Phase Transitions and Self-Organization in Electronic and Molecular Networks

Phase Transitions and Self-Organization in Electronic and Molecular Networks

FUNDAMENTAL MATERIALS RESEARCH Series Editor: M. F. Thorpe, Michigan State University East Lansing, Michigan

ACCESS IN NANOPOROUS MATERIALS Edited by Thomas J. Pinnavaia and M. F. Thorpe DYNAMICS OF CRYSTAL SURFACES AND INTERFACES Edited by P. M. Duxbury and T. J. Pence ELECTRONIC PROPERTIES OF SOLIDS USING CLUSTER METHODS Edited by T. A. Kaplan and S. D. Mahanti LOCAL STRUCTURE FROM DIFFRACTION Edited by S. J. L. Billinge and M. F. Thorpe PHASE TRANSITIONS AND SELF-ORGANIZATION IN ELECTRONIC AND MOLECULAR NETWORKS Edited by J. C. Phillips and M. F. Thorpe PHYSICS OF MANGANITES Edited by T. A. Kaplan and S. D. Mahanti RIGIDITY THEORY AND APPLICATIONS Edited by M. F. Thorpe and P. M. Duxbury SCIENCE AND APPLICATION OF NANOTUBES Edited by D. Tománek and R. J. Enbody

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

Phase Transitions and Self-Organization in Electronic and Molecular Networks Edited by

J. C. Phillips Lucent Technologies Bell Labs Innovations Murray Hill, New Jersey

and

M. F. Thorpe Michigan State University

East Lansing, Michigan

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SERIES PREFACE

This series of books, which is published at the rate of about one per year, addresses fundamental problems in materials science. The contents cover a broad range of topics from small clusters of atoms to engineering materials and involve chemistry, physics, materials science, and engineering, with length scales ranging from Ångstroms up to millimeters. The emphasis is on basic science rather than on applications. Each book focuses on a single area of current interest and brings together leading experts to give an up-to-date discussion of their work and the work of others. Each article contains enough references that the interested reader can access the relevant literature. Thanks are given to the Center for Fundamental Materials Research at Michigan State University for supporting this series. M.F. Thorpe, Series Editor E-mail: [email protected] East Lansing, Michigan, September 2000

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PREFACE

The problem of phase transitions in disordered materials is quite old, but until recently it has seemed too complex a subject for formal study. The advent of computers has changed matters in two important ways. First, it has become possible to implement formal methods for microscopic study of phase transitions in ordered materials, even in the quantum limit, in great detail. This work has been so successful that few qualitative

mysteries remain, and many microscopic details have been measured experimentally and derived theoretically from first principles. The second radical change brought about by computers is that scientists have been forced to recognize that even today phase transitions in disordered materials are very poorly understood. Apart from the inherent statistical problems raised by disorder, it is becoming clear that new fundamental concepts are needed to explain qualitatively new phenomena that arise in disordered materials that were absent in ordered crystalline materials, or even in such materials with disordered sublattices. This workshop addresses the need for fundamentally new concepts in three areas of physical science. The first is network glasses, simple mechanical systems in which important new phenomena (the intermediate phases, the reversibility window) have been discovered as a result of exploring stiffness transitions both experimentally and in

numerical simulations made possible by new computer algorithms. The considerable progress made here is most encouraging, but surprisingly it has turned out that these new mechanical phenomena are closely paralleled by new electronic phenomena. These are discussed for the second area, the metal-insulator transition in semiconductor impurity bands, in which an intermediate phase has also been identified. The third area is (mostly cuprate) perovskites, where an intermediate phase occurs which can have superconductive transition temperatures well above 100K. It appears very likely

that the electronic intermediate phases exist because of disorder, and that the electronic phase diagrams closely parallel the mechanical phase diagrams of network glasses. On a microscopic level, minimization of the free energy of a disordered system at moderate temperatures, followed by some kind of (mild) quenching, can produce selforganization. There are many indications of this in network glasses, but of course life itself is self-organized. Proteins can be described as self-organized disordered networks, and they are discussed briefly here, and in a special issue of Journal of Molecular Graphics and Modelling (edited by L.A. Kuhn and M.F. Thorpe, to appear early 2001). It turns out that several constraint-based concepts that have been developed for network glasses apply equally well to the apparently unrelated subject of protein folding. This focused workshop was held at Hughes Hall, Cambridge, England, July 10-14, 2000. We are grateful to Dr. Martin Dove for assistance with local arrangements, and Ms. Janet King and Mr. Mykyta Chubynsky for extensive editorial assistance. J.C. Phillips M.F. Thorpe

East Lansing, Michigan, September 2000

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CONTENTS

I. Some Mathematics Mathematical Principles of Intermediate Phases in Disordered Systems................................................................................................ 1 J.C. Phillips Reduced Density Matrices and Correlation Matrix.............................................................. 23 A. John Coleman The Sixteen-Percent Solution: Critical Volume Fraction for Percolation................................................................. 37 Richard Zallen The Intermediate Phase and Self-Organization in Network Glasses................................................................................................... 43 M.F. Thorpe and M.V. Chubynsky

II. Glasses and Supercooled Liquids Evidence for the Intermediate Phase in Chalcogenide Glasses........................................................................................... 65 P. Boolchand, W.J. Bresser, D.G. Georgiev, Y. Wang, and J. Wells Thermal Relaxation and Criticality of the Stiffness Transition....................................................................................................... 85 Y. Wang, T. Nakaoka, and K. Murase Solidity of Viscous Liquids................................................................................................ 101 J.C. Dyre Non-Ergodic Dynamics in Supercooled Liquids................................................................ 111 M. Dzugutov, S. Simdyankin, and F. Zetterling Network Stiffening and Chemical Ordering in Chalcogenide Glasses: Compositional Trends of Tg in Relation to Structural Information from Solid and Liquid State NMR ........................ 123 Carsten Rosenhahn, Sophia Hayes, Gunther Brunklaus, and Hellmut Eckert

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Glass Transition Temperature Variation as a Probe for Network Connectivity............................................................................................ 143 M. Micoulaut

Floppy Modes Effects in the Thermodynamical Properties of Chalcogenide Glasses....................................................................... 161 Gerardo G. Naumis The Dalton-Maxwell-Pauling Recipe for Window Glass .................................................. 171

Richard Kerner Local Bonding, Phase Stability and Interface Properties of Replacement Gate Dielectrics, Including Silicon Oxynitride Alloys

and Nitrides, and Film ‘Amphoteric’ Elemental Oxides and Silicates ...... 189 G. Lucovsky

Experimental Methods for Local Structure Determination on the Atomic Scale ....................................................................... 209 E.A. Stern

Zeolite Instability and Collapse.......................................................................................... 225 G.N. Greaves III. Metal-Insulator Transitions

Thermodynamics and Transport Properties of Interacting Systems with Localized Electrons.......................................................................... 247 A.L. Efros The Metal-Insulator Transition in Doped Semiconductors: Transport Properties and Critical Behavior............................................................ 263 Theodore G. Castner

Metal-Insulator Transition in Homogeneously Doped Germanium.................................................................. 291 Michio Watanabe IV. High Temperature Superconductors Experimental Evidence for Ferroelastic Nanodomains in

HTSC Cuprates and Related Oxides...................................................................... 311 J. Jung Role of Sr Dopants in the Inhomogeneous Ground State of La2-xSrxCuO4....................................................................................................... 323 D. Haskel, E.A. Stern, and F. Dogan

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Universal Phase Diagrams and “Ideal” High Temperature Superconductors: ..............................................................................................................331 J.L. Wagner, T.M. Clemens, D.C. Mathew, O. Chmaissem B. Dabrowski, J.D. Jorgensen, and D.G. Hinks Coexistence of Superconductivity and Weak Ferromagnetism in Eu1.5Ce0.5RuSr2Cu2O10......................................................................................... 341 I. Felner

Quantum Percolation in High Tc Superconductors............................................................ 357 V. Dallacasa Superstripes: Self Organization of Quantum Wires in High Tc Superconductors.................................................................................... 375 A. Bianconi, D. DiCastro, N.L. Saini, and G. Bianconi Electron Strings in Oxides.................................................................................................. 389 F.V. Kusmartsev

High-Temperature Superconductivity is Charge-Reservoir Superconductivity.................................................................. 403 John D. Dow, Howard A. Blackstead, and Dale R. Harshman

Electronic Inhomogeneities in High-Tc Superconductors Observed by NMR.................................................................................................. 413

J. Haase, C.P. Slichter, R. Stern, C.T. Milling, and D.G. Hinks Tailoring the Properties of High-Tc and Related Oxides: From Fundamentals to Gap Nanoengineering........................................................ 431 Davor Pavuna

V. Self-Organization in Proteins

Designing Protein Structures.............................................................................................. 441 Hao Li, Chao Tang, and Ned S. Wingreen

List of Participants.............................................................................................................. 447 Index................................................................................................................................... 451

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MATHEMATICAL PRINCIPLES OF INTERMEDIATE PHASES IN DISORDERED SYSTEMS

J. C. PHILLIPS Bell Laboratories, Lucent Technologies (Retired) Murray Hill, N. J. 07974-0636 ([email protected])

INTRODUCTION Intermediate phases are found in disordered systems that for a long time were supposed to exhibit simple connectivity transitions, similar to dilute magnetic transitions. The latter can be modeled by percolation on a lattice. The paradigmatic disordered offlattice systems that exhibit intermediate phases are network glasses, impurity bands in semiconductors [the metal-insulator transition (MIT)], and high-temperature doped (pseudo-)perovskite superconductors. The first two (relatively simple) examples show that self-organization of the flexibility inherent in disorder is what creates intermediate phases, and that these must be described by finite-size scaling methods. The third (very complex) example shows that high temperature superconductivity (HTSC) itself depends on glassy dopant disorder, and only indirectly on the crystalline matrix with its long-range order. The mathematical principles underlying the filamentary or percolative theory of such internally organized systems are fundamentally different from those of theories based on the effective medium approximation (EMA) or fully disorganized (randomly) diluted lattice connectivity transitions. These principles have been developed only in the last hundred years and are little known to most scientists. The counting methods used in the filamentary theory bear a striking resemblance to those used to prove Fermat’s Last Theorem or to factor efficiently large numbers using quantum computers. Examples of the intermediate phase for these three classes of materials are given that specifically identify the internal self-organized complexity that is responsible for the remarkable physical properties of each case. There is a growing realization that the physics of complex disordered systems differs qualitatively from that of simple crystalline systems with long-range order, especially in the vicinity of connectivity transitions. In this workshop both experimental and theoretical work illustrating this theme are discussed for a wide range of subjects, with special emphasis on three topics: network glasses, impurity band MITs, and HTSC. In each case we find that the single connectivity transition to which we are accustomed in simple

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F Thorpe, Kluwer Academic/Plenum Publishers, 2001

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systems is replaced by two transitions of quite different character. The first resembles the continuous transition expected from percolation theory, but with much simpler exponents,

while the second is a first-order transition with catastrophic character. Between these two transitions we find an intermediate phase which always has novel properties that are indeed qualitatively different from those of simple dilute lattice systems. In some cases these novel properties are of enormous technological value (window glass), and the study of intermediate phases has for the first time enabled us to understand quantitatively why these properties occur. In other cases, such as HTSC, the intermediate phase has properties that are so novel and so unexpected that so far almost all theories have failed to develop beyond the macroscopic or phenomenological stage. Within the physical sciences the level of interest in cuprate HTSC has greatly surpassed that of any other subject, with the sole exception of semiconductive materials (such as Si) basic to modern electronics. Initially the interest was stimulated by amazingly large values of the superconductive transition temperatures Tc, typically five (ten) times larger than the highest values found in compound (elemental) metals [1]. As expected, there were other anomalies as well: high sensitivity to doping by non-magnetic oxygen, and very little sensitivity to the presence of magnetic rare earths [2], both anomalies reversing the situation found in metallic superconductors. In the normal-state to superconductive phase transitions of metals, the superconductive properties are generally rather insensitive to the normal-state behavior, but in the cuprates the normal-state transport at high temperatures itself is anomalous. The anomalous behavior becomes most

characteristic at just those compositions that maximize Tc, even in cases where This tells us that a new electronic theory is needed to describe such “strange metals” [3].

This new theory must be very different from the Fermi liquid or Landau-Ginzburg theories used to describe normal metals, which are based on the effective medium approximation (EMA). The EMA cannot be even qualitatively correct here [4], as the Fermi liquid phase is separated from the intermediate phase by a first-order phase transition. Unfortunately, although the need for an alternative to Fermi liquid or LandauGinzburg theory is widely recognized [2,3], only the author’s own filamentary or percolative theory [5] avoids the EMA. This theory relies essentially on set-theoretic methods derived from number theory to establish quantitative results, and these methods are largely unknown to physical scientists. These methods have long been regarded as rather esoteric, even by most mathematicians, but their true significance, as a way of unifying results from algebra, analysis, geometry and topology, has become apparent recently from the proof [6] of Fermat’s Last Theorem (FLT). Several popular discussions

of set theory and FLT are available, but the connections with network glasses, impurity bands, and perovskite superconductivity are so simple and so direct that this paper will provide them as a matter of convenience to busy readers. We will then show how these novel mathematical ideas match the results of several recent decisive experiments in great detail. DISCRETE INTEGER AND CONTINUOUS REAL NUMBER FIELDS: FLT Physical scientists without a strong background in modern mathematics will find an excellent introduction to the subject, which carries them from its beginnings right through to an outline of the steps that led to the proof of FLT, in [7], amusing, anecdotal and thoroughly entertaining. For a long time number theory was regarded as a collection of

strange and rather accidental results of no general significance, but in the late 1800’s Cantor invented set theory and established an essential difference between integers and real numbers. Although both sets are infinitely large, the number (or order or cardinality) of

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real (irrational) numbers is larger than that of the integers and rational numbers, which have the same cardinality. He also hypothesized that there is no cardinality intermediate between those of the rationals and irrationals (continuum hypothesis); later Godel established a series of equivalent statements, including the axiom of choice, which showed that this axiom is independent of the rest of mathematics. These ideas are important in the present context because they highlight the idea that the methods of effective medium theories are fundamentally limited because they apply only to simple continuum systems

represented by real number fields. If the number field of interest is the integers or rationals, or a combination of these with the reals, then special methods need to be developed to prove theorems or derive results that do not exist for real number fields only.

The nature of these special methods is dramatically illustrated in the proof of FLT, which states that integer triples exist which satisfy the Pythagorean or Euclidean metric only for n = 2. Physical scientists, familiar with the example find Fermat’s conjecture quite plausible, especially as it has been confirmed by computer searches up to n = four million, but of course these searches do not constitute a proof. The proof involves two abstract mathematical tools, elliptic curves and modular functions.

An elliptic curve is not an ellipse: it is the set of solutions to a cubic polynomial in two variables, usually written in the form y2 = x3 + Ax2 + Bx + C. For number theory x and

y are integers. Modular functions are periodic and are a kind of integral generalization of sines and cosines. One can conjecture that all elliptic curves are modular. It then turns out that if this conjecture is valid, FLT follows. The proof of the latter began with a counter-example (Frey’s curve), which shows that should such an exist, it would generate an elliptic function with anomalous properties, in the sense that it would not be modular, as it is for integer triples with n = 2. To prove that this relation between elliptic and modular functions is necessary, Wiles counted the

number of both and showed that the two numbers were the same; thus the essential step was this counting [7].

Counting is a set-theoretic integral process. It is essential to our filamentary model of network glasses and the semiconductor impurity band transition [8-10] and to our

filamentary model of cuprate superconductivity [5]. In all cases the number of basis functions associated with cyclical vibrational states, or current-carrying states (or Cooperpaired current-carrying states in the superconductive case) is actually counted, as part of their separation from localized states in the neighborhood of the stiffness or metal-insulator transitions. Within the EMA and real number fields only, so far counting methods appear not to be feasible, and have not been used to discuss either the metal-insulator transition (MIT) or HTSC. All the EMA results that have been obtained are based on analytic (continuum) methods alone, which we believe are not well suited even to impurity band metal-insulator transitions and to the anomalous electronic properties of cuprates in the normal or superconductive states. It is obvious that in the network glass case continuum methods cannot identify floppy modes, which are obtained only by numerical solutions of matrix equations. BROKEN SYMMETRY, QUANTUM COMPUTERS, AND SHOR’S ALGORITHM The essential idea of our filamentary or percolative theory of random metals near the metal-insulator transition (MIT) is that in the limit such metals develop a new kind of broken symmetry even in the normal state. Electronic motion tangential ( ) to percolative filaments is phase-coherent, just as in normal metals, but normal to the filaments the motion is diffusive, as it is on the insulating side of the transition. This

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fractal behavior is what makes it possible for the MIT to be continuous, even in the presence of long-range Coulomb interactions, which could render the MIT first order in the

EMA, for example the MIT or Wigner transition of electrons in a box.

The presence of limited filamentary phase coherence makes many kinds of novel effects possible. Consider, for example, hypothetical quantum computers, which have attracted great interest recently among computer scientists [11]. These process complex integers (amplitude and phase) rather than merely real integers. Such hypothetical computers consist of quantum cells (cubits) connected by quantum wires which transmit

amplitude and phase information and thus are exactly the same as the quantum filaments discussed above. With such hypothetical computers Shor showed [11] how to factor large numbers in polynomial rather than exponential times by making use of fast Fourier

transforms and matrix methods, which rely on taking advantage of interference effects which occur conveniently for complex numbers but not for real numbers. COUNTING IN NETWORK GLASSES: AN EXAMPLE Counting is essential to our understanding of the remarkable properties of network glasses. Unlike the electronic cases, where the analysis is greatly complicated by both long-range Coulomb interactions and phase effects associated with complex wave functions, the properties of network glasses can be modeled by simple point mass-andspring systems. On the one hand, these matrix models with short-range forces and no quantum effects have proved to be (relatively) easily solved, compared to the electronic models (apparently insoluble). On the other hand, all the effects predicted by the network glass models are gradually being observed experimentally, and especially the properties of the intermediate phase are astonishingly similar to those observed for electronic materials. Thus, while it would have been easy to be skeptical of these mathematical analogies alone, it is apparent that they capture most, if not all, of the essential properties of intermediate phases. Recent work on intermediate phases in network glasses is discussed here by Boolchand, Thorpe, and Kerner, and the details can be found in their papers. Apart from the pivotal importance of counting in understanding the properties of network glasses, there is a second, and equally important, analogy between the methods Wiles used to prove FLT and the constraint theory of network glasses. The proof relies on establishing the connection between two sets, modular functions and elliptic curves, that at first seem to be unrelated, except that their numbers are the same. In constraint theory one compares the number of spatial degrees of freedom of the system to the apparently unrelated number of Lagrangian constraints associated with bonding interactions with localized vibrational frequencies (intact constraints). These constraints may involve n-body forces with (such as bond-bending forces, n = 3). In conventional continuum treatments, the relevant number is the number of interparticle forces (off-diagonal elements of the dynamical matrix, each of which contributes a different “interaction line” in diagrammatic perturbation theory). Constraint theory has shown that the relevant number is the number of intact potential interactions in potential space, not the number of forces implied by real-space derivatives of these potentials. In other words, spatial coordinates and interaction potential coordinates are treated as separate and distinct sets. The mean-field condition for the glass stiffness transition is that the numbers of elements in the two (apparently unrelated) sets are equal. This point is illustrated in Fig. 1, where the number of vibrational modes with zero

frequencies (cyclical modes) is plotted [12] against average coordination number r in as glassy network with bond-stretching and bending forces. At r = 2.40 the number of constraints equals the number of degrees of freedom, and the extrapolated number of

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Figure 1. The number of floppy modes as a function of r in bond-diluted models of three-dimensional glassy networks with stretching and bending forces [12]. The inset shows a blowup of the critical region. The second derivative of these curves shows a peak. This peak resembles the specific heat of a second-order phase transition, which shows that incomplete relaxation of the models generates the largest number of hidden linear

dependencies of constraints very near the connectivity transition.

cyclical modes is zero. (The smoothing of the kink at r = 2.40 is probably due to the fact that in the numerically simulated “random” network the topology is not ideally random.

This leads to redundancies among the constraints.) INTERMEDIATE PHASES: THERE ARE TWO STIFFNESS TRANSITIONS! For a long time we have all believed that in percolative problems there is only one connectivity transition. The first doubts began to appear in Boolchand’s measurements of

the critical coordination number, which was nearly always close to 2.40. However, even in cases where there was no evidence for nanoscale phase separation in the critical region, in other words, in cases where the theory should have worked, there were small discrepancies. Indeed, the numerical simulations shown in Fig. 1 predict small discrepancies, with a shift of the critical coordination number to 2.38 or 2.39. Experimentally, the shifts were in the other direction, to Those not familiar with constraint theory would probably say that such small discrepancies are insignificant - after all, in non-equilibrated glasses one should not expect better agreement between theory and experiment. But to us these discrepancies seemed significant. In particular, Boolchand’s ultraprecise Raman data also began to show evidence for a first-order transition, whereas all percolative models predict a continuous transition. The problems took definite form in 1998 workshop papers, where Boolchand et al. showed (see Fig. 2) that an apparent jump in the Raman frequency associated with corner-sharing tetrahedra in (Ge, Se) glassy alloys occurs at r = 2.46. This is the same critical value of r as occurs in the density and non-reversible part of the glass-transition enthalpy (discussed in more detail below). Yet still other Mossbauer experiments showed that some kind of transition, probably continuous, was happening at r = 2.40. I also found 5

Figure 2. Composition dependence in glasses (r = 2 + 2x) of corner-sharing Raman frequency, nonreversible enthalpy of glass transition, and molar volume.

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some marginal evidence for two transitions in elasticity data, with the compressibility transition occurring at r = 2.40, and the Young’s modulus transition occurring at r = 2.46. Today we are quite certain that these two transitions mark the boundaries of a new kind of phase in disordered systems, that we call the intermediate phase. To see what causes the intermediate phase, suppose we prepare an underconstrained network by bond dilution. Next we add bonds to the network at random until we reach the first connectivity transition. At this point the backbone begins to percolate from one face of the sample to the opposite face. It percolates as a “pure” filament that neither branches nor intersects itself. As we continue to add more bonds, two things can happen: we may get new “pure” filaments, or one of the old filaments can branch or cross. At the branching or crossing points locally the network is overconstrained and this increases the strain energy anharmonically compared to growing new filaments. Therefore the enthalpy, and initially the free energy, can be reduced by adding bonds selectively to avoid branching and crossing (“smart bonds”), and creating new filaments. However, as we add more and more bonds, and more and more filaments, at a certain point adding one more bond will lead to crossing or branching, no matter where it is added. This is the upper density limit for the second transition. It is intuitively plausible, and it is confirmed by numerical simulations, that the first transition is continuous, and the second is first-order (M. F. Thorpe et al., this volume). [It should be remarked that it is not surprising that the intermediate phase was overlooked for so long. It occurs only because the glassy network is not confined to a lattice. Whenever percolation occurs on a randomly diluted lattice with short hops (shortrange forces), there is only one transition, and it is continuous. It is the off-lattice selective relaxation character of the glassy network that makes “smart bonds” and a first-order transition possible.] COUNTING IN QUANTUM PERCOLATION THEORY: ANOTHER EXAMPLE In a d-dimensional sample with Nd randomly distributed impurities the formation of phase-coherent ballistic states is blocked (in the sense of Lagrange) by constraints [13,14]; note that from a counting viewpoint the microscopic nature of these constraints, orbital or spin, external or electron-electron interactions, is irrelevant; all that matters is their number. The central result of the filamentary theory of the MIT is the existence theorem, which states that these filaments can exist providing that the following condition on the log of this number is satisfied (Eqn. (7) of [8]):

This is a very amusing equation because of the way that it combines real and complex numbers. On the left hand side we have the exponent that represents the number of transverse degrees of freedom of our complex, current carrying basis states. Had these states been real standing waves, the factor 2 would have been absent. The first term on the right hand side is the number of real constraints generated by randomness. The second term is the (transverse) areal density of current-carrying filaments, an observable which of course is also real. Thus the left hand side measures complex quantum dimensionalities, while the right hand side measures real observable dimensionalities. In a simple but very fundamental way this equation describes all the implications of the quantum theory of measurement for the transport properties of random metals [14]. Quantum percolation theory explains all the experimentally observed critical exponents and prefactor sign reversals which are observed [15,16] in uncompensated random metals near the MIT such as Si:P and Ge:Ga. It applies to the scaling phase, which exhibits power-law behavior over

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a range 100 times larger than that found for magnetic critical phenomena. Very close to the continuous MIT, even the purest materials exhibit the effects of compensation, and the residual resistivity associated with scattering from compensating impurities dominates the transport properties, which revert to the conventional behavior predicted by the EMA.

COMPARISON WITH SCALING THEORY The dependence of (1) on dimensionality is strongly suggestive of scaling theories that have been developed to describe critical behavior of magnetic phase transitions [17]. It seems, however, that the results given in [8,9] differ fundamentally from those in the magnetic literature in two important respects. The latter rely on Bose, not Fermi, statistics, and hence contain no destructive interference effects. The interparticle forces of the latter are of short range, while electron-electron interactions are long range. Whether these two differences are necessary or sufficient to explain the large qualitative differences between the properties of random magnets and random metals has been unclear for a long time. It was even suggested [18] that the experimental data [15] may have been in error, a

suggestion which has recently been laid to rest [16]. Detailed comparison of filamentary quantum percolation theory with magnetic lattice percolation theory [17,18] has shown [10] that both Fermionic destructive interference and

long-range forces are necessary and sufficient to produce a consistent and successful theory of impurity band random metals such as Si:P and Ge:Ga. The destructive interference suppresses the divergence of the specific heat which is otherwise a characteristic of quasione dimensional ( d* = 1 ) Fermionic systems embedded in a d-dimensional matrix. This destructive interference is represented mathematically by a non-crossing condition for

semiclasical percolative paths similar to one that is already known for the integral quantum Hall effect at large n. (The elliptic curves which play an essential part in the proof of Fermat’s Last Theorem also satisfy a non-crossing condition [19], which suggests that

“arithmetic algebraic geometry” [briefly, “modern arithmetic”] may have a lot to offer in treating problems involving many Fermions.) In the limit Fermion statistics combined with long-range interactions cause d to be replaced by d + d* = d + 1. One can identify d* with the fluctuations of the component of the internal electric field that is locally tangential to the filament. Or one can introduce local times for Fermi-energy electrons moving along the filament. Then Newton is replaced by Einstein, and because of internal fields the filamentary paths fluctuate dynamically not only in space, but also in their local time It is amusing that the concepts of special relativity, originally developed to explain non-linear aspects of the Doppler effect, should reappear in the context of critical fluctuations in random metals. If the intermediate phase has a distinct topological character that is associated with off-lattice disorder, then one can immediately infer that it can occur in any disordered system that either has long-range forces, or can self-anneal. So I reasoned, and this led me to re-examine all the experimental data on strongly disordered systems near a connectivity

transition. Two such systems immediately spring to mind: the simpler system is the metalinsulator transition in semiconductor impurity bands, such as Si:P; following Shockley’s rule, we discuss it first.

There are two kinds of data on the metal-insulator transition in Si:P, both taken some 20 years ago. At that time everyone assumed that there was only one transition. This transition was supposed to occur continuously in both the conductivity and the coefficient of the linear term in the specific heat at the same value of uncompensated dopant density n and with related exponents, as both observables were supposed to be continuous functions of n.

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The trouble with this assumption was that when it was applied to the experimental data, the two sets of data, transport and thermal, did not lie on smooth, power-law curves (see Fig. 3 ). However, as everyone at that time was certain that there was only one transition, this problem passed unnoticed. Naturally, when I re-examined the by-now-allbut-forgotten thermal data some twenty years later [14], I immediately saw that the critical density for the thermal transition was actually much larger than that for the transport transition (by almost a factor of 2). All the effects of the intermediate phase that we have been discussing are connected with filamentary coherence and finite-size scaling. When a few minority dopants are present in the sample, two things happen, both based on interruption of coherence. Very close to the low-density continuous MIT, the conventional incoherent MIT occurs, which is smoother and has larger density exponents than the coherent transition. This conventional transition is of little interest here, except that it shows how different the intermediate phase is from a Fermi liquid. The second point of interest is that the presence of the minority dopants creates a natural length scale, associated with the average minority-minority spacing, that is larger than the majority-majority spacing. Physically this larger length scale is that associated with the residual resistance [8], a quantity that does not arise in many “modern” scaling theories of metallic behavior. The residual resistance is, of course, a very important quantity, as it eliminates divergences in the conductivity as It also provides a natural platform for filamentary counting. One constructs Voronoi polyhedra around each minority impurity, and counts the number of filaments crossing such polyhedra in various limits. Thus here counting shows up as a very basic operation. The omission of this construction has led some authors to the conclusion that the residual resistance of metals is not associated with impurities at all, but depends on “interactions of the electromagnetic field with the environment”, which is nonsense [8]. Counting has important implications for scaling in general. In continuum scaling

theory critical densities are irrelevant constants, and only the critical exponents are universal for a given class of interactions, independent of coupling strength, at least over limited ranges. Moreover, these critical exponents are in general irrational numbers. In

Figure 3. The electronic specific heat coefficient

for Si:P, showing both the continuous transport

transition at nc and the first-order thermal transition at ncb. The dashed line shows the value expected for a Fermi liquid.

The transition from the Fermi liquid to the filamentary metal occurs at n = n cb , and this

transition is first-order [14].

9

network glasses, for simple alloys, there are only a few classes of intact constraints, and the condition that their average number/atom be integral automatically leads to average coordination numbers that are simple small fractions (such as 12/5). The existence of such “magic” fractions is a direct consequence of treating the bonding interactions as a set separate from the set of spatial coordinates – in other words, the bonding interactions form a “space” different from real space. By connecting these two separate spaces we can identify one, and possibly even both, of the connectivity transition compositions. The process of connecting two apparently different spaces to prove a certain arithmetic result is much the same as in the proof of FLT. It has often been conjectured that the results obtained from continuum lattice scaling theories are “universal”. Specifically for given classes of interactions, for example, the critical densities for different diluted lattices – cubic, hexagonal, etc., will differ, but the critical exponents will be the same. This is no doubt correct, but it does not include the effects associated with intermediate phases in disordered solids, which are a new phenomenon that lies entirely outside the framework of continuum lattice scaling theory. The new phase transitions and the intermediate phase cannot even be described properly in terms of diluted lattices with their single connectivity transitions. A more flexible and more abstract description is required, that uses the methods and concepts of modem mathematics. In particular, one must be satisfied to describe the properties of sets, as the presence of disorder makes it impossible to describe fully the properties of the individual elements of the sets.

BASIS FUNCTIONS IN FILAMENTARY METALS

Suppose we have an impurity band in the intermediate phase. In this phase the metallic states are centered on arrays of filamentary, non-crossing paths that extend from electrode to electrode. These paths are similar in some respects to the Self-Avoiding random Walks (SAW’s) that are used in statistical mechanics to describe the mathematical properties of diluted magnetic lattices. There are also important differences. In the magnetic case the SAW’s are closed loops with pseudovector symmetry, whereas the electrical paths have

vector symmetry. In the magnetic case we are concerned with minimizing the free energy associated with magnetic susceptibility, essentially an equilibrium property. In the electrical case, the metallic conductivity contributes to dielectric screening of internal electric fields, thus it also can be varied to minimize the free energy. Because of quantum mechanics, the kinetic energy associated with transverse localization of charge carriers on filaments increases as the filaments become more closely packed, eventually delocalizing the electrons and leading to the transition to the Fermi liquid state at higher densities. This does not occur in the spin case, as spins are already localized objects with no intrinsic kinetic energy. The generalization of Fermi liquid wave functions, indexed by the continuous momentum p and represented by the EMA wave functions to the discrete case of filamentary arrays, is not difficult. One assumes that the real-space centers of density of each filament j are known, and denotes the corresponding path by Longitudinal wave vectors are oriented parallel to the local tangent to the path. There are no transverse wave vectors, only a local transverse localization length. In the normal state in the absence of a magnetic field these wave vectors can be used to construct basis functions for each filament. The actual wave functions near the Fermi energy will be time-dependent linear combinations of individual filamentary wave functions that minimize the free energy by screening internal electric fields.

10

BROKEN SYMMETRY IN THE SUPERCONDUCTIVE AND NORMAL STATES

The key feature of the BCS theory of metallic superconductivity is the formation of Cooper pairs, which become the Landau-Ginzburg order parameter with 1 = and As the volume of the system tends to the number of possible

choices for 2 (even when these are restricted by the isoenergetic constraint ) also tends to but from this infinitely large set only one state, the time-reversed one, is used to form the Cooper pair. In other words, the cardinality of the set of states consisting of the time-reversed state is lower than that of the set consisting of all the isoenergetic states. This is a characteristic feature of continuum or effective medium models in which scattering by some kind of disorder is added after the basis states have been chosen. In strongly disordered systems, such as random metals near the MIT, the situation is quite different. In order to explain the critical properties of such systems one must select the correct filamentary basis states at the outset. This is done variationally, and it affects both normal-state and superconductive properties in many ways that are radically different from normal metals, where all the isoenergetic states are essentially equivalent, as in Landau Fermi liquid theory. The special properties of filamentary metals are the result of atomic relaxation that leads to preselection of basis states even in the normal state, where there are only two states per filament, and with In other words,

Figure 4. There are two components in the filamentary model, the CuO2 planes (A), and the impurity bridges combined with secondary metallic planes (B). This Figure shows the density of electronic states near the Fermi energy, for with the energy scale set by the resonant bridging impurity width WR. There is a strong peak in due to the impurity resonance pinned by electron-ion polarization energies and the anti-Jahn-Teller effect, and a strong dip in NA (E) due to electron-electron Coulomb interactions. The product has peaks near At the optimal composition there are only extended states for The scattering rates are also shown for the optimal

composition. They are much larger for the localized states,

than for the extended states,

11

for the one-dimensional filaments the Fermi surface collapses to two points, and this has happened in the normal state because of quantum percolation. Another extremely unexpected feature of filamentary states is that they maximize (minimize) the conductivity (resistivity). This variational property means that the dynamically optimized filamentary states already include all many-electron interaction effects, including those of electron-electron scattering which give rise to T2 resistivities in Landau Fermi liquid theory. That such many-electron interactions are absent in impurity band random metals has been shown by a filamentary analysis [8,9] of critical exponents in Si:P [15] and neutron-transmutation-doped, isotopically pure Ge:Ga [16]. The dominant remaining source of scattering in the cuprates is thermal excitation into localized states with energies outside the WR resonance region [5] shown in Fig. 4. (This disappearance of the Coulomb interaction between discrete and dynamically optimized filamentary currentcarrying states is analogous to the existence of zero-frequency floppy modes in the intermediate phase of network glasses.)

