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Valence fluctuation phenomena

J M Lawrence?$, P S Riseboroughs and R D Parks8 / I t Physics Department, University of California, Imine, California 92717, USA § Physics Department, Polytechnic Institute of New York, Brooklyn, New York 11201, USA

Abstract

Valence fluctuation phenomena occur in rare-earth compounds in which the proximity of the 4f level to the Fermi energy leads to instabilities of the charge configuration (valence) and/or of the magnetic moment. We review the experimental results observed in the subset of such systems for which the 4f ions form a lattice with identical valence on each site. The discussion includes key thermodynamic experiments, such as susceptibility and lattice constant, and spectroscopic experiments such as XPS and neutron scattering. This is followed by a review of existing theoretical work concerning both the ground states and the isomorphic phase transitions which occur in such compounds ; the emphasis is on those aspects which make valence fluctuation phenomena such a challenging many- body problem.

This review was received in October 1980.

3: Supported in part by the National Science Foundation. 11 Supported in part by the National Science Foundation and the Office of Naval Research.

0034-4885/81/010001 f84 $06.50 0 1981 The Institute of Physics

2 J M Lawrence, P S Riseborough and R D Parks

Contents

1. Introduction 1.1. Prologue and scope 1.2. An introductory overview

2. Experimental properties 2.1. Phase diagrams for the valence transitions 2.2. Magnetic ordering 2.3. Thermodynamic and transport behaviour 2.4. Spectroscopic behaviour 2.5. Fermi liquidology 2.6. Common extrinsic effects

3.1. Introduction 3.2. Approximate solutions for the ground state 3.3. Charge fluctuations and spin fluctuations 3.4. The electron-phonon interaction 3.5. The electron screening interaction 3.6. Magnetic interactions and double exchange 3.7. The Kondo lattice 3.8. Theories of the valence transitions 3.9. The essentially localised model

References

3. Theory

4. Epilogue

Page 3 3 3

1 1 1 1 17 20 35 50 53 56 56 59 64 64 67 68 69 70 75 76 77

Valence fluctuation phenomena 3

1. Introduction

1.1. Prologue and scope

The valence fluctuation phenomena which occur in certain rare-earth compounds present a strong challenge to solid-state physicists. Several aspects of the problem play a central role in our understanding of condensed matter: magnetic moment formation in metals, Fermi liquid behaviour, electron screening and correlation, strong electron-phonon coupling, soft phonon modes, non-symmetry-breaking structural phase transitions, and magnetic phase transitions involving reduced-moment ground states and incommensurate modulated-moment structures. The physics is related to (but more intractable than) the Friedel-Anderson-Kondo problem as well as to the theory of the metal- insulator transition and of transition-metal magnetism. Several excellent review articles concerning valence fluctuation phenomena have already been published, both of a general nature and concerning specific experiments or theories ; these are indicated in the reference list. In this article we attempt a moderately thorough review of work done to date, both experimental and theoretical. Our primary prejudice and slant is that the physics involves a difficult and important many-body problem that is not going to vanish within one or two years.

The outline of the article is as follows. In a brief introductory overview of the field we relate a ‘standard picture’ of the physics, which will be subject to revision in the body of the article. Following this we review the key experimental properties of a large number of rare-earth valence fluctuation compounds. Since the number of such compounds is legion, we will discuss only those materials for which thermodynamic studies have been supplemented by microscopic studies such as photoemission or neutron scattering. We will also restrict our attention to rare-earth compounds even though the relationship to valence fluctuation effects in the actinides and transition metals is a fruitful area of study, and we will neglect related effects in ‘inhomogeneous mixed-valence’ compounds and in dilute alloys (e.g. Lal-2Cez) except for a few remarks. Throughout the review the terms ‘mixed valence’, ‘ambivalence’, ‘valence fluctuation compounds’, as well as other phrases and acronyms, will be used interchangeably and will embrace the complete class of rare- earth compounds for which mixing effects between the 4f electrons and conduction electrons play a vital role. As well as strongly non-integral-valent systems such as a-cerium and SmS this will include nearly integral-valent systems such as CeAlz or CeAb, where the mixing leads to important instabilities of the magnetic moments. We follow the experimental review with a theoretical discussion which will concentrate on those aspects of the ground state and associated phase transitions which make valence fluctuation phenomena such a challenging many-body problem. Theory which is primarily germane to the clarification of particular experiments will be discussed in the experimental sections.

I .2. An introductory overview

1.2.1. The valence transition. The archetypal valence instability occurs in FCC cerium metal (for a review, see Koskenmaki and Gschneidner 1978). In the PT plane a line of first-order phase transitions separates the low-density (ao= 5.15 A) y state from the high-


I I I I 1

Ce SmS 750 - -

- 8

-

1 I 1 I

density (a0=4.85 A) 01 state; it is an isomorphic phase transition (there is no change in the crystal symmetry) and the phase boundary terminates at a critical point (figure 1). The large (15 %) cell volume change associated with the lattice collapse arises from a change in electronic structure. In the y state the cerium ions primarily have the trivalent 4fl(5d6s)3 structure; application of pressure increasingly favours the tetravalent 4f0(5d6s)4 structure. There is a large decrease in radius for the tetravalent atoms because removal of the 4f electron decreases the screening of the nuclear charge so that the outer 5d6s valence electrons are sucked in closer to the nucleus. The valence (z) in the 01 state is not four, however. One form of evidence, based on the empirical correlations between valence and metallic radius which are found in the periodic table, suggests a non-integral valence, midway between z = 3 and z = 4 (Gschneidner and Smoluchowksi 1963). In a plot of metallic radius against atomic number (figure 2) a-Ce does not lie on the smooth extrapolated curve for tetravalent elements, but at an intermediate position, such that one would assign by linear interpolation an intermediate valence (IV), z = 3.67.

A similar isomorphic valence transition occurs in SmS (figure 1) which is an ionic solid with the rock-salt structure. In the low-pressure phase (B-SmS) it is a black, divalent 4f6(5d6s)2 semiconductor; under application of 6 kbar pressure the lattice collapses as the material undergoes an insulator-metal transition to a metallic phase (M-SmS) where the material turns golden as the plasma edge moves into the visible. The valence/radius correlations (figure 2) suggest that in the M phase the material is not fully trivalent 4f5(5d6s)3 but rather has a non-integral valence z=2.75 (for a review see Jayaraman et a1 1975b).

Valence transitions can also be driven at ambient pressure by alloying in Cel-2RE2, Sml-sRE5S or SmSl-%M,. (we will use the notation of RE to represent a rare earth or related solute; M represents a pnictide.) The phase diagrams are similar to figure 1 with x replacing P. Hence for x > xo z 0.15 at ambient conditions the alloy Sml-,GdzS is in an IV state. In addition, many compounds of cerium, samarium, europium, thulium and ytterbium exhibit non-integral valence at ambient conditions, e.g. CeN, SmBs, EuRh2, TmSe or YbAIz.

1.2.2. The mixed-valent state. A necessary condition for non-integral valence is that two bonding states 4f%(Sd6s)m and 4f%-1(5d6s)"+x of the rare earth be nearly degenerate. A


2.; 6.0

5.8 c

O S

0" 5.6

5.4

56 60 64 68 12 56 60 64 6% 12 Bo Ce Nd Sm Gd Oy Er Y b Hf Bo Ce Nd Sm Lid Cr Yb H:

La Pr Pm Eu Tb Ho Tm tu t o Pr Pm Eu nu' Ho Tm t u

Atomic Number

Figure 2. Metallic radius against atomic number for the rare-earth metals (data from Gschneidner 1961) and the lattice constants against atomic number of the rare-earth sulphides (data from Elliott 1965, Shunk 1969). The datum for M-SmS was taken in the high-pressure phase (Jayaraman et al1975b).

listing of the relevant rare earths is given in table 1. In the gas phase most rare earths are divalent, but in the solid state most are trivalent, due to the large cohesive energy gained by promoting a 4f into a bonding state. In the middle of the rare-earth row, however, Hund's rule correlation energy can be gained by converting to the divalent state and lining up the maximum number of spins, which explains the ambivalent tendency of Sm and Eu; at the end of the row (Tm and Yb), obeisance to Hund's rule creates a similar desire to complete the shell by conversion to divalence. In the case of cerium the ambivalent tendency arises from the fact that the 4f orbital is more spatially extended than for the other rare earths.

The materials of interest have been dubbed homogeneous inixed-valence compounds (Varma 1976); this is to distinguish them from such inhomogeneous mixed-valence compounds as Fe304 or Sm& where two distinct charge states occupy inequivalent

Table 1. VaIence states z and magnetic configurations in rare-earth ions which exhibit non-integral valence.

Higher lying multiplets

C Ion z f" S L J g p ( p ~ ) ( ~ ~ u K m o 1 - 1 ) Cubiccrystalfields J EJG)

3000 -

Ce 3+ f l 112 3 512 617 2.54 0.807 r7(2), rs(4) 712

Eu 2+ f7 712 0 712 2 7.94 7.91 rs(2) r7(2) 1x4)

Tm 2+ f13 112 3 712 817 4.54 2.58 1x2) r7w rs(4)

- - 4 + f O O 0 0 - 0 0 Sm 2+ f e 3 3 0 - 0 0 - 1 410t

3+ f5 512 5 512 217 0.84 0.090 r7(2), ~ 4 ) 712 1500t

3 + f 6 3 3 0 - 0 0 - 1 480;

3+ f12 1 5 6 716 7.56 7.17 rdl) W3), etc Yb 2+ f14 0 0 0 - 0 0

3+ f13 112 3 712 817 4.54 2.58 1-42)? r7w, 1x4) 512 l5000§

t Moon et al(1978). j: Bauminger et aZ(1973). 0 Ross and Tronc (1978).


T (K) Figure 3. Isomer shift for SmB6 against temperature. The isomer shift of SmFz is taken as representative

of Sm2+ while that of SmF3 is taken to represent Sm3+ (from Cohen et aZ1970).

lattice sites in a staticcharge ordered array. Evidence that there is only one kind of charge state present comes, for example, from Mossbauer isomer shifts; the case of SmB6 is shown in figure 3. Were two inequivalent charge states present there would be two lines in the Mossbauer spectrum, at positions corresponding to the 2+ and 3’ isomer shifts; however, only one line is observed at an intermediate position. On the other hand, the terminology ‘mixed valence’ or ‘ambivalence’ also suggests that in some sense both the 4 P and 4fn-1 configurations contribute to the intermediate-valence wavefunction. Evidence for this comes from x-ray photoemission experiments (XPS). When an electron is ejected from a 4fn (or 4fn-1) orbital it leaves the ion in a 4fn-1 (or 4fn-2) excited state; the excited state has a characteristic LSJ multiplet spectrum which serves as a fingerprint to identify the initial state from which it arose. In figure 4 we show the XPS spectrum of SmB6; two sets of lines are present corresponding to the presence of both 4f5 and 4f6 orbitals in the initial configuration.

1.2.3. Valence Puctuations, experimental time scales and hybridisation. This situation suggests a phenomenology, as follows: the mixed-valence state can be thought of as a mixture of 4fn and 4fn-1 ions, the energies of which are nearly degenerate. At any given site 4f charge fluctuations between the two configurations occur on a time scale Tvf, the

Energy below cF ieVl

Figure 4. Valence band XPS spectrum of SmB6. The theoretical curve was calculated as described in 52.4.1 (from Chazalviel et aZl976).


so-called valence fluctuation time or interconjiguration fluctuation time (hence also the acronym ICF). Experiments such as XPS which probe the sample on a time scale much shorter than TVf will see both configurations, but experiments which probe on a time scale longer than Tvf will see only one intermediate configuration. Lattice constant and isomer shift measurements fall into the latter category. This phenomenology was introduced in the early 1970s as an adaptation of ideas of Hirst (e.g. see Wohlleben and Coles 1973).

Quantum mechanically this situation must be understood in terms of the hybridisation of the two configurations ; schematically the hybrid wavefunction is

I $> =an1 f"> + an41 f"-l>.

However, the hybridisation involves the band electrons, and a more appropriate description is in terms of band structure. As indicated in figure 5 the f6 level Et of divalent B-SmS falls in the gap between the valence band and the empty 5d6s conduction band. Under pressure the bottom of the conduction band goes down until it overlaps the f6 level, at which point the latter empties electrons into the band. In the mixed-valent state the f G level is thus pinned to the Fermi level EF. It takes on a width A due to the hybridisation; the inverse of this width can be identified as the valence fluctuation time. The hybrid level is only partially occupied in the non-integral-valence state. A similar sequence holds in cerium, except that the 4f level is already degenerate with the sd band in the y state and hence already somewhat hybridised, although it is sufficiently far below the Fermi level to ensure nearly complete occupancy. This situation of a narrow level crossing a broader band might be expected to lead to a hybridisation gap, and we shall see that there is evidence that this indeed occurs in SmBe.

1.2.4. Non-magnetic ground state. In ordinary rare-earth compounds the 4f electrons are highly localised and possess well-defined moments obeying atomic spectral rules (Hund's rules). These moments exhibit characteristic crystal field splittings and also interact weakly with the conduction electrons through the sf exchange interaction. This latter

E,= f F

a-Ce

Valence U band sd band

B- Sin S

Ef = cF M-Sm S

Figure 5. Schematic density of states diagrams for cerium and SmS.


mediates the indirect (RKKY) exchange interaction between the 4f spins which leads to magnetically ordered ground states (Coqblin 1977). The magnetic properties of the ionic configurations of the elements exhibiting ICF are shown in table 1.

One of the key features of IV materials is that they exhibit non-magnetic ground states. Divalent B-SmS has a J=O ground state and exhibits a van Vleck susceptibility due to a low-lying J= 1 level; trivalent SmS ions would be expected to exhibit a diverging susceptibility at T=O due to Kramers degeneracy in the ground state (figure 6) . In the collapsed M phase at high temperatures the susceptibility interpolates between the expected f 5 and f6 susceptibilities as though the material were a mixture of the two types of ions. At T=O, however, the susceptibility saturates to a finite value; this led Maple

T (K) Figure 6. (a) Magnetic susceptibility of SmS in the divalent B phase and in the mixed-valent M phase

(8-12 kbar). The broken curve represents the susceptibility of hypothetical trivalent SmS (from Maple and Wohlleben 1971). (b) Susceptibility of Ceo.gTho.1 (thorium is added to pre- vent formation of the DHCP /3 phase) (Lawrence and Parks 1976).

and Wohlleben (1971) to the conclusion that the valence fluctuations render the mixture homogeneous and remove the divergence expected from the Sm3+ 4f5 ions; the f5 spin memory is lost when the configuration fluctuates. A similar sequence occurs in cerium metal; in the y state the susceptibility is of Curie-Weiss form, apropos trivalence, but in the cy. state the susceptibility is that of an enhanced Pauli paramagnet: much too large to represent simple tetravalence, but also non-diverging as T --f 0, indicating that the moments of the Ce3+ ions are quenched by the valence fluctuations. We will refer to this phenomenon as demagnetisation.

1.2.5. Fermi liquid behaviour and analogy to dilute magnetic alloys. The situation of a localised level degenerate with and hybridising with the conduction band and in close proximity to the Fermi level bears a strong resemblance to the problem of magnetic


moment formation in dilute alloys-the classic Friedel-Anderson-Kondo problem. Although there are important expected differences between the dilute and concentrated limits, many experimental properties of mixed-valence compounds can be understood on a phenomenological level through this analogy. In particular, in the dilute limit there is a crossouer from free moment behaviour at high temperatures to a strong-coupling Fermi liquid regime in the ground state (Krishna-murthy et a1 1977, Nozi&res 1974). In the concentrated ICF materials a crossover connects the high-temperature Curie-Weiss behaviour to the low-temperature, enhanced Pauli paramagnetic behaviour. In cerium this crossover coincides with the valence transition, but in most Ce and Yb intermetallics the crossover does not coincide with significant valence change. We will refer to the temperature scale of the crossover (which in the dilute limit is the Kondo temperature TK) as the spin jluctuation temperature T,f.

Further support for this phenomenological analogy to the dilute limit is that the resistivities of certain cerium intermetallics are Kondo-like (we will use the term ‘Kondoesque’) in that they exhibit resistance minima and regions of negative slope suggestive of a In T contribution, as in the Kondo effect in dilute systems (van Daal and Buschow 1970). Most of these materials are trivalent and several (e.g. CeA13) have non- magnetic ground states. It cannot be direct valence fluctuations which are responsible for the demagnetisation, since trivalence implies that the 4f level is too far below the Fermi level. Rather, the moment is quenched by spin fluctuations resulting from strong sf exchange; the latter presumably arises from virtual 4f charge fluctuations where the 4f hops to the Fermi level and returns with opposite sign, as in the dilute case. The condition for this to occur is that the 4f level be close to the Fermi level which means that the material is on the verge of a valence instability. We thus include such materials in this review as examples of (virtual) valence fluctuation compounds.

Certain trivalent cerium compounds (e.g. CeAlz and CeIn3) which show resistance minima and/or the reduced moments characteristic of spin fluctuations also magnetically order. This order is always anomalous, involving reduced ground-state moments and in some cases exotic spin density waves, The sequence tetravalence-intermediate-valence- non-magnetic trivalence-magnetically ordered trivalence presumably occurs as the 4f level moves from well above the Fermi level to well below.

It is quite clear experimentally that the ground-state properties of the non-magnetic valence fluctuation compounds are those of a Fermi liquid (Varma 1976, 1977). In addition to the finite susceptibilities, large linear coefficients of specific heat are also observed, suggesting a high density of states due to a narrow hybrid 4f level at the Fermi energy. The anomalous magnetically ordered compounds can also be understood in these terms by analogy to the classic Stoner theory of the magnetic-non-magnetic transition in a Fermi liquid. This latter analogy also brings out the fact that the charge fluctuation rates and spin fluctuation rates may well be substantially different. The relevance of this extended Fermi liquid analogy to the ICF compounds will be discussed in $2.5.

1.2.6. Theories of the ground state.

1.2.6.1. The Anderson lattice. The starting point for many theories of the valence instability and the mixed-valence ground state is the Anderson lattice: a periodic array of local moments at an energy A E = Ef - E F below the Fermi level E F of an sd conduction band of width W, hybridising through an interaction Vaf, and with strong intra-atomic Coulomb repulsion Ufp to provide the necessary local correlations. Direct ff hopping is generally ignored, i.e. the bare f-band width Wf satisfies Wf < Vdf since the 4f orbitals are so small


as to preclude any significant overlap. (For small but finite Wf the model becomes a two-band Hubbard model.) However, the hybridisation is expected to give rise to an f-level width of order A = N ( E F ) v d f 2 where N(EF) cc l /W; in the mixed-valence state the condition for pinning of the f level to the Fermi level is then lAel 5 A. A key point concerns the relative magnitudes of these microscopic energies, which are quite different froin those found in transition metals (Varma 1976); in particular, Uff - 5 eV is the large energy in the problem while A is the small energy, typically of order 0.01-0.1 eV for rare earths. Crystal field splittings (kTCFwO.OO1-O.O1 eV) are also much smaller than in the transition-metal case.

The relationship A<Uef points to a key theoretical difficulty, viz. how to obtain a non-magnetic ground state when magnetism appears to be so strongly favoured, i.e. small bandwidths and large local Coulomb interactions are usually considered to be the conditions for magnetic order, as in the Stoner criterion. Another way to express this is to note that, for the Anderson model, it is the strong negative resonant sf exchange scattering f s f which leads to demagnetisation ; but for the Anderson lattice such exchange should also lead to RKKY interactions, growing as f S P , and hence leading to magnetic order,

1.2.6.2. The Hirst model. In the situation A < Uff ordinary fermion decoupling schemes (e.g. RPA where one first hybridises exactly and then treats Uff as a small perturbation) should work very badly. This was first stressed by Hirst (1970, 1975) who suggested that the appropriate starting point is the ionic limit, where the intra-site Coulomb interactions are treated exactly by Hund’s rules, yielding spin-orbit-split LSJ multiplets of the 4f electrons; the hybridisation with the band states would then be treated as a small parameter. However, it is extremely difficult to hybridise these correlated 4f levels with the conduction band since the probability amplitude that a conduction electron hops on or off a 4f level depends strongly on the initial 4f occupancy and spin-orbit configuration of the ion.

1.2.6.3. Electron-phonon interaction and electronic screening. There are at least two important interactions which must be included, which are expected to significantly alter the dynamics in these models. The first is the strong electron-phonon interaction which can occur because the large cell volume change ( N 15 %) accompanying a valence change on a given ion can create significant distortion of the lattice. The second mechanism is the screening by the light 5d6s electrons o€ the 4f charge fluctuations, which we have seen to be quite slow; this leads to excitonic correlations tending to localise the sd electron that is emitted when the 4fn fluctuates to 4P-1.

1.2.7. Theories of the valence transitions. A basic element driving the transition in SmS is that the bottom of the conduction band lowers toward the 4f level under pressure, presumably because the crystal field splitting of the 5d tzg and e2g sub-bands increases while the centroid remains fixed. It is clear, however, that a (bootstrapping) interaction is required for the transition to be first order. Both screening and electron-phonon coupling have been implicated as driving the transitions. The screening mechanism forms the basis of the well-known Falicov model of the a-y transition in cerium metal (Ramirez and Falicov 1971). It can be understood roughly as follows: in order to thermally populate 4f levels in the a state, conduction holes must be created, which are then attracted to the 4f electrons, lowering their energy and thus favouring further 4f popula- tion; this bootstraps the metal into the y state. The importance of the lattice-mediated mechanism can be understood by a simple analogy to standard alloy theory. It is well

Valence juctuation phenomena 11

known that alloys of atoms of disparate sizes phase-segregate because there is a long- range attractive strain field between like-sized atoms. If we view the mixed-valent state as a static collection of 4fn and 4Pf l atoms, we then expect such segregation; furthermore, such segregation can be arrested while the material is still mixed valent by anharmonic forces (Anderson and Chui 1974). While this makes clear the importance of elastic forces, such a picture obviously requires modification for homogeneous mixed valence. Alternatively in SmS the electron-lattice coupling can be visualised as arising from a deformation potential which lowers the bottom of the 5d band when the lattice shrinks due to 4 f b 4 f 5 transitions; this favours further 5d-4f hybridisation and hence bootstraps the material into the metallic phase (Penney et a2 1975).

2. Experimental properties

2.1. Phase diagrams for the valence transitions

In this section we discuss the isomorphic phase transitions which occur in Ce, SmS and their alloys Ce1-,RE2, Sml-,RE,S, etc. Along the phase boundary PO( TO) in the PT plane (or XO( TO) in the x T plane) large discontinuities in the molar volume and entropy density occur; the phase boundary terminates at a critical point (P,, Tc) or (xC, Tc) beyond which the transitions are continuous. (We will use the symbols XO, To, PO to denote values at first-order transitions and xc, T,, P, to represent values at a critical point.) The situation is analogous to the liquid-vapour transition ; the compressibility, thermal expansion and specific heat diverge at the critical point and are consequently large in the vicinity of the critical point. Indeed the existence of the critical point has a profound effect on the thermodynamic properties in a wide surrounding region of the phase diagram, and the thermal behaviour of various thermodynamic coefficients such as the thermal expansion are strongly affected and can in large part be deduced once the phase diagram is known.

2.1.1. xPTphase diagrams.

2.1.1. I . Sml-,m,S. The phase diagram for SmS in the PT plane is shown in figures 1 and 7. The phase boundary is C-shaped; it has negative slope dTo/dP below 300 K and positive slope above; at high temperatures and pressures there is a critical point. Due to the large hysteresis, sample history effects and the effects of non-uniformity of pressure fields, the details differ somewhat from study to study: Tonkov and Aptekar (1974) report Tc=lOOO K, Pc=6.8 kbar and dTo/dP=240 K kbar-l at high T; Shubha et a1 (1978) report Tc = 1100 K, P, = 11 kbar and dT/dP= 170 K kbar-1; Raschupkin et a1 (1978, 1979) report Tc=700 K, P c = 7 kbar, dTo/dP=60 K kbar-1. The latter study (which involved conditioned samples which reproduce more faithfully than virgin samples) also gave dT/dP= - 50 K kbar-1 at low temperature. The transition with pressure appears to be first order at T=O (Bader et aZl973); hence the ‘re-entrant’ low-temperature transition which occurs at fixed P should also be first order.

For Sm1-,RE2S the two most widely studied rare-earth solutes have been gadolinium and yttrium. A schematic phase diagram at P=O for these two cases as deduced by Aptekar and Tonkov (1977, 1979) is shown in figure 7 ; the phase boundary is again C-shaped, terminating at a critical point for both Sml-,Y,S and Sml-,Gd,S near xcH % 0.15 and TcH% 550 K. In addition these authors analyse older thermal expansion data (Tao and Holtzberg 1975, Jayaraman et aZ1975a) and conclude that there is a second critical point at lower temperatures; this occurs near TcL=90 K and xCL=0.25 for Sml-sGdzS and near 160 K and x=0.25 for Sml-,Y,S. Pressure studies in Sml-,Gd,S

12 J A4 Lawrence, P S Riseborough and R D Parks

P (kbarl

Figure 7. The xPTphase diagram for Sml-,Gd,S. The boundaries for the pressure-driven B -+ M and M -f B transitions are given for various x (from Raschupkin et ~11979) ; the xTphase diagram at ambient pressure shows the limits of hysteresis and the upper and lower critical points (from Aptekar and Tonkov 1979). Combining these we arrive at the xPTequilibrium surface; lines ab and cd are critical edges.

alloys allow a determination of the three-dimensional xPT coexistence surface (Raschup- kin et al 1978, 1979). The equilibrium xPT phase diagram is shown schematically in figure 7; the co-existence surface terminates at a critical edge (i.e. line of critical points) at high temperatures (line ab) and for a limited range of x at a low-temperature critical edge (line cd).

Similar xPTphase diagrams are expected for other solutes. In table 2 we list the critical concentrations xo for the first-order transitions occurring at ambient conditions for a wide variety of solutes. (Since the hysteresis (figure 7) is large and complicated, these values are not highly accurate or reproducible. A sample with xSxcH cooled from the melting point will have a tendency to remain in a metastable B phase.) For Sml-,La,S the ambient transition is continuous, as is the low-temperature re-entrant transition (Holtzberg 1973). Pressure studies (Pohl 1977) show that the transition pressure PO (300 K) decreases with increasing x but the pressure-driven transition becomes continuous for xc=0.25, P, (300 K)=2 kbar. This suggests that the xPT coexistence surface terminates at a critical edge at finite pressure; the continuous lattice constant anomalies observed at P=O occur at the transcritical extension of the surface. Note also that for Eu, Yb and Ca substituents, dPo/dx is positive so that the alloys remain divalent for all x. Finally we note that the low-temperature critical point probably exists for other solutes than Y and Gd, e.g. it can be deduced from the susceptibility and thermal expansion data for SmS1-,P, (Henry et al 1979). For this system there is a first-order B-tM transition at 300 IC near x= O.05; for x=0.06 there is a first-order low-temperature M + B transition near 130 K, and for x=0.08 the low-temperature transition is continuous. Hence ~ ~ L z 0 . 0 7 , TcLz 130 K.

