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

58

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 << R2. Below the lower transition there are no lower resistance percolating paths, thus essentially all the potential drop is on the higherresistance bonds connecting lower-resistance clusters. This means the potential is almost constant within these clusters, and they may be considered as consisting of superconducting bonds. Thus we will have a smeared near-singularity in the conductivity at the lower transition. There will be a monotonic rise in conductivity throughout the intermediate phase, but it will remain low, as the low-resistance filaments are irrelevant by themselves in the thermodynamic limit. Above the second transition the low-resistance backbone by itself gives finite conductivity, so the high-resistance part of the network is irrelevant. Unlike the first transition, the second one is sharp, which is an artifact of change in the bond-insertion algorithm. Figure 14 shows the result for this model for R2/R1 = 200. Stability

We have already mentioned the extreme fragility of the backbone and thus of the very fact of percolation in the intermediate phase: removal of a single bond from it can destroy it. In fact, if at some point in the intermediate phase we start removing bonds at random, then since there are more than O(l) bonds in the backbone, percolation will be destroyed immediately. But maybe upon insertion of bonds it is re-created as easily? We can ask a more proper question. Namely, our networks turn out to be strongly biased [47]: at any given not all networks satisfying the condition of stresslessness are equiprobable. What if we build the networks, which are truly random with the only constraint of stresslessness? This question was studied for connectivity by variety of methods [47-49], and, formulating the result in our terms, regrettably, the intermediate phase is destroyed. We did not study this question in rigidity case yet, but it is clearly important and more work needs to be done on this issue. SUMMARY We have discussed the rigidity of random and self-organized networks. We find that there is a single transition from floppy to rigid in random networks, but an intermediate phase intervenes in the self-organized networks. This intermediate phase is rigid but contains no redundant bonds and so is stress-free. Some of this work, in the form of more extensive lecture notes has been presented at a NATO-ASI in Czech Republic in the summer of 2000, and will appear in the NATO-ASI

series. We should like to thank D.J. Jacobs for his many contributions to the “pebble game” algorithm and to the US National Science Foundation for support under grant numbers DMR-0078361 and CHE-9903706.

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64

EVIDENCE FOR THE INTERMEDIATE PHASE IN CHALCOGENIDE GLASSES

P. BOOLCHAND, W.J. BRESSER, D.G. GEORGIEV, Y. WANG AND J. WELLS Department of Electrical, Computer Engineering and Computer Science University of Cincinnati Cincinnati, OH 45221-0030

INTRODUCTION Progressive cross-linking of polymeric networks, whether crystalline or noncrystalline in nature, alters their physical behavior in remarkable ways. In crystalline solids the effect is strikingly illustrated if one compares the physical properties of group VI elemental solids (S, Se and Te that possess a coordination number, r, = 2) with those of group IV ones (C, Si, Ge that have r = 4). Thus, although the Pauling single-bond strengths [1] of a Se-Se bond (44kcal/mole) and of a Ge-Ge bond (37.6kcal/mole) are nearly the same, the thermal (melting temperature and heat of fusion), elastic (Young's modulus) and plastic (hardness) behavior of crystalline Ge (r = 4) completely overwhelms that of trigonal Se (r = 2) understandably because of its higher connectivity. More profound effects occur in non-crystalline solids. In the GexSe1-x binary glass system, for example, one has the luxury to change the network connectivity or mean coordination number in a continuous fashion by composition (x) tunning. And one finds that glass transition temperatures [2], T g (x), and bulk elastic constants, progressively increase with or the degree of cross-linking. Furthermore, random networks in contrast to crystalline networks can self-organize [3] with remarkable consequences on physical properties of glasses. In this review, we shall provide experimental evidence for self-organization in glasses. The broad consequences of these observations on percolative transitions in condensed matter, in evolutionary biology and in protein folding, are discussed in the contributions of J.C. Phillips [4] and M.F. Thorpe [5] in this volume. Self-organization in random networks apparently occurs when the global network stress nearly vanishes. The generic condition for a 3d-network to be stress-free is that the number of Lagrangian bonding constraints/atom, nc, exhausts the three available degrees of freedom per atom,

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

65

(1)

Historically, the counting algorithm (1) was first introduced [6] to describe optimization of the glass forming tendency in a covalently bonded random networks, and onset of rigidity [7] in covalently bonded random networks. These two conditions apparently coincide with the stress-free requirement of such networks. Glasses that are under- (nc < 3) and over- (nc > 3) constrained possess stress which appears to be, respectively, entropic and enthalpic in origin. Mean-field predictions of the rigidity transition and existing numerical simulations at T=0, treat enthalpic rigidity [8]. As ideas on entropic rigidity [9] evolve, one may be able to understand stress in under-constrained networks.

The count of Lagrangian bonding constraints (nc) in covalent systems is rather straightforward. In such systems the distinction between the intact nearest-neighbor (bondstretching and bond-bending constraints and the broken more distant (dihedral angle and van der Walls) ones is transparent because of the hierarchical nature of the

bonding interactions. On the other extreme, in metallic systems, the impediment to apply these ideas on constraint counting results from the long range nature of bonding interactions between mobile electrons and the positive ion cores that contribute to the cohesive energy of such solids. The case of ionic solids such as the oxides is a particularly fascinating one and is a subject of current discussions. The hierarchical separation between intact and broken Lagrangian bonding constraints for pure Silica has been facilitated by structure results. Although the bonding angle about cation atoms (Si) is close to the tetrahedral angle, the bonding angle about the anion atoms (O) displays wide excursions (100°-180°). These structure results suggest that while both the bond-stretching and bond-bending constraints on Si appear to be intact, it is only the r-constraints on O that is intact. Indeed that separation permits one to understand [10] why silica, the archetypal glass former satisfies equation (1) exactly, and thus forms a stress-free continuous random network. The counting algorithm(s) also show that window glass, a pseudo-ternary alloy of and in the approximate molar ratio 75:15:10 could well represent [11] yet another example of a stress-free oxide network.

In this article, we review recent optical and thermal experiments that shed light on the existence of the self-organized state in network glasses. The self-organized state apparently occurs in between the rigid phase and the floppy phase and is therefore also termed as the intermediate phase. Our experiments with chalcogenide glasses [12] reveals that glass

transitions (Tg) become almost completely thermally reversing in character in the selforganized state. The thermally reversing character of Tgs can be probed by a recently developed variant of DSC, known as T-modulated Differential Scanning Calorimetry

(MDSC). The non-reversing heat flow term near Tg accessed in an MDSC measurement provides a direct measure of network stress [13] in a glass, and represents a significant recent development in the field. Raman Vibrational spectroscopy in the bond-stretching regime provides a uniquely powerful probe of optical elasticity which displays specific power-laws as a function of in the floppy, intermediate and stressed rigid phases of network glasses. Before describing the results of these experiments, we provide an overview of principle theoretical ideas that have evolved in the field of enthalpic rigidity in the next section. PREDICTION OF A RIGIDITY TRANSITION

Mean-field Theory in Random Networks

66

In 1788, J.L. Lagrange [14] wrote the celebrated book "Mécanique Analytique", and thus laid the foundations of Lagrangian mechanics as we know it today. In particular he

introduced the notion of generalized coordinates and forces that can act as constraints (Lagrangian multipliers) in solving mechanical problems. About a century later, J.C. Maxwell [15] used the notion of constraints to examine mechanical stability of

macroscopic structures such as trusses and bridges. Phillips-Thorpe Rigidity Threshold. In 1979, a new beginning to develop a microscopic theory of glass formation in covalent solids was made [6]. It was suggested that valence forces such as bond-stretching and bond-bending forces can serve as atomic constraints in assembling covalent networks. It was conjectured that glass formation would be optimized when the number of constraints/atom equal 3, the degrees of freedom of an atom in a 3d network (Table 1). These ideas were advanced further in 1983 when a normal mode analysis of covalent networks showed [7] that the number of zero frequency solutions of the dynamical matrix, f (= floppy modes), equal nc – 3. These developments led to the recognition that a glass network will become elastically rigid when the number of zero frequency solutions f vanish, i.e., nc = 3. For 3d networks in which atoms possess a coordination number the number of and constraints/atom, are given by

(2) (3)

The onset of rigidity in such networks it then follows, would occur in a mean-field sense, when or

(4)

or the mean coordination, The Phillips-Thorpe mean-field rigidity threshold, has been refined by numerical simulations [8] using the standard model of a covalent glass as a random network. These calculations are discussed in the contribution of M.F. Thorpe et al., in this

volume. For completeness we summarize the principal results of these calculations here. The numerical simulations [8] start with a base network at r = 4, such as a diamond lattice or a 4096-atom computer generated amorphous Si network with periodic boundary

conditions. Bonds are depleted at random to produce networks of lower in the range insuring that one-fold coordinated atoms are not produced. A Kirkwood or Keating potential is used to describe the and forces between atoms. The number of floppy modes/atom are then calculated as a function of by performing a normal mode analysis at each The results show (Fig. 1) a linear decrease in in the range (as shown by mean-field theory) which is followed by a long exponential tail in the approximate range 2.35 The onset of rigidity transition is then localized by the second derivative, that reveals (Table 1) a sharp peak at for the case of diamond lattice and at for the case of a-Si base network. These results are remarkably close (within 1%) to the mean-field result of for the onset of rigidity in random networks estimated by constraint counting. One-fold Coordinated Atoms (OFC, r=1). The Phillips-Thorpe mean-field estimate for the onset of rigidity is restricted to covalent networks in which and constraints are intact, and all atoms possess a r of OFC atoms (r = 1) have to be handled separately because such atoms do not produce constraints but can only erase them according to the 2r – 3 estimate. Many glass systems include OFC-atoms, such as 67

68

Figure 1. Results of numerical simulations for the fractions of floppy modes per degree of freedom for the bond-depleted diamond network in the random case and the self-organized case. The filled circle gives the stress and rigidity transition for the random case. The open square gives the mean-field predicted rigidity transition at The open circle and triangle give, respectively, the rigidity and stress transitions for the self-organized case. The shaded region gives the intermediate phase. Figure is taken from ref.3a.

halogens in the chalcohalides, hydrogen in hydrogenated a-Si, and alkalis in alkali-silicates. In 1984 Boolchand and Thorpe [16] extended mean-field constraint counting algorithms to include OFC-atoms. They showed that equation (3) takes on a correction term in the presence of a finite fraction, n1/N, of such atoms, and rigidity percolates in such networks when (5)

In general, OFC-atoms play no role if the base glass network (without 1-fold coordinated atoms) is optimally constrained, on the other hand, these atoms will soften [10] an overconstrained base network, and conversely stiffen an underconstrained base glass network. The latter is illustrated in Fig. 2 which shows a plot of the negative of f, henceforth labeled h (= hardness index) as a function of the mean coordination of the base glass network. The base glass network is the network remaining once all 1-fold coordinated atoms have been plucked. For the case when a glass network has no 1-fold coordinated atoms, i.e., n1/N = 0, we recover the usual circumstance, h = -1 at r´ = r = 2 and h = 4 at r = 4, and rigidity onsets at h = f = 0 when In the presence of a finite concentration of 1-fold coordinated atoms, the slope of the line progressively decreases from its maximum value of 5/6 with increasing n 1 /N, with all plots passing through the point h = 0, Such a behavior constitutes evidence of a softening of an overconstrained network or conversely stiffening of an underconstrained network in the presence of OFC-atoms. These ideas provide a means to understand [17] the linear variation of the nano-indentation hardness results of diamond-like films as a function of as discussed elsewhere. Furthermore, these ideas have also provided a basis to understand the glass forming tendency in a wide variety of chalcohalide glasses as recently discussed in a review by Mitkova and Boolchand [18]. In dealing with networks possessing OFC-atoms, two circumstances need to be distinguished [19]. Case 1, when OFC-atoms bond to an atom possessing and case 2 when OFC-atoms bond to an atom possessing an r = 2. Equation 5 is valid for case 1 69

Figure 2. Graphical representation of hardness index h vs. mean coordination of base network showing the straight-line loci for different values of OFC-atom concentration n1/n=0.025 and 0.50. Figure is taken from ref. 17.

above. The more complicated case 2 has also been treated. In case 2, the rigidity threshold for the base glass network does not coincide with that of the (complete) glass network. For these reasons, in general, it is safer to always calculate mean-field constraints of the complete glass network with the OFC-atoms, rather than truncate it and work with the base

glass network. Bond-Bending Forces and Broken Constraints. Valence forces form a hierarchy with force strength exceeding force strength by a factor of 3 or more. And it is possible that as glass transition temperatures Tg increase, the constraint associated with the weaker forces intrinsically break. The structure manifestation of the broken -constraint is that the underlying bond-angle distribution becomes wide. Presumably this results because thermal energies (kTg) can overwhelm the underlying strain energies that would require a unimodal bond angle distribution. The circumstance is encountered in the chalcogenides [17] and oxide glasses [18]. It is useful to examine the underlying meanfield estimate of the rigidity transition in the presence of broken -constraint. Consider a glass network possessing finite concentration (mr/N) of r-fold coordinated atoms that have their -constraint broken. The rigidity transition in such a network will, in a mean-field sense, occur at

where m2/N, m3/N, m 4 N,... represent the fraction of 2, 3, 4,...fold coordinated atoms in the network that have their -constraints broken. A special case of equation (6) is that of a glass network possessing a finite concentration of 2-fold coordinated atoms that

70

have their -constraints broken. In binary GexSe1-x glasses, although the -constraint associated with 2-fold coordinated Se-atoms is intact in a Se-glass at x = 0, upon crosslinking with Ge, Tgs rapidly increase, and the underlying constraint is apparently broken as x increases to 0.20. For this reason, the rigidity transition [20] invariably shifts to higher values of due to the presence of the first correction term of equation (5), i.e.,

where m2/N designates the fraction of Se atoms in Sen-chain fragments in GexSe1-x glasses near x = 0.23. A second illustrative example of broken constraints determining the stiffness transition occurs in the Ge1-xSnxSe2 ternary, where Mössbauer spectroscopy reveals [21] evidence of Sn atoms to become all tetrahedrally coordinated when x increases to 0.35. The overconstrained tetrahedra, when admixed with the underconstrained tetrahedra, drive the network optimally constrained. Equations (7) predicts [22,23] the transition to occur at x = 2/5, indeed quite close to the observed value of 0.35. Two Rigidity Transitions in Self-Organized Networks In 1999, numerical simulations were extended to self-organized networks for the first time, and one found [3] rigidity to onset in two steps at r c(1), rc(2). The first of these transitions (rc(1)) is quite similar to the one described earlier for random networks in which rigid regions percolate at At r > rc(1), if the additional crosslinks are selectively placed in the floppy domains of the network, network stress will not accumulate. This situation must be contrasted to the usual case when crosslinks placed randomly drive some of the isostatically rigid-regions stressed-rigid (redundant bonds) thus building up network stress. One has found that such selective placement of crosslinks saturates at r = rc(2) = 2.395, when it is no longer possible to avoid forming redundant bonds. At (2) the network makes a first order transition to a stressed rigid phase (Fig. 1). The term isostatic or stress-free can be used to describe the self-organized state of a network in the r c (1) < r < rc(2) phase, residing in between the floppy (r < rc(1)) and stressed rigid (r > rc(2)) phase. The unstressed rigid phase is also termed as the intermediate phase. Calculations [24] show that the size of the first order jump in elasticity at rc(2) is controlled by the concentration of small rings (6-membered rings or smaller) where rigidity apparently nucleates. Raman scattering experiments on chalcogenide glasses provide evidence of two elastic thresholds (rc(1), rc(2)), as we shall describe in the next section. And even though there persists a systematic difference between theory and experiments on the values of rc(1) and rc(2), these new numerical simulations represent a significant advance. They not only narrow the gap between theory and experiments, but also provide a physical basis to characterize the order of the transitions and the self-organized state prevailing in between the two transitions. GLASS TRANSITION VARIATION AND NETWORK CONNECTIVITY Glasses are distinguished from amorphous materials in that they display a glass transition temperature, Tg, that represents a softening or melting transition of the solid glass into a metastable liquid glass. Across Tg, viscosity, thermal expansion, entropy, molar volumes and specific heat change [25] qualitatively. The basic nature of glass transition continues to be a subject of profound current discussions. There are undoubtedly 71

kinetic and sample thermal history effects associated with Tg. But, there are also much larger structure or network connectivity related effects that apparently control the magnitude of Tgs in glass forming materials. T-Modulated Differential Scanning Calorimetry (MDSC) In a conventional DSC measurement, the signature of softening of a glass is an endothermic heat flow usually measured with respect to an inert reference, as the temperature of the glass and reference is swept at a fixed scan rate. By programming a sinusoidal temperature variation over the linear T-ramp, it is possible to deconvolute [26]

the heat flow into two components, one that tracks the sinusoidal T-variation and is therefore called the reversing heat flow and the remainder that does not track the periodic T-variation and is called the non-reversing heat flow Figure 3 provides MDSC scans [23] of a GeSe2 glass and Ge0.23Se0.77 glass illustrating the deconvolution of heat flow rates into the non-reversing and reversing components. Two noteworthy features become apparent from these scans, first the apparent glass transition temperature deduced from the total heat flow is, in general, lower than the glass transition temperature (Tg) deduced from the reversing heat flow. Second, is found to be miniscule for the x = 0.23 sample but it is an order of magnitude larger for the stoichiometric glass, GeSe2. The shift, is only 3°C for the x = 0.23 glass sample when is nearly vanishing, but it is 12°C for GeSe2 glass when increases by almost an order of magnitude. The presence of a sizeable term in a glass will lower in relation to

Tg due to kinetic heat-flow effects.

Figure 3. MDSC scans (a) Ge0.23Se0.77 and (b) GeSe2 glasses revealing the deconvolution of the heat flow into the reversing and non-reversing components. Note the qualitative reduction in coordinated glass. See text for details. Figure is taken from ref. 23.

72

for the optimally

Network Stress and Non-Reversing Heat Flow. Important insights into the physical origin of the term in glasses have emerged from compositional trends. The case of the GexAsxSe1-2x ternary is particularly significant because for this system compositional trends in activation energies of Kohlrausch relaxation of an external stress were established in flexural studies [27]. Recently, compositional trends in for this ternary were measured [13], and the comparison of the two sets of results taken in two different laboratories, at different time frames, shows a remarkable similarity (Fig. 4). The results suggest a specific interpretation of the term; the heat-flow term provides a measure of internal stress in a network glass, while the activation energies, a measure of relaxation of an external stress.

Tg and Network Connectivity. The Tg deduced from the reversing flow provide a very useful datum characterizing the connectivity of a glass. At the outset, it should be mentioned that Tgs deduced from the reversing heat flow are virtually independent

of sample thermal history, although these will shift up in T as scan rates are increased. Therefore if Tgs are measured at the same low scan rate at different compositions in a glass system, one can obtain useful structure information. recent development in the field.

Figure 4.

The latter represents a relatively

variation in GexAsxSe1-2x glasses showing the thermally reversing window in the

< 2.42 range (a). Activation energy for Kohlrausch relaxation of an external stress in flexural studies (b). Figure in top panel taken from ref. [13], while the figure in bottom panel is taken from ref. [27].

73

Agglomeration theory [28, 29] developed by R. Kerner and M. Micoulaut provides a quantitative means to analyze compositional trends in Tg in the stochastic regime. One can, for example, predict the T g (x) variation in binary GexSe1-x and AsxSe1-x glasses near x

when cross-linking of Sen-chain fragments by 4-fold Ge and 3-fold As atoms

proceeds in a stochastic fashion, i.e. randomly. These predictions are well supported by experiments. The predictions also provide a quantitative means to distinguish domains of stochastic from non-stochastic network formation in these binary glasses [2].

In the GexSe1-x binary glass system, the prediction of agglomeration theory is that the slope

where T0 is the glass transition of Se glass. In the experiments one observes (Fig. 5a) T g (x) to increase linearly with x, with a slope of T0/ln2 = 44°C/10 at % of Ge up to x = 0.08 defining the stochastic regime. At x > 0.08, the experimental Tgs exceed the predicted Tgs, suggesting onset of extended range structures beyond Ge(Se1/2)4 tetrahedra and Sen-chain

segments, and signaling the onset of a non-stochastic regime (Fig. 5a). It is in this 129

compositional range that I Mössbauer spectroscopy measurements reveal [30, 31] that the oversized Te-tracer atom ceases to replace available Se-sites in a random fashion. Thus the onset of non-stochastic behavior at x > 0.08 in these thermal measurements is in excellent agreement with 129 I Mössbauer spectroscopy results published nearly two decades ago.

RAMAN ELASTIC THRESHOLDS AND THE INTERMEDIATE PHASE Although Raman scattering has been used as a probe of glass structure [32-35] for the past three decades, the application of the optical method as a probe of rigidity transitions in network glasses is a recent development [20, 36-39]. In many cases when

vibrational bands are resolved, it is possible to quantitatively follow mode frequency changes with glass composition and deduce power-laws describing optical elasticity changes. And although the scales of mode frequencies are set by the strength of and forces, variations in mode-frequencies with glass compositions result due to intertetrahedral couplings, i.e. network connectivity. Because of the presence of light-induced effects in glasses, the need to excite Raman scattering at low-laser power is paramount. For this reason, resonant enhancement of the scattering by tunning the laser energy close to the optical bandgap of the glass, is particularly desirable. It results in a large signal to noise ratio with very modest laser exciting power. Raman scattering results on several IV-VI glass systems have now been performed and provide evidence of the intermediate phase. The principal results on the GexSe1-x binary glass system are presented next. Bulk GexSe1-x Glasses. Glasses in the titled binary at several compositions in the 0 < x < 1/3 range were synthesized the usual way and characterized by MDSC [2, 20, 38]. The observed Tgs measured from the reversing heat-flow serve an important check on glass compositions. Raman scattering excited by 647.1nm radiation from a Kr+ ion laser was studied in a conventional back scattering set up using a T64000 triple monochromator system with a CCD detector. The laser beam was brought to a loose focus of about 500µm spot size, to keep the photon flux low (1017 photons/cm2/sec), in a set up [38] usually described as macro-Raman scattering.

74

Macro-Raman Scattering

Figure 6 shows Raman spectra of the glasses obtained at selective compositions.

The lineshapes show evolution of modes of corner-sharing (CS) tetrahedra (200 cm -1 ) and edge-sharing (ES) Ge(Se1/2)4 tetrahedra (215 cm-1) at the expense of the Se-Se stretch mode of Sen-chains (CM) at about 250cm-1 . In general CS and ES modes are symmetric and could be analyzed in terms of a Gaussian each. This was not the case of the CM, however, and it required at least 3 Gaussians for an adequate deconvolution at all x. Figure 7 provides the mode frequency variation for the CS and ES mode from a comprehensive lineshape deconvolution of the Raman spectra [38]. In Fig. 7a, one can discern a regime of a linear variation in v(x) in the 0.08 < x < 0.20 interval, and a regime of a power-law variation in the 0.26 < x < 1/3 interval. In between these two regimes, lies a transition region wherein the variation in v(x) is sub-linear. To extract the underlying power-laws, we have made a plot of against and obtained the slope pCS of the resulting

Fig. 5(a). Observed Tg(x) variation in indicated binary glasses. The Tgs are deduced from the inflection point of the reversing heat flow. (b) variation in GexSe1-x glasses deduced from the non-reversing heat flow showing a global minimum near x = 0.225. Figure taken from ref. 2.

75

Figure 6. Macro-Raman scattering in bulk GexSe1-x glasses at indicated glass compositions x. Note the

growth in scattering of corner-sharing (CS) and edge-sharing (ES) tetrahedra at the expense of Se n -chain mode (CM). See text for details.

line (Fig. 8a) and get a value of pCS = 1.54(6). A parallel analysis for the transition region (Fig. 8b) using xc(1) = 0.21, yields a slope pt = 0.75(15). For the ES mode, our results suggest that the linear variation in at low x (0.10 < x < 0.20) smoothly connects to the power-law variation at higher x (x > 0.21). Figure 8c shows a plot of against which yields a straight line with a slope pES = 1.32(06), defining the underlying power-law associated with elasticity due to ES units. The rational

for extracting the Raman elastic power-laws is discussed next. Stressed Rigid Phase. On general grounds one expects rigidity in the present

glasses to nucleate at moities that are overconstrained units. Changes in

and

such as the ES and CS

result from local or optical elasticities and can be expected

to display a power-law variation with

once

exceeds a critical value

when the

backbone becomes rigid, i.e.,

The justification for such a variation is suggested by the numerical simulations [40, 41] on random networks constrained by and forces (Keating potential) that show the elasticity to display a power-law power p = 1.4 or 1.5. The power-law variation of

(Fig. 7a), starting at x = xc(2) = 0.26 with pCS =

1.54( 10) constitutes direct evidence for onset of stressed rigidity. The power-law pCS= 1.54 is in excellent agreement with numerical estimates [40, 41] of elasticity in random networks. Furthermore, the stressed nature of the backbone at x > xc(2) follows from the

rapidly increasing 76

heat-flow term once x > xc(2) as illustrated in Fig. 5a. The small

Figure 7(a). CS- and (b) ES-Raman mode frequency variation with x in GexSe1-x glasses displaying the various phases as discussed in text. Figure is taken from ref. 38.

jump in v c between x = 0.25 and 0.26 of xc(2) is first order.

suggests that the rigidity transition near

Intermediate Phase. Our Raman experiments also show that rigidity first onsets at

x = x c (l) = 0.20(1) in which both ES and CS units partake. This is suggested by the kink in and the onset of a power-law behavior in both starting near x c (l). The optical elasticity associated with the CS and ES units yields a lower power-law of pCS = 0.75(15) and PES = 1.38(10). The sublinear power-law for the CS units, PCS (=0.75(15)) is

reminiscent of the finite-size scaling result [42] which is 2/d = 2/3 for d = 3 (3d-network). To obtain a power-law p < 1, one must invoke large-scale or long-range fluctuations as are

discussed in equilibrium-scaling theory [43]. The composition x c (l) coincides with the lower bound of the global minimum in suggesting that in the x c (l) < x < xc(2) interval glasses are largely unstressed. Furthermore, the transition at x c (l) appears to be second order as revealed by the continuous variation of mode frequencies and These signatures of the experimental results constitute evidence for glasses in the xc( 1) < x < xc(2) composition range to be unstressed rigid, and to be self-organized, constituting the intermediate phase. 77

Figure 8. Log-log plots of Raman mode frequency squared against glass compositions yielding power-laws

for (a) CS mode in the stressed rigid phase (b) CS mode in the intermediate phase (c) ES mode in the rigid phase. See ref. 38

78

Floppy Phase. The present experiments [38] also suggest that at x < xc(1), glasses are floppy. Both and display a linear variation with x in the 0.10 < x < 0.20 range with a small slope. However at x < 0.10 in the stochastic regime, the slope of

with x actually vanishes. Numerical simulations [40 ,41] using the bond-depleted diamond structure reveal the elasticity in the floppy region to vanish. Thus, theory and experiment agree rather well in the stochastic regime. At x > 0.10, both T g (x) variation and Mössbauer

site occupancy variation [30] with x, signal the emergence of a non-stochastic network consisting of extended rigid inclusions formed in a floppy background. Here the experiments show the elasticity to slowly increase as It would be of interest to see whether such an observed variation can be reproduced in numerical simulations on non-stochastic floppy networks. In summary, the present Raman and MDSC results on GexSe1-x glasses provide evidence of two rigidity transitions, a second order transition at xc(1) = 0.20 from a floppy to an unstressed rigid phase, and a first order transition at xc(2) = 0.26 from an unstressed rigid to a stressed rigid phase. These two rigidity transitions represent the lower (xc( 1)) and

upper (x c (2)) bounds of an intermediate phase that separates the floppy (x < x c (1)) from the stressed rigid (x > xc(2)) phase in the present glasses. Complete Raman and MDSC measurements are now available on the companion IV-VI binary glass system, Si x Se 1-x [36,37]. The results reveal strikingly parallel details of the intermediate phase in this binary glass system with x c (l) = 0.20 and xc(2) = 0.27. Taken together, the close similarity of results on the width and centroid of the intermediate phase in the two independent IV-VI binary glass systems provides for some selfconsistency in the underlying physics of the rigidity transition.

PHOTOMELTING OF THE INTERMEDIATE PHASE

Historically, the rigidity transition in GexSe1-x binary glasses was first investigated in micro-Raman measurements [20]. In these measurements, the exciting laser beam (647. 1nm) was brought to a sharp focus using a microscope attachment. This had the consequence that the photon flux (N f) of 1022 photons/cm2/sec, exceeded that in the macro-Raman measurements (Nf = 1017 photons/cm2/sec) by about 5 orders of magnitude. Changes in Nf reflect the 2 orders of magnitude smaller versus laser spot size in the micro-Raman measurements. In both experiments, the exciting laserbeam wavelength of 647.1nm or Eph = 1.92eV was kept the same. For glass compositions in the intermediate phase, possessing an optical gap Eg > 2.0eV, the exciting beam is thus weakly absorbing and penetrates to a depth of about 100µm. In sharp contrast to the macro-Raman results, the micro-Raman measurements revealed [20] a solitary rigidity transition from a floppy phase to a rigid phase near x = xc = 0.23(1). To facilitate a comparison, we have reproduced variation in GexSe1-x glasses from the macro-Raman (Fig. 9a) with the micro-Raman (Fig. 9b) measurements together. One is led to the suggestion that while the macro-Raman measurements probe the intrinsic rigidity transitions of the glasses, the micro-Raman ones probe a light-induced modification of the transitions. Certain features become transparent from the comparison of these results. The two transitions at x c (1) = 0.20 and xc(2) = 0.26 coalesce to a point at the centroid xc = (xc(1) + xc(2))/2 = 0.23 location in the micro-Raman measurements. Furthermore, the centroid location xc = 0.23 coincides with the global minimum in heat flow term (Fig. 1b). 79

Figure 9. Raman mode frequency variation of CS sharing Ge(Se1/2)4 tetrahedra taken observed in (a) macroRaman and (b) micro-Raman measurements compared. The intermediate phase collapses to a solitary point

at its centroid in the micro-Raman measurements due to the higher photon-flux of the exciting 647.1nm radiation. Figure is taken from ref. 38.

Our interpretation of these Raman results is that the collapse of the intermediate phase to a solitary point is that it represents a photo-structural effect. In the presence of a high flux of near band-gap radiation photomelting of the self-organized

state to a random network takes place. The underlying physical process probably consists of a transient self-trapped exciton [44, 45] that recombines non-radiatively leading to a light-induced dynamic state in which rapid bond-switching and diffusion occurs. The photodiffusion process, apparently is optimized when the network stress vanishes and photomelting is the consequence. It is for this reason that the photomelted state is easily formed at the global minimum of the term (xc = 0.23), when network stress vanishes. Parallel physical effects have recently been observed in Brillouin scattering

measurements [45] on these glasses. In these experiments, one found the longitudinal acoustic (LA) mode to soften reversibly and by a gigantic amount (25%) as a function of laser excitation power, but only for glass compositions near the center of the intermediate phase, i.e. x=0.225. These Raman measurements on GexSe1-x glasses provide an essential link between the two extremal descriptions of the rigidity transitions in glasses. One observes two 80

transitions [38] when the backbone is intrinsically self-organized in the melt-quenched

glasses and one observes a solitary rigidity transition [20] characteristic of a random network in the photomelted state. And one expects the solitary transition to occur in connectivity space when corresponding to the counting algorithm(s).

INTERMEDIATE PHASE AND GLASS STRUCTURE The discovery of the intermediate phase in several chalcogenide glasses [12] is beginning to shed light on its connection to glass structure. Apart from to the IV-VI glasses, there are several other glass systems [13, 39, 46, 47] in which the phase has now been observed (Fig. 10). In table 2 we provide a summary of results on the subject.

Perusal of the results in table 2 shows that the width of the intermediate phase changes by almost an order of magnitude in going from the Ge-S-I ternary [39] to the Ge-As-Se ternary with the results on binary glasses of Ge-Se, and As-Se lying in between 0.08(1)) these two extremes (Fig. 10). The Ge-S-I ternary appears [39] to be a paradigm of a random network. In this ternary, replacement of S in the base Ge0.25S0.75 glass by 1-fold coordinated I, proceeds to randomly depolymerize the network. Enumeration of constraints reveals that only one building block (m=l) out

of the five possible (m=0, 1, 2, 3, 4) mixed GeSe4-mIm tetrahedra is optimally constrained, and thus contributes to the formation of the intermediate phase. In Raman scattering, one can observe modes of these mixed tetrahedra and thus establish details of glass structure. Furthermore, the Raman mode frequency of Ge(S1/2)4 tetrahedra, v 0(x), is found to vary linearly with Iodine concentration (x) to show a kink (change in slope) at the rigidity transition, which coincides with the sharp minimum in The transition, according to mean-field theory, is predicted to occur at Remarkably, the present experiments (Fig. 10 show the transition to occur at Table 2 Experimental Results for the Intermediate Phase in Network Glasses, r c( 1) designates onset of

81

Figure 10.

variation as a function of glass composition revealing thermally reversing windows in

GexSe1-2x(O), GexAsxSe1-2X (z) and

bulk glasses. Note the large variation in the window

width from 0.15 for the Ge-As-Se ternary to < 0.01 for the Ge-S-I ternary.

In sharp contrast, the GexAsxSe1-2x ternary, often described as a prototype of a random network, turns out to be one of the best example of a glass system in which a high degree of self-organization occurs [13] as reflected by the extremely large width of the intermediate phase. We believe the result is intrinsically related to the multiplicity of optimally constrained units and combination of over- and under-constrained units that can form as part of the stress-free backbone. Some of these units include pyramidal As(Se 1/2)3, quasi-tetrahedral Se=As(Se1/2)3, in addition to a combination of Ge(Se1/2)4 tetrahedra (both CS and ES) with Sen-chain fragments. And it is possible that molecular dynamic simulations, in conjunction with the Raman elasticity power-laws can impose stringent restrictions on possible structures prevailing in this special region, for these to be identified in future.

CONCLUSIONS Raman scattering and T-modulated Differential Scanning Calorimetry measurements on several families of chalcogenide glasses have been performed. Comprehensive results are now available on the GexSe1-x and SixSe1-x binary glass systems, where rigidity is found to onset in two steps; a second-order transition at x c (l) =

0.20 in both binaries from a floppy to an unstressed rigid phase, and a first-order transition at xc(2) = 0.26 for Ge-Se, = 0.27 for Si-Se binary from an unstressed rigid to a stressed rigid phase. The two transitions (xc(1), xc(2)) define the bounds of an intermediate phase that separates the floppy from the stressed rigid phase. The near absence of the nonreversing heat-flow, constitutes evidence for the stress-free nature of the backbone of glass compositions in the intermediate phase. Light-induced melting of the intermediate

phase in micro-Raman measurements on the Ge-Se glass system has also been observed. 82

The observation constitutes evidence for photomelting of the self-organized state to a

random network. These results confirm that the breakdown of mean-field theory, results due to an intrinsic self-organization of the backbone, and provide the link between the two extremal descriptions of the rigidity transition in network glasses modeled in numerical simulations. ACKNOWLEDGEMENTS

It is a pleasure to acknowledge discussions with Mike Thorpe and Jim Phillips during the course of this work. This work is supported by NSF grant DMR-97-02189. REFERENCES 1. 2. 3.

Pauling, L. (1960) Nature of the Chemical Bond, Cornell University Press, Ithaca, NY pp 85. Boolchand, P. and Bresser, W. J. (2000) Structural Origin of Broken Chemical Order in GeSe Glass. Phil. Mag B 80, 1757 – 1772. a. Thorpe, M.F., and Chubynsky, M.V. (2000), Rigidity and Self-Organization of Network Glasses and the Intermediate Phase in M. F. Thorpe (ed.) Properties and Applications of Amorphous Materials,

Kluwer Academic Publishers, Dordrecht, (in press). b. Thorpe, M.F., Jacobs, D.J., Chubynsky, M.V., Phillips, J.C. (2000) Self – Organization in Network Glasses, J. Non-Cryst Solids 266-269, 859-866. 4. 5. 6.

Phillips, J.C. Mathematical Principles of Intermediate Phases in Disordered Systems, present volume. Thorpe, M.F., and Chubynsky, M.V. Rigidity and Self-Organization of Network Glasses and the Intermediate Phase, present volume. Phillips, J.C. (1979) Topology of covalent non-crystalline solids I: Short-range order in chalcogenide alloys, J. Non-Cryst. Solids 34, 153-181.

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Thorpe, M.F. (1983) Continuous deformation in random networks, J. Non-Cryst. Solids 57, 355-370.

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Zhang, M. and Boolchand, P. (1994) The central role of broken, bond-bending constraints in promoting

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Applications of Amorphous Materials, Kluwer Academic Publishers, Dortrecht, (in press). Wang, Y., Boolchand, P. and Micoulaut, M. Glass structure rigidity transitions and the intermediate phase in the Ge-As-Se ternary, Europhys. Lett, (in press). Lagrange, J. L. (1788) Mecanique Analytique, Paris. Maxwell, J. C. (1864) On the calculation of the equilibrium and stiffness of frames, Philos, Mag. 27, 294-299. Boolchand, P., and Thorpe, M.F. (1994) Glass-forming tendency, percolation of rigidity, and one-fold-

coordinated atoms in covalent networks, Phys. Rev. B 50, 10366-10368. Boolchand, P., Zhang, M. and Goodman, B. (1996) Influence of one-fold-coordinated atoms on mechanical properties of covalent networks, Phys. Rev. B 53, 11488-11494. Mitkova, M. and Boolchand, P. (1998) Microscopic origin of the glass forming tendency in chalcohalides and constraint theory, J. Non-Cryst. Solids 240, 1-21. Boolchand, P., Zhang, M., Goodman, B. (1997) One-fold coordinated atoms, constraint theory and nanoindentation hardness, in M.F. Thorpe and M. Mitkova (eds.) Amorphous Insulators and Semiconductors, Kluwer Academic Publishers, Dortrecht, pp. 339-348.

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Feng, X.W., Bresser, W.J. and Boolchand, P. (1997) Direct evidence for stiffness threshold in

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chalcogenide glasses, Phys. Rev. Lett. 78, 4422-4425. Stevens, M., Grothaus, J., Boolchand, P. and Hernandez, J.G. (1983) Universal structural phase-

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transition in network glasses, Solid State Comm. 47, 199-202. Phillips, J.C. (1983) Realization of a Zachariasen glass, Solid Stale Comm. 47, 203-206. Boolchand, P. (2000) Vibrational excitation in glasses: Rigidity transition and Lamb-Mössbauer factors, in P. Boolchand (ed.) Insulating and Semiconducting Glasses, World Scientific Press, Singapore, pp. 369-414. Thorpe, M.F., Jacobs, D.J., Chubynsky, M.V. and Rader, J.A. (1999) Generic rigidity of network glasses in M.F. Thorpe and P. M. Duxbury (eds.) Rigidity Theory and Applications, Kluwer Academic/Plenum Publishers, New York, pp. 239-277.

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Zallen, R. (1983) The Physics of Amorphous Solids, John Wiley and Sons, New York, pp.3.

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Modulated DSCTM Compendium (1997) Reprint #TA-210, TA Instruments, Inc., New Castle, DE http://www.tainst.com/ Böhmer, R. and Angell, C.A. (1992), Correlations of the non-exponentiality and state dependence of mechanical relaxation with bond-connectivity in Ge-As-Se supercooled liquids. Phys. Rev. B 45, 10911094. Kerner, R. and Micoulaut, M. (1994) A theoretical-model of formation of covalent binary glasses. 1. general setting, J. Non-Cryst. Solids 176, 271-279.

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Micoulaut, M. and Naumis, G.G. (1999) Glass transition temperature variation, cross-linking and structure in network glasses: a stochastic approach, Europhys. Lett. 47, 568-574. Also see contribution

of M. Micoulaut – Glass Transition Temperature variation as a probe for network connectivity in this volume. Bresser, W.J., Boolchand, P., and Suranyi, P. (1986), Rigidity Percolation and Molecular Clustering in Network glasses. Phys. Rev. Lett. 56, 2493-2497.

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Bresser, W.J., Boolchand, P., Suranyi, P. deNeufville, J.P. (1981), Direct evidence for intrinsically broken chemical ordering in melt-quenched glasses 46, 1689-1692.

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Galeener, F.L. (1990) The structure and vibrational excitations of simple glasses, J. Non-Cryst. Solids

33. 34. 35.

36. 37. 38. 39.

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44. 45. 46. 47.

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123, 182-190. A symposium was held in 1995 to honor the late Frank L. Galeener’s contributions to glass science. The interested reader may want to look at the proceedings of this symposium published in J. Non-Cryst. Solids 182, 1-212. Lucovsky, G. (1979) Chemical effects on the frequencies of Si-H vibrations in amorphous solids, Solid State Commun. 29, 571-576. Griffiths, J.E., Espinosa, G.P., Remeika J.P., et al. (1982) Reversible quasi-crystallization in GeSe2 glass, Phys. Rev. B 25, 1272-1286. Murase, K. (2000) Vibrational excitations in glasses: Raman scattering, in P. Boolchand (ed.), Insulating and Semiconducting Glasses, World Scientific Press, Inc., Singapore, pp. 415-463. Selvanathan, D., Bresser, W.J., Boolchand, P., and Goodman, B. (1999) Thermally reversing window and stiffness transitions in chalcogenide glasses, Solid State Commun. 111, 619-624. Selvanathan, D., Bresser, W.J., Boolchand, P. (2000) Stiffness transitions in SixSe1-x glasses from Raman scattering and temperature-Modulated Differential Scanning Calorimetry, Phys. Rev. B61, 15061-15076. Boolchand, P., Feng, X., Bresser W.J. (2000) Rigidity transition in binary Ge-Se Glasses and the intermediate phase, J. Non-Cryst. Solids (submitted). Wang, Y., Wells, J., Bresser, W.J., Boolchand, P. (2000) Stiffness Transition in a Zachariasen glass: Theory and experiment (unpublished). Franzblau, D.S., Tersoff, J. (1992) Elastic properties of a network model of glasses, Phys. Rev. Lett. 68: 2172-2175. He, H., Thorpe, M.F. (1985) Elastic properties of glasses, Phys. Rev. Lett. 54: 2107-2110. Josephson, B.D. (1966) Relation between the superfluid density and order parameter for superfluid He near Tc Phys. Lett. 21, 608-609. Chayes, J.T., Chayes, L., Fisher, D.S., et al. (1986) Fubute-size scaling and correlaton lengths for disordered systems, Phys. Rev. Lett. 57, 2999-3002, Fritzsche, H. (2000) Light induced structural changes in Glasses in P. Boolchand (ed.) Insulating and Semiconducting Glasses, World Scientific Press, Inc., pp. 653-690. Gump, J., Finkler I., Xia, H., Sooryakumar, R., Bresser, W.J., Boolchand, P. (2000) Direct evidence for photomelting of the Intermediate Phase in Network glasses (submitted to Phys. Rev. Lett.). Georgiev, D.G., Boolchand, P., Micoulaut, M. Rigidity transitions and molecular structure of AsxSe1-x glasses, Phys. Rev. B (in press). Georgiev, D.G., Mitkova, M., Boolchand, P., Brunklaus, H., Eckert, H., Micoulaut, M. Molecular Structure, Glass transition temperature variation, agglomeration theory, and network connectivity of binary P-Se glasses, Phys. Rev. B (submitted).

THERMAL RELAXATION AND CRITICALITY OF THE STIFFNESS TRANSITION

Y. Wang, T. Nakaoka, and K. Murase Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Osaka 560-0043, Japan

INTRODUCTION Covalent network glass system Ge-Se has been well studied and analyzed in terms of testing theories on materials (glasses) science. Mean-field constraint theory [1–4] of network glasses is proving to be a powerful tool in explaining numerous anomalous behaviors around the critical composition of rigidity transition threshold at an average coordination number, corresponding to x=0.20 for at which the number of constraints per atom is equal to the degree of freedom per atom [5–11 ]. Short- and medium-range order in glasses can be varied continuously to form both the floppy and rigid glasses in the sense of the constraint theory, because that for the composition range the coordination numbers of Ge and Se are 4 and 2, respectively. The character of a network glass undergoes a qualitative change, from being easily deformable at to being rigid at Recently, Boolchand et al. [5, 6] demonstrated that the results from Raman scattering, modulated differential scanning calorimetry, molar volumes and Mössbauer spectroscopy work provide evidence of a multiplicity of stiffness transition, an onset point near and a completion point near between which a transition region or an intermediate phase separates the floppy glass from the rigid one. Kamitakahara et al. [7] reported the dynamic density of state around 5 meV to prove the existence of the floppy mode (“zero-frequency” mode in floppy glass) by studying the inelastic neutron spectra of chalcogenide glasses, however, the structural explanation of the floppy mode is still underconstructed yet. In this paper, we report the Raman-scattering results for glasses in a temperature range of 10–1000 K which covers glassy, super cooled liquid (SCL), crystalline and liquid states. The thermal relaxations well below the glass-transition temperature, Tg, in the vicinity of Tg and above Tg differ critically between the floppy and rigid glasses. We propose a micro-structural explanation of the floppy modes in glasses that structural units Sen(n>2) play an important role in influencing the macroscopic properties. The criticality of the stiffness transition is discussed from the point of view of structural admixture of Sen

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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Figure 1. The preparation of thin film. (a, b) Se element or bulk source of GexSe1–x are held in the cell where a narrow gap of is confined by two paralleled quartz plates. (c) After sealing the cell in a 10–6 Torr vacuum, we heat up the cell to 200 °C higher than the melting point of GexSe1–x and make the temperature of GexSe1–x source slightly higer than the temperature in the narrow gap. Then, liquid GexSe1–x penetrates into the narrow gap. (d) After quenching, the thin film between the two quartz plates is obtained.

and GeSe4/2 units. We demonstrate that reversible structural changes with temperature are

associated with a medium-range order in GeSe2 glasses, involving significantly layered-like crystalline GeSe2 fragments and inhomogeneously distributed nanophases of Ge–Ge and Se– Se homopolar bonds. The intrinsic Ge–Ge bond that does break the chemical order assists the glass to construct the layered-like crystalline fragments. GLASS PREPARATION Bulk glasses GexSe1–x are prepared by quenching the melts after rocking the ampule of mixed compounds sealed in for at least 48 hours. All the measured bulk samples are

picked from the as-prepared broken pieces with a typical dimension of 5x5x1 mm, except the thin films for studying the temperature dependence at high temperature. The thin films of GexSe1–x, with thickness of about are prepared by quenching from melts held in

evacuated fused-silica cells with a narrow gap defined by two parallel plates. The preparation procedure is sketched in Fig. 1. Confinement of the films in the cells takes advantage of keeping the same composition of glasses even at the high temperature, at which considerable chalcogenide compounds might vaporize if the bulk are held in vacuum. Furthermore, the films are assumed to be quenched with less difference in cooling rate between the surface and the inside than the situation in melt-quenched thick bulks. RAMAN SCATTERING AND MEASUREMENT SETUPS

In noncrystalline materials, wavevector

is not a good quantum number for phonons

because that losing the translational symmetry yields the break-down of the usual wave-vector conservation law. Raman scattering processes are allowed to occur from essentially all the vibrational modes of the material. Usually the intensity of scattered light is proportional to the vibrational density of states

for Stokes process. 86

describes the coupling of the vibrational modes of frequency

Figure 2. Polarized Raman spectra for GexSe1–x at 80 K. The resolution of the spectra is better than 1 cm – 1 .

to the light, and is the Bose factor. Conventional backscattering configuration for incident and scattered lights is employed in our case. For avoiding light-induced events and restricting the sample temperature difference from the environment to less than 3 °C, a low power-density probing light of less than 3W/cm – 2 (< 10W/cm –2 for measuring the spectra of liquid phase around 1000 K) is applied for the Raman measurements. The back-scattered light is collected and analyzed with the triple grating polychromator (JOBIN YVON T64000) and charge-coupled-device (CCD) detector in polarized and depolarized configurations. Generally, we use a 680 nm line from an argon ion laser pumped Dye (DCM) laser due to the sensitivity of our analysis setup, mainly determined by the monochromator and CCD detector, peaking around 600 nm. Insofar as we eliminate the interference of Rayleigh wings in the low-frequency spectra, an 800 nm

line from argon ion laser pumped Ti:Sapphire laser is a better choice that extends our lower limit to 3 cm – 1 at room temperature. A longer wavelength of laser light used as the pumping source yields a higher resolution for Raman spectrum in our experimental setups as well. STRUCTURAL RELAXATION WELL BELOW Tg Stokes Raman spectra of GexSe1–x glasses for various composition at 80 K are shown in Fig. 2. Raman spectra are reduced by and normalized to the maximum intensity,

for low Ge concentration the one at 255 cm –1 or at 200 cm – 1 for high Ge concentration. In g-Se, the dominant peak at 255 cm – 1 and a shoulder at the lower frequency side are

relevant to the bond-stretching modes of Sen chains [12, 13]. The weight of Se-chains related intensity decreases with increasing Ge concentration, which qualitatively reflects the changes in the vibrational density of states. It follows the fact that the number of GeSe4/2 tetrahedra increases with increasing Ge content. A strong peak at lower (A 1 ) and a weak one at higher

87

Figure 3. Raman spectra for Ge8Se92 glass at various temperatures. The spectra are obtained in a T-increasing sequence.

energy side in the vicinity of 200 cm – 1 are classified to the breathing modes of GeSe4/2 tetrahedra which are in the corner-sharing (CST) and edge-sharing (EST) configurations, respectively [14]. It should be noted that even in the glass of Ge content of 8 percents a small amount of tetrahedra shares the edges. For GeSe2 and Ge35Se65, the Ge–Ge bond related band at 180 cm – 1 increases with Ge content. In GeSe2 glass, the homopolar Ge–Ge and Se–Se “wrong” bonds are assumed to release the exceeding tension or stress in the highly distorted network, since its mean coordination number is 2.67 much larger than 2.40. The Se–Se vibrational bands are able to be observed at 260 cm – 1 with the resonant Raman condition as well [15]. Further discussion on the Ge–Ge and Se–Se “wrong” bonds in GeSe2 glass will be done in the following sections and elsewhere. In GexSe1–x glasses, the basic structural units are Sen chains and GeSe4/2 tetrahadra. The former are floppy units that build floppy regions in the network; the latter are rigid units that construct rigid ones. Studying the temperature dependence of vibrational properties of Se–Se or Ge–Se bond for each unit will supply plenty information on the microstructure of the network. Figure 3 shows Raman spectra for Ge8Se92 glass from 20 K to room temperature. The spectrum at the successive higher temperature is plotted with the base line raised by the same value. To take systematic analyses, the line shape of the spectra in the range of 160–300 cm – 1 is fitted by one straight line and four Gaussian curves, corresponding to two breathing modes of GeSe4/2 tetrahedra and two Se-chains related modes. Here, we focus on temperature dependence of peak positions of GeSe4/2 breathing mode, A 1 , at 200cm –1 and Se-Se bond-stretching mode at 255 cm – 1 , as shown in Figs. 4(a– d). In the floppy glass of Ge8Se92, an anomalous peak shift of the A1 mode, a positive shift with temperature, is observed below 100 K. This anomalous feature is hard to be understood without introducing a structural changes at such low temperature. During we cool the glass down to 20 K (well below the Tg’s), in other senses which can be regarded as a second kind of quenching process, each structural unit freezes at the correspondent critical temperature below which the unit is forbidden to frequently change the topology in the network. In Ge-Se network, first the GeSe 4/2 -unit rich region freezes, and

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Figure 4. The temperature dependence of the A1 breathing mode of tetrahedra and the Se-Se bond-stretching mode. Both are scaled by the frequency of each mode at the lowest temperature we measured, (a) and (b) display the typical case in floppy glasses. (c) and (d) the rigid glasses.

at this time (or temperature) the tetrahedra might not be constraint or stressed. If all the atoms vibrate in limited spaces caused by the construction of volume with decreasing temperature, a large number of stress or tension then should be induced since the glassy sample is not

an ideal glass. Appending such stress or tension on the local structural units of GeSe 4/2 tetrahedra and Sen chains, the structure of each unit must be distorted more or less because in the microscopic region such kind of force will not be isotropic. The lowering frequency of A 1 mode during the cooling process suggests that the tetrahedra are departing from the regular structure. The similar stress or tension should influence the Se-chain structure as well. The reason why the Se-Se bond-stretching mode shows less temperature dependence than the A1 mode does is probably because that the bending force of Se-Se bond is much weaker than its stretching one. Most part of distortion to the Sen (n>2) chains causes a wide distribution of Se–Se–Se bond angle. Unfortunately, we can only detect the “rest” stretching mode of Se-Se bond. Structural Origin of Floppy modes in Ge x Se 1–x

In Ge8Se92 glass, Sen chains are the dominant structural units and GeSe4/2 tetrahedra are probably isolated from each other (few tetrahedra binding together should enhance the isolation). The structural units of Sen (n>2) chains, as shown in Fig. 5, are assumed as the key units for the heavily distorted tetrahedra “relaxing” to the regular structure. We suppose that the rotating motion of Se(2) atom on the axis through the Se(l) and Se(3) atoms will “completely” freeze at 20 K. [For the Se(3) atom, if there is a Se(4) atom next to it, the same discussion we did for the Se(2) atom holds.] With increasing temperature, the rotating mode of Se(2) atom could be thermally excited, since the relative force determined by the structure

89

Figure 5. Schematic diagram for atom arrangements of Se chains. The Se(2) atom rotates on the axis through n the Se(l) and Se(3) atoms. The rotating motion is constrained by two prior free dihedral angles consisting of the next nearest neighbor forces. The same rotating mode for the Se(3) exists with the Se(4) bonding to it.

consisting of two priori free dihedral angles, is fairly weak. Such thermally excited rotating

motion supplies channels to leak the stress or tension by the way of changing the topology of network. The heavily distorted network then relaxes and make the “remaining” (or observed) A1 mode strengthened. When temperature is approaching to 100 K, the rotating modes is totally excited, and thereafter the vibrational peaks start to shift in negative behavior due to anharmonic effects. In our measured samples, the most remarkable positive shift of the A1 mode occurs in the Ge8Se92. For x<0.08, although the positive shift (with the same origin) is expectant, it might be difficult to figure them up strictly because of a smaller population of tetrahedra to form the vibrational density of states (VDOS) that brings big errors in fitting the line shape of A1 mode. Or, the network could settle the stress or tension through the large floppy regions that consist of Sen (n>2) chains surrounding the GeSe4/2-rich region. While for Ge15Se85,

the floppy region that surrounds the rigid region of GeSe4/2 units is not large enough to provide space for the distorted tetrahedra “relaxing”. Nevertheless, we consider the rotating mode in Fig. 5 to be a probable micro-structural explanation for the floppy modes in Secontained covalent network glasses. The same kind of rotating modes has been discussed in the computing analysis for modeling the flexible and rigid regions in proteins [16]. The easy

excited dihedral angles that free the rotating motion of Se(2) and Se(3) in Fig. 5 are assumed as the bananas graphs proposed by Thorpe et al. [4]. In Se-contained glass system, the floppy mode was claimed in the vicinity of 5 meV from the neutron-scattering results of vibrational density of states. Weakly binding forces shift the floppy modes from zero frequency to 5 meV. In figure 4(a), the most positive shift of A1 mode occurs at 60 K. The relation between the frequency of 5 meV for floppy modes and the critical temperature of ~60 K for the thermal excitation of rotating motion contains rich hints to build a structural picture for the network glasses. It is natural to regard the soft constraint rotating modes as the floppy modes in the realistic floppy glasses. Up to now, most of properties of covalent glasses are discussed at room temperature or higher, at which such “constraint” (weakly binding forces) floppy modes are totally free. That is why the elegant constraint theory works so good [5–11], although the theory settles the problem of networks at 0 K.

Criticality of Stiffness Transition With increasing x, the

of floppy glass is approaching to the rigidity percolation

threshold of (x=0.20). The stiffness transition behavior clearly appears in the temperature dependence of the vibrational modes between 100 K and room temperature where the anharmonic effects determine the essential rules. As shown in Figs. 4(a) and (b), large drops 90

Figure 6. The slope of straight line obtained from the data of each modes in the temperature range of 100–250 K (figure 4). The A1 mode (square) has a fourth time large of the volume obtained from the Se–Se stretching mode( ) for While for they are similar. The solid lines are guide to the eye.

•

of the shifts for both the A 1 mode and the Se–Se mode at room temperature (RT) are observed in the floppy glasses. The RT is the temperature to which the samples were quenched (it should be 0 °C for an accurate discussion), and at which they rested for a long time before the measurement. The lower temperatures (
modes and get well linear fits. The slopes of lines as a function of x and mean coordination number for both the A1 mode and the Se–Se mode are plotted in Fig. 6. A nice convergence of the slopes, like the shifts themselves [Fig. 4(c)], comes to a common value at x=0.22. This is the stiffness transition at which the number of degrees of freedom is equal to the number of constraints [5]. Apparently, all the observed modes shift together at this transition, because the network becomes almost ideally random to realize a maximum mixing of the different local modes. For the rigid glasses, the fact that few Sen (n>2) units are included in the structure is the reason of admixture of all the local modes. Probably the same mixing of Se–Se bending motion (it should be the Se–Se–Ge bending motion) goes into both the A1 and Se–Se stretching modes. So although the local modes of the A1 breathing and the Se-Se stretching have different frequencies, 200 cm – 1 and 255 cm –1 , respectively, the CHANGES in those frequencies (scaled by their frequency at the lowest measured temperature) become the same. This kind of mixing should occur in a long-range scale and well detected by the light-probe technique. It should be stressed that the long-range mixing of vibrational modes is very analogous to critical fluctuations of random system in second-order phase transitions. Comparing the modes mixing (in Fig. 6) in Ge22Se78 and Ge30Se70 glasses, we find that the two modes in Ge3oSe70 glass mix better than those in Ge22Se78. The slope of A1 slightly differ from that of Se-Se mode at (x=0.22). These results should have respect to the transition window in which the glass is not ideally random due to a self-organization. Boolchand et al. has done a systematic study of the width of the transition window and pointed out the completion point ( or x=0.23). For the floppy glasses, the A1 mode shifts much more than the Se–Se bond-stretching mode does. The probable reason is that the mixing of Se–Se bending motion into the bending of Se–Ge–Se permits all the bending modes mixing in the floppy glasses. In other words, the A1 band is not a pure breathing mode because of the admixture of bending modes. Thus,

91

Figure 7. Dynamic phase diagram for Ge-Se glass system. The solid line is the liquidus taken from Ref. [18]. All the data are estimated from the Raman measurement.

the Se–Se stretching mode shows less temperature dependence than the A1 breathing mode in which the soft bending modes are mixed, since the stretching forces are stronger than the bending ones.

