Solid State Physics: Advances in Research and Applications, Vol. 41

SOLID STATE PHYSICS VOLUME 41

Founding Editors

FREDERICK SEITZ DAVID TURNBULL

SOLID STATE PHYSICS Advances in Research and Applications

Editors

HENRY EHRENREICH

DAVID TURNBULL

Division of Applied Sciences Harvard University, Cambridge, Massachusetts

VOLUME 41

ACADEMIC PRESS, I N C . Harcourt Brace Jovanovich, Publishers San Diego New York Berkeley Boston London Sydney Tokyo Toronto

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0 1988 BY ACADEMICPRESS, INC.

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NUMBER: 55-12200

Contents

CONTRIBUTORS TO VOLUME41 .............................................. PREFACE ................................................................

vii ix

Theory of Heavy Fermion Systems PETER FULDE. JOACHIM KELLER.AND GERTRUD ZWICKNAGL 1. Introduction ........................................ I1 . Formation of the Singlet State ........................ ................................. 111. Quasiparticle Bands ........... I V. Quasiparticle-Phono ................. V. Quasiparticle Interactions and Fermi Liquid De VI . Microscopic Theories ........................... VII . Superconductivity .................................................. Appendix A: Molecular Model for Strongly Correlated Electrons . . . . . . . . . Appendix B: Parametrization of the Model Hamiltonian . . . . . . . .

2 7 22 40 63 71 103 144 147

The Theory and Application of Axial king Models JULIAYEOMANS

I . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Experimental Applications ........................ . . . . . . . . . . . . . . . . . . .

151 180

Excitations in Incommensurate Crystal Phases R. CURRAT AND T . JANSSEN I. I1. 111. I v. V. VI . VII .

Introduction . . . . . . . . . Landau Theory of Mo ................................ Supersymmetry and Higher-Dimensional Space Groups . . . . . . . . . . . . . . . . . Microscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-Wavelength Excitations in Composite Systems ..................... Experimental Results .................................. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECTINDEX ........................................................... V

202 211 225 236 260 264 301

303 313

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Contributors to Volume 41

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

R. CURRAT, Institut hue-hngevin, Centre de Tri 156X, F-38042 GrenobleCedex, France (201) PETERFULDE, Max-Planck-Institutfur Festkorperforschung,D-7000 Stuttgart SO, Federal Republic of Germany (1) T. JANSSEN, Institute for Theoretical Physics, University of Nijmegen, nernooiveld, 6525 ED Nijmegen, The Netherlands (201)

JOACHIM KELLER,Fachbereich Physik, Universitat Regensburg, 0-8400 Regensburg, Federal Republic of Germany (1) JULIA YEOMANS, Department of TheoreticalPhysics, Oxford OX1 3NR England (151) GERTRUD ZWICKNAGL, Institut fur Festkorperphysik, TH Darmstadt, 0-6100

Darmstadt, Federal Republic of Germany, and Max-Planck-Institutfur Festkorperforschung,0-7000 Stuttgart SO, Federal Republic of Germany (1)

vii

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Preface

The recent discovery of high-temperature superconductors and that of quasicrystals have underlined the importance of theories that are able to account for the physical properties, including the phase transitions, of materials that are inherently complex for chemical or structural reasons. This volume discusses examples of such theories and theoretical models that are applicable to two classes of such materials: the heavy fermion (or heavy electron) metals, so named because of their extraordinarily large electron effective masses, and modulated structures that have magnetic or structural modulations superimposed on the underlying three-dimensional periodic structure. These modulations can be commensurate or incommensurate with the basic units and may have arbitrarily long wavelengths. Heavy fermion systems contain rare earth or actinide atoms (typically cerium or uranium) and exhibit highly unusual low-temperature properties including giant electronic specific heats, unexpected magnetic phenomena, and even superconductivity. The article by Fulde, Keller, and Zwicknagl discusses many of the important aspects of the theory of such systems. As the authors point out, their article does not represent a complete review. It emphasizes theory and deals relatively little with experiments. The article provides a clear exposition of many of the principal theoretical ingredients, with sufficient emphasis on simple physical arguments so that it is accessible to nonspecialists, and in particular, to graduate students. A number of theoretical features distinguish heavy fermion metals from ordinary metals. Conventional band theory does not work. It is instead necessary to introduce quasiparticle bands and to deal realistically with the interactions among quasiparticles. This requires modifying conventional Landau theory, as it applies to 3He, to anisotropic systems. The modifications of band theory are discussed in considerable detail, as are microscopic theories whose aim is to obtain the generalized Landau parameters appearing in the theoretical framework from ab initio considerations. The microscopic theories, based, respectively, on perturbation expansion techniques, mean-field approximations, and variational approaches are given a detailed and pedagogically clear exposition. The elastic properties of heavy fermion systems represent one of the emphases of this article. The electron-phonon interaction in such systems is therefore developed in some detail. This interaction too differs from that found in conventional metals because Migdal’s theorem does not hold when ix

X

PREFACE

the electron effective masses are so large. The superconducting properties are also novel. For example, UBe13containing a few percent thallium appears to exhibit two superconducting transitions of second order. This phenomenon is possibly associated with different parts of the Fermi surface acquiring superconducting properties at different temperatures. As in the case of the new high-temperature superconductors, the nature of the electron pairing or that of the mediating attractive interaction is still not settled. The articles by Yeomans and by Currat and Janssen discuss various aspects of modulated systems. Materials such as cerium antimonide exhibit modulated magnetic phases, whereas in silicon carbide the modulation is structural. More generally speaking, the materials of interest are polytypes, that is, compounds in which one or more structural units can be stacked in different ways to form several stable or metastable phases. The spinelloid structural family (AB204, where A and B are cations, such as nickel or aluminum) and the polysomatic series (for example, biopyriboles) are two examples discussed in the articles. The axial next nearest neighbor Ising (ANNNI) model, discussed in Yeomans’ article, is the most familiar of the discrete spin models whose phase diagram contains series of commensurate and incommensurate modulated phases. This richness of phenomena is associated with competing interactions between nearest neighbors on the one hand, and next nearest neighbor interactions along one particular lattice direction on the other. As pointed out, the model is readily generalized to include farther neighbor interactions, effects favoring chiral ordering, and the simulation of quenched impurities, which are of importance to the stability of the modulation. The ANNNI model is of interest because it provides a mechanism for the existence of polytypes as equilibrium or highly metastable structures. Its predictions will surely stimulate further experimental investigations of the stability and kinetics characterizing the phase transitions of these materials. Yeomans’ article is divided into two parts. The first half reviews the theory of the ANNNl model. It contains a general theoretical overview designed to make the article accessible to those experimentalists less concerned with theoretical detail. The second half of the article describes the experiments relevant to the various systems of interest and their interpretation in terms of the theoretical model. In a related article, Currat and Janssen discuss the excitations in incommensurate crystal phases from both theoretical and experimental viewpoints. The results of neutron and inelastic light scattering are discussed in some detail for a variety of materials (P-ThBr,, deuterated biphenyl, and K2SeO4),some of them related to those in Yeomans’ review. The excitation of spectrum of such systems exhibits a number of distinctive features, such as the persistence of soft phase fluctuations and the gradual opening of gaps in various parts of the Brillouin zone. However, experimental investigations of these efforts are not yet far advanced. Furthermore, the observation of long-wavelength phasons by either technique presents special difficulties.

PREFACE

xi

Many aspects of incommensurate crystal phases can be understood by means of the phenomenological Landau theory of phase transitions. As also stressed in the article by Fulde et al., there is an obvious need for microscopic models to explain the phenomenological parameters that appear in the Landau theory. The models discussed here include that of Frenkel-Kontorova or Frank-van der Merwe, originally proposed for dislocations or epitaxial monolayers, and the so-called Discrete Frustrated +4 Model (consisting of a one-dimensional chain embedded in a three-dimensional crystal), which is closely related to the ANNNI model. The discussion presented here extends that of the preceding article in its emphasis on dynamical properties. As the authors point out, the formalism used to describe excitations of incommensurate displacively modulated crystals is also applicable to other quasiperiodic systems, for example, quasicrystals. There are, however, significant differences because of strong diffusive and nonlinear effects that influence many of the elementary excitations. The study of the material systems discussed in the reviews appearing in this volume evidently pose many challenging opportunities for both theorists and experimentalists. HENRYEHRENREICH DAVIDTURNBULL

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SOLID STATE PHYSICS. VOLUME

41

Theory of Heavy Fermion Systems PETERFULDE Miiu-Plonc~-lngtrtirif u r Festkorperfimchuny. 0-7000Stuttgur f 80. Federal Republic of Germany

JOACHIM KELLER Fachhereic h Phvbrk. Unit~ervtaiReyenshury.

D-8400 Regen rburg. Federal Republic oj Germany

GERTRUD ZWICKNAGL

/......*../... ../&.. 0-6 100 Darmstadt. Federal Repuhlir 01 Germanv. and PI .A.r,>

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Maw-Planck-lnstrtuf fur Festkorperforschung. 0-7000 Stuttgart 80. Federal Republic of Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Formation of the Singlet State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Quasiparticle Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Renormalized Band Theory: Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Renormalized Band Theory: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Discussions and Model Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Quasiparticle- Phonon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Thermodynamic Relations . . . . . . . . . . . . . ........................... 5. Quasiparticle- Phonon Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 6 . Hydrodynamic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Quasiparticle Interactions and Fermi Liquid Description ...................... VI . Microscopic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. ExpansionTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Mean-FieldTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Varialional Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V11. Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Microscopic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Properties of Different Pair States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Molecular Model for Strongly Correlated Electrons . . . . . . . . . . . . . . . Appendix B: Parametrization of the Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . .

1

7 22 25 30 34 40 41 45 52 63 71 74 88 98 103 103 104 119 137 144 147

.

Copyright I( ,1988 by Academic Press Inc . All righls of rcproduciion In any rorm rcserved.

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PETER FULDE et al.

1. Introduction

Over the past few years a new branch of metal physics has emerged which is rapidly developing. It deals with systems that contain rare earth (predominantly Ce) or actinide (predominantly U ) ions and that have very unusual lowtemperature properties. Below a temperature T * , which is of the order of tens of kelvin, they behave like Fermi liquids. They have a linear specific heat and an (almost) temperature-independent Pauli spin susceptibility with values that correspond to huge electron densities of states. The latter are due to exceptionally large conduction electron masses. In fact the effective masses can become as large as a few hundred times the free electron mass. This behavior changcs completely for temperatures much larger than T*. In that region the systems can be described best by conduction electrons with conventional masses and in addition by well-localized f ’ electrons. The latter have a magnetic moment associated with them, and the susceptibility is therefore of Curie type. Both types of electrons, i.e., the “light” conduction electrons and the localized f’ electrons, interact by means of a standard exchange type of interaction or a generalization of it. Therefore, what apparently happens is that at sufficiently low temperatures the system gains energy by losing the magnetic moments associated with the .f electrons. A t the same time the f ’ electrons become part of the Fermi surface, which implies that they cannot be perfectly localized. Rather they have to be considered as part of the Fermi liquid. Due to the presence of low-energy excitations or quasiparticles with large effective mass at T << T * , these systems have been termed “heavy fermion” systems. We shall adopt this nomenclature in the following. Alternative names which have been suggested are heavy electron systems or strongly interacting fermion systems. A situation where, due to an energy gain, a magnetic moment is lost was first encountered in the Kondo effect. It relates to a single magnetic impurity embedded in a sea of conduction electrons. Due to strong many-body effects such an impurity can lose its magnetic moment and form a singlet ground state with the surrounding conduction electrons. The associated energy gain is characterized by kBTK,where the Kondo temperature TKcan vary over a broad range, i.e., from parts of a degree to hundreds of degrees. Connected with the singlet formation is an excess specific heat and spin susceptibility contribution or, alternatively, a contribution to the local density of conduction electron states. In the heavy fermion systems it seems that some of the features of the single Kondo impurity problem appear in an enhanced form simply because there is at least one magnetic ion per unit cell. For example, the huge density of states at T << T * which is connected with the large effective mass of the quasi-

THEORY OF HEAVY F E R M I O N SYSTEMS

3

particles is of the same order of magnitude as the one that would be obtained from a single Kondo ion multiplied by the ion concentration. Therefore, the heavy fermion systems have sometimes been referred to as “lattices of Kondo ions” or “Kondo lattice systems.” However, this requires one more comment in order to prevent misunderstandings. The original Kondo problem consists in treating the effect of an external (impurity) spin placed into a conduction-electron system. The formation of a singlet state takes place by a compensation of the impurity spin through the spins of the conduction electrons. The interaction between the two is described by the Kondo Hamiltonian. However, as the energy gain k”TK due to the singlet formation becomes larger, this picture becomes increasingly inadequate. It turns out that it is no longer permissible to consider the spin of the impurity as an external quantity, as is the case when the f electrons, which generate this spin, are perfectly localized. lnstead it is more appropriate to consider the .f electron number 17./ at the impurity site as noninteger. I t turns out that the larger kllTKis, the greater are the deviations from an integer number. Therefore, as the latter energy increases, it seems more appropriate to talk about a valence fluctuating system. The border line is, of course, not sharp. Even for small values of k,T, the f electron number is strictly not an integer. But the deviations are sufficiently small so that at least in the single impurity case one can neglect them and work instead with the Kondo Hamiltonian. I t is important to realize that in the heavy fermion or Kondo lattice case that is inappropriate even when k,T* is small. If we were to keep n f as an integer and, therefore, the .f electrons as perfectly localized, they would not be able to participate in the formation of the Fermi surface. Therefore, it is necessary to work with a Hamiltonian which allows for a variable f electron number at a given rare earth or actinide site. Such a Hamiltonian is the Anderson lattice Hamiltonian, which will be often used in the following. When the expression “Kondo lattice system” is used we have therefore in mind a system with noninteger f electron number. Also, it is not implied that T* is the same as TK.Rather one expects in most cases T* < TK.Due to magnetic interactions between different rare earth or actinide sites, the energy gain per site duc to singlet formation should be less than for a single ion. The reason is that one is also losing with the magnetic moments the magnetic interaction energy betwecw sites. This suggests immediately that the singlet formation, and hence heavy fermion behavior, should not occur when the antiferromagnetic intersite interactions are sufficiently strong. In that case the disappearance of the magnetic moment would lead to an overall energy loss instead of an energy gain. Systems for which this seems to be the case are, e.g., CeAI,, CePb,, or NpBe,,. Calculations on two Kondo impurities with ferromagnetic intersite coupling have shown that there the situation differs. If there is strong coupling

4

PETER FULDE ct

a/

between the two sites, then a quenching of the moment of the total twoimpurity complex is always accompanied with an energy gain. There is another important difference between the singlet formation for a single impurity ion and a lattice of ions, respectively. In the single ion case all physical properties scale with respect to the Kondo temperature TK,provided one is in the Kondo limit, i.e., nf is sufficiently close to an integer [for Ce3+ the criterion is (1 - n,) < 0.1 51. The Kondo temperature characterizes thereby all “material” properties, such as f level position and width of the ion. We cannot expect the Kondo lattice to be universal in that sense. Instead additional energy scales appear, as has been pointed out above. The scenario within which the heavy fermion systems should be viewed is shown in Fig. 1. It shows the different degrees of electron correlations in solids. The heavy fermion systems are the ones which come closest to the localization limit, in which electron correlations exclude any charge fluctuations within the j ’ shell. The latter increases as one goes from left to right. On the extreme righthand side electron correlations play no role except for the RPA type of screening, which takes place in metals and concerns the long-range part of the Coulomb interactions. Heavy fermion systems pose a number of new and basic questions which have been answered until now only to a small extent. We list a few of them in order to provide an overview of the topics which in the following sections will be discussed in more detail. One problem is the different energy or temperature scales. It has been pointed out above that one characteristic temperature of the system is T*. When multiplied by k , it is a measure of the energy gain per Ce or U site due to the singlet formation. When bands of heavy quasiparticles form at low temperatures with a width of order k,T*, the quasiparticle density of states

I heavy fermion regime

2simple

-

metals e g semiconductors

4

d

transition metals

FIG. 1. Schematic display of the strength of electron correlations. The on-site Coulomb repulsion I/ of electrons in units of their band width W can be used as a measure of the importance of electron correlations. Different types of electrons are roughly positioned as indicated. In the limit of localized electrons, charge fluctuations of electrons are completely suppressed.

THEORY OF HEAVY FERMION SYSTEMS

5

will show considerable structure and a number of peaks as for any other energy-band structure. These peaks result from the chemical structure of the compounds and from interference effects of scattered waves in the same way as, e.g., the peaks in a d-electron density of states of a transition metal. The widths of these peaks are generally much less than k,T*. When the Fermi energy falls into one of them of width k,T, there will, therefore, be another energy scale T, << T* in the problem. It results from lattice coherence effects. Whether or not these are the only energy scales is unclear at present. Other problems concern the determination of quasiparticle band structures in heavy fermion systems. Clearly, due to the strong electron correlations, conventional band-structure calculations which apply the local-density approximation to the density-functional approach for computing the exchangecorrelation potentials are not expected to work here. The question is, then, what will take their place? Of great importance is fw+ermore the problem of interactions between quasiparticles. They can be of different origin. First, there are the direct interactions which are usually described in terms of Landau’s Fermi liquid parameters. They describe the internal molecular fields set up by the quasiparticles. The conventional Landau theory of Fermi liquids has been worked out for isotropic one-component systems such as 3He. The question arises then how it must be modified in order to apply to strongly anisotropic systems with several types of quasiparticles (e.g., with strongly varying effective masses). In addition to the interactions described by the Landau parameters, there are indirect interactions taking place via phonons. The coupling of quasiparticles to phonons is an interesting problem in itself. It turns out that there are coupling mechanisms present that do not occur in ordinary metals. They result from a strong volume or pressure dependence of the characteristic temperature T* and affect not only the quasiparticles but also the phonons. Therefore, the elastic properties of heavy fermion systems turn out to be extremely interesting. Examples are the temperature and magnetic field dependence of the elastic constants and the ultrasound attenuation. In a number of cases they can be described by a hydrodynamic theory into which only static thermodynamic quantitites (e.g., specific heat, thermal expansion, etc.) and transport coefficients (e.g., electrical or heat conductivity) enter. It allows for relating different experimental observations with each other without having to specifiy the Hamiltonian. It turns out that the hydrodynamic fluctuations and the ultrasound attenuation mechanism are very different from their counterparts in ordinary metals. Because the elastic properties of heavy fermion systems are so fascinating and because they are usually underrepresented, we shall devote to them a sizeable fraction of this article.

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PETER FULDE

EI rtl

Another important subject within the theory of heavyfermion systems is the different microscopic theories which have been proposed for dealing with strongly correlated systems. They can be roughly divided into perturbational expansion techniques and variational methods. When applying expansion techniques, one would like to expand with respect to the hybridization between the .f‘ electrons of a rare earth or actinide site and the surrounding conduction electrons. For almost localized .f‘ electrons this seems a natural expansion parameter. Since in that case the large on-site Coulomb repulsions between the f electrons are contained in the “unperturbed” Hamiltonian, one is facing the problem that neither Wick’s theorem nor a linked cluster expansion holds. Dealing with this problem is the central issue of the expansion techniques. Of particular importance within the microscopic theories has been a mean-field approximation. Within that scheme the heavy fermion state is a broken symmetry state in analogy to the BCS state in superconductivity. In both cases the invariance of the Hamiltonian under certain phase changes is broken. In the BCS theory the symmetry breaking occurs because the BCS state is not an eigenstate to the total electron number. In analogy, the symmetry breaking in the heavy fermion system is due to the nonconservation of a quantity which states in the case of Ce that the .f shell is either occupied with one f’ electron or otherwise is empty. Naturally, there has been a considerable amount of discussion as to which interaction is responsible for the occurrence of superconductivity and as to the nature of the paired state. From the size of the jump in the specific heat at the superconducting transition temperature one can conclude that the Cooper pairs are formed from the heavy quasiparticles. As regards the interaction mechanism, it has been demonstrated (at least for CeCu,Si,) that the quasiparticle-phonon interaction is strong enough to lead to a T, of the observed size. That does not proue, though, that it is the interaction which leads to superconductivity. Other interaction mechanisms which have been invoked for the explanation of superconductivity include the exchange of overdamped spin fluctuations between quasiparticles. Coupled with the problem of the interaction is the problem of the nature of the pairing state. The suggestion has been made that in heavy fermion systems one is dealing with “unconventional” superconductivity. In this article we shall consider unconventional superconductivity as synonymous with an order parameter which has a lower symmetry than the Fermi surface. When used in this sense there is at present no proof for unconventional superconductivity in heavy fermion systems. Certainly it cannot be excluded either, at least in systems like UPt,, in which the mean free path of the quasiparticles is sufficiently long. Mean free paths shorter than the superconducting coherence length are expected to have a strong pair-breaking effect and in fact should prevent unconventional superconductivity from occurring. On the other hand, one can definitely state that superconductivity in heavy fermion systems has a

THEORY OF HEAVY FERMION SYSTEMS

7

number of new and interesting aspects. For example, no measured quantity has shown an exponential falloff at low temperatures, as one would expect if a sizeable gap for the elementary excitations extended over the entire Fermi surface. In UBe, doped with a few percent of Th there is evidence for two superconducting transitions of second order. Below the higher transition temperature the system acquired an order parameter A,, to which below the second transition a second order parameter Ab is added. The most plausible explanation is that with decreasing temperature first a part of the Fermi surface becomes superconducting before the remaining part joins in below the second transition. Phenomena like this are new and pose new questions independent of whether one is dealing with conventional or unconventional superconductivity in the above sense. The purpose of the present article is to discuss a number of the problems just outlined. Since we deal with the theory of heavy fermion systems, the allimportant experiments on that subject are only discussed very little. There are a number of reviews available of the experimental situation, to which we refer those readers who want to learn about the hard experimental facts which theory has to We also want to stress that the present article should be considered as one which concentrates on particular aspects of the theory of heavy fermion systems rather than giving a complete review. Partial reviews of the theory of heavy fermion systems were recently presented by Varma’ and Lee et a/.6The reader may also want to consult some of the classical reviews on the single-ion Kondo effect, e.g., Kondo,’ Gruner and Zawadowski,* and Wilson.9

II. Formation of the Singlet State An important step toward an understanding of heavy fermion systems is the study of the following problem: assume that an impurity with one electron in a well-localized, e.g., ,f orbital, is set into a metal. What will be the

’ G. R. Stewart, Rcu. Mod. f/7.v.s. 56, 755 (1984). ’ F. Steglich, in “Theory of Heavy Fermions and Valence Fluctuations” (T. Kasuya and T. Saso, eds.), p. 23. Springer-Verlag, Berlin, 1985. Z. Fisk, H. R. Ott, T. M. Rice, and J. L. Smith, Nature (London)320, 124 (1986). H. R. Ott, Proq. Low Tmlp. P/iy.s. 1 I , in press (1987). C. M. Varma, Con7/nen/sSolid S/CI/P Phys. 1 I , 221 (1985). P. A. Lee, T. M. Rice, J. W. Serene, L. J. Sham, and J . W. Wilkins, Cornrncw/.sCondens. h4rrllc.r P/rj,x. 12, 99 ( I 986). J. Kondo. in “Solid State Physics”(F. Seitz, D. Turnbull, and H. Ehrenreich, eds.), Vol. 23, p. 183. Academic Press, New York, 1969. G. Gruner and A. Zawadowski. Rep. f r o g . P/7y.s. 37, 1497 (1974). K. G. Wilson, Rev. Mod. Phvs. 47, 773 (1975).

’ ‘ ’

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PETER FULDE et d.

ground state of the system when the overlap of that orbital with the conduction electrons is small but nonzero? It turns out and will be shown below that the system gains energy by forming a singlet state. The alternative would be a degenerate ground state, i.e., a Kramers doublet or a higher multiplet, depending upon to what extent the f-electron orbital degeneracy is lifted by spin-orbit and crystal-field effects. The size of the energy gain due to the formation of a singlet depends, of course, on the details of the situation, in particular on the hybridization matrix element, the density of conductionelectron states, and the degeneracy of the localized orbital. It is customary to associate a characteristic temperature with it and to write it in the form kHTK,where k , is Boltzmann's constant. T K is called the single-ion Kondo temperature. Of particular interest are cases in which TKis of order T K N 1-10 K. Next one can go to the case of two impurities embedded in a metal. One would like to know how the energy gain due to the singlet formation changes as compared with that of a single impurity. What are the interaction effects and on what do they depend? Finally one wants to go to a system in which the ions with one electron in a well-localized orbital form a lattice. Our physical picture of heavy fermion systems assumes that a singlet formation takes place at each site. This is considered to be the reason why, e.g., the magnetic susceptibility at low temperatures is not Curie-Weiss-like, as expected for localized electrons with a magnetic moment, but instead Paulilike, as in nonmagnetic metals. In order to distinguish the energy gain per site in the lattice case from that of an isolated impurity, we denote the former by k,T*. Naively one expects T* < T K .Antiferromagnetic interactions between different sites, which are present in the lattice case, are quenched when the singlet is formed. When they are sufficiently strong a singlet formation would cause a net energy loss. In that case the system will become an antiferromagnet at sufficiently low temperatures instead of a heavy Fermi liquid. In the following we demonstrate the formation of the singlet state for the one-impurity problem. Thereby we have a Ce3+ ion in mind for which the f-electron occupancy nf is slightly less than one. The degeneracy of the f orbital is denoted by v f . We shall first neglect spin-orbit and crystalline electric field (CEF) effects but discuss them afterwards. This implies choosing v, = 14. We start from the single-impurity Anderson model

H

=

E(k)c:acka ko

+ E/

nd m

U +ndn;. 2 mfm'

kma

The operators c:,(ckr,) create (annihilate) conduction electrons with energy dispersion E ( k ) in momentum and spin states k, c. All energies will be mea-

9

T H E O R Y OF HEAVY F E R M I O N SYSTEMS

sured with respect to the Fermi level E,. The operators for the f ' electrons at the impurity site are f:(j,), respectively, with m = 1,. . . , vs, the number operators n i = fLfm. The position ef of the f level is supposed to be well below the Fermi energy E , . The Coulomb repulsion between two f ' electrons is U . Since we want to exclude that there be more than one f ' electron at the impurity site, we shall assume later that U + co. The last term on the righthand side describes the hybridization. The interesting physics stems from the coupling of the conduction electrons to the Ce impurity. We will therefore keep those degrees of freedom that couple directly to the impurity. This is achieved the following way: We observe that the hybridization matrix element Vmg(k)is a rather rapidly varying function of the angle i . This fact implies that the angular average

is rather small unless m Bringer and Lustfeld.".'

T

= m'.A

detailed discussion of this point is given by

' Following these authors we assume that

[z

KC(k)*vm,o(k) = V2(k)6mm,

(2.3)

Based on this assumption we introduce a new orthogonal electronic basis

fi, describes the dynamics of the conduction-electron degrees of freedom that do not couple to the impurity. They are irrelevant to the following discussion and will be omitted. We are mainly interested in the "local moment regime" where the f occupancy of the Ce ion is nearly one (despite broadening) and where the Ce ion possesses a magnetic moment at high temperatures. This regime is characterized by the condition

I ~ , >>~ vsr I = v f n ~ ( ~ ) v 2 , v = v(/+)

(2.6)

where N ( 0 ) is the conduction-electron density at the Fermi level per spin. I" 'I

A. Bringer and H. Lustfeld, Z. P ~ J YB: . Conrims. Mutter 28, 213 (1977). H. Lustfeld a n d A. Bringer, Solid Sratr C'omniirn. 28, 119 (1978).

10

PETER FULDE e/ r r l

IQO FIG.

)

f&CkrnI%))

2. States out of which the trial function It/jy:r=o)

[see Eq. (2.7)] is constructed

The ground state and thermodynamic properties, such as the enhanced specific heat and magnetic susceptibility of the Hamiltonian [Eq. (2.5)] and various related models, were successfully calculated using the numerical renormalization group’.’’ and the Bethe ansatz method (see, e.g., Kawakami and Okiji,13 S ~ h l o t t m a n n , ’and ~ the reviews by Andrei et a1.l’ and Tsvelick and WiegmannI6. These methods yield exact results. As such they serve as important references against which approximate solutions have to be tested. In the present section we focus on the ground state of the model Hamiltonian [Eq. (2.5)]. The derivation follows closely the one given by Varma and Yafet,I7 with the exception that we allow here also for degeneracies vf # 2.‘x-20 The important advantage is that it provides a physical understanding of its properties in simple terms. In addition, the mathematical scheme is sufficiently transparent; it is asymptotically correct for small values of 1 /vJ. We want to calculate the approximate ground-state wave function of the above Hamiltonian by limiting it to a space which is spanned by the states shown in Fig. 2. This leads to the following ansatz

H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phy.s. Rev. B: Cot2r/~ws. MutlerZI, 1003 and 1044 (1980). l 3 N. Kawakami and A. Okiji, in “Theory of Heavy Fermions and Valence Fluctuations” (T. Kasuya and T. Saso, eds.), p. 57. Springer-Verlag, Berlin, 1985. l 4 P. Schlottmann, in “Theory of Heavy Fermions and Valence Fluctuations” (T. Kasuya and T. Saso, eds.), p. 68. Springer-Verlag, Berlin, 1985. I s N. Andrei, K. Furuya, and J. H. Lowenstein, Rco. Mod. Phys. 55, 331 (1983). A. M. Tsvelick and P. B. Wiegmann, Adv. Phys. 32,453 (1983). ” C. M. Varma and Y. Yafet, PIIJS.R w . B: Condens. Mrr//er 13, 2950 (1976). I’ T. V. Ramakrishnan and K . Sur, Phys. Rev. B; Condens. Motrer 26, 1798 (1982). P. F. de Chitel, So/idStu/e Commun. 41, 8 5 3 (1982). 0 .Gunnarsson and K. Schonhammer, Phys. Reo. B: Condens. M n / / e r 28, 4315 (1983).

‘‘

THEORY OF HEAVY FERMION SYSTEMS

11

Here I@,) is the filled Fermi sea of the conduction electrons. It has total spin S = 0. Therefore, the linear combination of states f L c k m l O o has ) to be such that the singlet character is preserved. The energy of this state has to be compared with that of the multiplet

Ik)

(2.8)

= fLl@o)

The total electron number of the states I$s/s=o) and I$m) differs by one. This is of no importance if we use the Fermi energy to define the zero point of energy. The interesting and important point is that the energy of the singlet ($s=o) The latter is given by is always lower than that of the multiplet Emagn

=

EO + EJ

(2.9)

where E , is the energy of the Fermi sea (Qo). The energy of the singlet is written in the form (2.10) Here E has to be determined variationally by requiring that

6(H) 6 ( A M k( ) )

(2.1 1)

=o

This results in two coupled equations E =

-Ef+&VCM(k) (2.12)

k

EC((kj= J./ v - E ( k ) a ( k ) where the k dependence of V has been neglected for simplicity. The solution of these equations can be written in the form (2.13) By plotting the right- and left-hand side of this equation as a function of E (see Fig. 3), it is seen that three solutions exist when V is sufficiently small. The requirement is N ( 0 ) V 2<< 1 ~ ~ One 1 . of them has E < 0, which can be determined approximately from -cf = l y ~ ( 0 ) V In(D/JcJ) ’ (2.14) An energy cutoff D has been introduced which equals half the conductionelectron bandwidth. The energy by which the singlet is lower than

I$s =o)

12

PETER F U L D E ef

(I/.

FIG.3. Graphical solution of the integral equation, Eq. (2.13). One notices that there is always a solution with E < 0.

is therefore E =

-Dexp[-l~,.l/v,N(O)V~]

(2.15)

It is customary to associate with this energy gain a temperature TK,the Kondo temperature. It is usually written as

kH~‘,

=

Dexp(-nlEsl/vsr)

(2.16)

The energy gain due to the formation of a singlet state is the essence of the Kondo effect. Formally Eq. (2.16) reminds one of the energy gain in a superconductor due to the formation of Cooper pairs (see, e.g., de Gennes”). Clearly, Eq. (2.16)cannot be obtained from expansion in the hybridization I/. The form of TKderived here agrees with the one which follows from lowestorder scaling theory.22 Improved versions of scaling theory result in some modifications of the prefactor in Eq. (2.15). From Fig. 2 it is apparent that the normalization constant A in Eq. (2.7) satisfies IAI2 = 1 - n f (2.17) Thestate I@,) with theflevel unoccupied has thereforelittle weight in It,hs/s=o). This weight goes to zero as n , approaches 1. Nevertheless, it is vital for the singlet formation. The normalization requirement for It,bss=o) yields a relationship between TK and ( 1 - n,.). It implies A =1 / J m (2.18) where C is given by

= V,(N(O)V2/€) ’I

”

(2.19)

P. G. de Gennes, “Superconductivity o f Metals and Alloys.” W. A. Benjamin, New York, 1966. P. W. Anderson, J . Phys. C 3,2436 (1970).

13

THEORY OF HEAVY F E R M I O N SYSTEMS

By combining Eqs. (2.17)-(2.19) with (2.16), one finds that (2.20) The energy gain due to the singlet formation is therefore linear in (1 - ns) when V is kept constant. Another characteristic feature of the singlet ground state of Eq. (2.7) is a finite, though large, magnetic susceptibility ximpof the impurity. The lowtemperature saturation value of ximpcan be determined by computing the ground-state energy as a function of an external magnetic field and differentiating it twice with respect to the latter.20 In doing so, we must account for the fact that a magnetic field h removes the degeneracy of the f level. The energies of the split levels are given by 6s - gjp"Wlh,

-J I m5J

(2.21)

This implies that the energy difference AE(h) = E,(h) - E , is given by the solution of the following equation [compare with Eq. (2.1 3)]

Differentiating twice with respect to h yields the magnetic susceptibility ximP = -

-AE(h)l a2

ah2

h-0

(2.23) One notices that x i m p depends very sensitively on ns when ns approaches unity. The expression (2.23) does not contain enhancements which may result from quasiparticle interactions and which usually are described by a Landau parameter F ; # 0. In order to include them, one must go beyond the simple ansatz (2.7) and include corrections of order O(l/v,-) [see Eq. (2.29)]. We want to discuss how the above considerations have to be modified when the spin-orbit splitting of the f states and a splitting due to the presence of a crystalline electric field are taken into account. When there is one 4f' electron, spin-orbit interactions split the f levels into the J multiplets J = $ and $. The splitting energy Aso is typically of order 0.2 eV and, therefore, much larger than k , TK.The C E F splitting energy of Ce3 ions is usually of the order of hundred to a few hundred kelvin. Therefore it can be considered as being large as compared with TK.Let us first neglect the CEF splitting and consider the spin-orbit splitting only. Then the singlet state is +

14

PETER FULDE et a/.

constructed from a trial wave function of the form

(2.24) The .f states have been labeled according to J and M ( =J,), while the conduction electrons are classified according to k, 1 = 3, J and M . It is easily checked that Eq. (2.13) is now replaced by

The degeneracies of the J = $ and $ multiplets are denoted by vs,2 ( = 6) and v 1 / 2 ( = S), respectively. The solution of that equation iszo vl/zlvs/r

E =

-D($)

expl-

+/

I/.s/2rl

(2.26)

This result is interesting for the following reason. It demonstrates that one has to account for all excited states of the f-electron system, when one is calculating the energy gain due to the singlet formation. As long as the excitation energies are less than D,a condition which is certainly fulfilled by the spin-orbit splitting energy, they contribute substantially to E (see also S ~ h l o t t m a n n ~For ~ ) . example, for As,, = 0.2 eV and D = 3 eV the prefactor due to a high-lying J = $ multiplet is (D/Aso)(v1,2/vs,2) = 37. The population of this multiplet is small, however. At T = 0 the ratio of the occupation numbers of the two multiplets is given by (2.27) with E c Aso in most cases. When the CEF splitting of the lowest J multiplet is taken into account, the effect on the energy gain of the singlet state is similar to that of spin-orbit coupling. For example, when the ground state J = $ multiplet is split into three Kramers doublets with excitation energies and respectively, one

P. Schlottmann, 2. Phvs. B: Condens. Mutter 52, 127 (1983). P. Nozieres and A. Blandin, J. Phys. 41, 193 (1980). " K. Hanzawa, K. Yamada, and K. Yosida, J. Magn. Magn. Mafer. 478~48,357(1985) Z h D. Cox. Ph.D. thesis. Cornell University, Ithaca, New York, 1985.

23

l4

THEORY OF HEAVY FERMION SYSTEMS

15

In hcavy fermion systems one expects in most cases that A!&) >> /el, because ACEFis of the order of 100 K or more. This assumption has not been used in Eq. (2.28), in order to show how the expression reduces to Eq. (2.26) as the CEF splitting goes to zero. Having discussed the modifications which arise when the orbital degeneracy of the f' electrons is lifted, we want to direct our attentions toward possible extensions of the space of states within which the ground-state wave function is calculated. In the foregoing this space was spanned by the states IQ0) and ,fLocknIOO), respectively. The states which have to be included next for the construction of are C:mCkrrn~@O) (2.29a)

I$ s=o)

c ktm C k ' m C k " m ' f L ' I @ ) O )

(2.29b)

and in one-to-one correspondence (2.30a) C:mCk'm'fL'l@O>

(2.30b)

for the construction of I$,,). It turns out that with increasing orbital degeneracy ty these additional states become less important. In fact, in the limit vf -+ co with vf V 2 kept constant, the ansatz of Eq. (2.7) for It)ss=o)becomes exact.27 This follows from the observation that the hybridization part of the with vf different states fickm\@o). Hamiltonian of Eq. (2.5) connects I@,) But each of these states is connected with only state C L , , , C ~ . ~ ~ @ ~ ) .The relative weight of the latter state vanishes, therefore, like v;'. Since C ~ , , , C ~ , , , , ( @ ~ ) is through hybridization again connected with v, different states of the form of Eq. (2.29b), it follows that Eqs. (2.29a) and (2.29b) have to be considered is extended. Similar simultaneously when the ansatz in Eq. (2.7) for arguments hold for the extension of the ansatz for It),,,). The above arguments show that it is very useful to consider formally the orbital degeneracy vJ as an expansion parameter.'8,28To leading order in v?' the one-impurity problem becomes then very simple and, in fact, can be solved exactly. A reviewlike survey and discussion of the work utilizing v?' expansions has been recently given.28a With this insight, it is clear that the results of Eqs. (2.13)-(2.20) can be rederived by other methods than the variational one used above. In particular, Brillouin- Wigner perturbation theory with v, chosen as an expansion parameter is for that purpose a simple and elegant method18 (for early use of

'' 0.Gunnarsson and K. Schonhammer, Phys. Re{>.Lerr. 50,604 (1983). '' P. W. Anderson. in "Proceedings of the International Conference on Valence Fluctuations in Solids"(L. M . Falicov. W. Hanke, and M. B. Maple, eds.). North-Holland, Amsterdam, 1981. N. E. Bickers, Jr., Ph.D. thesis. Cornell University, Ithaca, New York, 1987.

16

PETER FULDE el ul.

FIG.4. Diagrams which are included in a Brillouin-Wigner perturbation theory. Different single lines denote various states in the absence of hybridization, while double lines denote the eigenstates in the presence of hybridizations. The state with an unoccupied j level is denoted by j”,and the one with an occupied level by ,f,. A solid upward (downward)running line denotes an electron (hole), respectively.

the Brillouin- Wigner method in that context, see Bringer and Lustfeld”). In that case one considers the 4f’configurations If,} and I f ” } of the impurity, i.e., the states with one and zero f’ electrons. In the presence of hybridization with the conduction electrons, the energies of these states change to lowest order in Brillouin-Wigner perturbation theory by (2.3 1a) (2.31b) The corresponding diagrams are shown in Fig. 4. One notices that Eq. (2.31b) agrees with Eq. (2.13) when one replaces A E by~ E + e l . The single-ion theory discussed above describes the behavior of extremely dilute alloys in which the magnetic ions are well separated. Typical heavy fermion systems, however, contain ions with localized electrons in every unit cell, and therefore the interactions between the magnetic ions cannot be neglected. As a consequence, the condensation energy as well as other physical properties (such as the susceptibility per ion) will be different from the single-ion case. We discuss in the following some results for the two-impurity problem.” It serves as a link between the one-impurity problem discussed above and the lattice case. In straightforward generalization of Eq. (2.7), an ansatz is made for the ground state of the two-impurity system which consists of a coherent superposition of the states shown in Fig. 5. They include the unperturbed Fermi sea of the conduction electrons (Fig. 5a) and states in which one electron has been transferred to a local .f level (Fig. 5b). In addition, we have to consider 29

G. Zwicknagl, 0.Gunnarsson, and T. C. Li, submitted (1987)

THEORY OF HEAVY FERMION SYSTEMS

ltrn km

1.m 2.m' +

@O

t f2.rnCkrn '*O )

)

1.m +

-

17

2.m

&

mtm'

FIG.5 . Basis states for calculating the approximate ground-state wave function of the twoimpurity Anderson model.

states in which f' levels on both impurity sites are occupied. Here we have to distinguish between two different cases. If the occupied f levels have different quantum numbers m # m' (Fig. 5c), they can form singlet states independent of each other. A repulsive interaction (which reflects a loss in condensation energy per ion) results from the states displayed in Fig. 5d in which the occupied levels have the same quantum numbers m = m'. The physical origin of the reduction in condensation energy is simply the fact that a conduction electron cannot hop simultaneously onto both impurities. We can derive a simple semiquantitative picture for the interaction energy between the ions from the following consideration, based on the behavior of a single impurity. The occupation of the f' level creates a hole in the sea of the conduction electrons as can be seen from Fig. 6 (or 5). The corresponding change in the conduction-electron density 6n(R)is simply given by 6n(R) = vr(l

-

j

nf) d3keik.RC?(k)

(2.32)

Here cr(k) is the same as in Eq. (2.7), and the prefactor (1 - nf) results from Eq. (2.17). The change 6n(R)implies that in the vicinity of one impurity fewer conduction electrons are available for hybridization with a second one. This leads to a reduction in condensation energy when two impurities with separation R are considered. To leading order in the inverse degeneracy, l/v,-, the interaction energy A E ( R ) is given as a product of the local density reduction,

18

PETER FULDE et al.

b

-

I

I

I

I

10

0 I1

LT LL JC

Y

05

W

a

\ c

CK

x ” 00

W

a

0

5

10

15

20

25

(kFR) Fit;. 6 . (a) Change in the conduction-electron density divided by (1 - n,) as a function of distance R from an Anderson impurity. (b) Variation of the interaction energy, A E ( R ) , between two Anderson impurities with separation R . The interaction energy has been calculated by subtracting twice the condensation energy of a single impurity from that of two impurities. The A E ( R )dala are normalized to the value for vanishing separation, A E ( R = 0).

6 n ( R ) ,and a “potential” A V ( R ) , AE(R)= Gn(R)AV(R) The potential

A V ( R )=

s

6 d3keik’R-A~o Snkm

(2.33)

(2.34)

THEORY O F HEAVY FERMION SYSTEMS

10-2L

19

/I

V f = l 4

1o+-

10-4 T:2-'mp' ---D

-

1o-~ -

10-~

10.~ T,/D

Fici. 7. Condensation energy per impurity T(KZ-imp' at vanishing separation ( R = 0) as a function of the single-impurity Kondo temperature, TK.The values for Tg-'"P'were obtained by numerical diagonalization of the two-impurity Anderson Hamiltonian in the basis displayed in Fig. 5.

is the Fourier transform of the ground-state energy changes which are caused by changes in the occupation of conduction-electron states. The quantitative results for the interaction energy of the two-impurity problem are illustrated in Fig. 6. At short distances R, the interaction between the impurities is repulsive. The repulsion between the two ions decays rather rapidly with their distance. The characteristic length scale turns out to be the Fermi wavelength. The spatial variation of the (reduced) interaction energy depends only weakly on TK. A remarkable feature of the interaction energy data is the existence of stable positions: the ion repulsion vanishes for distances R which are multiples of z/k,. This particular feature changes, of course, when higherorder corrections are taken into account. The condensation energy per impurity approaches a constant value as R 4 0. We denote that value by k,T~-i"'P'.It is related to the characteristic energy of the single impurity, k,TK, through

(2.35) In the integral valence limit this expression can be derived from fourth-order Brillouin- Wigner perturbation theory. Figure 7 shows that the relation (2.5) is valid over a rather broad range of the parameter TK/D.

20

PETER FULDE et al.

20

15

10

x F/x

SI NG LE

05

I

I

00 00

20

FIG.8. Magnetic susceptibilities per impurity as a function of the impurity separation R. The data are normalized to the single-impurity value. The Ferromagnetic and antiferromagnetic susceptibilities, xF and xAF, describe the response to magnetic fields which are parallel and antiparallel at the impurity sites, respectively.

The magnetic susceptibility measures the stability of the Fermi liquid state with respect to magnetic ordering. In Fig. 8 we compare the magnetic response in two cases which simulate ferromagnetic and antiferromagnetic situations. The ferromagnetic susceptibility per ion is reduced with respect to the single-ion case. A more striking feature of the susceptibility data is the strong enhancement in the antiferromagnetic response. It highlights the tendency toward antiferromagnetism, which, indeed, is observed in many heavy fermion compounds (e.g., CeAl,, CePb,, or NpBe,,). Finally we would like to mention some important exact results on the singlet formation in the two-impurity problem which are relevant for the have developed a thermodynamic scaling lattice case. Jayaprakash et theory which combines the “poor man’s scaling” approach of Anderson2’ and the renormalization-group ideas of Wilson.’ Their results for the twoimpurity Kondo problem, which are reproduced for the corresponding two~

30 3’

1

.

~

~

9

~

’

C. Jayaprakash, H. R. Krishna-murthy, and J. W. Wilkins, Phy.r. Reu. Le/t. 47,737 (1981) C. Jayaprakash, H. R. Krishna-murthy, and J. W. Wilkins, J . Appl. Pbys. 53,2142 (1982).

THEORY OF HEAVY FERMION SYSTEMS

21

impurity Anderson model in the local moment regime, can be summarized as follows: the ground state is always a singlet. Provided the RKKY interaction between the impurities is small compared to the Kondo temperature, both moments will be quenched separately and a single Kondo temperature is observed. For large ferromagnetic interaction, the moments first align and are then compensated in a two-stage process. This case is characterized by two Kondo temperatures. The underlying physics of this behavior is that in the two-impurity problem there are two exchange integrals J, and J,,between the total spin of the composite object and the conduction electrons. These two exchange integrals result from decomposing the conduction-electron states into even and odd channels about the midpoint of the line connecting the sites. The two-stage compensation process can be observed whenever the corresponding two Kondo temperatures are well separated-as is the case for strong ferromagnetic interaction. If the RKKY interaction is strong and antiferromagnetic, on the other hand, the two spins form a singlet, and there is no Kondo effect at all. The competition between RKKY-induced magnetic order and Kondo-singlet formation is expected to be an area of considerable future research (see, e.g., Ref. 31a). These results for the two-impurity Kondo problem are confirmed by the recent calculations of Jones and Varma.32These authors study the problem of two Kondo impurities by means of the numerical renormalization group. They apply the same techniques Wilson used in his numerical solution of the Kondo problem. For the technical details of the calculation we refer the interested reader to the literature. The results clearly demonstrate the importance of the RKKY interaction. Both the effective RKKY and Kondo couplings scale to strong coupling as the temperature approaches zero. This means that the low-temperature behavior is determined by the RKKY interaction and the Kondo effect, even if the initial magnetic coupling is small. This fact had been conjectured by Abrahams and Varma33from fourth-order perturbation theory. Also of great interest is the Fermi liquid description which Jones and Varma32 deduce from the low-temperature behavior of their solution. They derive an effective quasiparticle Hamiltonian which may serve as basis for an improved realistic model for quasiparticles in a heavy fermion lattice. A lattice Hamiltonian may be constructed from a sum of pairwise quasiparticle Hamiltonians which automatically includes intersite effects. This will probably become a new area of research in the near future. 3'a

32 33

S. Doniach, Phys. Rev. B35, 1814(1987). B. Jones and C. M. Varma, J . Magn. Magn. Mater. 63&64,251 (1987) E. Abrahams and C. M. Varma, unpublished.

22

PETER F U L D E c’t

ti/

111. Quasiparticle Bands There is strong experimental evidence that in the low-temperature limit heavy fermion systems are Fermi liquids. With this it is implied that there is a one-to-one correspondence between the low-energy excitations of the strongly correlated ./-electron system and those of a nearly free electron gas. Fermi liquid behavior is suggested from the observed low-temperature specific heat and spin susceptibility. But it should also be pointed out that there is no unambiguous proof for it. For example, even at low temperatures the electronic specific heat is not strictly linear in T. However, the relatively small deviations from ideal behavior have not seriously cast doubts on the assumption that one is dealing with a Fermi liquid. We will take here the same point of view. Assuming that the Fermi liquid picture holds for heavy-fermion systems, one can describe their low-energy excitations in terms of Landau quasiparticles. A characteristic feature of the quasiparticles is that their energy dispersion &(k) depends on how many other quasiparticles are present. Generally

Here 6n,(k’) describes the deviations from the ground-state occupation, the quasiparticle distribution. The matrix f,,,(k, k ’ ) characterizes the quasiparticle interactions. We shall be interested here in the case where only a single quasiparticle is present. In that case the interactions are irrelevant. The problem is then reduced to that of calculating the quasiparticle energy E(k) for realistic systems. Thereby o u r attention will be focused on the Ce compounds and in particular on CeCu,Si,. The difficulties which one encounters in the actinide heavy fermion systems will be pointed out. A determination of E(k) from a microscopic theory is not feasible at present. Therefore a phenomenological approach in the spirit of Landau will be taken. It consists of introducing a small number of adjustable parameters (here one parameter turns out to be sufficient) and determining with them the quasiparticle bands for a given substance. The key idea is thereby the following. As mentioned repeatedly before, heavy fermion systems behave in the low-temperature limit like ordinary nonmagnetic metals (the parameters such as the effective mass are, of course, very unusual). Therefore let us go ahead and describe them like any other metal. The quasiparticle energy dispersions for metals are conventionally determined by band structure calculations. Traditional band theory is based on the effective potential concept. The eigenstates of a system of interacting electrons moving in the field of the nuclei are computed approximately by solving a single-electron

THEORY O F HEAVY FERMION SYSTEMS

23

Schrodinger equation in an effective (not necessarily local) potential. This effective potential describes the field of the nuclei and the modifications resulting from the presence of the other electrons. The crucial assumption is that exchange and correlation effects can be accounted for by introducing suitable changes in the potential seen by the electron. The essential many-body aspect of the problem is then contained in the prescription for constructing the effective potentials, and one should expect that successful schemes for treating strong correlations-as encountered in the heavy fermion systemswill differ drastically from those suitable when the correlations are weak, as in conventional metals. In most present-day band structure calculations the potentials or phase shifts are calculated by applying the local-density approximation (LDA) to the density-functional theory. However, that concept cannot be applied without modifications to heavy fermion systems. This is easily understood by considering the interacting homogeneous electron gas for the purpose of illustration. In that case it is well known that excitation energies are given by k2 E(k) = - + C(k, E(k)) - p (3.2) 2m where C(k, 0)is the wave number and frequency-dependent self-energy of the electrons and p is the chemical potential. On the other hand, the local-densityfunctional eigenvalues are given by

k2 E(k) = - + C(kF, E ( k F ) ) 2m

-

p

(3.3)

Therefore a band structure calculation based on determining the densityfunctional eigenvalues will correctly describe the excitation energies only as long as Z(k, w ) is not varying strongly from its value at the Fermi energy. This variation can be characterized by the effective mass ratio

In a homogeneous (or nearly homogeneous) electron system this mass ratio is close to unity, but it is very large in heavy fermion systems. This is due to a strong variation of C(k,w) with frequency o.It results from breaking the singlets which are formed at each site. Therefore the same frequency variation of C(k, (I)) which is found in a homogeneous electron gas over an energy range of EF (i.e., a few eV) takes place here over an energy range of order kFT*, i.e., over a few meV. We conclude from this that it does not make much sense to identify the density-functional eigenvalues with the quasiparticle excitation energies of a heavy fermion system regardless of whether or not an LDA is made.

24

PETER F U L D E et al.

Band structure calculations within the L D A have been performed for CeCu, Si,,34*3sCeA1,,35 UBe,,,36 UPt,,37-40 CePb3,41and CeSn, .42.43 The general feature shared by all calculations is the fact that the Fermi level is pinned at the lower edge of a narrow f band. This pinning of the Fermi level is a result of the high degeneracy of thef band. The f-band width, which is typically of the order of 1 eV, is rather narrow by normal band theory standards but is still by far too large to explain the observed low-temperature behavior. This failure of the LDA can be readily understood from the explanations presented above. More subtle is the question of how much one can trust Fermi surface topologies obtained from conventional band structure calculations. The Fermi surface is a property of the ground state and therefore should be correctly described within density-functional theory. The LDA to density-functional theory, however, which has been very successful in describing conventional weakly correlated systems, introduces an uncontrolled approximation which may have great impact on the Fermi surface geometry in RE or actinide systems. The problem is connected with the fact that the LDA cannot reproduce the multiplet structures of the 4fand 5f shells.44 As a consequence, we cannot generally expect this scheme to properly describe the crystal-field splitting especially if there is more than one f electron per site. The CEF splitting, on the other hand, is usually much larger than the Kondo temperature. Higher C E F states should therefore make little contribution to the wave function of the singlet state (see Section 11). When a calculation is done within the LDA, one cannot expect that only the CEF ground state of the f electrons enters. For that reason one would expect that such calculations cannot reproduce the shape of the Fermi surface. Therefore it has been a surprise that recent de Haas-van Alphen measurements on UPt, by Taillefer et aL4' seem to be in good agreement with the predictions of conventional band structure t h e ~ r y . , ~ - The ~ ' measured areas and masses of the extremal T. Jarlborg, H . F. Braun, and M. Peter, Z . Phys. B: Condfvrs. M a i m 52, 295 (1983). J . Sticht, N. d'Ambrumenil, and J. Kubler, Z . Phjs. Bc Conden,r. Mutrcr 65, 149 (1986). 3 h W. E. Pickett, H. Krakauer, and C. S. Wang, Physicu B + C (Amsrerdurn) 135, 31 (1985). 3' J. Sticht and J. Kubler, SolidSfute Conirnun. 54, 389 (1985). T. Oguchi. A. J . Freeman, and G. W. Crabtree, Pizvs. Lctf.A 117,428 (1986). 3 y R. C. Albers. A. M. Boring, and N. E. Christensen, Phys. Rev. B; Condens. Mutier 33, 8116 (1 986). 4" C. S. Wang, H. Krakauer, and W. E. Pickett, Physicu B + C (Amsrerdurn) 135,34 (1985). 'I P. Strange and B. L. Gyorfy,J. P/i.v.~.F16, 2139(1986). 4 2 D. D. Koelling, Solid Stute Comrnun. 43,247 (1982). 4 3 P. Strange and D. M. Newns, J . Phvs. F 16, 335 (1986). 44 U . von Barth, Phys. Reti. A 20, 1693 (1979). 4s L. Taillefer, R. Newbury, G. G. Lonzarich, Z. Fisk, and J. L. Smith, J . Muyn. M a p . Muter. 63&64, 372 (1987). 3J

35

THEORY OF HEAVY FERMION SYSTEMS

25

orbits were compared with those obtained from three different LDA calculations (see Wang et d 4 O and references therein). The magnetic field direction was along the (010) axis in reciprocal space, which corresponds to the (100) direction in real space. The results are listed in Table I. The orbits are labeled by the center of orbit and the band. The calculated and measured areas can be seen to agree fairly well. All calculations show six closed orbits, in agreement with experiment, The orbital assignments, however, differ in some cases. Angle-dependent de Haas-van Alphen data are needed to settle this problem and to construct the Fermi surface. The measured effective masses are larger by a factor of order 20-30 than the calculated ones. It will be interesting to see whether such good agreement holds also for other heavy fermion compounds. In the following we want to demonstrate how one can avoid the difficulties in the calculation of quasiparticle bands. This is achieved by a “renormalized band theory” which is based on a phenomenological lattice ansatz for ions forming singlets. This ansatz was introduced in its simplest form by Razafimandimby et al.46 It was further developed by d’Ambrumenil and F ~ l d e by ~ ~including ” spin-orbit interactions and the crystal field with the aim to develop a semiquantitative description of quasiparticle dispersions, density of states, and Fermi surfaces of heavy fermion systems. The theory was applied to CeCu,Si, by d’ambrumenil et aL4’ (for recent refined results, see Sticht et and Z ~ i c k n a g l ~The ~ ) . same type of theory was used recently by Strange and Newns4, to describe the effective masses in the intermediate valence compound CeSn,. 1 . RENORMALIZED BANDTHEORY:FORMALISM The central goal of the renormalized band theory is to construct realistic quasiparticle bands for specific materials. The method emphasizes two aspects not included in the impurity model calculations: first, the coherence of the lattice is fully taken into account and, second, realistic Bloch states for the nonf band states are used. In order to outline the application of renormalized band theory, we limit ourselves to Ce compounds. The low-lying excitations, which result in the strong frequency dependence of C ( k , o ) and prevent the application of density-functional theory, are caused by the 4f electrons. They are responsible for the singlet formation which highlights the transition into a highly H. Razafimandimby, P. Fulde, and J. Keller, 2. Phys. B: Cundens. Matter 54, 11 1 (1984). N. d’Ambrumenil and P. Fulde, J . Magn. Magn. Mater. 47 & 48, 1 (1985). 4 7 N. d’Ambrumenil, J. Sticht, and J. Kiibler, unpublished results (1984). G . Zwicknagl, N. E. Christensen. and J. C. Parlebas, submitted for publication (1987).

4h

‘’

TABLE I. FERMI SURFACE TOPOLOGY OF UPt,”

Expt. Area 4.8 6.1 8.0 14.2 21.4 59.5

LMTO

LAPW

LMTO-CC

Mass

Band

Area

Mass

Band

Area

Mass

Band

Area

Mass

25

G-5 AL-1 L-2 MK-3 G-4 G-3

4.58(+2) 5.05(-2) 7.54 (0) 14.12(+1) 19.60(+ 1) 50.66(+2)

1.4 1.1 1.2 3.1 3.6 4.0

G-5 L-2 A-1 G-4 MK-3 G-3

4.67(+ 1) 5.72 (0) 9.01 (0) 15.70(+1) 21.95 ( + 1 ) 58.27 ( + 1)

1.4 0.8 1.6 3.2

L-2 G-5 A-1 MK-3 G-4 G-5

5.23 (0) 6.03 ( - 1 ) 9.07 (0) 13.58(9) 24.03 (0) 59.01 ( + 1)

1 .o 1.7 1.9 3.7 4.6 5.3

-

40 50 60 90

-

4.1

Extremal areas and masses from de Haas-van Alphen measurements compared with the predictions of three different LAPW, Wang et a/?’; LMTO-CC, Albers er conventional band calculations. The references are LMTO, Oguchi ef The numbers in parentheses are the adjustments of the Fermi energy (measured in mRy) used to obtain the quoted agreement.

THEORY OF HEAVY FERMION SYSTEMS

27

correlated state. The strong correlations are reflected in strong scattering off the Ce sites which electrons in the f channel experience in the vicinity of the Fermi surface. Therefore the f-electron phase shift which parametrizes these scattering properties must strongly vary near the Fermi energy, and it is this variation which cannot be accomplished by density-functional theory. Since an L I / ) iriitio theory for the ,f-electron phase shifts is lacking, they must be put into the theory by hand. On the other hand, one expects that all the other phase shifts either on the Ce or the other ions vary slowly with energy. For their description one expects that density-functional theory and the LDA are applicable, at least to a first approximation, as in ordinary metals. Whether that is indeed the case can be decided only by a comparison of the calculated band structures with experiments. For example, it is also possible45 that the strongly varying f-electron phase shifts modify the other phase shifts in the vicinity of the Fermi energy. This would correspond to a mass renormalization of the conduction electrons due to ,f' electrons as is known to occur in Pr metal."y~"OIn the following we shall discard such possibilities. The concept of doing renormalized band structure calculations is then the following. We start from weakly correlated non-f band states which are obtained as eigenstates of a single-particle band Hamiltonian Hband($k)

= E(k)I$k)

(3.5)

In a second step we modify the Hamiltonian Hband

-+

Heff= Hband

+ {scattering off Ce}

(3.6)

by phenomenologically introducing strong f scattering at the Ce sites. This step of introducing additional scattering will be referred to as "renormalization." The eigenvalues E(k) and eigenstates of this new phenomenological Hamiltonian Hcffare interpreted as quasiparticle energies and states. They are obtained by diagonalizing ITeff. Thus the problem is reduced to solving a (renormalized) band problem. This concept of calculations is illustrated in Table 11. The required input information is

(1) The band Hamiltonian H b a n d ; (2) A description of the f channel scattering. Let us first consider the .f scattering at the Ce sites before we turn to the descripfion of the band Hamiltonian. As pointed out before, the rapidly varying f phase shifts must be put in by hand into the theory. In the spirit of N o z i e r e ~ ' ~approach ' to Landau Fermi liquids, we expand these f phase shifts 49

''

R. White and P. Fulde, Phvs. RPJJ.Lc//.47, 1540 (1981). P. Fulde and J. Jensen, P h y . ~Rco. . B: Condens. Mrr/ter 27, 4085 (1983). P. Nozieres, Pro(,.Inr. C'onf. Lon, Temp. Pliys., 14th. 1Y75 5, 339 (1975);see also P. Nozieres, J . LoIt, Tcnrp. P h ~ s 17, . 31 (1984).

28

PETER FULDE et al T A I ~I IL ~S C H ~ M A TSI~CJ M M Aor K YTHF C ~ N C F P T UNDERLYING S RFNOKMALIZED BAND *OR STRONGLY CORRELATED ELECTRON SYSTEMS CALCULATIONS Physical picture

Technical procedure

(Conduction) band electrons

Band states from conventional band theory

Strong (local)scattering in the vicinity of the Fermi surface

Eigenstates of band Hamiltonian matrix Additional strong scattering off Ce sites Phase shifts

Formation of quasiparticle bands (in the vicinity of the Fermi surface)

Quasiparticle band Hamiltonian Diagonalization yields quasi particle bands

I

.1

around 6 = E,. This introduces expansion parameters into the theory, which eventually must be fit to experiments. In doing the expansion it is important to know how the degeneracy among the different f channels is lifted. Otherwise the number of adjustable parameters is too large. Fortunately, in the heavy fermion Ce compounds this information is available from inelastic neutron scattering. For example, experiments on CeCu,Si, by Horn et aL5, have shown that the crystal-field ground state of the 4f electron is a doublet r7of the J = 3 multiplet. It is thought to have the form = c\+$)

+ d(T9)

(3.7)

with c = 0.85 and d = 0.56. Hereby the notation IJ,) = IJ = + , J z ) has been used. [Recently a somewhat different ground-state doublet has been proposed by Hanzawa and M a e k a ~ a , but ' ~ until the problem has been fully resolved we will use the one given by Eq. (3.7).]The other four states of the J = 3 multiplet are separated from the ground state by a crystal-field excitation energy of at least 300 K (the structure seen by Horn er ~ 1around . ~ 150~K is apparently due to phonons, according to personal communications with M. Loewenhaupt). This energy is therefore much larger than T*, and thermal excitations of these levels do not play any role at temperatures T < T*. The assumption will be made that in the low-temperature Fermi-liquid state of the system the f-electron wave function has the same symmetry as the crystal-field ground state [Eq. (3.7)] above T*. The expansion around E , of the f-electron phase shift at the Ce site is then written as

52

S. Horn, E. Holland-Moritz, M. Loewenhaupt, F. Steglich, H. Scheuer, A. Benoit, and J. Flouquet, Phys. Rec. B - Condms. M a i m 23, 3171 (1981). K. Hanzawa and S.Maekawa, personal communication (1985).

THEORY 01; HEAVY FERMION SYSTEMS

29

Here T = 1 is a pseudospin which represents the two states of the doublet, Eq. (3.7). The phase shift at the Fermi energy ii(E,) and the slope of the phase shift at the Fermi energy (k,T*)-’ are two parameters which enter into the theory. The first is related to the ,f-electron count nf at a Ce site. The second defines a temperature T* which we will use in the following to characterize a given heavy fermion system. It can be considered as an analog of the single-ion Kondo temperature. We stress that a phase shift of the form of Eq. (3.8) does not imply a lattice of independent Kondo ions. The slope (k,T*)-’ in Eq. (3.8) is in no way simply related to that which one would have in the case of a single Kondo impurity.51 In that case the inverse slope defines a temperature To which can be related to the experimental or high-temperature value of the Kondo temperature T, through T, = 0.324T0.51Also 6(E,) will generally be different for a single Ce ion and a lattice of ions. Having discussed the 1-electron scattering at the Ce sites, we turn to a Its actual form depends, of course, on the representation discussion of Hband. of the Bloch states, i.e., on the (technical)method adopted for solving the band structure problem. The explicit expressions given below refer to the linear muffin-tin orbital method in the atomic-sphere approximation (LMTO-ASA) used by Strange and N e w n and ~ ~ Z~~ i c k n a g lA. ~detailed ~ description of the method is given in S k r i ~ e rFor . ~ an ~ elementary review see A n d e r ~ e n .The ~~.~~ most recent work of Sticht et used the augmented spherical wave (ASW) method. 5 7 The atomic sphere approximation assumes that the space can be filled with Wigner-Seitz spheres around each atom, the overlap of which is neglected. The atomic potential inside the spheres is approximated by its spherical average, and the wave functions are represented in a partial wave expansion. For a specific material with a given structure the energy bands are determined by volume- and energy-independent structure constants Sk’.(k) and by potential functions Pi(€) det[Pf(E)fiii.SLL.- S&*9(k)]= 0

(3.9)

where L = (1, rn) denotes angular momentum and i is a site index. The potential functions Pi(€) are functions of energy which depend only on the potential inside the atomic sphere i. In particular, they parametrize its scattering properties. Close to a scattering resonance (where the scattering is extremely

’‘ H. L. Skriver. “The L M T O Method.” Springer-Verlag. Berlin and New York (1984). ” 0. K.

Andersen. Europliys. NWS 12, 4 (1981). Andersen, in “The Electronic Structure of Complex Systems”(P. Phariseau, ed.), NATO AS1 Series. Plenum, New York, 1983. A. R. Williams, J. Kubler, and C. D. Gelatt, Jr., P h y . ~Reu. . B: Condens. M a f t e r 18,6094 (1979).

” 0.K. 57

30

PETER FULDE c't a/.

strong) they simplify to PI(E) = (I*fSZ)(E - c;,

(3.10)

The parameters C fand p f S 2 can be interpreted as band centers and reciprocal masses, respectively. More accurate parametrizations valid for a general value of the energy E outside the resonance region are listed in S k r i ~ e rIt. ~is ~these parameters, describing the potential function, which are readjusted during a self-consistency cycle. The (weakly correlated) band states as well as the quasiparticle bands are determined from equations such as Eq. (3.9). Strong spin-orbit interactions and crystalline electric field effects are included in a straightforward way. The Bloch functions are expanded in terms of the eigenfunctions of the total angular momentum J and the CEF basis, respectively (instead of the orbital angular momentum partial waves), and nondegenerate (atomic) channels are characterized by different potential parameters. We now turn to the question how the material-specific information, the potential parameters, is determined. It has been pointed out above that the weakly correlated conduction-band states are usually well described within the local-density approximation to the density-functional approach. We therefore determine the potential parameters for all orbitals except the Ce4f states by a standard self-consistent LDA band structure calculation. This amounts to determining the band Hamiltonian Hbandwithin the LDA. The renormalization is achieved by introducing an appropriate potential function P ( E )for the Ce sites, which parametrizes a resonant phase shift according to Eq. (3.8). The position of the band center relative to the Fermi level, C - EF [or 6(E,)] and the inverse band mass, ,us2, are of the order of the energy k , T *. These two parameters, however, cannot be chosen independently. The condition that the Fermi energy must not change during the renormalization imposes an additional condition, leaving us with a single-parameter theory.

2. RENORMALIZED BANDTHEORY: RESULTS Renormalized band calculations, as described above, have been performed for C ~ C U , S ~ and , ~ ~C ~~ S " ~I,.~, These two compounds belong to two different classes of systems, as evidenced by the combined spectra for the removal and addition of a 4f electron, p x p s and p H l s ,respectively. For CeCu,Si, the observed XPS 41' -+ 4f0 transition has weight nf 2: 1 and position E~ N -2 eV relation to the Fermi level, reflecting the energy of the 4f' configuration and its weight in the ground state. The BIS 4f0 4 4f" transition is expected to sit at T* = 20 K above the Fermi level with width of order nT*/vf (in CeCu,Si, the degeneracy is vf = 2). This extremely narrow peak cannot be resolved experimentally. Therefore, we refer to CeCu,Si, as a Kondo-like system. The combined

THEORY OF HEAVY FERMION SYSTEMS

31

4f spectrum of CeSn,, on the other hand, exhibits only one peak in the vi-

cinity of the Fermi level which is characteristic of a strongly mixed-valent system. This fact implies that the energies of the 4f' and 4f0 configurations are (almost) degenerate and that their weights in the ground state are comparable. All the f spectral weight is centered near the Fermi level around T* N 460 K, which again defines the low-energy scale for this material. We see that the two systems differ substantially with respect to the weight and the position of the 4f' configuration in the ground state. The presence of a 4f charge strongly influences the band states, since it screens the core potential seen by them. This fact implies that we have to proceed differently in constructing the band Hamiltonian Hbandin the case of a Kondo-type and a valence-fluctuating compound. Therefore we describe the calculations separately. The following discussions assume that the reader is reasonably familiar with present-day computational techniques for electronic band structures. The standard notation of the LMTO method can be found, for example, in Ref. 54.

u. CeCu,Si, The basic assumption of renormalized band theory is that the non-f electrons are adequately described by the LDA. In the starting calculation, however, which serves to determine the potential parameters for the band states, one must specify the f-charge distribution at the Ce site. The results reported in this section are obtained from the following model: one assumes that the f-charge distribution at the Ce site is basically that of a corelike state. In the starting calculation the Ce 4f state is treated as a core state which does not hybridize with the band states. Consequently, there is exactly one 4f electron per Ce site. Let us mention here that the final results for the renormalized bands do not sensitively depend on the initial choice of the 4 f charge distribution. A detailed discussion of this important point is given in Appendix B. Before describing the technical renormalization procedure, let us mention a few caveats. There are a variety of different equivalent potential parameter setss4 which are used by a standard LMTO program to calculate matrix elements, ground-state properties, etc. This fact implies that we must introduce the renormalization in such a way that all the potential parameters are transformed consistently and that the many interrelationships among the various parameter sets still hold. This is achieved by introducing the changes in the basic quantities which-in a standard LMTO program-are directly calculated by solving the radial Schrodinger equation for a given fixed energy inside the ASA sphere surrounding atom i. The index v is of historical origin and is not a running index. These quantities are the value of the radial function 4Ll at the Wigner-Seitz radius, its logarithmic radial derivative

32

PETER FULDE c! ul.

= Sqh:!l/4:l, and the logarithmic radial derivative of its energy derivative O i l = S4;l/4zl(in the following, the index i is left out). The width of the energy window around evlr within which the parametrization of the potential function is valid, is proportional to ( + : L ) - l ’ z , where

(3.11) [with $<[= (d$vl/dc)f=f;,] At this level, the renormalization is performed in the following way. We force the read-in value E$7 to be the band center Cy;, by defining the logarithmic derivatives for the r7channel at the Ce site through DCe vr7

=

-4;

DYF7 = 3

(3.12)

With this choice of the logarithmic derivatives Dv and Dv, the intrinsic band mass p S 2 is given by the value of the density at the ASA sphere, (3.13) Therefore, we define 2 / [ ( 4 $ , ) 2 S ] to equal the read-in value of the band mass. Finally, we define ((q5$,)2) = 0. This renormalization procedure guarantees that all matrix elements in the program are transformed consistently. The Fermi level E,(pS2, C )is determined by filling up ( N + 1)states per unit cell. In the last step we try to reach self-consistency, i.e., to achieve E(FO)= E , ( p S 2 , C)

(3.14)

where E‘,O’ denotes the Fermi energy of the starting calculation. This relation eliminates one parameter. We can determine n,.(T*) by integrating up the f partial DOS at the Ce site. We can view the above procedure as a prescription to construct the correct valence electron density. Figure 9 displays the quasiparticle density of states of CeCu,Si, (see Ref. 35 in which the authors use the ASW method). h. CrSn, Strange and N e ~ n determine s ~ ~ the band potential parameters for CeSn, by calculating a self-consistent LDA band structure within the LMTO-ASA framework. The 4f’ electrons are treated as band states being pinned to the Fermi level by the LDA. This treatment of the 4f electrons in the starting calculation seems to be adequate for CeSn, since it is claimed4’ that the LDA provides the correct Fermi surface topology, the strong f correlations showing up only in the enhanced effective masses. In the work of Strange and N e w n ~ ; ~ all 4f states are described by one set of potential parameters. Spin-orbit

THEORY OF HEAVY FERMION SYSTEMS

ENERGY / e V

x

33

lo3

FIG.9. Refined calculation of the density of quasiparticle excitations near E, for of the Ce r7phase shift (3.8) is given by C ~ C U ~ In S this ~ ~calculation, . ~ ~ .the~ parametrization ~ ~ d(E,) = n/2, T* = 13 K.

splitting is included perturbatively in the calculation of the band dispersion and thus affects the whole self-consistency process. The resulting band structure is similar to previous calculations of Y a n a ~ and e ~ ~K ~ e l l i n gThe .~~ calculated Fermi surface has ,f character, and the calculated areas are in fair agreement with the measured ones. In the second step, the f correlation is introduced by the following renormalization procedure: it is assumed that the dominant effect of the f correlations is to reduce the potential parameter S+i( -) associated with the intrinsic band width of the f bands. To calculate the quasiparticle dispersion this parameter is rescaled according to

(g) 2

=l-nl

(3.15)

The f electron number n1 < 1 is a parameter of the calculation to be determined by fitting the density of states at the Fermi level to the observed specific heat coefficient y . In the actual calculation, the authors choose n1 = 0.86 for CeSn,. The center of the quasiparticle band is assumed to coincide with that of the original LDA f states, C,. The authors claim that the good agreement in the Fermi surface topology is not spoiled by the renormalization, which improves the overall agreement between measured and calculated (cyclotron) effective masses.

e,

N. d’Ambrumenil and P. Fulde, in “Theory of Heavy Fermions and Valence Fluctuations” (T. Kasuya and T. Saso, eds.), p. 195. Springer-Verlag, Berlin and New York (1985). 5 8 A. Yanase, J . Muyn. M o g n . M a t ~ r31-34,453 . (1983). 57a

34

PETER FULDE et nl.

This renormalization procedure by Strange and Newns4j differs from the one described above in two respects. First, the renormalization affects all f states in the same way. Therefore, the widths of the bands derived from the (high-lying) J = multiplet are also narrowed by correlations. Second, the center of the quasiparticle band, lies high above the Fermi energy when compared with the low-energy scale T*. Since the self-energy C ( k , o ) of the electrons varies strongly with o over an energy of order k,T* from the Fermi surface, one would expect the same to hold true for the renormalization factor. A good test for the reliability of the renormalization scheme would therefore be to calculate the temperature dependence of the specific heat, the magnetic susceptibility, and the transport properties from the quasiparticle density of states and to compare the results with experiments.

c,,

3. DISCUSSIONS A N D MODELDESCRIPTIONS The renormalized band theory discussed before describes the quasiparticle excitations by a narrow band of predominantly f-electron character which is close to the Fermi energy. Therefore, with respect to the low-lying quasiparticle excitations, the ,f electron of, e.g., Ce in CeCu,Si, seems to sit right at the Fermi surface. But when a photoemission experiment is done it takes approximately 2 eV in order to promote the f electron to the Fermi surface. What seems to be a contradiction is, of course, none. The quasiparticle density of states N * ( o ) (Fig. 10a) and the f-electron spectral density pr(w)

W

W

FIG.10. (a) Quasiparticle density of states N * ( w ) as seen, e.g., in the low-energy thermodynamic properties of the system. The thick line parallel to the w axis is supposed to represent the density of states of bands with conventional band widths. The high density of states at E , has predominantly f character; (b) f-electron density of states as obtained from the imaginary part of the /-electron Green’s function and as seen in a photoelectron spectroscopy experiment. In, e.g., CeCu,Si, the /electron is found to be approximately 2 eV below E,.