2.1.1.2. Continuous transitions. In SmSe and SmTe a continuous valence transition occurs at rather high pressures (Jayaraman et al 1975b). In SmS1-,Se, the transition


Table 2. The effects of additives on the SmS phase transition; xo is the critical concentration for first- order transitions at ambient conditions in Smi-xREnS or SmS1-zMz; APolAx is the change of the B -+ M transition pressure with x ; where the latter isnot availableit is estimated as - 6 kbarlxo.

APolAX -61x0 Additive U&) xo (kbar) (kbar) References

LaS 5.8547 0.357 -- lo -17 Pohl(1977) PrS 5.743t 0.17 - 35 Gronau and Methfessel(l977) EuS 5.968-t c 35 Jayaraman and Maines (1979) GdS 5.574-t 0.155 -15 -38 Jayaramanetal(1975a) TbS 5.516% 0.21 - 28 Gronau and Methfessel(l977) DYS 5.482$ 0.237 -25 Gronau and Methfessel(l977) HoS 5.453$ 0.244 - 24.5 Gronau and Methfessel(l977) Y bS 5.6587 -!-I Jayaraman and Maines (1979) Cas 5.685 $. 30 Jayaraman and Maines (1979) YS 5.4667 0.15 -40 Tao and Iloltzberg (1975) ThS 5.6811 0.10 - 60 Campagna et aZ(1976) SmAs 5.9217 0.05-0.08 -92 Holtzberg et al(1977) SmP 5.767 0.03-0.06 - 130 HenryetaZ(1979) SmSb 6.271t Divalent Beeken et al(1978)

for all x

7 Gschneidner (1961). f Gronau and Methfessel (1977). 5 Jayaraman and Maines (1979). I j Campagna et a1 (1976). 7 Continuous transition.

pressure increases rapidly with x, the transitions becoming continuous for x > 0.80 (Bucher and Maines 1972); this again implies that a critical edge exists. The isostructural chalcogenides TmTe, TmSe and TmS are divalent, fractional-valent and trivalent, respectively, so that interesting xPT phase diagrams can be imagined. Kaldis et al (1979b) showed that a valence transition occurs with x in TmTel-,Se, and Wohlleben et a1 (1972) showed that a valence transition occurs with P in TmTe; the xPT phase diagram should thus be similar in some respects to that of Sml-,RE,S. There are also a number of systems such as CeIns-,Sn, (Lawrence 1979) where a continuous valence transition can be driven with alloy parameters in cerium- or ytterbium-based intermetallics.

2.1.1.3. Cel-&E,. The cerium 'y-cy. phase transition (figure 1) has been reviewed by Koskenmaki and Gschneidner (1978). A determination of the critical point by volu- metric means (Kutsar 1972) yielded Tc = 480 K and P , = 14.5 kbar ; earlier investigators found larger values. As in SmS the hysteresis at ambient temperature is quite large. In the case of the alloys an accurate determination of the P=O critical point of Cel-,Th,, showed that Tc= 148 K, xc=0.265 (Lawrence et a1 1977); hence the xPT phase diagram appears as in figure 8. Studies of Cel-,Th, alloys with pressure (Huang et al 1978) located the critical edge (for x=O.25, Tc=273 K and Pc=4.8 kbar) and showed that the slope dTo/dP decreases somewhat as x increases. For the system (Cel-,REz3.0.sTho.l (where the small fraction of thorium is included to circumvent formation of the DHCP ,6 phase) it can be seen (figure 8) that different RE solutes have markedly different effects on the slope dTo/dx of the P=O phase boundary (Manheimer and Parks 1979).

2.1.2. Thermodynamic behaviour near the critical points. The critical behaviour near

14

-20

J M Lawrence, P S Riseborough and R D Parks

z:2 - Eu 0

I I I

I 1

xcTc in Cel-,Ths obeys mean-field theory and the Landau free-energy functional in the vicinity of the critical point has been obtained (Lawrence et a1 1977). From the associated mean-field equation of state the main features of the thermodynamics can be obtained. The curves for the molar volume v (T) are s-shaped for x > x c and show first-order discontinuities for x c x,; for x= x,, AV =U( T ) - U( T,) is proportional to ( T - Tc)l/S where the critical exponent 6 has the value three for mean-field behaviour; hence the thermal expansion diverges at the critical point as ( T - Tc)-”3. This critical behaviour is shown in figure 9, where the resistivity p - p c is plotted in place of AV, the two quantities being proportional. The specific heat has a similar (AT)-2/3 divergence at the critical point (Markovics and Parks 1977) as does the compressibility (Croft and Parks 1977).

Similar behaviour is expected near the two critical points in the xTplane of Sml-zRE,S. In figure 9 we show data for two samples of Sml-,Y,S with x bracketing the critical concentration x c L of the re-entrant transition; these have negative aula T but otherwise the critical behaviour is similar to Cel-,Th,. Similar curves near the high-temperature critical point in Sml-,GdsS have been observed (Jayaraman et a1 1975a, b); the phase boundary (figure 7) is sufficiently steep that for x=O.15 the sample breaks up into domains of B- and M-SmS as the temperature is lowered through Tc. In addition to the thermal expansion divergence which can be inferred from figure 9, the compressibility of these alloys should diverge at both critical points. Penney et a1 (1975) have observed the compressibility to diverge as x+xo in the M phase at 300 K; such a divergence near a first-order transition should only occur as the limit of metastability of the M phase is approached.

2.1 -3. Discussion.

2.1.3.1. Shape of the SmSphase boundary. One of the first questions that arises concerns the peculiar C shape of the phase boundaries in SmS. The simplest explanation is advocated by Raschupkin et a1 (1978, 1979). The Clausius-Clapyron equation states that the slope of the phase boundary obeys the relation dT/dP=Av/As where AV and As are the discontinuities in molar volume and entropy at the BM transition. The observed changes in Av/v from 500 K to low temperatures are small; hence the change in sign in dT/dP near 300 K is associated with changes in the entropy. A major source of entropy is the spins; the divalent J=O ground state is separated from the J=l level by about


T ( K )

Figure 9. (a) Resistivity data for Cel-zTh, near the critical point for three samples bracketing the critical concentration xc (1, x=0.41; 2, x=0.269; 3, x=0.227). The resistivity is proportional to the lattice constant near the critical point, so these may be thought of as thermal expansion curves. The full lines are the predictions of the mean-field equation of state. (b) Plot of ( p - p p c ) / (dp/dT) against T for a nearly critical sample x=0.269; since ApccAu and Aucc(T-Tc)1/s the slopes in this plot estimate the critical exponent 6 which has the value three for mean-field transitions ((a) and (b) after Lawrence and Parks 1976). ( c ) Lattice constant data for Sml-2Y,S in the vicinity of the low-temperature critical point xcL (adapted from Tao and Holtzberg (1975) and Holtzberg (1973)).

400 K (Shapiro et a1 1975) while the trivalent ground state should show Kramers degeneracy. At low temperatures the entropy of a mixture of Sm2+ and Sm3+ ions, which would include the entropy of mixing ( - n ( 2 f ) In n(2+)-n(3+) In n(3+)) as well as the Sm3+ spin entropy, would thus be greater than in the divalent state where the spin entropy is frozen out. Hence, well below 400 K, dT/dP < 0, while above 400 K the effect reverses sign as the J= 1 entropy of Sm2+ is generated. Jefferson (1976) gave a similar argument. Of course, a proper treatment requires consideration of the Fermi liquid nature of the ground state of M-SmS as well as lattice entropy changes (Varma 1976). (Kaplan and Mahanti (1979) give an alternative explanation based on the essentially localised model discussed in 53.9.)

2.1.3.2. Role of the solute in the phase transitions. A second important issue concerns the role of the solute in the phase transition. Two factors have been singled out: the first is the 'chemical pressure' exerted on the lattice by a solute whose radius differs from that of the host atom; the second is the electronic structure of the solute. Of course, the radius and electronic structure are not unrelated, but the two effects do seem to compete. The clearest example comes from the studies by Manheimer and Parks (1979) of the dual effects of solute valence and radius on the slope dTo/dx of the phase boundary of (Cel-zmz)o.gTho.l alloys (figure 8). At fixed solute valence the slope decreases linearly with solute radius; hence the role of chemical pressure is simple: larger solutes stabilise the high-volume y state to lower temperatures. At fixed radius the slope increases with increased solute valence. This cannot be explained on the basis of a rigid band model in which the extra electrons donated by the higher valence solute would cause the Fermi


level to move upwards with respect to the 4f level, thus stabilising the y state to lower temperatures. A possible explanation is that for a higher valence solute, the extra conduction electrons increase the screening of the cerium 4f electrons, promoting delocalisation.

In SmS alloys a possible measure of the strength by which the solute stabilises or destabilises the B or M phase is the slope dPo/dx of the transition pressure against concentration at 300 K. While a careful study along these lines would require determination of both the B+M and M+B boundaries, for an estimate we list in table 2 the slope with PO determined only from the B+M transition; for those materials where no pressure studies have been performed we use the estimate dPo/dx = 6 kbar/xo which is the slope of the line from Po(x=O) to xo(P=O). It appears from table 2 that the dominant effect is due to electronic structure of the solute. For example, in the series YbS, PrS, ThS, all of which have comparable ao, the slope increases markedly as the solute valence increases ; on the other hand, there is no simple correlation with atomic radius, and for the trivalent series LaS through HoS dPo/dx appears to have a minimum near GdS. The extreme effectiveness of SmAs and SmP in driving the B+M transition is clearly an electronic effect, although in the case of SmS1-zSbz the extremely large lattice constant appears to be responsible for the fact that divalency is observed for all x. Wilson (1977b) showed that, for doping on to the Sin site, to lowest order the top of the valence band and the Sni 4f6 level remain invariant under alloying so that the main effect of the solute is to raise or lower the bottom of the 5d6s conduction band. The divalent RE S compounds generally have larger energy gaps than SmS so that the Sin ions can be expected to remain divalent independent of x, with positive dPo/dx as observed for Eu, Yb or Ca dilution (Jayaraman and Maines 1979) while the trivalent RE S compounds generally have smaller gaps than SmS, favouring the IV state and negative dPo/dx because the lowered 5d energy at the solute site promotes hybridisation with the Sm 5d electron. Wilson (1977a) also discusses the chemical physics behind these alloying effects ; the chemistry is quite subtle.

A similar question concerns the role of the solute or pressure in determining whether the phase transitions are first or second order. Apart from dilution effects, which trivially favour continuous transitions, an important factor is that strong anharmonicity favours continuous transitions (Anderson and Chui 1974). Hence it might be expected that the greater the pressure or coiicentration of solute required to drive the transition, the greater the anharmonicity present at the transition and hence the greater the tendency for a continuous transition. This would correlate the continuous transition observed in Sinl-,LazS with its weak dPo/dx. It also partially explslins the dependence of the critical concentration xo on metallic radius observed in (Cel-,La,)o.gTho.l; larger solutes, which encourage greater anharmonicity, have smaller xo (Manheimer and Parks 1979).

A third issue explored experimentally is the degree to which local environment effects (as opposed to the effects of the overall average band structure or average lattice constant) contribute to the lattice collapse. In the case of SmSl-,As, Holtzberg et al (1977) demonstrated that in the B phase ( x < 0.05) each As atom causes each of the six nearest- neighbour Sm atoms to become trivalent, but the 5d electrons thus generated remain localised. The phase transition (XO N 0.05) occurs when the Sm3+ concentration reaches 30% which is close to the threshold for site percolation. For ~ 2 x 0 the M state is homogeneous mixed valence with 2 ~ 2 . 8 . Similar local environment effects seem to occur in SmSl-,P, (Henry et al1979).

2.1.3.3. Critical behaviour. Mean-field critical behaviour is the exception rather than the


rule for phase transitions in solids, and the popular wisdom is that it results from long- range forces. For this reason the mean-field behaviour in Ce1-$Thl: was originally attributed to the strong electron-lattice coupling which occurs near valence instabilities and which is expected to have a long-range character (Lawrence et a1 1977). The critical behaviour of SmS would be expected to be mean field, for the same reason. If this were the correct interpretation of the mean-field behaviour, it would imply that short-range electron screening mechanisms are not sufficient to describe the transition.

Work by Cowley (1976) suggests, however, that for an isostructural phase transition on a cubic lattice the transition must always be mean field because ‘fluctuations of the [order parameter] will occur only for wavelengths comparable to the crystal dimensions and consequently deviations from Landau theory will only occur extremely close to T,, except for very small crystals’.

A second interesting feature of the critical behaviour is that in the vicinity of the critical point both the resistivity p and effective moment p2= TxjC couple linearly to the order parameter, i.e. they are proportional to the molar volume. This feature might provide a clue as to the microscopic mechanisms controlling p and x.

2.2. Magnetic ordering

While most mixed-valent compounds have non-magnetic ground states, TmS and several of the trivalent cerium compounds exhibit antiferromagnetic ground states which are unusual in that spin fluctuations play an important role in the ordering. The characteristic energy Tsr of these spin fluctuations can be determined from linewidths in inelastic neutron scattering experiments (42.4.4). When TBn is of the order of or larger than the NCel temperature TN competition occurs between the demagnetising tendency of the spin fluctuations and the moment-stabilising tendency of RICKY interactions. The most general result appears to be reduction of the ordered moment, e.g. below the value of 0.71 p~ expected for a cerium r7 crystal field ground state. Information concerning these transitions is listed in table 3.

2.2.1. Modulated-moment structures: CeAlz and TmS. In CeAlz a particularly exotic incommensurate modulated-moment antiferromagnetic ground state occurs (Barbara et a1 1979). The magnetic periodicity is Q = (+ + E , 4 - E , 4) where E 2 0.1 ; the structure (figure 10) can be described as follows : the moments point in the [ 1, 1, 1 J direction ; spins in successive (111) planes alternate sign; the magnitude of the moment is modulated sinusoidally along the [liO] direction with maximum value 0.89 k 0.05 p ~ , and the period of the modulation is incommensurate with the lattice; there is a phase difference of n/4 between the two cerium atoms at (0, 0,O) and ($., $., a) in the Laves phase crystal structure. A more recent study of the temperature dependence of an additional weak reflection at Qc=(+, 3, +) suggests that the actual structure is a trip1e-Q structure (Shapiro et a1 1979). In this structure three incommensurate spin density waves along Q l = (4 + E ,

$ - E , +), @ = ( & - E , $ , $ + E ) and Q 3 = ( - 3 , - $ + E , - + - E ) coexist and are weakly coupled through the commensurate component e,=($, +,+). The magnetic structure in real space is then quite complicated. Further evidence for the triple-Q structure comes from recent nuclear quadrupole resonance experiments ; the linewidth is too narrow to be readily explained by the distribution of hyperfine fields expected in the single-Q state but is consistent with a multiple-Q structure (MacLaughlin 1980, MacLaughlin et a1 1981). A key point is that it is the magnitude of the moment which is modulated, and not the direction (as in a helicoidal structure); furthermore, the modulation persists to


Table 3. Magnetic ordering in ambivalent materials.

Type of Ordered Compound TN(K) Tc(K) antiferromagnetism moment ( p ~ ) AS( Tx) References

p-Ce 12.5

CeAlz 3.9

Ce3Ah 3 .2

CeIns 10.2

TmS 8.9

TmSe 3

Doubling of hexagonal a axis; different TN on inequivalent sites.

Modulated-moment incommensurate triple Q

tripling of c axis 6 Modulated-monien t

Modulated-moment incommensurate FCC Type I

5.5

0.50-0.75

0.89 maximum

Two sites: 1 .5/0.35

0.5-0.7

4.0

1.7

0.8

R In 2 Koskenmaki and Gscheneidner (1978), Wilkinson et a1(1961), Tsang et al(1976)

$R In 2 Shapiro etal(1979), Bred1 et aZ(1978), Barbara et al(l979)

Chouteau et al (1978), Benoit et al(l980a)

R In 2 Lawrence and Shapiro (1980), Benoit et al(1980b), van Diepen et al (1971)

R In 1.6 Koehler et al(1979), Bucher et al(1975)

-$R In 2 Berton et al(1980), Batlogg et a1 (1979a, b), Bjerrum-Mnrller etal(1977) Batlogg et a1 (1979b)

very low temperatures and hence is not thermally induced, i.e. the structure does not ‘square up’ at T=O (Barbara et al 1979). (Such squaring up is seen, for example, in thulium metal, as discussed in Coqblin (1977).) This suggests that the sites at the nodes of the spin density wave have their moments reduced by Kondoesque spin fluctuations; and indeed a spin fluctuation temperature of order 5 K is suggested by neutron measurements (Steglich et a1 1979b). A further indication of the presence of spin fluctuations is the fact that only half the expected R In 2 entropy is liberated at T N ; the remainder is

I1111

Figure PO. Magnetic structure of CeAlz; the heavy sine wave shows the magnitude modulation along the (ITO) direction; the vector labelled Q is the magnetic reflection (I-+ E, 4- E , 0) (adapted from Barbara et al1979).


slowly generated up to about 15 K (Bred1 et a1 1978). Another requirement for the modulated structure is competition between positive and negative long-range interactions leading to a maximum exchange f ( Q ) at finite Q. Since the RKKY interaction is mediated through the Kondoesque sf exchange, this suggests a delicate cooperative mechanism, as though the position of the 4f level effectively varies from site to site. A further requirement is the existence of considerable anisotropy to hold the moments in the [l, 1, 11 direction so that a modulated-moment structure is favoured over a helicoidal structure. Such anisotropy has been observed by Croft et a1 (1978a, b) in thermal expansion and magnetostriction measurements; the HT phase diagram was also obtained in this work.

A similar modulated structure has been observed in trivalent TmS (Koehler et a1 1979). The observed superlattice reflections at (&+ E , $- E,+) show ~=0.080, independent of temperature. The structure is similar to that of CeA12, with moments in the Cl121 direction showing maximum amplitude 4.0 p ~ ; this compares to 7 p~ for the full Tm3+ manifold and 0 p~ for the expected rl singlet ground state. Hence an induced-moment ground state is indicated ; spin fluctuation effects are probably also important. While commensurate peaks indicative of triple-Q coupling do not appear to be present, a broad diffuse scattering maximum at (3, 3, +) is present above and below TN, and unexplained scattering streaks are also present.

2.2.2. Nearly mean-field behaviour: CeIna. The unique aspect of CeIn3 is that the spin fluctuation temperature, as deduced from neutron scattering, appears to be much larger than the NCel temperature: Tsrw 150 K, TS = 10.2 K (Lawrence and Shapiro 1980). The ordering is (n, n, 7) antiferromagnetism, which means that the cerium spins, which sit on a simple cubic lattice, merely alternate sign. Two estimates of the ordered moment (table 3) indicate a value somewhat smaller than that of the ground-state doublet; this conclusion is strengthened if we note that, as in CeA12, exchange-induced admixture of the l'8 quartet is expected. An additional feature which is also believed to be due to the strong spin fluctuations is that the critical exponent is nearly mean field. This is discussed further in $2.5; one way to understand it is that the critical fluctuations (large, slowly decaying antiferromagnetic patches above TN) are suppressed by the Kondoesque spin fluctuations, which tend to flip the individual spins. The critical fluctuations observed in the neutron experiment are, indeed, extremely weak and occur only very close to TK. Suppression of fluctuations is, of course, a well-known condition for mean-field behaviour.

2.2.3. TmSe and double exchange. The fact that TmSe is the only non-integral-valent material which magnetically orders appears to be related to the fact that, unlike Ce, Eu, Sm and Yb, both valence states of Tm have magnetic manifolds (table 1). This was first stressed by Varma (1976). The ordering is type I FCC antiferromagnetism with a small (1.7 p ~ ) ordered moment (Bjerrum-Maller et a1 1977). The HT phase diagram (first studied by Ott et a1 1975) is shown in figure 11. The HTP phase diagram has also been determined (Guertin et a1 1976); the T=O critical field for the metamagnetic transition increases rapidly with pressure. The small ordered moment is not well understood; presumably it reflects the presence of the Tm2f crystal field ground-state doublet, the Tm3+ J=O singlet with possible admixture of higher lying levels as in TmS, and moment reduction due to valence/spin fluctuations.

The FCC type I structure is tetragonal, with spins parallel within (001) planes but alternating signs between planes. This requires weak antiferromagnetic nearest-neighbour exchange Jnn and stronger next-nearest-neighbour ferromagnetic exchange Jnnn ; in


I r W X L 2 ~ ~ / ~ , ~ --- ---\

U

2 3 T I K )

Figure 11. Proposed phase boundary for TmSe. The line (I) of second-order transitions meets the line (111) of first-order transitions at a tricritical point (TCP); the broken line is the transcritical extension of the TCP. Line I1 is a domain reorientation boundary (from Bjerrum-Msller et nl1977).

stoichiometric TmSe it appears that J n n N - 0.003 meV, Jnnn = + 0.13 meV. Batlogg (1980) shows that in non-stoichiometric Tm,Se both Jan and Jnnn become increasingly positive as x increases; as we will see in 52.6, increasing x means increased valence mixing. The effect of increased pressure (decreased valence mixing) is to drive Jnn more negative leaving J n n n unaltered. In TmSel-,Te, the degree of valence mixing increases with x, and for x=0.17 ( ~ ~ 2 . 5 ) the materials order ferromagnetically (Batlogg et a1 1979b). Guertin and Foner (1979) show that if the HPT phase diagram of TmSe is extrapolated to negative pressures (increased valence mixing) a ferromagnetic transition would be expected at about the same temperature and with about the right ordered moment as observed in TmSe0.83Teo.l.i. All this is in line with predictions of Varma (1977, 1979) based on the double exchange mechanism (53.6); double exchange favours ferromagnetic nearest-neighbour coupling, and will be maximum for a maximum degree (50-50) of valence mixing.

2.3. Thermodynamic and transport behaviour

2.3.1. Magnetic susceptibility.

2.3.1.1. Samarium and thulium chalcogenides. In B-SmS the susceptibility is of van Vleck form and the J = 0, J = 1 splitting A deduced from the formula x(0) = 8N,u$/A agrees well with values observed directly in neutron scattering; indeed, B-SmS behaves as an archetypal non-ordering singlet-triplet system (Shapiro et a1 1975). As can be seen from figures 6 and 12, the susceptibilities x of mixed-valent M-SmS and SmB6 are quite similar: for T > 100 K, x interpolates between the Sm2+ and Sm3+ values; below 100 K, x saturates to a finite value. In stoichiometric SmB6 a broad maximum near 50 K can be resolved. From the intermediate value of the high-temperature susceptibility x = EX(^+) + (1 - E) x (2+) a value of the valence ( ~ ~ 0 . 7 ) can be deduced, in good agreement with lattice constant interpolations. While this high-temperature susceptibility is that expected of an inhomogeneous mixture of Sm2+ and Sm3+ ions, the finite value at T=O implies that the ground state is homogeneous. The crossover between high- and low-temperature behaviour occurs on a temperature scale ( N 100 K) which correlates with characteristic ground-state energies ; in SmB6 the temperature of the susceptibility maximum is comparable to the activation energy deduced from the resistivity (see $2.3.4) while in SmS the relevant ground-state energy can be estimated from the linear coefficient of specific heat to be 2-300 K. These overall features (high-temperature moment intermediate between


I I - I / '

- /

( a )

/ MI- - I

=l

0 100 200

1 \

10 -

I I I I - T 0 100 200

T (K) T (K)

Figure 12. (a) Inverse susceptibility of (stoichiometric) TmSe; the broken lines represent the free-ion behaviour for Tm2+ and Tm3+ (after Batlogg et a1 1979a). (b) Susceptibility of Sml-zV~Bs where V represents a vacancy; the broken curves represent the Sm free-ion behaviour (after Kasuya et all977).

that of the two relevant valence states, finite T= 0 susceptibility and crossover temperature in agreement with ground-state characteristic energies) are expected to be quite general in mixed-valence materials.

The susceptibilities of thulium chalcogenides were studied by Bucher et al (1975). TmTe is divalent and the moment p = 4.96 ,UB observed in its high-temperature Curie law is close to the free-ion value shown in table 1. The high-temperature behaviour of trivalent TmS is Ceff/(T+ 6) with moment p=7.19 p~ again close to the free-ion value; the Weiss parameter 0 =25 K reflects the antiferromagnetic interactions ( TN = 9 K) but possibly also spin fluctuation effects. Stoichiometry effects ($2.6) are responsible for deviations from the free-ion moments. In stoichiometric mixed-valent TmSe the susceptibility is also of Curie-Weiss form (figure 12) with 8-29 K and peffz6.39 peg; assuming that the high-temperature moment is that of an uncorrelated mixture of Tm2+ and Tm3f ions, Ceff= ~C(3+)+(1- E ) C(2+), a valence z=2+ E of 2.5 is deduced for stoichiometric samples (Batlogg et al 1979a, Holtzberg et a1 1979). At temperatures below about 50 K the susceptibility increases more rapidly than the extrapolated Curie- Weiss behaviour (figure 12); this increase in the moment is associated with a decrease in the quasi-elastic neutron linewidth observed below this temperature (52.4.4). These variations in x are doubtless also affected by the variations in nearest-neighbour and next-nearest-neighbour exchange constants ($2.2) with T and P. Thus, while the high- temperature susceptibility shows the general mixed-valence property of intermediate moment, the low-temperature susceptibility of TmSe is complicated by interaction effects.

2.3.1.2. Cerium and ytterbium compounds. The susceptibilities of a large class of ytterbium compounds have been studied (Sales and Wohlleben 1975, Klaasse 1977, Klaasse et a1 1976, 1977, Oesterreicher and Parker 1977). These include compounds such as YbAg2 and YbIn2 where the ytterbium ion is divalent and non-magnetic and compounds such as YbNi and YbAuz where trivalent Curie law behaviour is observed. Perhaps the most clear-cut case of a strongly non-integral-valent material is YbAlz (2-2.5); the susceptibility is finite at T=O and increases to a broad maximum near Tm,,=900 K. YbAls and YbCuAl, which are essentially trivalent, also show maxima at


Tmax= 150 IS and 28 IS, respectively; at temperatures above the maxima, Curie-Weiss behaviour x( T ) = Cm/( T+ 6) is observed, where the effective moments are slightly reduced from the free-ion value and the Weiss parameters (0 > 0) are comparable to the Tnlax (table 4).

As can be seen from figure 13 and table 4, a similar bchaviour is observed in cerium compounds. Strongly mixed-valent CeN shows a broad maximum near 900 K; a-Ce might show similar behaviour were it not terminated by the a-y phase transition. The nearly trivalent materials CeSn3 and CeBel3 show broad maxima near 150 IS, with Curie-Weiss-like behaviour at high temperatures and effective moments close to the trivalent free-ion value. Trivalent CeA13 and CeCuzSiz show Curie-Weiss-like mono-

Table 4. Susceptibility data for cerium and ytterbium intermetallics. For the inverse ground-state susceptibility C/x(O), C is the free-ion Curie constant; x(0) is chosen as ~ ( T x ) for materials which antiferromagnetically order. TnlaH is the temperature of the susceptibility maximum and perf, 6 are obtained from high-temperature Curie-Weiss fits.