CRYSTALLIZATION TENDENCY

An extensive study of the temperature dependence of the spectral shape for the GexSe1–x thin films [17] has been applied to understand the structural changes around and above the glass transition temperature, Tg . In general, the spectrum is accumulated for 10 minutes at each measured temperature which is fixed within and the rate of increase of the temperature between two successive measurements is 2–5 °C/min. For each composition, all the spectra are obtained within ~10 hours. In the vicinity of stoichiometic composition, or the medium range structures of Sen chain or GeSe2 fragments that are topologically similar to the layered crystalline GeSe2, respectively, dominantly contribute to the changes. In the glasses at intermediate composition, the two structural units mix well. With increasing temperature above the Tg, the low viscosity of the SCL enables us to detect the structural changes within our measurement periods. Thus, we define the Tg, obtained by Raman measurement, as the onset of the changes of spectral shapes. Heating the glasses thereafter, characteristic lines of crystalline

(c-) Se or c-GeSe2 may appear in the spectra and disappear at higher temperatures. We term the recrystallization temperature Tc and the melting point Tm as the temperatures at which the crystalline peaks appear and disappear, respectively, regardless of whether the sample is stoichiometic (x = 0, 0.33). The phase diagram of Fig. 7 summarizes the results for Tg (square),

92

and

The Tg (square) from Raman result is in a good agreement

with the result

from DSC measurement [17]. The present values for Tm agree with the

results of Ref. [18], where the liquidus line indicates the primary crystallization temperature for c-GeSe2 or c-Se. For x<0.04, as the small amount of GeSe4/2 tetrahedra fails to prevent the crystallization of Se, mixtures of c-Se and liquid Ge x Se1–x appear between the liquid and super-cooled liquid (SCL) states. While, for x>0.18, the large amount of GeSe4/2 tetrahedra builds a medium-range structure topologically similar to the layered c-GeSe2 to promote the crystallization of GeSe2. For since the time for the (Se)n chains forming the long-

range order is longer than the experimental time period, neither c-Se nor c-GeSe2 appears. Embryo of c-GeSe2 constructs in SCL state for x>0.10, however, only those of x>0.18 evolve into nuclei and crystallize. Here, we define a crystallization ability (CA) by a ratio of crystallizable temperature range to SCL range, using the Raman measurement results, and make the CA to be unit for the glass transferring directly from glassy to liquid phase. The crystallization tendency derives from the glass forming tendency (GFT), with the relation of Figure 8 indicates the crystallization tendency, obtained by the Raman measurement, which is very similar to the graph of glass forming difficulty summarized in Ref. [1]. It is worth to mention the recent result of GFT in GexSe1–xglasses from Boolchand’s group [5]. They synthesized homogeneous and strainfree bulk glasses by cooling melts at a very slow rate of 2.5 °C/minute in a T-programmed resistive furnace. The GFT is apparently optimized in the composition range of 0.20<x<0.24 (very similar to the stiffness transition window they found out) where no eutectic effects can be introduced. In accordance with the CA from the Raman result of heating glasses from RT to liquid phase, we do not observe a minimum of CA in the stiffness transition window. It should be noted that our samples are quenched from the temperature 200 °C higher than

the Tm inferred from the liquidus [18]. Thus, in our samples the local structure must quite differ from the structure in the melts during cooling at the rate of 2.5 °C/minute. The local structure, which is restricted to a nanoscale, attracts wide attention, such as the nanophase separation in GeSe2 glass being discussed in the following sections.

Figure 8. Crystallization ability calculated form the dynamics phase diagram (Fig. 7) by equation 2.

93

Figure 9. Three types of typical temperature dependence of the Raman spectra in GeSe2 glasses. Each spectrum is normalized at the most intense peak around 200 cm – 1 . Starting from the Raman spectra at RT, however, the glasses crystallize into (a) the phase, a layered crystal involving both CST and EST units; or (b) the phase, a 3D-network crystal consisting only CST unit; or (c) the phase, for which the Raman spectrum is the first reported.

TRIFURCATED CRYSTALLIZATION AND IN HOMOGENEITY IN GeSe2 GLASS

In this section, we concentrate on the structural changes in GeSe2 glass prior to the crystallization towards three crystalline phases (“trifurcated crystallization”). Thermally induced structural changes are reversible even well below the Tg of ~390 °C. These reversible changes provide an indication that the Ge–Ge wrong bonds bearing a nanophase play an important role on the network stability. Strong correlation between the degree of the structural change and the crystallized phase is discussed in terms of homogeneity in a nanoscale.

Structural changes and crystallizations Free-standing bulk GeSe2 samples are sealed in a silica tube in an argon gas atmosphere (~360 Torr) to reduce evaporation and oxidation effects. Raman spectra are measured using a 1.83 eV laser light with a very low power density of less than 1 W/cm–2. The laser light is focused onto the samples in a ~0.1x4 mm2 rectangular region. Figures 9 (a), (b), and (c) show the three typical temperature dependence of the Raman spectra of GeSe2 glasses from

RT to the crystallization temperatures, Tc. The intense Raman band located at 200 cm –1 at

RT represents an A 1 in-phase breathing vibration of the corner-sharing tetrahedra (CST) [9]. The A 1 companion band at 215 cm – 1 is related to a breathing vibration quasi-localized at edge-sharing tetrahedra (EST). The AG band at 176 cm –1 is interpreted as the stretching mode of Ge–Ge bonds involved in ethane-like units [19, 20]. No distinguishable difference is observed among the Raman spectra of the glasses at RT because all of the measured GeSe2 glasses are prepared at the same conditions. However with increasing temperature, notable spectral changes occur at 300 °C which is well below the Tg of 390 °C.

It was reported that once a crystallization process starts, some of them result in the same crystalline phase as they initially appeared, while some of them transform into other phases after held at the same temperature for an additional long period (~ 60 hours) [21]. In this work, we restrict the discussion of crystalline phase to the initially appeared crystalline phase. We checked the phase transition with keeping temperature for the same as the glasses initially

94

Figure 10. Integrated intensity ratios of the AG band to the A1 band (a–c) and the

band to the A1 band (d–f)

as a function of temperature. The ratios are normalized to those at RT. The GeSe2 glasses crystallize into the phase [(a) and (d)], the phase [(b) and (e)], and the phase [(c) and (f)]. Lines guide to the eye. The error

bars are within the symbols. Two measurement results for each case of crystallization are picked up from over 50 measurements (samples).

crystallizing and found that the phase (low temperature phase) is tending to transform to the phase (high temperature one). Figure 10 depicts the temperature dependence of the integrated intensity ratios of S(AG)/S(A1) and normalized to those at RT. The GeSe2 glass crystallizes into the phase (2D) when the intensity ratios decrease slightly with temperature, and they start to increase above Tg. The increases that relates to the glass transition will be discussed later. The phase (3D) is obtained when the decreases of the ratios are larger than the case of the phase occurring. The Raman spectra of the thermally formed and phases are in agreement with those of previous works [9, 22]. An unreported Raman spectrum of a crystalline phase [Fig. 9(c)] appears when the ratios show the largest decreases among the three types of the crystallization processes. This Raman spectrum is similar to that of the except the low-frequency vibrational modes (see, Fig. 11). We assign this crystalline phase to the phase which was reported to be appearing when Se-rich GexSe1–x glasses (0.15<x<0.32) were well annealed [23, 24]. According to the similarity of the position of Raman band relevant to the vibrations of CST in crystals around 200 cm – 1 , we conclude that the phase is composed of CST units. The details of the phase, such as the basic structural units, are future work. We confirmed that the spectrum of Fig. 9(c) was also observed in the cases of Ge28Se72 and Ge30Se70 glasses being well annealed. The two Se-rich glasses always crystallize into the phase in contrast to the trifurcated crystallization behavior in GeSe2 glasses. The crystallization to the phase always follows a strong growth of a Raman band around 260 cm – 1 (in Fig. 9(c) at 300 °C for example). This strong Raman band has been assigned to the stretching mode of Se–Se bonds [12, 13]. A weak growth of the same Raman band can be distinguished during the process to the a crystallization. The strong growth of the Se–Se mode is in accordance with the fact

95

Figure 11. Expanded view of the Raman spectra of thermally formed crystals shown in Fig. 9(a–c). The spectrum of the is similar to the around 200 cm – 1 but is different in the low frequency –1 range; for example, the Raman peaks at 25 cm and 60 cm – 1 of the are very weak in the spectrum of the and the peak at 90 c m – 1 of the is hardly observed in the spectrum of the A slight admixture of the different phases might happen.

that the

phase is formed by annealing Se-rich glasses as well.

A mixture of the and phases appears after the changes of the intensity ratios whose degree is between those towards the pure and -phases crystallization. Similarly, a mixture of and phases is also observed. Thus, the crystallized phase is strictly determined by the degree of the decreases of the integrated intensity ratios, and S(A G )/S(A 1 ). The close correlation between the changes of the integrated intensity ratios and the crystallized phase suggests the creation and annihilation of structural units besides the deformation

of the units, which also decrease the Raman intensity due to breaking symmetry of the structural units. Figure 12 shows temperature dependence of the full width at half maximum (FWHM) of A1 and bands. The FWHM for the AC band should be more valuable (interesting) to the following discussion if we can measure the extremely weak AG band at high temperature in a good quality. The width of the A1 band first increases with temperature, and then decreases while that of the band increases monotonically. The lower temperature at which the width of A1 band starts to decrease, the more the integrated intensity ratios decrease. Because the degree of the decreases of the S(A G )/S(A 1 ) is much larger than the in all the samples, the breaking of Ge–Ge wrong bonds should play a dominant role on the structural changes. The Ge–Ge wrong bonds are expected to be transformed by heating to energetically more favored Ge–Se bonds. By this transformation, the CST units are formed from ethanelike Ge-Ge units. We believe that a growth of the CST units at the expense of the Ge–Ge bonds promotes an ordering of the CST units to reduce the FWHM of the A1 band. The is consistent with the crystallization of the and phases after the larger decrease of the S(A G )/S(A 1 ) than the phase. The and phases, which are composed of only CST units, should be formed though the higher degree of CST formation.

96

Figure 12. Temperature dependence of the full width at half maximum (FWHM) of the A1 band (a), and the band (b) in the GeSe2 glasses which are crystallizing towards the phase the phase ( ), and the phase by further heating. Lines are added to guide the eye. The FWHM of the A1 band increases with temperature and then starts to decrease while that of the band increases monotonically.

•

Medium-range Order of GeSe2 Glass The phase is realized through the least structural changes among the three types of the crystallization processes. The fact stresses that fragments topologically similar to the crystalline phase are significantly involved in the network of GeSe2 glasses. These fragments are larger than the Ge–Ge and CST units, and they are randomly oriented. When GeSe2 glass is heated over the Tg, a lower viscosity yields the local structure to be rearranged more easily. In Fig. 9(a), the increases of the intensity ratios above Tg is assumed as the trend toward the nucleation for crystallization of Thus, the structure of GeSe2 glass is similar to the phase, hence, the large degree of formation of CST units is needed to crystallize into the or

phases. Boolchand et al. [25] have recently suggested that, in the Ge-rich Ge x Se 1–x glasses the presence of Ge–Ge signatures decreases the global connectivity, and it constitutes part of an ethane-like units bearing nanophase formed separately from the GeSe4/2

tetrahedra bearing backbone of the glass. Together with our results, it follows that the ethanlike units involving Ge–Ge bonds and like fragments coexist in the melt-quenched GeSe2 glass. We can observe the trifurcated behavior in GeSe2 and Ge35Se65 glasses, but, we can never observe it in Ge28Se72 and Ge30Se70 glasses. Instead, the Se-rich glasses always crystallize into the phase. Therefore, the Ge–Ge nanophase seems to be the physical origin of the trifurcated crystallization. Generally, disordered materials are more or less inhomogeneous or heterogeneous. For instance, the importance of the inhomogeneity to understand the dynamics at the glass transition has been indicated experimentally [26] and theoretically [27]. Here, we assume an inhomogeneous distribution of the Ge–Ge nanophases. The size of the Ge–Ge phase and the surrounding situation, such as the sorts of surrounding units and the surrounding stresses, are nanoscopically inhomogeneous. Since the laser spot size is 0.1x4 mm2, no evidence of

inhomogeneity is able to be detected among the samples at RT. With increasing temperature, 97

the ethane-like Ge–Ge units are selectively transformed to the CST units. A large size of Ge–Ge nanophase transforms to a large one of the CST region, and thereafter the CST region might grow at higher temperature. However, if the size of Ge–Ge nanophase is smaller than a critical volume, it is not transforming to the CST region. Because larger CST units is more stable due to the gain in volume free energy of a phase composed of CST units at high temperatures. Thus, the degree of formation of CST units spatially differs. When the gain in volume free energy of the CST phase can compensate a larger CST region for the cost in interfacial free energy, the CST formation will produce a nucleus to form the or phases.

The resultant crystalline phase, or seems to be determined by the degree of formation of the Se-chain clusters. Note that a stronger increase of the Se-chain vibrational mode lead the crystallization to the phase [Fig. 9(c)]. The difference in the degree of formation of the Se-chain clusters is also due to the inhomogeneity in an initial distribution of Se–Se bonds and the surrounding situation. On the other hand, if the degree of the CST formation is low and the size is smaller than the critical size above Tg, the like fragments will grow to form the phase. Thus, the appearance of three crystalline phases depends on the degree of the formation of structural units of CST and Se-chain. The nature of the trifurcated crystallization is attributed to the inhomogeneous network that consists of nanoscopically phase-separated structural units.

Figure 13. Integrated intensity ratio of the A G band to the A1 band (a, b) and the ratio of the band to the A1 band (c, d) as a function of temperature. The ratios are normalized to those at RT. The as-quenched sample is heated to and kept at 300 °C for five hours, and thereafter, it is cooled to RT. (a, c) show the first cycle of the heat treatment and (b, d) the second one. Lines are added to guide the eye. The error bars are less than the size of the symbols.

The reason why our samples show such a clear trifurcated behavior is that our meltquench temperature of 960 °C is 220 °C higher than the liquidus [18, 23]. The percentage of the Ge–Ge units in GeSe2 glasses increases with quench temperature [28], and the high temperature quenching will increase the inhomogeneity in the distribution of Ge–Ge units. We should note that the existence of the Ge–Ge nanophase and its inhomogeneity is intrinsic, because the annealed samples also show the trifurcated behavior. A higher temperature quenching process just enhances such properties. 98

Reversibility and stability

The reversibility and stability of the GeSe2 glasses are investigated by heating and cooling free-standing bulk glasses as follow. As-quenched GeSe2 sample is heated to and kept at 300 °C (below the Tg) for five hours, and thereafter, it was cooled to RT. Figure 13 shows the normalized integrated intensity ratios, and S(AG)/S(A1), in the first [Figs. 13(a,c)] and the second [Figs. 13(b,d)] heating cycles. The changes of the intensity ratios with temperature are nearly reversible in the first cycle, and they are fully reversible in the second one. The slightly irreversible parts in the first cycle are due to usual annealing effects. Changes of FWHM’s are also reversible. The reversibility confirms that the thermally transformed structures are stable at each fixed temperature during the experiment time scale (15 minute). The stable structure depends on temperature, and it is realized within a minute. At higher temperature, the structure similar to the phase or phase, which consists of CST units, is more stable than that at lower temperature. The recovery of the S(AG)/S(A1) ratio during the cooling process demonstrates that the network at lower temperature prefers to involve a larger percentage of the Ge–Ge wrong bond. In other words, the Ge–Ge homopolar bond in GeSe2 glass may reduce the total free energy of the network at RT, thus, it could not be wrong. It should be stressed that the temperature dependence of the S(AG)/S(A1) ratio is qualitatively the same as that of the ratio. It could be understood in the restricted sense of the local structure that rising the percentage of the Ge–Ge wrong bond are accompanied by a larger population of EST units. The layered fragments that consist of both EST and CST units must be constructed in GeSe2 glass with the help of the Ge–Ge wrong bond. Thus, the Ge–Ge wrong bond plays an indispensable role on the stability of the glassy structure. The reversible structural change from the as-quenched structure to a quasi-three dimensional structure composed of CST units can be caused by pressure [29]. The comparison of

the reversible changes (with temperature and pressure) between the amorphous-amorphous phase transition observed in SiO2 [30, 31], is interesting and it will be discussed elsewhere. ACKNOWLEDGMENTS It is a pleasure to acknowledge J.C. Phillips and P. Boolchand for helpful discussions and suggestions. This paper draws the recent results developed in our laboratory of collaboration and discussions with K. Inoue, O. Matsuda, M. Nakamura, and M.K. Nakamura. This work was supported by Grants-in-Aid for Scientific Research (B)(No. 09440117), Encouragement of Young Scientists (No. 11740173), and a grant for Scientific Research in the Priority Area “Cooperative Phenomena in Complex Liquids”, from the Ministry of Education, Science and Culture (Japan). Y.W. acknowledges support from the Inamori Foundation.

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Se glass-forming liquids in relation to rigidity percolation, and the Kauzmann paradox, Phys. Rev. Lett. 64, 1549–1552.

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Bridenbaugh, P.M., Espinosa, G.P., Griffiths, J.E., and Phillips, J.C. (1979) Microscopic origin of the

companion A 1 Raman line in glassy Ge(S,Se)2, Phys. Rev. B20, 4140–4144. 20. Jackson, K., Briley, A., Grossman, S., Porezag, D.V., and Pederson, M.R. (1999) Raman-active modes of a-GeSe2 and a-GeS2: A first-principles study, Phys. Rev. B60, R14985–R14989. 21. Sakai, K, Yoshino, K., Fukuyama, A., Yokoyama, H., Ikari, T., and Maeda., K, (2000) Crystallization of

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22. Nakaoka, T., Wang, Y., Murase, K., Matsuda, O., and Inoue, K. (2000) Resonant Raman scattering in crystalline GeSe2, Phys. Rev. B61, 15569-15572. 23. Azoulay, R., Thibierge, H., and Brenac. A. (1975) Devitrification characteristics of GexSe1–x, glasses, J. 24.

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4633.

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SOLIDITY OF VISCOUS LIQUIDS

J.C. DYRE Department of Mathematics and Physics (IMFUFA), Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark

INTRODUCTION The glass transition takes place when a liquid upon cooling becomes more and more viscous and finally solidifies to form a glassy solid [1-7]. Most liquids that are able to form glasses are supercooled and thus not in genuine thermal equilibrium, but this fact is probably not important for understanding the dramatic increase in viscosity preceding glass formation. Approaching the glass transition the viscosity becomes so large that molecular motion is arrested on the time-scale of the experiment. The fascination of this phenomenon from a theorist point of view lies in the fact that chemically quite different liquids – involving ionic interactions, covalent bonds, van der Waals forces, hydrogen bonds, or even metallic bonds – have a number of properties in common when they are in the very viscous regime, close to the glass transition temperature Tg. These general properties can be summarized into three “non’s”: • Non-Arrhenius temperature-dependence of the average relaxation time. • Non-Debye linear response. • Non-linearity of structural relaxation even with respect to quite small temperature changes (i.e., 1% of Tg). The last “non” is not very interesting, because non-linearity is unavoidable whenever the relaxation time is strongly temperature dependent. This is true independent of whether the relaxation time is Arrhenius or not, and independent of whether there is just one relaxation time or a whole relaxation time spectrum. Our focus here is on the two first “non”s. Before proceeding, let us briefly remind the reader of them. Almost all viscous liquids have viscosities and average relaxation times that are non-Arrhenius. This is perhaps not very puzzling, given the fact that it is likely a priori that not just one single energy barrier is involved in viscous liquid dynamics. What is puzzling, however, is the “sign” of the non-Arrhenius behavior: Whenever a range of activation energies is involved any naive model predicts an apparent activation energy which

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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decreases as temperature decreases; this is because at low temperatures the system prefers the “reaction channels” with lowest energy barriers. However, as is clear from Fig. 1, precisely the opposite behavior is observed:

Figure 1. Angell’s famous plot of the viscosity of a number of viscous liquids [8], giving the logarithm of the viscosity as a function of Tg/T. If the viscosity is Arrhenius this plot gives a straight line (diagonal). Almost all viscous liquids have apparent activation energies (= slope in the plot) which increase as temperature is lowered.

The non-Arrhenius behavior is one of the great mysteries of this research field. Another great mystery is the origin of the non-Debye linear response functions. Typical linear response quantities are dielectric relaxation, frequency-dependent shear or bulk moduli, or frequency-dependent specific heat. The Debye linear response is given by

Via the fluctuation-dissipation theorem this corresponds to a time autocorrelation function that is a simple exponential. Except for certain mono-alcohols Debye relaxation is never observed. Instead, one always observes loss peaks (imaginary parts of response functions) that are asymmetrically deformed towards the high frequency side (Fig. 2): Below the loss peak frequency the response function follows Eq. (1), above the loss peak frequency the loss typically decays as where is often close to 0.5 [see, e.g., Ref. 9 and its references]. Despite many years of research and thousands of measurements there is no consensus on the origin of the non-Debye behavior. The puzzle is not so much why Debye behavior is rarely observed, but rather why

not much broader response functions are observed [10]. If one again naively thinks in terms of energy barriers, the “correct” barrier distribution has a very sharp high energy cut-off and is an exponential towards low energies with a width which is only 5-10% of the maximum activation energy, the activation energy of the loss peak frequency itself. In this paper we shall discuss the physics of viscous liquids from a (perhaps) simplistic point of view. We shall argue that these liquids are more like solids than like ordinary liquids 102

Figure 2. Typical loss peaks for a viscous liquid. The figure shows the dielectric loss of triphenyl phosphite at 206.0 K, 208.0 K, and 210.0 K [10]. Below the loss peak frequency the behavior is as predicted by the Debye

expression Eq. (1)

but above the peak the loss follows an

-decay. At very high frequencies the

so-called relaxation is visible.

and that this “solidity” explains both the non-Arrhenius and the non-Debye behaviors. First, we consider how a viscous liquid flows at all, by asking: Is a viscous liquid like an ordinary liquid (e.g., water) or is it qualitatively different? ARE VISCOUS LIQUIDS “JUST” LIQUIDS THAT ARE VISCOUS?

Any medium described by ordinary hydrodynamics in the long wave-length limit has a shear viscosity, As shown by Maxwell, if denotes the instantaneous shear modulus the shear relaxation time is related to the viscosity by

According to this equation ordinary liquids like water have relaxation times in the picosecond range. In contrast, a liquid approaching the glass transition has a viscosity typically 1015 times larger than that of water and thereby a relaxation time of 1000 seconds or longer. This

enormous difference justifies asking whether there are qualitative differences between the two cases. The decoupling of relaxation times from phonon times as viscosity increases is reflected in a dramatic decoupling of diffusion constants: For “ordinary” liquids the molecular diffusion constant D is of the same order of magnitude as the transverse momentum diffusion constant, the dynamic viscosity of the Navier-Stokes equation is mass density]. With increasing viscosity D decreases – as according to a simple Stokes-Einstein type argument – while increases. At the glass transition is about 1030 times larger than D. The physical interpretation of the very long relaxation times of viscous liquids is simple: If s it takes roughly 100 s for a molecule to move one intermolecular distance. This interpretation is somewhat modified by the existence of dynamic heterogeneities and of the

103

relaxation, but basically there is no reason to question this picture. How can molecular movement be so slow? Again the answer is simple: Average velocities, of course, are determined by temperature, so molecular motion can only be slow because it is ineffective. It is ineffective because almost all motion is vibrational. The vibrations take place around a potential energy minimum in configuration space. For the vibrations to persist for billions and billions of times before anything else happens to the molecules, there must be rather large barriers for jumping into a neighboring minimum. The jumps are referred to as “flow events.” It is the existence of these rare flow events which eventually allows a viscous liquid to flow, but clearly the liquid looks like a solid on shorter time scales. It is interesting to note that this picture is almost as old as the research field itself. Thus Kauzmann referred to flow events as “jumps of molecular units of flow between different positions of equilibrium in the liquid’s quasicrystalline lattice” [1]. He thereby emphasized the fact that a viscous liquid most of the time is in a state of mechanical equilibrium. The flow event picture [11], which has never really been challenged, was recently confirmed by extensive computer simulations of a highly viscous Lennard-Jones mixture [12]. The “solid-like-ness” of viscous liquids – the fact that these liquids are virtually always in a state of mechanical equilibrium – we shall refer to as their “solidity” [13,14]. Solidity expresses the simple fact that on a short time scale there is only vibration. For instance, simulating a liquid with an average relaxation time around 1 s on the fastest computer available today would be a real disappointment: Only vibrations would be visible and one cannot possibly tell the difference between the viscous liquid and a solid. When a flow event finally

does take place, after a short time there is again mechanical equilibrium in the surroundings and the molecules vibrate over and over, now just around a slightly different potential energy minimum [15]. The effects of one flow event in the surroundings is discussed in detail in Refs. 13 and 14. There is a displacement field varying as where r is the distance to the flow event. This result is found by a straightforward application of solid elasticity theory. It is important to note that the solidity of viscous liquids has only a finite range [13]. Thus there is a “solidity length” l beyond which flow events effectively do not induce molecular motions at all. To estimate l we note that elastic displacements propagate with the velocity of sound, c. Consider a sphere with radius R. Within this sphere, if r0 is the size of one rearranging region, there are roughly possible locations for flow events. A molecule at the center of the sphere only feels the full effects from a flow event within the sphere if the following condition is obeyed: The displacement deriving from such a flow event must propagate throughout the sphere and elastic equilibrium be reestablished before another flow event takes place. Since is the average time between two flow events involving the same molecules, the time between two flow events within the sphere is This time must be longer than or equal to R/c. To estimate l we use equality for R = l and note that c is the sound velocity of the glass, cglass. This leads to the following expression for the solidity length l

The solidity length diverges slowly as

When

l

is several thousand Angstroms.

“SHOVING” MODEL OF NON-ARRHENIUS BEHAVIOR

According to the solidity picture, molecules in a viscous liquid have to overcome a relatively large energy barrier in order to move. The non-Arrhenius behavior means that the barrier is temperature dependent. For simplicity, we shall not distinguish between energy

barriers and free energy barriers. If the barrier is denoted by and prefactor (of order 1 picosecond), the average relaxation time is given by 104

is a typical

Experiments (Fig. 1) imply that to understand why.

increases as temperature decreases. The challenge is

Elsewhere we have briefly reviewed the two standard models for non-Arrhenius behavior, the entropy model and the free volume model, and critiques against these models [7]. The alternative “shoving” model is based on the following picture, which involves postulates similar to those of the free volume model: Molecules in a liquid are closely packed, so in order to be able to rearrange – for a flow event to take place – extra space is needed. This is the first postulate. A further input to the model is the well-known fact that intermolecular interactions are strongly anharmonic with harsh repulsions but only weak attractions. It is these harsh repulsions which makes it unfavorable (i.e., too energy costly) to rearrange at constant volume. To lower the barrier the molecules therefore shove aside the surrounding molecules. Let us quantify these ideas. Suppose the rearranging molecules constitute a sphere which at the transition state has increased its radius by The surrounding liquid behaves entirely like a solid during the flow event. Therefore, the energy needed for expanding the sphere is quadratic in The energy barrier to be overcome inside the sphere is given by some function so the total barrier is for some constant A given by Minimizing this leads to Thus, the ratio between the “shoving” work and the “inner” barrier, is given by

Because of the harsh intermolecular repulsions the logarithmic derivative of f must be numerically much larger than one (reflecting the fact that a lot is gained by not rearranging at

constant volume). Consequently, 1 and the “inner” contribution to the activation energy is small. The “shoving” model is characterized by ignoring this contribution. The next step is to determine the “shoving” work. What happens when a sphere in a solid is enlarged? Many people answer that the surroundings must be compressed in the process, but this is wrong. The deformation of the surroundings is a pure shear deformation. In fact, this is a standard exercise in elasticity theory, where one finds [16] that the radial displacement field varies as We know from Coulomb’s law that this field has zero divergence. Since the divergence of the displacement field is the relative volume change, there is no compression. This means that the relevant elastic constant for the “shoving” work is the shear modulus. Now, the shear modulus of any liquid is zero at zero frequency, but on short time scales the liquid behaves like a solid with shear modulus This quantity is crucial to the “shoving” model, because the activation energy is proportional to It is well-known that in viscous liquids increases strongly as temperature is lowered, and this is now our explanation for the non-Arrhenius behavior [7,17]. The final expression for is:

Here, Vc is a proportionality constant with dimension volume. In comparing to experiment Vc is always of order the volume of one molecule. The mathematical expression Eq. (6) may be traced back to 1943 (see, e.g., Refs. 5, 7, and 17).

As a model for the non-Arrhenius behavior the “shoving” model is not fundamental; it does not allow a calculation of the energy barrier because Vc is unknown. Instead, the model is phenomenological in the sense that it links the energy barrier to another macroscopic

physical quantity. In this respect the model is like the entropy model. The “shoving” model,

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however, has the unique feature of linking short time dynamics with long time dynamics:

reflects short time dynamics and may be measured by a fast experiment. Note also that may be calculated as a simple statistical mechanical canonical average (of the square of the fluctuating shear stress), so if the model is correct a considerable simplification has been achieved. Before comparing the “shoving” model to experiment, we briefly summarize its basic postulates:

• The main contribution to the activation energy is elastic energy. • This elastic energy is located in the surroundings of the reorienting molecules. • The elastic energy is shear energy. It is easy to compare the model to experiment. One simply measures [or equivalently, and to check Eq. (6). It is convenient to use an Angell type plot for this. Instead of having Tg/T as x-axis, however, we use normalized to one at T = Tg. Figure 3 gives two examples. The data confirm the model. More measurements are needed, however, before the model can be said to be well-established. Since the “shoving” model predicts that the rate of relaxation depends on the instantaneous shear modulus, the model should apply even for nonlinear relaxations. This has recently been tested in aging experiments on a silicone oil [19]. The liquid was subjected to

sudden temperature jumps from well annealed states, and was continuously monitored as a probe of the structural relaxation. By using the Tool-Narayanaswami formalism with a reduced time definition based on Eq. (6) it was shown that the model is able to fully account for the aging observed. A DIFFUSION-TYPE MODEL OF NON-DEBYE RELAXATIONS

We now proceed to discuss relaxation phenomena in viscous liquids. Only linear relaxations, i.e., those governing how infinitesimal perturbations decay to equilibrium, are considered. Moreover, the discussion focusses on the so-called a relaxation, the slowest and the dominant relaxation process. The relaxation process defines the average relaxation time which is linked to viscosity by Eq. (2). How does one construct a model exhibiting the kind of asymmetric relaxation seen in Fig. 2? We look for a model based on the solidity of viscous liquids, predicting a sharp cut-off at long relaxation time and an high frequency decay of the loss. Two points of views may be taken towards linear responses of viscous liquids: 1) There are a number of different response functions, all of which are more or less of equal status, probing different autocorrelation functions. Alternatively, 2) some response functions are regarded as more fundamental than others. We adopt the latter viewpoint: The condition of mechanical equilibrium is a zero-divergence condition referring to the stress tensor. This puts stress tensor fluctuations in focus. Consequently, we shall regard the frequency-dependent shear and bulk moduli as the two basic linear response function (the bulk modulus exists in two versions, the adiabatic and the isothermal – here only the isothermal bulk modulus is considered). Other response functions should somehow be derived from the mechanical ones. For example, a rotating sphere model for dielectric relaxation links the dielectric response to the frequency-dependent viscosity (or shear modulus). Consider the shear modulus. The -decay of the shear loss at high frequencies, which is to be reproduced by our model, means that at high frequencies one has (in dimensionless units): Since where is the shear stress, we find that at short times the stress autocorrelation function is given (again in dimensionless units) by 106

Figure 3. Comparing “shoving” model predictions to experiments on molecular liquids (reproduced from Ref. 17). (a) shows our own data, while (b) replots old data of Barlow and coworkers [18]. For both data sets the actual temperature dependence of the viscosity is plotted just as in Fig. 1. Also, the viscosity is plotted as a function of the variable normalized to one at T = Tg. The diagonal line gives the predictions of the “shoving” model, starting at high temperature at a physically reasonable prefactor (lower left corner).

The model we adopt is a diffusion model. The idea is that the stress averaged over some small volume of the liquid (containing, e.g., 100 molecules) – although it changes abruptly whenever a flow event takes place – changes by only a small amount. If so, it might be a good

guess to try to describe the process by a diffusion equation in stress space. This space has 6 coordinates, the 6 independent values of the stress tensor. The diffusion constant must be thermally activated if relaxation is to become strongly temperature dependent. We know that when a large shear stress is applied to a set of molecules

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they are much more likely to rearrange than otherwise (this is the mechanism behind fracture in its initial stage). Thus a large shear stress lowers the barrier, so the barrier must depend on the stress state. This means that in stress space the diffusion constant depends exponentially on the stress state. Denoting the 6 stress coordinates collectively by our first guess at a diffusion equation for the probability is

Here the diffusion constant is given by where is the stressdependent activation energy. All stress states are regarded as equally likely, but the allowed stress states define only a finite region of stress space. This region is bounded by the states with Diffusion out of this region is forbidden; in the model this is ensured by a suitable boundary condition. This model cannot possibly reproduce Eq. (7), however. For any activation energy that varies down to zero at large stresses, the model gives precisely the kind of very broad loss peaks that are naively expected (and not found in experiments) when a whole range of activation energies are involved. However, we have not yet taken solidity into account. How does this change things? Because of the mechanical equilibrium condition – the solidity – stress changes at one place cannot take place without minor stress changes throughout the viscous liquid. Both before and after a flow event there is mechanical equilibrium, so the stress change must also satisfy the zero-divergence condition. As shown in Refs. 13 and 14 this implies that the stress change in the surroundings varies as and is thus long ranged. Locally, stress may change either as a result of a local flow event or as a result of flow events in the surroundings. The latter changes are small, but quite common, and they are independent of the actual stress state. Thus, we find that the stress diffusion constant has two contributions and is given by

When a large range of activation energies is involved, depending on the actual value of this expression is dominated by either the first term or the last term. In stress space there is a rather sharp edge separating these two cases. On one side of this edge (the “slow” part

of stress space) is almost stress independent while on the other side of the edge varies rapidly and is generally much larger than D (the “fast” part of stress space). The linear size of the slow part of stress space, L, defines a characteristic time by As becomes clear below, is the average relaxation time of the relaxation. How does the model with Eq. (9) reproduce Eq. (7) and the asymmetry of the loss peak? First, we note that there is a cut-off at long relaxation times: Relaxation towards equilibrium can at most last the time it takes to diffuse across the slow part of stress space, The square root time dependence of the initial stress decay comes about in the following way: Stress states close to the edge, but in the slow part of stress space, preferably relax by moving to the edge and crossing it to utilize the fact that motion is fast on the other side. The limiting factor is the time it takes to move to the edge. For systems a distance d away from the edge this time is given by t = d2/D. When d is small the number of states a distance less than d away from the edge is proportional to d. Thus the number of states relaxing within time t varies as This gives us Eq. (7). It is interesting to note that for the model to work properly, there must be relaxations much faster than those of the a relaxation. We tentatively identify these fast relaxations with the Goldstein-Johari relaxation.

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SUMMARY

The solidity of viscous liquids is the key to understanding the physics of these liquids. Solidity is a direct consequence of the extremely high viscosity. Thus the similarity between chemically quite different liquids becomes less surprising. If this viewpoint is correct, understanding the physics of viscous liquids may be simpler than has hitherto been recognized. REFERENCES 1.

2. 3. 4.

5. 6. 7.

8. 9. 10.

11. 12. 13. 14. 15.

16. 17. 18. 19.

Kauzmann, W. (1948) The nature of the glassy state and the behavior of liquids at low temperatures, Chem. Rev. 43, 219-256. Harrison, G. (1976) The dynamic properties of supercooled liquids, Academic Press, New York. Brawer, S. (1985) Relaxation in viscous liquids and glasses, American Ceramic Society, Columbus, OH. Angell, C.A. (1991) Relaxation in liquids, polymers and plastic crystals - strong/fragile patterns and problems, J. Non-Cryst. Solids 131-133, 13-31. Nemilov, S.V. (1995) Thermodynamic and kinetic aspects of the vitreous state, CRC, Boca Raton, FL. Debenedetti, P.O. (1996) Metastable liquids: Concepts and Principles, Princeton University Press, Princeton. Dyre, J.C. (1998) Source of non-Arrhenius average relaxation time in glass-forming liquids, J. Non-Cryst. Solids 235-257, 142-149. Angell, C.A. (1985) Strong and fragile liquids, in K.L. Ngai and G. B. Wright (eds.) Relaxations in complex systems, U.S. G.P.O., Washington, DC, pp. 3-11. Olsen, N.B., Christensen, T., and Dyre, J.C. (2000) Time-temperature superposition in viscous liquids, e-print cond-mat/0006165. Voit, P., Tarjus, G., and Kivelson, D. (2000) A heterogeneous picture of relaxation for fragile supercooled liquids, J. Chem. Phys. 112, 10368-10378. Dyre, J.C. (1987) Master-equation approach to the glass transition, Phys. Rev. Lett. 58, 792-795. Schrøder, T.B., Sastry, S., Dyre, J.C., and Glotzer, S.C. (2000) Crossover to potential energy landscape dominated dynamics in a model glass-forming liquid, J. Chem. Phys. 112, 9834-9840. Dyre, J.C. (1999) Solidity of viscous liquids, Phys. Rev. E 59, 2458-2459. Dyre, J.C. (1999) Solidity of viscous liquids. II. Anisotropic flow events, Phys. Rev. E 59, 7243-7245. Goldstein, M. (1969) Viscous liquids and the glass transition: A potential energy barrier picture, J. Chem. Phys. 51, 3728-3739. Landau, L.D., and Lifshitz, E.M. (1970) Theory of elasticity, Pergamon Press, Oxford. Dyre, J.C., Olsen, N.B., and Christensen, T. (1996) Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids, Phys. Rev. B 53, 2171-2174. Barlow, A.J., Lamb, J., Matheson, A.J., Padmini, P.R.K.L., and Richter, J. (1967) Viscoelastic relaxation of supercooled liquids. I Proc. Roy. Soc. A 298, 467- 480. Olsen, N.B., Dyre, J.C., and Christensen, T. (1998) Structural relaxation monitored by instantaneous shear modulus, Phys. Rev. Lett. 81, 1031-1033.

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NON-ERGODIC DYNAMICS IN SUPERCOOLED LIQUIDS

M. Dzugutov, S. Simdyankin and F. Zetterling Department of Numerical Analysis Royal Institute of Technology 100 44 Stockholm, SWEDEN

INTRODUCTION The nature of the glassy state and the liquid-glass transition admittedly remains the most interesting unsolved problem of the solid state theory [1,2]. The interest is focused on two universal features observed in the liquids which remain in a (possibly metastable)

equilibrium when approaching the glass transition. In fragile glass-formers [3], a superArrhenius increase of the relaxation time is observed below a characteristic temperature TA, leading to an apparent solidification at a finite temperature Tg. Simultaneously, these liquids exhibit another feature, related to the first one, the stretched-exponential decay of the time correlation functions. Laboratory glass transition is defined as a point where the relaxation time of a supercooled liquid exceeds the experimentally accessible observation time. A liquid cooled below the glass transition temperature thus remains, on the experimental time-scale, in a nonequilibrium state, being unable to comprehensively explore the relevant area of its configurational space. Viewed from this point, Tg is commonly regarded as a crossover temperature separating the ergodic and non-ergodic regimes of the supercooled liquid dynamics. It is clear that this definition is not helpful for understanding the possible transformation in the behaviour of a liquid undergoing the glass transition. Indeed, the observation time limit affordable for an experimentalist involved in the study of the glass transition is not a relevant quantity for the description of relaxation processes in a supercooled liquid, and, therefore, both the glass transition and the ergodicity breaking as defined in the above way may be regarded as just artifacts of observation. In order to address this issue, we need an operational definition of ergodicity breaking that would indicate a criterion for discerning a possible change in the phase-space behaviour of a liquid in the the glass transition domain from the macroscopic observations. Moreover, this criterion should be formulated in terms of equilibrium liquid parameters. The ergodic aspects of the glass-transition can be considered in the context of a longstanding problem known as the Kauzmann paradox [4]. The latter is an observation that the extrapolation of experimental results for the specific heat in supercooled liquids leads

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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to zero value of the configurational entropy at finite temperature. This can be interpreted as indicating an underlying phase transition [5]. The existence, the nature, and the possible

location of such a transition are key issues for the problem of glass transition. Two possible scenarios can be considered. The mode-coupling theory [6] interprets the characteristic dynamics observed in the supercooled domain, and the eventual glass transition, as a purely dynamical equilibrium phenomenon arising as a result of an inherent feedback

mechanism - the so-called cage effect whereby the diffusive dynamics is coupled with the structural relaxation. Liquid dynamics remains in equilibrium as it slows down under cooling to complete structural arrest which in the idealized version of the theory occurs at a critical temperature Tc. In this context, the condition of ergodic equilibrium implies that the local structural fluctuations relax sufficiently fast so that the liquid remains spatially homogeneous on the time scale relevant for the diffusion. The theory thus circumvent the problem of Kauzmann paradox. A number of experimental observations, however, indicate a different scenario. It was found that, in the vicinity of Tg, structural relaxations in a number of glass-formers become significantly retarded as compared with the diffusive dynamics, breaking the Stokes-Einstein relation. This observation was interpreted as indicating that the liquid decomposes into subsystems which relax at different rate. Moreover, the slow dynamical states were concluded to be confined to structurally distinct spatial domains with the life-time much exceeding the time-scale of the diffusive dynamics. Formation of these domains, conceivably confining an energy favoured structure, may be regarded as an indication of an underlying phase transition which is possibly masked by the onset of the laboratory glass transition. On the other hand, the conjecture of a long-lived spatial heterogeneity in supercooled liquids implies that, on the time scale characterizing the relaxation dynamics of the faster subsystems, the liquid falls out of ergodic equilibrium already above Tg. In terms of the phase-space behaviour, this can be viewed as decomposition of the entire phase-space region corresponding to the thermodynamic equilibrium (the region of motion) into distinct components which are connected only on a time scale long as compared with the characteristic equilibration time within a component. To a considerable degree, the controversy that arises in discussing the ergodic aspects of the glass transition stems from the fact that there is currently no uniformly accepted definition of ergodicity. It is clear that a meaningful discussion of ergodicity breaking must refer to some qualitative change in the phase-space behaviour. In the following, we shall discuss a conjecture relating the characteristic anomalies arising in a supercooled liquid to the anomalies in its phase-space behaviour as well as the possible ways of detecting these latter anomalies from finite-time measurements of macroscopically accessible quantities. AN OPERATIONAL DEFINITION OF DYNAMICAL ERGODICITY

A statistical-mechanical system is regarded as ergodic if the time-average of a dynamical variable converges to its ensemble average as defined with respect of an invariant measure

The invariant measure here specifies the region of motion - the subspace to which the system is confined by the macroscopic constraints. A major problem involved in the above definition of ergodicity is that it refers to an

ensemble of identical systems and infinite observation time, whereas we, in fact, need to assess the ergodicity of a single system within finite time. Moreover, the definition (1) is usually interpreted in the spirit of the original Boltzmann’s ideas asserting that, to attain an

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ergodic equilibrium, the system’s phase trajectory has to cover with sufficient density the whole relevant phase-space region. This picture of the ergodicity restoring relaxation process is obviously wrong, as far as liquids are concerned. Indeed, the number of configurations the phase trajectory has to explore, and, consequently, the conjectured relaxation time, increase exponentially with the number of particles, whereas the relaxation times in real liquids are not size-dependent [12]. This problem can be resolved if we redefine the concept of statistical ensemble. We can do it using the fact that the entropy S of an equilibrium system of N particles is an extensive quantity: The extensivity implies that there exist a finite correlation length and a finite correlation time [12], such that regions separated by distances exceeding evolve independently and produce statistically indistinguishable time averages for time intervals exceeding An equilibrium liquid system can thus be viewed as an ensemble of independently evolving subsystems which are confined to regions separated in space by the distances exceeding In dealing with systems of particles, it appears reasonable to consider effective ergodicity based on the concept of mixing [12,13]. The latter requires that the measure of the points of a region R of the phase space which happened to be in any other region R´ after sufficiently long time t must be proportional to the volumes of these regions:

If a dynamical system satisfies this condition, its phase trajectory uniformly samples the coarse-grained phase space. Notice that although the property of mixing is sufficient to ensure ergodicity, it is not known whether this is also a necessary condition for ergodic behaviour [12,13]. The approach to effective ergodicity can be monitored by a measure based on the idea of statistical symmetry. The latter means that time-averages of the quantities associated with

independently evolving regions of a system (or its constituent particles) must become statistically indistinguishable when approaching ergodic equilibrium; this is an obvious result of the independence principle. If f i is a quantity associated with particle i, the respective measure

of ergodic convergence for a system of N identical particles is defined as [14]

where

is the ensemble average (average over all the particles of the system). In the domain of Arrhenius liquid dynamics above TA, this measure, defined for the particle energy e, was found [14] to decay with time universally as By contrast, in the glass-transition domain it asymptotically approaches a non-zero limit indicating ergodicity breaking. Thus, using the measure defined by Eq. 1 we can detect the divergence of the structural relaxation time-scale in a strongly supercooled liquid which is an indicator of the ergodicity breaking. Although such microscopic diagnostics can be useful in the analysis of computer simulations, its major drawback is that it requires full information on the phase trajectory which is not available in macroscopic experiments. The mixing behaviour is realized for the systems with the property that two phase trajectories, initially infinitely close, diverge exponentially with time [13,15]. For each phase space coordinate xi, this divergence is quantified by the respective Lyapunov exponent Notice that the connection between the exponential chaos and mixing, although intuitively clear, has been rigorously proved only for a few simple cases, like the gas of hard spheres [16]. Numerical simulations appear to be a natural way to extend these results to real systems. We discuss here a conjecture [17] that ergodicity of a dynamical system should be understood as global chaotic connectivity of its region of motion. If exponential chaos is confined to 113

subregions connected only on a sufficiently long time-scale, two trajectories, arbitrary close but belonging to different regions of chaotic behaviour, do not diverge exponentially. Obviously, such decomposition of a single chaotic domain results in (i) separation of the relaxation time-scales and (ii) slowing down the relaxation dynamics. In the following, we present arguments, supported by evidence from simulation, indicating that the discrepancy between the volume of a single stochasticity region and the total volume of the region of motion can be assessed by exploiting a recently found universal relation between the diffusion coefficient and the entropy [18]. ERGODIC DIFFUSION

The process of diffusion in liquids is controlled by the so-called cage effect whereby the diffusive motion of a particle surrounded by a cage of its neighbours is coupled with the relaxation of the local structural environment [19,20]. If a liquid is regarded as an ensemble

of independently relaxing regions as we pointed out above, the rate of diffusion is determined by the frequency at which these regions change their configurations. The phase-space picture of structural relaxation in the liquid state can be conveniently

discussed in terms of the energy landscape paradigm [2]. The topography of the energy landscape can be probed by mapping an instantaneous configuration of onto a local potential energy minimum by the steepest descent minimization [21]. The process of structural evolution of an independently relaxing region in a liquid can be viewed as a sequence of transitions between adjacent energy minima. Each point of the ensemble representing these regions explores adjacent minima positions by performing random trial moves at a rate which is determined by the general rate of the momentum and energy transfer. We assume that the probability that a random move in the configurational space results in a successful transition to a new minimum is proportional to the number of adjacent minima which, in its turn, is proportional to the total number of available minima. The latter scales with the thermodynamic excess entropy as The excess entropy represents the difference between full thermodynamic entropy and that of the perfect gas at the same conditions: Sex = S – SPG [22]. It has been recognized by Enskog [23] that the momentum and energy transfer in a dense hard-sphere fluid is mediated by binary collisions. The collision rate thus provides a natural time-scale for the dynamics. It can be assessed from the value of the radial distribution function, g(r), at the collision distance [23]:

where m is the particle mass, is the hard-sphere diameter and T is the temperature. At the same time, the hard-sphere diameter is a natural unit of length. If we express the diffusion coefficient in dimensionless form using these units, the diffusion coefficient in the hard-sphere liquid can be expresses, according to the conjecture suggested above, can be expressed as follows: with the empirically found value of the scaling factor A = 0.049. In real liquids, the hard-sphere diameter can be replaced by the position of the first peak of g(r), which allows the calculation of In this way, we can express D in different liquids in terms of universal units of time and length. Another approximation concerns calculation of Sex. The latter can be expressed as an expansion in terms of n-particle correlation functions [24]. In two-particle approximation, this gives:

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It was found [18] that, with the use of these approximations, eq. (6) universally describes the relation between S and D in a wide range of simple liquid systems. The scaling relationship between the thermodynamic entropy and diffusion rate is based on the assumption that the system explores its phase space at a rate proportional to The rate at which the state of an exponentially chaotic system is delocalized in the phase space can be quantified by the Kolmogorov-Sinai entropy [13]. This quantity, according to Pesin’s theorem [25], can be presented as the sum of all positive Lyapunov exponents:

Thus, the scaling relation between D and Sex implies the existence of a respective relation between hKS and Sex. Recently, it was indeed found [26] that the Kolmogorov-Sinai entropy in simple liquids, expressed in terms of is uniquely and universally related to the thermodynamic entropy. This implies that the collision frequency represents a universal time-scale for the liquid dynamics. Another conclusion, important in the context of the present discussion, is that the observed relation between D and Sex can be regarded as an indicator of exponentially chaotic behaviour of a liquid. Eq. (6) can be compared with other theories of liquid dynamics involving entropy. In the model of Adam and Gibbs [27] the central quantity is the minimum size of the cooperativity

rearranging region the configuration of which can be changed without interfering with the and the configurational

environment. The model conjectures that the the relaxation time entropy Sc are relates as:

Another model [28], suggested by Rosenfeld, relates the entropy and diffusion in liquids in a different way:

Recently, Di Marzio [29] proposed a relation between the relaxation time and the configurational free energy Fc: Notice that parameter A in Eq. (9) is the number of particles participating in a single jump. If that relation is interpreted in terms of hard spheres where with an additional assumption that A = 1 it becomes clearly consistent with (6). In that case, B would be expressed in terms of the collision frequency. An obvious advantage of relation (6) as compared with (8), (9) and (10) is that it does not involve free parameters. Therefore, a deviation from it is an unambiguous signatuture of a break down in the fundamental mechanism postulated for liquid diffusion which in the case of other relations may be masked by fitting.

NON-ERGODIC DIFFUSION

A central postulate in the model of liquid diffusion in ergodic domain presented in the previous section is that the probability of transition from a current energy minimum configuration to an adjacent one is entirely determined by the total number of these configurations

which can be expressed through the excess entropy. This postulate implicitly assumes that all adjacent configurations available are accessible within the same time-scale characterizing the local dynamics. The universal relation between the hKS and Sex also implies the validity of this assumption. We now consider a situation where this assumption is not valid, and analyze its possible consequences for the system’s behaviour in the phase space. A connection will be shown between this behaviour and the non-ergodic dynamics as it was defined in above, 115

Figure 1. A simple model demonstrating the impact of the valley structure of the phase-space on the relax-

ation dynamics. Energy barriers separating the valleys are depicted by solid lines. Squares and circles denote configurations belonging to different valleys. The crossover point between the valleys is marked by the cross.

and its possible diagnostics in terms of the macroscopically measurable parameters will be discussed. It was found that the average energy of the potential energy minima which remains constant above TA drops rapidly as the temperature decreases below TA [30]. This indicates that in the supercooled domain, the liquid resides on a different part of the energy landscape than the normal liquid, predominantly staying in deep valleys connected by narrow bottlenecks.

These connections are effectively used only on the long time-scale, while the short time-scale dynamics of a supercooled liquid unfolds in a limited subregion of the total region of motion. In order to illustrate the impact of such strongly profiled landscape on the liquid relaxation dynamics, we consider a simple model sketched in Fig. 1. The whole set of configurations comprising the region of motion is divided in two components (valleys), depicted by squares and circles. The filled symbols denote the configurations which are energetically forbidden. The components are separated by barriers, indicated by solid lines, and the single connecting pass is marked by the cross. Consider the probability wij of transition between the configurations i and j. At high temperature, where the components separation is not relevant, the average transition probability is entirely determined by the average probability that the destination configuration is allowed, and, in this way, by the entropy: Clearly, this leads to the relation (6) for the diffusion rate. In the case when the dynamics unfolds in the valley landscape, the average probability of transition can be estimated as where is the average probability that both i and j belong to the same valley. Obviously, in this case and, therefore the the diffusion rate is expected to show a negative deviation from the relation (6). This deviation, indicating that the liquid dynamics is not any more related to the static properties, can be used as a macroscopic diagnostics of the onset of the supercooled (non-ergodic) regime.

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Figure 2. Reduced diffusion coefficient in the binary mixture of hard spheres as a function of packing fraction (dots). Dashed line corresponds to the universal scaling law (6) relating the diffusion coefficient to the excess entropy, in the pair approximation, S2, with A = 0.049.

EVIDENCE FROM MOLECULAR DYNAMICS SIMULATION

In order to test the above conjecture, we investigated a two-component hard-sphere liquid simulated by molecular dynamics. The model consisted of 862 particles. The two species of hard spheres comprising the model (A and B) are characterized by the ratio of diameters and the ratio of the number densities The liquid phase of this system was found [31] to lose its thermodynamic stability when compressed beyond the critical value of packing fraction At higher densities, its stable phase was identified as the AB13 crystal, the unit cell of which includes 112 atoms. Due to the complexity of its crystallization pattern, this system possesses a pronounced glass-forming ability. In this simulation, it was found to remain in long-living metastable

equilibrium liquid state when compressed beyond the indicated critical packing fraction value. The absence of crystalline nucleation was thoroughly verified by monitoring the pressure and

the diffusion coefficient, both of which remained constant during the simulation run. In order to test relation (6) we calculated the diffusion coefficient and the excess entropy, in the pair approximation, S2 for the smaller atomic species B, exploring a wide range of both below and above the critical value. S2 was derived using (6) from two partial radial distribution functions gAB (r) and gBB (r). The results of this simulation are presented in Fig. 2. It is clear that the results agree well with relation (6) for which corresponds to the stable liquid domain. For higher values of the packing fraction corresponding to the metastable liquid domain, the diffusion coefficient demonstrates a negative deviation from the prescription of the scaling relation (6) which increases rapidly with increasing Another simulation demonstrating effects of non-ergodic diffusion that we consider here explores a simple monatomic liquid [32] with predominantly icosahedral short-range order induced by a properly constructed pair potential. Because of its structure, this liquid can, under supercooling, avoid crystallization and remain in a state of metastable equilibrium for a time sufficiently long for observation of its essential dynamics [33]. It was found that the 117

Figure 3. Reduced diffusion coefficient in a simple monatomic glass-forming liquid [32] as a function of temperature. Dashed line corresponds to the universal scaling law (6) relating the diffusion coefficient to the excess entropy, in the pair approximation, S2, with A = 0.049. In this liquid TA = 0.6

Figure 4. Time evolution of energy-based measure of ergodicity as defined by Eq. (2) calculated for the glassforming liquid [32] shown in Fig. 3 for three temperatures indicated in the inset.

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domain of temperatures where the liquid exhibits the characteristic anomalies of supercooled dynamics regime is bounded by TA = 0.6 [33]. In Fig. 3 we present its diffusion coefficient plotted as a function of temperature in a manner convenient for testing (6). As the liquid is cooled below TA, the diffusion coefficient displays a pronounced deviation from the scaling behaviour with respect to the excess entropy as postulated by relation (6).

OTHER EFFECTS OF ERGODICITY BREAKING

The latter model of the two discussed in the previous section can also be exploited to directly show the separation of the relaxation time scales that we associate with non-ergodic dynamics. Based on the microscopic information provided by molecular dynamics, the rate of the ergodicity restoring structural relaxations can be assessed using the ergodicity metric introduced by Eq. (3). Fig. 4 shows the evolution of this quantity as a function of the average mean square displacement at different stages of cooling. At sufficiently high temperatures, the results demonstrate an apparent universality in the relationship between the structural relaxation and diffusion. In the supercooled dynamics domain below TA, this universality breaks down as the structural relaxation gets strongly retarded as compared with diffusion. This observation, together with the results shown in Fig. 3, explicitly confirm our analysis

that conjectures non-ergodicity of the liquid diffusion below TA. The long-time decomposition of the region of motion can be viewed as reduction of the time-dependent effective entropy which measures the volume of the accessible phase-space region, as compared with the thermodynamic entropy. On the other hand, the entropy, can, as was pointed out above, be related to the correlation length. Therefore, on the time scale characteristic of the fast (itracomponent) relaxation, the liquid is expected to exhibit timelimited correlations the length scale of which would exceed the range of the static structural correlations which correspond to the ensemble average (or infinite time average). Therefore, the separation of time-scales in the relaxation dynamics of a supercooled liquid that we regard

as a signature of dynamical non-ergodicity, suggests the existence of large-scale clusters with the life time exceeding the time-scale of the “fast” relaxation processes. These time-limited clusters are conjectured to confine an energy favoured local structure incompatible with periodicity [9], and, therefore, their growth is limited by geometric frustration. In the case of the molecular dynamics simulation presented in Fig. 3, this structure corresponds to icosahedra packing. To detect such clusters, we looked for extended structures comprising connected icosahedra. Two icosahedra were regarded as connected if they shared at least three atoms. The results of this analysis are presented in Fig. 4 which depicts the largest clusters of interconnected icosahedra found in the liquid at different stages of cooling. One can indeed see a rapid increase of the cluster size as the temperature decreases below TA. Development of domain structure in supercooled liquids is consistent with a recent observation of long-range cooperativity in diffusive dynamics [34]. We also remark that the development of a network of icosahedral clusters that is observed here resembles the picture of bonding network that was discussed in the context of chalcogenide glass-formers [35]. Experimentally, the characteristic length of the described cooperative effects can be directly probed by measuring the rate of liquid dynamics in confined geometries [36], It has to be emphasized that the time-limited cooperativity is decoupled from the static structure and, therefore, from the thermodynamic entropy. Attempts to interpret the cooperative dynamics of supercooled liquids in terms of the Adam-Gibbs theory which refers to the (static) configurational entropy, although common, are logically incorrect. In order to detect the growth of time-limited domain structure in supercooled liquids, one has to compare the cooperativity range derived from the dynamical measurements in confined geometries with the range of the static structural correlations which, at least for quasi-simple liquids, can be assessed from the diffraction measurements. Ergodicity breaking implies that the former considerably exceeds

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Figure 5. Maximum size icosahedrally structured clusters observed in the glass-forming liquid [32] presented in Figs. 3,4 at different temperatures: (a) T = 1.0, (b) T = 0.5, (c) T = 0.45

the latter. DISCUSSION

We considered here the concept of dynamical ergodicity in the context of liquid dynamics. According to this concept, ergodic behaviour of a liquid is identified with globally connected chaotic behaviour in the phase space. Macroscopically, this behaviour is characterised by a single time-scale in the relaxation dynamics. It is shown that the proposed scenario of ergodic phase-space behaviour of a liquid is inherently consistent with the recently observed universal relationship between the dynamic properties, like transport coefficients or the KS entropy, and the static properties as quantified by the thermodynamic entropy. On the

other hand, decomposition of the phase space into separate regions of connected stochasticity results in long-time confinement of a phase trajectory. This leads to the appearance of characteristic anomalies, like separation of the time scales and spatial heterogeneity, which are commonly associated with the behaviour of a supercooled liquid. An important conclusion of the discussion presented here is that the long-time decom-

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position of the phase space into components developing in the supercooled liquid domain

breaks any conceivable relation between the rate of liquid dynamics and the thermodynamic entropy. In particular, this conclusion concerns the Adam-Gibbs relation which is commonly used for interpreting the behaviour of supercooled liquids approaching glass transition. ACKNOWLEDGEMENT

This work was supported by the following Swedish agencies: Natural Science Research foundations (NFR), Technical Research Foundation (TFR) and National Network for Applied Mathematics (NTM). We used the graphics software from ref. [37]. REFERENCES 1. 2. 3.

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Palmer, R. G. (1982) Broken ergodicity Adv. in Phys. 31, 669-735 Ma, S.-K., (1996) Statistical mechanics, World Scientific, Singapore (1985) Lichtenberg, A. J. and Lieberman, M. A. (1983) Regular and Stochastic Motion, Springer Verlag, NY Mountain, R. D. and Thirumalai, D. (1989) Measures of effective ergodic convergence in liquids Journ. of Phys. Chem, 93, 6975

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Krylov, N. S. Works on the Foundations of Statistical Physics, Princeton Series in Physics, Princeton 1979; see also Sinai, Ya. G. Development of Krylov’s Ideas, pp. 239-281 of the same volume. 16. Sinai, Ya. G. (1966) Izv. Akad. Nauk SSSR. Mt 30, 15-32 (in Russian) 17. Dzugutov, M., (1996) Dynamical diagnostics of ergodicity breaking in supercooled liquids J. Phys. Cond. Matter, 11, 253-259

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19. J. P. Boon and S. Yip, (1980) Molecular Hydrodynamics, McGraw-Hill, New York 20. Cohen, E. D. G. (1993) Fifty years of kinetic theory Physica A, 194 , 229-257 21. Stillinger, F. H. & Weber, T. A. (1984) Packing structures and transitions in liquids and solids. Science

225, 983-989 21. Hansen, J. P. and McDonald, I. (1976) Theory of Simple Liquids, Academic Press, London 23.

