Handbook of Magnetic Materials, Volume 9

Handbook of Magnetic Materials, Volume 9 Elsevier, 1995 Edited by: K.H.J. Buschow ISBN: 978-0-444-82232-1 by kmno4 PR...

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Handbook of Magnetic Materials, Volume 9 Elsevier, 1995 Edited by: K.H.J. Buschow ISBN: 978-0-444-82232-1

by kmno4

PREFACE TO VOLUME 9

The Handbook series Magnetic Materials is a continuation of the Handbook series Ferromagnetic Materials. The original aim of Peter Wohlfarth when he started the latter series was to combine new developments in magnetism with the achievements of earlier compilations of monographs, producing a worthy successor to Bozorth's classical and monumental book Ferromagnetism. This is the main reason that Ferromagnetic Materials was initially chosen as title for the Handbook series, although the latter aimed at giving a more complete cross-section of magnetism than Bozorth's book. Magnetism has seen an enormous expansion into a variety of different areas of research in the last few years, comprising the magnetism of several classes of novel materials that share with truly ferromagnetic materials only the presence of magnetic moments. For this reason the Editor and Publisher of this Handbook series have carefully reconsidered the title of the Handbook series and changed it into Magnetic Materials. It is with much pleasure that I can introduce to you now Volume 9 of this Handbook series. The magnetism of the majority of metallic systems can adequately be described by the well known concepts of localised or itinerant moment magnetism. These traditional concepts are, however, not able to describe the magnetism of a fairly large class of materials generally indicated as heavy-fermion systems. The magnetism of these strongly correlated charge-carrier systems has developed from two different sources, the Kondo-impurity concept and the intermediate-valence concept. The last decade has seen a strong proliferation in experimental and theoretical studies of such systems. Progress made in this field by means of inelastic neutron scattering was described already in Chapter 6 of Volume 7 of the Handbook. A more general account of the magnetism of heavy-fermion systems is presented in Chapter 1 of the present Volume. Towers of strengths to the understanding of the physics of magnetism are theory and experiment. In Volume 7 of the Handbook two different chapters were devoted to the former, emphasising results of electronic band structure calculations and their beneficial influence on the understanding of magnetism in many materials. As a counterweight, two novel experimental techniques will be described in the present Volume. The first one, Chapter 2, deals with muon spin rotation, the second one, Chapter 5, gives an account of the possibilities offered by photon beam spectroscopy. In both chapters it is shown how these sophisticated experimental methods can be

vi

PREFACE TO VOLUME 9

used to obtain experimental information not easily obtainable by conventional experimental methods. Interstitially modified intermetallic compounds of rare earth and 3d elements are described in Chapter 3. These materials can be obtained from the pure intermetallics by filling some of the available interstitial hole sites in their crystal structure with carbon, nitrogen or hydrogen atoms. Though the drastic changes of magnetocrystalline anisotropy and magnetic couplings are of substantial fundamental interest, a large part of the Chapter is devoted to practical consequences as found in modern permanent magnet technology. In one of the preceding volumes, Vol. 7, a major updating of the experimental results was presented for intermetallics in which rare earths are combined with 3d transition metals, while progress in ferrite research was presented in Vol. 8. Both groups of materials are fairly extensive, as are the many experimental results that have accumulated over the years. Of particular interest in these two groups of materials is the occurrence of field-induced phase transitions. These phase transitions are commonly treated in a rather phenomenological way, and at best, are described in terms of anisotropy and moment couplings. The last chapter of the present volume deals with the thermodynamic approach and shows how the understanding and description of these magnetic phase transitions can be considerably enriched. Volume 9 of the Handbook on the Properties of Magnetic Materials, as the preceding volumes, has a dual purpose. As a textbook it is intended to be of assistance to those who wish to be introduced to a given topic in the field of magnetism without the need to read the vast amount of literature published. As a work of reference it is intended for scientists active in magnetism research. To this dual purpose, Volume 9 of the Handbook is composed of topical review articles written by leading authorities. In each of these articles an extensive description is given in graphical as well as in tabular form, much emphasis being placed on the discussion of the experimental material in the framework of physics, chemistry and material science. The task to provide the readership with novel trends and achievements in magnetism would have been extremely difficult without the professionalism of the NorthHolland Physics Division of Elsevier Science B.V., and I wish to thank Joep Verheggen and Wim Spaans for their great help and expertise. K.H.J. Buschow Van der Waals-Zeeman Institute University of Amsterdam

CONTENTS Preface to V o l u m e 9 . . . . . . . . . . . . . . . . . . . . Contents .

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Contents o f Volumes 1-8

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v

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ix

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xi

1. H e a v y F e r m i o n s and R e l a t e d C o m p o u n d s G.J. N I E U W E N H U Y S . . . . . . . . . . . . . . . . .

1

2. M a g n e t i c M a t e r i a l s Studied b y M u o n Spin Rotation S p e c t r o s c o p y A. S C H E N C K and E N . G Y G A X . . . . . . . . . . . . . .

57

3. Interstitially M o d i f i e d Intermetallics o f Rare Earth and 3d Elements H. F U J I I and H. S U N . . . . . . . . . . . . . . . . . .

303

4. F i e l d I n d u c e d P h a s e Transitions in F e r r i m a g n e t s A.K. ZVEZDIN . . . . . . . . . . . . . . . . . . . .

405

5. Photon B e a m Studies o f M a g n e t i c Materials S.W. L O V E S E Y . . . . . . . . . . . . . . . . . . .

545

Author Index

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631

Subject Index

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679

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689

List o f Contributors

Materials Index

vii

CONTENTS OF VOLUMES 1-8 Volume 1 1. 2. 3. 4. 5. 6. 7.

Iron, Cobalt and Nickel, by E.P. Woh!farth . . . . . . Dilute Transition Metal Alloys: Spin Glasses, by J.A. Mydosh Rare Earth Metals and Alloys, by S. Legvold . . . . . . Rare Earth Compounds, by K. H.J. Buschow . . . . . . Actinide Elements and Compounds, by W. Trzebiatowski . . Amorphous Ferromagnets, by E E . Luborsky . . . . . . Magnetostrictive Rare Earth-Fe2 Compounds, by A. E. Clark

. . . . . . . . . .

1

and G.J. Nieuwenhuys .

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71 183 297 415 451 531

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

Ferromagnetic Insulators: Garnets, by M.A. Gilleo . . . . . . . . . . . . . Soft Magnetic Metallic Materials, by G. Y Chin and J. 14. Wernick . . . . . . . . Ferrites for Non-Microwave Applications, by P L Slick . . . . . . . . . . . Microwave Ferrites, by J. Nicolas . . . . . . . . . . . . . . . . . . . Crystalline Films for Bubbles, by A.H. Eschenfelder . . . . . . . . . . . . Amorphous Films for Bubbles, by A.H. Eschenfelder . . . . . . . . . . . . Recording Materials, by G. Bate . . . . . . . . . . . . . . . . . . . Ferromagnetic Liquids, by S. W. Charles and J. Popplewell . . . . . . . . . .

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1 .

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55 189

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243

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297 345 381 509

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Volume 3 1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry and Technology, by U. Enz . . . . . . . . . . . . . . . . . . . . . . 2. Permanent Magnets; Theory, by 11. Zifistra . . . . . . . . . . . . . . . . 3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R.A. McCurrie . . 4. Oxide Spinels, by S. Krupi&a and P Novdk . . . . . . . . . . . . . . . . 5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite Structure, by 14. Kojima

6. 7. 8. 9.

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Properties of Ferroxplana-Type Hexagonal Ferrites, by M. Sugimoto Hard Ferrites and Plastoferrites, by H. St~blein . . . . . . . . Sulphospinels, by R. P van S t @ d e . . . . . . . . . . . . Transport Properties of Ferromagnets, by I.A. Campbell and A. Fert

ix

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37 107 189

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305 393

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441

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603 747

x

CONTENTS OF VOLUMES 1-8

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. Sechovsky and L. Havela . . . . . . . Magneto-optical Properties of Alloys and Intermetallic Compounds, by K. H.J. Buschow

1

131 211 309 493

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

1

133 237 323 397 465

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

2. 3. 4. 5. 6.

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

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

by A. Szytula and L. Lundgren . . . . . . . M.E. McHenry . . . . . . . .

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85 181 289 453 511

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

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3. Density Functional Theory of the Ground State Magnetic Properties of Rare Earths and Actinides, by 34. S. S. Brooks and B. Johansson . . . . . . . . . . . . . . . 4. Diluted Magnetic Semiconductors, by J. Kossut and W. Dobrowolski . . . . . . . . 5. Magnetic Properties of Binary Rare-earth 3d-transition-metal Intermetallic Compounds, by J.J.M. Franse and R.J. Radwarlski

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

1

97 139 231 307

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

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503

Volume 8 1. Magnetism in Artificial Metallic Superlattices of Rare Earth Metals, by J.J. Rhyne and R. W. Erwin

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1

2. Thermal Expansion Anomalies and Spontaneous Magnetostriction in Rare-Earth Intermetallics with Cobalt and Iron, by A. V. Andreev . . . . . . . . . . . . . . . . . . 59 3. Progress in Spinel Ferrite Research, by V.A.M. Brabers . . . . . . . . . . . . 189 4. Anisotropy in Iron-Based Soft Magnetic Materials, by M. Soinski and A.J. Moses 325 5. Magnetic Properties of Rare Earth-Cu 2 Compounds, by Nguyen Hoang Luong and J. J.M. Franse . . . . . . . . . . . . . . . . . . . . . . . . . . 415

chapter 1 HEAVY FERMIONS AND RELATED COMPOUNDS

G.J. NIEUWENHUYS Kamerlingh Onnes Laborato~ Leiden University RO. Box 9506, 2300 RA Leiden The Netherlands

Handbook of Magnetic Materials, Vol. 9 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved

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

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

1.1.

S c o p e o f this c h a p t e r

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1.2.

The picture

1.3.

Archetypal heavy fermions

1.4.

The terms

1.5.

Other reviews

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3 3 3 5 6

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10

2. E x p e r i m e n t a l results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1.

122-compounds

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10

2.2.

111-compounds

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21

2.3.

U2T2X-compounds

2.4.

U-compounds with CaCu5 structure and related borides .........................

27

2.5.

334-compounds

34

2.6.

Gaps

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

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

26

36

2.7.

Magnetism and superconductivity ...........................................

39

2.8.

Miscellaneous compounds

40

2.9.

Yb-compounds

2.10. T h i n films 3. C o n c l u s i o n

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

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42 44 45 45 45

1. Introduction

1.1. Scope of this chapter As always, this review has to be restricted for many reasons, few of them being the interest and knowledge of the author. In this chapter some magnetic properties of heavy fermions and related compounds will be emphasized and presented mainly in tabular form. For other aspects of these very interesting new materials the reader will be referred to other reviews at the end of this introduction.

1.2. The picture Heavy fermions have been intensively studied during the last decade. The name 'heavy fermion' stems from the fact that these materials exhibit an anomalous specific heat. At temperatures much lower than the Debye temperature (and much lower then the Fermi temperature) the specific heat of a conductor can be described by Cp = 3`T +/3T 3,

(1)

where 3`T is the so-called linear term due to the excitations of the itinerant electrons and ¢3T3 is the low temperature approximation of the specific heat of the lattice. For normal conductors, 3' is of order 1 ... 10 mJ/(moleK2). However, a new class of materials - the heavy fermions - showed 3"-values up to 1000 mI/(moleK2). The expression for 3' in terms of the effective mass of the itinerant electrons reads

3" = (m*kF/Trzhz)(k2/3),

(2)

where kF is the Fermi wavevector and hB is Boltzmann's constant. The large value for 3' is then ascribed to a large value for ra*, hence the name heavy fermion. On the other hand, if the specific heat, Cv(T), is described in terms of the number of possible excitations at temperature T, Cp(T) c< nhB(T/To), then the characteristic temperature, To, is estimated at 10 to 100 K, much lower than the Fermi temperature of ordinary metals. Heavy fermions are typically found amoung the Ce and U containing compounds. The magnetic susceptibility of these compounds follows a Curie-Weiss law at high temperatures, with an effective moment approximately given by Hund's rule, but at low temperatures the magnetic susceptibility flattens off (sometimes after attaining a maximum) and becomes constant with decreasing temperature. This constant, T --+ 0, value is much larger than the Pauli susceptibility

4

G.J. NIEUWENHUYS

of normal metals. In fact, for those heavy fermions where no magnetic ordering is found, the enhancement of the magnetic susceptibility for T --+ 0 is as large as that of the specific heat. The crystal structure of most of the heavy fermions is tetragonal or hexagonal. A large anisotropy in the magnetic susceptibility is generally observed, e.g., in tetragonal URuzSi2 the magnetic susceptibility measured with the magnetic field directed along the c-axis shows the features mentioned above, with the field along the a-axis no temperature dependence at all is found for T ~< 100 K. The same is roughly true for hexagonal UPdzA13, however with a change in the role of the a- and c-axis. In both cases the susceptibility can be very well described by a simple crystalline electric field, CEF, model. A direct consequence of the large contribution of the itinerant electrons to the (linear term) of the specific heat is that at a temperature of only a few Kelvin the entropy gain of the itinerant electrons is already R ln(2), R being the gas constant, a value only expected for localized magnetic moments. Therefore, in spite of the linear term in the specific heat the question arises whether the f-electrons of the Ce or the U should be described in an itinerant model. Moreover, some heavy fermions are superconducting and the jump in the specific heat at Tc equals about 3'To indicating that the heavy electrons are involved in the superconductivity. Still the large entropy gain at low temperatures has to be explained as well, within the same model for heavy fermions. A priori it is not clear whether charge or spin degrees of freedom cause the entropy (Kagan et al. 1992). The same ambiguity between itinerancy and localization governs the role of the crystalline electric fields, CEF. Whereas inelastic neutron scattering can observe CEF levels in a number of Ce based compounds, the non-observation in U-based compounds is almost a rule rather than an exception. On the other hand, as mentioned above the strong magnetic anisotropy can easily be explained by CEF as is the case also for a number of other macroscopic properties such as specific heat and magnetic susceptibility as functions of temperature and external magnetic field. The electrical resistance of heavy fermions is anomalous too. At low temperatures it can be described by

p(T) = p(O) + A T 2,

(3)

where A was experimentally found by Kadowaki and Woods (1986) to be 10-53 '2 #~cm/(moleKZ/mJ)2, and where 3' is the coefficient of the linear term in the specific heat. In other words, A is (ra*/m) 2 larger than in normal metals. On increasing the temperature a maximum is attained followed by a logarithmic decrease towards higher temperatures. A number of different ground states has been found in heavy fermions. Some stay paramagnetic down to the lowest temperatures (20 mK), others order antiferromagnetically (ferromagnetism is seldom found), and a number of superconductors are observed. Finally, even a semiconducting ground state is possible. The transition temperature for the superconductivity is rather low, ~< 2 K, and the superconductivity coexists with antiferromagnetism in almost all cases (UBe13 seems to be the exception amoung the superconducting heavy fermions as it is with respect to its crystal structure, being cubic). The type of superconducting order parameter is not known with great certainty, but is not expected to be a simple

HEAVY FERMIONS AND RELATED COMPOUNDS

5

one. UNiaA13 and UPd2A13 have rather normal spontaneous moments as measured by neutron scattering ( ~ I#B) and the ordering is of long range. URu2Si2 has an ordered moment of only 0.02#B and the range of order parameter is restricted to only about 100 A. UPt3 and CeCu2Si2 show only short range correlations of small moments. These magnetic correlations have been observed, e.g., with magnetic Xray scattering (Isaacs et al. 1989, Mason et al. 1990 and Isaacs et al. 1994). In the case of CeCu2Si2, there is even growing evidence that the superconductivity and the antiferromagnetic order live in different parts of the sample, Feyerherm et al. (1995) and Luke et al. (1994). Another general feature of heavy fermions is the large sample dependence of the magnetic and superconducting properties. Low concentration replacements by other atoms or slight departures from the exact stoichiometry induce large effects. For example, small changes in the Cu content in CeCu2Si2 can alter the volume fractions mentioned above from zero to one hundred percent. Heat treatments can change the properties also in strange ways, without any evidence for a change the crystallographic structure. Neutron diffraction experiments (F5k et al. 1995) have shown that annealing URu2Si2 changes the temperature dependence of the ordered moment drastically without changing the Ndel temperature or the zerotemperature value of the ordered moment. This strong dependence on the samples indicates that heavy fermions live close to an instability. It makes experimental research in this field rather difficult and supports clearly the statement that the quality of the experimental results ~ the quality of the sample. The need for good sample analysis is evident. Maybe we are slowly reaching the point where we cannot improve our single crystalline samples, since starting materials have limited purity and thermodynamics tells us that the minimum of the free energy at final temperature (normal annealing temperature) is obtained for a finite number of imperfections.

1.3. Archetypal heavy fermions TABLE 1 Main features of archetypal heavy fermions. Compound

Tc (K)

TN (K)

"7 X(0) A (mJ/moleK2) (10-9m3/mole) (Iz~ cm/K2)

Crystal- Ref. structure

CeCuzSi2 UBel3 UPt3 URuzSi2 UPdzA13 U2Znl7 UCdl 1 CeA13 CeCu6 YbCu4Ag UA12

0.65 0.9 0.5 1.5 2.0 -

5 17 14 9.7 5 -

1000 1100 450 180 150 500 840 1620 1300 245 142

tetrag. cubic hexag. tetrag. hexag. rhomb. cubic hexag. orthor. cubic cubic

98 172 95 67 110 545 468 263 224 342 53

35 10.7 0.06

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

6

G.J.N~UWENHUYS

References: [1] [2] [3] [4] [5] [6]

Bredlet al. (1983) Ott et al. (1983) Stewart et al. (1984) Palstra et al. (1985) Geibel et al. (1991a) Ott et al. (1984)

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

Fisket al. (1984) Andreset al. (1975) Fujitaet al. (1985) Rosselet al. (1987) Stewart (1984)

1.4. The terms This section bears on the very well readable reviews by Lee et al. (1986), and by Bauer (1991).

1.4.1. Single impurity Since f-electron ions play the major role in heavy fermions and Anderson (1961) has invented a simple, nontrivial, successful and well understood model for an single impurity in a metal, this should be the starting point for a theoretical description. In its simplest form the three ingredients are the conduction electrons, an f-ion with a single orbital and a hybridization matrix element Vfk that couples this orbital with the conduction band. The Hamiltonian then reads

H:

+ ~-~ 6(k)nks q- 6f ~ nfs ~- Unfsnf$ + ~ (Wfkf+Ck~+ VfkekJ~), ks s ks

(4)

where U is the Coulomb correlation energy associated with double occupancy of the f orbital. The width, F, of the f level is given by ~rN(ez)lVfkFI2, where N(eF) is the single-spin density of states at the Fermi level. Ce has Nf degenerate orbitals with maximum total occupancy, nf of 1, since U is estimated to be 5 eV. The f orbitals have an energy of a few eV below the Fermi level. For Uranium the relevant configurations are f2 (angular momentum, J = 4) and f3 ( j = 9/2), The correlation energy is estimated to be 2 eV. The hybridization has often been taken as an adjustable, constant, parameter, but ab initio band structure calculations have been carried out, e.g., by Sheng et al. (1990). In order to obtain absolute information on this parameter Endstra et al. (1993a) have used a simple model to estimate Vek to explain trends in the magnetic ordering temperatures. The energy of the f level, cf, with respect to the chemical potential and the bandwidth, F, govern the equilibrium occupancy of the f levels, thus non-integer occupancies may occur, leading to socalled valence fluctuations. Additionally, because of these fluctuations, f electrons may exchange spin components with the conduction band, without an actual charge transfer. In that case one speaks of spin fluctuations. Crystalline electric fields, CEF, effects tend to lower the degeneracy, on the other hand, the hybridization can renormalize the strength of the CEF downwards. During the last decade many exact results have been obtained for the degenerate Anderson model taking into account the effects due to crystalline electric fields, spin-orbit coupling and external magnetic fields (Schlottmann 1989).

HEAVY FERMIONS AND RELATED COMPOUNDS

7

1.4.2. Kondo Under strict conditions (nondegeneracy and integer occupation) the Hamiltonian in eq. (4) can be transformed (Schrieffer and Wolff 1966) into H K = -23S.s(0),

(5)

which is exactly the Hamiltonian that Kondo (1964) used to describe the on site interaction of an S -- 1/2 single impurity interacting with the electron density s(0). Since 3 is negative, this interaction favours antiparallel alignment of the conduction electron spins with respect to the impurity spin at low temperatures, finally leading to a singlet state at T = 0. This moment compensation leads to a result for the magnetic susceptibility which agrees with the observations in heavy fermions. The compensation or screening also leads to a term in the electrical resistivity ~ ln(TK/T) in the vicinity of the Kondo temperature, TK. In the limit for U --+ oo the interaction 3 = - v Z / @ f - EF), where Vfk is replaced by a constant V. As a consequence of the singlet ground state of the impurity, the low-temperature properties of the system can be described within a Fermi liquid picture. In the degenerate case, the transformation from eq. (1) to eq. (2) can also be made for integer occupation (no charge fluctuations) and then leads to the Coqblin-Schrieffer model (Coqblin and Schrieffer 1969). Heavy fermions, consisting of a regular lattice of Ce or U atoms, are called Kondo lattices, but one should bear in mind that the Kondo Hamiltonian can only be obtained exactly in the single impurity case with integer occupation.

1.4.3. The compounds In a heavy fermion compound, we are dealing with a regular array of magnetic 'impurities'. In that case the interactions between the Ce or U ions cannot be neglected in general; the problem has acquired a translation symmetry and charge transfer between the f states and the conduction band can no longer be considered as a minor perturbation. The hybridization causes on the one hand the screening as in the single impurity model, but at the other hand induces an interaction between the magnetic moments mediated by the conduction electrons. The screening picture itself has to be reconsidered, since there are not enough conduction available for all magnetic moments. Charge transfer now renormalizes the chemical potential when a finite number of f electrons is promoted into the conduction band. As this process is temperature dependent, the chemical potential becomes temperature dependent, because of the conservation of the total number of electrons. On the one hand, this notion of the 'lattice' aspect of the Kondo problem has led to an item like correlation, which denotes the subtle antiferromagnetic correlations between the - partly - screened magnetic moments thought to cause the rapid decrease in the electrical resistance with decreasing temperature observed in heavy fermion compounds even without long range magnetic order. On the other hand, Strong and Millis (1994) showed that most properties (e.g., specific heat, magnetic susceptibility) of the archetypel heavy fermions CeA13 and CeCu6 c a n be beautifully explaned using crystal field and Kondo effects of single atoms only. The exception is the electrical resistance, where the agreement between calculation and experiment is less satisfactory. Ab initio calculations of the size of the magnetic moments and of the transition temperatures of

8

G.J. NIEUWENHUYS

magnetically ordering compounds have been carried out by Cooper et al. (Cooper 1992; Hu and Cooper 1993). In their calculation scheme, they have treated both hybridization and Coulomb exchange effects simultaneously in the presence of interconfigurational correlation effects. In this way, they are able to successfully compute the magnetic moments and transition temperatures of a number of U- and Ce-based compounds adopting the NaC1 structure. Band-structure calculations (Norman and Koelling 1992) are not able to correctly predict these magnetic properties because they include only those aspects of the valence fluctuations that can be captured by time averaging. Therefore, the true character of the 4f or 5f states is missed and has to be included by adding an additional on-site scattering term (Fulde et al. 1988) or interconfigurational correlation effects. Due to computational limits such calculations have not been carried out for the compounds we will discuss in this chapter. We will therefore limit ourselves to rough estimates of the interactions involved and use that as a guidline through the experimental results. To lowest order the strength of the conduction electron mediated interaction between the magnetic atoms, the RKKY interaction, is proportional to 32, and thus proportional to V 4. As long as this interaction does not dominate, the screening process is still effective and the physical properties of the concentrated systems can roughly be considered as a summation of the single impurity effects. The most apparent exception is seen in the electrical resistance of stoichiometric compounds, where coherent excitations of the f electrons become important. In the single impurity limit the electrical resistance increases as ln(TK/T) with decreasing temperature and saturates as T --+ 0. Coherency causes the resistance to decrease again for T << TK, which leads to the AT 2 term mentioned earlier. When the RKKY interaction dominates, the system will order magnetically, although sometimes with very small ordered moments. The competition between the 'screening', with a characteristic strength TK oc (1/N(cv) ) exp ( - 1/N(eF)3),

(6)

and the RKKY interaction with a characteristic strength TRKKY ~

32N(eF),

(7)

has lead to a phase diagram for the present type of compounds (Doniach 1977, Brandt and Moshchalkov 1984) depicted schematically in fig. 1. This phase diagram draws the boundaries between the magnetically ordering compounds, the 'Kondo-lattices' and the paramagnetic systems. Although it does not provide quantitative information for three dimensional systems, it has been shown by Endstra et al. (1993a) that it serves very well as a guidline for experimentalists to order the compounds within a given series based on a rude estimate of the hybridization parameter. Recognizing that :J ~x V 2 / ( e f

-

eF)

(8)

and assuming that: (i) the hybridization is mainly due to the d-itinerant electrons, so that 3 can be replaced by 3dr and (ii) that ef - eF is nearly constant over a given

HEAVY FERMIONS AND RELATED COMPOUNDS

9

2.0

1.5

i1.0 f /r.

PARAMAGNET e

0.5

0,5 d/W~

KONDO METAL

1.0

Fig. 1. The one-dimensional Kondo-necklace phase diagram as derived by Doniach (1977).

series of analogous compounds, 3 can by assumed to be proportional to V~. Using an approach put forward by Harrison and Straub (Harrison 1969, 1980, 1983, 1987, and Straub and Harrison 1985) the hybridization matrix elements can be written as ggl, m =(771l,mj~2/~'~e)[(T21--1~'2/'-l)1/2/dl+l'+1]

(9)

The input parameters are the atomic radii of the respective atoms (rl and rv), the interatomic distance d, the angular momenta l, l' (l = 0, 1, 2, 3 for s, p, d, f orbitals), and the symmetry of the bond m. Note that me in eq. (9) is the electron mass. r/u,,~ is given by (Harrison and Straub 1987) (-1) z'+l (l + 1')!(2l)!(2l')! ?]ll'm -- - -

6rr

× (-1)~ [

2z+Z' l!l, !

X

( 2 / + 1)(2/' + 1)

I 1/2

(1 + m ) ! ( l -- m ) ! ( l ' + m ) ! ( l ' -- m ) !

with m = 0, 1, 2, 3 for ~r, re, 5 and ~o bounds. This simple approach has enabled Endstra et al. (1993a) to order compounds of several series and to predict the behaviour of a number of pseudo binary compounds. This picture makes it clear why some compounds do order magnetically, while others don't. The typical heavy fermion compounds have to be placed near the right-hand side of the phase diagram, which partly explains their sensitivity for the exact composition.

10

G.J. NIEUWENHUYS

1.5. Other reviews Numerous reviews on several aspects of the heavy fermion phenomena have appeared in the literature. Reviews with emphasis on the development of the theory can be found in Lee et al. (1986), Gor'kov (1987), Fulde et al. (1988), Schlottmann (1989), Grewe et al. (1990) and Hewson (1993). An experimental review of the superconductivity is given in Fisk et al. (1986), while Fisk et al. (1988) deal mainly with the Fermi-liquid character. Other experimental reviews are: Stewart (1984), Ott (1987), Ott and Fisk (1987), Sechovsky and Havela (1988) (actinide compounds), Sigrist and Ueda (1991) (unconventional superconductivity), Bauer (1991) (CeCu and YbCu compounds), Sarma et al. (1992) (sound experiments), Aeppli and Fisk (1992) (Kondo insulators), Endstra et al. (1993b) (122-compounds), Schenck (1993) and Schenck and Gygax (this volume) (#+SR), Hess et al. (1993), Loewenhaupt and Fisher (1994) (neutron scattering), Knetsch et al. (1994) (URuzSi2), Sauls (1994) (phase diagram of UPt3), and Ott (1994) (superconductivity).

2. Experimental results The number of papers on experimental results on heavy fermion materials and related compounds amounts to a few thousands. Only in the last few years our data base grew with 2000 titles. Ineventable, a selection had to be made for this review paper. The consistency and readability of this chapter were the only criteria used. In view of the large number of papers, the idea about a world-wide data base for storing experimental data and there interpretation should be given some serious though, bearing in mind that the technical infrastructure is available. 2.1. 122-compounds There has been an extensive research on the MT2X 2 compounds, with M a magnetic (rare-earth or actinide) ion, T a transition metal and X being Si, Ge, Sn or Sb. Hundreds of compounds exist which makes these series very suitable for comparison along the periodic table. Almost all compounds crystallize in a tetragonal structure with a e/a ratio of order of 2.3. However, two different allotropic derivatives exist: The body-centered ThCrzSi2 structure (I4/mmm) and the primitive CaBe2Ge2 one (P4/nmm). These structures are depicted in fig. 2. Note the structures differ in the sequence of the atomic planes, being M-X-T-X-M-X-T-X-M for the bodycentered cell and M-T-X-T-M-X-T-X-M for the primitive cell. Since the extra X-ray diffraction lines from the primitive structure can be rather weak, it is not always clear from the literature which structure is the correct one for a given compound. Some compounds (e.g., UCo2Ge2) can be stabilized in either of the two structures. The magnetic properties appear to be strongly different in those cases. A general rule about the electrical resistivity, p, has been found by Steeman et al. (1990b). Compounds which crystallize in the ThCrzSi2 structure have Pt[c < P±c, while it is the other around for the compounds with the CaBezGe2 structure. Up to now the only exception is UNizSi2 (Ning et al. 1990). The family of 122-compounds has

HEAVY FERMIONS AND RELATEDCOMPOUNDS (a)

11

(b)

T I_ _T I

(~M ©T oX Fig. 2. A schematic drawing of (a) the ThCrzSi2 and CaBezGe2 crystal structure. The origin of the CaBe~Ge2 unit cell is shifted by (3/4, 3/4, --ZM)to facilitate a comparison with the ThCrzSi2 unit cell. M denote the rare-earth atoms, T are the transition-metal atoms and X the metalloids Si, Ge, Sn, etc. (after Endstra 1992). two famous members: CeCu2Si2, being the first heavy fermion materials found to become superconducting (Steglich et al. 1979), and URu2Si2 found to be an antiferromagnetically ordering (TN = 17.5 K) superconductor with moderately heavy electrons (Palstra et al. 1985). Extensive investigations have been devoted to these two compounds since the discovery of their peculiar properties. In both cases the physics is not straightforward and a large sample dependence has been found. In the case of CeCuaSi2 superconductivity can only be found if the material is melted with an excess amount of Cu (up to 10%). However, e.g., microprobe analysis shows the compound to be of the correct stoichiometric composition. The superconductivity is remarkable in the sense that the temperature derivative of the upper critical field is enhanced as much as the effective mass of the electrons and that the 'jump' in the specific heat at Tc divided by the now enhanced 3' is still of order 2. This means that the heavy electrons (being quasi particles) attain the superconducting ground state. Bearing in mind that the large effective mass of the quasi particles also means that the Fermi temperature, TF, is exceptionally low (~ 100 K), superconductivity is found now in a compound where TF is much smaller then the Debye temperature and that Tc is of order 1% of TF. Note that a similar situation exists in the high temperature superconductors. Recently (Feyerherm et al. 1995 and Luke et al. 1994) magnetic order of a spin glass type has been found in CeCu2Si2. However,

12

G.J. NIEUWENHUYS

depending on the starting composition, the volume fraction of the magnetically ordering part of the sample and the volume fraction of the superconducting part are both a function of temperature, in such a way that the total is 100%. Therefore, superconductivity and magnetic order do not coexist in CeCu2Si2 on a microscopic scale. We mention 'starting composition' on purpose, since microprobe analyse is not able to distinguish between the samples obtained. The other famous member of the family, URu2Si2, shows antiferromagnetic order at TN = 17.5 K, as evidenced by, e.g., specific heat (Palstra et al. 1985) and neutron diffraction (Broholm et al. 1987 and 1991). The latter experiments also showed that the magnetic order persists into the superconducting state below Tc = 1.3 K, and that the magnetic order must be of a peculiar type, since the magnetic moment in the ordered state is only 0.03#B per U atom. The smallness of the ordered moment seems not to be of any consequence, neither for the size of the anomaly in the specific heat at TN (being large), nor on the dependence of the magnetic properties in large external fields. For example, a transition to ferromagnetic alignment has been observed (de Boer et al. 1986) at B = 36 T, which would point, together with TN = 17.5 K, to a normal value of the ordered moment. A number of attempts has been made the explain the magnetic behaviour of URu2Si2 on the basis of CEF effects. The first one (Nieuwenhuys 1987) could describe the magnetic anisotropy and the general features of the specific heat and the magnetic susceptibility, but certainly not the small ordered moment, the last one (Santini and Amoretti 1994) ascribes the main features as being due to quadrupolar ordering and considers the staggered magnetic moment as a weak secondary effect. This, however, should assume that the quadrupolar ordering is not observable via neutron diffraction, since the detailed experiments by Walker et al. (1993) did not reveal any other order parameter than the magnetization along the crystallographic c-axis. A further remarkable effect is that the electrical resistivity first increases while cooling below TN, indicating the formation of a charge gap. We will come back to this point later. Pressure has an anisotropic effect on the properties of URu2Si2. Bakker et al. (1992) found for the magnetism dTN/dPa = 126 mK/kbar and dTN/dPc = -41 mK/kbar. The pressure dependence of the superconductivity can be described by dTc/dPa = - 3 5 mK/kbar and dTc/dPc = 25 mK/kbar. A hydrostatic high pressure experiment revealed a value of 36 K for TN at 80 kbar (Kagayama et al. 1994). The magnetic ordering is, in spite of the very small ordered moment, rather stable as a function of sample preparation. This in contrast to the superconductivity, where a large sample dependence can be observed (Ramirez et al. 1991). The large sample dependence has lead to conclusions pointing towards a double transition to superconductivity as observed in UPt3, but, e.g., the detailed specific heat investigation by Knetsch et al. (1992) has refuted this idea. Superconductivity in URu2Si2 is also not of a simple type. The power-law behaviour of the specific heat below Tc already indicates that the energy gap due to superconductivity does not exists over the whole Fermi-surface. Measurements of the penetration depth (Knetsch et al. 1992) support this conclusion. Very remarkable are the results of #+SR experiments (Luke et al. 1990): for fields along the c-axis a reasonable penetration depth (about 8000 A) is observed, however for fields directed perpendicular to the c-axis no sign of superconductivity, neither in the Knight shift

HEAVY FERMIONS AND RELATED COMPOUNDS

13

nor in the relaxation rate can be discerned. That all in spite of the observation by Knetsch et al. (1992) that Bcl is isotropic. For details on of the ~+SR results, the reader is referred to the review by Schenck and Gygax in this volume, more detailed information on URu2Si2 has been collected by Knetsch et al. (1994). The tetragonal character of the structure of the 122-series almost immediately leads to anisotropic magnetic behaviour. The investigation of single crystals is therefore sometimes a condition for the research. The materials can be grown into single crystal form via the tri-arc method (Menovsky et al. 1983). Quite a number of these systems show magnetic ordering (mainly antiferromagnetic) 1. The exact valency of the magnetic atom is of importance, particularly for Ce, since Ce 3+ has one 4f-electron and thus a magnetic moment and C e 4+ is nonmagnetic. In the CeTzX2 series the valency attains a maximum of 3.18 for T = Co and X = Si (Neifeld et al. 1985). The valency approaches 3 going to 4d and 5d transition metals and/or to X = Ge or Sn. Since magnetic C e 3+ is a Kramers ion with J = 5/2, the J-multiplet is split up by the crystalline electric field into 3 doublets. The energy difference between the two lowest doublets can be remarkably small (10-100 K) which is of the same order as the magnetic ordering temperature or the characteristic energy of the Kondo effect (TK). CEF effects in U based compounds are less clear, but they certainly play a role, e.g., in determining the magnetic anisotropy. For U, the ground state can be a non-magnetic singlet, assuming that U is in 4+ valence state.

2.1.1. CeTzX2 In table 2 the main structural and magnetic properties of the Ce based 122-compounds are lis(ed. In this table (and in the following tables) the structure is denoted by I for the inversion symmetric ThCr2Si2 tetragonal type and P for the primitive CaBezGe2 one. The lattice parameters are in ,~. The value of the linear specific heat coefficient, 7, is given in mJ/mole K: only if it is not disturbed by magnetic ordering and/or CEF effects. The magnetic ordering is denoted by A for antiferromagnetism, F for ferroor ferrimagnetism, C for complex ordering and G for spin-glass type ordering. The temperatures, TM and T~: (TM being the magnetic ordering temperature), and the energy of the first excited state in the CEF level scheme A 1 divided by Boltzmann's constant are in Kelvin. The crystal structure of CePt2Si2 is a unique type for REPt2Si2 compounds and has the P4/mm space group, and CePt2Ge2 crystallizes in a monoclinic deformation of the CaBe2Ge2 with space group P21, b -- 4.402 and fi = 90.83 °. CelrzSi2 can crystallize in two different structures, depending on the heat treatment (see discussion below on UCo2Ge2). CeNizSn2 adopts a structure which is slightly monoclinically distorted with respect to the CaBezGe2 structure (Liang et al. 1990). Most Kondo temperatures and crystalline electric field splittings have been obtained from the detailed neutron research by Severing et al. (1989a and b) and Loidl et al. (1992), others from have been estimated on the basis of bulk measurements such as specific heat and magnetic susceptibility. CeCuzGe: exhibits 1 We will not consider compounds with Mn, since Mn is the only transition metal carrying a magnetic moment in these type of compounds.

14

G.J. NIEUWENHUYS

TABLE 2 Structural and magnetic properties of CeT2X 2. The structure is denoted by I for the inversion symmetric ThCr2Si2 tetragonal type and P for the primitive CaBe2Ge 2 one. The lattice parameters are in ,~. The specific heat coefficient, % is given in mJ/(mole) K 2 only if it is not disturbed by magnetic ordering and/or CEF erects. The magnetic ordering is denoted by A for antiferromagnetism, F for ferro- or ferrimagnetism, C for complex ordering and G for spin-glass type ordering. The temperatures, T M and TK (TM being the magnetic ordering temperature), and the energy of the first excited state in the CEF level scheme A 1 divided by Boltzmann's constant are in Kelvin. Compound

Struc. a

c

7

Magn.

TM

TK

A1

Ref.

CeAg2Ge 2 CeAgzSi 2 CeAlzGa2 CeAuzGe2

I I I I I I P I I P I I P P I P I P I P I I I P * * P I I P I I

10.93 10.66 10.95 10.41 10.20 10.24 9.78 10.26 10.16 9.93 10.35 10.48 9.88 10.10 9.85 10.16 10.04 9.85 9.97 9.57 10.11 9.85 10.06 9.89 10.48 9.80 9.80 10.39 10.45 10.18 10.56 10.00 9.80

80 100 1000 350 400 600 89 -

A A F A A A P C A G A P P P P P A P P P A P A A A A P A A A A F P

6.3 9.5 9 15 10.1 29.5 7.7 4.15 0.8 2.1 4.1 1.8 5.1 10 0.5 2.2 0.88 14 36 0.47 7.91 -

3 2 6 10 2 28 30 7 10 2 2 60 1 33 2 15

0 100 127 200 171 20 190 145 150 3 17 110 230 194 29 500 220

[1-4] [5-8] [3] [2, 4] [5-10] [11] [5, 6] [12] [4, 13-16] [7, 9, 17-20] [21-23] [24] [25] [26] [27] [27] [22, 23] [28] [12, 21] [6] [21-23, 29] [27, 30] [31, 32] [7-9, 27, 31, 33] [22, 23] [34, 35] [10, 27, 36-39] [22, 23, 40, 41] [42--44] [8, 9, 27, 45] [22, 23] [2, 4, 31, 46, 47] [9, 27, 37, 48]

CeAu2Si 2

CeCozGe2 CeCozSi 2 CeCul.3 Sb2 CeCuzGe2 CeCu2Si2 CeCuzSn2 CeFe2Ge2 CeFe2Si2 CeIrzGe2 CeIrzSi 2 CelrzSi z CeIr2Sn 2 CeNizGe2 CeNi2Sb2 CeNi2Si2 CeNizSn 2 CeOs2Si2 CePdzGe2 CePd2Si 2 CePdzSn2 CePtzGe 2 CePtzSi 2 CePt2Sn2 CeRh2Ge2 CeRh2Si2 CeRh2Sn2 CeRu2Ge2 CeRu2Si2

4.280 4.250 4.203 4.367 4.310 4.099 3.953 4.341 4.140 4.105 4.433 4.070 3.995 4.246 4.143 4.085 4.499 4.150 4.416 4.036 4.409 4.162 4.369 4.237 4.554 4.400 4.252 4.581 4.161 4.089 4.472 4.256 4.197

References: [1] [2] [3] [4] [5] [6] [7] [8] [9]

Gignoux et al. (1988) B6hm et al. (1988) Rauchschwalbe et al. (1985) Loidl et al. (1992) Palstra (1986) Palstra et al. (1986a) Severing et al (1989a) Grier et al. (1984) Severing et al. (1989b)

[10] [li] [12] [13] [14] [15] [16] [17] [18]

Heeb et al. (1991) Fujii et al. (1988) Mentink et al. (1994) de Boer et al. (1987) Knopp et al. (1989) Jaccard et al. (1992a) Jaccard et al. (1992b) Bredl et al. (1983) Luke et al. (1994)

HEAVY FERMIONS AND RELATED COMPOUNDS [19] Jarlborg et al. (1983) [20] Feyerherm et al. (1995) [21] Kaczmarska et al. (1993) [22] Beyermann et al. (1991) [23] Selsane et al. (1990) [24] Felner et al. (1975) [25] Rogl (1984) [26] Francois et al. (1985) [27] Hiebl et al. (1986) [28] Knopp et al. (1988) [29] Takabatake et al. (1990) [30] Horvath et al. (1983) [31] Besnus et al. (1992) [32] Rossi et al. (1979) [33] Steeman et al. (1988)

15

[34] Sampathkumaranet al. (1991) [35] Das et al. (1991) [36] Ayache et al. (1987) [37] Gignoux et al. (1991) [38] Bhattacharjee et al. (1989) [39] Hiebl et al. (1985) [40] Mignot et al. (1993) [41] Shigeoka et al. (1992) [42] Thompson et al. (1994) [43] Venturini et al. 1988a [44] Venturini et al. (1998b) [45] Quezel et al. (1984) [46] Besnus et al. (1991) [47] Godart et al. (1987) [48] Regnault et al. (1988)

an accidental quartet as the excited state, while CeAg2Ge2 attains such a quartet as ground state. The calculated ordered moment obtained from CEF splittings is in excellent agreement with the measured values for Si and Ge compounds, except for CeCu2Ge2. This indicates that the ground state is not much influenced by the Kondo effect, i.e. A1 is much larger than TK, as shown in the table. However, for the Sn compounds these energies are of comparable magnitude. The ratio of the Kondo temperature and the CEF-splitting can therefore be used to classify these magnetic compounds. Small CEF splittings can easily lead to incorrect determinations of the linear specific heat contribution, e.g., CeNi2Sn2 with 7 K and CeNi2Sb2 with 3.2 K, have maxima in the low temperature specific heat at about 3 or 1.5 K, respectively, which could appear as an enormous "~ when C/T is plotted versus T over a restricted temperature range. CeCul.3Sb2 is a remarkable compound in the sense that it has the correct crystal structure, but that part of the Cu sites are not occupied. The magnetism in this compound is rather complex including field cooling effects, suggestive for spin-glass or some random ordering (Mentink 1994a). Polycrystalline CePt2Sn2 does show the magnetic ordering at TN = 0.47 K, however, a single crystalline sample exhibited at m a x i m u m only short range order (Mignot et al. 1993). We will come back to this point in connection with a detailed studied example: CePd2A13. Using eqs (8-10) and the structure parameters given in table 2 and in the references mentioned therein, an estimate can be given for the strength of the hybridization and thus of the magnetic ordering temperature, see section 1.4.3. Doing so, the CeT2Si2 and CeT2Ge2 can be placed in a phase diagram, as has been done by Endstra et al. (1993a) and displayed in figs 3 and 4. As can be seen from these figures, the Ce-based alloys can be classified within this magnetic phase-diagram. The Sn and Sb compounds have a smaller hybridization then their Si and Ge counterparts. This then would suggest that these compounds should be found at the left-hand side of the phase diagram, and thus order magnetically as well localized systems. However, it appears that the ordering temperatures are much lower or even zero, but at the same time these compounds have a considerable electronic contribution to the specific heat, which would point to the right-hand side of the diagram. On the other hand, for all transition metals there is a clear trend when going from Si --+ Ge --+ Sn and Sb, which suggest that a renormalization of the T/W axis would fit the Sn and

16

G.J. NIEUWENHUYS

CeT2Si 2

I--

Rh

Ru P t

Ir Lr

Ir~

Os

J._,/W Fig. 3. Schematic phase diagram for the Kondo lattice with magnetic ordering temperature of the

CeT2Si2 compounds (after Endstra et al. 1993a).

CeT2G %

I.-

Rh

ge

rid_/W

Fig. 4. Schematic phase diagram for the Kondo lattice with magnetic ordering temperature of the CeT2Ge2 compounds (after Endstra et al. 1993a).

Sb compounds without much difficulty. The fact that almost all above mentioned materials crystallize in a similar crystal structure implies that probably also pseudo binary mixtures can be made, which then opens the possibility to tune parameters like the Kondo temperature and/or the hybridization via small changes in the lattice parameters and the density of the d-electrons. For example, very rich phase diagrams have been found for Ce(Cul_~Ni=)2Ge2 by Steglich et al. (1990a, b), while Heeb et al. (1991) have shown that CePtAuSi2 forms a heavy fermion system between the local moment antiferromagnet CeAu2Si2 and the typical Kondo-type compound CePt2Si2. Another beautiful experiment was carried out by Jaccard et al. (1992a and b) on CeCu2Ge2, which is on the right-hand side of the phase diagram in fig. 4. By applying pressure the volume was decreased, thus the hybridization increased, pushing the compound towards the right. Indeed the magnetic ordering temperature decreased, a heavy fermion region was entered leading finally to superconductivity. Other pressure experiments were performed by Thompson et al. (1986) on CeM2Si2,

HEAVY FERMIONS AND RELATED COMPOUNDS

17

M = Ag, Au, Pd and Rh, where TN was determined as a function of the volume of the unit cell. Again these early experimental result are in good agreement with the phase diagram shown above.

2.1.2. CeCuX3 As an intermezzo, we will discuss here the properties of two new 113-compounds, which almost the same structure as the 122. CeCuA13 and CeCuGa3 have been prepared and investigated by Mentink et al. (1993a). CeCuA13 crystallizes in a disordered variant of the ThCr2Si2 structure, with a = 4.256 ,~ and e = 10.633 A, while CeCuGa~ adopts the atommically ordered primitive tetragonal BaNiSn3 structure (space group I4/mm), depicted in fig. 5, with lattice parameters a = 4.266 ,~ and c = 10.434 A. CeCuA13 orders antiferromagnetically at 3 K, while CeCuGa3 stays paramagnetic down to 0.4 K in spite of the fact the former one has a site disordered structure and the latter a atommically ordered lattice. From magnetic susceptibility and specific heat measurements an estimate can be given for the crystalline electric field splitting, being 13.1 K for the A1 compound and 3.7 K for the other one. The linear terms in the specific heat are 25 and 72 mJ/(moleK2), respectively. Based on the difference in the magnitude of the magnetisation a [ + 1/2) ground state is inferred for CeCuGa3 and a I + 3/2) ground state for CeCuA13. 2.1.3. UT2X2 Magnetism in UT2X2 have been studied quite extensively, partly because interesting phenomena are present, but of course also because of the presence the heavy fermion antiferromagnetic superconductor URu2Si2 in this series. Except for the so-called low-temperature phase of UIr2Ge2, again all compounds crystallize in a tetragonal

@M,

0 T,

•

Q X

Fig. 5. The BaNiSn3-type crystal structure (after Mentink et al. 1993).

18

G.J. NIEUWENHUYS

structure. The structural and main magnetic properties are given in table 3 based on the review paper by Endstra et al. (1993b). In those cases where neutron diffraction has been carried out, it was generally found that the ordered magnetic moment is directed along the crystallographic c-axis. The antiferromagnetic ordered state in general consists of ferromagnetic magnetic planes (being the crystallographic basal planes) stacked antiferromagnetically along the c-axis. This stacking is not always of the simple + - + - type, some compounds even change their stacking as a function of temperature, e.g., UNi2Si2 (Lin et al. 1991). This ge.neral anisotropy can be understood if we assume that the U ions are in the 4+ state. If we may apply Hund's rule, then the magnetic quantum number J equals 4, which leads to the possibility of non-magnetic singlet states. Konno (1993) has given algebraic expressions for the eigenvalues and eigenstates for the J = 4 case in a tetragonal surrounding. Assuming the CEF picture to be completely valid, the magnetism of the tetragonal U-based compounds is of the induced type and since the transition-matrix elements between the low-lying eigenstates are only non-zero for the Jz operator, the moments are confined to the c-axis (Nieuwenhuys 1987). Although this simple picture does explain major features, it should be noted that the calculated magnitude of the ordered moment is generally too large. Also, inelastic neutron scattering experiments have great difficulty to reveal the CEF levels, which of course might be due to a large broadening by hybridization. Nevertheless, the crystalline electric field plays a important role in determining the anisotropy of these compounds. In a number of cases, table 3 denotes the magnetic ordering as C, meaning complex, implying that more then one magnetic structure has been found depending on temperature. For example, UCu2Ge2 orders ferrromagnetically at 100 K and changes its structure to a + - - + type of antiferromagnetic order below 50 K. UCo2Ge2 and UIr2Ge2 have been mentioned twice in table 3 because of the different crystal structures. UCo2Ge2 forms at the melting point in the CaBe2Ge2 structure (P) and transforms at annealing at 1300 K into the ThCr2Si2 structure (I), thereby increasing its c-axis considerably. Details of this transformation have been studied by Drost (1995) via in situ neutron diffraction experiments. Note that this structural transformation is accompanied by a large change in the magnetism, the I structure orders antiferromagnetically at 175 K, while the P structure is a paramagnet down to 35 inK. UIr2Ge2 also adopts the P structure at its melting point, but transforms upon annealin~ into orthorhombic structure (space group Pmmm) with a = 4.054 A, b = 4.195 A and c = 10.25 A. Here the Nrel temperature increases. There are not many UT2Ge2 compounds with the 4d transition metal series. Only Rh and Pd exist, for Ru a compound U4RuTGe6 has been found. This is a ferromagnet with Tc = 6.8 K (Mentink et al. 1991, Endstra et al. 1992b). The magnetic properties of the superconducting compound URu2Si2 have already been mentioned above. As in the case of the Ce-based compounds, the trends of the magnetic ordering temperatures (including the absence of ordering) can be described in a simple model based on a calculation of the hybridization strength (Endstra et al. 1993a). In fig. 6 we display their results for the Si compounds, while fig. 7 shows the Ge compounds. As can be seen from these figures, the model indeed explains the trends, also, e.g., the absence of magnetic ordering in the P-phase of UCo2Ge2. Ab initio self-consistent density-functional band structure calculations in

HEAVY FERMIONS AND RELATED COMPOUNDS

19

TABLE 3 Structural and magnetic properties of UT2X 2. The structure is denoted by I for the inversion symmetric ThCr2Si2 tetragonal type and P for the primitive CaBe2Ge 2 one. The lattice parameters are in ,~. The specific heat coefficient, "3', is given in mJ/(mole K 2) only if it is not disturbed by magnetic ordering and/or CEF effects. The magnetic ordering is denoted by A for antiferromagnetism, F for fen'o- or ferrimagnetism, C for complex ordering and G for spin-glass type ordering. The temperatures, TM and TK (TM being the magnetic ordering temperature), and the energy of the first excited state in the CEF level scheme A1 divided by Boltzmann's constant are in Kelvin. Compound

Struc. a

e

3'

Magn. TM

TK

A1

Ref.

UAu2Si 2 UCo2Ge 2 UCozGe2 UCo2P2

I I P P I I I I I I P * P I I I I I P P I ? I I

10.29 9.88 9.30 8.97 9.61 10.50 10.23 9.95 9.96 9.53 9.77 10.25 9.83 9.48 9.51 9.68 10.23 10.19 9.75 9.69 9.76 10.06 9.59

62 24 18 105 22 32 305 180

C A P C A A C F P P A A A A C P A A A A P P A A

-

50

[1-4] [5, 6] [7] [8, 9] [3, 10] [3, 11] [7, 12, 13] [4, 10, 11] [7, 12, 14] [14, 15] [16] [16] [3, 17] [7, 10, 12] [4, 18, 19] [2] [20] [2, 20] [21] [22] [2] [23] [2, 20] [24]

UCozSi 2 UCr2Si2 UCu2Ge2 UCu2Si2 UFe2Ge2 UFezSi2 UIr2Ge2 UIrzGe~ UlrzSi 2 UNizGe2 UNizSi2 UOszSi 2 UPdzGe 2 UPd2Si 2 UPt2Ge2 UPtzSi 2 URezSi 2 URhzGe 2 URhzSi 2 URu2Si2

4.280 4.010 4.043 3.955 3.917 3.911 4.063 3.984 4.024 3.951 4.156 4.087 4.095 3.958 4.121 4.200 4.121 4.330 4.197 4.154 4.012 4.128

48 174 230 85 30 100 104 19 33 4.9 77 124 140 97 72 35 130 17.5

References: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Rebelsky et al. (1991) Palstra et al. (1986a) Buschow and de Mooij (1986) Torikachvili et al. (1992) Kuznietz et al. (1989) Endstra et al. (1991) Dirkmaat et al. (1990b) Reehuis et al. (1991) Trod et al. (1993) Chelmieki et al. (1985) Hiebl et al. (1990) Endstra et al. (1990)

the local approximation

[13] Kuznietz et al. (1990) [14] Szytula et al. (1988a) [15] Szytula et al. (1988b) [16] Lloret et al. (1987) [17] Dirkmaat et al. (1990a) [18] Lin et al. (1991) [19] Ning et al. (1991) [20] Ptasiewicz-Bak et al. (1981) [21] Endstra et al. (1992) [22] Steeman et al. (1990) [23] Dirkmaat et al. (1990c) [241 Palstra et al. (1985)

( t r e a t i n g t h e U 5 f s t a t e s as b a n d s t a t e s ) b y S a n d r a t s k i i a n d

Ktibler (1994) on the UT2Si2 compounds

correctly predict the occurrence

of mag-

n e t i s m i n t h e s e m a t e r i a l s . U R u 2 S i 2 f o r m s i n d e e d a n e x c e p t i o n i n t h e s e n s e t h a t it is p r ed i c t e d as n o n - magnetic,

whereas the experiment shows a very small magnetic mo-

20

G,J. NIEUWENHUYS tI

1"

T K iI

£--

tI

iI I

T~,Ky

i /

UT2Si 2

J

/ / I/ I/ H g

O

Ir

OS

jd_f/W

Fig. 6. Schematic phase diagram for the Kondo lattice together with magnetic ordering temperatures of the UT2Si 2 compounds. The dashed curves indicate the Kondo and the RKKY temperature. The thick curve indicates the effective magnetic ordering temperature (after Endstra et al. 1993a).

UT2Ge 2

P

C o ~r

j

Fe coRh

Jo_/W

Fig. 7. Schematic phase diagram for the Kondo lattice together with magnetic ordering temperatures of the UT2Gez compounds (after Endstra et al. 1993a).

ment. In the compound UCo2P2, the Co atoms also have a small magnetic moment so that this compound does not fit into the total picture. Another P containing compound is UCuzP2, which crystallizes in the hexagonal CaA12Si2 structure (Zolnierek et al. 1987) was reported to be a ferromagnetic with the record breaking Tc of 216 K (Kaczorowski and Tro6 1990). Magneto-optical spectroscopy has been carried out on this compound by Schoenes et al. (1989) and Fumagalli et al. (1988). They found a max~imum Kerr rotation of 3 ° and could characterize UCu2P2 as a semi-metal. Polarized neutron diffraction experiments (Delapalme et al. 1994) suggest that U is in the 3+ state in this compound and that the U 6d-electrons play an important role. In the course of the search for new boron-carbide superconductors, two new U compound have been found by Takabatake et al. (1994b): UNi2B2C and URh2B2C. Both crystallize in the ThCrzSi2 structure with a = 3.513 ,~, c -- 10.54 ,~ and a -- 3.782 ,~, c = 10.214 ,~, respectively. UNizBzC is a antiferromagnetic with the remarkable

HEAVY FERMIONS AND RELATED COMPOUNDS 200

U(C°'-xNix;2Ge2

150

\

- 1

21

0.28

t 0.26

¢

0.24

>

100 50 0

. . . . . . . . . . 0.0 0.5

1

0.22

1.0

x Fig. 8. Magnetic ordering temperature (n) and calculated Vdf (o) versus x for U(Col_~Ni~)2Ge2 compounds (after Endstra et al. 1993c).

high TN of 218 K and URh2B2C orders ferromagnetically at also a relatively high temperature of 185 K. The similarity between the crystal structures and the lattice parameters enables, as mentioned before, the study of pseudobinary compounds. This tuning of the various parameters makes comparison with Endstra's model more challenging. The U(Col_xNix)2Ge2 system has been studied by Endstra (1992) and Kuznietz et al. (1990b). The U(Col_xCux)2Ge2 system has been a subject of another paper by Kuznietz et al. (1990a), while the compounds U(Nil_zCux)2Ge2 were reported by Kuznietz et al. (1992a, b). Finally, the U(Co1_~Fe~)2Gez compounds has been investigated by van Rossum (1993). The U(Col_zNix)zGe2 is particularly interesting, since looking at fig. 7, one would expect the transition temperature to vary monotonously between the parent compounds. However, a minimum was found, which could be explained easily by the occurrence of a maximum in the hybridization strength (both parent compounds are on the right-hand side of the phase diagram). Figure 8 shows this result.

2.2. l l l-compounds 2.2.1. CeTX compounds The CeTX compounds crystallize in a large variety of structures, resulting in a broad spectrum of magnetic properties. It is therefore rather difficult to describe a general line. However, one thing is clear, there is always a relation between the volume of the unit cell and the valency of Ce. As pointed out above, the latter is of essential importance for the magnetic behaviour. Crystalline electric field effects do play a important role, they can be observed via inelastic neutron scattering. No extremely small splittings between the ground state and the first excited state have been found in this series. Remarkably, the influence of the CEF is mentioned always in the literature, but it is seldom taken into account, e.g., when analyzing the temperature dependence of the electrical resistance. Table 4 gives the main structural, magnetic en electrical properties of the CeTX compounds. The entries have their usual meaning; under the head 'conduc.' there is an M for metallic behaviour and an S for semiconducting behaviour. The latter may occur only at low temperatures.

22

G.J. NIEUWENHUYS

TABLE 4 Structural and magnetic properties of Ce 111-compounds. 'Crys.' denotes the symmetry of the lattice, 'Struc.' the structure type. The lattice parameters are given in ,~. Under the heading 'Magn.' an A means antiferromagnetism and F ferromagnetism. TM is the magnetic transition temperature in Kelvin. An M under 'Conduc.' means metallic and S semiconducting. Compound

Crys. Struc.

a

b

c

CeA1Ga CeAuIn CeCuGe CeCuSi CeCuSn CeIrGe CeNiAI CeNiGe CeNiIn CeNiSb CeNiSi CeNiSn CePdAI CePdGa CePdGe CePdln CePdSb CePdSn CePtGa CePtGe CePtIn CePtSb CePtSi CePtSn CeRhGe CeRhIn CeRhSb

hex hex hex hex hex o.rh hex o.rh hex hex tetra o.rh hex o.rh o.rh hex hex o.rh o.rh o.rh hex hex tetra o.rh o.rh hex o.rh

4.378 7.698 4.311 4.238 4.583 7.073 6.978 4.317 7.552 4.384 4.061 7.522 7.221 4.488 7.709 4.935 7.535 4.45l 7.657 4.533 4.202 7.463 7.430 7.547 4.609

4.374 7.388 4.555 7.300 4.700 7.344 4.628 4.466 7.416

4.329 A 4.256 A 7.933 A 7.988 A 7.865 A 7.575 4.020 7.201 3.986 4.110 F 14.0457.6024.233 A A 7.676 A 4.078 A 7.890 F 7.955 A A 7.616 A 4.069 8.058 F 14.4848.016 A 7.120 A 4.050 7.846 -

A1B 2 Fe2P Ni2 NizIn NizIn TiNiSn ZrNiA1 TiNiSn ZrNiA1 A1B2 LaPtSi E-TiNiSi ZrNiA1 TiNiSi CeCu2 ZrNiA1 GaGeLi c-TiNiSi TiNiSi CeCu2 FezP GaGeLi LaPtSi ~-TiNiSi TiNiSi Fe2P CeCu2

Magn. TM 6 5.7 10.2 14.9 8 7 2.7 2.2 3.4 1.7 17 7.5 3.5 3.4 4.5 7.5 9 -

References: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

Fremy et al. (1989) Pleger et al. (1987) Nakotte (1994) Fuming et al. (1991) Sakurai et al. (1993) Rogl. et al. (1989) Kuang et al. (1992) Buschow (1979) Singhal et al. (1993) Lee et al. (1987b) Satoh et al. (1990) Fujii et al. (1987) Fujii et al. (1989) Takabatake et al. (1990b) Pecharski et al. (1983) Skolozdra et al. (1994) Aliev et al. (1990) Takabatake et al. (1990)

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

Nakamura et al. (1991) Koghi et al. (1992) Yamaguchi et al. (1990) Kyogatu et al. (1990) Kalvius et al. (1993) Nohara et al (1993) Hulliger (1993) Schank et al. (1994) Kitazawa et al. (1994) Hovestreydt et al. (1972) Adroja et al. (1994) Rainford et al. (1994) Sakurai et al. (1989) Bruck et al. (1993) Adroja et al. (1991) BrUck et al. (1988) Maeno et al. (1987) Suzuki et al. (1990)

Conduc. Ref. M M M M M M M M M M M S M M M M M M M M M M M M M S

[11 [2] [3, 4] [3, 4]

[3-5] [6]

[3, 7-10] [3, 7] [3, 7, 11-14] [15, 16] [3, 7] [3, 7, 17-24] [25-27] [28-31] [6, 31] [11, 12, 32-37]

[38-40] [19, 20, 30, 31, 41, 42] [31, 43] [6, 31]

[11, 44] [45] [10, 46--49] [21, 31, 50, 51]

[6] [32, 52] [53-55]

HEAVY FERMIONS AND RELATEDCOMPOUNDS [37] Fujii et al. (1990) [38] Malik and Adroja (1991b) [39] Trovarelli et al. (1994) [40] Riedi et al. (1994) [41] Adroja et al. (1992) [42] Fujita et al. (1992) [43] Uwatoko et aL (1995) [44] Fujita et al. (1988) [45] Rainford and Adroja (1994) [46] Lee et al. (1987a)

23

[47] Krimmel et al. (1992) [48] Ott et al. (1992) [49] Mielke et al. (1993) [50] Adroja et al. (1988) [51] Takabatake et al. (1993) [52] Adroja et al. (1989) [53] Malik and Adroja (1991a) [54] Takabatake et al. (1994) [55] Nishigori et al. (1994)

Two compounds show the S for the conductance, CeNiSn and CeRhSb. The first one shows semiconducting behaviour for temperatures below 6 K, signalled by a rapid increase in the electrical resistance and a sharp drop in the specific heat. The size of the corresponding energy gap is only 2.4, 5.5 and 5 K for the current along the a-, b- and e-axis, respectively (Takabatake et al. 1990b). Also, nuclear magnetic resonance (Kyogatu et al. 1990) reveals the gap via a more rapid decrease of the relaxation rate, l/T1, as function of temperature. The Knight shift follows the magnetic susceptibility, exhibiting a maximum at 12 K. The experimental results could be described assuming that the structure of the gap is that of a pseudo gap, i.e. the density of states varies linearly as function of energy in stead of making a jump. Preliminary vacuum tunneling experiments (Aarts and Volodin 1995) confirm this picture. Within the CeNiX compounds, CeNiSn is situated at the end of a monotonic gradual trend, starting with A1, towards integer valency. CeNiA1 is a typical mixed valence compound, while CeNiSn for temperatures above 6 K should be characterized as a medium heavy fermion with 7 = 173 mJ/(moleK2). CeNiSn does not show magnetic ordering. If Ni is replaced partly by Pd, as done by Kasaya et al. (1991), then magnetic ordering starts at CeNi0.8Pd0.zSn; CePdSn is a ferromagnet with the same crystal structure as CeNiSn. CeRhSb shows the semiconducting behaviour at temperatures below 20 K, again a pseudo gap is found with a width of about 4 K. It should be noted, that in both cases, CeNiSn and CeRhSb, the corresponding nonmagnetic compound (with La in stead of Ce) has a normal metallic behaviour. We will come back to this point later. Also, 20% Ce replaced by La in CeRhSb recovers the metallic conductance. CePdSb is a ferromagnet with Tc = 17 K as found from magnetic measurements, which is a remarkable high temperature for Ce compounds in this series. The corresponding Nd, Gd and Sm compounds have ordering temperatures of similar values. Ce is here in the trivalent state, the CEF excitations are found at 240 and 560 K, respectively. The absolute value for the electrical resistance is quite high, 4 m~2cm. The specific heat, however, shows no sharp feature at Tc, only a swallow maximum around 10 K has been found. No clear explanation has been given yet. CePtIn shows no magnetic order. In contrast the electronic contribution to the specific heat has a coefficient of 1100 mJ/(mole K 2) at T = 0 K due to an upturn with decreasing temperature starting at 0.5 K. This would characterize this compound as extremely heavy. However, an external magnetic field of 5 T suppresses this -,/ value down to 500 mJ/(mole K2). The overall temperature dependence can be described by a Kondo contribution with TK = 11 K and a CEF splitting of the lowest two levels of A = 80 K. CePdIn,

24

G.J. N1EUWENHUYS

with 7 = 700 mJ/(moleK2), orders ferromagnetically at Tc = 1.7 K and CeRhln has an unstable moment. Mixtures of these two latter compounds show a gradual change of the Ce valency, accompanied by a related change of the volume of the unit cell (Brtick et al. 1993). Going from Rh towards Pd the hybridization and the mass enhancement (7) increase with a maximum for 3' at 800 mJ/(mole K 2) just before the ordering sets in at 80% Pd. Replacing Pd by Pt (Fujita et al. 1992) causes the magnetic order to disappear. CePtIn shows no magnetic order down to 60 mK. CePtSb and CeA1Ga both order antiferromagnetically at 4.5 and 6 K, respectively. In order to fit the data for the crystalline electric field, the addition of a 043 term is necessary. For CePtSb this could be related via detailed neutron diffraction to the GaGeLi structure, with is an atomically ordered version of the Cain2 structure. Probably, this implies that CeA1Ga is also not exactly in the A1B2 structure. CePtSi can be characterized as a heavy fermion system: specific heat measurements have shown a upturn in C/T starting at 10 K and resulting in a zero temperature extrapolation of about 700 mJ/(mole K2). In this case, an external magnetic field of 13 Tesla does not change essentially the results, ruling out short range magnetic order as a cause of the specific heat increase. Via replacing Pt by Ni, Lee et al. (1987b) have shown the gradual change from a heavy fermion system to a typical mixed valence behaviour. The magnetic order in case of antiferromagnetism is generally not of a simple type. For example, CePtSn (Sakurai et al. 1989) and CeCuSn (Nakotte 1994) show two peaks in the specific heat. Both anomalies are influenced by external magnetic fields in different ways. To explain these type of phenomena, detailed microscopic investigations should be carried out.

2.2.2. UTX-compounds The equiatomic UTX compounds have been studied quit extensively, starting with the work of Dwight (1974), Buschow et al. (1985) and Palstra et al. (1986b, 1987a, b). A review has been given by Sechovsky and Havela (1988) in vol. 4 of this book series and by Sechovsky et al. (1990). Therefore, I will restrict here to some new developments. As is the case with the Ce compounds, different crystal structures are found in the UTX system. The structure seems to be governed by the number of d-electrons of the transition metal atom (Nakotte 1994), starting with the hexagonal MgZn2 structure for Mn-, Tc- or Re-compounds, though the hexagonal Fe2P or orthorhombic CeCu2 structures towards the cubic MgAgAs and hexagonal Cain2 structures for the compounds containing Cu, Ag or Au. Compounds with the cubic MgAgAs structure are semiconductors, the other ones show metallic behaviour. Note that the corresponding non-magnetic Th compounds with the MgAgAs structure are also semiconductors, so that in these cases the magnetism is not a condition for the semiconducting behaviour. UNiSn is a rather famous compound crystallizing in the cubic MgAgAs structure. It undergoes an exotic phase transition from semiconducting paramagnetism for T > TN (= 43 K) to a metallic antiferromagnetic phase below TN (Palstra et al. 1986b, Fujii et al. 1989b). Neutron diffraction experiments have shown UNiSn to order in a type I antiferromagnetic structure (Kawanaka et al. 1989), while indications for a lattice distortion were found from far-infrared measurements (Kilibarda-Dalafave et al. 1993). The crystalline electric field scheme has

HEAVYFERMIONSAND RELATEDCOMPOUNDS

25

been deduced by Aoki et al. (1993) to consist for the U 4+ ion of a non-magnetic doublet ground state and a first excited triplet state at 180 K. From these scheme it can be shown that the phase transition in UNiSn should be of first order and includes the spins as well as the lattice, which was nicely confirmed by Suzuki et al. (1994) by powder X-ray and ultra-sonic measurements. The series URhIn, UPtIn and UPdIn (Nakotte et al. 1992, Havela et al. 1994a, Sugiura et al. 1990) can be classified as magnetic heavy fermions; the strength of the magnetism increases when going from Rh towards Pd, i.e. the moment increases from 1.2 to 1.6#B and TN increases from 7 to 20.4 K. This trend can be explained be the decrease in the hybridization. From the specific heat there are indications that the bandstructure changes at TN, optical and tunneling experiments should clarify this point. UPdSn and UAuSn both order antiferromagnetically at 37 and 40 K, respectively (De Boer 1992, Robinson et al. 1993a, 1994a, Nakotte et al. 1993a, b). From neutron diffraction it appears that both compounds have a very rich magnetic phase diagram. In both cases the magnetic unit cell of orthorhombic symmetry, while the crystallographic cell is hexagonal. This implies that different domains are possible. UPdSn has two transitions in zero external field. At both transition temperatures, 40 and 25 K, different Cartesian components of the magnetic moments order. Similar effects have been seen in the pure rare-earth metals. The electrical resistances of the two compounds is rather different, the one of UPdSn decreases by a factor of 4 below the magnetic ordering temperature, the resistivity of UAuSn increases gradually with decreasing temperature over the whole range from room temperature to the lowest temperatures. UPdGe, UPtGe and UNiGe also show quit complicated magnetic structures (Kawamata et al. 1992, Proke~ et al. 1994, Robinson et al. 1994a, b, Havela et al. 1994a and Sechovsky et al. 1994a, b). The electronic properties of UNiA1 have been described by Brtick et al. (1994). Theoretical or phenomenological predictions of this variety of magnetic structures are not (yet) possible, due to the delicate balance between the different interactions. The latter also implies that excellent sample quality is essential for this research. In all cases there is a huge magnetocrystalline anisotropy due to several reasons: crystalline electric fields, anisotropic interaction and/or anisotropic hybridization. As a general trend, as far as single crystals have been investigated, compounds crystallizing in the ZrNiA1 structure have their ordered moments directed along the c-axis, while the moments in the TiNiSi derived compounds prefer the b-c plane. Other structures reveal more complicated anisotropies. Evidence for anisotropic hybridization stems from detailed neutron diffraction studies on URhA1, URuA1 and UPdSn, where also a large orbital moment for the essentially itinerant 5f-electrons is found (Paixao et al. 1993, Nakotte 1994). Measurements of this kind (U-formfactor) as a function the external magnetic field - and thus for metamagnetic transitions as a function of the magnetic structure - should be able to shed light on the intriguing questions concerning the magnetic order in heavy fermion compounds (Wulff et al. 1990). The electrical resistance of the 111-compounds (and of heavy fermions in general) is rather high, several hundreds of # ~ cm. Also, it sometimes drastically changes as a result of magnetic order, although, as mentioned above, there are exceptions where the temperature dependence of the electrical resistance does not change at all at the ordering temperature. This implies that a simple conduction electron vs.

26

G.J. NIEUWENHUYS 1.6

/

|.0

v

/ 0.5

/

UNIGa T : 4.2

•

t

/ . t l v v

v

. . . .

15o, m ~ x ~ 100 E

V

50

UNiGa

~

A ~

T = 4.2K i //

~ //

c

0, 0 0.0

0.5 B

1.0 03

1.5

Fig. 9. Field dependence of the magnetization for BIle and of the electrical resistivity of UNiGa at 4.2 K for i[[BlJc and i±Blle. The open symbols represent the result obtained with increasing field, while the full symbols represent the one obtained with decreasing field (after Nakotte 1994).

local moment scattering picture is much too simple. There is in general a very strong interaction between the conduction electrons and the magnetic degrees of freedom of the 5f-electrons. The latter are also believed to take part in the electrical conductance. Then, the magnetic structure is going to play an important role, since the conduction electrons now also have to obey the symmetry of the magnetic unit cell. Conductivity calculations should now be carried out on the basis of band-structure calculations which include the various magnetic structures. We will come back to this point. For the case of the UTX compounds it is interesting to note that very large magnetoresistance effects have been observed, larger than in the multilayer materials, see, e.g., Havela et al. (1994a), de Boer et al. (1994) and Nakotte et al. (1994a, b). A typical example is UNiGa, which orders antiferromagnetically at 40 K (Havela et al. 1991), the ground state is characterized by the sequence + + - - + - . At low temperatures, an external magnetic field of 1 Tesla is sufficient to align the moments along the c-axis, which leads to a reduction of the electrical resistivity to only 15% of its zero-field value. In fig. 9 the magnetization and the resistivity are displayed.

2.3. UzT2X-compounds A new family of heavy fermion compounds have been synthesized with the composition UzTzX (Peron et al. 1994, Nakotte et al. 1994b, Mirambet et al. 1993, 1994, Nakotte 1994, Havela et al. 1994b). The compounds with T -= Co, Rh, Ni, Pd, Ir, Pt

HEAVYFERMIONSAND RELATEDCOMPOUNDS

c

27

C) U OT @X

Fig. 10. Schematicrepresentation of the unit cell of U2T2X crystallizingin the U3Si2 structure. The two shortest inter-uraniumdistances are indicated(after Nakotte 1994). and X = In, Sn all crystallize in the tetragonal U3Si2 structure, depicted in fig. 10. The shortest distances between the uranium atoms in this series of compounds ranges from 3.40 ,~ to 4.00 A, that is just around and above the famous Hill limit of 3.5 ,~ (Hill 1970). Therefore, these compounds are expected to cover the range between non-magnetic behaviour via heavy-fermion behaviour towards magnetic ordering. In table 5 the structure parameters, together with the magnetic ordering temperature and the linear coefficient for the specific heat are given. All magnetic ordering is antiferromagnetic. Note, that the value for 7 is still in mJ/(mole/.~. K2), but that the formula unit now contains two uranium atoms. High-field magnetization experiments (Nakotte et al. 1994b) have confirmed the antiferromagnetism by revealing metamagnetic transitions and confirmed the paramagnetism of the other compounds by showing that the magnetization is a linear function of the external field at higher fields. Calculations of the strength of the hybridization shows that indeed the magnetically ordering compounds have a lower value than the paramagnetic ones, placing this series of compounds to the right-hand side of the Doniach phase diagram.

2.4. U-compounds with CaCu5 structure and related borides A number of rare-earth intermetallic borides are excellent magnet materials with strong uniaxial magnetocrystalline anisotropy, see, e.g., Strnat (1988). Uranium borides, unfortunately, have not so many useful properties. The main structural and magnetic properties of a number of these compounds are given in table 6. The heading 'Magn.' now shows AF for antiferromagnetism, PP for Pauli paramagnetism and SF for spin fluctuator, meaning that the magnetic susceptibility is still rather temperature independent, but that the electrical resistivity shows a heavy-fermion like decrease for temperatures decreasing below ~ 150 K. Because of the similarity of the structure, also some 123-compounds are listed in table 6, including the new

G.J. NIEUWENHUYS

28

TABLE 5 Structural and magnetic properties of U2T2X. The lattice parameters are in ,~. The specific heat coefficient, "7, is given in mJ/(mole K 2) only if it is not disturbed by magnetic ordering and/or CEF effects. TN is the N6el temperature in Kelvin. Compound

a

c

U2Co2In U2Ni2In UzRhzIn UzPdzIn UzPtzIn UzCo2Sn U2Ni2Sn U2Rh2Sn U2Pd2Sn U21r2Sn U2Pt2Sn

7.361 7.375 7.553 7.637 7.654 7.654 7.263 7.524 7.603 7.566 7.670

3.431 3.572 3.605 3.752 3.725 3.725 3.691 3.630 3.785 3.601 3.698

TN 14.3 36.6 26.0 24.4 40.6 -

78 237 850 261 98 305

TABLE 6 U-compounds with CaCu5-type structure and related borides. 'Crys.' denotes the symmetry of the lattice, 'Struc.' the structure type. The lattice parameters are given in A. The specific heat coefficient, 7, is given in mJ/(mole K 2) only if it is not disturbed by magnetic ordering and/or CEF effects. Under the heading 'Magn.' A F means antiferromagnetism, S F spinfluctuator and P P Pauli paramagnet. TM is the magnetic transition temperature in Kelvin. Compound

Crys.

Struc.

a

U(Coo.25 Nio.75)4B U(Coo.5 Nio.5)4B U(Coo.75 Ni0.25)4B U(Ruo.7Rho.3)4B 4 UCosB 2 UCo4B UCo4B 4 UFesB 2 UlrsB 2 UNi2A13 UNi4B UOssB2 UOs4B 4 UPd2A13 UPd2Ga 3 URusB 2 URu4B 4

hex hex hex tetra hex hex tetra hex hex hex hex hex tetra hex hex hex tetra

CeCo4B CeCo4B CeConB LuRu4B 4 CeCosB 2 CeCo4B CeCo4B 4 CeCosB 2 CeCosB 2 PrNi2A13 CeCo4B CeCosB 2 LuRu4B 4 PrNi2A13 CeCo4B CeCosB 2 LuRu4B 4

4.931 4.924 4.910 4.947 4.895 5.027 5.047 5.373 5.204 4.950 5.511 5.365 5.295 5.473 7.460

c

3'

Magn.

TM

Ref.

6.964 6.954 6.928 3.065 6.933 7.030 2.997 3.181 4.018 6.962 2.971 4.191 8.505 2.965 15.000

73 114 48 52 32 64 37 55 21 88

AF SF SF PP SF SF PP PP SF AF AF PP PP AF AF PP SF

5 -

[1] [1] [11 [2] [3, 4] [5] [2] [3, 41 [3, 4] [6-81 [1, 9] [3, 4] [2] [6,7, 10, 11] [12] [3, 41 [2]

References: [1] [2] [3] [4] [5] [6]

Mentink et al. (1994e) Mentink et al. (1992) Yang et al. (1984) Mentink (1994) Nakotte et al. (1993c) Geibel et al. (1991b)

[7] Geibel et al. (1992) [8] Scr6der et al. (1994) [9] Mentink et al. (1994b) [10] de Visser et al. (1992) [11] de Visser et al. (1993) [12] Siillow (1994)

-

5.2 21 14 13 -

HEAVYFERMIONSAND RELATEDCOMPOUNDS

29

magnetic heavy fermion superconductors UPd2A13 and UNi2A13. Inspired by the anomalous ferromagnetism in CeRh3B2 (Dhar et al. 1981) with Tc = 115 K, CeT3B2 and UT3B2 materials have been investigated by Yang et al. (1984, 1985), Ku et al. (1980), Kasaya et al. (1990) and Mentink et al. (1994e). The compounds crystallize in CeCo3B2 structure, see fig. 11. All the uranium compounds of this series are either Pauli paramagnets or spin fluctuating materials with rather small temperature independent 7 values. This can be understood from the small inter-uranium distances (< 3.5 ]k) causing an overlap of the 5f-wavefunctions. UCo3B2 and UIr3B2 have somewhat larger distances and therefore exhibit the spin fluctuation phenomena as also found in typical heavy fermions. The 141-compounds order in the CeCo4B structure, depicted in fig. 12. The formation of a magnetic moment on the U-site

c I

3g la 2c

a @U,

© T,

eB

Fig. 11. The CeCo3Bz-typestructureas adoptedby the UT3B2compounds(afterMentink1994).

c ] ~

l

b

~-

-

~

z~-~--~__-_. - - = ~© :

6i

la ~:

a

u

2c

O U, © T, • B Fig. 12. The CeCo4B-typestructureas adoptedby the UT4B compounds. Notethe presenceof two differentU- and T-atomsites. The uraniumatomshave a triangulararrangementin the basalplane(after Mentink 1994).

30

G.J. NIEUWENHUYS

d

',,o/",o , ' \ o ,' ',,o/",o , ' \

,i_o

ok

/o,, O/o',,o ,,'o',, o ,,'0',, O/o',,o / ~

',, o .-' ,-,

.....

,~,

.... ~

,'t

oo ,'o, -.,,o , , ,' o'.,._ o . ' ,o , ° , , ' o \

'~' ~ ~ - - ~ d

/\7,,'

,,

', o / " , , o / \ o

,d

....~,

,,, ',, o f

o__k, o , /

Fig. 13. Magnetic structure of hexagonal UNi4B, projected on to the crystallographic basal plane. Given the FM coupling along the e-axis, every layer at z = 0 and z = 1/2 exhibits the same moment orientations. The thin solid line represents the magnetic unit cell. Note the presence of the 'free' uranium moments with two different magnetic environments (after Mentink 1994). in U(Co~Nil_~)4B can be understood since the d-band shifts downwards in energy with decreasing z (more Ni-rich), thereby decreasing its density at EF, while also the distance between the U-atom and the transition metal increases. Both effects will decrease the hybridization and enable the formation of moments on the Usite. At the end of this series, nature has placed a remarkable partially ordered antiferromagnet: UNi4B. In the ordered state, the magnetic moments are confined to the basal-plane of the crystal structure. According the neutron diffraction experiments (Mentink et al. 1994b) two-thirds of the magnetic moments order in a vortex-like structure in the basal plane, while being ferromagnetically aligned along the c-axis, see fig. 13. The other one-third of the magnetic moments are free to rotate in the basal plane. They are only coupled ferromagnetically along the c-axis, but a one-dimensional ferromagnetic chain does not order at finite temperature. Note that the nearest-neighbour coupling in the basal plane cancels for all magnetic moments, while the next-nearest-neighbour ones cancel for the 'free' moments. The unique magnetic structure is also reflected by the magnetic susceptibility, see fig. 14. The low-temperature upturn in small external magnetic fields represents the response of the ferromagnetic linear chains, which can easily be saturated by larger fields. A similar extra contribution is found in the specific heat. Also the large anisotropy is evident from these susceptibility results. This anisotropy of about a factor of ten remains present in the paramagnetic region, it is therefore not caused by a spatial anisotropy of the magnetic interaction between the moments. Probably, crystalline

HEAVY FERMIONS AND RELATED COMPOUNDS | o

300

m

i

Ooooo

*~. °o o

"6" O

31

~'~.o

200

"

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100 U~m

annjllmmIe ° °

,

0 0

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°

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•

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o ooe

n

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30

T (K) Fig. 14. Magnetic susceptibility of a UNi4B single crystal, measured in fields of 0.3 mT (o), 0.5 T (+) and 5 T (rn), perpendicular to the c-axis. The lower points (-) represent the result for a field of 0.5 T directed parallel to the c-axis (after Mentink 1994).

electric field effect and anisotropic hybridization dictate the magnetic moments to lie preferentially in the crystallographic basal plane. Small steps have been observed in the magnetization versus field curves at low temperatures for fields parallel to the a-axis and pm'allel to the b-axis. These could be explained by assuming a small 6-fold anisotropy in the basal plane (Mentink et al. 1994c). Remains the question why one-third of the spins behave differently from the others. The same neutron diffraction experiments also showed that UNi4B does not crystalize exactly in the CeCo4B structure, but that it adopts a superstructure with a unit cell of size av/3 with the axis rotated by 30 ° with respect to the CeCo4B cell. This superstructure persists to temperatures above TN. Presently the atomic positions are being investigated by single crystal X-ray diffraction. The new heavy fermion superconductors, UPd2A13 and UPd2A13 crystallize in the PrNi2A13 structure, which is depicted in fig. 15. This structure is an ordered variant of the CaCu5 type, where 3 of the Cu atom are replaced by A1. The existence of magnetic moments in these compounds is caused by the for the CaCu5 structure - relatively large volume of the unit cell, leading to interuranium distances larger than Hill's limit. The Ni system has an ordered magnetic moment of (0.24 + 0.1)#B (Schr6der et al. 1994), the transition temperature is 5.2 K and the compound becomes superconducting below a temperature of about 1 K. Its Pd counterpart has a larger ordered moment of 0.85#B (Krimmel et al. 1992b), an higher TN of 14 K and a superconducting transition temperature of ~2 K. The exact temperature of the transition to superconductivity depends on the sample preparation (Sakon et al. 1993). We will come back to that point. The magnetic structure of UPd2A13 consists of ferromagnetic layers in the crystallographic basal planes with

32

G.J. N1EUWENHUYS

c I

3g

la >

Et

U,

© Pd,

•

2c

A1

Fig. 15. The PrNi2A13-type structure adopted by UPd2AI3 and UPd2A13. The uranium-transition metal layers are separated by an aluminium layer at z = 1/2 (after Mentink 1994).

250

i

| 2.00

200 © 0

E o

150 100

x

50 i

0

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i

100

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200

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I

300

T (K) Fig. 16. Magnetic susceptibility of UPd2A13 along the a-axis (o) and c-axis (+). The solid lines denote a fit to a CEF model with a singlet ground state described in the text. The inset shows the highfield magnetization at T = 4.2 K parallel to the a-axis. Again, the solid line represents the calculated magnetization (after de Visser et al. 1993 and Mentink 1994). the m o m e n t s in the plane. The layers are stacked antiferromagnetically along the c-axis. For UNi2A13 the magnetic structure is slightly different, the ordering wave vector is n o w (½ + 3 , 0 , ½), with 5 = 0 . 1 1 0 + 0 . 0 0 3 . In both compounds the magnetic order persists into the superconducting phase as evidenced by neutron diffraction

HEAVY FERMIONS AND RELATEDCOMPOUNDS and #+SR experiments (see Krimmel et al. 1993 and Schenck and Gygax volume). The magnetic susceptibility of UPd2A13 is strongly anisotropic picted in fig. 16. The experimental results were explained on the basis of 1( model for the U 4+ ion with a singlet ground state, /"4 = ~ I + 3) - I -

33 in this as dea CEF 3)), a

first excited state, F1 = 10) at 33 K, a third state, /"6 = I + 1) at 102 K, a fourth state, F}2~ = a I q: 2) - bl + 4) at 152 K and other states at much higher energy (Grauel et al. 1993, Mentink 1994). The magnetic moment of UPd2A13 then is assumed to be due to an admixture of the first and second excited state as a consequence of an external and/or internal magnetic field. Note that this admixture will give rise to large moments in the basal plane. Any magnetic moment induced along the c-axis must come from an admixture between the/"4 and/"3 states, having an energy difference of about 560 K and is therefore relatively small. The maximum in the susceptibility versus temperature curve is due to the depopulation of the first excited state with decreasing temperature. The step in the magnetization as function of external field (inset to fig. 16) can be understood as a jump to ferromagnetic alignment of the moments due to the external field. Similar explanations for the magnetic susceptibility and the magnetization as a function of external fields were given in the case of URu2Si2 (see above). In that case (due to the tetragonal crystal symmetry) the level scheme is such that a field along the c-axis causes the largest admixture, and thus the moments are directed only along the c-axis, i.e. the basal-plane component simply does not exist. Note, however, that direct observations of the crystalline electric field levels by inelastic neutron scattering have not been successful up to now. The value of the ordered magnetic moment as well as the magnetic anisotropy have also been successfully explained on the basis of self-consistent density-functional calculations by Sandratskii et al. (1994), where the U 5f states are treated as band states. The magnetism in UPd2A13 is rather robust versus doping with other elements, this in contrast to the superconductivity (Geibel et al. 1994). In order to investigate the sample dependences further, Mentink et al. (1994d) have studied the sister compound CePd2A13. This compound does not become a superconductor so that the investigations can be focussed on the magnetism. In poly-crystalline form, after sufficient heat treatment, CePdaA13 orders antiferromagnetically at 2.7 K. However, as-cast samples as well as the best single crystals show at most only short range magnetic order. Such a discrepancy has also been observed in CePt2Sn2. Detailed nuclear quadrupole resonance experiments have shown that, e.g., in as-cast polycrystalline samples 11% of the A1 experience a different surrounding from the other 89%. In view of the number of nearest neighbours of the A1, this would imply that 2% of the A1 atom are wrongly placed. In the annealed polycrystal this number is about 1%. This experiment shows in a quantitative way the extreme sensitivity of the magnetism in heavy fermions compounds for the exact structure of the samples, and thus for the sample preparation. In other words this experiment emphasizes the notion that the ground state in heavy fermions is a result of a delicate balance between interactions.

34

G.J. NIEUWENHUYS

2.5. 334-compounds A remarkable set of compounds is formed by the U- and Ce-containing M 3 T 3 X 4 compounds, with T a transition metal and X is Sb, Sn or Bi. The existence of the Sb compounds had already been reported by Dwight in 1979, but it took a few years before the remarkable electronic and magnetic properties were studied in more detail (Buschow et al. 1985, Takabatake et al. 1990c and Endstra et al. 1990b). All compounds crystallize in the cubic Y3Au3Sb4 structure (space group 1 2~3d). The unit cell, depicted in fig. 17, contains 4 formula units. The inter-uranium distances are relatively large (~ 4.4 A), much larger than the distances between the magnetic atoms and the transition metal atoms (~ 3.3 A). Therefore, reasonable magnetic moments are expected, since no direct f - f overlap will occur. Most of the Sb compounds are semiconductors. In the case of U3T3Sb4 the gap in the density of states is rather large (,.~ 0.2 eV) in contrast to the tree Ce containing compounds mentioned, where gaps are found of the order of meV. The U compounds are therefore considered as 'normal' semiconductors, while the Ce compounds are most probably hybridization induced semiconductors. The normal semiconducting properties were confirmed by bandstructure calculation (Takegahara et al. 1990). Table 7 summarizes the structural, electronic and magnetic properties. It should be noted that in the case of U 3 C u 3 S b 4 electron-microprobe analysis shows that the Cu content was less than 3, so this is a Cu deficient material. From table 7 it is clear that the semiconducting compounds have a small coefficient of the linear term in the specific heat, i.e. no density of states at the Fermi level. Also, these compounds do not order magnetically, indicating that the conduction electrons are essential for the magnetic interaction. Again the trend in the magnetic properties in these system can be understood in terms of the strength of the hybridization. For example, replacing Ni by Cu in the Sn series

Fig. 17. The Y3Au3Sb4 crystal structure type. Small open circles indicate the Y atoms, small patterned circles are the Au atoms and the large open circles the Sb atom (after Takegahara et al. 1990).

HEAVY FERMIONS AND RELATED COMPOUNDS

35

TABLE 7 334-compounds. The lattice parameter is given in ,~. The specific heat coefficient, % is given in mJ/(mole K2) only if it is not disturbed by magnetic ordering and/or CEF effects. An M under 'Conduc.' means metallic and S semiconducting. Under the heading 'Magn.' an A means antiferromagnetism and F ferromagnetism. TM is the magnetic transition temperature in Kelvin. The effective moment/Zeff is given in/~B. Compound

a

Ce3Au3Sb4 Ce3Pt3Bi4 Ce3Pt3Sb4 U3Au3Sn4 U3Co3Sb4

10.058 9.998 9.820 9.824 9.284 9.415 9.522 9.393 9.380 9.684 9.683 9.675 9.531

U3Cu3 Sb4

U3Cu3Sn4 U3Ni3Sb4 U3Ni3Sn4 U3Pd3Sb4 U3Pt3Sb4 UPt3Sn4 URh3Sb4

3' 3 0 280 69 380 2 92 94 -

Conduc. S S S M M M M S M S S M -

Magn. F F A F

TM

#eff

Ref.

10 91 12 105

2.45 2.5 3.58 2.1 3.39 3.34 3.65 1.8 3.58 3.68 1.84 2.4

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

References: [1] Kasaya et al. (1991b) [2] Hundley et al. (1990)

[3] Takabatake et al. (1990c) [4] Endstra et al. (1990)

will d e c r e a s e the f - d h y b r i d i z a t i o n b e c a u s e the d - b a n d shifts a w a y from the F e r m i level and the lattice p a r a m e t e r increases. In U3Ni3Sn4 the h y b r i d i z a t i o n is too large to f o r m o r d e r e d m a g n e t i s m and the c o m p o u n d is close to the h e a v y f e r m i o n regime, while U3Cu3Sn4 is an antiferromagnet. In the s e m i c o n d u c t i n g U3T3Sb4 c o m p o u n d s a r e p l a c e m e n t o f Ni b y Pd or Pt ( s a m e c o l u m n in the p e r i o d i c system) does not c h a n g e m u c h in the g r o u n d state, e x c e p t for the size o f the gap. H o w e v e r , substituting Cu for Ni (different c o l u m n ) drastically changes the features o f the c o m p o u n d s as was shown in detail b y Fujii et al. (1992) b y investigating the transition f r o m s e m i c o n d u c t i v i t y to f e r r o m a g n e t i s m in U3(Nil_xCux)3Sb4. The increasing n u m b e r o f d-electrons with increasing first causes the b a n d gap to close and subsequently i m p r o v e s the m a g n e t i c ordering via a R K K Y - t y p e interaction b e t w e e n the m a g n e t i c atoms. Substituting Sn b y Sb in the series U3Ni3(Sbl_xSn~)4 has two effects: the b a n d gap is r e m o v e d and the h y b r i d i z a t i o n is g r a d u a l l y increased, reflected b y a m o n o t o n e o u s reduction o f the effective m o m e n t o b s e r v e d at high temperature (Endstra et al. 1992b). A m a x i m u m for 7 is f o u n d for z = 0.5. The z e r o - t e m p e r a t u r e resistivity changes b y a factor o f 105. A s m e n t i o n e d above, Ce3Pt3Bi4 has been c o n s i d e r e d as so-called K o n d o insulator (see A e p p l i and F i s k 1992). Note that in this case, as for C e N i S n , the charge gap is e x t r e m e l y small, 0.01 e V for Ce3Pt3Bi4 and 0.001 e V for CeNiSn. Interestingly, the charge gaps in Ce3Pt3Bi4 and in C e N i S n are a c c o m p a n i e d b y a spin gap o f c o m p a r a b l e m a g n i t u d e as o b s e r v e d b y neutron scattering ( S e v e r i n g et al. 1991, M a s o n et al. 1992). As m e n t i o n e d in the section on i l l - c o m p o u n d s , substituting L a for Ce b r e a k s the f-lattice s y m m e t r y and leads rapidly to a metallic like resistivity ( T h o m p s o n et al. 1993a, b). Severing et al. (1994) have shown that

36

G.J. NIEUWENHUYS

the spin gap is less sensitive to doping with non-magnetic elements. Probably the formation of a spin gap is a more local phenomenon. The charge gap in Ce3Pt3Bi4 has also been observed via reflectivity and optical conductance measurements by Bucher et al. (1994). These experiments showed that for temperatures below 100 K a charge gap develops. The spectral weight depends on temperature, its decrease scales with the decrease of the Ce 4f-moments, but the size of the gap is temperature independent. Electronic band structure calculations in Ce3Pt3Sb4 and Ce3Pt3Bi4 by Takegahara et al. (1993) show a gap of 300 K, while the experiment reveals 900 K for the Sb compound. In the Bi compound the calculated gap is 3.5 larger than the experimental one of 100 K. From these results the conclusion can be drawn that Ce3Pt3Sb4 the term 'Kondo insulator' is not appropriate. If the small value of the gap in Ce3Pt3Bi4 can be associated with impurity states, then the term Kondo insulator is also in doubt for this compound. Other band structure calculations by the same group (Takegahara et al. 1992, 1993) have shown that the non-magnetic compounds Th3Ni3Sb4 and La3Au3Sb4 have a narrow gap at the Fermi level. Therefore, the related compound U3Ni3Sb4 and Ce3Au3Sb4 are 'normal' semiconductors. The same is true for the 111-compounds crystalizing in the MgAgAs structure, where the nonmagnetic equivalents are also found to be semiconducting (Palstra et al. 1987a and b).

2.6. Gaps Several types of energy gaps in a dispersion relation can occur in metallic (magnetic) systems: • The gap in the conduction band due to superconductivity. • The gap in the conduction band, which leads the semiconducting behaviour. This gap is found in a number of magnetic and non-magnetic compounds, see above. • The gap in the spin wave spectrum of magnetically ordered systems. The resulting low-temperature electrical resistance has been described by Hessel Anderson et al. (1979, 1980), who showed that for temperatures smaller than the gap size the contribution to the resistivity should follow a behaviour given by: T T(l +-~)e -zVT

(11)

If the gap in the spin wave spectrum is zero than the normal contributions proportional to the second and fifth power of the temperature remain. It is rather difficult to distinguish the normal behaviour from the one described by eq. (11). • Gaps in the dispersion relation of the conduction electrons due to the lowering of the symmetry of the crystal lattice as a consequence of the magnetic ordering. Necessary conditions are that the symmetry is lowered (ferromagnets will not exhibit this phenomena) and that the itinerant electrons "see" the lowering of the

HEAVYFERMIONSAND RELATEDCOMPOUNDS

37

symmetry, thus a band electron - magnetic moment interaction must be present. Changes in the band structure can cause changes in the electrical conduction through a change in the number of electrons available at the Fermi surface, even sometimes in an anisotropic way. A textbook example is the c-axis resistivity of pure Er due to a cone-like structure between 20 and 80 K, see Miya (1963), Freeman (1972) and Coqblin (1977) • Gaps in Kondo insulators as described above, probably there are charge gaps as well as spin gaps in these materials. Note that only the last type of gap is typical for heavy fermions, the other ones are found in normal metals, semiconductors and magnets, and thus also in some heavy fermions. Of course it is very interesting how these 'normal' gaps behave when they are formed in a band of heavy quasi particles. Magnetic gaps in spin or in spin wave excitation spectra can be observed via inelastic neutron scattering, see Broholm et al. (1991), Severing et al. (1991), Mason et al. (1992). Charge gaps can be seen in optical experiments and in vacuum tunneling measurements (Wolff 1985). The latter type of experiments is rather new in the research on heavy fermions (Aarts and Volodin 1995), although the tunneling spectra of the most famous ones have been investigated. URu2Si2 does show a gap in the spin wave excitation spectrum, it shows the exponential term in the resistivity (Palstra et al. 1986a) and in the specific heat (Maple et al. 1986). URu2Si2 also shows an increase in the electrical resistivity when the temperature is lowered through TN, indicative of a change in the band structure. Indeed, optical data (Bonn et al. 1988) and vacuum tunneling experiments (Aarts et al. 1994) confirmed this conclusion. The latter experiment clearly showed that in the antiferromagnetic phase a gap exits for electrons tunneling in the basal plane, which gap decreased on increasing temperature and disappeared at TN. No such gap could be detected for electrons tunneling along the crystallographic c-axis. UPd2A13 has no gap in the spin wave spectrum (Mason 1994). Vacuum tunneling experiments (Aarts et al. 1994) show the same type of gap as observed in URu2Si2. This could be expected, bearing in mind that in both compounds the magnetic ordering consists of ferromagnetic layers in the basal plane stacked antiferromagnetically along the c-axis. Thus in both cases the symmetry of the crystal along the c-axis is broken be the antiferromagnetic ordering. On the other hand, the electrical resistivity of UPd2A13 shows no increase at TN, which can be explained by assuming that the effects on the number of conduction electrons near the Fermi level is accidentally too small to overcompensate the decrease in the electrical resistivity due to the decrease in magnetic disorder scattering. However, also optical experiments do not show a charge gap developing at TN (Degiorgi et al. 1994a) and that has still to be explained. Gaps have also been observed via optical experiments in UNi2Si2 (Cat et al. 1993) and UCus, but no gap was detected in U2Zn17 (Degiorgi et al. 1994b). Note that the resistivity of UCu5 also increases for temperatures below TN (Ott et al. 1985a). Finally, I show two other examples where strong indications emerge form electrical resistance that the antiferromagnetic order changes the band structure. The first one is a series of pseudo binary compounds U(Col_xFe~)2Ge2 (van Rossum 1993). The resistivity is shown in fig. 18.

G.J. NIEUWENHUYS

38 1.50

re

q.

1 .OO j,,¢.a~

oooo

o.

°° °

oo U(Co

0.50

I

O

1-X F e x i

)2Ge2

I

100

I

200 T

( K

300

)

Fig. 18. Normalized resistivity versus temperature for U(CoI_~Fe=)2Ge2 after van Rossum (1993).

500

J

~t--8.XlS

250

--

0

i//b-axis

--

i//c-axis

1.10

g. 5 o o 0

.-

,~

1.05 1.00

095

25O

0.90 i

0 4OO

i

i

0.85

35

40

45

50

T [K]

200 I

0

20

I

40 T (~<)

60

Fig. 19. Temperature dependence of the electrical resistivity of UNiGe for current along the three principal directions. The fits with an additional exponential term involving electron-magnon scattering are shown by solid lines. The right hand side figure shows an enlargement of the area around 41.5 K (after Proke~ et al. 1994).

HEAVYFERMIONSAND RELATEDCOMPOUNDS

39

Also U(Col_xNi~)2Ge2 (Endstra et al. 1993c) shows strong increases in the resistance. The other example is UNiGe, which exhibits a transition from a commensurate antiferromagnet to an incommensurate structure at T* = 41.5 K (Prokeg et al. 1994). The electrical resistivity as a function of temperature is depicted in fig. 19, where nothing can be seen when the current is directed along the a-axis, an increase at T* when the current flows parallel with the b-axis and finally a sharp decrease for currents along the c-axis. Clearly the change in the band structure has an anisotropic influence on the conduction and also it is clear that this change can induce an increase as well as a decrease in the number of available carriers. The same experiment gives indications for a gap in the spin wave spectrum, shown by fits to the expression given in eq. (11) (Proke~ et al. 1994). Another remarkable signature of the change of the band structure was found from M/3ssbauer experiments by Mulder (1994). He has determined the electric field gradient at the site of the magnetic ion - in the case Gd - by measuring the quadrupolar splitting of the 155Gd nucleus. The electric field gradient is governed by the symmetry of the crystal lattice and in particular its size by the band structure. It appeared that in some antiferromagnetic compounds, the gradient changes at the magnetic transition temperature, whereas in ferromagnetic compounds no change could be observed. Evidently, careful experiments is the area of spin and charge gaps will reveal more about the mechanisms in effect in magnetic and semiconducting heavy fermion materials.

2.7. Magnetism and superconductivity The interplay between magnetism and superconductivity plays an important role in heavy fermion physics. Almost all superconducting heavy fermions also show magnetic ordering, the only exception seems to be the cubic UBe13. In good samples of UPt3 two superconductivity transitions can be seen in the specific heat as a function of temperature. This double transition is though to be caused by the symmetry breaking effect of the antiferromagnetic order in the basal plane present below TN = 5 K (Aeppli et al. 1988). Extensive measurements as function of external magnetic field and external pressure have revealed a rich phase diagram where the different crossings of the phase separating lines merge in a tetracritical point at 2.6 kbar (Boukhny et al. 1994a, b; see also van Dijk 1994 and references therein). Keller et al. (1994) observed a clear dependence of the upper critical field, Bc2, on the angle between the magnetic field and the crystalline axis. As mentioned above URuzSi2 and UPdzAI3 show anisotropic gaps at the Fermi surface due to the antiferromagnetic order. The superconductivity therefore develops in this already partly gapped state and is evidently influenced by its symmetry. It is difficult to obtain really hard facts on this aspect, but, e.g., as mentioned earlier, the observation of superconductivity in URuzSi2 by #+SR is also highly anisotropic in this compound in spite of the apparent isotropic behaviour of the lower critical field. In the case of UPd2A13 it has been argued that the superconductivity and the magnetism are carried by two different f-electron systems (Caspary et al. 1993 and Feyerherm et al. 1994), being itinerant for the superconductivity and more localized for the magnetism. When U in UBe13

40

G.J. NIEUWENHUYS

is partly (few atomic percent only) replaced by Th, a rather exotic phase diagram is found, see, e.g., Ott (1994). Between 2 and 4 at.%, a second transition is observed from a second maximum in the specific heat (Ott 1989). Heffner et al. (1989a, b; 1990) have shown that below the second transition temperature the superconductivity coexists with a magnetic state, characterized at least by static randomly distributed internal magnetic fields as seen via #+SR. In all cases it is rather clear that the superconductivity cannot be described by an ordinary order parameter.

2.8. Miscellaneous compounds 2.8.1. CeCu6 and derivatives CeCu6 has been studied for a long time. Onuki et al. (1985b) found the first indication for its Kondo lattice or heavy fermion behaviour based on resistivity measurements on a single crystal. Rapidly other measurements followed (a.o. Ott et al. 1985b, Amato et al. 1987) showing that no long range magnetic order nor superconductivity was present down to temperatures of 15 mK. See also Bauer (1991) for a review. The compound CeCu6 crystalizes in an orthorhombic structure, space group Pnma, having 4 formula units per unit cell and lattice parameters a = 8.109 A, b = 5.098 A, and c = 10.172 A. Below a temperature of 220 K, the compound transforms into a monoclinic structure, space group P21/c, with gradually increasing angle /3 up to 91.36 ° and lattice parameters a = 5.080 A, b = 10.221 A and e = 8.067 A (Asano et al. 1986 and Gratz et al. 1987). With its 3' value of 1600 mJ/(molK2) it is now considered as an archetypal heavy fermion system. Neutron diffraction experiments showed the development of antiferromagnetic correlations below 1 K (Aeppli et al. 1986, Rossat-Mignod et al. 1988), while #+SR (Amato et al. 1993) reveals that the moment associated with these correlations should be smaller then 0.01#B. This then places CeCu6 really at the borderline between magnetic ordering and heavy fermion behaviour. Small amounts of Ag or Au (Gangopadhyay et al. 1992, Germann and von L6hneysen 1989) drive the compound into the long range ordered region, most probably by increasing the volume of the unit cell and thereby decreasing the Kondo temperature. Another beautiful experiment was carried out by Stroka et al. (1993) on CeCu6_~Au~ compounds showing that the Schottky anomaly due to the crystalline electric field splitting develops from a broadened maximum into the well-known free ion behaviour with increasing x from 0 to 0.9. At x = 0.9 the long range magnetic order is observed at 2 K. Non-Fermi-liquid behaviour has also been observed in Au doped CeCu6, at the composition CeCus.9Au0A by von LShneysen et al. (1994). CeCu6 is a famous heavy fermion without magnetic ordering, by increasing x in CeCu6_~Aux antiferromagnetism is rapidly obtained for small values of x, the transition is zero for the critical concentration x = 0.1. 2.8.2. UxYl-xPd3 UxYl-xPd3 forms a series of strongly correlated electron systems with a complex and rich magnetic phase diagram (Seaman et al. 1991, 1992, Andraka and Tvselik 1991). For x ~< 0.2 the specific heat, magnetization and electrical properties indicate non-Fermi liquid behaviour, while for 0.3 ~< x ~< 0.5 a spin-glass type ordering

HEAVYFERMIONSAND RELATEDCOMPOUNDS

41

is found. The lower concentration samples obey remarkably well the theoretical predictions for the two-channel Kondo effect (Cox 1987). The T -+ 0 behaviour is not completely clear, Andraka and Tvselik suggest a T = 0 second-order magnetic phase transition, while Cox assumes a collective Jahn-Teller instability. Stillow et al. (1994) have shown via detailed metallurgical analysis that compounds with z ~ 0.2 are intrinsically inhomogeneous. For example, for z = 0.2, the measured concentrations vary from 0.14 to 0.23. A long term heat treatment could only halve the range of concentrations. Nevertheless, the magnetic and electrical properties did not depend strongly on the homogeneity. Remains the question what the behaviour of a homogeneous alloy will be.

2.8.3. CeCu5 and UCu5 and derivatives CeCu5 crystallizes in the hexagonal CaCu5 structure, it orders antiferromagnetically at 3.9 K, and has a '7-value of 100 mJ/(mole K 2) and a logarithmic contribution to the electrical resistivity (Bauer 1991 and references therein). Inelastic neutron scattering experiments (Alekseev et al. 1992) and magnetic measurements have shown that the I + 1) is the ground state of the CEF levels and that the I + 23-}is at 17 meV (200 K) the first excited state. At the ordering temperature only the ground state plays a significant role. Remarkably, in spite of the [ 4- ½) ground state, from which a large magnetic moment in the basal can be expected, neutron diffraction (Bauer et al. 1994a) showed that the ordered moment is directed along the c-axis and has 1 a value of 0.36#B. The propagation vector of the magnetic structure is (0, 0, ~). The measured moment is smaller than the one calculated from the CEF scheme and also the entropy involved in the magnetic transition is smaller than R ln(2), indicating that the Kondo effect is present. The unexpected direction of the ordered moment points to a strongly anisotropic (in spin space) interaction between the Cemoments, i.e. the compound is Ising like (Bauer et al. 1994a). #+SR experiments (Wiesinger et al. 1994) showed some evidence for inhomogeneous magnetic order. These experiments also showed the strong anisotropic susceptibility from the Knight shift, confirming the Ising like behaviour. The study of CeCu5 is of particular interest because substituting A1 or Ga for Cu destroys the long range magnetic order (Bauer 1991, Bauer et al. 1990) and drastically increases "7 up to a value of 2.8 mJ/(mole K 2) for CeCu4A1. Presumably this large "7 should be interpreted as being due to short range magnetic correlations, since it appeared to be strongly dependent on external magnetic fields (Andraka et al. 1991). The electrical resistivity of CeCu4Ga is not pressure dependent (Eichler et al. 1994). UCu5 orders antiferromagnetically below 16 K. The crystal structure is of the cubic fcc AuBe5 type, and the magnetic structure has been determined via neutron diffraction (Murasik et al. 1974 and Schenck et al. 1990). The U sublattice orders ferromagnetically in the (111) planes, which are stacked antiferromagnetically. The ordered moments are aligned along the (111) direction and have a magnitude of ,.~ 1.55#B. The specific heat reveals a second transition at 1 K (Ott et al. 1985a, b), which is hysteretic in temperature, suggesting first order although no latent contribution could be observed. Schenck et al. (1990) showed that this transition could not be observed in the neutron diffraction, and that

42

G.J. NIEUWENHUYS

for the #+SR only the relaxation, not the frequency of the spontaneous precession changes at about 1 K. A possible explanation that remains is that besides the already preset antiferromagnetic order a second ordering takes place of small moments like in, e.g., UPt3 and URuzSi2 (Schenck 1993). If that interpretation is correct, then this would be a strong indication for the existence of to different f-electron systems. Nakamura et al. (1994) interpreted the 1 K transition as a spin reorientation on the bases of the an analysis of the nuclear magnetic resonance data. When part of the Cu atoms in UCu5 are replaced by Pd, the antiferromagnetism rapidly disappears. UCus_~Pd~ is an antiferromagnet only for z < 0.75 and becomes a spin glass for z > 2. For z = 1.5 the specific heat divided by temperature diverges logarithmically for T --+ 0 (Andraka and Stewart 1993). No Fermi liquid behaviour could be detected for this alloy down to 0.3 K, thereby characterizing this material as a non-Fermi liquid one, like Uo.zY0.8Pd3. The pseudo binary compounds mentioned in this section give the experimentalist in principle the opportunity to scan the zero-temperature transition between heavy fermion behaviour and long rang magnetic order. Such ideas are known for some time (Lawrence 1982) and elaborate theories are available (Millis 1993, 1994). Whether it will be possible to really investigate the critical region depends on our ability to engineer our samples or the use other external variables such as pressure or magnetic field sufficiently cleverly. See for a recent review: Andraka (1994) and Abrahams (1994).

2.9. Yb-compounds Not many Yb compounds can be considered as heavy fermions. Most of them are typical valence fluctuators. Yb 3+ is in the 4f 13 state and thus magnetic since it is off by 1 electron from the full 4f-shell. Yb 2+ is non magnetic. This should be contrasted with the Ce-compounds, where adding one f-electron leads to the magnetic state. Therefore, e.g., pressure dependences of the properties of Yb-compounds are just opposite from those in Ce-compounds. The counterpart of UBe13, YbBel3 is a simple antiferromagnet with an ordering temperature of 1.115 K and the crystal field splittings in this cubic material are rather small. The ground state is /"6, the first excited state a/"8 at 37 K and the last state is a/"6 at 51 K. These observation could be made by neutron scattering (Walter et al. 1985) and via specific heat measurements (Ramirez et al. 1986), indicating that hybridization and/or Kondo effects hardly play any role. YbCu5 acts as a parent compound for various substitution at the Cu site, see Bauer (1991) and references therein. A few of the resulting doped materials are heavy fermions. Two crystal structures seem to be possible for YCus, the simple hexagonal CaCu5 (Iandelli and Palenzona 1971) or the MgSnCu4 structure, related to the cubic AuBe5 (Hornstra and Buschow 1971). Substitution by A1 or Ga stabilize the CaCu5 and substitutions by Au, Ag, Pd or In stabilize the MgSnCu4 structure. Yb in YbCu5 is close to the divalent state and thus non magnetic. All the substitutions mentioned above drive the Yb ion into the trivalent state, thereby initiating magnetism. YbCu4Au orders antiferromagnetically below 1 K, the Kondo temperature is of order of 0.3 K (Bonville et al. 1992). The crystal field levels are /"7, /"8 and/"6 at 0, 46 and 81 K, respectively. YbCu4Pd behaves in the same way. The only compound in this series with a large -,/-coefficient for the specific

HEAVYFERMIONSAND RELATEDCOMPOUNDS

43

heat is YbCu4Ag. Rossel et al. (1987) found 200 mJ/(mole K2). A value of 245 was determined for the Pd sample, but this may be in error because of the magnetic order and the crystal field states. The magnetic and thermodynamic properties of the systems mentioned can be remarkably well described by the Coqblin-Schrieffer model (Besnus et al. 1990, Schlottmann 1993). The same can be said about YbCu2Si 2 (Rasul and Schlottmann 1989) and YbPdzSi2 (Schlottmann 1992). The interpretation of the neutron data obtained by Severing et al. (1990a) for YbCu4Ag is not unambigeous. The original authors could not find a consistent fit over the whole temperature range. A reanalyzis by Polatsek and Bonville (1992) revealed a characteristic temperature (approx. Kondo temperature) of 60 K and a crystal field level scheme such as that for YbCu4Au. On the other hand, Schlottmann (1993) concludes that the observed inelastic scattering is a result of transitions from the Kondo resonance to the Fermi level. YbCu4In is a remarkable compound. It exhibits a first-order phase transition at T~ = 41 K from the divalent (non-magnetic) state at low temperatures to a magnetic trivalent one at higher temperatures. The transition is marked by a sudden change in the magnetic, thermodynamic and transport properties (Felner and Novik 1986, Felner et al. 1987). Of course also the signature of the crystal field splittings in the inelastic neutron scattering (Severing et al. 1990b) disappears upon cooling below T~. The CEF scheme in the magnetic phase is Fs, F6 and F7 at 0, 38 and 45 K, respectively, almost the same as that for the stable trivalent YbNi4In. The pressure and magnetic field dependence of the electrical and thermal properties of YbCu4Ag have been studied by Graf et al. (1994), Bauer et al. (1994b, 1995), Thompson et al. (1994a) and Lacerda et al. (1995). In general the observations can be described by a decrease of the Kondo temperature with increasing pressure. A magnetic field of 8 T appears to be equivalent to 30 kbar. Thompson et al. (1994a) show that the relation between the quadratic term in the electrical resistivity and the linear term in the specific heat, A c< 72

(12)

holds when the pressure is the implicit variable. The same is not true for the external magnetic field (Lacerda et al. 1995). This is probably due to the small CEF splittings, so that external fields can effectively change the ground state of the system. The N6el temperature of YbCu4Au, being 0.6 K at ambient pressure, increases to more than 2 K at 50 kbar. The pressure dependence of the resistivity was described by a low Kondo temperature, consistent with the earlier observations (Bonville et al. 1992). A1 and Ga substitution also drive the YbCu5 system towards the magnetic state of Yb (Bauer et al. 1992, 1995) and the full trivalent state is reached around 40 at.% A1 or Ga. YbPtBi has received lots of attention due to its very large value of the extrapolation of Cp/T to zero temperature, being 8000 mJ/(mole K 2) (Canfield et al. 1991, Fisk et al. 1991). The specific heat exhibits a small maximum at 0.4 K, #+SR experiments showed spatially inhomogeneous disordered static magnetism below ~0.5 K, with a small moment of ~ 0.1#B in about 50% of the total sample volume. (Amato et al. 1992). In fact, the #+SR data suggest a spin-glass like behaviour for YbPtBi, although also a incommensurate spin density wave is possible (Heffner et

44

G.J. NIEUWENHUYS

al. 1994). The latter is in accord with the resistivity measurements by Movshovich et al. (1994a, b), who found an increase in p when cooling down below 0.4 K, indicating an ordering such that the band structure changes and at least the number of carriers at the Fermi level decreases. They also showed that a pressure of only 1 kbar is sufficient to suppress the low temperature state. The crystal structure has been solved by Robinson et al. (1994b). YbPtBi crystallizes in the cubic MgAgAs structure, Pt occupies the 4a sites, Bi the 4c, the 4b site is vacant and the 4d site is occupied by Yb. The same neutron diffraction experiment showed that any ordered moment should by smaller than 0.25#B. An inelastic neutron scattering experiment, also made by Robinson et al. (1993b) gives information about the CEF splitting. Most probably the order of the levels is FT, Fs and F6, the corresponding splittings however are of order of 1 K and about 65 K. The near degeneracy of the two lowest states is important. As Thompson et al. (1993b) pointed out, it is the fact that the CEF energies, the ordering temperature and possibly the Kondo interaction are of comparable strength which gives YbPtBi its peculiar properties. Remarkably, most of the properties are not influenced by doping with non-magnetic elements like Lu and Y (Lacerda et al. 1993): down to 50 at.% Yb the thermal and magnetic properties are proportional to the Yb content, even the temperature of the maximum in the specific heat does not depend on dilution of the Yb sub-lattice. Thompson et al. (1993b) also showed that the isostructural YbPtSb orders normally at T = 0.35 K and shows accordingly contributions from the nuclear specific heat at lower temperatures.

2.10. Thin films Thin films of heavy fermion materials have already been made occasionally for a number of years (e.g., Tedrow and Quateman 1986, Roessler and Tedrow 1990). The general conclusion is that the typical properties of the heavy electron persist in the thin film. Note, however, that these were thicker than any characteristic length of the heavy fermion state. A systematic effort was undertaken by the Darmstadt group: Huth et aI. (1993, 1994a, b, c, 1995), Hessert et al. (1995) (UPd2A13 films) and Lunkenheimer et al. (1994) (UCua+xA18_x films). The UPd2A13 films are highly c-axis oriented, the films show the antiferromagnetic transition at 14 K as well as the superconducting one at 1.8 K. The availability of these oriented films enables detailed investigation of the Hall effect and of the critical current in the superconducting state. The Hall coefficient exhibits a maximum around 55 K and a minimum at 6 K and saturates quadratically towards zero temperature. The temperature dependence can be understood as a superposition of skew scattering (dominating at high temperatures) and coherent effects at low temperatures. Critical current measurements in the superconducting state showed this magnetic heavy fermion superconductor to be able to carry 6.5 x 107 A]m2 at 0.32 K and 1 T external field. The activation energy for flux greep appeared to be 25 K in zero external field and to decrease with field along the c-axis to 5 K at 3 T. UCu4A18 crystallizes in the ThMn~2 structure and orders antiferromagnetically below 30 K. When z is increased in the series UCu4+xA18_x, then the antiferromagnetism decreases for z > 1.5 and a heavy fermion state emerges resulting in 7 = 800 mJ/(mole K2) at z = 1.9 (Geibel et al. 1990, K/3hler et al. 1990). Thermal and transport properties can be described

HEAVY FERMIONS AND RELATED COMPOUNDS

45

in a single ion Kondo effect model. Lunkenheimer et al. (1994) have prepared this system in amorphous thin film form. In this state the onset of magnetic order is suppressed with respect to the crystalline results at low z values and the coherent heavy-fermion behaviour is suppressed at large z values. From dc- and ac-resistivity measurements they conclude that all sample reveal a single ion Kondo behaviour, but that significant deviations from dipolar Kondo effect are found for the lowest temperatures. 3. Conclusion Heavy fermions and other related magnetic compounds have attracted lots of attention over the last decade. They still from the subject of many publications: the subject is alive. In spite of this efforts a number of questions have remained unanswered up to now, see, e.g., Coleman (1995) in the final talk of the Amsterdam conference on Strongly Correlated Electron Systems. Future attention will probably be focussed on the zero-temperature transition from the 'paramagnetic' state to magnetic order and the accompanying non-Fermi liquid behaviour. Also the band structure will become more important, via further ab initio calculations, and via spectroscopic experiments. 4. A c k n o w l e d g e m e n t s During the past ten years I have had the pleasure to work with a number of PhD students: T.T.M. Palstra, A.J. Dirkmaat, T. Endstra, R.A. Steeman E.A. Knetsch, S.A.M. Mentink, A. Drost and S. Stillow. Without their efforts this chapter would not have been written. I thank J. Aarts and J.A. Mydosh for many stimulating discussions. Without the cooperation and friendship of many colleagues around the world the research on heavy fermion would be dule, if not impossible.

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chapter 2 MAGNETIC MATERIALS STUDIED BY MUON SPIN ROTATION SPECTROSCOPY

A. SCHENCK and F.N. GYGAX Institute for Particle Physics of ETH ZBrich CH-5232 Villigen PSI Switzerland

Handbook of Magnetic Materials, Vol. 9 Edited by K.H.J. Buschow 01995 Elsevier Science B.V. All rights reserved 57

CONTENTS 1. Introduction

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

2. Muon spin rotation (/~SR) spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.

Parameters + phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60 62 62

2.2.

Muon site and local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

2.3.

More on spin-lattice relaxation in the paramagnetic and ordered state . . . . . . . . . . . . . .

76

3. Review of results in elemental metals and alloys

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

80

3.1.

Spontaneous dipole and hyperfine fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.

Critical phenomena

3.3.

Ferromagnetic 3d-element based alloys

3.4.

Chromium and its alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

3.5.

/~-SR in Fe and Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

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

4. Review of results in intermetallic compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96 101

104

4.1.

Compounds involving transition elements

4.2.

Intermetallic compounds containing rare earth elements . . . . . . . . . . . . . . . . . . . . . . . . .

114

4.3.

Intermetallic compounds containing actinide elements

175

5. Review of results in magnetic insulators 5.1.

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

84

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

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

Oxides with corundum-type structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.

Orthoferrites and RNiO3 perovskites

5.3.

Miscellaneous mostly Cu-based and layered oxides

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

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

104

199 199 209 212

5.4.

MnO

5.5.

Magnetic fluorides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

5.6.

COC12.2H20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219

5.7.

Solid oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. Review of results in layered cuprate (high Tc) compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.

215

220 221

La2CuO 4 and related compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

6.2.

YBa2Cu30= and related compounds

244

6,3.

Bi-based (2212)-compounds

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

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

58

269

MUON SPIN ROTATION SPECTROSCOPY 7. Study of magnetic order in organic compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.

(TMTSF)2X

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

7.2.

Ni2(C2HsN2)2NO2(C104) (NENP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3. p-NPNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of some of the used abbreviations and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 274 274 279 280 281 281 284

1. Introduction

This chapter contains a review of results on magnetic materials obtained by a single technique, namely Muon Spin Rotation (#SR) Spectroscopy. In this respect the present chapter is quite different from most other contributions in this series of volumes which focus on (all) the magnetic properties of certain materials as collected by different methods and techniques. The reader will find that the application of #SR-spectroscopy has in many instances just confirmed what was known already from other studies, but often has added a new flavour or twist to our knowledge and understanding of established magnetic phenomena. Beyond that, however, when using positive muons (#+) #SR spectroscopy has shown the potential to uncover new and unexpected magnetic features owing to its high sensitivity to very small magnetic fields and spatially inhomogeneous properties. Thus #SR-spectroscopy is most powerful in the field of small moment magnetism and in all instances when magnetic order is of a random or very short range nature and where neutron scattering will fail. Compared to NMR-spectroscopy #SR-spectroscopy will work also in the case of very broad lines (up to 100 MHz) and very short relaxation rates (down to 10 -8 s). On the other hand relaxation phenomena involving Ta > 100 #s are out of the time window accessible by #SR. In this respect #SR and NMR are rather complementary techniques. But note that generally the implanted positive muons (#+) are found at interstitial sites and, therefore, probe the magnetism from a different perspective. The local probe aspect of #SR and its sensitivity to inhomogeneous features has in particular brought the real nature of solids into light which are never perfect in lattice structure, stoichiometry and morphology, and hence also magnetic properties do not show up in an ideal manner. For example, in many intermetallic compounds magnetic order is found to be established in a spatially inhomogeneous way, leaving sometimes a fraction of the sample in the paramagnetic state even far below the phase transition temperature. There are even examples where magnetic order evolves so gradually in space and with temperature that the concept of a cooperative phase transition looses its meaning and consequently no anomalies in the specific heat are observed. It is in fact an open question whether such observations have to be correlated only with the sample quality (sample quality certainly matters) or whether they could also reflect more intrinsic properties. Such distinction, on the other hand, may be quite artificial: real solid compounds are what they are and their imperfections are an integral part of them. To perform #SR measurements positive muons have to be implanted into the material of interest. How innocent are the implanted #+? Do they, by their presence, 60

MUON SPIN ROTATIONSPECTROSCOPY

61

modify local properties, including magnetic ones? Note that practically only one muon will be present at a time in the sample (average # lifetime is 2.2 #s) and that the total number of #+ implanted during an experiment exceeds rarely 101° which number is to be compared with the usual density of host lattice atoms. In effect radiation damage will be of no concern whatsoever. In metals the muons's positive charge will be screened jointly by the conduction electrons piling up at and around the #+. The screening is usually, at normal conduction electron densities, accomplished within a distance of the order of the Bohr radius which is small compared with interatomic distances in a solid. Friedel oscillations will cause a ripple in the conduction electron density distribution outside of the screening cloud which will produce additional electric field gradients at the nearest host neighbor sites (also in cubic systems). Most importantly the presence of the #+ causes a local lattice dilatation: the nearest neighbor host atoms may be pushed away by a few % of their rigid lattice distance from the interstitial positions, changing also locally the interatomic distances. These are all well known effects when hydrogen is introduced into a metal in small concentrations (with #+ one probes the infinite dilution limit!). It is conceivable that the changed distances could have an effect on the magnetic coupling of the atoms in the vicinity of the #+, but so far there is no compelling evidence for such a possibility. In any case #SR-data have always reflected magnetic phase transitions at temperatures in agreement with bulk determinations of Tc or TN. The only clearly established effect of a #+ induced local modification was found in PrNi5 (see section 4.2.4) where the crystalline electric field (CEF) splitting of the ground state of the #+ nearest Pr 3+neighbor was significantly altered. The consequences of this observations for the interpretation of magnetic features observed by #SR in other rare earth or actinide based compounds are not clear yet but will require attention in all future investigations. In insulators the #+ is observed to be present in the form of the strongly paramagnetic muonium (#+e-) atom or is found to be bound chemically to one of the constituents (in particular to oxygen, if present). So far muonium has only been observed in the magnetic compound MnF2, while in all other investigated magnetic insulators the #+ appears to be in a diamagnetic state (see section 5). No indications for #+ induced modification in local magnetic features were ever manifest. So the answer to the above questions is that, as far as experience teaches, the positive muon is a fairly innocent magnetic probe, but that this has to be studied in each case, where it could matter, with care. Another problem in #SR studies is connected to the question of the #+ site in the lattice after implantation. To extract quantitive information from #SR-data knowledge of the site is a prerequisite, but often the site is not known. Therefore the reader will notice that the discussion of #SR results is almost always intimately interwoven with considerations of the muon's possible or actual site or sites. This review attempts to include all material on #SR studies of magnetic compounds up to the sixth international conference on #SR spectroscopy, held in June 1993 on the island of Maui, Hawai. However, we were forced by space and time limitations to skip one very important field of #SR-applications in magnetism, namely the very

62

A. SCHENCKand F.N. GYGAX

successful study of spin glasses. It is hoped that this very special subject will find coverage in some future review article.

2. Muon spin rotation (pSR) spectroscopy #SR spectroscopy is a variant of other well known hyperfine probe techniques such as NMR, 7"/PAC and PAD, and M6ssbauer spectroscopy. Therefore in #SR spectroscopy one measures basically the same parameters as in the other methods. In the following section 2.1 these parameters will be briefly recapitulated and some specialities, when using the #SR technique, will be pointed out. The technique itself will only be sketched briefly. The interested reader is referred to the many articles that describe the technique in detail (see, e.g., Chappert 1984, Schenck 1985, Chappert and Yaouanc 1986a, Cox 1987, Seeger and Schimmele 1992, Smilga and Belousov, 1994). Section 2.2 discusses the connection between the #+ site and local magnetic fields and section 2.3 provides some material on #+ spin relaxation due to the temporal fluctuations of the host magnetic moments. By far most of the investigations were done by using positive muons (#+). In a very few cases also negative muons (#-) have been applied. Positive muons are usually implanted at an interstitial site (the same site that is usually occupied by hydrogen in metals) and probe the magnetism from this point of reference. In contrast negative muons are captured into a ground state Bohr orbital of the host crystal atoms at substitutional positions. In this respect # - S R has more in common with NMR. However, the negative charge of the # - , close to the host nucleus, reduces the total nuclear charge seen by the electrons to effectively ( Z - l), thereby transforming the #--atom to an impurity atom, different in valency from the original one. In addition the captured # - has a much reduced polarization in the lowest Bohr orbital (< 1/6 of initial polarization) and a reduced effective free decay rate which renders # - S R much more difficult and limited in applications.

2.1. Parameters + phenomenology 2.1.1. Brief description of the #+SR technique #SR spectroscopy rests on the weak decay of the #+ : #+ --+ e + + ue + ~u which, because of parity violation in the weak interaction, leads to an asymmetric distribution of the decay positron with respect to the spin of the decaying #+: Are+(0) oc 1 + A cos 0,

(2.1)

where 0 is the angle between the e+-trajectory and the #+ spin. (Other relevant properties of the #+ are listed in table 2.1.) Hence by measuring the positron distribution it is possible to determine the original #+ spin direction. This, of course, requires the observation of many decays and implies that the participating #+ all possess initially the same spin orientation, i.e. that they are polarized. Polarized #+-beams with polarizations up to ~ 100% are available at the so called meson

MUON SPIN ROTATION SPECTROSCOPY

63

TABLE 2.1 Some properties of the muon where me is the electron mass, mp the proton mass and /~p the proton magnetic moment. Property

Values

Mass (m u)

206.76835(1 l)me = 0.1126096mp = 105.6595 MeV c -2 +e, - e ! h 2 3.1833455(5) 13.553879 (± 0.2 ppm) kHzG -1 2.002331848(17) ±l-y~lI (+ : #+, - : # - ) 2.19703(4)/~s

Charge Spin (I) Magnetic moment (/~) (in units of/~p) Gyromagnetic ratio (7~/2r) 9 factor (g~) Direction o f / ~ Lifetime (%)

TABLE 2.2 List of proton accelerators with #SR facilities (1993). Name

Country

Beam mode /z+-beamlines and/~SR-instrumentation

PSI

Switzerland

DC

ISIS(RAL)

UK

pulsed

JINR (Dubna)

Russia

pulsed or DC

PNPI (Gatchina) Russia LAMPF USA TRIUMF Canada

pulsed pulsed DC

BOOM (KEK) Japan

pulsed

2 surface (4 MeV)/~+ beamlines (1 dedicated to/~SR), 2 decay beamlines (20-50 MeV), 6 spectrometers available, including a low temperature set-up (dilution refrigerator) and a high pressure set-up. Good for ZF, LF, TF-measurements. Ultra slow muon beam line under development. dedicatedsurface/~+ beamline with 3 experimental ports, one equipped with a dilution refrigerator. Best for ZF and LF-measurements. 1 decay beamline with several ports (RIKEN-RAL), under construction. 1 decay beamline, several ports 1 general purpose spectrometer(MUSPIN). LF, ZF, TF-measurements possible. Surface muons also available in the future. 1 decay beamline, ZF, TF-measurements possible. 1 surface/z + beamline, best for ZF, LF-measurements. 3 surface/z + beamlines, 2 decay beamlines. Dilution refrigerator available, 5 T-SC-magnet, two general purpose spectrometers with various cryostats. 1 dedicated #+ surface beamline, dilution refrigerator available, if-resonance spectrometer, superconducting magnet for LF-measurements. Ultra-slow-muon beamline under development.

factories and a few other m e d i u m energy accelerator centers (see table a s y m m e t r y parameter A in eq. (2.2) is basically given by A = Pa,

2.2). T h e

(2.2)

where P is the b e a m polarization and a an intrinsic asymmetry parameter which is determined b y the weak interaction decay m e c h a n i s m . If all decay positrons,

64

A. SCHENCK and EN. GYGAX

irrespective of their energy 1, are detected 2 with the same efficiency, an average of = 0.3 will result. The total asymmetry is thus quite sizable and generally much larger than in nuclear 13-decays. If the spin polarized #+ are stopped or implanted in a target in which they are ~bjected to magnetic interactions, their polarization/5 may become time dependent: P(t). The evolution of/5(t) can be monitored by measuring the positron distribution as a function of elapsed #+ life time. In fact it suffices to monitor only the positron rate into a particular direction, say along the direction of the initial polarization /5(0). This geometry will be assumed throughout this section. It is straightforward to extend the discussion to other directions of observation. The positron rate dN~+ (t)/dt as a function of elapsed #+ life time is then given by the expression (see, e.g., Schenck 1985). dN~+(t) 1 d-----~ - No ~ exp(-t/%)(1 + A/5(t)./5(O)/P(O)),

(2.3)

where % = 2.2 #s is the average/z + life time and the exponential factor accounts for the decay of the #+. /5(t) •/5(0)/P(0) 2 can be identified with the normalized #+ spin auto correlation function

G'(t) -

(s(o) 2)

(2.4)

This function contains all the physics involved in the magnetic interaction of the #+ inside the target or sample. Henceforth we will define P(t) as the projection of fi(t) onto the initial polarization/5(0), choosen to be the direction of the positron observation, i.e. P(t) =/5(t)./5(O)/P(O) = G(t)P(O). P(t) is called the #SR signal, sometimes it is also referred to as the asymmetry since it determines the effective decay asymmetry in the distribution eq. (2.1). Figure 2.1 represents in a schematic way a typical experimental arrangement for measuring the distribution given by eq. (2.3) and to extract P(t). Since it involves the measurement of individual #+ life times (on an event after event basis in a continuous (dc) #+ beam; in a somewhat different fashion in a pulsed beam with pulse width << %) it is referred to as a time differential technique. For details see Schenck (1985). Also time integral detection schemes have been developed which we will not discuss here. Actual #SR experiments benefit from the availability of two types of #+-beams. The first, conventional type, is formed from pions decaying in flight (Tr+ -+ #+ + u~) and involves relatively high #+ energies (~ 40-50 MeV) and spin polarizations in the range (60-80)%. Muons in such a beam need first to be 1 Since the decay of the # + involves three particles (e +, Ue, r~,) the e + energy will vary continuously between zero and a maximum energy E ~ _~ 1/2muc 2 ,-- 52 MeV. On the average the e + energy amounts to ~ 30 MeV. 2 To detect both incoming # + and outgoing positrons plastic scintillators connected via light pipes to photomultipliers are commonly used.

MUON SPIN ROTATION SPECTROSCOPY

~÷

65

e÷

defector

® B

detector ,

)j*

,

,

~ 0,1~

PU

be'am-

= ~J*Larmorfrequency

i

,

i

,

"

,

,

,

,

,

,

POLARIZATION

~0.0

.

".'"

~'.=

01

target

asymmetric decay e+ distribut.

0 12,/3

/,

S

6

TIME[psec]

lysis Sfarf _I

Clock

I Hisfogramming memory

3

]_Stop

/

'1 ' ' ~' ' ' ' ~ / i , i J

IDispta y 0

~

0 1 2 3 4

S 6 7 TIHE[ysec]

Fig. 2.1. Schematic illustration of the/~SR-method pertaining to the transverse field technique, i.e. the initial /z+-polarization /3(0) is perpendicular to the applied field and the /z+ will perform a Larmor precession.

degraded by a suitable moderator (the stopping range at 50 MeV is about 15 g cm -2) before they are brought to rest in the sample of interest, which usually has a mass per unit area (facing the beam) smaller than the stopping range. The sample has to have a certain minimum thickness if all #+ are to be stopped. This minimum is related to the momentum resolution of the #+ beam and to range straggling effects and amounts to typically several g/cm 2, which prevents the use of thin samples. The advantage of this type of beam is that/z + will stop rather homogeneously throughout the sample and that one can be sure, therefore, to really probe bulk properties. The other type of beam is called a surface beam and originates from pions decaying at rest close to the surface of the primary pion production target 3. Those #+ have an energy of < 4.1 MeV, almost 100% spin polarization and a range in matter of 170 mg/cm 2. The main advantage of this type of beam is the possibility to use rather thin samples (some fraction of a mm), while required lateral dimensions are similar to those in a decay beam (,-~ (0.5-4) cm2). Occasionally it is observed that the top layer in which the 4.1 MeV #+ are stopped is different from the bulk so that rather misleading results are obtained. P(t) can be monitored under three different conditions: (i) no external field is present, P(t) evolves solely by the interaction of the #+-spin with internal fields. This is referred to as zero field (ZF)-#SR; (ii) an external field ~r is present perpendicular to /3(0). This is referred to as transverse field (TF)-#SR; (iii) an external field is present parallel to/3(0). This is referred to as longitudinal field (LF)-#SR. Accessible parameters are discussed below. 3 Pions are produced by high energy (> 180 MeV) protons impinging on a suitable target, e.g., graphite.

66

A. SCHENCK and EN. GYGAX

2.1.2. Larmor or precession frequency: wu The Larmor frequency or the splitting frequency of the #+ (ra = +l/2)-Zeeman states is usually measured by~ arranging a magnetic f i e l d / t to be perpendicular to the initial spin polarization P(0) of the implanted #+ (transverse field (TF)-pSR). The signal then observed is equivalent to the free induction decay signal in NMR. Alternatively the splitting can be measured by magnetic resonance techniques in which case /4 is directed parallel to /5(0). wu and H are related to each other through the gyromagnetic ratio %, i.e.

wu=%H.

(2.5)

wu is independent of the relative directions o f / ~ and/5(0). For arbitrary angles 0 between/~ and fi(0) the evolution of the #+ polarization component along/5(0) is given by (precession on the surface of a cone with aperture 0)

P(t)/IP(0)l

= cos 2 0 q- sin 2 0 cos wut.

(2.6)

The field H may be extemally applied or may be intrinsically present at the #+stopping site. Of course there could be different fields at different sites in which case several differently precessing components may be observed, i.e.

P(t)/IP(O)I = ~ Ai (cos 20i + sin 2 0i cos wit)

(2.7)

i

with

Ai = ~ Pi(O)/P(O) = 1. i

(2.8)

i

To make the splitting readily visible P(t) is often subjected to a Fourier analysis (Brewer 1982). The measurement of wi determines the local fields Hi.

2.1.3. Signal amplitudes: Ai Ai determines 4 the fraction of #+ or, since the #+ are implanted more or less homogeneously across the sample volume, the fraction of sites or of the sample volume which is associated with a particular Hi. Because the initial /5(0) is usually well known and because of the normalization condition, eq. (2.8), the Ai can be absolutely determined, including the identification of a so called missing fraction which may arise from extremely quickly depolarizing or from extremely rapidly precessing #+, introducing a time dependence too fast to be resolved by the spectrometer. The cos 2 0 term(s) in eqs (2.6) and (2.7) may also be invisible due to their time independence. The amplitude of the time dependent term, Ai sin 2 0i, allows to determine Oi and therefore the direction of Hi with respect to the initial polarization/~(0). This is, 4 The Ai may also be called (partial) asymmetries as follows from eqs (2.7) and (2.8) and the fact that P(t) determines the effective decay asymmetry.

MUON SPIN ROTATION SPECTROSCOPY

67

of course, restricted to one domain monocrystalline samples. In polycrystalline or monocrystalline samples with randomly oriented domain magnetizations all possible directions of Hi with respect to /5(0) may occur and one has to integrate over all possible directions in which case X

2

P(t)/lfi(O)l=~)--~A~ ~1 + -c ocos it3

1

/) = 3 +

2

~

Ai cos

wit.

(2.9)

The amplitude of the precessing component is reduced to 2/3. The l/3-term just reflects the fact that for a completely random orientation of/~i on the average 1/3 of all #+ will experience a n / t i parallel to their spin and hence will not precess. More complicated situations may be visualized in systems with complex or incommensurate magnetic structures. In any case a careful study of signal amplitudes as a function of sample orientation could be of great help in unravelling the structure of internal fields.

2.1.4. Knight shifl: Ku The Knight shift is defined as usual:

Hint-Next K• --

(47r - N ) X ,

(2.10)

Hext

where the second term corrects for the demagnetization and Lorentz field. N is the demagnetization factor of the sample and X the bulk magnetic susceptibility (in emu/cm3). For more details see section 2.2.

2.1.5. Spin lattice relaxation rate:

/~i =

1/T1

It describes the rate of repopulation between the #+-Zeeman states (m = ±1/2), if there is a deviation from thermal equilibrium, and involves spin flip transitions. The spin flip transitions are induced by fluctuating magnetic fields with components perpendicular to /5(0). Each spin flip transition requires an energy transfer of wh from or to the lattice heat reservoir. The initially polarized/z + (being in one of the two Zeeman states) will become depolarized according to

P(t) =

I~(O)l exp(-Alt).

(2.11)

In #SR, P(t) is measured either in zero external field (ZF-#SR): PZF(t) or in an external field (defining the z-axis) parallel to P(0) (longitudinal field (LF)-#SR): PLF(t). In ZF-field PzF(t) may be also affected by static field distributions (see below) and it may not be possible to distinguish dynamic and static features. In longitudinal fields one can decouple the #+ spin from static fields (see below) and PLF(t) will only reflect spin lattice relaxation. However spin lattice relaxation may be affected by the applied field (see, e.g., Abragam 1970, Slichter 1978). For more details on spin-lattice relaxation mechanisms see section 2.3.

68

A. SCHENCK and EN. GYGAX

2.1.6. Spin-spin or transverse relaxation rate: /~2 = ] / 7 2 (homogeneous line broadening) It describes the loss of polarization of the ~recessing spins, usually observed by applying a magnetic field perpendicularly to P(0) (cos 0 = 0): PTF(t). Equation (2.6) has then to be modified to PTF(t) = I/:3(0)le -'k2t COS totzt.

(2.12)

Relaxation is a result of dephasing of the precessing #+. It does not involve any energy transfer between the #+-spin system and the lattice. For very fast dynamics T2 -~ Ta (see, e.g., Slichter 1978). For arbitrary direction of H with respect to/5(0) eq. (2.6) can be written as

P(t)//5(9)1

= c°s2 0

e -*IT' + sin 20 e -t/T2 cos tout.

(2.13)

2.1.7. Inhomogeneous line broadening, Gaussian relaxation rate Inhomogeneous line broadening is a result of an inhomogeneous field distribution. #+ at different sites will feel more or less different fields. Such an inhomogeneity can be caused by nuclear or electronic dipole fields, which are static on the scale of the #+ life time. Inhomogeneous line broadening can be observed in a TF-#SR experiment. Choosing the applied field Hext to be much stronger than any fields arising from internal sources only the distribution of the internal field components H int along/~ext are of concern. If the spectral distribution of H int is given by F ( H int) with f + ~ --zHintb-'(/-/int'~dH' ; -i-nt--,--z - z = 0, eq. (2.6) (cos 0 = 0) can be written as

1F

PTF(t)/Ifi(O) I = ~

dto~ F(toz) cos (too + to~)t 0(3

=

= G ~ ( t ) cos coot,

too = %Hext.

(2.14)

The precessing spins suffer a loss of polarization (again by dephasing) which is described by the relaxation function GTF(t). The latter is given by the Fourier transform of the spectral distribution /;'(Hint). In favorable cases F ( H int) may be made visible directly by Fourier transforming the measured PTF(t). If F ( H int) is given by a Gaussian distribution: ( 1 / ~ ) e x p ( - H e z / 2 M 2 ) , GTF(t) will be given by a Gaussian as well GTF(T) = exp

( 1 2 2 ) -- -~ 7~Met

= exp

(

1 ) - - a2t2 2

(2.15)

where M2 is the second moment of the field distribution and ~r in this case is usually referred to as the relaxation rate. If F(H~) is given by a Lorentzian distribution GTF(t) assumes a simple exponential behaviour as in the case of homogenous line broadening.

MUON SPIN ROTATION SPECTROSCOPY

69

2.1.8. Inhomogeneous line broadening, zero field (ZF) case First realized with #+ and so far also restricted to #SR-applications this is a very powerful speciality in #SR-spectroscopy. No external field is applied but the implanted #+ are exposed to randomly oriented internal fields which are also statistically distributed in absolute values. If each cartesian component of the internal fields is Gaussian distributed with identical second moment A 2, P(t) (measured along/5(0)) assumes the form

P(t)/I/5(O) I --

1

2 (1 - ' y u2A 2t 2) exp ( - 1 2A2.2"~ ~ ")'/zZ3 t ) .

GGKT(t)=- ~ q- ~

(2.16)

This is the famous Kubo-Toyabe function first derived by Kubo and Toyabe (1967) and later redetected for #SR by Yamazaki (1979) and first observed by Hayano et al. (1979). If the components are Lorentzian distributed one derives instead 1 2 GLT(t) = ~ 3 + ~ (1 - %at) exp(-%at),

(2.17)

where a is the half width at half maximum of the Lorentzian distributions.

2.1.9. Decoupling from static internal fields in longitudinally applied fields (LF) The Kubo-Toyabe expressions can be generalized to include a longitudinally (i.e. parallel to fi(0)) applied field H. For Gaussian distributions Hayano et al. (1979) derive

G~w(t, H) = 1 - 2 H--~

H03 ao

1 - exp

i1

)

)

- ~ "/2AZt2 cos(%Ht) +

exp(-y2/2) sin(Hy/A)dy.

(2.18)

Figure 2.2 displays G~KT(t,H) for various H including the zero field Kubo-Toyabe function. The effect of H is to remove the time dependence and eventually for very large H (i.e. Hext/A >> 1) to restore P(t) to the initial IP(0) I even at long times, reflecting the decoupling of the #+ spin from the internal fields (the cone on which a #+-spin precesses shrinks to zero aperture). In case of very large A the rapid initial decay of GKT(t) may occur within the dead time of the spectrometer and only the asymptotic GKT(Oe) may be observed. Even in this case A can be determined by measuring the field dependence of GKT(ee, H).

2. i. i0. The effect of slow fluctuations (motional averaging) on the Kubo-Toyabe signal In the two previous sections static inhomogeneous field distributions were assumed from the outset. If there is also some time dependence involved, i.e. the internal fields

70

A. SCHENCK and EN. GYGAX 1.0

_5

0.8 0.6

L~

0.4

0.2

0.0 0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0

Time ((},uA) "1 )

Fig. 2.2. Display of the LF and ZF Gaussian Kubo-Toyabe function GGKT(t,H). The time is measured in units of (%,A) -1 and the applied field H in units of zl. For H/A >> 1 G~T stays close to 1 reflecting the decoupling from the internal fields.

become stochastic functions of time, the ZF Kubo-Toyabe function modifies to what is called a dynamic Kubo-Toyabe function. In general this function cannot be written in analytical form. Assuming a Gaussian-Markovian process Kubo and Toyabe in their seminal 1967 paper were the first to include also stochastic fluctuations. In #SR the dynamic Kubo-Toyabe function is usually calculated on the basis of a strong collision model, which implies a pure Markovian process (Hayano et al. 1979). The parameter introduced in the latter treatment is a fluctuation rate u of the local fields, whereby each fluctuation event destroys the correlation between the internal field distributions before and after the event completely. In order to produce an effect the spectral distribution in the static limit has to be Gaussian or such that it has a finite second moment. This excludes a Lorentzian distribution from the present considerations. Figure 2.3 displays the effect of u on a Gaussian Kubo-Toyabe signal. For very small u (i.e. u/TA << 1) only the 1/3-term is affected and decays like GzF(t) =

G~T(t, u)lt>3/za =

1 ( 2 )ut

~ exp

reflecting directly the fluctuations rate u. For very fast fluctuations (i.e. one gets

GzF(t) =

GP~T(t,u) =

(2.19)

-- ~

( 22)

u/%A

>> 1)

27u A

exp

--

/J

t

(2.20)

which is of the form of eq. (2.11) and we are back in the spin lattice relaxation regime.

MUON SPIN ROTATION SPECTROSCOPY

71

1.0 0.8

0.6

0

0.4

~\\ \

\

_

~

v/A = 0

0.2

0.0

0

2

4

6

8

10

Time ((~t/~) -1 )

Fig. 2.3. Display of the ZF dynamic Kubo-Toyabe function OGT(t, u), where u is the fluctuation rate of the local fields, u is expressed in units of 7 u A and the time in units of (%~A)-I.

Similarly the longitudinal field Kubo-Toyabe signal can be calculated. For very rapid fluctuations eq. (2.11) is again recovered. For extremely rapid fluctuations we enter again the spin lattice relaxation regime where depolarization in a longitudinal field is only arising from spin flip transitions. See also section 2.3.

2.1.1 I. Special features in dilute magnetic systems By dilute systems we mean on the one hand spin glasses, where the magnetic ions are indeed only present in small concentrations but on the other hand also perfectly ordered magnetic systems which contain a small number of randomly missing moments (magnetic holes) or which contain a few lattice defects or impurity atoms, perturbing the magnetic structure only slightly. Implanted/z + in such systems possess a wide distribution of distances to the magnetic ions or the defect centers and hence also a wide distribution of magnetic coupling strength with these centers. As is well known the spectral distribution of fields probed by the #+ (or any other hyperfine probe) assumes under such circumstances a Lorentzian distribution (Walstedt and Walker 1974) and the W-evolution of the #+ polarization follows eq. (2.12) or in ZF eq. (2.17), respectively. (For refined expressions in the case of real spin glasses see Uemura et al. 1985) If the internal fields are allowed to fluctuate in time with a unique rate u eq. (2.11) changes to PLF(t) = I/3(0)1 exp ( - x/~¢)

(2.21)

(McHenry et al. 1972). For very fast fluctuations GTF(t) and GGT(t) assume likewise this square root stretched exponential form. Note that for an ideal Lorentzian distribution the fluctuations should have no effect on the time evolutions of the #+ polarization. However, in reality even for very dilute magnetic systems the spectral

72

A. SCHENCK and EN. GYGAX

distributions always possess a finite second moment, facilitating motional averaging effects (see, e.g., Uemura et al. 1980). An obvious generalization of eq. (2.21) is the stretched exponential expression

IF(o)l exp ( - ( A l t ) ~)

PLF(t) =

(2.22)

which sometimes describes #SR-data quite well. Its implication may not always be clear. One cause could be a distribution of fluctuation rates u rather than an unique u. 2.2. Muon site and local fields Like any other hyperfine field probe the #+ interacts with magnetic atoms through the dipole-dipole interaction and more indirectly in metals through the RKKY mechanism by means of which a non zero spin density is induced at the #+ site leading to a contact hyperfine interaction. In insulators a non-zero spin density at the #+ may be caused by transferred hyperfine fields involving covalency effects. In general the effective magnetic field at the #+-site in the absence of an external field is given by /31oc =/3¢ + B~ip,

(2.23)

where/3c is the contact hyperfine field and Bdip the net dipolar field. /~dip can be expressed as

.