FILAMENTARY MODEL FOR HTSC

Even without measurements of the properties of the cuprates it was clear to crystal chemists and materials scientists that these multinary compounds would be extremely unusual from a structural point of view [1]. In addition to containing rare earths and

oxygen, these pseudoperovskites nearly always contain Cu, an element that in one oxidation state shows a greater diversity in its stereochemical behavior than any other element. This observation, together with the extremely anomalous transport properties, certainly bodes ill for any microscopic theory of the cuprates which is based on the EMA, as that approximation ignores the flexible material properties of Cu, and the ferroelastic properties of the perovskite family, altogether. (All materials with unusual properties, from Si to conjugated hydrocarbons to DNA, conform to the central principle of organic chemistry, “structure is function”.) Undeterred by these inescapable ground rules, almost all the theories developed so far, such as [2] and [3], are based on the EMA, augmented

only by good intentions and wishful thinking; given the richness and complexity of materials science, this is unlikely to suffice. The filamentary theory of HTSC diverges from EMA theories in two basic ways: it incorporates an extensive knowledge of the experimental data [1], and it has a sound mathematical foundation in the filamentary theory of the MIT in semiconductor impurity bands [14], which supersedes inadequate EMA theories of the Fermi liquid or LandauGinzburg type [2,3]. Because of interlayer ferroelastic interactions the “metallic” CuO2 planes are partitioned into metallic nanodomains separated by semiconductive domain walls. A specific filamentary path was envisaged [20] that connects metallic CuO 2 planes with secondary metallic planes (CuO chains or BiO planes) via resonant impurity states located in semiconductive planes (such as BaO) sandwiched between the metallic planes, as shown in Fig. 5. In addition to the bridging impurity points most samples contain two kinds of extensive defects which act as blocking lines or layers. Blocking macroscopic ab planar layers explain the usually semiconductive c-axis resistance; this aspect of the data has received too much attention [3], as these blocking layers are essentially extrinsic and can be avoided in some cases, by relieving interlayer strain energies [21], or by overdoping [22]. The interesting extensive defects are intraplanar semiconductive nanodomain lines in the metallic planes; these form grids, and each cell of a grid is connected to a square in an adjacent metallic plane by resonant (metallic) tunneling through a bridging impurity. The experimental evidence for the existence of such buckled cellular grids is discussed here by Jung. It is difficult to obtain evidence for spatial inhomogeneities on this scale, but the evidence available has been growing steadily if slowly. 12

Figure 5. The variationally optimized percolative filaments, shown in cross section, follow planar locally metallic CuO2 layers until they approach a domain wall which is locally insulating. The zigzag metallic path is continued by resonant tunneling through a state pinned at the Fermi energy associated with a defect, such as an oxygen vacancy. The next segment of the path is that of a chain, and this segment terminates at a chain O vacancy, where the zigzag path is continued by resonant tunneling back to a CuO2 layer, and so on. This model is designed for YBCO; in LSCO the tunnel paths simply connect CuO2 layers. Such filamentary paths should never be confused with “stripe phases” or pinned charge density waves, which are incidental

minority insulating phases.

There is also dramatic evidence for the existence of nanoscale spatial inhomogeneities in Debye-Waller factors measured by ion channeling, which are very sensitive to a few large out-of-plane atomic displacements and show striking precursive anomalies at Tc; these are absent from neutron diffraction data which measure EMA properties [23,4]. It is the electronic structure associated with Fermi-level pinning defects which experimentalists tune when they adjust oxygen concentrations, refine by annealing or observe as aging or quenching effects. To understand transport properties one must understand the topological

connectivity of these states, which is scarcely possible within the EMA. Within a single-particle picture the Fermi-level pinning metallic states can be represented as a narrow band of resonant states of width few meV the valence band width as shown in Fig. 4. Ordinarily one might expect that such states would be unstable against a Jahn-Teller distortion, and indeed it has been stated [3] without proof or citation that this is always the case. In fact, it is easy to find exceptions where such peaks are located self-consistently (with respect to both electronic and atomic coordinates) at EF, for example, in many total energy calculations. Moreover, Fermi-level pinning by impurity, surface or interfacial states at metal-insulator junctions (Schottky barriers) is one of the basic principles of semiconductor device physics. The error [3] arose from simplistic inclusion of only one-electron interactions and neglect of both core-core

and non-local electron-electron interactions. An amusingly similar error (sometimes called the Wentzel mistake), also on the subject of instabilities and superconductivity, led Wentzel to suggest [24] that the Bardeen-Frohlich attractive electron-phonon interaction was not the correct mechanism for simple metallic superconductivity. What is needed for the cuprates is a general mechanism for frustrating the Jahn-Teller effect. This is provided by a self-screening atomic relaxation mechanism which involves 13

long-range Coulomb interactions not representable as local single-particle energies [25]. The attractive self-screening energy of the already narrow band of resonant states is maximized by further narrowing the band and centering it on EF. This “anti-Jahn-Teller

effect” has the congenial feature that it is expected to be especially effective in strongly ionic materials where the long-range Coulomb forces are only weakly screened by a few metallic electrons. This is exactly the situation in the cuprates, which are close to a metalinsulator transition; it is also the case for impurity band random metals [8-10], where this

weak screening is responsible for the anomalously small exponent [15,16] which lies below the limit expected from scaling theory [18] with Boson statistics and shortrange forces only, and far below the value of 3/2 predicted by some one-electron EMA theories. The narrow resonance region of width WR was previously portrayed [5] as a peak in the density of electronic states centered on EF, but this need not be the case. All that is necessary is that in this region the scattering rate be extremely low compared to that at higher energies outside this region. Thus the picture we have now is that shown in Fig.4. The density of extended states is depressed relative to that of the localized states by Coulomb interactions, as happens for the random metal on the insulating side of the MIT [16,26]. This density of states would go to zero at E = EF and T = 0 were it not for the antiJahn-Teller effect, which leaves a residue of carriers at T = Tc which is about half that for T = T0 [27]. The pinning of the most polarizable filamentary states to EF by Coulomb interactions is similar to the energy-level reordering responsible for the pseudogap in

random metals [26]. NORMAL-STATE TRANSPORT

In a normal metal electron-phonon interactions typically contribute a temperaturedependent term to the resistivity proportional to with For large crystalline disorder, as in metallic glasses and thin films quenched at low temperatures, electronelectron scattering is strong and In certain cuprates, notably those without secondary metallic planes involving metallic elements other than Cu (Bi or Hg), the ab planar normalstate resisitivity is linear in T approximately from T0 to Th, where T0 is close to Tc and Th is the high-temperature limit of compositional stability [5]. This is a very remarkable result, as in cases where Tc is low, the ratio Th/T0 has been observed to be as large as 100. However, it holds only for those samples whose composition corresponds to a maximum in Tc; increasing doping causes to cross over to the Fermi liquid (strong electron-electron scattering) value of 2. This means that a satisfactory theory should contain some continuously tunable factor which will alter both anomalies at the same time, and this factor should be responsible for the MIT as well. As the reader will realize, these demands are very severe. He will probably not be surprised to learn that they are met by the author’s filamentary theory [5], but not by any theory based on the EMA, such as [2] or [3]. The limitations of EMA models becomes obvious when one examines the field theories developed by various authors [28] to implement Anderson’s suggestion that electric (holon) and magnetic (spinon) effects are somehow separated in the cuprates [3]. It is clear that such a separation is essential if the magnetic moments of the rare earths are not to quench superconductivity, but Anderson gives no microscopic explanation of how this can happen; it is merely one of his axioms, or, as he prefers, dogmas. Given this dogma, one is able to explain microscopically why normal-state transport anomalies exist and are loosely correlated with the optimization of Tc. However, one is unable to derive any functional form for the temperature dependence of the resistivity, much less to explain why 14

without assuming what was to be derived. Even the temperature scale ratio Th/T0 is merely the ratio of an inelastic high-temperature scattering rate to Tc, which is yet another assumption, which turns out to be incorrect, as we shall see. The separation (“spin bags” [2]) of magnetic and electrical effects, moreover, need not be axiomatic. It is derivable in the filamentary model simply by observing that the magnetic states are all localized, and that the separation of the extended current-carrying states from the localized states [14,5] automatically separates spin and Cooper-pair-forming states. Note, however, that this separation cannot be carried out correctly within the EMA because in that picture one is unable to count [6,7,19] the states that are being separated. The importance of counting is illustrated convincingly by the much simpler case of random impurity band metals, where the EMA has failed in calculating critical exponents [8,9], By contrast, the dogmatic holon-spinon separation [3] becomes, in the filamentary model, what one would naturally expect in optimized HTSC because of the success of the filamentary model for the closely related impurity band MIT. It is nothing more than intralayer nanoscale phase separation, driven by interlayer ferroelastic misfit forces.

Because the CuO2 planes are divided into an irregular checkerboard or grid of nanodomains by intraplanar domain lines which have semiconductive gaps currents can flow only along filamentary paths passing through interplanar resonant tunneling centers (impurity bridges). In YBCO, for example, such centers might be represented by the much-studied apical oxygen sites between Cu atoms in the CuO2 planes and the chains, selectively associated with vacancies on the later. For optimal doping there are two such centers per CuO2 planar domain, one source and one drain. When there are fewer than two, the sample is underdoped, and when there are more than two, it is overdoped (see Fig. 2 of [5]). Thus the average integral (bridge/domain) ratio is the continuously tunable factor mentioned above; there it was shown that when this factor is two all the normal-state anomalies are explained, as is the optimization of Tc. It was also

explained why overdoping depresses Tc and increases

from 1 to 2, at the same time producing the observed anomaly in the Hall resistance [29]. An important historical point is that the fact that all the normal-state transport anomalies can be explained by the existence of a narrow, high-mobility band pinned to EF was first explained in [29]. At that time the explanation was not generally accepted because it was not accompanied by a specific structural model that explained the origin of

the narrow band. Such a model is shown in Fig. 5, and it is the only such model that has been advanced. The narrow high-mobility band itself is the only way of explaining the normal-state transport anomalies, so that together with Fig. 5 it may be taken as the only satisfactory, perovskite-specific model for HTSC. What happens to underdoped samples? In the YBCO case, underdoping produces

more O vacancies on the chains that generate the crystalline orthorhombic symmetry. These chains are almost surely responsible for the phase shifts at twin boundaries where the chain orientations rotate by which experimentalists and many EMA theorists often like to describe as “d-wave superconductivity”. This is an EMA (or Fermi liquid) misnomer that is entirely inappropriate for the non-Fermi liquid intermediate phase. It implies a fundamental significance of what amounts to a non-bulk edge or surface

effect, which is seen to be trivial as soon as one realizes that the b-axis chains are essentially involved in constructing filamentary paths, so that the observed phase shift is unavoidable. The chain segments become shorter as x increases, and although probably only short chain segments are needed to bypass intraplanar CuO2 domain walls, it is clear that the ab planar resistance will increase as the chain segments shorten. Aside. Many experiments have shown that residual states exist within the pseudogap; these residual states are also often described as the result of “d-wave superconductivity”. In fact, the observed residual states are very similar to those predicted by [26]. Thus if

15

there are dopants in semiconductive domain walls that generate a pseudogap, then this would easily account for the experimental observations, without nonsensically using Fermi liquid terminology to describe the non-Fermi liquid intermediate phase. In fact, we expect increasing phonon-assisted currents across oxygen vacancies within the chains. The importance of these will increase with x and (the orthorhombic plateau in Tc(x)), it is possible that virtual phonon exchange at these vacancies will provide a stronger attractive interaction for forming Cooper pairs than phonon exchange at the caxis impurity bridge resonances does. The width and strength of the density-of-states peak of the latter may not depend on the oxygen mass, as they may well be associated with collective relaxation and optimization of many internal coordinates. This would explain the disappearance of the oxygen isotope effect for small x [30]. On the other hand, the electron-phonon interactions at chain vacancies promote phonon-enhanced coherent currents across these micro-weak links, enhancing the local energy gap. When this effect is linearized with respect to vibrational amplitudes, it may still be equivalent to a local Bardeen-Frohlich interaction and may thus give rise to what resembles a normal isotope effect for large After the above was written, a very important paper [31] appeared concerning the relaxation of Tc in YBCO after abrupt release of pressure. The relaxation was found to follow the form of a “stretched exponential”, with The key parameter of interest is the dimensionless stretching fraction which turns out to be highly informative. The stretched exponential can be derived from a microscopic model. The model involves diffusion of excitations in a configuration space of dimension d*p to

randomly distributed traps.

As time passes, all the excitations near the traps have disappeared, and only excitations distant from the traps remain. The latter must diffuse further and further. This leads to the stretched exponential and to

At first, it might appear that all that has been done is to replace one empirical parameter, with another, d*p. In fact,

for homogeneous glasses. (The dopants in a well-annealed and homogeneous HTSC presumably form a glassy array.) Here d = 3 is the dimensionality of Euclidean space. The key factor now is p. Comparison with experiment and several very accurate MDS showed that for homogeneous glasses p is nearly always 1 or 2; it measures the number pd of interaction channels involved in diffusion of excitations in d dimensions. In metals where phonon scattering dominates the resistivity, one of these channels is always e-p interactions. However, if other classes of interactions are present, there may be other diffusive interaction channels as well. It is easy to see that adding channels increases the stretching factor, which is Mathematically the simplest and most rigorous example with p = 2 is provided by quasicrystals, where the Euclidean coordinates r become and the Penrose projective coordinates are Motion in space (the first d channels) involves phonons and

produces relaxation, while motion in space (the second channels) involves phasons, which only rearrange particles without diffusion or relaxation. In the ideally random quasicrystal a given hop may tale place along either or Thus

16

where f p measures the effectiveness of hopping in pd channels, only d of which is associated with relaxation. For an axial quasicrystal, which is quasi-periodic only in the plane normal to the axis, the calculation is somewhat more complex. There are five channels, three in space, and two in space, so that fp = 3/5, and d*p = 9/5. Thus in excellent agreement with MDS15 which give From the value one can rigorously infer that p = 1, and thus only electronphonon interactions can cause HTSC. The proof is based on grouping the interactions involved in diffusive relaxation into classes of interactions that are effective (such as electron-phonon interactions) and ineffective (such as electron-magnon interactions) [32]. Because only the electron-phonon interaction is needed to explain all other interactions (such as electron-[magnon, plasmon, any-old-whaton] interactions) are excluded by experiment [31,33]. The remarkable aspect of this experiment and theory is that the conclusion transcends all the details of structure and large-scale relaxation around

dopants in these complex materials. Note that once again, the success of this approach rests on identifying two different sets, interaction space and real space, and then connecting them, just as in constraint theory and the proof of FLT.

CHAIN LENGTHS AND LOW TEMPERATURE CUFOFF T0 In the filamentary theory there is a close relation between average chain lengths L and the lower cutoff (or pseudogap) temperature T0 in for optimally and

underdoped samples:

(see (1,2) of [5]); here This relationship gave good results for the YBCO T0(x)], which are linear in x with T0 (0.1) 150K for Tc optimized 90K. It should be mentioned here that many samples appear to give T0 less than Tc, but these are probably inhomogeneously overdoped. For optimally doped (x = 0.1) samples the value T0 (0.1) 150K has been confirmed for single crystals and for thin films grown by several methods, and even fine-tuned with low-energy electron irradiation [27]. In a large magnetic field Tc is suppressed, “unmasking” or exposing the normal-state

resistivity at temperatures lower than Tc(H) for H = 0 in relatively “low Tc” cuprates ( T c

40K), such as [35], The central “unmasked” single-crystal results stressed by [34] are that for underdoped compositions pab(T) increases and becomes semiconductive below T0. Even more significant, however, is the disappearance of the pseudogap (the dip in normal-state resistivity between Tc and T0) in large magnetic fields. This disappearance is not discussed at all, as it is virtually inexplicable within the EMA. In terms of our topological model of the intermediate state, the explanation is immediate. The large magnetic field replaces the self-organized, non-crossing coherent filamentary basis states with vector symmetry by quasi-circular orbital states with pseudovector symmetry, thereby restoring Fermi liquid character, including strong electron-electron scattering, to states near EF.

Because chains represent the ideal local structure for filamentary currents, microscopic probes of the local chain structure in untwinned samples of YBCO are of great interest. Recently two experiments have done this, with results that cannot be explained by EMA models based on bulk energy bands and bulk phonons. The first experiment [36] revealed systematic changes in scattering strength with doping of longitudinal optic (LO) phonons

17

propagating in the basal plane normal to the chain direction. These phonon spectra contain a pseudogap which can be explained [33] as the result of short-range ordering of chain segments that alternate in oxygen filling factors. The changes in scattering strength are more interesting, as they turn out to be direct measures of phonon coherence along filamentary paths, and they change abruptly in as the composition passes

through the metal-insulator transition near x = 0.4. There is also a second abrupt change in scattering strengths centered on x = ¾ [the transition between the Tc = 60K and 90K plateaus] as the smaller filling factor passes through ½; this change is just what one would expect from percolation theory, and from it one can successfully predict the change in the Tc ratios of the two plateaus. In the second experiment [22], the longitudinal magnetoresistance in slightly overdoped untwinned YBCO was studied; in these samples the c-axis resistivity is linear in T, just as the ab planar resistivity is, which means that the coherent percolative paths are 3dimensional. Again the results show strong anisotropy that is connected with coherent current flow along the chains.

NMR RELAXATION AND THE SPIN PSEUDOGAP

Anderson has discussed (see [3], Fig. 3.27) the observation of anomalous nonKorringa planar Cu relaxation in various cuprates, and states that “this anomalous relaxation seems to be one of the common features of the high-Tc. state”, but “it is actually relatively more pronounced for somewhat lower Tc materials., so is not closely related to superconductivty.” This author concurs, and would add that in his opinion in most cases (LSCO is an exception) all that spin-scattering experiments are measuring is spin relaxation in pockets of insulating material with compositions which can be quite different from those of the superconductive bulk. For example, this effect is quite small in optimal with but is much larger in optimal Note that the oxygen diffusivity is very high in YBCO, but not the Sr diffusivity in LSCO, so that while it is possible to make very nearly homogeneous samples of the former, this is not possible for the latter. Thus in general there is no connection between T0 and the spin pseudogap Tsg, nor should we expect to find one, except for materials like LSCO, where magnetic microphase inclusions may be absent. In that case both T0 and the spin pseudogap Tsg can be related to the resonance width WR. COMPOSITION DEPENDENCE OF THE ENTROPY OF THE VORTEX PHASE TRANSITION

The physics of magnetic vortices in the mixed state is extremely complex, and it is nearly always treated from the point of view of the EMA, although it is clear that if the sample is spatially inhomogeneous the vortices will nearly always localize preferentially in regions of lower Tc. In the author’s view this is the most natural explanation of the “step kink - peak” phenomena in vortex lattice melting which have attracted much attention from experimentalists [37]. However, two-phase models have many adjustable parameters and it would appear that this greatly limits what can be gained from the analysis of such phenomena. Thus most of the discussion of these phenomena by experimentalists has focused on the fact that the entropy of vortex lattice melting is much larger than would be expected if the vortices are treated merely as point objects [38]. This can be easily explained by taking account of changes in the nonlocal structure [39] of the vortices near Tc. There is, however, 18

one very puzzling feature of these data which is explained quite easily by the present theory. This theory is a counting theory, and thus it is well-suited to studying the entropy of the vortex phase transition. In Bi2Sr2CaCu2Oy single crystals it is observed [37] that this entropy is several times larger for overdoped than for optimally doped samples, especially

at low T. Ordinarily in the EMA one would expect that Tc would reach its maximum value at the composition where N(EF), the density of electronic states in the normal state, has its maximum value, that is, at optimal doping EF coincides with a peak in N(E). In such a case

the entropy should reach its maximum at the same optimally doped composition. However, in the present model N(EF) is larger in the overdoped state than in the optimally doped state, so giving a larger entropy of melting. The transition temperature decreases in the overdoped state because the electrons at the Fermi energy spend more time in Fermi-liquidlike states in the CuO2 planes, where the electron-phonon coupling is weak, and less time at the discrete resonant tunneling centers, where it is very strong. See Eqn. [3] and Fig. 2(c) of [5].

INTERMEDIATE PHASES AND FIRST-ORDER PHASE TRANSITIONS

The mathematical character of intermediate phases is characterized by some discrete features (the filaments) and some continuous features (the off-lattice space in which both the network glasses and the spatially disordered impurities of the electronic examples are embedded). One felicitous consequence of mixing discrete and continuous features is that the lower-density (or first) transition from the disconnected (or insulating) phase to the intermediate phase is continuous, while the higher-density (second) transition from the filamentary phase to the effective medium (overconstrained, or Fermi liquid) phase is first order. This asymmetry is quite striking, and it cannot be explained by an entirely discrete

lattice model, or by an entirely continuous effective medium model. Both of the latter contain only one transition, and it was the similarity in this respect that has led many people to suppose (mistakenly) that there is a “universal” character of phase transitions that can be independent of the discrete/continuous dichotomy. In this workshop both Boolchand and Thorpe discuss these two phase transitions in convincing detail. In Figs. 3 and 6 the two transitions are sketched for impurity bands in Si:P. The two critical densities are separated by a factor of 2, and there is no doubt that the first one is continuous, and the second is not. The layered cuprates that form HTSC are complex multinary compounds, and

preparing samples that are microscopically homogeneous is not easy. Of course, unless such homogeneity is achieved, the second transition will be greatly broadened, and it will be difficult to show that the second transition is first-order. So far, three successful studies have reported first-order phase transitions: (1) near x = 0.21 in (after annealing for several months [41] at high T and constant O partial pressure), (2) near = 0.19 in

(also after annealing at constant composition [42]), and near x =

0.95 in (carefully designed chemical and thermal history, including slow cooling [43]). Normally ones observes only parabolic Tc(x)’s. Self-organization is not easily achieved, that is why only in a few experiments are the two HTSC transitions

separated to give trapezoidal Tc(x)’s.

19

Figure 6. A sketch of the thermal data on Si:P, showing both transition [40].

and

in relation to the transport

GENERAL CONCLUSIONS: ANALYTICITY AND CARDINALITY The essence of filamentary percolation theory is that it replaces the analyticity of field theory, which has been an excellent guide to the physics of the “old” metallic superconductors, by the “countability”, or cardinality of modern set theory. The justification for this in HTSC is an internally consistent theory of the crucial experimental properties, notably the normal state transport properties, as functions of both temperature and composition. Analytic models [3] can be constructed which account qualitatively for the temperature dependence, for example, but when these are examined in detail it soon becomes apparent that they are not capable of accounting precisely for the observed functional forms, either the resistivity linearity in T or the linearity and magnitude of the composition dependence of the low-temperature pseudogap linear resistivity cutoff T0(x) in YBCO7-x. The filamentary theory exhibits many similarities between the MIT in impurity bands and that in the cuprates, especially YBCO, which is the most homogeneous and macroscopic-defect free of the cuprates, because of its chains. In this sense the theory is self-proving (self-testing), because one would expect those similarities to be most pronounced for the best material. The disappearance of many of these properties from LSCO and (Y,Ca)BCO alloys shows that the theory is both selective and incisive, for it

successfully differentiates those properties which EMA models cannot explain because they are microscopic, from those which it cannot explain because they are macroscopic. Both electronic theories are reduced to their simplest form in the intermediate phases of network glasses. The overall similarity of the three systems shows that it is their shared

20

(discrete/continuous) topology that is responsible for their remarkable properties, from the reversibility window to HTSC itself.

THE BIG, BIG PICTURE

The differences between pure continuum models (or the effective medium approximation, EMA) and discrete off-lattice models (embedded in a continuum) are huge, not only conceptually, but also psychologically. Scientists who have been educated to think only in terms of continuum models, and who have developed their own concepts in that framework, often find themselves enslaved by that framework. (A concrete analogy, close to home, is that of the experimenter married to his equipment, or the computer scientist married to his software.) In this volume one can find many examples of problems and principles that certainly go beyond the limits of the continuum approach. Perhaps it is not surprising that many of these examples occur in the context of network glasses, as these materials are obviously unsuited to continuum treatments. On the other hand, that nanodomains should exist in perovskites and pseudoperovskites should come as no surprise, as this family of materials has been known to be ferroelastic (and prototypically so) for more than 50 years. Yet Jan Jung’s elegant survey in this volume of these nanodomains reaches conclusions that are

politically very unpopular, as anyone who has attended one of the numerous conferences on

HTSC can testify. One can ask oneself just why such an obvious consequence of one of the most general

principles in the materials properties of oxides should be deemed “politically incorrect”. More than 10 years ago, when the subject of HTSC was still in an embryonic stage, I believed that most people were still being conservative; perhaps there was not enough direct evidence for the existence of nanodomains, and their dimensions remained to be determined. As Jung shows, this is certainly not the case today. Another explanation is that most people feel that if their own experiments do not

directly exhibit nanodomain features, then those features are not needed to explain their results. This is not so naïve as it sounds: in the theory of critical exponents of continuous phase transitions, some very sophisticated theorists have postulated that (loosely speaking), all such transitions are equivalent. (This is the concept of universality.) Such on-lattice transitions never exhibit an intermediate phase. Thus, it is the existence of intermediate phases in self-organized disordered systems that causes the breakdown of universality. We are now at the crucial point, both conceptually and psychologically. It is widely admitted, even by almost all those who still adhere to continuum descriptions of HTSC, that the intermediate (often called “non-Fermi liquid”) phase is responsible for HTSC. If this is the case (and all the experimental evidence so indicates), then the fact that no microscopic continuum model is known that produces an intermediate phase between the insulating and Fermi liquid phase, becomes decisive. There is such a model in discrete theories, and it is very successful in describing intermediate phases in network glasses and semiconductor impurity bands, as discussed elsewhere in this volume. It follows that a discrete network model is the only practical model for HTSC. Of course, a successful continuum model based on the EMA may be developed someday for HTSC, about the time that there is pie in the sky.

REFERENCES 1.

J. C Phillips, Physics of High-Tc Superconductors, Academic Press, Boston 1989.

2.

J. R. Schrieffer, X. G Wen and S. C Zhang, Phys. Rev. B 39, 11663 (1989).

21

3.

P. W. Anderson, Theory of Superconductivity in the High-Tc Cuprates, Princeton Univ. Press, Princeton 1997.

4.

J. C. Phillips, Physica C252, 188 (1995).

5.

J. C. Phillips, Proc. Nat. Acad. Sci. 94, 12771 (1997).

6.

A. Wiles, Ann. Math. 141, 443 (1995).

7.

A. D. Aczel, Fermat’s Last Theorem, Four Walls Eight Windows, New York, 1996; S. Singh and K. A. Ribet, Scien. Am. (11): 68 (1997); D. Mackenzie, Science 285, 178 (1999). 8. J. C. Phillips, Proc. Nat. Acad. Sci. 94, 10528 (1997). 9. J. C. Phillips, Proc. Nat. Acad. Sci. 94, 10532 (1997). 10. J. C. Phillips (unpublished). 11. A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996); C. H. Bennett, Physics Today 48 (10), 24 (1995). 12. H. He and M. F. Thorpe, Phys. Rev. Lett. 54, 2107 (1985); M. F. Thorpe, D. J. Jacobs, N. V. Chubynsky and A. J. Rader, Rigidity Theory and Applications (Ed. M. F. Thorpe and P. Duxbury Kluwer Academic / Plenum Publishers, New York, 1999), p. 239. 13. Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 (1975). 14. J. C. Phillips, Solid State Commun. 47, 191 (1983). 15. G. A. Thomas, M. A Paalanen,. and T. F. Rosenbaum, Phys. Rev. B 27, 3897 (1983). 16. K.M.Itoh, E. E.Haller, J. W. Beeman, W. L. Hansen, J.Emes, L.A. Reichertz, E. Kreysa, T. Shutt, A. Cummings, W. Stockwell, B. Sadoulet, J. Muto, J. W. Farmer, and V. I. Ozhogin, Phys. Rev. Lett. 77, 4058 (1996). 17. M. F. Collins, Magnetic Critical Scattering, Oxford Univ. Press, Oxford (1989). 18. J. T. Chayes, L. Chayes, D. S. Fisher and T. Spencer, Phys. Rev. Lett. 57, 2999 (1986). 19.

K. A. Ribet, and B. Hayes, American Scientist 82, 144 (1994).

20.

J. C. Phillips, Phys. Rev. B 41, 8968 (1990).

21.

X. D. Xiang, W. A. Vareka, A. Zettl, J. L. Corkill, M. L. Cohen, N. Kijima, and R. Gronsky, Phys. Rev. Lett., 68,530(1992). N. E. Hussey, H. Takagi, Y. Iye, S. Tajima, A. I. Rykov, and K. Yoshida, Phys. Rev. B 61, R64 (2000). R. P. Sharma, F. J. Rotella, J. D Jorgensen,. and L. E. Rehn, Physica C 174, 409 (1991). G. Wentzel, Phys. Rev. 83, 168 (1951). J. C. Phillips, Phys. Rev. B 47, 11615 (1993). A. L. Efros and B. I. Shklovski, J. Phys. C 8, L49( 1975). S. K. Tolpygo, J.-Y. Lin, M. Gurvitch, S. Y. Hou and J. M. Phillips, Physica C 269, 207 (1996). N. Nagaosa and P. A. Lee, Phys. Rev. B 45, 966 (1992). H. L. Stormer, A. F. J. Levi, K. W. Baldwin, M. Anzlowar, and G. S. Boebinger, Phys. Rev. B 38, 2472 (1988).

22. 23. 24. 25. 26. 27. 28. 29. 30.

J. P. Franck, Physica C 282-287, 198 (1997); Phys. Scrip. T66, 220 (1996).

31. 32.

S. Sadewasser, J. S. Schilling, A. P. Paulikas and B. W. Veal, Phys. Rev. B 61, 741 (2000). J. C. Phillips, Rep. Prog. Phys. 59, 1133 (1996); J. C. Phillips and J. M. Vandenberg, J. Phys.: Condens. Matter 9, L251-L258 (1997).

33. 34.

J. C. Phillips (unpublished). G. S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, and S. Uchida, Phys. Rev. Lett. 77, 5417 (1996). T. Ito, K. Takenaka, and S. Uchida, Phys. Rev. Lett. 70, 3995 (1993). Y. Petrov, T. Egami, R. J. McQueeney, M. Yethiraj, H. A. Mook, and F. Dogan, LANL CondMat/0003414 (2000).

35. 36.

37. 38. 39. 40. 41. 42.

T. Hanaguri et al., Physica C 256, 111 (1996). A. I. M. Rae, E. M. Forgan, and R. A. Doyle, Physica C 301, 301 (1998). H. Darhmaoui and J. Jung, Phys. Rev. B 53, 14621 (1996). J. C. Phillips, Solid State Commun. 109, 301 (1999). H. Takagi, R. J. Cava, B. Batlogg, J. J. Krajewski, W. F. Peck, P. Bordet, and D. E. Cox, Phys. Rev. Lett. 68, 3777 (1996); H. Y. Hwang, B. Batlogg, H. Takagi, J. Kao, R. J. Cava, J. J. Krajewski, and W. F. Peck, Phys. Rev. Lett. 72, 2636 (1994). J. Wagner (this workshop).

43.

E. Kaldis, J. Rohler, E. Liarokapis, N. Poulakis, K. Conder, and P. W. Loeffen, Phys. Rev. Lett. 79,

4894 (1997).

22

REDUCED DENSITY MATRICES AND CORRELATION MATRIX

A. JOHN COLEMAN Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada [email protected]

I tackle a herculean task - attempting to wean our imagination from the 1-particle picture which, implicitly, we have all been using since our youth. I shall try to entice you to join a crusade for the creation of new concepts and images needed for problems in which interaction between 3 or more electrons is significant and which are appropriate for describing the information encapsulated in the second order reduced density matrix. (“2-matrix” for short) Perhaps the difficult part of our task is changing our language and mental images. It was to this task that we were called by Charles Coulson in private conversation, and in his speech [1] in Boulder in June 1959, urging us to look in the 2-matrix for correlation. Also, by H. Froehlich [2] when he bemoaned the fact that we have failed to exploit the deep import of the results [3] of C.N Yang on the 2-matrix. It is to this task that I have devoted much of my time and interest since 1952 culminating in the publication of REDUCED DENSITY MATRICES - Coulson’s Challenge [4]. I shall refer to this book, by Coleman and Yukalov, as “CY”. When we wrote CY, although it was known that the order parameter of one of the phases of He3 had p-symmetry, we were unaware that the existence of s,p and d symmetry has appeared in some of the high temperature superconductors. As a result there is only a brief reference in CY to the correlation matrix. I have made a modest effort to redress this lacuna in the present paper. ENERGY AND N-REPRESENTABILITY A reduced density operator (RDO), for a normalized pure state, (123... N), of a system of N identical fermions or bosons can be represented as an integral operator. For example, the kernel of a 1-RDO, D1 , is the first order reduced density matrix (1-RDM):

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

23

whereas, the 2-RDM is

Thus the operator, D 1 , acting on a symmetric or antisymmetric function f of N particles,

defines a function D1 f such that

More generally, the p-RDO, Dp, is an operator with unit trace such that

In the context of Second quantization, it is usual to employ RDO’s with somewhat different normalization introduced by Dirac and defined by

and

Thus, and

As far as I am aware, it was Dirac [5] who first made effective use of RDM’s. But he considered only states described by a single Slater determinant formed from N orthonormal

spin-orbitals

in which case

As can be easily verified, this operator is merely the Identity operator on the linear space, spanned by the N functions Dirac showed that all the physical properties of the Slater state, including the p-matrices, can be obtained from a knowledge of or,equivalently, of It is astounding that so much physics, including our understanding of the Periodic Table, has been built with what would seem to be a trivial tool - the identity operator on a linear space of dimension N. As we all know, the physics consists of a skillful choice of the spin-orbitals, or rather of It is precisely the purpose of Hartree-Fock theory to lead us to the “best possible” choice. A large thriving industry and much of the wealth of the pharmaceutical companies is based on the simple equation (7). Such is the power of mathematics! It was Husimi [6] who, apparently, first discussed the more general RDM’s in (1) and (2) .Indeed, he considered p-RDM’s for arbitrary p. When the hamiltonian is represented in the form

as a sum of N one-particle terms and two-particles terms, it is easy to see that, for both fermions and bosons, the exact energy, E, of the state is given by

where the reduced hamiltonian is defined by

24

In my view, these last two formulas are absolutely basic for understanding the quantum mechanics of many-particle systems in which interaction among the particles plays a significant role. From the form of (10) it appears that as N increases the relative importance of interaction becomes increasingly significant! Unfortunately, hitherto little attention has been given to the eigenstates of K and the role of N in determining its eigenvalues. I regard this as a key challenge for any analyst who is interested in making a significant contribution to the N-body problem. - cf. pp. 11 and 257 of CY. I discovered (9) in 1952 while trying to understand Frenkel’s exposition of so-called Second Quantization. Husimi had seen it at least ten years earlier! I immediately applied (9) to calculate the ground-state energy of Li by assuming a simple ansatz A for D2 such that

I did extremely well, indeed too well! The result was about 20% below the observed value! This was impossible and forced me to realize that imposing fermion statistics was more subtle than I had imagined. This led me to invent the concept of N-representability: The 2-matrix of a pure state,

must be representable in the form (2) in order to satisfy

fermion or boson statistics. Analogously for a p-matrix.

So, in 1952 I proudly announced to a group of able physicists at Chalk River that I had reduced the N-body problem to a body problem - we now merely had to solve the N-representability problem, which I assumed would be child’s play, and using (9) find the ground states using Rayleigh-Ritz. After 48 years there is no easy practical way of doing this in general. However, Carmela Valdemoro made a big break-through in 1992 which was quickly followed up Hiroshi Nakatsuji and then by David Mazziotti. They have devised an

effective method of calculating the energy levels, which I have dubbed the VNM method and which has been described [7] as “wave mechanics without wave functions”. For atoms and molecules with as many as 20 electrons, the VNM method competes favourably with FCI calculations of equal accuracy. Since RDM’s are initial values of Green’s Functions, a similar condition must be satisfied by GF’s. This has been generally unnoticed until quite recently and still has not been really absorbed by main-line physicists. However, the book [8] by Parr and Yang about Density Functional Theory (DFT) contains an early Section pointing out that N-representability is a dark cloud hovering over the validity of DFT. The usual methods of dealing with this problem is either not to be aware of it or to hope that it will go away! The latter method is not satisfactory. For small N, my experience with Li shows it is risky; whereas, for large N, the theorem of Hugenholtz [9] that, for an interacting system in the limit as N gets arbitrarily large, a single Slater determinant is orthogonal to the true wavefunction is rather dramatic. Perhaps this was in Froehlich’s mind [2] when he spoke to David Peat! BCS, MATTHIAS AND ALL THAT

I would be the first to admit that the BCS theory has been extraordinarily successful, making a contribution of immense value to Condensed Matter theory. Even so, as with the remarkable success of a single Slater determinant, I have always been amazed how the original BCS simple theory, managed to change and persist so long. In the cold light of current knowledge we now realize that the simple BCS theory had only two essential ingredients (i) The choice of a trial wavefunction formed with the same material as is needed to characterize one antisymmetric 2-particle function, or geminal. 25

(ii) An extremely simple ansatz for the potential as a step function exercising a positive attraction between electrons with energy close to the fermi energy.