S. Chapman and T. G. Cowling, (1939) The mathematical theory of non-uniform gases, Cambridge University Press

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25. Pesin, Ja. B. (1976) Lyapunov characteristic exponents and ergodic properties of smooth dynamical systems with an invariant measure Sov. Math. Dokl., 17, 196-203 (in Russian)

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Rosenfeld, Ya., (1977) Relation between the transport coefficients and the internal entropy of simple systems Phys. Rev. A, 15, 2545-2549 29. Di Marzio, E. A. and Yang, A. J. M. (1997) Configurational entropy approach to kinetics of glasses, Journ. of Res. of the Nat. Inst. of Stand. and Techn 102, 135-157 30. Sastry, S., Debenedetti, P., and Stillinger F. H. (1998) Signatures of distinct dynamical regimes in the

energy landscape of a glass forming liquid Nature, 393, 554-557; Angell, C. A. Liquid landscape ibid., 521-524 31. Eldridge, M. D., Madden, P. A., and Frenkel, D. (1993) Entropy driven formation of a superlattice in a hard sphere binary mixture Nature, 365, 35

32. Dzugutov, M. (1992) Glass formation in a simple monatomic liquid with icosahedral inherent local order. Phys. Rev. A 46, R2984-R2987 33. Dzugutov, M. (1994) Hopping diffusion as a mechanism of relaxation stretching in a stable simple monatomic liquid. Europhys. Lett. 26, 533-538 34. Donati, C., Douglas, J. F., Kob, W., Plimpton, S.J., Poole, P.H. & Glotzer, S.C. (1998) Stringlike cooperative motion in a supercooled liquid. Phys. Rev. Lett. 80, 2338-2341 35. Phillips, J.C., and Thorpe, M.F.,. (1985) Constraints theory, vector percolation and glass formation, Dynamics of glass-forming materials confined in thin films Sol. St. Comm. 53, 699-702 36. Jérôme, B. (1999) Dynamics of glass-forming materials confined in thin films Journ. Phys. Cond. Matter

11, 189-199 37. Humphrey, W., Dalke, A., and Schulten, K., (1996) VMD - visual molecular dynamics, Molecular Graphics 14, 33-38

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NETWORK STIFFENING AND CHEMICAL ORDERING IN CHALCOGENIDE GLASSES: COMPOSITIONAL TRENDS OF Tg IN RELATION TO STRUCTURAL INFORMATION FROM SOLID AND LIQUID STATE NMR

CARSTEN ROSENHAHN, SOPHIA HAYES, GUNTHERBRUNKLAUS, and HELLMUT ECKERT Institut für Physikalische Chemie; Westfälische Wilhelms-Universität Münster, 7, D-48149 Münster, Germany

INTRODUCTION During the past few years, there has been a resurgence of the interest in nonoxide chalcogenide glasses based on the sulfides and selenides of the group 13-15 elements. From a technological point of view, arsenic and germanium sulfide-based glasses in particular are promising low-phonon host materials for luminescent rare-earth dopants, with potential applications in the fiber optic laser industry [1-4], From a scientific point of view, non-oxide chalcogenide glasses represent interesting model systems for testing simple mean-field concepts used to discuss structure-property relations. It has been pointed out that certain physical characteristics of chalcogenide glasses, such as glass transition temperatures, melt viscosities, and molar volumes often depend non-linearly on their chemical composition [5-9]. An explanation has been offered on the basis of percolation theory, by considering the overall connectivity of the network [8,10-12]. For any two- or three-dimensional network one can define an average coordination number

where ni is the number of atoms of species i, ri is the number of bonds formed by it, and the summation extends over all of the different atomic species present in the network. Any covalent network constrained by bond-stretching and bond bending forces possesses a critical connectivity threshold at Glasses having average

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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coordination numbers higher than this threshold value represent single infinite clusters with no residual zero-frequency modes [12-16]. Accordingly, abrupt changes in physical properties are expected at such critical compositions. In this regard, much discussion has been focused on the dependence of the glass transition temperature on composition [1723]. While universal dependences of Tg on have been found in certain binary and ternary systems, there are numerous other glass systems in which no such universality has been observed. The applicability of the average coordination number concept implies that bond formation proceeds in a random fashion, i.e. without any chemical bond preferences. Thus, the concept works at its best when homo- and heteropolar bond energies are of equal magnitudes. In chalcogenide glasses, however, this situation is rarely encountered. Indeed, a considerable body of evidence proves that this situation is unrealistic in many

arsenic and germanium-based chalcogenide glasses, which have rather been found strongly chemically ordered. Because of significant differences in the corresponding bonding energies, heteropolar As-Se and Ge-Se bonds dominate over homopolar As-As, Ge-Ge or weakly polar As-Ge bonds in these systems. Tichy and Ticha were the first to recognize that in such chemically ordered systems the Tg value is well correlated to the overall mean bonding energy <E> [24,25]. Thus, maximum Tg values have been predicted (and found) at the composition of a chemical threshold, defined by R = 2z/(4x + 3y), where x, y, and z refer to the atomic fractions of Ge, As, and Se, respectively [26]. In systems with a strong dominance of heteropolar over homopolar bonds, the

chemical threshold R = 1.0 marks the minimum selenium content at which a chemically ordered network is possible without metal-metal bond formation. Driven by this interest, from both the technological and the scientific point of view, more detailed fundamental concepts concerning the local structure and intermediate range order of these materials are being developed. Amongst other techniques, solid state nuclear magnetic resonance (NMR) has played an important role in this endeavour. In particular, the favorable NMR properties of the 31P and 77Se isotopes have been exploited to advantage for a structural determination of binary and ternary phosphorus sulfide, selenide, and telluride materials [27]. In the solid state, magic angle spinning 31P NMR spectra have provided information about the microstructural units present, while dipolar spin echo decay methods have given information about the extent of P-P bonding. Complementary high-temperature liquid state NMR studies of the glassforming compositions have provided important insights into the chemical equilibria and the kinetics leading to glass formation [28]. These studies have been recently extended to arsenic and germanium selenide glasses [29]. It is the purpose of this contribution to review the current state of this field, and to discuss physical and chemical threshold effects on the compositional dependence of Tg in relation to the short range order in various binary and ternary chalcogenide glass systems. Based on this structural information we will show how the concepts of physical and chemical threshold behaviors can be unified within a more comprehensive model providing satisfactory correlations between elemental composition and macroscopic physical properties.

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BINARY PHOSPHORUS-CHALCOGENIDE GLASS SYSTEMS. Short Range Order

Both the binary systems P-S and P-Se have a strong tendency for glass formation, extending until about 50 at.% phosphorus [6,9,30], In spite of this similarity, the principles governing the structural organization of these glasses are fundamentally different from each other. This difference becomes immediately obvious when comparing the compositional dependences of the glass transition temperatures Tg(see Figure 1).

Figure 1: Compositional dependence of the glass transition temperature Tg in P-S and P-Se glasses. The curve is drawn as a guide to the eye.

In P-Se glasses, an increase in the phosphorus concentration is accompanied by a Tg increase up to about 40 at.% P, followed by an increase at a greater rate at higher

phosphorus contents. This behavior can be understood on the basis of percolation theory as applied to a polymeric network: the composition 40 at.% P corresponds to the rigidity percolation threshold, at which the average coordination number equals 2.4 bonds per atom. Such a change in slope, dTg/d, occurring at = 2.4 has been observed for various other binary and ternary chalcogenide glass systems, and it appears to be a general property of polymeric covalent networks. In contrast, the binary P-S glasses have a decidely different compositional trend [27]. At smaller phosphorus contents the Tg values are larger than in the P-Se glasses. There is a maximum near 25 at.% P, followed by a decrease at larger phosphorus contents. For glasses containing more than 40 at.% phosphorus the glass transition temperatures lie below room temperature. For this glass system, the rigidity percolation concept seems to be inapplicable. 31 P nuclear magnetic resonance studies have given substantial insights into the structural origins of these differences in behavior. Figure 2 compares the 31P MAS-NMR spectra of both systems. On the right, spectra of P-S glasses are displayed. Up to P contents of 15 at. % the spectra are dominated by a broad resonance centered near +110 ppm, which can be assigned to S=PS3/2 units. As the P content increases above 15 at.%, novel sharp resonances appear, signifying the formation of P4S10, P4S9 and P4S7 molecules embedded in the polymeric matrix [27,31]. The sharpness of these resonances

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indicates efficient averaging of the anisotropic interactions on the NMR timescale by virtue of fast molecular reorientation processes. At P concentrations above 30 mole% the glass can be considered an assembly of molecular (zero dimensional) P4Sn units As these molecules make no contribution to network connectivity, a dramatic decrease of Tg is observed for P-S glasses in this concentration region. Figure 2, left shows the 31P MAS-NMR spectra of the P-Se glasses. The rather broad resonances apparent in these spectra reveal a wide chemical shift distribution as is typical for polymeric chalcogenide networks. The two peaks at 130 and zero ppm at low P content have been assigned to three-coordinate PSe3/2 and four-coordinate Se=PSe3/2 groups, respectively [32]. At higher P contents, a gradual shift of the high-frequency peak signifies the appearance of other types of three-coordinate phosphorus species.

Figure 2: 31P MAS-NMR spectra of glasses in the systems P-S (left) and P-Se (right)

Complementary spin-echo NMR experiments reveal that P-P bonds make an increasing contribution to this resonance as the phosphorus content increases above 25 at% [33]. Altogether the structure can be described in terms of four different local units, namely Se=PSe3/2, PSe3/2, Se2/2P-PSe2/2, and (Se-Se)2/2 fragments. Figure 3 summarizes the results obtained by this analysis [33]. They indicate that in the local structure of these glasses there is competition between heteropolar P-Se bonding and the formation of homopolar P-P and Se-Se bonds, respectively. While a number of P-P bonds are being formed even at the smaller phosphorus concentrations, nevertheless their fraction remains sub-statistical within the entire region of glass formation. On the other hand the preference of heteropolar bond formation is not nearly as large as in arsenic or germanium selenide glasses. On the basis of this structural information the characteristic dependence of Tg on P content in the P-Se glasses can now easily be understood: with their efficient competition and balance of homo- and heteropolar bonds these glasses are the most ideal model systems for mean-field theory: the composition 40 at% phosphorus results in an average coordination number of 2.4, corresponding to the Phillips-Thorpe rigidity percolation threshold.

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Figure 3: Site speciation in P-Se glasses as deduced from the combined analysis of 31P spin echo and MAS NMR. A: Se-Se bonds, B: Se=PSe3/2 groups, C: PSe3/2 groups, D: P-P bonded units.

Medium-Range Order

Important complementary information about how these local units are 77 Se NMR studies [34,35]. In principle, several medium-range order scenarios are possible: The P-bearing units could be homogeneously interconnected was obtained from

dispersed (isolation model), they could be clustered together in domains, or all of the units could be randomly linked. These interlinkage scenarios make different predictions for the fraction fSe-Se of those Se atoms that are only bonded to other selenium atoms. Experimental values of fSe-Se can be estimated from an analysis of the 77Se chemical shift measured in the liquid state at temperatures above Tg [35]. Figure 4a shows a typical experimental data set of temperature dependent 77Se NMR spectra for a glass containing 20 at.% P. Above Tg, the P-bonded and non-P-bonded selenium atoms show initially distinct resonances, which are subsequently motionally narrowed and affected by chemical exchange as the temperature is increased. Above 210 °C the resonances collapse into a single peak which narrows continuously as the fast-exchange limit is approached at higher temperatures. A detailed inspection of the average chemical shift as a function of composition (Figure 4b) has shown that the data can be analyzed straightforwardly according to the expression:

where and refer to the chemical shift of Se-only and P-bonded Se atoms, f Se-Se estimates are also available for the glassy state, by differentiating between P-bonded and non-P-bonded selenium species based on their differences in the magnitude of the 31P77 Se magnetic dipole-dipole couplings [34]. As shown in Figure 5, the experimental data obtained by both methods are consistent with a random linkage scenario. This consistency is the result actually expected for a polymerized glass structure.

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Furthermore, the experimental data rule out a previously proposed model [36] based on phosphorus-rich P4Sen clusters (n= 3,4,5) in a selenium-rich matrix.

Figure 4a: Temperature dependent 77Se NMR spectra of a P-Se melt containing 20 at.% Phosphorus.

Figure 4b: (left) Temperature dependence of the 77Se chemical shifts in P-Se melts in the fast exchange limit. Reproduced from reference 35. Figure 5. (right) Fraction of Se-only bonded selenium species extracted from liquid state 77Se NMR and from 77Se-31P spin echo double resonance NMR (SEDOR). Predictions from various intermediate-range

order scenarios are shown for comparison. Reproduced from reference [35].

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BINARY ARSENIC SELENIDE GLASSES

Chemical Bond Distribution

Figure 6 juxtaposes the compositional dependence of Tg for the binary system PSe and As-Se, respectively. Again, in spite of the chemical homology, very different trends are observed: In both systems the composition of 40 at % pnictogen (=2.4, R=l) appears to be of special significance. The Tg maximum observed in the As-Se system indicates the dominance of chemical threshold behaviour at R=l whereas the steep upturn in the P-Se system, signifies rigidity percolation for =2.4. In view of the NMR results previously discussed this difference in the behavior is now well understood

in terms of the chemical bond distribution: the P-Se system is characterized by an efficient competition of homopolar and heteropolar bonds, whereas the As-Se system has a large degree of chemical ordering. Numerous structural studies have indeed suggested that these glasses are primarily constructed by pyramidal AsSe3/2 groups, however, the extent of As-As bond formation below the corresponding composition of 40 at.% As has been subject to some discussion. Likewise, such chemical disordering would imply the existence of Se-Se bonding at larger As contents. Furthermore, the open question remains as to whether the various structural building blocks are randomly linked or if any clustering into domains exists. While it has been demonstrated previously that hightemperature 77Se NMR is able to differentiate between Se- and As bonded selenium in arsenic-selenium liquids [37,38], new information on short- and intermediate range order in this system is also available. Figure 7 shows a typical temperature dependent set of 77 Se NMR spectra of binary AsxSe1-x melts containing 12.5 mole% As [29]. At low temperature (<200°C), we observe three well-resolved resonances in this sample, which we assign to three different types of selenium species. The good resolution indicates that

narrowing by rapid molecular motion is sufficiently complete to produce isotropic spectra, while, on the other hand, averaging by chemical exchange is still slow on the NMR timescale. The peak near 1361 ppm relative to a solid CdSe reference (species 1)

coincides with the spectrum of pure amorphous selenium and is therefore assigned to Se atoms that are part of selenium chains only. The peak near 1114 ppm (species 2) is found in all of those samples containing 20 mole% As or less. Since its fractional area increases

with increasing As content (data not shown), this resonance must signify selenium involved in bonding with arsenic. Based on arguments presented below, we assign it to a selenium species bonded to one arsenic and one selenium atom, i.e. the selenium atom in the midst of a As-Se-Se fragment. The assignment of the resonance at 1289 ppm (species 1`) is the most tentative one. Based on the fact that in samples with different arsenic contents, the area of this peak remains in a fixed relation to that at 1114 ppm, we assign this resonance to a selenium with a second-nearest As neighbor, i.e. to the selenium atom at the end of an As-Se-Se- fragment. A third, smaller feature near 980 ppm (species 3) becomes also evident at temperatures > 180 °C. We attribute this feature to selenium atoms bridging between two As species. Unlike for species 1, 1´ and 2, the temperature windows separating motional narrowing from chemical exchange averaging are not sufficiently distinct for species 3. Therefore, it is difficult to quantify its spectral contribution.

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Figure 6: Tg vs. correlation in the systems P-Se (crosses), As-Se (circles), and Ge-Se (diamonds).

Figure 7: Experimental temperature dependent 77Se NMR spectra of As12.5Se87.5 melt

As illustrated in Figure 7, signal coalescence owing to the onset of chemical exchange is observed above T = 180 °C, indicating that bond breaking and re-forming is occurring on the NMR timescale. At a sufficiently high temperature (>280°C) only a single resonance remains, whose frequency approximately corresponds to the weighted average of the individual peak positions. In glasses containing more than 20 % arsenic, the spectra in the slow-exchange limit are not observable at temperatures above Tg, most likely because molecular motion is slowed down by increased network rigidity, making it impossible to identify separate temperature regimes for motional narrowing and exchange narrowing. At appropriate temperatures >300 °C, these spectra show the lineshapes typical of the fast-exchange limit. Figure 8 illustrates that the average peak position depends on arsenic content and temperature. In the sample containing 12.5 mole% As, the chemical exchange process is revealed in the chemical shift trend shown (see above), while at larger As contents only the exchange-averaged spectra were studied. The position of the exchange averaged

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spectra does not coincide with the calculated center of gravity from the resolved spectra at low temperatures. This result indicates that part of the arsenic-bound Se species are not affected by motional narrowing and do not contribute to the sharp resonances seen

in Figure 7. This statement applies in particular to species 3 near 980 ppm, attributed to selenium atoms bridging between two As atoms.

Figure 8: Dependence of 77Se chemical shift on Composition and Temperature in As-Se melts.

The intrinsic linear temperature dependence observed in Figure 8 which is similar to the effect seen in Figure 4 for each composition probably arises from a thermal increase of the unpaired electron spin concentration in the liquid. The dependence of chemical shift on As content is also evident from Figure 9, which compiles data measured at a uniform temperature of 400 °C. As the As content is increased, the chemical shift decreases in a linear fashion up to a limiting value of near 1000 ppm at composition 40 at.% As. Within the concentration interval extending from 40 to 50 mole% arsenic, the 77Se chemical shift remains approximately constant. The nearly linear dependence of the average chemical shift within the composition region at % As, and its approximate invariance at larger As concentrations are consistent with a chemically ordered network structure composed mostly of AsSe3/2 groups. Thus, the change in chemical shift with As content is due to the progressive conversion of Se-Se-Se- to As-Se-Se to As-Se-As fragments. As predicted by such a chemically ordered model, above 40% As, the chemical environment of Se remains constant, and further introduction of As results in the formation of additional As-As bonds only.

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Figure 9: 77 Se NMR chemical shifts of AsxSe100-x glasses at 400°C.

Medium-Range Order

In analogy to the discussion in the P-Se system the detailed composition dependence of the 77Se chemical shifts enables an assessment of various medium rangeorder scenarios, relating to the linking of the As- and Se-bearing units in the network. Three basic scenarios are considered, from which different predictions with respect to the fraction of Se-only bonded selenium species are made. First of all, a molecular clustering model, based on the formation of As4Se4 molecules dispersed in a seleniumrich matrix can be considered. This model is of certain interest, because a similar proposal was previously made for P-Se glasses [36]. A second scenario (domain model) envisions the AsSe3/2 units congregating into domains, implying that the majority of the arseni oonded selenium species are bridging between two As atoms. The remaining scenario is based on a random dispersal of AsSe3/2 and Se2/2 groups, simulated either analytically or by means of a graph theoretical approach [29]. In the latter, AsSe3/2 groups are placed onto a two-dimensional triangular grid at random (according to the respective As concentration of the glass considered), and the number of Se-only bonded Se atoms is determined by a simple counting algorithm. No As-As bonds are allowed. The construction of the network from these units produces four distinct next-nearest neighbor (NNN) environments for each arsenic site, being linked via selenium to zero, one, two or three AsSe3/2 groups. The populations of these NNN environments are plotted in Figure 10 as a function of composition. Figure 11 indicates that the three medium-range order scenarios make different predictions concerning the fraction 'f1' of Se atoms that are exclusively bonded to Se. Experimentally, this number is available from the chemical shift data measured in the fast exchange region. In principle, the experimental shift corresponds to the weighted average of the three basic types of selenium species:

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where the fi are fractional contributions with f1 + f2 + f3 = 1. Each one of the three selenium species has a characteristic shift For species 1, the chemical shift of molten Se is a good approximation, while for species 3, the chemical shift of 1000 ppm observed at 400 °C in liquid As2Se3 (40 at.% As) can be taken as a reference. For the chemical shift of species 2 we assume that it lies close to the average (1200 ppm) of the two other species. The principal difficulty in extracting f1 from the average chemical shift, then, lies in the lack of knowledge of the relative contributions f 2 and f3 of the singly and doubly As - bonded selenium atoms. We are able to address this question quantitatively only in glasses containing 12.5 mole% As and 17.5 mole% As, where we know the ratio f1/f2 independently from the lowtemperature data (Figure 7). Figure 11 reveals that the f1 extracted from this additional information for the 12.5% As and the 17.5% As glasses are close to the f1-s predicted from a statistical linking scenario. For the other glasses in our study, only upper and lower limits for f1 can be specified, as represented by the vertical bars displayed in Figure 11. Clearly, the experimental data eliminate both the molecular and the domain scenarios from consideration, and show best consistency with the random linkage model. As such the results are in agreement with a continuous network structure of arsenic selenide glasses and are not in agreement with any scenarios involving clustering and phase separation processes, at least within the concentration region extending from zero to 40 mole% arsenic. On the other hand, the situation at larger As concentrations is less evident from 77Se NMR data. For such As-rich glasses, the formation of As-As bonds has been most clearly detected from 75As NQR spectroscopy [39] carried out on the glasses themselves. We anticipate that the combination of liquid state 77Se NMR with solid state 75As NQR spectroscopy offers new prospects for discussing chalcogenide glass chemical order and medium range order in future applications.

Figure 10: Compositional dependence of NNN environments as generated by random linking

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Figure 11: Comparison of the different scenarios with respect to the fraction f1 of the Se-only bonded Se atoms. Possible ranges of f1 deduced from the analysis of the experimental 77Se chemical shifts are indicated (see text).

BINARY GERMANIUM SELENIDE GLASSES

The compositional dependence of Tg for the binary Ge-Se glasses which has been included in Figure 6, shows the the clear manifestation of rigidity percolation and chemical threshold behavior occurring well-separated at the chemical compositions of

23 at% Ge (2.46) and 33.3 at.% Ge (R=l), respectively. The slight upward shift of the rigidity threshold to values above 2.40 has been noted previously by Boolchand [40] and will not be discussed here. Figure 12 shows temperature dependent 77Se NMR spectra of Ge-Se melts at variable compositions. In this case, no distinct Ge-bonded selenium species are detectable in the slow-exchange limit, presumably because the mobility of these species is too low to permit averaging of the anisotropic interactions. Therefore, in Ge-Se melts the effect of bonding to selenium is only visible in the fastexchange regime. Due to limitations in the temperature range available, the data currently available are restricted to low-melting glasses with Ge contents < 20 at.%. Since, consequently the limiting chemical shift of Ge-only bonded Se-species is not yet available, a quantitative analysis in terms of intermediate-range order is not possible on the basis of these data. Nevertheless, this plot illustrates the great sensitivity of 77Se chemical shifts to changes in the short range order in germanium selenide melts.

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Figure 12: Temperature dependent 77Se NMR spectra of glassy Ge-Se melts with variable Ge contents.

THE TERNARY SYSTEMS GE-P-X (X=S, SE, TE) Recently, ternary systems have been studied with Te as the only chalcogen component. Although no glassforming region exists in any of the binary systems P-Te, P-Ge, or P-Si, the formation of ternary P-Si-Te and P-Ge-Te glasses has been reported [5] within compositional limits which have been reproduced in our laboratory. 31P MAS and spin echo decay data obtained on such glasses indicate that in the Ge-P-Te glass the phosphorus environment is entirely dominated by phosphorus-phosphorus bonding. Heating these glasses above the glass transition temperature produces P4 molecules [41].

Thus, Ge-P-Te glass can be viewed as being composed of Pn clusters or polymeric bands embedded in a Ge-Te matrix. In contrast, the 31P chemical shifts of the P-Si-Te glasses occur in a region that is typical for Si-bonded phosphorus. The spin echo decays of these glasses are substantially slower suggesting weaker 31P-31P dipole-dipole interactions. From a detailed analysis of these results the extent of P-P bonding in this glass system has been quantified, and found to be approximately equal to that expected from statistical probability under explicit exclusion of P-Te bond formation [41]. A comparison of the spectroscopic behavior in the ternary systems P-Ge-S, P-GeSe, and P-Ge-Te reveals a striking influence of the chalcogen ion on the short-range

order of phosphorus. While in the sulfide system, the short-range order of P is dominated by S=PS3/2 groups [42], in Ge-P-Se glasses the formation of GeSe4/2 groups

takes precedence, and controls the balance of P-Se vs. P-P bonding (see below) [43]. Finally, in Ge-P-Te glasses only P-P bonding occurs. Most recent 31P MAS NMR results on phosphorus-germanium-sulfur glasses are shown in Figure 13. These glasses have recently attracted considerable interest as low-

phonon hosts for rare-earth luminescent ions, and the P-atoms are thought to play a crucial role in increasing rare-earth solubility. Structurally, these glasses can be viewed as polymeric networks with strong chemical ordering into GeS4/2 and S=PS3/2 units. Compared to binary P-S glasses (see Figure 1), the presence of germanium second nearest neighbors to the S=PS3/2 groups produces unusually large resonance

displacements, illustrating the great sensitivity of 31P MAS NMR to next-nearest neighbor effects in the structure of these glasses [42].

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Figure 13: 31 P MAS NMR spectra of two GeS2-P4S10 glasses.

Above all, the most interesting results have been obtained for the glass system GeP-Se [43]. Figure 14 shows that no chemical threshold behavior near R = 1 is observable

in this system. This result implies that in glasses with R<1 the selenium deficiency is balanced by the formation of P-P bonds, whereas Ge remains bonded in the form of GeSe4/2 groups. This conclusion has been confirmed by a detailed analysis of the 31P-31P dipole-dipole interaction in this system. Figure 15 plots the 31P-31P dipolar second moment M2, measured by spin echo decay spectroscopy, as a function of glass composition. As shown in reference [43] the second moment serves as a measure of the fraction of P-bonded phosphorus atoms. Addition of Ge to a glass of fixed P/Se ratio produces a marked increase of M2 owing to the tendency of Ge to attract selenium to form GeSe4/2 units. Thus, the increase of M2 reflects an increased fraction of P atoms engaging in P-P bonding as the germanium content of the glasses is increased. Figure 15 demonstrates a universal correlation of M2 with the compositional parameter

Here [Seeff] = [Se] -2[Ge] is the selenium content available for bonding to phosphorus after the amount needed for the formation of GeSe4/2 groups has been subtracted. The first factor in eq. (4) accounts for the competition effect in chemical bond formation and the second one for the overall dilution effect arising from the presence of Ge as a third component. Overall the universal correlation, which also includes the data on the binary P-Se glasses, provides strong evidence for a pronounced hierarchy of homopolar bond formation in P-Ge-Se glasses: Even in highly Se deficient glasses the formation of GeSe

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bonds takes precedence, while the Se deficiency is balanced by an increased formation of P-P bonds.

Figure 14: Plot of Tg vs. and R for glasses in the system Ge-P-Se. Note the absence of chemical threshold behavior

Figure 15: Correlation of the 31P-31P dipolar second moment with the compositional parameter X = P/Seeff) (P) (see text) for Ge-P-Se glasses. Open symbols denote data for binary P-Se glasses

THE TERNARY SYSTEMS GE-AS-SE AND GE-SB-SE. The ternary systems Ge-As-Se and Ge-Sb-Se are homologues of the phosphorusbased system discussed above. Tichy and Ticha noted that in spite of the strong chemical ordering observed in the binary As-Se and Ge-Se glass systems (strong Tg-maxima at R = 1), a plot of Tg vs. R does not yield a corresponding maximum in ternary Ge-As-Se glasses and only a weak effect in the Ge-Sb-Se system. New insight into this behavior has been recently obtained by 119Sn spectroscopy [44]. As previously demonstrated by Boolchand [45], the tin isotope substitutes isomorphically as Sn(IV) for Ge in glassy GeSe2. The formation of homopolar Ge-Ge bonds in Se-deficient glasses can then be monitored by the appearance of Sn(II) in the spectra. In contrast, our recent results show that Se-deficient As-Se-Sn glasses can be prepared over a fairly wide composition range (down to R=0.7) without Sn(II) ever appearing in the

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spectra[46]. Thus, the Se-deficiency in such glasses is balanced by As-As bond formation in analogy to the situation in Ge-P-Se glasses. The same balancing mechanism, then, is likely to occur in the Ge-As-Se ternary, explaining the absence of chemical threshold behavior. For Ge-Sb-Se glasses a similar explanation may hold for the attenuation of the chemical threshold effect, however, from 121Sb Mössbauer

spectroscopy only weak evidence for Sb-Sb bond formation has been found. Overall, these results suggest that in ternary Ge-X-Se glasses (X = P, As, Sb) there is a strong

secondary hierarchy governing the formation of homopolar bonds, with the tendency strongly decreasing in the order P-P, As-As, Sb-Sb, Ge-Ge. Over wide compositional ranges of Se-deficient glasses this hierarchy ensures that the number of Ge-Se bonds is

maximized and no ,,unnecessary“ Ge-Ge bonds are being formed as long as there are alternate possibilities of homopolar bond formation. In highly Se-deficient glasses the presence of Ge-Ge bonds is finally indicated by the appearance of divalent tin species in the 119Sn Mössbauer spectra. Figure 14 and 17 illustrate that universal Tg vs.

correlations are observed for all of the three ternary glass systems. In the system Ge-PSe this correlation includes all of the glasses studied. In the systems Ge-As-Se and GeSb-Se deviations from universality are only observed for those (highly Se-deficient)

glass compositions, for which the 119Sn Mössbauer NMR spectra indicate a fraction of tin to be divalent, hence suggesting that part of the germanium constituents are no longer part of GeSe4/2 units. CONCLUSIONS

In summary the results of the present study illustrate the utility of solid- and molten-state NMR spectroscopy to provide valuable information on short- and mediumrange ordering effects in non-oxide chalcogenide glasses. P-Se glasses with their efficient competition between homo- and heteropolar bond formation are the most ideal model systems for mean-field theory. Accordingly a dramatic change in dTg/d is

observable at the percolation threshold = 2.40. In contrast, As-Se and Ge-Se systems display a pronounced preference for heteropolar bond formation, resulting in chemical threshold effects that are superimposed upon the effects of physical percolation. These chemical threshold effects produce Tg maxima at compositions corresponding to R =1, where heteropolar bond formation is maximized. In ternary GeSe-X systems (X = P,As,Sb) these chemical threshold effects disappear because the formation of homopolar bonds is controlled by a strong secondary hierarchy. Over wide compositional regions of Se-deficient glasses this hierarchy serves to minimize the formation of Ge-Ge bonds and can be held responsible for the observation of universal Tg vs. dependences. Based on these results we predict that physical threshold

behavior will in general be observable not only in those glass systems having no bonding preferences, but also in those ternary glass systems in which chemical ordering is observed, but where there is a clear secondary hierarchy of heteropolar bond formation. The most striking example is the Ge-P-Se glass system; other examples can

be envisioned for tellurium-based chalcogenide glass systems Ge-X-Te. In contrast, no such secondary hierarchy is expected for sulfide-based glasses (such as Ge-X-S), where

we envision heteropolar Ge-S and X-S bonding to dominate the structure to such an extent that no secondary hierarchy effects are expected. The experimental varification of these ideas is currently under investigation in our laboratories.

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Figure 16: Plots of Tg vs. in the glass systems, Ge(Sn)-As-Se (top), and Ge(Sn)-Sb-Se (bottom). Open symbols indicate those glasses showing evidence of Sn(II) in the these glasses show deviations from a universal correlation.

spectra. Note that only

ACKNOWLEDGMENTS

Financial support of this research by the U.S. National Science Foundation (DMR 92-21197) and the Deutsche Forschungsgemeinschaft under grant Ecl68/l-2 is most gratefully acknowledged. Thanks are also due to the Wissenschaftsministerium Nordrhein-Westfalen for supplemental support. C.R. and G.B. acknowledge support by personal stipends awarded by the Verband der Chemischen Industrie, Stiftung

Stipendien Fonds. S.H. appreciates support by a University of California Presidential Dissertation Fellowship.

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GLASS TRANSITION TEMPERATURE VARIATION AS A PROBE FOR NETWORK CONNECTIVITY

M. MICOULAUT Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, Boite 121 4, place Jussieu 75252 Paris Cedex 05 France

INTRODUCTION The liquid-glass transition is one of the unsolved challenging problems [1] in actual solid state and materials sciences, and its basic phenomenology can be described as follows: a liquid which has been cooled sufficiently fast in order to avoid crystallization can be denoted as “supercooled” when its temperature lies below the melting temperature Tm. While decreasing the temperature, the supercooled liquid becomes so viscous that the structural relaxation time becomes of the order of the experimental time t exp. The temperature at which the glass transition occurs is denoted Tg and is best defined at which It corresponds to the temperature at which structural arrest occurs in the sense that the relaxations are too slow to observe. As a consequence, the glass transition seems to be primarily dynamic in origin, in contrast with usual thermodynamic phase transition. Another definition of Tg, denoted in the literature as the caloric glass transition temperature, is related to the thermodynamic behavior of the supercooled liquid close to the transition. At Tg, thermodynamic quantities such as enthalpy and volume (the first derivatives of the Gibbs energy) exhibit a first-order behavior while some of the second derivatives (such as the heat capacity) are continuous, but with an inflexion point at Tg. A standard measurement of the glass transition temperature consists therefore in following the endothermic heat flow through the transition in differential scanning calorimetry (DSC technique). We should note that the value of Tg depends on the heating rate (typically 10 K.min–1) in the DSC experiment, which signals a significant non-equilibrium behavior. However, the conventional technique to access the glass transition temperature has been recently improved by superposing a sinusoidal variation on the usual linear T ramp (modulated DSC) [2]. This permits deconvolving the total heat flow near the transition into reversing and a non-reversing contributions. The reversing component tracks the sinusoidal T variation and yields a measure of the heat capacity during glass transition, while the non-reversing heat flow is ascribed to the thermal history of the melt (kinetic processes and the enthalpic changes accompanying structural reorganization close to Tg). This means that the measure of Tg from the reversing

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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heat flow is almost expunged from kinetic effects and represents therefore a ”true” value of the glass transition temperature [3]. Surprisingly, little attention has been given to the variation of the glass transition temperature with respect to alloying, although it is fundamental for the understanding the nature of the glass transition, and for technological applications. For instance, it is well known that a few percent of sodium oxide lowers the glass transition temperature of vitreous silica from 1200°C to 600°C, thus permitting to obtain a glass with LT furnaces. Several proposals have been made in trying to relate the value of Tg with other physical or chemical measurable quantities. Kauzmann suggested for example that Tg scales with the melting temperature of the corresponding liquid (the so-called ”two-third rule”). Other authors have proposed relationships based on the boiling temperature, the Debye temperature of the phonon spectrum. The importance of structural factors has been stressed, in particular the valence of the involved atoms in the glass-forming material, but these Tg predictions remain mostly qualitative [4]. The main quantitative advances in this field are due to Varshneya and co-workers [5], who applied with success the Adam, Gibbs and Di Marzio theory [6] on glassy polymers to multicomponent chalcogenide network glasses. The chalcogenide glass systems can indeed be represented as networks of chains made of selenium or sulfur atoms, in which cross-linking units such as germanium atoms are inserted. The increase of Tg, identified with the second-order phase transition temperature in the initial theory, is produced by the growing presence of these cross-linking agents. The aim of this article is to show that the variation of the glass transition temperature upon alloying is mostly related to the connectivity of the glass network. First, we will show that the increase of the structural relaxation time can be translated in a very simple minded way, when considering the glass transition in covalent glass-forming liquids. We give the general ideas concerning the way one should define the glass transition temperature. Then we will apply it to binary and ternary glass formers for which the theoretical description is entirely solvable. This will give universal relationships that can be compared with experimental data. Last but not least, we explain why there is a systematic deviation of the prediction with respect to experiment in all investigated glass systems and how this is related to the constraint theory [7]. STOCHASTIC AGGLOMERATION THEORY

Stochastic agglomeration theory [8,9] provides a quantitative means to analyze compositional trends in the glass transition temperature. As we shall see below, this theory seems to be particularly well adapted for the description of covalent glass-forming systems, where the increase of the the structural relaxation time with decreasing temperature is Arrhenius-like: Another quantity of interest behaves similarly, which is the viscosity of the glass-forming liquid, via the relation represents the instantaneous shear modulus, which is almost constant. The increase of the viscosity with decreasing temperature can be viewed in a very simple way in covalent systems where the viscosity should be directly proportional to the number of dangling bonds remaining in the liquid structure. Each time, a new bond is created, the viscosity increases. Therefore, agglomeration between well-defined atomic or molecular entities (i.e. creating bonds between these entities) should be one of the most representative physical process taking place during the glass transition. Also, the activation energy appearing in the Arrhenius behavior of should be intimely related to the involved bond energy. In order to describe these agglomeration processes, we have to define more precisely the structural entities, which are going to be involved. The most simplest way of describing the short range order (SRO) in the liquid should rely on star-like entities sharing a certain number of dangling bonds (we shall call them local structural configurations (LSC) or singlets).

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Figure 1. A schematic representation of the realized experiment for N = 2. a) At time t = 0 (C) has a certain statistical distribution of LSC’s, which is modified if diffusive LSC’s are trapped by agglomeration during the time step (panel b). One can also imagine collective rearrangements where two bonds are created during the same time step (panel c). Note that the size of (C) can increase in order to avoid densification.

These LSC’s have a central atom and have a clear, unambigous experimental evidence. Their

coordination number is defined by e.g. X-ray or inelastic neutron diffraction techniques, which exhibit sharp and characteristic peaks at the corresponding bond length and give information about the number of nearest possible neighbors of the central atom. For instance, we can mention the tetrahedron which is the lowest possible short-range structure in silicate or germanate glasses (in this case the LSC’s are molecules) or or a four-valenced silicon atom in network glasses (here the LSC’s are atoms). An alternative choice could be the tetrahedron and a selenium atom. Of course, the results do not depend on the initial choice of the LSC’s. The idea is to start from an initial LSC, corresponding to the shortrange structure of an inital glass (or liquid-like) network and to see how the agglomeration is affected by the presence of another kind of LSC or several kinds of LSC’s.

Now, let us assume that we are able to realize the following experiment. In a supercooled liquid at T > T g, we consider always the same region of space (C), defined by its particular spatial coordinates. There, we can make at a time t = 0 a statistics of the different LSC’s,

yielding a probability distribution {pi(0)}, where i = 1...N and N is the total number of different LSC’s. If the liquid is homogeneous and at equilibrium, the value of the probabilities should be directly related to the macroscopic concentration. As long as the system is in the supercooled state, the low viscosity allows still intermolecular diffusion of remaining isolated LSC’s [10], passing through the particular region of space (C) we are considering. The probability distribution {p i (0)} at time t = 0 will be modified if these LSC’s are trapped after a certain finite time inside the region, by agglomerating on other existing LSC’s, leading to a distribution and a fluctuation for the LSC i which is We can repeat this experiment for different times ad decreasing temperature, and still looking at the quantity The next issue we have to addresss is the evaluation of a time dependence for the LSC probability distribution, during the well-defined discrete time-step and derive a master equation which imposes the minimization of local fluctuations at a certain temperature which will be identified with the glass transition temperature. We can write the master equation as:

where we have taken into account possible effects due to the cooling rate q = –dT/dt. Why should the LSC fluctations vanish at the temperature Tg ? If the viscosity is very high (of the order of e.g. 1013Poise), reaching the glassy state,

the diffusion of LSC’s inside the liquid should be enhanced, and agglomeration should not be possible anymore. One should therefore expect that after the same time step tion remains constant inside the considered region (C), and equals

the distribu-

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In other words, at glass transition temperature, the local probabilities reach a stationary value, which satisfy:

Equation (2) represents a set of (N-l) non-linear equations, which are closed by a normalization condition. What is the form of appearing in equ. (2) ? It represents the probability that an LSC i can stick onto another LSC j in the considered region during the time step TO, averaged over all j’s. is also related to the probability that a bond i – j is created. This probability should be proportional to the products of the probabilities of the LSC’s at time a Boltzmann factor which takes into account the energy of creation of the respective bond formation at temperature T, and a statistical factor which may be regarded as the degeneracy of the corresponding stored energy, because there are several equivalent ways to join together two coordinated LSC’s. If one has single bond formation, the statistical factor is simply the product of the coordination numbers of the involved LSC’s. For example, the number of equivalent ways to connect two LSC i and j with coordination numbers and is and can be related either to the number of outer shell electrons or to the coordination number (following in this case the so-called 8 – rule [11]).

Here, Z is a normalizing factor. One obtains the solution of (2) by solving the set of equations in terms of the variable

since the are found as functions of these quantities. It is easy to remark that such a set of solutions satisfying (2) varies with the glass transition temperature and depends strongly on (related to the connectivity of the network) and (related to the bond strength). The space of solutions of equ. (2) is a (N – 1) dimensional symplex whose vertices

correspond to the solutions identified with pure structural phases because all the are equal to zero, except one. Beside these trivial solutions, which do not depend on the temperature, it is possible to obtain other ones, lying either on the edges of the symplex (partially pure phases, at least one solution being equal to 0) or inside (mixture of all solutions Finally, the nature of the agglomeration process can be analyzed by studying the stability of the master equation in the vicinity of these stationary solutions. The set of solutions can correspond to possible structurally stable (if the solution is an attractor) or metastable states (if the solution is a saddle point) of the network. In most of the situations, the pure structural phases are attractive (or repulsive) and the solution found inside the symplex is a saddle point. The existence of the latter depends on the involved bond energies E ij, also on the value of Tg. In network glasses (e.g. the LSC’s are the atoms of the network. Thus the concentration x i of the atoms is directly related to the probability p i(0) of finding the LSC’s Bi inside the network, and pi = xi. Their respective derivatives with respect to the glass transition temperature are equal. Note that in this notation, A represents the atom of the initial network, when all xi are equal to zero (e.g. the selenium network in selenides). One interesting chemical parameter is the average coordination number < r >, which is widely used in these systems, since the introduction of this quantity by Phillips and Thorpe in constraint theory [7,12]. < r > is defined by:

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where ri is the coordination number of the atom Bi, given eventually by the 8 – second part of equ. (4) is obtained from the normalisation condition. Then:

rule. The

The expression (5) can be compared with numerous experimental data [3,13,14], which have been obtained since 1979 in order to check the validity of the constraint theory [7,12] and its mechanical threshold when < r > reaches the magic number of 2.4. In other glass systems, such as binary glasses, (made of a network former and a modifier), the LSC’s are molecules. For example, the silicon tetrahedra with one and two nonbridging oxygens are used in order to describe systems. Here, the relationship between concentration x and the solutions pi (0) of the LSC’s satisfying equation (2) can be obtained by recalling a charge conservation law [9]. In particular, the concentration of the modifier cation (such as or must be equal to the anionic contribution located on each different LSC. In binary glasses with formula (1 – x)network former+x modifier,

this conservation law reads:

where is the anionic contribution of the LSC with probability is a stoichiometric factor. Similarly to (5), we can then compute the variation of with respect to the reduced concentration R.

The expressions implicit functions

appearing in (5) and (7) are obtained by differentiating the obtained from (2), and by solving the system:

In the following, we will describe the simplest cases. To do this, we neglect the dependence of the cooling rate q = dT/dt in equation (1) because network chalcogenide glasses or binary oxide glasses form very easily and have critical cooling rates of the order of From fig. 2, we can also remark that the cooling rate dependence of equation (1) (and consequently on the value of Tg) is rather weak, although is changing from several orders of magnitude [The figure refers to the construction of the next section with N = 2]. Also, since we are interested in a variation of the glass transition temperature with respect to structural modification, if the preparation of the glass systems upon alloying is always realized with the same procedure, the effect of cooling rate should be negligible (in other words Then, we can solve the case of N = 2 and N = 3.

The simplest case: N=2

In this section, we consider the application of stochastic agglomeration theory to glasses having two different kinds of LSC’s, which will be denoted by A (the initial or regular LSC, with probability 1 – p(0) and coordination number and B (the modifier LSC with probability p(0) and coordination number The archetypal systems are the network former and the low-modified alkali silicate glass. For the latter, the building blocks of the

147

Figure 2. A plot of (equation (1)) for N = 2 as a function of temperature T for different cooling rates q. The system corresponds to a glass, with (solid line), (dashed line) and

K (dotted line). The glass transition temperature corresponds to the vertical value 0, where vanishes.

network are the Q4 (the tetrahedron in NMR notation, which is the short-range structure of vitreous silica) and the Q3 units (the tetrahedron), which appear when a small amount of sodium oxide is added in the network. Starting again from equation (2), one can see that there is only one single equation to solve, and for low modifier concentration, only two possible connections can occur during

glass transition (or the time step

A-A and A-B connections, neglecting B-B connections.

Z normalizes the whole, and the probability to find the LSC B among this distribution is We obtain the following equation at T = T g, from (2):

Equation (11) has two trivial solutions: p(0) = 1 and p(0) = 0 and a third one:

which defines a relationship between p(0) and Tg. We stress that only one energy difference is essential here: The relationship between Tg and the probability of finding the

LSC B depends therefore only on one parameter. The expression (12) will be physically acceptable only if which is true when the following condition is satisfied: 148

The condition (13) implies also that the border solutions p(0) = 0 and p(0) = 1 are unstable stationary solutions, which means that agglomeration out of A-A and B-B bonds only is prohibited. In this case, the bond statistics is given by the solution (12) and the expressions of (9) and (10). If the condition (13) is not satisfied, a stable attractive solution is found at p(0) = 0. There, the agglomeration tends to phase separate at a microscopic level. We note that the absence of B-B bondings automatically produces a repulsive solution for p(0) = 1. Equation (12) can be made parameter-free by considering the limit of the pure A network with glass transition temperature Tg = T0 and of course p(0) = 0. It is easy to check from (12) that:

Last but not least, we can see how the glass transition temperature changes in the very low concentration limit, by computing the derivative Tg with respect to p(0):

with the energy difference (14) established, this leads to a complete parameter-free equation:

Following the considered system, we will use either the equivalence p(0) = x or the charge conservation law in order to relate the probability of the local B configuration to the concentration. For network glasses the formula has been obtained in [8]:

The important and rather surprising point is that (17) depends neither on dynamical, nor on visco-elastic quantities, although they are used to define Tg and to track the glass transition. It becomes obvious that the bond strength between the LSC’s does not influence the variation of Tg with respect to T0. Connectivity plays here the major role. Equation (17) gives also the mathematical transcription of a well-known empirical rule extracted from numerous experiments in glass science, which states that the glass transition temperature Tg increases with the addition of a modifier that increases the network connectivity, i.e. when rB > rA. A more sophisticated case: N=3 The next step we can accomplish in agglomeration theory is also entirely solvable if one neglects again the effects of the cooling rate in equation (1). In solving this case, we will use the previous results obtained for N = 2. We consider now systems where there can exist three different kinds of LSC’s. This is of course the situation in ternary chalcogenide glass systems, where three different atomic species are involved (for instance, the As-Ge-S(Se) network glasses). Nevertheless, the construction will be also valid for binary glasses. Here, three different LSC’s can be found, although there are only two chemically different components (the network former and the

149

modifier). This is for example the situation in the glass in the concentration range [0,0.5], where three different types of tetrahedra can be observed: the aforementioned Q4 and Q3 units, but now also the Q2 unit, sharing two non-bridging oxygens, and being a part of the metasilicate chain structure at x = 0.5. The difference with the chalcogenides lies in the fact that both Q3 and Q2 contribute to the single concentration R (or x) of

alkali oxide. We will focus in this article only on chalcogenides. We shall denote the LSC of the two modified states by B and C, and the basic state by A. A represents then the local structure of the initial glass network. If we use again the previous examples, the selenium or the sulfur atom is identified with the A configuration in chalcogenides and the As and Ge atoms with the B and C LSC’s. We denote their coordination numbers by rA, rB, and rC and their corresponding local probabilities by 1 – pB – pC, pB and pC (for simplicity, we have removed the bracketts and the mention of t = 0). Again, we shall consider the situation of a ternary network glass in the case of low modification. This allows us to take only four possible connections into account, between the LSC’s A, B and C, and neglecting B – B and C – C bonds. In many situations,

these bonds occur only beyond the stoichiometric glass composition. The corresponding probabilities of A-A, A-B, A-C and B-C bonds can be written as follows:

where Z is the normalizing factor given by:

and represent the bond energies involved in such types of bonds. As one can see, the construction remains here very similar to that described in the previous section, but the main difference lies in the number of equations (2) to solve. Instead of a single equation with one variable p(0), as shown in (11), we obtain here a set of two nonlinear equations with two variables and These are obtained by defining the local concentration of B and C:

and writing an equation of type (1) for the B and C LSC’s at glass transition temperature, where the different (with i = B,C) vanish:

150

Figure 3. The solutions of the system (24). The solutions a), b) and c) correspond respectively to pure A, B and C LSC’s networks, the solution d) to a half mixture of B and C, the solution e) and f) to the related binary glasses (A,C) and (A,B). Finally, the saddle point solution g) corresponds to the ternary glass solution.

where we have set:

and

for a clearer presentation. It is worth mentioning that the system (24), which has to be solved in terms of the variables pB and pC, is simpler than it looks at the first glance, especially because of the symetric role of the LSC’s B and C. Also, the space of variables pB and pC is contained in a two-dimensional symplex (i.e. a triangle) All the meaningful solutions are contained in this triangle (see fig. 3). There are still some trivial solutions for the system (24), namely: pB = 0, pC = 0 (solution a), pB = 1, pC = 0 (solution b) and pB = 0, pC = 1 (solution c) which correspond to the vertices of the symplex (fig.3), also (middle of the edge 1 – pB – pC = 0, solution d). The remaing singular solutions correspond either to the related binary systems (A,C) (solution e) or (A,B) (solution f), i.e.:

and:

Solution a) represents a glass with only A configurations, identified with the network former (for example v – Se). Solution b) and c) correspond to a system made of 100% B or C configurations and are repulsive when their stability is analyzed. Solution d) is identified 151

with a half mixture of these LSC. It is interesting to note that the solutions e) and f) have the same expression as the obtained for the binary glass, with N = 2. Of more interest is the saddle point solution made of a mixture of A, B and C which is the ternary glass solution (solution g):

(27)

(28)

The latter solution is located inside the symplex, but its existence depends on the value of the involved structural and energetical factors and The other useful solutions are the binary (A,B) and (A,C) solutions e and f. If the system can be considered as phase separated, and composed of a mixture of the two binary systems. As a consequence, such kind of networks should display two glass transition temperatures, satisfying the ad-hoc normalized solutions (25) and (26). The application of stochastic agglomeration theory with N = 3 involves three independent parameters, which are the energy differences: and However, two bond energy differences can be computed by considering the related A – B and A – C binary systems, and performing on solutions (25) and (26) the limit or in a manner similar to what has been done for the case N = 2. We find again the energy difference (14) and: (29)

We have only one free parameter left, the Boltzmann factor involving the bond energy difference This parameter can be determined by considering the derivatives of with respect to and in the limit of low modification:.

(30)

and a similar form is obtained for the derivative with respect to y, when y = 0.

(31)

with:

(32) 152

where and and refer to the above mentioned bond energy differences, but either with glass transition temperature (when z = x) or to (when z = y). They have been inserted in order to avoid a cumbersome presentation. If one considers the limit values of the slopes (30) and (31) when x 0 and y 0, one should recover the results of the slopes for binary (A,B) and (A,C) glasses. This means that

these equations should have the following form, either identical to (17) or symetric in (B,C): (33) (34) where is still the limiting glass transition temperature of the pure A LSC network. In the limit x = y = 0), both equations are second order linear equations in since and There is only one solution satisfying simultaneously both equations (33) and (34), which is:

(35) The last energy difference to be computed is therefore: (36)

With all energy differences established, one is able to have access to the value of the glass

transition temperature with respect to for different concentrations of B and C LSC’s. Also, and will depend only on the Boltzmann factors and (37)

(38) An interesting point is the prediction of the glass transition temperature with the average coordination number of the network. The average coordination number of the network is

defined by: (39)

which leads to the final expression:

(40) with: and The relationship (38) is, again, parameter-free and can be plotted for any bonding numbers r and given the initial glass transition temperature of the chalcogenide network. We compare now the results of the stochastic agglomeration theory for N = 2 and N = 3 with experimental data on chalcogenide network formers. This is accomplished by using equations (17) and for network systems, and equ. (40) for the ternary glasses.

153

Figure 4. Glass transition ratio

in IV-VI network chalcogenide systems, as a function of the concentra-

tion of the element of Group IV. Data are taken from [3,15-18]. For a clearer presentation, the telluride systems

have been shifted upwards with an arbitrary value of 2. The solid lines represent the slope equation (17) with and

COMPARISON WITH EXPERIMENT

Figure 4 gives the glass transition temperature prediction in chalcogenide network formers of the type involving an element of Group IV and a chalcogenide (Group VI, Consequently, for all the systems displayed in the figure, following equation (17) the slope is We can observe that this value is in a very accurate agreement

with the experimental data, and for all and However, it is interesting to remark that only the sulphide and selenide systems exhibit a systematic deviation around the concentration of x = 0.2, to be put in contrast with the and systems for which the variation remains linear and for which the slope equation (17) agrees over the entire displayed concentration range. The reason for this difference is the following. Vitreous selenium or sulphur consist of polymeric chains beside which some eight-membered rings can exist. The addition of an atom of Group IV creates cross-linking between these chains in a random fashion and globally the whole network can be considered as random. The structural modification goes up to the characteristic concentration x = 0.2 which corresponds to the optimal glass composition where mechanical stability reaches its maximum. The latter behavior is very well understood in terms of the constraint theory developed by Pillips and Thorpe which predicts a rigidity transition when the average coordination number of the network reaches 2.4 [7,12]. For larger concentrations, the glass structure ceases to be random because it approaches the chemical stability composition (at x = 0.33). Close to this value, the local structure will be

stoichiometrically balanced and stable crystalline compounds or can be formed. Therefore, the local structure of these glass systems in the range [0.2,0.33] will be very close to the crystalline counterpart and each chalcogen will tend to be surrounded by two germanium atoms and vice-versa. In any case, the description in terms of random A-A

and A-B bondings will become inappropriate (more and more A-B connections are possible), and the glass transition temperature will vary superlinearly, in systematic deviation with the 154

Figure 5. Glass transition ratio in V-VI network chalcogenide systems, as a function of the concentration of the element of Group V. Data represent the systems P – S [20], P – Se [21], Bi – Se [22] and As – Se [23]. The solid line represents the slope equation (17) with and The dashed and dotted lines correspond to the slope (41) taking into account four-fold coordinated species with a fraction of 0.3 and 0.5.

predicted slope (17). In the telluride systems, there is no chemical stability composition at x = 0.33. The

reason is that there are neither crystalline nor compounds, but and GeTe instead, corresponding to the respective concentrations x = 0.4 and x = 0.5, But there is still a rigidity transition (mechanical stability) at x = 0.2 which can be detected either by observing the coalescence of the crystallization temperatures of the floppy and rigid parts of the network (for [15]) or the reversing heat flow window (for [19]). As a consequence, the chemical stability composition does not influence the local structure as it does for the selenide and the sulphide glasses, and therefore the network can remain random over a larger concentration range, as predicted by the linear increase from the slope (17). We are not aware of more data on these systems, but we believe that the deviation should occur close to x = 0.4. We now turn to the systems involving an element of Group V (a priori, supposed and one can clearly observe the same kind of agreement, at least for the bismuth selenide and the phosphorus sulfide glass. For the arsenic and the phosphorus selenides, there can be a reasonable prediction in the very low concentration limit, when the concentration x is less than 0.15. Nevertheless, the chosen coordination number is in contradiction with the observed structure (orpiment-like for As, i.e. three-fold coordinated units) of these glasses. Thus, one should reasonably expect which does not predict the trends at all. However, recent experimental results on both glasses (NMR [21] and modulated DSC [24]) suggest that at low concentration, there should be also four-fold coordinated units These units share a double As=Se (or P=Se) bond, which has to be taken into account in the stochastic agglomeration theory. Indeed, the double As = Se bond has a different statistical contribution than the single bonded P – Se. The slope (17) is then slightly modified [22]: (41)

155

Figure 6. Glass transition

in ternary network selenide systems, as a function of the average coordination

number. Data represent the systems Ge – As – Se [13,25] and Ge – Sb – Se [3]. The solid line represents the

inverse of equation (40) with

and

and

where represents the fraction of the four-fold species. If goes to zero, one obtains the slope for III-VI systems. The value of yields the correct trend for the glass transition temperature variation in glasses (dotted line in fig. 5), a result which is confirmed by NMR and Raman spectroscopy [22]. Similarly, one expects a rate of in the arsenic system (dashed line in fig. 5) and spectroscopic investigation is currently in progress. In the systems having really a five-fold coordinated LSC (such as bismuth), the deviation of the stochastic prediction of (17) is here also supposed to occur at the average ccordination

number of < r > = 2.4, which corresponds to the concentration x = 0.13. One can remark that the bismuth system exhibits the systematic deviation around this concentration. In the next figures, the prediction of in ternary glasses displays the same agreement as for the binary glasses, and the systematic deviation occurs, again, around the value < r > =

2.4. In fig. 6 we have represented data from the ternary IV-V selenides. In these compounds, the As and Sb atoms are three-fold coordinated, whereas the germanium is still four-fold coordinated, thus and Fig. 6 shows also that is a universal function of < r >, described by the inverse of equ. (40) up to and somewhat beyond the stiffness transition, a feature which was already pointed out by Angell and co-workers some ten years ago [13], The effects depending specifically on composition occur only below this value of 2.4. Stochastic description in terms of As-Se, As-Ge, Se-Se and Ge-Se bonds fails beyond this value, because of the chemical organization of networks in terms of stoichiometric balanced structures (e.g. or in the As-Ge-Se compound). Surprisingly, ternary Sb-Ge-Se glasses seem to support the random network picture up to < r > = 2.4 although the binary Sb-Se system is already chemically ordered at very low Sb concentrations (formation of clusters) [26]. Again, the similar behavior of the As and Sb glasses clearly confirms the major influence of connectivity, although the bond strength of these atoms is different. In the analogous IV-V telluride systems, displayed in fig. 7, we can note the same kind of behavior, although less data are available in the literature. Note that the stochastic description (and still is accurate in the low modified limit, when the average coordination number of the network is lower than 2.4. Deviation occurs also around the stiffness transition 156

Figure 7. Glass transition

in ternary network telluride systems, as a function of the average coordination

number. Data represent the systems Ge – As – Te, Ge – Sb – Te [27], Si – As – Te [28] and Si – P – Te [29]. The solid line represents the inverse of equation (40) with and and

and certain data become strongly composition dependent for < r >> 2.4. However, the solid lines represented here provide a useful information on the absolute magnitude of the glass transition temperatures, which should be measured in a manner similar to the As-Ge-Se system. Finally, equation (40) with and permits also to investigate germanium incorporated chalcohalides [9,26]. GENERATING MEDIUM RANGE ORDER AND SELF-ORGANIZATION

We have already mentioned that the systematic deviation of the stochastic agglomeration prediction in the glass transition temperature trends was due to the occurence close to the rigidity transition of larger structural correlations, so that a description in terms of only A-A and A-B bondings was not valid anymore. On a microscopic level, this means that the modifier LSC B will not be randomly diluted inside the A network. It seems that for a large class of systems, mechanical rigidity onsets progressively and that stressed-free (isostatic)

regions could emerge inthe network backbone. The signature of this phenomenon has been detected by experiments on different chalcogenide glasses [16,25] and by computer simulations [30]. The presence of rings may also play an important role and their existence should modify substantially the nature of the rigidity transition. How can agglomeration theory take into account such effects ? We can extend the construction described at the beginning of this paper to larger structures. One can indeed imagine that during glass transition, multiple bonds (two, as in the panel c of fig.l, but also three, four, etc.) will be formed during the time step τ0, in the region (C). The extension has several advantages. On the one hand, it will give a more realistic picture of what is really happening during the glass transition, where collective rearrangements (a number of bonds created during a finite time step) occur close to the transition and seem to be mostly responsible for the huge increase of the relaxation time. On the other hand,

we have a systematic way [31] to construct structures with medium-range order (fig.8), since each new multiplet created takes into account a corresponding energetical factor similarly to 157

Figure 8. From doublets (A-A and A-B) to quadruplets (A-A-A-A, A-A-A-B, etc.) in a IV-VI network glass. The numbers represent the statistical weight W of each structure.

equ. (3 ) and a statistical weight W that can be computed from the coordination number of the involved LSC’s. Last but not least, the construction permits to nucleate rings of growing size, which play the key role in the rigidity transition of IV-VI network glasses, such as [13]. We have performed the construction of multiplets in this type of glass up to structures having five atoms, allowing the creation of 4 and 5-membered rings. Both are present in

the celebrated chalcogen-rich ribbon layer structure [11], which has been used in order to explain the chemical and physical properties of v – One can remark that the statistical weight of rings and large B correlated structures is much larger than simple (A,B) crosslinked structures (2880 and 1920, against 256 or 768). We can again compute for each size of multiplets the probability and solve an equation of type (2). What we expect is that the growing size of the multiplets will enable us to give the prediction of on a increasingly larger concentration range. Several results of this analysis are interesting. First, one recovers for all the different sizes the energy difference (14) in the limit where p(0) = 0. Secondly, in case of dendritic

growth (no rings at all) the prediction diverges even more rapidly than for the A-A, A-B construction. Rings have therefore to be taken into account. The ring energy is the parameter which is used in order to fit the solution of (2) with the experimental data on From the determined value of we can then compute the ring statistics of this glass, as a function of the concentration (fig. 9). Both kinds of rings (4 and 5-membered rings) exhibit a monotonic increase with concentration. They should therefore be present in large proportions

at the stoichiometric glass composition. We believe that this construction is worth to be developed to larger structures where rings of larger sizes will be produced. Entropic (through the weights W) and energetical factors (through equation (14)) will select the dominant structures and give substantial information about medium range order and self-organization in these network glasses.