35

THEORY OF HEAVY FERMION SYSTEMS

(Fig. lob) are quantities which agree with each other only within the independent electron approximation. For strongly correlated systems such as the ones which we are considering here, they can be drastically different. This can be seen by examining the one-impurity system, where one conduction electron has been removed. To leading order in the inverse f-level degeneracy, 1 / ~ j the . (excited) hole states can be shown19*20,59 to have the simple form (3.16) The prefactor lA12 = 1 - !if is again given by Eq. (2.17), i.e., by ( 1 - n,), and therefore is very small in the K o n d o limit. Therefore the weight of the is very small. state C ~ , , , ~ . J @ ~ ) in In the lattice case the density of states of the quasiparticles is of the form shown in Fig. 1Oa. But the ,f-like weight measured spectrally at the Fermi energy, e.g.. via photoemission, varies like n,( 1 - n,) because it involves . in Fig. IOb the matrix element ck,m,l(~~~m’m.~.~m~$~=~)~2 = l A 1 2 n f / ~ yTherefore the structure at the Fermi energy has small weight in the K o n d o limit. Alternatively one can write for the f-electron Green’s function close to the Fermi energy

(3.17) where GinC(k, o)has n o pronounced structure in the frequency range under consideration. The E(k) are the quasiparticle energies and the renormalization factor ( 1 - n,) is in accordance with the above. The previously presented formulation of renormalized band theory referred to the limit of zero temperature. At finite temperatures quasiparticles (holes) are generated which affect the phase shifts according to Eq. (3.1). We want to discuss semiquantitatively these changes. We also want to derive a model Hamiltonian which contains the important ingredients of the renormalized band theory. It is required when we derive the quasiparticle-phonon interaction in Section IV. The hcavy fermion compounds known to date contain in most cases many atoms in the unit cell. As a consequence, the typical dimensions of the secular matrices of Eq. (3.9) are rather large, a fact which seems to exclude quick and qualitative discussions. The central goal of the present section is therefore to understand the formation of the heavy quasiparticle bands on the basis of a simple physical picture and to derive a model Hamiltonian. The latter should describe the essential physics while being simple enough to allow for semiquantitative discussions on a “back-of-the-envelope” level.

’’ T. C. Li, 0.Gunnarsson, K. Schonhammer, a n d G. Zwicknagl, J . Phy.~.C 20,405 (1987).

36

PETER FULDE

LV

ul

All heavy fermion systems have in common that the interesting physics is associated with an extremely narrow band with predominantly f character. The model therefore has to focus on this band and its dispersion. At first sight, it seems tempting to apply standard canonical band theory6' to model the heavy quasiparticle bands. Within this framework, the dispersion of a band derived from the 1 orbital of an atom at site i is obtained by simple diagonalization of the corresponding sub-block SE,.(k) of the structure constant matrix followed by rescaling plus shifting. The only input is the structure of the corresponding sublattice. As an example, the canonical r7 band in CeCu,Si, would be given by the structure constant SF$," itself. This canonical concept, which has proven to be extremely useful in numerous cases, cannot be applied here for the following reason: one characteristic feature of all heavy fermion systems known to date is the rather large separation of the Ce or actinide atoms. As a result, there can be no sizeable direct overlap between the f orbitals on neighboring sites in general. The structure constants for .f states scale like R - 7 , where R is the nearest-neighbor distance, and we would get an extremely weak dispersion for the heavy bands within canonical band theory. As a matter of fact, for CeCu,Si, the structure constants SF;.FS'(k) even vanish identically for all k values, yielding no dispersion at all. This is a general feature of canonical theory. There is no canonical dispersion for bands derived from orbitals corresponding to j = I - $. Apart from these rather formal arguments, there are more physical reasons indicating that the canonical picture does (probably) not provide the key to heavy fermion bands. The general behavior of heavy fermion systems is very complex, and the properties are not determined solely by the structure of the Ce or actinide sublattices, which are the only input to the canonical theory. The measured quantities also depend rather sensitively on parameters like composition in general, lattice constants, etc. These facts indicate that the coupling of the ,f states to the conduction states is important. The j' electrons become itinerant via their coupling to the conduction electrons. The model Hamiltonian describing this behavior is given by

This general form will be derived below, starting from the secular equation, Eq. (3.9). Let us first point out the characteristic features which will be used for model calculations in Section IV. The model of Eq. (3.18) describes a narrow (effective) .f band coupled to conduction electrons. The small band width of the J' band and the hybridization matrix elements are proportional

'' 0. K . Andersen, P/IJ's Rcu. B

Condenr. Matter 12, 3060(1975).

THEORY O F HEAVY FERMION SYSTEMS

37

to kBT* and (E,k,T*)’’2, respectively, where T* is the characteristic temperature. The proportionality factors, however, need not be of order unity. The quasiparticle states are obtained from a straightforward diagonalization of the effective hybridization Hamiltonian (see also Section IV). When written in diagonal form it is

The u!Jk), a,,(k) are the quasiparticle creation and annihilation operators, and I is their band index. We will now derive the model Hamiltonian, Eq. (3.18), from the secular equation (3.9).This involves many technical details which are not essential for understanding the subsequent development of the theory. The derivation of Herris based on block perturbation theory or Lowdin partitioning.61 Similar ideas have recently been developed and tested within the muffin-tin orbital (MTO) f r a m e ~ o r k ,where ~ ~ , ~the~ method is referred to as “down-folding treatment.” We exploit the fact that we are mainly interested in an extremely narrow energy range in the vicinity of the Fermi surface; i.e., we look for bands where the energy E in the secular equation (3.9) is restricted to a window of order k,T*. In the first step, we divide the set of MTOs, {i, L = (/,m)}, specified by site and angular momentum indices, i and L, into a “lower set” {i, LF’} and a “higher set” {i,LiH’}.For the latter subset of orbitals, the band centers C;!H), must be separated sufficiently well from the Fermi level, and the potential functions P+(EF) at the Fermi level have to be much larger than the width of the corresponding canonical band, i.e.,

Here ( .)Bz denotes an average over the Brillouin zone. Eliminating the set of higher MTOs from the secular equation (3.9) yields a renormalization of the structure constants _..,

S;&I

“

1

+

S;f+,>;!I.)

P. 0. Liiwdin, J . Chmi. Ply.!. 19, 1396 (1951). D. Glotzel, in “Highlights in Condensed-Matter Theory” (F. Bassani, F. Fumi, and M. Tosi, eds.). North-Holland, Amsterdam, 1985. W. R. L. Lambrecht and 0.K. Andersen, Phys. Reo. B: Cotidens. Murter 34, 2439 (1986).

‘’ 0. K . Andersen, 0. Jepsen, and ‘3

38

PETER F U L D E

C I ctl.

where

defines the “higher-block Green’s function.” The energy dependence of the renormalized structure constants which results from the energy dependence of the potential functions PidlH,can safely be neglected in our case. This is due to the fact that the propehies of conduction-band states vary only very weakly over the energy range of interest. In the next step, the potential functions P;;L,(E),which usually are nonlinear functions of the energy E , are replaced by suitable linear approximations. We subdivide the set of lower MTOs into the Kondo channel(s) and effective conduction electron orbitals, which we symbolically denote by “f and “c,” respectively. The secular equation in block-matrix notation now reads ”

(3.22) where PJ, P,, SJs, and S,, are square block matrices with P, and P, being diagonal. The abbreviations S,, and SCs,on the other hand, denote rectangular block matrices. In the case of CeCu,Si,, for instance, the index “ f ” refers to the two r, channels corresponding to the crystal-field ground state. We insert the resonant form, Eq. (3.10), for the potential function into the Kondo channel,

Pf(4 = (I*/S2)(E1

-

Cj)

(3.23a)

The conduction-electron potential functions, P,(E),on the other hand, are linearized around the Fermi energy E , PAC)

+

P,(E,)

$- (C -

EdPJEF)

(3.23b)

with

P,(E,)

=

-

dP,

(3.23~)

~

~ = E F

For the energy range of interest, 10 meV, this linearization introduces an error of 5 1%. With the replacement of Eqs. (3.23a) and (3.23b) the secular equation (3.22) simplifies to

The quasiparticle bands, which correspond to zeros of the secular equation,

39

THEORY O F HEAVY FERMION SYSTEMS

can be obtained as eigenstates of the model Hamiltonian in the LMTO basis

H

H,,

=

=

(

Hff(k)

H.fC(k))

H,,(k)

Hc,(k)

(EF) 1 EF1 - C + - S c , -

-

1

m m

We finally transform the Hamiltonian of Eq. (3.24b) onto the basis of unhybridized effective f- and conduction-electron states, Ik, rn) and lkna), in which the subblocks H,,(k) and H,,(k) are diagonal. The corresponding unitary transformation matrices are given by U,(k) and U,(k), respectively. The model Hamiltonian then reads H

=

C

km

+

',-m.fi!.m.Lm

c

+ nmka C [ Crn(k,n)f':mCkna

En(k)Ci!.noCkno

+ H.c.) (3.24~)

kna

with

g;m(k) = (U:(k)H,(k)U,(k))mm -Vma(k, 4 = (U:(k)H,,(k)u,(k))mna

(3.25)

%(k) = (Uf(k)H,,(k)U,(k)),,,, The more general model, Eq. (3.24), can be reduced to Eq. (3.18) under the following two conditions: (1)

The crystal-field ground state is only twofold degenerate; (2) Spin-orbit effects can be neglected for the conduction electrons.

In Section VI an effective Hamiltonian of the form of Eq. (3.24~)will be derived by starting from a microscopic model (i.e., by using an Anderson lattice Hamiltonian) and applying a mean-field approximation. Next we want to generalize the renormalized band theory to finite temperatures. The transition to T > 0 is simulated by decreasing the intrinsic band-mass parameter p S z or alternatively increasing T* in Eq. (3.8). The band center C is kept fixed, because at finite T the number of quasiparticles equals the number of quasiholes and therefore the last term on the righthand side of Eq. (3.8) can be neglected. Making the changes as described does not require additional calculations since we have to do these parameter scans to reach self-consistency E , = E F ) . Changing pusz is motivated by

40

PETER F U L D E et al.

the results from the single-impurity calculations. The self-consistent calculations for the spectral density p4f of the J-electron Green’s function show that the dominant effect of temperature is a dramatic broadening of the Kondo resonance. For dilute Ce alloys Coxz6 derived the following power law relating the width at T = 0, T(0) to that at finite temperatures, T(T) - T(TD= 0 ) + ,3.5(;)1.5 T(T) D

(3.26)

from Lorentzian fits to the fully calculated p4J curve. This relation holds over a rather broad temperature range and yields good fits of the transport coefficients. It should be used only for temperatures T < T*, where we can expect the quasiparticle picture to be valid.

IV. Quasiparticle-Phonon Interactions

In this section we want to demonstrate that the elastic properties of heavy fermion systems are almost as interesting as the electronic ones. They have been explored experimentally in particular by Luthi and c o - ~ o r k e r s , ~ ~ . ~ ~ Golding et d , h 7 and Goto et d6* We believe that they deserve Muller et more attention than they often obtain. Therefore we want to put special emphasis on them. The fact that they are so important is related to the strong volume R (or pressure) dependence of T*, the characteristic temperature of a heavy fermion system. We characterize this volume dependence by defining a parameter yIT=

dln T* dInR

-~

(4.1)

It is found to be typically of the order of 10-100 in heavy fermion systems65 and can be considered as an electronic Griineisen parameter.64 The large values of yT are plausible when one considers T* as the equivalent of the Kondo temperature TKof a single Kondo impurity. The latter depends in a

b4

R. Takke, M. Niksch, W. Assmus, B. Luthi, R. Pott, R. Schefzyk, and D. K. Wohlleben,Z. Phys. B: Conrkivw. M o t t i v 44,33 ( I98 1 ).

h5

B. Luthi, J. Magn. Magn. M a t w . 52, 70 (1985).

’’V. Muller, D. Maurer, K. de Groot, E. Bucher, and H. E. Bommel, Phys. Rev. Lett. 56, 248 (1986). ” B. Golding, B. Batlogg, D. J. Bishop, W. H. Haemmerle, Z. Fisk, J. L. Smith, and

’*

H. R. Ott, Proc.

I n / . Con/. Plionon Phjj.~.,.?id,p. 406 (1985).

T. Goto, A. Tamaki, T. Suzuki, S. Kunii, N. Sato, T. Suzuki, H. Kitazawa, T. Fujimara, and T. Kasuya, J . Magn. Mugn. Muter. 52,253 (1985).

THEORY OF HEAVY FERMION SYSTEMS

41

nonanalytic way on the hybridization V of the magnetic ion with the surrounding conduction electrons [see Eqs (2.15) and (2.16)]. When pressure is applied and the volume of the unit cell is changed, the hybridization matrix element changes, too. In most cases we expect it to increase with decreasing volume. Because of its exponential dependence on V, TKwill strongly vary with it. TKincreases as I/ increases, and, therefore, we expect TK to increase with pressure. Allen and Martin69 found that for a variety of Ce compounds the volume dependence of the quantity j = v f r / ( 7 c l c f l )in the exponent of Eq. (2.16) can be phenomenologically described by a relation of the form 4

J

= O.Z($)

(4.4

where R, is a reference volume (see also Ref. 69a). When this relation is used to derive dj/dR one finds that

(4.3) Sincej is of order 0.2, it is seen right away that ylT is large in that case. Therefore it is no surprise that it is also found to be large in heavy fermion systems. Needless to say that T* is not necessarily the same as the single-ion Kondo temperature. 4. THERMODYNAMIC RELATIONS Before we derive an explicit form for the interaction of the heavy quasiparticles with phonons, we want to demonstrate that a number of important relations between measurable quantities follow alone from thermodynamics and from the assumption of a large electronic Gruneisen parameter.64,70We want to assume in the following that the electron contribution to the free energy in the presence of an external magnetic field F,(T, H , R) can be written in the form

* ; ( )*:

F,(T, H,R) = - k B T N f

--

(4.4)

and that the volume dependence of it is solely due to that of the characteristic parameters T* and H*. N is the number of ions. We do not require that the functional form f ( x , y ) is the same for all the heavy fermion systems. Despite its 69

h9s 'O

J. W. Allen and R. Martin, P/iy.v. Rro. Lctr. 49, 1106 (1982). M. Lavagna, C. Lacroix, and M. Cyrot, f h y s . Lett. WA,210 (1982) P. Thalmeier and P. Fulde, Europliys. Letr. I , 367 (1986).

42

PETER FULDE rt ul

generality, one might question the above assumption. Lattice coherence effects result in quasiparticle bands with considerable structure in the high density of states (see, e.g., Fig. 9). Therefore one might expect that more than one characteristic temperature and magnetic field contribute to the volume dependence of the free energy. We shall ignore these complications here and study what information can be gained by making the above simplified assumption. Because F, depends on volume also through H * , this suggests introducing a second electronic Gruneisen parameter YH

=

dInH* -m

(4.5)

which is related to the scale of the magnetic field. There is no a priori reason that the two Gruneisen parameters ylT and qH should be the same. On the other hand, we do not expect them to be drastically different either, because we expect pBH*to be of the order of k,T*. We want to derive a number of interesting relations between thermodynamic coefficients which follow from Eq. (4.4) and the assumption about its volume dependence. For that purpose we write the total free energy in the form

F

=

Fo

+ 4 + F,

(4.6)

where F, is a background contribution and

4=

R 2

(4.7)

is the elastic contribution. The volume strain cQ = E,, + cyy + cZzis the trace of the strain tensor and can be written in terms of the volume change AR as E , = AR/R. c: is the isothermal bulk modulus which results from the elastic contribution to the free energy. We want to consider the electronic contribution to a number of thermodynamic coefficients. They arc bulk modulus: (4.8a) specific heat: (4.8b) magnetic susceptibility: (4.8~)

THEORY OF H E A V Y FERMION SYSTEMS

43

thermal expansion: (4.8d) volume- magnetostriction coefficient: (4.8e) magnetothermal coefficient: (4.8f) Furthermore, the electronic contribution to the magnetization is given by

(4.9) Therefore one can also write A,? = a A M / d T . By using Eq. (4.4) the thermodynamic coefficients can be expressed in terms of derivatives off(x, y)(x = T / T *, y = H / H * ) and the Gruneisen parameters qT and q H .Relations between thermodynamic coefficients can be obtained by finding combinations which cause the corresponding expressions in the derivatives of f(x,y) to vanish. The following three relations are found this way be, An

=

-q,c;TAu

-

q,HAa

= (2qH - v T ) HAx

+ qr,(qH

+ q T T AA

CO,ACC = q T A C + q , H A2 For H

=

-

q7)HAM

(4.10a) (4.10b) (4.1OC)

0 these relations reduce to cg ACC= qT A C

AcB =

-

qTcgT ACC

(4.1la) (4.11b)

Equation (4.1 la) can be used to determine the electronic Gruneisen parameter '1, by measuring Asl and A C . The second relation, Eq. (4.Ilb), can strictly not be used directly for determining q,,. In heavy fermion systems one is measuring in most cases the isoluted bulk modulus instead of the isothermal one (see Section 6). Therefore one would first have to correct for the difference between the two, before one can use Eq. (4.1 1b). Neglecting this difference, one can determine qT independently. For CeAI, good agreement is found between the two ways of determining y T . 6 4 The measured Gruneisen parameter shows temperature variation at low T. There is no theory yet available for it. From Eq. (4.1 la) it is seen that the thermal expansion in heavy fermion metals is

44

PETER FULDE

r t ctl.

enormous. In addition to the enhancement of the specific heat the large Griineisen parameter qT contributes as a multiplicative factor. For experimental results, see, e.g., Takke et Luthi,65and Ott.4 By setting T = 0 one finds from Eq. (4.10)for the magnetic field dependence of the bulk modulus

AcH(H)= -qH(2qfr - q T ) H 2Ax

+

qH(qH

-

qT)HAM

(4.12a)

which reduces to

AcR(H)= - q 2 H 2 Ax

(4.12b)

if qf, = qT = q. This relation was experimentally confirmed by Kouroudis et d.," who investigated UPt, and CeRu,Si,. It was found that to a good approximation qT = qH in those substances. They also found a minimum in one of the elastic constants which results from a maximum in the corresponding Ax(H). Interesting information is obtained from Eq. (4.10) when a low-field approximation is made, it., when AM = Ax( T, 0)H. In that case (4.13) That relation can be set into Eq. (4.10). We can directly introduce the volume magnetostriction &,,(H) = en(T, H ) - eR(T,0),which is related to the volume magnetostriction coefficient ACJthrough &,(H)

=

HAa/2c;

(4.14)

Then Eq. (4.10b) together with Eq. (4.13) results in k d H )=

H2

C(2YlH - V T ) Ax

+ V T T Ax']

(4.15)

i.e., the electronic contribution to the magnetic susceptibility Ax(T, 0) and its derivative determine the volume magnetostriction 8eJH). Systems with unstable 4f moment, i.e., heavy fermion or valence fluctuating systems, have a magnetic susceptibility which often can be reasonably well approximated by the equation

Ax(T,O) = C , / U

+ T*)

(4.16)

at least as long as T is not too small. C, is a Curie constant. From this form of 71

I. Kouroudis, D. Weber, M. Yoshizawa, B. Luthi, L. Puech, P. Haen, J. Flouquet, G. Bruls, U. Welp, J. J . M. Franse, A. Menovsky, E. Bucher, and J. Hufnagl, Phys. Reu. L c / / .58, 820 (1987).

T H E O R Y O F HEAVY F E R M I O N SYSTEMS

45

A x it follows immediately that JE~(T H ,) = ( A

+ B T )( T +HT*)i ~

(4.17)

In the case that the two Gruneisen parameters are identical, i.e., qH = qT = q, one obtains (4.18) with A = qCOT*/(2c;).This is indeed the behavior of the magnetostriction found for a large number of unstable valence corn pound^.'^ It demonstrates that in the temperature interval over which the data have been taken the starting assumption about the volume dependence of F, must be correct. When Eq. (4.16) holds, one can find another scaling relation from Eq. (4.10c), namely (4.19) This relation has not yet been checked experimentally. A third relation involving AcHand Au can be obtained from Eq. (4.10a), but we shall not write it down explictly. 5. QUASIPARTICLE-PHONON INTERACTION HAMILTONIAN In heavy fermion systems the description of the quasiparticle-phonon interactions is considerably difl-erent when T >> T* and T << T*. In order to demonstrate this let us consider a Ce compound. For T >> T* the Ce ions behave as though they had a well-localized 4f electron. Then the conventional magnetoelastic interactions apply. They rest upon the fact that a lattice deformation in the surroundings of a Ce ion changes the C E F which the 4f electrons see. This in turn modifies the eigenstates of the 4f electrons. An alternative way of stating the same is by saying that the lattice deformation (phonons) cause virtual excitations between the different C E F levels of the lowest J multiplet. As an example we want to write down the interactions at a 72

J. Zieglowski, H . U. Hiifner, and D. K . Wohlleben, f h y s . Reu. LEII.56, 193 (1986)

46

PETER FULDE et

a!.

Ce site in a cubic environment, The irreducible representations of the of the strain tensor uU8 symmetric part (cup)and antisymmetric part (on8) are denoted by Q:. Here CI is the representation index (in Bethe’s notation CI = 1,. . . , 5 ) and n is the index of degeneracy. They are

(4.20)

Similarly one can construct irreducible representations of the cubic point group 0, from the operators J,, Jy, Jz. We denote them by o l d . Again CI and n are the representation index and index of degeneracy, respectively. Furthermore d is the degree of representation. For the purpose of demonstration we list the ozd to second degree in J , i.e., for d = 2. This excludes the representations CI = 1,4 because for them the lowest degree in J is d = 4. They are 1

0 : 2

= -[3J,2

O:,

= - (1J ,

&

-J(J

2 -

+ I)]

J;)

Jz (4.21) 1

0522

= -(JzJx

Jz

+ JxJz)

THEORY OF HEAVY FERMION SYSTEMS

47

The interaction Hamiltonian is obtained by multiplying those representations Q; and OEd by each other, which have the same transformation behavior. For a Ce atom at site m this results in

The deformation tensor operator vro can be expressed in the standard way in terms of phonon creation and annihilation operators bi,, b,, ( p is the polarization index) (4.23) e,(qp) is the M component of the unit polarization vector of the mode qp. R, is the equilibrium position of ion m and A4 is the ion mass. Furthermore, wqpis the phonon dispersion. The Hamiltonian of Eq. (4.22) summed over all sites describes the interactions of the lattice deformations with the 4J'electrons. We mention in passing that, in order to properly ensure rotational invariance, one must also include interaction terms which are bilinear in the deformation tensor vao(m). The various effects caused by H,, have been discussed at length in a number of reviews, to which we refer the reader for detail^.^^.'^ The theory has also been successfully applied to Kondo-lattice systems such as CeAI,, in cases where the temperatures were sufficiently high. A number of magnetic field effects were observed for the elastic constants and have been explained not only qualitatively but also quantitatively. Even the existence of a bound state between a CEF excitation and a phonon was established for CeA1,.75s76These findings have resulted in considerable confidence that the interactions of 4.f electrons with phonons are well understood in Kondo-lattice systems as long as T >> T*. It should be mentioned that the interaction of phonons with 4f electrons must be supplemented by the interaction of phonons with the conduction electrons. This interaction can be described in terms of a deformation potential and is conventionally written as

P. Fulde, Hmrih. Ph.v.c..Chew!. Rurr Eur/h.c.2 (1978). B. Liithi, D w . Prop. Solids 1974-1980 3,245 (1980). 7s M. Loewenhaupt, B. S. Rainford, and F. Steglich, Phys. Rev. L e / / .42, 1709 (1979). 7 h P. Thalrneier and P. Fulde, P\J.v.s. Reu. Let/. 49, 1588 (1982).

73 l4

48

PETER FULDE

PI

al.

For a free-conduction-electron system the coupling constant takes the value C = -2E,/3, where E , is the Fermi energy. Let us now go over to temperatures T << T * . In that case the CEF ground state (e.g., r, in the case of CeCu,Si,) goes over into a heavy Fermi liquid. At the same time one must keep in mind that at the Fermi surface there are quasiparticles present with very different effective masses. The question arises then how the interactions with phonons are modified as compared with the case T >> T*. Those conduction electrons which do not hybridize with the partially delocalized f electrons remain unchanged as the temperature crosses T*. Therefore, their interaction with phonons also remains unchanged. Of particular interest is the coupling of phonons to the heavy quasiparticles. In order to evaluate it we consider the effective quasiparticle Hamiltonian of Eq. (3.18) from which the dispersion of a heavy quasiparticle can be derived. The quasiparticle-phonon interaction is obtained by utilizing the fact that the hybridization E(k) as well as the dispersion E;.(k) of the narrow band depends on T*, which is strongly volume dependent. In order to derive it we shall make some further simplifications. We keep only one of the conduction-electron sub-bands n. The corresponding hybridization matrix element is labeled by and it is assumed that it is k independent. By Fourier transforming into site representation, one writes

v,

(4.25) and similarly for f,(i), c;(i), and cr(i).We also transform Cf(k) into (4.26) The effective quasiparticle Hamiltonian of Eq. (3.18) is then rewritten in the form Hef' = He::,

+ 1rijf;(i)J( ijr

j)

+ 1[f'!(i)c,(i) + c;(i)J;(i)] ir

(4.27)

Heff

is the effective Hamiltonian of the conduction electrons. We assume that a volume strain cn(i) is present and that T * varies from cell to cell accordingly. Because ti,i T* we expand

-

(4.28) This relation is expected to hold as long as cn(i)changes over distances larger

THEORY OF HEAVY FERMION SYSTEMS

than the range of tij. Furthermore, since

P

-

49

it is

(4.29) Finally

T*(i) = T*

dT* + ---~,(i) a€,

(4.30)

The volume strain is expressed in terms of longitudinal phonon creation and annihilation operators h i , b, as

(4.31) By inserting Eqs. (4.28) and (4.29) into Eq. (4.27), one obtains the following interaction Hamiltonian

(4.32) where

is the electronic variable which couples to the phonon field. In deriving Hqppphwe have replaced dT*/&,(i) by the value in the presence of a homogeneous strain, in which case E,(i) = E , . The derivative dT*/dE, = -qT* is again related to the electronic Gruneisen parameter q [see Eq. (4.1)]. Next we want to express A, in terms of the quasiparticle creation and annihilation operators. For that purpose we diagonalize the quasiparticle Hamiltonian Heff.This is done by means of the canonical transformation

(:;;:I;) ( =

cos $k -sin$,

sin 'k) cos9,

(fir)

ckr

(4.34) The ai,(k), al,(k) are the creation operators for the two quasiparticle branches which are obtained from the simplified Hamiltonian of Eq. (4.27). We shall assume in the following that a f r ( k )describes the heavy and ai,(k) the "light" quasiparticle branch. Herr is diagonalized by those 9, which fulfill the

50

PETER FULDE

el

(I/.

condition (4.35) where, as before, E(k) is the energy dispersion of the conduction electrons. By applying the transformation of Eq. (4.34), one finds A,

=

C a:,(k + q)u,,(k)G(kq) + ...

(4.36)

kr

where the remaining terms describe the coupling of phonons to the second quasiparticle band and to scattering transitions between the two bands. We will leave out those processes for simplicity. The coupling function v"(k,q) is given by

The Hamiltonian

will be used in the following discussions. One notices that it is not of the deformation potential form because the variable k appears in the coupling function E(k, 4). For an estimate of G(k,q) we use the denominator in Eq. (4.35) to be of order - E,. Therefore we expand with respect to 9, and approximate 8, = - ?/&. By using that = U ( T * / E ~ )where " ~ , a is of order unity, one finds that v"(k,q) can be approximated by G(k,q) = -uT*

+ f[?;(k) + Zf(k + q)]

(4.39)

For q = 0, the term aT* results in a deformation potential, i.e., shift of the narrow quasiparticle band relative to the chemical potential p. The second term, which is k dependent, results in addition in a broadening and narrowing of the quasiparticle band which is modulated with the lattice oscillation (breathing effect). The effects of both couplings are schematically shown in Fig. 11. From the above it follows that the quasiparticle-phonon interaction is of order qT*. This has to be compared with the deformation potential coupling constant - f E , , which appears in the conventional conduction electron-phonon interaction Hamiltonian. Two remarks should be made at this stage. One is that there is obviously a partial cancellation between the two terms on the right-hand side of Eq. (4.39) when q = 0 and ( k (= k,. The other is that the interaction in Eq. (4.32) is not sufficient for a computation of, e.g., the bulk modulus. For that one must also

THEORY O F HEAVY FERMION SYSTEMS

51

P

N*(W) FIG. I I . Two forms of quasiparticle-phonon coupling. In case (a) the band of the heavy quasiparticles moves up and down with respect to p(Q) with volume changes en. In case (b) the band broadens and narrows with changing volume.

include interactions which are bilinear in the phonon operators. This topic will be discussed in detail in Section 8,d. In the conventional theory of conduction electron-phonon coupling, Migdal’s theorem plays an important role. It states that vertex corrections to the electron-phonon interaction as shown in Fig. 12 can be neglected because they modify the effective coupling constant y only by a factor [l + O ( ( r n / M ) ’ ’ 2 ) ]where , rn and M are the electron mass and ion mass,

FIG. 12. Vertex correction to the quasiparticle-phonon interaction. In ordinary metals such a correction can be neglected according to Migdal’s theorem.

52

PETER FULDE et al

respectively. In the heavy fermion systems, in which the quasiparticle mass is very large, an estimate shows that Migdal's theorem no longer holds. Thus vertex corrections are no longer negligible and in fact can give corrections to the coupling constant of order unity. At present there is no way to calculate them in detail. Therefore we will assume in the following that the coupling constant qv"(k,q) includes them already. The Griineisen parameter y~is then an effective parameter which is determined from ultrasonic experiments and not necessarily identical with qT of Eq. (4.1).

6. HYDRODYNAMIC DESCRIPTION When ultrasound of wavelength q and frequency Q is propagating in heavy fermion systems, the hydrodynamic conditions q1<< 1 and COT << 1 are usually well fulfilled. Because the sound velocity u p is only a little less than the Fermi velocity u: of the heavy quasiparticles, the above conditions are equivalent to requiring that the wavelength of the sound wave is much larger than the quasiparticle mean free path 1. In the hydrodynamic regime thermodynamic equilibrium is locally attained. Therefore, one can express the temperature-dependent velocity and attenuation of sound in terms of (1) static thermodynamic functions such as the specific heat, thermal expansion, etc., and (2) Onsager coefficients like the electrical or heat conduction. In the following we want to summarize the results for the ionic displacement correlation function (4.40) in the hydrodynamic limit.77 It contains all the information on ultrasound such as the sound velocity and attenuation. The longitudinal displacement operator cpq of the ions has the form (4.41) where b i , h, are the phonon creation and annihilation operators, respectively. The calculations are done by The phonon energy dispersion is given by oq. using the projection techniques introduced by Mori7' and Z w a n ~ i g After .~~

77 78 79

K. W. Becker and P. Fulde, Z . Pliys. B: Condens. Matter 65,313 (1987)and 67,35 (1987). H . Mori, f r o g . Theor. Phvs. 33,425 (1965). R. Zwanzig, Lecf. Theor. Ph.v.s. 3 (1961).

THEORY OF HEAVY FERMION SYSTEMS

53

relations have been established between, e.g., the longitudinal ultrasound attenuation and various thermodynamic quantities, one can try to calculate the latter in a second step from a microscopic theory. This will be done in Section 8. In setting up a theory of hydrodynamic fluctuations, the most important point is to decide which of the dynamical variables are strongly coupled with each other and therefore must be treated explicitly. The remaining degrees of freedom are then included in internal friction coefficients. A guideline for finding the right variables is provided by the following rule with respect to the number of hydrodynamic modes”: “The number of hydrodynamic modes equals the number of conserved quantities plus the number of broken continuous symmetries.” This rule has to be modified in the presence of longrange Coulomb interactions, as will be discussed later. We use it here and identify the phonon momentum, the total energy, and the total electron number N, as the relevant conserved quantities. The phonon momentum operator i$, is given by (4.42) The Fourier components q of the total energy and electron number operator are denoted by h, and p,, respectively. The translation symmetry is a continuous symmetry which is broken in a lattice system. The related operator is that of the lattice displacements of Eq. (4.41). Clearly the operators cpq and $, are strongly coupled to each other through the commutation relation

c$:,cp,,l-

= dqq*

(4.43)

But also h, and pq are coupled to ‘p, and $, because of the quasiparticlephonon interactions. Therefore we shall consider the following set of coupled variables

,I

.‘A’ L

=

bP,’$,’h,’P,~

(4.44)

and treat all other degrees of freedom by internal friction coefficients. It turns out as advantageous to work with those parts of h, and ps which are “orthogonal” 10 $, and q,, i.e.,

The relations ($,Ih,) = ($,Ipq) = 0 follow from time reversal symmetry. A Mori scalar product between any operators A and B has been introduced by

*’

P. C. Martin, 0. Parodi, and P. S. Pershan, P h j s . Rev. A 6, 167 (1972).

54

PETER FULDE

defining (A I B ) =

el a/.

IO1 ’T

d,? (e ‘‘I A e - ‘ H B )

(4.46)

For simplicity we set k , = 1 in the remaining part of this article. The bracket (...) denotes a thermal average. H is the Hamiltonian of the system. We assume it to be of the form = Hqp

+ Hph + Hqp-ph

(4.47)

Here Hq, is the quasiparticle Hamiltonian, Eq. (4.27), and H,, that of the phonons. Hqp-phdescribes the quasiparticle-phonon interactions, Eq. (4.38). The explicit form of the three contributions is not required in the following. In order to compute x,(q,w) one rewrites Eq. (4.40) in the form (4.48) Here 2’ is the Liouville operator. I t is a superoperator which acts on any other operator A through the relation l u A = [ H , A] ~. Equation (4.48) can also be written as (4.49) xlp,P(q, 4 = x,&) + w4,,(q3 w ) cb,,(q,to) is a diagonal element of the (4 x 4) relaxation matrix (4.50) The projection formalism (for reviews, see, e.g., Forster” and Fick and Sauermann82)gives a prescription for the calculation of the relaxation matrix. Formally, it is the solution of the following matrix equation

c(

-z$q

+;

[L,”(q)

-

M,,(q)lx;;

1

4 q y ( q ’ 4= xau((l)

(4.51)

x,,”,L,,,, and M,, are the static susceptibility, frequency term, and memory function matrix, respectively. They are defined by x,y(q)

‘I

’’

=

(A$q

D. Forster, “Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions.” W. A. Benjamin, Reading, Massachusetts, 1975. E. Fick and C. Sauermann, “Quantenstatistik dynamischer Prozesse.” Verlay Harri Deutsch, Frankfurt. Federal Republic ol Germany, 1986.

THEORY OF HEAVY FERMION SYSTEMS

55

where Q is a projection operator. It projects onto the operator space which is perpendicular to that spanned by the variables {A;}. The static susceptibility matrix contains scalar products of the form (cp,lcpq), ( h i l h ; ) , ( p + l p t ) , and ( h i lp;). The frequency-term matrix contains matrix elements of the form

Hereby we have assumed the special case of an isotropic stress tensor oil = - p S , , . In analogy to 9 the superoperator 2&-ph is defined through F q p - p h A = [H,,. ph,A]-. Finally, for the memory matrix the following approximation is made

lo

0

0

o \ (4.54)

\o

0

0

ole21

The matrix contains three Onsager coefficients. They are the internal friction coefficient D, and the thermal and electrical conductivities K and o, respectively. The internal friction coefficient contains the coupling of 'pqto all those variables which are not contained in the set 1.41;). The prefactor q 2 results from the conservation of the momentum, energy, and electron number. In the long-wavelength limit q + 0, the scalar products which appear in the static susceptibility matrix, as well as the matrix elements of the frequencyterm matrix, can be expressed in terms of static thermodynamic functions. For details we refer to Becker and Fulde.I7 The function xqp(q, o)is of the general form

xv&L4

=

1 -

+ iwq2D(q,w)

w 2 - (vTq)2

(4.55)

where uT is the isothermal sound velocity. With the ansatz of Eq. (4.54) for the memory function, D(q, w ) is of the following form D(q,W) = DR

' More precisely,

(i

+ i w +AiD,q , + i

B w + iDTq2

is the conductivity contribution of the heavy quasiparticles.

(4.56)

56

PETER FULDE

t't

ul.

The energy and quasiparticle diffusion constants DE and DT are given by

(4.57) where C , and xc are the specific heat at constant volume and the electronic density susceptibility, respectively. The prefactors A and B contain static thermodynamic functions. x,,(q, w ) has therefore a four-pole structure. The poles describe the hydrodynamic modes which are associated with density fluctuations. They result from the four variables { A;}(v = 1,. . . ,4). Two of the modes are propagating with w = u,q, where up is the phonon velocity (Brillouin and anti-Brillouin mode). The other two modes are overdamped. They are due to the energy and quasiparticle diffusion. The behavior of Im[X,Jq,w)/w] is shown in Fig. 13a for fixed value of (91. I t is quite distinct from that which one finds for ordinary metals or more precisely for a jellium model. There the function Im[x,,(q, O ) / W ] has a three-peak structure (see Fig. 13b). The quasielastic peak due to quasiparticle diffusion is missing. This is a consequence of the long-range Coulomb interaction which converts the fourth hydrodynamic mode into a plasmon oscillation. The density susceptibility is found to be xc q 2 in that case and therefore q 2 D , remains finite as q + 0. In a heavy fermion system (or more generally in a metal with two types of carriers with very different effective masses) the f electrons are screened by the conduction electrons and thus the quasiparticle interactions are short ranged. In the regime o < uFq, which is the one of interest here, one can therefore assume a finite density susceptibility even for q = 0. This subject will be taken up again, at the end of this section, when the nonhydrodynamic modes are briefly discussed. The second quasielastic line due to electronic charge fluctuations is a distinct feature of heavy fermion systems. But it should also be noticed that it is not limited to heavy fermion systems, because all that is required are two types of electronic carriers with very different effective masses. The excitations associated with the second quasielastic peak can be considered as overdamped acoustic plasmons (for a discussion of propagating acoustic plasmons, see, e.g., Gutfreund and Unnas3 and references cited therein). However, we shall prefer to speak of a quasiparticle difuusion mode which is causing the broad, second quasielastic line. In an ordinary metal it makes little difference whether the energy-density fluctuations are treated explicitly or not because their effect is much too small in order to lead to appreciable effects. Their influence shows up in the

-

x3

H. Gutfreund and Y . Unna, J . Phys. Chrm. Sblih 34, 1523 (1973).

THEORY OF HEAVY FERMION SYSTEMS

57

a

191 fixed

A - "p9

0

vPq

W

b

lql fixed

FIG. 13. Hydrodynamic modes (a) for a heavy fermion system and (b) for an ordinary metal. The intensities of the quasielastic peaks are a factor of 105-10" larger in heavy fermion systems than in ordinary metals. The quasiparticle diffusion mode is missing in the latter.

difference between the adiabatic and isothermal bulk moduli c i and c,T, respectively (Landau and L i f s ~ h i t z Vol. , ~ ~ V). (4.58)

Here a is the thermal expansion coefficient and C, is the specific heat at constant pressure per unit volume. At T = 0, ci(0) = c;(O), while at finite T H4

L. D. Landau and E. M. Lifschitz, "Lehrbuch der Theoretischen Physik," Vols. I-X. Akademie Verlag, Berlin, 1966.

58

PETER FULDE et nl.

the difference is proportional to (T/TF)2.Because TFis of the order of lo4 K in ordinary metals, this correction is negligible at low temperatures. Let us return to a discussion of the hydrodynamic modes in heavy fermion systems. When none of the diffuse modes overlaps appreciably with the Brillouin mode, one is in the isolated regime. As long as only the quasiparticle diffusion mode overlaps with the Brillouin mode, one is in the adiabatic regime. Finally, when the energy diffusion mode also overlaps with the Brillouin line, one is dealing with the isothermal regime. In heavy fermion systems one is usually in the isolated regime and almost never in the isothermal regime. To show that, one must compare the respective widths DEq2and D,q2 of the quasielastic lines with upq. Assuming the Wiedemann-Franz law to hold in heavy fermion systems, one finds that (4.59) where N*(O) is the (large) quasiparticle density of states. A particular feature of heavy fermion systems is that the density susceptibility (4.60) is of similar size as in ordinary metals. This has a simple physical reason. When electrons are added to the system, they d o not go into the narrow resonance near E , , but rather into the broad conduction band. Otherwise the f-electron count nf per site would change. This behavior of xc can be related to a large Fermi-liquid parameter F ; (see Section V). The latter enters xc through (4.61) where 1:’) is the density susceptibility of a free electron gas. The requirement xc = xLo’ is therefore equivalent to the one that F i N m*/m. From the above it follows that there is a subtle difference between the density susceptibility and the electronic compressibility. When pressure is applied, then T* changes according to Eq. (4.1) because the hybridization between f electrons and conduction electrons changes. This results in changes in the f-electron count nf and therefore in the positioning of the resonance relative to p . These changes are small, but they show that changing p by adding electrons or applying pressure leads to different responses. The “normal” size of xc in heavy fermion systems is a key input of the following analysis.

THEORY OF HEAVY FERMION SYSTEMS

59

From the above it follows that 1

DE -

D,

1

+ F i << 1

(4.62)

The quasielastic line due to electronic density fluctuations is by a factor of order m * / m broader than that due to energy density fluctuations. This is so because C, in DE contains the large density of states while xc in DT does not. For 10-MHz phonons with a sound velocity of up = 3 x lo5 cm/sec and v % = lo6 cm/sec, 1 = cm, one finds DTq

- N

UP4

-

2 x 10-4-m* << 1 m

(4.63)

implying that one is in the isolated regime. This may change when one moves into the GHz regime or uses samples with much longer mean free paths. In that case a crossover will take place into the adiabatic regime. In passing we note that in ordinary metals one is often in the isothermal regime. The presence of the quasielastic lines has an influence on the ultrasound attenuation a,.The latter can be directly determined from Eqs. (4.55) and (4.56), respectively. It turns out that the dominant contribution to a1 results from the particle diffusion pole. It is (4.64) Compared with it the two other contributions, i.e., the one from internal friction and from the heat conduction pole (4.65) are found to be small. Here pM = MN/R is the mass density and a is again the thermal expansion. The contribution of the heat conduction pole to a[ of, e.g., UPt, can be directly calculated since the thermal expansion, heat conduction, and specific heat are known. Of course, proper provision has to be made for the fact that the material is anisotropic. The experiments by Miiller et al. (1 986) on IJPt, show a pronounced peak in a,(T)around 12 K. When one computes the contribution of the heat conduction pole to a,(T),one also finds a peak, but its magnitude is too small by two orders of m a g n i t ~ d e . ~Similar ' reasoning applies to the internal friction term. The term in Eq. (4.64)cannot be is not known experimentally. Therefore directly evaluated because (dp/dR)T,Ne 85

C . J. Pethick and D. Pines, personal communication (1986).

60

PETER FULDE

el

ul.

one must try to determine it microscopically from a model Hamiltonian. In Section 8 such a model Hamiltonian is described. With its help one finds the following expression to lowest order in T

(%)T,Ne

=

A{

p

-"[.

+ 2N(O)

r

1

+ $($)2

+

(4.66)

The electronic Gruneisen paramter y has been defined in Eq. (4. l), while for the definition of r, we refer to Eq. (2.6). Within the same model one finds that the (T/T*)' contributions to cc,(T)which result from Eqs. (4.64) and (4.65) have the ratio (4.67) This is of order ( p / r ) ' >> 1 and therefore expected to explain the large observed attenuation in UPt,. Although one has obtained in this way only an expression for the ( T / T * ) 2increase of nl(T),it is obvious that the present theory results in a maximum around T*. For T > T* the quasiparticles change their character completely, and the interaction of phonons with delocalized .f' electrons disappears. Therefore this attenuation mechanism is no longer present, implying that q ( T )must have a maximum around T*. Such a maximum has been observed by Muller et ~ 1 for . UPt,, ~ and ~ the results of their experiments are shown in Fig. 14. The theory of sound absorption was

TEMPERATURE ( K ) FIG. 14. Longitudinal ultrasound attenuation in UPt, as function of temperature for different

frequencies. From Muller er ~

1

.

~

~

THEORY OF HEAVY FERMlON SYSTEMS

61

also considered by Yoshizawa et ~ 1 . , * but ~ ~ their attenuation mechanism differs somewhat from the present one. The ultrasound velocity is obtained within the above formalism as

(4.68) provided that the coupling term ( p + / h t )can be neglected, This is the case for sufficiently low temperatures. The T dependence of up is given by that of v T ( T ) and of the term in braces. That of v,.(T) can be measured independently, in principle, by measuring the isothermal bulk modulus c i . For a comparison with the available experimental data on uP(T)for, e.g., UPt,86, one should generalize the present theory to anisotropic systems. Such a generalization is straightforward. We want to discuss in more detail the diffusive excitations. For that purpose we assume isolated conditions. For small frequencies (i.e., o << upq)it is found that

xv,(%o) = x1

iDEq iDTq w iDEq2 + x2 w + iDTq2

For T < T* the susceptibilities

+

(4.69)

x1 and x2 are of the form

Here we have introduced the adiabatic sound velocity v, and the inequality holds, up > v, > u7. [see Eq. (4.68)]. The Landau-Placzek ratios are given by the intensities I, (or I,) of the quasielastic line and of the Brillouin lines 21,. One finds

(4.71) R,

+ R 2 =fV 2 - 1 UT

M. Yoshizawa, B. Luthi. and K . D. Schotte, Z . P/i~s. B: Con&n.s. M u / / r r 64, 169 (1986). M. Yoshizawa, B. Luthi, T. Cato. T. Suzuki, B. Renker, A. de Visser, and J. J. Frame, J. Magn. Magn. Motcv. 52, 413 (1985).

8s‘1

*’

62

PETER FULDE et a / .

R , is larger by a factor of 104-106 than in ordinary metals. This is seen best by considering Eq. (4.58) and how the difference (cL)-' - (c;)-' changes when one goes from an ordinary metal to a heavy fermion system. One notices that C,, is enhanced by m*/m but so is Act, the electronic contribution to the thermal expansion which dominates at low T. By setting H = 0 in Eq. (4.10~)and neglecting the difference in the specific heat at constant volume and constant pressure, one can write to leading order in T

(4.72) or alternatively, (4.73)

-

The prefactor of the T2-dependent term (note that AC T ) is therefore enhanced in heavy fermion systems by a factor of q2m*/m = lo5 as compared with that of ordinary metals. From Eqs. (4.71),(4.68), and (4.66) it is seen that the intensity of the quasiparticle diffusion peak remains finite as T + 0. This is of great advantage for a possible experimental verification of it, e.g., by light scattering. Indeed it seems that the quasielastic peak due to electron diffusion has been seen in a recent Brillouin scattering experiment on UPt, by Mock et This was the first time that such scattering due to density fluctuations was seen in a metal. Since 141 could not be varied in the experiment, there is still no conclusive proof that it is the hydrodynamic mode that is seen. At the end of this section we want to discuss briefly the density fluctuation modes that are not generated by hydrodynamic modes.77 Thereby one starts from a quasiparticle band structure as given by the effective Hamiltonian (3.18). Due to the dispersion E;(k) the Fermi energy is intersecting several quasiparticle bands. The electronic variables, which are treated explicitly by projection technique, are the partial densities belonging to the different quasiparticle bands as well as the transition operators between the bands. For q/<<1, one obtains, in addition to the hydrodynamic modes described before, an overdamped mode, w = - i/r. It results in an additional quasielastic peak in Im Xaa(q,w ) / o with a q-independent line width. As q increases and the regime q/ << 1 is left, the quasiparticle diffusion mode w = -iDTq2 and the mode w = - i / r go over into two propagating modes w,,,(q) =

kuaplq -

i/(24

(4.74)

R. Mock, B. Hillebrands, P. Baumgard, and G. Guntherodt, Verh. Dtsche. Phys. Gas. l(1987).

THEORY OF H E A V Y FERMlON SYSTEMS

63

where uaPl N ~ : ( r n * / ( 3 m ) ) ” ~ .This mode can be interpreted either as an acoustic plasmon or a zero-sound mode with velocity uapl.(The analogy to the zero-sound mode is seen from m * / m = 1 + F ; . ) In addition to the above modes, one finds two types of (optical) plasmon modes. One is the conventional plasmon of frequency wpl = (n,e2/m)”2, where n, is the conduction electron density. The other is a low-lying optical plasmon of frequency w;, = 6’”k,T* (see also Ref. 86b). When the scattering rate of the quasiparticles is very large, the low-lying optical plasmon mode can be g,oes over into w = - i ~ ( o , *It~ cannot ) ~ . be overdamped. In that case o,*] excluded at present that it is this overdamped mode that is seen by Mock et in UPt,.

V. Quasiparticle Interactions and Fermi Liquid Description

Landau’s approach to Fermi liquids starts from the basic assumption that the classification of the excited states of a (complex) interacting electron system is the same as that for an ideal Fermi gas. The elementaryexcitations of such a system are the quasiparticles. Because of the above assumption, the entropy associated with them is given by the same combinatorial expression as for a Fermi gas. Therefore the distribution function of the quasiparticles is that of an ideal gas (i.e., a Fermi function) but with renormalized parameters such as the effective mass. The deviations of the quasiparticle distribution function from a step function O(lkl - kF) are denoted by dn,(k). Because quasiparticles interact with each other, their energy depends on the distribution dn,(k) [see Eq. (3.1)]. The energy change of the system due to elementary excitations is therefore given by

The faar(k,k’) describe the quasiparticle interactions. This expression, together with the entropy expression for an ideal Fermi gas, constitute the essence of the Landau approach to Fermi liquids. Originally the Landau theory was derived for the description of isotropic quantum Fermi liquids like 3He. However, the approach has also been used for metals and more recently for the strongly anisotropic heavy fermion systems. The aim is thereby to determine the faar(k,k’) by relating them to measurable quantities and to predict with them other experiments.

86b

A. J. Millis, M. Lavagna, P. A. Lee, Phys. Reu. B: Condens. Matrer 36,864 (1987).

64

PETER FULDE rt al

In a homogeneous system the f0,,(k, k‘) depend only on the angle Y between G’.Therefore one can replace them by a 2 x 2 interaction matrix k, k’ and on cr

-

-

f(9) = J”(3)l + d crlf”(8)

(5.2)

and expand the coefficients in terms of Legendre polynomials 1

I7

(5.3) The F ; and F; are the Landau parameters. They characterize the interactions between the quasiparticles, and the assumption is usually made that only a few of them are of importance. The Landau parameters enter into measurable quantities and can be determined from them. Examples are: (1)

The spin susceptibility of the quasiparticles

where peffis their effective magnetic moment; (2) The density susceptibility xc as given by Eq. (4.61)into which F t enters; (3) The electronic specific heat

C ( T )= yT

+ 6 T 3In T + 0 ( T 3 )

(5.5)

where”

(5.6) and

(5.7) Here w, = 1 and w, = 3 are the respective degeneracies of the symmetric and antisymmetric states. The A : are related to the F : through Ad -

-1

F:

+ Ff/(21 + 1)

(5.8)

The form of Eq. (5.5) for the specific heat is valid only as long as there are no

87

C. J. Pethick and G. M. Carneiro, P/iy.s. Rcu. A 7, 304 (1977)

THEORY OF HEAVY FERMION SYSTEMS

65

scattering centers present. Otherwise important modifications do occur at low

If one assumes that only the Landau parameters FA, F : (2 = s, a) are different from zero, they can be determined from xs,x c , and 6, and the use of a forward-scattering sum ruleg0 CA:=O

(5.9)

1.1

In Galilean invariant systems there is an additional relationship between m* and the Landau parameter Ff;through m*/m = 1 + F”,3. However, in a crystal Galilean invariance is broken and this relation is modified and cannot simply be used to determine F ; .91-93 The above approach of a homogeneous quasiparticle system has been used by a number of authors for a description of heavy fermion systems, in particular UPt, (e.g., Bedell and Q u a d ~ r Valls , ~ ~ and T e S a n o ~ i c ,Pethick ~~ et a/.,96 and Pethick and Pines”). A thorough review of that work was recently provided by Pethick and p i n e ~ . ~Therefore, ’ we shall not discuss it here in more detail. Instead we want to concentrate on the modifications and changes which take place when one is dealing with inhomogeneous systems of quasiparticles. Heavy fermion systems are strongly anisotropic systems. Due to the presence of spin-orbit interactions, Hund’s rule coupling, and a crystalline electric field, the quasiparticles are characterized by momentum k and pseudospin z (see Section 111). The interaction parameters are therefore denoted by f,,,(k, k’). When the unit cell has an inversion center and in the absence of ~ ) , PTI$k,) = I$kr) have magnetic fields, the states I$kr), P I $ k r ) , ~ l t + b ~and the same energy. P and T are the parity and time-reversal operator, respectively. Then for fixed values of k and k’ the matrix f,,.(k, k’) has two independent components only (see, e.g., Pines and Nozieres”). In the case that

P. Fulde and A. Luther, f h y s . Reu. 170,570 (1968). P. Fulde and A. Luther, Phys. Rev. 175,337 (1968). ‘O B. Patton and A. Zaringhalam, f h y s . Lett. A 55, 329 (1975). 9 1 D. Pines and P. Nozieres, “The Theory of Quantum Liquids,” Vol. I. W. A. Benjamin, New York, 1966. 92 A. J. Leggett, Ann. f h y s . 46, 76 (1968). y 3 K. J. Quader, K. S. Bedell, and G . E. Brown, preprint (1986). y4 K. S. Bedell and K. F. Quader, fjiy,~. Reu. B; Conden.y. Mutter 32, 230 and 3296 (1985). ” 0. T. Valls and Z. TeSanovic, Phys. Reo. Left. 53, 1497 (1984). y 6 C. J. Pethick, D. Pines, K. F. Quader, K. S. Bedell, and G . E. Brown, fl1y.r. Rev. Left.57, 1955 ( 1 986). ” C. J. Pethick and D. Pines, Proc. I n / . Coj?/: Recent f r o g . Many-Body Thaor., 4 f h , in press ( I 987).

66

PETER FULDE r t ul

exchange interaction between the quasiparticles is much stronger than the spin-orbit interaction, the interaction matrix, when written in spin space (not pseudospin space), is again of the form

-

f (k, k’) = f”(k, k’) + t~ alf”(k,k’)

(5.10)

Here t~ and 0’ are the Pauli matrices which act on spins of the states /i,bkr) and I Ic/k.r.). In anisotropic systems, the quasiparticle scattering amplitudes f”(’”)(k,k’) depend on k as well as on k’. The point symmetry of the crystal, however, enables us to parametrize this quantity in close analogy to Eq. (5.3). The fundamental property which we exploit is the following invariance of the scattering amplitude

f”(”)(k,k’) = fs(a)(Rk,Rk’)

(5.11)

where R is an arbitrary element of the symmetry group. As a consequence, the following simple expansion is obtained

1fsca)(rj) c cp:”)(k)cp$’(k’) d(J)

f”’”’(k,k’) =

j

K =

(5.12)

1

The functions cp$(k) are the basis functions belonging to the kth row of the d(j)-dimensional irreducible representation r ( j )of the symmetry group. The parameters fsca)(rj) are the generalized Landau parameters. Generally there are many sets of basis functions belonging to a given representation. The expansion of the interaction fs(’”)(k,k’) should therefore contain all these sets. In Eq. (5.12) we have made the simplifying assumption that it is sufficient to use one set for each representation only, i.e., the basis functions of lowest order. In the more general case the Landau parameters f’”’”)(Tj)have to be replaced by matrixes. An important example is the interaction leading to anisotropic superconductivity, which will be discussed in Section 12,b. A formulation of Fermi liquid theory in terms of the generalized Landau parameters is still lacking. Therefore we have to proceed in a pragmatic way. Thereby we can either work in r space or in k space. Let us start with r space first. We use the renormalized band theory approach which describes the quasiparticle energies in terms of phase shifts (see Section 2). As has been pointed out in Section 1, the all-important phase shift of thef electrons at site i contains an additional term (5.13) when other quasiparticles in cell j are present. The interactions are described Equation (5.13) is already a simplified version of the by the parameter (D.’,: quasiparticle interaction contribution to the phase shift. In reality there are

THEORY OF HEAVY FERMION SYSTEMS

67

several different types of quasiparticles in cell j . They vary with respect to their s, p , or f character and correspond to different energy bands at the Fermi surface. This would require an additional label for the interaction parameters. For simplicity we discard here such complications. We want to discuss the effect of the quasiparticle interactions on the spin susceptibility xF. For this purpose it is useful to compute the so-called Sommerfeld- Wilson ratio R

rc2ki xs

=--

3dff

(5.14)

Y

Here pefris the effective magnetic moment of the quasiparticles. Because the pseudospin is not globally conserved, peffneed not be that of the CEF ground state of a single ion. Nevertheless, it is expected to be close to that of the singleion crystal-field ground state. It has to be determined from the KKR wave functions. This was pointed out, e.g., by Zou and Anderson.’8 Considering the ratio x J y has the advantage that the quasiparticle density of states has dropped out. This holds true at least as long as quasiparticle-phonon interactions are not important. When they lead to a contribution to the effective mass of the form meff = m*(l + A), one must keep in mind that the renormalization factor (1 + A) appears in y but not in xs.99 In that case it would be more appropriate to define

Reff = (1 + 2 ) R

(5.15)

Let us first consider the single-ion case. xs and y refer here to the excess susceptibility and specific heat coefficient due to the impurity. Neglecting spinorbit coupling and assuming S = +, the interaction constants in Eq. (5.13) reduce to the form . . = dijdr,-I,@ (5.16) and one is left with only one interaction parameter @ . 5 1 It is determined by requiring that the narrow resonance moves with the Fermi energy when the chemical potential changes. From the analog of Eq. (3.8)for the single-ion case one finds (5.17)

where N ( 0 ) is the conduction-electron density of states. In the presence of an applied magnetic field the difference in the phase shifts at the Fermi energy is d+(p)- d-(p) = 2pe,,HnN(O) 98

+ 2m,@

2. Zou and P. W. Anderson, Phys. Rev. Le/t. 57,2073 (1986).

’’D. Rainer, Winler Meet. LOW,Temp. PIIvs.,3 r d Huciendu COCJYJC (1982).

(5.18)

68

PETER FULDE et a\.

where the magnetization m is given by mH = (l/n)[6+(p)- &(p)]. This leads to the well-known result R = 2. For higher orbital degeneracies R = v,-(vf - 1)-’,24 which shows that in the limit of large vf the quasiparticle interactions do not affect xs. In the Kondo lattice case the situation is quite different. The number of interaction parameters is larger and the requirement of perfect pinning of the resonance to the Fermi level fixes only a certain combination of them, i.e., ~

1 = -N*(O) k , T*

1

(5.19)

@;r:j

jr’

Therefore it is no longer possible to derive a definite value for R. Instead one finds xs =

2PL,:“*(O)R

(5.20a) @

j)

(5.20b)

Measuring R determines therefore another combination of @:rTj. One notices that there are four Landau parameters when one is dealing with a cubic system with interactions between quasiparticles on neighboring f-electron sites (e.g., Ce or U sites) only. The number of parameters increases when the lattice symmetry is lower. Next we want to consider the problem of quasiparticle interactions in anisotropic systems by working in k space. In order to demonstrate some of the modifications that may arise as compared with the isotropic case, let us consider a Fermi surface which consists of two distinct parts only. On one part the quasiparticles have a heavy mass, while on the other part the quasiparticles are much lighter. For simplicity we shall again call the latter “light” quasiparticles. The heavy quasiparticles have mass m*,excitation energy Ek(h), and a density of states (per spin direction) N*(O).The light quasiparticles have mass m,excitation energy Ek(l), and a density of states N(0). The interactions between heavy and heavy, and heavy and light quasiparticles are denoted by fh and fi and are assumed to be independent of spin and of k and k’ for simplicity. The interactions between light quasiparticles are neglected. Only the spin symmetric part of the interactions is considered. The changes in the energy dispersions due to quasiparticle distributions 6nkr(h),6nkr(l)are (5.21a) (5.21b) The sums go over the respective parts of the Fermi surface of the heavy and light quasiparticles. We want to compute the electronic density susceptibility

69

THEORY OF HEAVY FERMION SYSTEMS

xc which has been defined in Eq. (4.60). For an ideal gas xc = x:

is given by the total density of states. For an isotropic Fermi liquid Eq. (4.61) holds, which relates xc with the Landau parameter F“, We want to study how this relation is generalized when the above simple model for an anisotropic system is used. With the help of Eq. (5.21) one can write for the change 6 p in the chemical potential (5.22a) (5.22b) (5.22~)

(5.23a) (5.23b) those equations can be rewritten as (5.24a) 2N(O)6p = 6nl

+ 6nhFl

(5.24b)

+

When one evaluates xc, thereby using that N - ’ S N , = 6nh 6nl, one notices that it depends on two parameters, i.e., F h and Fl. One can eliminate one of them by making additional physical assumptions. When electrons are added to the system with the volume kept constant, the energy gain per site due to the singlet formation kBT* changes. This is so because changes relative to p. In order to calculate that change we use to lowest approximation the single-ion relation, Eq. (2.16). Furthermore we use the relation, Eq. (2.20), to relate changes of T* to those of nh( =nf). For the lattice this is again an approximation. As pointed out in Section 111, the quasiparticle band-structure calculations should yield a more accurate connection between T* and n, for the lattice case. This way one finds

~E, I

(5.25) and with the help of Eq. (2.20) (5.26)

70

PETER F U L D E rt (11.

For simplicity we have set vf (5.24b), one finds

= 2. When

6n,(A

-

the last equation is combined with Eq.

F,) = dn,

(5.27)

) 2N(O)k,T*n,(l - nf)’. With the help of where A = 8 N ( 0 ) r 2 / ( n h n 2 k , , T * = this relation xc is found to be (5.28) For the special case Fi = 0 one can check easily that the above assumptions have the consequence that (5.29) Only when A = 1 does it follow that (1 + Fh) = m*/m, which is the assumption usually made for heavy fermion systems.’00-’02 By evaluating Eq. (5.29) one finds that Fh ( r / T * ) 2 Quite . generally one expects Fh >> 1. The reason for the large value of Fh(or alternatively Fb) is simply the pinning of the narrow resonance to the Fermi energy. This pinning takes place because the formation of the singlet depends only little on small changes of the Fermi energy. The pinning is not perfect, though, because of the weak dependence of T * on p. Equation (5.28) demonstrates, as did Eq. (5.20), that an anisotropic Fermi system requires more information about quasiparticle interactions than does an isotropic system. Therefore for practical purposes the Landau approach to Fermi liquids seems less powerful for heavy fermion systems than, e.g., for an isotropic liquid such as ’He. Finally we want to comment on another point concerning quasiparticle interactions. As pointed out before, the Landau quasiparticle theory starts out from Eq. (5.1)and the ideal gas expression for the entropy. This way one avoids using a particular Hamiltonian. Instead one relates the interaction parameters to measurable quantities and does not face the difficult problem of their microscopic determination. The above notwithstanding, a Hamiltonian has frequently been taken which describes the quasiparticles and their residual interactions. The “unperturbed” Hamiltonian H , is in that case given by Eq. (3.19), which is that of noninteracting quasiparticles. The quasi-

-

1[10

T . M. R’ice and K. Ueda, in “Theory of Heavy Fermions and Valence Fluctuations”

I”‘

(T. Kasuya and T. Saso, eds.), p. 216. Springer-Verlag, Berlin, 1985. T. M. Rice and K. Ueda, Plzys. Rcw. Le.rr.55,995 and 2093(E) (1985). C . M. Varma, W. Weber, and L. J. Randall, Plrys. Reo. B . Conc/c.ns.M a f l e r 33, 1015 (1986).

lo’

THEORY O F HEAVY FERMION SYSTEMS

71

particle interactions are added by an interaction part

For simplicity the different band indices have been left out. Clearly, caution must be exercised when H = H , + Hint is used for calculating different properties of a system. For example, it would be incorrect to use Hintin order to calculate from it contributions to the effective mass. All interaction contributions to the effective mass m* have been included in H , , provided that one is considering a single low-energy excitation out of the ground state only. On the other hand, static susceptibilities can be correctly calculated with the help of Eq. (5.30) provided one stays within RPA. Hintcan also be used to calculate, e.g., the quasiparticle lifetime, provided one uses low-order perturbation theory. It is reasonable to assume that the quasiparticle interactions are short ranged and essentially take place on the same lattice site only. Therefore the following form for Hinthas been used (e.g., Jichu et ul.’03 and Miyake’04)

Hint= ck,T*

nirnil

(5.31)

I

where nio = a:(i)u,(i) is the quasiparticle number operator at site i and c is a constant of order unity. A coupling constant of order T* is expected from the single-impurity case. There the effective local interaction of two quasiparticles , N ( 0 )is due to the presence of the impurity is of order { [ N ( 0 ) ] 2 k , T , } - ’where the conduction-electron density of states. In the lattice case one has to transform from the conduction electrons to the heavy quasiparticles. With T* replacing T, this yields the above coupling constant.

VI. Microscopic Theories In the preceding sections heavy fermion systems were described by a phenomenological theory, which had the advantage that it allowed for realistic quasiparticle energy-band calculations. In the following we want to discuss the different microscopic approaches which have been developed for their description. The ultimate aim is thereby to derive from a microscopic, i.e.,

‘04

H. Jichu, T. Matsuura, and Y. Kuroda, f r o g . Theor. f h y s . 72, 366 (1984). K. Miyake. in “Theory of Heavy Fermions and Valence Fluctuations”(T. Kasuya and T. Saso, eds.), p. 256. Springer-Verlag, Berlin, 1985.

72

PETER FULDE et al.

uh initio,theory the parameters which enter into the phenomenological theory. There is still some way to go, however, until this goal will be achieved. The starting point of the microscopic theories is the Anderson lattice Hamiltonian. It is conventionally written in the form

1

+ fi ~

c

imkno

~mo(k,n)[c~n,.fm(i)e-ik'R' + H.c.]

In distinction to Eq. (2.1) we have included also a band index n for the conduction electrons so that the k values are restricted to the first Brillouin zone. The site index of the N ions with f electrons is i, while R, denotes their positions. We have allowed for an f-electron degeneracy vf 2 2. The felectron energies cfmcan differ due to spin-orbit and crystalline-electric-field splittings. Energies are counted from the Fermi surface. The hybridization is described by the matrix elements Vmo(k,n).We could have introduced the Fourier transform

into the hybridization term but have preferred the above form for reasons which will become clear later. Most of the microscopic approaches consider the limit of large U. In that limit the f-electron number at a site is either zero or one. If one wants to have more than one f electron per site, one must let the difference between the Fermi energy E , and c f mbecome very large. By properly taking the limit of large U and ( E , - e f m )one can achieve sizeable band fillings also for large values of v f . The microscopic theories starting from the Hamiltonian of Eq. (6.1) can be roughly divided into three groups. One type of theory tries to solve the eigenvalue problem for H by expansion techniques. Thereby one would like to treat the hybridization by perturbation and include the large Coulomb interaction U in the unperturbed Hamiltonian H,. This causes the problem, though, that H , now contains a two-particle operator, so that Wick's theorem no longer applies when one attempts calculating thermodynamic expectation values with respect to the unperturbed ensemble. As a consequence, Feyman's diagrammatic techniques can no longer be used, and nonstandard expansion

THEORY OF HEAVY FERMION SYSTEMS

73

techniques must be d e v i ~ e d . ' ~ ~ -Wick's ' ~ ' theorem can be saved by eliminating the electron Coulomb repulsion term through the introduction of an auxiliary boson field.'08-'09 Feynman's techniques can then be applied with the expense that the boson field has to be included in the diagrams. Another way of saving Wick's theorem is by expanding with respect to the large Coulomb interaction. Against expectations such an expansion can be done and yields reasonable results.'lO.' '' The second type of theory is a mean-field theory. The mean-field approximation is not made in the original Hamiltonian of Eq. (6.1), however. Rather the Hamiltonian is first transformed into a form in which the Coulomb repulsion U no longer appears. The introduction of a boson field is one way of achieving this. It is with respect to that Hamiltonian that a mean-field approximation is introduced.' 1 2 , 1 1 3 The third type of theory we shall discuss is of a variational nature. It starts from a trial function for the ground state of the strongly correlated system and determines the inherent parameters by minimizing the energy. However, it is generally very difficult to calculate expectation values with these trial functions, and therefore further approximations are required. They can be of a form proposed by Gutzwiller' l 4 or similar ones'oo-'02*'15-117. Or they employ expansions with respect to the inverse of the f' orbital degeneracy vf.20 Using v/ as an expansion parameter has the advantage that in the limit of large L? the Anderson lattice problem reduces to one of independent ions. Therefore a v / expansion also has been used to justify approximations within the other two types of theories (see, e.g., Zhang and Lee118).

'

H . Keiter and J. C. Kimball, fhy.7. Rev. Lett. 25, 672 (1970). N. Grewe and H. Keiter, f h j , . 7 . Reo. B: Condens. M u / f e r 24, 4420 (1981). l o b N. Grewe, Z . W7~3. B: Conrken.s.Multer 53, 271 (1983). lo' Y. Kuramoto, Z . Phj'.v. B: Condens. Mufter 53, 37 (1983);see also H. Kojima, Y. Kuramoto. and M. Tachiki, 2. f h j s . B: Condens. Murter 54, 293 (1984); Y. Kuramoto and H. Kojima, 2. f I i , w . B: Condens. Matter 57, 95 (1984). S. E. Barnes, J . Phys. F 6 , 1375 (1976); see also S. E. Barnes, J . f h y s . F 7 , 2637 (1977). lo' P. Coleman, fhy.7. Rcu. B: Condens. Mutfer 29, 3035 (1984). ' l o K. Yosida and K. Yamada, f r o g . Theor. f h y s . Suppl. 46,244 (1970). I" V. Zlatic and B. HorvatiC, P/7j,s. Rev. B. Condens. Mutrer 28, 6904 (1983). ' I 2 N. Read and D. M. Newns, J . Phys. C 16, 3273 (1983). ' I 3 P. Coleman, J . Magn. Magn. Matrr. 478~48,323 (1985). ' I 4 M. C. Gutzwiller, Phys. Reo. A 137, 1726(1965). I" B. H. Brandow, f h y s . Rev. R: Condens. Mutter 33,215 (1986). 'I6 P. Fazekas. SolidS/o/eCommun. 60,431 (1986). "' P. Fazekas, preprint (1986). ' I * F. C. Zhang and T. K. Lee, fhjJ.7.Reti. B: Condens. Mutter 28, 33 (1983). Io5

lo''

74

PETER FULDE et

ctl

7. EXPANSION TECHNIQUES In applications of the Anderson lattice Hamiltonian of Eq. (6.1), the limit U rx, is of particular interest. As pointed out before, the occupation nf of an f state at a given site is then either zero or one. Such a state cannot be obtained from any finite-order perturbation expansion in U. This suggests treating the effect of U by restricting the Hilbert space to states with ns = 0,l. This is achieved by replacing the fermion operators ,fm, f i by standard basis operators X,, = IO)(rn( and X,, = lrn)(O1. Here I r n ) = fl/O). The Hamiltonian of Eq. (6.1) then becomes --f

=

‘n(k)C~nackna kna

1

+fi 1 ~

imkna

+ mi E f m X m m ( i )

+

Vma(k,n ) ( c ~ , , X , m ( i ) e - i k ~ R iH.c.)

(6.3)

One notices that the Coulomb interaction U no longer appears in it. The prize for that is the loss of Fermi statistics for the f states. The standard basis operators must fulfill the requirement

and therefore d o not obey simple commutation rules. This implies that Wick’s theorem or the linked cluster theorem cannot be used. Despite this it is possible to develop a perturbation theory with respect to Vmg(k,n). We first discuss the perturbation expansion technique and then consider the (slave) boson approach. a . Perturbation Expansion in the Hybridization

We want to compute the J-electron Green’s function matrix by a perturbation expansion in terms of the hybridization V. It is defined by Gfmm,(i>j ;

- 7 2 ) = - ( T X ~ m ( i , r ~ ) X ~ , ~ ( j , r 2 ) ) (6.5a)

The superscript H indicates the Heisenberg representation of the operators. For a perturbation expansion one splits the Hamiltonian into H = Ho H ‘ , where H’ contains the hybridization, and furthermore goes over to the interaction representation. Then Eq. (6.5a) reduces to

+

The bracket ( ...), implies that a thermodynamic average with respect to H , is taken. is the time ordering operator and the time evolution of any operator

THEORY OF H E A V Y FERMION SYSTEMS

75

O(z) is given by o ( T ) = ,HorOepHU‘

Finally

In ordinary many-body theory H , is a sum of single-electron operators, while H’ contains two-particle interactions. In that case Wick’s theorem can be applied. The numerator factorized then into contributions from so-called linked and unlinked graphs and the factor from the unlinked contributions cancels against the denominator. In the present case this scheme is no longer applicable, because the standard basis operators Xap(i) are not fermion operators. Consequently, the numerator and denominator of Eq. (6.5) must be evaluated separately. The denominator is just the partition function, because 2 = Tr (exp[ -P(H - pN)]) = Tr{expC-P(H,

-

PN)IS(P)l

(6.8)

Techniques for evaluating Eq. (6.5) perturbationally have been devised by Keiter and Kimball,105 Grewe and Keiter,”’” K u r a m ~ t o , ’ ~ ’and Coleman.L09They make use of the fact that in H , the f electrons are decoupled from the conduction electrons and furthermore that f electrons at different lattice sites are also decoupled from each other. When S(B) [see Eq. (6.7)J is expanded in terms of H‘ the computation of Eq. (6.5b) involves expectation values of the form (rCH:,(z,). . . H:,(r,)l>o

(6.9a) (6.9b)

Here H‘ has been decomposed into contribution HI from different sites, i.e., H’ = Xi HI [see Eq. (6.3)]. Because of the form of H , one can decouple those expectation values into products of conduction electrons and f electrons at different lattice sites. For the expectation value which involves conduction-electron operators only, Wick’s theorem may be used. The obvious way to associate a diagram with an expectation value of the form of, e.g., Eq. (6.9a) is shown in Fig. 15a. There is a “time” axis, and the different dashed lines correspond to different sites. In the figure we have assumed that z1 1 r2 > r3 > ... > z8. The solid lines represent conduction electrons, and it is seen that at each z, an f electron is converted into a conduction electron and vice versa. It is apparent that for an expectation value of the form of Eq. (6.9b) two “time”dots z1 and z2 must appear without having

76

PETER FULDE et ul

"time'

A

I

j

FIG. 15. Association of diagrams with expectation values of the form of Eq. (6.9a).Different sites are labeled by I , j , k , etc. (a) At different times T,, . . .,t8 conduction-electron lines leave or join diRerent sites. They change the f-electron number at that site. Conduction-electron lines must be paired in all possible ways. An example is shown in (b).

an electron line associated with them. The connection between different sites is established through the conduction electrons. For the latter Wick's theorem does apply, and therefore the conduction-electron lines must be paired in all possible ways. One particular example is shown in Fig. 15b. Each electron line starting at site i and terminating at site j corresponds to a free conduction-electron propagator Gc(i,j ; zi - xj). Clearly the simplest approximation one can make is to pair the conduction electrons in a diagram like that of Fig. 15a in such a way that they never connect different sites but always terminate at the same site at which they start. Then one is just dealing with a single-site problem. The f-electron Green's function matrix of Eq. (6.5) or its Fourier transform Gf(i,j;iw,,)reduces to matrix elements Gfm,.(i,.j; iw,) = dijdmm9c,-m(zw,,), where w,, = 27cTk,(n i), ( n = integer) are the Matsubara frequencies. We shall discuss below the properties of the one-site function ~fm(icon) in more detail. The next higher level of approximation is not easy to derive in a systematic way within the diagrammatic scheme. However, it can be derived in a simple, intuitive way by assuming that the conduction electrons connect different sites in a standard RPA-type fashion, i.e.,

+

+ IZi C G,-(iW,,)W(i, I ; i(o,,)Gf(l,j;ion)

Gr(i,j ; ion) = di,jG,-(i~,,)

(6.10a)

77

THEORY O F HEAVY FERMION SYSTEMS

where the elements of the matrix W are given by W(i, I ; iw,)

=

c V:,,(k,

n’)eik.R’Gc(k, n‘, iw,)e-ik.Rf Vmr,,(k, H’) (6.10b)

kn’u

The process I = i is excluded because it is contained in ~fm(icon). A similar form of intersite coupling is obtained when, e.g., rare-earth ions with perfectly localized f electrons are coupled through conduction electrons (see, e.g., F ~ l d e ’ ~Equation ). (6.10) can be easily solved in Fourier space. Writing (6.1 1 ) and similarly for the other Green’s functions, one can replace Eq. (6.10) by G,(k, ion) = C,(iw,,) + cf(iwn)W(k,iw,,)Gf(k,ion) -

w(io,)G,(k, ito,,)

(6.12)

The function w(iw,,) is defined through

c

1 w(iw,) = - W(k, iw,) N k

(6.13)

and enters because of the exclusion I # i in Eq. (6.10a). Solving Eq. (6.12) for Cf(k, iw,), one obtains G,’(k, icon) = Y - ‘(icon)- W(k, iw,)

(6.14)

where

Y ‘(iw,,)=

e;

+

(icon) w(iw,)

(6.15)

The function Y(iw,) plays the role of a local f-electron Green’s function. From Eqs. (6.13) and (6.lob) it follows that w(io,) is given by wmmz(iwn) = - irdmm. sgn(w,)

(6.16)

where r = xN(O)V2 as before. Thereby an approximation with respect to VmU(k, n) has been made which closely parallels the one leading to Eq. (2.3). At low temperatures the term w(i0,) in Eq. (6.15) compensates the selfenergy due to hybridization contained in cf(iw,). Of interest are the poles of the f-electron Green’s function in Eq. (6.12) because they are expected to describe the heavy quasiparticles. For that purpose one continuous analytically all functions to the real o axis and considers the low-temperature limit in order to avoid the problem of quasiparticlequasiparticle interactions. The problem reduces to that of G,,,(OI), which is a single-site problem.