Of these, I certainly consider (i) as more important. The BCS wavefunction is a Fock space equivalent of the wavefunction considered by Schafroth [10] and which, by a stroke

of luck, is as Yang proved [3], the type of wavefunction most likely to give rise to a large eigenvalue of the 2-matrix. To my mind this is the explanation of the success of the BCS

model. As for the nature of the force involved, we were told that the positive isotope effect definitely proved that it was phonon-mediated. So in my innocence, as a naive mathematician when a negative isotope effect was observed, I immediately inferred that this proved that the force could not be phonon mediated. But no! Since by this time the idea that the force was phonon-mediated had become firmly implanted in our collective consciousness, it was soon “proved” by an able theoretician that a negative isotope gave us even added evidence of our - by now - blind faith that the force was phonon-mediated! Also the myth was firmly established that “Cooper pairs” consist of two electrons with opposite spins.

Apparently

many phyicists still believe that this is essential to BCS theory. As suggested in Chapter 4 of

CY, this is not necessarily the case. For every new observation that contradicted the currently accepted theory our faith was saved by a small add-on or by a major or minor modification of the current formulation. The evolving BCS theory became more and more complex and subtle. But at that period, during

which I had the rare privilege of meeting and challenging Bernd Matthias every winter at Sanibel until his death, I developed the feeling that BCS theory had become, like Ptolemaic astronomy, a system of epicycles piled on epicycles! Bernd proudly proclaimed that he was anathematized by all theoretical physicists because for every new version of the theory proposed, he would go into the Bell Lab and emerge with a counter-example! Perhaps because he was a polite Swiss being kind to a Canadian or perhaps because he took pity on me as an innocent mathematician wandering among chemists and physicists, he carefully stroked my ego by stating that the ideas re. superconductivity that

I advanced were not contradicted by any known observation. I will pursue this below! However, in private conversation and in his lectures [11] at McGill in 1968, Matthias insisted that the truly interesting theoretical question is why do nearly all substances manifest

a form of Long Range Order (LRO) at sufficiently low temperature. He asserted that even gold would become a superconductor! I very much regret that he died before the discovery of HTSC. He would have so much fun bating theoreticians re. “anyons”, “RVB” and the other exotic ideas that have been bruited! When I asked John Harrison, the former Editor of JLTP and my colleague in Physics at Queen’s, to explain Matthias’s observation, his response was immediate. “It’s really not so surprising. At absolute Zero the entropy will vanish so we should expect total order”. Indeed, this is true and proves that my knowledge of thermodynamics is almost nil or I would have made this point to Matthias. So the interesting question becomes, not why there is LRO, but rather why is the LRO of the nature that actually occurs in a particular substance? I do not pretend to have a detailed answer to this question. I do think that I offer the basic set of the ideas essential for its answer. Another beef that I have with current physics practice is the error which Whitehead [12] calls “The Fallacy of Misplaced Concreteness” exhibited in such terms as p-electrons or Cooper pairs. If you understand the meaning of the word fermion or if you believe in democracy you know that all electrons are equal. They do not live in George Orwell’s Animal Farm

in which “all electrons are equal but some are more equal that others”. Only occasionally, have I noticed momentary indications of a bad conscience by chemists or physicists about this misuse of language. If challenged, as I am doing now, they excuse themselves with the same remark Bourbaki often uses “This is merely an innocent abus de language”. Whereas, I regard it as a noxious avoidance of our proper task of instilling in the minds of students a

26

set of valid concepts with which to explore the inner riches of Quantum Theory. It is not an electron which has p-symmety but a spin-orbital. In fact, it is a partially occupied eigenfunction of the 1-matrix! There are no such things as Cooper pairs, even if we think of them in the charming image, due I understand to Schrieffer, as partners dancing to Rock so that they can be at far ends of the floor yet fully synchronized, rather than breastto-breast in a gentle Strauss waltz. To even propose such an image almost makes the concept absurd. The functional unit is not a pair of electrons it is a spin-geminal. In fact, the key concept which we must learn to deploy is that of a partially occupied eigengeminal of the 2-matrix. CORRELATION MATRIX AND ORDER INDICES

I read somewhere that the nobellist, C.N. Yang, regarded the paper [3], in which he associated the onset of superconductivity with the appearance of a “large” eigenvalue in the 2-

matrix, as the most important paper of his distinguished career. My initial conjecture [13] was the obvious generalization of his observation and asserts that every type of LRO is associated with a large eigenvalue ot the 2-matrix. This was refined [14] by the definition of order indices and the correlation matrix. It is known that the least upper bound for the eigenvalues of the 2-matrix of a boson system is N(N – 1) and for a fermion system [15] the unattainable such bound is N. If we call the occupants of geminals pairons then we can say that the l.u.b. for the pairon occupation of a natural geminal is N(N – 1) for a system in which the constituent identical particles are bosons and N for a system if they are fermions. If a “Cooper pair” is anything it is a “pairon”. But the term pairon is more general and is not necessarily associated with superconductivity. Yang argued that superconductivity in a metal is triggered when

has an eigenvalue

of order N. Bloch [16] connected such an eigenvalue with flux quantization related to carriers with charge 2e, confirming Yang’s theory. From energy considerations sketched below, I inferred that eigenvalues of proportional to N were associated with eigengeminals describing a correlation which extends throughout the substance.- in other words, a Long Range correlation. The order index was then defined [17] as the largest value of such that has a finite non-zero value, in the thermodynamic limit. The correlation matrix is defined as By the above-mentioned [14] result, for systems of fermions For one or more eigenvalues of is proportional to N so LRO is present. For systems of bosons, could be as large as 2. As long as we conjecture that some form of mesoscopic [17] or local order is present. If we compare this with a percolation model for the onset of a new

phase of matter, cluster is infinite,

corresponds to the critical value of p when the diameter of an open close to 0 corresponds to the first moments at which nuclei of the new

phase are present. As

increases these nuclei become more widespread and larger. I am

thinking of the small bubbles becoming more widespread and larger which appear in water as it approaches the boiling point, or the complex systems of “fjords” of superconducting phase penetrating the whole of a cylindrical block of material which I had the privilege of viewing via polarized light as it was cooled by Martin Edwards in his Low Temperatre Lab at the Royal Military College of Canada many years ago. CONJECTURE. For all mono-particle systems, the appropriate order parameter (OP) is the correlation matrix From the structure of the 2-matrix it immediately follows that the order parameter can have spin character s, p or d and any combination of these. This is consistent with recent observations [18] that the order parameters of some HTSC’s exhibit s, p or d spin-symmetry or a combination of these - a phenomenon, which, apparently, BCS has difficulty accommo-

27

dating. My conjecture is that is the appropriate order parameter for all types of order in many-particle systems of one type of identical particles. Thus, I am making a bold generalization of Yang’s observation from superconducting to many other order transitions. I am encouraged by the fact that this conjecture is consistent with observations on the symmetry of the OP for HTSC and also for He3 in which p-type order occurs in at least one phase. I assume that for helical magnetism and many other types of order it will be necessary to study not only spin-symmetry but the total symmetry of the eigengeminals. I have called the above a conjecture rather than a theorem because a “proof” has not been obtained. This is because we do not yet have a sufficient understanding of the relation of the eigenvalues of to the occupation numbers of eigenstates of the reduced hamiltonian K. I regard this as an important urgent issue for theoretical research. Another is to explore the

dependence on the Order Index, of the physical properties of substances near the critical point. Note that if fermi-pairons were bosons, their occupation numbers could go to N(N – 1). This differs from the actual limit of N by a factor of (N – 1) which is infinite in the thermodynamic limit. Thus the universal practice in text-books, and in articles by writers who should know better, of saying that superconductivity arises as a result of a bose condensation of pairs is misleading talk which brings comfort, by creating the illusion that we know what we are talking about, but prevents us from coping with the real task of forging a set of meaningful concepts with which to understand condensed physics.

It is known that when Fock space is displayed with respect to a basis of a finite number, r, of orthonormal orbitals (i.e. 1-particle functions), the highest possible value for the eigenvalues n2i of is This is attained(CY,Chapter 3) only if the wavefunction is an antisymmetrized power of a single geminal - an AGP function and if the eigenvalues n1i of are equal. In this case(CY, p. 137) there is one large eigenvalue and the rest are equal to 2 N ( N – 2 ) / r(r–2). If we relax the condition that n1i be equal, it is possible to arrange that for an AGP function has several eigenvalues which are proportional to N. and thus model a variety of other situations including the co-existence of superconductivity and magnetic ordering. It is perhaps worth recalling here that r = N is a necessary and sufficent condition that

the wave function be a Slater determinant. In this case all the eigenvalues of are equal to 2 which is a long way from N. This corresponds to the fact that HF is accurate if and only if the effective hamiltonian has no 2-particle terms. We introduce some essential notation by recalling that in CY. Denote the eigenfunctions of Dp by

with corresponding eigenvalues

so

Setting

we obtain

and

28

Further, if the reduced hamiltonian (10) has eigengeminals, gi, such that then the total internal energy

where

Suppose that the are so numbered that they increase monotonically with i, and the numbering of n2i so that they decrease. Then in the ground state the system will choose so that pi for small i, and especially for i = 1, are as large as possible consistent with N-representability. The largest occupation of a natural geminal is n21. By the familiar theory of separation of eigenvalues of hermitian operators, 2 (N – 1)

In particular,

2(N – 1) p1 = n21 if and only if the eigenfunction g1 of K coincides with the first natural geminal, of the state. Exact coincidence is highly unlikely, but there will be a strong tendency towards this so it is possible that p1 will be of order N. In this case, we would expect that g1 describes a 2-particle correlation which extends throughout the sample, that is a LRO. For N electrons in a lattice if we neglect spin, we are led to study the hamiltonian

where i and j refer to electrons and k to nuclei; Z k is the charge on an ion at sk. For neutral systems This implies, in the notation of (8), that

Notice that, though we mentioned electrons in a lattice, if k assumes only one value, (23) would describe the hamiltonian of an N-electron atom with nuclear charge Z k = N, whereas, if k takes two values, a diatomic molecule. And so on. In fact, almost anything. By (10), associated with (23) is the Reduced Density Operator, K. However, for reasons which will become apparent, we introduce an additional parameter, t, and define K (t) by

If we divide by N2, set then (25) takes the form

and N2U (t) = K(t), and replace Nr i by ri and Nsk by sk,

29

For a neutral system,

For fixed N the spectrum of U(t) will depend continuously

on t. A famous theorem [19] of Zhislin assures us that when

the operator

(26) has an infinite number of bound states with energy levels crowding up to the limit of the continuous spectrum.

U (0) is a two-electron hamiltonian which approaches the hamiltonian of H – as N increases to infinity. On the other hand, for t = 0, and N = 2, (25) is the hamiltonian of the helium atom. According to (18) and (22) it is the spectrum of K = K(1) which is of real

interest in the study of energy levels of N-particle systems. Since K = N2U (l), it follows that the spectrum of K is obtained from that of U (1) by scaling by the factor N2.

It is known [20] that H– has only one bound state. It is a 1S state slightly below the continuum which accounts for an absorption line in the solar spectrum. The two lowest states of the helium atom are a 1 S and a 3S state. For a fixed system, (25) depends continuously on t so we expect that as t varies from 1 to 0 a correspondence will be established between the spectra of U (1) and U (0). However, while U (0) is an atomic hamiltonian, we shall expect the spectrum of U ( t ) , when t > 0, to be a series of bands, possibly narrow, each of which collapses, when and which could be named by an energy level of the atomic system which U (0) describes. If spin is neglected then it would be reasonable to expect that all levels of the lowest band would be 1 S.

In this case, for a system manifesting long-range order at low temperature, we would anticipate that the correlation matrix will depend on the eigenfunctions corresponding to the levels of the lowest band weighted by a distribution function depending on the inverse temperature, Unfortunately, little study has been made of the spectrum of K for solids or other condensed matter even though the fact that it must play a key role in understanding the energetics of condensed matter has been obvious for forty or fifty years. We noted above that the late Bernd Matthias, who probably discovered more superconductors than any three other experimentalists together, constantly insisted that an important

task for theoretical physics was to explain why nearly all fermion systems manifest longrange order of some type at sufficiently low temperatures - superfluidity, superconductivity, ferro- or antiferro-magnetism, charge density waves, coexistence of superconductivity and helical spin density waves, etc. To properly describe the electrons in condensed matter, our hamiltonian (23) would need to be supplemented by terms describing L·S coupling, spinspin effects, motion of the ions etc. However, the electric forces described by (18) would probably dominate the energy. If in fact the spectrum of K is similar to that of H— in having one eigenvalue, or a band of eigenvalues significantly below all others, then that level would tend to be occupied as fully as possible consistent with the statistics, the inter-particle forces and the temperature. For fermions, n21 could be of order N which, if it occurred, would manifest itself as long-range order. The nature of the particular LRO would be characterized by the correlation matrix. ANTISYMMETRIZED GEMINAL POWER

A theorem attributed [21] to Zumino states that a fermion geminal, in other words, an antisymmetric two-particle function, can be transformed by a unitary transformation into a canonical form in which each orbital is a member of a unique pair of orbitals. The reader should be aware that it is my custom to denote by the word orbital a function of a single particle including all relevant coordinates. Thus, depending on context, the word may denote the classical meaning of a chemist(if the particle is without spin), or what a chemist means by spin-orbital, or a function of spatial co-ordinates and two dichotomic variables for spin and isotopic spin. Thus if

30

is such that the normalized function g (12) = – g (21), with then r = 2s is even, and by a unitary transformation it is possible to find an orthonormal basis αi with respect to which

In (27), r is the rank of g and also the rank of the matrix c ij. It is well-known that the rank of an antisymmetric matrix is even. The antisymmetrized power of an orbital is always zero. Thus f(1)f(2) – f ( 2 ) f ( 1 ) = 0. However, the antisymmetrized power of a geminal, g, to obtain a function of N particles will vanish if and only if the rank, r, of g is less than N. When N = r, this N-particle function is a single Slater determinant formed with a basis of g. Perhaps inadvisedly, I have adopted the symbol gN to denote a normalized N – particle function obtained by antisymmetrizing an appropriate power of g. Several persons, of whom Nakamura [22] may have been the first, showed that the projection of the BCS function(which is a coherent ensemble in Fock space of functions of all possible particle number) onto a subspace of Fock space of particle number N produces an AGP function of rather special type. The importance of the AGP function is signalled by the fact that it has appeared in a variety of contexts with different names such as: Schaftroth condensed pair function; projected BCS function; correlated pair function; pairiing function; Generalized Hartree-Fock function. The mathematical concept goes back to Hermann Grassmann in the 1840’s since it arises naturally in Grassmann algebra. I prefer to use a name which suggests its mathematical nature and does not place it in a misleading context. For applications to physics and chemistry there is no need to insist that the occupants of a geminal are particles with opposite spin. Any kind of fermion geminal forces a natural pairing. Therefore it can be cogently argued that the apparent “pairing” in BCS is not forced by the physics but rather appears as a mathematical artifact forced by the assumption that the wavefunction is AGP. I realize that there is such a widespread commitment to the religious

belief that Cooper pairs are “real” that there is a high probability that I shall be accused of blasphemy, tried, condemned and burned at the stake!! Since the whole of Chapter 4 of CY is devoted to AGP, here I shall restrict myself to quickly mentioning what every young person should know about Grassmann algebra and fermions. 1) If is an N-particle fermion function and is an orbital such that the Grassmann product then there is an (N – l)-particle function, such that Further, these equivalent conditions are necessary and sufficient that be a natural orbital of with occupation unity.

2) Suppose that an AGP function formed from an arbitrary geminal, g, then if r is the rank of a) r < N implies that b) r = N implies that is a Slater determinant, c) r > N implies that can be expressed as a linear combination of ( ) Slater determinants consisting only of paired orbitals, where r = 2s, and N = 2m. 3) If N is even and the natural orbitals of

are evenly degenerate, with occupation

strictly less than unity, then there exists a geminal g such that This remarkable result, proved around 1965 by Erdahl, Kummer and myself, implies that for a manyparticle fermion state satisfying these conditions, all one-particle properties can be exactly described by an AGP function.

4) Further, suppose that N = p + q where q is even and precisely p natural orbitals have

occupation unity, then where S is a Slater determinant containing the indicated p natural orbitals and g is a geminal. Such functions have been called Generalized AGP functions.

31

5) On p. 139 of CY we indicate that an AGP function might have the possibility of having a 2-matrix with a finite number of “large” eigenvalues. It is therefore conceivable that observed co-existence of superconductivity and helical magnetism could be modelled by GAGP. 6) If N is even and if is of rank N + 2, then by making use of the so-called Hodge Correspondence between subspaces of dimension 2 and those of dimension N in a space of dimension N + 2, we can prove that is an AGP function. As the “cranking model”, AGP proved useful in nuclear theory. At first a theory with the fanciful sobriquet “superconducting nuclei” was introduced using the BCS coherent ensemble

equation subject to a condition that the expected value of the number operator be N. However, Nogami and others soon noticed that it was more accurate to use a projected BCS function, that is an AGP function. Chemists also found that AGP, as an ansatz for the wave-function, was more successful in modelling the dissociation of diatomic molecules than Haertree-Fock. It was observed that the Random Phase Approximation is self-contradictory i f , as is common, a single Slater is taken as the initial ground state, whereas the most obvious contradictions are avoided if the ground state is assumed to be a Generalized AGP.(For this, cf. p. 140 of CY). In view of these properties and the fact that GAGP can be a single Slater modeling a fermion system with no correlation or, on the other hand, a system with a with the largest possible eigenvalue and therefore modelling the highest possible correlation, it is apparent that the GAGP ansatz is of great scope and could be used to provide insight into a wide variety of fermion systems.

GRASSMANN AND THE FERMI SURFACE I come now to a little-known theorem of Grassmann for which I will present a proof, partly because I do not want to disappoint your expectation that proving theorems is my main

purpose in life, as a mathematician, but also because the result is unexpected and may be the “real” reason why we must replace “fermions” by “fermi pairons” in our thinking and

therefore the ultimate reason that “Cooper pairs” proved so serviceable. I announced this result [23] without giving the proof in 1961. In fact the proof was rather easy making use of the Hodge correspondence between sub-spaces of dimension p and those of dimension n – p in a linear space of dimension n. With a more complicated proof, the same result was proved later in the RMP by a theoretical physicist, but I have lost the reference. In fact, Whitehead [24] provides an almost trivial proof, attributing it to Grassmann - presumably from the 1840’s! Here in two equivalent forms, first Grassmann’s and secondly mine, is the

Theorem (i) A homogeneous element of order n in a Grassmann algebra of rank n + 1, is elementary. (ii) If is a pure N-particle state, then the rank of is not N + 1. Proof. I shall state the argument in the language of antisymmetric wavefunctions. Suppose the basis has N + 1 orbitals. Associated with a Slater determinant of order N is a unique subspace of dimension N spanned by the N vectors of the determinant or by any N linearly independent vectors in the same subspace. Changing these vectors does not change the subspace but may multiply the Slater by a constant. Suppose that is a linear combination of two Slaters. By a basic theory about subspaces, the dimension of the intersection of the subspaces associated to the two Slaters is N + N – (N + l ) = N – l. This intersection is common to both subspaces and is characterized by a Slater S. of order N – 1. Adjoin vectors and 32

to the intersection so that S and are N-th order Slaters respectively characterizing the two subspaces Then there are constants a and b such that

which is a single Slater of rank N. By induction we see that any linear combination of Slaters

of order N, in a space spanned by N + 1 orbitals, is again a Slater(i.e , in the language of Grassmann algebra, “elementary”) of N orbitals. It follows easily that version (ii) is implied by version (i) of the statement of the theorem. Hence if is not a Slater it must have rank at least N + 2. But it could have that rank as follows from Item 6 of the previous Section. The discussion of the energy of an AGP state in Section 4.6 of CY was used to estimate the change in energy if a Slater state of rank N is changed to a state of rank N + 2 by replacing two orbitals each of occupancy 1 by four orbitals each with occupancy 1/2. It was found (CY, p.155) that the change in energy of the state was

where κ and name distinct “pairs” of orbitals. The number denotes the interaction energy where and and has fixed phase. Whereas, denotes the interaction energy and has adjustable phase which was used to arrange the negative contribution in (29). Thus the Fermi surface is unstable with respect to pair formation unless is positive and numerically greater than If the so-called “pairing hamiltonian” is used, automatically so that for the pairing hamiltonian, the Fermi surface is always unstable. This seems to contradict the commonly expressed view that the

existence of superconductivity requires an attractive force which was part of the rational for the existence of Cooper pairs.

It is now widely recognized that HTSC is usually associated with phase separation. In the next section we find that a sufficiently strong repulsive Coulomb force is required to account for phase separation. PHASE SEPARATION AND SUPERCONDUCTIVITY

In his well-known survey [25], published in 1989, of the properties of HTSC, Phillips has 11 references to “Lattice instabilities”, a topic to which he devotes several pages at various points in the book. He even went so far as to suggest that lattice instability is the only factor that causes HTSC. I do not admit this since in 1991 Yukalov [26] surveyed 508 experimental and theoretical papers which dealt with evidence bearing on the incidence of phase transitions of what he called “heterophase fluctuations”. Yukalov, who is a Senior theoretical physicst in the Joint Institute for Nuclear Research in Dubna, is a remarkably competent and careful authority on Quantum Statistics . He agrees that instability of the lattice can be important but there are other significant factors. During the past ten years more evidence [26] has accumulated similar to the impressive collection which he assembled. In his basic paper referenced above, are foreshadowed many ideas that have recently become current. Chapters 5 and 6 of our book were laregly due to Yukalov since I know so little of the nitty-gritty of physics. In particular this is the case for Section 6.2 of CY in which we attempt to work out in a form relevant to HTSC the theory developed in his paper [26] when there are only two phases interpenetrating. Here I merely sketch the course of our argument directing the interested reader to Section 6.2 of CY and the references in Notes 2 and 3. We posit a situation which can be thought of as microsopic or mesoscopic nuclei of one phase (e.g. “superconducting”) scattered randomly through a host phase (e.g. “normal”) Experimental observation of this possibility was recently provided by a group from Dubna 33

with associates in a paper [28] entitled “Microscopic phase separation in

induced

by the superconducting transition”. We propose a simple model which takes into account the three interrelated factors: Coulomb interaction, phase separation, and lattice softening. We give a detailed analysis of the dependence of the critical temperature on parameters related to the attractive and repulsive interactions and to the superconducting phase fraction w.

Since the Hartree-Fock-Bogolubov - essentially AGP - approximation is used, our formulas look very much like those in usual presentations of BCS theory.. However, they have quite different meaning because they involve the parameter w in an intricate manner.We assume that the interaction between electrons is the sum of two components

- a direct part which is taken as a Debye-type shielded Coulomb force. and - an indirect part for which we assume the conventional Froehlich phonon term. Additional parameters are introduced by which it is possible to vary (i) the phonon

frequency, (ii) electron-phonon coupling, and (iii) the strength of the direct interaction. The resulting equations were solved numerically by Dr. E. Yukalova. The results are exhibited in CY as twelve graphs portraying the superconducting critical temperature against w, in three

groups corresponding to weak, moderate and strong softening of the lattice. We were pleasantly surprised by the wide variety of shapes of these graphs. Despite the rough approximations assumed for our model, the behaviour of the critical temperature in Figs. 3,4,7 and 8, has striking similarity to some corresponding experimental curves observed for HTSC. Even though one might expect the relation between doping intensity and w to be monotone, the actual relation is not known so a detailed comparison of our results with those of experiment is not possible. We were led to the following conclusions: - The presence of repulsive interaction is a necessary condition for mesoscopic phase separation. - Phase separation favours superconductiviy making it possible in certain het-

erophase samples when it would not occur in a pure sample. - The critical temperature as a function of the relative fraction of superconductive phase can exhibit the nonmonotonic behaviour characteristic of HTSC. FINAL REMARKS

1) I conclude that we need to develop a habit of thinking more comfortably about the second order reduced density matrix its eigenvalues and its eigengeminals. 2) For a system of a large number of identical particles which is all that I discussed,

the “large” component of the 2-RMD, denoted by parameter. If

is proposed as the appropriate order

there is no “order” present so this could correspond to what we usually

call “normal”. For fermion systems Long Range Order corresponds to

and for bosons,

to I have not expatiated on my conviction that neither bose nor fermi condensation, as normally understood, in the simple-minded sense derived from London(whose memory I honour!), actually occur in an interacting system and that we poison the innocent minds of our students if we persist in suggesting that they do. 3) More imporrtant, it is my view that because for the earliest discovered superconductors, Tc was so low and the isotope effect had a simple explanation, we were misled into thinking that the origin of superconductivity is exotic and/or subtle. However, I take seriously Matthias’s observation that LRO at sufficiently low temperature is universal and therefore should have a robust explanation.. Further the behaviour of various substances near the 34

critical temperature seems to have much in common. When charged particles are involved, I conclude that Coulomb forces are the real culprit. So I claim that the secret for this universal phenomenon is to be found in the second order reduced hamiltonian. The isotope effect implies that interaction with the lattice must play a role. My musings at the end of Section 3, suggest that contributions by the static coulomb interaction, specified by Zhislin’s theorem, involve quite minute energy differences for K jbetween the continuum level and a narrow energy band which is almost at the continuum limit. This means that lattice dynamics, L. S coupling and other spin-effects could also play a significant role accounting for the known variation of Tc across the Periodic Table which was noted by Matthias [11] in his McGill lectures. 4) I fully realize that some of my heterodox opinions are anathema to many. I shall try to face this with the equanimity of old age, welcoming all comments, questions and counterexamples at my email address on the title-page.

NOTES 1. 2.

Charles Coulson, in conversation with graduate students and Coleman in Oxford June 1975. Also in a speech in Boulder, Colorado: Rev. Mod. Phys. 32, 175 (1960). H. Froehlich, shortly before his death, in private conversation with David Peat.

3.

C.N. Yang, Rev. Mod. Phys. 34, 694 (1962).

4. 5. 6.

By A.J. Coleman and V.I. Yukalov, Vol 72 in Series published by Springer in Lecture Notes in Chemistry, April, 2000. P.A.M. Dirac, Proc. Cam. Ph. Soc. 26, 376 (1930); 27, 240 (1931). K. Husimi, Proc. Phys. Math. Soc. Japan 22, 264 (1940).

7.

Section 7.3 of CY.

8.

9.

R.G. Parr and W. Yang, Density Theory of Atoms and Molecules, Oxford University Press, 1980.

N.M. Hugenholtz, Physica 23, 481 (1957); L. van Hove, Physica 25, 849 (1958).

10.

M.R. Schafroth, Phys. Rev. 96, 1149, 1442 (1954); 100, 463, 502 (1955); 111, 72 (1958).

11.

B. Matthias, Three Lectures, in Superconductivity, Proc. Ad. Summer Study Institute, June, 1968, at

12.

McGill University, ed. P.R. Wallace, Gordon and Breach, New York. A.N. Whitehead, p. 64 Science and the Modern World, Cambridge U.P., 1933; p.11 Process and Reality,

13. 14. 15.

Macmillan Comp.,1929. A.J. Coleman, Can. J. Phys. 42, 226 (1964). A.J. Coleman, V.I.Yukalov, Nuovo Cimento B 108, 1377 (1993). A.J. Coleman, Rev. Mod. Phys. 35, 668 (1963).

16.

F. Bloch, Phys. Rev. A 137, 787 (1962).

17. 18.

A.J. Coleman, Jl. Low Temp. Phys. 74, 1 (1989). H. Srikanth et al., Phys. Rev. B 55, R14 733 (1997); K.A. Kouznetsov et al., Phys. Rev. Lett. 79, 3050 (1997); in Physica C 317-318, 410 (1999), van Hartington claims “unambiguous determination” of d-wave symmetry in HTSC cuprates; in Nature 396, 658 (1998), Ikeda et al. observe p-wave symmetry in a second HTSC. G.M. Zhislin, Trudi Mosk.Mat. Obsc. 9, 81 (1960), Th.III, p.84. R.N. Hill, J. Math. Phys. 18, 2316(1977).

19. 20.

21.

B. Zumino, J. Math. Phys. 3, 1055 (1963); see also Thm.6, Coleman, Bull. Can. Math. Soc. 4, 209 (1961).

22. 23. 24. 25.

K. Nakamura, Progr. Theor. Phys. (Kyoto) 21, 273 (1959). A.J. Coleman, Can. Math. Bull. 4, 209 (1961), Thm.7. A.N. Whitehead, Universal Algebra, Cambridge University Press, 1898. J.C. Phillips, Physics of High-Tc Superconductors, Academic Press, 1989.

26.

V.I. Yukalov, Phase Transitions and Hetrophase Fluctuations, Physics Reports 206, 395–488(1991).

27. 28.

Phys. Rev. B 54, 9054 (1996); Phys. Rev. Lett. 76, 439 (1996). V.Yu. Pomjakushin et al., Phys. Rev. B 58, 12 350 (1998).

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THE SIXTEEN-PERCENT SOLUTION: CRITICAL VOLUME FRACTION FOR PERCOLATION

RICHARD ZALLEN Department of Physics, Virginia Tech Blacksburg, VA 24061

INTRODUCTION The English call it “value for money” (vfm). The American equivalent is “bang for the buck”. The idea is simple: to provide a rough measure of the ratio of benefit to cost. For an author of scientific papers, one possibility for a vfm-type measure of “benefit” (impact) to “cost” (time and effort) is this: vfm = (number of citations)/(paper’s length in printed pages). In my case, the vfm winner is clear. It is a two-page paper by Harvey Scher and myself, published quietly as a note in J. Chem. Phys. [1], which has been cited over 350 times. Later work related to the central idea of that paper has also been widely cited [2, 3]. That idea is the concept of a critical volume fraction for site-percolation processes. NOSTALGIA One afternoon in mid-May of 1970, at my desk in the research building of the Xerox complex near Rochester, NY, I was poring over experimental Raman spectra, searching for significant peaks with my “spectroscopist’s eye” [4]. I was not having much luck, and I needed a break. So I left my office, walked down the hall, and went into the office of a colleague, Harvey Scher. Harvey was, as usual, good-natured and patient about the interruption of his own work, and he took the opportunity to describe an interesting problem that he was working on. A very approachable resident theorist, Harvey had been consulted by a technology group working on photosensitive layers in which photoconductor particles were dispersed in a resin. Their measurements had shown a dramatic threshold in the dependence of photosensitivity on photoconductor concentration. Elliott Montroll, then a frequent visitor to Xerox, had suggested to Harvey that he look at the literature on percolation theory. Harvey had assimilated that literature and made use of it, and he introduced me to percolation theory that afternoon. I was fascinated by this stuff,

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

37

and when I got back to my office I did not return to the strip-chart recordings (no PC’s in 1970). Instead, I worked on some geometry problems related to ideas that we had kicked around, and I became enthusiastic about writing up a short paper reporting what we had found. Ten days later the paper was circulated internally within Xerox, and it was submitted for publication in mid-June. This speed was then, and is now, uncharacteristic of both authors. The reason for the choice of J. Chem. Phys. is somewhat obscure. We did not want to send it to a math or math-phys journal, and we had seen a short paper in J. Chem. Phys. that mentioned percolation. It turns out that the referee for our paper was almost certainly a mathematician! He (or she) chided us for the empirical and approximate nature of our

critical density. (We knew it was approximate, and we were proud of “empirical”!) But he (or she) nevertheless pointed out to us an additional result for (an exact value for the two-dimensional Kagomé lattice) which fit our ideas very well. We added it to the Table in

our paper. The anecdote described above, dealing with the fruitfulness of an afternoon schmooze session at the Xerox lab in Webster, NY, was characteristic of a period now remembered by some as a “golden age” of industrial research [5]. The scientific issue arose in the context of a technological setting, which is of course a familiar tradition in condensedmatter physics [6]. The atmosphere was one in which it was OK to spend time on scientific

issues as well as on product-development and engineering ones. In the year 2000, that era is history and opportunities to do science are rare in present-day corporations. Globalization is sometimes given as the reason (or excuse) for this, but human herd-instinct considerations also enter: Everybody did it then (corporations supported research) because everybody else did it; nobody does it now because nobody else does it. This is a

cooperative phenomenon, so perhaps we can hope that a phase transition can happen again. CRITICAL VOLUME FRACTION In three dimensions, the percolation threshold

for site-percolation processes varies

from lattice to lattice by more than a factor of two [7]. For two-dimensional lattices,

varies by more than a factor of 1.5 [7, 8]. The Scher-Zallen construction for the critical volume fraction associates with each site a sphere (or circle, in 2d) of diameter equal to the nearest-neighbor separation. Spheres surrounding filled sites are taken to be filled. At the critical value of the site-occupation probability p , the fraction of space occupied by the filled spheres is taken to be the critical volume fraction The key point is this: From lattice to lattice (in a given dimensionality), is nearly constant, varying by just a

few percent. It is an approximate dimensional invariant. In three dimensions, 0.16; in two dimensions, is close to 0.45 [1, 3]. The relationship between and is

is close to

where f is the filling factor of the

lattice when viewed as a sphere packing. The values forming the basis of the 1970 paper correspond to familiar crystal structures. A structure that is not crystalline but is experimentally well defined is random close packing (rcp). The rcp structure corresponds to the atomic-scale structure of simple amorphous metals [3]. Since f is known for the rcp structure, predicts the value for this structure. Experiments carried out to determine the conductivity threshold (insulator-to-metal transition) of rcp mixtures of insulating and metallic spheres are in good agreement with this prediction [9, 10, 11]. One way to view is as an expression connecting the ease-of-percolation with the connectivity of the underlying structure. For bond percolation, such an

38

(approximate) connection had been found earlier, in 1960 [12, 13]. A reasonable measure

for the ease-of-percolation for a given structure is (1/ p c) , the reciprocal of the percolation threshold. For bond percolation,

(1/p c) is very close to (2/3)z in three dimensions and

(l/2)z in two dimensions. Here z is the average coordination number of the lattice. The proportionality between ease-of-percolation and coordination number shows that, for bondpercolation processes, the coordination number is the appropriate measure of the connectivity of the lattice. This, of course, makes sense. But for site-percolation processes, z does not work. Instead, shows that (1/p c) is proportional to f. This reveals that, for site-percolation processes, the sphere-packing filling factor is the appropriate measure of the connectivity of the underlying structure. This insight is a byproduct of the work on the critical volume fraction.