158

Figure 9. The ring statistics in

glasses as a function of concentration, obtained from the stochastic

agglomeration of quintuplets. The solid line corresponds to 4-merabered rings, the dashed line to 5-membered rings.

ACKNOWLEDGMENTS

LPTL is Unité Mixte de Recherche du CNRS n. 7600. It is a pleasure to acknowledge continued discussions with P. Boolchand, R. Kerner, G.G. Naumis, J.C. Phillips and M.F. Thorpe.

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Elliott, M. Balkanski Eds, World Scientific Press, Singapore; Micoulaut, M., Naumis, G.G. (1999) Glass transition temperature variation, cross-linking and structure in network glasses: a stochastic approach, Europhys. Lett 47, 568. This is another reason why SRO structures are more adapted than larger structures in this description. On can indeed reasonably assume that diffusion is only possible for atoms or small molecules.

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FLOPPY MODES EFFECTS IN THE THERMODYNAMICAL PROPERTIES OF CHALCOGENIDE GLASSES

GERARDO G. NAUMIS1 1

Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Apdo. Postal 20-364, 01000, Mexico D.F., Mexico.

INTRODUCTION One of the most fascinating problems in solid state theory is the glass transition (GT), where a non-crystalline amorphous solid (a glass) is formed by supercooling a melt. Not all materials are able to form glasses, and many different semi-empirical criteria have been proposed in order to explain the ability of a material to reach the glassy state [ 1 ]. This means that there are many factors involved in the process. Of these, an important one is the time available for crystallization of the material with respect to the cooling rate, since the crystal has a lower free energy with respect to the glass. Thus, the melt needs to be cooled fast enough to avoid thermodynamical equilibrium. Due to this reason, the GT is usually not considered as a true phase transition, although there are discontinuities in the specific heat (C p (T)) and the thermal expansion coefficient at constant pressure that lead to the idea of an underlying phase transition. Another criterion for the GT is the development of shear elasticity and a cessation of viscous flow [1]. As we will see, this increase in shear viscosity is often exponential with respect to the temperature (Arrhenius law) or follows the Vogel-Fulcher-Tammann law [2], The type of the viscosity behavior is known as fragility: glasses that follow the Arrhenius law are known as strong glasses, while the others are called fragile [3]. At a microscopic level, the development of shear viscosity is due to a slowing down of the atomic diffusion which is responsible of atomic rearrangements and structural relaxation. Below GT, the atomic structure is fixed so the contributions of structural relaxation to thermodynamical quantities is nearly absent [1]. One can say that the liquid stops exploring ergodically the available phase space (determined by the energy landscape) and stays in a region for a long-time. The temperature where this phenomena happen is known as the glass transition temperature (Tg ). A lot of attention has been devoted to correlate Tg with physical and chemical properties

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

161

in different glasses. Among the most studied systems, we can cite the chalcogenide glasses (formed with elements from the column VI of the periodic table, like Se or S), which are a

bench-mark test for the understanding of the GT. For example, Tg and (the jump in the specific heat during the GT) of a chalcogenide glasses can be raised or lowered by adding impurities like Ge or As, and the fragility of the glass can be changed from fragile to strong [4]. For the change of Tg with the composition, a method based on the statistics of agglomeration processes [5-9] succeeded in obtaining the empirical modified Gibbs-DiMarzio law, that allows to calculate the change of Tg as a function of the concentration of modifiers [9]. This stochastic method predicts the value of the characteristic constant that appears in the

Gibbs-DiMarzio law for almost any chalcogenide glass, and gives a topological explanation [5] for the origin of Tg. The changes of and fragility as a function of the composition has been less studied, but there are some experimental works about the subject [4,11,12]. A careful study of these experiments can reveal many interesting features of the GT as a function of the chemical composition.

The constraint theory introduced by Phillips [13] and further refined by Thorpe [14-15] has a very fundamental role in these problems. Phillips considered the covalent bonding as a mechanical constraint experienced by the atoms in order to explain the ability for making a glass. As we will see in the next section, when the number of constraints is less than the degrees of freedom given by the dimensionality of the space, the lattice has zero frequency

modes that are called floppy. However, altougth constraint theory has been very sucessfull in describing qualitative features of GT, not so much effort has been done in order to test the

theory in a quantitative way. In this work, we will discuss the implications of the constraint

theory approach in terms of the thermodynamical properties of the glass. The main idea is to connect the entropy of the liquid phase, with the range of possible disordered structures, since rigidity percolation operates more effectively in the liquid state of the chalcogenide systems [ 1 1 ] . This idea also allows to explain the change in fragility of the glass, which in some sense gives an idea of the “difficulty degree” for making the glass. The layout of this work is the following. In the next section we present a brief introduction to the constraint theory and floppy mode counting. In section 3 we present a discussion of the thermodynamical implications of the floppy mode approach and a comparison with the experimental data. Finally, in section 4 we give the conclusions. CONSTRAINT THEORY AND FLOPPY MODES

Covalent bonded atoms, like those found in chalcogenide glasses, follow the 8 – N rule, where N is the number of valence electrons and the coordination number of the atom r is given by 8 – N. For example, a Se atom has r = 2, since it is in the group VI of the periodic table. The coordination number of the atoms that are present in the glass, allows the construction of an average coordination number < r >, which in some sense is the coordination of a pseudoatom forming a structure whose topology is similar to that of the system [2]. For a ternary alloy of the type AxByC1–x–y (as GexAsySe1–x–y), < r > is,

while in general, if the total number of atoms is N and there are nr atoms with coordination r, the average coordination number is defined as,

In 1979, Phillips proposed a connection between glass-forming ability and the average coordination number [13]. The idea is that the tendency for glass formation is maximized

162

when the number of mechanical constraints Nc in each atom is equal to the number of degrees of freedom Nd given by the dimensionality of the space. As we will see, Nc is related with the average coordination number. A system with more constraints than Nd is overconstrained and cannot easily form a glass. In a covalent network, the atoms are constrained due to the fact that the interatomic lengths and angles are well defined, altougth there are small departures from the equilibrium position. The strain potential energy of these deviations can be expressed as sum of contributions from bond-stretching and bond bending forces,

where and are the bon-stretching and bond-bending force constants, and and are small deviations from equilibrium in bond-length and bond-angles respectively. Using this potential, one can solve the movement equations by finding the eigenmodes and eigenvalues of the dynamical matrix. Since the matrix is of size 3Nx3N, it has 3N eigenvalues which corresponds to the frequencies of oscillation of each eigenmode Of these modes, 6 are zero, since correspond to a symmetry in the translation of the center of mass and an arbitrary rotation in space of the whole system. However, for low average coordination the lattice can be deformed without cost in energy since there is no term in the potential that couples to the dihedral angles [15], and thus the rank of the dynamical matrix is reduced. These eigenmodes with zero frequency are called floppy. They give the number of possible deformations of the network without a cost in energy. Following the work of Phillips, Thorpe estimated the number of constraints by using the Maxwell counting [14], which is a procedure originally devised to determine the stability of a network of rods connected with pivot joints. In this counting, each r coordinated atom is associated with r/2 bond-stretching constraints, since there are r bonds, each of them are shared by two atoms. In each atom, there are 2r – 3 constraints that come from the angular forces. Using this counting, the fraction of floppy modes with respect to the degrees of freedom is,

Of special importance is the point < r >= 2.4, where the number of floppy modes is zero. This point was recognized by Thorpe to be a “rigidity transition” [14], between a floppy network and a rigid one; it corresponds to a strong tendency for making a glass. In some sense, this transition is akin to the percolation transition, but rigidity percolation is more difficult to study, since it is a vectorial and non-local problem. However, the counting presented here works remarkably well except near the transition point, where more sophisticated methods, like the pebble game are needed [16]. THERMODYNAMICAL EFFECTS OF FLOPPY MODES

In this section we will study of the effects of floppy modes in the thermodynamical properties during GT. We start by considering the effect of the constraints or floppy modes in

the internal energy of the glass if they were really at zero frequency. To make this calculation, we write the Hamiltonian of the glass in terms of the normal modes of vibration (Q i ) that are deduced from the dynamical matrix given by the potential,

where Pi and mi are the momentum and mass of the particle i. The most important point in the last equation, is that the sum in the potential energy is not carried over all the 3N 163

modes. In this case, each floppy mode does not contribute because in principle it has zero frequency. Using this Hamiltonian, we can obtain the internal energy of the glass by using the equipartition theorem of statistical mechanics, where at high temperature each “degree of freedom” contributes with 3 k T /2 to the internal energy,

( i t is worthwhile mentioning that the degrees of freedom in statistical mechanics correspond to constraints in rigidity theory). The first term is the contribution of the kinetic energy, and

the second is the contribution of the potential energy. The specific heat is,

As is known, the solids at high temperatures follow the Dulong-Petit law, which shows that

the specific heat is cal/mol – k. In the previous result, we have obtained a reduction of the specific heat given by 3Nkf /2. However, since the glass is a solid, it is difficult to imagine that it does not follows the Dulong-Petit law. Furthermore, different experiments for glasses show that Cp is 6cal/mol – k independent of < r > [11]. Thus, we are lead to the conclusion that floppy modes do not have zero frequency, instead they are shifted to small values close to zero due to the non-linearity of the potential or the residual Van-der Waals forces. This conclusion that comes from a thermodynamical argument, is supported by an inelastic neutron scattering experiment, where the density of states was measured in a GexSe1–x, and GexASySe1–x–y glass, and it was found a blue-shift of the floppy modes, peaked around 5meV [17]. Note that although at high temperatures the floppy modes does not have any effect in the specific heat, at very low temperature we can expect changes in the thermodynamical properties with the composition, since for a given temperature, the number of accessible states is bigger in the floppy glass. In the liquid melt, floppy modes are expected to be more important in the thermodynamical properties, since the system can explore a wider energy landscape due to its extra thermal energy. Floppy modes produce regions of less rigidity, and these regions are more suitable to produce a richer landscape. The idea that we are going to develop, is to relate floppy modes with the presence of local regions with less stress in the system, i.e., we relate low-frequency vibrational modes with disorder, since configurational modes have been shown to explain the additional specific heat of the melt [18], and these configurational modes are identified with the degree of stress. The rigidity of the network is related to how amenable is the glass to continuous deformations [15], and thus floppy modes correspond to motions in the system that require only very small energies. In principle, floppy modes are eigenvectors of the dynamical matrix with zero frequency in the glass, but in the supercooled and normal liquid phase, this matrix is not well defined,

except if we consider that the eigenrnodes can be identified with instantaneous normal modes [18]. However, rigidity is a static concept, involving virtual displacements, so while it is

useful to use a dynamical matrix for a given potential, any potential would give the same results for geometric aspects of rigidity [16]. To test these ideas, we start by evaluating the strain energy of the liquid near Tg as a sum of three contributions,

U(melt) = Ukin + Uharm + Uconf ,

(8)

where Ukin is the contribution from the kinetic energy of the atoms, Uharm comes from the

harmonic vibrations and Uanharm is the contribution from anharmonic configurational interaction. The contribution from the kinetic energy is given by the equipartition theorem which gives kT/2 for each degree of freedom. Since we have 3N particles, Ukin = 3 N k T / 2 . For the harmonic contribution, we must remember that liquids cannot withstand shear stress,

164

and thus they cannot sustain transverse modes of vibration, therefore, they have only N vibrational modes, corresponding to longitudinal phonons. The contribution from this term

is Uharm = NkT/2. Observe that we are counting all the 3N modes, instead of writing Uharm = (N – 3 N f ) k T / 2 , since we already noticed that floppy modes have finite frequencies. The third term represents the interaction which is responsible for nucleation and crystallization. One can think that near Tg, it is reasonable to suppose that Uconfig must be of the same order of Ukin= 3 N k T / 2, since the energetic barrier for configurational motion needs to overcome the thermal barrier. Then, one obtains for the internal energy,

and for the jump in the specific heat during glass transition (the difference between the specific heat of the glass compared to that of the melt),

Surprisingly, this result coincides with the value reported for GexSe1–x by Boolchand et. al. [12] and GexAsySe1–x–y by Tatsumisago et. al.[4] when < r >= 2.4, as can be seen in Fig. 1. In fact, when < r >= 2.4, from the following expression,

is minimum, and the configurational entropy Sc, calculated

where Tk is the Kauzmann liquid-crystal isoentropy temperature, is also a minimum. Thus, for < r >< 2.4 or < r >> 2.4, we are underestimating the contribution from relaxation or

configurational rearrangements. At this point, we can introduce the idea that constraints in

the lattice produce regions with different degrees of stress. For the underconstrained melt, it is reasonable to think that underconstrained regions have a bigger configurational entropy, since the existence of floppy modes means that there are many equivalent structures with the same energy. Here we propose to consider a hole activation model to describe the relaxation modes, with the number of holes determined by the number of floppy modes. As we will see, this model gives us a good fit of the experimental data. Before continue with this approach, we must comment the reason of separating anharmonic and floppy modes . As was said

previously, the shape of the pair potential does not affect the existence of floppy modes, it only changes the small finite frequency of these modes. In fact, these modes we expect to be more related with terms in the potential that produce stress, as for example happen in the modified soft-model, where a linear term is added in order to account for stress [18]. Anharmonic terms are related to melting, while we suppose that floppy modes are related with relaxation of strain. Here we will assume that Uflop is a function of the number of floppy modes. We suppose that each mode acts as a effective extra degree of freedom for relaxation in the region where < r >< 2.4, since the number of floppy modes gives the different possible deformations of the network. Thus, Uflop is proportional to f3NkT (where f = 2 – (5 < r > /6) is the fraction of floppy modes). Observe that this idea is connected with the work of Duxbury et. al., who showed that the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation [19]. In fact, a specific heat can be defined using the second derivative of the free energy [19]. By adding this contribution to the previous calculation of we find the specific heat to be,

165

Figure 1. as a function of < r > . The line corresponds to Eq. (12). Experimental data from GexAsySe1–x–y (taken from Ref. [4]) are shown with squares. The star corresponds to As2Se3. A best fit line is shown with a dashed line. The circles correspond to the data of GexSe1–x from

the work of Feng et. al. [12]. The dotted line corresponds to the best fit of their data.

This equation does not contain free parameters, and can be compared with the experimental data available. In Figure 1 we plot Eq. (12), and from the experimental data of AsxGeySe1–x–y and GexSe1–x–y; taken from the work of Tatsumisago et. al. [4] (squares, circles and diamond), and Feng et. al [12] (circles). AsxGeySe1–x–y and GexSe1–x systems are used because they are a bench-wood test for the constraint theory; specially the first system since a given average coordination number can be reached with many different compounds. We observe that although Eq.(12) does not have free parameters, it gives a good fit of the experimental data. One special exception is the binary As2Se3 (shown as a star), which is atypical in the sense that it does not follow the isocoordination rule in its properties. This is related to its ”raft”-like structures of two dimensional aspect [4]. If we exclude this point, then a linear regression of the experimental data from the work of Tatsumisago et. al. gives a slope of –5.07 cal/(mol – K), with a correlation of 0.993, which is very similar to the value of –4.96 cal/(mol – K) predicted by the floppy mode approach. This best fit is shown as a dashed line in Figure 1. The data from the work of Feng et. al. [12], give a slope for the best fit of –4.76, with a correlation of 0.985 (shown as a dotted line). More recently, the evolution of the modulated differential scanning calorimetry (MDSC) has made possible to obtain more information about the thermodynamical processes that ocurr due to rigidity [20]. In the MDSC technique, a sinusoidal variation of the heat is imprinted over the linear T-ramp that is used for decreasing the temperature in the usual differential scanning calorimetry. The response of the system can be deconvoluted in two contributions, one is the reversible heat flow and the other the non-reversing heat flow, which is the one that does not follow the variations in T. The importance of this technique is that the non-reversing heat flow

gives an idea of the non-reversing processes that ocurr

in the glass-liquid transformation [21]. An important experimental observation, is that is nearly zero for optimally coordinated glasses (< r >= 2.4), furthermore, the minimum in is due to the non-reversing component [20]. The fact that means that the glass stops exploring the energy landscape, and rest in a deep minimum [20]. The glass 166

transition is thermally reversing. However, as we change the composition begins to rise, as the glass explore the landscape. Observe that these results from MDSC are in agreement with the approach used in this work, since we assumed that rigidity is related with the number of accessible structures of the network via relaxation. In fact, this explains why the first estimation of cal/mol – k gave the correct estimation only for < r >= 2.4, since for other values of < r > there is an extra contribution as the system explores the energy-landscape. It is interesting that in some systems , the rigidity transition is not very sharp, since there are ”compositional windows” [21,22] where for a certain region of < r >. Another interesting feature of the thermodynamical behavior with respect to the composition is that the volar volume of the glass mimics the shape of i.e., the network packing is optimized at the rigidity transition [20], For the overconstrained region (< r >> 2.4), Phillips proposed the following formula for the total strain energy [13], (13)

where Nd is the dimension of the space, Nco the number of constraints. According to Phillips, a,b and c are constants that are estimated in the following way. a = 1/2 since it corresponds to the greatest relaxation allowed by atom movements during the configuration freeze at Tg. b = 1 since it corresponds to the harmonic contribution. The third term is the anharmonic

configurational interaction among the excess coordinates, and must be of the same order of magnitude than the first term. The expression for the strain energy can be written in terms of the average coordination number, (14)

However, from the data of Figure 1 we can see that seems to be linear in the overconstrained region. A liner regression of the data gives a slope of 4 cal/(mol – K) and correlation 0.991. The magnitude of the jump in Cp at the glass transition is also related with the fragility of the glass. Strong glass forming liquids are resistant to changes in the medium range order [2] because the amount of configurational entropy in the liquid is relatively small. Fragile glass forming liquids have a high entropy. In the present approach, from Eq. ( 1 1 ) is clear that Sc has a linear dependence on < r >; i.e., fragility is related with the number of floppy modes. Observe that this is the natural

link between the floppy mode theory and a quantitative property that measures the ability for making the glass [23]. A key quantity that allows to classify the fragility of the glass is the behavior of the viscosity. Fragile glasses forming liquids follow the Volger-Fulcher law [2], (15) where D and T0 are constants. Strong glasses follows an Arrhenius law. However, both

behaviors are to be expected from the Adam-Gibbs equation [2], (16) since if is small, from Eq.(11), S c is almost T-independent and Eq.(16) follows an Arrhenius form. The Vogel-Fulcher law is recovered from Eq. (12) when is bigger, with a functional form of the type B/T [2], where B is a constant that must be adjusted in order to account for the total value of during the transition. In this case, is known, and we get, (17)

167

Figure 2. D as a function of < r >. The squares are from reference [4]. The circles are the values obtained from Eq.( 18), and the line is a visual guide.

Tm is the temperature limit where Cp begins to descent towards Tg. Using this expression and Eq.(16), the constant D of the Volger-Fulcher law is given by: (18) which is a measure of the strength of the liquid. Higher values of D corresponds to strong glasses. The relation defined by Eq.(18) between D and < r > can be tested with the experimental data, if the constant C is fixed from one of the experimental points. For pure Se, the experimental data [4] shows that D = 10, TK = 240K and to give C = 38400 cal/mol. Using this constant and the values of TK and Tm from the experiment [4,11], from Eq.(18) we obtain the points that are shown with circles in Fig. 2. The experimental data of Tatsumisago et. al. [4] are shown with squares in Fig. 2. As it can be seen, there is a good correspondence between the prediction of Eq. (18) and the experimental data. Another quantity of interest is the excess expansion coefficient The present approach allows to obtain its functional form, although we cannot obtain the values of the constants. According to the free volume theory, we expect that constant, thus is of the form:

(19) where C1 and C2 are two constants. This can be corroborated in Fig. 3, where we show a plot of the experimental data fo r in the AsxGeySe1–x–y system [4] and the corresponding linear regression, which has the following form:

with a correlation

coefficient of 0.936.

SUMMARY

In this work, we made a discussion about the thermodynamical effects of the floppy mode theory. For the glass, we obtained that floppy modes have a finite frequency shift, since they follows the Dulong-Petit law. For the liquid melt, rigidity has a contribution to the 168

Figure 3. as a function of < r > from Ref. [4]. The line corresponds to the best fit.

configurational entropy, that depends on the number of floppy modes. Using this, the jump in Cp during glass transition can be estimated, and the change of fragility can be also obtained. This allows to find a quantitative measurement of the ability for making the glass in terms of the floppy mode theory. Acknowledgments

I would like to thank P. Boolchand for his comments and for bringing to my attention the MDSC measurements and the windows of rigidity. Thanks also to R. Kerner and M. Micoulaut for enlightening discussions. This work was supported by the projects DGAPAUNAM IN-108199 and CONACyT 25237-E .

REFERENCES 1. 2. 3. 4.

7.

Jäckle J. (1986) Models of the glass transition. Rep. Prog. Phys.49, 171-229. Elliot S.R.. (1989), Physics of Amorphous Materials, Wiley, New York. Zallen, R. (1998) The Physics of Amorphous Solids, John Wiley & Sons, New York. Tatsumisago M., Halfpap B.L, Green J.L., Lindsay S.M., Agnell C.A., (1990) Fragility of Ge-As-Se glass-forming liquids in relation to rigidity percolation and the Kauzmann paradox Phys Rev. Lett. 64 1549-1553. Kerner R. (1995) Two simple rules for covalent binary glasses, Physica B 215, 267-272. Barrio R.A.. Duruisseau J.P., Kerner R. (1995) Structural properties of alkali-borate glasses derived from a theoretical model. Philosophical Magazine B72, 535-550. Naumis G.G., Kerner R. (1998) Stochastic matrix description of glass transition in ternary chalcogenide

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Naumis G.G. (1998) Modelling of growth and agglomeration processes leading to various non-crystalline

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Micoulaut M., Naumis G.G. (1999) Glass transition temperature variation, cross-linking and structure in

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Sreeram A.N., Swiler D.R., Varshneya A.K. (1991) Gibbs-DiMarzio equation to describe the glass transition temperature trends in multicomponent chalcogenide glasses, J. Non-Cryst. Solids 127, 287-297. Senapati U., Varshneya (1995) Configurational arrangements in chalcogenide glasses: a new perspective

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systems. J. of Non-Cryst. Solids 231, 111-117. materials, J. of Non-Cryst. Solids 232-234, 600-606.

network glasses: a stochastic approach, Europhys. Lett. 47 (5), 568-574.

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on Phillips constraint theory, J. Non-Cryst. Solids 185, 289-296.

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12. Feng X., Bresser W.J., Boolchand P.( 1997) Direct evidence for stiffness threshold in chalcogenide glasses. Phys. Rev. Lett. 78,4422-4425.

13. Phillips J.C. (1979) Topology of covalent non-crystalline solids I: short-range order in chalcogenide al-

loys, J. Non-Cryst. Solids. 34, 153-181. Thorpe M. (1983) Continuous deformations in random networks J. Non-Cryst. Solids 57, 355. He H., Thorpe M. (1985) Elastic properties of glasses Phys. Rev. Lett., 54 2107-2110. Jacobs D.J., Thorpe M.( 1995) Generic percolation: the pebble game Phys. Rev. Lett, bf 75, 4051- 4054. Kamitakahara W.A., Capelleti R.L., Boolchand P., Halfpap B., Gompf F, Neumann D.A., Mutka H. (1991) Vibrational density of states and network rigidity in chalcogenide glasses Phys. Rev. B44 94-100. 18. Zürcher U., Keyes T., (1997) Soft-modes in glass-forming liquids: the role of local stress, in Fourkas J.T., Kivelson D., Mohany U., Nelson K. (eds.). Supercooled liquids: advances and novel applications, American Chemical Society, Washington D.C., pp. 82-94. 19. Duxbury P.M., Jacobs D.J., Thorpe M., MouzkarelC., (1999) Floppy modes and the free energy: rigidity and connectivity percolation on Bethe lattices Phys. Rev. B59, 2084-2092 20. Boolchand P., Feng X., Selvanathan D., Bresser W.J., (1999) Rigidity transition in chalcogenide glasses,

14. 15 16. 17.

in Thorpe M.F., Duxbury P.M., Rigidity Theory and Applications, Kluwer Academic/Plenum Publishers, p.p 279-295.

21. Selvanathan D., Bresser W.J., Boolchand P., Goodman B., (1999) Thermally reversing window and stiffness transitions in chalcogenide glasses. Solid State Communications 111 619-624. 22. Selvanathan D., Bresser W.J., Boolchand P., (2000) Stiffness transitions in Si xSe1–x glasses from Raman scattering and temperature-modulated differential scanning calorimetry, Phys. Rev. B61 15061-15076. 23. Naumis G.G., (2000) Contribution of floppy modes to the heat capacity jump and fragility in chalcogenide glasses Phys. Rev. B61 R9205-R9208.

170

THE DALTON-MAXWELL-PAULING RECIPE FOR WINDOW GLASS

RICHARD KERNER Laboratoire de Gravitation et Cosmologie Relativistes, Tour 22-12, 4-ème étage, Boîte 142 Université Pierre-et-Marie-Curie, 4 Place Jussieu 75252 - Paris Cedex 05, France

INTRODUCTION The aim of this article is to merge together two methods of understanding and modeling the structural and thermodynamical properties of covalent glasses: the constraint theory describing glasses in terms of rigidity transition [1,2], and the more recently created local

agglomeration model [3,4,10], on the particular example of silica-based glasses with alkaline oxydes as modifiers. Although these ternary glasses, with the molecular weight composition are the oldest and most widely used, [5,6] their local mediumrange structure is very complicated, and their modeling is much more difficult than what is to be achieved in the case of simple network glasses such as [7-9]. Nevertheless, the knowledge of local configurations which are stable enough at glass transition temperature, makes the task much easier. We shall also argue that the two approaches are complementary because they serve to probe the glass structure on different scales. The constraint theory gives its best predictions when applied to the atomic level, when the network is treated as a ball-and-stick structure, and taking into account all atoms and bonds present in the network, whereas the agglomeration theory, whose equations express the principle of minimal local fluctuations (i.e. maximal homogeneity), takes into account the characteristics averaged over clusters with a well defined medium-range structure (with chains and rings involving up to 25 to 30 atoms). We shall show that many predictions of the two models often coincide, their combination offering a better insight into the process of glass formation. New predictions can be given, in particular, concerning the glass transition temperature as a function of chemical composition (and local connectivity of the random network), the variation of thermal capacity during the glass transition, and many others. It seems very clear by now that in spite of high degree of randomness, glasses are characterized by well-defined local structures, such as stiff -tetrahedra and four-, five- or six-membered rings in silicate glasses, or stiff tripods and three-membered boroxol rings in borate glasses, etc.

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

171

The shape and energetics of these highly organized structures depend on the chemistry of the melt about to undergo the glass transition at the temperature

In silicate glasses, the alkaline or lime and alumina modifiers , must occupy certain topologically well-defined positions in the oxygen matrix pre-existing in the pure network, because no dangling oxygens can be observed later on in the resulting glass structure. This leads to the formation of stable building blocks containing these additives, which coexist with the silicon-centered tetrahedra. For example, the molecules split and form two truncated Si-centered tetrahedra, whereas the CaO molecules form stable bridges linking two Si-centered tetrahedra together, giving rise to stable six-coordinate blocks

With these elementary blocks, whose abundance is controlled by molar concentrations according to Dalton’s principles, we can perform computations based on two quite different

and independent approaches, which are the constraint theory introduced by J.C. Phillips and M.F. Thorpe, and the stochastic agglomeration model (R. Kerner, M. Micoulaut, R.A. Barrio, G.G. Naumis). The use of the constraint theory based on Maxwell’s principles requires a very careful analysis of the character of bonds formed in the network during glass transition. This is achieved with the help of Pauling’s theory of resonant chemical bonds, which attributes the (1 /n)-th part of chemical valence to each of the closest neighbors of a given covalent atom. The agglomeration model is able to predict the dependence of on the modifier concentrations not only close to i.e. pure but also at higher concentrations if one takes into account the ring-forming ability. The homogeneity of ring distribution in silica glasses become then an important factor for determining the optimal composition of the melt, combining the highest degree of homogeneity with minimal stress and possibly the lowest glass transition temperature. With the help of these principles, we are also able to predict optimal chemical compositions of other silica-based glasses, which combine the best proximity to the stiffness transition with maximal homogeneity. The article is organized as follows. First, we expose a simple version of the agglomeration model applied to binary and ternary glasses, and we show how this model is able to predict the correct behavior of the glass transition temperature as a function of modifiers’ concentration. Next, we recall the definition of spinodal and binodal curves that define the immiscibility region for binary liquids, and compare it with the immiscibility dome observed in the mixtures. We then draw certain conclusions concerning the average size of clusters agglomerating during glass transition. Finally, we show how the combination of these ideas with the application of the constraint theory to silicates makes it possible to derive the best compositions of silica-based window glasses.

THE AGGLOMERATION MODEL AND

IN SILICATE GLASSES

Let us first recall the simplest version of the agglomeration model set forth in Refs. [11,12,15,16]. We assume that two different stable building blocks are present in the liquid melt about to undergo glass transition. These elementary entities can be just atoms (e.g. Se and Ge, or Se and As in the chalcogenide glasses, or more complicated molecules or clusters, like e.g. the tripods and 4-coordinate boron atoms with a new oxygen bond appended by a -ion. The simple mathematical model of a network forming process during the glass transition consists in building up the probabilities of creation of new bonds between given stable build172

Figure 1.

in

and

glasses.

ing blocks, or microclusters, as the case may be. These probabilities contain the products of respective concentrations of these entities in the melt, the statistical factors acknowlidging for

the number of distinct ways in which a given new bond can be created (which in the simplest case is just the product of free valencies displayed by the building blocks under consideration), and finally, the factor which takes into account the energy barrier, or chemical potential relative to this bond creation, entering the corresponding Boltzmann factor, and perhaps an extra factor related to the kinetics of the bond creation process (e.g. when the cross-sections

for the encounters of various pairs of building blocks are very different). Let us analyze the first step of agglomeration process with two different elementary building blocks present, and at very low concentration of molecules. This should describe the decrease in the glass transition temperature via addition of some amount of to pure quartz This decrease is quite spectacular, because it is sufficient to add no more than 4 to 5% (molar) of in order to make the glass transition temperature fall down from 1480 K to about 1000 K (see Fig. 1). It turns out that there is no point in adding more pure because one enters the immiscibility domain, which extends up to the concentrations of about 15% of molar Even if the glass transitions occur in this range of concentrations, the result is an opaque material in which a very high degree of inhomogeneity is observed, due to the spontaneously formed micro-domains of two disctinct types: sodium-rich and sodium-poor. This problem will be analyzed in detail in the next section. After leaving the immiscibility region we arrive at the concentrations of which are higher than 15 – 16%. The resulting glass is again homogeneous, but it becomes chemically unstable, i.e. is very prone to water attack.. This is why some amount of CaO must be added to the melt, making increase again, but not very much; the positive slope of versus the CaO concentration y at x ~ 15% is much smaller than the initial negative slope of versus the concentration x at low values of x. The idea of maximal homogeneity of the network suggests that during the formation of the random glassy network from liquid melt all local fluctuations are close to the minimum. This enables us to put forth the equations expressing this principle and relate to local concentrations x and y. Let us start with binary mixture containing only two elementary

173

Figure 2. The elementary building blocks A and B and the doublets created by pairing AA, AB and BB.

building blocks : the four-coordinate A and the three-coordinate B with one dangling bond saturated with a Na atom, as represented in Fig. 1. If we denote by and the probability of finding one of these local configurations in the melt about to freeze and undergo the glass transition, at low concentration we can use the relations

Let us evaluate now the probabilities of creating the doublets, i.e. the new bonds between the elementary blocks. Supposing that the process is not very far from thermal equilibrium (slow quenching rate), we can use Boltzmann factors with respective energy barriers and The differences between these energies acknowledge the change in the rigidity of the respective tetrahedrons and the repulsion between the ions. The statistical weights attributed to each agglomeration are computed in an obvious way, taking into account the valency of each block. The common denominator Q normalizes the sum of the probabilities to 1. We get then:

where the normalizer Q is given by

Neglecting the BB-pairs at very low concentrations of evaluated on the doublets (denoted by reads as follows :

the local concentration

The agglomeration theory [3,4] provides us with a simple criterion for defining the glass transition, identifying it with minimal local fluctuations during the bonding of elementary building blocks creating amorphous random network. Close to x = 0 this leads to the following equation:

174

Explicitly, this gives the following equation:

Reducing this expression to the common denominator, in which we also neglect the term we obtain the following approximation for (equivalent to x ~ 0) :

containing

Besides the obvious solution state, given by

In the limit when

and

there is another one, which we can identify with glassy

the glass transition temperature of pure amorphous

we must have which leads to the relation

which fixes the difference between the energy barriers corresponding to the creation of an AA or an AB bond. It is equal to We may also evaluate the slope of the function describing the dependence of the glass

transition temperature on the concentration x. We should keep in mind that in the linear approximation, when the variables and x are proportional up to a multiplicative constant, the slope at the origin is given by the same universal expression, because for any continuous function one has

According to the formula (its derivation can be found in the refs. [3,4,10]), we obtain with respect to T, inversing it and going to the limit

when deriving

which has a negative value, as it should be, in agreement with the experiment. Inserting the

value of

we find that in the linear approximation,

the glass transition temperature goes down to 760° C = 1040 K when the concentration is only 5% high, which means that the glass transition temperature is reduced by as much as 105 K with the addition of one molar percent of which is in good agreement with the experimental data [19,20]. Further extrapolation of this linearized formula is of no use anyway, because when the concentration of gets close to 5%, the resulting glass is of no use, presenting strong opaqueness due to local phase separation - on the scale of many microns - into sodium-rich 175

and sodium-poor domains. We can now make an estimation of another slope, with respect to the variable y, the molar concentration of CaO (equal to in the limit y ~ 0,) at a given constant (but quite low, i.e. 0.05) value of x. In order to do this, we shall use again the “magic formula” from [3], which tells us that the initial slope of the function versus the concentration of a modifier in covalent glasses is given by

where is the concentration of the building blocks of the modifier with coordination number m´, and m is the coordination number of the dominant glass-former. In our case, m´ = 6, because as we know, each molecule of CaO creates a strong and rigid bond keeping together two tetrahedra, resulting in a six-fold C-unit, whereas m should be replaced by the average coordination number of a mixture of A and B blocks, defined (at and 1 as

Therefore, recalling that

we can write

Leaving only terms linear in x in Taylor’s expansion of this expression yields

so that after integration we get

Even with this very rough approximation, whose validity does not go beyond 10% concentration of the total of both modifiers, the basic behavior comes out right. In the very beginning, at x = 0 and y = 0, the glass transition temperature decreases with the addition of and increases even more considerably while a small amount of CaO is added. This is why the glass makers start always with adding more than 10% of first, and only then add some amount of CaO. Moreover, the positive slope with respect to y, which is very high at x = 0, y = 0, because after inserting the numerical values, we have

becomes much smaller after adding a relatively small amount of

which means that at x = 0.05 the glass transition temperature has become

1060 K instead

of 1480 K at x = 0, and the increase in temperature due to the addition of certain amount of CaO is now very gentle:

176

However, the above formulae are no more valid on the other side of the immiscibility region, where the concentration of is above 15%, and the glass transition temperature is well below 800 K. The slope of the curve (x) is now much lower than before, i.e. about 10 K per 1% of variation of x. It is not difficult to spot the source of eventual discrepancies between the linear approximation and the experimental curve. At the glass transition temperature close to 700 – 800 K, which is the case at higher concentrations on the sodium-rich side of the immiscibility gap, the approximation that local agglomeration in melt can be described by means of

freely floating tetrahedra can not be maintained. The real situation is more complex then, i.e. the melt certainly contains a lot of clusters with various sizes. As soon as bigger stable local configurations do appear, the most important growth mode is through the creation of rings, most of which are four-, five and six-coordinate, (this is different from what is observed in most of the crystalline phases of where the six-fold rings dominate) The creation of all these rings can be modeled quite successfully if doublets and triplets of tetrahedra become elementary building blocks, instead of simple Si-centered tetrahedra or NaO-modified tripods. The creation of rings has a much higher statistical weight than a single bond creation producing local chains of corner-sharing tetrahedra, although this mode of agglomeration should also be taken into account. Also the energies associated with the creation of various rings are now different, and should be adapted in order to give a good fit with the experimental curve; the nature and value of potential energies stored in rings have been discussed in the papers by F.L. Galeener [13] and others [14]. A model taking into account the ring-forming tendency has been developed recently, (see [17]), and it gives reasonable predictions for the glass transition temperature in ternary silica-soda-lime glass. Because of the complexity, no analytical formulae are available. We shall give here only the example how the influence of the CaO-additive can be evaluated with new statistical weights. If the ring-forming process is dominant, with three elementary building blocks A , B and C, the last one representing the O · Ca · O bridge with two Si-centered tetrahedra on both sides, we should replace the valencies m and m´ by statistical factors defining the ring-forming ability of each of the above building blocks. It is given by the corresponding binomial coefficients, and respectively. Therefore, the ring-forming ability of tetrahedron is equal to 6, the ringforming ability of a truncated (3-fold) unit with one NaO group is equal to 3, whereas the 6-forld unit C can give rise to 15 rings. For the relatively low concentrations of CaO we can still maintain the approximate equality Considering the binary mixture as a uniform substrate, and the CaO additive as modifier, we can introduce the average ring-forming ability of the binary as Then, inserting this value and the value 15 (the ring-forming ability of calcium-centered sixfold unit C) into the same universal formula for the slope of but with replaced now by we obtain the following result:

which leads to the following linear approximation fr the sodium-rich domain:

177

which is fairly well confirmed by the experimental data. IMMISCIBILITY, SPINODAL AND THE AVERAGE CLUSTER SIZE The existence of the “immiscibility dome” for the binary is well known by all glass-makers. It represents a convex parabola-shaped curve in the (x, T) -plane cutting the x-axis at x = 0 and at 0.2, attaining its maximum at about 1200K at x between 0.06 and 0.08. When the glass transition temperature goes below this curve (as it happens close to the low concentrations of at x as low as 4 – 5%), it becomes extremely difficult to obtain a homogeneous glassy network; one observes an opaque material in which partial demixtion

took place, producing microscopic inhomogeneous regions of two kinds, sodium-rich and sodium-poor ones. Let us demonstrate how the shape of this curve, called a spinodal, can be derived by means of simple thermodynamical arguments, and how its parameters (i.e. its width and the position of the maximum) give a clue about the average cluster size in the liquid melt close to the glass transition temperature. From our previous analysis it is clear that in a binary melt, at low concentrations of only two main kinds of clusters are important, whatever their average size might be: the agglomerates containing a NaO unit attached to one of the Si-centered

tetrahedra, and alternatively, the agglomerates composed of pure

this because at low

values of x, the probability of two NaO groups coming together is extremely low. This is why we shall pursue our analysis with this simplifying assumption. Assuming that the potential energy of interaction between two units (denoted symbolically by A and B) coming close enough to each other, is also proportional to the number of dangling bonds, we can evaluate its contribution to the internal energy of the hot liquid as follows. Let the average number of free bonds available in an A-type unit be and respectively, for a B-type unit, we shall suppose that the potential energies of interacting elementary agglomerates (but without a stable bond forming) are and These are not the same as the previously introduced activation energies and corresponding to bond formation during the glass transition; they might be different, and they are counted with the opposite sign. Then, if we denote by (1 – c) the concentration of the A-type clusters, and by c the concentration of the B-type clusters, the potential energy stored in the bonds created around and average elementary cluster is:

so that the total potential energy per building block in the mix is:

where the factor is put in front in order to count each bond only once. Now, in the case of pure A or pure B configurations present, we would have found

Therefore, the extra potential energy resulting from mixing is readily evaluated to be

178

The extra entropy resulting from mixing is given by the usual expression

so that the Gibbs’ free energy variation becomes

with

According to the laws of thermodynamics, which we believe to be still applicable here, the system is stable against the separation into two different phases if the following inequality is satisfied:

where denotes the usual chemical potential function. More explicitly, we have

If

and if T is above the critical value, there are two solutions to the equation

defining two inflexion points of the curve (c), called spinodal points, which we shall denote by and These points are found an equal distance from the central point due to the obvious symmetry of the above equation. The equation which gives the condition of vanishing first derivative of is also symmetric with respect to the substitution

When the temperature T is above the critical temperature it has also two solutions, called binodal points, which we shall denote by and With this in mind it is easy to draw the curves for a given temperature T (Fig. 3). The pairs of binodal and spinodal points, and respectively, enable us to draw two curves which give the implicit relation between these solutions and the corresponding values of temperature T. These curves are called the respectively the binodal and the spinodal; they meet at the critical temperature attained when (Fig. 3). According to the first principles of thermodynamics, in the range between the spinodal and binodal curves the system is stable against the phase separation, whereas inside the spinodal curve it becomes unstable, and local phase separation is preferred. The “immiscibility dome” for the binary as measured by different authors, is displayed in Fig. 4. below. It is easy to see that it is not perfectly symmetric around the maximum which is attained for ythe value 7 – 8%; upon extrapolation towards higher values of the molar concentration x we see that the extreme value of the spinodal curve (corresponding to c = 1 is attained at x slightly more than 20%. This provides us with another clue concerning the average size of entities which should serve as a basis for the calculus of the spinodal, and at the same time, as the elementary 179

Figure 3. a) The shapes of

b) The binodal and spinodal curves.

and meta-stable building blocks which to be used in the agglomeration theory calculations. Quite obviously, the hot melt can not be arbitrarily divided into just two types of clusters, with and without sodium ions attached, although at very high temperatures and with a few molar percent of the use of just two small entities in the agglomeration model, the Sicentered tetrahedra (building block A) and the truncated tetrahedra with one ion and one non-bonding oxygen (building block B) gave satisfactory predictions for the glass transition temperature behavior. However, using the same entities in the reconstruction of the spinodal leads to the obvious discrepancy with the experiment; indeed, when we set and the spinodal curve would attain its maximum at which corresponds to the molar concentration of whereas the maximum possible value c = 1 would correspond to x = 1/3 (which represents the molar rate of in the B-type building blocks, one half of an molecule versus one molecule). But if we assume that most of the elementary building blocks are represented by doublets (of AA or AB type), then the 100% concentration of AB doublets (one half of an molecule molecules will correspond to x = 0.2, which is much closer to the observed immiscibility dome; moreover, its maximum will occur at c = 0.5, which corresdponds now to x = 11.11%, which is also close to the reality.

We can go furher and postulate the presence of bigger elementary blocks, which will move the maximum even closer to the observed value of about 7 – 8%. The real curve is certainly the result of a superposition of several spinodal curves corresponding to the clusters of various size (see Fig. 4). The best fit can be obtained if we assume that the only two species of clusters present are just triplets of the type AAA and AAB (together with ABA, with proper statistical weights), in which case, setting (1 – c) equal to the concentration of the AAA-triplets, the maximum of the immiscibility curve at c = 0.5 will correspond to x =1/13 7.7% of molar It is worth noticing that the presence of doublets and triplets of Si-centers tetrahedra is enough to acknowledge the formation of rings of all sizes up to the 6-fold ones (6 Si atoms

connected via oxygen bonds).

180

Figure 4. The immiscibility dome, after [22].

RIGIDITY, CONNECTIVITY AND THE RING DISTRIBUTION

As argued in previous section, the knowledge of shapes and chemical properties of the elementary configurations that can be considered as the most stable ones at a given temperature is crucial for the subsequent construction of our model. Quite fortunately, the based glass, with the addition of certain amount (from 12 to 16%) of and about 7 to 14% of CaO, displays two well established tendencies:

* The addition of results in creation of saturated bonds, which amounts to local breaking of oxygen bonds in the random network. It is to be stressed that each molecule of creates two such bonds, which can be no longer connected to other bonds in the network, thus decreasing the average coordination number . Each tetrahedron is transformed into a tripod (or truncated tetrahedron) with coordination number = 3 :

** The addition of CaO results in creation of stable and practically undestructible units containing two tetrahedra, connected by a –O · Ca · O– bridge. Each of these new elementary building blocks can be considered as a stable unit,

with coordination number 6.

It should be stressed here that in both modified blocks the molecular concentration of the modifier (be it or CaO) is exactly the same, and equal to one molecule creating two modified blocks of and one molecule of CaO also creating a stable bridge between two such entities. Therefore, if we denote the elementary pure tetrahedron by A, the tripod with one saturated bond as B, and the six-coordinate doublet linked together by the O – Ca – O bridge by C, we can easily define the relation between x, y and : where and denote the respective probabilities to pick up an A, B or C building block in the network, with

181

The inverse relations read:

Obviously, if no other local configurations can be found in the network, the maximal molar concentration of each modifier, and CaO is equal to : which is more than enough in order to describe real silicate glasses in which the part of is always higher than 70%. To be more precise, consider a very huge sample composed of N building blocks. It contains, by definition,

A – blocks,

B – blocks and

Each B-cluster contains one half of an

C – blocks

molecule, while each C-cluster contains one

CaO molecule.

On the other hand, an A-block contains the equivalent of only one molecule a B-block contains the equivalent of one molecule and a half (one and half of finally, a C-block contains the equivalent of three molecules (two and one CaO). Therefore, the molar concentrations of (denoted by x) and of CaO (denoted by y) are computed respectively as:

Inserting common factor N and taking into account that

simplifying the above expressions by the we obtain

It is worthwile to notice that in the limit of very low concentrations, when both and tend to zero, x behaves as and y as whereas when is close to 1, one has and when is close to 1, one has We need two independent equations in order to fix a particular value for the molar concentrations x, y that is observed in the most currently used silicate glasses. We derive the first equation from the constraint theory, which sets the rigidity threshold at the average coordination number equal to 2.4 [1,2]. We shall evaluate this parameter on the atomic scale, counting all atoms,including the one-valenced ions. Although one may find in current litterature the discussions of radial distribution func-

tions obtained with various techniques such as the Mössbauer spectroscopy or Nuclear Magnetic Resonance, which tend to acknowledge the idea that sodium atoms are often surrounded by four or more oxygen atoms, we shall follow Pauling’s idea of resonating bonds, and base our counting of coordination number on chemical valence only. If we consider our random network as a conglomerate of single Si, O, Na and Ca atoms, then the computation of the average coordination number should be done in the following way. In terms of two independent probabilities and we can evaluate the average atomic coordination after observing what follows:

- an A block, found with the probability contains three atoms, one 4coordinate atom (Si) and two (four halves !) 2-coordinate (O) atoms;

- a B block, appearing with the probability

accounts for four atoms and a half, of

which one (Si) four-coordinate, one (Na) one-coordinate, and five halves of two-coordinate O atoms; 182

- finally, a C block (with the probability pC) contains two four-coordinate atoms (2 × Si), and six two-coordinate atoms (five O and one Ca). This enables us to write the following expression for the average atomic coordination number

More precisely, we acknowledge that is the total number of valencies divided by the total number of atoms in a randomly chosen huge sample, containing the total of N building blocks, out of which A-blocks, C-blocks. The total number of atoms in the sample is thus

B-blocks and

The total number of valencies (bonds that each atom is able to create with its immediate neighbors) is:

1 × 4 + 2 × 2 = 8 for an A-block;

1 × 4 + 2.5 × 2 + l × 1 = 10 for a B-block; 2 × 4 + 6 × 2 = 20 for a C-block. (therefore the total number of valencies (2 × the total number of bonds) is leading to the above expression for constraint theory predicts that the best glass-forming conditions occur for fore, we must have

The ; there-

which in terms of molar concentrations x, y reads

Now we need a second equation in order to fix the values of concentrations x and y corresponding to the “ideal” glass forming conditions. As we argued before, the principle of maximal homogeneity of the network should povide us with an independent relation. The most important point now is the choice of the right scale to which the homogeneity criterion whould be applied. The fact that with the new building blocks we are able to produce (mostly 4, 5 and 6fold) rings, which is certainly one of the most important characteristics of silicate glasses, suggest a criterion of homogeneity, which is the potential ability to form rings, evaluated on a given distribution and of elementary clusters. What is important here is the number of bonds coming out of the central unit, on which the rings will be constructed. We have the following situation: - the A - configuration can give rise to 6 different rings (because we can pick up 6 different couples out of 4 free bonds; - the B - configuration can give rise to only 3 rings;

- the C - configuration considered as a 6-coordinate unit, can participate in as many as 15 different rings (Fig. 5). If we want to keep the ring-forming ability constant on the average, and equal to 6, like in the case of pure then we get readily the equation:

183

Figure 5. The ring-forming ability of elementary clusters A,B and C.

leading to

Combining this extra relation with the previous result of fixing the following equation:

after inserting the relation

we obtain the

takes on the form:

resulting in

which corresponds to molar concentrations x = 15.79%, y = 10.52%, i.e. 73.79% of

15.79% of

and 10.52% of CaO.

It results from the above formulae that pure is not a very good glass former according to the constraint criterion; this can be seen from the fact that x = 0,y = 0 does not satisfy the above equation, which is quite obvious if we remember that for pure one has However, there is an evidence that in this case the bond-bending constraints are broken, so that the counting that leads to the rigidity threshold definition should be done in a slightly different way, suggested by P. Boolchand [7], The most important, and as a matter of fact, primary feature of a given network defining its rigidity threshold at which the glass forming tendency should be at its maximum, is the number of degrees of freedom per atom, which should be equal to 3 in a 3-dimensional network. Let us consider the ternary glass From the ring homogeneity condition we derive the relation that x and y should obey, i.e. 2x = 3y, or which corresponds to the cation ratio Na : Ca = 3 : 1, because each molecule destroys 6 rings, whereas each CaO molecule creates 9 new rings as compared with the average number of rings created around each Si-centered tetrahedron in a pure amorphous Therefore, we can write now :

The counting of constraints has to be performed carefully, taking into account the existence of non-bonding oxygen atoms, connected to the ions:

184

* Each molecule creates two non-bonding oxygens, which have only one constraint on them, and each ion has only constraint on it. ** Each silicon atom, with the coordination number (and valence) equal to 4, has of bond-stretching constraints (called constraints) and of bond-bending constraints (called constraints), the total *** Each Ca atom, as well as each bonding oxygen, have 2 constraints. Keeping this in mind, we can summarize the constraint counting per atom as follows:

In the above formula, we have substracted one non-bonding oxygen from the molecule, and added it to the molecule, because one of the non-bonding oxygens created is “bor-

rowed” from the surrounding

structure.

Then we can add up all the constraints, and compare the result with the average number of degrees of freedom per atom. Let us remind that the average number of atoms per mole is here: The two formulae combined yield the following equation:

Substituting i.e. the average number of constraints per atom equal to 3, so that the average number of remaining degrees of freedom is also equal to 3, we obtain the relation

leading to yielding the same solution again, i.e. and consequently, and 1 – x – y = 73.77%, the same solution as the previous one. This result seems satisfactory when compared with the composition of commonly used glasses : here are a few examples of most widely used chemical compositions :

Type Window glass:

71-73%

12-15%

8-10%

Bottle glass:

71-73%

12-14%

10-12%

1-4%

73.6%

16%

5.8%

3.6%

Glass for bulbs:

~

0.5-1.5% 0.5-1.5%

1%

In our computations we have neglected the presence of which is often added as a disorder-increasing agent. Some amount of is also added in order to enhance local tendencies of crystallization of with CaO; besides, its behavior is the same as that of CaO. Supposing that each molecule produces two new elementary blocks named D, whose relative abundance shall be denoted by whose ring-forming ability is as high as 36, because the three pending Si-centered tripods display now 9 oxygen bridges which must be inserted into the network, and there are 36 ways of picking up two bridges out of nine in order to start building a ring. Let us denote the molar concentration of by z, so that we have now the following constitutive relations:

185

obtain easily the following two equations:

acknowledging the fact that the average number of rings produced by building block of any

kind remains equal to six, as in the pure amorphous Si network, and

expressing the constraint theory requirement that Of course, these two equations are not enough in order to fix the values of all the three variables x, y and z. We can use them to check if they give right answers when one of the above variables is fixed at a given value. For example, if we consider the glass in which the amount of has been fixed as equal to 75%, we get :

If we consider ternary mixture with sodium and aluminium modifiers only, without any calcium, i.e. fixing and y = 0, then we get

showing that the maximal amount of is about 2%, which also does correspond to the current compositions. It is often argued that the Al atoms are four-coordinate in the silicabased glasses with alkali modifiers, because the ions stick to the Al atoms creating an extra oxygen bond. It is not difficult to check that even with such an assumption the above count will lead to a similar result, because it reinforces the ring-forming tendency provided by Al-centered groups, but at the same time it raises the number of constraints on them. ACKNOWLEDGEMENTS It is a pleasure to express my thanks to P. Boolchand, J.C. Phillips and M.F. Thorpe for invaluable help during the preparation of this article. The innumerable and enlightening discussions with R. Aldrovandi, R.A. Barrio, Ph. Jarry and M. Micoulaut are gratefully acknowledged.

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

9.

J.C.Phillips (1979) Journal of Non-Crystalline Solids, 34, 153. M.F.Thorpe (1983) Journal of Non-Crystalline Solids, 57, 355. R.Kerner(1991) Journal of Non-Crystalline Solids, 135, 155. R.Kerner (1995) Physica B 215, 267. J.Zarzycki (1979) Verres et Etat Vitreux, Ed. CNRS, Montpellier. S.R.Elliott (1990) Physics of Amorphous Materials, Longman, London. P.Boolchand etal. (1995) Journal of Non-Crystalline Solids, 182, 143; also: P.Boolchand and M.F.Thorpe (1994) Phys. Rev. B 50, 10366; also: W.Bresser and P.Boolchand (1986) Phys.Rev.Lett, 56, 2493 R.Kerner, G.G.Naumis (2000) Journal of Physics: Cond.Matter, 12 1641

P.Boolchand (2000), private communication; also: P.Boolchand and W.Bresser, Phil.Mag. B, (to appear in 2000)

10. M. Micoulaut (1998) European Journal of Physics, B 1, 277. 11. R.Kerner, D.-M. dos Santos (1988) Phys.Rev. B , 37, 3881. 12. R.Kerner (1991) Journal of Non-Crystalline Solids, 135, 155.

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13.

F.L.Galeener (1990) Journal of Non-Crystalline Solids, 123, 182.

14. 15.

R.Kerner (1995) Journal of Non-Crystalline Solids, 182, 9. R.A.Barrio, R.Kerner, M.Micoulaut and G.G.Naumis (1997) Journal of Physics : Cond. Matter, 9, 9219.

16. 17. 18.

R.Kerner, M.Micoulaut (1997) Journal of Non-Crystalline Solids, 210, 298. R.Aldrovandi, R.Barrio, Ph.Jarry and R.Kerner (2000) submitted to Phys.Rev.B. S.J.Gurman (1991) Journal of Non-Crystalline Solids, 125,191.

19. 20.

V.K.Leko et al (1977) Fizika and Khimiya Stekla, 3, 204. Ph.Jarry, private communication.

21. 22.

Z.Strnad, P.W.McMillan (1983) Physics and Chemistry of Glass, 24, 57. Yu.S.Tveryanovich et al (1998) Khimiya Stekol i Rasplavov, Ed. St.-Petersbourg University, (in Russian).

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LOCAL BONDING, PHASE STABILITY AND INTERFACE PROPERTIES OF REPLACEMENT GATE DIELECTRICS, INCLUDING SILICON OXYNITRIDE ALLOYS AND NITRIDES, AND FILM ‘AMPHOTERIC’ ELEMENTAL OXIDES AND SILICATES

G. LUCOVSKY Departments of Physics, Electrical and Computer Engineering, and Materials Science and Engineering, North Carolina State University Raleigh, NC 27695-8202, USA INTRODUCTION The scaling of silicon integrated circuits to smaller in-plane lateral dimensions of individual devices to increase speed and reduce cost through higher packing densities requires that insulating oxides and nitrides other than non-crystalline or amorphous SiO2 be incorporated into the gate stacks of field effect transistors, FETs. These replacement insulators must have dielectric constants, k, significantly larger than that of SiO2, e.g., in the range of 10 to 30, in order to increase the gate dielectric capacitance and thereby provide a sufficiently high density of channel charge under operating bias voltages to meet current drive requirements. The increased physical thickness associated with the higher values of k must reduce tunneling currents for the same effective capacitance as referenced to an SiO2 film. Finally, the alternative dielectrics must also have electronically-acrtive defect densities in the bulk, and at interfaces with crystalline Si that are respectively the same or less than those of thermally-grown SiO2 and Si-SiO2 interfaces. This paper provides a science base that underpins the technology challenges identified above. The approach is predicated on an empirical observation that any direct replacement for SiO2 must be a deposited, non-crystalline dielectric thin film. For example, the leakage current of films of non-crystalline deposited Al2O3 thin films increases by more than five orders of magnitude upon crystallization which occurs after a 900°C anneal. Since there are a large number of potential elemental and binary oxides to consider as replacement dielectrics, it is essential to develop a classification scheme that helps to narrow the process of identifying viable replacements to SiO2. This separates the potential replacement

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

189

dielectrics into two broad categories, i) silicon oxynitride and nitride films that retain a continuous random network structure and ii) the so-called high-k dielectrics that include elemental oxides, such as Al2O3 and ZrO2, as well as binary oxides, including transition metal silicates and aluminates.