78

PETER FULDE er d.

The single-site problem is solved best by means of the resolvent method.’” We consider again a Hamiltonian of the form of Eq. (6.3) but this time with the site index i limited to a single site which we place at the origin. By using Eqs. (6.8)and (6.7) and the form of H,, one can shown that the partition function Z can be written in the form Z = Z‘Z,, where Z, is the partition function of the noninteracting conduction electron system and 2, is given by Z,

= Tr/.R(r)

(6.17)

zrn

E , ~ X , , and , ~ H’ is the same as before but with the site index i Here H , = limited as pointed out above. The average (...), is with respect to the conduction electrons. The remaining trace Tr, is with respect to the j‘ electron only. Thus R(B) is an operator in which the conduction electrons no longer appear. Its matrix elements are R,,(T) = (ollR(z)lP), where a, are I . f o ) ( =0) and If1, m ) ( = m), respectively. By introducing the Laplace transform one can write

z

= (0

+ ill

(6.18)

The second equation serves as the definition of a self-energy operator C(z).In terms of R ( z ) the function Z, takes the simple form (6.19) where the contour C encircles counterclockwise all singularities of R(z). The commonly used approximation within which R ( z ) is calculated is the noncrossing approximation (NCA). When expressed in diagrams of the form shown in Fig. 15a, it corresponds to accounting only for those processes in which the paired electron lines do not cross each other. An example of such a diagram is shown in Fig. 16a. A diagram which is neglected is shown in Fig. 16b. In the NCA the matrix R,&z) becomes diagonal, i.e.,

(6.20) where cz equals 0 or efmrrespectively. The self-energies are solutions of the

T H E O R Y O F HEAVY F E R M I O N SYSTEMS

79

FIG.16. Diagrams contained (a) in the noncrossing approximation and (b) neglected in it. When a conduction electron (solid line) is propagating “up” in “time,” the dashed line in that time interval is an I f o ) state.

following coupled integral equations C0(z)= C m

Crn(Z) =

s

s

+ E)

dc N ( E ) Vl2f’(~)R,(z [

d€N(c)IV2C1 - . f ( ~ ) I R o ( z 6)

(6.21a) (6.21b)

The form of these equations is easy to understand. When a conductionelectron line is running up in time, the corresponding dashed line in that time interval corresponds to a I f o ) state (see the dashed enclosure in Fig. 16a). In order for the conduction electron to find an empty state a factor [l - f ( ~ ) ] is required. The resolvent of the I f o ) state is Ro(z). This explains the form of Zm(z). From similar arguments one can also understand the form of Eq. (6.21a). The theory can be also applied to multilevel system^."^ Before discussing the solutions of Eq. (6.21), we want to write down the Within connection between R ( z )and the f-electron Green’s function gfm(iwn). the NCA it is given by e ~ZRo(z)R,(z + ion)

(6.22)

This form is plausible when one goes back to Eq. (6.5) and sets j = i. At 7 2 a transition takes place from state 10) to Im) while at z1 the transition is from Im) to 10). Therefore a diagram can be drawn of the form shown in Fig. 17. The resolvents of the states 10) and Im) are Ro(z)and R,(z), respectively. This explains why Gfm(kon) contains a folding of the two resolvents. The additional frequency ionin R , is the conjugate of the time interval (7, - T ~ ) From . Eq. (6.22) it follows that the spectral density AJm(w)of G,,(o) ‘Iy

K. W. Becker and J. Keller, Z.P / i w . Bc Condms. Mu//er 62, 477 (1986).

80

PETER FULDE

1’1

ul.

I1

/-, I

/

\

\

Im>{ \

I to>

I \

/

‘*’

/

z2 FIG. 17. Graphical representation of Eq. (6.22). At ( f ’ m ) state.

T~

an

11’) state is transformed

into an

can be written as

where po(Z),pm(Z)are the spectral densities of the resolvents, i.e., 1 p,(o) = - - Im R,(w

n

+ id)

(6.24)

o) R(z) within the After having established the connection between (?,-,(and NCA, we continue with the discussion of the consequences of Eq. (6.21). For finite temperatures the coupled integral equations can be solved numerically. We shall consider directly the numerical solutions for A,-,(w). In the Kondo regime, i.e., when E,-, is sufficiently below the Fermi energy, A,-,(oi) has a broad peak at w = E,-” of width vrr = v,-nN(0)V2.But there is in addition a second peak at w = 0 which is often referred to as AbrikosovSuhl r e s ~ n a n c e . ” ~Its ” ~ ~presence is one of the essential features of the Kondo problem. One can trace it back to the feedback mechanism contained in Eq. (6.21). When I/ = 0 the spectral densities p,(o), p,(o) have &function peaks at w = 0 and at o = E,-, respectively. A finite hybridization does not only lead to a broadening but, e.g., due to Eq. (6.21a), p , ( o ) develops in addition a second peak at w ‘v efm - kBTK.As before, TK is the single-ion Kondo temperature [see Eq. (2.16)]. Folding the two spectral densities as in Eq. (6.23) results in a peak in A,,(o) at o ‘v kRTK. Its weight is of order ( 1 - nr), which is the probability that the f’ state is empty [compare with Eq. (2.17) and with Fig. lo]. A peak of the same weight and position follows also from the variational approach.” ‘‘“a

’

A. A. Abrikosov, Ph.v.sic~.c.2. 5 and 61 (1965). H. Suhl. in “Theory of Magnetism in Transition Metals,”Course 37(W. Marshall,ed.), p. 116. Academic Press, New York, 1967.

T H E O R Y OF HEAVY F E R M I O N SYSTEMS

81

For the later discussion of Gfm(k,o)it is useful to model the function Gfm(o) near Q N 0 by a simple expression, so that its spectral density agrees with A,.,(to), as obtained from the numerical procedure described above. Particularly convenient is an ansatz of the form (6.25) From the above it follows that a,. N (1 - n,.), Ef ‘v k , TKand ’v (1 - n f ) T , provided that o l k , , T-K TK. Because of its importance we want to consider Eq. (6.25) from yet atother point of view. One could have also started from the following form of GJ,(z), namely (6.26) Here cSmis the unrenormalized position of the ,f level, which is far below the Fermi energy, and C,(z) contains now the effects of hybridization and the Coulomb repulsion U. In the absence of the latter C,(z) = -iT. In order to take the Coulomb interaction into account, one can write z,.(Z)=

-ir

+ zin,(Z)

(6.27)

The real part C, (Q) = Re Cin,(z) must be large at o = 0 in order to produce a pole at w = 0. The imaginary part C,(w) = Im Zin,(z) must vanish like ozin order to produce Fermi liquid behavior at low T. By expanding Z,-(z) around o = 0, one obtains back the form of Eq.(6.25) with a,.=(l -1

-2)

(6.28)

Having a thorough understanding of Gfm(z),we are able to take up again the problem of the lattice f Green’s function [see Eqs. (6.10)and (6.12)]. From the representation of Eq. (6.26) of Gfm(z)and from Eq.(6.27), it is seen that the function Y ( o )[see Eq. (6.15)] has a sharp pole near o N 0 [the term -iT in Eq. (6.27) is cancelled by the term V 2 g , ( o ) in Eq. (6.15)]. From Eq. (6.14) it follows then that (6.29) The quasiparticle excitations are obtained from the poles of that function. They are of the form E(k),., = +{E(k) C,. T ,/[c(k) - C,.]’ 4a,.V2} (6.30)

+

+

82

PETER FULDE et al.

and therefore practically the same as in Eq. (3.18). One notices again the . that reduction in the effective hybridization by a factor k= U : ’ ~ VRemember u/. ‘v T * / r . The resonance form of Eq. (6.25)can be used in order to establish a link with Friedel’s virtual bound-state picture (see e.g., Friedel l Z o ) . In fact, one may consider the heavy fermion system as a classic example of a lattice of virtual bound states. In the case of a single f-state impurity in a metal the T matrix for the scattering of conduction electrons is expressed by the single-ion .f‘ Green’s function as

T ( w + iy) =

V2Gf(W

+ iy)

(6.31)

From the Lorentzian form of Eq. (6.25) it follows that Im T ( w + iy)

=

PzP-

-

(LO - E f ) 2

+ FZ

(6.32)

One can alternatively introduce a phase shift h(w)in order to characterize the scattering process. It is related to the T matrix through (see, e.g., Messiah l Z 1 ) sin2 6(w) = - ~ N ( o )Im ~ ( +w iy) -

(w

-

FZ 5,’ + T 2

(6.33)

The latter equation can be written as h(cr))= arctan

(6.34a)

(E; ~

This is the well-known result of Friedel’s bound-state theory. A t the Fermi energy the phase shift is of the form

h(0) = arctan,

PEf

=

n/. n-

(6.34b)

“f

The second equality follows when Friedel’s sum rule is applied. In general this sum rule relates the phase shift at the Fermi energy to the total number of accumulated electrons. It can be shown ‘ 2 2 * 12 3 that in the case of the Anderson Hamiltonian this electron number can be replaced to good approximation by the occupation number nf of the local state; i.e., the conduction electrons do not count, if the density of states of conduction electrons is constant around the Fermi energy. ”‘)

‘*I

”*

J. Friedel, Nuovo Cimcvz/o Suppl. 7, 287 (1958). A. Messiah, “Quantum Mechanics,” Vol. 2, Ch. 19.9. North-Holland, Amsterdam, 1961. D. Langrelh, Phja. Riw. 150, 516 (1966). A. Yoshimori and A. Zawadowski, J . f‘hys. C 15, 5241 (1982).

83

THEORY OF HEAVY FERMION SYSTEMS

When one expands Eq. (6.34a) around the Fermi energy, one obtains F

(6.35) This expression can be compared with that of the phenomenological theory Eq. (3.8). As one moves into the extreme Kondo limit (i.e., ns 1) and for vs = 2 the phase shift h(0)+ 71/2 and Cs + 0. In that case --f

7 1 . 1

S(0) =-

2

+r ,(I)

(6.36)

Let us return again to the lattice case. In deriving the ,f-electron Green’s function Eq. (6.29) from Eq. (6.25), we have treated the lattice as if it consisted of a collection of single-ion scattering centers with virtual bound states. Criticism might be raised that this picture is too simple. This is not the case, though, if one allows for values of Es and which differ from their respective single-impurity values. In the phenomenological theory of Section I11 we have treated them as adjustable parameters to be determined from experiments. Within a refined theory it should be possible to calculate the corrections due to the mutual influence of different sites (in the mean-field theory to be discussed in Section 8 such corrections are not yet contained, and essentially the simple picture developed here is obtained). The concept of a phase shift, however, loses its sense in a lattice, since it refers to asymptotically free conductionelectron states. Nevertheless, it is a useful tool for quasiparticle band-structure calculations. Phase shifts can be either defined through the relation (6.34a), which relates them to the virtual bound-state energy, or by characterizing the derivative of the electronic wave function on a muffin-tin sphere. Of course, with such a definition the value of the phase shift at the Fermi surface is no longer related in a simple way to the occupation of the localized f state. We have considered the perturbation expansion method in so much detail in order to show its relation to the phenomenological approach introduced in Section 111 and to the virtual bound-state concept of Friedel. As mentioned in the introduction of this article, one can save Wick’s theorem by eliminating the Coulomb repulsion between the f’electrons in the Hamiltonian, Eq. (6.l), and introducing instead an auxiliary boson field. When, within that formalism, perturbation theory is done in V,/mb,109 the results are the same as those obtained within the previously discussed methods. For that reason we shall describe the boson-field method later when the mean-field theory is discussed.

r

b. Perturbation Expansion in the Local Coulomb Interaction The perturbation expansion in terms of the local Coulomb repulsion U starts from the Anderson Hamiltonian, which is divided into an unperturbed

84

PETER FULDE

el

al.

efTective single-particle part H , and the remaining interaction H ' . The effective single-particle Hamiltonian H , is taken to be the nonmagnetic Hartree- Fock approximation to the full Hamiltonian, which includes the mean-field Coulomb potential. The resulting independent-electron problem can be solved exactly by the standard band-structure methods. The starting point for the U expansion, i.e., the Hartree-Fock band structure is qualitatively similar to a LDA result: the mean-field Coulomb potential shifts the position of the f level which has to be determined self-consistently. For the nondegenerate symmetric Anderson impurity ( E = - U J 2 ,vr = 2) the Hartree-Fock position of the ,f level coincides with the Fermi energy, whereas for U -+GO the .f' resonance lies above the Fermi level. The width of the f band is given by the bare hybridization strength r. The corrections resulting from finite U which are not contained in the Hartree-Fock part H , (i.e., the residual interactions) are expected to reduce the width and, in the asymmetric case, to shift the position of the .f' resonance toward the Fermi level. In that sense the U expansion techniques suggest a scheme to microscopically calculate the phenomenological parameters of renormalized band theory (see Currat and Janssen, this volume). The required corrections due to the residual interactions can be evaluated by standard field-theoretic methods since Wick's theorem is valid. The main reason that seems to make the method infeasible for heavy fermion compounds is that the relevant expansion parameter u = U J n r is very large for these systems. Estimates for Ce compounds yield a value as high as 100. Despite this, for the single-impurity problem an expansion in u has been known for a long time.'" It has been originally devised for the symmetric Anderson model for which cr = - UJ2, and for vr = 2. In that case it is nr = 1 exactly, and the narrow resonance is centered at the Fermi energy E , . A perturbation expansion in terms of u gave fast convergence even for u >> l.124-'26 More important, an exact relation for the SommerfeldWilson ratio R [see Eq. (6.44) below] could be derived for each order of the perturbation expansion, and therefore also for u + A comparison with exact results from the Bethe ansatz allowed for the which determination of all the coefficients of the series expansion,' showed that the radius of convergence is u = co. The theory has also been extended to orbital degeneracy vr > 2129.'30and to the asymmetric Anderson ' 1 9 1 2 7 3 1 2 8

K. Yamada, f r o y . Theor. f h y x 53. 970( 1975). K. Yamada, f r o g . Thcor. Phy.~.54, 316(1975). K. Yosida and K. Yamada, Pro
THEORY OF HEAVY FERMION SYSTEMS

85

m ~ d e l . ' ~ ' .In' ~the ~ following we shall briefly outline the results for the symmetric case with v f = 2 and then discuss attempts to apply the theory to the lattice case. We start by introducing dimensionless expressions for the local spin and density susceptibilities xs, x c , and the Sommerfeld specific heat coefficient y through (6.37)

2 xc

= -xc

nr

(6.38) (6.39)

They are chosen in such a way that for u = 0, x", = f c = 7 = 1, and the prefactor (7cT)-' is the local density of states at the Fermi surface for one spin direction. For and f c the following series expansion can be shown to exist

zs

fs

=

1 cnun

(6.40)

n

17,(u)=

c

( - l)"C,u"

(6.41)

n

The coefficients Cnobey thereby the recursion relations (6.42) with C, = C, = 1 . Detailed calculations show that for u + GO, x", -+ co,while f c -+ 0. For the specific heat coefficient the following exact result can be derived (6.43) L

so that the Sommerfeld- Wilson ratio is (6.44)

It varies between 1 (u = 0) and 2 ( u -+ m). The expression, Eq. (6.39), for the specific heat coefficient can be used to define a characteristic temperature of 13'

B. Horvatii. and V. ZlatiC, Phgs. S/u/us Solidi B 99, 251 (1980).

'" €3. Horvatik and V. Zlatic, Solid S/atc, Commun. 54, 957 (1985).

86

PETER FULDE c’t ul

the impurity system by

k,~,

=

.r/y

(6.45)

(in the notation of Yosida and YamadalZ6 the Kondo temperature TK is defined as TK = T0/4). The Yosida-Yamada theory can also be used to calculate the f-electron self-energy Ci,,(to) Gf(z) = [ Z

+ ir - C~,~(Z)I-’

(6.46)

In the low-temperature limit and for small values of w it is given by

+ i s ) = -(?

Cin,(UJ

- I)w

-i

2

(.rrT1)’ -

w 2 +;nT)’

(6.47)

The last term represents local quasiparticle interaction effects. It can be calculated from the graph shown in Fig. 18. Here Ttl is a vertex function. In the limit u >> 1 it is given by

rrr= 71ry

(6.48)

This form can either be derived from perturbation theory or by using Ward identities.’ 2 9 , 1 30 From GJ(z)a quasiparticle representation of the ,f-electron Green’s function near the Fermi energy E , is obtained. In analogy to Eq. (6.25) one can write (6.49) with a resonance right at EF. Its width is reduced by

7-I

which is also the

FIG. 18. Diagram for the computation of the 1-electron self-energy. Solid lines are ,f-electron propagators as given by Eq. (6.29). The vertex function r l is discussed in the text.

THEORY OF HEAVY FERMION SYSTEMS

87

weight of the bare ,f state in the quasiparticle state (renormalization factor). Remember that y'is a large number when u >> 1, i.e., y'-' = 1 - nf. As outlined in connection with Eq. (6.31),the quasiparticle representation of Eq. (6.49) of the impurity ,f-electron Green's function can also be used to calculate approximately the Green's functions [see Eq. (6.29)] for the Anderson lattice, i.e., of a heavy fermion system. This has been the starting point of several groups.1"4.'33-'39 For the calculation of transport quantities and in particular of the superconducting properties, it is necessary to treat the quasiparticle interactions explicitly. An approximate value for them in the Anderson lattice is obtained from rtl by introducing the appropriate quasiparticle renormalization factors. This yields

(6.50) which is of the order of TK.This value has been used as the on-site quasiparticle interaction to calculate the k dependence of the f-electron selfenergy in the Anderson lattice from a diagram of the form shown in Fig. 18.139 Inserted into the J-electron Green's function, Eq. (6.29), this leads to an additional k dependence of the quasiparticles energies E(k). The very important role of the quasiparticle interaction on the formation of Cooper pairs will be discussed in detail in Section 12,c. Recently general Fermi liquid relations have been derived for the Anderson lattice. 14" Also in this case the quasiparticle renormalization factor, the specific heat coefficient, and the susceptibilities are connected by Ward identities. But as one has to consider for the lattice also vertex functions for f electrons with parallel spins which are absent in the impurity case, the results are more general than the ones discussed above. They require going beyond the approximations made in Section VI, which are based on an adaption of the single-impurity results to the lattice. At present it is not possible to calculate these vertex functions accurately from an expansion in terms of u, but the general relations can be useful as a check for approximate calculations.

A. Yoshimori and H. Kasai, J . Mug!?. Moyn. Muter. 31-34,475 (1983). F. J. Ohkawa, J . P h j . ~Soc. . Jpn. 53, 1389, 1828, 3568, 3577. and 4344 (1984). 1 3 ' F. Ohkawa and H . Fukuyama, J . PIzj:v. Sot,. Jpn. 53, 4344 (1984). T. Matsuura, K. Miyake, H. Jichu, and Y. Kuroda, f r o g . Theor. Phys. 72,402 (1984). 13' T. Matsuura. K. Miyake, H. Jichu, Y. Kuroda, and Y. Nagaoka. J . Mugn. Mugn. Muter. 52, 239 ( I 985). H. Jichu, T. Matsuura, and Y. Kuroda, J . Mugn. Mrrgn. Muter. 52,242 (1985). H. Jichu, A. D. S. Nagi, B. Jin, T. Matsuura, and Y. Kuroda, Phys. Rru. B: Condens. Mutter 35, 1651 (1987). I4O K. Yamada and K. Yosida, Prog. Thror. Phy,r. 76, 621 (1986). 133

'34

'''

88

PETER FULDE c't d.

8. MEAN-FIELD THEORY The concept of mean-field (MF) theory has turned out to be a very useful one for treating strongly correlated electrons despite some shortcomings. As pointed out in the introduction of this section, it is not the large Coulomb repulsion U of the f' electrons, which is treated in mean-field approximation. Rather it is the hopping of an f' electron on and off Ce (or U ) sites which is treated as a mean field. The precise meaning of that statement will become clear in a moment. The content of the mean-field theory is best explained within the slave-boson field theory.10y~'13~141~142 It has been worked out by Read and N ~ w ~ sand~ by ~Coleman.'oy~'13~'4'~'42 ~ . ~ ~ ~ ,In this ~ ~ context, we want to mention also the work of Lacroix and C y r ~ t who ' ~ ~ started out from a Coqblin-Schrieffer Hamiltonian for the lattice and treated it in mean-field approximation. In the large-U limit a system described by the Hamiltonian in Eq. (6.1) has an f-electron occupation per site of either zero or one. Double or higher occupancies are forbidden in that case, provided E / . , remains finite as U -+ co. The interaction or U term can be eliminated from H by introducing a boson operator bt(i) at each site i. It can be considered as a creation operator for the empty f'state at that site. Accordingly bt(i)b(i)gives the probability that the f orbital at site i is empty. The operator

Q ( i ) = C ,fL(i)f'm(i) m

+ bt(i)b(i)

(6.51)

has the property that it is conserved with Q(i) = 1 in the large-U limit under consideration. Either the .f orbital is singly occupied, then C,fL(i)f,(i) = I or it is empty, but then b t ( i ) b ( i ) = l . In terms of the boson field the Hamiltonian in Eq. (6.1) is rewritten as H

= Hband

+ mi c / . m f L ( i ) f m ( i )

P. Coleman, J . Mugn. Mugn. Muter. 52,223 (1985). P. Coleman, in "Theory of Heavy Fermions and Valence Fluctuations" (T. Kasuya and T. Sam, eds.), p. 163. Springer-Verlag, Berlin and New York, 1985. '41 N. Read and D. M. Newns, J . Phys. C 16, L1055 (1983); see also D. M. Newns, N. Read, and A. C. Hewson, in "Moment Formation in Solids" (W. J. Buyers, ed.), p. 274. Plenum, New York, 1984. ' 4 3 0 N. Read and D. M. Newns, Solid Stcite Cornmiin. 52, 993 (1984). C. Lacroix and M. Cyrot, PI1y.s. Rev. B20, 1969 (1979).

14' '41

~

~

89

T H E O R Y OF HEAVY F E R M I O N SYSTEMS

where Hbandis the conduction electron part of H. The subsidiary conditions Q ( i ) = 1 can be accounted for by multiplying them with Lagrange parameters A(;) and adding them to the Hamiltonian. The above Hamiltonian is in a suitable form for a mean-field approximation with respect to the boson field. For that purpose one replaces b t ( i ) = ( b t ( i ) )+ Gbt(i)

(6.53a)

b t ( i )--+ ( b t ( i ) ) = r

(6.53b)

by

where r is the (site-independent) mean value of the boson field operator. The phase of the boson field can be chosen such that r is real. In the mean-field approximation the Hamiltonian goes over into

+ 1 (Vmg(k,n)rc:,,

fkm

+ H.c.) + A N ( r 2

-

1)

(6.54)

nmkn

The last term on the right-hand side is a c number and therefore adds a constant to HM,. Two effects of the mean boson field are noticable from HMF. One is a renormalization of the hybridization Vmn(k,n), which goes over into rVmo(k,n). The other is a shift of thef-electron energies E,-, by A. It will turn out that this shift brings them very close to the Fermi energy. The Hamiltonian in Eq. (6.54) is that of hybridizing bands. It can be readily diagonalized. Its eigenvalues can be identified with those of the effective Hamiltonian in Eq. (3.6) of renormalized band theory. Thus H,, can be considered as the Hamiltonian which describes the quasiparticle energy dispersion. One can also use the eigenstates of H,, in order to find out, e.g., how much conductionelectron character and how much f character a quasiparticle has. Since the large Coulomb repulsion U has been incorporated into Eq. (6.52) as well as in HMF, there is no problem of double occupancies of .f orbitals when HMF is used. However, one should bear in mind that the mean-field approximation for the boson field has eliminated most of the interactions and correlations between the quasiparticles. We shall return to this point later. First we specify how r and A are calculated. a. Mean-Field Equations The mean-field r is determined by requiring that the free energy F is minimized. According to Feynman’s inequality it is F

FMF

+ (H

-

HHF)MF

(6.55)

90

PETER FULDE

rt

a!.

where

(6.56) 1 FMF= --InZMF

B

HM,plays the role of a trial Hamiltonian with a parameter which is adjusted so that the right-hand side of Eq. (6.55) takes its minimum value. Minimizing the right-hand side of that equation results in the following condition (6.57) because for the mean-field solution (N - H,,),, equivalent to

=

0. The last equation is

It is supplemented by the requirement that the subsidiary condition is fulfilled,

Next we want to give a discussion of the Eqs. (6.58)and (6.59).This is done best in a quasi particle representation. The quasiparticles are the eigenstates of HMF.In diagonal form HM, is (6.60) where 1 runs over the different quasiparticle bands and T is a pseudospin index. We expand (6.61) The coefficients xne(k,I T ) and ym(k,IT) have to obey the normalization condition

They are the same as the functions sin 9, and cos 9, in Eq. (4.34) when we have

THEORY OF HEAVY F E R M l O N SYSTEMS

91

two bands only and are determined from the set of equations (cfm -

-

2 V:Jk3 m

El(k))ym(k,lz) +

1 Llma(k, n)-xno(k,

=0

(6.63a)

no

+ ( ~ , ( k ) Ei(k))x,,,(k, IT) = 0 E;., = cJIn + A. Those equations follows n)ym(k,

-

(6.63b)

with V,,,, = rV,,, and when the expansions in Eq. (6.61) are set into Eq. (6.54) and the diagonal form of Eq. (6.60) is required. The set of equations (6.63) can be reduced to a set of vf linear equations for the coefficients y,(k, l z ) which describe the probability amplitude that a quasiparticle in band I with momentum k is in the f state m,

Due to hybridization one finds that for most of the k values one quasiparticle branch has an energy close to CJm,while the other branches have energies close to c,,(k). Let us assume that the quasiparticle band structure has been determined numerically from Eq. (6.64). Then we can define the total density of states per spin direction N(w)=

26(0

-

E,(k))

(6.65)

kl

and the partial densities of .f states (per spin) om(0)

=

C Jym(k,lt)I2 6

( -~ E,(k))

(6.66)

klr

By substituting Eq. (6.61) into Eqs. (6.58) and (6.59), one can write the selfconsistency equations in the form {dmf(o))pm(m)(w

-

Zfm) + Ar2 = 0

(6.67a)

r

(6.67b) These equations are supplemented by the requirement that

s

d m f ( w ) N ( m )= n,

(6.67~)

where n, is the total electron number per site. The latter equation fixes the Fermi energy or chemical potential p, which enters into the Fermi function f ( o ) . Equation (6.67b) relates the f count per site, ns, to the renormalization factor r (i.e., r 2 = 1 - nJ). Finally, Eq. (6.67a) determines the energy shift A.

92

PETER FULDE el a1

The formalism outlined in this section allows within a mean-field approximation for a calculation of quasiparticle bands starting from an Anderson lattice Hamiltonian. Thereby three quantities have to be calculated selfconsistently, r, A, and p . So far such a program has not yet been realized. But similarities to the phenomenological approach outlined in Section 111 are clearly visible. The self-consistency equations have a solution r # 0 only for temperatures below a critical temperature T,. Further comments are in order with respect to the mean-field approximation. Setting (h(i)) = r breaks a symmetry of the Hamiltonian H . The latter is invariant with respect to phase changes 6 + beie, .f, +fmeis. The same holds true for the operator Q(i) [see Eq. (6.51)]. The assumption of a fixed phase, which is made when the c number r is introduced, breaks this symmetry. The shortcomings of mean-field theory are apparent. For T > T, r = 0 and the conduction and f electrons are therefore completely decoupled. In reality one would like to describe the system at high temperatures as one with localized moments coupled to the conduction electrons through a CoqblinSchrieffer type of Hamiltonian. The fluctuations 6b(i)[see Eq. (6.53a)I prevent the complete decoupling of the conduction and ,f electrons even at high temperatures. There is no second-order phase transition, as suggested by the mean-field theory. The phase fluctuations ( S O ( 7 ) SO(0))increase as In t and lead to a power-law decay of ( b ( 7 ) b t ( 0 ) ) 7 a . They destroy the broken symmetry state. It can be shown that with increasing orbital degeneracy vf the increase of the phase fluctuations with time is less and less. Therefore the mean-field theory becomes exact as vf + m. The similarity of the mean-field theory of heavy fermion systems with the BCS theory of superconductivity has been pointed out by C01eman.l~~ The BCS superconducting ground state is not an eigenstate to the total electron number, as the mean-field ground state of the heavy fermion system is not an eigenstate of Q(i). Because of the large total electron number, the fluctuations around the mean value are of little importance in the theory of superconductivity. The same holds true here if vf is very large.

-

h. Solution of the Self-Consistency Equution We want to solve the self-consistency equations, Eq. (6.67), in the zerotemperature limit for a simplified model. Thereby we assume that each of the vf orbitals hybridizes with one of the conduction bands in exactly the same way. The degeneracy of the conduction band is also assumed to be v!. Therefore the m index can be dropped and the renormalized hybridization IS just F. The c,(k) are assumed to be independent of n, and in order to simplify the calculations we use a constant density of states N ( 0 ) for the conduction

THEORY OF HEAVY FERMION SYSTEMS

93

Fici. 19. Quasiparticle energies E,,,(k) obtained for the simplified band-structure model described in the text. The horizontal lines denote the original position c, and the renormalized position ?, of the , / level, and 11, the Fermi energy, which cuts the lower branch at k = k , . E,, is explained in the text.

band. The above assumption may seem quite unrealistic in view that the degeneracy of the conduction-electron states is two (Kramer's theorem) for a general k value. The hybridization should therefore take place only with a subset of the f' states depending on the direction of k. When considered as an angular average over the Brillouin zone, the above assumption is nevertheless a useful approximation which has been frequently used in the literature (see, e.g., Rice and Ueda'00.'0'.145).From the above it follows that there are two branches of quasiparticle energies ( I = I , 2) of the form Edk) = N 4 k ) + Pf1 T W W ) ) )

(6.68)

where W(E(k))

=

{ [ ~ ( k )- E;]'

+ 4V2}1'2

(6.69)

and the -( +) sign corresponds to the bands I = l(2). In the following we assume that only the lower branch E,(k) is occupied up to a wave vector k , (see Fig. 19).This defines an energy c0 = c(kF)which would be reached if all electrons were placed into a conduction band. The Fermi energy p is obtained from El(/+) = p. This yields (6.70) The f-state occupation is given by (6.71) '45

T. M . Rice and K . Ueda, P/t!:c. Rw. B. Cimdcns. Mailer 34, 6420 (1986).

94

PETER FULDE et al

The coefficients y(k, 1 = 1) are obtained from Eq. (6.63) as (6.72) After integration of Eq. (6.57) this yields nf

=

+

V ~ N ( O ) ~ [W E (~e O) W(0)l

(6.73)

Similarly one finds

=

- v f N ( 0 ) V 2 In

€0

-.;

+ W(E0) + W(0)

- E;.

(6.74)

To leading logarithmic approximation one then obtains the following selfconsistency requirements [see Eqs. (6.58 and 6.59) or Eqs. (6.67a and b)] (6.75a) (6.75b) Let us define a characteristic temperature T* by k,T*

= pexp(

-$)

(6.76)

The renormalized position of the f level is then obtained from Eqs. (6.70) and (6.75a) as

-

cJ = p

+ k,T*

(6.77)

By using that r 2 = ( v / V ) and Eq. (6.75a), one can eliminate E~ from Eq. (6.75b) so that in close analogy to the single result in Eq. (2.20)

nJ = 1 - ( x k g T * ) / v l r

(6.78)

In a similar way the ground-state energy per site can be calculated. It is given by

THEORY OF H E A V Y FERMION SYSTEMS

95

This energy must be compared with the one in the absence of hybridization and for n,- = 1, i.e., E'" = +p(''(n, - 1) + ef (6.80) Note the difference in the chemical potentials p and p"), which can be calculated from Eq. (6.75b) as (6.81) where in this special model (6.82) The energy difference is therefore ,I-

E'o) = - k n T *

(6.83)

If the chemical potential instead of the electron number were kept fixed, the energy difference per site would amount to ( E - pn,)

-

(E'')

-

p~n;'))=

-

k,T*

(6.84)

with (6.85) n'p) = 1

+ pvfN(0)

In both cases the energy gain is of order k,T* and hence proportional to ( 1 - nr). c. Thermodynamic Quantities

From the quasiparticle excitation spectrum in Eq. (6.68)the total density of states (for both spin directions) at the Fermi surface is derived as dE,(k) &(k)

-' -

( )

p*(O) = V f N ( 0 ) -

nf

- k,T*

(6.86)

which leads to the large linear specific heat (6.87) The magnetic susceptibility is found to be (6.88)

96

PETER FULDE

rt

al.

The Sommerfeld-Wilson ratio is (6.89) This is the value expected in an exact theory for vf -+ cm and in accordance with the earlier statement that the mean-field theory is the classical limit of the heavy fermion problem. Of particular interest is the computation of the electronic contribution Ac, to the bulk modulus which at T = 0 is given by (6.90) The energy of the electrons, E , has been determined in the mean-field approximation, resulting in Eq. (6.83). From it one obtains two contributions to the bulk modulus, Ac, = A c f ' + Ac',". The first comes from changes in the spacing of the different k levels as the volume changes. This results in a bulk modulus of a free electron gas which is obtained from 6 2 E ' 0 ) / 6 ~and ~ , in this model is given by (adding a factor N/R, because in the previous section all energies are given per lattice cell): (6.91)

A c= ~ ~ , ( n ,- 1)N/R

The other, which is the one of special interest here, is due to changes in the hybridization with volume which enters the Kondo temperature T* in Eq. (6.76). It is obtained as (6.92) where we have introduced the Gruneisen parameter q defined in Eq. (4.1). If it were not for the volume dependence of T*, one would have Ac, = Acf'. Therefore the mean-field theory contains the large Fermi liquid parameter F g . It ensures that the f resonance remains close to the Fermi surface when the chemical potential is shifted. In order to obtain the other Fermi liquid parameters, one must go beyond mean-field theory and include the fluctuations. They vanish in the limit vf + co.

d . Electron- Phonon Interactions: Revisited In Section 5 a Hamiltonian was derived for the interactions between the (heavy) quasiparticles and phonons. The coupling between the two was based on the volume dependence of the singlet-condensation energy k , T*. The latter also enters the effective hybridization and the interaction Hamiltonian in Eq. (4.38) is derived from the variation of these quantities with volume changes [see Eqs. (4.29) and (4.30)]. As pointed out previously,

v,

THEORY OF HEAVY FERMION SYSTEMS

97

this Hamiltonian is incomplete for a computation of, e.g., the electronic contributions to the bulk modulus AcB. In order to derive the result of Eq. (6.92) from a quasiparticle-phonon interaction, one must also include interaction terms which are bilinear in the phonon operators (Becker et al., 1987). They result from changes in the boson mean field with volume. In order to derive the complete interaction Hamiltonian, we proceed as in Section IV but starting from the mean-field Hamiltonian H,, [see Eq. (6.54)]. Thereby we restrict ourselves to one twofold degenerate band. By taking the derivative with respect to volume, one obtains to first order in the phonon operators

+ a [ A ( r 2 - l)] NE,(q a% ~

(6.93)

= 0)

Here En(q) is the Fourier transform of en(i) [see Eq. (4.31)], i.e., (b,

=

J2NM0,

+ bt,)

(6.94)

The Hamiltonian H&-ph is the same as given by Eq. (4.32) except for the last term on the right-hand side. This term leads to changes in the equilibrium positions of the ions and disappears when the new positions are introduced. We keep it here because it is required when the next higher-order term in the phonon operators is calculated. As discussed in Section IV, the first two terms of the Hamiltonian in Eq. (6.93) can be re-expressed in terms of quasiparticle creation and annihilation operators a/,(k), a,,(k) with 1 = 1, 2. When only the band corresponding to I = 1 is kept, they reduce to the form of Eq. (4.38). As a side remark we want to point out that for q = 0 the coupling C(k,q) reduces to (6.95) where El (k)is defined by Eq. (6.68). Using the explicit results of the simplified band-structure calculation of Section 8,a, we obtain for k = k , (6.96) which is very small. This is a consequence of a near cancellation of the effect of the strain dependence of F’ and P in the quasiparticle energy El(k). The next higher-order term in the quasiparticle-phonon interactions is obtained when the volume dependence of the boson mean field is taken into

98

PETER FULDE et al.

account. It is ( b ( i ) ) = r(En

=

ar

0) + ---eQ(i)

a%

+

(6.97)

As a result one finds

(6.98) When d2E/iicA is calculated, only terms with q define the following two quantities

=

0 are required. One can

.99)

The second derivative d2E/&h can then be expressed as (6.100) where ( ( B ( ' ) I B ( ' ) ) )is the static response function of B"). When this response function is calculated, it is essential that interband transitions between the quasiparticle bands are taken into account. They cancel parts of so that the final result agrees with Eq. (6.92). For the bulk modulus we should calculate the derivatives in Eqs. (6.99) and (6.100) for constant N instead of constant p. In the present case the difference in Eq. (6.101) is of order T*', and therefore can be neglected. For details, see Becker et ~ 1 . 9. VARIATIONAL GROUND STATE

In the preceding section it was shown that the quasiparticle properties of a heavy ferrnion system can be described by an effective hybridization Hamiltonian. It contains an f level closely above the Fermi surface and a reduced hybridization matrix element p= rV, where V is the hybridization '46

K. W. Becker, J. Keller, and P. Fulde, unpublished.

l

~

~

THEORY OF HEAVY FERMION SYSTEMS

99

of the f electrons with the conduction electrons. In mean-field theory the reduction factor is r = (1 - n,)'". This result has a simple physical interpretation. The quantity (rV)' shows up as a prefactor in the effective probability that an f electron jumps from one site to another. In the limit of large Coulomb repulsion U a jump will take place only if the f level of the final site is empty. This results in a factor (1 - n,). It was first argued by Rice and Ueda 1 0 0 . 1 0 1.145 that this reduction factor should be replaced by (6.101) when the hybridization of the f state Im) with the conduction electrons is considered. Here T i is the ratio of the electron hopping probability onto an empty f site in the limits U + co and U = 0. The above form of ?; was suggested from experience gained with the Hubbard model, to which the variational method of Gutzwiller l4 had been successfully a ~ p l i e d . ' ~The ' arguments of Rice and Ueda were confirmed for vs = 2 by a recent investigation of Varma et ~ 1 . and ' ~ by~ Fazekas.I4' Before we outline the variational calculations, we want to summarize briefly the consequences of the factor 7; and its influence on the results. For that purpose we use the same simplified model as in Section 8, i.e., that of v,-fold degenerate ,f states hybridizing with v,-fold degenerate conductionelectron bands. In that case the consequence of choosing Fm as given by Eq. (6.101) are (1) For a fully polarized state, in which only one state rn, is occupied, one finds Tg = 1, which is a reasonable result. (2) In the paramagnetic state T2 = (1 - n,-)/(l - n,/v,). The correction is largest for vJ = 2, and it vanishes for vs -+ co. (3) The correction enters into the exponent of the characteristic temperature

v; N (0)V

(6.102)

For vs = 2 the exponent is smaller by a factor of than in Eq. (6.76). T* determines the ground state energy of Eq. (6.83),the effective position of the f level above the Fermi energy of Eq. (6.77) and the f occupation in n, in Eq. (6.78). The latter relation is changed into (1 - nJ) = (1 - l/v,)nk,T*/(v,T). (4) The magnetic susceptibility is found to be negative for vs = 2, implying that the ground state is unstable against ferromagnetic order. This is due to the spin dependence of the hybridization. 14'

14'

W. F. Brinkman and T. M. Rice, Ph:.v. Rev. B: Condens. Matrer 2,4302 (1970) P. Fazekas, J. Mayn. Magn. Muter. 638~64,545 (1987).

1 00

PETER FULDE et al.

In the following we want to outline the basic ingredients of the variational approach to heavy fermions.'00.'02,'15-' 7 , 1 4 5 , 1 4 8 Thereby we follow closely the treatment by Fazekas. The starting point is an ansatz for the ground-state wave function which has the form of a trial function. We shall consider the case vs = 2 only, because actual calculations have been limited to that case. The following ansatz, based on a generalization of Gutzwiller's ansatz is chosen' l 5

=

(6.103)

PI@,)

The state 10,) is a Slater determinant consisting of hybridized electronic Bloch states. It is of the form 10 ')

=

fl

+ a(k)f:nckcrl(FS)

(6.104)

kn

where (FS) is the filled Fermi sphere consisting of n, unhybridized conduction contains variational electrons per site. The radius of this sphere is k,. parameters a(k) which determine the degree of hybridization. They are related to the y(k, 1) of Eq. (6.61) through

I&,)

(6.105) where only the lower quasiparticle branch 1 is considered. The y2(k, 1) are the probabilities that a Bloch state with momentum k consists o f f electrons. The prefactor P in Eq. (6.103) is a projection operator. It eliminates from IQo) all those states in which two f electrons are at a given site i. This corresponds to the limit U + cc for the f-electron repulsions. Calculating expectation values like that of the energy, with respect to is not possible without additional approximations. This is so because (Qo) is easily formulated in k space while the projector has a simple form only in real space. When one uses for the f electrons a basis consisting of localized unhybridized f states, then I $ o ) is a sum of determinantal states, each of which contains local functions for the f electrons and Bloch states for the conduction-band holes. The coefficients with which those states are multiplied is decomposed are themselves determinants of Bloch factors when I$0) exp(ik Ri). The approximation employed by Fazekas replaces the norm of those determinants by their momentum and position-independent average values, which are combinatorial numbers. Furthermore, in the calculation of which contains contributions matrix elements use is made of the fact that I$,) from states with different f-electron numbers, is dominated by one state with the most probable f-electron number. As a result, one finds that whenever one calculates matrix elements with I&), one can do the calculations by using IQO)

I$o),

-

THEORY OF HEAVY FERMION SYSTEMS

instead of

I&)

101

but with the replacement a(k) + a"(k),where 1 - nf/2

a(k) = r"(k)

(6.106)

For expectation values of operators, which are off-diagonal in the number of ,f electrons, such as ctkUfjr an extra factor Fhas to be introduced. This agrees with the reasoning of Rice and Ueda. As a consequence one finds in the paramagnetic state for the number of f electrons

(6.107) and for the ground-state energy per site

E

2 N

=-

C

[c(k)x2(k, 1)

+ cfy2(k, 1 ) + 2 P ~ ( kl)y(k, , l)]

(6.108)

IklSkF

where P= FV as before. y2(k, 1) is given by Eq. (6.105) with a(k) replaced by a"(k).Furthermore, x2(k, 1) = 1 - y2(k, 1). We can interpret again y2(k, 1) and x 2 ( k , 1) as probabilities of finding an f electron and a conduction electron in state (kl < k , , respectively. The expectation value of the energy in Eq. (6.108) is minimized with respect to E(k) under the condition that nf and Fare kept fixed. This is done by adding a term A Ck,((f~,fk,) - n f ) to E, where A is a Lagrange parameter, so that

(6.109) From aE'/dZ(k)

=0

one obtains

1 Z(k) = -([Cf 2v

-

~(k)] {(Cf - c(k))'

+ 4P2)1'2)

(6.110)

where Cf = cS + A. One notices that the result is the same as that obtained from a hybridization Hamiltonian with a hybridization matrix element and an effective f-level position E;. . This position (or A) is finally determined by minimizing the energy E with respect to nf for fixed total electron number n,. Since E' is already stationary with respect to the parameters C(k), one obtains

(6.1 1 1 ) This equation is equivalent (for vf

=

2) to the self-consistency equation (6.58)

102

PETER FULDE

41

ul.

[see also Eq. (6.74)] obtained in the mean-field theory. However, the solution, which determines A and T*, will be different here because of the different dependence of Pon nJ due to Yrn [see Eq. (6.101)]. It is worth pointing out that the above theory can also be applied to a dilute alloy of f centers."' In the dilute limit the reduction factor F2reduces to the form r 2 = 1 - nffound in the mean-field theory. This shows that the difference between Fand r must be due to the interplay between different f' centers. At present it is not yet clear how the various approximations, which have to be made in order to compute 7, influence its dependence on nJm.Independent of that, one should also keep in mind that the angular dependence of the hybridization matrix element has been neglected. The angular dependence may considerably change the binding energy per site as compared with that found in the present calculations. The same also holds true, of course, for the mean-field results or other types of model calculations. In a recent paper Kotliar and Ruckenstein'49 have shown a way by which F can also be derived from an extension of mean-field theory. They introduce four boson fields, i.e., one for each state of the one-f level system (empty, occupied with spin up or down, doubly occupied). Unfortunately, the formulation of the hybridization matrix element in terms of the boson field is not unique. Therefore, a form is chosen which reproduces correctly certain limiting cases. For the Hamiltonian in Eq. (6.1) their method reproduces the r" renormalization factor. However, when applied to the single-impurity case a factor ?instead of r is again obtained, which is clearly incorrect. A hint on the arbitrariness of such a mean-field theory has been given by Kotliar and R u c k e n ~ t e i n ' ~It~ is : possible to extend the Hilbert space by introducing additional Bose variables in several ways such that the matrix elements of the bosonized Hamiltonian coincide with the matrix elements of the original Hamiltonian in the physical subspace. As in a mean-field theory the restriction to the physical subspace is only fulfilled on the average, different mean-field theories can give different results. Only a careful study of the influence of fluctuations can show which kind of mean-field theory is the best approximation for a given physical system. Finally a comment is in order on the Fermi liquid parameters which follow from the variational theory. Clearly the theory contains the large F i Landau parameter in the same way as the mean-field theory does [see Eq. (6.89) and the discussion following it]. But due to the spin (or rn) dependence of the renormalization factor Trnthe Fermi liquid parameter F: is no longer zero as in the mean-field theory. It can be derived from a calculation of the magnetic susceptibility xs. The latter is obtained from the variation of the ground-state energy as functional of an applied magnetic field. '41

G. Kotliar and A. E. Ruckenstein, Phvs. Rev. Lett. 57, 1362 (1982).

THEORY OF HEAVY FERMION SYSTEMS

103

VII. Superconductivity 10. INTRODUCTION

The discovery of superconductivity in the heavy fermion compound ’ the trigger for the rapid experimental and CeCu,Si, by Steglich et ~ 1 . ’ ~was theoretical development of heavy fermion physics. The discovery was contrary to all expectations and therefore met considerable scepticism and criticism at first. This was due to the fact that rare-earth ions were mainly known to act as pair breakers for the superconducting pairs. The superconducting transition temperature of LaAI,, for example, decreases rapidly when Ce ions are added. This decrease is associated with the magnetic moment of the Ce 4f shell, which gives rise to the conduction electron-f-electron exchange interaction

a(r)Si(Ri)G(r - R;)

Hint= -Jex

(7.1)

1

Si(Ri)is the spin of the 4,f electron at site i with position Ri and a(r) is the conduction-electron spin density. I t is seen that a given spin Si(Ri) acts differently on the two electrons in time-reversed states forming a (conventional) Cooper pair. Therefore the 4f electrons will act as pair breakers through this interaction. In the nonmagnetic compound CeCu,Si,, on the other hand, the 4f electrons must generate superconductivity since LaCu,Si, is not superconducting. The only, but essential, difference between the two compounds is the 4f electron at each Ce site. The superconducting pairs are formed by the heavy quasiparticles. This is demonstrated experimentally by the fact that the specific heat anomaly at the superconducting transition temperature T, is of the same order of magnitude as the strongly enhanced specific heat in the normal state. The discontinuity in the specific heat, AC, at T, is given by

AC N I.~CN(T,)

(7.2)

This should be compared with the BCS prediction for “conventional” superconductors, AC = 1.43CN(T,).The value of T, itself depends on details of the stoichiometric composition of the CeCu,Si, samples (see Steglich,). The heavy quasiparticles which have predominantly ,f character exhibit itinerant behavior, and there is no pair breaking associated with the f electrons. These two findings can be explained by the singlet formation and the formation of quasiparticle bands which were described in Sections I1 and 111. In heavy fermion metals, spin-orbit interaction and band-structure effects are very large and have to be accounted for properly in a discussion of the ”‘)

F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schafer, Pl7y.7. Rev. Left. 43, 1892 (1979)

104

PETER FULDE et al.

superconducting state. An important consequence is the fact that the possible forms of the order parameter are restricted by crystal symmetry. A detailed analysis of order parameters compatible with crystal symmetry was made by means of mathematical group theory. These important results will be reviewed in Section 11. We will briefly discuss some properties of so-called unconventional states. For superconductivity to occur, however, there must be an attractive interaction among the quasiparticles. The correct microscopic description of the interaction would yield, of course, the superconducting transition temperature as well as the detailed form of the order parameter. The origin of the attraction among the heavy quasiparticles is the subject of intense theoretical studies. It is one of the crucial questions in this field. In Section 12 we will review some of the models for the superconducting transition. A discussion of some of the properties of different pair states is given in Section 13. 11. GROUP THEORY

Many physical properties of the superconducting state are determined by the symmetry of the order parameter. The possible forms of the order parameter, on the other hand, are restricted by crystal symmetry. This fact provides a classification scheme for different superconducting states and, in addition, allows one to construct the superconducting classes by means of formal mathematical group theory. The group theoretical analysis described in the present section allows one to study the internal symmetry of superconducting states independently of the (currently unknown) microscopic mechanism for Cooper pairing in heavy fermion systems. a. Symmetry Properties of the Order Parumeter

Phase transitions usually lead to the appearance of long-range order in appropriate correlation functions. In superfluids such as superconductors, 3He, and neutron stars, the long-range order shows up in the two-particle density matrix’51

Here, r i , sidenote the fermion positions and spins, respectively. In systems with strong spin-orbit interactions, the indices sirefer to pseudospins. Below the superconducting transition, the two-particle density matrix p”’ does not 151

N. D. Mermin, in “Quantum Liquids” (J. Ruvalds and T. Regge, eds.). North-Holland, Amsterdam. 1987.

THEORY OF HEAVY FERMION SYSTEMS

105

vanish for large separations of the pairs of points r l , r 2 and r i , r > , (r1.~1;r2S21P(2)lr;s;;r;s;>

+

$:l,s2(rl >r2)$s;.sJr;

(7.4)

This type of ordering is usually called “pairing.” The ordered phase is characterized by the complex functions $ which-in a general system-depend on the center of mass and relative coordinates, R = $(rl + r2), r = r 1 - r2, and the (pseudo-)spins s, and s 2 , respectively. In the following discussions of this section, we will restrict ourselves to homogeneous systems, and therefore we will neglect the dependence of $ on the center-of-mass variable. The considerations presented in this section exploit symmetries and very general transformation properties of the superconducting order parameter. The fundametal property of the order parameter Y =(I))~,~*is that it behaves like a two-fermion wave function in many respects. This follows from its definition. First and foremost, it is antisymmetric under the interchange of particles $s,.s*(r) = -I)s*.s,(-r)

(7.5)

In addition, it transforms like a two-fermion wave function under rotations in position and spin space and under gauge transformations. These features characterize the order parameter of general pair-correlated states; i.e., they apply equally well to conventional superconductors, neutron stars, He, and heavy fermion systems. The above definition of the order parameter followed the one which is used in the literature for 3He. Later, in the section on microscopic theories, we shall use a gap function, AaP(r), which differs from the order parameter as defined above by multiplication with the quasiparticle interaction potential. We will briefly review the standard notation and its physical interpretation. Two particles with spinican occupy pair states with S = 0 (spin singlet) and S = 1 (spin triplet), where S denotes the total spin. The order parameter Y is expanded in terms of the four spin states

where

is the singlet state. We use an intuitive notation, where Irl) denotes a state in which the quasiparticles have spins up and down, respectively. The Ix,) are three orthonormal triplet states which-with the same notation-read (7.8a)

106

PETER FULDE et al.

(72%) (7.8~) The complex order-parameter matrix is then represented by the linear com bination Y = (4 + d z)iz, (7.9) where the T, are the Pauli matrices. This way of writing reflects the fact that an integer spin S is described either by a wave function with 2s + 1 components or by a spinor of order 2s (see, e g , Landau and L i f s ~ h i t zVol. , ~ ~ 111). The triplet states Ix,) transform like the components of a vector under rotation in spin space, and they obey the relations

-

(7.10) where S, are the components of the total spin S = i(n, + 6,).A general triplet pair state is characterized by a vector d in spin space. Let us now discuss the physical significance of the vector d. It is evident from the definition of a triplet state Vt = (d z)iz, that

-

+TrlVt(i)12= ld(i)(’

(7.1 1)

Evidently, the magnitude of d measures the total amplitude of condensation of the Cooper pairs at a given point on the Fermi surface. In addition, the vector d defines the expectation value of the spin of a triplet pair state through d * x d. This can be verified by directly calculating (YrtlS(W,). States with vanishing spin expectation value for which d * x d = 0 are called unitary states. Their order parameter matrices Y, satisfy Y p ”= 1

(7.12)

where 1 is the unit matrix. These states which are the most important ones are invariant under time inversion. For unitary states, the vector d defines a unique direction in spin space for every point on the Fermi surface. It is easy to verify that d-S=O

(7.13)

which implies that the Cooper pairs have vanishing spin projection on the direction d. The coefficients 4 and d , in Eq. (7.6) are complex functions of the relative variable or, if we perform a Fourier transformation, of the direction i on the Fermi surface. The antisymmetry of the order parameter Y, however, determines their parity. The singlet wave function (0) is antisymmetric under the exchange of the spins, whereas the triplet functions are symmetric. This

THEORY O F HEAVY FERMION SYSTEMS

implies that the orbital coefficient of the singlet part, function of It

107

4, must be an even

whereas the vector d must be odd, d(It) = -d(

-It)

(7.15)

It is obvious that the pair wave function is either a singlet or a triplet function if we have rotational symmetry in spin space, e.g., if we can ignore spin-orbit coupling. In this situation, we can speak of singlet and triplet superconductors, respectively. The symmetry conditions of Eqs. (7.14) and (7.15), however, suggest that this distinction can be made under more general conditions. At this stage, the point symmetry of the underlying crystal comes into play. We know that we can classify the pair states with respect to their parity if the crystal structure has an inversion center. This is the case for all heavy fermion superconductors known so far. The existence of an inversion center implies that we still can distinguish between the two types of states which are analogous to singlet and triplet states regardless of the strength of the spin-orbit interaction that might be present. Parity therefore provides a rather general and useful classification scheme, as first pointed out by A n d e r ~ o n . ' ~ Ev ~ .en' ~ ~and odd-parity states in systems with inversion symmetry are the counterparts of the (spin) singlet and triplet states mentioned above. An even-parity state is characterized by one complex function $(It) of the angle k, whereas an odd-parity state is described by three complex functions d,(It). Before examining the rather complex superconducting states that have been suggested for heavy fermion systems, let us review some familiar examples. The simplest even-parity state is certainly the isotropic state encountered in ordinary superconductors. This state is often referred to as an "s-wave state". The order parameter does not depend on the direction It and reduces to a complex constant 4. Its only degree of freedom is the phase. By far the most extensively studied examples of anisotropic pairing are the p-wave states realized in the superfluid phases of He. The corresponding order parameters are specified by 3 x 3 = 9 complex constants A P j , A

d,(k)

A

=

A,,kj;

p , j = I, 2, 3

(7.16)

where the indices p and j refer to directions in spin and coordinate space, respectively. The acquaintanceship with these pair states (mis)led many authors to the conclusion that the unusual properties of heavy fermion superconductors indicate the presence of p-wave pairing. The order parameter 15'

'53

P.W. Anderson, Phys. Rev. B: Condens. Mutrer 30, 1549 (1984). P.W. Anderson, Phg.s. Rev. B: Cundms. Mutter 30,4000 (1984).

108

PETER FULDE et d.

of ‘He, as defined above, is characterized by spin and orbital quantum numbers S = I , L = I . In addition to the phase degeneracy, it has a ninefold degeneracy which reflects the fact that spin and orbital degrees of freedom are (almost) decoupled. Finally, we consider the order parameter describing the superfluid phase in neutron stars. The motivation for including this example is that spin-orbit interaction is strong in these systems, a fact they have in common with heavy fermion superconductors. It is assumed that the pairing occurs in a 3P2 state. The order parameter is characterized by the quantum numbers S = I, L = I , the total angular momentum, however, being restricted to J = 2 as a consequence of the strong spin-orbit interaction. The vector d is again given by a matrix

d,

=

BPjkj

(7.17)

in close analogy to 3He. The restriction to the J = 2 manifold, however, implies that B must be a symmetric tensor with vanishing trace.’54 This order parameter, i.e., the matrix B, has five independent components as required for J = 2. It should be mentioned that a traceless matrix B can also define a scalar (even-parity) S = 0, L = 2 order parameter through

4 ( i )= iiBijij

(7.18)

for which we have again J = 2. The two states in Eqs. (7.17) and (7.18) are closely related since they are characterized by the same degrees of freedom. The order parameters, however, have different nodes. The fundamental concepts we have to introduce in this context are symmetry and symmetry breaking. We have to distinguish between the symmetry of a system (i.e., the symmetry of the underlying interactions as given by the Hamiltonian and the free energy) and the symmetry of the actual state. The occurrence of long-range order at a phase transition as described by an order parameter is aiways associated with symmetry breaking. The ground state is no longer invariant under the symmetry of the Hamiltonian; i.e., the symmetry is broken. In a superfluid characterized by an order parameter of the type of Eq. (7.4) gauge invariance is broken. Broken gauge symmetry is therefore the characteristic feature of superfluids. In addition to gauge invariance, however, additional symmetries can be broken. A typical example is the odd-parity state in a crystal with inversion symmetry. In such a state, the symmetry of the lattice under inversion is broken. We will call the simplest superconductors, i.e., those where gauge symmetry is the only broken symmetry, “conventional superconductors.” In those systems, the order parameter will have the same point symmetry as the underlying crystal. On the other hand, we will call a superconductor with additional broken symmetries I54

J. A. Sauls and J. W. Serene, Phys. Rev. D 17, 1524(1977)

THEORY OF HEAVY FERMION SYSTEMS

109

conventional

unconventional FIG.20. Total condensation amplitudes of the Cooper pairs in momentum space for (a) conventional and (b) unconventional superconducting states. The crystal is assumed to have cubic symmetry. In both cases the order parameter can have zeros on the Fermi surface (inner solid line).

an “unconventional superconductor.” This definition of conventional and unconventional pair states is illustrated in Fig. 20. In either case, the order parameter can vanish on the Fermi surface. The characteristic feature of unconventional superconductors in the sense introduced above is that the states have a nontrivial degeneracy which gives rise to very rich dynamics. The definition of conventional and unconventional superconductors refers to the symmetry properties of the order parameter. The transformation properties also provide a general classification scheme for unconventional pair states. According to this concept, a superconducting phase is characterized by its symmetry type, i.e., by the subgroup of the total symmetry group which leaves the order parameter invariant. Note that this classification scheme does not restrict the allowed order parameters. In particular, it can also be applied to phases where all symmetries of the original system are broken. The associated order parameters, like all functions, are invariant under the identical transformation, which, by itself, forms a subgroup of any group. In the following sections, however, we will be mainly interested in superconducting phases where not all symmetries are broken. The associated order parameters have a residual symmetry, i.e., they are invariant under a subgroup of order greater than 1 of the original symmetry group.

I10

PETER FULDE

el

al.

Subdividing the superconducting phases among the various distinct superconducting classes according to the symmetry types of their order parameters closely parallels the classification scheme for magnetically ordered states (Landau and L i f s ~ h i t z Vol. , ~ ~ 111). It is well known that the magnetic properties of a crystal on a macroscopic scale as, e.g., the presence or absence of a spontaneous magnetization, are determined by its magnetic class. The superconducting classes determine the possible node structures of the order parameters (and thus the low-temperature properties) as well as the magnetic properties of the corresponding phases.

b. Free Energy of Unconventional States: Lifiing of Degeneracies The possible ordered states, i.e., their order parameters, have to be determined by minimizing the free energy. The present section focuses on unconventional pair states in heavy fermion systems. Determining the order parameter for the fully developed superconducting state at temperatures sufficiently below T, can be a rather complicated problem. This is a consequence of the fact that unconventional states have nontrivial degeneracies (in addition to the phase degeneracy encountered in conventional systems). The higher-order terms in a Landau-Ginsburg expansion (i.e., quartic, sixth-order, etc., terms) can reduce some or all of the degeneracies of Y by picking a particular structure. We do not only have to determine an amplitude factor as in the case of an isotropic (singlet s-wave) superconductor but also the manifold in which the allowed order parameters are to be found. To further illustrate this point, let us consider the Landau-Ginsburg expansion for the L = 1, S = 1, J = 2 pair state realized in neutron stars. The problem was solved e x a ~ t 1 y . I ~ ~ The free energy must be invariant under gauge transformations and simultaneous rotations in spin and orbital space. The Landau-Ginsburg expansion for the case under consideration is given by AF = F N =

- FS

b2CrTrBoBg + b4[b,ITrBi12

+ f12(TrBoBg)2+ /j3TrBg2Bi] (7.19)

where the order parameter d,(i) = BPjCj is given in terms of an amplitude b and a normalized symmetric traceless matrix B,, B = bB,,

with TrB,B,*

=

1

(7.20)

For a 3P2 state, one can construct three linearly independent fourth-order invariants, and a general free energy will contain a linear combination of all of them. Stability, i.e., the requirement that the fourth-order term be positive 155

N. D. Mermin, Phy.c. Rcw. A 9, 868 (1974).

THEORY OF HEAVY FERMION SYSTEMS

111

definite, imposes some restrictions onto the otherwise arbitrary coefficients pi, i = I , 2, 3. In the present case, it implies that / j 2 be positive. Let us first consider the free energy as a function of the amplitude b, keeping the matrix B, fixed. The superfluid transition occurs whenever the coefficient of the quadratic term, a, becomes negative. Then the system can gain energy by ordering. A t temperatures beneath T,, the amplitude of the order parameter b ( T ) is determined by the quartic terms. Minimization with respect to b at fixed B, yields

This value of the free energy still depends on the matrix B,, which accounts for the five degrees of freedom in the J = 2 manifold. It is obvious that the minimum of the free energy in Eq. (7.21)is reached only for specific matrices B,, i.e., for very specific linear combinations of the five 'PZ basis functions. The degeneracy of the J = 2 manifold is lifted by the quartic terms. The relative stability of different states varies with the ratios &/flz and fl3/ljz which are u priori known only for weak-coupling systems. If strong-coupling corrections become important, they depend on the detailed microscopic structure of the system under consideration and cannot, of course, be determined from general symmetry arguments. To completely solve the problem in the general case, one has to determine the possible ground states as functions of the fourth-order coefficients. This, however, amounts to calculating the entire phase diagram. This can be achieved in the case described above, where one finds three different classes of superfluid states. We would like to mention at this point that this problem has not been solved so far in the case of 3He. A general guideline as to what the various ground states might look like is therefore highly desirable. The search for possible phases, however, is greatly simplified by general symmetry considerations. In this review, we will only state the fundamental facts and give some conclusions. We would like to emphasize at this point that the considerations described below are not restricted to superfluid phases nor to phase transitions in condensed matter physics. The same concepts are also very successfully applied in high-energy physics. The group theoretical background has been discussed by Bruder and Vollhardt 15" in the context of superfluid 3He. Determining the order parameter of an unconventional pair state by minimizing a free energy functional is an example of variational symmetry breaking. The general mathematical framework for dealing with variational symmetry breaking is given by Michel.ls6 Based on the rigorous theorems '55r

15'

C. Bruder and D. Vollhardt, Phys. Rev. B: Condens. Matter 34, 131 (1986). L. Michel, RCU.Mod. Phys. 52, 617 (1980).

112

PETER FULDE ct rrl

proved there, one can show, e.g., that for a real-valued order parameter and a finite symmetry group the minimum of a Landau-Ginsburg functional must always have residual symmetry.’ 5 7 Other examples of symmetry breaking and states with a residual symmetry have been studied within the context of highenergy physics. Last but not least, all stable phases of superfluid 3He known so far are characterized by an order parameter with a residual symmetry. There is some evidence that states with residual symmetry, as introduced in the preceding section, are more likely to correspond to a stationary point of the free energy than states where all symmetries are broken. Therefore the following approach suggests itself:

( I ) Find the subgroups of the given symmetry group of the system; (2) Construct order parameters which have a residual symmetry left; If, in addition, the free energy functional is known explicitly: (3) Insert these states and determine the one which corresponds to the lowest energy.

The important step in finding the possible ground states along these lines is the restriction of the order-parameter space to states with residual symmetries. The only input information required for finding the possible candidates are the symmetry operations which leave the free energy invariant. Once the form of the free energy is known (e.g., by a Landau-Ginsburg expansion or a more sophisticated expression), one must check which of the order parameters with different residual symmetries lead to a minimum of it. The approach outlined above is not limited to the vicinity of the transition temperature where a Landau-Ginsburg expansion is possible. The procedure also yields the superconducting classes that can be realized at low temperatures (provided that there are no additional phase transitions). All possible superconducting states with residual symmetry and their symmetry classification were presented by Volovik and Gorkov’ 5 8 for cubic, hexagonal, and tetragonal crystals. The latter are represented by UBe, 3 , UPt,, and CeCu,Si,, respectively. They assume that spin-orbit interaction is sufficiently strong, i.e., that the spins are “frozen” in the lattice and that therefore rotations of the crystal also rotate the spins. The total symmetry group of the problem G x T x U(1) (7.22) is the product of the crystal point group G, time reversal T, and the gauge group U (1). For this expanded group, all (discrete) subgroups are constructed. In distinction to ,He, this is, indeed, possible in the case of heavy fermion systems since the groups G and T are finite. In that sense, the heavy fermion 15’ 15’

M. V. Jaric, Phys. Rev. Lett. 48, 1641 (1985). G. E. Volovik and L. P. Gorkov, Sou. Phys.-JETP61,843 (1986)

THEORY O F HEAVY FERMION SYSTEMS

113

superconductors are much simpler and less complex systems than superfluid 3He. For the second step, i.e., the construction of the pair states with residual symmetries, a convenient basis consists of the order parameters that can be formed at a second-order phase transition. They are determined from the quadratic term of the free-energy functional which reads AF”’

=

C s1s2s;si

s

d 3 r d3r’$

s;si(r,r’)$s;s;(r’)

~ l s z ~ ~ ~ ~ s l s z

(7.23)

Here the variables rand r’ refer to the relative coordinates of the quasiparticles forming Cooper pairs. The space dependence of the order parameters is a consequence of the fact that superconductivity is associated with electron pairing. The matrix CI has the full symmetry of the Hamiltonian and its eigenfunctions r

(7.24) form a basis for the irreducible representations of the point group. Determining the irreducible representations of the symmetry group and their basis functions amounts to constructing the superconducting phases which can form at T, in a given structure of the normal state crystal. This approach corresponds to a soft-mode analysis for a structural phase transition. An analysis of possible order parameter forms at T - , T, was also given by A n d e r s ~ n , ” ~ ~Ueda ’ ’ ~ and Rice,’59 and Blount.’60 They assumed that the spins cannot rotate freely. The second authors also determined order parameter structures at low temperatures. The superconducting states for negligible spin-orbit interaction were worked out by Ozaki et Let us briefly discuss how these basis states of the irreducible representations are constructed. The symmetry-adapted even-parity states can simply be read off from the even representations of the crystal point group. This is a trivial consequence of the fact that scalar (pseudo-)singlet states transform according to k4(L) = 4 ( R L ) (7.25a) under the operations of the crystal point group. The determination of the vector order parameters d characterizing odd-parity states is only slightly more complicated. The crucial point to be observed here is that spin-orbit interaction is very strong. This fact implies that rotations of the crystal k also rotate the order parameter, R d ( i ) = Rd(RL)

16’

K. Ueda and T. M. Rice, Phys. Rev. B: Condens. Murfer 31,7114 (1985). E. Blount, Phys. Rev. B: Condens. Murter 22, 2935 (1985). M . Ozaki, K. Machida, and T. Ohmi, Prog. Theor. Phys. 74,221 (1985).

(7.25b)

114

PETER FULDE r! ol.

The spins are “frozen in the lattice” according to the terminology of Volovik and Gorkov.’ 5 8 The appropriate basis for the symmetry-adapted odd-parity states are constructed in two steps. First one expands the compo_nents of the order-parameter vector d,,(k^)in terms of the basis functions q i J ) ( k of ) the n ( j ) dimensional (odd) scalar irreducible representations which we symbolically denote by rv) (7.26) The vector functions constructed by this procedure transform according to the product representation T,, x rkj)which is a direct consequence of the strong spin-orbit interaction as seen from Eq. (7.25b). In the next step, these product representations are decomposed in terms of irreducible representations. TO finally obtain the proper basis functions, one has to project out the components corresponding to the different irreducible representations from the ansatz in Eq. (7.26). A list of the resulting states is given in the paper by Volovik and Gorkov.’ 5 8 Let us digress for a moment to make the role of spin-orbit interaction and crystal symmetry clearer. The odd-parity basis functions can be represented in terms of eigenstates of the total angular momentum J = L S of the Cooper pairs. In this way we can rewrite the basis functions belonging to the tith row of the n‘j)-dimensional irreducible representation rv)of the symmetry group

+

(7.27a) with 3

c&!,;~

=

1 (J,Mlr;),ti;p)

(7.27b)

p=l

The index p denotes the spin states introduced in Eq. (7.6). By this procedure we find the well-studied symmetry-adapted basis functions. This shows that the most natural way to classify the odd-parity states is in terms of the total angular momentum. To give a specific example, let us consider the pair states with total angular momentum J = 2 in a cubic lattice. As shown in the previous section, these pair states are characterized by symmetric traceless matrices B. It is well known that in a cubic environment the fivefolddegenerate d manifold splits into the doubly degenerate E and the threefolddegenerate T states. Order parameters with even and odd parity can be constructed for both even and odd symmetry. The basis functions of the twodimensional E representation are characterized by the matrices

115

THEORY OF HEAVY FERMION SYSTEMS

with E = e 2 n i / 3A. general order parameter of T2symmetry, on the other hand, is given as a linear combination of the three matrices

d; 1 :I;

0 1 0

BY’=

[o1

0 0

[: ,l 0 0 0

0 0 1

BY’=

BY’=

(7.29)

of the superconducting transition. The basic functions of the irreducible representations form a convenient basis for the construction of states with residual symmetry, i.e., of states which arc invariant under some subgroup of the full symmetry group. It is obvious that a state with a specific symmetry cannot contain basis functions from arbitrary irreducible representations. We can find, however, states with the same residual symmetries in different representations. A general order parameter with a specific residual symmetry can therefore be constructed from linear combinations of states belonging to different representations. Let us comment on this point. It was explicitely demonstrated by Monien et ~ 1 . for ~ ~ the ’ cubic system UBe,, that an odd-parity order parameter of D, x T symmetry, which can be constructed from states of the E representation, must in addition always include a component belonging to the A , , representation. The resulting order parameter is therefore given by

6)

d(c) = d(E)(c)+ d ( A l u )

(7.30)

Adding the A , , component docs not change the symmetry of the state since the latter is invariant under the higher group 0 x T. The most important result of the analysis of unconventional states concerns the node structure. In all symmetries investigated by Volovik and Gorkov’ 5 8 there is no odd-parity state with residual symmetry which vanishes on lines on the Fermi surface. Lines of zeros arc found only in even-parity states. In Fig. 21 we display unconventional order parameters of either parity for UPt, (hexagonal lattice). Quite generally we expect a singlet order parameter to vanish on a manifold of higher or equal dimension than a triplet state docs. This can be seen rather clearly for the unconventional states in a cubic crystal derived from the E symmetry. They arc all characterized by order parameters of the form

F‘ 0 0

:)

(7.31a)

-1-r

The nodes for the singlet state are determined by

c: + r i ; ”*

-

(1

+ r)Q

=0

(7.31b)

H. Monien, K. Scharnberg, L. Tewordt, and D. Walker, SolidSlute Commun. 61,581 (1987).

FIG.21. Typical examples of unconventional (a) even- and (b) odd-parity pair states for crystals with hexagonal symmetry (UPt,): order parameters with lines of zeros are found among the even-parity states. The sign of the order parameter alternates in the different segments. For the odd-parity state (b) with residual symmetry the modulus of the order parameter vector is shown. The order parameter vanishes only at points on the Fermi surface. Plots were performed with the help of the OBERFLIX Farbgraphik System developed by R. Nesper, B. Koerner, and U. Wedig.

THEORY O F HEAVY FERMION SYSTEMS

117

whereas those of the corresponding odd-parity state satisfy (7.31~)

c. Heavy Fermion Systems: Strongly Anisotropic Superconductors The character of the present section differs from the rest of the article. We will not review already existing theories. The central goal is to put forward the hypothesis that heavy fermion systems are conventional though highly anisotropic superconductors. We therefore focus on (pseudo-)singlet pair states the wave functions of which have the symmetry of the Fermi surface, i.e., the symmetry of the underlying lattice. In the present section, we will give some plausibility arguments why the pair states should be (highly) anisotropic. In Section 13, we will elaborate on special properties of conventional anisotropic states, as opposed to unconventional ones. The pair wave function of a conventional superconductor can be expanded according to

where O,, are orthonormal functions of the angle it which have the full symmetry of the lattice. They are conveniently expressed as linear combinations of the tesseral harmonics

z,,

= Yp

1

[Yrm

- -[Y;" 1 lm-J2

+ (-

1)"Y;"I

- (-

l)"Y;"]

(7.33)

where Y;l denote the usual spherical harmonics. The order parameter in Eq. (7.32) consists of a constant 4, plus an anisotropic part. It is generally agreed that the constant 4owhich yields an isotropic gap must be rather small in heavy fermion superconductors: first and foremost, low-temperature specific heat and transport data do not exhibit activated behavior. Second, a dominant constant term would be hard to reconcile with our present understanding of the quasiparticles and their interactions in heavy fermion systems, as we shall explain below. The order parameter +-by definition-describes the relative motion of the quasiparticles in a Cooper pair. Let us consider this motion (qualitatively) in real space. We refer to the Cooper problem of two quasiparticles attracting one another in the presence of a filled Fermi sea. The wave function of this pair

118

PETER FULDE el al.

satisfies the Schrodinger equation

where E is the energy of the pair state measured relative to twice the Fermi energy. We next assume that the effective attraction V(rl,r2) mainly depends on the separation r l - r2 = r. This assumption does not change the qualitative conclusion drawn here but greatly simplifies the formal argumentation. The pair wave function can then be represented in real space by

(7.35) in close analogy to Eq. (7.32). The radial dependence of &,,(r) is of particular interest. For separation r << ( = vT/kBK, $lm is just like the wave function of two free particles at the Fermi surface with angular momentum 1. For I = 0, it has its maximum at the origin k,r = 0 and falls off for increasing k,r, whereas it rises like (k,rj’ to its first maximum at r CL Ilk, for 1 # 0. This general behavior is displayed in Fig. 22. It is of central interest in our discussion of the pair states in heavy fermion systems. The instability of the normal state is determined by whichever pair state 4 yields the lowest energy E. The actual effective interaction V between the quasiparticles cannot be calculated explicitly from first principles. We should expect, however, that some characteristic features of the interactions between the bare particles (f electrons, conduction electrons) might persist in the effective interactions. The underlying physical picture is that the quasiparticles mainly consist of f’electrons which acquire itineracy through their coupling to the conduction electrons. The fact that we cannot have more than one 1’

I

41

ZIT 4rc

~ I T~ I T~

OK

kFr-

FIG.22. Qualitative behavior of the Cooper pair wave function for (a) isotropic and (b) anisotropic pair states.

