,
Fig. 15. Magnetization curve for the case of a two-sublattice ferrimagnet with a = 2, x = 0.75 and y = 0.6. This is an example of the linear regime that occurs when the system has only second-order sublattice anisotropy constants and the representative point P(x, y) belongs to the upper branch T of the hyperbola b x + a y - x y = O, shown in fig. 17. The equation of the linear part, in terms of reduced variables, is m = - h , which implies that susceptibility is a constant equal to X = - 1 / J . The transition to the linear regime occurs at a certain magnetization m 0 and is of second order. When m 0 < 1, as in the example shown, there are actually two superimposed magnetization curves in the range m 0 < rn < 1, corresponding to the same rn and the same energy but to different magnetic structures.
444
G. ASTI
a s e c o n d - o r d e r transition. In the case m 0 < 1, which occurs w h e n htl = - 1 , the m i n i m u m in the m a g n e t i z a t i o n implies that we have a range of h (Ih0l < Ihl < 1) w h e r e two different m a g n e t i c structures c o r r e s p o n d to the same m. Actually, there are in this range two b r a n c h e s of the m a g n e t i z a t i o n curve that c o r r e s p o n d to different angled-magnetic structures. Figure 15 shows an e x a m p l e for the case w h e n a = 2, x = 0.75 and y = 0.6. A w a y to elucidate the shape of the magnetization curve m o r e clear, is to consider it as the limit of a t w o - b r a n c h curve of the type given in fig. 16, c o r r e s p o n d i n g to a point just a b o v e line 1 [see eq. (3.40) and fig. 17]. Besides the s e c o n d - o r d e r transition, at the lower limit of the linear curve, w h e r e h = h 0 and m = m o , there is a first-order transition in the angles 01 and 02 that is not visible in the parallel c o m p o n e n t m , as defined by the last of the relations in eq. (3.13), but in the transverse c o m p o n e n t m I = a sin 0a + b sin 0b. A s a m a t t e r of fact, in the whole interval Ih0l < fhl < 1 the two magnetic states, besides having the same m, have equal free-energy density. In fact, it is easy to show that the free e n e r g y of the system in the linear regime d e p e n d s on 0a and 0b only t h r o u g h the variable m = a cos 0a + b cos 0b. F o r this p u r p o s e , it is sufficient to utilize relations (3.40) and (3.44) in the expression of F given by eq. (3.1) for the case of two sublattices, after substitution o f the r e d u c e d variables of eq.
1.05
.95
d
I
,9'5
. . . .
1
,
,
,
,
I
%05'
'-h ~
Fig. 16. Magnetization curve for the case of a two-sublattice ferrimagnet with a = 2, x = 0.745 and y = 0.595 corresponding to a point close to line '1' [see eq. (3.40) and fig. 17]. The case of fig. 15 can be considered as the limit of a two-branches curve of the type shown here. A FOMP transition is shown that brings the system from the intermediate saturated state into the spin-flop phase.
FIRST-ORDER MAGNETIC PROCESSES
445
(3.13). H e n c e , the system is d e g e n e r a t e a n d the two m a g n e t i c phases m a y coexist in the w h o l e r a n g e of h u n t i l h = h t l , w h e r e the p h a s e c o n n e c t e d with the i n t e r m e d i a t e s a t u r a t i o n state disappears. This t r a n s i t i o n occurs with n o discont i n u i t y i n the v a r i a b l e m a n d c a n b e c o n s i d e r e d as the limit of a F O M P of the type s h o w n in fig. 16. A n o t h e r possible l i n e a r s o l u t i o n of eq. (3.43) is 0a = 0 b = 0 w h e n the r e d u c e d a n i s o t r o p y fields of the two sublattices are e q u a l , i.e., w h e n the r e p r e s e n t a t i v e p o i n t P ( x , y ) b e l o n g s to the l i n e ' s ' (see fig. 17) which has the e q u a t i o n x = y. T h e n , eqs. (3.41) b e c o m e , (h + x cos 0) sin 0 = 0 , i.e., m = - h / x
=-h/y.
(3.49)
I n the case of f e r r o m a g n e t i c order, we m u s t have
.8 'O.
.6
.4
.4
.6
.8
X
1
Fig. 17. The phase diagram, shown in fig. 13 for the ferrimagnetic case, is shown here in greater detail in the area of the first quadrant which is crossed by various critical lines: h*, O'a+, 1, r, s and y~. For explanation of h*, ~r,+ and r see fig. 13. Hyperbola 1is mentioned in fig. 15 and, together with straight line s, is discussed in section 3.2.5. Line y~ is the asymptote of line h*. Line h* crosses line 1 and is tangent to it in a critical point G with coordinates x = a - ~aa2 - a, y = 1 - x. In G they are both tangent to straight line r.
446
G. A S T I
x = y < 0, while in the ferrimagnetic case x = y > 0. Obviously, the linear relationship b e t w e e n magnetization and field holds only up to htt. It is worth noting that we have no canting, even in the presence of 'competing' anisotropies (Kaa < 0 and Klb > 0 in the ferrimagnet). The various critical lines h*, o-+, 1, r, s and Y0, deduced in the analysis of t h e ferrimagnetic case, are shown in fig. 17. Line h* crosses line 1 and is tangent to it in a critical point G with coordinates x = a - ~/a 2 - a and y = 1 - x. In G, they are both tangent to the straight line r. Line 1 crosses line s in a point u, i.e., (1, 1). B e l o w line r the intermediate saturation state does not exist. A b o v e this line various magnetization curves are possible that, in general, are c o m p o s e d of two branches. These connect with continuity two couples of the four critical points: O (h = 0, m = 0), the origin; T1 (h = hta , m = 1) and T2 (h = ht2 , m = 1), the extremes of the intermediate saturation state; and T3 (h = ht3 m = a - b). In addition to these two branches, there is another equilibrium state represented by the line which connects T~ and T e (see, e.g., figs. 15, 16, 18 and 19) which covers
m
3
T3
2
T1 ~
1
o
1
2
2
-h
3
Fig. 18. The magnetization curve for the case of a two-sublattice ferrimagnet is, in general, composed of two branches connecting with continuity two couples of the four critical points: O (h = 0, m = 0), the origin; T1 (h = h,1, m = 1) and T2 (h = ht2, m = 1), the extremes of the intermediate-saturation state; and T3 (h = ht3 , m a - b ) the transition point to the forced ferromagnetic state. In addition to these two branches, there is a n o t h e r equilibrium state represented by line T 1 - T 2 which covers the range of intermediate-saturation state. The present graph, obtained for a = 2, x = 1.1 and y = 1.2, i s an example of a magnetization curve consisting of branch O - T 2 and branch T 1 - T 3 . T h e collinear state for h = 0 (0a = 0b = 17r) belongs to branch O - T 2 and is not the equilibrium state. The state of m i n i m u m free energy at h = 0 is, instead, on the other branch where we have 0a ~ 0b ~ ½~" and m ~ 0. This branch and its continuity is better evaluated w h e n considering it as a connection between T1 and the symmetric point of T3 with respect to the origin (i.e., for h = - h t 3 , m = - a + b ) . Note that in the present case, at h = 0 we have a noncollinear state with the resultant magnetization pointing out of the basal plane, but very close to it. =
F I R S T - O R D E R M A G N E T I C PROCESSES
447
m
T3
3
2
1
T1
~--._._... T2
1
Fig. 19. As in fig. 18 but for a = 2, x = 0.9700 and y = 0.9047. The two branches are of the type O - T 3 and T1-T2.
the range of intermediate saturation state. Besides figs. 15 and 16, other examples are reported in figs. 18 and 19 showing different connections. In the case of fig. 16 it is easy to predict a FOMP, because, with increasing field, there must necessarily be a jump from one branch to the other. Depending on the characteristics of the branches, we understand that we can have in the various cases a variety of transitions also involving the intermediate saturation state. A condition that can be favourable for observing these phenomena may be that of ferrimagnets in the neighbourhood of compensation points, because the field JM can be made small, and the various critical conditions are achieved at relatively low fields. We must remember that the examples given (figs. 15, 16, 18 and 19) are obtained in the limit of only second-order anisotropy terms (i.e., v = w = 0) and accordingly also lines h* and 1 in fig. 17 refer to this situation. Instead, the graphs and the expressions relative to critical fields h t ' s remain the same, even in the presence of higher-order anisotropy terms because they are independent of v and W.
3.3. The small-angle canting model 3.3.1. General remarks The magnetic behaviour of a multisublattice magnetic system, under conditions of small deviations from the collinear magnetic order, can be conveniently treated by the use of an approximate theory, the small-angle canting model (SAC) (Rinaldi and Pareti 1979). The SAC model appears as a simple extension of the classical phenomenological theory of magnetic anisotropy. By the admission of deviations from the collinear order, the system acquires a further degree of freedom that
448
G. ASTI
results in a relaxation of the free energy, which appears as a strong distortion of the anisotropy-energy surface. The opening of the angles between the interacting moments occurs at the expense of exchange energy, with a subsequent gain in anisotropy energy. The important point is that such a distortion can be readily accounted for, by appropriate corrections of the anisotropy constants. It means that the system can be described as a simple ferromagnet characterized by 'effective anisotropy constants' K1, K2, 1 £ 3 , . . . , that are functions of sublattice anisotropy constants and of the molecular-field coefficients. The effective anisotropy constants are not rigorously constant because there remains a magnetic field dependence to first order in the parameter h = H / J M , where H is the magnetic field and J M is a quantity of the order of the exchange field. High-order anisotropy constants appear, even if we attribute to the individual sublattices only second-order constants. For instance, /£1 can be very different from the simple algebraic sum of the individual Ka's. It can even have opposite sign. The small-angle canting model has general validity, as it has been shown (Asti 1981) that it applies to both ferromagnetic and ferrimagnetic order, as well as to any combination of anisotropy constants. The conditions for the application of the SAC model may refer: (i) To the orientation of the magnetic vectors within a narrow range close to the crystallographic symmetry directions (as in the case treated in section 3.2); (ii) To the whole range of orientations, when exchange interactions are much stronger than magnetic anisotropy forces in the system. The region well inside the spin-flop phase of ferrimagnets is obviously excluded (see section 3.2). In the case of magnetization processes close to the symmetry directions, the model provides exact expressions for important physical parameters, such as initial susceptibility and anisotropy fields, or relations concerning important phenomena such as the spin-reorientation transitions. The concept can be applied, in principle, to complex structures such as the case of a three-sublattice system (Asti 1987). In section 3.3.2, the case of a twosublattice system will be described and examples of applications will be given in section 3.3.3. 3.3.2.
The two-sublattice m o d e l
Let us consider a uniaxial system of two magnetic sublattices a and b having magnetizations M a and M b and anisotropy constants Kaa, 1£2o, K 3 a , . . . , and Klb, K2b, K 3 b , . . . . We refer the orientation angles of all vectors with respect to the symmetry axis, c. So, 0~, 0b, 0 and q~ are the angles of M a, M b , M = M a + M b and magnetic field H respectively. The free energy of the system is, E = - JMan b
-
-
M" H + ~ i=1
K ~ sin 2i 0a +
k Kjb sin2~0b ,
(3.50)
j=l
where - J M a • M b is the exchange energy and J the intersublattice molecular-field coefficient. The interaction is ferromagnetic if J > 0 and antiferromagnetic if J < 0. If 'a' denotes the major sublattice, one may have M b > 0 or M b < 0. The
FIRST-ORDER MAGNETIC PROCESSES
449
assumption that the minor magnetization M b c a n take negative values, allows to utilize the same definition for the angles, so as to extend immediately all the results obtained for ferromagnetic interaction to the ferrimagnetic case, simple by changing the sign of both J and M b (see section 3.2). We will consider only situations for which M~ and M b a r e nearly parallel, so that we can write e a = 0 , - 0 and 8 b -~-0b --O, where G and e b are infinitesimal deflection angles of M~ and M b with respect to M. T h e n eq. (3.50) becomes E = - J M a M o cOS(G - % ) - M a l l cos(q) - 0 - G ) - M b H cos(~ - 0 - %) 4- ~
giasin2i(o-} - Ca)'+ ~
i=t
Kjb sin2J(0 + % ) .
(3.51)
j=l
By putting p = M b / M ~ , M = M¢ + M b and e = eb, and taking into account that M~ e a + M b e b = 0, we obtain by expansion up to second order in e, E = E o + A e + B e 2,
with E o = - J M , , M b - M H cos(q~ - 0) + ~
(3.52)
(K~ + K~b).sin 2~ 0 ,
i=1
A = k
- 2 i ( P K i ~ - Kib) COS 0 sin 2/-x 0 ,
i=1
B = IjpM2
+ lpMH
i(Kiap 2 +
cos(q~ - 0) + ~
gib ) sin 2i-2 0
i=1
x [ ( 2 i - 1) cos 2 0 - sin 2 0] . H e r e E 0 is the ordinary expression for the total energy of the collinear system. The value of e is determined by the condition 3 E / O e = 0 , which gives e = -A/2B. Substitution in eq. (3.52) yields, E = E o - A2/4B ,
(3.53)
i,e.
E = Eo -
- i g i c s 2i-1 -
)7{
e +
ifde~-2[(2i i=1
}
1) - 2is 2] ,
(3.54)
with gi = PKia - Kib ,
c=cos0
and
~ = p2Kia q- Kib '
s=sin0.
e = ½[ p J M 2 + p M H cos(q~ - 0 ) ] ,
450
G. ASTI
If we suppose that gia and Kib are zero above a certain order N, eq. (3.54) can be expressed as,
)2/{
E = Eo- c
_
igis2i 1
e + fl
-
2N2fN
S2N
N--1 + ~
} [ ( j + 1)(2]
+ 1)f]+ 1 --
2]2fjls 2j .
(3.55)
j=l
This expression for the energy can be transformed in such a way that it assumes the usual form of a power series in sin 2 0. Hence, the coefficients K~ are the effective anisotropy.constants of the two-sublattice canted system. In the simplest case, i.e., for N = 17 it gives, c~
E = - M H cos(q~ - 0) + J M ~ M b + ~.
K i
sin2i 0 ,
i=1
with,
K1 = Kla + Klb -- g21/(e + fl) ,
K 2 = g21(e - f ~ ) / ( e
+ L)
2 ,
(3.56)
gn = gn-12fl/(e
-{-]el) =
2 , - n - 2 ~t e - f l ) / ( e + f ~ ) gaY1
n.
These results are the same as those reported in the work by Asti (1981) in which t h e p a r a m e t e r s a, b and h are related to the p a r a m e t e r s used here, by the following relations, a = 2g2/JpM 2
b
=
2fl/JPM 2 ,
h + 1 = 2e/JpM 2 •
Let us now consider the case N = 2, which means that we have sublattices with anisotropy constants of second and fourth order. Then from eq. (3.55) we obtain, 4-[4-k1 E= Eo-[g~/(e+
fl)] ~ k=l
r[(k+r),2]
r~=l - Pr-l[
j~--O __ ( k
_
r - , ] Jcelk-r-2j a 2j]12k JJa ,
j
(3.57) where [w] indicates the largest integer in w, with the assumption that [w] = 0 for w ~< 0, and Pa = - 1 + 4g2/g 1 ,
Po = 1 , 2
2
P3 = - 4 g z / g a ,
P2 = ( - 1 + g2/gl)4g2/ga ,
% = -(6f2 - 2fl)/(e + fl),
a2 = 8fz/(e + f l ) .
One can easily verify that for g2a = g2b = O, the coefficient K k of S 2k a r e coincident with expressions (3.56). It can be convenient to write down the first
FIRST-ORDER MAGNETIC PROCESSES
451
few terms of eq. (3.57) explicitly, if we limit the series to k = 4, it becomes, E = E o - [g~l(e + f~)]{s 2 + (Pl + 0/1)$4 -~- (0/21 -~ 0/2 q- P10/1 + P 2 ) $6
+ [0/31 + 20/10/2 + P,(°~21 + 0/z) + P z a l + P31ss + ' " "}.
(3.58)
The coefficients of s 2~ in eq. (3.57), including the terms contained in E 0, give analytical expressions for the effective anisotropy constants K k as functions of the parameters of the two sublattices, i.e., K~a, Kza, Klb , K2o , m a , m b and J. The dependence of K k on the magnetic field is through the parameter e. It is negligible in the transverse configuration, i.e., when the initial susceptibility is measured with H perpendicular to the c axis, both because H is small compared with J M and because q~ = ½7r and 0 is small, so that cos(q~ - 0) ~-0. Instead, when H is parallel to the c axis and this is a hard direction, the parameter e plays an important role. As a consequence, the anisotropy field has two different expressions depending on whether the axis under consideration is a hard or an easy direction. The same expressions given in sections 3.2.2 and 3.2.4 for the anisotropy field H A in a hard direction, can be obtained here from the definition, H A = 2Ka/M,
(3.59)
where K 1 is given by eq. (3.57) or (3.58), taking into account expression (3.52) for E 0. More easily, K 1 is directly taken from the first part of eq. (3.56) because H A depends only on second-order anisotropy constants. So, the condition for the spontaneous SRT has the usual form K 1 -= 0. The expression for the anisotropy field in easy direction is again deducible from eq. (3.59), here we have to put the term p M H cos(q~ - 0), which appears in the parameter e, equal to zero. All these considerations are also valid if we consider a planar direction instead of the c axis. In fact, if we make a K ~ R transformation (see section 2.2), we can exchange axial with planar directions, which means that all the expressions obtained can be interchanged once we replace all K constants by the corresponding R constants. Models including K 2 constants have been utilized successfully by Ermolenko (1979) and by Sinnema et al. (1987), for computing magnetization curves of R C % and R2Co17. This means that higher-order terms in the crystal-field interaction must be taken into account. The advantage of the present treatment is that it gives analytical expressions that are exact for magnetic processes that involve small angles between the sublattice moments, e.g., concerning magnetic transitions involving symmetry directions (see section 3.2). In the analysis of the magnetization curves, it is convenient to make reference to the ordinary K anisotropy constants when the applied magnetic field is along or near the c axis. On the other hand, when we are magnetizing along a direction close to the basal plane, we have a more accurate description of the magnetization curve in terms of the R constants, which are obtained from the above mentioned K<--->R transformation.
452
G. ASTI
3.3.3. Some applications The first application of the small-angle canting model was by Rinaldi and Pareti (1979) on PrCos, giving a straightforward explanation of the SRT easy-axis to cone transition observed at 110 K by Tatsumoto et al. (1971), Ermolenko (1976) and Asti et al. (1980), despite the fact that the s u m gla "~ Klb of the Co and Pr sublattice anisotropy constants is positive. In fact, as explained in sections 3.2.2 and 3.3.2, according to this model, the condition for the existence of a SRT is merely the vanishing of the effective/£1 constant, i.e., K 1 -- 0. By the use of the formulas given in section 3.3.2 it is possible to analyze, in a very simple way, the magnetization curves of both ferro- and ferrimagnetic materials. The approximation of the small-angle canting model is valid in a large number of cases and a variety of experimental conditions. As an example, we will report here the analysis carried out by Asti and Deriu (1982) of the initial susceptibility data, in the hard direction, obtained by Ermolenko (1979) on the series Yl_zGdzCos. In fact, the effective anisotropy constant K~ given by eq. (3.56) can be written in the form,
K1 = ~7(K~. + Klb),
(3.60)
where ~ is a correction parameter and K~a and Klb are the anisotropy constants relative to the Co and Gd sublattice, respectively. A convenient expression for 77 in terms of reduced exchange and anisotropy fields, as defined by eqs. (3.13) is r/= [1 + xy/(ax + by)]/(1 + bx + ay).
(3.61)
The case of the Gd sublattice is a special one, since Klb 0. It means that there is no competition between the anisotropies of the two sublattices. In spite of this, the canting effects exist because the competiton is with the applied magnetic field H. Hence, eq. (3.61) reduces to =
= 1/(1 + bx).
(3.62)
From the data on the initial susceptibility X as a function of the composition z, taken from the work of Ermolenko (1979), we can calculate the effective anisotropy constant K 1 = M2/(2X) as a function of z, and plot the experimental values of ~7= KI(z)/KI(O). Figure 20 shows the theoretical curve together with the experimental points. The agreement between the point and the curve supports the validity of the model, and shows how much the effective anisotropy is reduced due to the onset of a small deviation from collinearity. Indeed, in the absence of canting effects, the parmeter 77 ought to be independent of z. Another example is also taken from the work by Asti and Deriu (1982) and shows how general the validity of the Rinaldi and Pareti (1979) model is. As explained in section 3.3.2, it applies to any case of magnetic order (ferro- and ferrimagnetic) and sign of sublattice anisotropy constants. Besides that, for magnetic processes close to the symmetry directions, it is rigorously valid, even
FIRST-ORDER MAGNETIC PROCESSES
453
8
6
4
o
a
z'
,
Fig. 20. Plot of the values of the correction parameter ~7 [see eq. 3.60] calculated from the initial susceptibility data versus composition, taken from the work of Ermolenko (1979) for the series Yl_zGdzCos. The solid line is the theoretical curve deduced from eq. (3.62).
when anisotropy and exchange interactions are of the same order. Indeed, the case of ferrimagnetic order is that for which canting can produce dramatic effects. Peculiar examples are those of HozFe17 and Ho2Co17 , for which enormous discrepancies were obtained between anisotropy constants, as deduced from spin-wave spectra, and macroscopic magnetic measurements (Clausen and Lebech 1980). The apparent paradox is easily removed by the admission of canting, and the model presented above gives a complete account of the effect. Sarkis and Callen (1982), in order to study this problem, have developed an ad hoe calculation that led them to the following expression for the effective anisotropy constant, in terms of the constants (K R, KT) and the magnetizations (MR, MT) of the individual sublattices, Keff = (K 1 + 2K2)eff = (K R + K T + 2KRKT/L)/[1 + 2(KR M2 + KTMZ)/LM2],
(3.63)
where L = JMTM R and J is the intersublattice molecular-field coefficient. This result can be derived directly from eqs. (3.60) and (3.61). To do this, we have to use the K ~ R transformations described in section 2.2, which allow the immediate transfer of all the equations valid for the case HIIc axis to that of H_l_c axis and vice versa. So we can write the conjugate equation of (3.60), R 1 = 'rl(Rla + R~b),
(3.64)
where R~, Rla and Rlb a r e the anisotropy constants conjugate to K~, Kla and Klb , i.e., R I = - K 1 - 2 K 2 - 3 K 3 + ' " , and similarly for R1, and Rib. Taking into account expression (3.61) for ,/, eq. (3.64) becomes
R 1 = (Rla + R~b + 2RI,RIb/J')/[1 + 2(RaaM ~ + RlbMZa)/J'M2],
(3.65)
with J ' = JMaM b. This expression is coincident with eq. (3.63), once R1, Rla ,
454
G. ASTI
J', M a and M b are substituted by the corresponding quantities Keff, Ka-, KR, L, M r and M R. An approach based on the hypothesis of the small canting-angle has also been utilized by Cullen (1981) for the study of the problem of easy-magnetization directions lying along non-major axes in ternary cubic compounds of composition R~R~_xFe 2. This work also confirms that a classical model of canted magnetic moments can give rise to high-order anisotropy energy and phase instabilities. This type of approach is not as accurate as the direct numerical diagonalization of the total Hamiltonian containing crystal-field terms and the molecular field Hex. However, the classical treatment provides a transparent picture of the physical processes responsible for the observed phenomena.
Rib ,
3.4. Hexagonal ferrites having complex block-angled and spiral structures. Role of antisymmetric exchange During the investigation of magnetic properties of hexagonal ferrites, Enz (1961) encountered unusual properties in the system Bal_xSrxZn2Fe12022 (abbreviated as Bal_xSr~Zn 2 - Y ) . The compound has no spontaneous magnetization, but a field of the order of 1 kOe induces a first-order transition to a magnetization of about 85% of saturation. At a higher field, after a rather fiat plateau of the magnetization, another transition starts a further increase at a constant susceptibility up to saturation. Enz interpreted this behaviour as being due to a helical spin configuration having the axis parallel to the c axis and the magnetization vector rotating in the basal plane. The same behaviour has been observed in other hexagonal ferrites, both of the same Y structure [Ba2Mg 2 - Y (Albanese et al. 1975)] and of another structure, namely Ba3_~Sr~Zn2Fe24041 [Z-type, usually shortened as Ba3_~SrxZn2-Z (Namtalishvili et al. 1972)]. Like all hexagonal ferrites, also Y- and Z-type crystal structures are formed by the stacking of chemical blocks (usually labelled S, T and R) along the c axis (Smit and Wijn 1959). The starting point of the theory is the fact that no anomaly is observed in Zn2-Y and Zn2-Z containing Ba, and that the substituting Sr ions are localized in well-defined layers inside the T blocks for both structures. Hence, the suggestion of Enz was that substitution of Sr results in a weakening of the superexchange interaction patterns across the planes of the crystal. Then any further counteracting interaction across the same planes may result in a deviation from collinearity of the spin. In fact, neutron-diffraction investigations on a variety of hexagonal ferrites have shown the existence of a peculiar type of magnetic order that is known as 'block' structure (Aleshko Ozhewsky et al. 1969, Namtalishvily et al. 1972, Sizov and Zaitser 1974). The structure of these ferrites is such'that in it one can identify symmetric magnetic blocks, possibly comprising several chemical blocks: inside each magnetic block the localized spins stay mutually parallel, but the total magnetic moment of the block, taken as a whole, may assume a canted orientation with respect to contiguous blocks. The resulting configuration may be either a flat or conical spiral ('block conical spiral', or 'helix'), or an alternating structure
FIRST-ORDER MAGNETIC PROCESSES
455
when the rotation angle from block to block is 180°, thus producing an 'antiphase block' structure (also termed 'block angular' structure). As for the case of compounds containing Sr, in the case of Ba2Mg2-Y, the occupation by a nonmagnetic ion (Mg 2+ ) in the inner octahedral sites of the T block is expected to give rise to a similar weakening of the superexchange coupling between adjacent magnetic blocks. Besides the transition from spiral tO fan structure, these compounds having block-magnetic structures display another transition at a higher field. As a result, the magnetization curve appears as a two-step curve with a sharp knee at the end of the first fiat part. An example is given in fig. 10 of Chapter 6 of Vol. 3 of this handbook, p. 408 (Sugimoto 1982), for the case of Sr2Zna-Y. The figure has been taken from the publication by Enz (1961). Various theoretical models have been proposed for two-sublattice systems, indicating that different canted states and second-order spin-flop transitions are possible (Yamashita 1972, Morrison 1973). Mita and Momozava (1975) give an interpretation of the transition from helix to fan state on the basis of Nagamyia theory (1967). Just as in the interpretation given by Enz (1961), however, no explanation is given for the transition observed at higher fields. The existence of a critical field at the end of the first plateau in the magnetization curve has been interpreted by Sannikov and Perekalina (1969) in terms of a two-sublattice model, but only assuming different values for both the magnetic moments and the anisotropy constants of the sublattices. However, according to this model no spiral structure is possible, and moreover, the hypothesis is contradictory to the inherent symmetry of the two sublattices to be associated with the angled blocks. A step forward in the theory comes from an important qualitative consideration on the effects of the substitution. Indeed, the breaking of the symmetry centers inside the T block opens the way to the action of antisymmetric exchange as proposed by Koroleva and Mitina (1971). A complete theoretical study based on this line of thought has been given by Acquarone (1981). It provides the equilibrium configurations and the phase transitions of a magnetic model system, consisting of two sets of moments of equal magnitude, coupled by isotropic, anisotropic, and antisymmetric exchange, in the presence of uniaxial anisotropy and a transverse magnetic field. It applies both to two-sublattice and to spiral magnetic structures, and gives exhaustive explanations of a variety of two-step magnetization curves of the type described above. The model does not assume any restriction upon the relative strength of the different interactions. 4. Considerations on the experimental methods
4.1. Measurements in high magnetic fields First-order magnetization processes are normally observed at low temperatures, i.e., well below the Curie point, due to the rapid decrease of the anisotropy energy with temperature. It is clear from the graph shown in fig. 6, where an example of the ~l(l + 1) power law for the temperature dependence of anisotropy
456
G. ASTI
is given, that the effect of this behaviour is both to reduce the critical field and to move the state of the system towards the boundary with the normal state. In the majority of cases, at the higher limit of the temperature range, the phenomenon vanishes when the critical field is still relatively high. With only a few exceptions, as is the case with ferrites (Asti et al. 1978) and with Pra(Fe,Co)x 7 (Shanley and Harmer 1973, Melville et al. 1976), this means that magnetic fields above the limit of ordinary laboratory electromagnets are necessary to investigate this type of magnetic phenomena. On the other hand, the measurement of the critical parameters (field and magnetizations) provides useful data that allow deduction of important and precise information on high-order terms of the anisotropy energy. The use of single-crystal samples is obviously the most favourable condition for detecting the transition and a unique way to achieve precise evaluations of the critical magnetization mcr (or m s and m2, in the case of type-2 FOMP). However, the orientation of the crystal in the magnet appears, in most cases, to be quite critical for performing precise measurements, because of the strong sensitivity of Her to deviations from perfect alignment: e.g., a deflection by only two degrees of the applied-field direction from the basal plane in PrCo 5 gives rise to a change of Her by 2 T. Another source of error, which is important only for transitions occurring at low fields, is the demagnetizing field of the sample. If the parallel component M of the magnetization of the crystal is plotted versus the applied magnetic field intensity He, one gets a straight line at the transition that is not vertical, but has a slope equal to d M / d H e = 1 / N , where N is the demagnetizing factor of the specimen. So, in general, a better procedure is to plot directly M as a function of the internal magnetic field H = H e - N M . The use of spherical or ellipsoidal specimens offers obvious advantages: (i) accurate values of N can be determined, and (ii) one can avoid the broadening of the transition due to the non-uniformity of H. This non-uniformity produces distortions in the magnetization curve M ( H ) , particularly at the edges of the transition where the values of m 1 and m: must be identified. The use of continuous magnetic fields allows accurate and reliable determinations, if a large number of measurements are performed in the neighbourhood of the transition point. However, with the exception of a few magnets in the world giving constant fields above 20-25 T the practical limit is in the range around 15 T as is well-known, for both superconducting and water-cooled solenoids. Besides that, one has to take into account the cost of the equipment, which changes by orders of magnitude going through a threshold at about 8-10 T. As a matter of fact, rare earth intermetallic compounds having H~r above 15 T are fairly common. The only way to carry out magnetic measurements at very high magnetic fields is by using pulsed methods. The magnet laboratory at the Amsterdam University is unique, in that it provides the possibility of achieving 40 T, keeping the field intensity constant for a time interval of the order of one tenth of a second. Other installations in the world allow to go even above that limit in a reproducible way by using various pulsed-field techniques (Gr6ssinger 1982), which, in general, give sinusoidal pulse shapes. The main problems arising with the use of pulsed-field apparatus, in performing magnetic measurements, are
FIRST-ORDER MAGNETIC PROCESSES
457
connected with the inherent difficulty in overcoming the various noise sources. For instance, in using the induction method, the flux caused by the specimen has to be detected in the huge background signal coming from the applied field. Despite this, there are several reasons why a pulsed field could be used for magnetic measurements: (i) in a single shot it is possible to measure the whole magnetization curve from zero up to the maximum field; (ii) the continuous and regular variation of the field is a favorable condition for detecting sudden changes such as those associated with a transition; (iii) the experimental setup is relatively simple and economical.