THERMALLY-GROWN SiO2 – THE STANDARD THAT ALL REPLACEMENT DIELECTRICS MUST MEET

Crystalline silicon is unique among semiconductor materials since device-quality SiO2 layers can be prepared by thermal oxidation of silicon surfaces. The resulting oxides are stoichiometric to better than one part in a million, have bulk defect densities < 1016 cm–3 (or

equivalently < 1011 cm–2), and after annealing at ~900°C, they have interfaces with crystalline silicon with densities of interface traps and fixed charge, ~l-3×l0 10 cm–2, that are less than one part in 104. These densities of electrically-active defects define a standard that must be achieved by any deposited replacement gate dielectric. SiO2 films grown at temperatures between 800 and 1000°C are non-crystalline and remain so after annealing or other thermal exposures at 900-1100°C. Stoichiometric, device-quality SiO2 thin films have also been prepared by rapid thermal chemical vapor deposition, RTCVD, and by direct and remote plasma-enhanced CVD, RPECVD [1]. Deposited dielectrics require annealing at temperatures between about 900 and 1000°C to

have electrical properties similarly to thermally-grown SiO2 films grown and/or annealed at the same temperatures. The infrared absorption spectra of thermally-grown and deposited

oxides annealed at temperature of 900 to 1050°C are essentially the same as bulk SiO2 bulk glasses. Based on these comparisons with bulk glasses, the thermally-grown oxides are also chemically-ordered, continuous random networks with 4-fold coordinated Si-atoms bonded to bridging or 2-fold coordinated O-atoms. This includes a very weak bonding force at the 2-fold coordinated oxygen atoms and a large spread in the Si-O-Si bond angle from about 120 to 180 degrees [2]. There are several important aspects of the atomic scale bonding structure of SiO2 that provide a quantitative basis for a proposed classification scheme that can be applied to

potential replacement dielectrics. These are i) the bond-ionicity, Ib [3], and ii) the average bonding coordination or number of bonds per atom, Nav, and iii) the average number of bonding constraints per atom, Cav [4,5]. The simplest operational definition of bond-ionicity is the one originally introduced by Pauling [3], and based on atomic electronegativity. If X(O) is the Pauling electronegativity of oxygen, 3.44, and X(Si) is the Pauling electronegativity of silicon, 1.90, then the electronegativity difference, is 1.54. Using the definition of bond-ionicity proposed by Pauling, Ib for Si-O bonds is ~ 45 %. In the next section it is shown that the bond-ionicities for other good glass-forming oxides and chalcogenides are generally less than Ib for SiO2, i.e., the bonds are generally less ionic.

The network structure of SiO2 has been classified as a continuous random network, or crn. The randomness, which supplies the configurational entropy necessary for good glass formation, derives from two sources: i) a large spread in bond angle at the O-atom sites, and ii) a random distribution of dihedral angles. Phillips has shown that another important metric for characterizing a continuous random network is the average number of bonds/atom, Nav. using a valence force field

model, this in turn can be related to an average number of bonding constraints/atom, Cav [4]. In a series of seminal papers, Phillips has demonstrated that a simple criterion for good

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bulk glass formation is that Cav ~3. Based on bond-stretching and bond-bending valence forces, Cav is directly proportional to N av , and is given by Cav= 2.5 Nav –3. If m is the

coordination of a network atom, then the number of stretching constraints per atom is m/2, and the number of bending constraints per atom is 2m-3 for a large number of different bonding arrangements. Using this relationship for Cav, Nav for an ideal glass is given by 2.4, as for example in As2S(Se)3. One exception to the constraint counting is that if the atomic coordination is three or more, and the atom is in a planar bonding arrangement, e.g., B-atoms in B2O3, then the number of bending constraints per atom is reduced to m-1. For SiO2, Nav = 2.67, so that Cav = 3.67; however, since the bending force constant at the O-atom site is extraordinarily weak, it can be neglected and C av is then reduced to three, explaining the ease of glass formation for SiO2. Phillips has recently extended the bondcounting and bond-constraint counting relationships to silicate glasses which have networkmodifier ions [5]. This extension of constraint theory provides a basis for defining Cav for silicates with bonding structures that represent departures from the crn’s of oxide and

chalcogenide glasses. For example, at a metal atom ionic bonding site within a silicate network, such as a sodium atom in a sodium silicate glass, the effective coordination for bond counting and bond constraint counting is taken as the number of resonating bonds as defined by Pauling, or equivalently is the product of the number of actual bonding

neighbors and their effective bond-order. The success of this approach to silicates is illustrated by application to ‘ordinary’ window glass which is a ternary silicate alloy with a nominal composition of SiO2 (74%), Na2O (10 %) and CaO (16%). Based on the resonating bond model of Phillips, Nav = 0.74(2.67) + 0.1(1.33) + 0.16(2) = 2.43, so that Cav is very nearly equal to ideal value of three [7]. CLASSIFICATION SCHEME OF DIELECTRICS

The classification scheme is empirical and is based on several observations First, dielectrics with such as SiO2, B2O3, As2O3, P2O5, As2S(Se)3, are generally good glass formers (see Table I which includes values of and Ib). For example, they meet the Phillips criteria for Nav and Cav, with values Cav less than or equal to 3. There is a second glass of good glass formers that are fluorides rather than oxide or chalcogenides, and are considerably more ionic (see Table I). This class includes and Bond ionicities are greater than 70 %. Bond coordinations are essentially the same as SiO2 with the formal number of bonds/atom being 2.67. However, since the bonding is not covalent, a valence force field model is not appropriate, and the

constraint theory of Refs. 3 and 4 cannot be applied. A better way to describe the BeF2 (and ZnF2) glasses is to consider them as random closed packed ionic structures [8]. A third class of glasses with partial ionic character are the silicates or alternatively pseudo-binary alloys of SiO2 and metal oxides such as Na2O, CaO, Al2O3, ZrO2, PbO, etc. In these silicates, the metal atoms are network modifying ions whose charge is compensating by terminal negatively oxygen atoms that are covalently bonded to the Si-O network. The bonding in these silicates can then be characterized as being amphoteric with the Si-O network being the acidic or negatively charged component, and the metal ion the basic or positively charged component. Since these metals are less electronegative than silicon, the compositionally-averaged electronegativty difference will be greater than that

of SiO2, but less than that of the metal oxide. Since homogeneous silicate glasses and asdeposited thin films can be prepared with compositions up to 20-40 atomic % of the metal

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oxide, this means that

values for these silicates can be as large as 2. For example

for alloys of 70 atomic % SiO2 and 30 atomic % or ZrO2 and La2O3 are 1.74 and 1.81, respectively.

Based on i) the large differences in

and bond ionicity between the crn glasses,

exemplified by SiO2, and the rcp glasses, exemplified by BeF2, and ii) intermediate values of average bond ionicity in silicate glasses, we propose that non-crystalline oxides can then be grouped into three glasses according to their average bond ionicities: i) continuous random networks, crn’s, with and Ib < 48 %, ii) modified random covalent

networks, mrcn’s, with between 1.6 and 2.0m and Ib between about 48 and 65 %, and iii) rcp ionic structures with 2.0 and Ib > 65 % (see Table I). The boundaries between the classes are not to be construed as exact, but are representative of ionicities at which there are significant changes in bonding structure and microstructure. Additionally, and based on the bonding in the silicate alloys, there are a small number of elemental oxides, defined by electronegativities between approximately 1.6 and 2.1, that can also be characterized amphoteric or auto-compensating. For example, Table I includes several elemental oxides with for which is > 1.6, but less than 2: Al2O3 (1.83), Ga2O3 (1.63), Ta2O5, (1.94), and Nb 2 O 5 (1.84). The local atomic structure of at least two of these oxides, Al2O3 and Ga2O3, supports the validity of the classification scheme.

REPLACEMENT GATE DILECTRICS

The transition from thermally-grown SiO2 gate dielectrics to deposited dielectrics with higher static dielectric constants will likely proceed in two steps that are defined by socalled technology nodes associated with targeted values for scaling of feature sizes, see for

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example the National Technology Roadmap for Semiconductors, 1999 [9]. The first ‘generation’ of deposited gate dielectrics will be silicon oxynitride and nitride dielectrics, and the second will be metal oxides, silicates or aluminates. The boundaries between these generations of devices are marked by an equivalent oxide thickness, EOT, defined by the gate capacitance of a metal-oxide-semiconductor device structure, and a limiting value of direct tunneling current at an operating bias of about 1 volt above the threshold for field effect transistor operation. Based on a nominal criterion that the direct tunneling current be less than 1-5 A-cm–2 for high power, desk-top devices, these three generations are: i) thermally-grown SiO2 and lightly-nitrided SiO2 can be used for EOT down to ~1.6 nm, ii) silicon nitride and oxynitrides in stacked structures with nitrided Si-SiO2 interfaces can be

used for EOT down to about 1.1 nm, and iii) metal oxides, silicates and aluminates, etc., will be required for EOT of 1 nm and below. It is likely that this scaling will not extend below about EOT of ~0.6 to 0.8 nm, even though NTRS targets are set at about 0.5 nm. The limiting direct tunneling currents for devices incorporated into mobile devices, e.g., cellular phones, must be about three orders of magnitude less than for the desk top devices, and the three generations of devices are defined by proportionally larger values of EOT. In the discussion that follows the metrics for the high power devices will serve as the reference. Based on Table I, the most important boundary between these three generations of devices is the one between the crn’s, which included SiO2, and the silicon nitrides and oxynitride alloys, and the mcrn’s, which include amphoteric oxides, and silicate and aluminate alloys. The microscopic physics that is important in the technology applications is qualitatively different for the crn’s and the mcrn’s, and these differences will be addressed in more detail the remainder of this paper. LIMITATIONS IMPOSED BY NETWORK CONSTRAINTS IN CRN’s

The limitations on the performance and reliability of field effect transistors with SiO2, and Si nitride and oxynitride gate dielectrics with EOT < 2.5 nm have been shown to derive primarily from bonding defects at the Si-dielectric interfaces. These defects have their origin in mechanical bonding constraints. A recently published paper has addressed this limitation, and the results of that paper are summarized below [10], Several studies have

addressed chemical bonding arrangements at thermally annealed Si-SiO2 interfaces and have shown that optimized interfaces display transition regions ~0.3 nm thick with excess sub-oxide bonding arrangements different from those expected at abrupt Si-SiO2 metallurgical interfaces. These studies include: i) X-ray photoelectron spectroscopy, XPS,

on ultra-thin Si-SiO2 interfaces using monochromatic synchrotron radiation [11], ii) in-situ Auger electron spectroscopy, AES [12] and iii) in-situ Fourier transform infra red, FTIR [13]. From the perspective of constraint theory, these transition regions are at an interface between a ‘rigid’ crystalline Si material and an effectively ‘floppy’ amorphous material, SiO2, with a continuous random network structure [14]. Applications of constraint theory to non-crystalline solids have focused primarily on bulk glasses [4], thin films [15] and Si-SiO2/Si3N4 interfaces [16]. Recently, both theory [14] and experiment [17] have identified a new aspect of constraint theory by demonstrating that ‘rigid' or over-constrained’ and ‘floppy' or under-strained’ microstructural regions in an alloy glass are separated by monolayer scale, self-organized interface layers that are ‘over-constrained’ with respect to bonding coordination, yet not ‘mechanically-strained’. Experiments cited in Refs. 11-13, combined with the theory of Refs. 14 and 18, suggest that the interfacial transition regions in advanced Si FET gate

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stacks are intrinsic, and result in part from entropy effects. These transition regions constitute a basic limitation for the aggressive-scaling of CMOS silicon devices, as well as other semiconductor devices with similar ‘steps’ in interfacial bonding coordination.

For example, the data in Figs. l(a) and (b) have been explained in terms of model based on XPS data of Ref. 11 (see Figs. 2(a) and (b)). Analysis of the interfacial features in XPS data of Figs. 2(a) and (b) labeled Si l+ , Si2+ and Si3+, yields concentrations of Si-atoms in

Figure 1. Current density as function of gate voltage in (a) the Fowler Nordheim, and (b) the direct tunneling regimes for devices with plasma-grown interface layers, and plasma-deposited SiO2 bulk dielectric films. These devices have been subjected to a 30 s 900°C rapid thermal anneal in a He ambient.

suboxide bonding environments in excess of those that are required for an ideal and abrupt interface. The excess silicon atom concentrations have been equated to an average composition of SiO, thereby converting thye Si areal densities into equivalent transition region widths of 0.27 nm for the non-nitrided interface and 0.35 nm for the nitrided

interface [19]. Combining these XPS data with differences in optical SHG for interfaces with and without interfacial nitridation [20], band models have been generated for the interfacial transition

Fig. 8. Synchrotron XPS data for remote plasma processed interfaces without nitridation in (a) and with monolayer interface nitridation in (b). These interfaces have been subjected to a 30 s 900°C rapid thermal anneal in a He ambient.

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regions (see Fig. 3(a)). After completing the stacked dielectrics with films of SiO2 subject to the constraint of maintaining the same values of EOT, calculations of direct tunneling for substrate accumulation in Fig 3(b) have yielded the reductions in the tunneling current density for devices with nitrided interfaces essentially the same as shown in Fig. 3(b) [19]. It is important to note that the direct tunneling current is higher for an optimized interface with minimal suboxide bonding than for an abrupt interface, and that insertion of one

monolayer of nitrogen atoms at the Si(100) interface by remote plasma processing reduces

Figure 3. (a) Interfacial band structure for tunneling calculations for plasma processed and nitrided plasma

processed interfaces as derived from XPS data of Fig. 2. (b) Calculated direct tunneling for devices with EOT = 2 nm, for the interfaces of Fig. (a) and ideal Si-SiO2 interfaces with no suboxide bonding.

the direct tunneling current to approximately the same value as for an ideal abrupt Si-SiO2 interface. Constraint theory has also been invoked at semiconductor dielectric interfaces to explain differences in the defect densities between Si-Si3N4 and Si-SiO2 interfaces [16,21]. In this application, mechanical bonding constraints at the interface have been characterized in terms of the average number of bonds per atom in the interfacial region. Following Ref. 15, the interfacial bonding structure is defined by 0.5 molecular layers of Si (0.5 atoms and two bonds), and 1.5 molecular layers of the dielectric film (SiO2 or Si 3 N 4 . Interface nitridation has been taken into account by inserting one atomic layer of nitrogen between the Si substrate and SiO2 layer. The Si-SiO2 interface is used as a reference interface and is characterized by an average number of interfacial bonds/atom, Nav* = 2.86. Based on the model interface bonding model described above, Nav equals 3.47 for a Si-Si3N4 interface,

and 2.89 for a monolayer nitrided Si-SiO2 interface. As shown in Fig. 4, the concentration

Figure 4. Normalized defect density at interfaces as a function of ‘over-coordination’.

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of defects relative to the Si- SiO2 interface scales as (Nav - Nav*)2. This scaling is based on an empirical observation that bonding stretching force constants are significantly stronger than bonding bending force constants. It is therefore assumed that increased bonding coordination contributes predominantly to bond angle strain, so that (1) Strain energy, Es, is proportional to and if it assumed that the bond defect concentration [D] is proportional to strain energy, then the following scaling relationships

apply: (2)

Based this scaling, the defect concentration at an Si-Si3N4 interface is expected to be about three orders of magnitude higher than at a monolayer nitrided Si-SiO2 interface, consistent with experimental results [16,21], Constraint theory and the associated scaling neither predict defect concentrations, nor identify the defect bonding arrangements. Instead, they yield scaling relationships that provide a useful guideline for comparisons of the type discussed above. In the spirit of the scaling arguments, a value of interfacial Nav ~ 3 has been proposed in Ref. 16 as a demarcation between device-quality and increasing

defective interfaces, paralleling a similar criterion applied to defects in homogeneous amorphous thin films [15]. Scaling theory also provides an explanation for differences in fixed charge at internal

dielectric interfaces. In this application, the appropriate scaling variable is the difference in

the average number of bonds per atom, on either side of the internal dielectric interface. This model will now be applied to the C-V data of Figs. 5(a) and 5(a) for interfaces between interfacial SiO2 and Si nitride and oxynitride. The negative shifts of the

Figure 5. Capacitance-voltage, C-V, curves for (a) PMOS and (b) NMOS devices (EOT ~ 2 nm) with remote plasma oxidized interfaces (~0.6 nm) and remote plasma deposited oxide, oxynitride and nitride bulk dielectrics.

shifts of the C-V curves from the oxide, to the oxynitride alloy and nitride, correspond respectively to fixed charge levels. Qf , of ~2×l0 11 cm –2 and 7.5x1011 cm–2, respectively. The relative shifts in the C-V traces are essentially the same for NMOS and PMOS

capacitors indicating that they derive from fixed charge. Since there is no detectable fixed charge associated with the remote plasma oxidized Si-SiO2 interface, the charge must reside at the internal dielectric interface. These data are consistent with the scaling

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from constraint theory. The ratio of the defect concentrations, ~ 3.8, is approximately equal to the ratio in the values of

at these two internal dielectric interfaces, [4,10].

LIMITATIONS IMPOSED BY BOND-IONICITY IN MCRN DIELECTRICS

The silicon oxynitride and nitride continuous random network insulators of the last section have dielectric constants that are at most two times that of SiO2. Coupled with the necessity for nitrided SiO2 interface layers, the minimum EOT that can be obtained using silicon oyxnitride or nitride stacked gate dielectrics is about 1.1-1.2 nm [1]. Extension to smaller EOT requires dielectrics with larger values of k, preferably in the range of 15 to 25. This has created interest in both elemental oxides such as TiO2, Ta2O5, Al2O3, ZrO2 and HfO2, and silicate alloys, initial of Zr and Hf, and more recently extending to Y and La. This section of the paper addresses three issues relative to these potential alternative gate dielectric materials. Stability Against Chemical Phase Separation

An important issue in the replacement of thermally-grown SiO2 by an alternative deposited gate dielectric material is process integration. This aspect of manufacturing science and technology addresses the entire process sequence by which an integrated circuit is fabricated. The thermal stability of deposited silicon oxynitride and nitride dielectrics is sufficient for direct substitutions to be made. In particular, post deposition processing temperatures up to about 1050-1100°C can be utilized, and this is sufficient for dopant activation in both polycrystalline Si gate electrodes and the source and drain contacts of the PMOS and NMOS field effect transistors. This is not generally the case for deposited dielectrics. Many elemental oxides such as TiO2, ZrO2 and HfO 2 are either

polycrystalline upon deposition, or crystallize and relative low post-deposition temperatures. This places serious limitations on device structure and process integration, e.g., processing sequences may require all high temperature steps done prior to deposition of the gate dielectric, and then restrict the gate electrode to deposited thin film metals rather than doped polycrystalline Si. Our group has studied post deposition stability of remote plasma deposited dielectrics by three different techniques, Fourier transformation infrared spectroscopy, FTIR, X-ray diffraction, XRD, and high resolution transmission electron microscopy, HRTEM, lattice

Figure 6. FTIR spectra of (a) A12O3 and (b) Ta2O5 films as-deposited at 300C and after 30 s anneals in Ar at the temperatures indicated. The feature at 1100 cm–1 in (a) is a substrate artifact.

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imaging. Figures 6(a) and (b) illustrate FTIR results for A12O3 and Ta2O5 and as a function

annealing temperature. The Al2O3 films show evidence for crystallization at an annealing temperature of 900°C, and the Ta2O5 films show evidence for the onset of of crystallization at about 800°C. Figure 7(a) displays absorbance for Zr silicate films ([SiO2]x[ZrO2]1-x) prepared by remote plasma enhanced chemical vapor deposition [22,23]. Spectra of as-deposited films and those annealed at temperatures up to 800°C are essentially the same, whereas spectra of films annealed at 900°C are markedly different. After a 900°C anneal, changes in these spectra indicate a chemical phase separation into i) a non-crystalline low Zr-content silicate

alloy with ~1-2 atomic % Zr, or simply SiO2, and ii) non-crystalline (x~0.1 and ~0.23) or crystalline ZrO2 (x~0.5) [23]. Figures 7(b) and (c) compare the XRD and FTIR results for a film with x ~ 0.5, the compound silicate composition, ZrSiO4.

Figure 7. (a) Absorbance versus wavenumber for

three Zr silicate alloys with different ratios of of Zr:Si, as-deposited and after a 30 second, 900°C anneal in Ar. (b) XRD and (c) FTIR spectra of an alloy with x~0.5, for as-deposited and annealed films. The changes in XRD and FTIR at 900°C marking the onset of crystallization are evident.

Interface Properties

There are two aspects of interface properties that are important in high-k replacement

gate dielectrics: i) conduction band offset energies, Ecb, that determine the barrier for direct tunneling out of the Si substrate, and ii) interfacial defects in the form of traps, Dit, or fixed charge, Qf. The band offset energies for high-k oxides have estimated by Robertson [24], and are generally reduced by about 1 volt or more with respect to the 3.15 eV value of ECb the for the Si-SiO2 interface. The calculated values in Ref. 24 for Si3N4 and Al2O3, are

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larger by out 0.3 to 0.8 eV than those obtained from XPS studies [25]. The very low values of Ebc calculated for TiO2 and Ta2O5, ~0.0 and 0.36 eV, respectively, and confirmed by experiment, make these dielectrics unsuitable for replacement gate dielectrics. The low band offset energies of all of the transition metal elementary and binary oxides result from two effects, increased ionicity with lowers the valence band energies, and reduced energy gaps, which are associated with the d-state parentage of their conduction bands. The values of Ecb for the Zr and Hf silicates are about 1.5 eV, and it remains to be determined if these are sufficient for tunneling current reductions. The tunneling transmission probability is proportional to ~ exp ( -2 a tphys [Ebc m e * ] 0.5 ), where a is a constant, t phys is the physical thickness of the high-k replacement dielectric, ~[EOT]x[k/k(SiO2)], and me* is the tunneling mass of electrons. Reductions in Ebc will clearly offset gains in tphys, and this remains an important issue in gate dielectric replacement. A second issue relates to fixed positive charge. This interfacial charge which can have two different origins. For the stacked gate dielectrics with silicon oxynitride or nitride gate

Figure. 8 C-V characteristics for PMOS and NMOS capacitors with Al 2 O 3 gate dielectrics.

dielectrics this charge results from mechanical bonding constraints at the dielectric

interfaces. For the dielectrics with mrcn structures, the charge exists because of increased ionic bonding. C-V studies of NMOS and PMOS capacitors with plasma deposited Ta2O5, A12O3 and ZrO2-SiO2 silicate alloy high-k gate dielectrics demonstrated fixed charge at the Si-high-k gate dielectric interface. These dielectric films were formed by remote plasma processing. Post-deposition analyses indicated that detectable interfacial SiO2 layers were not formed during the processing. Fixed charge levels were determined from comparisons of flat band voltages for devices with different values of EOT, generally from 1 to 4 nm. For example, the C-V traces in Fig. 8 show positive flat band voltage shifts which increase with decreasing capacitance, or equivalently with increasing dielectric thickness. These charge levels are of order 1-5×1012 cm–2. The charge was positive for the test capacitors with Ta2O5 and ZrO2-SiO2 silicate alloy dielectrics, but negative for those with A12O3. Table II bond ionicity, ionic fraction, and fixed charge levels for devices with A12O3, Ta2O5 and Zr-silicate alloy (x ~ 0.3) gate dielectrics. The three dielectrics in Table 1 can be characterized as modified covalent random networks including a charged network and discrete ions. For example, a structural formula for A12O3 is given by 2 A12O3 =3 (AlO4/2)1– + A13+ [26]. The term (A1O4/2)1– represents a network structure similar to SiO2, wherein Al1– is isoelectronic with Si°. The A13+ ions occupy octahedral ‘interstitial’ sites of the network, and form three resonating or dative

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donor-acceptor pair bonds with bridging oxygen atoms of that network. The average

number of bonds/atom is 3. Similarly based on FTIR, Raman and EXAFS, the Ta atoms in Ta2O5, assumed to have a formal charge of +1, are in octahedral bonding sites. The octahedra are corner- and edge-connected, respectively in a network through 2-fold and 3-

fold coordinated O atoms. These O-atoms have a net negative charge that balances the positive charge of the Ta-atoms. Finally, the (ZrO2)×(SiO2)1-x alloys for values of x < 0.5

are in a prototypical silicate structure in which Zr4+ ions are nearest-neighbors to four or

more terminal and negatively-charged oxygen atoms of the network; hence the designation of these ions as network modifiers. Combining the results presented in Table 2, with these structural models, and with a propensity for covalent bonding of O to Si at the Si-dielectric interface, the respective bonding arrangements at the Si-Al2O3, Si-Ta2O5, and Si-Zr silicate (x=0.3) interfaces are, Si-O-Al1–, Si-O-Ta1+ and Si-O1–--Zr4+, suggesting a correlation between the sign of the fixed charge and the metal ion of the network component of the oxide in Al2O3 and, Ta2O5, and of the metal ion in the silicates. Studies of group IIIB silicates in which the bond-ionicity is higher, e.g., Y and La silicates, indicate fixed positive charge consistent with the incorporation of Y and La as 3+ ions rather than as network constituents. Finally, the magnitude of the fixed charge is more difficult to

account for. The convention has been to calculate an areal density of fixed charge defects of unit charge based on the macroscopic charge density obtained from the analysis of C-V data. This is not the only way to interpret the charge density. It can be described in terms of partial charges on atoms, or differences between charge distributions of both signs as in interfacial dipoles. Since the way the fixed charge is distributed influences the channel mobilities of charged carriers, analysis of mobility data from PMOS and NMOS FETs using different microscopic models for the fixed charge bonding arrangements may help to resolve some of these issues. Dielectric Constant Enhancement

MOS capacitors with SiO2-rich Zr and Hf silicates with 3 to 6 atomic percent Zr(Hf) have been reported to display increased dielectric constants and reduced tunneling currents [27-30]. Reported values of k, extracted from capacitance-voltage curves are ~8 to 11, and more than 50 % larger than values estimated from a linear extrapolation of k between SiO2, ~3.9, and the compound silicates, ~12 (Fig. 9). These enhanced values of k can not be reconciled with macroscopic dielectric theory that predicts a downward bowing of k between end-members in a mixed materials system [31]. Since macroscopic theory applies to mixtures in which chemical bonding of the constituents does not change with composition, it is important to determine if these SiO2-rich Zr(Hf) silicate alloys satisfy this condition.

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Chemical bonding has been studied in Zr silicates ( [SiO2]x[ZrO2]1–x ) by Fourier transform infrared spectroscopy, FTIR [23]. Similar bonding is expected in Hf silicates as

well. These results have already been discussed earlier in this paper. Several features in asdeposited films with x ~0.1 and 0.23 have been assigned to Si-O-Si groups in cornerconnected arrangements, as stretching modes at ~1150 cm–1, 1065 cm–1, and 810 cm–1, and a rocking mode at ~450 cm–1 [2,23]. Two other features are assigned to Si-O-Zr bondstretching vibrations, a terminal Si-O mode at ~950 cm–1 that is a shoulder on the 1065 cm–1

Figure 9. The dotted curve indicates the dielectric constant calculated from Eqn. 2 as a function of the percent Zr(Hf)O2 relative to the stoichiometric silicate compound composition, Zr(Hf)SiO4. Experimental points are from Refs. 27-30. The dashed line is a linear extrapolation between the dielectric constant of SiO2, and a nominal dielectric constant of 12 for Zr(Hf)SiO4.

band, and a broader Zr-O feature at ~450 cm–1 that is accidentally degenerate with a rocking mode of the Si-O-Si group. Since both stretching modes involve predominantly O-

atom motion, their frequencies reflect a significantly smaller force constant for the Zr-O vibration. This results from the increased Zr-O bond-length of 0.22 nm relative to 0.16 run for Si-O [9]. Far-IR spectra extending to 50 cm–1 show no additional discrete features. Broader spectral features of an as-deposited x ~0.5 film have been attributed to a random close packing of Zr4+ and SiO44– ions [8,23]. The 800 to 1200 cm–1 band includes internal SiO44– vibrations, and the ~450 cm–1 band is assigned to Zr-O vibrations. After a 900°C anneal, changes in these spectra indicate a chemical phase separation into i) a noncrystalline low Zr-content silicate alloy with ~1-2 atomic % Zr, or SiO2, and ii) noncrystalline (x~0.1 and ~0.23) or crystalline ZrO2 (x~0.5) [23]. As in bulk silicate glasses, introduction of oxides of electropositive Zr(Hf)-atoms into the SiO2 network results in a break-up or modification of that network [7,8]. Homogeniety in bulk silicate glasses quenched from high temperatures is limited by chemical phase separation, and in many instances homogeneous glasses are obtained only at relatively low metal oxide compositions, <5-10 mol percent. This is not a limitation in thin film silicates deposited at temperatures <500°C [22,23,27-30], The analysis presented below applies to these films, as well as films prepared at low temperatures, and subsequently processed at temperatures <800°C. The capacitors with Zr(Hf) silicate dielectrics in Refs. 27-30 have not been subjected to processing temperatures >600°C and meet these temperature constraints. The introduction of a Zr(Hf)O2 molecule into SiO2 is assumed to break two Si-O bonds of that network [8], so that the concentration of terminal Si-O terminal bonds is linear in alloy

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composition. The coordination of silicon to oxygen remains at four, and there are then five tetrahedral bonding groups with different distributions of O-atoms that are either i) connected to the network through bridging Si-O-Si bonds, or are ii) in terminal Si-O groups. Figure 10 gives the fractional concentrations of these tetrahedral groups as a function of alloy composition. Their concentrations are obtained from the appropriate terms of a binomial expansion, (3)

in which w is the fraction of bridging O-atoms, and z = 1 - w is the fraction of terminal Oatoms. This bonding model applies to both Zr and Hf silicates.

Figure 10. Relative fractions of five tetrahedral silicon-oxygen bonding groups with different numbers of bridging and terminal oxygen atoms plotted as a function of the percent of Zr(Hf)O2 relative to the stoichiometric silicate compound composition, Zr(Hf)SiO 4 .

Based on Fig. 10, the majority of Zr atoms (or Zr4+ ions) in alloys with x <0.1 are incorporated as network modifiers with four terminal negatively-charged O-atom neighbors in corner-connected arrangements [8] (Fig. 4). As the mol fraction of ZrO2 is increased, an increasing fraction of these groups must contain two or more terminal Si-O bonds. This causes the coordination number of the Zr-atoms to increase above four, including edge- as well as corner-connected bonding arrangements. In crystalline silicates, the Zr4+ ions have

a coordination of 8, with each Zr-atom making edge-connections to four different tetrahedral SiO44– groups [22,28]. The respective Zr-O and Si-O bond lengths are ~0.22 nm and 0.16 nm. This same bonding coordinations, four for Si and eight for Zr (or Hf) are assumed to apply in amorphous compound silicates. The contribution of a vibrational mode to the dielectric constant is proportional to the square of its transverse infrared effective charge, eT*, and is different for different bonding coordinations of the same atom pair [32,33]. The FTIR spectra indicate a broadening of the 950 cm–1 feature with increasing x. Based on Fig. 10, this broadening occurs at alloy concentrations where the ratio of edge- to corner-connected arrangements and Zr-atom

coordination have increased. If eT* scaled directly with increases in the number of terminal Si-O bonds, this broadening would also be accompanied by a marked increase in

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absorbance of the 950 cm–1 feature relative to the spectral peak from Si-O-Si network bonds at 1065 cm–1. To the contrary, the FTIR results indicate that relative absorbance of terminal Si-O groups does not scale in this way, and therefore decreases with increasing Zr-atom coordination. The bond-order a Zr-O bond is defined as the ratio of number of valence electrons available from each Zr-atom (four) to the number of nearest O-atom neighbors. Following this definition, four-fold coordinated Zr-atoms have the largest bond order, one. They also have the highest degree of covalency [34], so that dynamic contributions to eT* are expected to be larger than for higher bonding coordinations in which the bond order is reduced and the bonding becomes more ionic [32,33]. Scaling of local bond properties such as bond energies and stretching force constants with bond-order is well-established [34]. Consistent with the relative absorption strengths of single, double and triple carbon-oxygen bonds, this scaling can also be extended to eT*. Since infrared

active mode contributions to the dielectric constant are proportional to (eT*)2, the appropriate scaling variable is the square of the bond order.

Figure 1 1 . Transition in local bonding arrangements of Zr-atoms in Zr silicate alloys from low to high ZrO2 concentrations. At low concentrations the dominant bonding arrangements are between the Zr-atoms and four terminal O-atoms in a corner-connected geometry. At higher concentrations (fraction of ZrO2 >30-35 percent), there is a transition to bonding including more than one O-atom in edge-connected geometries.

Based on Eqn. (3) and an assumption that contributions of Si-O-Zr arrangements to the dielectric constant scale quadratically with the Zr-O bond order, the variation of k with alloy composition is approximated by Eqn. (2), (4)

in which the a ij are the product of i) the number of terminal Si-O bonds per group and ii) the square of an effective average bond order. The aij’s are approximated by a1,3 ~lx(4/4)2, a2,2 ~2x(4/(5-6))2 ~1.05, a3,1 ~3x(4/(6-7))2 ~1.15, and a4,4 ~4x(4/8)2 = 1, and are each of order 1. The curve is Fig. 1 is for ai,j = 1; values of a2,2 and a 3,1 >1 increase k for alloys with more than 45 % ZrO2, or x > 0.25. Since values of k for Zr(Hf) content > 7 atomic percent have not been reported, the application of the model is restricted to alloys with lower concentrations. The value of 3.9 fixes k for SiO2, and the value of the prefactor, 8.1, fixes

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k at 12 for compound silicates [1,4]. The experimental data in Refs. 27-30, with alloy content <7 atomic % (or <50% ZrO2 relative to ZrSiO4) fall close to calculated model indicating that the empirical relationship of Eqn. (4) provides a quantitative description of

the enhanced dielectric constant behavior for alloys in this composition range. There is also an additional, and smaller contribution to the dielectric constant from electronic transitions. A similar enhancement in the index of refraction, or equivalently the optical frequency dielectric constant, is expected to occur in Zr(Hf) silicate alloys. Our

initial experiments have indicated that this enhancement is present, and that it scales with bond order as well. In the spirit of the analysis presented above, this contribution is contained in the ai,j terms of Eqn. (4). It has been shown that the enhanced dielectric constants of the group IVB Zr and Hf SiO2-rich silicates are due primarily to the four-fold coordination of the Zr(Hf)-atoms in the alloy composition range below about 8 atomic %. Similar enhancements in k at low metal-atom concentrations are also anticipated for SiO2-rich silicate alloys with i) TiO2, ii) Y(La)2O3, but only for compositions up to the first silicate phase, La(Y)2SiO5 [35], as well as iii) and oxides of with other polarizable atoms such as Pb, Bi, Tl, etc. Interfacial properties of these SiO2-rich silicates are anticipated not to depart significantly form those of Si-SiO2 interfaces. Therefore these interfaces of Si with SiO2-rich silicate alloys are expected to have properties similar to Si-SiO2 interfaces, as well as providing significantly increased capacitance and reduced direct tunneling. INTERFACIAL LIMITATIONS FOR ALTERNATIVE GATE DIELECTRICS

Figure 12 provides a graphic description of the effect of introducing a nitrided SiO2 mechanical or chemical, interfacial buffer layer between an alternative gate dielectric and the Si substrate. Under these conditions EOT is given by:

Figure 12. Plot of EOT versus physical thickness of the high-k dielectric for k = 25 (e.g., Ta2O) and 15 (e.g., Zr or Hf silicate alloy) without and without a 0.35 nm EOT)int layer.

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where EOT)int is the equivalent oxide thickness of the interfacial layer, ~0.35nm for a remote plasma oxide film with a physical thickness ~ 0.5-0.6 nm and monolayer interface

plasma nitridation, tphys is physical thickness of the high-k dielectric [1]. In the absence of an interfacial layer EOT = tphys [k (SiO2) / k], so that the reduction in the physical thickness of the high-k dielectric that is associated with the interfacial layer is given by (6)

Since EOT)int ~ 0.35 nm, and k (SiO2) ~ 3.9, the reduction in physical thickness in nm, is approximately 0.9 k. For values of k ~ 10-20, this decreased physical thickness is equivalent to an increase in direct tunneling current of three to four orders of magnitude. This estimate is based on a dielectric constant of k~15, a 1.5 to 2 eV conduction band offset energy, Ebc, an electron tunneling mass of 0.5 m0, and an applied bias of one volt across the dielectric film. Therefore it is critically important to determine i) if interfacial nitride oxides are required for the high-k dielectrics, and ii) if there are any other interface options that would lead to a smaller vlaue of EOT)int. The nitrided interfaces required with

silicon oxynitride and nitride dielectrics limit the application of these layers to EOT ~ 1 . 1 nm, for a direct tunneling current limit of 1 -5 A-cm–2 .

SUMMARY

A classification scheme for gate dielectric materials based on bond ionicity as characterized by and I b has been introduced. Based on this scheme there are three

classes of deposited dielectrics that can be implemented into aggressively scaled silicon MOS devices. These are continuous random networks such as silicon nitride and silicon oxynitride alloys. The major limitations for the implementation of these dielectrics have been shown to be mechanical bonding constraints at Si-dielectric interfaces and internal dielectric interfaces in stacked structures. These limitations derive from increases in Nav which in turn promote values of Cav in excess of three and lead to defect concentrations increasing well into the 1011 cm–2 regime. The second class of dielectrics are based on modified continuous random networks. These dielectrics have a strong ionic contribution to their bonding, but retain aspects of network connectivity. Since the constraint theory of Refs. 4 and 5 is based on valence forces, it can only be applied to some members of this class that include a strong network component of bonding, e.g., Al2O3 and SiO2-rich silicate alloys. The major limitations in this class of materials come from their ionic character, e.g., fixed charge at Si-dielectric interfaces. The third class of materials are elemental metal oxides such as ZrO2, HfO2, La 2 O 3 where the bonding is ionic, i.e., the bonding in the non-crystalline state is best described by a random close packed array of spherical ions. The major limitation in the application of these materials is their instability against crystallization in a polycrystalline thin film, either as deposited or a relatively low post-deposition processing conditions. The discussion below is restricted to the crn and mcrn dielectrics. Continuous Random networks The paper has addressed three different aspects of interface bonding and defect structure that limit device performance and reliability of gate dielectrics that fall into the category of continuous random networks.. These are i) interfacial transition regions between the Si substrate and a dielectric film, ii) defect concentrations at over-constrained

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interfaces between Si and different dielectric films, and iii) defect concentrations at internal interfaces between dielectrics with different bonding constraints. It has been shown that the interfacial transition regions with suboxide bonding in excess of what is required at an abrupt interface between Si and SiO2 are typically about 0.3 nm thick. These regions are formed during oxidation of the silicon substrate and minimization of their spatial extent requires a post-oxidation anneal, e.g., 30 s to 1 minute at 900°C. It is particularly noteworthy that interfaces prepared by thermal oxidation at 900°C, also require an anneal at the same temperature to optimize interfacial smoothness or equivalently minimize suboxide bonding. The existence of an interfacial transition region between crystalline silicon and SiO 2 is anticipated on the basis of constraint theory. Constraint theory predicts that a transition region of the order of one molecular layer must be present at the interface between two materials in which the number of bonds/atom is different, or equivalently at which the number of bonding constraints per atom is different. The resulting interfacial region is over-constrained with respect to the lower constraint

partner of the interface, and is self-organized in a way that minimizes the development of mechanical strain. Interface layers of this type have been reported within glassy alloys with microstructure composed of a floppy-or under constrained constituent, and an overconstrained bonding partner. “Floppy” is defined in terms of a maximum average bonding

coordination of 2.4, except for SiO2 where, 2.67 is more appropriate due unusually small bonding forces at the oxygen atom sites. Values of Nav <2.4 (or 2.67) define a regime in which the average number of bond constraints per atom is lower than the network dimensionality. These concepts, originally applied to bulk glasses, have been extended to thin film amorphous materials and more recently to internal interfaces between crystalline and amorphous materials. It has been shown that control of interfacial bonding structure

by interfacial nitridation can modify interfacial transition regions between Si and SiO2 and produce significant reductions in tunneling currents, and thereby improve device performance. Experiments, combined with theory of suggest that the interfacial transition regions in advanced Si FET gate stacks are intrinsic, and result from entropy effects. Entropy is not a factor in the crystalline substrate, but configurational entropy is one of the most important factors in allowing for the formation of amorphous continuous random

networks. Therefore it is not surprising that it plays a role at the interface between a crystalline solid and an ideal amorphous covalent random network solid. Constraint theory has accounted for differences in the fixed charge at internal interfaces of stacked gate dielectrics. In particular, the quadratic scaling with has accounted for

differences in fixed charge at SiO2-Si3N4 and SiO2-(SiO2)0.5(Si3N)0.5 internal interfaces. This interfacial charge shifts flat band voltages and therefore will also change threshold voltages in FETs; however, it does not appear to degrade reliability. If the charge is high enough, and if the interfacial layer is sufficiently thin, it can also reduce effective channel mobilities as in p-channel FETs with bulk Si3N4 dielectrics. Modified Continuous Random Networks

These materials are qualitatively different than the cnr oxides, nitrides and oxynitride alloys. The increased ionic bonding of elemental oxides such as Ta2O5 and Al2O3, and the introduction of ionic metal oxides into SiO2, introduces local bonding groups were ionic in character. This markedly changes the nature of mechanical bonding constraints, since a description based on valence forces is no longer applies, accept for very dilute metal silicate alloys, for up to 10 to 15 atomic percent metal oxide content, or for group IVB Zr and Hf oxides, 3-5 atomic percent Zr or Hf. This regime of metal oxide doping provides

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increases in k, exceeding those of silicon oxynitride alloys and nitrides and may eventually

find its way into the technology. The paper has identified two significant issues relative to incorporation of these materials in device technology. These are thermal stability and fixed interfacial charge. Virtually all of the elemental oxides and binary silicate alloys, respectively either crystallize or chemically phase separate with or without crystallization at temperatures of at most 900°C, placing thermal budget restrictions on post deposition processing. For example, this limitation may require that high temperature processing steps such as source drain formation is done prior to dielectric depositions, and that dopant activation in polycrystalline silicon gate electrodes be replaced by in-situ doping during a low temperature deposition, e.g., at 800°C.

The interface limitation is equally challenging. There are several approaches that can introduce neutral interfacial bonding on a macroscopic scale, e.g., three-five analogs of

SiO2 such as AlTaO4. However, if the there are fluctuations in interfacial charge neutrality on a scale of 0.1 to 1 percent, corresponding the densities of scattering centers of the order of 1011 to 1012 cm–2, these may reduce the mobilities of holes and/or electrons in the

channel regions of field effect transistors. On the bright side, the identification of a mechanism for dielectric enhancement

discussed above opens up new opportunities to explore low concentration silicate alloys where both thermal stability and interface properties may be closer to SiO2. Acknowledgements Research support from the ONR, AFOSR, NSF and SEMATECH/SRC Front End Processing Center is acknowledged. The contributions from current and former graduate students and postdoctoral fellows in my research group at NC State University is gratefully

acknowledged. These include Sunil Hattangady, Hiro Niimi, Hanyang Yang, Yider Wu, Bruce Rayner, Bob Johnson and Bob Therrien. The author also acknowledges many important discussions and collaborations with Jim Phillips of Lucent Bell Laboratories and Mike Thorpe of Michigan State University.

REFERENCES [1] [2] [3]

Lucovsky, G. (1999), IBM J. Res. and Develop. 43, 301. R. Zallen, R. (1983) The Physics of Amorphous Solids, John Wiley and Sons, New York, pp. 49-72. Pauling, L. (1948) The Nature of the Chemical Bond, Cornel University Press, Ithaca, pp. 58-73.

[4] [5]

Phillips, J.C. (1979) J. Non-Cryst Solids 34, 153 ; (1981) J. Non-Cryst Solids 43, 37. Phillips, J.C. (2000) J. Vac. Sci. Technol. B 18, 1749.

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

Kerner, R., and Phillips, J.C., (unpublished). Zallen, R. (1983) The Physics of Amorphous Solids, John Wiley and Sons, New York, pp. 100-101. National Technology Roadmap for Semiconductors, (SIA, San Jose, 1999) Lucovsky, G., Yang, H., Niimi, H., Keister, J.W., Rowe, J.E. Phillips, J.C., and Thorpe, M.F. (2000) J. Vac. Sci. Technol. B 18, 1742. Keister, J.W., Rowe, J.E., Kolodzie, J.J., Niimi, H., Tao, N.-S., Madey, T.E., and G. Lucovsky G.

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(1999), J. Vac. Sci. Technol. A 17, 1340. Weldon, M., Queeny, K.T., Chabal, Y.J., Stefanov, B.B., and K. Raghavachai, K.. (1999) J. Vac. Sci. Technol. B 17, 1795.

[13]

Lucovsky, G., A. Banerjee, A. Hinds, B., Claflin, B., Koh, K.., and H. Yang, H. (1997) J. Vac. Sci. Technol. B 15, 1074.

[14] [15] [16]

Thorpe, M.F., Jacobs, D.J. and Chubynsky, M.V. (2000) J. Non-Cryst Solids 266, 859. Lucovsky, G. and Phillips, J.C. (1998) J. Non-Cryst Solids 227, 1221. Lucovsky, G., Wu, Y., Niimi, H., Misra, V., and Phillips, J.C. (1999) Appl. Phys. Lett. 74, 2005.

[17] [18]

Selvanathan, D., Bresser, W.J., Boolchand, P., Phys. Rev. B 61, 15061 (2000). Tu, Y and Tersoff, J. (2000) Phys. Rev. Lett. 84, 4693.

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[19]

Yang, H., Niimi, H., Keister, J.W., Lucovsky, G., and Rowe, J.E. (2000) IEEE Electron Device

[20]

Lucovsky, G., Niimi, H., Koh, K., Lee, D.R., and Jing, Z (1996) H.Z. Massoud, E.H. Poindexter and C.R. Helms (eds). The Physics of SiO2 and Si-SiO2 Interfaces-3, The Electrochemical Society, Pennington, N.J., p. 441.

[21]

Misra, V., Wang, Z., Wu, Y., Niimi, H., Lucovsky, G., Wortman, J.J., and Hauser, J.R. (1999) J. Vac. Sci. Technol. B 17, 1836.

[22]

Wolfe, D.,Flock, K., Therrien, R., Rayner, L. Günther, L., Brown, N., Claflin, B., and Lucovsky, G. (1999) Mat. Res. Soc. Symp. Proc. 567, 343.

[23] [24] [25] [26]

[27] [28] [29] [30] [31] [32]

Rayner. B., Therrien, R., and Lucovsky, G. (2000) Mat. Res. Soc. Symp. Proc. in press. Robertson, J. (2000) J. Vac. Sci. Technol. B 18, 1785. Hirose, M. and S. Miyazaki, S. unpublished. Lucovsky, G., Rozaj-Brvar, A., and R.F. Davis, R.F. (1983) P.H. Gaskell, J.M. Parker and E.A. Davis (eds) The Structure of Non-Crystalline Materials, Taylor and Francis, London, p. 193. Wilk, G.D. and R.M. Wallace, R.M. (1999) Appl. Phys. Lett. 74, 2854. Wilk, G.D., Wallace, R.M., and Anthony, J.M. (2000) J. Appl. Phys. 87, 484. Wilk, G.D. and R.M. Wallace, R.M. (2000) Appl. Phys. Lett. 76, 112. Misra, V., unpublished. Jayasundere, N and Smith, B.V. (1993) J. Appl. Phys. 73, 2462. W.A. Harrison, W.A. (1999) Elementary Electronic Structure,World Science, Signapore, Chap. 1 1 .

[33]

Burstein, E., Brodsky, M.H., and Lucovsky, G. (1967) International J. Quantum Chem. 1S, 759.

[34]

Cotton, F.A. and G. Wilkenson, G. (1972) Advanced Inorganic Chemistry, 3rd Edition, John Wiley and Son, New York, pp. 122-124.

[35]

Torpov, N.A. and I.A. Bondar, I.A. (1961) Izv. Akad. Nauk. SSSR, Ofd. Khim. Nauk 4, 547.

Letters 21, 76.

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EXPERIMENTAL METHODS FOR LOCAL STRUCTURE DETERMINATION ON THE ATOMIC SCALE

E.A. STERN Department of Physics, University of Washington, Box 351560, Seattle, WA 98195-1560

I. INTRODUCTION It is generally agreed that for a fundamental understanding of the properties of the condensed state it is necessary to know the composition and arrangement of the atoms in the material. In this article I will review the main experimental techniques for directly determining the local structure of materials in the condensed state with emphasis on the type of information each technique determines and the strengths and limitations in obtaining such information. The theory of the structure information obtained by the experimental techniques is already well known and only that aspect of it related to the focus of this review will be presented here. The most generally employed method is coherent scattering of either x-rays or neutrons. The most common application of the technique is on crystals where the scattering is peaked about discrete directions corresponding to Bragg peaks. By an appropriate analysis of the intensity and line shapes of only the Bragg peaks one can obtain the average periodic structure of the materials, e.g., by the Rietveld refinement method [1]. In recent years the interest in condensed matter physics has focused on phenomena such as high Tc superconductivity and colossal magnetoresistivity where the materials are relatively complex, and there are suggestions and experimental evidence that local deviations from the average periodicity for such materials may be important for understanding their properties. Even for the well-studied phenomenon of ferroelectricity, it has recently been shown that local deviations from periodicity occur which were previously not suspected [2]. This has resulted in a reassessment of the previous theories of ferroelectricity leading to a new theory which includes the local deviations [3]. This theory based on a new physical mechanism of ferroelectricity gives greatly improved quantitative and qualitative agreement with experiment. To obtain the local deviations, the standard Bragg diffraction method needs to be supplemented. If any such local disorder is present in the crystal, the standard Bragg diffraction method is able to detect this only as a decrease in intensity in the Bragg peaks, since such disorder diverts some scattering into angles between the Bragg peaks, so-called diffuse scat-

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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tering. Such a decrease in intensity would be interpreted only as a larger “thermal factor” of the vibration of atoms about their lattice sites. Detecting the local disorder requires including

in the analysis the diffuse scattering between the Bragg peaks. An analysis of both the diffuse scattering and the Bragg peaks leads directly to the local structure [4] in the form of the pair distribution function (PDF), as defined below. Disorder in the distribution of atoms can be classified as (a) local disorder which does not destroy the long-range order of the crystal or as (b) in amorphous solids and liquids where the disorder destroys long-range order. For the crystal with local disorder, only some of the Bragg peak intensity is transferred to the diffuse scattering, while in the sample without longrange order all of the scattering is diffuse. By neglecting the diffuse scattering the refinement

methods of Bragg reflection do not detect the local disorder. In this review I will focus on methods that detect the local disorder. For scattering techniques this means including the diffuse scattered intensity in addition to any Bragg peaks that may be present when analyzing the data. As discussed in more detail below, the local structure determines the actual distance between atoms while, when disorder is present, the diffraction structure may not correctly display the actual distances between atoms. For any fundamental understanding of the properties of materials, a knowledge of the actual distances between atoms is important, since the atom-atom interaction is a strong function of this distance. An example was mentioned above in relation to ferroelectricity. When an x-ray photon or neutron particle is scattered from atoms in materials there

is transfer of momentum and energy to the material. The and are given by the momentum and energy change, respectively, of the scattered particles. The intensity distribution of the scattering particles as a function of and can be analyzed to obtain the structure factor As discussed in Sect. II, the pair distribution function, also called the pair density or density correlation function, all being denoted by PDF, can be obtained from the structure factor. An alternate method to obtain the local structure including any local disorder is the x-ray absorption fine structure (XAFS) technique as discussed in Sect. IV. The techniques of both XAFS and diffuse scattering directly measure the local structure. Other techniques such Raman scattering and magnetic resonance are also employed to obtain information on the local structure, but their information is more indirect and of less general applicability. In this review I will focus on the two techniques that directly measure the local structure, namely diffuse scattering and XAFS. The outline of the review is as follows. Sect. II defines local structure in terms of the PDF, the partial PDF (PPDF) and the structure factor The relationship between the local structure and the average periodic structure will also be discussed in this section. Sect. Ill presents the aspects of the local structure determined by diffuse coherent scattering, discussing the strengths and limitations of using x-rays and neutrons. Sect. IV discusses the XAFS technique for determining the local structure, presenting its strengths and limitations. Sect. V presents a discussion, and a summary and conclusion are given in Sect. VI. II. AVERAGE PERIODIC AND LOCAL STRUCTURE

From measurements of the coherent scattering of x-rays or neutrons it is, in principle, possible to obtain directly the time-dependent pair distribution function of atoms, which gives the joint probability of finding an atom at point at time t = 0 and an atom at point at time t later. Because the typical experiment measures the intensity of the scattered radiation instead of the amplitude and phase of the scattered waves, absolute phase is lost

and only relative phase between two scatterers is retained. This gives a correlation between two atoms leading to In particular, diffuse coherent scattering gives directly the structure

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factor

which is given by [5] (1)

Equation (1) shows that

is the space-time fourier transform of The single particle density operator is given by:

(2)

where is the position of the jth particle at time t and the angle brackets in Eq. (1) indicate a thermal average at temperature T of the probability that a particle is at at and any particle is at at later time t. The average periodic structure determined by using only Bragg peaks is given by while the PDF (3) The most common samples studied by diffuse elastic scattering are either liquids, amorphous solids or polycrystals. In this case, the PDF is averaged over angle and is a function of where and is called the radial distribution, correlation or density function. To obtain the single particle distribution function requires determining the absolute phase of the scattered wave. For large unit cells as in biological macromolecules, additional

measurements are required, such as heavy atom substitution or multiwavelength anomalous diffraction (MAD) techniques [6]. For small unit cells the absolute phase problem can be solved by refinement methods, such as the Rietveld method [1]. These methods of solving for are only applicable to crystals with long-range periodicity and determine the average periodic structure, since these techniques use only the sharp Bragg peaks in their analysis. An important difference between and should be noted. In a spatially homogeneous amorphous solid or liquid (as expected for the condensed state in a single phase) is a constant. In a single crystal has a periodic array of atomic lattice sites determined by the space group symmetry of the crystal. The typical refinement of the structure accounts for the disorder due to vibrations (i.e., the Debye-Waller factor) by assuming a gaussian disorder about the lattice sites and then determines the gaussian-width about each lattice site. On the other hand, determines the relative distances between pairs of atoms and the disorder or distribution about this relative distance. The most striking difference between these two distribution functions is illustrated for the amorphous and fluid state. Whereas is a constant at the average density value, displays the short-range order of peaks at the first few nearest neighbor distances and then becomes blended into the constant average density as schematically shown in Fig. 1. In crystals, if there is local disorder present, then both and show peaks, but their relative spacings and distribution may not coincide. Coincidence occurs only if there is no local disorder. For example, PbTiO3, which is ferroelectric and has a tetragonal structure at low temperatures, displays a cubic perovskite structure by Bragg reflection measurements in the paraelectric phase above 760 K. However XAFS measurements [7] determine that the local structure in the paraelectric phase has a displacement of the Ti atoms from their cubic symmetry sites relative to the oxygen and Pb atoms, which are almost as large as occurs in the ferroelectric phase. The paraelectric phase occurs because the local displacements lose their long-range correlations above Tc, the ferroelectric to paraelectric phase transition temperature, i.e., they become disordered so that their average displacement is zero. When structure is the goal of measurements, one is not interested in the general but only the instantaneous snapshot of the structure given by the PDF. By referring to Eqs. (1) and (3) and using the relation that (3a) 211

Figure 1. A schematic of the radial pair distribution function

as a function of the radial distance R for a

liquid. The short-range order of peaks of the first few neighbors are discerned and then the distribution blends

into the constant average density of the liquid.

it is found that

(4) Thus, if the diffuse scattering measurement does not energy discriminate but detects all of the scatterings, both elastic and inelastic, then the spatial fourier transform of gives the PDF. In the above discussion it is assumed that the atoms cause all inelastic scatterings, which neglects electronic excitations. In practice, the inelastic scattering cross-section from electrons is much smaller for neutrons than the inelastic scattering from the nucleus of atoms, while the scattering of x-rays, though dominated by the intact atom without electronic excitations, still has a significant inelastic signal from the inelastic electron excitations of Compton scattering and the Raman effect. The inelastic energies for both neutron and x-ray scattering from intact atoms is typically of the order of phonon energies, namely the order of 15-100 meV, while the electron excitation energies are in the eV to many eV range. The experimental measurement of x-ray scattering is performed in a manner as to correct for both the Compton scattering and all other electronic inelastic loss contributions, but integrates over all of corresponding to the atomic motion contibutions. In short, if the experimental technique determines where then it accurately determines As we discuss next, in some cases neutron scattering measurements do not perform a correct integration over to accurately determine First, to understand the problem introduced by not integrating over a large enough frequency range we evaluate Eq. (1) when integrating over

(5)

where Note that the weighting factor of (sinu/u) in the integral over du gives its largest contribution over the range of and thus the term in brackets is integrated over the range 212

from

If atoms do not move appreciably in the time then and the integral in Eq. (5) over u gives the PDF, and is accurately determined. On the other hand, if atoms move a large distance in the time then the integral in Eq. (5) may give an inaccurate estimate of and the PDF. This is certainly the case for fluids where atoms can diffuse large distances from an initial starting point. In crystalline and amorphous solids where diffusion is negligible, and at low temperatures so that the thermal energy is small compared to the zero point energy and quantum effects dominate, the PDF is a gaussian independent of time and the integration over time or u gives a correct PDF even when However, when the thermal energy is much larger than zero point energy, classical physics dominate and the atoms typically vibrate harmonically about a fixed point with an average angular frequency that is denoted here by The net displacement is small when averaging over the time interval even when so that atoms classically vibrate many periods over the time interval . However, the PDF is still greatly distorted when It can be shown that what is determined for a one-dimensional oscillator of amplitude A about the origin is a distribution function that is a constant in space between while the correct distribution is Note that since typically for oxygen bonds where TR is room temperature and kB is the Boltzmann constant, then for temperatures below TR the quantum regime applies and the correct PDF would be determined for all values of Moreover, there is evidence that in some cases the motion of atoms may not be harmonic, e.g., a bifurcation of the planar Cu2 to apical oxygen (O4) relative distance has been detected [8] in with some indication that a snapshot of the distribution is not being measured. This shows up by the anomalous q-dependence of the bifurcation as shown in Fig. 2. If the snapshot distribution is being determined, then decreasing qmax (Qmax in Fig. 2) should simply broaden the two peaks proportionally to The change between –1 –1 –1 qmax = 31Å and qmax = 27Å should be about the same as between qmax = 27Å and –1 –1 qmax = 23Å . Such is not the case. At qmax = 23Å the distribution is not simply broadened but the splitting and positions of the peaks have greatly changed, placing into doubt any quantitative measure of splitting, or even if the splitting exists. In the presence of a dynamic non-harmonic motion, in order to determine the snapshot PDF, requires where is the characteristic time of significantly changing the distribution. For the neutron scattering measurements of Fig. 2, which corresponds to For the anharmonic dynamics of atoms it is quite possible that for this value of that the relation is not satisfied, even for oxygen atoms, and the PDF will not be accurately determined in this region, as indicated by the anomalous dependence of Fig. 2.

III. DIFFUSE SCATTERING BY NEUTRONS AND X-RAYS The coherent scattering of x-rays from a single atom is predominantly caused by the interaction of the electric field of x-rays with the average charge density of the atom when the x-ray photons have energy much greater than the most tightly bound electrons. A measure of the scattering of the incident x-ray’s electric field by the jth-type atom in the sample is given by its atomic form factor which is the fourier transform of the electronic charge density of the jth atom with respect to The value of f j(0) is Z, the number of electrons in the atom and the intensity of the scattered radiation from that atom is proportional to Z2.

The PDF determined by diffuse scattering of x-rays is a sum of the distribution of the relative distance between all pairs of atoms with the contribution from an atom proportional to its Z2. Thus, the scattering power of low Z atoms like oxygen compared to that of a heavy atom like Cu are less by a factor of (8/28)2 = 0.08, making it difficult to detect oxygens in the

background of the scattering from heavy atoms. For that reason, diffuse x-ray scattering is

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Figure 2. The dependence of the positions of the Cu2-O4 peaks of as presented in ref. 8. The maximum q (Qmax in the figure) is varied which should just decrease the resolution between the two peaks.