THEORY OF HEAVY FERMION SYSTEMS

119

electron per Ce site should translate into a quasiparticle interaction which is repulsive for short separations. The strength of this quasiparticle repulsion (it is of the order of k, T*, the characteristic energy of the quasiparticles) certainly reduces the probability of finding an isotropic contribution to the pair state. Such a component strongly weights the repulsive part of the interaction at short separations Irl. To gain energy by forming Cooper pairs we must assume that the dominant components &,,are small for short values of r but that they are large where the effective interaction potential is attractive. This fact implies that the dominant contribution to d, in Eqs. (7.32) and (7.35) should comc from the second anisotropic term. We assume that two .f electrons sitting on neighboring Ce or U sites attract each other. The large separations of the Ce or U atoms in these compounds make the high-1 pairing rather plausible. Let us make this more quantitative in the case of UBe,,: the cubic symmetry implies that the lowest 1 value contributing to is 1 = 4. The U-U distance, on the other hand, is 5.1 A and k, E 1 which makes an 1 = 4 state a good candidate. Finally we would like to comment on the possible conventional pair states in UPt,, the lattice of which has hexagonal symmetry. The bulk of experimental data seems to indicate that the order parameter vanishes on lines on the Fermi surface, a fact which already excludes odd-parity states. We shall show below that the desired node structure is a rather natural consequence for an order parameter which has the symmetry of the lattice. Let us assume that the hexagonal axis is directed along the z direction. The lowest-order contributions to the anisotropic order parameter in Eq. (7.3_2) result from Z,, and Z,,, which both depend only on the component k,. The variation with Sx and it, is introduced only by terms with 12 6. The latter, however, should not be the dominant ones, considering the nearestneighbor U-U distances. Let us summarize: the allegedly dominant contributions Z,, and Z,, yield conventional order parameters which depend only on it, and thus vanish on lines on the Fermi surface. This node structure which is consistent with experimental ob_servation_should not be strongly altered by including the higher-order k,- and k,-dependent terms: these contributions should mainly deform the lines into (still onedimensional) curves.

&',

12. MICROSCOPIC THEORY Microscopic theories of superconductivity in heavy fermion systems can be roughly divided into two groups, depending on what the basic mechanism is for electron attraction. In one group of theories the assumption is made that Cooper-pair formation is due to electron-phonon interactions. In contrast,

120

PETER FULDE rr d.

the second group assumes that in analogy to 3He Cooper-pair formation is the result of purely electronic interactions. In that case no lattice vibrations are required. We shall outline in the following both types of theories. Closely related with the basic interaction mechanism is the symmetry of the order parameter. This topic has been discussed in detail in the preceding part of this article. Most microscopic theories are restricted to a study of the superconducting transition. This requires working with the linearized self-consistency equation for the order parameter (or gap function) only. In its most general form this equation is

The om,w, denote again Matsubara frequencies. The Go(k,om) are normal-state Green’s functions. The above equation is the so-called strongcoupling form of the linearized gap equation. The interaction potential V,P,yP(k, k’; w,, om) contained in it is retarded. When the integration over the electron energy is performed, the k’ summation reduces to an angular average over the Fermi surface (FS). One obtains

where a sum over repeated spin indices is implied. The function Z(w,) is the quasiparticle renormalization factor defined through

(7.38) For a nonspherical Fermi surface the angular integration must be replaced by

(7.39) When the superconducting transition temperature is much less than the characteristic energy or temperature scales of the retarded potential (i.e., the Debye or spin-fluctuation temperature), a weak-coupling approximation can be made. In that limit the interaction is replaced by an instantaneous, i.e., nonretarded, potential. Furthermore Z(om)is replaced by its low-frequency

THEORY OF HEAVY FERMION SYSTEMS

121

value Z(0)= Z . This requires the introduction of a frequency cutoff o,in the Therefore summation over q,,.

It is seen that the gap function is no longer frequency dependent. When

kBT, << w, this equation reduces to

In the following the spin indices of I/,P.up(k, k') will often be left out. It will always be stated explicitly whether a spin-singlet or spin-triplet pair state is considered. Equations (7.40) and (7.41) are the basis for the following considerations. Of considerable interest is an understanding of the effects of impurities on the superconducting transition. It is known from experiments that in some of the heavy fermion superconductors the electronic mean free path is extremely short. Therefore the question arises how the different pair states are influenced by a finite mean free path of the conduction electrons. We shall devote attention to this problem in Section 12,d. a. Phonon-Induced Puiriny A microscopic theory of Cooper-pair formation through electron-phonon interactions must take into account the coupling of phonons to f electrons. Otherwise it would be difficult to understand why CeCu,Si, is a superconductor while LaCu,Si, is not. This point of view was first taken by Razafimandimby et uL4' Their theory uses the quasiparticle-phonon coupling described in Section 1V and shows that it is sufficiently strong in order to account for a superconducting transition temperature of CeCu,Si, of the observed size. Of course, that does not prove that phonon-induced pairing is indeed taking place in that substance. There are also electron-electron interactions present which could suppress phonon-induced superconductivity and favor superconductivity induced by electron-electron interactions. As discussed in Section 11, the order parameter must be very anisotropic in heavy fermion systems. A specific model which is based on the electronphonon interaction and which leads to an anisotropic order parameter was proposed by Ohkawa and F ~ k u y a m aand ' ~ ~by the Nagoya In particular the latter group has elaborated on the electron-phonon interaction

lh3

K. Miyake, T. Matsuura, H. Jichu, and Y. Nagaoka, Prog. Thror.Phys. 72, 1063 (1984).

122

PETER FULDE et ul.

proposed by Razafimandimby et al.46 Another early approach to phononinduced pairing, which does not attempt to specify the interactions, however, is due to Tachiki and M a e k a ~ a . ' ~ ~ The traditional way of treating superconductivity due to electron-phonon interaction is by starting from an effective Hamiltonian. It consists of a quasiparticle and an interaction part. The latter contains the quasiparticlephonon interaction and a Coulomb pseudopotential, which describes the short-range part of the Coulomb interaction projected onto the frequency range of the phonons. The difference between the attractive interaction due to phonon exchange and the repulsive (Coulomb) pseudopotential is responsible for the pair formation. In ordinary superconductors the following sequence of characteristic energies holds: k,T, << wD << E,, where wD is a characteristic phonon Debye energy. This has to be contrasted with heavy fermion systems, where the width of the quasiparticle band is k , T*. The corresponding sequence of energies is then k,T, << k,T* << wD. For virtual excitation energies w > kBT* the quasiparticles of a heavy fermion system change their character completely. They lose their heavy mass, and their coupling to the phonon changes, too (see Section IV). To a first approximation the situation in that energy regime resembles that of conduction electrons in the presence of well-localized moments. This implies a strong pair-breaking effect in that energy range. For a more detailed study we start out from the quasiparticle-phonon interaction in Eq. (4.32) with A , given by Eqs. (4.36) and (4.37). In order to estimate the effect of the interaction on the pair formation, we assume conventional pairing and apply the weak-coupling theory. In that case the effective interaction in Eq. (7.41) is given by (7.42) where D(q,O) is the static limit of the phonon propagator (7.43) with v, = 2nTn. Spin indices have been left out for simplicity, because we are considering singlet pairing, The renormalization factor Z has been set equal to one. It is important that the cutoff w, in Eq. (7.41) is now given by kBT* because of the strong pair breaking for higher excitation energies. Furthermore we direct attention to the fact that the angular average in Eq. (7.41) involves large momentum transfers q = k - k'. Therefore it would be incorrect to conclude that only a small fraction of the phonon contributes to the pairing. Instead one should realize that almost all phonons are involved, lh4

M. Tachiki and S. Maekawa, Phy.5. Reo B . Condens. Matter 29,2497 (1984).

THEORY OF HEAVY FERMION SYSTEMS

123

but with their "low-energy tails" only [i.e., the propagator D(q, 0) is generally far off the energy shell]. In order to evaluate Eq. (7.4l)further, we approximate v"(k,q)= -bT* [see Eq. (4.39)], where b is again of order unity. The superconducting transition temperature is then obtained in the form

T, = 1.14T*e-IiA

(7.44)

where (7.45) The bracket (...) implies a (weighted) average over the Brillouin zone. For CeCu,Si, we estimate (q2b2/2Mo,Z)RZ= 10-3/w,, with oDN 200 K. Furthermore, the electronic Griineisen parameter is q N 20-80 for that ~ ) . T* = 10 K this substance (for a discussion see Razafimandimby et ~ 2 1 . ~With yields a value of i = 0.5 for that substance and shows that the interaction is sufficiently strong in order to result in a transition temperature of the observed size. In the above estimate quasiparticle repulsions have been completely neglected. A complete theory must, of course, also incorporate them. Since their main effect is to prevent two f electrons from simultaneously occupying the same site, these interactions do not cause any serious problems. The attraction of quasiparticles due to phonon exchange must in any case involve either f electrons at different sites or f electrons at the same site, but at different times (retardation). The same holds true for alternative attraction mechanisms as well. No reasonable theory of superconductivity in heavy fermion systems should require the simultaneous presence of two f electrons at the same site. Formally, the quasiparticle repulsion results again in a pseudopotential. For virtual excitation energies larger than kBT* the quasiparticles are of predominantly conduction-electron character and experience the same Coulomb repulsion as in an ordinary metal. When reduced to the regime k,T*, this yields a pseudopotential p*. Generally 2 and p* are of the same order. Depending on which of the two is larger, the system will or will not become superconducting. Finally we want to stress again that the above estimates were done only in order to demonstrate that A is of the required size. They are by no means meant to imply that the gap function A(k) is isotropic. Accounting for anisotropies in a realistic way requires, among others, a good knowledge of the shape of the Fermi surface of the material under consideration.

b. Speciul Model of an Anisotropic Superconductor It was pointed out before that the constant term or I = 0 contribution in the expansion in Eq. (7.33) of q(k) is expected to be generally small or even vanishing. When the 1 = 0 contribution is not the largest one in the expansion, the gap will vanish on parts of the Fermi surface. In the following we want to

124

PETER FULDE rl

trl.

discuss a special model. It yields conventional superconductivity with a gap function, which vanishes along lines under quite general conditions. 1 3 5 * 1 6 5 We start from the following effective interaction Hamiltonian between quasiparticles (7.46a) where V(q) consists of two parts with u, 9 > 0

Viq) = u - RfW,

(7.46b)

Here u represents the strong on-site quasiparticle repulsion. The second contribution to V(q) is assumed to result from phonon exchange between ,f electrons on nearest-neighbor sites. For a cubic lattice (with lattice constant a) the form factor f(q) is f(q) = +(COS q,a

+ cos q,a + cos q,a)

(7.47)

The linearized gap equation, Eq. (7.41), can then be written as A,,.(k)

=

-N*(O)In(l.l4T*/T,)

V(k

-

k‘)A,,Jk‘)

(7.48)

A spherical Fermi surface has again been assumed and a cutoff o J k , = T* >> T, has been introduced. Furthermore we have set Z = 1. The gap function can be written as

A&) = [q(k)

+ S(k)r]iz,

(7.49)

in close analogy to the form of Eq. (7.9) for the order-parameter matrix y. The scalar function q(k) belongs to a spin-singlet state and is an even function of k \see Eq. (7.14)], while S(k) implies a spin-triplet state and is an odd function of k [see Eq. (7.15)l. A further classification of these functions according to the irreducible representations of the symmetry group of the Hamiltonian is possible. In order to solve Eq. (7.48), it is useful to decompose the interaction V(k - k‘) into contributions belonging to different irreducible representations. For a cubic system this yields (e.g., mi yak^"^) V(k - k’) = V,(k, k’) + V‘(k, k’) + V,(k, k’)

(7.50)

where V, belongs to the A , representation, while VEand V, belong to the E , and T,, representation, respectively. In Table I11 we list the forms of V,, V,, and VT when expressed in terms of products of functions depending on one variable only.

’”

F. J. Ohkawa, in “Theory of Heavy Fermions and Valence Fluctuations” (T. Kasuya and T. Saso, eds.), p. 242. Springer-Verlag, Berlin, 1985.

THEORY OF HEAVY FERMION SYSTEMS

125

OF THE QUASIPARTICLE INTERACTION POTENTIAL INTO TABLE111 DECOMPOSITION v,, v, A N D v, BLLONG~NC TO DIFFERENT IRREDUCIBLE REPRCSENTAT~ONS OF

CDNTRlBUTlONS

~

THL CUBIC GROUP

For an order parameter with the full symmetry of the Fermi surface, only the contribution V, is relevant. For q ( k ) the ansatz is made (see Table 111) V(k) = A0

+ h(k)A1

(7.51)

The subscripts 0 and 1 should not be confused with 1 = 0, 1 contributions to the gap functions! From Eq. (7.48) the following relations are obtained for A, and A, A. = N*(O)uln(1.14T*/T,)(Ao+ (fA)FsAl) A1

=

+ (f:)FSAl)

N*(O)yln(1.14T*/T,)((f,),sA,

(7.52)

The superconducting transition temperature is again of the form

T, = 1.14T*e-liA

(7.53)

where *

I

h. = AA = +(IL, -

Lo) + [+(A, - A , ) 2

+ A0(A2 - 4 ) ] 1 ’ 2

(7.54)

and Lo

= uN*(O),

4 = gN*(o)(f;,);s,

A,

= gN*(O)(f;),s

(7.55)

Note that always A, 2 A,. It is interesting to notice that superconductivity occurs even when 2, >> A2, i.e., when the repulsion dominates. In that case 2, ‘v L2 - A1 and the pair formation is solely the result of the variation of V, over the Fermi surface. Near T, the momentum dependence of the gap function is determined by

126

PETER FULDE

el a!.

It is seen that q(k) and hence A(k) vanish along lines at the Fermi surface provided that the variation of fA is sufficiently large. The restriction to a spherical Fermi surface can be lifted by redefining the average over the Fermi surface through the replacement of Eq. (7.39). The functions listed in Table 111 again form an orthogonal set on a Fermi surface of cubic symmetry, provided that the coordinate axes are oriented along the crystal axes. A nonspherical Fermi surface fixes the orientation of an anisotropic order parameter with respect to the crystal axes. We also want to mention briefly the cases where the order parameter has a lower symmetry than the Fermi surface. For a spin-singlet state of E, symmetry, e.g., the following ansatz can be made,

d k ) = fE,(k)A

(7.57)

In spherically symmetric systems this state corresponds to d-wave pairing. In that case only V, contributes to the pairing and T, is given by Eq. (7.53) with (7.58) , I= 1%= s
A

9x(k) = f T J 4 A in Eq. (7.53) is given by

(7.59)

(7.60) For a discussion of which of the three pairing states wins out in the present model, we refer to the original literature. In practice such a question cannot be answered without a detailed analysis of the quasiparticle band structure and the different interaction mechanisms. However, one can show within the present model that the states with E, and TI, symmetry are affected more strongly by impurity scattering through pair breaking than the A , state (see also Section 12,d). On the other hand, spin fluctuations are pair breaking for singlet states and pair forming for triplet states.

A = AT

=s
c. Pairing Induced by Electron- Electron Interactions

A formation of Cooper pairs can also be obtained from fermion interactions, i.e., without invoking phonons. A well-known example is the superfluidity of 3He. Here a quasiparticle interaction is generated by the exchange of spin fluctuations. It is attractive for pairs with parallel spin (see, for example, Leggett '66). The strongly repulsive short-range part of the interaction in 3He is avoided by forming pairs with finite angular momentum. Spin-triplet pairing in a I = 1 (i.e., p-wave) state is found to be most favorable.

'"

A. J. Leggetl, Rao. Mod. Phys. 47, 331 (1975).

THEORY OF HEAVY FERMION SYSTEMS

127

It has been suggested that a similar situation is encountered also in heavy . ~ is~ again . ~ ~a strong , ~ ~local ~ , quasipar~ ~ ~ , ~ ~ ~ fermion s y ~ t e m ~ . ~ ~There ticle repulsion present, which in the case of Ce systems prevents two f electrons from occupying the same Ce site. There are also spin fluctuations present. Local spin fluctuations result from breaking the singlet ground state. Magnetic fluctuations involving different sites are expected to be of antiferromagnetic character. The observation of a T 3In T term in the low-temperature specific heat of UPt, has been considered as strong evidence that spin fluctuations are important in that substance. The fact that a T 3In T contribution is also found in the specific heat of 3He has been considered as evidence that heavy fermion systems and 3He should be rather similar, also as far as their superconducting properties are concerned. For various reasons we do not share this point of view. But if one accepts it, then it is suggestive to use the same Fermi-liquid approach for the description of superconductivity in heavy fermion systems as has been used for ,He. We shall briefly discuss this approach. We also comment on a suggestion put forward recently, according to which the pairing interaction is generated by the exchange of “Kondo bosons” between the quasiparticles. This is a special form of quasiparticle interaction which is obtained if one goes beyond the mean-field approach discussed in Section 8. ( i ) Fermi-Liquid Approach. Let us assume that spin fluctuations are the origin of the pairing interaction. Then one is facing the problem that the characteristic energy of the bosons (i.e., spin fluctuations), which are exchanged between the quasiparticles, is of the same order as the width of the heavy quasiparticle band (or Fermi energy). This is in contrast to ordinary BCS superconductors, for which the characteristic energy of the exchanged bosons, i.e., phonons, is much less than E,. A theoretical method which tries to bypass these difficulties has been developed for ,He by Patton and Zaringhalam.” It is based on Fermi-liquid theory. Extensions to heavy fermion systems are due to Valls and TeSano~iC,~’ Bedell and Q ~ a d e rand ,~~ . ~ ~method establishes a link between superconductivity Pethick et ~ 2 1 The and Fermi-liquid theory. It requires, however, a number of model assumptions which are not free of arbitrariness. It is well known that the superconducting transition temperature is determined by a singularity in the scattering matrix of two quasiparticles with opposite momenta at the Fermi surface. In Fermi-liquid theory a two-particle yp(p1p2;p3p4) is introduced, which describes the scatterscattering matrix KO, ing of two quasiparticles with momenta p l , p2 and spins c1, /I into states with momenta p3, p4 and spins y, p on the Fermi surface (see Fig. 23a). Due to rotational invariance in spin space, the scattering matrix can be decomposed 16’

H. R. OK H. Rudigier, T. M. Rice, K. Ueda, and J. L. Smith, Phys. Rev. Lett. 52, 1915 (1984).

128

PETER FULDE e l al.

pa p_,+q_’

ha P>+S_Y +-*...* (b)

-
-

Vlq)

*

i -

< -

p,+g * 1

I

(?

-L-.-e%.L* _p,P B-S_P e p2-q‘ p,-g FIG.23. Scattering T matrix of two quasiparticles. (a) Labels used for incoming and outgoing quasiparticles. (b) Repeated scattering of two quasiparticles through an interaction potential given by Eq. (7.64).

into a symmetric (s) and antisymmetric (a) part T”6,,&l

Tap;yp =

+ TaflRyflpp

(7.61)

where oiare Pauli matrices (see, for example, Serene and RainerI6*). In the case of forward scattering, i.e., p3 = p l , p4 = p 2 , the scattering matrix depends on the angle 9 only, between the momenta p l , p2 of the ingoing particles. Using an expansion in terms of Legendre functions, one obtains TS.a(P,pz; P I , P2) = 7

c

A;*aPI(COS

6)

(7.62)

I

where the As,” are the Landau parameters defined in Eq. (5.8). Because of the Pauli principle, the scattering matrix must be antisymmetric with respect to the exchange of two particles

Xp,JP*

>

P2; P 3

7

P4) = - Tas;py(P1> PZ; P4, P3)

(7.63)

In order for this relationship to be fulfilled, the Landau parameters have to obey the forward scattering sum rule of Eq. (5.9). As mentioned above, the transition to the superconducting state is caused by a singularity in the scattering matrix of a quasiparticle pair with opposite momenta p, = -p2 = p into pairs with p3 = -p4 = p’. As only the forward scattering matrix can be related to Landau parameters through Eq. (7.62), one must make model assumptions as to how the former can be extrapolated to arbitrary values of momentum transfer q = p - p‘. A second model assumption which is required concerns the temperature dependence of the scattering matrix. It is needed for a determination of T, but is outside the scope of Landau’s Fermi-liquid theory. 168

J. W. Serene and D. Rainer, Phys. Rep. 101, 221 (1983).

129

THEORY O F HEAVY FERMION SYSTEMS

Concerning the extension to arbitrary momentum transfers, two different approaches have been often used. One is the s-p approximation of Dy and P e t h i ~ k , which ’ ~ ~ has been applied by Patton and Zaringhalam” to 3He and by Valls and TeSan0vi6~~ to heavy fermion systems. The other approach, which we will follow here, is the potential scattering Thereby it is assumed that the total scattering of two particles at the Fermi surface can be adequately described by repeated scattering due to an instantaneous, i.e., nonretarded, interaction which depends on the momentum transfer only, i.e., %~,,,(P,P’)=

~(IP

-

~’1)&,&, + j(lp - P ‘ I ) ~ ~ ~(7.64) ~ ~ ~

The repeated scattering is schematically shown in Fig. 23b. When the T matrix is calculated, the energy integration over the intermediate particle states requires the introduction of an energy cutoff w, as in Eq. (7.40) in order to obtain finite results. The requirements on the cutoff are k,T, << w, << kBTF, where k , TFis a characteristic energy over which the quasiparticle self-energy does not have any significant structure. Thus the requirements are similar to the one for going from Eq. (7.37) to Eq. (7.40). The T matrix can then be written as Ta,;,,(Pl?P2;P3,P4)

= 2N*(O)Cru,,,,(Pl,P2;P3,P4)

--Ta,;py(P1,P2;P4,P3)1 d(P1 + P2 - P3 - P4) where

(7.65)

r obeys the integral equation

rmp;yp(P1,P2;P3,P4) = &J;yp(Pl,P3)6(Pl+ P2

-

P3 -

P4)

%,;~p(PS,p6)~(pl + P2 - P5 - P6) ZP PS.P6

1

-

B

G(p5 wn)G(p6 7

9

-wn)r~p;

yp(P5

>

P6; P3 P4) 9

Ialnl<%

(7.66) In the limit p1 = p2 the kernel in the integral equation diverges like ln(w,/T) for T -+ 0. The second term in Eq. (7.65) is a consequence of the symmetry relation Eq. (7.63). In order to establish a connection between the potentials u(q) and j ( q ) and the Landau parameters As.”, one uses the fact that for k,T >> w, the T matrix is given by the lowest-order scattering process. In that case the T matrix is K. S. Dy and C. J . Pethick, Phys. Rev. 185, 373 (1969). K. Levin and 0.T. Valls, Phys. Rev. B: Condens. Mutter 20, 105 (1979). P. Wolfle, Rep. Pray. Ph-vs. 42, 269(1979). 1 7 2 J. A. Sauls and J. W. Serene, Phys. Rev. B . Condens. Matter 24, 183 (1981).

lhy 17’

I30

PETER FULDE et a / .

given by Eq. (7.65) with I- replaced by I/. For forward scattering, i.e., for p3 = p l , p4 = p 2 , the symmetric and antisymmetric parts of the T matrix are given by TS(P,7P2;P,,P2) = N*(O)C2f@) - Uf(PI - PZO T a ( P I , P 2 ; P l , P 2 ) = N*(O"j(O)

-

l'(IP1

-

-

3j((Pl

P21) + j ( l P l

-

-

P2l)l (7.67)

Pdl

Together with Eq. (7.62) these relations provide the desired connection between the potentials and the Landau parameters A;.". One must merely expand the potentials in terms of Legendre polynomials and compare COefficients. Let us return to the determination of T,. It is given by the temperature at which the T matrix diverges. The linearized gap equation, Eq. (7.40), is the equation for the eigenvector of Eq. (7.66) in the limit of a diverging T matrix. In the case of singlet pairing (S) and triplet pairing (T) only the following parts of the pair interaction enter the self-consistency equation V'(P,P')

=

~ ( I P- P'I)

VT(P?P') = UilP

-

P'l)

-

3j(lp - P'I)

+ A l p - P'l)

(7.68)

One expands the pairing interaction into Legendre polynomials with respect to the angle between p and p', i.e., x = p p',

.

(7.69) and uses a similar expansion for the pair wave function. The superconducting transition temperature for pairing in angular momentum states 1 is given by (7.70) iS,T =

-N*(O)V7*T/(21+ 1)

(7.71)

and the assumption has been made that o,>> k , T , . For singlet pairing 1 is even, while for triplet pairing 1 is odd. The corresponding 6 must be negative. When the relations between the scattering potentials v(q), j(q) and the Landau parameters are used, one can reexpress the in terms of the latter. One finds i 0s

-

-14

(

2:

=

1:.

=+(A;

(A; - 3.43

-+(A;

-

c (A;

-

3

4

l>O -

3.4;)/(21

+ 1);

+ A3/(21 + 1);

1 = 2, 4,. ..

(7.72)

1 = 1 , 3, 5 )...

These results are identical to those obtained in the s-p approximation, provided one restricts oneself to contributions with 1 I1.

THEORY OF HEAVY FERMION SYSTEMS

131

From the discussions in Section V, it is clear that in strongly anisotropic systems not enough experimental information is available in order to determine the Landau parameters. This is particularly so because one expects contributions from higher angular momenta to become important. An example was discussed in Section 12,b where a superconducting state with the total symmetry of the system was found in the presence of a strong, local quasiparticle repulsion and a nearest-neighbor attractive interaction. The anisotropic part of the order parameter in a cubic system requires 1 2 4. Therefore, from a practical point of view the above scheme is of little help. What can be said is that A ; = 1, because F ; is of order m*/m (see Section V). Despite these intrinsic difficulties, interesting attempts have been made to speculate on the size of the various 3, values (for a thorough review of that work see Pethick and Pines97).For example, one can require that AS;2 = 0, in which case one is left with four parameters only. One can then try to determine them by setting A ; = 1, by using the experimental information on the T31n T term in the specific heat [see Eqs. (5.6) and (5.7)], and the magnetic susceptibility xs [see Eq. (5.4)] and by applying in addition the sum rule, Eq. (5.9). For UPt, a value of A : N - 4 has been suggested as a result of such an analysis.97 One obtains then from Eq. (7.72) and the sum rule Eq. (5.9), a value A: = - ( A ; + A : ) which is positive and favors p-wave pairing. But besides the above mentioned approximations one is faced with additional problems. As pointed out in Section V, the Fermi-liquid theory does not account for effects resulting from impurity scattering. The specific heat contribution of the broad quasielastic peak in Fig. 13a is therefore not contained in it. Hence density fluctuations result in a djferent contribution to the specific heat than predicted by the A;-dependent terms in Eqs. (5.5)-(5.7). This, in turn, may have some influence on the evaluation of the experimental data. Although it has been suggested that from Fermi-liquid theory it follows that superconducting pairing in UPt, is of an unconventional type, we think that these conclusions are premature. Finally we want to mention that special models like paramagnon theory can be applied in order to calculate directly the momentum dependence of the interaction potentials u(q) and j ( q ) in Eq. (7.64). The Landau parameters are then computed, instead of determined by fits to experiments. This has been done by Fay and Appel 1 7 3 for UPt, also taking into account the electronphonon interaction. Thereby it is found that the coupling constant for p-wave pairing is attractive. ( i i ) Pairing Induced by the Exchange of Kondo Bosons. In Section 8 a mean-field approximation for the Anderson lattice was discussed in detail. D. Fay and J. Appel, PIzys. REV.B32, 6071 (1985)

132

PETER FULDE et u1.

This was done by introducing an auxiliary boson field b, where r 2 = (btb) = 1 - nf describes the probability that the f state is empty. In mean-field approximation the boson operators are replaced by their mean value, and fluctuations in the amplitude and phase of the boson field are neglected. They can be handled adequately by starting from a functional-integral formulation It has first been realized for the partition function.' 1 2 q 1 by Read and Newns for the single-impurity case that the inclusion of fluctuations in the calculations enables one to describe physical quantities correctly to leading order in a l/vf expansion. In particular, the right Wilson-Sommerfeld ratio is found, which in a Fermi-liquid theory is obtained only by including quasiparticle interactions. This implies that fluctuations in the boson field mediate interactions between quasiparticles. This idea has been extended to the Anderson lattice by TeianoviC and V a l l ~ , Auerbach '~~ and Levin,' 7 6 and Lavagna et al.177Thereby q-dependent fluctuations in the boson field are introduced in the form of boson propagators for the amplitude and phase. The interaction between quasiparticles is obtained by the exchange of such Kondo bosons. This procedure allows for the calculation of the scattering matrix of two quasiparticles and the evaluation of the Landau parameters As*"in a l/vf expansion. In this approximation it is found that A ; = 1 + A ; , where A ; and the other scattering amplitudes are of order l/vf. Furthermore, one has As = A ; for 1 > 1. This is a consequence of the fact that the boson field is carrying no spin. A spin-dependent interaction between quasiparticles is obtained only from the exchange-type diagrams of this boson-mediated interaction. The scattering amplitudes have been calculated explicitly for the simplified band-structure model discussed in Section 8. As a result it is found'77 that the A",.' are negative, while the A:a are positive. From Eq. (7.72) one may conclude that a superconducting state with d-wave pairing is favored in such theories. The way the quasiparticle interactions are handled by this method is certainly very attractive. Some of the results deserve special attention: 1 3 3 1 4 1 - 1 4 3 , 1 7 4

( 1 ) In a 1 /vf expansion the Landau parameter A ; always comes out to be small. This is in contradiction to the results of the preceding section, where a value of A ; N -4 has been suggested for UPt,. (2) The Landau parameter A ; equals 1 up to a correction of order 1/vf. This is in agreement with the content of Section V but differs from earlier work listed in that section (due to some missing diagrams in the treatment of

N. Read, J . Phys. C 18,2651 (1985). Z. TeSanoviC and 0.T. Valls, Phys. Rev. B: Condens. Mailer 34, 5212 (1986). "'A. Auerbach and K. Levin, Phys. Rev. Lei/. 57,877 (1986). 17' M. Lavagna, A. J. Millis, and P. A. Lee, Phys. Rev. Lrtt. 58, 266(1987). L75

THEORY OF HEAVY FERMION SYSTEMS

133

Auerbach and L ~ v i n , the ’ ~ ~size of l/vf correction is too large and gives incorrect results for the charge susceptibility).’78

A quasiparticle interaction can also be derived by starting from a perturbation expansion in the hybridization, as outlined in Section 7,a.’79 Then the vertex function for the two-particle interaction can be expressed by the propagators for the occupied and empty f states. In a recent paper Zhang et U I . ’ ~ ~ have investigated the pairing interaction in this model. For the propagator of the empty state [see Eq. (6.20)] a two-pole approximation is made. One pole is at the Fermi energy. The other pole is at E / - k,T* and is due to the manybody effects. It corresponds to the Abrikosov-Suhl resonance. When only the latter is kept, the pairing interaction resembles the one obtained when Kondo bosons are exchanged between the quasiparticles. When the pole at E, is also included, part of this interaction is canceled again. It is found that the calculated values of the Landau parameters depend sensitively on the band-structure model, and that the commonly used continuum approximation is not sufficient in order to decide which of the different pair states is stable. Though many questions remain open, this investigation shows that high-energy contributions to the vertex functions cannot be ignored in the calculation of superconducting and magnetic correlations. (iii) Other Pairiny Mechanisms Based on Electronic Interactions. Before we close this section on electron-induced pairing in heavy fermion systems, let us briefly mention some other work using nonphonon pairing mechanisms. In an earlier investigation of superconductivity in the Anderson lattice, Fedro and Sinha obtained a nontrivial solution of the self-consistency equation by a low-order decoupling of for a pair amplitude of the form (flr~!kL) the electron Green’s functions. This mean-field solution, however, is not stable against the pair-breaking effect of the magnetic moments which in their calculations are still present at T = 0. In a series of papers Schuh and G u m r n i ~ h ’ ~ ~ha-ve ’ ~studied ~ pairing of f-electron and conduction-electron-like quasiparticles by using a pairing interaction derived from the exchange part of the quasiparticle interactions.

’

17’ 17’

J. Keller and T. Hohn, submitted for publication (1987). N. Grewe and T. Pruschke, Z.Phys. B; Condens. Matter 60, 31 1 (1985). F. C. Zhang, T. K. Lee, and Z. B. Su, Ph,y.c. Rev. B: Condens. Ma/ter 35,4728 (1987). A. J. Fedro and S. K. Sinha, in “Valence Instabilities”(P. Wachter and H. Boppart,eds.), p. 371. North-Holland, Amsterdam, 1982. B. Schuh, PIiy.?. S/u/u.sSolid B 131,243 (1985). B. Schuh and U. Gummich, Z . Phys. B: Condens. Mutter 61, 139 (1985). U. Gummich and B. Schuh, Z. PIiy.v. B: Condens. Marrer 60, 345 (1985).

134

PETER FULDE e/ ul.

A variety of superconducting and magnetic phases were obtained. Whether those states really exist remains an open question, since the exchange interaction is only part of the total quasiparticle interaction. Similar ideas have been pursued by H ~ d a k , who ' ~ ~ introduced two bands of conduction electrons. One band couples weakly to the local moments leading to an RKKY interaction between the ions. The other band with strong antiferromagnetic coupling to the ions leads to Kondo singlet formation. The resulting effective Hamiltonian is then investigated with respect to different superconducting pair states. When searching for superconductivity induced by electronic interactions, one must also mention the work of Rietschel and Shamls6 (see also Grabowski and Sham18'), who studied the possible occurrencz of superconductivity in a homogeneous electron gas. They find that the exchange of high-frequency plasmons in the R P A approximation counteracts the static Coulomb repulsion and leads to superconducting pairing. Pairs do not form when (higher-order) vertex corrections to the exchange of plasmons are included. It is interesting to note that it is the frequency dependence of the pairing interaction which would result in superconductivity, were it not for the vertex corrections. The investigation shows how important the latter are, when the characteristic energy of the exchanged boson becomes comparable to the electron band width. In a series of papers, the Monte Carlo method has been applied to the Hubbard and the Anderson lattice model (see e.g., Refs. 188 and 189). In the Anderson lattice case, an effective Hamiltonian for the f electrons is first derived, which consists of a narrow band with renormalized width and an i :ninj). Here S, and n, are the spin interaction term of the form J C i + j ( S i S ,and density of thefelectrons at site i. Due to the interaction term an attraction is found between quasiparticles with antiparallel spin on nearest-neighbor sites. The model is investigated in for a small cluster of atoms on a cubic lattice by Monte Carlo techniques. Various correlation functions are calculated. An enhancement of the pair-correlation function is found for nearest-neighbor distances. It would lead to an anisotropic superconducting state of the form discussed in Section 12,b. At present, it is not clear how much of the effect is due to the finite size of t h e ~ l u s t e r . ' ~ ~The * ' ~method ~" is a very promising one and will make its impact on the field. 0. Hudak, JETP Let/. (Engl. Trunsl.) 42, 300 (1985). H. Rietschel and L. J. Sham, Phy.5. Rev. B: Condens. Mutter 28, 5100 (1983). I * ' M . Grabowski and L. J . Sham, Pliy.7. Rev. B; Condens. M a t t e r 28, 5100 (1983) I n *J. E. Hirsch, Pliy.s. Rro. B: Condens. M u t t e r 35, 1851 (1987). I n s J. E. Hirsch, Pliy.7. Rar. Le//.54, 1317 (1985). G. StollhoR, unpublished (1986).

IH5

''"

THEORY OF HEAVY FERMION SYSTEMS

135

A similar pairing mechanism has been proposed by C ~ r 0 t . He I ~ also ~~ derives an effective Hamiltonian for the f-like quasiparticles that consists of a (narrow) band part and an antiferromagnetic Heisenberg-like spin interaction part J C
d. Effects of' Impurity Scattering on the Transition With the exception of UPt,, the heavy fermion superconductors have electronic mean free paths which are of the order of a few Ce-Ce or U-U spacings, or even less. Therefore it is of great importance to understand the influence of impurity scattering on the superconducting transition. This is done in the following for elastic scattering. The main objection against anisotropic pair states has been that these states should be very sensitive to impurity scattering. Normal nonmagnetic impurities should act as pair breakers making pair condensation in those states rather unlikely when the superconducting coherence length 4 defined above is much longer than the mean free path 1. These arguments, however, usually assume that scattering off the normal impurities is predominantly isotropic. But the situation is very different when we come to systems where the Fermi surface as well as the scattering properties are strongly anisotropic. We might guess already that, for a given normal-state resistance, anisotropies in the scattering matrix elements lu(k, k')l might strongly reduce pair breaking as compared with pure s-wave scattering. This fact helps to reconcile the concept of anisotropic pairing in heavy fermion superconductors with their high electrical resistivities in the normal state. Let us make these arguments more quantitative. For that purpose we consider conventional, though highly anisotropic, pair states, i.e., the order parameter has the symmetry of the Fermi surface. The superconducting transition temperature T, is determined from the linearized gap equation, Eq. (7.40), which reads in the presence of nonmagnetic impurities

C

q(k) = -nk,T,

1 V(k,k")M-'(k",k';w,)q(k')

(7.73)

lwml 5 u c k ' k "

Provision has been made for an anisotropic Fermi surface and effective pairing interaction. The matrix M

(l

M(k,k';m,,,) = 6 k , k 9

~+ cn~ k" l lu(k,k")12

-

cn)u(k,k')12 (7.74)

is expressed in terms of the matrix element u(k, k') for scattering off a single impurity. M . Cyrot, So/irlS/u/e Conmiun. 601, 253 (1986)

136

PETER FULDE et nl

Let us first discuss the initial depression, dT,/dcl,,,. For small impurity concentrations c, we can expand M-' in terms of the impurity scattering which yields for the kernel of the linearized gap equation 1 C -V'(k,k')+c&T, 1 4

K(k,k')= -+,T,

lO,lSO,

x

(

C lu(k", ,'")I2

6k?f,k* k"'

i x 1

C IOmlSoc

om(

v(k,k")

k

(7.75)

- lu(k", k"')I2

We are looking for the change in the highest eigenvalue of this kernel,

E ( T )= E o ( T )

+ 6E(T)

(7.76)

where E , ( T ) is the eigenvalue of the pure system and d E ( T ) is the first-order change due to the impurities. The transition temperatures of the pure and the doped system, T,, and T,, are determined from the conditions E,(T,,) = 1;

E(T,)= 1

(7.77)

respectively. To lowest order in the impurity concentration, the highest eigenvalue E ( T ) of the linearized gap equation is given by

c

E(T)=AzT

1 ~-

I U ~5I0 ,

x c

l o r n 1

AnT

c

1 ~

2

1om15wcl o r n 1

1C(cp(k)lZl4k>k')l2- cp(k)lu(k,k')J2cp(k')1

(7.78)

kk'

where 3, and q(k) are defined as the largest eigenvalue and (normalized) eigenvector of the matrix V(k, k') { 7.79)

We assume here for simplicity that the As corresponding to different states q are suficiently different. The results, however, can easily be generalized. For p-wave pairing the Fermi surface averaging in Eq. (7.78) over k, k' yields the transport scattering rate T;' (e.g., Ref. 189c). The second term in Eq. (7.78) accounts for pair breaking. One notices that the reduction of the superconducting transition temperature is determined by weighted Fermi surface averages of the impurity-scattering matrix elements. It is obvious that the pair-breaking contribution vanishes identically for an isotropic gap, cpzo(k) = const, as expected. For a purely anisotropic gap qaniso with qaniso(k) = 0 (7.80)

C k

''''

1. F. Faulkes and B. L. Gyorffy, Phvs. Rev. 15, 1345 (1977).

THEORY OF HEAVY FERMION SYSTEMS

137

the situation is less trivial. In that case we have to account for the anisotropies in the scattering matrix elements. Maximal pair breaking is obtained for isotropic s-wave scattering. Then the last term on the right-hand side of Eq. (7.78a) which reduces the pair-breaking effect vanishes. The pair breaking is then simply related to the measured normal-state resistivity. If the scattering matrix element lu(k, k')l is strongly anisotropic then pair breaking is reduced. Anisotropic superconductivity can even be (almost) insensitive to sufficiently anisotropic impurity scattering. This is the case for the following example which we think is relevant for the discussion of heavy fermion superconductivity, e.g., in U, -,Th,Pt, alloys.'90 Assume that superconductivity nucleates only at certain parts of the (strongly anisotropic) Fermi surface. We assume that cp is almost constant on these different parts. If impurity scattering were to connect only those portions of the Fermi surface where cp is finite, superconductivity would be insensitive to impurity scattering. The above treatment of impurity effects can be easily extended to unconventional pair states. When the order parameter has odd parity, we simply have to insert the vector order parameter 9(k) instead of cp(k) into the preceding equations. The crucial point is that the second term in Eq. (7.78) which reduces the pair breaking will usually be smaller for unconventional states than it is for conventional ones. In conclusion, one may state that nonmagnetic impurity scattering always has a strong pair-breaking effect on unconventional pair states. For conventional, strongly anisotropic pair states the pair-breaking effect can be small. In particular it cannot be related to the normal-state resistivity. The effect of impurities on the density of states is discussed in Section 13. 13. PROPERTIES OF DIFFERENT PAIRSTATES Superconducting heavy fermion systems have attracted great interest because they are considered as possible candidates for unconventional pairing. An outstanding experimental finding is the power-law behavior of the lowtemperature specific heat, thermal conductivity, ultrasonic attenuation, and NMR relaxation rate in those systems. This seems to exclude the presence of an energy gap over the whole of the Fermi surface, because otherwise the lowenergy excitations die out exponentially with T. It is interesting, however, that a power law behavior of the specific heat C,(T) has been found even in superconductors like ZrV, or HfV,,I9l where it has been attributed to I9O

U. Rauchschwalbe, F. Steglich, G . R. Stewart, A. L. Giorgi, P. Fulde, and K. Maki, Europhys. Lett. 3, 751 (1987).

19'

H. Keiber, C. Geibel, B. Renker, H. Rietschel, H. Schmidt, H. Wuehl, and G. R. Stewart, Phys. Rev. B: Condens. Matter 30,2542 (1984).

I38

PETER FULDE et ul

strong-coupling effects. We consider it unlikely that this is the case in heavy fermion systems. The particular form of the power law depends on the particular system. All three compounds UPt,, UBe,,, and CeCu,Si, seem to have a linear contribution ysT to the specific heat. In UPt,, ys is roughly 40% of its normal-state value y N , while in UBe,, and CeCuzSiz it is much smaller, i.e., ys/yN 21 No linear term, however, was observed in recent specific heat measurements on extremely pure UPt, samples (G. G. Lonzarich, private communication, 1987).In addition, there is a T 2contribution found in UPt,, a T 3 contribution in UBe,,, and a T z . 8contribution in CeCu,Si,. These terms are sufficiently large, so that they must be attributed to excitations within the heavy quasiparticle system; i.e., impurity effects can be ruled out. For a detailed discussion of the experimental situation, we refer to the reviews by Stewart,' Steglich,2 and in particular Ott.4 A specific heat of the form C,(T) T 3 requires an electronic density of states in the superconducting state of the form N, 0'.Such an N J o ) follows from a gap function which vanishes at points of the Fermi surface (in ,He this corresponds to the axial p-wave state). Similarly, a specific heat of the form C,(T) T Zrequires a gap function which vanishes along lines on the Fermi surface (in ,He this is the polar state which has been found in narrow pores). As regards heavy fermion systems, the group theoretical approach, discussed in Section 11, sheds light on the possible order-parameter structures. The results of this theoretical analysis have important consequences for the interpretation of experimental data. According to Michel's theoremsIS6the states with all symmetries broken are less likely to correspond to stationary points and, hence to minima, of the free energy than states with residual symmetries left. Such states have been investigated by Volovik and G ~ r k o v "for ~ cubic, tetragonal, and hexagonal lattice symmetries. Thereby it was found that for strong spin-orbit interaction, no odd-parity states do exist for which the order parameter vanishes on lines. As pointed out above, strong spin-orbit interaction is defined by the reguirement that a rotation R of the lattice transforms the order parameter d(k) into Rd (I&)[see Eq. (7.24)]. States which vanish along lines can exist though with even parity. In rare earth and actinide systems, spin-orbit interaction is indeed very strong. Therefore from an observed T 2 variation of the specific heat in crystals of the above symmetries it must be concluded that the superconducting phase is not of p-wave type. It is indeed the strong spin-orbit interaction which makes the polar-type state unfavorable. Ozaki et al.'" find odd-parity order parameters with lines of zeros if the spins are not pinned to the lattice. A sizeable linear term ysT indicates a vanishing (or more precisely a sufficientlysmall) gap on a correspondingly large part of the Fermi surface. As discussed in Section 1l,c, this is definitely possible in strongly anisotropic

-

-

-

THEORY O F HEAVY FERMION SYSTEMS

139

systems. When there are quasiparticles at the Fermi surface with very different effective masses, it is possible that superconductivity is mainly within one quasiparticle subsystem with a small induced (or, for practical purposes vanishing) gap in the remaining part of the system. A finite density of states at zero excitation energy can also result from pair breaking due to impurities. It is well known that ordinary s-wave superconductors containing magnetic impurities can exhibit gapless behavior. The most important pair-breaking mechanism in anisotropic superconductors is momentum scattering off normal, nonmagnetic impurities. It has similar consequences as classical spin scattering in s-wave superconductors. A single impurity-which is an idealized description of a very dilute system-gives rise to well-defined bound states in the excitation gap or to virtual bound states depending on whether the spectrum of single-particle excitations of the pure system exhibits a gap or not. The energies of the bound states or the scattering resonances are given by the scattering strengths of the individual impurities. In the unitary limit, i.e., in the limit of extremely strong scattering, the impurity states show up at the Fermi level o = 0. In connection with heavy fermion systems, this limiting case was studied rather extensively. The main motivation was the prejudice that removing the Kondo ion should result in such strong scattering. The impurity states are broadened with increasing concentration, giving rise to the finite density of states at zero energy. If the individual impurities are sufficiently strong scatterers, one can find an excitation spectrum which is rather high at zero energy and still exhibits the features that characterize the spectrum of the corresponding pure superconductor. The first detailed study of these effects is due to Buchholtz and Zwick11ag1,’~~ who investigated the isotropic p-wave state. The theory was later Scharnberget ~ l . , ’and ~~ extended by Pethick and pine^,^' Hirschfeld et by Schmitt-Rink et al.195to the axial and polar states and by Monien et u1.’62 to d-wave pairing states. It has been proposed that the U, -xThxBe13alloys are examples of systems in which different parts of the Fermi surface behave quite distinctly with respect to pair f ~ r m a t i o n . ”These ~ systems show two second-order superconducting phase transitions in the specific heat.4 At the lower transition, the lower critical field H,, shows a discontinuous increase in slope when plotted as function of T 2 .Since H,,is proportional to the density of the superfluid, this implies the appearance of a second superconducting order parameter. The L. J. Buchholtz and G. Zwicknagl, Phys. Rev.B: Condens. Ma//er 23, 5788 (1981). P. Hirschfeld, D. Vollhardt, and P. Wolfle, Solid Sfate Commun. 59, 111 (1986). 194 K. Scharnberg, D. Walker, H. Monien, L. Tewordt, and R. A. Klemm, Solid Stale Commun. 19’

L93

60,535 (1986). 195

S. Schmitt-Rink, K. Miyake, and C. M. Varma, Phys. Rev. Let/. 57,2575 (1986).

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PETER FULDE et

(11

situation resembles very much that in Pb-Sn eutectics where, with decreasing temperatures, first the Pb layers become superconducting, while at a lower temperature the Sn layers also become superconducting.' 9 h (Note that, strictly speaking, the lower transition is not a thermodynamic phase transition, because by the proximity effect there is already a small amount of superconductivity in the Sn layers even above the lower critical temperature. However, this is of no practical relevance.) In U, _,Th,Be, there is no indication of a spatially inhomogeneous order parameter. But there is the remaining possibility of an inhomogeneous (anisotropic) order parameter in k space. Therefore the suggestion is that below the higher transition temperature K , one part of the Fermi surface (approximately 5 0 x ) becomes superconducting, while below the lower the remaining part becomes superconducting, too. transition temperature Tcb, A number of consistency checks are in agreement with this picture. The two order parameters A, and A,,, which are associated with the two transitions, need not be of the same symmetry. From details of the experiments it is unlikely, however, that they are of different parity. A more direct way to distinguish between different symmetries of the pair state is to look for anisotropies in the superconducting properties. Examples are the upper critical magnetic field or the ultrasonic attenuation. If, for example, a single crystal of the cubic system UBe,, should show an upper critical field which differs along different cubic axes, this would prove unambiguously the presence of an unconventional pair state. Unfortunately, large enough single crystals are not yet available in order to do those measurements. In the noncubic systems UPt, and CeCu,Si, the upper critical field already depends on the field direction with respect to the crystal axes due to the lattice anisotropy. A measurement of the upper critical field allows for learning something about the spin part of the pair-wave function. For spin-singlet pairing the upper critical field is limited by the pair-breaking effect of the magnetic field, which is trying to orient both spins of a Cooper pair in the same direction (Pauli limiting). In the case of a spin-triplet pairing and in the absence of a too strong spin-orbit coupling, the two parallel spins can orient themselves in the applied magnetic field without pair breaking. A very steep increase of the upper critical field with decreasing temperature close to T, has been found for all heavy fermion superconductors. Rauchschwalbe et al.' 9 7 have analyzed the upper critical field for CeCu,Si,, UBe,,, and UPt, by using computer programs developed by R a i ~ ~ e They r . ~ ~ compared the data with two

,

C . A. Shinman, J. F. Cochran, M. Garber,and G. W. Pearsall, Rev. Mod. Ph.vs.36, 127(1964). U. Rauchschwalbe, U. Ahlheim, F. Steglich, D. Rainer, and J. J. M. Franse,Z. Phys. B50,379 ( I 985).

IYh

'"

THEORY O F HEAVY FERMION SYSTEMS

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theoretical models, a "dirty" s-wave superconductor with pair breaking and spin-orbit coupling, and a p-wave model. They found that a spin-triplet pairing can be ruled out for CeCuzSiz,because of the low critical field at T = 0. Neither model gives a good fit for UBe,,. This is likely to be due to the occurrence of a second phase transition at T, = 0.55 K in analogy to the U, -,Th,Be13 alloys.'98 In the case of UPt, the data can be fitted approximately by both models. Another quantity of interest is the temperature dependence of the Landau penetration depth &(T). Within a two-fluid model it is proportional to ( n S ) - ' l 2 , where n, is the density of the superfluid. Measurements on UBe,, reveal a smaller ratio n,(T)/n,(T,)than the BCS theory predicts.'99 Indeed, the results are compatible with the presence of an axial p-wave state. The order parameter of an unconventional superconductor does not have the full symmetry of the lattice. Joynt and Ricezooconcluded that this fact should lead to strain distortions in these systems. The instability is driven by the fact that the systems will tend to maximize the density of states in the directions where the gap is large because that maximizes the condensation energy. Joynt and RiceZoocalculated the expected strain anomalies for the p-wave states of Ueda and Rice (1985). They conclude that the absence or presence of a strain anomaly and its sign contain information about the character of the pairing state. Let us return to pair states which are not eigenstates to the time-reversal operator and which occur for even- as well as odd-parity states. A physical consequence is that these states will have magnetic moments. On the other hand, the local moment inside a superconductor must vanish due to the Meissner effect. This suggests that there will be circulating surface currents on the surface of a single-domain superconductor and hence a magnetic field near the surface even in the absence of an external magnetic field. Magnetic unconventional superconductors can have domains with different orientations of the moments. These domains will be separated by domain walls. The structure of a domain wall was calculated by Volovik and G 0 r k 0 v . I ~ ~ An interesting aspect of their solution is that the magnetic moment estimated from the currents in the wall could exceed the lower critical field H,, at which the formation of vortices becomes favorable. A much-discussed test experiment of the parity of the pair state has been the Josephson current between two superconductors coupled by a weak link. U. Rauchschwalbe, C. D. Bredl, F. Steglich, K. Maki, and P. Fulde, Europhys. Lett. 3, 757 (1987). 199 D. Einzel, P. J. Hirschfeld, F. Gross, B. S. Chandrasekhar, K. Andres, H. R. Ott, J. Beuers, Z. Fisk, and J. L. Smith, Phys. Rev. Lett. 56, 2513 (1986). R. Joynt and T. M. Rice, Phys. Rev. B; Condens. Matter 32,6074 (1985).

'91

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PETER FULDE rt ul

With one of the two superconductors in a conventional even-parity state, one would like to be able to determine the parity of the other superconductor. We consider a weak link and discard for the moment the influence of the surface or the barrier on the Josephson effect. When the weak link connects an even and an odd pair state, the overlap of the two wave functions is zero. Therefore to lowest order in the transition (or tunneling) matrix elements, the dc Josephson current should vanish. However, when higher-order processes are taken into account a finite current is possible. The same holds true for the ac Josephson current.201 Because a finite ac current is the result of higher-order processes, the Josephson steps should have twice the frequency as, e.g., in a link between two even-parity states. The results hold for vanishing as well as finite spinorbit interaction as long as there is inversion symmetry present. A more realistic treatment of the Josephson effect in systems with unconventional pairing requires a more detailed treatment of the contact between the two superconductors (for a recent review, see Kurkijaervi et ~ 1 . ~ ' ~ ) . There are three different aspects of a contact which must be taken into account. One is that the contact layer, when viewed as a surface, itself breaks the symmetry of the lattice. As a consequence, the order parameter of an unconventional superconductor is depleted near the surface.192,203-205 Th'1s depletion is differently strong for the different components of an unconventional order parameter. Although the depletion or pair-breaking effect of a surface influences single-particle tunneling rather strongly, it has only a weak effect on the size of the maximum dc Josephson current. The coupling of two unconventional superconductors is only weakly influenced by this effect of a surface, which is a quite unexpected result. Another important feature of a contact is its surface roughness. The latter can be simulated by a normal conducting layer of thickness D. Surface roughness is very effective in destroying the phase coupling between two unconventional superconductors.206 When 1 is the mean free path in a superconductor, the maximum Josephson current decreases according to exp( - D / l ) . This assumes that the average of the order parameter over the Fermi surface vanishes when the momentum component perpendicular to the surface is kept fixed; i.e., ( A ( L ) ) k z = const = 0. The maximum Josephson current is therefore proportional to the inverse of the normal-state resistance R , of the contact. J. A. Pals, W. van Haeringen, and M. H. van Maaren, Phys. Rru. B: Condens. M a / / e r 15,2592 (1977). '"'J. Kurkijaervi, D. Rainer, and J. A. Sauls, submitted for publication (1987). '03 L. J. Buchholtz, Phq'.v. Rev. B: Condens. M a / / e r 33, 1579 (1986). '04 W. Zhang, J. Kurkijaervi, and E. V. Thuneberg, Ph.ys. L e t / . A 109,238 (1985). ' 0 5 W. Zhang, J. Kurkijaervi, D. Rainer, and E. V. Thuneberg, submitted for publication (1987). ' O h A. J. Millis, D. Rainer, and J. A. Sauls, submitted for publication (1987). 20'

THEORY O F HEAVY FERMION SYSTEMS

143

Finally there is the problem of spin-flip scattering when an electron passes through a contact. At present it is not yet clear whether this leads to an appreciable decrease of the phase coupling between the two supercond u c t o r ~ or ~ ~not.206 ' It has been also pointed out that a contact which breaks time-reversal symmetry (e.g., by containing magnetic impurities) leads to a dc Josephson current even when the two superconductors have order parameters of different parity.208 Unconventional pair states are anisotropic. Geshkenbein and Larkin,'' assume that the order parameters are oriented along symmetry lines of the (bulk)crystal. In that case, the Josephson current between unconventional and ordinary s-wave superconductors strongly depends on the angle between the surface normal and the basis vectors of the lattice. Finally we want to summarize briefly the present experimental situation. A dc Josephson current was found for CeCu2Si2coupled to Al, which is known to be a conventional even-parity superconductor.210From the observed large critical current one may conclude that CeCu,Si, is also an even-parity superconductor. Point-contact tunneling experiments on UPt,/Nb and UPt,/AI showed only gaps of Nb and Al. N o Josephson currents were observed between UBe13/UBel, and UPt,/UPt, weak links. For a UBel,/AI weak link a Josephson current was found below T, of Al due to the proximity effect. It increased with decreasing temperature down to the lowest measured temperatures which were below the transition temperature T, of UBe, 3 . 2 1 1 A different result was found by Han et aL2" for the system UBe,,/Ta. Here too a Josephson current was observed for T < T,(Ta). But, surprisingly, the maximum current decreased for temperatures T < T,(UBe,,). From this the conclusion was drawn that UBe,, must be in an odd-parity pair state. The superconducting order parameter which is induced in UBe,, below T,.(Ta) by the proximity effect has even parity. When bulk superconductivity sets in in UBe,, below T,(UBe,,), the induced order parameter is destroyed again, provided it has a different parity from that of bulk UBe,,. The mutual suppression of different pair states was investigated theoretically by various authors.2'3-215Th.IS would result in a maximum Josephson current which E. W. Fenton, Solid Slutr Cornniun. 60, 347 (1986). E. W. Fenton, Solid Sruk Commun. 54,709 (1985). '09 V. B. Geshkenbein and A. 1. Larkin, JETP Lerr. (Engl. Transl.)43, 395 (1986). ' l o U. Poppe, J. Muyn. Muyn. Mufer. 52, 157(1985). U . Poppe, unpublished results (1986). '" S. Han, K. W. Ng, E. L. Wolf, A. Millis, J. L. Smith, and 2. Fisk, Phys. Rev. Lett. 57,238 (1986). ' 1 3 K. Scharnberg, D. Fay, and N. Schopohl, J . Phys. 39, C6-481 (1978). 'I4 A. J. Millis, P/7ysicu B + C(Amsrerduni) 135, 69 (1985). ' I 5 B. Ashauer, G. Kieselmann, and D. Rainer, J. Low Tpn7p. Phys. 63,349 (1986). '07

'08

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PETER FULDE et al.

decreases as T is lowered below T,(UBe,,). This interpretation assumes that the effect is not due to the temperature dependence of normal-state properties of the weak link (see also Kadin and Goldman216). In conclusion, one may state that at present the nature of the pairing state in the heavy fermion superconductors is still uncertain. Personally we think that one is dealing with conventional, anisotropic superconductivity in these systems, with the possible exception of UPt,. The short electron mean free paths, which are present in CeCu,Si, and UBe,,, make it difficult to see how an order parameter with a symmetry lower than that of the Fermi surface should be able to survive. Also, strongly anisotropic conventional pair states are affected by impurity scattering. However, the effect is much less than for unconventional pair states, in particular when the impurity scattering is itself anisotropic. Like the pair state itself, also the interaction causing it is the object of considerable controversy. The fact that all heavy fermion superconductors show large elastic anomalies leads us to believe that at least in CeCu,Si,, but probably also in the other superconducting systems, the origin of the electron attraction is lattice degrees of freedom, i.e., phonons. That the coupling between quasiparticles and phonons is sufficiently strong in order to yield T, values of the observed size has been demonstrated for CeCu,Si,.

Appendix A: Molecular Model for Strongly Correlated Electrons

In the following we want to discuss the simplest possible model which contains the physics leading to Fig. 10 (a and b). The model must explain the formation of a singlet and furthermore the fact that there are lowlying excitations of predominantly f-electron character (see Fig. 10a). It also must show the feature that it takes a large energy to remove an f electron (see Fig. lob). We want to show that the essential physics is already contained in a “molecule” consisting of two atoms A and B with one orbital each (Fig. 24). We call the orbital on atom A an s orbital and the one on atom B an f orbital. The corresponding creation operators are c: and f! (for the s and f orbital), respectively. The orbital energies are E, and E,., and we assume that E, = E,. + AE, with AE > 0. An overlap or hybridization V is assumed between the two orbitals. Finally, it is assumed that two electrons on the B site interact with

A. M. Kadin and A. M. Goldman, Ph?;s. Rev. Lerr. 58,2275 (1987).

THEORY OF HEAVY FERMION SYSTEMS

145

v FIG.24. Two atoms A and B with one orbital each (s and f )coupled by a hybridization matrix element V . U is the interaction of electrons sitting in the f orbital.

an interaction energy U (see Fig. 24). The Hamiltonian is then of the form

where nfu = f 1f,. We consider two electrons and diagonalize H for two limiting cases, namely, for U = 0 and U + GO. In the case U = 0, we are dealing with a trivial problem of hybridizing electrons in an H, type of molecule. The single-particle eigenstates are the bonding and antibonding molecular orbitals (MO) (Fig. 25).

We assume that V << A€. In that case

pB = ; << 1 V

The bonding M O is predominantly of f-electron character, while the antibonding state is primarily of s-electron character. When two electrons are placed into the MOs (see Fig. 15) the resulting eigenvalues are as shown in Fig. 26a. I I

ES

-i' , I

I

I

I

i

AE

FIG.25. Eigenvalues of the bonding and antibonding molecular orbital.

146

PETER FULDE rr al

a FIG.26. Eigenvalues for the two-electron systems (a) for U = 0 and (b) for U the split-off of a singlet, which results in an energy gain of -2VZ/Ae.

+

cu.O n e notices

In the ground state 140)both electrons are in bonding states; i.e., 140) = B: B: 10). The excited doublet corresponds to the states I$$’) = B: A : 10) and 14:) = A i B : lo), respectively. The state corresponding to the excited singlet is IqL) = Af A: 10). Now we want to consider the limit U + m. In that case double occupancy of site B with two electrons is strictly forbidden, because of the infinitely strong Coulomb repulsion that would result. The eigenstates must be found within the space spanned by c : f : l O ) , c : f : l O ) , and c{c:(O). When H is applied to those states, it results in a 3 x 3 matrix. Diagonalizing it leads to the three eigenvalues shown in Fig. 26b. The corresponding eigenvectors are

where again V/AE << 1 has been assumed. One notices that the states I&,) and correspond to the doublet 14g)) and 14g)),respectively. The previous ground state consists predominantly of a state f:.f’:(O), which is forbidden in the limit U + 00 because it would correspond to an infinitely large energy. The crucial point is that the doublet in Fig. 26a has split into two singlets. The “moment” of the doublet has disappeared. The formation of a singlet is the essence of the Kondo effect. One can see here, within the simple model, how it occurs. The energy gain due to the singlet formation is V2/Ac, which should be contrasted with the exponential dependence on V of the energy gain kRTKin a solid [see Eq. (2.15)]. Assume that we had a solid consisting of uncoupled “molecules” of the form considered here. A specific

THEORY OF HEAVY FERMION SYSTEMS

147

heat experiment would show a giant low-temperature contribution due to the presence of the low-lying excited singlet state. From the form of and it follows that the excitation is within the f-electron system itself. This is what leads to heavy fermion behavior when the atoms A of different molecules are coupled together by hopping matrix elements. If, on the other hand, we ask for the energy which is required to convert an f electron into an s electron, this energy is clearly given by E = AE + 0 ( V 2 / A c ) .This makes the different appearance of .f' electrons in Fig. 10a and b very transparent. Finally, it should be mentioned that the inclusion of two-electron states with parallel spins changes the excited singlet state into a triplet state.

Appendix 6 : Parametrization of the Model Hamiftonian

As pointed out in Section 111, we have to adopt a model for the 4.f charge distribution at the Ce site in the starting calculation where we calculate the potential parameters of the non-f electrons. This fact seems to introduce some arbitrariness into the calculations since the f charge distribution contributes to the self-consistent Ce potential and, concomitantly, influences the dynamics of the non-f electrons. In the present appendix, we compare the results of two different starting calculations for CeCu,Si,. The sensitivity of the results with respect to the model assumptions allows us to estimate the uncertainty in the description of the band states. In the first calculation, one assumes that the ,f charge distribution at the Ce site is basically that of a corelike state. The 4f state is treated as a core state which does not hybridize with the band states. Consequently, there is exactly one 4f electron per Ce site. In the second calculation, one determines the f charge distribution from an ordinary LDA calculation. The 4f electrons are treated as band electrons which are allowed to hybridize with the band states. In the LDA thef-electron count is always larger than one. The results are summarized in two tables: In Table IV we list the sets of the four standard (LMTO) potential parameterss4 for the band electrons obtained in both calculations. (Such a set of potential parameters is all the potential-dependent information required for a band calculatjon within the rather small. LMTO-ASA scheme.) The deviations-except for o(-)-arc They are typically of the order of 1%. In Table V we compare the potential functions for the band states P{(E,) with j = Ce, Cu, Si and their derivatives with respect to energy, pi(&) = d P i / d c ( , = , , , evaluated at the respective Fermi energies E,. This is all the required potential-dependent information for th, calculation of quasiparticle bands. As mentioned in Section 111, the re iormalized band calculation

TABLE IV. SELF-CONSISTENT STANDARD LMTO POTENTIAL PARAMETERS FOR THE NON-f ELECTRONS IN CeCu,Si," Parameter

4f in core

Ad(-) d(-)/d( + )

e

o(-)

(is) 4f as band electron

w ( -)

Ad(-) + ( - ) i d ( +1

(4,')

Ce s

Ce P

Ce d

Si s

Si p

cu s

Cu P

cu d

0.31755 -0.45994 0.89207 0.08532

-0.08639 0.42822 0.65887 0.12431

0.41523 0.2701 1 0.39119 0.41970

-0.20920 0.53003 0.79808 0.11911

0.34538 -0.48520 0.56888 0.07200

0.05333 -0.58891 0.84321 0.05374

0.90120 0.56946 0.66635 0.02784

0.0058 1 0.13703 -0.0341 1 2.435 17

0.32156 -0.45879

-0.11345 0.42801 0.65083 0.12652

0.44014 0.27038 0.39825 0.41238

-0.19217 0.52796 0.79930 0.11858

0.35633

0.05794 -0.58845 0.84327 0.05381

0.91623 0.56974 0.66725 0.02767

0.0 1084 0.13581 -0.02787 2.44005

0.89175 0.08572

- 0.48435

0.56990 0.07195

The two sets of parameters are determined from the self-consistent band calculations treating the f electron as core and as band electron, respectively. The self-consistent calculations were performed at the experimentally determined volume of the unit The radii of the various Wigner-Seitz spheres are S,, = 3 . 9 4 5 3 3 ~S,~, ~ ;= Ss, = 2.63022a0, where o0 is the Bohr radius.

TABLE V. POTENTIAL FUNCTIONS AND ~

-

ENERGYDERIVATIVES AT THE FERMI LEVELFOR

THEIR

~

~

~

THE

NON-1'STATESI N CeCu2Si,"

~

Ce s

Ce P

Ce d

Si s

Si p

Cu s

Cu P

Cud

4f in core

P(E,) P(E,)

0.42169 2.15419

5.00637 16.72978

-6.64114 11.38561

2.64739 0.63132

-1.59564 14.21619

1.31530 2.15136

-5.002283 14.34230

49.51170 185.21040

41 as band electron

P(E,)

0.38826 3.52880

5.74230 24.27298

-6.68218 11.81083

2.76041 0.93296

1.60057 14.20650

1.31179 2.14990

5.04164 4.52288

57.34568 225.92620

\D P

~

P(E,) ~

The data are computed directly from the corresponding potential parameter sets listed in Table IV. Note that channels with large values of P ( E , ) are not kept explicitly in the model Hamiltonian. They are included by means of the down-folding procedure as described in Section 3.

I50

PETER F U L D E r t crl.

focuses on a narrow energy range around the Fermi level. Within this energy range one can safely linearize the potential functions of the band states. Again the deviations are found to be rather small. We therefore conclude that both starting calculations should yield very similar quasiparticle band structures. The final results should be rather insensitive to the assumptions made for the f charge distribution in the starting calculation. Let us finally add a technical remark: according to our experience, it is much easier to reach convergence within the first scheme which neglects the hybridization of the 4f electron with the band states. ACKNOWLEDGMENTS We would like to thank K. Becker, N. d'Ambrumenil, J. Kiibler, B. Luthi, D. Rainer, F. Steglich, and P. Thalmeier for numerous helpful discussions. We also thank E. Runge and T. Hohn for carefully reading the manuscript. Part of this work was supported by Sonderforschungsbereich 252 Darmstadt-Frankfurt-Mainz.

SOLID STATE PHYSICS, VOLUME

41

The Theory and Application of Axial k i n g Models JULIA YEOMANS Department

id Theoreiiral Physics,

Oxjord O X / 3 N P , England

1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Inlroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. The ANNNl Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Theoretical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Related Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Chiral Clock Model. . . . . . . . . . . . ............................... 11. Experimental Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Binary Alloys.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Polytypism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Magnetic Systems . . . . . . . . . . . . . . 9. Conclusion.. . . . . . . . . . . . . . . . . . . ......................

151 151 153 I59 I68 176 180 180 188 196 199

1. Theory 1. INTRODUCTION

The aim of this article is twofold: first, to survey theoretical work on discrete spin models where competition in the Hamiltonian results in modulated ordering, and second, to discuss the relevance of such models to experimental systems. Two compounds typical of those that will be considered are cerium antimonide and silicon carbide. Cerium antimonide is an Ising ferromagnet which locks in to a large number of different modulated magnetic phases separated by first-order phase transitions. In silicon carbide, on the other hand, the modulation is structural; the close-packed stacking sequence of the component layers can form long period patterns. The canonical spin model which shows similar behavior is the axial nextnearest-neighbor Ising, or ANNNI, model. The phase diagram of the ANNNI 151 Copyright fCj1988 by Academic Press. Inc. All rights of reproduction in any form reserved.

152

JULIA YEOMANS

model contains series of commensurate and incommensurate modulated phases of arbitrarily long wavelength. These are stabilized by entropic effects which dominate because of competition between short-range interactions. The first half of the article reviews theoretical work on the ANNNI and related models, and the second half describes relevant experimental work. In Section 2 the ANNNI model is defined and its phase diagram elucidated. Emphasis is placed on the physical reasons for the appearance of the modulated phases. This section aims to give a summary which will provide sufficient background to the experimental applications in Section I1 for those uninterested in theoretical detail. Section 3 reviews the details of the theoretical methods which have been developed to treat discrete spin models with modulated ordering. Mean-field theory, checked by low-temperature series expansions, has given the greatest insight into the phase diagram. In order to discuss the relevance of the ANNNI model to modulated ordering in real systems, it is important to consider the effect of new terms in the Hamiltonian on the phase diagram. Therefore, in Section 4, we review results on further-neighbor interactions, the inclusion of quenched or annealed defects, the application of a magnetic field, and the effects of lattice structure. The final theoretical Section 5 is devoted to the chiral clock model, in which a different mechanism, chiral interactions, provides the competition which leads to modulated phases. The novel interface properties of this model are discussed. Section 6 opens the experimental section of the article with a discussion of binary alloys where the structural ordering can form long period patterns for some compositions and temperatures. Beautiful high-resolution electron microscopy experiments on these compounds are described. The similar, but less well documented, properties of polytypes are reviewed in Section 7. Then, in Section 8, we discuss the magnetic properties of the cerium monopnictides and their relation to the ANNNI model. The concluding Section 9 points out directions for future research. A previous review by Bak gives a more general survey of commensurate and incommensurate phases. Shorter summaries of the behavior of the ANNNI model have been published by Selke2 and by yeoman^.^

' P. Bak, Rep. f r o g . f l i j ~ s 45, . 587(1982). W. Selke, NATO A S I Srr., Srr. E 83,23 (1984). J. M. Yeomans, fhy.sicu B + C (Anisterdurn) 127B, 187 (1984)

THEORY AND APPLICATION O F AXIAL lSlNG MODELS

153

2. THEANNNI MODEL Perhaps the simplest model where competing interactions lead to modulated structures is the axial next-nearest-neighbor Ising, or ANNNI, model.4 The model illustrates much of the essential physics we shall be interested in, and, in this section, we give a description of its phase diagram with particular emphasis on the features found generically in models with competing interactions. The ANNNI model is an king model with a two-state spin, Si = k 1, on each lattice site. Interactions are between nearest neighbors, together with a second-neighbor interaction along one lattice direction which we shall call the axial direction, z. Thus the ANNNI model is defined by the Hamiltonian

where i labels the layers perpendicular to the axial direction and j and J' are nearest-neighbor spins within a layer. It is useful to note that the ANNNI Hamiltonian, Eq. (2.1), is unchanged under the transformation S i , j+ - S i . j , i odd; J , -+ - J , (2.2) Hence the phase boundaries are symmetric about 5, = 0 and the phases for J , < 0 can be determined if those for J , > 0 are known by flipping every alternate layer of spins. To identify the regions of the phase diagram where the competition between J , and J , becomes important, it is helpful to look at the ground state. Note first that within the layers the ordering is always ferromagnetic and that therefore we need only consider the axial direction. The axial ground-state configurations are shown as a function of J1 and J , in Fig. 1. For J , > 0 there is no competition and the ordering is ferromagnetic or antiferromagnetic for 5, > 0 and J1 < 0, respectively. For J , < 0, however, the second-neighbor interaction prefers an antiphase configuration, ... t t l l . . . or (2), whereas the firstneighbor interaction prefers a simple ferro- or antiferromagnetic state. The former dominates if IJ,/J,I < 2 and hence there are three ground-state phases for J , < 0. The boundaries separating these phases have been termed multiphase line^^.^ because on each of them the ground state is infinitely degenerate. On J 1 / J 2 = - 2, J , < 0, between (2) and the ferromagnetic state, (a),any phase R. J. Elliott, Phys. Riw. 124, 346 (1961).

' M. E. Fisher and W. Selke, Phys. Reu. Lerr. 44, 1502 (1980). M. E. Fisher and W. Selke, Philns. Trmy. R. Snc. London 302, 1 (1981).

154

JULIA YEOMANS

FIG. I . Ground state of the ANNNI model. ---, multiphase boundaries.

containing bands of length two or more has the same energy. Here the term band is used to describe a sequence of layers of the same spin value, S, terminated by layers of value - S. This is tantamount to saying that any state that can be composed of lengths of the antiphase state, ( 2 ) , interspersed with lengths of the ferromagnetic state, is degenerate. Similarly, on the boundary J I / J z = 2, J , < 0, between the antiferromagnetic ground state, (l), and ( 2 ) , any phase which contains only- and two-bands is degenerate. In order to describe which of the degenerate ground states remain stable at finite temperatures, it is helpful to introduce a notation which distinguishes between the different axial orderings. We follow Fisher and Selke5-6in taking ( n , , n 2 , . . . ,n,) to represent a state in which the repeating sequence consists of rn bands of length n , , n 2 , . . . ,it,,,. For example,

... Trllrrlll...

(2.3)

will be denoted ( 2 2 2 3 ) or ( 2 3 3 ) . This ties in with our previous choice of (a), ( I ) , and ( 2 ) to describe the ferromagnetic, antiferromagnetic, and antiphase states, respectively. We now turn to a description of the phase diagram for finite temperatures. The most prominent phases are shown in Fig. 2. The results follow from a mean-field treatment of the ANNNI model on a cubic lattice first carried out by Bak and von Boehm7 and later extended by Selke and D ~ x b u r y . ' . The ~ modulated phases lie in a region of the phase diagram, bounded by (a),( 2 ) , and the paramagnetic phase, which springs from the multiphase point and increases in width with increasing temperature. The modulated region is

' P. Bak and J. von Boehrn, Phys. Rev. B.21, 5297 (1980). ' W. Selke and P. M. Duxbury, Z. Phys. B: Condens. Matter Quunta 51,49 (1984). P. M. Duxbury and W. Selke, J . Phys. A: M a t h . Gen. 16, L741 (1983).

155

THEORY AND APPLICATION OF AXIAL ISING MODELS paramagnetic

I

0

I

0.2

I

I

I

04

06

I

I

0.8

I

I

I

1.0

k=-Jz/JI

*

FIG.2. Mean-field phase diagram of the ANNNI model showing the main commensurate p l ~ a s e s . ’ ..., ~ ~ an estimate of the boundary above which incommensurate phases are found between the commensurate phases.I3

dominated by the shorter period phases, notably ( 3 ) . The modulated, paramagnetic, and ferromagnetic phases meet at a Lifshitz point.” The phase diagram in Fig. 2 probably best represents what would be seen experimentally: lock in to a few short-wavelength modulated phases separated by first-order phase transitions or by regions where the wave vector appears to vary continuously. However, with better resolution many more commensurate phases would be seen to have a finite field of stability, and it is now believed that infinite sequences of phases appear throughout the phase diagram. These rapidly become very narrow with increasing wavelength, but appear in regular patterns characteristic of models with competing interactions. We shall first consider the stable phases at low temperatures and then describe how new phases become stable as the temperature is raised. Low-temperature series expansions have enabled the phase sequence in the vicinity of the multiphase point to be ~ a l c u l a t e d . ~ -Results ~ ~ ” obtained by Szpilka and Fisher are shown in Fig. 3. An infinite, but very specific sequence of phases, ( 2 k 3 ) , springs from the multiphase point. (Throughout this article we shall take k to represent the sequence of non-negative integers, 0, 1, 2,. . .) The width of successive phases decreases exponentially with increasing lo ‘I

R. M. Hornreich, M . Luban, and S. Shtrikman, Pkys. Rea. Lerl. 35, 1678 (1975). A. Szpilka and M. E. Fisher, Phys. Rev. Le//. 57, 1044 (1986).

156

JULIA YEOMANS

1 k 2 FIG.3. The ANNNI phase diagram at low temperatures in the vicinity of the multiphase point for J , > 0." Mixed phases, (2'32""3), are stable in the shaded regions.

k. Phases with successively smaller k cutoff as the temperature is increased, and hence the boundary between ( 2 ) and the modulated region is weakly first order. Mixed phases, (2k32k' '3), appear at a temperature which decreases with increasing k , and more complicated combinations may also be stable.' As the temperature is increased, more phases appear in a systematic way through what has been termed in the literature "structure combination branching processes".s79 In all the cases tested so far, within mean-field theory and low-temperature series, the first new phase to appear between two neighboring phases ( a ) and ( h ) is always ( a b ) . Further increases in temperature will cause ( u 2 h ) and ( a h 2 ) to appear on the ( a ) : ( & ) and ( a h ) : ( h ) boundaries, respectively. Hence sequences of new phases are built up, a typical one of which is shown in Fig. 4. Selke and Duxbury8 checked several series of branching points, and in each case the sequence appeared to extrapolate to an accumulation point below the transition temperature to the paramagnetic phase, T,. Above the accumulation points one would expect to find commensurate phases with wave vectors corresponding to every rational number within a given interval. As T, is approached, the widths of the commensurate phases vanish as power Therefore, above some temperature incommensurate phases must become stable between the commensurate phases.' This behavior is referred to as an I*

A. Aharony and P. Bak, Phys. Riw. B 23,4770 (1981).

THEORY A N D A P P L I C A T I O N OF AXIAL ISING M O D E L S

157

(3)

* increasing temperature

FIG.4. Schematic representation of a typical structure combination branching sequence

incomplete Devil's staircase.' Some insight into the temperature at which incommensurate phases first appear follows from considering where the energy of interaction between domain walls (where a domain wall can be thought of as a 3-band in the (2) phase) dominates the energy, pinning the modulated structures to the l a t t i ~ e . ' ~The . ' ~ result of a mean-field estimation of this boundary is shown in Fig. 2. It is important to remember, particularly when using the ANNNI model to represent modulated structures in real systems, that there are an infinite number of metastable states at low temperatures near the multiphase point. Indeed, each stable phase persists as a metastable state when its free energy ceases to provide the global minimum. Free-energy differences between the different metastable states and the stable phase itself can be arbitrarily small. is This point, which shows up clearly in mean-field solutions of the of vital importance in the kinetics of modulated structures.

'' I4

M. H. Jensen and P. Bak, P / I ~ . F RCCL . B 27, 6853 (1983). P. Bak and V. L. Pokrovsky, Pkq's. Rev. Left. 47,958 (1981).

158

JULIA YEOMANS

4 la)

lb)

FIG.5. Possible variation of the interaction, I , , between domain walls with distance 1.

At this point, leaving details of the theoretical work on the ANNNI phase diagram to the following section, we emphasize some important aspects of the physics which leads to modulated phase sequences. We first stress the importance of entropic contributions to the free energy. Because energy differences are small in the vicinity of the multiphase point, the entropy plays the dominant role in determining the stable phases at finite temperatures. Entropic effects manifest themselves as fluctuations in the domain walls, where domain walls are most usefully considered to be the structures which destabilize a given commensurate phase (3-bands in the vicinity of (2); interfaces between up- and down-bands in the vicinity of (a)).As the walls fluctuate they impinge on each other's movement and hence interact.""5 Consider first only pair interactions between nearest-neighbor walls. Following Szpilka and Fisher,' it is useful to distinguish the two cases shown in Fig. 5: (a)the interaction is always repulsive; (b) the interaction has a unique minimum at some spacing, I,. If only pair interactions are taken into account, the walls are always equidistant at spacing 1. In case (a), although it may become favorable, because of a negative wall self-energy, to create walls, they Is

J . Villain and M. Gordon, J . Phys. C 13, 31 17 (1980).

THEORY AND APPLICATION OF AXIAL ISING MODELS

159

must overcome the repulsive interaction. Hence 1 decreases monotonically from 00, and the transition from the commensurate phase is quasicontinuous. In case (b), however, 1, provides the most favorable spacing, and the transition from the commensurate phase will be first order to a structure with wall spacing I,. The transition from (2) is Case (b) is relevant for the ANNNI weakly first order with l,, the spacing between the three-bands on the boundary, increasing with decreasing temperature to give the onion effect in Fig. 3. Similarly, near ( a ) ,where walls are boundaries between consecutive bands, I , = 3, giving an ( a ) : ( 3 ) transition (I, varies with increasing temperature and decreasing J, to give different transitions). Szpilka and Fisher' have pointed out that the appearance of mixed phases depends on multiwall interactions. As various combinations of these change sign, new phases appear through structure combination branching processes. It is also of interest to note that the domain wall interactions decay exponentially with distance. This results in the exponential decrease of phase width with increasing period. The calculation of the interaction between domain walls is discussed in Sections 3,b and 3,c.

'

3. THEORETICAL TECHNIQUES a. Mean-Field Theory Mean-field theories have proved to be a very successful tool in the study of models with modulated ~ r d e r . ~Phase , ~ , ~ diagrams obtained using such approximations appear to be correct in all but minor details. Indeed Szpilka and Fisher' have studied low-temperature series expansions for general coordination numbers and found that, for the ANNNI model, the mean-field limit agrees with results for lower dimensions. In this section we describe the more conventional theories and then, in Section 3,b, we summarize results due to Villain and Gordon,I5 who have used a mean-field approach to emphasize the role of interactions between domain walls in the formation of the ANNNI phases. Mean-field theory was first applied to the ANNNI model by Elliott4 He allowed only solutions where the magnetization varied sinusoidally and found that the wave vector varied continuously with K = IJ2 I/J1but was independent of the temperature. Bak and von B ~ e h m ' . ' were ~ the first to demonstrate the existence of a large number of commensurate phases. Later Duxbury and Selke8*9gave a more detailed picture of certain aspects of the mean-field phase diagram, emphasizing the role of structure combination branching processes.

'

l6

J. von Boehm and P.Bak, Phys. Rev. Lett. 42, 122 (1979).

160

JULIA YEOMANS

The essential difference between the mean-field theory of a model exhibiting must be modulated order and a simple ferromagnet is that the mean field Hi, allowed to vary from layer to layer. The mean-field free energy, F,, ,follows as usual from the Bogoliubov inequality' F,

= min(Fo

+(H

-

(3.1)

Ho)o)

where

In this equation the sum is over all spin configurations, { S i , j } ,and H , is a trial Hamiltonian chosen to be H,

=

-c

(3.3)

i,j

Hence the mean field, H i , appears as a variational parameter associated with the layer i. Minimizing Eq. (3.1)with respect to the Hi leads to mean-field equations for the average magnetization per layer Mj = tanh/?[4JoMj

where p

=

Np2F,,

+ Jl(Mjpl + Mj+l) + J2(MjP2+ M j + * ) ]

(3.4)

l/k,T, and for the free energy =

+ Mj)In(l + M j ) + (1 - Mj)ln(l - Mj)] - $1 [4JOMj2+ J , M j ( M j - , + M j + l ) -NkTln2 + $ T I [(l j

j

+ J,Mj(Mj-2 + Mj+2)1

(3.5)

where there are N spins in the system. The tanh in Eq. (3.4) may be linearized for small M ito obtain the transition line between the paramagnetic and ordered phase^.^ One obtains k,T,(K)

= 450 = 45,

+ (2 - 2K)Jo, K < S + ( 2 +~ 1/4~)J,, K 2 a 1

(3.6) (3.7)

for the boundary to the ferromagnetic and modulated phases, respectively. Near the boundary, for K > $, the modulated phase is well approximated by a sinusoidal variation of the magnetization with critical wave vector qc = C O S - ' ( ~ / ~ K )

(3.8)

Inclusion of umklapp terms indicates that the commensurate phases still " H.

Falk, Am. J . Phys. 38, 858 (1970).

THEORY AND APPLICATION OF AXIAL ISING MODELS

161

exist in this region with widths that vanish as power laws as T + T,.7 This conclusion has been confirmed using the renormalization group." At lower temperatures Eq. (3.4) admits, in general, a large number of solution^.^-^ That with the lowest free energy corresponds to the thermodynamically stable phase, with other minima corresponding to metastable states. To find the free energy of a structure of length L the mean-field equations are iterated on a lattice of length L (or 2L if there is an odd number of bands) with periodic boundary conditions. The initial conditions are typically taken to be the zero-temperature value of the magnetization in the phase under, consideration, and, in general, for stable or strongly metastable states, the fixed point achieved has the same periodicity but a reduced magnetization per layer which depends on the imposed temperature. The free energy of the trial structure is then calculated and the process repeated for as many structures as necessary. In theory, an infinite trial set is required to be sure of finding the stable phase. This is obviously impossible, and two main approaches have been used in the literature. In the original calculations7 all structures with period up to L = 17 were considered. This allowed a study of the overall features of the phase diagram and pinpointed the dominant phases that would be observed in an experiment. A second approach has been to assume, on the basis of the available numerical evidence and low-temperature series results,6 that new phases only appear through structure combination branching processes.6g9This allows a more systematic study of the fine detail in chosen regions of the phase diagram. Selke and D u x b ~ r yhave ~ . ~used these ideas to show that sequences of branching temperatures converge on an accumulation point. An interesting approach to the mean-field equations of the ANNNI model is to rewrite them as an iterated r n a ~ . ' ~ It ' ' ~is possible to approximate the equations using a two-dimensional mapping" or to study the full fourdimensional space (3.9) dire~t1y.I~ Limit cycles of the map correspond to commensurate ANNNI phases, and one-dimensional smooth invariant trajectories are identified as incommensurate phases. The physically stable states of minimum free energy correspond to unstable orbits of the map, rendering the numerical work difficult. Chaotic trajectories, which describe metastable states with randomly pinned domain walls, have also been observed. By searching for the appearance of one-dimensional invariant trajectories with increasing temperature, Jensen and Bak l 3 were able to approximate the line in the phase diagram above which incommensurate phases start P. Bak, Phys. Rev. Lett. 46,791 (1981).

162

JULIA YEOMANS

appearing. Their estimate is shown by a dotted line in Fig. 2. They argue that incommensurate phases can appear when the interaction between walls dominates the energy pinning them to a particular position on the l a t t i ~ e . ' ~ . ' ~ These quantities can be calculated numerically within mean-field theory, and equating them gives a boundary for the appearance of incommensurate phases in good agreement with that found from the iterated map. b. Interactions between Domain Walls

Villain and Gordon studied the ANNNI model using an approach which pinpoints particularly clearly the physical reasons for the existence of the modulated phases. At low temperatures, within a mean-field approximation, they were able to map the model onto a one-dimensional array of interacting domain walls. They showed that the interactions between the walls are long range and oscillatory [type (b) in Fig. 51 and used this to predict the behavior near the multiphase point. In particular, as outlined in Section 2, this approach gives a clear picture of the reason for the initial instabilities of the (co)and (2) phase boundaries. Very recently Szpilka and Fisher have shown that, with minor corrections arising from the effect of three-wall interactions, low-temperature series expansions give the same results. They point out that correction terms render the mean-field approach correct only in the intermediate temperature, anisotropic region qlJo >> k,T >> Icq,Jl, where qL and q1 are the coordination numbers for bonds in and between the layers, respectively. However, the more general low-temperature series analysis confirms that the results remain true as T + 0. The mean-field approximation is obtained as before by considering interactions between the average magnetization (per spin) in each layer, mi.15 This is allowed to depart from its zero-temperature value, a, = 1, by ui, giving a Hamiltonian

''

(3.10) i

i

where the last term ensures that the ui remain small. Equation (3.10) may be Fourier transformed to give, after some algebra, a trivial harmonic term together with an effective free energy I

F

=-

1Azf(j

-

i)aioj

(3.11)

i+j

where (3.12)

THEORY AND APPLICATION OF AXIAL ISING MODELS

163

with P(X) = X2

+ i(Jl/Jz)X - f - A/452

(3.13)

playing the role of a long-range oscillatory interaction between Ising spins. To render the physics of the problem transparent, Eq. (3.1 1) can be rewritten in a form where the variables in the free energy are the position, xp,of the pth domain wall, where we consider first the situation near (a)where a domain wall is the boundary between consecutive bands. The resulting free energy’ I

F

-

=

Fo

+ no,-

(3.14)

contains three terms. The first is the free energy of the ferromagnetic state. The second is the contribution from n walls each of free energy W

1 mf(m)

o1= 4A2

(3.15)

m=l

and the third with m

1 m f ( j + m) m= 1

u ( j )= -8A2

(3.16)

describes the interaction between rth-neighbor walls. The mechanism driving the instability of the ferromagnetic phase can be understood by considering just first-neighbor wall-wall interactions. In this case U(1) = - 8 J 2 > 0

U ( 2 )= - 16(Jl

+ J 2 ) J 2 / A> 0

U(3) = -8J$/A < 0

U(m)-0,

(3.17)

m> 3

at low temperatures. Therefore, we have case (b) of Fig. 5 with 1, = 3, and (a) destabilizes directly to (3) through a first-order transition. It is interesting to compare this argument with a similar analysis of the (2) phase boundary. The destabilizing walls are now three-bands, and Villain and GordonI5 show that Eq. (3.11) can be rewritten in a way analagous to Eq. (3.14) in terms of the free energy of such defects, together with the interaction between them. The nearest-neighbor interaction is positive for wall separations (3.18)

and becomes negative for larger 1 and then oscillates. Hence the (2) phase

164

JULIA YEOMANS

transforms through a first-order transition to a state with a value of 1, 1,, corresponding to the deepest minimum of the interaction potential which is found to beI5 21,

+1

N

2 n [ 5 ( 1 - +Jl/\J2\)]-1’z

(3.19)

Note that 1, increases as the multiphase point is approached, giving the sequential cutoff of the phase sequence ( 2 k 3 ) at the ( 2 ) boundary. In addition to determining the first instabilities of the (a)and ( 2 ) phases, it is possible to use Eq. (3.14) to establish the infinite phase sequence near the multiphase point. The analysis is similar in spirit to the FisherSelke6 low-temperature expansion with J J A playing the role of the expansion parameters. This calculation is not explicitly laid out in the literature, but similar models have been analyzed in this To conclude, we emphasize that, within a low-temperature mean-field theory, the ANNNI model can be rewritten in terms of a one-dimensional array of interacting domain walls. This point of view will be useful when we consider physical applications of the model in Section 11. c. Low-Temperature Series

A second theoretical method that has been important in establishing the phase diagram of the ANNNI model is low-temperature series expansions. These have been carried out near the multiphase point by Fisher and Selke.5-6 The difficulty in applying series techniques to this problem is that, because phases of all lengths are stable, all orders of the expansion are important. For example, the phases ( 2 k 3 ) and ( 2 k - ’ 3 ) are only distinguished by graphs of k + 1 spin flips. However, Fisher and Selke5*6showed that, by picking out the important terms at each order of the series expansion, it is possible to build up the sequence of phases inductively. Because of the degeneracy at the multiphase point, it is necessary to expand about all possible ground states. These are distinguished by a set of first-order structural variables, l k , the number of times per spin a band of length k appears in a given state. For example, for the structure ( 2 3 3 ) , 1, = $, l3 = $, and lk = 0, k # 2,3. The structural variables must be non-negative and are related through the constraint

1klk = 1

(3.20)

k

E , , the ground-state energy per spin, can be written in terms of the structural

l9

J. M. Yeomans, J . Phys. C 17,3601 (1984). M. Siegert and H. U. Everts, Z. Phys. B: Condens. Mutter Quanta 60,265 (1985).

THEORY AND APPLICATION OF AXIAL ISING MODELS

165

variables

t.

where 6 = K The low-temperature series expansion for the reduced free energy about a given ground state is, as usual,21

where N is the number of lattice sites and AZN(m,{Ik}) is the contribution to the free energy from configurations obtained from the ground state by flipping rn spins which, by the linked cluster theorem, will be linear in N . Defining the Boltzmann factors w = exp(-2Ko),

K i= BJi,

x i

= exp(-X,),

x-’’’-’

- exp(-2K2),

= 0, 1,2

(3.23)

and using Eq. (3.20) to eliminate l!,, the reduced free energy is given to first order by f{Ik}

= iq1KO

+

)K1

a2(6)12

K , 6/3

+ 1ak(6)kl!k,

+ 3(2 + X 3 + ”

) wq1

k24

(3.24)

where the structural coefficients

az(6) = 4K1 6/3 - 3(2 - 3 ~ ’ ” ’

+ x 3 + ” )w4i + O ( W ’ ~ ~ - ’ )(3.25)

kak(6) = -4K1 6 ( k - 3)/3 - [2(k - 3)/3 - (k - 4)x1-26 -2~’+~

x ~ + ” / ~+I0w( ~ ~’ 4L1 - ~ ) ,

k >3

(3.26)

The stable phases are found by maximizing f with respect to the l k . As there is a region in which both a2 and ak, k > 3, are negative, the free energy is maximized by taking the corresponding structural variables to be zero. Hence, from Eq. (3.20), 1, = +and the (3) phase is stable over a region O(w%)between (2) and (co). Fisher and Selke5v6studied the (3):( co) boundary in greater detail by taking the series to third order. They were able to ascertain its shape and, by showing that the surface tension on the boundary is positive, to confirm that the transition is first order. Note, however, that the situation on the (2):(3) boundary is completely different. Here, from Eq. (3.24), it is apparent that all 21

C. Domb, Adv. Phys. 9, 149 (1960).

166

JULIA YEOMANS

phases which contain only two- and three-bands remain degenerate. Hence it is necessary to proceed to higher orders in the series expansion to ascertain their stability. To proceed further, one defines a set of higher-order structural variables, l,, which denote the number of times per layer a band sequence v appears in a given structure. For example, for (233), 123 = &, I,, = 6, and so on. The higher-order structural variables are not independent but are related through equations like l2v3

+ /2v2

=

(3.27)

I2v

However, Fisher and Selke5-6showed that it is possible to choose an independent subset of the variables and that the free-energy expansion can be written as a linear function of these standard structural variables. To write down the free-energy expansion to second order, two standard structural variables are needed, I, and I,, . The second-order contribution to the reduced free energy is then given by f‘2’

= Z(@

+ a‘2’1 0 2 + a‘2’1 2 3 2 3)

(3.28)

ayi,

the structural coefficients evaluated to second order, where ab2’, af), and depend on the Boltzmann factors x, 6, and w. The important structural coefficient for our purpose is a2,(4

=

3(i - x 2 ) ( i

- x1+26)2w2ql

+0(~34~-2)

(3.29)

is positive on the boundary between (2) and (3) for x < 1. Hence the reduced free energy will be maximized by taking I, to be as large as possible and the (23) phase will be stable within a region O(wZ41)of this boundary. To proceed further, the new boundaries, (2) :(23) and (23) :(3), which remain infinitely degenerate, must be examined at higher orders of the series expansion. Let us now sketch the general order calculation. Consider the kth order of the series expansion, where the phase (2k-13) has just appeared as a stable phase between (2k-23) and (2). Fisher and Selke5p6were able to show that the first possible instability of the new boundaries (2k-13):(2k-23) and (2k-13):(2) are to (2kp’32k-23) and (2k3), respectively. Tocheck whether these phases are, in fact, stable, it is necessary to calculate the coefficient of the corresponding structural variable in the expansion of the free energy. If this is positive, the new phase will appear; if negative, the phase boundary will remain stable to all orders of the expansion. Calculating the coefficients is a matter of identifying the important graphs in the expansion of the free energy. The relevant graphs correspond to those one would intuitively expect: linear connected (that is, first- or secondneighbor) chains of spin flips along the axial direction, together with the

~23(6)

THEORY AND APPLICATION O F AXIAL ISING MODELS

167

corresponding disconnected configurations. A moment's thought shows that these are the lowest-order graphs which can distinguish between different axial orderings. The phase (2k3) is stabilized at order k + 1 because this is the number of connected spins that must flip to span the sequence. Because the important structural coefficients correspond to linear graphs, they are most easily calculated using a transfer matrix method,"'22 in which bonds are added one at a time, together with their Boltzmann factors and those of the corresponding disconnected configurations. For the ANNNI model one finds to leading order a 2 k - 1 3 2 k -=2 3-x'-26(1

) - 2W(2k-3)¶,

- x 1 + 2 6 2k

which is negative, and hence the (2k-'3):(2k-23) However, aZk3= ( k

+ 2)(1 - x2)(1- x l + ")k + l

(3.30)

boundary is stable. w (k+l)qi

(3.31)

+

is positive and, to order k 1, (2k3) appears between (2k-1 3) and (2). The analysis must then recommence about the new boundaries. Thus an inductive argument shows that the phase sequence (2k3) springs from the multiphase point of the ANNNI model. The width of the phase (2k3) is O ( W ( ~ + ' ) ~ , ) . It has recently been realized'' that rather subtle correction terms in Eqs. (3.30) and (3.31), which mathematically show up as a degeneracy of the eigenvalues in the transfer matrix used to calculate the coefficients, lead to oscillations for large k. Hence the (2k3) phase sequence is not infinite at any finite temperature and the (2):(2k3) boundary is weakly first order, as first pointed out by Villain and G0rd0n.l~Moreover, mixed phases, such as (2k32k+13) and possibly more complicated structural combinations, appear at low temperatures for large enough k. This is illustrated in Fig. 3. Finally we note that a calculation of the structural coefficients is tantamount to a calculation of pair ( ~ 2 k 3 )and three-wall ( a 2 k 3 3 2 k + 1 3 ) interactions, and the arguments given above can be reformulated in terms of such interactions.

d . Other Approaches ( i ) High-Temperature Series. It is possible to gain some insight into the behavior of the ANNNI model in the vicinity of the paramagnetic-ordered phase boundaries using high-temperature series. OitmaaZ3has recently analyzed series of l l terms for the three-dimensional ANNNI model extending

22 23

J. M. Yeomans and M. E. Fisher, Physica A (Amsterdam) 127, 1 (1984). J. Omitmaa, J . Phys. A : Math. Gen. 18, 365 (1985).

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JULIA YEOMANS

earlier work by Redner and S t a n l e ~ . ~ The ~ , ' ~transition temperature is decreased by about 25% from the mean-field value, and the Lifshitz point occurs at 1J2/J11 = 0.270 A 0.005, in close agreement with the mean-field result of b. Values for the susceptibility exponent, however, do not show the expected crossover from Ising to X-Y-like behavior as K is increased beyond the Lifshitz point.23 ( i i ) Monte Carlo. Montz Carlo work on systems with many modulated phases is beset with difficulties. The most evident of these lies in the impossibility of choosing a lattice size and boundary conditions along the axial direction26 which do not effect the periodicity of the modulation. Moreover, at lower temperatures, the kinetics are sluggish and the system is very likely to get stuck in a metastable state. Early Monte Carlo work went some way in considering the temperature dependence of the wave vector but was able to find no direct evidence of any discontinuous variation.26-28 At higher temperatures Monte Carlo a p p r o a c h e ~ ~have ~ , ~ 'been used to find the transition line to the modulated phase and the position of the Lifshitz point, giving results in good agreement with the high-temperature series expansion^.'^ The values of the critical exponents near the Lifshitz point have also been ~tudied.~' Although it is very difficult to use Monte Carlo algorithms to probe the details of the modulated structures in the ANNNI model, some consideration of the kinetics of phase changes between the different equilibrium phases may prove more fruitful.

4. RELATED MODELS

The main motivation in studying systems which are extensions and modifications of the ANNNI model has been to understand the extent to which the results are applicable to experiment. For example, calculations for the ANNNI model in a magnetic field were undertaken in an attempt to explain neutron scattering results for the cerium m o n ~ p n i c t i d e s .A~ ~study of the

S. Redner and H. E. Stanley, J . Phys. C 10,4765 (1977). S. Redner and H. E. Stanley, Phys. Rev. B 16,4901 (1977). 26 W. Selke and M. E . Fisher, Phys. Rev. B 20,257 (1979). 27 W. Selke and M. E. Fisher, J . Mayn. Magn. Mater. 15-18,403 (1980). 2 8 E. B. Rasmussen and S. J . Knak-Jensen, Phys. Rev. B 24,2744 (1981). 29 W. Selke, 2. Phys. B: Condens. Matter Quanta 29, 133 (1978). 'O K. Kaski and W. Selke, Phys. Rev. B 31,3128 (1985). V. L. Pokrovsky and G . V. Uimin, J. Phys. C 15, L353 (1982). 24 25

''

THEORY AND APPLICATION OF AXIAL ISING MODELS

169

effect of third-neighbor and defects34 aimed to understand whether the generic properties of the ANNNI phase diagram can be expected to apply to polytypic compounds and binary alloys where these features must be present. In this section we review results on ANNNI-like models. We whall first concentrate on the theoretical results. Applications to real systems will be discussed extensively in the second half of the review. a. The ANNNI Model in a Magnetic Field

ha ve studied the effect on the phase diagram of Several adding a field term, - HE,,Si,to the ANNNI Hamiltonian, Eq. (2.1).Because the field breaks the symmetry under spin reversal, it is necessary to refine the Fisher-Selke notation5 to describe the stable phases. Let bands of spin S = - 1 be identified by a horizontal bar above the number describing the band width. For example, the states ...T11TJ1...and * . . l t T J T T . . - are described by (12) and (i2), respectively. The ground state of the ANNNI model in a field can be determined by careful inspection35or by linear p r ~ g r a m r n i n gThe . ~ ~ result is shown in Fig. 6 for a value of H > 0. For J1 > 0, the phases (22) and (03) remain the sole ground states as the field is applied. For J1 < 0, however, the situation is more complicated with a new phase, (72), being stable even at zero temperature. Results for H < 0 follow immediately by reversing the sign of each spin. Yokoi et al.35 were the first to study the effect of a field on the ANNNI model phase diagram. They used mean-field theory and concentrated mainly on the region near the paramagnetic-modulated phase boundary, modeling the ordered state as a sine wave. They established that the transition temperature is depressed quadratically with the applied field and that the wave vector near the transition depends only on ic, not on field or temperature. It is interesting to note that, within this approximation, the boundary between the modulated and paramagnetic phases changes from second to first order as the field increases. Yokoi et aL3’ did not systematically study the lock-in to W. Selke, M. N. Barreto, and J. M. Yeomans, J . Phys. C 18, L393 (1985). M. N. Barreto and J. M. Yeomans, Physica A (Amsterdam) 134,84 (1985). 34 H. Roeder and J. M. Yeomans, J . Phys. C 18, L163 (1985). 3 5 C. S. 0.Yokoi, M. D. Coutinho-Filho, and S. R. Salinas, Phys. Rev. B 24,4047 (1981). 36 V. L. Pokrovsky and G. V. Uimin, Sou. Phys.-JETP (Engl. Transl.) 82,1640 (1982). ” H. C. Ottinger, J . Phys. A: Math. Gen. 16, 1483 (1983). 38 J. Smith and J. M. Yeomans, J . Phys. C 15, L1053 (1982). 39 J. Smith and J. M. Yeomans, J . Phys. C 16,5305 (1983). 40 G. V. Uimin, J . Stat. Phys. 34, 1 (1984). 41 A. M. Szpilka, J . Phys. C 18,569 (1985). 4 2 G. V. Uimin, Piz’ma-JETP 36,201 (1982). 32

33

170

JULIA YEOMANS

FIG.6. Ground state of the ANNNI model in a magnetic field for H > 0.

commensurate phases at lower temperatures. However, they determined that a large number of commensurate phases do exist and pointed out that those surviving to finite field are, in general, those with a finite magnetization. The mean-field theory of the ANNNI model in a field has also been studied by Ottinger.37 A more thorough study of the low-temperature region of the phase diagram using a low-temperature series expansion analogous to that applied to the zero-field problem by Fisher and Selke5g6has been undertaken by several a ~ t h o r s . ~ Pokrovsky ~-~' and Uimin36*42 have also addressed the problem of the phase diagram at low temperatures using a different expansion technique. They assumed the dominance of the coupling in the layers, J1,J, << J o , and then performed a cumulant expansion in the interlayer Hamiltonian. The behavior near each of the multiphase lines is c ~ m p l i c a t e d . ~For ~-~~ those readers interested in the details, we describe the behavior near each line below, but it is first useful to give a summary of the main resullts and to point out some unusual features of the phase diagram. In general the ANNNI model in a magnetic field behaves in a qualitatively similar manner to the zero-field case. In particular, there are several sequences of modulated phases near the multiphase lines which are generated by the usual structure combination branching processes. As one might have guessed, those of the zero-field phase sequences with a finite magnetization persist as the field is increased, giving phase sequences like ((22)k(23))and (T2(Z2)k).Uimin40 argues that, as the inplane interactions become less dominant, the harmless Devil's staircases are replaced by complete staircases with all commensurate phases appearing. It is

'

THEORY AND APPLICATION OF AXIAL ISING MODELS

171

interesting to note that unusual phases made up from 72 and 13 units appear between ( G O ) and (12). In more detail, results for specific phase boundaries are as follows: J1 > 0 ( i ) ( a ) : ( 2 2 ) Boundary. As a positive magnetic field is switched on, the multiphase sequence existing in zero field, (2k3), crosses over to ((22)k23).38-42The disappearing phases cut off at a field of order the zerofield phase ~ i d t h . The ~ ' sequence is replaced by a direct first-order transition between (GO) and (22) for J1 N 1J21. Uimin has argued that for sufficiently small Jo the harmless staircase becomes a complete ~taircase.~' J1 < 0 ( i ) (11):(12) Boundary. There is no splitting on this boundary for sufficiently large J,. However, for ).I2)> Jo, the phase sequence ((T2)k"T1) appears. (ii) ( 2 2 ) : (12) Boundary. The phase sequence (T2(22)k") is stable. Uimin4' argues that this crosses over to a complete staircase as Jo is decreased. (iii) (a):(12) Boundary. The situation on this boundary is not entirely clear. Structures made up from (12) and (13) band sequences are stable in certain regions of the parameter space.36440 The main application of these results, to explain the phase diagram of cerium antimonide, is described in Section 8. b. Further-Neighbor Interactions It is of interest both academically and in preparation for applying models with competing interactions to experimental systems to consider the effect of the further-neighbor interactions that must inevitably be present in reality. The most obvious extension of the ANNNI model is to an Ising model with both second- and third-neighbor interactions along the axial direction. This has been studied by Selke et al.32 using mean-field theory and by Barreto and yeoman^^^ using low-temperature series. The main features of the phase diagram are understood and are qualitatively similar to those described for the ANNNI model. However, there are quantitative differences in the particular sequence of phases stable near a given multiphase line. The ground state of the axial Ising model with second- and third-neighbor interactions is shown in Fig. 7 for J1 > 0.43Three of the ground-state phase

43

M. N. Barreto, Ph.D. thesis. Univ. of Oxford, Oxford, England, 1985.

172

JULIA YEOMANS

containing only 1- and 2- bonds

(W I

2 for J,

>o,

(3"'

J,IJq

L)

(1) FIG.7. Ground state of the axial king model with third-neighbor interactions for J , phase boundaries shown in bolder type are multiphase lines.

0. The

boundaries, shown in bold type in the figure, are multiphase lines close to which one might expect long-wavelength phases to be stabilized. Results for J1 < 0 can be obtained immediately from those for J1 > 0 using the transformation Eq. (2.2) together with J3 + -J 3 , and therefore we concentrate our attention on the latter case. We first describe the low-temperature series results,33which predict which phases spring immediately from the multiphase point. O n the boundary between ( 0 0 ) and (3) the phase sequence (3k4) is stable. For J2 > 0 the phases (3k43k+'4) also appear. Between (2) and (3) the sequences (23k'1) and (23k23k+1)are stable at the lowest temperatures. On the (12):(2) phase boundary many phases containing one- and two-bands, but no neighboring one-bands, spring from the multiphase point: here the situation is too complicated for the phase sequences to be identified using low-temperature series or mean-field theory. These results are summarized in Fig. 7. Mean-field theory has been used to obtain information about the phase diagram at higher temperature^.^^ In all the regions studied new phases were generated as the temperature was raised through the same structure combination branching processes seen in the ANNNI model. An interesting detail is that certain phase sequences cut off with increasing temperature and then reappear. For example, for small J3 < 0, the (3k4) sequence behaves in this way. The points where the phases disappear and reappear coalesce as J3 becomes more negative.

THEORY AND APPLICATION OF AXIAL ISlNG MODELS

173

c. Annealed Vacancies

Again with a view to assessing the relevance of models with competing interactions to real systems, Roeder and yeoman^^^ studied the effect of annealed vacancies on the phase diagram of the ANNNI model. Impurities are known to have a significant effect on the stability of long period

phase^.^^,^^ Vacancies were introduced through a variable ti = 0, 1 on each lattice site, giving the Hamiltonian

The last term is a chemical potential which controls the number of impurities present in the system. Using the transformation oi,j= ~ ~ , ~where t ~ ,oi,j ~ is , a spin-1 variable, the Hamiltonian in Eq. (4.1) can be rewritten as

where D' = D - k,T In 2. The phase diagram of this Hamiltonian was investigated using mean-field theory34for D' < 0, where the ground state is the same as that of the ANNNI model, with no layers with oi,j= 0. Within mean-field theory the shape of the boundary of the paramagnetic phase is independent of the value of D', although it moves to lower temperatures as expected as an increasing number of defects inhibit the ordering. Similarly the Lifshitz point remains at the same value of IC but moves to lower temperatures with increasing D'. Moreover, the appearance of the modulated phases is essentially unchanged by the presence of defects. The defects order in a way that reflects the modulation in the underlying structure, as shown in Fig. 8 for the phase (23233). The impurities prefer to lie on the boundaries of bands of width greater than two, where they increase the energy of the lattice least. A very good description of the distribution of impurities, especially at lower temperatures, is given by

ni = eXp(Ei/kBT )

(4.3)

where Ei is the difference between the energy of a spin and the energy of an 44 45

J. P. Jamet and P. Leaderer, J . Phys., Let6.44, L257 (1983). N. W. Jepps, Ph. D. thesis. Univ. of Cambridge, Cambridge, England, 1979.

174

JULIA YEOMANS

C, (23233)

0.25

0 2(

0.15

0.10

0 or

0 O(

FIG. 8. Distribution of annealed impurities in the (23233) phase of the ANNNI model. Ciis the percentage impurity concentration in layer i.46

impurity in the ith layer. At higher temperatures entropic factors tend to spread the impurities more uniformly throughout the lattice, but this effect causes only a small deviation from a Boltzmann distribution even close to the paramagnetic phase. By the addition of a term - K , C a ~ j 5 ~ + Iit, jis possible to introduce interactions between the vacancies in the system. However, this again leads to no qualitative change in the phase diagram.46 Further work to study the effects of different sorts of impurities on the modulated structures would be of interest.

d. Quenched Impurities Assessment of the stability of modulated structures in the presence of quenched impurities is a difficult problem, but some attempts have been made in this direction. Fishman and Yeomans4’ studied the effect of quenched 46 47

H. Roeder, Ph.D. thesis. Univ. of Oxford, Oxford, England, 1986. S. Fishman and J. M. Yeomans, J . Phys. C 18,857 (1985).

THEORY AND APPLICATION OF AXIAL ISING MODELS

175

random bonds or sites on modulated phases using domain arguments. Defects may locally favor a different phase which will be stabilized if the increase in free energy resulting from the creation of interfaces can be overcome. The domains then round the phase transition, and, if the rounding is greater than the phase width, the phase will be destroyed. Fishman and Yeomans4’ argue that this is indeed the case for 2 < d < 3 but that for d 2 3 the modulated phases remain stable against domain formation. Bak et however, look at a different mechanism which can affect the long period ordering in three dimensions. They argue that walls (for example, a band sequence 2 k p‘3 in ( 2 k 3 ) ) can lower their free energy by fluctuating in the presence of the impurities which behave like random fields. In three dimensions such walls become favorable, destroying the parent phase, for periods l>lc-l/c

(4.4)

where c is the concentration of defects. Note, however, that the critical length, I,, may typically be rather large. Nonequilibrium effects are likely to be more important in real systems. e. Lattice Structure Nakanishi and Shiba49 have considered a model with competing interactions on a lattice which comprises stacked, two-dimensional, triangular layers. Within the layers there are competing first- and second-neighbor interactions, J1 and J,; along the perpendicular chains the interaction, J,, is ferromagnetic. The ground state of this model in a magnetic field is shown in Fig. 9a. Note that structures with two-dimensional modulation are stable. Nakanishi and Shiba49 concentrated on the boundary between the (2 x 2) and (3 x 3) ground-state structures. O n this boundary it is possible to insert walls separating (3 x 3) domains with no additional energy. The walls form a triangular structure resulting in phases like that shown in Fig. 9b. As the temperature is increased, mean-field calculation^^^ show that a sequence of such structures with different domain wall spacings are stabilized. This is an interesting attempt to study two-dimensional modulated structures in three dimensions. In a second paper on the same model, Nakanishi5* considers the zero-field case. He discusses the possibility of lines of spins within the triangular layer which have zero average magnetization and their effect on the phase diagram. P. Bak, S. Coppersmith, Y. Shapir, S. Fishman, and J. M. Yeomans,J. Phys. C 18,3911 (1985). K. Nakanishi and H. Shiba, J . Phys. Soc. Jpn. 51,2089 (1981). 5 0 K . Nakanishi, J . Phys. SOC.Jpn. 52, 2449 (1983).

48

49

176

JULIA YEOMANS

FIG.9. (a) Ground state of the next-nearest-neighbor king model studied by Nakanishi and Shiba.49(b) Phase with a triangular domain wall configuration. Phases of this type with different wall spacings are stable at low temperatures near the multiphase line between the (3 x 3) and (2 x 2) phases.

5. THECHIRALCLOCKMODEL A second mechanism through which competition can be introduced into a spin Hamiltonian is by including a term favoring chiral ordering. This is the case in the p-state chiral clock m ~ d e l , ~ 'which * ~ ' is defined by the Hamiltonian H = - - :J,

1

C O S2-x( ~ ~ , ~-

ni,j,) - J

271 1cos-((~l~,~ - n i + l , j + A) P

p ij where the ni,j are p-state variables which take values 0, 1 , . . . , p ijj'

51

52

S. Ostlund, Phys. Rev. B 24, 398 (1981). D. A. Huse, Phys. Rev. B 24,5180 (1981).

-

(5.1)

1. The

177

THEORY AND APPLICATION OF AXIAL ISING MODELS

O

.

0

0

0

0 0

0

O 0 0

0

0 0

0

0

0 .

0

O

0

0

.

0

0

0

.

0

.

O

0

.

0

.

0 0

0 0

0

0 .

0

0

0

. 0

0 0

0

0

0

0

0

0

.

0

0

0

0 0

0

0

0

0

0

.

.

0

o o .

0

. 0

.

0 0

.O O

0 0 0

.

0

0

.

0

0 O

.

0

0

0

0

O

. . 0 . . 0

. 0

0

0

0

0 0

o o o . o o o o o . o J o ~ / o \ o o 0

0

O

.

0

0

O . . O . . O 0

0

0

O 0

.

0 0

O

.

0

0

0

0

.

0

0

0

.

0

.

.

.

. 0

0

O o O o . O

.

.

FIG.9. (Continued)

notation used to distinguish between in-layer and axial bonds follows Eq. (2.1). Note that the partition function is invariant under the transformation and reidentifications A+A'=l-A ni,j + ni,j, = (- ni,j

+ i)(mod p)

(5.2)

The phase diagram of the chiral clock model for p = 3 is shown in Fig. As A increases, 01 (and equivalently 12 and 20) bonds become energetically more favorable compared with ferromagnetic bonds, and, at A = $, the ground state crosses over from ferromagnetic to right-handed chiral ordering . . .012012... along the axial direction. A = $ is a multiphase point, and, as for the ANNNI model, small free-energy differences between the degenerate phases lead to an infinite sequence of commensurate phases springing from this point at finite temperatures.22

178

JULIA YEOMANS

(2)

.. .0011220011.. .

1 2

FIG.10. Low-temperature phase diagram of the three-state chiral clock model.’’

The existence of the phase sequences for a cubic lattice was first established by Yeomans and F i ~ h e r ” ,using ~ ~ low-temperature series. The stable phases can be described with a notation analogous to that used for the ANNNI model. For example, (23) represents

. . .0011122000112220011122.. .

(5.3)

For A = the phase (2) must be stable to satisfy the symmetry properties of the Hamiltonian, Eq. (5.2). Between (2) and (a)the stable phase sequence is ( 2 k 3 ) , whereas between (2) and the chiral phase, ( l ) , one obtains ( 12k+’). The transition from the (2) phase is weakly first order, indicating a wall interaction that behaves as shown in Fig. 5b. (Whereas for the ANNNI model the cutoffs of the phase sequence near the (2) boundary are tricky to establish using low-temperature series,” the effect follows immediately for the chiral clock model from an obvious change in the sign of the structural coefficients.z2) Yeomans 5 4 has reported the low-temperature series expansion for the chiral clock model for higher values of p . It is difficult to obtain an analytic expression for the structural coefficients as p increases, but they can be ” J . M. Yeomans and M. E. Fisher, J . Phys. C 14, L835 (1981). 5 4 J. M. Yeomans, J . Phys. C 15,7305 (1982).

THEORY AND APPLICATION O F AXIAL ISING MODELS

179

calculated numerically. The series show that the number of stable phase sequences springing from the multiphase point increases with increasing p , with the intervening sequences following the usual structure combination rule. The mean-field theory of the chiral clock model has been studied extensively. 19.20.5 5-57 1n particular, Siegert and Everts2' review and extend previous work. An expansion for small m a g n e t i ~ a t i o nshows ~ ~ that for

1 A < Amc = - c o s - ' ( ~ 271

-

Jo/J)

(5.4)

there is a first-order transition from the paramagnetic to the ordered state, whereas for A > Amc the transition is continuous. As the temperature is lowered the system locks in to a large number of commensurate phases in a way very reminiscent of the ANNNI model. The structure of the phases has been investigated using Landau theory, which includes umklapp terms,20finite lattice mean-field calculation^,^^ by viewing the mean-field theory as an iterated mapping,20,55 and by a method analogous to the Villain-Gordon theory reviewed in Section 3b.'9*20 An interesting point that has arisen from these calculations is that the lowtemperature series results in three dimensions do not agree in detail with mean-field results.20Szpilka and Fisher" have indeed demonstrated that the low-temperature series results are dependent on coordination numbers: mixed phases appear as these are increased from the value pertinent to the cubic lattice. Recently Huse et aL5*have pointed out that the chiral clock model exhibits novel interface properties. Consider imposing an interface perpendicular to the axial direction by fixing the left-hand side of the system in state 0 and the right-hand side in state 2. At A = 0 this will result in a 012 interface. However, as A is increased, the energy of a 012 interface increases relative to that of a 011 or 112 interface until, at A = b, it becomes energetically favorable for the 11122-..22,withatleast interface to wet togiveaconfiguration00..-00(11... one layer with spins n i , j = 1. Armitstead et aL5' recently showed that, at finite temperatures, the wetting takes place through a sequence of first-order layering transitions. The number n of layers with ni,j = 1 at the interface increases by one at each of the phase boundaries.

H. C. Ottinger, J. Phys. C 16, L257 (1983). H. C. Ottinger, J . Phys. C 16, L597 (1983). 5 7 H. C. Ottinger, J . Phys. C 15, L1257 (1982). 5 8 D. A. Huse, A. M. Szpilka, and M. E. Fisher, Physica A (Amsterdam) I21,363 (1983). 5 9 K. Armitstead, J. M. Yeomans, and P. M. Duxbury, J . Phys. A : Math. Gen. 19, 3165 (1986). 55

56

180

JULIA YEOMANS

These results were obtained from low-temperature series using a method analogous to that described for the ANNNI model in Section 3c. The behavior of the phase sequence as the temperature is raised remains an open questionthe phases may cutoff, the first-order boundaries could end in a sequence of critical points, or, if the roughening transition does not intervene, could reach the bulk phase boundary. The 4-state model has somewhat different features.h0 A 013 interface wets to 00...00/11...11(22...22133...33at A = 2(tan-'+)/n through a single firstorder transition from n = 0 to n = GO. A 0 I 2 interface, however, which wets at A = 0, has at least two layering transitions. Although experimental realizations of the chiral clock model in two no dimensions have been provided by absorbed layers of H on Fe( 1 similar correspondence has been established in three dimensions. It would certainly be of great interest to find such a system. Meanwhile the model stands as an example of an alternative mechanism leading to competition which can result in modulated ordering when the Hamiltonian contains only short-range interactions.

11. Experimental Applications 6 . BINARYALLOYS

Gradually evidence has been mounting that the ANNNI model is not just a theorist's playground but that it has considerable experimental relevance. Perhaps this is not so unexpected. It has long been acknowledged that a twostate system can be mapped onto an Ising model, often with just short-range interactiomh2If the effective coupling between nearest-neighbor spins turns out to be small, the second-neighbor interaction can be important, and, if it is antiferromagnetic, modulated phases can appear. Several authors have pointed out that the ANNNI model mirrors many of the features seen in binary alloy^.^^-^^ Long period structures are primarily observed in compounds containing noble metals, for example, Ag,Mg,66567

6o

P. J. Upton, Univ. of Oxford, Oxford, England (unpublished work).

I. Sega, W. Selke, and K. Binder, Surf. Sci. 154,331 (1985). 6 2 A discussion of results for /?-brassis given by J. Als Neilsen, Phase Transitions Crit. Phenom. 61

5A, 87 (1976). J. Kulik and D. de Fontaine, Muter. Res. Soc. Symp. Proc. 21, 225 (1984). 64 D. de Fontaine and J. Kulik, Acla Metall. 33, 145 (1985). " A. Loiseau, G. van Tendeloo, R. Portier, and F. Ducastelle, J . Plzys. 46,595 (1985). 66 J. Kulik, S. Takeda and D. de Fontaine, Acta Merail. 35, 1137 (1987). 6 7 R. Portier, D. Gratias, M. Guymont, and W. M. Stobbs, Acra Crystdlogr. A36, 190 (1980). 63

THEORY AND APPLICATION OF AXIAL ISING MODELS

t

t

t i

181

t

(bi FIG. 11. Atomic structure of a binary alloy: (a) the L l 2 structure; (b) the (2) phase.

CUAU,~'and A u , Z ~ . ~However, ' other examples, such as TiAl,, are well doc~mented.'~These compounds have a disordered face-centered cubic structure at high temperatures. As the temperature is lowered they lock in to the L1, structure, where, taking TiA1, as an example, planes of A1 alternate with mixed planes of Ti and A1 along the [OOl] direction. Within the mixed planes the two atomic species are ordered with each A1 being surrounded by four Ti as nearest neighbors and vice versa. This structure is illustrated in Fig. 1la. Modulation is introduced into the crystal structure by antiphase boundaries, as shown in Fig. 1Ib. Conservative antiphase boundaries, domain walls in the ANNNI model, correspond to a displacement of the Ti sublattice through [4,),0]. Hence each face-centered cube has two positions which we label t and 1. Long period phases can then be described using notation analagous to the ANNNI model.6 For example, the phase in Fig. 1 Ib is (2). 68

M. Guymont and D. Gratias, Acru Crystullogr. A35, 181 (1979). van Tendeloo and S . Amelinckx, Phys. Status Solidi A 43, 553 (1977). D. Broddin, G. van Tendeloo, J. van Landuyt, S. Amelinckx, R. Portier, M. Guymont, and A. Loiseau, Philos. Mug. A 54, 395 (1986).

69 G. 'O

182

JULIA YEOMANS

T 1200

1100

1000

900

800

700

I

I

I

I

I

70

71

72

73

74

I )

75

at.% A l

FIG. 12. Phase diagram of TiAI, .65

It is also possible to introduce nonconservative antiphase boundaries which change the stoichiometry of the alloy. These correspond to a displacement of [*,O,i] or [0,3,4] and introduce mixed planes which are either nearest neighbors or separated by three times the interplanar spacing. These structures will be mentioned only briefly in the following. Two beautiful sets of experiments illustrating the behavior of binary alloys have been performed on TiA1365and C U , P ~ . We ~ ' shall describe the results of these experiments and discuss their interpretation in terms of the ANNNI model. The section closes with a brief mention of the behavior of other binary alloys. Reference is also given to other theoretical appro ache^^^.^' which have been used to explain modulated ordering in these systems. a. TiAl,

Loiseau et al? used electron diffraction and high-resolution electron microscopy to study the phase diagram of TiAl, for 71-73 at.% A1 and temperatures between 700 and 1200 K. The phase diagram of this compound is shown in Fig. 12. Four phases are stable. Near stoichiometry, the TiA1, structure, which corresponds to an antiphase boundary every structural unit or (1 ), provides the stable phase. For less than 70 at.% A1 two TiAl, structures

'' H. Sato and R. S. Toth, Phys. Rev. 127,469 (1962). '* B. L. Gyorffy and G. M. Stocks, Phys. Rev. Lett. 50,374 (1983).

THEORY AND APPLICATION OF AXIAL ISING MODELS

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which contain nonconservative antiphase boundaries appear. Between these limits there are two regions of the phase diagram where different long period structures are stable, separated by a temperature range of about 100 K within which only the TiAl, and TiAl, structures are observed. In the low-temperature stability field of the long period phases, below 900 K, only three phases were observed, (12), (1 12), and ((12)’122). The former dominate the phase diagram. In the high-temperature range, however, a large number of different structures were found to be stable.65 These structures are listed in Fig. 13, which shows the temperature at which each phase was observed. The experiments suggest that the structures are each

9 4

1.8

-

1.7

09

. 8

1.6

-

d

3.

1.5

-

1.4

-

.

2.

.

1. 1.3

-

I 1

600

I

I

I

I

I

I

700

800

900

1000

1100

1200

m

T

FIG.13. Wave vector as a function of annealing temperature for the long period phases observed in TiAI, by Loiseau et ~ 1 . ~ ~

184

JULIA YEOMANS

stable over a small temperature range, with the shortest period structures having the largest range of stability: about 50 K for (122) and ((122)’12). At each annealing temperature different long period structures were found in separate areas of the sample, which may be due to variation in concentration across the sample or experimental limitations on the speed of quench, annealing time, or temperature control, producing metastability effects. Certainly these results are very reminiscent of the behavior of the ANNNI model with antiferromagnetic first-neighbor interactions. The ordering of the long period phases observed in TiAI, is identical to that in the ANNNI model and in both cases the short period phases occupy the largest portion of the phase diagram. Presumably improving the resolution of the experiment would reveal the missing ANNNI phases. At lower temperatures a few commensurate phases dominate, whereas for higher temperatures a larger number of phases appear in the phase diagram, as in the case of the ANNNI model. Whether all phases have a range of stability or not, that is, whether the Devil’s staircase is complete or harmless, remains an open question. To study the compound further, Loiseau et aL6’ used high-resolution electron microscopy, which provides a very powerful tool in the investigation of binary alloys. It enables study of the atomic positions and, in particular, a more detailed investigation of the nature of the antiphase boundaries. With modern instruments a spatial resolution of 0.2 nm can be achieved. In general, images are taken along a cubic axis oriented perpendicular to the axial direction. Along this direction only atoms of the same species overlap, and it is easier to see any shift of the antiphase boundary throughout the thickness of the crystal (typically 10 nm). The interpretation of the electron microscope image is not straightforward, and the contrast depends strongly on the film thickness and the properties of the lens. Computer simulation of the images is necessary for a correct interpretation. However, with care, photographs can be produced where a particular atomic species corresponds to the obvious white dots on the image. An example of such a photograph where the bright dots correspond to the Ti atoms in the structure ( 12,) is shown in Fig. 14. One of the features emphasized by Loiseau et aL6’ is the appearance of jogs along certain antiphase boundaries which cause a local change in the band structure. In the (l z 3 ) structure, for example, the jogs locally alter the ordering from 123123to 122124.Similar defects can occur through movement of the boundary on every seventh plane. Hence these planes, which are indicated by arrows in Fig. 14, are diffuse in the electron microscope image. The diffuseness increases with increasing annealing temperature. These jogs can be identified with spin flips in the ANNNI model and suggest strongly that entropic effects are important in stabilizing the modulated structures. Note that the diffuse boundaries neighbor one-bands, where spin flips are more favorable because they change the spacing of one-bands rather than creating a (13) configuration.

FIG. 14. High-resolution electron micrograph of the (12’) structure in TiAI,. The arrows indicate jogged antiphase b o ~ n d a r i e s . ~ ~

I86

JULIA YEOMANS

b. Cu,Pd Two differenttypes of conservative antiphase boundary have been observed in binary metal alloys with long period stable phases. In compounds such as TiAI, described above, the antiphase boundaries are predominantly straight,65 whereas in alloys like CuAu they are much more diffuse and not obviously related to a (001)plane.68de Fontaine et have suggested that the ability of the boundaries to wander depends on the magnitude of the inplane coupling Jo, Hence, well-defined antiphase boundaries would signal a compound corresponding to the low-temperature region of the ANNNI phase diagram and lockin to a series of modulated structures would be expected. Diffuse boundaries, however, would suggest higher temperatures (or equivalently smaller Jo), and the prediction that the wave vector would vary continuously, or perhaps quasicontinuously, with the external parameters and the concentration of the atomic species. In an attempt to investigate these ideas further, Broddin et uLfo have performed experiments on Cu,Pd in the regime 17-30 at. % Pd, where oneand two-dimensional modulated structures are observed with a wave vector which varies with composition and temperature. Again electron diffraction and high-resolution electron microscopy were the techniques used. The phase space can be divided into three regions: where the L1, or (a)structure is stable, where one-dimensional long period structures are observed, and where two-dimensional long period structures are stable. The experimental results indicate that within the one-dimensional long period regime Cu,Pd undergoes a crossover between two distinct behaviors. For low concentrations of Pd (18-21 at. %) the commensurate phase, (a),is stable at low temperatures. As the temperature is raised the wave vector becomes incommensurate and decreases continuously with increasing temperature. The antiphase boundaries are very diffuse and not obviously bound to (001) planes. These results suggest that the alloy is above the depinning temperature of the domain walls. At the other end of the concentration range considered (30 at. % Pd), however, the system locks in to short period commensurate phases. The wave vector is independent of temperature and varies discontinuously with concentration. For these concentrations the antiphase boundaries are sharp. This is typical low-temperature ANNNI behavior. For intermediate Pd concentrations the wave vector again locks in to commensurate phases. The period of ordering is strongly dependent on the concentration and shows a slight decrease with increasing temperature. Some of the antiphase boundaries are diffuse-which ones depends strongly on the 73

D. de Fontaine, A. Finel, S. Takeda, and J. Kulik, Noble Metals Symp. (1985).

THEORY A N D APPLICATION OF AXIAL ISING MODELS

187

stacking sequence itself. This seems to correspond to intermediate temperatures below depinning but at sufficiently high temperatures to allow substantial fluctuation in the softer domain walls. Comparison of the electron microscope images with computer simulations shows that the diffuseness of the boundaries results from compositional disorder in the neighboring atomic planes. In contrast to the situation in TiAl, ,the diffuseness appears to be independent of the annealing temperature. interpret their results in terms of the ANNNI model by Broddin et assuming that an increase in concentration corresponds to an increase in the value of ti. Indeed it was shown by Sat0 and Toth71 and by Gyorffy and that the composition has a strong effect on the atomic interactions. The behavior at low concentrations then corresponds to a vertical line at a value of ti < The low-temperature phase is the commensurate phase (a), and at higher temperatures there is a transition to a region of at least quasiincommensurate behavior. As the temperature is increased, the period would be expected to decrease with temperature, as observed experimentally. For higher concentrations, however, the stable phases correspond to larger values of ti. Presumably the effective temperature is also lower so that less softening of the domain walls is observed in the region where modulated phases are stable. Between the one-dimensional long period structures and the disordered face-centered cubic phase, two-dimensional modulated phases are thought to be stable over a small range of concentration. For the structure observed by Broddin et aL7' the antiphase boundaries were conservative in one direction and nonconservative in the other. The corresponding periods were 4.3 It 0.1 and 6.3 F 0.1, respectively. Other domain sizes have been observed by other authors.74 Since the two-dimensional modulated structures only exist over a small range it was not possible to study the evolution of their domain sizes as a function of temperature or composition. One might guess that these phases are a result of competing interactions in two directions. Little is known about a corresponding king model (but see Refs. 49 and 50), and any work on the nature of the phase diagram of such a model would be of great interest.

4.

c. Other Compounds and Theories

The appearance of long-wavelength phases in binary alloys is by no means limited to the compounds described above. Another notable example is Ag,Mg, where (12") with j = 2, 3,. . ., 7, 8, 12 have been observed together with the mixed phases (122123) and (123124).66-67A recent experimental

'' 0.Terasaki and D. Watanabe, Jpn. J . Appl. Phys. 20, L381 (1981).

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JULIA YEOMANS

study of this compound66 attempted to discern the concentration and temperature dependence of the period. However, this proved to be very difficult because of sluggish kinetics. (2), (233), and ( 3 ) have been identified in A u , Z ~ .In~ CuAu ~ the antiphase domain boundaries are much more diffuse and wavy and the modulation is probably incommensurate with the lattice.68 The existence of incommensurate phases in binary alloys has, until recently, been explained using ideas proposed by Sat0 and T ~ t h . They ~ ’ argued that incommensurate order could be stabilized by energy gained from the interaction between the Fermi surface and the new Brillouin zone boundaries resulting from the periodic modulation. Gyorffy and later performed band theory calculations supporting this picture for Cu,Pd. These ideas naturally explain the concentration dependence of the period of the modulated order but do not account for temperature effects nor predict lockin to cqmmensurate phases.75776The ANNNI picture allows the latter to be investigated but does not relate J1 and J2 to microscopic interactions. An amalgam of the two approaches in which band theory calculations are used to provide values for J1 and J2 and assess the effects of further-neighbor interactions would be very interesting.

7. POLYTYPISM We now turn to a second class of materials exhibiting modulated structures, the p ~ l y t y p e s . The ~ ~ , experimental ~~ situation here is far less clear, but, although metastability effects are very important, there is growing evidence to suggest that the modulated structures in these compounds can exist as stable phases which are well described by ANNNI-like Hamiltonians. 79-84 It is helpful in many cases to consider a compound to be constructed from one or more individual building blocks or structural units.85 If the units can be stacked in different ways to form several stable or metastable phases, the

’’K. Fujiwara, J . Phys. SOC.Jpn. 12, 7 (1957). ’‘ D. de Fontaine, J . Phys. A: Marh. Cen. 17, L713 (1984). ” A.

R. Verma and P. Krishna, “Polymorphism and Polytypism in Crystals.” Wiley, New York (1966). 7 8 P. Krishna (ed.),J. Cryst. Growrh Charact. 7 (Spec. Issue) (1984). S. Ramasesha, Pranwna 23,745 (1984). 8o J. M. Yeomans and G . D. Price, Bull. Minerul. 1W,3 (1986). 8 L G . D. Price and J. M. Yeomans, A m . Crysrullogr. 840,448 (1984). 8 2 J. Smith, J. M. Yeomans, and V. Heine, NATO ASZ Ser., Ser. E 83,95 (1984). 8 3 R. M. Hazen and L. W. Finger, in “Structure and Bonding in Crystals 11” (M. OKeeffe and A. Navrotsky, eds.), p. 109. Academic Press, New York, 1981. 84 G. D. Price, Phys. Chem. Miner, 10, 77 (1983). 8s J. B. Thompson, in “Structure and Bonding in Crystals 11” (M. OKeeffe and A. Navrotsky, eds.), p. 167. Academic Press, New York, 1981.

’’

THEORY AND APPLICATION O F AXIAL ISING MODELS

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resulting compounds are called polytypes. 7 7 * 7 8 Polytypism is surprisingly common in nature. Perhaps the best known examples are the classical polytypess6 such as silicon carbide and cadmium iodide, where the A , B, C stacking sequence of the close-packed layers can vary. Over one hundred different structures have been observed in silicon carbide with repeat periods up to 100 layers. Polytypic modifications are also found in the spinelloids, perovskites, micas; pyroxenes, chlorites, and many other mineral phases. The reasons for the appearance of numerous, but often closely related, polytypic phases and the extent to which the various phases are stable or metastable has been very controversial. Two main classes of theories have been mooted. Growth t h e o r i e ~ ' ~assume . ~ ~ that the modulated structure of the polytype results from growth around a screw dislocation with a period that reflects the step height of the dislocation. Equilibrium theories,80*88on the other hand, assume that the polytypic modifications can exist as stable thermodynamic phases, while admitting that equilibration is a problem in real systems. It has recently been pointed out that the ANNNI and similar models reproduce many of the properties of the polytypic phases. 79-84 Indeed, invoking short-range competing interactions gives a rather convincing explanation of their existence. We discuss two cases in detail: first, the spinelloids and second, the classical polytypes such as silicon carbide and cadmium iodide. Other theories of polytypism are briefly reviewed and the extension of the theory to treat polysomatic compounds is discussed.

-

a.

The Spinelloids

The spinelloid structural family89 is based upon an approximately closepacked oxygen framework and has an ideal stoichiometry of AB,O,, where A and B represent cations such as nickel and aluminum. Two-thirds of the cations occupy octahedrally coordinated sites within the 0 framework, while the remaining one-third are tetrahedrally coordinated. The cations define a basic structural unit within the oxygen framework, as shown in Fig. 15. All spinelloid structures are constructed from this unit, and its inverse, (J), and hence the structures can be mapped onto an array of structural variables which are Ising spins. In almost all spinelloids the ordering within two-dimensional layers corresponds to an Ising ground state which is ferromagnetic in one direction and antiferromagnetic in the perpendicular direction. Perpendicular to the

(r),

86

D. Pandey and P. Krishna, J. Cryst. Growfh Characr. 7,213 (1984).

89

H. Horiuchi, K. Horioka, and N. Morimoto, J. Mineral Sot. Jpn. 2,253 (1980).

'' F. C. Frank, Philos. Mag. 42, 1014(1951).

H. Jagodzinski, Neues. Jahrb. Mineral., Monafsh. 3,49 (1954).

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JULIA YEOMANS

roo1 1

I

/i- i O l O l I1001 lb)

FIG. 15. Atomic structure of the spinelloids: (a) a structural unit; (b) the (2) phase.

invariant layers, however, six different stacking sequences are observed in nature. These, using the ANNNI notation to describe them, are the spinel phase, (l), the phase, (2), the manganostibite structure, (3), and three structures found only in the Ni,SiO,-NiAl,O, system, (12), (12’), and (13). To model these systems we assume that the important interactions between the atoms in any pair of structural units can be represented by an interaction between the corresponding structural variables. 79--82 The correct in-plane ordering will result if the Ising spins interact through nearest-neighbor interactions which are ferromagnetic in one direction and antiferromagnetic in the other. However, the appearance of longer period structures along the axial direction suggests that the further-neighbor interactions are of the same magnitude as the nearest-neighbor terms. Hence one is led to represent the system using an ANNNI model.

THEORY AND APPLICATION OF AXIAL ISING MODELS

191

80

.

a

-

6-

W L

v 3

)

-

v) W

a

4-

2-

0

800

1000

1200

1600

1400

Temperature “C FIG. 16. Phase diagram of the system NiAIO, Ni,SiO,.’O

.

The assumption then is that, as external parameters such as temperature and pressure are varied, the atomic positions and hence interactions change slightly. This can result in a change in the ratio of the interaction parameters, 4, which, together with the variation in the temperature, moves the system through the phase space of the ANNNI model. Perhaps the most convincing evidence for these ideas results from the work ’ studied the phase relationships for the system of Akaogi et 4’who NiA1204-Ni2Si04 in the pressure range 1.5-13.0 GPa for temperatures between 800 and 1450°C. They found that, as the pressure was increased, the sequence of structures (3), ( 2 ) , (12’), (12), and ( 1 ) became stable, as shown in Fig. 16. This bears a striking resemblance to the stable phase sequence in the ANNNI model phase diagram. The missing long period phases do not pose a problem as they would not be expected to lie within the resolution of the experiment. The interaction parameters in the model Hamiltonian have been introduced on a purely phenomenological level to represent the energy difference between structural units which are aligned parallel or antiparallel. However, they are in M. Akaogi, S. Akimoto, K. Horioka, K. Takahashi, and H. Horiuchi, J . Solid State Chem. 44, 257 (1982).

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theory related to the microscopic interactions in the system which will depend on temperature, pressure, and chemical composition. In an attempt to obtain quantitative values for the parameters, Price et aL91 used lattice simulation techniques to calculate the interaction energies of the structural units in magnesium silicate spinelloids. Using two different model potentials and assuming pair interactions up to fourth-neighbor spins, they found J2/J1 = 0.45,

J3/J1 = -0.36,

J4/J1 = 0.23,

J1 < 0 (7.1)

= 0.56,

J 3 / J 1 = -0.00,

J4/J1 = 0.01,

J1 < 0 (7.2)

J,/Jl

These sets of results are both consistent with interactions lying within the multiphase region of the ANNNI model and support its use to interpret polytypism. Although further-neighbor interactions are inevitably present, they do not qualitatively change the nature of the phase sequences, as shown in Section 4,b. A problem which it would be interesting to resolve is the apparent stability of the (13) spinelloid phase, which does not fit into the ANNNI framework. The ( 1 3) phase, which has also been observed in other polytypic compounds, has not appeared in any of the models studied so far except the ANNNI model in a magnetic field.40 A similar mineral is wollastonite, where the phases (2), (3), (4), ( 5 ) , and (00) have been observed. The different symmetries of this compound suggest that it is best modeled by competing interactions between second- and fourthneighbor 1aye1-s.~’

b. Classical Polytypes We now return to the so-called classical polytypes.86 These have M X or M X , stoichiometry and are characterized by S i c and CdI,, respectively. Their structures consist of planes of M atoms, each of which is tetrahedrally coordinated with X atoms which are stacked in a close-packed array. The M X structures can be considered as a pair of interpenetrating close-packed sublattices with alternating layers of M and X atoms. Therefore, the entire crystal structure can be uniquely specified by the stacking sequence of the X layers. In the M X , structure, however, the M atoms occupy only alternate planes of tetrahedral sites between the layers of X atoms so that the structure is a stack of X - M - X “sandwiches.” Using the convention that the first layer in a stacking sequence is chosen to lie immediately to the left of an M layer, the polytype can again be defined by the stacking sequence of the X layers. 91

92

G. D. Price, S. C. Parker, and J. M. Yeomans, Acta. Crystallogr. B41,231 (1985). R. J. Angel, G. D. Price, and J. M. Yeomans, Acta Crystallogr. B41, 310 (1985).

THEORY AND APPLICATION OF AXIAL ISING MODELS

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It is conventional to describe the stacking sequence of a given polytypic compound using a notation introduced by Zdhanov and M i n e r ~ i n awhich ,~~ turns out to closely resemble that which we have used throughout to describe the long period phases in the ANNNI model. Zdhanov notation emphasises the fact that a close-packed stack of layers can be thought of as a two-state system because any given layer, say one in position A , can only be followed by layers in two possible positions, B and C . This is done by assigning T to represent the stackings A - B , B-C, and C - A and 1 to represent anticyclic ordering, B - A , C - B , and A - C . For example, a given stacking sequence is A B C A C B A B C A C B A B C A C B

t t t l l l t t t l l l r t t l l l

(7.3)

which as usual we denote (3). Consider first Sic. A very large number of polytypes have been observed,86994although in many cases the evidence for their existence as true thermodynamically stable phases is very slim. However, the stable and metastable phases which are documented have the following striking properties:

(1) The short period structures (3), (2), (23), (GO), and ( 1 ) are by far the most commonly observed. (2) Transformations have been observed between the short period p ~ l y t y p e s However, .~~ these often rely on the addition of impurities or the application of pressure and are hence far from being reversible. (3) Trivalent impurities, such as boron and aluminum, tend to favor two)). bands, whereas pentavalent impurities favor cubic stacking (( a (4) In the longer period structures three-bands predominate. Two-bands are also rather common, and four- and longer bands are seen occasionally. One-bands are only observed in the phase (1). (5) Long period structures can usually be formed from the simple phases through the usual structure combination rules. Indeed, Pandey and Krishna86 have suggested (for different purposes) that the observed phases fall rather neatly into ANNNI-like sequences. (6) Disordered structures, with no well-defined wave vector, are also common and may often have been documented as long period phases. These features are well explained by modeling silicon carbide with an ANNNI model with the Zdhanov variables taking the role of the Ising spin^.'^-^' A large value of Jo then ensures little disorder within the closepacked layers, whereas competing interactions along the axial direction allows 93 94

G. S. Zdhanov and Z. Minervina, Zh. Fiz.9, 151 (1945). N. W. Jepps and T. F. Page, J . Cryst. Growth Charact. 7,259 (1984).

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JULIA YEOMANS

the formation of a large number of polytypes. The best qualitative agreement is with the axial Ising model with third-neighbor interactions in the vicinity of the (2):(3) and (3):(00) multiphase lines.33 Here (2), (3), and (co) appear as ground-state phases and dominate the phase diagram, and (23) appears as an important finite-temperature phase near the (2): (3) boundary. The fifth common polytypic phase (1) would be stabilized by a change in the sign of J1. All the transformations between the short period phases94 correspond to obvious routes between neighboring phases in the ANNNI phase diagram. As was argued in Section 4,c, annealed impurities are not expected to destabilize the long period structure^.^^ The observed effect of impurities can be explained by assuming that the addition of acceptors tends to increase K , whereas donors tend to decrease it. Moreover, the long period phases observed in silicon carbide bear a striking resemblance to the phase sequences which are stable in this region of the phase diagram.86 Both the predominance of 2- and 3-bands and the arrangement of the bands within a given phase are suggestive of the same mechanism at work. It is of course the case that metastability effects are extremely prevalent in Sic. A batch of crystals grown under nominally the same conditions will contain many different polytypes, even within the same single crystal. The ANNNI picture is also able to explain why it is so difficult to obtain the true thermodynamically stable phase. In the vicinity of the multiphase lines the free energy of the long-wavelength phases which are degenerate on the line itself differ only by very small entropic contributions (typically less than one part in lo4). Moreover, transitions between the different energy states require a substantial rearrangement of atoms, and hence the energy barrier inhibiting the transitions is very large. Thus, once a compound has formed in a metastable state due, for example, to the effect of growth conditions, it is likely to stay there. The disordered structures often observed are also expected to be metastable. One should point out that a major defect of the ANNNI model as a theory of silicon carbide is that it does not accurately mirror the elementary excitations of the compound: an ANNNI spin flip would correspond to flipping a single line of atoms from the flip itself to the edge of the crystal, which is clearly unphysical. Note, however, that in the Villain and Gordon formulation of the meanfield theory of the ANNNI model,I5 described in Section 3,b, individual spin flips are replaced by the average deviation of the spins within a layer from their zero temperature value. Thus, it may be the case that small deviations throughout a plane of atoms caused by, for example, phonons, can stabilize the long period phases.

THEORY AND APPLICATION OF AXIAL ISING MODELS

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Cadmium iodide is usually quoted as an example of the M X , classical polytypes. For this compound over 50 phases have been observed. One- and two-bands predominate in the long period structures, which suggests that It is CdI, is best modeled by the ANNNI model with negative J1.81*82 interesting to note, however, that several of the long period phases observed in this compound (for example, (l"', 2"*), nl, n, positive integers) cannot be constructed using the structure combination branching rules. A third classical polytype with interesting behavior is zinc sulfide.95For this compound polytypism in mineralogical samples differs from that observed in laboratory grown crystals. In the latter almost any band length can occur and the screw dislocation mechanism is well documented. In mineralogical samples, however, which have had time to come to equilibrium, the observed phases are (a),( 5 ) , (4), (3), (23), ( 2 ) , (12), and (2), which correspond closely to the ANNNI phases, and (13) and (1 123), which do not. A study of many polytypic crystals obtained from a mine bore indicated that the structure was a function of depth.96 For comparison we briefly summarize other theories of polytypism in the classical polytypes. Jagodzinskis8 was the first to propose an equilibrium theory of polytypism. He argued that the vibrational entropy would provide a term in the free energy which would stabilize the long period structures. However, this theory is unable to correctly predict the fault distribution in the long period polytypes. More recently Hazen and Fingers3 and Price84 explained the existence of short period polytypes in terms of the ground state of the axial Ising model with third-neighbor interactions. Growth theories of polytypism, on the other hand, regard the long period polytypes as nonequilibrium structures which result from growth around screw dislocations. 8 6 , 8 7 The period of a given polytype is then determined by the step height of the growth spiral. The problem with these theories is that they cannot predict which of the long period phases actually occur in nature. Moreover, very large, and hence energetically unfavorable, steps would often be needed. A recent, interesting modification of this theory has considered the influence of low-energy stacking faults present near the surface of the basic matrix, (3), ( 2 ) , or (a),on the spiral g r ~ w t h . ~Consideration ~.~' of such afaulted matrix model enables the prediction of sequences of structures very close to those observed in silicon carbide. However, there is as yet no convincing way of deciding which of the possible series one should expect to occur in nature. 95

96 97

I. T. Steinberger, J . Cryst. Growfh Charact. 7 , 7 (1984). S. Hussuhl and G. Muller Beitr, Z . Miner. Petrogr. 9, 28 (1963). D. Pandey and P. Krishna, Curr. Top. Mater. Sci.9,415 (1981).

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There are, however, many interesting links between these ideas and the ANNNI picture. For example, Pandey and Krishna86 use a model in many ways similar to the ANNNI model at zero temperature in determing which of the stacking fault configurations are expected to occur most frequently. c.