(ff(ei) = ~

3(ei

-

~'.)(/7(,~i)

• (,~i -

e.)) "~

)

~dip (~/z -- ~ ) ' ~(ri),

(2.24)

(2.25)

i

where the sum runs over all magnetic atom positions ~ (lattice sum) and fi(~) is understood to be the static component (thermal average) of the total moment at a given site. The #+-position ~'u and the atomic sites are measured from some convenient origin. It is clear that Bdip depends crucially on the assumed #+-position +-~

and on the assumed orientations of the fi(Y0- The symbol Adip denotes the dipolar tensor with tr(Adip) = 0. In the ferromagnetic case all fi(~) will point in the same direction and are usually of the same magnitude. The evaluation of the lattice sum is split into two parts by separating the volume of the sample into a sphere around the ~+ (the Lorentz sphere) and the rest. Summing over the rest yields the Lorentz field BL = (47r/3))~rs and the +-+

demagnetization

field

/~dem

=

-- N

-

J~b, where -Ms is the magnetization of a single ++

domain (containing the #+) and .~rb is bulk magnetization of the whole sample. N is the demagnetization tensor. It is hereby assumed that the Lorentz sphere is smaller than a typical domain size. In non-magnetized ferromagnetic samples (all domain

MUON SPIN ROTATIONSPECTROSCOPY

73

magnetizations will add up to a zero total magnetization) Bde m = 0. Evaluation of the lattice sum inside the Lorentz sphere can be done by using the Ewald method (Meier et al. 1987) which assures a rapid convergence of the lattice sum. From symmetry considerations it will become immediately evident that in cubic systems with the #+ at a site of cubic point symmetry (e.g., tetrahedral and octahedral interstices in an fcc-crystal) the net dipolar field from the sources inside the Lorentz sphere will be zero. The tetrahedral and octahedral interstices in a bcc-crystal, on the other hand, which possess no cubic point symmetry, are associated with a non-zero net dipolar field. In this case the two types of sites each split into up to three magnetically inequivalent sub-sites with populations depending on the direction of the ordered moments (or the domain magnetization). The overall cubic symmetry then requires that the sum of net dipolar fields, weighted by the relative population of the sub-sites, yields zero again. Hence, if the #+ should happen to be implanted at tetrahedral sites in a ferromagnetic bcc lattice one may expect to find up to three different precession frequencies, the sum of which weighted by the signal amplitudes yielding indeed zero (see, e.g., the #SR-results on ferromagnetic iron, section 3.1). In antiferromagnetically ordered systems J~dem a n d /3L are necessarily zero. For simple antiferromagnets the lattice sum may be split into sums over ferromagnetic sublattices. Again symmetry considerations can help to identify those interstices at which the dipolar fields will cancel. Powerful programs have been developed which allow to calculate the net dipolar fields or the lattice sums for any kind of antiferromagnetic structure, whether single ~ or multiple q*, helical or with modulated moments. Commensurate structures will always lead to distinct net dipolar fields at a given type of site, but there may be many magnetically inequivalent sub-sites leading to a corresponding number of distinct precession frequencies. Incommensurate structures are more difficult to handle and usually are approximated by invoking very large magnetic unit cells. In any case implanting #+ in such systems a more complicated extended frequency spectrum can be expected, reflecting a more or less inhomogeneous field distribution over the #+-sites, which will lead to relaxation by dephasing (see section 2.1). If the local fields over the #+ sites vary sinusoidally in one dimension P(t) will be given by the Bessel function Jo(t). In metals the contact hyperfine field, or local spin density, respectively is induced by the magnetic moments via the RKKY mechanism which is linear in the moments. In ferromagnetic metals Bc is therefore proportional to the domain magnetization, i.e. /~c = Ac_~r~,

(2.26)

where Ac is a contact hyperfine coupling constant. Although Bdip may be zero at certain sites Bc will in general be different from zero. On the basis of the RKKY mechanism Ac may be expressed as (Kittel 1966, Schenck 1993) Ac -

87r g] - 1 __ Jsf(~,)r/(,Ftz), 3 gj

(2.27)

74

A. SCHENCKand EN. GYGAX

where gj is the Land6 factor of the electronic moment, Jsf(ft,) is an effective exchange coupling constant which depends on the #+ site via the distance to the neighboring electronic moments, and r/(Yu) is the so called spin density enhancement factor which incorporates changes in the local electronic density distribution due to the presence of the #+. r/(~'u) may be determined from Knight shift measurements in the paramagnetic state at high temperature (only the Pauli spin paramagnetism is left). In an antiferromagnetic structure Be may be written as ..,

Be(r,,)

8~

gj

: T rl(%) ~

-- 1

~

Jsf ([g~, - ~l)fi(~),

(2.28)

i

where we assumed that the spin density induced at the #+ by each electron moment fii at position ~ can be simply superimposed. 0Vsf(l~'u- ~l) is an effective exchange coupling constant for each moment fi(r~). This expression shows that in an antiferromagnetic structure Be can become zero at certain sites of high symmetry with r_eespect to the magnetic structure. From symmetry considerations follows that also Bdip will vanish at such sites quite in contrast to the ferromagnetic case. So far we assumed the #+ to be fixed at the geometrical center of an interstitial site. In reality the #+ performs zero point vibrations around this position. As Meier (1980) has shown this has no effect on Bdip as long as the #+-wave function is well confined to the interstitial site volume and does not overlap with the neighboring host magnetic moments. In contrast the contact hyperfine field/3c may be modified considerably by both the >+ zero point vibration as well as lattice vibrations (Manninen and Nieminen 1981, Estreicher and Meier 1982, 1984). Inclusion of such effects can account for small deviations between the temperature dependence of/3c and _~rs or the sublattice magnetization in antiferromagnetic systems, respectively. In the paramagnetic phase static magnetic moments can be induced by an external field Hext. At each magnetic atom site one has (2.29)

#+ : X++ a t " /~ext, ++

where Xat is an atomic susceptibility tensor. The moment arrangement corresponds to a ferromagnetic order. Bdip (eq. 2.25) can then be rewritten as follows /~dip(~'/~) = ~

~dip,i (7~/~ -- ~i)" ~ a t " / t e x t

(2.30)

i = Adip (r/z) " ++ Xat" /~ext,

(2.31)

..o.

where Adip (UIz) is again a traceless and symmetric tensor which depends on the assumed #+-site in a given crystal structure. Its calculation is straightforward. The total field (neglecting now BL and Bdem) at the #+ is

B. : I( xt + ETdip)l H xt + (&ip. B xt)/H xt + . . . .

(2.32) (2.33)

MUON SPIN ROTATIONSPECTROSCOPY

75

from which follows the dipolar Knight shift constant +4

+at

Kdip= (JtIext • Adip (r/z)' Xat" [text)/H2xt • Similarly the contact hyperfine field is given by

(2.34)

/3c(r~) = Ac Xat" /r~ext,

(2.35)

from which we obtain the contact hyperfine Knight shift constant 2 Hext)/Hext"

Kc = Ac (I-Iext ' ++ Xat" "

(2.36)

+-1.

This constant is isotropic as long as Xat is isotropic. Often one has to deal with axially symmetric systems in which case the total Knight shift is given by 1

/

\

K = ~ Ac ~(XII + 2X±) + 2(Xll - x±)Pz(cos 0))

+51 Adip ((Xll -- X±) + (X± + 2XII)P2(cos O)),

(2.37)

where 11, _1_refer to directions of/text parallel or perpendicular to the axis of axial symmetry. Specifically one finds Kit = (Ac + Adip)X[],

K± = (Ac- ~1 Adip)X±.

(2.38)

If X is taken to be the molar susceptibility in emu, Ac and Adip will be given in units of (mol/emu). If X is understood to be the atomic susceptibility measured in units of (#B/emu) Ac and Adip will be given in units of (emu/# B) per atom. Very often Ac and Adip are quoted in these latter units. Ac contains quite important information in that it depends on the exchange interaction between the conduction electrons and a local magnetic moment. The same exchange Hamiltonian is also responsible for the Kondo mechanism. However Ac depends also on the electronic structure established at and around the implanted #+. This feature has so far prevented, with some rare exceptions, a detailed analysis of measured' Ac-values. Kdip (or /~dip in the ordered state) is less affected by the presence of the #+ and reflects more directly intrinsic properties. Some lattice relaxation around the/z + will lead usually to a small reduction of Adip. Basically Adip is proportional to the atomic susceptibility (or the local moment) of just the nearest neighbours which are placed far enough away from the #+ to remain usually unaffected by its presence. Recently, however, some evidence has been obtained that the #+ could cause a change in the crystalline electric field splitting of rare earth atoms thereby changing their magnetic response (see section 4.24, Feyerherm et al. 1994b). Whether this could also have an effect on the size and orientation of ordered moments next to the #+ is not known. In any case one has never found any indications that the phase transition temperature measured locally by #SR is any different from its bulk value.

76

A. S C H E N C K a n d E N .

GYGAX

2.3. More on spin-lattice relaxation in the paramagnetic and ordered state #+ spin lattice relaxation or spin flip transitions are induced by fluctuations of the local field components perpendicular to the initial spin polarization. As in the static case the local fields may be of dipolar origin or of the contact hyperfine field typg. The fluctuations reflect the dynamics of the electronic moments or spins at the magnetic atom positions. Among other hyperfine probes #SR is special in that dipolar fields can be as strong as the contact hyperfine fields and contribute strongly to the #+ spin lattice relaxation rate. In contrast, in NMR and M6ssbauer spectroscopy dipolar field induced relaxation is usually negligible and not taken into account (see, e.g., Hohenemser et al. 1989)• Quasi-elastic neutron scattering is another method to investigate the spin dynamics. The differential cross section dcr2/dOd03 for magnetic scattering is proportional to the frequency and wave vector (momentum transfer) dependent Van Hove response function S(q, co) (Lovesey 1987), which can be approximated by

v(¢)

s(¢, w)= _1kBTx(0") 71"

(2.39) 032 q - / ' ( 0 " ) 2

X(0~) is the wave vector dependent susceptibility a n d / ' ( ( ) is inversely proportional to the lifetime of excitations with wave vector ¢. The #+ spin lattice relaxation on the other hand, can be expressed in the simplest case as A1 --- A 2 7r ~ ,S'(~, co), N ¢

(2.40)

which in zero field or for co/l'(() << 1 is simply

kBTA 2 -

x(q) q

(2.41)

r(¢)

A is proportional to the magnetic coupling strength. More complete expressions can be found in Yushankhai (1989), Lovesey et al. (1992a, b), Yaouanc et al. (1993a) and Keren et al. (1994b). Three important temperature regions have to be distinguished: (i) ordered regime (ii) critical regime just above the ordering temperature and (iii) high temperature paramagnetic regime• (i) In the ordered regime the dynamics arises mainly from spin wave excitations• However, since magnon energies are usually much larger than the #+ Zeeman splitting energy, conservation of energy forbids a direct (single step) process in which a single magnon is created or absorbed by a muon spin flip. Relaxation can only be affected in lowest order by a two step or Raman process. On this basis and describing the interactions of the host magnetic moments by a Heisenberg Hamiltonian and including a magnetic anisotropy term Yaouanc et al. (1991) have calculated in an

MUON SPIN ROTATIONSPECTROSCOPY

77

approximate way the #+-spin lattice relaxation rate in a ferromagnetic system. In the limit of Ean (= anisotropy energy) << k B T they arrive at

9 (.y,g,,B)2a k T2 In (k,r)

-

167r3

h:D 3

(2.42)

\ ~an ]"

G is a factor which depends on the #+ site and the lattice geometry (it may be zero) and D is the spin wave stiffness constant. Note that only the dipolar coupling of the #+ to the electronic moments leads to eq. (2.38) while the isotropic contact hyperfine interaction does not contribute. It is also found that )~1 is dominated by contributions from near the center of the Brillouin zone, i.e. by long wave length magnon excitations. Another remarkable feature is the fact that only longitudinal spin fluctuations (i.e. parallel to the easy axis) are effective in inducing/,+-relaxation also in zero external field - and can thus be studied by #SR under zero field conditions while polarized neutron scattering requires a non zero applied field. Two magnon induced relaxation in a ferromagnetic has also been considered by Lovesey et al. (1992a) with special reference to #+ stopped in EuO. Keren(1994) has considered a different, possibly quite efficient depolarization mechanism for the #+ with special reference to antiferromagnetic MnFe2 by noting that typical magnon energies could match well with the splitting of vibrational states of the #+. In this case absorption of a single magnon could promote the #+ to an excited vibrational level accompanied by a spin flip conserving both energy and the total angular momentum. It is expected that the temperature dependence of the thus induced relaxation rate will follow an Arrhenius law. (ii) In the critical regime just above Tc the spin dynamics is crucially affected by also the dipole-dipole interaction between the magnetic ions (Frey and Schwabl 1988, 1989b). This gives rise to the appearance of two separate terms which describe the spin dynamics along q (longitudinal mode, L) and perpendicular to ( (transverse mode, T). The two modes are differently effective in inducing #+ spin relaxation depending on the relative strength of the contact hyperfine and the dipolar coupling of the /~+ to the magnetic ions with angular momentum J and Land6 factor 9L. For an isotropic dipolar cubic (fcc) ferromagnet Yaouanc et al. (1993a, b) derive the following expression for zero applied field in the small q" limit (appropriate for temperatures close to Tc) =

-

v ao

dqq 2

2p 2 - + (1 _p)2 Ur(g) ~

'

(2.43)

where the weight factor p is given by

1 P=3+

n.H - -

47r

(2.44)

H is the contact hyperfine coupling constant (in units of 9 L # B / V where v is the volume of the primitive cell of the Bravais lattice) between the #+ and one of

78

A. SCHENCK and EN. GYGAX

the nu nearest neighbour magnetic ions and 1/3 arises from the dipolar coupling. If nuH/47r >> 1 (dominating contact hyperfine interaction) the transverse mode is dominantly contributing while in the other extreme of dominant dipolar interaction the longitudinal mode is most effective. xL'T(() and/~L,T(q) obey scaling laws in the critical temperature regime (dynamic scaling theory, renormalization-group calculations). Actual expressions may be obtained from the mode-coupling theory (Frey et al. 1988, 1989a, b). The susceptibility can be written in the Ornstein-Zernike form like xT(q ) _

T

q2 + ~-2

(2.45)

r q2 + q~ + ~ - 2 '

(2.46)

and xL(q ) =

where £ is the correlation length and qD is a dipolar wave vector (Yaouanc et al. 1993a). The correlation length has a power law temperature dependence T-Tc) =

~o

gc

-~ '

(2.47)

where u = 0.705 for a three dimensional Heisenberg magnet. I'L'T(() is more difficult to evaluate and the reader is referred to Yaouanc et al. (1993a) and Lovesey et al. (1992b). Using the scaling relation

F(¢) c
(2.48)

with 1

x = --

~q

(2.49)

(z is a critical exponent) and neglecting the dipolar interaction between the ionic moments (spin conserving dynamics, z = 5/2) and between the #+ and the ionic moments one finds (for z > 0) ~1 (3( ~3/2 --~ ¢ 2 + z - d - r t ,

(2.50)

where d = 3 is the dimensionality, r/~_ 0.03 for a 3D-Heisenberg system. Admitting now also non spin-conserving dipole-dipole interactions between the ionic moments one finds that z = 2 for the transverse mode and z = 0 for the longitudinal mode if the dipolar interaction is dominating over the Heisenberg exchange (i.e. q << qo)Consequently AT o( ~

(2.51)

MUON SPIN ROTATION SPECTROSCOPY

79

and k } is temperature independent. In the general case AI,ZF may be written as /kl,ZF = W [2p2/T(qo) + (1 - p)2IL(T)],

(2.52)

where ~ = arctan (qD~) and the functions IT'L(~o) are universal functions which have been evaluated by Frey and Schwabl (1988). A similar expression can be derived for a body centered cubic (bcc) lattice (Yaouanc et al. 1993a). Critical behaviour in an antiferromagnet has been considered by Lovesey (1992) (see also Keren et al. 1993), in the latter paper a comparison between #SR and neutron data has been worked out). For an isotropic antiferromagnet the #+ spin lattice relaxation rate diverges on approaching TN like (2.53)

)~1 (X {1/2

for both contact hyperfine and dipolar coupling. In an uniaxial antiferromagnet critical fluctuations are ineffective in inducing #+ relaxation if the coupling to the ionic moments is mediated by the contact mechanism. On the other hand, if the coupling is of dipolar origin one finds

(T-TN ) -~'

(2.54)

where ~z is the correlation length along the anisotropy axis. The temperature dependence in eqs (2.53) and (2.54) are precisely the same predicted for the NMR line width (l/T2). Equation (2.54) has been used to explain #SR results in MnF2 (see chapter 5). (iii) Spin lattice relaxation induced by paramagnetic fluctuations of local moments at high temperatures, i.e. far above any critical regime, have not been treated to any great extent. Assuming that the local moments fluctuate uncorrelated at high temperatures (F(q) = 1"(0) = r c 1, x(O) e< l / T ) one finds ),1 ~x rc,

(2.55)

where rc is a local correlations or relaxation time associated with the ionic moments. Two processes may be responsible: the RKKY interaction and the Korringa mechanism. The former induces a temperature independent, the latter a linear Tdependence of the ionic moment fluctuation rate %-z, i.e. (Hartmann et al. 1986)

7-c =

i ( c'Tc

J / ( J + 1)

+ -£ (IsfN(eF))

,

(2.56)

where e' is a constant, Tc is the ordering temperature (or the paramagnetic CurieWeiss temperature), Isf is the conduction electron - local spin exchange integral and N(eF) the Fermi energy density of states. Tc is proportional to the effective RKKY

80

A. SCHENCKand F.N. GYGAX

exchange coupling constant between a local moment and all other local moments which in turn is proportional to j2f. The RKKY part of eq. (2.52) follows also from a treatment by Lovesey et al. (1992b). Our discussion of the #+ spin lattice relaxation rate has been limited here to local moment magnets, ignoring also possible effects of the crystalline electric field (CEF) splitting of the ground state multiplet in f-electron ions. CEF-effects will be mentioned in some of the sections in chapter 4. Spin lattice relaxation in itinerant magnets is considered in section 4.1.

3. Review of results in elemental metals and alloys Since in general the magnetic properties of elemental metals are very well known, the results of #SR investigations on such systems have served primarily to develop a better understanding of technique and role of the implanted #+ as a magnetic probe. As a consequence the present section is organized somewhat differently than the following sections. While in the latter ones the results are listed compound by compound, in the present section the results on various elemental metals are collected in different paragraphs with respect to particular properties. The most salient features of crystal structures and magnetic properties for the magnetic elemental metals are sketched in table 3.1. Lists of essential data obtained in # + S R experiments, together TABLE 3.1 Schematics of the crystal structures and the magnetic orders for the magnetic 3d-transition metals and the lanthanides. The/z+ site is also indicated; it is in parenthesis if there is only a strong indication for that option. Element or alloy

Crystal structure

Fe Co Ni Cr

bcc hcp fcc fcc bcc

Pd (alloys) Pr Nd Sm Eu Gd Tb

fcc hex hex complex bcc hcp hcp

Dy Ho

orth hcp hcp

Er

hcp

T r a n s i t i o n Magnetic temperature o r d e r -+ 690 K

--+85 K

FM FM FM FM LIAF TIAF CAF AF AF AF AF FM FM AF FM AF FM AF FM AF

Critical /z+ temperature site -+ 1044 K ~ 1398 K --+ 627 K --+ 124 K -+ 312 K varying --+ 27 K -+ 19 K --+ 106 K --+ 89 K -+ 292 K -+ 223 K -+ 229 K -+85K -+ 178 K -+20 K --+ 132 K -+19K -+ 85 K

(O) O (O) (T±)

0 (0)

M U O N SPIN ROTATION SPECTROSCOPY

81

o N

r~

-~

#.

o

o

=o

~oN o

oo

~ ~ o •

o

~ ' ~ =.~

o~

u~ ¢0

O

09

o

odo

o

O9

~+

N

~,.~

V/

F'~NN

[-~

n~

+

O

9 "O

oo

o -I-

+

+

-l-

±

O

O

O

O

o

82

A. SCHENCK and EN. GYGAX

~ N - ~

~

. ~

~oz~z.~=&~ ed

~,~,~

~...~

~

~

~'-"

;~oe~'°

:~z

~.~

~ ~v,,

~<~

u

d

d

+

+

Z

~

~u

++

~mm

MUON SPIN ROTATION SPECTROSCOPY

o

..

o~

r~

o9

+

r~

t'¢3

0

"~

g N

~2

N

o

o

83

84

A. SCHENCKand EN. GYGAX

with the basically used experimental procedures are presented in tables 3.2 (3delements and their alloys) and 3.3 (lanthanides). These two tables are meant to give in the shortest form an overview with reference lists of the experimental #+ studies subject of the present chapter(sections 3.1 to 3.4). In section 3.5 we report briefly the results of investigations with negative muons (#-SR) in Fe and Ni.

3.1. Spontaneous dipole and hyperfine fields This section deals with the results obtained for the 3d-transition metals Fe, Co, Ni, Cr and for the lanthanides in the magnetically ordered states. Ferromagnetic Gd is presented as an introductory example in section 3.1.1. This choice is motivated by the fact that several problems related to the method can be treated in this case and because the #SR method has shown for Gd its best contribution to the understanding of magnetism in elemental metals.

3.1.1. Ferromagnetic Gd: an example The local field at the #+ in ferromagnetic Gd was first measured for two temperatures by Gurevich et al. (1975a), but the influence of the local dipole field was not considered in the interpretation. A thorough study was carried out in polycrystalline Gd by Graf et al. (1977). The results, together with the studies in Co, Dy, Fe and Ni are comprehensively reported in Denison et al. (1979). According to section 2.2 the local field at a #+ in an unmagnetized ferromagnet in zero field is given by /3.(~.) =/3L + B~ip(f.) +/3c(rn).

(3.1)

B~ip(/V~) is the lattice sum inside the Lorentz sphere. Gd has a hcp crystal structure. Between the Curie temperature Tc = 292 K and 230 K the Gd spins are aligned along the c-axis. Below 230 K the easy axis of magnetization deviates from the c-axis. The (non-pSR) literature lists various inconsistent temperature dependences of the angle 0 between magnetization and c-axis. Figure 3.1 shows ]B.[ versus temperature in zero field. Clearly this behaviour is very different from Ms(T), the domain magnetization in Gd. The interpretation (Denison et al. 1979) is summarized in the following points. (i) One starts with 4 7r~rs(T) '

(3.2)

where 2~rs is the domain or saturation magnetization. (ii) #+ sites: The hcp lattice offers tetrahedral (T) and octahedral (O) interstitial sites for the muon. A calculation of the dipole field tensor (assuming an undistorted lattice) indicates that the fields at T sites, O sites as well as at substitutional (S) sites are all parallel - but not parallel to the Gd moments - with of course different magnitudes.

MUON SPIN ROTATION SPECTROSCOPY

85

(iii) The complicated behaviour of B , versus T is essentially due to Bdip, more precisely to the variation of B~ip with 0. This is illustrated considering the behaviour of the field vector 3~, defined as the sum of/3L and/3c, left term of the equation (3.3) It is expected that/3c(T) and hence also J((T) will be approximately proportional to ~rs(T) - note that BL, Bc and )( are all parallel to Ms. The dependence O(T) can now be determined in the following way: For a given #+ stop site one draws a series of X(T) curves with 0 as parameter, using the measured values of B~(T) and the calculated B~ip(0 , r'tz)Such sets of curves are shown in fig. 3.2 a-c for substitutional, tetrahedral and octahedral p,+ sites. Since only the magnitude o f / 3 , ( T ) was determined experimentally, combinations using both B,(T) > 0 and B,(T) < 0 are retained. The boundary condition that X(T) has to follow the 0 = 0 ° curve between 230 K and Tc has to be considered - the points are indicated by solid circles in fig. 3.2 a-c. From the figure it is obvious that only the assumption of the octahedral site with Bu > 0 allows X(T) to be approximately proportional to Ms (monotonous dotted line in fig. 3.2 c). The O(T) dependence is so determined with an accuracy exceeding that of the neutron diffraction data, the resulting temperature dependence and the values obtained by other means are presented in fig. 3.2 d. The hyperfine field Be, obtained from I

I

I

I

I

I

1.5

20

1.0

v

z

g,

v

10

-t m

G_.~a 0.5 5

0

I

0

r 1 O0

tr LL

I

Temperature

I 200

i hl

m =

0

300

(K)

Fig. 3.1. Absolute value of the measured local field B~ vs. temperature in zero external field for polycrystaUine Gd. The right-hand scale corresponds tothe directly measured/zSR frequency. The solid line through the data points is meant to guide the eye (Denison et al. 1979).

86

A. SCHENCK and F.N. GYGAX

1.5 1.0

0 5 l Substitutional Site~

1~

~o.o

~ 0.0

X "0.5

Bl' < 0

-1.5

g . o , 5

-

i

-o.5 .,o..-.,. - 70"

0

-1.5

-2.0 0

-2.0

1O0

200

300

0

90

2.0

(b)

7O

1.0 '90"

60

0.5

50

~v 0.0

40

-1.o

B,, < 0

,"~f

1O0 '

(d)

8O

1.5

X -0.5 .

-I

B,,
20

-1.5

't

'

300 '

'|

1 I

°i#

30

F"

'

200

?""

?.',

r, ' t'ITo

10

-2.0

0

1O0 200 Temperature (K)

30(

0

0

1O0 200 Temperature (K)

300

Fig. 3.2. In a, b and c: field 2 =/3~, - / ~ d i p = /~c + BL plotted as function of temperature for three possible/~+ residence sites and for Bu > 0 and B~, < 0. The solid lines correspond to the indicated fixed values of the angle 0. For T > 230 K it is known that 0 = 0, the values of X deduced from the IB~,I measurements are displayed as solid circles for that temperature domain. Only the assumption of the octahedral/~+ site with B~, > 0 allows X(T) to be smooth and monotonous (dotted line in (c)) over the full temperature range. From that curve the function O(T)is determined and displayed in (d) as a solid line; in (d) the 0 values from neutron diffraction (full circles) and torque (full triangles) measurements are also indicated (from Denison et al. 1979). Be = X - B L (BL = (47r/3)Ms), is finally s h o w n in fig. 3.3. Bc is o p p o s i t e ( n e g a t i v e ) to Ms. T h e i n t e r p r e t a t i o n o f B c ( T ) for G d will b e briefly d i s c u s s e d t o g e t h e r with that for the o t h e r f e r r o m a g n e t i c e l e m e n t s i n section 3.1.3. N e w d a t a o b t a i n e d in a s i n g l e crystal G d s a m p l e b y H a r t m a n n et al. ( 1 9 9 4 ) call for slight a d j u s t m e n t s in the c o n c l u s i o n s w h i c h h a v e j u s t b e e n p r e s e n t e d - see s e c t i o n 3.1.2.

MUON SPIN ROTATION SPECTROSCOPY I

6 ~

,

I

~

i

I

n

°

I

r

m

4 o

I

87 I

)

\',

~

2

0

I

0

I

1oo

I

I

200

I

II

300

Temperature (K)

Fig. 3.3. Extracted hyperfine field Bc plotted as function of temperature. The dashed curve is the magnetization normalized to Bc(T = 0 K); Bc is opposite (negative) to the local magnetization. The particular interpretation of the Bu(T) measurements in Gd illustrates how new information about magnetism can be obtained from #SR. One has to stress, however, that since certain features of the #+ in the sample material are not always perfectly known, great importance has to be attached to the use of adequate assumptions, estimates of the importance of effects escaping control, and particularly to the consistency of the over-all picture obtained after analysis. Let's just mention two type of problems one can encounter in this context: - As far as the dipole field calculation is concerned, it matters of course to know the exact (interstitial) position of the #+. This achieved, the #+ zero-point motion, #+ tunnelling, departure from the ideal site position, e.g., because of bonding (in compound substances) or trapping at impurities may also play a role. Moreover, the muon can deform locally (mainly expand) the lattice possibly modifying the local structure anisotropy. - In case of sufficiently fast #+ motion (i.e. diffusion) a sampling of (magnetically) non-equivalent sites will average the v a l u e s / ~ ( ~ i) corresponding to the different sites i:

B.~eff= ~ a~/~u(~'.i),

(3.4)

i

where a~ stands for the #+ population fraction at site i. So it is possible, as, e.g., in Fe, to encounter an averaging, possibly to zero, of the different dipolar field contributions and thus to observe a single B~ff value instead of the characteristic multiplet corresponding to the number of non-equivalent #+ sites.

88

A. SCHENCKand F.N. GYGAX

3.1.2. Local fields Bu(~'u) and dipole fields in magnetic elemental metals Figure 3.4 a-d shows IBu[ as function of temperature for Fe, Co, Ni and Dy. As in the case of Gd (fig. 3.1), the complexity of these dependences is essentially due to Blip(T ). Where the dipole fields/3~i p vanish, in Ni because of its fcc structure, or in bcc Fe because of rapid #+ diffusion among interstitial sites (see section 2.1), Bu(T) (= BL + Be) is smooth and monotonous, whereas, e.g., in Co the drastic effect of B~ip is obvious (Graf et al. 1976b). In hcp Co (as in hcp Gd), the ordered magnetic moments produce a unique dipole field at each type of interstitial site. This is, therefore, not modified by diffusion, provided that diffusion takes place only among interstitial sites of the same type. Thus =

-

&

-

(3.5)

B~ip(~/~) can be calculated, e.g., as described by Denison et al. (1979). From the same ref. one see that/3 lip(T) deviates markedly from ~rs(T) if the direction of the axis of easy magnetization varies with respect to the crystal structure as a function of T. As for Gd, this is the case for Co, where in addition a modification from a hcp to a fcc structure at 690 K changes the picture even more. B~ip vanishes for the fcc structure. Measurements on Gd single crystal samples (Hartmann et al. 1990b and 1994, Kratzer et al. 1994a) confirm in essence the analysis presented in section 3.1.3 but show that the spin re-orientation between 245 and 220 K is steeper than anticipated and accompanied by a peak in the #+ depolarization rate. At a closer look a complex precession signal, which can be separated into two frequencies, is detected in this temperature range. Hartmann et al. (1994) conclude that the spin turning does not occur simultaneously in two different domains of the sample. The single crystal data imply that B~ and B~i obtained by Denison et al. (1979) cannot both be strictly correct. Either Bc un~fergoes a change around 230 K which is directly coupled to the spin turning, or the value of the dipolar field used in the earlier evaluation is slightly too low. It is noteworthy that so far all the #SR data obtained from the magnetic elements have been explained using the dipolar field calculation under the assumption of localized dipole moments. The pressure dependence of B , in Gd was measured by Hartmann et al. (1990a). The changes by pressure are much larger than those observed in the 3d-metais Fe, Co and Ni (Lindgren et al. 1987 and Butz et al. 1987). The essential contribution to dB,/dp comes from the turning of the axis of easy magnetization with pressure at low temperature and from the reduction of Tc with pressure at higher temperature. See section 3.1.4 for the effect of pressure on Bc. Dy shows rather interesting magnetic properties - see the schematic arrangement of the Dy spins in fig. 3.4 d. Between TN = 178 K and Tc = 85 K it is a helical antiferromagnet. The spins lie in the basal plane of the hcp lattice, the helix axis is parallel to the c-axis and the helix angle c~ between the ferromagnetically ordered planes is temperature dependent. Below 85 K a discontinuous orthorhombic distortion occurs, with a corresponding phase change to a ferromagnetic state. The spins

MUON SPIN ROTATION SPECTROSCOPY

89

are then along the a-axis. The striking features of the #SR measurements (Denison et al. 1979) are: (i) B u is observed in the two magnetic phases (i.e. the first time for an antiferromagnetic state in a metal), which can only be explained if the #+ is not diffusing through the lattice; (ii) a continuous variation of Bu occurs across the magnetic and structural phase transition. Interestingly the calculations of B~ip(T ) show also a smooth behaviour across the phase change for #+ at the octahedral sites and only a weak discontinuity of about 4% at this transition for #+ at the tetrahedral sites. From the measurements neither the muon site nor the sign of Bu could be determined. See section 3.1.3 and fig. 3.4 h for the deduced Be(T). Several studies were made about the influence of an elastic strain on Bu in Fe (Namkung et al. 1984, Kossler et al. 1985, Herlach et al. 1989, Fritzsche et al. 1990). The conclusion is that the observed effect stems essentially from a change in B~ip, due to a shift in #+ population between magnetic non-equivalent interstitial sites reduced by modified elastic energies. In this context see also the effect of impurity induced strain in section 3.3.1. 3.1.3. Temperature dependence of the hyperfine field For Fe, Co, Ni and Dy the dependence Bc(T) is deduced from B u at the #+, BL and the calculated/3~i p, using eq. (3.5), and is shown in fig. 3.4 e-h (bottom section). The hyperfine fields measured with t~- in Fe and Ni are treated separately in section 3.5. For Fe and Ni no knowledge of the #+ site is required, as discussed previously. For Co the octahedral interstitial site is assumed in order to obtain a smooth variation of Be with temperature (except for the discontinuity at the 690 K structural transition), following approximately the Co magnetization curve Ms(T). In that metal the sign of B~ (with respect to Ms) was determined to be negative for T < 500 K, below the spin reorientation between 500 and 600 K, and positive after completion of the easy axis reorientation for T > 600 K. This was obtained by measuring the precession sense of the muon (via the #SR signal phase information) in presence of an additional external magnetic field. (Another way to obtain the sign of Bu is to follow the measured ]B~[ as function of increasing external field values driving the sample beyond magnetic saturation - see the case of Ni, Denison et al. (1979), illustrated in fig. 3.5.) In a-Fe at low temperature Schimmele et al. (1990 and 1994) have observed an oscillating signal in a longitudinal #SR experiment on a magnetically saturated crystal between 30 mK and 600 mK. In the experiment the high-purity single crystal sphere was magnetically saturated in a (111) direction, say [111]. The dipolar magnetic fields at the interstitial sites of tetragonal symmetry (like O- or T-sites) are perpendicular to [111] (Seeger and Monachesi 1982). The fields at different sites transform into each other through rotations by 120 ° or 240 ° around [111], their /~(111) absolute value is labeled ~dip . The only remaining contributions to the local field are/~ext and/~o, both parallel or antiparallel to [111]. Therefore the local field value is given by Bu =

{ ~,(111)\2] 1/2 (Bext -t- Bc) 2 -~- I/Zldip )

j

(3.6)

90

A. SCHENCK and EN. GYGAX

(a)

(b)

I

I

t

[

I

Fe

~6 %

a

5O

qb

I

I

J

I l

600

800

1000

12

I

I I I I

~2 400

8 6

,¢

~ le"q

200

\

I

0 0

f

==

(e)

O3 2

I

200

~

I

I

I

600 1000 Temperature (K)

Temperature (K)

0 1400

(f) i

i

t

I

1.0

i

0.8

10

i

i

~

i

;

I

:

i

'

i

i

6

5 4

o.o

0.4

8

I

0.2

6

m

4

0.0

_o.o

I

200

I

[

I

400 600 800 Temperature (K)

I

1000

i

2

1

if-i-.i~we~J iI

-0.4

2 0

!

4

10 T~ ! 400

I

Co

600

3o ~ 2o

1

I

10

40

%

OlD 0

i 2oo

J

"I-

~2

o

I

800

Bdip h~p . ~ c ic~ci -

Bc 0.8 ~ 1.0 i 0 200

i

T,

0

tcc

I I

: ',

I I

i

'

I

i I I 600

-3 _, -5 -6 I

r 1000

I 140

Temperature(K)

Fig. 3.4. In (a), (b), (c) and (d): temperature dependence of the measured absolute value of B u for Fe, Co, Ni and Dy respectively, for polycrystalline (o) and single crystal ([3) samples in zero field. The spin structures in the ferro- and helical antiferromagnetic phase of Dy (d) are indicated. In (e), (f), (g) and (h): temperature dependence of the hyperfine field Bc for Fe, Co, Ni and Dy respectively, extracted from the IBul measurements (a~l). For Dy the two possibilities due to the unknown sign of Bu are indicated. For Fe and Ni the magnetization Ms(T) is indicated, whereas for Co the field contributions B~ip and BL as well as the deduced temperature dependence of Bu are displayed, together with a sketch of the spin turning and the structural phase transition (Denison et al. 1979).

MUON SPIN ROTATION SPECTROSCOPY

(c)

(d) 14

I

i

Ni 1.5 mllhl=l~

91

20

"Ib-,,,-

i

1

12

C i

•

I

Dy 150

c

t5! 28

qk

T

~oo

v

ell

u?. ,+ I

0.5

m 4 I I I

0

0

I

I

200

400

tl 600

5

0

2 -Ferromagnelic--b*-Ant=ferromagnetic1 I t I 100 150 5O Temperature (K)

0 0

Temperature (K)

(g) 0.8

5O

orthorhombic

1

0 200

(h) I

I

l

- - T

i

r

24 20

0.6

-Bc = 16

vO.4 Ferromagnelic-

8

0.2

4

I

I

I

200

400

600

Bc=

0 2 0

Temperature (K)

Fig. 3.4. (Continued).

100 150 5O Temperature (K)

200

co

92

A. SCHENCK and EN. GYGAX I

I

I

I

I 3

I 4

N

g= 21

t

I !

0

I 1

0

~s 2 Bext (ka)

Fig. 3.5. Local field [B~I as function of external field Bext for a single crystal Ni sphere. At the saturation field Bs = (4rr/3)Ms the /*SR frequency starts to increase with Bext, indicating that B u is positive, i.e. in the direction of the bulk magnetization (Denison et al. 1979).

95 AT= O T= O T= Jr T =

90 85 N 32 v

>=

80

30mK 70 mK 200 mK 300 mK

z~ rh T

x T = 400 mK O T = 600 mK

m

L~

R.~

75 70

m A

65 60 55

I

0.6

I

I

0.8

I

I

1.0

I

I

1.2

I

I

1.4

I

I

1.6

Bext (T) Fig. 3.6. /zSR frequency observed in longitudinal measurements on c~-Fe as function of the externally applied field at different temperatures - Schimmele et al. (1994).

Figure 3.6 shows the #SR frequency ut, = Bu%/(2rr) (% = gyromagnetic ratio of #+) obtained in the experiment for various values of Bext and temperature. The /zSR signal is moderately relaxed, which indicates that at such low temperatures the #+ hopping rate in Fe is slow compared to the observed #SR frequency, which is about 60 MHz at its minimum. The measurement of uu as function of the applied external magnetic field gives directly the dipolar magnetic field and Be, the latter value amounts to (-1.13 + 0.02) T, in close agreement with the earlier measurements by Denison et al. (1979) - see also table 3.4. For Dy, where, coupled to the structural change at 85 K (from orthorhombic to hcp), the magnetic order also changes (from ferromagnetic to a helical antiferromag-

MUON SPIN ROTATION SPECTROSCOPY

~

93

~o

o~

+.~

I

e~

I

~

~

i

I

I

I

I

H

.~ ~ . ~ ~ = g"Ng

. ~ ,'

o oo

o

o

,-~ o ~

rm

I

~=~ 8

I

-t-

-t-

~

-H

I

94

A. SCHENCK and EN. GYGAX

netic order), Bc can be determined along the same lines as described (Denison et al. 1979), calculating B~i p in the appropriate way for the antiferromagnetic phase, where of course Ms is zero. Over the full temperature range B~ip(T ) does not differ significantly if one assumes T- or O-sites for the #+. The fact that B u is observed in the antiferromagnetic phase of Dy can only be explained if the #+ is not diffusing through the lattice. In the experiment it was not possible to determine the sign of Bu, thus the two possibilities for Bc are indicated in fig. 3.4 h. For the ferromagnetic elements a compilation of Ms, Bj, and Bc is presented in table 3.4 for all data extrapolated to T - 0 K, basically following Denison et al. (1979). In addition the table shows hyperfine fields B ° derived from neutron diffraction results and values B the°r resulting from various theoretical calculations. The fact that Bc is not just caused by the undisturbed interstitial spin density, as measured by neutron scattering, is obvious - except for Ni. One common feature of the Be values is the negative sign, meaning that the magnetic densities at the #+ are opposite to the average magnetic density in the unit cell. This feature is explained by the Daniel-Friedel model (Daniel and Friedel 1963). The temperature dependence of both Bc(~'t,) for Fe, Co, Ni and Dy and the normalized domain magnetization Ms for Fe and Ni are shown in fig. 3.4 e-h and for Gd in fig. 3.3. Be is not strictly proportional to Ms. Nishida et al. (1978) defined the 'deviation' function A(T):

Be(T) Be(0)

Ms(T) -

A(T)

(3.7)

- - ,

Ms(O)

which is shown for Fe, Co and Ni in fig. 3.7. For Fe, Co and Gd Bc is below the normalized Ms for T # 0 K, whereas Bc is above Ms normalized in Ni. Schenck 0.10

I

I

I

Ni 0.05

'7, t-

0.00

-0.05

-0.10 0.00

i 0.25

i 0.50

p 0.75

1.00

T/mc Fig. 3.7. Temperature dependence of the reduced hyperfine field A ( T ) for Fe, Co and Ni: A(T) = {Bc(T)/Bc(O)}/{Ms(T)/Ms(O)} - Nishida et al. (1978).

MUON SPIN ROTATIONSPECTROSCOPY

95

(1985) lists possible reasons for such a behaviour and reviews various theoretical models dealing with that matter. For rare earth elements, Campbell (1984) has pointed to a complication one may encounter in the deduction of Bc from B , measurements. In such samples the electric field gradient (EFG) created by the #+ charge will affect the orientation of the neighboring lanthanide moments and so will lead to altered dipolar fields at the #+ site. Estimates are presented for Tb, Dy, Ho, Er and Tm in the cited paper. For Gd, which has no orbital moment contribution to the total moment, the effect is negligible.

3.1.4. Pressure dependence of the hyperfine field The possibility that the deviation between the temperature dependence of Bc and Ms originates from the different manner in which thermal volume changes affect the two quantities has been studied by measurements of the pressure dependence of Be (Butz et al. 1980, Lindgren et al. 1987, Butz et al. 1987, Hartmann et al. 1990a, and Kratzer et al. 1994a). Figure 3.8 show such measurements in Ni and Fe (Butz et al. 1980). Schenck (1985) discusses some implications. The emerging picture is not clear. Whereas for Ni half of the deviation between the temperature dependence of Be and Ms is explained by the volume change, the effect has the opposite sign for Fe. Other causes must be considered.

1.004

1.002

1.000

0 Ni • Fe

:=L t,n Q. v

0.998

m=

0.996

0.994

0.992

0

1

2

3

4

5

6

7

p (kbar)

Fig. 3.8. Pressure dependence of the local field B~ at 298 K for ferromagnetic Ni and Fe. B~ is normalized to unity at zero pressure (Butz et al. 1980).

96

A. SCHENCK and EN. GYGAX

3.2. Critical phenomena In their pioneering study in Ni and Fe, Foy et al. (1973) noticed that #SR could be used to examine critical behaviour near the magnetic critical temperature. Antiferromagnetic phase transitions were observed by Gurevich et al. (1976) in Dy and Ho and then by the same group (Grebinnik et al. 1979a) for the rest of the magnetic lanthanides. The experimental studies dealing specifically with critical phenomena are listed in tables 3.2 and 3.3 with the indication that the #SR measurements were performed near the critical temperatures Tc or TN or that the essential data (like the #+ depolarization rate, F, the spin fluctuation, SF, the Knight shift, K , , etc.) were obtained also near Tc or TN. The first #SR studies aimed at extracting quantitative information from experiments probing critical phenomena were performed by Barsov et al. (1983 and 1984) around the N6el temperature in Er, and in parallel by Nishiyama et al. (1983 and 1984) around the Curie temperature in Ni. In their study of Fe in the Tc region Herlach et al. (1986) stress some of the difficulties one encounters trying to extract static and dynamical exponents using the various #SR methods. This paper also reports K~(T) and transverse depolarizationrate versus temperature data obtained in FeCo and FeZr alloys. The results permit to follow the critical exponents deduced from the two temperature dependences as function of the increasing concentration of the elements alloyed to Fe.

3.2.1. Critical behaviour of the muon Knight shift In a first approximation we expect the measured muon Knight shift to follow the magnetization (or susceptibility) of a paramagnetic sample. This would give a #SR frequency shift (after correction for demagnetization field and Lorentz field) like ]¢Bext

Av~---(T - Tc) well above Tc, with possible deviation in the critical region. This is effectively so in Ni, where Gygax et al. (1980) have measured Ku over a wide temperature range above Tc. The strong T-dependence follows exactly that of the bulk magnetic susceptibility X, also in the critical regime. Figure 3.9 a shows the low temperature results, from which a critical exponent 3' ,-~ 1.28 is extracted, in good agreement with the value of 3' -- 1.35±0.02 obtained for the susceptibility in the same temperature domain. Figure 3.9 b - a Clogston-Jaccarino plot - shows that the high temperature #+ Knight shift is also strictly proportional to X. Since the temperature dependence of the total susceptibility is associated with the d-electrons' susceptibility term, the slope dK~/dx can be identified with the induced contact hyperfine field at the #+ per unpaired d-electron per atom, Ac,d (from K~,,d = Ac,d • x~t/#B , X~t is the atomic susceptibility). In this instance it is interesting to compare Ac,d with the induced hyperfine field per d-electron per atom in the ferromagnetic state of Ni, which is AcV,d = Bc(f'~,)/Ms (see Schenck 1985). The two values are -1.224(23), resp. -1.14(2) kG/(# B • atom), i.e.

MUON SPIN ROTATION SPECTROSCOPY

97

AT (K) 6 8 101215 2 0 2 5 3 0 4 0 5 0 6 0 80100

II

I II

I I II

/

I II

I I

"N'l•

,AKp (2~ mol = -0.223 AZ ' "emu

-1000

-500

X

X,

y = 1 . 2 8 /

V\ /

-

'\.

Q.

~ = 1.35

2

(fromz)

1 O0

\

-1000

-1500

\ \,

(a) -10 0.5

I 1.0

I 1.5

log (T-Tc)

-2000

I 2

-2500

(b)

\

'~",

I

I

I

40

80

120

I

160 xlO 6

Z (emu/g)

Fig. 3.9. Muon Knight shift for paramagnetic Ni, plotted against log(T -/Pc) in the low temperature region (a), and plotted against the bulk magnetic susceptibility with temperature as implicit parameter in the high temperature region (b). The critical exponent deduced for K u (solid line in a) is compared to the con'esponding value obtained from bulk susceptibility measurements (dashed line in a). The high temperature region extends to 270 K above TC (Gygax et al. 1980).

different by about 8%. A step has also been observed by Nishiyama et al. (1983). No explanation does exist so far - see Schenck (1985). However, the basic equality of the induced hyperfine field in the paramagnetic phase and in the ferromagnetic phase seems to be the rule - see, e.g., for Gd W~ickelg~rd et al. (1986). In this same paper it is further reported that closer to T¢, say in the critical region below 1.05T¢, the muon frequency shift deviates from the simple Curie-Weiss behaviour. Moreover, the frequency deviation observed indicates an exponent below 1.0 for (T - T¢), whereas the susceptibility for polycrystalline Gd has an exponent around 1.24 in the same temperature range. This deviation is however trivial, it is expected from the magnetization curves measured for Gd at the same applied field as in the /ZSR experiment (Karlsson 1990). 3.2.2. Muon-spin relaxation or depolarization rate in 3d-elements For Ni, Nishiyama et al. (1984) observed that above Tc the practically field independent longitudinal/z-spin relaxation rate (A1 = l/T1) measured at low field displayed loss of the critical behaviour when T approaches T¢ (fig. 3.10). It was ascribed to the establishment of short range ordering around the /z+, tending to diminish the dipolar field sum at the/z+ because of the cubic symmetry. The temperature dependences of the #+ depolarization rate (as well as of the muon Knight shift) have been studied in the giant moment alloys PdFe and PdNi,

98

A. SCHENCK and EN. GYGAX Illlll

t

I

I

III1111

,

,

,,,,,,I

I

I

IIIIII

,

,

,,,H,I

I

I

1 IIII

,

,

0.1 :=k

0.01

0.001 ,,,,,,I

0.1

1

,,t,

10

T-TC (K)

Fig. 3.10. Temperaturedependence of the/zSR damping rate for metallic Ni. The points are the data of Nishiyamaet al. 1984. The solid line is the result of the model from Yaouancet al. (1993a), taking the/z + dipolar interaction into account, whereas the dashed line gives the prediction when this latter interaction is neglected. which display ferromagnetic ordering, by Gygax et al. (1981). The measurements were performed in the paramagnetic state for both alloys and also below Tc for Pd0.97Fe0.03. The results are tentatively ascribed to the presence of slowly fluctuating ferromagnetically coupled clusters. One had to wait for Yaouanc et al. (1993a) who established a thorough scheme to compute the paramagnetic critical zero-field muon-spin relaxation rate, at first only for cubic ferromagnets (see section 2.3). This calculation includes the full (long range) dipolar interaction between the muon and the lattice dipole moments in a mode-coupling theory. It appears that the damping rate is determined by the relative weight of the hyperfine interaction and the dipolar interaction between muon moment and the lattice ions magnetic moments. The data from Nishiyama et al. (1983) are well explained by this model (fig. 3.10), as well as the #+-spin relaxation rates measured in paramagnetic Fe by Herlach et al. (1986).

3.2.3. Muon-spin relaxation in the lanthanides Compared with 3d magnetism, the magnetism of the lanthanide 3+ ions, with a well localized 4f shell, is characterized by much more pronounced effects of the orbital contribution and strongly localized magnetic moments. This last fact has important consequences for the positive muon occupying an interstitial site: (i) the hyperfine field (contact field) is small due to the rather low spin density of the conduction electrons, because of the indirect mediation of polarization from d-electrons, (ii) the local dipolar field is high and often the dominating field since the neighboring lanthanide moments are large and (iii) the symmetry of the dipolar coupling is low compared to that of the hyperfine coupling, which is isotropic with respect to the crystalline axes. The latter property means that the muon probes paramagnetic fluctuations along different axes with different weight. A judicious selection of the relative orientation of

MUON SPIN ROTATION SPECTROSCOPY

99

initial #+ polarization axis, crystal axes and applied magnetic field allows a selective weighting of the z-, y- and z-components of the spin correlation functions in the investigated material (Karlsson 1990). Due to its large gyromagnetic ratio (in comparison to nuclear hyperfine probes) the #+ can measure lanthanide relaxation times as short a s 10 -13 s. #SR measurements on lanthanide paramagnetism provide important information on the collective behaviour of coupled localized spins, particularly at high temperature. They are a valuable complement to other methods, mainly because of their sensitivity to short-lived correlations, their general applicability - independent of the choice of element - and their flexibility allowing to select among the different relaxation parameters. Gd. The #+ spin relaxation in Gd has been experimentally studied by W~ickelghrd et al. (1986 and 1989) in the temperature region from Tc to 2Tc; additional information and discussions are provided by Hartmann et al. (1990b), Karlsson (1990), and Dalmas de Rrotier and Yaouanc (1994a). The major results are that magnetic correlations extend to far above Tc, with a temperature evolution containing two different terms, for e > 0.03 (e = ( T - T c ) / T c ) and for e < 0.03. For single crystal measurements an anisotropy of the relaxation rate with respect to the crystalline c-axis is observed, at least up to a few K above Tc (Karlsson 1990). This behaviour, similar but weaker than in Er, is surprising for Gd which is a S-state ion. Let's mention that anisotropy has also been observed in the susceptibility of Gd (Geldart et al. 1989). Dalmas de Rrotier and Yaouanc (1994a) have adapted their #SR depolarizationrate model (Yaouanc et al. (1993a) - see section 3.2.2) to the special case of hcp Gd. Although their theoretical results are only valid for truly zero magnetic field, they assumed that the 10 mT transverse-field data obtained by W~ckelghrd et al. (1986) i

i

~L~II

I

i

i

i~11111

i

Ising

~

i l l ~ J

I

i

Heisenberg 4 dipolar q

isotropic

~ 1.o

0

average over single crystal measurements

\&

0.1 i

t

lliHI

t

1

i

irll~ll

i

10

i

l l r t H t

i

100

T-Tc (K) Fig. 3.11. Temperature dependence of the/~SR damping rate for Gd. The polycrystalline data are from Wackelgftrd et al. (1986) and the single crystal data from Hartmann et al. (1990b). The full line is the prediction of the dipolar Heisenberg model from Dalmas de Rrotier and Yaouanc (1994a); the different temperature regimes are indicated (see text).

100

A. SCHENCK and EN. GYGAX

on a polycrystalline sample could be fitted by the theory. The result (fig. 3.11) follows well the polycrystalline data, whereas the zero-field single-crystal data by Hartmann et al. (1990b) are less-convincingly described. In this theory the magnetic dipole-dipole interactions cause a crossover as the temperature is reduced in the paramagnetic region from an isotropic Heisenberg regime to a dipolar Heisenberg regime at about Tc + 15 K. As the temperature is reduced further, their is possibly a second crossover temperature to Ising behaviour at roughly Tc +4 K. For a discussion of the different features derived from #SR and susceptibility measurements as well as the discrepancies between #SR data the reader is referred to the original work. Er. The data obtained in partially oriented crystals by Barsov et al. (1983 and 1986)

indicated first the occurrence of an anisotropic paramagnetic relaxation rate. Later single crystal zero-field data were presented and commented by Hartmann et al. (1990c), Karlsson (1990) and W~ippling et al. (1993) for the temperature region extending from TN to 2TN. In contrast to the weakly anisotropic Gd, Er shows an extreme anisotropy of the relaxation rate which survives up to at least 2TN (fig. 3.12). The rate ),± (#+ polarization perpendicular to the c-axis) shows the usual strong divergence when approaching TN from above. This arises from the slowing down of the 4f-electron spin dynamics due to the evolution of paramagnetic spin correlations (Karlsson 1990). In contrast, All remains independent of temperature at the value corresponding to the spin fluctuation in a free paramagnet, i.e. the high temperature limit (fig. 3.12). Hence, solely the component of Er spin which undergoes magnetic ordering at TN develops also paramagnetic correlations and fluctuates with the c-axis as quantization axis. (Below the N6el temperature of TN ~ 85 K first the c-axis components of the Er magnetic moment orders in a sinusoidally modulated incommensurate arrangement!) For the perpendicular orientation the initial signal asymmetry drops sharply when decreasing T through TN and the signal is effectively lost at lower temperature, whereas the parallel signal remains altogether unaffected down to 58 K (Hartmann et al. 1990c). This confirms that magnetic order is totally restricted to the Er spin component parallel to the e-axis and that not even short range order develops for the perpendicular component.

6!

rN'

'

5-

2 I 0

Ln

0

0

I-1

ooooo¢ 0 gi ~ l og g 100

200

Temperature (K) Fig. 3.12. Muon spin depolarization rate in Er single crystal for two orientations: circles for the muon polarization parallel to the c-axis and squares for the muon polarization perpendicular to the c-axis (Hartmann et al. 1990c).

MUON SPIN ROTATIONSPECTROSCOPY

101

In their conclusion W~ippling et al. (1993) state that the long range nature of the dipolar interaction between the #+ spin and the lattice spins has to be taken into account to explain the measured relaxation rates. A correct evaluation of the data would have to proceed along the lines proposed by Yaouanc et al. (1993a) (section 3.2.2), but such calculations have so far not been applied to Er.

3.3. Ferromagnetic 3d-element based alloys 3.3.1. Fe, Co, Ni Several authors have measured the variation of Bu with impurity concentration in the elemental 3d-ferromagnets (see in table 3.2 the entries for Fe-, Co- and Ni-alloys). The goal was to find how the deduced Bc changes with concentration in relation to the corresponding changes in the bulk magnetization. However, it became clear that in such experiments the #+ did not randomly sample the lattice sites because of the attractive or repulsive interaction with the impurities. This showed up in the interplay of trapping and diffusion of the #+ in the 'dirty' host lattice, as reported by Nishida et al. (1979) and Kossler et al. (1979) in Fe- and Ni-alloys. Together with the studies of Stronach et al. (1979, 1981, 1983), and of Herlach et al. (1986) in Fe- and Coalloys, a total of over 15 different types of dilute alloys were examined. Because of the mentioned impurity dependent modification of the #+ behaviour these B~ dependences give only limited information on the intrinsic host magnetism. They can however provide an interesting contribution in conjunction with the results of other methods applied to study the magnetism in a specific alloy. Stronach et al. (1981, 1983) discuss the effect of the impurity induced strains upon Bu and Be. Herlach et al. (1986) have studied several Fe-Co alloys as well as an amorphous FeZr alloy. In particular the disturbing effects of sample inhomogeneities are discussed. Solt et al. (1992) have performed first #SR measurements in neutron-irradiated pure Fe and Fe-based copper containing alloys. This study was part of a general investigation undertaken with various methods on the radiation generated fast precipitation from supersaturated metastable FeCu solutions. #SR has shown its great sensitivity to both vacancy-like defects and to the Cu-depletion of the solvent matrix. The average and distribution-width of Bu changes with n-irradiation fluence as a consequence of the broad size distribution of created voids and other defects in the pure metal and of the induced Cu-precipitates in the alloys. Annealing after irradiation produces agglomeration of vacancies or defects, which again changes the field distribution sampled by the muons. A modeling of the experimental results is presented and the preliminary #SR study shows that due to the high sensitivity of the technique samples with higher purity and less irradiation damage are required for a better resolution of defect spectra. 3.3.2. Pd-based alloys Pure Pd is paramagnetic, but close to a magnetic instability. Alloying it, e.g., with Fe or Ni at atomic % concentration can strongly polarize the d holes on neighboring Pd sites, producing 'giant moments'. These moments can show ferromagnetic order or couple antiferromagnetically and exhibit spin-glass ordering, depending on the alloy concentration and the temperature.

102

A. SCHENCKand EN. GYGAX

Different #SR studies were performed on Pd-based alloys (see the references in table 3.2). The general meaning of the results is not at all clear. Again the peculiar effects of the impurity atoms on the #+ are difficult to assess. Even in 'pure' Pd the role of the muon is dubious. For example, Nagamine et al. (1977) interpret their data assuming a localized #+ at low temperature (in analogy to the behaviour of H in Pd), whereas Hartmann (1990) seems to believe that for very pure Pd the #+ diffuses fast for temperatures at least as low as 18 K. For weakly doped Pd, Dodds et al. (1983) assume a negligible #+ diffusion rate below 100 K (based on the muon diffusion measured in Pd lightly doped with Gd) and interpret their data in PdMn (2 at.%) assuming randomly distributed #+ locations, whereas the same group consider the muon as static in PdFe0.0o35Mn0.05 at low temperature, but physically excluded from the interstitial sites closest to the Mn sites (Gist et al. 1986).

3.4. Chromium and its alloys The three first #SR studies in Cr (Kossler et al. 1977, Grebinnik et al. 1979b, Weidinger et al. 1981) did not succeed in detecting spontaneous muon precession in this antiferromagnetic metal. The measurements yielded #+ depolarization rates as function of temperature or field and showed only a modest consistency. As Weidinger et al. (1981) noticed, it seems that the influence of impurities on the #+ behaviour partly biases the experimental outcomes. The question of the #+ localization was addressed but not conclusively answered. The Stuttgart group (Major et al. 1986, Templ et al. 1990) was able to see a #+ precession signal in a zero-field measurement in Cr below 12.5 K. The frequency of about 84 MHz shows up with a small amplitude, implying that only a small fraction of the muons see a distinct local field. Below 124 K Cr is in an incommensurate longitudinal spin-density wave (SDW) state. The Stuttgart group explain their data assuming that for T < 12.5 K the #+ stays at a tetrahedral interstitial site with tetragonal symmetry axis perpendicular to the crystalline [100] direction (labeled T ±site). The other tetrahedral site and the octahedral sites are excluded and #+ hopping has a low probability during the muon lifetime. In addition it is assumed that the #+ resides preferentially near a field maximum of the SDW. This situation can result either if the SDW maximum is pulled towards the #+ or if the muons thermalize and stop at the SDW antinodes (in Cr the SDW or better the accompanying charge density waves produce also strain waves - the effect of strain on muon trapping has already been mentioned in sections 3.1.2 and 3.3.1). In a further study (Grund et al. 1994) proton-irradiated Cr was used to vary the possible pinning/trapping conditions for muons and SDW. For the low temperature regime the data are now interpreted assuming a localization of the #+ at random position in the host; the surviving coherent signal at long times corresponding to the (small) fraction of the #+ stopping near the SDW maximum, as indicated by the calculation. Therefore, the muons neither cause SDW motion nor have to undergo long range diffusion. In a study by Noakes et al. (1992) it is shown that the zero-field coherent #SR frequency of 84 MHz previously observed in Cr has a dramatic impurity dependence.

MUON SPIN ROTATION SPECTROSCOPY

103

It is known that alloying Cr with small amounts of other metallic elements has strong effects on the magnetism (Fawcett 1988). In their experiment Noakes et al. (1992) were able to suppress, shift dramatically or split the 84 MHz #+-frequency observed in pure Cr by alloying with, e.g., V, A1 or Mn at atomic % concentrations and varying the temperature. So far only speculative explanations are suggested. Another contribution dealing with the condensation of incommensurate SDW and the onset of the AF state stems from a ZF #SR study in the Cr-like alloy Cr85Mo15 by Telling et al. (1994). As in other instances (see, e.g., section 3.3.2) the effect of impurities on the #+ behaviour in the Cr host is possibly pathological, i.e. very interesting for a defect therapist, but rather an annoying complication for the simple magnetically-minded pedestrian.

3.5. #-SR in Fe and Ni A # - stopped in a solid forms a muonic atom # - z X in which the bound # - behaves like a heavy electron (mu = 207me). The muonic radius % = aome/(muZ) is much less than the Bohr radius a 0, hence the pseudonucleus # - z X appears to the atomic electrons very similar to an isolated impurity nucleus of charge Z - 1 (Yamazaki et al. 1979 and Yamazaki 1981). However, for light elements (Z < 30) the muonic atom 1s-wave function extends noticeably outside the nucleus. For example, in the case of /z-Fe, % is about twice the nuclear radius Ro. In contrast to the previously discussed #+SR measurements, which yield information on the hyperfine field (i.e. the electron spin density) at the interstitial #+ position,/~-SR leads to inforrnations on the radial distribution of the electron spin density p(r) near the (pseudo-impurity) nucleus. This is much like NMR, M6ssbauer spectroscopy and time-differential perturbed angular correlation, all techniques studying the hyperfine fields acting upon the nuclei of adequately selected impurity atoms in the host. Calling B~f the average hyperfine field acting upon the # - bound to z X and B hf the average hyperfine field acting upon the impurity nucleus of charge Z - 1, one usually defines the hyperfine anomaly as AX = ( ~ h f _

BhNf)/Bhf.

(3.8)

Assuming that the muonic atom and the impurity atom occupy the same type of site in the lattice and have the same electronic structure, a non-zero A value will be due to the fact that the # - samples p(r) over a more extended region of space than the Z - 1 nucleus. Yamazaki (1981) has discussed some aspects of the hyperfine anomaly and their importance in the study of magnetic materials. Imazato et al. (1984) reported the first results in a ferromagnetic material, i.e. Ni, and Keller et al. (1987) extended the measurements to ferromagnetic iron. In Ni a comparison of the hyperfine field for #-Ni with that for 59Co yields nNi = -2.4(3)% to -2.8(5)% over the temperature range 23-303 K (Imazato et al. 1984). The results indicate that the electron spin density near the Ni nucleus decreases with increasing r more steeply than the s-state electron charge density, a behavior which is a consequence of core polarization (see Freeman and Watson 1965, Freeman et al. 1984). The values are in good agreement with the unrestricted Dirac-Fock calculation by Freeman et al. (1984).

104

A. SCHENCK and EN. GYGAX 25

I

I

I

I

1.0 2O

0.8 "o -~

15

o "a

10

0.6 0.4

E

& E "~ :~ "0

5

o/,

• 0 0

5SMn

NMR

0.2

rr

p Fe I 0.2

r 0.4

1 0.6

i 0.8

0.0 1.0

T/T c

Fig. 3.13. Magnitude of local field in Fe as a function of reduced temperature T/TC (TC = 1043 K). Solid triangle: data for 52MnFe (dilute limit) obtained with NMR on oriented nuclei from Hagn et al. (1982). Open circles and open triangles: 55Mn data for (1.5 at.%) 55MnFe from Koi et al. (1964) and Yamagata et al. (1983), respectively. The solid circles are the/~-Fe data for iron and the solid line represents a fit using a simple mean-field model (Keller et al. 1987). For comparison the reduced magnetization curve of pure iron is also shown (right scale). In Fe # - S R in zero applied field was used to study the hyperfine field in the temperature range 320-690 K (Keller et al. 1987). Bhuf departs from the magnetization curve of pure iron in the same way as the hyperfine field acting upon a 55Mn impurity in diluted (1.5 at.%) MnFe measured by N M R by several groups with different N M R techniques (Koi et al. 1964, Rubinstein et al. 1966, Yamagata et al. 1983), see fig. 3.13. This indicates that the electronic structure of # - F e is very similar to that of a Mn impurity in Fe. AFe is found to be - 0 . 9 ( 3 ) % and temperature independent over the temperature range investigated. The anomalous behavior of the impurity B~f has been discussed by Jaccarino et al. (1964), Low (1966) and Shirley et al. (1968), and can be understood on the basis of a mean-field approach. The A value measured by Keller et al. (1987) is significantly smaller than expected if core polarization plays a similar role in # - F e relative to Mn as in # - N i relative to Co. The reason for the small AFe is not known. In this case, in contrast to Ni, no unrestricted Dirac-Fock calculation are available.

4. Review of results in intermetallic compounds 4.1. Compounds involving transition elements Only a few intermetallic compounds with no lanthanides or actinide constituents have been studied up to now, by #SR. In this section we discuss the weak itinerant magnets MnSi, Y9Co7 and Nbl_=Zr=Fe2 (see table 4.1). Other Y containing compounds are included in section 4.2.

M U O N SPIN ROTATION SPECTROSCOPY

oo

~

oo o~

N~v e~

©

0

--1 r~

I.~ gT., .C~

M

©

c.i

~8 cq

o~< o~

g~

0 0

r..)

8

105

106

A. SCHENCK and EN. GYGAX

4.1.1. MnSi The interest in this system was motivated by the possibility to extend spin lattice relaxation rates (Ta l) measured by 55Mn NMR (Yasuoka et al. 1978a) to much lower temperatures and higher rates by using the #+. This extension appeared very important since detailed predictions on the temperature dependence of T1-1 on the basis the self consistent renormalization (SCR) theory for itinerant ferromagnetic electrons (Moriya et al. 1973, 1974, Moriya 1977) and applied to MnSi (Makoshi et al. 1978), could only sensibly be checked in this way. In fact MnSi is known to be a weak helimagnet with a period of ,-~ 180 A below Tc = 29.5 K (Ishikawa et al. 1976, Motoya et al. 1978) but since the helical structure has such a long period it was expected that T~-1 behaves almost like the spin lattice relaxation rate in an itinerant-electron ferromagnet (Hayano et al. 1978a). We start with a review of the measurements in the paramagnetic phase (T > Tc). In a first study the #+-Knight shift was measured in a transverse field experiment with Hext = 0.29 T (Yasuoka et al. 1978b). It was found to scale well with the magnetic susceptibility from which the coupling parameter Ac~+ could be determined (see table 4.2 which includes also the corresponding A Mn determined by NMR). According the SCR theory above Tc the spin lattice relaxation rate T1-1 (see also eq. (2.41)) is predicted to be given by Im X L(g wo) coo

2 ,u,,Mn 2 1/T1 (., Mn) = 271,(Mn ) (Ac ) T Z 1

(4.1)

T

= TI(~) T - Tc

(4.2)

,'~^,2 ( d,u,Mnh2 T C(~I/~(Mn) L~c ) T - Tc '

(4.3)

and hence 2 Mn 2 2 T(U)/T(Mn) =- "yfvln(A c ) /'yu(Ac/ ,)2 .

(4.4)

First results on (Ta~)-~ obtained in a longitudinal field of 700 G agreed with the SCR-predictions appropriate for an itinerant ferromagnetic system quite well. TABLE 4.2 Compilation of hyperfine fields and coupling parameters in MnSi (Tc = 29.5 K).

T > Tc T < TC

Acu (kG/#B)

B~f (T = 0) (W)

A Mn (kG//~B)

-4.8 ± 0.2 -6.94 -3.94

-0.273 -0.155

-138(1)

MUON SPIN ROTATION SPECTROSCOPY

107

Comparison with a few NMR results above 100 K proved also the validity of the scaling relation eq. (4.4) (Hayano et al. 1978a, 1980). Later it was realized that ZF-#SR measurements could not only be used to determine TI-t(# +) but also Tll(Mn) at the same time. This is a consequence of the fact that in ZF the #+ will also feel the 55Mn-nuclear dipole fields. If they are static the effect on the #+-polarization is described by the static Kubo-Toyabe relaxation function (see section 1.8). However, since the nuclear spins relax as well under the action of the spin fluctuations of the itinerant electrons a dynamic picture applies and the induced/z+-depolarization has to be described by a dynamic Kubo-Toyabe function GKT(t, u(Mn)), where u(Mn) = Ta-a(Mn) is the nuclear spin lattice relaxation rate. In this case the overall #+-depolarization function is then given by (two channel relaxation) CzF(t) = exp (--

t/T} ")) GKT(t, u(Mn)).

(4.5)

Analyzing zero field data with eq. (4.5) both 1/T~u) and 1/T~ Mn) could be determined over a much wider temperature range (Matsuzaki et al. 1987, Kadono et al. 1990). The results are compiled in fig. 4.1. The solid line through the 1/T1 (Mn)-data is a fit of the function 1 -

-

-

a

T1 (Mn)

-

T -

T + b

T - Tc

.

(4.6)

(T - Tc) 2

The second term is included to account for a somewhat modified temperature dependence in the vicinity of Tc following from numerical solutions of the SCR-theory I

I~

IIItl

I

101

I

I

I

I IIII

I

I

I

i

i iiii

I

i

_oc_ :,zedm_o_ ent.

10 0 "1 v

~.,._

10-1 'T'--

I1~:

=2 ",it--

10 -2 ~/j,, o

II

"+2,<, II

10-3

10-1

,,,,,,,

10 0

101

, ,,',',J,I

,

10 2

T-T c (K) Fig. 4.1. Log-log plot of both the #+ (open symbols) and the Mn-nuclear spin (filled symbols) lattice relaxation rates l/T1 versus temperature. The filled circles and diamonds are extracted from ZF-/zSR measurements, the filled squares are from NMR measurements. The solid lines follow eq. (4.6), the dashed lines represent various other theoretical models (Hayano et al. 1978a, Kadono et al. 1993).

108

A. SCHENCKand EN. GYGAX

(Moriya et al. 1973). The solid line through the 1/Ta(#)-data is obtained by just scaling the upper solid curve with the factor (UMnAcMn/l/IzAc#)2 = 5.0. Another nice verification of the SCR-theory follows from #SR measurements of the temperature dependence of the electrical field gradient (EFG) at the #+ nearest neighbour Mn-nuclei (Kadono et al. 1993). Applying the muon-quadrupolar level crossing technique (Kreitzman et al. 1986) the EFG showed a diverging behaviour as Tc is approached from above. This is shown in fig. 4.2 where the temperature dependence of the quadrupolar splitting frequency UQ of the 55Mn-nucleus (I = 5/2) is displayed. Also shown are results from NMR-measurements performed at higher temperatures (Yasuoka et al. 1978a). The two data sets do not agree too well (and do not scale either) in the temperature range where they overlap. The reason for this is not known at present. In the SCR-model the temperature dependence is explained by the coupling of charge density to spin density fluctuations (Takahashi and Moriya 1978). In this model the net EFG at the Mn-nucleus is given by q/ = q=qo+qel qo+ ( 1 / x ) + d ,

(4.7)

where q0 is the contribution from the positive ion cores including the #+ and qel arises from the deviation of the conduction electron charge density from a cubic distribution. The parameter q~ and d are related to the EFG in the absence of electron-electron interactions and of the effect of exchange enhancement. Inserting the Curie-Weiss law for the susceptibility the temperature dependence of UQ can be

1.5

i

j

i

t

MnSi

1.25 1 N

-1-

.75

o

.5

°Ooo

O O 0000

.25 0

0

I

I

I

I

I

100

200

.300

400

500

Temperoture

600

(K)

Fig. 4.2. Temperaturedependence of the quadrupole coupling parameter //Q in MnSi. Triangles are from the #+ level crossing measurements, octagons stem from NMR measurements (from Kadono et al. 1993).

MUON SPIN ROTATION SPECTROSCOPY

109

expressed as 1

3e2Q

[

q~

]

UQ(T) -~ 27r 2 M ( 2 I - 1) qo + (T - T c ) / c + d

.

(4.8)

This equation can indeed be used to fit the #SR data well (Kadono et al. 1993). Using c = 0.6 (emu/mol)K and Q = 0.55 barn one obtains q0 = 6.75(3) x 102z cm -3, q' = 1.30(1) x 1025 c m - 3 e m u - l m o l and d = 1.68(1) x 102 mol/emu. In summary the SCR-model can describe very well the temperature dependence of both the 55Mn and/z+-spin lattice relaxation rate and the EFG at the Mn-nuclei in the paramagnetic phase. We now review results obtained below Tc (Takigawa et al. 1980, Kadono et al. 1990). ZF-measurements reveal two distinct precession frequencies (with relative populations Pl/P2 = 0.77(9)) the temperature dependence of which is shown in fig. 4.3 (Kadono et al. 1990, Takigawa et al. 1980). The data are compared with

0.25

[

I

I

I

I

I

0.20 -

0.15

0.10 -

--.

\iO

"

IP

0.05 qb A . f = "6.94kg/~ B I Ahf = -3.94kg/l~B 0.00

0

I

5

I

10

I I 15 20 25 Temperature (K)

\~

1I 30

35

Fig. 4.3. Temperature dependence of local fields Btz in MnSi below T C. The solid lines represent theoretical calculations of MQ(T)/MQ(O)and the dashed lines show MQ(T)/MQ(O)obtained from neutron scattering (from Kadono et al. 1990).

110

A. SCHENCK and EN. GYGAX

the temperature dependence of the saturation magnetization MQ(T) (determined by neutron scattering, Ishikawa et al. 1976) and a perfect scaling is observed implying (after correcting for the Lorentz field) Bhf(T) = A2 + MQ(T).

(4.9)

The resulting coupling constants A2 are to be found in table 4.1. The weighted average Ac~ = pl Acu(1) + p2Acu(2) = -5.63 kOe//z B

(4.10)

is 17% larger than the coupling constant in the paramagnetic state (see table 4.2). It is suggested that this discrepancy may be due to the ordered helical structure causing two magnetically inequivalent sites and also dipolar field contributions. Note that in the paramagnetic state only one signal showed up in the TF-data. Possible #+-sites in MnSi were not discussed. 400

/

¢) Ahf = -6.9kg/g B / 0 Anf =-3.9kg/g g

/

~)

300

o o~

200 e4

=t.

100

0

5

10

15 20 25 Temperature (K)

30

35

Fig. 4.4. Temperature dependence of/z + spin lattice relaxation rates, multiplied with the respective u 2 from the oscillating components below T C in MnSi. The SCR theory predicts a linear dependence (from Kadono et al. 1990).

MUON SPIN ROTATIONSPECTROSCOPY

111

The solid lines in fig. 4.3 represent the predictions of the SCR-theory for Me(T) below To As can be seen the data do not follow very well the predicted Me(T ). This is discussed further in Takigawa et al. (1980). Information on the spin lattice relaxation rate below Tc is obtained from the exponential decay of a third nonprecessing component in the #SR-signal. This component, actually consisting of two contributions, arises from those #+ whose spins happen to be parallel to the static internal fields in the randomly oriented multi-domain sample but which spins are affected by fluctuating field components perpendicular to their direction (see eq. 2.13). In the analysis of these non-precessing components it is assumed that 1/T( 0 o( w2%, where wi are the frequencies of the corresponding precessing components and % is a common correlation time arising from the Mn-spin fluctuations. Figure 4.4 shows the temperature dependence of (wi/21r)2/T( O. According to the SCR-theory one expects 1/T1 c( T / [ M o ( T ) ] 2

(4.11)

and hence w~i)/T1 oc T. This linear dependence is indeed seen for temperatures T < T c / 2 but it breaks down for larger T. Takigawa et al. (1980) suggests that the deviation of 1/T(O(T) from eq. (4.15) may be a consequence of the helical structure: "Since it has a finite Q, although very small, the spin fluctuations in a small region around Q may play an important role in the relaxation process, at least near Tc ". In this case an antiferromagnetic SCR-model would be more appropriate which predicts 1/TI o( T / M e ( T ). At least qualitatively this can describe the trend of the data closer to Tc. 4.1.2. Y9Co7 This compound shows a magnetic phase transition at Tc ~ 6 K and becomes superconducting below Tc ~- 2.7 K in coexistence with the magnetic order (Sarkissian 1982). The ordered magnetic phase below 6 K is believed to be an itinerant ferromagnetic state (Sarkissian 1982, Huang et al. 1983a, b). In the latter reference it is reported that application of pressure suppresses the magnetic order and increases the superconducting transition temperature and Hc2. It is further concluded that the long range ferromagnetic- and superconducting order parameter coexist but vary spatially. NMR measurements (Takigawa et al. 1983b) suggest that only at the Co-sites in the 2b positions (for notation see Grover et al. 1982), forming a linear chain along the c-axis, a sizable spin density is to be found associated with a ferromagnetic moment of 0.2#B per atom. ZF-pSR measurements were performed with the aim to collect further information on the peculiar magnetic behaviour of this compound (Ansaldo et al. 1985). Figure 4.5 displays ZF-spectra for temperatures around and below To At temperatures above Tc a well developed static Kubo-Toyabe signal was visible which arises from the 59Co-nuclear dipole fields and proves the absence of #+ diffusion up to at least 220 K. The results shown in fig. 4.5 could only be adequately fitted by invoking two components, one of which is the afore mentioned Kubo-Toyabe signal

112

A. SCHENCK and F.N. GYGAX

.30 ~ 6 . 1 K

Co 7

,

.20 O3 < .10

0

-

0

10 20

-

30 40 50 60 70 Time (#s)

80

Fig. 4.5. ZF-/~SR signal in Y9Co7 at various temperatures. The solid lines represent fits with two components. See text (from Ansaldo et al. 1985).

.30

Y9 0 ° 7 •

O0 ~ Slow Signal

~>, .2o E <

a)

.10

©

O

Fast Signal

4.0

go • 1 I I I L 2.0 3.0 4.0 5.0 6.0 Temperature (K) ~)

3.0

I

0 0

~

°~c__~

'~

o

I 1.0

b)

O

~-'- 2.0 .o_

Fast Signal

_~ ¢ 1.0

0

q 0

1.0

slow Signal

•

•

O

OC~DO-

2.0

3.0

i

4.0

5.0

Temperature (K) Fig. 4.6. Temperature dependence of (a) signal amplitudes (asymmetries) and (b) relaxation rates from a two component fit to the ZF-#SR signal in Y9Co7 (from Ansaldo et al. 1985).

MUON SPIN ROTATIONSPECTROSCOPY

113

(below Tc to be replaced by a dynamic Kubo-Toyabe function) which shrinks with decreasing temperature while a second more rapidly relaxing signal grows by the same proportion. The data could be equally well described by choosing for the second 'fast' signal a heavily damped oscillating function or alternatively a dynamical Kubo-Toyabe expression with a much more rapid overall relaxation as compared to the first signal. Using the first approach one finds at 2 K an average field of ,,o 100 G at the #+. The fit results from the second approach are shown in fig. 4.6. Figure 4.6a displays impressively the shrinking of the 'paramagnetic' component and the rise of the new 'fast' component below Tc. Qualitatively the same result is found in the first approach. Figure 4.6b displays the fluctuation rates following from the dynamic Kubo-Toyabe fit of the 'fast' signal. The static line width for this signal amounts to A ~ 4.5 #s -1 corresponding to a field width of ~,, 53 G which must arise from the ordered Co-moments. These results are quite different from the ones in MnSi and do also not show any resemblance with what is observed in spin glasses. Similar behaviour, however, is found in some of the compounds classified as heavy fermion materials (see sections 4.2.6 and 4.3.4). The most remarkable feature of the present data is the temperature dependence of the two component structure. It could imply that the paramagnetic phase persists well into the ordered state in part of the sample volume. It would be interesting to find out whether superconductivity is associated with only this fraction of the total volume. TF-#SR measurements on monocrystalline samples should allow to check on this conjecture (see also the discussion on CeCu2Si2, section 4.2.6). The other noteworthy result is the lack of any indication for a truly long range magnetic order. The present data indicate a rather random or extremely short range magnetic order. In contrast to ordinary spin glasses no slowing down of spin-fluctuations is seen when approaching Tc from above.

4.1.3. Nbl_~ZrxFe2 NbFe2 is considered to be a weak itinerant antiferromagnet (TN = 13 K) with a rather small ordered moment (~ 0.1#B) (Yamada et al. 1990). The antiferromagnetic state is very sensitive to the exact stoichiometry and weak ferromagnetism is observed in compounds with either enhanced Fe or Nb content. The mixing of dynamic ferromagnetic and antiferromagnetic correlations in NbFe2 may be the result of topological frustration associated with the Kagome lattice like structure of the Fe sites in NbFe2. The development of itinerant ferromagnetism is also observed in compounds in which Nb is partially replaced by Zr up to a concentration of 20%. At larger Zr-concentrations the system converts back to weak antiferromagnetism (Yamada et al. 1984) #SR-measurements were started with the aim to study the phase diagram of Nbl_xZr~Fe2 in more detail. So far only preliminary results on a powdered sample of Nb0.9Zro.lFe2 (Tc -~ 43 K) were presented (Crook et al. 1994). These results from ZF- and LF-measurements seem to imply that already below 90 K (the temperature at which the susceptibility X, on cooling, tends towards the Curie-Weiss form) some static behaviour of the atomic moments develops in coexistence with dynamical fluctuations. It is also found that the static field distribution shows a trend from a Gaussian shape to a more Lorentzian one as the temperature is lowered down from

114

A. SCHENCKand EN. GYGAX

90 K. This could suggest the development of dilute inhomogeneities in an otherwise regular spin arrangement, perhaps as a consequence of the topologically frustrated lattice of Nbl_xZrzFe2. Below Tc the amplitude of the ZF-#SR signal drops rapidly to 1/3 of its initial value, marking clearly the onset of the ferromagnetic state. From the relaxation of the residual signal it can be concluded that also in the ordered state some spin fluctuations must persist. No oscillating signal below Tc was seen which could be due to the limited time resolution at the pulsed #+-source at ISIS. These preliminary results together with magnetization data suggest that Nb0.9Zr0.1Fe2 cannot be viewed as a simple weak itinerant ferromagnet. 4.2. Intermetallic compounds containing rare earth elements 4.2.1. Rare earth aluminides: RAI2 (except R = Ce) The rare earth aluminides with the exception of CeA12 all show a ferromagnetic state with Curie temperatures ranging from 6 K for TmAI2 to 160 K for GdA12 (see table 4.3). Their crystalline structure is of the cubic Laves phase type MgCu2 (C15). The magnetism is believed to be associated with well localized 4f-moments. Whereas it is known that in certain itinerant magnetic systems short range correlations among the spins persist to temperatures well above the transition temperatures the situation with respect to local moment systems is less clear. #SR-measurements were performed with the specific aim to search for such correlations by studying the spin dynamics of the 4f-moments via #+-relaxation measurements in the paramagnetic phase (Chappert et al. 1981, 1986b, Kalvius et al. 1984, Gradwohl et al. 1986, Hartmann et al. 1984, 1986, Dalmas de R6otier et al. 1990b). The results on CeA12 will be treated in section 4.2.7 (heavy fermion compounds). From TF-measurements in the nonmagnetic compound LaAI2 it was concluded that the #+ most likely resides at the so called (2-2) site, having two R and two Al-atoms as nearest neighbours (see fig. 4.7). ZF- and LF-measurements in DyAI2 below Tc showed a drop in the signal amplitude to 1/3 of its value above Tc which is typical for a ferromagnet with randomly oriented domain magnetizations (Gradwohl et al. 1986, Asch et al. 1986). The remaining signal displays an exponential relaxation indicating that fluctuating field components are still present below Tc. Detailed measurements in the other RA12 compounds below Tc are lacking. Above Tc mostly TF-measurements were done, supplemented later by ZF- and LF-studies (Kalvius et al. 1984, Gradwohl et al. 1986, Chappert et al. 1986b, Dalmas de Rfotier et al. 1990b). From an analysis of the Knight shift from TFmeasurements it was established that the contact hyperfine field is vanishingly small with the exception of GdA12, and that dipolar fields from the 4f-moments5 must be responsible for the dynamically induced ~+-depolarization (Hartmann et al. 1986). With respect to the extraction of the #+ spin lattice relaxation rate from the TFspectra a correction for powder broadening had first to be applied. Some room temperature relaxation rates A2, AzF of the #+-polarization are collected in table 4.4. It is seen that grosso modo the corrected )~2 and the )~zF agree reasonably well with each other. Az,zF can be compared with predictions based on the RKKY interaction 5 Dipolarcouplingparameters for all RA12are quoted in Hartmannet al. (1986).

MUON SPIN ROTATION SPECTROSCOPY

r~

mm

~

u

H

~o ~

115

116

A. SCHENCK and EN. GYGAX

001 projection

,,~

®

f

®

®

® Rare Earth o A~iuminum 2-2 muon position ,, 3-I muon position Fig. 4.7. Crystal structure of the cubic laves phase compounds RA12. The most probable # + location is the so called (2-2) site. TABLE 4.4 Compilation of contact hyperfine field constants Ac, room temperature relaxation rates A2 (from TF measurements, extrapolated to Hexp = 0) and AZF (from ZF-measurements), calculated values Aff based on the RKKY mechanism and assuming the/z + at the (2-2) site, calculated values Aj at 300 K including the Korringa mechanism, and calculated total fluctuation times ~-j (ct = 1.477 × 1011 s - 1 K - 1 , eq. (2.52)) (Hartmann et al. 1986).

PrA12 NdA12 GdAI2 DyA12 HoA12 ErAlz TmA12

[Ac I

>'2 (300 K)

AZF(300 K)

),ff

~j (300)

(kG//~B)

(#s -1 )

(/zs -1 )

(/zs -1 )

(/zs -1 )

(10-12s)

-~ 0 ~- 0 ~_ 0.46 -~ 0 "~ 0 "~ 0 "~ 0

0.05 0.02 0.38 1.05 1.16 0.84 0.43

0.133 0.059 0.11 0.96 2.14 3.76 4.42

0.09 0.04 0.11 0.71 1.17 1.36 1.02

0.39 0.19 0.089 0.34 0.57 0.81 0.98

(3) (3) (6) (7) (8) (5) (4)

0.22 (6), 0.23 (1) 1.4, 0.67 (8) ~ 1.05 ~ 1.4

~'j (300 K)

among the 4f-moments and on the Korringa mechanism (see eq. (2.52)), assuming the #+ to reside at the (2-2) site. The exchange interaction constant Isf, entering into the RKKY interaction, is calculated from the Curie temperature T¢. The Korringa mechanism involves the quantity [IsfN(EF) I which, according to L6wenhaupt et al. 1983, is of the order of 0.08(4) and includes de Gennes' factor. For PrA12 and NdA12, A2 are rather small and nearly temperature independent. The data are consistent with a temperature independent 4f-moment fluctuation rate of the order of several 1012 s -1 (see table 4.4) which is caused by the RKKY interaction alone. In the heavier HoA12 AzF is much larger but shows also a relatively weak temperature

MUON SPIN ROTATION SPECTROSCOPY II I

I

I

i

I

I

I [1111111111

I

o

0.20 -,$ 0.15 E E 0.10 c,O

<

0.05

117

0 0 0 0 (30

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HoAI 2 2.0 b,

0' /%oooo Ooo o

1.5 ¢,.

1.0

o

oo

o

0.5 rr

I

0.0 III

20

I

Tc t

I

I

I Illlflllt~l

100

200

Temperature (K)

Fig. 4.8. Plot of signal amplitude (asymmetry)and relaxation rate of the ZF-/zSR signal in HoAI2 versus the logarithm of temperature. Note the collapse of the amplitude below Tc (from Dalmas de R6otier et al. 1990b). dependence (see fig. 4.8). The RKKY prediction amounts to an even larger rate. Inclusion of the Korringa mechanism brings the value of the calculated )`ZF,calc. down to nearly the measured one at room temperature and reproduces roughly the temperature dependence. The same seems to be true for TmA12. The remaining compounds GdAI2, DyA12 and ErAI2 in contrast show a very pronounced variation of ),2 and/or )`ZF with temperature (see fig. 4.9) displaying an almost divergent behaviour as Tc is approached from above. Hartmann et al. 1986 take this as evidence for the evolution of short range dynamic correlations as Tc is approached. Dalmas de Rdotier et al. 1990b argue that some of the temperature dependencies are strongly influenced by crystal field (CEF) effects. (Not, of course, in GdAI2.) Data on a single crystal DyAI2 show an even more peculiar behaviour. There a maximum in )`1 shows up at T ~ 100 K, clearly above Tc, f o l l o w e d by a shoulder at ~ 125 K. Attempts were made to explain this behaviour i n terms of #+ trapping and diffusion (Gradwohl et al. 1986). Asch et al. 1986 h a v e also found differences when investigating differently prepared polycrystalline DyA12-samples, but all samples showed a clear temperature dependence of )`1. Finally Kaplan et al. (unpublished data) investigated another polycrystalline sample

118

A. SCHENCK and EN. GYGAX l

I0 -, v

I

I

I

+

i

o~

f

8

I

I

I I I llllllll~

I

ErAI 2

6 tO

4

x

~

2

rr

0

O

o o Oooom I

20

I

I

i

I

I

I

I [ I I I I I [ I I I I

1O0

200

Temperature (K) Fig. 4.9. Plot of the relaxation rate of the ZF-/zSR signal in ERA12 versus the logarithm of temperature. Note the diverging character of ~ZF when the temperature approaches TC. The signal is lost already about 10 K above TC (from Dalmas de R6otier et al. 1990b).

which displayed a near temperature independent )~1 ,-,o 1 #s -1. From all those results it must be concluded that sample dependent features are involved and the question whether dynamical short range correlations persist above Tc in some of the RAlz-compounds or whether CEF-effects are responsible cannot be answered with certainty. Systematic investigations in the LF-mode may remedy our present ignorance. 4.2.2. Other cubic Laves phase (C15) compounds RM2 with M a 3d transition metal 4.2.2.1. RM2 (M = Ni, Fe, Co). All compounds with M = Ni, Fe, Co, considered here (see table 4.5), show ferromagnetic order within the local moment rare earth sublattice and the more itinerant 3d-element sublattice. The coupling of the two sublattices is ferromagnetic for light R elements (Ce-Sm, no such compounds have yet been investigated by #SR) and antiferromagnetic for the heavier ones. Included here is also the compound YFe2 in which, as well as in LuFe2, only the Fe-moments, of course, can order. Below Tc, in zero external field, well developed #+-precession signals were observed in GdFe2, GdCo2, YFe2, LuFe2, TmFe2 and ErFe2, using polycrystalline samples (Barth et al. 1986a, Graf et al. 1981). In each case only one distinct precession frequency was observed. The temperature dependence of these frequencies is in part rather peculiar (see, e.g., fig. 4.10-4.13) and does not scale with the macroscopic magnetization. The analysis is made difficult by the presence of #+ diffusion and probably trapping at defect sites and the presence of two sublattices. #+-diffusion, already quite rapid at the lowest temperatures (4.2 K), has to be invoked in order to explain the occurance of just one frequency.

M U O N SPIN ROTATION S P E C T R O S C O P Y

119

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120

A. SCHENCK and F.N. GYGAX

B. (T) 0.9

120 M(T)/M(O) ~

,

0.8

100 0.7 "1-

0.6

80 o e,,,,

0.5

13"

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-~GdFe2-B~xt= 0 T

+

40

0.3 0.2

20 0.1 0

0

e

I

=

I

=

I

200 400 600 Temperature(K)

i

800

Fig. 4.10. Temperature dependence of the spontaneous field/3~ from ZF-/~SR measurements in GdFe 2. The solid lines show the temperature dependence of the saturation magnetizations (from Barth et al. 1986a).

Adopting again the (2-2) site as the site of #+ residence (this site is also found to be occupied by hydrogen or deuterium at low concentrations (Didisheim et al. 1980)) one finds that there are 96 interstices of this type in the crystalline unit cell of the C15-structure (see fig. 4.7) which are not all magnetically equivalent. This leads to the appearance of different dipole fields at these sites which should have lead to a well resolvable splitting of the #SR-signal into several distinct frequency components. Averaging over all (2-2)-sites results, of course, in a cancelation of the dipolar fields in accord with the overall cubic symmetry. The absence of any splitting suggests that rapid #+-diffusion provides such an effective averaging mechanism. The remaining static field at the #+ is then simply given by (see eq. (3.5)) 471

B u = B c + -~- Ms,

(4.12)

MUON SPIN ROTATION SPECTROSCOPY ,

i

i

121

l

• q,

B~ (T)

0.8

-

100

•

Gd 002 T

80

- 0.7 - 0.6

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>" = m EL

0.4 40

0.3 0.2

20

\T o= 404K

0

I 100

0

I 200

i 300

0. 1

t 400

Temperature (K) Fig. 4.11. Temperature dependence of the spontaneous field B~ from ZF-/~SR measurements in GdCo2. The solid lines show the temperature dependence of the saturation magnetizations (from Barth et al. 1986a).

0.4

5 0 l l e ~-~ l ~

N

'

40

,

, YFezl

i _ B. (T) 0.3

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•

-

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20

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0.1

u..

10 0

I

0

100

I

I

I

200 300 400 Temperature (K)

t d

500

= si]5t

600

Fig. 4.12. Temperature dependence of the spontaneous field B~ from ZF-/~SR measurements in YFe2. The solid lines show the temperature dependence of the saturation magnetizations (from Barth et al. 1986a).

122

A. SCHENCK and EN. GYGAX '

140

•

'

'

'

'

t

•

1.0 O~

TmFe2

120

0.9 B~,~= 0 T

~" -r

100

>,

80

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O

•

0.8 0.7

•

t-

0.6

•

0

•

o" ,~

B~ (T)

0.5

60

•

0.4

40 -

0.3 0.2

200.1

I 0

100

200

300

400

500

600

Temperature (K) Fig. 4.13. Temperaturedependenceof the spontaneous field B~ from ZF-/zSRmeasurements in TmFe2. The solid lines show the temperature dependence of the saturation magnetizations (from Barth et al. 1986a). where the contact hyperfine field is the only microscopic field contribution. The temperature dependencies of the extracted Bc are shown in fig. 4.14. The contact hyperfine field seems to derive mainly from the M-sublattice. In YFe2 and LaFe2 this is a trivial conclusion. It is found from the study of the dependence of B~, on an external field that/3c is antiparallel to the Fe-moment direction or the Fe-sublattice magnetization. In GdFe2 and GdCo2, /3c is determined to be parallel to the total magnetization. Since the total magnetization is determined by the Gd-sublattice magnetization and in view of the antiferromagnetic coupling of the two sublattices one arrives again at the conclusion that Be and the Fe or Co-sublattice magnetizations are antiparallel. Finally in TmFe2 the total magnetization drops to zero at ~ 225 K (see fig. 4.13), where the two sublattice magnetizations happen to cancel, but this is not reflected at all in the #SR-data suggesting again that Bc is induced by the M-sublattice alone. Extrapolating Bc(T) in fig. 4.14 to zero temperature and plotting the Bc(0) versus the value of the ordered moment on the M-atoms one finds indeed a reasonable linear scaling as can be seen in fig. 4.15. No scaling is observed if Bc(0) is plotted versus the value of the ordered moment on the R-atoms. To the extend that #+-diffusion and trapping can be ignored fig. 4.14 displays then basically the temperature dependence of the M-sublattice magnetization in qualitative agreement with results from neutron scattering experiments (Bargouth et al. 1971). ZF-studies were also performed on DyNi2 and GdNi2 in the paramagnetic phase (Chappert et al. 1986b, Dalmas de R6otier et al. 1990b) with the aim to study

MUON SPIN ROTATION SPECTROSCOPY I ]

!

I

t

..,,.To

Bhf 0.4 ~ (T) 0.0

I

l

123

I

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I

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t

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GdFe2

• • leIO

0.4n0

,

-

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t 4

400 600 Temperature (K)

800

Fig. 4.14. Temperature dependence of the contact hyperfine field Be(0 K) in various RMa-compounds (from Barth et al. 1986a).

I

1.0-

I

I

T = OK

"--

TmFe2 e ~ _~.41~GdFe2 ~ ' ~ e 2 eC°' ErFe2 GdC°2

-z m___ 0 . 5 - ~ 0.0 ~" 0

I

t 0.5

i I 1.0 1.5 I~ of M atom (~ta)

I 2.0

2.5

Fig. 4.15. Correlation of/3c(0) with the M-atom magnetic moment in various RM2-compounds (from Barth et al. 1986a).

124

A. SCHENCKand EN. GYGAX

the spin dynamics in this range. The observed ZF #+-relaxation rate in GdNi2 is very similar to the corresponding one in GdA12. At room temperature one finds ,kzF(GdA12)/)~zF(GdNi2) = 0.23/0.42 ~_ 0.55(10) which agrees reasonably well with a calculated ratio of 0.32 invoking only the RKKY mechanism. The temperature dependence of ~zF(DyNi2) is also quite similar to the one in DyA12 but the room temperature ratio ,kzF(DyA12)/),zF(DyNi2)= 0.67/0.37 = 1.8(4) is in severe disagreement with a calculated value of 0.21. Various possible explanations for this disagreement are discussed in Chappert et al. (1986b).

4.2.2.2. RMn2 (R = Y, Tb, Dy). In contrast to the compounds covered in the two preceedings sections the cubic Laves phase compounds with M = Mn are antiferromagnets (see table 4.5). Of particular interest is YMn2 which displays a first order strongly hysteretic phase transition to an antiferromagnetic, long wave length helical state at ~ 100 K built from local moments (/~m ~ 2.7#B), while its behaviour above TN is that characteristic for a weak itinerant electron Pauli paramagnet and is described by the self consistent renormalization (SCR) theory of Moriya (1985). The collaps of the local moment at TN by ,,o 30% is accompanied by a decrease of the unit cell volume of 5%. The local moment can be destabilized by external pressure or by replacing Mn by an element with a smaller ionic radius. On the other hand substitution by an element with a larger ionic radius (providing 'negative' chemical pressure) is expected to stabilize the local moment on the Mn site. Similar effects can be induced by substitution of Y. Neutron scattering work revealed that strong antiferromagnetic correlations persist to at least 6TN. #SR work on this system was started with the aim to learn more about the spin dynamics and the change from local to itinerant moment behaviour (Asch et al. 1990, Cywinski et al. 1990a, 1991, Cywinski and Rainford 1994, Weber et al. 1994). Pure YMn2 has been investigated by both ZF and TF-techniques. TF-results (Hext = 22 mT) of Cywinski et al. (1991) are displayed in fig. 4.16 for both ascending and descending temperature scans. The hysteretic nature of the phase transition is clearly visible from the shifted drop in the signal amplitude, which signals the onset or the destruction of the ordered phase. The loss of asymmetry below TN reflects the appearance of a very wide internal field distribution over the #+ sites which induces an extremly rapid #+ depolarization within the dead time of the spectrometer. The appearance of a wide field distribution is in line with the long wave length helical magnetic structure in the ordered state. The approach of the transition temperature from above in both scans is accompanied by a quasi divergent behaviour of the TF-relaxation rate )~2 = l/T2. )k2 could be affected by various contributions and effects which were not considered in detail: powder broadening, nuclear dipole fields, fluctuating fields from the electronic moments, #+-diffusion etc. Weber et al. (1994) by a combination of TF, LF and ZF results find a somewhat modified temperature dependence of the signal amplitude A as shown in fig. 4.17. Most interesting is a drop of A starting below ~ 150 K. This drop results from the appearance of a rapidly damped component which is not taken into account in the analysis leading to fig. 4.18 and which signals the evolution of another magnetic phase in part of the sample volume above TN. Interestingly this fast component is

MUON SPIN ROTATIONSPECTROSCOPY 0.10

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125

I I I I I I

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(a)

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-

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300

0.25

II

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llll~

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0.20

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<

cooling

22

0.10

0.05

0.00

(b)

II I IIIII IIIII II II IIIII IIII1 0

1O0 200 Temperature (K)

300

Fig. 4.16. Temperaturedependence of signal amplitude (asymmetry) and relaxation rate ),2 from TF measurements (Hext = 2.2 mT) in YMn2. The solid lines are guides to the eye. Not shown explicitely are the ),z-data points on cooling. Note the hysteretic behaviour (from Cywinski et al. 1991). best fitted by a stretched-square root-exponential decay function (see eq. (2.21)) as appropriate for a dilute spin glass above the freezing temperature. LF-measurement show that this component is purely dynamical in origin and independent of the applied field strength (Hext < 0.2 T). The evolution of the fast component could be interpreted as following from the development of spin glass like clusters, starting well above the transition into the long range ordered state. The same and even more pronounced behaviour was also found in Tb doped samples of composition Yo.9Tbo.lMn2 (Asch et al. 1990) and Y0.95Tb0.osMn2 (Weber et al. 1994). Here the damping rate in the spin glass like phase could be extracted (~1 ~- 1 #s -1) and

126

A. SCHENCK and EN. GYGAX I

I

I

I

100

50 13.

E <

0 0

1 O0

200

Temperature (K) Fig. 4.17. Temperature dependence of the signal amplitude derived from a combination of TF, LF and ZF measurements in YMn2. For the slight drop at 150 K see text (from Weber et al. 1994).

~ 3.014 >

3.012 I

4-"

,5~

o

~ 3.013 I

I

~' I

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0.20

b)

2• 015

1

0.20

015

,~ o.lo

o.lo .~

0.05

0.05

0.0

t

0

t

t

100 Temperature

t

200

t

300

(K)

Fig. 4.18. Temperature dependence of signal amplitude (asymmetry), relaxation rate ),2 and precession frequency from TF-measurements (Hext = 22 roT) in Y(Mn097Fe0.03)2 (from Cywinski et al. 1990a). was found to have no significant temperature dependence. Some preliminary studies were also undertaken on Sc-doped (substituting for Y) samples (Weber et al. 1994). Due to the limited amount of data the results will not be discussed here. Further TF-#SR studies were performed on Fe substituted Y(Mnl_=Fex)2 (Cywinski et al. 1990a). Neutron diffraction and susceptibility measurements (Cywinski et al. 1990b) imply that as little as 2.5% substitution of Fe for Mn suffices to suppress magnetic order altogether and to induce Pauli paramagnetism. This was basically confirmed by #SR results on a sample with z = 0.03. However, neutron polarization analysis measurements (Cywinski et al. 1990a) show the persistence

MUON SPIN ROTATION SPECTROSCOPY

127

TABLE 4.6 #+-hyperfine fields extrapolated to T = 0 K and contact hyperfine coupling parameters Ac. Also listed is the ordered moment on the M-ions (Barth et al. 1986a).

GdFe2 GdCo2 YFe2 TmFe2 ErFe2

/~hf(0)

Ac

(T)

(T/LtB)

0.7 0.41 -0.6 0.8 0.6

-0.43 -0.40 -0.41 -0.47 -0.35

/ZM/#B 1.62 1.02 1.45 1.7 1.7

of antiferromagnetic correlations in the paramagnetic state over a wide temperature range. The neutron results could not distinguish between the possibilities that these correlations are static, reflecting perhaps topological magnetic frustration of localized Mn moments on a tetrahedral lattice, or are dynamic on the time scale of longitudinal fluctuations of the Mn spins. The TF-#SR relaxation function is very well described by an exponential function implying a dynamical origin and the temperature dependence of the relaxation rate A2 is displayed in fig. 4.18. It is well reproduced by the expression A2(T) = C T - x with z = 0.75, C = 1.12 # s - l ( K ) x. In contrast no temperature dependence in the precession frequency (Knight shift) could be detected in agreement with the picture of a Pauli paramagnet. The increase of A with decreasing temperature signals a slowing down of the Mn spin dynamics. It is argued that it does not arise from a slowing down of transverse spin fluctuations but instead from a slowing down of longitudinal spin fluctuations reflecting the localization of the moments on the Mn(Fe) sites. In any case the #SR-data are inconsistent with the evolution of static correlations and thus help in the interpretation of the neutron results. Substituting Mn by A1 instead of Fe leads to a 'negative' chemical pressure and the expectation that it stabilizes the local moment on the Mn site. Indeed a slowing down of spin fluctuations has been claimed in A1 substituted Y(Mnl_~AI~)2 (Motoya et al. 1991) and for z > 0.06 the volume collapse is suppressed and a spin glass like state is found at low temperatures. #SR-measurements on a sample with z = 0.1 clearly confirms spin glass order below Tg = 60 K (Cywinski et al. 1994b). Powder samples of DyMn2 and TbMn2 were studied by a few TF-#SR measurements only (Cywinski et al. 1992). Below TN the #SR-signal is lost completely. Above TN Fourier transforms of the #SR-spectra show the development of a pronounced asymmetric line shape as the temperature approaches TN from above. This feature which, in part, may be due to powder broadening has so far prevented a detailed interpretation of the results. Very preliminary ZF-measurements on TbMn2 (Weber et al. 1994) show a rather complex behaviour suggesting the coexistence of static and dynamic correlations already at relatively high temperatures (250 K).

128

A. SCHENCKand EN. GYGAX

4.2.3. Binary compounds with NaCl and CsCl-structure A list of compounds studied by #SR is found in table 4.7. The magnetic behaviour of all these compounds is rather complex and is characterized by a delicate interplay of various exchange and anisotropy mechanisms.

CsAs. W-measurements (0.02 T) in this compound display a strongly temperature dependent exponential relaxation rate ),2 below 9 K with a sharp singularity at 7.6 K, as shown in fig. 4.19 (Asch et al. 1988, Litterst et al. 1990). This temperature is identified with TN. At the same temperature the onset of a spontaneous spin precession is observed (at 4 K: u, -~ 24 MHz or B , "-~ 0.177 T). ZF-measurements reveal additional subtle features. Here a rapidly damped component appears already below 9 K and can be followed down to 7.6 K. At 7.3 K the spontaneous oscillating component undergoes a drastic change in spectral appearance (see fig. 4.20). The following scenario is suggested by these results: above 100 K the Ce-spins fluctuate freely (paramagnetic fluctuations); below this temperature spin correlations develop which, in part of the volume, lead to short range order below 9 K; at 7.6 K long range order sets in with a change in magnetic structure at 7.3 K. The change in structure at 7.3 K may reflect a transition between the single ( a n d triple q structure which both may have been seen in CeAs (H~tlg and Furrer 1986, Buffet et al. 1984) and which theory predicts to be nearly degenerate (Prelovsek and Rice 1987). Some interesting correlations are found in another neutron scattering study (H~ilg et al. 1987), where magnetic Bragg peaks appear already at 8.5 K and where short range ordered critical spin fluctuations peak at 7.3 K. A problem is connected with the observation of a rather broad spontaneous precession signal. If, as usual for these cubic compounds (see also section 4.3.2) the #+ is placed at the center of the cube formed by four Ce and four As atoms (see fig. 4.21) dipolar field calculations lead for both the single ~ and the triple q"structures to zero net dipolar fields at this site. As a way out it has been suggested that the presence 0.30

b~

0.25 -0

v

0

0.20 t-

._o 0.15

..t-., tl:l .BN i_ O Q.

a

0.10

O(9oO° CO oO

0.05 0.00

I

0

I

20

I

I

40

I

I

60

0 I

0 I

80

I

I

O I

I

O I

I

I

100 120 140 160

Temperature (K)

Fig. 4.19. Temperaturedependenceof transverse fieldrelaxationrate ,k2 ( n e x t Litterst et al. 1990).

=

20 mT) in CeAs (from

MUON SPIN ROTATION SPECTROSCOPY Ox O~

o~ ,,~

Ox

~d ~

:

~o~=

R R

~

H 0

H I~

~0

O~

T~

< t~

tr~

r~ BT~ t~,~

..'~

~

Z

Z

Z

~

o

o

<<<

129

130

A. SCHENCK and EN. GYGAX 2.0

1.0

13K

Ok2.0 1.0 ~

~

8K

0 2.01.0~

7.5K

2.o o

7.4K

1.0 o u_ 0

0

4

8

12

16 20

1.0 0

0

7.OK ~

lb

1~ 2b 2~

3'o

1.0 o

0

4K 10 20 30 Frequency(MHz)

40

Fig. 4.20. Fourier power spectra from ZF-measurements in CeAs. Note the change of the spectral distribution around 7.4 K (from Litterst et al. 1990).

of the #+ alters locally the spin structure which may not be unreasonable in view of the delicate balance of the various competing exchange interactions etc. CeSb. A total of 6 different AF phases are observed in this compound (Fischer et al. 1978). The magnetic structure results from a temperature dependent stacking of paramagnetic and ferromagnetic (001)-layers with the Ce 3+ spins pointing along +(001). The observed spin modulations can be described by a single wave vector q'0 = (27r/c)(0, 0, q) where q falls between 1/3 and 2/3 in fractions of integers and hence indicates a commensurate behaviour of the spin modulations (see table 4.8).

MUON SPIN ROTATION SPECTROSCOPY

)o

•

( ®

I

131

( @

o

~:

Fig. 4.21. Crystal structure of fcc RX-compounds (X = pnictides, chalcogenides). Indicated is the most likely/z+-site.

Placing the #+ again at the previously assumed site (see fig. 4.21) three different magnetic environments can be identified with populations depending on the exact value of q. At one type of site the net dipolar field vanishes ( u u = 0), at the two other sites non zero dipolar fields are calculated corresponding to precession frequencies of u~ = 16.5 MHz and u~ = 33 MHz. ZF-pSR measurements show the following behaviour (Klauss et al. 1994): Below TN two signals are found, one reflecting spin precession at 29.4(5) MHz and a second non oscillating signal displaying a stretched square root exponential behaviour. The relative amplitudes mirror nearly the expected populations for the sites associated with the u~, = 0 and u u = 33 MHz sites (see fig. 4.22). The sum of the amplitudes do not add up to the full asymmetry, seen above TN. The missing amplitude is assigned to the uu = 16.5 MHz site. Its absence is explained in terms of a very rapid damping of the corresponding #SRsignal arising from relatively slow fluctuations of the spins in the paramagnetic plane adjacent to this site. Consistent with the first order nature of the transition at TN the spontaneous frequency of 29.5 MHz is nearly constant in the whole antiferromagnetic temperature regime. In contrast above TN a rising relaxation rate (following from both TF and ZF measurements) with decreasing temperature proves the existence of TABLE 4.8 List of AF phases and related quantities in CeSb (Klauss et al. 1994). Phase

Temperature range (K)

q

I II III IV V VI

15.9-16.1 15.3-15.9 13.7-15.3 11.0-13.7 8.9-11.0 2.2-8.9

2/3 8/13 4/7 5/9 6/11 1/2

Population (in %) of sites with u=0

u=16.5MHz

u=33MHz

33.3 38.5 42.8 44.4 45.4 50

66.7 46.1 28.6 22.2 18.2 0.0

0 15.4 28.7 33.3 36.4 50.0

132

A. SCHENCK and EN. GYGAX 0.05

I

I

I

I 10

I 15

0.04

E E

0.03

I

I

0.02 0.01 0 0

5

20

Temperature (K) Fig. 4.22. Temperature dependence of the amplitude (asymmerty) of the 29.5 MHz signal (from ZF and 5 mT-TF measurements) in CeSb. The horizontal solid lines represent the predictions according to table 4.8 (from Klauss et al. 1994).

spin fluctuations which slow down upon the approach of TN. This has also been observed in diffuse critical neutron scattering (Halg et al. 1985).

DySb. This compound shows a MnO-like antiferromagnetic (AF II) structure (Felcher et al. 1973). Dipolar field calculations predict a single frequency of 210 MHz at the #+-site considered before. In contrast ZF-#SR measurements (Klauss et al. 1994) reveal two signals with nearly the same amplitudes, one displaying a spontaneous precession with uu _~ 240 MHz and zero frequency the other one. The latter signal displays again a square root exponential relaxation with A ,-~ 10 #s -1. The relaxation seems to be purely of dynamical origin since it could not be quenched in LF-runs up to 0.6 T. The appearance of two signals can be understood if instead of the MnOlike magnetic structure a CoO-type structure is considered. One then predicts two magnetically inequivalent sites with the same statistical weight and zero frequency at one site and ~300 MHz at the other site. Such a structure, however, seems to be severly at odds with earlier neutron results (Felcher et al. 1973). Again the observed spontaneous precession frequency is nearly temperature independent consistent with the first order nature of the transition. However, the ZF relaxation rates show a strong increase as TN is approach from both above and below. TF-measurements above TN reveal an anomalously strong increase of the (negative) Knight shift below 20 K on approaching TN (Klauss et al. 1994). If the CoO-type structure is adopted it can be explained by the onset of a quadrupolar anisotropy some degrees above TN which would be in line with earlier structure studies (Chen and Levy 1971).

CeAg, CeAgo.97Ino.03.CeAg as well as other RX (X = Cu, Ag etc.) compounds with cubic CsC1 structure have attracted much attention due to their magnetic and structural properties which are governed by one ion magnetoelastic couplings and two

MUON SPIN ROTATION SPECTROSCOPY 0.20

133

0.20

CeAg l_xlnx(x=O.03)

O CeAg O

-...,- o

0.15

b

o

c (/)

o o

,$

c9

E 0.10

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0.2 E O

O

t.9

,<

0.05

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X (1)

0.2

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m

•

~IID,

'

20

40

T (K)

•

0

60

0

I

I

20

40

0

60

T (K)

Fig. 4.23. Temperaturedependence of ZF signal amplitude(asymmetry)and relaxation rate in CeAg and CeAgo.97Ino.o3(fromHyomiet al. 1988). ion quadrupolar interactions (Morin and Schmitt, 1990a). CeAg shows a martensitic transition into a tetragonal phase at TQ _~ 15 K which is followed at Tc = 5 K by a second order transition into a ferromagnetically ordered state. The structural phase transition is driven by quadrupolar ordering (Morin, 1988). Substituting In for Ag it is found that TQ increases strongly with In concentration (Ihrig and Lohmann, 1977). There is some debate as to the nature of the structural instability in substituted compounds CeAgl_xIn= with z > 0.01 which does not seem to arise from quadrupolar ordering. ZF- and TF-#SR measurements on polycrystalline samples of CeAg and CeAg0.97In0.03 were aimed at a study of the structural phase transition (Hyomi et al. 1988) while the magnetically ordered phase was not investigated. Previously the #+ Knight shift was studied in the nonmagnetic series LaAga_=In= for 0 ~< z < 0.3 (Wehr et al. 1983). Figure 4.23 shows ZF relaxation rates ),ZF and the signal amplitudes as a function of temperature. In CeAg0.97In0.03 Azv starts to rise below ~ 25 K, reflecting the slowing down of the spin dynamics as the second order magnetic phase transition is approached, but no sign of the structural phase transition is manifest in these data. In contrast the behaviour of AZF in CeAg is totally different. Again below about 25 K AZF starts to rise. But concomitantly the signal amplitude starts to drop pronouncedly. This is due to the development of a very rapidly decaying component which could not be resolved in these experiments performed at the pulsed #+ source at KEK (Tsukuba, Japan). These results suggest that static spin correlations develop already around and below 25 K, leading to a wide distribution of static fields at the #+-sites quite in contrast to the situation in the In doped compound. Whether this different behaviour is connected to the nature of the structural phase transitions remains to be studied. Measurements with much improved time resolution would be extremely interesting.

DyAg. This compound displays antiferromagnetic order below TN = 60 K but no structural instability. Neutron diffraction has established that the spin order just

134

A. SCHENCK and EN. GYGAX

below 60 K is an incommensurate sinusoidally modulated structure which below 51 K transforms to a type I non-collinear triple (structure (Kaneko et al. 1987). The presence of two transition temperatures is also indicated by bulk magnetic and transport studies (Morin et al. 1990b, Sousa 1990). These findings are nicely corroborated by ZF and LF-#SR measurements on poly and monocrystalline samples (Asch et al. 1984, Kalvius et al. 1986, 1987, 1990). Approaching TN from above one observes a strongly rising relaxation rate as expected for a second order phase transition. At 300 K one finds )~ = 0.9 #s -1 which is of the same order of magnitude as in DyAI2 (see table 4.4). At TN the relaxation function changes from exponential to a stretched exponential without any drastic change in the overall relaxation rate. A further change occurs at ,,~ 51 K, below which a dynamic Lorentz-Kubo-Toyabe function describes the ZF-signal best. With further decreasing temperature a more and more static Lorentz-Kubo-Toyabe signal develops. Figure 4.24 shows the temperature dependence of the width A and of the fluctuation rate u extracted from the dynamic Lorentz-Kubo-Toyabe fits. No oscillating signal is seen below TN. This finds its explanation in the fact that at interstitial sites of high symmetry the sum of all dipolar fields from the ordered moments is zero. The Lorentzian shape of the residual field distribution at the #+ implies that the magnetic structure is nearly perfect and that any distortions (e.g., magnetic holes) are dilute and not too close to the #+. A surprising and unusual result is the increase of the width ,4 with increasing temperature. It is interpreted as being due to an increasing de-locking of the triple (magnetic structure and concomitantly, a progressive increase of a slight 10

,

I

,

,

,

4

DyAg 3 -1-

6 ¢-

,

Width/

2

,,,'

rr

4 o

E 1 J

P Rate

,

I ~O ~

00

I

10

I

I

20 30 40 Temperature (K)

L

50

0

Fig. 4.24. Temperature dependence of 4f-moment fluctuation rate u and coupling constant (= width of the field distributions in the static case) A from dynamic Lorentz-Kubo-Toyabe fits to ZF-data in DyAg (from Kalvius et al. 1990).

MUON SPIN ROTATIONSPECTROSCOPY

135

incommensurability (invisible to neutrons) as one moves toward the incommensurate phase commencing at ~ 51 K (Kalvius et al. 1990). The stretched (power) exponential relaxation behaviour between ~ 51 K and 60 K, which should be associated with the incommensurate sinusoidally modulated structure, has not been rationalized yet. Some #SR-measurements have also been performed on amorphous DyAg which orders ferromagnetically at Tc = 18 K (Kalvius et al. 1986, 1987). A very rapid relaxation is seen to develop below 100 K leading to a loss of the signal below 90 K consistent with M6sslauer measurements (Chappert et al. 1982). This signals a massive slowing down of the spin dynamics and a wide distribution of local fields due to random distortions of the local symmetry in the amorphous state.

4.2.4. RNi5 compounds The RNi5 compounds crystallize in the hexagonal CaCu5 structure (see fig. 4.25) and for heavy R (Gd,...) undergo a transition into a ferromagnetic state at temperatures below 37 K (see table 4.9). PrNi5 is a Van Vleck paramagnet which shows hyperfine enhanced nuclear spin ordering at 0.4 mK (Kubota et al. 1980). ZF and TF-#SR measurements on the nonmagnetic reference compound LaNi5 indicate that the #+ is immobile up to 150 K and most likely occupies the f-site (1/2, 0, 0) which is also found to be occupied by hydrogen in the a-phase of LaNisHx (Dalmas de R6otier et al. 1990c). In PrNi5 the i-site, located near the f-site, was unambiguously identified to be the #+ site from the angular dependence of the #+Knight shift [Feyerherm et al. (to be published), a previously claimed multiple site occupation in PrNi5 (Kaplan et al. 1989, Hitti et al. 1990a) could not be confirmed in subsequent studies on several new high quality samples]. As reviewed below, the magnetic properties as sensed by the #+ are largely determined by the crystalline electric field (CEF) split ground state multiplet of the 4f-electrons affecting both the static and dynamic behaviour.

)

) OU

~Ni ~AI

Fig. 4.25. Crystal structure of the hexagonal CaCu5 type. Indicatedare various possible #+-sites: b, f, h, o, m with multiplicities of 1, 3, 4, 12, 6 per unit cell, respectively. The i-site (not shown) is found along the c-axis above or below the f-site.

136

A. SCHENCK and EN. GYGAX

~-

o

¢->

.~=

.= o

,/

~',~ II

"~ ¢0

©

~-~'N (",1 t"q

b

~

:8"

b

o

0 0 N

~ m

~ ~.~.~.~ ~.~.~ .... Z Z ~

~N ~'~

MUON SPIN ROTATIONSPECTROSCOPY

137

GdNis. Zero field measurements on a single crystalline sample with fi(0) perpendicular to the c-axis at 3.3 K, i.e. below Tc, revealed two spontaneous precession signals with Ul = 5.3 MHz and u2 = 220.9 MHz with an amplitude ratio al : a2 of ~ 3 : 2. Obviously two different #+ sites must be involved. If fi(0) is chosen parallel to the c-axis no oscillating components are seen (Gubbens et al. 1994a). These results imply that the spontaneous local field at the #+ site is oriented parallel to the c-axis and suggest further that the c-axis is the easy axis. This conclusion is inconsistent with magnetization measurements (Gignoux et al. 1976) but agrees with M6ssbauer data (e.g., Tomala et al. 1977). The ZF-relaxation ),ZF has been measured for fi(0)H c-axis from room temperature down to 0.2 K. The results are shown in fig. 4.26. At high temperature AzF is temperature independent as expected for a Heisenberg magnet, neglecting the Korringa mechanism (see also section 4.2.1). The sharp increase of AzF in the vicinity of Tc reflects the critical slowing down of the spin dynamics. Since the temperature reading was not very accurate these data cannot be used for the determination of critical exponents. Relaxation can also be seen below Tc, involving the full asymmetry, which, below ~ Tc/2, varies approximately like T 2 ln(T) (solid line in the insert of fig. 4.26). This behaviour points to a two magnon scattering process (Dalmas de Rdotier and Yaouanc, 1994b). Fitting with the appropriate function a spin wave stiffness constant of D = 4.2(2) meV ,~2 is deduced. Note that this analysis does not require knowledge of the #+-site (Gubbens et al. 1994a). I''''

I''''1''''1''''

.

....

i ....

~ ....

i ....

I ....

] _

:

0.10 F 0,08 [-

4

.

*

o.12~° "

-~

0.06 0.04

{D

~-

.~

0.02 3

ooo

L_

.c Cl.

,i

0.14

5 N •T"

I''''1'''' |

• , 2

~ ....

I ....

5

,o

lO

i ....

15

,

t . . . . . . . .

20

25

R

30

•

•

•

1 j

GdNis

0 !

0

I

I

I

I

[

50

I

i

I

I

[

. . . .

100

I

,

150

,

,

,

I

,

,

,

200

i

I

. . . .

250

T e m p e r a t u r e (K) Fig. 4.26. Temperature dependence of the ZF-relaxation rate in GdNi 5 for P(0)llc-axis. The insert shows the data below Tc On an expanded scale. The solid line is a fit to a two magnon scattering process (from Gubbens et al. 1994a).

138

A. SCHENCK and EN. GYGAX

TbNi5. This compound displays a strong magnetocrystalline anisotropy and the ordered moments point along the a-axis. Information on the CEF-levels of Tb 3+ can be found in Gignoux and Rhyne (1986). ZF-#SR measurements on a single crystal sample were performed below and up to Tc (Dalmas de R6otier et al. 1992). A single spontaneous precession signal was observed the temperature dependence of which is displayed in fig. 4.27. According to section 2.2 /~

:

4~r

(4.13)

~ - ]~fs -}- Bdip -}- Bc

~: gj/ZB (3~f)

(4.14)

assuming that the contact hyperfine field/3c c~ (~f). As shown in section 3.1 this is not generally true, but may be a reasonable assumption here by noting that Tc << (90. The temperature dependence of {~f) was calculated in a mean field approximation taking into account the CEF splitting of the Tb 3+ ground state multiplet (Gignoux Illllllllllllll

IIlllllll

80

~" 6o 4O

g (~

20 0

TbNi~

30 rid

20

c E

10

+++

-

0 IlllJltq~ll,lllltllllll 5 10

0

15

20

Temperature(K) Fig. 4.27. Temperature dependence of the spontaneous precession frequency and associated relaxation rate below T c in TbNi5 (from Dalmas de R6otier et al. 1992).

MUON SPIN ROTATION SPECTROSCOPY J

t

~

v

uul~

n

t

i

u ill]

139 ;

t

100

CPOo

~,S~,

£/

d~ o

N

-i10 '1,...

r° ~

E £3

o8

? ~o~ • . •o°~5~o,, °

ErNis o

PSI

•

ISIS

0.1 i

i

i

t

~lJl

I

10

~

i

I

i

i ill

100

Temperature (K) Fig. 4.28. Log-log plot of the ZF relaxation rate versus temperature for P(0)llc-axis and/~(0)_Lc-axis, respectively in ErNi5. Data shown by filled (open) symbols were obtained at the ISIS facility of the Rutherford-Appleton Lab. (at the Paul Scherrer Institute) (from Gubbens et al. 1994a).

and Rhyne, 1986). It is represented by the solid line in fig. 4.28 and reproduces the data very well. The dashed line is a Brillouin function obtained by ignoring the CEF interactions. Obviously CEF-effects are important and cannot be ignored. However, choosing a different set of CEF parameters, one obtains about the same results as represented by the solid line (Dalmas de Rdotier et al. 1992). ZF relaxation rates measured above Tc (Dalmas de R6otier et al. 1990c) as well as the relaxation of the precession signal below Tc have not yet been explained but, at least for T > Tc, seem to be also significantly affected by the CEF-splitting.

DyNis. Gubbens et al. (1994a) report that their ZF-results on DyNi5 resemble basically those on TbNis.

ErNis. Like in TbNi5 a huge magneto crystalline anisotropy is found in ErNi5 which leads in the ordered state to an easy axis parallel to the c-axis below Tc = 9.2 K. ZF-/zSR measurements on sing~ crystals display strongly anisotropic features (Gubbens et al. 1992, 1994a) For P(0)±c-axis no signal is found below Tc. Above Tc the ZF signal is well described by a single exponentially damped component. The relaxation rate Az~c rises steeply with decreasing temperature and becomes so short below ~ 55 K that the signal is altogether lost (see fig. 4.28). For f(0)llc-axis a single exponentially damped signal is found above about 7 K while below 7 K two components are revealed, one of which is temperature dependent. However, already at around 12 K, the signal amplitude drops by ~- 25% indicating that a corresponding fractions of #+ are relaxed extremely fast, reflecting perhaps the on-set

140

A. SCHENCK and EN. GYGAX

of static spin correlations in part of the volume already at around 12 K. The temperature dependence of the relaxation rate ),lzl~ is displayed in fig. 4.28 too. It shows a maximum at around 12 K (!) and is much reduced from ,kzX~up to ,,~ 200 K. The observed anisotropy thus persists to temperatures T ~ 20Tc. Gubbens et al. 1992 could explain qualitatively the high temperature anisotropy in terms of a mean field picture including the CEF splitting in a very approximate way. For P(0)_Lc-axis an exponential temperature dependence of )~ZF at low temperatures is predicted while )~lzl~ is only weakly temperature dependent. According to a 166Er-MSssbauer study the spin fluctuations are very slow (quasi static) below 50 K (Gubbens 1989). This may be in line with the up turn of ) ~ below around the same temperature. The slow fluctuations are thought to arise from an indirect process between the two levels of the Kramers ground state doublet. The temperature dependence of ,klZlFbelow Tc is very well described by the expression A(T) = a ctg ( A / kBT) + bT 7

(4.15)

with a = 0.15 MHz, A/kB /> 5 K and b = 1.1 Hz/K 7 (solid line in fig. 4.28). This functional dependence is a clear signature for a phonon induced relaxation mechanism which is produced by the magnetoelastic coupling of the 4f spins to lattice vibrations. The first term corresponds to a one phonon process and the second to a two phonon process, respectively. The fitted parameters have not been quantitatively explained yet. TmNis. ZF-measurements on single crystal samples above Tc (Gubbens et al. 1994a) yield spin fluctuation rates which for both orientations/5(0)1 ]c-axis and/5(0)_Lc-axis show identical temperature dependencies and agree very well with 169TinM6ssbauer

lo

"t

=.E.

t ~''

.E

"4-'~_

°

.o

}+C~

1

4)0

;,2

bauer

TmNi s

o

0.1 I ....

0

I,,~

50

JL,~

JlT~,,I

....

100 150 200 Temperature (K)

I~,~L

250

Fig. 4.29. Temperature dependence of the Tm-4f moment fluctuation rates in ZF normalized to corresponding M/~ssbauer data in TmNi5 (from Gubbens et al. 1994a).

MUON SPIN ROTATIONSPECTROSCOPY

141

results. Figure 4.29 shows the temperature dependence of the corresponding correlation times "r (= reciprocal fluctuation rate). Since the #+ site and related coupling parameters to the Tm 3+ moments are not known fig. 4.29 shows essentially ~IzlF(T) and )~-~(T) normalized to the M6ssbauer results. The observed functional form of the temperature dependence has not been explained yet.

YbNis. YbNi5 has a very low ordering temperature of only ~ 0.5 K (Hodges and Bonville, unpublished results). ZF-#SR and M6ssbauer measurements on a polycrystalline sample were performed from 0.1 K up to room temperature (Bonville et al. 1994). The temperature dependence of the #+ relaxation rate agrees very well with the temperature dependence of the spin fluctuation correlation time "r extracted from the M6ssbauer data. It is found that the fluctuation rate increases linearly with temperature (i.e. AzF oc 1/T) pointing to a Korringa type mechanism as the driving force. Compared to ErNi5 and TmNi5 the relaxation rates are much larger and much less temperature dependent. It seems that CEF effects are less important in YbNi5 than in the two other compounds. PrNis. Somewhat out of order we will also briefly review some TF-#SR results on this singlet ground state compound which does not show electronic magnetic order. The previous discussion of results in TbNis, DyNis, ErNis, TmNi5 and YbNi5 rested to some extent on CEF-induced effects. It was hereby assumed that the implanted #+ does not modify the CEF-Hamiltonian governing the nearest neighbour rare earth ions. That this assumption may be grossly in error is demonstrated by #+ Knight shift measurements in monocrystalline PrNi5 where the #+ site is unambigeously identified with the interstitial i-site (Feyerherm et al. 1994b). It is found that while the Knight shift at high temperatures (i.e. above 50 K) scales very well with the bulk susceptibility this scaling is totally lost at lower temperatures. The result can 0.45

_~l

0.40 \

i

;¢y

I

I

i

i

I

I

!

. . . .

"5 0.35

y

0.25

T

.~ 0.20

=o_ o.15 0

0.10

m

0.05 0.00

0

20

40

60

80

1O0

Temperature (K) Fig. 4.30. Temperature dependence of the local susceptibility in the basal plane of PrNi 5 sensed and modified by the/z + via the Knight shift (dipolar and contact) for two different directions defined in the

insert (fromFeyerhermet al. 1994b).

142

A. SCHENCK and EN. GYGAX

PrNi 5 E( 1 )

5

48 K -

-

r6

39 K -

-

23 K - -

v4

o

p e r t u r b e d by IJ+

\

-

-

45K

-

-

37K 23 K

7.5K 0

Fig. 4.31. Crystal field splitting of the 3H4-ground state multiplet of Pr3+ in PrNi5 without (undisturbed) and with (disturbed) the/~+ at the interstitial i-site (only nearest neighbour Pr3+ are considered) (from Feyerherm et al. 1994b and private communication). be used to extract an effective local susceptibility which in the basal plane becomes anisotropic as shown in fig. 4.30. The solid lines represent a fit to a CEF-model in which the CEF parameters B2°,/322 and the exchange parameter A were allowed to vary independently. A best fit was obtained for B ° = 0.54 MeV (0.51 MeV) B 2 = 0.08 MeV (0 MeV) and A = 2.7 mole/emu (4.73 mole/emu). The values in parenthesis refer to bulk parameters as obtained from inelastic neutron scattering (Amato et al. 1992a). Figure 4.31 compares the CEF level schemes with and without the #+ induced perturbation. Of particular significance is the appreciable lowering of the first and second excited levels. Point charge model calculations failed to reproduce the fit results but it is clear that the #+ introduces an additional electric field gradient and destroys the local symmetry on the nn-Pr sites. The #+ induced perturbation may be particularly important in the case of singlet ground state ions but may be also non-negligible in other cases showing a CEF splitting. In GdNi5, however, no #+ induced effects would be expected (see also the discussion in section 3.2 and Campbell 1984, Schillaci et al. 1984).

4.2,5. Ternary compounds with tetragonal ThCr2Si2 structure Table 4.10 presents a list of investigated compounds. Two more compounds with the same structure are treated in section 4.2.6 on heavy fermion systems (see table 4.11). ZF-#SR measurements in the nonmagnetic compounds YCo2Si2 and LaPd2P2 served to get an idea on the #+-site. Possible #+-interstitial sites in the ThCr2Si2 type structure are indicated in fig. 4.32. The observed zero field Kubo-Toyabe signals point to an occupation of site No 2 (1/2, 0, 0) or site No 4 (1/2, 1/4, 0) (Dalmas de R6otier et al. 1990a). The question is, of course, whether in all compounds with ThCrzSi2 structure the same site is necessarily occupied (see also discussion on CeRuzSi2, section 4.2.6 and URu2Si2, section 4.3.4). #+ diffusion is seen to set in above ~ 130 K (Dalmas de R6otier, 1990a).

MUON SPIN ROTATION SPECTROSCOPY

143

TABLE 4.10 List of ternary compound with tetragonal ThCr2Si2 structure investigated by #SR (see also table 4.11). Compound Magnetic structure

TN, TC (K)

#SR

Samples

References Dalmas de R6otier et al. (1990a) Dalmas de R6otier et al. (1990a) Dalmas de R6otier et al. (1990a) Yaouanc et al. (1994) Matokawa et al. (1990), Nojiri et al. (1992) Gubbens et al. (1994b)

YCo2 Si2

-

-

ZF

sintered powder

LaPdzP2

-

-

ZF

sintered powder

~ 35

ZF

sintered powder

CeRhzSi2

AF:single ~',/7[[c-axis

PrRu2Si2 PrCo2Si2

FM,/Tile-axis ~ 15 ZF s.c. AF, 4 metamagnetic 30 (9,17) ZF s.c. transitions NdRhzSi 2 AF: single ~,/Tllc-axis 57 ZE LF s.c., sintered powder

~

)

#

6

#3 i

.#7

5

© o • Ce Cu Si Fig. 4.32. Crystalstructureof tetragonaltenarycompoundsof the ThCr2Si2 type. Starsindicatepossible interstitial/~+ locations.

CeRh2Si2. Z e r o field data confirm the onset o f m a g n e t i c order at ~ 35 K ( D a l m a s de R6otier et al. 1990d), p r e v i o u s l y identified b y neutron scattering (Quezel et al. 1984a). T h e onset o f o r d e r manifests itself b y a drastically increased relaxation rate with no loss in the signal amplitude. M e a s u r e m e n t s at 22.5 K reveal an additional p r e c e s s i o n signal c o r r e s p o n d i n g to a spontaneous field o f ~ 11 m T at the # + position. Since no m e a s u r e m e n t s were m a d e in the range 22.5 to 33 K no information is available on the e v o l u t i o n o f the precession c o m p o n e n t which, as far as the limited data can tell, is not present yet a c o u p l e o f degrees b e l o w TN. The a p p e a r a n c e o f two signals a s s o c i a t e d with zero net field and 11 mT, suggest the o c c u p a t i o n o f two different sites. In fact, using the m a g n e t i c structure o f Q u e z e l et al. (1984a, b), one calculates d i p o l a r fields o f zero and 13 m T at the sites No. 2 and No. 4, respectively.

144

A. S C H E N C K and E N . G Y G A X

0o

co

°i b

~D

r.~

d

O

O e'~

O ~

O e'~

O e'~

q

q

[..U

PU

b -6

d

,-d N

b ~

q

t'q

t"q N N

,...]

[-.., r.s.,

;.i.7 g . ,

N 0

8 t'¢3

P

T

s

,

.4-

.~-

~

t~ t"q

O

t"q

t"q

t"q

-d

,4

tt~

~D I

t'q

t.-q.

o=

~m

t~:r, ©

t'q

t"q

,4

,4

<

<

E

t"q e,~

X

a.7o

O

z

O

i <

<

o eq

o

,..~

~r.)

2 ?

eq

¢q

eq

< 0

o

6

.< r)

666

eq

g~r,.) #

e"~ v

, 0

~

r,.) "~

u6

r,.)

~-.

MUON SPIN ROTATION SPECTROSCOPY

145

c~

r,~

[-.-,

,.d

[--.,

O t"-'-

•4

.4

.4

¢-..1

.4 eb

,4

.4

,.-.-k

t',l

r..)

u-!. o o

r..)

° °

146

A. SCHENCK and EN. GYGAX

PrRu2Si2. In contrast to the other compounds in table 4.10 PrRu2Si2 shows ferromagnetic order below Tc = 15 K and has a giant magnetic anisotropy (anisotropy field Ha = 400 T (Shigeoka et al. 1992)) which seems to be much larger than in any other 4f-electron system reported so far. CEF effect are certainly very important but cannot account exclusively for the magnetic properties of this compound. ZF-/~SR measurements below Tc yield a spontaneous precession signal if fi(O)±caxis and only an exponentially relaxing signal if P(0)l]c-axis (Yaouanc et al. 1994). It follows that the local field B~ at the #+ must be oriented parallel to the c-axis (at T = 4.5 K, Bu = 0.0393(2) T). The relaxation behaviour and rates obtained below and above Tc are partially very puzzling and have not yet found an adequate quantitative explanation.

PrCo2Si2. This compound orders antiferromagnetically below TN = 30 K and shows metamagnetic transitions at 1.2 T, 3.8 T, 6.7 T and at the saturation field 12.2 T if an external field is applied parallel to the c-axis. In zero field two more transitions in the AF-structure appear at 9 K and 17 K (Shigeoka et al. 1987, 1989a, 1989b, Nojiri et al. 1992). ZF and LF-#SR measurements on a monocrystalline sample with/5(0) and the applied field parallel to the c-axis were initiated with the goal to monitor changes in the #+ spin lattice relaxation rate at the 12.2 T metamagnetic transition, exploiting the possibility to combine pulsed magnetic fields with pulsed #SR-spectroscopy at the pulsed muon source at KEK (Tsukuba, Japan) (Motokawa et al. 1990, Nojiri et al. 1992). Indeed an enhanced (by a factor of ,.o 10) exponential relaxation rate (~ 0.5 #s -1) compared to ZF-results at 4.2 K and 12 K was observed for longitudinal fields varying from ,-~ 8 T to 13 T at effective temperatures < 12 K. However, in view of the very large error bars, +(30-50)%, on the relaxation rates and possible systematic uncertainties it is premature to draw some physics conclusions from these results. More reliable are ZF-results. From 50 K down to 4.2 K exponential relaxation is observed without any loss of asymmetry. The temperature dependence of the relaxation rate AzF is displayed in fig. 4.33. It shows a pronounced and rather wide 1.0

1

&-0.8 •

E++

0.6

0

cO

0.4

rr

o

0

X

0.2

oo i

0.0 0

10

t 20

i

i 30

I 40

50

T e m p e r a t u r e (K)

Fig. 4.33. Temperature dependence of ZF relaxation rate in PrCo2Si2. The three different transitions temperatures are indicated by the dashed vertical lines (from Nojiri et al. 1992).

MUON SPIN ROTATION SPECTROSCOPY

147

maximum at ,-~ 25 K with essentially no clear hints on the transitions at 9 K, 17 K and 30 K. In particular no divergent type of behaviour is visible at TN. Nojiri et al. (1992) speculate that this behaviour arises from strong spin fluctuations in the long period AF structures between 30 K and 9 K, which can be viewed as a soliton like motion of domain boundaries.

NdRh2Si2. Antiferromagnetic order sets in at 57 K and, differently from CeRh2Si2, ferromagnetic (001) planes are coupled antiferromagnetically along the c-axis. ZFand LF-#SR measurements on monocrystalline samples yield the following results I ....

%-- 0.06 -i- 0.05 0.04

m

÷,~~ 0+ I ' ' ' ' 1

....

I ' ' ' ' 1

....

I ....

I ' ' ' '

NdRhzSi2 ÷

0.03 ~s~

o. 0.02 E r~ 0.01

• 0 mT

÷~

0.00 0 I

e 200 mT 50

. . . .

100 150 200 250 300 Temperature (K)

I

. . . .

{ . . . .

I

. . . .

I

. . . .

I ' ' ' ' 1

l

NdRh2Si2

.

ZE

zero field

v

:

:

r" .m

E

~,s.

0.1

G

a

k~°°°

|o

•

•

•

S S S l L I S : I * I t ' * ' I , * * ' I * * * L I ' * ' * I

0

50

100 150 200 250 Temperature (K)

300

Fig. 4.34. Temperature dependence of the ZF relaxation rate in NdRh2Si2. The upper part refers to /5(0)llc-axis, the lower one to /5(0)_Lc-axis. The dashed line in the upper part below TN follows a T 2 In(T) dependence, pointing to a two magnon mechanism for the relaxation (from Gubbens et al. 1994b).

148

A. SCHENCKand EN. GYGAX

(Gubbens et al. 1994b): ff/3(0) and Bext is applied along the c-axis an exponentially relaxing, non oscillating signal is seen, the relaxation rate )H of which is essentially independent of [/3extl, implying a dynamical origin of the relaxation. The data are shown in the upper part of fig. 4.34. The temperature dependence of A1 below TN seems to follow a T 2 ln(T)-law which points to a two magnon (Raman) process (Dalmas de R6otier and Yaouanc 1994b) as already indicated in the weakly anisotropic ferromagnet GdNis (see section 4.2.4). However, no quantitative analysis of this behaviour has been presented so far. On the other hand if P(0) is perpendicular to the c-axis the #SR signal is lost below TN, probably because the limited time resolution at the pulsed muon source of ISIS prevented the observation of a fast oscillating precession component. Taken together with the nonvanishing, nonoscillating component for/3(0)llc-axis it can be concluded that the spontaneous field at the #+ is oriented parallel to the c-axis. The relaxation rate for fi(0)_l_c-axis above TN is displayed in the lower part of fig. 4.34. In contrast to the other orientation a strongly divergent behaviour is revealed which signals a slowing down of the spin dynamics as TN is approached. Note that the rate close to TN is about two orders of magnitude larger than for the other orientation. Combining the results from both orientations it can be concluded that the fluctuating components are also essentially only present along the c-axis pointing to longitudinal spin fluctuations along the same axis. Due to limited precision in the temperature control the precise form of the critical slowing down could not be determined.

4.2.6. Rare earth based heavy electron and related compounds Intermetallics treated in this section possess quite different compositions, crystal structures and magnetic properties but have in common certain low temperature properties by which they are classified as heavy electron or heavy fermion compounds (see, e.g., Steglich et al. 1993). Included are also some systems which show nearly heavy fermion or Kondo system behaviour. The study of magnetic properties by #SR spectroscopy has been reviewed in detail by Schenck (1993). Short reviews have been presented by Amato (1994) and Heffner (1993). Table 4.11 lists all rare earth based heavy electron and related compounds which have been studied by #SRspectroscopy up to date (1993). Quite unique information could be obtained which was not available from other techniques. The most important pieces of information are compiled in table 4.12. CeAl2. This CeA12 moderately heavy electron compound possesses a rather complex incommensurate and modulated antiferromagnetic structure to be described by a multiple 0~arrangement (Shapiro et al. 1979). Interestingly the transition temperature is found to vary for different samples and seems to cluster either around 3.4 K or 3.9 K, respectively (Gavilano et al. 1993) #SR measurements were performed on both monocrystalline and polycrystaltine (powder) samples (Hartmann et al. 1989, Ott et al. unpublished (1993)). TF-#SR measurements (Hext = 15 mT) on a single crystal served primarily the purpose to determine the #+-site in this cubic Laves phase compound which was identified as the (2-2) site (see fig. 4.7) as in the other rare earth aluminides (see section 4.2.1) (Hartmann et al. 1989). #+-diffusion was shown

M U O N SPIN ROTATION SPECTROSCOPY

149

.,--t

..-t

0",

o o

ii .r.

=o~

~

~:

=

< 0

o'.,"-I. ooo I

o

~d kO

" ~

~

~

oi

0

H [e

e~

v~

v~

~-~

,I °e,i

"-I,

oV/

vzS

-~~ g ~ 0

~

~'~ ,.~

~,~

o m

A

0

0

0

~

II

H

~'~. ~.~.~

"0

2 ,-o o

.,< o

r~

'~ ~

~

o~

o+ r..) ~

~

o ~o

[,,e o

o ~

~o m

~.~

.<

r.j 0

~

150

A. SCHENCK and EN. GYGAX £ O~

,....4

0

Ob

,~

<~

<

I

b~

C~

~d m

~6 ~

v~'

e-, o

S

~.~

"~ - ~ •-

o o

r..,)

~

.

° o~

o:

~

~

~

~

_~

o~

~~

#-

MUON SPIN ROTATION SPECTROSCOPY I

'

i

L

n

151

I

0.20

ooo~O Z" "$ 0.15 E E < 0.10

°

°

oe

0.05

i

i

r

i

i

3

i

i

4

Temperature (K) Fig. 4.35. Temperature dependence of the paramagnetic and the magnetically ordered volume fractions in monocrystalline CeA12 as reflected by the amplitudes (asymmetries) of the two components in the TF-#SR signal below TN = 3.9 K (from Hartmann et al. 1989). not to be important below ~ 80 K. The same measurements revealed the appearance of a two component structure below 3.9 K one reflecting a paramagnetic behaviour, the other one showing a very rapid damping rate reaching ~ 40 #s -1 at ~ 3 K. The temperature dependence of the amplitudes of the two components is shown in fig. 4.35. The rapidly damped component reflects the evolution of a wide static field distribution, consistent with an incommensurate, modulated antiferromagnetic structure. But, as fig. 4.35 implies, this order does not involve at once the total volume of the sample, but the ordered volume grows gradually from zero at 3.9 K to ~ 80% of the total volume at ~ 3.2 K. Is this unexpected behaviour somehow related to the observation of two different transition temperatures? Zero field measurements on a powder sample yielded essentially the same picture (Ott et al. 1993). The latter sample was also investigated by 27A1-NQR (Gavilano et al. 1993). No mentioning of a two component behaviour is found in this paper. The two component structure of the #SR signal in CeA12 around TN is not unique but can be found in several other rare earth based heavy fermion compounds (e.g., CeA13, CeCu2Si2, YbBiPt etc.) giving the impression that it may be a characteristic feature of this class of materials.

CeA13. Long thought to be paramagnetic down to at least 60 m K (Murani et al. 1980) it came as a real surprise that this archetypical representative of fermion systems revealed the presence of an ordered state below ~ 1 #SR measurements (Barth et al. 1987, 1989a). Similar to CeA12 a two three) component structure developed below 1 K, but unlike in CeA12, the

the heavy K in ZF(or rather additional

152

A. SCHENCK and F.N. GYGAX 0.05 0.00

~\ + -f,,, , -...,/ - - : ~ - " ~ ' ~ - - ¢ ~ ' ~ " ~

-'--'--~"7

'$ -0.05 E E -0.10 <

-o15 a

-0.20

-o.%1o

I

0.5

I

1.0

I

I

1.5 2.0 2.5 Time (psec)

I

I

3.0

3.5

0.05 0.00

"-"~"~"~"

.. - - - 2 - ~

-0.05 E E -O.lO 69 ,< -0.15

b

-0.20 -0.25 0.0

I

I

I

I

I

I

I

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Time (psec)

Fig. 4.36. ZF-#SR signal in CeAI3 at 0.01 K (upper part) and at 0.2 K (lower part). It can be fitted by a sum of three components: one fast relaxing but nonoscillatingwith amplitude Af, one slowly relaxing and nonoscillating with amplitude As and an oscillating component with essentially a node near zero time (from Amato et al. 1994a). component displays a precession pattern corresponding to a local field of -,~ 22 mT rather than a wide field distribution with zero average. Figure 4.37 displays the temperature dependence of the amplitudes of the two signals, derived from more recent ZF-measurements (Amato et al. 1994a). To fit the data in fig. 4.36 actually a three component structure had to be adopted consisting of one oscillating, one very fast relaxing and one slowly relaxing component. The oscillating component showed an unphysically large phase shift (ignored in Barth et al. 1987, 1989a) which was later attributed to a precursor state manifest by the fast relaxing component (Amato et al. 1994a). It now seems that the sum of a fast relaxing and of a phase shifted oscillating components is under certain circumstances more representative of a spin density wave phenomenon (see Le et al. 1994, and section 7.1). The amplitude in fig. 4.37 associated with the ordered state signal represents the sum of the amplitude of the fast relaxing and of the oscillating components extrapolated to zero time. There is another important difference to CeA12. While the phase transition in CeA12 can also be monitored by the specific heat, which shows the characteristic cusp at TN, no such feature is seen in CeA13. Hence it seems that the magnetic order in CeA13 evolves in a non cooperative manner. This may be also indicated by the precession frequency which shows very little variation up to the temperature where the ordered volume vanishes (see fig. 4.38).

MUON SPIN ROTATION SPECTROSCOPY

153

25 ' 7 '

Amax

'

' /

20

'

//

/ /

15

"As

o:l

E E <

10

\

OI

0

0.1

I

I

I

0.2 0.3 0.4 Temperature (K)

I

0.5

0.6

Fig. 4.37. Temperature dependence of the zero time amplitudes (asymmetries) Af and As in CeAI3, Af reflects the volume fraction associated with magnetic order and As the (complementary) volume fraction remaining paramagnetic (from Amato et al. 1994a). 3.5 3.0 3-O-O -I- 2.5 2.0 'i1) 1.5 1.0 LI.

0.5 0.0 0.0

~

I

i

0.1

0.2 0.3 0.4 0.5 Temperature (K)

0.6

0.7

Fig. 4.38. Temperature dependence of the spontaneous precession frequency in CeA13. The solid line is intended to guide the eye only (from Amato et al. 1994a). S o m e o f the o b s e r v e d features are s a m p l e dependent. The first a p p e a r a n c e o f a m a g n e t i c a l l y o r d e r e d v o l u m e m a y vary f r o m 2 K d o w n to 0.6 K. In Barth et al. (1987, 1989a) it is shown that the m a g n e t i c order m a y be r a n d o m or spin glass like a b o v e 0.8 K up to 2 K, no such feature is seen in A m a t o et al. (1984a). A t the

154

A. SCHENCKand EN. GYGAX

lowest temperature Barth et al. (1987, 1989a) find nearly no persisting paramagnetic volume while Amato et al. report on a residual paramagnetic volume of ,,o 25% (see fig. 4.38). NMR-measurements have provided conflicting results on the presence or non-presence of magnetic order in CeA13 (Nakamura et al. 1988, Wong and Clark 1992). Later NMR studies by Ott et al. (1993) (priv. communication) seem to be consistent with the #SR results. From the local field of 22 mT at the #+, considering only dipolar fields and ignoring a possible contact hyperfine contribution, an upper limit of ~ 0.1 I#B was estimated for the ordered moment (Schenck 1993).

CePb3. TF-#SR measurements on a polycrystalline sample at 10 K revealed a split four-component signal with distinct Knight shifts ranging from -4200 ppm to +1000 ppm (Uemura et al 1986a). No further results are available. CeCus, Ce(Cul_xAIx)5, Ce(Cul_xGax)5. The system CeCu5 is not considered to be a heavy fermion compound according to a commonly adopted classification scheme, although its low temperature Sommerfeld constant 7 is nearly the same as for CeA12. Replacing some of the Cu by A1 or Ga (only the 3g-sites in the hexagonal CaCu5 structure are involved) the system seems to acquire heavy fermion properties as reflected by vastly increased 7 constants which seem to peak for z around 0.20 (Bauer 1991). Undoped CeCu5 orders antiferromagnetically below ,-~ 4 K (propagation vector ( = (0, 0, 1/2), ~orallc-axis), with rising z a suppression of TN is observed and for z > 0.2 long range order seems to be suppressed (Bauer 1991). The question was whether some type of short range or spin glass order could persist though. To answer this question ZF-#SR was applied to undoped CeCu5 and to doped compounds with z (A1, Ga) = 0.1 and 0.2 (all polycrystalline) (Wiesinger et al. 1994a, 1994b and priv. communication). In CeCu5 a spontaneous precession signal appeared below TN in an increasing part of the volume, however, never accounting for more than ~ 30% of the total volume, as can be deduced from the signal amplitude displayed in fig. 4.39 (insert). This figure shows also the temperature dependence of the spontaneous precession frequency. The non-oscillating component shows appreciable exponential relaxation down to the lowest temperatures (A _~ 2.2 #s -1 at 1 K). The observed behaviour seems to be incompatible with the magnetic structure indicated above, since for that structure only one signal is predicted with Bu = 0 for the #+ at the b-site, which site is deduced from the ZF Kubo-Toyabe signal observed above TN. The temperature dependence of the two signal amplitudes, moreover, suggest that two different kinds of magnetic domains could coexist like in CeA12 and CeA13. Preliminary data on the doped compounds show that the oscillating signal, which indicates long range order, has disappeared (or nearly disappeared in the sample with z = 0.1) and is replaced below ~ 2 K by a wide static field distribution of Gaussian shape, which now accounts for 75% of the total volume (Wiesinger et al. 1994b, priv. communication). The rms width of the field distribution amounts to ~ 23 mT which is about half the field resulting from the spontaneous precession frequency in CeCus. The data reveal clearly that indeed some spin glass like order is present even in compounds with z = 0.2. This raises the possibility that the enormous "),-values

MUON SPIN ROTATIONSPECTROSCOPY G

155

O

6 °

'

'

C 'e C u 5

o ~ 0

0 30

~ ° ~

4

,

,

,

i

~ 2O

~

0

o oc.c io~ oc ~ ,~

°°

o

q)

0 0

i

i

i

1

2

3

Temperature

I ~V

4

5

(K)

Fig. 4.39. Temperature dependence of the spontaneous precession frequency in CeCu5. The insert shows the behaviour of the amplitudes. Open circles refer to the precessing component, closed circles to the nonprecessionbut rapidly relaxing (),ZF ~ 2.2 #s for T --+ 0 K) component(from Wiesinger et al. 1994a). observed around z = 0.2 are of magnetic origin and arise from a high density of excited states at small energies which are known to exist in spin glasses.

Cefu6. This is one of the few remaining heavy fermion compound which have not shown any hints for some type of order at low temperatures. ZF-#SR measurements performed on a single crystal down to 40 mK likewise did not reveal any indication for static magnetic order placing an upper limit of ~ 10-3#B on any ordered moment if such should exist (Amato et al. 1993c). CeB6. This cubic compound shows antiferroquadrupolar ordering below TQ = 3.2 K and antiferromagnetic order below TN = 2.3 K, described by a double 0" structure (ql = (1/4, 1/4, 1/2), 42 = ( 1 / 4 , - 1 / 4 , 1/2)) (Effantin et al. 1985). A possible temperature dependence was not studied. In the quadrupolar phase antiferromagnetic order can be induced by an external field. Concerning the structure of the induced order controversial results are obtained from UB-NMR (Takigawa et al. 1983a, Kawakami et al. 1982) and neutron scattering (Efffintin et al. 1985; Erkelens et al. 1987). TF-#SR measurements on monocrystalline samples above TQ proved unambiguously that the #+ is located at the site (1/2, 0, 0) (or equivalently at the sites (0, 1/2, 0) and (0, 0, 1/2)) (see fig. 4.40) (Feyerherm et al. 1994a). ZF-measurements below TN displayed a very complex signal with 8 different spontaneous precession frequencies ranging from ~ 2.3 MHz to ~ 77 MHz as can be seen from

156

A. SCHENCK and F.N. GYGAX

(~)Ce

0 B

Fig. 4.40. Crystal structure of cubic CeB6. The/z+-site is indicated.

a Fourier-transform of the ZF-#SR signal at 10 mK (fig. 4.41). The temperature dependencies of the spontaneous frequencies are shown in fig. 4.42. Their partially very strange behaviour could not be explained yet. It may point to a temperature dependent reorientation of some of the Ce-moments. On the basis of the magnetic structure suggested by Effantin et al. (1985) dipolar field calculations predict no more than seven different fields, associated with the sites (1/2, 0, 0), (0, 1/2, 0) and (0, 0, 1/2), out of which six appear as slightly splitted triplets (sites (1/2, 0, 0), (0, 1/2, 0)). The calculated fields (a contact contribution is included) range from 30 mT to 90 mT (or 4.06 MHz to 12.2 MHz), far lower than the maximum observed field of ,-~ 0.57 T (Feyerherm et al. (1994a) and to be published). In particular the high frequency components are principally not explainable if the ordered moment is as small as 0.28#B as claimed by Effantin et al. (1985). It must be concluded that the AF-magnetic structure in CeB6 as well as the size of the ordered moments are drastically different from the neutron scattering conclusions or that the pres-

100 "

i

i

i

100[

i

i

i

,~,

=,

r,,,,. ILl

60l

3= 6O

T

= 60mK

13.

0

2

4

6

8

10

~ 2O 0

0

40

60

'

80

FREQUENCY (NHz) Fig. 4.41. Fourier power spectra of the ZF-/zSR signal in CeB6 below TN (from Feyerherm et al. 1994a).

MUON SPIN ROTATION SPECTROSCOPY

157

80 ¸

• 0 @ @0@ @ O

@ 0@0@O

@O

70

@0 O •

&AAAA

@

AA • A

50 40,oe

ooo , o

30 20.Pure mare un

I

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A O

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[] •

[]

10 O C CD D Cr CD

A

CeB 6 8.~<~

m

%,

•L z ~

6

•

o

IOO OOi O O 4'

2

spontaneous muon precession in zero magnetic field

@

•

~ m mt~u N D

~ O@0@0 O @

OO@

O

m D m ~ D

mmDmD

O

O0~

[] m D l ~ D

A A~i~ ....

0.0

, ,

0.5

. . . . . . . . . . .

1.0 1.5 Temperature (K)

*

2.0

2.5

Fig. 4.42. Temperature dependence of the eight spontaneous precession frequencies in CeB6 below TN. Open symbols: /5(0)11(1, 1, 0), filled symbols: /5(0)11(1, O, O) (from FeYerherm et al. 1994a).

ence of the #+ alters the magnetic structure locally. However, even when admitting the latter possibility, the appearance of more than seven frequencies is difficult to understand. Knight shift measurements in the quadrupolar phase are also at odds with the suggested induced AF-structure in this regime in that no further splitting of the two signals, seen above TQ, is observed below TQ. The observation of two signals above TQ reflects the fact that the three sites (1/2, 0, 0), (0, 1/2, 0) and (0, 0, 1/2) which are crystallographically equivalent, are magnetically inequivalent in the paramgnetic state (Feyerherm et al. 1994a). Despite considerable efforts to understand CeB6 it remains a mysterious system and is a challenge for further studies.

CeCu2Si2. This is probably the most famous and probably the most thoroughly investigated heavy fermion compound due to the fact that it was the first HF system displaying superconductivity. It was known that some kind of AF-magnetic order develops in nonsuperconducting Cu-deficient CeCuz_xSi2 or in Cel_~LaxCuzSi2, but no magnetic order was found in superconducting (Cu-enriched) CeCuz+xSi2 (Grewe and Steglich 1991). It therefore was a surprise when ZF-#SR measurements on a superconducting CeCu2.1Si2 sample revealed the onset of some random magnetic order below 0.8 K, i.e. at a temperature above the superconducting transition temperature

158

A. SCHENCK and EN. GYGAX

0.8 ~,

v__-~ "~x!

CeCu2 Si2

SC

"

-

0.4

"\ ~

"./ I"° 0

2

\

o

\"° 4 B (Tesla)

\

\" 6

8

Fig. 4.43. Magnetic and superconducting phase diagram of CeCu2Si 2. The solid lines represent He2 for HextHa-axis and llc-axis, respectively, the lower dashed curve reflects anomalies in the elastic constant Cll, the upper dashed curve separates a not well identified ordered magnet phase from the pararnagnetic phase (adapted from Steglich et al. 1991).

Tc ~ 0.6 K (Uemura et al. 1989). Since then NMR and other measurements have provided indications for a magnetic phase diagram in the temperature versus field representation as shown in fig. 4.43. For small applied fields the phase boundary is close to Tc somewhat inconsistent with the ZF-#SR result. An important question, never answered satisfactorily before, concernes the nature of the coexistence of magnetic order and superconductivity: is it carried by the same electrons, is it microscopic or macroscopic? ZF-measurements on a CeCu2.2Si2 sample (Luke et al. 1989b, 1994) and on a CeCu2.05Si2 sample (see Amato 1994, Feyerherm et al. 1995) seem to show that magnetic order and superconductivity do not coexist on a microscopic scale, that they are even mutually exclusive. The ZF-#SR signal starts to develop a two component structure as the temperature is lowered below ,-~ 1.2 K. Above this temperature relaxations is only induced by random dipolar fields from the Cu-nuclear moments (Kubo-Toyabe signal). One of the two components below 1.2 K shows a fast Gaussian decay with a strongly rising cr as the temperature decreases (fig. 4.44). The other component show basically the same behaviour as above 1.2 K with a somewhat enhanced relaxation rate at around To. The relaxation rate of the fast component corresponds to a static field spread of about 140 G at the lowest temperature. Assuming truly random order and considering all possible and reasonable interstitial sites an upper limit of 0.25#B is estimated for the ordered moment. The #+-site in CeCu2Si2 is not known. From the splitting of the TF-signal above 1.2 K in a polycrystalline sample it may be concluded that two different sites are involved (unpublished results). Most interesting is the behaviour of the amplitudes. As fig. 4.45 shows the ordered volume fraction, as measured by the amplitude of the fast signal, increases to about 70% at T~. Just below Tc it decreases again, signaling a depression in the ordered volume as superconductivity sets in. Superconductivity seems to be confined to the volume fraction not showing magnetic order, but this needs to be studied in more detail. There is essentially no difference

MUON SPIN ROTATIONSPECTROSCOPY 4

i

i

I

r

i

i

i

i

p

i

i

159

i

i

i

CeCu2osSi 2 ZF "

121"~

lO I

81 3

6: 4. T¢ C

0

0.2

0.4

0.6

0.8 r (K)

t.0

t.2

1.4

Fig. 4.44. Temperature dependence of the Gaussian decay constant of the fast relaxing component in CeCu2.05Si2 (from Amato 1994). 1.0

CeCu2.05Si2 0.8

pm

0.6 <E 0.4

0.0

i

0.0

i

0.2

i

i

i

i

i

i

0.4 0.6 0.8 Temperature (K)

i

1

1.0

i

1.2

Fig. 4.45. Temperature dependence of the amplitudes of the two components in the ZF-/zSR signal in CeCu2.osSi2. A1 (open symbols) and A2 (filled symbols) reflect the paramagnetic and the magnetically ordered volume fractions, respectively. The latter is identified by the fast Gaussian decay (fig. 4.44) (from Amato 1994). in the temperature dependence of the two amplitudes measured in CeCu2.2Si2 and CeCu2.05Si2. The appearance of two magnetically inequivalent domains in CeCu2Si2 resembles to a great extend the results obtained in CeAI2, CeA13, Ce(CUl_x(Ga, A1)=)5 as shown above. Once more one gets the impression that this behaviour is characteristic for this class of compounds.

160

A. SCHENCK and EN. GYGAX

CeRu2Si2. Conventional measurements did not reveal any hint for magnetic order in this compound. However, a metamagnetic like transition at Hext -~ 8 T for T < 10 K suggests that CeRu2Si2 is close to a magnetic instability (Haen et al. 1987). Inelastic neutron scattering show indeed (also for CeCu6 !) the evolution of intersite antiferromagnetic correlations below 60 K (Rossat-Mignod et al. 1988b). It is argued that quantum fluctuations prevent the thermal divergence of the correlation length thereby preventing long range static order. This picture was changed by ZF- and LF#SR measurements (Amato et al. 1993a, 1994b, Amato 1994). These measurements on two different monocrystalline samples revealed the onset of some static order at around 2 K (see fig. 4.46). The only other signature for a phase transition is seen in the thermoelectric power S which displays a minimum at ,-~ 2 K (see fig. 4.46). Ill,

I

i

i

i i Illl|

I

I

I I IIII

I

I

I

,1, ~

I I IIII

I

0.03

I

S.C. 2 0.02

'1/) v

CeRu2S'2 ZF

=L

II)

0.03

_.~_~_~_@~ ...... 0.01

¢-

.o 0.02 rr 0.01 a ml

I

l

I

f,,,,

l

t

illl

0

1.5

II

l

I

,

, i ,,,,

i

I

I I IIII

i

i

i i loll

i

i

i

AT_Lc i

1.0 ~-

0 @

0 0

@

-

0

0.5

o -

b

o 0

o

,,,I

10 .2

,

,

%°

o

,,i,I

10 1

,

,

iJl,,,I

~

10 °

'

i t'~'l'l

I

I

101

T (K) Fig. 4.46. Temperaturedependence of (a) the ZF-relaxationrate and (b) the thermoelectricpower S in CeRu2Si2 (from Amato 1994).

MUON SPIN ROTATION SPECTROSCOPY

161

This feature was previously attributed to the onset of a coherent regime in a Kondo lattice (opening of a pseudo gap (Martin 1982)) but this interpretation may now need revision. The width of the static internal field distribution at the #+ in the ordered state is extremely small and of the order of 0.02 mT. From this an ordered moment of I0-3#B is estimated (upper limit 0.7 x 10-3/ZB). This is the smallest moment ever encountered in an ordered state. The #+-site in CeRu2Si2 is known from the angular dependence of the Knight shift measured at higher temperatures in a transverse field (Amato, priv. communication and to be published). It is the (1/2, 1/2, 0) position (site No. 1 in fig. 4.32). Note that measurements in the non magnetic isostructural compounds Y(Co)2Si2 and LaPd2P2 did not favour this site. TF-field measurements in a very small applied field (7.4 mT) served to monitor the #+ spin lattice relaxation which is induced by the 4f-electron spin fluctuations (Amato, priv. communic.). The TF-relaxation rate A2 = l/T2 is mainly reflecting a 1/Tl-mechanism since static line broadening is practically absent due to the absence of sizable nuclear moments (a small correction for those is applied) and the small applied field. The relaxation time T2 is displayed in fig. 4.47 together with the onsite (Fss) and inter site (/]s) line width from inelastic neutron scattering (Fss is proportional to the single site fluctuation rate and /is to the correlated inter site fluctuation rate) (Regnault et al. 1990). Since the fluctuations are fast we have T2~- T1 oc F.

(4.16)

As can be seen T2 scales closely with /'ss as expected. The proportionality factor between T2 and /'ss can be used to extract a rough estimate on the size of the

3.5

~

3.0

C e R u 2 Si2

2.s "-"

L.-

150

,

1

2.0 ~_~L,k_.,L-i,JS..~ '

'

'

~

'

~

-

loo

I

~" .-.s

1.5

I--~

1.0

0.5

50 -is

-

0.0

,

0

I

20

i

I

,

40

1

60

i

I

80

i

0

100

T (K)

Fig. 4.47. Temperaturedependenceof the/z+ TF relaxation time 7"2(open circles) and the on site (Fss) and intersite (/]s) line width from inelasticneutron scatteringin CeRu2Si2 (from Schenck 1993).

162

A. SCHENCK and EN. GYGAX

fluctuating moment: ~ I#B. The ordered moment at low temperatures is extremely reduced compared to this value. Since such small moments are found in other heavy electron systems (UPt3, Ul_xThxBel3, UCus, see section 4.3.2) it seems that the establishment of very small moment magnetic order (random or long range) is another characteristic feature of heavy fermion systems. Note that generally almost all ordered moments are much reduced in comparison to the Hund's rule moment of Ce3+: # = 2.54#B. The system CeRu2Si2 shows more conventional AF-magnetic order when other elements are placed on the Ce, Ru or Si sites. For instance Ce(RUl_~Rhx)2Si2 has shown AF like transition for 0.05 < z < 0.3 by means of specific heat and susceptibility measurements (Sakakibara et al. 1992). For z = 0.15 a transition temperature of TN --~ 5.5 K is estimated. ZF-#SR measurements on polycrystalline Ce(Ru0.asRh0.15)2Si2 confirmed the onset of magnetic order below 5.5 K, which appears to be random in nature as deduced from the Gaussian character of the relaxation behaviour below 5.5 K (Murayama et al. 1993). However, as found in other systems only a fraction of the volume takes part in the order. From the static field spread of N 15 m T at 4 K the ordered moment is estimated to be ~ 0.2#B.

CeTSn (T = Ni, Pd, Pt). These orthorhombic crystallographically rather complex compounds are viewed as Kondo lattice systems with no heavy fermion properties. However substitution of a few % Cu for Ni transforms the system CeNiSn to a HE compound. CeNiSn is a Kondo semiconductor with as small pseudo gap ( ~ 14 K) at the Fermi surface which is thought to arise from AF-magnetic correlations between quasi particles. In high magnetic fields ( ~ 12 T) the semiconducting behaviour disappears and gap formation is suppressed (Takabatake et al. 1992). Various experimental results (see, e.g., NMR results of Kyogaku et al. 1992) hint at a magnetic phase transition at low temperatures (< 0.2 K) but an actual phase transition could not be established. In contrast CePdSn and CePtSn are Kondo metals which display antiferromagnetic order below N 8 K, which for both compounds shows an incommensurate, modulated structure according to neutron results (Kohgi et al. 1992, Kadowaki et al. 1993). TF-#SR measurements on monocrystalline CeNiSn reveal an unmeasurably small relaxation rate above 2.5 K. Below this temperature the relaxation rises in a pronounced manner but remains small (A2 "-~ 0.065 #s -1 at 33 mK). The observed temperature dependence, following a power law A2 e( T -1/3, is not untypical for a system approaching a phase transition but down to 33 mK no phase transition was encountered (Kratzer et al. 1992). It can be concluded that CeNiSn is at all temperatures (i.e. down to 33 inK) a fast spin fluctuator. This may be correlated with the other notion that CeNiSn is a valence fluctuating system, as has been deduced from susceptibility data (Takabatake et al. 1990). ZF measurements in CePdSn and CePtSn reveal the onset of spontaneous coherent spin precession below 7.5 K and 8.2 K, respectively (Kalvius et al. 1994). In CePdSn the #SR signal displays clearly two well resolved frequencies and the relaxation is relatively slow implying relatively narrow field distributions at the #+-sites. The latter are not known yet. This observation of narrow and distinct field distributions

MUON SPIN ROTATION SPECTROSCOPY

163

0.2 a

0.2

2.. ....

20G LF ,,,.

.........

-" . . . .

"'

'

" '~ '

t

-J

i

t

'"t

0.1

~ t

0.1

CePtSn 9K

0

2

~a~ 0.2

CePtSn 7K 4

6

8

0

200G LF

~

0

lOG LF

o

i

0.4

~

3

O. I

~

,

2

b

CePtSn 8.2K

0.1

I

ZF

I

o.a

,

I

t.2

,

I

1.8

CePtSn 4.5K

~

-0.

2

I

I

o

i 0.5

I

ZF I

t

I

Time (#s) Fig. 4.48. ZF-#SR signal in CePtSn at various temperatures reflecting the appearance of static random order (b) and coherent long range order (c and d). The static order at 8.2 K is verified by the decoupling in longitudinal fields (b) (from Kalvius et al. 1994).

appear to be inconsistent with an incommensurate modulated spin structure which is expected to produce rather wide field distributions. CePtSn displays a rather unexpected behaviour just above the temperature where coherent spin precession first appears. As can be seen from fig. 4.48 the ZF-signal changes from a weak decay above 8.5 K to a static Kubo-Toyabe behaviour below 8.5 K, reflecting the development of a random static field distribution with a width of about 45 G (Kalvius et al. 1994), followed by the appearance of the coherent spin precession pattern just below 8.2 K. Obviously long range order is preceeded by a precursor spin glass like state. It could imply that the fast paramagnetic fluctuations above 8.5 K are slowed down to rates below 1 MHz, creating a quasi static situation with respect to the #SR time window. A further change in the coherent precession pattern is observed below ~ 5 K (fig. 4.48d), indicating a transition to a different magnetic structure in agreement with neutron scattering results (Kadowaki et al. 1993). Again there is a problem to reconcile the appearance of distinct and relatively narrowly peaked frequency distributions with the claimed incommensurate structures (also below 5 K in CePtSn).

YbBiPt, Ybo.sYo.sBiPt. YbBiPt possesses a record high Sommerfeld constant of 7 -~ 8 J/(mol-Yb K 2) at low temperatures which has lead to quite some interest in this

164

A. SCHENCKand EN. GYGAX

cubic compound. Specific heat and susceptibility measurements suggest some kind of phase transition at ~ 0.4 K (Fisk et al. 1991). Recent resistivity measurements suggest a partial gapping of the Fermi surface at 0.4 K and the appearance of a spin density wave (SDW). The opening up of a gap can be suppressed in fields >0.3 T (Movshovich et al. 1994). ZF-, TF- and LF-#SR measurements were performed on crushed powder and crystalline samples (Amato et al. 1992b, Amato et al. 1993b, Heffner et al. 1994). The ZF-measurements identified the transition at 0.4 K to be a magnetic phase transition. Since no precession pattern was seen but only a wide Gaussian like distribution of fields (~ 8 mT) with zero average it was concluded that a spin glass like random order or some complicated incommensurate structure must be established again in a spatially inhomogeneous way. Some information on the #+ site was obtained from TF-measurements at higher temperature, which indicated that the #+ is located somewhat off center at the position (1/2, 1/2, 1/2) (Feyerherm, priv. communication). Adopting this site and the random order picture an ordered moment of ~ 0.1#B is estimated. This is much smaller than the moment following from the Curie behaviour of the susceptibility below 10 K (3.6#B). However, the small value of the ordered moment is consistent with the absence of a nuclear Schottky anomaly in the low temperature specific heat (Thompson et al. 1993). Like in other heavy fermion systems the ZF-signal showed a two component structure which developed already much above 0.4 K, i.e. below ~ 1.2 K. Above 0.4 K both components showed an exponential relaxation behaviour. At 0.4 K one of the two components changed abruptly to a Gaussian behaviour with a rapidly increasing decay rate, while the other component continued to show exponential relaxation (see fig. 4.49) just as in the paramagnetic phase. Interestingly the Gaussian component changed in appearance, below ~ 0.1 K, indicating a change in magnetic structure or spin dynamics (Heffner et al. 1994) consistent with similar evidence from susceptibility and specific heat data (Thompson et al. 1993). The Gaussian field distribution did not change in longitudinal fields up to 2 T. This is quite significant since it proves that the magnetic order, sensed by the #+, cannot be associated with the SDW carried by the conduction electrons, which is destroyed in fields above 0.3 T. It therefore seems that the magnetism in YbBiPt is of twofold origin: Yblocal moment order and a conduction electron SDW, implying that the electronic ground state is made up of rather independent electronic subsystems. In this respect there is a certain parallelity with the results from UPd2A13 and UCu5 (see section 4.3.2). ZF-measurements in Yb0.sYo.sBiPt lead to rather similar results. Interestingly, despite a 50% dilution of the Yb-moments, the phase transition or spin freezing temperature was very little reduced. The dilution effect was clearly seen in a corresponding decrease of the static field spread (Amato et al. 1993b).

Sm3Se4. Low temperature susceptibility measurements revealed a huge linear specific heat term with 3' = 0.79 J/(K 2 mol-Sm3+) despite the fact that Sm3Se4 shows semiconducting behaviour with no free carriers at low temperatures (Fraas et al. 1992). This system has been discussed within the context of "heavy electrons without free

MUON SPIN ROTATION SPECTROSCOPY 8

,

I

I

I

I

I

I

I

I

I

I

165

7 6

4" s ,.~ 4

o

[]

3

2 1

0 0.6

I

I

I

I

r

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

b

0.5 r " 0.4 0.3 ,..=

0.2 0.1 0.0

I

I

I

I

I

IIII

II

I l t l

I I

10080I lOi i IoDA~A,,Ii o 6o

0°°~

40

° 0 o

I

I

0.2

I

o

0

[]

c

o o

o

2O 0/ 0.0

o

I I 1 1 1 1

0.4 0.6 0.8 1.0 T (K)

I

1.2

Fig. 4.49. Temperature dependence of the amplitudes and the relaxation rates from a two component analysis of the ZF-/zSR signal in crystalline YbBiPt. The rapidly relaxing signal (Af, ¢rf, Af) changes from an exponential behaviour (Af) to a Gaussian behaviour 0rf) at 0.5 K. The slowly relaxing signal (As) displays an exponential behaviour with only a weakly temperature dependent ~,s (from Amato et al. 1993b).

carriers". Sm appears in this compound in the two valence states Sm2+ and Sm 3+ with a relative occupation of 1 : 2, and no charge ordering. Relatively slow valence fluctuations are observed above 100 K (Takagi et al. 1992, 1993). Nothing was known about the low temperature magnetic properties of this compound. ZF-#SR measurements on a polycrystalline sample revealed the development of static magnetic order below 20 K by the appearance of a fast damped Gaussian component, again in only part of the volume (see fig. 4.50). The low temperature static field spread seen by the #+ amounts to ~ 3.5 mT, from which, making some reasonable assumption on the #+ site, a randomly ordered moment of 0.05#B can be estimated (Takagi et al. 1993 and unpublished results). A spin glass type structure of randomly frozen moments (perhaps related to the statistical distribution of Sm 3+ ions

166

A. SCHENCK and EN. GYGAX

3.5 3.0~ + + + ~ =k v 2.5 I

i

illllh

I

i

i

lll,,q

i

"7

tr tO

,

Ulllll

I

Sm3Se4 ZF

o Gauss.

• Exp.

2.0

'~ ,~

1.5

DLF6kOe i

X

m 1.0

rr

0.5

• oOdpe

0 100

oo

i

0

i Illlll

F

I

I IIIIII

I

'

'

'

'

'

'J'"l

•

~ I

0 I

m~

I IIllll

l

;I-'.-,~'"-U-~ "-

''H'l

, i

80

"o

t i

_o_e_o ~ . . . . .

60

.-_= ~. 40 E ,<

oe'" i

.... %.

"if,

20 0

-1

10

I

,,,,,,I

i

I

iiiiiii

100

i

10~

i

iiiiill

102

T (K) Fig. 4.50. Temperaturedependence of ZF relaxation rates (exponential, Gaussian) and amplitudes (asymmetries) from a two component analysis of the ZF-#SR signal in Sm3Se4 below ~ 20 K. Above 20 K only one component, decaying exponentially, is seen which is unchanged in a longitudinal field (from Takagi et al. 1993 and unpublished results). on the Sm sublattice) may lead to a high density of low energy magnetic excitations - as in a conventional spin glass - and could explain the large linear specific heat term, as discussed in the case of YbBiPt.

4.2.7. Ternary rhodium borides and chevrel phase compounds The interest in the ternary compounds RRh4B4 (R = rare earth) stems from their property to show superconductivity in coexistence with AF magnetic order as in SmRh4B4 or reentrant ferromagnetism as in HoRh4B 4 and ErRh4B 4 (see, e.g., Maple et al. 1982, Kakani and Upadhyaya 1988). The Chevrel phase systems RMo6S8 (RMo6Se8) show coexisting AF-magnetic order and superconductivity for R = Nd, Gd to Er, Yb (Fischer 1978). The situation in EuMo6S8 and EuMo6Se6 is controversial. Recent investigations claim the establishment of some complex order below 0.4 in EuMo6S8 and below 0.85 K in EuMo6S8 (Takabatake et al. 1984, Quezel et al. 1984b). A structural phase transition from rhombohedral at high temperatures to trictinic a low temperatures was detected at ~ 110 K in EuMo6S6 (Baillif et al. 1981). At the same temperature a change from a metallic to a nonmetallic behaviour was found in resistivity measurements (Meul et al. 1982). The system Eu]_~SnxMo6S8 (z > 0.2) shows again superconductivity. No structural phase tran-

MUON SPIN ROTATIONSPECTROSCOPY

167

sition is seen in this pseudo ternary compound and the rhombohedral structure is observed at all temperatures which is apparently a prerequisit for the development of superconductivity. It has attracted particular attention because of the observation of field induced superconductivity due to the Jaccarino-Peter effect (Meul et al. 1984). Rare earth rhodium borides. TF and ZF-#SR measurements on the nonmagnetic homologues LaRh4B 4 (Boekema et al. 1982) and YRh4B4 (Huang et al. 1983b) showed that the #+ is immobile up to about 200 K. Possible #+ sites were discussed in detail by Noakes et al. 1987. Measurements on the magnetic compounds yield the following results.

SmRh4B4, ErRh4B4. #+ relaxation

observed in ZF and TF displayed a pronounced temperature dependence (Noakes et al. 1987, 1985) from which the rare earth moment fluctuation rates could be determined. Although the Er 3+ moment is considerably larger than the Sm 3+ moment the fluctuation rates are very similar in their magnitudes and temperature dependencies (as an example see fig. 4.51). No effect of the superconducting transition on the fluctuation rates are seen. Since the #+ relaxation rates became excessively large in ErRh4B4 when lowering the temperature below 50 K the region around the onset of ferromagnetism could not be studied. ZF-measurements below TN in SmRh4B4 did not reveal any precession signal. Instead the data could be best fitted with two exponentially relaxing components, one .-(1) --~ 10 #S -1) and the other one of approxidisplaying a very fast relaxation rate (Zzv mately equal amplitude showing a rather slow relaxation (x(2) ~ 0.17 #s -1) (Huang ~-"ZF et al. 1983b). It is argued that this most likely reflects an occupation of different sites with rather different magnetic environments in the ordered state. In any case the absence of coherent spin precession and the appearance of exponentially damped I

I

I

I

6 o g"J 0

-~ rr

4

._o ¢3 II

0

0

I

I

i

I

50

1O0

150

200

250

Temperature (K) Fig. 4.51. Temperature dependence of the Sm 4f-moment fluctuationrate in SmRh4B4. The solid line is a fit to a model which includes the RKKY, the Korringa mechanism and phonon induced transitions between CEF levels (from Noakes et al. 1987).

168

A. SCHENCKand EN. GYGAX

signals points to a rather complicated antiferromagnetic structure such as has been seen in NdRh4B 4 and TmRh4B4 (Majkrzak et al. 1982, 1983). There is no other information on the AF structure in SmRh4B4. It also needs to be checked to what extend the #SR-data below TN still reflect dynamic features. The spin relaxation rates, shown in fig. 4.51, were explained in terms of phonon induced transitions between the CEF levels of both Er 3+, Sm 3+ and the RKKY- and the Korringa mechanism (Noakes et al. 1987). A fit of this model (solid line in fig. 4.51) leads to values for the RKKY exchange rate P'RKKY'the Korringa constant UKorr/T and the strength of the CEF related spin lattice relaxation which are in rough agreement with theoretical estimates for SmRh4B6 (Kumagai et al. 1981) (t,fi~K Y ,--o 1.2 x 10 s S- 1 , UfiKtorr/T~ 3.4 x 10 6 K - I s - l ) . For ErRh4g 4 the apparent RKKY rate and the Korringa rate are determined to be URKKynt~-- 3 X 10Ss -1 and U~torr/T -~ 4 x 105 K - i s -1, i.e. the RKKY-mechanism dominates the spin fluctuation rate below ~ 50 K and the Korringa mechanism is essentially absent. This is not understood in detail. Some excess slowing down in the spin fluctuation rate is seen in SmRh4B4 below ,-~ 4 K when approaching TN (Noakes et al. 1987).

HoxLul_xRh4B4 (x = 1, 0.7, 0.02, 0,005). The lowest state of the J = 8 518 Ho 3+ground state multiplet in HoRh4B4 is a degenerate nearly pure I + 8) doublet with no or only very small off-diagonal matrix elements of J, inhibiting or suppressing relaxation within this doublet at low temperatures by, e.g., the Korringa mechanism. In this respect HoRhaB4 is quite distinct from SmRh4B4 and ErRh4B 4 which both possess degenerate CEF ground-state level which are connected by IAmjI = 1 transitions. Consistent with the ground state properties of Ho 3+ in rhodium boride a severe slowing down of the spin fluctuations is observed in ZF- and LF-#SR measurements. In the system HoxLul_xRh4B4 this is seen for all concentrations x to the extent that even in the very dilute compounds with x = 0.02 and x = 0.005, which do not show magnetic order, a quasi static situation develops below T "-~ 9 K for x = 0.02 and below T _~ 6.5 K for x = 0.005 (described as an isolation of the ground state doublet). Since the fluctuation rates are essentially independent of x below ~ 10 K a single ion relaxation mechanism must prevail which points to the Korringa mechanism, although quite suppressed, as the responsible one (Heffner et al. 1985, 1984, MacLaughlin et al. 1983, Boekema et al. 1982). The onset of magnetic order in HoRh4B4 (Tc = 6.6 K) and in HO0.TLU0.3Rh4B4 (Tc = 4.1 K) is not reflected visibly in the ZF-#SR relaxation data. However, very similar to the dilute compounds, quasi static behaviour develops below (9-10) K (fluctuation rates < 1 MHz). Above ~ 10 K the fluctuation rates change to a thermally activated behaviour reflecting the transition from intra ground state doublet relaxation to a much faster activated behaviour involving excited CEF levels as in SmRh4B 4 and ErRh4B 4. The onset Of superconductivity in Hoo.TLuo.3Rh4B4 has no apparent effect on the fluctuation dynamics.

GdRh4B4. The spin dynamics in compounds such as SmRh4B4, ErRh4B4 and HoxLUl-xRh4B4 is obviously very much determined by the CEF-split ground state

MUON SPIN ROTATION SPECTROSCOPY

169

features. In contrast Gd 3+, which is in a pure spin s t a t e (857/2) , is unaffected by the CEF. This is clearly reflected in ZF- and LF-#SR data which reveal an essentially temperature independent spin fluctuation rate (~ 5 x 101° s -1, depending on the assumed site) down to the ferromagnetic ordering temperature (_~ 5.6 K). The temperature independence points to an RKKY exchange mechanism. In comparison to S m R h 4 B 4 and E r R h 4 B 4 the rate is enhanced by two orders of magnitude (MacLaughlin et al. 1983). No slowing down of the fluctuation rate close to Tc shows up when approaching Tc from above (the critical region may be very narrow) but the #+ relaxation rate decreases steeply below Tc reflecting the freezing of fluctuations in the ordered state (small amplitude fluctuations). The rather sharp break in the #+ relaxation rate at Tc is rounded off remarkably in a longitudinal field of 0.1 T. This feature has not been explained. No spontaneous coherent spin precession signal is seen below Tc. This together with the small relaxation rate at 2 K implies that the #+ resides at a site where, in the ordered state, the internal fields must cancel rather well.

EuMo6S7.5Seo.5, Euo.75Sno.25Mo6S7.6Seo.4.The first compound does not become superconducting but shows, as pointed out before, a structural phase transition at ~ 110 K. The second compound becomes superconducting below 5.15 K. The partial substitution of Se for S was chosen in order to create a particular phase diagram in the field

80 a ) ~

100

'~

I

I

I

I

I

60

oO O

40 >

20

~ oo I I I I I I I f I ~ I I I I ] I I I I I L I I I I I I

0

50

100

i

80 '~

100 150 200 Temperature (K) i i i

I

250

300

I

b)

60

(~

O

~.~ 40 20 0

i

i

i

i

i

50

i

i

i

i

i

i

100

i

i

i

i

i

150

i

1 f

i

i

200

i

i

i

+ ~ i

250

i

I

300

Temperature (K)

Fig. 4.52. Temperature dependence of the Eu-4f moments fluctuation rates u in (a) EuMo6S7.5Seo+5 and in (b) Euo.75Sno.25Mo6S7.6Seo.4 extracted from ZF-relaxation rates (from Birrer 1991).

170

A. SCHENCKand EN. GYGAX

induced superconducting state which, however, is of no relevance for the present discussion. ZF, LF and TF-measurements show consistently the presence of rather slow spin fluctuations in the range ~ (1-300) K and some kind of magnetic order below or around 1 K (Birrer et al. 1990, 1989b, Birrer 1991). The temperature dependence of the Eu-4f moment fluctuation rate u is shown in fig. 4.52 for both compounds. It was deduced from the muon spin relaxation rate AZF (= A1) by adopting a reasonable #+ site and assuming an Eu 3+ moment of # = 7.5#B (S = 7/2). The effect of the structural phase transition is clearly manifest in the data shown in fig. 4.52a. Interestingly u rises linearly with T in the metallic phase (also in Eu0.75Sn0.25Mo6Sv.6Se0.4 from 10 K up to at least 130 K, fig. 4.52b), indicating a Korringa mechanism, but is temperature independent in the non metallic phase down to 10 K. In both compounds a change in behaviour is observed below 10 K. While u in the Sn doped compound becomes now temperature independent at the same value as found in the non metallic phase, u in the undoped compound starts to rise down to ~ 0.8 K, where it saturates. However, LF measurements show that below 0.8 K in EuMo6S0.vSe0.5 and somewhere below 3.5 K, but above 1.4 K, in Eu0.75Sn0.25Mo6ST.6Se0.4static random fields are established signaling the onset of some magnetic order (Birrer 1991). Whether it is spin glass like or of a complex long range structure could not be determined. The fluctuation behavior displayed in fig. 4.52b, in particular also the linear temperature dependence below 130 K, agrees closely with M6ssbauer results in Eu0.25Sn0.75Mo6S8(Dunlap et al. 1979). The spin dynamics in Eu0.75Sn0.25Mo6S7.6Se0.4is unaffected by the superconducting transition at 5.15 K. 4.2.8.

R2Fel4B

This class of rare earth intermetallic compounds has found an important application in the construction of hard permanent magnets. Their usefulness in this respect derives from very large magneto-crystalline anisotropies which are related to CEF effects and strong spin-orbit interactions of the rare earth constituents (see, e.g., Buschow 1986) From a more fundamental point of view these compounds are interesting since both the R- and the Fe-sublattice display magnetic order which sets in below a common Curie temperature indicating a strong interplay of the localized 4f electrons and the more itinerant 3d electrons. The sublattice order is ferromagnetic for all compounds but the two sublattices couple ferromagnetically for the light rare earth up to Sm and antiferromagnetically for the heavy rare earth. In each case a collinear arrangement is found. In some compounds spin canting without destruction of the collinearity is observed. The crystal structure is quite complicated as shown in fig. 4.53. From studies of hydrided compounds with small hydrogen concentrations it was deduced that hydrogen occupies the so called tetrahedral 4e sites with two R and two Fe nearest neighbours (see fig. 4.53) (Ferreira et al. 1985). There are four crystallographically equivalent 4e sites within the unit cell. They are also magnetically equivalent as long as the ordered moments are parallel to the c-axis. It is reasonable to assume that also implanted #+ will reside at the 4e site. ZF-#SR measurements were under taken with the aim to gain additional information on the low temperature magnetic structure of the RzFex4B system (Yaouanc et al. 1987, Niedermayer et al. 1990).

MUON SPIN ROTATION SPECTROSCOPY

171

/,e site

[I

010~,

R2Fell.B Fig. 4.53. Crystal structure of tetragonal R2Fel4B (R = rare earth). Indicated is a tetrahedral 4e interstitial site which is known to be occupied by dissolved 'hydrogen (from Niedermayer et al. 1990).

In all investigated polycrystalline compounds (see table 4.14) a single spontaneous coherent spin precession signal was seen in ZF-measurements. In the compounds Y2Fe14B and Pr2Fel4B the spontaneous field Bu at the #+ site decreases smoothly with temperature with no further structure consistent with no spin canting in this two compounds. B,(T) scales roughly but not precisely with the macroscopic magnetization. Note that in Y2Fel4B Bu originates solely from the Fe-sublattice. The upper part of fig. 4.54 shows the temperature dependence of z~u in Nd2Fe14B together with the macroscopic M(T). Below 150 K, uu(T) decreases while M(T) continues to increase. The break in ut,(T) coincides with the start of the spin canting known from other experiments. Here the macroscopic magnetization changes continuously from an orientation parallel to the c-axis to an orientation parallel to the [110] axis. Interestingly uu(T) below 150 K scales with the magnetization component along the c-axis (dashed curve in fig. 4.54). The spin canting should lead to the appearance of two magnetically inequivalent types of 4e sites and one would expect to see two distinct precession signals. The absence of a splitting below 150 K and the scaling of u~(T) with the magnetization component along the c-axis points to some averaging mechanism by which planar internal fields are averaged to zero. It is suggested that /~+ diffusion, which must be fast in this complicated compound even at temperatures as low as 4.2 K, is responsible for this mechanism. However, in the absence of any strong temperature effect on the relaxation rate of the precession signal #+-diffusion

172

A. S C H E N C K and E N . G Y G A X

m

m~

~

0

O0

~

~

oo.~.~

0 0 0 0 0 0 0 ~

~ M M M ~

r,¢?

~-

,--7. ~ ~ ~.

,~.

o

~V

t~ o0

1

~0 ~0 7

~ V~V~

VV m

•~ . ~

~.~.

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0

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~

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0

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MUON SPIN ROTATION SPECTROSCOPY

173

tz~

0

~ o o ~ ~ g~

o ©

ca

O

I ::i. t~

::i.

<

m I

m

I O

m O

O

o

~

O 0 o

Z~

0 0

174

A. SCHENCK and EN. GYGAX I

180

I

I

Nd2Fe14B

_

M (30M'~-~'f0) el"~e ~

"1'~ 170 o rI1}

"

160

I

100

400

200

02Fe14 B

300

~8

o>" 350

,,--.

t"-

O N

u.

14

300 0

f 100

I 200

T 12 300

Fig. 4.54. Temperature dependence of the spontaneous precession frequency and of the saturation magnetization M (solid line~ in Nd2Fe]4B (upper part) and in Ho2Fel4B (lower part). The dashed lines show the projection of M on the c-axis (from Yaouanc et al. 1987, Niedermayer et al. 1990).

appears somewhat unlikely and it rather seems as if some aspect of the magnetic structure in the spin canting regime is not properly understood. Very similar results are also observed in H o 2 F e l 4 B where again uu(T) shows a break at ~ 50 K below which temperature spin canting develops in this compound (see lower part of fig. 4.54). Interestingly in this compound the macroscopic magnetization decreases with temperature due to the AF coupling of the two sublattices, while u(T) increases till ~ 50 K because in this case the dipolar fields from the Fe- and Ho-moments add constructively at the 4e #+-site. Table 4.14 lists the low temperature (15 K) B~ and the difference B~ - BL (BL is calculated from the macroscopic saturation magnetization at low temperatures) which is given by +[/~c Figure 4.55 shows that B , - BE scales very well with the rare earth moment or its component along the c-axis, respectively. This figure teaches that the Fe-sublattice contribution to the microscopic local field at the #+ is rather independent of the rare earth element present.

+/~dipt-

MUON SPIN ROTATION SPECTROSCOPY I

I

I

I

175

1

I

I

I

I

I

I

I

3.5

3.0 2.5

D Tb y

2.0 i

m= 1.5 •

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

0.5 0

I

I

I

I

-10 -8 -6 -4 -2 0 2 4 gnE(~B)

Fig. 4.55. Linear correlation of (Bu - BL) (low temperature limits) with the value of the rare earth

moment (from Niedermayeret al. 1990).

4.3. IntermetalIic compounds containing actinide elements Some of the results to be discussed below have been reviewed before by Asch (1989, 1990) and Schenck (1993). Tables 4.15 and 4.16 list all the compounds studied so far by #SR spectroscopy.

4.3.1. Cubic laves phase compounds: UAI2, UMn2, NpAl2 UAI2. Some times considered to belong to the class of heavy fermion systems (7 = 90 mJ/mol K 2) UA12 seems to be a fast spin fluctuator (TSF ~ (25-30)K) down to the lowest temperatures with no disposition towards magnetic order. The importance of spin fluctuations is revealed by a T 3 In T behaviour of the low temperature specific heat. This picture is fully in line with ZF- and LF-#SR measurements on a polycrystalline sample (Kratzer et al. 1986, Asch et al. 1987). LF-measurements in particular reveal an extremely small relaxation rate ), ~< 0.05 #s -1 which translates into a U-5f electron spin fuctuation rate of 1/Tc > 1013 s -1, assuming the U-5f moment to be given by ~ 4#B, as deduced from the magnetic susceptibility above 100 K (see, e.g., Fournier et al. 1985). This rate estimate agrees well with results from a neutron scattering measurement of the dynamic susceptibility (Loong et al. 1986). UMn2. ZF- and LF-#SR measurements on a powder sample (Cywinski et al. 1994a) prove UMn2 to be a fast spin fluctuator like UAI2 and no indications for long range order involving either the 5f and for the 3d moments are seen in agreement with neutron diffraction measurements. This seems to rule out expectations that AF order could develop below ~ 240 K, somewhat above a structural transition in the range

176

A. SCHENCK and EN. GYGAX

_9.o -~t ,,:~N

•

•

.

.

N

~

~

N

.-~ ,~ ,~

~ ~

o

o0

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0

T~

~o~

~

©

O~DZ

ZZ

ZZo O9

O9

0 dn

MUON SPIN ROTATION SPECTROSCOPY

177

o.q~

d

o9

"0

b

b~

o

~ d

oO

N

N

N

~

N

g~2 gT.2

NN

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e-

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v~

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oo

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t¢3

e-,

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t¢3

e~~

~

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d

t'~ ~

o

eq

eq

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II

''~

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t~ P,

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.~

o

e~

o

0

<

o

Z

U

178

A. SCHENCKand EN. GYGAX

(210-230) K in analogy to U F e 2 where a rhombohedral distortion coincides with a ferromagnetic transition at 165 K (see, e.g., Fournier et al. 1985).

NpAl2. NpA12 is a ferromagnet which orders below Tc = 56 K. Only prelimenary ZF-#SR results are so far available (Aggarwal et al. 1990). In contrast to all other cubic Laves phase compounds studied by ZF-#SR in the ordered state, NpA12 is the only system in which a spontaneous spin precession signal is seen with more than one precession frequency. Indeed if the #+ is placed at the (2-2) site, as found in CeA12 (see fig. 4.6, section 4.2.6) several distinct local fields should be visible, but a detailed analysis is as yet missing. TF-#SR measurements (0.02 T) show a strongly rising relaxation rate or a slowing down of the spin dynamics, respectively as Tc is approached from above, starting at ,-~ 75 K, which is typical for a second order phase transition.

4.3.2. Binary compounds with NaCl structure Like the rare earth monopnictides the corresponding U compounds show complex magnetic structures of multiple q spin arrangements. The U-monochalcogenides are ferromagnets (see table 4.15).

UAs. #SR-measurements in ZF, TF and LF on a mosaic of unoriented crystalline platelets mirror very nicely the different magnetic phases and provide some additional information (Asch et al. 1987, 1989, Kratzer et al. 1990). ZF-measurements below 62 K in the type IA, double 0"AF phase reveal spontaneous coherent spin precession involving two frequencies and one non-oscillating component (see fig. 4.56). The temperature dependence of the frequencies is displayed in fig. 4.57. Assuming the #+ to occupy the same site as in CeAs (see fig. 4.21) one calculates for the given AF structure a zero net dipolar field and by symmetry arguments also a zero contact hyperfine field. The fact that non-zero frequencies and a splitting is seen is a result of a tetragonal lattice distortion occuring at 62 K (c/a < 1) (also at 124 K (c/a > 1)). This distortion is a necessary prequisite for the establishment of the type IA double ~' spin structure (Sinha et al. 1981). It changes the point symmetry at the #+ site with the effect that crystallographically equivalent sites become magnetically inequivalent (--+ splitting of signal) and that the dipolar fields from the neighbouring U-moments do no longer cancel (--+ non zero frequencies). In fact the distortion explains the ratio of the signal amplitudes associated with the two oscillating and the one nonoscillating signal. Dipolar field calculations predict a ratio of x/2 for the two frequencies independent of the size of the distortion. The fact that at the lowest temperature a ratio of ~ 2 is observed points to additional contact hyperfine fields, which are also needed to account for the strange temperature dependence of the lower frequency in fig. 4.57. A detailed understanding, however, is lacking. It should be emphasized that neither neutron nor X-ray diffraction could detect the tetragonal distortion, providing only an upper limit of ]c/a - 11 < 2 x 10 -11 (Sinha et al. 1981, Knott et al. 1980). The #SR-results provide the only direct evidence for the distortion by the change of the point symmetry but no quantitative analysis was presented.

MUON SPIN ROTATION SPECTROSCOPY

179

0.15 0.10 0.05 0 -0.05 -$ E E

I

I

I

I

I

I

1

2

3

4

5

6

Time (gs)

(n

0.15 0.10 0.05 0 -0.05

I 0.5

I 1

Time (~s)

Fig. 4.56. ZF-/zSR signal in UAs below 62 K in the type IA, double ~"AF phase. The lower part shows an expanded early section of the signal (from Asch et al. 1987). I

12.0

o o

I

I

O0

I

I

I

03 O 0 o

0

~, 11.5

o"

6.0

E

o8

8 5.5 LL

og

o

5.0

o

oO o o 00

4.5

0 0

or

I

I

I

I

I

10

20

30

40

50

60

Temperature (K)

Fig. 4.57. Temperature dependence of the two spontaneous precession frequencies in UAs in the type IA, double q AF phase below 62 K (from Asch et al. 1987).

Above 62 K the coherent spin precession signal vanishes abruptly and only a weakly damped Lorentzian Kubo-Toyabe signal is seen with no temperature dependence up to TN ----- 124 K. This is consistent with the prediction that even in the presence of a tetragonal distortion the type I single q" spin structure will not lead to

180

A. SCHENCK and EN. GYGAX

non-zero net dipolar (and contact) fields at the #+-site. The extremely small residual field spread of N 0.8 m T at the #+-site, following from the static Lorentzian K u b o Toyabe behaviour implies a very perfect and static spin structure in the vicinity of the #+ and that disturbances are dilute and more distant. Spin excitations do not seem to play a role. The transitions at TN = 124 K into the paramagnetic state shows up very markedly in W-measurements. The relaxation rate drops by more than a factor of two and the precession frequency displays a jump by more than +2500 ppm (Asch et al. 1987). The relaxation rate above 124 K is essentially temperature independent up to ,-~ 180 K, where another down jump to a very small rate is observed (see fig. 4.58). In particular no increase of the relaxation rate (no slowing down of spin fluctuations) is seen when approaching TN from above consistent with the notion of a first order phase transition (Sinha et al. 1981). The change at 180 K is accompanied by a transition from a Gaussian damping function above this temperature, reflecting the nuclear (75As) dipolar field distribution, to a more exponential one below this temperature (but still reflecting static features, Kratzer et al. 1990). The modified relaxation behaviour below 180 K is ascribed to the onset of a magnetic precursor state in which the 5f-moments no longer act as free paramagnetic spins. This is in accord with diffuse neutron scattering results slightly ( ~ 10 K) above TN which reveal strongly anisotropic spin fluctuations tending towards an incommensurate sinuoidally Tt

TN

°o

0.4

UAs

oo

0.2

(a) O O

,

0 0.8

, ,

~ ....

I ....

Tt

I

,

, ,O,

I ,Q

TN

....

UP

_~o~p o o o oo o

0.4

91

(b)

o ,,,I

0

....

I , I~'~,OI , O ,

,q

, ,

T N

1911 I CeAs

0.2

(c)

0.1

~b~:~

0 .... 0

50

,o, ,o,, o, ,o,

,9,,,,

100

250

150

200

300

Temperature (K)

Fig. 4.58. Temperaturedependence of TF-relaxation rates A2 in (a) UAs, (b) UP and (c) CeAs. The first order nature of the phase transition at TN in UAs and UP is manifest by the discontinuousjump of ),2- In contrast ),2 in CeAs reflects a critical slowing down of the spin dynamics near TN and signals a second order phase transition (from Asch 1989).

MUON SPIN ROTATION SPECTROSCOPY

181

moduled spin structure (Sinha et el. 1981). The #SR data show that the precursor state sets in already about 50 K above TN (Asch et el. 1989). USb. Like in UAs in the type I single q phase, ZF-measurements on USb do not exhibit any coherent spin precession below TN (the spin structure is of type I, triple q-) but display a Lorentzian Kubo-Toyabe behaviour with full amplitude. In contrast to UAs the internal fields acting on the #+ are not totally static below TN and both the static width A and the spin fluctuation rate u show a pronounced and very interesting temperature dependence (see fig. 4.59) (Asch et el. 1990). A displays a maximum at ~ 140 K at which temperature the fluctuation rate u starts to rise dramatically with increasing temperature. Neutron scattering measurements signal at ~ 140 K the collapse of spin waves, well resolved at lower temperatures, and the development of a broad inelastic peak centered at zero frequency (Hagen et el. 1988). The #SR results are interpreted in terms of a phase de-locking of the magnetic components in the triple ( structure just as in DyAg (see section 4.2). Phase de-locking, induced thermally, produces irregularities (defects) in the spin structure and hence increases the field inhomogenity sensed by the #+. This explains the rise of A with increasing temperature. At around 140 K a motionat narrowing effect sets in which arises from a slow diffusive motion of the thermally induced perturbances or defects in the spin structure and explains the step rise in u and - at least qualitatively - the decrease of A above 140 K. Note that the disturbances/defects are still quite dilute so that a Lorentzian Kubo-Toyabe picture is applicable. By comparison with other compounds it is found that a dynamic Lorentz-KuboToyabe behaviour with a temperature dependent A is typical for a multiple 0"structure which, in contrast to a single 0~structure, seems to be more easily distorted due to de-locking of the phases between the Fourier components of the moments (Asch et el. 1994). UP. This compound displays rather similar properties as UAs (see table 4.15). A subtle but important difference is that in the ordered phase, associated with the double

USb

9 i

1.3

i

o/°\

~ 1.2 ~"

~ 1.1 1.0

6.o

/

,-Width

o

) ...... n

-

/

~.

50

Rate---- 4.0 ~ 3.0

~"

/0%.u

2.0

O "~ B "/"

• ..... I

5.0

I

I

100

150

Temperature (K)

1.0 I

200 I

250

TN

Fig. 4.59. Temperature dependence of the width A and the spin fluctuation rate v from the dynamic Kubo-Toyabe signal seen in USb (from Asch et el. 1990).

182

A. SCHENCKand F.N. GYGAX

structure, the propagation vector is given by q = (0, 0, 1) (type I) while in UAs ~ = (0, 0, 1/2) (type IA). Only in the latter case one calculates non vanishing dipolar fields at the #+ site, if also a tetragonal distortion is present. Consequently in UP no coherent spin precession signal is seen in ZF-#SR measurements (Aggarwal et al. 1989). TF-measurements show abrupt changes in the relaxation rates at both transitions temperatures (see fig. 4.58b) but no indication for critical behaviour in line with the first order nature of the phase transitions. In contrast to UAs also no magnetic precursor state is seen in UP above TN = 122 K (Aggarwal et al. 1989).

UN. UN orders antiferromagnetically into the type I single q structure at 53 K. The transition is accompanied by a tetragonal distortion with c/a > 1. From the pressure dependence of TN and #u strong evidence for itinerant magnetism was deduced (Fournier et al. 1985). The phase transition is believed to be of second order. ZF-#SR measurements below TN do not reveal a spin precession signal but a static Lorentzian Kubo-Toyabe behaviour consistent with similar results in the single qphase in UAs and UP (Mtinch et al. 1993). TF-measurements (Mtinch et al. 1993) show a drastic increase of A2 at TN but no indications for a critical slowing down of the spin dynamics when approaching TN from above just like in UP (see fig. 4.58b). This is unexpected in view of the assumed second order nature of the phase transition. A2 above TN is in fact very small (~ 0.03 #s -l like in UP) and temperature independent and implies a very fast fluctuation of the U-moments. From the point of view of #SR there is no difference in the magnetic behaviour of UP and UN above and below TN (except that UP possesses another phase transition at lower temperatures). UTe. In contrast to the U-monopnictides this monochalcogenide shows ferromagnetic order below Tc = 104 K. ZF-#SR measurements below Tc could not detect any coherent spin precession signal, probably because of a too rapid relaxation due to a very wide field distribution associated with lattice irregularities (Aggarwal et al. 1989). TF-measurements (0.1 T) above Tc yield a frequency shift which scales very well with the susceptibility. The exact form of the temperature dependence of the TF-relaxation rate could not be extracted but its diverging trend when approaching Tc from above is consistent with the second order nature of the ferromagnetic phase transition. 4.3.3. Binary compounds with cubic AuCu3-structure: USn3, Uln3, U(Ino.5Sno.5)3 UIn3 shows antiferromagnetic order below TN = 88 K with a type I triple q spin structure. USn3 on the other side is a paramgnet exhibiting strong spin fluctuations. In the mixed system U(Sn~Inl_~)3 long range magnetic order is suppressed for x/> 0.4 (Zhou et al. 1985). Prelimenary #SR-measurements applying the ZF-, TFand LF-technique yielded results which generally confirmed the present picture on the magnetism of these compounds (Zwirner et al. 1993, Kratzer et al. 1994b, Asch et al. 1994). The results on USn3 imply indeed a very fast spin fluctuation of the order of 1013 s -1 or faster. The same is found in the paramagnetic state of UIn3. As in the other compounds with type I triple-~f spin structure (USb, DyAg) the ZF-#SR

MUON SPIN ROTATION SPECTROSCOPY

183

signal in UIn3 below TN = 88 K is well described by a dynamic Lorentz-KuboToyabe function. The temperature dependence of A and u resembles closely the results on USb displayed in fig. 4.59 and is discussed there. In the present case the dynamic range extends only for about 10 K below TN. A possible first-order nature of the phase transition is reflected by a temperature independent frequency shift and relaxation rate in TF-measurements when approaching TN from above. New information is gained on the mixed compound U(Sn0.sIn0.5)3 which is expected not to show long range magnetic order. AF- and TF-measurements reveal, however, the onset of spin glass order below ~ 30 K. This is evidenced by a change of the ZF-relaxation function from a Gaussian to a Lorentzian behaviour (well described by the spin glass relaxation function of Uemura et al. 1980) and a strong increase of the W-relaxation rate below 50 K. Previously it was found in U(Sn0.5In0.5)3 that the specific heat c(T)/T shows a slight up turn and the resistivity p(T) a break in slope and a weak decrease at or below 30 K (Lin et al. 1987). These features were not explained before but find now in the light of the #SR results a natural explanation.

4.3.4. Tenary compounds with tetragonal ThCr2Si2 or CaBe2Si2 structure URh2Si2. The antiferromagnetic local moment (#u = 1.95#B ) order in this compound is identical to the one in NdRh2Si2 (see table 4.10), yet ZF- and LF-#SR measurements on a polycrystalline sample display quite a different behaviour (Yaouanc et al. 1990, Dalmas de R6otier et al. 1990d, 1994c). Below TN a one frequency coherent spin precession signal is seen which at low temperatures implies a local field of 0.305 T. Obviously only one type of site is occupied by the #+. The site (1/4, 1/4, 1/4), No. 7 in fig. 4.32, can be excluded since it involves a zero net internal field). The corresponding information is missing for NdRh2Si2 due to the limited time resolution available at the ISIS facility. In CeRh2Si2, which has a

URh2Si2

-

0.08 u_

N 0.06

"+ ÷ ÷÷t

0.04 tO

"~

0.02

X Q

rc

Oo

0.00 I

10

I

I

I

I

I

I I II

f

100 Temperature (K)

Fig, 4.60. Temperature dependence of the relaxation rate ~ZF in URh2Si 2. Below TN )~ZFis really a )q (see text) (from Dalmas de R6otier et al. 1994c).

184

A. SCHENCK and F.N. GYGAX I

b~

v=

~T (ff

'

i

,

I

'

~

i

I

l

,

i

I

'

uRh2Si2

0.06

'

'

I

o

O

i

~

I

30K

Z~ 9 6 K

0.04

• t-

i

121K

0.02

X t~

~ cc

0.00 I

0

k

i

i

I

2

,

,

,

I

4

,

,

,

I

6

~

,

I

I

8

I

I

I

I

10

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Fig. 4.61. Dependence of the spin lattice relaxation rate -~1 on the strength of a longitudinally applied field (/~(0)llnext) below TN in URh2Si 2 (from Dalmas de R6otier et al. 1994c).

slightly different magnetic structure, two different internal fields were seen below TN, suggesting that the #+ is located at two different interstitial sites. Very different is the dynamic behaviour below TN in URh2Si2 (see fig. 4.60). The longitudinal relaxation rate )`1 (the relaxation function is well described by an exponential decay) revealed by those #+ whose spins happen to be parallel to the static internal field is essentially temperature independent while in NdRheSi2 a quadratic temperature dependence is seen for P(0)llc-axis (see fig. 4.34). Moreover, again in contrast to NdRh2Si2, )`1 below TN is dependent on the strength of a longitudinally applied field (see fig. 4.61). This feature is not understood at present. Above TN, ),1 drops quickly with rising temperature similarly to what is seen in NdRh2Si2 for fi~(0)_Lc-axis or in GdNi5 (see figs 4.34, 4.26), reflecting some slowing down of the spin fluctuations as TN is approached from above. A more quantitative analysis of the data is as yet missing.

U(Rho.35Ruo.65)2S@ While URh2Si2 is characterized as a local moment (# _~ 1.4#~) antiferromagnet URu2Si2 is a magnetic heavy fermion superconductor exhibiting ultra small moment (# _~ 0.02#B ) magnetic order below 17 K (see section 4.3.5). It is of considerable interest to study the transition from one type of behaviour to the other one by investigating the mixed compounds U(Rh~Rul_=)2Si2. ZF- and LF-#SR measurements were conducted on the compound U(Rh0.35Ru0.65)2Si2 [polycrystalline sample for which the magnetic structure is not known (Yaouanc et al. 1990, Dalmas de R6otier et al. 1990d)]. The ZF-results are shown in fig. 4.62. A single exponentially damped signal is observed. Its amplitude starts to drop smoothly at about 150 K to 1/3 of its initial value below ~ 40 K. This drop signals the development of a very fast relaxing (or precessing) component which, since these measurements were performed at the pulsed muon source of ISIS, could not be resolved. In any case the low temperature value of the amplitude reveals unambigeously that the #+ are exposed to a very wide static field distribution or, alternatively, to a rather high

MUON SPIN ROTATIONSPECTROSCOPY [

i

i

i

i

i

i

i1111111111

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o

0.15

185 i

o

oOOO

so

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o Oq~O 0

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U ( R h . 3 5 Ru.6 5 )28i2

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0.08

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0.04

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(~

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_m oo 0.00 or "

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,

I

I

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100 200 Temperature (K) Fig. 4.62. Temperaturedependence of the ZF-~SR signal amplitude (asymmetry) and relaxation rate AZF in U(Rho.35Ruo.65)zSi2. Note the loss of amplitude between ~ 150 K and ~ 42 K, indicating the transition into a magnetically ordered state (from Dalmas de Rdotier et al. 1990d). local field leading to a rapid coherent spin precession, too rapid to be resolved. In other words magnetic order is established in this compound somewhere below ,-~ 40 K. Whether it is long range or short range (e.g., spin glass like) cannot be deduced from the present #SR data. The drop in the amplitude could indicate that the development of magnet order proceeds in a spatially inhomogeneous fashion, a not uncommon feature in heavy electron systems (see sections 4.2.6, 4.3.5). Very interesting is also the temperature dependence of the ZF-relaxation rate AzF (see fig. 4.62). LF-measurements prove that AZF (= At) is more or less entirely of dynamic origin in the whole temperature region studied. AzF peaks sharply at ~ 42 K, similar to what is seen in polycrystalline NdRh2Si2 (Dalmas de Rtotier, 1990d) and is a clear signature for a magnetic phase transition. Note that AzF is generally quite small implying that above TN --~ 42 K spin fluctuations must be rather fast. The relaxation rate below TN is independent of a longitudinally applied field, therefore reflecting still some dynamics as in NdRh2Si2 and UPt2Si2, quite in contrast to the field dependence seen below TN in URh2Si2. In summary magnetic order is seen in U(Rho.35Ruo.65)2Si2 below ~ 40 K but the dynamical behaviour in the ordered state is different from the one in the parent compound URh2Si2.

186

A. SCHENCK and EN. GYGAX '

,,-;-

I

'

I

I

UPt2Si2

0.08

=1. v 0.06 "~ ¢._o

i

o 0mT z~ 1 0 r o T • 200 mT

o

0.04

7! q

0.02 rr

0.00

I

20

i

I

40

t

I

60

i

I

80

Temperature (K)

Fig, 4.63. Temperature dependence of the ZF and LF relaxation rate in UPt2Si2. Note the sharp cusp of AzF, A1 at the transition temperature of 35 K (from Dalmas de REotier et al. 1994c).

UCo2Si2. Some preliminary ZF-#SR measurements imply very fast spin dynamics above the ordering temperature TN = 85 K. No slowing down of the dynamics is seen on approaching TN from above. Below TN 2/3 of the signal is lost as is expected for a polycrystalline sample, given the limited time resolution at the ISIS facility (Dalmas de Rdotier et al. 1994c). UPt2Si2. Again only preliminary ZF- and LF-#SR data are available (Dalmas de R6otier et al. 1994c). The observed temperature dependence of AzF and A1 is displayed in fig. 4.63. The small field dependence is ascribed to a decoupling from the Pt-nuclear dipole moments. The data are quite similar to the results in U(Rh0.36Ru0.35)2Sia and other isostructural rare earth compounds. They reflect mostly the dynamics of the spin system and indicate a certain slowing down, which starts already much above TN. 4.3.5. Actinide based heavy electron and related compounds All systems investigated by #SR are listed in table 4.16. For a more detailed discussion of the #SR results see Schenck (1993). A compilation of some of the most important #SR results is presented in table 4.17.

UPt3. The heavy fermion superconductor UPt3 is certainly one of the most fascinating compounds among all heavy fermion systems possessing a rather intriguing low temperature phase diagram (for a recent status see de Visser et al. 1993). First indications for magnetic order at ~ 5 K, preceeding the superconducting transition at Tc ~ 0.5 K were provided by ZF- and TF-#SR measurements on a polycrystalline sample (Heffner et al. 1987, 1989a, Cooke et al. 1986). Figure 4.64 displays the measured ZF-relaxation rate AZF which exhibits a significant increase below 5 K. LF-measurements proved that this - sample dependent - increase is associated with the development of small static fields of order 2 G at the #+ site or sites. Subsequent neutron diffraction measurements confirmed the onset of an ordered state involving

MUON SPIN ROTATION SPECTROSCOPY

187

+

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188

A. SCHENCK and EN. GYGAX

UPt 3 Gaussian relaxation 0.20

_ T¢tJ :zL

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+

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0.5

~

1.0

2.0

5.0

10,0

20.0

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tiny moments of the order of 0.02#B (Aeppli et al. 1988). Taking the #SR-and neutron results together it follows that the /~+ must be located at a site at which the dipolar fields from the ordered U-moments, assuming the structure suggested by Aeppli et al. (1988), cancel perfectly for a perfect stoichiometric and defect free lattice (Schenck 1993). Note that no anomaly in the specific heat is seen at 5 K. Prior to the discovery of magnetic order in UPt3 antiferromagnetic order had already been found in the doped compounds Ua.95Th0.05Pt3 and U(Ph_=Pd~)3 (0.02 ~< :c ~< 0.07) (Goldman et al. 1986, Frings et al. 1987). Interestingly the magnetic structure is the same as in undoped UPt3 with TN also of the same magnitude (TN ~ (3.5-6.5) K) but the ordered moment is much larger (,-~ 0.5#~) and the magnetic phase transition can also be seen in transport and thermodynamic data (Ramirez et al. 1986). A few low statistics ZF-#SR measurements on the compound U1.95Th0.05Pt3 confirmed the onset of magnetic order at TN -- 6.5 K (Heffner et al. 1989a): below this temperature a two component precession signal appears reflecting spontaneous local fields of 0.06 T and ~ 0.009 T. However, the total amplitude of the two components accounts for only 20% of the implanted ~+ implying either a severe loss of polarization due to an extremely inhomogeneous distribution of internal fields in most of the sample volume or a zero field site for most of the implanted #+. Since the two spontaneous fields are also associated with a large field spread of ~ 0.007 T it seems that AF-order in the investigated polycrystalline specimen is not so well developed, probably as a result of poor sample quality. Nevertheless the appearance of well resolved precession signals in U1.95Th0.osPt3 and the absence of any precession signal in pure UPt3 is somewhat of a mystery given the belief that the AF-structure for both compounds is identical (e.g. scaling the field of 0.06 T

MUON SPIN ROTATIONSPECTROSCOPY

189

down by the moment ratio 0.02/0.5 _~ 0.04 one should have seen a local field of 24 G instead of the 2 G mentioned above). ZF- and W-measurements on high quality mono- and polycrystalline UPt3 samples revealed another increase in relaxation rate starting at 490 mK, i.e. near the temperature at which a second phase transition is seen in specific heat data some 60 mK below the transition into the superconducting state (see fig. 4.65) (Luke et al. 1993a, b). Obviously the lower of the double transition around 0.5 K is associated with a further increase in the static field spread ( ~ 0.01 mT). Judging from TF-measurements at 0.18 T the field spread is rather isotropic. The appearance of a split transition into the superconducting state is believed by many investigators to be a consequence of the antiferromagnetic state below 5 K which provides a symmetry breaking field. This couples to the superconducting order parameter and splits the transition into an otherwise degenerate ground state. A coupling between the magnetic and the superconducting order parameter seems also to be indicated by neutron scattering data (Broholm 1989). Blount et al. (1990) have suggested that the neutron data below Tc could be explained by a reorientation of the small antiferromagnetic moments. Luke et al. (1993b) showed that a rotation of the moments in the basal plane by ~- 30 ° could reproduce the increase in local fields below 490 mK. Other explanations, based on the speculation that the lower transition leads to a time reversal invariance violating state, are discussed in Luke et al. (1993b). The low temperature features seen by ZF-#SR in UPt3 are similar to results obtained in U1L=Th~Be13, in which also a double transition is seen for 0.019 ~< z ~< 0.043. However, in the latter case no magnetic ordering seems to precede the transition into the superconducting state (see below) and the idea of a symmetry breaking

0.065

I

UPt~ ZF-/~SR 1

l

0.060

l

P ,.k c

'T

0.055

I i

o x O

0.045 0.00

0.25

I 0.50 0.75 Temperoture (K)

1 1.00

1.25

Fig. 4.65. Temperaturedependenceof the ZF relaxationrate assuming exponentialrelaxation in a single crystal of UPt3 with/5(0)Zc-axis. The positions of the two transition temperatures are indicated by the arrows (from Luke et al. 1993b).

190

A. SCHENCK and EN. GYGAX

field responsible for a split transition may not be applicable. The occurence of ultra small ordered moments in other nonsuperconducting HF-compounds (see, e.g., CeRu2Si2, UCus) moreover may suggest that the small moment magnetic order is perhaps a phenomenon unrelated to superconductivity.

UCus. This moderately heavy electron compound orders antiferromagnetically below TN = (15-16) K. The magnetic structure has been studied by neutron diffraction and NMR spectroscopy. The data were interpreted in terms of two different structures. Murasik et al. (1974) (also Schenck et al. 1990b) explain their data in terms o f ferromagnetic (1, 1, 1) planes and an antiferromagnetic coupling between neighbouring planes (single q structure) while Nakamura et al. (1990) on the basis of NMR data propose a quadruple q structure. Specific heat and resistivity measurements reveal a second phase transition with hysteretic features at ~ 1 K (Ott et al. 1985). #SR-measurements were aimed in particular at a better characterization of the 1 K transition. The studies started on a sample which did not show the 1 K phase transition. ZF-measurements revealed the onset of three different signals below TN, one associated with zero average field, but non-zero static field spread, the other two displaying coherent spin precession corresponding to low temperature local fields of 0.146 T and 0.1 T, respectively (Barth et al. 1986b, 1988). Figure 4.66 shows the temperature dependence of the spontaneous precession frequencies. The solid lines in fig. 4.66 represent the temperature dependence of the ordered moment deduced from neutron diffraction experiments (P. B6ni, priv. communication, Schenck et al. 1990b). As can be seen, only the 0.146 T signal scales with the order parameter. The crystal structure of UCu5 is displayed in fig. 4.67. The two possible #+ interstitial sites are indicated. Dipolar field calculations (#ord = 1.55#B) show that irrespective of whether the single-q or the quadruple-0" structure is adopted, Baip at the site (1/2, 1/2, 1/2) is zero. For the single-0'structure one finds Bdip = 0.23 T at all equivalent sites (3/4, 3/4, 3/4), while for the quadruple-q structure one finds t3di p = 0 T 2O

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2

4

6

8

I0

12

14

0

i

16

18

20

T(K) Fig. 4.66. Temperature dependence of the two spontaneous precession frequencies in UCus. The solid lines represent the temperature dependence of the ordered moment (from Schenck 1993).

MUON SPIN ROTATIONSPECTROSCOPY

UCu 5

• ~,o

191

u Cu

I

"-'-- p+ (2)

I

" ' ' - " #+ (1)

w

W

Fig. 4.67. Crystal structure of the cubic AuBe5 type. Two likely #+ interstitial sites are indicated. at site (3/4, 3/4, 3/4) and B d i p = 0.27 T at sites (3/4, 1/4, 1/4), (1/4, 1/4, 3/4), (1/4, 3/4, 1/4), respectively. The occurence of two precession frequencies cannot be explained by either structures. Also the signal amplitude ratios (A(0.146 T) : A(0.1 T) : A(0 T) _~ 3 : 2 : 2) do not correspond to the relative site populations. Since for each signal the damping rate is found to be quite small and could be explained more or less by just the random Cu nuclear dipole fields the local fields are quite narrow and would point to a rather perfect commensurate structure. These results together with the strange temperature dependence of the 0.1 T signal could indicate that the magnetic structure is still more complex than a single-q or a quadruple-q structure or that the #+ is found also at defect sites or changes by its presence the magnetic structure locally. In view of the fact that basically the same results are also obtained in a second high quality sample, which showed the 1-K-phase transition, and that all the lines are rather narrow ( ~ 0.5 mT) we rather tend to believe that what is seen reflects intrinsic properties. The first sample investigated showed, as pointed out before, rather small damping rates which were temperature independent from 3/4 TN down to some 10 mK. ZFmeasurements (limited to T < 2 K) on a second sample, which showed the 1 K phase transition, produced the same temperature dependence of the spontaneous frequencies with no break at ~ 1 K. In contrast the relaxation rates for both the zero frequency signal and the 0.1 T and 0.145 T signals showed a dramatic increase

192

A. SCHENCK and EN. GYGAX 3.0

Z52~o

~,i (c~) 30 20

1.5 1.0

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(

~

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T e m p e r a t u r e (K) Fig. 4.68. Temperature dependence of the relaxation rates associated with the 0.145 T signal (~z, o1), the 0.1 T signal (,X2,o'2) and the 0 T signal (,~4, o.4) in UCu5. Around 1.1 K the relaxation function changes from Gaussian (o-) above 1.1 K to exponential (X) below. A further signal with approximately the same frequency as the 0.145 T signal but an order of magnitude larger relaxation rate ,~s, also above 1.1 K, is attributed to disturbed magnetic domains (from Schenck et al. 1990d).

M U O N SPIN ROTATION SPECTROSCOPY

193

at ~ 1.1 K, as shown in fig. 4.68 (Schenck et al. 1990a, b). In parallel neutron diffraction showed neither a change in the nuclear nor in the magnetic Bragg peaks when changing the temperature across the 1 K-transition (Schenck et al. 1990b). Obviously the antiferromagnetic structure is unchanged by the 1 K-transition, but the local fields at the #+ sites show a pronounced broadening (NMR results point to the opening up of a gap in the spin excitation spectrum at ,-~ 1 K, Nakamura et al. 1990). The specific heat jump at ~ 1 K suggests that the 1 K phase transition is associated with the heavy electrons. The appearance of obviously quasi random static fields below 1.1 K may than be linked to the heavy electrons as well and it is suggested that these electrons enter into some ordered state (random or in form of a spin density wave) coexisting with the antiferromagnetic order established below 16 K and that these electrons or quasi particles must be distinct from those electrons responsible for the 'conventional' antiferromagnetic order below TN. In other words it is suggested that the low temperature behaviour of UCu5 is determined by the presence of two rather independent electronic substates in the ground state ('heavy' and 'light' mass states, involving different sections of the Fermi surface?) (Schenck et al. 1990b). The idea that more than one kind of electrons has to be considered has come up also for other compounds (see, e.g., Caspary et al. 1993, Feyerherm et al. 1994c). Assuming a random picture the ordered moment of the heavy quasi particles in UCu5 has been estimated to be ,-~ 0.01/zB. Subsequently Nakamura et al. (1994), on the basis of NMR results, have suggested that the 1 K transition reflects a transition from the quadruple c7structure above 1 K to the single q structure below 1 K. This interpretation is clearly inconsistent with the #SR results.

UCd11. This nonsuperconducting HF compound orders antiferromagnetically below 5 K but the magnetic structure is not known. Neutron scattering investigations place an upper limit of ~ 1.5#B on the ordered moment (Thompson et al. 1988). In fact the only direct information on the development of magnetic order at 5 K stems from #SR-measurements on a polycrystalline sample (Barth et al. 1986c, Barth 1988). ZF-, TF- and LF-measurements reveal a loss of the signal below 5 K which indicates the onset of a very wide static field distribution exceeding several 0.01 T. UCd11 crystalize in the cubic BaHgll-structure which is rather complex and provides a large number of magnetically inequivalent possible sites for the #+. In any case the ordered moment must be of the order of lPB in order to produce a field spread of several 0.01 T. Of interest is the LF-relaxation behaviour above TN. As fig. 4.69 shows a field independent #+ spin lattice relaxation is observed which follows a power law and indicates a 5f-spin fluctuation rate of l]5f O( ( T - T N ) 0"4+0"1 o(

v/-T,

T > TN.

(4.17)

This dependence is typical for a system with Kondo resonance behaviour and is consistent with the theory of Cox et al. (1985). A fluctuation rate of usf ~ 10 l° s -1 is estimated for T = 2TN. No such behaviour is seen in the other heavy electron U-compounds. The fact that spin fluctuation become manifest in #SR-measurements in UCdll is probably related to a relatively small distance between the p+ and a nearest U-neighbour providing a strong hyperfine coupling (Schenck 1993).

194

A. SCHENCKand EN. GYGAX I

I

I

I

I

UCdll

longi1'udinolfield

4 -r 5

I000 G o 2000 G

::k

° 4000 G

v

"7 2 F--

0

I

I

I

I

I

I

2

5

4

5 T/T N

6

7

8

Fig. 4.69. Temperature dependence of the spin lattice relaxation rate )~1 = T11 in UCdll. Note that )~1 is independent of the applied field. The solid line represents a fit of eq. (4.17) to the data (from Barth et al. 1986c).

U2Znl7. Neutron diffraction shows the onset of a rather simple antiferromagnetic structure in U2Znl7 below 9.7 K (Cox et al. 1986). Nearest neighbour U-moments both in the basal plane and along the c-direction in this rhombohedral structure of the ThEZn17 type are coupled antiferromagnetically and are oriented parallel to the basal plane. On the basis of this structure one calculates vanishing dipolar fields at the two high symmetry interstitial sites available to the #+. From the splitting of the TF-#SR signal in the paramagnetic state, using a single crystal sample, and from the angular dependence of the frequency shifts it seems that both sites are indeed occupied (Barth et al. 1986b, 1989b, and unpublished results). However, when cooling the sample through TN both ZF- and TF-measurements show a partial (~ 25%) loss of the signal implying that in part of the sample volume a large static internal field spread develops below TN. This is found in both a polycrystalline and a monocrystalline sample. From LF decoupling measurements it is deduced that the with of the field spread amounts to -,~ 0.1 T (Barth et al. 1986b). Interestingly the fraction of implanted #+ exposed to the large field spread below TN shows also a highly peculiar behaviour above TN. In TF-measurements on a single crystal above TN this fraction is associated with two (or even four) components distinct by their different frequency (Knight) shifts and by a highly unusual angular dependence of the shifts (Schenck et al. 1992, and unpublished results). The angular dependence involves higher order (> 2) Legendre polynomials and is at present not understood at all. It is speculated that the mysterious behaviour of the 25% fraction is a result of competing interactions which could also explain the development of same sort of random order in part of the volume. The question o~f course is what makes this 25%-fraction of the sample volume so different from the rest both in the ordered and in the paramagnetic regime. It does not seem to be sample dependent, more-over its size renders it unlikely

MUON SPIN ROTATION SPECTROSCOPY

195

that crystal defects or the presence of foreign phases are responsible. The onset of the random order coincides precisely with the independently determined N6el temperature. In view of all this it seems as if the inhomogeneous magnetic features seen by #SR in U2Znl7 are an intrinsic property.

UBel3, Ul_~Th~Be13, U1_~Th~Be13_vBv. UBe13 is one of the few remaining U-based heavy electron systems which have not shown signs for a magnetically ordered state. Rumors that a transition occurs at ,-~ 9 K could not be substantiated in later studies. In particular ZF-#SR on UBe13 did not reveal any evidence for a magnetic state with ordered moments > 10-3#B (Luke et al. 1991b, Heffner et al. 1990). If more than 1.9% of U is replaced by Th (but less than 4.3%) a second phase transition at Tc2 somewhat below the superconducting transition temperature Tel is observed. The actual phase diagram is shown in fig. 4.70. It is argued that below Tc2 a different type of superconducting phase could be entered (see, e.g., Sigrist and Rice 1989, Sigrist and Ueda 1991). ZF-#SR measurements established that the phase below To2 is associated with the development of static random intemal fields of electronic origin (~ 0.18 roT) (Heffner et al. 1987, 1989a, 1989b, 1990). This follows from an increased damping rate A of the ZF-signal (given by a static Gaussian Kubo-Toyabe function) below Tc2. As an example see fig. 4.71, which collects results from specific heat, ac susceptibility Xae and ZF-#SR measurements on a U0.965Th0.035Bea3 sample. Clearly visible is the rise of ,4 at Tc2, which is determined from the strong anomaly in the specific heat. Tel is determined by the diamagnetic response of Xac. The rise of ,4 below Tc2 is well described by a spin 1/2 Brillouin function. Assuming a random moment order and making some educated I

I

I

I

I

I

I

0.9

-~

Ul"xThxBe13

0.5

h

f 0.0 0.I 0

MAGNETIC I

2.0

~

4.I 0

I

I

6.0

x (%)

Fig. 4.70. Phase diagram of U1_xThxBe13 (from Heffner et al. 1990).

196

A. SCHENCK and EN. GYGAX 0.30

a) l

I

-! ::L

I

I

I

I

S" 0')

I

I

I

I

U 0.965 T h 0.035 B e 13 -

0.28

Ho = 0

Oe

-

13 0.26 -

0.24 Y

& -6

I

I

I

I

I

I

1

1

I

I

I

I

I

I

I

I

I

I

b)

,

E 2.0 < w i 0 u_

1.0

-

0 o3

o 0""

c) U

-2

-

-4

-

-6

--

"8

•e.~t~l

k

m

- -

-lO

I 0

I 0.2

I

1• 0.4

I

I

I

0.6

TEMPERATURE

l 0.8

t 1.0

(K)

Fig. 4.71. Temperature dependence of # + ZF relaxation rate O'KT, specific heat cp and ac susceptibility X~c in Uo.965Th0.035BeI3 (from Heffner et al. 1989b).

guess on the #+-site the ordered moment is estimated to amount to ~ 5 x 10-3/ZB (Schenck 1993). As we have seen a similar behaviour was also observed in UPt3. But ultra-small moment ordering is also seen in non superconducting heavy electron systems (e.g., CeRuzSie). The question is then whether different mechanisms have to be involved to explain these results. In the superconducting systems it has been suggested that the phase below Tc2 (in UPt3 and UI_=Th=Bel3) could violate time reversal invariance and would be associated with spontaneously appearing magnetic fields (Sigrist and Ueda 1991, Heffner et al. 1990). Alternatively the small moment

MUON SPIN ROTATION SPECTROSCOPY

197

magnetism is unrelated to superconductivity and is a phenomenon not uncommon for a wider class of heavy-electron systems. Finally we mention that substitution of B for Be, although it has an effect on the entropy released at Tel, does not interfere with the small moment order induced by Th doping (Heffner et al. 1991).

URu2Si2. Also this moderately heavy fermion superconductor displays small moment magnetic order below TN = 17.5 K. The phase transition is accompanied by a relatively huge jump in the specific heat but the ordered moment amounts to only 0.04#B as determined by neutron and X-ray diffraction measurements. (Broholm et al. 1987, Mason et al. 1990, Isaacs et al. 1990). The same studies revealed an antiferromagnetic structure of type I with the AF propagation vector given by ( = (0, 0, 1) and the static moment aligned along the c-axis. ZF-#SR measurements reflect clearly the transition into the ordered state (McLaughlin et al. 1988, Luke et al. 1990c, Knetsch et al. 1993). Figure 4.72 displays results on the ZF-relaxation rate in a monocrystalline sample for P(0)±c-axis. At TN the relaxation rate rises steeply within a few degrees followed by a much weaker increase as the temperature is reduced further. This temperature dependence does not follow the temperature dependence of the ordered moment as measured by neutron scattering (Broholm 1989, Mason et al. 1990) a feature not understood at present. Adopting the proposed antiferromagnetic structure one calculates dipolar fields of the order of 5 mT, which are parallel to the c-axis. Only at the site (1/4, 1/4, 1/4) one finds Bdi p ~ 0. The internal fields should thus be perpendicular to/5(0) and, except for the (1/4, 1/4, 1/4)-site one expects to see a clear spin precession signal for P(0)±c-axis. However, no such signal is observed. If the slow relaxation seen instead is interpreted as the beginning of a cos oJr modulation on extracts B , _~ 0.2 mT. The neutron scattering data and 0.20

"'I"

l""l";'l~"'l""l't"l

'''

URu2Si 2

0.16

-.>

ZF, P I ~ 09

0.12

TN

tO

0.08 ¢1

n" 0.04

¢~

¢ 0.00

,,,I,, 0

5

l,,,,l,,,,l,t,,l,,,,l,,,,~P,,, 10

15

20

25

30

35

40

T (K) Fig. 4.72. Temperature dependence of the ZF-relaxation rate ),ZF in URu2Si 2. Note the discontinuity at TN (from Knetsch et al. 1993).

198

A. SCHENCK and F.N. GYGAX

the #SR-data, therefore, appear totally inconsistent. Several possibilities are implied by this inconsistency: (i) the #+ resides at the site (1/4, 1/4, 1/4) where B~ip = 0: this possibility is questionable in view of the results in URh2Si2, which clearly rule out the site (1/4, 1/4, 1/4) in this homologous compound (see section 4.3.4); (ii) the #+ quenches somehow the 5f-moments at the nearest neighbour U-sites (MacLaughlin et al. 1988); again this possibility is not supported by the #SR results in URh2Si2 which seem to be fully determined by the known AF-structure; (iii) magnetic structure and the magnitude of the ordered moments are not determined correctly by the neutron and X-ray scattering experiments. Also this is not a very likely possibility. Hence the #SR-results provide an unsettled problem. It should be noted that the small moment magnetic order is embedded in a strong paramagnetic background as evidenced by the magnetic susceptibility and the #+ Knight shift below TN (Knetsch et al. 1993, and unpublished results).

UNi2AI3. Similar to UPt3 and URu2Si2 this heavy electron system orders first magnetically below ,-~ 5 K and becomes superconducting inside the antiferromagnetic phase below ~ 1 K. ZF-#SR-measurements were the first to demonstrate that indeed magnetic order was established and that it was unaffected by the transition into the superconducting phase. (Amato et al. 1992c). The same study provided an estimate of the ordered moment of --~ 0.1#B. A typical ZF-signal is shown in fig. 4.73. This signal was interpreted in terms of a multisite occupancy and a simple antiferromagnetic structure with the moments aligned parallel to the c-axis of this hexagonal system of the same type as the RNi5 compounds. Alternatively Uemura et al. (1993) proposed that the #SR-data could also reflect an incommensurate spin

0.0

c.O m

-10.

N

Q_

-20.

r 0

I 1

i

I 2

i

I 3 Time

r

I 4

= 5

(l.~sec)

Fig. 4.73. ZF-/zSR signal in polycrystalline UNi2A13 at 0.3 K (from Amato et al. 1992c).

6

MUON SPIN ROTATIONSPECTROSCOPY

199

density wave order. Indeed later neutron scattering measurements revealed a long range incommensurate AF-order with a wave vector 0'= (1/2 + 3, 0, 1/2) (3 = 0.11) and a maximum ordered moment of # ~_ 0.24#B (Lussier et al. 1994). Whether the #SR-data and the neutron results are consistent with each other has to await a better understanding of the #+ site in this system.

UPdzAI3. Among all heavy electron superconductors this system is special in that it displays the highest superconducting transition temperature (~ 2 K) in coexistence with the largest ordered moment (,-~ 0.85#B ). The magnetic structure has been determined by neutron scattering (Krimmel et al. 1992). ZF-#SR measurements imply that superconductivity and magnetic order coexist microscopically (Amato et al. 1992d, Feyerherm et al. 1994c). However, no information on the magnetic structure could be determined from #SR-measurements due to the fact that the #+ resides at the high symmetry b-site (see fig. 4.25) at which the net internal field in the ordered state is zero. The #+-site has been determined unambigeously from the angular dependence at the #+ Knight shift and the A1 nuclear dipole field induced relaxation rate (Feyerherm et al. 1994c). 5. Review of results in magnetic insulators In contrast to the #+ in a metal, a positive muon implanted into an insulator represents a more complex system. Not only is it possible to encounter the formation of muonium and the concomitant - more complex - #SR signal (or the corresponding missing fraction), but other chemical effects may be of particular importance and influence the #+ in its localization and its coupling to the environment. For detailed descriptions of the situation see, e.g., Schenck (1985) and Patterson (1988). A reference list and an overview of the magnetic insulator systems studied by #SR spectroscopy is presented in table 5.1. (This table and the present chapter do not embrace the substances related to high-temperature superconductors, which are the subject of chapter 6.) Because the #SR studies are more complicated for magnetic insulators, they are also more fragmentary and less systematic than the ones on elemental metals. Most of the examined substances are oxides, and we divide these schematically in crystals with corundum-type structures, rare-earth orthoferrites with perovskite-type structures and rare-earth (R) perovskites RNiO3, CuO or related compounds, and MnO. The studied non-oxide magnetic insulators are fluorides and the COC12.2H20 compound. Finally the case of the antiferromagnetic molecular crystal a-O2 is also presented. Disregarding perhaps Cr203 and MnO, which have not been particularly well studied in this respect, it seems established that for all other oxides below approximately 500 K the occurrence of oxygen-muon bonds ('muoxyle bridges') plays an essential role in the localization of the positive muons - see, e.g., Boekema (1984), Boekema et al. (1985), Chan et al. (1986) and Lin et al. (1986).

5.1. Oxides with corundum-type structure ~-Fe203. In this compound a spontaneous #SR signal has been observed for the first time in an antiferromagnet (Graf et al. 1978). Consecutively the same research

200

A. SCHENCK and EN. G Y G A X

0

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~

{~

,,_., o o

.~

-~•

~2,

~

.~" .i~"

.

. ~

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II ~

II ~

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~.~

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o.~ o~" ~E~~ =o~o~ ~ c~

~

8~ ~° c?

MUON SPIN ROTATION SPECTROSCOPY

~ ,..,

__

o~

8

201

z~

"--" ~

V

-.~

•

8

"

•~ : ~ . ~ -

.~

--~

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~

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=

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t¢3 ..~

~

°

d

II ~

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~t

II ~

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,::? r,.)

6

=

rj

r~

rj

>

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A. S C H E N C K and EN. G Y G A X

o, O~

~v

~

8

r....,

r~

,-c,

ca

~ E...-r'' ..-,...T,

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4

4~v~ "

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z

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0

t.,q

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MUON SPIN ROTATION SPECTROSCOPY

203

group (Rtiegg et al. 1981) noticed that below 120 K the spontaneous signal was in fact split into three components - see fig. 5.1. Between 120 and 500 K a single frequency signal remains and the corresponding Bu follows the magnetization curve. At the Morin temperature, TM = 263 K, the signal shows that the local field at the #+ changes by a factor 2.15, due to the iron spin reorientation in the host, but the signal shows no discontinuity in the relaxation rate. The collapse of the three lines into one at about 120 K seems to correspond to a transition of three distinct muon states into a single one. This could be an indication for the onset of jumps between different potential minima for the muon in the unit cell. Potential energy calculations in c~-Fe203 (and in the non-magnetic compound c~A1203) favour particular sites for the #+, all located about 1 A away from their nearest neighbour, an oxygen ion. In such an oxygen-muon bond configuration the contact hyperfine field at the #+ may not only result from a direct overlap of the wavefunction tails of the magnetic electron and the #+, but supertransfer of hyperfine fields from a magnetic cation via the oxygen anion to the #+ can take place, depending on the geometry and the wave functions of the ions. This field/3c will add to the dipolar field to produce the local field/31oc acting on the #+: (5.1)

Bloc = /~c + J~dip.

/~dip@*/~) has to be calculated, e.g., as described by Denison et al. (1979) (notice that for an antiferromagnetic sample J~dip = /~dip)" Thus, a comparison of the results of hyperfine field calculations in this covalency-effect scheme, assuming the various proposed/z + sites as starting guides, will help to find the #+ states by requiring overall consistency (Boekema 1984). So far, however, the data allow more than one unique interpretation for the #+ localization. 250

'

'

'

1230'

200 ~

'

'

220

15 16

~"

150 -

I,..-,,-,.'4,. 50 100

1 2 1 0

150 200 :::L m

>=- 100

50

TM

0

I 0

I 200

,,\T~

i

I 400

i

I 600

I

..... 3 "I- -I--800

0

Temperature (K) Fig. 5.1. Temperature dependence of the zero-field /zSR frequencies and the local fields measured in single-crystal a-Fe203. The low temperature data are expanded in the insert (Rtiegg et al. 1981).

204

A. SCHENCK and EN. GYGAX

The direction of the local field/31oc with respect to the crystalline axes was determined by applying an external field with various orientations relative to the crystal and measuring IB, I. Since the antiferromagnetic structure is not destroyed if the applied field is not too strong, the demagnetization field remains zero and the/z + sees a field given by the vector equation /~/z ~-~ /~loc 4- /~ext.

(5.2)

With the correct assumption for the orientation of/31oc relative to ]~ext the absolute value deduced from eq. (5.2) will match the experimental value. Below the Morin temperature it was observed that each frequency splits into two according to

I~.1

IS~l =

Bext 4-

IB~o~l

(5.3)

for/~ext parallel to the c-axis, showing the alinement of/~lo~ along the c-axis. At room temperature, well above TM, B]o~ is found to be perpendicular to the c-axis, probably parallel to the crystalline a-axis. The disappearance of the remaining spontaneous signal at 500 K and the reappearance of a signal when applying an external field around 750 K is explained by means of #+ diffusion (Rtiegg et al. 1981). Examining the scheme for supertransfer of hyperfine fields it appears that Bo is parallel to the 3d-moments and that in contrast to Bdip the field/3c is not changing its value at the Morin temperature (for a discussion see Boekema 1984 and references therein).

Cr203. In zero field two #SR frequencies are observed at low temperature, one of them disappears around 150 K and the other is present up to TN. Both follow approximatively the behaviour of the lattice magnetization (fig. 5.2). Part of the ~ i I8 • Single crystal _ o Powder

1 O0 80

g

-

6

60 4 m= 4o

2

20 0

I 0

I 1 O0

I

I 200

I

I

0

300

T e m p e r a t u r e (K)

Fig. 5.2. Temperature dependence of the zero-field /~SR frequencies and the local fields measured in Cr203. The powder-sample data scatter more than the single-crystaldata (Rilegg 1981).

MUON SPIN ROTATION SPECTROSCOPY '

I

i

I

t

I

'

I

i

I

i

I

I

205

I

120 110 ~.100 9o

80 70 60 i

1

2

3

I

i

4

Bext (kG)

Fig. 5.3. Field dependence of the transverse-field /zSR frequencies measured in single-crystal Cr203 at 130 K. /3ext is applied parallel to the crystalline c-axis. The solid lines are results of calculations assuming various values for the angle a between/31oc and the basal plane. The lines correspond (from top to bottom) to values a = +9 °, +1 ° and - 9 ° for the upper triplet, and a = +24 ° and -24 ° for the lower doublet (Rilegg 1981). muons observe a vanishing local field (see Rtiegg et al. 1979). The Cr203 /zSRsignals differ sensibly from those observed in c~-Fe203: in the former system they are weaker - with smaller asymmetries and higher relaxation rates. Due among other things to the different spin structures of Cr203 and ~-Fe203, the local fields at the #+ sites have different orientations. Transverse field #SR measurements in Cr203 (see, e.g., fig. 5.3) show that/31oc is at an angle o~ with respect to the basal plane. This angle amounts to ± 2 4 ° for the lower frequency line, and to + 9 °, +1 ° and - 9 ° for the upper one (see the splitting of the two lines showing up when/3ext is applied parallel to the c-axis). The azimuthal orientations of Blo~ are also determined for the two frequencies and the various values of the angle ~ (Rtiegg 1981). As for c~-Fe203, the question of the muon site is not solved in a unique way. Also #+ trapping at a defect site cannot be ruled out - the lower #SR frequency in Cr203, e.g., can best be reproduced assuming a trapped muon near an non-magnetic A13+ ion substituting a Cr 3+ ion (Rtiegg 1981).

FeTi03. Transverse-field #SR measurements (Boekema et al. 1983) showed that above TN free muon-like behaviour was observed, whereas below TN the muons are experiencing local internal fields. Zero-field measurements below TN showed two weak but observable signals (fig. 5.4), following approximatively the magnetization curves for the Fe z+ ions. The values extrapolated to T = 0 K of the internal fields amount to 2.0 and 3.4 T respectively. Relatively large covalent contributions are

206

A. SCHENCKand EN. GYGAX i

500

.~- . . . .

400 ~" 300

i

i

i

i

i

i

@

, O____O_O...._.

O',,,,

%" 200

•

~'.~% %%

100 0

I

I

I

I

10

20

30

40

I "~

50

I

I

60

70

Temperature (K) Fig. 5.4. The frequenciesof the zero-field#SR signal in FeTiO3 at low temperature. For comparison, calculated magnetizationcurves for Fe2+ ions are drawn(Boekemaet al. 1983). expected and a rough estimate, taking also dipolar contributions into account, yields a field interval corresponding to #SR frequencies extending from 200 to 500 MHz, in agreement with the measurements (Boekema et al. 1983). Precise calculations were found difficult because possible effects of local #+ motion occurring at low temperature.

Fe304. This is a ferrimagnetic oxide (TFN = 858 K) that undergoes a semimetal-toinsulator transition at the Verwey temperature (Tv) near 121 K. In the course of a series of #SR studies (Boekema 1980, Boekema et al. 1985 and 1986) an anomalous change in local field and depolarization rate was observed at 247 K. The temperature dependences of the zero field data (spontaneous frequency and relaxation rate) are shown in fig. 5.5. The frequency follows essentially the bulk magnetization, but a small offset is clearly visible for the interval starting at Tv and ending at about 247 K. The local field is observed to be directed along the (111) direction, which is the easy axis of magnetization. The experiments performed with an external field greater than the demagnetization field and applied along the (110) direction (Boekema et al. 1986) showed that as the temperature is decreased below 247 K, the #SR frequency line splits, indicating the onset of two local fields, i.e. two magnetic inequivalent sites. This supports the model of a phase transition involving the onset of a short-range order already nearly 130 K above the well known Verwey transition. In that sense the anomaly at 247 K can be viewed as a precursor of the semimetal-to-insulator transition. V203. This oxide, which possesses interesting electronic properties, has been the latest member of the group of the corundum structured sesquioxides studied by #SR. Although V203 has the basic corundum structure in the high temperature phase where it is a paramagnetic metal, it undergoes a combined structural, magnetic and electrical phase transition below 155 K to a monoclinic antiferromagnetic insulating phase. At low temperature the V 3+ moments are aligned at +71 ° with the corundum c-axis, in alternating ferromagnetic planes normal to the monoclinic [0 1 0] or hexagonal [1 1 0]

MUON SPIN ROTATION SPECTROSCOPY 40

i

I

I

I

i

I

207

i

(a) 30

70 ,.(

60

O "O.~ '"'"~ to

50

20

40

|%.

-'-I

|

".

10

30

"q .o,

""lit,.. _

0

I

0

200

400

20

"'

"O,,

I

I

600

-1-

10

"... I.

800

I

0 1000

Temperature (K)

70

I

I

i

(b)

65 -1-

I

4.5

60

(.9 v :=L 1213

55

~4.0

rv 50

5.0

I

50

I ~ 100

I

I

I

150

200

250

Temperature (K)

Fig. 5.5. Temperature dependence of the frequency and the relaxation rate of the zero-field /~SR signal observed in Fe304 single crystals (a), and detailed frequency data for the Verwey-phase-transition temperature region (b) - from Boekema et al. (1985). direction. In V2O 3 one can ignore the covalent contributions to the field observed by a muon bonded to an oxygen and treat the internal field as purely dipolar to a very good approximation (Chart et al. 1986). Zero-field #SR data were obtained first by Uemura et al. (1984) and then by Denison et al. (1985). Below TN one oscillating signal following the trend of the magnetization curve appears in addition to the signal continuing to have a zero frequency (fig. 5.6). Data taken (Denison et al. 1985) with an external field applied along the corundum c-axis show that the oscillating signal splits into four, corresponding to orientations of the internal field at 7 °, 66 °, 116 ° and 174 ° (±5 °) with respect to the c-axis in the absence of an applied field (fig. 5.7). The magnitude of this internal field is about 0.11 T. The measured relaxation rates clearly indicate the absence of diffusion, even above

208

A. SCHENCK and EN. GYGAX 20.0

i

i

i

i

100.0

150.0

200.0

15.0 "° "o'-'@-O..o4...i,~

g 10.0

5.0

0.0

50.0

Temperature (K)

Fig. 5.6. Temperature dependence of the zero-field #SR frequencies measured in V203. Note signals at zero frequency below the transition temperature at 134 K. This transition temperature shows a well known hysteresis, not marked on this figure (Denison et al. 1985). 32.0 '

i

i

i

i /

.o_

S /

24.0 -

g 200 16.0 12.0 8.0 4.0 0.0 0.0

I

I

I

I

I

0.25

0.50

0.75

1.00

1.25

.50

Bext (kG)

Fig. 5.7. Transverse-field /zSR frequencies measured as function of Bext applied along the hexagonal e-axis of a V203 single crystal at 65 K. The solid lines represent the calculated frequencies for the net field I#~1, where 0 is the angle between the original/3]oc and the c-axis (Denison et al. 1985). the transition temperature of 134 K. Let's mention at that point that the transition temperature around 150 K shows a well k n o w n hysteresis, observed also in the # S R signal ( U e m u r a et al. 1984). In V203 only a subset o f the expected #+ sites in the c o r u n d u m structured oxides is occupied ( B o e k e m a et al. 1986). This is viewed as a signature for the vanadium pairing interaction suggested by G o o d e n o u g h (1963) as the mechanism responsible

MUON SPIN ROTATIONSPECTROSCOPY

209

for the structural, magnetic, and electrical phase transition in V203. In this picture the #+ cannot occupy the interstitial sites of reduced volume resulting from the V 3+ ion displacement in the corundum to monoclinic structural transition. Hence, #SR provides corroborative evidence for the mentioned phase-transition model.

5.2. Orthoferrites and RNi03 perovskites The rare-earth (R) orthoferrites (RFeO3) and the RNiO3 compounds crystallize all with the perovskite-type orthorhombic Pbnm (D 16) structure. The sublattice of the Fe or Ni ions, octahedrally surrounded by oxygen ions, is nearly cubic. Though crystallographically similar, the orthoferrites show a rich variety of magnetic properties (White 1969). The iron spins are coupled essentially antiferromagnetically, with Ntel temperatures in the range 740-620 K, whereas the spins of the rare-earth ions remain unordered down to typically 4 K. Due to a small spin canting, these compounds behave as weak ferromagnets. A systematic #SR study has been performed in six different orthoferrites (table 5.1) and reported by Holzschuh et al. (1980, 1981), and in the comprehensive article by I

I

I'

100 F ~ " SmFeO

50 25 0

-

1"2"41-]74

i

=

I

I

• --~1~--~0~0--

Cor "

I

,,-,._,,_.,

=

25

3

I

I

IQ---O-'O'-[ "~

EuFe03 (F4) 4

I

}-£4 " - * " " ~ "

I

I

I

I

DyFe03

2 5 I }-F4 1 I I I I t I 0 >=. 50 _ . _ ~ .~ . _ . YFeO 3 (F4) 25 I I I I I 0 50 HoFeO 3 25 Fr4 I I I I I O. 100" ErFeO a 75 50 25 r2~4Fr4 I I I l I 0 100 200 300 400 500 600 0 Temperature (K) Fig. 5.8. Measuredzero-field/~SRfrequenciesfor the rare-earth orthoferriteseries. The solidlines only connect the data points. Fe-spin configurationsare indicated(Holzschuhet al. 1983).

210

A. SCHENCK and EN. GYGAX

Holzschuh et al. (1983). In fig. 5.8 the temperature dependences of the zero-field #SR frequencies are summarized. At room temperature only one signal is observed, whereas at low temperature two or three signals are seen - except for SmFeO3 and DyFeO3. As in the case of a-Fe203, and possibly as in Cr203 and FeTiO3, this suggests that different sites in the unit cell will be occupied by the muon at 0 K. At higher temperature local #+ motion or hopping between these sites occurs. The Fe spins of the studied orthoferrites are at the higher temperatures in the so-called F4 configuration (notation of Bertaut 1963), i.e. basically aligned along the crystalline -t-z-direction, disregarding small canting angles of the order of 0.5 ° . At the lower temperatures the spin configuration is different, except for EuFeO3 and YFeO3. There, for SmFeO3 and ErFeO3 the 1"2 configuration is encountered, with now spins parallel to the -t-z-direction, however again with small canting angles. The temperature domains characterized by/"2 and/"4 configurations are marked in fig. 5.8. The discontinuities in the #SR frequencies in SmFeO3 and ErFeO3 are obviously related to the/"2-/"4 transition. In the other compounds the disappearance of the higher frequency components at certain temperature cannot be related to any of the known properties of the substances. Holzschuh et al. (1983) state that the higher frequency lines are associated with metastable #+ states, which decay with raising temperature to a ground state configuration associated with the lower frequency line. For this latter state a #+ site has been determined. This was done comparing the results of refined dipolar field calculations with a thorough set of zero-field as well as transverse-field #SR data, including the measurement of field and single-crystal orientation dependences - see, e.g., fig. 5.9. This most stable #+ site is located in the rare-earth-oxygen plane (z = 1/4 mirror plane), at 1 A of the O-ion, practically at the center of the slightly distorted square formed by the rare-earth ions. The distance to the O-ion corresponds to the bond length in the (OH)- molecule, 271oc is only of dipolar origin, Bc and hence the supertransfer of hyperfine fields can be neglected.

66 t

0

I I I YFeO3 Bext = 4kG T = 296 K

I

60

120

CI

I

180

I

I Bext

t

240

Angle ~ (deg.) Fig. 5.9. Measured orientation dependence of the transverse-field/zSR frequencies for YFeO 3 with an applied field of 0.4 T in the a-b plane. The solid lines are from a fit of a field-induced canting model. The results are used for the/*+-site determination (Holzschuh et al. 1983).

MUON SPIN ROTATION SPECTROSCOPY

211

Lin et al. (1986) have re-evaluated the data in these compounds (Holzschuh et al. 1983) and pushed further the muon site search. They first confirm the findings of Holzschuh et al. (1983) concerning the most stable site. They observe, however, that the proposed site splits into two different subsites (in terms of magnetic structure), leading to the same frequencies in SmFeO3 and ErFeO3, but to the two slightly different frequencies in YFeO3 and HoFeO3 (the sites are now called 1 for the ground state and 2 for the first metastable state). In addition Lin et al. (1986) find a new site candidate, corresponding to the remaining unaccountable frequency line in EuFeO3, YFeO3, and ErFeO3. This site ~site 3) lies symmetrically, approximatively above and below site 2, at the usual 1 A bond distance of the O-ion also bonded to site 2. New for site 3 is the observation that neglecting supertransfer, effects is not nearly as good as for sites 1 and 2, situated in the mirror plane. A precise re-evaluation including this covalent hyperfine field has not yet been performed. Torrance et al. (1992) have performed a systematic study of the insulator-metal and the magnetic transitions in the perovskites RNi03 (R = Pr, Nd, Sm, Eu) - fig. 5.10.

Rare Earth Ionic Radius (angstr6ms)

1.10 ~ O Eu

.00 i i Transitions in

500

Mago:

.-.

~- 4 0 0 - h , k (Z) •¢ [-],m (n-soatt.) 'm

i _ ~ ~ ~ \

1.20 i

t@Srn " ~ ~ r ~ ( ~ / ~ _ ~ O1~(~)~

¢~ 3 0 0 {3..

INSULATOR c

.o

. ~ " ~"J- ~'~ I~ Nd

"

2OO -L

y

/

METAL

Eu

-

o0 ¢-

r

,~ 100 0 0.86

ANTIFERROMAGN. INSULATOR 0.88

~ 0.90

~ 1

I 0.92

\

I 0.94

Tolerance Factor Fig. 5.10, Insulator-metal-antiferromagnetic phase diagram for RNiO3 as a function of the tolerance factor and (equivalently) the ionic radius of the rare earth (R). The observed insulator-metal transitions (resistivity or differential calorimetry measurements) are indicated by large open symbols, whereas the magnetic transitions are represented by three different solid symbols, according to the used method: susceptibility, n-scattering or /xSR measurements (left inset). Additional ~SR measurements of the AF-metal transition (not shown) are in excellent agreement with those from conductivity and neutron measurements. The structure of the RNiO3 compounds is shown schematically in the right inset - see text (from Torrance et al. 1992).

212

A. SCHENCKand EN. GYGAX

For the Eu and Sm compounds as well as for the solid solutions Sml_xNdxNiO3 the antiferromagnetic ordering-temperature data are obtained from TF-#SR measurements only. For the Nd and Pr compounds and for the solid solutions Ndl_xLa~NiO3 and Prl_~La~NiO3 the transition temperatures obtained from #SR and neutron scattering or conductivity measurements are in perfect agreement. The GdFeO3 structure of the RNiO3 compounds (Demazeau et al. 1971, Lacorre et al. 1991) is shown schematically in the inset of fig. 5.10. Regular NiO6 octahedra share comers to form a three-dimensional array, with the R ions occupying the space between these octahedra. In this structure the Ni-O-Ni bond angle, 0, plays an important role, since the electronic bandwidth and the magnetic exchange interaction are closely related to cos 0 (see, e.g., Sawatzky et al. 1976). This angle is generally less than 180 ° because of the orthorhombic distortion, which is conventionally discussed in terms of the tolerance factor, defined as t =_(dR-o)/X/~(dNi-O). If the rare-earth ions were large enough to give t = 1, the rare-earth-oxygen bond lengths (dR-o) and nickel-oxygen bond lengths (dNi-o) would be compatible with the ideal cubic perovskite structure, i.e with 0 = 180 °. Since the rare-earth ions are too small to satisfy this criterion, the structure becomes distorted as the NiO6 octahedra tilt and rotate in order to fill the extra space otherwise present around the rare-earth ion. The distortion tends to be slightly reduced as the temperature is increased. Torrance et al. (1992) draw the general phase diagram (fig. 5.10) for the RNiO3 series as function of the tolerance factor t. It appears clearly that the transitions observed form a coherent pattern. These transitions separate three distinct regimes: an antiferromagnetic insulator, a paramagnetic insulator and a metal. The observed insulator-metal transition depends strongly on R. For small R it occurs a at higher temperature than the antiferromagnetic ordering. The observations are discussed in the framework of a picture developed by Zaanen et al. (1985) and (1990), according to which there are two general types of band gaps possible: the Mott-Hubbard gap due to the Coulomb correlation energy U and the charge-transfer gap associated with an energy A. The insulator-metal transition in RNiO3 is most probably caused by the closing of the charge-transfer gap, induced by an increase in the electronic bandwidth either as a function of increasing temperature or ionic radius of R.

5.3. Miscellaneous mostly Cu-based and layered oxides Various #SR studies have been undertaken on CuO and other copper-oxide compounds related to high-To superconductors. The review concerning this latter class of materials is however mainly included in chapter 6.

CuO, BaCu02, BaY2Cu05. (See also BaCuO2 and BaY2CuO5 in section 6.2.6.) Weidinger et al. (1988) have found magnetic ordering in CuO, BaCuO2 and BaYzCuO5 ('green phase' of Y-Ba-Cu-O) indicated by well defined muon spin precession frequencies in zero-field measurements. A single precession frequency was observed in BaCuO2, whereas 5 frequencies were seen in both CuO and BaYeCuOs, corresponding to different muon stopping sites. A TN of 226 K was found in CuO,

MUON SPIN ROTATIONSPECTROSCOPY 0

213

Frequency (MHz) 10 2O 30 i

(a) 0 O_ .r0 I.t_

250

(b)

30 200 1-

"r"

15o E

~v20

e

>,.,, ,o ,t"-

rrl

100

0"

~10

t.l_

50 ,, ', ~ Ii ; ', ', ', Ii ~ ', ~ ', I ', ,,,I, .....

(c)

[

I ....

I,,

...... i 6 o 200 Temperature (K) Fig. 5.11. Zero-field #SR measurements in CuO: (a) typical Fourier-transform spectrum at 43 K; (b) temperature dependence of spontaneous frequencies compared with S = 1/2 molecular-field model; (c) muon depolarization rates. The five distinct lines, characterized by five corresponding point symbols in (b) and (c), originate from different muon sites (from Niedermayer et al. 1988).

whereas ordering temperatures of only 11 K for BaCuO2 and 15 K for BaY2CuO5 were observed. A more complete report on the magnetic properties of CuO, studied by #SR as well as 57Fe M~3ssbauer-source spectroscopy (MS) and l°°Rh perturbed angular

214

A. SCHENCK and EN. GYGAX

correlation (PAC), is given by Niedermayer et al. (1988). For the interpretation of the zero-field #SR results in magnetically ordered CuO, it is assumed that the fields at the muon sites are due to a sum of dipolar fields and supertransferred hyperfine interactions via the # + - 0 2 - bond, as, e.g., in c~-Fe203 (see section 5.1). As shown in fig. 5.11, below 60 K four distinct signals are seen, two disappearing at 60-80 K, and above 100 K, an additional signal is populated. The various components of the signal sum to the full original muon polarization, indicating that the precession of all implanted muons is observed. Clearly, antiferromagnetic order is detected by #SR below TN ,-~ 226 K; this is also indicated by MS and PAC. The muon-depolarization anomalies detected below TN (fig. 5.11c) are interpreted as indicating a change in muon localization produced by diffusion from metastable traps at low temperature to stable traps at higher temperature. The #SR fields that survive to TN are reasonably well approximated by a S = 1/2 molecular-field model. A high-quality polycrystalline CuO sample has also been studied with transverseand zero-field #SR by Duginov et al. (1994a). The data confirm most of the zero-field features reported by Niedermayer et al. (1988). However, the highest frequency line observed by Niedermayer et al. (1988) above 100 K is not seen and Duginov et al. (1994a) claim that no #+ polarization is missing in the signals they observe. In the temperature region of the incommensurate antiferromagnetic structure detected with neutron diffraction measurements by Forsyth et al. (1988), between 213 K and TN, no spontaneous #+ precession is seen. A peak in the relaxation rate of the slowly relaxing component of the zero-field signal at 219 K is supposed to be connected with the strong development of dynamic correlations near the incommensurate-commensurate transition.

Cao.86Sro.14Cu02, Sr2Cu03, La2fu04_ v, Sr2fbt02Cl2. The infinite-layer compound infinite-chain compound Sr2CuO3 have been studied by #SR by Keren et al. (1993) and compared to the La2CuO4_y and Sr2CuO2C12 systems (section 6.1). In Cao.86Sr0.14CuO 2 spin precession is observed below 360 K above that temperature a rapid #+ depolarization due to the onset of #+ diffusion occurs. The measured #SR frequency permits to extend nicely the sublatticemagnetization curve observed with neutron scattering to lower temperature. Below 225 K the frequency curve splits into two lines approaching 17.9 MHz and 14.3 MHz respectively for T --+ 0, and Keren et al. (1993) discuss also the possible corresponding #+ sites. The sublattice magnetization Ms shows a slower decay with increasing temperature in Cao.86Sr0.14CuO2, compared with that observed in La2CuO4_u and Sr2CuO2C12 (see fig. 6.19 in section 6), indicating that a wider CuQ-layer separation results in more two-dimensional magnetic behavior. Calculations show that Cao.86Sr0.14CuO 2 (dcu O = 3.20 A) is more appropriately described by a 3D model whereas Sr2CuO2C12 (dcuo = 7.76 A) is a very good example of a quasi-2D Heisenberg system. In SrzCuO3 ZF #SR oscillations were seen below 4.15 K, and the Ntel temperature determined as 4.15K < TN < 6 K; the sublattice magnetization was followed down to ,-~ 10 mK (Keren et al. 1993). Again two frequencies are present. The exchange interaction, inferred from susceptibility measurements, is of the order of J = 103 K, Cao.86Sr0.14CuO2 and the

-

MUON SPIN ROTATIONSPECTROSCOPY

215

implying a remarkable suppression of the ordering temperature with kBTN/J ~ 0.01 in SrzCuO3. The result demonstrates a clear signature of low-dimensional magnetic behavior in the CuO chains. For further discussion see also Uemura et al. (1994).

Y2Cu205. Duginov et al. (1994b) have performed ZF and TF #SR measurements in Y2Cu205 ('blue phase' in the Y-Ba-Cu-O family). In ZF the temperature dependence of Bu has been measured. In addition to the well established AF ordering of the Cu 2+ moments at TN1 = 13 K the authors find the indication of a second transition at TN2 = 7.5 K. This can be interpreted as a steady change in orientation of the magnetic copper moments with decreasing temperature, starting at 7.5 K. c~-Bi203. NQR and #SR measurements have been performed in c~-Bi203 by Duginov et al. (1994c). ~-Bi203 is usually considered as diamagnetic, but a splitting of the NQR spectral lines and the internal fields observed by #SR can be explained by the bonds in c~-Bi203 being of partially covalent nature. According to this explication not only 6s- and 6p-electrons but also 5d-electrons take part in the bond formation, hence the electronic shell can get a small magnetic moment of the order of 0.1#B, producing the observed field at a temperature of 135 K. SrCrsGa4019 (Kagomd lattice). The layered oxide SrCrsGa4019 (frustrated Kagomrlattice system) has been examined by LF #+ spin relaxation technique by Keren et al. (1994a). The results have been discussed by Uemura et al. (1994) in the context of frustrated and/or low-dimensional spin systems. SrCraGanO19 shows an unconventional spin-glass like behavior with very strong dynamical spin fluctuations persisting for T/Tg --+ O.

5.4. MnO Muon spin precession was observed in zero field in the ordered state of MnO by Uemura et al. (1984). The single component #SR signal found below TN underlines the fact that the sublattice magnetization Ms, proportional to B,, deviates quite noticeably from a S = 5/2 Brillouin function expected for conventional antiferromagnets. Such an anomalous behaviour of Ms was also observed in neutron scattering (Shull et al. 1951) and ESR (Sievers and Thinkham 1963) measurements. The #SR data, however, provide much more accurate information, especially near TN; the internal field of 0.68 T at T = 117 K suddenly disappears at T = 119 K. B~, extrapolated to T = 0 K (1.14 T) is reasonably well understood assuming a #+ placed at the body centered interstitial site of the simple cubic MnO lattice.

5.5. Magnetic fluorides Several papers have been published on #+ in magnetic fluorides. First, De Renzi et al. (1984a) and (1984b) intended to use the magnetically well known CoF2 and MnF2 insulators as a test case for #SR. In their extended reports they present a strong indication for an octahedrallike #+ location in CoF2, for which they could observe a spontaneous /~+ frequency in the antiferromagnetic state (fig. 5.12). However,

216

A. SCHENCK and EN. GYGAX

Fig. 5.12. Summary of measured angular dependences of the #SR frequency shifts in CoF2 at 41 K (circles and triangles) and at 13 K (squares). The crystal was rotated such that the external field of 0.3 T scanned the a--c plane (left side of the vertical axis on the figure) or the a-b plane (right side of the vertical axis). The solid lines are from calculationsfor the octahedral #+ site (De Renzi et al. 1984b). in both fluorides only a limited fraction of #+ contribute to the #SR signals. This was especially unfavorable for the MnF2 case - with no signal observed in the antiferromagnetic phase - resulting in an uncertain #+ localization. Nevertheless, De Renzi et al. (1984b) present several conclusions on the host magnetism, particularly for the CoF2 sample. The occurrence of a muonium signal in MnF2 and in site-diluted (Mn0.sZn0.5)F2 was discovered by Uemura et al. (1986b) (fig. 5.13). The formation of a F - : #+ : F 'hydrogen'-bonded center was considered for MnF2 by Kiefl et al. (1987), in analogy to the clearly established muon-fluorine bonds in various non-magnetic fluorides studied by Brewer et al. (1986) (such a location for the muon is compatible with the octahedrallike site favoured by De Renzi et al. (1984b) in CoF2); this type of state is now generally considered for #+ in fluorides - Noakes et al. (1993) and references therein. Thus, the #+ situation in fluorides is somewhat complicated, since the muons can appear in 3 different electronic states: (i) as (partly screened?) bare #+, (ii) in muonium form and/or (iii) as part of a ( F - # + F - ) - ion. This latter state especially has a rather noticeable effect on the host: by pulling two F - together (e.g., in MnF2 the nominal distance of 1.76 A between the two fluorine ions is reduced to 1.21 A by the intercalate #+ - Kiefl et al. (1987)) the crystal structure is quite affected in the #+ vicinity. Also in the (F#+F) - ion the #+ interaction with the quite close 19F nuclear moments will compete with the interaction between #+ and the possibly much larger anion moments, but also more distantly situated (Noakes et al. 1993). To add to the complexity, it is observed that the occupation of the three #+ states is a function of temperature (Noakes et al. 1993). Luckily the observation of a single

MUON SPIN ROTATION SPECTROSCOPY 1.0

A

217

A

•

O

O

~ I-

0.5

• v ( 0 ) = 153 MHz o v ( 0 ) = 1.3 GHz 0.0 0.0

I

I

I

I

I

d.5

I

I

I

.0

v/v N

Fig. 5.13. Temperature dependence of the zero-field #SR frequencies measured in MnF2 below TN = 67 K; a low frequencymuon signal and a high frequencymuoniumsignal are observed (Uemura et al. 1986b). The low frequency signal originatesfrom a (F/~+F)- ion, as proposed by Kiefl et al. (1987). spontaneous non-zero #+ frequency (other than a muonium frequency) in zero-field measurements in the antiferromagnetic phase (MnF2, Uemura et al. (1986b), CoF2, De Renzi et al. (1984b)) or of the characteristic oscillating zero-field muon spin relaxation signal generated in (F#+F) - above the N6el temperature (MnF3, CuF2, Noakes et al. (1993)) indicate a static behaviour of the muon in all studied fluorides, at least certainly for T ~< TN. So far the following informations on the magnetism of fluorides was deduced from #SR measurements:

CoF2 (De Renzi et al. 1984b). In the zero field #SR experiment the spontaneous field B u is proportional to the sublattice magnetization. The temperature dependence of B~, reflects the presence of the strong magnetic anisotropy. In particular at the lower temperature the behaviour follows the predictions of the spin-wave theory, while on approaching TN the observed simple power-law dependence agrees with the three-dimensional Ising model. MnF2 and site-diluted (Mno.sZno.5)F2. The/z-spin relaxation rates 1/T1 measured in zero field in both muonium and (F#+F) - states decrease rapidly with decreasing temperature below TN (Uemura et al. 1986b). The mechanism of the spin relaxation is explained above TN by the exchange fluctuations of the Mn moments, and below TN by the Raman scattering of spin waves. With the magnetic Mn atoms of MnF2 randomly substituted by the non-magnetic Zn atoms, the diluted antiferromagnet (Mn~Znl_~)F2 system can be used to study the effect of randomness on the spin fluctuation and ordering. The rate 1/T1 for the diluted (Mn0.sZn0.5)F2 is significantly

218

A. SCHENCK and EN. GYGAX

I 0.001

I

I

I

T= 20K

,L~

"

I

0 T = 33K

b "6 E E

<

,I ....

0

I,. II " --~ -,i~"ll.r-WW'-l~

T = 48K ~ l l

i,ll

0

"8 0.002 0.001 C3

0

i I '

-0.001

~

Illl T= 6511~1 1

I

-0.002 0

I

I

I

I

0.1

0.2

0.3

0.4

0.5

Longitudinal Magnetic Field (T) Fig. 5.14. The muonA9F level-crossing spectra in MnF2 as a function of temperature below TN = 67 K. The upper (A = 1) resonance is off scale at lower temperatures. The curves are fits of a theoretical difference signal; the positions of the resonances scale approximativelywith the sublattice magnetization (from Kiefl et al. 1987). larger than the corresponding values for pure MnF2 at the same normalized temperature (Uemura et al. 1986b). The difference between pure and dilute systems is related to the large spectral weight of low-energy magnons in (Mn0.sZn0.5)F2 found by neutron scattering (Uemura and Birgeneau 1986c). A model for #+ spin-lattice relaxation in an antiferromagnet is presented by Keren (1994) and used successfully by Keren et al. (1994b) to describe the 1/Ta measurements in MnF2. The observation of muon-nuclear level-crossing resonance (LCR) in the antiferromagnetic state of MnF2 (Kiefl et al. 1987) demonstrates that this technique can be used to obtain informations both on the local field at the muon and at neighboring nuclear spins in magnetically ordered systems (fig. 5.14). The observed shift of the local field at the 19F nuclei next to the #+ (with respect to 19F NMR data in MnF2) , is attributed to the disturbing influence of the muon.

MnF3 (Noakes et al. 1993). Below TN no fast oscillation can be detected in the zero-field measurements. At 10 K a large #+ spin-relaxation rate is observed for the major fraction of the signal, indicating a distribution of local fields at the (F#F) sites with a width of 0.1 T or more. Such apparent local magnetic disorder in a relatively simple magnetic structure may be due to the complicated crystal structure. With 12 formula units per monoclinic unit cell, the muons in the (F#F)- ions may be placed at a large number of slightly different positions with respect to the magnetic structure.

MUON SPIN ROTATION SPECTROSCOPY

219

CuF2 (Noakes et aL 1993). In the antiferromagnetic phase, the relatively low spontaneous frequency (corresponding to a local field of 0.15 T at low temperature) and the relatively low #+ depolarization rates of the signals indicate that the host is a rather weak-field magnet. The large bulk susceptibility above TN (Fischer et al. 1974) is interpreted as indicating substantial short range order. The muons as local probes can be sensitive to local ordering, so the coherent frequency need not necessarily go to zero at TN in this case. Unfortunately, the lowest temperature paramagnetic-state data obtained so far in CuF2 were taken well above TN, at 100 K, and no evidence for ordered moments was seen. 5.6. CoCl2.2H20 This compound has been comprehensively studied by proton NMR in its antiferromagnetic phase (Narath 1969). A unique magnetic field at the proton site in the water molecule has been found, indicating that all proton sites are magnetically equivalent. The field at the proton amounts to 0.42 T at the T = 0 K limit, and follows over a wide temperature range a power law of the form {Bp(0) -- B p ( r ) } oc T 6'5.

I

/

I

1.0

o

Vl0wer

x

Vupper

/

0.5 I-

'2. 0 v

:=L

T 4 . law

0 ._1

-0.5

-1.0 I

0.7

X

I

I

I

0.9

1.1

1.3

Log (T)

Fig, 5.15. Double logarithmic plot of the temperature dependence of the frequency shifts for the two zero-field/zSR signals in antiferromagnetic COC12-2H20 - Brewer et al. (1981).

220

A. SCHENCK and EN. GYGAX

One expected to find the #+ at one of the proton sites, but the zero-field #SR measurements (Brewer et al. 1981) showed surprising results: t w o frequencies are found, corresponding to internal fields of 0.273 T and 0.283 T at the T = 0 K limit, and both fields follow over the studied temperature range (from 4 to 17 K) a power law with a quite different exponent: {B#(0) - B~(T)} o( T 4"0. This temperature dependence, plotted in fig. 5.15, is totally different from the behaviour in the other antiferromagnetic insulators, where generally Bloc follows the sublattice magnetization. No explanation of the results is available so far. 5. 7. Solid oxygen

Storchak et al. (1994) present a ZF #+SR study of a-O2 (AF phase of solid oxygen) in the temperature range 10-24 K. Solid oxygen is one of the most unusual molecular crystal, as the 02 molecule possesses an electronic spin S = 1 in the ground state. Strong direct coupling of the 02 molecules' 7r-orbitals is realized on the background of the weak intermolecular Van der Waals interaction, closely connecting magnetic and lattice properties. At equilibrium vapor pressure solid oxygen exists in three crystalline structures; the low-temperature a-phase is known to be antiferromagnetic. Informations on the/z+-solid oxygen complex is given by Storchak et al. (1992). Below the a - f l transition temperature (T~-~ = 23.8 K) Storchak et al. (1994) observe long-lived #+ spin oscillations in ZF, manifesting the existence of an ordered state. Figure 5.16 shows the local magnetic field at the muon as function of temperature. For a comparison the normalized Brillouin curves for TN = T~-~ and TN = 40 K (about the estimate given by Bhandari and Falicov (1973) and Slyusarev et al. (1979)), respectively, are displayed. The abrupt drop of B~ in the vicinity of T~-~ shows that the ordinary second order phase transition does not occur in solid 1.6

I

I

I

I

1.2 (..9 v

0.8 'l I

0.4

0.0

I

5

10

I

I

15 20 Temperature (K)

I

25

30

Fig. 5.16. Temperature dependence of the magnetic field at the /~+ in c~-O2; circles: experimental points. Brillouin curves for spin S = 1, B0 = 1.27 kG, TN = Ta_;~ (dash-dotted line) and TN = 40 K (dashed line), respectively. The solid line is a fit of a 2D-Heisenbergmodel with an anisotropyparameter c~~ 10-2 - see text (from Storchak et al. 1994).

MUON SPIN ROTATIONSPECTROSCOPY

221

oxygen. Storchak et al. (1994) fit their data with the phenomenological model presented by Le et al. (1990a) for a 2D-Heisenberg spin-1 system with TN ----40 K and the anisotropy parameter c~ = 0.01; a good agreement is obtained up to T = 23 K (solid line in fig. 5.16). The behavior is similar to that found for Sr2CuOzC12 by Le et al. (1990a) (see section 6.1). It is known that in the/3-phase of solid oxygen a magnetic order, at least of short range nature, is present (Stephens and Majkrzak 1986). Storchak et al. (1994), however, do not observe the presence of a magnetic order above T~-~. 6. Review of results in layered cuprate (high To) compounds #SR spectroscopy has contributed significantly to our knowledge of the magnetic properties of the high temperature oxide superconductors and their magnetic parent and related compounds. First evidence for magnetic order in the (123)- and the (2212)-families and in NdzCuO4 (the parent compound of the electron high Tc superconductors) was in fact provided by #SR (Nishida et al. 1987a, b, De Renzi et al. 1989, Luke et al. 1989a). The potential of #SR to detect short range and random order has been particularly of value and helped in the elucidation of phase diagrams. Some of the work has been reviewed by Budnick et al. (1990a), Nishida (1992a, b) and De Renzi (1992).

6.1. La2Cu04 and related compounds Table 6.1 presents a list of the (214) cuprates and related compounds studied so far by #SR. The possible #+ site or sites were considered in several papers (Le et al. 1990a, Hitti et al. 1990b, Torikai et al. 1993a). According to the most recent and detailed investigations by Torikai et al. (1993a) possible #+ sites are restricted to small areas on the (100)- or (011)- and the (ll0)-plane near to an apical oxygen at a distance of ,-~ 1 A as shown in fig. 6.1. The site proposed by Hitti et al. (1990b) is close to the 'a' area. The 'a' area is also close to a theoretical prediction by Sulaiman et al. (1993). As in other oxide materials (see section 5) the #+ seems to bind to an oxygen ion forming a kind of an O-H bond (see also Boekema 1988).

La2Cu04. The appearance of antiferromagnetic order in La2CuO4+6 depends very sensitively on the exact oxygen content. Some oxygen deficiency (~ < 0) stabilizes the antiferromagnetic state (e.g., ~ = -0.3 --+ TN --~ 300 K), while an oxygen excess (~ > 0) (introduction of holes into the system) suppresses the antiferromagnetic order and eventually leads also to superconductivity with critical temperatures nearly as high as in the Sr-doped compounds. The presence of antiferromagnetic order in LazCuO4_6 is readily detected by #SR by the appearance of a spontaneous precession signal below TN in a ZF-experiment. It is found that the spontaneous field at the #+ extrapolated to zero temperature falls within a rather narrow range of B u _~ (38.0-43.0) mT independent of TN or the oxygen deficiency, respectively. Also the exact La/Cu ratio seems to be of no effect.

222

A. SCHENCK and EN. GYGAX

TABLE 6.1 List of La(214)-cuprates and related compounds studied by /~+SR (and #-SR). For references see table 6.2. Compound

Crystal struct,

Mag. struct.

TN (K)

Tc (K)

/~+ SR

La2_eCuO4_ ~

orthorhombic for T < 530 K orthorhombic for T < 530 K orthorhombic for T < 530 K orthorhombic for T < 530 K

AF

<290 6-dep. ~<260

-

ZF (#-SR) ZE ZF, LF

La2(CUl_~Zn~)O4 0 ~< :c ~< 0.10 La2(CUl_x Coz)O4+ 6 La2_zSr~CuO4_6 0 ~ x ~< 0.15 Lal.875Bao.125CuO4 Lal.875Bao.075Sro,050Cu04 La2_~_ySr~NdyCuO4 La2NiO4+6 Nd2CuO4_6 Pr2CuO4_ 6

Sm2CuO4_, Nd2_~CezCuO4_ ~ 0 ~< x ~< 0.17 Nd2_xSrxCuO4_6 0~<x ~<0.20 Lal.2Tbo.sCuO4

Sr2CuO2C12 La2MCu206+ 6 M = Ca, Sr

Lal.9Y0.tCaCuO6+6

tetragonal tetragonal tetragonal tetragonal, T~ tetragonal, tetragonal, tetragonal, tetragonal,

T~ T~ T' T~

tetragonal, T t

AF, random SG

ZF

AF (x ~< 0.05) SG (x ~ 0.13) AF AF AF AF, different from La2CuO4 AF (La2NiO4) AF (La2NiO4) AF (La2NiO4) AF(x ~ 0.14)

38 <50 < 30 320 ~ 245-275 ~ 250 ~ 250 ~<250

AF(x ~< 0.10)

<280

AF(La2CuO4) /~ = 0.46#B random tetragonal (KzNiF4) AF 2 layer perovskite AF

~- 170 <280

tetragonal, T*

2 layer perovskite

-

AF

> 0 for x/>0.6 0 20.5 -

ZF, LF, TF (#-SR) ZF ZF, TF ZF ZF

~< 24 (x > 0.14) -

ZF ZF ZF ZF, TF

-

ZF

260 >300 <45 for (M = Sr) 250 for (M = Ca) ~ 250

ZF

ZF ZF, TF

ZF

A c o m p i l a t i o n o f results is p r e s e n t e d in table 6.2. T h e temperature d e p e n d e n c e o f Bu is d i s p l a y e d in fig. 6.2 for various 5. It can be r e p r o d u c e d b y adopting a 2DH e i s e n b e r g m o d e l with a small anisotropy to allow for 3D long range o r d e r (Le et al. 1990a, b). A n i m p o r t a n t p a r a m e t e r is g i v e n b y A = gpBHA/4JMs w h e r e HA is the a n i s o t r o p y field, J the H e i s e n b e r g e x c h a n g e p a r a m e t e r and Ms the sublattice m a g n e t i z a t i o n in a p p r o p r i a t e units. In order to reproduce, Bu(T ) o(Ms(T), e.g., for = 0.03, o n e finds A _~ 10 . 3 ( L e e t al. 1990a, b). T h e n e a r l y constant B u ( 0 K ) in all the different L a 2 C u O 4 - s a m p l e s i n v e s t i g a t e d is in m a r k e d c o n t r a s t to results f r o m neutron scattering. F i g u r e 6.3 shows the sublattice m a g n e t i z a t i o n d e r i v e d from the m a g n e t i c B r a g g p e a k intensity o f three different m o n o c r y s t a l l i n e s a m p l e s with different TN ( Y a m a d a et al. 1987). A s can b e seen the sublattice m a g n e t i z a t i o n Ms for T --+ 0 K drops significantly with TN. It c o u l d be s h o w n that the r e d u c e d Ms(0 K) does not arise f r o m only partial m a g n e t i c order in

MUON SPIN ROTATION SPECTROSCOPY

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(c)

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Fig. 6.3. Temperaturedependence of the sublattice magnetization in La2CuO4_~ for various ~ as obtained from neutron diffraction by Yamada et al. (1987) (figure taken from Uemura et al. 1988). with maximum possible amplitude, rules out the occupation of more than one type of interstitial site by the #+. Finally Torikai et al. (1993b) have also measured the direction o f / 3 , ( 0 K) with respect to the crystal axes (see table 6.2). The direction found together with the suggested site implies that /3,(0 K) is not only arising from dipolar fields from distant moments but in addition also from the contact hyperfine and a dipolar field associated with a local (non spherical) spin density distribution in the vicinity of the/z +. La2(CUl_xZnx)O 4. Quite interesting results were obtained from ZF-measurements

in Zn substituted compounds Laz(Cul-=Zn=)O4 (Lichti et al. 1991b). The idea was to study the effect of Zn substitution on the magnetic properties in a system where the substitution takes place only at known Cu-sites. Since La2CuO4 only

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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Zn Concentration (x) Fig. 6.4. Magneticphase diagramof La2Cul_zznxO4. Opensymbolsare from/~SRmeasurementsand closed symbolsare from dx/dt and Xmaxdata (fromLichtiet al. 1991b). possesses CuO2-planes Zn substitution can only take place there, quite in contrast to the case in YBazCu3Oz. The substitution of Zn 2+ creates a spin hole (3d-shell is filled up in Zn) and removes a 3d charge hole from the planar network and disrupts locally the Cu-O hybridization. It suppresses not only the magnetic order but also superconductivity in the Sr-doped compounds. The ZF-#SR shows the presence of an ordered phase in all samples with 0 ~< x ~< 0.10. However, just below TN the magnetic order appeared random or of very short range only as evidenced by a fast decaying but nonoscillating PzF(t). Only at significantly lower temperatures, T ~, was a spin precession signal observed indicating the evolution of also 'long range' order. The results allow to construct a phase diagram as shown in fig. 6.4. It suggests a total suppression of magnetic order for z /> 0.11. The 'flat' part extrapolates to an x value which is quite close to 0.41, the site percolation threshold for Zn substituted sites. This could suggest that simple dilution controls this part of the phase diagram. The faster decrease of TN and T p for small x may point to additional frustration involving the coupling beyond the nearest neighbors. Quite remarkable is also the observation that the development of first the random order and then the 'long range' order is restricted to a fraction of the volume which grows from zero just below TN to essentially 100% at the lowest temperature. The remaining volume stays paramagnetic. This behaviour is reminiscent of what has been observed, e.g., in CeA13 and also in HoBazCu307 (see later). Whether, in the present sample, it could reflect a very inhomogeneous distribution of the substituted Zn, implying a wide distribution of local TN or T p, or an evolution of magnetic order through a percolative mechanism or something else remains to be seen. A comparison with results on YBa2(Cul_~Zn=)Ox will be presented below.

La2_xSrxCu04. Doping La2CuO4 with Sr (i.e. substituting Sr2+ for La 3+) introduces holes into the CuOz-planes just like in the case of an excess oxygen content. The additional spin carrying holes, probably located at the oxygen sites, are believed

MUON SPIN ROTATION SPECTROSCOPY

229

to be coupled ferromagnetically to the neighboring Cu-moments and thus interfere with the antiferromagnetic coupling of the Cu-moments, causing frustration and eventually a suppression of the Cu-moment order. At higher concentrations the holes will become the carriers responsible for superconductivity. It is suspected that the mechanism leading to Cooper pair formation may be magnetic in nature. This possibility has triggered quite some research into the question whether magnetic order and superconductivity can coexsist on a microscopic scale or whether they are mutually exclusive. One important model in this respect has been suggested by Aharony et al. (1988) which predicts a spin glass type of state as a result of magnetic frustrations. The #SR-results on La2_xSr~CuO4_6 for x < 0.5 resemble very much the results obtained in the pure La2CuO4_6 system when varying ~. In the ordered state the low temperature spontaneous field at the /~+, B~ (~ 0 K), is rather independent of x although TN drops quickly with increasing x. Also the width of the local field, AB (~ 0 K), increases with increasing x and thus reflects nicely the disorder introduced by the hole doping induced frustration (see table 6.2). The #SR results on poly as well as monocrystalline samples show that long range antiferromagnetic order is lost for a critical xc > 0.5 (since ~ is usually not determined the Xc-values from different investigations vary somewhat) (Kitazawa et al. 1988, Weidinger et al. 1989, Grebinnik et al. 1990a, Torikai et al. 1993a). However, this does not mean that no magnetic order is present at all. Rather a different type with spin glass like properties emerges which can be observed up to z _~ 0.15 (Weidinger et al. 1989). Although this is still somewhat controversial (see the comments by Harshman et al. 1989 and Heffner and Cox 1989 to Weidinger et al. 1989, and also the zero result by Kiefl et al. 1989) there is mounting evidence that at least up to x = 0.13 random static order can be observed. The evidence is the following. (i) The phenomenon is observed in high quality single crystals, (ii) the whole sample volume participates, (iii) the typical features of #+-relaxation in a classical spin glass are seen. As an example fig. 6.5 shows the ZF-#SR signal in a single crystal with x = 0.11 (Torikai et al. 1993c). At 20 K a Gaussian damping is observed which is induced by the random dipolar fields from the Cu- and La-nuclei. At a slightly lower temperature (To) the Gaussian decay is changed to an exponential one signaling a sudden influence of the Cu-3d moments as a result of the onset of correlations and possibly a slowing down of their dynamics. Below 8 K (= Tf) static random order is indicated by the development of a Kubo-Toyabe signal. Very similar results have been obtained by Sternlieb et al. (1990) who also demonstrated the static nature of the random field distribution at low temperatures in LF-decoupling measurements. There is another distinct difference between samples displaying long range magnetic order (x < 0.1) and samples with random order (0.1 < x < 0.13). While in the former samples above TN the 3dspin dynamics is so fast as to be ineffective in inducing #+ spin lattice relaxation (--+ 3d-spin fluctuation rate ~> 1012 s - l ) the dynamics is much slower in the latter samples and seems to freeze out smoothly on approaching the freezing temperature. Grebinnik et al. (1990a) find for a polycrystalline sample with x = 0.05 that PzF(t) above the freezing temperature Tf ~ 4 K is well described by eq. (2.18), i.e. PzF(t) = Po exp ( - (At)~).

(6.1)

230

A. SCHENCK and F.N. GYGAX

0.15 20K 0.10 EE <

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Time (IJS) Fig. 6.5. ZF-#SR signals in La1.89Sr0.11CuO4 for different temperatures. Note the appearance of a nearly static Kubo-Toyabe signal at 3.5 K (from Torikai et al. 1993c).

Corresponding fit results are displayed in fig. 6.6. A displays a divergent type of behaviour, and the exponent changes from 1 to 1/3 near Tf. This is a typical behaviour for a not so dilute spin glass (see, e.g., Campbell et al. 1994) and has also been seen in a Co substituted (214)-compound: La2Cu0.25Co0.7504. (Lappas et al., to be published.) Figure 6.7 presents a phase diagram for monocrystalline La2_=Sr=CuO4 proposed by Torikai et al. (1990, 1993a) on the basis of their #SR-data. As is evident the spin glass type of phase overlaps more or less completely with the superconducting phase. However, long range order just disappears approximately where superconductivity first shows up. It is also very interesting to note that Tf and To assume a maximum at an z-value which corresponds to the concentration where the low temperature tetragonal (LTT) phase is observed in polycrystalline samples. However, in the present case the sample with z = 0.11 shows 100% bulk superconductivity (Torikai et al. 1993c). The fact that both Tf and To tend to zero when z approaches 0.15 has led Torikai et al. (1993c) to suggest that a phase boundary exists at z corresponding to the maximum To and that the spin state above that z is qualitatively different from conventional antiferromagnetic or spin glass order. Finally we mention that also TF-#-SR-spectroscopy has been applied to the system La2_=Sr=CuO4. Quite visible and distinguishable are two components arising from # - captured by the planar and apical oxygen (Nishiyama et al. 1993, Torikai et al. 1993b). From these measurements the temperature and angular dependence of the /z--Knight shift in the lowest muonic Bohr orbital in oxygen has been determined. One expects that the results should correspond to equivalent data from t70-NMR insofar as the same site is probed. This is not the case and also the assignment

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232

A. SCHENCK and EN. GYGAX

of the two # - S R components to the two oxygen sites is not clear. The difference between the 170-NMR (Ishida and Kitaoka 1991) and the (O#-)-SR results may arise from the fact that (O#-) z-1 corresponds to nitrogen, thus being an impurity ion in the lattice, or that the oxygen (or nitrogen) electronic shell modified during the #--cascade to the lowest Bohr orbital is left in an unrelaxed paramagnetic state.

La2_:~(Sr,Ba)~Cu04. The system La2_~Ba~CuO~ displays a rather similar phase diagram as La2_~Sr~CuO4, except in the vicinity of z = 0.125 where superconductivity was found to be almost completely suppressed (Moodenbaugh et al. 1988). This appears to be correlated with an additional structural phase transition at To ,-~ 60 K which changes the system from the low temperature orthorhombic (LTO) phase to a low temperature tetragonal (LTT) phase and which is only seen in the Ba doped compounds (Axe et al. 1989). At much higher temperatures La2_~Ba~CuO4 like La2_~Sr~CuO4 undergoes a structural transition from a high temperature tetragonal (HTT) phase to the LTO phase. The structural phase diagram is shown in fig. 6.8. The question that arose immediately was, of course, whether the disappearance of superconductivity was accompanied by the appearance of magnetic order. ZF-#SR measurements on polycrystalline samples with z in the vicinity of 0.125 revealed indeed the presence of spontaneous #+ spin precession as can be seen from fig. 6.9 (Luke et al. 1991a). The low temperature spontaneous B~,(,,~ 0 K) amounts to 0.025 T which is considerably reduced from the maximum value of ~ 0.043 T found in La2CuO4_& Assuming that the antiferromagnetic structure is the same as 500

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in the parent compound the reduced Bu(,~ 0 K) can be used to estimate the magnitude of the ordered moment to be about 0.3#B. This value is in good agreement with estimates resulting from heat capacity measurements near z = 0.12 (Wada et al. 1990). B~,(~ 0 K) turns out to be rather independent of the exact value of z (see table 6.2). Following the temperature dependence of B~,(T) the transition temperatures TN could be determined. The results are also plotted in fig. 6.8. It is seen that To and TN follow the same z-dependence (Kumagai et al. 1993). The LTO-LTr transition was also found in the compound La2_~_vSr=NdyCuO4. Again magnetic order could be detected by ZF-#SR in a sample of composition Lal.775Sr0.125Nd0ACuO4 with B,(0 K) ~ 0.025 T (Kumagai et al. 1993). The results indicate that (at least long range) magnetic order and superconductivity are mutually exclusive. The role of the LTO-LTT transition is not yet clear (new #SR studies of the system Laz_=Sr=CuO4 have also revealed a long range ordered AF-state in the range z = 0.105 - 0 . 1 2 without a transition to the LTT phase (Watanabe et al. 1994)). It may involve a reduction in hole concentration in the CuO2-planes, shifting the system closer to the Lal.sSr0.aCuO4-situation. On the other hand the reduced value in the ordered moment needs to be explained as well. The latter suggests a change in the electronic structure within the CuOz-plane not just a dilution of the hole concentration. To study these problems further investigations of the mixed system La2_=(Sr, Ba)=CuO4 have been started. Neutron diffraction and TF- and ZF #SR measurements were performed on a Lal.a79Bao.075Sr0.050CuO4 sample (Lappas et al. 1994a, Lappas 1993, and unpublished results). This sample revealed clearly the presence of a superconducting transition at Tc - 22 K, detected by resistivity and magnetization measurements. The neutron measurements showed a gradual increase of the LTI" phase out of the LTO-phase below about 40 K, saturating at about 80% of the sample volume below about 25 K. TF-#SR measurements revealed two signals: one with a

234

A. SCHENCK and EN. GYGAX 100 • Neutron e /J, +SR

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MUON SPIN ROTATION SPECTROSCOPY

235

displayed spontaneous spin precession with B , 000(~ 0 K) _~ 0.024 T. Interestingly the onset of the spin precession signal, implying some 'long range' order, took place only below N 27 K while the W-measurements imply an onset temperature of 50 K (compare figs 6.10 and 6.11). In any case the #SR-measurements together with the neutron results prove that the occurence of superconductivity is restricted to the LTO-phase, while magnetism is associated with only the LTr phase. These findings are tentatively explained in terms of coexsisting Sr-rich and Ba-rich microdomains. The temperature dependence of the relative fractions remains unexplained. La2NiO 4. Very similar in structure to La2CuO4 this system has attracted attention with

respect to the possibility that it may show superconductivity like its Cu counterpart. Although no superconductivity was unambiguously found, magnetically it behaves very similar to La2CuO4. In stoichiometric La2NiO4 long range 3D antiferromagnetic order is observed below TN =320 K by neutron diffraction (Rodriguez-Carvajal et al. 1992). Its magnetic structure can be derived from the one in La2CuO4 by flipping the spin of the center Cu atom in fig. 6.13(b). This is confirmed by ZF-#SR which shows the presence of spontaneous spin precession below room temperature (Martinez et al. 1992). In contrast to La2CuO4_a two frequencies are identified corresponding to low temperature local fields of B(~I)(0 K) _~ 0.26 T and/3(2)(0 K)= 0.015 T, respectively. O ' O O qrl) • N

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(R = Nd, Pr, Sm). Also these compounds have a crystal structure similar to La2CuO4. In contrast to La2CuO4 they remain tetragonal at all temperatures (the R2CuO 4

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AC~A¢~A

1

g.ie

00

o AO A -->

B~_L~

I

I

I

25

50

75

I

100

o4 125

Temperature (K) Fig. 6.14. Temperature dependence of the spontaneous/z + spin precession frequency and the direction of/~u in Nd2CuO 4. Clearly manifest are the spin reorientation transitions at 35 K and at 75 K (from Luke et al. 1990a).

phase is called T') and the Cu-ions are only four fold coordinated (there is no apical oxygen) (see fig. 6.13c). They are the parent compounds of the electron doped (214) superconductors. ZF-#SR spectroscopy was first (Luke et al. 1989a) to demonstrate that all these compounds show Cu-moment magnetic order below typically 250 K (see table 6.2). However, the magnetic structure appeared more complex as a function of temperature, as is best seen from the results on a Nd2CuO4_6 single crystal sample. The ZF-spontaneous #+ precession frequency from this crystal is shown in fig. 6.14 as a function of temperature. As can be seen/3t,(T) changes not only its value but also its direction abruptly at certain temperatures. In accordance with neutron diffraction results (Endoh et al. 1989) this is traced back to spin reorientation transitions at 35 K and 75 K, respectively. Above 75 K the antiferromagnetic structure is the same as in La2NiO4, it changes between 35 K and 75 K to the magnetic structure established in La2CuO4_~ and reverts back to the La2NiO4 structure below 35 K. The ordered moment amounts to generally ~ 0.5#B. The reduced value of Bu(O K) as compared to Bu(OK) in La2CuO4 is a consequence of the different magnetic structure. Note that B , (0 K) is of similar order of magnitude as the lower field B(~2) in La2NiO4 (see table 6.2). A spin reorientation transition has also been observed by/zSR in Pr2CuO4 at ~ 40 K (Akimitsu et al. 1994).

Nd2_zCe=CuO4_a. Replacing

some of the trivalent Nd 3+ (or Pr 3+) by tetravalent Ce4+ mobile electrons are introduced into the Cu-O planes. For a critical concentration z ~> 0.14 superconductivity is observed with Tc,m~x~- 24 K (up to z ~- 0.18). Another necessary condition for superconductivity seems to be a certain oxygen deficiency (6 > 0.01) (Takagi et al. 1989). Thus electron doping has the same effect as hole doping in the La2CuO4-system. The question then was whether also the magnetic order, present in the undoped parent compounds, is affected in the same

238

A. SCHENCK and EN. GYGAX

300 250

i

La2.xSr,Cu04

Nd2.,CexCuO4

-~ 200 AFM

(11 n

E 150 cO

,*2-100 ffJ t'-

AFM

SC

0 0.2

~C

I

0.( 0.1 DopantConcentration

0.1

0.2

Fig. 6.15. Magnetic and superconducting phase diagram of hole doped La2_=Sr=CuO4 and electron doped Ndz_=CexCuO4. Solid symbols are deduced from/~SR-data (from Luke et al. 1990b).

way by electron - as by hole doping. #SR spectroscopy on Ce-doped Nd2CuO4 and Pr2CuO4 found indeed a rather abrupt suppression of Cu-magnetic order for z > 0.14 (Luke et al. 1989a, 1990a, b, Davis et al. 1990, Akimitsu et al. 1994) in agreement with later neutron diffraction work (Rosseinsky et al. 1989, 1991a). These measurements served to track the ordering temperature as a function of :c. The results are displayed in a phase diagram, combining the systems Lae_~Sr~CuO4 and Ndz_=CexCuO4 (fig. 6.15). It is again found that magnetic order is not established at once in the whole sample volume. A sizable paramagnetic fraction survives to temperatures much below the ordering temperature TN. It is also found that a very wide field distribution is sensed by the #+ - rather than a unique field, indicating that the magnetic order is disturbed and probably not of very long range nature. Figure 6.15 shows that TN declines much more rapidly with rising electron doping than with rising hole doping. This is understood in the following way: electron doping will change Cu 2+ to Cu 1+, the latter being nonmagnetic like a Zn ion substitute, diluting effectively the Cu-moment system. In contrast holes are mainly located at the oxygen ions, changing O -2 to O-, the latter carrying spin and inducing effectively a ferromagnetic coupling between Cu-moments and hence introducing frustration into the antiferromagnetic order. Frustration is known to have a much stronger effect than dilution on the suppression of magnetic order. On the other hand the maximum of Tc appears at approximately the same level of hole or electron doping. In contrast to the LazCuO4-system no intermediate doping regime is found where superconductivity and magnetic order of spin glass type coexist. Rather there seems to be a common sharp boundary separating the superconducting and nonmagnetic phase from the magnetic and semiconducting phase.

MUON SPIN ROTATIONSPECTROSCOPY

239

Nd2_~SrxCu04_6. It is also possible to introduce holes into the T~-phase Nd2CuO4 by substituting Sr2+ for Nd 3+ (Rosseinsky et al. 1992). Due to a slight oxygen deficiency (3 >t 0.03) holes were only created for x/> 0.1. ZF-#SR measurements showed indeed a magnetic behaviour for x <~ 0.06 which was little changed from the undoped parent compound. For a sample with x = 0.06, TN dropped to 240 K. Also the spin reorientation transitions where hardly affected in this doping regime (Rosseinsky et al. 1991b, 1992). However, for x = 0.10 (formal Cu-ionic charge: +2.02) the #SR data revealed a drop of TN to 60 K and no signs for the spin reorientation transitions were any longer visible. Also the decay rate of the #+ spin precession signal was considerably faster than in the samples with smaller x, implying a certain amount of disorder. Nevertheless over all long-range-order seemed to prevail (at least in part of the volume) as in the other samples, as was evident from the observation of magnetic Bragg peaks in neutron scattering (Rosseinsky et al. 1991b). Increasing x to 0.20 (formal Cu-ionic charge: 2.04) no precession signal could be found any more and no Cu-moment could be detected at 1.5 K by powder neutron diffraction. Obviously long range magnetic order was now suppressed. But the ZF#SR signal still displayed the presence of a wide static field distribution with zero average typical for a spin glass, which starts to develop below ~ 100 K in part of the sample volume. Both in the x -- 0.01 and the x = 0.02 sample pronounced relaxation was also visible in the paramagnetic regime indicating a much slowed down Cumoment dynamics in comparison to the hole free samples. The phenomenology of magnetism and its dependence on hole doping in Ndz_xSr~CuO4_6 has thus very much in common with the magnetic behaviour in the La2_xSrxCuO4_6 system. However, no superconductivity was observed in Ndz-xSrxCuO4. Suppression of magnetism by hole doping is thus not necessarily followed by a superconducting or even a metallic phase. L a 2 _ x T b x C u O 4 (x -~ 0.8). The tetragonal crystal structure of this compound is shown

in fig. 6.16. It arises from a fusion of half the T' (Nd2CuO4) and half the O/T (La2CuO4)-unit cells, forming the so called T*-phase in which Cu is five fold coordinated. Lal.zTh0.8CuO4 is a parent compound of yet another system which becomes superconducting upon hole doping (Sawa et al. 1989). Little is known sofar about the phase diagram of this system. ZF-pSR spectroscopy was first to reveal the presence of magnetic order in the parent material of the T*-phase (Lappas 1993). Subsequently the strength of the staggered Cu-moment (= 0.46(6)/zB) and the antiferromagnetic structure could be determined by neutron scattering. The magnetic structure was found to be identical to the one in La2CuO4 (Lappas and Prassides, to be published). High resolution ZF-#SR spectroscopy displayed the onset of a spontaneous spin precession signal below ~ 170 K, but already below ~ 220 K evidence for some sort of random static order appeared as evidenced by a strong relaxation. Static random order may have even survived in a small fraction (_~ 15%) of the sample volume up to room temperature (AB, _ 2.3 mT). Below 220 K one notices a steady increase of the magnetically ordered volume fraction which levels off around 80% below 100 K. The temperature dependence of the spontaneous

240

A. SCHENCKand EN. GYGAX

T

@

Fig. 6.16. Crystal structure of T*-phase (La, Tb)2CuO4. Cu is fivefoldcoordinatedby oxygen. B u is displayed in fig. 6.17. law:

B.(T) = B,(O) [1 - -T-~N] ~

It can be fitted below 150 K by a simple power

(6.2)

The parameters TN and /3 depend on the temperature range covered by the fit. Two best fit curves are indicated in fig. 6.18 with/3 = 0.36(3), TN = 130(1) K and /3 = 0.49(4), TN = 146(1) K, respectively. The persistence of magnetic order above these temperatures could point to the appearance of a different magnetic phase in part of the sample volume. An exponent of/3 = 0.36 is in close agreement with the prediction (/3 = 0.367) for a 3D S=1/2 Heisenberg antiferromagnet. An exponent of /3 = 0.5 would better correspond to a mean field model. A transition temperature closer to 130 K is also indicated by the exponential relaxation behaviour ()~zF(T)) in the paramagnetic volume fraction. It shows a pronounced divergence at around 120 K (see fig. 6.18), a temperature at which about 50% of the sample is still in the paramagnetic phase. This feature signals a considerable slowing down of the spin dynamics in the paramagnetic volume at ~ 120 K. Strangely below this temperature the dynamics appears to speed up again. The decay of the spin precession signal does not show any anomaly at ~ 110 K and seems to arise only from the spread in static local fields. The temperature dependence of ~ZF follows an Arrhenius behaviour between 120 and 170 K yielding an activation energy of ~-, 0.06 eV. From the value

MUON SPIN ROTATION SPECTROSCOPY

350 j , , "'-...'~. | 300 - -,r~.~.

,

241

, , , , ...... b = 0.36(3) -b = 0.49(4)

250 E. ~ . . "'"'" I ''"~.,., ~,~"" 200

i N~el

m= 1 5 0 .

""",.

'°°t 50

Oil

i

I

0

50

i

I

=

I

=

100 150 Temperature (K)

200

Fig. 6.17. Temperature dependence of the spontaneous B# in Lal.2Tb0.sCuO4. The dashed and solid lines represent fits of eq. (6.2) to the data, see text (from Lappas 1993).

3.0

I

I

I

i

2.5 if)

2.0 tO

1.5 1.0

CC

0.5 0

0.0

I

0

50

I

100 150 200 Temperature (K)

-O-

0

i

250

300

Fig. 6.18. Temperature dependence of the ZF-relaxation rate in the paramagnetic volume fraction of La]2Tb0.8CuO4. The sharp cusp seems to coincide with TN, determined from B#(T) (see fig. 6.17) (from Lappas 1993, Lappas et al. 1994b). o f AZF at 20 K a fluctuation rate o f u ~ 109 s -1 can be estimated ( L a p p a s 1993, L a p p a s et al. 1994b). O n s e t o f m a g n e t i c o r d e r o f the Tb 3+ 4 f - m o m e n t s is o b s e r v e d b e l o w 20 K b y a drastic i n c r e a s e o f the d e p o l a r i z a t i o n rate o f the spin precession signal ( L a p p a s et al. 1994b).

242

A. SCHENCK and EN, GYGAX

Sr2Cu02Cl 2. Also this compound is a layered perovskite with the tetragonal K2NiF4

structure. Compared to La2CuO4 all apical oxygen ions are replaced by CMons (see fig. 6.13a). This leads to an increase of the unit cell along the c-axis by 20%. The Cu-O coordination is thus four fold and in this sense Sr2CuO2CI2 resembles the T'-phase (Nd2CuO4). Neutron scattering and magnetic susceptibility measurements have revealed that SrzCuOzCI2 orders antiferromagnetically like La2CuO4 (Vaknin et al. 1990). Magnetic order has been also established by ZF-#SR measurements on a single crystal sample (Le et al. 1990a, b). A spontaneous spin precession signal was found which disappears sharply at 260 K (see fig. 6.19). It consist of one frequency down to 60 K and splits into two components with equal weight below 60 K. By using different orientations of the single crystal sample with respect to fi(0) it was determined that the spontaneous fields are directed parallel to the a - b plane. Note that in La2CuO4 Btz is pointing out of the plane. The low temperature frequency average (N 16 MHz) corresponds to a field of 0.12 T which is much larger than B u (0 K) in La2CuO4. These differences are ascribed to a different #+-site which, since there is no @ical oxygen, must be located near an oxygen in the CuO2 plane. The splitting of Bu for T < 60 K is believed to be of intrinsic origin since there is also a peak in the magnetization near 50 K when the external field is applied parallel to the c-axis. The splitting may arise from a possible spin reorientation transition not seen so far in any neutron scattering work. The temperature dependence of B~, resembles the results obtained in La2CuO4. As in the latter compound it has been explained on the basis of a 2D-Heisenberg model with A -~ 10 -5 (see section 6.1). The much smaller value of A in the present case implies a very weak anisotropy field as a consequence of the tetragonal structure (Le et al. 1990a). 20

I

I

I

I

I

61: ..... H I I c '= N

z

15

~2

~Lm~ ~ Aax LX

o c-

10

I

5 _

I

i

I

I

T"--r

O 50 100 150200250300 Zk Temperature (K) "-'

o P# I l c a x i s

12r

u_

0/

Z~

~LX A

zx P # / caxis

Zk

0

0

I

I

I

I

I

50

100

150

200

250

300

Temperature (K) Fig. 6.19. Temperature dependence of the spontaneous /z+ spin precession frequency (or frequencies) in monocrystalline Sr2CuO2C12. A splitting develops around 50 K which correlates with a peak in the magnetic susceptibility (see insert) (from Le et al. 1990a).

MUON SPIN ROTATION SPECTROSCOPY

243

La2MCu206+6 (M = Ca, Sr). The hole-doped double-layer system La2MCu206+6 (see, e.g., Fuertes et al. 1990) has shown superconductivity for M = Ca and for Sr doped La2_xSr~CaCuO6+6 but not for the compound with M = Sr. Prior to #SR measurements no magnetic order in the parent material had been identified. But ZF-#SR spectroscopy (Ansaldo et al. 1991, 1992) proved again the presence of magnetic order by the appearance of a spontaneous spin precession signal allowing also to determine the transition temperature TN. In the system La2SrCu206+6 it was found that TN decreases from more than 300 K for a sample with 6 < 0.06 to ,-, 250 K for a sample with 6 = 0.2. A TN -~ 250 K was measured in a La2CaCu206+~ with 6 -~ 0.037. Common for all samples was a two frequency precession at low temperatures corresponding to local fields of B~, (0 K) = 0.036 T (main signal) and B~ (0 K) = 0.13 T. The lower field is probably associated with #+ bound to an apical oxygen while the high field is associated with #+ near a CuQ-plane. Note that B~ (0 K) = 0.13 T is close to the corresponding value in Sr2CuO2C12 (see table 6.2). The high frequency component becomes blurred above ~ 250 K. Very peculiar is the temperature dependence of Bu(T) in the LaaCaCu206+6 sample (see fig. 6.20). In contrast to the M = Sr samples both B~, drop by roughly a factor of 1.75 when going from the lowest temperatures up to ~ 30 K (Ansaldo et al. 1991, 1992). The authors suggest that this drop may be associated with the onset of superconductivity due to the mobility and segregation of the excess oxygen intercalated in a Ca(La) layer between two CuO2-1ayers. As the temperature is lowered the excess oxygen may cluster into domains producing locally a high enough hole concentration to facilitate superconductivity and depleting other regions from holes thereby stabilizing the antiferromagnetic order there and restoring the moments to their full staggered value. The presence of superconductivity below 45 K in about 15% of the sample volume was indeed interfered from weak W-measurements (Ansaldo et al. 1992).

18

I

I

I

I

I

I

I

I

16 14 1

12 10

¢.-

Ill•

8

6 LL

4

0

2 0

I

0

I

i

0 I

I

0 I

I

100

I

I

0

I

200

Temperature (K)

Fig. 6.20. Temperaturedependence of the two spontaneous/z+ spin precession frequencies in La2CaCu206.037. The pronounced increase of both frequenciesbelow (20-30) K is attributed to a segregation into oxygen (hole) rich and oxygen depleteddomains (from Ansaldoet al. 1992).

244

A. SCHENCK and EN. GYGAX

6.2. YBa2Cu30x and related compounds A list of compounds studied by #SR spectroscopy is given in table 6.3. Possible #+-sites have been considered in a number of papers. Theoretical work has been presented by Boekema (1988), Dawson et al. (1988), Li and Brewer (1990), Halim et al. (1990), Lichti et al. (1990a, b) and experimental work has been discussed by Dawson et al. (1990, 1991a), Nishida et al. (1990a), Brewer et al. (1990) and Weber et al. (1990). There is a general consensus that, like in the L a 2 C u O 4 system, the #+ will be bound to an oxygen, forming a kind of an O - H bond with a bond length of ~ 1 ,~. More specifically, in the fully oxygenated (123)-system (x = 7) the #+ is preferentially found near a chain oxygen at a position (0.14, 044, 0.071) (Weber et al. 1990, see fig. 6.21). The same site is also taken by hydrogen in hydrogenated YBa2Cu307 (Weidinger et al. 1994). With less probability the #+ is also found near an oxygen in the CuO2-plane. This position is not very precisely known. In the oxygen depleted (x = 6) system all the chain oxygen is absent and the #+ is now found near an apical oxygen.

6.2.1. YBa2Cu30~ The very first indication for magnetic order in an unspecified member of the superconducting YBaCuO-family was provided by Nishida et al. (1987a), using ZF-#SR. Subsequently the same group (Nishida et al. 1987b) was first to prove the existence of long range magnetic order in the tetragonal phase of YBa2Cu3Ox, the parent

b/

1',

r. sa

/

___k

Fig. 6.21. Lower part of the crystallographic unit cell of YBa2Cu3Oz with possible/z + sites indicated by olden circles and filled circles. The cluster of points to the right (center is at (0.14, 0.44, 0.071) about 1.0 A away from the chain oxygen 0(4)) and equivalent positions is found for x ----7, the more spread out distribution of sites close to the apical oxygen O(1) (distance is again of the order of 1.0 ,~) and equivalent positions for x -- 6 (from Weber et al. 1990).

MUON SPIN ROTATION SPECTROSCOPY

245

[.. F

c~

d ,,d V

v~

o

v~ ,,,.4,

v~

I

+

~ ~ v~

e- c4 ~: II II II

~t

::::1.

=L&:&

0

<~ 0 o

~ tl u u q~ tt~

N

~v

#

c5 II

"~

o--

o

N ? ~ A ~ V

o

t~

0

v~v/~v/~v/v~V/V ~v/

~ o~

v/Zv/ ~v/av/vav/~/~.v/

~¢~

246

A. SCHENCKand EN. GYGAX 500

I

I

i

I

I

I

f

I

I

I

YBa2Cu3Ox gSR

400

[2] quenched [2] annealed

n.s. 300

[41 [51

I

g AFM

~

[]

!

200

100

~

~ellj~ annealed~ I

Ortho-I

I 'l

Ortho.ii r ~ )O00"-

i~

i

i

i

i

7.0 X Fig. 6.22. Magnetic and superconductingphase diagram of YBa2Cu30=. The magnetic part stems from/zSR and neutron scattering measurements: (1) Nishida et al. (1988b), (2) Brewer et al. (1988), (3) Tranquada et al. (1988), (4) Rossat-Mignodet al. (1990), (5) Rebelsky et al. (1989) (from Nishida 1992b).

6.0

6.5

material for the high temperature superconductor YBa2Cu307. Long range antiferromagnetic order was confirmed shortly after by neutron diffraction measurements on an oriented powder sample (Tranquada et al. 1988). Since then #SR experiments on the YBazCu30= system investigated the dependence of magnetic order on the oxygen content and in particular tried to verify a possible coexistence of magnetic order and superconductivity in a certain range of z like in the La2CuO4-system (see table 6.3). Results on the transition temperatures following from these experiments are summarized in fig. 6.22, including data from neutron scattering (Rossat-Mignod et al. 1990, Rebelsky et al. 1989, Tranquada et al. 1988) and superconducting transition temperatures (Nishida 1992a, b), As in the LazCuO4 system TN decreases with the hole concentration (oxygen content), most rapidly in the range z = 6.2-6.4, depending on the sample preparation method: samples prepared by an annealing procedure show a drop of TN at smaller z-values while in samples quenched from high temperatures the drop of TN is shifted to slightly larger z-values (Brewer et

MUON SPIN ROTATION SPECTROSCOPY

247

al. 1988). Weidinger et al. (1990) even found TN _~ 60 K in a quenched sample with z = 0.5. The difference in behavior is ascribed to differences in oxygen ordering in the Cu-O chain planes for nonstoichiometric oxygen concentrations. By very slow and careful annealing the orthorhombic phase can be stabilized down to z = 6.2. For quenched samples the tetragonal phase is observed which converts to the orthorhombic phase at higher x-values (for a detailed discussion and references see Nishida 1992b). Independent of preparation technique TN is rather constant up to z = 6.2. It is argued that the addition of oxygen into the chain position up to z = 6.2 does not introduce holes into the CuO2-planes and thus cannot disturb the intra plane antiferromagnetic exchange coupling (Rossat-Mignod et al. 1990). Figure 6.22 shows that magnetic order (not necessarily of long range nature) and superconductivity seem to coexist around and below z = 6.5. Actually all available data show that magnetism and superconductivity exist in the same sample, if Tc is below approximately 50 K (see Weidinger et al. 1990). Again the question arises on what scale this coexistence takes place. A macroscopic separation into a magnetic and a superconducting phase can be ruled out since TN and Tc vary in a continuous way with the oxygen content. Although the fraction of #+ sensing magnetism declines with rising z or Tc, respectively, the corresponding volume fraction is not 100 90

80

70

o~ 60 t-

0.57

O

50 .2 '$

\7

tt~

40

3;

0.67

30 20

\

YBa2Cu306+x

\ \ \

10

\ \

0

I

10

I

}

20 30 Tcrnidp°int (K)

I

I

40

50

60

Fig. 6.23. Fraction of magnetically ordered volume in YBa2Cu306+= as a function of z. The fraction is obtained from the amplitude of the corresponding/~SR-signal (from Weidinger et al. 1990).

248

A. SCHENCKand EN. GYGAX 320

I

I

c I P.

300

I

I

Site#1

280 L9 260 .._= o~ C 0

240

i

I

[-

I

,ev ~

1300 0 ._J

I

8 MHz

1260 1220 1180

ii 0

i 50

l

i~~

1O0 150 Temperature (K)

200

Fig. 6.24. Temperature dependence of the two spontaneous /z+ precession frequencies in an oriented powder sample of YBa2Cu306.0 (from Brewer et al. 1989). associated with a constant TN, as one may suspect if macroscopic magnetic domains would be present. The #SR data also show that the spontaneous static internal fields at the #+ decrease very rapidly when the superconducting phase is entered (Weidinger et al. 1990, Brewer et al. 1989). Also this is inconsistent with a macroscopic phase separation picture. However, one cannot exclude the coexistence of microdomains of orthorhombic and tetragonal structure which are so small that they will influence each other, explaining the continuous variation of TN and To. The presence of a spontaneous #+ spin precession signal or of at least coherent short range magnetic order places a lower limit on the tetragonal domain size of ~ 50 .~. This is still small enough to not inhibit superconductivity~ due to a proximity effect vis-a-vis a superconducting coherence length of (20-30) A. Of course the amplitude of the spin precession signal must be reduced (as is observed, see fig. 6.23) and #+ in the orthorhombic domains will still feel some random fields, spilling over from the tetragonal domains, causing depolarization (as is also observed). It has been also suggested that an electronic phase separation could produce hole rich and hole depleted microdomains (Emery et al. 1990). The general picture provided by #SR on the phase diagram of YBa2Cu30~ fits well with other results, notably from neutron scattering experiments. We now discuss some more specific #SR results. ZF-#SR in the ordered state displayed the presence of clearly two frequencies (Brewer et al. 1989, Budnick et al. 1990a, Barsov et al. 1994). The dominant frequency (B, (0 K) ~_ 0.03 T) is seen

MUON SPIN ROTATION SPECTROSCOPY

18

~illl

249

YBa2Cu30x

•

17

il

16

+ ~ Q x

g

LL

• x 0 •

2

I

II~

I

I

•

~OAcL °

6.08 8.16 6.31 6.32

40

~'

+

4~ (~

I

I

I

I

I

I

I

l

I

I

80 120 160 200 240 280

• 6.08 × 6.16

4

a

/i /~jt J

-- 2

b

n- 1

0

I

t

I

I

I

I

I

I

I

I

1

I

I

I

I

40 80 120 160 200 240 280 Temperature(K)

Fig. 6.25. Temperature dependence of (a) spontaneous /~+ precession frequencies and (b) associated relaxation rates in samples with various oxygen concentrations (from Barsov et al. 1994).

for :c up to 6.5 (depending on the preparation procedure). A second high frequency component ( B , (0 K) = 0.13 T II c-axis) is only seen in samples with x < 6.3. Occasionally a third frequency is seen corresponding to a Bu (0 K) -- (0.0080.015) T (Nishida et al. 1990a). The three frequencies correspond to the three sites discussed above and are consistent with the antiferromagnetic structure determined by neutron diffraction measurements (Tranquada et al. 1988). The dominating B , (0 K) is rather insensitive to the oxygen concentration for z < 6.2-6.32 (Kiefl et al. 1989, Weidinger et al. 1990, Barsov et al. 1994). In contrast and analogous to the La2CuO4-case neutron scattering showed a pronounced decrease of the magnetic Bragg peak intensity when z was changed from 0.15 to 0.38 (Rossat-Mignod et al. 1990). The temperature dependence at B~ = 0.03 T and B~ = 0.13 T in an oriented z ~_ 6.0 powder is displayed in fig. 6.24 (Brewer et al. 1989). Very peculiar is the rise of Bu below ~ 80 K (T/TN ~- 0.2) which is more pronounced for the dominant 0.03 T-signal which arises from ~+ located near the apical oxygen. This

250

A. SCHENCK and EN. GYGAX

anomaly is also found in sample with z -- 6.08-6.32 (Nishida et al. 1988b, Barsov et al. 1994). Interestingly a similar anomaly at T/TN ~- 0.2 is also found in the (2212)-family (see below). At 10 K another anomaly shows up in that Bu(T ) drops sharply for the 0.13 T-signal and slightly for the 0.03 T-signal. This drop has not been seen in the data of Nishida et al. 1988b (z _~ 6.2) and of Barsov et al. 1994 (z = 6.08-6.32, see fig. 6.25a). Its origin in the z = 6.0 oriented powder sample is not explained. The 80 K anomaly shows also up in the staggered magnetization determined from the intensity of the magnetic Bragg peaks in neutron scattering for samples with z > 6.2 but apparently not in samples with z < 6.20 (Rossat-Mignod et al. 1990). In contrast to the #SR results the magnetization decreases below ~ 80 K. Interestingly also the spread ABe(T) of the local field B,(T) increases below 80 K (Barsov et al. 1994, see fig. 6.25b). Taken together it seems that long range order gets disturbed below 80 K (at least in samples with z > 6.08) with the effect that the Bragg peak intensity decreases and the field spread at the #+ site is increased. That also Bu(T) increases more rapidly below 80 K may be a subtle result of forming the vector sum of dipolar fields at the #+ position (the position near an apical oxygen may be particularly sensitive to slight changes in the moment order since it is situated almost half way between the CuO2-planes). The appearance of a certain disorder below 80 K may be associated with a freezing of holes at the oxygen sites. Finally it has to be emphasized strongly that from the perspective of the implanted #+ the onset of magnetic order never involves the total sample volume at once. Depending on the sample preparation magnetic order in most of the sample volume may only be obtained at temperatures significantly below TN. This may be explained in terms of a distribution of transition temperatures due to an inhomogeneous distribution of the chain oxygen ions, but this explanation cannot very well be applied to samples with z ~< 6.2, where TN varies only little. Exploratory TF-#-SR-spectroscopy on YBa2Cu30~ (z = 6,6.5,7) has been reported by Nishida et al. 1991, Nishida 1993).

6.2.2. Yl_vPrvBa2Cu30~ (x = 6,7) Replacing Y in Y B a 2 C u 3 0 7 by other rare earth atoms the superconducting properties of the (123)-O7 system are hardly effected. A notable exception is a replacement by Pr which - in contrast to Y or other rare earth substitutes - is believed to have an ionic charge of ~ 4 + instead of 3 + and hence donates electrons 6. Therefore one may expect that Pr-substitution has a similar effect on the hole concentration in the CuO2planes as removing oxygen. Indeed PrBa2Cu307 does not show superconductivity but antiferromagnetic order of the Cu-moments in the CuO2-planes instead (Felner et al. 1989). A remarkable difference to YBa2Cu306 is the fact that PrBazCu307 retains the orthorhombic structure. ZF-#SR measurements on polycrystalline samples served to determine the phase diagram of Yl_vPrvBa2Cu307 (Cooke et al. 1990a-d). Long range antiferromagnetic order, again in evidence via a spontaneous 6 This view is challenged by X-ray absorption (XANES) results which show that Pr in the (123)compounds is in the trivalent state (Pr 3+) (U. Staub, priv. communication). The appearance of Cu 2+moment order for y > 0.5 and the destruction of superconductivity must, nevertheless, imply that the holes in the CuO2-planes are removed and instead are probably located close to the pr3+-ions.

MUON SPIN ROTATIONSPECTROSCOPY

251

spin precession signal, was observed for 0.5 < y ~< 1. The temperature dependence of the spontaneous frequency u~ is displayed in fig. 6.26 for various y. The abrupt drop in u, below 20 K for y = 1 and below 6.5 K for y = 0.8 is probably arising from the onset of magnetic order of the pr4+-moments. This phase transition at TN2 has also been seen in susceptibility, specific heat and neutron scattering measurements. It seems, however, that pr4+-moment ordering alone is not sufficient to explain the drop of ur, satisfactorily. Rather it appears necessary to invoke in addition a small (0.02#B) ordered moment on the Cu-chain sites (Dawson et al. 1991b). Such a small moment may have escaped detection in NMR/NQR measurements (Ltitgemeier 1989). The spontaneous field at the #+, extrapolating the values above 20 K to zero temperature, B~ (0 K) = 0.016 T, is clearly smaller than the corresponding value in YBa2Cu306 (see table 6.4). The difference arises from the different sites that are taken by the #+ in the 06- and the O7-compounds and is not the result of different values of the ordered Cu-moments. The value of 0.016 T is close to what is found in hydrogenated YBa2Cu307 (see below) and occasionally in YBa2Cu306+6 (5 < 0.5) and confirms a #+-site assignment close to a chain oxygen. As before in YBa2Cu306+~, B~(0 K) seems to be rather independent of the Pr-content y or the hole concentration, respectively. At y = 0.5 no long range magnetic order could be detected. Instead a fast relaxing signal was seen which could be well described by a function typical for spin glasses with nearly static features (Cooke et al. 1990d). From the temperature dependence of B~ for y > 0.5 and of the relaxation rate for y = 0.5 the N6el temperatures or a spin freezing temperature TN1 could be determined which are displayed in fig. 6.27 as a function of y. Included are also superconducting transition temperatures Tc for compounds with y < 0.6 as well as the additional magnetic transition temperatures TN2 at low temperatures. The resulting phase diagram is very similar to the YBa2Cu306+6-phase diagram including a narrow regime of coexisting superconductivity and spin glass order. This feature has to be commented in the same way as for YBa2Cu306+6. i

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252

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In PrBa2Cu306 two spontaneous frequencies showed up, one corresponding to the frequency seen in PrBa2Cu307 the other one being close to (in fact 16% larger than) the value seen in YBa2Cu306+6. This points to the presence of an appreciable amount of chain oxygen ions so that #+ can be found both near chain oxygen and apical oxygen sites, respectively.

6.2.3. HzYBa2Cu307_6 Doping YBa2Cu307 with hydrogen has the same effect as replacing y3+ by Pr 4+ ions, namely it donates electrons to the CuO2-planes and reduces the hole concentration. This conjecture was first fully proved by ZF-#SR-measurements which showed the presence of long range antiferromagnetic order in HzYBa2CuaO7 for z > 0.5 (GliJckler et al. 1989, 1990a, 1990b, Niedermayer et al. 1989), following a first indication for some sort of local magnetic order by NMR (Goren et al. 1989). Persistence of superconductivity up to hydrogen concentrations of z = 0.2 was observed before by Reilly et al. (1987). The ZF-#SR results revealed two spontaneous precession signals corresponding to B (1) (0 K) = 0.015 T and _/~R (1) (0 K) -- 0.03 T. The temperature dependence of R (i) for a sample with z = 0.9 (6 = 0) is reproduced in fig. 6.28. The higher frequency signal appeared only weakly while in oxygen deficient samples it was the dominant component. This fact indicates that the low frequency signal is associated with #+ bound to the chain oxygen and the high frequency signal with #+ bound to the apical oxygen site. The near perfect agreement of the frequency values with the corresponding values in YBa2Cu306 and Y l _ y P r y C u 3 0 7 (see table 6.4) proves that the antiferromagnetic structure of the ordered Cu-moments in the CuO2-planes as well as the moment value is not affected to any significant degree by H- or Pr-doping. The ZF-#SR measurements show that, depending on the level of hydrogen doping, only a fraction of the sample volume contributes to the spontaneous precession

MUON SPIN ROTATION SPECTROSCOPY 5

i

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50

i 100 150 200 250 300 350 400 Temperature (K)

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30 25 o~ 20

g

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2.5

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Fig. 6.29. Fraction of the magnetically ordered volume in YBa2Cu3OTHz as a function of z. The fraction is determined from the amplitude of the spontaneous spin precession signal (from Niedermayer et al. 1989).

signals. As fig. 6.29 reveals this fraction is zero below z = 0.5 and saturates at about 20% above z ~_ 1.5. The data show that the remaining volume fraction is associated with a nonoscillating, slowly and exponentially decaying signal, reflecting random internal fields of a few 0.1 roT. Niedermayer et al. (1989) speculate that this signal could be associated with a third site with very small residual fields in the ordered state. More likely may be an explanation in terms of a non random distribution of the dissolved hydrogen atoms (e.g., a phase separation into a hydride phase and a largely hydrogen free (z < 0.13) phase; Reilly et al. 1987) leaving a large part of

258

A. SCHENCKand EN. GYGAX

the sample volume in a non magnetic but also non superconducting state since no superconductivity is indicated for z > 0.8, as follows from TF-#SR-measurements (Gltickler et al. 1990a, b). Consistent with the picture of hydrogen as an electron donor, compensating holes in the CuOz-planes, is the observation that magnetic order becomes manifest already at smaller z-values in oxygen deficient compounds (see table 6.4) (Gltickler et al. 1989). 6.2.4.

YBa2(Cul_vTy)3Ox,T

= Zn, Fe

The replacement of Cu by trivalent ions, such as Fe 3+, mainly involves the Cu chain sites while divalent Zn 2+ substitutes preferentially for Cu in the CuO2-planes. Both types of substitutions reduce Tc and eventually suppress superconductivity altogether. In this respect the 'nonmagnetic' Zn is much more effective than even the magnetic substituents Co, Ni and Fe. Zn substitution does not alter the orthorhombic structure, the oxygen stoichiometry and the hole concentration while in the case of Fe substitution the hole concentration is reduced and the orthorhombic structure changes to the tetragonal one for y/> 0.04. This structural change can be avoided by employing a special thermal treatment (Katsuyama et al. 1990). The interest in studying the effects of substitutions on the Cu sites is motivated by the expectation to learn more about the possible role of magnetic interactions in the pairing mechanism. While the effects of substitutions on the superconducting properties have been studied in some detail much less is known about the magnetic features induced by the substitutions. #SR-spectroscopy was applied to gain more information in this respect. Zn-substituted compounds were studied by Garcia-Munoz et al. (1991) and Mendels et al. (1994) with respect to magnetic properties. Superconducting properties were studied by #SR by Albanese et al. (1992). Zero field #SR and neutron polarization analysis were applied in a search for localized moments in a YBa2 (Cu0.96Zn0.04)306.95-sample. No evidence for static or slowly fluctuating local atomic spins were detected. However, Tc was reduced to ,-~ 47 K. More extended ZF and TF-#SR measurements by Mendels et al. (1994) on samples with y = 0.04 and varying oxygen deficiencies allowed to draw a complete phase diagram which is shown in fig. 6.30. The interesting part of it is found in the range 6.4 < z ~< 6.67 where a disordered magnetic structure is clearly revealed by the ZF-data which extends right up to the metallic threshold but does not seem to overlap with the superconducting phase. Note the difference to pure YBazCu3Oz. Obviously Zn-doping has a destructive effect on long range magnetic order but stabilizes a random magnetic order to a higher level of hole doping. The latter is ascribed to the effect of Zn-doping on the hole dynamics which is slowed down as the missing Cu ions limit the spatial extension of the wave function of the holes. Comparing Zn-doped YBazCu307 with Zn-doped La2CuO4 it appears as if Zn-doping interferes differently with the magnetic order in the two systems. One reason may be that, although to a lesser extend, Zn replaces also Cu in the Cu-O chains. Another remarkable result of Mendels et al. (1994) is the observation of a Curie behaviour of the transverse field relaxation rate in the paramagnetic regime for 6.43 ~< z ~< 6.88. This points strongly to the presence of local moments also in the paramagnetic phase.

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,

7

Fig. 6.30. Magnetic and superconducting phase diagram of YBa2(Cu0.96Zno.04)30=. Magnetic critical temperature TN and Tf are determined from NMR- and/zSR-data. For comparison the phase diagram of pure YBa2Cu30= is also schematically shown (from Mendels et al. 1994). These moments must be created due to the presence of the Zn-substituents. Again this may be a consequence of a modified hole dynamics. Fe-substituted compounds were studied by Matsui et al. (1990), Okuma et al. (1990), Kossler et al. (1990) and Baggio-Saitovitch et al. (1990). Previously MSssbauer spectroscopy (for a list of references see Okuma et al. (1990) and Matsui et al. (1990)) and neutron time of flight measurements (Mirabeau et al. 1989) had already provided evidence for a freezing of the Fe-moments, the latter experiment suggesting a type of spin glass order in an y = 0.06 sample below 18 K in coexistence with superconductivity. In contrast MSssbauer data obtained by Baggio-Saitovitch et al. (1990) seem to indicate some non random but probably very short range order in a semiconducting sample with g = 0.05. All #SR-data, mostly taken in the zero field configuration, are consistent with a completely random order of the Fe-moments: PZF(t) did not show any oscillation in samples with y >/0.04 but was well described by a stretched exponential function (see eq. (2.22)) above the spin freezing temperature Tf and by a modified spin glass Kubo-Toyabe function below Tf (Okuma et al. 1990). The spin glass order below Tf was not completely static but showed a fluctuation rate of the Fe-moments of uFe ~-- 2 x 105 s -1 at 4 K in a tetragonal sample with y = 0.08 (Okuma et al. 1990). The ZF-#SR data allowed to determine the freezing temperature Tf with some precision. Figure 6.31 displays a phase diagram of YBa2(Cu~_uFeu)30= which includes M6ssbauer, susceptibility and /zSR data, and shows nicely the over all consistency on the spin freezing temperatures (Matsui et al. 1990) (see also table 6.4). As mentioned above, by a proper heat treatment YBa2(CUl_uFey)307 can be prepared to remain in the orthorhombic phase for y > 0.04. Such an 'ortho' sample with y = 0.08 was also investigated by Okuma et al. (1990). It was found from TF-/zSR measurement that about 1/3 (increasing to about ~ 45% at ~ 25 K and decreasing again to ~ 20% at ,,~ 4 K) of the volume

260

A. SCHENCKand EN. GYGAX t

t

t2

YBa2(CUl_y Fey)30x 0 Tc (Meissnereffect " ~ _ N(CW) x Tg susceptibility x 4 . . • ~+sR Z~ Nasuet al g 50 S U~BTamakietal O % F-100

~

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0.10

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,

0.15

Y (Fe) Fig. 6.31. Magnetic and superconducting phase diagram of YBa2(Cul_yFe#)30= (from Matsui et al. 1990). remained superconducting with Tc = 90 K and no signs of magnetic order, while the remaining volume fraction displayed random magnetic order below ~ 33 K. These observations are explained in terms of the formation of Fe-rich domains, which enter into a spin glass phase at low temperatures, and almost Fe-free domains which become superconducting below 90 K by forming a percolative network. The observed complementary temperature dependencies of the two volume fractions are not explained.

6.2.5. Lanthanide substituted (123)-compounds As mentioned in section 6.2.2 the full replacement of Y by lanthanide ions, with the exception of Ce, Pr, Tb and Pm, does not affect superconductivity in the (123)-system with Tc staying close to 90 K in fully oxydized samples. At low temperatures antiferromagnetic ordering of the lanthanide 4f-moments is observed without any adverse effect on the superconductivity, although the lanthanide moments are sandwiched by the superconducting CuO2-planes. Apparently the superconducting electrons are very narrowly confined to the CuO2-planes, as also evidenced by the extremely short coherence length along the crystallographic c-axis (~c -~ 3 A), and do not overlap with the local 4f-electrons. The absence of conduction electrons at the rare earth sites is also implied by M6ssbauer data (Alp et al. 1987, Smit et al. 1987). In view of these features it is not so clear by what mechanism the observed 3D magnetic order is driven. It is argued that dipole-dipole interactions may be responsible (Felsteiner 1989). Nevertheless, it is found that the amount of oxygen deficiency can have a pronounced effect on the magnetic structure of the 4f-moments. For example, in ErBa2Cu30= the antiferromagnetic structure changes from 2D for x < 6.5 to 3D for z > 6.5 (Maletta et al. 1990a). Also #SR-measurements show that the 4fspin dynamics is strongly affected by the amount of oxygen deficiency (see below). These results indicate that at least the coupling between the rare earth-planes is of the super exchange type involving the chain oxygen ions and then probably also

MUON SPIN ROTATION SPECTROSCOPY

261

oxygen ions in the CuO2-planes without inducing any pair breaking. The system HoBa2Cu30~ is special in that Ho 3+ possesses a non magnetic singlet ground state level and the observed magnetic order requires the formation of moments out of this singlet ground state by either overcritical exchange or nuclear hyperfine effects (Roessli et al. 1993).

GdBa2Cu30:~. Zero field/zSR measurements on two GdBa2Cu307_a samples with Tc = 60 and 90 K revealed two spontaneous spin precession signals below TN = 2.3 K, confirming the onset of antiferromagnetic order of the Gd-4f moments (Golnik et al. 1987). The temperature dependence of these frequency is shown in fig. 6.32. The higher frequency in only marginally visible in the 90 K-sample, but is well developed in the 60 K-sample. Both frequencies can be consistently accounted for by considering the antiferromagnetic structure of the Gd-sublattice and assigning the #+ to the apical oxygen site (high frequency component) and to the chain oxygen site (low frequency component). Very interesting is the observation of/z+-relaxation (see fig. 6.33) above TN in the 60 K-sample as well as in a z = 6.2 sample, indicating some slowing down of the Gd-4f spin dynamics but not in the 90 K sample (Budnick et al. 1990a, Niedermayer et al. 1993). This implies that the spin dynamics in the 90 K sample is much faster than in the 60 K-sample. This phenomenon is particularly intriguing because the N6el temperature seems to be rather independent of the oxygen deficiency. It will be interesting to study these features in more detail. A possible interaction between the Cu(II)-sublattice order and the Gd-sublattice order was investigated by ZF-fSR in a GdBa2Cu306.3 sample (Niedermayer et al.

Gd BCl2 Cu 30T_v 7 6

C"

~3

0

1 2 3 Temperature (K)

Fig. 6.32. Temperature dependence of the two spontaneous precession frequencies in GdBa2Cu3OT_~. The circles are from a Tc = 60 K sample and the squares from a Tc = 90 K sample. Lines are guides for the eye (from Golnik et al. 1987).

262

A. SCHENCK and EN. GYGAX

15 I

7-

t/\,

G Bo :u3O ,

•

_ 10 0

CI2 .....2

o0_5

10

100

Temperature (K) Fig. 6.33. Temperature dependence of the ZF-relaxation rate in three different GdBa2Cu306_ = samples

above TN of the Gd-sublattice (z = I, Te -~ 90 K; z = 0.7, Tc = 60 K; ~c= 0.2, not superconducting) (from Budnick et al. 1990a).

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--~ 4 o 3 u. 2

• GdBa2Cu306+x(tetragonal) [] YBa2Cu306+x(tetragonal)

............ "~r'[]-----~D--_r-~

0

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i

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l

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~ ~ T N (Gd)

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oloo T N (Cu(II))

Temperature (K) Fig. 6.34. Temperature dependence of the single spontaneous /z+ precession frequency observed in . For comparison a few data points from a YBa2Cu306+y-sarnple are also shown (from Niedermayer et al. 1993).

GdBa2Cu306.3

1993). A single spin precession signal was found below ~ 300 K, the temperature dependence of the frequency u~, is displayed in fig. 6.34. Analysis of these data show that below the Gd-sublattice ordering temperature T~ d = 2.3 K the measured frequency can be understood by simply considering the vector sum of dipolar fields originating from the ordered Cu(II)-sublattice and the ordered Gd-sublattice, employing for the magnetic structures the information gained from neutron scattering work

MUON SPIN ROTATIONSPECTROSCOPY

263

(McK. Paul et al. 1988, Mook et al. 1988, Rossat-Mignod et al. 1988a). Hence it seems that below TN Ga the two magnetic sublattices are not affected by the presence of the other one. At high temperatures, i.e. above --, 150 K, u,(T) follows closely the behaviour seen in YBa2Cu306.3 (see fig. 6.34) indicating that also in this temperature range the magnetic structure of the Cu(II)-sublattice is not modified due the presence of Gd-moments. Quite a different behaviour is observed in the intermediate range TN Gd < T < 150 K. Clearly the Cu(II)-magnetic sublattice is now drastically affected by the Gd-4f moments. Whether this is a result of the development of static correlations among the 4f-spins, induced by and perhaps also modifying the Cu(II)-magnetic sublattice, or of the slowed down Gd-4f-spin dynamics in oxygen deficient material, destabilizing to a certain degree also the Cu(II)-moment ordering, is not known. It should be noted that again TN Gd and TCu(II) are not affected to any noticeable degree.

DyBa2Cu307_6. Attempts to measure a Dy-4f moment induced #+-Knight shift above Tc produced a zero result within the achieved precision (Schenck et al. 1990a). No studies at low temperatures have been performed so far.

HoBa2Cu3Ox. This system has been repeatedly investigated by #SR-spectroscopy. Generally it was observed in ZF-#SR measurements that the /z+ relaxation rate increases drastically below 10 K and saturates below ~ 2 K (Nishida et al. 1988a, Kuno et al. 1988, Birrer et al. 1989a, Grebinnik et al. 1990c). The low temperature relaxation rate in samples with z ~ 7, if interpreted as dephasing, would correspond to an internal field spread of (5-13) mT, pointing to a random freezing of the Ho4f moments already around 2 K, while specific heat and neutron diffraction data indicate a transition to long range antiferromagnetic order around (140-190) mK (Dunlap et al. 1987, Fischer et al. 1988, Roessli et al. 1993). More detailed ZF/~SR measurements by Birrer et al. (1989a), extending down to 40 mK, revealed also some perhaps restricted, long range order starting already below 5 K in a small fraction Of the polycrystalline sample volume. Their data can be analyzed in terms of a two component structure of the ZF-#SR-signal, which develops below ,-~ 50 K. In the range 50 K to 5 K random order appears in a fraction of the sample volume growing from ~ 0% at 50 K to ~ 30% at 5 K, the remaining volume staying paramagnetic. Below 5 K the random order transforms to some coherent, but short range order, as evidenced by the emergence of an oscillating but heavily damped signal. The temperature dependence of the corresponding frequency u, is displayed in fig. 6.35. In parallel, the ordered volume fraction grows at the expense of the paramagnetic fraction and reaches essentially 100% around 2 K. The oscillating signal shows a pronounced and rather temperature independent Gaussian relaxation, implying a static field spread of 8.2 roT, which explains why the oscillating behaviour has been overlooked in the earlier work. A clear oscillation is only directly evident at the lowest temperatures (see insert in fig. 6.35). As fig. 6.35 shows ut, changes rather abruptly at ~ 100 inK. This change was attributed to a spin reorientation transition from fiHo being parallel to the e-axis below 100 mK (also predicted theoretically by Misra et al. 1992) to being parallel to the e-axis above 100 inK. The ordered

264

A. SCHENCK and EN. G Y G A X

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I

I I @11111 101

I

I I I

III

10 2

Temperature (K) Fig. 6.35. Temperature dependence of the spontaneous # + precession frequency in HoBa2Cu307_~. The insert in the upper right corner shows the same on a linear temperature scale. The insert in the lower left corner shows the ZF-/~SR signal at 39 mK (from Birrer et al. 1989a).

moment was estimated to be #Ho ~- 2"6#B" The most remarkable feature of the #SRresults is the absence of a clear cut cooperative phase transition which is inconsistent with both the specific heat and neutron results. In view of this one is tempted to associate the anomaly in uu at 100 mK with the true phase transition to long range antiferromagnetic order (implying that the temperature calibration was in error) and to interpret the results above "100 inK" as an extremely short range precursor phenomenon. More recent neutron diffraction measurements revealed a clear onset of long range antiferromagnetic order at (190 4- 10) mK with a propagation vector q = (0, 1/2, 1/2) and #Ho = 2.8#B aligned parallel to the c-axis (Roessli et al. 1993). The long range order is limited to a correlation length of ~ 30 A along the c-axis due to the occurence of stacking faults along this direction. It was further established that the nuclear hyperfine interaction is the driving force in the long range magnetic order of the Ho3+-moments which evolves out of a singlet level of the Ho3+-518 ground state multiplet. The magnetic structure proposed by Roessli et al. (1993) is in marked disagreement with the #SR-results. On the basis of this structure one calculates a net dipolar field of 0.64 mT at the #+-site while the #SRmeasurements reveal a field of 18.5 roT. It is impossible to explain the difference in terms of an additional contact hyperfine field in view of the zero Knight shift result in the paramagnetic phase (see below). The unique and indisputable information provided by the #SR results is the random freezing of the Ho-moments already far above the Nrel temperature. With respect to this several interestingquestions can be asked: what causes frustration; which mechanism is responsible for the coupling of

MUON SPIN ROTATIONSPECTROSCOPY

265

the moments (dipole-dipole, super exchange); which mechanism is responsible for the formation of a local moment out of the singlet ground state level also at higher temperatures (nuclear hyperfine, overcritical exchange, population of higher excited levels); etc? Answers to these questions are not yet available. Also, the oxygen content seems again to be of importance since the freezing of the Ho-moments is less visible in a HoBaeCu306.2 sarnple (Kuno et al. 1988). Dilution of Ho by Y has also a negative influence on the freezing process (Grebinnik et al. 1990c). TF-#SR measurements on a HoBa2Cu307_~ (To = 90 K)-sample produced no visible Ho-induced #+-Knight shift (Weber 1991). Earlier claims (Schenck et al. 1990a,c, Maletta et al. 1990b) to have seen a very sizable Knight shift in HoBa2Cu307 are incorrect and resulted from neglecting effects due to powder broadening.

ErBa2Cu30~. ZF-#SR investigations in a series of ErBa2Cu3Ou-compounds with different z (see table 6.5) yielded in all cases a spontaneous spin precession signal below the respective Ntel temperatures (Maletta et al. 1990c). Of course, for z /> 6.6, only the Er-sublattice enters into a (3D) antiferromagnetic phase below TN = 0.6 K (Maletta et al. 1990a). The corresponding #SR-signal displayed a very small single frequency v~, the temperature dependence of which is shown in fig. 6.36. Interestingly the precession pattern appeared to persist up to temperatures of at least 1.2 K. At the transition temperature TN a significant increase of v~ is observed, which is more pronounced in the z = 6.6 sample. The shift of TN and of v~ (~ 0 K) with the oxygen content z is well in line with neutron diffraction experiments on the same samples (Maletta et al. 1990a). In fact using the magnetic structure and #Zr = 4"4#B (1[b'axis) derived from the neutron work one predicts a field of ~ 0.6 mT at the/z + position near a chain oxygen in excellent agreement with the experiment. The persistence of a precession signal above TN is also well in line with neutron results which reveal short range magnetic correlations up to ,,o 1 K. Unfortunately the #SR-measurements did not extend beyond ,-~ 1.2 K. (But see also Grebinnik et al. 1990b, c.) In the whole covered temperature range the damping of the precession signal is very small (and temperature independent below TN) (compare with the case in HoBa2Cu3Ox !) but this has not to be taken as evidence for a very perfect long range magnetic structure. It merely reflects the fact that the net dipolar field at the #+ site is close to zero (this is a consequence of fiZr being aligned along the b-axis). The difference in the relaxation behaviour (see fig. 6.36) between the z = 7.0 and z -- 6.6 sample below 1 K is puzzling and not explained yet. For samples with x ~< 6.4 the onset of magnetic order of the Cu(II) sublattice is seen again by the appearance of spontaneous spin precession in the ZF-#SR signal. No such signal was seen in a compound with z = 6.53 down to 3 K. The temperature dependence of the single frequency v~,, seen in compounds with z -- 6.11, 6.34 and 6.40, is displayed in fig. 6.37. In particular the data from the x -- 6.34 sample indicate that, if it were not for the presence of the Er-moments, v~ would approach ~ 4 MHz as in YBa2Cu306. A sample with z = 6.20 was also investigated below 1 K. No evidence for magnetic order of the Er moments was found. The anomalous increase of u~, below 10 K, particularly in the z = 6.34 sample, is somewhat reminiscent of a similar behaviour in GdBa2Cu306+~

266

A. S C H E N C K and EN. G Y G A X

0

,--Z

0 ~,

0

i"

t",l o

+ o

=

•

o~

o

I~

8

~ = ~ o 0

~ o

~

_

~

~

=

~'~

"~

~

~ ~

~

~'~

~

~

~ o . ~

0

=~

0

,~

~,~,~

0

o I

v~

I

I

0

I

0c q

9 o

0 0

O~

II c~

cS

8 0 ~.~ 0

t'N o~

m~

0

7

irJ "0

r~

oo

C R

0

0

MUON SPIN ROTATION SPECTROSCOPY

267

v~vvv~

0

,.,.,

~o

~,~ ~ ~

o

~ o o

<~ on

~

V

on

A

oc

C o

..... ~

o

r',!,,.q.

o,..o

268

A. SCHENCK and F.N. GYGAX

ErBa2Cu30 x 0.09 o,.,o

-1-

0.06 g~a 0.03 ¢._ U_

0.00

"7

I

I

I

I

I

I

I

I

I

I

I

I

I

!

I

::L

0.30 1:3 I._.

0.20 .$... X

0.10 C~

0.00

I

I

I

I

f

f

I

|

[

0.5 1.0 Temperature (K)

Fig. 6.36. Temperature dependence of the spontaneous /z+ precession frequencies and the associated relaxation rates in two superconducting ErBa2Cu30= samples (z = 6.6, 7.0) (from Maletta et al. 1990c).

(Niedermayer et al. 1993) and reflects clearly some effect of the Er-moments on the Cu(II)-magnetic ordering. Neutron diffraction measurements on the same samples showed the onset of magnetic correlations among the Er-moments in the a-b plane below 1 K which essentially saturate below 230 mK without the formation of a 3D long range antiferromagnetic structure (Maletta et al. 1990a). Since the Er-moments do not display a 3D-antiferromagnetic order the increase of uu below ,-~ 10 K cannot be understood in the same way as in GdBa2Cu306+6. It rather seems that the Cu(II) magnetic sublattice is somehow modified in an as yet undetermined fashion. It has to be emphasized that the Cu(II)-long range magnetic order is again not seen in the full sample volume. Particularly in the z = 6.4 sample quite a fraction of it is associated with a very rapidly but nonoscillating signal. This fraction decreases with increasing temperature and disappears at around 300 K. It seems to reflect spin glass behaviour in part of the sample. Whether this behaviour reflects a nonhomogeneous oxygen distribution or is induced by the Er-moments remains to be seen. It is not known whether this sample could become superconducting. ZF-#SR measurements on a ErBa2Cu306.2 sample by Lichti et al. (1990b) revealed two frequencies in the spin precession signal appearing below ,-~ 300 K. In addition

MUON SPIN ROTATION SPECTROSCOPY

269

o 6.11

_ ~.#...... o ~ _ _ e"..........-,,, ~ ~

,.~

-r "

e'O-~

"

3

~

•

-

• 6.34 ,', 6.40

"'......~ _

~

°.

2

i I 0

I

I

I

I

I

I1[

t

I

I

101

I

I

I

I I

I

10 2

T e m p e r a t u r e (K) Fig. 6.37. Temperature dependence of the spontaneous/z + precession frequencies in ErBa2Cu30= with x = 6.11 (tetragonal), 6.34 (orthorhombic) and 6.40, The latter sample shows a weak superconducting response below 32 K (from Maletta et al. 1990c).

to the familiar 0.036 T-component a second frequency corresponding to a field of 0.055 T was seen. The latter may be associated with a site near the CuOz-plane and could correspond to the 0.13 T signal seen in YBazCu306+~ (5 < 0.3) by Brewer et al. (1989) (see table 6.4). The same authors performed also TF-#SR measurements. From an anomaly in the W-relaxation rate, seen at 65 K, they conclude that a second magnetic phase transition may be present (Lichti et al. 1990b). A search for a Er induced #+ Knight shift in samples with z = 6.53 and z = 7 (T > To) produced again a zero result (Schenck et al. 1990a). Earlier reports to the contrary (Schenck et al. 1990c, Maletta et al. 1990b) are again unsubstantiated. Knight shift results below Tc are discussed by Lichti et al. (1991a).

6.2.6. Other compounds Long range antiferromagnetic order was detected for the first time by ZF-#SR in the compounds BaCuO2 and BaYzCuO5 (green phase of YBaCuO). From these data the transition temperatures could be determined (see table 6.3) (Weidinger et al. 1988).

6.3. Bi-based (2212)-compounds For an overview of investigated compounds (all have an orthorhombic crystal structure) see table 6.6. In fact magnetic order in insulating or semiconducting members of the (2212)-family7 was first demonstrated in #SR experiments almost simultane7 Substituting trivalent y3+ for divalent Ca2+ or Sr2+ the carrier (hole) concentration is reduced (Tamegai et al. 1988). BizSr2YCu208 and BizSrCaYCu208 are insulators. Holes are primarily introduced by some oxygen excess 5 > 0.

270

A. S C H E N C K

and EN. GYGAX

oo

~o

oo

"G ©

0 :t

r~

~o P~

p~

p~

O

r~

+

0

~

~

"~

0

r~

©

:~

d d d d

O

(-q

c5

c5

e

t'N

~ A

N

N

A

~ A A~

b..,

[..-,

t"q

t'q t'q

.1

© )

r..) e.i

eq

t..)

~1

M U O N SPIN ROTATION SPECTROSCOPY

271

0

O

© o ;~ ee3 o

~

~

rm o o

~n

O

~.

0

¢xl e q

¢¢3

II

II

O o

A~ [.r.. N

r.~ N

u

O

[q

© o

0 0

272

A. SCHENCK and EN. GYGAX

ously by Yang et al. (1989), Nishida et al. (1988c) and De Renzi et al. (1989). ZF-measurements at ~ 5 K in compounds Bi2Sr2YI_~Ca~Cu20~8 with z ~< 0.2 revealed three oscillating components (see table 6.6) in addition to a nonoscillating one. The temperature dependencies of the three frequencies or local fields in a Bi2Sr2YCu20~8 sample are displayed in fig. 6.38 together with the relaxation rate of the low field (0.4 MHz) component (Nishida et al. 1990b). As can be seen, the 0.027-T-component disappears at 80 K. At the same temperature ~r drops abruptly by a factor of ,.o 3. These features are not understood at present. In Bi2Sr2YCu20~8 a fourth high frequency component was seen with, however, an extremely fast relaxation rate (,.o 10 7 s - l ) . The appearance of several components may not be a surprise in these more complicated compounds and points to the existence of several #+-sites. Possible #+-sites have not yet been considered in detail but the pair of signals centered at 0.03 T may indicate sites near an apical oxygen similar as in YBa2Cu306. The high frequency signal may correspond to the 0.12 T signal in YBa2Cu306+6, i.e. a site close to a CuO2-plane may be occupied. The low frequency signal may be associated with a Bi2Sr2YCu20Y i i

0.4 bo 0.3 -

~mo

• •

:b o (Pa

0.2 o.1 Im-,nlliIRm-, mm

m mmm

0

l

t

m[

1.0

j

i

i b

0.8 ~m

0.6 F WI i 0.4

¢,... .'¢ 0 0

,

I

I

I

1 O0

200

300

400

T (K) Fig. 6.38. Temperature dependence of (a) the 3 spontaneous local fields at the /~+ observed in BizSr2YCu20~ 8. The 0.027 T local field disappears above ~ 80 K. Below the same temperature the relaxation rate (b) associated with the lowest local field signal increases abruptly (from Nishida et al. 1990b).

MUON SPIN ROTATION SPECTROSCOPY

273

site near an oxygen in the Bi-O plane, which has no counterpart in the (123)-system. In any case the close correspondence of some of the frequencies with observed frequencies in the (123)- and the LaCuO-systems indicates that the Cu-moment order must be quite similar in all the cuprate layered compounds. In particular the magnitudes of the Cu-moments must be quite comparable. The ZF-#SR data obtained in Bi2Sr2Y0.6Cao.4Cu208, Bi2Sr2.5Y0.5Cu20~8 and Bi2SrCaYCu2Os+6 also show the presence of magnetic order but it is now of a random (spin glass like) structure. This phase is followed, when further decreasing the Y-content, by the appearance of superconductivity (see fig. 6.39). Like in the (123)- and the LaCuO-families the development of magnetic order starts first in only a tiny fraction of the sample volume and even at very low temperatures may not involve all of the sample volume. Figure 6.40 presents an example for how the paramagnetic fraction of the sample volume decreases with decreasing temperature in compounds BizSr2Yt_xCa~Cu208 with z = 0, 0.1, 0.2 and 0.3. These data are obtained from TF-#SR-measurements which allow to identify the 400

I

Bi2 Sr2Y1 -x CaxCu2Oy

3O0

b-

+

2OO

100 [] AF

FIND

SG

SC D

0 0.0

i

I

I

I

I

I

I

0.5

I

I

1.0

X

Fig. 6.39. Magnetic and superconducting phase diagram of Bi2Sr2_xY~Cu208. The magnetic part is deduced from/~SR-data showing AF-order for x<0.3 and spin glass like order for 0.5 > z > 0.3. The arrow for z = 0 indicates that TN is above the instrumental T limit (from Nishida et al. 1990b).

274

A. SCHENCK and EN. GYGAX

10 gO.8

."-~ _u

~-0.6

~

Bi2 Sr2Yl_xCax Cu20, i i ~'x:o.3

~ ,~

0.4

x:o-2 I

-

~X:0.1

f •

o

-

a_ 0.2 0(3 0

t

I

100

200 T (K)

300

400

Fig. 6.40. Fraction of paramagnetic volume in various Bi2Sr2Yl_~CaxCu20~t samples as a function of temperature. The fraction is determined from TF-/~SR measurements (from Nishida et al. 1990b).

fraction of #+ in a paramagnetic environment. It can be seen that, e.g., in a sample with z = 0.2 magnetic order sets in at around 300 K but at ~ 250 K more than half of the volume is still paramagnetic and this fraction does not drop below 30% at the lowest temperatures. Even for a more or less stoichiometric sample with z -- 0 magnetic order develops in a spatially inhomogeneous fashion. The latter result, assuming that the oxygen content is close to 8 per unit formula, could suggest that the inhomogeneous magnetic ordering is an intrinsic property of these layered systems. #SR-spectroscopy should be in a position to explore this property in a more systematic way. Interpreting the onset temperatures of magnetic order as N6el or freezing temperatures a phase diagram can be drawn as shown in fig. 6.39 (Nishida et al. 1990b). By comparison with fig. 6.22 it is seen that all layered cuprate families behave in the same way. In the insulating or semiconducting phase the CuO2-1ayers show antiferromagnetic order. With the introduction of holes into the CuOz-planes magnetic order becomes frustrated and eventually a spin glass phase is entered. On further increasing the hole concentration magnetism is lost altogether and a superconducting phase evolves. Whether a small region exists where spin glass order and superconductivity coexist in a truly microscopic manner is still a disputed matter.

7. Study of magnetic order in organic compounds All compounds studied are listed in table 7.1.

7.1. (TMTSF)2X The family of the tetramethyl-tetraselena-fulvalene (TMTSF)2-X compounds displays many intriguing properties such as superconductivity, anion ordering, quantum Hall effect, spin density wave (SDW) ordering and field induced spin density waves

M U O N SPIN ROTATION SPECTROSCOPY

eee

~

e

+ d

¢q ¢q

¢q

,,~

~

a

a

c~

c~

~c;~ z r..)

275

276

A. SCHENCK and EN. GYGAX

owing to their highly anisotropic electronic structure (low dimensionality) (J6rome and Schulz 1982, Ishiguro and Yamaji 1990). The spin density wave properties have been studied by a variety of techniques but many detailed features, e.g., elementary excitations, sublattice magnetization, SDW wave vector and the order of the transition are not too well determined. The usefulness of #SR-spectroscopy in addressing some of these problems rests again in the possibility to perform the experiments in zero applied field.

(TMTSF)zPF6. Antiferromagnetic SDW order is observed in this compound at ambient pressure below TN = 122 K. The anion PF6 is located in a centrosymmetric position. ZF and LF-#SR measurements were aimed at a determination of the exact SDW structure (commensurate versus incommensurate) and the order of the phase transition (Le et al. 1991, Le et al. 1994). Previous NMR measurements (Takahashi 0.14 ~N~" ~ 0.14 ~ b 0.14: ~ 0.12

~ .

%

(TMTSF)2PF 6 H =0 ~,xt "~ " ~ ' ~

I

12.25K

% E 0

(a)

'

,? ?' '

oo,

0.00

0

I

I

I

2

4

6

Time (/zs)

(b) 13_

2.0

E .<

1.0 n-0.0

I

0.0

I

0.2 0.4 0.6 Frequency (MHz)

/~

0.8

Fig. 7.1. (a) ZF-/~SR signals in (TMTSF)2PF 6 for various temperatures (b) Fourier transform of the signal taken at 3.7 K (from Le et al. 1994).

MUON SPIN ROTATION SPECTROSCOPY

277

et al. 1986, Delrieu et al. 1986a, b) had detected the SDW-state, but the applied field was rather high and the system was in the spin-flop state. Figure 7.1a displays some typical ZF-#SR spectra around the transition temperature. As can be seen below 12.2 K a precessional pattern develops but the actual frequency spectrum, as revealed by Fourier transforming the signal, is quite broad, showing a shoulder on the low frequency side and a peak at a frequency of ~ 0.55 MHz, corresponding to an internal field of 0.004 T (see fig. 7.1b). This distribution appears incompatible with a commensurate structure (provided only one type of site is occupied by the #+) but can be well reproduced by a distribution of dipolar fields arising from an incommensurate sinusoidally modulated intermolecular spin density wave with wave vector ~) = (0.5, 0.24, -0.06) and maximum amplitude of 0.1#B/molecule (Takahashi et al. 1986). The shape of the frequency spectrum does not depend very much on the assumed #+ site and also not on the exact value of Q as long as the modulation is incommensurate. In this picture the peak frequency uu,p (see fig. 7.1b) provides a good measure of the SDW amplitude M or the order parameter. The temperature dependence of Uu,p is displayed in fig. 7.2 together with the corresponding results on the two other (TMTSF)2-X compounds with X = NO3 and X = C104. The data show little temperature dependence below TN/2 and quite an abrupt change at TN. The latter is taken as evidence for a first order phase transition. It may be ascribed to a cross over from a 1D to a 3D behaviour whereby the already existing, albeit fluctuating order parameter in the 1D-regime is frozen out at a finite value at the transition temperature. The temperature dependence of Uu,p below ~ TN/2 is, although weak, much stronger than one would expect from a mean field treatment of the SDW transition 0.7

(TMTSF)2-X 0.6

"T-

Hext=O

0.5

',

0.4 >, 0 c

~"4 p;6"~

NO 3

Ci04

0.3

4

"

E)a) 0.2 t, k_

0.1 o.o

0

t ,

5

I

I0

15

Temperature (K) Fig. 7.2. Temperature dependencies of the spontaneous iz+ precession frequencies in (TMTSR)2-X with X = C104, NO3, PF6. The solid lines are guides to the eye (from Le et al. 1994).

278

A. SCHENCKand EN. GYGAX 1.02

I

i

i

i

1.00 0o0 0.98 0

0.96

I.-- 0.94 0.92 0.90

Heisenberg (NN)

0.88

1

0

2

I

I

4 6 Temperature

I

8

10

(K)

Fig. 7.3. Temperature dependence of uj,(T)/u~(O) = M(T)/M(O) in (TMTSF)2PF6 below TN/2 and various model predictions. The data are very well described by a 2D-spin wave excitationmodel (from Le et al. 1994). (Yamaji 1982). The opening up of an energy gap A should largely suppress single particle excitations and consequently the magnetization or Uu,p should be largely temperature independent below ,,~ TN/2. It is therefore argued (Le et al. 1994) that the variation of U~,p or M at low temperature is determined by collective spin wave excitations. Indeed the temperature dependence of tJ~,p(T)/u#,p(0) = M(T)/M(O) can be very well accounted for by such an approach (solid line in fig. 7.3) leading to an average spin wave stiffness constant of 13 = 200 K. This value is much larger (by a factor 3-5) than what is expected from a local moment nearest neighbour Heisenberg model. However, the large value can be understood, at least qualitatively, on the basis of an itinerant electron model (Le et al. 1994).

(TMTSF)2N03, (TMTSF)2CI04. In these compounds the anions NO3 or C104 are located at non-centrosymmetric positions with the consequence that an anion ordering transition is found at temperature TAo (= 41 K in the NO3-compound). The ground state of these systems depends on the degree of anion order-disorder produced by the cooling rate through TAO. In the NO3-compound a SDW state below 12 K can be obtained by slow cooling, while in the C104-compound a SDW state is achieved below ~ 4 K by very rapid cooling (quenching). ZF-#SR measurements revealed spectra very similar to the ones found in the PF6compound indicating an incommensurate spin density wave state in each case (Le et al. 1994). The temperature dependence of the peak frequency Uu,p for both compounds is included in fig. 7.2. As can be seen the low temperature u~,v (or the spin density wave amplitude M) is essentially the same in all three compounds despite the fact

MUON SPIN ROTATION SPECTROSCOPY

279

that rather different transition temperatures are observed. This behaviour is taken as support for a nesting model for the SDW condensation (Takahashi et al. 1989). According to this model the SDW gap ,4 and the order parameter M are mainly determined by the on site Coulomb energy U and the on-chain coupling t, which are independent of the anion X. The decrease of the transition temperature, on the other hand is a result of the loss of perfect nesting as the dimensionality of the systems increases (i.e. becoming more 2D), e.g., as a function of pressure or type of anion, but this has no effect on M as long as the SDW ground state is established.

7.2. Ni2(C2H8N2)2N02(CI04) (NENP) NENP is a nearly ideal realization of a spin-1 linear-chain Heisenberg antiferromagnet. For such a system Haldane (1983) has predicted the existence of a quantum energy gap between the nonmagnetic singlet ground state and the lowest triplet excited state. No such gap occurs for half integer spin systems. Admitting also some inter chain coupling above a certain critical value long range magnetic order may be expected (Affleck 1989). However, the inter chain coupling in the NENP system appears too weak to allow for the development of long range magnetic order. In fact magnetization measurements place an upper limit of 1.2 K on the transition temperature to an ordered magnetic phase (Renard et al. 1987). Nevertheless the low temperature properties of NENP remain an important issue. ZF- and LF-#SR measurements on a polycrystalline specimen reveal indeed interesting new information which indicates that NENP could be very close to static magnetic order (Sternlieb et al. 1992). Applying a small (0.01 T) longitudinal field in order to decouple the #+ spin from nuclear dipolar fields the LF-relaxation follows well a stretched exponential (exp(-iv/~-/at)) which is indicative for a broad fluctuating local field distribution. Figure 7.4 displays the temperature dependence of ),1. As can be seen ),1 rises steeply between 3 K and 2 K followed by a saturation down to 20 mK. The fast rise of ),1 below 3 K points to a slowing down of the spin dynamics. However, no static limit is reached. This follows from a persisting field 1.4 ,~

1,2

I

p

o

I

I

I

I

o 0

~7 1.o

I

NENP LF=IOOG

_

o.8

~O

0

0.6 0.4

rr

O

0.2 0.0 0

O

O

I

~

t

I

I

2

3

4

5

6

7

Temperature (K) Fig. 7.4. Temperature dependence of A1 (measured in an external field of 0.01 T) in NENR Note the rapid rise below 3 K (from Sternlieb et al. 1992).

280

A. SCHENCK and EN. GYGAX

dependence, even at 20 mK, which is roughly described by ALF O( 1/co~ and implies very small fluctuation rates of the order of 40 MHz at 20 mK. The local fields are of the order of Bloc ~ 90 G. The latter value is too large to be caused by loose chain ends or impurities but, on the other hand, too small to be representative of all Ni-moments. Therefore, if A1 and Bloc are reflecting a truly intrinsic behaviour, it must be concluded that only a limited portion of the Ni-spin fluctuation-spectrum is affected by a slowing down of the spin dynamics (Sternlieb et al. 1992).

7.3. p-NPNN p-NPNN (p-nitrophenyl nitronyl nitroxide, C13H16N304) is a chemical radical which has been identified to show ferromagnetic order (Awaga et al. 1989, Takahashi et al. 1991). Although various techniques have been used in the study of this system the most fundamental parameter, namely the spontaneous magnetization has remained uninvestigated, but has later been measured in a ZF-#SR experiment on t-phase monocrystalline material (Le et al. 1993). The onset of magnetic order below 0.7 K is clearly reflected in the appearance of a long lived spontaneous coherent spin precession signal with only one frequency uu implying a unique #+ site in this complex system. The temperature dependence of uu, which is proportional to the spontaneous magnetization M is displayed in fig. 7.5. The Curie temperature following from this plot is 0.67 K and indicates by comparison with other samples a slight sample dependence. The magnetization has been previously measured at 0.44 K with the result M = 0.5#B/molecule (Tamura et al. 1991). Rescaling this value with the present u(T) one gets M(0 K)= 0.6#B/molecule. Expected is M = l#B/molecule. The difference is ascribed to systenaatic uncertainties in the magnetization measurements. The #SR data indicate that the internal spontaneous field is nearly parallel to the crystalline b-axis. 2.5

I

I

I

I

I

I

I

H,~=0 "1-

2.0-

> ",

4.5

a~

1.0

u_

0.5

0.0

I

0

i

w

I

J

I

T i

100 200 300 400 500 600 700 800 Temperature (mK)

Fig. 7.5. Temperature dependence of the spontaneous /~+ precession frequency in p-NPNN. The solid line represents a fit of eq. (7.1) to the data (from Le et al. 1993).

MUON SPIN ROTATION SPECTROSCOPY

281

The temperature dependence of uu c( M is very well fitted (solid line in fig. 7.5) by the expression M ( T ) oc (1 - (T/Tc)'~) ~

(7.1)

with c~ = 1.86 and/3 = 0.32. The low temperature behaviour can be expressed as (M(0) - M ( T ) ) c< T '~ which is close to the magnon induced temperature dependence in a 3D system with ~ = 1.5. Near Tc one has M ( T ) ~x (Te - T ) ~ which, with /3 = 0.32, is in excellent agreement with the value /3 = 1/3 expected for a 3D Heisenberg ferromagnet. The behaviour of/3-phase p-NPNN is thus consistent with that of a 3D Heisenberg system. This is in contrast to susceptibility and magnetization measurements in the ?-phase of p-NPNN which could be well explained in terms of a quasi 1D-ferromagnet (Takahashi et al. 1991). Acknowledgements We are indebted to the many colleagues who have sent us re- and preprints, who responded quickly to questions and even provided us with high quality figures. Many thanks go to Dr. Alex Amato who read carefully the manuscript and was patient enough to provide advice and help whenever needed. Finally we have to thank Mrs. M. Sekolec and Mrs. R. Bachli for preparing skilfully the typed manuscript and Mrs. I. Kusar of PSI for drawing or redrawing a major fraction of the figures. List of some of the used abbreviations and symbols ABBREVIATIONS (#)LCR #SR AF anneal bcc CAF cor

fcc FI FM hcp hex ins LCR LF-pSR LIAF met mon

(muon) level crossing resonance muon spin rotation, relaxation and resonance antiferromagnetic annealing body centered cubic commensurate spin density wave corundum face centered cubic ferrimagnetic ferromagnetic hexagonal close-packed hexagonal insulator level crossing resonance longitudinal field #SR longitudinal incommensurate spin density wave metal monoclinic

282

A. SCHENCK and EN. GYGAX

Mu

n-irrad O, oct OPC orth P PC per p-irrad rut SC, s.c. scu

SDW sem

SF SG T T± TF-#SR TIAF USC WA WF ZF-#SR

muonium (#+e-) neutron irradiated octahedral oriented polycrystals orthorhombic paramagnetic polycrystalline, polycrystal perovskite proton irradiated rutile single crystal simple cubic spin density wave semimetal spin fluctuations spin glass tetrahedral tetrahedral with tetragonal axis perpendicular to [100] direction transverse field #SR transverse incommensurate spin density wave state unoriented single crystals weak antiferromagnetic weak ferromagnetic zero field #SR SYMBOLS

F,A /~ % z~ 2 AB u 0

~1 = 1/TI /~2 = 1/T2 AZF /./ uu - u x u u (= w u / 2 7 r ) o

~-c

(= 1/.)

muon depolarization or dephasing rate, general #+ depolarization - several components muon gyromagnetic ratio (see table 2.1) second moment of one cartesian field component in ZF rms width of local field distribution strain orientation or angle between field and c-axis spin lattice relaxation rate spin lattice relaxation rate in ZF transverse relaxation rate ZF-relaxation rate from exponential decay magnetic moment fluctuation rate level crossing resonance frequency muon - nucleon X muon Larmor or precession frequency Gaussian depolarization rate (cr2 = M2) correlation time orientation or angle between field projection on the basal a-b plane and the a-axis

MUON SPIN ROTATIONSPECTROSCOPY

283

X Xat A Ac

susceptibility tensor (usually in emu/mol) atomic susceptibility tensor (emu/atom) amplitude or asymmetry of #SR signal contact hyperfine coupling constant

Adip Ai AB u B~

dipole coupling tensor amplitude or asymmetry of i-th component of #SR signal rms width of local field distribution total local magnetic field at the muon field at the muon - several components or sites (contact) hyperfine field at the muon demagnetization field dipolar field applied magnetic flux density Lorentz field field at the #+ in the Mu-state Kubo-Toyabe relaxation function for Gaussian field distribution Kubo-Toyabe relaxation function for Lorentzian field distribution depolarization function in transverse field applied magnetic field muon Knight shift second moment of field distribution (usually in MHz2), related to AB~

Bc Bdem Bdip Bext BL BMu

a~T

% GTF Hext

K. M2 4-4

N p, press

P(t) PLF(t) PTF(t)

PZF(t) T

T1 T2 Tc TFN TM TN Tv G Tf

demagnetization tensor (factor) pressure time dependence of muon spin polarization, projected on P(t = 0) P(t) measured in longitudinal field (/~ext]lfi(0)) P(t) measured in transverse field (/lext±/3(0)) P(t) measured in zero field temperature longitudinal spin-lattice relaxation time transverse (spin-spin) relaxation time Curie temperature ferrimagnetic ordering temperature Morin temperature N6el temperature Verwey temperature superconducting transition temperature freezing temperature in a spin glass-like system

284

A. SCHENCK and EN. GYGAX

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chapter 3 INTERSTITIALLY MODIFIED INTERMETALLICS OF RARE EARTH AND 3D ELEMENTS

HIRONOBU FUJII Faculty of Integrated Arts and Sciences Hiroshima University Higashi-Hiroshima 739 Japan

and

HONG SUN Materials Science Research Department Research and Development Center Sumitomo Metal Industries, Amagasaki 660 Japan

Handbook of Magnetic Materials, Vol. 9 Edited by K. H.J. Buschow 01995 Elsevier Science B.V. All rights reserved 303

CONTENTS 1. Introduction

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2. Formation of the interstitially modified intermetallic compounds . . . . . . . . . . . . . . . . . . . . . . . 2.1. Arc-melting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Melt-spinning method

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

305 307 308 309

2.3. Gas-phase interstitial modification method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

310

2.4. Solid-solid (or liquid) reaction and plasma nitriding methods

314

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2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Interstitial compounds of the 2:17-type structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Crystallographic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

314 315 315

3.2. Curie temperature and exchange interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

328

3.3. Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

332

3.4. M6ssbauer and NMR studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341

3.5. Substitution effect

349

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4. Interstitial compounds of the l:12-type structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

353

4.1. Location of N atoms in the tetragonal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

354

4.2. Structural and intrinsic magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

356

4.3. Substitution studies

364

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4.4. Interstitial modification study on compounds with various structures . . . . . . . . . . . . . . . . 5. Electronic band structure calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

364 367

5.1. Calculations of the 2:17-type interstitial compounds R2Fel7Z3 ( Z = N or C) . . . . . . . . . .

367

5.2. Calculations of the l:12-type interstitial compounds RFe12_~TzZ u ( Z = N or C) . . . . . .

375

5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Applications

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

380 382

6.1. Improvement of the thermal stability

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

382

6.2. Development of permanent magnets

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

383

7. Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction

One of the methods for fabricating new magnetic materials is to interstitially introduce non-metallic atoms like H, B, C or N with small atomic-radius into host metals or compounds. As interstitially modified compounds, hydrides have been well known since 1960s and the magnetism has been covered in chapter 6 of volume VI in this series (Wiesinger and Hilscher 1991). Generally speaking, no significant change in the magnetism was observed upon hydrogenation. On the other hand, the possibility of dramatic improvement of magnetic properties was pointed out in the nitride FeI6N2 twenty years ago, which had been accidently found in the Fe-N thin films formed by evaporating Fe in a Nz-gas atmosphere (Kim and Takahashi 1972). This compound is regarded to be one of the interstitially modified compounds. The deduced Fe moment was reported to be 2.9#B, named 'giant magnetic moment'. Metallic iron crystallizes in the body centered cubic structure below 912°C and has an atomic moment of 2.2#u at 0 K, which is smaller than 3#B/Fe-atom expected for a localized d-state in an Fe atom with seven 3d-electrons. This moment reduction is due to the hybridization of the 3d-3d electron states or the overlap between the 3d-electron wave functions. If the hybridization or overlap is reduced or removed by lattice expansion due to interstitial modification, we can expect an increase of the moment per Fe atom. Since a confirmation of the 'giant moment' (~3.0/zB/Fe-atom) was made by Sugita et al. (1991) in a single crystalline Fex6N2 thin firm, much attention has been paid to developing Fel6N2 in bulk form, which is technically promising material for the applications as magnetic medium in high performance metal recording tapes or as soft magnetic materials. However, data of the magnetic properties of Fel6N2 phase widely scatters, and there are still controversies as to whether the giant moment exists in the Fel6N2 compound or not. Under these circumstances, it is very interesting to apply the interstitial modification technique to rare earth (R) transition metal (TM) intermetallic compounds for searching new promising magnetic materials. Before introducing the magnetism on interstitially modified R-TM intermetallic compounds which have been discovered in recent years, we will briefly trace the historical background of rare earth permanent magnets. Until now, the intermetallics composed of rare earth and 3d elements have been mainly developed as high-performance permanent magnets. In the 1960s, hexagonal SmCo5 compound with the CaCu5-type structure appeared on the stage as the 305

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H. FUJII and H. SUN

first rare earth high-performance magnet. The compound has quite favorable magnetic properties as permanent magnets, especially, (1) the extremely large uniaxialmagnetocrystalline anisotropy (#0Ha "~ 28 T) originating from the single-ion anisotropy of the Sm sublattice, (2) the relatively large saturation magnetization due to ferromagnetic coupling between Sm and Co moments (Ms '-~ 1.14 T) and (3) the high Curie temperature (To = 1000 K) (Strnat 1967). The development of the liquid-phase sintering technique made fully dense and stable SmCo5 magnet possible (Das 1969, Benz and Martin 1970). In alloys of Sm with Co, the saturation magnetization of Sm2Co17 (Ms N 1.25 T) is larger than that of SmCo5 and the Curie temperature (Tc = 1193 K) is also higher. Although the uniaxial anisotropy field of Sm2Co17 is only 7 T which is smaller than that of SmCos, Sm2Co17 has the possibility of leading to better permanent magnets than SmCo5 magnet. As a result of much effort, a nice combination between the high saturation magnetization of Sm2Co17 and the high magnetic hardness of SmCo5 was realized by controlling the kinetics of the precipitation reaction in a system with the approximate composition SmCo7.4, in which some of the Co was replaced by Fe and small amounts of Cu and Zr (Ojima et al. 1977, Yoneyama et al. 1978). The highest energy product which has been achieved is more than 240 kJ/m 3 (Mishra et al. 1981). As the main components Sm and Co are particularly expensive, it is desirable to use iron-based compounds in place of cobalt-based compounds. Unfortunately, the rare earth iron compounds with the CaCus-type structure do not exist. The R2Fe~7type compounds which are isomorphous to R2Co17 exist, but the Curie temperature is too low to be used as permanent magnets and the magnetic anisotropy at room temperature is not uniaxial which is not suitable for permanent magnets. In the 1980s, scientific research on rare earth iron compounds containing a small amount of non-metallic elements had been done quite intensively. As a result of continuing efforts, a novel type of permanent magnet material was discovered in 1984, which was based on the ternary compound Nd2Fel4B with the tetragonal structure (Sagawa et al. 1984, Croat et al. 1984). The magnetic characteristics of Nd2Fe14B are Ms = 1.60 T, #0Ha = 7 T at room temperature and Tc = 588 K. The achieved energy product was 360 kJ/m 3, which significantly exceeded all previous values. Because of its lower cost and superior properties, the Nd-Fe-B magnets have rapidly replaced the Sm-Co magnets and the spectrum of applications has continuously expanded. The main problem of the Nd-Fe-B magnets is the poor temperature stability due to the relatively low Tc which leads to a practical temperature limit of 150°C. Also their corrosion resistance is weak. Thus it is still necessary looking for new magnetic materials with better thermal and magnetic properties. Here, it should be noted that B is not an interstitial atom in the tetragonal structure and NdzFel4B does not belong to the interstitially modified compounds. In 1990, the interstitially modified compound Sm2Fea7N3 was discovered by applying the gas-phase interstitial modification technique to Sm2Fel7 (Coey and Sun 1990). This interstitial nitride was prepared by heating SmzFea7 at 450-500°C under a nitrogen or ammonia gas atmosphere. The crystal lattice of Sm2Fe17 expands more than 6% to accommodate three nitrogen atoms at the interstitial sites. The Curie temperature Tc increases dramatically from 398 K to 752 K. The saturation

INTERSTITIALLYMODIFIED INTERMETALLICS

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magnetization of Sm2Fel7N3 (Ms = 1.54 T) is comparable to that of NdzFel4B and the uniaxial magnetic anisotropy (#0Ha = 26 T) is three times as strong as that of NdzFel4B. By applying a similar method, the nitrides of Nd(Fel_zTz)lzN with T = Ti, V and Mo, which crystallize in the tetragonal ThMnlz-type structure, were also successfully formed (Yang Y.C. et al. 1991, Anagnostou et al. 1991a, Wang and Hadjipanayis 1991). The Curie temperature and uniaxial anisotropy field of these Nd-containing 1:12 nitrides are comparable with those of NdzFel4B, but the magnetizations are somewhat lower. Similarly to the R-TM intermetallics with the 2:17-type and l:12-type structures, many other structure type compounds can also be interstitially modified, and the structural and magnetic properties can be drastically altered. Hydrocarbon gas can be substituted for nitrogen to make interstitial carbides. Therefore interstitial modification has opened a wide field for scientific research besides the generation of great technical interest in the application of SmzFe17 interstitial nitride or carbide as permanent magnets. Since the first scientific publication of the discovery of the 2:17-type nitrides in early 1990 (Coey and Sun 1990), world wide efforts have been made and are still being devoted to the study of interstitial modifications of various kinds of R-TM intermetallic compounds. The study covers not only every aspect of the structural and intrinsic magnetic properties, but also the development of the promising interstitial compounds into hard magnets. In addition, much attention has been paid to the study of the gas-phase interstitial modification (GIM) process itself. In this chapter, we will review the studies of the interstitially modified intermetallic compounds in the following order: the study of the GIM process; the effects of interstitial modifications on the structural and magnetic properties of the 2:17-type compounds, l:12-type compounds and also compounds of other structure types; the electronic band structure calculations of the interstitial compounds and finally the research aimed at applications of the interstitial compounds as permanent magnets. The emphasis is on the 2:17-type and 1:12-type interstitial nitrides and carbides.

2. Formation of the interstitially modified intermetallic compounds Hydrogen, nitrogen and carbon are the well known elements that have been successfully used for interstitial modifications. The interstitial hydrides have been known for a fairly long time (see the review of Wiesinger and Hilscher 1991). In the field of magnetism, besides the improvement of magnetic properties (mainly the Curie temperature) by hydrogenation, the HDDR (Hydrogenation-DecompositionDesorption-Recombination) process has been used as a novel technique for the production of permanent magnets. Another field where the hydrides are of great interest is their applications as hydrogen storage and hydrogen purification materials, and furthermore as hydrogen batteries. The interstitial carbides and nitrides were discovered and developed in recent years. C and N were found to have much stronger effects than H on the magnetic properties. While the hydrides and nitrides can only be produced by the Gas-phase Interstitial Modification methods (GIM), the carbides

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H. FUJII and H. SUN

can also be made by arc-melting and melt-spinning in addition to the GIM process. In this section, various kinds of processing methods for producing the interstitially modified materials will be described. The emphasis is on the GIM method.

2.1. Arc-melting method Detailed investigations on the conditions under which the tetragonal R2Fel4C phase is formed have shown that the corresponding ternary systems contain another new phase with the composition RzFea7Cy (Liu N.C. et al. 1987, Gueramian et al. 1987, de Mooij and Buschow 1988), which is the interstitial modified 2:17 phase. As shown in fig. 2.1, the solid line defines temperature regions below which the tetragonal R2Fe14C phase is stable and above which the hexagonal or rhombohedral R 2 F e l 7 C v phase is stable. Because of the high melting point of carbon, Fe3C is made first by arc-melting. Then it is mixed together with appropriate amount of R and Fe and melted again to form the as-melted ingot. High temperature and long time (above 1000°C for a few weeks) annealing is necessary for forming the R2Fe17C u phase. In order to avoid the phase transition into R2Fel4C during cooling (fig. 2.1), samples have to be quenched into water from the annealing temperature. The carbon content y is continuously, variable, but the maximum amount is less than 1.6 for heavy rare earths and less than 1.0 for light rare earths. Both the maximum contents are much less than 3, which is the available interstitial sites in the hexagonal or rhombohedral structures (Sun et al. 1990a, Coene et al. 1990, Zhong et al. 1990). The substitution of Ga for Fe can stabilize the 2:17 structure with higher interstitial carbon content. Carbon concentration as high as 2.5 was achieved in R2Fe17_zGazCy(z = 2 and 3, 0 ~< y ~< 2.5) by arc-melting (Shen et al. 1993, I

I

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R2Fe17Cy~ ~ v

t--- lOOO R2Fe14 C

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LaCe Pr NdPmSmEuGd.TbDyHoEr Fig. 2.1. Transformation temperature Tt of the R-Fe-C systems. The R2FelTCv phase is stable above the solid line and the RzFel4C phase is stable below the solid line. The broken line indicates the temperature range in which the reaction rate is too low for formation of the R2Fel4C phase from the non-equilibrium phases of the as-cast melt. Replotted from de Mooij and Buschow (1988).

INTERSTITIALLYMODIFIEDINTERMETALLICS I

309

I

#_ Sm2Fe1702.0

r"

Sm2Fe14Ga3C2.0 o 1~

30

[

1

40

50

60

20 ( degree ) Fig. 2.2. X-raydiffractionpatterns of Sm2FelTC2.0and Sm2FeI4Ga3C2.0. The compoundswereprepared by arc-meltingand annealedin vacuumat 1270 K for 12 hours (after Shen et al. 1993). 1993b). Shen et al. also reported similar stabilization effects by the substitutions of Si or A1 for Fe. Annealing at temperatures higher than 1000°C is still necessary, but the time period requested is reduced to less than 24 hours. The effect of Ga can be clearly seen in the X-ray diffraction patterns of Sm2FelTCz0 and SmzFe14Ga3C2.0 in fig. 2.2. While the later compound is of nearly pure 2:17 phase with the ThzZn17 rhombohedral structure, the former one is a mixture of a-Fe and probably the 2:17 phase.

2.2. Melt-spinning method Interstitial compounds R2Fe17Cv with y = 0-2.8 can be prepared by arc-melting and subsequent melt-spinning (Shen et al. 1992, Kong et al. 1992, 1993a, b, Cat et al. 1992a, b, 1993a, b). The alloys with the corresponding stoichiometric compositions were first arc-melted, and then by using a suitable quenching rate (1020 m/s for Y and Tb), almost pure 2:17 phases were obtained. The maximum carbon content y increases as the rare earth atomic radius decreases (or as the atomic number increases). Only the preparation of carbides containing heavy rare earths has been reported. Common features of the arc-melted and melt-spun interstitial carbides are their high temperature stability (contrary to the metastability of the gas-phase interstitially modified interstitial materials) and the structure transition from the hexagonal Th2Ni17-type structure to the rhombohedral Th2ZnlT-type structure with the increasing carbon content for heavy rare earth compounds. For interstitial compounds made by the GIM process, no such structure transformation has been reported. These features will be discussed later.

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H. FUJII and H. SUN

2.3. Gas-phase interstitial modification method 2.3.1. Gas-phase interstitial modification process It has been known for long that rare earth-iron intermetallics can absorb prodigious quantifies of hydrogen, and that their magnetic properties are altered (Wallace 1978, Buschow et al. 1982). This idea was extended to nitrogen and carbon absorptions by using gasses containing N or C (Coey and Sun 1990, Coey et al. 1991a, b, Sun et al. 1990b and 1992). The gas absorption characteristics were studied by the latter authors using a thermopiezic analyzer (TPA), which involved the measurement of the gas pressure variation as a function of temperature. The basic principle is that when a sample which absorbs or desorbs gas is heated in a closed volume at a constant rate, the pressure-temperature curve p(T) and its derivative dp(T)/dT provide a fingerprint of the material, and allow the amount of gas-envolving or gas-absorbing phase to be determined. The TPA traces of Y2Fel7 in H2, N2, NH3 and Chill0 are shown in fig. 2.3. Y2Fel7 acts as a catalyst and it induces breaking of the strong N -= N bond even at temperature as low as 400°C. When making carbides using this method, normally a hydrocarbon gas is used. At the same time of the formation of the carbides, hydrogen gas also forms, which has to be pumped out at temperatures around 500°C in order to avoid the formation of hydrides during cooling to room temperature. As can be seen from the TPA trace of Y2Fe]7 in H2 shown in fig. 2.3, the H2 gas pressure at about 500°C is roughly the same as at room temperature, which indicates that nearly no hydrogen remains in the compound. Therefore, the hydrogen content in the carbides is negligible. The gas-phase interstitial modification process can be carried out using fixed or flow gas atmospheres (nitrogen or ammonia for nitrogenation, and hydrocarbon gasses for carbonation) in conventional furnaces. Sometimes a certain fraction of hydrogen gas is added in order to activate nitrogenation or carbonation by hydrogenation which occurs at lower temperatures (fig. 2.3). Liu J.P. et al. (1991) interpreted '

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Fig. 2.3. Thermopiezicanalysisof Y2Fe17 heatedin H2, NH3, N2 and C4H10at 10 K/min.

INTERSTITIALLYMODIFIEDINTERMETALLICS

311

the nitrogenation reaction process using the same model as for the ternary hydrides by considering the reaction enthalpy (AHf) of the following two transformations, 2R2Fe17 + 3N2 --+ 2R2Fe17N3,

(2.1)

R2Fe17 + N2 -+ 2RN + 17Fe.

(2.2)

and

The reaction enthalpy AH~ of the interstitial nitride formation process is sufficiently negative, which makes the formation of the 2:17 nitrides possible, once the temperature has been raised high enough to overcome the activation energy for the absorption process. Although the enthalpy change AH~ of the decomposition process is more negative, the activation energy is also much higher. Thus the nitride phase can be formed in a limited temperature range as a metastable phase. If the nitrogenation process is performed at too high temperatures or for too long times, degradation takes place and the harmful impurity phases appear in the nitrides. By carefully controlling the nitrogenation condition, the decomposition process can be depressed, but it can not be completely eliminated. Many studies have been devoted to the preparation of high quality nitrides. High pressure nitrogenation, pre-hydrogenation treatment etc. have been found to be helpful. Fujii et al. (1992a, b) have claimed that one of the reasons for the decomposition of the nitride phase into SmN and ~-Fe is that the temperature at the surface of the Sm2Fe17 particles increases as a result of the heat formation during the reaction process and might exceed the decomposition temperature. Thus, the relative better thermal conductivity of the high pressure (up to 100 atm.) N2 gas could help to reduce the particle surface temperature and depress the decomposition process. Fukuno et al. (1991, 1992a) reported that hydrogen treatment before nitrogenation significantly increased the gas-solid reaction area by inducing many cracks in the Sm2Fe17 particles. This hydrogenation process thus promoted nitrogenation at a lower temperature and the formation of impurity phases could be minimized. Lower temperature and longer nitrogenation time is preferred in order to obtain high quality nitrides. 2.3.2. Reaction mechanism

The nitrogenation process has been observed and studied by many different experimental techniques, which include the Kerr microscopy observations by Mukai and Fujimoto (1992), in situ neutron powder diffraction studies by Isnard et al. (1992a), micrograph studies applying metallography and EPMA techniques by Colucci et al. (1992, 1993a, b) and Fujii et al. (1994b). The nitrogenation process is understood to proceed preferentially through extended defects such as phase and grain boundaries, dislocations and dislocation arrays. Nitrogen diffuses into the particles along one such path and then bulk diffusion occurs perpendicular to this path inside the particle. The bulk diffusion coefficient is very small compared with the short circuit diffusion through defects.

312

H. FUJII and H. SUN

The reaction between the gas phase and the rare earth intermetallics was studied by Skomski and Coey using the lattice gas model (Coey et al. 1992, Skomski and Coey 1993a). The net reaction energy U0 consists of three parts; the energies which are necessary to dissociate the gas molecules, the energy for expanding the lattice and the energy gain due to the gas-lattice interaction. For the Sm2Fel7N v system, it was estimated that U0 = - 5 7 + 5 kJ/mol, and the diffusion parameters Do = 1.02 mmZ/s, Ea = 133 kJ/mol. Diffusion constant at a certain temperature T can be calculated by the expression DT = Do exp(-Ea/kT). The reaction kinetics of N2 absorption by Sm2Fel7 were experimentally investigated using volumetric and gravimetric measurements by Uchida H.-H. et al. (1993) and they found that Ea was in the range 100-163 kJ/mol, in agreement with Skomski and Coey (1993a). Skomski and Coey (1993a) also investigated the stress and strain in the inhomogeneously nitrogenated grains in the linear elastic approximation. The nitrogen diffusion progresses from the grain surface into the center. At the initial stage of nitrogenation or in the case of not fully nitrided materials, the stress and strain lead to an expanded nitrogen-free particle center with an increased Curie temperature but negative anisotropy constant/(1. This soft center could act as a nucleation center for domain walls and destroy coercivity. Fujii et al. (1994b) studied the nitrogen absorption process in SmzFel7 particles at 733 K under various Nz-gas pressures. The observation of EPMA-line profiles of the N-element in the grains led these authors to claim that nitrogenation process consists of the following two mechanisms: The nitrogen diffusion process and the phase transformation from the nitrogen poor Sm2Fel7Nx to the fully nitrided Sm2Fel7N3. Both mechanisms progress concurrently under Nz-gas pressures above 0.1 MPa. However, the diffusion process becomes dominant under low N2-gas pressures below 0.05 MPa and at this pressure a homogeneous phase with any value of z is stabilized, suggesting that the SmzFeI7-N solid solution exists above 733 K. 2.3.3. Nitrides with intermediate nitrogenation content It has been an open question whether the metastable nitride is a simple gas-solid solution with continuous range of intermediate nitrogen contents or whether it is a two-phase mixture of nitrogen-poor and nitrogen-rich phases. The attractive interatomic long-range interaction energy arising from the lattice deformation around the interstitial site is important for answering this question (Coey et al. 1992). Below a critical temperature Tcri, the attractive interaction dominates and the interstitial atoms form macroscopic clusters, and consequently a two-phase mixture is formed. If the nitrogenation is carried out above Tcri, then a solid solution phase should be formed. For Sm2FeavNv, Tcn was estimated to be about room temperature. As the nitrogenation is usually performed at temperatures much higher than room temperature, the nitrides should be present as a gas-solid solution phase with a continuous range of intermediate nitrogen contents, according to the estimation by Coey et al. (1992). However, not all the experimental results support this conclusion. Katter et al. (1992a) reported that the Sm2Fel7 nitrides existed over the whole concentration range of 0 ~< y ~< 3. SmzFel7Nu with intermediate nitrogen contents (0 ~< y ~< 2.94) was successfully prepared and the nitrogen content dependence of the

INTERSTITIALLYMODIFIEDINTERMETALLICS

313

unit cell volume, the Curie temperature, the saturation polarization, the anisotropy field and the thermal stability was examined. All these properties showed a continuous increase with increasing nitrogen content y. Christodoulou and Takeshita (1993e) also reported that the Sm2Fe17 nitride is not a line compound but exhibits a large solubility of nitrogen. However, it was difficult to form homogeneous interstitial compounds with y between 0 and 3, which requires either partial nitrogenation followed by long time heat treatment in argon atmosphere or nitrogenation in a predetermined amount of nitrogen. Mukai and Fujimoto (1992) observed the domain patterns of SmzFelyNy and concluded that nitrides with an intermediate nitrogen content exist. They found that the magnetic domain patterns were sensitive to the nitrogen content and the domain width became narrower as the nitrogen content decreased. The possibility of the existence of an intermediate nitrides was also suggested by Uchida H. et al. (1992) by measuring volumetrically the pressure-composition isotherm of the SmzFelv-N system at 823 K. No plateau region was observed in the P - C - T curve at nitrogenation concentrations lower than 3, which was taken as an indication of the existence of solid solutions between Sm2Fe17 and N. On the other hand, there are many publications in which it is reported that nitrides with intermediate nitrogen content do not exist. By X-ray diffraction experiments, Coey et al. (1991a) examined the reaction products after heating YzFe17 in the TPA to temperatures ranging from 300 to 850°C. The results showed that at temperatures lower than the temperature at which fully nitrided sample could be formed (600°C), the products were always mixtures of the pure 2:17 phase and the fully nitrided phase. Isnard et al. (1992a) reached the same conclusion by in situ neutron powder diffraction study of the nitrogenation process of NdzFel7. They found that even at the beginning of the reaction, the nitride was a highly charged interstitial phase, no progressive filling of the interstitial site was observed and only the ratio of the nitride phase increased as nitrogenation progressed. Similar results were obtained in a metallography study of the nitrogen diffusion patterns in Nd2Fe17 by Colucci et al. (1993b). No formation of a nitrogen solid solution phase was observed, and the fully nitrided phase precipitated directly from the phase free of nitrogen.

2.3.4. Formation of carbonitrides and other multiple interstitial compounds of H, C and N Carbonitrides were formed by subsequent nitrogenation of the arc-melted carbides (Coey and Sun 1990, Nagata and Fujii 1991, Kou et al. 1991a, b, Yang Y.C. et al. 1992a). This is a combined arc-melting and GIM process. Altounian et al. (1993) and Chen et al. (1993a, b) made the carbonitrides by the GIM process only. The carbonitriding process can be done either using mixtures of N2 gas and hydrocarbons or sequentially one followed by another. A mixed gas of nitrogen and methane proved to be successful for obtaining carbonitrides of R-Fe intermetallics with various nitrogen to carbon ratios. The reaction was written as follows, 1

yN2 + zCH4 + R2Fe17 ~ R2Fe17CxN v + 2xH2.

(2.3)

Similarly to the preparation of carbides by the GIM process, H2 should be pumped out to avoid the formation of the undesired hydrides. For the sequential process, it was

314

H. FUJII and H. SUN

found that nitrogenation should be done first, followed by the carbiding reaction. If carbonation was followed by nitrogenation, then only carbides were produced because the carbon layer formed on the surface of the particles prevented nitrogen atoms from entering into the particles. Christodoulou and Takeshita (1993f) reported the preparation of SmzFelv-carbonitildes, carbonhydrides, nitrohydrides and carbonitrohydrides. The carbide was made first, by direct melting or gas phase carbonation, then it was nitrogenated and finally hydrogenated. The nitrohydride can be produced directly by nitrohydrogenation in a mixture of N2 + H2 or N}-I3 gasses. The SmzFely-carbonitrohydrides can also be prepared by gas phase reaction of Sm2Fel7 powders with a mixture of hydrocarbon, N2 and H2 gasses.

2.4. Solid-solid (or liquid) reaction and plasma nitriding methods A new route for the synthesis of metalloids interstitials is solid-solid (or liquid) reactions. This has been tried for synthesizing carbides using heavy hydrocarbon organic compounds, such as benzene (C6H6) and toluene (C7H8) etc. (Fruchart et al. 1994). Interstitial modifications of Y2Fel7 by reaction with solid boron, silicon and sulfur were tested by Skomski et al. (1993), but were not very successful. Compared to the gas-phase interstitial modification, the solid-solid reaction has the following advantages; (1) it can be conducted at relative low temperatures and thus the decomposition process could be depressed, and (2) liquid and solid materials are easier for handling. In order to avoid the formation of hydrides, the reaction should be conducted at temperatures higher than 400°C. For this purpose, Fruchart et al. (1994) suggested to use hydrocarbon (and hydrocarbonitrogen) materials of heavier formula weight, which have both a higher thermal stability and a lower H/C(N) ratio. Plasma nitriding of SmzFe~7 was done by Machida et al. (1993). It was performed by glow-discharge between the electrodes under a differential pumping condition at 2 Torr of a N2-H2 mixed gas with a molar ratio of 1:2. The advantage is that the reaction temperature was only 423 K, which is much lower than the conventional thermal gas-solid or solid-solid reaction.

2.5. Summary Sm2Fe17nitride is a most interesting material for permanent magnet applications. For making the nitride, GIM is the most suitable way. High quality (fully nitrided and free from impurity phases) nitrided material is essential for making the final magnet products with useful magnetic properties. The nitrogenation process is a relative complicated process which is not only affected by the nitrogenation conditions, such as the temperature, reaction period, gas pressure and composition of gasses, but is also affected by the initial Sm2Fel7 particle size and surface condition. The basic features and diffusion patterns of the GIM process have been fairly well understood. However, further studies are necessary for improvement and better controlling of the GIM process.

INTERSTITIALLYMODIFIEDINTERMETALLICS

315

3. Interstitial compounds of the 2:17-type structure Among binary rare earth iron intermetallics, compounds with the 2:17-type structure are the most iron rich ones and they are particularly stable. The R2Fel7 compounds exist across the whole lanthanide series, from Ce to Lu, except for La itself. The magnetic properties of the series have been studied in great detail (Buschow 1977, Wallace 1985). The magnetization is fairly high because of the high iron content. However, none of these compounds exhibits an easy-axis anisotropy at room temperature, and only Tm2Felv does so in the liquid nitrogen temperature range. Furthermore, the Curie temperatures are surprisingly low (240-480 K) for compounds containing so much iron. Therefore the R2Fe17 compounds have been disregarded as potential permanent magnet materials. It has been found that elements with small atomic radius can occupy interstitial sites of the R2Fel7 host lattice. These interstitial atoms cause displacements of the metal atoms from their regular sites, and the resulting crystal lattice distortions give rise to various kinds of changes in physical properties which are interesting from both fundamental and applied points of view. Interstitial modifications by H, C and N have been studied intensively and proved to be successful for improving the magnetic properties of R2Fe17. In this section, various studies of the structural and magnetic properties of interstitial R2Fe17 carbides and nitrides will be summarized. Studies of the R2Fe17-hydrides have been already reviewed by Wiesinger and Hilscher (1991). For this reason we will omit them here. 3.1. Crystallographic structures The RzFe17 compounds crystallize in the rhombohedral ThzZn17-type structure (space group R3m) for rare earths lighter than Gd and in the hexagonal ThzNily-type structure (space group P6/mmc) for rare earths heavier than Tb. The compounds with R = Gd, Tb and Y can exist in both forms depending on the high temperature annealing condition. Hexagonal form of Ce2Fel7 has also been reported to coexist with the rhombohedral one (Buschow and van Wieringen 1970). Both of the structures are derived from the CaCus-type structure by the ordered substitution with a (dumbbell) pair of Fe atoms for each third rare earth atom in the basal plane (Florio et al. 1956, Zarechnyuk and Kripyakevich 1962). When these substituted layers are stacked in the sequence abcabc, the Th2Zna7-type rhombohedral structure is realized. If the stacking sequence is, instead, ababab, then the ThzNi17-type hexagonal structure is formed. The structures are illustrated in fig. 3.1, both of which are represented by hexagonal cells with two and three RzFe17 formulas for the hexagonal and rhombohedral structures, respectively. 3. I.I. Structure modification by the GIM process The modification of the R2Fe17 structures by interstitial nitrogen and carbon atoms introduced by the GIM process is essentially to expand the unit cells, without changing the rhombohedral or hexagonal symmetry of the parent compounds. This retaining of the crystallographic symmetry is the main difference when compared to the arcmelted carbides, where a structure transition from the hexagonal symmetry to the

316

H. FUJII and H. SUN

v

v

v

_

I

C

a v

v

a

v

R 06C

R (~)2d 02b

Fe

~6c

~gd

Z

•

o18g

9e

Th2Znl7

(~)18f

O18h

Fe ~ 4 f

@6g

Z

0121

•

6h

@12J

4D12k

Th2Nl17

Fig. 3.1. Crystal structuresof R2Fe17,left: rhombohedralTh2ZnlT-type;right: hexagonalTh2Ni17-type, showing the rare earth sites (R), iron sites (Fe) and the interstitialsites (Z). rhombohedral symmetry was observed with increasing carbon content (for detail see the following section). Lattice parameters of nitrides and carbides of all the RzFel7 compounds are collected in table 3.1, in which also included are the carbonitrides and carbonitrohydrides for some of the rare earths. As shown in table 3.1, the total number of nitrogen and carbon atoms per formula unit is close to or slightly less than 3 in nearly all the interstitials. The expansion of the cell volume upon nitrogenation or carbonation, or the combination of them, is 6-7% for all the R2Fea7 compounds except for Ce, where the volume increase is more than 8%. It is also noticeable that nitrides have a slightly larger volume than carbides in general. Figure 3.2 compares the lattice parameters of the GIM nitrides R2Fel7N v and carbides R2Fel7Cv, the arcmelted carbides R2FeivC and the host R2Fel7 compounds (2/3e and 2/3V for the rhombohedral compounds are plotted). The normal variation in lattice parameters associated with the lanthanide contraction is shown excepting the Ce compounds. The anomalous position of Ce suggests a near tetravalent (4f°) configuration for Ce in CezFe17, which changes to an intermediate valence configuration between 4f ° and 4f I in the nitrides and carbides.

3.1.2. Structure studies of the arc-melted and melt-spun carbides The expansion effect of the unit cell volume due to interstitial carbon atoms introduced by the arc-melting method is similar to those introduced by the GIM method,

INTERSTITIALLY MODIFIED INTERMETALLICS

317

TABLE 3.1 Crystallographic structure data and magnetic properties of R2Fel7 and their interstitial compounds. V is the crystal cell volume calculated from V =

-~a2c. Saturation magnetization as at room temperature

and 4.2 K are expressed in the unit of/zB/formula. Compound

a (]k)

C (/k)

V (•3)

AV/V (%) Tc (K)

as (#B/f.u.) RT

Ce2Fel7 Ce2Fe17N2.8 Ce2FelvN2.5 Ce2FeI7N25 Ce2FeI7N3. 6 Ce2FelTCu CezFe17C2.8 Ce2Fe17C2.5 Ce2Fel7C2 Ce2FeI7CxN/,

8.48 8.73 8.743 8.743 8.73 8.73 8.74 8.73 8.72 8.73

12.38 12.65 12.673 12.688 12.81 12.56 12.65 12.58 12.64 12.68

773.5 834 838.9 847.9 845.5 829.5 836 830

8.8 8.24 9.43 9.3 8.3 8.0 8.2

Pr2Fel7 Pr2Fel7N2. 5 Pr2FelvN25 Pr2FelTN2.86 PrzFelTCy Pr2Fel7C2. 7 PrzFel7C2. 5 Pr2Fel7C 2 PrzFelvCxNu

8.574 8.77 8.794 8.795 8.80 8.79 8.85 8.78 8.79

12.324 12.64 12.668 12.659 12.59 12.63 12.61 12.65 12.66

790.7 841.8 848.4 847.9 843.3 845 855

6.5 6.65 6.68 6.7 6.8 8.3

Nd2Fel7 Nd2Fel7N2.3 Nd2Fel7N2.5 Nd2Fel7N2. 4 Nd2FelvCu NdzFel7C2.2 NdzFel7C2.5 Nd2Fel7C2 NdzFel7CxN u

8.563 8.76 8.783 8.776 8.79 8.76 8.80 8.76 8.76

12.444 12.63 12.663 12.661 12.60 12.62 12.60 12.64 12.65

790.2 838.8 845.5 844.5 842.1 839 845

6.2 6.45 5.85 6.6 5.9 6.7

Sm2Fel7 Sm2Fe17N2.3 SmzFeI7N2. 5 Sm2FelTN2. 2 Sm2Fel7N2 SmzFe17N3.1 SmzFel7N2. 9 Sm2FelTN3.o Sm2Fe17N2.7 Sm2Fel7Ns.2 SmzFelvN2.9 SmzFeI7N2. 6 Sm2Fel7 N2.94

8.55 8.73 8.741 8.730 8.732 8.74 8.76 8.728 8.778 8.838 8.743 8.71 8.7425

12.43 12.64 12.666 12.630 12.631 12.65 12.76 12.638 12.74 12.82 12.703 12.61 12.659

786.9 833.7 838.2 834.1 834.1

6.3 6.36 6.01 6.2

833.73

837.9

6.2

210 713 700 712 589 716 608 589 721 290 728 720

4.2 K 29.3

36.4 32.2

36.3

31.6

39.9 33.8

25.1

30.0

18.1 37.9 36.5

653 735 690 654 737

27.0

330 732 740

17.1 40.5 34.9

41.3 34.5 30.8

43.9

659 739 662 658 740

29.0

31.8

389 749 750

22.4 31.9 32.2

34.1

745 752

35.1 34.0

748

35.7

740

34.3 35.5 36.2

746

Ref.

35.1

38.2

38.1

35.0

[13, 24] [1] [2, 3] [4] [13, 24] [7] [21] [22] [8, 9] [8, 9] [1] [1] [2, 3] [4] [7] [21] [22] [8, 9] [8, 9] [1] [1] [2, 3] [4] [7] [21] [22] [8, 9] [8, 9] [13, 24] [1] [2, 3] [4] [5] [13, 24] [14] [15] [16] [16] [17] [18] [19]

318

H. FUJII and H. SUN TABLE 3.1 (Continued)

Compound

a (,~)

c (-~)

V (,~3) AV/V (%) Tc (K)

Sm2Fel7Cy Sm2Fel7C 2 Sm2Fe17C2.5 Sm2FelvC2.5 Sm2Fe17C2 Sm2Fel7C:rNy Sm2Fe17CN u Sm2Fe17Ny Sm2Fe17Co.4Ny Sm2FelTCo.TN u Sm2FelTCo.9N u Sm2Fel7N3.oHo.8 Sm2FelTCo.sN2. 4 Sm2FelTCo.5N2.4Ho.8 Sm2FeÂTC2.6 Sm2FeI7C2.6H1.1 Sm2Fel7 C2.6No.1 Sm2FelTC2.6NoA HI.O

8.75 8.749 8.73 8.77 8.73 8.75 8.712 8.742 8.736 8.742 8.765 8.739 8.742 8.735 8.744 8.753 8.747 8.754

12.57 12.595 12.67 12.61 12.65 12.65 12.604 12.651 12.650 12.633 12.683 12.733 12.652 12.715 12.572 12.622 12.584 12.634

833.0

Gd2Fel7 Gd2Fel7N2. 4 Gd2Fe17N2.5 Gd2Fe17N2.5 Gd2Fe17C u Gd2FelTC2.5 Gd2Fe17C2 Gd2FeI7CzN u Gd2FelTCNy

8.508 8.69 8.713 8.715 8.70 8.68 8.68 8.70 8.695

12.432 12.66 12.652 12.653 12.61 12.69 12.66 12.67 12.603

779.4 827.6 831.8 832.3 825.8 828

Tb2Fe17 Tb2Fe17N2.3 Tb2Fe17N2.5 Tb2Fel7Cy Tb2Fe17C2 Tb2FelTCxNy Tb2Fel7CNy

8.484 8.66 8.683 8.67 8.65 8.65 8.678

12.410 12.66 12.666 12.64 12.66 12.71 12.602

773.6 823.1 827.0 823.2

Dy2Fel7 Dy2Fe17N2.8 Dy2Fe17N2.5 Dy2Fel7Cy Dy2FelTC2. 7 Dy2Fe17C2.5 Dy2Fel7C 2 Dy2FelTCxNu Dy2Fe17CNv

8.445 8.64 8.670 8.65 8.63 8.69 8.63 8.66 8.643

8.304 8.45 12.666 8.42 8.42 8.46 8.42 8.45 8.461

512.9 545.9 824.5 545.2 543 553

836 840

828.5 837.3 836.0 836.0 843.8 842.12 837.34 840.14 832.48 837.49 833.85 838.54

6.2 5.9 6.5 7.1

6.2

6.2 6.89 6.01 5.9 5.9

825.2

6.4 6.54 6.4

821.9

547.4

6.4 6.52 6.3 6.1 8.5

668 670 760 679 680 758 778 753 759 753 760 748 752 752 673 674 678 678 477 758 740

~s (/zB/f.u.) RT

4.2 K

26.6

34.5 30.1

26.2

29.2

31.2 32.3

35.1

Ref.

[7] [20] [21] [22] [8, 9] [8, 9]

[lO] [11, [11, [11, [i1, [15] [15] [15]

36.3 36.3 35.0 35.2 29.3

12] 12] 12] 12, 15]

[15] [15] [15] [15]

10.4 26.7 17.9

711 763 712 764 813

27.3 20.1

28.1

404 733 730 680 680 748 778

11.6 22.4 23.6

367 725 720 674 733 681 683 724 758

11.4 27.1 25.6

25.0 21.3

24.3

17.9

23.0 17.1 14.2

23.1

[1] [1] [2, 3] [4] [7] [21] [8, 9] [8, 9] [lO] [1] [11 [2, 3] [7] [8, 9] [8, 9] [10] [1] [1] [2, 3] [7] [21] [22] [8, 9] [8, 9] [10]

INTERSTITIALLY MODIFIED INTERMETALLICS

319

TABLE 3.1 (Continued) Compound

a (A)

e (.A)

V (A 3)

AV/V (%) Tc (K)

as (#B/f.u.) RT

Ho2Fel7 Ho2FelTN3.0 Ho2Fe17N2.5 Ho2Fel7N2.1 Ho2Fel7Cy

8.439 8.62 8.632 8.609 8.61

8.280 8.45 8.472 8.480 8.43

510.7 543.8 546.7 544.3 540.8

Er2Fel7 Er2Fel7N2.7 Er2FelTN2.5 Er2Fe17N2.45 Er2Fel7C u

8.272 8.46 8.476 8.469 8.41 8.438 8.44 8.46 8.44 8.49 8.460 8.472 8.469 8.469 8.480 8.478 12.628

508.0 542.8 544.9 540.5 538.5

Er2Fel7 C2.4 Er2FelTC2.5 Er2FelTCy Er2FelTCxN u Er2FelTCNy Er2Fel7Ny Er2Fel7Co.4Ny Er2Fel7Co.6Ny Er2FelTCo.sNy Er2Fe17Cl.0Ny Er2Fel7Cl.sNy

8.421 8.61 8.622 8.584 8.60 8.630 8.61 8.68 8.63 8.63 8.615 8.633 8.643 8.643 8.651 8.647 8.652

543.8 546.8 547.9 547.8 549.6 549.0 818.6

Tm2Fel7 Tm2Fel7 N2.7 Tm2FeI7N2.5 Tm2Fel7C u Tm2Fel7Ny TmzFel7Co.zNy TmzFeI7Co.4Nu Tm2Fe17 Co.6Ny Tm2Fel7Co.8Nv TmzFel 7C 1.oNy

8.397 8.58 8.583 8.60 8.569 8.584 8.590 8.586 8.586 8.584

8.276 8.47 8.482 8.43 8.481 8.486 8.478 8.485 8.484 8.485

505.4 540.6 541.1 539.6 539.3 541.5 541.8 541.7 541.6 541.4

Yb2Fe17 Yb2Fel7N2.8

8.414 8.5701

8.249 8.495

505.7 540.0

6.48

280 675

Lu2Fel7 Lu2Fe17N2.7 Lu2Fe17N2.5 Lu2FelTCy

8.390 8.57 8.576 8.57

8.249 8.48 8.475 8.42

502.9 539.4 539.8 535.4

7.1 6.68 6.3

255 678 675 657

Y2Fel7 Y2Fel7

8.477 8.51

8.264 12.38

514.3 776.4

Er2Fe17C2

541 553

6.5 6.56 6.27 5.9

6.9 6.78 5.84 6.0 6.7 6.5 7.3

7.2

7.0 6.59 6.8 6.3

327 709 710

11.2 27.2 25.2

672 296 697 690 663 675 708 671 675 700 723 683 712 706 709 701 698 260 690 690 656 700 701 703 705 701 703

325 300

24.5 17.5

7.4 31.7 27.4

20.5 17.9

22.0

12.8

27.4

32.5 32.9

30.2 21.2

[1] [1] [2, 3] [4] [7]

[1] [1] [2, 3] [4] [7] [20] [21] [22] [8, 9] [8, 9] [10[ [11] [11] [11] [111 [111 [11] [11 [1] [2, 31 [7] [11] [11] [111 [11] [11] [11] [23] [2, 31

34.2

39.0 36.4

[11 [11 [2, 31 [71

32.8 34.0

[1] [24]

35.2

18.6

Ref.

4.2 K

320

H. FUJII and H. SUN TABLE 3.1 (Continued) c (A)

V (~3)

AV/V (%) Tc (K)

O's (p,B/f.u.)

8.44 8.465 8.484 12.69 8.40 8.414 8.46 8.40 8.40 8.48 8.415 8.488 12.606

6.4

34.2

YzFelvN3.1 YzFeI7Cy Y2FeITC2 Y2Fel7C2.8 Y2FelTC2.5 Y2FelTC2 Y2Fel7CzN u YzFel7CNy Y2Fel7Co.6Ny YzFe17C1.2Ny

8.65 8.637 8.646 8.67 8.66 8.689 8.64 8.65 8.67 8.66 8.635 8.634 8.673

547.3

543.4 547.9 821.2

701 668 673 722 678 668 717 723 722 724

Th2Fel7 Th2Fe17Nu

8.572 8.798

I2.472 12.703

794 853

320 747

Compound

a (h)

RT YzFeI7N2.6

Y2Fe17N2.7 Y2FeI7N2.8

References: [1] Sun et al. (1990b) [2] Buschow et al. (1990) [3] Liu J.P. et al. (1991) [4] Isnard et al. (1992d) [5] Katter et al. (1990) [6] Jacobs et al. (1991) [7] Sun et al. (1992) [8] Altounian et al. (1993) [9] Chen et al. (1993) [10] Yang Y.C. et al. (1992a) [11] Kou et al. (1991a) [12] Kou et al. (1991b)

549.2 826.1 546.1 547 544

6.70 6.4 6.2 6.8 6.6 6.2

694 690

Ref.

4.2 K 38.5

28.9

39.8 35.8 34.1

27.9

32.5

35.8

30.4 38.1

[11 [3] [4] [24] [7] [20] [21] [22] [8, 9] [8, 9] [10] [11] [11] [6] [6]

[13] Fujii et al. (1992) [14] Machida et al. (1993) [15] Christodoulou and Takeshita (1993f) [16] Uchida H. et al. (1992) [17] Wei et al. (1993) [18] Mukai and Fujimoto (1992) [19] Katter et al. (1992a) [20] Hu and Liu (1991) [21] Liao et al. (1992) [22] Tang et al. (1992) [23] Buschow (1972) [24] Fujii et al. (1995a)

but the overall expansion is much less in the arc-melted carbides (table 3.2), which is obviously associated with the smaller amount of carbon atoms introduced in. As mentioned in section 2, the solubility y of carbon in R 2 F e l T C v obtained from the melt and subsequent high temperature annealing is limited to y ~< 1.6 for heavy rare earths and y ~< 1.0 for light rare earths. From the study of the dependence of the unit cell volume V on carbon content y for R = Sm (Popov et al. 1990, Gr6ssinger et al. 1991, Wang and Hadjipanayis 1991a), Gd (Dirken et al. 1989), Ho (Haije et al. 1990), Tm (Gubbens et al. 1989, Gr6ssinger et al. 1991), Er (Kou et al. 1991c), Y (Coene et al. 1990, Sun et al. 1990a) and Th (Isnard et al. 1992e, Jacobs et al. 1991), a linear relationship between V and y was found until y ~ 1. In the case where the host R2Fe17 takes the Th2Ni17 structure, there is a structure transition from the hexagonal to the rhombohedral structure with increasing carbon concentration (Sun et al. 1990a, Coene et al. 1990, Haije et al. 1990, Kou et al. 1991c). This structure transformation is reminiscent of the structural transformation

INTERSTITIALLY MODIFIED INTERMETALLICS

321

8.7

,< ._... 8 . 6 - o

r~

R ~ ~'~

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

/ 0 - - 0 ~'

'~'o.~ ~0~

8.4 ~

+

~ D

~

8.5A

o

8.3

o , . n l o ° . . . o°

o\/

8.2

,..

°

-o

o

_li-i~_,

560 "-" 540

""~

>

O~C[/ - 0 ~ ° ~

520

o°--o° o°__~o oo-

~L~....~,,,~.~,,,~.

0... / 0

o... 0"-0

[]~ 0

~o

~0~

°-°'-'-°

500 Y

Pr Ce

Nd

Pm Eu Tb Ho Tm t u Sm Gd Dy Er Yb

Fig. 3.2. Lattice parameters and cell volumes of R2FeI7, the arc-melted R2FelTC, the GIM R2FelTCy and R2Fel7Ny (after Sun et al. 1990b and Zhong et al. 1990).

observed in R2Fel7 when passing in the opposite direction through the lanthanide series (from Lu to Pr). In both sequences the hexagonal structure is observed for relatively small lattice constants and the rhombohedral structure for relatively large lattice constants. The lattice parameters as a function of carbon concentration y for Y2FelTC v are shown in fig. 3.3, where a discontinuity in the lattice constants near the transformation point can be seen. It is also clear that the change in crystal structure is accompanied by an expansion and contraction of the lattice in the a and c directions, respectively, with ahex < arhomb, but Chex > Crhomb at the transition point. It should be noted that the value of y at which the transformation occurs was found to be different by different authors. This is due to differences in heat treatment condition and difficulties in the accurate controlling of the carbon content in the 2:17 structures. The structural and magnetic properties are very sensitive to the heat treatment condition because the dissolution of carbon in the matrix depends strongly on the annealing temperature and time (Sun et al. 1990a, Wang and Hadjipanayis

322

H. FUJII and H. SUN

TABLE 3.2 Crystallographic structure data and magnetic properties of the arc-melted carbides R2FeI7C. V is the crystal cell volume calculated from V = '/~aZc. Saturation magnetization ~s at room temperature and 2 4.2 K are expressed in the unit of ~B/formula. Compound

a (A)

c (,~)

V (,~3) AV/V (%) Tc (K)

Ce2FeI7C Pr2FeI7C Nd2FeI7C Nd2Fe17C Sm2Fel7C Sm2Fe17C Sm2Fe17C Gd2Fe17C Tb2Fe17C Dy2Fe17C Ho2Fe17C Er2Fe17C TmzFelvC Lu2Fel7C Y2Fe17C ThzFel7C1.2 Th2Fel7C1.2

8.540 8.604 8.630 8.6299 8.644 8.6297 8.624 8.627 8.602 8.585 8.572 8.538 8.524 8.487 8.589 8.697 8.694

12.424 12.466 12.474 12.4739 12.476 12.4614 12.459 12.470 12.462 12.454 12.453 8.331 8.321 8.321 12.448 12.518 12.509

784.5 799.5 804 804.5 807 803.7

2.3 1.1 1.7

804 798 795 792 526 524 519 795 820 819

3.1 3.1 3.3 3.3 3.5 3.8 3.0 3.1 3.1

as (/zB/f.u.) RT

References: [1] Zhong et al. (1990) [2] Weitzer et al. (1990) [3] Weitzer et al. (1991)

2.9

297 370 449 435 552 495 516 582 537 515 504 488 498 490 502

24.8 462

4.2 K 32.8

31.0

Ref.

40.6 36.5 34.0 35.0 35.1 23.6 19.0 17.0 17.4 18.8 24.4 35.2 35.5 35.8 34.5

[1] [1] [1] [2] [1] [2, 3] [4] [1] [1] [1] [1] [1] [1] [1] [1] [5] ND [6]

[4] Popov et al. (1990) [5] Isnard et al. (1992e) [6] Jacobs et al. (1991)

1991a, b). In the R2Fe17C series, the stability range o f the rhombohedral structure extends towards the Lu end and the structure transformation from ThzZnl7 to Th2Nil7 takes place at Er, instead o f at Gd in the pure R2Fel7 c o m p o u n d s (Zhong et al. 1990). A novel type o f stacking, in which h o m o g e n e o u s sheets o f dumbbell pairs o f iron atoms and h o m o g e n e o u s sheets o f rare earth atoms along particular ordering planes are stacked at random, has been reported for Y2FeI7Cy when y is close to 0.6 at which t h e structure transition occurs (Coene et al. 1990). These stacking faults can locally change the R site s y m m e t r y (Gubbens et al. 1989) and m a y also form an intrinsic barrier in these materials for the attainment o f high coercive forces (Buschow et al. 1990). With increasing o f carbon content, a similar structure transformation from the hexagonal to the rhombohedral symmetry was also found in the interstitial carbides prepared by the melt-spinning method (see for example Cao et al. 1993a). On the other hand such a phase transition was not observed in the G I M interstitial carbides and nitrides because o f the low reaction temperature. For R2Fe17 with the ThzZnl7 structure (R = Sin, Gd and Th), the expansion o f the cell in the a-axis direction upon carbonation is much larger than that in the c-axis direction, which could be related with the crystallographic site occupation o f the interstitial atoms. In ThzFe17Cv, when y > 1.2, the appearance o f a BaCdal type

INTERSTITIALLY MODIFIED INTERMETALLICS

0

323

8.8 I I I (a/ 8.7 a 8.6 8.5 o ~° ~ ° ~°/°-°-°-°-° 8.4' 8.3 - - ° ~ ° ~ ' ~ ° " ° - ° - ° - ° ' ° 8.2 c 8.1 8.0 550

I

I

I

(b)

600 Tc

oso-o--o-

54O

o

o

550 50O

53O

450 ,.t

520

4O0

>

350

51C 0.0

I

I

I

0.5

1.0

1.5

30O 2.0

Carbon Content, y Fig. 3.3. Carbon concentration dependence of (a) the lattice parameters and (b) Curie temperatures and cell volumes of Y2FelTCv (after Sun et al. 1990a).

phase was reported and the fraction of this new phase was found to increase with increasing carbon content (Jacobs et al. 1991).

3.1.3. Structure properties of the nitrides with intermediate nitrogen content Unlike in the arc-melted carbides, a continuous variation of the nitrogen content in the GIM nitrides was first reported not to be possible. However, more detailed studies have later demonstrated that nitrides with nitrogen content in between 0 and 3 do exist (see section 2.3.3 and the references cited there in). There are a few reports that the nitrogen content y can be as high as 6 or 8 in Sm2Fe17Nv (Iriyama et al. 1992, Wei et al. 1993), where the best magnetic properties can be obtained at y -- 3. Structural and magnetic properties of SmzFelTNu as a function of y have been investigated by Katter et al. (1992a) and they found that the SmzFel7 nitrides existed over the whole concentration range 0 ~< y ~< 3. The lattice parameters and cell volume dependences on the nitrogen content y are redrawn in fig. 3.4. The unit cell volume has increased by 6.2% at y = 3.0, but most of the expansion has occurred at y < 1.99. The gas-phase interstitial modification process can be simulated by the lattice-gas model which has been considered to bear strong resemblances to the magnetic Ising model (Skomski and Coey 1993a). The corresponding parameters are the critical temperature Tcrit in the lattice-gas system and the magnetic ordering temperature Tc in the Ising system. Terit describes a second-order phase transition; below Tcrit there is the possibility of two coexisting phases, the nitrogen-poor a-phase and the

324

H. FUJII and H. SUN 8.80

,<

,

~

8.70

~

,

/ f ' f

,

......

"

8.60 i , / i

8.5o

I

I

12.60

/

,,< v 0

12.50

820840[-F >

I

I

,~"-"-*"-*~

~,....~,/"

12.40

v

I

I

I

I

1

I

I 2.0

I 2.5

j./.~*~'-'--

800~....~ . . . . 780[760~I I 0.0 0.5 1.0

I 1.5

3.0

Nitrogen Content, y Fig. 3.4. Nitrogen concentrationy dependenceof lattice parameters a and c, and the unit cell volume V for Sm2Fe]7Ny (after Katter et al. 1992a). nitrogen-rich/3-phase. For Sm2Fel7Ny, Tcrit is not yet known, but it does not depend on the gas pressure and there are strong evidences indicating that Terit < 400°C (Fujii et al. 1994c). This indicates that above 400°C, Sm2FelvNy forms a gas-solid solution phase rather than a two-phase mixture of a and/3 nitrides. The lattice-gas model provides theoretical support for the existence of the nitrides with intermediate nitrogen concentration. 3.1.4. Location of interstitial atoms The expansion of the cell volume due to nitrogen or carbon additions has led to the conclusion that the added atoms occupy interstitial hole sites in the 2:17 structure. In order to understand the drastic changes of structural and magnetic properties accompanied by interstitial modifications, various crystallographic studies have been conducted. The interstitial carbides were studied first and later work was focused on the nitrides after they had been discovered. By X-ray and neutron diffraction studies, carbon atoms in the arc-melted carbides RzFelvCy were first reported to be located on the 3a site, which is in between the dumbbell iron atoms and which is too small to accommodate one carbon atom (Luo et al. 1987a, b). It was later established by neutron diffraction studies that the carbon atoms fill voids of nearly octahedral shape formed by a rectangle of Fe atoms and two rare earth atoms at opposite comers (Helmholdt and Buschow 1989, Haije et al. 1990), which are the 9e sites in the ThzZnl7-type structure in fig. 3.1. Similar conclusions had been drawn earlier by Block and Jeitschko (1986 and 1987 ) from

INTERSTITIALLY MODIFIED INTERMETALLICS

325

the structure refinement of X-ray data on single crystal of R2Mn17C3_y. While full occupation of the 9e (6h for the Th2Ni~7-type structure) sites by carbon atoms corresponds to the formula R2Fel7C3, only less than half of the full occupancy has been achieved in the arc-melted carbides. Neutron diffraction results on the crystallographic position and occupancy of nitrogen atoms in the interstitial nitrides R2Fe~7Ny for various rare earths are summarized in tables 3.3a and b. Nitrogen has a large neutron scattering length pi = TABLE 3.3 The occupancy factor n and the atomic position parameter x of the interstitial sites occupied by interstitial nitrogen atoms in (a) the rhombohedral Th2ZnlT-type R2FelTNy (R = Ce, Pr, Nd and Th) and (b) the hexagonal Th2Ni~7-type Y2FelTNy compounds. a 9e (0.5, 0, 0)

18g (x, 0, 0.5)

Compound

n

z

Nd2Fel7N2. 6 Nd2Fe17N2.56 Nd2FelvN2.52 NdzFe17Nk26 NdzFe17Nv NdiFe17N3 NdzFelvN2.85 NdzFel7N2.91 NdzFelTN4.5

0.86(2) 0.85 0.60(2) 0.14(5) 0.860(8) 0.984(8) 0.95(4) 0.95(4) 0.8

Ce2Fe17N3 Pr2Fe17N2.9 ThzFelvN3

0.992(12) 0.972 0.988(4)

0.806(6) 0.805

0.12(1) 0.14(5)

0.811(57) 0.854

0.01(2) 0.35

References: [1] Ibberson et al. (1991) [2] Yang Y.C. et al. (1991a) [3] Jaswal et al. (1991) [4] Yelon and Hadjipanayis (1992) [5] Miraglia et al. (1991)

Compound Y2Fe16.5N2.9 Y2FelTN2.5 Y2Fe18N2 YzFels.sN2

n

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

12i (x, 0, 0)

x

X

0.8329(5) 0.833 0.8319 0.8292(13)

References: [1] Ibberson et al. (1991) [2] Yang Y.C. et al. (1991a)

0.77(1) 0.83 0.67(2) 0.56(3)

e (A)

Ref.

8.7760(1) 8.762 8.763(1) 8.629(1) 8.776(1) 8.786(1) 8.7746(5) 8.7773(4) 8.763

12.6366(8) 12.631 12.644(2) 12.512(2) 12.661(1) 12.676(1) 12.6570(6) 12.6602(5) 12.688

[1] [2] [3, 41 [3, 4] [5] [6] [7] [7] [8]

8.737(1) 8.771(1) 8.8020

12.702(1) 12.629(1) 12.737

[6] [9] [10]

Isnard et al. (1992b) Kajitani et al. (1993) Yan et al. (1993) Isnard et al. (1992c) Isnard et al. (1993)

6h (x, 2x, 0.25) n

a (,~)

0.1465(19)

n

a (h)

c (£)

Ref.

0.10(1)

8.6393(1) 8.653

8.4749(2) 8.451

8.6622(11)

8.4702(13)

[11 [2] [3] [4]

[3] Jaswal et al. (1991) [4] Yelon and Hadjipanayis (1992)

326

H. FUJII and H. SUN

0.930 x 10 -12 cm which makes it easy to determine the location of nitrogen accurately. All the studies indicate that nitrogen predominantly occupies the 9e(6h) octahedral sites and the best fits have been obtained with exclusive occupancy of these sites in Pr2Fe17N2.9 (Isnard et al. 1992c). For Nd2Fe17, some authors also suggested a partial occupation of a second interstitial site, the 18g sites (see table 3.3a). Jaswal et al. (1991) have reported that nitrogen fills the 18g site first, to its occupancy limit n = 1/6 and then fills the 9e site to its limit n = 2/3. They also found that the cell volume increased almost linearly with increasing 9e site occupancy, but not with the total N content. In fitting the neutron diffraction spectra, Kajitani et al. (1993a, b) positioned a small amount of nitrogen atoms on the 18g site. However, they suspected that these 18g-site interstitial atoms might be hydrogen instead of nitrogen, since some hydrogen atoms coexisted with nitrogen in the samples because the nitrogenation was performed in a NH3-H2 gas mixture. A different model was given by Yan et al. (1993) who reported a nitrogen content as high as 4.5 atoms per formula of Nd2Fe17. From their structural analysis these authors concluded that nitrogen filled the 9e and 18g sites simultaneously and achieved a final occupancy of 0.80 and 0.35, respectively. The occupancy of nitrogen on the 18g site was limited to 50% of the nitrogen hexagon at alternating vertexes and consequently a full occupation of 6 nitrogen atoms per formula was suggested in the model. From the abnormally large thermal factor of nitrogen on the 18g sites, the authors concluded that vibrating ellipsoids of nearest neighbors overlapped, so that the nitrogen atoms should be able to move freely from one site to another nearest-neighbour site. The cluster consisting of the nitrogen hexagon at 18g sites and two rare earth atoms at 6c sites above and below the nitrogen hexagon was considered to be responsible for the metastability of the nitride, because the nitrogen atoms at the 18g sites would have sufficient kinetic energy to move to other sites at high temperatures. For the Y2Fel7 compound, Jaswal et al. (1991) reported that the hexagonal host compound showed considerable compositional variation as well as disorder associated with less than complete dumbbell substitution of one-third of the rare earth sites. These features remained after nitrogenation and complicated the structure refinement of Y2Fel7Ny. Nevertheless, N atoms were located at the 6h octahedral sites with 2/3 occupancy and a final composition of Y2Fe18N2. Another type of partial disorder consisting of the replacement of some of the rare earth atoms at the 2b site by a pair of Fe 4e atoms in the host Y2Fe17 was taken into account by Ibberson et al. (1991). In the nitride, they placed 12% of the N atoms on the 12i site and this led to a final formula of Y2Fe16.sN2.9 was deduced. Neutron diffraction studies of the Sm compounds are not possible because of the large absorption cross section of Sm atoms. The site occupancy of nitrogen atoms in SmzFeITN u was studied by EXAFS experiments using the SmLni edge (~ = 0.184 nm) (Coey et al. 1991a, Capehart et al. 1991). The magnitude of the Fourier transformation of the EXAFS is shown in fig. 3.5(a) and (b), where the experimental (solid line) and calculated (dashed line) radial distribution of atoms surrounding the Sm atom, before and after nitrogenation, were plotted. Figure 3.5(a) shows only the first shell of iron neighbors with 7(2) atoms at a distance of 0.310(3) nm.

INTERSTITIALLY MODIFIED INTERMETALLICS

327

(a)

N c

_c

0.0 Fig. 3.5.

0.2

0.4 0.6 0.8 Radlal dlstance (nm)

1.0

Radial distribution functions deduced from Sm2FelTLIn edge EXAFS of (a) before and (b) after nitrogenation (after Coeyet al. |991a).

" 18,-Fo

/~

9e-z / / O~c-R ,~__ ~ •

,~

"

~,

/,,°/

~(

o

/,,, O 18h-Fe / 9d-Fe /

•

~/ ~/

,~____¢

//

zo0 p~''Q''..

/

/

/- """". " '("o---3'/ Fig. 3.6. Atomic site position of z = 0 and z = 1/6 planes of the Th2Zn17 unit cell, where the interstitial 9e and 18g sites are showed.

328

H. FUJII and H. SUN

In fig. 3.5(b), the R-Fe distance has expanded to 0.316(3) nm, and another closer shell appears, corresponding to 2(1) nitrogen atoms at 0.252(2) nm. This distance, which equals to the sum of the samarium metallic radius (0.180 nm) and the nitrogen single bond radius (0.074 nm), allows the nitrogen atoms on the 9e octahedral interstitial site to be located. Yang C.J. et al. (1993) reported a detailed X-ray diffraction study of the Sm2FelTnitride made by nitrogenation of the melt-spun Sm2Fel7 compound. The site occupancy by nitrogen atoms were found to be the same as those derived from neutron diffraction and EXAFS experiments. However, they reported that the volume expansion was less for the rapid quenched compound than the cast compound. Figure 3.6 shows the atomic site position of the z = 0 and z = 1/6 planes of the rhombohedral unit cell, where the interstitial 9e and 18g sites are included. As the 9e sites are in the same e-plane with rare earth sites and have the rare earth sites as the nearest neighbors, the electronic environment of the rare earth atoms will be modified severely upon introducing N and/or C atoms into the 2:17 structure and the magnetocrystalline anisotropy is expected to be changed drastically.

3.2. Curie temperature and exchange interactions The magnetic ordering temperature Tc (Curie temperature) is governed by three kinds of exchange interactions: (1) the 3d-3d exchange interactions, which are the direct exchange interaction due to overlapping of the 3d-electron wave functions and these are strong enough to dominate Tc of the 3d rich intermetallic compounds; (2) the 3d-4f exchange interactions, which couple the 3d and 4f moments; and (3) the interactions between the rare earth spins, which are assumed to be weak and negligible in comparison with the interactions mentioned above. Exchange interactions in the R2Fe17 compounds are weak compared to those in the elemental Fe. The Curie temperatures are around room temperature for R2Fel7, whereas Tc for Fe is about 1043 K. This has been assumed to be due to the very short Fe-Fe interatomic distances at the dumbbell sites (Givord and Lemaire 1974). It is well known (the N6el-Slater curve) that 3d transition metal atoms interact with each other positively, i.e. couple ferromagnetically, at large interatomic distances, but that at too short distances they interact negatively or couple antiferromagnetically. In the RzFel7 compounds the Fe-Fe distance at the dumbbell 4f or 6c sites is typically about 2.4 A, which is shorter than 2.42 A below which negative exchange interaction occurs. This weakens the overall positive exchange interactions and lowers the Curie temperature. As a direct effect of the volume expansion by interstitial modifications, the Fe-Fe interatomic distances increase, which induces the enhancement of the exchange interactions and thus the Curie temperature increases enormously. Using the mean field model, Anagnostou et al. (1994) fitted the temperature dependence of the reduced magnetizations obtained from the M6ssbauer hyperfine fields and estimated the exchange interactions between the four different Fe sites in YzFe17 and Y2Fe17Nu. The results have shown that the exchange interaction between the 4f sites is strongly negative (antiferromagnetic) in Y2Fel7, and it increases and becomes

INTERSTITIALLY MODIFIED INTERMETALLICS I

I

i

I

I

I

329

I

800 e ~ . e .~ ,b..~ e - - ' " - " e " - - ' - " e ~ e . ~ ,

R2Fel7Ny

\/

600

..Fe,.C,

•

v

nn~ n""~" ran...,m~,..~_~.__ nn

O

"..........

400

R2Fel7

200

Y

Pr Ce

Prn Nd

Eu Sm

Tb Gd

Ho Dy

Tm Er

Lu Yb

Fig. 3.7. Curie temperature of R2Fel7, the arc-melted R2FelTC, the GIM R2Fe17Cy and R2Fe17N u (after Sun et al. 1990b and Zhong et al. 1990) through the rare earth series.

120

-

I

'

'

'

I

'

'

I

'

'

'

I

•

100

/,

'-

O

A N

80 v

u

60

o

U F'-

<1

0

40

•

I ~

0

20

o.-" -

Y2Fel7Zy

#

#

I 0

l

l

l

l

l

l

,

l

l

2

l 4

t

l

l 6

A V I V (%) Fig. 3.8. ATc/Tc as a function of AV/V for Y2Fel7 and its hydrides, carbides and nitride. The dashed line was calculated from the pressure dependence of the Curie temperature d(Tc)/dp = - 4 7 K/GPa (Nikitin et al. 1991) and the bulk modulus 13 = 128 GPa (Beuerle et al. 1991).

330

H. FUJII and H. SUN

weakly negative in Y 2 F e l 7 N v. Data on Curie R2Fel7C, the RzFelTNy are listed in table 3.1 and 3.2 and hancement of Tc are 150 K, 332 K and 386 K carbides and nitrides, respectively.

R2Fel7, the arc-melted carbides

temperatures for the host compounds GIM carbides R 2 F e l 7 C u and nitrides plotted in fig. 3.7. The average enfor the arc-melted R2Fe17C, the GIM

3.2.1. Relation between Curie temperature and the unit cell volume The volume-dependence of Te for hydrides, carbides and nitrides of Y2Fel7 is plotted in fig. 3.8. The dashed line was calculated by using the pressure dependence of Curie temperature d(Tc)/dp = - 4 7 K/GPa (Nikitin et al. 1991) and the bulk modulus B = 128 GPa (Beuerle et al. 1991). The common trend indicates that the volume effect on Tc outweighs the chemical particularities of the individual interstitials. Figure 3.9 plots Tc versus the corresponding unit cell volume for various interstitials of Sm2Fe17, which includes the data obtained by Katter et al. (1992a) for the intermediate nitrogen content interstitials Sm2Fe17N u (0 ~< y ~< 2.94), and data obtained by Christodoulou and Takeshita (1993f) for hydrides, carbides, nitrides and the combinations of them. Here a linear relationship between the two quantities is demonstrated again. Data by Fujii and coworkers (Nagata and Fujii 1991, Fujii et al. 1995a) have shown that the increase of Tc in the interstitial compounds is related linearly to the increase of the lattice constant a rather than to the unit cell volume V. The a-axis expansion induces a reduction of the hybridization between the Fe 3d and R 5d electron states. This in turn reduces the density of states of the 3d-electron at the Fermi energy and leads to the increase in Te according to the spin fluctuation theory (Mohn and Wohlfarth 1987). Most of the published results agree that the maximum interstitial nitrogen solubility in RzFe17 is 3 nitrogen atoms per formula unit and that the occupancy is exclusively on the 9e sites (Isnard et al. 1994). In this interstitial range (0 ~< y ~< 3), the Curie 600

,

,

,

800

820

840

5OO 40O ,,t 1v (.~

300 20O 100

780

Cell Volume V ( A3)

Fig. 3.9. Relationship between the Curie temperature Tc and the crystal cell volume V for the Sm2Fel7 nitrides, carbides, hydrides, carbonitrides, carbonhydrides, nitrohydrides and carbonitrohydrides. Data were taken from Katter et al. (1992a) and Christodoulou and Takeshita (19931").

INTERSTITIALLY MODIFIED INTERMETALLICS

331

temperature increase can be well explained by the magnetovolume effect, as shown in figs 3.8 and 3.9. However there are publications which claim that y could be as high as 8, but the relation between Curie temperature and volume breaks down for y > 3. Wei et al. (1993) reported that the strength of exchange interactions reaches a maximum at y = 3 and then T~ decreased with further increase of y.

3.2.2. Estimation of the strength of exchange interactions The exchange interactions can be analyzed by the molecular field model, which is commonly used to describe the variation of the Curie temperature in the R-Fe intermetallic series, under the assumption that the localized 3d-electron model is applicable. The exchange interactions take place between all unpaired spins in the 3d-4f system and they are generally considered to be of the Heisenberg type. Applying the two-sublattice molecular field model to the paramagnetic state (Belorizky et al. 1987), the following expression can be obtained,

To=1 [TFe + TR -k- i ( T F e -

TR)2 -I- 4T2Fe],

(3.1)

where

rFe = nFeFeCFe,

(3.2)

TR = ~2nRRCR,

(3.3)

and

TRFe = I ' Y [ n R F e ~

: V/(Tc -- TFe)(Tc - TR).

(3.4)

Here nij are the molecular field coefficients (with the same dimension as #o),

CR = NRg2j(j + 1)/~2/3kB, NR is the number of rare earth atoms per unit volume, CFe = NFe4SFe(SFe + 1)#2/3kB, NFe is the number of Fe atoms per unit volume and 3' = 2(9 - 1)/9. Taking SFe = 1 and neglecting TR, Tc is given by,

Tc=1 (TFe q- V/T2e + 4rR2Fe)

(3.5)

and, nFeFe and nRFe can be calculated from

TFe nFeFe = CFe

(3.6)

and

1 /Tc(Tc - TFe),

nRpe--I'Y~V ~C~e respectively.

(3.7)

H. FUJII and H. SUN

332

TABLE 3.4 Molecular field coefficients n ~ e and nFeFe of R2Fel7, R2FeI7Ny and R2Fel7Cv with y ,,~ 2 to 3.

Compound

np,Fe (/z0) Ce Pr

R2Fe17 R2Fe17Nv

---

R2Fel7Cy

--

345 695 --

nFeFe (#0)

Nd

Sm

Gd

Tb

Dy

Ho

Er

Tm

355 476 87

328 352 131

251 199 158

227 197 123

227 220 127

215 221 149

204 227 124

----

181 515 496

Taking the Curie temperature of the Lu compounds as Tve, nFeFe can be deduced and then nRFe can be obtained by substituting the appropriate Tc data of each rare earth into eqs (3.5), (3.6) and (3.7). Values of nFeFe and nRFe for R z F e I 7 and their carbides and nitrides are listed in table 3.4. As can be seen from the table, the Fe-Fe interaction is more than doubled for the nitrides and carbides compared with the parent compounds, whereas the R-Fe interaction nRFe is slightly weakened for the nitrides on average and is more signifcantly reduced in the case of carbides. This feature is reflected in fig. 3.7, in which the Tc curve of R 2 F e l 7 C y is much flatter than that of R2Fel7. Another method for the estimation of nRFe is by high-field magnetization studies (Liu J.E et al. 1991). At a critical field strength Beri, the antiparallel configuration between the R-sublattice magnetization and the Fe-sublattice magnetization in the heavy rare earth compounds is broken and the two sublattice moments start to bend towards the parallel direction with each other. In this situation the quantity nRFe can be derived from the slope of the M versus B curve above Bcri. Conclusions achieved by this method are similar to those obtained by the molecular field method. The reduction of the magnetic intersublattice coupling strength has also been concluded from the inelastic neutron scattering study of Gd2Fel7 and its nitride , Gd2FelTNu (Loewenhaupt et al. 1994), from the fitting of the temperature dependence of 166Er and 169Tin hyperfine fields in Er2Fe17C and Tm2FeayC (Gubbens et al. 1994) and from the fitting of the spin reorientation temperatures of Tm2FelvCy in terms of the crystal field and the T m - F e exchange interactions (Zhao et al. 1993).

3.3. Magnetic anisotropy In hard magnetic materials, the anisotropy energy originates from both the rare earth and 3d sublattices. The 3d anisotropy can be deduced from compounds with nonmagnetic rare earth elements. The rare earth contribution to the magnetocrystalline anisotropy is dominant at low temperatures whenever the 4f ions have non-zero orbital moments, but it rapidly decreases with increasing temperature. In the case when there are competitions between the rare earth and the 3d sublattice anisotropies, temperature-induced spin reorientation phenomena can occur as a consequence of the cancellation of the anisotropy contributions from the 4f and 3d sublattices. The easy magnetization direction of the R 2 F e l 7 compounds lies in the basal plane in the whole temperature range except for Tm2Fe17, where a change of the easy magnetization direction from basal plane to e-axis occurred at around 80 K with

INTERSTITIALLYMODIFIEDINTERMETALLICS

333

decreasing temperature (Givord and Lemaire 1974, Gubbens et al. 1976). This has been understood as the result of a competition between the easy-plane Fe and the easy-axis Tm sublattices anisotropies. Interstitial modification by nitrogen or carbon has led to a radical change of the magnetic anisotropy. In the interstitially modified compounds, spin reorientation phenomena were observed not only for R = Tm, but also for R = Er. In the case of Sm compounds, the room temperature anisotropy was found to be of strongly uniaxial character instead of planar, as in the host SmzFea7. This strong easy axis anisotropy combined with the reasonable high Curie temperature and high magnetization, makes the Sm2Fea7 interstitials to be very promising candidates for permanent magnet applications.

3.3.1. Theoretical background The main contribution to the magnetic anisotropy in permanent magnet materials is the crystal-field induced single-ion anisotropy on the magnetic rare earth-atoms. The 4f moment of the rare-earth atoms will prefer a certain magnetization direction which is determined by the properties of the rare-earth atoms themselves and the crystal electric field at the particular crystallographic sites. In compounds consisting of rare-earth and 3d metals, the easy magnetization direction is controlled by the 4f sublattice magnetic anisotropy owing to the strong exchange interaction between 3d and 4f moments. The macroscopic anisotropy energy can be expanded phenomenologically as EA : E E K~ sin~ 0 cos m ¢, n

(3.8)

m

where 0 and ¢ are the polar and azimuthal angles of the magnetization with respect to the [001] and [100] crystallographic directions, respectively. Considering the lowest order term only, the above equation can be expressed as EA = KI sin 2 0. In the absence of an external field, the spontaneous magnetization direction favours the crystallographic c-axis when satisfying the following condition: ~EA/~O = 0 and O2EA/O02 > 0, at 0 = 0. This means that a positive value of K1 will lead to an easy-axis anisotropy. On the other hand, within the ground state multiplet, the crystal field Hamiltonian on the rare earth atom is written as

HCF = E E B'~Om~(J)'

(3.9)

m

where the parameters Og~ are Stevens equivalent operators (Hutchings 1964) and B ~ are the crystal-field parameters. The thermal averages of O~ vary initially as a power n(n+ 1)/2 of the rare earth magnetization (Callen and Callen 1966). Hence the higher the order of the anisotropy, the lower its contribution at higher temperature. B ~ can be separated into terms related to the surrounding charges (A~) and terms related to the 4f ions only (On(r~)),

B'~ = O~{rn)A~,

(3.10)

334

H. FUJI1 and H. SUN

where A T are known as the crystal field coefficients, 0n is a constant depending on the rare earth known as the Stevens factor (c~j,/3j, 7J for n = 2, 4, 6, respectively) (Stevens 1952), and (r n) is the mean of the nth power of the 4f radius, which has been calculated on the basis of Dirac-Fock studies of the electronic properties of the trivalent rare earth ions (Freeman and Desclaux 1979). The second order Stevens factor a j has a fixed value for a given rare earth; the sign is related to the asymmetry of the charge cloud of the 4f electrons. The shape of these charge clouds could be either like a pancake (o~j < 0, Ce, Pr, Nd, Tb, Dy, Ho) or like a cigar (c~j > 0, Sin, Tm and Er). In the case of Gd, aa = 0, because the 4f charge cloud has spherical symmetry. For both the hexagonal and rhombohedral symmetries, the relation between the macroscopic anisotropy parameter K1 and the crystal field-related parameters can be given as (Lindgard and Danielsen 1975), K

(T = O) = - 3

d(r2)(2j2

j)AO '

(3.11)

considering only the ground state multiplet, which is usually a good approximation for most of the rare earths. However Sm is a typical exception, where the involvement of higher multiplets in the calculation is necessary (Sankar et al. 1975). The above equation means that a positive value of K1k requires that aj and A° are of opposite sign. A ° represents the lowest order deviation from sphericity of the electrostatic potential and it can be split into two terms; the contribution from the charges on other atoms in the lattice, A°(lat), and the contribution from the charges of the valence electron of the rare earth atoms themselves, A°2(val). Recent band structure calculations of crystal field parameters for rare earth intermetallic compounds have showed that the asphericity of the valence electron charge density of the rare earth itself forms the dominant contribution to the lowest order crystal field parameter A° (Zhong and Ching 1989, Coehoorn et al. 1990). The sign of A ° is negative in the pure 2:17 compounds. In the nitrides and carbides, N and C atoms occupy the interstitial sites around rare earth atoms within the basal plane (fig. 3.6), which causes the increase of the rare earth valence electron charge density in the plane in order to match those of the interstitial atoms, and A° decreases to more negative values (Coehoorn 1991). Consequently, for rare earths whose a j are positive (Sm, Er and Tm), the easy axis anisotropy of the rare earth sublattice is enhanced by interstitial modification. For the Sm interstitial compounds, the easy axis Sm sublattice anisotropy predominates that of iron even above room temperature. In the case of Er and Tm, the easy plane iron sublattice anisotropy is still dominant at high temperatures, but the spin reorientation temperature increases upon interstitial introduction.

3.3.2. Summary of experimental data The easy magnetization direction at room temperature for all compounds in the RzFe17 series lies in the basal plane. The easy magnetization direction at room

INTERSTITIALLY MODIFIED INTERMETALLICS

335

temperature can be deduced from X-ray diffraction patterns of magnetically-aligned powders (fig. 3.10). When the alignment direction is parallel to the the scattering vector of the X-ray beam, the enhanced (001) or (hk0) reflections indicate that the easy magnetization direction is parallel and perpendicular to the crystallographic c-axis, respectively. As shown in fig. 3.10, the Fe sublattice anisotropy retains the same sign of the host R2Fe17 compounds in nitrides and carbides, in which the easy magnetization direction lies in the basal plane in the whole magnetically ordered regime. The Sm compounds are the only members in the interstitial nitrides or carbides which exhibit easy e-axis anisotropy at room temperature. There are many studies on the magnetic anisotropy of the SmzFcI7 nitrides and carbides. In the arc-melted carbide Sm2FelvCv, the easy c-axis anisotropy at room temperature was observed when y ~> 0.4 by X-ray diffraction studies on magnetically aligned powder (GrSssinger et al. 1991), by magnetic measurements (Zhong et al. 1990, Popov et al. 1990) and by 57Fe MOssbauer measurement (Ding and Rosenberg 1991). When y < 0.3, SmzFe17Cv has easy plane anisotropy and when 0.3 < y < 0.45 it exhibits easy cone anisotropy (Popov et al. 1990). The room temperature anisotropy field yoHa was found to be 5.3 T for y = 1.0 (Kou et al. 1990). In the nitrides Sm2FelvN v, #oHa is also strongly related to the nitrogen content y (Katter et al. 1992a). Due to different measuring methods and different maximum nitrogen content attained by different authors, #0Ha at room temperature has been reported to be in a wide range between 11 and 26 T. By fitting the magnetization '

~oo I i~20'

'

I

I

'

'

$m2Fe 17

Y2Fel7

600

600

440

olo61-7--T ---T-v T-v-r---r$m2Fe17N2,3

Y2Fe17N2,6

5" c

3O0

,,-4,,

?7 600

;-i

440

003

V

i

L ,

,

] '

OO6

, '

, '

I I

(pog, ' '

"

I I

:

I0012 ~,--

Sm2Fe 17C2,2

Y2Fe17C2.2

30O .

.

.

.

.

.

600 I

40

60

2e (')

BO

00.3 , I

, .1_

.~_

I

....

,.009,

I

,

,

I

40

60 2e

BO

(')

Fig. 3.10. X-ray diffraction patterns of magnetically aligned Y2Fe]7, Sm2Fel 7 and their nitrides and carbides. Sm2Fel7 and all the Y compounds show easy-plane anisotropy, while SmzFelvN2.3 and Sm2Fel7C2. 2 show easy-axis anisotropy.

336

H. FUJII and H. SUN

curves of magnetically aligned powders, Katter et al. (1992a) estimated the total anisotropy constants K1 and K2 for the Sm2FelTNu series. Nitrogen concentration dependences of #0Ha, K1 and K2 are shown in fig. 3.11. It can be seen that K1 increases almost linearly with nitrogen content y, whereas K2 is not much affected when y > 0.8. The nitrogen concentration at which the easy magnetization direction reaches the c-axis can be estimated from the change of the sign of K1, the value of which has been found to be y = 0.55. An easy-cone concentration range was found at y ,-~ 0.4. In a study of the magnetic properties of Sm2Fe~7C=N v (x = 0, 0.4, 0.7 and 0.9, x + y ,-~ 3), Kou et al. (1991a) reported that the anisotropy fields were higher for x = 0.4, 0.7 and 0.9 than for x = 0, and they concluded that the effect of carbon on the crystal electric field acting on the Sm ion was slightly higher than that of nitrogen. On the other hand, Chen et al. (1993a, b) claimed that the effects of nitrogen and carbon were almost the same. Most of the reported #0Ha values at room temperature for the GIM carbides, carbonitrides and carbonitrohydrides were around 15 T (Christodoulou and Takeshita 1993f, Wei et al. 1993, Sun et al. 1992, Hu and Liu 1991). The temperature dependence of the anisotropy field for the Sm interstitial compounds has been studied by Kou et al. (1990), Katter et al. (1990), Chen et al. (1993a, b), Miraglia et al. (1991). The easy magnetization direction (EMD) remains parallel to the c-axis up to the Curie temperature and #oHa increases with decreasing temperature. From high field magnetization data, Liu J.P. et al. (1991) derived that 30

~ . . @ f

20

.j.jo.--'~"~"

"-r"

i

°

10

J

j" 0

¢ I

I

t

I,,,,

[

I

I

I

8 co

KI~ 6

J

,=J

.J

4 x,'o J

"

I

2

K2

.J ....

j ' ~ : 7

°

e - - e ~ ,

o

.2, r ~

• ,

0

,

,

,

I

1

,

,

,

,

I

2

Nitrogen Content, y

Fig. 3.11. Nitrogen concentration dependences of the anisotropy field/z0Ha , anisotropy constants K] and K2 at room temperature for Sm2FelTN u (after Katter et al. 1992a).

INTERSTITIALLY MODIFIED INTERMETALLICS

337

#oHa at 4.2 K is larger than 35 T in Sm2FelTN2.7. In a fitting of the high field magnetization processes by Kato et al. (1993), #0Ha of Sm2Fe17N3.o at 4.2 K was found to be extremely large and the field of saturation along the magnetic hard direction was estimated to be higher than 70 T. By using the singular point detection (SPD) technique, Chen et al. (1993a, b) measured #0Ha of Sm2Fe17N2.3 and Sm2Fe17C2, and Sm2Fe17C~Nv made by two different methods, in the temperature range from 370 K to 670 K. The data for the carbides and nitrides are replotted in fig. 3.12. The other two members in the 2:17 series which are interesting for the study of magnetic anisotropic properties are the compounds with R = Er and Tm, which show spin reorientation phenomena. At low temperatures, the anisotropic properties are dominated by the rare earth sublattice and the magnetization lies along the c-axis direction. At higher temperatures, the iron sublattice anisotropy dominates and the magnetization lies in the basal plane. The temperature at which the easy magnetization direction turns from the c-axis to the basal plane is the spin reorientation temperature Tsr. It is often measured by thermomagnetic scan in a low magnetic field, or by the temperature dependence of the a.c. susceptibility. It can also be deduced from analyzing the 57Fe M6ssbauer spectrum as will be described in section 3.4.2. Spin reorientation studies of various compounds made by different methods are listed in table 3.5. Some of the data taken from table 3.5 are plotted in fig. 3.13. In the arc-melted ErzFe17C v system, spin reorientation occurs when y >/0.8 and Tsr increases with increasing y (see the references in table 3.5). Two spin reorien12

10

I

[

I

oil

I

Sm2Fe,~TN2.3

kv

Sm2Fe1702

2 300

I

I

I

400

500

600

700

T (K) Fig. 3.12. Anisotropy field/~0Ha as a function of temperature for the GIM Sm2Fel7C2 and Sm2FelTN2.3. The values were obtained by the SPD technique and have not been corrected for the demagnetizing field (after Chen et al. 1993a, b).

338

H. FUJII and H. SUN

TABLE 3.5 Investigation of the magnetic spin reorientation properties of the interstitially modified R2Fel7 (R = Er and Tin) carbides and nitrides by various experimental techniques. M-T is the magnetization versus temperature curve. Compound

Method

Ref.

M/3ssbauer M-T a.c. susceptibility M-T, M6ssbauer

[1] [4] [11] [2, 3]

TmzFel7 Cu (0 ~< y ~ 1.4), arc-melted Tm2FelvCv (0 ~< y ~ 1.0), arc-melted TmzFel7Cv (0 ~< y ~< 1.4), arc-melted

M-T M-T a.c. susceptibility

[4] [5] [12]

Er2Fel7N2.7, GIM Er2Fel7N2. 7, GIM Er2Fel7N2.7, GIM

Mtissbauer M-T, MOssbauer a.c. susceptibility

[6, 7] [8] [9]

Tm2Fel7N2.7, GIM Tm2Fel7N2.7, GIM

M-T, M0ssbauer a.c. susceptibility

[8] [9]

Er2Fel7C2.2, GIM Er2Fel7C2, GIM TmzFeI7C2.2, GIM

M-T, M0ssbauer a.c. susceptibility M-T, Mtissbauer

[10] [14] [10]

Er2Fel7CxNy (0 ~ x ~ 1.5) Tm2Fel7CxN u (0 ~ x ~ 1.0)

a.c susceptibility a.c. susceptibility

[13] [13]

Er2Fel7Cy (0 ~ y Er2Fel7Cy (0 ~ y Er2Fe17Cu (0 ~ y Er2Fe17Cy (0 ~< y

~ 1.5), arc-melted ~ 2.0), arc-melted ~< 1.5), arc-melted ~ 3.0), melt-spun

References: [1] Zhou R.J. et al. (1992) [2] Kong et al. (1992) [3] Kong et al. (1993c) [4] Ohno et al. (1993) [5] Gubbens et al. (1989) [6] Gubbens et al. (1991) [7] Gubbens et al. (1992)

[8] Hu B.E et al. (1990) [9] Liu J.P, et al. (1991) [10] Qi et al. (1992) [11] Kou et al. (1991c) [12] Grtissinger et al. (1991) [13] Kou et al. (1991a) [14] Hu B.E and Liu (1991)

tation temperatures were observed when y /> 1.0 due to the existence of both the rhombohedral and hexagonal modifications in the specimen (Kou et al. 1991c). It was found that Tsr(hex) > Tsr(rhomb), as the hexagonal form is more contracted in the a-direction than the rhombohedral form. As can be seen from fig. 3.13, data obtained by different researchers are quite scattered, which could be due to the difficulty in the accurate determination of the concentration of the interstitial atoms.

3.3.3. Estimation of A ° of Sm2Fel7Ny By fitting the experimental magnetic anisotropy field data on the basis o f a twosublattice model including the exchange and crystal-electric-field (CEF), the values of the crystal field parameters A T can be deduced. Data obtained in this way for the SmzFel7 interstitials are listed in table 3.6. All the calculations included not only the ground state J = 5/2, but also the first and second excited states J = 7 / 2 and 9/2.

INTERSTITIALLY MODIFIED INTERMETALLICS 200

'

'

'

I

. . . .

I

'

'

'

339

'

Er2FeleCy 150 w

._..L---~.

,¢,

/:

100 I.-,-

50 ~'} ErzFe~TN2.7 I I j I Tm2Fe17Cy 200

~'~~ t---

150

I

I

I

I

t

O

~

A

I

I

I

I

I II

•

°

~ ) Tm2Fel7N2.7

:/1

IO0

50 0

I

i

i

]

I

i

i

i

i

2

Carbon Content, y Fig. 3.13. The spin reorientation temperature TSR as a function of carbon concentration y for Er2Fel7Cy and Tm2FelTCy. Data were taken from references in table 3.5. Values for Er2Fe17 and TmzFe17 nitrides are also included. TABLE 3.6 Crystal field parameters A~m of the Sm2Fel7 nitrides and carbides. /z0Ha (in unit of Tesla) is the experimentally measured anisotropy field at room temperature and 0 K. Compound

A20 (Kao 2)

SmzFe]TNa.94 Sm2FelTNu SmzFelTNv SmaFelTN3 Sm2Fe17C2.2

-242 - 160 :t: 5 -340 -600 -134 :k 20

A40 (Kao 4)

A°6 (Kao 4)

#oHa (T) RT

References: [1] Li and Cadogan (1991) [2] Li and Coey (1992) [3] Li and Cadogan (1992a)

8.0 + 1.0 200 -20 18.0 + 5.0

-3.0 + 1.0 - 4 4- 2.0

22 14 13.7 26.0 13.5

Ref. OK

31.2 > 70

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

[4] Zhao et al. (1991) [5] Kato et al. (1993) [6] Li and Cadogan (1992b)

Different values c a l c u l a t e d b y different authors can at least partially be attributed to the fitting o f different e x p e r i m e n t a l data. M a g n e t i z a t i o n curves o f m a g n e t i c a l l y a l i g n e d Sm2Fe17N3.0 parallel and p e r p e n d i c u l a r to the easy m a g n e t i z a t i o n direction at 4.2 K and 296 K are replotted in fig. 3.14, together with the fits, w h e r e the i m p o r t a n c e o f the inclusion o f the excited J multiplets is d e m o n s t r a t e d (Kato et al. 1993).

340

H. FUJII and H. SUN I

!

40

i

I

(c) talc. 4.2K

(a) obs. 4.2K

H//HaJ~n

__

H//c-axis....,.......-"k~.~q/

/ ~ ~ / \

H _Lc-axis

--t

20 ' ~

0

2~)0

--

}

I

i

0

200

400

i 600

800

i

I

(d) calc. 296K

(b) obs. 296K 40

20 /

i;rCol~:i2gm:~;i:td2~;tiplets

/..,.

HJ-Ha~ign

A

.......

..,,,

%=88o HJ.c-axis

I 200

O~

H (kOe)

,, i

I

200 H ( kOe )

Fig. 3.14. Observed and calculated magnetization curves of Sm2Fel7N3. 0. Solid lined in (c) and (d) represent the results in which ground, first excited, and second excited J multiplets are taken into account, while broken lines are those including the ground J mulfiplet only. The thinner solid line in (d) denotes the case when the angle 0/~ between the c-axis and the field direction is 88°, simulating a situation of incomplete alignment (after Kato et al. 1993). The saturation magnetization of the hard direction was found to be lower than that of the easy direction. This anisotropy resulted from the ferrimagnetic coupling of the Sm and Fe moment when the external field was applied along the hard direction, whereas they coupled ferromagnetically when the field was in the magnetic easy direction (Kato et al. 1993, Zhao et al. 1991, Li and Cadogan 1991). The reason for this field induced ferrimagnetism has been explained by the intermultiplet mixing which originated primarily from the molecular field interaction rather than the CEF interaction. As the relative contribution from the molecular field becomes larger than that from the CEF at elevated temperatures, this ferrimagnetic coupling is more pronounced at 296 K than at 4.2 K. Xu et al. (1993) analyzed the data obtained by Iriyama et al. (1992) for Sm2Fe]7N v (0 < y < 6) and estimated the effect of different interstitial nitrogen sites on A T. They concluded that nitrogen atoms on the 9e site had a negative contribution to A °, and hypothetical nitrogen atoms on the 3b and 18g sites contributed a positive value to A ° and reduced the easy-axis anisotropy, which was reported to be in agreement with the experimental data. The observed spin reorientation temperatures of Tm2Fe]7Cv (0 ~< y ~ 2.2) was fitted by Zhao et al. (1993) and the average A ° value of the two Tm sites was derived (fig. 3.15). The absolute value of the average A ° increases with increasing carbon content in a nearly linear relationship. Similar results have been obtained by Li and Cadogan (1992b) for Sm2Fe]7C v and Sm2Fel7Ny.

INTERSTITIALLYMODIFIED INTERMETALLICS 500

. . . .

,

. . . .

341

,

Tm2FelTCy 400

,~" ,/ /

J"

300

200

/

/ /"

1oo

O0

,

,

,

,

I

I

,

,

,

,

I

2

,

Carbon Content, y

Fig. 3.15. Carbon concentration dependence of the average second-order CEF parameter A° for Tm2Fe17Cy (after Zhao et al. 1993). A0 was obtained by fitting the data of the spin reorientation temperatures.

3.4. MSssbauer and NMR studies MSssbauer spectroscopy is an effective experimental technique for obtaining information of the magnetic properties on an atomic scale. Both the 57Fe M6ssbauer and some of the rare earth M6ssbauer effects have been studied for the interstitially modified compounds.

3.4.1. 57Fe MOssbauer studies As there are four different crystallographic sites for iron in both the hexagonal and the rhombohedral-type R2Fe17 compounds, the observed spectrum must be a superposition of at least four sextets. Point charge calculations showed that under the combined effect of the dipolar field and quadrupole interaction, only the dumbbell 4f(6c) sites remain equivalent (Steiner and Haferl 1977). When the magnetization direction is along the c-axis, the angle between the hyperfine field and the electric field gradient is the same for all crystallographic equivalent sites so that there is no additional splitting and the sites remain equivalent for both the structures. When the magnetization direction is in the basal plane, for the rhombohedral structure, the 9d, 18h and 18f sites each splits into two groups with an intensity ratio 2:1, while in the hexagonal structure the 6g and 12k sites split into two groups with an intensity ratio 2:1, and the 12j site splits into three groups with an intensity ratio 1:1:1. Since the dipolar fields at two of the three groups of the 12j sites are very close, they can be treated in the same way as the 18f site in the rhombohedral compounds by splitting them into two groups with an intensity ratio 2:1. Thus the spectra of all the RzFelv compounds can be fitted to seven independent sextets.

342

H. FUJII and H. SUN

In the case of the interstitial modified carbides and nitrides, it is possible that the interstitial sites are not fully occupied. Then the 12j(18f) and 12k(18h) iron sites, which have one neighboring octahedral interstitial site, can have either one or zero interstitial neighbor. For most of the nitrides, as the occupancy of the interstitial sites is nearly full, the probability for one neighbor is much higher than that for zero neighbor. Hence it is reasonable to fit the spectra in the same way as for the host compounds and the variation in their near-neighbor environments may be considered as the reason for the broadening of the absorption lines. Ten subspectra were used by Long et al. (1994) in the fitting of the Th2FelTN2.6 spectra because the easy axis of magnetization was found to be in a general basal direction and not oriented along one of the basal axes. For much lower interstitial content, especially in the case of the arc-melted carbides, a further splitting of the subspectra arising from the influence of the substoichiometric concentration has to be taken into account, which complicated the fitting procedure. Some authors simply ignored this influences or derived the parameters of the overall averaged hyperfine field and the distinctly separate hyperfine field of the 4f(6c) dumbbell sites only (Zhou R.J. et al. 1991 and 1992). As to the site assignment of the spectra, most of the analyses were based on the hyperfine field and intensity considerations. The idea is that the strength of the hyperfine field on each site is predominantly determined by the number of iron and rare earth near-neighbours of the site. The higher the number of iron-neighbours, the larger is the hyperfine field, whereas the higher the number of rare earth neighbors, the smaller is the hyperfine field. According to the above considerations, Hu B.P. et al. (1991) decided that the hyperfine fields were in the order of 4f(6c)> 6g(9d)> 12j(18f)> 12k(18h), which agreed with by many other studies, while Kong et al. (1993c) fitted their spectra in a different order of 4f(6c)>12k(18h)>12j(18f)> 6g(9d). Another way of spectra assignment has been used by Long et al. (1992), who took isomer shift as a main clue for the assignment. The model was based on the Wigner-Seitz size cell environment of each iron site and they also considered the orientation of the magnetization and the magnetic moments as determined from either neutron-diffraction measurements or band structure calculations. The data of 57Fe Mrssbauer hyperfine fields are summarized in table 3.7, where data for the 6g(9d), 12j(18h) and 12k(18f) sites are the weighted averages. It can be seen that although the fitting and assignment procedures are different, the values of the overall averaged hyperfine field (Bhf) does not differ too much. However, the individual hyperfine field values at the various sites scatter a lot. The overall average hyperfine field of RzFel7, RzFelvN v and R2Fel7Cy at 15 K are plotted in fig. 3.16. The general feature is that the average hyperfine field (Bhf) across the series increases by about 4 T in the nitrides, but it decreases slightly in the carbides. These results could be understood by considering the different polarization effects on Fe atoms by their nitrogen and carbon neighbors. The hyperfine field in metals is largely due to the Fermi contact term Bs, which is proportional to the unpaired spin density at the nucleus. The increase of the hyperfine field indicates a larger polarization of s electrons. It is considered that the 4s band of Fe is more highly polarized by nitrogen atoms in nitrides than by carbon atoms in carbides,

INTERSTITIALLY MODIFIED INTERMETALLICS

343

TABLE 3.7 57Fe M0ssbauer hyperfine field (in units of Tesla) of each crystallographic site (averaged over subspectra) and the overall weighted average (Bhf) for the RzFe17 interstitial compounds at various temperatures. The superscripts a, b and c correspond to different methods of making the carbonitrides as described in section 2.3.4. Compound

6g(9d)

12k(18h)

12j(18f)

4f(6c)

(Bhf)

T (K)

Ref.

Ce.2FelTNu Ce2FeI7C v PrzFelvN u Pr2Fe17Cv Nd2Fe17Nv Nd2Fe17Cy Sm2Fel7N v SmzFet7Cy SmzFe17 SmzFelyNo.4o SmzFelvNo.81 Sm2Fe17N1.2o Sm2Fe|7N1.99 Gd2Fe17Ny Gd2Fe17C v Gd2Fe17C2.o GdzFe17 Gd2Fe17Co.5 Gd2FelvC1. 0 Gd2Fel7C1.5 Tb2Fe17Ny Tb2Fe17C v DyzFe17N u Dy2Fel7C u Ho2Fel7Nv HozFe17Cy Er2Fe17N v ErzFe17C v Er2Fel7C2.5 Er2Fel7 ErzFelvCo.5 Er2FeI7Co.8 Er2Fel7Cl.o Er2Fe17C1.2 Er2Fe17C1.4

36.7 30.7 35.6 33.5 36.1 33.7 39.4 36.2 32.2 34.3 37.4 38.2 39.5 36.7 34.3 25.5 26.6 27.4 26.2 26.0 37.2 34.9 37.3 34.8 36.8 35.1 40.7 36.5 24.7

31.4 25.4 30.8 25.1 30.5 25.1 31.1 26.2 27.4 28.7 30.3 31.4 31.8 31.5 25.5 31.1 30.0 31.2 32.0 31.5 31.6 26.0 32.6 26.4 32.3 26.4 33.5 26.1 28.4

34.2 31.6 33.3 28.5 33.3 29.0 35.6 30.2 30.1 31.4 34.1 35.4 35.7 34.8 29.3 32.1 32.0 32.8 33.0 32.5 34.6 31.4 35.3 31.3 35.8 30.9 35.9 30.7 35.4

39.9 36.4 37.1 34.5 36.8 34.7 32.9

33.1 25.3 31.7 25.5 31.6 26.2 28.2

35.3 31.1 35.5 30.0 35.2 30.7 29.7

38.3 34.7 37.7 35.4 38.8 35.2 41.6 36.5 36.1 37.6 41.9 42.3 42.1 38.8 36.5 36.7 36.9 37.0 37.1 36.6 39.5 37.3 40.1 36.6 40.7 36.5 42.5 37.4 38.0 37.4 37.5 41.4 40.9 40.9 39.7 40.0 41.7 37.4 40.4 36.8 40.2 36.2 34.6

34.1 29.7 33.3 29.0 33.5 29.2 35.4 30.6 30.3 31.7 34.3 35.3 35.7 34.5 29.7 31.1 30.9 31.8 31.9 31.4 34.6 30.8 35.3 30.8 35.3 30.7 36.8 30.9 31.4 31.6 32.3 33.0 32.9 32.7 32.4 32.6 36.2 30.7 35.0 30.0 34.8 30.5 30.3

15 15 15 15 15 15 15 15 4.2 4.2 4.2 4.2 4.2 15 15 12 12 12 12 12 15 15 15 15 15 15 15 15 12 4.2 4.2 4.2 4.2 4.2 4.2 4.2 15 15 15 15 15 15 15

[1] [2] [1] [2] [1] [2] [1, 3, 4, 6] [2] [3] [3] [3] [3] [3] [1] [2] [15] [15] [15] [15] [15] [1] [2] [1] [2] [1] [2] [1] [2] [14] [13] [13] [13] [13] [13] [13] [13] [1] [2] [1, 6] [2] [1, 4, 5] [2, 5] [5]

Er2Fe17C1.5 TmzFe17N u Tm2FelvCu LuzFe17Ny Lu2Fe17Cy YzFe17Nv YzFelvCy YzFelTH2.7

344

H. FUJII and H. SUN TABLE 3.7 (Continued)

Compound

6g(9d)

12k(18h)

12j(18f)

4f(6c)

(Bhf)

~/~(K)

Ref.

Pr2Fel7N2. 6 Nd2Fel7N2. 6 SmzFelvNv Sm2Fel7 C:~N,~ Sm2FelTC~Nv~ Sm2FelTCxN~ Sm2Fe17C2 Sm2Fe17H3.7 Er2Fel7C2.5 Lu2FelTNu Y2FelTN2.3 Y2FelTC2 Y2FelTC~N~ Y2Fel7CxNub YzFelvC~N~ Th2FeI7N2. 6 Th2FelTN2. 6

37.4 37.6 39.7 33.7 33.2 38.9 33.2 32.1 24.6 36.9 35.3 29.8 31.3 32.5 34.9 35.4 30.3

31.5 32.8 31.1 26.7 28.2 30.4 25.7 29.1 28.2 31.7 31.4 25.9 27.5 28.7 30.8 33.1 32.6

31.8 30.4 35.7 28.9 31.6 34.9 27.9 30.7 35.1 34.5 34.9 28.4 30.4 31.6 33.8 32.7 35.9

35.9 36.0 41.7 36.7 39.5 41.2 37.6 34.0 37.5 39.7 40.1 35.9 36.3 37.6 39.2 35.7 35.8

33.2 33.3 35.5 29.9 31.6 34.8 29.2 30.8 31.1 34.3 28.6 30.2 31.4 33.6 33.7 33.8

85 78 77 77 77 77 77 77 70 77.4 77 77 77 77 77 85 78

[10] [8, 9] [4, 6, 7] [7] [7] [7] [7] [7] [14] [6] [7] [7] [7] [7] [7] [11] [12]

Ce2FelTNy Ce2Fel7Cy Pr2FelTNy PrzFelvN2.6 PrzFelTCy NdzFe17N u Nd2Fel7N2. 6 NdzFe17Cu SmzFel7N~ Sm2Fe17 SmzFe17No.4o Sm2FelTNo.81 Sm2Fe17Nl.20

33.6 29.7 34.2 35.5 31.2 34.1 35.9 30.8 37.3 23.0 26.5 30.6 33.6 36.8 32.8 34.6 32.7 23.8 22.7 24.1 24.5 23.9 34.6 32.8 33.8 31.7 33.8 31.7 33.5 30.4

28.9 20.7 29.2 30.2 22.8 28.7 31.4 22.8 29.5 19.5 21.4 23.9 26.3 29.0 24.1 29.6 24.0 26.9 24.3 26.0 27.0 27.0 29.2 23.7 29.2 23.8 29.0 23.5 28.5 22.8

31.9 26.1 31.3 29.9 26.5 31.2 29.0 24.1 33.3 21.9 24.2 27.6 30.7 32.9 25.0 32.0 26.8 30.3 25.6 27.6 29.0 29.5 32.3 28.4 32.6 28.4 32.2 28.4 31.9 26.9

35.9 33.5 36.6 34.2 33.0 36.9 34.3 31.8 39.0 26.4 29.1 36.7 37.7 38.9 34.1 36.7 32.9 33.6 30.2 32.0 32.7 32.8 37.2 32.8 37.3 33.7 37.2 33.0 37.0 32.6

31.6 25.7 31.7 31.5 26.8 31.5 31.7 25.7 33.3 21.8 24.2 27.9 30.5 32.9 27.1 32.2 27.6 28.3 25.2 26.9 28.0 28.1 32.2 28.1 32.2 28.0 31.9 27.8 31.6 26.7

293 293 293 295 293 293 295 293 293 RT RT RT RT RT 293 293 293 293 293 293 293 293 293 293 293 293 293 293 293 293

[1] [2] [1] [10] [2] [1] [8, 9] [2] [1, 3, 4, 6] [3] [3] [3] [3] [3] [2] [1] [2] [15] [15] [15] [15] [15] [1] [2] [1] [2] [1] [2] [1] [2]

Sm2FelvN1.99 Sm2FelvCu Gd2FelTNy Gd2Fe17C u Gd2Fe17C2.o Gd2Fel7 Gd2Fe17Co.5 Gd2Fe17Cl.o Gd2Fe17Cl.5 Tb2Fel7N u TbzFe17Cy DyzFe17Ny DyzFel7C u Ho2FelTNu Ho2Fe17Cy Er2FelTNy Er2Fel7Cy

INTERSTITIALLY MODIFIED INTERMETALLICS

345

TABLE 3.7 (Continued) Compound

6g(9d)

12k(18h)

12j(18f)

4f(6c)

(Bhf)

T (K)

Ref.

Er2Fe17C2.5 Tm2FelTNu Tm2FelTCy Lu2Fe]7N u Lu2Fe17N u Lu2FezTC v Y2Fe]7Ny Y2Fe17Cu Y2Fe17H2.7 Th2FeITN2. 6

22.2 34.5 31.2 34.2 32.7 31.8 34.0 30.5 14.8 33.0

24.0 28.7 23.3 28.4 28.1 23.2 28.5 23.1 3.0 30.8

29.9 32.0 26.9 31.9 31.3 27.5 32.2 26.1 10.8 30.7

33.1 36.5 33.2 36.6 35.5 33.5 36.9 33.3 18.5 33.7

26.8 31.8 27.1 31.6

300 293 293 293 300 293 293 293 293 295

[14] [1] [2] [1] [6] [2] [1, 5] [2, 5] [5] [11]

References: [1] Hu B.R et al. (1991) [2] Qi et al. (1991) [3] Zhou R.J. et al. (1993) [4] Kapusta et al. (1992) [5] Qi et al. (1992b) [6] Zouganelis et al. (1991) [7] Chen et al. (1993b) [8] Long et al. (1992)

[9] [10] [11] [12] [13] [14] [15]

. o. R2Fel . . 7 I x R2Fel7CY • R2Fel7Ny

38 - 36

27.4 31.8 26.7 9.7 31.5

Pringle et al. (1992) Long et al. (1993) Long et al. (1994) Jacobs et al. (1991) Zhou R.J. et al. (1992) Kong et al. (1993c) Kong et al. (1993d)

. . . .

I

. . . . •

~- 3 4

• o

~-32 m 30

28 26 --o

(a)

I!l!l o R2FeI7 x R2Fel7Cy

(b)

• R2Fel7NY

0.1

Q

E E v

x

0.0

X X x N x x oo

x oo

X--.

o o

-0.1

~ l l l l

CePrNd

I l l l ~ l l l Sm

Gd Tb Dy Ho Er Tm

Lu

Fig. 3.16. Overall average (a) hyperfine fields and (b) isomer shifts of R2Fe]7, R2Fe17N u and R2FelTCy at 15 K (after Hu B.R et al. 1991, Qi et al. 1991).

346

H. FUJII and H. SUN

and consequently the magnetic hyperfine field is higher in nitrides than in carbides. Qi et al. (1992b) have attributed the difference of the hyperfine field for nitrides and carbides to the transferred hyperfine field from interaction with neighboring atoms, which is sensitive to the chemical nature of the interstitial impurity. The importance of this transferred hyperfine field can also be seen from the fact that the proportionality of the incremental moment and hyperfine field is not valid in interstitially modified intermetallics. 57Fe Mrssbauer studies of interstitial compounds with intermediate interstitial concentration have been carried out for the arc-melted ErzFel7Cy (0 ~< y ~< 1.5) by Zhou R.J. et al. (1992), for the arc-melted and melt-spun Gd2Fe17Cu by Kong et al. (0 ~< y ~ 2.0) and for Sm2Fe17Nv (0 <~ y ~< 2.94) by Zhou R.J. et al. (1993). Hyperfine field data for the above three series of compounds are included in table 3.7 and the data for Sm2Fe17Nv at 4.2 K are replotted in fig. 3.17. Also listed in table 3.7 are the 77 K magnetic hyperfine fields of the carbonitrides of Y and Sm compound made by Chen et al. (1993b), where the superscript a, b and c indicate the different synthesizing routes, for detail see ref. [7] of table 3.7. There is a general increase in the overall isomer shifts for the interstitially modified R2Fe17 compounds. The increase is 0.12 mm/s in the nitrides, which is higher than the value if considering the volume effect only. The situation in the carbides is the opposite, the 0.05 mm/s increase in the average isomer shift is lower than the contribution from the cell volume increase. This suggests that the interband charge

45

40

I-~'~

'

f

~

~

I-•

6c

9d

_

t

18f

35

m=

18h

~

30

~

25

~

-

~

I

~------------

I

< Bhf >

I-

35

rn V

30

"'

A

0

I

1

°

I

2

3

N i t r o g e n Content, y

Fig. 3.17. Dependence of the hyperfine field at the four Fe sites and the average hyperfine field (Bhf) on the N concentration for Sm2Fe17Ny at 4.2 K (after Rosenberg et al. 1993).

INTERSTITIALLY MODIFIED INTERMETALLICS

347

transfer occurs in the opposite sense in nitrides and carbides. As the isomer shift increases with decreasing 4s occupation, the increase of 3d occupation will also increase the isomer shift because the 4s electron density at the nucleus will be reduced owing to the expansion of the 3d shell. Thus it was suggested (Qi et al. 1991) that there could be a 4s-+3d transfer in the nitrides and a smaller 3d--+4s transfer in the carbides, which is consistent with the consideration of the average electron density at the boundary of the Wigner-Seitz cell. The other possibility is a greater 4s-+2p interatomic charge transfer in the nitrides, on account of the greater electro-negativity of nitrogen than carbon. However, definite conclusion can not be drawn from the M6ssbauer data alone.

3.4.2. Spin reorientation studies by 57Fe M6ssbauer measurements Besides the Fermi contact term Bs, the magnetic hyperfine field Bhf also contains an anisotropic orbital contribution Bo created by the electronic current around the nucleus. When there is a temperature induced change of the magnetization direction (the spin reorientation), there is a discontinuity in the temperature dependence of the magnetic hyperfine field. The temperature at which the hyperfine field anomaly happens corresponds to the spin reorientation temperature Tsr. Thus Tsr c a n be determined by studying the temperature dependence of the 57Fe M6ssbauer hyperfine field. In the R2Fel7 compounds and their interstitial compounds, spin reorientation occurs when R = Tm and Er. The hyperfine field discontinuity results mainly from the reduction of the orbital contributions when the iron magnetic moments rotate from the direction perpendicular to the c-axis to parallel to the c-axis with decreasing temperature. The anomaly is most pronounced for the dumbbell 4f(6c) site. An example is given in fig. 3.18, which shows the spin reorientation transition of ErzFe17Cy. Tsr determined by this method has been proved to be in good agreement with those obtained by a.c. susceptibility measurements and thermomagnetic scans. 3.4.3. Rare earth M6ssbauer spectroscopy and NMR studies The principal component of the electric field gradient Vz~ at the nucleus can be deduced directly from M6ssbauer quadrupole splitting when the electric field gradient tensor has axial symmetry with Vx~, = Vvv. The M6ssbauer spectroscopy of 155Gd nuclei is of special interest because the spherical 4f shell of Gd does not itself contribute to V~z at the nucleus. In other types of rare earth M6ssbauer spectroscopy, for instance 166Er and 169Tin, the crystal field contribution to V~ can be deduced by taking the difference of the total V~ and the free ion contribution (the asymmetric 4f ion contribution). Besides rare earth M6ssbauer studies, NMR is another experimental technique from which V~ can be obtained. Within the point charge model, which is based on considering the electrostatic charges due to surrounding ions as point charges and performing lattice summations over a sufficient number of neighbors, the following relation was often used, e V ~ z = - 4 C A °,

with

1 -- ")1oo

C - - - ,

1--o'2

(3.12)

348

H. FUJII and H. SUN I

I

35

I~

O ~ Q •

I

~

4

f

i-30

tn

_, . . . .

25

~ 2 k I

I

" *1

34 I-30

A rn

"~'0~0

V

26 0

I

I

I

100

200

300

T(K)

!

I

!

I

i

I

,

r.0 v

/

zJ g 0 co

~xxxx~ !

,

I

110

l

I

a

120

I n l l 130

140

T(K)

Fig. 3.18. Hyperfine fields at the four Fe sites, the average hyperfine field (Bhf) and the a.c. magnetic susceptibility of Er2FelvC2. 2 as a function of temperature(after Qi et al. 1992a, b). The spin reorientation temperature is indicated. where "Too is the Sternheimer antishielding factor and ~r2 is the screening constant. If the semi-empirical value of C can be determined, then A ° can be calculated from V~. Data of V~ and A ° found in the literature are summarized in table 3.8. It can be seen that there is a nearly three-fold increase in V~ for the nitrides with respect to the V~ value of the host compounds. If the ratio of A ° to V~, is preserved, there is in turn a three-fold increase in the absolute value of A °. The value of V~ for Gd2ColTN v is somewhat larger than that for Gd2FelTNu. This could mean that the rare earth sublattice anisotropy in the Co compounds at 4.2 K is larger than in the corresponding Fe series (Mulder et al. 1992). In contrast to carbon or nitrogen insertions, hydrogen was found to lower V~ at the Gd nuclei and reduce the crystal field induced anisotropy.

INTERSTITIALLY MODIFIED INTERMETALLICS

349

TABLE 3.8 The electric field gradient V,z at the rare earth nucleus deduced from the M6ssbauer quadrupole splitting and the second order crystal field parameter A ° estimated from Vzz for various R2FeI7 and their interstitial compounds. Compound

Isotopes

Gd2Fel7 Gd2Fel7CI. 2 Gd2FelTN3 Gd2Co17 Gd2Co17N3 Gd2Fe17 Gd2Fe17H3 Gd2FelTH5 Er2Fel7 Er2Fe17C ErzFel7N2.7 Tm2FeI7C Tm2Fel7N2.7 Sm2Fe17 Sm2Fe17N3

155Gd 155Gd 155Gd 155Gd 155Gd 155Gd 155Gd 155Gd 166Er 166Er 166Er 169Tm 169Tm NMR NMR

References: [1] Dirken et al. (1991) [2] Mulder et al. (1992) [3] Isnard et al. (1994b) [4] Gubbens et al. (1991) [5] Gubbens et al. (1992)

Vzz (1021V/m 2) 4.3(1) 9.3(3) 12.6(2) 4.8 14.9 4.02(13) 2.66(11) N0 9.9 4. 1.5 9.7 4- 1.5

A° (Kao 2)

Ref.

-200 -430 -580

[1] [1, 9] [1] [2] [2] [3] [3] [3] [4, 51 [4, 5, 101 [4, 5] [6, 5, 10] [5] [7, 8] [7, 8]

-351 -233 ,-~0 -50 4- 100 -290 4. 50 -400 -t- 50 -300 4- 50 -300 4- 50

10.2 33.9 [6] Gubbens et al. (1989) [7] Kapusta et al. (1992) [8] Kapusta et al. (1992c) [9] Dirken et al. (1989) [10] Gubbens et al. (1994)

The validity of the general proportionality relation between Vzz and A ° has been questioned by Coehoorn and B u s c h o w (1991) from band structure calculations. Problems arise from that Vz~ (or the quadrupole splitting) measures the a s y m m e t r y of the electric charge distribution at the rare earth nucleus, whereas A ° is a measure of the a s y m m e t r y charge distribution experienced by the rare earth 4f electrons. Although contributions from 6p and 5d electrons are of equal importance to A °, V** is determined almost entirely by the p electron charge density. However, within a series of structurally related compounds, experiments have shown that the A°/Vz~ (Kao2/1021Vm -2) ratio is m o r e or less a constant, which is - 4 6 + 3 for the 2:17 c o m p o u n d s (Dirken et al. 1991). It should be mentioned that A ° estimated in this w a y has a relatively large error due to the uncertainty of the factors 7o~ and cr2. As an example, unreasonable large value of A ° ~ - 1 0 0 0 K a o 2 has been estimated for the Gd2FeITC v c o m p o u n d (Dirken et al. 1989, Jacobs et al. 1990).

3.5. Substitution effect Similarly to the various substitution studies on the host R2Fel7 compounds, the effect of substitutions for both the Fe sublattice and the R sublattice by other metal or nonmetal elements on the structural and magnetic properties have been studied for the R2Fe17 interstitial compounds. The results will be summarized in this section.

350

H. FUJII and H. SUN

3.5.1. Substitution of Co for Fe The unit cell volume of R2Co17 is smaller than that of R2Fel7 and thus the spatial size of the interstitial sites is smaller in the Co compounds, which makes it more difficult to introduce interstitial atoms into the 2:17 lattice. Early studies on the Y2(Fe1_xCo~)17N v series by Hurley and Coey (1991) have showed that the 2:17 nitride phase exists only when x < 0.85. They found that the suitable nitrogenation temperature increased with the Co content x and at x > 0.85 the required nitrogenation temperature became higher than the nitride phase decomposition temperature. The amount of absorbed nitrogen atoms decreased with increasing Co content, which changes from 2.6 for x = 0 to 1.6 for x = 0.8. Similar conclusion has be reported by Xu and Shaheen (1993a, b) on the R2(Fel_xCo~)17Ny series with R = Ce, Pr and Nd, where the 2:17 nitrides were formed in a limited range of x ~< 0.6. The situation was more critical for the Ce compounds and reasonable pure nitride phase was successfully synthesized only in the range of 0 ~< x ~< 0.3 (Xu and Shaheen 1993b). However, later work by Katter et al. (1992b) has proved that by using lower nitrogenation temperature (~< 450°C) and longer nitrogenation time the nitride phase can be formed for the whole substitution range from x = 0 to 1.0 in S m 2 ( F e l - x C o x ) 1 7 N 2 . 7 , although the decomposition temperature of Sm2Fe17Nv is reduced by the substitution of Co and the diffusion of nitrogen in Sm2Co17 is slower than in S m 2 F e l 7 . In fact, the nitrides R 2 C o l 7 N u of all the rare earth members have been synthesized by Liu J.R et al. (1993) and their structural and magnetic properties were studied. Data obtained by Liu et al. (1993), and other authors are summarized in table 3.9. After nitrogenation, the Curie temperature is lower than that in the parent compounds. TABLE 3.9 Crystallographic structure data and magnetic properties of R2ColTNy. V is the crystal cell volume calculated from V = -~a2c. Saturation magnetization Ms and anisotropy field/t0Ha are data pertain to room temperature and are expressed in units of Tesla. R

y

a (A)

c (]k)

V (~3)

A W E (%) Ms (T)

Ce Pr Nd Sm Sm Sm Gd Tb Dy Ho Er Tm Y

2.7 2.7 2.1 2.6 2.7 2.2 2.2 2.0 1.7 1.6 2.4 2.5 2.0

8.58 8.63 8.62 8.57 8.591 8.584 8.55 8.48 8.47 8.46 8.42 8.42 8.48

8.30 12.40 12.31 12.40 12.473 12.462 12.37 8.44 8.38 8.35 8.35 8.35 8.32

529.4 800.1 795.6 789.6 794.9 795.3 784.4 525.9 521.3 517.5 513.7 512.2 518.6

8.0 5.9 6.0 6.1 6.2 7.1 6.1 6.2 6.0 6.0 5.8 6.4 5.7

References: [1] Liu J.R et al. (1993) [2] Katter et al. (1992b) [3] Hu B.R et al. (1992b)

1.03 1.02

Tc (K)

840 811

#0Ha (T)

16.4 11.8

Ref. [1] [1] [1] [1] [2] [3] [1] [1] [1] [1] [1] [1] [1]

INTERSTITIALLYMODIFIED INTERMETALLICS

351

This has been explained as a result of the weakened Co-Co exchange interaction and the reduced Co moment. From the drastic increase of the high field differential susceptibility at 4.2 K, Liu et al. suggested that the R-Co exchange interaction was strongly reduced by nitrogenation. Nitrogenation was found to have a strong influence on the magnetic anisotropies of both the 3d and rare earth sublattices. The easy axis anisotropy range of 0.5 < z ~< 1 in Y2(Fel-xCox)17 was extended to 0.15 < z ~< 1 in Y2(Fel_xCo~)ITNu (Hurley and Coey 1991). As Y is a nonmagnetic ion, this reflects the modification of the interstitial nitrogen on the 3d anisotropy. Ce in Ce2Fel7 can roughly be considered to be in a nonmagnetic state because of the strong hybridization between the Ce 4f-electron states and the ligand 3d-electron states. Experiments have demonstrated that in Ce2(Fel-:cCox)17 the transition from easy plane to easy axis anisotropy with increasing z is shifted from z ~ 0.5 to z ~ 0.15 upon nitrogenation, which is in agreement with the result on the yttrium compounds (Xu and Shaheen 1993b). The maximum of the concentration dependence of the room temperature saturation magnetization has been found to be located at z = 0.2 in Y2(Fel_=Coz)17Nu, which is close to z = 0.35 in the host compounds (Hurley and Coey 1991). The combined effects of nitrogenation on the anisotropy and magnetization make it possible to maximize the saturation magnetization while simultaneously achieving uniaxial anisotropy of the 3d sublattice by choosing appropriate z values. Some of the results obtained by Katter et al. (1992b) in a systematic study of the Sm2(Fe1_~Co=)17N v series are replotted in fig. 3.19. It is showed that Tc first .

1200

.

.

.

Sm2(Fel.xCo,)17

I

.

"

v

/

.

.

o..._...--- -~

o~

~1000 v 800 .....,.~. o ~ e ~ ° 600

.

o/

~

Srn2(Fe~-xCOx)lTNy

400, I

I

/

/

Srn2(Fet.,eOx)lTNy 1.6 ~

I

I

I

[

--°~°~'~O~o~

1.2 0.8

1-

I

I

I

I

I

Sm2(Fe1.xCOx) lTNy

25. t o ~ o "

]

L

I

I

[

e ~ o ~ e ~

m

20 ::~

15 10 0.0

l

l

l

l

l

l

l

r

l

0.5

1.0

Nitrogen Content, y

Fig. 3.19. Composition dependence of the Curie temperatureTc, the saturation polarization Js and the anisotropy field /~0Ha (determined by the SPD method) for Sm2(Fel_xCo=)17Ny (T = 293 K). Also included are Tc of Sm2(Fel_=Co=)]7 (open circle) (after Katter et al. 1992b).

352

H. FUJII and H. SUN

increases with increasing Co content x reaching a maximum at x = 0.5 and then decreases again, which is different from the monotonical increase with x in the host series. Nitrides for the entire x range show a strong uniaxial anisotropy at room temperature, whereas the easy axis of magnetization of the host compounds lies in the basal plane for z < 0.45. Most excellent intrinsic magnetic properties for hard magnetic applications were achieved for Sm2(Fe0.sCo0.2)lTN2.8 with Ors = 1.55 T, #0Ha = 23.7 T at room temperature and Tc = 842 K. 3.5.2. Substitutions of elements other than Co for Fe The effects of substitution for Fe of Si, Ga, A1, Ti, V, Ni and Nb etc. have been studied by Hu B.E et al. (1992), Tang et al. (1992), Li X.W. et al. (1993), Valeanu et al. (1994) and Middleton and Buschow (1994) on Rz(Fel-xMx)a7Ny compounds, where M is one of the above mentioned elements. The largest stability range of the 2:17 nitride phase was reported for A1, with 0 ~< z ~< 0.4, for R = Sm (Li X.W. et al. 1993) and with 0 <~ x < 0.6 for R = Ce (Middleton and Buschow 1994). For all other M elements, only data with x < 0.2 have been reported. Although many of the substitutions raise the Curie temperature of the host RzFe17 compounds, the general effect of substitution in RzFelvNu has been found to reduce To. Exceptions are Ti and V, for which Hu B.E et al. (1992b) showed that Tc of Sm2(Fe0.982Ti0.018)ivN2.3 is 764 K, being higher than that of Sm2Fel7N2.6 (743 K). The Tc values of Smz(Fel_~V~)a7N v (x = 0.03 and 0.059) are almost the same as in the un-substituted nitrides. It has been accepted that an increase in the unit cell volume of the RzFeI7 compounds results in an increase in Tc due to the magnetovolume effect. Nitrogenation might have maximized Tc and further substitution by non-magnetic atoms can only act as magnetic dilution which leads to a decrease of Tc, in spite of the additional unit cell expansion. No comments were given as to the slight increase of Tc when M = Ti and V. Saturation magnetizations and magnetic anisotropy fields are generally reduced by substitutions. When M = Ti and V, either Ms or #0Ha were raised slightly. Unfortunately both parameters did not increase simultaneously for a given substitution element. 3.5.3. Substitutions for rare earths For the technical important Sm2Fe17Nu interstitial compound, replacements of Sm by mischmetal and Ce (Huang et al. 1991b and c), by Nd (Huang et al. 1991c, Liang et al. 1991, Yu et al. 1992, Katter et al. 1992c), by Tb (Huang et al. 1991b), by Dy and Er (Tegus et al. 1992), and by Y (Huang et al. 1991b, Lu et al. 1992) have been studied. The system (Sm1_xR~)2Fe17Nu exists for 0 ~ x ~< 1.0, whereas for mischmetal the single phase region depends on the amount of La contained in the mischmetal. As La2Fe17 does not exist, higher La concentration would results in a narrower x range. Modification of the Curie temperature by substitutions of other rare earths for Sm is not significant, which is in the range of the Tc variation across the R2Fel7Ny series. Saturation magnetization Ms increases by Nd substitution owing to the higher magnetic moment of the Nd atoms than the Sm atoms. Substitution of nonmagnetic

INTERSTITIALLYMODIFIED INTERMETALLICS

353

(Sm 1-xNdx)2(Fel-z C°z)17 N~2.7 ~l i ~ ~~ , , -

1.0

I

0.8 -

~

\\

1.25

~ - -

~

N

E

0.6

C O

0

O

'-'~'~

~

~~

~

\~ 5

0.4

o

0.2 o o

1 45

; /

iI 0.2

1.55.

/ //

I(-~

0.4 0.6 0.8 Nd Content, x

Js(T)

---

.0

I.t0HA (T)

Fig. 3.20. Contour line diagram of the saturation polarization Js and the anisotropy field #0Ha for (Sml_zNd~)2(Fel_zCoz)17Nv at room temperature (after Katter et al. 1992c). Y also results in an increase of Ms, suggesting an antiferromagnetic Sm-Fe coupling, as has been discussed in section 3.3.3. The value of Ms decreases with z in (Sm]_~R~)zFeaTNu when R is a heavy rare earth, because of the ferrimagnetic configuration between the heavy rare earth sublattices and the Fe sublattices. The strong uniaxial anisotropy of Sm2Fe17N u is significantly reduced by substitution of other rare earths R for Sm. When R = Y, #0Ha decreases linearly with increasing Y concentration z. In (Sm1_~Nd=)2Fea7Nu, the easy axis anisotropy is retained up to z ,-~ 0.6, while the uniaxial anisotropy can be preserved up to z > 0.8 in most of other rare earths. Substitution of Nd or mischmetal for Sm is of interest because their price is much lower than that of Sm and thus there are possibilities for the fabrication of inexpensive permanent magnets. Huang et al. showed that (Smo.6Mm0.4)zFelvNv exhibited an anisotropy field of 6.8 T, which was close to that of NdzFe14B, but the Curie temperature is 150°C higher than that of Nd2Fe]4B (Huang et al. 1991c). The (Sml_~Nd~)z(Fel_~Co~)N2.7 series were investigated by Katter et al. (1992c). By substitution of Co for Fe, the decrease of the anisotropy field due to Nd substitution was partly balanced and the saturation magnetization and the Curie temperature were further increased. The contour line diagram at room temperature for (Sml_xNdx)z(Fel_zCoz)N2.7 is drawn in fig. 3.20. For (Sm0.vNd0.3)2(Fe0.8Coo.z)N2.7 intrinsic magnetic properties of/z0Ha = 14.8 T, Js = 1.57 T at room temperature and Tc = 835 K were obtained.

4. Interstitial compounds of the l:12-type structure Intermetallic compounds with the composition RFel2_xT:c have been of interest in the field of magnetism for quite a few years. Although the binary compound RFe]2

354

H. FUJII and H. SUN

.,.-®

,

0 j'=' I o,,~:~

0 ''=' I

I c~l,-~

I

~--~'v

I I° ~

I

a

R

C) 2a

Fe(T) qD 8f

(~)8i

~8j

N • 2b Fig. 4.1. Unit cell of RFe12_=T=Ncompoundshaving the ThMn12-typebody centered tetragonal structure.

does not exist, the 1:12 phase can be stabilized for T = Si, A1, Ti, V, Cr, Mo and W. It has been well established that this series of compounds crystallizes in the tetragonal ThMn12-type body-centered tetragonal structure (space group I4/mmm), which is directly related to the hexagonal structure of the R2Fe17 compounds (Mose et al. 1988). As shown in fig. 4.1, there is one single crystallographic site (2a) occupied by rare earth atoms and there are three different sites (8i, 8j and 8f) occupied by Fe and T elements. Among various RFe12_=T= compounds, SmFe11Ti has been considered to be a suitable candidate for potential permanent magnet applications. However, it can not compete with Nd2Fe14B owing to the following practical reasons: Although its Curie temperature is close to that of Nd2Fe14B, its saturation magnetization is substantially lower than that of Nd2Fel4B, which leads to a much lower theoretical maximum energy product. In addition Sm is a much more expensive rare earth than Nd. Stimulated by the success in the 2:17 series, interstitial modification studies were naturally extended to the 1:12 materials. Similarly to the case of the 2:IT-type intermetallic compounds, interstitial modifications lead to remarkable changes on the magnetic properties also in the l:12-type materials. Upon introducing N or C atoms into the tetragonal structure, the Curie temperature increases by about 200 K and the second order crystal field parameter A ° changes its sign from weakly negative to relatively strongly positive. This leads to a strong uniaxial anisotropy in compounds containing Nd, for which the second order Stevens coefficients c~j is negative. In this section, we will summarize the experimental results on interstitial modifications o f the 1:12 compounds. 4.1. Location of N atoms in the tetragonal structure

It appears to be more difficult to make fully nitrogenated or carbonated 1:12 interstitial compounds than 2:17 interstitial compounds. The partially interstitially-modified

INTERSTITIALLY MODIFIED INTERMETALLICS

355

materials consist of compounds with a range of interstitial atom contents. Consequently there are significant spreads in the lattice parameters and the magnetic properties observed by various researchers. The structural and magnetic inhomogeneities broaden the neutron diffraction peaks and reduce the reliability of the crystallographic parameters deduced from them. Fortunately, reasonably high quality (homogeneous phase) interstitial compounds can be obtained for T = Mo. Therefore most of the neutron diffraction studies have been done for RFe12_~Mo~ (R = Nd and Y). The lattice parameters and the occupancy of the 2b interstitial sites deduced for various compounds are listed in table 4.1. Besides neutron diffraction, site occupation of the nitrogen atoms in SmFex0MozNu has been studied by Psycharis et al. (1991) using X-ray diffraction method. All the studies led to the same conclusion that nitrogen atoms are located in the octahedral 2b interstitial sites, which are equivalent to the 9e or 6h sites in the 2:17 structures. Full occupation of this 2b site leads to the composition RFe12_~T~Na. Upon nitrogenation, the crystallographic symmetry of the host is retained, but the lattice is expanded by more than 3%. Two nitrogen atoms become the nearest neighbors of the rare earth site, which is responsible for the radical changes of the magnetocrystalline anisotropy. Most of the results indicated that Ti and Mo shared the 8i site with Fe, while Sun et al. (1993b) obtained a better fits of the diffraction pattern when placing a small amount of Mo on the 8f sites. Vanadium atoms were reported to be located on the 8j site by Yelon and Hadjipanayis (1992). In many cases, the experimentally estimated nitrogen content is more than 1 nitrogen atom per formula unit. As decomposition takes place concurrently with nitrogenation, most of the authors believed that the extra amount of nitrogen is contained in the decomposition products consisting of rare earth nitrides or iron nitrides, whereas Suzuki et al. (1992c) suggested that not only the 2b site but also other sites are occupied by nitrogen atoms. A neutron diffraction study of the nitride NdFesCo3Ny performed by Fujii et al. (1995b) showed that the nitrogen occupancy TABLE 4.1 Nitrogen occupancy y of the 2b interstitial site in interstitial nitrides with the tetragonal ThMnl2type structure, a, c and V are the lattice parameters and tetragonal cell volume. All the data were deduced from room temperature neutron diffraction diagrams. Compound

a (,~)

c (]~)

YFell TiNy YFel0V2Ny YFel0.zMol.sNu YFel0.6Mol.4Nu YFellMoN u YFe9Mo3Ny NdFel0Mo2Ny NdFel0MozNy

8.611 8.5436 8.646 8.6648 8.62925 8.7022 8.6593 8.660

4.821 4.7834 4.8519 4.8012 4.79663 4.82734 4.8295 4.869

References: [1] Yang Y.C. et al. (19910 [2] Yelon and Hadjipanayis (1992) [3] Wang et al. (1993a)

V (]~3)

Y

Ref.

349.16 362.69 360.47 357.18 365.57 362.13 365.2

0.5338 0.44 0.8 0.8 0.96 0.94 0.48 0.49

[1] [2] [5] [5] [4] [4] [2] [3]

[4] Sun et al. (1993b) [5] Tomey et al. (1993)

356

H. FUJII and H. SUN

of the 2b site is only 0.5 although the nominal nitrogen content was y > 1.0. Considering the periodicity loss along the crystallographic c-axis when y > 1.0, Fujii et al. suggested that the remaining N-atoms occupy some of the tetrahedral interstitial sites near Ti, possibly the 16m or 8h site.

4.2. Structural and intrinsic magnetic properties Interstitial modification of RFea2_xT~ with T = Si, Ti, Mo and V has been reported. The structure and magnetic properties of the interstitially modified RFelz_zTxZy (Z = N and C) compounds are summarized in table 4.2. Combining high temperature annealing (1100°C) with water quenching, Fujii and coworkers (Fujii et al. 1992b, Akayama et al. 1994) prepared for the first time the RFellTi compounds with R = Ce and Pr. It had been reported so far to be difficult or impossible to synthesize them. Also included in the table 4.2 are the data of RFel0Si2Cy compounds formed by arc-melting. Although carbon atoms were introduced into the interstitial sites of RFeloSi2 by melting, the amount was much less than that obtained by gas-solid reactions and the influences were smaller (Xu X.F. et al. 1994). The structure and the magnetic properties of RFe12_~T~Zv are sensitive not only to the interstitial atom content y but also to the concentration z of the stabilizing element T, as shown in the studies on structure and magnetic properties of NdFe12_xMoxNv (Wang et al. 1993b). As both :c and y can differ from the nominal concentration due to the compositional changes occurring during sample preparations, the data listed in table 4.2 scatter over quite a wide range. Nevertheless, the modification effects due to interstitial atom additions are significant and clear.

4.2.1. Curie temperature The averaged increase in Curie temperature is about 200 K for the nitrides and 160 K for the GIM carbides. The mechanism has been understood to be a volume effect, as in the 2:17 series: the unit cell volume is expanded by interstitial modification, which induced an enhancement of the Fe-Fe exchange interaction, and the Curie temperature was greatly increased. An almost linear relationship between the Curie temperature and the corresponding cell volume for YFeaaTiC v and RFe1~TiNv has been reported by Li Z.W. et al. (1993) and Akayama et al. (1994), respectively. Based on the mean-field theory, the Fe-Fe interaction was estimated to be 35% stronger in RFe11TiNy than in RFelaTi, which corresponds to an increase from 5.5x 10 -22 J in the hosts to 7.4x 10 -22 J in the nitrides (Li Z.W. et al. 1992). Some of the data taken from table 4.2 are plotted in fig. 4.2. The familiar variation of Tc across the rare-earth series showing maximum at R = Gd is not always demonstrated in the interstitial modified compounds. This could be related to the different interstitial content for different rare earths. 4.2.2. Magnetic anisotropy In the host RFel2-xTx, when z is small, the easy magnetization of the Fe sublattice is along the crystallographic e-axis. If we consider only the lowest order crystal field interaction, the rare-earth sublattice anisotropy prefers the e-axis for R = Sm,

INTERSTITIALLY MODIFIED INTERMETALLICS

357

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358

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360

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Fig. 4.2. Curie temperature Tc across the rare earth series for RFellTi (Hu B.P. et al. 1989 and Akayama et al. 1994), RFea0MozN v (Anagnostou et al. 1991b), RFel]TiN v (Yang Y.C. et al. 1991d) and RFellTiC v (Hurley and Coey 1992).

Er and Tm whose second order Stevens coefficients ad are positive because the second order crystal field parameter A° is negative. For RFellTi, all the members show easy axis anisotropy at room temperature due to the preponderant contribution of the iron sublattice, except Pr and Tb (Hu B.E et al. 1989, Akayama et al. 1994). At low temperatures, spin reorientation takes place in the compounds with R = Nd, Tb, Dy as a result of the competition between the Fe and rare earth sublattices and in the compound with R = Er also due to the effect of high order terms of the crystal field parameters. The Fe sublattice anisotropy remains uniaxial after interstitial modification, although it is slightly weakened by carbonation (Hurley and Coey 1992). Drastic changes take place in the rare earth sublattice anisotropy. In the 1:12 interstitial compounds, the rare earth atoms have two nearest interstitial neighbors located along the c-axis direction instead of within the plane configuration perpendicular to the c-axis in the 2:17 interstitial compounds. While the in-plane interstitials have a strong negative contribution to A ° in the 2:17 compounds, interstitials along the c-axis contribute positively to A2° in the 1:12 structure. Consequently, A2° is changed from negative in the host compounds to positive in the interstitially modified compounds and therefore the rare earth sublattice anisotropy changes completely. In the 1:12 interstitial compounds, Sm, Er and Tm have easy-plane anisotropy and all the other magnetic rare earths should have easy-axis anisotropy. X-ray diffraction patterns on aligned powders of SmFellTi and its nitride (Coey and Otani 1991) demonstrate clearly the

362

H. FUJII and H. SUN

effect of interstitial nitrogen atoms on the magnetic anisotropy (fig. 4.3). Only the Sm compound shows easy-plane anisotropy at room temperature. Spin reorientations at 45 K for ErFealTiNy and 50 K for ErFel0.sMol.sNv, have been reported by Yang Y.C. et al. (1991c, 1993). It is likely that also the Tm compounds will exhibit spin reorientations as a function of temperature or interstitial atom content when the easy-axis anisotropy of the iron sublattice is outweighted by the opposing rare earth contribution. However, no publications were found for the spin reorientation studies of the Tm compounds. Using the single-ion model, Yang Y.C. et al. (1991e) estimated A° = 85 Kao 2 for RFealTiNu in contrast to A° = - 3 0 Kao 2 for RFel~Ti. The value of A° in NdFe11TiNywas calculated by Li and Cadogan (1992c) using their 'bonding charge model' and A° = 170 Kao 2 was estimated for NdFe11TiN0.5. They also predicted an A ° value of 370 Kao 2 for NdFe11TiN1.0, which corresponds to an anisotropy field of 13 T at room temperature. Technically important materials are the Nd compounds. In fact, most of the studies have been devoted to them as can be seen from table 4.2. Although the Mo or V containing materials might have higher anisotropy fields than the Ti containing compounds, their magnetizations are low. Hence NdFellTiN v is still the most promising r

1

W"'--

002 SmFellTi

004 ,.s

40O SmFellTiN0.8

40

60 Two Theta (degree)

80

Fig. 4.3. Cu Ks X-ray diffraction patterns of SmFellTi and SmFellTiN0.8 oriented in an applied magnetic field of 1.2 T.

INTERSTITIALLY MODIFIED INTERMETALLICS

363

material. Of equal importance is PrFellTiN v as reported by Fujii et al. (1992b) and Akayama et al. (1994). We can say, that when compared with Nd2Fe14B, their anisotropy fields are about the same, their magnetizations are slightly lower, but their Curie temperatures are much higher.

4.2.3. 57FeMOssbauer effect and magnetization studies Although there are only three different crystallographic iron sites in the 1:12 structure, the subspectra needed for fitting the 57Fe Mrssbauer spectra are more than three as Fe in a given crystallographic site can have different nearest neighbor environments because of the existence of the T atoms. The situation becomes more complicated when nitrogen or carbon occupy the 2b interstitial sites partially, which is true in most of the cases. Details of the fits and the number of the subspectra used by various authors were different, but the order of Bhf(8i) > Bhf(8j) > Bhf(8f) was generally adapted for both the host and interstitial compounds (Li Z.W. et al. 1992 and 1993, Qi et al. 1992c, Ge et al. 1992, Sun J.J. et al. 1993, Wang Y.Z. et al. 1993a). The room temperature hyperfine fields of the nitrides and carbides of YFe11Ti are listed in table 4.3. The agreement between data from different sources is not a very good one, even for the overall averaged hyperfine fields. Probably the composition deviation is the main reason for the data scattering. The 3d sublattice magnetization of YFellTiZy (Z = N and C) was investigated in detail by Qi et al. (1992c). Magnetization and 57Fe hyperfine fields were measured and compared with the results of band structure calculations. The magnetization at 4.2 K increases by about 14% for both the nitride and carbide, which was considered to be essentially due to a narrowing of the 3d-band caused by the volume expansion. The chemical effect of N or C was considered to reduce the iron moment slightly by hybridization, especially on their nearest neighbor 8j-sites. The average hyperfine field increases by 12% for the nitride, whereas it hardly changes for the carbide (1%) although the volume expansion and moment increases are virtually identical for the nitride and carbide. This was understood originating from differences in the contribution due to the transferred hyperfine field from neighboring atoms. TABLE 4.3 57Fe M0ssbauer hyperfine field (in units of Tesla) of each crystallographic site (averaged over subspectra) and the overall weighted average (Bhf) for the YFe11Ti nitrides and carbides at room temperature. Compound

8i

8j

8f

(Bhf)

Ref.

YFe11TiNo.8 YFel 1TiNu YFel 1TiNy YFel 1TiCo.9 YFel 1TiCo.7

32.9 35.5 32.7 28.9

29.9 32.8 29.8 24.9

25.1 28.7 24.7 21.1

29.0 32.0 28.7 25.1 30.2

[1] [2] [3 ] [ 1] [4]

References: [1] Qi et al. (1992c) [2] Li Z.W. et al. (1992)

[3] Ge et al. (1992) [4] Li Z.W. et al. (1993)

364

H. FUJII and H. SUN

4.3. Substitution studies Effects of substitution of Co for Fe have been studied for RFell_~Co~TiNy (R = Nd, Pr) (Suzuki et al. 1992c) and NdFel0.v_xCo~Til.3N u (Kim et al. 1992). The nitrogen absorption ability decreases with increasing Co concentration because the crystal lattice contracts as the Co content x increases. The nitrogen content y nearly vanishes when x > 5. Similar properties have been observed in the 2:17 compounds. The anisotropy field #oHa is not significantly influenced by Co substitution, but the Curie temperature and magnetization increase. The best magnetic properties have been obtained for NdFesCo3TiN1.3, for which Is = 1.50 T, #0Ha = 7.5 T at room temperature and Tc > 600°C (Suzuki et al. 1992c). Substitution of Dy for Nd in Ndl_~Dy~Fe11TiN v was studied by Kong et al. (1993e). They showed that the nitrogen content y and the magnetic anisotropy field at room temperature increase by Dy substitution for Nd. Compounds with a small Dy concentration showed better intrinsic magnetic properties than the unsubstituted nitride. 4.4. Interstitial modification study on compounds with various structures Stimulated by the success of interstitial modification of intermetallic compounds with the 2:17 or l:12-type structures, similar studies have been extended to materials with other crystal structure types. Published research works on the interstitial modification of various compounds are summarized in table 4.4. 4.4.1. R-rich compounds It is difficult to avoid small amounts of Sm rich impurity phases (SmFe2 and SmFe3) when preparing SmzFel7 because of the peritectic formation of the SmzFel7 compound in the binary Sm-Fe phase diagram. The nitrogen absorption behaviors of SmFe2 and SmFe3 have been investigated by several groups (Christodoulou and Takeshita et al. 1993 a and 1993b, Ishizaka et al. 1993, Yau et al. 1993). Unfortunately, all the results showed that SmFe2 and SmFe3 reacted with nitrogen to form the decomposition products SmN and a-Fe, without formation of any interstitial nitrides. The produced soft magnetic a-Fe phase deteriorates the magnetic properties of Sm2FelyNy magnets. Immediate decomposition by nitrogenation was also observed in the Nd5Fea7 compound, forming NdN and a-Fe (Helmolt et al. 1993). It has been suggested that in intermetallic rare earth-iron compounds with a high rare earth content, the formation of the metastable nitride is impeded by the high Fe diffusivity, but the stable equilibrium phases are formed quickly. Nitrogenation and carbonation of RFe2 with R = Y, Tb and Dy were studied by Singleton et al. (1993). The nitrides and carbides were formed at low heat treatment temperature (lower than 350°C) and the crystal lattices expanded slightly. However, the Curie temperature hardly changed. Similar results were obtained by Yamamoto I. et al. (1993) in the nitrogenation of YFe2, YFe3 and Y6Fe23. These authors even observed a negative volume expansion for the 1:2 and 1:3 compounds

INTERSTITIALLYMODIFIED INTERMETALLICS

365

TABLE 4,4 Various rare earth-transition metal intermetallic compounds that have been subjected to the interstitial modification studies. Formula

Composition

Structure type

Space group

Ref.

RFe2 RFe3 GeFe3

Sm, Y, Tb, Dy Sm, Y, Tb

RsFe17 R6Fe23 RFe5 R2FeI4B Nd2(Fe,Ti)19 RFeloSi La(COl_xFe~)t3 LaFel3_zSi~

Nd Y Y, Sm Y, Nd, Dy Nd Ce, Pr, Nd, Sm, Y 0 <~ z ~< 0.4 1.5 <~ • ~< 2.4 3.6 ~< x <~ 5.0

MgCu2,fcc Laves phase PuNi3, rhombohedral Ni3Sn, hcp (high temperature) Cu3Au, fcc (low temperature) hexagonal Th6Mn23, fcc CaCus, hexagonal tetragonal TbCuT, hexagonal BaCdlb tetragonal NaZnl3, fcc NaZnl3,fcc Ce2Nil7Si 9, bct

Fd3m R3m P63/mmc Pm3m P63/mcm Fm3m P6/mmm P42/mnm P6/mmm I41/amd Fm3c Fm3c I4/mcm

[1--4] [2, 4-7] [8] [8] [9] [4] [10] [11-14] [15] [16] [17] [18] [18]

References: [1] Christodoulou and Takeshita (1993b) [2] Ishizaka et al. (1993) [3] Singleton et al. (1993) [41 Yamamoto I. et al. (1993) [51 Yau et al. (1993) [6] Cheng et at. (1993) [7] Christodoulou and Takeshita (1993a) [8] Xu Y. et al. (1992) [9] Helmolt et al. (1993)

[10] Yang EM. et al. (1992) [11] Zhang X.D. et al. (1992) [12] Onneby et al. (1993) [13] Leccabue et al. (1992) [14] Barrett et al. (1993) [15] Cadogan et al. (1993) [16] Singleton and Hadjipanayis (1993) [17] Huang et al. (1992) [18] Tang et al. (1993c)

after nitrogenation, which could be related to the change of magnetostriction and thermal expansion properties. Nitrogenation of the highly magnetostrictive material TbFe3 was studied by Cheng et al. (1993). Nitrogen absorption was found to be an amorphization process in which the nitrogen atoms diffused into the TbFe3 lattice but did not enter the interstitial sites of the structure. The amount of nitrogen absorbed in the amorphous phase was a function of the annealing time.

4.4.2. Nd2Fe14B-type compounds N d - F e - B magnets have drawbacks like the poor thermal stability and inferior corrosion resistance. If similar effects on the Curie temperature as in 2:17 and 1:12 compounds could be realized in the 2:14:1 compounds by interstitial modification, the thermal stability of Nd2Fel4B could be greatly improved. The only successful formation of interstitial nitrides, YzFet4BNy and NdzFel4BN v (y N 0.3), has been reported by Yang Y.C. and coworkers (Yang Y.C. 1991g, Zhang X.D. et al. 1992). Upon nitrogenation, the tetragonal structure of the R2Fe14B compounds was retained, but the unit cell volume slightly increased. While Tc increased by about 60 K, the magnetocrystalline anisotropy and the saturation magnetization decreased somewhat. Neutron diffraction study showed that nitrogen atoms occupy the 4f interstitial sites in the tetragonal structure, and full occupation would correspond to the composition RzFelaBN.

366

H. FUJII and H. SUN

However, the above work has been criticized by Barrett et al. (1993). In a detailed study of the effects of nitrogen uptake in Y2FeI4B compound, they observed only a decomposition of the starting alloy into c~-Fe and RX v. Conclusions drawn by Barrett et al. (1993) in Y2Fel4B are in agreement with those obtained by ()nneby et al. (1993) on NdzFel4B and by Leccabue et al. (1992) on DyzFe14B. Thus it is still an open question whether RzFel4BNu quaternary phases exists or not. 4.4.3. CaCus-type related structures Compounds with the composition R10Fe79Si11 (R = Y and Sm) and their nitrides were studied by Yang EM. et al. (1992). The X-ray diffraction patterns were indexed on the basis of the CaCu5 structure type. After nitrogenation, the unit cell volume expanded by more than 6% and the Curie temperature increased by 108 K and 127 K for the Sm and Y compounds, respectively. Furthermore the easy magnetization direction of the Sm compound changed from planar to uniaxial upon nitrogenation with a room temperature anisotropy field of 12.5 T. Sm10Fev9Si11Ny shows interesting intrinsic magnetic properties for permanent magnet applications. A new ternary phase Nd2(Fe,Ti)19 has been reported by Cadogan et al. (1993). X-ray diffraction and Mrssbauer data indicated that the 2-19 phase is related to the hexagonal TbCu7 superlattice structure based on the CaCu5 structure. Nitrogenation induced a volume expansion of 5.4% and a Curie temperature increase of about 200 K. The average 57Fe hyperfine field at room temperature became enhanced by 42% primarily due to the increase in Curie temperature. Later, the exact composition of this phase was determined to be Nd3(Fe,Ti)29, which crystallizes in a monoclinic structure of the space group P21/c (No.14) (Li H.S. et al. 1994, Hu Z. and Yelon 1994, Yang EM. et al. 1994). 4.4.4. Compounds with the NaZn13-type structure Two systems have been studied for compounds of the 1:13 structure, namely the La(Col_~Fex)13 (0 <~ x ~< 0.4) series (Huang et al. 1992) and the LaFel3_~Si~ (1.5 ~< x ~< 5.0) series (Tang et al. 1993a, b, c). La(COl_zFex)13 absorbs substantial amounts of nitrogen and the unit cell volume increases by 8.3-8.9%. However, both the Curie temperature and the magnetic moment decrease and therefore no beneficial effects were obtained by nitrogenation. The LaFe13_xSi~ compounds were found to have two types of crystal structure; the fcc NaZn13-type for x ~< 2.4 and the bct Ce2NilTSi9-type for 3.5 ~< x ~< 5.0. For the cubic phase, nitrogenation expands the unit cell volume by 6.2-10.8% and this cell expansion led to an increase of the Curie temperature by 52 to 80 K. At the same time the spontaneous magnetization increases by ~10%. On the other hand, the structural and magnetic properties of the tetragonal phase were not significantly affected by nitrogenation. 4.4.5. RFeloSiCv and the Fe3GeNu compounds Using the GIM process, Singleton and Hadjipanayis (1993) reported the formation of the BaCdll-type structure compounds RFel0SiC u with R = Ce, Pr, Nd, Sm and Y, where the C atoms were known to fill the octahedral voids in the tetragonal cell. This

INTERSTITIALLYMODIFIEDINTERMETALLICS

367

success demonstrated that the GIM method can become a new route for synthesizing compounds not readily formed by standard metallurgical techniques. The simple transition metal-based Fe3Ge and the nitrides were studied by Xu Y. et al. (1992). Fe3Ge crystallizes in two different type of structures. The high temperature phase exhibits a hcp crystal structure of the Ni3Sn-type with the space group P63/mmc and the low temperature phase is the face centered cubic Cu3Au-type structure with the space group Pm3m. The fcc phase absorbs nitrogen up to z = 0.24 in Fe3GeNy, the cubic structure being retained, but the effect of nitrogenation is to decrease both Tc and the room temperature magnetization. The hcp phase transforms to either the fcc or the tetragonal phase or a mixture of them upon nitrogenation. 5. Electronic band structure calculations

As described in sections 3 and 4, experimental results have clearly indicated that the structural and magnetic properties of the rare earth-iron intermetallic compounds can be strongly modified by the introduction of nitrogen or carbon atoms into the interstitial crystallographic sites. Interstitial modifications of the binary RzFe17 and the psuedo-binary RFelz_:~Tx are of particular interest and have been studied in great details. In order to understand the basic physical origin of the strong influences of the interstitial atoms on the various aspects of magnetic properties, electronic band structure calculations have been carried out for the RzF17Z3 and RFel2_xTxZa (Z = C and N) interstitial compounds by many research groups.

5.1. Calculations of the 2:17-type interstitial compounds R2Fel7Z3 (Z=N or C) 5.1.1. Compounds of the nonmagnetic rare earth element Y The calculated magnetic moments on each atomic sites of YzFel7 and Y2Fe17Z3 (Z = C and N) and the methods used for the calculations are summarized in table 5.1a. For comparison, the experimental results on the magnetic moments are also given there. The non self-consistent spin-polarized electronic energy band calculations using the orthogonal linear combination of atomic orbital (OLCAO) approximation were carried out for hexagonal Y2Fel7 and its nitride YzFel7N3 (Li Y.E et al. 1991), with the nitrogen atoms on the 6h-site. The calculated total moments for YzFel7 and the nitride are qualitatively in agreement with the values obtained by the magnetization measurement at 4.2 K. Although the calculation is not self-consistent, general feature of the influence of the interstitial N on the electronic band structures is clarified. In this calculation, it was stressed that the enhancements of the Fe average moments and Curie temperatures mainly come from volume expansion due to interstitial modification. Beuerle et al. (1991) first calculated self-consistently the local magnetic moments on all Fe and Y sites in Y2Fel7 and its expanded form as would be obtained by nitrogenation, but without nitrogen atoms on the interstitial site. The self-consistent tight-binding, linear-muffin-tin-orbital method (LMTO) in the atomic-sphere approximation (ASA), the so-called self-consistent LMTO-ASA method, was used and the

368

H. FUJH and H. SUN

TABLE 5. la Calculated and measured values of the magnetic moments on different atomic sites and of the total moment per formula unit/>r in Y2Fe17 and Y2FelTZ3 (Z = N or C) with the hexagonal Th2Ni17-type structure. #z is the magnetic moment of the interstitial atom Z. Compound lAR (lAB) 2b Y2Fe17

2d

lAFe(lAB) 4f

12j

12k

lAZ (lAB) tAT (lAB/f'u') Method

-0.41 -0.42 2.41 -0.47 -0.45 2.53 -0.25 -0.23 2.96

2.35 2.12 1.91 2.25 2.00 1.95 2.23 2.14 1.78

Y2FelTN3 -0.20 -0.45 2.65 -0.06 -0.09 3.46 2.50 Y2Fel7Ny(exp.)

2.01 2.57 2.53 2 . 2 0 1.69 3.11 2.08 2.32 2.66

Lu2Fe17(exp.) Lu2Fe17Nv(exp.) YzFe17C3 Y2FelTCz(exp.)

2.07 1.87 2.23 2.35 2.17 2.51 1.65 2.06 2.58

Y2Fel7(exp.)

References: [1] Beuerle et al. (1991) [2] Jaswal et at. (1991) [3] Li Y.P. et al. (1991) [4] Qi et al. (1992)

2.47 2.69 2.42

Ref.

6g

-0.06

36.8 35.5 37.0 32.8 34.7 40.3 39.2 39.4 38.1 40.3 35.3 40.0 34.8 36.1

scLMTO-ASA[1] scLMTO [2] nscOLCAO [3] MOssbauer [4] Magnetization [5] scLMTO [2] nscOLCAO [3] scLMTO-ASA[7] MOssbauer [4] Magnetization [5] M0ssbauer [6] M0ssbauer [6] scLMTO-ASA[7] M0ssbauer [4]

[5] Huang et at. (1990) [6] Zouganelis et al. (1991) [7] Beuerle and Fahnle (1992)

calculated results showed that the Fe moments on all the sites at 0 K increased by about 10% due to the experimentally observed volume expansion. Extending the above study, these authors calculated the local magnetic moments and hyperfine fields on the Fe, Y and Z atoms in Y2Fe17 and Y2Fel7Z3 with Z = H, C or N by means of the local-spin-density approximation and self-consistent LMTO-ASA method (Beuerle and Fahnle 1992). They have found that there are two counteracting effects due to the interstitial atoms, namely a geometrical effect (volume expansion and local relaxation) which increases the Fe moments and the hyperfine fields, and the hybridization effects of the Z atoms with the neighboring Fe atoms, which reduce the Fe moments and hyperfine fields. The calculated moments and average hyperfine fields are slightly larger for the nitrogenated system, but they are smaller for the carbonated system than those for the pure Y2Fe17. This is not in agreement with experimental observations, as can be seen in table 5.1a. This may be related to stoichiometric deviations of the carbon content in the compound. Since the carbon content does not experimentally exceed y = 2.0 in Y2Fel7Cy, the geometrical effect rather than the hybridization effects may strongly contribute to the experimental enhancement of #T- Theoretically, it has been deduced that, when y reaches 3.0, the magnetic moments in Y2FeI7C3 becomes smaller than in the host compound Y2Fe17 because of the strong hybridizations between C p- and Fe d-states. In their calculations, the ratios of the atomic sphere radii were chosen as ry/rFe = 1.35 and rz/rFe = 0.7.

INTERSTITIALLY MODIFIED INTERMETALLICS

369

Furthermore, F~ihnle and Beuerle (1993) suggested that fluorine could be the optimum interstitial dopant atom for achieving large magnetic moments in R2Fe17. The reason for this is that, when going from N to O or even to F, the potential around the dopant atoms becomes steeper, the p-state appears at lower energies and the hybridization with the Fe states becomes weaker. As a result, it was deduced that the geometrical effect due to the increase of the unit cell volume becomes essential, leading to a strong increase in magnetic moments and high Curie temperatures. In addition, it has been pointed out that the strong electronegativity of the F atom possibly induces a highly aspherical valence charge density around the R atom, resulting in large crystal electric field (CEF) parameters and strong rare earth contribution to the magnetocrystalline anisotropy. Based on the semi-relativistic LMTO method in the local-density and ASA approximations, Jaswal et al. (1991) and Jaswal (1992) conducted the self-consistent spin-polarized electronic structure calculations for the hexagonal Y 2 F e l 7 and the nitride Y2Fe17N3. The lattice constants which were used in the calculations were the experimental obtained values, a = 8.48 (8.66) A and c = 8.26 (8.51) ,~ for Y 2 F e l 7 ( Y 2 F e l 7 N 3 ) respectively. The ratio of the Wigner-Seitz radii was chosen as ry : rFe : r N = 1.4 : 1.0 : 0.75 for the Y, Fe and N atoms. The spin-polarized

Y2Fe~7'

o 0.

.

.

'

'

.

.

/'

A ' A

.

.

v

\

'

0

2

_._

I

-10

_ (~))

E O

I

I

I

I

I

I

-4 -2 Energy ( eV ) '

^ / ' ~ 1I

Y2Fe~7N3

I

/~./

SpinUp /

~

-

\i

0

~ F eY o D

*

-6

> •

*

-8

-1

,

-10

SpinDown ~

V

........N I

-8

I

-6

I

I

-4

*

I

-2

~

I

0

2

Energy ( eV ) Fig. 5.1. Spin-polarized partial density of states for (a) Y2Fel7 and (b) Y2FelTN3 calculated by using the linear-muffin-tin-orbitals (LMTO) method in the scalar-relativistic approximation (after Jaswal et al. 1991, Jaswal 1992).

370

H. FUJII and H. SUN

partial densities of states (DOS) for Y2Fel7 and the nitride are given in figs 5.1 (a) and (b), respectively. As expected, the DOS are dominated by the Fe d-bands near Fermi level. Upon nitrogenation, the Fe d-bands shift to the higher binding energy side owing to the reduction of the overlap between the Fe d-wave functions and hybridizations between Fe d- and N p-states. This leads to a decrease in the values of both the up and down-spin DOS at the Fermi level. The structure around 6 eV in the DOS of Y2Fel7N3 mainly originates from the N p-states. Since the charge transferred away from the N atoms is estimated to be about 0.25, being quite small, there are no strong chemical bonding effects of N with its neighbors. The interesting aspects of the calculated results on Y2Fe17 and Y2Fel7Z3 are summarized as follows: i) The magnetic moments decrease for the Fe atoms that are nearest neighbors to N atoms, owing to the strong hybridizations of the Fe 3d-states with N 2p-states. ii) The magnetic moments of the Fe atoms which are farther from the N atoms increase because of the reduction in overlap between the Fe 3d-states due to lattice expansion. This effect also enhances the interatomic exchange interactions between the Fe atoms and leads to an increase in the Curie temperature. iii) There is an overall increase in the total magnetization upon nitrogenation, which is in excellent agreement with the magnetization values measured at 4.2 K. iv) Small negative moments are induced on the Y atoms. The reason is that the Fe 3d-bands are on the average at higher binding energies than the Y 4d-bands. Since the exchange splitting raises the energy of the minority Fe 3d-band with respect to that of the majority band, an increase in the hybridization of the minority Fe 3d-band with the Y 4d-bands occurs, and it enhances the DOS of Y 4d-electrons with the same spin direction as the minority Fe 3d-band. Thus, the Y moments are induced and are antiparallel to the Fe moments. This also explains the fact that in all the R-T compounds the R spins with non-zero 4f-electrons always couple antiferromagnetically to the spins of the 3d transition metal atoms when having a more than half filled 3d-band. The Curie temperature is almost doubled by interstitially introducing Z atoms into R2Fel7. It is of interest to discuss this problem using the energy band structures. The quantitative predictions of the Curie temperatures are rather difficult because we have not yet any reliable and available method to evaluate the effects of the spin fluctuations. However, the general trends of the Curie temperature, the strength of the coupling between the magnetic moments on each atomic site, can be well described when using the results of the band structure calculations. According to the spin fluctuation theory of Mohn and Wohlfarth (1987), the Curie temperature Tc is given by

(5.1~

Tc o~ Mg/Xo,

where Mo is the magnetic moment per atom at 0 K and Xo is the enhanced susceptibility given by X° 1 -

1 ( - -1 2# 2 2N¢(EF)

+

1 2NI.(Ez) -

-

I

)

.

(5.2)

INTERSTITIALLYMODIFIEDINTERMETALLICS

371

Here, Ny(EF) and N~(EF) are the spin-up and spin down DOS at the Fermi level EF and I is the Stoner parameter. Jaswal et al. (1991) estimated the increase in Curie temperature upon nitrogenation for YzFel7 from the results of the band structure calculations, leading to the ratio Tc(Y2Fe17N3)/Tc(YzFe17) = 2.34. This suggests that an increase in M0 and a substantial decrease of N,(EF) and N+(EF) upon nitrogenation are essential for the almost doubling of Tc. From these analyses, Jaswal et al. have concluded that the Tc increase mainly originates from the reduction in overlap between the Fe 3d-wave functions due to lattice expansion upon nitrogenation. It seems likely that the spin-fluctuation theory is a reasonable model for understanding the magnetism of this class of compounds at finite temperatures.

5.1.2. Compounds of the magnetic rare earth elements R = Nd, Sm and Gd In a series of papers, Lai and coworkers (Gu and Lai 1992, Zeng et al. 1992 and 1993, Gu et al. 1993) have reported first principle calculations of the rhombohedral R2FelTZ3. The electronic band structures of Nd2Fe17 and NdzFel7N3 with N atoms on the 9e, 3b or 18g sites were calculated using the first principle spin-polarized OLCAO method (Gu and Lai 1992, Gu et al. 1993). For the case that the N atoms are located on the 9e sites, the total DOS, site-projected partial DOS and the magnetic moments on the four non-equivalent Fe sites were given. The highest Fe moments are found on the Fe 6c sites in NdzFe17 and on the Fe 9d sites in Nd2Fe17N3, while the smallest ones are on the Fe 9d sites in NdzFe17 and on the Fe 18f sites in the nitride. When the N atoms occupy the 3b or the 18g sites, the maximum local moments were deduced to be on the Fe 6c sites and the minimum moments to be on the Fe 18h sites. The result of these calculations gives the impression that the N atoms preferentially occupy either the 3b or the 18g sites, but not the 9e sites. However, without doubt the N atoms have been confirmed to occupy the 9e sites in the rhombohedral RzFe17N3. Therefore, the results of calculated band structures of Nd2Fe17N3with the N atoms on the 9e sites are suspect because self-consistent iteration is not performed in the calculation. Electronic structures of R2Fel7Z3 (R = S m or Nd; Z = N or C) were studied by the self-consistent local spin density functional method using several cluster models (Zeng et al. 1992 and 1993). The distribution of electrons and the magnetic moments on different Fe sites, Sm(Nd) and N(C) sites were calculated. The calculated moments are given in table 5. lb, which are in fairly good agreement with the experimental results obtained by Kajitani et al. (1993b), suggesting the validity of the method employing cluster-treatments. The work of Woods et al. (1993) is the only published photoemission study of Sm2FelvNu for y = 0 and y ~ 2.6. These authors also calculated the electronic band structures by the self-consistent spin-polarized LMTO method in the scalar relativistic approximation, and compared the computational results with the experimental results obtained by photoemission studies. The major features of the ultra-violet photoemission spectra include the Fe 3d band with a strong peak at 0.8 eV and a small peak at 2.9 eV below the Fermi energy, which are in good agreement with the theoretical DOS calculation. There is a subtle difference in the experimental energy distribution curves (EDC) of Sm2Fe17 and Sm2Fel7N2.6 near the Fermi level. The systematic shift of the EDC to higher energy by 0.05 eV in the nitride is clearly

372

H. FUJII and H. SUN

TABLE 5. lb Calculated and measured values of magnetic moments on different atomic sites and of the total moment per formula unit/zT in R2Fel7 and R2FelTN3 (R = Nd, Sm and Gd) with the rhombohedral Th2ZnlT-type structure. Compound

/ZR (/ZB) 6c

Nd2Fe17 Nd2FelT(exp.) Nd2FelTN3 Nd2FelTNu(exp.)

1.22 3.0 0.48 1.95 3.7 3.8 1.03

Sm2Fel7 Sm2FelT(exp.) Sm2FelTN3 2.11 Sm2Fe17N3(exp.) Gd2Fe17 -7.58 -7.17 -7.38 Gd2FelT(exp.) Gd2Fe17N3 -7.53 -7.07 -7.21 Gd2FelTN3(exp.)

#Fe (/ZB)

/zN (/ZB) /zT (#B/f.u.) Method

6c

18f

18h

9d

2.68 2.54 2.6 2.48 2.67 3.0 3.4 2.02

2.02 2.33 1.5 1.50 2.24 2.2 2.2 2.39

2.34 2.58 1.4 2.19 2.62 2.5 1.9 2.57

1.93 2.01 0.7 2.66 2.46 2.4 2.1 2.13

2.61

2.27

2.58 2.41

2.38 2.61 2.62

2.29 2.29 2.28

2.06 2.10 2.20

2.43 2.69 2.75

2.15 2 . 4 4 2.41 2.13 2 . 3 8 2.44 2.12 2.38 2.48

References: [9] Gu and Lai (1992) [10] Zeng et al. (1993) [11] Kajitani et al. (1993b) [12] Isnard et al. (1992b) [13] Coehoorn and Daalderop (1992)

0.07 -0.11

-0.03

2.15 1.77 1.71 0.06 0.05 0.05

37.3 43.1 30.7 36.3 45.4 48.8 45.3 42.3 34.1 45.8 38.1 22.0 22.5 22.5 22.9 24.2 25.6 25.6 27.3

nscOLCAO scLMTO neutron 14K nscOLCAO scLMTO neutron 14K neutron 4.2K scLMTO Magnetization scLMTO Magnetization scLMTO FLAPW FLAPW Magnetization scLMTO scLMTO FLAPW Magnetization

Ref. [9] [10] [11] [9] [10] [11] [12] [10] [15] [10] [15] [13] [14] [14] [16] [13] [14] [14] [17]

[14] Yamaguchi and Asano (1991) [15] Fujii et al. (1992) [16] Verhoef (1991) [17] Liu J.P. et al. (1991)

found in the expanded binding energy scale, which agrees with the calculated D O S at small binding energies. Apart from the increase in the magnetization and Curie temperature, the third exciting p h e n o m e n o n asso~ iated with interstitial modification is the strong enhancement o f the magnetocrystalline anisotropy energy, which mainly comes from the crystal electric field acting on 4 f electrons of the R site. A few attempts have been made to calculate the crystal field parameters at the R site starting from a realistic charge density distribution determined theoretically from band calculations. The crystal electric field (CEF) parameter A ° and the electric field gradient V,z at Gd sites in Gd2Fel7, Gd2Fe17N3 and Gd2Fe17C3 were first calculated by Coehoorn and Daalderop (1992) using the augmented spherical wave method with atomic sphere approximation ( A S W - A S A ) . The results o f their calculations of A°(val) c o m ing from valence electrons and Vzz are given in table 5.2. The calculated A°(val) values show an increase upon doping with N and C. The trend is in fairly g o o d agreement with the C E F estimations based on the experimental results which has been discussed in sections 3.3.3 (table 3.6) and 4.2.2. Furthermore, they stressed

INTERSTITIALLYMODIFIEDINTERMETALLICS

373

TABLE5.2 Crystal electric fieldparameter A° (Kao2) at the Gd site deducedfrom electronic band structure calculation(the FLAPWmethod) in GdFel2, GdEel2N, Gd2Fel7, Gd2FelTN3 and Gd2Fel7C3. Compound A~(val) A~(val) A~(lat) A~(tot)

GdFel2 GdFeI2N Gd2Fe17 -303 64 1454 -421 -89 -439 135 -25 1015 -286

Gd2Fel7N3 Gd2Fel7C3 Ref. -475 -613 [1] - 1174 [2] 226 [2] -948 [2]

References: [1] Coehoornand Daalderop (1992) [2] Yamaguchiand Asano (1994) that especially the carbon interstitials lead to a stronger anisotropy than the nitrogen interstitials, though the increase of magnetic moment in carbides is much lower than in nitrides. Yamaguchi and Asano (1994) have calculated the electronic structures and the charge distributions of Gd2Fe17 and Gd2Fe17N3 by the self-consistent full-potential linearized augmented plane wave method (FLAPW) within the framework of the local spin density approximation (LSDA). In fig. 5.2, the contour maps of the electron density of Gd2Fe17 and Gd2FelTN3 are drawn for planes parallel and perpendicular to the c-axis. The electron density distribution around the Gd(6c) ion seems to be rather uniform in the c-plane (fig. 5.2(a)), while in the plane parallel to the c-axis, there is a low density area between the Gd(6c) ions (fig. 5.2(b)). This anisotropic electron distribution produces an appreciably negative A° at the Gd site. When nitrogen is introduced and occupies the interstitial 9e site, the electron density around the Gd(6c) ion changes, as in figs 5.2 (c) and (d). Nitrogen strongly hybridizes with the Gd(6c) atoms and increases the electron density around the Gd(6c) ion along the direction of the N(9e) ion. Since the anisotropy of the density distribution is much larger than that in Gd2Fe17, it gives rise to a larger negative A° at the Gd(6c) site. From the calculated charge distributions, the second order crystal electric field parameters A° at the Gdsite have been carefully determined, the results of which are included in table 5.2. A°(val) is negative and large in Gd2Fe17, due to the valence electrons within its own muffin-tin (MT) sphere at the Gd site. Though an appreciable part of the A°(val) is cancelled by the positive A°(lat) contributions coming from other charges in both the other MT spheres and the interstitial region, the calculated value of the total A° is still negative and much larger than the experimental one determined in Sm2Fel7 (Li and Cadogan 1991). In Gd2Fe17N3, the calculated value of the total A° is negative and about 3 times that in Gd2Fel7, which is also much larger than the experimental one for Sm2Fe17N3 (see table 3.6) (Kato et al. 1993). The origin of these discrepancies is an open question, and requires further investigations of theoretical approaches to the 4f-magnetic anisotropy. Yamaguchi and Asano (1994) also estimated the electric field gradient V~z at the Gd nuclei from the charge distributions. Results are listed in table 5.3 together with the value obtained by the analysis of the 155Gd M6ssbauer spectrum. It can be seen that the calculated values of V~z are in good agreement with the experimental ones.

374

H. FUJII and H. SUN 100

10.0

50.0

50.0

-I0.0 "10.0

-10.0 0.0 a.u.

10.0

-10.0

(a)

10.0

(b) 1010

10.0

500

cd

-10.0

-10.0

-I0.0

0.0 a.u.

0,0

a.u.

(c)

10.0

-10.0

0,0 a.u.

100

(d)

Fig. 5.2. Electron density distribution of Gd2Fel7 ((a) in a plane perpendicular to the c-axis, (b) in a plane parallel to the c-axis) and of Gd2Fe17N3 ((c) in a plane perpendicular to the c-axis, (d) in a plane parallel to the c-axis). The lowest contour value is 0.0125 a.u.-3, the contour spacing is 0.0125 a.u.-3, and the highest contour value is 0.1 a.u.-3. The lowest density regions in (b) and (d) have densities between 0 and 0.0125 a.u.-3. The density regions with densities higher than 0.1 a.u.-3 near each atomic site are not shaded (after Yamaguchi and Asano 1994). Finally, Yamaguchi and Asano calculated the electronic structures and the magnetic properties o f Gd2Fel7 and the nitride by the self-consistent L M T O - A S A method. In their calculations, the lattice constants were determined by minimizing the total energy. The increases in the lattice constants and the magnetic moments due to introduction o f the N interstitials can be well explained when the ratio o f the atomic sphere radius is chosen as rGd/•Fe = 1.35. The calculated magnetic moments are included in table 5. tb, together with the values estimated by the self-consistent F L A P W - L S D A method. The results are in g o o d agreement with each other, indicating that the choice of rGd/rFe = 1.35 in the self-consistent L M T O - A S A method is

INTERSTITIALLYMODIFIED INTERMETALLICS

375

TABLE 5.3 Calculated and experimentalelectric field gradient Vzz (1021 Vim2) at the Gd nucleus in GdFe12, GdFe12N, Gd2Fe17 and Gd2Fel7N3. Compound Vzz(cal) Vzz(exp)

GdFe12 GdFeI2N Gd2Fe17 1.1 15.5 5.8 6.4 4.3 4.4

Gd2Fe17N3 Method Ref. 11.0 FLAPW [1] 10.5 scLMTO-ASW[2] 12.6 [3] [4]

References: [1] Yamaguchi and Asano (1994) [3] Dirken et al. (1991) [2] Coehoorn and Daalderop; Daalderop et al. (1992) [4] van Steenwijk et al. (1977) quite reasonable. The calculated Fe magnetic moments are in fairly good agreement with those determined for Nd2Fe17N3 by the neutron diffraction studies (Isnard et al. 1992b, Kajitani et al. 1993b). 5.2. Calculations of the l :12-type interstitial compounds RFel2_xT~Zy (Z=N or C) Electronic band structure calculations have been carried out for the l:12-type compounds RFe12_~T~Z v with T = Ti and Mo, R -- Nd and Y, and also for the hypothetical compound YFe12Z. Results are summarized in table 5.4 and will be discussed here. 5.2.1. Calculations of RFelz_xT~N v with R = Y or Nd and T = T i The spin-polarized electronic structures of YFe11Ti and YFel aTiN were calculated by Li Y.R and Coey (1992) using the non self-consistent OLCAO approximation and the calculations succeeded in describing some physical aspects of this system. However, it has been pointed out later by others that the results possibly lack the stability of the electronic system because of the non self-consistent character of the method. The results of the calculation of Li Y.E and Coey showed that in YFellTiN there are differences on the different sites as to the opposite effect of hybridizations of Fe 3d-states with N 2p-states and the effect of volume expansion due to interstitial nitrogen atoms on local magnetic moments. The competition of these opposite effects leads to a slightly increase of the total iron moment per formula from 20.3#B to 20.5#8. Self-consistent spin-polarized band structure calculations have been performed by the LMTO-ASA method with a ratio of atomic radii ry : rFe " rTi ---=1.35 : 1.0 : 11.1 in the frame of local spin density functional formalism by Sakuma (1992). To examine separately the lattice expansion effect due to introduction of N atoms, Sakuma also calculated the electronic structures of YFe11TiN0 with the same atomic structure and lattice constant as in YFe11TiN but with no N atoms. The result indicates that the total moment of YFellTiN (23.6#B) is larger than that of YFe11TiNo (22.4# B) and YFexlTi (21.7#B), which is different from the results of the non self-consistent spin-polarized band calculations by Li Y.R and Coey (1992). The asphericity parameters Anp and And were estimated by Sakuma to gain insight into what caused the increase of the moment due to the inclusion of N atoms in addition to the lattice

376

H. FUJII and H. SUN

TABLE 5.4 Calculated and measured values of magnetic moments on different atomic sites and of the total moment per formula unit pq- in RFellT, RFel2 and their interstitial compounds RFeI1TN, RFel2N with T = Ti and Mo. #z is the magnetic moment of the interstitial atom Z. Compound YFellTi YFell Ti(exp) YFe11TiN

#R (#B) /ZFe (/ZB)

]Az (/ZB) /zT (#B]f.u.)Method

2a

Fe(8f) Fe(8i) T(8i) Fe(8j) Z(2b)

0.27 -0.46

1.77 1.70

1.91 2.37

0.21 1.76 -0.83 2.23

0.19 -0.36

1.73 2.18

2.31 2.67

0.15 1.73 -0.83 1.98

1.67

2.46

-0.86 2.32

YFe11TiN(exp) NdFell Ti 3.2 NdFe11Ti(exp) NdFe11TiN 3.3 NdFe11TiN(exp) YFe11Mo -0.04 YFel 1Mo(exp)

1.93

2.55

-0.89 2.14

1.64 2.15

2.45 2.36

-0.48 2.15 - 1.0 2.42

YFellMON -0.41 YFe11MoN(exp)

2.21 2.38

2.62 2.43

-0.45 1.92 - 1.0 2.32

YFe12 YFelzN YFeI2C

1.80 2.30 2.30

2.45 2.61 2.60

2.40 2.03 1.64

-0.43 -0.30 -0.39

References: [l] Li Y.P. and Coey (1991) [2] Sakuma (1992) [3] Yang Y.C. et al. (1991) [4] Jaswal (1993) [5] Akayama et al. (1994)

0.10 0.04

-0.02

20.3 21.7 18.6 20.5 23.6 21.8 26.9 25.7 28.1 27.0 21.6 26.0 25.8 23.7 27.4 27.0 26.2 27.5 25.6

nsc OLCAO scLMTO Magnetization nsc OLCAO sc LMTO Magnetization scLMTO Magnetization sc LMTO Magnetization sc LMTO neutron Magnetization scLMTO neutron Magnetization scLMTO scLMTO scLMTO

Ref.

[1] [2] [3] [1] [2] [3] [4] [5] [4] [5] [6] [7] [8] [6] [7] [8] [9] [9] [9]

[6] Ishida et al. (1994) [7] Sun H. et al. (1993a) [8] Sun H. et al. (1993b) [9] Asano et al. (1993)

expansion. Here, Anp and And give the degree to which the p and d charge densities are prolate or oblate (Coehoorn 1990). The results of the calculations show that Anp > 0 and And > 0 for the nitride, suggesting that the valence orbitals elongate along the c-axis. The asphericity, therefore, predicts that the N atoms attract electrons from the Y atoms rather than from the Fe atoms and in turn release the Fe atoms from bonding with the Y atoms. This effect leads to a further increase o f the Fe magnetic moments in addition to the increase associated with the expansion o f the lattice upon nitrogenation. As general features, it has been deduced that the m o m e n t s on the Fe atoms which are nearest neighbors to the N atoms are lowered upon nitrogenation, while the Fe atoms which are farthest from the N atoms have the largest moments. The calculated total Fe magnetic moments after nitrogenation are much larger than the experimental values obtained so far. This might be related to the incomplete nitrogenation and to the fact that the occupancy of the 2b site by nitrogen is only about 50% in YFelxTiNy. Experimental data by A k a y a m a et al. (1994) have shown that the total m o m e n t per formula unit for NdFellTiNa.5 and N d F e l l T i are 27.0#B and 25.7#B, respectively. Assuming Nd behaves as a free

INTERSTITIALLYMODIFIEDINTERMETALLICS

377

ion with a moment of 3.6#B in NdFe11TiN1.5, the moment on the Fe-sublattice is deduced to be 23.4#B, which is close to the above calculated value of 23.6#B for YFe11TiN. The self-consistent spin-polarized electronic structure calculations have been also carried out for NdFeI1Ti and NdFe11TiN0.5 by Jaswal (1993). The calculated magnetic moments are comparable with the experimental data by Akayama et al. (1994) (see table 5.4). Based on the spin-fluctuation theory of Mohn and Wohlfarth (1987), the enhancement of Curie temperature by interstitial modification was discussed by Sakuma (1992) and Jaswal (1993) independently. Using eqs (5.1) and (5.2) in section 5.1.1, the ratio Tc(NdFeHTi)/Tc(NdFealTiNo.5) is estimated to be 1.32 from the band structure parameters. As the corresponding experimental value is 1.31, it has been concluded that the Curie temperature enhancement upon nitrogenation is again primarily due to the lowering of the DOS for both spin directions at the Fermi level. Electronic structures of NdFea 1Ti and its nitride were studied by the photoemission spectroscopy and also by the self-consistent spin-polarized band structure calculations using the LMTO-LDA method (Fernando et al. 1993). We notice that there is an overall agreement between the experimental and calculated DOS. 5.2.2. Calculations of RFel2_xT~N with R = Y and T = M o

The electronic structures of YFelz_xMo~ and their nitrides have been calculated by Ishida et al. (1994) using the self-consistent LMTO-ASA method in the nonrelativistic approximation to examine the effect of N and Mo atoms on the magnetic properties. The local DOS of the Fe 8f, 8i and 8j-atoms in YFellMO are similar to those of YFel2 which will be discussed later. The calculated magnetic moments are listed in table 5.4, which are not inconsistent with the results of neutron diffraction study by Sun H. et al. (1993b) for YFeaIMo and the nitride. Regardless the Mo content, the moments on the different Fe sites increase in the order of #(Sf) < #(8j) < #(8i) for YFe12_~Mo~, and nitrogenation leads to a decrease of the moment on the 8j site that is nearest to the N atoms, and an increase of the moments on the sites 8i and 8f that are further from the N atoms. These changes are explained by the strong hybridization between the N 2p-states and the Fe d-states. The substitution of Mo atoms for Fe atoms is also reflected in the values of the Fe moments as well. The calculated Fe moment on all sites decrease gradually with increasing the Mo concentration, which originates from the hybridization between the Mo d-states and Fe d-states near the Mo atoms. Similar results have been obtained by Jaswal et al. (1990) in YFel0T2 (T = V or Cr). These authors found that the moments of V or Cr couple antiparallel to the Fe moments, leading to a reduction in the average moment per Fe atoms. The overall tendency of the concentration dependence of the calculated moments in YFe12_~Mox and the nitride is similar to the tendency of the experimental values obtained by Sun H. et al. (1993a) for z < 0.2, but the discrepancy is large for x > 2. Besides the experimental error, this discrepancy might be due to the assumption in the calculation that the Mo atoms are located only at the 8i-site. In real compounds, it is possible that a small amount of Mo atoms might locate at the 8f-site as well (Sun H. et al. 1993b).

378

H. FUJII and H. SUN

5.2.3. Calculation of the hypothetical RFe12Z The results of electronic structures calculations for Y F e l 2 (Coehoorn 1990) and YFeloT2 (T = Cr or V) (Jaswal et al. 1990) have indicated that the characteristic features in the DOS are commonly conserved for all the YFe12_xT~systems. Thus, it is valid and of interest to study the electronic band structures of RFe12Z (Z = N or C) theoretically. The self-consistent spin-polarized electronic structures of the hypothetical RFe12 and RFe12Z (R = Y, Ce or Gd; Z = N or C) have been calculated by Asano et al. (1993) using the LMTO-ASA method within the frame work of the LSD approximation in order to examine the effect of the Z atoms on the magnetic properties. In the calculation, the ratio of the atomic radii was chosen as rR/rFe = 1.4 and the total energies of RFe12 and their interstitial compounds were calculated as functions of the lattice constants. The theoretical lattice constants determined by minimizing the total energies were used as initial parameters for further calculations. In figs 5.3 and 5.4, are shown the local DOS of the Fe d-states, Y s-, p-, d-, f-states and N 2p-states in Y F e l 2 and YFe12N, respectively. For Y F e I 2 , the up-spin bands in the Fe d-states (indicated by full line in fig. 5.3) are almost fully occupied for all the three Fe sites, but the unoccupied peak in the down-spin states (shown by dotted line in fig. 5.3) is smaller for the Fe 8f-sites than for the Fe 8i- or 8j-sites, indicating/~Si o r #sj >/Z8f" From the local DOS of the s-, p- and d-states of Y atom, we notice that the up-spin and the down-spin bands are separated, but the number

30

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0.0

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-0.2

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0.2

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Fig. 5.3. Spin-polarized local density of states of the Fe(8f), Fe(8i) and Fe(8j) d-states and of the Y s-, p-, d- and f-states in YFe12 calculated using the LMTO-ASA method in the non-relativistic approximation (after Asano et al. 1993).

INTERSTITIALLY MODIFIED INTERMETALLICS

i

379

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Fig. 5.4. Spin-polarized local density of states of the Fe(8f), Fe(8i) and Fe(8j) d-states, N(2b) p-state and of the Y s, p, d and f-states in YFe12N calculated using the LMTO-ASA method in the non-relativistic approximation (after Asano et al. 1993).

of occupied states in the s- and p-bands is almost the same for both the spin states, while the occupied states in the d-bands are more full for the down-spin than for the up spin. This indicates that the spin polarization is small for the s- and p-states, but the magnetic moment due to the d-spin polarization is appreciable and antiparallel to the Fe moments, which leads to negative exchange interactions between the Fe spins and Y spins. For YFe12N, the general DOS shapes of the Fe 3d-states are similar to those of YFe12. However, there are some essential differences as follows: The main peaks of the N p-states are located below -0.6 Ryd, where the d-band tails of the Fe 8j-atoms (the nearest of the N atoms) becomes larger than those of RFe12. This shows that the Fe d-states of the 8j-sites well hybridize with the N 2p-states. Consequently, the tail of the up-spin band above EF becomes larger for the Fe 8j-sites but smaller for the Fe 8f- and 8i-sites upon nitrogenation. On the contrary, we notice that the unoccupied DOS peaks of the Fe down-spin state becomes smaller for the Fe 8j-sites, but larger for the Fe 8f-sites compared to those of RFe12. These variations of DOS upon nitrogenation lead to a decrease in the moment on the Fe 8j-sites but to an increase in those on the Fe 8f- and 8i-sites. Paying attention to the DOS of the N p-states, we can see that there is a peak just below EF in the up-spin states that is located at the shoulder in the Fe d-bands. There is a similar peak above EF located at the center of unoccupied peaks of the down-spin Fe d-states (in the energy range between +0.1 Ryd). These peaks mainly come from the antibonding states between

380

H. FUJII and H. SUN

the N p-states and Fe d-states. We notice, moreover, that the sharp peaks at EF for the Fe 8i- and 8j-sites in YFe12 disappear in YFel2N, indicating a substantial decrease of the DOS at EF upon nitrogenation. As described above, it has been shown that the Curie temperature is enhanced by nitrogen uptake because the spin fluctuation is restrained, according to Mohn and Wohlfarth (1987) model. The calculated values of the magnetic moments on the different sites for RFel2 and RFel2Z (Z = N or C) are summarized in table 5.4. We notice that the calculated moments of the Fe atoms become larger in the sequence the 8f-, 8j- and 8i-sites in YFe12, while the Fe moment on the Fe 8f-sites is larger than that on the Fe 8j-sites in YFe12N. This tendency is common to all the cases, irrespective of the rare earth R. The band calculations also predict that the total magnetic moment per formula unit in YFe12 increases from 26.2#B to 27.5#B upon nitrogenation, but decreases to 25.7#B by carbonation. For YFel2C, the DOS peaks of the C p-states due to the antibonding states shift to the higher energy region above EF. This makes the effect of moment enhancement due to hybridization weak compared to the nitride. Therefore Asano et al. (1993) claimed that N is the best among H, N, C and B to improve the magnetic properties of RFe12. Unfortunately, since RFe~2 dose not exist, we can not directly compare the calculated moments of RFe12 and RFe12N with experimental results. However, we notice that the calculated moments are in fairly good agreement with the extrapolated values to z = 0 of the moments of YFe12_xMox and YFe12_xMoxN (Sun H. et al. 1993a). In order to shed light on the appearance of uniaxial anisotropy in Pr- and Ndsystems, Yamaguchi and Asano (1994) determined the CEF parameter A° at the Gd site in GdFe12 and GdFe12N from the electronic structures and charge distributions calculated using self-consistent FLAPW-SDA method. In figs 5.5(a) and (b), are shown the contour maps of electron density distribution in GdFe12 and GdFe12N parallel and perpendicular to the c-axis. The density distribution around the Gd(2a) ions in GdFe12 seems to be rather uniform, reflecting the small value of A°. When the N atoms are introduced and occupy the interstitial 2b sites between the Gd(2a) ions, the electron distribution is changed as is seen in fig. 5.5(b). Here, we can see that N is strongly bonded to the Gd(2a) atoms, which seems to be much stronger than the Gd-Fe bonding. Those bonding states between the Gd and N atoms increase the electron density around Gd along the direction of the N atoms. This type of anisotropic density distribution around Gd produces the large positive value of A°. The calculated values of A ° are shown in table 5.3 for GdFe12 and GdFe12N. In GdFe12, the calculated net value of A°2 = A°(val) + A°(lat) is small. In GdFelzN, while A°(val) is very large and positive, A°(lat) is negative and cancels an appreciable part of A°(val). Nevertheless the net value of A° is positive and still large.

5.3. Summary We can qualitatively understand the origin of the improvement of the magnetic properties in both the interstitially modified 2:17 and 1:12 systems from the calculated band structures. However, there are obvious discrepancies between the experimental results and the results of electronic band structure calculations. It is necessary to

381

INTERSTITIALLY MODIFIED INTERMETALLICS 10.0

:5 0.0

-10.0 -10.0

0.0

10.0

a.u.

(a) 10.0

:50.0 ai

-10.0 -10.0

0.0

10.0

a.u. (b)

Fig. 5.5. Electron density distribution in a plane parallel to c-axis: (a) of GdFel2, (b) of GdFel2N. The lowest contour value is 0.0125 a.u. -3, the contour spacing is 0.0125 a.u. -3, and the highest contour value is 0.1 a.u. -3. The lowest density regions in (a) and (b) have the density between 0.0125 and 0.025 a.u. -3. The regions with densities higher than 0.1 a.u. -3 near each atomic site are not shaded (after Yamaguchi and Asano 1994).

382

H. FUJII and H. SUN

perform more careful experimental studies using high-quality interstitially modified samples. On the other hand, band structure calculations involve various atomicsphere approximations which can be not true in the real materials. We believe that both the experiment and calculation are need to be improved. By quantitatively comparison with more reliable experimental data, the validity of the various approximations can be clarified and subsequently the electronic band structure calculation method can be further sophisticated.

6. Applications Owing to their excellent intrinsic magnetic properties, interstitially modified materials have been regarded as promising candidates for permanent magnet applications. Especially Sm2Fe17 in the 2:17 series and NdFel2_~T~ (T = Ti, V, Mo, etc.) in the 1:12 series are of great interest. Unfortunately these materials are metastable compounds and disproportionate at temperatures higher than 600°C in general. Thus the conventional high-temperature powder metallurgy is not applicable to the interstitial materials. The problem now is to either improve their stability or develop a suitable processing route for making bulk permanent magnets. By partially substitution of other elements for Fe, the stability has been more or less improved. However, the metastability still remains as a problem to be solved. Various techniques have been applied to the interstitial compounds for producing both isotropic and anisotropic magnets. An extremely high coercivity of more than 4 T has been obtained by combined mechanical alloying and pressure-assisted zinc bonding (Kuhrt et al. 1992). An energy product (BH)rnax of 206 kJ/m 3 has been realized in an isotropic nanocrystalline exchange-coupled two-phase magnet composed of the hard SmzFelyN v phase and the soft a-Fe phase, although the coercivity is still low (Ding et al. 1993). It should be noted here that the interstitial compounds are stable at room temperature for a period of the order of 1015 years if we take into account of the diffusion of the iron atoms only (Skomski and Coey 1993a, b). So, it is not necessary to consider deterioration of the permanent magnetic properties due to aging at room temperature. In this section, studies concerning the stability improvement will be first summarized, followed by different processing techniques and their achievements in making permanent magnets. The studies on the magnetic domain structure and the coercivity mechanism will also be reviewed. The emphasis is on the SmzFel7 nitride, for which most of the progress has been made.

6.1. Improvement of the thermal stability 6.1.1. Effects of substitutions Due to differences in the definitions involved with the decomposition and with the various experimental techniques, substitution effects on the thermal stability of Sm2Fe17-nitride have been reported to be slightly different by different authors. Hu B.E et al. (1992) studied the thermal properties of Sm2(Fel_~M~)17Nu (M = Co,

INTERSTITIALLYMODIFIEDINTERMETALLICS

383

Ni, A1, Ti and V) by a thermopiezic analyzer (TPA) and using thermal magnetic analysis (TMA), and concluded that there was no significant effect of M substitutions for Fe. Sugimoto et al. (1992a, b) investigated the decomposition temperature of Sm2(Fel_xM~)lTNy with M = A1, Si, Ti, V, Cr, Mn, Fe, Co, Ni, Ga, Zr, Nb and Mo by detecting the phase changes using X-ray diffraction and SEM-EDX techniques. These authors have found that Cr is effective for improving the thermal stability, the decomposition temperature of Sm2(Fe0.9Cr0.a)17Ny being about 100°C higher than that of the pure Sm2Fe17N v phase. Zhou S.Z. et al. (1992) reported that the replacement of Fe by Co (10 at.%), or Ga (1.7 at.%), or Zr (1.7 at.%) enhanced Ton, the starting temperature of the decomposition being defined as the temperature at which the magnetization starts to increase in the thermo-magnetic curve.

6.1.2. Effects of carbonation There are two routes for synthesizing carbonitrides. The interstitial modified carbides Sm2FelvCx with x < 1.5 can be made first by melting. The carbonitrides SmzFel7CxN u are then formed by the subsequent gas phase nitriding process (Kou et al. 1991a, b, Yang Y.C. et al. 1992a). The other route is to synthesize carbonitrides by the gas phase interstitial modification process only. Nitrogenation is done first, followed by carbonation in C2H2 for a short time (Chen et al. 1993a, Altounian et al. 1993). Carbonitrides made by the second method were reported to have better thermal stability than the pure nitrides. The N2 out-gassing temperature was increased by 180 K mainly as a result of the low diffusion rate of nitrogen in the outer-most carbon layer formed during the carbonation process. The other advantage is that there is less segregation of c~-Fe in the carbonitrides made by the second method, because of the shorter annealing time required for carbonation and because the formation of cementite Fe3C. The magnetic properties of SmzFelTC~Ny are similar to those of the SmzFel7N v, but with a somewhat better thermal stability. 6.2. Development of permanent magnets Although the thermal stability can be improved to some extent, the metastability of the nitrides (or carbonitrides) remains to be a problem and dictates that the magnets have to be made by a low temperature process and thus the solution falls naturally on bonded magnets produced from coercive powder. Anisotropic magnets can be made from single crystal powders either by metal or epoxy bonding, while nanocrystalline fine-structured powders produced by mechanical alloying, mechanical grinding, rapid quenching or HDDR, are used for making isotropic magnets. Most of the published data are still at the stage of making high coercive powders, that is, a stage preceding the manufacture of bulk magnets. The processing procedures of the various techniques are shown in fig. 6.1 to 6.4.

6.2.1. Bonded anisotropic magnets based on the Sm2Fel7 nitride The procedure for making anisotropic bonded magnets is as follows (fig. 6.1). The host ingots are produced by melting of the starting materials (either arc-melting or induction melting), which are then subjected to a homogenization treatment and are

384

H. FUJIIand H. SUN melting Sm, Fe metals

I

meltingandtomogenization SmzFe17compound

pulverization

I powdersI nitroginatton

InitridepowdersI

furtherpulverization l finepowders I |

bonding and

I

ma.~etacgil nment

I

Fig. 6.1. Schematicdiagramof the meltingprocess. crushed into powders with the size of a few 10 #m in diameter prior to the gas phase interstitial modification process. Nitrogenations are performed in N2, NH3, or mixtures ofN2, NH3 or H2 at temperatures typically of 450°C. The bulk magnets are then made by epoxy bonding or metal bonding of the nitrided powders. A magnetic field is applied during the bonding process for magnetic alignment. Sometimes, the nitride powders are further pulverized down to a few #m in diameter before bonding. In the case of metal bonding, a heat treatment performed at temperatures slightly higher than the melting point of the bonding metal is necessary. Every step in the processing procedure is important for the finally obtained hard magnetic properties. First, a high purity Sm2Fe17 phase in the host compounds used for nitrogenation is essential. As Sm is an element with a relative high vapor pressure, Sm evaporates during melting, more or less depending on the melting condition, and the Sm content in the melt becomes lower than that in the starting mixture of the pure elements. Oxidation can also cause losses of Sm. Thus an excess amount of Sm is needed for compensation of the losses. However, it is difficult to

INTERSTITIALLYMODIFIED INTERMETALLICS

385

apply the exact amount of Sm excess because of the difficulties in controlling the Sm losses during melting. If not enough Sm is present, c~-Fe appears as a second phase in the Sm2Fel7 compounds. This soft magnetic c~-Fe phase remains present after nitrogenation and certainly is harmful to the hard magnetic properties of the material. On the other hand, if too much Sm is present, SmFe2 or SmFe3 will appear as impurity phases. They directly decompose into StuN and c~-Fe upon nitrogenation (Christodoulou and Takeshita 1993a, b, Rodewald et al. 1993, Fukuno et al. 1992) and again the soft magnetic c~-Fe phase appears. In order to avoid the harmful c~Fe phase, the composition of the Sm2Fe17 ingot after homogenization must match the stoichiometric composition as exactly as possible. The nitrogenation process is always accompanied by the decomposition process of the 2:17 phase into c~-Fe, SmN and possible some other unknown phases. Thus carefully controlling of both the alloy preparation and the nitrogenation process is important for reducing the formation of the soft magnetic phases and in turn for making high quality magnetic powders. Nitrided powders of SmzFe17Ny having the highest value reported so far for the maximum energy product of 244 kJ/m 3 were obtained by Suzuki et al. (1993). The influence of the Sm content on the magnetic properties of Sm2Fe17N v was studied by Fukuno et al. (1992) and these authors succeeded in obtaining (BH)max values as high as 240 kJ/m 3 and a coercivity of 1.0 T for a Sm content of about 11 at.%, which is slightly higher than 10.5 at.% Sm in stoichiometric SmzFea7. Fukuno et al. performed a pre-hydrogenation treatment at 250°C for 4 hours before nitrogenation. The 250°C pre-hydrogenation treatment was reported to be effective also by Rodewald et al. (1992). Nitrogenation in a mixed gas of NH3 and H2 resulted in a (/3H)max value of 170 kJ/m 3 for Sm2Fel7N v powders (Iriyama et al. 1992, Kobayashi et al. 1992). The magnetic properties of the nitride powders are found to be closely related with the particle size. The coercivity increases almost exponentially with decreasing average particle size. The dependence of coercivity on particle size was studied in detail for SmzFelvN v by Suzuki and Miura (1992) and for Smz(Fe0.8Co0.2)IvN3.03 by Yamamoto H. et al. (1993). Particle sizes in between 2-4 #m were considered to be close to the size of single magnetic domain particles (Iriyama et al. 1992) and the single domain theory was used to explain the sharp increase of the coercivity upon further pulverization after nitrogenation. A sensitive effect of particle size on coercivity was also demonstrated by Wendhausen et al. (1992). The authors found that the squareness of the hysteresis loop could be improved just by controlling the particle size in a certain range by sieving. The nitrided powders can be processed into real magnets in bulk form by two kinds of binders, epoxy and metal. Suzuki et al. (1993) obtained 2.5 wt.% epoxy bonded SmzFe17Nv magnets with a (BH)max of 145 kJ/m 3 and a density of 6.1 g/cm 3. They also studied the dependence of the magnetic properties on the compaction pressure and found that pressures in the range of 800-1000 MPa were necessary for reaching a high (BH)max. However, the coercivity decreased slightly with increasing pressure for unknown reasons. The (BH)max value of 125 kJ/m 3 and a density of 5.93 g/cm 3

386

H. FUJII and H. SUN

were obtained in Sm2(Fe0.sCo0.2)17N3.03 by Yamamoto H. et al. (1993) by adding 3 wt.% of epoxy and applying a pressure of 784 MPa. Non-magnetic metals with a melting point much lower than the decomposition temperature of the nitride are considered to be suitable candidates that could be used as binders for making bulk nitride magnets. Zn (melting point 419 °C), Bi (271 °C), Sn (231 °C), A1 (620°C) and In (157°C) have been tried experimentally (Otani et al. 1991, Huang et al. 1991a). Zn was found to be the only suitable one and has been studied most intensively. Sn is also of some use but it is much inferior to Zn (Huang et al. 1991a, Rodewald et al. 1992). Bi, A1 and In were found to be unsuitable. The bonding processes were performed mostly at temperatures slightly higher than the melting point of the bonding metal. However, Rodewald et al. (1993) reported that annealing of the mixture of Sm2Fel7Ny and Zn powder at temperatures above the melting point of Zn (419°C) resulted in the decomposition of Sm2Fel7N v by the reaction Sm2Fe17Ny + Zn --+ Sm2FeZn2 + c~-Fe + N2, and the magnetic properties deteriorated due to the appearance of ~-Fe. The effect of Zn content on coercivity, remanence and magnet density has been investigated by Suzuki and Miura (1992), and Rodewald et al. (1993). Coercivity values in excess of 2.5 T were reported by using up to 50 wt.% Zn powder (Suzuki and Miura 1992). Because of the dilution of the magnetic phase by the non-magnetic Zn metal, the remanence is low and the energy product is also low. Rodewald et al. (1993) suggested that 10-15 wt.% Zn powder should be used in order to reach a balance between the remanent polarization and the coercivity. In table 6.1 we summarize the magnetic properties of the metal and resin bonded magnets, as well as the magnetic properties of the nitride powders. The mechanism of the strong positive effects of Zn bonding on the coercivity has basically been understood on the basis of the formation of the secondary paramagnetic Zn-Fe phase (Zn7Fe3 or Zn4Fe etc.) between the nitride particles. The surfaces of the nitride particles were smoothed by the Zn-Fe phase and as a result, the interparticle stray fields were reduced. At the same time, the deleterious o~-Fe could be eliminated or at least be reduced. A similar effect was found for Sn and attributed to the formation of SnFe (Huang et al. 1991a, Hu J.E et al. 1993b). On the other hand, a microstructural study of the Zn bonded SmzFea7N3 magnet by transmission and analytical electron microscopy revealed an alternative mechanism for the high coercivity (Hiraga et al. 1993). The reaction between the matrix Sm2FeayNv phase and Zn metal produced the Smz(Fe, Zn)17Nu (assumed to be non-magnetic) and excess a-Fe. The main part of the produced c~-Fe combined with Zn and in turn the Zn-Fe phase was formed. The remaining c~-Fe existed in the form of Fe-Zn primary solid solution. The Sm2(Fe, Zn)lyN v coated the main nitride phase and prevented contact between the ~-Fe and the main nitride phase, improving the coercivity. The temperature coefficients of Br and #0iHc for the epoxy bonded magnets were reported to be -0.07 ~ -0.08 %/K and -0.44 ~ -0.53 %/K, respectively (Suzuki et al. 1993, Yamamoto H. et al. 1993), and the temperature coefficient of #0iHc for the Zn bonded magnet was -0.45 %/K (Rodewald et al. 1992). In general, the temperature stability was better than for NdFeB magnets, and also a better environmental stability was reported by Suzuki et al. (1993).

INTERSTITIALLY MODIFIED INTERMETALLICS

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H. FUJII and H. SUN

6.2.2. Mechanical alloying of the 2:17 compound The mechanical alloying method starts from elemental Sm and Fe powders and leads to a two-phase mixture of amorphous Sm-Fe and c~-Fe after milling in a protected atmosphere, which is Ar in most of the cases (fig. 6.2). The Sm2Fe17 phase with an optimized microstructure forms during vacuum or argon annealing at around 700 °C. Subsequent nitrogenation in the temperature range between 400-500°C transforms the soft easy plane anisotropy Sm2Fe17 phase into the hard Sm2Fe17N v phase with a very strong uniaxial anisotropy. Hard magnetic properties of the epoxy bonded or cold pressed isotropic powders obtained by various authors are summarized in table 6.2. An extremely high coercivity of 4.36 T (Kuhrt et al. 1992) has been obtained by mechanical alloying and subsequent bonding with 20 wt.% Zn at a high pressure of 270 MPa. The remanence of 0.4 T was low because of the dilution by the non-magnetic Zn and also because of incomplete densification in spite of the high pressure applied. The high coercivity was understood as arising from pressureassisted formation of the paramagnetic Zn7Fe3 phase at the grain boundaries inside the powder particles. The effect of compaction pressure on the coercivity and on the magnet density was also studied. Although the density increased with increasing compaction pressure, pressures much higher than 270 MPa induced the decomposition of the nitride phase and resulted in a lower remanence. A method analoguous to mechanical alloying (MA) is mechanical grinding (MG). Instead of using elemental Sm and Fe powders, the Sm-Fe ingot is first synthesized by melting and then subjected to ball milling in an Ar atmosphere (Ding et al. 1992b, Wang K.Y. et al. 1993, Majima et al. 1993). The as milled two-phase

mechanical alloying

Sm,Fepowders ] mil~ng cx-Fe+ Sm-Feamorphousphase I annealing nitrog;nation finelycrystallized nitride powders

bonding l bulkmagnetI Fig. 6.2. Schematicdiagram of the mechanical alloyingprocess.

INTERSTITIALLYMODIFIED INTERMETALLICS

389

TABLE 6.2 Hard magnetic properties achievedby nitrogenationor carbonationof the Sm-Fe compoundsproduced by mechanical alloying(MA) or mechanical grounding (MG). #0iHc and Br are the intrinsic coercivity and remnant magnetization, respectively. (BH)max is the maximum energy product. Composition

Method

#0inc (T)

Br (T)

Sm12.sFe87.5 Sm13.sFe86.5 Sm14.6Fe85.4 Sm13Fe87

MA MA+Zn bonding MA MA MA MG MG anisotropic MA carbide

2.94 4.36 3.13 2.95 3.61 3.13 1.94 2.32

0.71 0.4 0.51 0.624

Sm14Fe86 Sm15Fe87 SmlsFe85

Sm13.5Fe86.5

References: [1] Schnitzke et al. (1990) [2] Schultz et al. (1991) [3] Kuhrt et al. (1992a) [4] Kuhrt et al. (1993) [5] Ding et al. (1992a)

0.7 0.785 0.61

(BH)rnax(kJ/m3) Ref. 86 101 89 83 167 58

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

[6] Ding et al. (1992c) [7] Wang K.Y. et al. (1993) [8] Ding et al. (1992b) [9] Kuhrt et al. (1992b)

product containing a-Fe and amorphous Sm-Fe is similar to that obtained by MA. The followed two-step heat treatment (annealing and nitrogenation) is also similar and again the high coercivity is considered to be related to the ultrafine-grain microstructure. The advantage of MG when compared with MA is that the handling of the S m - F e melted ingot is much easier than the handling of the Sm and Fe metal powders. Ding et al. (1992b) reported the successful manufacture of anisotropic powder by MG with hard magnetic properties of Br = 0.785 T, #0iHc --= 1.94 T and (BH)max = 170 kJ/m 3. The key point is the control of the milling time which has to be kept below a certain critical period, depressing the formation of the amorphous phase. The latter would have transformed into randomly oriented nanocrystals during the two-step heat treatment and resulted in isotropic magnets. The effect of the milling atmosphere during MA and MG was studied by Calka et al. (1992). While MA in neither N2 nor NH3 resulted in the formation of the 2:17 phase alloy, MG of pre-arc-melted Sm-Fe ingot in N2 or HH3 could be of interest. An alternative way of MA is the chemical reduction via mechanical alloying as reported by Liu Y.N. et al. (1992). The starting materials are Sm203, SmF3, Fe and Ca, which are much cheaper and easier to deal with than the pure Sm and Fe powders. The 2:17 nitride phase is formed together with CaO or CaF2 by milling and the two-step heat treatment mentioned above. The maximum coercivity obtained is 2.5 T. Apart from the task of developing magnetic anisotropy, which is common to all other MA and MG materials, separation of the magnetic phase from the impurities CaO or CaF2 is a main task in this kind of materials.

6.2.3. Rapid quenching of the 2:17 compounds The single-wheel melt-spinning method is frequently used for rapid quenching (see fig. 6.3). The pre-melted stoichiometric alloy is placed in a quartz crucible with an orifice at the bottom. After being induction melted, the alloy is ejected by argon

390

H. FUJH and H. SUN

gas through the orifice onto a rotating Cu wheel, which is coated by Cr in some cases. With a wheel surface velocity of around 30 m/s, a mixture of the modified 2:17 phase (possible the 1:7 phase) and an amorphous phase can be obtained. In order to achieve hard magnetic properties, a two-step heat treatment after meltspinning, not only nitrogenation, but also a short time annealing (less than 1 hour) in argon or vacuum at temperatures between 650°C and 750°C, has to be done. The microstructures developed in the materials have subtle dependence on the spinning condition, as well as on the annealing condition and the nitrogenation condition. As the hard magnetic properties (the coercivity) are closely related to the microstructures, the melt-spinning, annealing and nitrogenation conditions under which the best hard magnetic properties can be obtained are quite different by different authors. Table 6.3 summarizes the hard magnetic properties reported in the literature. The interstitial carbides included in table 6.3 were made by arc-melting and meltspinning, without the gas-phase interstitial modification process. Sm2FelnGa3C1.5 was melt-spun at 30 m/s and the as-spun material showed a high coercivity of 1.5 T (Shen et al. 1993a), while Sugimoto et al. (1992b) obtained #0iHc = 1.26 T by

melt-spinning

I SmzFe17compound melt-s

inning ¢

l ribb )ns

I

pulverization

E powders I annelling nitrogSnation

finelycrystallized nitride powders

bon~ing .

I bulk magnet I Fig. 6.3. Schematicdiagram of the melt-spinning process.

INTERSTITIALLYMODIFIED INTERMETALLICS

391

TABLE 6.3 Hard magnetic properties achieved in the nitrides and carbides of Sm-Fe compounds producedby melt-spinning. /~oiHcand Br are the intrinsic coercivity and remnant magnetization, respectively. (BH)max is the maximum energy product. Composition

/z0iHc (T)

Br (T)

(BH)max (kJ/m3)

Ref.

Sm10.6Fe89.4Ny

0.616 2.10 2.23 1.77 0.85 1.26 1.5

0.86 0.73 0.72

69 65 68 95

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

Sml2Fe88Ny SmlsFe85Nu Sm2.TFelTNu Sm2(Feo.9Gao.1)17C1.6 Sm2(Fe0.85Ga0ACo0.05)17CI.6 Sm2Fe14Ga3Ck5 References: [1] Katter et al. (1991) [2] Pinkerton and Fuerst (1992) [3] Pinkerton and Fuerst (1993)

62

[4] Christodoulou and Takeshita (1993c) [5] Sugimoto et al. (1992b) [6] Shen et al. (1993a)

performing a 30 rain vacuum annealing at 600°C to 800°C after melt-spinning of Smz(Fe0.9Gao.l)17C1.6 and Sm2(Fe0.85Ga0.1Co0.05)17C1.6. Similarly to MA nitrides, pressure-assisted zinc bonding raises the coercivity of the melt-spun nitrides. Kuhrt et al. (1992a) reported a coercivity increase of 75% from 1.17 T to 2.06 T in melt-spun and nitrided Sm2Fe17 by zinc bonding at a very high pressure, whereas the remanence decreased from 0.7 T to 0.4 T at the same time. Pinkerton and Fuerst (1992) reported that a pre-nitriding heat treatment on finely grounded particles (<25 #m) was effective for improving the magnetic properties, while heat treatment on coarser particles (<45 #m) under the same condition had little effect. Again, similarly to the MA nitrides, the hard magnetic properties are attributable to the favorable fine-grained microstructure generated by melt spinning and annealing, in addition to the excellent intrinsic magnetic properties of the 2:17 nitride. The microstructure of the as-spun Sm-Fe ribbons (35 m/s) was studied by TEM by Pinkerton and Fuerst (1993) who observed that the grain size was less than 40 nm in diameter.

6.2.4. HDDR of the 2:17 compounds HDDR starts with pre-melted Sm2Fel7-based alloys. The alloy absorbs hydrogen at temperatures lower than 200 °C and the 2:17 hydride forms. Upon further increasing the temperature in the H2 atmosphere up to about 500°C, the formed hydride decomposes into Sm hydride and a-Fe by the following reaction, SmzFe17H6 + (2 + z 6/2)H2 --+ 2SmH2+~+ 17c~-Fe. Heating the above formed SmH2 and a-Fe in vacuum or argon atmosphere at around 750 °C results almost simultaneously in hydrogen desorption and recombination of the 2:17 phase, where the Sm hydride dissociates into elemental Sm and H2 and where a subsequent reaction of the Sm and a-Fe reforms the SmzFe17 phase. The hard magnetic properties are then obtained by nitrogenation of the HDDR powder (fig. 6.4). The data of isotropic fixed powders are summarized in table 6.4. The HDDR process leads to microstructures of very fine grains with

392

H. FUJH and H. SUN

HDDR

I SmzFe17compound [

pulverlzation I owder I

"I°"

finelycrystallized2:17 phase

nitrog~ation Initride powders.,, I b°nling Jbulkmagnet I

Fig. 6.4. Schematic diagram of the HDDR process. TABLE 6.4 Hard magnetic properties achieved by nitrogenation of the Sm-Fe compounds produced by the HDDR method. P,oiHc and/3r are the intrinsic coercivity and remnant magnetization, respectively. (BH)max is the maximum energy product. Composition

#0iHc (T)

Sm2Fe]7 Sm2Fei7 Sm2(Feo.95Cr0.o5)17 Sm2(Fe0,983Gao.ol7) 17 Sm2Fel7 Sm3.2Fel7

0.82 1.6 2.0 2.5 0.87 1.6

References: [1] Christodoutou and Takeshita (1993d) [2] Takeshita (1993) [3] Zhou S.Z. et al. (1992)

Br (T)

(BH)max (kJ/m3)

Ref.

111

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

0.75 0.71 0.73

[4] Nakamura (1992) [5] Sugimoto et al. (1992c) [6] Okada (1992)

submicron size, that are held responsible for the hard magnetic properties (Takeshita 1993).

6.2.5. Other methods for manufacturing the 2:17-type magnets An explosion technique was used by Hu B.E et al. (1993) for sintering of the arc melted, nitrided and subsequently ball milled Sm2Fea7 powders. This technique mainly involved the application of a high pressure shock wave. The powders were c o m p a c t e d to a high-density magnet by the shock wave. The achieved hard

INTERSTITIALLYMODIFIEDINTERMETALLICS

393

magnetic properties by the above authors are /3r = 0.83 T, # 0 i H c = 0.57 T and (BH)max = 88 kJ/m 3 with density of 6.0-7.4 g/cm 3. The demagnetization curve of the explosion sintered magnet has a better rectangularity than that of the resin bonded magnet. Compared with NdFeB magnets, the explosion sintered Sm-Fe-N magnet was reported to have better temperature stability, the temperature coefficients c~ of remanence and ~ of coercivity being -0.076 %/K and -0.51%/K, respectively. Nano-structured two-phase magnetic system, being composed of small soft magnetic grains which are strongly exchange-coupled to a hard magnetic phase, has regained interest. The idea is to achieve an enhanced remanence due to the existence of the soft magnetic phase with high saturation magnetization and result in a remarkable high energy product. The melt-spun NdzFeI4B and Fe3B (Coehoorn et al. 1989, Kneller and Hawing, 1991), NdzFe14B and c~-Fe systems (Manaf et al. 1993a, b) were reported to show the above mentioned characteristics of the twophase magnet. Schrefl et al. (1993) simulated and calculated the dependence of the coercivity on the size of the soft magnetic grains embedded in a hard magnetic matrix for a two-dimentional system. The result showed that in order to induce magnetic hardness from the hard magnetic phase into the soft magnetic grains, the size of the magnetically soft regions has to be under 10 nm or smaller than twice the domain wall width of the hard magnetic phase. A remanence enhancement was reported by Ding et al. (1992c, 1993) for mechanically alloyed isotropic SmvFe93-nitride, with the hard magnetic phase being SmzFea7Ny and the soft magnetic phase being c~-Fe. The magnetic properties reported are Br -- 1.127 T,/z0iHc = 0.39 T and (BH)max = 207 kJ/m 3. TEM examination on the as-milled powders showed that nanocrystalline c~-Fe grains of about 5 nm size were embedded in an amorphous Sm-Fe matrix. If a higher coercivity could be generated, a much higher (BH)max is expected. 6.2.6. Magnets based on the l:12-type interstitial nitrides Compared with the 2:17-type nitrides, there are much less studies in the field of the l:12-type nitrided materials. Hard magnetic properties obtained by various methods are listed in table 6.5, the main interest being the Nd containing compounds. Induction melted and nitrided NdFe9_xCo3Ti~Ny compounds were studied by Suzuki et al. (1992a, b). The highest coercivity 0.59 T, was obtained by 50 wt.% Zn bonding for x = 1.2. Good environmental resistance (high temperature and humidity) was reported. Nitrides and carbides of Nd(Fe,M)ae alloys (M = Mo, V, Ti or combination of Ti and V) were investigated by Tang et al. (1993a, b) using melt spinning and annealing in order to achieve suitable microstructures for hard magnetic properties. Only the nitride NdFel0Mo2Nvshowed a relative high coercivity (0.88 T), while nitrides and carbides of other compositions had much lower coercivities although the intrinsic magnetic properties were similar. Preliminary results of hard magnetic properties including a coercivity of 0.3 T for PrFel0.sMOl.sN u were obtained by Yang Y.C. et al. (1992b) by using the hydrogenation disproportionation and desorption (HDD) method. In an investigation of mechanical alloyed 1:12 compounds, Gong and Hadjipanayis (1992 and 1993) reported an increase in coercivity by 0.2-0.3 T through replacement

394

H. FUJII and H. SUN

TABLE 6.5 Hard magnetic properties of the interstitial nitrides and carbides of the 1:12-type compounds processed by various techniques. /.t,0inc and Br are the intrinsic coercivity and remnant magnetization, respectively. (BH)max is the maximum energy product. Composition

Method

/zoiHc (T)

NdFe7.8Co3Til.2N1 +6

Zn bonding melt-spinning HDD MA MA MA+A1 bonding MA MA

0.59 0.88 0.3 0.75 0.85 0.95 0.4 0.64

NdFel0Mo2Nu PrFelo.sMOl.sNy Nd10Fe75VlsNv Nd10Fe75Mo15Nu Nd10Fe75MOlsNv Nd10Fe75MolsCy NdllFe74Mo15Nu

References: [1] Suzuki et al. (1992a, b) [2] Tang et al. (1993a) [3] Tang et al. (1993b) [4] Yang Y.C. et al. (1992b)

Br (T)

(BH)max

Ref. [1] [2, 3]

0.4

[4] [5, 6] [5, 6] [5, 6] [5, 6] [7, 8]

[5] Gong et al. (1992) [6] Gong et al. (1993) [7] Hirosawa et al. (1992) [8] Itsukaichi et al. (1993)

of Nd by 1.5 at.% Dy forming Nd8.sDy1.sFe82MosN u. The same authors the metal bonded MA materials and found that small amounts of A1 increased the coercivity by 0.1-0.2 T. The use of Zn was found to be due to the absence of the Fe-Zn phases, which were observed in the Zn-bonded magnets.

also studied (0-5 wt.%) not suitable Sm2FelTNy

6.2. 7. Coercivity mechan&m Kerr microscopy observations of the domain structure of SmzFel7Ny were reported by Mukai and Fujimoto (1992) and by Hu J.F. et al. (1993a). In close correspondence with the intrinsic magnetic properties, a sensitive nitrogen content dependence was also found for the intrinsic magnetic domain parameters, such as the domain wall energy 7, the average exchange constant A, the domain wall width 3B, and the critical diameter of the single domain particles De. Kou et al. (1993) calculated the magnetic domain parameters of Sm2Fe17N u from its Curie temperature and the magnetocrystalline electric-field anisotropy constants K1 and K2. Values of the magnetic domain parameters obtained by various authors and various methods are listed in table 6.6 for comparison, together with those of Nd2FeI4B. By calculating the temperature dependence of the average exchange constant A, Hu J.F. et al. (1993b) also studied the temperature dependence of the above mentioned domain parameters in the temperature range from 200 K to 580 K. The results showed that with increasing temperature, A, 7 and De decrease, while ~B increases slightly. The coercivity mechanism in hydrogen decrepitated, metal bonded, explosion sintered and mechanical alloyed Sm2Fe17Nx magnets were studied in a series publications by Hu J.E et al. and Kou et al. (Hu J.E et al. 1992, 1993b, 1993c, 1993d, Kou et al. 1993) on the basis of the micromagnetic theory developed by Kronmtiller (1978) and Friedberg and Paul (1975), all the conclusions supporting a nucleation field controlled coercivity.

INTERSTITIALLYMODIFIED INTERMETALLICS

395

TABLE 6.6 Intrinsic magnetic domain parameters of the Sm2Fe17 nitride. In the table, are included those of Nd2Fe14B. W is the observed domain width, 3' domain wall energy, A the average exchange constant, De the critical diameter of single domain particles and 6B is the domain wall width. Composition

V (/zm)

7 (10-3 J/m2) A (10-11j/m)

Dc (#m)

6B (nm)

Ref.

SmzFe17Ny 0.85-1.6(1.0) 62.5-27.9(39) 2.98-0.59(1.16) 0.3-0.6(0.36) 6.03-2.68(3.75) [1] SmEFel7N2.6 0.76 28 0.27 [2] Sm2Fe17Ny 33 1.19 0.32 4.3 [3] Nd2Fe14B 0.55 24 0.2 3.9 [4] References: [1] Hu J.E et al. (t993a) [2] Mukai and Fujimoto (1992)

[3] Kou et al. (1993) [4] Durst and Kronmtiller (1986)

TABLE 7.1 Comparison of the intrinsic magnetic properties at room temperature for some well-known permanent magnets. Compound

Tc (K)

Ms (T)

/~0Ha (T)

SmCo5 Sm2Co17 Nd2Fel4B Sm2FelTN3 NdFel 1TiN

1000 1193 588 749 729

1.14 1.25 1.60 1.54 1.45

28 6.0 8.0 26 12

7. Prospects The discovery that metastable intermetallic compounds can be produced by a process of gas-phase interstitial modification has significantly widened the range of rare earth intermetallics with intrinsic magnetic properties suitable for use as permanent magnets. In order to have a sufficiently high magnetic hardness as well as a reasonable temperature stability, high magnetization and high Curie temperature are essential for new materials suitable for permanent magnet applications. Table 7.1 lists the intrinsic magnetic properties at room temperature for some well known permanent magnet materials. As can be seen from the table, Sm2Fe17N v is the first new magnetic material since 1983 that has a chance of rivalling Nd2Fex4B in permanent magnet applications. It combines a Curie temperature that is high for iron-rich intermetallics, with a large uniaxial anisotropy field and a respectable spontaneous magnetization. The theoretical upper limit on the attainable energy product is 472 kJ/m 3. In fig. 7.1, the energy product of commercialized permanent magnets is plotted as a function of cost per unit mass. Permanent magnets with different levels of (BH)max have their own market according to different requirement in various kinds of applications. Substantial progress has already been made in developing hard magnets from the interstitial compounds. The latter materials are particularly interesting for the applications where anisotropic bonded magnets with (BH)max around 200 kJ/m 3 are desired, as indicated by the shaded X area in fig. 7.1. The (BH)max of Sm2Fe17N3 based nucleation-type magnet has reached 140-180 kJ/m 3 by improving

396

H. FUJII and H. SUN 400

30C ............................................................................ Nd-Fe-B (s)

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

200 ~E

"lrn

lOO ................................................................................

Sm-Co (B) ......................................

t Nd-Fe-B (B) 0.2

0.4

i 0.6

0.8

1.0

Cost / g ( arb.unit )

Fig. 7.1. The maximumenergy product as a function of cost per unit mass for various permanent magnets (provided by S. Suzuki). the nitrogenation processes (Iriyama et al. 1992, Fukuno et al. 1992, Suzuki et al. 1993), but the temperature coefficient is still large compared to the Sm-Co bonded magnets. Obviously the Sm-Fc-N magnets have to be further improved, especially the pinning-type anisotropic magnets with nano-structurc are of much interest. The appearance of a new candidate for permanent magnets provides us with new hopes and also presents new challenges as to the realization of a material that can be practically used. In solving the problems and making progresses, we believe that better understanding of the basic magnetism in the interstitial modified compounds and sophistication in the permanent magnet processing techniques are important issues.

Acknowledgements The authors would like to express their sincere thanks to Mr. M. Akayama, Mr. Y. Miyazaki and Mr. K. Tatami for their kind assistance in the preparation of this manuscript. Prof. S. Asano is thanked for providing us some of his original figures for being used in this manuscript. H. Fujii would like to take this chance to thank the NEDO, Japan, for the financial support in the past three years of an International Joint Research Programme on Development of Functions and Applications of Interstitially-Modified Rare Earth Intermetatlics from 1991 to 1993. We are grateful to our colleagues of the NEDO team, especially Prof. J.M.D. Coey, Prof. S. Asano and Dr. M. Sagawa, for the kind cooperations and many helpful discussion.

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397

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chapter 4 FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

A.K. Zvezdin General Physics Institute Russian Academy of Sciences Vavilov st. 38 117942 Moscow Russia

Handbook of Magnetic Materials, Vol. 9 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved 405

CONTENTS 1. Introduction

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

408

2. H - T phase diagram of isotropic ferrimagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411

3. Non-cotlinear structures in a weakly anisotropic ferrimagnets. Basic equations . . . . . . . . . .

419

3.1.

Therlnodynamical potential of the non-equilibrium state . . . . . . . . . . . . . . . . . . . . . . . .

419

3.2.

Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

421

3.3.

Extreme conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

422

3.4.

Low magnetic field approximation

423

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

4. H - T phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.

Uniaxial magnetic anisotropy

4.2.

Cubic anisotropy

424

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

424

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

430

5. Field induced phase transitions in rare-earth-ferrite garnets . . . . . . . . . . . . . . . . . . . . . . . . . .

439

6. Single crystal ferrite garnet films

450

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

7. Some general features of field induced phase transitions

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

454

8. Magnetization and susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

456

8.1.

Differential susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2.

The temperature hysteresis of the magnetization. Hall and Faraday effects

9. Thermal properties in the vicinity of the spin-reorientation phase transitions 9.1.

Magnetocaloric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.2.

Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10. Magnetoelastic anomalies

456 ........

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

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

462 465 465 466 469

10.1. Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

469

10.2. Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

472

10.3. Young's modulus, sound velocity change (AE-effect) and sound absorption . . . . . . . .

473

11. Non-collinear phases and domain structure

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

11.1. 'Break' of symmetry in the canted phase and formation of domain structures with twins, triplets and quadruplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Nucleation of new phases from domain walls. The hysteresisless first order phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Canted phase domains in ferrite garnet single crystals and films . . . . . . . . . . . . . . . . .

406

475 475 477 478

FIELD I N D U C E D P H A S E TRANSITIONS IN FERRIMAGNETS 12. Hexagonal ferrimagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Free energy and equilibrium conditions

482

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

12.2. Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Non-collinear magnetic structures in the intermetallic compounds DyCo 5 13.1. Crystal and magnetic structure

407

482 485

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

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

13.2. Domains in the basal plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

491 491 491

13.3. Magnetization and magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

493

13.4. Magnetizations of the sublattices, exchange field and anisotropy constant . . . . . . . . . .

495

13.5. H - T phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

497

14. Spin-flop and spin-reorientation phase transitions in the anisotropic ferrimagnet HoCo3Ni2 with Tcomp = TSR1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1. Spin reorientation in HoCo3Ni2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. The H - T phase diagrams

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

499 499 500

14.3. The magnetization measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

502

14.4. Magnetostriction and spin-flop transitions

503

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

15. Surface anisotropy effects and surface phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

506

16. Phase transition at the local defect. Dislocations and FIPT's in Gd3Fe50~2 . . . . . . . . . . . . .

508

17. Free-powder samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

512

18. Spin-flop transitions in itinerant metamagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

521 532 532

19. Appendix. Microscopic calculation of the thermodynamic potential of the non-equilibrium state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

532 537

1. Introduction

This review is concerned with high magnetic field magnetization processes in ferrimagnetic compounds of the f-d type, i.e. in compounds consisting of rare-earth and transition metal elements of the Fe group. Attention will be mainly focused on such peculiarities of these processes which are connected with the variation of mutual orientation of their magnetic sublattices and which reveal themselves through characteristic kinks and jumps in magnetization curves. According to the present views, magnetic symmetry of a crystal changes when the orientation of magnetic sublattices spontaneously varies. In other words, a phase transition accompanied by a change of the magnetic symmetry takes place. In the present terminology such phase transitions are referred to as field induced phase transitions (FIPT) which may be of 1st or 2nd order. Yet this notion comprises a more general class of magnetic phase transitions (Date 1990, Franse 1990). It is evident that such phase transitions are accompanied by an anomalous behavior of several physical properties and characteristics of ferrimagnets (susceptibility, thermal properties, magnetostriction, magnetoelastic anomalies, Halleffect and magnetoresistance, magnetooptics etc.). The role of the magnetic anisotropy in field induced phase transitions and in corresponding phase diagrams has to be discussed in this review. It is natural that the range of properties related to a compensation point of ferrimagnets (in the vicinity of which the effects of magnetic anisotropy are most conspicuous) is worth considering here in detail as well. Field induced phase transitions are traditional subjects for the physics of magnetic phenomena. The first examples attracting considerable attention were spin-flop transitions. The concept of these transitions was proposed by Nrel (1936) and they were found later in CuClz-H20 by the Dutch group (Poulis et al. 1951). In these transitions the collinear phase turns into a canted phase and further into a ferromagnetic one in an increasing field. A distinctive feature of a spin-flop transition in an ideal Nrel antiferromagnet is that the differential susceptibility of the material in the angular phase is not dependent on the magnetic field and temperature. Later similar phase transitions were investigated in ferrimagnets whose behavior tumed out more sophisticated and interesting to be studied. The ferrimagnetic structure may happen to be non-collinear in a certain range of magnetic fields and temperature. Such structures are the result of a competition among negative exchange interactions within the sublattices, that force their magnetic moments to be antiparallel to each other. They may also arise as a consequence of the interaction of sublattices with the applied field. 408

FIELD INDUCED PHASETRANSITIONSIN FERRIMAGNETS

409

That non-collinear magnetic structures may exist was first demonstrated theoretically by Tyablikov (1956, 1958, 1965). He also defined the conditions of their existence in the case of isotropic ferrimagnets and calculated anomalies of the magnetization, arising in the vicinity of such transitions. Further theoretical analysis of non-collinear structures in two- or three-sublattice ferrimagnets was carried out by Gusev (1959), Schltimann (1960), Gusev and Pakhomov (1963). An important contribution to this problem was made by Clark and Callen (1968). These authors gave a summary of earlier results and applied them to the study of rare-earth-ferrite garnet compounds. Due to a negligible exchange interaction between the rare earth and Fe sublattices, the induced non-collinear structures in garnets were shown to occur at experimentally attainable values of the field. Perhaps the first experimental manifestations of these phenomena were observed in rare-earth-iron garnets (Rode and Vedyaev 1963, Kharchenko et al. 1968). We shall consider the basic theoretical and experimental results in section 2. At a compensation point T c a ferrimagnet behaves as an antiferromagnet. For the ideal N6el ferrimagnet the critical fields of transition in a canted phase are approximately equal to zero. To comprehend the physical properties of a ferrimagnet and to obtain its phase diagrams, it is of prime importance to take the real magnetic anisotropy of the material into account. Consequently, various interesting phase diagrams result for ferrimagnets close to the compensation temperature. They include the lines of 1st and 2nd order phase transitions, critical points of the liquid-vapor type, and tricritical points, where the 1st order phase transition line goes over to the 2nd one on the line separating the phases, etc. (sections 3-7). Thus a big number of the anomalies detected experimentally in the vicinity of the compensation temperature can be explained. These points are considered in sections 8-10. So far the field induced phase transitions have been investigated more closely in rare-earth-ferrite garnets where the critical fields of transition into the canted phase are not too large. Gadolinium-iron garnet was studied in great detail owing to the fact that the application of Ndel's model is well justified since the Gd +3 ion, whose ground state is an orbital singlet 8S, has a comparatively small magnetic anisotropy. The rare-earth sublattices in other rare-earth-ferrite garnets possess a substantial magnetic anisotropy. Phase diagrams and the character of phase transitions differ essentially from those of Ndel ferrimagnets. A forthcoming paper will be devoted to these aspects. There are several studies in which domain structures have been investigated in the context of field induced phase transitions (Zvezdin and Matveev 1972a, b, Dikstein et al. 1980). Later such domains were experimentally observed in ferritegarnets in strong magnetic fields (Kharchenko et al. 1974, Lisovskii and Schapovalov 1974), and in epitaxial ferrite-garnet films (Lisovskii et al. 1976 a, c, Gnatchenko and Kharchenko 1977, Dikstein et al. 1980) and in the YIG in megagauss fields (Druzhinin et al. 1981) (see section 11). The discovery of domains in strong magnetic fields essentially enriches the knowledge of domain forming in magnets.

410

A.K. ZVEZDIN

For the last few years considerable attention was devoted to intermetallic compounds RE-TM, where RE -- rare earth, and TM = transition metal (such as, for instance, RCos, R2Fe17, R2Col7, R2Fe14B, and REe2 etc.; Buschow 1980, 1988, Radwanski et al. 1989a, de Boer et al. 1989). Of various interesting phenomena, inherent in these materials, field induced phase transitions from a collinear into the canted phase are of prime importance because the critical fields of these transitions allow us to estimate directly the value of an exchange interaction between rare-earth and transition metal sublattices (Franse et al. 1990b). Until now polycrystal and powder samples have been the main objects for measurements. With progress in both single crystal growth and strong magnetic field production, crystal field effects and the role of magnetic anisotropy will be obviously of particular interest. Berezin et al. (1980) experimentally explored monocrystals of the type DyCos+a. Some general features of phase diagrams of these intermetallic ferrimagnets with hexagonal symmetry and of rare-earth-ferrite garnets were found. This is considered in sections 12 and 13. Section 14 is devoted to the field-induced phase transitions for which a unique situation occurs when the compensation To and the spontaneous spin-reorientation Ts~ temperatures practically coincide (Zawadzki et al. 1993, 1994). Investigation of this interesting subject gives an opportunity to understand clearly the interrelation between the magnetic anisotropy and the f-d exchange. Theoretical activity in the latter was devoted to the elucidation of the role of the surface and the role of point-like, linear, and other defects of a crystal lattice in processes that determine phase transitions. The study of a spin-flop transition in dislocation-containing single crystals may be quite important from this viewpoint. Similar study was made by Vlasko-Vlasov et al. (1983), Vlasko-Vlasov and Indenborn (1984). These problems are considered in sections 15 and 16. Sections 17 and 18 are devoted to field-induced phase transitions in free-powder samples and itinerant metamagnets. Microscopic calculations of the thermodynamic potential are given in the Appendix. In the final section we emphasize the applicability of the material presented in this survey. Namely, we operate herein with weak anisotropy ferrimagnets for which the Ndel model "works" sufficiently well. Innovative trends concerning field induced phase transitions investigated for strong anisotropy ferrimagnets and itinerant ferrimagnets with an unstable d-sublattice are briefly discussed. Two-sublattice, more precisely, two-subnet amorphous systems with an antiferromagnetic interaction between the subnets and ferrimagnet superlattices with antiferromagnetic interaction between layers have been also investigated. They are still under investigation. Though each of these systems possesses its own characteristics, a conceptual connection with the FIPT is definitely present in all of these systems. Referring to the question about the term 'field induced phase transition' let us note that a broader class of phase transitions is described by this name: • Ferrimagnetic - canted - ferromagnetic transitions which are under consideration here. Occasionally these are referred to as spin-flop transitions by analogy with antiferromagnets;

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

411

• Metamagnetic transitions in itinerant ferrimagnets of the YCo2 or RCo2 type (Wohlfarth and Rhodes 1962, see also experimental work by Goto et al. 1989); • Crossover level transitions which may be due to a field-induced crossing of ground state levels of f- or d-ions. It should be stressed that this crossing is accompanied by the transformation of the magnetic structure of a crystal, i.e. a magnetic analog of the Jahn-Teller effect takes place (Zvezdin et al. 1976); • First order magnetic processes (FOMP) which can be due to a competition among crystal-field contributions of different order to the free energy (see, e.g., Asti and Bolzoni 1980, Asti 1990). We would like to emphasize that there are no distinct boundaries between these field-induced phase transitions. Moreover, it is often possible to find some features, inherent in one of the type of transitions, in the others as well. For example, the mechanism of spin-flop transitions considered in this survey is closely connected with crossover type transitions. Such interrelation of different mechanisms of fieldinduced phase transitions is of great significance for a strongly anisotropic ferrimagnets. So in (HoY)3FesO15 iron garnets the transformation of ferrimagnetic structure into ferromagnetic structure does not proceed smoothly via the canted phase (as it does, e.g., in Gd3Fe5O12) but follows the pattern of jumps of the magnetization, originating from the crossing of the levels of Ho +3 ions (Demidov and Levitin 1977, Zvezdin et al. 1977). A similar example is (TbY)3Fe5012 (Lagutin et al. 1990a, b). 2. H - T

phase diagram of isotropie ferrimagnets

Let us begin by considering the basic regularities of non-collinear magnetic structures in a ferrimagnet. For this purpose we can use the mean field approximation. The equations, describing the state of the two sublattices of isotropic ferrimagnet in this approximation can be represented in the form

]~1 = /-11 M~,

(2.1)

M2 = __/~2 M2, //2

(2.2)

Ht

where ~q and ~r2 are the sublattice magnetizations and Mx and M2 are their absolute values. The relations /t~ = / ~ - A-/~2,

(2.3)

&

(2.4)

= / t - .X2v~rI

412

A.K. ZVEZDIN

describe the effective fields acting on the 1 and 2 sublattices and A is the exchange interaction parameter. Equations (2.1) and (2.2) denote that the sublattices magnetizations are oriented along effective field directions under thermodynamic equilibrium. Generally speaking in eqs (2.1)_. and (2.2) one needs to take into account the intra sublattice molecular fields AllM1 a n d )~22]~2 . However, it is clear that taking them into account leads only to renormalization of the parameters Xi. Furthermore, in the case most important for us, i.e. the case of 3d-4f compounds when All >> A >> A22, the validity of the approximation under consideration here is practically valid. In the free ion approximation the functions MI(T, H1) and Mz(T,/-/2) are reduced to the well-known Brillouin functions. Let us consider the change of the vectors with the increasing field H at a transition from theferrimagnetic ordered state (M1 is antiparallel to M2) to the ferromagnetic state (M1 is parallel to 2~r2). First of all we define the canted (angular) solutions. The equations for the magnetization components perpendicular to magnetic field are of the form: (M1)± + / ~ x I ( M 2 ) . L = 0, ) ~ x 2 ( M 1 ) ± q- ( M 2 ) ± = O,

where

M~

X i - Hi'

i = 1,2.

These equations have the trivial solution ( M 0 . = 0 corresponding to the collinear phase non-trivial solution, (Mi)± ¢ 0, is realized at the condition

(2.5)

1 -- )~2/~lX 2 = O.

The equations for the components parallel to t h e / t are given by

(2.6)

(M1)II + AxI(M2)II = x 1 H , ~x2(M1)II + (M2)II =

(2.7)

X2H.

Relations following from these equations are (1 -

AZx1x2)(M1)II =

XI(1 -

AXz)H,

(1 --

A2X1X2)(M2)It =

X2(1 - / ~ x 1 ) H .

(2.8) (2.9)

Hence it follows that 1 X1 = -

1

(2.10)

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

413

H

M2

Fig. 2.1. The orientations of the sublattice magnetizations kT/1, K/2 relative to the external magnetic field/t.

in a canted phase, i.e. H1 = AM1 and H2 = AM2. Let 01 and 02 be the angles defining the deviations o f ml and 2hr2 from/~. It is obvious t h a t / 1 and the vectors/~ri are in the same plane (see fig. 2.1). These angles can be derived from eqs (2.3) and (2.4). Here H1 = ( H 2 q- ,~2M22 - 2HAM2 cos 02)

1/2

,

(2.11)

H2 = (H 2 + )~2M12 - 2HAM1 cos 01) 1/2,

(2.12)

and MI(T) and M2(T) are known functions. By solving these equations we find (-A2(M22 - M12) + H 2) cos 01 =

2HAM1

,

(2.13)

( - A 2 ( M 1 2 - M2 2) + H 2)

cos 02 :

2H.XM2

(2.14)

The overall magnetization in a canted (angular) phase can be defined from eqs (2.1) and (2.2) directly; summarizing them and with regard to X1 = )~2 = 1//~ we obtain

+&

: !17. A

414

A.K. ZVEZDIN

0~,02

HI

//2 /-/

Fig. 2.2. Field dependence of the sublattice rotation angles for isotropic ferrimagnet.

Equations (2.13), (2.14) describe a continuous rotation of the magnetization from the ferrimagnetic to the ferromagnetic phase (fig. 2.2). The critical fields restricting the canted phase can be found from the eqs (2.13), (2.14) under assumption 0 = 7r and 0 = 0. Hcl = )~IM1 - M2 l,

(2.15)

He2 = A(MI + M2).

(2.16)

Thus three phases exist (see fig. 2.3): 1. Ferrimagnetic phase: [0a - 02[ = zr, 2. Ferromagnetic phase: 01 = 02 = 0, 3. Canted phase: 0i are defined by equations (2.13), (2.14). The boundaries between the phases determined by the expressions (2.15), (2.16) are the 2nd order phase transition lines. Let us discuss these phase transitions from a different point of view. We consider RE-TM ferrimagnets (for example rare-earth-ferrite gamets R 3 F e 5 0 1 2 , o r RnTm intermetallics, where T -- transition metal) in which the magnetization of the d-sublattice Md does not depend on the magnetic field, and the magnetization of the f-sublattice Mf is determined by the action of the effective field: qeff =

-- a

rd.

The absolute value of Heff decreases to 0 when H goes to H = AM (see the line OP at fig. 2.3) and increases when H > AMd. The value H = AMd is the crossover point. Here we have the field-induced crossing of levels of the f-ions, i.e. the ground state of the f-ions is degenerate at the line OP. It leads to the instability of the collinear magnetic structure when the temperature decreases, i.e. to the phase transition into the angled (canted) phase at T < T* (fig. 2.4). It is clear that the free energy of the system decreases due to magnetic splitting of the energy levels and increases due to deflection of the ~rd from magnetic field /~.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS H

415

H

MR'~.,,,4Mr ~ 0

0

\

\

\ \ o rk

"*

T

T

a)

b)

Fig. 2.3. Phase H - T diagrams of isotropic ferrimagnets: a) with a compensation point and b) without a compensation point. On the line O P the magnetization of one of the sublattices ('paramagnet' sublattice) is equal to zero.

E

i\,\ x I

!/

./"

"

"\i

I

H

H

lit

XMd

a)

He,

He2

b)

Fig. 2.4. Dependence of the ground state levels on the magnetic field (schematically): a) T > T*, b) T < T * . This field induced magnetic transition that is related to a crossing of energy levels can be considered as a "magnetic Jahn-Teller" phase transition (Zvezdin et al. 1976, 1977), because there is a direct and profound analogy of this effect with the wellknown Jahn-Teller effect and related phase transitions. We emphasize that crossing of the energy levels is accompanied by a transformation of the magnetic structure (in the case that a canting of the spins is initiated) in full accordance with concept of the cooperative Jahn-Teller effect. At T < T* the collinear structure is unstable in the whole region adjacent to the H = AMd line in the H - T diagram. This means that the instability occurs already before the degeneracy of the ground state is lifted.

416

A.K. ZVEZDIN

The theoretical H - T diagrams of isotropic ferrimagnets shown in fig. 2.3 are in good agreement with experimental diagrams for the polycrystal samples of rareearth-ferrite garnets. These compounds can be discussed in terms of two magnetic sublattices: 'Fe' and 'rare earth'. A more detailed description of the structure and magnetic properties of the rare-earth-ferrite garnets is given in section 5. The magnetic moments of the Fe sublattice are coupled together by a strong exchange interaction (Hexch ~ 106 Oe). The RE-Fe interaction is weaker (Hexch ~ 105 Oe), and the much more weak f - f interaction plays an essential role only at the lowest temperatures. Proceeding from this, it can be assumed that magnetization of the Fe sublattice is independent of the external magnetic field (at H << 106 Oe) and the state of the rare-earth sublattice. The intersublattice exchange interaction between the rare-earth moments may also be neglected. In this approximation the rare-earth sublattice represents an 'ideal paramagnet', placed in external field and exchange field. Calculations show that at low temperatures the critical fields of transitions into noncollinear phases are of the order of 105-106 Oe. The possibility to induce the noncollinear magnetic structures by the field is comparatively high, close to compensation point. Figure 2.5 illustrates experimental magnetic phase diagrams of some rare-earth-ferrite garnets obtained from the measurements of various physical properties (Levitin and Popov 1975). It also shows theoretical dependencies for isothermal and adiabatic regimes of the measurements. Satisfactory accordance of experimental and theoretical dependencies of the critical fields is observed. A lot of experimental work was done to study field induced phase transitions in the rare-earth-transition-metal compounds (for instance, on RFe2, RT3, R2T7, RCos, R2Co7B3, R6Fe23, R2T17, R2Fe14B, R2FelaC, R2Fel7C, RT12, RCol2B6, R2(FeMn)laC). Most of the experiments were performed on fine (single crystal) powder particles, being free to orient themselves in the applied magnetic field. Since they are free to rotate inside the sample holder, the particles orient their total magnetic moment parallel to the external field. This approach enables in all cases to circumvent the difficulties connected with magnetic anisotropy and to use the simple isotropic model for experimental data treatment (for details see the papers by Radwanski et al. 1989, Franse et al. 1990, de Boer et al. 1990, Liu et al. 1994). It is of interest to study the complete magnetization curve up to full saturation. The required magnetic fields, however, are typically too large to make such a study feasible. Matsuuva et al. (1979) have measured the magnetization of a crystal of Mn(CH3COO)2.4H20 in an external field up to 400 kOe at 1.1 K. This system is ferrimagnetic below He1, ferromagnetic above Hc2 and a spin-canted state appears at He1 < H < Hc2, where He1 = 125 kOe and Hc2 = 288 kOe for/~ parallel to c-axis. This is a clear example of a full magnetization process in a Heisenberg ferrimagnet. Another clear example of a full magnetization process was reported by Gurtovoy et al. (1980) in (GdY)3FesO12 iron garnet (for details see section 8). Full magnetization curves were also measured by Demidov and Levitin (1977) in (HoY)3Fe5012 garnets in fields up to 300 kOe and by Lagutin and Dmitriev (1990), Lagutin and Druzhinina (1990) in (TbY)3Fe5012 garnets in magnetic fields up to 2 MOe. In contrast with a continuous magnetization process, they observed a series

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

150

-o~

,

150

100

I

i

280 ",,,

50

I

0

300 T K

o~ '~,,

I

I

220 200

0

100

Tb3FesO12

I

260

150

dCl

50 Gd3FesO12

200

o,

100

50

417

I

240

"'°",,,,,,o I

I

260 T, K

I

150

~

m',

100 V

~,~

50

,,

i

'.

Ho3FesO12 0

0 t

t

i

i

190 210 230 T, K 100 120 140 T, K Fig. 2.5. Magnetic phase diagrams of polycrystalline samples of rare-earth-ferrite garnets: light circles correspond to longitudinal magnetostriction measurements, black circles - to transversal magnetostriction measurements, light squares - to magnetization measurements, dark squares - to Faraday effect measurements. Theoretical dependencies for regimes of isothermal (solid lines) and adiabatic (dotted lines) measurements are presented; after Levitin and Popov (1975). of j u m p s in the magnetization curves during the transition from the ferrimagnet to the ferromagnet state. However, (HoY)3Fe50]2 and (TbY)3Fe50]2 are strong anisotropic ferrimagnetics and demand a special consideration. In the studies of Guillot and Le Gall (1976), Miura et al. (1977), Pavlovskii et al. (1979, 1992), Druzhinin et al. (1981, 1992) the Faraday effect was used. The critical fields of the spin-flop transitions from the collinear ferrimagnetic phase to the canted phase in yttrium-iron, y t t r i u m - i r o n - g a l l i u m (Y3Fe5012 and Y3(FeGa)5012)

418

A.K. ZVEZDIN

IO 8

6 \

~4

\ \

\ \

2

\ \

0

I

I

0.2

0.4

I

I

I \

0.6 0.8 1.0 r = TK/600 Fig. 2.6. H - T phase diagram of Y3Fe5012 iron garnet. The diagram has been calculated by the use of the exchange parameters fitted to Faraday rotation measurements in explosion generated magnetic fields up to 10 MG; after Druzhinin et al. (1981). and in other garnets are in the megagauss field range. In Y3FesO12, according to Druzhinin et al. (1981), the critical field at 300 K is 2.6 MOe (fig. 2.6). These authors have also observed a strong scattering of light in this field region. They connect the origin of this effect with the appearance of a domain structure in the canted phase (see section 11). Great attention was paid to intermetallic compounds of rare earth and d-metals in which the d-subsystems represent itinerant metamagnets and in which the f-subsystem could be described in the framework of the localized electron model. Typical ferrimagnets of this family are the materials RCo2, (YR)Co2 and (YR)(CoA1)2 (for details see, e.g., Levitin and Marcosyan 1988), where magnetic field induced transitions in the electron d-subsystem from the paramagnetic to ferromagnetic state were observed. Such metamagnetic phase transitions were predicted by Wohlfarth and Rhodes (1962). To explore magnetization processes and magnetic field induced phase transitions in such ferrimagnets, it is necessary to abandon the 'rigid' d-sublattice approximation used above and to include into the theory the possibility not only of changes of the sublattice magnetization direction but also variations of its magnitude. The phase transitions and phase diagrams for such system were investigated theoretically in the framework of the isotropic approximation (Zvezdin 1993, Zvezdin and Evangelista 1995). The new feature of the H - T diagrams is the occurrence of tricritical points at which three phases can coexist: paramagnetic, ferrimagnetic and canted. In contrast with the well-studied tricritical points in DyA1G and FeF2, three real phases can coexist in these systems near tricritical points. Thus there is an analogy between these tricritical points and those found in 3He-4He mixtures.

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

419

Of interest is also the possibility of a jump-like transition from the paramagnetic into the canted phase in the isotropic ferrimagnet system. Another uncommon feature of these diagrams is the occurrence of a reentrant-like phase transition (Zvezdin and Utochkin 1992).

3. Non-collinear structures in a weakly anisotropic ferrimagnets. Basic equations

3.1. Thermodynamical potential of the non-equilibrium state The model outlined in the previous section reflects the principal features of noncollinear magnetic structures in ferrimagnets and the character of their phase diagrams. The main weakness of this model is the neglect of magnetic anisotropy. The anisotropy effects are of principal importance in the vicinity of the compensation temperature Tc where the critical fields for the isotropic model become equal to zero (fig. 2.3). Furthermore we shall see below that the presence of anisotropy has a strong influence on the character of sublattices rotation in the non-collinear phase and on the type of phase transitions. Anisotropy effects on the phase diagram of ferrimagnets can be represented qualitatively (for uniaxial anisotropy) in the following way. It is evident that at Hllg (g is the easy axis direction) anisotropy hinders the rotational process. Therefore, close to Tc the rotation of the sublattices (meaning that the canted phase appears) may be expected to start at the threshold field Hc -= v/H-AHexch, where HA is an anisotropy field, Hexch - the intersublattice exchange field. As a result, the first order phase transition takes place in the vicinity of Tc and is accompanied by different other types of anomalies here. At H_l_r~ as well as in antiferromagnets the canted phase exists starting from the zero field values. At T = Tc the ferrimagnet behaves as antiferromagnet, i.e. the field increasing results in a monotone spin-flop trend of sublattices. Ferrimagnet remains in the canted phase when H changes its value from zero to a value in the ferromagnet phase. Slightly below and above Tc the transition from the ferrimagnetic to the ferromagnetic phase could proceed in two steps when the magnetic field increases. In low magnetic fields the spontaneous magnetic moment of the ferrimagnet compound declines from the easy axis and can attain an intermediate position between H and 4. In this case the magnetic moments of the sublattices become non-collinear, though the angle between their directions has to be close to 7r. Such a low-field canted phase exists up to the value of the external field that makes the overall magnetic moment to be parallel to /~. The ferrimagnet remains in the collinear phase upon further increase of the field up to the values at which the ferrimagnetic configuration becomes energetically unstable due to the competition between the external and exchange fields, and the system passes into the canted phase where further field increase leads to a spin flop.

420

A.K. ZVEZDIN

Investigation of the H - T phase diagram for anisotropic ferrimagnets is a cumbersome and complicated problem requiring numerical computations in a number of cases. In order to clarify qualitative peculiarities of the phenomena associated with the anisotropy and noncollinear structures in rare-earth ferrimagnets we first discuss a simplified model. When describing this model and its consequences we shall follow the results already reported by Goranskii and Zvezdin (1969b), Zvezdin and Matveev (1972a, b). When studying anisotropic ferrimagnets it is more convenient to employ the thermodynamic approach instead of using the molecular field eqs (2.1) and (2.2) directly. We take as basic model an f-d ferrimagnet, considered in the previous paragraph, where the magnetization of one of the sublattices (let it be the d-sublattice) is supposed to be saturated by the intrasublattice exchange field and where the second (rare-earth) sublattice is an ideal paramagnet placed in the external field and the exchange field created by the d-sublattice. The thermodynamic potential for the system under consideration can be written in the form (Goranskii and Zvezdin 1969b, Zvezdin and Matveev 1972a, b) =

-MdH

,

-

f0 Hen Me(x) dz + K,

(3.1)

where the first term is the Zeeman energy associated with the interaction of the dsublattice with magnetization Md and the external field/7. The second term is the thermodynamic potential of the paramagnetic rare-earth ions in the effective field Heff which is equal to /7eff = / 7 - AMd.

(3.2)

A particular dependence of Mf on Heff does not play an essential role here. In some cases we shall approximate it by a corresponding Brillouin function. The last term in (3.1) is the anisotropy energy. It should be stressed that magnetic anisotropy in the considered model is taken to be low in comparison with the exchange energies

IKI << ~M~. Only when this condition holds can the energy of magnetic anisotropy of rare earth ions be represented as an additional term in the thermodynamic potential. In a general case the anisotropy energy depends on the magnetization direction in both sublattices. We restrict ourselves to the approximation where this energy K depends only on the magnetization direction of one sublattice, K = K()~rd). In connection with this assumption we note the following. In the paper of Zvezdin and Matveev (1972) it was shown that the general form of the anisotropy energy, if it depends on the magnetization directions in both sublattices due to the paramagnetic character of the f-sublattice and the smallness of the anisotropy field compared to the exchange field, can be reduced to the above mentioned form on condition that H << AMd.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

421

But this weak field region is of importance mainly when considering anisotropy effects (K << AM~). On the contrary, in strong fields (H ~ AMd) the anisotropy effects in the weakly anisotropic ferrimagnet are small. Although each of the thermodynamic potential terms in (3.1) has an obvious physical origin in the framework of the discussed model, they can be also justified microscopically (see Appendix section and the paper by Zvezdin and Popkov (1980)). Formula (3.1) rep~sents the thermodynamic potential of the non-equilibrium state, depending only on Md. It is assumed here that the paramagnet sublattice is in equilibrium with the effective field. In such form the thermodynamic potential can be employed for the analysis of equilibrium properties and low-frequency non-equilibrium processes (whose characteristic frequencies are much smaller than the frequencies of precession of paramagnetic ions in the effective field). For studying high frequency phenomena the thermodynamic potential has to be taken in a more general form which is dependent on the both sublattices magnetization (see, for example, Zvezdin and Popkov 1974, Popkov 1976b).

3.2. Magnetic anisotropy The explicit form of the function K(2~ra) is defined by symmetry. Let i,j,k be unit vectors of the Cartesian reference frame such that klIH, and let 0, ~ be the polar and azimuthal angles determined in the usual way.

3.2.1. Uniaxial anisotropy F o r / t l l g (g is an easy axis direction) and/~[1~ we have K(O, ~) = - K c o s 2 0,

K > 0.

At H_l_g in coordinate system

K(O, ~) = - K

COS2 ~

sin 2 0,

(3.3)

(g, [h, g], h), where h = H/H), K(O, (y) is K > 0.

(3.4)

So the general formula for uniaxial magnetic anisotropy may be written as

K(O, ~) = -KF(O, ~), where the explicit form of the function f(O, ~) follows from the eqs (3.3) and (3.4). It is apparent that there is no pure uniaxial magnetic anisotropy in crystals. It needs in any case to take into account the dependence of the anisotropy energy on the azimuthal angle. For a crystal with tetragonal and hexagonal structure the following expressions for K(O, ~) are used:

K(O, ~) = K1 sin 2 0 + K2 sin 4 0 + K3 sin 6 0 + / ( 4 sin 4 0 sin &p, for the tetragonal structure, and

K(O, ~) + K1 sin 2 0 + K2 sin 4 0 +/t" 3 sin 6 0 +/~24 sin 6 0 cos 6%

422

A.K. ZVEZDIN

for the hexagonal structure, where 0 is the polar angle between the tetragonal or hexagonal axis and the magnetization of sublattice; ~ is the azimuthal angle between the component of the magnetization in the basal plane and the [100] axis. The magnetic anisotropy of rhombohedral crystals can be represented in the form K = -KI

C0S 2 0 --

K2 cos 4 0 + / ( 3 cos 0 sin 2 0 cos 3g),

where 0, ~o are the polar and azimuthal angles, if the z-axis is parallel with the c-axis of the crystal. The important case of hexagonal anisotropy is considered in detail in section 12.

3.2.2. Cubic anisotropy The anisotropy energy in this case is defined by the expression 2

2

2

2

= Kl(Cq a 2 + % %

2

2

K20el ct2o~3.

We will restrict ourselves to only the first invariant of this formula, i.e. we put /(2 = 0 so that

(3.5)

K(O, ~) = -Klf(O, ~). Let us define the function f(O, ~) as:

f(O, qo) = - ( 1 / 4 ) ( s i n 2 20 + sin 4 0 sin 2 2qo),

(3.6)

if/7111001] in the coordinate system (i,~, k) = ([100], [010], [001]);

f(O, ~o) =

3

cos 0 sin 3 0 sin 3qo - (1/3) cos 4 0 - (1/4) sin 4 0,

(3.7)

if/711 [111] in a coordinate system (i, 7, k~) = ([1 i0], [115], [111]);

f(O, ~) = (1/4)(sin 2 0 cos 2 g) - cos 2 0) + sin 2 0 sin 2 qo(1 - sin 2 0 sin 2 ~),

(3.8)

I7111110] in a coordinate frame (i, j, k~) = ([110], [001], [110]). For K < 0 the directions of the easy axes are along directions of the type [111] and for K > 0 along [100]. In rare-earth-ferrite garnets the constant K is commonly negative. if

3.3. Extreme conditions The extreme conditions of the thermodynamic potential relative to 0 and ~p have the form d~ -

-

=

sinOMdH(1

-

dO

O0

d~

(3.9)

OK -

d~

O# A~(O,~a) + - - = O,

- -

8~

-

O,

(3.10)

FIELD INDUCED PHASE TRANSITIONSIN FERRIMAGNETS

423

where ~ is Mf(Heff) ~,(0, ~)

-

(3.11)

Heff The analysis of these equations and the determination of the conditions for their stable solutions in particular cases will be discussed in sections 4, 12, 14, and 18.

3.4. Low magnetic field approximation The most important area of anisotropy effects pertains to H << AMd. The thermodynamic potential (3.1) and eqs (3.9), (3.10) can be represented in a more simple form. Then up to second order terms in H/~Md we get - K l f ( O , qg) + Q cos 2 0,

q5 = - ~ I H M d

(3.12)

where 77 =

Q = -A~'MH 2,

AXo,

1 -

~,=(

d

Xo = ~ ( H = 0),

_

<0.

For T --+ Tc 1

X0-+ ~ ,

1/A X t --+

Xo

-

),Ma

so that in the vicinity of Tc we have, M°_-__ u o ,

H2

1

(3.13)

(3.14)

where Mf° is the magnetization of rare-earth sublattice at H = 0 and Xo is its

susceptibility. In such approximations the problem of the determination of the ferrimagnet equation of state is reduced to the well-known equivalent problem of a uniform fen'omagnet reversal of magnetization Mt = 7/Md. It should be stressed that the magnetic field induces here an additional effective anisotropy of the 'easy plane' type aKeff = Q cos 2 0.

424

A.K. ZVEZDIN

The energy expressed in eq. (3.14) has a simple physical interpretation. As well as in antiferromagnets, here 1/), is the transverse susceptibility of a system and X0 is the longitudinal susceptibility, i.e. X± "~ l/A,

XI[ ~ X0.

Since usually XII << X± the energy Q is a prerequisite (with enhancing H) for spin reversal in a state with a maximum susceptibility value, i.e. in the canted phase. The second and third terms in (3.12) are typical for spin-flop transitions in antiferromagnets. The competition between these terms defines the threshold field of such transitions. Specific properties due to ferrimagnetism are reflected in the first term of eq. (3.12). In this approximation, x(O) can be replaced by

(3.15)

~(0) = Xo + x'HcosO,

where X0 and X' are defined below eq. (3.12). From the above reasoning it is convenient to express formula (3.12) for the thermodynamic potential in the following form ()C± -- Xll)(/~- - (/~l*)~)2 ¢, = -

2

-

M°(gri)

-

K~I(O,

~) +

const,

(3.16)

w h e r e / ' = Md/Md. There is a clear analogy of this formula and similar expression for the free energy of antiferromagnets (see, e.g., Belov et al. 1979) that can have a definite heuristic significance. Notice that at T = 0 the thermodynamic potential (3.1) takes the form = - - m d ~r -- M O ( H 2 -Jr-)`2M2 -

2HAMo cos 0) 1/2 + K(O, ~),

where Mf° is the magnetization of the f-sublattice at T = O. 4. H - T p h a s e d i a g r a m s

4.1. Uniaxial magnetic anisotropy 4.1.1. Case 1: It& E A (EA is the easy axis of magnetization) In that case the phase diagram has a most simple form. Substituting formula (see subsection 3.2.1 and eq. (3.3))

df

--

dO

= 2sin0 cos0,

(4.1)

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

425

H A

-

~3

"~ :rr/4

II~

lm

ro

r

Fig. 4.1. Phase diagram of a uniaxial ferrimagnet in a field perpendicular to the easy axis. The dashed curves are isoclines 0 = O(H,T). The solid curves pertain to the second order phase transitions; after Goranskii and Zvezdin (1969b). into eqs (3.9), (3.10) we get two solutions 0 = 0 and 0 = 7r o f these equations corresponding to collinear phases. Boundaries o f the stability region o f these collinear phases (in fig. 4.1 it is AA' and BB' lines) are determined by the equations 1-A~(0)

1 - A~(Tr) +

2K MdH

---0, 2K

MdH

(4.2)

- 0.

(4.3)

The canted phase is defined by 1 - A~(0)

- 2- K

Man

cos 0 cos 2 F = 0,

2 K sin 2 0 cos ~ sin ~ = 0.

(4.4) (4.5)

It is easy to verify that the values T = 0 and T = 7r form the m i n i m u m conditions o f a t h e r m o d y n a m i c potential so that the values of a corresponding thermodynamic potential are equal. Such 'degeneracy' of solutions is a characteristic property o f the angular structures, It can physically be realized by dividing a crystal into domains when transitions to a canted phase occur (see section 11).

426

A.K. ZVEZDIN

The curves A A t and B B ~ lines in fig. 4.1 correspond to the phase transitions of the second order. It is not difficult to check this by expanding F into a series of A0 in the vicinity of a the corresponding collinear phase. For example, close to AA' the expansion of F in A0 has the form

¢' = ¢'o + a(H, T) 02 +/3(H, T) 04 + . . - ,

(4.6)

where a(H, T) = 0 and/3(H, T) > 0 on A X ( it is clear here that A0 = 0). The expansion of F close to the curve B B p has an analogous form with the only difference that the expansion parameter (order parameter) in that case is A0 --- ~- - 0. In a small field region where H << AMd, equations for the curves AA' and B B t can be represented in the following form (in first order of H/AMd): T - Tc 2K T ~ - ,~fx'IH + Md----H'

(4.7)

where the T signs correspond to the curves A W and B B ~, respectively. Here we have used the relation

1 - ,~Xo(T) ,,~ (T - Tc)/Tc,

(4.8)

which is valid if IT - Tel << Tc; X0 and X' are defined by formulas (3.13). A distinctive feature of phase diagram (fig. 4.1) in that case is the presence of 'narrow throat' joining the low field and high field areas of the canted phase. The 'throat' coordinates (Tth, H*) are

H* = ~/

Tth,~Tc,

2K MA[X'I"

(4.9)

Its semi-width is AT

2~ / 2K/~IX']

rc

V Md

(4.10)

In the low-field region the phase transitions are such that the continuous spinreorientation is accompanied by two phase transitions of the second order. In this region the angle between the sublattices is a little bit different from ~. In the high field region the transition FI ++ C ++ FE takes place with an essential sublattices 'inflection', i.e. the angle between the sublattices substantially differs from ~. If the paramagnet sublattice is close to saturation,

Mf a2M~

Ix'l ~ - -

(4.11)

Substituting this value into (4.9), (4.10) and taking into account that in the 'throat' region &~" ~ 1 yield

H* ~ (HAHex) ~/2,

A T ~ 2(HAHex)I/2,

Tc

(4.12)

where Hex = AMd. If the paramagnetic sublattice is far from saturation the magnitude of H* strongly increases and AT/Tc decreases.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

427

/~ll

4.1.2. Case 2." EA (EA is the easy axis of magnetization) The boundaries o f the collinear phases {0 = 0} and {0 = 7r} are defined in that case by the equations

1 - A~(0)+

2K

-0,

(4.13)

MdH 1 - A~(~)

2K

- -

-

0.

(4.14)

MdH Curves A X and B B ~ in the phase diagram (fig. 4.2) correspond to these boundaries. The system possesses axial symmetry. Therefore the azimuthal angle ~ at 0 = 0 is arbitrary here. The canted phase is defined by equation 2K 1 - A~(0) + - cos 0 = 0. MdH

(4.15)

The main peculiarities in the corresponding phase diagram are the following. There is area in phase diagram that is restricted by the curve X Q R . . . B ~ where both H A

\P/ H*

~t MF~

,

AL.j-

05"

MR ~ MF~

R 7

T

Fig. 4.2. Phase diagram of an uniaxial ferrimagnet in a field parallel to the easy axis. BP, AR are curves of the 2nd kind of phase transitions, AIR, PB ~ are curves representing the loss of the collinear phase stability. PQ is a curve representing the loss of metastable canted phase stability, PRTc is a curve describing the first kind of phase transition and P is the tricritical point; after Zvezdin and Matveev (1972a).

428

A.K. ZVEZDIN

collinear phases are stable. RT is the curve of the first order phase transition where ~ ( 0 = 0) = ~ ( 0 = ~).

As in the case of /7_l_g, where g is parallel with the easy axis, we write the thermodynamic potential as a series expansion in/tO approaching the AA' and BB' lines, the points where the coefficient at 04 in the formula (4.6) (i.e. fl(H, T)) go to zero (point Q and P) appear on these curves; /3 > 0 above those points, i.e. transition from collinear phases to the canted phase become the second order phase transition. According to modem terminology these points are referred to as tricritical points. Familiar examples of similar critical objects are tricritical points in the anisotropic antiferromagnets Dy3A15Oa2 and FeF2 (see, e.g., Landau et al. 1971, Blume et al. 1974, Giordano and Wolf 1977). In the vicinity of the tricritical point P the thermodynamic potential expansion should be considered up to the sixth order term. The coefficient of this latter term can be shown to be positive. In the approximation quadratic of H/.~Md (see eq. 3.14 of section 3) the points P and Q coincide. In this case

/3(H,T) = 2 K

H .2 - 1

(4.16)

in a vicinity on the curves A X and BB'. H* in (4.16) is defined by formula (4.9). The line corresponding to the first order phase transition between the collinear phase {0 = ~r} and the canted phase (line PR in fig. 4.2) can be derived by eliminating 0 from the system of equations ~/i(00) = ~/i(Tr), = 0. -~

(4.17) (4.18)

0=00

The canted phase loses its stability on curve PQ and the system transforms {0 = or} in a jump-like way. The jump of the angle 0 increases from A0 = 0 at point P to A0 = 7r at point Q. The field- and temperature-dependence of the PQ curve can be found from the system of equations

0=0o

820

0=0o

This curve belongs to the family of curves O(H,T) = const.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

429

The intersection points of the curves A A ' and B B ~ associated with the loss of collinear phase stability has the coordinates

H*=IHa

~qX'l '

Tth ~ Tc.

(4.21)

The equation of the curves A A ~ and B B ' can be represented in the low-field limit by the formula T - Tc F q

T~

2K

- ~Ix'IH- Md-----H'

where the ~: signs correspond to the curves A A ~ and B B ~, respectively. The magnitudes of the magnetic field defining the points P and Q slightly differ from H*. In particular, if we take Hex = AMa = 2.5 x 105 Oe,

K = 7 x 103 erg/cm 3,

Tc = 300 K.

(parameters of gadolinium-ferrite garnet) then He - H* : H* - HQ ~ 400 Oe. It should be borne in mind that in the vicinity of the point H* the effective anisotropy energy (3.12) changes its sign. Some additional details of the phase diagrams in the vicinity of the 'narrow throat' were studied by Baryakhtar et al. (1976). To gain greater insight into effects of the magnetic anisotropy on the critical fields, let us consider the case T = 0. It follows immediately from the eqs (4.2) and (4.3) that 2K Hcl = )~(Mf - Md) T

M f - Md m~ 2K

He2 ~-/~(mf -}- md) 4-

Mf

mf

mf + md md'

where the signs - / + corresponds to the case H]]EA (H_I_EA). Notice that in the center of the region of the canted phase (more precisely at the line O(H, T) = 7r/2) there is no influence of the magnetic anisotropy on the magnetization curve. Here, the susceptibility of ferrimagnet should be equal to 1/A similar to the isotropic case. This follows immediately from the eq. (4.4) (see also section 8).

430

A.K. ZVEZDIN

4.2. C u b i c a n i s o t r o p y 4.2.1. C a s e 1 : / t ] ] [ 0 0 1 ] , K1 < 0

The phase diagram for this case is shown in fig. 4.3. Three phases are present: two collinear (A and B) and one canted (C) phases. Lines A A ~ and B B ~ are secondorder transition lines. The equations for the curves A A ~ and B B ~ are (4.2), (4.3) at K = - K 1 . The canted phase C is determined by the equations

1 - A;~(0) +

K1

mdH

cos 0 (2 cos 20 + sin 2 0 sin 2 2~) = 0,

(4.22)

K 1 sin 4 0 sin 2~ cos 2~ = 0.

(4.23)

The second of these equations and the stability conditions simultaneously define the equilibrium values of the angle ~: = ~/4,

3~/4,

5~/4,

7~/4,

i.e. the rotation of sublattices takes place in the (110) plane (~ = 7r/4, 5~r/4) or in the (150) plane (~ = 37r/4, 77r/4). A fourfold degeneracy of the tilted phase follows from the symmetry of the system: the [001] axis is fourth order axis. The canted phase splits into two different canted phases: (C, C t) in the low field region. It means that in this range the solution of eq. (4.22), i.e. the function H '

A

•

== ,i =,o)ot

,1o M

To

, T

Fig. 4.3. Phase diagram of a cubic fe~imagnet for HH[001], KI < 0. A A ~, B B I are curves of the 2nd l~nd phase transitions from the collinear phases A and B into the canted phase C. OTc is a curve of the first-order phase transition between the canted phases. O is the critical point; after Zvezdin and Matveev (1972a).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

431

O(H, T), is not unique and the rotation of the sublattices takes place discontinuously. For phase C one finds 0 < 0 < ~r/2 and for phase C' 7r/2 < 0 < 7r. The area restricted by the curves O M and O M ~ is the area of coexistence of the phases C and C . These curves are defined by the system of the following equations

K1 1 - A~(0) - - cos 0 (3 cos 2 0 - 1) = 0, MaN

(4.24)

- MdH)~x'(O) + K1 sin0(9cos2 0 - 1) = 0.

(4.25)

In the approximation H / A M d (See (3.12)) the equations of the curves O M and O M ' becomes T-To

~-c

2 IK1] ( 2H2) --T~Md--~ 1 - H,---~ •

(4.26)

The coordinates of point O in the plane H - T are HO = H*

T ° ~ Tc,

where H* is the coordinate of the 'narrow throat' in the phase diagram (see formula (4.9)). The curve TcO is the locus of the first order transitions between the phases C and C'. There is a jump in angle from zero in point O up to A0 = 2 arccos(2/3) for H-+0. The critical point O is the limiting point of the curve on which there is an equilibrium of two low field phases. For high magnetic fields (H > H ° ) the crystal is homogeneous and for H < H ° it is divided into domains with different orientations of the magnetic sublattices. From this viewpoint the critical point O is analogous to the critical point of vapor-liquid phase transitions. Analogous to the density in that case is the rotation angle. There are many peculiarities in the behavior in the system near the critical point: anomalies of specific heat, propagation in the sound and scattering of light. (For details see sections 7-10.) Alben (1970a, b) was the first who paid attention to the possibility that this critical point existed in ytterbium-ferrite garnet. 4.2.2. Case 2: / I l l [ I l l ] , K1 < 0

The phase diagram for this case is shown in fig. 4.4. Four phases exist: two collinear phases, A and/3 and two canted phases C and D. The canted phases are three-fold degenerated (it is clear since axis [111] is of the third order). Curves A A ' and/3/3' refer to the stability loss of the collinear phases A and B. They are defined by the equations 4t(1 - - 0, 3Mall

(4.27)

4K1 1 - A~(Tr) + - - 0. 3Mall

(4.28)

1 - ),;~(0)

432

A.K. ZVEZDIN

H

H 1

2

© B D

MR**Mve I v

ro

r

r

a)

b)

Fig. 4.4. Phase diagram of a cubic ferrimagnet for HIl[lll], /41 < 0. a). Curves of the first order phase transitions: Tc(1), To(2), To(3). b). Curves describing the loss of stability: phase A-AA', phase C-CC' and F F ' , phase D-GG' and DD ~, phase B-BB'; after Zvezdin and Matveev (1972a).

The canted phases are determined by

1 -

A:~(0)

- -

-

-

K1

Md----H

( 4 cos2 0 - sin 2 0 cos 0 -

x/2 sin3 0sin3~ + v/2cos2 0 sin 0 sin 3cp) 3 -

z

0,

x/2K1 cos 0 sin 3 0 cos 3~ = 0.

(4.29) (4.30)

Equations (4.30), combined with the stability conditions, define the equilibrium values of the angle ~: PhaseC: ~=vr/2,

7rr/6, llrr/6,

P h a s e D : ~ = 31r/2, ~r/6, 57r/6,

0 < 0 < rr/2; rr/2 < 0 < rr.

The crystallographic planes (110), (011), (101) are easily seen to correspond to the angles ~o = rr/2, 3rr/2;

~ = 77r/6, rr/6;

~ = 1Dr/6, 5r~/6,

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

433

respectively. The equilibrium values of O(H, T) at a given value of the azimuthal angle ~ are defined by eq. (4.29). The boundaries of the stability for the canted phases C and D do not coincide with the corresponding stability curves of the collinear phases. These boundaries are determined by the curves C C and F U for phase C and by the curves D D ' and GG ~ for phase D. In that case all phase transitions A +4 C ++ D ++ B are of the first order. This follows from the fact that terms of the third order in A0 are present in the thermodynamic potential (3.1), (3.7). A stereographic projection of the trajectories of the d-sublattice moment 2~rd at phase transitions A - C - D - B is shown in fig. 4.5. Let us bring forward analytical formulas for the characteristic curves of the phase diagrams (fig. 4.4), which are valid for H << AMd. In the vicinity of A A ' the series expansion of the thermodynamic potential in 0 has a form r -- T c 0 2

= Mall ~

2

V/2

04

3 -IKll03 + IKII((H/H*)2 - 3) ~- + - . . ,

where H* is defined by the formula (4.9) at K = IK1 t. Along the curve A A ' the magnitude TA(H) is defined by eq. (4.7) for K =

-(2/3) IKll. [0011 D

[0101

Fig. 4.5. Trajectory of the vector 3~td on the unit sphere for the transitions A-C-D-B, /~11[111]. Section AC corresponds to phase C, DB to phase D, the points A, B to phases A and B. A jump of the vector M takes place between the points G' and D.

434

A.K. ZVEZDIN

The equation of the curve of the 1st order phase transitions (3To) for H > v/-3H * is --TA + T

41K~I

TA

9 M d H ( ( H / H * ) 2 - 3)

The jump in angle 0 along this curve is equal to

AO=

3 ( ( H / H * ) 2 - 3)

These results are valid for the area close to B B ' as well (with the replacements (--TA + T) --+ ( - T + TB), 0 -+ 7r- 0 ). For H < v~H*, the coefficient of 04 changes its sign. Therefore, it is necessary to take it into account in the analysis of terms of higher order in 0. The curves F F ' and GG' are defined by equations

v~lK, I

1 - t~'(7r/2) -4- - -

3Mall

- O,

or

(

T-Tc

Tc

~-

H

)2

~

v~IKI[ T 3Md-------H

The jumps in angle 0 are equal to

AO:--(H/H*)

3

2

atH>H*.

Let us consider the asymptotes of the first order phase transitions curves as at H --+ 0 (fig. 4.4a):

To(3):

T - Tc

T~

4H

3~Md'

Tc(2): T = Tc,

r~(3):

T - Tc

T~

4H

3),Md "

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

435

4.2.3. Case 3: /-IN[ll0], K < 0 The phase diagram is shown in fig. 4.6. There are five phases: two collinear phases (A a n d / 3 ) and three canted phases (G, D, E). The stability curves o f the collinear phases (AA' and /3/3') are defined by the equations

K1 1 -

;,~(o)

+

1-A~'(zr)

MdH

-

o,

K1 - - - 0 .

MdH

The canted phases C and D are described by equations 1 - ),~(0) +

K1 - -

MdH

cos 0 (3

COS 2 0 - -

2) = 0,

qv = ±¢r/2. For phase C one has 0 < 0 < 0o and for phase D 7r - 0o < 0 < ~r, where 0o is defined by equation sin 2 0o = 3/5.

3

A

rl/ Te

9

T

Fig. 4.6. Phase diagram of a cubic ferrimagnet for /~[l[ll0], t£1 < 0. AA', BB', 2-0, 3-0' are the curves of the first order phase transitions; 7-0-8, 6 - 0 ' - 9 are curves describing the boundaries of the existence range of the metastable phase; O, O ~ are the tricritical points; after Zvezdin and Popkov (1977).

436

A.K. ZVEZDIN

A A I and B B ~ are curves pertaining to the second order phase transitions A - C and B - D . The stability of the canted phases C and D are described by the curves 29 and 36, which are defined by the equations 25/2K1 1 - ),~(00)

53/2MdH

-

0,

25/2K1 1 - A~(Tr - 00) +

5312MdH

-- 0.

The canted phase exists for sin20 > 3/5. It follows from eqs (3.8), (3.9), (3.10) that sin 2 (p = sin -2 0 - 2/3. After substituting this formula into eq. (3.9), we obtain an equation for angle 0 in phase E: 4K1 1 - A~'(0) + - cos 0 (1 - 4 COS 2 0 ) 3MdH

=

0.

The stability condition of this solution is of a form

Heff

Heff

~Heff

- (4/3)K1(1 - 12cos20) > 0.

It is seen from the last inequality that for sufficiently high values of H when

H

> H1 =

V/5/2H *

the stability condition is satisfied so that transitions C - E and D - E are continuous in this region (transition curves 2 - D ' and 3 - D ) H* is defined by formula (4.9). For H < H1 the transitions C - D and D - E are of the first order (curves OTc, O'Tc). Points

O(H = H1, T = Tc - 0, 43AT), O'(H = H1, T = Tc + 0, 43AT) are tricritical points, AT is determined by formula (4.10).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

437

~1 ira]

C [112]

B

Fig. 4.7. Trajectoryof the vector /17/aon the unit sphere at the transitions A - G - E - D - B , ~11[110]. The points A and B correspond to A and B phases, the trajectorysections A G and AGt - to phase G", GD and G~D ~ - to phase E, D B and D~B ~ to phase D (see fig. 4.6). The curves 7 - 0 - 8 and 6 - 0 ' - 9 in fig. 4.6 define areas where metastable phases exist. These curves are defined by equations of the type (4.19, 4.20) (for more details see Zvezdin and Popkov 1977). In fig. 4.7 a stereographic projection of the trajectories at the sequential phase transitions A - C - E - D - E is given (see fig. 4.6)

HI[[001],

C a s e 4: K1 > 0 The character of the phase diagrams changes substantially when the anisotropy constant changes its sign. This will be illustrated by means of an example where In both cases (K1 > 0 and K1 < 0) the phase diagrams have a most simple form for s u c h / 1 orientation. The phase diagram for the case/(1 > 0 is represented in fig. 4.8. Three phases exist: two collinear (A and B) and one canted (C) phase. The boundaries of the collinear phase stability regions are determined from eqs (4.2), (4.3) (reference frame is identical to the one used in subsection 4.2.1). They define lines A A ' and B B ' in fig. 4.8. The canted phase is described by the equations

4.2.4.

/~ll[001].

1 - A~(0)

2Ka -cos 0 (1 - 2 COS 2 0 )

:

0,

(4.32)

MdH

= 0, 7r/2, 7r, 37r/2. The equilibrium values of the azimuthal angle define the rotational in the planes (010) and (001). It is easy to verify that the second derivative of • with respect to ~o satisfies the condition ~ t! > 0 here.

438

A.K. ZVEZDIN

/-/

~M~o

n, M~

0

r~

'

0

4o

r

Fig. 4.8. Phase diagram of a cubic ferrimagnet for ErH[001], K1 > 0; after Zvezdin and Popkov (1977).

The condition for a stable canted phase is

()~MdH)2(

Heft

) +2Kl(1-6cOs20)>O"

OHeffi~Mf

The canted phase is seen from this equation to be stable in all ranges of 0 (from 0 up to re) for sufficiently high values of H > H1, where H1 ~ x/-5H*. For H > HI the transitions A - C and B - C are the second order phase transitions (lines AO, BO'). For H < H1 the stability regions of the phases A, B and C overlap. The curves O T and O'T represent the first order phase transitions from A - C and from B - C . Points O and O' are tricritical. Their coordinates are

O(H = HI, T = Tc - (1/2)AT), O'(H = H1, T = Tc + (1/2)AT), where AT is defined by formula (4.10). The curves 1-O-A' and 2 - O ' - B ' form the boundaries of the stability region of the canted phase which are determined by simultaneous solution of eq. (4.32) and by ~"o0 = 0. Overlapping between stability range of the canted phase and the metastable regions, the collinear phases 0 = re and 0 = 0 take place for

H < H2 = x/T-/2H*.

FIELD INDUCED PHASETRANSITIONSIN FERRIMAGNETS

439

H

M~ e

S r~

r

Fig. 4.9. Phasediagramof a cubic ferrimagnetfor/~ll[ll 1], K > 0; afterZvezdinand Popkov(1977).

In fig. 4.9 the phase diagram of a cubic ferrrimagnet is presented for the case of K1 > 0 and Htl [111]. All phase transitions described here as well as those described in 4.2.2 are the first order phase transitions. Popkov (1976a, b) investigated phase diagrams of rhombohedrical ferrimagnetic crystals and films and discussed the problems connected with pressure effects.

5. Field induced phase transitions in rare-earth-ferrite garnets There is a number of investigations devoted to the study of the field induced phase transitions and phase diagrams for the magnetic crystals with cubic and uniaxial magnetic anisotropy. In most of these, rare-earth-ferrite garnets and ferrite garnet single crystal films were investigated. Single crystals of ferrite garnets have natural cubic magnetic anisotropy (Ndel 1954, Pauthenet 1958a, b, Pearson 1962); the films have an induced uniaxial magnetic anisotropy (Le Craw et al. 1971, Callen 1971, Rosencwaig et al. 1971, Akselrad and Callen 1971, Stacy and Rooymans 1971, Gyorgy et al. 1971, Kurtzig and Hagedorn 1971). Let us start by considering the single crystals of ferrite garnets. The next paragraph will be devoted to ferrite garnet films. Rare-earth-ferrite garnets with R3Fe5012 as a chemical formula (R is a rare earth element) are isomorphic in structure to the natural mineral orthosylicate garnet Gd3A12(SiO4)3 which has a cubic crystal structure (O~° - Ia3d space group, Geller and Gilleo 1957)). Magnetic garnets have been synthesized in the fifties (Forestier

440

A.K. ZVEZDIN

and Guiot-Gullion 1950, 1952, Bertaut and Forrat 1956, Geller and Gilleo 1958a, b, 1960, Geller 1960). The ferrite garnet unit cell consists of 8 formula units: 64 cations (40 Fe +3 ions and 24 R +3 ions) and 96 anions (oxygen ions). The ions of Fe and rare earth are arranged in the node-to-node sets of the oxygen matrix. When characterized according to their coordinate number they occupy tetrahedral [d], and octahedral (a) and dodecahedral {c} positions (Geller 1960). The R +3 ions are in {c} positions, 24 Fe +3 ions are in [d] positions and 16 Fe +3 ions are in (a) positions. Therefore the ferrite garnet formula is often written in the form {R 3+3}(Fe 3+3)[Fez+3]O12. The magnetic ions of the same type located in equivalent cell nodes have the same moment direction and form a magnetic sublattice. Therefore the three sublattice model is in common usage to describe the properties of these garnets (N6el 1954). The overall exchange interactions in the sublattices and the exchange interactions between the sublattices are of the antiferromagnetic type: the strongest one is the a-d interaction (the effective exchange field H a d ~ 2 x 106 Oe) as a result of which the Fe +3 ions in (a) and [d] positions are oriented in an antiparallel to each other. The exchange interaction of the rare earth with the Fe +3 ions is one order of magnitude lower than the Fe(a)-Fe[dl interaction and the main contribution to the rare earth - iron exchange is the c-d interaction (Hc-d --~ (1-4) x 105 Oe). Therefore, a non-compensated magnetic moment arises (..~ 5#B/per formula unit at 0 K). It should be noted that the rare earth sublattice consists of six different sublattices, which is due to the orientation of the surrounding crystal field. However, this is important only for rare earth ions with a nonzero orbital moment in the ground state. This point will be considered in the second part of this review. Generally speaking, the (a) and [d] positions in the garnet structure should also be subdivided into two types of different nonequivalent positions. But by virtue of the fact that the ions Fe +3 are scarcely affected by the crystal field, this inequality plays a considerably lower role than the inequality of the {c} positions, occupied by rare-earth ions. The magnetization vector of the rare earth sublattice is antiparallel to the magnetization vector of the Fe sublattices. The intra sublattice interactions in the a and d sublattices are not large compared with the a-d interaction (Hc--d ~ (1-4) x 105 Oe). Furthermore, the exchange interaction within the c sublattice is very small, so for T > 10 K the rare earth ions can be considered as a system of paramagnetic ions placed in a strong effective field produced by the Fe ions (Pauthenet 1958a, b, Aleonard 1960, Anderson 1964, Clark and Callen 1968). The predominant a-d exchange interaction between the Fe sublattices is destroyed at the Curie temperature Tc which is approximately equal to 560 K for all rareearth-ferrite garnet materials (Pauthenet 1958a, b). The temperature dependence of the magnetic moments of the rare earth and Fe sublattices is different which leads to the existence of a compensation temperature Tc (Pauthenet 1958a, b) for many of the magnetic garnets. At T < Tc the rare earth sublattice magnetization prevails and at T > Tc the net magnetization of the Fe ions dominates.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

441

In this section we consider only Gd3FesO]2 iron garnet since this material most closely corresponds to the above considered theoretical model of weak anisotropic ferrimagnets. Actually, the ground state of the Gd +3 ion is 8S, i.e. it has L = 0, S = 7/2. Therefore, the spin-orbit interaction is only a small perturbation for this ion and the anisotropy energy is one or two orders of magnitude less than the Gd-Fe exchange energy. This is the very condition for the applicability of the weak anisotropy model. Some other rare earth garnets will be considered in the second part of the review. A great number of papers are devoted to investigation of induced noncollinear magnetic structures in ferrite garnets by using the Faraday effect (Kharchenko et al. 1968, 1974, 1975a, b, Bernasconi and Kuse 1971, Gnatchenko and Kharchenko 1976, Lisovskii et al. 1976a-c, 1975, Smirnova et al. 1970). The simplest compound, also from optical aspect, is Gd-ferrite garnet. Therefore, the major part of investigations by means of the Faraday effect on induced noncollinear magnetic structures have been carried out on this ferrimagnet. The % min 296 K

100 80 60 40 20

7

0

$ H"

285.3

84 -20 -40 -60 -80 -100 0

I

i

I

I

I

10

20

30

40

50

Fig. 5.1. Magnetic field dependencies of the Faraday rotation in Gd3FesO12 in the vicinity of the compensation temperature; (X = 6328 ,~; the broken line shows the Faraday rotation of the optical system; after Kharchenko et al. 1975a, b).

442

A.K. ZVEZDIN ~o,deg 1.0 0 -l.0

k?o ' ~~ttttqttttt i

i

i

i

i

i

r l

1.0 0

$j

$ ~ " 7 T 1 ~ 1~6 kOe

I t l l l l / l l l l l [ l l l l lI l I I

-1.0 1.0 I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

P

-1.0 -~~ _ . t . . . , . I . ~

~

1.0 0 -1.0 i

i

280

'

2 -

-

"

:

i

285

i

i

i

290

r

i

i

i

295

T,K

Fig. 5.2. The temperature dependencies of the Faraday rotation in Gd3Fe5012 in the vicinity of the compensation temperature; after Kharchenko et al. (1975a, b).

dipolar contribution of the Gd sublattice to the circular birefringence is negligibly small in Gd ferrite garnet. If we consider the two Fe sublattices as one, the dependence of the Faraday rotation angle ~b on the direction of sublattices magnetic moment may be represented in the lonagitudinal geometry in the form (Kharchenko et al. 1975a, b) (magnetic intensity H is collinear to the light propagation direction f:): )9 = ~0 COS 0Fe -k FH.

Here )90 is a spontaneous Faraday rotation of the total Fe sublattice. The term FI-I accounts for the effect of the magnetic field on the excited energy states of the crystal and this enables us determine the angle 0Fe between the direction of the

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

443

ZII °e

0±0 t ttt'~tlt#lw'~ 260

t

r~m~

285

t~lSlk 5.17

290

T, K

Fig. 5.3. The ellipticity of the circular polarized light in the canted phase of Gd3FesOl2, B[I[lll]; after Kharchenko et al. (1975a, b).

H, kOe

I I

20

II II 15

•

lO

,'o

t

I

275

280

•

285

I

l

290

295

I

T, K

Fig. 5.4. Phase diagram of the Gd-ferrite garnet for /~11[100]. Thick solid lines are the calculated boundaries of the existence regions of stable and metastable phases, (o) are experimental points of the transition into uncollinear phase, (o) and thin lines indicate the observed existence region of the magnetic inhomogeneities, the dashed line is extrapolation; after Kharchenko et al. (1975a, b). m a g n e t i c m o m e n t o f the F e sublattice and the direction o f the field H b y using the e x p e r i m e n t a l data o f F a r a d a y rotation. T h e d e p e n d e n c e o f the F a r a d a y rotation at t e m p e r a t u r e close to the c o m p e n s a t i o n t e m p e r a t u r e is shown in figs 5.1 and 5.2. The e x p e r i m e n t a l d e p e n d e n c i e s are in

444

A.K. ZVEZDIN H, kOe 15

10

5

283

284

"I

a)

285

286

287 T, K

H, kOe

1

2

3'~"

•

•

x,,~ •

I

I

I

2

1

I

0

1

4, ~3

I

1

2 T-Tc, K

b) Fig. 5.5. Phase diagram of Gd-ferrite garnet for HII[lll]. a) Experimental boundaries of the existence region of the magnetic phase (corresponding magnetic phases are indicated in the circles). b) Theoretically calculated stability curves (points - experiment); after Kharchenko et al. (1975a, b). a c c o r d a n c e with theoretical curves c a l c u l a t e d for a three-sublatticed model. I n f o r m a t i o n a b o u t the sublattices rotation can b e d e r i v e d b y e x p l o r i n g the birefring e n c e effects ( K h a r c h e n k o et al. 1975a, b, Pisarev et al. 1969, 1971, G r j e g o r j e v s k i y and Pisarev 1973). F o r the case it was shown b y K h a r c h e n k o et al. (1975a, b) that the linear b i r e f r i n g e n c e in G d ferrite garnet is equal to

f~ll~

A n = A n • sin 2 0Fe,

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

445

ATmin

H*V~15

O=Tr /I iI ~

" Mad

4 /d

Mre

o/

/ 0

N

0=0

i

A'

~

//MGd-

I

tel

270

290

\x o

Cd

\\N I

Fe

"N'

I

TK

310 T,K

Fig. 5.6. H - T phase diagram of Gd3FesOl2 iron garnet for/211[100]. The thin solid curves A A ~ and B B ' are theoretical curves of the 2nd order phase transitions, calculated according formulas (4.2), (4.3). The thick solid curves A X and B B ' are experimental curves, obtained from the measurements (o) - the specific heat, (×) - the sound absorption at a frequency 30 MHz with the wave vector ffl/2; OTe is a curve representing first order phase transitions, O N and O N ' are the boundaries of the existence region of the metastable phase, O is the critical point; after Kamilov et al. (1975).

H, kOe

200

+

100

+ +

0

I

50

100

I

150

T, K

Fig. 5.7. Magnetic phase diagrams of a Ho3Fe3O12 iron garnet single crystal (o) obtained for/211[111], (e) obtained for/2111110], (+) - polycrystalline data; after Hug (1972).

446

A.K. ZVEZDIN M,, p~/molecule

M,,/zB/molecule

2 1.5"-

8

Y

o

~2

0

I H, kOe I

20 40~"'-

6

J

4

22

o

t

I 100

I

I

/ 0

~

0

1

I

t

I

100

200

I

f

4 2

I

I

I

I

100

200 f

i ~

I 0

a)

1 100

I

/ I

0 6 '

I

200 ~ -

J

I

200 H, kOe

I

I

4

P

I

100

I

0

I

100

I

t

200 H, kOe

b)

Fig. 5.8. Experimental (a) and theoretical (b) plots of the magnetization of Hoo.41Y2.59FesO12 garnet against the field: solid curves - for /~11[111], dashed curves - for nll[ll0], dash-dot curves - for ~11[100]. The insets show MII(H) in weak fields; after Silant'ev et al. (1980).

where An0 is the spontaneous linear birefringence in the direction perpendicular to the optical crystal axis, and 0Fe is the angle of the Fe-sublattice moment orientation relative to the field direction. Thus, the development of a noncollinear structure and concomitant change of the Fe sublattice direction results in the change of birefringence as well. Figure 5.3 shows a typical example of the temperature dependence of the birefringence in the vicinity of the compensation point. The magnetic phase diagram of Gd-ferrite garnet (Kharchenko et al. 1974, 1975a, b, Gnatchenko and Kharchenko 1976, Kamilov and Schachschaev 1972, Kamilov et al. 1975) has been studied in detail on single crystals. These investigations show that the theory presented above can qualitatively explain the main features of the Gd ferrite garnet phase diagram in the vicinity of the compensation temperature. This includes the character of temperature depen-

H E L D INDUCED PHASE TRANSITIONS IN FERRIMAGNETS Mjr,

447

7~--~ , molecule

10

f /

I

0

a)

I

100

I

I

200

I

I

0

I

100

I

-

I

200

H, k O e

b)

Fig. 5.9. Experimental a) and theoretical b) plots of the magnetization of HOl.05Y1.95FesOI2 iron garnet against the field: solid curves - for/t11[111], dashed curves - for n l l [ l l 0 ] , dash-dot curves for Hll[100]; after Silant'ev et al. (1980).

dencies of the critical fields, the number and the order of different magnetic phase degenerations and the type of the phase transitions. However, a quantitative agreement between the experimental and theoretical data in the two-sublatticed model is not reached. This is due to the fact that even small changes of magnetization of the total Fe sublattice in a field, owing to the finite value of the antiferromagnet exchange interaction between Fe sublattices, will result in a substantial change of the critical field value when the temperature of compensation is approached. Kharchenko et al. (1975a, b) calculated several phase diagrams of cubic ferrimagnets with three sublattices. Qualitatively these are similar to those considered above, the only difference being renormalization of characteristic points and temperature dependencies of critical fields of the phase diagrams. Phase diagrams constructed on the basis of theoretical formulas reported by Kharchenko et al. (1975a, b) are satisfactorily consistent with the experimental data for Gd-ferrite garnet (figs 5.5, 5.4, see also figs 4.3 and 4.4) though complete agreement is not reached. Apparently this is due to the influence of the magnetoelastic energy on phase diagram (Kharchenko et al. 1975a, b, Gnatchenko and Kharchenko 1976), dividing of a sample into domains (see section 11 below). Kamilov et al. (1975) investigated the specific heat and sound propagation in G d 3 F e s O 1 2 n e a r the compensation temperature. Figure 5.6 shows the H - T phase diagram of this garnet for HI1[001] according to the experimental results of these authors (for details see sections 9 and 10).

448

A.K. ZVEZDIN

H, kOe 200 150

H, ~Oe

IV

200

A

~

Q

150 II

lOO

100

I

50

~r

I

I

I

0

10

20

i~

30

F

1 4 1

I

T,K

0

a)

10

20

I

I

30

40

I

~K

b) H, kOe 200

I]I

150

100

50

I

0

I

20

I

Jr

I

40

I

T, K

c) Fig. 5.10. H - T - x phase diagrams for (HoY)IG when x = 0.67: a) HII[lll]; b) /~l[[ll0]. The solid lines: theory. The open circles were obtained with increase of the field during the measurement process, the dark circles with decrease of the field; after Babushkin et al. (1983). Some features of the phase diagrams of a Ho ferrite garnet single crystal were obtained for a field orientation along the [111] and [110] axes (fig. 5.7) (Hug 1972). It is seen that these diagrams are in good qualitative agreement with the theoretical phase diagrams presented in section 4. Detailed investigations of the FIPT in (HoY)3FesO]2 and (TbY)3Fe5012 iron garnets were made by Levitin and Demidov (1977), Zvezdin et al. (1977), Silant'ev et al. (1980), Babushkin et al. (1983), Lagutin and Dmitriev (1990), Lagutin and

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS H, kOe

~ 9 ~ ? ~

200

449

H, kOe

~

200

150

150 L

100

~ ~ ~ ~

i

~,

i

0

r

1t"NI/

20

a)

100

i

~t

To40

/ K/t~ r

I

60 T,K

30 b)

i 50

i 60

i 70 LK

H, kOe 200

~Q

100

5o

~I

I

20

r$

30

'

~

B II

Te

Tcr~M'

t

50

,1.

I

60

I

70 T, K

e)

Fig. 5.11. H - T - z phase diagrams for (HoY)IG when z = 1.05: a) HII[lll]; b) /qll[ll0]. The solid lines: theory. The open circles were obtained with increase of the field during the measurementprocess, the dark circles with decrease of the field; after Babushkin et al. (1983). Druzhinina (1990) and Lagutin (1993). These ferrimagnets have a very strong (Isinglike) magnetic anisotropy. It seems possible to find some common features between the phase diagrams of the strongly and weakly anisotropic ferrimagnets. However, strong anisotropy drastically changes phase transitions and phase diagrams. A distinctive feature of the strong anisotropic case is that the transitions follow the pattern of jumps of the magnetization, i.e. the first order transitions. These drastic distinctions can be seen in figs 5.8-5.11 in which the magnetization curves and H - T phase diagrams of the (HoY)3Fe5012 garnets are displayed.

450

A.K. ZVEZDIN

There is an elaborate investigation of the phase diagrams of the Yb-ferrite garnet (Alben 1970a, b, Feron et al. 1971, 1972, 1974). However this garnet is a strongly anisotropic ferrimagnet and it also demands special considerations.

6. Single crystal ferrite garnet films There are several investigations of field induced phase transitions in single crystal ferrite garnet films with uniaxial magnetic anisotropy (Avaeva et al. 1975, Antonov et al. 1976a, b, Gnatchenko et al. 1977, Dikstein et al. 1980, 1983, 1984, Lisovskii et al. 1976a-c, 1980). Such films are prepared by means of liquid phase epitaxy on Gd-Ga garnet substrates (see, e.g., Eshenfelder 1981). The structural properties of these films are well characterized in terms of the described above model. These films possess high optical transparency which makes it easy to perform optical investigations. However, their complex composition, and their intrinsic non-uniform strain are the cause of unstable compensation temperatures over the film thickness which creates additional peculiarities in the phase diagrams. An important manifestation of such inhomogeneities in these films is the existence of so-called compensational domain walls near the compensation point (Hansen and Krumme 1973, Krumme and Hansen 1973). Such walls complicate the picture of the phase transitions. It should be noted that investigations of field induced phase transitions particularly are the most effective method for determining the compensation temperature profile along the film thickness and other parameters. As an example we show in fig. 6.1 the phase diagram of a Y2.6Gd0.4Fe3.9Gal.lO12 film grown from the liquid phase on to a Gd-Ga garnet substrate which had been cut parallel to the (111) plane (Gnatchenko et al. 1977). The film thickness and compensation temperature were of the order of 6 #m and 180 K, respectively. The critical temperatures (or fields) represented in the phase diagram have been determined by recording the temperature at which various magnetic phases arise and vanish in the field. The temperature dependencies of the Faraday effect presented in fig. 6.2 have been employed as well. From well defined linear rotation angles studied as a function of the temperature near the compensation point Tc the appearance of the canted phase was determined. The observations show that the sample in weak fields (lower than the threshold field Htn ~ 0.7-0.85 kOe in the case) can exist only in two states with spins to be collinear with magnetic field. Further processing of these results indicated the presence of a considerable gradient of the compensation temperature over the thickness of the film, the averaged value of which is equal to 2 × 103 K/mm. The temperatures T( and T~ in figs 6.1 and 6.2 are defined as the temperatures at which corresponding collinear phases (low and high temperature phases) become unstable at least in one of the film layers. Since investigated films possess a compensation temperature gradient along the thickness direction, transitions from the collinear phases into the canted phase show

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

451

H, kOe

10

170

180

190

T, K

Fig. 6.1. The H - T phase diagram of an epitaxial film of Y2.6Gdo.4Fe3.9Gal.lO12in the vicinity of the compensation temperature (H < 15 KOe). The easy axis of magnetization is parallel to the normal ff of this film and ~rlla. The symbols (o) correspond to appearance and disappearance of the low-temperature coUinear phase during heating of the sample. The symbols (o) - to appearance and disappearance of the high-temperature collinear phase during cooling. The data (e) and (o) were determined visually, the data (z~) were determined by Faraday effect measurements, the rectangles correspond to threshold fields; the solid lines correspond to theory; after Gnatchenko et al. (1977). corresponding inhomogeneous behavior. At T[ and T~ the whole sample switched into the canted phase. Different phases distributions in the film thickness correspond to different temperatures and magnetic field ranges. For H > H* the film has adopted the canted phase with the orientations of the sublattice magnetizations varying along the thickness direction. In the temperature ranges from T~ to T~' and from T~ to T~' when going from one range to another, the collinear and canted states arise alternatively. For He < H < H* the magnetic structure of the film represents a mixture of collinear and canted phases. For H < He only the two collinear phases (low and high temperature) exist and the transition between them is of the first order. Figure 6.3 shows phase diagram of a Y2.3Gd0.4Fe3.9Gal.lO12 film in a wide field range up to 60 kOe. The critical fields here are obtained by means of extrapolations of the observable temperature dependencies of the manetooptical rotation angle. Examples of such extrapolations are depicted on curves 2 and 3 of fig. 6.2. The

452

A.K. ZVEZDIN 1.0

~o,deg

0.5 0

0/

-0.5

i

i

i/

~

i

1160170,//190200 _1.0 ~ _ _ . ~ 2 1

t

-1.0 I

I

I

I

I

I

I

160

170

180

190

200

210

220

T, K

Fig. 6.2. The temperature dependencies of the Faraday rotation of an epitaxial film of 1 - H = 1 kOe, 2 - 10 kOe, 3 - 45 kOe; after Gnatchenko et al. (1977)

Y2.6Gdo.4Fe3.9GaI.IO12in different magnetic fields:

H, kOe 60

\

\

12

,0 " ,\\ ,\ ',\

/

/// /

// //

40

20

0

c~ ~ a

/1

\

,

/

-10

0

10

d

f

-30

-20

20

(r-ro),K

Fig. 6.3. The 'reconstructed' high-field phase diagram of the epitaxial film Y2.6Gdo.4Fe3.9Gal.1012 with Tc = 183 K; broken lines correspond to different theoretical models; after Gnatchenko et al. (1977). critical fields are defined by the bend points o f the extrapolation curves and they correspond to the following film parameters averaged over the thickness: Tc = 183 K and Hc = 2.5 kOe. Let's note that Gnatchenko et al. (1977) e m p l o y e d three sublattices model o f Nrel ferrimagnet for processing the experimental data. Lisovskii (1980) and Dikstein et al. (1980) have studied F I P T in single crystal films with the aim to elucidate the effects o f its layered nature. Experiments were produced in the high magnetic stationary fields up to 150 kOe. A number o f the films

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

453

with different thickness and crystallographic orientations having a different number of layers (from 5 up to 20 ) were studied. Figure 6.4 shows the H - T phase diagram of the film (YGdYbBi)3(FeA1)5OI2 made up from 19 layers at ~lln where ~ is the easy axes (see also fig. 4.2). The compensation temperature changes in the layers 3-18 from 201 K up to 207 K (the layers 1, 2 and 19 don't have compensation temperature). External boundaries of the canted phase are shown in fig. 6.4 by the solid lines (H > 36 kOe). Broken lines show schematically the same boundaries for the layers 3 and 18. The pattern of vertical lines (compare with the curve P T c in fig. 4.2) at this diagram is caused by the distribution of the compensation temperature in the layers. This pattern was observed by means of domain structure transformations. The threshold field, where three phases - two collinear and one canted - coexist, is spread out here into the shaded area in the diagram (in the region of H* ~ 34.836 kOe and T ~ 201-207 K). In the studied films of (YGdYbBi)3(FeA1)5012 one has Ha ~ 2 K u / M s "~ 300 Oe, HE '-~ 106 Oe and H* ~ 30 kOe which agrees sufficiently well with the experimental data. Figure 6.5 shows H - T phase diagram of a film of (YGdYbBi)3(FeA1)5012 for /1_LEA (easy axis) which should be compared with the theoretical diagram shown on fig. 4.1 (see for details section 11). A peculiar feature of the film geometry here is that the second order transitions from collinear state into the canted phase H, kOe

I

I !

/ //// /

I

60

i

~,,.

/

\\

~ 20

0

/

II\

80

///

4

I

I

/

"a~" 0~"i~ , ~ * . "

I

I

~

!

I

I

I

I

I

192 196 200 204 208 212 216 T, K Fig. 6.4. H - T phase diagram of a ferrimagnetic film of (Y,Gd,Yb,Bi)3(Fe,AI)5012 made up from 19 layers in a magnetic field Erll,~, where a is the normal to the film. The easy axis of magnetization is parallel with ~ in this film. High field solid lines are the external boundaries of the canted phase (at H > 36 kOe). Broken lines are the same for the layers 3 and 18. The pattern of vertical lines is caused by the distribution of the compensationtemperaturein the layers (comparewith fig. 4.2); after Lisovskii (1980).

454

A.K. ZVEZDIN

H, kOe D, n l m -1

150

300

~

100

50 -100

0 200

_

~_

250

300

350

400 T, K

Fig. 6.5. The H - T phase diagram of an epitaxial iron garnet film of (Y,Gd,Yb,Bi)3(Fe,A1)5012at H_I_g, where ~ is the normal to the film being parallel with the easy axis of magnetization of this film. The lines He(T) (A) are the boundaries of the canted phase (see fig. 4.1). Temperaturedependencies of the inverse period of the domain structure Do I at H = 0 are displayed together with the inverse critical period of the domain structure Dc I (o) and the critical field Hc (A) of the film; after Dikstein et al. (1980). follow behavior of the 'soft mode' (Dikstein 1991). The soft mode in this case is the spin-density wave transformed into an ordinary domain structure away from the transition point. There are interesting investigations of H - T phase diagrams of amorphous D y - C o films in the vicinity of the compensation point in stationary magnetic fields up to 150 kOe (Fisch et al. 1986, Khrustalev et al. 1989, 1993a, b). The topology of these phase diagrams qualitatively well agrees with the theoretical ones, considered in section 4.1. However, there are some features inherent in the amorphous nature of these films (e.g., the possible existence of an asperomagnetic state).

7. Some general features of field induced phase transitions The phase transitions considered above are typical transitions with a magnetic symmetry change. For instance in the first case of section 4.2 (H_I_EA (easy axis)) during the I-III transition the symmetry is relative to a rotation around the z-axis with an angle 7r, i.e. it is the symmetry element C~ (in values). This symmetry element is absent in phase III but the double number of equilibrium states in comparison with phase I is present here: ~ = 0 and ~ -- rr, which are transferred from one to the other by the 'broken' symmetry element. Identical values of the free energy ('degeneration') correspond to these two states. It is accompanied with a division of the sample into domains of a low symmetry phase. In the case Hllg (section 4.1.2) distortion of the continuous symmetry C ~ takes place during the transition into the canted phase. Such distinction of a 'break' of

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

455

the continuous or discrete symmetry elements are especially important for dynamical properties of the system. A change of sign of coefficient a(H, T) in the Landau expansion of the free energy (see, e.g., section 4.1) causes peculiarities in the behavior of many physical properties near the transition point: • the susceptibility goes to c~. In this case the susceptibility describes the response of an order parameter to the thermodynamically conjugated field (see section 8). • the occurrence of anomalies thermodynamical magnitudes quantities such as kinks, jumps and )~-curves. The heat capacity, specific heat, magnetocaloric effect (see section 9), Young modulus and sound velocities, magnetostriction (section 10), and magnetooptical phenomena. • the conversion of the order parameter oscillation frequency to zero (soft mode) and the hindering of its relaxation. • the increasing order parameter fluctuations and their correlation radius • the expansion of domain walls and the rearrangement of domain structure in the sample Many theoretical studies devoted to the field induced phase transitions have been carried out using the mean field theory (or using equivalent approximations as the Landau theory). These theories when constructing the free energy of a system neglect, to some extent, the fluctuations of the order parameter. In the region of the transition temperature, i.e. in the region where the system stability is lost, the fluctuations increase strongly and these theories become inapplicable. A characteristic feature of the studied phase transitions is that the Landau theory can be used for their description with practically no limitations. The region of inapplicability becomes extremely narrow AT ~ 10-6-10 -8 K. This is a consequence of the fact that the fluctuations that occur in the region of the transition have a very large value of the correlation radius. We will consider this problem using the example of the orientation transition investigated above. To study critical fluctuations we use thermodynamical potential (3.1), taking additionally into account the energy of magnetic nonuniformities. For small values of 0 we have

F = f dv{~O 2 q-/304 + A(grad0)2),

(7.1)

where the constant o~(T) can be represented in the form ~(T) = at, t = (T - Tc)/Tc. Suppose the fluctuation in the angle 0 is 30(~, and its Fourier transform is 30~. It is well known that the mean square fluctuations are given by

1~0~12 =

T , V(at + Aq 2)

t>0;

T , t
456

A.K. ZVEZDIN

An inverse Fourier transformation of this equation gives the correlation function of the fluctuation of the order parameter

9(r3 = 60(0) 0(r) -

T e_r/p, 47tAr

where

p =

( A / a t ) /z

t>0;

(A/Zat)U2,

t<0,

is the correlation radius, which tends to o0 at the phase transition point. The unlimited increase in the uniform fluctuations when approaching the phasetransition point indicates that the Landau theory is not applicable in the immediate vicinity of the transition point. The Levanyuk-Ginsburg criterion, which defines the region in which the Landau theory is applicable, has the form

3)2 TcZ/ 2 t>>¢=

A3a

(the temperature is measured in energy units). An estimate of the value of ( using this formula for typical values of the parameters p / a ~ 0.1, Tc ~ 100 K, A ~ 4 x 10 -7 erg/cm, and a ,~ 104 erg/cm3 gives ( ~ 10 -8. For comparison we give the value of the corresponding quantity for a transition at the Curie point (c g 10-1-10-2. We will give once more the characteristic values of the correlation radii for field induced phase transitions Pzi and transitions at the Curie point Pc (with the same parameters) 3 × 103 PFI ~' - A,

(3-5) PC ~ A.

The 'orientation' fluctuations are more long-wave than the fluctuations near the Curie point. This explains, in particular, their small contribution to the free energy (the statistical weight of the long-wave fluctuations is small, while the short-wave fluctuations are strongly suppressed).

8. Magnetization and susceptibility 8.1. Differential susceptibility

The magnetization of ferrimagnets in collinear and noncollinear phases is defined in a usual way as Mt -

--

OH

- ~(O)H + md (1 -- A~(0)) cos 0,

(8.1)

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

O(H,T) is

where ~ is determined by formula (3.1) and In the collinear phases 0 = 0, 7r we have Mt = Md - Mf sign(AMa - H), Mt = Mf -

Md,

457

taken from eqs (3.9), (3.10).

0 = 0,

O = Tr.

In order to obtain the value of Mt in a canted phase let's express function x(O) in terms of A, K, f(O) from the section 3.2 and substitute x(O) into (8.1). Then we obtain

Mr=

Mf-Md H

K

~+(H-AMdcos0)~

( f° "]

for0:Tr,

\sin0J

for0<0<~r,

Md - Mr sign(H - AMd)

for 0 = 0.

This formula is simple for the case when the transition into the canted phase takes place in a strong magnetic field H > AMd and the influence of anisotropy can be neglected. Then

{ mf- md M~= A mf+md

f o r H < Hcl, for Hcl ( H < Hc2 , for H > Hc2,

where Hcl = )~(mf - md) , He2 = /~(mf q- md) (see section 2). The terms of the

order K/HMd << 1 are neglected here. The longitudinal differential susceptibility is equal to

dmf

for 0 = 0, 7r,

dH ;

(8.2) 1 + MOOd'-H

H

sinO ,/J

for0<0
In the case of the strong critical fields considered above we have Xd = 1/A in the canted phase. The susceptibility Xd displays a jump at the transition from the collinear into the canted phase 1 AXd- A

Xf.

Thus, the isotherms of the magnetization Mr(H) show to a kink and the longitudinal susceptibility a jump at transitions into noncollinear phases. This fact can be

458

A.K. ZVEZDIN M

I

I

I

I

21OK// f

iH1DY3Fe5012 I

0

I

50

M

I

I

I

100 150 200 H, kOe I

I

I

100

Ho3Fe5012 I

0

I

50

M

///

I

6j

I

7

I

100 150 200 I4 kOe I

~

I

[

IH1 _ Er3FesO12

0 50 100 150 200 /4 kOe Fig. 8.1. Field dependenceof the magnetizationof Dy3Fe5012iron garnets; after Levitin and Popov (1975). employed for the experimental determination of the critical fields. Particularly, by such methods the critical fields in ferrite garnets of Yb, Gd, Ho, Dy, Er (Clark and Callen 1968, Feron et al. 1971, 1972, 1974, Levitin and Popov 1975, Fillion 1974) and in some hexaferrites (Sannikov and Perekalina 1969) have been found. In fig. 8.1 the fields dependence of the magnetization in ferrite garnets of Dy, Ho and Er is shown. The linear parts Mt(H) describe the magnetization in the collinear phase and the curved parts describe the magnetization in the noncollinear phase. Bend points (or the kinks) in the magnetization curves determine the values of the transition fields from collinear into the noncollinear phases.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

459

A large body of research was devoted to the high field magnetization of ferrimagnetic f - d intermetallic compounds. Many results were obtained by using orientedp o w d e r samples because single-crystalline materials are not always available (see, for instance, Sinnema et al. 1984, Buschow et al. 1985, 1989, de Boer et al. 1987, 1988a, b, 1990, Radwanski et al. 1989a-c, Verhoef et al, 1989, 1990a-c, Z h o n g et al. 1990a-c, Date 1990, Franse 1990, Liu et al. 1991a-c, Zhou et al. 1992a, b). Figure 8.2 shows, e.g., the typical curves for several 2 - 1 7 c o m p o u n d s clearly displaying the transitions from the collinear into the canted phase (Verhoef 1990). The effect of anisotropy on the Mt and especially on the Xd behavior can be essential in the low field part of the phase diagrams and can give rise to additional anomalies. The region of canted phase narrowing (the 'narrow throat' of phase diagram, see, e.g., fig. 4.1) is concerned to be the most 'dangerous' on that count. In this area the angle rapidly changes with variation of the field and temperature which results in anomalies in Xd. In fig. 8.3 temperature dependencies of Xa calculated by formula (8.2) for and

,qll~

/~_l_g when H << AMd are depicted. The following analytical formulas correspond to them. F o r m u l a (8.2) for Xd when 2K/Md << H << ),Md can be rewritten in the form

1 ~

1

2K . ___ (3+ H2- H 2

H2 '/

H 2 + H .2

H2

3-

cos01

(8.3)

cos 20

for/~_l_g,

where H* is defined by formula (4.9). For cos 0 we used eqs (4.4) and (4.7) for /~IIEA and H_LEA, respectively. 100

~

I

~

I

D

t ........ ~'7" .,.;:'~,,''" ....

80 "~ 60

Q . g l . e . t 3 . ~.t~.fi]..Ei.E]..[]......~......l~.~ } .......... ~ . , , _ t n ................

~

e ............ [].................,..~.....

•

,.o"

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

_:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::..............~,.............

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

.,~.............. ....."~'.......... a ~x..a..z~.~..,--~-~..a.-~..-...~.....-~..~...........,s.......... ,~............. aH°2C° 17 _~.e.,o.~,~,~,.~,.e,,~.~,.--.-.c,.-...-e,,e-.........~, ....-~>'0.............. vHo2Col4Fe 3 20 °Er3C°17 c~Er2Fel7 ODy2COl7 0 I i I I I i I 0 10 20 30 40

40-

B [T]

Fig. 8.2. High-field magnetization curves at 4.2 K of several R2M17 single-crystalline spheres that are free to orient themselves in the applied magnetic field; after Verhoef (1990).

460

A.K. ZVEZDIN

Xd

x IIEA

±

\

IEA ,

0

Te

T

Fig. 8.3. Temperaturedependence of the differentialmagnetic susceptibilitycalculatedaccordingto the formulas (8.2), (8.3); after Bisliev et al. (1973). Notice that the canted phase exists only when H > H* and HI]g. Therefore it is reasonable to consider the case H > H* in the expression for Xdll" The solution of these equations for AM >> H > H* can be represented in the form T-

cos 0 =

Tc H m d

H .2

Tc 2 K H 2 _ H .2 T - Tc H M d H .2 - . Tc 2 K H 2 + H .2

for/1H EA,

(8.4) for/~_I_EA.

It should be noted that the anisotropy part of to Xd changes its sign at H = H * / v ~ . Formulas (8.3), (8.4), and fig. 8.3 form a representation of the effect of the anisotropy on the longitudinal differential susceptibility behavior Xd. We do not present here analogous (but more cumbersome) formulas for Xd for a case of cubic anisotropy. Let us note only that in this case anomalies are possible not solely on the boundaries separating canted and collinear phases but inside canted phases as well. For example, Xd -+ oo in the vicinity of the critical point 0. In fig. 4.3 for HH[001] when H tends to a critical point from above (since dO/dH ~ c~ in that case). Susceptibilities Xd(T) of intermetallic compounds ErFe3, HoFe3 measured on the polycrystal samples (Bisliev et al. 1973) are shown in fig. 8.4. The anomalous behavior of Xd(T) in the vicinity of the compensation point is attributed to the noncollinear structures in this field. It should be noted that magnetic anisotropy in intermetallic compounds is much larger than the anisotropy of rare-earth-ferrite garnets (close to Tc it accounts for about ~ 106 erg/cm 3) (Clark et al. 1974). Therefore the effects of anisotropy on magnetic phase diagrams and the anomalies of physical properties are of greater significance in these compounds than in rareearth-ferrite garnets. Opposite, to some extent, is the example represented in fig. 8.5 (Gurtovoy et al. 1980). They studied the differential magnetic susceptibility of the Y3-xGd~FesO12 iron garnet system (0.01 ~< z ~ 0.2) in magnetic fields up to 50 T at temperatures

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

461

Xd,10-3 cma/g / (2) (1,4)[

3 ",,,x\

(3)

a ~~ x / x~ .

1.0

4

0.1 -0.15

0.5

0.05- 0.1

0

0 -0.05

0 300

350

400

450

500 T, K

Fig. 8.4. Temperaturedependence of the differentialmagnetic susceptibilityof HoFe3 in the vicinity of the compensation temperature for: 1) H = 11 kOe, 2) 9 kOe and 3) 1.5 kOe; and 4) of ErFe3 for H = 11 kOe; after Bisliev et al. (1973).

X.IO 4

~__.4 3

I MFe MGd

41

jl

'v,/ I

3" 40 50 H, T 10 20 30 The field dependence of the differential magnetization of (GdY)3FesO12 iron garnet; (o) Fig. 8.5. experiment, solid lines - theory; after Gurtovoyet al. (1980). between T = 186 K and 4.2 K. Transitions from the ferrimagnetic to the noncollinear phase and from the noncollinear to the ferromagnetic phase are observed. These examples illustrate that the 'universal law' Xd = 1/A for the canted phase is appropriate in the strict sense only in the case of isotropic ferrimagnet. In section

462

A.K. ZVEZDIN

16 an example will be given where this law also holds in the anisotropic situation (in the cone-canted phase).

8.2. The temperature hysteresis of the magnetization. Hall and Faraday effects A temperature hysteresis of various physical quantities may arise during a transition into the noncollinear phase. Diagrams depicted in figs 4.1-4.4, 4.6, 4.8, 4.9 show that the transitions between some ferrimagnetic phases become first order phase transitions when the anisotropy is included. This can result in a temperature hysteresis of the Faraday effect and the magnetization, which is attributed to the presence of a phase coexistence area in the vicinity of the first order phase transition line. Such effects become the strongest in weak magnetic fields close to the compensation temperature Tc. Let us consider the ferrimagnet magnetization behavior close to Tc in the scope of model described above. Let HrlEA. It is seen from the phase diagram (fig. 4.2) that a first order phase transition occurs on curve TcP (Belov and Nikitin 1970, Zvezdin and Matveev 1972a, b). As the temperature decreases, the transition I-II proceeds in a jump like on the curve A'Q and reverse transition proceeds on the curve PBq Consequently the overall magnetization in these phases depends on T in two different ways: Md -- Mf(AMd - H) = -Ms(T) + xfH Mt = ]. Mf(AMd + H) - Md Ms(T) + xfH

for 0 = 0, for 0 = ~r,

where Ms(T) = Mf(AMd) - Md is saturation magnetization of the ferrimagnet for H = 0, Mf(AMd + H) is magnetization of the rare earth sublattice in an effective field AMd + H, Xf is its susceptibility. Taking into account that Ms(T) can be represented in the vicinity of the compensation temperature Tc in the form

T-To

Ms(T) = Md - - ,

Tc

we obtain (Zvezdin and Matveev 1972)

--Md Mt =

IT-Tel Tc

-I- xfH

for T > TI(H),

IT_ Tc[

Md

Tc

+ XfH

for T < T2(H),

where Tl(H) and T2(H) are determined by the stability lines A'Q and B'P in fig. 4.2. In the case of weak magnetic fields they can be approximately defined in the following way:

HMd T1,2(H) = Te 1 q: 2K //

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

463

Ms

\

J

J

T

Fig. 8.6. Temperature hysteresis of the magnetization of an uniaxial crystal for HIIEA (easy axis) in the vicinity of the compensation temperature; after Zvezdin and Matveev (1972b). cr~ Gcm3/g 0.8

0.6

0.4

0.2

I

\

\

I

^

480 t

-0.2 ~-

~

-\

500

d

520

~K

"o"

Fig. 8.7. Temperature dependence of the magnetization of the ErFe2 compound in the vicinity of the compensation temperature; after Belov et al. (1972b). or more precisely by equations (4.13), (4.14). The butterfly-like temperature hysteresis loops are shown in figs 8.5 and 8.6. Such loops have been observed experimentally in the temperature dependence of the remanent magnetization in ErFe2 near Tc (fig. 8.7, Belov et al. 1972b). Notice that the very unusual two-peak temperature dependence of the coercive force of

464

A.K. ZVEZDIN He, kOe C

+MR

a)

i

+Mo

i t---~l, . . . . ! t

10

20 I

'

i i

I

i

J30 J

x.x~

r, K

'

Fig. 8.8. The temperature dependence of the coercive force a) and magnetization b) of the Er0.sDY0.mFeO 3 in the vicinity of the compensation temperature Tc ,-~ 25 K; after Belov et al. (1979).

polycrystalline Gd3Fe5012 iron garnet observed by Belov and Ped'ko (1960) can be explained by similar model descriptions (Goranskii and Zvezdin 1969a). The butterfly-like hysteresis curves are rare in nature. Figure 8.8 shows very distinctive butterfly-like hysteresis of the magnetization near the compensation point (Tc ~ 25 K) in the Erbium and Terbium orthoferrites ErFeO3, TbFeO3 (Derkachenko et al. 1974, 1984, Belov et al. 1979). There is an analogy of the hysteresis of magnetization considered here and the butterfly-like hysteresis of the linear effects in the antiferromagnets and weak ferromagnetics (linear magnetostriction and piezomagnetism (Borovik-Romanov 1959, 1960, Zvezdin et al. 1985), linear magnetoelectric effect, linear birefringence (Kharchenko et al. 1978, Rudashevsky et al. 1977, Merkulov et al. 1981)). All these effects are caused by the hysteresis of the antiferromagnetic vector/~. A similar vector in the case of ferrimagnets considered here is /~ =- ]~ff - f~d. In f-d ferrimagnets the orientation of the vector/~ is fully determined by Md (see section 3). Therefore the vector /~rd can be used for the analysis of behavior of different physical properties in the vicinity of Pc. The hysteresis of the vector/~ or Md in ferrimagnets near the compensation point leads to many anomalies in the behavior of the physical values which are proportional to L, e.g., of the kinetic effects. This is the situation with the galvanomagnetic effects (Hall effect and magnetoresistance) in the vicinity of the compensation point. These effect have been studied in ferrites (Belov et al. 1960b, 1961), in amorphous rareearth-Co(Fe) alloys (Asomoza et al. 1977, McCuire et al. 1977, Okamoto et al.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

465

1974, Ratajczak and Goscianska 1980), in the intermetallic compounds MnsGe2 (Levina et al. 1963, Novogrudskii and Fakidov 1964, Vlasov et al. 1976, 1980, 1982), and in amorphous Dy-Co films (Khrustalev et al. 1989, 1993a, b). The temperature hysteresis and other features of the Faraday-effect in ferrite garnets in the vicinity of Tc have been investigated by Smimova et al. (1970) (see also Belov et al. 1979, Krinchik and Chetkin 1969, Eremenko and Kharchenko 1979). The effects connected with this first order phase transition in amorphous films in the vicinity of the compensation temperature have been studied by Makarov et al. (1980), Kandaurova et al. (1982, 1985), Fisch et al. (1986), Popov et al. (1990). Theoretical papers by Turov et al. (1964), Schavrov and Turov (1963), Schavrov 1965, Turov (1987), Zvezdin and Matveev (1972a, b) are devoted to this problem.

9. Thermal properties in the vicinity of the spin-reorientation phase transitions 9.1. Magnetocaloric effect The entropy of ferrimagnetic system (in the scope of the model studied) can be expressed by

S-

~¢T - OT ~ fo He" Mf(x)dx, O

(9.1)

where the quantity ¢ is given by formula (3.1), and where O(H,T) is defined by eq. (3.9). The assumption of a saturated d-sublattice has been made here, i.e. OMd/OT = 0 and ~ / ~ 0 = 0 in equilibrium conditions. Equation (9.1) in conjunction with (3.9) determines the isoentropic region in the (H, T) plane and henceforth the magnetocaloric effect, i.e. the variation of the sample temperature during adiabatic magnetization. It is defined usually by the isoentropy slope, i.e. by the value

(OS/~T)H

CH

(

OMH ~ ,

where C H is the specific heat of the s_ystemfor H = const, MH is the projection of an overall magnetic moment on the H direction. Substituting the value MH = M(H) from (6.4) into the above formula, we get

T DMf CH1 8T

CH2 AHMa dT T

for 0 = 7r,

(H - AMd cos O) --sinf°O

~Mf H - A M a

CH3 ~T I H -

AMd]

for 0 < 0 < 7r, for 0 = O.

466

A.K. ZVEZDIN

These formulas and (3.9) and (3.10) simultaneously determine the magnitude of Furthermore, it is easily seen from (9.3), to be equal to zero in the canted phase in the isotropic case since K = 0. It follows at once from the fact that in this case the magnetization Mt = H/A is independent of T (see section 2). Notice that Belov et al. (1972a) have obtained more detailed formulas for ferrimagnets with cubic anisotropy. By contrast, (dT/dH)s > 0 in the low temperature phase {0 = 7r), i.e. an increase of the temperature of the sample during magnetization takes place whereas (dT/dH) < 0 in the phase {0 = 0}, i.e. cooling takes place. This property has an evident physical meaning. The external field is directed in parallel with the exchange field acting on H rare earth ion in the phase {0 = 7r) so that the Zeeman splitting of its ground multiplet is enhanced with increasing field. As a result more of the lower levels become occupied (entropy of the system decreases). The energy released at such a transition heats up the sample. The situation becomes reversed in the phase

(dT/dH)s.

{0

=

0}.

A characteristic property of the second order phase transitions is the kink in the isoentropic curves, i.e. the jump of (dT/dH)s at the boundary between the phases. Experimental data for some compounds are shown in fig. 9.1 (Belov et al. 1970a). It is seen that (dT/dH)s = 0 with a good accuracy in some phase of Gd-ferrite garnets. This value perceptibly differs from zero in the noncollinear phase of Ho and Dy ferrite garnets. This is attributed to the large value of the anisotropy energy inherent in these materials. The sign change of (dT/dH)s is also well-defined at the transition between the phases {0 = 0} and {0 = 7r} in fig. 7.1. Similar dependencies of T(H) have been observed in Gd-ferrite garnet, Yb-ferrite garnet and in the mixed Gd/Yb-ferrite garnet as well in the work by Clark and Callen (1969). The isoentropic curves show discontinuity on the boundary of the first order phase transitions.

9.2. Specific heat The magnetic part of specific heat may be derived by differentiating formula (9.1):

dS C H : T ( - ~ ) H.

(9.3)

In order to obtain analytical expressions let us define a particular form of the dependence of the magnetization of the rare-earth sublattice on temperature and magnetic field

(9.4) Such an approximation is valid in the vicinity of the compensation temperature (and also above) and can be considered as an expansion of the Brillouin function.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS A7 10-2 K

A7 10-2 K 5.0

Gd3Fe5012

rr°"o 283.21 K

~ 220.04 K ,.~_~:,- 228.19

Dy3Fo,o12f

2.0

~/j~-E85.n , ¢ , ~ ~

2.5

~-.~'_ -

285.47 285.65 285.89 288.07

_ _1~_ _ _ _ ~ _

~ - - ~ 7 °~~ _

H, KOe

288.47 288.71 r 287.12

-2.5

467

220.34

~ : ~ : ~ 1.0 0

~ ~

220.47 220.60

I

I

I

H, KOe -1.0

~ 287.32 -5.0

-2.0

a)

b)

AT, 10-2 K 7.5

.72

a/137.57 K

Ho3FesO12/ 1.9

2.5

x~_ ~,.~-o~ooo

0

~

"

~ 10

15

-~-~--o~,<>~a~ -2.5

~

138.65 138.78 138.87 KOe

138.96

noq39.13

-5.0 -7.5

%xab'Q'~-,Ri ~ ~

c) Fig. 9.1. Magnetocaloriceffect in a) Gd3FesOl2, b) Dy3FesO12, c) Ho3FesO12 in the vicinity of the compensation temperature; after Belov et al. (1970a). For example, we have a = 3/7, b = 67/1029, /zf = 7/ZB, Mg = 21#B for Gd-ferrite garnet. Substituting (9.4) into (9.1) yields

aMf°~,f

S - - 2T 2

(2H)~Mo cos e - H 2 - k2M~).

(9.5)

468

A.K. ZVEZDIN

Only the first term of the expansion in (9.4) is presented here. The error allowable in such an approximation for Gd3FesO12 at T ~ Tc is lower than 5%. Replacing this value in the formula for the specific heat Cr~ we obtain Mf°#f a- (H -/~md) 2,

for 0 = 0,

T2

M% a

CH=

-

(H 2 2HAMdcos 0 + AZM~+

-

\

T2

+AMdHT dcos______O0) dT

M%f

a - T2

(H + AMd) a,

for 0 < 0 <

J ' for 0 = 7r.

The magnitude O(H,T) is determined here by the equation of state (3.9) for the corresponding canted phase. Let us consider for example the case of cubic anisotropy HIll100]. The specific heat changes in a jump like during the transition from the collinear phases into the canted phases on the curves AA' and BB' (lines of the second order phase transition, see fig. 4.1) a2 M ° AC = - -

H2

2b /zf H a + 4 H , 2

,

(9.6)

where H* is given by formula (4.9). Formula (9.6) predicts extremely curious behavior of the specific heat CH in a canted phase. Schematically the CH(T) behavior at various values of H is represented in fig. 9.2 (HII[001]). cn _

_

~

H > H* H - H*

T Fig. 9.2. Schematic representation of the temperature dependence of the magnetic contribution of the specific heat of a ferrimagnet of cubic symmetry in the vicinity of the compensation temperature for HIll001] (see fig. 4.3); after Belov et al. (1979).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

469

Cp, 1N -2 g.K ca] ~" w~ coo ' d

,,'

,

It

"'"'"'"'"'""'""'"'"'"'"'""'"

14

ttl

,

,,,

I¢¢s

,.,

,

H=20.4k0e %

,.........,,,,,.,,,,,,v,.....

/,

,;

°¢r # I~

It .,~. ,,,# i t , .......... "'". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

,i¢•II'

14

,,,",I

~',

16.4

' .......... ' " .......

.,.~ '1fl. t. . . ,l ,,'

*,

' ,~Ii ,~ ,1

'\ ,.,,s ¢S I I~

11o11%6 ¢tllllllll I• .¢ I

IB fill III"

"'" ..... " " " ......

16

1,

l

s,

,"'

i • •,~

14

o

'tl°o

•1•o¢p

I

Ii i iii I i • " lllll

0

II

Ill

6.0

llllllli•llllllllllllll~l

Ii IIIDII I#I011111III •

t I ii • |llle•l•

2.0

~1111

........... ". . . . . .

0

14 I

I

I

I

280 290 300 310 T, K Fig. 9.3. Temperaturedependencies of the specific heat of the Gd3Fe5012 at different magnetic fields, grll[100]; after Kamilov et al. (1975). The peak of the specific heat in the vicinity of the critical point O is of interest (see fig. 4.1). When this point is approached we have d cos 0 -

-

--+

OO.

dT Let us note that the peaks of the specific heat in the vicinity of the critical curves and critical points represented in fig. 9.2 have not a fluctuational nature but have to be attributed to strong temperature dependence of the order parameter O(T). Generally, the region situated close to the phase diagrams 'throat' is of the most interest from the point of view of observation of unusual specific heat dependencies (and other physical parameters) since there is the strongest O(T) dependence. Anomalies of the specific heat in Gd ferrite garnet for HI[[100 ] in the vicinity of Tc have been observed by Kamilov and Schachschaev (1972), Kamilov et al. (1975), as shown in fig. 9.3. The data are in qualitative conformity with the phase diagram represented in fig. 4.3. We note that the comparatively weak magnetic fields considered in these investigations are the most difficult ones for theoretical analyzes since domain structures play an essential role here. Kamilov et al. noted that (dT/dH)s ~ 0 in the canted phase. This experimental result is also consistent with the theory. 10.

Magnetoelastic

anomalies

I0.I. Magnetostriction The noncollinear magnetic structures in rare-earth-ferrite garnets are accompanied by anomalies of the magnetostriction (Belov et al. 1969, 1970b, 1972a, Levitin

470

A.K.ZVEZDIN

et al. 1970, Popov 1971, Levitin and Popov 1975). If the magnetostriction is measured along the field direction then only longitudinal component of the anisotropic magnetostriction and the bulk magnetostriction (magnetostriction of paraprocess) contribute to the measured value in the collinear phases (if the magnetic moments of the sublattices are oriented parallel or antiparallel with field direction). Transverse components of the anisotropic magnetostriction also arise. Since longitudinal and transverse components of magnetostriction have the opposite signs, the derivative of anisotropic magnetostriction, with respect to the field, changes its sign during the transition from the collinear phase into the noncollinear one. Besides, the bulk magnetostriction of the paraprocess changes in anomalous way as well during formation of the noncollinear structure since it is dependent on the magnetization of the sublattices and as it was shown above, the magnetization of the rare earth sublattice depends in a different way on the field for collinear and noncollinear phases. As a consequence of this anomalous magnetostriction, its variation in ferrite garnets during the transition into the noncollinear phase has a much sharper character than the variation of the magnetization, and minimums (or maximums) appear in the field dependence of the magnetostriction curve at fields corresponding to the critical fields of the transitions. Let us consider in more details the field dependence of the magnetostriction of ferrite garnets at the transition into noncollinear phase. The anisotropic magnetostriction is supposed to be a single-ion effect and the bulk magnetostriction is to be attributed to exchange interaction between the rare-earth and Fe sublattices. Then the overall magnetostriction of ferrite garnet polycrystal can be represented by means of usual relations for magnetostriction (Belov et al. 1979) 1

All - 21 /~Fe(3COS2~)Fe_ 1)+~ /~R(BCos2~R- 1)+ (10.1)

+ aA(MRMFeCOS(0R+ 0Fe)+ MRMFe). Here the first two terms describe anisotropic magnetostriction of Fe and rare-earth sublattices, ~bFeand ~bR are the angles associated with the directions of the sublattice magnetizations and the direction of the magnetostrictive deformation measurements. It is necessary to take into account the field dependence of magnetostriction constant AR of rare earth sublattice attributed to the paraprocess. In the single ion approximation such dependence has a form of AR =

A°]5/2[L-I(MR/M°)],

where i5/2(x) is the reduced Bessel function, L -1 is the reciprocal Langevin function. The third term in expression (10.1) is the bulk magnetostriction of the paraprocess. The coefficient ax is related to the dependence of the exchange interaction between the sublattices A upon the tension strains cri~ as follows dA d~xx

dA d~yy

dA d~zz

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

471

OR and 0Fe are the angles between the sublattices magnetizations and the field, which are equal to 0 or 7r in the collinear phases and can be found from relations given in section 2. In sufficiently strong fields when the isotropic ferrimagnet approximation is valid one may write (HclHc2 -[- H 2) COS0R =

2H MR MFeA ' (-HclHc2 + H 2)

COS 0Fe =

2HMRMFe~

If the magnetostriction is measured along the field direction then ~Ve = 0Fe, ~R = OR and ,~111 :

/

T

111

1 1 /~Fe(3 COS0Fe -- 1) + ~ -~R(3 COSOR -- 1)+ 2

+ a,X(MRMFe COS(0R + 0Fe) -[- MRMFe). It is easy to obtain the field dependence of the various contributions to the magnetostriction (Levitin and Popov 1975) by means of presented above formulas. Let us note that the magnetic field dependencies of the rare-earth sublattice magnetostriction in the ferrimagnetic phase above and below the compensation temperature are different. This difference is attributed to the decreasing rare earth sublattice magnetization by the field above the compensation temperature and to the increasing magnetization below the compensation temperature. Figures 10.1, 10.2 show field dependencies of longitudinal magnetostriction in Gd, Tb, Dy and Ho-ferrite garnets close to the compensation temperature in fields up to 250 kOe (Levitin and Popov 1975). The magnetostriction of explored garnets anomalously depends on the field in this temperature range: maxima (and minima) appear in the A1/l(H) curves for some values of the fields; these fields correspond to the critical fields of transition into noncollinear phases. Similar field dependencies of the magnetostriction in the vicinity of Tc have been observed for the intermetallic compound ErFe3 (Nikitin et al. 1975). Popov (1971) provides numerical computations of the theoretical field dependence of the magnetostriction in Ho-ferrite garnet (formulas (10.1), (10.3)) when fieldinduced noncollinear magnetic structure arise. Comparison between the computational results and the experimental field dependence of Al/l is made in fig. 10.2. Taking into account approximate character of calculations the obtained agreement between theoretical and experimental dependencies Al/l(H) can be considered as satisfactory.

472

A.K. ZVEZDIN (A///)n

(A///)n I

(xl0 -6) 4

I

t

~

0

I

'

292 K 289

36K

(×10-6) 40

240

~286

30

20 -16 10 -24

v 0

100

200 h kOe

Tb3FesO12

o'

' 100

2 0 H kOe

(A///)II

(AV0H

i

'

'

'

(xl0 -6) 20

(×10 -6 ) 20

0

0

-40

-40

-80

-80

-120

-120

DY3Fe5OI~

~//Ho3Fe5012N~~ " I

0

100

i

~24 t

200/-J kOe

0

I

100

I

T

200 H, kOe

Fig. 10.1. The longitudinal magnetostrictionof the rare-earth-iron garnets in the vicinity of the compensation temperature(after Belov et al. 1970b, Levitin and Popov 1975). 10.2. Thermal expansion The appearance of noncollinear magnetic structures leads also to anomalies of the thermal expansion coefficient, which is magnetostrictional deformations (Nikitin et al. 1975). The existence of these anomalies follows directly from the expression for the magnetostriction (10.2) and this is due to the fact that only the magnetostriction constants ~Fe, ,~n, a;~ vary with temperature in the collinear phases while also 0R and 0Fe vary with temperature in the noncollinear phases. Figure 10.3 illustrates the thermal expansion of Dy-ferrite garnet along the [111] axis. It is seen that thermal expansion has no peculiarities at H = 0. However, the anomalies in A~/,k and ~(T) are observed to become sharper with increasing field strength for a sample placed into a magnetic field. These anomalies are connected with the formation of noncollinear magnetic structure in the field near the temperature of magnetic corn-

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS 0

50 I

100

150

200

I

I

I

473

H, kOe

(A///)II Ho3FesO12 T = 110 K

-50

-100

-150 2 (xl0 ~ ) (-200) I

I

P

I

Fig. 10.2. The theoretical (1) and experimental (2) magnetic field dependencies of the longitudinal magnetostriction of Ho3FesO12 iron garnet at T = 110 K (after Popov 1971), AHI 1

or, 10-6 K-1

(xl0 -6 ) 100

20

50

10 4

e/, 210rC/,fl 2~

Tk 220

230

T,K

0

-50 Fig. 10.3. The temperature dependencies of the thermal expansion of DY3FesO12 iron garnet along [111] in the vicinity of the compensation temperature (To) at different magnetic fields: 1 - 0, 2 - 50, 3 - 15 kOe; 4 - temperature dependence of the coefficient of thermal expansion at H = 15 kOe; after Nikitin et al. (1975). pensation. S i m i l a r a n o m a l i e s h a v e been o b s e r v e d in the vicinity o f the c o m p e n s a t i o n p o i n t o f ErFe3 (Nikitin et al. 1975, fig. 10.4).

10.3. Young's modulus, sound velocity change (AE-effect) and sound absorption A d d i t i o n a l d e f o r m a t i o n s caused b y magnetostriction arise in the m a g n e t i c a l l y ord e r e d c o m p o u n d s u n d e r effect o f the tension strains. It leads to Y o u n g ' s m o d u lus c h a n g e s at transitions into the m a g n e t i c a l l y ordered state and to d e p e n d e n c e o f Y o u n g ' s m o d u l u s on external fields (AE-effect). A n o m a l i e s o f Y o u n g ' s m o d u l u s ought to a p p e a r in ferrimagnets as well since external strains affect not only the

474

A.K. ZVEZDIN

150

1

100 30

x

50

20 "7 lO

£230 ~50

.2¢-"

230

T,K 10

P /

Fig. 10.4. Temperature dependence of the thermal expansion of the ErFe3 intermetallics in the vicinity of the compensation temperature (Tc) (1 - H = 0, 2 - H = 50 kOe, 3 - H = 15 kOe) and the thermal expansion coefficient (4 - H = 15 kOe, 5 - H = 50 kOe); after Nikitin et al. (1975). AE/E I

(xl0 -3 ) 6

o - 100 K

I

I

_ ° ~

O

I

''m~

4

2 0

-2 0 50 100 150 200 H, kOe Fig. 10.5. Magnetic field dependence ofthe AE-effect ofthe Ho3Fe5012 iron g~net; aher Levitin and Popov (1975). magnitude o f the sublattice magnetizations in the collinear phases but also their directions in non-collinear phases. Measurements of AE-effect in Ho-ferrite garnet in the fields up to 220 kOe (Levitin and P o p o v 1975) corroborate Young's modulus j u m p existence during the transition into non-collinear phase (Fig. 10.5). K a m i l o v et al. (1975) observed the anomalies of the sound propagation in the G d - i r o n garnet near the compensation temperature which are in reasonable agreement with the H - T phase diagrams o f this material. Figure 10.6 shows the sound absorption coefficient in the Gd3FesO12 at frequency 30 M H z for the magnetic fields H = 0, 2, 4, 10 k O e

(~qll[001]).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

or, dR/cm

475

a, dR/cm •

3.8 -

3.8 k

::

H=akOe o

-, -

°°o

' ,~,~,

~oo

e °ee°

H = 2 kOe

o

3.6 k9

0

L9

~

GO0

0000 0

I

q

10

3.6

20

3.7

° •

oH=0

0

:

0

~

3.8

3.4

;

oo oo~o

4~

--': I

I

30

40 T, oc

3.4 3i51 0

f 10

I 20

r 30 T, oc

Fig. 10.6. The temperature dependence of the coefficient of sound absorption in the Gd3FesO12 garnet at the frequency 30 MHz; 0"_LH[[[1001;after Kamilov et al. (1975).

Similar anomalies were obtained also for the sound velocity, for which there are the pattern of the minima near the compensation temperature (see also figs 4.3, 5.6, and 9.3). Notice that magnetostriction of the rare-earth sublattice depends strongly on the magnetic fields in the region of temperatures and fields where magnetization is saturated (Vedernikov et al. 1988, Kadomtseva et al. 1989). This effect should be taken into account for adequate determination of critical fields. 11. Non-collinear

phases and domain

structure

There are two specific phenomena in the context of field induced phase transitions: i) splitting of a sample into twinned domains in the canted phase, and ii) domain wall expansion and nucleation of the new phase from the domain wall during the first order phase transition. The latter mechanism may be responsible for the fact that the hysteresis of these transitions is often absent or very small.

11.1. 'Break' of symmetry in the canted phase and formation of domain structures with twins, triplets and quadruplets A noteworthy feature of the phase diagrams treated above is the presence of original 'degeneracy' of the canted phases. It means that in every case considered there exist several solutions of the thermodynamic potential minimization problem corresponding to one and the same value of this potential. Thus, for H l l [ l l l ] the degeneracy is equal to three since three physically equivalent rotational planes are presented here: = ~r/2 (37r/2),

~ = 7~r/6 (7r/6),

~ = 117r/6 (57r/6)

476

A.K. ZVEZDIN

for canted phases D(C) in fig. 4.4(a) (K < 0). In a case when/~U[100], the degeneracy ratio is equal to four, w h e n / t l l [ l l 0 ] it is equal to two. The degeneracy does not depend on the starting model but is a result of the system's symmetry. In the c a s e / I l l [ I l l ] , the [111] axis corresponds to the third order symmetry axis in the symmetrical (collinear) phase. During transition into an angular phase the symmetry decreases. The symmetry element C is absent in the canted phase. This broken symmetry element has transferred equivalent solutions (rotational planes) from one to another after symmetry has been restored. The interesting situation arises when HII [110]. In this case the 'break' of symmetry element C2 i.e. rotation around the [110] axis takes place at a transition from the collinear phases A and B into the canted phases C and D and the two-fold degeneracy of the angular phases develops. In going from the canted phases C and D into phase E one more symmetry element - the mirror plane disappeares. It is also evident that the number of phases becomes doubled here. The degeneracy of a given phase may be revealed by the fact that the crystal becomes divided into domains (twines, triplets, quadruplets). Let us consider the character of such domain structure for the case/~11[110] (see fig. 4.6). Let's fix the magnetic field and change the temperature, going over all phases A, C, E, D, B sequentially. At the transition A - C domains of the type (~ = 7r/2) and (qo = -7r/2) appear. With increasing temperature 0 decreases from 7r down to the value 0 = - arcsin v/3-/5 after which each of the domains of phase C in the phase E splits into two parts with =~+ and~=~_. The domain walls strongly expands close to the point of phase transition between phases C and E. As this point is moved away the distinction between the split domains is enhanced, i.e the difference between ~+ and ~_ increases and attains its maximum value in the center of phase E, after which the magnetizations in the separated domains 'tend' to each other again and at the of E - D transition point the split domains merge together. The maximum expansion of the domain walls occurs close to the critical points O and O'. Two types of domains (0, ~ = 7r/2) and (0, ~ = -7r/2), exist in the phase D. The difference between these domains vanishes at the D - B transition point and they collapse in the collinear phase. This picture of domain structure transformations during transitions over angular phases practically does not depend on model assumptions, and is defined by the symmetry properties of the system. Finer details, particular those pertaining to domain wall behavior may be obtained by the methods of qualitative theory of differential the equations (see below).

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

477

11.2. Nucleation of new phases from domain walls. The hysteresisless first order phase transition Let us note that first order phase transitions here can be accomplished in a nonhysteretic way. The mechanism of such transition is continuous growth of a new phase from the domain wall. This mechanism has been put forward by Mitsek et al. (1969), Mitsek and Serebryanik (1976) for the spin-flop transitions and later by Belov et al. 1974, 1975, 1976 to explain the absence of the temperature hysteresis at the first order transitions in DyFeO3 and (YTb)3FesO12. Similar behavior of the domain walls at the first order phase transitions was observed also in RE-TM intermetallics (see for details the review by Asti 1990). There are direct visual observations of this picture of the first order phase transitions (Dillon et al. 1974, King and Paquett 1973, Belyaeva et al. 1977, Lisovskii 1980, Gnatchenko 1989, Szewczyk et al. 1985, Szymczak et al. 1983, 1987). Mathematically we can illustrate this process easier by using an example diagram as represented in fig. 4.8 (/~]l [100]). Let us consider the series of transitions A - C - B at a fixed value of field H < / / 3 . In an area lying under curves OA', O'B' the phases A and B can coexist in the form of domains separated by 180 ° domain wall. It is easy to show that rotation of the angle 0 in such walls occurs in (010) and (001) oriented planes. We shall follow the work by Zvezdin and Popkov (1977) in the further analysis. Free energy allowed for the exchange energy has a form = f { [(grad 0)2 + sin 2 O(grad ~)2] + F(O, ~0)} dV,

(11.1)

where A is an exchange stiffness constant. ~(0) is thermodynamic potential determined by formula (3.1). The first integral of the differential Euler-Lagrange equations defining the O(ec) dependence in a domain wall leads to the following equation (ec is coordinate in a perpendicular to domain wall direction).

dO/dz = +A-1/2(~(0)

-

~0) 1/2,

(11.2)

where ~0 is equilibrium value of ~(0) at given H, T and qo = const. This equation can be integrated at once but to clear up the main features of phase transformations it is sufficient to restrict ourselves by a qualitative analysis of the equation obtained. Figure 11.1 illustrates the series of the integral curves represented by the equation (11.2) at different values of T. Obviously, the singular points on these curves where dO/dec = 0 correspond to domains and the whole curve between these points conforms to domain wall. Minima in the curves dO(ec)/dec are consistent with the bends of domain wall where rotation of O(ec) slows down. This slowing down signals the origination of the new phase from the wall. The domain of new phase come into existence at the first order point (e.g., the curve O'To in fig. 4.8). The integral curve at fig. 11. lb) nearly touches the abscissa axis at this point.

478

A.K. ZVEZDIN

--7l"

7"t"

0

-zr

01

zr

0

0

--71"

71"

0

Fig. 11.1. Integral curves in the plane 0~,0 governed by equation (11.2). They describe the domain walls depending on the temperature at H = const (Hill001], KI > 0). The points of contact with the axis 0~= 0 correspondto the domains, the lines connectingthese points describe the domain walls; after Belov et al. (1979). It is seen from the fig. 11.1 that bends in the domain wall grow when the temperature decreases and approaches to the first order phase transitions (the transition A --+ C in fig. 4.8). In the canted phase C domains of the 'old' phase A transform into bends of domain walls separating the twinned canted domains (fig. 11.1(b)). Corresponding bends of these domain walls convert into domains of phase B at transition C -+ B and the canted domains collapse. Obviously, there is a continuous conversion of domains, i.e. the process is completely reversible. We have explored the transition along the line H = const. The discussion holds completely for the other transition trajectories (T = const, for instance).

11.3. Canted phase domains in ferrite garnet single crystals and films The first indirect experimental indication of the possibility of domains existing during the transition into the noncollinear phase induced by the applied field has been given by Kharchenko et al. (1968). The jumps and hysteresis phenomena were revealed in the field dependence of the Faraday effect in gadolinium-ferrite garnet during the transition into noncollinear phase. It was shown that observed peculiarities cannot be explained by a rotation of the overall ferrite magnetic moment. The domain structure arising at a transition into the noncollinear phase in D y ferrite garnet was visualized by use of the Faraday effect by Lisovskii and Schapovalov (1974).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

•~ "

479

j:

. . . . . . .

.

,

~

-

. ~

Fig. 11.2. Domain structure in a plate of a Gd3FesO12 single crystal (the dark and bright areas are the domains of different phases): a) H = 7 kOe, T = 284.8 K, b) 285.1 K, c) 285.2 K, d) 285,3 K; after Kharchenko et al. (1974). The domain structure in Gd-ferrite garnet in high fields with/~ll[100] has been observed by Kharchenko et al. (1974), Gnatchenko and Kharchenko (1976) at various temperatures (fig. 11.2). In this case sample segregates into a mixture of the high and low temperature noncollinear phases close to the point of magnetic compensation. Suitable materials for experimental investigations of the domain structure in the canted phase are the epitaxial ferrite garnets films. Dikshtein et al. (1980) investigated epitaxial films of magnetic garnets (Y, Gd,Yb,Bi)3(Fe,A1)5012, of thickness 5-15 #m, grown on substrates of Gd3GasO12 cut along (111). These authors have shown that in the thin films thermodynamically stable domain structure can exist in strong magnetic fields, up to the flip field of the magnetization. The domain structure of the films was observed in polarized light, by means of a microscope, using the Faraday effect. The films were placed in a magnetic field directed approximately parallel to the chosen plane. The light was propagated along [111] axis. The source of the magnetic field was an electromagnet of the 'Solenoid' installation of the General Physics Institute of the Academy of Sciences, Russia, which allows experiments to be performed in stationary magnetic fields up to 150 kOe (Veselago et al. 1968).

480

A.K. ZVEZDIN

Notice that the magnetic anisotropy of films of magnetic garnets differs from uniaxial (cubic and rhombic components also present; see, for example, Eshenfelder (1981)), therefore a second-order phase transition, for a chosen orientation of the film, occurs within a narrow interval of angles between H and the developed surface of the film (Dikshtein et al. 1980). The results of the experiments are shown in figs 6.5 and 11.3 for one of the films investigated, of 5 # m thickness, with compensation point Tc = 310 K and Curie temperature 420 K. The easy magnetization axis in the film was inclined to the normal (the [111] axis) by 1 deg. The uniaxial anisotropy field, at any temperature in the range 80 K < T < 420 K exceeded the saturation magnetization 47rMs and therefore the domain structure that existed in the film was of the 'open' type (Kittel 1949). At H = 0, within the temperature intervals T < 250 K and 375 K < T < 420 K, an ordinary maze (spike type) domain structure was observed in the film. With increase of the magnetic field, which was oriented so that the disappearance of the domain occurred via the second-order phase transition, the period of the domain structure decreased according to a linear law (see fig. 11.3). At the instant of the disappearance of the Faraday rotation between domains with opposite sings of 37/ the period remains finite. Near the compensation point when H = 0, a single-domain

9 T=212K

3

5

I

I

I

6

7

8

~

H, kOe

Fig. 11.3. The magnetic field dependenceof the period of the domain structurein the film; after Dikstein et al. (1980).

D c

=

~

D e

~-

Fig. 11.4. Distribution of the magnetization in the uniaxial film of two sublattice ferrimagnets in the vicinity of the second-orderphase transition with the magnetic field H_I_EA(easy axis); after Lisovskii (1980).

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

481

interval (250 K < T < 375 K) was observed but with increasing of H, a domain structure was generated near the critical values of the magnetic field He. In fig. 6.5 there are plotted temperature variations of the critical field Hc and of the reciprocal critical dimension of the domains in the critical field D~-~ = D-:(Hc) and in the zero field D -1 = D-l(0). It is evident that the form of the curves Hc(T) agrees well with the form of the theoretical diagram of fig. 4.1. Figure 11.4 shows schematically the distribution of the sublattice magnetizations in the vicinity of the second-order transition in the films studied. When approaching to the compensation point Dc and Hc increase; the singledomain interval with respect to 'high-field' domains is 45 K (290 K < T < 335 K). For T = 290 K, coarse domains (~ 100 #m) were observed for H = 110 kOe. For observing a domain structure in strong magnetic fields, it is necessary to maintain carefully the conditions required for the occurrence of a second-order phase transition (or of fist-order transition close to it). When appreciable departures from these conditions occur, a domain structure is not observed at all, and an increase of the field leads only to a replacement of one phase by the other via motion of the interphase boundaries. It is important to emphasize that in these experiments a domain structure in uniaxial ferrimagnets in strong magnetic fields was observed during the phase transitions, which are not accompanied by a jump of the resultant magnetization. This domain structure can, occasionally, be thermodynamically stable (Khalturin 1976, Dikstein et al. 1980). Dikstein et al. (1983, 1984) discovered that regular domain structures in the ferrite garnet films may become amorphous near the second-order phase transition. Moreover, the process of amorphization is independent of crystal defects and exists even in perfect, defect-free films. These observations can be understood, when the possible formation of dislocation- and disclination-like magnetic defects are taken into consideration. This idea has been confirmed by direct observations of the domain structure at the points of second-order phase transitions. By contrast with the common domain structures in ferrimagnets which are advantageous energetically, domains arising at a transition into noncollinear phase are not energetically advantageous. Presumably the reasons for noncollinear ferrimagnetic ordering during phase transition in the vicinity of compensation point ought to be analogous to those giving rise to the appearance of antiferromagnet domain structures. Particular domain structure can arise in a real crystal with defects, inhomogeneities, internal strains and other impurities as a result of which one of the domains becomes more preferable than others in that crystal region. Though the nature of domain structure in ferrimagnets in strong field is not completely understood, its discovery itself essentially extends our knowledge of criteria of domain formation in magnets. The presence of metastable or energetically degenerate states in a certain range of temperatures and magnetic fields should be considered as a main criterion but not the smallness of the external field. Nowadays lots of objects are already known (not only ferrimagnets) where similar domains can exist over wide ranges of strong magnetic fields and temperatures.

482

A.K. ZVEZDIN

12. Hexagonal ferrimagnets 12.1. Free energy and equilibrium conditions This paragraph is concerned with the phase transitions and the phase diagrams of hexagonal ferrimagnets near the compensation point Te. The basal plane is assumed to be that of easy magnetization. In this plane there are three easy and three hard axes. We shall consider the two cases where a) an external field H is parallel with one of the easy axes, and b) with one of the hard axes. The case in which the hexagonal axis is the easy one has actually been examined in section 4.1 (uniaxial anisotropy). I f / t is parallel with the hexagonal axis the phase diagram is the same as for uniaxial anisotropy where H is perpendicular to the easy axis (or easy plane). The most important objects for which the developed theory may be applicable are hexagonal compounds of d-f type, such as RCos, R2Fel7, R2Co17, Rz(FeMn)17, RzFe14B and so on (Deryagin 1976, Kirchmayr and Poldy 1979, Buschow 1980, 1988, Sinnema et al. 1987, Givord et al. 1988, Yamada et al. 1988, Radwanski and Franse 1989, Franse et al. 1990). There are compounds DyCo5 and TbCo5 in which the basal plane of the crystal is the easy plane at low temperatures. Compounds of this class are usually described in the two (d and f) sublattice approximation. The magnetic moments of the d-sublattice are coupled via a strong exchange interaction, Hex ~ 107 0 e . This produces the high Curie temperatures (of the order of 1000 K) of these magnets. The d-f interaction is weaker Hex 106 0 e , and the f - f interaction is much weaker. On this basis we can say that the magnetization of the d-sublattice does not depend on the value of the external field or on the state of the f-sublattice. The exchange interaction between rare-earth ions can also be neglected. In that approximation the rare-earth subsystem can be treated as an 'ideal paramagnet' in an external field and in the exchange field generated by d ions. The presence of rare-earth atoms in these compounds leads to an appreciable anisotropy in the basal plane at low temperatures including the magnetic compensation temperatures (120-150 K). This leads us to expect here a strong influence of the hexagonal anisotropy on the occurrence of field induced non collinear magnetic structures. Although the formula for the free energy of ferrimagnets has been discussed above (section 3) we shall present the main arguments again to emphasize by this important example the area of the application of the theory and its asymptotic behavior (small parameters, etc.) On the basis of the properties of the above model we shall say that the magnetization of the d-sublattice does not depend on the directions of magnetization of d- and f-sublattices, i.e. the thermodynamic potential of the d-subsystem can be defined by the single vector "7 oriented along the magnetization of the d-sublattice. The rare earth sublattice must be in equilibrium with the d-sublattice, whatever the direction of "7. So the thermodynamic potential of the system can be written as ~("7) = qSd("7) + ~bf("7).

(12.1)

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

483

The potential ~d('7) is the sum of the magnetic energy -(lhrd/~ ) and the anisotropy energy, i.e. d~d('~) = - ( M d H ) + Kd('7) = -MdHCOS ~ - (1/6)K cos 6(~ + a), where ~ is an azimuthal angle which defines the direction of ~ in the basal plane relative to H. We are making use of the fact that/~ and/l~rd lie in the basal plane; a is the angle in the basal plane, between one of the easy axes and/~. The second term in (12.1) ~f('7) is the equilibrium thermodynamic potential of the f ions in any effective field/teff /'teff = /r~ _}_/-IM'

in which /~M = --A/~rd is a molecular field due to the d-sublattice experienced by the of f ions. We shall divide ~f('7) into two parts ~f('7) = ~S(Heff) - (1/6)Kf cos 6(~f + a), • S(Heff) is the axially symmetrical part of the function ~f('7); the second term is the anisotropy energy of the f-sublattice in the basal plane, qaf is the azimuthal angle of the vector/7. Evidently ~S(Heff) does not depend on a because of the axial symmetry. Function qSf(Heff) c a n be given as ~sS(geff) = -

f0 He~ Mf(x) dx,

where Mf(Heff) is the magnetization of the f-ion when H_l_~', in the axial symmetry approximation for the crystalline field. The explicit form of the function Mf(Heff) is of no great significance. For qualitative conclusions we shall approximate it by the Brillouin function (see, e.g., Li et al. 1988). Thus

4)(~) = - M d H cos ~ -

fo ~

Mr(x) d x (12.2)

- (1/6)Kd cos6(~ + a) - (1/6)Kf COS6(qof + a). Here Heff = ( H 2 + H 2 - 2HHM COS qo)1/2, mf(x) is a known function of x. The functions ~f, ~ are defined as follows (fig. 12.1): sin ~f --

COS ~ f =

HM

Heff

sin ~,

H - H M COS ~o neff

(12.3)

(12.4)

484

A.K. ZVEZDIN

e.a.

Fig. 12.1. Orientation H , 37/d and -Oeff relative to the easy axis (e.a.) in the basal plane of a hexagonal magnet.

Minimizing (12.2) with allowance for eqs (12.3) and (12.4) we can now determine the equilibrium phases in the ranges in which they exist. However, we shall simplify eq. (12.2) further. Expanding (12.3) and (12.4) in powers of H/HM we get H

~ f = 71" ÷ ~ ÷

HM sin~p.

(12.5)

The non-collinearity of the d and f sublattices depends on the angle ~b = qof - 71 - ~9 (see fig. 12.1). When H/HM << 1, z9 << min(~f, cp), so in the last term we can substitute ~f = 7r + ~ and then

~5(~p) = - M d H cos T - fo n ~ Mf(x) dx - (1/6)K cos 6(~p + a),

(12.6)

where K = Kd + Kf is constant reflecting the basis anisotropy of the crystal. This is a reasonable approach when K << )~M~ because in this system the part of the H - T phase diagram which is essentially dependent on anisotropy is concentrated in the range H << HM (see below). If necessary, the equilibrium values of 9~ and ~f can be found in the range H << HM by the method of successive approximation with the small parameter

Here we have actually used the lowest approximation of this parameter. In RCo5 one has ),Md2 ~ 10 9 erg/cm 3 and if K ~ 10 5 erg/cm 3, then ~ ~ 10 -2. The minimization of eq. (12.6) thus defines ~(H,T), and eq. (12.5) defines qof(H, T) and the difference in sublattice orientations ~b. The minimum condition for

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

485

• (~) is - - = M o l l sin~(1 - A~(Heff)) + K sin6(~ + a) = 0, 0~

(12.7)

~}2~

= MdHcOs ~(1 - A~(Heff)) + 6 K cos 6(~ + a ) (12.8)

H2 H 2 d~ sin ~ >/0. ~ff

d~ff

Here Mr(Neff) ~ ( g e f f ) -- _

(12.9)

_

neff

12.2. Phase diagrams 12.2.1. Case 1: HIIEA (easy axis) The analysis of (12.7) and (12.8) for a = mr/3, n = 0, 1,2 . . . . . reveals three phases: two collinear phases A and B, and one canted phase C (fig. 12.2). The stability

H1

0

-- ~

/

J

To

T

Fig. 12.2. Magnetic phase diagram of a hexagonal ferrimagnet with a compensation point; H is parallel with the easy axis in the basal plane of the crystal. Letters in boxes indicate phases of the magnet. The curves A A t, B B t etc. indicate where the stability of the corresponding is lost. K and K ~ are tricritical points, (Tc, H1) is a critical point; after Zvezdin and Popkov (1980).

486

A.K. ZVEZDIN

curves of the collinear phases A (~ --= 0) and B (~ = 70 are described by 1 -- AX(HM

q: H) -4-6K/MdH = O.

The stability curves of the various phases have been indicated in the figure by and BB' (see fig. 12.2). These lines cross at a point with coordinates

AA p

Tl = Tc, ( 6K ~1/2 nl =

H---~X~j

( 6KA ) 1/2, ~

(12.10)

(1SA--Xf

where

aMf Xf=

~H

We shall find the canted phases by expanding (7) over in the form (0 < ~ < 70

~HMd

H2(1 - AXe)

6K

6KA

H/HM.

It is then described

1 cosqo = - cosqo(4cos2 ~ - 1 ) ( 3 - 4cos3 ~) 3

(12.11)

in which r / = 1 - AXo;

)~o = x ( H

= 0).

This equation is best investigated graphically (fig. points of intersection of the lines fl(x) =

(1/3)x2(4x 2 -

12.3). The abscissa of the

1)(3 - 4x 2)

~flxl +1

-11\ / ' - ' \ o " . . . ( \ /i \j

X

-1 Fig. 12.3. Diagram explaining the solution of eq. (12.11).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

a)

cos ~o

[]

b)

cos 9

J

487

[]

Im

[]

[]

[]

[]

Fig. 12.4. Temperature dependence of cos ~,(T) for H parallel with the easy axis: a) - H > //3; b) H1 < H < / / 3 ; c) - H2 < H < H1; d) - H < HI. Letters in boxes indicate the phases corresponding to the diagram in fig. 12.2; after Zvezdin and Popkov (1980).

and

HMdrl

H2( 1 -- )~Xf)

6K

6KA

f2(z) --

x,

where z = c o s ~ is the solution of eq. (12.11) for a given H and T. The slope of f2(z) and the points at which it crosses the axis z vary with the H and T. For instance, if T = const and the field H is increased from H = 0 we obtain the following: five intersection points, of which three satisfy the stability conditions; then three points two of which are stable; and finally one point. Some solutions of c o s ~ ( H , t ) are shown in the diagrams of fig. 12.4. The canted phases are twice degenerate. Equation (12.14) obviously has two energetically equivalent solutions: 0 < ¢Pl(H,T) < zr

and

~2(H,T) = 27r- ~I(H,T).

This degeneracy is a consequence of the symmetry, because the easy axis is a second order axis. It is a characteristic feature of canted structures. Physically, it is manifested by the fact that the crystal breaks up into domains on changing to the canted phase. In sufficiently large fields the transitions A - C and B - C are second order phase transitions. The expansion of the thermodynamic potential in power of 3qo = p - ~0 where ~0 = 0 near the curves A K and B K ' is actually given by ~b ~__ ~ 0 q-

1

a,(~gg) 2 -I-

1

b(aT) 4 + ....

(12.12)

488

A.K. ZVEZDIN

in which a = H M d [1 - Axf(Heff)] sign cos T0 + 6K; 3H2H 2 b_

- -

x}(Heff) - 216K - HMa(1

H ff

-/~xf(Heff ) sign cos ~0,

where signx = z / l z [. a ( H , T ) = 0 on the curves A A ' and B B ' (see (12.9)). On the segments A K and B K ' of these curves b(H, T) > 0. According to the theory of Landau this means that those are the second-order phase transitions. The points K and K ' are tricritical. Their T, H coordinates are found from the equations a(H,T)=O;

b(H,T)=O

The approximate solution of these equations is

70KA

)1/2 (12.13)

( K 7 = Tc

1 +

\ 1/2 (1 -

xf))

) sign cos

o

.

(12.14)

The ranges of collinear and canted phases overlap when H < //3. So those are first-order phase transitions they occur on the curves of K T and K T ' which depend on the conditions of equality of the thermodynamic potential of the boundary phases. The curves of K C and I£~C ~ bound the range of metastability of the canted phases. These lines are described by (leaving out ~p(H, T))

d~ --

d~

d2~i = 0;

d~ 2

= 0.

(12.15)

When H < H1 a further ambiguity appears near ~ = 7r/2, 37r/2 on the curve cos ~(H, T) (see fig. 12.4 b), d)) with corresponding jumps in ~. This means that the canted phase C is split into two different canted phases C' and C". In phase C~: 0 < ~ < 7r/2 and 37r/2 < ~ < 27r, but in C": 7r/2 < ~ < 7r and 7r < ~ < 37r/2. A first-order transition C ~ to C" occurs on curve of TcO in fig. 12.4 a). The equation of the curve TcO: F(~c,) = F(~c,,). Curves O M and O M ~ in fig. 12.2 are the of stability curves of phases C and C"; they are defined by (12.19). The point O in fig. 12.4 is a critical point of the vapor-liquid type. When H < H2 (see fig. 12.2) the metastable ranges of the collinear phases so greatly increase that a direct transition can occur between them (see fig. 12.4 d)).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

489

We can derive one more asymptote of the first-order phase transition curves where H --+ O. The lines KTc and K'Tc:

T-To

3

Tc

H

4 AMd"

Line OTc

3 8

T - Tc T~

12.2.2. Case 2: KrIIHA (HA is the axis of hard magnetization (a = re~6 + nre/3; n = O, 1,2 .... )) Assuming that ~ = re/6 + nre/4 in the thermodynamic potential, we can see that the only difference in comparison with that examined above (for H parallel to with the easy axis) is the sign of the anisotropy constant K. The phase diagram is shown in fig. 12.5. There are three basic phases, two collinear phases, A{~ = 0} and B { ~ = re}, and one canted phase, C. The stability curves A(AA') and B ( B B ' ) are described by eq. (9) in which K must be substituted for - K . Here we have second-order phase transitions curves because the coefficient b in the expansion of (12.12) is positive everywhere. In the canted phase the angle is defined by eq. (12.11) in which - K has been substituted for K . The dependence in the canted phase is unambiguous if H > H~ but ambiguous if H < H~', as we can see from the graphical method (see

H

i"/ "i r,,..m

HI* H:*

P : ~---" ~ 0

VP'~ ro

-- ~ - 0 ' _ r

Fig. 12.5. Magnetic phase diagram of a hexagonal ferrimagnet with H parallel to the hard axis; after Zvezdin and Popkov (1980).

490

A.K. ZVEZDIN

fig. 12.3). When H < H~, the first-order phase transitions C'-C" and C " - C ' " may occur along MTc and M'Tc. The coordinates of the critical points M and M ' are approximately

7 ( 6K,~ ~1/2 H = H i ~ ~ 1 - )~Xf,]

T=Tc

{

21 1 :t: ~-d

2K

'

],2}

(1 -- AXf)

•

The equations of the lines MTc and M'Tc have the form of

T - Tc - -

Tc

v/3 H -

+

4

- -

tMd

(1 -

Axf).

MO, MO' and MP, M P ~ are the curves along which the stability of the adjacent canted phases C ~, C " and C " is broken. They are given analytically by a system of the (12.15) type. The ranges of coexistence of the phases C" and C m overlap considerably in very weak fields H < H~. In the canted phase, there may be direct phase transition when the temperature changes avoiding the intermediate canted phase C", as it is shown in fig. 12.6 c). The anisotropy constants in rare-earth materials usually increase rapidly with temperature and magnetic field (where T < Tc). Here we have assumed that K -- const. This is sufficient for determining qualitatively the behavior of hexagonal ferrimagnets in a magnetic field near the compensation point, including the number and sequence of phases, topology of phase diagrams, order of magnitude of critical fields in the region of the compensation point, type of phase transition. Without affecting the qualitative pattern of phase transitions near Tc, a sudden change in the anisotropy energy with temperature will make the phase diagram a)

cos ~o

b)

[]

c)

,,

[]

cos ~o

L__

[]

1 0

[]

[]

[]

Fig. 12.6. Plots of cos ~(T), where H is parallel with the hard axis: a) - H > H~ ; b) - H~ < H < H~ ; c) - H < H~. Letters in boxes indicate the phases corresponding to the diagram in fig. 12.5; after Zvezdin and Popkov (1980).

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

491

highly asymmetrical, relative to the compensation point. In particular, the critical points M in fig. 12.5 and K in fig. 12.2 will be displaced in downward direction on the temperature scale. Preliminary evaluations show that the shift may be so great that the transformation to a high-field canted phase can no longer be achieved by a second-order phase transition.

13. Non-collinear magnetic structures in the intermetallie compounds DyCo5 13.1. Crystal and magnetic structure Suitable objects for experimental investigation of field induced non-collinear magnetic structures in hexagonal ferrimagnets are the intermetallic compounds DyCo5 and TbCos, which possess anisotropy of the 'easy plane' type at below room temperature. The influence of hexagonal anisotropy on the occurrence of non-collinear magnetic structures, induced by an external magnetic field, has been investigated in the intermetallic compound DyCo5 near the compensation temperature (To = 124 K) by Berezin et al. (1980). Let us consider the main results of this study. The compound DyCo5 possesses hexagonal structure of the CaCu5 type (space group P6/mmm), in which some of the Dy atoms have been replaced by pairs ('dumbbells') of Co atoms (Dworschak and Khan, 1974). The cobalt excess in this case made it possible to obtain a single-phase compound, whereas the alloy of the integral composition DyCo5 contained the Dy2Co7 phase (Velge and Buschow 1968). In a structure of the CaCu5 type the Co atoms occupy two non-equivalent positions with, according to neutron-diffraction data, slightly different values of the magnetic moments (Ermolenko et al. 1975). We shall, however, neglect this difference and treat DyCos.3 as a ferrimagnet that consists of two (Dy and Co) magnetic sublattices. Characteristic of all RCo5 compounds are the huge values of the uniaxial anisotropy constants, often comparable with the R-Co exchange interaction, which may be of different signs (Ermolenko et al. 1975, Druzhinin et al. 1977). In DyCos.3, the negative anisotropy of the Dy sublattice leads to the result that at temperatures below 320 K, the magnetic moments of the Dy sublattices lie in the basal plane, despite the fact that for the Co sublattice the favored orientation of the spins is along the hexagonal axis. The resultant anisotropy at low temperatures according to Berezin et al. (1980), reaches 108 erg/cm 3. This prevents departure of the magnetic moments of the sublattices from the basal plane when the crystal is magnetized along any direction in this plane. 13.2. Domains in the basal plane The high degree of symmetry in the case of hexagonal anisotropy results in the existence of three equivalent axes of easy magnetization in the basal plane. Therefore in the demagnetized state, there can exist in the specimen six types of domains, oriented at the angle of 60 ° with one another, which must be taken into account in the analysis of the experimental data.

492

A.K. ZVEZDIN

The presence of several easy axes and hard directions in the basal leads also to the result that in the non-collinear phase, during rotation of the magnetic moments of the sublattices in this plane, they should lag at the easy directions and pass more rapidly through the hard directions. Then wave-like singularities appear in the fielddependencies of the magnetization and magnetostriction. Near the compensation temperature, the magnetic moments of the sublattices may pass through the hard direction discontinuously; that is, a phase transition of first order occurs. For an external field applied along an easy direction, a jump of the magnetic moments may be observed on transition from the collinear phase to the non-collinear (then the moments of the sublattices are 'thrown over' to the nearest easy axes, oriented at an angle of 60 ° with respect to that along which the specimen was magnetized). Along a hard direction, transitions of the first order are possible only in the non-collinear phase (from the easy axes oriented at 30 ° to the field direction to the easy axis perpendicular to the field). The diagrams in the upper part of fig. 13.1 may explain this.

tH~~,May b

Z

Mc~~~,- a %, G-cm3/g

era, G.cm3/g 20 1 10

20

~

1

I0

2

0 5 0

5 5

5 0 5 0 10 5 0

a)

0

J I

I

60

120

I

5

0 10 5 0

180 H, kOe

60

120

180 H, kOe

b)

Fig. 13.1. Experimental field dependencies of the magnetization of DyCo5. 3 near the compensation temperature, a) field directed along a-axis: 1 - 86 K; 2 - 105 K; 3 - 120 K; 4 - 152 K; 5 - 163 K; b) field directed along g: 1 - 78 K; 2 - 97 K; 3 - 115 K; 4 - 157 K; 5 - 166 K. The dotted lines show magnetization curves without allowance for the occurrence of a non-collinear magnetic structure; after Berezin et al. (1980).

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

493

13.3. Magnetization and magnetostriction The lower part of fig. 13.1 shows the experimental variation of magnetization with the intensity of the external field, for temperatures above and below the compensation point, in the fields directed along an easy axis a) and along a hard axis b) in the basal plane. It is seen from the figure that after technical saturation of the specimen, there is linear increase of magnetization, caused primarily by the paraprocess in the Dy sublattice (curves 1, 2 , 4 and 5 in fig. 13.1). With increasing field, the character of the curves changes: magnetization starts to increase considerably more rapidly on the whole; one can see alternating sections of rapid and slow increases of magnetization ('waves'). At temperatures close to the compensation temperature (curves 3 in fig. 13.1) considerable increase of magnetization occurs over a whole field interval. It is also seen that the determination of the critical transition field is subjected to a large error. In order to confirm independently the presence of rotation of the magnetic moments of the sublattices in the basal plane, and in order to determine the critical field more accurately, measurements were made of the magnetostriction along the a- and b-axes (in the latter case, the field was directed perpendicular to the measurement direction). Characteristic field dependencies of the )~aa and )~ab are shown in fig. 13.2. It is seen from the figure that singularities due to the occurrence of the noncollinear phase and under the influence of hexagonal anisotropy show up considerably more clearly in the field-dependence curves of magnetostriction than in those of magnetization. Consider the curves at 4.2 K in an increasing magnetic field (curves 1 in fig. 13.2; the direction of the field is shown by an arrow). In weak fields (up to 25 kOe), the change of magnetostriction along the axis a is caused by domain processes; along axis b by both the displacement of boundaries and rotation of the magnetic moments along the hard axis. On further increase of the field (in the non collinear phase), saturation is observed for both directions (the slight linear drop of magnetostriction is due, as was shown by measurements that we made on YCos, primarily to isotropic magnetostriction of the paraprocess in the Co sublattice). In fields exceeding a certain critical value Hal, there is an abrupt change of character of the field dependence of magnetostriction (along the b-axis, for example, it changes sign), resulting from appreciable departure of the magnetic moments of the sublattices from the direction of the field. Along the b-axis in the non-collinear phase (H > H1) there can be also seen wavelike anomalies, similar to those observed on the field-dependence curves of magnetization. With increase of temperature, the critical field of the transition decreases; near the magnetic compensation temperature, no saturation is observed (curves 2 in fig. 13.2). On further increase of the temperature, a transition from the collinear phase to the non collinear is detected again; the wavelike anomalies become less pronounced (curve 3 in fig. 13.2). At high temperature, a noticeable role is played by isotropic

494

A.K. ZVEZDIN

~au~a

1

H~'r

T

I

I

[ [ I

I

I I I

2 U

i 240

v

7, °

,

,

H, kOe

a)

b)

Fig. 13.2. Experimental field dependencies of the longitudinal magnetostriction )~a and transversal magnetostriction A~b of DyCos. 3 in a field applied along axes a and b respectively in the basal plane: 1 - 4.2 K; 2 - 124 K; 3 - 162 K. The arrows show the direction of change of the external rid& H~r is the critical field for transition to the noncollinear phase during increase, Hc~ - during decrease of the external field; after Berezin et al. (1980).

magnetostriction of the paraprocess; this leads to a linear increase of magnetostriction with increase of field in the collinear phase (curve 3, fig. 13.2). The magnetostriction measurements revealed still another peculiarity that was scarcely noticeable on the field-dependence curves of magnetization; namely, appreciable hysteresis, that is not coincidence of the values of the strain during increase and subsequent decrease of the external field. Correspondingly, the critical fields of the transition were strongly dependent on the past history of the magnetic state of the specimen. The hysteresis phenomena decrease with the rise of temperature and disappear completely at temperatures above 170 K. The variation of the critical transition fields and of the anisotropy fields (the H - T diagram) is shown in fig. 13.3, where open circles show the critical fields during increase, filled circles - during decrease of the external field. It is seen from the figure that near the compensation temperature (Tc), the collinear phase is absent for both directions; the critical fields increase with the distance from the compensation point. For the field dependencies of magnetostriction and at 4.2, 124 and 162 K, and also for the magnetic phase diagrams along the a- and b-axes, the corresponding theoretical relations were plotted.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS H, kOe

I

I

I

%,,,,>0,/ nlla

180 120

495

[

/

60 irol

H, kOe

50

100

150

~

i

i

180

120600

r

Hll

D~I

b

,

200 T K

A

CD~I~o,

1

K

Fig. 13.3. Experimentaland theoretical phase diagrams of DyCos.3 in a field directed along the easy axis a and along the hard axis b in the basal plane: (o) - experimental critical field (H~r) and anisotropy field during increase of the external field; (e) - critical field (Hc"r)during decrease of the external field; (zx) - the critical fields coincide. The heavy lines are experimentalphase diagrams; the thin solid lines are theoretical lines of phase transitions of the second kind; the dotted lines are theoretical stability curves of the collinearphase, the dashed-dottedcurves are those of the non-collinear(high-field)phase; after Berezin et al. (1980).

13.4. Magnetizations of the sublattices, exchange field and anisotropy constant The thermodynamic potential (12.1) and equations of critical fields (12.9) and (12.11 ) were used to calculate the phase diagrams. The parameters that occur in eq. (12.1) were specified as follows. Values of the magnetic moment of the Co sublattice as a temperature function were taken from the paper of Nowik and Wernick (1965); with allowance for the fact that our specimens had a slightly different structural formula, the value of magnetization M was taken Moo = 8.2# B at 4.2 K. The magnetic moment of the Dy sublattice was found as the sum (below the compensation point) or difference (above the compensation point) of the spontaneous moment and the moment of the Co sublattice. As it has already been mentioned, the value of the intersublattice exchange interaction in an isotropic ferrimagnet can be found from the measurements of the

496

A.K. ZVEZDIN

susceptibility in the non-collinear phase. In the presence of hexagonal anisotropy, as is easy to show that the susceptibility in the non-collinear phase at the compensation temperature should be 1/A in a field applied along either the a- or the b-axis, if the angle of 'bending under' of the magnetic moments of the sublattices is not too large. Actually, it follows from the measurements of magnetization at 124 K that the susceptibility in both cases, within the limits of accuracy of the experiment, is the same and equal to (0.92-t-0.07)x 10 -3 hence HM = (9004-50) kOe. This value agrees well with the value HM = 900 kOe obtained by Ermolenko et al. (1975) on the basis of the analysis of the temperature variation of spontaneous magnetization of DyCo5 in the spin-reorientation range. This value disagrees with the value HM = 1570 kOe determined from Mrssbauer measurements (Nowik and Wernick 1965). The anisotropy constant K6 was determined at low temperatures from measurements of the anisotropy field when the crystal was magnetized along the hard direction (/£6 = (42 -4- 7) x 104 erg/cm3 at 4.2 K), and at high temperatures from torque curves in the basal plane. On the basis of experimental magnetization data, allowance was also made for the dependence of the values of the magnetic moments on the external field. Using the formula for the value of magnetostriction of a hexagonal crystal in the basal plane, Aaa = 1/2A'~'2(T) cos 2qODy, "~ab = --'~aa

(here A"Y,2is the magnetostriction constant responsible for the strains of the crystal in the basal plane) one can plot the theoretical Aaa(H) and Aab(H) relations. It is assumed here that at low temperatures magnetostriction is caused principally by the Dy sublattice (see below). The angle FCo was found from the relation (12.3) sin ~Oy --

HM sin ~Co Heff

The theoretical field dependence curves of magnetostriction are given in fig. 13.4. In the calculations it was assumed that in the initial state, the specimen is demagnetized and the magnetic moments are uniformly distributed among the three easy axes in the basal plane; the isotropic paraprocess magnetostriction was also taken into account. In fig. 13.4 phase transitions of the first order are shown dotted (with allowance for the maximum possible hysteresis); increase and decrease of the field are denoted by arrows. It is evident from a comparison of fig. 13.2 and fig. 13.4 that the theory describes a number of peculiarities that are observed on the experimental curves, such as the abrupt change of the field dependence of magnetostriction on transition to the noncollinear phase and the change of sign of magnetostriction. On the theoretical curve there are wavelike sections (curve 1, fig. 13.4 b)) coinciding qualitatively with the experimental ones. The difference between theory and experiment takes place chiefly due to the presence of appreciable hysteresis in the experiment relations and will be discussed below.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

497

~ab

I

I

f I

/ 1 0

iI

t,

I

II

?

I

l~ 6' 170 1 0 ---''-

~

H, kOe

3 50

a)

I

0

240 H, kOe

b)

Fig. 13.4. Theoretical field dependence of the magnetostriction; the magnetostriction behavior shown is that for the maximum possible hysteresis. The notation is the same as in fig. 13.2; after Berezin et al.

(1980).

13.5. H - T phase diagram Theoretical magnetic phase diagrams for DyCos, obtained by using formulas (12.7) and (12.8), are shown in fig. 13.3. The thin solid lines correspond to phase transitions of the second order; the dotted lines show the critical fields during increase, the dashed-dotted ones - during decrease of external field for phase transitions of the first order. Transitions of the first order that occur in the non-collinear phase in a field applied along the hard direction as well as those that occur during technical magnetization of the specimen, are not shown. It is seen from fig. 13.3 that there is a partial agreement of the theoretical and experimental phase diagrams. For a field applied along the a-axis, the phase transition of the first order predicted by theory observed at low temperatures. For a field applied along the b-axis, both theoretical and experimental diagrams have a 'throat' form, characteristic of hard directions. Agreement of the experimental data with the theoretical shows up also in the fact that for each fixed temperature the critical fields of transition those along the a-axis are larger than those along the b-axis.

498

A.K. ZVEZDIN

The greatest discrepancies between the theoretical and experimental phase diagrams occur at low temperatures and near the compensation temperature. Without discussing the quantitative differences, which to some degree can be explained by inaccuracy in the choice of the parameters in calculations, we shall consider the qualitative difference between the theoretical and experimental results. First, appreciable hysteresis is observed experimentally in a field directed along the hard axis, whereas theoretically along this direction the transition from the collinear phase to the non-collinear one should be of the second order, and the occurrence of hysteresis is in principle impossible. Second, along the easy direction the collinear phase is absent within the interval of 85-135 K; according to the theory, it is unobservable within a considerably narrower temperature interval (110-135 K). To the qualitative differences we can be add the fact that the experimentally observed hysteresis along the a-axis exceeds the calculated hysteresis by an order of magnitude (curve 1 in figs 13.2 a) and 13.4 a)). It is possible that the presence of such hysteresis can be explained by taking into account the dependence of the hexagonal anisotropy constant on the external field. According to the single-ion theory at low temperatures, on increase of the field the low-field non-collinear structure transforms directly to the high-field structure. Then on the field-dependent curves of magnetization and magnetostriction approximately horizontal sections can occur, imitating the phenomenon of saturation. If it is so, then the above-stated value of the anisotropy constant K is too low (by more than an order of magnitude), and the observed hysteresis phenomena are actually connected with phase transitions within the uncollinear phase. This assumption would also explain the appreciable hysteresis in a field directed along the a-axis. A final explanation of this problem requires additional experimental data. The presence in DyCo5 of hexagonal anisotropy considerably exceeding that measured by Berezin et al. (1980) would lead to a substantial broadening of the range of metastability near the compensation temperature. In this case, a collinear structure may not be observed in a field directed along the a-axis, for the following reason. In the absence of an external field, as has it already been mentioned, several types of domains exist in a hexagonal crystal. If it turns out that the fields for displacement of domain boundaries exceed the critical field for stability loss of the collinear phase. Then the transition to the high-field non-collinear phase occurs from a multi-domain (nonuniform) state, avoiding the uniform collinear phase. Such situation is especially probable near the compensation temperature, where, because of the weak interaction of the magnetic moments of the domains with the external field, the field for displacement of domain boundaries may be large. There is also another reason explaining the absence of a collinear phase along the a-axis. In a real crystal there may be homogeneous regions (grains) with somewhat different values of the magnetic moment per elementary cell; that is, with different compensation points. Such possibility exists because of the replacement, which may not be completely statistical, of dysprosium atoms by cobalt. Then what is observed experimentally will be a mean value of magnetostriction produced by asynchronous rotation of the sublattices in each such grain. The phase transition may then turn out

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

499

to be so 'smeared out' that determination of the critical field becomes impossible. Along the a-axis this will lead to an apparent absence of a collinear phase near the compensation temperature; along the b-axis to an increase of the range of existence of the non-collinear phase. The presence of grains with different magnetic moments in the crystal may also explain other anomalies. For example the fact that the spontaneous magnetization does not vanish exactly at the compensation point (the 'residual' magnetization is approximately 10 G) - a result already noted by Ermolenko et al. (1975), Nowik and Wernik (1965) - and the unusual behavior of the field dependence of magnetostriction at Tc in a field applied along the b-axis (curve in fig. 12.2 b)). The increase of magnetostriction observed here experimentally indicates that at least some of the magnetic moments rotate toward the field direction, whereas theoretically there should occur a change-over of domains forming an angle of 60 ° with the field direction to domains oriented perpendicularly to the field (curve 5 in fig. 13.4 b)).

14. Spin-flop and spin-reorientation phase transitions in the anisotropic ferrimagnet HoCo3Ni2 with Tcomp : ~SR1 14.1. Spin reorientation in HoCo3Niz The aim of this section is to describe the field-induced phase transition in the ferrimagnetic compound HoCo3Ni2. Ferrimagnet HoCo3Ni2 is a unique member of the hard magnet family of RCos-type compounds. It is characterized by the compensation of magnetic moments of 4f and 3d sublattices occurring at T = 160 K. This temperature marks also the start of the process of spontaneous change of easy axis direction (spin-reorientation) from the basal plane (a-axis) to the hexagonal axis (c-axis) in the temperature interval from TSR 1 = 160 K up to TsR2 = 200 K (Drzazga 1981, Drzazga and Drzazga 1987). This coincidence, i.e. (Teomp ~ TSR1), is the reason why the critical fields are very low (Zawadzki et al. 1993). This fact leads to a rich variety of possible magnetic phase transformations induced both by temperature changes and by magnetic fields applied along the basic crystallographic axes. It allowed us to verify experimentally the main theoretical results. Ferrimagnet HoCo3Ni2 crystallizes into the hexagonal CaCus-type structure. It is well established that in compounds of this type the exchange and crystal-field interaction do not differ very much in magnitude and that the SR transition is governed mainly by two (K1 and K2) anisotropy constants (Asti 1990). Therefore, to obtain magnetic phase diagrams of HoCo3Ni2, it was necessary to take into account the exchange interaction as well as second and fourth-order contributions of the anisotropy energy. The anistropy energy has been expressed as / ( = t(1 sin 2 0 + / ( 2 sin 4 0 where /(I(T) changes the sign at T = Tsm (/(a > 0 for T > Tsm) and /£2 > 0. This is a typical temperature dependence of the magnetic anisotropy energy for the occurrences of spin-reorientation processes (Belov et al. 1979).

500

A.K. ZVEZDIN

The thermodynamic potential (3.1), after taking into account this magnetic anisotropy energy has been minimized with respect to the polar 0 and azimuthal qa angles of the 3d sublattice (Krynetski et al. 1993, Zawadzki et al. 1994). The exchange constant, the temperature dependencies of both sublattice magnetization as well as the anisotropy constants have been taken from Zawadzki et al. (1993).

14.2. The H - T phase diagrams The obtained magnetic phase diagrams are presented in figs 14.1 and 14.2. The magnetic phases are labeled as follows: FO - ferromagnetic, FI - ferrimagnetic, C - canted. The numbers following these labels denote possible sub-phases. The subscripts [] and _k indicate the direction of magnetic field in relation to c-axis. The subscript 0 denotes H = 0.

14.2.1. Case 1: Magnetic field perpendicular to the c-axis 1) Low temperature range (T < TSR1): For T < TSR1 the magnetic moments of both sublattices are lying in the basal plane (0 = 7r/2). The anisotropy energy is constant which results in the-well known case of isotropic ferrimagnet. Three magnetic phases exist now ferrimagnetic FII± (or FI10), canted CI± and ferrimagnetic FO±. They are separated by phase boundaries described as follows Hcl,2 = ,~(mf(Heff) m md).

H, 10xkOe

1

I"

]

T

T

100 FO±

CI± 50

lOO

200

300

T,K Fig. 14.1. The magnetic phase diagram for the field parallel to the basal plane. The meaning of the symbols can be found in the main text in section 14.2 (after Zawadzki et al. 1994).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

H, 10xk0e

i

J

t

J

501

J

100 FOIl 50

Fllll

Cll~ C20~/] I

0

B

I

100

~

200

FI2I, I

300

r,K Fig. 14.2. The magnetic phase diagram for the field parallel to the hexagonal c-axis. The meaning of the symbols can be found in the main text in section 14.2 (after Zawadzki et al. 1994).

The signs ' - ' and '+' correspond to the low and high-field limits of the canted phase region, respectively.

2) High temperature range (T > TSR2): In this case three phases are stable, too. The low field stability region of the collinear ferrimagnetic phase is reduced to the line H = 0 (FI20). For magnetic fields lower than He, where

He=

2(K1 + K2) Md(2 - X(Heff)A)

canting of ferrimagnetic structure is induced, and the magnetic moments are bent away from the c-axis creating the C2± phase. The angle of deflection increases with the field, and finally, in He the magnetic moments are oriented perpendicularly to the c-axis. In fields higher than Hc the canting vanishes and the collinear phase appears again. Further increase of the field reduces the net moment of the Ho-sublattice. It reaches zero for H = AMa, and then grows along with the field. As a result, the high-field collinear phase is divided into two sub-phases: ferrimagnetic FI3± and ferromagnetic FOx.

3) SR transition temperature range (TsR1 < T < TSR2): The noncollinear magnetic structure (C20) is stable in the SR range even at H = 0. This weakly canted phase appears as a result of competition between anisotropy and exchange energies was investigated both theoretically and experimentally (Decrop et al. 1982, Irkhin and Rozenfeld 1974). In non-zero magnetic fields, apart from the above described C2±

502

A.K. ZVEZDIN

and FOx phases, a new canted phase C3± appears in the SR range. Such cone-canted phase is allowed if the condition sine 0 -

I~1

2K2

is fulfilled. This phase is characterized by constant susceptibility X(Heff) = 1/A. The region of stability of the phase is limited at high temperature by the following critical field

) 1/2 Hcl,2 = AMd sin0 :t: AMf 1

M2

Mff(H~ff)

cos 2 0

The low temperature limit of the C3± phase is T = TSR1. The properties of the new cone-canted phase C3± resemble the properties of CI± in the isotropic case. Nevertheless, due to the conical character of anisotropy, magnetic moments of the 3d and 4f sublattices during the canting progress are lying on the easy cone, not in the basal plane. Comparing the magnetic structures of the cone-canted phases C2± and C3± it is worth noting that in both phases, magnetic moments of the 3d and 4f sublattices are not collinear (~rf /]]Md). However, their components perpendicular to the e-axis remain collinear in C2±, whereas in C3± they are noncollinear (~rf,± is not parallel with ~rd,±). Moreover, the susceptibility of C2± phase, contrary to that of C3±, is constant.

(.~rf,±ll~d,±)

14.2.2. Case 2: Magnetic field parallel to the c-axis

In this case the magnetic phase diagram is simpler. Phase boundaries are given by

MdH 1 - A

Mf(H~ff)) Heft

T 2K1

O.

The curve corresponding to the ' - ' sign surrounds the stability region of collinear ferrimagnetic phase (FIIlI). In the region bounded by both these lines exists a canted phase. This region is divided into two parts, the low-field part (ClU,), where rotation of the weakly canted structure from the easy plane (easy cone) to the c-axis takes place, and the high-field part (Clnll), where a change of magnetic order from ferrimagnetic to ferromagnetic is induced. It can be seen in fig. 14.2, that the link between low-field (ClUll) and high-field (c1H") regions of the canted phase is very narrow. 14.3. The magnetization measurements

In order to verify the existence of the newly proposed phase $3± the magnetization measurements have been carried out on high quality single crystals using both SQUID (up to 5 T) and ballistic (up to 7 T) magnetometer.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

1.0

0.8

I

I

I

I

T= 170 K

. , I I

503

I

/

0.6

0.4

0.2

W~,~,~,

;, . ;, ~,2.~.~-0

I

I

l

2

[

I

3 4 H, 10×kOe

I

I

5

6

7

Fig. 14.3. Longitudinal (1) and transversal (2) components of magnetization at 170 K as function of magnetic field applied parallel to the a-axis (after Zawadzki et al. 1994).

Longitudinal and transverse magnetization components have been measured simultaneously (Zawadzki et al. 1993, 1994). It allowed us to examine precisely the magnetic structure evolution during the magnetization process. The application of a ballistic magnetometer allowed us to observe phase transitions occurring in higher magnetic fields. A good agreement between calculated and experimental magnetic phase diagrams has been found in the attainable field range. The transition from C2± to C3± can be deduced from the field dependence of the longitudinal Ma and transverse Mc magnetization components, in the case of the field applied along the a-axis. An example of such a field dependence measured at 170 K in presented in fig. 14.3. The linear relation Ma(H) = ~ - I H for fields higher than the critical field (Hc ~ 4 T) is similar to that often reported for field-induced phase transitions in easy-plane in ferrimagnetic intermetallics (Ballou et al. 1989). Moreover, the relation Mc(H) supplies still more information on the magnetization process. The beginning of the phase transition is now more distinct. Besides, it is easy to notice that canting of magnetic structure takes place on the easy cone, not on the basal plane (C2± -+ C3± transition). Otherwise, the transverse component Mc should vanish above Hc. . . . . •

14.4. Magnetostriction and spin-flop transitions

,2-

~;

To obtain further information on magnetic phase diagrams of HoCo3Ni2 Krynetski et al. (1994, 1995) have measured its linear magnetostriction since it is known (Belov et al. 1979) that magnetostriction is a more sensitive tool for magnetic

504

A.K. ZVEZDIN

phase diagram study than magnetization. Furthermore, the study of magnetoelastic properties of intermetallic ferrimagnet HoCo3Ni2 is a very interesting problem by itself, particularly when the role of rare-earth sublattices in magnetostriction of 3d-4f compounds is concerned.

14.4.1. Magnetic field parallel to the a-axb Spin-reorientation temperature range (160 < T < 200 K). In this temperature range the non-collinear magnetic structure caused by the competition between anisotropy and 4f-3d-exchange energies is stable even at H = 0. Therefore, the transformation of this phase induced by the external field has a very complex character. In fig. 14.4 the longitudinal magnetostriction )~a~ of HoCo3Ni2 at T -- 162.5 K is shown as an example. The low-field step-like anomaly at about 0.3 T is due to the domain process. The high-field anomaly on this curve results from the spin-flop transition. The value of the critical field agrees with magnetization data (see insert to fig. 14.4). It is seen from fig. 14.4 that magnetostriction measurements allow to determine the values of the critical fields of induced phase transitions really more precise. The additional mid-field anomaly seems to be due to the fact that at this temperature the primary direction of the weakly canted 4f and 3d magnetic moments does not coincide Zaa(10-5) T= 162.5 K

HoCo2Ni 3

f

M(10_l~ "

T= 170K

1

Bc5 0

2 0

1

2

3

4

5

I

I

1

2

6

Bcr ,F 3

I 4

5

B(T) Fig. 14.4. The longitudinalmagnetostriction)~aa of HoCo3Ni2 at temperature T = 162.5 K. The insert shows the magnetization of HoCo3Ni2 along a-axis versus magnetic field parallel to a-axis at temperatureT = 170 K.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

505

with the a-axis due to the spin-reorientation process. It must be emphasized that transformation of magnetic structure of HoCo3Ni2 caused by a spin-flop transition is accompanied by the relative positive deformation along the a-axis that is equal to 3 × 10 -5. High temperature range (T > TSR2). At these temperatures the stable zero-field phase is a collinear ferrimagnet with magnetic moments directed along hexagonal c-axis. The magnetic field along the a-axis causes both canting of ferrimagnetic structure and continuous rotation of the magnetic moments towards the basal plane. Finally, in Her the magnetic moments are directed perpendicularly to the c-axis. In fig. 14.5 the field dependence of longitudinal magnetostriction )~aa for T = 210.2 K is shown. The process of gradual spin reorientation to the basal plane is finished at the critical field equal to 10 T. The appropriate magnetostriction deformation equals 2.2 × 10 -5. The additional high-field anomaly is due to the field-induced canting suppression and the collinear ferrimagnetic phase recovery again. The essential reason for such conclusion is the fact that total magnetostrictive deformation (fig. 14.5), within the limits of experimental accuracy, agrees with the thermal expansion anomaly caused by the reorientation of magnetic moments from hexagonal c-axis to the basal plane Aa~ = 3 x 10 -5 (Krynetski et al. 1994). The process of canting suppression has the significant field hysteresis, so when the external field decreases to zero the residual strains of the crystal lattice of HoCo3Ni2 exists. &aa(lO-s)

HoCo2Ni3 T = 210.2 K

/

r1 I

I

I

I

4

8

BerI1 I

I

12

~(T) Fig. 14.5. The longitudinalmagnetostriction)~aa of HoCo3Ni2 at temperature T = 210.2 K.

506

A.K. ZVEZDIN

14.4.2. Magnetic field parallel to the c-axis Rather unexpected results are obtained at temperatures just above TSR2. At these temperatures the stable zero field phase is collinear ferrimagnet with magnetic moments directed along the hexagonal c-axis. The magnetostrictive anomaly (Her equal to 11 T for T = 207 K) indicates that the field induced phase transition occurs. This transition has the significant hysteresis, for instance, at T = 207 K it is equal to about 5.5 T. The nature of this transition needs further investigation. The magnetostriction measurements show that the ferrimagnet HoCo3Ni2 has not only the magnetic moment compensation occurring at T = 160 K but the magnetostriction compensation, too. The magnetostriction due to the domain process goes passes zero point at T = 120 K with rare-earth contribution being positive while d-sublattice portion has the negative sign.

15. Surface anisotropy effects and surface phase transition The magnetic behavior near the surface of a crystal or film may drastically differ from that of the inside. An attempt to describe the influence of the surface inevitably would lead to consideration of the surface shifts of energy levels and to modification of the s-d hybridization. This is beyond the scope of this review. Here we would like to focus our attention on only one aspect of surface magnetism, the surface anisotropy (a more complete discussion of surface magnetism including the state of the art of this problem can be found in a detailed review by Kaneyosi (1991a, b)). There are many experimental data which confirm its existence in a large class of crystals and films. This surface anisotropy is due to the various causes associated with material, preparation techniques as well as with aging processes (oxidization of RE ions, crystallization). Further we will consider surface anisotropy constants as phenomenological constants. Let us come back to the expression for total free energy with allowance made for the homogeneous exchange energy A((grad0) 2 + sin2 0 (grad~)2), where A is the exchange stiffness constant (see section 7) ~St = ~bulk q- ~surf.

(15.1)

Here ~bulk is given by the eqs (3.1) and (7.1). The uniaxial anisotropy follows from the most common surface energy term ~surf = - K s cos 2 0s,

(15.2)

where 0s is the angle between the easy axis (z-axis) and the spontaneous magnetization M, the subscript s denotes that this angle refers to the surface of the film. Let us consider a plate of thickness 2d. For the sake of simplicity we assume the surface anisotropy values on the both sides of the film to be equal: Ks(d) = Ks(-d). Besides, we imply the film to be homogeneous in the xy plane. Then we can state that O(z) = O(-z)

and

--dzdO~=o = O.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

507

A simple calculation yields us the Euler-Lagrange equation d20

()~3eff

dz 2

00

2A-----

'

and the boundary conditions d0 I

0,

d z z=0

dO z=±a Ks d-~ = - 2---Asin20s, where #eff is defined by eq. (3.1). There are two approaches to this problem: a) to solve the first integral of eq. (15.3) and do numerical calculations, and b) to use the bifurcation theory to study the stability of eq. (15.3). For our purposes it is sufficient to investigate the stability of this equation (Kaganov 1980, Kaganov and Chubukov 1982). We have three solutions of eq. (15.3): two collinear phases {0 = 0, 7r} and a canted inhomogeneous phase {0 < 0(r-) < 7r}, their stability regions depending now on the surface anisotropy. In particular, the stability curves of collinear phases are of the same shape as in the bulk case, the only difference being the uniaxial anisotropy (Zvezdin et al. 1991) Ku - + K = Ku(1 - h),

where Ku is the bulk uniaxial anisotropy and h is a dimensionless parameter defined by the equation ,=(~)l/2arctanh(h)

1/2,

K,a 0"--

(~--

KuA' A As appears from the first approximation above, the presence of the surface leads to changes in the bulk anisotropy. Thus, the characteristic points and curves in the phase diagrams depend on the surface anisotropy. It is quite easy to determine the influence of the surface anisotropy on the coercive force of the film (Zvezdin et al. 1991). The presence of domain wall-like solution of eq. (15.3) is another interesting property of this equation. It differs from the domain wall equation only by the boundary conditions. In our case such equivalence leads to the existence of surfaceinduced domain walls (in the canted inhomogeneous phase) which divide the film into surface and bulk domains. This statement agrees well with Mrssbauer data (Kaneyosi 1991a, b). Here it means that a surface anisotropy can induce surface phase transition, when a new field induced phase is generated only near the surface (Kaganov 1980, Zvezdin et al. 1991). So, the surface anisotropy affects considerably the critical curves and points of phase diagrams, particularly in thin and ultrathin films. The presence of surface anisotropy can lead to the existence of surface domain walls and to the surface field-induced phase transition.

508

A.K. ZVEZDIN

16. Phase transition at the local defect. Dislocations and FIPT's in Gd3Fe~O~

The problem of elucidation of the role of point-like, linear, and other defects of the crystal lattice in processes that determine phase transitions is of considerable interest for solid state physics from a basic point of view (Levanyuk et al. 1979, Ginsburg 1981). Exceedingly useful may be experimental research involving a considered here spin-flop transitions, due to the change of direction of the magnetization vectors of magnetic sublattices with changing external magnetic field or temperature. Dislocations are particularly appealing to the study of this problem. The effect of strains induced by dislocations may be expressed in the free energy (3.16) by a contribution to the magnetoelastic interaction

X_I_

XII

2

H 2 sin 2 0 - M ° H cos 0 - K](0, ~) + e(~, 0, ~o),

where 0 is the angle between the 3~rd a n d / t , e is the energy increment due to the elastic stresses of dislocation, f" is the distance from dislocation. This contribution can be considered as the density of magnetic anisotropy induced by dislocation deformations. The value of this induced magnetic anisotropy in the vicinity of the dislocation can be superior to the magnetic anisotropy of the perfect crystal. This additional magnetoelastic anisotropy causes a considerable deflection of Md from its orientation away from dislocation. Figure 16.1 illustrates schematically this effect. It is of interest to study the FIPT near the dislocation. A particular feature of this case is the dependence of critical fields on distance from the dislocation. Vlasko-Vlasov et al. (1981, 1983), Vlasko-Vlasov and Indenbom (1984) investigated singularities in the course of spin-flop phase transition near an individual dislocation in single crystal gadolinium iron garnets. The samples used in their experiments were obtained from a single crystal of gadolinium iron garnet cut into wafers parallel to the (110) plane, Their thickness after mechanical and chemical polishing was 30-50 #m. Because thin wafers of this material are transparent to visible light, one can use methods based on polarized light to study simultaneously the distribution of internal stresses (by the photoelastic effect) and the magnetic domain

----~'lllllj 81 ~ & l l l 4 ~V7 . . . . . . . "" ~- -- '-, - ' ~ PI | ' | l l ~ l l it* i l ' ~ . ~. .-.-.-.-. .. - - ~

--,,

, t

,,

....

--~-~-.~k, ¢ ] i r ~"

,,,,v, I I It

,,v,;l,

!

T!|2~, V v,, v t I t It

. . . . . ,~,_--

,,,,,,

Fig. 16.1. Domains arisen near a dislocation in a cubic ferromagnet with K l < 0 (i.e. the easy axes are parallel to [001]) (after Ditchenko and Nikolaev 1979, Dichenko et al. 1983).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

509

structure (by the Faraday and Cotton-Mouton effects). This circumstance affords a unique possibility for direct experimental observation of the influence of elementary dislocations on the rotation of magnetic moments, nucleation of domains of different phases, and motion of phase boundaries during spin-flop transitions. The spin-flop transitions in constant field ~ll[ll~] of 177 Oe, near a single 60 ° dislocation with a glide plane (111) and axis [110], is illustrated in fig. 12.2. Approaching the compensation point Tc 282.5 K from below, in the vicinity of the dislocation a low-temperature canted phase D, in the form of a light three-lobed rosette appeared against the background of the collinear phase B. In the latter the magnetization MII [1 l i] was in the plane of the plate (fig. 16.2 a)). One of the lobes of the rosette was considerably smaller than the other two. The region occupied by

Fig. 16.2. Change of the shape of the interphase boundary near a dislocation in the course of a spin-flop transition in an external magnetic field H I I [ l l i ] (the Nicol prisms are slightly uncrossed): a) T = 280.6 K; b) T = 281.0 K; c) T = 281.7 K; d) T = 282.1 K; H = 177 Oe (after Vlasko-Vlasov et al. 1983).

510

A.K. ZVEZDIN

phase D in fig. 16.2 a) is brighter than surrounding dark region of phase B, since the magnetization of the iron sublattices in phase D does not coincide with the plane of the light polarization. With increasing temperature, the rosette of the canted phase near the dislocation increased and merged with the macroscopic region of the same phase advancing from the bulk of the sample (fig. 16.2 b). In the course of the subsequent successive (fig. 16.2 c)) redistribution of the volume fractions of phases B and D, their sectoral arrangement in definite sections of the dislocation field of microstresses was rigorously preserved. The collinear phase B, decreasing with temperature formed a rosette (fig. 16.2 d) symmetric relative to the dislocation axis and to the phase-D rosette that existed at low temperatures (fig. 16.2 a)). The phase B vanished at 284 K. A similar process took place when approaching Tc from above. The picture of replacement of the high-temperature collinear phase C is practically identical to that considered above when replacing the corresponding values of T (see fig. 16.2) by 2Tc T. Figure 16.3 shows the H - T diagrams of the phase transitions in four points that are symmetric with respect to the dislocation and are located at different distances from the dislocation axis. Each plotted line corresponds to the values of T and H at which the size of the lobe of magnetic dislocation rosette (see fig. 16.2) in the [111] direction retained a fixed value of R. It can be seen that in local sections, the H - T diagram does not undergo fundamental changes as the dislocation is approached, -

/-/ kOe 8

4

3

i

I I I I I I 282

283

T, K

Fig. 16.3. Local phase diagrams in the vicinity of the dislocation at a distance R from its axis in the [111] direction. 1, 1') R = 13/zm; 2, 2') R = 35/zm; 3, 3') R = - 3 5 / z m ; 4, 4 ' ) / ~ = 13/zm. Unprimed and primed numbers refer to the low- and high-temperature transition from collinear into canted phase, respectively (after Vlasko-Vlasov et al. 1983).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

511

whereas from the opposite side (R < 0) the diagram changes qualitatively when the distance to the dislocation is decreased. At the point where the transition curves change from a diverging type (curves 1-3) to an intersecting type (curve 4), the dislocation microstresses and the stresses from other sources cancel each other, as discussed below. The form of the phase diagram obtained at different points of the crystal depends substantially on whether the induced anisotropy at these points contributes to the appearance of a canted phase, as in the case of curves 1-3 of fig. 16.3, or hinders it, as at the point corresponding to curve 4. Both cases are shown in fig. 16.4. In fig. 16.4 b), in the metastable existence region of the high-temperature phases at T < Tc = 283.1 K, there is no experimental phase-transition line, because in the corresponding magnetic field the temperature of this high-temperature transition is lower than the temperature of nucleation of the compensation boundary in the region with the larger value of To. At H > 5 kOe, in all cases, the asymptotes of the experimental points are straight lines (shown dashed in fig. 16.4) passing through Tc and having a slope IdH/dTI ,~ 12.8 kOe/deg. The compensation-boundary motion which causes the transition between the low- and high-temperature phases, took place through the chosen sections of the crystal at different field-independent temperatures T = T~. The good agreement, not only qualitative but also quantitive, between the experimental and theoretical data uncovers prospects for using of the determined relations H, kOe 20

/

I

20

I I I

/ p

15

15

I I

10

10

O

i\'

281

a)

282

283

282

284

283

A_

284

T, K

b)

Fig. 16.4. Phase diagrams plotted at crystal points where the induced anisotropy enhances a) and hinders b) the formation of the canted phase: (zx) - transition between low-temperature and canted phase; (v) transition between high-temperature canted and collinear phase; (o) - transition between canted phases (180 ° rotation of sublattice magnetization), (.) - transition between low- and high-temperature collinear phase (after Vlasko-Vlasov et al. 1983).

512

A.K. ZVEZDIN

for the spin-flip phase transitions for the purpose of studying the defect structure of rare-earth-iron garnets. In particular, investigations of the magnetic rosette produced near the dislocation make it possible to determine the direction and magnitude of its Burger's vector. From the distribution of the striction - is nonequivalent magnetic phases in the inequivalent crystal one can determine more reliably anisotropy and the weaker internal stresses the induced than by the photoelasticity method. Thus, by accurately measuring the transition-temperature shift due to the stresses, e.g., to TO. 1 K we can record a stress level of the order of 2 kgf/cm 3. 17. Free-powder samples Single crystals are not always available for a study of the field induced phase transitions and the intrinsic magnetic properties. Free-powder samples give a good alternative. Verhoef et al. (1989, 1990a-c), de Boer and Buschow (1992) reported the elegant high-field free-powder method (HFFP) which has been used to determine the intersublattice-coupling strength in a fairly large number of different intermetallic compounds. The particles of the free-powder sample, having a size of about 40 #m, are assumed to be small enough to be regarded as single crystallines and are free to rotate in the sample holder during the magnetization process, so that they will be oriented by the applied field with their magnetic moment in the field direction. Verhoef et al. (1989, 1990a-c) analyzed magnetization curves of Er2Fe14_~MnxC compounds by assuming i) the powder particles to be free to rotate in the sample holder, ii) the magnetization of the rare earth sublattice to be fixed in the easy axis direction, iii) the magnetic anisotropy of the transition metal sublattice (Kd) to be zero, and iv) both sublattices to be spontaneously magnetized to saturation. In this model the magnetic anisotropy does not influence the magnetization measured on a free-powder sample. These magnetization curves are the same as in the isotropic case (Verhoef et al. 1989, 1990a-c)). Only if both sublattices display magnetic anisotropy, an influence on the free-powder magnetization can be observed. de Boer and Zhao (1994) and Zhao (1994) studied this problem assuming both sublattices to be spontaneously magnetized to saturation. The above considered model of two-sublattice anisotropic ferromagnet (see sections 2 and 3) allows to abandon the last-mentioned assumption and to obtain the magnetization curves and the phase diagrams of free-powder samples taking into account the real dependence of the rare earth magnetization on temperature and magnetic field. The thermodynamic potential (3.1) can be rewritten as F(O, a) = - M d H cos ¢ - f0 H~ Mf(h) dh + Kd sin 2 0 + Kf sin2(0 + a),

where Kd and Kf are the magnetic anisotropy constants for d- and f-sublattices, ~b is the angle between magnetic field and the d-sublattice magnetization, 0, Of = 0 + a are the angles between the easy axis and the d- and f-sublattice magnetization, a is

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

513

EA

oe

M

Mf

Fig. 17.1. The orientations of the sublattice magnetizations relative to the external field and the easy axis.

the angle between the two sublattice magnetization (see fig. 17.1). The angles 0 and determine the direction of the d- and f-sublattice magnetization. It can be shown that in the case of freely suspended sample the angle ¢ is uniquely determined by the angle a: Mf COS 't/) - - - -

COS ~ ,

M(~)

where M(oO = ( M 2 + M 2 + 2MdMfCosa)U 2. This means that the angles 0 and a can be adopted to define the magnetization process in this case. It is appropriate at this point to recall that the condition Kf << )~MdMf is herein taken into account as well as above in section 3. The angles 0 and a can be found by minimizing the thermodynamic potential Off O---'O= Kd sin 20 + Kf sin 2(0 + c~) = 0,

O~

(17.1)

mdmf

- - = Kf sin 2(~ + 0) - AMdMf sin a + H - 0c~ M

sin c~ = 0

(17.2)

Here we make use of the following formula for Here: H~ff = H c o s ( a - ¢ ) - Amd cos ¢ .

After some algebra, eq. (17.1) can be brought to the form (Zhao 1994) ]K I sin[2(0 + c~)] = Kd sin 2c~,

(17.3)

514

A.K. ZVEZDIN

where

IK[ = [Kg +

+ 2 K d K f c o s 2o~11/2.

Equation (17.3) can be used to eliminate 0 from eq. (17.2), which becomes sinc~

+ I K B M d M f cosc~ = 0.

--M - ~

(17.4)

In contrast with the theory elaborated by de Boer and Zhao (1994) and Zhao (1994), the approach presented here is unbounded by the severe requirement Mf = const. The magnetization of the rare-earth sublattice in this approach depends on the temperature and Heff. 2.5 2 1.5 1 0.5 ,,,I,,,,I, 0.5

a)

J

1

1.5

2.5 2

2

2.5

3

3.5

1.5 1

0.5

~

l lllllllllrllll~

b)

0.5

1

1.5

2

2.5

Fig. 17.2. Representation of the uniaxial anisotropy influence on the magnetization curves of the free powder sample of the uniaxial ferrimagnet. These magnetization plots are calculated by formula (17.6) at a = 3 and: a) e = -0.1, b) e = +0.1, c) e = +0.2. The straight line /~ = h on the graphs refers to the 'isotropic' case Kd = 0. The portions of the magnetization curves having negative first derivative d#/dh correspond to the unstable phases. The first-order transitions points of the jump-like magnetization curves should be determined using the Maxwell rule, i.e. the thermodynamic potentials of the corresponding phases should be equated at the point of the transitions.

FIELD INDUCED PHASE TRANSITIONSIN FERRIMAGNETS

515

Figure 17.2 schematically shows the influence of the magnetic anisotropy on the high-field magnetization curves of free-powder samples. Critical fields Hcl and Hc2 can be written as (Zhao 1994)

Hc ,2 = IMd

Mfl( + za),

where A=

2KdKf

MaMelKa + Kfl and the upper signs being valued for Hcl and the lower ones for Hc2. An additional term A appears in the formula for Hcl and Hc2. This additional term will be zero when either Ka or Kf is zero. For simplicity assume that Kf >> Ka then 2Kd A ~ - =e MdMf

(17.5)

does not depend on the angle a. Notice that the same result is true when the rareearth sublattice is described in the Ising approximation, i.e.

where ~ is the unique vector of the Ising axis. This approximation often has a considerable utility for the description of large magnetic anisotropy of rare-earth ions. Magnetization curves in the canted phase follow immediately from eq. (17.4). It is convenient to express these curves in parametric form: h = (a + 2cos o01/2(1 - ecosc0, # = (a + 2 cos oL)1/2,

(17.6)

where M

H

--

h -

(MdMf)l/2 '

A(MdMf)I/2'

Me ma a= m+__ md Me' O < a < rr,

e = 2Ka/ MaMf.

Evidently, the portions of magnetization curves having negative first derivative d#/dh correspond to the unstable phases. The first-order transitions points of the jumplike magnetization curves should be determined using the Maxwell rule, i.e. the

516

A.K. ZVEZDIN

Fig. 17.3. The h-¢ phase diagram of the free powder sample for a uniaxial anisotropy); h = H / A ~ , ~ = 2 K a / A M a M f . A C , B D are the second order curves, C - E - D is the first order curve (see figs 4.1, 4.2 as well).

A

z--N I O'

~

D

E

B f

lit

~4 Fig. 17.4. The h-¢4 phase diagram of free powder sample for tetragonal symmetry;A6', BC' (CD, O 0 ~, C ' D , D E ) are the second (first) order curves (see figs 4.3, 4.8). thermodynamic potentials of the corresponding phases should be equated at the point of the transitions. Figure 17.3 show the typical magnetization curves for the different values of the parameter ~ defined by the formula (17.5) and the associated phase diagrams. By the same procedure as in the case of uniaxial anisotropy, one obtains for the tetragonal and hexagonal structure similar equations in the parametric form. Figures 17.4 and 17.5 show schematically the phase diagrams of the tetragonal and hexagonal ferrimagnets. As above, it is suggested here that IKfl << IKdl. There is a striking similarity of topological features of these phase diagrams with phase diagrams of monocrystals discussed in the section 4. In the same way it is possible to construct H - T phase diagrams for these free-powder samples, but this is beyond the scope of this chapter. The associated magnetization curves can be determined by the following parametric equations taking into account the corresponding equation for the magnetic anisotropy energy of the tetragonal and hexagonal ferromagnets:

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

O~~o

2

517

~E

~6

Fig. 17.5. The h - e 6 phase diagram of free powder sample for hexagonal symmetry; Bt3', AA' ( O 1 0 ~ , 020~, BD, AD, 03D, DE) are the second (first) order curves (see figs 12.2, 12.5). 20 16

DyCo12B6 ..o"'" .~.........-""'

12

...........o ..........

.,0.....,0.0 ..... 8 ,...C'" ~000.0.0.0-0O"

0

i

0

I

I

I

10

20

t

l

30

I

40

/~(r; Fig. 17.6. Free-powder magnetization for D y C o l z B 6 at 4.2 K. The circles represent measurements in quasi-continuous field and the dotted line represents a measurement in a field varying linearly with time (after Zhou et al. 1992a). Tetragonal s y m m e t r y : # = (a + 2COS o01/2, h=(a+2c°sc0]/2

(sine°) l-e4

sinc~

"

Hexagonal symmetry: # = (a + 2 c o s cO 1/2, h = (a + 2 c o s a ) 1/2

)

sin 6 a 1 - e 6 sin------~ '

w h e r e 0 < a < 7r, e 4 = K4/mdmf for tetragonal symmetry, and ~6 = K 6 / M d M f for h e x a g o n a l symmetry. T h e constants K4 and K6 are d e t e r m i n e d in subsection 3.2.1.

518

A.K. ZVEZDIN

Figure 17.6 shows the free-powder magnetization for D y C o 1 2 B 6 a t 4.2 K. This is a good example of the complete bending process in a ferromagnet. The magnetization curve in the canted phase is nearly linear indicating that the Dy sublattice anisotropy is much larger than that of the Co sublattice. It is seen that the effect of the magnetic anisotropy of the Co sublattice is rather small. From the slope in the field region where the canted phase exist, the interaction between the rare-earth and cobalt sublattices can be derived to amount JDyco/k = --4.1 K (Zhao 1994). Figure 17.7 show the free-powder magnetization curves of Ho2Fe17_~Al~ compounds in which only the first critical field Hcl is attainable. The second critical field Hc2 is very large here. In all measurements in figs 17.6 and 17.7 a quite good 20

'

I

'

i

'

I

16

...... ~ifOD::'~:'a'~ • O ............ 0~:::::~...... <>

[]

ooo~.ooOOO-O.."o'"'o'o''""°'"'°'c" ........ ,"3.

12 1200 0 m O [ ] • O 0

[]

rn•

~<>oooooo<>

o o•

o[]o

§

$$

o x=l [] x = 2 <> X = 3 A X=4

A e~ mm

vv

v

X=5

V VVV ~V

0

i

0

l

i

10

16

'

l

I

20

I

'

1

I

30

I

~

40

I

' A ............. ~'......... ~'" ......,"

12

A o .............

..... @-'''÷'ae ............ 4-................. '~" ....... ,+...÷-q: .... /~ ............... ,

_., +..... ~:..~÷'*"

8

o~ -.,~,~'"''""~"..x~-.-'X~A

.~,~'V'"V'~'~AZxA • 'xA"zxzx

4

O

I~ Ov

A x= 8

Du [] o o oo

v x=9 + x = 10

o I

0

o x=5 o x = 7

8

moo• ° ° 0

-

[] x = 6

o o 08

_ ^OOO O

eO<>C'v unS8

"' """

.....

I

10

i

I

20

i

I

30

i

40

Fig. 17.7. Free-powder magnetization curves of Ho2Fel7_xAlz compounds measured at 4.2 K. Circles stand for the step-field measurements and dotted lines represent measurements in which the field decreases linearly with time (after Jacobs et al. 1992b).

FIELD INDUCEDPHASE TRANSITIONS1N FERRIMAGNETS

519

linear magnetization in the canted phase is found with little influence of the magnetic anisotropy. A number of ferrimagnets have free-powder magnetization curves which are more complicated in the canted phase. Figure 17.8 shows the free-powder magnetization of Erz_~YxCoTB3 for z = 1.9 and 1.7. The occurrence of the first order magnetic phase transitions is clearly seen in these curves. 12.0

i

I

i

I

•. 0,," .....

i

......... O . ............... o

..::y"

,-, 8.0

..~i?.6%:,Cr..O'..::%7.::';'% "Q~"

<,7 4.0

/ / ~ 0 0 . 0 0 "0'0

0.0

T=4.2 K

I

0

Er0.sY2.sC°l 1B4 Free powder

""

I

I

I

10

I

I

20

i

30

40

B(r)

15.0

I

Er0.75Y2.25CoI 1B4 Free powder

/ i

50

/

:"i

I

i

/o-----

u

~o-

/

T=4.2K

,-,10.0

'

i

i'

/ <,

''''U'' rvO.00.,...~ 0.0

'°°°'~i

0

I

10

i

I

20

I

I

30

I

40

/~ (/3 Fig. 17.8. Free-powder magnetization at 4.2 K of Er3_xY~CollB4 compoundswith z = 2.0 (top) and 1.9 (bottom) (after Zhao 1994).

520

A.K. ZVEZDIN 16.0 _.--.---O

.....::....oi" ...- ." /0 //

ErY2Co 11B4 ,_owder Free 13

12.0

T=4.2 K

~

// /o / - /

~ 8.0

~J

/

4.0

000.0.0.0. 0 O0

i

0.0

0

I

/ / -

OD':f''" =

10

f

i

20

I

i

30

40

(73

L

I

~

I

i

I

i

I

I

16.0 Erl.IY1.9COllB4 Free powder T=4.2 K

~, 12.0

f -i¢ .//

//

8.0

// .//

~ 0 0 0 . 0 , 0 , 0 .....0....0,..., 0,0" ........

4.0

0.0

=

0

I

10

f

I

20

i

30

40

/~ (73 Fig. 17.9. Free-powder magnetization at 4.2 K of Er3_~Y=CoHB4 compounds with z = 2.5 (top) and 2.25 (bouom) (after Zhao 1994).

Figure 17.9 shows the magnetization curves for (Tm,Y)Co7B3 and (Er, Y)CollB4, (Tm,Y2)CoIIB4 compounds. All these results illustrate that when the magnetic of the d-sublattice in the basal plane can not be neglected in comparison with the anisotropy of the rare-earth sublattice the magnetization process is rather complex.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

521

18. Spin-flop transitions in itinerant metamagnets Spin-flop transitions in ferrimagnets have been considered above in detail both theoretically and experimentally. The most common materials, which have been the setting for most of the research on these phase transitions so far, are compounds of rare-earth elements and transition metals of the iron group: the iron garnets R3FesO12 and the intermetallic compounds R,~Tm, where R is a rare-earth element and T a transition metal. Such systems are generally described in the approximation of a 'rigid' d-sublattice, in which the magnetization of the d-sublattice is assumed to be independent of the external magnetic field and the exchange field exerted on it by the rare-earth sublattice. Strong interest has been caused by itinerant f-d metamagnets, which in a sense are the opposite of the materials mentioned above. In them, the d-subsystem is a weak itinerant ferromagnet and can undergo a metamagnetic phase transition with a substantial jump in magnetization. Some typical members of this family are YCo2, LuCo2, and RCo2, which exhibit metamagnetic transitions in the electronic d-subsystem from a paramagnetic state to a ferromagnetic state under the influence of a magnetic field. Magnetic field-induced transitions of this sort were predicted by Wohlfarth and Rhodes (1962). According to the theory of Bloch et al. (1975), Cyrot and Lavagna (1979), Yamada et al. (1984), such transition is possible in YCo2; it was indeed observed (Goto et al., 1989, Murata et al. 1991) in fields on the order of HM ~ 106 0 e and is explained by the peculiarities in energy dependence of the density of states N(F_,) of the d-electrons near the Fermi level. The critical field decreases in the compounds RxYl-zCo2, since metamagnetic transition arises in an effective field which is the sum of the external field and the molecular field exerted on the d-ions by rare-earth ions (Wohlfarth and Rodes 1962, Cyrot and Lavagna 1979, Duc et al. 1988, Steiner et al. 1978, Ballou et al. 1992). Another possibility for a decrease in the critical field of a metamagnetic transition is realized in compounds of the type (R,Y)Co2_xAI~ (Aleksandryan et al. 1985). Our purpose in this section is to discuss magnetic-field-induced phase transitions and the H - T phase diagrams in itinerant metamagnets with an unstable d-sublattice. This problem is important to the magnetism of f-d intermetallic compounds in general, since according to the present understanding of itinerant ferromagnetism the d-subsystem in such compounds is frequently an unsaturated (weak) ferromagnet with all the characteristics of this situation. We first consider the ground state of an f-d itinerant ferrimagnet at T = 0. The thermodynamic potential of the d-subsystem of a weak itinerant ferrimagnet is customarily expressed as an expansion of magnetization power (Wohlfarth and Rodes 1962) 1

1

1

F = ~ a~T~2 -4- ~ b~r~4 -[- g cfn 6,

(18.1)

where a, b, and c are coefficients whose temperature dependence is determined by the particular band structure and/or by spin fluctuations. Let us assume, as

522

A.K. ZVEZDIN

is customarily done in the theory of itinerant metamagnetism, that the following relations hold: a > 0, b < 0, c > 0. We consider the case in which the d-subsystem is itself paramagnetic down to T = 0. It would seem at first glance that in this case the f-d system as a whole should also be paramagnetic, since the molecular field created by the d-subsystem at the f-ion must be zero. However, we will show that this is not the case. Actually, the paramagnetic state (i.e. a state with a zero spontaneous magnetization) of such a subsystem is unstable in the presence of an f ion with a degenerate ground state, and a spontaneous magnetization arises in the f-d systems. The onset of spontaneous magnetization below the threshold., i.e. formally, in the paramagnetic phase, can be seen directly on the magnetization curves reported by Ballou et al. (1992). The physical meaning of this phenomenon can be explained as follows. We consider a spin fluctuation in a system of d ions surrounding the f ion of interest. We assume that this fluctuation causes a splitting of the ground state of the f ion. The meaning here is that the f ion becomes magnetized, and in its turn causes further magnetization of the surrounding ions by virtue of the f-d interaction. A spontaneous magnetization thus arises in the d-f system. In other words, the symmetry under time reversal is spontaneously broken. (In the case of an isolated ion, of course, quantum fluctuations should lead to the restoration of symmetry under time reversal, but in the case of cooperative instability discussed below this symmetry breaking has a completely real meaning.) This conclusion can be drawn as a consequence of a more general position - as a manifestation of crossover instability in magnetically ordered systems (Zvezdin et al. 1976, 1985). The lowering of the energy due to the splitting of the ground state of the f ion is equal to

-.XmMe, where Me is the magnetic moment of the f ion, m is the amplitude of the magnetization of the cloud of magnetized ions around the f ion, and A is the f-d exchange constant. When the paramagnetic d-subsystem becomes magnetized, its energy increases by an amount

v m Z /2xcl, where Xd = a - 1 is the magnetic susceptibility of the d-subsystem, and V is the volume of the magnetized region around the f ion. The instability of the uniform state (m = 0) is obvious even in an arbitrarily weak interaction, since the improvement in terms of energy is linear in m, while the degradation is quadratic. If the concentration of m ions is high enough, i.e. if magnetization clouds around the f ions overlap, there will be a cooperative spontaneous magnetization of the f-d-subsystems in the systems. In this case the total energy per molecule is

E=-~1 m Z / x d _ t A m M ,

(18.2)

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

523

where t is a relative concentration of f ions. Minimizing (18.2), we find (18.3)

m = XdtAmM.

This discussion can easily be generalized for nonzero temperatures Hell. The thermodynamic potential # of this system is

1 a m 2 + -~1 bm 4 + -~1 cm 6 ~5 = -~

-

-

t ~o n*~ Mf(h) dh,

(18.4)

where Mr(h) is the magnetization of the f-sublattice expressed as a function of the effective field H e f f = H - Arfi (in this case, the external field H is assumed to be zero), of temperature T, and of the magnetic moment # of the f ion. In particular, for the Gd +3 ion we have Mr(h) = # B 7 / 2 ( # h / T ) , where B ( x ) is the Brillouin function, and/z = 7#B. Minimizing (18.4) with respect to m and ¢, we find a m + b m 3 + c m 5 - t A M f ( A # m / T ) = O.

(18.5)

This equation determines two phase-transition points: the magnetic-ordering temperature Te, which is determined by the condition XdXfA2t = 1, and the (usually lower) temperature of the metamagnetic transition to a 'strong ferrimagnetic' state,

TM. The threshold concentration for the t* transition to a cooperative behavior of the system can be estimated on the basis of the following qualitative considerations, which are based on arguments concerning the size of the magnetized cloud around the f ion. This is determined by the correlation radius of fluctuations in the dsubsystem. To find it we work from the Ginzburg-Landau energy E = f (A(gradm) 2 + m2/2Xd) dV,

(18.6)

where A is a nonuniform-exchange constant (exchange stiffness) of the d-subsystem, and the integration covers the entire volume except a small volume around the f ion with a diameter of an interatomic distance. We assume m ( r ) = rn at the boundary of this small volume, and at infinity we naturally assume m ( r ) ~ O. Using these boundary conditions and Euler-Lagrange equations for functional (18.6) we then find re(r) = (ma/47rr) e x p ( - r / p ) ,

(18.7)

where p = (AXd)I/2 is the correlation radius which we are seeking. Setting A = 5 x 10 -9 esu/ion and Xd ~ 2 × 10 -7 esu/ion, we find p = 3A. This value is of course too small for the continuous approximation to be reliable, but it will do for a qualitative argument. The magnetization distribution described by (18.7) is obviously degenerate under the direction of the spins of the f ion and the d-subsystem (the matrix). The actual

524

A.K. ZVEZDIN

wave functions of this composite formation is therefore a linear combination of all possible wave functions of this system which are included in the degeneracy space, i.e. which are characterized by different spin orientations. A wave function of this sort describes a localized spin-wave mode which includes the joint motion of f and d spins. A detailed description of this mode goes beyond the scope of the present chapter, but we would like to point out that this picture of an isolated f ion in a dmatrix may be observed at a low concentration of f ions, at which local spin modes do not overlap. The mode overlap determines the interaction of centers, and if the concentration of centers is sufficiently high, it will lead to cooperative effects. The threshold concentration t* can be estimated with the help of a percolation model. If the distance between the two nearest f ions is smaller then p, then these ions perturb each other substantially. As a result, a correlation arises in the directions of spins. The percolation threshold can then be found approximately from N * p 3 = q, where q is a numerical factor of the order of one; for our purposes we can set it equal to one. Thus the critical threshold concentration is t* ~ ( a / p ) 3 ~ 0.1,

where a is the average interatomic distance in the compound. We turn now to phase transitions induced by a magnetic field and to phase diagrams, assuming that the concentration of f ions is high enough that the behavior of the system can be assumed to be cooperative. In other words, we assume t > t*. In this case we can ignore the nonuniform-exchange energy and we can write the thermodynamic potential in the form (Zvezdin and Evangelista 1995) 1 1 q5 = L a m 2 + - b m

2

1 4 + L c m 6 -- m ( H 2 + A 2 M 2 - 2 A M f H c o s ¢ ) I / 2 -

4

6

- M f H cos ¢ - T S ( M f ) , w h e r e ~ d is the energy of the d-subsystem found from (18.1), and ¢ is the angle between the magnetic field H and magnetization Mf of the f-sublattice; S(Mf) is the entropy of f-subsystem, Let us first consider the case T -- 0, when Mf = const. Minimizing (18.8) with respect to m and ¢, we find

(18.9)

hd(m) = Heff(Mf, ¢), M f H sin ¢(1 - A m / H e f f ) = O, ~)S

-T

cos ¢(1

c)Mf --

where/~eff = / t - A/~ff.

-

Am "~eff ) --

(18.10)

mA2mf ~eff

-- O,

(18.11)

FIELD INDUCEDPHASETRANSITIONSIN FERRIMAGNETS

525

Let us approximate function m(h) in the form

( Xdh,

m(h)

mr,

h < Hp, h > Hp,

(18.12)

where Hp is a threshold field for metamagnetic transition in the d-subsystem. From (18.12) we can directly determine the functional dependence h(m). It is simple to verify by direct calculation that the values of magnetization on the descending slope of the m(h) curve (or the h(m) curve determined by eq. (18.12)), i.e. the values corresponding to a negative differential susceptibility 8m/Sh < 0, correspond to unstable states of the system. For this reason, the interval of magnetization values from XdHd to ml is of no interest for our purposes, in our approximation of re(h). For simplicity we will also ignore the hysteresis in the m(h) curve. We should also mention that in a real metamagnet the magnetization of the dsubsystem is not constant in ferromagnetic phase. Instead it increases slightly with an increasing field. This point is simple to deal with mathematically, but taking it into account makes the analysis more complicated without leading it to any qualitatively new facts. Since it tends to obscure the overall picture, we will ignore it. Equations (18.9), (18.10), and (18.11), along with the stability conditions (82~ > 0), determine the following solutions (phases) and regions in which they exist: W :¢=0, Fsl : ¢ = 0 ,

m=Xd

Fs2 : ¢ = 7r, C :0<¢<7r,

m=ml, m=ml, O=0, m=ml,

Fe : ¢ = 0 ,

0=Tr,

(H-AMf), m=ml,

O=0,

where a is the angle between the magnetic field and rS. Phases W (Fsl and Fs2) can be weak (strong) ferrimagnetic collinear phases, Fe is a ferromagnetic phase, while phase C is a canted (angular) phase. To plot H - T phase diagrams, we need to equate the energies of coexisting phases and find the curves of first-order phase transitions from these equations. As ~d(m) in our approximation we should use the quantity qSd =

/?

hd(m) dm.

Phase diagrams in fig. 18.1 give a general picture of magnetization curves and the critical field of itinerant metamagnets. The nature of the H-Mr phase diagrams depends on the relation between Hp and Aml. If Hp < Aml/2, the phase diagram will have, in addition on the 'ordinary' lines of second-order phase transitions between collinear and angular phases (which are determined by the known expressions for critical fields Hcl = Alml -Mf[ and Hez = Alml +Mfl), lines of first-order transitions, which are determined by the equation (under the condition AXd << 1) HM --

Hp - AMf

1 -- 2Mf/m

(18.13)

526

A.K. ZVEZDIN H. N Xml

~.m 1

H

"2~O N #

[] X-1Hp

mI

~-lHp m1

M

a)

M

c)

H

H

Y

~,n1

H

~.-IHp

m1

x-lG

M

b)

M

d)

Fig. 18.1. Phase diagrams of an itinerant ferrimagnet with an unstable d subsystem, a) /fp < Am1/2; b) Hp = ),ml; c) Am1~2 < Hp < k,ml; d) Hp > k,ml. The phases W, Fsl, Fs2, Fe are labeled A1, A~, B2, A2. The inserts are schematic diagrams of the corresponding magnetization curves (after Zvezdin 1993).

In the interval Aml/2 < Hp < Am1 the phase diagram (fig. 18.1 b)) has two interesting features. Not only W --+ Fs2 phase transitions can occur here, but also first-order transitions from the weak ferrimagnetic phase A1 to the canted phase C. Transitions of this sort have not previously been seen in isotropic systems. Lines QL and Q'L* are determined by eq. (18.13), and lines Q N and Q'N* by H = AIMf + ETI~1 l,

where e =

H = ,~[Mf - ETn 1 l,

((2Hp/Aml)- 1) 1/2. The points Q and Q* have coordinates Mf = l m l ( l +

e), H = 1Am1(1 q:e). The second unusual feature of this diagram is that there can be an 'inverse transition' from a strong ferrimagnetic phase to a weak ferrimagnetic phase. This transition would occur as the magnetic field is increased (Zvezdin and Utochkin 1992). The

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

527

same feature is seen, even more clearly, at Hp > )~ml (fig. 18.1 c)), where an inverse transition occurs abruptly, without an intermediate canted phase. Figure 18.2 shows the H - T phase diagrams of itinerant ferrimagnets deduced from the minimization of the thermodynamic potential (18.8) (Zvezdin and Evangelista 1995). Substituted intermetallic compounds of the Laves Phases Y1-tRt(COl-~Alx)2 (R is a heavy rare earth element) with a negative intersublattice exchange interaction may serve as representatives of the itinerant ferrimagnets with one unstable magnetic sublattice. One, stable, magnetic subsystem of these two-sublattice ferrimagnets is formed by the localized moments of the 4f-shells of rare earth ions. The second itinerant, magnetically unstable, subsystem is formed by magnetic 3d-electrons of Co hybridized with 5d(4f) electrons of rare earth ions. For this subsystem, the density of d-states at the Fermi level and the magnitude of d-d exchange are such that the Stoner criterion for the appearance of itinerant ferromagnetism is not fulfilled (Bloch et al. 1975, Yamada et al. 1984). Because of this, YCo2 and LuCo2 compounds are exchange-enhanced itinerant paramagnets (Lemaire and Schweizer 1966). In RCo2 compounds with magnetic rare-earth ions, both magnetic subsystems are magnetically ordered (Bloch and Lemaire 1970, Levitin and Marcosyan 1988). In this case, the magnetic ordering of the itinerant subsystem is extrinsic and due to a magnetizing molecular field acting on the d-subsystem. The investigations of the f-d exchange on the properties of the d-subsystem have been carried out on (RY)Co2 compounds for R = Gd, Tb, Ho, and Er (Lemaire and Schweizer 1966, Levitin et al. 1984a, b, Duc et al. 1988a, b, 1989, Gratz et al. 1986, Baranov et al. 1989, 1990). In order to study the influence of the f-d interaction on the magnetic order of the unstable d-subsystem, Ballou et al. (1992) investigated the magnetic properties of intermetallic compounds Yl-tGdt(Col-xAlz)2, where 0 ~< t ~< 0.2, 0 ~< :c ~< 0.105. Partial replacement of cobalt by aluminum in YCo2 leads to a decrease in the field of metamagnetic transition and to the appearance of itinerant ferromagnetism in Y(COl_~AI~)2 compounds for z/> 0.12 (Aleksandryan et al. 1985). Figure 18.3 shows the field dependence of magnetization at 4.2 K for certain compounds belonging to the system Yl-tGdt(Co0.095A10.05)2. It is clear that for small replacements by gadolinium (t < 0.12) there is no spontaneous magnetization. Increasing the gadolinium content leads to increased weak-field susceptibility; the magnetization curves of compounds with Gd become nonlinear, and exhibit a tendency toward saturation in strong fields. Compounds with Gd content t /> 0.12 possess a spontaneous moment. The value of this spontaneous moment decreases as the gadolinium content increases, passing through a minimum at tcomp ~ 0.17-0.18 and then increasing once more (i.e. this concentration marks a balance point with respect to compensation). In compounds near tcomp, kinks in magnetization curves take place, which are characteristic of a transition from a collinear ferrimagnet to noncollinear phase. Figure 18.4 shows the magnetization curves of several compounds of the Y1-tGdt(Coo.915Alo.oss)2 system at 4.2 K. It is clear that the original compound Y(Coo.915Alo.oss)2 is an itinerant metamagnet with a critical magnetic transition field

528

A.K. ZVEZD1N

a) 2 1.5 1

F~2 0.5 i

I

i

~

i

i

r

50

100 150 200 Temperature (IO

250

300

50

100 150 200 Temperature (K)

250

300

b)

1.5

0.5

e) 2.5

,•1.5

W 1

0.5 I

50

100 150 Temperature (K)

I

200

Fig. 18.2. H - T phase diagrams of an itinerant ferrimagnet with unstable d-subsystem, a) Hp < ,kin1/2; b) )~ml/2 < Hp < Aml/2; c) Hp > )~ml (after Zvezdin and Evangelista 1995).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

529

1M p~/f.units

a) $++ 1 4-+

1

xX

0.8

.Z~Zi ^L~

0.6 AZ~ ~ ZIA

~,~l

+4-

++ +4/ ++

xx

0.4 0.2

xx XX xx

xX~X~~xx 2

++ /

++++++

xx ++ ~o~x+++ +++ x~.+ +

x ~.>f.+

,'~<". x~-+ ~xX~+ "{ "''1" T

0

i

I

50

I

I

I

t

100 150 200

250

b) 0.5

-

4 + + 4.4. + + .t.+ +.t. 4.-I-+ 4- -I-4.-I-'1-+ "1"t"+ + + ~

0.4 0.3 0.2

7 .~IA

0.1 Nxx

x×6

I

t

P

I

I

I

0 50 100 150 200 250 H, kOe Fig. 18.3. Magnetization curves of the compounds (Yl-tGdt)(Co0.95A10.05)2 at 4.2 K and t = 0.0 (1), 0.04 (2), 0.1 (3), 0.12 (4), 0.15 (5), 0.18 (6), 0.2 (7) (after Ballou et al. 1992). HM = 225 kOe. As the gadolinium concentration increases, the metamagnetic transition field HM decreases, and for concentrations t/> 0.04 these compounds possess a spontaneous magnetization. At comparatively small gadolinium concentrations (0.04 ~< t ~< 0.06) the spontaneous magnetization is small in the magnetically ordered region, and the application of a field leads to a metamagnetic transition from a weakly ferrimagnetic to a strongly ferrimagnetic state. For a larger gadolinium content (t >~ 0.06) metamagnetic transitions are not observed: these compounds are in a strongly ferrimagnetic state even at a zero field. The saturation of magnetiza-

530

A.K. ZVEZDIN M 0.6

/as/f.units 0.6

DoDoDo~

0.5

xXXXXXxXxXX

0.4

um~

0.3

=

n=

XXX

0.3 ~o~

ooooo 2 5

0.2 0.2

:÷

x:JS

D °

0.4

,~

++~,~

nmO~O~~ ' ~

o :natc~az~a ; ,

0.1

0

50

aa~aaaaaaA1 aaaa '

100

150

,

6:

0.1

,

200

0

a)

I

I

50

100

I

I

150 200

I

H, kOe

b)

Fig. 18.4. Magnetization curves for the compounds (Yl-tGdt)(Coo.915Alo.o85)2 at 4.2 K and t = 0.0 (1), 0.02 (2), 0.04 (3), 0.10 (4), 0.15 (5), 0.18 (6), 0.20 (7) (after Ballou et al. 1992). M, t~/f.uniis

0.8

0.8

.~

0.6

0.6

-

0.4

0.4

-

0.2

0.2

J

0

a)

0.05 0.1 0.15

0.2

b)

0.8

0

0.05 0.1 0.15 0.2

0

0.05 0.1 0.15

0.8

0.6 0.4

L

0.6 0.4

0.2

e)

I

0.2

0

0.05 0.1 0.15 0.2

d)

0.2 t

Fig. 18.5. Dependence of the magnetization on gadolinium content t for various compounds from the system (Yl_tGdt)(Col_xAl~)2 at 4.2 K. (o) - spontaneous magnetization of the weakly ferrimagnetic compounds, (A) - magnetization of the weakly ferrimagnetic compounds in the field of 270 kOe, ([]) spontaneous magnetization of the strongly ferrimagnetic compounds: a) x = 0.105, b) 0.075, c) 0.07, d) 0.05 (after Ballou et al. 1992).

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

531

HM, kOe

300 25O 200 150 100 50

\~~,~.

i k z~ I

0.02 0.04 0.06 t Fig. 18.6. Dependence of the metamagnetic transition field on gadolinium concentration t for the system (Yl_tGdt)(Col_~Al~)2; x = 0.07 (1), 0.085 (2), 0.105 (3). The straight line is calculated using eq. (18.13), and the points are experimental data (after Ballou et al. 1992). tion decreases with increasing t, and this causes it to increase once more. As with compounds with a low aluminum content, near this concentration a transition is observed from the collinear ferrimagnetic phase to a noncollinear phase in the external magnetic field. A decrease in the metamagnetic transition field is also observed in other systems, i.e. Y~_tGdt(COl_~Al~)2 with a high aluminium content (x + 0.07 and 0.105), as the gadolinium content increases. This decrease is followed by the appearance of a weakly ferrimagnetic phase, which is then replaced by a strongly ferrimagnetic phase. The only difference is that the increase of the aluminium content causes the metamagnetic transition field in these compounds to increase. It also causes the concentration region, where paramagnetic and weakly ferromagnetic metamagnetic phases can exist, to shrink. At the same time, the values of magnetization and gadolinium concentrations, at which a compensation of the magnetic moments of f- and d-subsystems is observed, are close for all these systems, although with certain differences. Furthermore, the spontaneous magnetization of a low-aluminiumcontent system with x = 0.05 is also close to the magnetization of those compounds corresponding to systems with larger amounts of aluminium (x = 0.07, 0.085, and 0.105). All of these features are easy to see in figs 18.5 and 18.6, where the basic characteristics of all systems under discussion are plotted. Figure 18.5 presents data on magnetization as a function of gadolinium concentration for systems with various x values. Here the spontaneous magnetization of weakly ferrimagnetic and strongly ferrimagnetic samples are shown as well as magnetization in the field of 270 kOe for samples with metamagnetic transitions. Figure 18.6 shows measured dependence of the metamagnetic transition field on gadolinium content for systems with various

532

A.K. ZVEZDIN

aluminium contents. A comparison of the experimental data with calculations for f-d magnetic systems shows that they agree in most cases, at least qualitatively.

Conclusion Field induced phase transitions connected with breaking of a collinear alignment of sublattices in ferrimagnetics have been considered here. It is assumed that the magnetic anisotropy of these materials is small in the sense that the energy of a magnetic anisotropy is smaller in comparison with the energy of the exchange interaction of sublattices. The magnetic anisotropy in such materials is best manifested near a compensation point, where critical fields of transitions to a canted phase tends to zero. In this connection H - T phase diagrams are very complicate near Tc and composed of curves of the first and second order phase transitions and of critical, tricritical points, etc. From this standpoint the anomalous behavior of many physical properties near Tc can be explained. Many rare earth ferrimagnets should be referred to as strongly anisotropic materials, i.e. crystal field interactions are much stronger than the exchange interaction between sublattices. The field induced phase transitions in these materials are more complicated and of interest. We shall address these questions elsewhere with a special attention paid to mechanisms that can lead to jumps in a magnetization curve: field induced crossing levels and the magnetic Jahn-Teller effect, crystal field related transitions, metamagnetism of 3d-sublattice and other. Finally, mention should be made of field induced phase transitions in an amorphous rare-earth-transition metal system where strong local anisotropy effects are of importance.

Acknowledgements I acknowledge fruitful interactions with A.S. Andreenko, K.R Belov, A.S. BorovikRomanov, S.L. Gnatchenko, L.R. Evangelista, N.E Kharchenko, A.M. Kadomtseva, I.K. Kamilov, B.E Khrustalev, I.B. Krynetskii, R.Z. Levitin, EV. Lisovskii, V.M. Matveev, A.A. Mukhin, S.A. Nikitin, R.V. Pisarev, A.E Popkov, A.I. Popov, R. and H. Szymczak, V.K. Vlasko-Vlasov, Y. Zawadzki. In particular I would like to thank Prof. Dr. K.H.J. Buschow for his valuable remarks and kind help in preparing this report.

19. Appendix. Microscopic calculation of the thermodynamic potential of the non-equilibrium state We shall find the thermodynamic potential of the non-equilibrium state as follows (according to Leontovich (1944)). The non-equilibrium state of a magnetic crystal containing interacting d and f electrons will be defined by the magnetization of the d-subsystem, i.e. by the value of ~rd. As we have mentioned above, this means that

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

533

the f subsystem is in equilibrium with the subsystem of d ions whatever the value of 2Qd. We shall also introduce an auxiliary (imaginary) field h which is conjugated with l~rd and only affects d ions. The Hamiltonian of the d-f system interacting with this field is (A.1) where H is the Hamiltonian of the d-f system where ft = 0 and includes interaction with the external field, the exchange interaction and the crystal field; the summation is carried over all d ions; #d and Sd are the magnetic moment and spin of the ion. The free energy depending on h, T and H is F(f0=-TlnSpexp

-~-

.

(A.2)

For the sake of brevity we shall omit variables T a n d / t in the functions F and ~b (see below). We wish to obtain the potential in variables 2krd, T, H. It can be found from (A.2) by using the Legendre transformation: +(l~rd) = F(h) + fiM'd;

(A.3)

]~rd _

(A.4)

OF

The latter relations define the function ~(l~rd) in a parametric form (h is the parameter). It can be taken as a thermodynamic potential of a non-equilibrium state with a given value l~rd. The auxiliary field h 'prepares' this state. It is easy to see that • (Md) reaches the minimum in equilibrium. So the calculation of the potential ~(Md) is reduced to a calculation of the equilibrium function F(h) with eq. (A.2) followed by the elimination of the auxiliary field with the Legendre transformation. This procedure is e~ily generalized to obtain potential q~(Md,Mr) in anon-equilibrium state with any Md and Me. We shall calculate ~(Md) for a specific d-f system with the Hamiltonian

(A.5) d

2 d,f

in which

Itf= E

H i.

f,

i

= V° + V° + V° + V $ - 9j

.Z

;

534

A.K. ZVEZDIN

V~ are the operators of the crystal field. H should be given in the form g---~ H d - - h E # d ~ - ~ d

~1 ~ i d f ( ~ f ) ~ -'}-Hf + d,f

(A.6)

+ ~1 Z Iaf<~a/<Jf/+ 1 E If(Sd - (Sf))(Sf- (Sf)) d,f

d,f

- ~ro + V. Here V= 1 d,f

(A) =

Sp A exp(- FIo/T) Sp exp(- £ro/T)

Expanding F(ft) defined by eq. (A.2) over V, we get F(h) = - T l n Sp exp

~

- / / d - h~--~#aSd + ~ ~ Idf(Sf)Sd d

- T In Sp exp

- ~

d,f

-/

(A.7)

Hf + ~ d,f

1

2 ~ / d f ('°qf)('~d/ + o(g2)" d,f

Putting F(flt) into (A.3) we calculate (A.8)

~(~rd) = ~d(~rd) + Ff(Beff),

~ d ( ~ ) = - T l n Sp exp

- y

Ha d

1

2 E fdf<~'~d)('J~f) -b h]~fd, d,f

/~rd -

8F Jz'

a J ~ + ~ Z ~fg~<Jf> d,f

-

(A.9)

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS Ff0qeff) = - N T

In Sp exp

-

1 (vo + vo + v O _ g j ~ B O j , ) } ,

535

(A.10)

where N is the number of f ions;

~qoff=/~ + ~%, /~M --

g j-

- 29j

1

E fdf(ffd) = --/~fd; d,f

~d(Md) is the actual thermodynamic potential of an anisotropic ferromagnet in a magnetic field. In particular, if the magnetic energy ( - M H ) , the anisotropy energy and the energy of f-d exchange are lower than the d-d exchange energy and if we expand the thermodynamic potential Oa(Md) defined by eq. (A.9) in these energies then

(A.11)

Od(Md) = ~i~0(Md) -- MdH ~- Hd(Md).

The small quantity of this expansion is

where a is atomic spacing, R - radius of d-d-interaction, z - number of nearest neighbors of the d-type in the interaction sphere of a d-ion. The function Ff(Heff) is also expanded over V6 (llV~ll << IlV°ll, ~ = 2, 4, 6) (A.12)

Ff(/~eff) = Fc(/]eff) -}- (V6), FC(Heff) = - N T l n Sp p~,

(V6) : Sp V6p°, p°:exp

-~

(VO + VO + v60-gJ#BJHeff)

.

Of course, the function FC(/~eff) does not vary with rotations of Heff around the hexagonal axis c because it depends only on two variables:

0qeff~

I/4effl, 0 = a r c c o s -

Heft

If/~effJ-~ then Fc(/]eff) = FC(IHeffl) = _ [ H e , Mf(x) dz,

J0

(A. 13)

536

A.K. ZVEZDIN

(V~) = ~ (M[V66]M')(M']p°[M), MM ~

where IM) are the eigenfunctions of Iz. In a crystallographic system of coordinates in which the plane ZOX is the symmetry plane, the matrix elements (~r[V66]~r') are known to be real. The sign ..~ indicates that wave functions [M) are taken in the crystallogr~iphic coordinates system. However, it is best to calculate (V66) in a system coordinate rotating around the 2 axis relative to the crystallographic system, so that the new axisJ?'l[/goff. Wave functions in the new IM) and the original system of coordinates [M) are related by [M) = ei~brM[J~r), in which ~bf is the angle in the basis plane between 3~ and/~eff. In view of this we get exp[i(M - M')g)e](M[V661rM')(M'lp°[M ')

(V6) = ~ MM'

: e i64~r ~ ( 2 ~ f l V 6 6 1 ~ f

q- 6) +

M

(A.14) ( M t -[- 6[V6IM')

-1- e -i6~bf Z

(M'[p°IM' + 6)

M~ 1

= -- - g f ( H e f f ) c o s 6(~bf + a ) ,

6

where 1

g Kf = ~ b 2 ' a

cos 6c~ sin 6a

v ~ + l)2 b

-

b2'

(~rlv661~r+ 6)(M + Sip°[M),

a = 2Re ~ M

b = 2Im Z

( ~[V6[~r + 6 ) ( M + 6[p°lM).

M

Adding together (A. 1 1), (A. 13) and (A. 14), we then find the required thermodynamic potential of the nonequilibrium state ~ ~ ( ] ~ d ) = ~iSd(J~d) --

[Herr

MdH -

Mf(x) d z J0

1

-- - Kf(Heff) cos 6(qgf+ a). 6

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS

537

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Radwanski, R.J., R. Verhoef and J.J.M. Franse, 1990a, J. Magn. Magn. Mater. 83, 141. Radwanski, X.P. Zhong, ER. de Boer and K.H.J. Buschow, 1990b, Physica B 164, 131. Ratajczak, H. and I. Goscianska, 1980, Phys. Status Solidi A: 62(1), 163. Rode, V.E. and A.V. Vedyaev, 1963, Sov. Phys. - JETP 18, 286. Rosencwaig, A., W.J. Tabor and R.D. Pierce, 1971, Phys. Rev. Lett. 26(10), 779. Rudashevsky, E.G., C. Legcuras and H. Le Gall, 1977, Solid State Commun. 24(11), 587. Sannikov, D.G. and T.M. Perekalina, 1969, Zh. Eksp. Theor. Fiz. (Sov. Phys. -JETP)56, 730 (in Russian). Schavrov, V.G. and E.A. Turov, 1963, Zh. Eksp. Theor. Fiz. (Soy. P h y s . - JETP)45(2), 349 (in Russian). Schavrov, V.G., 1965, Zh. Eksp. Theor. Fiz. (Sov. Phys. - JETP) 48(5), 1419 (in Russian). Silant'ev, V.I., A.I. Popov, R.Z. Levitin and A.K. Zvezdin, 1980, Sov. Phys. - JETP 51, 323. ShRimann, E., 1960, in: Solid State Physics, Electronics and Telecommunications, Vol. 3, Part I (Academic Press, New York) p. 322. Sinnema, S., R.J. Radwanski, J.J.M. Franse, D.B. de Mooij and K.H.J. Buschow, 1984, J. Magn. Magn. Mater. 44, 333. Sinnema, S, J.J.M. Franse, R.J. Radwanski, A. Menovski and ER. de Boer, 1987, J. Phys. F: 17, 233. Smirnova, E.L., V.I. Smirnov, Ju.I. Ukhanov and V.A. Petrusevich, 1970, JETP Lett. 11(9). Smit, H.H.A., R.C. Thiel and K.H.J. Buschow, 1988, J. Phys. F: 18, 295. Stacy, W.T. and C.J. Rooymans, 1971, Solid State Commun. 9(10), 2005. Steiner, W., E. Gratz, H. Ortbauer and W. Gaben, 1978, J. Phys. F: 5, 1525. Szewczyk, A., A. Dabkovski, S.L. Gnatchenko, N.E Kharchenko and H.M. Krsymanska, 1985, Acta Phys. Pol. A 67(5), 877. Szymczak, R., 1983, J. Magn. Magn. Mater. 35(1-3), 243. Szymczak, R., 1987, J. Magn. Soc. Jpn 11 Supplement S1, 39. Tagirov, R.I. and A.A. Glazer, 1978, Fiz. Met. Metalloved. 46(1), 75. Taylor, K., 1975, Adv. Phys. 24, 681. Taylor, A.S. and A. Gangulee, 1976, J. Appl. Phys. 47, 4666. Toxen, A.M., T.N. Geballe and R.M. White, 1988, J. Appl. Phys. 64(10), 5431.

Turov, E.A., V.G. Schavrov and Yu.P. Irchin, 1964, Zh. Eksp. Theor. Fiz. (Sov. Phys. JETP) 47(1), 296 (in Russian). Turov, E.A., 1987, Zh. Eksp. Theor. Fiz. (Soy. Phys. - JETP) 92(5), 1886 (in Russian). Tyablikov, S.V., 1956, Phys. Met. Metalloved. 3(1), 3. Tyablikov, S.V., 1958, Phys. Met. Metalloved. 8(I), 152. Tyablikov, S.V., 1965, Methods of Quantum Theory of Magnetism (Nauka, Moscow, in Russian). Tyablikov, S.V., 1967, Methods in the Quantum Theory of Magnetism (Plenum, New York). Valiev, U.V., G.S. Krinchik, R.Z. Levitin and K.M. Mukimov, 1979, JETP Lett. 29, 214. Velge, W.AJ.J. and K.H.J. Buschow, 1968, J. Appl. Phys. 39, 1717. Veselago, V.G., L.P. Maksimov and A.M. Prokhorov, 1968, Vest. Akad. Nauk USSR 12, 58 (in Russian). Vedernikov, N.E, A.K. Zvezdin, R.Z. Levitin and A.I. Popov, 1988, Zh. Eksp. Theor. Fiz. 94(8), 358 (Sov. Phys. - JETP 66(6), 1233). Verhoef, R., 1990, Thesis, Magnetic Interactions in R2Fe14B and Some Other R-T Intermetallics, Universiteit van Amsterdam, The Netherlands. Verhoef, R., ER. de Boer, J.J.M. Franse, CJ.M. Denissen, T.H. Jacobs and K.H.J. Buschow, 1989, J. Magn. Magn. Mater. 80, 41. Verhoef,, R., R.J. Radwanski and J.J.M. Franse, 1990a, J. Magn. Magn. Mater. 89, 176. Verhoef, R., EH. Quang, J.J.M. Franse and R.J. Radwanski, 1990b, J. Magn. Magn. Mater. 83, 139. Verhoef, R., RH. Quang, J.J.M. Franse and R.J. Radwanski, 1990c, J. Appl. Phys. 67, 4771. Visnovsky, S., V. Prosser, E. Zvara and P. Polivka, 1974, Phys. Status Solidi A: 26(2), 513. Vlasko-Vlasov, V.K., L.M. Dedukh and V.I. Nikitenko, 1981, Fiz. Tverd. Tela 23, 1857 (Soy. Phys. Solid State 23, 1085). Vlasko-Vlasov, V.K., L.M. Dedukh, M.V. Indenbom and V.I. Nikitenko, Sov. Phys. JETP 57(1), 1983. Vlasko-Vlasov, V.K. and M.V. Indenbom, 1984, Zh. Exsp. Theor. Fiz. (Sov. Phys. -JETP) 86(3), 1084 (in Russian). Vlasov, K.B., B.N. Novogrudskii and E.A. Rosenberg, 1976, Fiz. M e t . Metalloved. 42(3), 513. Vlasov, K.B., E.A. Rosenberg and A.M. Burchanov, 1980, Fiz. Met. Metalloved. 49(1), 1005.

FIELD INDUCED PHASE TRANSITIONS IN FERRIMAGNETS Vlasov, K.B. and E.A. Rosenberg, 1982, Fiz. Met. Metalloved. 53(3), 493. Vonsovsh."i, S.V., 1971, Magnetism (Nauka, Moscow). Wohlfarth, E.P. and P. Rhodes, 1962, Philos. Mag. 7, 1817. Yamada, H., J. Inoue, K. Terao and H. Komatsu, 1984, J. Phys. F: 14, 1943. Yamada, M., H. Kato, H. Yamamoto and Y. Nakagawa, 1988, Phys. Rev. B 38, 620. Zawadzki, J., R. Ballou, R. Gorges and R. Szymczak, 1993, Acta Phys. Pol. A 83, 209. Zawadzki, J., R. Szymczak, A.K. Zvezdin and I.B. Krynetski, 1994, IEEE Trans. Magn. 30, 866. Zhao, Z.G., 1994, Thesis, Magnetization Processes of Ferrirnagnets, van der WaalsZeeman Laboratorium, Universiteit van Amsterdam, The Netherlands. Zhong, X.P., R.J. Radwanski, ER. de Boer, T.H. Jacobs and K.H.J. Buschow, 1990a, J. Magn. Magn. Mater. 86, 333. Zhong, X.P., ER. de Boer, R.J. Radwanski, T.H. Jacobs and K.H.J. Buschow, 1990b, J. Magn. Magn. Mater. 92, 46. Zhong, X.P., F.R. de Boer, D.B. de Mooij and K.H.J. Buschow, 1990c, J. Less-Common Met. 163, 123. Zhou, G.E, X. Li, ER. de Boer and K.H.J. Buschow, 1992a, J. Magn. Magn. Mater. 109, 265. Zhou, G.F., ER. de Boer and K.H.J. Buschow, 1992b, J. Alloys Comp. 187, 299. Zvezdin, A.K., 1993, JETP Lett. 58(9-10), 719. Zvezdin, A.K. and V.M. Matveev, 1972a, Zh. Eksp. Teor. Fiz. 62, 260 (Sov. Phys. -JETP 62, 140).

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Zvezdin, A.K. and V.M. Matveev, 1972b, Izv. Akad. Nauk SSSR, Ser. Fiz. 36, 1441 (in Russian). Zvezdin, A.K. and S.G. Kalenkov, 1972, Fiz. Tverd. Tela (Sov. Phys. Solid State)14(10), 2835 (in Russian). Zvezdin, A.K. and A.E Popkov, 1974, Fiz. Tverd. Tela (Sov. Phys. Solid State)16(4), 1082 (in Russian). Zvezdin, A.K., A.A. Mukhin and A.I. Popov, 1976, Pis'ma Zh. Eksp. Teor. Fiz. (JETP Lett.) 23(5), 267. Zvezdin, A.K., A.A. Mukhin and A.I. Popov, 1977, Zh. Eksp. Theor. Fiz. 72, 1097 (Sov. Phys. - JETP 45, 573). Zvezdin, A.K. and A.E Popkov, 1977, Fiz. Met. Metalloved. 44, 689. Zvezdin, A.K. and A.F. Popkov, 1980, Fiz. Met. Metalloved. 50(2), 11. Zvezdin, A.K., Zorin I.A., Kadomtseva A.M., Krynetskii I.B., Moskvin A.S. and Mukhin A.A., 1985, Soy. Phys. - JETP 61(3), 645. Zvezdin, A.K., V.M. Matveev, A.A. Mukhin and A.I. Popov, 1985, Rare Earth Ions in Magnetically Ordered Crystals (Nauka, Moscow, in Russian). Zvezdin, A.K., G.V. Sayko and S.N. Utochkin, 1991, Fiz. Tverd. Tela (Soy. Phys. Solid State) 33, 3175 (in Russian). Zvezdin, A.K. and S.N. Utochkin, 1992, J. Magn. Magn. Mater. 104-107, 1479. Zvezdin, A.K. and L.R. Evangelista, 1995, J. Magn. Magn. Mater. 140-144.

chapter 5 PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

Stephen W. Lovesey ISIS Facility Rutherford Appleton Laboratory Oxfordshire OX11 OQX U.K.

Handbook of Magnetic Materials, Vol. 9 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved 545

CONTENTS 1. Prologue 2. Orientation

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

3. Survey of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

547 550 558

3.1.

Dichroism .............................................................

559

3.2.

Diffraction

565

3.3.

Elastic resonant scattering

3.4.

Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

571 576

4. K r a m e r s - H e i s e n b e r g amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

579

5. Scattering by orbital m a g n e t i s m

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6. Scattering by spin m a g n e t i s m

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

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

7. Dichroisrn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.

Circular dicbroism

7.2.

Linear dichroism

8. Diffraction

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

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

585 592 595 598

8.1.

Unpolarized primary b e a m

8.2.

Linear polarization

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

604

8.3.

Circular polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

609

9. Elastic resonant scattering

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

583

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10. Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Scattering by free charges 10.2. B o u n d electrons

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

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

10.3. C o m p t o n scattering

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

11. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements .............................................................. Appendix. Polarization effects and magnetic scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . Polarization states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of important s y m b o l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical values of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

546

599

610 613 613 616 617 618 619 619 619 622 625 626 626

1. Prologue The past decade has seen a surge of activity in the use of photon beam techniques to study magnetic properties of materials. By and large, in this period the experimental work has been accomplished with beams produced by electron synchrotron facilities. At the time of writing, several new major facilities are ready for routine operation, while other facilities are being steadily improved. So, it seems, the application to magnetic materials is likely to remain a burgeoning activity for some more years to come. To date, it is fair to say that the surge of activity is underpinned by improvements in instrument performance, enjoyed at synchrotron sources, rather than outstanding intellectual advances. Improvements over conventional laboratory X-ray generators include (Margaritondo 1988, Gerson et al. 1992): a high brightness with the concomitant option of superior resolution; a high degree of linear polarization; tuneability of the primary photon energy; and the provision of good beams of circularly polarized photons. Observable effects due to magnetic properties of a sample are usually relatively small compared with charge induced effects, e.g., in diffraction experiments magnetic intensities are typically five orders of magnitude smaller than Thomson scattering. In consequence, improvements in the intensity at the sample, and the provision of good beams of polarized photons, which enable polarization induced discrimination effects to be exploited as a means of increasing the signal-to-noise ratio, are particularly significant in the use for magnetic studies of photon beam techniques. Broadly speaking, it is useful to consider applications of photon beam techniques in one of two regimes of the primary photon energy. These are the limits of high TABLE 1 Possible (El) dipole transitions and X-ray energies for some elements occuring in materials of current interest. The corresponding wavelength (]~) = (12.40/E) with/~ in units of keV.

3d 4d 5d 4f 4f 5f

Fe Rh Pt Gd Ho U

Transition

Edge

/i/ (keV)

Edge

p-d p-d p-d d-f p-d d-f

L2 L2 L2 M4 L2 M4

0.72 3.14 13.27 1.22 8.92 3.73

L3 L3 L3 M5 L3 M5

547

E (keV) 0.71 3.00 11.56 1.19 8.07 3.55

548

S.W. LOVESEY

photon energies, when the primary photon energy lies above the excitation energy of an absorption edge, and near resonance, when the primary energy is in the vicinity of an absorption edge. The tuneability of synchrotron sources is essential for the exploitation of a resonant enhancement to magnetic signals. As a guide to the energy scale, we refer the reader to table 1 and remark that the L absorption edges of the rare earth elements are in the range 6-10 keV, and the M absorption edges of the actinides are 3-5 keV. The corresponding wavelengths are convenient for Bragg diffraction by crystals, since 1 keV = 12.40 A. At M-edges in the actinides resonant enhancements can reach many orders of magnitude, leading to magnetic diffraction satellite intensities of order 0.1% of the charge Bragg peaks. In the second, high energy, regime Compton spectroscopy emerges in the spectrum of secondary photons when the primary photon energy is in the hard X-ray region, and typically larger than 30 keV. As a general point, we mention that for diffraction experiments there is a gain in scattered intensity on moving to harder X-rays due to the attendant increase in the penetration depth, e.g., for salts changing from 10 to 80 keV X-rays increases the penetration depth by some three orders of magnitude. As we have already noted, the magnetic contribution to the scattering amplitude is generally weak compared to the charge contribution. This discrepancy in size can be interpreted as due to the magnetic scattering being a (small) relativistic correction (Grotch et al. 1983, Bhatt et al. 1983, Sakurai 1987). In practical terms, it is usually necessary to worry about achieving adequate discrimination in the two contributions to data sets. Discrimination in elastic scattering is possibly provided naturally by magnetic order if it differs from the chemical order, e.g., antiferromagnetic materials and a spiral structure displayed by some rare earth magnets. Resonant enhancements of elastic scattering have been observed; particular attention has so far been given to large enhancements found for primary photon energies near the M4 absorption edges in actinides, and near the L3 absorption edges in rare earth and transition metals. In scattering from ferromagnets it is possible to discriminate between charge and magnetic scattering by inducing interference in the two contributions. The interference generated by circular polarization is convenient for making difference experiments in which the polarity of the interference is reversed either by reversing the easy axis, by application of a magnetic field, or reversing the handedness of the polarization. This experimental technique has been applied to ferromagnets to obtain diffraction and spectroscopic (Compton) data. The same basic technique has been used in measurements of the attenuation coefficient, where the quantity isolated in the difference data is the contribution to magnetic dichroism picked out by circular polarization. Circular dichroism, in some ways, is more intimately related to the magnetic properties of the sample than its counterpart observed with linear polarization. There is no analogue of linear and circular magnetic dichroism in neutron beam attenuation experiments, in as much that the corresponding scattering amplitude does not usually contain a significant elastic resonant contribution (Lovesey 1987a, Balcar and Lovesey 1989, Byrne 1994). Other differences between neutron and photon beam techniques applied to magnetic materials include, a possible superior spatial resolution with photon beams, and the clean separation of spin and orbital magnetism

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

549

in photon scattering whereas only the total magnetic moment appears in the neutron scattering amplitude. Notwithstanding these useful attributes of photon beam techniques, neutron beam techniques are the preferred choice for many investigations and, currently, provide the only means of measuring in full dispersion curves and lifetimes for magnetic collective excitations in magnetic salts, metals and alloys. Absorption and scattering experiments can be interpreted in terms of a KramersHeisenberg formula (Hayes and Loudon 1978, Lovesey t993). It provides a description of the contribution from orbital magnetism to the diffraction (Bragg) pattern, and the interpretation of dichroism and elastic resonant scattering from magnetic materials. But, for such materials, it must be extended to include events that arise from the spins of unpaired electrons in the sample. While the derivation of the formula is straightforward, by either standard perturbation theory in quantum mechanics or, more convincingly, as an exercise in covariant perturbation theory applied to quantum electrodynamics, bear in mind that the original formula, not including electron spins, was derived before the advent of quantum mechanics, by use of the correspondence principle. For the present purpose, the framework of standard perturbation theory is adequate to recount the basic structure of the Kramers-Heisenberg formula (Sakurai 1987, Hayes and Loudon 1978). To this end, the photons are described by a vector potential, A, which is a linear combination of photon annihilation and creation operators. A one-photon scattering event in the scattering amplitude is quadratic in A, since this function admits the absorption of a primary photon and the simultaneous emission of a secondary photon. Thus, photon-matter interactions linear in A must be taken to the level of second-order in perturbation theory, where on account of the intermediate states and characteristic energy denominators they generate contributions which can provide resonant processes. Turning now to the photon-matter interaction, it contains the familiar (p. A) term, from the kinetic energy operator {(p - ~ A)2/2m}, and a Zeeman interaction between an electron spin s and the photon magnetic field =curlA; these are the only terms linear in A and, as such, their matrix elements provide the weight of the resonant processes in the scattering amplitude. To date, the evidence is that the (p. A) term accounts for the observed magnetic dichroism and elastic resonant scattering, i.e. the magnetic content of these effects apparently stems from the orbital, or geometrical, features of the atomic wave functions rather than the direct magnetic interaction, via the Zeeman term, involving electron spins. Finally, in the scattering amplitude there are terms quadratic in A coming from first-order perturbation theory. One such term is the A 2 in the kinetic energy. A second, similar, contribution arises from the spin-orbit interaction s. (A x E) where the electric field is proportional to the time derivative of A. At this juncture, we can make a few useful general observations about the structure of the photon scattering amplitude. First, the explicit dependence of the amplitude on electron spin operators arises both from terms linear and quadratic in A, which are treated by second-, and first-order perturbation theory, respectively. After the algebraic details of the calculation of the amplitude are put in place, therefore, it is probably not a surprise to find that the full form of the spin-dependent contribution to the scattering amplitude is really quite complicated. Another observation is that

550

S.W. LOVESEY

a dependence of the amplitude on orbital magnetism arises solely from the (p • A) term, treated by second-order perturbation theory. Since spin and orbital magnetism in the amplitude are not on an equal footing there is the possibility in an experiment to obtain independent information for them. Clear-cut examples of this arise in diffraction when polarization effects are exploited. The aim of this chapter is to survey both a variety of recent experiments, and a basic framework for their interpretation in terms of variables at the atomic level of description. If the inclusion, in what is a relatively short chapter, of a theoretical framework needs defence, largely, it is that in the eyes of the author its omission engenders sciolism. However, little is given by way of the details for derivations of results which, in many instances, are provided in their general form, rather than for specific instrument settings, and hitherto not published by the author. 2. Orientation Three main topics are covered in this chapter, namely, attenuation of a photon beam, and elastic (including resonant) and inelastic scattering. In all topics, the influence of the magnetic properties of the target sample on observed quantities - attenuation coefficient and elastic and inelastic scattering cross-sections - is the principal focus of attention. These quantities have a unifying factor, in so much as they are calculated from the same scattering amplitude operator. Also, the quantities all depend on the polarization states of the photon beams used in the experiments. These few observations lead one to anticipate that the three main topics share some essential features. Our goal in this section is to identify the features in the underlying physics and the formalism necessary for the interpretation of experiments. To some extent, this exercise gathers material appearing in later sections but there, with more attendant detail, common features in the topics might not be so easy to discern. At the same time, the material given here provides the important service to newcomers to the subject of an orientation to the main concepts and ideas. All the observable quantities of interest are described by a common scattering amplitude. We will denote the scattering amplitude operator by G; it is a quantum mechanical operator, and a 2 x 2 matrix in the space spanned by the polarization states, for which our convention is described in an appendix. Elastic processes are determined by matrix elements of G diagonal with respect to the states of the target. Moreover, the total attenuation coefficient is determined from a knowledge of these matrix elements averaged over the polarization states of the primary beam, whereas elastic (diffraction) cross-sections are proportional to products of diagonal matrix elements averaged over the polarization states. Similarly, inelastic cross-sections are determined by products of off-diagonal matrix elements of G averaged over the polarization state of the primary beam. The averaging process, couched in terms of a photon density matrix, is also covered in our appendix on polarization states. Let us denote by fu(E) the diagonal matrix element of G, with respect to a target state labelled #, averaged over the polarization states of a primary beam with energy E. To describe the attenuation of a beam passing through a sample one

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

551

needs this amplitude evaluated without change in the wave vector of the photon, i.e. the (elastic) forward scattering amplitude. If E = hcq, the total cross-section, O'tot, which includes all possible elastic and inelastic processes for a given initial state of the photon, and the imaginary part of the forward scattering amplitude are related in the optical theorem by (Newton 1982), Otot = (4~/q) Im fu(E).

(2.1)

When the density, n o, of scatterers is sufficiently small their contribution to the attenuation coefficient, "7, is simply, "7 = n0Crtot.

(2.2)

Note that, since O'tot has the dimension of (length) 2, 7 has the dimension of a wave vector. Magnetic dichroism refers to the contribution to 7 that depends explicitly on the polarization (circular or linear) of the beam, and the magnetic state of the sample. A clear account of non-magnetic absorption effects is given by Templeton and Templeton (1994). We continue the topic of magnetic dichroism with a few more comments on circular dichroism. Experimental data for the total absorption and magnetic circular dichroism at the L2 and L3 edges of Ni are displayed in fig. 1 together with a theoretical interpretation based on tight-binding theory. The diminution in the intensity of a beam on passing through a thickness R of dilute scatterers is proportional to exp(-TR). If all aspects of the experimental geometry in two measurements are kept fixed apart from changing the handedness of the circular polarization the change observed in the intensity is proportional to, { exp ( - (7 + 7O)R) - exp ( - ('7 - 7O)R) } ,- -27OR exp(-'TR), where 7O, the circular dichroic component of the attenuation coefficient of immediate interest, is assumed to satisfy I%IR << 1. The other component, '7, is generated by multiple (geometric optics) scattering, and it is often related in a simple manner to the imaginary part of the index of refraction for the sample. By contrast, a satisfactory interpretation of 7o is achieved in terms of a single (Born) scattering approximation. Indeed, in most cases, Y0 is dominated by electric dipole events in which the magnetic state of the sample is present in matrix elements through partial, or total, removal of magnetic quantum number degeneracies by the internal, molecular magnetic field, and crystal field perturbations. For the next topic, we tum to elastic scattering; by way of an illustration, form factors for Fe determined by neutron and photon diffraction experiments are shown in fig. 2 together with results from a spin-polarized band structure calculation. Let {...) denote a thermal average of the enclosed quantity with respect to the states of the unperturbed target sample. (Recall that the equivalence of thermal and time averaging is a tenet of statistical mechanics.) For a real sample, with some impurities and disorder, and coherent (elastic) scattering, we include in (G) an average of the

552

S.W. LOVESEY I

d +

lOO

>-

80

b*

p-

zuJ p-

60

~

40

I

I

I

NICKEL L2, 3 EDGES - -

MEASUREMENT

-

........ CALCULATION

~o. 2o 0 m en

~

(o)

J

0

'i ...................L..2....... L I I

i

I

J

I

i

-__.__mL~~ ......... (b)

I

>"

l

-5

z -10 I.z o °15 u ~ -2o

Lz MEASUREMENT

-~

........ CALCULATION

850

870

890

PHOTON ENERGY (eV) Fig. 1. Comparisons between soft X-ray data (full curves) and tight-binding band structure calculations (dashed curves) of the L2 and L3 white lines in ferromagnetic Ni (Chen et al. 1991). (a) The total absorption, (b) the magnetic circular dichroism (MCD). The raw MCD spectrum shown has been multiplied by a factor of 1.85 to account for incomplete polarization and sample magnetization.

scattering amplitude with respect to all other parameters needed to describe the sample, e.g., Bragg diffraction occurs only when strict geometrical conditions are satisfied in scattering by a sample with perfect translational symmetry, which in reality is a crystal averaged with respect to all forms of disorder. The corresponding cross-section for radiation observed in an element of solid angle dO is (Lovesey 1986),

~d./dO)-- Tr{.I12},

(2.3)

where the trace operation is taken with respect to the polarization states described by a density matrix/~. The formula applies to resonant and non-resonant scattering. The total coherent elastic amplitude for photon scattering is the sum of pure charge and pure non-resonant magnetic contributions, and a contribution from dispersive and absorptive processes (de Bergevin and Brunel 1986). The latter contain both charge

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

553

.

°0 ~

6

iI

O kk.

0 0

IN

3-

W--

E k.

2

0

W.-

._o O'

t

T i

a) co

:~-2 0.0

'0.2 '0.~¢ 'OJ5 '0.8 '1.0 '1.2 'l.~l. Momentum transfer (~-1)

Fig. 2. Magnetic X-ray results on ferromagnetic iron for the atomic form factor (error bars), together with polarized neutron data (squares) and band structure calculations (triangles) (Collins et al. 1992).

(Templeton and Templeton 1994) and magnetic interactions. Figure 3 features results for the energy dependence of the intensity at a satellite in UAs through the M3, M4 and M5 edges, and a fit to a model based on the coherent sum of three dipole oscillators. In the experiments reported to date the observed magnetic intensities and polarization dependencies are accounted for by the charge (electric) contribution to resonant processes. In this case, the magnetic character of the scattering arises from the magnetic character of the electronic orbitals that enter the calculation of the matrix elements of the multipole operators. For holmium in its ordered magnetic state, the significant events are virtual dipole-allowed transitions, coupling 2p core electrons with 5d-derived conduction band states, and quadrupole transitions, coupling 2p core electrons to 4f atomic like states. It is found for the spiral magnetic phase that some magnetic satellites arise solely from quadrupole transitions, while the remaining observed satellites are mixtures of dipolar and quadmpolar transitions. At this stage, let us enlarge on the question of the dependence of cross-sections on the polarization states of the primary beam. There is a sense in which the question can be answered in general form. To begin with, any 2 x 2 matrix, such as G, can be expressed as a linear combination of the unit matrix, I, and Pauli matrices, o'. Let us use the notation, G =/31 + or. o',

(2.4)

where/3 and ot are determined from a detailed knowledge of G, namely, the generalized Kramers-Heisenberg formula to be provided later on. Next, the polarization states of the primary beam are described by a Stokes vector P = (P~, P2,/)3)- For

554

S.W. LOVESEY

I 012 3 0 x 106

i

I

I

I 4.4

I

15x10 6

O0 I'-" Z

0

::k

0

10~o

4.2

x >-

Z FZ 121 I---

W FZ

o oo - - z . . 0 106 3.4

I 3.6

I 3.8

I 4

I 4.2

ENERGY

I 4.4

I 4.6

J 4.8

5

(KEY)

Fig. 3. The energy dependence of the intensity at a satellite in UAs through the M3.4,5 edges, which display a strong interference effect between the two closely spaced edges. The full curve through the data is a fit to the coherent sum of three dipole oscillators. Inset is an expanded view of the M3 edge (McWhan et al. 1990). See also table I. the moment, we will not dwell on the physical significance of the three parameters (they are discussed in the appendix) other than mention, by way of an example, that P2 is the mean helicity in the beam, i.e. the degree of circular polarization. Armed with this knowledge of the polarization states, the trace operation can be performed leaving the cross-section expressed in terms of/3, ot and P. In fact, since the cross-section, of course, is purely real, and a scalar object one can anticipate that the expression is a linear combination of the terms,/3+/3, ~+c~,/3+(~x • P), and P . (c~+ x c~), so the detailed work is simply to determine the coefficients of such terms in the expansion. One finds, Tr{izG+G} = (or + . m +/3+/3 + / 3 + ( P . o0 + ( P . o~+)/3 + i P . (o~+ x oz)). (2.5) The elastic cross-section, mentioned above, is obtained from this expression by replacing oL and/2 by their average values. Let us further note that, the expression provided applies also to inelastic scattering, to which we turn later, and a similar

PHOTON BEAM STUDIES OF MAGNETICMATERIALS

555

0.4

0.2 ~ 1 0 , 0 0

6,6,0

E

~-0.2 (n ~... -0,4

t .NxX~8,18,0 14,14,0

-0.6 -0.8 0

0.5

1 k(A'+~

1.5

2

Fig. 4. The spin/orbital form factor ratios for holmiumin HoFe2 at room temperature. The solid line represents the results of a relativisticspin-polarizedband structurecalculationwhichhas been normalized to the Hund's rule ratio of 1/3 at k = 0 (Collins et al. 1993). procedure can be used to generate general expressions for the Stokes vector of the secondary beam and the forward scattering amplitude. Moving on, in section 1 we mentioned that (a) spin and orbital magnetism in the target sample influence scattering in different ways, and (b) the magnetic contribution to scattering by a ferromagnet can be isolated through use of a circularly polarized primary beam. Both points feature in the diffraction data for HoFe2 shown in fig. 4, which are the ratios of the spin to orbital form factors derived from the interference scattering induced at mixed charge and magnetic Bragg reflections by circular polarization. For the description of polarization states of the primary, and secondary, beams we advocate use of a formalism based on Stokes vectors (McMaster 1961, Berestetskii et al. 1982, Lovesey 1987b). The main argument in support of this choice is generality. For, in reality, one is unlikely to have a perfectly polarized beam, and partially polarized states encountered in experiments are fully described by allowing for all three Stokes parameters {Pi}. The alternative is to provide G couched in terms of polarization vectors, e and e' for the primary and secondary beams, respectively, and relate these to the geometry of a particular experiment. To illustrate this aspect, let us consider the cross-section for Thomson scattering by electrons located at positions {Rj}. The variable measured in scattering is the spatial Fourier transform of the charge distribution,

n(k) = E exp (i k . R j), J

(2.6)

556

S.W. LOVESEY

where k = (q - q') is the change in the wave vectors of photons in the primary and secondary beams. The Thomson cross-section is, (d~/dS2) = r2(e.

(2.7)

in which re is the classical radius of an electron. Looking at the dependence of the cross-section on the polarization vectors, there are several, more or less, standard representations for (e • e,)2 in terms of angles that describe the elastic scattering geometry. However, with the advocated formalism, in terms of Stokes vectors, (e. e,)2 is replaced by its value averaged over the possible distributions of partial, or total, polarization in the primary beam, viz., ( g . ~t)2

.....>

1 { 1 + cos 2 0 q- P3 sin 2 0},

(2.8)

where 0 is the angle through which the primary beam is deflected to the detector. Note that the result does not depend on P1 or P2. The values P3 = +1, describing complete linear polarization perpendicular (a-polarization) to the plane of scattering, and complete linear polarization in the plane (rr-polarization), are unlikely to be achieved in practice. To conclude this slight digression on the formalism for handling less than complete polarization, consider values of the Stokes parameters for the secondary beam {P'}. Sticking with the example of Thomson scattering, one finds (Berestetskii et al. 1982); for i = 1,2, P" = 2Pi cos0/(1 + cos20 + P3 sin2 0),

(2.9)

and, P~ = ( sin 2 0 + P3(1 + cos z 0))/(1 + cos 2 0 + P3 sin 2 0). It is interesting to note that, for 0 = (rr/2) the Stokes parameters of the secondary beam, for any primary polarization, are/'/ = (0,0, 1), i.e. 90 ° charge scattering produces complete e-polarization of the secondary beam. This well-known effect has been exploited to discriminate between charge and magnetic components in the secondary beam (Gibbs et al. 1985). Inelastic magnetic scattering experiments using photon beams from synchrotron sources have so far focused on deriving the momentum distribution, or Compton profile, of unpaired electrons in metallic magnets (Cooper 1987, Sakai 1992). These studies exploit the charge-magnetic interference scattering induced by circular polarization in the primary beam to extract the magnetic component of the Compton scattering process, while accurate total Compton profiles can be obtained with laboratory 7-ray spectrometers (Anastassopoulos et al. 1991). Figure 5 contains data and theoretical predictions for the Compton profile of unpaired electrons in ferromagnetic nickel. An experimental investigation has shown that Compton scattering is not sensitive to orbital magnetism in the sample (Timms et al. 1993), i.e. the profile extracted from data is the momentum density of the unpaired electron spins.

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

557

'o F Ni

8

110 FLAPW

"~., 6

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

EXPT

e O~ 0

hE4

2 -

0

I

I

I

I

1

2

3

/-.

5

6

7

8

pz (o.u.) Fig. 5. Magnetic Compton profile for Ni in the (110) direction. The solid line is derived from a band structure calculation, and the experimental data are denoted by the dotted line (Kubo and Asano 1990).

A compact expression for the partial differential cross-section, which includes all elastic and inelastic processes, is achieved with the help of correlation functions, standard in the interpretation of a wide range of other experiments, including NMR, #SR, electron and neutron beam scattering (Lovesey 1986, 1987a). To this end, let G(t) be the Heisenberg operator at time t, formed from the scattering amplitude operator; for simplicity of notation, we will write G(0) =_ G. The correlation function required to calculate the cross-section is (G+G(t)). If No is the energy transferred from the primary beam to the target sample, the partial differential cross-section which gives the fraction of photons of incident energy E scattered into an element of solid angle d£2 with an energy between E ~ = (E - No), and E' + dE ~, is, (d2cr/d~?dE ') = (E'/E)(1/2~rh)

F dt exp(-icot)Tr{~(G+G(t))}.

(2.10)

The trace operation, with respect to the polarization states of the primary beam described by the density matrix ~, can be accomplished to the extent of creating an expansion in terms of correlation functions formed with the operators fl and a introduced earlier on. The strictly elastic cross-section, already encountered, is derived from the partial differential cross-section by taking t -+ c~ in the correlation function, since,

(G+G(t =

:/G+/(6(t

:

: I/G/I 2,

558

s.w. LOVESEY

where the first equality follows from the law of increase in entropy, or loss of information, which requires that, for a bulk assembly, there is no correlation between processes well separated in time. To reach the final expression use the result, correct for any Heisenberg operator, (G(t)) = (G(0)) which is a consequence of the condition for a stationary system that a correlation function is independent of the origin chosen for the time variable. Returning to the cross-section, when a time-independent value of the correlation function is inserted in the Fourier integral the latter reduces to a delta function with w as its argument, i.e. the cross-section vanishes except for w = 0 which corresponds to purely elastic scattering. The result for the partial differential cross-section when integrated with respect to E ~ is identical with the previous elastic cross-section, as required. In view of the foregoing analysis, the partial differential cross-section formed with the correlation function,

(-II=) <{G + - *){G(t)- ) >,

(2.11)

is exclusively inelastic in its content; this is the cross-section which described all forms of spectroscopy, e.g., Compton and Raman processes. Photoemission has a role to play in the exploitation of photon beams to study magnetic materials (Thole and van der Laan 1991, Williams et al. 1980, Halilov et al. 1993, van der Laan 1994a) but it is not included in the scope of this chapter. A simple picture of photoemission entails three independent processes; photoabsorption, propagation of an excited electron to the surface, and the escape of the photoelectron into the vacuum. To underscore the intrinsic complexity of photoemission, we mention that linear response theory does not give rise to the photoelectric process, which puts it in a different category of experimental methods to many others used to study magnetic materials, including, #SR, NMR and neutron and photon scattering. Another feature which merits comment is that, in angle-sensitive photoemission one must be aware of macroscopic refraction and reflection effects that can be modelled by Fresnel equations. However, this is just one of several near-surface effects involved in a full analysis of photoemission data. Circular dichroism in photoemission is treated by Thole and van der Laan (1994).

3. Survey of experiments The recent flurry of activity with applications of photon beam techniques to address magnetic properties of materials seems to have really got underway around 1985 (de Bergevin and Brunel 1986, Cooper 1987, Gibbs 1992). At the risk of being invidious, we mention two pieces of work published at this time that played a part in raising the awareness of researchers in magnetism to the potential value of synchrotron-based techniques. Magneto-optic effects, e.g., the magneto-optical Kerr effect in the visible region, appeared to have minimal value as an investigative tool prior to predictions by (Thole et al. 1985) for 3d absorption edges of rare earth materials based on atomic multiplet calculations. The relatively strong magnetic

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

559

X-ray dichroism was first observed in terbium-iron garnet (van der Laan et al. 1986). At more or less the same time, interesting science was revealed in X-ray scattering experiments on the rare-earth metal holmium (Gibbs et al. 1985). Direct high-resolution measurements of the nominally incommensurate magnetic satellite reflections revealed lock-in behaviour which was successfully explained in terms of a simple spin-discommensuration model (Bohr et al. 1989). Looking much further back in time, theoretical work was ahead of experimental investigations of magnetic effects in photon scattering, with the first explicit calculations published in 1938 (Tolhoek 1956, Evans 1958). While these works focused on the basic nature of the photon-matter interaction, Platzman and Tzoar (Platzman and Tzoar 1970) explored a potential value for fundamental investigations of magnetic materials, successfully demonstrated by de Bergevin and Brunel (de Bergevin and Brunel 1981, 1986) in a series of experiments on various different materials. Skipping forward looking for particularly significant findings, over the 1985 developments mentioned in the beginning, one lights on the successful observation of resonant elastic scattering (Gibbs et al. 1988, Hannon et al. 1988, 1989), which is now a small industry in part because resonant enhancement ameliorates technical problems faced in measuring intrinsically weak scattering events. Recent reviews of experimental investigations of magnetic systems include (de Bergevin and Brunel 1986, Cooper 1987, Gibbs 1992, Sette et al. 1991, Chen 1993). For our part, we look at experiments which have used circular and linear dichroism, diffraction from ordered magnetic structures (Bragg scattering), including elastic resonant scattering, and spectroscopy. Basic concepts for the interpretation of the experimental investigations are gathered in subsequent sections. 3.1. Dichroism

Here, and in section 3.3, we consider the resonant regime and discuss magnetic X-ray dichroism and elastic magnetic resonance scattering. The relation between absorption and scattering is the standard optical theorem, discussed in section 2 and section 7. Within a simple one-electron picture of electronic structure illustrated in fig. 6, in resonant scattering the incident photon promotes by a virtual transition an inner shell electron to an unoccupied orbital above the Fermi energy, which subsequently decays through the emission of photon. The amplitude for resonant scattering then depends on the matrix elements which couple the initial state and the intermediate magnetic states allowed by the Pauli exclusion principle. The scattering ampfitude (6.4) contains charge, linear momentum and spin interaction operators. To date, the experimental data on resonant scattering and dichroism have been successfully interpreted in terms of the momentum interaction operation. In this instance, the magnetic character of the observed electron-photon events in magnetic materials stems entirely from the nature of the wave functions used to calculate the dipole, quadrupole . . . . . matrix elements. The underlying physics is common to a range of magneto-optic effects, including dichroism, the Faraday effect and the magnetooptical Kerr effect. Reviews in (Kao et al. 1993, van der Laan 1990) of these effects use the formalism outlined in sections 7 and 9.

560

S.W. LOVESEY L///Edge s-p 4f

EF s-p-d

4f

s-p E1 : 2p3/2-~5ds~ E2: 2p312--)4f7/2

2P3/2 Fig. 6. Schematic, one-electron view of resonant magnetic scattering at a n L 3 absorption edge (Gibbs 1992). The linearly polarized primary photon promotes a 2p3/2 core electron into an empty state above the Fermi level, EF. In the rare earth elements, on which this simple example is modelled, there are localized 5d-states available in dipole-allowedtransitions (El), and un-filled4f states available through quadrupole transitions (E2). Scatteringresults when the virtually excited electron decays, thereby filling a core hole and emitting a photon. Magnetic dichroic effects make the near-edge, inner-shell absorption of polarized photons a useful tool for investigating the magnetism of transition metal, rare earth, and actinide elements and compounds. Table 2 is a summary of representative examples of experimental studies. The absorption and dichroic effects at the L2,3 edges in ferromagnetic nickel are displayed in fig. 1. The relatively high precision with which the intensity ratios can be determined provide good tests of models of magnetism. Indeed, data for the L2,3(2p --+ 3d) and Mz,3(3p --+ 3d) magnetic dichroism and X-ray photoemission of nickel have been subject to various theoretical interpretations (Chen et al. 1991, Jo and Sawatzky 1991, van der Laan and Thole 1992, van der Laan 1994b), with attention to electronic correlations. The data in fig. 1 for nickel are an example of circular dichroism observed with soft X-rays. In contrast, the prediction (Thole et al. 1985) of strong magnetic dichroism and experimental proof (van der Laan et al. 1986) was for linear dichroism in the M4,5(3d --+ 4f) absorption edges of rare earth materials. Calculations (Thole et al. 1985, Goedkoop et al. 1988a) are made on the basis of atomic multiplet configurations, illustrated in fig. 7 for the simple case of yb3+(4f13), and outlined in

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

561

TABLE 2 Representative examples of materials investigated by magnetic linear (MLD) and circular (MCD) diehroism, see also table 1. Material

Ref.

Edges

Probe

Comment

Tb3FesO12 Ni

[1] [2, 5, 9, 12, 14]

M4,5 L2,3M2,3

MLD MCD

Gd3Fe5012 Gd Co 5d impurities in Fe 3d impurities in Ni rare earth intermetallics

[2, 4] [3] [6, 7] [8, 11] [10] [13]

L2,3M4,5

MCD MCD MCD, Kerr effect MCD MCD MCD

applied field film on Cu; single crystal applied field

L1,2,3 L2,3 L2,3 L L2,3

multilayer

References: [1] [2] [3] [4] [5] [6] [7]

van der Laan et al. (1986) Tjeng et al. (1991) Schtitz et al. (1988) Rudolf et al. (1992) Chen et al. (1990) Kao et al. (in press) Kao et al. (1993)

[8] [9] [10] [11] [12] [13] [14]

Schtitz (1990) Koide et al. (1991) BOske et al. (1994) Schtitz et al. (1993a) Vogel and Sacchi (1994) Krill et al. (1993) O'Brien and Tonner (1994)

sections 7 and 9. Circular and linear dichroism probe, respectively, the magnetization and mean-square magnetic fluctuations, and the effects in question are much greater than those observed in the visible region. Circular dichroism can only be exploited with single domain magnetically oriented samples, e.g., ferro- and ferri-magnets in an applied field; however, with linear polarization, it is possible to examine single crystal antiferromagnets as well (Kuiper et al. 1993) provided the magnetic moments are aligned in preferential crystallographic directions. The usually neglected electric quadrupole transition are predicted to be as important as the dipole terms, considered in the foregoing discussions, for the interpretation of magnetic dichroism at rare earth L edges and transition metal K edges (Carra and Altarelli 1990, Carra et al. 1991, Wang et al. 1993, Jo and Imada 1993). Various types and levels of theoretical work are applied to the interpretation of magnetic dichroism. Atomic calculations are appropriate for 3d absorption in rare earths (Thole et al. 1985, Goedkoop et al. 1988a) and 2p absorption in 3d transition metals (van der Laan and Thole 1991). Group theory has been exploited to derive a general model for spin polarization and magnetic dichroism in photoemission, and applied to the 2p, 3s, 3p and 3d photoemission from divalent Cu, Co d 7 and F e d 6 (Thole and van der Laan 1991). A relatively simple model of circular dichroism at the L2,3 edges of rare earth atoms has been successfully used (Jo and Imada 1993) to interpret data on atoms from Ce through to Tm in (RE)2COl7. This theoretical work is based on the 2p ~ 5d dipole (El) transition, and the tolerable accord with the data is seen to vindicate the neglect in the calculations of electric quadrupole (E2) transitions. Also included in (Jo and Imada 1993) are well discussed applications of sum rules for dichroic signals, which is a topic taken up in more detail later in this section. For heavy rare earth metals, circular dichroism evaluated from first-principles relativistic

562

S.W. LOVESEY

+ magnetic

yb ~

field M'

/

* I

J'=5/2

-~_

i

I

,

& ',;

A' ' '

4 ' I I

J

',

,

~ I

~:

"r

I

I I

I I

--i I I

I I

, ,,,, I I I

{

I

I

I

I I I

;

....

i

I I I

I

i

&

I

;,,

[ I -3/2

i

I

A

i

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~

j_L.

~M=O

z~ M=-I

,,

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i i

i

i

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I I I I t

I I i i i i I

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•

J =7/2

=11

, i:i

i,

,

,, ;,,,

ii i

,,, '

I ~

-5/2

I I

]

:i,'

i

I

-v,

. ,, t

-1/2

k' I

I

/',M= 1

3/2

,

&;

: ;;

i

I

A, '

I

j_j.~.,__~_

zxJ =-1

I

5/2

t

I

I

M

'

7/2

,i

312 1/2

]

[I

--1/2

, I

-5/2 -7/2

\x

/,

? Fig. 7. Energy level diagram of the 3d]°4f 13 --+ 3d94f 14 transition in Yb 3+ without (left) and with (right) a magnetic field (Goedkoop et al. 1988b). The vertical arrows indicate the dipole selection rule allowed transitions. Their relative intensities are given by the dots. (In the text, the label m in the 3j symbol is denoted by q.) The required polarization is indicated at the bottom of the figure.

spin-polarized band-structure calculations (Wang et al. 1993), applied to the dipolar contribution, provides an adequate account of data. Several calculations of the soft X-ray magnetic circular dichroism at the L2,3 edges of ferromagnetic nickel have been reported, based on the Anderson impurity model (Jo and Sawatzky 1991, van der Laan and Thole 1992, van der Laan 1994b) and a tight-binding band structure model (Chen et al. 1991). The latter analysis isolates features in the data which lie beyond the physics included in one-electron band-structure models, and are attributed to a many-body shake-up or shake-off process accompanying the creation of a core

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

563

hole. These interesting findings are supported by subsequent work (Jo and Sawatzky 1991, van der Laan and Thole 1992, van der Laan 1994b). A theory of dichroism in iron is provided in (Smith et al. 1992). The dichroic effects observed in the cited experiments are really not the same as expected from the Fano effect (Fano 1969, Thole and Van der Laan 1991). Fano predicted that polarized electrons can be produced by photoionization of unpolarized atoms with circularly, linearly and unpolarized light. The photoelectron polarization is caused by spin-orbit interactions in the continuum state, or in the atomic states. These are not influenced by a polarized valence shell, leaving the core-hole - spinorbit coupling, for example, as the remaining interaction. But, this picture is not appropriate when the electron is excited to a more localized state, such as the valence d states in transition metal compounds or the f states in rare earth and activide materials. For intra-atomic excitations there are electrostatic interactions between the spin and orbital momenta of the valence state and core-hole, and the magneto-optical effect, referred to as magnetic dichroism, is much larger than for spin dependent photoabsorption. While it was recognized many years ago that the structure of the scattering amplitude afforded the possibility to separate spin and orbital magnetism in the analysis of diffraction patterns, it was not until 1992 that a similar advantage became available in interpretations of circular dichroism (Thole et al. 1992). In this case, a sum rule for the integral of the signal over a given edge allows one to determine the ground state value of the orbital angular momentum, and it has been successfully applied to data collected on ferromagnetic nickel. Various other sum rules have since been derived (Carra 1992, Thole and van der Laan 1993, Carra et al. 1993a, b) which afford the possibility to extract from measurements of dichroic effects the ground state expectation value of the magnetic field operators (orbital, spin and magnetic dipole) of the valence electrons. While the original work used a local atomic picture, similar results have also been deduced from a simple single-particle band structure picture (Shtitz et al. 1993a, b). To illustrate the form taken by so-called sum rules, consider the integrated strength of dipole transitions. Referring to section 7, the strength of such a transition is proportional to (AL<wlRqI~>I2),where A is the difference in energy between the ground state labelled by /z, and an intermediate state labelled ~/, and q = 0, 5:1 denotes spherical components of the position operator. The state [#) is that of an ion with an incomplete outer shell, angular momentum g, and h holes. The integrated strength of dipole transitions is proportional to,

and, for instance, the ratio, D :

(I1 - I _ , ) / ( I 1

+ I o + -[-1),

measures the circular dichroic signal. To proceed, one argues that for all transitions the radial integral in the dipole matrix elements is the same, and A = (En - E~) is

564

S.W. LOVESEY

replaced by some average value and taken outside the sum. In consequence, D is assumed to be independent of the radial integral, and energy level separation. After algebra that entails the reduced matrix element (7.12), one finds (Altarelli 1993), D = -(~lLzl~)/eh,

which is the sum rule first derived in (Thole et al. 1992). The derivation of this sum rule and its application are further discussed by van der Laan (van der Laan 1994b). By way of an apparently simple example of the use of sum rules we mention results obtained from magnetic circular dichroism at the L2 and L3 edges in a remanently magnetized Ni (110) single crystal (Vogel and Sacchi 1994). The expectation values of orbital and spin moments (in units of #R) in the d shell are found to be, 0.06 4- 0.01 and 0.27 :k 0.03, respectively. Mixed magnetic systems have been studied by the authors of reference (Shtitz et al. 1993b). They disclose a tolerable agreement between the analysis of experimental data and band structure calculations for the average spin and orbital moments of 5d elements dissolved in iron. Equally interesting is a comparison of two methods of analysing the data. One method is an application of sum rules (Thole et al. 1992, Carra et al. 1993b), derived originally for an atomic model. The other method, developed by the authors, is a two-step model based on the Fano effect (Fano 1969) and a single-electron band structure picture. Significant differences are found between the two sets of results, e.g., for Pt dissolved in Fe the deduced orbital moments are of a similar magnitude but opposite in sign. On the other hand, for Os the methods give similar results for the average spin moment but, the near common value is very different from the value derived from band structure theory. These and other related (Jo and Imada 1993, van der Laan 1994b) findings, for the moment, post a cautionary tale about the application of sum rules to analyse dichroic signals, although it seems that the sum rule for the orbital moment, discussed in the preceding paragraph, is more robust than those for the spin moment and magnetic dipole. Beside the dichroic effects already mentioned, it is possible to probe local magnetic moments from absorption of unpolarized X rays in unpolarized electron shells using branching ratio analysis (Thole and Van der Laan 1988a, b, van der Laan and Thole 1988). The core hole spin-orbit branching ratio is extremely sensitive to the angular momentum of the valence electrons. The sample requirements for detection of magnetic dichroic effects (a single crystal and magnetic alignment) are less severe in analysis of core level X-ray absorption line shapes (Alders et al. 1994). Moreover, the information, on interatomic exchange interactions and short range spin autocorrelation functions, is element specific and can be used to study ferro-, ferri-, or antiferromagnetic materials in single crystals, thin film and powder film. To date, the effect has been demonstrated in a study of the Ni L2,3 edge in a layer by layer grown NiO film. Measurements of dichroic effects in systems with reduced spatial dimension have proved fruitful. Examples include, magnetic films (Idzerda et al., in press; Tjeng et al. 1992, Heinrich and Cochran 1993, O'Brien and Tonner 1994), single crystal thin films (Idzerda et al. 1993a), heteromagnetic multilayers (Chen et al. 1993, Idzerda et al. 1993b), and near surface magnetism of ferromagnetic nickel (van der Laan et al. 1992).

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

565

3.2. Diffraction When compared with neutron magnetic diffraction, synchrotron-based photon diffraction has several intrinsic strong points. These include;

(a) High spatial resolution In some applications, the spatial resolution obtained in photon diffraction is significantly finer than that available in a corresponding neutron diffraction experiment. This point is illustrated for critical scattering from holmium by the data displayed in fig. 8; the resolution obtained in the two experiments differs by an order of

Holmium (O,O,2-'f) Sample#1 2.0 3-

~-~

1.5

X

C)

~-~

1.0

X_ra, Face, la haft-width o Int. Intensity

-i-

0.5 -6

0 0.025

~ID

"X~ • XX~

•

~-

/

XX~

-

- e--e--oJ-'-rt~

,

I

,

I

o i \

~4"01

0

0 h

0.020

,~

o.o15

"r"

,~ o.olo 0.005 • 0

130.0

i

I

130.5

~ I

u

u

I

131.0 131.5 Temperature (K)

132.0

Fig. 8. Integrated intensity and half-width (HWHM) of scans taken at the (0, 0, 2 - t) peak position of holmium metal in the vicinity of the N6el temperature ~ 131.2 K (Thurston et al. 1994). Low resolution neutron scattering data, and X-ray data are displayed. The solid lines represent power law fits; integrated intensity ~ (TN -- T) 2~, and HWHM ~ (T - TN) ~' where ~ and/3 are the standard critical exponents, and v = 0.54 -4-0.04, whereas the exact meaning of/3 depends on the interpretation of the experiment.

566

S.W. LOVESEY

magnitude. With the current and planned high-brightness synchrotron sources, it has become possible to investigate ordering phenomena with correlations extending over micron length scales with high accuracy.

(b) Polarization analysis There is usually more scope to benefit from polarization analysis with photon beams than with neutron beams. This stems from both differences in the intrinsic properties of the radiations and production methods, e.g., photon beams from a synchrotron source have a high degree (,-~ 90%) of linear polarization while neutron beams from reactor and spallation sources are unpolarized. The scope afforded by polarization dependent properties is illustrated by the material gathered in fig. 4 and fig. 9 on the spin and orbital moments in HoFe2 and Ho obtained, respectively, by use of circular and linear polarization properties in diffraction.

(c) Extinction-free scattering Because of the relative weakness of magnetic photon scattering, an interpretation is appropriate within the first Born approximation, so there is no extinction correction.

(d) Static approximation The quite broad energy resolution typical of many photon scattering observations (5-10 eV) means that inelastic events are integrated over, to a good approximation, i.e. the total cross-section is observed. For diffuse charge scattering this means that the observed intensity is described by (2.7) which relates the scattered intensity to the instantaneous value of spatial Fourier transform of the spatial distribution of scatterers.

10~ ~eL 0.5

x•

L=6

"-..

, •

0

-0.5

L=0 ..

~',,

""

""- ..

-1.0 Fig. 9. Data obtained for holmium (Gibbs eta]. 1991). The solid square at 8 = 0 corresponds to

the degree of linear polarization of the primary beam. The open circles show the degree of linear polarization measured for the charge scattering at chemical Bragg reflections at a temperature where the

magnetic configurationis conical T < 20 K). The full curve is the degree of linear polarizationfor charge scattering calculatedfrom (2.9) with P3 = 0.77. The solid circles show the degree of linear polarization measured for magnetic (satellite)reflections. The broken curves are calculations,based on results given in section 8.2, for three different values of the total spin, S, and total orbital angular momentumL.

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

567

(e) Relatively small samples are adequate The diffraction intensity arising from charge scattering is usually much larger than from pure magnetic scattering. For ferromagnetic systems, in which chemical and magnetic order coincide, one of two methods to discriminate charge and magnetic contributions to Bragg peaks suggest themselves. Both methods rely on creating an interference between charge and magnetic amplitudes, which has the attractive feature of giving access to the sign and magnitude of the magnetic scattering amplitude. In one method, interference is generated by the imaginary part of the charge scattering amplitude (de Bergevin and Brunel 1981, Vettier et al. 1986) that is available in noncentrosymmetric samples or through tuning the X-rays from a synchrotron source to an adsorption edge. Alternatively, use is made of circular polarization in the primary beam (Brunel et al. 1983, Collins et al. 1992); data for the magnetic form factor of iron obtained from polarized X-ray and neutron diffraction are contrasted in fig. 2. Table 3 lists some of the experiments mentioned in the previous paragraph together with studies of rare earth metals. In the latter work, attention is given to satellite reflections that can be purely magnetic in character. Before mentioning more experiments, it is useful to note some simple properties of the cross section (2.5) constructed from the four components of a and/3 listed in (A.8). Looking at the magnetic contributions in these four components, which are identified by the small parameter ~- = (E/mc2), it is seen that ~l and c~z are purely magnetic while c~3 and/3 are linear combinations of charge and magnetic operators. It is significant that, in ~3 and/3 the magnetic contributions differ from the charge contributions by a phase factor i = x/S] . Assume in the first instance that the diagonal matrix elements of the atomic quantities in ot and 13, namely n(k), S(k) and Z(k), are purely real, and the primary beam is unpolarized, P = 0. The cross section for diffraction is proportional to {](a)l 2 + [(/3)12). When the matrix elements in (a) and (/3) are real it follows that TABLE 3 Representative examples of magnetic X-ray diffraction studies Material

Ref.

Zn0.sFe2.504 Fe Ho Tb Er Tm HoFe2 Gd-Y superlattice Ho15-Y12 superlattice

[51 [4] [1, 8, 9] [8] [71 [61 [101 [3] [2]

References: [1] [2] [3] [4] [5]

Gibbs et al. (1985) Bohr et al. (1989) Vettier et al. (1986) Collins et al. (1992) Brunel et al. (1983)

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

Bohr et al. (1990) Gibbs et al. (1986) Tang et al. (1992a) Gibbs et al. (1991) Collins et al. (1993)

568

S.W. LOVESEY

the magnetic contribution to the cross section is proportional to r 2, i.e. there is no term proportional to r. Since r << 1 (otherwise use of the amplitude (6.5) is not valid) the magnetic content of a mixed (charge and magnetic) Bragg reflection is dominated by the charge contribution. However, pure magnetic reflections can occur in magnetic materials for which the chemical and magnetic structures are different, i.e. materials other than simple ferromagnets. If the condition on the matrix elements in (c~) and (/3) is relaxed and they possess imaginary charge components, which is possible in non-centrosymmetric materials or from anomalous scattering, then the diffraction cross section may contain terms proportional to ~- that are probably larger than terms proportional to ~.2. Polarization of the primary beam has a significant effect on the cross section. We refer to (6.5) for the diffraction amplitude, and figs 10, 1 la, and 1 lb together with (6.7) evaluated for elastic scattering (a = b = 1). Our choice of axes for the scattering geometry is given in fig. 12. With regard to the influence of linear polarization, note that the matrix B, given in (6.7), which represents the polarization dependence of spin scattering, contains off-diagonal elements. In consequence, cr - 7r and 7r events occur through scattering by the spin magnetization. There are also finite off-diagonal elements in (¢' x ¢) which represents the polarization dependence of orbital scattering. By contrast, (e' • e), which represents the polarization dependence of charge scattering, is diagonal. From these observations it follows that (cr-Tr) and (Tr-cr) events revea ! purely magnetic scattering. To appreciate the role of circular polarization in photon scattering by magnetic materials, recollect that circular polarization can be represented by complex polarization vectors. If the matrix element in (a) and (fl) are purely real, the phase difference between their charge and magnetic components can be negated by the imaginary part of the complex polarization vectors leading to purely real interference terms. So, circular polarization induces interference between charge and magnetic amplitudes, and hence terms of order ~- in the cross section. Furthermore, changing from right-to left-hand polarization, or vice versa, changes the sign of these interference terms leading to the appealing possibility of a difference measurement to isolate the relatively small linear magnetization contribution. This scheme has value only for materials that have mixed (charge and magnetic) reflections, as in a ferromagnet. Experiments that confirm many of these properties of photon scattering by magnetic solids have been performed by de Bergevin and Brunel; their 1986 paper contains a review of their findings and theoretical interpretations. These researchers observed magnetic diffraction of order T2 from bulk antiferromagnetic NiO using

I1(~)

_a.(a)

I1(~)

GII

GI2

G21

G22

Fig. 10. The representation chosen for the polarization-dependent elements of the scattering amplitude operator G expressed as a 2 × 2 matrix eq. (2.4).

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

a)

(¢'. ¢)

(~)

b)

1

0

o

(~ .~')

(¢' x s) A_(~)

±(cr)

I1(~)

o

-~'

I1(~)

c)

(~' × ~)

(~' • m)(¢ • m)

±(0)

±(~r) I1(~

569

I

11(70

,n~

m±(¢ll, m)

(e~l - m ) m ±

(e'll.,n)(Sli.m)

Fig. 11. Matrix representations of the functions ~' • s, ¢ ' × ~, and (~' • m)(¢ • m) formed with the (real) polarization vectors of the primary, ¢, and secondary, ¢', beams. 9 and q ' are unit vectors in the directions of the primary and secondary beams, respectively, and 9 • q ' = cosO, c f fig. 12. Regarding the vector m, r a ± is its projection perpendicular to the plane of scattering defined by q and q '. Polarization vectors parallel and perpendicular to the plane are denoted by ~ll and ~ ± , respectively, and m ± = ( m . ~ ± ) : ( m . ¢~_); cf. fig. 12. Note 9 = ~ ± × ~LI' and a similar relation for the secondary beam.

an X-ray tube operated at 1 kW. Photon beams from synchrotron sources are predominantly linearly polarized within the plane of the electron orbit, and elliptically polarized above and below the plane. The resolution in reciprocal space can be superior to that currently available in neutron diffraction (Tang et al. 1992a); fig. 8 illustrates this feature for critical scattering by holmium. Magnetic diffraction from ferromagnetic iron and HoFe2 of circularly polarized photons has been reported by Collins et al. (1992, 1993). They used a white beam and single crystal samples, and reversed the direction of the magnetization by an applied field to affect isolation of the polarization induced charge-magnetic interference contribution. The experimental technique offers several attractive features, including

570

S.W. LOVESEY

ll" I

Fig. 12. Coordinates (, r/, ~ used to define the scattering geometry. The (-axis is perpendicular to the plane of scattering, defined by q and q ~, k = q - q ~, and the secondary beam is deflected through an angle 0. a fixed scattering geometry, while several harmonics of a given reflection can be resolved with an energy dispersive detector and measured simultaneously. The possibility in photon diffraction to separately measure spin and orbital contributions to the amplitude has been realized in studies of holmium (Gibbs et al. 1991) and HoFe2 (Collins et al. 1993); some results are shown in figs 4 and 9. In principle, a scattering geometry can be chosen at which the magnetic scattering is either purely orbital or spin in nature cf. section 8. With regard to magnetic neutron diffraction, the observed intensity is always a mixture of spin and orbital contributions that are not separately obtainable through a choice of scattering geometry (Tang et al. 1992a). As mentioned in the previous section, analysis of dichroic effects also provides spin and orbital moments of the ground state configuration. Photon diffraction studies of magnetic fluctuations around the (100) Bragg reflection of the diluted antiferromagnet Mn0.ysZn0.25F2 have clarified its complex properties (Hill et al. 1991, 1993a, b). The compound has been extensively studied by various experimental techniques because when subject to a magnetic field it models a system with both quenched disorder and competing interactions; here, the underlying canonical model is the random field Ising model. Neutron scattering measurements of long range magnetic order are severely affected by extinction, while other experimental techniques are limited in value by the large disorder. Magnetic photon scattering has been proven to be free from these shortcomings. In applications to a sample with the rutile structure, the (100) Bragg reflection has the great advantage of being a special point at which the charge structure factor vanishes. Lastly, there is mention in table 3 of studies of superlattices. Advances in molecular beam epitaxy deposition techniques have led to production, an atomic plane at a time, of single crystal superlattices composed of altemating layers of magnetic rare earth. These, and other, samples have been successfully investigated by magnetic

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

571

X-ray diffraction experiments, as often as not accompanied by corresponding neutron scattering studies (Bohr et al. 1989, Vettier et al. 1986, Majkrzak et al. 1991). 3.3. Elastic resonant scattering

When the primary photon energy is tuned near an absorption edge, large resonant enhancements of a Bragg intensity can be observed. Figure 3 shows the energy dependence of the integrated intensity at the first-order (0, 0, 5/2) satellite to the (0, 0, 2) charge reflection of UAs which occurs for the simple linear antiferromagnetic configuration of moments established below 127 K. The intensity depends on the specific absorption edge, photon polarization states and the magnetic state of the sample. Of course, non-ferromagnetic materials, with purely magnetic Bragg reflections, are most easily studied. An interpretation of elastic resonant scattering, based on an atomic model of the magnetic state and a one-electron process, has been shown to adequately explain the features observed in antiferromangetic UAs and more complicated magnetic configurations. Within an intuitive one-electron picture of the electronic structure, fig. 6 depicts the resonant enhancement process at the L3 edge of a model appropriate to rare earth materials. For this model, there are localized 5d-states available in dipole-allowed transitions, and unfilled 4f-states available through quadrupole transitions. Some materials studied by elastic resonant scattering are listed in table 4. Reviews of the work include (Gibbs 1992, Stirling and Lander 1993, Vettier 1994). Elastic resonant scattering was brought to the fore of synchrotron-based scattering studies of magnetic materials by work on holmium (Gibbs et al. 1988). Just below the N6el temperature, 131.2 K, the magnetic moments order ferromagnetically TABLE 4 Representative examples of materials studied by elastic resonant scattering, see also table 1. Material

Ref.

Edges

Er Ho TmSe UAs Fe Tb Dy UO 2 USb NpAs Hoo.sEr0.5

[4] [1, 2, 10] [5] [3] [6] [7] [8] [9] [9] [11] [12]

L3 L3 L3 M3,4,5 L2,3 L3 L3 M4,5 M4,5 M4 L3

References: [1] [2] [3] [4] [5] [6]

Gibbs et al. (1988) Gibbs et al. (1991) McWhan et al. (1990) Sanyal et al. (1994) McWhan et al. (1993) Kao et al. (1990)

[7] Gehring et al. (1992) [8] Isaacs et al. (1989) [9] Tang et al. (1992b) [10] Thurston et al. (1994) [11] Langridre et al. (1994) [12] Pengra et al. (1994)

572

S.W. LOVESEY

within the hexagonal planes of the h.c.p, crystal structure to form a spiral magnetic configuration with a turn angle per plane ~b ~ 50 ° along the c-axis. The magnetic diffraction pattern consists of pairs of satellites offset from each of the chemical Bragg reflections by w = (0, 0, t), where t = (2~b/c0) and c0 = 5.62 /k at room temperature (the satellite position changes with temperature). The resonant effect near the L3 edge at 8.07 keV, for which the corresponding wavelength = 1.54 A is quite well suited to diffraction studies, increases the scattered intensity by a factor of about 50. The observed intensities and polarization properties are consistent with a model (Harmon et al. 1988, 1989) based on one-electron El, E2 processes (Gibbs et al. 1988, 1991). Unlike a simple linear antiferromagnetic configuration, displayed by UAs (McWhan et al. 1990), dipole (El) transitions for a spiral configuration produce first- and second-order magnetic satellite reflections. The model proposed in (Hannon et al. 1988, 1989) is outlined in section 9 for the case of dipole-allowed transitions. Of course, there is a very close connection between absorption and resonant scattering, since both processes are described by the same part of the scattering amplitude. However, the absorption coefficient is proportional to the imaginary part of the amplitude while the Bragg intensity is proportional to the square of the absolute magnitude of the amplitude averaged over thermal, concentration and defect fluctuations. On performing the average of the scattering amplitude both magnetic and lattice properties are effected so, for example, it contains Debye-Waller factors that arise from lattice vibrations. Just as for magnetic dichroism, the thermodynamic properties of the magnetic state of the sample influences elastic magnetic scattering through the averaging of matrix elements of the position operator with respect to magnetic quantum numbers. Regarding the dependence of quantities measured in absorption and scattering experiments on the polarization of the primary beam for the former experiment one averages the amplitude, and for the latter one averages its absolute square. The Bragg scattering amplitude is usefully developed in powers of the (unit) vector which described the configuration of the average magnetic moments in the magnetic structure. For example, a simple antiferromagnetic configuration is described by re(R) = (0, 0, exp(iw. R)) where the phase factor has values -4-1 when R coincides with moments on one or other of the two sublattices. The dipole (El) contribution to the amplitude (9.5) is found to be a linear combination of the factors (s' .¢), m.(s' x ¢) and ( # • m)(¢. m), i.e. terms of order 0, 1 and 2 in m. Figure 11 shows these three factors in our 2 x 2 matrix representation of polarization states. If the moments are contained in the plane of scattering, the term of order one is seen to have no diagonal elements so, for example, cr-polarization is rotated to ~r-polarization. This rule, and several others, has been confirmed by experimental studies (McWhan et al. 1990, Tang et al. 1992b). An incommensurate spiral configuration of magnetic moments, realized in Ho and some other rare earth metals; is described by, re(R) = (cos(w- R), sin(w, g), 0), and the E1 amplitude can contribute to a main Bragg reflection and satellites to it defined by w and 2w, coming from the terms of order m and m 2, respectively. The intensity of the first satellite in Ho approaches zero at the N6el temperature in a manner characteristic of a continuous phase transition (Gibbs et al. 1991). This temperature

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

573

dependence of the intensity is consistent with the model under discussion. For, on referring to section 9, the products of dipole radial integrals that multiply m . (e t x e) in the scattering amplitude - the same combination of radial integrals which arise in the circular magnetic dichroic contribution to the attenuation coefficient - when evaluated within a localized model of rare earth magnetism (Thole et al. 1985) are proportional to the ordered magnetic moment. The model predicts a temperature dependence for the magnetic intensity at the main Bragg peak and associated 2wsatellite which stems from thermal fluctuations in the mean-square magnetization. The one-electron atomic model of elastic resonant scattering has been explored for rare earth, actinide and 5d transition-metal compounds (Carra et al. 1989, Hannon 30

25 " T = 7 8 K 2o

15 ~ 10 5 0 30 25 ' T = g 4 K •

.

.

.

,

.

.

,

.

,

~2Ol -~, 5 0

:

30

A_ = 50K

25

~ 20 ~ 15 -~ 10 5 0 30 25

;T-

.~ 20 g 15 -~ 10 5 0 30 25 T

I

,

I

l

I

'

I

'

I

34 K

.-~ 20 g 15 -~ 10 5

0

8300 8320 8340 8360 8380 8400 Energy [eV] Fig. 13. Integrated magnetic scattering intensities (arbitrary units) of the scattering at the (0, 0, 2 + t) satellite of erbium at various temperatures, T > 34 K, corrected for absorption. The solid lines are a fit to a single oscillator at the L 3 = 8.36 keV absorption edge (Sanyal et al. 1994).

574

S.W. LOVESEY

et al. 1988, 1989, McWhan et al. 1990, Tang et al. 1992b), and applied to scattering from surfaces (Fasolino et al. 1993). In Luo et al. (1993) it is extended to inelastic resonant scattering, and sum rules are developed to separate out contributions from spin and orbital magnetism. Tang et al. (1992b) report ab initio atomic calculations of the scattering amplitude at the M4 and M5 edges of uranium in UO2 (U 4+) and U S b ( U 3 + ) . With the magnetic moments arranged to lie in the plane of scattering, the observed cr - 7r scattering is proportional to the m. (e' x e) contribution to the amplitude (9.5). Radial integrals in FaM(E) were obtained from a Hartree-Fock scheme, including relativistic corrections. Fits to the experimental data are good for UO2, modelled by U 4+, indicating that an atomic picture is useful. Some discrepancies are found between experimental data and calculations for USb, which might indicate the possible need to go beyond an atomic picture and include hydridization between f states and band states. In the atomic model the magnetic order is brought about by use of a magnetic field. For erbium at low temperatures an unusual behaviour of the magnetic scattering as a function of energy has been observed. Looking at figs 13 and 14, the integrated magnetic scattering intensity for the (0, 0, 2 + t ) reflection at temperatures above 34 K is quite different from that seen at 12 K (Sanyal et al. 1994). Neutron diffraction studies of erbium have identified three magnetic configurations. Below the Ntel temperature of about 89 K and above 52 K, the moments are believed to be ordered along the c-axis and longitudinally modulated. For temperatures between 52 and 18 K, an additional component of the magnetization develops within the basal plane, forming a magnetic structure with a unique chirality. Below 18 K, there is a firstorder transformation to a commensurate magnetic structure which is believed to be a conical phase. Looking at fig. 14 for T = 12 K, the cross-section is unlike the single peak at the L3 absorption edge observed at the higher temperature. Instead, what is found is a series of sharp peaks and broad humps extending as far as 2 keV below the absorption edge. 0.20

0.15 t--

"=0.10

~0.05~ 0

e e

.

we~,,,,, e

ee

c

0

~,~

~'~*" t

8000

~ . I

,

I

8150 8300 Energy [eV]

,

..

I

8450

Fig. 14. As - measured integrated intensity (arbitrary units) for the (0, 0, 2 + t) satellite of erbium at 12 K, as a function of the primary photon energy. The vertical solid line indicates the position of the L3 absorption edge (Sanyal et al. 1994).

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

575

,10 -= I

I

"m

I

I

I

t.50

IIt O

T=175.3K

I.O0 E

0.50 "~ I.,J I..-

10 -~

z

3 0.00 II ¢-

o

~2 -

% oo o o o o o o

>-

T=175K

o10-,

_J

5

4~ go

,"..

Beg oo

C 0

T=177K

•

E

•

>-

p-

.,,,

Z

•. - ' : . . ' . . . . •

-

'z

"".'"'.:'i

g go

o

3.0t

T=178K ,~

o o

~

2o,o J/ o 1.C

I

0,20

,

I

0.22

,

0.24

0.26

0.28

(0 K 2) Fig. 15. Examples of the critical scattering observed for NpAs as a function of temperature at the (0, 0.23, 2) peak (Stirling and Lander 1993, Langridge et al. 1994).

576

S.W. LOVESEY

Magnetic critical scattering has been observed in Ho (Thurston et al. 1993, 1994) in the vicinity of the transition to a spiral antiferromagnetic phase at 131.2 K, and NpAs (Langridge et al. 1994) in the vicinity of an incommensurate antiferromagnetic phase at 173 K. Figure 15 contains examples of the critical scattering observed for NpAs. In both cases, the measurements were performed with the radiation tuned in energy close to a resonant level; for holmium a primary photon energy ~ 8.07 keV is the dipole maximum of the resonant magnetic scattering at the L3 absorption edge, while for the experiment on neptunium arsenide the M4 edge (~ 3.85 keV) of the Np ion was utilized. The consensus view from the experiments is that the observed critical behaviour is significantly different to that observed with conventional neutron scattering techniques. There are apparently two length scales, and the new, longer one may be related to the surface or be a result of long-range order nucleated by impurities. Magnetic correlations with two length scales have been observed (Hill et al. 1993a) in photon diffraction studies of the random field Ising antiferromagnet Mn0.75Zn0.zsF2, and in this case the longer length correlations have been directly related to surface preparation. It is early days for this type of work, and the use of two length scales in data analysis might be a notion with a short lifetime.

3.4. Spectroscopy It is well established experimentally that the cross-section for light scattered inelastically by ordered magnetic materials displays pronounced features due to the collective spin oscillations, known as spin waves and magnons (Hayes and Loudon 1978), and in insulating materials their maximum energy is of the order of several meV. Antiferromagnetic materials contribute two magnon effects (Cottam and Lockwood 1986, Rosenblum et al. 1994), in addition to one magnon events seen also with ferromagnets. The intensities of these events are too large to be consistent with excitation mediated by the magnetic dipole operator. Instead, a satisfactory account of the data, on one and two magnon events, is provided by the electric dipole operator when account is taken of the spin-orbit interaction. In this instance, the photon field couples to the orbital degrees of freedom of the unpaired electrons, and thus through the spin-orbit interaction to the atomic spins. These, in turn, interact through an exchange interaction, of the Heisenberg type, which supports collective spin oscillations. The two magnon events mentioned appear in spectra for antiferromagnetic materials and are insensitive to the application of a magnetic field. Their contribution is in the form of a broad band of intensity, from two magnon sum and difference events, and bound states condensed out of the two magnon continuum. The two magnon and one magnon events have similar intensities, and the former, in fact, are for some materials the dominant feature. These aspects of two magnon events preclude second-order one magnon scattering as the underlying mechanism, since this mechanism does not discriminate between ferro- and antiferromagnetic materials. Instead, the key lies in considering processes that involve two neighbouring magnetic ions, coupled by the exchange interaction. Moreover, just as the Heisenberg exchange arises from the electrostatic interaction, between neighbouring ions, and

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

577

the requirements of the Pauli exclusion principle so the same mechanism transfers in opposite directions spins on neighbouring ions. For the N6el state, the transfer of spins leads to an excited state without change in the total spin of the system which, of course, is zero in both cases. Since there is no net magnetization the process is not influenced by a magnetic field. In a ferromagnet, the creation of two spin excitations requires a change (of two units) in the total spin and this is not allowed in the proposed scheme. Finally, let us mention that there is a similarity between the schemes proposed for two magnon events in magnetic materials and interaction induced (sometimes called collision-induced) events in light scattering from nonmagnetic materials. For example, it is observed with a dilute fluid of noble gas ions that scattered light is significantly depolarized, yet if the scattering is by individual atoms the intensity is proportional to (e. e~)2 which vanishes for orthog0nal primary and secondary polarization states. Analysis of the scattered intensity is consistent with the idea that scattering comes from two, or more, correlated ions. At the lowest level of approximation, the correlation is created by the electric dipole-dipole interaction between neighbouring ions, responsible for the Van der Waals attraction between neutral atoms. The recent development of high resolution X-ray spectrometers, for the measurement of phonon dispersions, plasmon peaks, particle-hole continuum, etc., is reviewed by Burkel (1991). In the remaining part of this subsection, attention is directed to measurements at synchrotron sources of Compton profiles for magnetic materials. Background theory for the interpretation of measurements is covered in sections 2 and 10. At present, relatively few magnetic materials have been investigated by Compton scattering; table 5 lists most materials for which Compton profiles are available. This situation is likely to change in the near future as new facilities, such as the TABLE 5 Representative magnetic materials for which Compton profiles have been measured. In all but one example, circular polarization of the primary beam is used to isolate the profile associated with unpaired electron spins. Reference [2] establishes by investigation of several materials that the Compton profile is not sensitive to orbital magnetism. Material

Ref.

Comment

Ni Fe Mn ferrite Ni Fe Gd HoFe2

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

unpolarized 7 rays circularly polarized 7 rays synchrotron source

References: [1] [2] [3] [4] [5]

Anastassopoulos et al. (1991) Timms et al. (1993) Sakai et al. (1991a) Cooper et al. (1986) Tanaka et al. (1993)

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

Sakai Sakai Sakai Kubo

and Ono (1977) and Sekizawa (1987) et al. (1991b) and Asano (1992)

578

S.W. LOVESEY

°

..-

••

%

•

°

.;

• • •

"1

•. ;

.."

.

;. •

"%

:-

•

..

¢' ,.-;

0

. %

.. "..•..-., "--.,..,_~I ;" .. ".. ". '..

....

-'- ..

t/'l ¢-

..

• • •

.d

~

Y,

•"

-"

ul

-

t.-

"•

.

~e,,

,,,,,,,, " ~ ~ 2 2

°.% " •

•

...... _-. "-,,,

,_.

". - . , . J~

,...."-'"

•

"•

~"

".° :

'" ".

"~3

• .. -

"~__

31

110

,,_.-.".....

-'-I":"" _

",,~.~.~211 %,

' • : a ° :.

•

c-

"

" ;"

""

,,-" '-. - ": ~"'-,~__2~ o i

°. °°,

I0

"%'-,,,,,,,,,..,,~ ~I'IQIII~ e

II

i

0

1

2

3

/-,

5

6

7

8

9

Pz (a.u.)

Fig. 16. Magnetic Compton profiles of ferromagnetic Fe ÷ 3 wt.% Si along fourteen crystal directions. The evaluated momentum resolution is 0.76 a.u. (Tanaka et al. 1993).

PHOTONBEAMSTUDIESOF MAGNETICMATERIALS

579

elliptical multipole wiggler at the National Laboratory for High Energy Physics, Tsukuba, Japan (Tanaka et al. 1992), attain full capacity. Reviews of experimental and theoretical studies of Compton profiles of magnetic materials are provided in Cooper (1987), Sakai (1992), while the well-established field of non-magnetic materials is reviewed in Cooper (1985). Most of the, relatively few, Compton profiles available for magnetic materials have been subjected to detailed theoretical interpretation (Cooper et al. 1986, Kubo and Asano 1992, Sakai et al. 1991a). It is likely that, even more interest will be forthcoming with the advent of three-dimensional Compton profiles reconstructed from many one-dimensional profiles (Tanaka et al. 1993). Figure 16 shows the magnetic Compton profiles of ferromagnetic iron along 14 crystal directions, which are the raw material for the reconstruction procedure proposed in Suzuki and Tanigava (1989). Recent work has used circularly polarized hard X rays, e.g., at the Tsukuba elliptical multipole wiggler the energy of emitted photons is 60 keV. For materials with a net magnetization, the Compton profile can be extracted from the polarizationinduced charge-magnetic interference scattering. This contribution to the total scattering can be isolated by a differencing method; the sign of the interference scattering is reversed by reversing the polarity of the net magnetization, by application of an external field, or changing the handedness of the circular polarization in the primary beam. Turning to the data for ferromagnetic iron displayed in fig. 16, the measured magnetic scattering intensity is typically 1% of the charge scattering and the counting statistics give an accuracy of about 1% at the Compton peak. The good statistical accuracy was found to be essential for a meaningful reconstruction of the three-dimensional Compton profile. All the main features of the data, including the diminution of intensity at the origin (usually ascribed to a negative spin density), are reproduced by a one-electron theory based on the full-potential linearized augmented-plane-wave method (Kubo and Asano 1990).

4. Kramers-Heisenberg amplitude The results discussed here have applications in the interpretation of resonant and nonresonant events. Effects due to the spin of an electron are not included, although it is not very difficult to do so (Sakurai 1987). Instead, we delay our discussion of spindependent events to section 6 where our most general expression for the scattering amplitude is recorded. One reason for giving here a discussion of a less general result for the amplitude is that it is possible to get an appreciation for the key ideas and approximations while pursuing an algebraically simpler problem which, none the less, is of use in the interpretation of experiments. In particular, the KramersHeisenberg formula provides a basis for the contribution from orbital magnetism to the diffraction cross-section, and the interpretation of dichroism and elastic resonant scattering. Photon scattering consists of the absorption of a primary photon and the simultaneous emission of a secondary photon. The wave vectors of the primary and secondary

580

S.W. LOVESEY

photons are q and q', and the scattering vector, (4.1)

k=q-q'.

The concomitant change in energy is, (4.2)

hw = hc(q - q') =-- E - E'.

The real polarization vectors are ¢ and #, and e . q = e' • q l = O. To provide a compact expression for the scattering amplitude operator it is prudent to define a momentum density, J(q)=~pj

exp(iq . R j ) ,

(4.3)

J

where the sum is over all the charged particles in the sample, and p and R are conjugate variables. Note that the Hermitian conjugate, (4.4)

J+ (q) = J ( - q ) + h q n + (q),

so ¢. J+(q) = ~ . J ( - q ) . With this notation, the scattering amplitude operator, first derived in 1925 by Kramers and Heisenberg, in units of re, is, G = - ¢ . # n ( k ) - (1/m*){¢. J(q) ( E , - E '

-

~t.~)-I ~ t j + ( q , ) + (4.5)

+ e'. J+(q')(E.

+ E

-

7-~)-1 e . J(q)},

where ra* is the mass of the charged particles, the Hamiltonian 7-£ describes the target sample, E u is the energy of the initial target state (7-/1#) = Eul#) ) and E = ~ q , E ' = hcq t. The resolvent operators (Eu - E / - 7-/)- 1 and (E u + E - 7-/)- I are defined with an infinitesimal negative imaginary part added to 7-/ (not shown explicitly) for pole avoidance. The avoidance rule is important when the poles of (4.5) with respect to the energies of intermediate states are in the region of the continuous spectrum; e.g., if the initial state is the ground state of an atom this would occur for E exceeding the ionization threshold of the atom, whereas in a molecule the threshold for dissociation into atoms takes the place of the ionization threshold. With the result (4.5) for G, the differential cross-section is found to be, d~r/df2 = ( m / m * ) 2 ( q / / q ) [ ( # 'IGI~)

[2.

(4.6)

This expression does not contain effects due to the spin magnetic moment of the electrons, as discussed earlier in this section. But it does contain magnetic scattering in the form of a contribution that can be identified as due to orbital angular momentum. The occurence of orbital angular momentum is taken up in the following section.

PHOTONBEAM STUDIESOF MAGNETICMATERIALS

581

To conclude this section, let us consider the explicit form of one of the two 'resonance-type' contributions in G. The infinitesimal negative imaginary part added to 7 / i s denoted by (-i~//2). The choice of notation is meant to convey the notion that physically this contribution arises from damping, just like one has in the purely classical theory of scattering. Selecting to examine the third term in (4.5), and inserting a complete set of states for 7/labelled by quantum {~7} including all states from the discrete and continuous parts of the spectrum, one finds,

6'. J+(q')[~)(~16. J(q) (1/m*) E

~-~ ~ ~ - -

~-7 +-i7~

•

(4.7)

in practice, % be neglected except when E "~ E, 7 - E u. Most often, the resonant amplitude is much larger than the sum of non-resonant amplitudes. This arises because the magnitude of the resonant amplitude, is of the order k while the magnitude of the non-resonant amplitude if of the order of r~. If the condition for resonance with a particular intermediate state is almost satisfied, so E _~ E,7 - Et,, and the intermediate state is nondegenerate, the single-level inelastic resonance formula is,

d~r/dO = ( m r ~ / m * 2 ) Z ( q ' / q )

I ( U ' [ 6 " J + (q')177) (r/16 . J(q)l/~)12

(Eu + E - E,7)2 + (%/2) z

(4.8)

If q and q' are small the matrix elements which appear in (4.8) can be calculated using J(0) = P where P is the total momentum. The result, (~'[PD) = ( i m * / h ) ( E ~ , - Ev)(dIRI~7 },

(4.9)

in which (/z'lRl~7) is an off-diagonal matrix element of the dipole moment operator, is often utilized. The next term in an expansion of J(q) in q produces the total orbital angular momentum operator interacting with a component of H = curiA, and the quadrupole operator. Hence, beyond the dipole approximation, resonant scattering contains processes that are explicitly magnetic in character.

5. Scattering by orbital magnetism The object of the present discussion is to expose the orbital magnetic moment in the Kramers-Heisenberg scattering amplitude, and demonstrate its simple relation to the appropriate operator in the amplitude for magnetic neutron scattering. This identification has an immediate practical benefit since it enables us to utilize the knowledge of the properties of the operator in the literature on neutron scattering. In fact, the non-resonant limit of G is shown to contain the operator Z defined in (A.7) which is related to orbital magnetism. The experimental evidence is that orbital magnetism is observed in Bragg diffraction but not in Compton scattering.

582

S.W. LOVESEY

Turning to the definition of the photon scattering amplitude operator (4.5), it is evident that orbital angular momentum can stem only from the last two terms. When E, E I are much in excess of the energies of states in the spectrum of N, the Hamiltonian that describes the target, the resolvent operators in G can be safely expanded in ( I / E ) and (1/E'). Keeping the first-order terms in such an expansion leads to a non-resonant scattering amplitude operator,

G=-re

[e . e ' n ( k ) + - - 1 { 1 e' . J + ( q ' ) e . J ( q ) - ~ _ E'I . e .j(q)e,.j+(q,)}](5.1 ) m* -E

Since J(q) is the spatial Fourier transform of the linear momentum, the second two terms are manifestly quadratic in the momenta of the particles. But, because p~ and Rj do not commute there is, in fact, a term linear in the momenta and proportional to k:

# . j+(q,) ~. j(q) = ~ e-iq'.R3 ¢1 .pj eiq.R~, ¢ . p y jj'

-: Z e iq'R~'-iq''R~ e I. (pj + hq~jj,)¢.pj,, j3,

(5.2)

in which q can be replaced by k because e'. q' = 0. In the subsequent development, terms quadratic in p are dropped in favour of linear terms since the latter are the most significant. Two features of the final result contribute to the licence for this decision. First, the terms quadratic in p will generate smaller effects than those from terms linear in p, given that, as we shall see, the latter produce a finite result. The argument here is akin to that which leads to allowed electric dipole transitions dominating magnetic and higher-order electric transitions. Secondly, the quadratic terms assembled in G arise with a factor (E, - E I ) / E E ~ which vanishes for elastic scattering (Bragg aad the static limit); terms quadratic in p are analysed in Grotch et al. 1983. Hence, in future we shall use the approximate result,

e' . J+(q') e . J(q) = h Z exp(ik. Rj) (e' . k)(e .pj), J

(5.3)

where k = q - q l , as elsewhere in this chapter. On inserting this value in (5.1),

G = -re [e" ¢'n(k) - b-(m/ra*)(# x e). Z(k)].

(5.4)

Here, r = (hq/mc) and (5.4) is valid for ~- << 1 (Grotch et al. 1983). The operator Z is defined in (A.7). It is closely related to an operator used in the theory of neutron scattering to describe orbital magnetism (Balcar and Lovesey 1989). Since the operator T is extensively studied in neutron scattering theory it is prudent to import the knowledge to the present case of photon scattering. To this end we note that, for elastic scattering, Z(k) = - ( k / q ) 2 T(k),

(5.5)

PHOTON BEAM STUDIESOF MAGNETICMATERIALS

583

and (T(k)) = --(1/2#B) f dr exp(ik, r){k x ((Mz(r)) x k) },

(5.6)

in which Mt(r) is the orbital angular momentum operator, and k = k/k. Matrix elements of T(k) for atomic 3d and 4f states, of a single multiplet, have been tabulated by Balcar and Lovesey 1989.

6. Scattering by spin magnetism The amplitude for spin-dependent scattering of photons by bound electrons has been considered in great detail (Bhatt et al. 1983, Grotch et al. 1983). The expression quoted here is derived in the cited papers by a systematic treatment of the quantum electrodynamic Hamiltonian for a bound electron, and an expansion in powers of the small parameter r = (primary energy/electron rest mass). In section 1 we indicated how the result can be derived by addition of spin-dependent terms to the non-relativistic interaction. While such a scheme is quite similar to the workings reported by Grotch et al. (1983) it is more difficult to be confident about making an internally consistent calculation. We find to order T = (hq/mc) that results from the two approaches are identical. Hence, to this level of approximation it is not necessary to employ the mathematical apparatus used by Grotch et al., but their scheme is a safe route by which to obtain higher-order terms, containing products of spin and momentum operators for example. If very hard X rays are used, "r is no longer a good expansion parameter. In which case, an alternative route to follow is to expand the amplitude in terms of the fine-structure constant. Following on the development provided in section 4, we employ a scattering amplitude operator G in terms of which the differential cross-section is (m = ru*), d~r/df2 = (q'/q)l(#'lGl#)12.

(6.1)

Here, # and #~ label the initial and final states of the target, and hcq and hcq~ are the energies of the primary and secondary photons. Polarization of the primary beam is accounted for in (2.10) in terms of a density matrix, and the result (2.5) explicitly displays all possible polarization-induced processes. In place of the momentum density (4.3) we need in the general case,

J(q) = Z ( p j + ihsj x q) exp(iq • Rj), J

(6.2)

for which, {J(q) • e} + = J(-q). e,

(6.3)

where e is the real polarization vector for the primary photon, ~. q = O. In (6.2), s5 is the spin operator of the electron labelled by j.

584

S.W. LOVESEY

The operator G calculated to first-order in ~- is obtained from eq. (2) in (Grotch et al. 1983). Expressed in our notation, G : - r ~ [e. ~' n(k) + (i/2)7-(1 + ( q ' / q ) ) S ( k ) . (~ x ¢') + + (l/m){~'. J(-q')(Eu + E

-

7~)-1

J(q) +

fir •

(6.4)

+ ~. J ( q ) ( E u - E' - 7{)-1 ~,. j ( _ q , ) } ] . Here, n(k) is the spatial Fourier transform of the charge density, and S(k) is the spatial Fourier transform of the spin density defined in (2.6) and (A.6), respectively. Examination of (6.4) shows that spin arises in the resonant and non-resonant events, and the structure of (6.4) is similar to (4.5) - on setting m = ra* - apart from the second term which is due solely to the spin density. The matrix representations of G and the three combinations of polarization vectors are displayed in figs 10 and 11. The evidence to date is that the result (6.4) together with (2.4), (7.1) and (6.1) are cornerstones for the interpretation of photon absorption and scattering by magnetic materials. The partial differential cross section derived from (6.1), in terms of .correlation functions, is given in (2.10). The scattering amplitude operator G in (6.4) enables us to discuss resonant processes and nonresonant processes. For the latter, resolvent operators are expanded in ( l / E ) and (1/E'). The expansion is justified when E, E' are much in excess of the energies of states in the spectrum of 7~, the Hamiltonian which describes the charge carriers in the sample. On keeping the leading term in such an expansion, we find, G : -r~ [~. 6' n(k) - iT{S(k). B + (~' x ~). Z(k)}].

(6.5)

The operator Z(k) is defined in (A.7); it is intimately connected to the spatial Fourier transform of the orbital magnetization, cf. (5.5) and (5.6). The quantity B in (6.5) is a rather complicated function of polarization and photon wave vectors. Working directly from (6.4),

8: !

x

×

x

+

2

(6.6) -

×

The first term in (6.6) is explicit in (6.4), whereas the three remaining terms arise from the use of momentum and spin commutation relations in the components in (6.4) that are quadratic in J. The result (6.5) is the basis of our discussions of X-ray scattering by magnetic materials. Evaluated for q = q' the amplitude (6.5) permits a discussion of Bragg diffraction. The inelastic scattering process of main interest is Compton scattering. For all these experiments there are benefits to be gained in utilizing the polarization of

PHOTON BEAM STUDIES OF MAGNETICMATERIALS

585

primary and secondary photons. A compact presentation of polarization phenomena, employing a density matrix to handle states of partial polarization, is given in the appendix. The result (6.5) for G determines the four components c~ and #, listed in the appendix, from which we obtain the full polarization dependence of the cross-section and the state of polarization of the secondary photon beam. An outline of the steps leading to our expressions for c~ and/3 begins with the definition,

G=l#+a.o, in terms of the unit and Pauli matrices. The next move is to express the polarization dependence that occurs in (6.5) in terms of 2 x 2 matrices using the basis defined by figs 10 and 11. Of the three combinations of e and # appearing in G the most complicated is B defined in (6.6), for which we find,

(a-b)~+(bcosO-a)~')

a(~x~') B:

(a-cos0)~+(1

- a)~'

(b+l

- a)(~ x ~')

"

(6.7)

with b = (q'/q) and a = (1 + b)/2. If the matrix elements of n(k), S(k) and Z(k) that appear in the cross-section are real then the latter, being related to the absolute square of the matrix element of G, is quadratic in T. Hence, the magnetic contribution to the cross-section, weighted by T2, is small compared with the charge contribution. Another factor reduces even further the ratio of magnetic to charge contributions. Charge scattering engages all electrons whereas magnetic scattering arises from the few which are unpaired. If, on the other hand, the charge matrix element, say, is complex (anomalous scattering) the cross-section contains an interference between charge and magnetic contributions, i.e. a term weighted by T. Such a term is likely to dominate a term weighted by ~_2. Another means by which to create an interference between charge and magnetic amplitudes is to employ primary photons with (complex) elliptic polarization. Terms of order "r2 are most likely to be measured at purely magnetic reflections that occur in non-ferromagnetic materials. 7. D i c h r o i s m

The diminution of intensity of a beam caused by a target foil is described by exp(-RT), where 3' is the attenuation coefficient and R is the distance through the foil in the direction of the incident beam. It is found that 7 depends on the state of polarization of the primary beam and properties of the target; the polarization dependence is usually called dichroism. In many cases of practical interest, the imaginary part of the index of refraction, n", and 7 are simply related, ~, = 2qn". The bulk parameters 7 and n can be correlated with properties of the constituents of the target sample. A simple result is obtained for a low density target, in which the particle size is small compared with the average distance between particles. In

586

s.w. LOVESEY

this case, if n 0 is the density of particles in the target, each having the same total cross-section atot, one finds, = noO'tot. Recall that, the total cross-section includes all possible elastic and inelastic processes for a given initial state of the photon. Perhaps it is worth mentioning that, a target sample can be absorptive without its individual constituent scatterers being so. Since, if the latter only scatter, they thereby remove energy (intensity) from the forward direction. In view of the significance of Crtot as measuring the removal of energy from a beam, it is also sometimes referred to as the extinction cross-section. The foregoing relation between the attenuation coefficient and O'tot stems from the so-called optical theorem, which says that Crtot proportional to the imaginary part of the forward scattering amplitude, denoted by f". Here, the forward scattering amplitude means without change in wave vector and polarization. To be concrete, we turn to our standard representation of the scattering amplitude G in terms of a and/3, cf. eq. (2.4), where, a and/3 are now diagonal matrix elements evaluated for = ~ . With this notation, the attenuation coefficient,

,,/= (41rno/ q) f " ( E) = (4~rno/ q) Ira{/3 + P . a},

(7.1)

where the Stokes vector P describes the state of polarization of the primary beam. It is a matter of straightforward algebra to show that there is no contribution to "7 from the non-resonant magnetic scattering amplitude; specifically, from (6.7) and (A.7) the matrix B = 0 and orbital operator Z = 0 for the condition of elastic forward scattering, q = qt. In consequence, the magnetic contribution to 7 arises from resonant processes in the scattering amplitude. Of course, there is no explicit account in our result of the geometric optics part of the scattering amplitude, modelled by Fresnel equations, which describes the absorption caused by the magnetically inert features of the target. The proper incorporation of this with the magnetic scattering is not a simple task. For one thing, the magnetic scattering is here treated by the Born approximation for single scattering events, whereas geometric optics, and the refractive index, are intrinsically multiple scattering properties. One avenue for a consistent treatment is to consider scattering from the sum of two systems, one defined as the geometric properties of the sample and the second the magnetic constituents. The corresponding amplitude is the sum of the exact treatment, using the refractive index and geometric optics, of scattering from the sample's shape, and the amplitude for the magnetic component. The latter entails a matrix element of the magnetic interaction operator evaluated not with plane waves but primary and secondary waves distorted from plane wave forms by the sample's shape. However, in our approximate treatment, we neglect all aspects of the geometric scattering to focus on the magnetic contribution calculated with the (first) Born approximation. An improvement to our treatment, ia the form of explicit account of the reflectance, can be accomplished in the framework of the distorted wave Born approximation, familiar in the interpretation of particle scattering by nuclei.

PHOTON BEAM STUDIESOF MAGNETICMATERIALS

587

In the event that the target sample contains several scattering species, which includes the possibility of particles of different orientation, the attenuation coefficient is the sum of individual attenuation coefficients. A few more words are in order on the topic of the optical theorem. If atot(v,~) is the total cross-section, which, in general, arises from elastic and also inelastic processes, for a primary beam of polarization v and incident direction ~, and G~,~(~, ~) is the corresponding scattering amplitude evaluated in the forward direction, the optical theorem is embodied in the relation, O'tot(u, q) =(47r/q)ImG~,~,(~,'~). Bear in mind that IG~,~,(~,~')I 2 determines the cross-section for the scattering of the radiation in an unchanged state, i.e. the cross-section for elastic scattering. Hence, the optical theorem relates the imaginary part of the elastic scattering amplitude in the forward direction to the total cross-section, which is the sum of elastic and inelastic parts. As we have just demonstrated, the attenuation coefficient is calculated from the imaginary part of the forward scattering amplitude, which in turn is determined by the so-called resonant contributions to the total amplitude (more specifically, the contributions generated by second-order perturbation theory applied to that part of the photon-target interaction which is linear in the vector potential). The quantity actually observed is derived from the elastic forward scattering amplitude averaged over the polarization states, denoted by f", as in (7.1). The influence on f " of the magnetic state of the target appears in matrix elements of the current operator,

J(q) = ~ ( p j + ihsj × q) exp(iq • Rj),

(7.2)

J

in which p, R and s are, respectively, the linear momentum, position and spin operators for an electron, and the sum is over all electrons. Let the states I/z) and 1~7) describe initial and intermediate states, respectively; the matrix element of interest is,

Q(q) = (r/IJ(q)l/z >.

(7.3)

Atomic states are appropriately labelled by total angular momentum J, and the corresponding magnetic quantum numbers M, e.g., I/z) = IJM;t}, where £ is the orbital angular momentum of the atomic shell, and Jzl#} = MI/Z). The degeneracy of Q(q) with respect to magnetic quantum numbers is partially, or perhaps wholly, lifted by the internal molecular magnetic field which results in the magnetic splitting of multiplets of the order of 0.1 eV. In consequence, f " reflects the target's bulk magnetic state. This result is not derived from the explicit spin dependent term in J(q); indeed no account of the spin term is included in the subsequent discussion since, usually, it is a small correction to the contribution made by the linear momentum. So, the magnetic character of f" arises from the distribution

588

S.W. LOVESEY

of magnetic quantum numbers generated by the target's internal magnetic structure, which changes with temperature, applied magnetic field, etc. The (exchange) molecular field is different from an applied field in that it acts only on the valence electron spins and not the orbital moment or on the magnetic moment of the core hole. Let us make the dependence of Q(q) on magnetic quantum numbers explicit. To this end, we focus on the dipole approximation for the current operator, achieved with q = 0, i.e. the current operator is replaced by the momentum operator (as already mentioned, spin is neglected in the present discussion). In this limit, Q(q = o) =

(imA/h)(rllRl~),

(7.4)

where the energy difference A = ( E n - Eu). Of course, the dependence on magnetic quantum numbers of the dipole matrix element (~IRI~) is readily derived by a straightforward application of the Wigner-Eckart theorem (Cowan 1981, Weissbluth 1978), (J'M';g"]RqlJM;£)=(-1)J'-M'

J'

-M'

1

q

J)

M

(J'IIRIIJ)"

(7.5)

Here, q = O, + 1 labels spherical components which are related to Cartesian compo- " nents through, R+] = - ( 1 / v ~ ) ( R ~ + iRy), Ro = Rz, R_ 1 = (1//V~)(Rx

(7.6) -- i R u ) .

Hence, the magnetic quantum numbers actually factor out of the matrix element, in the guise of a 3j-symbol. Later on, we will discuss the other key factor in the matrix element, (J' IIR[I J), which is a so-called reduced matrix element. The 3j-symbol vanishes unless the arguments satisfy specific conditions that in effect select some processes and forbid others. These selection rules are illustrated in fig. 7. First, the triangular condition limits J' to the values J' = J, J + 1. Secondly, the 3j-symbol vanishes unless the magnetic quantum numbers in the argument satisfy M ' = q + M . From this result it follows that the matrix element of the z-component of R has no off-diagonal elements with respect to magnetic quantum numbers, whereas the x- and y-components contain only M' = M + 1. A selection rule on the orbital angular momentum quantum numbers g, g' is contained in the reduced matrix element (J'[[R[[J). This vanishes unless g' = g-4- 1. While the latter result appears immediately in a full calculation of the dipole matrix element, for the moment, we will justify it using an argument based on parity considerations. Evaluation of the matrix element entails an integration over all directions of the vector R. The integral vanishes if the intergrand has an odd parity, i.e. if the integrand changes sign under the operation R --+ - R . Evidently, R has

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

589

an odd parity, so a necessary condition for a non zero integral is that the product of the two wave functions also has an odd parity. The parity of a state with angular momentum g depends on whether ~ is an even or odd integer. Therefore, the parity of a product of wave functions, labelled by ~ and 6, is odd if g' differs from ~ by an odd integer. An additional constraint imposed by the triangular condition, which stems from coupling three spherical vector components, leads to the specific values ~' = ~4- 1. Not surprisingly, experimental conditions also impose selection rules on events observed in the attenuation of a beam passing through a magnetic material. For example, if the target has a net magnetization, and the primary photon beam is not perpendicular to the easy axis, the absorption coefficient is changed by reversing the direction (polarity) of the easy axis, by application of a magnetic field. The effect is absent when the beam has no circular polarization, and polarity reversal is found to be equivalent to changing the handedness of the circular polarization. Usually called circular dichroism, the effect is a manifestation of the interaction between the photon spin and the net magnetization (circular dichroism is not observed in paramagnets or antiferromagnets not subject to a magnetic field). The photon spin, also referred to as helicity, is the angular momentum along the direction of the wave vector. It has the values +-1, i.e. the helicity of a photon is a conserved quantity and on absorption there is a change of + h in the component of the angular momentum of the target in the direction of the beam. (With our notation, the component P2 of the Stokes vector is the mean value of the helicity of the primary beam.) So, circular magnetic dichroism is described by the transverse components of the dipole matrix element q = +1, for these, and not the z-component, possess off-diagonal matrix elements in which M' - M + 1. (It is amusing to observe that, if the helicity equals 4-2 then the dipole matrix element would vanish.) While linear polarization does not induce a coupling to the net magnetization, and therefore provides no selection rules, nevertheless it is a useful external experimental parameter. As one might expect by physical intuition, linear dichroism yields information on the mean square fluctuations in the magnetization, present in ordered (ferro-, ferri-, anti-ferromagnets) and paramagnetic systems. As an example of the absence of a selection rule present in circular dichroism we mention that, in general, linear dichroism is present in the ordered magnetic state for all orientation of the beam relative to the easy axis. The foregoing treatment has been couched in terms of matrix elements evaluated in the IJM) basis. Had we opted to use basis [gm)[sms) for the discussion we would have noted that, the matrix element is diagonal with respect to spin quantum numbers, and also derived the selection rules g' = g + 1 with m' = q + m. To keep the mathematical features of the immediate discussion to a minimum, we consider a simple physical model in which one electron in a core orbital, labelled by #, is promoted to a single intermediate orbital, ~/, after absorption of a photon of energy E = hcq. For an atomic model of the type used here, the various transitions in Yb 3+ ( 4 f13 ) are illustrated in fig. 7. Atomic structure and spectra calculations are reviewed in (Cowan 1981). A more realistic model contains many intermediate states individually weighted by the probability that they are vacant in the atomic ground

590

S.W. LOVESEY

state, cf. (7.18). However, the one electron model is a standard approximation. The intermediate orbital has a total decay width F (of the order of a few eV for L edges in rare earth ions) due to both radiative and nonradiative (Auger and Coster-Kronig) processes, and it consists of the initial configuration with a hole in a core level and an additional electron in a higher energy level (E, 7 > Eu). A nice example of the use of the one electron model, to L2,3 absorption by ferromagnetic rare earths, is given by Jo and Imada (Jo and Imada 1993). These authors also give an interpretation of the working of the sum rules, and discuss the vexed question of the overall sign of the circular dichroic signal derived from experimental data. The starting point in the calculation of the forward scattering amplitude is (6.4). For elastic scattering, we need the diagonal matrix element of the scattering amplitude operator taken with respect to the state labelled #. It is convenient to employ the notation (7.3) and A = ( E , 7 - E u ) . The relevant matrix element is,

(~IGI~) : - r e [(~l~" tin(k) + i~-S(k). (~ x d)]~ ) + (7.7) + (1/ra)(z' • Q*)(e. Q ) / ( E - A + iF/2)], and for forward scattering k = 0. It remains to average the matrix element with respect to the polarization states. To this end, use the matrix representations of the three combinations of polarization vectors as shown in fig. 11. The choice of axes to describe the beam relative to the magnetic properties of the sample is displayed in fig. 17. With the notation, Tll = [Q~I z,

T2z = IQu cos~p - Q~ sin cp[2,

Fig. 17. Geometry of a photon beam attenuation experiment. The wave vector q lies in the y - z plane, and makes an angle ~o with the z-axis.

PHOTON BEAM STUDIESOF MAGNETICMATERIALS

591

and

Here, Z is the number of electrons per atom, S is the average spin for the state #, and T~2 and T(~ are the real and imaginary parts of T12, respectively. The first two terms in f(E) are purely real, and do not contribute to the attenuation coefficient, for which we find from (7.1), on taking the limit F ~ 0,

Later, this expression is developed with the addition of two simplifying assumptions; first, use of single atomic orbitals and, secondly, the dipole approximation to the matrix elements Q(q) that make up Tij (the spin term in J is neglected). Before doing so, we remark on the relation between the real and imaginary parts of f(E) which stems from causality. The relation is the bridge between dichroism and the Faraday effect, f(E) depends on E through the simple denominator in the resonant contribution, and F > 0. One sees that f(E) has no poles in the upper-half complex plane, and thus f ' and f " are one another's Hilbert transforms; in truth, it is not f(E) which satisfies these relations but, on account of the terms in Z and S, the subtracted function, (I(E) - Y(Eo))/(E

- Eo),

where E0 is arbitrary. Let us continue the discussion of 7 making use of the dipole approximation (7.4) and the Wigner-Eckart theorem (7.5). If the states # and r/in Q(q) have the same magnetic quantum numbers the components Q~ and Qy vanish, while if they have different magnetic quantum numbers Qz vanishes. These rules bring significant simplifications to T~j since there are no terms involving Q~ mulitplied by either Q~ or Qy. In particular, T12 is purely imaginary, so V is independent of P1. Furthermore,

592

S.W. LOVESEY

and

r(& = -51 cos (mA/h) 2 { I< lml I.>12 - r< JR_l I >r 2 } . Making use of the simplifications in (7.9) the attenuation coefficient reduces to, 47r2nore 7 = - mq -

t, + 1 T l l ( 1 _1_c0s2~ @ p 3 - P2T~2

(~(E - A )

sin2 ~) + (7.11)

+ ~ (l - P3)[Q0] 2 sin 2 ~ .

Note that, for ~ = 0 (7r/2) the attenuation coefficient is independent of P3(P2)- To complete the specification of the model we give the reduced matrix element in the Wigner-Eckart theorem, 3 j,

(J'IIRI]J) = ( - 1 ) 2 (~ x

_1

(R)g,e{(2g + 1)(2g' + 1)(2J + 1)(2J' + 1)} 2 x 1 0

~){J' g

1 1/2

J} gl ,

(7.12)

in which (R)ee is the radial dipole matrix element. The last factor in the reduced matrix element, a 6j-symbol, contains four triangular conditions; these apply to the coupling of (J', 1, J), (J', 1/2, g'), (J, 1/2, g) and (g, 1, g~) and the 3j-symbol vanishes unless (g + g ' + 1) is an even integer. By way of orientation to the value of (R)e,e we consider its value for 2p -+ 3d in nickel. A Hartree-Fock calculation (Cowan 1981, van der Laan 1994b) gives a value (R)zp,3d = 0.125%. It is interesting to compare this value with the value obtained by use of hydrogenic wave functions, namely, 4.748(ao/Z ) which gives 0.170% for Z = 28 (Ni). The tolerable agreement between these two estimates of the 2p --+ 3d dipole matrix element of nickel is by no means a typical situation. For one thing, the hydrogenic estimate is always positive, whereas the use of non-hydrogenic wave functions, as often as not, produce negative values. In arriving at the result (7.11), on which much of the subsequent discussion is based, atomic states have been represented by single orbitals. The validity of this license is open to scrutinity when the crystal field perturbation is large, since the atomic states are most likely to be tolerably described by a linear combination of orbitals with different magnetic quantum numbers.

7.1. Circular dichroism As explained in section 2, circular dichroism is often detected by performing measurements of beam attenuation with two opposite values of the circular polarization. For the model under discussion, the difference in 3, for the extreme values P2 = 4-1

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

593

is proportional to Y~ which, in turn, is proportional to the difference of the radial integrals [(~[R±I I#)l2. The following model calculations of the matrix elements make clear their dependence on the magnetic state of the sample. This entails averaging T{t2 with respect to the magnetic quantum number M explicit in the 3j-symbol in the dipole matrix elements. For the averaged T(~ we introduce the notation,

_

!

c o s ~ ( m A / h ) 2 ( j ' l l R I I d ) 2 F ( j , j'),

2 and calculate F(J, ji) using the molecular field approximation. We find that, F(J, jr) is proportional to the average magnetic moment, and therefore vanishes in the paramagnetic state. The atomic magnetic states involved for Yb3+(4f13), for example, are illustrated in fig. 7. Our choice of axes, and the definition of the angle ~, is provided in fig. 17. Magnetic dichroism in the M4 and M5 absorption edge structure of rare earths leads to consideration of the transition 3d -+ 4f. The initial state 3dl°4f'~(J) is determined by Hund's rule. Dipole transitions are to the configuration 3d94f'~+l(J I) with J - J' = 0, :k l; there are a plethora of states in the configuration that satisfy these selection rules, many of which will not be resolved in energy. Even so, in a study of linear dichroism taken up in the next subsection, Thole et al. (1985) correctly predicted that significant structure exists in the absorption spectrum (van der Laan et al. 1986). The dichroic effect is more pronounced when the final states with different Jt have significantly different mean energies. This can occur in the rare earth M4,5 edges due to the electrostatic interaction between the 3d and 4f electrons. As a guide, in the M4.5 edges the J' = J + 1 states are 2 - 3 eV lower in energy, and the J~ = J - 1 states 2 - 3 eV higher than the J' = J states. Referring to the matrix elements that occur in T(~, the (2J + 1) degeneracy of the initial state is lifted in an ordered magnet by the Weiss molecular field. The interionic interaction responsible for this field couples pairs of ions at sites labelled by m and n. If Z(n) is the exchange interaction, defined to have the property Z(n = 0) = 0, the Heisenberg exchange interaction between spins is described by the Hamiltonian,

7-I =

-

~Z(n)Sm

" Sm+,~ = - ( g - 1)2 ~ Z ( n ) J m

??rt, , T~

"Jm+n,

r P . ~n

where the second equality is valid when the exchange is smaller than the spin orbit splitting, and g is the Land6 factor. Within the molecular field approximation, the magnetic moment is, J

(jz) = ( l / Z ) ~

M exp(2Mu),

M=-J

in which Z is the partition function, and

u = ( J Z ) ( g - 1)2 ~ Z ( n ) / k s T . n

594

S.W. LOVESEY

The critical temperature Tc at which the magnet orders satisfies, (9 - I)2 ZZ(n)

= 3ksTc/2J(J

+ I).

n

From,

Z = E exp(2Mu): sinh {u(2J + 1)}/sinh(u),

(7.13)

M

it follows that,

(J~) -

1 d Z = JBj(u), 2Z du

(7.14)

where the Brillouin function satisfies,

2JBj(u) = (2J + 1)coth {u(2J + 1)} - coth (u).

(7.15)

From the definition,

F(J,J') = ( 1 / Z ) Z e x p ( 2 u M ) { ( _ _ J r 1 11 j ) 2 M

(7.16)

1 -M

-I

one finds,

F(J, J - 1) = -<JZ)/{J(2J + 1)}, F(J, J ) = _ < j z ) / { j ( j + 1)(2J + 1)},

(7.17)

and

F(J, J + 1) = ( j z ) / { ( j + 1)(2J+ 1)}. Hence, F(J, J') is proportional to the magnetic moment. The complete expression for T(~ which is the weighted sum of many terms proportional to F(J, JO, must reflect the thermodynamic properties of the rare earth magnet, e.g., Y[t2 continuously decreases as the temperature approaches Tc, and it vanishes in the paramagnetic state T > Tc. We conclude this section by recording expressions for the matrix elements which are more general than those obtained so far, and also the expression used by Carra et al. (1991) in their study of E1 and E2 magnetic dichroism in gadolinium metal.

PHOTON BEAM STUDIESOF MAGNETICMATERIALS

595

Electrons participating in the absorption process are labelled by j. The probability that the intermediate orbital ~7is vacant for a given initial state # is denoted by pu(o), and the (Boltzmann) probability distribution for the initial states is denoted by pu (at absolute zero, pu = 1). Let (L = 1,2 . . . . . and - L ~< M ~< L), WL -- q(27ce)2 ((L + 1)/L) Zp~pu(~)5(Et~ + E - Ev) x 2

(7.18)

J where JL(X) is a spherical Bessel function and y L ( ~ ) is a spherical harmonic (Weissbluth 1978). For small arguments,

JL(X) ~ xL/(2L + 1)!!. The dipole (L = 1) and quadmpole (L = 2) contributions to the circular dichroic part of the attenuation coefficient "/are,

7 (') = (67rno/q 2) cos~ { W11 - wL1 },

(7.19)

and, 3'(2)= (101rno/qZ) c o s ~ { [ w Z - w 2 _ 2 ]

sin2~ + [W2 - W_21] cos2~}.

(7.20)

Note that W L is dimensionless. Using the small argument expansion of the Bessel function, it is quite easy to demonstrate that (7.19) is consistent with the previous result obtained for a simple model with single component initial and intermediate states with non-vanishing dipole matrix elements. In their study of Gd, Carra et al. (1991) use relativistic Bloch functions to evaluate E1 matrix elements with 2p core states. For comparison with experiment, they added a core-hole lifetime (about 4 eV) and an energy-dependent intermediate-state lifetime. The calculation of the quadrupole spectra is made with an atomic model based on Hartree-Fock theory with relativistic corrections.

7.2. Linear dichroism In order to exhibit the salient features of linear dichroism obtained by Thole et al. (1985) and Goedkoop et al. (1988a), we continue to use their model of the absorption process developed in section 7.1. Figure 18 contains their prediction of the dichroic effect in Dy. The starting point of our calculation is (7.9) for the attenuation coefficient. Synchrotron radiation is predominantly linearly polarized in the plane of the electron beam orbit, and described by the Stokes vector P -- (0, 0 , - 1 ) . Inserting this value in (7.9), the attenuation coefficient is given by,

7 = (47r2nor~/mq)6( E - A) T22.

(7.21)

596

S.W. LOVESEY |

I

I

Dy

,I.~

>.-

T >T c

l-.J

ISS 0 n Z

o

T=0

b

Z

//

C I

I

1295 EXCITATION ENERGY (eV}

1285

T =0 1305

Fig. 18. Calculated spectrum for linear dichroism of Dy (a) for T > To, and for T = 0 with the polarization (b) perpendicular and (c) parallel to the direction of the molecularfield (Thole et al. 1985). If the orbitals I/z) and 1~7) are characterized by a single magnetic quantum number, as described in the text leading up to 7.11), then some products of dipole matrix elements vanish leaving, T22 = (IQ~I 2 c o s 2 ~ ÷ IQ~I 2 sin 2 ~P

(77/,A/h)2[~c0s2~{I('QIRq-ll/Z)]2÷I(T]IR-II/Z)[2} ÷ + sin 2

(7.22)

~l<~lml/z>12],

where ~ is the angle between the photon beam and (magnetic) z-axis, as depicted in fig. 17. We find that linear dichroism is sensitive to thermal fluctuations in the mean square magnetization of the sample which is finite at all temperatures including, of course, the paramagnetic state. After averaging over the (2J + 1) degeneracy of the initial state we find, I(O[Rol#)l a ---4

(J'llnllj)2Fo(J, J')

PHOTON BEAM STUDIESOF MAGNETICMATERIALS

597

where, Fo(J, J') =

1J) 2

M~ exp(2uM)

0

M

"

Evaluating the 3j-symbol leads to, 1)= (J2-((J~)2))/{J(4j2-

Fo(J,J-

1)}, (7.23)

Fo(J, J) = ( ( j z ) z ) / { j ( j

+ 1)(2J + 1)},

and F0(J, J + 1) is obtained from the first result by making the substitution J --+ (J + 1). Further algebra yields,

(7.24) > (J'[[R[[j)2{ (1/(2J + 1)) - Fo(J, J')},

for J' = J, J + 1. Hence, for the chosen model, the thermodynamics revealed in linear dichroism appears solely in ((j~)2) which appears in Fo(J, J'). We conclude with some remarks on the value of ((jz)2). In a pure paramagnet, the result, 1 <(jz)2> _- 3 j ( j + 1), follows because J . J = J ( J + 1). The corresponding values of Fo(J, J') are, Fo(J, J') = 1/[3(2J + 1)].

Inserting this in (7.22) one finds that 2"22 is independent of ~, as it should be in the absence of a preferred magnetic axis. On the other hand, in the ordered state (T ~< Tc) the molecular field approximation developed in section 7.1 provides the estimate, ((j~)2)

-

1 d2 Z = J ( J + 1) - (J~} cothu. 4Z du 2

(7.25)

The result for <(j~)2) is consistent with the standard identity for spin 1/2 operators ((j~)2) = (jz)2 = 1/4. In the limit T -+ Tc, u -+ 0 and from (7.25), ) ~1 J ( J + l ) { 1 =- 3

J(J +

1)

1 + ~2 u Z ( 2 J - 1)(2J+ 3) }

1-

(2J+3)(2J-

1) T - T c ]

598

S.W. LOVESEY

where the second line is achieved with results provided in section 7.1. In the opposite extreme, T --+ 0, only one initial state is accessible and ( ( j z ) 2 ) = j 2 . In this instance, Fo(J, J - 1) = 0 while,

Fo(a, J) = J / { ( J + 1)(2J + 1)},

Fo(J, J + 1) = 1 / { ( J + 1)(2J + 3)}.

So, for T = 0 and J ' = (J - 1), the matrix element 7"22 is proportional to cos 2 ~, while for J ' = J, J + 1 it is a weighted sum of cos 2 ~ and sin 2 ~, although this angular dependence vanishes for J = 1/2. In terms of the more general formalism surveyed at the end of section 7.1, the E1 attenuation coefficient is, %(')(E)-- (6rrno/q 2)

{1

-~ [WJ

+ VV"I1] COS2 ~0 q" W 1 sin 2 ~

}

.

(7.26)

This result describes the so-called white line in the absorption spectrum, and it is the companion to (7.19) for circular dichroism. In the appropriate limit, (7.26) is compatible with 7.22). 8. Diffraction

From the discussion provided in section 2 it follows that strictly elastic scattering (diffraction) is described by (2.5) and (A.5) when products of operators are taken as products of thermal averages of the operators. So, for example, the term/3+(a • P) in (2.5) is (/3+)(a) • P for strictly elastic scattering, achieved in Bragg diffraction. The same interpretation holds for terms in (A.5) for the polarization of the secondary beam. As a first example of the use of (2.5) and (A.5) consider diffraction from a nonmagnetic material. Referring to (A.8), we have for this case, O~1 = O~2 = O,

and 1 % = - ~ re(1 - cos O)n(k), 1 /3 = - - re(1 + cosO)n(k). 2

(8.1)

Let,

(n(k)) = N(k),

(8.2)

which means that N(k) is the average particle density for the sample, and it is usually a complex quantity, N = N' + iN".

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

599

For the cross-section we find the value, 1

do-/dS'2 =

2

elN(k)12{1 + cos 2 0 + P3 sin2 0}.

(8.3)

It is interesting to observe that (a) the cross-section is independent of circular polarization and of the Stokes parameter P1, and (b) the cross-section for photons polarized perpendicular to the scattering plane (~r polarization, P3 = + 1), defined by q and q~, is larger than for photons in the scattering plane Qr polarization, P3 = - 1 ) . Stokes parameters for the secondary beam follow from (A.5) and (8.3). The results are given in (2.9).

8.1. Unpolarized primary beam As the second case we choose to examine diffraction of an unpolarized photon beam by a magnetic crystal. When q = qr the operator Z(k) reduces to,

exp(i k. Rj) (k x pj),

tug(k) = (i/q 2) ~

(8.4)

J and for elastic events, Z(k) is related to the orbital magnetization through (5.5) and (5.6). Equation (8.4) shows that k • Z = 0. The quantities a 1, a 2, a 3 and fl are obtained from (A.8) by taking a = b = 1; in units of re they are, O~1 =-

(i/2)~-(1 - cos 0 ) ( ~ - ~ ' ) . S(k),

O~2 ~-~ - - - ~1

"r('~+ "~'). (Z(k) - (1 - cos O)S(k)),

(8.5)

1

a 3 = - ~ (1 - cosO)n(k) + (i/2)'r(~ x ~ ' ) . Z(k),

and,

1

= - ~ (1 + cos O)n(k) - (i/2)~-(~ x ~ ' ) . (Z(k) - 2S(k)). Here, the small parameter ~- = (incident energy/electron rest mass) = (E/mc2). We choose to write, (s(k))

-

l

fdr

e x p ( i k . r ) (Ms(r)).

(8.6)

For a perfect magnetic crystal, (Mz(r)) and (Ms(r)) are periodic functions that can be expressed as Fourier series in reciprocal space.

600

S.W. LOVESEY

Following Lovesey (1987a), the position of a unit cell in the crystal is defined by a lattice vector l and the cell volume is denoted by %. An atom position Rj = l + d, where d defines the position within a unit cell. The site d = 0 coincides with the comer of the cell, so there are (r - 1) non-null vectors d. Vectors of the reciprocal lattice {-r} satisfy exp(il, r ) = 1

for all 1.

(8.7)

Because (Mz(r)) is periodic its spatial Fourier transform satisfies, (2~) 3 1 f d r exp(ik, r) (Mz(r)) = E 5(k - "r)Fz('r), 2# B at2 Uo ~.

(8.8)

and a similar expression for the spin density. The delta function in (8.8) expresses the fact that diffraction from a perfect crystal does not occur unless strict geometric conditions are satisfied, namely, Bragg's law. We refer to Ft(~-) and F~(T) as magnetic unit-cell structure factors for orbital and spin magnetizations, respectively. Inverting (8.8) we get, exp(-i r . r)Fz(r),

( - 1/2#s)(Mz(r) ) = 1 ~

(8.9)

T

and the conjugate relation,

FL(T) = fcell dF exp(i -i-.

r)

(Mz(r)}/(--2#s),

(8.10)

together with analogous relations for the spin magnetization. Consider a simple magnet with one atom per unit cell. At least for moderate values of k, the atomic structure factors for spin and orbital contributions to the total moment are almost the same. Let f(k) be the atomic form factor, with the standard normalization f(0) = 1. So, to a good approximation,

FL(k)=

l(L)f(k),

and

(8.11)

Fs(k)=(S)f(k), where L and S are the (total) orbital and spin angular momentum operators for the magnetic ion. For a 3d-transition metal ion it is usual to define a gyromagnetic factor go by, (L) = ( g o - 2)(S),

(8.12)

PHOTON BEAM STUDIES OF MAGNETICMATERIALS

601

in which case the ratio of Ft to F , is simply (g0 - 2)/2. On the other hand, a rare earth ion is characterized by the total angular momentum J -- L + S. For a given J-manifold, L = 9zJ and S = 9 s J with, 9t = (2 - 9),

9s = (9 - 1),

(8.13)

in which 9 is the Land6 factor. Finally we note that all the cross-sections involve products of densities of the form (Lovesey 1987a),

( - 1/2#B ) {N(k)}* /

d r exp(ik,

r) (Mz(r)) (8.14)

-----No ((2rr)3/Uo) E

5(k - r ) F ; ( r ) F z ( r ) ,

where No is the number of unit cells and Fc(k) is the charge unit cell structure factor. If the chemical and magnetic structures are different there exist r for which F t ( r ) = 0 and/or F , ( r ) = 0 while Fc(r) ¢ 0, and vice versa, i.e. {r} generally comprises pure charge and pure magnetic reflections, and mixed reflections. In subsequent expressions for diffraction cross-sections the factor, (2rr)3 No--g°

E

5(k - r ) ,

T

on the right-hand side of (8.14) is omitted, simply to minimize the notation. The factor cancels in expressions for the Stokes vector for the secondary beam. Thus, subsequent expressions for cross-sections and P' are understood to refer to a Bragg reflection defined by {'r}. With our chosen notation, (oe} and (~3) are linear combinations of the unit cell structure factors for charge (Fc), spin (F,) and orbital (Fz) densities. Working directly from (8.5), (OZl) ~-- r ~ ( i / 2 ) r ( 1 - cos 0)(~ - ~ ' ) . F s ,

(1)

(c@ = r~r(1 - c o s O ) ( ~ + ~ ' ) • F, + ~ F , (O~3)

=-

-r~(1 - cos0)

{'

,

~ Fc + ir(~ x ~ ' ) . Ft

(8.15)

}

,

and (/3) = - r ~ 1 (1 + cos O)F~ + r~iT(~ X ~ ' ) . {(1 -- COSO)fl + f~}.

602

S.W. LOVESEY

In general, Fc, Fz and Fs are complex quantities. For a centrosymmetric crystal structure, Fl and F~ are purely real, as is Fo unless the photon energy is tuned to a resonance for an atom in the unit cell. In some materials the spatial anisotropy of the atomic environment is sufficient, for wavelengths near an absorption edge, to give rise to dichroism, birefringence and anisotropic charge scattering. The example of tetrahedral anisotropy, realized in germanium, is discussed by Templeton and Templeton 1994. An example of a structure for which Fz and F~ can be complex is the hexagonal close-packed structure adopted by rare-earth metals. It is interesting to observe that, if the spin and orbital moments are perpendicular to the plane of scattering defined by q and q~ then (OZl) = (OZ2) = 0, while if F~ and Ft lie in the plane (c@ and {fl) contain only the charge density. Approaching the forward scattering position, 0 -+ 0, all the components of (a) tend to zero, and {/3) contains only the charge density. With the component of Fs in the plane of scattering arranged perpendicular to k it follows that (a 1) = 0, whereas when the components of Fs and Ft in the plane are arranged parallel to k one has (a2) = 0. With an unpolarized primary beam the diffraction cross-section for the scattering geometry depicted in fig. 19 is, (d~/d£2)o = r~ { sin4(0/2)[T2[ ( ~ - ~ ' ) . F~[2+ -+- T2](q - [ - a t ) . ( 2 F t

1

+ Fs)]2 + [Fc + 2iT sin 0F~[ 2] +

(8.16)

}

4- ~ [(1 + cos O)F~ - 2iT sin 0((1 - cos O)Flu + F~)12 .

Fig. 19. Coordinate system adopted in section 8 for the description of diffraction experiments. The x - z plane coincides with the r/- ( plane (fig. 12), k = q - q/ and (~ x ~1) is parallel with the B-axis (not shown) which is aligned opposite to the ~-axis. An alternative coordinate system is used by Blume and Gibbs 1988.

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

603

In this expression, the superscript y denotes a component perpendicular to the plane of scattering in the direction opposite to the (-axis. An interference between charge and magnetic amplitudes is possible at mixed Bragg reflections for which there are components of F~ and Ft perpendicular to the plane. For a pure magnetic reflection Fc = 0 and in general both Fs and Ft are different from zero. Above the ordering temperature Ft = F~ = 0, and the cross-section is consistent with the classical expression for coherent scattering by a charge distribution. For small 0 in (8.16) the leading-order magnetic contribution is F~. It is interesting to examine the linear polarization of a purely magnetic diffraction signal. In (8.15) take Fc = 0, and for P~, (da/dY2)P~ = -T2(1 - cos 0) 2 x x

Re{ (2

cos(O/2))2F?(F~ + (1 - c o s O)F{)* +

+(~-~').V*(~+~').

V,+~V~

,

(8.17)

where the cross-section on the left-hand side is given by (8.16) with Fc = 0. Observe that when the moment is perpendicular to the plane, P~ is proportional to the orbital magnetization, and for the extreme case in which there is no spin magnetization P~ = - 1 for all qualifying reflections. Conversely, with Fc = Fz = 0 and F~ perpendicular to the plane P~ = 0. The last finding is a special case of the result, - sin 0 sin 2T

(8. 1 8)

{ 1 - cos 0 cos 2W + 2(tan ¢ cot ½0) 2 } ' in which Fs makes an angle (~ 7r - ¢) with the y-axis, perpendicular to the plane of scattering, and the projection of Fs in the plane makes an angle p with the scattering vector k; see fig. 19. Note that P~ = 0 when the projection of F~ in the plane is parallel or perpendicular to k, as is evident from the general expression (8.17) with Fz -- 0. These various results for P~ at a purely magnetic reflection serve to illustrate the value in analysing the linear polarization and the striking differences in photon scattering between the spin and orbital magnetization components. Turning now to the cross-section for mixed reflections, the appropriate expression evaluated to leading-order in r is 1 d¢/dY2 = r 2 ~ (1 + cos 2 0)IGI 2 + T sin 0 (1 + cos 0) Im(Fc*F~u) + (8.19) + T sin 20 (1 -

cosO)Im(FcF{) }.

604

s.w. LOVESEY

The interference contribution is seen to involve just the components of Fz and Fs that are perpendicular to the plane of scattering. At least with regard to experiments performed at synchrotron sources, where the photons are predominantly polarized in the plane of the orbit, (8.19) is actually misleading. The complete expression (8.21) shows that, the orbital term in the cross-section vanishes for P3 = 1 and, to a good approximation, this is realized at a synchrotron source when the plane of scattering is perpendicular to the plane containing the electron beam orbit. The calculation of pI reveals that P~ and P~ are proportional to the magnetic amplitude, and hence they are small quantities. On the other hand, P~ contains both pure charge and mixed amplitudes and the pure charge contribution is, of course, consistent with (2.9).

8.2. Linear polarization In this section the focus is on features due to linear polarization in the primary photon beam, and P = (0, 0, P3). The cross-section of interest is, der/dD = (d•/df2)0 + 2P3 {Re((/3)*(a3) ) - Im((al)*(a2) ) },

(8.20)

where (dcr/df2)0 is the cross-section for unpolarized photons given in (8.16), and the components of (a) and (/3) are provided in (8.15). Perhaps the first thing to observe is that, ((/3)* (a3)) involves only those components of the magnetization that are perpendicular to the plane of scattering (the y-axis in fig. 19) while ((Cel)*(a2)) involves only those components of the magnetization that lie in the plane. Furthermore, ((al)*(a2)) vanishes if the components of Fs and Fz in the plane are either parallel or perpendicular to the scattering vector k. The product ((al)*{oe2)) is proportional to r 2, and consequently it does not contribute to the leading-order interference between charge and magnetic amplitudes. The results (8.19) and (8.20) produce the expression, correct to order r, da/dY2 = r 2

{(1

)

1 + ~ ( P 3 - 1)sin20 IFc[2 +

+ ~-(1 - P3) sin 20 (1 - cos 0) Im(Fc*FzU) +

(8.21)

+ Tsin0((1 + P3) + cos0(1 - P3))Im(F£ F~ ) . Observe that, the pure charge contribution is consistent with (2.8) for the classical limit of scattering by a free electron. The orbital contribution to (8.21) vanishes for or-polarization when the linear polarization of the primary beam is perpendicular to the plane of scattering, and oppositely aligned to the y-axis, cf. figs 4, 9. In consequence, linear polarization permits a separate determination of the spin and orbital contributions to the cross-section. Some general features of the linear polarization of the secondary beam, described by the Stokes parameter P~ can be deduced from the structure of the magnetic

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

605

scattering amplitude displayed in (A.2) and fig. 10. For Bragg scattering the relevant amplitude is,

Evidently, if (o~1) = (0~2) = 0, e.g., the magnetization is perpendicular to the plane of scattering, ~r-Tr and 7r-~r events are forbidden, and P~ = 4-1 for/:'3 = + 1. On the other hand, if (%) = (/3) = 0 then cr-cr and 7r-Tr events are forbidden, and P~ = ±1 for/:'3 = q: 1. (N.B. P~ = 0 for both of the two cases.) These features of the linear polarization of the secondary beam are nicely illustrated in result for Mn0.75Zn0.25F2 (Hill et al. 1993b) shown in figs 20, 21. There is an exact cancellation of the charge structure factor of the ruffle crystal structure at the (1 0 0) Bragg reflection. (This is a fortunate result since the chemical and antiferromagnetic unit cells are identical which, in general, precludes the possibility of observing pure magnetic scattering.) For this reflection, (%) = (/3) = 0 if the magnetization is in the plane of scattering. Turning attention to the data displayed in fig. 20, the polarization analysis demonstrates that the polarizations of the secondary and primary beams are opposite when the moment is in the plane of scattering, and the same when it is perpendicular to the plane of scattering. Also included, for reference, is data taken at the (200) Bragg reflection which is nominally purely charge. Figure 21 shows the observed variation of the linear polarization of the secondary beam with the projection of the moment on the plane of scattering. The linear polarization of the secondary beam is determined by the expression,

Here, (dc~/dg2)0 is the cross-section (8.16) and the cross-section on the left-hand side is given by (8.20). In discussing the latter, we have noted the salient features of ((/3)*(a3)) and ((al)*(a2)). The only point that needs to be added with regard to (8.22) is that I(%)12 and [(a2)l 2 are both of order r 2, and both vanish if the magnetization is perpendicular to the plane of scattering. Evaluated for pure charge scattering, (8.22) agrees with (2.9). For the case of mixed reflections,

606

S.W. LOVESEY i

25 20 15 10 5 0 150 125

'~"

E

C2} Q

tJ

I

,,, ~, ,,, t-'=O

!

i

(200)

__j •

•

.

,

.

,

.

•

75 50 25 0 7

II

~lL~u{T{(((i,l(l~t~k&E~k~l;IfY~rCflhtft~ddtfUltltl~__

i ,

.

. --

••

,

• ,

. .

• .

. .

, .

. .

l ,

(100) @=90°

100

@

J

I

i -

,

-

.

i

.

*

.

,

.

I

,

6

5

(100)~-0°!I~

02

_:_%.,J

34

.

.

.

.

.

.

.

.

-0.02 0 0.02 0.04 -0.02 0 0.02 0.04

t~ [Degrees] Fig. 20. Polarization analysis of charge (200) and magnetic (100) Bragg reflections in Mn0.YsZno.25F2 (Hill et al. 1993b). The primary polarization is largely perpendicular to the plane of scattering (~rpolarization, P3 ~ 1) and is unchanged by charge scattering; the observed ratio (I_k/Ill) is due to the inherent polarization of the synchrotron beam. At the magnetic Bragg reflection, (100), the same ratio is observed with ~b = 90 ° (magnetic moment perpendicular to the plane of scattering) while for ~b ----0 it is inverted, as expected for magnetic scattering. For charge scattering (top panel) the measured value of the secondary polarization P~ = 0.9 4-0.1, and for magnetic scattering with the moment in the plane of scattering (bottom panel) P~ = -0.9 4- 0. I.

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

'

1.5f ' 1.0 ~_ :~

o {100)' * (200)

]i

f

607

I

0.5 O-

-0.5 -1.0 -1.5 I

0

!

!

I

30

i

,

I

60

i

i

I

90

[degrees] Fig. 21. The secondary linear polarization measured at the magnetic (open circles) and charge (full circles) Bragg reflections in Mno.75Zno.25F2 studied by Hill et al. (1993b). The inherent polarization of the primary beam is preserved in charge scattering. On the other hand, linear polarization at a purely magnetic Bragg reflection (Fc = 0) is rotated as the magnetic moment is rotated into the plane of scattering. If the primary beam contains only linear polarization, which is almost true in the present case, and the magnetic moment is due to spin magnetization (the manganese ion spectral state is 6S5/2) the variation of the secondary linear polarization with respect to the orientation of the moment is provided by eq. (8.29). with the result (8.21) for the cross-section on the left-hand side. Note that (8.21) and (8.23) obey the requirement P~ = -4-1 for P3 = -4-1. Expressions for the cross-section and secondary polarization are algebraically quite complicated even for the special case of pure magnetic reflections, i.e. reflections for which Fc(k) = 0 and the magnetic structure factors are finite. In view of this, we elect to consider special limits of the expressions beginning with the magnetization perpendicular to the plane of scattering, defined by q and q ' . The cross-section for this case is, do-/da2 = (rr~ sin 0) 2 { 2(1 - P3)(1 - cos 0) x (8.24) x [(1 - cos0)lFtVl a + Re(F~F,U*)] + IFsYI2}, and the corresponding result for P~ is, (da/d£2)P~ = (rr~ sin 0)2{2(P3 - 1)(1 - cos0)[(1 - cos 0)IFtul 2 +

(8.25) + Re(F~FYs*)] + P3lF~]2}.

608

S.W. LOVESEY

Hence, when the magnetization is perpendicular to the plane, and the primary radiation is ~r-polarized, the cross-section measures the spin magnetization. A similar finding holds for charge-magnetic scattering. The pair of results (8.24) and (8.25) obey P~ = + 1 for P3 = 4-1. The last case we consider is that of pure magnetic reflections (Fc = 0) with the magnetization confined to the plane of scattering. In this instance, (a3) = (¢3) = 0. These results lead directly to the finding P~ = + 1 when P3 = :~ 1. The foregoing result is manifest in the following pair of relations, dcr/dg2 = {l(C~l)[2 q- ](c~2)l2 - 2P3 Im((c~l)*(~2) ) },

(8.26)

(d,/df2)P~ = -{P3(1(%)12 ÷ [(%)1 2) - 2Im((%)*(%)) }.

(8.27)

and,

On developing the expression for the cross-section one arrives at the result, do-/dI2 =

('rre sin2 ~ 0 ) 2 { l ( q - * q ' ) ' F s

12+ I(q+ q') • (2Ft + Fs)12 +

(8.28) + 2P3 R e [ ( ~ - ~ ' ) . F: (~ + ~ ' ) . (2F, + Fs)] }, from which the corresponding result for P~ quite readily derived. Inspection of these expressions leads to the view that, the geometry for which (8.26)-(8.28) apply is not particularly useful for a comparative study of spin and orbital magnetism in solids. For most transition metal compounds the orbital magnetization is very small, due to the action of the relatively large crystal field. This is motivation to continue the study of polarization effects for Fl = 0 begun at the end of the last subsection. At a purely magnetic reflection (Fc = 0) and P = (0, 0, P3), the secondary linear polarization is found to be,

_{ t + (s - r)P3 } P3 =

r + 8+~73

'

(8.29)

where r, s, t are functions of the angle 0 through which the photons are deflected, the angle q5 that defines the spin moment relative to the y-axis, (1 7r - ~b), and the angle ~ between the direction of k = (q - q') and the projection of the moment on the plane of scattering, 1

2

r=2(tan~bcot~O) , and t = sin 0 sin 2~.

s = (1 - cos0 cos 2~),

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

609

The formula (8.29) agrees with (8.18) for P3 = 0, and the corresponding result for pure charge scattering is (2.9). Two special cases are ~b = 0 and 1 7r which correspond, respectively, to the spin moment in the plane of scattering and the spin moment aligned along the y-axis perpendicular to the plane. Taking ~b= 0 in (8.29),

s + tP3

from which one finds P~ = + 1 for P3 = • 1. For the second special case, 4) = ~1 7r, P~ =/:'3. So, if the spin moment is perpendicular to the plane of scattering linear polarization is preserved, as in pure charge scattering at fixed 0. On the other hand, if the spin moment lies in the plane of scattering linear polarization is rotated by magnetic scattering. 8.3. Circular polarization

Unlike linear polarization, circular polarization induces interference between the charge and magnetic scattering amplitudes. In consequence, use of circular polarization is a valuable method for the measurement of magnetic scattering, providing as it does both the sign and magnitude of the amplitude, good signal-to-noise ratios by a difference technique, and potentially better flexibility than use of anomalous charge scattering. However, it is restricted to the investigation of mixed charge-magnetic reflections. Examples of diffraction data obtained in this way are provided in fig. 2 and fig. 4. The existence in the cross-section of charge-magnetic interference induced by circular polarization is apparent when we look at terms to be added to the crosssection (8.20). From (2.5) it is found that the additional terms are, 2P2{Re((/3)* (c@) + Im((c~l)*(c@)}.

(8.30)

Turning to (8.15), (c~1) and (~2) are purely magnetic while (fl) and (c@ are combinations of charge and magnetic terms. One general observation is that (oq) and (c~2) vanish if Fs and Ft are perpendicular to the plane of scattering. Hence, chargemagnetic interference induced by circular polarization does not contribute to the cross-section unless there is a component of the magnetization in the plane, precisely as for scattering by a polarized electron. On developing the quantities in (8.30) to leading-order in 7-, the contribution to the cross-section induced by circular polarization is found to be, -r~'r(1 - cos O)P2Re{Fc ((1 + cos 0)(~ + ~ ' ) . Ft + (~ cos 0 + ~ ' ) . Fs) }. (8.31) The cross-section for a primary beam described by a Stokes vector P = (0, P2, P3) is the sum of (8.21) and (8.31). It is interesting to observe that the spin contribution in (8.31) is precisely the same as the spin contribution for the classical non-relativistic cross-section for scattering by a single free electron that is at rest before the collision.

610

S.W. LOVESEY

The contribution (8.31) can be measured by forming the difference of data for P2 = 1 and P2 = - 1 , since the contribution (8.21) will cancel in forming the difference. Keeping P2 fixed and switching the polarity of an external magnetic field, strong enough to saturate the magnetization, achieves the same result provided the charge-magnetic terms in (8.21) are negligible. The occurence of different geometric factors with the spin and orbital contributions to (8.31) offers the possibility to separately measure the two contributions. For example, if the magnetization is aligned along the scattering vector the orbital contribution will not participate in the diffraction event. Taking the standard case P = (0, O, P3), P~ is small being proportional to the magnetic component of the amplitude. 9. Elastic resonant scattering The polarization dependence of resonant scattering, satellite selection rules, and basic features of line shapes are quite elementary consequences of a model based on electric multipole contributions to the resonant scattering amplitude (Carra et al. 1989, Hannon et al. 1988, 1989, Luo et al. 1993). The nontrivial calculation required to confront experimental and theoretical findings is the calculation of radial integrals which arise in matrix elements of multipole operators (Cowan 1981). Where this has been attempted, ab initio calculations based on an atomic model have provided a good account of experimental data (McWhan et al. 1990, Tang et al. 1992b). Reviews of experimental work include (Gibbs 1992, Stirling and Lander 1993, Vettier 1994). The description of elastic resonant scattering has much in common with dichroism, considered in section 7. Referring to eq. (7.7), the resonant component of the elastic scattering amplitude is, f~(E) -

re (#lg'J(-q')l~)(~Tl~'J(q)ll~)

re

(E - A + iF/2)

(9.1)

Here, the intermediate state, labelled r/, is taken to be a single electronic orbital; formulae for a more general case are provided later in this section. Neglecting the contribution to J from the electron spin, and treating the contribution from the linear momentum in terms of the dipole approximation (7.4), f~(E) does not depend on the wave vector of either the primary (q) or secondary (q'). Near resonance, the amplitude (9.1) reduces to, f,.(E) = -(eq) 2

where, Q = (7?lRl#),

(s'. Q*)(e • Q) (E - za + i t / 2 ) '

(9.2)

PHOTON BEAM STUDIES OF MAGNETICMATERIALS

611

is a dipole matrix element; the present value of Q differs from (7.4) by a simple factor proportional to A ,-~ E, near resonance. The observed cross-section is proportional to the absolute square of fr averaged over thermal, concentration and defect fluctuations, i.e. the square of the scattering length of the perfect crystal. Thermal fluctuations will include both those arising from the magnetic and lattice vibrational properties of the sample. Following the development pursued in section 7, we adopt a simple representation of the orbitals which reduces the product of matrix elements in (9.2) to an appealing form. Let the initial and intermediate orbitals be represented by states described by a single magnetic quantum number. In this instance, Q*Q~, (u = 0, +1) vanishes unless the spherical component labels satisfy the condition u = u ~, and so,

(e'" Q*)(e" Q) = Z e ~ , ,. ~,IQ~,I2. = ~1

((~,, ~)(IQ÷112 + IQ-II 2) +

1/

+ i r a . (e' x e)(lQ_ll 2 - I q + l l +

2) +

(9.3)

(~'. m)(~. m)(210012 -IQ+~I 2 - IO_a[ 2) }.

The three terms in the second equality are arranged in ascending powers of the (unit) vector m that defines the magnetic quantization axis. It comes as no surprise that, the combinations of dipole matrix elements in (9.3) arise also in the description of E1 circular and linear dichroism. The atomic model described in section 7 can be used to estimate the temperature dependence of the products of radial integrals in (9.3). This reveals that, the first and third terms involve thermal fluctuations of the mean square magnetization, while the second term is proportional to the magnetization. With regard to the dependence of the scattering amplitude on polarization states, the representations of e ~• e, e' x ~ and (e'. m)(e. m) in fig. 11 could be useful. The notation of cr and 7r polarization states appears in the literature; a photon polarized perpendicular to the plane of scattering is said to exhibit ~r polarization, while a photon polarized in the plane has 7r polarization. From fig. 11 we see that, for the scattering amplitude formed with (9.3) there are three forbidden events, namely, c~-Tr events in the term which is independent of m, and cr-cr events in the term proportional to m. When the magnetic moment lies in the plane of scattering, rn± = 0, the only event in the term of order rn 2 is 7r-Tr scattering, while for rn± = 1 the only event allowed is a-or scattering. To illustrate the contribution made to Bragg intensities by resonant events, consider the term of order m in the resonant amplitude. Inspection of (9.3) shows that this term is of the form X" re(R) where the vector R defines the position of the magnetic atom. For coherent scattering from an array of atoms the total amplitude is, exp(ik • R) X" ra(R), R

612

S.W. LOVESEY

where, as usual, k = q - q'. A simple magnetic configuration that gives rise to satellite reflections is a spiral structure in which the moments are perpendicular to an axis, and rotation about the axis varies sinusoidally with position along the axis. Such a spiral configuration of magnetic moments is realized in holmium in the temperature interval 20-132 K. Let,

re(R) =

(cos(w • n), sin(w • R), 0),

where the moduation wave vector, w, might change with temperature. The scattering amplitude is then of the form, 1

~--~exp {iR • (k + R

from which we conclude that Bragg scattering occurs when,

k+w=r, where r is a reciprocal vector for the magnetic lattice. Consideration of all three terms in the resonant amplitude shows that, dipole (El) resonant scattering can contribute to a main Bragg reflection and incommensurate satellites to it defined by w and 2w, i.e. satellites of order 1 and 2. The E2 amplitude reported by Hannon et al. (1988, 1989) contains terms in m from zero up to fourth order, and there are thirteen distinct contributions. Hence, E2 resonant scattering can contribute to a main Bragg reflection and satellites to it of order 1 through 4. The model for E1 events used up to now in this section is based on single atomic orbitals for the initial and intermediate states. We conclude by recording the expression for the amplitude in which more than one initial and intermediate state is required. But, we retain the simplifying assumption that each orbital contains one M component, e.g., consideration of crystal-field perturbations are not included. The notation for the more general model follows that introduced in section 7. Let the dimensionless quantity,

I@ EjJL(qRy)yL(Rj) #> : , t'7/

(9.4)

E,, + E - E~ + i / 7 2

in terms of which the E1 amplitude is,

f~(E)-

3 {(¢,.e)(Fll+Fl_l)+im.(s,x¢)(Fll_F~l) + 4rrq + ( e . m ) ( , . m)(2Uo' - U+ l -

(9.5)

PHOTON BEAM STUDIESOF MAGNETICMATERIALS

613

The physical significance of the three terms in this expressions, and the connection of fr(E) to the interpretation of the magneto-optical Kerr effect, has recently been reviewed in an application to data for Co (Kao et al. 1993). The Debye-Waller factor is not shown explicitly in (9.5). The corresponding E2 amplitude contains linear combinations of F ~ . For a given magnetic reflection, the amplitude is the coherent sum of contributions from El, E2 . . . . events. Line shapes, proportional to the absolute square of the total amplitude, as a function of photon energy can display marked asymmetries, which result from the interference of the resonant and non-resonant contributions.

10. Spectroscopy Inelastic photon scattering, here referred to as spectroscopy, is described by the partial differential cross-section (2.10) and the concomitant result, based on (A.5), for the polarization of the secondary beam. Observed events include the excitation of collective charge oscillations, also known as plasmons, and interband (particle-hole) transitions (Calaway 1991). These events occur in metals for energy changes, hw, of the order of 10 eV. At larger values of co the cross section approaches the Compton limit, which gives access to the electron momentum distribution. For a magnetic material there is also a contribution from the density of unpaired electrons (Platzman and Tzoar 1970). As might be expected circular polarization can be of assistance in efforts to discriminate magnetic from charge induced events in the scattered beam (Gibbs 1992, Sakai 1992). Here, we first recall exact results for scattering by free charges, after which there is a discussion of cross-sections for bound electrons and the corresponding Compton limit.

I0.1. Scattering by free charges We begin with non-relativistic scattering by free charges. The corresponding crosssection is (2.7) extended to dynamic events, as demonstrated in (2.10). The relativistic result, usually referred to as the Klein-Nishina formula (Berestetskii et al. 1982), is discussed at the end of the subsection. In free space a free electron cannot emit or absorb a photon without violating energy or momentum conservation. Therefore, there are no first-order processes involving the p .A terms in the interaction provided by (p - -~A) 2. However, there are first-order processes involving the A 2 term, which is quadratic in photon operators. The scattering of photons, which is of interest here, is a process in which one photon is destroyed and another created. The quantum theory of scattering applied to the A 2 term shows that, within the first Born approximation (equivalent to the use of Fermi's Golden Rule for transition rates), d2cr/df2 dE' = Nr2e(q'/q)(e • e')2S(k, w),

(10.1)

614

S.W. LOVESEY

with the Van Hove response function for non-relativistic charge scattering (Calaway 1991, Lovesey 1986),

S(k, w) = (1/2~rhN)

F dt exp(-iwt)(n+(k) n(k, t)).

(lO.2)

oo

Here, n(k) is the spatial Fourier transform of the microscopic particle density of N electrons, and an explicit expression is found in (2.6). The expressions (10.1) and (10.2) reduce (2.7) in the limit of a large primary photon energy, since then it is appropriate to replace the correlation function in (10.2) by its static (t = 0) value. The latter step is based on the observation that, for E ~ ~ the duration of the scattering event is vanishingly small; this line of reasoning is not restricted to free charges, of course. The corresponding cross-section, (2.7) in the case of charge scattering, is an estimate of the total scattering. Indeed, a complementary line of argument is to say that, when E --+ e¢ all allowed events contribute to scattering and, so, the observed response approaches (10.1) integrated over all energy transfers, hw. Perhaps the most direct way to calculate S(k, w) is to employ a second quantized representation for n(k) (Lovesey 1986). If the carriers obey Fermi statistics, correct for spin- 1/2 particles,

(n +(k)n(k, t)} = 2 ~ re(1- fk+p)exp { it___~h(k 2 + 2k .p)}, p 2ra*

(10.3)

where fp is the Fermi distribution function (/3 = 1/kBT), (10.4)

fp = (exp {fl(Ep - #)} + 1)-',

and the chemical potential, #, is determined, in the usual way from the number of carriers. Note that, the one-particle energy Ep = (hp)Z/2ra * is independent of the spin state. From (10.2) and (10.3),

2 E S { h w + Ep - Ek+p}fp(1 - fk+p) P

={l+n(w)}

(lO.5) 2

a{r

+ uj, -

k+p }(/j, -

P

where, in the second form of the result, the new factor is defined by n(w) = {exp(hwfl) - 1}- 1. The Boltzmann limit is recovered from (10.5) when the chemical potential satisfies exp(#/3) << 1, and in this case it has the value, n0

exp(#p) = -~- ( 2 r r h 2 ~ / m * ) 3/2

<<

1,

(10.6)

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

615

which is a condition on particle density, n o, temperature and mass. Another interesting limiting case of (10.5) is when the scattering vector k is large. For sufficiently large k, fp >> fk+p, whence S(k,w) in (10.5) approaches the value,

S(k, w) ~- { 1 + n(w)} 2

Zp

+ Ep

(10.7)

-

which describes the Compton limit of scattering. The structure of the right-hand side is a delta-function, expressing conservation of energy for free particles, weighted by fp which is the momentum distribution function for free particles. The Compton limit of the response function for bound particles has precisely the same structure, but the momentum distribution is the one appropriate to the energy surface of the binding potential. As the second topic in this subsection, let us turn to the Klein-Nishina formula for relativistic scattering a photon by a free electron that is at rest before the collision. The latter is polarized, and the average value of the spin is (s). The primary radiation is assumed to contain linear and circular polarization, represented by Stokes parameters P2 and P3. The values P3 = + 1 and P3 = - 1 correspond to complete linear polarization and, respectively, labelled a- and rr-polarization states by many authors. The parameter P2 represents the degree of circular polarization; with our convention, the probability that the primary photon has right-hand or left-hand circular polarization is respectively (1 + P2)/2 and (1 - P2)/2. In scattering, the primary photon is deflected through an angle 0, and the relation between relativistic photon energy change and the scattering angle is, 1

1

q'

q

- (h/me)(1 - cos 0).

(10.8)

Having dispatched these necessary definitions, the exact cross-section is, do'/d~ = ~1 r2~

q + ~ + (P3 - 1) sin 2 0 ~7

(10.9) - 2rP2(1 - cos0)(s) • (~ cos0 +

q'/q) }.

Here, the dimensionless quantity "r = (/'N/me). The contribution from the electron spin vanishes if (s) is perpendicular to the scattering plane, defined by q and q '. When this condition holds, the cross-section is that for scattering by unpolarized electrons; photons polarized perpendicular to the scattering plane (P3 = 1) have a larger crosssection than photons in the plane (P3 = - 1 ) . To engage in scattering the spin of the electron must have a projection on the scattering plane, as already mentioned, and the primary photon must be circularly polarized (P2 ¢ 0). The spin-dependent term can be thought of as an interference between charge and magnetic (spin) scattering

616

S.W. LOVESEY

induced by circular polarization. There is no orbital angular momentum for a free electron, of course. The classical, non-relativistic result is obtained for hq << rac, i.e. r << 1. In this instance q' ~ q, and the cross-section (10.9) reduces to,

do"

d[? = r e

{

'

2 1-4- (t:)3-1)sin20- rP2( 1-c°sO)(s)" (q c°sO +q 2

(10.10)

For/:)2 =/:)3 = 0, (10.10) is the same as (2.7) and (2.8) evaluated for one electron (N = 1), i.e. Thomson's formula is recovered, as it should be.

10.2. Bound electrons The~expression developed in this section includes circular polarization in the primary beam, P = (0, P2, 0), and magnetic contributions to leading-order in r = E/rnc 2. The starting point is (2.10) taken together with (2.5). Examination of (A.8) for the components of a and/3 shows that % and ~2 are proprotional to r so the terms ce+ce, 1 l and a+% are not included in our expression. Probably the most compact formalism is achieved with correlation functions introduced in section 2. Recall that the frequency change in scattering is denoted by co = c(q - q') and the concomitant wave vector change k = q - q'. For ease of notation, the k-dependence of quantities will not always be displayed, e.g., the charge variable n(k), defined in (2.6), is written as n in subsequent working. The partial differential cross-section is, d2o-

d[2dE'

_

q'

1

q 27rh +

i7

dt exp(-icot)

~

{<~+a3(t) +/3+/3(0) +

(10.11)

P2(/3 +O~2(t) 4- Ol;/3(t)÷ i [O~3+OZl(t) -- 0~+0~3(~)]) },

where the term proportional to P2 is magnetic in character in as much that it vanishes for r - + 0. On using the expressions listed in (A.8) and developing results for correlation functions to leading-order in -r, the dynamic quantity in (10.11) for a ~ 1 becomes

1 (1 + cos 2 o)(,~+n(t)) + ir

+-~ (~ × ~'). { cosO (n+z(t)- z+n(t)) -

a(1 + cosO)(n+S(t)

-

S+n(t))} .4-

•4"~ /92 (q(1-- a + (cos 0 -- a) cos 0) + + ~ ' ( 3 a c o s 0 - 2 cos 0 - a ) ) .

(n+S(t) + S+n(O) +

+ (~ + ~' cos0). @+z(t)+ Z+n(t)>}.

(lO.12)

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

617

Here, a = (1 + qt/q)/2 and the spin and momentum operators S and Z are defined in (A.6) and (A.7). The first term in (10.12) is the pure charge contribution, evaluated in section 10.1 for free particles. In the subsequent work, the various correlation functions in (10.12) are evaluated in the Compton limit, discussed in section 10.1 for the particular case of charge scattering.

10.3. Compton scattering The Compton limit is achieved with a high primary energy and large scattering vectors. Scattering in this limit is dominated by incoherent events that involve individual particles. Compton profiles of unpaired electrons in Ni and Fe are shown in fig. 5 and fig. 16. Experiments (Timms et al. 1993) have shown that Compton scattering from magnetic materials measure the spin density and supporting arguments are discussed by Lovesey (1993). This subsection addresses the structure of the spin correlation functions in (10.12) evaluated in the Compton limit. We argue that the Compton limit of the mixed correlation function (n+S} is,

(n + (k) S(k, t)) ..~ h3 / dQ ~((~(BQ - pj)sj) x J

(10.13) x exp {it(Ek + h2k • Q/m)/h},

where Ek = (hk)2/2m is the recoil energy. On taking the time Fourier transform of this result, to obtain the appropriate response function, the exponential term produces a delta function which expresses conservation of energy, c.f. (10.7). An argument which produces the estimate (10.13) rests on the use of R(t) = (R(0)+ tp/m) where p is the momentum conjugate to R = R(0); this expression for R(t) is manifestly a short-time expansion. The first few steps leading to (10.13) are,

(n+(k) S(k, t)) = ~ ( exp(-ik • Ry) exp (ik. Rj(t))sj(t)) jj'

~ ( exp(-i k. R3) exp {i k. (R3 + tpj/~)} J = ~

( exp { ( i t k . p j / m ) + i tE~/h}sj(t)),

J

where the last line is reached by using a standard identity for the product of two exponential operators. Note that, just the self part of the correlation function is retained, i.e. the incoherent approximation is made. Before collecting the results for the partial differential cross-section in the Compton limit, let us remark that in the extreme limit E -+ oe scattering is elastic, and the

618

s.w. LOVESEY

scattered intensity is proportional to the static limit of the correlation function. The static limit of (n+S) in the incoherent approximation is simply the average value of the spin density (S). In subsequent working, the static correlation function in (10.13) is assumed to be purely real, in which case the polarization-independent spin contribution to (10.12) vanishes. Starting from (10.12), and gathering results from the foregoing discussions, the Compton limit of the cross-section is, d2~r/dY2 dE' --

× {

1

h3(q'/q)r 2 ×

(l+cos 2 O)N . /

dQ p(Q)~(Energy) +

(10.14)

+ ~-P2 (~(1 - a + (cos 0 - a) cos 0) + ~'(3a cos 0 - 2 cos 0 - a)) ×

x J dQ E (~(hQ-pj)sj)~(Energy)), J where the momentum density

p(Q) = I ( ~ ~(hQ-pj) I,

(10.15)

and 3(Energy)

= ~(hw- Ek -- h2k • Q/m).

If the inelasticity is very small, a = (1 + q'/q)/2 ,-~ 1 and the orientation factor in the polarization dependent term is identical with that found for the spin contribution to the diffraction cross-section given in (8.31). The orientation factor in (10.14) is not the same as the one derived from the Klein-Nishina formula. This result is attributed to the different models; the Klein-Nishina result is the exact cross-section for a free electron that is at rest before the collision, whereas (10.14) is derived from an approximate amplitude, appropriate for bound electrons, derived by perturbation theory.

11. Concluding remarks Frenetic activity over the past few years by researchers at synchrotron light sources has established the value of photon beam techniques to investigate magnetic properties of materials. At the moment, the most productive area of work is with methods using linear and circular dichroism. Like elastic resonant diffraction, the capabilities of these methods are not shared by techniques based on neutron beams. Nonresonant

PHOTON BEAM STUDIESOF MAGNETICMATERIALS

619

scattering, including Bragg diffraction and Compton scattering, suffer in full the disadvantages that stem from the intrinsic weakness of the magnetic photon scattering interaction. Looking to the future, one can readily anticipate improvements in methods and techniques, and the interpretation of data. In light of the rapid pace of the expansion in experimental activities over the past decade, current instrumentation might not yet be optimal. Increased sensitivity in scattering experiments will open the way to more work on ferromagnetic materials, for example. Different approaches could prove useful. For example, the use in diffraction of high energy photon beams (,-~ 80 keV) improves penetration, which increases the scattered intensity, and decreases surface sensitivity thereby making diffraction closer to a bulk probe as in neutron beam diffraction (Brtickel et al. 1993). Furthermore, techniques will be refined to the stage where some measurements are routine. Understandably, inelastic magnetic photon scattering has yet to be exploited. While sum rules are proving to be valuable in the interpretation of data on magnetic dichroic effects there is more scope to relate observed signals to physical variables built from atomic quantities. Probably, it will not be too long before the domain of validity of currently known sum rules is firmly established through the confrontation of data with model calculations based on one-electron pictures of atomic and band states. Turning to another issue, the quantitative value of elastic resonant scattering depends on the quality of absorption corrections. Several techniques are apparently sensitive to near-surface structure and defects. An understanding of these effects is important, as currently witnessed in the efforts to fully account for the two, or more, length scales appearing in the analysis of critical magnetic scattering.

Acknowledgements The work of preparing this chapter contributes to an EC funded project SCI 0467M(SMA). Discussions and correspondence with the following researchers is gratefully acknowledged: U. Balucani, J. Bohr, E Carra, C.T. Chen, S.E Collins, D. Gibbs, G. van der Laan, S. Manninen, D.B. McWhan, G.D. Priftis, N. Sakai, G.A. Sawatzky, and G.T. Trammell. Dr. G. van der Laan commented on the first draft of the text. Preparation of the text for publication was made by Shirley Fortt (ISIS Facility). Figures 1-9, 13-16, 18, 20, and 21 are reproduced with the consent of the authors, and the American Physical Society, Institute of Physics, Springer-Verlag, FrancisTaylor, and Gordon-Breach.

Appendix. Polarization effects and magnetic scattering amplitude Polarization states

The scattering amplitude operator, and observable quantities derived from it, depends on the polarization of the primary radiation, described by a real unit vector e. In this

620

S.W. LOVESEY

appendix we describe a formalism for handling the consequences of this dependence, and a more general example in which the beam of secondary radiation has also a definite polarization. Additional material is provided in Lovesey (1993), Newton (1982), Berestetskii et al. (1982). Let us begin by recapping the properties of a classical light wave propagating in the z-direction, say. The electric E and magnetic H field vectors are perpendicular to the direction of propagation of the light, whence E = (E~, Ey, 0) and for a planewave H = ( - E u, E~, 0) or H = (~x E) where ~ is a unit vector along the z-direction. It is often convenient to use a complex notation for the electric field, remembering that in the end the real components lead to observable quantities. The polarization state of the light wave is directly related to the E vector. Examples of particular interest include: (a) E v = 0, plane polarized in the x-direction, (b) E~ = 0, plane polarized in the y-direction, (c) E~ = Ey, polarized at 7r/4, (d)

Ey = e i ~ / 2 E x , the y-component lags the x-component by 90 °, and the wave is right circularly polarized,

(e) Ey = -iE~, the wave is left circularly polarized. Let us choose to employ real polarization vectors; e for the primary and e' for the secondary radiation. By definition, 6 . q = e , . q t _- 0. The properties of partially polarized radiation are best described in terms of a density matrix. Although several polarization effects are adequately treated by elementary methods, slightly more formalism is required to provide a complete description of polarization contributions to the cross-section and secondary beam. Fortunately, it is possible to derive master formulae for the cross-section and polarization of the secondary beam from which special cases are readily obtained. Hence, at the end of the day there is no need to grapple with the formalism. The master fomulae to which we refer are (2.5) and (A.5). First some material to define concepts and notation, beginning with results that provide a physical interpretation of the density matrix. Any polarization ~ can be represented as a superposition of two mutually orthogonal polarizations, chosen in some specified manner. These polarizations can be taken to be two mutually perpendicular linear polarizations, or two circular polarizations having opposite directions of rotation. For the moment, we choose to denote complex vectors of right-hand and left-hand circular polarization by ~(+) and ¢(-), respectively. Seen head-on right-hand circular polarization is counter-clockwise. In coordinates (, r/, i, shown in fig. 12, with the if-axis in the direction of the photon q, we take,

i(i )

,

and ¢(-) = (¢(+))*. The spin operator for the photon is usually taken to be,

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

621

and, S¢ (+) = e (+),

right-hand,

S¢ (-) = - e ( - ) ,

left-hand.

The component of angular momentum along the direction of the momentum vector q (the C-axis), called the helicity of the photon, is a conserved quantity. The vectors e (+) and ~(-) correspond to the helicity values +1 and - 1 , respectively. Thus, the component of the photon angular momentum along the direction of its motion can have only the two values 4-1; the value zero is not possible. We employ a density matrix to describe states of partial polarization; applications of this method to photon scattering are reviewed in Newton (1982), McMaster (1961 ), Berestetskii et al. (1982), Tolhoek (1956). The polarization density matrix is a tensor / , ~ of rank two, in a plane perpendicular to the wave vector q which is the ~--~plane with our notation and conventions. The tensor is required to be Hermitian # ~ = (/,~)* and normalized such that the trace is unity. The probability that the photon has any given polarization e is determined by the quantity,

Writing the density matrix in the form,

z

(~11, ~21,

/3'12/ ~22 '

with/z21 = (1Z12)*, the components//,11 and//,22 are the probabilities of linear polarizations along the ~- and rj-axes. Choosing s to be the vector e (+) we find that the probability for right-hand circular polarization,

(~(q-))*~(+) : ! {1 "+ i(~12 --/Z21)}. 2 Next let us consider the two extreme cases of unpolarized and completely polarized photons. For unpolarized photons all directions of polarization are equally probable, and, #,~ = 6,~/2.

A photon beam with complete polarization is described by the tensor,

]l/, = ~ * , for then the probability for the state e is unity, as required.

622

s.w. LOVESEY

In general, it is convenient to describe partial polarization by means of three real Stokes parameters P1, P2 and P3, which form a vector P. The latter is a purely formal step and is done only for the sake of notational convenience. The density matrix t* can be written,

/z=

1 ( 1+P3 k, P1 +iP2

Pl-iP2) 1 -P3

a

(A.1)

=

In the second equality I is the unit matrix, and ~r are Pauli matrices. Referring back to our comments on the physical significance of the elements of/z we see immediately that the parameter/:'3 defines the linear polarization along the ~- and ~/-axis. For the scattering geometry depicted in fig. 12, P3 = 1 corresponds to complete polarization perpendicular to the scattering plane and P3 = - 1 is complete polarization in the scattering plane. Very often, P3 = 1 and P3 = - 1 are referred to as states of ~r- and 7r-polarization, respectively. The parameter P2 represents the degree of circular polarization; the probability of fight-hand polarization is (1 + P2)/2, and the probability for left-hand polarization is (1 - P2)/2. Since these two polarizations correspond to helcities :~1, it is clear that P2 is the mean value of the helicity of the photon. With our conventions, which depend on our use of a right-handed Cartesian co-ordinate system, the state of complete fight-hand circular polarization (seen head-on this is counter-clockwise) is described by the polarization vector ¢(+), and the density matrix for the state with P2 = 1 (positive helicity) is t* = ~(+)(¢(+))*. The probability for finding the state ¢(+), by definition, is, (~(+))./z¢(+) = 1 (1 + P2). 2 Our conventions for photon polarization states and those in Berestetskii et al. (1982) concur. A complete discussion of helicity states can be found in section 16 of Berestetskii et al. (1982). The reader should be constantly wary about conventions used to describe the polarization and helicity states of a photon beam because there appears to be a plethora of conventions and plenty of scope for inconsistent conventions. Other frequently used consistent conventions are described in (Shore 1990) and references therein. Choosing ¢ = (1, ~ I ) / V ~ reveals that P1 describes the linear polarization along directions at angles ±7r/4 to the ~-axis. Complete linear polarization of the photon is described by P2 = 0, p2 + p2 = 1. In the unpolarized state P1 = P2 = P3 = 0, while for a completely polarized photon p2 + p2 + p2 = 1.

Scattering amplitude The scattering amplitude operator G is written as a 2 × 2 matrix by constructing four elements for the two values of ~ and #. Our scheme for this is defined in fig. 10, in which the diagonal elements Gll and G22 are amplitudes for both ~ and # perpendicular or-or and parallel 7r-Tr to the scattering plane, respectively. The element

PHOTON BEAM STUDIES OF MAGNETIC MATERIALS

623

G21 is the amplitude for the event in which the initial polarization e is perpendicular to the scattering plane and the final polarization e' is parallel to the plane a-Tr, while G12 describes the event in which the polarizations are in the reverse order 7r-or. It is convenient to write the matrix G in the form, G=/31+c~.cr=

/3 -t- 0:3 0:1+i0:2

0:1 - i0:2 /3 0:3 ,/'

(A.2)

where/3 is a scalar quantity, cz = (0:1, 0:2, 0:3), and I is the unit matrix. The convenience arises when we calculate the cross-section, (da/d$2) = Tr{ttG+G},

(A.3)

where the trace operation is taken with respect to the photon polarization states. A similar situation arises in the calculation of Stokes parameters for the secondary photon beam pI obtained from the relation, (dcr/df2)P' = Tr{ttG+crG}.

(A.4)

The trace operations in (A.3) and (A.4) entail quite a bit of algebra. The trace operation yields (2.5) for the cross-section, while for P ' we obtain,

(da/df2)P' = [/3+a + oz+/3 - i(c~+ x a) +/3+/3P + a + ( a . P) +

(A.5) + (oz+- P)o~ - P ( a + . a) + i/3+(c~ x P) + i(P x a+)/3]. In deriving (2.5) and (A.5) we have assumed that/3 and c~ are quantum-mechanical operators that do not commute. The expressions for (da/dO) and P ' are in the form of products of operators, and to obtain results for the interpretation of experiments (Bragg diffraction, Compton scattering, etc.) appropriate matrix elements have to be formed, which are topics taken up in sections 8-10. For the moment, we emphasize that (2.5) and (A.5) provide general statements with regard to the polarization dependence of the cross-section, and the polarization state of the secondary beam. The expressions for 0:1, 0:2' 0:3 and/3 appropriate for diffraction and spectroscopy can be expressed in terms of three atomic quantities that have immediate physical interpretations. One quantity is the spatial Fourier transform of the particle density, n(k), defined in (2.6). The remaining two quantities relate to magnetic properties, and they are the spin density,

S(k) = E s j exp(ik. Rj),

(A.6)

J

where sj is the spin operator for the jth electron, and,

Z(k) = (i/hq) E exp(i k . Rj){ (~ - ~') x pj }, J

(A.7)

624

S.W. LOVESEY A!

where p is the momentum operator conjugate to R, and ~ and q are unit vectors. Note that, S(k) and Z(k) are dimensionless and Z(k) vanishes in the forward direction, and that the sums in (A.6) and (A.7) are over all electrons in the target assembly. Matrix elements of S(k) and Z(k), needed in the cross-section for a magnetic material will, however, involve only the fraction of electrons that are unpaired, e.g., for the localized atomic ion model for a rare earth magnet, core electrons will not contribute to the matrix elements in the first approximation. The expressions for c~ and/3 are found from the results (6.5) and (6.7) together with figs 11 a) and 11 b); the following expressions are in units of r~, and the small parameter 7- = E/mc 2,

i

~-S(k) • { ( 2 a - b - cos 0)~ + (1 + b cos 0 - 2 a ) ~ ' } +

i

+ ~ 7-(~-~'). z(k), 0 : 2 -_-

_ 1 7-S(k). {(cosO - b)~+ (bcosO - 1)~'} 2 (A.8)

2 0:3

~

m

= _

~

7-(~ + ~'). Z(k), i

(1 - cos 0)~(k) - ~ 7-(~ × ~ ' ) .

{(b + 1 - 2a)S(k) - Z(k)},

(1 + cos o)~(k) - ~ ~-(~ x ~ ' ) .

{Z(k) - (b + 1)S(k)},

where b = q~/q and a = (1 + b)/2, and a = b = 1 for elastic events. Notice that a 1 and 0:2 are purely magnetic in character (0:1 = 0:2 = 0 for T = 0), and the expressions for c~ and/3 are correct to first-order in 7- (Grotch et al. 1983). We have remarked that for elastic scattering Z(k) is proportional to the orbital magnetization in the sample. Looking at a and/3 one is struck by the lack of any symmetry in the spin and orbital contributions. The origin of this state of affairs can be traced back to the perturbative calculation of G, (6.4). It is found that, the orbital contribution arises from second-order processes, the same ones that generate Raman scattering. On the other hand, the spin contribution is the sum of terms that arise in first-and second-order processes. A consequence of the lack of symmetry in the spin and orbital contributions is that scattering geometries exist were one or the other contribution is absent, a feat not possible in neutron scattering (Vettier 1993).

P H O T O N B E A M STUDIES OF M A G N E T I C MATERIALS

625

List of important symbols

A ao c

E

E, E' e

Fc Fs, Fl f(E) G g H

kB J(q) k=q-q' m fr~*

N(k)

~(k) P Pi, Pi' q

R re

S(k) s(k,w) T(k)

°}

7 X #e P(q) o

"r" 03

photon vector potential Bohr radius velocity of light electric field primary and secondary photon energies unit of charge for an electron charge unit cell structure factor unit cell structure factors for spin and orbital magnetism scattering amplitude in the forward direction (2.1), = f ' + if" scattering amplitude operator (2.4) gyromagnetic factor magnetic field Hamiltonian operator for electrons in a solid Boltzmann's constant current density scattering vector mass of an electron effective mass of a charge carrier average particle density Fourier transform of the single particle density function linear momentum operator conjugate to R Stokes parameters, i = 1,2, 3, for primary and secondary beams (2.8) wave vector of photon; q, q' wave vectors for primary and secondary photons position vector classical radius of an electron spin density operator response function orbital density operator components of the scattering amplitude operator, eq. (2.4) attenuation coefficient (2.2) polarization vector of a photon, ¢ • q = 0, ¢* = e wavelength density matrix for polarization states (2.5) Bohr magneton momentum density distribution cross-section (2.3) ratio of primary photon energy to electron rest mass = (N//mc) = (E/mc 2) reciprocal lattice vector frequency change in scattering event

S.W. LOVESEY

626 Numerical values of units

Quantity

Symbol

Value

Bohr radius Planck's constant/27r Rydberg Fine-structure constant Rest mass energy of the electron Bohr magneton Classical radius of the electron

% : h2 / m e 2 h Roo = mc2a2/2 = e2/2% a = e2/hc me 2 #B = e h / 2 m c re = e2/mc 2 hZ/2m

0.529 ,~ 0.658 meVpsec 13.606 eV = 911 ,~ 1/137.04 0.511 MeV 0.579x 10-1 meV T -1 0.282x 10-12 cm 3.810 eVA 2

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Subject Index A°(lat) 380 A°(val) 380 absorption edge 571 adiabatic magnetization 465 alloys 3d-element based 101 anion ordering 274 anisotropic ferromagnet magnetic field 535 anisotropic magnetostriction 470 anisotropy constants 336, 500 anisotropy energy 420, 441,483, 499 anisotropy field 419, 420 anomalous scattering 568 antibonding states 379 antiferromagnetic correlations 124 antiferromagnetic vector 464 antiferroquadrupolar ordering 155 Arrhenius behaviour 240 asperomagnetic state 454 atomic form factor 553,600 atomic model 589 atomic picture 574 attenuation coefficient 592, 595 Auger process 590 average particle density 598

bound electrons 616 boundary conditions 507 Bragg diffraction 552, 598, 619 Bragg intensity 572, 611 Bragg law 600 Bragg scattering 572, 605, 612 break of symmetry 476 Brillouin function 139, 412, 420, 483, 523, 594 Brillouin zone 77 bulk anisotropy 507 bulk diffusion 311 bulk magnetostriction 470 Burger's vector 512 CaCu5-type structure 499 calculated magnetization 340 canted phase 413,414, 419, 425,429, 430, 436438, 453, 454, 518 canting 505 carbonitrides 313 causality 591 CEF 168, 340 - effects 13, 118, 139, 146, 170 Hamiltonian 141 - levels 168 - parameter 380 picture 18 splittings 15 centrosymmetric crystal 602 charge gaps 35 charge ordering 165 charge screening 61 chemical radical 280 chemical reduction via mechanical alloying 389 Chevrel phase systems 166 circular dichroism 561,589, 592 circular polarization 568, 609, 620 -

ballistic magnetometer 503 band structure calculations 375 basal plane 500 Bessel function 73 bifurcation theory 507 birefringence 446 - effects 444 Bohr orbital 230 Bohr radius 626 Boltzmann limit 614 bonded magnets 383 Born approximation 586, 613

-

-

679

680

SUBJECT INDEX

classical radius of the electron 626 Co sublattice 491,495, 518 coercive force 463, 507 coercivity mechanism 394 coherence length 260 collinear phases 412, 438 compensation-boundary motion 511 compensation point 409, 415, 416, 446, 450 compensation temperature 409, 419, 440, 442, 443,447, 453, 462, 475 compensational domain walls 450 competing exchange interactions 130 compounds - 2:17-type compounds 459 Compton limit 613, 617 of scattering 615 Compton profile 556, 577, 579 Compton scattering 617, 619 cone-canted phase 502 contact hyperfine coupling 77 contact hyperfine field 73 contact hyperfine interaction 72, 78 contact hyperfine Knight shift constant 75 continuous symmetry 454 copper-oxide compounds 212 Coqblin-Schrieffer model 7 correlation functions 557 correlation length 78, 79, 160, 264 correlation radius 455, 456, 523 Coster-Kronig process 590 Cotton-Mouton effects 509 Coulomb correlation energy 6 covalency effect 203 critical behaviour 79, 182 critical concentration 40 critical exponent 78 critical field 416, 447, 493 critical fluctuations 79, 455 critical phenomena 96 critical point 430, 431,445, 460, 476, 485 critical scattering 575 critical temperature 78, 312, 313 critical transition fields 494 cross-section 552, 554, 623 crossover instability 522 crossover level transitions 411 crossover point 414 crystal electric field parameter 373 crystal field 7, 440 - parameters 333, 338, 339, 349, 361, 372, 380 - perturbations 612 crystalline electric field (CEF) 6, 13, 135 - effects 21, 117 splitting 61, 75 -

-

crystallographic parameters 357 crystallographic structure data 317, 322, 350 Cu-O hybridization 228 cubic anisotropy 422, 430 cubic Laves phase 148 phase compound 148 Curie point 456 Curie temperature 328, 330, 354, 356, 370, 377

-

d-d-interaction 535 d-f interaction 482 d-sublattice 482, 506, 512 Daniel-Friedel model 94 Debye temperature 3 Debye-Waller factors 572, 613 defects 508 degeneracy 425, 475 degeneracy space 524 degeneration 454 demagnetization field 67, 72, 96 density matrix 552, 620 dichroism 559, 579, 585 differential cross-section 580 differential susceptibility 456 diffraction 565, 598 - cross-section 579, 601,602 experiments 602 diffusion constant 312 dilute magnetic systems 71 dipolar interaction 78 dipolar Knight shift constant 75 dipole-allowed transitions 571 dipole approximation 581,588, 610 dipole-dipole coupling 264 dipole-dipole interaction 72, 77 dipole field calculation 87 dipole transitions 547 dislocation 508, 509 dodecahedral 440 domain magnetization 93 domain of 180° 477 domain structure 453, 475, 478 domain wall 476, 507 - energy 395 - width 395 DOS 370 Dy sublattices 491,495 dynamic Kubo--Toyabe function 70, 107, 113 dynamic Lorentz-Kubo--Toyabe behaviour 181 dynamic Lorentz-Kubo-Toyabe function 183 dynamic scaling theory 78 -

E1 attenuation coefficient 598 E1 circular dichroism 611

SUBJECT INDEX E1 linear dichroism 611 E1 magnetic dichroism 594 E2 magnetic dichroism 594 easy axis 419, 424, 425, 427, 482, 485 easy magnetization direction 336 easy plane 423, 482 elastic cross-section 557 elastic resonant diffraction 618 elastic resonant scattering 579, 610 elastic scattering amplitude 610 electrical field gradient (EFG) 108, 142, 347 created by the #+ 95 electrical resistance of heavy fermions 4 electron density distribution 374, 381 electron doping 237 electronic band structure calculation 367 electronic phase separation 248 elemental metals 80 ellipticity 443 energy product 382 enthalpy change the decomposition 311 entropy 465 epoxy bonded magnets 386 equilibrium conditions 482 Euler-Lagrange equations 477, 507, 523 Ewald method 73 EXAFS experiments 326 exchange - a - d 440 - f - d 527 exchange constant 500 exchange-coupled two-phase magnet 382 exchange energies: 4f-3d exchange energies 504 exchange-enhanced itinerant paramagnets 527 exchange field 420, 466 exchange field and anisotropy constant 495 exchange interaction 75, 328, 331, 379, 416, 440, 482 parameter 412 exchange stiffness 477, 523 constant 506 expansion of the domain walls 476 explosion sintered magnet 393 extinction 566, 570 -

-

-

F - :/~+ : F - 'hydrogen'-bonded center 216 f-d exchange 527 f--d interaction 522, 527 f~l intermetallic compounds 521 f-d system 522 f-f interaction 416, 482 Fano effect 563, 564 far-infrared 24

681

Faraday effect 441,450, 462, 478, 591 measurements 417 Faraday rotation 441--443, 480 Fermi Golden Rule 613 Fermi liquid picture 7 Fermi surface 193 ferrimagnetic 419 compounds 408 phase 414 ferrimagnets 408 ferromagnet: 1D-ferromagnet 281 ferromagnetic iron 553, 569 ferromagnetic Ni 552 ferromagnetic phase 414, 419 field hysteresis 505 field induced phase transition 408 fine-structure constant 626 first order magnetic processes (FOMP) 411 first order phase transitions 419, 430, 434, 438, 439, 445, 448 first order transitions 431,488, 526 fluctuations 455 forward scattering amplitude 587, 591 four-fold degeneracy 430 Fourier analysis 66 free energy 424, 482, 506 free-powder magnetization 518 free-powder samples 512 freely suspended sample 513 Fresnel equations 558, 586 Friedel oscillations 61 frustrated Kagom6-1attice system 215 frustration 223, 236, 238, 264 f-sublattice 482, 512, 523

-

-

-

galvanomagnetic effects 464 gapping of Fermi surface 164 gas-phase interstitial modification (GIM) 307, 310 gas-solid solution phase 312 Gaussian distribution 69 Ganssian Kubo-Toyabe function 195 Gaussian Kubo-Toyabe signal 70 Ganssian relaxation 68, 263 Gd sublattice circular birefringence 442 Gd-Fe exchange energy 441 Gd-Y superlattice 567 giant moment alloys 97 GIM process 313 Ginzburg-Landau energy 523 ground state 414 ground state multiplet 168, 264 gyromagnetic factor 600

682

SUBJECT INDEX

Hall effects 462, 464 Haldane system 279 Hamiltonian 533 Hamiltonian of the d-f system 533 handedness of the circular polarization 589 hard permanent magnets 170 HDDR method 392 HDDR process 307, 391 heat capacity 455 heavy electron compounds 148 heavy electron superconductors 199 heavy electron systems 185 heavy electrons without free carriers 165 heavy fermion 3 heavy fermion compounds 148 heavy fermion superconductor 184, 186, 197 Heisenberg antiferromagnet 240 Heisenberg exchange interaction 593 Heisenberg exchange parameter 222 Heisenberg ferrimagnet 416 Heisenberg ferromagnet: 2D-Heisenberg ferromagnet 281 Heisenberg Hamiltonian 76 Heisenberg model 278 - 2D-Heisenberg model 222 Heisenberg operator 557 helicity 589, 621 helimagnet 106 hexagonal anisotropy 422, 482, 491,498 hexagonal axis 482 hexagonal c-axis 501 hexagonal ferrimagnets 482 hexagonal structure 422, 516 hexagonal structure of CaCu 5 491 hexagonal symmetry 517 hexagonal Th2NilT-type structure 315 high coercivity 388 high-field domains 481 high-field magnetization 336 high-field free-powder method (HFFP) 512 Hilbert transforms 591 Ho-sublattice 501 Hois-Ylz superlattice 567 hole doping 237, 239 hole dynamics 259 holmium 553, 566, 569, 570, 612 holmium metal 565 homogeneous line broadening 68 H-T diagrams 418, 510 Hund's rule 593 hybridization between Fe 3d and R 5d electron states 330 hybridization effects 368 hybridization matrix element 6

hydride phase 257 hyperfine anomaly 103 hyperfine field 84, 85, 93, 103, 203, 204, 342, 363 - 57Fe M6sbauer 363 hysteresis 462--464, 497, 498, 525 - loops 463 phenomena 494, 498 -

impurity induced strains 101 incoherent approximation 617 incommensurate 134 magnetic structures 67 modulated spin structure 163 spin density wave 199 - sinusoidally modulated 277 structures 73 index of refraction 585 induced anisotropy 511 induced contact hyperfine field at the/z + 96 inelastic neutron scattering 142 influence of an elastic strain 89 inhomogeneous line broadening 68, 69 inhomogeneous magnetic ordering 274 integral curve 477, 478 interaction - f~l 522, 527 - f-f 416, 482 inter-chain coupling 279 intermediate nitrogen content 323 interstitial atoms: 18g-site interstitial atoms 326 interstitial site: 2b interstitial site 355 intersublatice-coupling strength 512 intersublattice exchange field 419 lntra sublattice interactions 440 intrinsic magnetic domain parameters 394 inverse transition 526, 527 irradiated 101, 102 Ising axis 515 isoentropy slope 465 isomer shifts 346 lsotropic ferrimagnet 411,416 lsotropic magnetostriction paraprocess 494 itinerant electron model 278 itinerant f-d metamagnets 521 itinerant ferrimagnet 526, 528 itinerant ferromagnet 521 itinerant ferromagnetic system 106 itinerant ferromagnetism 527 itinerant metamagnets 410, 418, 521

-

-

-

-

Jaccarino-Peter effect 167 Jahn-Teller effect 411,415

SUBJECT INDEX jump-like magnetization 514, 515 jump-like transition 419 Kagom6 lattice 113 Kerr effect 613 kinks in the magnetization curves 527 Klein-Nishina formula 613,615, 618 Knight shift 67, 114, 132, 141, 198, 199, 264, 265, 269 Kondo effect 7, 13 Kondo insulator 36, 37 Kondo lattice 8, 161, 162 Kondo mechanism 75 Kondo metals 162 Kondo-necklace phase diagram 9 Kondo resonance 193 Kondo temperatures 13 Korringa constant 168 Korringa mechanism 79, 116, 141, 168, 170 Kramers-Heisenberg amplitude 579 Kramers-Heisenberg formula 549 Kubo-Toyabe function 69, 107 Kubo-Toyabe signal 111, 142, 158, 179, 230 Landau theory 455, 456 Land6 factor 593, 601 Larmor precession frequency 66 lattice dilatation 61 lattice gas model 312 lattice parameters 316, 321 of nitrides and carbides 316 lattice relaxation 75 lattice sum 72, 73 lattice vector 600 Laves phases 527 layered oxides 212 left-hand polarization 622 Legendre polynomials 194 Legendre transformation 533 Levanyuk-Ginsburg criterion 456 line shapes 613 linear birefringence 444, 464 linear dichroism 561,589, 595 linear magnetoelectric effect 464 linear magnetostriction 464 linear polarization of secondary beam 605 linear term 4 of specific heat 166 linear-muffin-tin-orbitalmethod (LMTO) 367 liquid phase epitaxy 450 local DOS 378 local field 203 - a t / z + 84,93

-

-

683

local phase diagrams 510 localized model of rare earth magnetism 573 location of interstitial atoms 324 location of N atoms 354 longitudinal field (LF)-/~SR 65, 67 longitudinal magnetostriction 472, 494, 504, 505 measurements 417 longitudinal spin fluctuations 127 longitudinal susceptibility 424 Lorentz field 67, 96, 110 Lorentz sphere 72, 73 Lorentz-Kubo--Toyabe signal 134 Lorentzian distribution 69, 71 Lorentzian Kubo-Toyabe behaviour 181 LTO phase 235 LTO-LTT transition 233 LTT phase 235 -

m* 3 magnetic anisotropy 146, 333, 335, 356, 416, 421,512 constants 512 magnetic correlations 268 magnetic critical scattering 576 magnetic domain patterns 313 magnetic energy 483 magnetic fluorides 215 magnetic frustrations 229 magnetic insulators 199 magnetic Jahn-Teller 415 magnetic moments 367, 376, 377, 379 magnetic phase diagrams 445, 503 magnetic properties of elemental metals 80 magnetic susceptibility 522 magnetism of insulators 200 magnetization 456, 457, 459, 462, 493,498, 502 - process 503, 512 - of sublattices 495 magnetocaloric effect 455, 465 magnetocrystalline anisotropy 139 magnetoelastic anomalies 469 magnetoelastic energy 447 magnetoelastic interaction 508 magnetometer 502 magnetooptical phenomena 455 magnetooptical Kerr effect 558 magnetoresistance 464 magnetostriction 455, 469, 493, 497, 498, 503 - deformation 505 - of paraprocess 470 magnon energies 76 magnon excitations 77 martensitic transition 133 matrix element 588

-

684

SUBJECT INDEX

maximum energy product 387, 396 Maxwell rule 514, 515 mean field approximation 411 mean field theory 455 mean field treatment 277 mean square magnetization 596, 611 mechanical alloying (MA) 388, 389 mechanical grinding 388 mechanical grounding (MG) 389 melt spinning 389 melt-spinning method 309 melt-spun interstitial carbides 309 metamagnetic phase transition 521 metamagnetic transition 160, 411,525, 527, 531 metamagnetic transition field 529 metastable canted phase stability 427 metastable phase 435, 437, 443 rnicrostresses 510, 511 missing fraction 66 mixed reflections 603, 605 mixed valence 23 mode-coupling theory 78 modulated structure 134 molecular crystal 220 - o~-O 2 199 molecular field 420, 483, 522, 527, 593 - approximation 593, 597 - coefficients 331 molecular magnetic field 587 momentum density 580 momentum distribution 556 Morin temperature 203 M6ssbauer 140, 259 - data 260 - 57Fe 341, 363 - spectroscopy 62, 76 motion of phase boundaries 509 motional averaging 69 muon Knight shift 96 muon-nuclear level-crossing resonance 218 muon-quadrupolar level crossing 108 muon-spin relaxation in 3d-elements 97 muon-spin relaxation in lanthanides 98 muon-spin rotation - #+-beams 64 - # + diffusion 117, 118, 124 #+-Knight shift 106 - / z - - K n i g h t shift 230 - #+ motion 87 + site 61 - / z + S R 103 62, 103 - #SR frequency shift 96 - / z S R signal 64 -

- / z

-

# - S R

#+SR technique 62 - / ~ + trapping 117, 118 #+ tunnelling 87 - / ~ + zero-point motion 87 muonic atom 103 muonium 61, 199, 216, 217 -

-

nano-structured two-phase magnetic system 393 N6el antiferromagnet 408 negative differential susceptibility 525 nesting model 279 neutron diffraction 227, 236, 238, 246, 264, 268 neutron diffraction studies 375 neutron magnetic diffraction 565 neutron polarization analysis 126 neutron scattering 110, 124, 128, 155, 181, 189, 239, 248, 250, 262 nitrided materials of l:12-type 393 NMR 62, 66, 76, 106, 107, 111, 155, 158, 190, 193, 219, 230, 256 NMR/NQR 251 non-centrosymmetric materials 568 non-collinear phase 494 non-collinear structure 446 - magnetic 409 non-Fermi liquid behaviour 40 non-magnetic absorption 551 non-resonant scattering 618 amplitude operator 582 nonuniform-exchange constant 523 NQR 151, 215 nuclear hyperfine interaction 264, 265 nuclear Schottky anomaly 164 nuclear spin lattice relaxation 107 nucleation 477 - of domains 509 - of new phase 475 -

O-H bond 244 observed domain width 395 octahedral 440 one-electron atomic model of elastic resonant scattering 573 one-electron model 590 operators of the crystal field 534 optical theorem 551,586, 587 orbital angular momentum operator 583 orbital contributions to the cross-section 604 orbital magnetism 581, 582 orbital magnetization 584, 599, 624 ordered moment 15 orientation transition 455 overcritical exchange 265 oxygen-muon bonds 199

SUBJECT INDEX PAC: 3'~' PAC 62 PAD 62 paramagnetic fluctuations 79 paramagnetic state 522 paramagnetic sublattice 426 paraprocess 470 parity 588 - violation 62 partial differential cross-section 557, 616 partially polarized radiation 620 particle-hole continuum 577 Pauli paramagnetism 126 percolation threshold 524 percolative network 260 permanent magnet 396 - applications 382 phase - HTT 232 - LTO 232 - LTT 230, 232 phase diagram 419, 428, 430-432, 437, 439, 443,444, 485, 495, 526 - H-T 411,420, 424, 497, 500, 521, 525, 528 phase transitions - of 1st kind 427 - of 2nd kind 427 - of 2nd order 445 phonon dispersions 577 phonon induced relaxation 140 photoelastic effect. 508 photoemission 558 piezomagnetism 464 plasma nitriding 314 plasmon peaks 577 plasmons 613 plots of the magnetization 447 point charge model 142 polarization - analysis 566 - 7r 556, 622 - c~ 556, 622 states 619 - vectors 580 polarized/~+-beams 62 positron distribution 64 powder broadening 114, 127 power law 219, 220 pre-hydrogenation treatment 385 precursor phenomenon 264 pressure 16 pressure dependence 12 - of Bt~ 88 - of hyperfine field 95 proximity effect 248

685

quadrupolar anisotropy 132 quadrupolar interactions 133 quantum fluctuations 160, 522 quantum Hall effect 274 quasi-elastic neutron scattering 76 R-Co exchange interaction 491 radial dipole matrix 592 radiation damage 61 Raman process 76 Raman scattering 624 random order 228 range in matter 65 rare-earth M6ssbauer 347 rare-earth orthoferrites 199, 209 rare-earth sublattice 416, 423, 466 reaction enthalpy of interstitial nitride formation 311 reaction enthalpy 311 reciprocal lattice 600 recoil energy 617 reduced matrix element 592 reentrant ferromagnetism 166 reentrant-like phase transition 419 related alloys 81 remanent magnetization 391,463 renormalization-group calculations 78 reorientation 505 rest mass energy of the electron 626 rhombohedral structure 322 rhombohedral Th2ZnlT-type structure 315 rhombohedral crystals 422 right-hand circular polarization 621 RKKY 8 - exchange mechanism 169 - exchange rate 168 - interaction 79, 114, 116 - mechanism 72, 73, 124 rotational process 419 Rydberg 626

-

s--d hybridization 506 satellite reflections 612 saturation magnetization 529 scaling laws 78 scattering amplitude 622 scattering amplitude operator 568, 580, 619 scattering by free charges 613 scattering of light 431 scattering vector 580 SCR theory 107, 108, 110, 111 SCR model 109

686

SUBJECT INDEX

SDW condensation 279 second-order phase transition 428, 436, 438, 480, 481,488 secondary linear polarization 608 selection rules 588 self-consistent renormalization (SCR) theory 106, 124 semiconducting 24, 35, 36 semiconductors 34 shake-off process 562 shake-up process 562 short circuit diffusion 311 short range correlations 118 short range magnetic order 223 short range order 128, 228, 236 signal amplitudes 66 single crystals 502 single particle excitations 278 singlet ground state ions 142 singular point detection 337 site percolation threshold 228 small moment magnetic order 197 soft mode 455 solid oxygen 220 Sommerfeld constant 154, 163 sound absorption 445, 473 sound velocities 455 sound velocity change 473 specific heat 3, 445, 455, 465, 466 - -y-values 3 specific heat anomaly 188 spectroscopy 613 spherical components 588 spin canting 170, 174 spin contributions to the cross-section 604 spin density 584, 600, 623 spin density enhancement 74 spin density wave (SDW) 152, 164, 274 spin dynamics 76, 77, 114, 280 - 4f 260, 261, 263 spin flip transitions 67 spin fluctuations 175, 182, 330, 370, 521,522 spin gap 35 spin glass 62, 125, 127, 153, 165, 229, 230, 259, 274 - Kubo-Toyabe function 259 - order 251 - type 11 spin glass-like order 154 spin hole 227 spin lattice relaxation 79, 106, 111, 230 spin lattice relaxation rate 67 spin magnetization 600 spin operator 620

spin polarization 65 spin reorientation 337, 347, 361, 362 - temperature 339 - transition 237, 239, 242 spin wave excitations 76, 278 spin wave spectrum 36 spin wave stiffness 137 - constant 278 spin-1 linear-chain Heisenberg antiferromagnet 279 spin-orbit branching ratio 564 spin-spin or transverse relaxation rate 68 spin-canted state 416 spin-flop 419, 499 - state 277 - transitions 410, 424, 503, 504, 508, 521 spin-lattice relaxation 76, 218 spin--orbit interaction 441 spin-polarized partial densities of states (DOS) 370 spin-reorientation 496, 499, 504 phase transitions 465 - processes 499 spontaneous linear birefringence 446 spontaneous magnetization 496, 506, 522, 531 spontaneous spin-reorientation 410 square root stretched exponential 71 SQUID 502 stability conditions 430, 436 stability lines 462 stability regions 507 stacking faults 264 static approximation 566 static correlations 263 static limit 618 static Lorentzian Kubo-Toyabe behaviour 182 stereo graphic projection 433 Stevens equivalent operators 333 Stokes vector 553 Stoner criterion 527 stretched-square root-exponential 125 strongly ferrimagnetic phase 531 structural phase transition 166 structural transitions 175, 236 structure modification 315 structure of Th2Ni17 320 structure transformation 320 structure transition from hexagonal Th2NilT-type structure to rhombohedral Th2Znl7-type structure 309 sublattice magnetization 411, 511 substituted intermetallic compounds 527 sum rules 563, 619 superexchange coupling 264 superconducting coherence length 248 -

SUBJECT INDEX superconducting order parameter 189 superconductivity 5, 36 superconductors 4 superlattices 570 supertransfer of hyperfine fields 203, 210 surface anisotropy 506 surface beam 65 surface field-induced phase transition 507 surface phase transition 506, 507 susceptibility 423, 429, 456, 457, 496 symmetry breaking 522 symmetry breaking field 189 symmetry of crystal lattice 36 synchrotron sources 547, 618 T*-phase 239 technical magnetization 497 temperature dependence of the hyperfine field 89 terbium-iron garnet 559 tetragonal structure 421,516 tetragonal symmetry 517 tetragonal ThMnl2-type body-centered structure 354 tetrahedral 440 TF-/~-SR-spectroscopy 230, 250 thermal expansion 472 thermal properties 465 thermal stability 382 thermodynamic potential 419, 420, 423, 424, 482, 512, 523 of the non-equilibrium state 421, 532 thermopiezic analyzer 310 Thomson cross-section 556 Thomson formula 616 three sublattice model 440 threshold concentration 523, 524 threshold field 419, 424 time reversal invariance 196 time reversal invariance violating state 189 topological frustration 113 magnetic 127 total scattering 614 transferred hyperfine field 346 transition metals: 3d-transition metals 81 transversal magnetostriction 494 measurements 417 -

-

-

687

transverse field (TF)-/~SR 65, 66 transverse magnetization 503 transverse spin fluctuations 127 transverse susceptibility 424 tricritical points 418, 427, 428, 435, 436, 485 tunneling spectra 37 twins 475 twinned canted domains 478 twinned domains 475 two-channel relaxation 107 two-fold degeneracy 476 two-magnon effects 576 two-magnon scattering 137 two-sublattice anisotropic ferromagnet 512 two-subnet amorphous systems 410 ultrasmall ordered moments 190 ultrathin films 507 uniaxial anisotropy 421, 506, 507 field 480 magnetic 421,424 uniaxial ferrimagnet 425 unit cell 600 structure factors 600 unstable d-subsystem 527, 528 -

-

-

vacuum tunneling experiments 23 van Hove response function 76, 614 Verwey temperature 206 weak itinerant antiferromagnet 113 weak itinerant ferrimagnet 521 weakly ferrimagnetic phase 531 white line 598 Wigner-Eckart theorem 588 X-ray absorption (XANES) 250 X-ray dichroism 559 X-ray scattering 198 Young modulus 455, 473 - AE-effect 473 Zeeman energy 420 zero field (ZF)-/~SR 65 zero point vibrations 74

Materials Index amorphous Bi2Sr2CaCu20 v 271 amorphous Dy-Co films 454, 465 amorphous DyAg 135

CeCo2Ge2 14 CeCo2Si2 14 CeCu2 5, 11-16, 22, 24 CeCu5 41, 144, 149, 154, 155 CeCu6 5, 7, 40, 144, 149, 155, 160 CeCuA13 17 CeCu4A1 41 CeCu6_xAuz 40 Ce(Cul_xAl=)5 144, 149, 154 Ce(CUl_=Ga=)5 144, 149, 154 Ce(Cul_z(Ga, A1)=)5 159 Ce(CUl_zNi=)2Ge2 16 CeCuGa3 17 CeCu4Ga 41 CeCuGe 22 CeCul.3Sb 2 14, 15 CeCuSi 22 CeCu2Si2 144, 151, 157, 158 CeCu2+=Si2 149, 159 CeCuSn 22, 24 CeCuX3 17 CeFe2Ge2 14 CeFe2Si2 14 CeFelo.sVl.sNy 360 CeFellTi 357 CeFel 1TIN1.5 357 Ce2Fel4 B 173 Ce2Fel7 317 Ce2Fe17C 322 Ce2FeI7C2.5 317 Ce2Fe17Cu 343 Ce2Fel7N2.5 317 Ce2Fel7N3.6 317 Ce2Fe]7Ny 343 CelrGe 22 Celr2Ge 2 14 CeIr2Si2 13, 14 Celr2Sn2 14

BaCuO2 201,212, 245, 269 BaY2CuO 5 201,212, 245, 269 a-Bi203 201, 215 Bi2SrCaYCu208 269 Bi2SrCaYCu2Os+ 6 273 Bi2SrYCaCu2Oy 270 Bi2Sr2Yo.6Cao.4Cu208 273 Bi2Sr2Yl_xCaxCu208 273 Bi2Sr2Yl_zCazCu2Ou 271,272, 274 Bi2Sr2YCu208 269 Bi2Sr2YCu2Ov 270, 272 Bi2Sr2_~YzCu208 273 Bi2Sr2.sYo.sCu20~8 273 BizSr3_=YzCu208 270 Cao.86Sro,14CuO2 201, 214 Ce: ternary equiatomic compounds 22 CeAg 129, 132, 133 CeAgl_=In= 129, 132, 133 CeAg2Ge2 14, 15 CeAg2Si 2 14 CeA12 114, 115, 144, 148, 149, 151, 159 CeA13 5, 7, 144, 149, 151-153, 159, 228 CeA1Ga 22, 24 CeA12Ga2 14 CeAs 129, 130, 178, 180 CeAu2Ge 2 14 CeAuIn 22 CeAu2Si 2 14, 16 Ce3Au3Sb4 35, 36 CeB 6 144, 149, 155, 156, 157 CeCo3B 2 28, 29 CeCo4B 28, 29, 31

689

690

MATERIALS INDEX

CeM2Si2 16 CeNiA1 22, 23 CeNiGe 22 CeNi2Ge2 14 CeNiln 22 CeNio.8Pdo.2Sn 23 CeNiSb 22 CeNi2Sb2 14, 15 CeNiSi 22 CeNizSi2 14 CeNiSn 22, 23, 35, 144, 150, 162 CeNizSn2 13-15 CeOs2Si2 14 CePb3 144, 154 CePdA1 22 CePdzA13 15, 33 CePdGa 22 CePdGe 22 CePd2Ge2 14 CePdIn 22, 23 CePdSb 22, 23 CePd2Si2 14 CePdSn 22, 23, 144, 150, 162 CePd2Sn2 14 CePtAuSi2 16 Ce3Pt3Bi4 35, 36 CePtGa 22 CePtGe 22 CePt2Ge2 13, 14 CePtln 22, 23, 24 CePtSb 22, 24 Ce3Pt3Sb4 35, 36 CePtSi 22, 24 CePtaSi2 13, 14, 16 CePtSn 22, 144, 150, 162, 163 CePt2Sn2 14, 15, 33 CeRh3B2 29 CeRhGe 22 CeRh2Ge2 14 CeRhln 22, 24 CeRhSb 22, 23 CeRh2Si2 14, 143 CeRh2Sn2 14 CeRu2Ge2 14 Ce(Ruo.85Rh0.15)2Si2 144, 150 CeRu2Si2 14, 142, 144, 150, 160-162, 190, 196 CeSb 129-132 CeT2Ge2 15 CeT2Si2 15 CeTX 21 CeT2X2 13, 14 Co 81, 93, 101, 561 Co alloys 81 COC12.2H20 219, 202

CoF2 202, 215 Chromium 82, 102 Chromium alloys 82, 102 compounds - 3d-4f 504 ternary equiatomic 22 Cr203 200, 204 CsAs 128 CuF2 202, 217 CuO 201, 212 -

Dy 83, 93, 96, 571,595, 596 DyAg 129, 133, 134 DyAI2 114-117, 124 DyA1G 418 Dy3AIsO12 428 DyBa2Cu307 266 DyBa2Cu3OT_~ 263 DyCo5 482, 491,496, 498 DyCos+~ 410 DyCos.3 492, 494, 495 DY2Co7 491 DyCo12B6 518 DY3Fe5012 458, 467 iron garnet 473 Dy2FeI4B 173 Dy2Fe17 318 Dy2FeI7C2.5 318 DyzFel7Cu 343 DyFelo.sMol.sNu 359 Dy2FelvN2.5 318 DyzFel7Ny 343 DyFeO3 200, 209, 477 DyFeloSizCo.3 360 DyFell Ti 357 DyFe11TiCu 358 DyFell TiNo.5 358 DyFell TiNy 358 Dy-ferrite garnets 471,472, 478 DyMn2 119, 127 DyNi2 119, 122, 124 DyNi5 136, 139 DySb 129, 132 -

epitaxial ferrite-garnet films 409, 479 epitaxial film Yz.6Gdo.4Fe3.9GaHO12 451,452 erbium 573 Er 83, 96, 100, 567, 571 ERA12 115-118 ErBa2Cu3O6.2 268 ErBa2Cu3Ox 260, 265, 267-269 Ero.8Dyo.2FeO3 464 ErFe2 118, 119, 123, 127, 463 ErFe3 460, 471,473,474

MATERIALS INDEX Er2Fel7 319 Er2FeI4B 173 Er2Fe17C 322 Er2Fel7C2.5 319 Er2Fel7fy 337, 338, 343, 346, 347 ErzFelTCI.oNy 319 Er2Fel7CxNv 338 Er2Fel4_xMn~C 512 ErFelo.sMOl.sNy 359, 362 ErzFelvN2.5 319 Er2FelvN2.7 338 Er2Fe17Ny 343 ErFeO3 201, 209, 464 ErFel 1Ti 358 ErFel 1TiCy 358 ErFel 1TiNo.5 358 ErFej1TiNu 358, 362 ErFel0.sV1.sNy 360 ErNi5 136, 139 ErRh4B4 166, 167, 172 Er2_zYzCoTB3 519 Er3_zYxCOllB 4 519, 520 (Er,Y)COllB4 520 Eu 83 EuBa2Cu307 266 EuFeO3 200, 209 EuMo6S8 166 EuMo6S7.sSe0.5 169, 172 EuNiO3 201, 211 EuO 77 Euo.75Sno.25Mo6S7.6Seo.4 169, 172 EUl_zSnxMo6S 8 166 Fe 81, 89, 93, 95, 96, 101, 104, 567, 571,577, 617 Fe alloys 81 FeCo alloys 96 FeF2 418, 428 Fe3Ge 367 Fe3GeNy 366 c~-Fe203 199, 200 Fe304 200, 206 FeTiO3 200, 205 FeZr alloys 96 gadolinium-iron garnets 409, 508 gadolinium-ferrite garnet 429, 478 Gd 83, 84, 88, 93, 97, 99, 561,577, 595 GdA12 114-117, 124 Gd3A12(SiO4)3 439 GdBa2Cu307 266 GdBa2Cu3Oz 261 GdBa2Cu306_ z 262

-

691

GdBa2Cu307_~ 261 GdCo2 118, 119, 121-123, 127 GdFe2 118-120, 122, 123, 127 Gd2Fel7 318 Gd2Fel4B 173 Gd2Fel7C2.5 318 GdzFel7Cy 343 GdzFe17N2.5 318 Gd2Fel7Ny 343 Gd3FesOl2 411, 441,447, 464, 467, 469, 474, 479, 508, 561 (Gd,Y)3Fe5OI2 461 garnets 411,416, 417, 448 Gd-ferrite garnets 443, 444, 466, 471,479 GdFelo.sMol.sNu 359 GdFeloSizCo.3 360 GdFeloVzNv 360 GdFe11Ti 357 GdFell TiCu 357 GdFel 1TiNo.5 357 Gd-Ga garnet 450 Gd3GasO12 479 GdNi2 119, 122 GdNi5 136, 137 GdRh4B4 168, 172 germanium 602 hexagonal 421 compounds of d-f type 482 ferrimagnets 482 HyYBa2Cu306.6 254 HuYBa2Cu306.7 254 HuYBa2Cu307 254 HzYBa2Cu307_~ 245, 256 Ho 83, 96, 567, 571,576 HoA12 115-117 HoBa2Cu306.2 265, 266 HoBa2Cu307 228 HoBa2Cu3OT_~ 264, 266 HoBa2Cu3Oz 261, 263 HoCo3Ni2 499, 503-506 Hoo.5Ero.5 571 Ho-ferrite garnets 471 HoFe2 555, 567, 569, 570, 577 HoFe3 460, 461 Ho2Fel7 319 Ho2Fel7_xAl~ 518 Ho2Fe14B 173, 174 Ho2Fe17Cy 343 HoFemsMol.5Ny 359 Ho2Fe17N2.5 319 Ho2Fe17Nv 343 HoFeO3 201, 209 Ho3FesO12 467

-

-

692

MATERIALS INDEX

- garnet 445, 473, 474 HoFell Ti 358 HoFell TiCy 358 HOFelo.5V1.5Nv 360 Hoo.oosLuo.995Rh4B4 172 Hoo.02Luo.98Rh4B4 172 HoxLUl_xRh4B4 168, 172 HoRh4B4 166, 172 Hoo.sYo.sBa2Cu3OT_, 266 HOo.41Y2.59FesO12 garnet 446 Ho1.05YL95Fe5012 garnet 447 (Ho,Y)3FesO12 449 (Ho,Y)IG 448, 449 iron 579 lantanides 83 LaAgl_~Inx 133 LaA12 115 La1.875BaoA25CuO4 222, 233 La2_xBa~CuO4 232 La2_xBazCuO4_ ~ 225 La2_~Ba~CuOx 232 Lal.875Bao.o75Sro.osoCuO4 222, 225, 233, 234 La2CaCu206+ ~ 226, 243 La(Col_~Fex)13 365, 366 La1.85CuO4_, 224 Lal.95CuO4_~ 224 La2_~CuO4_, 222 La2CuO4 221, 223, 236, 239, 258 La2CuO4_, 214, 224 La2.olCuO4_~ 224 La2Cuo.25Coo.7504 230 La2(CUl_xCox)O4+6 222 La2(Cul_~Zn~)O 4 222, 224, 227 LaFel3_~Si~ 365, 366 La2MCu206+ ~ 222, 243 LaNi5 135, 136 La2NiO4 235 La2NiO4+a 222, 225 LaPd2P2 142, 143, 161 La2_xSrxCaCuO6+ , 243 Lal.85Sro.olCuO4_ ~ 224 La1.89Sro.ÂlCuO4 229 Lal.95Sro.osCuO4 231 La2SrCu206_ 6 226 La2SrCu206+ ~ 243 La2_~(Sr, Ba)xCuO4 232 La2_z_uSrxNduCuO 4 222, 233 La1.775Sro.125Ndo.lCuO4 225, 233 La2_xSrxCuO 4 228, 230, 231,233, 238 La2_xSrxCuO4_ 6 222, 225 Lal.2Tbo.sCuO4 222, 239, 241

Lal.2Tbo.8CuO4_6 226 La2_xTb~CuO4 239 (La,Tb)2CuO4 240 Lal.9Yo.lCaCuO6+, 222 LuCo2 521,527 LuFe2 118, 119 Lu2Fe17 319 Lu2Fe17C 322 Lu2FelTCv 343 Lu2Fe17N2.7 319 Lu2Fe17Nv 343 LuFell Ti 358 LuFe11TiCu 358 LuRh4B4 172 Mn(CH3COO)2.4H20 416 Mn ferrite 577 MnF2 61, 79, 202, 216 MnF3 202, 217 MnFe2 77 MnGe2 465 MnO 202, 215 MnSi 105, 106, 108-110 Mno.75Zno.25F2 216, 570, 576, 605--607 NbFe2 113 Nbo.9Zro.1Fe2 113 Nbl_xZrxFe2 105, 113 Nd 83 NdAI2 115, 116 Ndi+yBa2_vCu207+~ 245 Nd1+vBa2_yCu3OT_a 266 Ndl.88Ceo.12CuO4_~ 226 Nd2_~CezCuO4 238 Nd2_zCexCuO4_, 222, 226, 237 NdzCuO4 221,236, 237 Nd2CuO4_, 222, 226, 237 Nd~_~Dy~FeI1TiNu 364 Nd2Fe17 317 NdsFe17 364 Nd2FeI4B 171, 173, 174 NdzFel4BN~ 365 Nd2FelTC2.5 317 Nd2Fe17Cy 343 Nd2Fel7N2.5 317 Nd2Fel7N4.5 325 Nd2FelTNu 343 NdFelo.sMol.sNu 359 NdFelo.75Mol.25Nu 358 NdFeloMo2Nu 355, 359 NdFellTi 357 Nd2(Fe,Ti)19 366 Nd3(Fe,Ti)29 366 NdFell TiCu 357

MATERIALS INDEX NdFe11TiNo.5 357 NdFeH TiNu 357 NdFeloV2N 360 NdFelo.5V1.sNy 360 NdNiO3 201, 211 NdRh4B4 168 NdRhzSi2 143, 147, 183, 185 Nd2_zSrxCuO4_a 222, 226, 239 NENP 279 Ni 82, 89, 93, 95, 96, 101, 103, 561, 577, 617 Ni alloys 82 Ni(CzHsN2)2NO2(C104) (NENP) 275 Ni2(C2HsN2)2NO2(C104) (NENP) 279 NiO 568 NpA12 175, 176, 178 NpAs 571, 575 p-NPNN 275, 280 Pd alloys 82 Pd-based alloys 101 PdFe 97 Pdo.97Feo.o3 98 PdFeo.oo35Mno.o5 102 PdMn 102 PdNi 97 Pr 83 PrA12 115, 116 PrBa2CuO6 254 PrBa2Cu307 250 PrBa2Cu3Oz 266 Pr2_zCexCuO 4 226 PrCo2Si2 143, 146 PrzCuO4 237, 238 Pr2CuO4_~ 222, 226 Pr2Fe17 317 PrzFeI4B 171, 173 Pr2Fel7C2.5 317 PrFelo.5MOl.5N~ 358 PrzFe17N2.5 317 PrFell Ti 357 PrFellTiN1.5 357 PrFellTiNy 363 PrNi5 61, 135, 136, 141, 142 PrNiO3 201,211 PrRuzSi2 143, 146 rare-earth-Co(Fe) alloys 464 rare-earth-ferrite garnets 409, 414, 416, 422, 439, 440, 460 RBa2Cu30~ 245 RCo2 411,418, 521, 527 RCo5 410, 416, 482 R2Co17 410, 482

RCo12B6 416 R2Co7B3 416 R2Co17Ny 350 RE-TM intermetallic compounds 410, 477 {R3+3}(Fe+3)[Fe2+3]O12 440 RF~ 364, 410, 416 RFel2 378 R2Fel7 315, 316, 330, 342, 410, 482 R6Fe23 365, 416 R2Fe14B 170, 410, 416, 482 R2Fe14C 416 RzFeI7C 316, 330, 416 R2FelvCy 316, 330, 342 R2(Fel_xCox)17Ny 350 RFel l_~Co~TiNy 364 R2(Fel_xMx)lTNy 352 R2(FeMn)17 482 R2(FeMn)14C 416 R2Fe17Nu 316, 330, 342 R3Fe5012 439 RFeloSiCu 366 RFe12_~TxNu 375 RFel 1TiNy 362 RFe12Z (Z -- N or C) 367, 378 R2Fel7Z3 (Z = N or C) 367 R2M17 459 RNiO3 201,211 perovskites 199, 209 RT3 416 RT12 416 R2T7 416 RnTm intermetallic compounds 521 (RY)CO2 527 (RY)CO2_x Al~ 521 -

single crystal ferrite garnet films 450 Sm 83 Sm2CuO4_~ 222, 226 SmFe2 364 SmFe3 364 SmzFel7 317 SmzFe14B 173 Sm2Fe17C 322 Sm2Fe17C2.5 318 Sm2Fe17Cy 335, 343 SmzFe17Cz.6H1.1 318 SmzFelvCNu 318 Sm2Fel7CxNy 336, 383 SmzFel4Ga3C1.5 390 Sm2Fe17N2.5 317 SmzFel7N3.0 317 Sm2Fe17N5.2 317 SmzFe17Nv 323, 335, 343, 385, 388, 394 Sm2Fe17 nitride 313

693

694 Sm7Fe93 nitride 393 Sm2Fel7N3.0Ho.8 318 Sm2(Fel_zCox)17N2.7 350 Sm2(Fel_zMx)17Ny 382 SmFeO3 200, 209 SmloFe79SillNy 366 SmFeH Ti 357 SmFe11TiCu 357 SmFel 1TiN 357 Sm2 (Feo.982Tio.ol8) 17N2.3 352 Sm2(Fel_xVz)17Ny 352 SmFeloV2Nu 360

(Sml-zNdz)2(Fel_zCOz)N2.7 353 SmNiO3 201, 211 (Sm1_xR:c)2Fe17Nu 352 SmRh4B4 166, 172 Sm3Se4 145, 150, 164 Sr2CuO3 201, 214 Sr2CuO2C12 214, 222, 226, 236, 242 SrCr8Ga4019 201,215 Tb 83, 567, 571 TbCo5 482, 491 Tb-ferrite garnets 471 TbFe3 365 Tb2Fe17 318 Tb2Fe14B 173 TbFel0.sMoi.sNv 359 Tb2Fel7N2.5 318 TbFeO3 464 Tb3 Fe5012 561 TbFeli Ti 357 TbFetl TiNo.5 357 TbFelo.sV1.5Ny 360 TbMn2 119, 127 TbNi5 136, 138 (TbY)3Fe5OI2 411,417, 448, garnets 416 Th2Fe17Cv 322 Tm 567 TmA12 114-116 TmFe2 118, 119, 122, 123, 127 Tm2Fe17 319 Tm2Fe14B 173 Tm2Fe17Cy 338, 340, 343 Tm2Fe17C1.oNy 319 Tm2Fel7CxNy 338 Tm2FeI7N2.5 319 Tm2Fel7N2.7 338 Tm2Fel7Ny 343 TrnFell Ti 358 TmFellTiC~ 358 TrnNi5 136, 140 -

MATERIALS INDEX TmRh4B4 168 TmSe 571 (Tm,Y)Co7B 3 520 (Tm,Y2)COll B4 520 (TMTSF)2C104 275, 278 (TMTSF)2NO3 275, 278 (TMTSF)2PF6 275, 276, 278 (TMTSF)2X 274 (TMTSR)2-X, X = C104, NO3, PF6 277 UA12 5, 175, 176 UAs 176, 178-180, 571 UAu2Si2 19 UAuSn 25 U3Au3 Sn4 35 UBel3 4, 5, 39, 42, 177, 195 UCdll 5, 177, 193, 194 UCo3B2 28, 29 UConB 28 U(Col_zCuz)2Ge2 21 U(Col_zFe~)2Ge2 21, 37, 38 UCo2Ge2 10, 18, 19 U2Co2In 28 U(Coo.25Nio.75)nB 28 U(Co0.sNi0.5)4B 28 U(Co0.75Nio.25)4B 28 U(Col_zNiz)2Ge2 21, 39 UCo2P2 19, 20 U3Co3Sb4 35 UCo2Si2 19, 176, 186 U2Co2Sn 28 UCr2Si2 19 UCu5 37, 41, 42, 162, 177, 187, 190, 192 UCu4A18 44 UCu4+~A18_x 44 UCu2Ge2 18, 19 UCu2P2 20 UCus_xPd~ 42 U3Cu3Sb4 34, 35 UCuzSi 2 19 U3Cu3Sn4 35 UFe2 178 UFe3B2 28 UFe2Ge2 19 UFezSi2 19 UIn3 176, 182 U(Ino.sSno.5)3 182 UIr3Br2 28 UIrzGe2 17, 18, 19 UIrzSi2 19 UzIr2Sn 28 UMn 2 175, 176 UN 176, 182 UNiA1 25 UNi2A13 5, 28, 29, 32, 177, 187, 198

MATERIALS INDEX UNi4B 28, 30, 31 UNi2B2C 20 U(Nil_zCuz)2Gez 21 U3(Nil_xCuw)3Sb4 35 UNiGa 26 UNiGe 25, 38, 39 UNi~Ge2 19 U2Ni2In 28 U3Ni3Sb4 35, 36 U3N!3(Sbl_zSnx) 4 35 UNi2Si2 10, 18, 19, 37 UNiSn 24, 25 U2Ni2Sn 28 U3Ni3Sn4 35 UO2 571, 574 UOs3B2 28 UOs4B4 28 UOs2Si2 19 UP 176, 180, 181 UPd2AI3 4, 5, 28, 31-33, 37, 39, 44, 177, 187, 199 UPd2Ga3 28 UPd2Ge2 19 U2Pd2In 28 U3Pd3Sb4 35 UPd2Si2 19 UPdSn 25 U2Pd2Sn 28 UPt3 5, 10, 12, 35, 39, 42, 162, 177, 186-189, 196 UPt2Ge.2 19 U2Pt2In 28 U(Ptl_~Pdz)3 188 U3Pt3Sb4 35 UPt2Si2 19, 176, 186 U2Pt2Sn 28 uranium 574 URe2Si2 19 URhA1 25 URh2B2C 20, 21 URh2Ge2 19 URhln 25 U2Rh2In 28 U(Rho.35Ruo.65)2Si2 176, 184, 185 URh3Sb4 35 URh2Si2 19, 176, 183, 198 U2Rh2Sn 28 URu3B2 28 URunB4 28 U4RuTGe6 18 U(RUo.TRho.3)4B4 28 URu2Si2 4, 5, 10-13, 17-18, 19, 33, 37, 39, 42, 142, 177, 187, 197 USb 176, 181,571, 574

695

U3Si2 27 USn3 176, 182 U(Sno.sIno.5)3 176 UT3B2 29 UT4B 29 UT2Ge2 16, 18, 20 U3T3Sb4 34, 35 UT2Si2 16, 19, 20 UTX 24, 26 UT2X2 17, 19 U2T2X2 26--28 UTe 176, 182 Uo.965Tho.o35Be13 195, 196 U1_~Th~Be13 162, 177, 187, 189, 195, 196 Ul_~ThzBel3_yBu 177, 195 Ul_xTh~Pt3 177 UxYI_~Pd3 40, 42 U2ZnI7 5, 37, 177, 187, 194, 195 V203 200, 206 YBa2Cu306 248, 251 YBa2Cu306+,~ 247, 252, 262, 263, 269 YBa2Cu30~ 244-246, 249, 259 YBa2(Cul_u Fey)30~ 245, 254, 259, 260 YBa2(CUl_yTu)3Ox 258 YBa2(CUl_~Zny)2Ox 245, 254 YBa2(Cul_yZny)307 254 YBa2(Cuo.96Zno.o4)30~ 258, 259 YCo2 411, 521, 527 YCo5 493 Y9Co7 105, 111, 112 Y(CoI_zAI:~)2 527 YCo2Si2 142, 143, 161 Y2Cu205 201, 215 YFe2 118, 119, 121, 123, 127 YFe3 364 Y2Fe17 319 Y6Fe23 364 Y2FeI4B 171, 173 Y2Fe14BNv 365 Y2FeI7C 322 Y2Fel7C2.5 320 Y2FelTCy 321, 323, 343 Y2Fe17Cl.2Nu 320 Y2(Fel_~Coz)lTNy 350 Y2FeI7H2.7 343 YFelo.sMol.sNu 359 YFelo.2MOl.sNy 355 YFell.sMoo.5N 359 YFell MoNu 355 YFe9Mo3N 359 YFe9Mo3Nv 355 Y2Fe17N3.1 320

696 Y2Fel7Ny 343 YFeO3 200, 209 Y3Fe5012 garnet 417, 418 Y3(Fe,Ga)5 O12 417 (YTb)3FesO12 477 YFellTi 358 YFe11TiCy 358 YFellTiNo,5 358 YFell TiNv 355 YFellTiZy (Z = N and C) 363 YFeloVaNv 355 YFelo.sV1.sNu 360 Y2Fel7Z3 with Z = H, C or N 368 (YI _ t Gdt )(Coo.915Alo.o85)2 530 (YI_ t Gdt)(Coo.95Alo.o5)2 529 (Yl_tGdt)(Col_~Alz)2 530, 531 film 451 Yl_tGdt(COl_~Alx)2 527, 531 Y2.6Gdo.4Fe3.9Gal.lO12 450, 452 Y3_zGd~Fe5O12 garnet 460 (Y,Gd,Yb,Bi)3(Fe, A1)5012 453,454, 479 (YR)Co2 418 (YR)(CoA1)2 418 YMn2 119, 124-126 Y(Mnl_zAI~)2 119, 127 Y(Mno.97Feo.o3)2 126 -

MATERIALS INDEX Y(Mnl_zFe~)2 119 Yl_uPr~Ba2Cu307 253 Yl_yPryBa2Cu3Ox 245, 250 Yl_uPryCu307 256 (Yl_zPrx)Ba2Cu307 251 Yl_tRt(COl_xAlx)2 527 YRh4B4 172 Yo.9Tbo.lMn2 119, 125 Yo.95Tbo.osMn2 119, 125 Yb--ferrite garnet 450, 466 YbBe13 42 YbBiPt 43, 44, 145, 150, 151, 163, 165 YbCu5 42, 43 YbCunAg 5, 43 YbCu4Au 42, 43 YbCu4In 43 YbCu4Pd 42 YbCu2Si2 43 Yb2Fe17N2.8 319 YbNi5 136, 141 YbNi4In 43 YbPtSb 44 Ybo.sYo.sBiPt 145, 150, 163 ytterbium-ferrite garnet 431 Zno.sFe2.504 567

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