LIMITATIONS

Thanks to the piece of information provided by the unknown referee, we knew immediately that is only approximately invariant. The site-percolation threshold is known exactly for two two-dimensional lattices, the triangular lattice (2d close packing) and the Kagomé lattice, so that

is exactly determined for each. The two values differ by

2%. A few people in the critical-phenomena community took an instant dislike to It wasn’t exact. It wasn’t rigorous. It wasn’t even an exponent, so why care about it? [One can imagine one of them having the following reaction to the experimental discovery of a new superconductor: “So Tc is 450 K, so what? What are the exponents?” But maybe that’s unfair.] The value of

can be estimated from a plot of (1/p c) versus f [3]; the slope is

Here a question arises at the low end of the plot, where the proportionality

between the ease-of-percolation and the filling factor has to eventually fail because (1/ p c) cannot be less than 1. This consideration is unimportant in three dimensions in which (1 / pc ) does not closely approach unity; the values cluster in the region from about 2.3 to 5.0. In two dimensions, typical (1/p c) values are closer to 1.0, lying between 1.4 and 2.0. Within this region, the proportionality of (1/p c) to f holds very well [1]. However, Suding and Ziff [8] have recently considered very-low-connectivity two-dimensional lattices with (1/p c) values down to 1.24. Their results show that at these very low connectivities, the deviation from becomes appreciable. Suding and Ziff offer a revised, nonlinear relation between pc and f that improves the fit in the very-lowconnectivity region. Most structures of physical interest are far from this region. APPLICATIONS The notion of a critical volume fraction insensitive to the details of local structure, as

suggested in the 1970 paper, is an attractive one. But it is heuristic, empirical, approximate. It had been my original plan for this paper to review its success (or failure) in relation to experimental literature on metal/insulator composites. This has turned out to be too

mammoth an undertaking for the presently available space and time, and will have to be deferred. The experimental literature is vast; one extensive compilation can be found in a 1993 article by Ce-Wen Nan [14]. The experimental studies span an enormous variety of systems and differ greatly in depth and quality.

39

Figure 1. The conductivity threshold in graphite/boron-nitride composites [19].

At a later time I may attempt a plot of frequency-of-occurrence versus value, but here only some less-than-satisfactory observations will be offered. For three-dimensional

composites a value close to 0.16 is very often encountered, and it is interesting that this occurs for some of the most carefully studied systems. Examples are the carbonblack/polymer composites studied by Heaney and co-workers [15, 16, 17] and the graphite/boron-nitride composites studied by Wu and McLachlan [18, 19]. Figure 1 displays the very clean experimental results of Wu and McLachlan, showing a conductivity threshold spanning many orders of magnitude. Graphite and boron nitride are structural

and mechanical isomorphs, but differ in conductivity by a factor of 1018. The points are

measured values; the curves are scaling-law fits that closely determine

(0.15 for this

system). But there are many systems for which is quite different from 0.16; this value is not universal. The reason is unclear, though different classes of topology have been suggested. One of these is the “Swiss-cheese” void-percolation topology analyzed by Halperin and coworkers [20] and studied experimentally by Lee et al. [11] ACKNOWLEDGMENTS

I wish to thank Wantana Songprakob for crucial help in preparing this paper. I also wish to thank Harvey Scher for thirty years of friendly interaction.

40

REFERENCES 1. 2.

3. 4.

Scher, H. and Zallen, R. (1970) Critical density in percolation processes, J. Chem. Phys. 53, 3759. Zallen, R. and Scher, H. (1971) Percolation on a continuum and the localization-delocalization transition in amorphous semiconductors, Phys. Rev. B. 4, 4471. Zallen, R. (1998) The Physics of Amorphous Solids, John Wiley and Sons, New York. pp. 183-191. I first heard this apt term mentioned in a talk given by Manuel Cardona.

5.

In the seventies, the Xerox lab in Palo Alto was the site of some now-famous computer-science

6.

Harvey Scher, now at the Weizmann Institute, has commented on technology as a rich source of

examples: Hiltzik, M.A. (1999) Dealers of Lighting, Harper, New York.

7. 8.

9. 10.

scientific questions in his recent Festschrift article: Scher, H. (2000) Reminiscences, J. Phys. Chem. B 104, 3768. Reference [3], p. 170. Suding, P.N. and Ziff, R.M. (1999) Site percolation thresholds for Archimedean lattices, Phys. Rev. E 60, 275. Fitzpatrick, J.P., Malt, R.B., and Spaepen, F. (1974) Percolation theory and the conductivity of random close packed mixtures of hard spheres, Physics Letters 47A, 207. Ottavi, H., Clerc, J.P., Giraud, G., Roussenq, J., Guyon, E., and Mitescu, C.D. (1978) Electrical conductivity of conducting and insulating spheres: an application of some percolation concepts, J. Phys. C: Solid State Phys. 11, 1311.

11. 12.

13. 14. 15. 16.

17. 18.

19. 20.

Lee, S.I., Song, Y., Noh, T.W., Chen, X.D., and Gaines, J.R. (1986) Experimental observation of nonuniversal behavior of the conductivity exponent for three-dimensional continuum percolation systems, Phys. Rev. B 34, 6719. Domb, C. and Sykes, M.F. (1960) Cluster size in random mixtures and percolation processes, Phys. Rev. 122, 170. Shklovskii, B.I. and Efros, A.L. (1984) Electrical Properties of Doped Semiconductors, SpringerVerlag, Berlin, p. 106. Nan, C.W. (1993) Physics of inhomogeneous inorganic materials, Prog. Mater. Sci. 37, 1. Viswanathan, R. and Heaney, M.B. (1995) Direct imaging of the percolation network in a threedimensional disordered conductor-insulator composite, Phys. Rev. Letters 75, 4433. Heaney, M.B. (1995) Measurement and interpretation of nonuniversal critical exponents in disordered conductor/insulator composites, Phys. Rev. B 52, 12477.

Heaney, M.B. (1997) Electrical transport measurements of a carbon-black/polymer composite, Physica A 241, 296. Wu, J., and McLachlan, D.S. (1997) Percolation exponents and thresholds obtained from the nearly ideal continuum percolation system graphite/boron-nitride, Phys. Rev. B 56, 1236. Wu, J., and McLachlan, D.S. (1997) Percolation exponents and thresholds in two nearly ideal anisotropic continuum systems, Physica A 241, 360. Halperin, B.I., Feng, S., and Sen, P.N. (1985) Differences between lattice and continuum percolation transport exponents, Phys. Rev. Letters 54, 2391.

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THE INTERMEDIATE PHASE AND SELF-ORGANIZATION IN NETWORK GLASSES

M.F. THORPE and M.V.CHUBYNSKY Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824

INTRODUCTION The study of the structure of covalent glasses has progressed steadily since the initial work of Zachariasen [1] in 1932 that introduced the idea of the Continuous Random Network (CRN). Zachariasen envisaged such networks maintaining local chemical order,

but by incorporating small structural distortions, having a topology that is non-crystalline. This seminal idea has met some opposition over the years from proponents of various microcrystalline models, but today is widely accepted, mainly as a result of careful diffraction experiments from which the radial distribution function can be determined. The CRN has been established as the basis for most modem discussions of covalent glasses, and this has occurred because of the interplay between diffraction experiments and model building. The early model building involved building networks with ~ 500 atoms from a seed with free boundaries in a roughly spherical shape [2]. Subsequent efforts have refined this approach and made it less subjective by using a computer to make the decisions and incorporating periodic boundary conditions. The best of these approaches was introduced by Wooten, Winer and Weaire [3] and consists of restructuring a crystalline lattice with a designated large unit supercell, until the supercell becomes amorphous. The large supercell contains typically ~5000 atoms. Both the hand built models and the Wooten, Winer and Weaire model are relaxed during the building process using a potential. The final structure is rather insensitive to the exact form of the potential and a Kirkwood [4] or Keating [5] potential is typically used. Despite this success in understanding the structure, some concerns remain. Perhaps the most serious of these is that the network cannot be truly random. Even though bulk glasses form at high temperatures where entropic effects are dominant, it is clearly not correct to completely ignore energy considerations that can favor particular local structural arrangements over others. A simple example of this is local chemical ordering, where, for example, bonding between certain same-type atoms is unfavorable. This can lead to chemical thresholds that appear at certain concentrations, at which unfavorable bonding can no longer be avoided. A more interesting and subtle effect of interest to us here is how the structure itself can incorporate non-random features in order to minimize the free

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

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energy at the temperature of formation. Such subtle structural correlations, which we refer to as self-organization, will almost certainly not show up in diffraction experiments, but may have other manifestations, as discussed in the paper of P. Boolchand in this volume. Here we focus on the mechanical properties and critical mechanical thresholds, as this is where it is easiest to make theoretical progress at this time. How can such an idea be developed theoretically? A proper procedure might be to consider a very large supercell and use a first principles quantum approach, like that of Car and Parrinello [6], to form the glass. The problem with this is that the relaxation times at the appropriate temperatures are very large, so full equilibration is impossible. The structure thus obtained would be unreasonably strained. This situation is made worse as only small supercells with about 100 atoms can be used at present and in these the periodic boundary conditions produce unacceptably large internal strains. Using the fastest linearscaling electronic structure methods or even molecular dynamics with empirical potentials is still much too slow. We therefore need to look at other ways of generating selforganizing networks. One promising approach is that of Mousseau and Barkema [7] who explore the energy landscape of a glass by moving over saddle points. In network terms, this corresponds to selective (thus non-random) bond switching. In these lecture notes, we look at even more simplified approaches that show what kinds of effects self-organization, and the resulting non-randomness, can lead to. The layout of this paper is as follows. In the next section we review ideas of rigidity percolation that lead to a mechanical threshold in networks as the number of bonds per atom in them (related to the mean coordination number) changes. Then we describe our

model of self-organization of glassy networks, which has two thresholds instead of one and

thus exhibits an intermediate phase, and study the properties of this model. We also consider a similar model for random resistor networks, based on the analogy between the usual (connectivity) percolation and rigidity percolation. Throughout much of this paper we focus on both central force networks in two dimensions and bond-bending networks in three dimensions that have central and noncentral forces as these are more relevant to glasses. The reader should be aware that we do flip back and forth between these model systems, as appropriate, in order to illustrate various points.

RIGIDITY PERCOLATION Going back more than a century, Maxwell was intrigued with the conditions under which mechanical structures made out of struts, joined together at their ends, would be stable (or unstable) [8]. To determine the stability, without doing any detailed calculations (that would have been impossible then except for the simplest structures), Maxwell used the approximate method of constraint counting. The idea of a constraint in a mechanical system goes back to Lagrange [9] who used the concept of holonomic constraints to reduce the effective dimensionality of the space.

The problem under consideration is a static one – given a mechanical system, how many independent deformations are possible without any cost in energy? These are the zero frequency modes, which we prefer to refer to as floppy modes because in any real system there will usually be some weak restoring force associated with the motion. Sometimes it is convenient to look at the system as a dynamical one, and assign potentials or spring constants to deformations involving the various struts (bonds) and

angles. It does not matter whether these potentials are harmonic or not, as the displacements are virtual. However it is convenient to use harmonic potentials so that the system is linear. It is then possible to set up a Lagrangian for the system and hence define a dynamical matrix, which is a real symmetric matrix having real eigenvalues. These eigenvalues are either positive or zero. The number of finite (non-zero) eigenvalues defines 44

the rank of the matrix. Thus our counting problem is rigorously reduced to finding the rank of the dynamical matrix. The rank of a matrix is also the number of linearly independent rows or columns in the matrix. Neither of these definitions is of much practical help, and a

numerical determination of the rank of a large matrix is difficult and of course requires a particular realization of the network to be constructed in the computer. Nevertheless the rank is a useful notion as it defines the mathematical framework within which the problem is well posed.

The rigidity of a network glass is related to how amenable the glass is to continuous deformations that require very little cost in energy. A small energy cost will arise from weak forces, which are always present in addition to the hard covalent forces that involve bond lengths and bond angles. These small energies can be ignored because the degree to which the network deforms is well quantified by just the number of floppy modes [10] within the system. This picture of floppy and rigid regions within the network has led to the idea of rigidity percolation [11,12]. When new constraints are added to an initially floppy network and it crosses the rigidity percolation threshold, a single rigid region percolates through the network and it becomes stable against external straining (elastic moduli become non-zero). There are two important differences between rigidity and connectivity percolation. The first difference is that rigidity percolation is a vector (not a scalar) problem, and secondly, there is an inherent long-range aspect to rigidity percolation. These differences make the rigidity problem become successively more difficult as the dimensionality of the network increases. In two dimensions, Figure 1(a) shows four distinct rigid clusters consisting of two rigid bodies attached together by two rods connecting at pivot joints. Now the placement of one additional rod, as shown in Figure l(b), locks the previous four clusters into a single rigid cluster. This non-local character allows a single rod (or bond) on one end of the network to affect the rigidity all across the network from one side to the other.

Using concepts from graph theory, we have set up generic networks where the connectivity or topology is uniquely defined but the bond lengths and bond angles are

arbitrary. A generic network does not contain any geometric singularities [13], which occur when certain geometries lead to null projections of reaction forces. Null projections are caused by special symmetries, such as, the presence of parallel bonds or connected collinear bonds. Rather than these atypical cases, their generic counterparts as shown in Figures 1(b) and (c) will be present. This ensures that all infinitesimal floppy motions carry over to finite motions [13-15].

Figure 1. The shaded regions represent 2D rigid bodies. The (closed, open) circles denote pivot-joints that are members of (one, more than one) rigid body. (a) A floppy piece of network with four distinct rigid clusters. (b) Three generic cross links between two rigid bodies make the whole structure rigid. If the bonds were parallel, the structure would not be rigid to shear. (c) A set of three non-collinear connected rods connecting across a rigid body is generic and contains one internal floppy mode. If they were collinear (along

the dashed line), then there would be two infinitesimal (not finite) floppy motions, and under a horizontal compression buckling would occur.

By considering generic networks, the problematic geometric singularities are completely eliminated. Therefore, the problem of rigidity percolation on generic networks leads to many conceptual advantages because all geometrical properties are robust.

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Moreover, real glass networks have local distortions, and are modeled better by generic networks.

Constraint Counting The genius of Maxwell [8] was to devise the simple constraint counting method that allows us to estimate the rank of the dynamical matrix and hence the number of floppy modes.

The number of floppy modes in d dimensions is given by the total number of degrees of freedom for N sites (equal to dN ) minus the number of independent constraints. A dependent (redundant) constraint does not change the number of floppy modes. It can only add additional reinforcement and it cannot be accommodated without changing the natural bond lengths and angles of the network, so stressed (over-constrained) regions would be created. A key quantity is the number of floppy modes, F , in the network, or normalized per degree of freedom, f = F/dN. By defining the total number of constraints per degree of freedom as nc and the number of redundant constraints per degree of freedom as nr , we can write quite generally,

It is straightforward to find the total number of constraints (and consequently nc) for each

given network. Neglecting redundant constraints [n r in Eq. (1)] as first done by Maxwell [8], we come to Maxwell counting:

Now the idea is to associate the rigidity percolation transition with the point where fM goes to zero. The Maxwell approximation gives a good account of the location of the phase transition and the number of floppy modes, but it ultimately fails, because some constraints are redundant and also because, as we will see soon, there are still some floppy pockets inside an overall rigid network. We now describe Maxwell counting for specific cases. Central Force Network in Two Dimensions. The elastic properties of random networks of Hooke springs have been studied over the past 15 years [11,16-20]. This system can be viewed as a network of Hooke springs in 2 dimensions, which is built from a regular (say, triangular) lattice, whose bonds are represented by springs, by removing the bonds at random, so each one is present with probability p (bond dilution). The site diluted version of the problem was also considered (see, e.g., [21]). For constraint counting it is convenient to introduce the mean coordination as an average number of bonds stemming from a site. It is given by where p is the probability of the bond being present, z is the coordination of the underlying regular lattice (6 for the triangular lattice, for example). If the total number of sites is N , the number of bonds is Each of these bonds represents one constraint, as always in central force networks, and therefore the number of constraints per degree of freedom is given by

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Therefore, according to Eq. (2), Maxwell counting gives

This quantity goes to zero at

which we associate with the rigidity percolation transition. For the triangular lattice this corresponds to pc =2/3. Bond-Bending Glassy Networks in Three Dimensions. We start by examining a large covalent network that contains no dangling bonds or singly coordinated atoms. We can describe such a network by the chemical formula GexAsySe1–x–y , where the chemical element, Ge, stands for any fourfold bonded atom, As for any threefold bonded atom and Se for any twofold bonded atom. Each atom has its full complement of nearest neighbors and we consider the system in the thermodynamic limit, where the number of atoms There are no surfaces or voids and the chemical distribution of the elements is not

relevant, except that we assume there are no isolated pieces, like a ring of Se atoms. The total number of atoms is N and there are nr atoms with coordination r (r = 2, 3 or 4), then

and we can define the mean coordination

We note that (where ) gives a partial but very important description of the network. Indeed, when questions of connectivity are involved the average coordination is the key quantity. In covalent networks like GexAsySe1–x–y , the bond lengths and angles are well defined. Small displacements from the equilibrium structure can be described by a Kirkwood [4] or Keating [5] potential, which we can write schematically as

The mean bond length is l,

is the change in the bond length and is the change in the bond angle. The bond-bending force is essential to the constraint counting approach for stability, in addition to the bond stretching term The other terms in the potential are assumed to be much smaller and can be neglected at this stage. If floppy modes are present in the system, then these smaller terms in the potential will give the floppy modes a small finite frequency. For more details see Ref. [16]. If the modes already have a finite frequency, these extra small terms will produce a small, and rather uninteresting, shift in the frequency. This division into strong and weak forces is essential if the constraint counting approach is to be useful. It is for this reason that it is of little, if any, use in metals

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and ionic solids. It is fortunate that this approach provides a very reasonable starting point in many covalent glasses.

To estimate the total number of zero-frequency modes, Maxwell counting was first applied by Thorpe [16], following the work of J.C. Phillips [22,23] on ideal coordinations for glass formation. It proceeds as follows. There are a total of 3N degrees of freedom. There is a single central-force constraint associated with each bond. We assign r/2 constraints associated with each r-coordinated atom. In addition there are constraints associated with the angular forces in Eq. (7). For a twofold coordinated atom there is a single angular constraint; for an r-fold coordinated atom there are a total of 2r–3 angular constraints. The total number of constraints is therefore

Using Eqs. (5) and (6), their fraction nc can be rewritten as

thus, according to Eq. (2),

Note that this result only depends upon the combination

which is the relevant

variable. When (e.g. Se chains), then fM = 1/3 ; that is, one third of all the modes are floppy. As atoms with higher coordination than two are added to the network as crosslinks, fM drops and goes to zero at and network goes through the rigidity percolation transition. This mean field approach has been quite successful in covalent glasses and helps explain a number of experiments. Also in later sections, we discuss the results of computer experiments and show that they are rather well described by the results of this subsection.

We note that Eq. (8) holds only when there are no 1-fold coordinated atoms. Their presence leads to the threshold being shifted down [24-26].

The Pebble Game

Until recently it has not been possible to improve on the approximate Maxwell constraint counting method, except on small systems with up to ~ 104 sites using brute force numerical methods. Now a powerful exact combinatorial algorithm, called the Pebble Game, has become available. This algorithm, first suggested by Hendrickson [13] and implemented by Jacobs and Thorpe [12,27,28], allows systems containing more than 106 sites to be analyzed in two-dimensional generic central-force networks and in threedimensional networks with both central forces and bond-bending forces. The crux of the Pebble Game algorithm in two dimensions is based on a theorem by Laman [14] from graph theory. We note first that if for a two dimensional network Maxwell counting gives less than 3 floppy modes (3 modes are always there, as they correspond to rigid motions of the network), the counting cannot be exact and thus a redundant bond (or bonds) are present. One says that the Laman condition for the network is violated in this case. But if the opposite is true (the Laman condition is satisfied), this is not sufficient for redundant bonds to be absent, as the network can have more than 3 floppy modes and redundant bonds simultaneously. The statement of the theorem is that non48

violation of the Laman condition for every subnetwork is sufficient for not having redundant bonds. This statement does not generalize to dimensions higher than two. We do not go into details of the algorithm, which can be found elsewhere [12,27,29]. For this consideration it is enough to know that one starts from an “empty” lattice (having no bonds, only sites) and adds bonds one at a time. Each newly added bond is tested for independence and each independent bond decreases the number of floppy modes by one. Besides providing exact constraint counting, the algorithm is able to identify all rigid

clusters (and thus whether or not rigidity percolation occurs) and find all the regions, in which redundant bonds introduced stress (over-constrained regions).

Figure 3. The topology of a typical section from a bond-diluted generic network at p = 0.62 (below percolation) and at p = 0.70 (above percolation). A particular realization would have local distortions (not shown), thus making the network generic. The heavy dark lines correspond to over-constrained regions. The open circles correspond to sites that are acting as pivots between two or more rigid bodies.

Sections of a large network on the bond-diluted generic triangular lattice are shown in Figure 2 after the pebble game was applied. Below the transition the network can be macroscopically deformed as the floppy region percolates across the sample. Above the rigidity transition, stress will propagate across the sample. However, below the transition there are clearly pockets of large rigid clusters and over-constrained regions, while above the transition there are pockets of floppy inclusions within the network. While Laman’s theorem does not generally apply to three dimensions, it is possible to generalize the Pebble Game algorithm for a particular class of networks, namely, the bondbending networks with angular forces included as in a Kirkwood or Keating potential [Eq. (7)]. Fortunately, the bond-bending model is precisely the class of models that is applicable to the study of many covalent glass networks. A longer discussion of the three dimensional Pebble Game is given in Refs. [28,30].

Two Dimensional Central Force Network. In this subsection, we review some results for central-force generic rigidity percolation on the triangular net. A more detailed account can be found in Ref. [27]. We begin by finding the number of floppy modes and comparing it to the Maxwell counting result. The exact value of f is very close to fM far enough below the mean-field estimate for the rigidity transition

but then starts to deviate significantly and does not reach zero (until full coordination, is reached). The quantity f looks quite smooth, but the second derivative of it with respect to (shown in the insert) does in fact have a singularity. This singularity corresponds to the rigidity percolation threshold, as can be checked by detecting the percolating rigid cluster directly. Using finite-size

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scaling, the position of the transition was found to be This is amazingly close to the mean-field value of 4. The behavior of the second derivative suggests that the number of floppy modes is an analogous quantity for rigidity and connectivity percolation. In the case of connectivity percolation, the number of floppy modes is simply equal to the total number of clusters, which corresponds to the free energy [16,31-33]. It would be nice if a similar result holds for rigidity percolation. It turns out that the second derivative of the total number of clusters changes sign across the transition, thus violating convexity requirements. Noting that typically rigid clusters are not disconnected, it was suggested that the number of floppy modes generalizes as an appropriate free energy [16,31-33]. With this assumption, the exponent is estimated in the usual context of a heat capacity critical exponent, even though no temperature is involved here. Again analogously to connectivity percolation, the fraction of bonds in the percolating rigid cluster serves as the order parameter for this system. The critical exponent is

defined as the rigid cluster size critical exponent. Another order parameter is also possible, namely, the fraction of bonds in the percolating stressed cluster, which is defined as a percolating stressed subset of the percolating rigid cluster. It was found (and this is an important point) that both and go to zero at the same point – the percolation transition. This will be different in the next chapter on self-organization and will lead to the existence of the intermediate phase, and two phase transitions. The results of study of this model [27] lead to the conclusion that the rigidity transition in this system is second order, but in a different universality class than connectivity percolation. It has been suggested by Duxbury and co-workers [21] that the rigidity transition might be weakly first order on triangular networks. While we think this is unlikely, it

cannot be completely ruled out at the present time. Three Dimensional Bond Bending Networks. It can be shown [28] that the only floppy element in a three dimensional bond-bending network is a hinge joint. Hinge joints can only occur through a central-force (CF) bond and are always shared by two rigid clusters – allowing one degree of freedom of rotation through a dihedral angle. Note that in two dimensional central force generic networks, sites that belong to more than one cluster act as a pivot joint, and more than two rigid clusters can share a pivot joint. Because of this difference between CF and bond-bending networks, the order parameters analogous to and of the previous subsection, have to be defined as a fraction of sites in respective percolating clusters and not bonds, as bonds can be shared between a percolating and a non-percolating clusters. For purposes of testing rigidity in generic three-dimensional bond-bending networks, it is only necessary to specify the network topology or connectivity of the CF bonds, since the second nearest neighbors via CF bonds define the associated bond-bending constraints. Here, we have considered two test models. In the first model, a unit cell is defined from our realistic computer generated network of amorphous silicon [34] consisting of 4,096 atoms having periodic boundary conditions. Larger completely four-coordinated periodic networks containing 32,768, 262,144 and 884,736 atoms are then constructed from the amorphous 4,096-atom unit cell. The four-coordinated network is randomly diluted by removing CF bonds one at a

time with the constraint that no site can be less than two-coordinated. That is, a CF bond is randomly selected to be removed. If upon removal either of its incident atoms becomes less than two coordinated, then it is not removed and another CF bond is randomly selected from the remaining pool of possibilities. The order of removing CF bonds is recorded. This process is carried out until all remaining CF bonds cannot be removed, leading to as low an

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average coordination number as possible. All CF bonds that were successfully removed are marked. This method of bond dilution gives a simple prescription for generating a very large model of a continuous random GexAsySe1-x-y type of network. For comparison, a second test model, a diamond lattice, was diluted in the same way and contained 32,768, 262,144 and 106 atoms. The results of simulations of both models are qualitatively similar to those for 2D central-force networks. Both have a rigidity transition slightly below the Maxwell counting estimate of 2.4. Again, the rigidity transition can be accurately found from the sharp peak in the second derivative of the fraction of floppy modes. In particular, for the diamond lattice and for a-Si, Remarkably, the Maxwell counting estimate is accurate to about 1% in locating the threshold in both cases. A more detailed account of the results can be found in Ref. [35]. SELF-ORGANIZATION AND INTERMEDIATE PHASE Self-Organization in Rigidity Percolation

Description of the Model. We have mentioned that starting from an “empty” lattice

(without bonds) and adding one bond at a time, we can use the pebble game to analyze whether the bond we are adding is independent of those already in the network or redundant. We also know that redundant bonds create stressed (over-constrained) regions. Thus within the present approach we have a rather unique opportunity to construct stressfree networks without a huge computational overhead. The idea is to start, as before, from an “empty” lattice and add one bond at a time to it, applying the pebble game at each stage. If adding a trial bond would result in that bond being redundant and hence create a stressed region, then that move is abandoned. Thus the network self-organizes in such a way that there is no stress in it at all. Note that the pebble game now serves not only as a tool to analyze the network, as before, but also as a decision-making mechanism when building the network. It is not possible to keep adding bonds beyond a certain point, without introducing stress (this is considered in more detail below). How should we proceed then? While going on with some sort of self-organization would be reasonable (as some bonds would create less stress than others), it is impossible to analyze this within our model, so we start inserting bonds completely at random, once avoiding stress becomes impossible. General Properties. First of all, how long is it possible to keep adding bonds to a network without introducing stress? It is certainly impossible to have more independent constraints then there are degrees of freedom in the network. Now recall that in the Maxwell counting approximation, the rigidity transition occurs when the numbers of constraints and of degrees of freedom balance. Thus it is certainly not possible to have an unstressed network with the mean coordination above where Maxwell counting predicts the transition (that is, above for central-force networks in 2d and for glassy networks in 3d). This provides an upper limit (still not always reachable, as we will see) for the unstressed networks. Note, though, that since the Maxwell counting percolation limit is not exact, this does not mean that rigid networks are necessarily stressed! The actual rigidity transition may occur below the point where Maxwell counting puts it. This is a very important point that leads to possibility of an intermediate phase, as described below. Secondly, we know that the Maxwell counting result for the number of floppy modes would be exact if all constraints in the network were independent. But this is exactly what we have in our case! Thus the number of floppy modes in Maxwell counting is exact for as

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long as we are able to keep the network unstressed. Hence we follow the Maxwell result for the number of floppy modes in the floppy and intermediate phases. We now analyze some specific cases in more detail. Intermediate Phase in 2D Central-Force Networks. Let us first prove that it is indeed possible to reach the Maxwell counting limit without any stress in this case (and for any CF networks), provided that the fully coordinated (undiluted) network has no floppy modes (which is the case for triangular networks). As we have seen before, generally speaking, we should distinguish carefully between constraints and bonds. A constraint can be thought of as one algebraic relation for the coordinates of atoms; stress appears whenever one or more of such relations are not satisfied. A bond can have several associated constraints, as in bond-bending networks. In the case of CF networks, though, each bond has only one associated constraint (the distance between the sites it connects), so “bonds” and “constraints” are identical. Recall once again that every single constraint can be either independent (in which case it reduces the number of floppy modes of the network

by 1), or redundant (so it does not change the number of floppy modes). At the point where stress becomes inevitable any trial bond would cause stress (be redundant). So all the bonds, which will be subsequently inserted, are redundant. Thus f will remain constant up to the very end (which is the full lattice), therefore f = 0 at this point. Since Maxwell counting is still exact there, the proof is complete. We would like to emphasize that equivalence of “bonds” and “constraints” was essential for this proof (we used these terms

interchangeably). See the next subsection for comparison. Secondly, it is possible to establish a relation between the self-organized networks and those obtained by usual completely random insertion (to which we for simplicity refer as “random” in contrast to “self-organized” in what follows). Indeed, assume we are using the same random list of M bonds to build a random network and a self-organized one, trying to insert bonds as they are listed. For the random network, all the M bonds will get in; for the self-organized network, some of them will be, generally speaking, rejected, so that will be inserted. The bonds rejected in the self-organized network will be redundant in the random one; they do not influence the number of floppy modes, the configuration of rigid clusters (and thus whether or not rigidity percolation occurs) and the redundancy or independence of all the subsequently inserted bonds. Thus all these characteristics will be identical for the two networks. The consequence is that there is a

correspondence between self-organized and random networks having the same number of floppy modes; in particular, rigidity percolation occurs at the same number of floppy modes. This analysis allows us to make a very important conclusion. Since in random networks rigidity percolates at a non-zero f and the same has to be true for self-organized networks (because of the just mentioned consequence), yet stress appears exactly at f = 0, we conclude that there exists an intermediate phase, which is rigid (i.e. the infinite rigid cluster exists), but unstressed (so, evidently, there is no stress percolation). This is different from the situation with random insertion, where the rigidity and stress percolation thresholds always coincide (see Figure 3). It could be possible that stress does not percolate immediately after it is introduced; we will see from simulation results that this is not the case, so the upper boundary of the intermediate phase (the stress transition) may be defined as either the point where stress first appears, or equivalently, the point where it percolates. As is seen from our consideration, it lies at As we have mentioned, the fractions of bonds in percolating rigid and stressed clusters (denoted and respectively) can serve as order parameters. Now, since there is an intermediate phase where rigidity percolates, while stress does not, these two parameters 52

turn zero at different points, between which the intermediate phase lies. Besides, since the number of floppy modes is zero above the stress transition, the whole network is rigid, and thus is identically 1. These facts are illustrated in Figure 3.

Figure 3. Order parameters and for self-organized and random triangular networks. It is seen that the intermediate phase (shaded) is formed in the self-organized case, extending from 3.905 to 4, while in the random case the two thresholds coincide and there is no intermediate phase. All results are averages over two realizations on 400×400 networks.

Given the discussion of the floppy modes in the random and self-organized networks, it is tempting to suggest that the same relation holds for the just defined rigidity order

parameter. The subtlety is that the relation is defined in terms of sites (i.e., same sites are in the percolating cluster and same sites are pivot joints on its border), while the order parameter is defined in terms of bonds. Of course, there is no direct correspondence between bonds, as there are different numbers of bonds in related random and selforganized networks. Still it might be safely assumed that the rigid cluster size critical exponents are the same for rigidity percolation in random and self-organized networks. Other critical exponents may be different, though. It is interesting to note that since f given by Maxwell counting is exact in the whole unstressed region, in both the floppy and the intermediate phase f is a perfect straight line and the rigidity transition does not show up in f .

Results of our simulations of this model are shown in Figures 3 and 4. The simulations were done for networks with periodic boundary conditions in both directions. There are several facts to be inferred (besides confirming all the results we have obtained

so far). We see that stress percolates immediately after it appears at (this fact was mentioned above). Second, the cluster size critical exponent for the stressed cluster is quite small (smaller than the one for the rigid cluster). In random networks, the stressed cluster exponent is larger than the rigid cluster exponent, which is because the stressed percolating cluster is smaller than the rigid cluster (the former being a subset of the latter) and the two thresholds coincide.

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Figure 4. Number of floppy modes per degree of freedom for self-organized and random triangular networks. Thresholds are shown with different symbols. The intermediate phase in the self-organized case is shaded. Note that rigidity percolation occurs at the same f in the random and self-organized cases. The self-

organized plot is strictly linear up to

and coincides with Maxwell counting.

Intermediate Phase in 3D Bond-Bending Networks. In case of glassy networks there is a slight problem with implementing our general algorithm of self-organization. In the CF case we were starting from an empty lattice to ensure that it had no stress initially. In the present case the initial dilution can only go as far as to the point where any further dilution would create a 1-coordinated site. At this limit there are no bonds with both ends being sites of coordination 3 and higher, so that further dilution is impossible. It is

generally not true that this final network is unstressed. For smaller networks (~104 sites and less), it is possible to pick those that are unstressed; for larger ones such cases are rare, and it is reasonable to assume that the fraction of constraints that are redundant is a constant in the thermodynamic limit. This constant seems to be very low, though (in our simulations, typically about 0.05% of constraints were redundant). Besides, the number of redundant constraints does not grow when new bonds are inserted according to our algorithm (up to the stress transition), so this problem is largely irrelevant. Unlike the case of CF networks, BB networks have more than one constraint associated with each bond. When a new bond is added, not only the distance between the sites it connects is fixed, but the angles between the new bond and those stemming out of the two sites at either end of that bond are fixed as well. Any bond that has at least one redundant constraint associated with it would cause stress. Some of the stress-causing bonds have only part of the associated constraints redundant and the rest independent, and such a bond will change the number of floppy modes. This makes some of our conclusions made for CF networks invalid in this case. Firstly, this invalidates the proof of the reachability of the Maxwell counting limit ( in this case). This is because even when at the upper reachable limit all the as yet uninserted bonds would cause stress, some of these bonds may further decrease the number of floppy modes and thus this number is not necessarily zero at this point.

Secondly, the nice relation between random and self-organized networks no longer holds, because out of the redundant bonds by which the two differ, some (namely, the partially redundant ones) change f , rigidifying the network and changing the 54

configuration of rigid clusters. Still the equality of critical exponents for rigid cluster sizes in random and self-organized cases probably holds. At the same time, some facts are unchanged. In particular, f given by Maxwell counting is still exact in the unstressed region. Most importantly, the intermediate phase still exists. The results of simulations done for the diluted diamond lattice are given in Figures 5 and 6. As in the previous subsection, we use periodic boundary conditions in all directions. We note in addition to the graphs that, as in the CF case, stress percolates immediately

after it appears. The intermediate phase extends from to 2.392 (not reaching 2.4). Again, the stress transition is sharper than the rigidity transition. Our results are consistent with the second order transition with the very small critical exponent or a first order transition is more likely. Another feature of the plot in Fig. 5 is that the rigidity order parameter is not exactly unity in the stressed phase (which is expected, as some floppy modes remain in the stressed phase) and the second transition shows up as a kink in the rigidity order parameter. In conclusion to this section, we would like to mention that it is possible within our approach to establish a hierarchy of stress-causing bonds (by the number of associated

redundant constraints) and when stress becomes inevitable, first put those having one redundant constraint, then those having two, and so on. Exactly at only those bonds having no associated independent constraints will remain uninserted. It is unlikely, though, that there is a good correlation between the number of redundant constraints and the actual increase in stress energy, as the distribution of stresses caused by different bonds is quite wide, so this complication seems unreasonable.

Figure 5. The order parameters and for the self-organized diluted diamond lattice. The intermediate phase is shaded. Circles are average over 4 networks with 64,000 sites, triangles are averages over 5 networks with 125,000 sites. The dashed lines are the power law fit below the stress transition and for guidance of the eye above. Note the break in the slope at the stress transition.

Elastic Properties of Self-Organized Networks. So far our study of self-organized networks was limited to their geometrical properties. Of course, this work becomes really meaningful when we turn to what the physical consequences of self-organization are. The simplest quantity to look at is the elasticity of the networks of springs. Unfortunately, the 55

pebble game, being concerned with the geometric properties only, is unable to help us find the numerical values of elastic constants, so we have to do a usual relaxation using, for

example, the conjugate gradient method [36] and consider particular configurations, and not just the connectivity. So far in this preliminary study, we have only considered the 2d case. The first and quite surprising fact is that in case of periodic boundary conditions in all directions the elastic constants are exactly zero in the intermediate phase, regardless of the

size of the supercell and despite the existence of the percolating rigid cluster. Indeed, periodic boundary conditions mean that positions of images of same site in different supercells are fixed with respect to each other. The network is built stressless with these additional constraints taken into account. The exact specification of these constraints beyond stating what sites are involved is determined by the particular size and shape of the supercell, but is never taken into account (just as particular bond lengths never matter in determination of stressed regions). So straining the network by changing this size and shape leaves it stressless. The important thing here is that straining does not add any new

constraints. We confirmed this result numerically by doing exact diagonalization of the dynamical matrix (similar to [37]), rather than by relaxation, which ensures better precision.

Figure 6. The fractions of floppy modes per degree of freedom for the diluted diamond lattice (both selforganized and random cases). Different thresholds and the Maxwell prediction for the rigidity threshold are shown with different symbols. The intermediate phase in the self-organized case is shaded. The Maxwell

counting line is seen only above the stress transition point in self-organized networks, as below this point it coincides with the self-organized line.Note that the rigidity transition in the two cases no more occurs at the same f . Instead, the values of are close, which is probably coincidental.

Of course, for different boundary conditions the elastic constants may be non-zero for finite samples, but are expected to vanish in the thermodynamic limit. We consider the busbar geometry, in which busbars are applied to two opposite sides of the network and it is strained perpendicular to the busbars. The network is built assuming open boundaries at the busbars and periodic boundary conditions parallel to the busbars. The first and the last rows of sites are assumed belonging to the respective busbar (i.e., attached rigidly to it). In addition, when building the network, we consider the sites belonging to each busbar as being fixed with respect to each other, connecting them with fictitious bonds and considering these bonds as belonging to the network. This makes the open boundaries “less open” and eliminates certain boundary effects, as will be clear from an analogy in the next 56

section with connectivity percolation. The arguments of the previous paragraph do not apply here, as the network is built not assuming a fixed distance between the busbars (as if it is allowed to relax) and straining changes and fixes it thus imposing an additional constraint.

Figure 7. An example of the triangular self-organized network 150×150 in the intermediate phase (at ). The thickest bonds belong to the applied-stress backbone, those of medium thickness are in the percolating rigid cluster (but not in the backbone), the thinnest ones are not in the percolating cluster. The busbars are shown schematically.

When introducing the boundary conditions as described above, we will have non-zero stress when an external strain is applied, and some of the bonds will be stressed. These bonds are said to belong to the applied stress backbone [21] (which we refer to as simply backbone in what follows). It can be found easily by the pebble game using a method proposed by Moukarzel [38], which in our case consists in putting an additional bond across the network emulating the external strain, and finding those bonds in which stress is induced. A typical result is shown in Figure 7. It is seen that the backbone has filamentary structure. We note that stress in this backbone was created by putting just one extra bond and thus it is enough to take any one bond out of the backbone for it to be destroyed, so it is extremely fragile. Also, since the backbone always has only one redundant bond (when the bond across is added), it does not grow throughout the intermediate phase after it appears at the rigidity transition, because growth can only occur by adding new redundant bonds. This means that for any given sample the elastic constants are the same throughout the intermediate phase (here we mean finite samples, of course, as in the infinite limit the elastic constants are zero).

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Figure 8. The elastic modulus c11 for self-organized triangular networks. Each point corresponds to one sample (their linear sizes are specified by different symbols). The intermediate phase is shaded. The dashed line is the mean-field linear dependence, reaching 1 at the full coordination.

We found the elastic modulus c11 numerically in both the intermediate and stressed

phases. The triangular lattice was distorted by random displacement of atoms. For displacements along each axis uniform distribution on an interval (–0.1; 0.1) in units of the lattice constant was chosen, but the results are only slightly sensitive to the width of the distribution. Equilibrium lengths of springs were chosen equal to the distance between the atoms they connect, so the initial network is unstressed. Thus subtraction of two large energies when finding elastic constants is avoided. The results are shown in Fig. 8. Predetermining the applied stress backbone speeds up the relaxation greatly, as was first pointed out in Ref. [21]. Still, we were unable to reach full relaxation in the intermediate phase in all but the smallest samples (up to 30×30). The values in the intermediate phase are very low and are assumed to go to zero in the limit of large samples. We are currently doing finite size scaling to test this. Above the stress transition, the modulus seems to grow linearly, but, of course, it is hopeless to try and determine the critical exponent with reasonable precision from our data. Self-Organization in Connectivity Percolation

The model. It is interesting and useful to see if similar phenomena are possible in the more familiar case of connectivity percolation, especially as connectivity percolation is easier to study and understand. The essence of our algorithm of building self-organized networks in the rigidity case is rejecting stress-causing bonds (or those having redundant constraints). As we have seen, in the CF case, when “bonds” and “constraints” are the same, we may equivalently formulate this as rejecting redundant or irrelevant bonds. In bond connectivity percolation we also can build the networks by inserting bonds one by one; most importantly, there is a clear analog to redundant bonds. The relevant property now is connectivity, by which we mean the presence or absence of paths connecting any two sites of the network. Redundant bonds are those which connect sites already connected, that is would close a loop in the network. Thus the analog of self-organization is building loopless networks. There are other equivalent ways to draw this parallel. The first is based on the fact that connectivity percolation can be considered as rigidity percolation with the sites having one

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degree of freedom regardless of the lattice dimensionality. Each site thus has one coordinate and each bond is a relation between the coordinates of the sites it connects. Then the concepts of rigid clusters and clusters in the usual connectivity sense coincide. The number of floppy modes f is now the number of clusters. A redundant bond in the rigidity sense is the one that does not change f, it is also stress-causing, as it would introduce a relation between coordinates that cannot generally be satisfied. On the other hand, viewed from the connectivity perspective, such a bond connects the sites belonging

to the same cluster and closing a loop, and our model is again recovered. Yet another way is to recall that rigidity percolation with angular constraints in 2D (or with angular and dihedral constraints in 3D) is equivalent to connectivity percolation. Then stresslessness is equivalent to looplessness. Connectivity percolation and related phenomena were studied so extensively in all imaginable flavors that it would be strange if this and similar models were not studied before. Indeed exactly this model was proposed as far back as 1979 [39] and rediscovered in 1996 [40]. Besides, there was an extensive study of loopless graphs (trees) in relation to various phenomena ranging from resistance of a network between two point contacts (considered by Kirchhoff in mid nineteenth century [41]) to river networks [42] to certain optimization problems [43,44]. In many of these and other papers the algorithm for building trees was equivalent to ours. Still, we consider this model from a different perspective. Given that connectivity percolation can be considered as rigidity percolation with one degree of freedom per site, we can apply the usual two-dimensional pebble game with the simple modifications. Of course, the essence of our self-organization algorithm is still the rejection of bonds that are not independent. The pebble game allows the determination of all analogs of the quantities considered for rigidity. The Intermediate Phase. In this section we carry out the same kind of analysis as was done for rigidity percolation.

First of all we describe Maxwell counting, as this, although simple, is rarely discussed in relation to connectivity percolation. For a network with N sites the number of degrees of freedom is now simply N, the number of constraints is, as before, so the number of floppy modes per site is and this becomes zero at Since, as we have seen, connectivity percolation is nothing but a kind of rigidity

percolation on a CF network with 1 degree of freedom per site, all of the general analysis for CF networks in the previous section is valid. Specifically, Maxwell counting is exact in the “unstressed” (this now means loopless) phase; the limit is reachable without creating loops; the relation between random and self-organized networks also holds. The order parameters are defined analogously to the rigidity case. The first parameter is (by analogy) the size of the percolating (connectivity) cluster. However, the difference is that now the clusters (including the percolating one) can be defined in terms of either bonds or sites (there are no “pivot joints” that would be shared between several clusters). Therefore, there is a possibility to define this order parameter as the fraction of sites (instead of bonds) in the percolating cluster. This makes the relation between the order parameters of self-organized and random networks with the same number of “floppy modes” (clusters) exact. Yet, to be consistent, we ignore this possibility and define the order parameters as fractions of bonds, not sites. The second order parameter is, logically, the fraction of “stressed” bonds (bonds in loops). We do not show the results of simulations (which were done for the square lattice) as they are very similar to those in rigidity case, except that the “stressed” cluster critical exponent is larger, not smaller than the connected cluster exponent. Existence of the

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intermediate phase is confirmed in the range from to 2 for the square net. The lower transition coincides with the result obtained in Ref. [40]. Conductivity. Similarly to the elasticity case, we consider the busbar geometry here in two variants, with and without fictitious bonds making sites at the busbars rigid with respect to each other (we refer to these two cases as boundary conditions A and B respectively). As in rigidity, it is possible to find the conductivity backbone by Moukarzel’s method [38] (for B all the fictitious busbar bonds have to be put prior to placing the bond across, while they are already in the network in case A). Two examples corresponding to A and B are shown in Fig. 9. For A the backbone consists of just one path, while for B it is tree-like with branching near the busbars. Analog of these “boundary effects” in B is what was eliminated in study of elasticity, when the boundary conditions

analogous to A were chosen. The simulations for conductivity in 2D can be done very efficiently with the FrankLobb algorithm [45], whose only limitation is that it is applicable for the open boundary conditions only.

Figure 9. Examples of self-organized square networks in the intermediate phase with boundary conditions A (left panel) and B (right panel), as described in the text. The thickest bonds are in the conducting backbone,

those of medium thickness are in the percolating cluster (but not in the backbone), the thinnest are not in the percolating cluster. The busbars are shown schematically.

It is known from work on a river network model built in the same way as our network [42] that the backbone branches (in fact, all network branches) are fractal and the fractal dimension is

The only essential difference between the river network model and

our one is that they consider spanning trees (i.e., all sites are in the connecting cluster), which in our case corresponds only to This should not matter, though, since it is the dimensionality of the network that the cluster actually spans (i.e., of the connecting

cluster) that is important and this dimensionality is 2 everywhere in the intermediate phase. Thus we come to a conclusion that the fractal dimension of backbone branches is the same throughout the intermediate phase and equals 1.22. We confirm this fact in our simulations. We note that this differs from both the random walk result (d = 2) and that for selfavoiding random walks (d = 4/3) – in our case branches are more “straight” than both of these walks. Then for boundary conditions A it is obvious that the fraction of bonds in the backbone is and the conductance is Our simulations confirm this result, for both variants of boundary conditions. The effective conductivity in 2D is

equal to the conductance. Thus we come to the conclusion that the conductivity does indeed go to zero in the thermodynamic limit for the intermediate phase.

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The results in both the intermediate and the stressed phase are shown in Fig. 10. Just as for elastic constants, the dependence in the stressed phase is linear, but now much larger sizes are available, so this linearity may be exact, but we know of no reason for this to be so. Note the finite value of the conductivity in the intermediate phase, which is a finite size effect. This value would be constant for boundary conditions A, as the conducting backbone consists of just one stem not changing across the intermediate phase. Here this value changes slightly across the intermediate phase. We mention here briefly that our preliminary results in 3 dimensions show that the conductivity is also zero in the thermodynamic limit in the intermediate phase, and it goes to zero with increasing size even faster than in two dimensions.

Figure 10. Conductivity for resistor networks with present bonds having resistance R1 = 1 and missing having resistance

(diamonds); superconducting networks (R 1 = 0, R2 = 200, circles); mixed

networks (R1 = 1, R2 = 200, triangles). All results are averages over 10 square networks 100×100 with open boundary conditions parallel to the busbars, the busbar sites are treated as in case B (see text).

Superconducting Networks. We have seen that in the thermodynamic limit the

conductivity is zero in both the disconnected and intermediate phases (just as elastic constants were zero in both the floppy and intermediate phases in the rigidity case). These results make us wonder if the lower transition shows up in any physical quantities for infinite networks. One possibility is to consider superconductor networks instead of resistor networks. In this model all the existing bonds are replaced with conductors of zero resistance (“superconductors”), while all the absent bonds are equal resistors with finite resistance. It turns out that the same kind of correspondence between random and self-organized networks with the same f we had for clusters is valid for the conductance in this case. Indeed, these networks differ by redundant bonds that connect sites already connected. All the connected sites have zero potential difference (as they are connected with superconductors), so putting redundant bonds does not change the distribution of the potential and thus does not influence the conductance. It is known [46] that in the random case the resistivity is zero above the threshold and non-zero below it, with the critical exponent the same as for the conductivity of resistor networks (1.30). Thus in the self-organized case the resistivity will turn zero in the point related to the percolation threshold of random networks by the above relation, i.e., at the 61

lower transition. The critical exponent will be the same as in the random case (1.30), but this is now different from the value for of resistor networks Mixture of Two Sorts of Resistors. We can now “combine” the resistor and superconductor models by introducing two sorts of resistors, with resistances R1 and R2, R1 < R2, and putting R1 resistors in place of present bonds and R2 resistors in place of missing bonds. Assume now that R1 n c, n=nc, and nEc, but their number will be very small and will be neglected here. One of the important ideas in the MIT field was the notion of a minimum metallic conductivity σmin developed by Mott [16 ]. This idea is based on the Ioffe-Regel (IR) criterion that where kF is the Fermi wavevector and is the mean free path and that the Boltzmann conductivity was only meaningful for where d is the atomic spacing (the donor spacing in n-type Si). Mott employed the Boltzmann result to obtain

Invoking the IR criterion and

Mott obtained (4)

where the coefficient C depends on the number of valley v of a multivalley semiconductor. The Mott notion was that would drop discontinuously to zero for nnc) for finite T is the e-e interaction theory of Altshuler and Aronov (A-A) [36] for diffusing electrons in strongly disordered metals. A very large variety

of disordered metals exhibit the correction This correction, plus a second T1/2 correction from ionized impurity scattering, will provide an excellent description of the doped Si and Ge data for n>nc. The A-A correction for the d=3 case takes the form

(9a)

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Figure 3. Extrapolated T=0 values of versus the uniaxial stress S from Ref.[42]. The solid line is the region which is reproducible in 3 samples and yields s= 0.49. The inset shows data from Ref. [41] from individual samples. (Copyright by the American Physical Society)

where (9b)

where x = (2k F/κ)2 where this κ is the screening wave vector. The factor is known as the exchange-Hartree factor, or the singlet and triplet (spin=1) scattering amplitudes. The form of F= x-1 ln(1+x) is for a Fermi liquid. Because the A-A theory is only first order in the disorder many believe the theory is not valid very close to nc. An early perturbative form of the theory used (4/3-2F) for the exchange-Hartree factor and this was used by Rosenbaum et al. [37] to explain their data using x = 0.2x(n/1018)1/3 based on free electron-like expressions for kF and k. Had they used for x in the range 0.3 to 1.0 they would not have been able to explain their negative m(n) values for n>1.05nc unless they reversed the sign of the exchange-Hartree factor. An examination of the A-A integral and final result in Eqs 5.3 and 5.4 of their review indicates the sign is negative for d=3 and opposite that for d=1. However, the integral diverges for all and analytical continuation cannot be used from d=1 to d=3. Choosing an appropriate cutoff frequency for the integral of (10TC) depending on oxygen concentration and sample preparation. Here Tw/Tc ~4, a trend which is contrary to that observed in the intermetallic systems. The SC charge carriers originate from the CuO2 planes and the W-FM state is confined to the Ru layers. SC survives because the Ru moments probably align in the basal planes, which are practically de-coupled from the CuO2 planes, so that there is no pair breaking. Scanning tunneling spectroscopy and muon spin rotation experiments have demonstrated that all materials are microscopically uniform with no evidence for spatial phase separation of superconducting and magnetic regions. That is, both states coexist intrinsically on the microscopic scale [3]. In the Ru-2122 system, the W- FM state, as well as irreversibility phenomena, arise as a result of an antisymmetric exchange coupling of the Dzyaloshinsky-Moriya (DM) type [3] between neighboring Ru moments, induced by a local distortion that breaks the tetragonal symmetry of the RuO6 octahedra. Due to this DM interaction, the field causes the adjacent spins to cant slightly out of their original direction and to align a component of the moments with the direction of applied field. Below the irreversible temperature (Tirr,), which is defined as the merging temperature of the zero-field (ZFC) and field-cooled (PC) curves, the Ru-Ru interactions begin to dominate, leading to reorientation of the Ru

moments, which leads to a peak in the magnetization curves.

The most remarkable magnetic properties of the Ru-2122-samples are: (a) The negative magnetic moments in the ZFC branches measured at low applied fields (H) (b) The ferromagnetic-like hysteresis loops and strong enhancement of coercive field which appear only in the SC state at T< TC. (c) The so-called spontaneous vortex phase (SVP) model, which permits magnetic vortices to be present in equilibrium without an external field. The vortices in the SC planes, are caused by the internal field (higher than Hc1) of a few hundreds of G of the FM Ru sublattice. (d) No diamagnetic signal, in the FC branch (the Meissner state (MS) – the conventional signature of a bulk SC), has been

observed. The absence of the MS, may be a result of the SVP, and/or the high Ru magnetic moment induced by the external field at TM, which masks this SC signature. On the other hand, when Ru is partially replaced by Nb, the small positive contribution of the W-FM Ru sublattice decreases the internal field and the MS is readily observed. Hole (or carrier) density in the CuO2 planes, or deviation of the formal Cu valence from Cu2+, is a primary parameter which affects TC in most of the HTSC compounds. The concentration of charge carriers (p), which may be measured as the effective [CuO2]p charge, can be varied by removal or addition of oxygen. It is well accepted that addition of hydrogen reduces p in a way very similar to that caused by removal of oxygen, and at high hydrogen concentrations SC is suppressed and the materials a become semi-conducting and magnetic. In Ru-2122 the hole doping in the CuO2 planes can be achieved with appropriate variation of the oxygen concentration which is obtained by annealing the as prepared samples (ASP) under oxygen pressure up to 150 atm, or by loading the materials by various amount of hydrogen. The effect of oxygen treatment is to shift both TC and TM up to 49 and 225 K respectively (when annealed under 150 atm.). On the other hand, when hydrogen atoms are loaded, they occupy interstitial sites and suppress SC and enhance the W-FM properties of the Ru sublattice. This effect is reversible: namely, by depletion of

hydrogen, SC is restored and TM drops back to its original value. This paper is organized as follows: (a) We first show that in the Ru-2122 system, both TC and TM depend strongly on the oxygen concentration, (b) We present a systematic study of the effect of hydrogen on both states. (c) We show experimental evidence for the existence of the SVP by means of magneto-optical imaging. It is shown that below TC, at 342

zero applied field, magnetic flux is present in equilibrium in the sample and disappears above TC. (d) The magneto-SC mixed (Ru,Nb)-2122 system is introduced, in which the MS is readily observed. EXPERIMENTAL DETAILS

Ceramic samples with nominal composition (Ru-2122) and (Ru,Nb)-2122) were prepared by a solid state reaction technique as described elsewhere [3-5] Parts of the ASP sample were re-heated for 24 h at 800° C under various pure oxygen pressures up to 150 atm. and will be identified according to applied pressure. Determination of the absolute oxygen content in the ASP material and in the samples annealed under oxygen pressures, is difficult because CeO2 is not completely reducible to a stoichiometric oxide when heated to high temperatures.

Figure 1. XAS spectra at the K edge of Ru of Ru-2122 and reference compounds.

Thermo-gravimetric measurements show that the materials are stable up to 600°C and no oxygen weight loss is detected. Above this temperature a small weight decrease begins and our analysis indicates that the sample annealed at 150 oxygen atm. (150 atm.) contains ~4 at % more oxygen than the ASP sample. Hydrogen charging with several concentrations (up to 0.28(1) at.% per formula unit) was accomplished by direct contact with high pressure hydrogen gas at 300° C in a calibrated volume chamber. The hydrogen loaded samples (Ru-2122HX) will be referred according to their hydrogen content. Removal of the hydrogen was made by re-heating the hydrogenated sample for 10 hours at 250° C at ambient pressure. Powder X-ray diffraction measurements confirmed the purity of the compounds (~97%) and indicate within the instrumental accuracy, that all samples studied, have the same lattice parameters as the ASP material, a=3.846(1)Å and c=28.72(1)Å. ZFC and FC dc magnetic measurements in the range of 5-300 K were

performed in a commercial (Quantum Design) super-conducting quantum interference device (SQUID) magnetometer. Magneto-optical (MO) flux imaging studies have been carried on polished ceramic sample, using an indicator (iron garnet film) with in-plane anisotropy and high Faraday rotation angle, which was attached to the sample. The magnetic flux density in the material (the bright regions) was deduced from the light intensity depending on the Faraday rotation angle of the indicator, by using a polarizing microscope.

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Mossbauer spectroscopy performed at 90 and 300 K on 151Eu show a single narrow line with an isomer shift =0.69(2) and a quadrupole splitting of 1.84 mm/s, indicating that the Eu ions are trivalent with a nonmagnetic J=0 ground state. This is in agreement with Xray-absorption spectroscopy (XAS) taken at LIII edges of Eu and Ce that shows that Eu is trivalent and Ce is tetravalent. The local electronic structure in several Ru based compounds was studied by XAS at the K edge of Ru, and the results obtained at room temperature are shown in Fig. 1. Since the valence of Gd3+, Sr2+ and O2- are conclusive, a straightforward valence counting for GdSr2RuO6 and SrRuO3 yields Ru, as Ru +5 and Ru +4 ions respectively. The similarity between the XAS spectra of Ru-2122 and GdSr2RuO6 indicates clearly, that in Ru-2212 the Ru ions are in a pentavalent state. It is apparent that SC in the M-2122 system exists only for pentavalent M ions such as Nb, Ta and Ru. EXPERIMENTAL RESULTS

(I) The Effect of Oxygen on the SC and magnetic behavior of Ru-2122

The temperature dependence of the normalized resistance R(T) for the ASP and 22 atm. samples (measured at H=0 ) is shown in Fig. 2. The onset of the SC transition for the ASP (TC = 32 (0.5) K) is shifted to 38 K. At high temperatures, a metallic behavior is observed, and for the ASP sample, an applied field of 5 T smears the onset of SC and shifts it to 28 K. The SC transition for the ASP sample is more easily seen in the derivative dR/dT plotted in the inset. At TC = 32 K the derivative rises rapidly and does not fall to zero until the percolation temperature around 19 K is obtained. This behavior is typical for under-doped HTSC materials, where inhomogeneity in oxygen concentration causes a

Figure 2. Normalized resistivity measured at H=0 of the as prepared ASP Ru-2122 and the sample annealed under 22 oxygen atmosphere. The inset shows the derivative of the resistivity for the ASP sample.

distribution in the TC values. This distribution is also reflected in the broad range of gap values observed in our STS data, as shown below. The dependence of TC on the applied oxygen pressure obtained from resistivity measurements, is presented in Fig. 3, exhibiting a monotonic increase from 32 to 49 K. R(T) curves of the 75 atm. sample measured at various applied fields are shown in Fig. 4. In contrast to the ASP sample, an applied field of 5 T only smears the onset of SC at 46 K, but does not shift it to lower temperatures. The temperature dependence of dR/dT at 344

Figure 3. The effect of the annealing oxygen pressure on TC.

H=0 T, shows two peaks (Fig. 4 inset). This provides clear evidence for the two major SC phases having TC at 32 and 46 K, where the latter is below the percolation threshold. This is consistent with the STM data shown in Fig. 5. The spatial distribution of the SC gap on the surface of the ASP and 75 atm. samples are exhibited by the histograms in Fig. 5(a) and (b). The gaps were extracted by fitting the Dynes’ function [8] to tunneling I-V curves acquired at various lateral tip

positions. In the ASP sample, the I-V curves show an ohmic gap-less structure, and the values of range mainly between 3 and 5.5 meV. This broad distribution probably results from spatial variations in hole-doping, and is consistent with the broad SC transition exhibited in Fig. 2. In Fig. 5 (b), two peaks are clearly observed in the distribution, showing the existence of two SC phases, (a trace of the higher TC phase is already present

Figure 4. Normalized resistivity measured at various magnetic applied field of Ru-2122 sample annealed under 75 oxygen atmosphere. The inset shows the derivative of the resistivity curve at zero applied field.

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in the ASP sample). The ratio between the large and small gap values is 1.45, in agreement with the ratio between the two peaks extracted from Fig. 4 (inset). Note that the small-gap phase (lower T C ) is dominant, consistent with the fact that the higher TC phase should be below the percolation threshold.

Figure 5. Histograms showing the spatial distribution of the SC energy gaps for the ASP sample (a) and the sample annealed at 75 atm. oxygen (b). Inset: Two tunneling dI/dV vs. V curves obtained on the annealed

sample, one taken on a region of small gaps (dotted), the other on a large-gap region (solid).

Generally speaking, in Ru-2122, the temperature dependence of the magnetization =M/H) at low applied fields is composed of three contributions: (a) a negative moment

below TC due to SC state, (b) a positive moment due to the paramagnetic effective moment of Eu (or Gd) and (c) a contribution from the ferromagnetic-like behavior of the Ru planes. ZFC and FC magnetic measurements for all samples were performed over a broad range of applied magnetic fields, and typical M/H curves measured at 50 Oe., of the

Figure 6. ZFC and FC susceptibility curves measured at 50 Oe for the typical ferromagnetic hysteresis loop at 5 K.

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The inset shows

ASP and 75 atm. are shown in Figs.6-7. At 50 Oe. the diamagnetic signal due to high shielding fraction (SF) of the SC state dominates, and the net moment at low temperatures is negative. The weak ferromagnetic component of Ru and the high paramagnetic effective moment of Eu3+ do not permit a quantitative determination of the SF from these curves. TC can also easily be determined from the deflection points in ZFC curves. The two curves merge at Tirr=92 and 137 K, respectively, indicating the effect of oxygen on the magnetic behavior of the Ru sublattice. Note, that TM(Ru) is not at Tirr. The curves do not lend themselves to an easy determination of TM(Ru), and TM(Ru)=122 and 168 K, were obtained directly from the temperature dependence of the saturation moment (Ms),

discussed below. Isothermal magnetization measurements at various temperatures indicate that the Ru moment saturates around 5 kOe, therefore at this applied field, both, the anomalies and the irreversibility are washed out [3].

Figure 7. ZFC and FC susceptibility curves for Ru-2122 sample annealed under 75 oxygen atmosphere measured at 50 Oe.

Since, SC is confined to the CuO2 planes; therefore, all the magnetic anomalies in Figs. 6-7 are related to the Ru-O planes. The irreversibility at Tirr arises as a result of an antisymmetric exchange-coupling of the DM type between neighboring Ru moments, induced by a local distortion that breaks the tetragonal symmetry of the RuO6 octahedra. Due to this DM interaction, the external field causes the spins to cant slightly out of their original direction and to align a component of the moments with the direction of H. At low temperatures, the Ru-Ru and/or Eu(Gd)-Ru interactions begin to dominate, leading to reorientation of the Ru moments, and the peak in the ZFC branch is observed. The exact nature of the local structural distortions causing this reorientation is not presently known. and we assume that the magnetic DM exchange coupling (as well as the SC behavior) in the Ru-2122 system are extremely sensitive to oxygen concentration. The isothermal magnetization (M(H)) curves measured at various temperatures can also be divided into three parts. Fig. 8 shows the low field part for both the ASP and 75 atm materials measured at 5 K. The negative moments of the virgin curves increase up to 50 and 170 Oe, and the estimated SF (taking into account contributions from Ru and Eu3+) are ~30% and 65%, for the ASP and 75 atm samples respectively. All M(H) curves below TM,, are strongly dependent on the field up to 4-5 kOe, until a common slope is reached (Fig. 9 inset). M(H) can be described as: where Ms (the saturation moment) corresponds to the W-FM contribution of the Ru sublattice, and is the linear paramagnetic contribution of Eu and Cu. Ms decreases with increasing T, and becomes zero at TM(Ru)=122(2) and 168(2) K for the ASP and the 75 atm. samples respectively

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Figure 8. The isothermal magnetization as a function of the applied field in the low fields limit for the ASP

compound and the sample annealed at 75 atm.

(see Fig. 11). For the ASP sample the Ms values for the APS sample at 5 K) are larger than for the 75 atm. material. These values, are smaller than the fully saturated moment expected for low-spin state of Ru5+, i.e., for g=2 and S=0.5. This means that in the ordered state, some canting on adjacent Ru spins occurs, and the saturation moments at low temperatures are not the full moments of the Ru5+ ions. In the intermediate applied field region, a ferromagnetic-like hysteresis loop is opened (Fig. 6 inset) from which the two characteristic parameters: the coercive field (HC) and the remanent moment (Mrem) can be deduced. Fig. 9 shows, that for both materials, HC disappears around TC, (and not at Tirr and/or at TM) which strengthen our experimental evidence for the spontaneous vortex phase, described below. Mrem(T) for both samples disappears at Tirr (not shown). Below 20 K, the Mrem values for the ASP material are a bit higher than those of the 75 atm sample, (0.37 and 0.26 at 5 K), but for 20 0.07, whereas for the Ru-2122H0.03 these values are close to those of the APS sample. In contrast to the ASP sample, the ZFC signal has a large contribution from the positive moments due to the W-FM state, which mask the negative contribution due to the SC state. Hydrogen induces high porosity in these materials, affecting the macroscopic transport measurements, therefore, we could not extract the SC state properties, neither directly from the magnetic ZFC curves, nor from

four point resistivity measurements. These features were studied by the STM technique.

The picture emerges from the STM measurements [6] is that hydrogen doping indeed leads to phase separation. Even at very low doping (Ru-2122H0.03), insulating regions start to form. As doping is increased, the density and size of the insulating regions increase, until

they coalesce and the sample becomes globally insulating. Typical hysteresis loops opened below 5 kOe and are shown in Fig. 13. For Ru2122H0.07 both Ms, and Mrem values are higher than for the ASP

Figure 13. The low range of the hystersis loops at 5 K for the ASP material and for Ru-2122H0.07

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Figure 14. ZFC susceptibility curves for various hydrogen loaded samples.

sample. In the limit of uncertainty the HC values do not change much. Fig. 14 presents the ZFC curves obtained for all hydrogen loaded samples studied. For the ASP and the Ru2122H0.03 samples, the peak is around 80 K, and for the samples with H>0.14 at. the peaks are shifted to about 160 K. For the intermediate hydrogen concentration the (Ru-

2122H0.07), a superposition of both peaks is observed which leads to a somewhat flat curve.

Regeneration was made on the sample with the Ru-2122H0.14 and the magnetization measurements (carried out on powdered sample) prove that: (i) SC is restored, (ii) the peak in ZFC curve is shifted back to 80 K and (iii) Tirr ~92 K. Thus, all the “enhanced” parameters of the Ru-2122HX presented in Figs. (9-11), are reduced to the original values of the ASP compound [6].

Figure 15. Magneto-optic images of Eu-2122 measured at 10 and 90 K and H=0 and 50 Oe. Note the bright

area below TC at H=0.

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(III) Spontaneous Vortex Phase in Ru-2122

Fig. 15 shows MO images for Ru-2122 measured at 10 K (below TC) and 90 K at external fields of zero and 50 Oe. The triangular dark regions in the left and right hand sides of the pictures are due to magnetic domains of the indicator with reversed magnetization direction, and are not related to the flux density of the Ru-2122 sample. The bright area at 10 K and H=0 observed in the central part of the picture, indicates clearly that magnetic flux (vortices ?) is present in the sample in equilibrium. The internal field (of a few hundreds G) induces these flux lines in the SC planes (the mixed state), without an external field. At 90 K (TC 90 K, there is no sign whatsoever of magnetic order, indicating a weak coupling Fe -

Ru. Those measurements are completely different from the data presented in Figs 6-7. In addition, our STM topography and spectroscopy measurements, described above, are not consist, to say the least, with the mixed granular magnetic/SC picture. The physical behavior of the oxygen and hydrogen charged Ru-2122 YBa2Cu3O7 and YBa2Cu4O8 which have been studied extensively. Hole (or carrier) density in the CuO2 planes, or deviation of the nomimal Cu valence from Cu2+, is a primary parameter which governs TC in most of the HTSC compounds. Changes in the SC properties of YBa2Cu3O7 can be induced by either (I) removing oxygen or by (ii) hydrogen loading which is a destructive and irreversible. By depletion of hydrogen the crystal structure is destroyed and SC is not restored. It appears that in Ru-2122, the ASP compound, is under-doped, due to the fact that (a) annealing under high oxygen pressure shifts TC to higher temperatures (Figs. 2-3), and (b) the effect of an applied field is to reduce T C . It is not clear yet whether optimum doping is obtained with the 150 atm. sample. On the other hand, the influence of hydrogen on Ru-2122 is reversible and not destructive, which means that hydrogen changes the hole density of the Cu-O2 planes, either by increasing or decreasing the ideal effective charge of the planes. Depletion of hydrogen leads to the original charge density and SC is restored. This is reflected in both the macroscopic magnetization studies and in the SC gap distribution extracted from STM result. Data for the regenerated sample are not presented here.

Oxygen pressure, as well as hydrogenation, enhance TM and changes other W-FM characteristic features of the APS material. This effect, which was also observed in several rare-earth based intermetallic hydrides, is probably an electronic effect. As described above, in addition to the change in the hole density of the Cu-O planes, there is a transfer of electrons from hydrogen (or oxygen) to the Ru 4d sub-bands, resulting in an increase of the exchange interactions between the Ru sublattice and hence to an increase in TM of the materials. An alternative way is to assume that the change (enhancement in the case of hydrogen) in Msat, and Mrem arises from an alternation of the anti-symmetric exchange

coupling of the DM type between the adjacent Ru moments, which causes the spins to cant out of their original direction with a smaller (or larger) angle and as a result, a different component of the Ru moments forms the W-FM state. However, this scenario cannot reconcile the higher TM observed in all oxygen and hydrogen loaded materials. The exact nature of the local structure distortions causing the W-FM behavior in this system, as well as the oxygen and/or hydrogen location in the matrix, are not presently known and neutron diffraction studies are now being carried out to address these points. Since hydrogen loading affects both (SC and W-FM) phenomena, and the original behavior is restored when hydrogen is depleted, we tend to believe that H atoms occupy interstitial sites close to these planes, presumably inside the Sr-O planes. In conclusion, we have shown that both SC and weak-ferromagnetism coexist in RU2122 and are an intrinsic property of this system. In contrast to other intermetallic magnetic-SC systems, the present materials exhibit magnetic order well above the SC transition (TM/TC ~4). We attribute the magnetic order to the Ru sublattice, whereas SC is confined to the CuO2 planes. Both sites are practically decoupled from each other.

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ACKNOWLEDGMENTS

This research was supported by the BSF(1998) and the Klachky Foundation.

REFERENCES 1.

Cava, R.J., Krajewski, J.J., Takagi, H., Zandbergen, H.W., Van Dover, R.B., Peck Jr, W.F. and Hessen, B. (1992) Superconductivity at 28 K in a cuprate with niobium oxide intermediary layer, Physica C 191, 237-242.

2.

Bauernfeind, L., Widder, W. and Braun, H.F. (1995) Ruthenium-based layered cuprates, RuSr2LnCu2O8 and RuSr2Ln1+xCe1–xCu2O10 (Ln=Sm, Eu and Gd), Physica C 254, 151-158. Felner, I., Asaf, U., Levi, Y. and Millo, O. (1997) Coexistence of magnetism and superconductivity in

3.

R1.4Ce0.6RuSr2Cu2O10 (R=Eu and Gd), Phys. Rev. B 55, R3374-R3377. 4. 5.

Sonin, E.B. and Felner, I. (1998) Spontaneous vortex phase in a superconducting weak ferromagnet, Phys. Rev. B 57, R14000-R14003. Felner, I., Asaf, U., Goren, S. and Korn., C. (1998) Reversible effect of hydrogen on superconductivity and weak ferromagnetism in R1.4Ce0.6MSr2Cu2O10 (M=Nb and Ru), Phys. Rev. B 57,

6.

550-556. Felner, I., Asaf, U., Levi, Y. and Millo, O. (2000) Tuning of ferromagnetic behavior by oxygen and

7.

Pringle, D.J., Tallon, J.L., Walker, B.G. and Trodahl, H.J. (1999) Oxygen isotope effect on the critical

hydrogen in R1.5Ce0.5RuSr2Cu2O10, Physica C (in press). and Curie temperature and Raman modes in the ferromagnetic superconductor RuSr2GdCu2O8, Phys. Rev. B 59, R11679-R11682. 8.

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Dynes, R.C., Narayanamurti, V. and Garno, J.P. (1978) Direct measurement of quasiparticle-lifetime broadening in a strong-coupled superconductor, Phys. Rev. Lett. 41, 1509-1512.

QUANTUM PERCOLATION IN HIGH Tc SUPERCONDUCTORS

V.DALLACASA Laboratory for materials analysis, Department of Science and Technology, University of Verona, Strada Le Grazie, Verona, Italy

INTRODUCTION Order and periodicity of perfect crystals have been the leading aspects of solid state physics since 1970. The band theory, elaborated as an outcome of quantum mechanics, explains in a nice way the metallic character of copper as well as the one of semi-metal as graphite and of semiconductors as germanium and silicium. This theory asssumes that there is a regular arrangment of atoms in space and the potential experienced by any electron is periodic in space. However, disordered solids are the rule and not the exception and the science of materials and researchers have to deal massively with methods of treatment both theoretical and experimental appropriate. The corresponding electronic properties of such materials have for a long time discouraged researchers who have preferred to reduce, when possible, their study to methods employing some remnant form of order. This is even more true if one thinks to the immense impact played by crystalline materials of high purity in the electronic industry, but this does not mean that the disordered systems can be neglected. Despite their complexity arising from the spatial inhomogeneity of the potential seen by each electron, one can argue that the strongest the disorder the more uniform the crystal will appear in the average, since a global property like, say the conductivity, may be thought to result from repeated motion of electrons through inequivalent sites. This has as a consequence that the conductivity will have eventually comparable order of magnitude in all systems, a distinct feature with respect to a crystalline and pure system where it may easily vary by many orders of magnitude from metals, to semiconductors to insulators. The reason may be traced back to the fact that electronic states tend to be localized by disorder in stochastic positions in space and hence the movement will take place predominantly to nearest sites where localization occurs. It is a sort of quantum space on which electrons are permitted to move. On the contrary, the infinite extension of the wavefunction in a regular system will give the electrons chance to hop to more distant sites, compatibly with the scattering mechanisms involved. There is a consequence on the transport properties: in a regular medium the presence of phonons will be an adverse mechanism of movement, while in conditions of localization the opposite will take place, with the phonons aiding the proceess by supplying the energy necessary to overcome the energy difference between any two localized states. In an otherwise perfect medium there will be a diffusive motion

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

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through space, while in a disordered space only difficult hops will occur only with the assistance of phonons.

For an ensemble of sites of localization arranged stochastically in space, one basic problem is then be to understand the elementary hop between two sites, this may be modeled as a tunnelling process assisted by phonons. But actually, the more difficult problem is to follow the evolution of iterative and succesive hops. Percolation theory has proved of great value, since it has been shown that a problem like this is dominated by critical paths corresponding to easy transit between sites and since necessarily this problem if of statistical nature, we can understand why there will be at a macroscopic scale, greater independence on hopping details like molecular structure, crystal structure etc. On the contrary, great dependence on the dimensionality is expected since the hopping diffusion is expected to be quite different in one, two or three dimensions. LOCALIZATION AND PERCOLATION

When the degree of disorder is sufficiently strong, the electron wavefunctions will be localized, i.e they will decay exponentially from some point in space where is the localization length and r the distance from the point. The existence of localized states in presence of disorder is a consequence of the Ioffe-Regel criterium. In order that a state described by a wave packet be extended troughout the whole system a necessary condition is that the distance over which it loses coherence, i.e the mean free-path, be

longer than the interatomic spacing , i.e otherwise the state should be localized. As a function of energy then, states must change their character and the critical energy Ec at which this occurs is the mobility edge. Thus the mobility edge marks the transition from a metal to an insulator because for extended states the conductivity will have metallic character, while for localized states it will vanish at zero temperature. The scaling theory of localization of non interacting electrons and numerical estimates [1] have established that for dimensions d =3 there is transition between extended and localized states as the strength of disorder increases and correspondingly there is a

localization threshold On the contrary all states are localized in d=l and d=2 , which may be restated saying that the carrier density has a threshold at pc = The localization length is predicted to diverge when going from the localized side to the extended one, i.e on crossing the mobility edge, as where the critical exponent In general, as the carrier density increases, the Fermi energy approaches the threshold and hence the localization length tends to diverge in the vicinity of a MIT.

The theory of percolation aims to obtain quantitative estimates for the properties of a disordered system. In classical percolation [2] one considers a periodic lattice of sites each

of which can be randomly occupied with probability p or empty with probability 1 – p. Clusters, i.e a group of occupied sites containing neighboring occupied sites, will then be present. As the concentration increases from zero, larger and larger clusters appear. The mean size of these clusters grow with p and diverges at a well defined critical concentration pc. For there exists an “infinite cluster, which connects the two sides of an arbitrary large sample. In the limit of an infinite lattice the value of the percolation threshold pc is sharply defined. The infinite cluster percolates through the lattice with finite probability. For p geater than pc the infinite cluster coexists with smaller finite clusters which join it as p is further increased. If the above probabilities are referred to occupancy of a nearest bond, rather than a site, we have bond percolation. Although the percolation problem is easily defined it cannot be solved exactly. However, it has quite interesting properties, i.e universality (independance of details) a

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result of self-similarity, that is invariance under the change of length scale. These features are closely related to the special geometric structure of the infinite cluster at pc, which exhibits self-similarity, that is invariance under the change of length scale. The main appeal of percolation theory is based on the possibility of correlating the geometrical and topological properties with transport properties such as the electric conductivity, noise and optical properties. Quantum percolation can be formulated in terms of a tight binding one electron

hamiltonian [1] on a regular lattice. As in the classical case, we can define site and bond percolation .The main concern is the location of the percolation threshold pc below which all eigenstates of the hamiltonian are localized. The quantum threshold is greater than its classical counterpart since the existence of an infinite cluster is a necessary but not a sufficient condition for states to be extended. The most fundamental question of percolation theory is to predict the critical value of the percolation threshold, namely the value of the concentration at which an infinite network is first formed in the infinite lattice. Results for some type of lattices are reported [2] in Table 1. The numbers in parenthesis refer to quantum site and bond percolation thresholds [1] which in the case of two-dimensional lattices diverge according to the scaling hypotesis.

The mean cluster size S , i.e the average of the size of clusters around randomly selected occupied states in the lattice and the correlation length defined as the root mean square average distance between two randomly selected occupied sites in the same cluster diverge on approaching the threshold from below as and for p very close to pc, pc> p. Values of the critical exponents are reported [2] in Table 2.

The exponents reported are the same for all two-dimensional and three-dimensional lattices. This universality is what makes percolation theory appealing; there will be universality behaviour of all systems irrespective of their details. One notes that both the localization length and the correlation length have exponents of order 1. If interactions between particles are taken into account these exponents turn out to 1/2. When discussing percolation due to localization and to electron grains in granular metals, this difference will be evidenced.

TRANSPORT PROPERTIES It is now widely accepted that the parent state of the high Tc materials is an insulator showing long-range antiferromagnetism and that the doping process on such a primitive structure introduces holes/electrons either by cation substitution or by oxygen intercalation

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or by a combination of them [3]. When holes are sufficient in number, usually already at very low doping level of order 0.05 holes per Cu in the CuO2 planes, they destroy the antiferromagnetic state and superconductivity appears. Despite the disappearance of the long-range order, strong spin correlations persist up to quite large doping levels, even when the maximum Tc is obtained, the so-called optimum-doped region, and further. The passage

to superconductivity occurs at a critical doping where an insulator-to-metal transition first occurs followed by the superconducting state. The critical temperature increases with doping from the value zero up to a maximum value and then decreases with the further doping until its total disappearance again at un upper critical value of the doping. In the electron-doped system like Nd2–xCexCuO4–y the phase diagram is quite similar, the disappearance of the critical temperature at the higher doping still occurs, whereas there can be some coexistence of antiferromagnetism and superconductivity [3]. It has become costumary to call underdoped those specimens with doping lower that the optimal doping at which the maximum Tc is observed and overdoped those with higher number of carriers. It is generally agreed that the overdoped region is most similar to a Fermi liquid while the underdoped phase is dominated by insulation of the Mott-Hubbard type at the lowest doping levels. A notable feature of the phase diagram is that the superconducting phase occurs close to the insulating phase and that on increasing the number of carriers one has a transition from an

insulator to a metal-superconductor. Also noticeable is the fact that the appearance of superconductivity takes place at the lowest Tc values already in the insulating state. Similar data have been obtained in the La 2–xSrxCuO4, in YBa2Cu3O7–y, Bi2Sr2Ca1–xYxCu2O8+y,

Nd1+xBa2–xCu3O7–y and many others. The system appears to bifurcate between an insulating ground state and a superconducting state with no normal metallic state in between. The report by Bednorz and Muller [4] of superconductivity in Ba xLa5-xCu5O5(3–y) (x=1

and x=0.75,y=0) showed that superconductivity could be obtained in an insulating state. They attributed the onset of superconductivty in the 30K range to granularity and percolation and concluded that grains of dimension 100A should exist in their sample. The

resistivity in these samples has metallic character at higher temepartures, at temperatures slightly higher than Tc onset shows an upturn and on lowering further the temperature it undergoes a substantial drop into the superconducting state. It would become clear later on, through extensive transport studies in a variety of cuprates, that in fact superconductivity evolves from an insulating state in which localization induced by disorder and electronphonon interactions seem to play a major role. In the low-doping region [5,6,7,8]the temperature dependence of the resistivity in the normal state can be fitted by the law:

where the exponent

usually

assumes values ranging from ½ to ¼ (Table 3.). By increasing the number of holes there is a progressive tendency towards a “metallic state” with the resistivity losing the semiconducting behaviour and acquiring a monotonic tendency to rise with temperature. Close to optimum doping the resistivity shows a linear dependence in the normal state which may persist even at temperatures as high as T=1000K At still increased doping the transition superconducting temperature disappears while the resistivity tends to acquire a superlinear dependence of the form T1…5–2[9] which is interpreted as the appearance of a conventional Fermi liquid. A number of studies of transport properties as a function of temperature and doping [10,11] have indicated the progressive decrease of the characteristic temperatureT0 as a function of doping on approaching the metal-insulator transition (Table 4) In the underdoped and optimum doped region (with the exclusion of the hard insulating phase, where it can be shown that the exponential law, with and with

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suitable changes of T0 , is a fair representation of data even close to the superconducting state, and including also the linear behaviour.

These results are usually interpreted in terms of phonon-assisted hopping models, either of the Mott type or of the Efros-Shklovskii type (see next paragraph), enphasizing thus the

role of phonons. In such a parametrization the observed decrease of the characteristic temperature T0 can be traced back to the increase of the localization length As an order of magnitude one finds that in the insulator with a density of states at the Fermi level in the metallic state and close to the MIT values can be found. Another possible parametrization of data can be achieved through the theory of

granular metals (see nest paragraph); in this case the role of the localization length is assumed by the grain radius and similar orders of magnitudes for the latter are found.

In underdoped materials there are striking deviation from the linear behaviour. The resistivity assumes values less than linear and eventually can turn upwards at the smaller temperatures if superconductivity does not set in. We refer to ref. [9] for a summary of

behaviour in La2–xSrxCuO4. At the lowest temperatures the resistivity shows “semiconducting“ behaviour, a signature of localization. At intermediate temperatures the resistivity increases superlinearly up to a certain temperature , where a break occurs in the slope and the temperature dependence becomes almost linear. A reduced resistivity in YBa2Cu4O8, in YBa2Cu3O6+x and underdoped Hg1223 showing similar deviations from the Tlinear law have also been observed [3]. There is now some consensus on the fact that these

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deviations can be attributed to the opening of a pseudogap in the normal state, to which corresponds a reduced scattering rate. It is interesting to refer ref. [12] which reports the temperature obtained through various probes including Hall coefficient, static susceptibility and Knight shift and of course the resistivity and infrared measurements.

PERCOLATION IN HIGH Tc SUPERCONDUCTORS The idea of a quantum percolation model can be traced back to the most peculiar property of high Tc cuprates, namely that small changes in composition or structure can

change the material from semiconducting to metallic in the normal state and that there is a metal-insulator transition in the transport properties evolving from a hard insulating phase as the doping level is increased towards the critical point. Although being a probe of the average charge structure, the resistivity as a function of temperature and eventually of a magnetic field, is certainly a decisive parameter and in fact it is usually the first characterization made of samples. Phillips suggested a filamentary model [13] in which the metallic cuprate planes are broken up into metallic domains and interlayer defects provide electrical bridges which give the CuO2 layers metallic character. In their absence the layers would be striclty two dimensional and hence insulating, as predicted by the scaling theory, as a result of localization from disorder in 2D. In this quantum percolation theory the density of electronic states near the Fermi level is the contribution of localized and extended parts and only the extended states can become superconducting. In this modified electronic structure, as compared to a normal metal, called the “X” phase superconductivity is the result of the electron phonon interaction. As a result of a sufficient density of defects, the coupling in the planes can be sufficiently high to produce a high Tc, yet the lattice instabilities that would accompany such coupling if only extended states were present is in reality restrained by the “cage” of the localized states in the planes [14]. Within the filamentary model, normal state resistivities, optimized superconductivity with Tc and the lowest temperature for which linearity holds are explained [15]. Direct evidence of charge domains can be obtained from direct imaging in diffraction or similar studies. Early investigation of local structure through electron microscopy indicated i.e, inhomogeneous distribution of oxygen in the form of blocks, of typical linear size 100A, with different oxygen concentration in YBa2Cu3O7–x (x>0.5) single crystals [16]. Vacancy-ordering effects and the existence of their domains have been predicted theoretically and revealed experimentally. Computer model simulations have suggested the formation of oxygen-ordered domains in various cuprates including YBa2Cu3O6+x [17]. Electron diffraction studies in YBa2Cu3O7–x [18] and electron microscopy in YBa2Cu3O6.7 [19] have confirmed the presence of short-range oxygen-vacancy ordering within the CuO2 planes with dimensions of the order 100A . Mesot et al. [20], on employing inelastic neutron scattering in ErBa2Cu3Ox(6<x Tx with In

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and where

for T

Tx and

forT

Tx. Recently it was

shown that Efros-Shklovskii’s and Mott’s activation energies can be viewed as two terms of a multipolar expansion of the polarization charge on the sites, the R–1 term being the charge term and the R–3 term the quadrupolar term [38]. A R–2 term, which is viewed as the counterpart of the cubic term in two dimension, can be identified with the dipolar term of the expansion. A discussion of the cross-over including also the dipolar term has been given, showing that universality is maintained on rescaling Tx [38].

GRANULAR METALS The granular metals are materials composed of metals and insulators. These materials, known as cermets, were originally used as resistors due to their high resistivity at low temperatures, but they are now widely studied for their unique properties even at the submacroscopic scale 10-100A. There are three regimes of interest. The metallic regime occurs when the metal fraction x is large, the metallic grains touch and a metallic continuum exists with dielectric inclusions. The dielectric regime occurs when there is an inversion of the former where small metallic particles are dispersed in a dielectric continuum. Finally there is a transition regime corresponding to an intermediate state. In the dielectric regime with metallic islands dispersed in the dielectric the electrical conduction results from tunneling processes from one island to the other and thermal activation. The carriers are generated on removing an electron/hole from a metallic island and moving it to the other. Thus a pair of charged grains is created with a cost in energy of the order where d is radius of the grains and s is the wall thickness among grains and where the factor F takes account of the particular form and distance of

the grains. For suffciently large distances F = 1 leaving a purely coulombic barrier, but at closest distances the effects of the wall thickness among the grains may radically change F . Sheng et al. [39], taking the charging energy into account , have proposed that the resistivity at low applied fields should obey a law of the form In The hypotesis underlying this result is a “homogeneity” rule implying that to ensure spatial homogeneity of the samples it should hold that d/s=C=const. The dielectric regime thus is characterized by small radius of the grains and corresponding small intergrain wall thickness. This law for the conductivity is the consequence of a distribution of insular radius and an average of the conduction paths over the grain dimensions. In the transition region the metallic particles grow and there appear interconnections between them. At a critical composition it first appears a metallic continuum and conductivity is due to an infinite diffusive process. We review the theory of Sheng et al. [39], Our presentation of the problem makes use of concepts of the formulation of the tunnelling procesess introduced by Neugebauer and Webb [40]. The model assumes that there are a large number of metal islands with a relatively small number of them charged. The equilibrium concentration of these charges is maintained thermally. The probability that an electron will tunnel from one negatively charged island i to a neighbouring neutral island j is proportional to the density of states at each island and to the transmission coefficient, i.e (8)

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where D is a diffusion constant and the Fermi functions f and 1 – f take account that the initial state is full and the final one is empty. Both the forward and backward (with respect t o the direction of the field) probabilities are taken into account as well as the energy shift induced by the field energy eV and the charging energy Ec. The net transition probability will be P = P + – P – . The conductivity is related to P by the relation On assuming that both NF and D have a weak energy dependence, P can be obtained from a straighforward integration of (...). One finds at low fields (9)

We shall first discuss results valid in the case when the barrier thickness is large. The transmission coefficient can be obtained from the tunneling across the walls as

on assuming thus that where is a barrier parameter. The charging energy is given by ,using the homogeneity condition, and the distribution probability of the sizes of the islands can be taken of the form [39] P(s) = Asn exp(–s/s0) with n a suitable power (Sheng et al. assume n = 1 ) and s0 the average wall size. At low tempeartures Ec / kT

1 one then finds from eq. (9)

(10)

where

The upper limit can be pushed to

at small temperatures and the

integral involved in this expression can be reduced to a form calculable in terms of Bessel functions whith v = n . Thus at low T such that T0 / T 1 we find the resistivity (11)

At high temperatures the integral of the conductivity can be reduced to the the form, for sufficiently high n (12)

Thus, for T T0 the lower limit of the integral can be push down to zero and the result is again expressable in terms of Bessel functions. We get

(13)

where f is a number depending on the order of the Bessel function. There is a close mathematical analogy between the hopping model and the model of granular metals: in the former carriers make percolative transitions between localized states , in the latter the transitions occur between grains. In both models there is a tunneling factor as well as a thermal factor arising from an energetic barrier to overcome. Both models are

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based on random motion with the conductivity being an average through a distribution of

hopping sites or grain dimensions.. The conclusions for the two models are identical with the obvious significance of terms.In partciular we note the linear relation (13) for the resistivity with an intercept different from zero at T=0. Numerical simulation performed by Zhou et et. [41] indicates that the ½ power of the exponential law is obtained as an intermediate behaviour of the high T simply activated behaviour and the ¼-like Mott’s exponential law. This result indicates that while in hopping

processes usually the low temperature region is dominated by the T1/2 law, for the granular metals it is dominated by the T 1/4 law. In the transition region we expect and and one can assume that to account for power-law fall of the correlations with distance and emphasizing the increased probability of percolating paths with small s . At sufficiently small T we shall again find the conductivity behaving as eq. (..) i.e dominated by the exponential dependence; but at large T the leading term of the integral will be (14)

Since, due to normalization requirements for the distribution function, one has

we end with a superlinear behaviour of the resistivity with T. The physical origin of it is the spatial variation of the percolation probability at small s. SUPERCONDUCTIVITY

The anomalous phenomenology of high temperature superconductors forced a reconsideration of Fermi liquid behaviour, casting doubts on the the assumption of pairing resulting from phonon coupling and suggesting the investigation of a number of purely electronic or mixed models. In the light of the quantum percolation model, however,

phonon coupling occurring in a finite volume is worth of attention. As a result of the small coherence length in the cuprate superconductors there can be much more sensitivity to structural changes, local structures and even imperfections. Such a coupling may arise in localization regions either within the localization radius or in metallic clusters or grains within the metallic grains of finite volume. It turns out that, as experiments suggest, the relevant scales for optimal superconductivity temperatures are of the order 100A. In order to understand superconductivty coupling in these conditions, under the assumption that superconductivity is an effect of the domain interior and not of its surface, attempts were done to assimilate domains in first approximation to a small metallic particle and on concentrating mainly on the influence of the domain size on the electron system. The idea leading to this assumption is two-fold. First, the phonon indirect coupling of electrons is inversely proportional to the volume, secondly within the BCS theory adapted to the case of the finite dimension of localized states (or percolating clusters) the cutoff of momenta for which results in the separation of the levels of the finite cluster approximating such finite space and if this exceeds the attraction range, only levels at the Fermi enrgy will contribute, enhancing Tc The change of the phonon frequency, as compared to the bulk, can be taken into account by the introduction of an effective Debye frequency.

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The validity of the BCS theory rests upon the fact that the fluctuatios of the order parameter within the domain is negligible. The minimal allowed size and turns out to be of order 25A.

Another minimal length is the coherence length, typically 10 A. Thus for values of interest 100A of the domain size, these conditions are met. No domain interaction is considered likewise; predictions then are made only on the Tc onset corresponding to superconductivity in single uncorrelated domains and not on the lower Tc at which coherence between domains occurs. One finds increased critical tempeartures as compared to the bulk, due to the inverse dependence of the pair interaction on the volume and to quantization of levels. We present results for the Cooper problem of non interacting carriers. The study of the Cooper pairing problem in the finite system can be approached via a numerical procedure, assuming the usual form of the interaction

where

is the step

function and are the energy levels of the box representing the domain, while M 2 and are the electron-phonon matrix elements and the phonon cut-off frequency. The form of the coupling, essential for the results, i s M 2 =M 2 / N where N is the number of cells within the domain and M2 refers to the coupling in the bulk . For a system confined to a cube of length L there will be a set of degenerate states whose separation is

aroung

the Fermi surface.

The Cooper problem leads to the eigenvalue equation (15)

This equation can be solved using the form of the levels of a box.. The outcome of this is a single bound state and positive eigenvalues. For sufficiently dense levels within the attraction range (high L),the numerical result go over to the analytical Cooper result. The limit depends on the strength of the coupling. An increase of Tc on decreasing the length L is obtained, the effects becoming stronger at smaller lengths. In general, one finds a region at large L in which Tc is almost constant and approaches the Cooper limit, and a region at small L in which a rapid increase of Tc occurs as L decreases. At the minimum length of the numerical procedure one can find Tc of the order of some 100K under conditions of weak coupling The cross-over can be estimated to occur when the separation of levels becomes countably small within the attraction range.

The asymptotic behaviour at small and large lengths can be obtained analytically. The procedure can be implemented directly on the BCS equations rather than on the Cooper problem. The limiting results for Tc are (16)

(17)

where the limiting length is given by Similar calculation can be carried out for the gap The second of these equations indicate a scaling of the critical temperature with the inverse dimension of the percolating clusters and a direct relation with the Fermi level. These are two competing factors: when

370

the carrier density increases, the Fermi energy increases while N decreases. The result will be a compromise with the Tc exhibiting a maximum at some carrier density.

Results for interacting electrons within the coulomb gap can be obtained by solving the BCS equations with the density of states of the gap: for and for with where n is the static dielectric constant. For

the results indicate a proportionality of the critical temperature to the

coulomb gap, i.e

which displays a similar dependence on the carrier density as in

the non-interacting case at small L.

It appears quite natural to assume that L correspond to the coherence length of the quantum percolation states, i.e the mean square radius of the percolating-localized clusters. We then can take below the percolation threshold in which is a critical exponent. Thus assuming that the Fermi energy scales as for a two-dimensional electron gas we get the formula

as a convenient parametrization

formula of the critical temperature. This equation predicts in particular that in the underdoped region where m* is the carrier effective mass, a result in agreement with the findings of [42]. At larger n the critical temperature will be a dome-shoped curve with a maximum at and Analisis [42]of avalilable data in and systems as well as and Cheverel phases have indicated that the critical exponent can be accurately determined. The data strongly indicate an extrapolation of to zero for a critical number of holes in the planes, i.e the existence of a threshold. From the fitting procedure values are obtained with the exclusion of the Chevrel systems for which is obtained. The percolative threshold value established by these data is per plane. Thus, there is a threshold which indicates a three-dimensional character of the percolative network This means that although the planes are expected to provide the superconducting carriers, the critical temperature is influenced by out-of-plane effects[13]. Low Tc systems as the SrTiO and Chevrel systems appear to display characteristics similar to the high materials, perhaps indicating that the mechanism analized here is not peculiar to cuprates. The value of the critical exponent indicates coulomb effects, i.e an interacting carrier gas. This result is in agreement with quantum percolation with interaction and an analysis of Kaveh and Mott [43]who have indicated how this critical exponent changes from the value 1 to the value ½ in the vicinity of a superconducting state as a result of coulomb interactions. For the Cheverel systems we get indicative, on the contrary of a non-interacting carrier gas. The value of the threshold agrees with the one expected for simple cubic strucures although the complicated nature of the unit cell of cuprates is difficult to reduce to a simple known lattice, for which percolation thresholds have been calculated. The typical percolation sizes can be estimated around L=100A depending of the values typically used for the Fermi energy. Previous results of the effect of altered wavefunctions as a result of reduced dimensions have been considered in connection with superconductivity in thin films; the gap parameter is found to increase with decreasing film thickness [43]. The application of the BCS theory to finite sysytems has been considered in connection with its mathematical limit when the size becomes infinite [44].

371

SUMMARY AND CONCLUSIONS

We have reviewed a quantum percolation model in which charge clusters below the percolation threshold undergo phonon-asssited hopping processes. We have indicated theoretical and experimental evidence of the existence of such clusters and discussed two possible mechanisms which appear to be consistent with the phenomenology of high Tc superconductors. These are: a phonon-assisted hopping model relying on Anderson’s localization by disorder and a granular metal model in which hopping occurs on charged grains. In both models the role of coulomb interactions occurring during hops between localized wavefunctions or charged grains has been emphasized. The principal parameters of quantum percolation is the localization length/grain radius, which defines the value of the gap parameter whose increase on increasing the carrier concentration describes the evolution of transport properties in the whole region of the phase diagram. At very low doping, the insulating phase can be described by variable range hopping of the Mott’s type. For larger doping, in the overdoped and optimum doped region eqs. (6) and (7) relative to coulomb hopping or (11) and (12) for the granular metals description apply. One finds a low temperature regime eq.(7) with an insulating exponential variation at the lowest temperatures, followed by a superlinear behaviour at moderately higher temperatures with the curves appear to rise linearly with temperature with zero intercept at the origin; a typical high limit to this behaviour being 300K. In the high temperature range the curves are described by eq. (6) and are dominated by the pre-exponential so that a linear relation with T occurs, but with an intercept at the origin proportional to as evidenced by eq. (13), so the curves appear as if there were a lower slope at high T. Since decreases as the relevant length increases, on increasing the carrier concentration the linear behaviour will be observed in a larger temperature interval down to zero and it will progressively lead to the disappearance of the “superlinear” behaviour These features are found to describe quite accurately the superlinear and linear behaviour of the resistivity in the underdoped an optimum doped regions of the phase diagram and give a plausible origin to the pseudogap as observed in transport properties.. It follows that the existence of the coulomb gap in the underdoped region may also explain why a gap is detected in the normal state by external probes, like photoemission, giving the impression of persistence of the superconducting gap inside the normal state. The high critical temperatures are understood within the quantum percolation model as a result of electron-phonon coupling in the microscopic clusters of charge. Two competing factors affect it, namely the Fermi energy leading to its increase and the length of the percolating cluster, leading to a decrease with carrier density. The result is a maximum allowed value. The relevant cluster size at which is of the order required in the cuprates is L=100A. The proportionality of to the coulomb gap establishes a situation in which there is a coincidence of the gap in the superconductor, arising from the Cooper pairing, and in the normal state, in which it corresponds to the vanishing of the density of states at the Fermi level. REFERENCES 1.

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SUPERSTRIPES Self organization of quantum wires in high Tc superconductors

A. BIANCONI1, D. DI CASTRO1, N. L. SAINI1 and G. BIANCONI2 1

Unità INFM, Dipartimento di Fisica, Università di Roma La Sapienza, 00185 Roma, Italy

2

Department of Physics, Notre Dame University, 46556 Notre Dame, Indiana

INTRODUCTION High Tc cuprate perovskites provide an exotic superconducting phase at half way between absolute zero temperature and room temperature. Conventional superconductivity appears in metals with a very high charge density and near absolute zero temperature. In these materials the electrons in the normal phase, above the critical temperature Tc can be considered as free particles following the Fermi statistics (fermions) being in the highdensity limit and at low temperature. The electrons are described by a single particle wavefunction that gives the probability to find an electron in the point r. Below Tc electron pairs condense into a single quantum state. The condensate is described by the order parameter where gives the density of condensed pairs and is the phase. This macroscopic quantum state is characterized by exceptional manifestation of the quantum order: zero resistivity [1], perfect diamagnetism [2], quantization of magnetic flux [3,4] and quantum interference effects [5]. The wavefunction of the condesate decays exponentially as we go from the surface of the material to the vacuum with the Pippard coherence length [6] and the magnetic field decays exponentially [7] as we go from the surface into the material with the London penetration length The formation of the condensate made of electron pairs has been described by the BCS theory [8]. The key point of the BCS theory is that the formation of the condensate is due to the fact that electrons actually are not free particles but they are interacting; however the interaction is much smaller than the Fermi energy. In this weak coupling limit the interacting electrons are replaced by Landau quasiparticles. The very small electronelectron attraction triggers the formation of pairs of quasiparticles, with zero momentum and zero spin. The standard BCS theory assumes that the electron-phonon interaction provides the mechanism for the pairing however the pairing can also be mediated by electronic excitations (excitonic or plasmon mechanisms) in the low density limit.

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

375

In the weak coupling limit the critical temperature Tc is related with the energy needed to break the pair (1) where is the superconducting energy gap. The critical temperature (and the gap) is given by: (2)

where T F is Fermi temperature,

is the wavevector of electrons at the Fermi

level, is the coherence length of the condesate that is related with the size of the pair and f is a measure of the deviation from the weak coupling limit The BCS approximations are valid for a 3D metal with critical temperature close to zero Kelvin. In the strong coupling limit the critical temperature for the many body superconducting phase remains low since the pairs form Bose particles at high temperature but the phase coherence of the Bose condensate occurs only at low temperature. The discovery of high Tc superconductivity [9] in copper oxide perovskites, with a

record of TC~150K in Hg Ba2Ca2Cu3O8+y [10] has clearly shown that the superconducting condensate can be formed beyond the standard BCS approximations. The mechanism driving the superconducting state from the range 0 -over the associated next nearest sites j. The matrix elements of the electron-phonon

interaction is equal to (2)

For the acoustical phonons (the dispersion relation

where s is a velocity of the function const while for the Jahn-Teller phonons The weak dispersion is associated with the elastic interaction between neighboring octahedra. To describe these multi-electron strings we employ a variational Hartree-Fock(HF) manybody wave function and consider adiabatic and antiadiabatic approximations, separately. sound) and for dispersionless optical (Holstein) phonons

ADIABATIC APPROXIMATION

First for an adiabatic limit it is convenient to approach this problem working in the first quantization form (see, for example, in Ref.[8,9] To build up such HF approximation for the ST phenomenon we have to find relevant single particle wave functions. The discrete Schrödinger equations describing a single electron(hole) interacting with a lattice potential matrix associated with the strain deformation, and Jahn-Teller distortions in a tight-binding model on a hypercubic lattice has the form: (3)

where is a hopping integral matrix, for similicity we will consider here only a diagonal matrix The operator is a lattice version of the Laplacian operator which for the hypercubic lattice is defined as: (4)

391

where the summation is carried out over all the nearest-neighbor sites around the n-th site; is the wave function of the p–th self-trapped electron(hole) on the n–th site associated with the orbital and we use the units where a = 1. JAHN-TELLER PHONONS AND STRING SOLUTIONS

When an electron or a hole is interacting with Jahn-Teller distortions of the lattice there may arise the similar electron strings. The Jahn-Teller distortions are typical for oxides and manganites such as La2CuO4 and La2MnO3 or other their relatives. In both cases there are oktahedron configurations of CuO6. At the presence of the cubic symmetry for the ions Cu2+ they are associated with the degenerate Jahn-Teller term Eg. The presence of atoms La descreases the cubic symmetry to tetragonal, due to Jahn-Teller effect and, in principle, removes the degeneracy. In this case the Jahn-Teller Hamiltonian of the electron-phonon interaction has the form[31,32]: (5)

where this matrix acts on the coefficients and of the two-component electron wave function expanded with respect to the symmetry basis The value D is the constant of deformation potential created by the defortmation

where are unit basis vectors of the hypercubic lattice. This model is very different from Holstein-Hubbard model used, for example, in Ref.[33]. With the use of this Hamiltonian, the Shrödinger equations have the form: (6) (7)

With the aid of these Shrödinger equations and taking into account the Hamiltonian of the elastic and Jahn-Teller lattice distortions which has the form:

(8)

where in the elastic interaction between neighboring octahedra the values are defined as and < n, m> are the nearest neighbor pairs of octahedra along the direction and (see, for example in Ref.[32]). We may built up the total adiabatic potential of the lattice, which also includes the Coulomb and exchange repulsion between the self-trapped electrons(holes) as an extra term VHF which will be explicitely given lately:

(9)

where the Hartree-Fock term must be modified properly to take into account the two component many body wave function. 392

In adiabatic approximation neglecting the atomic kinetic energy after a minimization of H with resepect to Q3,n and Q2,n we have (10) (11)

and by a minimization of H with respect to the deformation we obtain the system of discrete equations describing elastic strain deformations created by M electrons(holes) in an approximation of isotropic elastic medium: (12) where the index p indicates a summation carried over all M trapped electrons(holes). After the solution of these equations with respect to uknown deformations and next substitution these found expressions into the Shrödinger equations obtain a new system of complicated nonlinear equations. The system is immediately simplified if we put K' = 0. Then we have the following system of nonlinear equations:

(13) (14)

where

and another

couple of equations for the complex conjugation of these wave function. This system of four

equations may be simplified to the case to a single component nonlinear Shrödinger equation. So, for example, if we put while we recover the conventional nonlinear Shrödinger equations[8,9]. Another self-consistent substitution is which symplifies this system to the same conventional discrete nonlinear Shrödinger equations discussed in Refs.[8,9]. In general these two solutions are following from a self-consistent substitution

which remains the invariant the strain tensor: and The value of the parameter may be found by a minimization of the total energy associated with JahnTeller distortions which gives that the value After the complete exclusion of phonon variables the extremal points (minima and maxima) of total adiabatic potential, the Hamiltonian Htotal are determined then with the aid of the following nonlinear Schrödinger equations(NSE): (15)

where the operator is defined as and the coupling constant, c', is defined as c' = The described wave functions associated with fermions self-trapped into the string correspond to the following eigenvalues: (16) where d is the dimension of the hypercubic lattice containing the string and the constant JN is a dimensionless integral. When the string is embedded into a 3D atomic lattice the integral, JN, takes the form: (17) 393

where

is a 3D lattice Greens function for a Jahn-

Teller deformations and the value In the limit and b 1 the NSE allows the following assymptotically exact string solutions, in which the M electron(holes) are trapped by N neighboring sites with equal probability, 1/N: (18)

where kx is the momentum of the p–th electron. We assume that the string is oriented in the x direction and is located on the sites starting from nx = 1 to nx = N. Inside the string each trapped electron(hole) has a free motion along the string with the momenta k oriented in the direction of the string. Such plane waves localized inside the string correspond to the following eigenvalues, E, of NSE: (19)

where d is the dimension of the hypercubic lattice containing the string and in this solution we have to define the new coupling constant The electron (hole) momentum k is simply quantized if we assume that the trapped current carriers are spinless fermions and that their wave function satisfies periodic boundary conditions (PBC) along the string. If we employ other boundary conditions for the trapped electrons (for example, open boundary conditions) the main result will not be changed drastically, although in this case the electron density along the string may become inhomogeneous. With the use of PBC the electron momenta along the string are quantized: k nx = /(aN). With the use of these eigenvalues and the Pauli exclusion principle we calculate the assymptotically exact expression for the adiabatic potential A N,M , describing M trapped electrons(holes): (20)

From this expression for AN,M(d) one sees that for a single electron (hole), i.e. M = 1, the lowest energy corresponds to the string with one site, i.e. N = 1. The existence of such a single site state was noticed by Rashba and Holstein[15-22] but from different arguments. With the increase of the number of trapped particles, M, while the deformational energy increases ~M2, the value of the adiabatic potential AN,M(d) decreases ~ – M 2 . This indicates on the electronic phase separation. However, such an electron(hole) phase separation is strongly prevented by Coulomb forces between the trapped current carriers. The energy of the Coulomb repulsion is minimal for the maximal separation of individual electrons (holes). The adiabatic potential of M separated particles is equal to (21)

From the comparison of these eqs. for J and Asep we may conclude that the energy of the single string AN,M(d) coincides with Asep if the number of trapped particle, M, is equal to the length of the string, N. In the case if we would have spinful particles and the double occupancy will be allowed the energy of a string with a greater density of particles, with M > N, will be lower than Asep since AN,M(d) decreases faster with M than the energy of the individual self-trapped particles Asep. This indicates that the separate individual self-trapped particles may be unstable and so an electron string may be created. The less dense strings (with M < N) correspond to metastable minima. All these results are based on the assymptotically exact solutions obtained in the limit 394

LONG-RANGE COULOMB FORCES

A Coulomb repulsion between the electrons(holes) increases when the interparticle distance decreases and, therefore, is acting against the density increase, i.e. against the phase separation. However, it may not completely overcome the phase separation but only stabilizes the strings of a finite length. With the Coulomb interaction taking into account the model of non-interacting electrons, whose effective interaction was initially introduced only by lattice vibrations, is modified by an addition of a new term in the Hamiltonian associated with the two particle Coulomb interaction. The new modified model (with the Coulomb interaction) may be treated within the Hartree-Fock approximation. Then the extremal points minima and maxima) of adiabatic potential A is determined with the aid of appropriate Hartree-Fock equations, which consist of the eqs (6) with the addition of appropriate Hartree-Fock terms. At large number of the particles trapped (Mc/t >> 1) these modified equations have exact string solutions, in which the M electrons are trapped by N neighboring sites with equal probability, 1/N. The solution (the many-body wave function) has the form of a Slater determinant of free single particle wave functions. The eigenvalues of these new equations modified by Hartree-Fock terms are different, of course, from those presented in eq.(8), have more complicated and tedious form. However, the total energy (adiabatic potential) is modified only by an addition of the Hartree-Fock term VHF obtained self-consistently with the use of the obtained, assymptotically exact, solutions of Hartree-Fock equations. This term depends solely on N and M and is given by: (22) where and the paramethter is the effective dielectric constant which may be equal to a static or a high-frequency dielectric constant depending on the ionicity of the solid. So for solids like metallic oxides we may take as due to a strong participation of polar (longitudinal optical) phonons into the screening of the electron-electron interaction. Then, the function VHF may be approximated as

(23)

and behaves similar to that obtained in the electrostatic approximation[9]: (24)

The total energy consisting of the adiabatic potential ANM(d) and the energy of the Coulomb repulsion VHF equals: (25)

where ANM(d) is defined in eq. for adiabatic potential. From the presented expression for ES, one sees that for a string of fixed length N the total energy always has a minimum given by (26)

The optimal number of particles trapped in the string of fixed length N is determined by a minimization of ES with respect to M and is given by the eq.: (27)

395

ANTIADIABATIC APPROACH

The similar expression for the length of the string valid even beyond adiabatic limit may be obtained[9] with the use of Lang-Firsov unitary transformation[34] which transforms the Hamiltonian, H, into the form:

(28)

where S= i ni[wi(q)bq–h.c.], the hopping integral = t exp( [w i (q)–w j (q)]b q –h.c.), the polaron shift and the effective inter-particles interaction is (29)

Then, with the use of this expression the total energy of the string when M = N, and when t 0, is equal to (30) The minimization of this expression with respect to the value M gives the equation for the

number M The same result may be obtained for the dispersionless optical (Holstein) phonons. However for Jahn-Teller phonons (in a contrast with acoustical phonons) there arises a weak dispersion like (with q – independent matrix element there a weak attraction between particles on next-neighboring sites will be generated. This will give an extra contribution into the total energy, as: (31) The minimization of this expression with respect to the value M gives the length of the string N = M as (32)

This equation gives much longer value for the length of the string than it was recently estimated in the same model in Ref.[35]. The different value for the estimation of N in Ref.[35] originates in incorrect approximation for the total energy, where the contribution from the polaron shift equal to cM/2 has been missed. From these two approaches (adiabatic and antiadiabatic) we obtain that the minimum of the total energy corresponds to a string of arbitrary length N and with M trapped particles, the logarithm of which is proportional to the electron-phonon interaction, c, or to the polaron shift Ep and is inversely proportional to the intersite Coulomb repulsion between holes, εc. On the other hand the total energy ES(M) at large values of M decreases strongly with M and increases with N. In the case when the double occupation of the sites is prohibited, the number M can not be larger than N. Then the minimum energy ES corresponds to the relation N = M. With the use of this relation and eq. for M the expression for ES–min, is simplified to the form: (33) The comparison of these equations indicates that a string with M trapped charged particles may have a lower energy than the total energy of M separated self-trapped particles if < c. However, the optimal length becomes very small N < 1, that indicates that in this 396

case there is realized a marginal extremum with N = 1. In the opposite case when the smooth minimum of the total energy associated with string does exist but corresponds to a

metastable state. THE STRING ENERGY FOR JAHN-TELLER PHONONS

With the use of the many body wave function obtained in the course of exact solution in the limit of strong coupling with phonons here we have estimated an expectation value of the Hamiltonian H. These calculations have been done in two steps. Using this many-body wave function, first, we calculated the one body and the pair correlation functions. Then with the use of the adiabatic approximation we have excluded slow (classical) phonon variables to get an expression for adiabatic potential ES including the Coulomb and exchange energies (see, for details, Refs[8,9]. The calculated expression of the total energy ES per particle has the form: (34) where d is a dimension of the hypercubic lattice, the value n is the electron(hole) doping inside the string: n = M/N and the value with a as an interatomic distance; in the Hamiltonian the coupling constant of interaction with acoustical phonons c = D2/K where D is a deformational potential and K is an elastic modulus or for Jahn-Teller phonons we define c = / ( c 1 1 – c12). The first three terms in the r.h.s. of this eq. are associated with electron kinetic energy while the last two terms in the r.h.s. of the same eq. are associated with the energies of electron-phonon and electron-electron interactions, respectively (see, for comparison, in Ref.[8]). This expression represents a variational estimation of the total energy of M particles self-trapped into a string of length N valid for a wide range of values of c/t since it was obtained on the basis of an exact solution found in the limit of very strong coupling c/t 1. Therefore, in the framework of this variational approach we may get a reliable estimation of the number of particles, the length and the energy of an electron string valid for a wide range of the parameters of the Hamiltonian such as a coupling constant

c, the bandwidth t and the characteristic Coulomb energy Here the values M and n are variational parameters. The optimal number of particles trapped into the string of fixed length N is determined by a minimization of ES /M with respect to M and is given by: (35)

After a substitution of this expression into eq. for ES we get the dependence ES = ES(n) on the doping of the string n = M/N. Depending on the relation between the values of t, c and ec there may exist one or two types of solutions which correspond to two different types of

strings: when n = 1 we define an insulating string and when n < 1 we define a conducting string. When c ~ t > the conducting string may be in a ground state. Then the number of particles trapped into the string is described by last eq. and the value of the string doping must be determined numerically by next minimization of the total energy with respect to the value n. When the coupling constant c is very large (c t and c> ) the obtained eq. is not applicable, since the associated solution describing a conducting string disappears while the other solution associated with the marginal extremum n = 1 and describing insulating strings still exists. The insulating strings have been already discussed in previous sections. ANTI-FERROMAGNETIC CORRELATIONS

Additional factors which may enhance the string formation are the polaron effect[9] 397

which arises in polar semiconductors and the exchange next-neighbor spin-spin interaction

which arises in doped antiferromagnet. The metastable minimum associated with the deformational string may become an absolute minimum in the doped antiferromagnet. For a single hole in the antiferromagnet there is an increase in the exchange energy equal to 2dJex, where Jex is an exchange constant. For M separated holes this energy increase is equal to 2dMJex. On the other hand for M holes trapped in a string such increase in exchange energy is equal to Jex(2dM –M+1). Therefore, the total energy of the deformational string in a doped antiferromagnet is described by the eq.:

(36) where the value M is defined by eq. for M. The comparison of this expression with the total energy of M separated self-trapped particles indicates that the strings may have a lower energy if the following inequality holds:

(37) Thus, the exchange interaction significantly improves the physical conditions required for string formation in doped antiferromagnets. Therefore, if this conditions holds at low temperatures the M separated particles will condense into a string configuration. For an arbitrary number of particle in the system there may be created many strings or an array of these electron strings. This array may be in ordered or in disordered state. Probably, at some critical concentration there arise a percolation between these strings and the strings may be ordered into charged stripes. In general this percolation picture reminds the filamantary microstructures suggested by Phillips[36] with the difference, however, that there the filaments are conducting, like conducting strings decribed in Ref.[9].

STRINGS IN IONIC SOLIDS In previous sections we have discussed the string solutions found for deformational type of strings and also in the case when the electron(hole) is interacting with Jahn- Teller phonons [31]. The obtained results are applicable, in general, for any type of short-range electronphonon interaction as, for example, with Holstein optical phonons. The number of particles in the string is defined above and nearly equal to the number of sites in the string. The string

length depends on the type of the string and for conducting strings must be estimated by a minimization of ES(n) with respect to n, numerically. For each type of phonons which have a short-range interaction with electrons(holes) the coupling constant in eqs. above must be defined, respectively, while the main eqs. for a number of particles and the length of the string remain the same (for more details, see Ref[8,9]) The case when a single electron or hole is interacting with polar phonons, i.e. with longitudinal optical phonons with frequency coo (and with the constant of the electron- phonon interaction ( see, for example, in Refs.[15-22]) is relevant and important to most oxides having a considerable amount of ionic bonding. Here the value of total energy including the Coulomb and exchange contributions from the long-range Coulomb forces between fermions may be calculated analogously to the case of short-range electronphonon interaction presented above (see, also for example, in the Refs[8,9] That is, first, with the aid of the Hartree-Fock many-body wave function of the M self-trapped particles (1,2,...,M) (see, eq. for the many body function) we have calculated the pair and offdiagonal correlation functions, and then with the use of these functions the dependence of the total energy on n and M having the form:

(38) 398

where we have introduced the notations The first two terms in the r.h.s. of this eq. are associated with the electron kinetic energy while the other terms in the r.h.s. of this eq. are associated with the energies of electron-phonon (~ Ep) and electronelectron (~ Ec)interactions, respectively. A minimization of this expression with respect to M and n gives an estimation for the length of the string N and the number of particles M trapped into the string. For the value of M we get the analytic expression: (39)

The value of the doping n may be calculated numerically. In the limit of a low density n the values of M and N or n may be presented by the analytic formulae:

1

(40) where The total energy of the string per electron equals jstring= 2d – 2 + 2/N – nEc. To be in a ground state this string energy must be smaller than the energy of an individual

polaron jp equal to 2d–Ep. The comparison of these two energies gives the precise criterion for the string formation. The conducting string corresponds to the ground state iff (41) which roughly means that the polaron shift must be smaller than the string bandwidth 2t. Thus, we arrive at the conclusion that in oxide compounds with ionic bonding the formation of highly conducting electron strings created by a polarization potential is possible. The

string length is typically much larger than the number of self-trapped holes, which is determined by the dielectric constants of the solid. Below in this table we present the parameters for the electron strings calculated only taken the longitudinal optical phonons into account. Table 1. Approximate number of electrons M trapped by a string consisting of N lattice sites in Oxides due to a polarization Compound La2MnO3 La2CuO4 TiO2 SrTiO3 WO3 M 3 7 15 34 63 N

5

40

40

40

160

APPLICATION TO HTSC

Thus, we arrive at the conclusions that in polar oxide materials, like HTSC there may arise electronic strings which are linear multi-particle “electronic molecules”. At low temperatures the electron strings may be ordered in CuO planes creating a nematic liquid crystal. The striped phase in HTSC and in manganites observed in numerous experiments[41-47] may correspond to such a liquid crystal of conducting strings. With the doping of an antiferromagnet La2CuO4 there arises only the change in the distance between the strings while the structure of the strings (like, the string doping n or the length N) is not changed. The metal-

lic stripe phase arises due to a correlated percolation over these strings when a density of such strings will be larger than the percolation threshold. For square lattices the percolation threshold is well known and is equal to xc ~ .5 . Then, using this value and our estimation for the string doping in La2CuO4 as n = 7/40 we may readily get the hole doping = nxc .09 of the antiferromagnet La2CuO4 at which the metallic stripe phase may arise. The spin-spin 399

and hole-spin correlations will of course slightly modify this result. It seems that our conclusion about the important contribution of the phonons into the origin of the stripe phase is confirmed in recent experiments which discover a huge influence of isotope effect on the critical temperature of the stripe ordering and strong lattice fluctuations in YBCO which may be associated with the dynamics of the strings. With the isotope changes the structure of individual strings is changed (for example, the strings become shorter) and, therefore, the critical temperature of the stripe ordering must change. In summary, we find that in oxides HTSC there may arise highly-conducting electron strings which are linear electronic molecules. Note that to find such molecules we have to treat the kinetic and potential energies of electrons on equal footing. A single electronic molecule has a cigar shape with the length of the order of 10-20 nanometers and consisting of 7-10 holes. For other oxides the string parameters will not be changed as much. It is also very natural that such “polymeric” electron molecules may form a liquid crystal which may be associated with the stripe phase of HTSC. The described linear strings with M = N have a much lower energy than 2D sheet configurations like a single circular or square spot consisting of M sites with M self-trapped electrons (holes). It is clear that while in the single sheet spot of arbitrary shape with the same number of particles, the phonon contribution remains the same the contribution of the Coulomb interaction increases compared to the string shape. The string may not only have a linear form but may also be bent, curved or even create a closed loop. Such curved configurations will probably correspond to low energy excitations of the string. Thus, we arrive at the conclusion that in narrow band doped antiferromagnet there may arise an electronic phase separation in a form of linear electronic defects-strings. That is the motion of free particles becomes unstable with the formation of strings which are linear multiparticle objects. These strings are created by a deformation potential and have a length equal to the number of self-trapped particles, which is determined by the elastic and deformational constants of the doped antiferromagnet. Our findings are probably relevant to stripe formation observed in HTSC[37-47] While here we have discussed insulating strings the stripes may have an origin in other type of the hole conducting strings arising for a small hole doping as described in Ref[9]. With an increase of the hole doping the concentration of these strings increases and they may form either a nematic liquid crystal or a superlattice of stripes. STRINGS IN MANGANITES Although the precise qualitative and quantitative picture of physics in manganites must

be worked out in the framework of a double exchange model a qualitative wave-hand scenario of the Colossal Magnetoresistance effect may be readily given. Indeed, these materials consist of magnetic ions embedded into a 3D atomic perovskite lattice which may be described by a double exchange model with a strong Hund’s coupling. From a general point of view there the parameter and the onsite Coulomb repulsion and/or Hund’s interaction are very large so that as an plausible approximation we may consider only spinless fermions associated with the charge degrees of freedom induced by an infinite strong Hund’s coupling. Let us estimate the length of these strings in the manganite materials. If we take in a conventional way the elastic module is such a material of the order c11 = 111010erg/cm3, the interatomic distance a 4Å then we get K ~ 4.1eV. The deformation potential may be conventionally approximated as D e2/a = 3.4eV then for the electron-phonon coupling we obtain that c = D2/K = 2.5eV. If we take the dielectric constant for LaMnO3 found by an analysis of experimental data in (see, Refs[10,l 1]) as =5, then we get that V e2/ = 0.68eV. This estimation gives that the length of the string will be of the order 10 interatromic distances. Then, if we suggest that there is a coexistence of a droplets with Fermi liquid of the

polarised strongly-correlated individual particles with the regions of the phase consisting of

400

insulating strings similar to that which was observed in Ref.[12], then CMR phenomenon could be described as follows. At lower temperatures the ground state energy may correspond to a droplets with Fermi liquid of the polarised strongly-correlated individual particles. Each droplet is in highly conducting state coexisting with insulating domains consisting of the described strings. When the temperature rises the area associated with the domains consisting of insulating electron strings with M = N increases since they are associated with the metastable minima. The conductivity drops since such strings are not conducting. At higher temperatures the strings begin to evaporate and the conductivity associated with mobile polarons[15-22] increases. A magnetic field will remove the barriers between conducting domains so improve the conductivity. It is also interesting to note that a qualitatively similar instability of small polarons and the formation of a string-type trap has been found in another system of soft polymers[48-50] but in the framework of a very different model of electrons interacting with rotating dipoles. They found that the highly conducting strings termed “superpolarons” are more stable than single polarons or bipolarons. The criteria for the “string-superpolaron” formation are that (a) the polymer has very different static and high-frequency dielectric constants (we also need such a condition) and (b) it is very soft (shear elastic constants are very small) so that an excess charge orientates all dipoles around the string. In a summary the multi-electron self-trapping phenomenon of the electronic stringdefect creation has been described in the framework of a simple self-consistent electronphonon models of interacting spinless fermions. This is a many-body (Hartree-Fock) generalization of the Pekar-Rashba model for the single particle self-trapping. The electron phase separation is associated with the appearance of the multi-electron cigar-shaped localized string- droplets. Note that for the first time we have shown that in systems with a narrow band the electron-phonon interaction may play a very significant role in the electronic phase separation associated with the string’s.

Acknowledgments I am very grateful to A. Bianconi, C.H. Chen,C. Di Castro, M. Grilli, S. Kivelson, P. Littlewood, D. Edwards, G. Gehring, V. Emery, E.I. Rashba, Danya Khomskii, A.S. Alexandrov, V.V. Kabanov, H.S. Dhillon and other participants of the workshop ECRYS-99 and on strongly correlated electrons in Isaac Newton Institute (Cambridge) for illuminating discussions. The work has been supported by Isaac Newton Institute, University of Cambridge. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

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402

HIGH-TEMPERATURE SUPERCONDUCTIVITY IS CHARGE-RESERVOIR SUPERCONDUCTIVITY

JOHN D. DOW1, HOWARD A. BLACKSTEAD2, and DALE R. HARSHMAN1,3 1

Department of Physics, Arizona State University, Tempe, Arizona 85287-1504 U.S.A.

2

Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556 U.S.A.

3

Permanent address: Physikon Research Corporation, P.O. Box 1014, Lynden, Washington 98264 U.S.A.

INTRODUCTION Surprisingly, although high-temperature superconductivity is now virtually fifteen years old, no agreement has been reached concerning the nature of either the phenomenon or the theory that describes it. In this paper, we present evidence that such superconductivity originates from holes in the charge-reservoirs of the crystal structures and does not necessarily require cuprate planes, as has been widely assumed. An overview of high-temperature superconductivity and the nature of its origin is presented. The following critical facts will be discussed here: (i) PrBa2Cu3O7 superconducts at 90 K when its crystal structure has no PrBa defects; (ii) the chemical trends for superconductivity in PrBa2Cu3O7, Pb2Sr2Pr0.5Ca0.5Cu3O8, and Pr1.5Ce0.5Sr2Cu2NbO10 indicate that the holes responsible for the primary superconductivity reside in the charge-reservoir layers and not in the cuprate-planes (and so materials without cuprate-planes can be hightemperature superconductors); (iii) there are no n-type high-temperature superconductors; (iv) in the Nd2–zCezCuO4 homologues, the superconductivity proceeds through interstitial oxygen ions paired with Ce ions; (v) the magnetic rare-earth ions in the superconductors are potential Cooper pair-breakers, but for many materials only the L=0 magnetic ions in contact with the superconducting condensate are pair-breakers, because the L 0 rare-earth ions often experience crystal-field splitting of their electron energy levels; and (vi) Cu-doped Sr2YRuO6 is a high-temperature superconductor having no cuprate planes.

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

403

Figure 1. The c-axis parameter of (rare-earth)Ba 2 Cu 3 O 7 compounds against rare-earth radius. Note that the traveling solvent floating zone (TSFZ) grown PrBa2Cu3O7 superconducts, but flux grown PrBa2Cu3O7 and CmBa2Cu3O7 (which have short c-axes) do not.

PrBa2Cu3O7

PrBa2Cu3O7 had long been regarded as a material that did not superconduct, until (i) it was predicted to superconduct [1,2]; (ii) it was shown to exhibit granular superconductivity [3–5]; and (iii) bulk superconductivity was demonstrated in it as well [6–13]. Despite this evidence, some scientists still believe that PrBa2Cu3O7 does not superconduct, although the PrBa2Cu3O7 materials that do not superconduct all have anomalously short c-axes, and hence do not follow the trends for c-axis length versus ionic radius obeyed by

superconducting (rare-earth)Ba2Cu3O7 compounds. (See Fig. 1.) The short c-axes are symptomatic of PrBa2Cu3O7 material containing PrBa (antisite Pr-on-a-Ba-site) defects. (See Figs. 2 and 3.) The PrBa2Cu3O7 materials that do superconduct have longer c-axes than nonsuperconducting PrBa2Cu3O7 [14,15]. The important facts for superconducting PrBa2Cu3O7 (and NdBa2Cu3O7) compounds are that perfect PrBa2Cu3O7 (and NdBa2Cu3O7) are 90 K superconductors, while the same materials with numerous PrBa or Nd Ba defects have relatively short c-axes and do not superconduct (or have depressed transition temperatures). Defects such as PrBa (and NdBa) destroy superconductivity because they are magnetic pair-breakers. The failure of PrPr and NdNd to destroy superconductivity, and the location of the PrBa and NdBa defects, namely in or near the charge-reservoir (CuO and BaO) layers, indicates that the holes responsible for the primary superconductivity do not reside in the cuprate-planes, as widely assumed. TRENDS

Further evidence against the usual picture which places the superconducting holes in the cuprate-planes is provided by examining the trends in critical temperatures for PrBa2Cu3O7 (Tc 90 K [13]), for Pb2Sr2Pr0.5Ca0.5Cu3O8 (Tc 60 K [16]), and for Pr1.5Ce0.5Sr2Cu2NbO10 404

Figure 2. Crystal structure of imperfect (non-superconducting) PrBa2Cu3O7 with a defect PrBa and an O(5) oxygen. Another PrBa is needed to balance charge.

Figure 3.

The crystal structures of (a) perfect PrBa2Cu3O7 with Tc

90 K; (b) Pb2Sr2Pr0.5Ca0.5Cu3O8

(Pr0.5Ca0.5-PSYCO) with Tc 60 K, and (c) Pr1.5Ce0.5Sr2Cu2NbO10 with Tc 30 K. Note that the layers surrounding the cuprate-planes in all three crystal structures are almost the same — suggesting that the cuprateplanes are not the primary superconductors.

405

Figure 4. Crystal structure of Nd 2–z Ce z CuO 4 with an interstitial oxygen. The interstitial oxygen is needed to provide a potential sufficient to ionize Ce to Ce+4.

(Tc 30 K [17]) — all of which have similar crystal structures adjacent to their cuprateplanes, with a rare-earth on one side of a cuprate-plane, and either SrO or BaO on the other.

(See Fig. 3.) Consequently we must conclude that either (i) the local chemistry of the cuprateplanes is unimportant in determining Tc or (ii) the primary superconducting layers in hightemperature superconductivity are not the cuprate-planes. Based on arguments given below, we conclude that the cuprate planes do not determine Tc — because they are not the primary superconducting layers.

THERE ARE NO n-TYPE HIGH-TEMPERATURE SUPERCONDUCTORS Pb2Sr2(rare-earth)Cu3O8, namely (rare-earth)-PSYCO, has the property that it should be a p-type high-temperature superconductor for most rare-earth ions co-doped with Ca: (rareearth)1–xCax, for x 0.5. In fact such p-type materials do indeed superconduct [16,18,19]. It is widely believed (based on questionable analyses of the superconductivity of Nd2–zCezCuO4) that Pb2Sr2(rare-earth)Cu3O8 should be an n-type high-temperature superconductor once the crystal-field potential at the rare-earth site is strong enough to ionize the rare-earth to the (rare-earth)+4 charge-state, as does occur for Am and Ce, whose ionization potentials to Am+4 and Ce+4 are small enough in magnitude. In fact, perfect Am-PSYCO and Ce-PSYCO must be n-type but do not superconduct at all, although they apparently do conduct [16,20]. Indeed, the claims that Nd2–zCezCuO4 (Fig. 4) is an n-type superconductor, although widely believed, also have serious problems associated with them: (i) The computed potential at the rare-earth site is too weak (by 7 V) to ionize Ce to Ce+4[21]; (ii) p-type high-temperature superconductivity has been observed in Pr2–zCezCuO4 by Brinkmann et al. [22]; and (iii) attempts to make rectifying p-n junctions of Nd2_zCezCuO4/YBa2Cu3O7 failed [23]. These problems, which compromise claims of n-type superconductivity, caused us to propose that interstitial oxygen ions paired with Ce ions actually dope the Nd2–zCezCuO4

406

Figure 5. Crystal structure of (a) Gd2–zCezCuO4 (Tc=0 for this Gd compound; but Tc for Pr2–zCezCuO4 is 24 K) and (b) Gd2–zCezSr2Cu2NbO10 which is a superlattice of Gd2–zCezCuO4 and /SrO/NbO2/SrO/CuO2/ layers.

homologues p-type, causing (rare-earth)2–zCezCuO4 compounds to superconduct: the superconducting Nd2–zCezCuO4 homologues are actually p-type [24,25].

Gd2–zCezCuO4 COMPARED WITH Gd2–zCezSr2Cu2NbO10 Gd2–zCezSr2Cu2NbO10 is a superlattice of Gd2–zCezCuO4 with the additional layers /SrO/NbO2/SrO/CuO2/. (See Fig. 5.) Consequently from a cuprate-plane theory viewpoint, one would expect that Gd2_zCezSr2Cu2NbO10 will superconduct only if Gd2_zCezCuO4 also superconducts (and vice versa). However, Gd2–zCezCuO4 does not superconduct, while Gd2–zCezSr2Cu2NbO10 does superconduct [26]. Indeed, for all the rare-earth ions that form both (rare-earth)2–zCezCuO4 compounds and the corresponding (rare-earth)2–zCezSr2Cu2NbO10 superlattice materials, except for the

Gd-based (and Cm-based) materials, both superconduct. The sole exception occurs for the L=0 ions Gd (and Cm): Gd2–zCezSr2Cu2NbO10 superconducts but Gd2–zCezCuO4 does not. (Cm2–zThzCuO4, which involves L=0 Cm, also does not superconduct.)

If we assign the different behaviors of Gd2–zCezCuO4 and Cm2–zThzCuO4 (which do not superconduct) and Gd2–zCezSr2Cu2NbO10 (which does superconduct) to the L=0 character of J 0 Gd and Cm, then we must propose that pair-breaking Gd in superconducting Gd2–zCezSr2Cu2NbO10 is not a nearest-neighbor to the superconducting layer, although Gd in non-superconducting Gd2–zCezCuO4 is — which is why Gd breaks pairs in the latter material and not in the former. This means that SrO is the nearest potentially superconducting layer to Gd in Gd2–zCezSr2Cu2NbO10, while Gd2O2 is the layer that would superconduct in Gd2–zCezCuO4 if Gd were replaced by an L 0 ion such as Nd [27]. Putting a layer of magnetic L=0 ions in or adjacent to the SrO layer can kill the superconductivity, provided the superconducting condensate is in the charge-reservoir layers (as we propose), not in the cuprate-planes (as widely believed currently). Magnetic L=0 ions, such as Gd and Cm are not crystal-field split and are Cooper pair-breakers which destroy superconductivity. In contrast L 0 ions are often crystal-field split, which can cause them to lose their pair-breaking ability. 407

Figure 6. Idealized crystal structure of one-quarter of the unit cell of Sr 2 YRuO 6 . Copper doping of the Ru-site

produces superconductivity. Not shown are the rotations of the oxygen octahedra.

Ba2GdRu1–uCuuO6 COMPARED WITH Sr 2 YRu 1–u Cu u O 6

Ba2GdRuO6 and Sr 2 YRuO 6 (both doped with Cu on Ru sites) are interesting compounds [28–33] because (i) they have the same crystal structure, but (ii) Cu-doped Ba2GdRuO6 does not superconduct, while S r 2 YRuO6 does. (For the crystal structure, see Fig. 6.) We propose that this is because the two isostructural compounds are two-layer compounds (like Nd 2–z Ce z CuO 4 ) and so a magnetic ion that is not crystal-field split, such as L=0 Gd or Cm, will destroy the superconductivity in both layers: Cu-doped Ba 2 GdRuO 6 consequently does not superconduct, but Sr2YRuO6 does (at Tc 45 K [31]). Sr2YRu1–uCuuO6: CUPRATE-PLANE-LESS SUPERCONDUCTIVITY

Sr 2 YRu 1–u Cu u O 6 superconducts at 45 K, but also has two antiferromagnetic ordering temperatures: one at 86 K which has been identified as due to Cu, another at 23 K to 30 K due to Ru [31]. This means that when the material superconducts, (i) its Cu is already antiferromagnetic, and (ii) below 23 K the Ru is also antiferromagnetically ordered. (The material is too pure to permit the assumption that the Cu dopant forms cuprate planes.) The conventional concept is that magnetic layers do not superconduct. Therefore Sr 2 YRu 1–u Cu u O 6 , with its magnetic Ru and Cu ions on its Ru sites, must superconduct in the non-magnetic layers, and not in the magnetic YRu 1–u Cu u O 4 layers. Namely, the holes carrying the superfluid density must reside in the SrO layers. Recent muon spin relaxation (µSR) studies [31-34] lend further support to the assignment of the superconducting hole condensate to the SrO layers. The Sr2YRu1–uCuuO6 material contains oxygen ions in both its (SrO)2 and YRu 1 _ u Cu u O 4 layers. Since positive muons tend to stop near the negatively charged oxygen ions, one would expect to observe one magnetically distinguishable muon site for each of the various different local environments capable of trapping a µ + particle: we observed two such sites, each with a radius of order 0.5 Å. Above 30 K, the µSR spectra (acquired in a field of 500 Oe transverse to the muon beam) 408

Figure 7.

Muon spin rotation relaxation rate (in (µ s) –1 ) versus temperature (in K) for the two muon sites,

measured in a 500 Oe field (perpendicular to the initial muon spin polarization direction), after reference [31]. The data were analyzed assuming exponential forms for the relaxation functions. (The open triangles correspond to data taken at 500 Oe after cooling in zero field.) We attribute the fast relaxation to the YRuO4 layer, and the slower one to the SrO layer.

exhibit a single slowly relaxing component with a Larmor precession frequency corresponding to the applied field. However, below 30 K the Ru ions begin to order, revealing two distinct components, one fast-relaxing and the other slow-relaxing. (See Fig. 7.) The fastrelaxing signal, which accounts for about 90% of the muons, also shows a dramatic increase in its muon precession frequency, corresponding to a local field of about 3 kOe. The remaining component, with the markedly slower relaxation rate, also exhibits a slight diamagnetic shift of the average muon precession frequency. These data indicate that the muon site associated with the slowly relaxing component experiences a near-zero net local magnetic field in the ordered state. (See Fig. 8.) Neutron powder diffractometry measurements, acquired on the same samples as the µSR data, indicate that the Ru moments order ferromagnetically in the YRu 1–u Cu u O 4 planes, with the net polarization reversing direction from one YRu1–uCuuO4 layer to the next along the c-axis. This polarization reversal produces a net zero field in the SrO layers. In contrast, the local field in the YRu 1–u Cu u O 4 layers is necessarily non-zero due to the Ru and the Cu moments. From this, we can unambiguously attribute (i) the fast-relaxing component observed in the µSR spectra to muons stopped in the YRu 1–u Cu u O 4 layers, and (ii) the slowly-relaxing signal to muons stopped in the SrO layers. These identifications are consistent with the bondvalence sum calculations [35], which show that (i) the oxygen ions in the SrO layers have stretched bonds (which implies that those oxygen ions are positively charged with respect to O –2 ), and (ii) the oxygen ions in the YRu 1–u Cu u O 4 layers are virtually fully charged to O –2 . The positive muons tend to stop near the more negatively charged oxygen, which is why 90% stop in the YRu 1–u Cu u O 4 layers. The relaxation rate increase and the diamagnetic shift for the SrO component are both consistent with the presence of vortices, as determined by first cooling in zero magnetic field, and then by applying a 500 Oe magnetic field. (See 409

Figure 8. Muon spin rotation frequency versus temperature for the SrO layers of Sr2YRuO6 (doped with Cu on Ru sites) after reference [31]. The large error bars below 30 K indicate detection of a flux lattice (because the detected spot sometimes is somewhere in a vortex). The open triangle represents a zero-field datum.

Fig. 7.) The observed increase in relaxation rate at low temperatures indicates the presence of vortices. Moreover, the flux lattice was observed to be only weakly pinned, which is consistent with a very short c-axis flux-line correlation length, such as what one would expect for a set of isolated sheets of “pancake” vortices. Interestingly, bulk superconductivity is not evident in the muon data until the Ru ions order, suggesting that the fluctuating paramagnetic Ru spins may act to suppress the superconductivity above 30 K.