Polysomatism

A similar approach has been applied to explaining the occurrence of polysomatic series.98 These materials correspond to families of structures which can be obtained by stacking in varying proportions two or more chemically distinct units, A and B, say.99 An example is the biopyriboles, where A represents mica and B pyroxene layers."' The observed phases are then ( A ) , ( A B ) , ( A A B A B ) , ( A A B ) , and ( B ) . Replacing A and B by T and 1, respectively,one again obtains ANNNI-like phases. A chemical potential term is needed to control the relative abundance of the two species, and hence the appropriate spin model is the ANNNI model in a magnetic field. One hopes that using ANNNI-like models to describe ordering in polytypes will encourage the view that long period modulated structures can exist as stable thermodynamic phases in compounds where the atomic interactions are predominantly short ranged. Hence it is hoped that experimental effort will be directed toward the difficult task of establishing phase diagrams for polytypic compounds. The model systems are useful in predicting which phase sequences will be stable and the expected distribution of bands within a given phase. Moreover, predictions can be made which hopefully can be verified experimentally, about defect distributions and the relative probability of occurrence of different types of dislocation. It would be of great interest to pursue further calculations which relate the phenomenological energies 4, i = 0,1,2,. . ., to the atomic interactions in the polytypic c ~ m p o u n d s . ~ ' ~ ' ~ ' 8. MAGNETIC SYSTEMS The best candidates for ANNNI systems where the Ising variables correspond to magnetic spins are found among the cerium monopnictides. In cerium antimonide strong uniaxial spin anisotropy constrains the spins to point along the [loo] direction. Within the (100) planes the ordering is ferromagnetic: most planes lie in a state with saturated magnetization along or G . D. Price and J. M. Yeomans, Min. Mag. SO, 149 (1986). J. B. Thompson, Am. Mineral. 63,239 (1978). l o o D. R. Veblen and P. R. Buseck, Am. Mineral. 64, 687 (1979). l o ' C. Cheng, R. J. Needs, V. Heine, and N. Churcher, Europhys. Leu. 3,475 (1987).

98 99

THEORY AND APPLICATION O F AXIAL ISING MODELS

10

0

197

20 Temperature K

FIG.17. Phase diagram of cerium antimonide.'"

antiparallel to the axial direction. However, at first sight, rather surprisingly, planes with zero magnetization also appear for temperatures 2 TN/2,where the Ntel temperature, TN = 17 K. The ferromagnetic planes form modulated structures with a wave vector along [1001,which locks into different values as a function of temperature and magnetic field. Cerium antimonide has been extensively studied by neutron scattering102-106and specific heat m e a s ~ r e r n e n t s . 'The ~ ~ experiments give consistent results, although there are small differences between samples. The resulting phase diagram, which comprises 14 commensurate phases, is shown in Fig. 17.'" We give the layer configurations rather than the Zdhanov notation for each phase to emphasize the appearance of the layers with zero magnetization. It is apparent from Fig. 17 that the stable phases can rather B. Lebech, K. Clausen, and 0. Vogt, J. Phys. C 13, 1725 (1980). P. Fischer, B. Lebech, G. Meier, B. D. Rainford, and 0.Vogt, J. Phys. C 11,345 (1978). '04 G . Meier, P. Fischer, W. Halg, B. Lebech, B. D. Rainford, and 0. Vogt, J . Phys. C 11, 1173

lo'

lo3

(1978). lo'

lo6 lo'

J. Rossat-Mignod, P. Burlet, J. Villain, H. Bartholin, T.-S. Wang, D. Florence, and 0. Vogt, Phys. Rev. B 16,440 (1977). P. Burlet, J. Rossat-Mignod, H. Bartholin, and 0. Vogt, J . Phys. 40,47 (1979). J. Rossat-Mignod, P. Burlet, H. Bartholin, 0.Vogt, and R. Langier,J. Phys. C 13,6381 (1980).

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naturally be grouped into three classes: (1) 1-7, which form the zero-field sequence as the temperature is lowered; (2) 7-10, which appear as the magnetic field is increased at low temperatures; and (3) 10-14, which appear at higher temperatures and fields. All the phase transitions are first order. lo'

von Boehm and BakI6 were the first to point out that the behavior of cerium antimonide could be explained by invoking short-range competing interactions. In an attempt to obtain a more quantitative fit to the experimental data, Pokrovsky and Uimin3'336studied the ANNNI model in a magnetic field (see Section 4,a) in the regime Jo >> J1 with small third- and fourthneighbor couplings. They obtained a phase diagram which topologically rather closely resembles that of cerium antimonide, and they identified the three sets of phases as belonging to the sequences (1) (1222kf2) (2) (122k+') (3) (12(13)2), (13), (14),...

They pointed out that the zero magnetization layers could not be disordered but gave no explanation of their existence. One possible explanation of the zero magnetization layers in sequence (1) is that they lie in the same position as the fluctuating domain boundaries observed in the binary alloys.65This immediately suggests that the sequence is (12k+') (although (122) is replaced by (12(122)2)) with the boundary between the one- and two-bands fluctuating to give local order (12k12kf2)on a time scale short compared to that of the experiment. Another possibility is that higher order magnetic interactions are important in this material.'07a Two other features of the phase diagram which require explanation are, first, why the transition to the paramagnetic phase does not proceed via a region of incommensurate order and, second, why the longer period, rather than the shorter period, phases appear to be stable over wider ranges of temperature. Cerium bismuth is a similar compound where modulated magnetic phases havebeenobserved.108Thephases(l), (2), (125), (13), (133),and(13313) are stable, and no layers of zero magnetization appear. This system has been

lo*

9.Halg and A. Furrer, Phys. Rev. B34,6258 (1986). H. Bartholin, P. Burlet, S. Quezel, J. Rossat-Mignod, and 0. Vogt, J. Phys., Colloq. 40,C5 (1979).

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studied by Uimin"' using an axial king model with first-, second-, and thirdneighbor interactions in a magnetic field. His results reproduce the experimental phase diagram rather well.

9. CONCLUSION The experiments described above give convincing evidence for the applicability of the ANNNI model to natural phenomena. The model provides a mechanism through which polytypes can exist as equilibrium or highly metastable states and therefore challenges conventional growth theories and encourages experiments on the stability and kinetics of phase tranformations in these compounds. It provides an explanation for commensurate modulated order in binary alloys and ferrimagnets and for the phase sequences observed and their dependence on temperature. Moreover, the formalism is sufficiently simple that the effects of defects and changes in the Hamiltonian can be assessed. Having espoused the cause of the ANNNI model, it is important now to point out the drawbacks of this approach and areas where further research is needed. First, the interaction parameters, 4, i = 0,1,.. ., are introduced phenomenologically and their magnitude is inferred by fitting to the experimental results. Initial attempts have been made to calculate the effective interactions from first principles using band theory"' and atomic modeling techniques," and the calculations, although difficult, are well worth pursuing. A second point that warrants emphasis is the role of long-range interactions. It was shown in Section 3,b that the ANNNI model can be mapped In onto a system of domain walls with long-range oscillatory interacti~ns.l'*'~ the ANNNI model the interactions are a result of entropy: of local fluctuations in the wall positions at finite temperatures. However, other physical mechanisms, for example elastic interactions or, in metals, electronic terms, could play the same role. Bruinsma and Zangwill describe one approach which invokes longrange interactions in an interesting paper which considers magnesium-based, Friauf- Laves-phase ternary alloys. These compounds lock in to an ANNNIlike sequence of phases-(122), (1'212), (12), (1212'), (12'), (123), ( n 4 ) , (2), (a),(3), (2)-as a function of the number of valence electrons per atom.'" Bruinsma and Zangwill' l o calculate the free energy of a domain

lo9 'lo

'I1

G. V. Uimin, J . Phys., Left.43, L665 (1982). R. Bruinsma and A. Zangwill, Phys. Rev. Lett. 55,214 (1985). Y. Komura and Y. Kitano, Acta Crystallogr. B33,2496 (1977).

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wall and nearest-neighbor interactions between walls from the pair potentials of pseudopotential theory. These interactions stabilize the polytypes (2), (a),and (12). The degeneracy on the boundaries between these phases is then lifted to stabilize the longer period phases by invoking elastic interactions. This and similar calculations112emphasize that either temperature or longrange interactions can stabilize modulated phase sequences. Evidence for the former is found in fluctuations in the domain walls. These have been seen in binary alloys and ferrimagnets but not yet in polytypes like S i c or the FriaufLaves phases. This is an important problem, and more experimental and theoretical work is needed. We conclude by considering the transition from cubic to hexagonal close packing. Within the ANNNI formalism this corresponds to a transition, (a) -+( l ) , at which J1 changes sign. J, will therefore be important, and, if it is antiferromagnetic, competition can result in modulated phases. This suggests that polytypism should be rather common in the vicinity of structural phase transformation^."^ There is some evidence for this in metals and in recent experiments which show the appearance of tweed-incommensurate regions -near a martensitic transformation."4 ACKNOWLEDGMENTS I should like to thank my colleagues for enjoyable collaborations and discussions on the topics treated in this article. Thanks are also due to NORDITA, Copenhagen, for kind hospitality while part of the review was written.

'I2 'I3

'I4

P. Bak and R. Bruinsma, Phys. Rev. Lett. 49,249 (1982). A. Zangwill and R. Bruinsma, Comments Condens. Matter Phys. B 13, 1 (1987). L. E. Tanner, A. R. Pelton, and R. Gronsky, J . Phys., Colloq. 43, C4-169 (1982).

SOLID STATE PHYSICS, VOLUME

41

Excitations in Incommensurate CrystaI Phases R. CURRAT lnstitut Laiie-Langevin, Centre de Tri 156X, F-38042 Grenoble-Cedex, France

T. JANSSEN Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, 6525 E D Nijmegen, The Netherlands

I. Introduction . . . . . . . . . . . . . ............................. 1. Incommensurate Cryst riodic Structures .... 2. Modulated Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Real Incommensurate Solids. . ........................... 4. Systems and Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Landau Theory of Modulated Systems . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . 5. Normal-Incommensurate Phase Transitions . . . .................. 6. Discommensurations . . . . . . . . . . . . . _. . . _ . . . ___ 7. Excitations in the Incommensurate Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 8. Debye-Waller Factor . . . . . . . . . . . . . . . 9. Light Scattering from Incommensurate 111. Supersymmetry and Higher-Dimensional Space Groups . . . . . . . . . . . . . . . . . . . . . . 10. Symmetry of Quasiperiodic Structures . . . . . . . . . 11. Excitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........................... s..................

IV. Microscopic Models 14. Frenkel-Konto

...

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

...

16. Excitations in Incommensurate Phases: Simple Models. . . . . . . . . . . . . . . . . . . . . . .. . . 17. Dynamics of the DIFFFOUR Model. . . . . . . . .. 18. Dynamics of the Frenkel-Kontorova Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Optical Properties ................................ V. Long-Wavelength Excit ...... 20. Structure and Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Experimental Results.. . . ............................ .. . . . . . . . rystals . . . . . 21. Neutron Scattering 22. Optical Studies . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Results from Other Techniques ...................... V11. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

202 202 204 205 207 21 1 21 1 213 217 22 1 223 225 225 228 23 I 234 236 236 238 242 249 256 251 260 260 264 264 287 291 301

20 I Copyright 01988 by Academic Press. Inc. All rights o f reproduction in any form reserved.

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1. Introduction 1. INCOMMENSURATE CRYSTAL PHASESAS QUASIPERIODIC STRUCTURES Several classes of solids are known to lack translational periodicity, in the conventional sense, while retaining a high degree of structural order. The interference pattern from such systems is characterized by well-defined discrete “Bragg” spots, which, in a standard diffraction experiment, may appear just as sharp as for a crystalline solid. The basic difference between the periodic and nonperiodic solids lies in the labeling of the spots. In the latter case, this can no longer be performed in terms of three integer indices, and additional indices (i.e., basis vectors) must be introduced. A structure for which diffraction spots can be labeled with a minimum number of r indices is called quasiperiodic with rank r. A simple such example is given by a three-dimensional crystal in which a static distortion of wave vector q has been frozen in. The diffraction condition then reads:

K

= ha*

+ kb* + Ic* + mq

(1.1)

where (a*,b*,c*) are the three reciprocal basis vectors of the undistorted lattice and (h,k, I, m) are integers. As long as the distortion wave vector q is not reducible to a linear rational combination of the other three basis vectors, all four integers are required in order to index the complete diffraction pattern. Incommensurately modulated crystals, as in the above example, are quasiperiodic structures for which the rank is related to the number of independent frozen modulations. In intergrowth compounds the rank is determined by the number of chemical subsystems participating in the common structure. A quasiperiodic structure of space dimensionality d and rank r > d can be viewed as the intersection of an r-dimensional periodic structure, with ddimensional space. This property is general and independent of the physical nature of the system. To illustrate this, let us consider a 2D periodic density function: f(X,Y) =f

( x + 1, Y ) = S(x, Y

Its intersection with a line of irrational slope:

generates a nonperiodic 1D function:

+ 1)

(1.2)

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FIG.1. Relationship between superspace periodicity and physical space quasiperiodicity: (a) displacivelymodulated I D crystal as intersection of 2D periodic structure (wavy lines) with direct physical space (rEaxis); (b) same in Fourier space: diffraction spots are obtained as projections of 2D reciprocal lattice onto physical Fourier space (QEaxis); (c) 1D quasicrystal as intersection of 2D periodic arrangement of linear segments with physical space (rEaxis); (d) same in reciprocal space.

With the help of Eq. (1.2) the Fourier transform of g(x) is readily obtained as:

aij6(K - 2nia - 2nj/3)

G ( K )= i J

which identifies g(x) as a 1D quasiperiodic density function of rank 2. The set of K values defined by the 6 function in Eq. (1.5) is called a quasilattice. It is generated by projecting the nodes of a 2D square lattice onto a suitably oriented line (cf. Fig. 1). The above picture can be extended to a 2D or 3D quasiperiodic physical system.'-3 The rank of the corresponding density function is called the superspace dimensionality of the system. The relationship between the quasiperiodic physical system and its periodic superspace counterpart follows in the same way as in the 1D example above: the quasiperiodic structure is defined as the intersection of the superspace structure with direct physical space (or as its projection, in Fourier space).

' P. M. de Wolff, Actu Crystallogr. A30,777 (1974). A. Janner and T. Janssen, Phys. Rev. B 15,643 (1977). A. Janner and T. Janssen, Physicu A (Amsrerdum)99,47 (1979).

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As any periodic function, the density function of the rD structure is invariant under a group of symmetry operations, which form an rD space group. Equivalently, one may say that the periodicity and space-group symmetry of the incommensurate crystal is restored in higher-dimensional space. The concept of superspace symmetry enables one to make use of standard crystallographic and group-theoretical techniques: the labeling of normal modes and electronic states in terms of irreducible space-group representations, the derivation of selection rules and systematic extinctions can all be performed rigorously in rD space. The projection onto 3D Fourier space is generally straightforward. 2. MODULATED CRYSTALS

Many incommensurate systems can be viewed as modulated crystals. This implies the existence of a basic 3D periodic structure to which the modulation is applied. The word modulation is used here in a generic sense and stands for any distortion or static perturbation characterized by its own independent periodicity. In the simple case of a single or “one-dimensional” modulation (r = 3 l), the diffraction pattern is given by Eq. (1.1). The spots with m = 0 are referred to as fundamental reflections, and the spots with m # 0, as mth-order satellites. The basic structure may be defined in several different ways. For example, the fundamental reflections, once identified, may be back Fourier transformed to generate a periodic “average” structure. Another useful reference structure is the “undistorted” structure, which is obtained by reducing the modulation amplitude to zero in a continuous way. This procedure, as shown later, is not always equivalent to spatial averaging. Fortunately, in many modulated systems, the basic structure has physical reality, being the stable configuration of the system in a certain pressure and the modulated state temperature range, say T > q to simplify. Below develops, with the amplitude of the modulation increasing progressively as the temperature is lowered. Oftentimes the value of the modulation wave vector is observed to drift, and a lockin transition to some commensurate superstructure is observed at lower temperature ( T < TL). The existence of a continuous or quasicontinuous phase transition at suggests a symmetry relationship between basic and modulated structures. This relationship can be formulated in the language of the Landau theory of continuous phase transitions. The modulation amplitude (or more precisely its first Fourier component) plays the role of the order parameter associated with the transition. As such it is expected to transform according to a specific

+

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irreducible representation of the basic space group. The knowledge of this representation (wave-vector and small-group representation) contains most of the relevant symmetry information on the modulated structure. All the selection rules, systematic extinctions, etc., that follow from the consideration of superspace symmetry can also be obtained from Landau theory4: to each invariance property in rD space corresponds a specific transformation law in 3D space. Although for modulated crystals both approaches yield equivalent results, the superspace approach is more general and remains applicable even when no basic structure can be defined. O n the other hand, Landau formalism has the advantage of being a complete phenomenological theory, where symmetry and thermodynamical aspects are treated simultaneously within the same framework. 3. REALINCOMMENSURATE SOLIDS

So far we have referred to incommensurate (crystal) structures as quasiperiodic, ordered systems and we have seen how simple mathematical models can be constructed which meet both of the above criteria. The next question to be addressed concerns the relevance of quasiperiodic models to the description of real physical systems. There is a wide variety of systems for which, (at least) in first approximation, the Fourier spectrum is discrete, with peak positions given as in Eq. (l.l), or more generally as: r>3

K=

hja) j= 1

where the hjs are integers and the a7s are rationally independent basis vectors. The spectrum described by Eq. (3.1) is discrete in a mathematical sense; i.e., it consists of a countable set of sharp peaks. Discreteness in a practical sense implies that the individual peaks in the diffraction pattern may be resolved and observed separately. This imposes the additional requirement that the extra indices hi ( j > 3) should not assume arbitrarily large integer values, a condition which is very generally fulfilled in real systems due to the rapid falloff of the diffraction spot intensities with increasing hj ( j > 3) values [m values in Eq. (1.1)]. Even when this is not the case, as for example in quasicrystals, the spots generally remain discrete. The precise rate at which this falloff takes place is controlled by several factors. In the case of modulated crystals. the most directly relevant factor is T. Janssen and A. Janner, Physica A (Amsterdam) 126, 163 (1984).

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the amplitude of the modulation, i.e., the magnitude of the deviations between basic and modulated density functions. A small modulation amplitude is clearly associated with a fast decay rate for higher-order satellite intensities. Another relevant factor, as originally pointed out by Overhauser,’ is thermal fluctuations. At a given temperature these are governed by the excitation spectrum of the modulated structure. The characteristic features of excitation spectra in modulated crystals will be examined in the following sections, and the special role played by long-wavelength phase fluctuations will be emphasized. The relationship between fluctuating displacement amplitudes (atomic Debye-Waller factors) and satellite intensities is complex, even in the simple case of a sinusoidal displacive modulation, as discussed by Axe6 and Adlhart.’ In the general case where the modulation must be expanded into a series of harmonic waves, one hardly expects simplifications. In Section IV, we shall attempt to shed some light on this question, that is, the behavior of satellite intensities in the presence of thermal fluctuations, in the context of a simple dynamical model of a displacively modulated crystal. In parallel with thermal fluctuations, static fluctuations, as originating from the presence of imperfections, are known to influence the macroscopic properties of real modulated systems. A typical situation, as often described8 in the context of charge-density waves (CDW),is one in which the phase of the CDW is locally distorted due to the interaction of the modulation with either a single impurity (strong pinning case) or a local fluctuation in the impurity concentration (weak pinning case). In many cases these static phase fluctuations are sufficiently severe as to destroy the long-range ordered nature of the modulated system,’*1° the latter breaking up into finite coherent domains, whose average radius defines a “phase-coherence’’ length.’ From satellite width measurements typical domain sizes are found to be of the order of a few hundred cells, although substantially larger values have been deduced from the analysis of dark-field electron microscopy results.I2 The presence of phase distortions and the absence of true long-range order obviously limit the applicability of quasiperiodic models to the description of real modulated systems. As long as the modulation periodicity is not sharply defined, a commensurate superstructure approximation is always possible. The deviation of the real system from the commensurate reference structure

’

A. W. Overhauser, Phys. Rev. B 3,3173 (1971). J. D. Axe, Phys. Rev. B 21,4181 (1980). W. Adlhart, Acta Crystalloyr. A38,498 (1982). H. Fukuyama and P. A. Lee, Phys. Rev. B 17, 535 (1978). Y. lmry and S. K. Ma, Phys. Rev. L e f f .35, 1399 (1975). l o L. J. Sham and B. R. Patton, Phys. Rev. B 13,3151 (1976). ‘ I P. A. Lee and T. M. Rice, Phys. Rev. B 19,3970 (1979). J. W. Steeds, K. K. Fung, and S. McKernan, J. Phys., Colloq. 44, C3-1623 (1983)

’

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is then d e ~ c r i b e d 'in ~ terms of localized phase defects, which are either metastable or stabilized by the presence of crystalline defects. Such phase defects can be viewed as a generalization of the concept of phase solitons or discommensurations often referred to in the context of long-range ordered, quasicommensurate solids. l4 Many physical properties do not depend on the precise value of the modulation periodicity nor on its coherence range. As far as collective excitations are concerned, one expects that effects associated with the finite phase-coherence range will manifest themselves mostly in the long-wavelength phason spectrum. The situation is wholly analogous to that encountered in short-range ordered magnetic systems' where spin waves with wavelengths shorter than the magnetic correlation length are not affected by the finite range of the magnetic ordering. In the present case, with typical coherence lengths of a few hundred cells, only a very small fraction of the modulated system's normal modes will be sensitive to the presence of phase defects. These predictions are consistent with available results16 on molecular dynamics simulations on pure and doped systems. In principle, long-wavelength phasons are also sensitive to commensurability effects, and the occurrence of a gapless phason branch is a direct consequence of the irrationality between modulation and basic lattice periodicities:

q # G/n where n is an integer and G a reciprocal basic-lattice vector. However, microscopic lattice dynamical models' '-I9 of high-order commensurate systems indicate that phason gaps become numerically very small with increasing n values, and cannot be computed reliably (and even less measured) for n 2 10. In that sense, the distinction between incommensurate and high-order commensurate phases (n 2 10) becomes largely academic. 4. SYSTEMS AND MECHANISMS

In the preceding paragraphs we have been concerned with the more general aspects of incommensurate crystal phases, without explicit references to specific systems. There is a wide variety of condensed matter systems which

l4

l9

K. Nakanishi, J . Phys. SOC. Jpn. 46,1434 (1979). W. L. McMillan, Phys. Rev. B 14, 1496 (1976). J. W. Lynn, Phys. Rev. B 11,2624 (1975). K. Parlinski, Phys. Reu.'B 35,8680 (1987). T. Janssen and J. A. Tjon, Ferroelectrics 36,285 (1981). T. Janssen and J. A. Tjon, Phys. Rev. B 24,2245 (1981). T. Janssen and J. A. Tjon, Phys. Rev. B 25,3767 (1982).

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qualify as incommensurate, at least in some respect and within a limited range of external parameters. The corresponding physical situations are necessarily very diverse and cannot all be described within the same framework. For example, systems such as adsorbed monolayers or intercalation compounds exhibit a number of specific features connected with their strong 2D character. These aspects are covered in several major reviews2’-’’ and will not be discussed here again. As quasiperiodic structures, incommensurate structures can be defined in terms of two, or more, mutually incompatible periodicities. These periodicities are sometimes inherent in the nature of the system and can be readily identified. In the case of rare gas monolayers adsorbed on a crystalline surface, the adsorbed atoms experience a periodic potential from the substrate, the periodicity of which provides one of the length scales (a) of the system. The other length scale (b) corresponds to the “natural” equilibrium distance between adsorbate atoms. This is defined as the interatomic distance in the absence of a periodic potential, i.e., in the limit of a perfectly smooth substrate. In the opposite limit of a strong substrate potential the adsorbed atoms are drawn toward the bottom of the potential wells and form a registered (or commensurate) structure. Incommensurate structures are obtained for intermediate potential strengths and general values of the ratio a/b. In addition to epitaxial and intercalated compounds, a number of nonstoichiometric intergrowth systems are known to show compositional incommensurabilities. example^'^-'^ are Hg3-,AsF,, TTF,-,I,, and the “chimney-ladder’’ structures or Nowotny phases -rX, (T = transition metal; X = Si, Ge, Sn). In all cases the competing periodicities are associated with different chemical subsystems. (See Fig. 2.) The Peierls mechanism, which gives rise to charge- or spin-density wave modulations in low-dimensional metals,27has also been discussed along these lines, since one of the periodicities, the CDW wavelength, is supplied by the electronic subsystem. A rather different situation arises if one of the length scales is determined as a result of a balance between competing interactions. To illustrate this point, M. W. Cole, F. Toigo, and I. Tosatti (eds.), Surf. Sci. 125, (1983). S. K. Sinha (ed.),“Ordering in Two Dimensions.” North-Holland, Amsterdam, 1980. F. Safran, in “Solid State Physics” (H. Ehrenreich and D. Turnbull, eds.), Vol. 40. Academic Press, Orlando, Florida, 1987. 2 3 J. P. Pouget, G . Shirane, J. M. Hastings, A. J. Heeger, N. D. Miro, and A. G . McDiarmid, Phys. Rev. B 18,3645 (1978). 24 I. U. Heilmann, J. D. Axe, J. M. Hastings, G . Shirane, A. J. Heeger, and A. G. MacDiarmid, Phys. Rev. B 20,751 (1979). ” C. K. Johnson and C. R. Watson, J . Chem. Phys. 64,2271 (1976). 26 R. De Ridder, G . van Tendeloo, and S. Amelinckx, Phys. Stutus Solidi A 33, 383 (1976). ” P. Monceau, (ed.), in “Electronic Properties o f Inorganic Quasi-one-Dimensional Materials,” pp. 139-268. Reidel, Dordrecht, The Netherlands, 1985. ’O

”

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FIG.2. Structure of Nowotny phases TXJT = Ti, Zr, V, Mo, Cr, Mn, Rh; X = Si, Ge, Sn). Right: sketch of unit cell for two stoichiometric compounds in the MnSi, senes(x = 1.73 f 0.02).26 Left: basic subcells associated with T- and X-sublattices (aT= a,; cT = xcx).

let us consider a simple 1D model of interacting particles in a set of equally spaced double-well potentials. The potential energy of such a system may be written as: A v=C-X; n 2

B +-x$ 4

+ C ~ ~ x n +- 1

Dxnxn-2

(4.1)

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where x, denotes the displacement of the nth particle away from the center of the nth double well ( A < 0; B > 0). The sign of the coefficients C and D determines the nature of the nearest-neighbor and next-nearest-neighbor interactions. For D = 0 and C > 0 the potential energy of the chain is minimum for a simple “antiferromagnetic” configuration of the type ... RLRLRL..., where R and L label the positions of the particles at the bottom of the right- or left-hand half of each double well. Such a configuration corresponds to a chain periodicity equal to twice that of the potential. On the other hand, in the limit where C = 0 and D > 0, the minimum energy is obtained for the configuration * . . LLRRLLRRLLRR . . .,corresponding to a repeat unit of 4 double-well spacings. Finally, if C and D are both positive and of comparable magnitudes, the system is frustrated and the ground state may correspond to a modulated structure of intermediate incommensurate periodicity. Mechanisms based on competing interactions of different characteristic ranges have been p r o p o ~ e d to ~ ~account ,~~ for the occurrence of incommensurate structures in systems ranging from magnetically modulated rare earth metals to structurally modulated insulators. Crystalline biphenyl is an example of a modulated insulator where a simple model with competing interactions, as in Eq. (4.l), can be applied: the biphenyl double-ring molecules display a finite torsional angle, the value of which is determined by a balance between an intramolecular (double-well) potential and intermolecular electrostatic interaction^.^' The molecules are arranged in layers and the coupling between adjacent layers favors opposite torsional angles (C > 0). The same is true for the interaction between layers twice as far apart (D > 0). The frustration leads to a modulated phase where the torsional angle is a periodic function of space along the direction normal to the layers and the periodicity is incommensurate with the interlayer distance. Analogous mechanisms involving competition between long-range Coulomb interaction and short-range overlap forces are believed to be responsible for the occurrence of incommensurate phases in ionic crystals such as NaNO,, Na,CO,, K,SeO,, and many other compounds in the A,BX, (See Fig. 3.)

28

29

R. J. Elliott, Phys. Rev. 124, 346 (1961). H. Bilz, H. Biittner, A. Bussmann-Holder, W. Kress, and U. Schroder, Phys. Rev. Lett. 48,264 (1982).

V. Heine and S. L. Price, J . Phys. C 18, 5259 (1985). K. H. Michel, Phys. Rev. B 24,3998 (1981). j 2 C. M. Fortuin, Physica A (Amsterdam)86,224 (1977). 3 3 M. S. Haque and J. R. Hardy, Phys. Rev. B 21,245 (1980). 30 3’

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21 1

modulation period = 9.220

FIG.3. Mixed order-disorder-displacive modulation in NaNO,. Na+ and NO, ions are found in two alternate positions (full and broken lines)with occupation probabilities proportional to shaded areas. Probabilities and ionic shifts vary sinusoidally along a with a (temperaturedependent) periodicity of 8 to ~ O U . ~ ~

11. Landau Theory of Modulated Systems 5 . NORMALINCOMMENSURATE PHASE TRANSITIONS

Theories for incommensurate crystal phases exist on various levels. Many aspects of incommensurate crystal phases arising from a structure with space group symmetry can already be understood on the basis of a phenomenological theory, the so-called Landau theory of phase transitions. Its application to incommensurate phases poses two problems. In the first place the symmetry group of the incommensurate phase is either a very small subgroup of the symmetry group of the high-temperature phase, which is a three-dimensional space group, or not a subgroup at all. Second, in the usual theory the solutions are in general unstable with respect to nonuniform deformations if a so-called Lifshitz invariant is present. In the original formulation this was a criterion for the case where the application offered problems. For incommensurate phases, however, this Lifshitz invariant is essential. The method for dealing with these problems has been given a long time ago on the basis of work of D z y a l ~ s h i n s k by i ~ ~Levanyuk and S a n n i k ~ vby , ~ Ishibashi ~ and Dvorak,37 and by Bruce, Cowley, and M ~ r r a y . Excellent ~ ~ , ~ ~reviews exist on this

D. Kucharczyk and W. A. Paciorek, Actu Crystallogr. A41,466 (1985). I. E. Dzyaloshinski, Sou. Phys.-JETP 19,960 (1964). 3 6 A. P. Levanyuk and D. G . Sannikov, Sou. Phys.-Semicond. (Engl. Trunsl.)18,245 (1976). 3 7 Y. Ishibashi and V. Dvorak, J . Phys. SOC. Jpn. 44,32 (1978). 38 R. A. Cowley and A. D. Bruce, J . Phys. C 11,3577 (1978). 39 A. D. Bruce, R. A. Cowley, and A. F. Murray, J . Phys. C 11,3591 (1978).

34

35

212

R. CURRAT AND T. JANSSEN

t h e ~ r y ~ to '.~ which ~ the reader is referred for details. Here we shall only be concerned with it to describe the excitations in incommensurate phases, as far as this can be done in the context of the phenomenological theory. For simplicity we shall deal mainly with the case of a one-dimensional displacive modulation in an orthorhombic crystal, with modulation wave vector along one of the axes. This means that the order parameter of the phase transition is two dimensional: there are two points in the star of the wave vector and the small representations are one dimensional. As order parameter one can consider the average of the normal coordinate of the mode that gives rise to the instability in the high-temperature phase: vl =

(Qki),

vl* = ( Q - k i )

(5.1)

In general, the free energy may be expanded in a power series in the order parameter:

F

= F,

+ $A(ki, T)lq12+ 1i Fi4'(v,T ) + ...

(5.2)

where F14' denotes the ith fourth-order invariant. In the case of an incommensurate wave vector there is only one such invariant and

F

= F,

+ $A(k,, T)IvI2+ $Blqi4 + ...

(5.3)

We assume that B > 0. The instability of the high-temperature phase occurs for T = T for which A(k,, TJ = 0. For temperatures close to we have A(ki, T ) = H ( T- 7J

(5.4)

Consequently for T > the minimum of F is obtained for '1 = 0. For T < T there is a modulation with wave vector k,, the displacement of the j t h particle in the unit cell n being given by . = +ki.n nJ

e(ki,j)

+ C.C.

(5.5)

where e(k,,j) is the polarization vector for the j t h particle in the mode belonging to the unstable branch. Because the phase of the order parameter does not appear in the expression for the free energy, its value is arbitrary. For temperatures below T secondary order parameters may appear due to the coupling to q. A higher harmonic in the displacement wave with wave vector nk, may occur if an invariant term

is present for some mode v. The symmetry of (Qnkiv) is determined by the 40 41

V. Dvorak, Lecr. Notes Phys. 115,447 (1980). R. BIinc and A. P. Levanyuk (eds.), "Incommensurate Phases in Dielectrics," 2 Vols. NorthHolland, Amsterdam, 1986.

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

213

requirement that the free energy is invariant under the action of elements of the high-temperature space group. For g = {Rlv} in the group of k, the order parameter transforms according to ~

eiki

(5.71

"X(R)?

where x ( R ) is a phase factor, the character of R in the small representation. The secondary order parameter couples to q if it transforms according to Minimization of the free energy including this coupling F

= Fo

+ja(T

-

'&)lq12

+ aBIvl4 +

C,(?,'"(Q?k,,) n

+ C.C.) + ."

(5.9)

then gives the higher harmonics of the modulation wave. The superposition of these higher harmonics yields the displacements unj = fj(ki. n

+ 4)

(5.10)

where fj(x) is a periodic function of x. Also, coupling to other degrees of freedom, such as the strain, or the electric polarization may on one hand lead to changes in the modulation wave, and on the other hand to macroscopically observable properties. 6. DISCOMMENSURATIONS

As discussed by Landau, the solutions of the minimization problem should be uniform, i.e., independent of the spatial variables. This requires the absence or vanishing of the Lifshitz invariant. In the preceding section we have chosen the wave vector in such a way that the Lifshitz invariant 1

-D(k)(q iQ*/az - q* dq/dz) 2

(6.1)

vanishes. However, when for the chosen wave vector and parameters the Lifshitz invariant does not vanish, it may shift the value of the instability wave vector away from the chosen one. One may use the existence of a Lifshitz invariant to describe the incommensurate phase in another way. One takes the wave vector at a fixed value k, on the axis in the neighborhood of ki.A natural choice would be the wave vector of the commensurate phase below the lock-in transition, when this does exist. In general the coefficient D will not vanish at k = k,. Since the order parameter q,(z) = )qc(z)lei4@) is now spatially varying, the free energy becomes a functional of the function q,(z) which differs in two respects from the form of the function (5.3): there is a nonvanishing Lifshitz term and, because k, is now commensurate, there may be additional

214

R. CURRAT AND T. JANSSEN

invariants. If k, = G / p for some reciprocal lattice vector G and some integer p one has an invariant C -(q; c.c) m

+

with m = p or m = 2p depending on the order parameter symmetry. These terms are called umklapp terms because the sum of the wave vectors involved is not equal to zero but equal to a reciprocal lattice vector. The free energy then can be written as

Here it is supposed that B > 0 and ti > 0. Then it is clear that D = 0 implies a uniform order parameter: dq,/dz = 0. Introducing new variables by qc = pe@,where p and $ are spatially varying, Eq. (6.3) may be written as

(6.4) To find the minimum of this functional, one writes down the Euler-Lagrange equations: t

a2p i

az

2 =

Ap

+ B p 3 + 2CpmP'cosm$ + a

a$ aZ aZ

~-p'-

+ D -aZap 2

=

-2Cp"sinm4

(6.5b)

In the neighborhood of the value of p is small, and one may neglect higherorder terms. Assuming that p is constant, one obtains a24/dz2= 0 + $ = Lz. Conseqeuntly the free energy as a function of 1is:

F

= F,

B + -A2p 2 + -p4 + D p 2 A + -ti2p 2 A 2 4

This expression is minimal for

(6.6)

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

So the wave vector of the modulation is ki = k , incommensurate. In this case A

= CX(T -

-

215

D/u, which is, generally,

T )+ D 2 / u

(6.8) In general, however, the solution of Eq. 6.5 will not be sinusoidal with a single wave vector. M ~ M i l l a n ' ~ has . ~shown ~ that there is another domain of the parameter space where the solution has a different character. To get an analytically soluble problem, one makes the assumption that the spatial dependence of the order parameter is in the phase 4 only, not in p. In that case Eq. (6.5) reduces to (6.9a) 0

=

A

(")'

+ Bp2 + 2Cpm-2cosrnq5 + 20-aZ + -2 az

By choosing variables t = pz and $ may bring Eq. (6.9a) into the form d2 dt2

--II/

=

(6.9b)

rn4

with p2 = (2rnC/~)p"'-', one

-sin$

(6.10)

=

which is the equation of motion for the pendulum. It has as integral of motion (6.11) There are three different types of solution for this equation, depending on the initial conditions. For E < 1 the function $ is oscillatory, for E = 1 one has a motion taking $ from --n at t = -co to + n at t = +a: $

= 4tan-l , ( t - t o )

-

n

(6.12)

This solution corresponds to a kink in the function 4(z) which is concentrated on t n / p and in which the phase changes an amount 2n/rn. Finally for E > 1 one has a regularly spaced array of kinks which corresponds to a so-called lattice of discommensurations: localized regions where the phase changes rapidly over 2n/rn, separated by regions of almost constant 4. The latter have locally the wave vector k , . Integrating Eq. (6.11) for E > 1 gives. t = to

+k

ln'" 41 -

42

k 2 sin2$

W. L. McMillan, Phys. Rev. B 12, 1187 (1975).

,

2 with k 2 = E+l

(6.13)

216

R. CURRAT AND T. JANSSEN

FIG.4. Discommensurations. The function +(z) in the discommensuration regime. The corresponding displacements (solid line)compared with those in the commensurate phase (broken line) are shown in the lower part of the figure.

or in terms of the elliptic functions

*

sin2

=s

n ( y , k)

(6.14)

For E tending to infinity one has k tending to zero and sn(x, k) to sin(x). Hence then

*

2

= -(t - t o )

k

(6.15)

This is the solution discussed before with one incommensurate wave vector. For E tending to 1 one obtains the one-kink solution. In between the number of discommensurations decreases to the point where the last kink disappears. The displacements as a function of coordinates and the corresponding discommensuration solution d ( z ) are exemplified in Fig. 4. Here we have used the approximation that p is not space dependent. Nakanishi and Shiba43 investigated the effect of this approximation by solving numerically Eq. (6.5). They find that the amplitude shows a dip at the positions of the discommensurations. According to these calculations the phase-modulation-only approximation is rather well justified. 43

K. Nakanishi and H. Shiba, J. Phys. SOC.Jpn. 45, 1147 (1978).

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

217

7. EXCITATIONS IN THE INCOMMENSURATE PHASE Up to this point only the statics of incommensurate phases were considered. In order to get the dynamics one has to study fluctuations. In Landau theory one takes as a starting point the expression of the free energy in terms of normal coordinates corresponding to the high-temperature symmetry. In the incommensurate phase the wave vector is no longer a good quantum number because of the lack of lattice translation symmetry. The free energy expressed in a series expansion is F

c

= 1 siV&.&Q:1

dk

f

+ n = 3 11,1*,... ~ ~ " ( k i ~ i ~ ~ . . ~ ~ n ~ n ) Q k l ~ ~ . . . Q (7.1) ~~~~dki~.. Considering only the lowest branch of long-wavelength excitations (long with respect to the IC phase, i.e., with wave vector close to ki),this can be simplified. Assume a parabolic dependence of 0;around the minimum at ki: cot,+, = A K q 2 . Under the assumption that V(4) for k vectors in the neighborhood of k A. is a constant B, the fluctuations are given up to fourth order by

+

6F

=$

s

dqm;a+qQk,+qQz+q

which in the IC phase may be approximated by

obtained by putting in a product of four Q s two of them equal to their average value in the IC phase. Since Eq. (7.3) is again bilinear, it may be diagonalized via a unitary t r a n ~ f o r m a t i o n .Because ~ ~ ~ ~ . of ~ ~the assumed parabolic form of the dispersion curve around ki, one has the degeneracy q i f=qW-ki+',. Then the 44 45

J. D. Axe, Proc. Neutron Scattering Con$, Gatlinburgh, p. 353 (1976). A. D. Bruce and R. A. Cowley, J . Phys. C 11,3609 (1978).

218

R. CURRAT AND T. JANSSEN

unitary transformation is simply

By this transformation, Eq. (7.3) takes the form

s

6 F = 3 dq(oy'2A,A:

+ C O ~ " ~4,m

(7.5)

where the new frequencies are

w Y )=~ a(T

T )+ K q 2 = rX(T - T ) + K q 2 -

+ 3BqT = -2rX(T

-

T ) -t K q 2 (7.6)

+ Bq? = K q 2

The displacements associated with the new normal coordinates A , and for q = 0 are, respectively, given by: 1

6u . = -Aoeik'"e(k, j ) + c.c., lI.l

Jz 1

= -(i4,)eik'"e(k,j)

Jz

+ c.c.,

@,

4,

=0

(7.7a)

A, = 0

(7.7b)

Comparing these expressions with Eq. ( 5 . 3 , this means that Eq. (7.7a) corresponds to an oscillation of the amplitude of the modulation. Its frequency is zero at T , but increases proportional to .-/, The other oscillation, Eq. (7.7b), has an eigenvector that describes a shift of the phase of the modulation. Its frequency is zero not only at but also below q.The two oscillations are called amplitudon and phason, respectively. The modes with nonzero value of q belong to the amplitudon and phason branch, respectively. The value of q should be small, because the diagonalization is based on the degeneracy of the modes with k , q and k , - q owing to the parabolic form of the original dispersion near k i . The phason branch has a linear dispersion near ki. The excitation spectrum of more complex incommensurate structures, when the star of kihas more than two arms, has been discussed by Cox et al.,45a Poulet and and L a u n o i ~for~ the ~ ~case n = 4 and by Walker and et ~ 1 . for ~ ' n~ = 6. G ~ o d i n and g ~ Vallade ~~

+

45a

D. E. Cox, S. M. Shapiro, R. A. Cowley, M. Eibschutz, and H. J. Guggenheim, Phys. Rev. B 19, 5754 (1979).

H. Poulet and R. M. Pick, J. Phys. 12 C6,701 (1981). P. Launois, Thesis, University Paris-Sud, p. 92-121 (1987). 4 5 d M. B. Walker and R. J. Gooding, Phys. Reo. B32, 7412 (1985). 4 5 e M. Vallade, V. Dvorak, and J. Lajzerowicz, J. Phys. 48, 1171 (1987). 45b 45c

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

219

WAVE VECTOR FIG.5. Dispersion curve of the phason branch in the phase-modulation-only approximation. The gap occurs at n/d,where d is the interdiscommensurationdistance.

In the discommensuration regime one can calculate the excitation spectrum for small q analytically in the phase-modulation-only approximation. The equation of motion for a system with energy given by Eq. (6.4) is then

a 2 4 = -2Cpmsinrn4 Miat

a24 az2

-~p'-

Here M is some effective mass. The function 4 ( z , t ) has a static part &(z), which is a solution of the equilibrium condition and an oscillating part which, in the harmonic approximation, may be written as Oexp(iwt). Expanding Eq. (7.8) for small values of 0 gives

d20

~p 7= ( M u 2 - Cp"mcosm~,)O

dz

(7.9)

As shown by S ~ t h e r l a n d this , ~ ~ equation may be integrated, yielding the frequencies w which still depend on a parameter that may be identified with the wave vector. The spectrum also starts here at q = 0 with o ( q ) = 0. The eigenvectors of this mode describe a vibration of the discommensuration lattice where all discommensurations move in the same way. It is the same mode as the original phason mode in the sinusoidal region. There is one gap at qD = n/d,where d is the distance between the disc~mmensurations~~ (Fig. 5). Nakanishi and Shiba4j have solved the equations of motion numerically, also taking into account the amplitude modes. Their result is shown in Fig. 6,

46 47

B. Sutherland, Phys. Rev. A 8,2514 (1973). M. Horioka and A. Sawada, Ferroelectrics 66, 303 (1986).

220

R. CURRAT AND T. JANSSEN

0 i 2 6. Phason and amplitudon dispersion curves in the discommensuration region when also the amplitude has spatial dependen~e.4~ (-), incommensurate phase in the discommensuration regime; (---), commensurate phase. Horizontally, the mode wave vector q (with respect to ki)in units $(kp - kc), with ko = ki(T)and k, = a*/3. FIG.

which shows two branches and more gaps, at wave vectors q = mq,. The lower mode at q = 41,corresponds to the antiphase motion of the discommensuration lattice. The dielectric relaxation phenomena associated with this type of mode have been discussed by Horioka and S a ~ a d a . ~ ’ According to Eq. (7.6) the dispersion of the phason branch is linear around k i . These modes therefore recall acoustic modes. There is, however, an important difference between phason and acoustic modes. For the latter the damping goes to zero when the wave vector tends to zero and the modes remain well defined. The reason is that for long-wavelength acoustic modes all particles in a neighborhood move in the same way and their mutual distances do not change much during a vibration period. The zero frequency for q = 0 of the phason mode is also the consequence of an energy degeneracy: the potential energy of the configuration does not depend on the phase. A change of the phase therefore leads from a configuration to another with the same energy, but in this motion the relative positions of the atoms do change, which gives rise to d i s s i p a t i ~ n . Consequently ~~-~~ the phason mode always becomes overdamped if the wave vector q is small enough. This is probably the reason why in only very few materials the phason branch has been observed. This is only possible when the damping is small enough to follow the dispersion

49

’’

W. Finger and T. M. Rice, Phys. Reo. B 28, 340 (1983). V. A. Golovko and A. P. Levanyuk, Sou. Phys.-JETP 54, 1217 (1981). R. Zeyher and W. Finger, Phys. Reu. Lett. 49, 1833 (1982).

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

22 1

curve to sufficiently low wave vectors. Because the frequency of the phason is zero as a consequence of the existence of a group of continuous transformations (the shifts of the phase) leaving the energy invariant, this mode has been called a Goldstone mode. There are two objections to this use. In the first place the mode is not a propagating mode.48 In the second place it is not the consequence of the breaking of a continuous symmetry. Above the modulation amplitude vanishes and hence the concept of phase translation invariance has no physical meaning. Therefore, one cannot find a continuous symmetry group as required for the applicability of the Goldstone theorem. The continuum theory of incommensurate phases gives definite predictions. There are, however, a number of properties that cannot be dealt with, such as plateaus in the dependence of the modulation vector on temperature and possible gaps in the phason spectrum. Some of these are due to the discrete nature of the crystal. These discreteness effects will be discussed when treating microscopic models in Section IV. They have been studied in a discrete version of the theory of Section 5 by Bruce." He finds a slightly different form of the discommensurations induced by the discrete lattice and a gap in the phason spectrum.

8. DEBYEWALLER FACTOR The modes that are typical for an incommensurate phase, the phason and amplitudon modes, do not form new degrees of freedom of the crystal. The total number remains three times the number of particles in the crystal. However, because the phason branch goes to zero in frequency, it may play a role that differs from that of the ordinary modes. Their influence on the Debye-Waller factor was first studied by Overhauser' to see whether satellites could be observed. He considers a monatomic crystal with a sinusoidal modulation. The phasons are assumed to be long-wavelength fluctuations in the phase 4 of the modulation. When the equilibrium positions of the atoms are

u," = A sin(ki. n

+ 4),

the motion of the particles in one phason mode is given by u,(t)

- + & sin(q .n - at)] (8.2) z u," + $A&{sin[(q + ki) - n w t ] + sin[(q - ki). n - ot]}

N"

A sin[k, n

-

This agrees with Eq. (7.4):a phason with wave vector q is a mixture of phonons with wave vectors q + ki and q - ki. According to Overhauser such a phason 51

D. A. Bruce, J . Phys. C 13,4615 (1980)

222

R. CURRAT AND T. JANSSEN

mode does not influence the main reflections, but it gives a Debye-Waller factor = e-(l/2)mz(q52>

e-W(G+mk,)

(8.3)

Here G is a reciprocal lattice vector. This calculation of the DW factor was repeated by Axe,6 who compares two different approaches. First he considers the Gaussian phase approximation (GPA). The structure factor for a monatomic crystal with a sinusoidal modulation is F(K) = C e i K . n ( e i K . m n ) (8.4) n

with

8,,-ki.n-8,

u,,=Acos(f3,,-4,),

(8.5)

The GPA assumes that the phase 4,, fluctuates with a Gaussian distribution: the average for a function f ( & ) is then given by

(f(4,)>= const

s

f ( 4 n ) e - ( ' / 2 ) ( 0 ~ / ( qd4,, 5~))

(8.6)

The structure factor in Eq. (8.4) then becomes for the crystal with positions as in Eq. (8.5):

-

F(K) = C ime-imeoJm(K A)e-m2(@2)12 A(K + mk,) m

(8.7)

where the function A is a sum of delta functions on the reciprocal lattice of main reflections. Equation (8.4) is in agreement with the Overhauser result. The phase fluctuations are independent of the position. This does not mean that the position fluctuations are everywhere the same: phase fluctuations induce position fluctuations that are larger near the nodes of the modulation wave than near the maxima. This implies that the mean square position fluctuations are periodic in space with a wave vector of 2k,. Also the mean position is influenced by the phase fluctuations. For the modulation of Eq. (8.5) one has (u,)

= A(cos4)cosOn

+ A(sin4)sin8,

q

=A(cos~)

-

(u,) = A[cos 8,(cos 4,, - (cos

and dun = u,

= qcos8,,,

4)) + sin 8, sin $,,I

(8.8a)

(8.8b)

The influence of the phase fluctuations now is a sum of two effects.

-

q K ) = C j m e i ( K + m k J '"Jm(K q)(eiK'dun) nm

(eiK.&)

=

1-1 2m(m - 1)(42)

+ ...

(8.9)

223

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

So the amplitude is renormalized and fluctuations around this new average have a Debye- Waller factor W(G + mki) = $m(m - 1)(42)

(8.10)

This approximation is expected to hold for small values of (4'). Another approximation is used by Axe for the general case: the Gaussian displacement approximation (GDA). Using the new normal coordinates A , and 4, of Eq. (7.4), the displacement of the particle at n in a superposition of amplitude and phase modes can be written as 1

-

6% = - [cos(ki n - @,)A, + sin(ki. n - @&,]eiq"' f

i

1

(8.1 1)

q

If one assumes that the modes are independent, one obtains for the mean square displacement

(6~:)

=

(6~:) cos26,

+ ( 6 ~ $ )sin29,

(8.12)

Here (6~:) and ( 6 ~ 2 )are the mean square position fluctuations associated with amplitude and phase fluctuations, respectively. Assuming a Gaussian distribution of the displacements, one may calculate the structure factor:

F(K)

=

1

(8.13)

eiK.neiK.
n

For the reflection G

+ mki this gives in the case of

Fm(K)= A(K

+ mki)imePw'1(-

sinusoidal modulation

I)'Jm-zl(K * q)Il(wrr)

(8.14)

I

with

-

-

Here J, and If are (modified) Bessel functions and K = K A/IAI. For K q < 1 and n 2 ( 4 ' ) << 1 the results for the GDA are almost identical to those for the GPA. However, when one of these limitations is not fulfilled, the two approximations lead to different results.

9. LIGHTSCATTERING FROM INCOMMENSURATE PHASES The theory for light scattering from incommensurate crystal phases in the context of the phenomenological theory and especially for the "new" phase and amplitude modes was developed by Dvorak and P e t ~ e l t ,Poulet ~ ~ , ~and ~ 52

53

V. Dvorak and J. Petzelt, J . Phys. C 11,4827 (1978). J. Petzelt, Phase Transitions 2, 155 (1981).

224

R. CURRAT AND T. JANSSEN

Golovko and L e v a n y ~ k , ~ 'and ? ~ Berenson and B i r m a ~ Inelastic ~.~~ light scattering may occur because of two mechanisms: first-order Raman or Brillouin scattering from the new modes or via coupling to an acoustic mode. In the Poulet and Pick theory the diffracted intensity for wave-vector transfer K and energy transfer w is calculated for modes in the amplitudon or phason branch:

where winand E correspond to the incident light and noris the polarization of the scattered light. The tensor iapyd(K,m)=

1 nn'

s

dte'"'(6Eaa(rn,t)6Ey*d(rn,,0)> e-ik'(rn-rn')

(9.2)

is determined by the fluctuations in the dielectric tensor E caused by the phase and amplitude fluctuations.

and the displacements u(nj, t ) are given by the new normal coordinates A , and &, Eq. (7.2). The expression for iaaY6(K,w) then contains products of four normal coordinates. Two of them are taken to be those of the static lattice distortion with wave vector ki, whereas the two others describe longwavelength excitations of this structure. This leads to first-order Raman scattering. It gives

where the amplitudon (respectively, phason) Raman tensor is given by

+ K ) + R(-ki, ki + K)] R @= 2[R(ki, - ki + K ) - R(ki + K, - ki)]

R A = 2[R(ki, -ki

(9.5)

For K tending to zero the amplitudon Raman scattering gives a normal contribution. The phason Raman tensor tends to zero for K going to zero. 54

55

H . Poulet and R. M. Pick, J . fhys. C 14,2675 (1981). V. A. Golovko and A. P. Levanyuk, in "Light Scattering near Phase Transitions" H. Z. Cummins and. A. P. Levanyuk, eds.), pp. 169-226. North-Holland, Amsterdam, 1983. R. Berenson and J. L. Birman, f h y s . Rev. B 31,3993 (1985).

225

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

However, its development in powers of K contains a linear term which may account for Brillouin scattering. When the first-order term is Rap

=2

c R&L,KY

(9.6)

Y

and the phason dispersion for small values of K is given by o$(K) = C A a p K a K pthe (integrated) intensity is

Because of the factor 1’ the scattering intensity is proportional to - T and will vanish at the normal-incommensurate transition. However, far away from this transition the sinusoidal approximation of the modulation function is no longer valid. For a damped phason, moreover, Eq. (9.7) gives only the integrated intensity, and for sufficiently strong damping the phason will not be observable. Therefore, from this theory it is not clear whether the phason is observable. If it is, it is so only in certain geometries. Poulet and have determined these from the symmetry of the Brillouin tensor for materials with basic structure Pnma (the symmetry of K2Se04, for example).

111. Supersymmetry and Higher-Dimensional Space Groups 10. SYMMETRY OF QUASIPERIODIC STRUCTURES A quasiperiodic structure can be seen as the intersection of a higherdimensional periodic structure with a three-dimensional hyperplane. 1 * 2 3 7 * 5 8 Its Fourier transform then is nonzero only at points that belong to the projection of the higher-dimensional reciprocal lattice. The latter is easily found from the diffraction pattern. The diffraction spots are located at wave vectors 3 fd

K

=

1 h,a:

(10.1)

i= 1

Considering a: as the projection of af = (a:, a;), K is the projection of the vector (K, K,) that belongs to the reciprocal (3 + d)-dimensional lattice generated by a:. The additional components can be determined using grouptheoretical argument^.^ In the simple case of a one-dimensionally modulated crystal, Eq. (10.1) is of the form

K 5’

’*

= ha*

+ kb* + lc* + mq

P. M. de Wolff, Acta Crystallogr. A33,493 (1977). P. M. de Wolff, T. Janssen, and A. Janner, Acta Crystallogr. A37,625 (1981).

(10.2)

226

R. CLJRRAT AND T. JANSSEN

In this section the modulation wave vector is denoted by q and not by ki as in the preceding section. The four basis vectors in the 4-dimensional space are (a*, 0),(b*, 0),(c*, 0),and (9, 1). The corresponding direct lattice is generated by a1

= (a,

- a), - c),

-9

a3 = (c, -q

= (b,

a,

- 9 . b),

(10.3)

a4 = (0~27~)

To see the 4-dimensional structure which has this periodicity, consider a displacively modulated crystal for which the position of thejth particle in unit cell n is given by n rj fj(q n) ( 10.4)

+ +

-

where f(x)has periodicity 2n. Introducing a variable z, one defines a set of lines with coordinates (n

+ rj + fj(q - n + z), z)

(10.5)

Obviously this structure is left invariant by the 4 basis vectors, Eq. (10.3), and consequently it has four-dimensional lattice periodicity and therefore also four-dimensional space group symmetry. The elements of this space group (for distinction called superspace group) are pairs of elements (g,g,), where the elements g form an ordinary 3dimensional space group and g1= {E(u,},

with E

= _+ 1

(10.6)

+

acts on the variable z: z + EZ 24. (For a higher-dimensional modulation gris a &dimensional Euclidean transformation.) When g = {Rlv) the corresponding element g1satisfies

Rq

= eq(mod A*),

(10.7)

where A* is the 3-dimensional reciprocal lattice generated by a*, b*, c*, and

u,=-q.v+6

(10.8)

The origin may be chosen such that 6 = 0 for E = - 1 and 6 = 0, n, f2n/3, n/2, or fn/3.58 The symbol of the superspace group is then given by the symbol of the 3-dimensional space group formed by the elements g, in parentheses the coordinates of q with respect to a*, b*, c*, and for each generator of the 3D group the corresponding value of 6 indicated by 0 (6 = 0), s (6 = n), t (6 = &2n/3), q (6 = +n/2), or h (6 = +n/3). As an example consider the group Pcrnn(00y)OsO. Here q = yc* and, besides the lattice translations of Eq. (10.3), there are other generators that are given by their action on an arbitrary point of the four-dimensional space: rs = (xa

+ yb + zc,2ns) = (x,y,z,s)

(10.9)

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221

as follows (c, 1): (x,y,z,s)+(-x

(m,1): (x,y,z,s)+(x,

+ 7,y, z + f,s - -:Y) -y + 9, z, s + f) 1

(10.10)

(n,i):(X,y,z,s)+(x + + , y + $ , - z + + , - s - ~ 1Y ) Notice that the three first translations from Eq. (10.3) leave the 3dimensional hyperplane given by (r, -q r + zo) with fixed zo invariant. Therefore, on this hyperplane one has a structure with a three-dimensional lattice translation group and consequently a three-dimensional space group symmetry. For the given example Pcmn(00y)OsO these are Pc2,n if zo=O or n, P2,Jcll for zo = 7112 or 3x12 and Pc for other values of zo. This can be seen by determining all elements of the superspace group that leave the hyperplane (x, y, z, zo - yz)invariant. As explained by Heine, the modulated structure can be viewed as a (nonuniform) superposition of two structures, called C , and C2 by him, with space groups Pc2,n and P2,Jcll, respectively. As seen in the preceding section, incommensurate structures arising from a structure (at higher temperature) with space group symmetry via a continuous phase transition may be described using representations of that space group. For an one-dimensional modulation there is a simple relation between the superspace group symmetry of a structure and the representation of its basic space group according to which it transform^.^ To see the relation consider the displacively modulated structure given in Eq. (10.4). The displacement field (one displacement vector for each atom) can be decomposed into components that belong to the irreducible representations of the basic space group Go, which leaves the basic structure {n + rj} invariant. Because fj is periodic, one has

.

fj(q. n

+ z) =

-

+m

A,jeif(q'"+')

(10.11)

I=-m

Suppose that AIj belongs to one irreducible representation of the little group of the vector lq. Then Afj = Qiq,dlq,v;j) (10.12) where e(lq, v ) is the polarization vector of the mode v. Under an element {Rlv) of the little group, Qlq,"and also AIj get a factor exp( -ilq * v)xI(R) = exp( - ilq v idI),where x f ( R )= x ( R ) is the character of the representation of the little point group. Therefore, the combined action of {Rlv} on the space variables and { E = llv,} on the variable z transforms Eq. (10.11) into

. +

fi - (q - n + T) =

AIjexp[il(q n 1

+ T - u,)

-

-

- + id,]

ilq v

(10.13)

which is identical to Eq. (10.11) if l(u, + q v) = 6,, for all 1. This means that 6, = 16 for some value 6. This is exactly the requirement that the Zth harmonic

228

R. CURRAT AND T. JANSSEN

transforms with the Zth power of the representation to which the basic satellite belongs (cf. Section 11,5). So the internal shift 6 is -ilog[~(R)]. Then the superspace group for a structure arising from a periodic structure with a space group Go via a second-order phase transition should have the group Go as basic space group. The character of the associated representation then follows from the internal shifts 6. This does not necessarily hold for higherdimensional modulations. For a one-q structure (d = 1) arising from a cubic crystal, the basic space group may at most be tetragonal. The same difference between basic space group and high-temperature group may occur for a onedimensional modulation caused by a soft mode that belongs to a degenerate level. 1 1. EXCITATIONS

Excitations may generally be characterized by irreducible representations of the symmetry group of the system. In the present case this group is a superspace group. One can use this symmetry to treat the excitations in a way that is rigorous within the harmonic a p p r ~ x i m a t i o n . ~ ~ . ~ ’ In four dimensions there are four possible directions: four degrees of freedom per particle. A modulated crystal in superspace, however, consists of lines, for which the displacements of the points can be considered’to be “horizontal.” Therefore, three coordinates are sufficient to describe the displacements. If one chooses the three components of the displacement perpendicular to the additional direction, the displacement of the j t h particle in unit cell n, with internal coordinate z, is unj,(t). In the following we shall mostly use this convention, but sometimes it is useful to consider other possibilities as well. Then one should be aware of the fact that displacements along the fourth axis may also be described as displacements in ordinary space (Fig. 7). In the harmonic approximation the potential energy of a displacement field unj, is $(z) =

1 unjr- +(’)(nj, n‘j’; z) - un,jrr

(11.1)

njn’j’

So the linearized equations of motion are wzmjunj,=

1 4(’)(nj, n’j‘;z) - unPjrr

(1 1.2)

n‘i’

The tensor $(’) has as symmetry group the superspace group of the IC phase.

’’ M. B.Walker, Can. J . Phys. 56, 127 (1978). 6o

T.Janssen, J . Phys. C 12, 5381 (1979).

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229

T

-f(r)+U(d

---u(T)

FIG.7. Displacements of vibration modes: the function U ( T )as a deviation from the equilibrium modulation function f ( ~ ) .

In particular it has translation symmetry. @')(nj, n'j'; z) = +(n - n', jj', z

+q

0

(11.3)

n)

where II/ is periodic (with period 27c) in its last argument. Because Eq. (11.2) has lattice periodicity, its solutions are in the form of Bloch waves: (1 1.4) unJr. = m:112 exp( -ik .n - ik,z)Uj(z + q n) J Substituting Eq. (11.4) into Eq. (1 1.2), taking into account Eq. (11.3), the term exp( - ik,t) drops out and one obtains: 02Uj(z)= e-ik'"(mjmf)-1'2+(n, j j ' ; z) Uf(z+ q n) (11.5)

1 nj'

-

.

One sees that the excitations do not depend on k,, which means that the excitations may be labeled by the wave vector k in the Brillouin zone of the basic space group. Notice that translation symmetry in four dimensions does not lead to a finite-dimensional problem as it does in three dimensions. Here we still have an infinite number of coupled equations, corresponding to the fact that in superspace one has an infinite number of degrees of freedom in the unit cell: the degrees of freedom of the points on the lines. Although every mode may be labeled by a vector k in the basic Brillouin zone and a branch label A, the labeling is not unique. If Uj(t)belongs to kR, the function UJ(z) = exp(imt)Uj(z) gives the same displacements for t = 0 if one takes k' = k + mq. nJO . = mTl/ze-"""Uj(q. J n) =

l/2e-i(k +mq).neimq

*"Uj(q. n) = mj1/2e-i(k+mq)'"UJ(q. n) (11.6)

Therefore, there is a label 1' such that k'1' denotes the same mode as k1.

230

R. CURRAT AND T. JANSSEN

Consequently the wave vector is only determined modulo q and modulo the reciprocal lattice, i.e., modulo the quasilattice M*, Eq. (10.1). The solutions of Eq. (11.5) have for each eigenvalue w l n a displacement function U:’(t) for the line corresponding to the j t h particle in the unit cell of the basic structure. Because Uj(t) and $(n, jj’; t) have periodicity 2n in the variable z, one may use the Fourier decomposition

Ajmexp(imt)

(11.7b)

C C D(k - mq, jj‘, m - m’)Ajrmf m’ j ’

(11.8)

Uj(t) = m

to get WLAjm =

where D is the Fourier transform of

D(k, jj’, m)

6:

-

(mjmjz)-1’2$m(n; jj’) exp( - ik n)

= n

(11.9)

beWhen the modulation amplitude goes to zero, the interaction comes t independent and consequently D(k, j j ’ , m) = D(k, jj’)hmo.In that case one recovers the equations for an unmodulated crystal, and the number of solutions is three times the number of particles in the unit cell. The three k = 0 acoustic modes remain solutions of the equations of motion for modulated structures as well, because obviously a uniform displacement does not change the distances. On the other hand, one can consider a uniform displacement in the fourth direction. As said before, such a description is equivalent to one where the displacement is given by (11.10) when the modulation is given by Eq. (10.4). Now, the potential energy is invariant under four-dimensional lattice translations, Eq. (10.3). The internal components of all translations generated by Eq. (10.3) form the set -q (nla n,b n3c) 2nn,, which is dense on the real line. This means that in an arbitrarily small neighborhood of z there is a number from this dense set for which the energy of the configuration is equal to that for t = 0. So, if the energy is a smooth function of the phase, this means that an arbitrary displacement along the fourth axis leaves the energy invariant. Consequently the restoring force is zero, and this mode has zero frequency. It is the k = 0 phason mode. It turns out, however, that the energy is not always a smooth function of the phase. In the sinusoidal region the modulation function is smooth and the energy does not depend on the phase. Consequently in this region the k = 0 phason

+

+

+

-

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23 1

I

I

f (TI

u (TI

f IT1

u (TI

FIG.8. Modulation function f ( t ) and phason displacement function U ( T in ) (a) the sinusoidal regime, (b) the sharp discommensuration limit.

has zero frequency. Its displacement is indicated in Fig. 8a. In the discommensuration region the discommensurations are, generally, not precisely equally spaced, but tend to be centered on (or between) two lattice sites. The modulation function and the displacement function are schematically given in Fig. 8b. Three features can be observed. In the first place, the displacement is only substantial for the neighborhood of the discontinuities, i.e., for atoms near the discommensurations. Second, since the position of the discommensurations is important, there is a restoring force, which is the reason for a finite frequency for the k = 0 phason mode. Finally, the phason mode involves large displacements for atoms near the discommensurations. In the limit of a really discontinuous function, this means jumping from one atom position to another via an energy barrier. Because in this case nonlinear interactions become important, the mode will be strongly damped or even d i f f ~ s i v e . ~It ’,~~ has generally been shown 5 0 that excitations belonging to the phason branch are always diffusive for small enough wave vector k. There is a crossover from diffusive to propagating behavior. This means that at long wavelength (light) scattering is mainly in the diffusive region, whereas neutrons may probe propagating modes.

12. INELASTICSCATTERING The inelastic coherent neutron scattering from excitations in incommensurate crystal phases is given by the differential cross section, for which the expression can be derived analogously to the usual case.61The displacements 61

W. Marshall and S. W. Lovesey, “Theory of Thermal Neutron Scattering.” Clarendon, Oxford, England, 1971.

232

R. CURRAT A N D T. JANSSEN

of the particles are given by

where akVis the annihilation operator for the mode kv. The sum over kv is over all modes, i.e., over all different couples. [Recall that for each (k)v there is a (k mq)v' that denotes the same mode.] Assuming independence of the harmonic excitations, one finds for the differential cross section

+

Here the summation over L = G + mq is over all vectors of the quasilattice, Eq. (10.1).The matrix elements are given by 1

S V (0) ~ , = -Cnkv

+ @kv)

f

(%v +

- %v)l

(12'3)

20kv

where atv is the occupation number for the mode with labels k and v: the usual Bose factor. Apart from this temperature-dependent factor there is the element

6

where is the coherent atomic scattering factor, Y(K) the Debye-Waller factor, and fj the modulation function of the j t h particle in the unit cell. Because f j is periodic with period 271, it can be expanded in a Fourier series: (12.5) For a sinusoidal modulation the coefficients B are simply Bessel functions with, as argument, the scalar product of K and the polarization vector. For a momentum transfer K = G - mq - k, where G - mq belongs to the quasilattice, the matrix element may be expressed as (12.6)

So the contribution to the scattering is determined by terms Blj that are given by the sratic deformation, and terms in A F given by the dynamics. Two processes are here simultaneously at work. For I = m there is direct scattering from the phonon in an unmodulated structure, for 1 = 0 the scattering is for a uniform excitation of the structure belonging to the mth-order satellite. In between the processes work together.

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233

In the expression for the differential cross section given in Eq. (12.2) appears the Debye- Waller factor, which describes the reduction of scattered intensity associated with the-effect of thermal and quantum ( T = 0) fluctuations. For incommensurate phases this factor has been calculated by O ~ e r h a u s e rAxe,6 ,~ and Adlhardt under the assumption of a sinusoidally modulated monatomic structure and within the phenomenological theory (Section 11,8). Jaric6’ has discussed the long-wavelength phasonlike excitations in quasicrystals. Actually in these works the whole DW factor was not discussed but only the part due to the “new” modes, phase and amplitude fluctuations. In the sinusoidal regime the phasons are of course important, because of their low frequency, but actually one should treat these excitations on the same footing as the “ordinary” phonons. Using the formalism of Section II1,ll one may derive the generalized expression for the DW factor. Quite generally, the Debye-Waller factor for the nth particle of a structure is, in the harmonic approximation

’

K(K)= M K - U A 2 >

(12.7)

The displacements for an incommensurate crystal are given in Eq. (11.4). Substitution gives Wnj,(K) = H$(K; z

+q

9

(12.8)

n)

with

In the limit of high temperatures the Bose factor becomes proportional to T and Wj simplifies to (12.9) This relation reduces to the standard expression for vanishing amplitude of the modulation. The function Wj(7)is periodic. In the sinusoidal regime it may be approximated by a constant plus a function with period K, as explained in Section 111,4: In this limit the structure factor becomes Wj(z) = Woj + W2jcos(2~). F(K) =

62

7

1

P2n

Jo

.

.

exp[iK rj + iK Ajsin(z) - W,(K,z)

M. V. Jaric, J . Phys., Colloq. 47, C3-259 (1986).

+ imz] dz

(12.10)

234

R. CURRAT AND T. JANSSEN

Using the Jacobi-Anger relation one obtains ( j = 1)

F(K)

1(-

= 6(K - G - rnq)imCwo

-

l)fJ-m-21(K A)Zl(W2) (12.11)

1

where JI and I,,, are (modified) Bessel functions. Equation (12.11) is the same as Eq. (8.14). The amplitude A occuring in Eq. (12.11) is the thermodynamic average and corresponds to the renormalized value q in the phenomenological theory.

13. TRANSFORMATION PROPERTIES AND SELECTION RULES Excitations of systems with a symmetry group transform according to representations of that symmetry group. For an ordinary crystal the symmetry group is a three-dimensional space group, for which the irreducible representations are labeled by a star of k vectors and a branch label that indicates a representation of the point group of the group of one of the k vectors in the star. For incommensurate crystals the symmetry group is a superspace group. Such a group is isomorphic to a higher-dimensional space group. The wave vectors characterizing the excitations have a vanishing fourth component. Therefore, they transform trivially under the lattice translations in the additional dimensions. These translations form an invariant subgroup of the superspace group, and this implies that the excitations belong to irreducible representations of the factor group, which is an ordinary threedimensional space group. The excitations have displacements u n j r ,Eq. (11.4). Under a transformaof the superspace group these displacements transform tion ({ Rlv}, according to

{EIu~))

U;,j,r, = R u n j r ,

+ u, This implies as transformation law for the periodic function q.(t): Ui,(q .n' + T')e-ik'."' - RV,(q. n + t)ePik'" where

R(n

+ rj) + v = n' + rf;

t' = €7

(13.1) (13.2)

Here k' = Rk. If one introduces for each particlej and each superspace group element g a lattice translation vector w(g,j) = ri. - Rrj - v, Eq. (13.2) leads to the following transformation of the Fourier components A$: A'Ji, R ACIJ,ei(Rk+IP).W(Q,j)+ilul (13.3) ~

In particular, when g is a lattice translation a, R = E = 1 and j' = j , one has A,= ~ ~ ~ ~ ~ i [ ( k + W . a= + i AI I~j e~i kl ' a because in this case uI = -q

a. Equation (13.3) means that an excitation

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

235

belonging to wave vector k is transformed into one belonging to Rk. Therefore, excitations belonging to a vector k span a representation for the group of k, i.e., all elements g with Rk = k. A displacement with wave vector k which has Fourier components Alj different from zero for 1 = k lo and j = j , span a two-(lo # 0) or one-(/, = 0) dimensional representation. Therefore, the excitations with wave vector k span a representation with character

x(d = x@)

1 C exp[i(k + l

*

w(s,j) + W

(13.4)

j

j.=j

where xv is the character of the 3-dimensional vector representation of the point group. Special excitations are the phason and the amplitudon. For the latter the displacement function U(z) is a (small) factor times the modulation function. Because the latter is invariant under the symmetry group, the same is true for Uj here. Therefore, the k = 0 amplitudon transforms according to the fully symmetric k = 0 representation. The displacement function for the phason is proportional to the derivative of the modulation function. Therefore, it belongs to a k = 0 representation with character E for all point group elements. We exemplify the procedure on the modulated y-phase of Na,CO,. This has superspace group C2/rn(aOy)Os as symmetry group. The translation group has basis (a, - 2x4, (b, 0), (c, - 2ny), and (0,2n). There is a centering translation (+a i b , -nu). Furthermore there are elements ({Rlv}, {E~U,})= ({2,,,0}, { - 1lo}), and ({rn,,,O}, { 1In}).Under R = 2, two out of the four Na atoms remain invariant, under my all atoms remain invariant ( j ' = j ) . For k = 0 Eq. (13.4) then leads to the character

+

+

~ [ 1 , 2 ~ , m , , T ]= [3 x 6 x (2N + l), - 1 x 2 x 1, 1 x 6 x (2N

+ l),-3

x 2 x 11

(13.5)

Here N denotes the number of pairs k1 that occurs. For a commensurate structure this number is finite, for an incommensurate structure infinite. Expression (13.5)is the result in the approximation of rigid nonlibrating ions. When librations are taken into account there is an extra contribution obtained from Eq. (13.3) by replacing xy(R)by x V ( R )det(R). This gives for the total character, Eq.(13.5):[24 x (2N l), -2,4 x (2N + l), -61. Thus therepresentation is reducible into a sum of 7(2N + 1) - 2 Alg, 5(2N 1) 1 A1,, 5(2N 1) - 1 B,,, and 7(2N 1) 2 B,, representations. In principle all excitations belonging to one irreducible representation are coupled. For vanishing modulation amplitude this coupling tends to zero. Then the 5 A,, 6 A,, 4 B,, and 9 B, excitations remain for k = 0. Among them are the 3 acoustic modes. The phason and amplitudon have main Fourier components A,j for 1 = 1, and belong to B, and A,, respectively.

+

+ + +

+ +

236

R. CURRAT AND T. JANSSEN

IV. Microscopic Models

14. FRENKEL-KONTOROVA MODEL The Landau theory allows one to understand quite well a number of features found in incommensurate phases. A step further requires attention to discreteness effects and to the microscopic origin of the phenomenological parameters. A number of simple models has been studied to clarify these effects. Most of them are essentially one dimensional. They are applicable to 2 D and 3D systems if one treats these in the mean-field approximation. The models consider a linear chain with linear and sometimes nonlinear interactions between the particles and eventually an interaction with a background potential. Spin models have also been studied, but since these are less adapted to the study of the dynamics, we shall not consider them here (see the review by Yeomans in this volume). When the interaction between particles with mutual distance s times the lattice constant is denoted by Osand the interaction with the background by Q0, one has for the potential energy (14.1) where x, is the position of the nth particle or its displacement from an equidistant array. A well-studied model, originally proposed for dislocations of epitaxial monolayers, is the Frenkel-Kontorova or Frank-van der Merwe Here Oo is a sinusoidal potential and Ol a harmonic nearest-neighbor interaction. In this case the variables x , are just the positions. The potential energy is -

x,-1 - b)2 -

a

(14.2)

The model describes, for example, a layer adsorbed on the face of a crystal. The equilibrium configuration satisfies

+

(3.

a(2x, - x , + ~- x n p 1 ) A - sin=0 (14.3) 2:n For long-wavelength solutions, this equilibrium may be approximated by a differential equation. For new variables y, = x,/2nu and t = z / p with 63 64

Y. I. Frenkel and T. Kontorova, Zh. Eksp. Teor. Fiz. 8, 1340 (1938). F. C. Frank and J. H. van der Merwe, Proc. R. SOC.London, A 198,205 (1949).

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

p2

= (u/27~)~cr1 one

237

obtains for slowly varying y , d2 y ( t ) = sin y ( t ) dt

(14.4)

which is the pendulum equation which appeared already in Section 11. This continuum approximation has been studied by Frank and van der M e r ~ e . ~ ~ There is a kink solution or a discommensuration lattice with mutual distance I, which depends on the so-called misfit parameter, the ratio b/a. Theodorou and Ying,66 and Snyman and van der M e r ~ have e ~ studied ~ discreteness effects and found a phase diagram with several phases. Aubry and Le Daeron 6 8 3 6 9 proved rigorously several properties of this model and generalizations of it. For il = 0 the solution of Eq. (14.3) is an equidistant array with x, = xo + nb. When the background potential does not vanish there is a conflict between the periodicity of the substrate and the natural periodicity of the chain. When the misfit parameter b/a is rational it can be proved that the ground state is a commensurate superstructure: if b/a = L / N the period is Nu. If x, is a solution with minimal energy x; = x, + a is also one. A configuration with a higher energy now is one that coincides with x, for n --t - oc) and with x; for n -+ co. The transition between the two regions is formed by a kink, a topological excitation. A shift of this excitation costs energy: the kink has to surmount an energy barrier, the Peierls-Nabarro barrier. When the misfit parameter is irrational, the equilibrium configuration is given by x, = xo

+ nl + f ( n 1 + xo)

(14.5)

where the modulation function f ( x ) has periodicity a. The behavior of f ( x ) depends on the coupling A. For almost all irrational values of the misfit b/u there is a critical value of 1such that for 1< Ac the function is smooth, whereas for 1 > Ac the function is discontinuous with an infinite number of discontinuities. The transition between the two regimes coincides with the appearance of a nonvanishing value of the Peierls-Nabarro barrier. The transition has been called a breaking of analyticity.” The critical properties in the neighborhood of the transition have been studied by Peyrard and A ~ b r y . ~ ’ m G. Theodorou and T. M. Rice, Phys. Rev. B 18,2840 (1978).

S. C. Ying, Phys. Rev. B 3,4160 (1971). J. A. Snyman and J. H. van der Merwe, Surf. Sci. 42, 190 (1974). 6 8 S. Aubry, J . Phys. (Paris)44, 147 (1983). 6 9 S. Aubry and P. Y. Le Daeron, Physica D (Amsterdam)8,381 (1983). ’ O S. Aubry, in “Solitons and Condensed Matter Physics” (A. R. Bishop and T. Schneider, eds.), pp. 264-277. Springer, Berlin-Heidelberg, Federal Republic of Germany, 1978. ” M. Peyrard and S. Aubry, J. Phys. C 16, 1593 (1983). 66

67

238

R. CURRAT AND T. JANSSEN

REGISTERED PHASE

b/2a

FIG.9. Devil’s staircase in the Frenkel-Kontorova model. When one follows 1 as a function of b / a for, for example, C/o;= 36 one obtains a Devil’s staircase. Indicated are commensurate phases. In the gray regions one has higher-order commensurate phases.” The constants C and wo are related to the interparticle spring constants and the strength of the background potential, respectively.

The average distance I of the chain depends on the parameters b and A. For fixed A the value of 1 is a monotonous function of the natural lattice constant b (or of the misfit parameter). The behavior is, however, rather spectacular. It is a function that has an infinite number of flat plateaus at commensurable values. There is even a related exact solvable model7’ where the function has everywhere a zero derivative and nevertheless is increasing (Fig. 9). Such a function has been called a Cantor function, or a Devil’s staircase. All these models consider an elastic chain in an undeformable background potential. The case of deformable potentials was discussed by I ~ h i i . ’ ~ 15. DIFFFOUR MODEL

Another (semi-)microscopic model for IC phases in dielectrics with a soft mode has been studied in detai1.’7-’9,74 We call it the discrete frustrated 44 (DIFFFOUR)model. The model is a one-dimensional chain, embedded in a 372

73 74

S. Aubry, J. Phys. C 16,2497 (1983). T. Ishii, J. Phys. SOC.Jpn. 52, 168 (1983). T. Janssen and J. A. Tjon, J . Phys. C 16,4789 (1983).

239

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

dimensional crystal. The chain consists of identical atoms with nonlinear firstneighbor and linear second- and third-neighbor interaction. Indicating the positions of the atoms by nu u,, where a is the lattice constant, the potential energy reads

+

V

=

In-a2(.,

- u,-1)2

Y + -(u, 4

- u,_1)4

B + -(u, 2

- u,-2)2

6 + -(u, 2

- u,-3)2

(15.1)

The model allows one to study the properties of IC phases, in particular those effects that are called discreteness effects. It is certainly an oversimplification but it turns out to be very rich in structure. In contrast to the FrenkelKontorova model it has continuous translation symmetry and, therefore, acoustic excitations. The equilibrium positions of the atoms are found as solutions to

+ B(2u,

- u,-2 - u,+2)

+ 6(2u, -

u,-3

- u,+g) = 0

(15.2)

Two methods can be used to determine the solutions of this infinite set of coupled nonlinear equations. One is to look for periodic solutions: x , + ~ = x, for some (large) value of N . Then the problem is reduced to finding the solutions of a finite set of coupled nonlinear equations, for which algorithms can be given. Another method is to introduce the difference variable x, = and to write Eq. (15.2) as u, - u, (a

+ 28 + 36)x, + yx,3 + ( B +

26)(x,-,

+ x , + ~ )+ 6(x,-, + x , + ~ )= const (15.3)

This is a recurrence relation relating x, + and (x, + x,, ,x, - ,x, - 2). This leads to a (volume-preserving) mapping in four-dimensional space. The orbits correspond to extremal points of V. The problem then is to find among them the configurations that minimize V. The model always has the original equidistant configuration as a solution with extremal energy. The harmonic oscillations around this configuration have frequency given by

+ Bsin’k + 6sin2-

(15.4)

For fixed B and 6 all frequencies are real for a greater than a critical value ai, but for a = aithe mode at the minimum of the dispersion becomes unstable. If one considers a as T-dependent, this is a soft mode. For b/6 between - 6 and

240

R. CURRAT AND T. JANSSEN

+ 2 and positive 6 the instability occurs with wave vector k, given by (15.5) which is, generally, incommensurate. The transition occurs for a = ai= - p + p2/46. Below ai the ground state is (generally incommensurate) modulated. In terms of the variable x , the potential energy may also be written as (15.6) where A = a + 2p + 36; C = y ; B = p + 26; D = 6. For negative values of A this is the potential energy for a chain of particles, each in a double well potential, with harmonic first- and second-neighbor interaction. When a is sufficiently negative the wells will be so deep that a commensurate structure will be the ground state. So besides the normal-to-incommensurate phase transition the model has a lock-in transition at a = a,. One may say that the stability of the IC phase is the effect of the competition between the interactions with constants B and D . The phase diagram, i.e., the ground-state configuration as a function of the parameters, and the structure of the solutions have been studied in detail (Fig. 10). In general, most of the properties agree with the picture arising from the phenomenological theory. There is a soft mode leading to an instability at ki given by Eq. (15.5). Below the transition there is a region where the modulation is to a good approximation sinusoidal with an amplitude that varies as (ai- a)'". For lower values of a there is a lock-in transition, and in the region close to this transition the structure may be described as consisting of nearly commensurate domains separated by discommensurations. Here the structure is very well indicated by a series of integers giving the length of intervals where x , has the same sign. As an example consider the sequence (3333433334), abbreviated to ((344)2). This is a periodic structure with period N = 32. The solution starts with 3 values of x of the same sign, followed by 3 of opposite sign, etc. After 12 sites there is an interval with 4 consecutive values of x of the same sign. Without the intervals of length 4 the period would be 30/5 = 6 and the wave vector commensurate: k = 2n/6. The two longer periods give rise to a structure with wave vector k = 5n/ 16. The longer intervals correspond to discommensurations with distance b = ~ ( -l ki/k,)-'

between them. In the example b

=

16a.

(15.7)

STAGE ORDERING IN INTERCALATION COMPOUNDS

24 1

thickness, di. Ulloa and K i r c ~ e n o w ' ~ ~used ~ ' ~ *a Morse potential to characterize the interlayer potential in their analysis, which also included the full expression for the bending energy (not just the small-slope approximation). They found that the energy barrier is a strong function of d , , due to the anharmonic terms in the interlayer potential. Most intercalates in graphite fall in a region where the energy barriers for domain coalescence are smalldue to the large values of d , and the resulting small interlayer interaction. However, some species fall in a region of large energy barriers. Ulloa and Kirczenow argue that this behavior should occur for Na and Tm (di = 1.20 and 1.27 A, respectively) and that this explains the observed difficulties in the intercalation reactions of these materials. Their results imply that the slowing of intercalation kinetics by the elastic energy barriers is a very nonuniversal phenomenon and depends in some detail on the intercalant species. OF STAGING AND INTERCALATION 10. KINETICS

The calculations of the energy barriers that prevent the merging of domains in intercalation compounds signify that kinetic effects may be important in understanding staging and the intercalation process in real materials. In this section, models of staging and intercalation kinetics are reviewed, along with a discussion of the small amount of experimental data on the kinetics of the staging transition. The kinetics of staging and intercalation is a problem of current research; the material reviewed here represents a first step. The existence of finite-size domains in intercalation compounds is in itself a kinetic effect. In true equilibrium, the material would minimize all elastic strains by forming macroscopic intercalant layers. However, as the early experiments of Daumas and H e r 0 1 d ~showed, ~ the intercalation compound is most likely composed of finite-size domains. Otherwise, restaging on relatively short time scales would involve the motion of macroscopic layers-a highly unlikely process! Recent theoretical studies have shown that, indeed, the kinetics of intercalation do result in a domain structure. The earliest study was that of S a f r a ~who ~ , ~used ~ the simple lattice-gas model described in Section 111as the starting point for a kinetic calculation of the evolution of a dilute stage-1 compound (with 50% in-plane vacancies) to a stage-2 material with a saturated in-plane density. The kinetic constraints, which prohibit diffusion of the intercalant atoms through the host planes, result in the Daumas-Herold S. E. Ulloa and G. Kirczenow, Phys. Rev. Lett. 55,218 (1982). S . E. Ulloa and G. Kirczenow, Phys. Rev. B: Condens. Mutter [3] 33, 1360 (1986).

242

R. CURRAT AND T. JANSSEN

in the limit just the expression of the ANNNI model. Since in the present model there is one parameter more, there is more room for locating certain structures. For example, if the parameters are such that for T = 0 the parameters have a value corresponding to an IC ground state, there will be no lock-in transition. In the ANNNI model the T = 0 structure is always commensurate. 16. EXCITATIONS I N INCOMMENSURATE PHASES:SIMPLEMODELS

The lack of lattice periodicity in IC phases means that the usual treatment of lattice vibrations breaks down. There is, strictly speaking, no 3-dimensional Brillouin zone, and therefore no labeling by vectors in this zone or gaps on the boundaries. The consequences of this have been studied on very simple model systems. Although these do not describe actual systems, they may point out what are the differences in dynamics between ordinary periodic crystals and incommensurate crystal phases. One of the simplest is the modulated spring model.75 It is a linear chain of particles with harmonic interactions. There is, however, no lattice periodicity because the spring constants are periodic with a period that is incommensurate with the chain. It could represent a quasi-one-dimensional system embedded in a periodic host lattice to form an incommensurate composite system. As potential energy one takes (16.1)

The equation of motion for this harmonic system leads to the eigenvalue equation mw2un = an(un - un-1)

+

%+1(un

- un+1)

+ Pnun

(1 6.2)

Here the constants are quasiperiodic. The most simple assumption is a, = a[l

+ ECOS(qn + +)I;

p,

=0

(16.3)

When the model represents a chain in a host lattice, q = 2n/a, where a is the lattice constant of the host. Equation (16.2)forms an infinite set of coupled linear equations. Because of the quasiperiodicity, the solutions are of the form u, = eik"U(qn),

75

U ( z )= U ( z

C. de Lange and T. Janssen, J. Phys. C 14,5269 (1981).

+ 271)

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

243

This property, however, does not lead to further simplification. The system remains an infinite set for incommensurate q. One may try to get an insight in its properties by considering commensurate cases with q = 2 n L / N for values of N that tend to infinity. Since each irrational number may arbitrarily well be approximated by rationals, one may draw conclusions if the behavior of the spectrum tends to a limit. This is the case here. Consider as an example the Fibonacci series: a series of rational numbers that form the successive truncations of the continued fraction expansion of the golden number l ) ; $ , + , j , $ , & ,..., i.e.,F,,/F,,+,withF,+, =F,+F,-,.Forqapproximated by 2nLIN the spectrum has N bands. The extreme limit of the modulation of the spring constants is the case E = 1. In order to avoid springs of strength zero one takes (p = n / N . Even then the situation is rather unphysical because the springs may become arbitrarily weak when N goes to infinity. This limit, however, shows most clearly the behavior of the spectrum that remains in weakened form also for E < 1 . For the Fibonacci approximants to$(fi - 1)the N = F, spectral bands are given for the first approximations in Fig. 11. From the calculation it is clear that the spectrum becomes self-similar. This becomes even more clear if one takes all rational numbers L I N with N below a certain limit. For example, this has been done in Fig. 1275 for all N smaller than 50. Also here the selfsimilarity is clear from the diagram: the same butterfly-type pattern occurs all over the diagram on different scales and usually slightly deformed. When N tends to infinity the total length of the bands tends to zero: at the end it is a pure point spectrum consisting of discrete points that are nowhere dense. It should be stressed that this conclusion here is only based on numerical calculations. For other models there are rigorous statements. There it has been shown that not all irrational numbers have this behavior but the ones that behave differently form a set of measure zero. From the calculated spectra one may deduce an algorithm to determine the gap structure. It turns out that it is determined by the continued fraction expansion. This was observed for the first time by Hofstadter for a related problem.76 There is also a relation with a number theoretical concept: the Farey numbers. Starting from 011 and 1/1, one adds to the collection of rational numbers pn/qn the rational numbers ( p , p n +,)/(q, qn+ In this way the Farey numbers of successive degree are obtained. When L I N is a given Farey number the spectrum of the chain is a combination of the spectra of the two constituent Farey numbers. For example, the spectrum for q / 2 n = 215 is obtained from the spectra of 112 and 113. In one dimension the density of states becomes infinite at the gap boundaries. Therefore, one considers the integrated density of states. This

+(a-

+

'' D. R. Hofstadter, Phys. Reu. B 14,2239 (1976).

+

244

R. CURRAT AND T. JANSSEN

1 W*

I

. . . . . 1

1

I

:

i

:

:

:

:

I

-1 2 1 5

8 13213L 3 5 8 13 2 1 3b 55 89

FIG. 1 1 . Spectra of the modulated spring model for consecutive approximations to 4 / 2 n =

+(Js- 1).

monotonically nondecreasing function is rather pathological for the modulated spring model with E = 1. In the gaps the function is constant. Because the measure of the spectrum goes to zero for N tending to infinity, the IDS is everywhere constant but increases nevertheless from zero at o = 0 to 1 at

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

245

FIG. 12. Spectra of modulated spring model for all commensurate values L / N with N < 50 l).75For each value of q/2n = L / N , a vertical broken line indicates the spectrum, i.e., the values of o2in a band. The q-axis runs horizontally from 0 to 271.

(E=

246

R. CURRAT AND T. JANSSEN

0.0

1.o

0.5

1.5

2.0

w

FIG. 13. Integrated density of states for the modulated spring model with L / N

= 21/34.

o = om,,.This is again an example of a Cantor function (Fig. 13) as encountered already in Section 4,14. When E < 1 the measure of the spectrum for an irrational number becomes nonvanishing. Only the strongest gaps survive and the fine structure vanishes. The strong gaps may be obtained from a perturbation calculation. The interesting self-similarly also vanishes if the modulation function is no longer sinusoidal. There are, however, also other cases where the self-similarity reappears. A related model has been studied extensively in the literature. It is actually a tight binding model where electrons may hop from one site to the next with a fixed hopping probability and modulated site energies, but it is equivalent to a chain with potential energy I/ =

- C [ ~ ( U-,u , - ~ ) *

-

n

(2 + V , ) U , ~ ] ; V, = A c o s ( ~ ~+ 4 ) (16.4)

This is Eq. (16.1) with CI, = - 1, Bn = 2 + V,. The related equation of motion is called the Harper equation, or the almost Mathieu equation because it is a discrete version of the Mathieu differential equation. It has been studied

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

247

numerically by H o f ~ t a d t e rand ~ ~ Aubry and Andre,77 and many theorems about its spectral properties have been obtained by Avron and and Bellissard et u1.81-83 Also the eigenvectors of the incommensurately modulated chain are different from those of the periodic chain. It has been shown that a quasiperiodic potential allows localized states. In the almost-Mathieu equation all states are delocalized for R < 2 and localized for 2 > 2, for almost all irrational numbers q/27c.77*80s82 In the modulated spring model this can be seen from Fig. 14 where the 34 eigenvectors are given for L / N = 21/34. Some of the eigenvectors describe delocalized excitations, but others are localized in the 34-fold unit cell. Since the calculation was made in a superstructure approximation, the displacements are in fact periodic, but it turns out that for N tending to infinity there are modes with only one concentration point in each unit cell. Therefore, these can be considered as localized. This is not the place to discuss this type of localization. The remarks made on the models were meant to illustrate the differences one may encounter in the lattice dynamics of incommensurate phases compared with that in periodic crystals. There are many effects, but usually they are subtle and will be hard to detect. The sinusoidal modulation of the force constants in Eq. (16.3) is the most simple one and leads to spectra that show a manifest self-similarity. This gets lost when one considers modulation functions with more harmonics. One can also use a function that is determined via a (sinusoidal) modulation from a more realistic interaction. In Ref. 84 a Lennard-Jones potential is taken. Then it is possible to find phase and amplitude modes also in this model. Because there is no coupling between the displacements and the force constants in the model of Eq. (16.1), there is no way to find phason- or amplitudon-like excitations there, which is a weakness of the model. For quasicrystals similar models have also been considered. Kohmoto and Banavara5 have studied the modulated spring model with periodic spring - 1) and f ( x ) is a periodic constants a, given by a, = f ( n z ) , where z = function of period one that is equal to a for 0 5 x < z and equal to a’ for z g x < 1. The spectrum is very similar to that of Fig. 11 and the eigenvectors

*(a

” S.

Aubry and G. Andre, Ann. Isr. Phys. SOC.3, 133 (1980).

’’J. Avron and B. Simon, Commun.Math. Phys. 82, 101 (1981).

J. Avron and 9. Simon, Duke Math. J . 50, 369 (1983). B. Simon, Adv. Appl. Math. 3,463 (1983). J. Bellissard, R. Lima, and E. Scoppola, Commun.Math. Phys. 88,465 (1983). J. Bellissard, R. Lima, and D. Testard, Commun.Math. Phys. 88,207 (1983). 83 J. Bellissard and B. Simon, J . Funcf. Anal. 48,408 (1982). 84 N . V. Cohan and M. Weissmann, J. Phys. C 16,5581 (1983). 8 5 M. Kohmoto and J. R. Banavar, Phys. Rev. B 34,563 (1986). 79

‘O

’’ ’’

248

R. CURRAT AND T. JANSSEN

W2

1 126

-

1 I25

I

:::: 0 318

0 318

0 305

0 30I

-

,

0 301

-7

0 080 0 077

-

0 077

,

- ,

0 018

0 017

P 0 001 IIIII.,,,,,,,

,, ,

IIIIIIII

I I I I I I I l l I 1 I l l I I l l , , I , I I I t , ,

0001

I 1 I l l ,

0000 FIG. 14. The 34 eigenvectors for k = 0 of the modulated spring model ( L / N = 21/34). The length of the vertical bars is proportional to the atomic displacements.

to Fig. 14. The same model has been investigated by Lu et dE6 By means of a transfer matrix technique the spectrum and integrated density of states for the Fibonacci chain were determined numerically. The self-similarity of the spectrum found here has been proved by Kohmoto and B a n a ~ a r . ~Odagaki ’ and NguyenE7have considered the two-dimensional case of a Penrose tiling with harmonic springs between the vertices. For all the values of the ratio of the (two) spring constants there is a rich gap structure and an integrated density of states that resembles a Cantor function.88 86

*’

J. P. Lu, T. Odagaki, and J. L. Birman, Phys. Rev. B 33,4809 (1986). T. Odagaki and Nguyen, Phys. Rev. B 33,2184 (1986). T. Janssen, in “Incommensurate Phases in Dielectrics” (R. Blinc and A. P. Levanyuk, eds.), Vol. 1, pp. 67-142. North-Holland, Amsterdam, 1986.

249

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

17. DYNAMICS OF THE DIFFFOUR MODEL Although the DIFFFOUR model has not been developed to describe actual crystals, it is a simple model that nevertheless has many of the features of incommensurate phases and gives one the possibility of studying physical properties in a tractable model. Equilibrium configurations for the chain are obtained from the solutions of the coupled nonlinear equations (15.3) and (15.4). The harmonic oscillations around these equilibria are determined by the linearized equations of motion. Here one can distinguish two cases. First the model may be considered in the u variables. Then the linearized equations of motion for small deviations e,(t) = E , exp(iwt) are mo2E, = (2a + 2P -

+ 26 + P n + -

P n + 1)En

-

-

- (a

+ P n + 1)%+

1

- (a

+Pnkn-

1

with p n = 3(u, - u , _ ~ ) ’ (17.1)

The solution u, is stable if all eigenvalues w 2 are positive. In a commensurate approximation the solution u, has periodicity N . Then the eigenvibrations are labeled by a wave vector k in the Brillouin zone ( - n / N < k < n / N ) and a branch label v (1 5 v 5 N ) . In this case the model is translationally invariant and has an acoustic branch tending to w = 0 for k = 0. In the x variables the equilibrium conditions are the same, but the linearized equations of motion are different: mw2en= ( a

+ 2B + 36 + pn)e, + (j+ 2 6 ) ( ~ , +- ~

+ d(En-2 + %+Z)

(17.2)

In this case the model has no translational invariance and no acoustic modes. So the same model may be used to study the behavior of incommensurate structures originating from a soft mode in either an acoustic or an optic branch and to study the coupling with either of the two types of modes. For a commensurate approximation one may draw the dispersion curves wk2y as a function of the wave vector k. An example is given in Fig. 15 for oscillations around an equilibrium with period N = 11. Here the case of the translationally invariant model is considered. The modulation function in this case is almost sinusoidal. One observes two branches that go to zero for k tending to zero. The one with the lowest slope corresponds to the phason branch, the upper one to the acoustic phonons. As discussed in the previous section, the scattering of, for example, neutrons, is determined by the static modulation function and the dynamical coefficients

250

R. CURRAT AND T. JANSSEN

3

t

1.5

-k FIG. 15. Dispersion curves for the translationally invariant DIFFFOUR model; sinusoidal solution with L j N = 4/11 ( a = - 1.5, = + 1, 6 = 1.614).19

Ak,Yj.Here there is only one particle per original unit cell. Therefore, j = 1 and the index j is omitted. The coefficients A t are the Fourier coefficients of E , . It contributes to the scattering at k mki (modulo the reciprocal lattice of the basic structure). Therefore, one may plot the absolute value of A t at the in the (k,w)plane. This has been done in Figs. 16-19. position ( k mki, okv) The intensity is indicated by the length of the dash at the given position. For a periodic structure with period 1 this would give a line of delta peaks coinciding with the dispersion curve. In Figs. 16 and 17 two examples are given for the translationally invariant model. One example is chosen in the sinusoidal region. Then the original dispersion curve is still observable, and this curve has two minima at frequency zero: the acoustic mode at k = 0 and the phason at k = k i , where a gap exists between phason and amplitudon branch. The second example is in the discommensuration region. The spectrum now shows many gaps. There is a small gap in the phason branch at k = k i :its minimum is not equal to zero. The same phenomena can be seen in Figs. 18 and 19 for the not-translationallyinvariant model. One example is chosen in the sinusoidal region, and the optical branch dispersion curve can again be recognized. The messy picture of

+

+

EXCITATIONS I N INCOMMENSURATE CRYSTAL PHASES

25 1

Frequency

Wave vector 16. Function AX,, translationally invariant model, sinusoidal regime. At ( k + mki, ukv) dashes of length proportional to IAZI are drawn (kJ2n = 8/27, G( = 1.2, p = -1, 6 = 1.1726). FIG.

the example in the discommensuration region indicates that scattering will occur for the same energy transfer at many wave vectors. In all pictures one sees that the intensities of the phason and amplitudon branch are not symmetric with respect to the point k = ki:left of it the phason branch is stronger, right of it the amplitude branch is more important. This is a consequence of the fact that to the left the phason branch is closer to the Frequency

t

3.0

....

.

-_ --

.

. .

. .

I

0

_-_

0.25

I

I

0.50

Wave vector FIG.17. Function AX,, translationally invariant model, discommensuration regime (ki/2n = 8/27, G( = 0.95,p = - 1, s = 1.1726).

252

R. CURRAT AND T. JANSSEN

Frequency

1 .o

0

0.50 Wave vector

0.25

FIG. 18. Function A:, not translationally invariant model, sinusoidal regime (ki/2n = 6/37, - 1, 6 = 0.2441).

a = 1.99, =

original dispersion curve, whereas it is the amplitude branch to the right of ki. A further discussion will be given in Section 21. One may also calculate the Debye-Waller factor in the model using Eq. (12.9). For small amplitude, i.e., in the sinusoidal region, the main contribution is the constant W, which is independent of z. Next there is an oscillating part W,(z) which is periodic with period n. In the discommensuration regime the function Wj(z> has many more harmonics. Frequency 2 .o

1 .o

I 0

1.

I

0.25

I

I

0.5 Wave vector

FIG.19. Function A ; , not translationally invariant model, discommensuration regime (ki/2n = 6/37, a = 1.8, = - 1 , 6 = 0.2441).

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

.

1

.*.

..

Un

253

I

f IT)

.. .. *

.

FIG.20. Left: displacements of a solution to the DIFFFOUR model with L / N = 3/19 and the displacements with respect to this solution in the phase (a) and amplitude (b) mode, respectively. Parameters are in the discommensuration regime. Right: corresponding modulation function f ( ~ ) and eigenvectors V(z).

Because the exact equations of motion in the model are nonlinear, the eigenmodes will have a finite width. In this respect, however, the model is rather pathological, in the sense that the damping of the modes is extremely small due to the fact that there is only a fourth-order nonlinearity in the potential. Perhaps this is the reason that the phason branch in biphenyl could be observed: it can be modeled with the DIFFFOUR modeLS9 Within the context of the model one may investigate the character of the eigenmodes of an incommensurate crystal phase and compare it with that found in phenomenological theories, as discussed in Section 11. The eigenvector of the k = 0 mode belonging to the lowest branch in the sinusoidal region may be described as a shift of the modulation function. In Fig. 20 are given the eigenvector and the function U ( z )for that mode, which is the phase mode. In the sinusoidal region its frequency is zero, even for commensurate structures 89

T. Janssen, fpn. f . Appl. Phys. 24, (Suppl. 2) 747 (1985).

254

R. CURRAT AND T. JANSSEN

with sufficiently high denominator N . Even for N as low as, for example, N = 1 1 , the numerically found frequency is zero within the accuracy (Fig. 15). This value o 2= 0 allows the wave vector to vary. When only the length of ki changes, the unit cell in superspace changes, as is the case for a threedimensional crystal when the lattice constants change. The symmetry remains the same. When the direction changes, the symmetry may also change. In that case one has a soft phason. For example, the 47 K phase transition in TTFTCNQ, in which the unit cell of the crystal in superspace is deformed without change of the unit cell volume, may be considered as a case where this occurs. In the discommensuration region the lowest branch, apart from the acoustic one, at k = 0 has another type of eigenvector. It is to be described as an oscillation of the discommensuration lattice with k = 0. Owing to the discreteness effects, this oscillation has a nonvanishing frequency: a phason gap opens. Increasing the temperature from the lock-in temperature o 2decreases and vanishes at the transition to the sinusoidal region. This transition may be considered as the melting of the discommensuration lattice (Fig. 21). A mode for which the frequency is zero at the normal-to-incommensurate phase transition and increases below it is the amplitude mode, as can be seen from its eigenvector and function U ( z ) (Fig. 20). The other phonons, not associated with the soft mode, are not strongly affected near TI. However, for lower temperatures, gaps open up and the bands become flatter. This is due to the increasing localization of the modes. An example of such a

a=l, p = -1 , 6 = 0.23 0.02

-

0.01 -

Tc

Ti

FIG.21. Phason frequency as function of ternperat~re.’~

255

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

pseudolocalized mode is given in Fig. 22. It describes the vibration of the discommensuration lattice, where consecutive discommensurations move with opposite velocity. This mode was also predicted in the phenomenological theory. There, however, a distinction is usually made between the “new” modes and the ordinary phonons. Here it is argued that they are on the same footing, but may have different character. The temperature dependence of the model may be considered to be hidden in the parameters a, p, and 6, but may also be calculated explicitly, for example in a mean-field appro~imation.’~ Then the quasiharmonic frequencies for the not translationally invariant model follow from the equations of motion (17.3) W O ~= ~ g.6, , ( p 26)(efl-1 €,+I) B ( E , - ~ E,+Z)

+ +

+

+

+

where the force constants g. are given by the mean square of the thermal fluctuations: gn = k,T/@X,2) (17.4) Using this expression, one may again determine the vibration spectrum. In this way one finds the thermal dependence of the phason frequency as given in Fig. 21. The phason frequency may be influenced not only by the presence of discommensurations, but also by defects. A special case is that of periodically distributed defects. This situation may be found in systems with mobile

I

En

0-

p=-1 , 6 = 0 23,

azl,

T = 0 067

I I

’

l

l

, .

I

.

.

.

.

.

.

.

.

1

3

,

I

.

I

c

5

7

9

11

1 3 1 5

1 7 1 9

I

I iI 2

L

6

FIG.22. Eigenvector of a localized k = 0 mode. It corresponds to an oscillation of the discommensuration lattice with wave vector n/d,with interdiscommensuration distanced = 16.19 (Top) Displacements u, of static structure. (Bottom) Displacements E , in the given mode.

n

256

R. CURRAT AND T. JANSSEN

impurities. When the material is quenched at a certain temperature the defects are thought to distribute themselves periodically with the period of the temperature of quenching. When the temperature is changed the system not only experiences the (competitive) forces between the particles but also an external force due to the defects. The influence may be investigated with the DIFFFOUR model by considering a periodic background potential in addition to the potential energy [Eq. (15.2)] and looking for periodic solutions with a period that is a common multiple of the period associated with the given parameters and the period of the background. It turns out that the phason frequency is extremely sensitive to the amplitude of the background potential, i.e., the defect concentration. For very small concentrations the phason gap opens and increases rapidly. This is, however, not a general phenomenon (see Section I). Only long-wavelength oscillations are affected by impurities. 18. DYNAMICS OF

THE

FRENKEL-KONTOROVA MODEL

Although the DIFFFOUR model and the Frenkel-Kontorova model describe different physical situations, their properties are in many respects similar. This is particularly true for the dynamics. Small oscillations around the equilibrium positions satisfy the following equations of motion: m o 2 ~ ,=, [2a

+ A(%).sin?]q,

+

- a ( ~ , - ~ E,+J

(18.1)

where the set {x,,} forms an equilibrium configuration. The spectra have been calculated for small 1values and for commensurate values of the misfit parameter b/a = L / N , varying over all values with N < 25. The result is rather similar to that of the modulated spring model (Fig. 23). Novacog' calculated the excitations of the Frenkel-Kontorova model and determined the Fourier components Ak, in this model. The results are very similar to those of Figs. 18 and 19. Theodorou and Rice6' studied a twodimensional system consisting of two interpenetrating lattices, an incommensurate composite structure, and studied its dynamics. They show the existence of a sliding mode with zero frequency. In the continuum approximation the phason branch has a gap at the zone boundary of the discommensuration lattice.46 This free sliding mode was also found in the numerical calculations of Sacco and S o k ~ l o f f .The ~ ~ *latter ~ ~ also studied the excitation spectrum and the eigenvectors for vibrations near the A. D. Novaco, Phys. Rev. B 22, 1645 (1980). J. E. Sacco and J. B. Sokoloff,Phys. Rev. B 18,6549 (1978). 92 J. B. Sokoloff, J. E. Sacco, and J. F. Weisz, Phys. Rev. Left.41, 1561 (1978). 90

91

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

257

1.0

3.0

2.0

1.0

0

0.5

0

1.0

b/a FIG.23. Spectra for the Frenkel-Kontorova model for rl

= &.75

commensurate-incommensurate t r a n ~ i t i o nIn . ~ agreement ~ with the results for the DIFFFOUR model, in this region the spectrum has many gaps with a hierarchical structure and flat bands associated with localized or pseudolocalized vibrations. Radons et al. studied the detailed form of the peaks in the dynamic structure factor S(k, 04.'~ 19. OPTICAL PROPERTIES

The far-infrared activity of lattice vibrations is determined by oscillator strengths of these oscillations. For a mode with labels k and v these are for a crystal with s particles in the basic unit cell with masses mi and charges zj:

.

mj1/2zjeik'nU~v(ki n)

fkv = ni

= A(k

+ mk,)

mj1/2zjAyi

(19.1)

jm

Therefore, only modes k, v that have components Ajm that satisfy simultaneously k + mk, = 0 and Ajm # 0 are infrared active. In the sinusoidal region 93 94

J. B. Sokoloff, Phys. Rev. B 25,5901 (1982). G . Radons, J. Keller, and T. Geisel, 2. Phys. B: Condens. Matter 61, 339 (1985).

258

R. CURRAT A N D T. JANSSEN

most of the coefficients Ajm are negligibly small for a given wave vector and branch label. Therefore, in this region the number of active modes is not much larger than that in the unmodulated phase, although there is an infinite number of modes with wave vector equivalent to zero modulo the quasilattice, and with the proper symmetry. The same reasoning holds for Raman activity. In the discommensuration region the number of coefficients Ajmfor a mode becomes larger and consequently the number of active modes may increase. One may study the dependence of the number of active modes as a function of temperature in a simple model, for example, the DIFFFOUR model. To calculate the infrared activity one assumes that the chain consists of charged particles of the same mass but with alternating charge. If one assumes that the interaction does not depend on the charge, one can use the eigenvectors of the k = 0 modes to calculate the oscillator strength. For a periodic solution (with even period N) of Eq. (15.3), the oscillator strength of the vth mode is f, = Cj”= (- 1)’ e(Ov,j ) . The Raman activity can also be studied with a variation of the same model. Each atom is replaced by a core and a shell of opposite charge. The potential energy is given by

(19.2)

where yn is the position of the nth core and xn the position of the center of the nth shell. If the mass of the shell is small with respect to that of the core, the dynamics can be found immediately from the dynamics of the original model, and it is possible to calculate the inelastic scattering of light by the lattice

vibration^.^^ Calculated Raman and infrared spectra based on these models are shown in Fig. 24 for various temperatures. The number of active modes is small, especially in the sinusoidal region. The k = 0 phason is inactive in both infrared absorption and Raman scattering. The amplitudon is Raman active. It can be seen as the mode that changes rapidly in frequency, starting from zero at T . In real crystals the situation may be more complicated. In the onedimensional model only longitudinal vibrations arf present and the point group symmetry is trivial: only the identity leaves the modulation wave vector invariant. The light propagation itself in incommensurate structures may also be different from that in ordinary crystals. Glass and M a r a d ~ d i nhave ~~

’’T. Janssen and J. A. Tjon, Ferroelectrics 53, 255 (1984). 96

N. V. Glass and A. A. Maradudin, Phys. Rev. B 29, 1840 (1985).

tI

I l l .10.0

111. 2.5

[arbrunits:

I/T=5.0

L l/T=E.O

l l T = 20.0

I 1.

0.

1.

2.

3.

b

2.

3.

L.

0

FIG. 24. Raman (top) and infrared (bottom) spectra for the DIFFFOUR model as function of temperature. The rapidly moving Raman-active mode is the amplitude mode with k = 0.

260

R. CURRAT AND T. JANSSEN

considered coupling of light to surface polaritons for a modulated structure. van Beestg7has predicted additional propagation modes for incommensurate systems with a modulation vector that differs only little from a simple commensurate value. V. Long-Wavelength Excitations in Composite Systems

In the preceding sections the emphasis was on incommensurate displacively modulated insulators. A class of incommensurate structures next to the modulated is formed by incommensurate composite structures which consist of generally modulated substructures with average lattices that are mutually incommensurate. Examples are mentioned in Section I: TTF,I, --x, where the iodine forms a structure inside channels in the TTF host lattice,” Hg, -xAsF,, where two perpendicular mercury systems are contained in the host lattice consisting of AsF, octahedra’, and Nowotny phases.’, The dynamics of these systems show some peculiar properties that have been investigated by Heilmann et ~ 1 . ’ using ~ inelastic neutron scattering techniques. 20. STRUCTURE AND EXCITATIONS

The difference between composite and modulated structures is that the latter have a basic structure that is periodic: there is a lattice of main reflections that corresponds to an unmodulated structure. For a composite structure even the basic structure is not commensurate. Neglecting the modulation, one still has a quasiperiodic and not a periodic system of interpenetrating, mutually incommensurate, crystal structures (Fig. 25). Nevertheless, the structure and symmetry of these materials can also be described using a higher-dimensional space.98The description of the excitations then is a generalization of that discussed in Section 111. As an example consider a system of two interpenetrating systems, each consisting of identical atoms A and B, respectively. Suppose that the lattice constants in the x and y directions are the same for both systems, but that there is incommensurability in the z direction. In the basic structure the z components of the atoms A are given by z$ + nu and those of atoms B by z: + mb, where u/b is an irrational number. Because of the local differences in environment, the A chain is modulated with period b and the B chain with period a. Therefore, the z

97

98

B. W. van Beest, Phys. Rev. B 33,960 (1986). A. Janner and T. Janssen, Acta Crysrallogr. A36,408 (1980).

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES G

i)

J 0

c

0

i

a

0

a

0

c

c

0

L

0

i

0 IJ

a

0

c

I 0

a

0

0 0

I 0

0

26 1

a

FIG.25. Two-dimensional model of an incommensurate composite structure.

components in the modulated structure are ;

z:

+ mb + g (20.1)

respectively. The functions f and g are periodic with period 27c. This structure may be seen as an intersection of a line with a two-dimensional pattern of lines given by - z:

+ nu) + z (20.2)

where z denotes the internal coordinate. In general, a three-dimensional incommensurate composite crystal may be obtained as the intersection of a (3 d)-dimensional patterm of &dimensional (eventually nonplanar) hyperplanes with a three-dimensional hyperplane. The pattern of Eq. (20.2) is left invariant by a two-dimensional lattice generated by vectors (0,27c) and (a, -27calb) and has, therefore, space group symmetry (Fig. 26). It should be noted that the embedding given here is not unique, which is clear from the asymmetric treatment of A and B chains. One can, however, give a more symmetric formulation. Excitations may be described analogously to Section 12, as a displacement field in the unit cell.

+

The system has the usual 3 acoustic modes with frequency zero which consist of equal shifts of r t and r:. In addition, there is a mode which consists of a

262

R. CURRAT AND T. JANSSEN

@

0

\\ \

\

\, a,

FIG.26. Embedding in higher-dimensional space of a composite structure. (a) Projection on the X - T plane of the embedding of the system in Fig. 25. (a) Basic structure. (b) Unit cell of the modulated structure (dashed lines correspond to the unmodulated cell).

sliding of one chain system with respect to the other in the incommensurability direction: r; is shifted in the z direction with respect to r:. When the modulation functions f and g are analytic this shift does not require energy and the mode has zero f r e q ~ e n c y . However, ~ ~ . ~ ~ when the particles are charged, a gap may open up.99 The excitations of such a system have been studied by Axe and Bak”’ in the long-wavelength limit. In this approximation the displacements are given by the fields ui(r), where i = A or B. The interaction between the chain systems then can be written as a sum of 3 terms: uij

=

ug + u i + uij,

i,j = A,B

(20.4)

which correspond to uniform relative sublattice displacements, spatially varying relative displacements and the Coulomb interaction.

uij - 1 dr 1uiDij u j a aP P’ D - 2 1

m, p

= x, y, z

aP

(20.5)

99

G. Theodorov, Solid State Commun. 33, 561 (1980). J. D. Axe and P. Bak, Phys. Rev. B 26,4963 (1982).

loo

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

263

In this approximation the equations of motion take the form

(20.6) where M is the dynamical matrix. When there are no free electrons: (20.7)

There are 3 acoustic gapless modes at q = 0. For dimension a' of the modulation (the number of incommensurate directions) there are 3-d optic modes with displacements perpendicular to the incommensurate directions that have a gap at q = 0. The remaining d optic modes form, together with the acoustic modes, the hydrodynamic degrees of freedom. When A" = 4nzizjpipj/codoes not vanish additional gaps open, as stated above. The formalism has been applied by Axe and Bak"' to uniaxial systems and to the mercury chain system. For the former the dispersion of the 4 hydrodynamic variables u, = *(u: + u,") and v, = $(u: - u:) have been calculated. For the u variables there is no gap at q = 0, for the optic variable the gap vanishes for q perpendicular to the discommensurability direction. The corresponding dispersion curves are sketched in Fig. 27. Finger and Rice4* have calculated the dynamic structure factor for incommensurate composite structures in the long-wavelength limit, using Green's function techniques. They show a crossover from a (relatively) highfrequency regime with two propagating modes in each of the two sublattices to the true long-wavelength limit where there is one propagating mode with

UNSCREENED

-3 CT

-q

I1 x

FIG.27. Phonon dispersion for a uniaxial incommensurate polar material. The incommensurability direction is 2.l"

264

R. CURRAT AND T. JANSSEN

1

Ci=L I

I

I

w/ Y .2

6

1

FIG.28. Dynamic structure factor S(q, o)for the system of Fig. 25; q in units y/c,(c, is the sound velocity, y the relaxation rate of the relative momentum).48

damping vanishing as q 2 for q + 0 and a diffusive mode, the relative motion of the two sublattices (see Fig. 28). The diffusion pole behaves as w&) = - iDq2. This crossover has already been discussed in the dynamics of modulated structure^.^' The dynamic structure factor for incommensurate composite systems has also been discussed by Brand and Bak."' VI. Experimental Results

2 1. NEUTRON SCATTERING RESULTS ON MODULATED CRYSTALS

a. Inelastic Scattering Channels In an ordinary crystal normal modes are characterized by a wave vector defined in the first Brillouin zone and a branch index v = 1,2,.. ., 3s,where s is the number of particles in the unit cell. The corresponding eigenfrequencies generate dispersion curves o,(k) with gaps at the Brillouin zone boundaries. In an inelastic scattering experiment performed at a fixed momentum transfer K, the probe, say, the neutron beam, interacts with excitations of wave vector k defined through K+k=G where G is a reciprocal lattice vector. H. Brand and P. Bak, Phys. Rev. A 27, 1062 (1983).

(21.1)

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

265

In an incommensurately modulated crystal the 3 D Brillouin zone has collapsed to zero volume since an infinite number of points within the basic structure’s Brillouin zone has become reciprocal lattice points (i.e., zone centers). Due to the infinity of zone boundary points which have thus been introduced, an infinite number of frequency gaps should appear in the excitation spectrum. In practice, however, most of the new zone centers correspond to high-order satellite peaks. Rewriting Eq. (21.1) as:

K

+ k = G + rnki

(21.2)

the condition for the wave vector k to be arbitrarily close to zero, at fixed K, implies arbitrarily large values of rn and IGI. In the limit of small modulation amplitudes, the higher-order satellite intensities converge rapidly to zero with increasing rn values and the frequency gaps at the corresponding zone boundaries will scale accordingly. It is thus possible to draw the dispersion of the new normal modes in the Brillouin zone of the basic structure. The wave vector associated with each new eigenfrequency is defined by continuity, using the correspondance between normal modes in the two structures. The new dispersion curves look continuous except for a few gaps at low-order zone boundary points, given by:

kG,,,= $(G

+ rnk,),

rn = 0, f 1, f 2 ,...

(21.3)

For the situation discussed in Section 17 (cf. Fig. 18), the main gaps are located at k,,z = ki = 0.162, and k l , z = f - ki = 0.338. They result from the mixing of soft-mode normal coordinates of wave vector k and - G - mk, k which are degenerate at k = kG,,,. The mixing term is proportional to the lrnlth power of the modulation amplitude. The spectrum reconstruction is more substantial near ko,z as the interacting soft-mode coordinates, Qk and Q - Z k , + k remain quasidegenerate over a range of wave vectors about k = k,,z = ki (the position of the soft-mode minimum). An excitation with nominal wave vector k can be described as a linear combination of normal modes of the basic structure, with wave vectors k, k f ki, k f 2k,,. . . . As such, it can participate in inelastic scattering processes at K = G - k k,, G - k f 2ki, etc., with scattering strengths proportional to the appropriate mixing coefficients. This point is illustrated in Fig. 18, where the scattering due to the phason branch extends beyond k = k,, while conversely the amplitude branch is observed for k < k , . Strictly speaking (i.e., in order to conserve the same number of normal modes in the modulated and basic structures), the phason dispersion should be restricted to the range Ik( < ki(lkl > k , for the amplitude branch). The phason intensity observed for k > ki corresponds in fact to an excitation of nominal wave vector k - 2ki, and renormalized normal coordinates f$kk-2ki. Both

+

266

R. CURRAT AND T. JANSSEN

#k(k < ki) and #kk2k,(k > ki) are linear combinations of soft-mode coor, both are seen near k N ki via their Q k component. dinates Qk and Q k k 2 k ,and As k increases, the relative weight of the Qk component decreases on the phason branch (increases on the amplitude branch), thereby leading to a progressive transfer of intensity from the lower to the upper branch. The observability of an excitation (kv) for a momentum transfer K = G - mki - k, is given by Eq. (12.6). The quantity H,, is analogous to the one-phonon structure factor in ordinary crystals, where the summation X I replaces the usual K e(k, v) factor. Each term in the summation corresponds to a specific scattering process. The mode-mixing mechanism discussed above corresponds to the term 1 = m, with a scattering amplitude proportional to:

-

.

K A$B,,(K),

j = 1,s

(21.4)

The normal mode (kv) with eigenvector Us"(z)given by Eq. (1 1.7) is observed at G - mki - k, through the mth-order Fourier component of its eigenvector, A%. The term 1 = 0 in Eq. (12.6) gives a contribution:

EK * A!; Bmj(K)

(21.5)

This latter process may be called a direct process, since the excitation is observed through its fundamental component, A!;. It is akin to the direct scattering of light by k N ki phasons and amplitudons (cf. Section 9) and is sometimes referred to as "two-phonon" scattering, one of the phonons being the static modulation [here, its mth harmonic Bmj(K)]. In the superspace formalism of Section 111, this term is nonetheless included in the one-phonon cross section. Terms with 1 # 0, m can be viewed as combination of the above two types of processes. All terms of index 0 < 1 < m are of comparable magnitude (i.e., cc the mth power of the modulation amplitude). b. Spectral Response Function

Most of the recent experimental effort has been focused on attempting to detect phase and amplitude modes in displacively modulated systems. Insulators are better suited for this purpose, as additional complications arise in CDW systems due to the infrared activity of the q = 0 phase mode53~'02-'04 (Frohlich mode). Simple Landau theory [cf. Eq. (7.6)] predicts a classic soft-mode behavior for the amplitude branch, while the phason branch appears as a temperatureindependent acoustic-like dispersion branch, emanating from first-order satellite reflections. P. A. Lee, T. M. Rice, and P. W. Anderson, Solid State Commun. 14,702 (1978). P. A. Lee and H. Fukuyama, Phys. Rev. B 17,542 (1978). K. Carneiro, G . Shirane, S. A. Werner, and S. Kaiser, Phys. Rev. B 13,4258 (1976).

267

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

When anharmonicity is included, the spectral response [cf. Eq. (12.3)] from low-q phasons may be written as:

(21.6) where, as in Section 7, q is referred to the satellite position (q = k - ki): o&) = V&

Unlike for acoustic modes, the damping coefficient rdis a slowly varying function of q, comparable in magnitude to the soft-mode damping coefficient (T > 7;). At low q(q < r,/&,), the response function, Eq. (21.6) is approximately Lorentzian.

(21.7) with a q-dependent half-width given by:

(21.8) At larger q values the diffusive response, Eq. (2 1.7), becomes progressively underdamped, with intensity maxima at: w=

&G&)

with (21.9) In order to verify Eq. (7.6) directly, it is essential to find a system with low soft-mode damping, where the phase-mode response remains underdamped over as wide a q range as possible. This criterion eliminates many well-known compounds, l o O7 such as NaNO,, Rb,ZnBr,, and BaMnF,, for which the dynamics of the normal-to-incommensurate transition is not clearly displacive in character. Of course, it is always possible to extend the arguments developed in Section 7 to order-disorder systems and define relaxation times for phase and amplitude fluctuations.'08 However, both types of fluctuations then contribute together to quasielastic scattering, and the separation must be performed via a detailed analysis of the experimental spectra.

'-'

lo6 lo'

J. Sakurai, R. A. Cowley, and G. Dolling, J . Phys. SOC.Jpn. 28, 1426 (1970). C. J. de Pater, J. D. Axe, and R. Currat, Phys. Rev. B 19,4684 (1979). D. E. Cox, S. M. Shapiro, R. J. Nelmes, T. W. Ryan, H. T. Bleif, R. A. Cowley, M. Eibschuetz, and H. J. Guggenheim, Phys. Rev. B 28,1640 (1983). I. Hatta, M. Hanami, and K. Hamano, J. Phys. SOC.Jpn. 48, 160(1980).

268

R. CURRAT AND T. JANSSEN

The order of magnitude of the phason velocity u4 [which is related via Eq. (7.6) to the curvature of the soft-mode dispersion about its minimum at ki] is the same as for acoustic velocities. Thus in general it may be difficult to distinguish experimentally between the phason branch and an acoustic branch originating at a satellite position. Three types of situations have been described so far, for which the phason-branch assignment could be made: (1) In biphenyl, the acoustic slopes are sufficiently different from that of the phase mode that there can be no confusion between them, at least in some propagation direction^^^'-^"; (2) In K,SeO, the soft-mode damping is large and the phase-mode response is identified on the grounds that it is overdamped' 12,' 1 3 ; (3) In ThBr, none of the above criteria is applicable, but the assignment can be made on the grounds that the intensity of the observed branch is much too large to be of acoustic origin."4,"5 c. Experimental Results on P-ThBr,

Here we only summarize the main results (see Ref. 115 for a more complete review). The structure of P-ThBr, at room temperature is shown in Fig. 29. The space group is 0:: and the primitive cell contains two formula units. The 8 bromine ions are located at equivalent sites and can be chosen in such a way as to make up the first and second coordination shells of the Th4+ ion at the origin, as shown in the figure. The symmetry of the thorium site is D,, (32m), and the inversion center is halfway between the 2 thorium ions. has Raman work' l 6 followed by neutron diffraction studies' ' 4 * 1 ' 7 * ' revealed the occurrence of a structural phase transition at N 95 K toward a H. Cailleau, F. Moussa, C. M. E. Zeyen, and J. Bouillot, Solid State Commun. 33,407 (1980). H. Cailleau, in "Incommensurate Phases in Dielectrics" (R. Blinc and A. P. Levanyuk, eds), Vol. 2, pp. 72-100. North-Holland, Amsterdam, 1986. ' I 1 H. Cailleau, J. C. Messager, F. Moussa, F. Bugaut, C. M. E. Zeyen, and C . Vettier, Ferroelectrics 67, 3 (1986). 112 M . Quilichini and R. Currat, Solid State Commun. 48, 101 1 (1983). 1 1 3 M. Quilichini and R. Currat, manuscript in preparation. 'I4 L. Bernard, R. Currat, P. Delamoye, C. M. E. Zeyen, S. Hubert, and R. de Kouchkovsky, J . Phys. C 16,433 (1983). R. Currat, L. Bernard, and P. Delamoye, in "Incommensurate Phases in Dielectrics"(R. Blinc and A. P. Levanyuk, eds.), Vol. 2, pp. 162-204. North-Holland, Amsterdam, 1986. 116 S. Hubert, P. Delamoye, S. Lefrant, M. Lepostollec, and M. Hussonnois, J . Solid State Chem. 36, 36 (1981). 117 R. de Kouchkovsky, M. F. Le Cloarec, and P. Delamoye, Muter. Res. Bull 16, 1421 (1981). l ' * L. Bernard, J. Pannetier, R. de Kouchkovsky, P. Delamoye, C. M. E. Zeyen, and R. Currat, manuscript in preparation. 'lo

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

269

FIG.29. Primitive unit cell of b-ThBr,. The two Th4+ ions are located at (O,O,O) and (O,f,i). The conventional body-centered tetragonal cell (a = b = 8.93 A; c = 7.96 A) contains 4 formula units. After Bernard er a1.'I4

modulated structure with modulation wave vector:

ki = [;c*

= 0.310 f 0.005 C*

(21.10)

The displacive nature of the transition was confirmed by the observation of a soft optic phonon branch by inelastic neutron scattering (cf. Fig. 30). From the parametrization of the soft-mode dispersion, approximate values for the constants appearing in Eq. (7.6) could be obtained. The symmetry representation of the soft branch (T > 7.1) and the detailed description of the modulated ionic displacements ( T < 7-1)were established on the basis of first-order satellite intensity measurements:

unj = ~ [ c o Be, s (j)cos(k, .rij + +)

+ sin O e , ( j )sin(ki - rij + +)]

(21.11)

where unj is the average displacement of atom j in cell n away from its hightemperature equilibrium position rij; e , and e, are two normalized basis

270

R. CURRAT AND T. JANSSEN I

N

I k

.5

[o.o,c

1

FIG.30. Dispersion curves for the longitudinal acoustic (LA), transverse acoustic (TA), and 300 K; ( A )150 K; (0) 120 K; ( 0 )101 K; soft-optic modes, propagating along [O,O,c]: (0) = 95 K. After Bernard et al.""

vectors with B,,and B,, symmetry, respectively; 4 is an arbitrary phase angle; the index j runs over the 8 bromine ions in the cell (cf. Fig. 29). Figure 31 illustrates the ionic displacements involved in the el and e, components (rotations and twists of the bromine shells about the 2 axis). Their relative weights in the soft-mode eigenvector is determined by the value of the mixing angle 0 (0 N 200). The only temperature-dependent quantity in Eq. (21.11) is the modulation ~ 0 K, where a is the amplitude r, which grows from q = 0 at IT.I to q 2: 0 . 0 2 at cell edge. The value of the modulation wave vector, liin Eq. (21.10), remains constant. Even at low temperatures no higher-order satellite reflections are observed. Hence the modulated displacements can be described as a small perturbation on the basic structure and the sinusoidal approximation of Eq. (21.11) should be applicable throughout. Figure 32 shows the inelastic neutron spectra observed near the strong (2, 3, 1 - l i )satellite reflection at T = 81 K = Ti - 14 K. Three excitation branches are observed:

(1)

An acoustic-like branch originating at the (2,3,0.69)satellite position

(l = 0.31 in the figure); (2) An optic-like branch with a minimum at the satellite position; (3) A transverse acoustic branch originating at the (2,3,1) fundamental reflection.

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

27 1

C X

31. Projection of unit cell along theaaxis showing(a) the B,, and (b) the&, components of the soft-mode eigenvector. All arrows have equal length, corresponding to equal amplitudes for the 8 Br- ions. The Th4+ ions remain at rest. The labeling convention is the same as in Fig. 29. After Bernard et nl.'L4 FIG.

The slope of the first branch is close to that of the TA mode. However, its intensity is much larger than expected for an acoustic mode near a first-order satellite reflection. From Eq. (12.6) with rn = 1, the inelastic structure factor for a TA mode of wave vector k (lkl << ki) at K = G - ki - k is written as (to leading order in q):

+ (K

(21.12) A:;)Boj(K)] The second term in Eq. (21.12) reflects the modulation-induced mixing between the TA mode (k, a) and the soft-mode coordinate Qki+k.For kllkillc* this mixing is forbidden due to rotational selection rules.' l 4 Hence Eq. (21.12) reduces to: x [(K * A:;)BIj(K)

H,,(G

-

ki - k) % j= 1

*

g,rnT1/2e-Wj(K)eiK"j J J (K Ag)B,j(K)

= (K * ak,)F,(G - ki)

.

(2 1.13)

272

R. CURRAT AND T. JANSSEN

I

400!

L

0

- 17

.""

-.37

c 3

P

A- 5

.55

Am

1 .2 .3 .4 .5 .6 .7 .8 .9

-

=.6

Frequency(THz) FIG.32. Constant-q scans at (2,3,1 - 5) for T = 81 K. For each scan the baseline is indicated by a horizontal arrow pointing toward the relevant value of <.After Bernard et ~1."'

where, for a long-wavelength acoustic mode, the mass-weighted eigenvector m,:"' A t ; can be replaced by a j-independent polarization vector Mku. F, is the elastic structure factor of the satellite at K = G - ki. With the help of Eq. (21.13) one readily shows that: Hk,(G

-

ki - k)

1 =I

Fl(G - ki)

I

(21.14) k, FdG) From the observed intensity of the TA branch near (2,3,1) and the known ratio of the (2,3,1) versus (2,3,0.69) Bragg intensities, one concludes that the acoustic-like dispersion originating at (2,3,0.69) is approximately two orders Hku(G

-

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

273

f I

t

0.5

FIG.33. Phase ( 0 )and amplitude (0)mode dispersions in fi-ThBr, at 81 K. The measured frequencies are obtained from the analysis of the spectra in Fig. 32. (-) are calculated using a generalized version of Eq. (7.6),with parameter values determined on the basis of Raman data116 [for the value of the q = 0 amplitude mode at 81 K) and neutron data above (cf. Fig. 30)J After Bernard et al."'

of magnitude too intense to be of acoustic origin, and is thus identified with the phason branch. Similarly the optic branch is identified with the amplitude mode. Its frequency at = 0.31 is consistent with the frequency of the q = 0 amplitude mode, as deduced from the data of Hubert et ~ 1 . ' ' ~ The analysis of the spectra in Fig. 32 by damped-harmonic-oscillator functions [cf. Eq. (21.6)] yields quasiharmonic frequencies for the phase and amplitude modes, as shown in Fig. 33. As expected from the asymmetry of the soft-mode dispersion (cf. Fig. 30) and from simple zone-folding arguments, the phase-mode structure factor decreases with (, while the reverse is true for the amplitude mode (note the analogy with the behavior of the model system shown in Fig. 18). Similar spectra at lower temperature (cf. Fig. 34)"' show a rapid hardening of the amplitude mode upon cooling while the phase mode becomes sharper, particularly at small wave vectors (cf. Fig. 34a), but its integrated intensity (corrected for the effect of thermal population) remains essentially constant. Most of the above observations are consistent with elementary theoretical predictions. Thus fl-ThBr, appears as a good model system for further dynamical studies on displacively modulated solids. L. Bernard, R. Currat, and P. Delamoye, manuscript in preparation.

274

R. CURRAT AND T. JANSSEN I

I

I

I

I

(b)

0 -0.1 -0.2

o

-03

o -on

-02 -a3 -0.4 -0.5

-02

-02 -03 -0.4 -a5

FREQUENCY (THr)

FIG.34. Constant-q scans at K = G , , , - ki - q for several temperatures between 83 and 20 K: (a) q/c* = 0.06; (b) 0.12; (c) 0.16. ( 0 )T = 83 K; (0) 60 K; (0) 40 K; ( A ) 20 K. Vertical scales have been adjusted to account for the effect of the thermal population factor. After Bernard etal.'19

d. Deuterated Biphenyl (C12Dlo) Figure 35 shows the room-temperature structure of biphenyl (phase I). As already mentioned, this is one of the few systems where the competing interactions which lead to the incommensurate state have been The modulated variable is the torsional angle 4 (cf. Fig. 35a) between molecular phenyl rings. There are in fact two successive modulated phases (I1 and 111),corresponding to wave vectors of the form ki

=

&(6,a* - 6,C*)

= +(1

-

6,)b*/2,

TI < T < 7;

(1 - 6b)b*/2, T<

(21.15)

for phases I1 and 111, respectively. The temperature dependence of the quantities a,, 6b, 6, is shown in Fig. 36, for the deuterated compound. In phase I1 the star of ki has 4 arms, corresponding to a 4-component order parameter and a superspace dimensionality of 3 2 = 5. As far as inelastic neutron scattering is concerhed, the situation is complex, since, in principle, two phason branches and two amplitudon branches can be simultaneously observed. Studies in the partially commensurate phase I11 are comparatively much simpler. Also the lower temperature range considerably reduces the mode linewidths.

+

FIG. 35. Structure of biphenyl in phase I: (a) the molecule and its torsional angle 4 due to ortho-hydrogen repulsion; (b) the P2,/a unit cell; (c) the crystalline axes orientation (a = 8.11 A; h = 5.56 A; c = 13.61 A; /I= 92"). After Cailleau.l'o

0.1!

I

I

I

mo.ll

z

-I-

0

f > w

n

O.O!

i" (

I

LgLg-- ,

20 0 TEMPERATURE(K1 FIG.36. Temperature dependence of components of the modulation wave vector: k, = *(&a* - 6,c*) (1 - 6,)b*/2. After Cailleau et al."' 10

*

276

R. CURRAT AND T. JANSSEN I

I

I

I

I

I

1

I

1

~100K 37K

r

I

I

I

I

I

I

I

I

.1

.2

.3

.4

.5

.4

.3

.2

z

I

.l

r

AS FIG.37. Dispersion of soft phonon branches in the b* direction (TZ). The figure is drawn in an

S

extended zone representation. The branches have, respectively, symmetric (S) and antisymmetric (AS) character, with respect to the screw axis 2,. They are pairwise degenerate at the zoneboundary point Z (Lifshitz invariant). Points are at ( A ) 100 K and (*) 37 K. After Cailleau.''o

Figure 37 shows the dispersion of the soft-phonon branch along the two-fold axis b* in phase I. The true frequency minimum is slightly displaced from the high-symmetry direction. The soft-mode linewidth is in the range 100- 150 GHz near 7; (as compared to 70 GHz in the case of p-ThBr,). Inelastic spectra obtained in phase I11 near a strong first-order satellite reflection are shown in Fig. 38,"' where a propagating excitation is clearly seen. The corresponding dispersion relation is plotted in Fig. 39, as a function of reduced wave vector along a*. The observed slope is substantially lower than the slope of the lowest acoustic mode, as obtained from Brillouin'21 and neutron"' scattering measurements in that temperature range: the distinction between phase and acoustic mode is here straightforward. The amplitude branch is already too far up in frequency at 10 K and is not observable in the above measurements.

-

O''

H. Cailleau, F. Moussa, C. M. E. Zeyen, and J. Bouillot, J. Phys. 42, C6-704 (1981). C. Ecolivet, M. Sanquer, J. Pellegrin, and J. De Witte, J . Chem. Phys. 78,6317 (1983).

,

+

FIG.38. Constant-q scans near the satellite reflection (2,+(1 6,),0), for several reduced wave vectors q along a*, at T = 10 K (phase 111). After Cailleau et a1."'

0.5

I

-

-

I

-I

/

0.4k

/

/

T=lOK

N

r

c

I

/

/

0.3

/ /

/

:

/ /

io.1 w

LA K

/

0.1- /

0

//

-

/ / / /

1

I

I

278

R. CURRAT AND T. JANSSEN

Similar results have been reported along b*, which is the hard direction as far as the soft-mode dispersion is concerned. There, the slope of the phase mode is not significantly lower than that of the acoustic branches and the assignment must rest on intensity arguments, using the a* direction as reference. The intermolecular interactions which favor a coplanar molecular configuration, as in phase I, can be enhanced by hydrostatic pressure. Figure 40 shows the ( P , T ) phase diagram for deuterated biphenyl, as deduced from neutron diffraction data’22:as expected, the modulated phases I1 and I11 are progressively destabilized with increasing applied pressure. Cailleau and coworkers’” have taken advantage of this situation to study the low-frequency dynamics in all 3 phases, at low temperature ( 3 K) where damping effects can be further reduced. Below the critical pressure separating phases I and 11, the splitting of the soft mode into a phason branch and two amplitudon-like excitations could be e. Potassium Selenate (K,SeO,)

As prototype of the A2BX4 family, K,Se04 is one of the best studied dielectric materials exhibiting an incommensurate phase.lZ3At room temperature this compound has orthorhombic symmetry (Pnarn-D;,6)with 4 formula units per cell. At q = 129 K it undergoes a continuous transition toward a modulated state with modulation wave vector:

k,

= f(1 - 6)a*

(21.16)

The incommensurability parameter 6 is temperature dependent and vanishes rapidly on approaching T, = 94 K (cf. Fig. 41), leading to a commensurate 3a superstructure below T,. Inelastic neutron scattering measurements’ 24 indicate that the transition at is of displacive character (cf. Fig. 42).l 2 5 The microscopic lattice-dynamical origin of the instability has been discussed in detail by Haque and Hardy33 and Bussmann-Holder et a1.lz6 H. Cailleau, A. Girard, J. C. Messager, Y. Delugeard, and C. Vettier, Ferroelectrics 54, 597 (1984). 1 2 2 a F. Moussa, P. Launois, M. H. Lemee, and H. Cailleau, Phys. Rev. B, in press (1987). P. Launois, M. H. Lemee, H. Cailleau, F. Moussa, and J. Mans, Ferroe[ectrics,in press (1987). ”” D. Durand, Thesis, University Paris-Sud, p. 104 (1987); D. Durand, A. H. Moudden, and D. Petitgrand, in preparation. J. D. Axe, M. Iizumi, and G. Shirane, in “Incommensurate Phases in Dielectrics”(R.Blinc and A. P. Levanyuk, eds.), Vol. 2, pp. 1-48. North-Holland, Amsterdam, 1986. M. lizumi, J. D. Axe, G. Shirane, and K . Shimaoka, Phys. Rev. B 15,4392 (1977). 1 2 ’ J. D. Axe, M. Iizumi, and G. Shirane, Phys. Rev.B 22,3408 (1980). A. Bussmann-Holder, H. Biittner, and H. Bilz, Ferroelectrics 36,273 (1981). I”

”’

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

279

FIG.40. (P, T ) phase diagram of deuterated biphenyl. After Cailleau et al.lzz

K, Se 0, -qg=(++S)Ti *

TI

0.01

0

modulation FIG.41.-=.Temperature .wave GO vector here dependence Tc is100 given as of kiincommensurability = 110 (3 + @a*. The 120 points parameter Tcorrespond ( K ) 130 6 intoK,SeO,. >experimental The

results, the line to a theoretical curve [ h ( T - TJ-1.169

5.0-

I

Z.B.

I

OO

0.5 q ( I N REDUCED UNIT)

1

I .o

FIG.42. Phonon dispersion of the soft branch in K,Se04 (extended zone scheme).After Axe et a1.125

A detailed x-ray diffraction study by Yamada and Ikedalz7indicates that at 113 K the modulated ionic displacements are adequately described in a quasisinusoidal approximation. Near T,, however, a number of experimental r e s ~ l t s ~favor ~ ~ a' description ~ ~ ~ ' ~ in ~ terms of discommensurations (DC). Figure 4 3 shows the expected temperature behavior for the k = k, softmode frequency. Below it splits into a pair of q = 0 phase and amplitude modes (w+ and w, in the figure). In a simple sinusoidal model, cob and w,, vary according to Eq. (7.6),namely wg renormalizes upwards while cob remains at zero frequency until T,. Below T, the q = 0 phase mode continues as a zonecenter optic mode of the new 3a superstructure, with finite frequency. It can be observed by Raman scattering in appropriate geometries.'30-' 34 N. Yamada and T. Ikeda, J. Phys. SOC.Jpn. 53,2555 (1984). M. Fukui and R. Abe, J. Phys. SUC.Jpn. 51, 3942 (1982). T. Kobayashi, M. Suhara, and M. Machida, Phase Transitions 4,281 (1984). I 3 O M. Wada, A. Sawada, Y. Ishibashi, and Y. Takagi, J. Phys. Suc. Jpn. 42, 1229 (1977). M. Wada, H. Uwe, A. Sawada, Y. Ishibashi, Y. Takagi, and T. Sakudo, J. Phys. SOC.Jpn. 43, 554 (1977). 13' P. A. Fleury, S. Chiang, and K. B. Lyons, Solid State Commun.31,279 (1979). 1 3 3 H. G. Unruh, W. Eller, and G. Kirf, Phys. Status Solid A 55, 173 (1979). 134 M . Quilichini, unpublished results.

'"

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

5

RAMAN

---*--NEUTRON

80 100 120 140 TEMPERATURE ( K ) FIG.43. Temperature behavior of k = k, soft mode ( T > T ) and q mode ( T < q).After Axe et a1.125 "0

20

40

60

28 1

160 =0

180

200

phase and amplitude

If the DC model is at all relevant, deviations from the above behavior should be detectable in the vicinity of T,. Following the discussion in Section 7, two effects are expected: (1) A finite value of o+(q= 0) due to pinning of the DCs by the lattice5' (discreteness effect); (2) A gap in the o+(q)dispersion at q = qD = n/d, where d is the average distance between DCs (cf. Fig. 5).

In principle all these delicate features can be explored by inelastic neutron scattering, and comparisons with results from other techniques are possible. For example, the lower and upper gap modes at qD(ol and o2in the figure) are both optically active: o2is predominantly Raman active. For a continuous lock-in transition, it would go smoothly into the commensurate phason frequency as 41, + 0 (02is the oscillation of the phase of the modulation in the quasicommensurate regions between DCs). On the other hand, o1(DC dimerization mode) is predominantly infrared active, and its frequency should vanish as q D -+ 0. Its softening is responsible for the strong dielectric anomaly observed in K2Se04 (and in other A2BX4 compounds) at the lock-in 36 transition.' ' 9 '

13'

K. Aiki, K. Hukuda, H. Koga, and T. Kobayashi, J . Phys. SOC.Jpn. 28,389 (1970). A. Levstik, P. Prelovsek,C. Filipic, and B. Zeks, Phys. Rev. B 25,3416 (1982).

282

R. CURRAT AND T. JANSSEN

-

0.6

- 0.4

- 0.2

0

0.2

0.L

FREQUENCY ( T H z ) FIG.44. Soft-mode line shape in K'SeO, at T = 139 K: (0)constant-q scan at (1.31,0,2); (-):damped harmonic oscillator (DHO) fit. Best-fit parameter values are: m+ = 250 f 5 GHz; r, = 350 & 20 GHz. After Quilichini and Currat."'

A major obstacle in the study of the low-frequency dynamics in the incommensurate phase of K,SeO, is the large value of the soft-mode damping coefficient r,.Figure 44 shows the soft-mode line shape at k = k6 and T = 139 K (= 7; + 10 K), as obtained from high-resolution neutron scattering measurements"2: the line shape is already critically damped, with a DHO-fitted value of 2.350 GHz for r,. Typical line shapes in the incommensurate phase are shown in Fig. 45 (qlla*) and Fig. 46 (q(lc*),for T = 120 K (= - 9 K). The spectra correspond to the phase-mode response alone, since at that temperature the amplitude mode is already outside the frequency range of the measurements. In Fig. 45 a comparison is made with the TA-mode response, measured at equivalent reduced wave vectors near a fundamental reflection: the two modes have very different line shapes and can hardly be confused. Phase-mode spectra of the type shown in Figs. 45 and 46 have been collected for several temperatures in the incommensurate phase and analyzed by fitting to a DHO function [cf. Eq. (21.6)]. For overdamped spectra, the fitting procedure becomes unreliable because strong correlations occur between best-fit values for co+ and r' and, in practice, additional assumptions must be introduced. ~ , ~ ' that As discussed in Section 7, several theoretical a r g u m e n t ~ ~indicate the function IJq, T ) is slowly varying with q and T, and is closely related to the soft-mode damping coefficient T,(k,, T ) . Hence, if one neglects the q

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

283

400

E 300

1200

z

z?

x

\

m

+

0

5 200

800

7 111 c

U 0

z 2

2 x

100

LOO

0

0

250

500

200

LOO

1 1 5 0

300

m

8

\ Il -n

l-

=

f 0

l r

z

8

100

200

50

100

0

0

2

U

FREQUENCY ( T H z )

FIG.45. Phason and TA-phonon line shapes at equivalent reduced wave vectors ( T = 120 K): (a) 4 = 0.046 a*; (b)4 = 0.096 a*; TA data (0)and phason data ( 0 )are collected near (O,O, 2) and (1 + k,,O, 2), respectively. After Quilichini and Currat."'

dependence of r,(q,T ) in a small neighborhood of q = 0: T+(q, T ) = r,co, T ) = T,(T)

(21.17)

the function r,(T ) may be obtained by interpolating between T,(k,, T ) ( T > TJ and the Raman-determined values of the commensurate phasemode damping for T( T,. As shown in Fig. 47, a straight line interpolation appears to be justified. The resulting values for w,(q)(T, < T < T),are shown in Fig. 48 for q11a* and in Fig. 49a for qllc*. As before, the origin of wave vectors is the first-order satellite reflection at (1 + k g ,0,2). Also shown in Fig. 48 are the Raman data of Inoue and I ~ h i b a s h i ' ~corresponding ' to the upper gap mode w2 at q = q D ( T ) for 3 temperatures in the incommensurate phase. The value of q D 13'

K. Inoue and Y. Ishibashi, J . Phys. SOC.Jpn. 52, 556 (1983).

a

80C

z

F 600 0 7

\

m + z 3 0

u LOO

2 00

I

I -

0.6

I

I

I

-0.2

I

0

I

I

I 1

0.2

FREQUENCY ( T l i z ) FIG. 46. Phason line shape in K'SeC, at T = 120 K for q = 0.03 c*. Best DHO-fit parameters 5 GHz and r, = 310 GHz (fixed). After Quilichini and Currat."' are = 190

0.1

N

I

t +

z

0.3

Lu

U -

L L U

W

0

0.2 U

a z

x

Q

0

0.1

0

FIG.47. Soft-mode ( T > TJ and commensurate phase-mode ( T iT,) damping coefficient as deduced from the analysis of neutron and Raman spectra: 0,"' @,130 0,133 0.13, 284

0

- 0.1

- 0.05 0 0.05 REDUCED WAVE VECTOR q (a*)

0.1

FIG.48. Phason dispersion in incommensurate K2Se04 for qlla: neutron results at 120 K (0,8 ) ;110.5 K (0);100 K (0); 96 K (0).112,113 Raman results ( A ) corresponding to 95, 100, and 106 K from left to right.I3’

(4

(b) 0.4--

\

\ -

\

-2 -

\ \

A,

385K

\

0.3‘

\@

‘ 9’

-

d

I-

Y

\\ 120K \

6

\@

23 -

0-

60K

0.2 =’

\

/‘
/

J\

\

E -

/

/

-

96/K

\

/

\

91K

\

-

A-

0.1--

.

/.

92K&’/

P’/.,

-

/

Tc = 94.1K (1+%+ 5 ,o, 2 1

(1+ C I S , 0 2 - 5 ) t

I

1

I

/

I

1

I

I

I

I

286

R. CURRAT AND T. JANSSEN

is readily calculated from the relationship:

d ( ~ * / 3- k,)

(21.1 8)

= AI#J

where d is the average spacing between DCs and AI#Jis the phase jump per DC. From Eqs. 6.2 and 6.9, with m = 2p = 6, we get: Ad = 1 ~ 1 3whence, , using Eq. (20.16):

(21.19)

qD = n / d = ~ ( T ) u *

The zone boundary of the DC lattice thus corresponds to a second order: k,

+ 41,= ( ~ * / 3 ) ( 1+ 26) =

U*

-2

k,

(21.20)

or fourth order: ks - q D = ( ~ * / 3 ) ( 1 - 46) = -u*

+ 4ks

(21.21)

satellite position in the basic structure reciprocal lattice. The data in Fig. 48 suggest a temperature-independent dispersion for the phase mode, with a gap of 70-100 GHz at q = 0. Good agreement is found l ~ ~the upper gap mode with the Raman data of Inoue and I ~ h i b a s h i for at 6(T). The lower gap mode is not detected. From dielectric relaxation measurements Horioka and S a ~ a d predict: a ~ ~ 0,(96 K) N 30 GHz. This should give rise to an additional quasielastic component of width [cf. Eq. (21.8)]: y1 =

w:/rl

4 G H ~

too narrow to be identified experimentally. The data in Fig. 48 are probably not sufficiently detailed to draw conclusions about the occurrence of a gap at 6(T).They do, however, point toward the presence of a gap at q = 0. Such a gap is expected close to T, (cf. Fig. 21) but not at 120 K, when the sinusoidal approximation should be valid. At the moment there is no satisfactory explanation for this result, although the possibility of a lineshape distortion due to coupling between phasons and acoustic modes has been recently pointed out by Gooding and Walker.I3'" On the experimental side, the neutron technique is seriously hampered by the heavy damping of the excitations and by possible systematic errors due to finite instrumental resolution (particularly momentum-space resolution). This last restriction, however, does not apply to the Raman technique, and it seems unlikely that the good agreement between the results from both techniques should be fortuitous. This consistency is also seen below T, (cf. Fig. 49b), where the neutron and Raman determinations of the commensurate phason frequency are in excellent agreement.

137a

R.J. Gooding and M. B. Walker, Phys. Reo. B, in press (1988).

EXCITATIONS IN IKCOMMENSURATE CRYSTAL PHASES

287

Additional neutron and dielectric measurements are in progress. New dielectric results’ 3 8 indicate that the relaxation frequency associated with the w1 mode is not electric-field dependent, in contradiction with the predictions from phenomenological DC-type rn~dels.’~’Attempts to detect the q N 0 phase-mode response by Brillouin scattering, if conclusive, could also provide some crucial information on the dynamical aspects of lock-in transitions. Obviously, the last word on this topic has yet to be said.

22. OPTICAL STUDIES Results from light scattering studies on incommensurate systems have been mentioned on several occasions in the preceding sections. Optical techniques (Raman and infrared spectroscopy) have been applied extensively to the study of modulated structures, and the literature on this subject has been reviewed by several authors.53~55~’40-’42 When going from a normal crystalline structure into a modulated phase, optical spectra are modified in a rather similar way, as for an ordinary structural phase transition. This is illustrated in Figs. 50 and 51, which show the temperature evolution of polarized Raman spectra in 8-ThBr, and biphenyl. The dominant feature in Fig. 50a is a sharp line at -120 cm-’ (1 THz = 33 cm-’), which appears to split into several sharp peaks below 90 K, as commonly seen at ferroelastic phase transitions. In general, however, the new lines observed in the modulated phase are not directly associated with a particular feature in the parent-phase spectrum. In phases I1 and I11 of biphenyl (cf. Fig. 51)’43 several modes become activated in a 10-40 cm-’ frequency range where no lines are observed above Tl. As already stated, the new lines, in fact, originate from excitations with wave vector k = & ki, f2ki, etc.. ..,in the normal phase. The apparent splitting of the 120 cm-’ mode in Fig. 50a is only a consequence of the weak dispersion of the corresponding phonon branch. As in the case of inelastic neutron scattering (cf. Section 21,a), two types of activation processes are to be considered: (1) Activation of modes of wave vector k

=

fki, f2ki, etc.. . ., due to

mixing with optically active k = 0 modes. This may be looked upon as an A. A. Volkov, G. V. Kozlov, J. Petzelt, and 0.Hudak, preprint. 0.Hudak, J . Phys. C 16,2659 (1983). I4O M. V. Klein, in “Light Scattering near Phase Transitions” (H. Z. Cummins and A. P. Levanyuk, eds.), pp. 503-530. North-Holland, Amsterdam, 1983. l4I H. Poulet and R. M. Pick, in “Incommensurate Phases in Dielectrics” (R. Blinc and A. P. Levanyuk, eds.), Vol. 1, pp. 315-335. North-Holland, Amsterdam, 1986. 14’ P. Gervais and P. Echegut, in “Incommensurate Phases in Dielectrics” (R. Blinc and A. P. Levanyuk, eds.), Vol. 1, pp. 337-364. North-Holland, Amsterdam, 1986. L43 M. Wada, A. Sawada, and Y. Ishibashi, J. Phys. SOC. Jpn. 50,737 (1981).

13*

39

288

R. CURRAT AND T. JANSSEN

Th Br4

I

0

r\

Y(xxlz

100

200

1

20

I

I

I

40

%crn-l) FIG. 50. Temperature dependence of polarized Raman spectra in 8-ThBr,: (a) part of the EB spectrum above and below the phase transition (T L 92 K):(b) soft-mode behavior below T .After Hubert et

indirect process. It has been discussed in detail by Dvorak and Petzelt5' and P e t ~ e l t Within . ~ ~ Landau theory, the mixing coefficients are obtained via a linearization of the relevant anharmonic terms in the free-energy expansion (quasiharmonic approximation; cf. Section 7). (2) Direct activation mechanisms, derived from a linearization of the twophonon cross section, one of the phonons being identified to a Fourier component of the static modulation. This is the process which is discussed in ~" Section 9 above, as originally proposed by Johnston and K a m i n ~ w ' ~for ordinary structural transitions and by Poulet and Pick 5 4 for incommensurate phase transitions. Both channels lead to the same qualitative predictions, notably concerning selection rules,54but may differ quantitatively.

W. D. Johnston, Jr. and I. P. Kaminow, Phys. Rev. 168, 1045 (1968).

289

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES I

a I

100

x

)

4

0

3

0

FREOUENCY

2

0

SHIFTS

1

0

I

0

I

b

5

0

4 0 3 0 FREOUENCY

lcm-9

2 0 SHIFTS

I

0

I0

I cmP1

FIG. 51. Temperature dependence of polarized Raman spectra in biphenyl: (a) x(zz)y polarization; (b) z ( y x ) y polarization (the x and y axes are parallel to the directions of the crystal a and b axes; the z axis is perpendicular to the x y plane). Peaks marked by arrows have been ascribed to soft modes associated with the phase transitions at TI and T,.After Wada et al.14'

Optical selection rules can be derived on a rigorous basis through the use of superspace symmetry, as discussed in Section 13. Experimental studies by Rasing et a1.'44*145on Rb,ZnBr, and Meekes et on Na,CO, have demonstrated the usefulness of this approach. Some of the Raman results from Rb,ZnBr4 are shown in Fig. 52. This compound is isostructural with K2Se04 in the normal phase ( T > 5 = 355 K) and follows a very similar sequence of incommensurate and commensurate (T,= 200 K) phases. The soft mode is overdamped above 7i: and gives rise to quasielastic scattering.'06 Below the amplitude mode becomes progressively underdamped and is observed in the frequency range 15-20 cm-' ( T < 250 K). The phase mode remains overdamped throughout. In addition to the amplitude mode, three new lines are observed below 5,which have been identified as k = ki modes ~

2

1

.

~

~

~

9

~

~

T. Rasing, P. Wyder, A. Janner, and T. Janssen, Solid State Cornrnun. 41,715 (1982). T. Rasing, P. Wyder, A. Janner, and T. Janssen, Phys. Rev. B 25,7504 (1982). 146 H. Meekes, K. Hanssen, A. Janner, T. Janssen, P. Wyder, and T. Rasing, Ferroelectrics 53,285

144 145

(1984). 14'

H. Meekes, T. Rasing, P. Wyder, A. Janner, and T. Janssen, Phys. Rev. B 34,4240 (1986).

~

290

R. CURRAT AND T. JANSSEN

c(bb)a scattering Ag b(c b)a %I Pcmn P ss7 Pcmn

Rb$nBr,

0

240 -

220

P12,l? Pc2,n

I#

I I

I

I

I I

I

-

IL

I m

100-

>

V

z

W

80-

60-

IL

40 -

20-

0

1 ,

I I

1

\ '

I

1

FIG.52. Frequency shifts for part of the A, and B,, Raman spectrum in Rb,ZnBr,. The new activated modes in the incommensurate phase are indicated by numbers. After Rasing et

[ki = *(1 - S)c*].For example, the line at 206 cm-' (labeled 1 in the figure) is assigned to an internal vibration of the ZnBr, tetrahedra. Both modes at 200 and 206 cm-' originate from the same dispersion branch in the normal phase, and correspond to k = 0 and k, modes, respectively. To interpret the observed spectra, Rasing et aZ.'45 have developed a simple lattice dynamical

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

29 1

6 1

117 01L

0.13

I18 0 12

li 0.11

9

1

A-

!

I

1

1

I

T

180 ’95 (K FIG.53. Temperature dependence of the modulation wave vector 6c* in thiourea (cooling and heating runs). After Durand ef

model of the parent structure. In the incommensurate phase, the spring constants between adjacent Rb ions and ZnBr, tetrahedra are assumed to be sinusoidally modulated. The amplitude of the modulation varies with temperature as the actual modulated displacements. The model is an adaptation of the ID modulated spring constant model discussed in Section 16 above. An interesting situation arises in incommensurate proper ferroelectrics, with regard to the optical activity of the q = 0 amplitude mode. Materials belonging to that symmetry class are sodium nitrite (NaNO,) and thiourea [SC(NH,),]. In thiourea,’48 the modulated state sets in at T4 2: 200 K, with modulation wave vector k, parallel to the c * axis of the Pbnrn(D:,6) parent 148

F. Denoyer and R. Currat, in “Incommensurate Phases in Dielectrics” (R. Blinc and A. P. Levanyuk, eds.), Vol. 2, pp. 129-160. North-Holland, Amsterdam, 1986.

292

R. CURRAT AND T. JANSSEN

W

B2"

et

FIG.54. Schematic temperature dependence of the soft optic branch in thiourea. After Wada al.i50

structure: k,

(22.1)

=~(T)c*

As shown in Fig. 53,149 6 ( T )decreases from a value close to 3 at T4 (T in the figure) to 6 at T,(T,)with a narrow intermediate lock-in phase at 6 = Below TI, 6 vanishes and the structure becomes ferroelectric (Pb2,m) with macroscopic polarization along b. The soft mode belongs to a polar optic branch which is IR active (&) at the point (cf. Fig. 54).l5O In the modulated regime (Tl < T < T4), selection rules53indicate that the q = 0 phase and amplitude modes ( k = k, in Fig. 54) are not IR active. At T = TI, the translational periodicity of the parent structure is restored and the phason branch disappears altogether. The q = 0 amplitude mode survives as a k = 0 polar optic mode, with an oscillating dipole moment parallel to the spontaneous polarization (llb). Thus the mode is both Raman and IR active below Tl. The LO-TO splitting associated with its polar

4.

'41 I5O

D. Durand, Thesis, University Paris-Sud, p. 100 (1987). M. Wada, A. Sawada, Y. Ishibashi, and Y. Takagi, J. Phys. SOC.Jpn. 45, 1905 (1978).

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

40

293

t

>. V

z W

20W

a

LL

10

0

1 I00

150

200

TEMPERATURE ( K 1 FIG.55. Frequency shifts of the q 1 0 amplitude-mode peaks for several scatteringgeometries, 0",( A ) 45", and (0)90". After Wada as a function of temperature. Angles between q and b are (0) et u l . l S 0

character can be obtained from the angular dependence of the observed Raman frequency. This is shown in Fig. 55 for 3 directions of photon momentum transfer q with respect to the b axis. Above TI, the macroscopic electric field associated with the vibration disappears and all 3 geometries yield the same ( T O )frequency. The LO frequency (and the TO frequency below TI) are in good agreement with IR reflectivity results from Brehat et al.lsl and Khelifa et a l l s 2 The observability of the q = 0 phase mode by light scattering techniques has been often discussed in the literature, sometimes in controversial terms. 15' 15'

F. Brehat, J. Claudel, P. Strimer, and A. Hadni, J. Phys., Left.37L, 229 (1976). B. Khelifa, A. Delahaigue, and P. Jouve, Phys. Status Solid B 83, 139 (1977).

294

R. CURRAT AND T. JANSSEN

There are in fact three distinct questions to be answered, which concern, respectively: (1) The selection rules; (2) The order of magnitude of the integrated spectral intensity; (3) The spectral line shape.

The answer to point (1) is now fairly clear: the q = 0 phase mode is always active in some Brillouin spectra, and group theoretical methods have to derive selection rules appropriate to each crystal been developed symmetry. To try to answer point 2 let us apply Eq. (9.7) to a crystal in the symmetry class of K,SeO,. We assume a standard scattering geometry, where the incident and scattered beams travel along principal directions in the crystal and are both polarized along the same direction, CI. Equation (9.7) reads: 53-56360

(22.2)

where we have made use of the selection rule (cf. Ref. 54, Table 2): Raa.0 1

=L

R a a1, ,

(22.3)

Equation 22.2 is analogous to Eq. (18) in Ref. 55. It illustrates two basic points: (1) The anisotropic character of the Brillouin cross section. The scattered intensity vanishes when the momentum transfer K is perpendicular to the modulation wave vector ( K , = 0). (2) The scattered (integrated) intensity i a ( K ) is likely to be much lower than for an acoustic mode of same frequency, due to the 1’ factor in Eq. (22.2). This prediction assumes that the tensor elements Rala,, are of same order of magnitude as the Pockels coefficients which enter the ordinary Brillouin cross section, as there appears to be no general argument for why they should be much larger.

The spectral line shape of the q = 0 phase mode [point (3)] is expected to be overdamped [cf. Eq. (21.7)], and much of its spectral weight may be lost in the Rayleigh line. Inserting realistic values for u?, r,, and q = K in Eq. (21.8), the quasielastic half-width y$ is estimated to lie in the range from 10 to 100 MHz, which is comparable to the best attainable instrumental resolution^.'^^ The observation of the q = 0 phase mode by Brillouin scattering is thus a major experimental challenge, much more so than for inelastic neutron scattering. This point was first stressed by Golovko and Levanyuk.,’ 153

P. A. Fleury and K. B. Lyons, in “Light Scattering near Phase Transitions” (H. Z. Cummins and A. P. Levanyuk, eds.), pp. 449-502. North-Holland, Amsterdam, 1983.

295

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

The possible occurrence of a gap at q = 0 is a mixed blessing since the increased linewidth is accompanied by a loss in integrated intensity. Let us assume the presence of a 70 GHz gap at q = 0 in the phase mode spectrum of incommensurate K,SeO,, as suggested by the neutron data in Fig. 48. With m4 = mg = 70 GHz and r, = 250 GHz, the linewidth y4 is raised to 20 GHz, for both neutron and light scattering. However, the loss in integrated intensity, as compared to the gapless case, is quite severe:

-

("->

a* (3)' = v4a* K 0

4

2

=

(5 2 x 103 103)

=

103

(22.4)

Infrared and microwave spectroscopy offer an alternative approach for the experimental observation of the q = 0 phase mode. Although selection rules indicate that IR activity is limited to systems which are piezoelectric in the normal phase,53there is a whole class of materials where the q = 0 phason is IR active, independently of any symmetry considerations: these are the CDW compounds for which the IR activity is associated with electron transport (Frohlich conductivity), as opposed to ionic vibrations. Figure 56 shows the 3 H 2 0 (KCP) as calFIR conductivity spectrum from K,Pt(CN),Br,,, ' ~ ~sharp resonance around culated from reflectivity data by Briiesch et ~ 1 . The 15 cm-' has been interpreted as the pinned q = 0 phase mode, the pinning frequency arising from impurities or lattice imperfections. This interpretation has been challenged by P e t ~ e l t who , ~ ~ argues that the mode in Fig. 56 corresponds to an antiphase vibration of CDWs on adjacent Pt chains, the q = 0 phason (in-phase vibration) being IR inactive. The finite mode frequency would then arise from Coulomb interactions between CDWs rather than pinning by impurities. On the other hand, it seems intuitive that the in-phase q = 0 phason carries current and hence should be IR active, regardless of the symmetry of the ionic displacements involved. Hence, if Petzelt's assignment is correct, another resonance corresponding to the in-phase mode should be seen at lower frequencies. The present authors are not aware of any dielectric measurements in the millimeter-wave range. Conductivity measurements at 1 GHz and below have been performed on a number of other CDW compounds'55,156and have revealed an overdamped response corresponding to a relaxation frequency y4 in the 100 MHz range. Recently, conductivity measurements in the intermediate frequency range (5 to 100 GHz) have been performed in (TaSe,),I, another quasi-1D CDW compound.' 5 7

-

lS4 '55

15'

P. Briiesch, S. Strassler, and H. R. Zeller, Phys. Rev. B 12,219 (1975). A. Zettl and G. Griiner, Phys. Reo. B 29,755 (1984). A. Zettl, C. M. Jackson, and G. Griiner, Phys. Rev. B 26,5773 (1982). D. Reagor, S. Sridhar, M. Maki, and G. Griiner, Phys. Rev. B 32,8445 (1985).

296

R. CURRAT A N D T. JANSSEN

0

+O

20

40 50 w (cm-')

30

60

70

80

FIG.56. Frequency-dependent conductivity in KCP, as calculated from far-infrared reflectivity data. T = (---), 295 K; (- .. -), 206 K; (...), 135 K; (- . -), 88 K; and (-), 4.2 K. After Bruesch et ~ 1 . ' ~ ~

A weakly damped resonance was observed (cf. Fig. 57) which could be fitted to a single DHO-type response function: Reo(w) = o,

w2r: (w2-

+ w2r;

(22.5)

yielding cob N 35 GHz and r, N 20 GHz. The frequency cob correlates well with a low-lying TA-mode frequency measured by inelastic neutron scatteringI5*at the CDW satellite position. 158

P. Monceau, L. Bernard, R. Currat, F. Levy, and J. Rouxel, Physica B+ C 136B,352 (1986).

297

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

1.C

k-

b” ox

\

b

0.2 50

10

90

w l 2 7 r (GHz) FIG.57. Frequency-dependent conductivity for (TaSe,),l at 150 K (qcierls = 263 K). (-)is the best fit to Eq. (22.5),with m$ = 34 GHz and r, = 21 GHz. After Reagor et ~ 1 . ’ ~ ’

23. RESULTS FROM OTHER TECHNIQUES Indirect evidence on soft-mode frequencies and lifetimes may been derived from resonance spectroscopy and ultrasonic attenuation. Both techniques are applicable to the study of low-frequency excitations in structurally modulated solid^.'^^,'^^ In particular, the NMR spin-lattice relaxation rate T;’, which increases critically at a second-order phase transition, should be sensitive to the existence of a gap in the phason spectrum. Figure 58 shows the NMR signal corresponding to the *’RbL2 + -L2 transition, as observed in the (1) normal and (2) incommensurate phases of Rb2ZnC1,. In the latter spectrum the resonance line is broadened by the distribution of local fields at the Rb sites (inhomogeneous broadening). The line shape is well described, within the sinusoidal approximation, assuming a linear relation between the resonance frequency at each site, vl, and the local amplitude of the modulated displacements: v, = vo

+ v1 cos(k, - r + 40)

(23.1)

This simple description accounts for the observed line shape over most of the incommensurate region. Deviations from this sinusoidal or “plane-wave” limit are confined to a narrow temperature range above the lock-in transition at T,. R. Blinc, in “Incommensurate Phases in Dielectrics”(R.Blinc and A. P. Levanyuk, eds.), Vol. 1. North-Holland, Amsterdam, 1986. 160 R. Zeyher, Ferroelectrics 66,217 (1986). 159

298

R. CURRAT AND T. JANSSEN

0 0 0 0

experimental theory

FIG.58. NMR line shape of the 8 7 R b f _ t -$ transition in the (a) normal and (b) incommensurate phases of RbJnCI,. b IH ; angle between a and H , = 120";vL = 29.5 MHz. After Blinc er

The temperature dependence of the s7Rb spin-lattice relaxation time is shown in Fig. 59. A typical soft-mode behavior is observed near IT; with TI decreasing as 7; is approached. In the incommensurate phase TI is constant and anomalously low over a wide temperature range. Tl increases sharply again on approaching T,. These results suggest the presence of a low-lying optic branch with temperature-independent frequency behavior, as expected for the phason branch. Further evidence for a phason-induced relaxation mechanism may be obtained from the study of TI for specific Rb sites. The data in Fig. 59 are broadband measurements where Tl is averaged over the whole frequency distribution in Fig. 58b,16' i.e., over all sites. Narrow-band measurements at two positions within the band are shown in the inset: Tl is seen to depend R. Blinc, S. Juznic, V. Rutar, J. Seliger, and S. Zumer, Phys. Rev. Leu. 44,609 (1980).

299

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

200

100

I>[

1

- 50

0

50 -

I

Lo ""-

20 -

v

10 -

Tc I

I 5~11°C)

'I

/r

o a a 0 >

I

C

. . I"

I

P

TI

I

I

I

strongly on position within the band, the lowest Tl values being obtained at the center of the band. The Rb nuclei which contribute to the center of the band are such that:

el = k i - r l + &, = * x / 2 The corresponding thermal amplitudes are dominated by phase fluctuations [cf. Eq. (8.12)] while the Rb nuclei which contribute to the edge singularities are mostly sensitive to amplitude fluctuations. The temperature dependence of TI in the latter case thus reflects the renormalization of the amplitudemode frequency below 7;. Conversely, the constant Tl value observed at the center of the band follows from the temperature independence of the phason branch. In principle, the dependence of TI over the Larmor frequency (vL) enables one to draw conclusions about the presence of a q = 0 phason gap and on the phason damping r,. Here Tl is found not to vary with vL, which implies the presence of a gap larger than v, = 88 MHz. However, no definite statement can be made about r,.

300

R. CURRAT AND T. JANSSEN

Lbz80.172 MHz

87 rnA

FIG. 60. ”Rb NMR spectra with and without bias current and for two current values above threshold. The central peak arises from motional narrowing in the parts of the sample where the CDW is unpinned. After Segransan et

Another interesting aspect of resonance spectroscopy in incommensurate systems is the possibility of observing motional narrowing of the inhomogeneously broadened transition lines. This will occur whenever the modulated displacements fluctuate at a rate which is fast compared to the characteristic time scale of the measurement (inverse inhomogeneous linewidth). A mixed spectrum of fast and slow fluctuations has been reportedI6’ in Rb’ZnBr, just below the ordering temperature 7;. It has been interpreted as reflecting the coexistence within the specimen of pinned (i.e., static) and free (“floatingphase”) regions. In CDW systems, depinning of the modulation can be achieved by applying a dc electric field, E . The threshold value for the field, E,, is, in principle, related to the phason gap. For E > E , motional narrowing of the resonance lines should occur. Recent NMR results on NbSe3’63 and R ~ , , , M o O , ’ ~ ~

162

163

R. Blinc, D. C. Ailion, P. Prelovsek, and V. Rutar, Phys. Rev. Lett. 50, 67 (1983). J. H. Ross, Z . Wang, and C. P. Slichter, Phys. Rev. Left.56, 663 (1986). P. Segransan, A. Janossy, C. Berthier, J. Marcus, and P. Butaud, Phys. Rev. Left. 56, 1854 (1986).

EXCITATIONS IN INCOMMENSURATE CRYSTAL PHASES

30 1

(see Fig. 60) show a correlation between the CDW current and the partial narrowing of the resonance line shapes. Ultrasonic attenuation, like resonance spectroscopy, yields indirect information on the phason spectrum. In particular, ZeyherI6' has shown that the presence of a small phason gap should give rise to a characteristic frequency dependence of the acoustic attenuation (see Ref. 160 for a review of the experimental situation). A phason contribution to the low-temperature specific heat has been looked for in alkali metals.'65 Conclusive results have been reported recently166 in (TaSe,),I, where a specific heat anomaly was detected in the expected temperature range.

VII. Concluding Remarks

While, in theory, the distinction between ordinary and incommensurate crystal phases appears to be of a fundamental nature, physical properties in the two classes of systems differ only in subtle ways. This is particularly true for macroscopic behavior, where, with the outstanding exception of Frohlich conductivity in CDW systems, the absence of translational invariance has very little impact. The excitation spectrum of modulated systems exhibits a number of distinctive features, such as the existence of a soft phase-fluctuation spectrum and the progressive opening of gaps at specific locations, inside the Brillouin zone of the parent structure. The spectroscopy of these effects is still at an early stage and only a few loworder gaps have been investigated. The practical difficulties encountered in attempting to detect low-q phasons have been underlined in the context of neutron and light scattering studies. The role of defects has not been discussed in detail in this review. Defects are known to influence the static properties of incommensurate phases, to a much larger extent than in ordinary crystals. Doping with non-symmetrybreaking defects has little influence on the normal-to-incommensurate transition, but may destroy completely the lock-in transition. The role of defects for the dynamics is less well understood, and hardly discussed in the

J. Van Curen, E. W. Hornung, J. C. Lasjaunias, and N. E. Philips, Phys. Rev. Lett. 49, 1653 ( 1 982). L66 K. Biljakovic, J. C. Lasjaunias, F. Zougrnore, P. Monceau, F. Levy, L. Bernard, and R. Currat, Phys. Rev. Lett. 57, 1907 (1986).

302

R. CURRAT AND T. JANSSEN

literature. As a rule dilute defects are expected to couple to long-wavelength excitations only. However, the extent to which “defect-induced central-peak’’ t h e o r i e ~ , ’ ~ ’ . developed ’~~ in the context of ordinary phase transitions, remain applicable in the case of low-q phase modes, needs to be clarified. The formalism used to describe excitations in incommensurate displacively modulated crystals may also be used for other quasiperiodic systems, for example for quasicrystals. There, however, phasons play a role that is very different, because of the strong diffusive and nonlinear effects. At the moment of writing, the dynamics of these systems has only been studied in extremely simplified models that do not fully take into account their specific features. Nonlinear forces are at the very basis of the incommensurate phase transitions. The dynamics, however, have in the present contribution only been treated in the linear regime. Nonlinear excitations are expected to play a more important role in modulated systems than in ordinary crystals.

16’

B. I. Halperin and C. M. Varrna, Phys. Rev. B 14,4030 (1976). A. P. Levanyuk, V. V. Osipov, and A. A. Sobyanin, in “Theory of Light Scattering in Condensed Matter” (B. Bendow, J. L. Birrnan, and V. M. Agranovich, eds.). Plenum, New York, 1976.

Author Index

Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text.

A Aarts, J., 102 Abe, R., 280 Abrahams, E., 21 Abrikosov, A. A., 80 Adlhart, W., 206 Aharony, A., 156, 161(12) Ahlheim, U., 140 Aiki, K., 281 Ailion, D. C., 300 Akaogi, M., 191 Akimota, S., 191 Albers, R. C., 24, 26(39) Allen, J. W., 41 Amelinckx, S., 181, 182(70), 186(70), 187(70), 188(69), 208, 209(26), 260(25)

Andersen, 0. K., 29, 36, 37 Anderson, P. W., 12, 15, 20(22), 67, 107, 113(152,153), 127(152,153), 266

Andre, G., 247 Andrei, N., 10 Andres, K., 192 Angel, R. J., 131 Appel, J., 131 Armistead, K., 179 Ashauer, B., 143 Assmus, W., 40, 41(64), 43(64), 44(64) Aubry, S., 237, 247 Auerbach, A., 132, 133(176) Avron, J., 247 Axe, J. D., 206, 217, 222(6), 233(6), 208, 260(24), 262, 263(99), 267, 278, 280(125), 281(125)

Bak, P., 152, 154, 156, 157, 159, 161, 162(13,14), 175, 198(16), 200, 262, 263(99), 264 Banavar, J. R., 247 303

Barnes, S. E., 73 Barreto, M. N., 169, 171, 172(32), 194(33) Bartholin, H., 197, 198 Batlogg, B., 40 Baumgard, P., 62, 63(86a) Becker, K. W., 52, 55(77), 62(77), 79, 98 Bedell, K. S., 65, 127(94,96) Bellissard, J., 247 Benoit, A., 28 Berenson, R., 223 Bernard, L., 268, 269(114), 270(114), 271(114), 272(119), 273, 274(119), 296, 301 Berthier, C., 300 Beuers, J., 141 Biljakovic, K., 301 Bilz, H., 210, 278 Binder, K., 180 Birman, J. L., 223, 248 Bishop, D. J., 40 Blandin, A., 15, 68(24) Bleif, H. T., 267 Blinc, R., 212, 297, 298, 299(161), 300 Blount, E., 113 Bommel, H. E., 40, 60(66) Boring, A. M., 24, 26(39) Bouillot, J., 268, 276, 277(120) Brand, H., 264 Brandow, B. H., 73, 99(115) Braun, H. F., 24, 148(34) Bredl, C. D., 102, 141 Brehat, F., 293 Bringer, A., 9, 16(10) Brinkman, W. F., 99 Broddin, D., 181, 182(70), 186(70), 187(70) Brown, G. E., 65, 127(96) Bruce, A. D., 211, 217 Bruce, D. A., 221, 281(51) Bruder, C., 111 Briiesch, P., 295, 296(154) Bruinsma, R., 199, 200 Bruls, G., 44

304

AUTHOR INDEX

Bucher, E., 40, 44, 60(66) Buchhholtz, L. J., 139, 142 Bugaut, F., 268, 275(111), 278(111) Burlet, P., 197, 198 Buseck, P. R., 196 Bussmann-Holder, A,, 210, 278 Butaud, P., 300 Biittner, H., 210, 278

C Cailleau, H., 268, 275(110,111), 276, 277(110,120), 278, 279(122)

Carneiro, G. M., 64 Carneiro, K., 266 Chandrasekhar, B. S., 141 Cheng, C., 196 Chiang, S., 280 Christensen, N. E., 24, 26(39) Churcher, N., 196 Claudel, J., 293 Clausen, K., 197, 198(102) Cochran, J. F., 140 Cohan, N. V., 247 Cole, M. W., 208 Coleman, P., 73, 83(109), 87, 92(141) Coppersmith, S., 175 Coutinho-Filho, M. D., 169 Cowley, R. A., 211, 217, 267 Cox, D. E., 267 Cox, D., 15 Crabtree, G. W., 24, 26(38) Currat, R., 267, 269(114), 270(114), 271(114), 272(119), 273, 274(119), 283(112), 284(112), 285(112,113), 291, 296, 301 Cyrot, M., 41, 88, 135

D d’Ambrumeni1, N., 24, 25, 29(35), 30(47), 32(35), 33

de Chltel, P. F., 10, 35(19) de Fontaine, D., 180, 186, 187(66), 188, 188(66)

de Gennes, P. G., 12 de Groot, K., 40, 60(66) de Kouchkovsky, R., 268, 269(114), 270(114), 271(114) de Lange, C., 242, 243(75), 245(75)

de Pater, C. J., 267 De Ridder, R., 208, 209(26), 260(25) De Witte, J., 276 de Wolff, P. M., 203, 225, 226(58) Delahaigue, A., 293 Delamoye, P., 268, 269(114), 270(114), 271(114), 272(119), 273, 274(119), 288(116) Delugeard, Y., 278 DCnoyer, F., 291 Dolling, G., 267 Domb, C., 165 Doniach, S., 21 Ducastelle, F., 180, 182(65), 183(65), 184(65), 186(65), 198(65) Durand, D., 291(149), 292, 302 Duxbury, P. M., 154, 156(8,9), 161(8,9), 179 Dvorak, V., 211, 212, 223, 288(52) Dy, K. S., 129 Dzyaloshinski, I. E., 211

E Echegut, P., 287 Ecolivet, C., 276 Eibschuetz, M., 267 Einzel, D., 141 Eller, W., 280, 284(133), 285(133) Elliott, R. J., 153, 210 Everts, H. U., 164, 169(20)

F Falk, H., 160 Faulkes, I. F., 136 Fay, D., 131, 143 Fazekas, P., 73, 99 Fedro, A. J., 133 Fenton, E. W., 143 Fick, E., 54 Filipc, C., 281 Finel, A., 186 Finger, L. W., 188, 189(83), 195(83) Finger, W., 220, 231(48,50), 263(48), 264(50), 281(50)

Fischer, P., 197 Fisher, M. E., 153, 154(5,6), 155, 156(11), 158(11), 159(11), 162(11), 164(5,6), 165(5,6), 166(5,6), 167, 168, 170(5,6), 177(22), 178, 179, 199(11)

305

AUTHOR INDEX

Fishman, S., 174, 175(47), 175 Fisk, Z., 7, 24, 27(45), 141, 143 Fleury, P. A., 280, 294 Florence, D., 197 Flouquet, J., 28, 44 Forster, D., 54 Fortuin, C. M., 210 Frank, F. C., 189, 195(87), 236, 237(64) Franse, J. J. M.,44, 63, 140 Franz, W., 102 Freeman, A. J., 24, 26(38) Frenkel, Y. I., 236 Friedel, J., 82 Fujimara, T., 40 Fujiwara, K., 188 Fukui, M., 280 Fukuyama, H., 86, 121(135), 124(135), 206, 266 Fulde, P.,25, 27, 33, 41, 47, 52, 55(77), 65, 77(73), 98, 121(46), 122(46), 137, 141 Fung, K. K., 206 Furrer, A., 198 Furuya, K., 10

G Garger, M., 140 Geibel, C., 137 Geisel, T., 257 Gelatt, C. D., Jr., 29 Gervais, P., 287 Geshkenbein, V. B., 143 Giorgi, A. L., 137 Girard, A., 278, 279(122) Glass, N. V., 258 Glotzel, D., 37 Goldman, 144 Goldring, B., 40 Golovko, V. A., 220, 223(49), 231(49), 281(49), 287(55), 294(49) Gordon, M., 158, 159(15), 162(15), 163(15), 164(15), 194(15), 199(15) Gorkov, L. P., 112, 114(158), 115(158), 138(158), 141(158) Goto, T., 40 Gratias, D., 180, 181, 186(68), 187(67), 188(69) Grewe, N., 73, 133 Gronsky, R., 200 Gross, F., 141 Griiner, G., 7, 295, 297(157)

Gummich, U., 133 Gunnarsson, O., 10, 13(20), 14(20), 15, 16, 35, 80(20) Guntherrodt, G., 62, 63(86a) Gutfreund, H., 56 Gutzwiller, M. C., 73, 98(114) Guymont, M.,180, 181, 182(70), 186(68,70), 187(67,70) Gyorffy, B. L., 24, 136, 182, 187(72), 188(72)

H Hadni, A., 293 Haemmerle, W. H., 40 Haen, P., 44 Hafner, H. U., 45 Halg, B., 198 Halperin, B. I., 302 Hamano, K., 267 Han, S., 143 Hanami, M., 267 Hanzawa, K., 15, 28 Haque, M. S., 210 Hardy, J. R., 210 Hasai, H., 86 Hastings, J. M., 208, 260(23,24) Hatta, I., 267 Hazen, R. M., 188, 189(83), 195(83) Heeger, A. J., 208, 260(23,24) Heilmann, I. U., 208, 260(24) Heine, V., 188, 189(82), 190(82), 193(82), 195(82), 196, 210 Hillebrands, B., 62, 63(86a) Hirsch, J. E., 134 Hirschfeld, P. J., 139, 141 Hofstadter, D. R., 243, 247(76) Hohn, T., 133 Holland-Moritz, E., 28 Horioka, K., 189 Horioka, M., 219, 280(47), 286(47) Horiuchi, H., 189, 191 Horn, S., 28 Hornreich, R. M., 155 Hornung, E. W., 301 HorvatiC, B., 73, 84 Hubert, S., 268, 269(114), 270(114), 271(114), 273(116), 288(116) Hudak, O., 134, 286, 287 Hufnagl, J., 44

306

AUTHOR INDEX

Hukuda, K., 281 Huse, D. A., 176, 179 Hussonnois, M., 268, 273(116), 288(116) Hussuhl, S., 195

I Iizurni, M., 278, 280(125), 281(125) Ikeda, T., 278 Imry, Y., 206 Inoue, K., 283, 285(137), 286(137) Ishibashi, Y., 211, 280, 283, 284(130), 285(130,137), 286(137), 287, 289(143), 292, 293(150) Ishii, T., 238

J Jackson, C. M., 295 Jagodzonski, H., 189, 195(88) Jamet, J. P., 173 Janner, A., 203, 205, 225, 226(58), 227(4), 260, 262(98), 288, 289, 290(145)

Janossy, A,, 300 Janssen, T., 203, 205, 207, 225, 226(58), 227(4), 228, 238, 241(74), 231(50), 242, 243(75), 245(75), 248, 250(19), 253, 254(74), 255(19), 258, 260, 262(98), 288, 289, 290(145) JariC, M. V., 111, 233 Jarlborg, T., 24, 148(34) Jayaprakash, C., 20 Jensen, J., 27 Jensen, M. H., 157, 161(13), 162(13) Jepps, N. W., 173, 175(47), 193, 194(94) Jepsen, O., 37 Jichu, H., 71, 86, 121(136), 121 Jin, B., 86 Johnson, C. K., 208, 260(25) Jones, B., 21 Jouve, P., 293 Joynt, R., 141 Juznic, S., 298, 299(161)

K Kadin, 144 Kaiser, S., 266

Kaski, K., 168 Kasuya, T., 40 Kawakami, N., 10 Keiber, H., 137 Keiter, H., 73 Keller, J., 25, 79, 98, 121(46), 122(46), 133, 257

Khelifa, B., 293 Kieselrnann, G., 143 Kimball, J. C., 73 Kirf, G., 280, 284(133), 285(133) Kitano, Y., 199 Kitazawa, H., 40 Klein, M. V., 287 Klernrn, R. A., 139 KnakJensen, S. J., 168 Kobayashi, T., 280, 281 Koelling, D. D., 24, 32(42), 33(42) Koga, H., 281 Kohmoto, M., 247 Kornura, Y., 199 Kondo, J., 7 Kontorova, T., 236 Kotliar, G., 102 Kouroudis, I., 44 Kozlov, G. V., 286 Krakauer, G., 24, 25(40), 26(40) Kress, W., 210 Krishna, P., 188, 189, 192(86), 193(86), 195, 196(86)

Krishna-rnurthy, H. R., 10, 20 Kubler, J., 24, 25, 25(35), 29, 30(47), 32(35), 33(35)

Kucharczyk, D., 211 Kulik, J., 180, 186, 187(66), 188(66) Kunii, S., 40 Kurarnoto, Y., 73, 78(107) Kurkijaervi, J., 142 Kuroda, Y., 71, 86, 121(136)

L Lacroix, C., 41, 88 Lanbrecht, W. R. L., 37 Landau, L. D., 57, 106(84), 109 Langier, R., 197 Langreth, D., 82 Larkin, A. I., 143 Lasjaunias, J. C., 301 Lavagna, M., 41, 132

307

AUTHOR INDEX

Le Cloarec, M. F., 268 Le Daeron, P. Y., 237 Leaderer, P., 173 Lebech, B., 197, 198(102) Lee, P. A., 7, 132, 133, 206, 266 Lee, T. K., 73 Lefrant, S., 268, 273(116), 288(116) Leggett, A. J., 65, 126 Lepostollec, M., 268, 273(116), 288(116) Levanyuk, A. P., 211, 212, 220, 223, 231(49), 281(49), 287(55), 294(49), 302 Levin, K., 129, 132, 133(176) Levstik, A,, 281 Levy, F., 296, 301 Li, T. C., 16, 35 Lieke, W., 102 Lifshitz, E. M., 57, 106(84), 109 Lima, R., 247 Loewenhaupt, M., 28, 47 Loiseau, A., 180, 181, 182(65,70), 183(65), 184(65), 186(65,70), 187(70), 198(65) Lonzarich, G. G., 24, 27(45) Lovesey, S. W., 231 Lowdin, P. O., 37 Lowenstein, J. H., 10 Lu, J. P., 248 Luban, M., 155 Lustfeld, H., 9, 16(10) Luther, A., 65 Luthi, B., 40, 41(64), 43(64), 44, 47, 61 Lynn, J. W., 207 Lyons, K. B., 280, 294

M Ma, S. K., 206 Machida, K., 113, 138(161) Machida, M., 280 Maekawa, S., 28, 122 Maki, K., 137, 141 Maki, M., 295, 297(157) Maradudin, A. A., 258 Marcus, J., 300 Marshall, W., 231 Martin, P. C., 53 Martin, R., 41 Matsuura, T., 71, 86, 121 Maurer, D., 40, 60(66) McDiarmid, A. G., 208, 260(23,24) McKernan, S., 206

McMillan, W. L., 207, 215 Meekes, H., 289 Meier, G., 197 Menovsky, A., 44 Mermin, N. D., 104, 110 Meschede, D., 102 Messager, J. C., 268, 275(111), 279(122) Messiah, A,, 82 Michel, K. H., 210 Michel, L., 111, 138(156) Mihaly, L., 84, 86(130) Millis, A. J., 63, 132, 142, 143 Minervina, Z., 193 Miro, N. D., 208, 260(23) Miyake, K., 71, 86, 121(136), 121, 139 Mock, R., 62, 63(86a) Monceau, P., 208, 296, 301 Monien, H., 115, 139 Mori, H., 52 Morimoto, N., 189 Moussa, F., 268, 275(111), 276, 277(120), 278(111) Muller, G., 195 Muller, V., 40, 60(66) Murray, A. F., 211

N Nagaoka, Y., 86, 121 Nagi, A. D. S., 86 Nakanishi, K., 175, 175(49), 207, 216, 219(43) Needs, R. J., 196 Nelmes, R. J., 267 Newbury, R., 24, 27(45) Newns, D. M., 24, 25(43), 29(43), 30(43), 32(43), 34(43), 73, 87 Ng, K. W., 143 Nguyen, 248 Niksch, M., 40, 41(64), 43(64), 44(64) Novaco, A. D., 256 Nozikres, P., 15, 27, 65, 67(51), 68(24)

0 Odagaki, T., 248 Oguchi, T., 24, 26(38) Ohkawa, F. J., 86, 121(135), 124 Ohmi, T., 113, 138(161)

308

AUTHOR INDEX

Okiji, A,, 10 Omitmaa, J., 167, 168 Osipov, V. V., 302 Ostlund, S., 176 Ott, H. R., 7, 40, 44(4), 127, 138(4), 139(4), 141 Ottinger, H. C., 169, 170(37), 179 Overhauser, A. W., 206, 221(5), 233(5)

Q Quader, K. F., 65, 127(94,96) Quader, K. J., 65 Quezel, S., 198 Quilichini, M., 268, 280, 283, 284(112,134), 285(112,113) Qzaki, M., 113, 138(161)

P Paciorek, W. A., 211 Page, T. F., 193, 194(94) Pals, J. A., 142 Pandey, D., 189, 192(86), 193(86), 195, 196(86) Parker, S. C., 192, 196(91), 199(91) Parlinski, K., 207 Parodi, O., 53 Patton, B. R., 65, 127(90), 129(90), 206 Pearsall, G. W., 140 Pellegrin, J., 276 Pelton, A. R., 200 Pennetier, J., 268 Pershan, P. S., 53 Peter, M., 24, 148(34) Pethick, C. J., 59, 64, 65, 127(96,97), 129, 131(97), 139(97) Petzelt, J., 223, 286, 287(53), 288(52,53), 195(53)87 Peyrard, M., 237 Philips, N. E., 301 Pick, R. M., 223, 225(54), 287, 288(54) Pickett, W. E., 24, 25(40), 26(40) Pines, D., 59, 65, 127(96,97), 131(97), 139(97) Pokrovsky, V. L., 157, 162(14), 168, 169, 170(36), 198(31,36) Poppe, U., 143 Portier, R., 180, 181, 182(65,70), 183(65), 184(65), 186(65,70), 187(67), 198(65) Pott, R., 40, 41(64), 43(64), 44(64) Pouget, J. P., 208, 260(23) Poulet, H., 223, 225(54), 287, 288(54) Prelovsek, P., 281, 300 Price, G. D., 188, 189(79,84), 190(80,81), 192, 193(80,81), 195(81,82,84), 196, 199(91) Price, S. L., 210 Pruschke, T., 133 Puech, L., 44

Radons, G., 257 Rainer, D., 67, 128, 129(168), 140, 142, 143 Rainford, B. S., 47 Rainford, R. D., 197 Ramakrishnan, T. V., 10, 15(18), 16(18) Ramasesha, S., 188, 189(79), 190(79), 193(79) Randall, L. J., 70, 99(102) Rasing, T., 288, 289, 290(145) Rasmussen, E. B., 168 Rauchschwalbe, U., 137, 140, 141 Razafimandimby, H., 25, 121(46), 122(46) Read, N., 73, 87, 132 Reagor, D., 295, 297(157) Redner, S., 168 Renker, B., 137 Rice, T. M., 7, 70, 92, 99, 100, 113, 127, 141, 206, 220, 231(48), 237, 256(65), 262(65), 263(48), 266 Rietschel, H., 134, 137 Roeder, H., 169, 173(34), 174, 194(34) Ross, J. H., 300 Rossat-Mignod, J., 197, 198 Rouxel, J., 296 Ruckenstein, A. E., 102 Rudigier, H., 127 Rutar, V., 298, 299(161), 300 Ryan, T. W., 267

S Sacco, J. E., 256, 262(91) Safran, F., 208 Sakado, T., 280 Sakurai, J., 267 Salinas, S. R., 169 Sannikov, D. G., 211 Sanquer, M., 276

309

AUTHOR INDEX

Sato, H., 182, 187(71), 188(71) Sato, N., 40 Sauermann, G., 54 Sauls, J. A., 108, 129, 142 Sawada, A., 219, 280, 284(130), 285(130), 286(47), 287, 289(143), 292, 293(150) Schafer, H., 102 Scharnberg, K., 115, 139, 143 Schefzyk, R., 40, 41(64), 43(64), 44(64) Schlottmann, P., 10, 14 Schmidt, H., 137 Schmitt-Rink, S., 139 Schonhammer, K., 10, 13(20), 14(20), 15, 35, 80(20) Schopohl, N., 143 Schotte, K. D., 61 Schroder, U., 210 Schuh, B., 133 Scoppola, E., 247 Sega, I., 180 Segransan, P., 300 Seigert, M., 164, 169(20) Seliger, J., 298, 299(161) Selke, W., 152, 153, 154, 155(5,6), 156(8,9) 157(9), 161(8,9), 164(5,6), 165(5,6), 166(5,6), 167, 169, 170(5,6), 171(32), 172(32), 180 Serene, J. W., 7, 108, 128, 129 Sham, L. J., 7, 134, 206 Shapir, Y., 175 Shapiro, S. M., 267 Shiba, H., 175, 175(49), 216, 219(43) Shiffman, C. A., 140 Shimaoka, K., 278 Shirane, G., 208, 260(23,24), 266, 278, 280(125), 281(125) Shtrikman, S., 155 Simon, B., 247 Sinha, S. K., 133, 208 Skriver, H. L., 29, 30(54), 31(54) Slichter, C. P., 300 Smith, J. L., 7, 24, 27(45), 40, 127, 141, 143 Smith, J., 169, 170(38,39), 171(38,39), 188, 189(82) Snyman, J. A., 237, 238 Sobyanin, A. A., 302 SokceviC, D., 84 Sokoloff, J. B., 256, 257, 262(91) Sridhar, S., 295, 297(157) Stanley, H. E., 168

Steeds, J. W., 206 Steflich, F., 7, 28, 47, 102, 137, 138(2), 140, 141 Steinberger, I. T., 195 Stewart, G. R., 7, 137, 138(1) Sticht, J., 24, 25, 25(35), 29(35), 30(47), 32(35), 33(35), 33(35) Stobbs, W. M., 180, 187(67) Stocks, G. M., 182, 187(72), 188(72) Stollhoff, G., 134 Strange, P., 24, 25(43), 29(43), 30(43), 32(43), 34(43) Strassler, S., 295, 296(154) Strimer, P., 293 Su, Z. B., 133 Suhara, M., 280 Suhl, H., 80 Sur, K., 10, 15(18), 16(18) Sutherland, B., 219, 256(46) Suzuki, T., 40 Szpilka, A. M., 155, 156(11), 158(11), 159(11), 162(11), 169, 170(41), 171(41), 178(11), 179, 199(11)

T Tachiki, M., 122 Taillefer, L., 24, 27(45) Takagi, Y., 280, 284(130), 285(130), 292, 293(150) Takahashi, K., 191 Takeda, S., 180, 186, 187(66), 188(66) Takke, R., 40, 41(64), 43(64), 44(64) Tamaki, A., 40 Tanner, L. E., 200 Terasaki, O., 187 Tesanovid, Z., 65, 127(95), 129(95), 132 Testard, D., 247 Tewordt, L., 115, 139 Thalmeier, P., 41, 47 Theodorou, G., 237, 256(65), 262(65), 264 Thompson, J. B., 188, 196 Thuneberg, E. V., I42 Tjon, J. A., 207, 238, 231(50), 250(19), 254(74), 255(19), 258 Toiho, F., 208 Tosatti, I., 208 Toth, R. S., 182, 187(71), 188(71) Tsvelick, A. M., 10

310

AUTHOR INDEX

U Uanna, Y., 56 Ueda, K., 70, 92, 99(100, 145), 100, 113, 127 Uimin, G. V., 168, 169, 170(36, 40, 42). 171(40, 42), 192(40), 198(31, 36). 199 Unruh, H. G., 280, 284(133), 285(133) Upton, P. J., 180 Uwe, H., 280

V Valls, 0. T., 65, 127(95), 129, 132 van Beest, B. W., 260 Van Curen, J., 301 van der Merwe, J. H., 236, 237 van Haeringen, W., 142 van Landuyt, J., 181, 182(70), 186(70), 187(70) van Maaren, M. H., 142 van Tendeloo, G., 180, 181, 182(65, 70), 183(65), 184(65), 186(65, 70), 187(70), 188(69), 198(65), 208, 209(26), 260(25) Varma, C. M., 7, 10, 21, 70, 99(102), 139, 302 Veblen, D. R., 196 Verma, A. R., 188, 189(77) Vettier, C., 268, 275(111), 278(111), 279(122) Villain, J., 158, 159(15), 162(15), 163(15), 164(15), 194(15), 197, 199(15) Vogt, O., 197, 198, 198(102) Volkov, A. A., 286 Vollhardt, D., 111, 139 Volovik, G. E., 112, 114(158), 115(158), 138(158), 141(158) von Barth, U., 24 von Boehm, J., 154, 156(7), 157(7), 157(7), 159, 161(7), 198(16)

W Wada, M., 280, 284(130), 285(130), 287, 289(143), 292, 293(150) Walker, D., 115, 139 Walker, M. B., 228 Wang, C. S., 24, 25(40), 26(40) Wang, T. -S. 197 Wang, Z., 300 Watanabe, D., 187

Watson, C. R., 208, 260(25) Weber, D., 44 Weber, W., 70, 99(102) Weissrnann, M., 247 Weisz, J. F., 256 Welp, U., 44 Werner, S. A., 266 White, R., 27 Wiegmann, P. B., 10 Wilkins, J. W., 7, 10, 20 Williams, A. R., 29 Wilson, K. G., 7, 10, 20(9) Wohlleben, D. K., 40, 41(64), 43(64), 44(64), 45 Wolf, E. L., 143 Wolfle, P., 129, 139 Wuehl, H., 137 Wyder, P., 288, 289, 290(145)

Y Yafet, Y., 10 Yamada, K., 15, 73, 84, 85(126), 87 Yamada, N., 278 Yanase, A., 33 Ying, S. C., 237 Yoemans, J. M., 152, 164, 167, 169, 170(38, 39), 171(32, 33, 38, 39), 172(32), 173(34), 174, 175, 177(22), 178, 179, 188, 189(80, 82), 190(80, 81, 82), 194(33, 34), 195(81, 82), 196, 199(91) Yokoi, C. S. O., 169 Yoshimori, A,, 82, 84, 86 Yoshizawa, M., 44, 61 Yosida, K., 15, 73, 84, 85(126), 87

Z Zangwill, A., 199, 200 Zaringhalam, A,, 65, 127(90), 129(90) Zawadowski, A., 7, 82, 84, 86(130) Zeks, B., 281 Zeller, H. R., 295, 296(154) Zettl, A., 295 Zeyen, C. M. E., 268, 269(114), 270(114), 271(114), 275(111), 276, 277(120), 278(111) Zeyher, R., 220, 231(50), 264(50), 281(50), 297, 301(160)

AUTHOR INDEX

Zhang, F. C., 73, 133 Zhang, W., 142 Zieglowski, J., 45 Zkhanov, G. S., 193 Zlatid, V., 73, 84 Zou, Z., 67

311

Zougmore, F., 301 Zumer, S., 298, 299(161) Zwanzig, R., 52 Zwicknagl, G., 16, 25, 29(48), 30(48), 35, 139, 142(192)

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Subject Index

A Abrikosov-Suhl resonance, 80, 133 Ag,Mg, ANNNI model, 187-188 Almost Mathieu equation, 246-247 Amplitude mode dispersions, 273 Amplitudon, 220, 235 Anderson lattice, 87, 134 Anderson model, 8, 17-19 Annealed vacancies, 173-174 ANNNI model, 241-242 annealed vacancies, 173-174 average magnetization per layer, 160-161 axial ground-state configurations, 153-154 binary alloys, 180-188 Boltzmann factors, 165-166 branching points, 156-157 CdI,, 195 classical polytypes, 192-196 Cu,Pd, 186-187 domain wall interactions, 162-164 entropic contributions to free energy, 158 first-neighbor wall-wall interactions, 163 free energy, 160, 162-163, 165-166 further-neighbor interactions, 171-172 ground state, 169-172, 175-177 Hamiltonian, 153, 160, 162 high-temperature series, 167-168 iterated map, 161 lattice structure, 175-176 limitations, 194, 199 low-temperature series, 164-167, 172 magnetic field, 169-171 magnetic systems, 196-199 mean-field theory, 159-162, 172-173, 194 Monte Carlo method, 168 multiphase lines, 153-154 nearest-neighbor wall interactions, 158-159 phase boundaries, 171 phase diagram, 154-155 polysomatism, 196 polytypism, 188-196 313

quenched impurities, 174-175 Sic, 193-194 spinelloids, 189-192 structural coefficients, 165-167 TiAI,, 182-185 zinc sulfide, 195 Atomic-sphere approximation, linear muffin-tin orbital method, 29 Augmented spherical wave method, 29 Axial Ising model, see ANNNI model

Band structure, calculations, 24-25, 27, 97 BCS theory, 6, 92 Bethe ansatz method, 10 Binary alloys, see also Cu,Pd; TiAI, ANNNI model, 180-188 atomic structure, 181 electron microscopy, 184-185 Biphenyl crystalline, 210 deuterated, 274-278 polarized Raman spectra, 287, 289 structure, 275 Bloch waves, 229 Bogoliubov inequality, 160 Boltzmann factors, 165-166 Boson field, q-dependent fluctuations, 132 Bragg spots, 202 Brillouin-Wigner perturbation theory, 11-12, 16 Bulk modulus, 42-44 electronic contribution, 95-96 isothermal, 62

C Canonical band theory, 36 Canonical spin model, 151 Cantor function, 238

314

SUBJECT INDEX

C,,D,,, neutron scattering, modulated crystals, 274-278 CdI,, ANNNI model, 195 Ce impurity, conduction electron coupling, 9 quasiparticle-phonon interactions, 45-46 Ce compounds, f scattering, 27, 29 CeCu,, renormalized band calculations, 30-32 CeCu,Si, Josephson current, 143 potential functions and energy derivatives, 149 renormalized band theory, 30-32 self-consistent LMTO potential parameters, 148 spin-triplet pairing, 141 superconductivity, 103 upper critical field, 140-141 CEF, see Crystalline-electric-field Ce’ ions, CEF splitting energy, 14 Cerium antimonide, 151, 197-198 Cerium bismuth, magnetic phases, 198-199 CeSn,, renormalized band theory, 32-34 Charge-density waves, 206, 295-296, 300 Chemical potential, 69, 94 Chiral clock model, 176-180 Commutation relation, 53 Conduction electrons, 17, 27, 48 Continuum theory, incommensurate phases, 221 Cooper pair, 106, 118-121 Correlation function, ionic displacement, 52 Coulomb interaction, local, perturbation expansion, 83-87 Coulomb pseudopotential, 122 Coupling function, 50 Crystalline-electric-field effects, 24, 30, 48 Cu,Pd, ANNNI model, 186-187

Damping coefficient, 267, 282-284 DC lattice, zone boundary, 286 DC model, 280-281 Debye-Waller factor, 221-223, 232-233, 252 Deformation tensor operator, 47 Degeneracy, lifting, 110-117 Density fluctuation modes, 62 Density-functional theory, local-density approximation, 23-24 Density matrix, two-particle, 104

Density of states, 90, 95 incommensurate crystal phases, 243-244, 246 specific heat and, 138-139 Density susceptibility, 58 Devil’s staircase, Frenkel-Kontorova model, 238 DIFFFOUR model, 238-242 coefficients, 250-252 Debye-Waller factor, 252 dispersion curves, 249-250 dynamics, 249-256 eigenvectors, 253-255 phase diagram, 241 phason branch, 251-252 phason frequency, 254-256 Raman and infrared spectra, 258-259 Diffraction spots, 225 Diffusion constant, 56 Discommensurations, 213-216, 240, 251 Dynamic structure factor, 264

Electron correlations, strength of, 4 Electron-electron interactions, pairing induced by, 126-135 Fermi-liquid approach, 127-131 Kondo boson exchange, 131-133 mechanisms based on electronic interactions, 133-135 Electron microscopy, binary alloys, 184-185 Electron-phonon interaction, 96-98, 121-122 Electrons, strongly correlated, molecular model, 144-147 Energy, 101; see also Free energy Equation of motion, 218, 242 composite structures, 263 Frenkel-Kontorova model, 256 Harper equation, 246 linearalized, 249 pendulum, 215, 237 superspace group, 228 Euler-Lagrange equations, 214-215 Expansion techniques, 74-87 f electron Green’s function lattice, 76-77, 81-83 matrix, 73, 75 near Fermi energy, 86 poles, 77-79, 81 single-ion, 82

315

SUBJECT INDEX

noncrossing approximation, 78-79 perturbation hybridization, 74-83 local Coulomb interaction, 83-87 phase shift, 82-83 spectral density, 79-80 vertex function, 86-87 Yosida-Yamada theory, 85

F Farey numbers, 243

f charge distribution,

147 $electrons, 118-119 bandwidth, 24 Green’s function, 35, 40 heavy fermion systems, 36 hybridization, 6 number, 3 paramagnetic state, 100 phase shift, 28-29, 66 wave function, 28 4f electrons, 28, 30-31 Fermi energy, phase shift, 29, 82-83 Fermi level, 32 Fermi liquids, see also Quasiparticle interactions Landau theory, 5 theory, 66, 127-131 Fermi surface, topologies, 24-26 Ferromagnetic intersite coupling, Kondo impurities, 4 Feynman’s inequality, 89 Forward-scattering sum rule, 65 Fourier components, 234 Fourier decomposition, 230 Frank-van der Merwe model, 236 Free energy, 42, 113, 162-163 electron contribution in presence of external magnetic field, 41 entropic contributions, 158 mean-field, 160 orthorhombic crystal, 212-214 reduced, 165-166 series expansion, 217 unconventional states of superconductors, 110-117 Frenkel-Kontorova model, 236-238, 256-257 Frequency-term matrix, 54-55 Friedel’s bound-state theory, 82 Friedel’s sum rule, 82

/states, 91, 93 spin-orbit splitting, 13-14 structure constants, 36 Further-neighbor interactions, 171-172

G Galilean invariant system, 65 Gap equation, 136 Gap function, 124-126, 130 Gaussian displacement approximation, 223 Gaussian phase approximation, 222 Goldstone model, 220-221 Green’s function, 35 $electron, 40 lattice, 76-77, 81-83 matrix, 73, 75 near Fermi energy, 86 poles, 77-79, 81 single-ion, 82 Fourier transform, 76-77 higher-block, 38 normal-state, 120 Ground state, variational, 98-102 Group theory, see Superconductivity, group theory Griineisen parameter, electronic, 40, 42-43, 60

H Hamiltonian Anderson lattice, 72, 74, 134 annealed vacancies, 173 ANNNI model, 153, 160, 162 chiral clock model, 176 conduction electron-$electron exchange interaction, 103 effective diagonalization, 49-50 hybridization, 36-37 quasiparticle, 48 hybridization, 144-145 interaction, 49-50 model, 39, 60 interaction parameters, 191-192 parametrization, 147-150 quasiparticle-phonon interaction, 45-52, 54, 96-97 singlet state, formation, 9-10

316

SUBJECT INDEX

in terms of boson field, 88 unperturbed, 70-71 Harper equation, 246 ’He, 105, 107, 127 Heavy fermion systems, see also Expansion techniques; Singlet state, formation Abrikosov-Suhl resonance, 80 adiabatic regime, 58 Anderson lattice Hamiltonian, 72, 74 association of diagrams with expectation values, 75-76 BCS theory, 6 bulk modulus, electronic contribution, 95-96 canonical band theory, 36 chemical potentials, 94 Coulomb repulsion, 87-89 density susceptibility, 56, 58 electronic mean free paths, 135 energy gain, 3-5 expansion parameter, 84 f electrons, 36, 118-119 Friedel’s bound-state theory, 82 f-state occupation, 93 Galilean invariant system, 65 Green’s function, Fourier transform, 76-77 ground-state energy per site, 94 Hamiltonian, 36-37, 60 hydrodynamic modes, 56-58 inhomogeneous systems, 65-66 isolated bulk modulus, 43 isothermal regime, 58 Kondo lattice case, 68 magnetic susceptibility, 95, 99 mean-field theory, see Mean-field theory Michel’s theorems, 138 microscopic theories, 71-73 narrow band withf character, 36 nonphonon pairing mechanisms, 133 partial densities off states, 91 perturbational expansion techniques, 6 plasmon modes, 63 quasielectric line width, 58-59 quasiparticle band, see Quasiparticle bands quasiparticle energies, 92-93 quasiparticle interactions, see Quasiparticle interactions; Quasiparticle-phonon interactions rotations, 113 Schrodinger equation, 118

self-consistency equations, 91 Sommerfeld-Wilson ratio, 85, 95 specific heat, 84, 95 spin fluctuations, 127 spin-orbit interaction, 112 strongly anisotropic superconductors, 117-1 19 superconductivity, see Superconductivity total density of states per spin direction, 90 ultrasound attenuation, 59-60 variational ground state, 98-102 Wick’s theorem, 72-73, 75-76 Hybridization matrix, 9, 144-145 Hydrodynamic modes, 56-58

I Impurity distribution, 173-174 scattering effects o n transition, 135-137 Incommensurability parameter, temperature dependence, 278-279 Incommensurate crystal phases, see also Neutron scattering, modulated crystals amplitude mode, 254-255 composite structures, 260-264 Debye-Waller factor, 232-233 DIFFFOUR model, see DIFFFOUR model displacement function, 230-231, 235 equation of motion, 218 excitations, 217-221, 228-231, 260-264 Fourier components, 234 free energy, series expansion, 217 free sliding mode, 256 Frenkel-Kontorova model, 236-238, 256-257 Goldstone model, 220-221 inelastic scattering, 231-234 Landau theory, see Landau theory Larmor frequency, 299 light scattering, 223-225 matrix elements, 232 mechanisms, 210-211 modulated crystals, 204-205 modulated spring model, 242-248 natural equilibrium distance, 208 NMR spin-lattice relaxation, 297-299 Nowotny phases, 208-209 optical properties, 257-260

317

SUBJECT INDEX

optical studies, see Optical studies, incommensurate crystal phases phase mode, 253-254 phase transition, 204 phasons, 207, 219-220 potential energy, 209, 236, 239-240, 258 as quasiperiodic structures, 202-204 symmetry, 225-228 Raman tensor, 224 real incommensurate solids, 205-207 selection rules, 234-235 structure factor, 222-223, 233-234 systems, 207-210 in terms of mutually incompatible periodicities, 208 tight binding model, 246 transformation properties, 234-235 ultrasonic attenuation, 301 Inelastic scattering, 231-234, 264-266, 278, 280 Inelastic spectra, 270, 272, 276-277 Infrared spectra, DIFFFOUR model, 258-259 Interaction energy, 17-18 Internal friction coefficient, 55 Inversion center, 107 Ionic displacement correlation function, 52

J Josephson effect, 142-143

K KCP, frequency-dependent conductivity, 295-296 Kondo bosons, pairing induced by exchange, 131-133 Kondo channel, 38 Kondo effect, 2-4, 12, 146 Kondo Hamiltonian, 3 Kondo-lattice systems, see Heavy fermion systems Kondo temperature, 4, 12, 21, 29 dependence on hybridization of magnetic ion, 40-41 single-ion, 8 Kramer’s theorem, 92 K,Se04 damping coefficient, 282-284

DC lattice zone boundary, 286 incommensurability parameter, 278-279 inelastic scattering, 278, 280 neutron scattering, modulated crystals, 278-287 phason and TA-phonon line shapes, 282-284 phason dispersion, 283, 285-286 phonon dispersion, 278, 280 temperature behavior of soft-mode frequency, 280-281

L Landau-Ginsburg expansion, 110, 112 Landau paramenters, 64-65, 102, 128 Landau penetration depth, 141 Landau theory, 63, 70, 266 Fermi liquids, 5 modulated systems, 211-225 Debye-Waller factor, 221-223 discommensurations, 213-216 Euler-Lagrange equations, 214-215 excitations in incommensurate phase, 217-221 Gaussian approximations, 222-223 Lifshitz invariant, 211, 213 light scattering, 223-225 normal-incommensurate phase transitions, 211-213 one-kink solution, 216 Landau-Placzek ratios, 61-62 Larmor frequency, 299 Lattice structure, 175-176 Lifshitz invariant, 211, 213 Light scattering, incommensurate crystal phases, 223-225 Linear muffin-tin orbital method, 31, 37 atomic-sphere approximation, 29 potential parameters, 147-148 Local-density approximation, 23-25, 32 Longitudinal displacement operator, 52 LO-TO splitting, 292-293 Low-temperature series expansions, 164-167, 178-179

M Magnetic field, ANNNI model in, 169-171 Magnetic susceptibility, 13, 20, 42, 44, 95, 99 Magnetic systems, ANNNI model, 196-199

318

SUBJECT INDEX

Magnetization, electronic contribution, 43 Magnetothermal coefficient, 43-44 Mass ratio, 23 Mean-field theory ANNNI model, 159-162, 172-173, 194 chiral clock model, 179 heavy fermion systems, 73, 87-98 boson field, 88-89 electron-phonon interactions, 96-98 equations, 89-92 Feynman’s inequality, 89 Hamiltonian, 88, 91 phonon operators, 96 self-consistency equations, solution, 92-95 thermodynamic quantities, 95-96 Memory function matrix, 54-55 Michel’s theorems, 138 Migdal’s theorem, 51-52 Misfit parameter, 237 Modulated crystals, incommensurate crystal phases, 204-205 Modulated spring model, 242-248 density of states, 243-244, 246 eigenvectors, 247-248 with periodic spring constants, 247-248 spectra, 243-245 Modulation function, 230-231 Molecular model, strongly correlated electrons, 144-147 Molecular orbitals, bonding and antibonding, 145 Monte Carlo method, 134, 168 Mori scalar product, 53-54

Na2C0,, modulated y-phase, 235 NaNO,, mixed order-disorder-displacive modulation, 210-211 Neutron scattering, modulated crystals, 264-287 deuterated biphenyl, 274-278 inelastic scattering channels, 264-266 inelastic spectra, 277 phase diagram, 278-279 phase-mode dispersion, 277 phason velocity, 268 potassium selenate, 278-287 spectral response function, 266-268

structural phase transition, 268-269 P-ThBr,, 268-274 wave vector, 264-265 NiAI2O4-Ni,SiO4,phase diagram, 191 Nowotny phases, structure, 208-209

Optical properties, incommensurate crystal phases, 257-260 Optical selection rules, 289, 294-295 Optical studies, incommensurate crystal phases, 287-297 activation processes, 287-288 FIR conductivity spectrum, 295-297 LO-TO splitting, 292-293 modulation wave vector, temperature dependence, 291- 292 optical selection rules, 289, 294-295 polarized Raman spectra, temperature dependence, 287-289 scattered intensity, 294-295 Optic branch, soft, temperature dependence, 292 Order paramenters, 140-141 complex matrix, 105-106 orthorhombic crystal, 212 symmetry properties, 104-110

P Pair states, properties, 137-144 Pair wave function, 117-118 Peierls mechanism, 208 Peierls-Nabarro barrier, 237 Phase-coherence length, 206 Phase fluctuations, 299 Phase mode dispersions, 273 Phase transitions, normal-incommensurate, 211-213 Phason, 207, 220, 267 displacement function, 230-231, 235 line shape, K,Se4, 282-284 velocity, 268 Phason branch, 251-252 dispersion curve, 219-220, 283, 285-286 Phason frequency, as function of temperature, 254

SUBJECT INDEX

Phason mode, particle mode, 221 Phonon branch, dispersion, 276 Phonon-induced pairing, 121-123 Phonon momentum operator, 53 Phonon operators, 96 Phonon propagator, static limit, 122 Plasmon modes, 63 Polysomatism, ANNNI model, 196 Polytypism, 188-196 classical, 192-196 defined, 188-189 theories, 195 Potassium selenate, see K,SeO, Potential energy, 228-229, 236, 239-240, 258 Potential functions, 37-38 Potential scattering model, 129 p-wave pairing, 131

Q Quasiparticle diffusion mode, 56 dispersion, quasiparticle density of states, 34 effective Hamiltonian, 48 energies, 92-93 heavy and light, 68 Landau theory of Fermi liquids, 5 scattering T matrix, 127-129 spin susceptibility, 64 Quasiparticle bands, 22-40 conduction-electron potential functions, 38 energy dispersion, 22 excitation energies, 23 f-bandwidth, 24 f-electron, 27, 34-35 higher-block Green’s function, 38 Kondo channel, 38 model descriptions, 34-40 model Hamiltonian, 36-37 potential functions, 37-38 power law, 40 quasiparticle density of states, 34-35 renormalization of structure constants, 37 renormalized band theory, 25, 27-34 structures, heavy fermion systems, 5 Quasiparticle interactions, 132-133 anisotropic systems, 68

319

chemical potential change, 69 effective, 118 electronic specific heat, 64 energy change due to elementary excitations, 63 energy dispersion changes, 68 f-electron phase shift, 66 Fermi liquid description, 63-71 forward-scattering sum rule, 65 Hamiltonian, 70-71 homogeneous system, 64-65 interaction constant, 67 Landau parameters, 64-65 phase shifts at Fermi energy, 67-68 potential, decomposition, 125 scattering amplitude, 66 Sommerfeld-Wilson ratio, 67 superconductivity, 103-104 Quasipariticle-phonon interactions, 40-63, 122 coupling function, 50 cubic environment, 45-46 deformation tensor operator, 47 diffusion constants, 56 electron contribution, external magnetic field, 41 electronic Griineisen parameter, 40, 42-43, 60 forms, 50-51 frequency-term matrix, 54-55 Hamiltonian, 45-52, 54 hybridization, 41, 48-49 hydrodynamic description, 52-63 isothermal bulk modulus, 62 Landau-Placzek ratios, 61-62 magnetic susceptibility, 42, 44 magnetothermal coefficient, 43-44 memory function matrix, 54-55 Migdal’s theorem, 51-52 Mori scalar product, 53-54 relaxation matrix, 54 specific heat, 42 static susceptibility matrix, 54-55 thermal expansion, 43 thermodynamic relations, 41-45 ultrasound velocity, 61 vertext correction, 51-52 volume magnetostricton, 43-44 volume strain, 49 Quasiperiodic structure, 202-204, 225-228 Quenched impurities, 174-175

320

SUBJECT INDEX

R Radial function, 31-32 Raman scattering, 224 Raman spectra, 258-259, 287-290 Raman tensor, 224 "Rbl/2 -1/2 transition, 297-298 Rb,2nBr4, Raman spectra, 289-290 Rb,ZnCI,, 297-300 Reciprocal lattice vector, 264 Renormalization factor, 122 Renormalized band theory, 25, 27-34 crystalline-electric-field effects, 30 expansion parameters, 28-29 generalization to finite temperatures, 39-40 Hamiltonian, 27, 30 potential functions, 29-30 quasiparticle dispersion, 33 radial function, 31-32 structure constants, 29 underlying concepts, 27-28 RKKY interaction, 21

-

S Scattering T matrix, 127-130 Schrodinger equation, 118 Self-consistency equation, 91-95, 120, 130 Sic, 151, 193-194 Singlet state, formation, 7-21 Bethe ansatz method, 10 Brillouin-Wigner perturbation theory, 11-12, 16 CEF splitting, 14-15 conduction-electron density, 17-18 energy, 8, 11, 13 Fermi sea of conduction electrons, 11, 16-17 Hamiltonian, 9-10 interaction energy, 17-18 ion repulsion, 19 Kondo temperature, 12, 21 magnetic susceptibility, 13, 20 normalization constant, 12 RKKY interaction, 21 single-impurity Anderson model, 8, 16 space of states, 15 thermodynamic scaling theory, 20-21 two-impurity Anderson model, 17-19 Slater determinant, 99-100 Slave-boson field theory, 87 Sommerfeld specific heat coefficient, 84

Sommerfeld-Wilson ratio, 67, 85, 95 s-p approximation, 129 Specific heat, 42, 103, 127 electronic, 64 power law behavior, 137-138 Sommerfeld coefficient, 84 Specific heat coefficient, 85-95 Spectral density, 79-80 Spinelloids, ANNNI model, 189-192 Spin-flip scattering, 143 Spin fluctuations, 127 Spin-orbit interaction, 138 Spin-orbit splitting, 32-33 Spin susceptibility, quasiparticles, 64 Spin-triplet pairing, 126 Static fluctuations, incommensurate crystal phases, 206 Static susceptibility matrix, 54-55 Structure combination branching processes, 156 Structure factor, inelastic, 271 Superconductivity, 6 angular integration, 120 anisotropic, 123-126, 137 coherence length, 135 conduction electron-$electron exchange interaction, 103 free energy of unconventional states, 110-117 Green's function, 120 group theory, 104-119 broken gauge symmetry, 108 classification, 108-109 inversion center existence, 107 order parameter symmetry properties, 104-110 pairing, 104 s-wave state, 107 symmetry, 108 Landau-Ginsburg expansion, 110, 112 microscopic theory, 119-137 d-wave pairing, 126 effective interaction Hamiltonian, 124 form factor, 124 gap function, 124-126, 130 impurity scattering effects on transition, 135-137 momenta, 128 Monte Carlo method, 132 pair breaking, 136-137 pairing induced by electron-electron interactions, 126-135 phonon-induced pairing, 121-123

321

SUBJECT INDEX

potential scattering model, 129 p-wave pairing, 131 quasiphonon interaction, 122-123 renormalization factor, 122 superconducting transition temperature, 123, 125, 127, 130, 135 odd-parity basis functions, 114 order parameter, 117 pair state properties, 137-144 density of states, 139 isotropic pwave state, 139 Josephson effect, 142-143 order parameters, 140-141 spin-flip scattering, 143 upper critical field, 140-141 parity determination, 142 quasiparticle renormalization factor, 120 self-consistency equation, 120, 130 specific heat, 103, 137-138 spin-orbit interaction, 138 strongly anisotropic, 117-119 symmetric traceless matrix, 110-111, 114-115 symmetry breaking, 111-112 tesseral harmonics, 117, 119 transition to superconducting state., 128 vector functions, 114 Superconductor, conventional and unconventional, 6, 108-109 Superspace group, 226-231 Superspace periodicity, relationship with physical space quasiperiodicity, 203 s-wave state, 107

T (TaSe,),I, frequency-dependent conductivity, 296-297 Tesseral harmonics, 117, 119 8-Th Br4 ionic displacements, 270-271 neutron scattering, modulated crystals, 268-274 phase and amplitude mode dispersions, 273 polarized Raman spectra, temperature dependence, 287-288 structure, 268-269 Thermal expansion, 43 Thermal fluctuations, incommensurate crystal phases, 206 Thermodynamic coefficients, 42-43

Thermodynamic relations, quasiparticlephonon interactions, 41-45 Thermodynamic scaling theory, 20-21 Thiourea, temperature dependence of soft optic branch, 292 TiA13, ANNNI model, 182-185 Tight binding model, 246 Transformation law, 234 Transition temperature, superconducting, 123, 125, 127, 130, 135-136

U UBela, 140-141, 143 Ultrasonic attenuation, 59-60, 301 Ultrasound velocity, 61 UPti, 26, 140-141 Josephson current, 143 order parameters, 115-116 pair states, 119 superconductivity, 144

V Vibration modes, displacements, 229 Volume magnetostriction, 43-44 Volume strain, 49

W Wave vector, 186, 224 Brillouin zone, 229, 249 excitations. 264-265 as function of annealing temperature, 183-1 84 temperature dependence, 274-275, 291-292 Wick’s theorem, 72-73, 15-16 Wigner-Seitz spheres, around atom, 29

Y Yosida-Yamada theory, 85

Z Zinc sulfide, ANNNI model, 195

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