4.2. The singular-point detection technique The inherent high sensitivity of the pulsed-field method for detecting transitions is more and more enhanced if one observes the successive derivatives of the magnetization M with respect to the field H, or to time. This is indeed the principle at the basis of the singular-point detection technique (SPD) (Asti and Rinaldi 1974a), an experimental method that allows to reveal the singularities associated with the field-induced magnetic transitions, utilizing polycrystalline specimens. The SPD theory was originally developed for measurements of the hard-direction anisotropy fields, and has been subsequently extended to the case of FOMP (Asti 1981, Asti and Bolzoni 1985). The magnetization curve of a ferromagnetic crystal shows, in general, a singuarity when it reaches saturation at the anisotropy field H A. The transition is, in principle, of second order and occurs only if the magnetic field is perfectly aligned with the hard direction. For field orientations other than hard and easy directions, the reversible magnetization curve is, in general, a regular function for H = HA, never achieving complete saturation for finite values of H. When we apply a magnetic field to a polycrystalline material, we obtain, in general, a smooth curve M(H) which, at first sight, seems to give no indication of the singularity at H = H A. However, in reality the averaging over all the crystal orientations does not completely cancel the singularity, which is due to the contribution of the crystallites oriented in such a way that their hard directions are nearly parallel to H. The singular point can be detected by observing the successive derivatives d~M/dH n. The shape of the singularity and the order of differentiation n at which it becomes apparent depends on the symmetry of the hard axis and on the ratios of the anisotropy constants. Asti and Rinaldi (1974a), besides giving explicit expressions for the singularity in the most important cases of both cubic and uniaxial symmetry, have determined general rules for obtaining the order of differentiation at which a discontinuity appears at the singular point, both for the cases of longitudinal and transverse susceptibilities. The principle allows precise and reliable measurements of the anisotropy fields using polycrystalline specimens. The most important case is that of uniaxial materials having hard directions in the basal plane; the singularity is observed in the derivative of the differential susceptibility, d2M/dH 2, and has the shape of a cusp on the left-hand side (H < HA) of the peak at H A. Usually, SPD measurements are
458
G. A S T I
easily performed in pulsed fields by direct differentiation of the signal from the pick-up coil. However, also continuous-field techniques can be used by computing the derivatives of the measured M(H) curve (Obradors et al. 1984), or by applying single or double modulation, or triangular waves. The grain size has no effect, while the grain orientation distribution only affects the amplitude of the observed peak, in a way similar to X-ray diffraction in powders. Indeed, this inherent sampling characteristic of the technique, to select the crystallites oriented with their hard axes nearly parallel to H, allows studying distributions of anisotropy fields, and even to determine the angular distribution function of the crystallites (Asti 1987), an application that could be of interest for characterization of permanent magnets and for the study of coercivity mechanisms. The detection of the singularity in the transverse susceptibility configuration is difficult because domain-wall displacement is superimposed to the reversible rotation process for H < H A. However, this high sensitivity to domain walls offers a convenient method for the study and characterization of single-domain particles, as shown recently by Pareti and Turilli (1987). SPD measurements have been performed on a wide variety of hard magnetic materials, such as ferrites, various types of alloys such as MnA1, PtCo, CoCr, and rare earth intermetallic compounds. In all cases, the SPD technique proved to be particularly convenient for extensive studies on solid solutions, on the effects of additives and various chemical modifications, as well as on materials obtained via melt spinning and similar methods. An interesting case is that of the system Zr(Fel_xAlx) 2 which is hexagonal over a wide range of x values (Hilscher and Gr6ssinger 1980). Among the most recent applications are measurements of anisotropy fields made on new classes of rare earth intermetallics, i.e., on Nd2Fe14B with various substitutions (Asti et al. 1987, Pareti et al. 1988, Pareti 1988) and on SmFe,,Ti and similar compounds (Li et al. 1988). Also worth noting is the case of NdCos, for which SPD measurements have been performed of anisotropy fields relative to the c axis as well as to in-plane anisotropy, in the whole range of temperatures where the compound changes its easy direction, from axis to cone and from cone to plane (Marusi 1988). When applied to materials that exhibit FOMP transitions, the SPD technique provides exact determinations of the critical field Her. In fact, it was demonstrated, in the case of uniaxial materials (Asti 1981, Asti and Bolzoni 1985), that Her increases continuously if the crystal is rotated out of the exact orientation, e.g., the c axis parallel (A-case) or perpendicular (P-case) to the magnetic field. As a consequence, the magnetization curve of a polycrystalline aggregate shows a singularity exactly located at Her (see fig. 21). For a random orientation, in the case of a P-type FOMP, there is a discontinuity in the differential susceptibility, dM/dH, equal to, A X = (Ms/Hcr)(m
2
ml)2/(~/-]
- m 1
(4.1)
where m I and m e are the critical magnetizations as defined in section 2.2 [(eq. (2.9)]. In the case of A-type FOMP the discontinuity is in the slope of the
F I R S T - O R D E R M A G N E T I C PROCESSES
459
M/Ms
Her
H
Fig. 21. (Top) Magnetization curve M(H) of an easy axis uniaxial single crystal at various orientations close to ~p = 90 °, where q~ is the angle of magnetic-field direction with respect to the c axis. The F O M P transition is progressively displaced to higher fields with decreasing ~. From left to right the various curves in the figure refer to ~p = 90 °, 86 °, 82 °, 78 °, 74 ° and 70 °. (Bottom) M(H) and first derivative dM/dH for a polycrystalline aggregate having crystallites oriented at random.
differential susceptibility, d 2 M / d H 2, and turns out to be, A X ' = ( M J H c2~ ) ( m 2 - m 1)3/(W~- _ m 2I - wry-_ m22)2 .
(4.2)
For the P-type case Asti and Bolzoni (1985) calculated the slope of AX just above the critical field, i.e., d A x / d H , a quantity that is related to the peak width appearing in d M / d H at H = Her. Perfect agreement was found with computed curves based on an extended Stoner-Wohlfarth model that includes anisotropy terms up to sixth order and arbitrary distribution functions for crystallite orientation. Figure 22 shows an example of a computed curve M ( H ) simulating the case of an oriented polycrystalline sample of PrCo s (Asti et al. 1980). The computed curve turns out to be in perfect agreement with the experimental curve obtained by a pulsed-field experiment at a temperature of 78 K. For comparison the theoretical curve for a single crystal is also shown. The accurate measurement of Her by the SPD technique, together with the value estimated for M~r, allowed precise determinations of the ratios x = K 2 / K 1 and y = K3/K1, through c o m parison with the phase diagram of FOMP (see fig. 4). Above 155 K, the FOMP vanishes and the same technique gave for the same sample the anisotropy field, H A = 2(K, + 2K 2 + 3 K 3 ) / M ~. As regards the determination of the anisotropy constants K 1, K 2 and K 3 from the magnetization curve M ( H ) in oriented polycrystalline materials, there is another remarkable phenomenon, described below, that is important to point out, because it gives strong support for a suitable best-fit procedure. In fact, it is found (Asti et al. 1980) by computer simulations with the above-mentioned extended
460
G. ASTI
M
s
..:-=---.-----"
~
me r Hcr
;
0
;
;
I
6
I
I
I
I
12
I
I
I
I
18
I
;
;
"-
-"
H (Tesla)
Fig. 22. Computed magnetization curve M(H) based on an extended Stoner-Wohlfarth model that includes anisotropy terms up to sixth order and arbitrary distribution function for crystallite orientation. The case shown in the figure is a simulation of an oriented polycrystalline specimen of PrCo 5, after Asti et al. (1980). Solid line: computed curve, dashed line: single-crystal curve with magnetic field in the basal plane, (a) experimental points.
Stoner-Wohlfarth model, that the magnetization curve of an oriented polycrystalline material is reproduced very well by the single-crystal curve oriented at an angle p, which represents the average angle of misalignment of the crystallites. The coincidence of the two curves is surprisingly good up to values of M near saturation, and for ~p of the order of 10-20 °, which means that it is insensitive to changes both of the type and of the width of the angular distribution function of the crystallites. As a consequence, it is possible to use a very simple and rapid best-fit program based on the analytical expression of M(H) for a tilted single crystal, in which the angle enters as a further adjustable parameter just as K1, K 2 and K 3. Even when there is a FOMP, this procedure can be conveniently used in the part of the curve below Her. Besides PrCos, there are other cases where SPD techniques proved to be very effective in giving evidence of the existence of FOMP. Figure 23 shows the progressive change with decreasing temperature of the SPD signal, d2M/dH 2, from a magnet of Nd2Fe14B. As is evident, the downward cusp located at H A is modified until, at 200 K, a sharp positive peak appears that grows more and more with further decrease of the temperature. Clearly, the positive peak is an approximation to the delta function representing the derivative of the discontinuity in the differential susceptibility given by eq. (4.1). Other examples are intermetallic compounds of ThMn12 structure (Li et al. 1988, Deriu et al. 1989, Solzi et al. 1988). Finally, it is worth noting that there is a surprising resemblance between the shape of the SPD peak in d2M/dH 2 for a random polycrystal of a uniaxial easy-axis ferromagnet, and the thickness, 6(H), of a domain wall in a magnetic field perpendicular to the c axis and opposite to its magnetic moment. Expressions for 8 have been obtained by Drring (1966) and Wasilewski (1973), and fig. 24 shows the graphical representation of 8. By using the energy density criterion explained in section 2.6, it is very easy to deduce the expression for 8. In fact, if we assume that all anisotropy constants of higher than second order are vanishing,
FIRST-ORDER MAGNETIC PROCESSES
I
d=M/dt2
f
' T - 290
461
K
l
f
.~
184
0
I
50
100
H(KOe)-~-
Fig. 23. Progressive change with decreasing temperature of the SPD signal,
L
dEM/dH 2, for a magnet of
Nd2Fe14B. Evidently, the downward cusp located at H = H A is modified until, at 200K, a sharp positive peak appears that grows continuously.
E i
i
i
i
I
i
I
~1.4 1.2 1 i
0
i
4
i
6
H(Koe)
I
12
Fig. 24. The thickness, 3(H), of a domain wall of a uniaxial easy-axis ferromagnet, under the action of a magnetic field perpendicular to the c axis and opposite to its magnetic moment [see eqs. (4.4) and (4.5)], after Wasilewski (1973).
we have,
A(dO/do')~ = K , ( 1 - sin 2 OH) + HMs(1 + sin OH),
(4.3)
w h e r e OH is the equilibrium angle within the domains. R e m b e r i n g that h = HMs/2K ~ and sin 0H = h, f r o m the usual definition of the wall thickness, we obtain, 3 = (7r + 2 0 H ) / ( d 0 / d r r ) = (A/K~)I/2(~r + 20H)/(1 + h ) ,
for h < 1 ,
(4.4)
462
G. ASTI
and
= (A/K1)~/2~'/V~,
for h > 1
(4.5)
Expressions (4.4) and (4.5) are in agreement with the expressions given by Wasilewski (1973). References Acquarone, M., 1981, Phys. Rev. B 24, 3847. Acquarone, M., and G. Asti, 1975, J. Magn. & Magn. Mater. 1, 48. Albanese, G., G. Asti, M. Carbucicchio, A. Deriu and S. Rinaldi, 1975, Appl. Phys. 7, 227. Aleshko-Ozhewskii, O.P., R.A. Sizov, I.I. Yamzin and V.A. Lubimtsev, 1969, Sov. Phys.-JETP 28, 425. Andreyev, A.V., A.V. Deryagin, N.V. Kudrevatykh, V.A. Reimer and S.V. Terentiev, 1985, JETP 90, 1042. Asti, G., 1981, IEEE Trans. Magn. MAG-17, 2630. Asti, G., 1987, in: Proc. 5th Int. Symp. on Magnetic Anisotropy and Coercivity in RETM Alloys, Bad Soden 1987, eds C. Herget, H. Kronmuller and R. Poerschke (DPGGMBH, D-5340 Bad Honnef 1, FRG) p. 1. Asti, G., and F. Bolzoni, 1980, J. Magn. & Magn. Mater. 20, 29. Asti, G., and F. Bolzoni, 1985, J. Appl. Phys. 58, 1924. Asti, G., and A. Deriu, 1982, in: Proc. 3rd Symp. on Magnetic Anisotropy and Coercivity in RE-TM Alloys, Vienna 1982, ed. J. Fidler (Technical University of Vienna, Vienna) p. 525. Asti, G., and S. Rinaldi, 1974a, J. Appl. Phys. 45, 3600. Asti, G., and S. Rinaldi, 1974b, in: Proc. 3rd Eur. Conf. on Hard Magnetic Materials, Amsterdam 1974, ed. H. Zijlstra (Bond voor Materialenkennis, Den Haag, The Netherlands) p. 302. Asti, G., and S. Rinaldi, 1977, AIP Conf. Proc. 34, 214. Asti, G., F. Bolzoni, F. Licci and M. Canali, 1978, IEEE Trans. Magn. MAG-14, 883. Asti, G., F. Bolzoni, F. Leccabue, R. Panizzieri, L. Pareti and S. Rinaldi, 1980, J. Magn. & Magn. Mater. 15-18, 561. Asti, G., F. Bolzoni, F. Leccabue, L. Pareti
and R. Panizzieri, 1984, Nd-Fe Permanent Magnets - Their Present and Future Applications, Rep. and Proc. of a Workshop Meeting, Brussels, October 25, 1984, ed. I.V. Mitchel (Commission of the European Community) p. 161. Asti, G., F. Bolzoni and L. Pareti, 1987, IEEE Trans. Magn. MAG-23, 2521. Atzimony, U., M. Dariel, E. Bauminger, D. Lebenbaum, I. Novik and S. Ofer, 1973, Phys. Rev. B 7, 4220. Barbara, B., M.F. Rossignol, H.G. Purwins and E. Walker, 1978, J. Phys. (USA) C 77, L183. Bickford Jr, L.R., 1960, Phys. Rev., Suppl. llS, 1000. Birss, R.R., G.R. Evans and D.J. Martin, 1977, Physica B 86-88, 1371. Bolzoni, F., and L. Pareti, 1984, J. Magn. & Magn. Mater. 42, 44. Bolzoni, F., O. Moze and L. Pareti, 1987, J. Appl. Phys. 62, 615. Boucher, B., R. Buhl and M. Perrin, 1970, J. Phys. & Chem. Solids 31, 2251. Bozorth, R.M., 1936, Phys. Rev. 50, 1076. Buschow, K.H.J., 1980, Rare earth compounds, ch. 4, in: Ferromagnetic Materials, Vol. 1, ed. E.P. Wohlfarth (North Holland, Amsterdam) p. 299. Buschow, K.H.J., 1986, Mat. Sci. Rep. 1, 1. Callen, H.B., and E. Callen, 1966, J. Phys. & Chem. Solids 27, 1271. Casimir, H.B., J. Smit, U. Enz, J.F. Fast, H.P.J. Wijn, E.W. Gorter, A.J.W. Duyvesteyn, J.D. Fast and J.J. de Jong, 1959, J. Phys. Radium 20, 360. Clark, A.E., and E. Callen, 1968, J. Appl. Phys. 39, 5972. Clausen, K.N., and B. Lebech, 1980, J. Magn. & Magn. Mater. 15-18, 347. Cullen, J.R., 1981, J. Appl. Phys. 52, 2038. Cullen, J.R., and E. Callen, 1984, J. AppI. Phys. 55, 2426.
FIRST-ORDER MAGNETIC PROCESSES Cullen, J.R., and E. Callen, 1985, Physica B 130, 289. Deriu, A., G. Leo, O. Moze, L. Pareti, M. Solzi and R.H. Xue, 1989, Hyperfine Interactions, 45, 241. Dillon Jr, J.F., 1963, Domains and domain walls, ch. 9, in: Magnetism, Vol. 3, eds G.T. Rado and H. Suhl (Academic Press, New York) p. 415. D6ring, W., 1966, Mikromagnetismus, in: Encyclopedia of Physics, ed. S. Flfigge, Vol. XVIII/2, Ferromagnetismus, ed. H.P.J. Wijn (Springer, Berlin) p. 405. Eibler, R., R. Gr6ssinger, G. Hilscher, H.R. Kirchmayr, O. Mayerhofer, H. Sassik, X.K. Sun and G. Wiesinger, 1984, Nd-Fe Permanent Magnets - Their Present and Future Applications, Rep. and Proc. of a Workshop Meeting, Brussels, October 25, 1984, ed. I.V. Mitchell (Commission of the European Community) p. 167. Enz, U., 1961, J. Appl. Phys., Suppl. 32, 22S. Ermolenko, A.S., 1976, IEEE Trans. Magn. MAG-12, 992. Ermolenko, A.S., 1979, IEEE Trans. Magn. MAG-15, 1765. Ermolenko, A.S., 1982, in: Proc. 3rd Symp. on Magnetic Anisotropy and Coercivity in RETM Alloys, Vienna 1982, ed. J. Fidler (Technical University of Vienna, Vienna) p. 771. Ermolenko, A.S., and A.F. Rozhda, 1978, IEEE Trans. Magn. MAG-14, 676. Forlani, F., and N. Minnaja, 1969, J. Appl. Phys. 40, 1092. Graetsch, H., F. Haberey, R. Leckebusch, M.S. Rosenberg and K. Sahl, 1984, IEEE Trans. Magn. MAG-20, 495. Gr6ssinger, R., 1982, J. Phys. (USA) D 15, 1545. Gr6ssinger, R., and J. Liedl, 1981, IEEE Trans. Magn. MAG-17, 3005. Gr6ssinger, R., P. Obitsch, X.K. Sun, R. Eibler, H.R. Kirchmayr, F. Rothwarf and H. Sassik, 1984, Mater. Lett. 2, 539. Gr6ssinger, R., X.K. Sun, R. Eibler, K.H.J. Buschow and H.R. Kirchmayr, 1985, J. Phys. (France), Colloq. 46(6), C6-221. Hathaway, K.B., and G.A. Prinz, 1981, Phys. Rev. Lett. 47, 1761. Hilscher, G., and R. Gr6ssinger, 1980, J. Magn. & Magn. Mater. 15-18, 1189. Hiroyoshi, H., H. Kato, M. Yamada, N. Saito, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, Solid State Commun. 62, 475.
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Huang, Ying-Kai, C.H. Wu, Y.C. Chuang, FuMing Yang and F.R. de Boer, 1987, J. Less-Common Met. 132, 317. Kajiwara, S., G. Kido, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, J. Phys. Soc. Jpn. 58, 829. Kazakov, A.A., and N.A. Litvinenko, 1978, Fiz. Met. & Metallloved. 45, 940. Khan, W.I., and D. Melville, 1983, J. Magn. & Magn. Mater. 36, 265. Kido, G., S. Kajiwara, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, IEEE Trans. Magn. MAG-23, 3107. Koroleva, L.I., and L.P. Mitina, 1971, Phys. Status Solidi A 5, K55. Krause, D., 1964, Phys. Status Solidi 6, 125. Kudrevatykh, N.V., E.W. Lee and D. Melville, 1986, Fiz. Met. & Metalloved. 61, 898. Li, H.S., B.P. Hu, J.P. Gavigan, J.M.D. Coey, L. Pareti and O. Moze, 1988, J. Phys. (France) 49, 541. Lotgering, F., 1974, J. Phys. & Chem. Solids 35, 1633. Lotgering, F., P.R. Locher and R.P. van Stapele, 1980, J. Phys. & Chem. Solids 41, 481. Marusi, G., 1988, Thesis (University of Parma). Melville, D., W.I. Khan and S. Rinaldi, 1976, IEEE Trans, Magn. MAG-12, 1012. Melville, D., K.M. A1 Rawi and W.I. Khan, 1981, Phys. Status Solidi A 66, 133. Melville, D., J.M. Machado da Silva and J.F.D. Montenegro, 1987, Port. Phys. 18, 49. Meyer, A.J., 1964, C.R. Acad. Sci. 258, 4935. Mita, M., and N. Momozava, 1975, AIP Conf. Proc. 24, 406. Mitsek, A.I., N.P. Kolmakova and D.I. Sirota, 1974, Fiz. Met. & Metalloved. 38, 35. Morrison, B.R., 1973, Phys. Status Solidi B 59, 551. Mukamel, D., E.M. Fisher and E. Domany, 1976, Phys. Rev. Lett. 37, 565. Nagamiya, T., 1967, Solid State Phys. 20, 305. Namtalishvili, M.I., O.P. Aleshko-Ozhewskii and I.I. Yamzin, 1972, Sov. Phys.-JETP 35, 370. Obradors, X., A. Collomb, M. Pernet, J.C. Joubert and A. Isalgue, 1984, J. Magn. & Magn. Mater. 44, 118. Odozynski, R., and W.J. Zietek, 1977, Physica B 86-88, 1373. Paoluzi, A., F. Licci, O. Moze and G. Turilli, 1988, J. Appl. Phys. 63, 5074.
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Pareti, L., 1988, in: Proc. Int. Conf. on Magnetics, Paris 1988. Pareti, L., and G. Turilli, 1987, J. Appl. Phys. 61, 5098. Pareti, L., F. Bolzoni and O. Moze, 1985a, Phys. Rev. 32, 7604. Pareti, L., H. Szymczak and H.K. Lachowicz, 1985b, Phys. Status Solidi a 92, K65. Pareti, L., O. Moze, M. Solzi, F. Bolzoni, G. Asti and G. Marusi, 1988, Final C.E.A.M. Meeting, Madrid, April 12-16, 1988. Radwanski, R.J., 1986, Physica B 142, 57. Radwanski, R.J., J.J.M. Franse and S. Sinnema, 1985, J. Phys. (USA) F 15, 969. Rebouillat, J.E, 1971, J. Phys. (France) 32, C1-547. Rinaldi, S., and L. Pareti, 1979, J. Appl. Phys. 50, 7719. Rozenfeld, Y.V., 1978, Fiz. Met. & Metalloved. 45, 911. Sannikov, D.G., and T.M. Perekalina, 1969, Sov. Phys. JETP 29, 396. Sarkis, A., and E. Callen, 1982, Phys. Rev. 26, 3870. Shanley, C.W., and S. Harmer, 1973, AIP Conf. Proc. 18, 1217. Sinnema, S., 1988, Doctoral Thesis (Amsterdam University, Amsterdam). Sinnema, S., J.J.M. Franse, A. Menovsky, F.R. de Boer and R,J. Radwanski, 1987,
3rd Conf. on Phys. of Magnetic Materials, Szczyrk-Billa 1986 (World Scientific, Singapore) p. 324. Sizov, R.A., and K.N. Zaitsev, 1974, Sov. Phys.-JETP 39, 175. Slonczewski, J.C., 1961, J. Appl. Phys., Suppl. 32, 253S. Smit, J., and H.EJ. Wijn, 1959, Ferrites (Philips Technical Laboratory, Eindhoven). Solzi, M., L. Pareti, O. Moze and W.I.F. David, 1988, J.Appl. Phys. 64, 5084. Sugimoto, M., 1982, Properties of ferroxplanatype hexagonal ferrites, ch. 6, in: Ferromagnetic Materials, Vol. 3, ed. E.P. Wohlfarth (North Holland, Amsterdam) p. 408. Tatsumoto, E., T. Okamoto, H. Fujii and C. Inoue, 1971, J. Phys. (France), Colloq. 32, C1-550. Tiesong Zhao, and Jin Hanmin, 1987, Solid State Commun. 64, 103. Wasilewski, W., 1973, Acta Phys. Pol. A 43, 729. Wohlfarth, E.P., 1980, J. Magn. & Magn. Mater. 20, 77. Wohlfarth, E.E, 1983, First order magnetic transitions in some metallic materials, in: High Field Magnetism, ed. M. Date (North Holland, Amsterdam) p. 69. Yamashita, N., 1972, J. Phys. Soc. Jpn. 32, 610.
chapter 6 MAGNETIC SUPERCONDUCTORS
Oystein FISCHER D#partement de Physique de la Mati#re Condensee Universit# de Gen#ve Quai E. Ansermet 24, 1211 Geneve 4 Switzerland
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Early investigations . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Magnetic impurities in superconductors . . . . . . . . . . . . . . . . 2.2. Multiple pairbreaking theory . . . . . . . . . . . . . . . . . . . 2.3. Early studies of magnetic ordering and superconductivity . . . . . . . . . . 3. M a g n e t i s m and superconductivity in ternary c o m p o u n d s . . . . . . . . . . . . 3.1. T h e Chevrel phase compounds, ( R E ) M o 6 X 8 . . . . . . . . . . . . . . 3.2. T h e ternary r h o d i u m borides, ( R E ) R h 4 B 4 . . . . . . . . . . . . . . . 3.3. O t h e r ternary c o m p o u n d s . . . . . . . . . . . . . . . . . . . . 4. Re-entrant superconductors . . . . . . . . . . . . . . . . . . . . . . 4.1. T h e two c o m p o u n d s HoMo6S 8 and E r R h a B 4 . . . . . . . . . . . . . . 4.2. O t h e r re-entrant c o m p o u n d s . . . . . . . . . . . . . . . . . . . . 5. Antiferromagnetic superconductors . . . . . . . . . . . . . . . . . . . 5.1. Ternary m o l y b d e n u m chalcogenides, R E M o 6 X 8 . . . . . . . . . . . . . 5.2. Ternary r h o d i u m borides, primitive tetragonal (RE)Rh4B4 . . . . . . . . . 5.3. Ternary r h o d i u m borides, body centered tetragonal ( R E ) R h 4 B 4 . . . . . . . . 5.4. Other antiferromagnetic superconductors . . . . . . . . . . . . . . . 5.5. Analysis of the upper critical field curves . . . . . . . . . . . . . . . 6. Pseudoternary systems . . . . . . . . . . . . . . . . . . . . . . . 6.1. (MI_xREx)Mo6S 8 systems . . . . . . . . . . . . . . . . . . . . 6.2. ( R E I _ x R E ' ) R h 4 B 4 systems . . . . . . . . . . . . . . . . . . . . 6.3. (RE)(Rhl_xTx)4B4 systems . . . . . . . . . . . . . . . . . . . . 6.4. Some other pseudoternary systems . . . . . . . . . . . . . . . . . 7. Magnetic-field induced superconductivity: the J a c c a r i n o - P e t e r effect . . . . . . . . 8. Coexistence of superconductivity and m a g n e t i s m in the RE-oxide superconductors . 9. S u m m a r y and conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
466
467 469 469 470 474 476 477 482 487 489 490 501 503 504 508 509 510 511 516 516 518 522 527 529 534 539 541
1. Introduction
Both magnetism and superconductivity are important and common collective phenomena of solids. However, a quick inspection of various lists of elements, alloys and compounds shows that, as a rule, the two do not normally occur in the same material at the same time. Considering the large number of both superconducting and magnetic substances this mutual exclusion is clearly not merely accidental, but has a deeper reason, reflecting the antagonistic nature of the two phenomena. The first person to notice this, and to discuss the possible reasons for it was Ginzburg (1957). He considered the interaction of the magnetic induction B = /x0M, due to magnetization of the magnetic ions, with the superconducting state and pointed out that, for most cases (he considered at the time only what since then has become known as type-I superconductors), the magnetic induction would be higher than the critical field of the superconductor. Thus, superconductivity would not coexist with ferromagnetism unless special geometries, like thin films, were considered. The first experiments addressing this question were carried out by Matthias et al. (1958a,b). They introduced magnetic impurities into superconductors in order to obtain materials that would show both phenomena. The result of these experiments showed that the antagonistic nature of the relation between the two phenomena went beyond the electromagnetic coupling considered by Ginzburg. In fact, very small amounts of magnetic impurities were found to destroy the superconducting state, and Matthias and co-workers suggested this to be a result of the exchange interaction between the conduction electrons and the localized magnetic moments. This led Abrikosov and Gorkov (1961) to develop their ground-breaking theory of magnetic impurities in superconductors. These initial investigations left the experimentalists with the challenge of finding materials where the two phenomena could be confronted directly and the theorists with an equally interesting challenge to describe how the two would mutually interact. In the 30 year interval up till today, numerous papers have been published on this topic, and many new and striking phenomena have been found. In the first half of this period several important results were obtained, but the investigations were made difficult by the absence of materials where the 467
468
O. FISCHER
phenomena could be properly studied. The discovery in the mid-seventies of the ternary superconductors containing a regular lattice of magnetic ions, like (RE)Mo6S 8 (Fischer et al. 1975a), (RE)Mo6Se s (Shelton et al. 1976) and (RE)Rh4B 4 (Matthias et al. 1977) changed this situation. With these compounds the ideal materials for this problem were found, and since then considerable progress has been made in our understanding of the interplay of magnetism with superconductivity. This chapter, after a short summary of the early investigations in section 2, concentrates on the experimental results obtained in this second period. Since other reviews have appeared earlier on this topic (Fischer and Maple 1982, Maple and Fischer 1982, Ishikawa 1982, Machida 1984, Bulaevski et al. 1985a) I shall, while trying to give a complete picture, put special emphasis on work carried out in the last few years. In section 3 an overview is given of the various classes of ternary compounds with their magnetic and superconducting transitions. Section 4 deals with the re-entrant superconductors, a phenomenon triggered by ferromagnetic ordering in the superconducting state. The coexistence of antiferromagnetism and superconductivity is discussed in section 5, followed by a presentation of the lowtemperature phase diagrams of the various pseudoternary systems in section 6. The magnetic-field induced superconductivity due to the Jaccarino-Peter effect is presented in section 7. The essential property of these compounds is that the magnetic 4f-electrons are just sufficiently weakly coupled to the conduction electrons that magnetism and superconductivity can coexist to a certain extent, and that they are nevertheless sufficiently strongly coupled so that rather dramatic effects occur in the interplay of the two phenomena. Whereas the rule is normally that the two phenomena couple too strongly,, the discovery of the new oxide-superconductors (Bednorz and Muller 1986, Wu et al. 1987) have given us the example of the (RE)Ba2Cu307 compounds, where the two phenomena coexist with practically no coupling at all. These recent results will be summarized in section 8. The materials considered here are all materials in which the electrons giving rise to magnetism are localized and spatially separated from the conduction electrons. The resulting small overlap of the wave functions gives a small exchange interaction and thus makes the confrontation of the two phenomena possible. In opposition to this, there are many interesting and important situations where there is a competition between magnetism and superconductivity within the conduction band, or where it is possible that magnetic interactions somehow contribute to the pairing. Examples of such systems are organic superconductors (see, e.g., Jerome and Schultz 1982), heavy-fermion systems (see, e.g., Ott 1987), the compound Y9Co7 (Kolodziejczyk et al. 1980) and the new oxide-superconductors (see, e.g., Johnston et al. 1988). Intense research is presently carried out all over the world to understand the superconducting state of such compounds and, in particular, the oxide-superconductors. However, in the present chapter I shall limit myself to the former compounds where the magnetic electrons can be clearly distinguished from the superconducting electrons.
MAGNETIC SUPERCONDUCTORS
469
2. Early investigations
2.1. Magnetic impurities in superconductors In their initial investigations, Matthias et al. (1958a) studied the superconducting critical temperature T~ of La as a function of the concentration of rare earth magnetic impurities. They found that T~ was reduced linearly with the concentration and that superconductivity was completely destroyed beyond a critical concentration x~ of the order of one percent. Furthermore, in comparing the Tc reduction for the various rare earth elements, they found that it scaled with the spin S and not with the total moment J. This result strongly indicated that the exchange interaction between the localized moments and the conduction electrons was responsible for this effect. In its most simple form, this exchange interaction can be written as Hcx = ( l / N ) E I(Sis),
(1)
where I is the exchange integral, Si the spin of the N localized moments, and s the spin of the conduction electrons. The complete solution of this problem was given by Abrikosov and Gorkov (1961) (AG), using the Green function technique, and this work later became the basis for numerous investigations of pairbreaking phenomena in superconductors. Whereas scattering off nonmagnetic impurities does not destroy superconductivity (Anderson 1959), magnetic impurities tend to lift the degeneracy of the singlet Cooper pair. In a scattering process this leads to pair breaking and superconductivity will be destroyed when the scattering time r~ becomes of the order h/A, A being the superconducting order parameter. They showed that the critical temperature is given by the expression, In(I/t) = ~(½ + ½p/t)- ~ ( ½ ) ,
(2)
is the digamma function, t = TJ Too, and Tc and T~0 are the critical temperatures of the superconductor with and without impurities, respectively, p is known as the pairbreaking parameter and is given by, p = Am = ( q r T s L 0 ) -1 =
xN(O)
I2S(S + 1 ) / 8 k B T ~ 0 .