SUMMARY By placing the charge-carrying holes in the charge-reservoirs, rather than in the cuprateplanes, we predicted that PrBa2Cu3O7 would superconduct — and it does. We also predicted the superconductivity of three more compounds that have since been found to have at least granular superconductivity: Gd1.6Ce0.4Sr2Cu2TiO10 [36,37], Pr1.5Ce0.5Sr2Cu2NbO10 [21,38], and Eu 1.5 Ce 0.5 Sr 2 Cu 2 TiO 10 [36,39]. The chemical trends for PrBa2Cu3O7, Pb2Sr2Pr0.5Ca0.5Cu3O8, and Pr1.5Ce0.5Sr2Cu2NbO10) suggest that the cuprate-planes are not the main generators of superconductivity — while the fact that Sr2YRuO6 doped with Cu superconducts (and essentially at the same temperature as GdSr2Cu2RuO8 and Gd1.5Ce0.5Sr2Cu2RuO10) lends credence to the idea that the superconducting holes reside in the SrO layers of all three of these compounds, which is certainly true for Sr2YRuO6. The evidence is that there are no superconductors that can be made both n-type and ptype, because n-type high-temperature superconductors (at least of this class of materials) do not exist. Although Gd2–zCezSr2Cu2NbO10 is a natural superlattice of Gd2–zCezCuO4 and layers of /SrO/NbO2/SrO/CuO2/, the fact that Gd2–zCezCuO4 does not superconduct, while its superlattice does, indicates that the superconductivity originates in the charge reservoirs, not in the cuprate-planes. Ba2GdRuO6 doped with Cu does not superconduct because 410

of the Gd, which is an L=0 magnetic pair-breaker. Finally, Cu-doped Sr2YRuO6 is a superconductor with an onset temperature of Tc 45 K and with the main superconductivity being in its SrO layers. Logical extension of this idea to (rare-earth)Sr2Cu2RuO8 and (rareearth)1.5Ce0.5Sr2Cu2RuO10 (which both superconduct at 45 K), strongly suggests that the superconducting holes are also carried by the SrO layers of these materials as well.

ACKNOWLEDGMENTS

We are grateful to the U. S. Office of Naval Research (Contract N00014-98-10137), for their support. REFERENCES 1.

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ELECTRONIC INHOMOGENEITIES IN HIGH-TC SUPERCONDUCTORS OBSERVED BY NMR

J. HAASE1,2, C.P. SLICHTER1, R. STERN1,*, C.T. MILLING1, and D.G. HINKS3 1

Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080. 2 2. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany. 3

Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439.

INTRODUCTION The pairing mechanism for the electrons in high-temperature superconductors (HTSC) is still elusive. The original ideas which led to the discovery focused on strong electronlattice interactions like Jahn-Teller distortions [1]. However, certain properties (like swave pairing or the isotope effect) expected for such processes within traditional pictures are lacking. Although various theories of superconductivity emerged, none of them seems widely accepted up to now. It is quite clear that dynamic antiferromagnetic spin fluctuations [2, 3] are present in the materials, since they derive from antiferromagnets by hole doping. The study of these spin fluctuations and their relation to superconductivity is an active field of research [4]. Since the early days of HTSC it was argued that the hole doping in the two-dimensional antiferromagnetic background (the exchange interactions are much smaller between Cu-O planes) may not lead to a homogeneous electronic fluid [5-7]. Instead, conducting domains of holes might alternate with antiferromagnetic domains (e.g., one-dimensional arrays of charge in the two-dimensional antiferromagnetic background) [8]. Over the years more and more experiments turned up evidence for inhomogeneous charge and spin structures [8-17], including atomic distances, static and dynamic spin or charge variations, so-called stripes. When superconductivity is suppressed

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

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by co-doping with other atoms static stripes appear [8], similar to static stripes that are incommensurate with the underlying crystal structure in non-conducting mixed valance compounds. The question arises whether they could be present as dynamic stripes. If so, they are candidates for a new pairing agent for HTSC since they provide low energy excitations. We will review some nuclear magnetic resonance (NMR) data and show new evidence that electronic inhomogeneities exist, even for samples with the highest critical temperature, Tc. Nuclei that posses magnetic dipole and electric quadrupole moments interact with local magnetic fields and electric field gradients, respectively, causing a splitting of the nuclear spin states which is detected by NMR. In non-magnetic materials the electric quadrupole interaction can dominate in zero magnetic field and nuclear spin transitions can be observed (nuclear quadrupole resonance, NQR). Application of an external magnetic field introduces a well defined Zeeman splitting for nuclei without quadrupole moments. For quadrupolar nuclei the resonance frequencies depend, in addition, on the internal electric field gradient and its orientation with respect to the magnetic field [18]. Time dependent local fields can induce transitions among the nuclear levels and cause relaxation, the so-called spin-lattice relaxation (for NMR and NQR). Local field changes due to nearby nuclear dipole moments (nuclear spin-spin interactions), often used for distance determination, can also be used to study the electronic properties if the inter nuclear coupling is amplified by the electron spin excitations.

From the NQR frequency one can determine the static electric field gradient at the nuclear site and from its spin-lattice relaxation one obtains information about the fluctuations of magnetic or electric fields. Similar information is available from NMR measurements in strong magnetic fields, where the electric quadrupole interaction causes shifts of the Zeeman levels, however, NMR experiments give even greater insight and allow the application of various techniques which have made NMR so successful. A most important NMR parameter is the chemical or orbital shift KL. It is caused by the unquenching of orbital angular momentum of the electrons due to the external field Bext. Electron spin excitations of low energy can also cause a change of the local magnetic field (static or dynamic). These effects can be rather strong if the electronic coupling to the nucleus (hyperfine coupling) is large. These so-called Knight shifts or spin shifts will be denoted with KS. Both shifts measure the deviation of the local magnetic field Bloc at the nucleus from the external field and are quantified by comparison with a reference sample of preferably small shifts. If the nucleus of the reference sample has the frequency vref in the field Bext, we write v = (1 + K)vref, where K = KL + KS is the sum of both shift tensors. The contribution of each component is not known a priori. One has to invoke models and measure, e.g., the temperature dependence of the shift, in order to decompose it. The NMR shifts measure the orbital and spin susceptibilities if the local field changes are proportional to the polarization created by the external field. Both parameters have proven to be very useful tools in determining chemical bonding and electronic structure. The presence of a quadrupole interaction complicates the understanding of the NMR, but by the same virtue, it is another source of information about the structure of the material. The strength of the quadrupole interaction does not depend on the external field. By choosing various laboratory fields (up to 20 T) one can change the size of the Zeeman term. The two limiting cases of a vanishing external magnetic field (NQR) and a strong magnetic field (such that the quadrupole interaction is only a small perturbation of the Zeeman interaction - quadrupole perturbed NMR) can be treated easily. We will mostly be

interested in the latter case since one can obtain all information from the high field spectra. In such a case, the single Zeeman transition for an I>l/2 nucleus splits into 2I transitions.

414

Figure 1. (a) Zeeman levels of an I = 5/2 nuclear spin in a strong magnetic field with perturbation from quadrupole interaction. The unperturbed distance between neighboring levels is given by the frequency v0. The quadrupole splitting vQ,z depends on the orientation of the electric field gradient with respect to the magnetic field (z-direction). Corresponding NMR spectra for a distribution of (b) electric field gradients and (c) local magnetic fields. The latter affects all transitions equally, whereas the former broadens the outer satellites twice as much as the inner ones and leaves the central transition unaffected to first order.

The splitting depends also on the orientation of the electric field gradient (EFG) tensor relative to the applied magnetic field. Most nuclear spins are half integer (I = 3/2, 5/2, 7/2) and in such a case there is the so-called central transition (m=±l/2) which is only affected by the quadrupole interaction in second order. We will simplify the situation by using single crystals or oriented powder (oriented along the crystal c-axis). With the external magnetic field in c-direction, the Hamiltonian is axially symmetric so we can neglect the 2nd order quadrupole interaction and find a simple formula for the 2I transition frequencies, (1) where n denotes the particular transitions and has values for I = 3/2 of n = -1, 0, +1, and for I = 5/2 of n = -2, -1, 0, +1, +2. The central transition, n = 0, measures the magnetic shifts only, whereas the satellite transitions are affected by both the magnetic shifts and the quadrupole splitting vQ. This situation is depicted in Fig. 1. Higher order effects from crystal misalignment or other changes of the EFG orientation can be detected by field dependent measurements: while the magnetic shifts are proportional to the field, the second order quadrupole effects are inversely proportional to the field. In order to have a feeling for the properties that can be measured with NMR, we mention that an electron in the ground state will produce an electric field gradient in zdirection of (2)

415

where is the polar angle and r the distance from the nucleus. Also, external charges from the crystal electric field will influence the electric field gradient. The calculation of the EFG can often be obtained with quantum chemical methods. The orbital or chemical shift measures the availability of excited states (3)

where is the electron gyromagnetic ratio, L z is the z-component of the angular momentum and the difference En-E0 measures the energy to the excited state n [19]. The orbital shift is therefore typically temperature independent. It is large if unoccupied states are nearby in energy.

Figure 2. Structure of La2-xSrxCuO4. The Cu-O a-b planes are formed by Cu and planar oxygen. The distance from Cu to the apical oxygen is much larger than that to the planar oxygen. The La atoms are replaced by Sr

to achieve conductivity/superconductivity. There is only one unique Cu-O plane separated by apical oxygen and La/Sr atoms.

Lastly, we write down an expression for the spin shift. (4)

where Azz denotes the hyperfine coupling constant and the electron spin polarization in z-direction of the electron to which the nucleus couples. The latter can be strongly temperature dependent, as for an isolated electron spin or for a superconductor with spin singlet pairing, or, almost temperature independent as for a metal. The spin polarization in a magnetic field is often given by the spin susceptibility. We see that all three parameters give very important information about the electronic

structure of the material. In a similar fashion one can write down equations for the nuclear spin relaxation. The time dependence of the processes and the coupling matrix elements will determine the prevailing relaxation mechanism.

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EXPERIMENTAL ASPECTS OF NMR OF HIGH-TC SUPERCONDUCTORS A typical high-temperature superconductor, La2-xSrxCuO4 (LSCO), is shown in Fig. 2 (for structural details see [20, 21]). For x = 0 this material is an antiferromagnet with a

Neel temperature of about TN = 320 K. At a Sr concentration of x = 0.02 the Neel order vanishes and around x = 0.05 the material becomes superconducting at very low critical temperature Tc. For x = 0.15, Tc has a maximum of 38 K and starts to decrease further with increasing x. La2-xSrxCuO4 is a particularly simple since the Cu-O planes are unique and there are no so-called Cu-O chains in this material. The relevant nuclear parameters for NMR in La2-xSrxCuO4 are shown in Tab. 1. One realizes that all nuclei have quadrupole moments. The NMR sensitivity is high for a large resonance frequency (v = B) and high abundance of the particular isotope. We see from Tab. 1 that both Cu isotopes provide good sensitivity. NMR on the I7O isotope is not readily performed due to the low abundance and one works preferably with enriched samples. 139La NMR is easy to perform in contrast to that of Sr. From the low crystallographic symmetry of the materials one expects strong quadrupole interactions. Early experiments showed that the quadrupole splitting is around 30 MHz for Cu, a few hundred kHz for O and a few MHz for La. Therefore, zero field NQR measurements are readily performed on both Cu isotopes and on La, but not on O. Table 1. Selection of nuclear parameters for NMR in La2-xSrxCuO4.

Isotope

Spin

Gyromagnetic ratio in MHz/T

63

Cu

65

Cu

17

Quadrupole moment Q in 10-24cm2

Natural abundance Pin %

3/2

11.28

-0.210

69.09

3/2

12.09

-0.195

30.91

5/2

5.77

-0.026

0.037

139

La

7/2

6.01

0.20

99.91

87

Sr

9/2

1.84

0.15

7.02

O

For NMR experiments various complications arise: (1) The rather large quadrupole coupling for Cu which causes an orientation dependent splitting would result in huge distributions for powder samples. Therefore, if single crystals are not available, a magnetic grain alignment is performed. The powder is mixed with epoxy which is then cured in the magnetic field. Clearly, such a process which is based on the magnetic anisotropy of the grains will not be very accurate and will produce a distribution of angles between the crystal c axis and the external magnetic field. (2) Even for aligned material there are a great many resonance frequencies due to (i) the various transitions of the quadrupole perturbed NMR, cf. Eq. (1), (ii) the 2 different isotopes for Cu, (iii) the 2 different oxygen sites, and (iv) the presence of non-equivalent sites (see below) for a given isotope due to doping. (3) As we will see later on, the distribution of local fields can be quite large in superconductors. This adds to the complication with the various resonances and the difficulty of alignment so that severe resolution problems exist in cuprates. In order to overcome these problems we have introduced new NMR methods [22] which are indeed very helpful. The basic idea is simple: For quadrupole perturbed NMR

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the various transitions belong to nuclear spin flips between the 2I eigenstates, cf. Fig. 1. Before the experiment starts, these will be populated according to the Boltzmann factor. Given that the splittings are very small, the level population will be a linear function of the magnetic quantum number. Now, with a pulse that precedes the normal experiment one can change the level population at will, e.g., a selective inversion pulse on the upper most transition (Fig. 1) will change the intensities observed in a succeeding (before relaxation sets in) usual NMR experiment. This is of great importance since the effective interactions are different for the different nuclear transitions (the central transition is only affected by the magnetic shift, cf. Fig. 1). This way one can study the correlation between magnetic shift and electric field gradient, obtain single isotope spectra (Fig. 3), or, measure the magnetic shift for overlapping apical and planar oxygen signals. The difference of these methods compared to ordinary NMR is that we irradiate two different frequencies, a special kind of NMR double resonance [18].

Figure 3. (a) Upper satellite spectrum of La1.85Sr0.15CuO4 at 8.3 T and 300 K. (b) Difference between the regular spectrum and one where the NMR experiment was preceded by an inverting pulse on the 63Cu central

transition. One notices the absence of 65Cu signal intensity and the lower frequency tail from not well aligned grains.

BASICS OF NMR IN THE NORMAL STATE Application of NMR to the newly found HTSC [23] revealed many important details. We will focus on the superconducting materials (but should perhaps mention that, e.g., La NQR is a wonderful probe of the phase transition to Neel order [24]). We will start with summarizing results that seem to be unique to the various materials, and, one can guess that the study of the low energy electron spin excitations (from electrons near the Fermi surface) through spin shift and relaxation measurements are of special interest, as for the classical superconductors [25]. In order to separate the spin shift from the magnetic shift one has to do temperature dependent studies of the magnetic shift assuming that the model of a temperature independent orbital shift is correct. For such experiments, preferably the central transition is observed, n = 0 in Eq. (1). We start with the Cu NMR. The observed magnetic shift for c||B0 (crystal c-axis parallel to the magnetic field in z-direction) turns out to be rather large but temperature independent. For c B0 the magnetic shift decreases with temperature, this effect is stronger for doping levels below that with highest Tc (optimal doping). Roughly speaking, the total magnetic shift is about twice as big with c||B0 at 300 K. An appealing way to disentangle

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both shift contributions is by assuming that the spin shift accidentally vanishes for c||B0. For c B0 one assumes that the temperature independent component (inferred from

comparison with, e.g., Y as an internal standard) represents the orbital shift. This analysis, especially the vanishing spin shift for c||B0, may seem somewhat artificial, however, it yields 3 results which fit very well other experiments, (i) The ratio of the orbital shift for both orientations thus obtained is about 4 and therefore agrees with a simple model for a Cu2+state (3d9) . From the mixing of the orbitals xy and yz with x2-y2 due to L2 or Lx, respectively, we would expect a factor of 4 from Eq. (3) for vanishing orbital energy differences between xy and yz [19]. (ii) The temperature dependence of the spin shift (c B0) thus obtained is proportional to that of the static spin susceptibility measured with a magnetometer [26, 27]. (iii) For the planar oxygen resonance where one expects small orbital shifts the temperature dependence of the Knight shift agrees with that determined for Cu. The question arises whether one can understand the observed spin shift data? Without going into details about the history of the explanation of the electron nuclear coupling [23, 28, 29], we present the hyperfine Hamiltonian [30, 31] which is believed to be correct for the cuprates. It assumes a single electronic fluid to which all nuclei couple such that we have the following expressions for the spin shift, cf. Eq. (4). The Cu nucleus couples to the onsite electron spin as well as to that of the four Cu neighbors, (5) The planar O nucleus couples to two neighboring Cu electron spins, (6)

where B, are the hyperfine coupling coefficients for the alignment of the sample (note that B is isotropic). For La2-xSrxCuO4 we can also write down a similar expression for the apical O nucleus, (7) For the latter material we have the following values for the hyperfine coupling constants, Ac = -41.8, B = 11.5, Cc = 7.44, Ec = 2.22 in units of 107 Hz [32, 33]. For a uniform spin polarization we have (8)

where is the uniform spin susceptibility. As explained above, one can fit the shift data with Eq. (8) and determine the hyperfine coupling constants. It is interesting to note that for the various materials one finds Ac + 4B 0, which comes as a surprise. Another interesting aspect is the decrease of the spin shift as the temperature is lowered at temperatures which are far above the critical temperature for underdoped materials. For a metal one would expect a temperature independent spin polarization and thus a temperature independent spin shift. Classic superconductors show a drastic decrease in spin shift only below Tc where the electron spins pair up in singlets with vanishing magnetic moment (freezing out of spin degrees of freedom). The fact that the spin excitations decrease already above Tc has been named the “spin gap” or “pseudo gap”

419

effect. Other methods also find a decrease in spectral weight already far above Tc, a phenomenon which is not understood. At this point we could talk about spin lattice relaxation or spin-spin relaxation. However, these topics are somewhat more involved and not very well understood. So we will only give a brief summary [23, 33-35]. First, one finds that the dominant relaxation mechanisms for Cu and planar oxygen are magnetic (but not exclusively [36]). It is interesting to note that due to the different structure of the Zeeman vs. quadrupole Hamiltonian the transition matrix elements for nuclear spin flips (allowed transitions) are different. Assume we have created a nonthermal equilibrium population in the nuclear spin system, similar to the one mentioned earlier (a selective inversion pulse on one transition). Then, the actual time dependence of all the population numbers will depend on the allowed transitions. One can perform a mode analysis for various initial conditions and thus find out whether the relaxation process is magnetic or quadrupolar in origin. Second, the relaxation rates for Cu and planar O are very different. The time dependent local magnetic field fluctuations are much stronger at the Cu nuclei. One can

understand such differences by assuming different form factors for both sites and antiferromagnetic spin fluctuations. From Eq. (5) and (6) it becomes obvious that an antiparallel electron spin polarization for neighbors will shield the planar O nucleus from fluctuating magnetic fields but not the Cu. Indeed, a nearly quantitative understanding could be reached [3, 37]. However, since it seems to be proven lately that the low energy spin fluctuations are incommensurate with the underlying lattice, the oxygen relaxation should be faster than actually observed. This problem is still not quite understood. Third, the pseudo gap effect for the shifts is also present for the spin-lattice relaxation, as it should, since the scattering rate of electrons which is involved in magnetic nuclear relaxation will be diminished as well if electronic spin degrees of freedom freeze out.

INHOMOGENEITIES IN La2-xSrxCuO4 FROM NMR It was clear from the very beginning that the HTSC must be inhomogeneous to some extent since they derive from the stoichiometric parent compounds by doping. In systems where the dopant can move, like oxygen doped La2Cu it was soon observed that phase separation can occur into hole rich and hole pure regions. It is still a matter of debate what the details of such ordering are (e.g., for oxygen ordering in the chains of YBa2Cu3 The situation is quite different in La2-xSrxCuO4 (LSCO) since the Sr atoms are distributed in the lattice at high temperatures and are immobile certainly below ambient temperature. Apart from the doping induced lattice changes there is a phase transition from a high temperature tetragonal phase to a low temperature orthorhombic phase [21]. This phase transition temperature decreases with the doping level, but, it does not seem to be related to the onset of superconductivity. Let us review some NQR results. Early experiments [38,39] in LSCO already revealed that there were 2 inequivalent Cu sites in the Cu-O plane, cf. Fig. 3, they were called the A site and B site. It was noticed that the number of B sites is roughly the same as that of Sr atoms. It was concluded that this site has to be related to the dopant, perhaps a Cu site close to it. However, this interpretation was challenged [40] in the following years when it was discovered that the oxygen doped material also showed two lines but with different intensities. The idea was put forward that the second line might be a result of a certain amount of trapped holes the number of which increases with doping. However, with the new NMR methods we found that the magnetic linewidth of both lines in LSCO is similar, the same is true for their shifts [41]. Together with theoretical calculations [42] it seems

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quite convincing that the Cu B site is the Cu position which is bridged by an apical oxygen to a Sr atom (we also find an apical oxygen B site). The second site in the oxygen doped material must be related to the oxygen dopant and accidentally similar in frequency to that in LSCO.

Figure 4. Sketch of the doping dependence of the NQR frequency vQ (or NMR satellite splitting) and the full width at half height of the resonance line at 300 K.

We will now focus on the intense Cu A site only, since we think that the debate over the origin of the Cu B site is settled (we will give more evidence below). We now look at

the NQR parameters at 300 K as a function of doping in Fig. 4. The electric field gradient increases smoothly with doping, 21MHz [43]. On the other hand, the distribution of field gradients increases drastically from about 60 kHz for the undoped material to about 2.2 MHz near the onset of superconductivity. As the doping increases further the linewidth remains nearly constant. Although one might expect an increase in the distribution of the EFG upon doping due to lattice inhomogeneities, the actual data are surprising: (i) One would expect a doping dependence of the EFG distribution for x > 0.05. (ii) From the doping dependence of the mean frequency one concludes that a 2.2 MHz linewidth would correspond to a variation in doping of about 0.1 which for small dopings is bigger than the average doping level. From NMR satellite transition measurements for c||B0 and c B0 one can estimate the asymmetry of the EFG since its largest component is along the crystal c-axis. We find that the EFG remains axially symmetric which seems to contradict random lattice distortions. Very recently, it was discovered [13] that the Cu NQR intensity for the underdoped materials is lost below a given temperature which is similar to the charge ordering temperature observed with inelastic neutron scattering in similar co-doped materials. If the local electric field gradient undergoes slow (on the time scale of measurement, i.e., ~1 ) variations with large amplitudes, this can wipe out the NQR signal. Another possibility for the intensity to disappear is by slowly fluctuating electronic moments, e.g., the transition into a spin glass state, since the hyperfine fields at the nucleus are very large. NMR experiments at sufficiently low temperatures, such that the low energy excitations are frozen out, can reveal the order in the system. At the present time it seems likely that both fluctuations are involved, but it is not yet certain [44]. Above the doping level of x = 0.12 no loss of intensity is observed, instead, we observe temperature dependent changes in all the local fields. One of the strongholds of NMR is that we can measure the local fields at various locations in the unit cell and

421

compare them with each other. We will illustrate the foregoing. First let us compare the local field distributions, in terms of the second moments, of the central transitions of the planar Cu and oxygen nuclei in optimally doped LSCO. This is shown in Fig. 5. As the temperature is lowered from 300 K the inverse root of the second moments decreases. This increase in linewidth is similar for both nuclei down to about 80 K where the oxygen line starts to narrow. That is indeed to be expected if the broadening is caused by spin effects

(pseudo gap effect). We also realize that the Cu linewidth keeps increasing, which tells us that it cannot be dominated by spin effects. Could long range doping variations, or grain effects cause this behavior? In order to investigate that problem one can perform a SpinEcho-Double-Resonance (SEDOR) experiment [18] which makes use of the short-range inter nuclear magnetic coupling (dipolar and indirect nuclear coupling). Here, one looks at a part of the broad magnetic 63Cu lineshape with a usual spin echo experiment. For comparison, in a similar, second experiment one irradiates in addition a selective inversion pulse to some of the 65Cu nuclei, i.e., of the other isotope. (Both isotope’s lineshapes are similar and the difference in the resonance frequencies of the 2 isotopes’ central transition is much larger than their linewidth.) This additional pulse flips some 65Cu nuclear moments and will change the local field at all its neighbors, thus preventing them from contributing to the 63Cu spin echo (destruction of the spin echo for the neighbors). If the large Cu linewidth were caused by long wavelength magnetic field variations, neighboring Cu nuclei would feel the same local field and thus in the SEDOR experiment described, the destruction of the 63Cu spin echo would only occur for the part of the 63Cu lineshape that corresponds to the position of the selective inversion pulse of the 65Cu. We observed experimentally that no matter where the selective inversion pulse is irradiated, the destruction of the spin echo occurs at all parts of the 63Cu line. This tells us that the strong local field distributions occur over rather short distances and not at long distances such as between different grains in the powder.

Figure 5. Inverse root second moments of the 63Cu and 17O central transition lineshape at 8.3 T for optimally doped LSCO (full lines are guides to the eye).

Before we draw conclusions about the origin of the field distributions, we compare the planar oxygen data with those for the apical oxygen. We use Eq. (6) and Eq. (7) and assume that a variation in the spin polarization can be written as

422

(9)

where has zero mean and varies between the Cu sites and creates the linebroadening. Introducing Eq. (9) into Eq. (6) and Eq. (7), we see that the root second moments of the spin shifts contain the variance and correlation function between neighbors, i.e., and While the variance can be obtained from the apical oxygen linewidth,

by plotting the second moments of the planar oxygen versus that of the apical oxygen we can determine the electronic correlation function for neighbors. This is shown in Fig. 6. From the slope of the line in Fig. 6 we find,

(10) independent on temperature. This result comes as a surprise since one would expect a ratio near minus one for a predominantly antiferromagnetic electronic spin response. We can write down equations for the second moment of the Cu lineshape which will include correlations between more distant neighbors, cf. Eq. (5). If we only measure one more parameter, the Cu linewidh for this alignment, we cannot solve for all the correlation functions. However, we can try to find the maximum possible spin contribution to the local

field at the Cu from that measured at the planar and apical oxygen position. By doing so, we find that the Cu local field from spin effects is much too small to explain the observed width at any temperature. This fact, together with the lack of narrowing of the Cu line at lower temperatures clearly states that the spin effects are not responsible for the (short

wave length) Cu local field variations.

Figure 6. Second moment of the central transitions of the planar 17O line versus that of the apical 17O line (contributions from nuclear dipole interactions subtracted) for temperatures between 100 K and 300 K at 8.3 T (the solid line is linear fit).

The only explanation left for the Cu linebroadening is thus a large scale orbital shift variation, which comprises almost the total orbital shift range at lower temperatures. Such effects are very unusual since the orbital structure typically remains unaltered for these 423

small temperature changes. (Contributions from second order quadrupole coupling are ruled out by studies of the magnetic field dependence of the linewidth.)

Figure 7. Apical oxygen NMR at 8.3 T (c||B0). Shown are the two satellite transitions with n = +2 at 300 K and n = –2 at 80 K. This reveals the temperature independent broadening which is also the same for both transitions (the satellites with n = ±lwhich are not shown are half as wide, cf. Fig. 1; the sharp features to the right of the main line are caused by incomplete subtraction of the planar oxygen resonance).

The large distributions of the EFG for the Cu, which can be measured with NQR and satellite NMR, are found to be temperature independent at optimal doping. At the apical oxygen site, cf. Fig. 7, one also finds a temperature independent distribution of the EFG’s but much larger, 12 % of vQ,z for oxygen (only 3.2 % for Cu). The origin of these distributions are still unclear, but suggest the involvement of the 3 z2-r2 orbital for Cu. For the planar oxygen the situation is quite different as can be seen in Fig. 8. We notice that the linewidths of the various satellite transitions are not the same but increase as the temperature is lowered, however, the apparent asymmetry of the whole set of lines remains unchanged. From comparison with Fig. 1 it is obvious that the found spectrum is neither caused by a mere quadrupolar nor magnetic shift distribution. The full oxygen spectrum must result from an interplay between variations of both the local magnetic field and electric field gradient such that, e.g., for the n = -1 satellite both shifts oppose each other whereas they add for the n = +1 line. It turns out that we can understand the experimental findings with a simple theory where we assume that KS and the EFG at the planar oxygen are linear functions of a “hidden” parameter h, such that (11)

and h is distributed over a range of values. Then one can reproduce the data and determine the ratio R/M. It is found that between 300 K and 100 K the ratio R/M is about 2 for 8.3 T. It is inversely proportional to the applied field, since M is proportional to the field. As one approaches Tc, M decreases rapidly (it describes the broadening of the central transition, cf. Fig. 5) and the full spectrum become symmetric. This shows that at the planar oxygen site the electric field gradient is a linear function of the spin shift (a correlation of spin and charge). Also, one notes from Fig. 8 that these effects are quite strong. For other doping levels such detailed analysis is not yet available. A few more results on the planar oxygen modulation are shown in Fig. 9 for other Sr dopings.

424

Figure 8. Total planar oxygen lineshapes at 300 K and 100K at 8.3 T.

It is seen that the correlated modulation of the EFG and the spin shift at the planar oxygen site depend on the doping level x. The clear asymmetry found for the optimally doped sample, shown in Fig. 8, is not observed for the x = 0.10 sample. One also sees a slight increase in the local field distribution. For the overdoped sample, x = 0.20, we observe a tremendous increase in linewidth as the temperature is lowered, the asymmetry seems to increase as well, indicating an enhanced correlation between EFG and spin shift.

Figure 9. Total planar 17O spectra at 300 K and 80 K of La2-xSrxCuO4 for x = 0.10 and 0.20 at 8.3 T.

DISCUSSION

A direct consequence of Sr doping of La2CuO4 is the appearance of the planar Cu and apical oxygen B sites. Together with the facts that the Cu B site shows similar local field

distributions as the Cu A site (as well as modulations of the spin-spin interaction which we only mention here), and, that the measured shifts (and spin-lattice relaxation rates) can be explained by quantum-chemical calculations [45] we believe that the B sites are caused by changes of the location of the apical oxygen which bonds to Sr and Cu. This result is perhaps not surprising but has to be taken into account for the interpretation of other structural data which find ambiguous mean Cu-apical O distances [16, 46].

425

The tremendous increase of the EFG distribution (300 K) at Cu which is nearly independent of doping for x > 0.05 is surprising. The distribution is well approximated by a Gaussian even for low dopings where one might expect two components from Cu near the dopant and those very far from it. The found distribution of apical oxygen positions [20, 21] which does not seem to depend on temperature agrees with these findings. The loss of Cu NQR intensity and the formation of static magnetic moments below a certain temperature for underdoped LSCO suggest a transition into a spin and/or charge ordered state and the freezing of stripes was proposed [13] to be a likely scenario. For optimally doped LSCO (x = 0.15) there is no transition into a charge or spin ordered state. However, we showed evidence for correlated modulations at the various points in the unit cell. Most surprisingly it involves a short range orbital shift modulation which is strongly temperature dependent and increases as the temperature is lowered. It seems to be correlated with a modulation of the electron spin susceptibility which was measured at the oxygen positions. This modulation also increases as the temperature is lowered (apart from the pseudo gap effect). The electron spin correlation function is almost zero (this does not mean that there are no correlations, an additional wavelength for the spin fluctuations apart from the antiferromagnetic one could give a similar result). Lastly, the total NMR spectrum of the planar oxygen reveals very convincingly the correlated modulation of the susceptibility with that of the charge (electric field gradient). Neutron scattering [9, 14] finds incommensurate spin fluctuations for all dopings. On the underdoped site of the phase diagram the incommensurability is proportional to the

doping x and saturates for the highest Tc. However, elastic peaks [47] are only found for the underdoped materials. This change in the behavior from the presence of static spin structures to an incommensurate susceptibility might be connected with the NMR/NQR results. A modulation of the static susceptibility will not result in elastic neutron scattering. Such a modulation could be observed only if a magnetic field were present. One might think about alternative concepts for the interpretation of the local field distributions in the optimally doped material. The presence of localized holes (< 0.5 %) whose presence has been suggested by various authors could have an effect on the local fields. They should represent a polarizable medium which changes the electronic spin susceptibility [48, 49]. For small concentrations of such holes one expects a magnetic spin shift broadening (if the correlation length of the electron spin fluctuations is not too large) particularly near such holes. While such a scenario might explain the doping independent EFG distribution at the Cu and the temperature dependence for the oxygen linewidth, there are various problems: (i) The correlation function between neighbors should still be close to –1 and not near zero as measured, (ii) It does not explain the correlation between the oxygen magnetic shift and the temperature dependent EFG at the planar oxygen. (iii) It does not explain the Cu orbital shift distribution and its temperature dependence. One can find more arguments against an explanation of the results which has localized holes as the only agent. It should be said that the local field distributions differ among the various cuprates. All the doped La2CuO4 materials seem to have on average larger distributions than the YBa 2Cu3 family of materials, in particular the stoichiometric compound YBa2Cu4O8. Nevertheless, the temperature dependences of the linewidths we find are similar to those presented for LSCO. Although a detailed analysis of the local field distributions in other cuprates is not available, we found also short wavelength spatial modulations which seem to exhibit similar properties as for LSCO. The planar 17O spectrum for an (optimally doped) YBa2Cu3O6 . 95 sample, shown in Fig. 10, reveals the similarities most clearly. An appealing explanation for the fact that the local field distributions are different in various samples could perhaps start with the idea of the pinning of stripes: The degree of crystallographic disorder might produce pinning potentials for otherwise rapidly

426

fluctuating dynamic electronic structures from which NMR would only measure the time average.

Figure 10. Planar 17O NMR spectrum at 100 K and 8.3 T for YBa2Cu3O6.95. The solid line is a fit using a shift and EFG correlation according to Eq. (11), R/M = 1.8 (there are 2 non-equivalent planar oxygen sites in this material).

NMR IN THE SUPERCONDUCTING STATE

The superconducting state bears consequences for NMR [25] due to the changes in low energy excitations and flux quantization, e.g., the spin shift disappears and additional local field variations are acquired for NMR. This allows the study of quasiparticle excitations (which was so important for the proof of the BCS theory [50]) and of fluxoid properties [51]. At present, only the basic features have been investigated (singlet pairing, d-wave symmetry). Nevertheless, there are indications for unusual behaviour, e.g., the local field distribution for Cu is bigger than that expected from the magnetic field distribution due to the fluxoid lattice. More experiments are needed and a better understanding of the normal state has to be reached. CONCLUSIONS We reviewed recent experimental data and showed new evidence in support of the idea that structural disorder due to doping is not the mere cause of the large local field variations observed with nuclear magnetic resonance. The above data showed that there is also disorder present among the electrons which engage in conductivity (superconductivity) and that these inhomogeneities are of short range. The data do not allow us to give a detailed picture of the underlying causes nor can we draw conclusions from them about the possible connection between inhomogeneous electronic structure and superconductivity. More experiments and a better understanding of the results obtained with other methods will help to better identify the cause and role of inhomogeneous electronic structures.

427

Acknowledgement This work was supported by The Science and Technology Center for Superconductivity under NSF Grant No. DMR 91-20000 and the U.S. DOE Division of Materials Research under Grant No. DEFG 02-91ER45439. We would like to thank N.J. Curro, T. Imai, P.C. Hammel, S. Kos, A.J. Leggett, P.F. Meier, K.A. Müller, D.K. Morr, D. Pines, R. Ramazashvili, J. Schmalian for helpful discussions. CPS would like to thank D. MacLaughlin for a fruitful discussion of tests of the modulation of the orbital shift. J.H. acknowledges the support by the Deutsche Forschungsgemeinschaft and D.G.H support from DOE - Basic Energy Sciences under Contract No. W-31-109-ENG-38.

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