Instead the

plot shifts the peak positions and increases the splitting, indicating that the instanta-

neous PDF is not being measured in this region as discussed in the text,

not the method of choice for materials where oxygen plays an important role, such as the Cu-O high Tc materials. As discussed below, the scattering of neutrons is more favorable for detecting oxygen atoms. When x-ray photons have energies near absorption edges, so-called anomalous or resonance scattering occurs where the scattering strength of the atom is energy dependent. Utilizing this anomalous scattering allows the determination of the partial PDF (PPDF), i.e., the PDF about the atom with anomalous scattering [9,10]. The PDF is the sum of the PPDF about each type of atom of the material so the PPDF gives more detailed information about the structure and is preferred when available. X-rays used for diffuse scattering are in the hard x-ray region, greater than 3KeV, because soft x-rays have too long wavelengths to resolve atomic dimensions. Unless heroic efforts are used to obtain very high energy resolution, the inelastic losses due to atomic motion, typically 20 meV, are not resolved and the detected x-rays then are integrated over the full w range of the atomic motion and determine In order to be able to utilize anomalous scattering to obtain the PPDF with atomic resolution, the absorption edge energy of the atom of interest must be in the hard x-ray range. This precludes the use of light atoms as anomalous scatterers. In addition, the need to use x-rays near absorption edge energies limits the qmax that is obtainable, except for the very heaviest atoms. More details are given in this Section below, in Sect. V and in the extensive reviews of Waseda [9,10].

Neutrons scatter from the nucleus of the atoms and the scattering strength from atoms depend only on their nuclear properties and not on the atomic Z. The scattering of neutrons from a nucleus of charge Z is strongly spin- and isotope-dependent. Even for a sample composed of a single isotope, the scattering depends strongly on the projection of the nuclear spin on the neutron spin and, unless the nuclear spin is zero, the scattering from these atoms will differ dependent on this projection. This introduces a disorder in the scattering that produces both coherent and incoherent scattering, where the coherent scattering amplitude is proportional to the average of the scattering factor of the nuclei at each site, and the incoherent

scattering by the rms fluctuation of the scattering factor from the average value. Only the coherent scattering defines the PDF. When it is intense enough the coherent scattering can be

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separated from the incoherent scattering. For some nuclei, e.g., vanadium and hydrogen, the coherent component is weak compared to the incoherent one and it is difficult to obtain their contribution to the PDF. However, for most nuclei such as Cu and O their coherent signal

can be used to obtain the PDF. Neutrons have a great advantage over x-rays for determining the O contribution to the PDF, since its coherent scattering is comparable to that of heavy atoms such as Cu. For that reason neutrons are the particles of choice for determining both the average periodic structures and the PDF of the Cu-O high Tc materials. Neutrons can also be used to obtain the PPDF by isotropic substitution. For instance, a recent diffuse scattering investigation of YBa2Cu3O6.92 determined the PPDF about Cu atoms by utilizing the contrast in the neutron scattering cross-section between 63Cu and 65Cu in two isotropically-pure samples with identical composition [8]. However, isotope substitution is limited by the availability of suitable isotopes and by how identical the two isotope samples can be made in composition and perfection. Neutrons have another advantage over x-rays in that the scattering amplitude from a nucleus is essentially constant as a function of permitting the determination of the scattering –1 signal to high values of up to 35Å and beyond. This feature is a consequence of the small size of the nucleus which for practical q-values appears as a delta-function in space. In contrast, the scattering amplitude fj of x-rays decreases significantly with q for values of where the radius of the atomic mean charge, causing a practical limit for measuring scattering of In addition, for atoms lighter than Tc, anomalous scattering requires lower q x-rays, lowering below 20Å–1 . Thus, neutrons can determine up to significantly larger values of q which translates into a higher spatial resolution in the PDF. To illustrate this point, consider a solid that has a structure containing an atom pair whose relative distance is split. To detect the split requires having enough spatial resolution in the measurement. Similar to the Rayleigh criterion for distinguishing the separation

between two points, the criterion for distinguishing the splitting from another distribution, such as a gaussian broadening, is that (6)

This criterion guarantees that the measurement detects the unique beat in the interference between the scattering from the two sites involved in the splitting. At q-values below the beat, the signal for a gaussian and split is quite similar and only near the beat occurring at is the distinction clear. Thus, to resolve a splitting of requires including –1 q-values up to at least qmax = 30Å To obtain such large q-values in neutron scattering requires a source of higher energies than obtainable from a continuously emitting thermal reactor. Pulsed sources of neutrons are used to obtain such energetic neutrons, e.g., the Intense Pulsed Neutron Source (IPNS) of the Argonne National Laboratory. The scattered neutrons are analyzed by a time of flight measurement determining both their speed and angle of scattering, allowing the determination of by these two quantities. A detector at the IPNS used to determine PDFs is the Glass,

Liquids, and Amorphous Materials Diffractometer (GLAD), The detected by GLAD is integrated over energy with an energy cutoff at about 20 meV. However, this spread in causes the q not only to depend on the angle of scatter, but also on since the final magnitude of the scattered neutron momentum depends on Thus and the integral over is not the correct one to determine S(q), since that requires that remains constant as is integrated. This problem does not occur for x-rays because is orders of magnitude smaller than the photon energy while it is not negligible compared to the neutron energy. This problem is additional to the issue discussed in the previous section which required that the cut-off energy be large compared to phonon energies to guarantee a correct value of A procedure has been developed to correct the dependence only when extended phonons are involved [11], but not for localized modes. In practice, it appears that errors introduced 215

in the PDF by these effects are limited to temperature ranges around phase transitions where dynamics in splittings of metal-oxygen distances apparently occur with time scales that are sensitive to both the -dependent effects, causing the anomalous behavior shown in Fig. 2 and discussed above and in ref [8]. The result is that some possibly very interesting behavior

of the PDF for high Tc and colossal magnetoresistance phenomenon detected by neutron scattering may be occurring near the transition, but unfortunately one cannot be confident that diffuse coherent neutron scattering is correctly determining this behavior. In any experimental determination of the PDF it is very important to assess the uncertainties in the result. Both x-ray and neutron scattering can obtain the PDF with reasonable accuracy. However, the PPDF accuracy is not as favorable as it is derived from a smaller difference between larger measured spectra, which introduces greater uncertainties in the final result compared to that in each measured spectrum. This is especially severe for x-ray scattering where the anomalous differences for high concentration atoms are typically 10% or less, thus increasing the uncertainties by more than an order of magnitude. On the other hand, anomalous x-ray scattering determines its various spectra on exactly the same sample, whereas neutron scattering requires fabricating two different samples with different isotopes introducing systematic uncertainties which are hard to assess. This important issue of uncertainties is discussed further in Sect. V. In summary, neutrons have significant advantages over x-rays for determining the PDF (better spatial resolution and better sensitivity to low Z atoms) but, because of experimental limitations, neutrons measurements sometimes do not have a rapid enough time response

nor the correct dependence to obtain an accurate snapshot determination of the PDF of anharmonic motions in temperature ranges near some phase transitions, which, unfortunately, are physically very interesting regions. IV. X-RAY ABSORPTION FINE STRUCTURE (XAFS)

XAFS is another technique that determines directly the PPDF using different physics than diffuse coherent scattering. XAFS is the fine structure that occurs in the x-ray absorption coefficient on the high-energy side of absorption edges [12]. The absorption edges occur as the x-ray photons attain enough energy to just dislodge electrons bound in atoms. The fine structure occurs only when the atoms are in the condensed state and the dislodged electron can backscatter from surrounding atoms and interfere with the outgoing portion of its wave function. By appropriately analyzing this XAFS, the arrangement of atoms about the x-rayabsorbing atom, the probe atom, can be determined [12]. Recent advances in theory [13] and analysis [14] have extended the range of reliable detailed structural information from the first neighbor to four or more neighboring shells of atoms about the probe atom. Among other information, XAFS directly obtains the PPDF of the relative distance between the probe atom and its neighbors with a high spatial resolution (typically the former for light atoms and the latter for heavy atoms) and with high sensitivity. Since the XAFS signal is produced by the scattering of the photoelectron from its neighboring atoms, this scattering does not have the decrease in signal from light atoms compared to heavy atoms that x-ray scattering suffers from. The electron scattering from light atoms has a comparable signal to heavy atoms but differs in the energy dependence of this signal. This dependence allows the identification of the surrounding atoms, while the x-ray absorption edge energy identifies the probe atom. XAFS measures the PPDF for the most dilute concentrations of impurities by far than the coherent scattering techniques because it measures this feature directly. As mentioned above, since the other techniques of x-ray and neutron diffuse elastic scattering measure the PPDF by subtracting two large numbers from one another to obtain a very small result for dilute impurities, the answer has a much larger relative uncertainty than either measurement. XAFS, by using the atom specific experimental technique of detecting

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its characteristic signal by x-ray fluorescence or electron emission, obtains the spectra from only that atom, giving the PPDF about that atom directly without adding noise from the signal coming from the PPDF of the other atoms, leading to its much greater sensitivity for detecting PPDF about dilute impurities, down to ppm. In addition to determining the PPDF, which is a two-body correlation function, a unique feature of XAFS is its capability to determine three-body correlations with high accuracy in special cases, namely, when three atoms are collinearor nearly so, due to a focusing effect of the intermediate atom on the propagation of the photoelectron [12,15]. Compared to neutron scattering, XAFS has the same advantage of x-ray scattering of a stronger interaction with matter allowing the use of much smaller samples. The XAFS measurements are much quicker and less tedious to complete than the diffuse scattering techniques. Orientation information on the PPDF is obtained in the special cases of anisotropic single crystals by varying the angle between the x-ray polarization (which is horizontal for synchrotron radiation sources) and the crystal axes. To observe the instantaneous PPDF, the characteristic time scale of the measurement must be much shorter than the characteristic time in which the arrangement of atoms can change. Since its characteristic time is typically less than 10–15 seconds, XAFS easily satisfies this criterion. For comparison, the indirect experimental techniques of magnetic resonance and Raman scattering have characteristic times of 10–10 seconds or longer and are limited in obtaining a snapshot of dynamic distortions to ones that are relatively slow. For example, in second-order structural transitions the dynamics of local distortions become fast somewhat above Tc, and these techniques typically lose their ability there to determine any disordered local distortions, e.g., as occur in ferroelectricity [2]. Two other aspects of the information content of XAFS should be mentioned. The XAFS analysis software, e.g., UWXAFS [14], contains reliable objective estimates of the uncertainties of the PPDF determination. This is important in assessing the reliability of the results. The present analysis procedures for the diffuse elastic scattering techniques do not present objective estimates of the uncertainties of their PDFs [8,9] as discussed in more detail in the next section. In addition to the PPDF information, the XAFS spectrum within 15 eV of the edge (the XANES) contains electronic structure information, e.g., the valence state of the probe atom, and in some cases its coordination geometry [12]. The XAFS spectrum that extends far beyond the edge (typically from 15-1000 eV), the EXAFS, contains the PPDF information. XAFS has two limitations. The region around the probe atom where XAFS data can be analyzed is limited by the mean free path of the photoelectron and the complications of multiple scattering effects to typically four nearest neighbors. Also, because of both the physics of the photoelectron excitation process and difficulties in distinguishing the XAFS fine structure from the rapidly increasing total absorption near the edge, XAFS presently does not obtain the values below q = 6Å–1 . This latter deficit translates into XAFS not being able to measure distribution functions with widths greater than about 0.4Å. In practice this typically limits XAFS to determining the PPDF of only the first neighboring atoms in disordered liquids, glasses and amorphous solids as illustrated in the next Section. Fortunately many interesting physical phenomena occur with local disorder in crystals with long-range order where XAFS can probe the PPDF to four nearest neighbors. Examples are structural phase transitions, including ferroelectricity and strongly interacting materials such as colossal magnetoresistance and high temperature superconductors. For these materials XAFS is the premier technique for determining the local structure up to 5 – 6Å. Even in the highly disordered materials XAFS has decided advantages for the PPDF of the first neighbors, namely, more accurate determination of ligand distributions and greater sensitivity to dilute components.

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V. DISCUSSION By determining the relative distribution of local displacements of atoms from the average structure, the pair distribution function (PDF) supplements diffraction measurements, which

determine only the average periodic structure. In cases where structural disorder is induced by compositional disorder, this information allows the determination of the actual structure, including visualizing the displacements of atoms from the periodic structure [16]. In cases of crystal structures with no local disorder and the location, of atoms within the unit cell is not completely defined by the space group crystal symmetry, refinement techniques on the diffraction measurements, such as the Rietveld [1] method, are required to solve the structure. When, for experimental limitations, not enough diffraction data are available for refinement, e.g., measurements under high pressure, adding PPDF using XAFS, allowed the solving of the structure of the intermediate pressure phases of AgCl [17]. Information contained in the PPDF is important for understanding the atomic interactions that are present in solids because they determine the actual distances between atoms which may differ from the average periodic structure distances when local disorder is present

as mentioned in Sect. I in regard to perovskite ferroelectrics [2,3]. In a sense coherent scattering and XAFS are complementary in determining the PDF. The much stronger interaction of the photoelectron with its surroundings compared to x-rays and neutrons limits the spatial range of XAFS for probing the PDF to the much smaller range of about 4-5 Å. In addition, the inability of XAFS to determine low q information limits the

detection of spatially slowly varying distributions. On the other hand, XAFS is able to more accurately determine the PPDF in the range it covers. Figure 3 shows the difficulty that the lack of low q data introduces for determining the PPDF beyond the first coordination shell about Cu ions in water solutions under hydrother-

mal conditions, and also illustrates the clean signal XAFS gives for determining the first shell to high accuracy in spite of its relative dilution. There are just two discernable peaks above the noise for all four plots which have been measured by Fulton et al. [18] at the three temperatures indicated in Fig. 3. The highest temperature 350°C is at supercritical conditions. Besides varying the temperature and its concomitant pressure the solutions had varying con-

centrations of Cl ions added by dissolving various amounts of NaCl and HCl besides CuCl. Without going into the details of how the Cl concentration relative to the Cu concentration was varied, the issue to focus on is what can XAFS detect in such a disordered medium. The first and largest peak is the first neighbor ligands which for the T = 25°C sample is six water molecules. The second and third plots have two Cl atoms as first neighbors without any detectable water molecules in that same shell. The last plot contains one Cl atom and one oxygen atom without any other atoms detected in the first shell. Although the Cu atoms have only a fraction of a molarity concentration (0.2-0.4), there is a very clean signal allowing the determination of distances to better than 0.01 Å and coordination number to about 10%. Note that the second smaller peak is roughly twice the distance of the first peak. The locations of the peaks are not at the actual distances, but the theory can interpret the data to give the necessary corrections to determine the true distances to the stated accuracy. It turns out that the small second peak is not due to an actual position of a peak in the PPDF but is due to multiple scatterings of the photoelectron due to a pair of first neighbors being collinear with the Cu probe atom, as shown in the drawings in the upper right-hand corner of Fig. 3. The relevant multiple scattering consists of the photoelectron scattering backwards from one neighbor, scattering forward through the Cu atom to the opposite neighbor and finally backwards to the Cu atom. The total distance the photoelectron travels is twice

the single scattering path from the Cu to its neighbor and back again to the Cu, causing the small peak at the larger distance. This multiple scattering signal is greatly enhanced by the forward scattering and would not be visible if the atoms were not collinear. Thus the presence of the multiple scattering peak and the first peak are evidence of both the collinearity of

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Figure 3. The magnitude of the transform of the EXAFS signal from Cu ions in water solution under various

temperatures and Cl ion concentrations as presented by ref. 18. This plot is a pseudo-PPDF which, after appropriate analysis, determines the correct PPDF relative to the Cu ions, leading to the three ligand structures shown in the diagrams in the upper right-hand corner. The text explains the origin of the two peaks that are significantly above the noise level in each plot as being caused by the first neighbors of the Cu ion. No other neighbors are detected because of their broad distribution in the PPDF.

pairs of first shell atoms, and the strength of their bonding as verfied by its rms disorder of 0.02Å, sharp enough to be fully measurable by XAFS. However, no other shells are visible indicating that all the rest of the surrounding atoms are not tightly bound to the Cu atoms and have a distribution too broad to be detected by XAFS, rms > 0.2Å. Thus, because of the high sensitivity of XAFS to determining the PPDF it is able to determine the near portion of the PPDF to higher accuracy and for far more dilute atoms than can the coherent scattering techniques. In addition, in the special cases when three atoms are near collinearity, XAFS has good sensitivity for measuring three-body correlations, in particular, it determines directly, where is the angular deviation from collinearity. Finally, the XANES portion of the XAFS (not shown) determines the valence of the Cu ion and independently verifies the coordination number about the Cu atoms. The complementarity between XAFS and diffuse coherent scattering was first exploited by Raoux and Flank [19] who combined the results of the two techniques to obtain a better model of the PPDF of a series of amorphous Cu-Y alloys than could be obtained from either separately. More recently di Cicco et al. [20] combined the two techniques to obtain an improved PPDF of silver-halides melts. The addition of XAFS to the coherent scattering results improved the short-range atom-atom correlations in both cases. It is striking that based on the anomalous x-ray scattering measurements of Saito and Waseda [10] a direct numerical solution for the three partial structure factors of molten CuBr, namely, the q-dependent interference pattern due to scattering between the Cu-Br, Cu-Cu, and Br-Br pairs of atoms which contain the information leading to the three PPDFs for these

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pairs of atoms, contain large errors as shown in Fig. 4. The cause is apparently due to the required solution of three simultaneous linear equations being ill-conditioned at various q

values because of the quite small difference of the anomalous scattering and the magnitude of the inversion determinant in the denominator becoming small which greatly magnify unavoidable experimental errors. Worse yet is that apparently the errors are not random, but are systematic, since a direct fourier transform to the corresponding PPDFs gives physically unreasonable results!

Figure 4. The three partial structure factors of molten CuBr at 810K as presented in ref. 10. The solid plot corresponds to the values calculated by the reverse Monte Carlo method while the vertical lines denote the uncertainties estimated after a direct determination from the experimental data.

To overcome this fatal problem the reverse Monte Carlo simulation technique [21] was employed to obtain a best fit to the original experimental data. This technique is a method to vary a distribution of a cluster of atoms with the experimental composition and determine a best calculated fit to the experimental data. The result of this best fit is shown in Fig. 5 and

the resulting fit to the three partial structure factors are shown by the solid lines in Fig. 4. Unfortunately, there is no objective way to estimate the uncertainties in the PPDF determined in this way. Some uncertainties are estimated in the publication though no description is given of how this is determined. In the recent neutron scattering paper [8] discussed in Sect. III there is no quantitative estimate of uncertainties given at all. This is a serious defect since one cannot judge whether small changes in the modelled PPDF are real, i.e., whether they are outside of experimental errors. For example, it is claimed ”that two types of Cu1-O4 (Cu chain and O apical) bonds are likely to be present locally” which differ in their bond distances by 0.027 Å. Using the criterion of Eq.(6) for resolving a splitting of from another distribution with the same rms 220

Figure 5. The three experimentally measured interference functions as presented in ref. 10 are plotted with the solid lines. The top is determined from the anomalous scattering from the Br atoms, the middle from the anomalous scattering from the Cu atoms while the bottom plot is determined far from any anomalous scattering region. The fit of the reverse Monte Carlo method is shown by the dotted lines. From these three seta of data the partial structure factors of Fig. 4 are obtained by direct solution giving the large uncertainties and then by the reverse Monte Carlo method giving the solid plot there.

variation such as a gaussian distribution, leads to the requirement of a Since –1 the measurement had a qmax = 36Å the existence of the splitting is not credible. In addition, the errors introduced by the limited range of the integration of for neutrons are not understood when dynamic effects are present for anharmonic motion, as discussed in Sect. III in relation to Fig. 2.

It should be emphasized that the above discussion on uncertainties introduced in determining the PPDF by diffuse coherent scattering applies to samples where each component is a substantial fraction of the total composition. When a component is dilute, as per impurities or dopants, then it is not possible to obtain any reliable PPDF for that component. The fact that XAFS can measure the PPDF directly without the signal and the concomitant noise from the rest of the atoms in the sample is a decided advantage, not only because of the greater sensitivity for measuring more dilute atoms, but also for obtaining objective estimates of uncertainties. The introduction of uncertain sources of systematic errors is significantly less of a problem for XAFS. It is possible to separate random and systematic errors since the signal and noise comes from only the atoms of interest as opposed to the diffuse scattering measurements where the major source of noise comes from the other atoms while the signal comes predominantly from the atoms of interest. This latter situation leads in most cases to the need to smooth the result by special techniques such as maximum entropy [22] or

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the reverse Monte Carlo method [21] which introduce uncontrolled uncertainties. For example, in XAFS it is possible to define the total information content of the data and to control the fitting procedure to guarantee that the number of variable parameters is less than the relevant number of independent points, NI, the data contain (the information content), which allows a meaningful estimate of the errors. Diffuse scattering also has inherently a limited amount of information due to the limited q-range of the data, as per the Nyquist sampling rate criterion [23], namely, where NI is the maximum number of variable paramters allowed and is the range of real space being fit. When the smoothing procedure is introduced, what corresponds to NI is obscured and how objectively to assess uncertainties is a problem not generally addressed in the diffuse coherent scattering literature. VI. SUMMARY AND CONCLUSIONS

Refinement of diffraction data allows the determination of the average periodic structure of crystals, while diffraction gives no information on materials without long-range order since no diffraction peaks occur for these materials. However, the PDF and PPDF contain local structure information on both of these classes of materials. The information from PPDF or PDF and diffraction are complementary. The local structure of PDF and PPDF give the actual distances between atoms while diffraction gives the average position on crystal sites. When local disorder is present, the actual distances between atoms need not agree with that

determined by diffraction. Since the interaction of atoms, neglecting coulomb interactions, is typically of the order of interatomic distances, the PPDF is most important for accurately determining this local interaction which is strongly distance-dependent. Two types of experimental techniques are presently utilized to determine the PDF and the PPDF. They are diffuse coherent scattering and XAFS. For diffuse scattering, use of neutrons instead of x-rays is the technique of choice for determining the PDF of solids containing light elements, such as the Cu-oxide high temperature superconductors, because of the ability to better detect the distribution of the light elements. Neutrons have the added advantage of obtaining data to higher q, resulting in higher spatial resolution of the PDF. However, neutron measurements do not integrate over a large enough frequency range to be able to freeze the motion of rapidly moving atoms as required for an accurate determination of the PDF, which sometimes results in spurious results. Also, pulsed neutron sources, as needed for high spatial resolution, do not allow the clean separation between and as required for obtaining reliable PDF for some anharmonic atom motions. XAFS, similar to neutrons, has the advantage of a favorable signal from light atoms comparable to heavy atoms. This occurs because the particle being scattered in XAFS is the photoelectron. XAFS also has similar spatial resolution to neutron scattering. It has decided advantages compared to diffuse coherent scattering in ease and speed of measurement, and in accuracy and sensitivity of determining the partial PDF, especially for dilute atoms, because it can measure the partial PDF directly. However, this advantage is typically limited to the first neighbor atoms in non-crystalline materials and up to about the fourth neighbor atoms in

crystalline materials. Beyond those distances diffuse coherent scattering has the advantage. Another advantage of measuring the partial PDF directly is that XAFS can estimate the errors objectively as discussed in the previous section. Unfortunately, recent diffuse coherent scattering papers on the partial PDF do not present methods to estimate objectively the uncertainties in their results, a serious defect. An important future goal should be the development of such a method for error estimation. When unit cell dimensions are less than the range of XAFS, combining XAFS results with diffraction allowed the solving of the actual structure of the crystal, including local disorder, if present [16,17]. Knowing this structure then allows the calculation of coulomb

interactions in addition to the local interatomic interactions. 222

XAFS contains more information than the PPDF. The near edge fine structure contains

information on the electronic state of the probe atom and, in some cases, geometry of the near neighbor angular distribution. In the special case of three atoms which are within 20deg of being collinear, XAFS can measure the three-body correlations corresponding to the average of where is the angle deviation from collinearity [12,15]. Thus, the PDF, and particularly the PPDF, is an important complement to the average periodic structure determined by diffraction. When local disorder is present it is important to determine the PPDF in addition to the average structure, in order to have the complete information necessary for a correct theory, as illustrated for the case of ferroelectricity. For distances up to about the first four neighbors, XAFS is usually the premier technique to determine the PPDF in crystals. For non-crystalline matter, XAFS typically can only obtain first-neighbor information. Only XAFS can determine the PPDF for dilute atoms while only diffuse coherent scattering can determine the PDF or PPDF for intermediate distances beyond the range of XAFS.

ACKNOWLEDGEMENTS

I am grateful to Dr. Daniel Haskel for many discussions, that have helped to crystallize my thoughts on the topics presented here. Research involved in the writing of this article was partially supported by DOE grant no. DE-FG03-98ER45681. REFERENCES 1.

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Kelly, S., Ingalls, R., Stern, E.A., Voronel, A. (2000) (unpublished). Fulton, J.L, Hoffman, M.M., and Darab, J.G. (2000) Copper (I) chloride coordination structure under hydrothermal conditions up to 325°, submitted to Chem Physics Letters. Raoux, D., Flank, A.M. (1984) Local Ordering of Metallic Glasses by EXAFS and X-ray Scattering: Cu-Y alloys, in EXAFS and Near Edge Structure III K.O. Hodgson, B. Hedman and J.E. Penner-Hahn (eds), Springer-Verlag, pp. 321-26. Di Cicco, A., Taglienti, M., Minicacci, M., and Filipponi, A. (2000) Local structure of liquid and solid silver halides probed by XAFS, The 11th International Conference on XAFS, to be published. McGreevy, R.L. and Pusztai, L. (1988) Mol. Simul. 1, 359-367. Skilling, J. and Bryan, R.K. (1984) Maximum entropy image reconstruction: general algorithm Mon. R. Astron. Soc. 211, 111-125. e.g.,Brigham, E.O., (1974) The fast fourier transform Prentice-Hall, NY; Also see Stern, E.A., (1993) Number of relevent independent points in XAFS data. Phys. Rev. B48, 9825.

ZEOLITE INSTABILITY AND COLLAPSE

G.N. GREAVES Department of Physics, University of Wales, Aberystwyth, Ceredigion, SY23 3BZ, UK

INTRODUCTION It is well known that zeolites are metastable materials and that retaining their lowdensity friable structures is dependent upon maintaining the majority of the connections within the expanded alumino-silicate network. However, it is the accessibility of the massive internal structure of zeolites for heterogeneous chemistry that is responsible for their huge industrial versatility [1,2,3]. Zeolites and their microporous analogues such as ALPOs, find diverse application as ion exchange materials in agriculture, water softeners in detergents, the primary catalysts for cracking heavy oils, hosts for nanoparticle synthesis, and as the most commonly used molecular sieves and drying agents in the laboratory. Many of the processes involved in these wide-ranging applications require the chemistry to be altered locally, for instance through ion exchange, hydroxylation and calcination, but these modifications can in turn result in silicon and aluminium being removed from the network. Even bound oxygen in the network can exchange with gasphase oxygen molecules. The removal of any of the essential linkages in the network depletes its rigidity. As-prepared zeolites generally contain molecules within the structure which must be removed by modest heating (300°C - 400°C) in order to generate porosity. Water is the most common species but larger molecules are often also engaged as templates in synthesising the open network. Porosity is accomplished at modest temperatures. The removal of intra network molecules, however, puts the mechanical integrity of the whole atomic scaffolding under threat. The microporous structure can be destablisied by heating in the vicinity of the glass transition temperature, Tg, of the corresponding alumino-silicate glass [2,4]. This can also be accomplished at ambient temperature by the application of pressure [5]. When sufficient linkages are broken the low density microporous structure collapses to an amorphous alumino-silicate of higher density – a crystalline-to-amorphous transition. STRUCTURE OF ZEOLITES All zeolites share the same general formula: Mx/mm+AlxSi2-xO4.nH2O, where Mm+ is the charge compensating cation for tetrahedral AlO4+ units in the framework and AlO 4+ and SiO4 tetrahedra are linked through common bridging oxygens (BO’s) to form a completely compensated alumino-silicate network. The different zeolite structures, rather 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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like the analogous feldpsars which share the same stoichiometry, depend primarily on the Si:Al ratio but are also influenced by type of charge compensating cation, M m+ , present. This paper concentrates on Na zeolite-A (Na 12 Al 12 Si 12 O 48 ) which is one of the most wellstudied zeolites. For Na zeolite-A Si:Al = 1 and the structure is shown in Figure 1. The length of the cubic unit cell is 24.61 Å [6,7] and contains eight Na12Al12Si12O48 units. In

common with other zeolites, the low density results from the stacking together of large polyhedral units [2]. For zeolite-A these comprise sodalite or cages and double four ring cubes (D4R). Together cages D4R cubes generate the large 26-hedra or cages which can be clearly seen in Figure 1. The windows into this open atomic framework are also shown: 4.2Å for cages and 2.2Å for cages. Water molecules, whose maximum molecular dimension is < 2Å, can easily pass through the channels in zeolite A whereas

methanol (CH3OH) molecules, with an effective diameter of ~ 4Å, are adsorbed only slowly. Ethanol (C2H5OH) and glycerol (C3O3H6) are excluded, even though they could be readily accommodated within the girth of the cage. This is the action of a molecular sieve. The diameters of the and cages are in fact 11.4Å and 6.6Å respectively and these voids are largely responsible for the low density of the structure shown in Figure 1. For Na zeolite-A this is 1.52 gm.cm-3, compared to 2.62 gm.cm-3 and 2.50gm.cm-3-the densities of nepheline (NaAlSiO4) and nepheline glass. When Na zeolite-A amorphises with temperature or pressure, the density increases by more than 60%.

Figure 1. The open framework structure of zeolite A. The comers of the various polyhedra consist of Si or Al atoms whilst bridging oxygens lie along the edges linking adjacent tetrahedral SiO4 or AlO4 - units. The sizes of the channel openings are also shown [2].

AMORPHISATION

Amorphisation is the transformation of a crystalline phase to an amorphous phase without melting and vitrification. It is a progressive process that results from chemical, thermal or pressure-induced disruption of the crystalline structure [8,9]. There are

numerous examples ranging from bombardment of minerals with -particles, neutron and electron beams [10] to continuous grinding [11]. Ion implantation processes used in fabricating very-large-scale integrated circuits can also result in sufficient radiation damage so that the surface layers amorphise, which must then be annealed to regenerate electronic activity [12]. 226

Temperature-induced amorphisation of Zeolites

The temperature-induced amorphisation of zeolites is well known [2, 4,13,14, 15, 16]. Collapse occurs over a wide range of temperatures, starting with phillipsite (K2Ca1.5NaAl6Si10O32) which amorphises at ~ 250°C, extending to faujusite (Na12Ca12Mg11Al58Si134O384) whose collapse occurs at ~500°C. The most resilient are

zeolite A and ultrastable zeolite Y which amorphise at ~900°C and ~1000°C respectively [2]. It is useful to define Tamorph as the temperature at which half of the zeolite has amorphised. This is because, as will be shown later, the onset of amorphisation and its completion are both protracted and difficult to judge. The precise value of Tamorph, even for the same zeolite structure, can vary widely and is influenced by a variety of factors.

Figure 2. DSC scans at 10 deg.min-1 tracing the calcination, amorphisation and devitrification of Na zeoliteA (a), Zn exchanged zeolite-A (b) and Cd exchanged zeolite-A [4,19]. There are large endotherms between 130°C and 220°C which relate to the removal of water from the zeolite structures. Calcination is complete by ~400°C. Amorphisation can be identified by the first of the sharp exotherms, the remainder relating to recrystallisation. Note the large differences in the amorphisation temperature, Tamorph, from 900°C in Na zeolite-A to 825°C in Zn exchanged zeolite-A.

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The rate at which water is removed during calcination can lead to hydrolytic splitting of the Si-O-Al bonds (dealuminisation) and therefore a weakening of the network. This effect can be exaggerated with steam treatment which can lower Tamorph for zeolite-A

to ~350°C [15,17]. The type and size of the charge compensating cation, M m+ will also influence the temperature of collapse [18]. Figure 2 compares the DSC scans for Na zeolite-A (a), Zn (b) and Cd (c) exchanged zeolite-A. Not included are broad endotherms from 130°C to 220°C which vary with composition and which coincide with where the major water loss occurs. These are followed by sharp exotherms which mark temperatures of collapse, the peak approximating to Tamorph. For Na zeolite-A this occurs at 900°C but collapse is lowered to 825°C for Zn exchanged zeolite-A [4]. Zn2+ clearly destabilises the zeolite-A framework. Tamorph is lowered to 849°C for Cd exchanged zeolite-A, suggesting that the destabilisation caused by Cd2+ is less severe than Zn2+ [19]. Beyond the narrow amorphisation exotherm in each of the DSC scans in Figure 2 are broad endotherms which are decorated at higher temperatures with exotherms identifying devitrification processes. The amorphisation and recrystallisation process is well-illustrated by the in situ X-ray Powder Diffraction patterns illustrated in Figure 3(a) which follow the course of events for Zn exchanged zeolite-A between 595°C and 811°C

Figure 3. In situ combined XAFS-XRD experiments following the heat treatment of Zn exchanged zeolite-A [4]: XRD powder patterns (a) and Fourier transforms of zinc K-edge XAFS (b). The coordination numbers of oxygen around zinc versus the variance in Zn-O distances analysed from XAFS (c). For a heating rate of 1.7 deg.min-1, the temperature of amorphisation, Tamorph, for Zn exchanged zeolite-A is 692°C.

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[4]. Amorphisation at 692°C can be clearly identified. Whilst amorphised Na zeolite-A recrystallises totally to the feldspar nepheline (NaAlSiO4) [14,15], Zn exchanged zeolite-A converts to the spinel gahnite (ZnAl2O4) in a glass matrix [4]. By contrast, Cd exchanged zeolite-A devitrifies to an anorthite (CdAl2Si2O8) glass ceramic [19]. These differences in recrystallisation are illustrated in Figure 4 where the X-ray powder diffraction pattern for

the starting zeolite-A at room temperature is contrasted with patterns for Na zeolite-A, Zn and Cd exchanged zeolite-A after heat treatment at 1100°C [19].

Figure 4. XRD powder diffraction profiles for Na zeolite-A and exchanged with Zn and Cd with heat treatment [19]: zinc exchanged zeolite-A (a) and after treatment at 1200°C showing the formation of the

gahnite (ZnAl2O4) phase (b).

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Figure 4 (cont). The formation of nepheline (NaAlSiO4) from Na zeolite-A at 1100°C (c); the formation of an anorthite phase (CdAl2Si2O8) from Cd exchanged zeolite-A at 1000°C (d).

In situ XAFS enables the environment of exchanged cations to be followed from the starting zeolite, through amorphisation to recrystallisation [4,18,19] as illustrated in

Figure 3(b) and (c). These measurements reveal, for example, how Zn switches from octahedral coordination in the zeolite to tetrahedral coordination as the collapse commences [4], identifying the diffusion pathway of zinc from charge compensating sites in the vicinity of the cage to network sites displacing host tetrahedral cations and hence

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disrupting the network. The eventual nucleation of ZnAl 2 O 4 (Figure 3(a) and Figure 4(b))

suggests that zinc initially destabilises the zeolite-A network by preferentially removing aluminium and therefore distorting the Si:Al ratio in the surrounding crystal. In situ Cd XAFS also reveals cadmium transforming with rising temperature from 6-fold extraframework sites to tetrahedral network sites [19]. The subsequent nucleation of an

anorthite composition, however, means that cadmium destabilises the network by removing silicon and aluminium in equal numbers, leaving the local zeolite Si:Al ratio in tact. Cadmium therefore should have less effect compared to zinc, which explains the marked differences in Tamorph illustrated by the DSC scans in Figure 2.

Figure 5. The decline in the strength of zeolite-A diffraction peaks with risng temperature (a) and increasing

time (b) showing the reduction in the crystalline fraction through the process of collapse for Zn and Cd exchanged material. The solid curves show the result of fitting to the Avrami-Erofe’ev expression (Eq. (1)).

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The fact that zeolites are cheaply made in terms of energy, collapse and recrystallisation has been exploited in devising energy-saving processes for synthesising

ceramics [20,21,22]. Devitrification from these amorphous alumino-silicate sources avoids the much higher temperatures required for fabricating ceramics by annealing quenched glasses above the glass transition temperature, Tg, or by sintering crystalline powders [23]. As we have seen the capacity for altering the composition of the starting zeolite by ion exchange, can facilitate the formation of specific ceramics. Refractory dielectrics of commercial importance include anorthite itself (CaAl2Si2O8) [21] – a refractory dielectric for electronics applications and cordierite (Mg 2 Al 4 Si 5 O 18 ) [22] – the preferred support for exhaust catalysts. SOLID STATE ASPECTS OF COLLAPSE

The zeolite crystalline fraction, x, given in Figure 5(a) as a function of temperature, can also be plotted as a function of time (Figure 5(b)). Although the in situ XRD measurements are non-isothermal, the temperature rise through which the major collapse occurs is <5% of Tamorph. The overall shape of the decline of the crystalline phase and the growth of the amorphised phase (1-x) with time is characteristic of solid state nucleation and growth processes [24]. These commence with an induction or nucleation phase followed by an acceleration in growth. An inflection point at x~0.5 heralds a deceleration in growth and the process terminates on completion at x=l. In this well-developed area of

solid state chemistry there are proven rate equations for each of the above regions of which

the Avrami relation (1)

models the sigmoidal region. In Eq. (1) n equals the number of steps involved in nucleus formation plus the dimension of growing nucleating sites. Typically n lies between 2 and 4. The solid curves in Figure 5(b) are the result of fitting XRD amorphisation data for Zn and Cd exchanged zeolite-A to eq.(l). The values of n for Zn (3.4) and Cd (3.2) exchanged zeolite-A are consistent with approximately three dimensional growth following an initial nucleating period. We can equate k-1 with the characteristic

amorphisation time. is approximately the time for half the zeolite to amorphise. This is shorter for Zn (2635s) than for Cd (3361s) and reflects the greater efficiency of Zn in disrupting the zeolite-A network, implicit in XRD and XAFS experiments. In terms of solid state nucleation and growth kinetics we might expect

to

decrease as the collapse temperature, Tamorph, increases, possibly in an Arrhenius fashion. For a given composition Tamorph is indeed strongly affected by the thermal conditions. This can be clearly seen by comparing the rates of collapse for Zn and Cd exchanged zeolite-A in Figure 2 with those in Figure 5(a). With the heating rate of ~ 1.5 deg.min-1 used for in situ XRD, collapse is lowered to 692°C for Zn and 772°C for exchanged zeolite-A, compared to 825°C and 849°C respectively for the DSC heating rate of 10 deg.min-1 (Figure 2(b) and (c)). For Na zeolite-A Thomas, Lutz and others have shown that Tamorph can be reduced to 700°C after isothermal heating for 80 hours [16]. These values of Tamorph

should be compared with a collapse temperature of 900°C when Na zeolite-A is heated at 10 deg.min-1 (Figure 2(a)). As Tamorph decreases with heating rate the collapse time, rises sharply. This can extend to many hours when TamorphTg (Figure 2). These different aspects of the dynamics of temperature-induced amorphisation are drwan together in Figure 5, which takes results for and T amorph for Na zeolite-A and Cd and Zn exchanged zeolite from the literature and from unpublished data, plotting

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them as log10 versus 1/T. The differentials in the values between the three zeolite-A compositions are clearly maintained over a wide temperature range. The approximately Arrhenious behaviour yields an activation energy for collapse of ~2.5eV, a value intermediate between the magnitude expected for bond breaking processes compared to diffusion processes.

Figure 6. Comparison of amorphisation time, and the temperature of collapse, Tamorph, for Na zeoliteA [13,15,4], Cd exchanged zeolite-A [Greaves and Sankar, unpublished results] and Zn exchanged zeoliteA [4]. Error bars indicate the extent of non-isothermal conditions. Log 10 versus Tamorph-1 plots yield activation energies of ~2.5eV.

AMORPHISATION AND MELTING The huge depressions in the temperature of zeolite collapse, Tamorph, as the heating rate is reduced, however, are reminiscent of the behaviour of the glass transition temperature, Tg, with thermal history. In this context it is worth noting that Tg (at 1012 Pa.s) for nepheline glass (NaAlSiO4) is 804°C [25], approximately 100 deg less than the Tamorph of Na zeolite-A (Na12Al12Si12O48) measured with DSC at 10 deg.min-1. Conversely the viscosity of molten nepheline at Tamorph = 900°C is ~ 109 Pa.s, representative of a viscous, albeit fragile fluid. The dependence of for different compositions of zeolite-A with temperature is contrasted in Figure 6 with Toplis’ careful measurements of the viscosity, of nepheline melts [25]. It is tempting to consider the collapsing zeolite as a conventional fluid and to equate the viscosity of nepheline glass, with the viscosity of the collapsing zeolite during amorphisation. In this case the rigidity or shear modulus, Gs, can be related to the relaxation time, by Maxwell’s relation: (2)

Taking parameters from Figure 6 for the different zeolite-A compositions where Tamorph>Tg, we obtain Gs increasing from ~2MPa for Na zeolite-A to ~500MPa for Zn and Cd exchanged zeolite-A, which together with the viscosity values are indeed consistent with the rheological properties of viscous fluids. 233

Figure 7. Scanning electron micrographs of cubic zeolite-A crystals at room temperature (top) and after

amorphising at 750°C for 2 hours (bottom) [19].

The allusion of amorphisation to melting can be clearly seen in electron micrographs of zeolite-A before and after amorphisation illustrated in Figure 7. The cubic morphology of the initial zeolite prior to heat treatment is transformed to the globular shapes following annealing at 750°C, well below Tg but for 2 hours [19]. For isotropic solids Gs becomes identical with c44, the shear elastic constant. However, the much stronger temperature dependence of compared to means that for conditions where Tamorph>Tg, Eq.(2) leads to unrealistically high values of C44. Accordingly, whilst amorphisation of zeolites is clearly a Theological phenomenon, it is complicated by the very large difference in density between the zeolitic and the amorphised phases of >60%

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which results in a massive negative expansion coefficient in the vicinity of Tamorph. This is in contrast to the viscosity properties of a conventional glass where the density changes

only modestly and continuously through Tg. In common with melting, though, amorphisation can be viewed as a heterogeneous process promoted by instabilities which start at surface nucleating sites and advance inwards [8]. The micrographs shown in Figure 7 are a graphic illustration of this. In particular, even the initial stages of amorphisation in zeolites are expected to have serious consequences for the porosity of the remaining crystalline phase. Indeed Thomas and coworkers detected the onset of amorphisation in Na zeolite-A by a reduction in water

absorption [13] at temperatures as low as 550°C. With successive heating this falls rapidly to zero by 670°C. However, subsequently grinding the partially amorphised material results in much of the original porosity returning. It would appear that from the start the amorphised phase forms a film around the intact crystal, as is clearly the case when amorphisation is complete (Figure 7). Whilst this amorphised shell blocks the sorption capacity, Thomas et al report that this can be recovered if the shell is then broken by

grinding. A further increase in porosity can also be achieved with subsequent sintering, suggesting an additional “closed macoporosity”. These interesting results are reproduced

in Figure 8, where “new product phases” include devitrified as well as amorphised material.

Figure 8, Changing macroscopic sample composition during heat treatment of Na zeolite-A [13]. Proportions are determined pro rata from water absorption capacity for specimens heated for 2 hours at each of the above temperatures and subsequently ground and sintered. See text and ref. 13 for details.

COMPRESSIBILITY

Compressibility, K, is the inverse of the bulk modulus, B, and is defined to be (3)

Inorganic solids respond to the application of pressure by the contraction of interatomic distances, by changing the configuration of cation polyhedra and by tilting polyhedral linkages. The latter two processes coincide with first order phase transitions, whereas polyhedral compression can often be accommodated without a change of symmetry. 235

Indeed the compressibility of polyhedral has been shown to be independent of structure type [26], with (4)

where is the mean action-anion distance and zc and za are the cation and anion charges, respectively. This is essentially a relationship between compressibility, K, and volume and is different for different families of solids. By using scaling factors S for each anion system (S=0.5 for silicates, for instance), polyhedral compressibilities for a huge range of mineral families can be projected onto the common universal relationship (eq.(4)) plotted in Figure 9.

Figure 9. Polyhedral Compressibility (Kp-1) - Volume relationship for 4-fold 6-fold (x) and 8-fold (o) sites in a wide range of minerals [26]. The scaling factor S=0.5 for silicates, 0.75 for halides, 0.4 for chalcogenides, 0.25 for pnictides and 0.2 for carbides. Note: 1Mbar-1 = 0.0lGPa-1 .

Compressibility of zeolites

At ambient temperature the application of external pressure to zeolites can lead to “new” phase transitions with high compressibility [27,28] and eventually to collapse [5]. The changes in cubic phase are accompanied by the discontinuities in unit cell volume illustrated in Figure 10 [28] but also by “dramatic decreases in diffraction intensity” indicative of amorphisation. For example, in this classic series of high-pressure experiments on zeolite A, Hazan and Finger demonstrated that compressibility was critically dependent on the molecular size of the hydrostatic pressure medium [27]. If water was used, they found that the cubic structure of zeolite A was retained up to 4 GPa

with a compressibility of 0.007 GPa-1, comparable to that of -quartz – found in the bottom corner of Figure 9. Water molecules are sufficiently small and rigid to fill the zeolite cages. However, if an alcohol mixture was used instead, penetration was not optimum. A series of “cubic transitions” occured, starting around 2GPa, and the zeolite exhibited compressibilities in the range 0.014 – 0.028 GPa-1 – a third of the way up Figure 9. By comparsion, 0.03GPa-1 is the compressibility of silica and silicate glasses. Finally when 236

glycerol, which is impervious to zeolite A, was employed as the hydrostatic fluid much damage was done to the starting crystal and the compressibility reached 0.046 GPa-1 – half way up Figure 9 and in the zone of open-framework silicates and octahedral oxides. Accordingly as the sizes of the hydrostatic liquid molecules increase they find it less easy to enter the zeolite micropores through the channel openings (Figure 1), internal support for the open network reduces and, importantly, amorphisation becomes less reversible.

Figure 10. Compression of Na zeolite-A in different hydrostatic fluids [27]. The unit cell volume, V, is plotted as a function of pressure and the compressibility, K, given by eq. (3). K increases for water (0.007GPa-1 ), which is completely absorbed by the microporous structure, to glycerol (0.046GPa-1) which is virtually impervious.

PRESSURE-INDUCED AMORPHISATION

Amorphisation induced by pressure – pressure melting – was first discovered by Mishma in 1984 when ice I was observed to “vitrify” at 77K and 1 GPa [29]. By 1990 the

room temperature amorphisation of -quartz at 25 GPa [30] and of berlinite (A1PO4) at 15 GPa [31] had been reported. Since then the number of examples has grown and reviews of this new field can be found in refs. 8 and 9. There is a broad relationship between the pressure of amorphisation, Pamorph, and the density of the crystalline precursor at ambient pressure. Figure 11 illustrates this for a range of minerals [8] where Pamorph can be seen to scale from ~5 GPa for the natural zeolite, scolecite (Ca8Al16Si24O80.24H2O), to ~55 GPa for the dense perovskite, forsterite (Mg2SiO4). The upshot of this empirical relationship is that amorphisation demands the exisitence of higher density more stable phases.

Conversely a dense and stable ambient phase kinetically hinders the route to amorphisation. It will be already be clear that, with their intrinsically low densities and with their manifold of higher density crystalline and amorphous phases, zeolites, like scolcerite, fall in the bottom corner of Figure 11. They are clearly metastable structures. The gradual destruction of the periodic structure during the amorphisation of dense materials like ice and quartz is presumed to be initiated by mechanical instabilities

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[33,34,35,36,37]. For microporous materials these can be identified with the chemical and thermal instabilities described above in connection with temperature-induced collapse.

Figure 11. The range of pressure-induced amorphisation in silicates as a function of their ambient densities (SCO:Ca8Al16Si24O80.24H2O; SP:Mg3Si2O5(OH)4; QZ:SiO2; AN:CaAl2Si2O8; MS: KAl 2 (Si 3 Al)O 10 (OH,F) 2 ; WO: CaSiO3; FO: Mg2SiO4; EN: Mg2Si2O6) [8].

Reversibility of Amorphisation Unlike temperature-induced amorphisation, when amorphisation takes place under pressure at ambient temperature it may be reversible. The crystalline-to-amorphous transition, however, is only one-way for ice I and for -quartz [29,30]. Indeed nonhydrostatic processes during compression are believed to play an important role in engendering this irreversibility, -quartz being particularly studied in this respect [38,39,40]. Never the less, routes back to the starting crystalline state have been developed for some systems. In the case of ion-beam damaged silicon, for example, thermal epitaxial processes have been perfected to reactivate implanted dopants in vlsic’s [12]. Reversibility of amorphisation has also been reported for A1PO4 [31], giving rise to the notion of a “Memory Glass”. Molecular Dynamics calculations of AlPO4 point to the retention of rigid PO4 tetrahedra within the pressurised amorphous alumino-phosphate structure which then provide a key for reestablishing the crystalline phase when the pressure is released [32,41]. For zeolites [27,28] and other low density materials [42] it is also claimed that amorphisation can be reversed or even prevented. In these cases the pre-requisite for recovering or retaining the starting crystalline phase is that the microporous structure is internally supported by non-deformable molecular units. Figure 12 illustrates the benefits of this for a model of the microporous clathrasil dodecasil-3C or D3C [42]. In the calculations the open silicate structure is filled with “spherical guests” to mimic nondeformable molecules (a). As the pressure is increased amorphisation is fully established by 10 GPa (b). Further compression to 15 GPa densifies the amorphous structure (c), but, as the pressure is gradually released, the original crystalline structure is recovered. Irreversible Amorphisation in Zeolites If the hydrostatic fluid in the pressure cell cannot penetrate the microporous structure of the zeolite, irreversible amorphorphisation will occur. The diffraction experiments relating to the location of scolecite (Ca8Al16Si24O80.24H2O) on the Pamorph versus Density relationship (Figure 11) are reproduced in Figure 13. These are high pressure

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energy dispersive XRD measurements and used non-penetrating solid KBr as the pressure

transimitting medium. Irreversible amorphisation takes place at 6.9 GPa and this is corob-

Figure 12. Molecular Dynamics simulation of reversible amorphisation in the clathrasil D3C [42]. The microporous silicate structure is “supported” by 8 spherical guests replicating internally located molecules: ambient conditions (a); 10 GPa (b); 15 GPa (c); and after gradual depressurisation to 5 GPa.

orated by in situ Raman measurements, where spectra for amorphised material bear strong similarities to spectra for aluminosilicate glasses [5]. The amorphous phase of scolecite is retained when the pressure is removed. Similar behaviour is reported for mesolite (Na16Ca16Al48Si72O240.64H2O), another of the naturalite series. The bars in Figure 11 for scolecite and each of the other minerals recorded depict the range of Pamorph observed, when amorphisation takes place. Typically values are around 30% of the amorphisation pressure, Pamorph reaching 100% for the microporous scolecite. is attributed to pressure inhomogeneities that develop within the pressure cell as the crystalline-amorphous transition takes place [5,8]. These are also a problem during decompression and are very likely to be a major cause of the hysterisis reported in reversible systems like berlinite [32].

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Figure 13. Energy dispersive diffraction patterns for scolecite at the pressures shown [5]. Amorphisation of this natural zeolite is irreversible in the non-penetrating hydrostatic medium, solid KBr.

Being lower in desnity than scolecite, zeolite-A should collapse at even lower pressures than 5-10 GPa. Preliminary results for Zn exchanged zeolite-A are plotted in Figure 14 [43]. Unlike Gillet and co-workers’ experiments [5], these XRD patterns are of higher resolution and were obtained in angle-dispersed geometry using image plate detection. The reduction in the intensity of the diffraction lines with increasing pressure indicates the onset of amorphisation. This is matched by the increase in the diffuse background resulting form the accumulating amorphous phase. By 4.8 GPa the Zn exchanged zeolite-A is fully collapsed and remains so on decompresion to ambient pressure. It is clear from Figure 15 that the diffraction lines shift to higher angles between ambient and 2 GPa, indicative of crystalline compressibility. Indeed K ~ 0.07 GPa-1 for Zn exchanged zeolite-A pressurised in silicone fluid, similar to the value reported by Hazan and Finger for Na zeolite-A pressurised in glycerol [28]. The decline in XRD diffraction intensity for pressure-induced amorphisation is comaparable in profile to that for temperature-induced amorphisation with time (Figure 5(b)). This points to a similar sequence under compression, viz: nucleation, accelerated growth, an inflection point, decelerated growth and termination when all of the zeolite has collapsed. Likewise an Avramiesque expression can be employed to fit the process of pressure-induced amorphisation. Defining Pamorph by analogy with Tamorph, as the pressure when half the crystal is amorphised (x=0.5), Pamorph can be easily interpolted from the experimental points. A value of 2.5 GPa is obtained for Zn exchanged zeolite-A and, interestingly, a higher value of 3.9 GPa for Cd exchanged zeolite-A [43]. These differences in Pamorph mirror differences in Tamorph discussed earlier, suggesting that the different instabilities created by the two charge compensating cations at high temepratures may well be present as different mechanical instabilities under compression at ambient temperature. The massive difference in density between the starting crystal and amorphous feldspar mean that pressure-induced amorphisation is accompanied by a huge rise in the compressibility of the combined system from the crystalline value as Pamorph is approached. This rises from the “unsupported” zeolite-A value of ~0.05 GPa-1 and returns to ~0.02 GPa-1, typical of a aluminosilicate glass [43].

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From the picture of solid-state melting developed earlier for temperature-induced amorphisation, the effect of pressure is expected to impinge at the surface of zeolite crystals and work inwards, as the electron micrographs in Figure 7 suggest for amorphisation with temperature. Accordingly, considerable pressure inhomogeneities are expected from the huge differences in density between the crystalline and amorphous state.

In particular, if the amorphised shell is rigid it will serve to reduce the pressure in the crystalline core. The initial positive compressibility, evident from the XRD patterns in Figure 14, switches sign as the pressure advances [43], which serves to explain the pressure imhomogeneities observed by Gillet and others [5,8]. The large uncertainties,

evident in Figure 11 [8] will of course be compounded if Pamorph is associted with complete amorphisation, the sigmoidal tail-off as the growth of the amorphous phase is completed is difficult to pin-point.

Figure 14. Image plate XRD patterns for Zn exchanged zeolite-A at the pressures shown [Greaves, Sapelkin and Sankar, unpublished results]. The hydrostatic fluid silicone fluid does not enter the microporous structure so amorphisation is irreversible.

Finally it is worth looking again at the “reversible compressibility” results for Na zeolite-A (Figure 10) [28], where discontinuities in crystalline compressibility occur at pressures in the range 1 to 3 GPa, pressures over which Zn and Cd exchanged zeolite-A are collapsing [43]. Hazen reports considerable time variations in the high pressure crystallography of up to 100h in traversing between the different phases. These could well

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reflect the gradual removal of pressure inhomogeneities between the external amorphised shell and the internal crystal.

MECHANICAL RIGIDITY AND ZEOLITE COLLAPSE

Zeolite collapse exhibits all the hallmarks of the ridity percolation threshold [44]. This coincides with the elastic constant c\\ of a solid falling to zero and with an abrupt rise

its compressibility, K. As we have seen, when zeolites collapse by temperature-induced amorphisation, c44 for the crystalline-amorphous composite falls to fluid-like values. When collapse occurs under pressure at ambient temperature, K rises sharply with

pressure. Since c 11 = 1/K + 4c44/3 for isotropic materials, then, when K approaches infinity and c44 zero , c11 must also fall to zero. In the context of an atomic framework, a material becomes a fluid when the number of constraints per atom, nc, falls below the number of degrees of freedom that it enjoys, nd.

In a solid nc generally exceeds nd and the rigidity percolation threshold occurs when the nc = nd [45]. When nc < nd, zero frequency or “floppy” modes are present, the fraction being given by (nd - n c ). As nc can be re-expressed in terms of the average coordination number , the fraction of zero frequency modes, f, can be written [44] (5)

where the average coordination number is defined to be (6) At the rigidity percolation threshold f=0, so the average coordination number

(7) For the 3D aluminosilicates we are considering – zeolites, fully amorphised zeolites or recrystallised feldspars like nepheline etc. – the average coordination number is equal to that of -quartz, which equals 8/3 [45]. However, for networks which contain large numbers of 2-fold coordinated anions (oxygen and the chalcogenides, S, Se, Te) and/or cations whose coordination number is less than 4 (like the pnictides, P, As, Sb), can fall to 2.4 or less. For “incomplete networks” which include terminal bonds or singly

coordinated atoms, the average coordination of the remaining atoms in the network can also reduce to 2.4 or less. Hydrogenated amorphous semiconductors like a-Si:H or a-C:H are a case in point. Here, covalent linkages are “broken” by singly coordinated hydrogens. In this event Thorpe has introduced the concept of the “skeleton” network, where only complete linkages are included in counting constaints [44]. By ignoring non-bridging linkages, this approach enables the above criterion for defining the rigidity percolation threshold ((eq. (5,7)) to be extended to incomplete networks. Applying the skeleton approach to an incomplete silicate or aluminosilicate

network, i.e. one that incorporates terminal or non-bridging oxygens, we get from eq.(6)

where x4 = fraction of Si/Al sites, x2 = fraction of bridging oxygens and x1 = fraction of non-brdging or terminal oxygens. Recognising that x4 + x2 + x1 = 1, the average coordiantion number for network atoms is given by (8) 242

In most minerals and stable aluminosilicate glasses, oxygen bridges are complete and x1 = 0 and = 2.67 which is also the value for -quartz. (When bon-bending constraints are removed, as has been proposed for silica glass, this will fall to 2.4 [46].) For modified crystalline silicates, however, where < 8/3. In particular for disilicates, of which -Na2Si2O5 is an example, x4 = 2/7, x4 = 3/7 and x1 = 2/7. From eq.(8) = 2.4, which marks the rigidity percolation threshold (eq.(7). For these compositions, there is one non-bridging oxygen per tetrahedral site on average, i.e. all silicons occupy Q3 configurations. Crystalline disilicates are 2D layered structures and the melting points of disilicates are massively reduced compared to molten quartz or indeed silica [46], i.e. nc<3. As oxygen linkages are gradually removed from a zeolite structure by nucleation, thermal or mechanical instabilities, we expect that the 3D rigidity of the starting structure will be irreversibly lost when the equivalent proportions for a disilicate structure composition are achieved locally. Taking the collapse of Na zeolite-A as an example, for some intermediate stage of amorphisation we can write: (9)

where (1 – x) is the fraction of low density zeolite that has collapsed to a high density amorphous nepheline. During the nucleation process local oxygens from each component will be shared. It can be shown that the ruptured zeolite reaches the point where each tetrahedral site has a Q3 configuration when x ~ 0.2.