(3)
In this expression N(0) is the density of states at the Fermi level and x is the concentration of magnetic impurities. If the magnetic moment is not a pure spin but also has an angular momentum part one has to replace S(S + 1) by the de Gennes factor, G = (g - 1)2J(J + 1).
(4)
The x dependence of T c has since been investigated in numerous systems. Figure
470
O. FISCHER ~.0 ~ o.~ ~
--A-G T h e o r y
• Lal_xGdxA1a
b- 0.6 ~
upper limit of
~- o. 4 ~
temperature
O.2 ~ 0
~ ,
i 0.2
,
i 0.,4
,
i 0.6
,
i 0.8
,
A ,
~.0
r "1.2
,
r
t,4
n/nov Fig. 1. Critical temperature versus concentration of magneticimpurities in the Lal_xGdxA12system (after Maple 1968).
1 shows the theoretical result, obtained from eq. (2), together with experimental results obtained by Maple (1968) on the Lal_xGdxAl2 system. Among the many dilute systems of this kind that have been studied, many show deviations from the A G curve in the high-concentration region. Such deviations may result from the Kondo effect. Mfiller-Hartmann and Zittarz (1971) predicted that under certain circumstances the Tc versus x curve may fall below the AG curve and show re-entrant superconductivity in a narrow concentration range due to the temperature dependence of the scattering time zs. This effect was observed by Riblet and Winzer (1971) in the Lal_xCexA12 system. On the other hand, crystal-field effects on the magnetic moments will lead to a slower reduction of T~ than predicted by AG (Keller and Fulde 1971, Fulde and Peschel 1972). For a review of these properties see Maple (1976). 2.2. Multiple p a i r b r e a k i n g theory
The destruction of superconductivity by a magnetic field results from the pairbreaking nature of the interaction of the field with both the conduction-electron orbits and the spins. For most superconductors the upper critical field Hc2 is determined essentially by the orbital interaction, which is also responsible for the Meissner effect and the vortex lattice in type-II superconductors. In a dirty superconductor, where the mean free path l is shorter than the superconducting coherence length ~, the orbital critical field He2 is determined by eq. (2) (de Gennes 1964, Maki 1964) with, p = ev2rH/37rks T c ,
(5)
where z is the transport scattering time and t = T~ T c. The resulting temperature dependence is the well-known parabolic like curve of the critical field versus temperature.
MAGNETIC SUPERCONDUCTORS
471
The interaction of the field with the conduction-electron spins, the Zeeman term, leads, when treated alone, to a first-order transition at the paramagnetic limit Hp0 (Chandrasekhar 1962, Clogston 1962), /zoHpo = 1.84T~ [TI.
(6)
Fulde and Ferrel (1964) showed that a clean superconductor should enter a new phase when submitted to a field (acting only on the conduction-electron spins) approaching Hp. This new state has a finite momentum pairing and is found to have a lower energy than the uniform BCS state in this field range. However, scattering by nonmagnetic impurities will destroy this state and so far no system has been found where the presence of such a state could be observed. The observation of this state, as well as the paramagnetic limit in nonmagnetic superconductors is made difficult by the fact that the orbital effects described above usually destroy superconductivity before Hp0 is reached. Furthermore, spin-orbit scattering will reduce the paramagnetic pairbreaking and thus enhance the paramagnetic limit. In the limit of strong spin-orbit scattering the paramagnetic critical field becomes, Hp = 1.33V~o Hp0,
(7)
where hso = 2h/37rkBTjs o and Zso is the spin-orbit scattering time. In a real superconductor, both orbital and paramagnetic effects have to be taken into account when describing the critical field. This was worked out in detail by Werthamer et al. (1966) and by Maki (1966). In high-field superconductors the influence of the paramagnetic pairbreaking is seen as a change in the temperature dependence of the critical field with respect to the one found for orbital effects only [eqs. (2) and (5)] [for a review see, e.g., Decroux and Fischer (1982)]. In a superconductor containing magnetic moments, the theory has to include both the magnetic scattering and the polarization effects due to the exchange interaction. When the spins align and produce a finite magnetization, either as a result of a magnetic order or as a result of the action of an external field, the interaction given in eq. (1) corresponds, to first order, to the action of an effective exchange field Hj given by, H j = X[( S z )/gel.~B = (I/ggeNld,2)M(H, T ) ,
(8)
where M(H, T) is the magnetization and N the number of spins per unit volume. This field can become very high in the magnetic superconductors and even dominate the behaviour of the critical field in certain cases. Fulde and Maki (1966) worked out the theory for the critical field of a paramagnetic superconductor taking into account these various pairbreaking effects. The result, using the notation of Werthamer et al. (1966) can be written as (Fischer 1972),
472
O. FISCHER
In(l/t) = [½ + ¼i(Aso - Am)/]/] x ~0[ 1 + ½(h + m +/~m
+ ½(/~so --
/~m) -I-iy)/t]
+ [½ - } i ( X ~ o - X m ) / y ]
x $[2I- + ½(h + m +Am + ½(A~o-- A m ) / 2 - i y ) / t
]
- 6(½), y = [ a 2 ( h + m + h , ) 2 - ¼(A,o
-/~m)211/2"
(9)
In this equation, we have used h = 0.281Hc2(T) /He2(0 * ), hj = 0.281Hi(T)/H~2(O )* and m =0.281M(H, T)/Hc2(O ). a is the Maki parameter given by V2Hc2(O)/ Hp0. An important result, obtained by Fulde and Maki, is that for a dirty superconductor, in the limit of strong spin-orbit scattering, eq. (9) reduces to eq. (2) with the pairbreaking parameter p given by the sum of the various contributions, /9 = /~m -t- (h -I- m ) + (a2/rso)(h + h,) 2 •
(10)
The first term on the right-hand side describes the magnetic scattering, the second the orbital pairbreaking due to the external field and the magnetization and, finally, the last term the polarization effects due to both the external field and the effective exchange field. Using the solution of eq. (2) one can then write an equation for the critical field in terms of the orbital critical field Hc2 defined as the solution of eq. (2) with p given by eq. (5) (Fischer 1972, Decroux and Fischer 1982), H~2(T ) = H~2(T ) - M(Hc2, T) - 3.56AmHc2(0 )
- 0.22(a/%oT~o)[H~2(T ) + M(nc2, T) + Hj(Hc2 , T)] 2 .
(n)
Although for a quantitative comparison with experiment it is necessary to use eq. (9), the approximative eq. (11) allows an easy discussion of the various effects that determine the critical field of a magnetic superconductor. Note that we have here neglected the demagnetization effects. In experiments, the latter may influence the results in an important manner and, as usual, the field H has to be replaced by H - n M in order to get a complete description of the results, n being the demagnetization factor. Let us first consider the case where both the scattering and the polarization effects can be neglected (i.e., the case of very weak exchange interaction). We then have,
H~2(T ) = H~2(T ) - M(H~z, T).
(12)
This is basically the situation considered by Ginzburg (1957), and it tells us that
MAGNETIC SUPERCONDUCTORS
473
superconductivity will be destroyed whenever the magnetization exceeds the orbital critical field, i.e., the critical field the superconductor would have had in absence of magnetic moments. Thus, if the magnetization increases abruptly as a result of ferromagnetic order, one expects that the critical field decreases abruptly and superconductivity may possibly disappear altogether. However, in type-II superconductors it is rather plausible that H~*2> M0 (M0 = saturation magnetization) and thus this electromagnetic interaction between the magnetic moments and the superconducting electrons does not necessarily exclude coexistence. In the re-entrant superconductors to be discussed in section 4, the orbital critical field is found to be either somewhat smaller (HoMo6Ss) or somewhat higher (ErRh4B4) than the saturation magnetization. The third term on the right-hand side of eq. (11) describes the reduction of Hc2 due to magnetic scattering. This term is a constant in the AG approximation; it becomes, however, temperature and field dependent in other cases as, e.g., the Kondo effect. This term is also expected to be somewhat modified in the case of magnetic ordering, as discussed in section 5. Let us now consider the case where the electromagnetic and the scattering term can be neglected, the critical field is then given by, H¢2(T ) = H¢2(T ) - 0.22(a/r~oT¢)[H¢2(T ) + M(Hc2, T) + H~] 2 .
(13)
In most materials discussed here, H~2(0 ) and M(nc2 , T) are of the order of 1 T or less, whereas the saturation value of the exchange field is of the order of several tens of T, so that in the second term on the right-hand side we can neglect H¢2 and M compared to Hj. This term then becomes proportional to the magnetization squared (since H j is proportional to M) and we can write, H~z(T ) = H*c2(T ) - AMZ(H¢2, T ) .
(14)
This equation is thus similar to eq. (12), with the difference that the term subtracted from He2 in eq. (14) may become much larger than the one in eq. (12). This is the term which in most ferromagnetic materials makes a coexistence with superconductivity impossible. In a paramagnetic material, the field and temperature dependence of the magnetization may result in a very unusual temperature dependence of Hc2(T ). If one follows He2 while decreasing the temperature, the second term in eq. (14) will be very small if T is close to T c. However, as T decreases and the critical field increases, the second term rapidly grows and finally dominates, so that the critical field is determined by the condition that the second term does not become too large. This leads to a maximum in H~z(T ) and, thus, to a decrease of H~2 at low temperatures. The first indication of such an effect was observed by Crow et al. (1967) in La3_xGdxIn. In the ternary superconductors, several examples of this behaviour have been found. Figure 2 shows the example of ErMo6S 8 (Ishikawa and Fischer 1977b). The round maximum is not related to magnetic ordering but to the gradual polarization of the spins by the external field. The minimum at low
474
0 . FISCHER
0.25 2.6
Erl.2 M°6 58
2.4 "7 2.2 X 2.0 1.8
0.20
w
1.6 1.4
•
.,... p 0 0.1 0.2 0.3 0J
015
j, :3:
9 u.
,~ 010
_o
0.05
0
0.5
1.0 TEMPERATURE
1.5
2.0
(K)
Fig. 2. Critical field versus temperature for ErMo6S s (after Fischer et al. 1979).
temperature reflects an antiferromagnetic ordering to be discussed in section 5. The full line is the theoretical curve using eq. (9) (Fischer et al. 1979). A spectacular suggestion was put forward by Jaccarino and Peter (1962). They noted that the exchange interaction between the localized moments and the conduction electrons is in some cases negative, resulting in a polarization of the conduction electrons opposite to the polarization of the localized moments. They therefore suggested that it should be possible in a ferromagnet to compensate the exchange polarization of the conduction electrons by a very high external magnetic field and thus induce superconductivity by the field. This can be discussed using eq. (13). A negative exchange interaction means Hj < 0 and thus the second term can be canceled by applying an external field (plus the magnetization) equal to - H j . Superconductivity is then possible if He*2 is high enough. This effect, which has recently been observed, will be discussed more in detail in section 7.
2.3. Early studies of magnetic ordering and superconductivity Matthias and co-workers discovered early some alloy systems, Cel_xGdxRu 2 (Matthias et al. 1958b) and Yl_xGdxOs2 (Suhl et al. t959), where the magnetic moment of the Gd atoms would only lead to a weak reduction of T c. In a certain concentration range, they observed both superconductivity and a kind of magnetic
MAGNETIC
SUPERCONDUCTORS
475
order, with apparently no special influence of the magnetic order on superconductivity compared to the paramagnetic state. This left researchers with two important problems, one experimental and one theoretical. The experimental problem was how to be sure that the magnetic order and superconductivity occurs in the same volume element. The study of the alloy systems considered here was made difficult both by the possible presence of secondary phases and by the possible inhomogeneous distribution of magnetic moments. The theoretical problem was how to understand the apparent coexistence. Gorkov and Rusinov (1964) extended the AG theory to include magnetic ordering. They concluded that ferromagnetism would destroy superconductivity due to polarization effects. For certain concentrations one would observe superconductivity, but when lowering the temperature a return to the normal state would occur when the system enters the ferromagnetic state, with possibly a narrow interval of coexistence. They pointed out that for very strong spin-orbit scattering one would expect a phase diagram similar to the one observed experimentally. Careful investigations by Wilhelm and Hillenbrand (1970) allowed the preparation of single phase material and they investigated several phase diagrams. Figure 3 shows the phase diagram obtained for Cel_xTbxRu 2 (Hillenbrand and Wilhelm 1970). A coexistence region was found, but specific-heat studies (Peter et al. 1971) and neutron-scattering measurements (Roth et al. 1973, 1974, FernandezBaca and Lynn 1981) showed that the ordering is a short-range spin-glass type. Lynn et al. (1977, 1980) also found similar results in the Cel_xHoxRu 2 system. The magnetic correlation length in these systems is much shorter than the superconducting coherence length and, therefore, the superconductor does not distinguish the magnetically ordered state from the paramagnetic state, which explains the shape of the phase diagram shown in fig. 3. More recently, H/iser et al. (1983) observed an anomalous behaviour in another spin-glass system of this type, Thl_xNdxRu 2. In a narrow concentration interval they observed re-entrant behaviour, probably reflecting that in this system the magnetic correlation length is closer to the superconducting coherence length. /
Cel. x
Tbx Ru2 /
Z ~: 10
e
10
20
/
30
40
Concentration[%] Fig. 3. Phase diagram for the system Cel_,Tb=Ru 2. 0c reflects the maximum in the susceptibility and the hatched area shows the region where short-range magnetic order coexists with superconductivity (after Hillenbrand and Wilhelm 1970, Roth 1978).
476
O. FISCHER
When superconductivity and ferromagnetism are confronted with each other, the mutual interaction will, in case of coexistence, modify each of the two states. Anderson and Suhl (1959) pointed out that since the susceptibility of a superconductor goes to zero as q ~ 0, magnetic order in a superconductor, if mediated by the RKKY interaction would not be ferromagnetic, but oscillatory with a period of the order (a2~) 1/3, a being of the order of interatomic spacings and ~ being the coherence length. Oscillatory states were not observed in the early investigations, but they have indeed been seen in the ternary compounds and this has led to a detailed investigation of this phenomenon. In particular, it has been found that the electromagnetic interaction also leads to oscillatory ordering (Blount and Varma 1979). We shall discuss this in section 4. In an antiferromagnet, the polarization effects are not present as long as the antiferromagnetic period is much shorter than the superconducting coherence length. Baltensperger and Strassler (1963) showed that superconductivity can coexist with antiferromagnetism. However, a modification of the superconducting state is expected at the magnetic transition. They also investigated the magnonmediated electron-electron interaction and found that it is repulsive for singlet pairs, but that it is not likely to destroy superconductivity completely, a result later confirmed by Klose et al. (1974). Since then, several ternary compounds have been found where superconductivity coexists with antiferromagnetism. Section 5 contains a description of these results. Further results from these early investigations can be found in reviews by Fischer and Peter (1973) and Roth (1978).
3. Magnetism and superconductivity in ternary compounds The difficulty with the pseudobinary systems studied prior to 1975 was that the magnetic atoms were distributed at random and that they were present in relatively small quantities. In fact, superconductivity was destroyed at low concentrations of magnetic impurities as a result of the large exchange interaction J between the magnetic moments and the conduction-electron spins. Consequently, in these materials it was difficult to separate clustering and short-range order effects from the effect of long-range magnetic ordering on superconductivity. In addition, long-range magnetic order would usually appear only at concentrations of magnetic atoms where superconductivity had already been destroyed. The discovery of superconducting ternary compounds containing magnetic rare earth atoms changed this situation completely, and in these materials many of the predicted properties, as well as many new phenomena, have been observed. The first, and also the most, studied compounds so far are the Chevrel phases (RE)Mo6Ss, (RE)Mo6Se 8 (Fischer et al. 1975a, Shelton et al. 1976) and the ternary rhodium borides (RE)RhgB 4 (Matthias et al. 1977). The characteristic properties of these materials are: (1) They contain a regular lattice of magnetic rare earth atoms. (2) Their crystal structure is of the cluster type, leading to a separation of the conduction electrons from the rare earth 4f-electrons. As a result, the exchange
MAGNETIC SUPERCONDUCTORS
477
interaction between the two is low and superconductivity is possible, in spite of the high concentration of magnetic atoms. Other classes of rare earth ternary and multinary superconductors have also been found, and in some of these, the interplay of the two phenomena also leads to unusual behaviour. In the revolutionary new oxide-superconductors, the (RE)Ba2Cu3Cu30 7 series constitute an extreme example of decoupling between superconductivity and magnetism. Most of these compounds have been found to order magnetically, and their study has allowed a detailed investigation of the interplay between magnetism and superconductivity. In this section we shall give an overview of the various compounds and a discussion of how their properties vary across the RE series. For a more detailed discussion of the chemistry and the crystallography of these compounds, as well as other related classes of compounds, we refer to various reviews (Chevrel and Sergent 1982, Yvon 1982, Johnston and Braun 1982).
3.1. The Chevrel phase compounds (RE)M06S s and (RE)Mo6Se s The Chevrel phase materials belong to a large class of superconducting compounds with the formula MxMo6X 8 (M = rare earth, Pb, Sn, Cu, A 1 , . . . ; X = S, Se, Te) (Chevrel et al. 1971, Matthias et al. 1972, Odermatt et al. 1973, Fischer et al. 1975a). Among these compounds one finds some, like PbMo6Ss, SnMo6S s and LaMo6Ses, with very high upper critical fields Hcz [for a review see Decroux and Fischer (1982)] and the first two are presently being investigated in view of very high-field applications (Seeber et al. 1988). The special properties of these materials can be understood to a first approximation by considering their crystallographic structure, which is shown in fig. 4. This structure may be viewed as a slightly rhombohedrally deformed CsC1 structure where the RE replaces the Cs and a pseudomolecule Mo6S 8 substitutes for C1. The latter is approximately a cube with the S atoms on the corner and the Mo atoms at the face center. The latter atoms thus form a slightly deformed
Fig. 4. The crystalstructure of M M o 6 X s (x = S, Se) compounds. (a) One M o 6 X 8 unit. (b) Stackingof the M o 6 X 8 units and the M atoms (after Yvon 1979).
478
O. FISCHER
octahedron, often referred to as a cluster [see also the reviews by Yvon (1979) and Chevrel and Sergent (1982)]. Various experimental investigations (Fischer 1974, 1978) as well as band-structure calculations (Mattheiss and Fong 1977, Andersen et al. 1978, 1982, Jarlborg and Freeman 1980, Freeman and Jarlborg 1982) show that the conduction electrons are mainly of Mo 4d character. An electron transfer from the Mo to the S atoms brings the Fermi level into a region where the density of states may be relatively high and where the bands have mainly 4d character. The RE atom has, to a first approximation, the role of stabilizing this structure and to transfer its valence electrons to the Mo cluster. The superconducting properties, independently of any magnetic effects, will therefore depend upon the valence of the RE atom. It turns out that in the sulphides, divalent RE atoms will give a rather high Tc and a high Hc2 like in PbMo6S8, whereas a trivalent RE will give a low Tc and a low He2. In the selenides this difference is less pronounced. All compounds REMo6S 8 and (RE)Mo6Se 8 have been synthesized, and nearly all of them are superconducting. Figure 5 shows the variation of T c across the rare earth series for the sulfides [for a review see Fischer et al. (1979) and Ishikawa et al. (1982a)]. Most of them also order magnetically at a temperature TM below the superconducting temperature, as indicated in fig. 5. One of them, HoMo6Ss, becomes ferromagnetic at a temperature T~ and just below that temperature, at Tc2 , the superconducting state is destroyed, a phenomenon referred to as re-entrant superconductivity (Ishikawa and Fischer 1977a). Several other compounds order antiferromagnetically without destruction of the superconducting state (Ishikawa and Fischer 1977b). However, the onset of the antiferromagnetic
11
(RE) Mo6 X 8 o
Tc
10,
\ 7
zx T m
(RE) Mo6Se 8
gI- 6 5 4 3 2 1 0 La Ce Pr Nd Pm Sm
Gd Tb Dy Ho Er Tm Yb Lu Rare Earth
Fig. 5. Critical temperatures Tc (open and filled circles) and magnetic transition temperatures Tm (open and solid triangles) for REMo6X 8 (X = S, Se) compounds [after Ishikawa et al. (1982a) with some modifications].
MAGNETIC SUPERCONDUCTORS
479
order is reflected in the superconducting properties as, e.g., anomalies in the upper critical field. Tables 1 and 2 give an overview of the superconducting and magnetic transition temperatures in this class of compounds. Three compounds appear anomalous with respect to their superconducting properties, compared to the others in each of the series: the Ce, the Eu, and the Yb compounds. The Ce compounds are nonsuperconducting, presumably as a result of the higher exchange interaction between the 4f level, which is close to the Fermi level, and the conduction electrons. This results also in a higher magnetic ordering temperature than in the other compounds (Pelizzone et al. 1977). The europium compounds both have a structural phase transition below room temperature (Baillif et al. 1981, Rossel et al. 1983) and are semiconducting at low temperatures (Maple et al. 1977). For EuMo6S8, a hydrostatic pressure of about 12 kBar suppresses the phase transition and the rhombohedral compound is found to be superconducting with a Tc of 12 K (Chu et al. 1981, Harrison et al. 1981, Decroux et al. 1983). This high value of Tc compared to the other RE compounds is a result of the divalent state of Eu. This compound is thus closer to PbMo6S 8 in its nonmagnetic properties than to LaMo6S s or LuMo6S 8. The same argument applies to YbMo6S8, in which possibly also some of the Yb atoms are trivalent. Overall, the superconducting critical temperature decreases as one goes through the series from LaMo6X 8 to LuMo6X 8. This reduction is to a large extent due to nonmagnetic volume effects related to the lanthanide contraction. In TABLE 1 Superconducting and magnetic transition temperatures of REMo6S 8 compounds [data also taken from Fischer (1978), Ishikawa et al. (1982a)]. Compound CeMo6S s PrMo6S s NdMo6S s SmMo6Ss EuMo6S 8 GdM06S 8 WbM06S8 DyM06S 8 HoM06S 8 ErM06S 8 TmM06S 8 YbM06S 8 LuM06S s
Tel (K)
To2 (K)
4.0 3.5 2.9 (12 K at 12 kbar) 1.6 2.1 2.1 2.2 2.2 2 9
* References: [11 Pelizzone et al. (1977). [21 Alekseevski et al. (1985). [3] Quezel et al. (1984). [4] Majkrzak et al. (1979). [5] Thomlinson et al. (1981).
0.79
T M (K)
Magnetic order
Ref.*
2.3
AF(?)
[11
0.3
AF
[2]
0.3
AF complex
[3]
0.84 1.05 0.4 0.75 0.2
AF AF AF F (oscillatory) AF
[4] [5] [61 [7, 8] [5]
(2.7?)
[6] [7] [8] [9]
Moncton et al. (1978). Ishikawa and Fischer (1977a). Lynn et al. (1978b). Bonville et al. (1980).
[91
O. FISCHER
480
TABLE 2 Superconducting and magnetic transition temperatures of REMo6Se 8 compounds [data also taken from Shelton et al. (1976)]. Compound CeMo6Se s PrMo6Se 8 NdMo6Se s SmMo6Se s EuMo6Se 8 GdMo6S % TbMo6S % DyMo6Se 8 HoMo6Se 8 ErMo6Se 8 TmMo6Se 8 YbMo6Se 8 LuMo6Se 8
T c (K)
TM (K)
9.2 8.2 6.8
(2.3?)
5.6 5.7 5.8 5.6 6.2 6.3 5.8 6.2
Magnetic order
Ref.*
[11
0.75
AF
[2]
0.53 1.1
AF (oscillatory) AF (complex)
[3] [4]
* References: [1] Alekseevskii et al. (1985). [2] Maple et al. (1980a). [3] Lynn et al. (1984). [4] Lynn et al. (1978a).
addition to this, there is a Tc reduction due to the Abrikosov-Gorkov type exchange scattering in the compounds with magnetic moments. In order to determine the latter, one should know how large the nonmagnetic T c reduction is. A first estimate is to assume that the nonmagnetic effect interpolates linearly between the La and the Lu compounds, or use the Lal_xLu, Mo6S 8 series as a nonmagnetic reference (McCallum 1977). We find then that the reduction of Tc due to exchange scattering, A To, is roughly proportional to the de Gennes factor G for the heavy rare earths as illustrated in fig. 6 (Ishikawa et al. 1982a). From the AG formula [eqs. (2) and (3)] we estimate, using N(0) = 0.2 states/eV atomspin a value of 25 meV for 1. A large uncertainty is still associated with this value, which is presumably too high. A study of T~ versus volume in these compounds and pseudoternaries suggests that the nonmagnetic Tc reduction is, in fact, larger than the above estimate (Fischer et al. 1979). Magnetic atoms in a metal interact via the RKKY interaction. In the mean-field approximation Tr~ is proportional to N(O)GI 2. A plot of TM versus G for the heavy rare earth molybdenum sulfides, as displayed in fig. 6, shows that the proportionality roughly holds, and suggests that the R K K Y interaction is at least partly responsible for the magnetic order in these compounds (Ishikawa et al. 1982a). However, the exchange constant F deduced from this plot is 55 meV, and a more detailed analysis is clearly necessary to account in detail for the magnetic order. Other mechanisms, like the dipolar interaction, must also be considered (see, e.g., Bulaevskii et al. 1985a). In the above analysis, free rare earth ions are assumed, and, in fact, the magnetic moments as measured by the high-temperature susceptibility are nearly
MAGNETIC SUPERCONDUCTORS
481
2.8 ( RE ] Ho6S a 2-/~
/./(
y2.0
// /
I
1.~
z/ -- z~Tc
~_E
/ //
1.I Ho / ,~//
0.~
//
2
Gd
Tb
/
/ ' o/
°o
/"
o By
Er/// 0.4
//
J
.
Tm
;
;o G=
4,.
16
(g-1}2j(J+l)
Fig. 6. Reduction of the critical temperature A T, and magnetic transition temperature T~ versus de Gennes factor G for REMo6S s compounds (after Ishikawa et al. 1982a).
all close to the theoretical free-ion values. However, at lower temperatures one observes deviations characteristic of crystal-field effects. In reality, a more detailed analysis of the magnetic properties shows that the rare earth moments in these compounds have rather strong crystal-field splittings, in particular large compared to the magnetic energy associated with the magnetic ordering. It is, therefore, crucial to take this into account when the magnetic order is being discussed. The fact that the susceptibility of polycrystalline samples shows Curie-. like behaviour, is probably due to a mutual cancellation of deviations for different directions with respect to the field, as found in the rhodium borides (Dunlap et al. 1984). In the selenides the splittings are found to ~be of the order of 20 to 30 K (Lynn 1980, Lynn and Shelton, 1979, 1980). As an illustration, fig. 7 shows the crystal-field transitions in ErMo6Se 8 giving levels at 2.5, 5.5 and 7.3 eV. Crystalfield splittings of similar magnitudes have also been observed in HoMo6Se 8 and TbMo6Se 8. 200
d E ""
Erl,oMo6Se8 I~l= 1.2o ~"
L50
T= 4.2 K Ei=14.75 meV
I00
c
g
51 ÷ -200
ODO
200
4.00
6.00
8.00
E (meV)
Fig, 7. Inelastic neutron-diffraction data for ErMo6Se8 (after Lynn 1980).
482
~ . FISCHER
The sulfides seem to have much larger splittings. In HoMo6S8, no transition was observed up to 20 meV in these early measurements. Later on, levels at 150, 230 and 500 K have been reported (Burlet et al. 1987, Bonnet et al. 1989). An analysis of these results in terms of a cubic crystal-field gives a magnetic moment which is too small. The noncubic environment of the rare earths has to be taken into account in order to understand the magnetic properties of the Ho atoms. The ground state is a doublet with a moment of about 9/za which is pointing in the [111] direction.
3.2. The ternary rhodium borides, (RE)Rh4B 4 The rhodium borides, with the general formula (RE)Rh4B4, have been studied extensively in relation to the interplay of superconductivity and magnetism. The crystallographic structure of these compounds has been shown to be of the CeCo4B 4 type (Vandenberg and Matthias 1977, Yvon and Gr/ittner 1980). This is a primitive tetragonal structure shown in figure 8. It can be viewed as a tetragonally distorted NaC1 structure where one site is occupied by the rare earths, thus forming a distorted face centered (nearly cubic) network and the other site is occupied by a distorted Rh4B 4 cube. The Rh atoms form a tetrahedron and the B atoms form B 2 dimers. There are two orientations of the Rh 4 tetrahedra. As indicated in fig. 8, the tetrahedra form sheets in the a - b plane with the orientation of the tetrahedra alternating from one sheet to the next. As in the Chevrel phases the RE atoms form a regular network and the transition metal atoms can, to a certain extent, be considered extended clusters. Band-structure calculations show that the electrons close to the Fermi level have essentially Rh 4d character and that the conduction-electron density at the RE
C) Ce
@ Co
• B
CeCo4B 4
Fig. 8. Representation of the CeCo4B 4 crystal structure (after Woolf et al. 1979a).
MAGNETIC SUPERCONDUCTORS
483
site is small, thus giving a small exchange interaction between the conduction electrons and the localized 4f-electrons (Jarlborg et al. 1977, Freeman and Jarlborg 1982). These compounds could initially be made with all the rare earths except La, Ce, Pr and Eu. Later on, the compounds with Ce and Pr were synthesized under pressure (Kumagai et al. 1987). The superconducting and magnetic transition temperatures are given in table 3. Three compounds show coexistence of superconductivity and antiferromagnetism: NdRh4B 4 (Hamaker et al. 1979a), SmRh4B 4 (Hamaker et al. 1979b) and TmRh4B 4 (Hamaker et al. 1981a). Coexistence is possibly also realized in PrRh4B 4 (Kumagai et al. 1987). One compound, ErRh4B4, shows re-entrant behaviour similar to that in HoMo6S 8 (Fertig et al. 1977). Finally, four compounds, GdRh4B4, TbRhnB4, DyRh4B 4 and HoRh4B4, are ferromagnetic and do not display superconductivity (Matthias et al. 1977). Figure 9 shows the magnetic and superconducting transition temperatures across the rare earth series. For the superconducting compounds, it is expected that the observed Tc is reduced by exchange scattering compared to the Tc of the equivalent nonmagnetic compound. To get information about this problem, MacKay et al. (1980) studied the (Lu1_xREx) RhaB 4 system. In fig. 10 the T~ depression for the different RE atoms is shown. Using the La-Lu line as a base line, it was found that the T~ depression follows approximately the de Gennes factor G as expected from the AG formula. This is especially the case for the heavy RE elements, for which an exchange constant of approximately 23 meV per atom was calculated from these data. More recently, Dunlap (1986) reanalyzed these results and included crystalfield effects based on the theory of Fulde and Peschel (1972). As was pointed out by the latter authors, the crystal-field effects significantly influence the AG TABLE 3 Superconducting and magnetic transition temperatures of primitive tetragonal RERh4B 4 compounds [data also taken from Maple et al. (1982)]. Compound
Tel (K)
CeRh4B 4 PrRh4B 4 NdRh4B 4
0.29 4.6 5.3
SmRh4B 4 GdRhaB 4 TbRh4B 4 DyRh4B 4 HoRh4B 4 ErRh4B 4 TmRh4B 4 LuRh4B 4
2.7
8.7 9.8 11.5
* References: [1] Kumagai et al. (1987). [2] Majkrzak et al. (1982). [3] Sinha et al. (1982). [4] Majkrzak et al. (1983).
To2 (K)
0.75
T~ (K) (1.6?) 1.31 0.89 0,87 5.8 7.4 10.7 6.7 0.9 0.4
Magnetic order
ReL*
AF, complex
[1] [1] [2]
AF F F F F F or oscillatory AF complex
[3] [4]
484
O. FISCHER 14
I
12
I
I
I
i
I
I
I
]
I
]
/
I
RE Rh4B4 o Tc
]0
•
TM
/\
6
o
•
0
9
i
[
,
L
I
t
i
,
[
L
?
I
Lo Ce Pr Nd Pm Sm Eu Gd TD Dy Ho Er Tm Yb Lu RE
Fig. 9. Magnetic and superconducting transition temperatures of the series of primitive tetragonal RERh4B 4 (after Maple et al. 1982).
.~
20.0
•~ 15.0
4 "'~,
,~ 5.00 0.00 LaCePrNdPmSm EuGdTbDy HoErTm YbLu
Fig. 10. Rate of depression of To, ATe, per atomic fraction for the various RE in the series (LUl_xREx) Rh4B 4 (after MacKay et al. 1980, Dunlap 1986).
formula only when the crystal-field splittings are considerably higher than the critical temperature, because of virtual transitions over energies large compared to Tc. The dashed curves show the curves corrected for crystal-field effects. The discrepancies still present in fig. 10 have been interpreted as a result of a variation of the exchange interaction across the RE series. That this might be a general trend is illustrated by the fact that a similar behaviour has been found for the REAl 2 series by Maple (1970). Inspection of the Tc values for the Chevrel phases, shown in fig. 5, suggests that the same might be true here, although a detailed analysis remains to be made.
MAGNETIC SUPERCONDUCTORS
485
Crystal-field effects are clearly of importance for the interpretation of the properties of these compounds. Schottky anomalies have been observed in the specific heat in a number of them. In fig. 11, as an example, the anomaly found by Woolf et al. (1979a) in ErRh4B 4 is shown. The fit to a crystal-field energy-level scheme as indicated, is shown by the solid line. A detailed investigation of a single crystal of this compound allowed Dunlap et al. (1984) to refine the level scheme. Extrapolating these results through the Stevens factors to the other heavy R E atoms, they were able to give a reasonable account of the crystal-field properties of these compounds as well. One important result of this work is that although the susceptibility and the magnetisation curves for a single crystal may be very different from the flee-ion behaviour for specific orientations, the curves obtained for a polycrystal are usually very close to the free-ion behaviour. This apparent free-ion results from a compensation of the very different behaviour of different orientations. For this reason, preferential orientation in a polycrystalline sample may strongly influence its magnetic behaviour. This result emphasizes the need for single crystals for a complete description of the properties of these materials. Absolute measurements of the Ho magnetic moment in HoRh4B 4 by Li et al. (1987) have given a value of 8.61p~ instead of the 10/~ expected from crystalfield calculations by Dunlap et al. (1984), and show that further investigations of the crystal-field properties are necessary. Let us finally comment about the trend of the magnetic ordering temperatures across the RE series. If the magnetic state resulted from a RKKY interaction between free RE moments we would expect T M to be proportional to N(O)GI 2, and to show a maximum at RE = Gd. Contrary to this, the maximum is found for Dy. This behaviour can partly be traced back to crystal-field effects (Dunlap et al. 1984). However, this does not explain completely the observed behaviour, and so
0.9 0.8-0.7 _.
/..~"
0.6
,¢..T.,:.~.~ -~,.,,
ErRh4B4
/
"'.
0.5 n" 0.4
j ,i t
0.3
t~ :
~ 34.3 K - - - - 33.6 K ~ 32.4K 32.0 K
" "
0.2 0.1 0.O
4
8
12
16
20
24
28
32
36
T(K)
Fig. 11. Specific heat AC in units of the molar gas constant R versus temperature for ErRh4B 4. The solid line represents a fit of the Schottky anomaly with the crystal-field level scheme indicated in the figure (after Woolf et al. 1979a).
486
0 . FISCHER
other effects must also play a role. Dipole effects probably have some influence on T~ but they are unlikely to have a dominant effect on the highest T~. An interesting property of the magnetic compounds is that of nearly perfect meanfield behaviour, as demonstrated by the specific-heat measurements (Ott et al. 1980b, MacKay et al. 1980). Johnston (1977) found that a body centered tetragonal (bct) modification of the CeCo4B 4 phase can be stabilized by substituting a few percent of Ru for Rh. This structure is also found for the RERunB 4 compounds. Johnston solved the structure for LuRu4B4, and found that it is similar to the CeCo4B 4 structure with only a different arrangement of the two orientations of the Rh 4 tetrahedra as shown in fig. 12. This structure can also be obtained for some of the RERh4B 4 compounds. Table 4 contains some examples of compounds with this structure. Contrary to the CeCoaBn-type compounds, bct ErRh4B 4 and bct HoRh4B 4 are superconductors and become antiferromagnetic at a temperature well below Tc (Iwasaki et al. 1983, Iwasaki and Muto 1985). Thus, the rearrangement of the Rh 4 clusters leads to a reduction and a change of sign of the exchange interaction. Johnston (1981) studied the Tc systematics across the RE series for RE(RhossRuo.15)B 4. A plot of Tc versus the de Gennes factor G (fig. 13) suggests that the Tc reduction due to exchange scattering is about a factor of two lower than in the CeCo4B 4 type compounds. Several of the RERu4B 4 compounds are ferromagnetic and show no superconductivity except for the nonmagnetic compounds with RE = Sc, Y, Lu.
•
RU
o B
Fig. 12. Representation of the LuRu4B 4 type crystal structure (after Johnston and Braun 1982).
MAGNETIC SUPERCONDUCTORS
487
TABLE 4 Superconducting and magnetic transition temperatures of some body centered tetragonal RERh4B 4 compounds. Compound
Tc (K)
TM (K)
HoRh4B 4 ErRh4B 4 NdRu4B 4 GdRu4B 4 TbRu4B 4 DyRu4B 4 HoRu4B4 ErRu4B 4 LuRu 4
6.5 7.8
1.62 0.65 1.62 4.55 4.30 2.65 2.58 2.16 2.0
Magnetic order
Ref.*
AF, metamagnetic AF, metamagnetic
[1] [1] [2] [2] [2] [3] [4] [5] [2]
F, complex F, complex
* References: [1] Iwasaki and Muto (1986). [2] Johnston (1977). [3] Motoya et al. (1984). [4] M/iller et al, (1984). [5] Horng and Shelton (1981).
I
I
- o Lu
~
I
'
I
I
'
R E(Rh0.85Ru0.15)4B4
_i
To(K)
4 Pr o
2
Nd
0
I 0
,
l, 2
Sm
,
I l~
4
1
6
(g-1)2j(j
,
,
,
[
8
,
,
I ~,/
10
12
* 1)
Fig. 13. Superconducting transition temperature versus de Gennes factor for the LuRu4B 4 type compounds, RE(Rh0.gsRu0.15)4B4 (after Johnston 1981).
3.3. Other ternary compounds Several o t h e r s u p e r c o n d u c t i n g t e r n a r y rare earth c o m p o u n d s have b e e n found, in which superconductivity and m a g n e t i s m are confronted. H o w e v e r , c o n t r a r y to the two main g r o u p s of c o m p o u n d s described above, these are usually n o t cluster c o m p o u n d s in a strict sense, which m a y explain w h y only a few c o m p o u n d s have b e e n f o u n d w h e r e a coexistence or a re-entrance occurs.
The ternary stannides, (RE) TxSny (T = Transition element) These c o m p o u n d s were first investigated by R e m e i k a et al. (1980), w h o s h o w e d that the c o m p o u n d E r R h l 1Sn3 6 is re-entrant with Tel = 1.2 K and Tc2 = 0.5 K.
488
~. FISCHER
Structural investigations revealed that two structure types, I and II, occur in this class of materials. The type-I is a primitive cubic structure with composition RE3T3Sn13 (Vandenberg 1980, Hodeau et al. 1980). Characteristic for this structure is that the Sn atoms occupy two different crystallographic sites, so that the formula should rather be written as Sn(1)RE3T4Sn(2)I 2. This structure forms with the early rare earths (RE = L a - G d ) and with Yb as well as RE replaced by Ca, Sr and Th. The compounds here are either superconducting (for nonmagnetic RE) or magnetic. The class II (and II*) has a tetragonal structure with the composition (Snl_xREx)(1)RE4T6Sn~s. Here also, we find that the Sn atoms occupy two different sublattices with a partial substitution of Sn for RE on sublattice 1. Early investigations mention also a class-III structure; however, later studies have shown this to be a micro-twinned version of class II and is currently referred to as class II* (Hodeau et al. 1984). Two compounds with this structure type have been found to show re-entrant superconductivity, ErRhl.lSn3.6, and ErOsxSny (Lambert et al. 1981). The compound TmRhl.3Sn 4 is re-entrant in the presence of a magnetic field. The relationship between structure, composition and re-entrance has been studied by Miraglia et al. (1984). The magnetic orders of these compounds have been studied by various groups and are found to be of short-range nature of spin-glass like. This will be further discussed in section 4.
The rare earth ternary silicides, (RE)zFe3Si 5 The series (RE)2Fe3Si 5 was first described by Braun (1980). In this series one finds superconductivity for RE = Sc, Y and Lu, whereas for most of the other RE the compounds are magnetic. The role of the Fe atoms is particularly interesting in these compounds, since very few Fe compounds show superconductivity. Cashion et al. (1981) showed through Mrssbauer experiments that the Fe ions are nonmagnetic and thus it seems possible that the Fe 3d-electrons play an important part in the superconducting state. The structure of these compounds is primitive tetragonal with two different positions for the Fe atoms, one associated with isolated squares in the basal plane and one with linear chains running perpendicular to the basal plane (Bodak et al. 1977, Braun 1981). Of particular interest is the compound Tm2Fe3Sis, which was found by Segre and Braun (1981) to show re-entrant superconductivity. This was confirmed by Vining and Shelton (1985), but they found that this happens under pressure, their sample being only antiferromagnetic in zero pressure. The striking fact here is that the re-entrance is caused by an antiferromagnetic transition as discussed in section 4. The Heusler-type compounds, (RE)Pd2Sn Ishikawa et al. (1982b) synthesized several new compounds with the composition (RE)PdzSn. These compounds crystallize in the cubic MnCu2Al-type structure (also called Heusler alloys). Table 5 contains a list of some compounds in this series. Superconductivity has, so far, been found for RE = Y, Er, Tm, Yb and Lu. Magnetism is found for RE = Gd, Tb, Dy, Ho, Er and Yb. Malik et al. (1986) investigated the magnetic scattering from the rate of depression of Tc in the YI_xRExPd2Sn series for the different rare earths. Although the product N(0)I 2
MAGNETIC SUPERCONDUCTORS
489
TABLE 5 Superconducting and magnetic transition temperatures of some REPd2Sn c o m pounds. Compound YPd~Sn GdPd2Sn TbPdzSn DyPdzSn HoPd2Sn ErPdzSn TmPd2Sn YbPd2Sn LuPdzSn
Tc (K) 3.72
1.17 2.6 2.46 3.1
TM(K)
Magneticorder
3.6 3.3 4.7 2.0 -5.0 1.0
AF AF
0.23
AF?
Ref.* [1] [1] [1] [1] [1, 2] [3] [1] [4] [1]
* References: [1] Ishikawa et al. (1982b). [2] Li et al. (1988, 1989). [3] Stanley et al. (1987). [4] Kierstead et al. (1985). in the A G formula is found to be small, the scattering is strong enough to destroy superconductivity in the middle of the R E series, but is weak enough to allow superconductivity for the heavy REPd2Sn compounds. Coexistence between superconductivity and antiferromagnetism has later on been established in ErPd2Sn (Shelton et al. 1986, Stanley et al. 1987, Shinjo et al. 1987) and in YbPdaSn (Kierstead et al. 1985). For TmPd2Sn, the crystal-field effects produce a nonmagnetic ground state, explaining the relatively high value for T~ (Malik et al. 1985b). The crystal-field levels have been determined, using neutron scattering, for HoPdzSn and ErPd~Sn by Li et al. (1989). 4. Re-entrant superconductors One of the most striking phenomena observed in relation with the interplay of magnetism and superconductivity is the destruction of superconductivity at a temperature To2 well below the temperature where superconductivity first occurs upon cooling down, Tel. This phenomenon, which is related to the appearance of a ferromagnetic state, was discovered simultaneously and independently in ErRh4B 4 by Fertig et al. (1977) and in HoMo6S 8 by Ishikawa and Fischer (1977a). Later on, re-entrant behaviour has also been found in ErRhl.lSn3. 6 by Remeika et al. (1980), and in Tm2Fe3Si 5 by Segre and Braun (1981) and confirmed (under pressure) by Vining and Shelton (1985). Since these initial discoveries, numerous investigations have been carried out in order to understand the detailed interplay between ferromagnetism and superconductivity. Interest has concentrated around three different questions: (a) The behaviour in the paramagnetic region, especially how the superconducting properties evolve as T approaches T M from above. (b) The behaviour in the coexistence region in ErRh4B 4 and HoM06S s where it
490
~. FISCHER
has been found that the magnetic phase transition occurs at a temperature Tu above the re-entrant transition Tcz. (c) The magnetic state below To2 and the occurrence of (metastable?) superconductivity in this temperature region. 4.1. The two compounds HoMo6S 8 and ErRh4B 4 The re-entrant superconductivity is most readily seen in the resistance versus temperature. Figure 14 shows this for HoMo6S 8 (Ishikawa and Fischer, 1977a). The transition at Tel is, as in most Chevrel phases, relatively large and the value of Tea may vary from sample to sample due to slight variations in stoichiometry. In contrast to this, the re-entrant transition occurs between 0.6 and 0.7 K upon cooling and is usually sharper than the first one. Figure 15 shows the AC 7
6 5
, ,'
"~E 3~'' 24
.! ~e)
'
/
0
/ ~ o--o-. . . .
! . . . . 0.5
0
. . . .
i
I
1.0
.
~
.
,
T(K/
.
i
1.5
.
2.0
Fig. 14. DC resistance versus temperature of HoMo6S 8 in magnetic fields of 0, 70, 100, 200 and 300 mT (after Ishikawa and Fischer 1977a).
.I00
m
H~(_oe/ o
0
° _100
100
o
300
-200 .
.
.
.
0..~ i
.
,
,
i
1.0T(K) 1.5 I
,
i
,
i
I
i
,
l
i
2.0 I
,
,
Fig. 15. AC magnetic susceptibility of HoMo6S 8 in magnetic fields of 0, 10 and 30 mT (after Ishikawa and Fischer 1977a).
MAGNETIC SUPERCONDUCTORS
491
susceptibility which also displays the re-entrant behaviour. DC susceptibility and low-temperature magnetization measurements indicated that the destruction of superconductivity was due to a ferromagnetic transition. However, the unusual shape of the magnetization curve (fig. 16) as well as long relaxation times suggested that the order might be more complicated, possibly large ferromagnetic domains coupled antiferromagnetically. Figure 17 shows the resistivity and AC susceptibility for ErRhaB 4 as obtained by Maple et al. (1980b). Upon cooling down, the system first becomes superconducting at Tel = 8.7 K. Then, at To2 = 0.92 K the system returns to a normal, but
(b)
H01.2 MO6 S 8
T= 0.10K
"~c
.6 b
-16 J
-12
-8
S
/'
-t.
I
r
4
I
8
12
16
20
H [kOe )
Fig. 16. Magnetization versus applied magnetic field at T = 0,1 K for HoMo6S 8 (after Ishikawa and Muller 1978).
I
l
I
f ~
I
I
I
Er Rh4 B4
1,
,,..; ~<
_
f
L.__.
-2
:t. z
150 100
o-J
50
_!
I
3
4
5
6
7
8
9
TEUPERATURE(K)
Fig. 17. AC susceptibility and A C electrical resistance v e r s u s temperature for ErRh4B 4 in zero applied field (after Maple et al. 1980b).
492
~ . FISCHER
magnetically ordered, state. Note that there is a thermal hysteresis at Tc2 , and that the resistance in the magnetically ordered state is lower than above Tcl.
The paramagnetic and superconducting temperature region This region has been studied in quite some detail in E r R h a B 4. Magnetization measurements on polycrystalline bulk samples (Ott et al. 1978) and resistivity measurements on polycrystalline thin films (Cantor et al. 1980) showed a critical field He2 with a maximum around 4 K and a gradual decrease as T approached Tc2 from above. Crabtree et al. (1982) studied the magnetization and the upper critical field in a spherical shaped single crystal. After correcting for demagnetization effects, their results are given in terms of the internal field H = He,,t - nM in figure 18. For the field in the direction of the c-axis (the hard magnetization direction) the critical field behaves as He2 for a nonmagnetic superconductor, and only minor corrections from magnetism are present. When the field is applied parallel to the a-axis (the easy magnetization direction), quite a different behaviour is seen. The development of a large magnetization results in a maximum of He2, and the critical field decreases gradually as T approaches T~2. Shenoy and Malik (1986) found that this behaviour can be reproduced by the multiple pairbreaking theory given by eq. (9) by using the measured magnetization for the two directions. Thus, the strong anisotropy of the upper critical field is a direct consequence of the strong anisotropy of the magnetization resulting from the crystal-field effects. Both electromagnetic effects, represented by the term proportional to the magnetization M in eq. (11) and the exchange effects resulting from the exchange field//i are important. However, close to the magnetic transition, the exchange effects dominate. Lambert et al. (1985) were able to synthesize single crystals of several RERhaB 4 and they confirm the H~2 curves reported by Crabtree et al. Schossmann and Carbotte (1986) have developed a theory for the i
i
t
i
i
i
12 II ,o
i
i
A,, DC Hc2 o AC Hcz • DC Hm
Single Crystal
2.4 2.2
2/0 1.8
ll~ L4 _~ 1.2 L~ l.O 0,8 0.6 04 0.2 0
I
2
3
4
5
6
7
8
;o
T (K) Fig. 18. The upper critical field He2 for the a- and the c-directions of ErRh4B 4 as a function of temperature (after Crabtree et al. 1982).
MAGNETIC SUPERCONDUCTORS
493
upper critical field that goes beyond the weak-coupling theory used to obtain eq. (9). To which extent the dynamical effects, considered in this work, are important in the actual compounds remains to be investigated. Equation (9) assumes the dirty limit (l < ~) and a second-order phase transition. The excellent fit obtained by Shenoy and Malik (1986) shows that this explains the overall behaviour very well. However, a closer inspection of the data shows that this is not the whole story. Crabtree et al. (1982) and Behroozi et al. (1983) found also that below 3.3 K the magnetization shows an abrupt decrease with an infinite slope at He2, as shown in fig. 19, indicating a first-order transition and that there is a tricritical point at T = 3.3 K (Crabtree et al. 1986). Furthermore, below 1.5 K the mixed state disappears and the system becomes a type-I superconductor as predicted by Tachiki et al. (1979a). In fig. 18 are also shown the results for He1 joining the t/c2 curve at T = 1.5 K. Behroozi et al. (1988) have analyzed these data in more detail by giving the temperature-dependent Ginzburg-Landau parameters, confirming the tendency towards a type-II to type-I transition above T M. Evidence for a type-II to type-I transition in thin films has also been reported by Gray et al. (1983). The first-order transition at He2 below 3.3 K is probably a result of the strong exchange effects (Sakai et al. 1983, Koyama et al. 1983, Matsumoto et al. 1984). If the spin-orbit scattering is not too strong (~so < 8) and orbital effects can be neglected, a first-order transition should occur at a temperature determined by ~tao and the temperature dependence of H j (see Fulde and Keller 1982). The transition to a type-I superconductor is also expected since the Ginzburg-Landau coherence length increases as one approaches TM and the penetration length
//
-0.5 /
,¢,"/~/ .
~
Er Rh4B4
MelssnerSlope Sing!eCrystal "0.5 a axis
0
0
o
A
0.5
0.5
# .o_ =
.8_ .N
1.5 .5
2
___QZ
2
2.5
0.5
1 1.5 2 Internal Field (kOel)
Fig. 19. Magnetization versus internal field for the a-axis direction in ErRh4B 4 (after Behroozi et al. 1983).
494
O. FISCHER
decreases. The exchange field leads to a reduction of the coherence length gig:~n by ::
=
-
where ~0 is the G - L coherence length in absence of an exchange field and Iz = 2.76ma/hsoTc (m is the electron mass). On the other hand, the penetration depth will be modified by the electromagnetic interaction as follows (Matsumoto et al. 1982), A = Ao/(1 + X) 1/2 Thus, as X gets large enough, the G - L parameter K = a/~ will become so small that the system becomes a type-I superconductor. Maekawa et al. (1979) considered the case where exchange effects are not important and found that the interaction between the flux lines would become attractive before the transition to a type-I superconductor and a first-order transition would occur at He1 , characteristic of a type-II/1 superconductor. Kumagai et al. (1980) interpreted their NMR data in terms of such a transition. However, the magnetization curves of Crabtree et al. and Behroozi et al. do not show direct evidence for this effect, possibly due to the dominating role of the exchange interaction. Another interesting question which arises in this temperature interval is the possibility of a Fulde-Ferrell state (Fulde and FerreU 1964) close to He2. Since the superconductivity is largely destroyed by the exchange field in ErRhaB4, one would expect this state to be realized if the superconductor is in the clean limit (~ < l). For a more detailed discussion of this point see Bulaevskii et al. (1985a). Contrary to what was found in ErRhaB4, the behaviour of He2 in polycrystals of HoMo6S 8 in the temperature interval Tc2 < T < Tel (Fischer et al. 1979) shows very little influence of the magnetism until just above T~2. This is probably because HoMo6S 8 has a unique easy-magnetization axis, and a measurement on a polycrystal reflects the two hard-magnetization directions having the higher H~2. Giroud et al. (1987) measured He2 in a single crystal. Figure 20a shows the results in terms of the internal critical field for three different orientations; parallel and perpendicular to the (easy) [111] direction, and with an angle of 35 ° to this direction. The overall behaviour is similar to the behaviour seen in ErRh4B4, with a strong reduction of He2 for the easy-magnetization direction compared to the hard direction. Closer inspection of the data shows, however, that exchange effects only have a minor influence on He2 in this compound and that H~2 can be described by eq. (12) after correcting it for demagnetization effects. This is illustrated in fig. 20b which shows that B~2 =/z0[Hc2 + (1 - n)M] is isotropic. The reason for the small effect of the exchange is probably a combination of a low value for the exchange constant I and a large spin-orbit scattering parameter hso in these crystals. In agreement with this, there was also no first-order transition at H¢2.
MAGNETIC SUPERCONDUCTORS
495
#oHc2 (roT) 300
200
100
015
1
1.5
0.5
1
1.5
T(K)
Be
(mT) 30C
b
20£
lO0
0
T(K)
Fig. 20. (a) Internal upper critical field txoHc2versus t e m p e r a t u r e for H o M o r S ~. T h r e e different orientations of the c-axis of the single crystal with respect to the applied field are shown, a = 0 ° (©), 35 ° ( A ) and 90 ° (m). (b) Internal critical induction BeE =/x0(Hc2 + 1 - n)H(Hc2) for the same data (after Giroud et al. 1987).
The coexistence region To2 < T < T M
The transition at To2 has been the object of intense research. Early investigations suggested that there could be an overlap between the magnetic and the superconducting state, i.e., magnetism would occur at a temperature TM> T~2. This assumption has since then been confirmed experimentally, and so we have a situation in these compounds where ferromagnetism and superconductivity directly confront each other. Numerous theoretical investigations have dealt with this problem. Simplifying the argument somewhat, it appears that the system has basically two possibilities: either the magnetic order is uniformly ferromagnetic and the superconducting state is oscillating in space (self-induced vortex state), or the superconducting state remains uniform and forces the ferromagnet into an oscillatory state. Note, however, that in nearly all real situations the spatial oscillations of one order parameter will lead to oscillations in the other, due to the coupling between the two. Experiments have, so far, confirmed the second possibility of a long-wavelength oscillatory magnetic state. As mentioned in section 2, one mechanism for producing an oscillatory state was proposed by Anderson and Suhl (1959). This prediction was based on a change in the RKKY interaction as a result of the reduction of the low-q part of the conduction-electron susceptibility. Consequently, the effective exchange interaction between the R E moments has a maximum at a finite value of q, leading to an oscillatory magnetic state. The wavelength of this
496
O. FISCHER
sinusoidal oscillation close to T M is found to b e (a2~) 1/3, where a is the magnetic stiffness, a distance of the order of the interatomic spacing. At lower temperatures it is expected that the order might become more domain-like with a basic wavelength given by (a~) 1/2 (Bulaevskii et al. 1983). Blount and Varma (1979) and Matsumoto et al. (1979) developed a different theory, and showed that an oscillatory state may also occur as a result of the electromagnetic coupling between the two phenomena. Qualitatively, the argument goes as follows: If a spontaneous magnetization occurs in a superconductor, the latter will respond with a Meissner effect that compensates the average magnetization. This is energetically costly, and in order to avoid this, the spins tilt slightly and form a long-wavelength oscillatory state. If the wavelength is shorter than the superconducting penetration depth no Meissner effect will result. Expressed in other words, the Meissner effect will screen out the long-wavelength components of the magnetic interaction and thus produce an oscillatory magnetic state. The wavelength of this state is found to be of the order of (aAL) 1/2 Further theoretical work investigating the electromagnetic interaction has shown that different magnetic states may develop depending on the values for the penetration depth and the coherence length as well as the anisotropy: (i) the spiral state (Blount and Varma 1979, Ferrell et al. 1979, Matsumoto et al. 1979, Greenside et al. 1981); (ii) the laminar state (Tachiki 1981a); (iii) the linearly polarized state (Greenside et al. 1981) and (iv) the spontaneous vortex state (Tachiki et al. 1979a, Kuper et al. 1980, Greenside et al. 1981). For a review of theory, see Fulde and Keller (1982) and Bulaevskii et al. (1985a). Let us now consider the experimental results for HoMo6S s in more detail. That this compound is basically a ferromagnet with TM close to To2 was confirmed by neutron-diffraction experiments by Lynn et al. (1978b), who also found that the Ho moments are pointing in the [111] direction. Subsequent investigations on powder samples revealed the existence of an oscillatory state between T M = 0.75 K and To2 = 0.69 K upon cooling (Lynn et al. 1981a,b). The wavelength of this sinusoidal modulation was found by Lynn et al. to be about 200 A. They also observed that upon warming there is no intensity for the corresponding q-vector and that T¢2 is then equal to T M. Figure 21 shows the peak in the neutron scattering at a finite wave vector. An hysteresis in T~2 was also reported by Woolf et al. (1979b) and the spike-like structure they observed in the specific heat at T~2 would be consistent with the first-order transition implied by the hysteresis. The neutron data also showed that below T~2 the ferromagnetic peaks remained rather broad, implying the possibility of the presence of ferromagnetic microdomains. These questions have been further investigated by neutron scattering (RossatMignod et al. 1985, Burlet et al. 1987) on single crystals (Horyn et al. 1985, Pena et al. 1985). In these samples, the oscillatory state occurs with a wavelength that is about 3 times larger than in the powder samples investigated previously. It could be shown that this state is a transverse sine wave modulated (TSWM) ordering with the wavelength varying from 380 .A close to 0.75 K to 570A at
MAGNETIC SUPERCONDUCTORS
497
500 Q:O.050 A F=0.75 K
~,.
5C >,
g E
2OO
o~ 0 0.65
0.75 T~
0.85
I00 "
." .." 0
,."'*.~T=0.74
K
".£1..~T =0.80 K .I..~..._.I. I 0.04 0.06 0.08
--d:.*... I.............
0
0.02
Q(~-~)
Fig. 21. Net scattering intensity, after subtraction of the scattering at 2 K, versus scattering vector Q for HoMo6S 8 (after Lynn et al. 1981b).
0.70 K. The modulation wave vector is in the [110], and equivalent, directions perpendicular to the [111] direction of the magnetic moments. At 0.70 K, while the sample is still superconducting, there is a phase transition to a different magnetic state. The modulation with a wavelength of 400 A to 500 A disappears and gives way to what appears to be a ferromagnetic state. Closer inspection shows that the width of the ferromagnetic peak corresponds to a domain size of 820 A and Burlet et al. (1987) were able to observe a corresponding modulation at 0.715 K by applying a magnetic field of 120 Oe and thereby transforming the TSWM state into this new state which has been interpreted as a transverse antiphase-modulated (TAPM) ordering corresponding to the ordering predicted by Tachiki (1983). In this state, superconductivity can still exist along the antiphase boundaries. Burlet et al. (1987) observe that the zero-resistance state remains down to 0.685 K and then R increases linearly down to 0.55 K, which is interpreted as a gradual destruction of superconductivity as the antiphase wave becomes more square like (see fig. 22). Thin HoMo6S 8 films have been prepared by Przyslupski et al. (1985) and Maps et al. (1985, 1987). A striking result is that the resistivity shows no re-entrance down to well below 0.1 K. Przyslupski et al. found that they could restore the re-entrant behaviour in a field of 0.1 T parallel to the film plane, but not in a perpendicular field. On the other hand, Maps et al. observe very long relaxation times at low temperatures, but observe that the critical current decreases rapidly with external field. These findings can be understood globally in the framework of Bloch-wall superconductivity. The results have been compared with the theoretical predictions by Tachiki et al. (1979b), Takahashi and Tachiki (1983), Buzdin et al. (1984) and Bulaevskii et al. (1985b). The ferromagnetic nature of the low-temperature phase in ErRh4B 4 was
498
~. FISCHER J
J HoMo6S8
100 -
t -~= [0.005, 0.0"05,1]
z
>
Tm2 ,~-~--IP-
_ .2 2.0 .~
0.2
- -
g
\
~ 0.1-
Tc?-
6\
~_~ ~
Tm-1
~
i
~1.4
o
j12 I
0.65
1
0.70 Temperature(K)
0.75
Fig. 22. Variation of the resistivity, the HWHM of the (001) peak and the neutron count at Q = (0.005, 0.005, 1) for HoMo6S 8 (after Burlet et al. 1987). confirmed by Moncton et al. (1977). Subsequent work (Moncton et al. 1980) focused on the region around To2, and showed a broad ferromagnetic transition up to 1.4 K with significant thermal hysteresis between 0.8 and 1.4 K. Low-angle scattering on a sample whose re-entrant transition occurred at 0.7 K revealed a peak at an angle of about 1°, present between 1 and 0.7 K. This was interpreted as fluctuation into an oscillatory state as proposed by Blount ,and Varma (1979). These measurements were followed by investigations on a single crystal by Sinha et al. (1982), who showed that there is, as in HoMo6Ss, a real oscillatory state with a wavelength of about 90 ,~ that coexists with superconductivity. They interpreted their data in terms of a transverse linearly polarized long-range magnetic structure. Figure 23 shows the satellite intensity obtained upon cooling and warming. The hysteresis noted above is also present here, but contrary to the HoMo6S 8 case, the oscillatory state also forms upon warming. The intensity of the main ferromagnetic peak is also shown in fig. 23. As can be seen, a ferromagnetic component coexists with the oscillatory state. Due to the large width of this peak, Sinha et al. (1982) interpreted this as small ferromagnetic clusters growing out of the oscillatory and coexisting phase. Whether a kind of transverse antiphase ordering, as observed in HoMo6Ss, also occurs here is not resolved yet. However, below To2 the ferromagnetic peaks become narrow and a true long-range ferromagnetic order seems to develop. Specific-heat measurements on the single crystals used in the above investigations, confirmed the hysteresis at To2 (DePuydt and Dahlberg 1986). These investigations showed superheating of the ferromagnetic state into the supercon-
MAGNETIC SUPERCONDUCTORS
499
T(K) 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
ErRh4B 4 single crystal
6000
-
Ferromagnetic Intensity IFM
"~ 5000
40oo
• cooling • warming
t
3000 z Z 2000 1000
8O (,3 >, Z w'~
40
Satellite Intensity Is
~ /'~
•cooling ,warming
II ~ I ,~
20
D.C, Resistance
40
•cooling
ac 20 c,~ 3.0 × "-~._~2.0
~ ~
=,~
Intensity Ratio-~ ,cooling }
"~ 1.0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 T(K)
Fig. 23. Temperature dependence of the ferromagnetic and satellite intensities, the bulk resistivity and the intensity ratio of the satellite and the ferromagnetic peaks (after Sinha et al. 1982). ducting state upon heating, consistent with a type-I superconductivity induced by the coupling to the magnetic moments as discussed above. Lin et al. (1983) studied the tunneling characteristics of a junction produced on a ErRh4B 4 thin film, and found evidence for an abrupt transition at Tea also consistent with a type-I superconductivity. The theoretical prediction of a spatially oscillatory magnetic phase has thus been verified in both HoM06S 8 and ErRh4B 4. The anisotropy of the crystal field leads to a linearly polarized state, and the electromagnetic coupling assures that the polarization is transverse. However, the comparison between theory and experiment leaves several points to be clarified. The oscillatory states are in all
500
O. FISCHER
cases found to be sinusoidal and so far no higher harmonics have been found (see, e.g., Sinha et al. 1983, Lynn et al. 1984). The question of the relative importance of the exchange mechanism and the electromagnetic mechanism is still not completely settled. It was originally argued that the exchange mechanism would give a wavelength much smaller than the observed ones, and that only the electromagnetic mechanism could explain the results. However, Bulaevskii et al. (1985a) argue that a careful inspection of the parameters for the two compounds shows that the exchange mechanism gives the correct wavelengths. On the other hand, the more recent observation of very long wavelengths in H o M o 6 S 8 (Burlet et al. 1987) have been interpreted as resulting from the electromagnetic mechanism. Clearly, both mechanisms are present, and in order to progress with this question for H o M o 6 S 8 , further investigations, with a detailed comparison between theory and experiment, are necessary.
Observation of superconductivity below To2 In H o M o 6 S 8 as well as in E r R h 4 B 4 it occurs that the resistance below To2 is significantly lower than the resistance above T~I. This has been interpreted as evidence that a small part of the sample remains superconducting down to 0 K. Genicon et al. (1984) (see also Giroud et al. 1987) investigated this question for H o M o 6 S 8 , and discovered that by first applying a field at low temperature (<0.1 K), then removing the field and finally heating, the sample would go into a zero-resistance state at around 0.12 K and then return to the normal state at about 0.2 K. The zero-resistance state could also be induced by applying a field in the opposite direction instead of heating. Figure 24 illustrates these results. They interpreted these effects as being evidence for superconductivity along Bloch walls, as predicted theoretically by Tachiki et al. (1979b). Later, Burlet et al. (1987), repeating the experiments of Genicon et al. (1984), found that the integrated intensity of the (001) peak in the neutron scattering is considerably reduced in this low-temperature superconducting state. They interpreted this
1.2
T= 25 mK
Ho Ho6 S8 dl
1
¢
-
.8
E
,6
3 I
1
.2
I
I
~
t
I
-5
-4
-3
-2
-1
I 1
I 2
I
I 4
I 5 H(~e
Fig. 24. Variation of the electrical resistance with magnetic field at T = 25 mK for HoMo6S 8. The different curves are traced in the order 1, 2, 3 (after Genicon 1984).
MAGNETIC SUPERCONDUCTORS
501
103 HoMo6S8 H=0
102 o
Q
101
o o o o o
100
10-1
o
~O OOOO
0.2
0.4 T (K)
0.6 0.8
Fig. 25. Temperature dependence of the relaxation time in HoMo6Ssfor the coexisting(oscillating) phase when cooled below To2 (after Koike et al. 1987). result as evidence that superconductivity occurs in the whole sample, and not just in the Bloch walls. This can be understood if we assume that exchange effects are weak and that M 0 > He2 = 0.32 T. Demagnetization effects may lead to situations where M(T)< He2, and thus allow superconductivity (Giroud et al. 1987). This effect was already predicted by Ginzburg (1957) in his original paper. The fact that thin films show no re-entrance in zero field would agree with this interpretation. It has also been found that the oscillatory TSWM state can be quenched in below 0.70 K by cooling rapidly (Burlet e t a . 1986). The system then relaxes back to the TAPM state with a temperature-dependent time constant. The latter has been determined by Koike et al. (1987) and is shown in fig. 25. Clearly, further investigations on this system must carefully separate time-dependent effects from effects that are temperature and field dependent. In ErRhaB 4 one also finds that the resistance is lower below T~2 than above Tcx as seen in fig. 17. This probably reflects Bloch-wall superconductivity. However, up to now a shape-induced superconductivity below T¢2, similar to the one observed in HoMo6S8, has not been observed. This may be due to a larger pairbreaking effect due to the exchange interaction. Magnetoresistance studies by Dunlap et al. (1985) have shown that there is substantial spin disorder in the low-temperature magnetic phase.
4.2. Other re-entrant compounds Re-entrance due to magnetic order has been seen in several other compounds. Remeika et al. (1980) observed re-entrant behaviour in ErRhl 1Sn3.6 as shown in
502
~. FISCHER
fig. 26. This compound becomes superconducting at T~ = 1.2 K and returns to the normal state at T~2=0.4K. This compound belongs to the solid solution (Snl_~Er~)Er4Rh6Sn18 (Hodeau et al. 1984), and re-entrance has been found for 0 . 2 5 < x <0.6. Contrary to the two compounds discussed above, there is no hysteresis at T~2. No overlap of the superconducting and magnetic state with corresponding oscillatory behaviour has been reported. Superconducting order is apparently destroyed as the magnetic order is established. The magnetic order has been found to be short range in nature (Remeika et al. 1980), possibly due to the disorder on site I for Er. More detailed neutron investigations by Bordet et al. (1986) revealed a complicated magnetic structure, with a ferromagnetic component along the c-axis and with relatively broad magnetic reflections due to short-range order. Van de Pasch et al. (1987) observed re-entrant behaviour in the (Er~_~Y~)Rhx.xSn3. 6 system for 0.6 < x < 1.0 (in slight disagreement with the above results). Specific-heat measurements by Willekers et al. (1987) on this system also revealed a broad peak in the region of the magnetic transition, reflecting the short-range order in this system.
~ 16
Er Rhl.ISn3. 6
p
-I
0
i
0
0.2
0.4
0.6
i
i
0.8
i
1.0
1.2
1~4
TEMPERATURE(K)
Fig. 26. Resistivity, critical field, AC susceptibility, intensity of the (111) peak (neutrons) and specific heat versus temperature for ErRhl.lSn3. ~ (after Remeika et al. 1980).
MAGNETIC SUPERCONDUCTORS 4.0
i
i
1
T m 2 Fe 3 Si 5
503
I
] Tel o TC~
~.0
~ T/,/ , oc su~c. -
222a ~
TN. diffractior
~ 2.0 I--
NA
I
,
5
I
I
I0
t5
Pressure
/
20
25
(kbar)
Fig. 27. Temperature-pressure phase diagram of Tm2Fe3Si5 showing the normal paramagnetic (NP) phase, the superconducting paramagnetic (SP) phase and the normal antiferromagnetic (NA) phase (after Vining and Shelton 1985, Gotaas et al. 1987).
Segre and Braun (1981) reported re-entrant superconductivity in the silicide Tm2Fe3Si5. Vining and Shelton (1985) confirmed this, but found that (in their sample) superconductivity and re-entrance only occur under pressure. The temperature-pressure phase diagram is shown in fig. 27. The remarkable property of this compound is that superconductivity is destroyed by antiferromagnetic order. A neutron study (Gotaas et al. 1987) has confirmed this, and showed in addition that no ferromagnetic component is present. Apparently, the Fermi-surface effects (partial gapping) or the change in electron-electron interaction due to change in the pairing conditions in the antiferromagnetic state (see next section) are strong enough to destroy superconductivity. No coexistence region has been reported. Re-entrance has been observed in numerous pseudoternary compounds and most of these will be discussed in section 6. Let us just recall here that re-entrance has also been observed in the pseudobinary compound (Thl_xNdx)Ru z, in a narrow concentration range near the transition to a spin-glass state (Hiiser et al. 1983).
5. Antiferromagnetic superconductors The possibility of coexistence of antiferromagnetism and superconductivity was first considered theoretically by Baltensperger and Strassler (1963). They showed that coexistence is possible, but that the superconducting state would be modified by the antiferromagnetism. Today, a number of compounds have been found where the two phenomena coexist. Although coexisting with the magnetic state, superconductivity is found to be influenced in various ways by the antiferromagnetism. In this section we shall describe some of these compounds and their anomalous behaviour.
504
O. FISCHER
5.1. Ternary molybdenum chalcogenides, (RE)Mo6X8 As mentioned in section 3, most compounds of the two series (RE)Mo6S 8 and (RE)Mo6Se 8 order antiferromagnetically at low temperature. Coexistence with superconductivity was first demonstrated in the compounds DyM06S8, TbM06S8, ErMo6S 8 and GdMo6S 8. The antiferromagnetic transition was initially observed as an anomaly in the superconducting upper critical field He2 (Ishikawa and Fischer 1977b, Ishikawa et al. 1978). These compounds become re-entrant when a sufficiently high magnetic field is applied. Figure 28 shows the resistance of DyMo6S 8 as a function of temperature for different applied magnetic fields. In fields above 100mT there is an abrupt increase in the resistance at 0.4K. Magnetization and susceptibility measurements suggested that an antiferromagnetic ordering takes place at that temperature. A neutron-scattering investigation on this compound confirmed the antiferromagnetic ordering at 0.4 K (Moncton et al. 1978). The ordered state was found to be of the A1 type, composed of (100) planes with the spins parallel within the plane and the spin orientation alternating between planes. The magnetic unit cell is thus nearly tetragonal. In fig. 29 is shown the intensity of the (½,0, 0) peak together with the Hc2 anomaly to be discussed in section 5.4. The H~2 reduction is seen to be directly related to the increase in the sublattice magnetization proving thereby the coexistence of the two phenomena. The magnetic order of TbM06Ss, ErM06S 8 and GdM06S 8 is equivalent to that of DyMo6S 8 (Thomlinson et al. 1979, 1981, Majkrzak et al. 1979). Figure 30 shows the powder neutron-diffraction data for TbMo6S 8 above (4.2 K) and below (0.07 K) the antiferromagnetic transition (T M = 1.05 K). The magnetic peaks are hatched for clarity. Figure 31 shows the nearly tetragonal antiferromagnetic structure of all these compounds. The angle ~b between the magnetic moment and
DYl.2 Mo 6 S 8
/f '
:: °O
/
e 1000 a 1250 m
i
0
,
i
i
I 0.5
i
i
i
i
i
)0
i
i
,
)
i 1.5
i
,
i
l 2.0
)
2000
i
,
i
I 2.5
TEMPERATURE( K ) Fig. 28. D C electrical resistance versus temperature of DyMo6S 8 in various magnetic fields (after Ishikawa and Fischer 1977b).
MAGNETIC SUPERCONDUCTORS
505
Dy MO6 SB 1.5
0 ~ 0-" " 0 -- -- ~...0~ I I __..... I I
ZOO0
2 •- P O 0 - - - - , . 0 ~ _ 0 _ ~ 0
._c E
~
1.0
Iz
:z
1000 HEATING • COOLING
/
o
0,5
I 0,2
I 0.4 TEMPERATURE (K)
I 0.6
Fig. 29. The temperature dependence of the (½,0, 0) antiferromagnetic peak in DyMo6Ss, shown together with the critical field anomaly (after Moncton et al. 1978).
3500
V
I
I
I
I
I
I
I
I
3000
2500
--
TbI.:~Mo6S 8
T= 0 . 0 7 K -- 2 0 0 0 E ea ¢n 1 5 0 0
I000
--
z w Iz
500
2,,}
{,oo}
M {00"]
{ 2)
3
JI
i
--
%
J ~ :
0
soo
~I 5
~
~U~_..~-~..J
2,o)(i
{100} ! TbL~MO6S 8 T=4.2 K
I •
I
I
I
IO
15
20
~
..
L
I
-
~
'
~
?.5 30 35 SCATTERING ANGLE
~
W
I
I
I
I
40
45
,50
55
6O
(28)
Fig. 30. Powder neutron-diffraction data for TbMo6S 8 above ( 4 . 2 K ) and below ( 0 . 0 7 K ) antiferromagnetic transition of T M = 1.05 K (after Thomlinson et al. 1981).
the
506
~ . FISCHER
¢
i
.) J
,/ // //
Fig. 31. TOe antiferromagnetic structure for the Dy, Er, To, and Gd molybdenum sulfides and GdMo6Se 8. q~ is the angle between the spin direction and the [100] axis. The spin direction shown is for DyMo6S 8 (after Thomlinson et al. 1982).
the (001) direction was found to be 55 °, 60° and 45 ° for DyMo6Ss, TbMo6S 8 and ErMo6Ss, respectively. Using the splitting of the (1, 0, 1) peak resulting from the slight trigonal distortion, it was possible to determine the moment direction. For DyMo6S 8 the moment was found to lie in the [111] direction, whereas in TbMo6S 8 the magnetic moment is closer to the [100] direction. ErMo6S 8 was found to be an incipient antiferromagnet, where the magnetic order only develops at low temperature after an external magnetic field has been applied (Thomlinson et al. 1981). Low-temperature specific-heat measurements have been carried out for several of these antiferromagnetic superconductors (McCallum et al. 1977, McCallum 1977, Azevedo et al. 1978, Maple et al. 1980a, Alekseevskii et al. 1985). A pronounced lambda-type anomaly was found at the magnetic phase transition. For the Gd compounds it was found that the entropy was roughly equal to the Hunds-rule value, i.e., R ln(2J + 1). However, for the other RE the entropy is lower, illustrating the importance of crystal-field effects. In most of these investigations other anomalies are found in the temperature range between 1 and 10 K. So far one has not been able to connect these with other properties of the Chevrel phases, and no satisfactory explanation has been found. In view of the difficulty of making very pure material, it is not excluded that these anomalies are connected with impurity phases. The influence of the superconductivity on the magnetocaloric effect has been studied in some of these systems by Rogacki et al. (1985, 1987). In the (RE)Mo6Se 8 series, GdMo6Se8, ErMo6Se 8 and HoMo6Se 8 have been studied in some detail. In GdMo6Se 8 early specific-heat anomalies at low temperature indicated magnetic ordering, but the probable presence of impurity phases made definite conclusions difficult (McCallum et al. 1976, 1977). Azevedo et al. (1978) reported a lambda-type anomaly at 0.75K, and neutron measurements by Maple et al. (1980a) confirmed an antiferromagnetic order similar to
MAGNETIC SUPERCONDUCTORS
507
the one found in the sulfides. Early specific heat (McCaUum et al. 1977) and neutron studies (Lynn et al. 1978a) of ErMo6Se8, showed a magnetic transition at 1.1 K. However, the complex magnetic structure could not be resolved, and it could not be ascertained that 'the ordering really took place in the Chevrel phase. Later investigations have, without resolving the problem completely, brought more evidence that the ordering really takes place in the Chevrel phase, with an oscillatory state characterized by a modulation vector along the [111] direction (Thomlinson et al. 1982). For a review of the neutron-scattering investigations we refer to Thomlinson et al. (1982). A very interesting result was found by Lynn et al. (1984) in HoMo6Se 8. This compound becomes superconducting at 5.6 K and crystal-field studies show that the magnetic moment in the ground state is about 6/z B. Magnetic order is 1
i
1
i
i
160
l/~ T=0.286K
I\ !
120 .E E
o
a!
80
z ~ 40
0
40
0
l
;
I
I
~
;
I
I
I
I
~
;
;
;
0.030 0.060 0.000 WAVE VECTOR 0(~ -1)
,;
t ._c 30
tl/
~" >.
T:O.2 6
,,=,
"'
I
/i
z 10
/ s,! ~ 0.03 observed
b
,~~.~ 0.460K = l/ t r;t
20
The
;
~/
I--
Fig. 32. (a)
0.120
neutron
counts
a.
0.500K
~0.520K
0.06 0.09 WAVE VECTOR Q(~-I)
as a f u n c t i o n
of wave
0.12 vector
above
and
below
the magnetic
transition in HoMo6Se 8. (b) Temperature dependence of the scattering at several temperatures (after subtraction of the nonmagnetic background) (after Lynn et al. 1984).
508
O. FISCHER
observed below TM = 0.53 K and the system remains superconducting down to below 40 mK. The magnetic order reveals itself in the neutron diffraction in a unique peak at low angle, corresponding to a sinusoidally modulated magnetic structure with a wavelength of 101 ,~. No higher order satellites and no ferromagnetic component was observed. Figure 32 shows the observed peak at various temperatures. The wave vector of the modulation qc increases as T approaches TM from below. These results were interpreted in terms of a ferromagnetic interaction modified by superconductivity into a long-range oscillatory state. Contrary to the two re-entrant superconductors HoMo6S 8 and ErRh4B4, where the oscillatory state only exists in a narrow temperature interval, here the oscillatory and coexisting state survives down to the lowest temperatures. This is consistent with the higher Tc and Hc2 in this material, as compared to HoMo6S 8. The observed increase in q~ was interpreted in terms of a weakening of the superconducting state as temperature is reduced below TM. Further studies of this compound would be of considerable interest in order to deepen our understanding of the interplay of ferromagnetism and superconductivity.
5.2. Ternary rhodium borides, primitive tetragonal (RE)Rh4B 4 Three compounds in this class of materials have been found to show coexistence of antiferromagnetism and superconductivity, NdRh4B4, SmRh4B 4 and TmRh4B 4. NdRhaB 4 becomes superconducting at Tel = 5 . 3 K (Matthias et al. 1977). Investigation of the properties of this compound at lower temperature by Hamaker et al. (1979a) revealed very interesting features. Figure 33 shows the 800
I
I
I
I
I
NdRh4B 4 7
0
0
500 Z
,< 400 300 200 100 -
00
'
/
0.45/ 0.2
0.3 L./, 1
2
3 4 TEMPERATURE (K)
)
0.1
0 5
6
Fig. 33. AC electrical resistance versus temperature for NdRh4B 4 in various applied magnetic fields between 0 and 0.ST (after Hamaker et al. 1979a).
MAGNETIC SUPERCONDUCTORS
509
resistance as a function of temperature for different applied magnetic fields. In zero field the compound stays superconducting down to the lowest temperature. However, for fields above 0.3 T, an abrupt increase occurs at 1.31 K followed by a sharp drop at 0.89 K. Specific-heat measurements in zero applied magnetic field showed two lambda-type anomalies at these temperatures indicating that two magnetic phase transitions are related to the anomalies in the R(T, H) curves in fig. 33. The presence of thermal hysteresis in the lower transition suggests it to be of first order. The magnetic order is found to be complex and antiferromagnetic in both cases (Majkrzak et al. 1982). The Nd moments are aligned along the unique c-axis with a sinusoidal modulation in the [i10] direction in the high-temperature phase and along the [110] direction in the low-temperature phase. Recently, surface impedance studies revealed an anomalous increase in the penetration depth at TM1 (Hou et al. 1988). SmRhaB 4 becomes superconducting at Tc = 2.72 K. A sharp kink in the H~2 versus T curve at TM = 0.85 K originally indicated the occurrence of a magnetic transition (Hamaker et al. 1979b). Heat-capacity data reveal a lambda-type anomaly at that temperature. Thermal-conductivity measurements by O t t e t al. (1980a) showed that this compound remains a bulk superconductor below TM. The surface impedance measurements of Hou et al. (1988) showed a reduction of the penetration depth at TM, just contrary to the result found in NdRh4B 4. Magnetic ordering has been found at 0.4K in TmRhaB 4. This compound, which becomes superconducting at 9.8 K, stays superconducting down to the lowest temperature in zero field. However, in fields above 0.3 T, a hysteretic re-entrant transition occurs (Hamaker et al. 1981a,b). This has been interpreted as evidence for a metamagnetic transition in finite fields from antiferromagnetism to ferromagnetism with a possible narrow coexistence region of the metamagnetic state and the superconducting state. Neutron investigations have shown that this compound has a complex antiferromagnetic structure in zero applied field (Majkrzak et al. 1983). The data were interpreted in terms of a modulated antiferromagnetic structure with a wave vector q = 0.24 ~ in the [101] direction and with the moments pointing in the [010] direction.
5.3. Ternary rhodium borides, body centered tetragonal (RE)Rh4B 4 ErRh4B 4 and HoRh4B 4 have been synthesized in the body centered tetragonal phase (Iwasaki et al. 1983, Tournier 1984, Iwasaki and Muto 1985). Both these compounds are superconducting (T c =7.8 and 6.5K, respectively) and both become antiferromagnetic at temperatures well below T~ (T M --0.65 and 1.4 K, respectively). Specific-heat measurements on a single crystal of ErRh4B 4 revealed a sharp lambda-type anomaly, with an entropy suggesting that the ground state of the Er moments is a doublet (Iwasaki et al. 1987). A detailed study of the magnetization of this single crystal has been carried out by Iwasaki et al. (1986). The compound is found to be metamagnetic (Iwasaki et al. 1986). Figure 34 shows the magnetization as a function of field at 0.1 K together with the resistive transition. Superconductivity apparently coexists with the first metamagnetic state
~. F I S C H E R
510
5
bct ErRh4B4 1 HI/[110] ~"~ ~ T=O.1K . . ~ 1
Cl)
a
i~
I
2
3\
4
Hc2
•
o
_
H II [100]
1 ½ 3 4 5 b) ['~'----
E
i / -- 1 -
2
3
H(kOe)
4
- 0-
H(kOe)
Fig. 34. Magnetization and electrical resistance of bet ErRh4B 4 versus magnetic field at T = 0.1 K. (a) Hll[aao]. (b) Hill100] (after Iwasaki and Muto 1985).
(field in the [100] direction) but is destroyed by the second. In HoRh4B 4 only one transition was observed, and the hysteresis in M leads to a hysteresis in Hc2 at low temperatures. Tournier (1984) reports the observation of a rectangular-shaped magnetic hysteresis, characteristic of a ferromagnetic superconductor with a self-induced vortex state (Avenhaus et al. 1969, Tachiki 1981b).
5.4. Other antiferromagnetic superconductors In the Heusler alloys series (RE)Pd2Sn , two compounds have been found to order magnetically in the superconducting state. Kierstead et al. (1985) confirm the observation of magnetic ordering in YbPdzSn , reported by Ishikawa et al. (1982b). This compound, which becomes superconducting at T c = 2.36 K, undergoes a magnetic phase transition at T m = 0.23 K. This transition was observed in specific-heat measurements as a cusp-like anomaly with an entropy corresponding to a doublet. The antiferromagnetic nature of this transition was inferred from susceptibility measurements showing a negative Curie-Weiss temperature of - 0 . 3 6 K . The system remains superconducting down to 0.16K, and it was concluded that the two p h e n o m e n a coexist. Shelton et al. (1986) studied ErPd2Sn and found a magnetic phase transition at T M = 1 K, just below the superconducting transition at T c = 1.17 K. The system remains superconducting below T M and it was inferred from the negative C u r i e Weiss temperature that the magnetic order is presumably of antiferromagnetic nature. Neutron-diffraction studies have been reported by Stanley et al. (1987). Antiferromagnetic correlations are found to develop even above To, and a transition to an antiferromagnetic state takes place at around 1 K. The structure is an fcc type-II, where the fcc E r cell doubles in each direction. Indications for the existence of a modulation with a wavelength of about 50 ,~ in a direction perpendicular to the [111] direction were also found. However, since the basic
MAGNETIC SUPERCONDUCTORS
511
structure is antiferromagnetic, the origin of this modulation is probably not the same as in the re-entrant ferromagnetic superconductors or in HoMo6Se 8. A similar magnetic structure has also been found in HoPd2Sn where Li et al. (1989b) found that a considerable smearing apparently occurs in the transition between 2 and 5 K. One system has been found where antiferromagnetism destroys superconductivity, i.e., Tm2Fe3Si 5 (Gotaas et al. 1987). This compound has been discussed in the previous section under re-entrant superconductors.
5.5. Analysis of the upper critical field curves The upper critical field of several antiferromagnetic superconductors was studied by Ishikawa and Fischer (1977b). Two examples, ErMo6S 8 and DyMo6S8, are shown in figs. 2 and 35, respectively. An analysis of these curves based on the multiple pairbreaking theory described in section 2 has been carried out (Fischer et al. 1979, Ishikawa et al. 1982a). The solid curves are the result of this calculation, including the internal field, the exchange field and the temperatureindependent Abrikosov-Gorkov exchange scatteringl Consider first the case of ErMo6S 8 where this theory accounts very well for the whole anomalous He2 curve. The maximum in He2 at 1 K and the decrease below that temperature is a result of polarization effects combined with a temperaturedependent magnetization and does not reflect the onset of a magnetic ordering. The magnetic order manifests itself in the minimum in He2 at 0.2 K, which follows
'
'
'
I
. . . . O
I
. . . .
I
'
DyMo6S 8
~, H o M o 6 S 8 0.20
b,.-
I -.J <~
0.1C
O.OE
0
I1~ 0.5
~
~
I 1.0
TEMPERATURE
1.5 (K)
Fig. 35. Upper critical field versus temperature for DyMo6S s Fischer 1977b).
and HoMo6S 8
(after Ishikawa and
512
O. FISCHER
directly, within the above calculation, from the maximum in the susceptibility at TM. Similar //ca curves have been found for NdMo6S 8 (Alekseevskii and Narozhnyi 1984), GdMo6Se 8 (Yamaguchi 1984) and TmRhaB 4 (Hamaker et al. 1981a). For DyMo6S 8 the critical field above TM can be understood in terms of the multiple pairbreaking theory; the spin-polarization effects cause Hc2 to flatten out above TM. However, at TM there is a drop which cannot be explained within the multiple pair-breaking theory, unless one assumes that there is an additional temperature-dependent pairbreaking parameter. A similar situation occurs in TbMo6S 8 and in GdMo6S 8 (fig. 36). The critical field of NdRh4B4, shown in fig. 37 together with the critical field of other ternary borides, shows a similar abrupt drop at TMI and then, in addition, an increase at TM2 (Hamaker et al. 1979a). The re-entrant behaviour at TM of Tm/Fe3Si 5 may be viewed as an extreme case where superconductivity is destroyed altogether, possibly by the same mechanism. As a first approach to the analysis of this anomalous behaviour, Fischer et al. (1979) assumed an additional pairbreaking parameter to be present and determined it from the difference between the measured and the calculated upper critical field shown in figs. 35 and 36. The resulting additional pairbreaking parameters are shown in fig. 38. Both for TbMo6S 8 and DyMo6Ss, one finds a temperature dependence that closely resembles the magnetic order parameter. In these two compounds, one finds no hint of a fluctuation effect close to Tc. However, in GdMo6S 8 the additional pairbreaking parameter has a sharp maximum, indicating the presence of such fluctuations. This is also supported by the
'
'
'
'
I
.
.
.
.
i
.
.
.
.
I
'
TbMo6S8 0.20
O GdMo6S8
_~ 0.15
_o b-
0.5
1.0
1.5
TEMPERATURE (K)
Fig. 36. Upper critical field He2 for TbMo6S 8 and GdMo6S s (after Ishikawa et al. 1978, Fischer et al.
1979).
MAGNETIC SUPERCONDUCTORS
0.6 0.5 0.4
i
i
l
NdRhs Bs
\
i
i
t
i
i
0 04 i
015
L~
rRh4B4
~-' 0.4 %
SmRh4B,I
0.08
\_
0
~ ,.
0.12
',~.
oz
~0.1 J
0.2
i
0.16 " " ' / ~ \
0.3 E
i
513
1.0[
'
1.0
.>-"~
o8t //
% 02
1.5 '
"~
t/
2.0
\\
I
"~
0.2
\ I
3.0
'Tm'Rh4B4 '
0.4t';
°"°
2.5
~.
I
I0
TEMPERATURE(K} Fig. 37. Upper critical fields for NdRh4B 4, SmRh4B 4, ErRh4B 4 and TmRh4B 4 (aRer Maple et al. 1982).
~P\\
24 2O o
Gdl, 2 Mo6 S 8
16
__o_O_~O
q
8
'\'b
ic
f I L l < <
0
24 ~ - 0
20
Tbl.2 Me 6 S 8
- --0~
TM
X
'
Z
12
Wr~
L+
m
"¢
\
1 \\
i
i
i
i
I
i
i
~
i
O o ~ ~
I -o
2z,~
o_ f e
Tc
,
,
,
L
I
,
O0
0
0fi
L
0
1.0
1.5
TEMPERATURE(K) Fig. 38. Additional pairbreaking parameter versus temperature for GdMo6Ss, TbMo6S s and DyM06S s (after Ishikawa et al. 1982a).
514
O. FISCHER
observation that the AC susceptibility shows an anomaly at Tu in this compound, contrary to the other AF superconductors (Ishikawa et al. 1978). Thomlinson et al. (1981) and Lander et al. (1984) studied the influence of a magnetic field on the magnetic ordering. They found, in particular, that DyMo6S s develops a strong ferromagnetic component in the neutron scattering in fields below Hc2 , confirming earlier magnetization measurements by Ishikawa and Muller (1978). Whether this is a result of a spin-flop phase or of an induced ferromagnetism in the center of the vortices (Krzyszton 1980), or that it reflects a real coexistence of superconductivity and ferromagnetism (self-induced v o r t e x state) or, more trivially, that it somehow reflects a compositional inhomogeneity, remains open. In any case, these results imply that the data, displayed in fig. 29, show a close correlation between the zero-field sublattice magnetization and He2 , measured in a field where a strong ferromagnetic component is present. This suggests that the anomaly seen in Hc2 is largely a field-independent effect of the magnetic state on the superconducting state. This view is supported by tunneling experiments on TbMo6S s and GdMo6Se s carried out by Poppe and Schr6der (1984). They found anomalies in the tunneling conductance at T~, suggesting a reduction in the energy gap below Tu. Several theoretical investigations have dealt with the explanation f o r the anomalous H~2 behaviour. Ramakrishnan and Varma (1981) showed that the magnon-mediated electron-electron repulsion studied by Baltensperger and Str~issler (1963) can be viewed as a continuation into the antiferromagnetic state of the Abrikosov-Gorkov paramagnetic state exchange scattering. A small increase or decrease of the pairbreaking may take place at the antiferromagnetic transition. They suggested that if the antiferromagnetic transition is triggered by a Fermi-surface nesting, the anomaly could be explained by a modification of the pairbreaking. This needs, however, a very special Fermi surface, and the phenomenon seems to be too general for this to be the case. Machida et al. (1980) and Machida (1984) pointed out that a partial gapping of the Fermi surface in the antiferromagnetic state will lead to a reduced phase space and thus to a reduction in the superconducting order parameter. Zwicknagl and Fulde (1981) used the original Baltensperger-Str~ssler pairing in a strong-coupling theory to show that a reduction of the phonon-mediated electron-electron interaction occurs in the antiferromagnetic state, and they were able to describe the anomaly using experimentally determined parameters. Fulde and Keller (1982) argue that this effect is normally more important than the phase space reduction due to the antiferromagnetic gaps. Ashkenazi et al. (1983) have taken a similar point of view and have found good agreement with experiment. Fenton (1988) argues that the influence of static antiferromagnetism can be interpreted as a change in the electronic density of states from the nonmagnetic to the antiferromagnetic state. For reviews on these questions, see Keller and Fulde (1982) and Buzdin and Bulaevskii (1986). Not all antiferromagnetic superconductors show this anomaly. ErMo6S s and NdMo6S 8 show an increase of Hc2 that can be understood in terms of the multiple pairbreaking theory as a result of the reduced magnetization in the anti-
MAGNETIC SUPERCONDUCTORS
515
ferromagnetic state. Iwasaki and Muto (1986) studied He2 on a single crystal of bct ErRh4B4, and found that both the anisotropy in the basal plane as well as the large anisotropy between the c-axis and the basal plane (shown in fig. 39) can be understood in terms of the multiple pairbreaking theory as given by eq. (11). As in primitive tetragonal ErRh4B4, this analysis shows that exchange effects are essential to explain the observed behaviour. The same conclusions apply to bct HoRh4B 4 where the hysteresis in Hcz shown in fig. 40 can be directly related to a hysteresis in M(H). Such a hysteresis in He2 was first observed in bct Ho(Rhl_xRux)4B 4 (Adrian et al. 1984, Thom~i et al. 1986), and was explained as a result of two competing interaction mechanisms, possibly at the origin of the metamagnetic behaviour.
EPRh4B4 (bct) o H H [001]
40
• H II
[t00]
H II [110] 3O
11z 20
~.0
2
4
5
8
T /K) Fig. 39. Temperature dependence of the upper critical field for a single crystal of bct ErRh4B4 with the field along the easy axes ([100], [110]) or along the hard axis [001] (after Iwasaki and Muto 1986).
6 - bct T
T.:
HoRh4B4
oHc2[l')
= 6.50 K
oH~(1)
~.40K
:22 2
0
0
2
4
T
{K)
6
Fig. 40. Upper critical field of bct HoRh4B 4 (after Iwasaki and Muto 1985).
516
O. FISCHER
SmRh4B 4 (fig. 37) has a temperature dependence of the upper critical field that is apparently very different from what is found in the other antiferromagnetic superconductors (Hamaker et al. 1979b). Various theoretical explanations can be invoked for this anomaly (Ro and Levin 1984, Bennemann et al. 1969, Machida 1979), it is, however, unclear whether these apply to the present case. Lambert et al. (1984) investigated this question experimentally by the study of the pseudoternary system (Sm~_xYx)Rh4B 4. They found, using eq. (9), that the influence of the magnetization, either directly or through the exchange field, is small above TM. They conclude that the kink seen in He2 at T M is the result of a reduction of the exchange scattering Am in the antiferromagnetic state. This is supported by the observation of a reduction of the resistance below T• in a field of 0.2 T (Hamaker et al. 1979b). The reason why this is seen in SmRh4B 4 and not in other compounds is certainly the fact that /~m is very large in this system (accounting essentially for the Tc reduction from 11.5 K in LuRhaB 4 to 2.7 K in SmRh4B4) and that other pairbreaking effects are weak at or above TM. In most other compounds studied here, one finds that /~m is smaller and other effects are larger at the magnetic transition, possibly masking changes in Am. Note that the additional pairbreaking parameter, discussed above for other antiferromagnetic compounds, probably does not have its origin in the exchange scattering. Zasadzinski et al. (1984) studied T~ as a function of the residual resistance ratio (RRR) in thin films of SmRh4B 4. They observed an anomalous decrease of T~ with decreasing RRR, and they suggested a disorder-enhanced spin-flip scattering to be responsible for this effect. This result is further supported by radiation-damage studies (Terris et al. 1984). In the (Smx_xYx)Rh4B4 system Tra decreases rapidly as x increases; at the same time H~2 increases and this results in an increased importance of polarization effects. H~2(T ) thus becomes re-entrant, a behaviour which Lambert et al. (1984) find is well-described by multiple pairbreaking theory.
6. Pseudoternary systems A large number of pseudoternary systems have been synthesized, and some of them studied in great detail. These investigations have revealed numerous new and interesting features and these systems have proven to be a rich ground for systematic studies of the interplay of superconductivity and magnetism. In what follows, we shall briefly discuss some of the low-temperature phase diagrams that have been determined.
6.1. (RE~_xRE'x)Mo6S 8 systems Snl_xEUxMO6S8 This system has turned out to be important because of the particular properties of EuMo6S 8 and the observation of magnetic-field induced superconductivity in this system for x = 0.8. Eu is divalent in this system, and was initially expected to have superconducting properties similar to PbMo6S 8. However, EuMo6S 8 is not super-
MAGNETIC SUPERCONDUCTORS
517
-o-- - -- -o-- - o - - o t~
"
o
5
8
Ft.>
./ // i
i
I
[
[ 0.5
I / t
CONCENTRATION
E
X
Fig. 41. Superconducting critical temperature Tc (O) and structural transition temperature T~ (&) versus Eu concentration in Snl_xEuxMo6S 8 (after Decroux et al. 1982). conducting at all (Fischer et al. 1975a) and, as described in section 3, it was found by Baillif et al. (1981) that this compound undergoes a structural phase transition and becomes semiconducting at low temperature. The low-temperature phase diagram, including the superconducting transition and the structural transition, is shown in fig. 41 (Decroux et al. 1982). Tc decreases very slowly as E u is substituted for Sn, showing that the exchange scattering in this series is very weak. Extrapolation to x = 1 shows a T~ reduction of about 1 K for the nontransformed superconducting EuMo6S 8. The suppression of the structural transition with pressure, and the observation of a critical temperature of T c = 12.2 K (Chu et al. 1981, Harrison et al. 1981, Decroux et al. 1983) for the rhombohedral phase confirms this very weak exchange depression of T c. Between x = 0.6 and 0.8, T~ falls abruptly to zero. This T~ depression was found to be directly related to the appearance of the structural phase transition. The corresponding transition temperature decreases linearly with decreasing x and reaches zero at x = 0.65. EuMo6S s has been found to order magnetically at T M = 0.3 K (Bolz et al. 1977, Ishikawa et al. 1981). Neutron-scattering studied by Quezel et al. (1984) confirmed this observation and revealed a complex modulated magnetic structure. It is expected that this state would exist also for x < 1, but no systematic study has been carried out in this range.
Hol_xEuxMo6S8 This system is of particular interest, because of the unusual properties of both HoMo6S 8 and EuMo6S 8. The low-temperature phase diagram was studied by Ishikawa et al. (1981) and is shown in fig. 42. In the Ho-rich region T~I ( T c in fig. 42) decreases rapidly when E u replaces Ho, whereas Tc2 and T M stay roughly constant. The system remains re-entrant up to x = 0.3, where a coexistence region is found. In this region, T c has increased again, probably as a result of a reduction of the charge transfer since E u is in a divalent state. The nature of the magnetic order has not been determined with precision, but the magnetization measurements showed a finite remanent magnetic moment, indicating that the state
518
O. FISCHER --;
.
.
.
.
.
,
I
,
,
,
I"
H°I.2-xEUxM°6 $8 Para
- 2.0
2.0
E Super
1.0
~ ~
1.0
Tm
• Antiferro 0
1.2
I
I
1.0
Eu
I I
i
Super+"F" b i 0.8 06i 04'. i
X
i
Ferro '02,.
,.
0
Ho
Fig. 42. Low-temperature phase diagram for (Hol_xEux)Mo6S 8. To2is approximately equal to TMin the concentration region marked by Ferro and decreases below the lower limit of the measurement in the region marked Super + "F" (after Ishikawa et al. 1981). has a ferromagnetic component. This system is a candidate for some kind of coexistence of ferromagnetism and superconductivity. Further experiments are necessary, however, to clarify the exact interplay of the two phenomena. In the Eu-rich part of the phase diagram, superconductivity is destroyed as a result of the structural transformation and the magnetic transition temperature is roughly independent of x.
Other RE~_xRE" Mo 6S8 systems McCallum et al. (1981) studied the YI_~Eu~Mo6S8 system, and found a more pronounced depression of T c with increasing x than observed in Snl_~Eu~Mo6S 8. A Mossbauer-effect study revealed a hyperfine-split spectrum below 0.4 K which was explained in terms of a complex paramagnetic relaxation of the E u moments. Ishikawa et al. (1978) investigated the HOa_xLUxMo6S , system in the dilute limit, and observed some evidence that the system might show a transition back to the superconducting state at a very low temperature, a p h e n o m e n o n possibly related to the low-temperature ( T < To2) superconductivity observed by Genicon et al. (1984) in HoMo6S 8. The Lax_xCe~Mo6S 8 system was investigated by Okuda et al. (1981). They found that T c is reduced linearly as Ce is substituted for La and a low critical concentration of x = 0 . 1 3 was deduced from the experiments. CeMo6S 8 shows a peak in the magnetoresistance around 2 K, where it becomes magnetic.
6.2. (REI_xRE')Rh4B 4 systems (Er1_xHox)RhaB4 This solid solution of the re-entrant ferromagnetic superconductor ErRh4B 4 and the ferromagnet H o R h a B 4 has been shown to display several interesting
MAGNETIC SUPERCONDUCTORS
519
phenomena. The phase diagram was initially determined by Johnston et al. (1978). Since then, it has been somewhat refined with the most recent version, as given by Pringle et al. (1985) and Maple et al. (1985), shown in fig. 43. As x increases from zero, Tea , T M and To2 all decrease linearly until at x = 0.3, T M and To2 coincide. Above this value of x , both TM and To2 increase linearly until they meet T~I in a tricritical point at x = 0.9. It has been found that below x = 0.3, TM is determined by the ordering of the Er moments in the basal plane. On the other hand, above x = 0.3 one finds that TM is determined by the ordering of the Ho moments with their spins in the direction of the c-axis. Mook et al. (1980) studied (Er0.4Ho0.6)Rh4B 4 by neutron diffraction, and found that only the Ho moments ordered ferromagnetically at T M -- 3.6 K. A subsequent investigation (Mook et al. 1982) of a sample having the composition (Er0.7Ho0.3)RhaB 4 showed that the Ho moments order along the c-axis at TM and that the Er moments order at a lower temperature in agreement with specific-heat data showing two peaks (MacKay et al. 1979). The dash-dotted curve in fig. 43 indicates the ordering temperature for the Er moments. This independent ordering of the two species results from the different crystal-field effects on the two. The fact that the Er moments are being confined to the basal plane and the Ho moments to the c-axis clearly leads to a decoupling of the two and to the minimum observed in TM. Pringle et al. (1985) showed that the sinusoidally modulated phase, which coexists with superconductivity and with (normal?) ferromagnetic domains, is also found at x = 0.2 with notably the same modulation wavelength of about 100 A. The shaded area in fig. 43 shows the region where the modulated phase is found. 9
d3..~o I
I O- 0
8 --
I
[
I
0o.o~
]
I
I
l
[
PARAMAGNETIC
_
o~ 7 --
6
( E r l _ x H o x ) R h 4 B4
O ~
0~0~ 0 Os0 Oo ~ o e"
-
SUPERCONDUCTING
74
-
,,6°0
7: j -3 --
0 --
SC SINUSOIOALLY MODULATED & N O R M A L FM REGIONS
FM (.ulc)
i
0
?
t
•
4 / / J
FERROMAGNETIC (p II C)
MIXED
i
0.2
I
--
/ 0
" ""11, ~ .
t
0.4
I
]
0.6
~
I
-1:3--. _
I
0.8
I
i
1.0
X
Fig. 43. Low-temperature phase diagram for (Erl_xHox)Rh4B 4. The open circles (0) correspond to Tc~ and the solid circles (@) to To=. The solid triangles (A) show the ferromagnetic transition and the squares (E3) the magnetic ordering of the Er moments for x > 0.4 (after Maple et al. 1985).
520
O. FISCHER
In the region x < 0.9, it is found that the ferromagnetic transition is reduced compared with what one would expect if there was no superconductivity, as indicated by the dotted line in fig. 43. Such behaviour has been attributed to the reduction of the RKKY interaction in the superconducting state. The observation of a spike in the specific heat at T M was attributed to the sha W rise in the magnetization and the destruction of superconductivity at TM = To2. Direct evidence for a first-order transition was provided by a calorimetric investigation of Lachal et al. (1982). A study of the Ho-rich part of this phase diagram by Lynn et al. (1983, 1985) confirms the reduction of T M for x = 0.75. However, these authors present evidence for a certain coexistence region with T~2 < T M. No oscillatory behaviour was observed, contrary to what is found in other cases when ferromagnetism and superconductivity are confronted. Possibly the wavelength is much larger than the experimental resolution or/and the system develops a vortex structure with an oscillating superconducting order parameter. This narrow coexistence domain has been confirmed by Pringle et al. (1988). However, they suggest that this coexistence is due to an inhomogeneous mixture of superconducting and ferromagnetic domains. Unusual behaviour of the ultrasonic attenuation has been reported by Sun et al. (1985) in this concentration region.
(Y1_xErx)Rh4B4 The low-temperature phase diagram for this system was established by Okuda et al. (1979) and is shown in fig. 44. The interesting part is that of high-Y concentration, where the magnetic ordering is found to be antiferromagnetic and superconductivity coexists down to the lowest temperature. In this region, I
~
lo.o ~ . .
f
i
1
i
r
!
i
~
~:~9.5 ~
9.0
w
~_ 8.a
t Tc2~
•-
-
0.5 f,lv
0
0.2
0.4 0.6 Concentrationc
01.8
1.0
Fig. 44. Low-temperature phase diagram showing Tcx and To2 in (Y1 ~Erx)Rh4B4 (after Okuda et al. 1979).
MAGNETIC SUPERCONDUCTORS
521
Radousky et al. (1983) reported anomalous behaviour of the upper critical field. At about x = 0.2 the exchange polarization becomes strong enough to destroy superconductivity when the magnetization approaches saturation. The critical field is then determined by the condition M(H~2) < M 0. As a result, H~2 drops abruptly when x becomes larger than a certain value (i.e., 0.2). Since the magnetic-ordering temperature is very low, it is possible that the magnetic state is modulated, due to the presence of superconductivity. Specific-heat measurements by Kierstead and Dunlap (1987) confirm the magnetic transition at about 0.3 K for x = 0.3, but neutron investigations would be necessary to understand the coexisting phase.
(Sml_~Erx)Rh4B 4 and (Sml_xY~)Rh4B4 The phase diagram for (Sml_~Erx)RhnB 4 was established by Woolf and Maple (1981) using AC magnetic susceptibility and specific-heat measurements and is shown in fig. 45. The antiferromagnetic order disappears already at x = 0.15, and above x = 0.3 ferromagnetic order and re-entrant superconductivity is found. The (Sml_xYx)Rh4B 4 system was investigated by Lambert et al. (1984). They found that there is a linear decrease of TM as x increases from 0, and above x = 0.40 the system apparently stays paramagnetic down to the lowest temperatures investigated. The critical-field measurements reported in this work are discussed in section 5. 9
a g~
(Sml_xEr x) Rh4B4
Paramagne £
~- 4
//~
.~ A ~
1
/
Superconducting
2
~i ~.+_~_A~,,.~.-~NORMAL+OMAGNETICALLYORDERED
Oo o:~io,, o;~ o.4' o15 o;~ oJ7 o18 oi~ ~.o X
Fig. 45. Low-temperaturephase diagram for the system(Sml_~Erx)Rh4B4(after Woolf and Maple 1981/ .
Erl_xGdxRh4B4 and (Gdl_xYx)Rh4B 4 The low-temperature phase diagram of Er l_xGdxRh4B4 is shown in fig. 46 (Wang et al. 1978, Kohn et al. 1979). It is interesting in that it shows a classical behaviour with an AG-type decrease of T c, and a linear variation of TM between ErRh4B 4 and GdRh4B 4. The system has a tricritical point at x = 0.28. The rapid decrease of Tc reflects the large de Gennes factor for Gd as compared to, e.g., Ho. The linear variation of T~a is probably related to the S-state of Gd for which crystal-field effects are much less important than for other RE. Critical fields for this system have been reported by Wang et al. (1978).
522
0. FISCHER I
I
I
I
I
I
I
I
I
/
t
laJ
k-
•z
2
~o~o
j
~
o~O ~ 0
I
I
0.2
I
0,4
1
0.6
I
I
0.8
I
1.0
x
Fig. 46. Low-temperature phase diagram for the system Erl_xGdxRh4B4(after Wang et al. 1978).
Adrian et al. (1981) studied the phase diagram of (Gd]_xYx)Rh4B 4. An AG-type T¢ decrease is observed, with some deviations close to the critical concentration (x = 0.32) where T c and T M meet. Just below the critical concentration, the system is re-entrant and evidence for a transition from type-II to type-I superconductivity was found.
6.3. ( n E ) ( R h l _ x T x ) 4 n 4 systems Y(Rhl-xRux)aB4 Several pseudoternary systems of the type RE(Rhl_xRUx)4B 4 have been studied and will be described below. In order to have a nonmagnetic reference compound for these series, Johnston (1982) and Shelton et al. (1983) have investigated the Y(Rhl_xRUx)4B 4 system. As found initially by Johnston (1977), a small amount of Ru substituted for Rh stabilizes the bodycentered tetragonal structure. This structure is stable in the 0.1 < x < 1.0 range. The T c variation in this range is given in fig. 47. Tc is constant up to about x = 0.5. Then an abrupt decrease occurs, and for x > 0.5 no superconductivity is found except for x = 1.0, where a T~ of 1.42 K was observed. In fig. 47 two series of samples prepared with different heat treatments are shown. The abrupt drop d e a r l y depends somewhat on the preparation technique. In a recent publication, Iwasaki et al. (1987) report evidence that the abrupt drop in T~ occurs rather at x = 0.3 than at x = 0.6. The origin of the abrupt drop in ]re is subject to various speculations. Johnston (1982) found that although the lattice parameters vary smoothly the interatomic distances show abrupt changes at around x =0.5. H e suggested an abrupt redistribution of electronic states. H e also noted that the drop always occurs when the c/a ratio is equal to 2. However, Shelton et al. (1983), in an Auger spectroscopy study, found no evidence for sudden electronic changes close to
MAGNETIC SUPERCONDUCTORS Tc(K)
i
i
Y(Rh=-x
i
523
i
RUx)4
B4
I0 9
8 7 6 5
2. I 0.4
2
[ 0.6
-
±
I 0.8
1.0
X
Fig. 47. Critical temperature versus Ru concentration x for the system Y(Rh1_xRUx)4B 4. The open and the solid circles correspond to two different sample series with different heat treatments (after Shelton et al. 1983).
x = 0.5. Ku et al. (1980), who observed a similar behaviour of Tc when Ir is substituted for Rh, suggested that in order to have a high Tc in these compounds, it is necessary to have integer R h 4 clusters.
Dy(Rhl_xRUx)gB4 The low-temperature phase diagram of this system has been determined by Hamaker and Maple (1981a,b, 1983). The part with the body centered tetragonal structure 0.1 < x < 1 is shown in fig. 48. In contrast to the primitive tetragonal DyRh4B4, which becomes ferromagnetic at T M = 10.7 K and shows no superconductivity, substitution of 0.1 Ru gives the body centered structure, and the compound becomes superconducting at T¢ = 4 K. Anomalies in the specific heat t
I
I
I
L
I
I
I
I
by (RuxRhi_x) 4 B4 5
l-and moqreticolly I ~rdereo r 0| I I I'~ 0
0.I
0.2
0.3
Mognetically ordered t
I
I
I
0.4
05
06
0.7
I 0,8
I 0,9
1.0
x
Fig. 48. Low-temperature phase diagram for the system Dy(Rh~_~Ru~)4B 4. The circles (©) show the superconducting transition and the squares ([7) the magnetic transition. The vertical dashed line indicates the probable behaviour of the superconducting transition temperature (after Motoya et al.
1984).
524
0. FISCHER
and in the AC susceptibility led to the conclusion that a magnetic transition takes place around 1.5 K in the Rh-rich part without destruction of superconductivity. Close to x = 0.3, the system becomes re-entrant at T~. A neutron-scattering investigation by Motoya et al. (1984) showed the magnetic transition to be of antiferromagnetic nature, but with a TM roughly one degree above the temperature given in fig. 48. An anomalous temperature dependence of the magnetization, with a high slope where the specific heat has its maximum, explains this discrepancy and also the anomalies observed in the H~2 data reported by Hamaker and Maple (1983). The superconducting transition shows the same abrupt decrease as in Y(Rh1_xRUx)4B4, but here it happens at x = 0.3. Clearly, this has a nonmagnetic origin with possibly some influence of the magnetism close to x = 0.3. No superconductivity is observed for x > 0.33. The magnetic order in the intermediate region 0.3 < x < 0.85 was found to be of short range, presumably as a result of frustration between competing antiferromagnetic and ferromagnetic interactions. Finally, DyRuaB 4 was found to have a complex magnetic structure with both ferromagnetic and antiferromagnetic components.
Ho(Rh~_xRUx)4B4,Ho(Rhl_xOSx)4B4 The phase diagram of this system has been established in several papers by Knauf et al. (1984a), Adrian et al. (1984) and Thom~ et al. (1986). The result, which resembles very much the phase diagram found for Dy(Rhx_xRux)4B4, is shown in fig. 49. Superconductivity is found up to x = 0.4. The presence of antiferromagnetic phase transitions around 1 K were deduced from anomalies in the upper critical field. In particular, it was found that He2 show a hysteresis below 0.5 K, probably related to a hysteresis in the magnetization. Irradiation experiments on thin films (Mohle et al. 1987) show that Tc is much more sensitive to disorder than TM, as judged from the hysteresis in/-/ca. Knauf et al. (1984b) showed by photoemission studies that the decrease of Tc can be related to a decrease in the density of states N(0) at the Fermi level. For 0.4 < x < 0.8 it has been concluded T IK)
65
J
~
HO(Rhl_~Ru~)c.BL.
Z, SC + PM
PM
3
--4k. . . . . . . . . . ~
1
0
O
FM
D
SC + AF
0.0
0.2
0.4 0.6 Ru - concentration x
0.8
1.0
Fig. 49. Low-temperature phase diagram for the system Dy(Rhl_xRux)4B4. The solid circles (O) indicate the superconducting transition. The open squares ([3) show the antiferromagnetic transition and the filled squares (11) show the ferromagnetic region (after Thom/i et al. 1986).
MAGNETIC SUPERCONDUCTORS
525
that only magnetic short-range order occurs, whereas for x = 1, two magnetic transitions were found at TM1 = 2.6 K and TM2 = 2.3 K. The magnetic structure of the lower phase has been determined by Mfiller et al. (1984). Mohle et al. also investigated the Ho(Rhl_xOS~)aB 4 system, and found a similar superconducting behaviour as for the R h - R u system.
Er(Rh~_xRux)4B4 Homg and Shelton (1981) determined the phase diagram for this system. Superconductivity is found below x = 0.4-0.5 and magnetism for ruthenium-rich compounds. So far, no direct confrontation of magnetism and superconductivity has been found in this system.
Ho(Rh,_xIrx)4B4 When substituting Ir for Rh the structure stays primitive tetragonal up to x = 0.8. In spite of the fact that Ir is isoelectronic with Rh and that the magnetic sublattice remains unchanged in this system, dramatic changes both in the magnetic and in the superconducting properties are found as x varies across the series. Ku et al. (1980) were the first to study this phase diagram and an improved version was given by Yang et al. (1982). Their result is shown in fig. 50. For small x values, the ferromagnetic transition is rapidly reduced, and above x = 0.1 one finds re-entrant superconductivity. The T¢ curve in this region suggests that HoRh4B 4 would have been superconducting at about 5 K in absence of ferromagnetism. At x = 0.2, there is a transition to antiferromagnetism and coexistence at low temperatures. The abrupt drop in Tc at x = 0.5 is believed to be of nonmagnetic origin, similar to the situation in the R h - R u systems discussed above. Knauf et al. (1984a) found from photoemission studies that there is a broadening of the density of states at this Ir concentration that could qualitatively explain this I
r
;~o
I
t
]
~..o..¢~.~.z ./z~./~ " ~
6
~'~
I
• \
2
FERRO- • MAGNETIC%
• •~•
a/
/ =/
? ~ 0
0
I
I
0.2
=
i
H°(Rhl _xlrx)4B4 " A~EA'~O Tc Tm • c(m) o • Xac(T)_
5
u~ k-
t
o" ~ ANTI~'*" ~ FERRONO J . ' ~ A G NETIC ;eCo4B4 ~ ^ TYPE & ~ PHASE AF+S
i
I
0.4-
l
I
0,6
l
I
0.8
I
1,0
X
Fig. 50. Low-temperaturephase diagram for the systemHo(Rhl_xlr~)4B4 (after Yang et al. 1982).
O. FISCHER
526
anomalous behaviour. In the interval 0.6 < x < 0.8 a new situation arises; here, the antiferromagnetic transition occurs above the superconducting one, as confirmed in a more detailed study (Woolf et al. 1983). The ferromagnetic (x < 0.2) and antiferromagnetic ( 0 . 2 < x < 0 . 8 ) order has been confirmed by neutronscattering experiments (Hamaker et al. 1982). Dy(RhI_xlG)4B 4 and Tb(Rhl_Jrx)4B 4 The superconductivity in the RE(Rhl_~IG)4B 4 system has been investigated in detail• For an overview, the reader is referred to a report by Ku (1984). Here we shall just mention the two cases RE = Dy and Tb, in addition to the Ho case discussed above. The low-temperature phase diagram for Dy(Rhl_~IG)4B4, as obtained by Ku and Acker (1980), is shown in fig. 51. It resembles the one
12-'
I
I
1oil 8-
I
'
r
D y ( ] r x Rhl.x)4B 4 Arc-melted (bulk a powder)
-t
PARAMAGN ETIC
1
I
-ff
i
6
s
M ~
/
J
T/
ySoeCO,s, 0.0
~I 0.2
I
\
g t
y
]
~ ~rdered(M)(?) I o
nducting
I
I 0.4
~/ X
"
S+M
I 0.6
~
I 0.8
I
1.0
Fig. 51. Low-temperature phase diagram for the system Dy(Rhl_xIrx).B 4. The magnetic transition is shown by solid circles (Q) and the superconducting transition by open circles (O) (after Ku and Acker 1980).
Phase A
Phase B
Tb(IrxRhl.x)4B 4
"FE..O"~ ANTIFERRO
1
/v.\ 0.0
0.2
0.4
0.6
X
O.S
1.0
Fig. 52. Low-temperature phase diagram for the system Tb(Rhl_xIrx)4B 4. The magnetic transition is shown by solid circles ( 0 ) and the superconducting transition by open circles (©) (after Acker et al. 1982).
MAGNETIC SUPERCONDUCTORS
527
discussed above for the Ho compound. Also, here is found a region where the magnetic transition occurs above the superconducting one. The low-temperature phase diagram for Tb(Rhl_,Ir~)4B4 has been reported by Ku et al. (1982) and Acker et al. (1982). In contrast to the Dy case, here one finds an antiferromagnetic transition above the superconducting one in the whole domain where superconductivity exists (0.2 < x < 0.7) (fig. 52).
RE(Rhl_xCOx)4B 4 (RE = Er, Ho, Lu) An investigation of these systems has recently been reported by Ku et al. (1987). Upon substitution of Co for Rh, T c is reduced roughly linearly with x with the critical concentration between 0.3 and 0.4. For higher x values an antiferromagnetic normal phase is reported for Er and Ho. Figure 53 shows the phase diagram for Er, including the Er(Rhl_~Ir~)4B 4 system. The ferromagnetism disappears for x > 0.1 and is replaced by an antiferromagnetic state that coexists with superconductivity. This agrees with the anomalous temperature dependence of Hc2, reported earlier by Isino et al. (1983). ErCo4B 4 i
ErRh4B4 i
i
i
1
i
I
i
i
Etlr4 B4 i
i
i
i
i
i
~
i
9 Er(Rhloylry),~ B 4
Er(Rhl'xC°x)4 B4
8
o Ts 7
• TM {TNor Tc )
6 T(K) PARAMAGNETIC
PARAMAGNETIC SUPERCONDUCTING IS)
4
3
2 FERROMAGNETIC
[
ANTIFERROMAGNETIC (AF)
o'--T LO
5 0.8
,--? 0.6
~
AF+ S
'/
AF+$
, x
0.4
0.2
0,0
0.2
0.4
y
0.6
0.8
LO
Fig. 53. Low-temperaturephase diagram for the systemsEr(Rhl_~Cox)4B 4 and Er(Rhl_xIr)4B4 (after Ku et al. 1987).
6.4. Some other pseudoternary systems (LUl_xErx)RuB2 Several superconducting or magnetic compounds have been found in the class of ternary borides with composition MTB 2 (T = Ru, Os) (Ku and Shelton 1980, Shelton et al. 1980). Superconducting compounds were found with M = Sc, Y, Lu and magnetic compounds with M = Tb, Dy, Ho, Er, and Tm. The pseudoternary system (Lul_xEr~)RuB 2 has been investigated by Shelton and Horng (1986). The corresponding phase diagram is shown in fig. 54. Between x - - 0 and x = 0.3 the system is superconducting and paramagnetic. In a narrow interval, i.e., 0.3 < x < 0.5, re-entrant superconductivity is observed. Finally,
528
~. FISCHER I
i
i
i
i
i
(LUl_xErx}RUB2 1(3
i TC1 • TC2
II
~"
8
g Z
~5 z 4
2 0
I
t
i
i
I
I
0
0.2
0.4
0.6
0.8
1.0
X
F i g . 54. Low-temperature phase diagram for the system (Lul_=gr=RuB2) (after Shelton and Horng 1986).
above x = 0.5 the system is magnetic, presumably ferromagnetic, and normal. So far no coexistence has been reported, but in the Er-rich region two magnetic transitions are observed.
(Tml_xLu=)RuB2 This system was investigated by Ku and Shelton (1981). The phase diagram is shown in fig. 55. Re-entrant superconductivity is observed for 0.5 < x < 0.6. The system is magnetically ordered for x < 0.5 and superconducting for x > 0.6. Tim (a/o) 25 12
'
20 1
'
15 I
T
10 I
5 i
'
{Tml_x LUx) RuB 2 o
lO
Tc(Tcl)
• TM(Tc2)
r
/S
!
8
PARAMAGNETIC
/
~6 4
~
SUPERCONDUCTING MAGNETICALLY ORDERED
2
0
i 0.0
1 0.2
i
I 0.4
I
1 0.6
%1
I 0.8
I
i 1.0
X
F i g . 55. Low-temperature phase diagram for the system ( T m l _ = L u x ) R u B 1981).
2 ( a f t e r K u and Shelton
MAGNETIC SUPERCONDUCTORS
529
(Yl_,~Dyx)Pd2Sn The Heusler-type compounds REPd2Sn are good candidates to form interesting pseudoternary systems. M a l i k et al. (1986) have studied the system (Yx_xDyx)Pd2Sn. Figure 56 shows the low-temperature phase diagram obtained by these authors. A very narrow region of possible coexistence is found. The re-entrant behaviour of the upper critical field in this concentration range and its deviations from the expected free-ion value suggest the possibility of antiferromagnetic ordering coexisting with superconductivity. However, further investigations are necessary to make definite conclusions.
PhQ
~5
~m = ~oPd2(YI_xDYx)Sn ~-
5
PaPamagnet ~c ,uperonducting .
0
x
__ !ntifn_tife~ro_m_a ...... gT_net i~e_t~__.
0.5
l.O
Fig. 56. Low-temperaturephase diagram for the system (Yl_~Dyx)Pd2Sn(after Malik et al. 1986). 7. Magnetic-field induced superconductivity: the Jaccarino-Peter effect A very striking prediction was made in 1962 by Jaccarino and Peter. They proposed that a weak ferromagnet could become superconducting if a strong magnetic field is applied. The basis of their idea was that the conduction-electron spin polarization in such a system can, in certain cases, be directed opposite to the externally applied field. Consider a ferromagnet made out of localized magnetic moments. In zero applied field the conduction-electron spins will be polarized due to the exchange interaction [eq. (1)]. As described in section 2, this polarization can be thought of as arising from an exchange field H j acting on the conductionelectron spins. In normal ferromagnets, this field is much stronger than the paramagnetic limiting field, the Chandrasekhar-Clogston limit Hp (Chandrasekhar 1962, Clogston 1962) of any superconductor and no superconductivity will be observed. The idea of Jaccarino and Peter was based on the fact that for some systems the exchange constant I is negative. The conduction electrons will then be polarized opposite to the localized magnetic moments. In zero applied field this makes no difference for the superconductivity. However, when a field is applied, the localized magnetic moments, having the larger magnetization, will align in the magnetic field, and the conduction-electron polarization will be oriented opposite to the external field. On the other hand, the direct interaction between the conduction electrons and the external field will tend to align the
530
O. FISCHER
conduction-electron moments in the same direction as the field. That is, the external field will oppose the internal exchange field and for H = H j, a compensation will occur so that the total field acting on the conduction-electron spins is zero. At this value of the external field, the polarization of the conductionelectron spins is, to first order, zero and superconductivity is possible. The condition for observing this magnetic-field induced superconductivity (MFIS) is, of course, first that the ferromagnet in absence of magnetism would have been a superconductor. In addition, it is necessary that the orbital critical field of the superconductor is high enough so that the very high externally applied field does not destroy superconductivity as a result of its interaction with the conduction-electron orbits. Since in a ferromagnet none of the relevant parameters can be determined by experiment beforehand, it turned out to be very difficult to find a compound where this effect could be observed. However, the effect can also be observed in a paramagnet since the strong external field will in any case polarize the localized magnetic moments at low temperature, and thus produce the necessary ferromagnetic alignment (Schwartz and Gruenberg 1969, Fischer 1972). The advantage in this case is that in zero field the material is superconducting and the relevant parameters can be determined. The Chevrel phases, with their very high critical field and with their possibility to have a regular lattice of magnetic moments and still maintain superconductivity, are very good candidates for observing this Jaccarino-Peter effect. A first investigation of the pseudoternary series EuxSnl_xMo6S 8 (Fischer et al. 1975b) revealed an anomalous behaviour of the upper critical field He2 as a function of temperature that was interpreted in terms of the Jaccarino-Peter compensation mechanism. The negative sign of the exchange interaction necessary for this interpretation to be correct was confirmed independently, both by experiment (Fradin et al. 1977, Odermatt 1981) and by theory (Freeman and Jarlborg, 1982). Similar anomalies have also been reported in the Lal_xEUxMo6S 8 system by Torikachvili and Maple (1981). Analysis of the multiple pairbreaking theory including the polarization effects leads to the possibility of a field-temperature phase diagram containing two separate superconducting regions (Fischer 1972). The first region is situated at low fields, where the magnetization is sufficiently small so that polarization effects are weak enough for superconductivity to exist. The other superconducting region is found at high fields where the exchange field is compensated by the external field. This behaviour is described by eq. (9), or, qualitatively, by eq. (11), for a negative H j and a certain choice of parameters. A more detailed discussion of this magnetic-field induced superconductivity is given by Fischer (1972), Decroux and Fischer (1982) and Rossel et al. (1985). After careful investigations of the EuxSnl_~Mo6S 8 system, Meul et al. (1984a,b) were able to observe such a phase diagram. Figure 57 shows their results for a sample of composition Eu0.75Sn0.25Mo687.2Se0.8. Two well-separated domains can be seen. When, at low temperature, the magnetic field is increased from zero, the sample is first superconducting. Then, at about 1 T, an abrupt transition occurs into the normal state, due to the rapidly rising magnetization. At
MAGNETIC SUPERCONDUCTORS
531
25
%
Euo.75 5n0.25 Mo6 $7.2Se0.8
20
t
• Hc2-meClsurement using He3- He4 dilution refrigerator • superconducting magnet (12 tesla)
U3 Ld 15
• Hc2-measurement using He3refrigerator ÷ polyhelix magnet (25 tesla)
-r-
kd U_
-- Calculation of HC2from multiple
U
pairbreaking theory
n~ U
1
I 1 2 3 TEMPERATURE T[K~
~"%1
4
Fig. 57. Upper critical fieldversus temperature for the compoundEU0.TsSU0.25Mo6ST.2Se8 (after Meul et al. 1984a). higher fields the magnetization, and thus the exchange field Hj saturates, and due to the negative exchange interaction, the total polarization of the conduction electrons slowly decreases with increasing field and the sample returns to the superconducting state until the orbital pairbreaking effects finally provoke a return to the normal state. Figure 58 shows the resistance as a function of field for different temperatures, displaying clearly this superconducting-normal-superconducting-normal ( S - N - S - N ) behaviour. A similar S - N - S - N behaviour of the AC susceptibility was reported by Remenyi et al. (1985). The complicated formula of the sample shown in figs. 57 and 58 results from the necessity of fine-tuning the parameters in order to have the fight conditions to observe this effect. In the Eu-rich samples of the Snl_~EUxMO6S8 pseudoternary system, the values of the parameters turn out to be such that T c has to be about 4 to 7 K for this S-N-S-N-type phase diagram to occur. In view of the x dependence of Tc (fig. 41), the observation should be possible close to x---0.8, where Tc is rapidly reduced and this is also where the anomalies in He2 were initially observed. A first indication of the possibility of field-induced superconductivity was reported by Isino et al. (1981) and Wolf et al. (1982). However, due
532
~ . FISCHER
1.89K
~
/
~
0.5
OK
5
10
15
20
25
Applied magnetic field #.oH [tesl~ Fig.
58. DC
electrical resistance versus field at various temperatures for the compound Eu0.75Sno.25MO6ST.2Se0.8 (after Meul et al. 1984a).
to the stringent conditions for this effect to occur on the one hand and the broad transitions resulting from the rapid decrease of Tc with x on the other hand, a clear interpretation was difficult. In order to circumvent this difficulty, Meul et el. (1984a,b) chose a lower Eu concentration and replaced some S by Se in order to adjust Tc to the value necessary for this effect to occur. One prediction of the theory is that the shape of the phase diagram is very sensitive to the exact value of T~. For a somewhat higher T~, the two domains melt together and become one (see fig. 59), and for a somewhat lower T~, the field-induced domain disappears. This was verified by Meul et el. (1984a). By changing the Se concentration, T~ could be varied without varying the other parameters very much. Another prediction is that the Tc required to observe the
I
a)
24~
~ 24~ ~
'c = 3'9 K
b) T¢: 4.3K
jvortices
9
161-'/
jj.==0
I
/
1, p--q
'=F---.-J 8
antivortices
4
4
vortices
I u~
_\fvo. ces
0
\antivortices 1
2
3
4
5 6 0 1 2 TEMPERATURE [K]
3
4
5
6
Fig. 59. Critical field versus temperature as obtained from eq. (11) for parameters corresponding to the Eul_=Sn=Mo68 system. The occurrence of vortex and antivortex structures are indicated schematically (after Fischer et el. 1985).
MAGNETIC SUPERCONDUCTORS
533
MFIS increases with increasing Eu concentration (Rossel et al. 1985). Capone et al. (1985) studied a sample with composition Eu0.9Ho0.1Mo6S 8. They induced superconductivity with pressure and observed, for a pressure of 8 kbar and T~ = 6.4 K, field-induced superconductivity, although the resistance did not reach zero in the field-induced region. They were able to obtain a very good fit to eq. (9) with parameters similar to those reported by Meul et al. (1984a,b). Initially, there was some doubt whether the anomalous H~2 behaviour could be related to the structural phase transition occurring at x = 0.8 in the Sn-Eu pseudoternary system. In order to check this, it was important to investigate this effect in the rhombohedral phase of E u M o 6 S 8 away from the structural transition. Measurements under pressure of H~2 on a sample having a T~ = 12 K (under pressure) by Decroux et al. (1984, 1988) showed an anomaly in H~2 versus T, but failed to show field-induced superconductivity. An analysis of the data (Rossel et al. 1985) showed that for this system to show MFIS, T~ would have to be lower than 12 K. This question was subsequently investigated by Cors et al. (1987), who studied Se-doped EuMorS s under pressure. MFIS was verified on a sample having the composition EuMorST.sSe0. 5. At 15 kbar, this sample had a T~ = 7 K. The higher T~ and the higher Eu concentration than studied before, resulted in a shift to higher fields of the field-induced superconducting region. In this sample, the MFIS domain was centred around 20 T, whereas it was centred around 12 T in the first measurements by Meul et al. (1984a). Kawamata et al. (1987) have recently repeated the original measurements of Meul et al. and confirm these results. They also investigated the low-temperature specific heat and found only a small crystal-field splitting as expected for an S-state moment. An interesting alternative mechanism to produce magnetic-field induced superconductivity is the freezing of spin fluctuations by a magnetic field. This question was investigated theoretically by Maekawa and Tachiki (1978). However, so far no evidence for this mechanism has been found in t h e materials considered here. The magnetic properties of the superconducting state have been predicted to be very unusual (Fischer et al. 1985). Using Ginzburg-Landau theory, it was found that the vortex lattice changes its nature in the field-induced state. At fields where dHc2/dT= % the local superconducting magnetization M s vanishes, i.e., there is no oscillating field associated with the flux lines. Below this field, the currents in the vortices change sign and the field is enhanced in the superconducting regions, leading to antivortices and a paramagnetic superconducting magnetization. This situation is illustrated in fig. 59 where two different He2 versus T diagrams are shown. In the single-connected domain of fig. 59b, we have two lines with M s = 0 dividing the superconducting domain into three regions. A measurement of the superconducting magnetization as a function of field is predicted to give a positive value in the lower half of the field-induced domain and a negative value in the upper half. The observation of this effect is made difficult by several facts. First, the superconducting contribution will be a small part of the total magnetization dominated by the contribution of the Eu moments. Second, pinning of flux lines will tend to mask the effect. Third, the stringent conditions on homogeneity will tend to smear out the effect. The first observation
534
~. FISCHER 5 i
10 15 / i MIFII. S i i P. . . . ogn ,~ Diamagn
[
20 I
i
[ [
'~ 252
t/.~s ./~-/
LtJ 25t 27 _o ,,~ 25C _N
< 24E
2520 EM u/g /
. / , , ' ~ " " 25.07 E.M U/g
./ I
I
__1
I
MAGNETIC FIELD [Tesla]
Fig. 60. Magnetization versus field for EUo.75Sno.esMo6S7.73Seo.27 at T ~ 1.2 K (after Fischer et al. 1985).
of such a magnetization is shown in fig. 60. On top of the Brillouin-like magnetization of the Eu moments a small deviation corresponding qualitatively to the expected behaviour is seen. Further investigations are under way to explore the properties of the vortex lattice in this system (Birrer et al. 1988).
8. Coexistence of superconductivity and magnetism in RE-oxide superconductors With the new oxide high-temperature superconductors the question of the relation and interplay of magnetism and superconductivity has come strongly into focus. The fact that insulating Cu-oxides like LazCuO 4 and YBazCu30 6 are antiferromagnetic, has led to the idea that superconductivity in the hole-doped compounds Lal.85Sr0.asCuO 4 and YBazCu30 7 might also be related to the magnetic interactions. These questions are investigated in numerous groups all over the world. However, a discussion of this topic would go beyond the scope of this chapter. For a broad overview on the various ideas in this direction, the readers are referred to the Proceedings of the Conference on Materials and Mechanisms of Superconductors and High Temperature Superconductors, edited by Muller and Olsen (1988). Furthermore, the experimental results on the antiferromagnetism have been recently reviewed by Rossat-Mignod et al. (1988) and by Lynn and Li (1988). The interplay of superconductivity and magnetism can also be studied in these compounds in a different manner. Soon after the discovery of YBa2Cu30 7 and its superconducting critical temperature of 92 K (Wu et al. 1987, Cava et al. 1987), it was found almost simultaneously by a large number of groups that Y can be replaced by nearly all the rare earth elements (see, e.g., Hot et al. 1987, Willis et al. 1987, Murphy et al. 1987, Kitazawa et al. 1987). The remarkable fact is that Tc remains practically unchanged, i.e., there is no measurable effect of the magnetic moments on superconductivity. Obviously, we have here an extreme case of decoupling between the rare earth atoms and the conduction electrons. Note, however, that the Tc reduction due to exchange scattering is not directly
MAGNETIC SUPERCONDUCTORS
535
dependent on Tc itself. If the former would be of the same order of magnitude as in the Chevrel phases (about one degree) it would be difficult to prove experimentally since the nonmagnetic effects of the substitution are evidently of the same order of magnitude. Table 6 contains a list of the REBa2Cu30 7 compounds and their various ordering temperatures. In this section, we shall briefly summarize the results obtained so far in the relation between the rare earth magnetism and superconductivity. Several of the REBa2Cu307 compounds have been found to order antiferromagnetically at low temperature. The ordering temperatures are indicated in table 6. In view of the very weak interaction between the conduction electrons and the rare earth atoms, an interesting question is related to the origin of the magnetic order and the possible role of the RKKY interaction. Dunlap et al. (1988) studied the specific-heat anomaly of this transition in superconducting GdBa2C%O 7 and semiconducting GdBa2Cu306. Their result is shown in fig. 61. Clearly, the transition does not depend on the presence of the conduction electrons, and the RKKY interaction seems to play no role in the magnetic ordering. Dipole interactions are certainly important, but they may not be sufficient to explain everything. Figure 62 shows the structure of these compounds as determined by Dunlap et al. (1987) for HoBa2Cu30 7. There is one RE atom per unit cell in this structure and the RE lattice is simple orthorhombic with R E - R E distances about 3 . 8 5 A in the a-b-plane and about l l . 7 A in the c-direction. The distances in the a-b-plane are much larger than in the RE elements, but much smaller than in the Chevrel phases, whereas the distances in TABLE 6 Superconducting and magnetic transition temperatures of REBa2Cu30 7 compounds [data also taken from Neumeier et al. (1988)]. Compound
T c (K)
PrBa2Cu307 NdBa2Cu30 7 SmBazCu307 EuBa2Cu307 GdBazCu307 TbBa2Cu307 DyBazCu307 HoBa/Cu307 ErBazCu307
92 91 94 94 35 91 93 92
TmBa2Cu307 YbBa2Cu307 LuBa~Cu307
91 90 90
T M (K) 15
* References: [1] Gering et al. (1988). [21 Maple et al. (1987). [3] McK Paul et al. (1988). [4] Brown et al. (1987). [5] Goldman et al. (1987).
Comments
Ref.*
Antiferromagnetic?
[1]
0.5 2.25 1 0.17 0.5 0.14
[21 Antiferromagnetic Different crystal structure Antiferromagnetic Two-dimensional ordering? Two-dimensional order at 0.5 K Three-dimensional order at 0.14K
0.35
[6] [7] [8] [9] [10]
Dunlap et al. (1987). P. Fischer et al. (1988). Lynn et al. (1987). Chattopadhyay et al. (1988). Hodges et al. (1987"1.
[3] [1, 4] [5] [6, 7] [8, 9]
[lO]
536
O. FISCHER 15 eo ~o o E
~ o
-'3
~
5
0
o"
I
0
2
3
4
5
Tempe~atuPe (K) Fig. 61. Specific-heat anomaly at the antiferromagnetic transition for superconducting GdBa2Cu307 (fflied circles) and insulating GdBa2Cu306 (open circles) (after Dunlap et al. 1988).
)
9
Ba
Ho
03 j f 04
Cu2
a Fig. 62. Structure of REBaaCu307 (after Dunlap et al. 1987).
the c-direction are clearly very large. It is, therefore, plausible that a quasi two dimensional magnetic behaviour should be observed. It has been pointed out by van den Berg et al. (1987) and by Nakazawa et al. (1987) that the shape of the specific-heat anomaly suggests a two-dimensional nature of the magnetic order. However, neutron measurements have shown that a three-dimensional order occurs in GdBa2Cu30 7 (McK. Paul et al. 1988) and in DyBa2Cu30 7 (Goldman et al. 1987, Fischer et al. 1988). On the other hand, Lynn et al. (1987) have reported ErBazCu3Ov to show two-dimensional order at T = 0.5 K, where specific-heat measurements (Dunlap et al. 1987, Brown et al. 1987, Maple et al. 1987, Nakazawa et al. 1987) show a pronounced lambda-type anomaly. Data by Chattopadhyay et al. (1988) have shown that a three-dimensional order sets in at
MAGNETIC SUPERCONDUCTORS
537
a lower temperature, T = 0.14K. For HoBa2Cu307, Dunlap et al. (1987) concluded that a magnetic order could occur below 0.17 K. A neutron study by Fischer et al. (1988) showed that magnetic correlations similar to the ones observed in ErBa2Cu30 7 develop in HoBazCu307 below 0.14 K. Magnetic order has also been reported to occur in YbBa2Cu307 at T = 0.35 K (Hodges et al. 1987). The systematics of T~ in these compounds does, at first sight, seem to follow the de Gennes factor G with a maximum at RE = Gd, as expected for a RKKY interaction (Ramirez et al. 1987). However, as pointed out above, the RKKY interaction does not seem to play a significant role in these compounds. A more detailed investigation of the magnetic interactions is, therefore, necessary. Dipolar interactions might account for the ordering in the a-b-plane, but it is too weak to account for the coupling between the planes in compounds like GdBa2Cu30 7 and DyBa2Cu30 7. One must assume that some kind of superexchange must operate in these compounds to explain the three-dimensional ordering. This conclusion is supported further by recent neutron investigations by Chattopadhyay et al. (1989). They found that, whereas in GdBa2Cu30 7 the Gd spins order antiferromagnetically in all directions, in GdBa2Cu306.5 the Gd spins order antiferromagnetically in the a-b-plane, but that the nearest-neighbour spins in adjacent planes order ferromagnetically. This is shown in fig. 63. For PrBa2Cu3OT, Gering et al. (1988) report evidence for a magnetic ordering temperature of 15 K, whereas they find 10 K for PrBa2CuaO 6. Contrary to the Gd compound, here there seems to be a clear difference between the insulating and a
b
¢tl
I
Fig. 63. Spin arrangement of antiferromagnetic (a) GdBa2Cu306.5 and (b) GdBaaCu307 (after Chattopadhyay et al. 1989).
538
0. FISCHER
the metallic compound. Crystal-field studies have been carried out by several groups, and it appears that the type of magnetic order is strongly influenced by these effects, except for RE = Gd. Holmium has been found to have a singlet ground state (Dunlap et al. 1987, Furrer et al. 1988) and the magnetic order observed at low temperatures might be a combined nuclear and electronic ordering. PrBa2Cu30 7 and NdBa2Cu30 7 have been investigated by Walter et al. (1988). Whereas the crystal-field levels could be determined for the Nd compound, the Pr compound showed typical features of a valence instability, offering thereby a reason for the absence of superconductivity. Similar results were reported by Gering et al. (1988), who also concluded that TbBa2Cu307, which forms in a different crystal structure, shows evidence for valence fluctuations. In all superconducting REBa2Cu30 7 compounds, superconductivity remains below TM, and so far no influence of the magnetic order on superconductivity has been reported. Clearly, ~the situation is very different from the case of the ternary compounds considered before. In these oxide compounds, the superconducting condensation energy is typically two orders of magnitude larger, and the upper critical field is two orders of magnitude higher than the saturation magnetization of the rare earth. Furthermore, the paramagnetic limit is of the order of 200 T, probably much higher than the exchange field Hj. It is, therefore, not surprising that no destruction of superconductivity occurs, even in magnetic fields sufficiently strong to completely polarize the RE moments. Because of the very high values of He2 in these compounds, no study has been made of He2, so far, in the temperature range where magnetic order occurs. The very weak exchange interaction between the rare earth moments and the conduction electrons has been shown to result from a low density of conduction electrons at the RE site (Massidda et al. 1987). Nevertheless, it is remarkable that the RE atom is situated between the two CuO 2 planes which are believed to be essential to superconductivity, and that it has 8 oxygen atoms as nearest neighbours whose 2p holes are thought to form the conduction band. The necessity to introduce an exchange-like coupling to explain the three-dimensional ordering, must somehow be reconciled with the apparent absence of exchange coupling to the conduction electrons. Possibly, there is a suppression of the RKKY interaction in the superconducting state (Dunlap et al. 1987, Walter et al. 1987). Magnetic rare earth atoms have also been introduced into the (La,Sr)CuO 4 3ystem. Contrary to the 92 K superconductor, one observes here a reduction of Tc with the substitution of RE for La (Takagi et al. 1988). At first sight, this could result from an exchange-scattering process since the maximum T~ reduction occurs for Gd. However, closer inspection of the data led Takagi et al. to conclude that the observed Tc reduction does not have a magnetic origin. Thus, in this system too, the RE substitution does only lead to a very weak, if any, exchange depression of To. In this connection, it is interesting to notice that superconductivity has also been observed in Ce (Tokura et al. 1989) and Th (Markert and Maple 1989) doped Nd2CuO4, Pr2CuO 4 and Sm2CuO4,with Tc as high as 24 K. In these compounds, the charge carriers are apparently electronlike, contrary to the (La,Sr)2CuO 4 system which has hole-like carriers. A direct
MAGNETIC SUPERCONDUCTORS
539
comparison is, therefore, not possible, in particular also because the crystal structure is not exactly the same. Nevertheless, the presence of superconductivity in these systems is remarkable, and suggests that a more detailed study of their magnetic properties could be rewarding. 9. Summary and conclusions
This chapter has dealt with the interplay between magnetism and superconductivity in rare earth compounds, where the magnetic moments are produced by the 4f-electrons of the rare earths and where the conduction electrons are in most cases d-electrons, well-separated from the magnetic 4f-electrons. A key element of these compounds is that the exchange interaction between the conduction electrons and the magnetic moments is weak. This assures that the unavoidable pairbreaking due to exchange scattering can be kept on a low level, and, thus, make the confrontation of the two phenomena possible. In the two most prominent classes of materials, the Chevrel phases (REMo6Ss) and the ternary borides (RERh4B4) this weak exchange interaction can be related to the particular cluster-type crystal structure which provides a spatial separation of the two kinds of electrons. A similar, but even more extreme situation is found in the oxide superconductors, REBa2Cu307, where the coupling is practically absent between the RE and the conduction electrons. On the other extreme, compounds with magnetic 3d-atoms have, so far, turned out to have too strong exchange interactions, so that scattering effects destroy superconductivity in compounds able to develop long-range order. Furthermore, the low exchange interaction in the RE compounds has made it possible to have a regular lattice of magnetic moments, avoiding thereby clustering effects and short-range magnetic order.
Antiferromagnetic superconductors A large number of compounds where antiferromagnetism coexists with superconductivity has now been found, and there is clearly no fundamental reason that these two phenomena should mutually exclude each other. However, superconductivity is influenced in various ways by the magnetic order. This is most readily seen experimentally in anomalies in the upper critical field. In some compounds, like ErMorSs, NdM06S 8 or TmRh4B4, superconductivity seems to be very little modified by the magnetic order. However, the latter can be seen in the upper critical field as a positive curvature with a tendency to an increase of the critical field below T M. This can be understood as a result of a decrease of the susceptibility, and a corresponding reduction of the polarization in the antiferromagnetic state in the applied field. In other compounds, like DyM06S 8 and NdRh4B4, a rather abrupt change in Hc2 occurs at the magnetic transition. This effect, most likely, seems to result from a change in the effective electron-electron interaction due to the modification of the pairing condition in the antiferromagnetic state, or it may result from a reduction in the phase space available for pairing due to gaps at the Fermi surface, which open up at the antiferromagnetic transition. To which extent these
540
0. FISCHER
two mechanisms are equivalent remains to be fully clarified. One example is known where antiferromagnetism destroys superconductivity completely, i.e., Tm2Fe3Si 5. This is possibly an extreme case of the above effect. The case of SmRh4B 4 is somewhat different from the others. At the antiferromagnetic transition, a reduction in the exchange scattering seems to occur, enhancing superconductivity in the magnetic state. A somewhat surprising fact is that there are nearly no indications of enhanced pairbreaking in the vicinity of the magnetic transition due to fluctuation effects. Only in one case, GdM06Ss, have such effects deafly been seen. The antiferromagnetic transition is apparently only weakly, if at all, influenced by superconductivity. In most cases investigated so far, the magnetic transition occurs below the superconducting transition so it is difficult to observe directly the magnetic order with and without superconductivity. In certain pseudoternary systems, the antiferromagnetic transition occurs above the superconducting transition, but, in these cases, the antiferromagnetic state apparently does not change at T~. However, in some cases the magnetic order is partly or fully of longwavelength oscillatory nature, where one might suspect that the order would have been different (ferromagnetic?) in the absence of superconductivity.
The interplay of ferromagnetism and superconductivity Two compounds, HoM06S 8 and ErRh4B4, and several pseudoternary compounds, have a ferromagnetic transition T M below the superconducting one, Tel, and are re-entrant, i.e., superconductivity is destroyed below a second critical temperature To2. In the two compounds, superconductivity coexists with magnetic order in a narrow temperature interval Tc2 < T < T M. However, the interaction between the two phenomena leads to a long-wavelength oscillatory magnetic state. In some of the pseudoternary systems, an oscillatory order has not yet been observed, although evidence exists for a coexistence region. The nature of this coexistence is not yet clarified, but it could be a coexistence between ferromagnetism and superconductivity in the form of a self-induced vortex state. There has been much discussion in the literature about the origin of the interactions leading to the magnetic order, to the destruction of superconductivity and to the oscillatory state. In principle, both electromagnetic effects and exchange effects can be invoked for the explanatio n . No simple and general answer can be given and, most probably, the answer will be different for the different cases mentioned above and for the different compounds. In HoM06S s there is strong evidence that superconductivity is destroyed by the internal field /xoM whereas in ErRh4B 4 exchange polarization seems to play the dominant role. In fact, one expects that electromagnetic effects will dominate in compounds where He2 is small, the exchange interaction is particularly small and/or the magnetization is small. In other cases, more usually met, the exchange effects are expected to dominate. An especially striking example of the influence of the exchange is seen in the magnetic field-induced superconductivity due to the Jaccarino-Peter effect.
MAGNETIC SUPERCONDUCTORS
541
Another striking property of these systems is the transition towards a type-I superconductor as T approaches T M from above. This occurs because the magnetic penetration depth decreases as the susceptibility increases and at the same time the coherence length increases as the spin polarization increases. Thus, the Ginzburg-Landau parameter K decreases as the magnetic transition is approached. Some compounds, where superconductivity coexists with magnetism down to lowest temperatures, have an oscillatory magnetic state. This is in particular the case for HoMo6Se 8. It appears plausible that this compound is, in fact, a ferromagnet that has been turned into an oscillatory magnetic state by superconductivity, and where the coexistence region with the oscillatory state is stable in the whole temperature range below T M. In other compounds, oscillatory components of the magnetic order have also been observed. However, further investigations are necessary to clarify to which extent this is related to superconductivity.
Concluding remarks The problem of the interplay between magnetism and superconductivity and the question of a possible coexistence of the two phenomena, remained for a long time largely unanswered. However, the last ten years have seen a considerable progress in our understanding of this problem. Although many detailed questions remain to be clarified, today we do understand the basic mechanisms governing the interplay of magnetism and superconductivity in these types of compounds. However, this does not mean that all the aspects of the interrelations between superconductivity and magnetism are understood. Apart from the many challenging questions that remain to be answered for the type of compounds considered here, the different and intriguing problems met in other types of compounds, concerning the competition between the two phenomena within the conduction band and the possible contributions of magnetic interactions to the pairing interaction, are presently of central interest. Questions of this nature, met in compounds like the heavy-fermion compounds, the organic superconductors and particularly in the oxide-superconductors, are still largely open. The advances made in the now more conventional ternary compounds considered in this review have elucidated some of the basic questions concerning the interrelation of the two phenomena and may constitute a basis for future investigations in this field.
Acknowledgements I thank Dr. M. Decroux and Dr. M. Karkut for numerous discussions and for comments on the manuscript.
References Abrikosov, A.A., and L.P. Gorkov, 1961, Sov. Phys.-JETP 12, 1243. Acker, F., L. Schellenberg and H.C. Ku, 1982,
in: Proc. Int. Conf. on Superconductivity in d- and f-band Metals, eds W. Buckel and W. Weber (KFK, Karlsruhe) p. 237.
542
0. FISCHER
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SUBJECT INDEX absorption coefficient, optical 146, 158, 208 absorptive part of off-diagonal conductivity 150 AC susceptibility of ErRh4B4 491 additional pairbreaking parameter 512 AFQ interactions 74 AG curve 470 amorphous R E - C o thin films 173 angled block structures 401 angular phase 428 anharmonic electron-phonon coupling 82 anisotropic bilinear coupling 24, 50, 52, 93 anisotropic corrections to thermal expansion 378, 380 anisotropic ferrimagnets 399 anisotropic magnetoelasticity 79 anisotropy - coefficients 411 constants 401, 402 - corrections 378 - field 405 fields 429 anisotropy, magnetic 427, 492 anistropic bilinear interactions 10 antiferromagnet 273, 476 antiferromagnetic chain 401 antiferromagnetic Invar 265, 266, 269, 291, 317 antiferromagnetic ordering 474 antiferromagnetism 468 antiferroquadrupolar orderings 20, 38 antistructure atoms 355, 385, 387 antisymmetric exchange 401,454 antivortices 533 Arrott plots 342, 345, 355
-
atomic coupling effects on final state configuration 153 atomic coupling schemes 153, 179, 182 atomic wavefunctions 151 atomistic interpretation of the Faraday rotation 147
band structure calculations 151, 230, 478, 482 bilinear interaction 8 birefringence 160 Bloch-wall superconductivity 497 block-angled structures in ferrites 454 blue shift 216 boundary lines in magnetic phase diagrams 404, 406 bulk modulus 290 bulk reflectivity 146 canted phase 428 canting 401, 422 canting angles 400 CEF Hamiltonian 7, 419 CEF parameters 118 Chandrasekhar-Clogston limit 529 Chevrel phases 476, 530 circular polarization 139 circularly polarized wave 139 clean limit of superconductors 494 cluster 370, 372, 476, 478, 482 clustering-range order 476 Co-Elinvar 246 cobalt ions influence on anisotropy 414 coercivity 423
-
-
577
578
SUBJECT INDEX
coherence length 470, 475, 494 collinear magnetic order 400 collinear phase 428 compensation points 447 competing interaction in multisublattice systems 400 competition between exchange and anisotropy 427 complex Faraday effect 144 complex indices of refraction 139, 143 complex phase of interacting light 145 concentration dependence of Curie and N~el temperatures 276, 346 concentration dependence of magnetic moment 277 conduction-electron spin-polarization 154 conductivity tensor 137, 139, 143 configuration in magneto-optical detection 136 configurational entropy 386 contributions to excess linear thermal expansivity 372 corrosion 160 Cotton-Mouton configuration 136 Coulomb interaction 153 coupling - bilinear 24 - quadrupolar 24 critical behaviour 52 critical concentration 469 critical exponent 185, 351, 353 critical field for spin rotation 176, 183, 408 - in superconductors 473, 492, 504 critical magnetization 404, 408 critical phenomena in magnetic phase transitions 421 critical region - due to fluctuations 351 crystal-field effects 470, 481, 483, 485, 492 Hamiltonian 7, 419 splitting 182, 194, 481 studies 538 crystalline electric field Hamiltonian 7, 419 crystallographic structure of magnetic superconductors 477, 482 cyclotron frequency 154 cyclotron resonance frequency 167 -
-
DC resistivity 156 de Cennes factor 469, 480, 483, 486 Debye temperatures 294, 295, 298 degenerate magnetic system 445
demagnetization effects 472, 492, 494 density of states in magnetic superconductors 478 derivatives field derivatives of magnetization 457 diagonal weight of magneto-optical transitions 152 diamagnetic - Faraday effect 148 line 197 line shape 166, 197, 214, 225 dielectric tensor 137, 139 differential susceptibility 457 dilute rare earth systems 38 dipolar interaction 480 direction of easy magnetization changes 69 dirty superconductor 470 discontinuity - of magnetostrictive effects 42L 457 of slope of differential susceptibility 421, 457 domain structures 425 domain wall 423, 460 - 180 degree wall 424 mobility 426 double FOMP 404 Drude term 167 Drude treatment of free electrons 154 dynamic domain-wall structure 426 -
-
-
-
-
easy direction 403 easy-plane 401, 417 effective anisotropy constants 401, 448 effective Griineisen parameter 376, 381 elastic constant 10, 14, 26, 32, 54, 113, 285 elastic energy 10, 26 elctronic (thermal) Grfineisenparameter 341 electral resisitivity 156 electro-reflectance 185 electromagnetic coupling 467 electromagnetic effects 492 electromagnetic interaction 476 electron-electron interactions 339 electron-phonon enhancement 340 electron transfer 478 electron-phonon interaction 327, 337 electronic and magnetic contributions to the coefficient of'thermal expansion 361 electronic Griineisen parameter 329, 367 Elinvar 240, 245-247, 285 energy criterion for domain wall behaviour 425 energy gaps 77 entropy effects 76
S U B J E C T INDEX equatorial magneto-optical Kerr effect 137, 141 equipartition principle 425 europium monochalcogenides 176 EuS 189 exchange - competition with anisotropy 427, 428 constant 187, 480 effect 492 - energy 153 - field 471-473, 529 - force 151 integral 176, 469 interaction 182, 186, 467-469, 474, 483, 538 scattering 480 splitting 152, 184 experimental set-up for determining reflectances 158 experimental set-up for polar Kerr-effect measurements 159 extended Griineisen relations 337 extraordinary FOMP 404 -
-
-
-
579
fluctuations 50, 52, 334, 346, 362, 364 FOMP 399 forced ferromagnetic order 429 forced magnetostriction 359, 371, 375 forced volume magnetostriction 271, 358 free energy description of FOMP 400, 402 Fresnel coefficients 142 l~¥esnel equations 141 Fulde-Ferrel state 494 garnets 441 generalization of Ohm's law 141 giant moment 354, 357, 369, 374 Ginzburg-Landau parameters 492 Gr/ineisen parameter 365, 368~ 370, 377 Griineisen relation 335, 336
-
f-d hybridization 210 Faraday - configuration 136 - effect 142, 145 ellipticity 145 rotation 136 Fermi-liquid model 340 Fermi-surface nesting 514 ferrimagnetic compounds 401 ferrimagnetic order 433 ferriquadrupolar ordering 20 ferromagnet 399, 421 ferromagnetic 372 ferromagnetic clusters 498 ferromagnetic contribution 373 ferromagnetic order 433, 473 ferroquadrupolar ordering 20, 34 field-induced magnetic phase transition 399, 428, 429 figure of merit of magneto-optical performance 231 films of magnetic superconductors 497 final-state coupling 207 final-state effects 183, 220 finite momentum pairing 471 first-order - magnetization process 399 - moment reorientations (FOMR) 401 transition 44, 76, 93, 399, 401, 428, 494 flopside state 74 -
-
Hall effect 139, 156, 214, 224 Hall resistivity 156 hard magnetization direction 430 heat capacity 242 heavy-fermion system 339, 376, 379 Heisenberg-type exchange interaction 24 Heisenberg-type Hamiltonian 7 Hexagonal ferrites 454 high-field susceptibility 271, 273 high magnetic fields 401, 415 high-magnetization state F O M P 432 high-order terms in the free-energy expression of magnetic states 400 high-spin 316 high-spin-low-spin state transition 305,309, 310, 314 hysteresis FOMP 402 hysteresis loops 399 hysteresis of magnetic transitions 423, 496, 498, 510 improper Griineisen parameter 337 incipient antiferromagnet 506 indirect exchange 186 inequivalent minima in the free-energy surface 4O2 inhomogeneity influences on Arrott plots 347 interband transition 150, 151 interference effects 161 intermediate saturated states 429 intermediate saturation 435 intermediate-valent state 196 intermetallic compounds 5, 162, 243, 385, 401, 465 intraband transitions 153 Invar 244, 245, 247, 266, 285, 296, 305, 306
580
SUBJECT INDEX
Invar effect 240 Invar theories 299, 300 irradiation experiments 524 irreversible magnetization process 399 irreversible rotation of magnetization 423 isotropic Heisenberg exchange 176 isotropic magneto-elasticity 79 itinerant spin glass 277, 308, 309 j - j coupling 153, 208 Jaccarino-Peter effect 468, 529, 530 joint spin-polarization 151 Jones matrix formalism 159 K - R transformation 405 Kerr effect 145, 146 Kerr effect engineering 138 Kerr effect for arbitrary orientation Kerr ellipticity 142 Kerr rotation 142 Kondo effect 470, 473 Kondo peak 383 Kramers-Kronig equations 156 integral equations 156, 158 - relation 137
142
-
-
L~S coupling 153, 208 laminar state 496 Landau free-energy expansion 414 Landau-Ginzburg formalism 331 Landau-Ginzburg model 351 Landau-Ginzburg parameter 332 lanthanide contraction 479 lattice instability 82, 85 line shape 148 linear magnetic birefringence 136 linear regime of magnetic phase stability 440 linearly polarized state 496 local-environment theory 310 local moments in a polarized matrix 354 localized magnetic moments 467 long range contribution to magnetic volume 370, 372, 373 long-range magnetic interactions 357 long-wavelength oscillatory magnetic state 495 longitudinal coherence length 348 longitudinal fluctuations 334 longitudinal (= meridional) Kerr effect 137, 141 low-energy extrapolation of conductivity 156 low-spin 316
magnetic - anisotropy 399 - circular birefringence 136 - circular dichroism 136, 145 clusters 363 - correlation length 475 - excitations 77, 78 - field induced superconductivity 468, 516, 529, 530 - Griineisen parameter 340 - impurities 467, 469 - oxides 422 - phase diagram 248, 403 phases 421 pressure 326, 333, 335 red shift 181, 187, 191, 217 - scattering 472 - susceptibility 13, 17 - transition 20, 43, 399 - volume 327, 358, 369, 371,373 magnetization 273, 278 magnetization jumps 399 magnetization measurements 492 magnetization processes 67, 402 magneto-absortion 185 magneto-caloric effect 506 magneto-elastic coefficients 114 magneto-elastic Hamiltonian 8, 25, 123 magneto-optical effects 146 magneto-optical Kerr effect 137 magneto-reflectance 185 magnetostriction 11, 27, 56, 61, 64, 268, 271, 358, 421 magnetostriction constants 32 magnetovolumeparameter 327, 329-331,344, 346, 358, 359, 389 magnitude - of Faraday and Kerr effects 147 - of the Kerr effects 160 of the plasma-edge splitting 155 magnon-mediated electron-electron interaction 476, 514 martensic transistion 90 martensitic-type transition 85 Maxwell equations 140 mean free path 1 470 Meissner effect 470 meridonal Kerr effect 141 metamagnetic state 509 microwave absorption 135 modulated spin arrangement 76 molecular field 440, 441 moment-volume instabilities 301
S U B J E C T INDEX M6ssbauer experiments on magnetic superconductors 488 multi-axial spin structures 72, 74, 95 multi-layer thin films 142, 155 multiple pairbreaking theory 470, 492, 511 multisublattice system 400, 440 n-type InSb 155 nearly-ferromagnetic systems 339 neutron-diffraction on magnetic superconductors 510 neutron scattering on magnetic superconductors 475, 496 non-absorbing magneto-optical material 142 non-linear moment configurations 401 normalized magnetovolume parameter 330 off-diagonal conductivity 150 off-diagonal weight 152 Ohm's law 140 one-ion magneto-elastic Hamiltonian 9 optical functions 139 optical refiectivity 141 orbital critical field 470 orbital pairbreaking 472 ordering 496 antiferromagnetic 474 antiferroquadrupolar 20 ferriquadrupolar 20 - ferroquadrupolar 20 - model for atomic ordering 386 quadrupolar 20 - transverse sine-wave modulated 496 origin of the quadrupolar interactions 117 oscillator strength 151 oscillatory magnetic state 476, 496 oxide superconductors 468, 477 -
-
-
581
perfectly ordered stoichiometric compound 389 permanent magnets 458 permeability 140 perturbation theory 13, 28, 124, 125 phase diagram of magnetic systems 402 phase transitions between magnetic states 421 phenomenology of first-order magnetization processes 402 photoemission studies 524 plasma edge 138, 199 plasma frequency 153 polar Kerr effect 141 polar Kerr ellipticity 137 polar Kerr rotation 137 polarizability 140 polarization clouds 357 polarization effects 471,472, 511 polycrystalline materials - measurements of anisotropy fields by SPD 457 Ports model 400, 421 pressure dependence of Curie (N@el) temperatures 273, 345 pressure effects on magnetic transitions 79, 81 proper Grfineisen parameter 328 proper Grfineisen relation 336 pseudobinary systems 476 pseudoternary systems 468, 516 quadrupolar excitations 78 quadrupolar field-susceptibility 13, 19 quadrupolar Hamiltonian 12 quadrupolar ordering 20, 34, 95 quadrupolar pair-interaction coefficients quadrupolar pair interactions 119 ab initio calculations 120 quadrupolar strain susceptibility 13 quadrupolar transitions 20, 43
116
-
p-f hybridization 215, 225 p-f mixing 222 pair contribution to magnetic volume 370 373 pairbreaking 469 pairbreaking parameter 469, 472 paramagnetic Faraday rotation 148 paramagnetic limit 471 paramagnetic line shape 148, 166, 197, 225 paramagnetic pairbreaking 471 parastriction 56, 113 partial gapping of Fermi surface 514 penetration depth 494, 509 perfectly ordered compounds 392
radial overlap integral 152 P~aman scattering 135 re-entrance 488 re-entrant behaviour of magnetic superconductors 475,483, 488 re-entrant superconductivity 468, 470, 473, 478, 489, 503, 524 RE-oxide superconductors 534 reconstruction of the Arrott plot 355 red shift in magneto-optical transitions 182, 188, 216
582
SUBJECT INDEX
reduced field 404 reduced magnetization 404 reflected intensity 141 reflection 146 relaxation of conduction electrons 153 renormalization group theory 43 resonance-like enhancement of the Kerr effect 155 reverse magnetostriction 19, 61, 63, 64, 108 RKKY interaction 119, 476, 480, 485, 495 rotation of the magnetization 104, 399 SAC, see small-angle canting selection rules 142, 148, 182 self-induced vortex state 495, 514 semiconductor metal transition 168, 200 Senarmont principle 160 SEW model 326, 340, 345 short-range order effects in magnetic superconductors 476 sign convention for magneto-optical transitions 138, 178 sign of the Faraday rotation 148 sign of the Kerr rotation 142 silver - Kerr effect 154 single-ion susceptibilities 13 singular-point detection 457 singularity in field derivative of magnetization 457 size of the Faraday, effect 146 skew-scattering 154 skew-scattering frequency 153 small-angle canting 401 small-angle canting model 447 softening of elastic constants 54 Soleil-Babinet compensator 160 spatial oscillations of order parameters 495 specific heat 292, 296, 298, 475, 485, 486, 498, 506, 509 spin correlation function 187 spin-flip transition 151, 184 spin-flop 401 spin-flop transition 455 spin fluctations 332, 379 spin-glass 372, 475 spin-glass formation 370 spin-polarization effects 512 spin-polarization of states 150, 153 spin-polarized band-structure calculations 151 spin reorientation transistion 400, 429 spin structures 401 spin-wave 280, 362 - corrections 363
damping 283 - linewidth 283 stiffness 280, 283 spin orbit - coupling 148 - energy 149 - interaction 153, 182 - scattering 471, 475, 492 spiral state 496 splitting of the plasma edge 154 spontaneous spin reorientation 429 spontaneous volume magnetostriction 268270, 313 spontaneous vortex state 496 SRT 429 stainless-Invar 245 static strain wave 76 Stoner enhancement 340 Stoner enhancement factor 328 Stoner interaction 327 Stoner model 342 Stoner temperature 328, 346 Stoner-Edwards Wohlfarth model 326 strain susceptibilities 17 structural defects 385 structural phase transition 479, 517 structural transitions 41 structures of magnetically ordered systems 454 sublattices 418, 428 sum rules 157 super-Elinvar 246 super-Invar 245 snperlattice splitting 184 surface impedance 509 susceptibility 441, 476, 480 symmetries 24 symmetrized elastic constants 10, 26 symmetrized Stevens operators 122 -
-
ternary compounds 476 ternary superconductors 468 tetragonal symmetry 24, 414 thermal-conductivity 509 thermal expansion 30, 240, 264, 267 thermal fluctuations 348 thermal Gr/ineisen parameter 340 thermo reflectance 185 thickness of domain walls "460 thin films of magnetic superconductors 492 third-order magnetic susceptibility 14, 17, 57, 113 Th3P4-structure 169 total weight of interband transition 151
SUBJECT INDEX transformation of anisotropy constants into conjugate quantities 411 transitionmetal-rare earthintermetallics 135 transitions from LS to HS states 309, 316 transmission of light 146 transverse - antiphase modulated 497 - (= equatorial) Kerr effect 136, 141 - fluctuations 334 - or linear dichroism 136 - sine wave modulated ordering 496 - susceptibilities 457 tricritical point 492, 521 tricriticality 44 trigonal symmetry 414 tunneling characteristics 499 tunneling experiments 514 two-dimensional order 536 two w-state model 309 two-ion magneto-elastic Hamiltonian 10 two-ion magnetoelasticity 50 two-ion quadrupolar Hamiltonian 8, 25 two-suhlattice model 433, 448 type-1 FOMP 405 type-2 FOMP 405 type-II superconductors 470
583
ultrasonic attenuation in magnetic superconductors 520 uriiaxial magnetic anisotropy 428 uniaxial symmetry 402 upper critical field He2 470 velocity of domain walls 426 very large magneto-optical effects 154 very weak itinerant ferromagnetism (VWIF) 328 Voigt Cotton-Mouton configuration 136 vortex lattice 470 VWIF, see very weak itinerant ferromagnetism
Walker limit 427 wavelength-independent Faraday rotation wavevector 141 weak-coupling model 202 weak itinerant ferromagnet 274, 301 Wilson ratio 329, 340 Wohlfarth correction 364 Zeeman effect 147 Zeeman Hamiltonian
7
189
MATERIALS INDEX
A gR alloys 106, 107 amorphous Gd-Co 195 Au 135 AuCu3-type compounds A_uuEr 108 A uR alloys 107 A__~uRcompounds 107 AuTb 108 AuYb 108
CeX 96 CeZn 46, 84, 86 Co 135, 172, 173, 194 Co Fe 256 Co-Fe-Cr 263 Co-Mn 1-33 Co-Ni 256 CoCr 458 CoFeCr 245 CoMn 265 Cr 76 Cr-Fe 256, 268 Cr-Mn 257, 268 CsCl-type structure compounds C u_uEr 108 CuR-alloys 107
95
Ba3_~SrxZn2Fe24041 454 Bal_xSrxZn2Fe12022 454 Ba2 (Znl-xCox)Fe12022 414
CaCu5-type structure compounds 102 Ce 109, 110, 162 Ce-Chalcogenides 166 CeAg 37, 42, 50, 82, 83, 89, 90 CeAl2 76, 95, 99 CeAs 96, 166 CeB6 38-40, 42, 46,50, 74, 112, 538 CeBi 166 CeCd 85, 88 Cel-xGdxRu2 474 Cel-=HoxRu2 475 C e x L a l - x B i 61 CeMg 46, 92 CeN 163 CeP 96, 166 CePb 3 95, 98 CeS 166 CeSb 94, 165-167 CeSbxTel_= 164 Cel_xTbxRu2 474, 475 CeTe 164~ 166
Dy 76, 106, 109, 162, 203 DyAg 70-73, 85, 89, 90, 92 DyA12 52, 53, 99, 421 DyAs 51, 52, 96 DyBa2Cu30 7 536 DyBi 52, 96 DyCd 70-72, 85, 88 DyCo5 104, 105 Dy2Col7 418 DyCu 70 73, 90-92, 116 DyFe2 102 DyMo6Ss 504, 511 DyNi2 100 DyP 51, 52, 96 DyRh4B 4 483 Dy(Rhl-xIr=)4B4 526 Dy(Rhl-=Rux)4B4 523 DySb 50 52, 54, 55, 93, 94, 116
585
83
586
MATERIALS INDEX
DyVO4 116, 117 DyZn 70-72, 80, 84 86 Er 76, 77, 106, 109, 110, 162 ErAg 63, 85, 89 ErA12 53, 99 ErBa2Cu30 7 536 ErBi 52 ErCd 88 ErCo 5 105 Er2 (Co1_zFe~)17 421 ErCu 90-92 ErFe2 102, 205 Erl_xGdxRh4B 4 521 ErMo6Ss 473, 504, 511 ErMo6Ses 481, 506 ErNi 2 71, 72, 100 ErOsx Sny 488 ErPd2Sn 489, 510 (Erl_~)Rh4B4 518 ErRh4B4 485, 486, 489, 492, 494, 500, 509 Er(Rhl-xRux)4B4 525 ErRhl.lSn3.6 488, 501 ErSb 52, 94 (Erl_xYx)Rht.lSn3. 6 502 ErZn 69, 84, 86, 92 Eu 162 Eu 2+ 205 Eu 2+ diluted in CaF2 190 EuF2 189 Eu monochalcogenides 175 EuO 89, 175, 179, 181-183, 187, 207 Eu203 190 (EuO)l_x (FeO)x 189 EuO films deposited on mirror substrates 190 EuS 175, 178, 179, 183, 189 EuSe 168, 175, 178, 181 183 EuSiO3 191 Eu2SiO4 191 Eu3SiO 5 191 EuxSnl_xMo6S8 530 EuTe 89, 175, 178, 179, 183, 187 fcc Fe 301, 302, 304, 308, 309, 311, 312 Fe 135, 308 Fe-Co-Mn 261 Fe-Mn 251 Fe~Ni 249 Fe-Ni-Co 260 Fe-Ni-Cr 258 Fe-Ni-Mn 259 Fe-Pd 250 Fe-Pt 249
FeCo 276 FeMn 270, 292 Fe60 Mn4o 291 Fe14Nd2B 244 FeNi 241, 242, 245, 246, 248, 273, 276, 287, 293, 308 FeaNi 305, 307, 309 Fe65Ni35 240, 244, 264, 272, 275, 278, 280, 283, 284, 287, 290, 294, 299, 308 Fe68Ni32 314 FeNiCo 245, 276 FeNiCoCr 246 Feso_xNixCr2o 244 FeNiMn 247 Fe5o (NixMnl_x)50 244 FesoNixMnso-x 272, 297 Fe5o (NiMn)5o 293 Fe5oNi35Mn15 294, 295, 297 Fe65Ni~ Mn35-x 264 Fe65(NiMn)35 273, 293 FeO 189 FePd 273 FePt 273 Fe3Pt 283, 284, 289, 305, 307, 310, 311 Fe32 Pt 313 Fe72Pt2s 272, 275, 276, 278, 280, 287, 290, 295, 314 Fe17RE2 205 Fe2 (Zro.7Nbo.3) 281 garnets 135 Gd 162, 192-194, 204 Gd-Co 203, 204 Gd-Fe 195 GdAg 80 GdA12 79, 80 GdAs 79, 96 GdBa2Cu306 535 GdBi 79, 93, 96 GdCd 88 GdCo2 194, 195 GdCo3 194, 195 GdCo5 194 Gd2Co17 194 Gd2Col4B 196 Gd compounds 79 GdCu 90, 91 Gdo.26Feo.74 196 GdFe~ 79, 205 Gd2Fe14B 196 (Gdo.26 Feo.74)o.s9 Bio. 11 196 Gd ions diluted in LaSb and LaBi GdMg 92 GdMo6Ss 504, 512
108
MATERIALS INDEX GdMo6Se s 506 G d P 96 GdRh4B4 483 GdS 96 GdSb 79, 94 GdSe 96 GdTe 96 GdX pnictides 52 (Gdl_~Yx)Rh4B4 521 GdZn 79, 80, 86 Ge-Si or InSb 135 Heusler-type compounds 488 Ho 76, 106, 109, 162 HoAg 85, 89 HoA12 99, 100 HoAs 95, 96 HoBa2 C u 3 0 7 535, 537 HoBi 95, 96 HoCd 88 HoCo2 I00 HoCo5 104, 105 Ho2Co17 453 HoCu 91 Hol_xEuxMo6Ss 517 HoFe 2 102 Ho2Fe17 453 HoMo6Ss 478, 482, 483, 489, 494, 500 HoMo6Se s 481, 507 HoNi2 100 HoNi0.5Cu0.5 76 HoP 74, 75, 96 HoPd2 Sn 511 HoRh4B4 483, 485, 486, 509 Ho(Rhl_xIr~)4B 4 525 Ho(Rhl_xOsx)4B4 524 H o ( R h l - x R u x ) 4 B 4 524 HoSb 52, 94, 116 HoZn 70, 84, 86 LaAg 82, 83, 89, 90 L a A g x I n l - x 82 LaAgTb 110 LaA12Tb 110 LaBiCe 62-64 LaBiGd 111 LaBiR series 108, 110 Lal_xCexMo6Ss 518 La2CuO4 534 Lal_xEuxMo6Ss 530 LaFe2+Fe3+ O19 422 Lal_xGdxAl2 470 La3_xGdxIn 473 Lal_xLuxMo6Ss 480
587
LaMn2Ge2 230 LaMn2Si 2 230 LaMo6Ss 479 LaMo6Se s 477 LaMo6Xs 479 La3S 4 171 LaSb 94 LaSbCe 62-64, 108 LaSbDy 108 LaSbGd 111 LaSbR 114 LaSbR compounds 1 0 ~ 1 0 9 LaSbSm 108 Lal.s5Sr0.15CuO 4 534 Laves phase compounds 95 Ln Co films 172 LuCd:Tm 64 (LUl_xErx)RuB2 527 LuMo6Ss 479 LuMo6Xs 479 (LUl_~RE)Rh4B4 483 LuRu4B 4 486 LuZn:Tm 64 MxMo6X 8 477 MgR compounds MnA1 458 MnNi 292 Mns5Ni15 291 MnSi 331, 361
110
NaCl-type structure compounds Nd 109, 162 Nd 3+ 167 N d - A u 173, 175 Nd Au thin films 173 Nd-Co 173, 175 Nd-Fe 173, 175 NdAg 89 NdA12 95, 99 NdB6 46, 112 NdBa2CuaO7 538 NdCd 85, 88 Nd chalcogenides 163 N d x C o l - x 173 NdCo5 104, 105, 458 N d i C u O 4 538 Nd2FelaB 196, 418, 458 Nd metal 171 NdMo6Ss 514 NdN 163, 164 Nd pnictides 163 NdRh4B 4 483, 508 NdS 162, 167, 171, 175, 230
93
588
MATERIALS INDEX
Nd2Sa 168 Nd3S 4 162, 170, 171, 173, 175, 230 NdSb 78, 94 NdZn 73, 84-86 Ni 135 Ni-Co Mn 262 Ni-Mn 1 27 Ni3A1 330, 331,346, 350, 359, 360, 362, 363, 365, 368, 387 Ni74A126 343, 349 Ni75A125 361 NiCo 293 NiCr 273 NiaGa 331,343, 345, 356, 363, 387 Ni75+~Ga25-x 344 Ni3Ga + Fe 354 Nir5_xGa25Fe~ 357 NiMn 273, 293 (Nil_xPdx)raA125 347 NiPt 331, 359, 360, 368 Ni42.9Pt57 A 352 NiUSn 230 noble metals compounds (silver, gold) containing magnetic rare earth 61 PbMo6Ss 477, 479 Pd 331 PdMn 368 Pr 30, 81, 109, 111, 112, 162 PrAg 72, 82, 85, 89, 90 PrA12 95, 99 PrB 4 112 PrB 6 46, 74, 112 PrBa2CuaO6 537 PrCd 85, 88 PrCo5 104, 105, 452, 456, 459 PrCu2 41, 42 Pr2CuO4 538 Pr2(Fe Co)it 456 Pr2Fe~4B 419 Pr0.2Fe0.4C0. 4 172 PrMg 92 PrMg2 49, 98, 100 PrNi2 30 PrNi5 102, 103 PrPb 3 40-42, 59, 60, 95, 98 PrRh4B4 483 PrSb 52, 55, 80, 81, 94 PrZn 64, 84, 86 Pt 135 PtCo 458 RAg 115 RAg compounds
82, 83, 85, 89
RA12 100 RA12 compounds 52, 60, 95, 99 RB 6 76 RB4 compounds 112 RB6 compounds 112 RCd compounds 85, 88 RCo 3 111 RCo 5 79, 103, 172 R2 (Co-Fe)17 415 RCo2 compounds 100 RCo5 compounds 103, 105 RCu compounds 90, 91 (RE)-Fe2 204 (RE)-TM compounds 135 (RE)Ba2CuO7 468 (RE)Ba2Cu307 477 (RE)2Fe17 270 (RE)2Fe14B 270 (RE)2Fe3 Si5 488 (RE)Mo6Ss 468, 476, 478 (RE)Mo6Se s 476, 478 (RE)Pd2Sn 488 (RE)Rh4B4 476, 482 (RE)(Rhl_xCox)4B 4 527 (RE)(Rh0.s5Ruo.15)B 4 486 (RE)Ru4B4 486 RFe2 compounds 64, 100, 102 RGa 3 76 RMg compounds 92 RNi2 compounds 100 RNi 5 compounds 102, 103 RSb compounds 94, 114 RZn 58, 79 RZn compounds 57, 84, 86, 115 rare earth-ferrites 135 Rhodium borides 476, 482, 508 silver(0.5%) rare earth alloys 62 Sm 109, 162 Sm 2+ 168 (Sm-Nd)Co 5 400 SmCo5 105~ 172 Sm2CuO 4 538 (Sml_x)ErxRh4B 4 521 SmFe2 102 SmFellTi 458 SmIn 3 95 SmRh4B4 483, 508, 516 SInS 168 SmSb 94 SmTe 168 (Sml-~Yx)Rh4B4 516, 521 SmZn 84 (Snl_X~Erx)Er4 Rh6 Snls 502
MATERIALS INDEX Snl_xEuxMo6S8 516 SnMo6S 8 477 (Snl-x REx) (1)RE4T6Sn18 Sn(1)RE3T4Sn(2)12 488
488
Tb 106, 109, 110, 162, 203, 204 Tb 3+ 205 Tb-Co 203, 204 Tb-Fe 204 (Tb-Gd)x (Fe-Co)l-z 203 TbAg 89, 90 TbA12 95, 99, 100 TbAs 47, 48, 79, 96 TbB4 113 TbBa2Cu30 7 538 TbBi 48, 79, 96 TbCd 88 TbCo2 100 TbCo5 104, 105, 229 TbCu 90, 91 TbFe2 33, 64, 65, 79, 101, 102, 204, 205 (TbxFel-x)l-yUy 229 ( T b x F e l - x ) y U l - v 225 Tb0.olLa0.99Ag 110 TbxLal_~A12 59 TbMo6Ss 504, 512 TbMo6Se s 481 TbNi2 100 TbP 47, 48, 52, 96 TbRh4B4 483 Tb(Rhl-xIrx)4B4 526 TbS 47, 96 TbSb 47, 48, 79, 94 TbSe 47, 96 TbTe 47, 96 TbVO4 116, 117 TbX 116 TbZn 64-66, 80, 84, 86 ternary silicides 488 ternary stannides 487 Th3As4 222 ThMn12-type compounds 460 Thl_xNdxRu2 475, 503 ThaP4 222 thulium monochalcogenides 162 TiCo 331 TiFe 331 Ti(Fe-Co) 345, 389 TiFe0.5Co0.5 331, 355 Tm 76, 106, 109, 162 TmAg 44, 45, 85, 89, 90 TmA12 99 TmCd 37, 50, 59, 60, 85, 88, 114, 538 TmCo5 105
589
TmCu 44, 45, 55, 56, 68, 76, 90, 91, 114 Tmo.5Euo.5Se 170, 200 TmFe2 102 Tm2Fe3Si5 488, 489, 503 TmGa3 39-42, 73, 92, 95, 98 (Tml_xLux)RuB 2 528 TmxLul_xZn 78 TmMg 92, 114 TmNi5 103 TmPO4 59 TmPd2Sn 489 TmRhaB4 483, 508 TmRhl.3 Sn4 488 TmS 167, 170, 196, 199, 200, 230 TmSb 55, 80, 94, 116 TmSe 167, 170, 196, 199, 200, 230 TmSeo.32Teo.6s 196, 197, 200 TmSeo.sTTeo.13 196 TmTe 96, 196 TmVO4 116, 117 TmX compounds 55, 113, 114, 116 TmZn 33, 40, 42, 50, 55, 57, 66-68, 78, 79, 84-86, 114 U 206 UAs 135, 213, 215, 216 U3As4 222 UAsxSel_x 216 UAs0.7Se0.3 216 UAsSe 222 UCo5+x 225, 229 UCo5.3 229 UCuAs2 227 UCu2As2 222, 227 UCuP2 222, 227 UCu2P 2 222, 227, 229 UFe2 229, 351, 352 UMn2Ge2 222, 230 UMn2Si2 222, 230 UNiAs2 227 UO2 207, 220 UP 206, 213, 215 UaP4 222 UPd 3 42 UPt3 378, 379 US 206, 209, 211, 213, 215, 227 USb 213, 215, 216, 220 USbxTel_x 217, 221 USbo.s5Teo.15 213, 215, 217, 220 USbo.gTeo. 1 217 USe 211, 213, 216 UTe 162, 209, 211, 213, 216 U~YI-xSb 216, 220, 221, 229 Uo.15Yo.s5Sb 220
590 Uo.4Yo.6Sb 220 Uo.TYo.3Sb 220 UbxTea_x 216 Y-Co 203 (Y-Nd)Co5 400 YBa2Cu306 534 YBa2Cu307 534 YCo5 79, 103, 106 Y9 Co7 468 YDy 111 (Yl-xDyx)Pd2Sn 529 YEr 110, 111 (Yl_xErx)Rh4B 4 520 YGd 111 Yl-zGdzCo5 452
MATERIALS INDEX Yl-zGdxOs 474 YI_xMo6Ss 518 YR 110 YI-xRExPd2Sn 488 Y(Rhl-xRuz)B4 522 YTb 110, 111 Yb 109, 162 YbBa2Cu307 537 YbMo6Ss 479 YbPd2Sn 510 YbSe 201 YbTe 201, 202 Ydy 110 Zr(Fel_xAlx)2 458 ZrZn2 331, 361