C/X(O) perf 8 Tma, Compound (K) (PB) (K) (K) References

P-Ce a-Ce y-Ce a-Ce0.73Th0.27 y-c~.73Th0.27 a-Ceo.7aLao.14Th0.1 a-Ca. 7oLao. zoTho .I CeAla CeAk CeBe13 Ceo.esLao.ssBe13 Ceo.39Lao.slBe13 Ceo.loLao.90Bela CeCuzSip CeIn3 CeN

CePds CeRuz CeSn3 CeSnz.7Ino.3 CeSnz.sIn0.6 CeSnz.sIn0.7 CeSndn CeSn1.sInl.s CeSnIn YbAlz YbA13

Y bCuAl YbCuzSiz YbNizGez YbInAuz

40 2.61 41 1550 - - - 2.42 44 1152 - - 2.59 150

-

- 272 - 134 - - 17 2.52 32 21 2.63 46

372 2.65 200 211 2.65 120 168 2.54 90 123 2.54 60 100 2.62 164 67 2.54 50

2956 - - 2305 - 538 ? ?

1179 - - 474 2.54 220 290 2.54 128 235 2.54 120 170 2.54 80 95 2.54 58

135 2.54 65 140 2.54 68

554 4 . 2 225

-

- 6287 -

- lo00 - - 2000 -

103 4.35 34 152 4.38 120 172 ? ? 515 ? ?

- - - - - - - -

0.7 140 80 50 20 - - 900 900 150

140 80 62 40 20 33 42

850 125

200

400

28 ? 35

100

-

Burgardt et al(1976) Koskenmaki and Gschneidner (1978) Stassis et al(1978) Lawrence and Parks (1976) Lawrence and Parks (1976) Parks et aI(1980) Parks et aZ(1980) Zoric et al(1977) Wallace (1973), Andres et al(1975) Kappler and Meyer (1979) Kappler and Meyer (1979) Kappler and Meyer (1979) Kappler and Meyer (1979) Sales and Viswanathan (1976) Lawrence (1979) Danan et aI(1969) Olcese (1979a) Gardner et a1 (1972) van Daal and Buschow (1 970) Buschow eta1 (1979), Lawrence (1979) Dijkman et a1(1980b), Lawrence (1979) Dijkman et a1(1980b), Lawrence (1979) Dijkman et aI(1980b), Lawrence (1979) Dijkman et al(1980b), Lawrence(l979) Dijkman et a1(1980b), Lawrence (1979) Dijkman et al(1980b), Lawrence (1979) Klaasse et a/ (1 976) Klaasse et a1(1977), Majewski et U/ (1 978) Klaasse et aZ(1977), Majewski et al (1978) Klaasse et al(1977), Majewski et al (1978) Mattens et al(1977) Sales and Wohlleben (1975) Oesterreicher and Parker (1977) Oesterreicher and Parker (1977)

Valerzce fluctuation phenomena 23

0 500 1000

T IKI T (lo

Figure 13. (a) Susceptibility for three concentrations x of CeIns &Snz (after Bkal-Monod and Lawrence 1980). (b) Susceptibility of CeN (after Danan et aZ1969).

tonic susceptibilities down to very low temperatures; CeAb shows a maximum at 0.7 K while CeCuzSiz becomes superconducting at 0.5 K. The other trivalent materials (P-Ce, CeA12, CeIn3) order antiferromagnetically at low temperatures ; the susceptibilities show Curie-Weiss behaviour above TN.

The existence of a maxima suggests that the magnetic trivalent configuration is thermally excited and the large positive Weiss parameters suggest the presence of intrinsic spin fluctuations. This has led to various phenomenologies which fit the data rather well : Sales and Wohlleben (1975) (see also Sales and Viswanathan 1976, Sales 1977) use the formula Np2 nf( T)/( T+ Tsn) where the valence fraction nf( T) is computed using a pseudo- Boltzmann statistic for an ionic two-level system (4f@, 4P-9 in which the trivalent magnetic level is higher than the non-magnetic level by an energy Eex (of order T,,,) and T,f, which is essentially equal to 8, is interpreted as the inverse of the valence fluctuation lifetime. While this leads to a finite susceptibility at T=O, the formula violates Nernst’s law (Klaasse et al 1977). An alternative phenomenology (de Chatel et a1 1977), utilising an exact thermodynamic treatment of an ionic two-level system with hybridisation between the ionic levels, yields similar susceptibilities. Additional support for these models comes from the fact that the observed susceptibility maxima tend to coincide with maxima in the thermal expansions (92.3.3); on the other hand, the models often suggest valences which are more non-integral and more temperature-dependent than seems justified by the lattice constant measurements, and indeed the lattice constants and Curie constants of CeSns, YbCuAl or YbAL are sufficiently close to the trivalent value as to rule out any explanation that requires either that the 4f level be above the Fermi level or that requires substantial valence change as a function of temperature.

On the other hand, the finite susceptibilities observed at T=O suggest Pauli paramagnetism, and the weight of the evidence suggests that the low-temperature susceptibility is that of a Fermi liquid ($2.5). Indeed the susceptibilities of CeSns, YbA13, etc, are very similar in all their features to that of the transition metal palladium, which also shows finite x(0) comparable in magnitude to that of a-cerium, a broad maximum near


100 K, and Curie-Weiss behaviour at higher temperature (Jamieson and Manchester 1972). Recently BCal-Monod and Lawrence (1980) have shown that the low-temperature susceptibility of CeSn3 obeys the law x( T ) = x(0) (1 + aT2); the quadratic temperature dependence, with its innuendo of Sommerfeld expansion, gives further support to a Fermi liquid description. We note here that in a Fermi liquid, the magnetic level can be below the Fermi level (rather than above, as suggested by the ionic phenomenologies) and still give rise to a maximum, due to thermal excitation of magnetic holes; this is certainly the case in palladium. In table 5 we list the Fermi temperatures which would be deduced from the T- 0 susceptibilities in a free-electron model ( TF = $C/x(O)); these range from several thousand to less than one hundred degrees Kelvin.