For temperature-induced

amorphisation, it is clear from Figure 5 that amorphisation accelerates around this point. The same is true for pressure-induced amorphisation [43]. It is also clear that amorphisation induced either way decelerates when x ~ 0.8. There are, then, two sides to the amorphisation process - “collapse” of the low density crystal and “re-assembly” of the higher density amorphous phase. In between, the average coordination number will be less than 2.4, nc will fall below nd and zero frequency modes will emerge. This provides a natural explaination of the sharp exotherms that mark amorphisation (Figure 2) and which are so conter-intuitive with respect to

conventional melting, which is strongly endothermic. In particular, if is the enthalpy of zeolite collapse, we can equate the fraction of floppy modes created, f, with Hcollapse/3RTamorph. From eq.(5) the minimum value of average coordination number, min, will approximately be given by

(10) If falls to 2.2, Hcollpase/3RTamorph = 0.17 and taking Tamorph = 900°C gives

kJ.mol-1. This value is close to the enthalpies associated with the exotherms shown in Figure 2 for Na, Zn and Cd zeolite-A. Each is followed by a broad endotherm which is expected as f returns to zero and rises back to 2.4, eventually reaching 2.67 as the fully connected alumino-silicate is formed. Interpretting this endothermic re-assembly process is complicated, however, by the entropic nature of the crystalline-to-amorphous transition and by the exothermic devitrification processes that are beginning to start. EXPERIMENTAL POSTSCRIPT

Many of the results described in this review have come about very much in the wake of the development of new light sources and new experimental techniques. The brilliance of synchrotron radiation enables diamond anvil cells and multi-anvil presses to be penetrated so that X-ray diffraction can be measured routinely at high pressures and also

at high temperatures [8,9,47]. High temperature X-ray measurements have necessitated the development of furnaces with sufficient window area for angle dispersed XRD and also for accessing other X-ray probes [48]. With tunability XAFS measurements can be made 243

at the high temperatures encountered in temperature-induced amorphisation and these can be combined with XRD [4,18,22,49].

As we have seen, XAFS enables dopant

environements in zeolites to be followed through the course of collapse and recrystallisation, the crystallography being monitored with XRD. In situ Small Angle Xray Scattering (SAXS) combined with XRD [50] facilitates the observation of the microstructural changes that accompany zeolite amorphisation [51,52]. Because amorphisation is a complex phenomenon and often embroiled with the uncertainties of hysteresis, multiple experimental probes offer the most consistent approach. Laboratory techniques like Raman and IR [8] are especially complementary to X-ray technqiues as is ionic conductivity [53]. Bringing some or all these together is technically demanding but the returns would be immense. ACKNOWLEDGEMENTS

Thanks are particularly extended to G. Sankar and A. Sapelkin for their skill in applying combined X-ray techniques and high pressure techniques to the study of zeolite collapse. L. Colyer and C. Alétru are also acknowledged for their willingness to try new things and to analyse the consequences.

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deformable units in pressure-induced amorphization of clathrasils, Nature, 369, 724-726 43. Greaves, G.N., Sapelkin, A. and Sankar, G., unpublished results 44. Thorpe, M.F. (1995) Bulk and surface floppy modes, J. Non-Cryst. Solids, 182, 135-142 45.

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THERMODYNAMICS AND TRANSPORT PROPERTIES OF INTERACTING SYSTEMS WITH LOCALIZED ELECTRONS

A.L. EFROS Department of Physics, University of Utah, Salt Lake City, UT 84112 USA

INTRODUCTION The problem of strongly correlated electrons is a focus of modern condensed matter physics. Nobody doubts that the fractional quantum Hall effect results from electron-electron interaction. Many people thing that this interaction is also the origin of the high temperature superconductivity and the two-dimensional Insulator-Metal transition. In this paper we concentrate on a problem of localized in space interacting electrons in a disordered system. The impurity bands of lightly doped semiconductors, the two-dimensional electron gas in different structures in an insulating regime are common examples of such systems. The study of electron-electron interaction in localized regime has been initiated by M. Pollak [1] and G. Srinivasan [2], Efros and Shklovskii [3],[4] have argued that the single particle density of states (DS) tends to zero at the Fermi level due to the long range part of the Coulomb interaction which, in a sense, remains non-screened. They proposed the following universal soft gap of the DS near the Fermi level at T = 0 which is called the Coulomb gap. (1) (2)

The reference point for the energy is the Fermi level. The universality means that the above DS is constructed from the energy and from the electron charge e. Note that this is the only combination of and e with a proper dimensionality both in 2D and 3D cases. We assume here and below that the dielectric constant of a lattice is included into electron charge. The lack of screening is connected with vanishing of the DS at the Fermi level. In this case the screening radius should depend on the magnitude of the electric field which is subjected to screening. Both the Coulomb gap and the non-linear screening result from the Coulomb law and from the discrete nature of electron charge. The question of screening in a

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

247

system of localized interacting electrons has been considered in details by Baranovskii et al [5]. The main tool for a quantitative study of the Coulomb gap is the computer simulation

[6-14], which mostly confirm the above results on the DS. However, some deviations from the universal behavior have been reported [9,13] and even the main concept has been equivocal [15]. We think that in 2D case these deviations can be understood in terms of the work [14], which shows that the universal physics of the Coulomb gap in the 2D case is valid at large disorder only. The apparent proximity to the universal result has been observed earlier because the simulations were restricted by a relatively high values of a disorder. A renormalization group calculations, performed by Johnson and Khmelnitskii [16], confirmed Eqs.(1-2) for strong disorder and the results of Ref. [14] on the crossover from strong to weak disorder. The experimental manifestations of the Coulomb gap are mostly variable range hopping conductivity (VRH) and tunneling experiments. Without interaction the VRH conductivity obeys the Mott law [17]

(3) Here s = 1/4, 1/3 for three-dimensional (3D) and two-dimensional (2D) systems respectively, where G 0 is the DS at the Fermi level, is localization length, is a numerical coefficient which depends on the space dimensionality D. It is obvious that Eq. (3) can not be valid if the Coulomb interaction makes G 0 = 0. In the next section we present the derivation by Efros and Shklovskii [3], which gives the law

(4) for D = 2, 3. Here the space dimensionality.

where

is another numerical coefficient which depends on

In some cases, like in neutron-transmutation-doped germanium [18], one can be sure that experimental data closely follow the Efros-Shklovskii (ES) law while in some other

cases, where conductivity changes only by a few orders of magnitude, it is difficult to distinguish between the two laws. We think that the most important results have been obtained

recently by groups of Jiang and Dahm [19] and Adkins [20]. They have proved that the transport they observed in 2D case reflects the crucial feature of the Coulomb gap, the lack of screening in a system of localized electrons. To do that they used a metallic electrode (gate) parallel to the 2D electron gas at a distance d from the gas. This electrode provides an extrascreening of the Coulomb interaction. Namely, at a distance, larger than d, the interaction

between electrons becomes dipole-dipole due to the image charges in the metal. Then, the Coulomb gap becomes smeared and the DS is energy independent at Therefore, the conductivity obeys the ES law at high temperatures and the Mott law at low temperatures. Note, that without an external screening the Mott law may be valid at high temperatures where the relevant electronic states are outside the Coulomb gap while the ES law is valid at low temperatures. The experimental observation of the completely different and very unusual temperature dependence in the gated structures confirms that the the long-range interaction plays an important role in the VRH. The study of the d- dependence of the VRH gives even stronger arguments in favor of the Coulomb gap. The direct observation of the Coulomb gap in tunneling has been claimed by Lee and Massey [21-22]. The glassy properties are another important manifestation of the electron-electron interaction in a system with localized electrons. Davies, Lee and Rice [8] where the first to rise

this problem theoretically. They have coined a term “electron glass” which have survived until now, sometimes with a transformation to “Coulomb glass”. These terms have to stress

248

the relation of an electron system and a spin glass system. Davies et al. introduced an analog of the Edwards-Anderson order parameter, but their calculations already showed that this parameter is, probably, non-zero at any temperature. The absence of such parameter in the system with an external (or built-in) disorder has been pointed out by many researches (See [23] and references therein). However, the glass transition may exist even without EdwardsAnderson order parameter. The new experiments, started by the group of Ovadyahu in 1993 [24,25], definitely show the relation of this system and the ordinary glasses. In these experiments one could control the electron density in the films of indium oxide by the gate voltage Vg. In some density range

the mechanism of a conductivity is the VRH, which means that electron states are localized. The most important result is the discovery of a memory in this range. The sample, slowly cooled down at some Vg = V0, memorizes the value of V0. Conductance at low temperature, measured as a function of Vg, has a small minimum at V0. The minimum disappears with time very slowly. The characteristic time is up to 15 hours and increases in a magnetic field.

Similar phenomena have been observed by the group of A. Goldman [26] on ultrathin films of metals near the superconductor-insulator transition. Slow relaxation has been demonstrated by Don Monroe et al. [27] in compensated GaAs. The analogy with the ordinary dielectric glasses can be seen from the data of Osheroff group [28], where similar effect in the capacitance of silicon oxide has been reported. As the first step toward the understanding of the glassy effects we have studied the thermodynamic properties of the Coulomb glass[29]. The paper is organized as follows. In the second section we give a simple derivation of the Coulomb gap and of the VRH of interacting electrons. In the third section we concentrate on the glassy properties of an electron system.

COULOMB GAP AND VRH CONDUCTIVITY

Hamiltonian and Some Exact Properties We consider here a standard classical Coulomb glass Hamiltonian [4]

(5) The electrons occupy sites on a lattice, ni = 0, 1 are the occupation numbers of these sites and rij is the distance between sites i and j. The quenched random site energies are distributed

uniformly within the interval [ – A , A ]. To make the system neutral each site has a positive background charge ve, where v is the average occupation number, i.e. the filling factor of the lattice. If v = 1/2, the chemical potential µ is zero at all temperatures due to electron-hole symmetry. Hereafter we take the lattice constant l as our unit length and e2/l as our unit energy. Using these units, the single particle energy at site i is given by

(6) An important result for the Coulomb glass Hamiltonian is that the average occupation number of a site with energy is given by the Fermi function. This result can be obtained from a self-consistent equation derived in Ref. [30]. It has been also mentioned as an exact result in

Ref. [31]. Due to the strong electron-electron interaction this is not obvious, and we provide a short proof here, which has never been published before. The average occupation number 249

of sites with energy can be calculated by considering a single site i = 1 and calculating it’s average occupation number < n1 > subject to the constraint that it has the required energy By definition, this is given by

(7) The Hamiltonian Eq. (5) can be written in the form

where

does not depend

on n1. This enables us to separate out Tr1 which is the trace over the variable n1 = 0, 1, thus obtaining

(8) Here Tr´ and stand for the trace and sum over all ni except n1. From Eq. (8) we readily obtain the Fermi function

(9)

Coulomb Gap in a Single-Particle Approximation

The Coulomb gap in the DS around the Fermi level can be derived in a simple manner [32]. At zero temperature the distribution of electrons over the lattice sites is determined by the condition of minimum H at a given total number of electrons. Equivalently, one can look for the unconditional minimum of the functional (10) The single-particle energies

have the reference point at the Fermi level µ

(11)

The functional

should be minimized with respect to simultaneous changes of any amount

of occupation numbers ni. It is easy to see that the change of one occupation number gives the condition and which is equivalent to the regular definition of the Fermi level µ in a non-interacting Fermi gas. In the next approximation we consider the transfer of one electron from the site i, occupied in the ground state, to the site j, which is vacant in the ground state. The energy increment of is positive if for any pair of such sites

(12)

where Rij = ri – rj. The last term in Eq. (12) reflects a simple fact that the ground state energy includes the potential of electron, which is initially at site i. One can see the origin of the Coulomb gap from Eq. (12). Since and the first two terms give a positive contribution, while the third term is negative. Thus, the

sites with energy close to the Fermi level should be separated in space. Consider sites whose energies fall in a narrow band around the Fermi level. According to Eq. (12), any two sites in this band with energies on the opposite sides of the Fermi level must be 250

separated by a distance Rij not less than

Therefore the concentration of such sites

cannot exceed where space dimensionality D = 2, 3. Then, the DS must vanish at at least as fast as It is clear that in the approximation based on Eq. (12), no faster law can arise than Indeed, if it were the case, the interaction energy between the sites would be much less than their energies Such a week interaction could not be responsible for lowering of the DS. It is obvious from the above derivation that the long range Coulomb interaction is a crucial condition for the Coulomb gap. The DS at the energy is determined by interaction at a distance It has been show by computations that if the localized electrons are embedded into some conducting medium which is able to screen interaction, the Coulomb gap disappears [10]. This consideration leads to Eqs. (1,2), where numerical coefficients are obtained from the mean-field equation [4,10]. The Eqs. (1,2) are valid if where G0 is a bare DS without interaction. In our model it is 1 / A l D , where l is the lattice constant. Thus, at large A the width of the Coulomb gap is Eg = ( e 2 / l ) D ( 1 / A ) D – 1 . At the DS is close to G0. The applicability of the above consideration is limited by the condition or

in dimensionless units

The crossover between large and small disorder has been

considered in Ref. [14] for D = 2. It has been shown that A = 1 can be considered as a large

A, since substantial deviation from the law Eq. (1) appears at very low energy only. The dimensionality D = 1 is a marginal for the Coulomb gap. In this case one gets [33] (13) Thus, DS tends to zero at

as

This result is a starting point for an

expansion

used in Ref. [16]. At finite temperature, it has been shown [8,10] that the gap is smeared at energies smaller than the temperature. Roughly speaking, Interaction of Excitations Until now we have taken into account only conditions of the minimum of with respect to change of one and two occupation numbers. What about many electron transitions? Do they present an extra restrictions on the DS or everything is already taken into account? The analysis of this questions can be done in terms of interaction of dipole and charge excitations. It shows that in the 2D case the single-electron approach, used above, is, probably, good,

while in the 3D case physics is more difficult. All single-electron excitations are electron transitions from the sites, which are occupied in the the ground state, to the empty sites. The energy of the transition of one electron is given by Eq. (12).

For any transition from the ground state the energy should be positive. The result of each transition is an “electron-hole pair”. If the pair is just two independent single-particle excitations: an extra electron on a site j and an extra hole on a site i. They participate in the VRH as the carriers and they are described by the single-particle DS. If the pair is rather an exciton with a large binding energy. From electrostatic

point of view this is a small dipole. A dipole can be also formed by transitions of many electrons in a small region. Such a soft excitation has many-electron nature. At small these dipoles are spatially separated from each other and they do not participate in dc. However, they contribute to the ac and they are responsible for the low temperature thermodynamics of the system. The interaction of these dipoles has been carefully studied (See review [10]) and has been found not very important even in the 3D case. Assuming that the DS of the dipoles

251

is a weak function of energy, one gets that the specific heat provided by these excitations should be approximately linear in temperature. The typical size of a pair is r0 = e2/Eg. The crucial question now is the interaction of the single particle excitations (carriers) with this dipoles. Suppose, an extra electron enters the system or a pair with a very large R is excited. In both cases an extra electron appears at some point which we take as the origin. This electron creates an electric field E = e/r2. The field causes electron transitions leading to a new ground state. At a large distance the field is small and transitions occur within the pairs with a small excitation energy. The transition occurs if Here eR is the dipole moment of the pair with the excitation energy As a result, a pair becomes polarized by the

electric field. It is easy to show that the total number of dipole pairs polarized by this field at a distance r is of the order of (r/r0 ) (D – 2) where D is space dimensionality. In these estimates we use for the DS of pairs g = G0 and we ignore the logarithmic factor in the DS, which may be due to the interaction of pairs in 3D case [10]. At D = 2 the interaction between carriers and dipoles is not very important. We think,

however, that it creates a persistent drift between the pseudoground states observed recently [29] (See the next section). However, at D = 3 the number of polarized pairs is large and they create a polaronic

shift of the order of the width of the Coulomb gap Eg [10]. That is why in 1980 we came to conclusion [34] that carriers have very different nature in 3D- and 2D-cases and that in 3Dcase they are rather “electronic polarons” than individual electrons. It follows straightforward [10] that for D = 3 the single-particle DS has a form instead of Eq. (2).

It is obvious now that the concept of “electron polarons” does not correspond to reality. The most accurate computational results [9] are inconsistent with exponential behavior of the DS in 3D-case. Experimental data also do not show any signature of the polaronic effect. Thus, we have a contradiction in our understanding of the 3D-case. We would like to propose a new picture of the low energy excitations in the 3D-case [35]. The basis of this approach is a generalized universality principle. Assume that at large disorder the DS of pairs with given energy and length is universal. We are considering now the DS of all low energy excitations at a given length R, in the interval of which are either carriers or dipoles. In some way this description takes into account interaction between pairs of all sizes, and the proposed DS is a result of this interaction. In 3D-case only the expression

(14) is universal and obeys the proper dimensionality(1/energy.volume). Universality means that

Eq. (14) contains only the charge of electron. Here

and q are some numerical constants,

and is a positive energy of an excitation. Eq. (14) is valid if both and e2/R are less than Eg. At large R we get parabolic DS for carriers, which coincide with Eq. (2). At small R one gets for the DS of short pairs Then, specific heat CV/T ~ Tq.

To choose q we should assume that the polaronic effect is absent. Otherwise Eq. (14) would be inconsistent because of the polaronic shift in a single-particle DS. It follows from Eq.(14) that the total number of pairs Np with the length R polarized by one extra electron at a distance smaller than r is

(15)

The concept of constant DS for small dipole pairs and of a strong polaronic effect corresponds to q = 0. For the new scenario we choose q = 1/2. Then the dipoles are not interacting with electrons in all scales or, to be precise, q = 1/2 is a marginal exponent for the interaction. In this sense the picture is the same as in 2D case. One should mention at this point that we try to avoid the linear divergence in Np at large distances and, on this stage, we do not care about possible logarithmic factors.

252

Finally, the new scenario for 3D-case gives that the DS of short pairs The pairs with R = r0 = e2/Eg give the major contribution to thermodynamics. The DS of carriers obeys Eq. (2). An appealing feature of the proposed scenario is the unification of all strong interactions. Consider all dipoles and carriers with the energy less than The distance between the dipoles is of order of At small ε this distance is much less than the distance between carriers However, the interaction between the dipole with length R and the nearest carrier is of the order of which is of the same order as interaction between the nearest carriers. Thus, all important interactions become of the same order. If this scenario is correct, one should expect that specific heat CV ~ T 3/2. It might be possible to check it experimentally in lightly doped semiconductors. It is possible also to check it by a standard computer modeling at finite T. Such a modeling has been done before by different methods [23,36-37]. The results show superlinear temperature dependence. Mobius and Pollak [37] got CV ~ Tp with p 1.8 ± 0.2. This is close to the result we expect. The main problem for the modeling is a large size effect due to the long-range interaction. As far as we know, the size effect in the specific heat has never been studied systematically. It can be described in the framework of the above scenario. The macroscopic regime

appears when the size of the system is much larger than the average distance between the pairs

We predict that in a sample L × L × L

(16)

where f(x) is some constant at

and f(x) ~ x at

The scaling law, given by Eq.

(16), is the best check of the proposed scenario. For this purpose one should neither go very

low in temperature no1/2increase the size very strongly. One should just show that CV/T is a universal function of T L. In the same finite temperature computer modeling one can find directly the form of the function An experimental evidence for this scenario may come from the measurements of the ac conductivity of disordered systems. The theory of ac conductivity goes back to the famous papers by Pollak and Geballe [38] and Austin and Mott [39]. It has been reconstructed by Efros and Shklovskii [10] taking into account interaction. However, the constant DS of short pairs has been assumed. If the q = 1/2-scenario will be proved, all the theory should be reconstructed again. This reconstruction will substantially change the temperature and frequency dependencies of the conductivity. We understand that the Nature might be more sophisticated than any scenario. Say,

another possibility for the 3D case would be a glass transition at finite temperature. Finally, we think that the problem of the DS in 3D case is very important and still unsolved. Variable Range Hopping Conductivity (VRH) In 1968 N. F. Mott [17] found out that at sufficiently low temperatures hopping conduction results from the states, whose energies are in a narrow band around the Fermi level. With decreasing temperature the width of the band decreases and the hopping length increases. That is why this mechanism is called VRH. Mott assumed that the DS in the relevant band near the Fermi level is energy independent. To repeat his calculations assume that the hops are allowed within the band (µ + w, µ – w) only. The concentration of sites within the band is Nw = 2wG0 and the average distance between them is Hopping rate has a form (17)

253

where R is the distance between the two sites and is the localization length. The first term in the exponent is responsible for the tunneling, while the second one describes activation. We do not care now about numerical factors which cannot be obtained by this way. To estimate the logarithm of effective conductivity due to the hops inside the chosen band one may substitute an average distance between the sites instead of R. Then (18)

The tunneling term is small at large w, while activation prefers small w. The exponent has a maximum at This is the width of the VRH band. At this value of w one gets the Mott law Eq. (3). To calculate numerical coefficient one should solve the corresponding percolation problem [40]. To take into account the Coulomb gap near the Fermi level one should use (19)

Repeating similar calculations, one gets Eq. (4). Note that in this case Therefore we may not to take into account the smearing of the Coulomb gap due to the finite temperature. The first principal computer simulation of the VRH in the interacting system is an extremely difficult problem because of a strong size effect and a huge dispersion of transition rates given by Eq. (17). We think that all attempts to solve this problem, which we know, [7,41,42] are not satisfactory due to different reasons. In Ref. [7] we artificially slow down the transition rates of small pairs, while in Ref.[41,42] the size of a system is too small. The experimental situation is described in the Introduction. THERMODYNAMIC FLUCTUATIONS OF SITE ENERGIES AND OCCUPATION NUMBERS In this section we are trying to understand the nature of a glassy properties observed experimentally in Ref. [24-27]. This part is based upon our works [29,43]. Another important effect, coming from the interaction, is that the phase space of the many particle system has the so-called pseudoground states (PS) which were first described by Baranovskii et al. [6] and then studied by many authors [44-46] in connection with the long range relaxation. The PS are the states with the total energies very close to one another but with very different sets of the occupation numbers. Thus, many electrons have to be transferred to go from one state to another. Each state presents a local minimum of the total energy and transition from one state to another requires passing huge barriers, if electrons are moving by small groups one after another. We understand the slow relaxation of a glassy system as traveling around different PS which is hindered by barriers between them. We study the system of interacting electrons in thermal equilibrium, where it has a possibility (long enough time) to wander around many PS. Such a wandering is accompanied by fluctuation of site occupation numbers. The main manifestation of these fluctuations is that the configuration of occupied sites within the Coulomb gap changes with time, even though the shape of the gap itself is time independent. This persistent change of the configuration of occupied sites occurs even at temperatures which are much lower then the Coulomb gap width. A related effect is the fluctuations in the site energies, the magnitude of which is much larger than the temperature. A crucial ingredient of the above picture is the assumption that there is no finite temperature thermodynamic glass transition in the system. In the thermodynamic limit, such a 254

transition would prevent the system from reaching thermal equilibrium below the transition temperature, thus limiting the validity of our results to finite sized samples. In fact, no such transition has been observed either experimentally or numerically in the two dimensional (2D) Coulomb glass. Furthermore, our system has much in common with various 2D spin glass models, where there is a strong numerical evidence that no finite temperature thermodynamic transition occurs [47]. In the work [43] we present results which support such a conclusion for the Coulomb 2D glass as well, While we believe that the 2D Coulomb glass can always reach thermal equilibrium,

the experimentally observed glassy dynamics [24-25] indicate that at low temperatures the equilibration time of the system becomes very long. This results from the large energy barriers that need to be traversed in order for the system to drift amongst the different PS’s. As the aim of the current work is to study thermodynamic fluctuations, it is necessary to design the simulations so that the system equilibrates within a computationally feasible time scale. Since the standard Coulomb glass Hamiltonian itself does not contain any dynamics, we employ dynamics which differ from the physical dynamics of a typical system, but are significantly faster. Namely, we assume that the transition rate is independent of a distance between sites. Therefore, any non-equilibrium phenomena, and particularly transport, cannot be studied directly by the methods discussed here.

Spectral Diffusion Now we turn to the central topic of this Section, which is the study of the thermodynamic fluctuations within the Coulomb gap. These fluctuations can be seen in a few ways. One is

the time dependence of the single particle energies, which we call spectral diffusion. Our computer simulations use the standard Metropolis algorithm, where the rate of a hopping transition depends only on the energy difference between initial and final config-

urations. The simulations were performed on a square lattice of L × L sites with periodic boundary conditions. In this torus geometry, the distance between any two sites is taken as the length of the shortest path between them. The filling factor v = 1/2. All results obtained were averaged over P different sets of the quenched random energies Unless stated otherwise the value P = 100 was used throughout. Furthermore, it was verified that all our results saturate as a function of the system size. To study the spectral diffusion, we first equilibrate the system for tw MC sweeps. ( We define a single MC sweep as a series of N = L2/2 consecutive MC attempts). Then we mark all the sites whose single particle energies are in a narrow interval [Ec – W, Ec + W] within

the Coulomb gap as ”test sites”. We follow the evolution of the distribution of these energies as the simulation proceeds. We observe that after some number of MC steps, this distribution becomes time-independent. We have also checked that the final form of the distribution does not depend on the initial time tw. This asymptotic form is shown in Fig. 1 for Ec = 0, W = 0.1, at two different temperatures. Note that the final energy distribution covers most of the Coulomb gap, although the initial distribution (shown by arrows in Fig. 1) is centered in a small region at the center of the gap. This is in spite of the fact that the temperature is much smaller than the gap width. We have also studied the dependence of the final distribution on W, and have found that the results are independent of W for W < 0.1. Also shown in Fig. 1 is an example where the initial distribution is asymmetric, namely In this case, the asymptotic distribution is also asymmetric, however it is nearly as broad as the symmetric distributions. Moreover, while the initial asymmetric distribution consists of only sites with positive energies (unoccupied

sites), the final distribution also includes sites with negative energies (occupied sites). Another way to observe spectral diffusion is to measure the time average of the singleparticle energy at site i, and the standard deviation at the same site, We perform this calculation for all sites and create a function This function is shown 255

Figure 1. Final energy distribution of sites initially in the energy range [Ec – W, Ec + W]. Diamonds are for Ec = 0, W = 0.1, T = 0.05. Squares are for Ec = 0.3, W = 0.1, T = 0.05. Triangles are for T = 0.1, Ec = 0 and W = 0.1. The arrows mark the positions of the two i n i t i a l distributions of test sites which were used. All results are for A = 1.

in Fig. 2 for A = 1 and several temperatures. It is found that for all sites, the standard deviation of the single-particle energies is much larger than the temperature. Moreover, for

sites with energies near the Fermi level the standard deviation is up to two times larger than for other sites. We understand this picture in the following way. Sites with large are expected to be “active” sites which change their occupation frequently, as their energies cross the Fermi

level. The changes of the occupation numbers of these sites are accompanied by a reorganization of the local configuration of occupied sites, which in turn is responsible for the larger value of in a polaron manner. On the other hand, the sites with smaller are “passive” sites, and they change their energy only in response to the random time dependent potential

created by the active sites. In the context of passive and active sites, it is instructive to define the quantity Ew as the energy at which The meaning of this is that sites which satisfy have energy fluctuations larger then their average energy, and therefore are active. From Fig. 2 it is also apparent that these sites have larger value of our understanding that these are indeed the active sites of the system.

thus supporting

The width of the maximum in Fig. 2 may indicate that the active sites are predominantly within the Coulomb gap. This is reasonable, since the occupation number of sites within the

gap is strongly affected by interactions. However, at A = 1 all characteristic energies, including the gap width, are of the same order. To check whether the active sites are indeed within the gap, we estimate the dependence of on A for A > 1 and compare it with simulations. The width of the gap Eg decreases with A as Eg ~ 1/A. The electron density within the gap is If active sites make up a finite portion of all sites in the gap, they create time dependent potential with the mean square value We cut off the logarithmic integral at the screening radius which is proportional to the reciprocal density of states A. We make simulations for and plot the results of simulations in the scale against A. The temperature for each value of A is T = 0.05/A, keeping it constant in units of the gap width. If our hypothesis is correct we can chose parameter b in such a way that all curves collapse in one at least for passive sites. One can see

256

Figure 2. Site energy standard deviation as a function of site average energy for various values of A and T. For A > 1 the temperature is given by T = 0 . 0 5 / A . Only positive energies are shown due to particle hole symmetry.

from Fig.2 that indeed all curves collapse in one for b = 1.5. The temperature dependence of the results of Fig. 2 is presented in Fig. 3. The curves marked and show the maximal and minimal values of the standard deviation as a function of temperature. The curve marked Ew shows the width of the function which was previously defined. All the quantities shown in Fig. 3 appear to be much larger than the temperature for the entire temperature range shown. A large number of active sites we understand as a result of wandering of the system around different PS. Each PS is characterized by a unique set of sites forming the Coulomb gap. So when the system passes from one PS to another the number of sites that change

their occupancy is a finite fraction of the number of sites in the Coulomb gap. The energy separation between different PS decreases with increase of the sample size [43]. If the ther-

modynamic fluctuations of the total energy

(C is the heat capacitance of the system)

is larger than this energy separation, then the described mechanism of the spectral diffusion

gives a temperature independent contribution to we use for simulations,

estimated above. For small samples, that

goes to zero with temperature when

becomes smaller than the

energy separation between PS, and the above estimate does not work at too low temperatures. To get a non-zero result at the limit one should increase the size of the sample while decreasing temperature. We are not able to perform this procedure in full scale but we check that there is no size effect in the temperature range presented in Fig. 3. We think that the temperature dependence in Fig. 3 results from soft dipole excitations

(See previous section). The density of states of the dipole excitations is 1/A, so that the concentration of active excitations is ~ T/A. These excitations not only contribute themselves to the spectral diffusion but also induce an energy change of passive sites leading to linear temperature dependence of in Fig. 3. Indeed, each dipole creates the potential r0/r2 at a passive site, where r is the distance between the site and the nearest dipole and r0 ~ 1 /Eg ~ A is the size of the dipole [10]. This estimate leads to a slope of dependence close to that in Fig. 3.

257

Figure 3. Temperature dependence of quantities quantities.

and Ew for A = 1. See text for a description of these

Correlation Function

Thus, the spectral diffusion shows that the configuration of occupied sites within the Coulomb gap persistently changes in thermodynamic equilibrium. To obtain more information about this motion, one can study the correlation function of occupation numbers. We do this by constructing a vector D(t w ) after tw MC steps have been performed. The components of D(t w ) are the occupation numbers ni of all sites within a given energy range [–W, W]. The vector is normalized so that D(tw) · D(t w ) = 1. As the simulation proceeds, we check the occupation number of these same sites, construct the vector D(tw + t), and calculate the correlation function C(tw, t) = D(t w )·D(t w + t). Correlation functions analogous to C(tw, t) are commonly used to measure the similarity between different configurations in systems such as spin glasses [47]. For two identical configurations C(tw, t) = 1, while if there is no correlation C(tw, t) = 0.5. Basically, we are interested in which is a measure of the similarity of two arbitrary states of the system at thermal equilibrium. For a non-interacting system, (20)

where

is the Fermi function. Thus, for the non-interacting system at and at In order to evaluate from the simulation, we measure C(tw, t) as a function of t for a given tw, and wait long enough so that C(tw, t) becomes independent of t. We denote this saturated value as In the inset of Fig. 4 we show an example of such a saturation. We then increase tw until becomes independent of tw, and thus obtain our estimate of The results for for the interacting system are shown in the main part of Fig. 4 as a function of temperature, for A = 1, W = 0.3 and W = 0.6. The corresponding functions for the non-interacting system, calculated directly from Eq. (20), are also shown. We observe that the correlation for the interacting system is much weaker than for the corresponding

258

Figure 4. The correlation function

as a function of temperature for A = 1 and W = 0.3 (solid circles)

and W = 0.6 (open circles). The solid line and dashed line show the correlation function for the non-interacting system, as given by Eq. (20), for W = 0.3 and W = 0.6 respectively . The inset demonstrates the saturation of the correlation function with time.

non-interacting system at the same temperature. Note that by increasing W we include more sites in the correlation function However, our results for the spectral diffusion indicate that most of these additional sites remain passive as Thus, should increase as W increases, as we indeed observe comparing the results for W = 0.6 and W = 0.3 in Fig. 4. The most interesting result coming from the study of the correlation function is the possibility of a finite value for Clearly, such an extrapolation cannot be considered conclusive, however, one has to keep in mind the following points: First, while

we have included in the definition of the correlation function only sites that were in the initial energy range [–0.3, 0.3], it is clear that many passive sites are still included in this definition. These passive sites mask the behavior of the active sites and tend to increase the correlation.

Second, a finite value of

means that thermal motion continues down to

zero temperature. This conclusion is consistent with the results obtained from the spectral diffusion in the previous section, and we understand it in the same way: Namely, we expect that a nonzero value of may be obtained only if the size of the sample grows with the decrease of temperature so that remains larger than the energy separation between

different PS. It is important to point out that our results cannot be explained by assuming that the excitations of the system are separated pairs of sites, with electrons hopping back and forth between the sites of each pair. This assumption would mean that electrons are effectively localized in space. Since the energy density of such excitations is constant at low energies, meaning the number of available excitations decreases linearly with temperature, one immediately obtains that like in the non-interacting system. The same temperature dependence is obtained even if excitations involve a few electrons that change their positions simultaneously (so called many electron excitation [49-50]). In fact, any picture based upon confined separated excitations which do not interact with each other would mean that Since our data definitely contradicts this temperature

259

dependence, we conclude that such excitations cannot explain our results. Thus, a nonzero value of

can be explained only by a thermal motion

of such electron configurations in the whole system that cannot be separated into thermal motion of independent clusters. This means that there is a finite portion electrons that are not localized in some region of space but move around the whole system. We view this motion as another manifestation of wandering of the system around different PS. The conclusions of this section are as follows. We have presented strong computational evidence that in a disordered 2D system of localized interacting electrons, the configuration of occupied sites within the Coulomb gap persistently changes with time. This effect persists down to temperatures well below the Coulomb gap width, and it causes a large time dependent random potential. This result is an exclusive property of the interacting system. Without interaction only electrons in a small interval of the order of T change their occupation. We argue that this effect may exist at zero temperature, if the size of the sample increases with decreasing temperature so that the separation between local minima of the total energy is much smaller than the thermal fluctuations of the total energy. However, our computational abilities are not enough to prove this point. Nevertheless, our simulations have been done at low enough temperature to compare them with the experimental data. We have obtained these results for equilibrium state only. Our program does not limit the length of the electron transition in order to reach equilibrium by the easiest way. In a real systems hopping distance is limited by tunneling so that applicability of the above results to a real system arises questions. We have shown [43] that equilibration time saturates at some value L = Rc, which is approximately proportional 1 / T . One can draw a conclusion that equilibration time is sizeindependent if localization length is large. Such a condition can be satisfied at very low T near metal-insulator transition. However we are not able to discuss transport in this situation because our computations still do not take into account such important factors as narrowing of the Coulomb gap with increasing (See [22]) and decrease of the random potential because of the quantum mechanical averaging at large CONCLUSIONS

We have given a critical review of the theory and experiment on the interaction of localized electrons and hopping conductivity. There is another very interesting field of quantum mechanical description of the same system which includes such important problems as the metal-insulator transition. The simplest Hamiltonian of this problem can be obtained by adding the non-diagonal term to the classical Hamiltonian Eq. (5). Many efforts have been made in this field, but we do not review them in this paper. We would mention only that quantum theory should include all the problems of classical theory mentioned above… and something else. Coming back to the classical theory of the Coulomb gap, we would make the following conclusions: 1. We believe that the theory of the 2D systems is basically complete and correct. We are not so optimistic about the 3D systems. We believe that the physics of long-range interaction is valid in this case as well, we believe that there is a gap near the Fermi level, we even think that the VRH law Eq. (4) is also valid, but we are not sure about the structure of elementary excitations in this case and about the possibility of thermodynamic glass transition. However, we propose in this paper a reasonable scenario which leads to a pretty simple picture, similar to the 2D case. 2. We think that the pseudoground states are responsible for the glassy effects which has been observed recently. We claim that in 2D case there is no thermodynamic glass transition, but the equilibration time strongly increases at low temperatures. We argue that the VRH

260

in the interacting system might be a result of non-ergodic behavior due to the long-time relaxation at low temperatures.

AKNOWLEDGEMENTS I am grateful to my long-term friend and coauthor Boris Shklovskii for many long discussions of the problems, considered here. I am grateful to David Menashe, Boris Laikhtman,

and Ofer Biham, with whom I was working on the problem of thermodynamic fluctuations. I appreciate helpful discussions with Zvi Ovadyahu and A. Vakhin. The work was funded by BSF Grant 9800097.

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Rev. 122, 1742-1751. Austin, I.G., Mott, N.F. (1969) Polarons in crystalline and Non-crystalline Materials, Adv. Phys. 18, 41-50. Ambegaokar, V., Halperin, B.I., Langer, J.S. ( 1 9 7 1 ) Hopping conductivity in disordered systems, Phys.

Rev B4, 2612-2630. 41. 42.

Tenelsen, K., Schreber, M. (1995) Low-temperature many-electron hopping conductivity in the Coulomb glass, Phys. Rev B52, 13287-13293. Perez-Garrido, A., Ortuno, M., Cuevas, E.,Ruiz, J., Pollak, M. (1997) Conductivity of the twodimensional Coulomb glass, Phys. Rev B55, R8630-R8633.

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Menashe, D., Biham, O, Laikhtman, B.D., Efros, A.L. (2000) Thermodynamic Fluctuations of Site Ener-

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Kogan, Sh. (1998) Electron glass: Inter-valley transitions and the hopping conduction noise, Phys. Rev B57, 9736-9744.

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Diaz-Sanches, A., Möbius, A., Ortuno, M., Perez-Garrido, A., and Schreiber, M. (1998) Coulomb Glass Simulations: Creation of a set of Low-Energy Many-Particle States, phys. stat.sol (b) 205, 17-19. Yu, C.C. (1999) Time-Dependent Development of the Coulomb Gap, Phys. Rev. Lett. 82, 4074-4077. Marinari, E, Parisi, G., Ruiz-Lorenzo, J.J. (1998) Numerical simulations of spin glass systems, in P.

gies and Occupation Numbers in the two-dimensional Coulomb Glass, unpublished

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Young (ed), Spin Glasses and Random Fields, World Scientific, Singapore.

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Knotek, M.L., Pollak, M. (1972) Correlation effects in hopping conduction: Hopping as multielectron transition, J. Non-Crystal. Solids 8-10, 505-515.

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Knotek, M.L., Pollak, M. (1974) Correlation effect in hopping conduction. A treatment in terms of multielectron transitions, Phys. Rev B9, 644-655.

262

THE METAL-INSULATOR TRANSITION IN DOPED SEMICONDUCTORS: TRANSPORT PROPERTIES AND CRITICAL BEHAVIOR

THEODORE G. CASTNER, emeritus Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627

INTRODUCTION Most graduate level texts in condensed matter physics devote very little attention to metal-insulator transitions (MIT) in solids even though magnetism (ferromagnetism and antiferromagnetism) are covered thoroughly. The subject is usually treated in special volumes, review articles and monographs. Sir Nevill Mott, the individual most responsible for the development of the field, provided a monograph [1] in 1974 (second edition 1990) that provided a comprehensive discussion of this field. Mott notes there are many different types of MIT ranging from the vanadium oxides, nickel sulfide, metal-ammonia systems, alkali-rich alkali halides, compressed solid rare gases, hydrogen, and iodine, liquid mercury, and the doped semiconductors. Since 1986 one would add to that list the doped cuprate high temperature superconductors (HTS), the superconductor-insulator transition, the apparent twodimensional (2D) MIT in Si MOSFETS and other 2D systems such as GaAs/AlGaAs heterostructures. In this paper the emphasis is on the doped semiconductors. Doped Ge and Si have been more carefully studied than other doped systems and therefore this review will focus on these two materials, although there has also been work on 3-5 materials. The feature of the MIT in the doped systems distinguishing them from the vanadium oxides, etc., is the disorder. However, many of the features of MIT for ordered systems need also to be considered for disordered systems. Mott has termed the MIT for these disordered systems an Anderson transition. Anderson [2] described the lack of diffusion of carriers in a random potential of sufficient strength. The Anderson transition has become known as the MIT of non-interacting electrons. However, Belitz and Kirkpatrick (BK) [3] in their 1994 review have noted that interactions are always present, although not necessarily relevant. They have noted that neither Anderson’s nor Mott’s picture based on screening from interactions, is separately sufficient to understand the MIT in the disordered doped semiconductors such as Si:P. In this review the focus will be on the transport in the critical regime (near the critical density nc) and what the transport data tells us about the relative importance of interactions and disorder. Theories taking account of both interactions and disorder are inherently difficult. BK [3] have provided the most comprehensive review of the theoretical progress and outstanding problems. How-

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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ever, the experimental data provides some very important clues on the relative importance of disorder and interactions. Some concepts from classical semiconductor physics will be shown to be crucial in explaining the critical behavior of weakly compensated doped Si and Ge. Doped Si(Ge) systems consist of random distributions donors (n-type) or acceptors (ptype). Since the hosts are insulators as with a filled valence band and empty conduction band any itinerant carriers must arise from the dopants. In the early period of semiconductor physics there was substantial interest in the scattering mechanism of the itinerant carriers and

what scattering mechanisms determined the mobility. Shockley’s text [4] provided an early discussion of both ionized impurity scattering and phonon (lattice) scattering and provided a discussion of the early experimental results. The experimental results and theory demonstrated that ionized impurity scattering (iis) [Rutherford scattering] was the dominant scattering mechanism at low T and doping densities comparable to nc. This result is of vital importance in understanding the critical behavior of the conductivity of The random distribution of dopants accounts for the random potential necessary for an Anderson MIT. The magnitude of the disorder potential can be enhanced by compensation that leads to ionized acceptor-donor pairs (n-type). Compensation has an important effect on not only the nature of the random potential, but also on the critical behavior of Mott’s classic 1949 paper [5] showed that Thomas-Fermi screening (the result of interactions) provided a remarkably accurate criterion for the critical dopant density nc for the onset of metallic behavior at T=0. The nature of the screening for the dopant density n just above nc, presumed to be ThomasFermi by many theorists, has not yet been documented experimentally. However, it is the

temperature-dependent that provides definitive evidence for the dominant role of carrier-carrier interactions at finite T. The dominance of interactions in describing the T-

dependent demonstrates that the Anderson transition without interactions is insufficient to describe the behavior of Si:P, etc.. One of the central themes of the present review is an effort to reconcile the T-dependent transport that is dominated by interactions with the critical behavior of where the role of interactions is less well established. Since the MIT phase transition is viewed as a T=0 transition, or a quantum phase transition (QPT) it is of particular interest to consider the conductivity at the critical point, namely There are new theoretical and experimental results on the critical conductivity that will be discussed. One of the dominant features on the insulating side is variable range hopping (VRH) conduction, both Mott VRH [6]and Efros-Shklovskii (ES)VRH [7]. Since the latter features the Coulomb gap resulting from long range Coulomb interactions, while Mott VRH apparently ignores interactions one wants to understand the role of interactions for n
IV will discuss the T-dependent conductivity for n>n c, n=nc, and n
The earliest model that was useful for predicting the occurence of the MIT was the Herzfeld model [10] which was based on the Clausius-Mossotti equation for a dielectric. It can be written for a doped semiconductor in the form (1)

where is static dielectric constant, is the host value, N is the density of polarizable impurities of polarizability diverges as The right hand side is usually written as R/V where R is the molar refractivity and V the molar volume. Materials with R/V<1 are insulators and those with R/V>1 are metals. The Herzfeld criterion works well for the halogens, the solid rare gases and the alkaline earth and transition metals. Although it has been applied to doped semiconductors where gets very large as there are serious doubts as to whether the Clausius-Mossotti result is applicable for a random set of donors, because the Lorentz local field may not be valid as the localization length diverges as The Herzfeld criterion has been reviewed by Ross and Barbee [12]. The Mott criterion (1949), namely where nc is the critical density (of donors) and a* is the Bohr radius of the donor atom in the dilute limit has been remarkably successful in predicting the nc for doped semiconductors and other systems over more than 7 orders of magnitude in nc, as demonstrated by Edwards and Sienko [13]. The Mott derivation, based on a Yukawa potential of the form where ks is the screening wave vector given by for Thomas-Fermi screening. This potential no longer binds an electron as ksa* approaches 1. Despite the success of the Mott criterion, the situation is considerably more complicated because donor electrons becomes itinerant in the impurity band at energies well below the conduction band edge. Other theories and experimental results will suggest smaller values of ks of order 1/d=nl/3. However, the empirical success of the Mott criterion is perfectly clear. A more sophisticated Mott criterion calculation needs to take account of the screening, which isn’t going to be Thomas-Fermi just above nc and the much smaller energy required to excite an electron to the mobility edge of the donor impurity band. The Anderson model and Anderson localization are described by the Hamiltonian (2)

where the ai+ and ai are creation and annihilation operators and the are the site energies which are randomly distributed and the distribution function is characterized by a width W. The first term is the tight-binding term for transfer from site j to site i and is frequently simplified to 265

Figure 1. The DOS versus energy E (where Ecb=0) for the 1s-bands for Si:As for n near n c based on the valleyorbit splittings from the dilute limit. The 1s-bands are Gaussians with widths of 6, 8, 8, 12 meV for the 1s-A1, 1sT2, 1s-E, and UHB respectiively. EF and Ec are near a minimum (or inflection point) in the DOS . Interaction contributions are not included.

nearest neighbors with tij = t for i and j neighbors. For large enough W/t all states are localized. For 3D there is a critical Wc below which there is a mobility edge separating localized from ex-tended states within a single band as shown in Fig. 1. The extended states are in the band center and the localized states are in the band tails. For the multivalley conduction bands of Si and Ge the situation is complicated by overlapping bands associated with the 1s donor states. Fig. 1 shows a typical density-of-states (DOS) for Si:As for 8.6x1018 donors/cc. Including the overlapping 1s-A1, 1s-T2, 1s-E, and upper Hubbard bands. One should stress the DOS magnitude of the impurity bands is much larger than that for the lower part conduction band. The other important feature is the large valley-orbit splitting between the 1s-A1 and the 1s-T2 (1s-E

bands). The spin-orbit split 1s-T2 band is responsible for the strongly donor-dependent Orbach spin-lattice relaxation process in the dilute limit and for the strong donor dependence of the metallic conduction electron spin resonance (CESR) linewidth for Si:P, Si:As, and Si:Sb, Shown on Fig. 1 is the Fermi energy EF(n) and the mobility edge Ec(n). For this density EF(n)

is near Ec(n). A characteristic feature of an Anderson transition is the smooth passage of EF(n) thru the mobility edge Ec(n) as n is increased thru nc. A second crucial feature of the Anderson transition is the two-component model. For a given n and T there are both localized and itinerant electrons of density and na(T) respectively, subject to the condition + na = n

266

where n is the net doping. At T = 0 these densities are related to the DOS in Fig. 1 by (3)

Of some importance for the transport at higher temperatures well above 1K is the Tdependence of the chemical potential µ(T) [13]]. It is straight forward to calculate EF at T=0 for the overlapping bands. This overlap is important and it yields empty states in the upper part of the 1s-A1 band and filled states in the lower part of the 1s-T2, 1s-E, and upper Hubbard band.

In the past there has been controversy [14,15] about the existence of localized and extended states at the same energy E. Auxiliary minima in the random potential above Ec can produce localized states in the itinerant regime E>Ec, 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 n
where is the energy required to excite a donor electron to the conduction band, while is the energy to excite an electron to a neutral donor (i.e. to the upper Hubbard band, UHB). is the activation energy to a nearby neighbor. This last process depends on compensation and is successfully explained by the Miller and Abrahams (MA) theory [17] of hopping. Of particular interest for the MIT is which scales to zero as Pioneering studies of the density dependence of these activation energies in Ge were made by Fritzsche [18] and Davis and Compton [19]. Experimental results in amorphous semiconductors led Mott to develop a new mechanism of hopping namely VRH with a T-dependent hopping energy and mean hopping distance. This was first observed in doped Ge or Si in Ge:Sb by Alien and Atkins [20]. A second type of VRH in the presence of long range Coulomb interactions was calculated by Efros and Shklovskii [7] and observed in many semiconductors. There were also substantial studies of metallic samples at LT and a particularly comprehensive study was that of Yamanouchi et al. [21] for Si:P of both and the Hall coefficient RH(n, T=4.2K) over more than two decades in n. Fritzsche [22] gave an excellent review of the early transport properties. This period also featured some controversy. Mott’s σmin was criticized by Cohen and Jortner [23], who argued there was an “inhomogeneous” regime near nc. They suggested class-ical percolation theory would describe conduction along metallic channels. would continuously drop to zero as the width of the metallic channels decreased as Mott [24] responded that one could not divide the sample into insulating (“opaque” regions) and metallic

267

Figure 2. versus n=N D -N A for Si:Sb, Si:P, and Si:As from [27]. The increase in with n is more rapid than predicted by Eq. (1). The scaling exponent for Si:P is 1.15 from [70-f] and is 1.20 for Si:As from [66g].The results give (Copyright by the American Physical Society)

regions and gave theoretical reasons for this. However, at this time there was no data that clearly demonstrated that As a result the Cohen-Jortner idea didn’t gain much support. Nevertheless, the issue of inhomogeneity has always been an important concern

close ton c and is addressed in section V. One of the important features of the MIT is the screening, as reflected in the dielectric response. The long range Coulomb interaction between two electrons is directly affected by Mott frequently mentioned the importance of behavior for MIT systems. There were far fewer measurements of than for however early measurements of at 10 Ghz for Na-ammonia solutions [25] and doped Ge [26] demonstrated an enhancement of as

268

n approached a critical density. A comprehensive study of Si:P, Si:As and Si:Sb at low fre-

quencies and low temperatures by Castner et al. [27] demonstrated the onset of a polarization catastrophe as [see Fig.2)]. (n) increased more rapidly than predicted by Eq. (1). 1976 -1986 - The golden era of scaling and interactions Using renormalization group methods Wegner [28] obtained an expression for the dynamical conductivity where t is a measure of distance from the critical point and for many cases t=(EF - Ec)/Ec. Wegner’s result leads to a DC conductivity (6)

where v is a correlation length exponent and d is the dimensionality. For d = 3 the conductivity

scaling exponent s = v. From the definition of the correlation length the conductivity can be written as An influential paper by Abrahams et al. (AALR, Gangof-Four) [29] used the -function approach with (g) = dlng/dlnL. Their approach, a oneparameter scaling theory for noninteracting electrons, assumes is only a function of the dimensionless conductance g [g=G(L)/(e2/2h)]. Using for obtain

= s ln(g/gc) and integrating AALR

(7)

where the prefactor was identified with Mott’s σmin and with 1/s = v their result resembles Wegner’s result. However, a different interpretation of the AALR result is given below. AALR noted that

which for d=3 yields Ohm’s law. For d = 2 AALR concluded that

for all g and that all states were localized and there was no MIT in 2D. In the last decade there has been substantial experimental evidence for a MIT in 2D [30]. There are numerous more sophisticated scaling theories. The scaling equations and scaling exponents depend on

the symmetry (universality classes) and these are discussed in BK. Of relevance for this discussion are the orthogonal class (no symmetry breaking) and the unitary class (breaking of time-reversal symmetry by a magnetic field).The scaling theory for interacting electrons was

initially discussed by McMillan [31], but was followed by by Finkel’shtein [32] and others. These more complex scaling theories may be of interest theoretically, but apparently are not required to explain the experimental data for Si:P, etc. An independent theoretical approach developed by Gorkov et al.[33] and by Bergmann [34] known as “weak localization” based on coherent back scattering gives a DC conductivity (8)

where the const is of order unity. Weak localization has been reviewed by Lee and Ramakrishnan [35] and is also discussed by BK. For doped Si and Ge the experimental results can be explained without weak localization contributions. The most important theory (n>nc) 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)

269

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 (10
Figure 4. Combined data showing

versus S-Sc (upper scale n/nc-1) from Fig.3. The

open circles are from Ref. [41], the solid circles are from Ref. [42]. The combined results give s=0.48 ±0.07. (Copyright by the American Physical Society)

It will be demonstrated below that an alternative interpretation based on a T-dependent diffusivity which arises from an overlooked Einstein relation [39,40] between D and the mobility µ, can explain the features of using the A-A result and a new AT1/2 term from iis. The most significant experimental results of this period were for Si:P in a series of papers [41,42] that established that the scaling exponent s for was s~0.51±0.05. This work featured dilution refrigerators (DR), and particularly the uniaxial stress experiments reached the lowest T (~3mK) and probed closer to nc (n/nc-1<10-4) than any previous or subsequent measurements. Uniaxial stress tuning allowed a detailed study of with 10-40.88nc was a good fit to the Mott VRH law and the prefactor appears to be nearly independent of T. The 271

Figure 5. Normalized versus n/nc-1 for Si:B [49], Si:P [49), Si:As [48], and Ge:Sb [47]. In each case g<s and is given by g=0.7±0.07 s. The inset shows normalized Hall number A/nceRH versus n/nc-1. (Copyright by the American Physical Society)

Mott characteristic temperature To scaled rapidly toward zero as The data for 1-n/nc<0.05 suggested 2.3<3v<2.9 and that the localization length exponent v was closer to 1 than to ½ and should have suggested that s was not equal to v. A second feature of this data is the very rapid increase in To(n) for 1-n/nc>0.06 that is much faster than This is not consistent with the Mott VRH theory. This data also pinpointed nc for Si:As to the interval 8.55n c as

272

Figure 6. Scaling behavior of the CESR excess linewidth

Si:P from [50J.

versus n/nc -1 for Si:As and

is a minimum at n~n c at low enough T. (Copyright by the American Physical Society)

(10)

where

is the second term in (8), with Li the inelastic diffusion length and where LT is the thermal interaction length LT = (hD/kT)1/2 seen in (9a). The last term in (10) is just the A-A term. Eq. (10) illustrates the common trend for the 3D MIT of employing where Lcj is the jth characteristic length. In sections III and IV it will be argued the data can be explained with only the first and last terms on the RHS of (10). Associated with is a length that has previously been ignored. In the late 80’s the 3D semiconductor MIT effort was eclipsed by HTS. However Hall coefficient results [47,48,49] began to appear for these systems. These results showed results scaling to zero as where g is in the vicinity of 0.7s. The fact that g<s is important because is confirms the mobility µ is well behaved and also scales to zero as Koon and Castner [48] attempted to claim their results were flattening in the vicinity of nc , but their Si:As results yielded g~0.35 further above nc. The Hall results, as summarized in Fig. 5, were particularly important in my own thinking because with the simplest Kittel-level interpretation determines the density of itinerant electrons, implying as This notion, however, was at odds with the interpretations given in the early 80’s. This provided the motivation for many of the new developments described herein.

Another quantity exhibiting scaling is the excess conduction electron spin resonance (CESR) line width as discussed by Zarifis et al. [50] (see Fig. 6). The theory for is given by an Elliot-Yafet [51,52] expression

273

(11) where is the spin-orbit splitting associated with the 1s-T2 excited state, is the valleyorbit splitting [the 1s-A1 - 1s-T2 splitting], and is the scattering rate associated with the valley-orbit process. It must be stressed that this is a different quantity than the conductivity τc in Eq. (12) below. It is believed the ratio is very weakly dependent on n suggesting that the scaling of originates for For the scaling of can be approximately explained by the scaling of and a constant cross section. However, there is no rigorous derivation of a constant cross section for this case. Note that the scaling exponent for is slightly smaller than that for D(n,0). Recent developments and controversy There was substantial frustration that the Wegner and AALR results yielded s~1.0 while the Si:P results gave s~0.5 and there were many efforts to explain the reasons for this “exponent puzzle”, however none of the different theoretical approaches to explain s~0.5 produced any concensus. This problem led to new experimental results and to a controversy that still remains today. In 1993 Stupp et al.. [53] presented new Si:P results that purported to show a crossover from s-0.55 to s~1.3 when n/nc-1<0.1 in the region where m(n)>0. They defined the “critical regime” as that for which m(n)>0. Their approach lowered nc from 3.74x1018 to

3.52x1018, namely a 6% drop in nc, which is large for this type of phase transition. This interpretation was challenged by Rosenbaum et al. [54] because of homogeneity considerations when and by myself [55] based on the fact that 7 samples were claimed to be metal-

Figure 7.

versus T-1/2 for 10 insulating Si:As samples [43] showing Mott VRH for T<10K for Strong deviations from Mott VRH are found for below a certain Tc(n). (Copyright by

the American Physical Society)

274

lic were actually a much better fit to Mott VRH for 0.1
n-type Si and Ge samples in that the m(n) dependence on n was very weak. In 1996 Itoh et al.[8] reported the NTD Ge:Ga results showing s=0.50 for weakly compensated samples. The particular importance of these new Ge:Ga results is that the samples have a compensation K<0.001, which is much better than in earlier Ge studies. One of the important claims for the NTD approach is the more homogeneous dopant distribution obtained as compared to thermal doping.

Figure 8. Data from [53] replotted in [80] to compare with Mott VRH. The results for T>0.09K are a good fit to the Mott law (vertical dashed line T= 0.09K). For T<0.09K there are sample dependent upward deviations (see 3.63) from thermal decoupling. (Copyright by the American Physical Society)

275

III. A NEW APPROACH FROM TRADITIONAL SEMICONDUCTOR PHYSICS Although iis was the focus of enormous attention in the 50’s, 60’s and 70’s it never played a significant role in MIT theory and the effort to explain the scaling results for 1)s [s=0.5 for Si:P, Ge:Ga, etc.], nor was it used in efforts to explain the results. It was believed that Boltzmann transport expressions could not explain the scaling behavior. This is typified by the approach of Bhatt and Ramakrishnan [59] who attempted to explain the scaling of with localization and interaction corrections, but employed the Boltzmann result to explain the prefactor σo. The pioneers in early Ge and Si transport studies knew that iis was the

dominant scattering mechanism at LT and high doping densities. An excellent review of iis is given by Chattopadhyay and Queisser [60]. A general expression for σ(n,, T)iis , valid for arbitrary degeneracy, employed by Mansfield [61]for scattering from a random distribution of impurities, namely (12) where N(E) is the DOS and f is the Fermi function. Mansfield intended his expression for electrons in the conduction band (E>ECB) but it applies equally well for itinerant electrons in the impurity band with E>Ec. For EF-Ec/kT»1 Eq. (12) leads to the standard Boltzmann result where na = N(E)f dE. In the present case na is the density of itinerant electrons above Ec. The collision rate for iis is where Ni is the density of ionized

impurities, The phase shifts

and the angle-averaged cross section

are restricted by the Friedel sum rule [62 ] where Z is the valence difference [Z = 1 for Si:P,etc.]. For an arbitrary N(E) = C[(E-Ec)/Eo]p it can be shown [63] for where the phase shift sum can be removed from the integral, that (13)

where and is the Fermi integral The charge neutrality condition (see [4] p.238) as is na = Ni for no compensation, because the itinerant electrons can only originate from the donors. For a compensation K = NA/ND (n-type) one has Ni = na + 2KND. Evaluating the Fermi integrals in (13) keeping the leading terms for and using the charge neutrality condition for K«1 one obtains (14)

There are various points to be made about this result: 1) the result is the square root of the Wegner result; 2) the second [ ] is simply kF (measured with respect to Ec) and is

where is the well known de Broglie wavelength; 3) this result lacks the extra in Mott’s σmin, thus demonstrating the Ioffe-Regel criterion is irrelevant for σB,iis; 4) this result arises from incoherent scattering; 5) for EF - Ec = Eo(n/nc -1) this result explains the scaling exponent s = ½; 6) the form of the DOS has no effect on the exponent s, but does affect the magnitude of the prefactor σo which can be calculated using k F(2nc); 7) this result is consistent with an Anderson transition with EF crossing Ec smoothly; while screening effects are buried in the phase shifts and screening affects the prefactor but not the scaling exponent s as long as the system is sufficiently degenerate; 8) the compensation dependence of σB,iis is the result in (14) multiplied by na/(na+2KND). It is worth emphasizing (14) doesn’t depend on the itiner-

276

ant electron density na, but depends only on λi,h and the E-dependence of Basically, the derivation of (13) has used Friedel scattering developed for impurities in metals [64], The MIT in doped Si and Ge is a special case of alloy theory. The difference is that the host is an insulator as but also kF in an alloy changes only a small amount because Ni«n. In the MIT at A A second difference is that for an alloy while for the MIT The same formalism, starting with (11), has been shown [65] in the nondegenerate limit to lead to a new contribution AT1/2, where A is proportional to (m*/m)1/2 and the ratio of a scattering integral In this case the phase shift sum term cannot be removed from the integral and one has E- and T-dependent phase shifts that still satisfy a generalized Friedel sum rule for arbitrary degeneracy valid for doped semiconductors. For finite T if one considers k(E,T) [or the velocity as (15)

This emphasizes the importance of low temperatures and sufficient degeneracy to obtain the scaling behavior with s = ½. Paalanen et al. [42] achieved the lowest T (~3mK) corresponding to an energy difference of 0.26µeV. For a characteristic energy E, = 12.5meV for Si:P the results in [42] extended to n/nc-1<10-4 and still maintained The above derivation should be compared with that of Phillips [15,44,66]. The common features are the use of the two-component model, and the notion that “weak localization” is not applicable to these MIT systems. It seems highly likely that both models feature conducting filaments for n barely above nc. However, the crucial difference in σB,iis is the use

of the charge neutrality condition na=Ni, so that the kF dependence comes from

whereas in

the Phillips case it comes from the density of itinerant electrons na. The Phillips approach

yields while the Hall data yields with g~0.7s. Regrettably, physicists are more familiar with iis (Rutherford scattering) and Boltzmann transport

than they are with set theory, topological constraints, and Voronoi polyhedra. Besides the scaling of and [and the Hall mobility

the

scaling of the diffusivity also needs to be considered as well as some of the characteristic lengths like the mean-free-path Some of these transport quantities are related by Einstein relations. Two of these in the strongly degenerate limit are (16)

The first one between and D is due to Kubo [67], while the second relation is the original Einstein relation between D and µ which has been overlooked for many years. For many years

it was thought that and D scaled with the same exponent because of Lee’s identification [68] of with the thermodynamic potential Lee argued that since the specific heat cv(n,T) for Si:P varied smoothly thru nc then

also varied smoothly thru nc. However, as

it will be shown that the in (16) is unrelated to cv and the new intepretation shows This quantity diverges as as Classically which using becomes in the degenerate limit. This result is important because D(n,0) gives

a direct measure of Ioffe-Regel factor even though the treatment of yields a result independent of From (15) one finds so that The scaling exponents for these transport parameters are summarized in Table 1 where the experimental values are given for Si:P, Si: As, Si:B, and Ge:Ga, namely for the most weakly compensated samples. The major focus in experiments has been on the scaling of the conductivity, but there are

277

other quantities as that show scaling with . The scaling of RH has been discussed above and the mobility µ [µ and µ H are expected to scale with the same exponent at low fields as H approaches a small value of order 0.1 tesla] is given by the product The diffusivity D has not been measured directly but is inferred from from the A-A theory and the

experimental results as discussed below. The exponent for D from the Einstein relation scale with is 1+s-g. For a constant or slowly varying effective mass m*(n) D and the Ioffe-Regl factor the same exponent. The mean-free-path scaling can be inferred from that of D and kF and scales with a positive exponent ½+s-g. The scaling of to zero as requires more discussion since the traditional viewpoint has been that could not be less than the donor spacing. On the insulating side the DC conductivity The interesting scaling quantities on the insulating side besides the dielectric constant are the Mott and ES characteristic temperatures To and associated with VRH conduction. Several authors [69] have discussed the fact the exponent and Kawabata has considered the product to be a

universal quantity for noninteracting electrons. The data in Table 1 for Si:P and Si:As show to 2.4 times s. For the amorphous S-M alloy Al0.3Ga0.7As:Si (a persistent photoconductor) Katsumoto [71] obtained the scaling exponents s~1.0 and hence the ratio is similar to the n-type Si results. A slightly different explanation for the scaling behavior of the quantities in Table 1 was given in [40]. However, that explanation depended on the scaling of m* and gave a different scaling relation of kF(n). The scenario given above with k F (EF Ec)1/2

(n/nc-1)1/2 is a far more convincing solution.

The new theory for kF, the Einstein relations, and µ H = appear to give a self consistent set of scaling relations describing the transport behavior just above nc. A problem not yet resolved is what form of DOS N(E) in the vicinity of Ec will explain the scaling exponent g~0.33 to 0.4 using Eq. (3). This value of g is in reasonable agreement with the 3D percolation prediction by Kirkpatrick [72] for 1/RH, although this might be fortuitous since the exponent s=½ for is less than 1/3 the percolation prediction of 1.6. The experimental result g<s appears to be firmly established and is in agreement with the Cohen et al. [73] hypothesis that the mobility must be well behaved (non diverging as This requires All of the known data, including that for compensated cases like Ge:Sb [47] satisfies g ~ 0.70 ±0.07 s. One issue still to be resolved is the apparent conflict between the 3D AALR result (s~1) and the result s~½. This requires a second look at the AALR calculation. The AALR = s ln(g/gc) is only valid very close to the critical point and becomes

278

much larger than 1 for g»gc. However as one must have for 3D and a better choice for is = sx/(1 + sx) where x=(g-gc)/gc. This has the same form as the AALR choice for small x, but approaches 1 as This choice permits an exact result for the integral and yields, using g(L) = where L is a macroscopic length (sample dimension) (17)

where the last term results from sx term in the denominator of the new The significance of this extra term might seem unimportant for s~1 but all the data for the 3d MIT is in the regime and the result in (17) is basically independent of the magnitude of s. For a fixed L this is basically Ohm’s law The 3D data, from the standpoint of the AALR result is very far from the critical point, even though it shows scaling. In this regime g»gc one should not compare (17) with the Wegner result which contains the correlation length exponent v. The result is independent of v and isn’t obviously related to the correlation length Lee and Ramakrishnan [35] employed a Taylor series expansion of in terms of powers of EF-Ec which then yielded a result analogous to the Wegner result for 3D and v=1. This result, using EF-EC (n/nc-1) yielding s~1 explains the scaling exponent for the a-S-M alloys. However, a different Taylor series expansion in the quantity kF-kc yields a result (18)

Since k is measured with respect to the mobility edge at Ec this implies kc = 0 and because kF (EF-Ec)1/2 the first term in (18) yields the exponent s= ½.Because for 3D e2/hLc where Lc is a characteristic scaling length this suggests (18) becomes In this case the first derivative in (18) is a constant and all higher derivatives dng/dkn are zero for This is a compelling result because it demonstrates that in agreement with Eq. (13). This form of the AALR result not only explains the exponent s=½, but also explains the large width of the scaling regime. This provides a satisfactory resolution of the “exponent puzzle”. What is the physical meaning of in the Einstein relation in Eq. (16)? If one were to use the Mott result for one obtains the result This free electron-like expression is familiar for good metals. On the other hand when Eq. (14) [for p=½] is used one finds which in this form is an apparently unfamiliar result. However, this can be rewritten, using as (19)

(19) is a free electron-like expression, since na is the density of itinerant electrons and EF-Ec is the Fermi energy measured with respect to the mobility edge. However, unlike normal, good metals both na and = EF - Ec scale to zero as The fact that na scales to zero more slowly than EF - Ec explains the divergence of The simple dependence of in a free electron manner also strongly suggests the MIT system in Si:P is that of a Fermi liquid. IV. THE T-DEPENDENCE OF Metallic samples

The most important new data in the last decade is the NTD Ge:Ga data [8,9] which will be reviewed by Watanabe, includes measurements to 20mK. This is not as low as the 3mK

279

achieved by Paalanen et al., but is sufficient for most samples, but may not be low enough for n/nc-1 < 0.002 to maintain the strong degeneracy thruout the T-range of the measurements. Watanabe et al. [9] have used the T1/3 prediction of Altshuler and Aronov [74] to analyze the data for the samples closest to nc. This T1/3 result, purportedly valid very close to nc where is based on the assumption (see Eq. (16)) is independent of T, since the A-A derivation involves the removal of D from Eq. (9a) using As will be demonstrated below and D(n,T) have different n- and T-dependences. This can be seen from the general Einstein relation [75] between D and µ , namely

where

and the

is the Fermi integral. Series expansions for

and

yield.

(20a) and (20b) At T=0 D scales with an exponent nearly twice that that for The T-dependence of D(n,T) is linear in Tµ(n,T) in the nondegenerate regime whereas in the degenerate regime D(n,T) consists of a constant D(n,0) plus a quadratic correction in T2. It will be shown below that the observed deviations from T1/2 behavior can be interpreted with the A-A term in Eq. (9a)

with a T-dependent D(n,T). The features in the Watanabe et al. Ge:Ga data are similar to the earlier data in Ge:Sb [38], Si:P, and Si:As. The most important features of the data are: 1) for large values of n/nc-1 with m(n)<0 the deviation from T1/2 behavior is upward (a positive change in 2) for very small values of n/nc-1 [<0.01 for Si:As] with m(n)>0 the deviation from T1/2 behavior is downward (a negative change in in 3) there is a special concentration n* where no deviations from T1/2 behavior are observed. This can only be explained if m(n)=0. This occurs for n*=1.6x1017 for Ge:Sb and near

1.912x1017 for Ge:Ga, but has not been as accurately identified for Si:P and Si:As. However, the T1/2 magnitude is not zero at n* but is positive. This suggests there is a second AT1/2 contribution which has been found [61] to arise from iis. The fact that at n* there is no deviation from T1/2 behavior rules out the localization correction The features 1) and 2) are illustrated in Fig. 9 showing results versus T1/2 for Ge:Sb, Si:P [41] and Si:As [43,76] samples. The two samples with positive slopes show downward deviations from T1/2 behavior while the four samples with negative slopes show upward deviations. For Si:P and Si:As these deviations are much smaller than for Ge:Sb and Ge:Ga. The Newman

and Holcomb results give the most accurate picture of these deviations for n-type Si because the data shown extends to T~20K. It should be stressed that the deviations are in conflict with a localization correction BT, but are well described by inserting (20a) into the A-A expression

in Eq. (9a). For Ge:Sb and Ge:Ga the deviations from T1/2 behavior are easily observed below 1K reflecting the much smaller values of EF-Ec for Ge systems. This second contribution AT1/2 provides an additional reason why the A-A derivation of the T1/3 term is not correct. A new analysis [77] of the of the data for Si:P and Si:As based on

= (A +m(n))Tl/2 has yielded a more satisfactory agreement between theory and experiment. The detailed fitting procedure will be discussed elsewhere. The essential new features of this analysis are: 1) the addition of the AT1/2 from iis; 2) the use of k f2 Ef-Ec (n/nc-1) in the expression for x in Eq. (9b) which guaranties that as thus requiring 3) the use of a screening wavevector κ consistent with the two-component model and with the 280

Figure 9. a) Typical data for

versus T1/2 for Ge:Sb [38], Si:P [41], and Si:As [43,76] show upward

(downward) deviations for At<0 (m(n)>0); b) the ratio

versus T. For large n/nc-1

shows a

quadratic variation in (kT/EF)2 .

screening radius rs identified with the insulating side; 4) neglect of intervalley effects [78] and the use of the single valley Hartree-exchange factor taking account of the A-A sign error for the d=3 case. The parameters for Si: As and Si:P are the same, but will differ for Ge cases because of the different values of n/nc-1 where At = 0. The results of this new fit are shown in Fig. 10 showing At(n) versus n/nc-1. An excellent fit is obtained with p~½ [D(n,0)=Do(n/nc-1)2p] over the range 0.0008
Figure 10. A t =A+m(n) versus n/nc-1 for SiP [41,42] and Si:As [43,76]. The solid (•) symbols for the Si:P stress data [42] don’t join well with zero stress data where At=0. The inset show D(n,0)/Do versus n/nc-l for Si:P and

Si:As yielding scaling exponent 1.0±0.1

(21)

where mexpt = At,expt - A and the K= (Ce2/h){ ]. With the exception of the region near 1.01nc where m(n) is near zero and the experimental uncertainties in m(n) are larger D(n,0)/Do is very close to linear in n/nc-1 over more than three orders of magnitude in n/nc-1. There may be a small increase in the slope above 1 for n/nc-1<0.06. These results establish the result that D(n,0) scales with an exponent at least twice that for in good agreement with the Einstein relations in (16) and in (14). This should leave no doubt about the significance of the Einstein relations in interpreting the MIT.

282

The T-dependence of

for insulating samples just below nc

The temperature dependence of insulating samples is more complex than that for metallic samples and consists of at least 3 different contributions in the range 0.75nc
(22) where the first term is the activated component the second term is Mott VRH and the third term is ES VRH. The activated term is characterized by a T-dependent prefactor and an activation energy Ea(n,T) that is itself a function of T for n close to nc. As and this term becomes the critical conductivity AT1/2 at nc calculated [61] earlier which suggests q~½. Because Ea(n) increases rapidly with (1-n/nc) this activated term will be unimportant for T<1K for doped Si and Ge. Castner and Shafarman [13] have done a deconvolution of the first two terms in (22) for the Si:As data for 4 insulating samples for 10
(23) where is the host dielectric constant, is the static dielectic constant which can include a large hopping contribution if T isn’t low enough the susceptibility as is a screening length given by where k~4 which yields rs~d~ n-1/3. In the critical regime where one obtains for while for r»rs one has This feature is crucial in explaining the Ge:Ga ES VRH and scaling of SD of changes the MA matrix elements leading to a prefactor for the MA resonance energy W that is independent of R. This leads to a minimum impedance after employing the Mott ansatz The parameter is the crossover parameter between conventional LT Mott VRH for zm>3 and the new HT Mott VRH for zm<1. The Mott exponent ¼ changes to 2/7 for zm<1/3. The principal results of this theory are in Table 2. The 283

characteristic length L is a macroscopic length independent of T, such as the voltage lead spacing. The prefactor is T-dependent for the zm>3 usual case, but significantly is independent of T for the new HT Mott regime where To
Figure 11. Characteristic Mott To (Si:P [80] and Si: As [43]) and ES values versus 1-n/nc. The rapid increase in To for Si:P and Si:As for 1-n/nc> 0.04 represents a crossover regime where the coefficient D in To increases rapidly. The Ge:Ga is linear in 1-n/nc . (Copyright by the American Physical Society)

284

the Watanabe et al. [9] for the case with SD. For 0.9nc0.14, however the dramatic deviation from the theoretical prediction occurs for 1-n/nc<0.07 where σo, expt is nearly flat and appears to approach a constant value. Although there is more scatter the Ge:Ga of is also nearly flat in this range. One possible explanation for this flattening is doping inhomogeneity. There is a second instance where SD of is of substantial importance, namely in the width of the ES Coulomb gap which has measured by tunneling measurements by Massey and

Figure 12. The conductivity prefactor versus 1-n/nc for Si:P [80], Si:As [43], and Ge:Ga [8,9]. The dashed line shows theory prediction for zm< 1. In all cases expt. values of as (Copyright by the

American Physical Society)

285

Lee [84] for Si:B. The Coulomb gap width is given by Massey and Lee reported data at T~2K that showed a decrease in from 0.85 to 0.93nc, but the trend reversed and increased from 0.93 to 0.96nc. When is increasing [the second term in (24) is dominant] the is decreasing with increasing n because of the rapid increase in with n. However, at some point becomes so large that the first term becomes more important with now closer to This manifests itself in a broadening of the Coulomb gap as the first term in (23) takes over in importance. V. INHOMOGENEITY NEAR THE CRITICAL POINT

Those involved in magnetic phase transition studies have always understood the importance of uniform temperatures near the critical temperature Tc and uniform magnetic fields in magnetic resonance experiments. In MIT studies with doped Si and Ge one is always confronted with doping inhomogeneities that need to be understood. It is of particular importance to understand how doping inhomogeneities (IN) might affect the scaling exponents. It was recognized long ago that there were several methods to tune thru the critical point and both magnetic fields and uniaxial stress have been utilized in tuning the MIT in doped Si and Ge. The former removes time reversal symmetry and changes the universality class and the scaling exponent s.Uniaxial stress does not change the universality class, but nevertheless it may

change the scaling exponent because of the introduction of relatively large stress inhomogeneity from sample bending in compression experiments with bars with large slenderness ratios. This has now led two groups to suggest a larger exponent s for both Si:B [85] and Si:P [86] this despite the fact that the data exhibited features similar to that of Paalanen et al. [42]. For a completely homogeneous system Doping IN will introduce a normalized dopant distribution of the form where is the mean dopant density and is a measure of the width of the variation in n. Strictly speaking the distribution should be spatially dependent with an n(r) and the integral should be over the volume of the sample, but for simplicity we consider a case neglecting spatial correlation of n(r). The average conductivity will be given by For the Mott VRH case for Tonc the important exponent is t=½. For this case the integration from nc to [note that for n
286

Thus it appears possible, neglecting correlations in n(r), to have a crossover from s = ½ to s = 1 if the inhomogeneity is large enough. If one can independently identify nc experimentally then provides a direct measure of the inhomogeneity. The Paalanen et al. results [42] yield scaling with s~½ for n/nc-1>3xl0-4. This either suggests or that the result neglecting correlation in n(r) in (25) is based on an assumption that isn’t valid.

Stress inhomogenity (SI) from sample bending is a simpler case to treat because the stress distribution is accurately known and is perfectly correlated spatially. When the mean compressive stress is S at the median plane S varies linearly across the beam cross section [axb,a
where the z integration is between the two voltage leads at z2 and z1, while. The effect of SI for a cross section at a given z is to shift Sc to but the scaling exponent t+1 in this regime is independent of z. A numerical integration of (25) for several positions of the dmax relative to z1 and z2 confirms the result that scaling exponent

changes from t to t+1 [t~0.5 to 0.6} in excellent agreement with the uniaxial stress results of Bogdanovich et al. [85] for Si:B. For larger values of their Si:B results are in good agreement with the other weakly compensated cases showing s~0.5. SI from bending with samples with L/a>20 is large and unavoidable. It would be useful to repeat these uniaxial stress experiments with tension rather than compression. Although this review has emphasized the weakly compensated cases where s~½ it is worth noting that heavily compensated samples seem to show substantial differences in the compensation dependence of nc(K). Recent results by Rentzsch et al. [87] on NTD 74Ge-70Ge crystals (Ge:As,Ga) show a strong exponential dependence of nc(K) [nc(K)=nc(0)exp(K/Ko)s, Ko=0.40, s=1.67] that is much stronger than chemically doped Ge:Sb [38] and Si:(P,B) [88]. Rentzsch et al. suggest it is far more likely in the chemically doped cases to have spatial correlation between donors and acceptors than in NTD systems. VI. DISCUSSION, SUMMARY AND ACKNOWLEDGEMENTS

Many theorists in the MIT field have commented on the difficulty of the problem of treating both disorder and interactions in a self-consistent manner and there has been criticism of perturbation theories like that of A-A because of the claim it should break down when In addition the prevailing viewpoint had been that any Boltzmann result could not explain the scaling of Furthermore, it was not realized, that unlike magnetic phase transitions characterized by a correlation length the MIT in doped Si and Ge features a second divergent length, namely the deBroglie wavelength The Boltzmann result from iis yields It must be emphasized that the theory of iis (based on Rutherford scattering) yields the same result in both classical mechanics and QM for an individual ion. The treatment in section III using the alloy-like scattering theory and the phase shift scattering approach (valid for kd<1 where d is the range of the potential) with the Friedel sum rule is essential in obtaining the exponent s~½ in the degenerate limit. The notion 287

of the MIT in doped Si and Ge being a special, novel case of alloy theory is one that needs more attention. The screening implicit in the Mott approach and the Mott criterion is buried in the phase shifts which have no affect on the scaling exponent s, but do affect the magnitude of the prefactor This is consistent with the Anderson statement [89] that a McMillan type of theory is inescapable at T=0. At T=0 interactions can do nothing but screen the existing random potential. However, the McMillan approach ignored the Einstein relation between D and µ, and the charge neutrality condition that cancels na and Ni that leads to σB,iis

kF. The irony in the explanations for is that the A-A theory, only first order in the disorder, appears able to explain all the features of the T-dependence, once a second contribution AT1/2 from iis is added to the mix. The metallic data can be explained without any contribution from weak localization theory. Specifically, the features of the data rule out any weak localization contribution. This not surprising since coherent back scattering and quantum interference seem to be ruled out when the extent of itinerant electron wavepacket is comparable to and is orders of magnitude larger than the impurity spacing in the critical regime. Very close to nc (very small kF) Heisenberg uncertaintly principle supports the notion the carrier wave packets are huge relative to the mean donor spacing and one shouldn’t talk about scattering from individual impurities. This apparently explains the lack of importance of scattering by neutral donors (localized electrons) and at the same time explains why the IoffeRegel criterion is not relevant for is independent of the mean-freepath The scaling of to zero gives strong evidence that as a result at odds with the older conventional wisdom. In the degenerate regime the experimental data comparison with the A-A result yields D(n,0) scaling with exponent closely twice that for consistent with the two Einstein relations [Eq. (16)]. Deviations from T1/2 behavior demonstrate for the first time the temperature dependence of D(n,T), which differs from that for in the same T-regime. This behavior is consistent with the classical relation and clearly demonstrates the importance of the Einstein relations in understanding the MIT. This same relation and the data demonstrate that the A-A derivation of a T1/3 dependence for is incorrect because of an erroneous assumption. The data demonstrate that as one is leaving the degenerate regime and approaching the regime where is comparable to kT. The insulating T-dependence of is more complex because of both Mott and ES VRH, each of which can exhibit T-dependent prefactors as well as exponentials. In addition, there is an activated component from itinerant electrons thermally excited above Ec. Whereas Mott VRH dominates for n>0.9nc for Si MIT systems (for T>0.1K) in the Ge:Ga and Ge:As cases ES VRH is dominant. The new theory, featuring spatial dispersion of extending VRH into the critical regime where To
five years the author has enjoyed stimulating and invigorating discussions with J.C. Phillips and D.F. Holcomb. The opportunity to take a semiconductor physics course from J. Bardeen at the University of Illinois played a substantial role in affecting the new developments in Section III. Finally, the author remembers with fondness various communications with Sir Nevill Mott and Sir Nevill’s trip to Rochester in 1982 to present the first David L. Dexter Lecture.

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Castner, T.G. (2000), Phys.Rev.Lett.84, 1539-42. Kittel, C. (1963), Quantum Theory of Solids, John Wiley and Sons, New York, Ch. 28. Castner, T.G. (2000), Phys.Rev.Lett.84,2905-08. Phillips, J.C. (1992), Phys.Rev.B45, 5863-5867; (1998) Proc.Nat.Acad.Sci.95, 7264-69. Kubo, R. (1957), J.Phys.Soc.Jpn.12, 570-86. Lee, P.A. (1982), Phys.Rev.B26, 5882-85. Imry, Y., Gefen, Y. and Bergmann, D. (1982), Anderson Localization, edited by Y. Nagaoka and H. Fuku-

yama, Springer, Berlin, 138-149; Kawabata, A. (1984), J.Phys.Soc.Jpn.53, 1429-1433. a-Castner, T.G. (1995), Phys.Rev.B52, 12434-38; b-Ref. [40]; c-Ref. [47]; d-Dai, P, Zhang, Y. and Sarachik, M.P. (1991), Phys.Rev.Lett.66,1914-17 ; e-Ref. [8,9]; f-Hess, H.F.; DeConde, K.; Rosenbaum, T.F. and Thomas, G.A. (1982) Phys.Rev.B25, 5578-80; g-Brooks, J.S.; Symko, O.G. and Castner, T.G.(1987), JJour.Appl.Phys.26, Suppl.26-3,721-22; h-Ref. [50]. Katsumoto, S. (1987), J.Phys.Soc.Jpn.56,2259-62. Kirkpatrick, S. (1973), Rev.Mod.Phys.45, 574-88. Cohen, M.H., Economou, E.N. and Soukoulis, C.N. (1984), Phys.Rev.B30, 4493-500. Altshuler, B.L. and Aronov, A.G. (1983), Pis’ma Zh. Eksp.Teor.Fiz.37, 349-51 [JETP Lett.37,410-413)]. Smith, R.A. (1968), Semiconductors, Cambridge Univ. Press, Cambridge, p.237. Newman, P.P. and Holcomb, D.F. (1983), Phys.Rev.B28,626-28. Castner, T.G. (2000), private communication Bhatt, R.N. and Lee, P.A. (1983), Solid State Commun.48 755-59. Dai, P., Bogdanovich, S, Zhang, Y. and Sarachik, M.P. (1995), Phys.Rev.B52, 12439-40. Castner, T.G. (2000), Phys.Rev.B61, 16596-609. Harris, A.B. (1974), J.Phys.C7, 1671-92. Chayes, J., Chayes, L., Fisher, D.S. and Spencer, T. (1986), Phys.Rev.Lett.57, 2999-3002. Belitz, D. and Kirkpatrick, T.R. (1992), J.Phys.Condens.Matter4, L37-L42. Massey, J.G. and Lee, M. (1996), Phys.Rev.Lett.77, 3399-402. Bogdanovich, S., Sarachik, M.P. and Bhatt, R.N. (1999), Phys.Rev.Lett.82, 137-40. Waffenschmidt, S., Pfleiderer, C. and v. Lohneysen, H. (1999), Phys.Rev.Lett.83, 1305-08. Rentzsch, R., Müller, M., Reich, Ch., Sandow, B., lonov, A.N., Fozooni, P., Lea, M.J., Ginodman, V. and Shlimak, I. (2000), phys.stat.sol.(b)218, 233-36. Thomaschefsky, U. and Holcomb, D.F. (1992), Phys.Rev.B45, 13356-62. Anderson, P.W. (1985), Localization, Interaction and Transport Phenomena, edited by B. Kramer, G. Bergmann, and Y. Bruynseraede, Springer, Berlin, p. 12.

METAL-INSULATOR TRANSITION IN HOMOGENEOUSLY DOPED GERMANIUM

MICHIO WATANABE

Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

INTRODUCTION The metal-insulator transition (MIT) in doped semiconductors with a random distribution of impurities is a unique quantum phase transition in the sense that both disorder and electron-electron interaction play a key role (see for example Refs. 1 and 2). The metallic phase of the transition is characterized by a finite electrical conductivity at T = 0, while the conductivity in the insulating phase vanishes in the limit of zero temperature. From a theoretical point of view, the correlation length in the metallic phase and the localization length in the insulating phase diverge at the critical point with the same exponent v, i.e., they are proportional to in the critical regime of the MIT, and the value of v provides important information about the MIT. Here, N is the impurity concentration and Nc is the critical concentration for the MIT. Since direct experimental determination of v is extremely difficult, researchers have usually determined, instead of v, the value of the conductivity critical exponent defined by immediately above Nc Here, is the conductivity extrapolated to T = 0 and is a prefactor. Values of v are then obtained assuming the relation which is valid for three-dimensional systems without electron-electron interaction. With a number of nominally uncompensated semiconductors has been obtained [2]. One of the best studied nominally uncompensated semiconductors is Si:P. Rosenbaum et al. studied both the doping-induced MIT and the uniaxial-stress-induced MIT, and showed that Eq. (1) describes of Si:P with a single exponent over a wide range Here, uniaxial stress was used for “tuning” because fine control of N/Nc – 1 is difficult. In the same material, however, was claimed about ten years later for a narrow regime Nc < N < 1.1N c by a different group [5]. Moreoever, was reported recently for the MIT driven by uniaxial stress [6]. As the origin of the discrepancy, inhomogeneous distribution of

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

291

the impurities has been pointed out [7]. For the case of melt- (or metallurgically) doped samples, which have been employed in

most of the previous studies including Refs. 4 – 6, the spatial fluctuation of N due to dopant striations and segregation can easily be on the order of 1% across a typical sample for the four-point resistance measurement that has a length of ~5 mm or larger (see for example Ref. 8). For this reason, it will not be meaningful to discuss physical properties in the critical regime (e.g., ), unless one evaluates the macroscopic inhomogeneity in the samples and its influence on the results. In order to rule out the ambiguity arising from the inhomogeneity, we prepared 70Ge:Ga samples by neutron-transmutation doping (NTD) of isotopically enriched 70Ge single crystals. The NTD method inherently guarantees the random distribution of the dopants down to the atomic level [9,10]. We show from the conductivity measurements at T = 0.02 – 1 K that of the NTD 70Ge:Ga samples is described by Eq. ( l ) with over a wide range In order to determine v without assuming we have analyzed the temperature dependence of the conductivity on the insulating side of the MIT in the context of variablerange-hopping (VRH) conduction within the Coulomb gap [12]. Low-temperature ( T < 0.5 K) conductivity of the insulating 70Ge:Ga samples obeys which is predicted by the VRH theory [13], with an appropriate temperature dependence in the prefactor Magnetic field and temperature dependence of the conductivity of the samples are subsequently measured in order to determine directly the localization length and the impurity dielectric susceptibility as a function of N in the context of the theory. This kind of determination of and

was performed for compensated Ge:As by Ionov et

al. [14] They found and with and respectively, for samples having N up to 0.96Nc. The significance of their result is the experimental verification of the relation that had been predicted by scaling theories [15]. However, the critical exponents of compensated samples are known to be different from those of nominally uncompensated samples [11]. Therefore, our determination of and in nominally uncompensated samples is important. The previous effort to measure the impurity dielectric susceptibility as a function of N has also contributed greatly. Hess et al. found in nominally uncompensated Si:P [16]. Since was determined for the same series of Si:P samples [4], the relation was again valid. Katsumoto has found and for compensated Al 0.3 Ga 0.7 As:Si, i.e., again, applies [17]. Thus, in these cases the conclusion

was reached indirectly, by assuming

We, on the other hand, determine v directly,

i.e., we do not have to rely on the assumption

in order to study the behavior of the

localization length near the MIT. According to theories [2] on the MIT which take into account both disorder and electronelectron interaction, the critical exponents do not depend on the details of the system, but depend only on the universality class to which the system belongs. Moreover, there is an inequality which is expected to apply generally to disordered systems irrespective of the presence of electron-electron interaction [18]. Hence, if one assumes which is derived for systems without electron-electron interaction, violates the inequality. This discrepancy has been known as the conductivity critical exponent puzzle. Kirkpatrick and Belitz have claimed that there are logarithmic corrections to scaling in universality classes with time-reversal symmetry, i.e., when the external magnetic field is zero, and that found at B = 0, should be interpreted as an “effective” exponent which is different from a “real” exponent satisfying Therefore, comparison of with and without the time-reversal symmetry, i.e., with and without external magnetic fields becomes important. We study the MIT of 70Ge:Ga in magnetic fields up to B = 8 T and show that changes from 0.5 at B = 0 to 1.1 at The same exponent is also found in the magnetic-field-tuned MIT for three different samples, i.e.,

292

where Bc (N) is the critical magnetic field for concentration N. Moreover, an excellent finitetemperature scaling [2]

where x/y is equivalent to is obtained with the same value of The phase diagram on the (N, B) plane is successfully constructed, and we find a simple scaling rule which would obey and derive from a simple mathematical argument that as has been observed in our experiment.

EXPERIMENTS Sample preparation All of the 7()Ge:Ga samples were prepared by neutron-transmutation doping (NTD) of isotopically enriched 7()Ge single crystals. The basic idea of NTD is as follows. Suppose that a nucleus in a crystal of a semiconductor captures a thermal neutron. After the capture, the nucleus is not necessarily stable. If it is stable, the element remains unchanged, but if it is not, it decays and transmutes into a new element which may act as a dopant. This is NTD. Practically, a crystal is placed in a nuclear reactor which produces thermal neutrons. Since the neutron field produced by a reactor is large enough to guarantee a homogeneous

flux over the crystal dimensions and the small capture cross section (typically 10 –24 cm 2 ) of semiconductors for neutrons minimizes “self-shadowing”, NTD is known to produce the most homogeneous, perfectly random distribution of dopant down to the atomic level [9], As for Ge, there are five stable isotopes: 70Ge (20.5%), 72Ge (27.4%), 73Ge (7.8%), 74

Ge (36.5%), and 76Ge (7.8%). The numbers in the parentheses represent the natural abundance. These five stable isotopes of Ge undergo the following nuclear reactions after captur-

ing a thermal neutron. (4) (5) (6) (7) (8) Here,

is a neutron, EC and denote electron capture and decay, respectively, and the time in the parentheses represents the half life. Note that the natural Ge form both acceptors (Ga) and donors (As and Se) after NTD. Empirically, the impurity compensation is known to affect the value of the critical exponent [11]. To avoid the impurity compensation, we use isotopically enriched 70Ge. The Czochralski grown, chemically very pure 70Ge crystal has isotopic composition 70 [ Ge]=96.2 at. % and [72Ge]=3.8 at. % [10]. The as-grown crystal is free of dislocations, p type with a net electrically-active-impurity concentration less than 5 × 1011 cm –3 . The thermal neutron irradiation was performed with the thermal to fast neutron ratio of The small fraction of 72Ge becomes 73Ge which is stable, i.e., no further acceptors or donors are introduced. The post-NTD rapid thermal annealing at 650 °C for 10 sec removed most of the irradiation-induced defects from the samples. The short annealing time is important in order to avoid the redistribution and/or clustering of the uniformly dispersed 71Ga acceptors. The concentration of the electrically active radiation defects measured with deep level transient spectrometry (DLTS) after the annealing is less than 0.1% of the Ga concetration, i.e., the compensation ratio of the samples is less than 0.001. The dimension of most samples used for conductivity measurements was 6×0.9×0.7 mm3. Four strips of boron-ion-implanted regions on a 6×0.9 mm2 face of each sample were coated with 200 nm Pd and 400 nm Au pads 293

Figure 1. Electrical conductivity as a function of T 1/2 for NTD 70Ge:Ga. From bottom to top in units of 1017 c m – 3 , the Ga concentrations are 1.853, 1.856, 1.858, 1.861, 1.863, 1.912, 1.933, 2.004, 2.076, 2.210, 2.219, 2.232, 2.290, 2.362, and 2.434, respectively.

using a sputtering technique. Annealing at 300 °C for one hour activated the implanted boron and removed the stress in the metal films. The concentration of Ga acceptors after NTD is determined from the time of thermalneutron irradiation. The concentration is proportional to the irradiation time as long as the same irradiation site and the same power of a nuclear reactor are employed.

Low-temperature measurements The electrical conductivity measurements were carried out down to temperatures of 20 mK using a 3He-4He dilution refrigerator. All the electrical leads were low-pass filtered at the top of the cryostat. The sample was fixed in the mixing chamber and a ruthenium oxide thermometer [Scientific Instrument (SI), RO600A, 1.4×1.3×0.5 mm 3] was placed close to the sample. To measure the resistance of the thermometer, we used an ac resistance bridge (RV-Elekroniikka, AVS-47). The thermometer was calibrated against 2Ce(NO 3)3 ·

3Mg(NO 3 ) 2 ·24H 2 O (CMN) susceptibility and against the resistance of a canned ruthenium oxide thermometer (SI, RO600A2) which was calibrated commercially over a temperature range from 50 mK to 20 K. We employed an ac method at 21.0 Hz to measure the resistance of the sample. The power dissipation was kept below 10 –14 W, which is small enough to

avoid overheating of the samples. The output voltage of the sample was detected by a lock294

Figure 2. Conductivity as a function of (a) T1/2 and (b) T1/3, respectively, near the metal-insulator transition. From bottom to top in units of 1017 cm –3 , the concentrations are 1.853, 1.856, 1.858, 1.861, 1.863, and 1.912,

respectively. The upper and lower dotted lines in each figure represent the best fit using the data between 0.05 K and 0.5 K for the first and the third curves from the top, respectively. Each fit is shifted downward slightly for easier comparison.

in amplifier (EG&G Princeton Applied Research, 124A). All the analog instruments as well

as the cryostat were placed inside a shielded room. The output of the instruments was detected by digital voltmeters placed outside the shielded room. All the electrical leads into the shielded room were low-pass filtered. The output of the voltmeters was read by a personal computer via GP-IB interface connected through an optical fiber. Magnetic fields up to 8 T were applied in the direction perpendicular to the current flow by means of a superconducting solenoid. RESULTS AND DISCUSSION

Temperature dependence of conductivity in metallic samples and the critical exponent

for the zero-temperature conductivity

The temperature dependence of the electrical conductivity mostly for the metallic samples is shown in Fig. 1. The temperature variation of the conductivity of disordered metal is governed mainly by electron-electron interaction at low temperatures [1], and can be written as (9) where (10)

Here, is a dimensionless and temperature-independent parameter characterizing the Hartree interaction and D is the diffusion constant [1], which is related to the conductivity via the Einstein relation (11) 295

Figure 3. (a) Conductivity as a function of From bottom to top in units of 1017 cm–3, the concentrations are 1.858, 1.861, 1.863, and 1.912, respectively. The solid lines denote the extrapolation for finding (b) Zero-temperature conductivity vs the dimensionless distance N/Nc – 1 from the critical point on a double logarithmic scale. The dotted line represents the best power-law fit by where

where

is the density of states at the Fermi level. In various reports such as Refs. 5 and

10, was obtained by extrapolating to T = 0 assuming dependence based on Eq. (9). One should note, however, that it is sound only in the limit of where D can be considered as a constant, i.e., m is constant, and that the inequality is no longer valid as N approaches Nc from the metallic side since also approaches zero. In such cases m in Eq. (9) is not temperature independent and may exhibit a temperature dependence different from To examine this point in our experimental results, we go back to Fig. 1. We see there that of the bottom five curves are not proportional to

while of

of the other higher N samples are described by

for the six samples with positive

in the scale of

The close-ups and T1/3 are shown

in Figs. 2(a) and 2(b), respectively. The upper and lower dotted lines represent the best fit using the data between 0.05 K and 0.5 K for the samples with N = 1.912 × 1017 cm – 3 and

N = 1.861 × 1017 cm – 3 , respectively. Each fit is shifted downward slightly for the sake of clarity. From this comparison, it is clear that a T 1/3 dependence rather than a dependence

holds for samples in the very vicinity of the MIT. The opposite is true for the curve at the top.

This means that the dependence in Eq. (9) is replaced by a T1/3 dependence as the MIT is approached. A T1/3 dependence close to the critical point for the MIT was predicted originally by

Al’tshuler and Aronov [21]. They considered an interacting electron system with paramagnetic impurities, for which they obtained a single parameter scaling equation. At finite temperatures, they assumed a scaling form for conductivity according to the scaling hypothesis: (12) where

is the correlation length and

is the thermal diffusion length. When which is equivalent to Eq. (9). In the critical region,

where

Eq. (12) should be reduced to (13)

296

Figure 4. The logarithm of the conductivity as a function of T –1/2 for insulating samples. the concentrations from bottom to top in units of 1017 cm–3 are 1.717, 1.752, 1.779, 1.796, 1.805, 1.823, 1.840, 1.842, 1.843, 1.848, 1.850, 1.853, 1.856, and 1.858, respectively.

Combining this equation and Eq. ( 1 1 ) , they obtained The T1/3 dependence was also predicted from numerical calculations that consider solely the effect of disorder [22]. It is not clear whether the argument in Refs. 21 or 22 is applicable to the present system or not, but there is an experimental fact that dependence of conductivity changes to

T 1/3 as Nc is approached. It is important that we find a consistent method that allows the determination of for both the cases. For this purpose, we follow Al’tshuler and Aronov’s manipulation [21] of eliminating m and D in Eqs. (9)–(l 1) and obtain

(14)

where

which is temperature independent. In the limit of

this equation gives the same value of as Eq. (9) does. When it yields a T1/3 dependence for Thus, it is applicable to both and T1/3 dependent conductivity. From today’s theoretical understanding of the problem, Eqs. (9) and (14) are valid only for and their applicability to the critical region is not clear, because the higher-order terms of the function [23] which were once erroneously believed to be zero do not vanish [24]. Nevertheless, we expect Eq. (14) to be a good expression for describing the temperature dependence of all metallic samples because it expresses both and T1/3 dependences as limiting forms. Then, based on Eq. (14), we plot for the four close to Nc samples in Fig. 3(a). The data points align on straight lines, which supports the adequacy of Eq. (14). The zero-temperature conductivity is obtained by

extrapolating to T = 0. The curve on the top of Fig. 3(a) is for the sample with the lowest N 297

Figure 5.

Conductivity multiplied by T–1/3 vs (a) T–1/2 and (b) T–1/4. From bottom to top in units of

10 17 c m – 3 , the concentrations are 1.848, 1.850, 1.853, and 1.856, respectively.

among the ones showing dependence at low temperatures, i.e., this sample has the largest value of among samples. The value of obtained for this particular sample using Eq. (14) differs only by 0.6% from the value determined by the conventional extrapolation assuming Eq. (9). This small difference is comparable to the variation arising from the choice of the temperature range in which the fitting is performed. Therefore, the extrapolation method proposed here is compatible with the conventional method based on the extrapolation. Based on this analysis the MIT is found to occur between the first and second samples from the bottom in Fig. 3(a), i.e., 1.858 × 10l7 cm – 3 < Nc < 1.861 × 1017 cm – 3 . Thus, unlike the case for Si:P [5,7], Nc is fixed already within an accuracy of 0.16% and the evaluation of the critical exponent µ will not be affected by the ambiguity in the determination of N c. Figure 3(b) shows the zero-temperature conductivity as a function of N/Nc – 1 on a double logarithmic scale. A fit of Eq. (1) (dotted line) is excellent all the way down to ( N / Nc – 1) = 4 × 10 –4 . The fitting parameters are and We note that is obtained even when we use only the four samples closest to Nc for the fitting. Variable-range-hopping conduction in insulating samples and the critical exponents for localization length and dielectric susceptibility The temperature dependence of the conductivity of insulating samples is shown in Fig. 4. The electrical conduction of doped semiconductors on the insulating side of the MIT is often dominated by variable-range hopping (VRH) at low temperatures. The temperature dependence of for VRH is written in the form of (15)

where p = 1/2 for the excitation within a parabolic-shaped energy gap (Coulomb gap), and p = 1/4 for a constant single-particle density of states around the Fermi level [13]. The temperature dependence of contributes greatly to the temperature dependence of 298

Figure 6. T0 determined by 1 – N/Nc.

as a function of the dimensionless concentration

near Nc because the factor T0 / T in the exponential term becomes very small, i.e., the temperature dependencies of the prefactor and that of the exponential term become comparable. Theoretically, is expected to depend on temperature as (16) but the value of r including the sign has not been derived yet for VRH conduction with both

p = 1/2 and p = 1/4. As we have seen in Fig. 2(b), the temperature variation of the low-temperature conductivity of the 70Ge:Ga samples within of Nc is proportional to T1/3. Since both the 1/3 T dependence of the conductivity and the VRH with p = 1/2 are results of the electronelectron interaction in disordered systems, they can be expressed, in principle, in a unified form. Moreover, the electronic transport in barely metallic samples and that in barely insulating samples should be essentially the same at high temperatures so long as the inelastic scattering length and the thermal diffusion length are smaller than, or at most comparable to the correlation length or the localization length. So, the temperature dependence of conductivity at high temperatures should be the same on both sides of the transition. Such behavior is confirmed experimentally in the present system, i.e., as seen in Fig. 2(b) the conductivity of samples very close to Nc shows a T1/3 dependence at irrespective of the phase (metal or insulator) to which they belong at T = 0. Based on this consideration we fix r = 1/3. Figure 5 shows with r = 1/3 for four samples (N/N c = 0.993, 0.994, 0.996, and 0.998) as a function of (a) T–1/2 and (b) T–1/4. All the data points lie on straight lines with p = 1/2 in Fig. 5(a) while they curve downward with p = 1/4 in Fig. 5(b). This dependence is maintained even when we change the values of r between r = 1/2 and 1/4. Thus we conclude that the conductivity of all samples on the insulating side for N up to 0.998Nc is described by the theory for the VRH conduction where the excitation occurs within the Coulomb gap, i.e., Eq. (15) with p = 1/2. Based on these findings, we evaluate the N dependence of T0 in Eq. (15) with p = 1/2 and r = 1/3. Figure 6 shows T0 as a function of 1 – N/Nc. The vertical and horizontal error bars have been estimated based on the values of T0 obtained with r = 1/2 and r = 1/4, and the values of l – N/(l.858 × 1017 cm–3) and l – N/(1.861 × 1017 cm–3), where 1.858 × 1017 cm–3 is the highest concentration in the insulating phase and 1.861 × 10l7 cm – 3 is the lowest in the metallic phase, respectively. According to theory [13], T0 in Eq. (15) is 299

Figure 7. (a) Logarithm of

vs B2 at constant temperatures for the sample having N = 1.840 × 1017 cm – 3 .

From top to bottom the temperatures are 0.095 K, 0.135 K, and 0.215 K, respectively. The solid lines represent the best fits. (b) Slope

vs T

–3/2

for the same sample. The solid line represents the best fit.

given by (17)

in SI units, where is the dielectric constant, and is the localization length. Here, we should note that the condition T < T0 is needed for the VRH theory to be valid, i.e., T0 has to be evaluated only from the data obtained at temperatures low enough to satisfy the above condition. This requirement is fulfilled in Fig. 6 for all the samples except for the one with N = 0.998Nc. Concerning this latter sample, we will include it for the determination of

(Fig. 8) and and (Fig. 9) but not for the calculation of the critical exponents. Our next step is to separate T0 into and in the framework of the theory of VRH conduction with and without a weak magnetic field [13]. For the magnetoconductance is expressed as (18) where is the magnetic length in SI units. According to Eq. (18), the magneticfield variation of ln at T = const, is proportional to B2, i.e.,

(19)

and the slope C2(T) in the above equation is proportional to T –3/2. In order to demonstrate that these relations hold for our samples, we show for the N = 0.989Nc sample vs B2 in Fig. 7(a) and C2(T) determined by least-square fitting of vs T – 3 / 2 in Fig. 7(b). Since Eq. (18) is equivalent to (20)

is given by (21) In this way we have determined as a function of T0 for nine samples (Fig. 8). The value of is almost independent of T0 , and if one assumes the form of one obtains a small 300

Figure 8.. Coefficient

defined by Eq. (20) as a function of T0.

Figure 9. (a) Dielectric susceptibility vs 1 – N/N c.

arising from the impurities vs Nc /N – 1. (b) Localization length

value of from least-square fitting. We determine and from Eqs. (17) and (21), and show them in Fig. 9 as a function of 1 – N/Nc and Nc /N – 1, respectively. Here, is the dielectric constant of the host Ge, and hence, is the dielectric

susceptibility of the Ga acceptors. We should note that both and are sufficiently larger than the Bohr radius (8 nm for Ge) and respectively. According to the theories of the MIT, both and diverge at Nc as respectively. We find, however, both and

and do not show such simple dependencies

on N in the range shown in Fig. 9, and that there is a sharp change of both dependencies at On both sides of the change in slope, the concentration dependence of and are expressed well by the scaling formula as shown in Fig. 9. Theoretically, the quantities should show the critical behavior when N is very close to Nc. So and may be concluded from the data in 0.99 < N/Nc. However, the other region (0.9 < N/Nc < 0.99), where we obtain and is also very close to Nc in a conventional experimental sense. As a possible origin for the change in slope, we refer to the effect of compensation.

301

Although our samples are nominally uncompensated, doping compensation of less than 0.1 % may be present due to residual isotopes that become n-type impurities after NTD. In addition to the doping compensation, the effect known as “self compensation” may play an important role near Nc [26]. It is empirically known that the doping compensation affects the value of the critical exponents [11]. Rentzsch et al. studied VRH conduction of n-type NTD Ge in the concentration range of 0.2 < N/NC < 0.91, and showed that T0 vanishes as with 3 for K = 38% and 54%, where K is the compensation ratio [27]. Since [Eq. (17)], we find for our NTD 70Ge:Ga samples = 3.5 ± 0.8 for 0.99 < N/Nc < 1 and = 0.95 ± 0.08 for 0.9 < N/Nc < 0.99. Interestingly, = 3.5 ± 0.8 agrees with 3 found for compensated samples. Moreover, we have recently proposed the possibility that the conductivity critical exponent µ 1 in the same 70Ge:Ga only within the very vicinity of Nc (up to about +0.1 % of Nc) [28]. An exponent of µ = 0.50 ± 0.04, on the other hand, holds for a wider region of N up to 1.4N c as we have seen in Fig. 3(b). Again, µ 1 near Nc may be viewed as the effect of compensation. Therefore, it may be possible that the region of N around Nc where v 1 and µ 1 changes its width as a function of the doping compensation. In the limit of zero compensation, the part which is characterized by v 1 and µ 1 vanishes, i.e., we propose v = 0.33 ±0.03, = 0.62±0.05, and µ = 0.50±0.04 for truly uncompensated systems and that the relation µ = v [3] is not satisfied. In compensated systems, on the other hand, µ = v may hold as it does in the very vicinity of Nc. However, the preceding discussion needs to be proven experimentally in the future by using samples whose compensation ratios are controlled precisely and systematically. Metal-insulator transition in magnetic fields

Figure 10 shows the temperature dependence of the conductivity of the sample having N = 2.004 × 1017 cm – 3 for several values of the magnetic induction B. Application of the magnetic field decreases the conductivity and eventually drives the sample into the insulating phase. This property can be understood in terms of the shrinkage of the wave function due to the magnetic field. In strong magnetic fields and at low temperatures, i.e., when gµ BB k B T , the conductivity shows another dependence [ 1 ] (22)

where (23)

One should note that Eqs. (9) and (22) are valid only in the limits of (N,0,0) or It is for this reason that we have observed a T1/3 dependence rather than the dependence at B = 0 in Fig. 2 as the critical point [ (N,0,0) = 0] is approached from the metallic side. However, Fig. 10 shows that the dependence holds when B 0 even around the critical point. Hence, we use Eq. (22) to evaluate the zero-temperature conductivity (N,B,0) in magnetic fields. Since mB is independent of B, the conductivity for various values of B plotted against should appear as a group of parallel lines. This is approximately the case as seen in Fig. 10 at low temperatures (e.g., T < 0.25 K). The zero-temperature conductivity (N,B,0) in various magnetic fields obtained by extrapolation of (N , B , T ) to T = 0 based on Eq. (22) is shown in Fig. 11. Here, (N,B,0) is plotted as a function of the normalized concentration: (24)

Since the relation between N and (N,0,0) was established in Fig. 3(b) as (N,0,0) = (0)[N/N C (0) – 1] 0.50 where NC(0) = l.860 × 10l7 cm –3 and (0) = 40 S/cm, n is equivalent to N/Nc(0) – 1. Henceforth, we will use n instead of N because employing n reduces 302

Figure 10. Conductivity of the sample having N = 2.004 × 1017 cm – 3 as a function of T1/2 at several magnetic fields. The values of the magnetic induction from top to bottom in units of tesla are 0.0, 1.0, 2.0, 3.0, 4.0, 4.7, 5.0, 5.3, 5.6, 6.0, 7.0, and 8.0, respectively.

Figure 11. Zero-temperature conductivity ( N , B , 0 ) vs normalized concentration N/Nc(0) – 1, where (N,0,0) is the zero-temperature conductivity and (0) is the prefactor both at B = 0. From top to bottom the magnetic induction increases from 1 T to 8 T in steps of 1 T. The dashed curve at the top is for B = 0. The solid curves represent fits of (N,B,0)

[n/n c (B) – 1]µ(B).

For B

6 T, we assume

µ = 1.15.

303

Figure 12. Zero-temperature conductivity (N,B,0) vs magnetic induction B. From bottom to top, the normalized concentrations defined by Eq. (24) are 0.04, 0.09, and 0.22, respectively.

the scattering of the data caused by several experimental uncertainties, and it further helps us concentrate on observing how ( N , B , 0 ) varies as B is increased. Similar evaluations of the concentration have been used by various groups. In their approach, the ratio of the resistance at 4.2 K to that at a room temperature is used to determine the concentration [5]. The dashed

curve in Fig. 1 1 is for B = 0, which merely expresses Eq. (24), and the solid curves represent fits of (25) The exponent µ(B) increases from 0.5 with increasing B and reaches a value close to unity at B 4 T. For example, µ = 1.03 ± 0.03 at B = 4 T and µ = 1.09 ± 0.05 at B = 5 T. When B 6 T, three-parameter [ (B), nc(B), and µ(B)] fits no longer give reasonable results because the number of samples available for the fit decreases with increasing B. Hence, we give the solid curves for B 6 T assuming µ(B) = 1.15. We show (N,B,0) as a function of B in Fig. 12 for three different samples. When the

magnetic field is weak, i.e., the correction (N,B,0) small compared with (N,0,0), the field dependence of

( N , B , 0 ) – (N,0,0) due to B is (N,B,0) looks consistent with

the prediction by the interaction theory [1], (26) In larger magnetic fields, (N,B,0) deviates from Eq. (26) and eventually vanishes at some magnetic induction Bc. For the samples in Fig. 12, we tuned the magnetic induction to the MIT in a resolution of 0.1 T. We fit an equation similar to Eq. (25), (27)

to the data close to the critical point. As a result we obtain µ' = 1 . 1 ± 0.1 for all of the three samples. The value of µ' depends on the choice of the magnetic-field range to be used for the fitting, and this fact leads to the error of ±0.1 in the determination of µ'. In Fig. 13 we show that µ' = 1 . 1 yields an excellent finite-temperature scaling [Eq. (3)]. Note that the data both on the metallic side and on the insulating side are included in this scaling plot. Here 304

Figure 13.

Finite-temperature scaling plot for the field-induced metal-insulator transition in the 70Ge:Ga

sample having n = 0.09.

we employ Bc obtained by fitting Eq. (27), x = 1/2 from the fact that dependence holds in magnetic fields even around the critical point, and y = x/µ' = 0.45, i.e., none of them are treated as a fitting parameter. Hence, Fig. 13 strongly supports µ' = 1.1. From the critical points nc(B) and Bc(n), the phase diagram at T = 0 is constructed on the (N,B) plane as shown in Fig. 14. Here, nc(B) for B 6 T shown by triangles are obtained by assuming µ = 1.15. The vertical solid lines associated with the triangles represent the range of values over which nc(B) have to exist, i.e., between the highest n in the insulating phase and the lowest n in the metallic phase. Solid diamonds represent Bc for the three samples in which we have studied the magnetic-field-induced MIT. Estimations of Bc for the other samples are also shown by open boxes with error bars. The boundary between metallic phase and insulating phase is expressed by a power-law relation: (28)

From the eight data points denoted by the solid symbols, we obtain C3 = (1.33 ±0.17) × 10–3 and = 2.45 ±0.09 as shown by the dotted curve. The shift of N c in magnetic fields was studied theoretically by Khmel’nitskii and Larkin [29]. They considered a noninteracting electron system starting from

(29) where

is the correlation length. They claimed that the argument of the function f2 should 305

Figure 14. Phase diagram of 70Ge:Ga at T = 0. The solid circles and the open triangles represent the critical concentrations nc, and the solid diamonds and the open boxes represent the critical magnetic induction Bc.

be a power of the magnetic flux through a region with dimension

This means (30)

where

is the magnetic length, and hence,

= 1 /2. In order to discuss the shift

of the MIT due to the magnetic field, they rewrote Eq. (30) as (31) based on the relation in zero magnetic field (32) Here, t is a measure of distance from the critical point in zero field, e.g., (33)

The zero point of the function equals

gives the MIT, and the shift of the critical point for the MIT (34)

Thus,

= l/(2v) results. In the present system, however, this relation does not hold, as long

as we assume µ = v [3]. Experimentally, we find 70 Ge:Ga at B = 0.

= 2.5, while l/(2v) = l/(2µ ) = 1 for

Based on the phase diagram we shall consider the relationship between the two critical

exponents: µ for the doping-induced MIT and µ' for the magnetic-field-induced MIT. Suppose that a sample with normalized concentration n has a zero-temperature conductivity at B 0 and that [n/n c (B) – 1] 1 or [1 – B/Bc(n)] 1. From Eqs. (25) and (27), we have two expressions for (35) 306

and (36) On the other hand, we have from Eq. (28) (37)

in the limit of (1 – B/BC)

1. This equation can be rewritten as (38)

Using Eqs. (35), (36), and (38), we obtain

(39) Since Eq. (39) has to hold for arbitrary B, the following relations (40) and

(41) are derived. In Fig. 15 we see how well Eq. (41) holds for the present system. We have already shown in Fig. 12 that µ ' = 1.1 ± 0.1 is practically independent of n. Concerning the exponent µ, however, its dependence on B has not been ruled out completely even for the highest B we used in the experiments. This is mainly because the number of available data points at large B is not sufficient for a precise determination of µ. In Fig. 15 the results of the doping-induced MIT for B

4 T (solid symbols) and the magnetic-field-induced MIT for three different

samples (open symbols) are plotted. Here, we plot vs [n/n c (B) – 1]/ with = 2.5 and µ = 1.1 for the doping-induced MIT, and vs [1 – B/Bc(n)] for the magnetic-field-induced MIT. Figure 15 shows that the data points align exceptionally well along a single line describing a single exponent µ = µ' = 1.1. We saw in Fig. 11 that µ apparently takes smaller values in B 3 T, which seemingly contradicts the above consideration. We can understand this as follows. We find that the critical exponent µ in zero magnetic field is 0.5 which is different from the values of µ in magnetic fields. Hence, one should note whether the system under consideration belongs to the “magnetic-field regime” or not. In systems where the MIT occurs, there are several characteristic length scales: the correlation length, the thermal diffusion length, the inelastic scattering length, the spin scattering length, the spin-orbit scattering length, etc. As for the magnetic field, it is characterized by the magnetic length When is smaller than the other length scales, the system is in the “magnetic-field regime.” As the correlation length diverges at the holds near the critical point, no matter how weak the magnetic field is. When the field is not sufficiently large, the “magnetic-field regime” where we assume µ = 1.1 to hold, is restricted to a narrow region of concentration. Outside the region, the system crosses over to the “zero-field regime” where µ = 0.5 is expected. This is what is seen in Fig. 11.

CONCLUSION We have measured the electrical conductivity of NTD 70Ge:Ga to study the metalinsulator transition, ruling out an ambiguity due to inhomogeneous distribution of impurities. The critical exponent µ 0.5 in zero magnetic field for doped semiconductors without impurity compensation has been confirmed. On the insulating side of the MIT, while the relation 307

Figure 15. Normalized zero-temperature conductivity (N,B,0)/ (n) and (N,B,0)/[ (B)] as functions of [1 – B/Bc(n)] and [n/n c (B) – 1]/ respectively, where = 2.5 and µ = 1.1. The solid line denotes a powerlaw behavior with the exponent of 1.1. The open and solid symbols represent the results of the magnetic-fieldinduced metal-insulator transition (MIT) in the range (1 – B/BC) < 0.5 for three different samples (n = 0.04, 0.09, and 0.22) and the doping-induced MIT in constant magnetic fields (4, 5, 6, 7, and 8 T), respectively.

308

2v predicted by scaling theories [15] holds for 0.9 < N/NC < 1, the critical exponents for localization length and impurity dielectric susceptibility change at N/NC 0.99. The small amount of doping compensation that is unavoidably present in our samples may be responsible for such a change in the exponents. We have also measured the conductivity in magnetic fields up to B = 8 T in order to study the doping-induced MIT (in magnetic fields) and the magnetic-field-induced MIT. For both of the MIT, the critical exponent of the conductivity is 1.1, which is different from the value 0.5 at B = 0. The change of the critical exponent caused by the applied magnetic fields supports a picture in which µ varies depending on the universality class to which the system belongs. The phase diagram has been determined in magnetic fields for the 70Ge:Ga system.

ACKNOWLEDGMENTS

This work has been performed in collaboration with K. M. Itoh, Y. Ootuka, and E. E. Haller. We are thankful to T. Ohtsuki for fruitful discussions, and to S. Katsumoto, B. I. Shklovskii, M. P. Sarachik, and J. C. Phillips for valuable comments. The author is supported by Research Fellowship of Japan Society for the Promotion of Science for Young Scientists. REFERENCES 1. 2. 3.

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Itoh, K.M. Haller, E.E. Beeman, J.W. Hansen, W.L. Emes, J. Reichertz, L.A. Kreysa, E. Shutt, T. Cummings, A. Stockwell, W. Sadoulet, B. Muto, J. Farmer, J.W. and Ozhogin, V.I. (1996) