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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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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
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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
. . .
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
1
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.
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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 . . . . . . . .
1
85 181 289 453 511
Volume 7 1. Magnetism in Ultrathin Transition Metal Films, by U. Gradmann . . . . . . . . . 2. Energy Band Theory of Metallic Magnetism in the Elements, by V.L. Moruzzi and P.M. Marcus
. . . . . . . . . . . . . . . . . . . . . . . . . . .
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
. . . . . . . . . . . . . . . . . .
503
Volume 8 1. Magnetism in Artificial Metallic Superlattices of Rare Earth Metals, by J.J. Rhyne and R. W. Erwin
. . . . . . . . . . . . . . . . . . . . . . . . . . .
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
.....................................................
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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o ooe
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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
I
i
100
I
200
i
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
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_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
0.00
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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0
0
,,,el
,-'
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120
A. SCHENCK and F.N. GYGAX
B. (T) 0.9
120 M(T)/M(O) ~
,
0.8
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0.6
80 o e,,,,
0.5
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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
-r
0.5 60
>" = 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
Bext = 0 T
•
-
-130 --
v
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ro tCT
20
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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
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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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~
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Oom~ i
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i
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-
/
0.4 [-0.0~
• T. I
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~°
t
I -
GdFe2
• • leIO
0.4n0
,
-
" " " ''.~L,. 200
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
I
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125
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(a)
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-
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300
0.25
II
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llll~
..iil.I.II ~ ~warming
.I6 ....
0.20
~" 0.15 E E >.,
o~
<
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
I
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/
Q ,..z-~3£X300.O o °_~ ' ' - - P-- - - 0 - - 0 - 0 - 0 - 0 - . . 0 - - 0 -
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
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v
0
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._o 0.15
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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~ ,,~
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129
130
A. SCHENCK and EN. GYGAX 2.0
1.0
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8K
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7.5K
2.o o
7.4K
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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
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CeAg l_xlnx(x=O.03)
O CeAg O
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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
,
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rr
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P Rate
,
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00
I
10
I
I
20 30 40 Temperature (K)
L
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0
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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
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.= o
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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 ....
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:
0.10 F 0,08 [-
4
.
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. . . .
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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
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139 ;
t
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CPOo
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i
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t
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i
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i
i ill
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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
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MUON SPIN ROTATION SPECTROSCOPY
145
c~
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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
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=
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A. SCHENCK and EN. GYGAX £ O~
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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
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-o15 a
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I
1.0
I
I
1.5 2.0 2.5 Time (psec)
I
I
3.0
3.5
0.05 0.00
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.. - - - 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
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~
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o oc.c io~ oc ~ ,~
°°
o
q)
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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 "
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4
6
8
10
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60
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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
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10 O C CD D Cr CD
A
CeB 6 8.~<~
m
%,
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6
•
o
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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
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i
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CeCu2osSi 2 ZF "
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lO I
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0
0.2
0.4
0.6
0.8 r (K)
t.0
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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
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,
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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.-
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,
1
2.0 ~_~L,k_.,L-i,JS..~ '
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'
~
'
~
-
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
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~a~ 0.2
CePtSn 7K 4
6
8
0
200G LF
~
0
lOG LF
o
i
0.4
~
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O. I
~
,
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b
CePtSn 8.2K
0.1
I
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I
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,
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~
-0.
2
I
I
o
i 0.5
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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
,
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165
7 6
4" s ,.~ 4
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c
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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
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i
i
lll,,q
i
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tr tO
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o Gauss.
• Exp.
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m 1.0
rr
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oe'" i
.... %.
"if,
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10
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,,,,,,I
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iiiiiii
100
i
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i
iiiiill
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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
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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
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~.~ 40 20 0
i
i
i
i
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50
i
i
i
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i
i
i
150
i
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i
i
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i
i
i
+ ~ i
250
i
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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
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~
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MUON SPIN ROTATION SPECTROSCOPY
173
tz~
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~ o o ~ ~ g~
o ©
ca
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::i.
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m
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174
A. SCHENCK and EN. GYGAX I
180
I
I
Nd2Fe14B
_
M (30M'~-~'f0) el"~e ~
"1'~ 170 o rI1}
"
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I
100
400
200
02Fe14 B
300
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,,--.
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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
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MUON SPIN ROTATION SPECTROSCOPY
177
o.q~
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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
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03 O 0 o
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og
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0 0
or
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I
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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
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(a) O O
,
0 0.8
, ,
~ ....
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_~o~p o o o oo o
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91
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o ,,,I
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50
,o, ,o,, o, ,o,
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100
250
150
200
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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
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~ 1.1 1.0
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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
-
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f
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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
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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 [
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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 '
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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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A. SCHENCK and EN. GYGAX
UPt 3 Gaussian relaxation 0.20
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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
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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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MUON SPIN ROTATIONSPECTROSCOPY
UCu 5
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191
u Cu
I
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I
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w
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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
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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
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v
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0
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I
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I
2
5
4
5 T/T N
6
7
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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
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Ul"xThxBe13
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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
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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
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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
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r 0
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i
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i
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r
I 4
= 5
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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
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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 electro