Table 5. Specific heat data for valence fluctuation materials. TF(C) represents an effective Fermi temperature deduced from the linear coefficient of specific heat y using T F = + R / ~ ~ ; this is compared to TF(x) obtained from the susceptibility (table 4) using TF=~C/~X(O). (These assume a free Fermi gas with c=A2kZ/2m*.) Tm,, represents the temperature of the specific heat maximum and AS(Tmax) is an estimate of the entropy under the peak.

~~~ ~~ ~~

Y TF(c)/TF(X) Tmax Compound (mJ mol-1 K-2) (K) (K) AS( Tmax) References

a-Ce

cc-Ce0.7Th0.3 CeAlz CeA13

CeBel3 CeCuzSiz CeIn3 CeN

CePds CeRuz CeSn3 YbAlz Y bAl3 YbCuAl YbCuzSiz SmB6 M-SmS M-Smo.7Yo.sS

12.8

19

1620

115 1000

8

37 14 53 17 45

255 135

145 50

-

-

6.8

320312325

21 5711728 40 +RIn6/2

25/31 25 Rln 612

3561558 41/150? 3.5 0 . 9 R I n 2

60 RIn 612 51 2313450-

4434 11201807 292712070 773171 1

241019430 9101841 1601155 28 R I n 8 3041228 - 40 1.75R 2821 > 0 + 7 I n 2 8201

Koskenmaki and Gschneidner (1978) Elenbaas et aZ(1979) Wallace (1973) Wallace (1973), Andres et a2 (1975) Cooper and Rizzuto (1971) Steglich et aZ(1979a, b) Wallace (1973) Danan et al(1969)

Wallace (1973) Joseph et a2 (1972) Cooper and Rizutto (1971) Havinga et aZ(1973) Havinga et aZ(1973) Elenbaas (1980) Sales and Viswanathan (1976) Kasuya et aZ(1979) Bader et al(1973) von Molnar and Holtzberg (1976), von Molnar et nl(1976)

While the ionic phenomenologies imply that two independent energy scales Eex and TSp are needed to describe the susceptibilities, recent work suggests that in the nearly trivalent compounds only a single energy scale TSp is required, and that the susceptibilities of these materials obey an approximate scaling law. Klaasse et a2 (1977) noted that in YbA12, YbAL and YbCunSi2 there is a correlation between Tmrtx and ~ ( 0 ) ; the smaller Tmax the higher ~ ( 0 ) . A similar effect was noticed in Ybl-,Sc,Al3 alloys (Majewski et al 1978). In CeIns-,Sn, it was shown (Lawrence 1979) that only a single function of a scaled variable T/Tsf is needed to describe the susceptibility for a broad range of x and T, i.e. the effective moment , u 2 = Tx/C satisfied p2(x; T ) =f(T/T&)) (figure 14). An alternative way of describing this scaling behaviour is as follows: for T B Tmax the susceptibility obeys x( T) = C/(T+ 0) where Cis the trivalent Curie constant;


T (K) Figure 14. Effective moment against temperature in CeIns--zSnz for four x (values given in the drawing).

The full curve represents a single curve shifted sideways to fit the data; for a semilogarithmic plot such a procedure demonstrates that the effective moment is a function of a scaled temperature T/Tsf. The values T* at which p2(T*)=f are: x=O, T * = 6 2 K ; x=1.85, T * = 6 0 K ; x=2.5, T*=115 K; x=3.0, T*=200 K. The breakdown of scaling in CeIn3 is discussed in the text (after Lawrence 1979).

the maximum occurs at T m , x Z 2 6 / 3 and the T=O susceptibility is x ( O ) ~ C / 2 8 , i.e. the ratio 6: Tmax: C/x(O) is fixed independent of x (roughly 1 : 3 : 2) and the high-temperature effective moment is the trivalent moment (see also Dijkman et a1 1980b). Strict definition of the spin fluctuation energy Tsi is not required to demonstrate this scaling behaviour, but if it is defined implicitly as the temperature T* at which the effective moment reaches half the free-ion value (p2( T*) 3) then it is found to coincide with the Weiss temperature 8(x). In CeIns-&~~ such scaling is observed for a range of x where T,r(x) varies from 50 to 200 K. From table 4 it can be seen that the same ratio 8 : Tmax : C/x(O) is observed approximately in CeBel3, Cel-,LasBe13, CeN, CePd3 and CeSns. Similarly the ratio C/x(O) : Tmax = 5 : 1 is observed for the whole class of ytterbium compounds; the high-temperature scaling is, however, less clear. It is also not clear to what extent the scaling holds for strong mixed valence. We will discuss this scaling behaviour further in $2.5.

For trivalent materials with sufficiently small spin fluctuation temperatures, crystal fields can markedly affect the susceptibility. These have been resolved, using neutron spectroscopy, e.g. in CeAls, as discussed in $2.4.4. However, as pointed out there, the crystal field excitations in cerium intermetallics are themselves anomalous in comparison to ordinary rare earths, being strongly broadened due to spin fluctuation effects; hence, ordinary theories of the effects of crystal fields on x cannot be very accurate. Neverthe- less in CeAh, and perhaps CeCuzSiz, a reduction in the moment is expected for T< A,f where Ace is the level splitting. This alters the relationship between low- and high- temperature behaviour observed for materials with larger TSe, e.g. the scaling law for the ratio Tma, : C/x(O) should no longer hold. In /We, CeAlz and CeIn3 the RKKY interactions which lead to antiferromagnetic order should also affect the susceptibility. While antiferromagnetic fluctuations should decrease the moment in the bulk, they simultaneously stabilise the local moment against spin fluctuations and the measured effective


moment p 2 = TxjC then becomes larger than expected on the basis of the scaling curve as observed (figure 14) in CeIn3 (Lawrence 1979).

2.3.2. Specific heat.

2.3.2.1. SmBg and SmS. The specific heats C of mixed-valence materials reflect the presence of spin entropy, the configurational entropy of mixing of the two types of ions and possibly the effects of lattice softening. Consider first the specific heat of SmBs (figure 15); when the lattice contribution is estimated and subtracted out, a large broad specific hcat maximum near 40 K is observed; the integrated entropy under the maximum (table 5) is equal to the value 1.75 which is the total configurational entropy for 0.6 Sm3f and 0.4Sm2+ ions plus the spin entropy in the Sm3+ J = 4 level (Kasuya et al 1979). The maximum occurs at a teinperature quite comparable to the energy gap inferred from the activated conductivity (52.3.4) and the linear coefficient of specific heat (table 5) is quite small; hence, the specific heat is consistent with the picture whereby SmB6 is a semiconductor with a small hybridisation gap.

T ( K )

Figure 1s. Specific heat of SmB6 (full curve). The broken line is obtained from the measured curve by subtracting the lattice contribution (dotted curve) where the latter is obtained by averaging the specific heats of La& and EuBe (after Kasuya et a[ 1979).

The specific heat of M-SmS under 15 kbar pressure was studied by Bader et aZ(l973); up to 20 K it is quite comparable to that of SmBs. A large linear coefficient y~ 145 mJ mol-1 K-2 was deduced in this work and historically this was taken as an indication of a large density of states at the Fermi level due to the narrow 4f band. Given the similarity of the specific heat to that of SmBe and our discussion (42.6) of the fact that M-SmS may also be a semiconductor whose small gap is masked by extrinsic effects, it is possible that this large y is not intrinsic. Despite this, the key feature in both M-SmS and SmBs is the rapid generation of entropy on a temperature scale comparable to the crossover in the susceptibility, suggesting considerable variation of N ( E ) near EF. In the M-Sml-,Y,S alloys, which are already metallic due to the Y-donated electrons, large y are also observed (table 5).

2.3.2.2. Ce and Yb cotnpounds. In the cerium and ytterbium intermetallics, large linear coefficients of specific heat are consistently observed. We list these in table 5, along with Fermi temperatures deduced under a free-electron assumption ; these are seen to correlate quite well with the Fermi temperatures deduced from the susceptibilities. We will discuss this Fermi liquid behaviour further in 42.5.


In addition to the large linear coefficient, the specific heat of YbCuAl (Elenbaas 1980) shows a maximum, as in SmBG. When the lattice contribution is estimated from that of its non-magnetic counterpart YCuAl the resulting magnetic specific heat shows a maximum at Tmax=28 K which coincides with the susceptibility maximum. The estimated entropy under this peak is close to the R In 8 expected for a J = 1 Yb ion. Hence, the total spin entropy is generated on a scale comparable to the spin fluctuation energy. Similar results are expected for CePd3, CeBel3, etc, but the subtraction cannot be reliably performed because in the latter materials TI,,, is comparable to the Debye temperature OD so that the lattice contribution is not small at T,,,,. A further complica- tion is that the lattices of these materials are often softer than those of their non-magnetic counterparts; indeed, this is true even for YbCuAl where OD = 21 1 K compared to 350 K for YCuAl.

In the more strongly trivalent compounds antiferromagnetic transitions and/or crystal fields affect the specific heat. For P-Ce, CeAlz and CeInn large specific heat anomalies occur at TN with integrated entropies not very different from R In 2 , suggesting that ordering occurs primarily in a ground-state doublet (see 92.2). At higher temperatures the entropy of the higher lying multiplets must be generated. The resulting specific heats are not simple Schottky curves, however, since the crystal field levels are strongly broadened by spin fluctuation effects ($2.4.4). What is observed in CeAlz and CeIns are broad maxima centred near 40 K and 60 K, respectively, containing most of the remaining spin entropy R In 6 - R In 2.

Amazingly large y coefficients are observed in CeCuzSin and CeAl3 (table 5) . In CeCuzSiz the linear variation of C occurs below 1 K, and a maximum in C occurs at 3.5 K, with integrated entropy close to R in 2. In CeAl3 a broad maximum occurs near 25 K with entropy on the order of R (In 6 -In 2) ; below 0.3 K the specific heat is linear. (In the region 2-8 K the intrinsic specific heat is obscured due to the presence of CeA12 and CesAlll impurity phases.) In both cases the J = 2 level is probably split into three doublets ; the very-low-temperature behaviour involving the ground-state doublet, and a maximum occurring at higher temperatures due to the remaining four levels. We note here that the temperature scale for entropy generation in the ground-state doublet (1-2 K) is comparable to the quasi-elastic linewidths observed in neutron scattering at very low temperature (52.4.4).

2.3.3. Lattice constants, thermal expansion and compressibility. Lattice constant interpolation is one of the most frequently used methods to estimate the valence of non- integral-valence materials. The technique assumes the validity of Vegard’s law, i.e. linearity between lattice constant and valence, as well as knowledge of the lattice constants ao(n) and ao(n - 1) of compounds where the rare earth is in the relevant configurations 4fn and 4P-1. The latter are readily obtainable for SmS, SmB6 and TmSe by extrapolation from the lattice constants of existing di- and trivalent compounds, as can be seen in figure 2 for M-SmS. The valences so deduced are listed in table 6.

In attempting to make valence estimates for cerium-based materials an important difficulty is encountered : the hypothetical tetravalent lattice constant is unknown. (In addition, the curve of a0 against atomic number is often highly non-linear for the light rare earths, making the trivalent lattice constant hard to determine, e.g. see figure 16(0).) For cerium metal Gschneidner and Smoluchowski (1963) estimated the valence using observed metallic radii; the hypothetical tetravalent radius was estimated by extrapolation from hafnium metal (figure 2). Other authors (Olcese 1979b, Sereni et al 1979) attempt to correlate valence with ionic radii, which is plausible for ionic materials such


Table 6. Valence z ( T ) deduced from lattice constant and thermal expansion data as described in the text and compressibilities K.

K Compound z(0) ~(300) z(8W) kbar) References

a-Ce

y-Ce

CeAlz

CeA13 CeBe13

CeCuzSiz CeIn3 CeN CePd3

CeRuz CeSn3

YbAlz YbAla

Y bCuAl YbCuzSiz B-SmS

M-SmS SmBa

TmSe

3.0

3 .O 3.25

3.0 3.6 3.5

3.3

2.4 3 -0-

3.0

2.70

3.67

3 -06

3.0

3.0 3.2

3.0 3.0 3.6 3.4

3.6 3.15

2.4 3 .O-

3.0 3.0 2.0

2.75 2.75

2.75

2-8

5-10

0.8

2

3.5 2.2 1 .o

2.3

2.75

2-6

0.91

4

Koskenmaki and Gschneidner (1978), Voronov et a1 (1979) Koskenmaki and Gschneidner (1978), Voronov et aZ(1979) Wallace (1973), Croft and Jayaraman (1979) Andres eta1(1978),TakkeetaZ(1980) Krill et al(1980b), Borsa and Olcese (1 973) Rieger and Parthe (1969) Harris and Raynor (1965) Olcese (1979a, b), Danan et al(1969) Takke et a1(1980), Gardner etaZ(1972), Harris eta1(1972), Krill et aZ(1980a) Smith and Harris (1967) Beille et al(1977), Harris and Raynor (1965), Dijkman etal(1980a) Iandelli and Palenzona (1972) Havinga et al(1973), Iandelli and Palenzona (1972) Klaasse (1977), Mattens et a1(1980a, b) Sales and Viswanathan (1976) Smirnov and Oskotskii (1978), Jayaraman et al(1975b) Smirnov and Oskotskii (1978) Niihara (1971), Kasaya eta1(1980), Geballe et al(1970) Batlogg et a1(1979a, b)

as CeN but less so for metals. Similar difficulties arise for Yb compounds, where the hypothetical divalent lattice constant ao(2 +) is often unknown. To circumvent this Klaasse (1977) proposed utilisation of the lattice constant of the isostructural Pr compounds to estimate ao(2+), since the ionic radius of Yb(2+) and Pr(3 +) are comparable. This procedure has been criticised by Buschow (1979) who points out that there are known materials where it fails rather markedly. In addition to these difficulties, we note that the linear relationship between radius and valence is largely untested on a microscopic level. Presumably a probe such as photoemission should be used, but as pointed out in $2.4.1 this technique at present is not highly reliable for determining valences. It is thus clear that large uncertainties in the valences deduced from lattice constants are to be expected.

Given this situation we have proceeded as follows in arriving at the estimates of valence given in table 6 : we have used the observed ao(2+) - ao(3 +) for divalent Eu in


Atomic number 58 60 62 6L 66

Figure 16. (U ) Lattice constants for RES^ compounds (data from Harris and Raynor 1965). The broken curve is for hypothetical tetravalent compounds. (b) Temperature dependence of the lattice constant for CeSm (data from Umlauf et ul 1980). The broken curves are the hypothetical curves for trivalent and tetravalent CeSn3, assuming identical thermal expansion as in LaSns. (c) Thermal expansion coefficient a= (l/uo)duojdT for CeSns. The dotted curve is the thermal expansion for LaSna and the broken curve is the difference between the two (after Umlauf et ull980).

EuSns as an estimate of Uo(3+)-Uo(4+) for CeSn3 (figure 16); the lattice constant anomaly in CeBel3 is quite comparable to those of CeSn3, so we have assigned a similar valence (2- 3.1-3.2). Similarly we have utilised the observed a0(2+)-a0(3 +) in EuAh to estimate the valence for YbA12. For CeN we have estimated Uo(3+)-uo(4+) by comparison to ao(2 +) - Uo(3 +) in the isostructural RE chalcogenides. We have dubbed CeRus as 3.6 valent because its metallic radius is quite comparable to that of a-cerium. We note that our estimates deviate substantially from those of other authors, particularly in the cases of CeN, CeRu2 and YbAIz. However, our assignments have the merit that they are consistent with susceptibility data ($2.3.1). Compounds with small spin fluctuation temperatures Tsr are for all intents and purposes trivalent ; this includes CeA12, CeAL, CeCu2Siz and CeIns. Materials such as y-Ce, CeSn3, CeBel3, all of which have Tsf- 100-200 K, have valences on the order 3.1-3.2. The strongly mixed-valence compounds (2-3.4-3.6) are then, in our analysis, wCe, CeN and YbAl2; these all have Tsf- 1000 K. An important exception appears to be CePd3 which has Tar,- 100-200 K but which appears to be strongly mixed valent. CeRuz has a low-temperature susceptibility quite comparable to that of a-Ce and we speculate that it, too, is strongly mixed valent rather than tetravalent as suggested by van Daal and Buschow (1970).

The thermal expansions of non-integral-valence materials are often anomalous, e.g. as can be seen in figure 16(c) the thermal expansion coefficient a(T) of CeSn3 is substantially larger than that of LaSn3 and shows a broad maximum at a temperature 130 K comparable to that of the susceptibility maximum (table 4). For CeInSn2 where the susceptibility maximum occurs at a much smaller temperature, the peak in the thermal expansion also occurs at much smaller temperature (Maury et a1 1979, 1980). Similar


expansion anomalies occur in CeBe13, CePd3, CeN and YbAl2 and they again appear to occur on the same temperature scale as the susceptibility maximum. Recently Mattens et aZ (1980b) have observed a minute negative thermal expansion below 50 K (again comparable T,f) in YbCuAl and it is possible that a similar effect occurs in YbAl3 (Iandelli and Palenzona 1972). A shallow minimum below 150 K in the lattice parameter of SmB6 has also been reported (Geballe et a1 1970).

An obvious interpretation of these anomalies is that the valence changes with temperature. In table G we have indicated the magnitude of the anomaly by estimating (by interpolation) the valence at T=O and/or T=800 K, which can be compared to z(300 K). It can be seen that the magnitude of the valence change so deduced is not large (Az-0.1 in CePd3, CeSns, CeBel3 and CeN); it is negligibly small in YbCuAl (YbA12 seems to be an exception). Kappler et aZ(1980) mention an alternative explanation for the thermal expansion anomaly in CeBel3 : the Debye temperature (OD - 360 IC) is very small compared to that of other RE B13 materials (00-700-900 K). Similarly, for YbCuAl On = 21 1 K while for YCuAl BO = 350 K (Elenbaas 1980). Thus the anomalous expansion appears related to a softening of the lattice and whether this involves a straightforward valence change is an open question. A particularly fascinating case is that of trivalent CeAl3; this shows a negative thermal expansion peak near 0.5 K (Ribault et a1 1979) which again is on a scale comparable to the spin fluctuation temperature for the ground- state doublet; the linear thermal expansion observed at the lowest temperatures was attributed to Fermi liquid behaviour.

Softening of the lattice can be observed directly in the compressibilities reported in table 6 . The compressibility of Tm,Se is maximum for stoichiometric samples (x= 1); it is four times larger than expected on the basis of extrapolation from other chalcogenides, and three times larger than the compressibility of trivalent Tmo.sSe (Batlogg et al 1979a). We note that the compressibilities of CeSn3 and CeN are about twice that of their isostructural rare-earth counterparts, while CePd3 and CeA12 have normal compressibilities. Thus there appears to be no general rule for the degree of lattice softening in the mixed-valent state. This conclusion is reinforced by examination of the compressibilities of cerium and SmS. Of course, in these inaterials the compressibility must diverge at the critical point, and will show precursive softening at the first-order transitions. Apart from such effects, it has been shown that in cerium metal, the compressibility is largest in the y phase, increasing to large values as the pressure increases to Po-$ kbar; for larger P in the a phase the compressibility falls off markedly, i.e. strongly mixed-valent a-Ce appears to possess a much harder lattice than nearly integral- valent y-cerium (Voronov et a2 1979). A similar effect appears to hold in SmS (Jayaraman et al 1975a); the strongest softening occurs for P<Pc in B-SmS, and the lattice stiffens considerably in the mixed-valent state. On the other hand, in Sml-2Y,S it is the mixed- valence alloys which exhibit anomalously soft lattices (Penney et al 1975); as pointed out in 82.1.2 this softening may reflect the approach to the metastable limit of the M+B transition.

2.3.4. Transport properties. The transport behaviour (resistivity p , Hall coefficient RH, magnetoresistance Ap(H) and thermopower S) falls into two categories. Most ambivalent cerium- and ytterbium-based compounds are metallic, while SmBs, TmSe and possibly also M-SmS exhibit activated conductivities in the ground state. This presumably reflects the difference that in the former category the conductivity arises from pre- existing background spd electrons, while in the latter category the conduction electrons are only made available by depopulation of the 4fn level.

Valence Jluctuation phenomena 31

2.3.4.1. Cerium and ytterbium compounds. The resistivities of several cerium compounds are shown in figure 17. (An older review is that of van Daal and Buschow (1970).). In one subset (‘s-shaped resistivities’) p( T ) increases monotonically with temperature but tends to saturate and show negative curvature above an inflection temperature Tin!. Included in this set are both materials which are trivalent (YbCuAl, Tinf=30 K (Mattens 1980)), nearly trivalent (YbAls, Tint= 100 K (Havinga et a1 1973); CeSn3, Tinf= 100 K (Stalinski et a1 1973); CeBe13, Tin!= 150 IC (Krill et a1 1980b)), strongly ,mixed-valent (CeN, Tinf = 600 K (Cornut 1976)) and possibly tetravalent (CeRug, Tinf = 150 K (van Daal and Buschow 1970)). A second subset (‘Kondoesque resistivities’) includes materials exhibiting regions of negative a p / a T suggestive of In T behaviour as in

T ( K )

Figure 17. Resistivities of several cerium-based compounds, adapted from the following sources: (a) CeSns (Stalinski et al 1973), (6) CeAlz (van Daal and Buschow 1970), (c) CePds (Scoboria et a1 1979), (d) CeIns (Elenbaas ef al 1980), (e) S-Ce (Burgardt et a1 1976), (f) CeCuzSiz (Franz et aZl978).

the dilute Kondo problem. Since the resistivity must vanish at T= 0 in a periodic metal, resistance maxima at temperature Tmax are observed and in most such materials resistance minima at Tmin are also present. These materials are all trivalent (e.g. CeAl2 and CeIn3) with the exception of CePds (figure 17).

The magnitude of these resistivities at high temperatures or at Tmax is often large ( N 100 pC2 cm), and this is usually taken to represent spin disorder scattering associated with a large (negative) Kondoesque exchange constant ysn. A related explanation is that the electrons scatter incoherently from uncorrelated virtual bound states, with a maximum possible resistivity given by the unitarity limit. Andres et a1 (1975) show for CeA13 that the maximum observed resistivity ( p ( Tmax) = 140 pi2 cm) is quite close to the calculated unitarity limit (188 pi2 cm); a similar result holds for CeCu2Si2 (Franz et a1 1978).


The temperature scale of these effects (viz. Tini or Tmax) is often quite comparable to the spin fluctuation temperatures Tsr deduced from the magnetic susceptibility. This can be seen by comparing the inflection temperatures given above to the temperatures given in tables 4 and 5 (a notable exception is CeRu2). This supports the interpretation in terms of spin scattering mechanisms.

The Kondo scattering mechanism invoked for CePds, CeAls, etc, is expected to be valid only at high temperatures (T> Tsf). Below Tmax correlations set in amongst the 4f spins and the resistivity falls to zero (Haldane 1977a). For materials with non-magnetic ground states the behaviour is expected to be that of a Fermi liquid (Varma 1977), a typical feature being a quadratic temperature dependence p = AT2. This is observed in CePds (A = 0.1 $2 cm K-2; Scoboria et al1979), possibly in CeSn3 (A =0.001 pQ cm K-2; Stalinski et al 1973); and in CeAls ( A = 35 pQ cm K-2; Andres et al 1975). By analogy to paramagnon theories (e.g. Jullien et aZl974) it is expected that A varies inversely with some power of Tsp; this would explain why no T2 contribution is resolvable in materials with large TSp such as a-cerium (see Koskenmaki and Gschneidner 1978) or YbAl2 (Havinga et al 1973). The truly enormous T2 coefficient in CeA13 suggests a Tsp-2 K, consistent with neutron scattering (52.4.4).

In other materials the correlations lead to magnetic order, as in CeA12, CeIn3 or P-Ce; for these the resistivity plummets at the NCel temperature as the spin scattering freezes out. In /3-cerium this decrease begins somewhat above the ordering temperature, suggesting the importance of short-range order (Burgardt et aZ 1976).

Crystal fields also affect the behaviour. The resistivity is expected to drop when the temperature becomes smaller than the crystal field splitting as the 4f degrees of freedom freeze out (Cornut and Coqblin 1972). The resistivity maximum at 40 K observed in CeAl3 (van Daal and Buschow 1970) may arise this way, since the crystal field teinpera- tures are comparable to Tmax while the spin fluctuation linewidth of the lowest lying doublet is much smaller (52.4.4). The two maxima observed in CeCuzSi2 (figure 17) might be explained as follows : the high-temperature maximum represents crystal field freeze-out, while the low-temperature maximum corresponds to the spin fluctuation temperature of the lowest multiplet. (Two maxima are also observed in Ce3Al (see van Daal and Buschow 1970).) The knee observed near 80 K in the resistivity of CeAl2 is also thought to be associated with crystal fields (Croft and Levine 1980).

In addition to strong spin scattering, anomalous phonon scattering due to the large electron-phonon coupling may also be important. This is analogous to the situation in AI 5 transition-metal superconductors where s-shaped resistivities similar to those discussed here are observed and which are often attributed to electron-phonon coupling (see Fisk and Webb 1976). Since the resistivities of such lanthanide superconductors as LaAlz (van Daal and Buschow 1970) are large and s-shaped, it is possible that the electron-phonon scattering already dominates the resistivity in the non-magnetic counterparts of the cerium compounds. The resistivity of CeRuz ,which may also arise this way, raises the additional question concerning the role of the transition metal in the anomalous resistivities of CeRu2, CePds, etc. Finally we note that the various contributions to p( T) are not simply additive when the total resistivity is a substantial fraction of the maximum possible value which occurs when the mean free path becomes of the order of a lattice spacing (Fisk and Webb 1976). This breakdown of Matthiesen’s rule is often ignored in analysing resistivities, leading to an overemphasis on Kondoism.

The thermopower of metals is highly sensitive to rapid energy dependences of the scattering cross section in the vicinity of the Fermi surface; hence the anomalous thermopowers observed in ambivalent cerium and ytterbium compounds reflect the presence


Table 7. Thermopowers. The temperature of the thermopower maximum Tmax is given together with the magnitude of the maximum.

Tmax S(Tmsx) Compound (K) (uV K-l) References

CeAh CeBela CeCuzSiz

CeIn3 CePd3 CeSns Y bAl3 YbCuzSiz

50 120 20

170 75

150 200 300 50

40 60

- 33 + 23

40 125 30

-90 40

van Aken et al(1974) Cooper and Rizzuto (1971) Franz et a2 (1978)

Gambino et al(1973) Gambino etaZ(1973) Cooper and Rizzuto (1971) van Daal(l974) Sales and Viswanathan (1976)

of the 4f resonance at the Fermi level. Typically the cerium-based compounds exhibit a positive thermopower maximum centred at a temperature comparable to the spin fluctuation temperature and of large magnitude. Examples are given in table 7. Crystal field and interaction effects also occur, e.g. in CeAl3 the temperature of the thermopower maximum corresponds to the crystal field splitting. For YbAl2 and YbAl3 the thermopower has negative sign. In CeCuzSiz the thermopower shows positive and negative maxima at temperatures comparable to those of the resistivity maxima.

2.3.4.2. Sm&, SmS and TmSe. The resistivity and Hall coefficient of stoichiometric SmBa (figure 18) provides clear evidence that the material is a small-gap semiconductor (Allen et al 1979, Allen and Martin 1979). The gap, deduced from the activated conductivity, is of the order of 3 meV, while the large negative Hall coefficient suggests a carrier density of less than 5 x 1017 cm-3 at 4.2 IS. At room temperature the material behaves as a poor metal with - 1022 cm-3 activated carriers. This behaviour is believed to represent the existence of a hybridisation gap arising from the 4f/5d level crossing.

In high-pressure M-SmS for samples closest to stoichiometry combined measurement of p and RH suggests that the carrier density increases from 1020 cm-3 to 1022 cm-3 but the Hall mobility decreases by a factor of twenty as the pressure is raised past PO at ambient temperature (Bzhalava et a1 1976). As the temperature is lowered at 10 kbar

T ( K 1

Figure 18. Resistivity and Hall coefficient of SmBe. The open circles correspond to positive (and the closed circles to negative) values of the Hall coefficient (after Allen et aZ 1979).

3


negative ap/aTis observed and a large value of resistivity (~(4.2)- 1.3 mS2 cm) is observed at low temperatures. The interpretation is ambiguous, due to the extrinsic effects discussed in $2.6; either M-SmS is a small-gap semiconductor as is SmB6, with SmS+ impurities forming a donor band and hence reducing the T=O resistivity; or it is a metal, and the Sm3+ impurities act as Kondo scattering centres, increasing the T = 0 resistivity.

The thulium chalcogenides provide an isostructural series where the valence and concomitantly the transport behaviour can be altered over a wide range. TmTe is a divalent semiconductor, with an activation energy of the order of 0.3 eV; TmS is trivalent and metallic with a resistivity (figure 19) reminiscent of the Kondoesque cerium- based metals discussed above (Bucher et al 1975). Stoichiometric TmSe is mixed-valent and its transport behaviour is quite unusual (figure 19). The resistivity increases from its room-temperature value to a value at 4.2 K which Flouquet er a f (1980) show to be

1 M 100 T I K I

Figure 19. Resistivity of stoichiometric TmSe with pressure and magnetic fields: 1 , H=O, P=6 kbar (Flouquet et al1980); 2, H=O, P=O (Haen e? all979); 3, H=3 kG, P=O and 4, H=6 kG, P=O (Andres et a1 1978). Curve 5 is the resistivity of TmS (Andres et all978).

quite close to the unitarity limit; between 4-40 K the resistivity depends linearly on In T (Batlogg et a1 1979a). Taken together with the observed magnitude of p, which is that of a poor metal, this suggests that a Kondo scattering mechanism is responsible for the negative ap/aT. The Hall coefficient, however, is large and negative at 4.2 K, suggesting fewer than 1018 carriers per cm3, a fact which is difficult to reconcile with a metallic model. Indeed, examination of figure 19 suggests that the large low-temperature resistivity is tied up with the antiferromagnetic order occurring near 3 K: there is an enormous negative magnetoresistivity associated with the metamagnetic transition to the high-field aligned state (figure 11); in addition, the Hall coefficient decreases three orders of magnitude in a 50 kG field (Haen et all977). Recent work (Haen et aZl979) suggests that the residual T=O resistivity diverges as stoichiometry is approached in Tml-,Se (see §2.6), and under application of 6 kbar pressure the residual resistivity increases by an order of magnitude. This provides strong evidence that in the ground state TmSe is an insulator. While a gap at the Fermi surface can be opened up simply by the new (antiferromagnetic)


periodicity, it is quite reasonable to suppose that it reflects an important feature of the mixed-valent ground state; Haen et a1 (1979) suggest it is a Kondo lattice effect.

In doped SmS the solute strongly affects transport. In B-Sml-2Y,S the Hall carrier density arises entirely from the yttrium atoms (Penney and Holtzberg 1975), suggesting that the 5d electrons at Sms+ sites remain localised. For ~ 2 x 0 the resistivity is larger in the mixed-valent state than in the (re-entrant) divalent ground state; this reflects a strong decrease of the scattering time which has also been observed in optical conductivity measurements ($2.4.3). This is consistent with the above-mentioned decrease of mobility in M-SmS at 8 kbar; the effect arises from strong scattering of sd electrons from the 4f resonance at the Fermi level.

2.4. Spectroscopic behaviour

2.4.1. Photoemission. Photoemission experiments, both x-ray (XPS) and ultraviolet (UPS), have been of central importance for understanding mixed-valence materials. An excellent review of the early work is given by Campagna et a1 (1976).

In rare-earth photoemission, when the photon hv ejects an electron from the 4fn shell it leaves behind a 4fn-1 configuration, hence the kinetic energy distribution curve (EDC) of the emitted electron measures the spectra of the final-state hole. The final state 4fn-1 has characteristic LSJ multiplet splittings which serve as a fingerprint, and these are accurately resolved and calculable in rare-earth photoemission ; by identification of the final-state hole the initial state can be inferred. Furthermore, the energy of the final- state hole includes both intra-atomic and extra-atomic (screening by the conduction electrons) relaxation effects, hence the position of the final-hole multiplet closest to the Fermi energy represents the binding energy of the 4fn configuration (i.e. eF-E(ffi)= E(f@-1 +e) - E(P) = - A- where e is an electron at the Fermi level). There is an inverse spectroscopy called bremsstrahlung isochromat spectroscopy (BIS) where an electron of known kinetic energy is captured in an empty state above the Fermi level, emitting a photon whose energy is measured. For rare earths this process corresponds to 4fi2-+4fN.+1 and hence allows measurement of the spectra of unoccupied final states above the Fermi level ; the corresponding excitation energy is E(F+1) - EF = E(fn+l+ h) - E(P) = A+ where h is a hole at EF.

Recently Lang et aZ(l979) measured A+ and A- for all the rare-earth metals in a combined apparatus for XPS and BIS. The results (figure 20) are in excellent agreement with calculations of A* using the renormalised atom scheme (Herbst et aZ 1972, 1976, 1978). This treatment avoids the formidable calculation of the additional energy of the fn(5d6s)z metal when in the presence of a single core hole and extra conduction electron by finding the Hartree-Fock energy per cell of a fn-l (5d6s)Z+1 metal; it thus assumes a fully screened final-state hole, and it further assumes that the main correlation effects in the metal are the same as in the free atom, i.e. entirely local. The excellent agreement of this theory with experiment, as seen in figure 20, attests to the importance of screening in the photoemission excitation. Furthermore these data show that the difference A+-A-, which is the Coulomb correlation energy (Hubbard U ) , is fairly constant (about 6 eV) across the rare-earth row.

For mixed-valence materials, such as SmBe, the XPS spectra show two sets of final- state multiplets, corresponding to 4f6-+4f5+e and 4fb4f4+e transitions; this is shown in figure 4. The multiplet intensities shown as bars are calculated using the ‘coefficients of fractional parentage’ and provide unambiguous identification of the final-state holes ; the theoretical curve is generated by convoluting these bars with instrumental resolution

36 J M Lawrence, P S Riseborough and R D Parks .

i n Pr Pm Eu Tb Ho Tm tu Ce Nd Sm Gd O y Er Yb 58 60 62 64 66 68 70

Atomic number

Figure 20. Comparisons of theoretical (open circles) and experimental (closed circles) positions of 4f levels with respect to the Fermi level EF. Levels below EF (designated A-) were obtained by XPS; levels above EF (designated A+) were obtained by BIS (after Lang et aI1979).

and with Doniach-Sunjic asymmetric lineshapes with lifetime broadening. Several key features of figure 4 are noteworthy. First, the existence of two final-state spectra implies that both 4f6 and 4f5 configurations are present in the initial state. Next, the lowest binding energy transition in the 4 fL4f5+e manifold is at the Fermi level; this is the condition for homogeneous mixed valence : 4f6+4f5 transitions cost zero energy. Third, the separation between the lowest members of the two multiplets again corresponds to the Coulomb correlation energy U and is seen to be 7 eV. Fourth, the ratio of the areas under the two multiplets gives a measure of the average valence which is in decent agreement with estimates from Mossbauer or susceptibility measurements. Finally the lowest binding energy member of the Sm2+ multiplet shows a linewidth equal to instrumental resolution, implying that the natural linewidth due to valence fluctuations is less than 0.08 eV. (The lifetime for the deeper transitions is a result of decay of the final-state hole through Auger processes where valence electrons fall into the hole and simultaneously a second valence electron is photoemitted.) These key features are present in the spectra of other mixed-valence materials such as Tml-,Se (Wertheim et aZ 1980), EuRhz (Nowik et a1 1977), EuCuSi2 or YbAls (Buschow et a1 1977a). (Note, however, that the lattice constant anomaly shown in table 6 for YbAls suggests that it should be trivalent rather than mixed-valent as indicated by XPS.)

In divalent samarium chalcogenides XPS spectra show only the Sm2+ excitation, about 1 eV below the Fermi level (Campagna et a1 1976). This splitting is much larger than that deduced from optical absorption (52.4.3). Wertheim et aZ(1978a) suggest that the final state in low-energy absorption is a bound exciton, at much smaller binding energy than the hole-plus-free-electron seen in XPS; similarly Guntherodt (1976) comments that ‘the difference in 4f binding energy presumably originates from 5d final states in optical absorption, but plane-wave-like states in photoemission’. Wertheim et aZ(1978a) studied the alloys Sm1-,RE2S (RE=Y, Gd, Thy Ca) for small x (divalent Sm). For Gd and Y doping A- rapidly approaches zero with increasing x but for Th doping A- remains nearly 1 eV up to the critical concentration for lattice collapse. That the instability is reached well before the f level reaches the Fermi level suggests that the binding energy shift is not the cause of the phase transition. In the Iv phase of Sm1-&E,S (RE=Y, Th,


Gd) or SmS1-,Ass (Campagna et a1 1976) both Sm2+ and Sm3+ multiplets are resolved with the f6 -f f5 excitation at the Fermi level; when Smo.sY0.2S is cooled through the low- temperature re-entrant transition the f 5 -+ f4 multiplet is no longer observed-this being consistent with the thermal expansion which shows the material becoming divalent at low temperatures.

In addition to these valence band spectra, two sets of lines are observed to appear in core level spectra of IV materials. This is demonstrated for CeN in figure 21 ; the 4d10 --f 4d9+e spectra can be understood as a superposition of a broadened version of the 4fl spectra as represented by trivalent CeSb and the 4fo spectra as represented by LaSb; the latter is easily identified by its characteristic spin-orbit splitting 6 (Baer et al 1978). Similar two-configuration 3d and 4d spectra are observed in SmB6 (Chazalviel et a1 1976), a-cerium (Baer and Zurcher 1977), CeBels (Krill et al 1980b) and CePd3 (Gupta et al 1980, Krill et a1 1980a). Herbst and Wilkins (1979) show that the 10 eV binding energy

LaSb q 100 d 6 - 120 110

Energy below cF

Figure 21. XPS spectra of the 4d core levels of LaSb, CeSb and CeN; the spectra of mixed-valent CeN consists of a broadened version of the trivalent CeSb structure plus a spin-orbit split (8) tetravalent component (after Baer et al1978). Note the shift of scale for LaSb.

difference between the 4fn-14d10(5d6s)z -+ 4P-l4d9(5d6s)z+ e and the 4P4d10(5d6s)z-1-+ 4fit4d9(5d6s)z-1+ e transitions reflects the increased screening of the nucleus provided by the additional 4f electron. The area ratios of the components in SmB6 are in fair agreement with other estimates of the valence, but in a-cerium, CePds and CeN, area ratios suggest valences (- 3.1) much smaller than those given by other means. The temperature dependence of the valence of CeBe13 deduced from these spectra are in reasonable agreement with bulk studies, but in CePd3 the core spectra suggest that the 4f0 component disappears at low temperatures while the thermal expansion suggests the 4f0 component should increase.

The core level spectra also exhibit a ‘shakedown’ structure corresponding to 3d10 4fn(5d6s)z -+ 3d9 4fn+1(5d6s)z--l+e transition. This appears as a shoulder on the low binding energy side of the 3d10 -+ 3d9+e transition and implies that the final state captures an electron from the valence band in order to fully screen the 3d hole, the captured electron going into a 4f orbital. It is thus a many-body effect. Crecelius et al (1978) demonstrated the existence of such lines in the light rare-earth metals; more


recently it has been observed in the intermetallics CePds, CeCu2Si2 and CeAl2 (Lasser et aZl980). These studies also show the absence of mixed-valent splitting of the 3d core levels in trivalent CeCuzSi2 and CeA12.

A recent advance in our understanding of mixed-valence photoemission comes from the recognition that surface spectra can be markedly different from those of the bulk. The escape depth of a photoelectron with kinetic energy of a kilovolt is only about 15 A; recent work has utilised the variation of the take-off angle (electron angle with respect to the surface normal) to enhance surface sensitivity. A key finding is that the valence at the surface can be very different from that in the bulk. In samarium metal (Wertheim and Campagna 1977, Wertheim and Crecelius 1978) and YbAu2 (Wertheim et a2 1978b), both of which are trivalent in the bulk, large divalent components are present in surface photoemission. This has been confirmed by photoemission using the softer ( N 100 eV) radiation available from synchrotron sources ; for these energies the escape depth is only about 4t%, hence only the surface layers are probed. These latter studies (Gudat et a1 1978, Allen et aZ1978a, 1980) lead to another interesting finding, namely that the binding energy of the 4f6 -+ 4f5+e transition is greater than in the bulk. This decrease of relaxation energy of the order of 1 eV might arise due to decreased screening by conduction electrons near the surface or surface lattice relaxation effects. It has the effect of pulling the 4f6 level from above the Fermi energy to below in Sm metal; and since the 4f6 -+ 4f5 + e transition is pulled below the Fermi energy, both SmB6 and Sm show inhomogeneous mixed valence at the surface. An important observation (Wertheim 1979) is that the older photoemission results need to be reinterpreted in the light of these surface studies, e.g. area ratios need to be corrected for surface effects, and when this is done for Tml-,Se better agreement with other estimates is obtained (Wertheim et a1 1980).

An interesting experimental feature of the synchrotron studies is that they take advantage of a 4d104P -+ 4d9 4P+1 absorption resonance near 100-200 eV which decays into (and hence resonantly enhances) the 4d104fn-1 +e channels. This (Fano) resonance has been exploited to address the question of the position of the 4f level in y-cerium. Older XPS studies of y-cerium (Baer and Busch 1973) show that 4f emission is weak and strongly obscured by 5d valence band photoemission. A tentative 4f excitation energy A-% 1.8 eV was assigned. By utilising the resonance to enhance the 4f transition, Johansson et a1 (1978) show that the 4f level indeed lies 1.8 eV below the Fermi energy; this is confirmed by UPS studies which use oxidation to suppress the valence band photoemission (Platau and Karlsson 1978). However, in cognisance of the above-mentioned larger binding energies seen at the surface (i.e. in UPS and XPS with soft synchrotron x-rays), relative to the bulk (hard x-rays), the actual bulk binding energy in y-cerium might be considerably smaller. We will discuss this large value of A- further in $2.5.5.

It is worth stressing some of the problems associated with this technique. As we have seen, valences deduced from XPS are often strongly at odds with those obtained by bulk means, and the temperature dependence of the valence fraction deduced by core XPS for CePd3 is even backwards; there are also discrepancies between binding energies deduced by XPS and those observed in optical studies. In part this may reflect the extreme difficulty of preparing unstrained, clean, stoichiometric surfaces in rare-earth compounds ; in part it may reflect the high surface sensitivity of XPS. Indeed, it is plausible that the binding energies and valences within 30 A of the surface (as observed by photoemission) differ substantially from those in the bulk; one way this might come about is through the enormous sensitivity to strain which many of the compounds exhibit, making surface relaxation effects substantial. Given this caveat it is clear that photoemission experiments hold enormous promise for our future understanding of mixed valence.


2.4.2. X-ray absorption andfluorescence. X-ray absorption can be thought of as a ‘fast’ spectroscopy, in that both 4fn and 4P-1 components are resolved in the experiment. Experiments to date have studied the LIII absorption edge which represents excitation of a 2p core level into the conduction band. There is a chemical shift of the edge to about 7-8 eV higher binding energy when the 4f is trivalent as opposed to divalent; since this shift is both larger than instrumental or valence fluctuation linewidths, in a mixed- valence material two peaks are observed and the valence can be estimated from the relative intensities. Valences so estimated are: SmB6, z= 2.65 (Vainshtein et nZ 1965); Smo.75Y0.25Sy z=2.44 (Martin et al 1980); TmSe, z=2.58 (Launois et a2 1980). Similar effects have been reported for EuCuzSiz (z z 2.6 at 300 K, z z 2.8 at 77 K) and YbCuzSiz ( 2 ~ 2 . 9 ) by Hatwar et al(1980).

In the studies of Sm0.75Y0.25S and TmSe the EXAFS associated with the LIII edge was also measured; this experiment probes the instantaneous distribution of nearest-neighbour distances which affects the fine structure which results from back-scattering of the out- going electron. In the mixed-valence state the measured EXAFS is the sum of two contributions from the two valences with different thresholds, but the nearest-neighbour distance is the same for both components : all Tm (Sm) atoms have equivalent environments. This is interpreted as meaning that there are no polaronic effects whereby atoms in the two different valence states distort the local environment differently : the lattice finds a single nearest-neighbour distance corresponding to the hybrid wavefunction.

Chemical shifts of the x-ray fluorescent lines arising from inner shell (K, and Kg) transitions have also proven useful to measure valences as a function of temperature, pressure and dopant. The situation has been reviewed by Sumbaev (1978). The shifts are small ( G 1 eV), indeed smaller than the natural linewidths (- 30 eV) ; Sumbaev shows that in this situation the line in the mixed-valent state is a single line, with natural linewidth and with chemical shift intermediate between those of the two relevant valence states, allowing an estimate of the valence. Valences so obtained are 3.05 for y-cerium, 3.35 for a-cerium, 3.6 for SmS at 9 kbar or for Smo.sGd0.2S. The intermetallics CeA12, CeNiz, CeFe2 and CeCo2 were also studied; while the former appears trivalent, the latter three show increasing valence mixing. This technique, of course, requires knowing chemical shifts in the relevant P and P-1 standards and is subject to the same sort of errors as found in Mossbauer or lattice constant determinations of valence, namely that varying chemical shifts are expected within a class of univalent standards.

2.4.3. Optical response. The optical properties (i.e. dielectric function as measured by optical absorption and/or reflectivity) have been measured in the range 0.1-10 eV for several of the mixed-valence rare-earth chalcogenides. There are two recent review articles concerning this work (Guntherodt 1976, Suryanarayanan 1978).

In semiconducting SmS, SmSe and SmTe optical transitions from the 4f6 initial state into LSJ multiplets of the 4f55d final state are observed, giving a similar spectral fingerprint as seen in XPS; at higher energies a valence band/conduction band absorption edge is observed (Batlogg et al1976a). A key parameter for mixed valence is the lowest energy absorption edge Eg which should be related to the distance of the 4f6 level below the conduction band. Values for Eg taken from Batlogg et a1 (1976a) are 0.2 eV for B-SmS, 0.62 eV for SmTe and 0.46 eV for SmSe; these compare with the values 0.9 eV (SmS), 1.25 eV (SmTe) and 1.1 eV (SmSe) observed in XPS (Campagna et aZl976). As mentioned in $2.4.1 this may be due to a difference in final states in the two processes, the final state in optical absorption being excitonic. On the other hand, Batlogg et a1 conclude, on the basis of photoconductivity measurements, that the 4f6 -f 4f5(6H)5d tzg transition goes


into a conduction band of primarily 5d character, as opposed to going into an exciton which then decays into 6s band states; a similar observation is made by Suryanarayanan (1978). Since the energy level structure of SmS plays a key role in any model of the valence transitions, the discrepancy between XPS and optical spectra needs clarification.

The shift of the 4f6+ 4f55d transition with pressure has been measured in B-SmS (Batlogg et al 1976a). The tzg level of the final 5d state shifts downward at a rate 10 meV kbar-1 while the eg levels shift upwards in such a way as to keep the centre of gravity constant. Hence it is the changing crystal field splitting which is responsible for the reduction of the energy gap with applied pressure.

When SmS is squeezed to 8 kbar pressure, the reflectivity changes from that of a semiconductor to that of a metal (Kirk et a2 1972) and the reflectivity is virtually identical to that of GdS. Similar results are obtained when the metallic state is induced by mechanically polishing the surface of a single crystal (figure 22(a)) or by doping in Sml-$Y,S (Guntherodt 1976). A key result in Sm0.72Y0.28 S is that the plasma edge shifts by 0.3 eV to lower energies as the temperature is lowered through the low-temperature phase

Photon energy lev1 Photon energy (eV1

Figure 22. (a) Reflectivity of a metallic M-SmS film (Batlogg et al 1976a). (b) Dielectric constant for SmBa obtained by Kramers-Kronig analysis of reflectivity data (Allen et al1978b).

transition; this gives quite definitive evidence for the role of the valence change in determining the plasma edge in the alloys. Using the formula TDC N O D C / W ~ the DC scattering time was calculated in the alloys; it decreases by a factor of five on entering the mixed- valence state. This decrease was discussed in 42.3.4; the 4f resonance in the IV state increases the conduction electron scattering rate.

Similar results hold in thulium chalcogenides (Batlogg et a1 1976b). In TmTe a semiconducting absorption spectra (divalent) is observed which is quite sensitive to surface oxidation (see also Suryanarayanan 1978). TmS shows a metallic reflectivity with plasma edge and interband transitions similar to figure 22(a). Tml-,Se is metallic, with a reflectivity edge which shifts about 0.5 eV as x varies from thulium deficiency to excess (Batlogg et a1 1977); such studies as a function of stoichiometry give convincing evidence that the intrinsic TmSe behaviour is metallic.

Recent reflectivity experiments in SmBe have proven the existence of significant structure in the dielectric constant in the far infrared (Allen et a1 1978b). The reflectivity above 1 eV is quite similar to that of metallic SmS, with a steep plasma edge at 1.5 eV and interband transitions at higher energies. If the behaviour were that of a simple metal the real part of the dielectric constant EI which crosses zero near 1.5 eV would


continue to go negative at lower photon energies; instead, it crosses zero a second time near 0.2 eV and is positive down to 0.02 eV. This is shown in figure 22(b). This behaviour is consistent with the transport behaviour which shows SmBe to be semiconducting; a small gap at the Fermi level would not affect the high-energy dielectric response, so that the plasma edge at 1.5 eV has no simple relation to the carrier density involved in DC transport, whereas the large positive e1 observed below 0.5 eV can be interpreted as that of an insulator. In addition, preliminary infrared absorption experiments show that below 0.1 eV the optical conductivity falls from its high, metallic value to a much smaller DC value, as in a small-gap material (Batlogg et al1980). Similar low-frequency behaviour has been observed in metallic SmS (obtained by polishing single crystals); there is a zero crossing near 0.2 eV, and EI is positive down to 0.03 eV (Allen et al1978b).

2.4.4. Magnetic neutron scattering.

2.4.4.1. Form factor measurements. The neutron form factor f(Q) is a measure of the wavevector dependence-and hence the spatial distribution M(r)-of the magnetisation. Form factors are measured in one of two ways: in an inelastic scattering experiment, the energy transfer is set at a value larger than the phonon cutoff and the scattering studied as a function of wavevector Q; or, in an elastic scattering experiment using polarised neutrons, Q is set to different Bragg peaks and the polarisation ratio is measured and deconvoluted to give f ( Q ) . The latter technique is capable of generating highly detailed maps of the magnetisation density; as an example we mention the recent form factor studies of CeAiz (Barbara et a1 1979) where the 4f crystal field ground-state wavefunctions are accurately determined as well as a background, spatially diffuse (5d6s) component of M(r) .

In most of the non-integral valence systems the measured form factor appears to be that of only one of the relevant integral valence states. For example, when f(Q) is measured using polarised neutrons in B-SmS it agreed within the error with the calculated Si++ form factor, as desired ; however, when the intermediate-valence state was obtained by alloying (Sm0.7sYo.24S) or under application of pressure, the measured form factor did not change from that observed in the divalent phase, even though large, easily resolved differences were expected (Moon et a1 1978). That the Sm3f configuration makes no contribution to the form factor is a very surprising, anomalous result. Similarly, the measured form factor in a (stoichiometric and thus mixed-valent) single crystal of TmSe agrees with the calculated Tm3+ form factor even though more than 20% of the sample is believed to be in the Tm2f configuration (Shapiro et al 1978). In CeSn3 crystals the form factor agrees with the predicted Ce3f value for temperatures greater than 40 K, but at 4.2 K there are significant discrepancies at small Q(- I A-1) which also indicate considerable anisotropy in the spatial distribution of the induced magnetisation (figure 23). On one sample the form factor could be fit by assuming that 38 % of the magnetisation at 4.2 K is of Ce 5d electronic character of eB symmetry. This is an extremely nice result in that it exhibits the presence of valence mixing in a rather direct fashion (Stassis et al1979b, c); however, the result is not entirely reproducible (see $2.6). The measured form factor of single crystals of y-cerium at 300 K also agrees with the calculated 3f values (Stassis et al 1978); as does the form factor of polycrystalline y-Ce0.74Tho.zs (Shapiro et a1 1977). In a-Ce0,74Tho,~6 the measured form factor is enhanced over the predicted Ce3f value for Q s 2 A-1, which is interpreted as arising from polarisation of the conduction electrons (Moon and Koehler 1979).

2.4.4.2. Inelastic magnetic scattering. The frequency-dependent neutron cross section

I I I I I I I I I 1 1 1 1

8- 3

4.2 K - e

- e

40 U - -

reveals the underlying dynamics of the fluctuating 4f moments as well as yielding crystal field splittings where relevant. The inelastic scattering cross section for magnetic scattering of neutrons obeys the proportionality

where x is the susceptibility. In the mixed-valence materials y-cerium (Stassis et a1 1979a), y- and a-Ce0.74Th0.26 (Shapiro el al 1977), CeSn3, CePd3 and CeBen (Loewen- haupt and Holland-Moritz 1979) and YbCuAl (Mattens et al 1980a) it is found that the spin dynamics are relaxational (i.e. decay exponentially with time) so that

This is referred to as quasi-elastic scattering (figure 24(a)). The Q dependence of the cross section, for Q > 1 A-1, agrees with /f(Q)12 so that

x(Q, w ) shows no Q dependence. This implies that the spin fluctuations are local in character, as opposed to highly itinerant. However, this statement is subject to a caveat: first of all: a rigorous sum rule for the Anderson lattice model shows that the second moment of Imx(w) is independent of Q so that any Q dependence appears only in higher-order moments, which may be unresolvable given the experimental statistics (Goncalves da Silva 1979). Similarly, an RPA treatment of the Anderson lattice (Rise- borough and Mills 1980) suggests that Q dependence should only be visible for much smaller Q values ( 5 0.1 A-1) than have been studied experimentally to date.

Local, relaxational spin dynamics is also a characteristic of the dilute Kondo problem, where the linewidth r is observed to be comparable to the Kondo temperature TK. We here take the observed linewidths in the ambivalent materials to be a direct measure of the spin fluctuation energy T8f. The values of the characteristic energy r / 2 are given in


100

- w -L-

'E 50

? E! f

-5 SC 50

c - -

0 20 40 60

t c

3 e 40 .. e

0 10 B E ( m e W

20 10 0

Figure 24.

A E (meV1 A E i m e V i

(a) Imaginary part of the susceptibility of Ce0.74Th0.26 as determined by inelastic neutron scattering. The full curves are best fits to the quasi-elastic form for the cross section. The two temperatures are above and below the critical temperature (TO- 146 K) for the a-y transition (after Shapiro et a2 1977). (6) Neutron scattering intensity against energy transfer in CeAlz at low temperatures (after Shapiro et a1 1979). (c) Neutron scattering spectra of Ce0.76La0.14Th0.1 at 110 K. The solid lines represent the best fit to a form for the cross section given by the sum of a central quasi-elastic component and a broadened crystal field level (broken curves) (after Parks et a1 1980).

table 8; for the materials quoted they are typically 10-20 meV with the exception of a-cerium where r /2 -70 meV. It can be seen by comparison with table 4 that the characteristic energy F/2 corresponds closely to the inverse DC susceptibility C/2x(O). This is an example of a sum rule which arises from the conservation of total spin; for a localised paramagnet of Nf spins

For relaxational spin dynamics at IOW temperatures the w integral is readily evaluated and gives x(0) N Nf(gpB)'J(J+ l)/r N nf C/r, where nf is the valence fraction. However, the applicability of such a paramagnetic sum rule to a homogeneous mixed-valent system stands in need of theoretical justification. Despite this fact, a similar evaluation near the phase transition in Cel-,Ths showed the integrated scattering roughly tracking the valence as estimated from the lattice constant (Shapiro et a1 1977).

The temperature dependence of the quasi-elastic linewidths has been discussed by Loewenhaupt and Holland-Moritz (1979). In CePda, CeSns, YbCuzSiz, etc, the observed linewidths are about two orders of magnitude larger than those of similar rare-earth compounds with stable 4f shells; presumably these latter obey a Korringa law W/2= AkT where A is much smaller than unity. For valence fluctuation materials r/2 is finite as T -+ 0, and nearly temperature-independent (figure 25). This contrasts with the behaviour


b) - -

- - 4

- I

Table 8. Neutron scattering: quasi-elastic linewidths r j 2 and crystal field levels ACP.

12

- 0 - 4 Q

4 L

ACF (mev) References

y-Ce yCe0.74Tho.26

a-Ceo.7eLao.14Tho.l a-Ceo.7oLao.~0Th0.1 CeAlz CeA13

CeBe13

CeCuzSiz

CeIna

CePd3

CeSna

YbCuAl YbCuzSiz

a-Ceo.74Th0.26

17 20 72 11.3 6.0

0 . 5 a t T = 4 K 0.7 at T=2 K

24

2 a t T = 4 K 9 at T=300 K Broad maximum near 15 meV, 10 meV half-width 20

22

8.8 at T=4 K 6 a t T = 4 K

21 16 9, 16 5 , 8

?

?

Stassis eta1(1979a, b, c) Shapiro et al(1977) Shapiro etal(1977) Grier et aI(1980) Grier et af(1980) Lowenhaupt et al(1979) Alekseev et a1(1976), Murani et ai

Loewenhaupt and Holland-Moritz ( 1 979) Loewenhaupt and Holland-Moritz ( 1 979) Lawrence and Shapiro (1980), Gross et al(1980) Loewenhaupt and Holland-Moritz ( 1979) Loewenhaupt and Holland-Moritz (1 979) Mattens et al (1980a) Holland-Moritz et al(1978)

(1980)

of dilute Kondo systems for which the linewidths are also finite as T -+ 0 but increase rapidly with temperature above TK.

The characteristic energies r/2 observed in CePds, etc, are larger than typical 4f crystal field splittings ACE. and no crystal field excitations are resolved. By contrast in CeAl2 and CeA13, r/2< ACF and the inelastic scattering reveals a quasi-elastic component with F/25 1 meV and well-defined crystal field levels at higher energies (Alekseev et al 1976, Steglich et al 1979b, Murani et a2 1980). Although the crystal fields are well resolved, they possess anomalously large and strongly temperature-dependent linewidths, especially at high temperatures. When ACF N r/2 the crystal field excitations are no

I I I 100 200 300

T i K )

Figure 25. (a) Quasi-elastic neutron linewidths r / 2 against temperature for CeSn3, CePd3 and YbCuSiz. The full diagonal line represents r / 2 = kT. (b) Quasi-elastic linewidths (full circles) and position A of the inelastic line for TmSe (after Loewenhaupt and Holland-Moritz 1979).


longer well defined ; the cross section for such materials as YbCuzSiz (Holland-Moritz et a1 1978) or CeIn3 (Lawrence and Shapiro 1980, Gross et a1 1980) is a complicated combination of quasi-elastic scattering and strongly damped crystal field scattering. In (Cel-zLa&.gTho.l (Parks et a1 1980) r / 2 can be tuned to small values as x increases, and the crystal field levels which are unresolvable for small x emerge from the soup to give a striking double-humped appearance (figure 24(c)). In CeAlz the excited quartet shows additional splitting: two magnetic lines appear near 9 and 19 meV (figure 24(b)). This has been ascribed to a dynamic Jahn-Teller effect (Loewenhaupt et a1 1979).

In TmSe the lineshape is quasi-elastic above 100 K with constant linewidth r /2 - 7 meV. Below about 80 K an inelastic line appears near 6 meV, whose position increases strongly as the temperature is lowered (figure 25(b)) ; simultaneously the quasi-elastic linewidth decreases rapidly with temperature (Loewenhaupt and Holland-Moritz 1979). The origin of the inelastic linewidth is a mystery; if it represents a crystal field excitation, its temperature dependence is highly unusual. It correlates with deviations from the Curie-Weiss law observed in the susceptibility (52.3.1.1).

In B-SmS the J=O -+ J = 1 singlet-triplet excitation of SmZ+ gives strong inelastic scattering (Shapiro et a1 1975); the results for the singlet-triplet dispersion as a function of temperature are in excellent agreement with predictions of a mean field theory. Mook et a1 (1978b) measured the temperature dependence of the intensity of this excitation and also found good agreement with the theory. In S ~ O . ~ ~ Y O . Z ~ S , where the fraction of Smz+ increases as the temperature is lowered, the intensities appeared to decrease proportionally to the valence. The excitation seems to disappear completely in the high- pressure phase (McWhan et a1 1978) and there is no observable scattering from Sm3+ crystal field transitions.

2.4.5. Magnetic resonance. The principles involved in understanding NMR and EPR in valence fluctuation compounds are very similar to those needed in the study of dilute magnetic alloys; we refer the reader to Narath (1972) for a review.

In the simplest case the 4f electron gives rise to a transferred hyperfine field Hf at a neighbouring nucleus mediated through the 4f/conduction electron exchange PsfQ *Sf. This yields a Knight shift Kf = HfXe which adds to the usual temperature-independent shift KO to give a total Knight shift KT=Ko+K~ which is linear in the 4f susceptibility. This linearity is realised at high temperatures in most of the ICF materials studied. For example, in the series CeNMzSiz where NM = Cu, Ag, Au, the magnitude of Kf scales with x in such a way as to suggest H f is constant for the series (Sampathkumaran et al 1979a). The 4f hyperfine field is in the simple case proportional to KoBSf and hence when the linear relationship is observed f s f can be estimated; for CeAlz and CeAb Niculescu et a1 (1972) so estimate ysf to be -0.2 and -0.33 eV, respectively.

At lower temperatures this linear relation often breaks down for a variety of reasons. An explanation which fits the data in CeAlz where such a breakdown occurs below 100 K is that the hyperfine coupling of the r7 crystal field doublet is stronger than that of the FS quartet, which effectively makes the hyperfine field strongly temperature-dependent as the FS level is frozen out (MacLaughlin and Hewitt 1978, MacLaughlin et a1 1981). In YbAb breakdown occurs only at very low temperatures and is ascribed to an impurity effect (Buschow et a1 1979); this again can be understood in terms of different hyperfine couplings Hi and Hf for impurity and matrix atoms so that K~=Ko+Hfxi+Hixi. In CeSns the measured KT is proportional to the susceptibility above 40K, showing a well-defined broad maximum near 140K; below 40K, KT remains constant while x blows up (Malik et al 1975). Buschow et al (1979) suggest that this may represent a


similar impurity effect as in YbA13; an alternative explanation suggested by the neutron form factor measurements (82.4.4.1) is that a 5d contribution arises at low temperatures with a different hyperfine coupling to the nucleus. This would be a genuine mixed-valent effect. Similarly it has been speculated that the breakdown of the linear relationship observed in YbCuAl below the temperature of the susceptibility maximum Tmax may result from the onset of coherence in a hybridised 4f band and concomitant temperature- dependent hyperfine field (MacLaughlin et a1 1979).

Important information can also be extracted from the nuclear relaxation time l/T1, which measures fluctuations in the hyperfine field driven by the 4f spin fluctuations. It is hence proportional to H P ~ (&*Se) w L where the correlation function, evaluated at the Larmor frequency, is the same as that measured by neutrons. For relaxational spin dynamics with r, kTB WL this reduces to a Korringa law l/T1= H?XfT/r. Hence, if Hf is known from the Knight shift, F can be obtained. In CeBen, l/T1 is nearly linear with T up to 300 K and then decreases somewhat as the susceptibility decreases; the estimated I? =4.2 x IOl3 rad s-l-28 meV is quite comparable to the value 24 meV obtained from neutron scattering (table 8). For CeSns, TIT is nearly linear (Malik et al 1975) with a value quite close to that of metallic tin, suggesting that the conduction electrons dominate the relaxation ; in CeIns-,Sn, the 4f relaxation mechanism begins to dominate for x z 2 (Welsh and Darby 1972). Korringa behaviour is also observed for Gd EPR in CePd3 where the transferred mechanism is similar (Gambke et a1 1978). In YbCuAl the spin fluctuation energies obtained from l/T1 measurements are in excellent quantita- tive agreement with those obtained by neutron scattering; F increases with T above 100 K (which may represent Korringa relaxation of the 4f spins driven by conduction electron fluctuations) but between 100 K and Tmax=28 K I' decreases with T, a fact which is not at all understood (MacLaughlin et a1 1979). Finally, in the case of CeA12 there is a big discrepancy between NMR- and neutron-measured fluctuation rates (MacLaughlin 1980, MacLaughlin et a1 1981). This can in part be explained by the onset of antiferromagnetic correlations, which repress the independence of spin fluctuations on differing 4f sites, but the discrepancy also leaves open questions concerning the correct interpretation of NMR relaxation rates. In the ordered state of CeAl2 l/T1 measurements yield an exponentially activated contribution, probably due to magnons, while an additional contribution at low temperatures is believed to arise from a Kondo- esque mechanism in which 4f moments and relaxation rates vary from site to site in the modulated structure.

Quadrupole resonance (NQR) has also proven useful ; because it measures the electric- field gradient at the probing nucleus it is sensitive to the 4f charge configuration and hence can be used in a similar way as lattice constants, Mossbauer isomer shift or x-ray chemical shift to establish valences. This has been demonstrated for the REBs series and SmB6 by Aono and Kawai (1979), and for the RECuzSiZ series by Sampathkumaran et ul (1979b, c).

2.4.6. Mcssbauer efect. A highly pedagogical discussion of the application of the Mossbauer effect to mixed valence is given by Coey and Massenet (1977). Essentially, since the isomer shift measures the s-electron density at the probing rare-earth nucleus, the isomer shifts Sn and Sn-1 for the 4 P and 4P-1 configurations differ because the extra 4f electron increases the screening of the nucleus seen by the outer-shell s electrons. For a hybridised 4f wavefunction in the IV state, a shift intermediate between Sn and Sn-1 is observed, allowing an estimate of the valence when Sn and Sn-1 are known. This is demonstrated for SmBs in figure 3 ; valences for various materials deduced in this fashion

Valence Juctuation phenomena 47

Table 9. Mossbauer data. Valences as deduced from isomer shifts.

Compound z(0K) z(3ooK) References

M-SmS (10 kbar) M-Sml -zYzS SmB6 Y bAla Y bAlz EuRhz EuCuzSiz EuFe4Als EuIra-xPtx Eul-xLaxRhz EuNis -zCuz i

2.8-2.9 2.6-2.7

2.65 2.65 -3 -3 < 3 < 3 - 3 2.8(500K) 2.9 2.36 (800 K) 2.96 2.85 Inhomogeneous mixed valence/local environment effects

Coey et aZ(1976), Coey and Massenet (1977) Coey et aZ(1976), Coey and Massenet (1977) CohenetaZ(1970) Ross and Tronc (1978) Percheron-Guegan et aZ(1974) Nowik et a1(1977), Bauminger et aZ(1976) Bauminger et a1(1973), Nowik (1977) Felner and Nowik (1978,1979) Bauminger et aZ(1976) Bauminger et a2 (1 974) Bauminger et aZ(1978)

are shown in table 9. We note that large temperature dependences are observed in the europium compounds. This technique for estimating valences suffers from the same difficulty as lattice constant interpolation, namely that the hypothetical integral-valence shifts are uncertain. Even in the most favourable case of the Eu compounds SZ varies between - 8.7 and - 14.5 mm s-1 relative to EuzO3 while trivalent compounds have shifts S3 varying between +0.1 and +0.3 mm s-1 (Bauminger et a1 1978); thus it is difficult to pinpoint SZ and S3 precisely. In a study of Sm compounds Eibschutz et a1 (1972) showed that in the metallic state Sm3+ shifts S3 vary from 0.1 to 0.3 mm s-1, while trivalent ionic compounds have 5'3-0.0; in the ionic divalent salts -0.6,< SZ < -0.9 and SZ is proportional to the electronegativity difference.

Only a single line is observed in the isomer shift measurement because the 4f charge fluctuations are faster than the characteristic measuring time, which is proportional to 1/ISn- and is typically 10-9 s ; if the charge fluctuations were slower than this, two lines would be observed. This is the phenomenon of motional narrowing and implies that only a lower limit on the ICF rate can be so obtained. Several papers (Goncalves da Silva and Falicov 1976, Lopez and Balseiro 1978, Balseiro et a1 1979) show how such motional narrowing can occur in model systems where quantum-mechanical hybridisation (as opposed to classical stochastic fluctuations) occurs. In a Fermi liquid the linewidth would measure the low-frequency spectral density of the charge fluctuations resulting in a Korringa-like broadening proportional to ISn-Sn-11 TITFL where TFL is the bandwidth; but such broadening is too small with respect to the natural bandwidth to be observed (Hirst 1977).

In several alloys listed in table 9 two lines are observed in the Mossbauer spectra with shifts approximately equal to SZ and S3. This occurs, for example, when divalent EuPtz is alloyed with trivalent EuIrz; the interpretation is that the Eu valence is determined by local environment effects, viz. the number of Pt nearest neighbours. The lines often show considerable temperature dependence and there is evidence that some of the Eu atoms in the alloys possess genuine fractional valence.

Information can also be obtained from studies of the hyperfine field at the nuclei. In SmB6 no hyperfine splitting is observed at 1.2 K in fields of 12 kG (Cohen et a1 1970); this result can be interpreted in motional narrowing language as meaning that the Sm3+ relaxation time is short (< 10-11 s) compared to the Larmor frequency. Similar studies in YbAL show a similar result (Ross and Tronc 1978); the field-induced hyperfine splitting is far too small to be understood in terms of a local moment but can be understood as a Pauli paramagnetic Knight shift arising from a 4f band. In the ordered state of TmS

48 J M Lawrence, P S Riseborough, and R D Parks

and TmSe hyperfine splittings are resolved, with small splittings which are consistent with the small moments observed in neutron scattering (92.2) (Triplett et al 1977). As T approaches TN from below in these materials, thermally induced relaxation broadening is observed, implying that slowly relaxing higher lying magnetic levels are being excited (Dixon et a1 1980).

2.4.7. Phonon spectra. Since large electron-phonon coupling is expected in IV materials, the phonon spectra can be anomalous. The phonon modes which are most affected by the valence fluctuations are those which most directly affect the cell volume, hence the transverse modes are relatively unaffected. The crystal symmetry then determines which longitudinal mode sees the maximum coupling to the 4f electrons, e.g. in SmS, with rock- salt structure, the maximum coupling is expected for longitudinal phonons in the [I 11 ] direction : for such modes the sulphur atoms in (1 11) planes move into the gaps in the close-packed structure of neighbouring (1 11) sulphur planes and create the maximum compression on the 4f electrons (Grewe and Entel 1979). A second point is that for the systems Ce, SmS and their alloys the bulk modulus B = + ( C I I + C ~ ~ ) must vanish at, and be small in a substantial neighbourhood of, the critical points in the xPT phase diagram. No single phonon mode goes soft, but all branches whose sound velocities depend on C11 and Clz must soften, particularly near the zone centre. Since the ground- state properties and phase transition properties are strongly connected, it is a moot point whether the observed anomalous elastic properties represent critical behaviour or a property of the mixed-valence ground state. Nevertheless, studies in such systems as TmSe where there is no valence transition would be particularly welcome.

To understand the phonon dispersion in mixed-valent Sml-,Y,S we refer the reader to figure 26 where the phonon spectra for Sm0.75Y0.25S are compared to those of B-SmS and YS. Neutron scattering results in B-SmS (Birgeneau and Shapiro 1977) demonstrate that the LA, TA and TO branches are essentially identical to those of EuS. The LO branch is, however, somewhat softer than in EuS; this was first suggested by lattice dynamic shell models (Guntherodt et a1 1977) on the basis of the larger compressibility of B-SmS and has been confirmed by defect-induced Raman scattering (Guntherodt et al 1978) which sees primarily zone boundary LO phonons of AI, symmetry. This is a fully symmetric breathing mode in the [I 11 1 direction; the fact that it is not observed in GdS,

r5oo1 I r x r x r x

15551 Figure 26. Schematic phonon dispersion curves for (a) SmS, (b) Sm0.7jY0.25S and (c) YS (after Bilz er al

1979).


together with its [ l l l ] symmetry and concomitant large coupling to the 4f radius, suggests that it is the large polarisability due to virtual 4f6+4f5 5d transitions which is responsible for the AI, contribution and soft optic phonons in B-SmS. The case of YS has been discussed by Gupta (1979); the soft LO mode is not understood. Finally we turn to mixed-valent Smo.'i5Y0.2jS where neutron scattering reveals several distinctive features (Mook et al 1978a). There is little effect observed on the transverse phonons, which interpolate smoothly between B-SmS and YS, whereas the longitudinal phonons are soft, particularly in the [l 1 13 direction where the LA branch is actually lower than the TA. The anomalous [ill] softening is consistent with the symmetry arguments given above. The data are also quantitatively consistent with the elastic constants observed in ultrasonics measurements (Penney et a1 1975); however, the neutron results show that the softening is not restricted to the zone centre. Indeed, the LO phonons are sufficiently soft that the LO phonon lies below the TO at the L point; this can be viewed as an extreme version of the LO(L) softening seen in B-SmS.

The [l 1 1 ] LA branch softens further as the temperature is lowered, showing maximum softening at 200 K corresponding to the continuous low-temperature re-entrant transition (Mook et al 1978a); it hardens sufficiently at yet lower temperatures that the LA branch no longer falls below the TA. Studies of the Debye-Waller factor of Sm0.7Yo.aS (Dernier et a1 1976) show behaviour consistent with this. In that experiment the mean square displacements (U& of the sulphur atoms arise mainly from the optic phonons where the lighter sulphur atoms dominate. The observed US^) are anomalously large, due to the softening of the LO phonons, and increase further as the continuous re-entrant transition temperature is approached.

The linewidths of the phonons in Smo.7jYo.2jS were studied by Mook and Nicklow (1979). The transverse phonons are narrow and show no unusual features. On the other hand, unusually large phonon widths are associated with the soft longitudinal branches ; these widths are maximum at 200 K. The maximum phonon widths are associated with the greatest softening, as a function of both T and Q.

Similar results occur in other mixed-valence materials. Recently neutron scattering results have been obtained for single crystals of y-cerium (Stassis et al 1979a). Com- parison with the spectra for FCC thorium shows that the spectra of y-Ce are softer than expected, particularly for the branches whose slopes in the elastic limit involve the elastic constants C11 and C12. They correlate this with precursive effects of the valence transition. On the other hand, the phonon spectra of CeSns at ambient conditions (Pintschovius et al 1980) are notable for the absence of phonon anomalies, either in the form of softening or linewidth effects. Despite this, ultrasonic measurements (Edelstein et al 1980), which are capable of high precision for measuring relative changes as a function of temperature, show a pronounced, albeit small ( N 2 %), minimum in the bulk modulus near 135 K, which correlates well with the temperature of the susceptihility maximum and the thermal expansion reported above for CeSns. Ultrasonic measurements in CePd3 (Takke et all980) show that the bulk modulus is large and its temperature dependence is quite normal

In Tm,Se first-order defect Raman scattering (Treindl and Wachter 1979) shows that the zone boundary LO(L) phonon softens as x increases, linearly as the valence increases. The fully symmetric AIg component increases with x, reflecting as in SmS the contribution of the valence fluctuations to the polarisation process responsible for the scattering. In SmB6 where (by symmetry) first-order Raman scattering is allowed, only the T2, mode is anomalous and its frequency falls between that expected for Sm3+B6 and Sm2+B6 (Guntherodt et a1 1978). Such linearity of the mode frequency with thc valence suggests

4


that no additional softening, occurs; this is not surprising since the borons form an extremely hard cage.

2.5. Fermi liyuidology

The valence fluctuation compounds fall into two classes. In SmBs, TmSe and M-SmS there is evidence that the ground-state behaviour is semiconducting with a small gap; the evidence is strongest for SmBe, is confused by extrinsic effects in SmS (92.6) and for TmSe the gap is correlated with the antiferromagnetism. In most of the relevant Ce and Yb compounds the valence fluctuation ground states are metallic Fermi liquids. This was first stressed by Varma (1976). The primary evidence for this is the finite ground- state susceptibilities x(0) and large linear coefficients y of specific heat (tables 4 and 5). Further evidence comes from the quadratic temperature dependence observed in the low- temperature susceptibilities and resistivities of various materials, and from the large linear thermal expansion observed at very low temperatures in CeAls. In this subsection we examine the Fermi liquid phenomenology in greater detail.

2.5.1. Fermi liquid behaviour in 3He: a brief summary. For purposes of comparison we first summarise some key features of the behaviour of the archetypal Fermi liquid 3He. The degeneracy temperature (TF) is of the order of 5 K; the susceptibility is (Stoner) enhanced from the Fermi gas value xo=3C/2T~ to its measured value x(0)=3C/2Tsf where T8f=TF/SzOo.5 K. The effective moment p2=TxIC can be empirically fitted to 5 % or better over the whole temperature range by replacing TF by Tsf in the standard formula for p2(T/Tp) in the free Fermi gas (Goldstein 1964). This continues to hold even when Tsf is decreased by a factor of two by application of pressure (Ramm et a1 1970). This represents a particularly simple form of the spin fluctuation phenomenology : as far as the spins are concerned, Tsf plays the role of TF. A striking feature is the crossover from enhanced Pauli paramagnetism to free spin behaviour for T8f < TGTF; thus, the spins behave classically well below the degeneracy temperature. A second important aspect, implicit in the phenomenology, is that the effective moment scales with T = T/TBf(P). This scaling is most apparent at low temperatures where the susceptibility varies as x(O)[l- P(x(0) T/C)2] where x(0) varies as 1/TSf(P) but p N 0.5 is nearly independent of pressure.

The quasi-particle interactions which distinguish the liquid from the gas are con- ventionally modelled by a Hubbard-like repulsion Unt(x)nl(x), leading to a Stoner enhancement S= (1 - UN( E F ) ) - ~ . The crossover at Tsf = ( 1 - ( EF)) TF and the strict Curie behaviour observed for Tsf < T < TF have never been well established theoretically, but at low temperatures paramagnon theory (Btal-Monod et a1 1968) correctly predicts the quadratic temperature dependence, and recent work employing functional integration shows that the first few terms in a Landau-Wilson expansion exhibit T/Tsr scaling (Mishra and Ramakrishnan 1978). Such theories are valid close to the T=O critical point Uc for the magnetic instability of the Fermi liquid, for it is just in this regime that the slow spin fluctuations dominate the magnetic behaviour. (This critical point is reached before the Stoner criterion is satisfied, i.e. UcN(&g) < 1, due to renormalisation.) It is reasonable to speculate that T/Tsf scaling is valid to all orders for U s Uc. The T=O phase transition has been shown to be rigorously mean field (Btal-Monod and Maki 1975, Hertz 1976); this ‘quantum critical behaviour’ is valid for Tc(U)<Tsf, for slightly larger T,, nearly mean-field behaviour if expected, and for T,. 2 TJf ordinary Wilson exponents are expected. The (U, T ) phase diagram appears as in figure 27.

Valence fluctuation phenomena

TF Classical moments \

51

t h

\ \

Tsf \ \

Pauli paramagnetism

U, \ Mean-field critical behaviour

ucff I r F -- Figure 27. Schematic diagram showing the speculation concerning the Fermi liquid behaviour of valence

fluctuation metals. See text for explanation.

2.5.2. Spin fluctuation phenomenology in IV compounds. In ambivalent materials the degeneracy temperature TF is expected to be of the order of the hybridisation width A of the 4f band. The analogy to 3He suggests that the spin fluctuation temperature Tsf might be considerably smaller than A due to quasi-particle correlations induced by the strong intra-site Coulomb interaction. The actual 4f bandwidths have not been resolved directly by any experiments to date, e.g. the presently attainable resolution in XPS cannot resolve widths smaller than about 0.1 eV. Many authors estimate TF from x(0) and y using a free-electron assumption, as we have done in table 5 , but until it is shown that Tsp= TF it is clear that the temperatures so deduced should be interpreted as spin fluctuation temperatures. The characteristic spin fluctuation energy is directly measured as the quasi-elastic linewidth r / 2 in inelastic neutron scattering, and we have seen that ~ ( 0 ) and r/2 are related by a sum rule. Since in the nearly trivalent cerium-based compounds the Curie-Weiss temperatures 8, the temperature Tmax of the susceptibility maximum and the temperature T* where p2( T") = g are all related to C/x(O) by the scaling relation discussed in $2.3.1, it is clear that any of them may be chosen as order of magnitude estimates of Tsf.

For strong non-integral valence the susceptibility can be reduced by reduction of the 4f occupation number nf even when the Tsf as measured by neutron linewidths remains constant. For cerium and ytterbium compounds this suggests that the susceptibility should obey x N nfC/( T+ Tsi) at high temperature and x(0) -nrC/2Tsp in the ground state. This is at the basis of the phenomenology of Sales and Wohlleben (1975). It is consistent with the reduced Curie constants observed in Yb compounds (table 4), as well as with the neutron sum rule discussed in $2.4.4. Experiments near the critical point of the valence transition in Ceo.73Th0.27 also are consistent with this dual role of np and Tsf in reducing perf; the lattice constant and effective moment vary rapidly in a broad interval (k25 K) around T,, but the neutron linewidth remains constant except in a smaller interval ( & 5 K) around Tc where it varies from 20 meV in the y state to greater than 70 meV in the a state (Shapiro et a1 1977).

2.5.3. The (Ueff , T)phase diagram. The ICF materials may exhibit an analogous phase diagram to that of figure 27. Ueff is to be thought of as an effective quasi-particle inter-

4*


action in the narrow hybrid 4f band, not necessarily equal to the bare Hubbard Unf. In this analogy the T/Tsf scaling observed in the nearly trivalent cerium compounds reflects proximity to the T=O critical point (Uef f s Uc). Strongly mixed-valent materials such as a-cerium with larger Tsr fall closer to the Ueff =O axis and the scaling might break down ; this is unclear experimentally. Antiferromagnetism occurs in the ordered phase, and the reduced ordered moments observed in several ambivalent materials suggest that U 2 Uc and that the T=O transition is second order. The nearly mean-field critical exponents observed in CeIns reflect the ‘quantum critical behaviour’ expected when TN < Tsf ; Wilson exponents are observed in CeAlz because TN Tsn.

The low-temperature quadratic susceptibilities x( Tj = x(O)[l+ /3(x(O) T/C)2] recently reported for CeIn3-,Snz are also consistent with this analogy; it was also shown that the paramagnon theory can account for both the positive value of ,5 and its weak dependence on x through bandshape effects (BCal-Monod and Lawrence 1980). Recently it has been reported that CeCuzSiz is a superconductor below 0.5 K (Steglich et a1 1979aj. Since LaCuzSiz does not exhibit superfluidity this suggests the f electrons are responsible for the pairing. One speculation that has been put forward is that p-wave pairing occurs to circumvent the large spin fluctuations which should kill s-wave pairing; this strengthens the analogy to 3He. Alternatively it is possible that (e.g. due to strong electron-phonon coupling) Ueff is strongly reduced from Uff to negative values.

This whole analogy hinges on the generality of the depicted phase diagram, i.e. on the assumption that it holds regardless of the details of the Fermi liquid. It allows us to view much of the magnetic behaviour of ICF materials as arising from the proximity of a T=O fixed point.

2.5.4. Experiments pertaining to the microscopic character of the Fermi liquid. The spin fluctuations in ICF materials are highly localised; this is shown by the fact that in most cases the Q dependences in inelastic neutron scattering follow the 4f form factor (52.4.4). In Ybl-sY,CuA1 and Cel-,La,Sna it is found that the susceptibility per mole of ICF atoms remains unchanged (i.e. constant Tmax, x(0) and 0) down to small concentrations of Yb or Ce (Mattens et a1 1979, Dijkman et a1 1980b). That the susceptibilities in the dilute limit are virtually identical to those of the fully periodic system is strong evidence for the high degree of locality of the spin fluctuations.

Experiments in C e I n ~ - ~ s n ~ (Lawrence 1979) show that Tsr tracks the background conduction band density of states Nsd(&F) as inferred from experiments in LaIn3-,Sn,. While this might be taken as evidence for equality of charge and spin fluctuation rates (Tsf=Nsd(&F) V2= A) it is also consistent with a Stoner model (Tsf=(l -“U) T p ; T F ~ A, N f - l/A). In any case it demonstrates that Tsf arises from 4f coupling to the conduction electrons.

The Tsf appear to correlate with the valence: strongly mixed-valent a-Ce, CeN and YbAlz have large Tsf- 1000 K. Similarly in the a state of (Cel-,La,)o.gTho.l the small Tsf found for x>O.l correlate with valences of order 3.2 while the large Tsf found for smaller x correlate with valences of order 3.5 (Parks et a2 1980).

In cerium alloys the Tsi also correlate with phase (valence) transition temperatures. In (Cel-,La,)o.eTho.l it is found that C/x(O)=11.2 To (Parks et a2 1980) while in y-Cel-,Th, the Curie-Weiss 0 tracks the first-order y --f a transition temperature, which is roughly the metastable limit of the y state (Lawrence and Parks 1976).

Highly localised spin fluctuations arising from coupling to conduction electrons is observed in the dilute Kondo problem, which is known to have a Fermi liquid ground state (Nozibres 1974); here Tsr tracks Nsd(&F) through Tsfccexp (- ~/N(&F) $). As


mentioned in $2.4.4 the neutron lineshapes in dilute Kondo systems are also relaxational and local, but they appear to be more strongly temperature-dependent than their ICF counterparts. If the observed Fermi liquid behaviour respresent Kondoism, the weak ordered moments would arise from compensation by the conduction electrons, as in the Kondo lattice theory (43.7).

2.5.5. The 4f bands in the cerium compounds. The evidence concerning the character (i.e. position, width and occupation) of the 4f bands in cerium is somewhat confusing. The lattice constants and bulk magnetic properties suggest valences as in table 6, and the usual picture is that the bands are quite narrow (-0.01 eV). Other evidence seems to contradict this picture.

One such kind of contradictory evidence comes from cohesive energy arguments (Johansson 1974). Comparing cohesive energies of trivalent lanthanide metals (AH3 N

105 kcal mol-1) and of tetravalent lanthanide metals (AH4- 145 kcal mol-1) to the difference between the lower 4f15d16s2 and the higher 4f05d26s2 energies in free aLomic cerium (hH34atom- 105 kcal mol-l) it is seen that the difference between tri- and tetravalent configurations in the metal AH34metal= A&+ hH34atom- AH4 is of the order of 65 kcal mol-1. Other estimates using heats of formation of oxides and halides give 40 kcal mol-1 (Johansson 1978). With such a large value of the transformation energy it is difficult to understand how cerium could ever become tetravalent or even strongly fractional valent. More recently de Boer et a1 (1979) have utilised the scheme of Miedema (1976a) to compute heats of formation of intermetallics of cerium with transition and noble metals. This scheme quite accurately correlates the valence states with the heats of formation of ytterbium and europium compounds (Miedema 1976b), but for such cerium compounds as CeRu2 and CeRh3, where the cerium ions are ordinarily classified as tetravalent, the scheme indicates that a model in which the cerium 4f occupation number is zero cannot apply.

The angular correlations of the annihilation radiation observed in positron annihilation experiments in all three phases (y , a and a’) of cerium agree better with the assumption of trivalence than tetravalence (Gempel et a1 1972). The Compton profiles measured in the y and a states (Kornstiidt et a1 1980) similarly suggest one f electron in both phases (Felsteiner et a1 1979). The XPS measurements (42.4.1) in y-cerium place the 4f level 1.8 eV below the Fermi level (41 kcalmol-l) which, we note, agrees with cohesive energy estimates of AH34metal; the 4f level width suggested by this experiment is 1-2 eV. Recent valence band photoemission experiments in CeA12, CeIn3 and CePd3 also seem to indicate 4f bands well below EF and with widths of the order of 1-2 eV (J Allen, private communication). Core level spectra in a-cerium suggest rather small valence mixing ( z - 3.1, $2.4.1) while the x-ray fluorescence chemical shifts in a-cerium (42.4.2) suggest strong valence mixing.

Given this situation we feel that the question of the nature of the 4f bands in cerium compounds remains open.

2.6. Coininon extrinsic efects

In this subsection we review work concerning effects which, while not intrinsic to the pure materials, can strongly alter the measured properties. A widespread feature which is unique to ambivalent materials is that one and/or the other integral-valence state of the rare earth can be stabilised by any of several mechanisms : non-stoichiometry, neighbouring vacancies, impurities or lattice defects, neighbouring interstitials, cold-


work-induced defects and so on. This relates to the extreme sensitivity of the valence state to local charge distributions and strain fields. Ions so stabilised act as ‘impurities’, e.g. Sm3+ impurities in divalent or homogeneously mixed-valent SmS. These ‘impurities’ can then act as donors in a semiconducting matrix or as magnetic centres in a non- magnetic metallic matrix, strongly affecting the low-temperature susceptibility or transport measurements.

2.6.1. TmSe. The properties of TmS, TmSe and TmTe are highly sensitive to stoichiometry (Bucher et al 1975). The homogeneity range of Tm,Se is extensive at room temperature. The Se-rich phase boundary occurs near x= 0.87-0.90; there is, however, disagreement in the literature as to whether the Tm-rich phase terminates at x = 1.04 (Kaldis et al 1979a, b) or x=l.OO (Holtzberg et al 1979). Kaldis et al (1980) suggest that in Tm-rich samples the excess Tm fills the Se vacancies; furthermore, the high- temperature phase diagram is complicated and the stoichiometric composition may well be unstable. There is also a large (- 1 %) concentration of Schottky defects. The thermodynamic properties-in particular, the Tm valence-are strong functions of stoichiometry. This is shown both by variation of the lattice constant and of the Curie constant with x (Batlogg et al 1979a); samples with xzO.87 appear to consist entirely of Tm3+, while stoichiometric samples have a valence of the order of 2.75. The low- temperature type I antiferromagnetic ordering is only well defined for xk 1.0; Se-rich samples appear to exhibit only short-range order (see also Shapiro et all978). Anomalies in thermal expansion, magnetoresistance, etc, associated with the transition are thus well defined only close to stoichiometry. The transport properties are also affected; Se-rich samples behave as poor metals, but stoichiometric TmSe shows a negative ap/aT, between 40 and 4 K (Batlogg et al1979a); the variation with x of the resistivity in the ordered state suggests that a perfectly stoichiometric sample might become insulating at T=O K (Haen et a1 1979).

2.6.2. SmB6. In SmB6 it is known that vacancies form on samarium sites, but not boron sites (Niihara 1971); the compounds Sml-,V,B6 (where V represents a vacancy) are stable in the range 0 < x < 4. The T= 0 resistivity varies by nearly an order of magnitude with x , the stoichiometric compounds behaving more nearly as semiconductors ; and the susceptibility, which for x = O is finite and exhibits a weak maximum at higher temperature, shows a large low-temperature rise for x=O.3 as though the vacancies stabilise Sm3+ impurities (Kasuya et al 1977). I t appears that truly stoichiometric SmB6 is a small-gap semiconductor (Allen and Martin 1979).

2.6.3. SmS. The compound Sinl+,S forms in the homogeneity range O<x<O.17; the excess Sm atoms are believed to occupy interstitial positions (Sergeeva et al 1972, Zhuze et al 1973). For x < O the samples are two-phase mixtures of SmS and Sm&; the SmSSm4 precipitates have been observed even in nominally stoichiometric SmS single crystals by electron microscopy (Ryabov et al 1979) and are associated with dislocation networks. At high temperatures the samples decompose via the reaction 4SmS --f Sm& + Sm. The transport properties of semiconducting SmS are extremely sensitive to stoichiometry. The room-temperature carrier concentration and conductivity are smallest for x=O and increase rapidly with x ; and the thermal variation of the resistivity indicates that, while stoichiometric SmS is semiconducting, Sm excess as small as x = 1.02 is sufficient to render the materials semimetallic. Zhuze et a1 (see also Shadrichev et al 1976) suggest that the excess Sm atoms serve as trivalent donors and form an impurity


band. This leads to two activation energies (an extrinsic gap from the impurity band to the conduction band, and an intrinsic gap from the 4f6 level to the conduction band) affecting the temperature dependence of the resistivity and leading to conflicting estimates of the optical gap (Batlogg et a1 1976a). Walsh et al(l975) observed an EPR signal in B-SmS which they attributed to trivalent Sm3+ ions in sites of tetragonal symmetry (although the g factor was somewhat larger than expected). Furthermore, when defects are created via neutron irradiation, the conductivity increases markedly, and it has been argued that this is a result of local Sm2+ -+ Sm3+ conversion (Morillo et al 1979). One group (Chouteau et a2 1976) has studied the magnetisation of B-SmS crystals and Sml-,La,S for small x < X O ; they argue that 1-3 % Sm3+ is present in all samples and associate the negative a p / a T observed at low temperatures with Kondo scattering from the Sm3+ impurities.

In order to circumvent the need for high-pressure apparatus several groups have studied the properties of polished SmS; either the surfaces of single crystals or the full body of thin films can be converted to a golden metallic phase by polishing. The basic idea is that the pressure applied during polishing converts the sample to M-SmS which is then metastable at ambient pressure due to substrate effects. Electron microscopy (Ryabov et all979) shows that when a SmS single crystal is polished, the induced metallic phase has a large mosaic, i.e. is not an ideal single-crystal film, but a set of small crystal- lites disoriented relative to one another within a given range of angles. The metallic regions coexist on the surface with up to 15 % of the black phase (Batlogg et al1976a, b), and consequently two lattice constants are observed by x-ray diffraction. Smirnov and Oskotskii (1978) argue for the full identity of the M-SmS prepared via polishing and application of hydrostatic pressure by noting that in both cases the lattice constants are identical (5.70 A) and the absolute value and temperature dependence of the resistivities are similar. The optical reflectivity between 0.1 and 10 eV at room temperature is also similar in the two cases (Batlogg et al 1976a, b, Volkonskaya et al 1975), but at 4.2 K the polished films show no shift in reflectivity edge, while low-temperature measurements under hydrostatic pressure indicate a substantial shift (Guntherodt et a l 1977). This latter fact serves as a warning that the two cases are not completely identical. A further warning comes from studies which show that SmS films of any desired lattice constant between 5.97 A (Sm2+) and 5.62 A (Sm3+) can be obtained directly (without polishing) by co-evaporation under proper conditions; one of the conditions is the presence of excess samarium in the condensate (Shul'man et a2 1976). This indicates that the metallic films are highly sensitive to stoichiometry effects.

The presence of extrinsic, stabilised Sm3+ impurities (and also Sm2f) are expected in the metallic phase of SmS, arising both from stoichiometry effects and from inhomogeneous strain fields in high-pressure experiments or polished films. Apart from a single study (Bzhalava et all976) which seems to indicate that the room-temperature resistivity and Hall coefficient are not as sensitive to stoichiometry in the high-pressure M phase as in the semiconducting B phase, no systematic studies have been carried out at high pressure. The low-temperature resistivities of polished films have been studied (Chenevas- Paule et a1 1977); ap/aT is negative and apparently very sensitive to the presence of Sm3& impurities, and the authors suggested that the negative ap/aT observed both in these films and in single crystals under pressure is an extrinsic effect due to Kondo scattering from the Sm3+ impurities. We feel that a competing explanation is not ruled out by the data, namely that in analogy to SmBa, M-SmS is actually a small-gap semiconductor. In support of this we note that polished M-SmS films show the same low-energy optical anomalies as SmB6 (Allen et a1 1978b) and that the resistivities of films are highly


stoichiometry-sensitive; indeed, the resistivities of those with intermediate lattice constant are not enormously different in magnitude from those B-SmS crystals slightly off stoichiometry which behave semimetallic (Goncharova et a1 1976). We feel that the questions of whether the observed negative ap/aT in M-SmS is intrinsic and whether there is an excitation gap remain open.

2.6.4. Ce and Yb compounds. The presence of Ce3" (or Yb3+) impurities in mixed-valence cerium (or ytterbium) intermetallics is also widely suspected. These are believed to contribute an extrinsic low-temperature Curie 'tail' to the susceptibility which does not reproduce from sample to sample. The true temperature-independent susceptibility x( T ) is then modified to a measured susceptibility xmeas( T ) = x( T') + Ximp( T ) , where Ximp( T ) includes the effect both of foreign impurities and extrinsic Ce3+ (Yb3+) impurities. When x ( T ) is constant at low temperatures, as in many mixed-valence materials, Ximp can be easily subtracted out under the assumption that it is of the form Cimp/T. This impurity contribution can be saturated in high fields; the saturated part of the magnetisation corresponds to the same number of impurities as Cimp and the slope of the high-field magnetisation corresponds to x( T ) , for example, Dijkman et a1 (1980a) demonstrate this for CeSn3. It should be pointed out, however, that some workers believe the low-temperature tail in CeSn3 to be at least in part intrinsic; it correlates with a low-temperature magnetic form factor anomaly (Stassis et aZ1979b, c). The latter effect is not completely reproducible, however, and it is plausible that the form factor anomaly at low T is itself an extrinsic effect, due to trivalent Ce3+ impurities sitting on their proper lattice sites but stabilised by neighbouring vacancies or interstitials.

An important extrinsic effect arises from hydrogen absorption. Buschow et a1 (1977b) have shown that the Eu ions in EuRh2 change valence under hydrogen absorption; the Eu and Rh ions remain on the EuRh2 lattice. Similarly Tessema et a1 (1979) showed that when nonmagnetic CeRu2 is hydrogenated to CeRuaH4, the cerium ions become trivalent; the CeRuz retains its crystal symmetry with expanded lattice constant when hydrogenated. Rare earths are known to soak up a substantial amount of interstitial hydrogen (often 1-2 %) at ambient conditions; this suggests that the 'impurity tails' may partly arise from cerium atoms surrounded by interstitial hydrogen which are stabilised in the trivalent configuration.

The effect of non-stoichiometry on the low-temperature resistivity of CePds+% has been studied by Scoboria et aZ(l979). Close to stoichiometry the resistivity vanishes at T=O and increases quadratically with temperature; but for Pd deficiency as small as x = - 0.1 the resistivity at T= 0 grows at almost 300 $2 cm and decreases monotonically with increasing temperature. Ceo.99Lao.olPd3 samples behave similarly to CePde.9 samples. Edelstein et al (1977) have studied the effect of radiation-induced disorder in CeA13; irradiation increases the low-temperature resistivity, the increment decreasing with In T. Both effects would be consistent with a mechanism whereby trivalent Ce impurities act as Kondo scattering centres.

Extrinsic effects in cerium metal are discussed in the review by Koskenmaki and Gschneidner (1978); in the a state, a severe problem is incomplete transformation (metastable retention of y- or ,&phase material). This possibly arises from inhomogeneous strain fields.

3. Theory 3.1. Introduction 3.1.1. The ionic limit: the Hirst model. Since the properties of mixed-valence materials


are similar in many respects to those of transition-metal compounds and their dilute alloys, it is natural that several concepts which evolved in the study of transition-metal physics should be applied in the context of mixed valence. As a result, the almost traditional dichotomy of localised electrons versus itinerant electron theory has reap- peared in the study of mixed-valence phenomena. At one end of the spectrum of the theories one has the localised electron or ionic limit, in which account is taken of the strong intra-ionic correlations between the electrons in the 4f shell. The coupling between the rare-earth ions and the sea of conduction electrons is then introduced as a perturbation. This view, advocated by Hirst (1970, 1971), is expected to be apt for most of the ]nixed-valence materials, with the possible exception of the Ce compounds. Hirst’s picture is that of several ionic configurations 4P, with n electrons in the 4f shell, bathing in a sea of conduction electrons. As found in Hubbard’s (1963, 1964a, b) work on transition metals, the relative energy of these configurations is given by the sum of a Coulomb term, which represents the energy of interaction (Uff) between the electrons in the 4f shell, and a binding energy E1 due to the potential of the screened nucleus. The relative energies of the ionic configurations are then given by the formula

E(4fm) = +Uff n(n - 1) + Efn

at the integer values of the 4f shell occupation number n. These energies lie on a parabola. In this model, the phase of integer valence 4fm is stable for values of Ef and Uff such that the excitation energy required to promote a 4f electron or hole into the conduction band is much greater than a measure of the coupling between the 4f ions and the conduction band. For certain values of Ef and Uft two ionic configurations may be nearly degenerate at the bottom of the parabola. If the excitation energy required to promote a 4f electron or hole into the conduction band is of the order of the residual ionic conduction band coupling it is possible to have an appreciable admixture of two ionic configurations in the ground state, which would correspond to the mixed-valence state. This state once formed has a remarkable stability, as first pointed out by Coqblin and Blandin (1968), since in the concentrated 4f systems a promotion of 4f electrons into the conduction band is accompanied by an increase in the Fermi level of the conduction band. Thus a rise in the energy of the 4fm configuration which tends to cause electrons to spill into the conduction band gives a concomitant increase in the energy of the 4P-1 configuration with an electron at the Fermi level in the conduction band. This mechanism tends to pin the Fermi level in the sharp 4f structure of the one-electron excitation spectrum.

3.1.2. Hybridisation in the ionic limit. Hirst (1977) advocates that one should use the lowest LSJ spin-orbit ground levels of the two configurations as the ionic basis states for the calculation of the electron excitation energy spectrum. The resulting levels should then be coupled to the conduction band via a mixing process. This mixing process stems from many different sources, but the most commonly used one is a one-electron process which removes one electron from the 4f shell and places it in the conduction bands and vice versa. This mixing process is not simple to treat, since the removal of an electron from an ionic configuration will necessitate a reorganisation of the many electrons on that ion in order to minimise the intra-ionic correlation energy. Mathematically, this difficulty manifests itself as a failure of the usual form of Wick’s theorem and the linked diagram expansion of conventional perturbation methods. Due to this difficulty, Hirst (1977) has established a set of sum rules for the various intensities of the peaks in the one-electron excitation spectrum. This facilitates the introduction of a phenomenological width to the ionic levels. The resulting scheme has a certain simplicity which is


useful in interpreting experimental data dominated by the one-electron excitation spectra, since one only needs to know the ionic spectra and one free parameter, the level widths. However, it is not expected to be apt for experiments that are sensitive to collective electronic excitations that are extended over many lattice sites. Such collective excitations as spin fluctuations or valence fluctuations are usually only well-defined quasi- particles at low temperatures, losing their identity at higher temperatures.

3.1.3. High-temperature expansion. Jefferson and Stevens (1976) recognised that at high temperatures a perturbation expansion in terms of the ionic conduction band mixing is expected to converge rapidly. Using a formalism developed by Hubbard (1964a, 1967) to treat such ionic basis states they established the first two terms of a perturbation series for the magnetic susceptibility of SmS. They found that the susceptibility was of the form of a weighted superposition of the ionic susceptibilities together with a paramagnetic, temperature-independent contribution from the conduction electrons. The first correction term played the role of a slowly varying temperature-dependent Curie-Weiss constant. This was also the result of a calculation by Hewson (1977), who included a temperature-dependent shift of the chemical potential due to the hybridisation process. This shift in the chemical potential is important in the mixed-valence regime, when the chemical potential is roughly pinned to the 4f level. Hewson also calculated the one-electron excitation energy spectrum using Hubbard’s cumulant expansion method for the one-electron Green function; he found that if the magnetic degeneracy was suppressed, a gap, due to the hybridisation, appeared in the spectrum; but if the incoherent magnetic scattering was reintroduced, the gap was lost due to a self-energy that essentially broadened the 4f part of the spectrum. This loss of coherence, due to the magnetic scattering, is suggestive of an often used simplification, namely of treating the system as an incoherent superposition of non-interacting 4f impurity ions.

These high-temperature studies show that perturbation series do not converge at low temperatures, due to divergent terms of similar nature to those found in the Kondo effect. The absence of either magnetic order or Curie-like susceptibilities in the mixed- valence materials help reinforce the suggestion that the mixed-valence ground state has much in common with the Kondo ground state. In both problems, the coupling between the ions and the conduction electrons become very important at low temperatures, producing fluctuations in the direction of the ionic magnetic moments. In the mixed- valence problem there are also fluctuations in the occupation number of the 4f shell that have to be treated, as well as the interference between these processes on the various ions.

3.1.4. The Anderson lattice. Due to the enormous complexity of the problems involved, many authors have made diverse approximations in modelling the systems. These simplifications have, in general, led to models which are also intractable. The most common type of simplification assumed leads to the model of the Anderson lattice (Varma and Yafet 1976). The Anderson lattice contains two types of states, the localised, highly correlated 4f states, and the itinerant uncorrelated conduction band states. The degeneracy of the 4f sites is ignored, leading to the Hamiltonian

i o i U

in which&+ andf;:,, respectively, create and destroy an electron of spin U in the f orbital of site i. The first term represents the binding energy of a single f electron to the screened


nucleus. The second term represents the Coulomb interaction between two f electrons, of opposite spin, located on the same ion. Frequently, the Coulomb interaction Vir is assumed to be infinite in accordance with Hirst’s (1970) ionic picture. Although the neglect of the seven-fold degeneracy of the 4f orbital is not realistic, such an approxinia- tion may be useful in contributing to the understanding of the phenomena of mixed valence. The conduction band is assumed to be composed of uncorrelated itinerant states, mainly of d character. These are described by the Hamiltonian

in which dkb+ and dko, respectively, create and destroy an electron of spin LT in the Bloch state labelled by wavenumber k. These two types of states are assumed to be coupled via a one-electron transfer term :

f i f a = 1 [Vfd(k)fi,+ d k , exp (ik.rdSHC1.

The first term takes an electron out of the conduction band and places it in the 4f shell at site i, and the second term represents the reverse process. This hybridisation term is spin conserving.

ika

3.2. Approximate solutions for the ground state

3.2.1. The equation of motion method. An early attempt at an approximate solution of the Anderson lattice model was carried out by Varma and Yafet (1976). They calculated the one-electron Green function using an equation of motion technique. In this method one finds that the one-electron Green function

can be expressed in terms of higher-order correlation functions which themselves satisfy an equation of motion of the same type. Varma and Yafet truncated the resulting hierarchy of equations in a manner which preserved the intra-atomic Coulomb correlations by approximating the two-particle correlation functions which appear together with a factor of V f d , such as

<<h-u’ f i -odka;hu+)> < f a - U + f i - u > <<dko;ho+>)*

On taking the limit Uir --f 00, they obtain a one-electron excitation spectra which exhibits an energy gap of the order of (41 V12/W) (fa,.fi-,+) and is located at the energy of the unhybridised 4f level &. In the mixed-valence phase, the chemical potential p should lie in the near vicinity of the 4f level. Varma and Yafet’s calculation yields completely different results, depending on the exact position of the chemical potential. If the chemical potential lies in the hybridisation gap, the resulting susceptibility and specific heat at T=O will be zero, while if the chemical potential lies in a region with a non-vanishing density of states, the resulting susceptibility and specific heat is finite and of the order of W/Vz(1 -nf) at T=O. In summary, this work leads to a Fermi liquid picture for the ground state, in which all the properties of the system are governed by a ‘renormalised’ one-particle density of states. This approximate solution to the equations of motion should be treated with caution. Firstly, similar decoupling schemes also based on the


original treatment used by Hubbard (1963) have lead to spurious results in half-filled one-band systems as well as in impurity problems. Secondly, there is no systematic way of estimating the effects of higher-order corrections.

3.2.2. The coherent potential approxiination. Other authors, such as Martin and Allen (1979) and Leder and Czycholl (1980), have undertaken further higher-order decoupling schemes, equivalent to the Hubbard I11 (1964b) treatment of one-band systems. The basic approximation involves the decoupling of the higher-order equations of motion, in a manner consistent with the simple picture of the system as being an alloy. The subset of terms kept is formally equivalent to the coherent potential approximation (Velicky et al 1968, Kirkpatrick et al 1970) used in the context of random systems. It utilises a single site approximation which separates the scattering of electrons from the ions such as f O or f1 at a single site from the rest of the medium which is instrumental in specifying the initial state of the electron. The rest of the system is then replaced by an effective medium which is determined by the condition that the ensemble average (over f O and f1 configurations) of the scattering produced by the ions at site i is zero. The result of these calculations is a density of states in which the hybridisation gap still survives for a large range of positions of the 4f level EP. For a range of values of Ef in the centre of the band the hybridisation gap disappears. However, Martin and Allen (1979) regard this dis- appearance of the gap (when it is degenerate with the Fermi level) as being spurious on the basis of the Luttinger theorem (Luttinger 1960).

Allen and Martin (1979) then use this model to interpret the behaviour of SmS and SmB6. They conclude that in SmBa the Fermi level will lie in the hybridisation gap, and that in accordance with experiment the properties should be that of a narrow-gap semiconductor. In the Sm chalcogenides, there is a vanishing of the hybridisation matrix element due to symmetry requirements. This would wipe out most of the gap and leave the properties as being more metallic than SmB6. Although the agreement with experiment is impressive there do remain theoretical questions regarding some details of the solution. One such detail is that the approximation allows the electron-electron scattering near the Fermi level to be large and even unlimited by Fcrmi statistics. This is a consequence of neglecting corrections such as Hubbard’s (1 964b) ‘resonance broadening’ corrections which allow the ions which scatter to change their state of occupation, and can allow motional narrowing to occur in the one-particle excitation spectrum. A second point is that of the CPA replacement of the proper self-energy by a site diagonal one. This completely washes out any spatial correlations, and in essence gives all properties of the solution a local character. Leder and Czycholl (1980) utilised this analogy with a self- consistent alloy to calculate the magnetic susceptibility and the specific heat. In the limit Uff --f 00, they found no evidence of magnetic instability to a ferromagnetic phase, the only type of ordering allowed by the framework of the approximation. They have recently extended this calculation to real disordered mixed-valence systems with two types of ions at the rare-earth lattice sites such as Yb,Yl-,CuAl. The qualitative features of the solutions remain unchanged, as one might have expected.

3.2.3. Variational methods. An alternative treatment based on the variational principle has been used by Brandow (1979, 1980). His calculations are undertaken from the view- point that the system cannot be described as that of non-interacting impurities, but rather that the interactions are of great importance and interfere destructively in a manner which eliminates the Kondo-like divergence. Brandow differentiates between the case of one electron per ion and the case of two electrons per ion. In the former case he assumes


a trial wavefunction of the form

I$) =n %(hC++z akj, dkof)>I O> j = k

and in the latter case

I $) {fi t+h &++E akj ( d k f + f 3 J ' + f 3 t + dk 1'))1 O)' j k

These wavefunctions correspond to the limit Ufn --f 00, since in this case only two ionic configurations contribute to the ground state. Brandow's (1979) Bloch state representation of the conduction band in the trial expression requires care to insure that these states are not multiply occupied. The second variational wavefunction, originally suggested by Stevens (1976, 1978), does contain a subtlety since the singlet nature of the expression needs to be modified if a non-trivial response to an external field is to be obtained. Brandow (1980) calculates the magnetic susceptibility and, for the one electron per ion case, he finds that the results at T=O can be interpreted as the sum of a Pauli paramagnetic term from the d band together with a term that is proportional to the nuin- ber of f electrons and inversely proportional to the effective f bandwidth V z / W. The case of two electrons per ion gives a result indicative of a zero susceptibility at T=O. This result is in accord with Martin and Allen's (1979) picture of the two electron per atom system being an insulator, where the Fermi level lies within a hybridisation gap. Although these results do agree with those of the equation of motion method, there is an unresolved controversy about the implications, namely the destructive interference removing the Kondo divergences. Other variational calculations were performed by Stevens and Arnold (1979). A variational treatment of the isolated mixed-valent impurity (Varma and Yafet 1976) exhibited the transition from Kondo behaviour to mixed-valent behaviour as the f level approaches the Fermi energy.

3.2.4. Havtree Fock and RPA. When the Coulomb interaction Urf may be regarded as small compared to the effective f bandwidth, it seems more reasonable to use a weak coupling theory to treat the Anderson lattice. This might be the case for the cerium compounds where it has been argued that Ufn is comparable to the effective f bandwidth (52.5.5). In these weak coupling cases it is assumed that the mixed-valence state is made of an extended, coherent ground state and that the low-lying excitations can well be described by Fermi liquid theory. This picture prompted Leder and Muhlschlegel(l978) to treat the Anderson lattice within the Hartree-Fock approximation. In this work the Coulomb interaction terms, proportional to Uif, were linearised, replacing their effect on the f electrons by a spin- and site-dependent potential Uff (fi-C+fi-,). The f occupation numbers were calculated self-consistently but the calculation was restricted to the sub- space which allowed either ferromagnetic or antiferromagnetic instabilities to occur. On diagonalising the resulting approximate Hamiltonian they found that the one-electron excitation spectrum exhibits a gap in the ferromagnetic or paramagnetic phases. Further- more, in the case of one electron per ion they found that an instability to either a ferromagnetic or antiferromagnetic phase could occur, while for two electrons per ion only antiferromagnetic instabilities could occur and the system was always semiconducting. In accordance with Anderson's kinetic exchange arguments for antiferromagnetism, the Nee1 temperature became largest for systems in which the effective number of f electrons per ion approached unity. This correlates well with the behaviour observed in the non- stoichiometric samples and alloys of the antiferromagnetic substance TmSe. Coqblin et a1 (1979) used the Hartree-Fock solution to describe the behaviour observed in the resistivity of TmSe, where an antiferromagnetic gap appears to dominate the resistivity

62 J Ad Lawrence, P S Riseborough and R D Parks

of the stoichiometric samples, but increasingly less in the region of non-stoichiometry. Riseborough and Mills (1980) extended the calculation of Leder et a1 to that of the dynamical wavector-dependent susceptibility, and thus were able to show that, within the framework of the Hartree-Fock approximation, no other types of instability such as helical ordering could occur. They also showed that the prominent features of the neutron scattering spectrum of CePds could be interpreted by their calculation as due to scattering with low-energy, extended spin fluctuations.

3.2.5. Fermi liquid treatments. BBal-Monod and Lawrence (1980) have also proposed that spin fluctuations play an important role in the static susceptibility. These authors assume that the low-temperature properties are well described by Fermi liquid theory (82.5) and that there is a well-defined f density of states between which one can introduce an ff Couloinb repulsion. Thus BBal-Monod and Lawrence are able to apply a spin fluctuation calculation apt for a one-band Hubbard model. Thus they showed that the first consistent and conserving expression for the static susceptibility x( T ) gives a result which scales to order (T/T,r)2. This scaling behaviour has been observed in CeIn3-,Snz.

Another Fermi liquid approach to the problem has been advocated by Newns and Hewson (1980). It is based on the earlier observation of Hewson, that the magnetic scattering leads to a loss of coherence in the excitation spectrum. Thus they were led to a description of the mixed-valence state as a local resonance and showed that many experimentally observed properties followed directly. Mattens (1 980) showed that the properties of many materials, but most notably YbCuAl could be put within the framework of Newns and Hewson’s model.

However, it does remain a mystery as to how it is possible that the effective 4f bandwidth can be assumed to be so large compared with the Coulomb interaction U p One possible explanation can be found in the work of Schrieffer and Mattis (1965) on the single-impurity Anderson model where they showed that, although Uti may be large, in the ladder approximation it may be replaced by an effective interaction or T matrix, which is small. The net result of their calculation was that of completely precluding any net Curie moment at zero temperature, a result to be found much later in Wilson’s numerical renormalisation group calculations.

3.2.6. ‘Poor man’s scaling’ and related methods. Jefferson (1977) and subsequently Haldane (1978) have taken other approaches to the large Uii limit of the single-impurity Anderson model in deriving effective Hamiltonians that could eliminate many of the logarithmic terms that appear in the high-temperature perturbation theory. Both authors’ approaches are based on Anderson’s (1970) derivation of scaling laws for the Kondo problem-the so-called ‘poor man’s scaling’. The method is constructed around the concept that excited states with energies much greater than the temperature are irrelevant since they only contribute to the physics as virtual intermediate states in the lower energy states. Thus the method consists of eliminating the highest particle and hole excited states by treating these states (near the conduction band edges) by low-order perturbation theory. In this manner these states are eliminated at the expense of producing a new effective Hamiltonian with a reduced conduction bandwidth. The ultimate aim is to scale down the bandwidth to an energy that will eliminate the logarithmic divergences such that the new Hamiltonian has a simple perturbation expansion. In perform- ing such a calculation the authors used first-order perturbation theory to eliminate the states near the band edge, thereby introducing new couplings and retardations not included in the original Hamiltonian. These were neglected. On reducing the effective

Valence JEuctuation phenomena 63

bandwidth down towards the temperature, Jefferson (1977) found that the position of the 4f level was strongly renormalised, and that the effective f bandwidth was slightly changed. Haldane (1978) performed a similar calculation, omitting several irrelevant and cutoff-dependent terms included by Jefferson. Haldane also completed the identification of the limits of validity of the scaling treatment, the scaling procedure stopping when the 4f level becomes decoupled from the band states, separated by a gap, as intimated by Jefferson. The other limit, that corresponding to mixed valence, is that in which the bandwidth becomes comparable to the effective f bandwidth A, which implies A. At these points the irrelevant terms grow and scaling breaks down. In the Kondo problem it has been established that the scaling method is equivalent to summing up the most divergent terms of the high-temperature perturbation expansion. Such a partial summation of the most divergent terms of each order in the perturbation series was done for the Anderson impurity problem by Bringer and Lustfeld (1977) and Lustfeld and Bringer (1978). The results of their calculation can be interpreted in terms of a temperature-dependent renormalisation of the position of the f level, with only trivial differences from that obtained by the scaling method. The authors consider terms that represent interactions between neighbouring impurity ions; these are shown to be less divergent than the most divergent single-impurity terms when the f level is not too far from the Fermi level. However, they are of the order of less-divergent single-site terms that were neglected. The condition for these neglected terms to be small is very similar to the condition derived for the validity of the scaling method.

3.2.7. Numerical renormalisation group methods. Below the characteristic temperature TK one has to rely on the numerical renormalisation group work on the single-impurity Anderson model. In this method, Krishna-niurthy et aZ(1980a, b) divide the conduction band into discrete intervals with energies between A-(n+l) and A-&, so that these intervals may give contributions, of equal order, to the logarithmic divergences found in perturbation series. Basis sets are defined for each of these intervals, but only the lowest Fourier harmonic is kept. The resulting approximate Hamiltonian is then transformed into a form with similar structure to the tight binding Hamiltonian, in which the 4f orbital couples to the conduction band via a local term. This Hamiltonian is solved numerically by iteration in the number of ‘sites’ involved in the system. In order that the iteration process may be truncated to arbitrary accuracy, the number of iterations must be increased in proportion to the logarithm of the temperature. The lowest energy levels of the system for a finite number of iterations is then compared with the energy levels of fixed-point Hamiltonians. This allows the system to be treated analytically in the vicinity of these fixed points. The effective Hamiltonian for a given temperature, i.e. for a given number of sites, may be expressed in terms of a corresponding fixed-point Hamiltonian and the eigenvalues and eigenvectors of the linearised iterative transforma- tions about these fixed points. For a detailed description of the results and for a fuller account of the method the interested reader is urged to read the original papers. Although the results for the Anderson impurity model do bear a striking resemblance to those of mixed-valence compounds, there remain some important questions concerning the interactions between the impurity ions. There exists strong evidence, both theoretically and experimentally, that the impurities produce a compensating spin cloud that has a large spatial extent such that interaction effects can be observed to be important down to extremely small concentrations. Therefore in a concentrated compound these effects should not be negligible, especially at low temperatures where a coherent gap appears in the ground state.


3.3. Charge Jktuations and spin Jluctuations

Thus far we have only considered either the single-impurity Anderson model or the Anderson lattice. These models contain the hybridisation mechanism which allows both the local 4f electron spin and the local 4f charge to change. The intra-ionic Coulomb interaction Uff affects the dynamics of the 4f spin and the 4f charge in different ways. It tends to stabilise the local 4f moment, and tends to produce slowly time-varying spatial regions which exhibit short-ranged magnetic order. Thus as Uff is increased, thermal spin fluctuations are expected to become more numerous and the system is expected to produce a bigger response to an applied magnetic field. A measure of the fluctuations of the spin about its average value is given by the spin fluctuation propagator

where S ( q ) is the spatial Fourier transform of the f spin density. Similarly, we can define a charge fluctuation propagator by

where p(q) is the Fourier transform of the f charge density operator. In the absence of Ufr, spin and charge fluctuations are trivially related. As Ufn is increased, charge fluctuations become suppressed, being significant only at higher energies. This can easily be seen once the partition function has been expressed in terms of contributions from various fluctuations. Morandi et aZ(l974) have reviewed this technique as applied to the single- impurity Anderson model, while Wio et a1 (1978) have applied the technique to the case corresponding to mixed-valent behaviour.

The first step in these calculations is the derivation of a time-dependent Landau- Ginzberg free-energy functional, from which one can immediately recognise the existence of spin fluctuation modes that may exhibit a phase transition as Uff is increased. The spin fluctuation modes become soft and, in the Gaussian approximation, the propagator is enhanced by a factor [ l - Uff xO(q, w ) ] - ~ . The charge fluctuations do not exhibit such an instability; the modes become harder and the charge fluctuation propagator is suppressed by a factor of [1+ Ufp xO(q, w)] - l as Uff is increased. Thus it is clear that in order to have a simple state with appreciable charge fluctuations it is necessary to introduce further interactions in the model. The interactions that have been mostly considered are the Coulomb interactions between the 4f electrons and the other charges in the system, either the conduction electrons or the combination of electrons and ions that form the phonons of the system. In the next subsection we shall consider the effects of electron- phonon coupling on the dynamics of the system and in the following subsection we shall consider the effects of electronic screening.

3.4. The electron-phonon interaction

3.4.1. Efect of the coupling on the electron dynamics. The effects of the electron-phonon interaction are very strong in most intermediate-valence compounds, unlike the corresponding case in the physics of transition metals. The rare-earth ions exhibit large changes in the ionic radii which accompany changes in the state of ionisation. The removal of a 4f electron from an ion will result in the spatial contraction of the higher lying electronic orbitals, inducing a lattice distortion centred on that ion. This is supported by many


differing experimental facts, such as the large changes of volume associated with the transitions from the phase of integral valence to that of mixed valence. This coupling could give rise to many dynamical effects, such as elastic relaxation shifts similar to the Franck-Condon relaxation, which might be correlated with the apparently large excitation energies seen in XPS and photoemission experiments. The coupling could also be the cause of many anomalies of the lattice dynamics, namely the unusual shape of phonon spectra and Debye-Waller factors.

The effect of electron-phonon coupling on the dynamics of the electrons was first considered by Sherrington and Von Molnar (1975). They considered a single-impurity problem described by the Hamiltonian H ,

.@= f i e + H p h -k Heel-ph

where are described as spinless fermions, governed by f i e :

He = &fo+fo f

describes a single 4f level hybridised with a conduction band. The eiectrons

E&) dk+dk 4- k k

(V(k)fo+ d k + HC). The phonons are described by a localised Einstein oscillator :

B p h = ( a o + a o + ~ ) hw

where ao+ and a0 are the boson creation and destruction operators. The electron-phonon interaction which couples the state of occupation of the 4f level with the lattice distortion is described by the Hamiltonian

Hel-gh = A(ao+ + ao) (fo'fo - (fo+,h>). Sherrington and Von Molnar showed that this electron-lattice coupling could strongly affect the dynamics of the electrons, as can be seen by considering the canonical transformation

H -f I?= D+HD

where D+ is a unitary operator D = exp [(hlhwo) (ao+ - ao) ( fo+fo - (f~+f~))] that eliminates the electron-phonon interaction at the expense of introducing emission or absorption of phonons, albeit coherently, that occurs concomitantly with the hopping of the f electron:

A simple treatment of the transformed Hamiltonian yields an exponential reduction in the 4f hopping rate to ( V z / W ) exp (- h2/kwo) and the 4f level also exhibits a Franck- Condon type of relaxation from Ef to Ef-(h2/hwo).

Hewson and Newns (1979) carried out a detailed analysis of this model and found that the reduction in the width of the 4f excitation spectrum only pertained to the limit h2/tiwO> W, which is not expected to be applicable for mixed-valence materials. The authors also investigated a two-site model and found that, although the 4f excitation spectrum did not show any polaronicnarrowing, under the conditions W > h2/hwo > Vz/ W, the rate at which the 4f electrons hopped between the two sites did show the exponential reduction. This indicates that the two sites might behave as isolated impurities, for a large temperature regime. However, the calculation is based on a method for the x-ray edge problem which involves electrons far from the Fermi surface and it is not clear that such


a method is reasonable to describe processes at the Fermi surface as those involved in mixed-valence phenomena.

Sherrington and Riseborough (1976) derived a more realistic model that was of a periodic array of 4f sites which was coupled to a phonon spectra that showed dispersion. Their investigation of this model showed that in the polaronic regime, the lattice mediated an interaction between the 4f sites of the form

which might cause spatial ordering in the 4f occupation. Analysis of the non-polaronic regime shows that even there the electron-phonon interaction leads to anomalies in the 4f excitation spectra through logarithmic energy shifts.

All the above calculations were performed with specific assumptions of the position of the f level with respect to the Fermi level, despite the expectation that these effects should be extremely sensitive to the average 4f occupation number. Khomskii (1978) investigated such effects in the context of studying polaronic effects on the phase transition to the mixed-valence state. Since the model was of a local character, some of the effects of the dependence on valence were not included. In particular, it is simple to visualise the system with a low density of 4f electrons or holes, where each quasi-particle is surrounded by a lattice distortion, giving rise to polaronic effects ; however, when every other 4f level is occupied the lattice distortion can no longer be regarded as distinct and dressing individual 4f quasi-particles. These types of interference effects have not been investigated and may be important in strongly mixed-valence materials.

3.4.2. Efect of the coupling on the lattice dynamics. Up to now, we have discussed the effects of the phonons on the electron dynamics; now we shall discuss the reverse effects, the effects that the electron-phonon coupling has on the lattice dynamics. Grewe et a1 (1978) calculated the polarisation part of the phonon Green function in a perturbation series in the electron-phonon coupling constant A. They found that the phonons, when mixed to the 4f electrons, produced two distinct modes. They suggested that in the mixed-valence state the upper branch of the excitation spectrum would be mainly electronic and the lower branch mainly of phonon character. They showed that in the mixed- valence regime, when the hybridisation matrix element is large compared to &wo(qs), the longitudinal phonon spectrum may exhibit a softening and concomitantly develop a shorter lifetime, i.e. a larger width. Ghatak and Bennemann (1978) reached the same conclusion in a similar investigation. Furthermore Ghatak and Bennemann considered the mean square lattice displacement that occurs in the Debye-Waller factor. They were able to show that valence fluctuations had the effect of enhancing the mean square displacement, becoming largest near the transition between the integer- and intermediate- valence phase. These works provide a good explanation of the anomalies observed in both Raman scattering and Debye-Waller factor measurements on Sml-,Y,S (52.4.7). The inelastic neutron scattering study of Mook et a1 (1978a, b) of the phonon spectrum of Sm0.75Yo.25S provided a stimulus for further theoretical work. In their experimental work they found strong anomalies for phonon wavevectors in the (1 11) direction. The longitudinal acoustic phonon branch was found to be lower than the transverse acoustic mode, except at the Brillouin zone boundary where they exhibited a degeneracy. A similar softening of longitudinal optic mode was observed at the zone boundary. These anomalies were fitted by the theories of Bennemann and Avignon (1979), Entel et a1


(1979), Bilz et a2 (1979), Matsuma et al( l980) and Seitz et al(1980), all using the same basic physical interpretation. In Bennemann and Avignon’s interpretation, the symmetry and the q dependence of the electron-phonon coupling played the central role; they also considered the Coulombic suppression of the valence fluctuations. Entel and Grewe (1979) and Entel et a2 (1979) also describe the renormalisations of the transverse phonon modes by keeping higher-order terms in the electron-phonon interaction. Bilz et a1 (1979) use a phenomenological breathing shell model including both Sm dipolar and S quad- rupolar deformabilities. Bilz et a1 systematically apply the model to EuS, SmS and YS and provide an excellent fit to the experimental data for these materials using five parameters. Matsuma et al(l980) use a more microscopic theory and are able to fit the data for Sml-,Y,S using only two parameters describing the electron-phonon coupling.

In conclusion, we note that in mixed-valence systems, the electron-phonon interaction leads to many anomalies, the effects being most marked and apparent in lattice dynamical properties. The effects on the electron dynamics are not so immediately apparent and may not be easily distinguishable from similar effects due to the electron screening which we consider in the following subsection.

3.5. The electron screening interaction

Ramirez et a1 (1970) first introduced the Coulomb interaction between the 4f electrons and the conduction electrons into the theory of mixed-valence phenomena in the context of valence transitions. From charge neutrality considerations, in the single-impurity Anderson model, Haldane (1977b, c) independently arrived at the conclusion that the screening produced by the 4f electron-conduction electron interaction is important in the mixed-valence phase. Haldane (1977~) proposed that the Anderson model should be extended to include such screening mechanisms that involve states other than those contributing to the resonant scattering. Using the Tomanaga approximation, Haldane re-expressed the model in terms of a single Anderson impurity coupled to a boson field. Haldane (1977b) examined the resulting model in the Hartree-Fock approximation obtaining a phase diagram which, when coupled to the pinning of the Fermi level to the 4f density or states, contained a mixed-valence regime. He also examined the stability of the collective excitation spectrum in the random phase approximation. The inclusion of the electron-boson coupling was found to give rise to an enhancement of the valence fluctuation spectrum of magnitude

a modification similar to that found by Riseborough (1977) in an electron-phonon model. Khomskii and Kocharjan (1976) also formulated a one-impurity Anderson model in

which an fd Coulomb interaction was included. The interaction u f d Z;f~+f-,dk+dk was allowed to act on the states to which the f electrons hybridise, in contrast to Haldane’s model. Khomskii and Kocharjan’s Hartree-Fock treatment included some off-diagonal averages that lead to spurious results for the phase diagram even in the exactly soluble limit of zero hybridisation as shown by Hewson and Riseborough (1977). De Menezes et a1 (1978), in their consideration of Eu(Ir~-,Pt,)z and the smooth valence transitions observed in experiments, used a similar model. These authors introduced an electron- phonon coupling to the electrons, which produced a phonon-mediated attraction between the d electrons. This in turn enhanced the charge cloud screening of the f shell produced by Ufd. It seem obvious that these large polarisations induced by valeiicc


fluctuations in the conduction band will interfere and strongly affect the behaviour of the 4f levels in a concentrated system.

As Haldane (1 977a) has emphasised, single-impurity Anderson models cannot be regarded as a good description of mixed-valence materials, since at low temperatures coherence effects and interactions between neighbouring 4f ions become increasingly important. There have been many investigations of the interactions between dilute Anderson impurities such as the high-temperature work of Bringer and Lustfeld (1977) using infinite-order perturbation series or Schlottmann’s (1980) low-temperature Fermi liquid theory based on Ward identities. The introduction of the fd screening gives the problem an added dimension. Varma (1977) has extended Haldane’s Hartree-Fock calculation to the problem of two interacting Anderson impurities, similar to the calculations of Alexander and Anderson (1964). The slow charge fluctuations were found to lead to a suppression of the magnetic interaction between sites when one of the valence states has no magnetic moment ($3.6). Schlottman (1978) on the other hand has shown that the fd Coulomb interaction can give rise to a non-local oscillating interaction between the f electrons, similar to an RKKY interaction. In the Bethe-Peierls approximation he finds that short-range order can be stable at low temperatures.

3.6. Magnetic interactions and double exchange

In periodic mixed-valence systems one would expect that the conduction band would mediate interactions between the magnetic moments on the 4f ions. Goncalves da Silva and Falicov (1972b) have shown that terms of fourth order in the hybridisation V4

mediate such interactions. A 4f electron can mix into a band state at site i , propagate, and then mix into a 4f site at s i te j yielding an effective RKKY interaction of order V4, or an Anderson-like superexchange interaction. The relative importance of thesc are determined by the number of conduction electrons. From considerations such as these one might expect magnetic ordering to occur, such as happens in many trivalent Ce compounds ($2.2).

TmSe is the only non-integral-valent material that magnetically orders. Varma (1 979) noted that Tm is the only mixed-valent element that has a magnetic moment in both the ionic configurations that appear in the mixed-valence phase, and this fact may be responsible for the existence of both well-defined moments and their tendencies for magnetic ordering. The existence of well-defined moments in the mixed-valence phase can be understood on the basis of Hund’s rule couplings which tend to align the spins of the electrons in the 4f shells. For instance a Tm3+ ion has a magnetic moment, and in the mixed-valence phase an electron of opposite spin may hop onto the 4f shell, thereby reducing the moment. The moment of the resulting Tm2+ ion will be correlated parallel to that of the original Tm3+ moment. Hund’s rules will favour only the further fluctuations Tin2 --f Tm3+ which occur by omitting electrons of spins opposite to the Tm ions moment. This results in stable magnetic moments. The interactions between the moments may then cause magnetic ordering, such as the antiferromagnetic behaviour of TmSe and the ferromagnetic behaviour of such compounds as TmSe,Tel-, (Batlogg et a1 1979a, b). Varma predicted the change from antiferromagnetism when x = 1 to a ferromagnetic phase as x is reduced, by arguments concerning the competition of RKKY interactions and double-exchange interactions. In this model, the RKKY interactions are responsible for the antiferromagnetic ordering. However, when the average valence per ion tends toward a half-integer value, double exchange becomes increasingly important. The double-exchange mechanism also occurs through the action of Hund’s rule correla-


tions (de Gennes 1959). An electron which hops off a Tm2+ ion will have its spin correlated with that of the initial ion. The electron may then lower its energy by hopping onto neighbouring Tm3f ions which are aligned parallel to the initial Tm2+ ion, such that the resulting Tm3f configuration satisfies Hund’s rules. This kinetic type of exchange thus favours ferromagnetic correlations between neighbouring Tm2+ and Tm3+ ions and should be most important for Tm compounds with half-integer valence. Lustfeld’s (1 980) calculations have confirmed this picture.

3.7. The Kondo lattice

Doniach (1977a, b) formulated a model which can be derived from that of the Anderson lattice by using a Schrieffer-Wolf (1966) transformation. If one assumes one 4f electron per site, and Ufi -+ CO, so that ff hopping terms become negligible, then one obtains the Kondo lattice Hamiltonian :

where Jkkt = v k i v k i (Et - &&))-I. St is the 4f spin at site i and Q has its components given by the Pauli matrices U”, uu and uz. The first term represents the Hamiltonian of the itinerant conduction electrons, while the second term represents the antiferromagnetic coupling between the itinerant conduction electrons at site i and the spin of the f electron localised at that site.

For convenience Doniach (1977a) restricted this problem to the case of a one-dimensional lattice with one conduction electron per site. Also he replaced the Kondo lattice Hamiltonian by an analogous Hamiltonian which he termed the Kondo necklace. The replacement was motivated by the equivalence, in one dimension, of a spinless band of conduction electrons to a chain of pseudo-spins under the Jordan-Wigner transformation. The 4f spin was then assumed to couple directly to the pseudospins T( via a form

coupling the spin S6 to the boson charge density modes. As Doniach showed, the problem of one localised 4f spin coupling to the chain of pseudo-spins showed the same weak coupling scaling behaviour as that of Kondo spin coupling to the true conduction electron spins.

This one-dimensional, analogous problem was then treated in a mean-field approximation. A phase diagram was found, which for small J/ W showed an antiferromagnetic ordering of the 4f spins, while for larger J / W another phase was found in which each 4f spin is tightly bound to a conduction electron, forming a Kondo singlet. As Doniach (1977b) noted, the mean-field approximation is notorious in one dimension, due to the suppression of the mean-field phase transition at T=O by phase fluctuations of the order parameter. Nevertheless, these results were shown to be reasonable as a result of complicated numerical renormalisation group calculations on the Kondo necklace problem (Jullien et aZ1977a, b). When this method was applied to the Kondo necklace Hamilto- nian, two fixed points were found corresponding to J/ W g 0.4. For small J, the iteration process reaches a fixed point corresponding to both the localised spins and the pseudo- spins forming an antiferromagnetic lattice, while for larger J values each local spin strongly binds to a pseudo-spin of opposite direction as in the Kondo problem. Since in the Kondo necklace problem there is one conduction electron per spin, the resulting Kondo state is expected to be insulating.

5


Since the rotational invariance of the Kondo lattice Hamiltonian has been broken in the analogous problem of the Kondo necklace, it is not obvious that the two models will show the same behaviour. However, the results are only slightly different (Jullien et al 1979a, b). The mean-field treatment of the Kondo lattice does show a transition from a metallic phase for small J/ W to a phase in which there is a gap in the density of states. For one conduction electron per site, this would lead to an insulating phase. However, different one-dimensional mean-field treatments give the small J/ W metallic phase different magnetic properties, and thus the above result must be regarded as suspect. The renormalisation group treatments of the one-dimensional Kondo lattice have focused mainly on the case of one conduction electron per site. In this case the energy level spectra of the renormalised Hamiltonian is consistent with an energy gap of the Fermi level for all finite values of J/ W, a feature attributed to the one-dimensional nature of the problem. There may, however, still be a transition between a magnetic insulator and a non-magnetic Kondo insulating phase, since the renormalisation group calculations have not yielded decisive results on the magnetic character of this case.

For the case of a fractional number of conduction electrons per site, the T=O state is found to be metallic for all values of J/ W. There are no definitive results on the existence of a magnetic to non-magnetic phase transition.

Hoshino and Kurata (1979) used a sophisticated equation of motion technique on the case of a disordered, three-dimensional Kondo alloy. They find a reduction in the density of states at the Fermi level, which they indicate as being consistent with a gap occurring at the Fermi level in the ordered Kondo lattice case.

The model of the Kondo lattice has been applied to the compounds CeA12, CeA13 and TmSe. However, it is not so well justified to use the model for the case where the occupation numbers are non-integral, as in TmSe, or for values of J approaching W. Under these conditions, the Schrieffer-Wolff transformation breaks down and other processes (e.g. charge fluctuations) become important ; the Anderson lattice then becomes a suitable alternative description.

3.8. Theories of the valence transitions

The spectacular isomorphic valence transitions observed in Ce, SniS and alloys thereof (82.1) are believed to be brought about by both electronic and lattice-mediated mechanisms. While the lattice mechanisms are clearly present (as evidenced by the observed anomalous phonon spectra, lattice constants, thermal expansion and compressibilities), there are no agreed upon estimates of the relative magnitudes of the lattice and electronic mechanisms. Indeed, reasonable agreement with the observed phase diagrams are obtainable from either mechanism. In order to elucidate the workings of the various mechanisms we discuss the two cases separately.

3.8.1. Electronic models of the phase transitions.

3.8.1.1. The Falicov model in the mean-jield approximation. The archetypal theory for an electronically driven phase transition is that due to Falicov and Kimball (1969). The model assumes the existence of two types of states: localised and highly correlated 4f levels and the itinerant conduction band states. The electrons in those states are coupled via a Coulomb repulsion, which acts as the driving mechanism for the transition. The Coulomb interaction 4f is assumed to be so heavily screened that it only acts between a localised 4f electron and a conduction electron located at the same lattice site. This


feature of the interaction is somewhat unrealistic, and the results found in this model may be changed by including a self-consistent determination of the screening of the Coulomb interaction such as in the works of Mott.

The phase diagram of the Falicov-Kimball model was found in a mean-field approximation by considering the free energy

where the energy is calculated as

E = s’? dwfiwp(w)f(w) + NmEr + Nnr s? d i d i d p(w)f(w)

F = E - T S

in which p ( o ) is the conduction electron density of states. The entropy is given by the sum of the band entropies and the 4f ion entropies. The minimum of the free-energy functional gives the stable phase. Ramirez et al(l970) considered the case when the 4f level lies below the conduction band and one electron per 4f ion. For small values of u r d compared to the conduction bandwidth W, the minimisation procedure yielded only one solution, which showed a gradual condensation as T -+ 0, of all the electrons into the localised 4f levels, giving an insulating phase. For larger values of the ratio i&/w multiple minima of the free energy were found at low temperatures. As T decreased the system was found to exhibit continuous and discontinuous transitions from a metallic to an insulating phase. For even larger values of ufd/w the system showed a metallic phase to be stable at all temperatures. As discussed by Ramirez et a1 (1970), this mean- field theory is expected to break down for ufd/w>o.3, which is the criterion for the possibility of excitons to form, and which might be accompanied by the formation of excitonic phases.

3.8.1.2. CPA and other treatments of the Falicov model. Plischke (1972) has suggested that discontinuous valence transitions exhibited by the mean-field approximation might not hold in the exact treatment of the model. In particular, he treated the model in the coherent potential approximation, by assuming the 4f electrons to be distributed ran- domly over the lattice. These then acted as a random potential Ufd acting on the conduction electrons. The number of f electrons was then determined self-consistently, For the same values of Ufd/ W as used by Ramirez et a1 (1970), Plischke (1972) found only smooth transitions between metallic and insulating phases as T decreased. On the other hand, Goncalves da Silva and Falicov (1972a) performed a similar CPA calculation, and on parametrising the results of the CPA for the density of states of the conduction band they were able to investigate the model over a larger region of parameter space, This calculation again showed first-order transitions as the temperature was lowered. This discrepancy, within the framework of the CPA, was investigated by Ghosh (1976) who only found an insulating behaviour in the unsplit band regime and did not find the stable metallic phase contrary to Plischke’s calculation. Ghosh’s solution was confirmed independently by Mazzaferro and Ceva (1978). However, Ghosh concluded that the discontinuous transitions and the stable metallic phase might be found in the split band regime which would correspond to the regime in which excitonic phases could occur. This conclusion seems reasonable, in view of Falicov’s work, since the parametrisation of the density of states near the band edges could result in a underestimate of the value of Ufd/W at which these transitions occur, in a similar way to which a similar parametrisation of the density of states would underestimate the criterion for exciton formation.

Schweitzer (1978) treated the same model using a decoupling scheme which treated the intra-site correlation exactly. Using this method, which is closely related to CPA, he also found discontinuous transitions to a metallic intermediate-valence phase at T= 0.


As mentioned by Schweitzer, the onset of the discontinuous transitions is closely related to the onset of exciton formation. Similar transitions were also found by Rossler and Ramirez (1976) using a generalisation of one-dimensional technique. Their results suggested that, if the positions of the f electrons was truly random, then the Coulomb interaction Ufd would give a distribution for the f-electron energies. This distribution results in a sizeable fractional valence on either side of the discontinuous transition. This raises the question as to whether the positions of the f electrons (at T=O) are truly random (Sakurai and Schlottmann 1978). Schlottmann (1978) has identified that the Coulomb interaction Ufd may give rise to a phase in which short-range order is present. Also, the hybridisation process, neglected in the Falicov model, will help in producing a translational invariant state with uniform non-integer valence.

3.8.1.3. Virtual bound-state models: the role of hybridisation. The effect of hybridisation on the Falicov-Kimball model has been investigated by several authors. The results of these calculations indicate that not only does the bare conduction band strongly influence the systems, but so does the way in which the hybridisation processes are modelled.

The differences can be seen by considering the simple models in which the excitonic- like interactions are completely absent. The first we consider is due to Iglesias-Sicardi et a1 (1975). They assume that the degeneracy of the electronic system will destroy any coherence effects in the density of states. They choose the f density of states to be described as a set of non-interacting virtual bound states

where the width of the 4f level A is given as in the Anderson impurity model by

N E ) = Y 2 po(E)

where po(E) is the conduction electron density of states. As shown by Iglesias-Sicardi et a1 (1975) as Er approaches the bottom of the conduction band there is a jump in A(Ef) that gives rise to a discontinuous transition from a phase of integral to intermediate valence, In a non-degenerate model, the behaviour as Er is varied is quite different. For the system considered by Doniach (1974),

the canonical transformation

trivially diagonalises the Hamiltonian. The 4f density of states is given by

where

and


On varying Er through the bottom of the conduction band, only continuous transitions occur to the intermediate-valence phase.

Since the details of the conduction band density of states play a vital role in the determination of excitonic correlations these models display varied behaviour as u f d is turned on.

3.8.1.4. Hybridisation in the Falicov model. Avignon and Ghatak (1975) considered the effect of the Coulomb interaction on the virtual bound-state model. They made a further assumption that A(E) may be replaced by a constant, consistent with a smoothly varying density of states. The interaction Ufd was treated in a mean-field approximation and this resulted in both continuous and discontinuous valence transitions at T=O on varying Ef. Kanda et a1 (1976) included the energy dependence of A(E) but neglected the effect of hybridisation on the conduction band. The result of their mean-field calculation resembles those of Avignon and Ghatak (1975). Goncalves da Silva and Falicov (1975) also considered the same model within the mean-field approximation. These authors then used the limit of zero conduction bandwidth, W --f 0, for computational ease. Again they found discontinuous and continuous transitions.

Leder (1978) has considered the phase transition using the non-degenerate coherently hybridised model. This mean-field treatment included excitonic correlations of the form urd( f f d ) of the form considered by Khomskii and Kocharjan (1976). In contrast with the virtual bound-state models he only found continuous transitions, which corresponds to the smooth variation in the character of the states which make up the lower hybridised band.

The properties of the Falicov-Kimball model of valence transitions are not only sensitive to the form of the non-interacting density of states, but are also sensitive to modifications of the fd Coulomb interaction. Lin Liu (private communication) has introduced realistic screening of the Coulomb interaction, rather than postulating a ‘heavily screened’ localised interaction u f d ; again, this modifies the phase diagram in a drastic way.

3.8.2. Lattice models.

3.8.2.1. Thermodynamic treatments. As we mentioned previously, mechanisms other than that of electronic screening play a role in the determination of the phase diagram. Coqblin and Blandin (1968) were the first to discuss the effect of the electron-lattice coupling in this respect. They introduced the ‘compression shift mechanism’ to describe the a-y phase transition in Ce. They added a large elastic contribution to the energy and introduced a volume dependence to the position of the 4f levels. By minimising the enthalpy H = E+PV they calculated the P-V relation for Ce.

The effect of adding these terms is to enhance an otherwise electronic transition. For a purely electronic transition the criterion for a valence instability, at T=O, can be written as

while the more stringent criterion, found from dP/dV> 0, can be expresed as

a2Elanf2 > o

a2E 2 a2E -1

Z>(a,aV) (m) * Typically the energy is assumed to be composed of an electronic and a lattice term:

E h , V > = Eednf, V ) + Elatt (nr, VI


where the electronic energy can be written as a contribution from the conduction band and the f orbitals :

E e h V)=['O,dw wp(w)+Em

and the lattice energy is assumed to be

Then, if the energy of the f level is assumed to vary linearly with volume, the criterion becomes

C E < 2 (83)". ani2 Bo aV

Physically, this criterion corresponds to a process in which the application of pressure causes a lattice compression, which increases the energy Ef of the f levels, causing the electrons to spill out in the conduction band, The associated decrease in the volume causes a further drop in the energy and the discontinuous transition.

Hirst (1974) introduced the anharmonicity of the lattice energy in the form of the phenomenological Birch formulae :

This form acts as a non-linear process which finally stops the discontinuous valence transition at a fractional valence. Varma and Heine (1975) performed a similar calculation; they pointed out that the most significant volume dependence in the electronic energy may stem from the width of the conduction band. They also used an elastic energy which depends both on the volume and the average f-electron occupation. In effect, this accounted for the different ionic volumes of the ionic states together with a term representing an interaction between the ions of different sizes.

Jefferson (1976) extended these theories to finite temperatures. He incorporated the effects of the higher excited ionic levels, the electronic entropy and the lattice entropy. His results were successful in describing the thermal expansion of Sm, Sml-sGd,S and the Sml-sYsS compound.

3.8.2.2. Microscopic treatments. Most of these theories are, at best, semiphenomeno- logical. Sherrington and Riseborough (1976) formulated the problem using a Born- Madelung model to derive a lattice energy and an electron-lattice coupling. Entel and co-workers have also derived an electron-phonon Hamiltonian, the small electronic terms being described by the Anderson lattice Hamiltonian. By treating the ff Coulomb interaction Uff in the Hartree-Fock approximation and a similar approximation for the phonon displacements, Entel and Leder (1978) are able to describe the phase boundary and the anomalous thermal expansion coefficient in both phases. These models are also able to describe the anomalies in the phonon spectra as we have discussed previously.

3.8.2.3. Effects of anharmonicity. Anharmonic properties such as that introduced phenomenologically by Hirst (1974) may even be able to drive the phase transitions by themselves. Anderson and Chui (1974) took a different approach to the problem by first considering the effects of the large lanthanide contraction. The size mismatch of the rare-earth ions results in a long-ranged interaction between the ions like the Friedel interaction between impurities in a lattice. The interaction energy is composed of a


harmonic and an anharmonic term. The harmonic interaction is found to be attractive while the anharmonic energy can be either attractive or repulsive. For SmSe and SmTe the anharmonic energy is larger than the harmonic interaction and is repulsive. For SmS both interactions are attractive, indicating a discontinuous transition in SmS, but not in the other monochalcogenides. This interaction plays a role not unlike the Falicov- Kimball interaction; the similarity extends further since in both cases the models have been formulated in the representation appropriate to inhomogeneous mixed-valence states in which each rare-earth ion has a time-independent 4f integral occupation number.

3.9. The essentially localised model

Kaplan and Mahanti (1975) have proposed a model for the mixed-valence state of the Sm compounds and have consistently worked out most of the experimental properties expected to be given by this model. The model appears to be fundamentally different from that of the Anderson lattice since most of the states involved in the model are localised. The main states of the system are localised f electrons and localised d electrons. These localised states are assumed to be mixed via an inter-atomic electric dipole interaction. In the absence of the dipole interaction, the two electrons per site would combine in the localised orbitals to produce a singlet ground state in accordance with the Hund’s rules appropriate to Sm2f and Sm3+. Kaplan et a1 (1977, 1978) treated the model in a variational manner and found a mixed-valence state for a large enough dipole interaction. In this manner the transport properties become decoupled from the mixed- valence phase transition as had been suggested by Nickerson et a1 (1971) on the basis of experimental data. The model was later extended to include a coupling to the lattice (Mahanti et a1 1976) and the compression shift mechanism. The phase diagram obtained does agree with that of the Sm monochalcogenides, and further predicts a second valence instability to an integral-valence state at a pressure of 40 kbar (Kaplan et a1 1978). This might be correlated with an anomaly in the lattice constant which occurs at 20 kbar (Guntherodt et a1 1977). The optical and transport properties of SmS require that these localised d electrons coexist in the same energy range as a broad sd conduction band. Kaplan et a1 assert that this band is relatively unpopulated with only 0.1 electrons per ion. Schweitzer (1976) has investigated the stability of the Kaplan-Mahanti ground state against a tight-binding transfer process, and he did indeed find that the system can be stable against the formation of delocalised electron-hole pair excitations. Although this model has been widely criticised, it is not as diametrically opposed to the Anderson model approach as one might first expect, since an inclusion of an fd Coulomb interaction needed in the Anderson lattice is expected to introduce excitonic correlations into states near the conduction band edges. These would tend to introduce atomic-like correlations in the d band and effectively reduce the itinerant nature of these states. Similar concepts have appeared in the theory of transition metals proposed by Stearns (1976a, b) and so the model of Kaplan et al is not without precedence. With this concept of virtual bound excitons Kaplan et a1 (1978) are able to describe the experimentally observed optical data, DC conductivity, Hall effect and Mossbauer data. It is clear that the model offers a powerful alternate theory of Sm compounds to the Anderson lattice. However, there remain many unanswered questions concerning this model, such as whether the two-electron hybridisation mechanism necessarily requires concomitant electric dipole ordering in the mixed-valence state as suggested by Robinson (1979), and also whether the theory is able to describe successfully the magnetic susceptibility of the samarium compounds, a question which has not been tackled quantitatively.

16 J M Lawrence, P S Riseborough arid R D Parks

4. Epilogue

While the body of work on valence fluctuation phenomena is large and rapidly increasing, important questions remain unanswered. A central problem is that the bare energetics- i.e. the 4f level positions, widths and occupation numbers-are largely unknown. Con- comitantly there is often substantial disagreement amongst estimates of the valence obtained by different techniques. A related key problem is whether charge and spin fluctuation rates are equal or differ. Another related unresolved issue concerns the microscopic character of the Fermi liquid behaviour; the degree and kind of coherence of the charge and spin fluctuations in the 4f band is unknown, as are the appropriate renormalised parameters TF, Uefi, etc.

The most powerful and direct experiment for measuring the bare energetics seems to be valence band photoemission, but certain difficulties with the technique require resolution. Theoretically the degree of surface sensitivity of the measurement needs to be clarified, including surface to bulk level shifts, relaxation and reconstruction; experimentally, improvements in resolution as well as in control of surface strain, cleanliness and stoichiometry would be desirable. Studies of the optical response in the far infrared might be expected to help establish the unknown charge fluctuation rates; Fermi surface studies such as de Haas-van Alphen experiments would be welcome. Neutron experiments at very small Q directed towards establishing the degree of coherence of spin fluctuations might prove valuable. Cleverly designed transport experiments might also address the question of the coherency of the charge fluctuations, but this will hinge strongly on the development of the transport theory. The recent discovery of super- conductivity in CeCuzSi~ will doubtless be an important impetus for clarification of the Fermi liquid microscopics. Finally it is clear that future experimental advances will depend strongly on improvements in our understanding of the underlying metallurgy and solid-state chemistry of the relevant materials.

On the theoretical side existing work has focused either on the magnetic properties of the ground state or on the phase diagrams of mixed-valence materials. For each of these areas many differing models and simplifications have been used. One resulting problem is that seemingly small differences, such as in the non-interacting density of states or the degree of coherence of the hybridisation, can lead to qualitatively different results. It is thus encouraging when qualitative agreement is obtained in calculations based in the extreme ends of parameter space of a given mathematical framework as, for example, in the observation of a hybridisation gap in many differing treatments of the non-degenerate Anderson lattice. A central difficulty is that the interactions involved in the problem (hybridisation, intra-site correlation, electron screening and electron- phonon interaction) are sufficiently strong that they should be treated on equal footing. Most theories sacrifice one or the other interaction in an effort to explain a single isolated experiment, whereas comparisons between a given model and many different experiments on a given material are necessary in order to separate the good simplifications from the bad. In this regard, it is encouraging that the same model has been successful in dealing with the anomalies in the phonon spectra, the phase diagram and the magnetic susceptibility of the Sm compounds. Such comparisons to differing experiments will, of course, be facilitated by the experimental observation of correlations amongst several measurements, such as susceptibilities, neutron linewidths, NMR rates, T2 coefficients in the susceptibility and resistivity and phase transition temperatures. A unified model of the physics of valence fluctuation compounds would represent a significant advance for condensed matter physics.

Valence puctuation phenomena 77

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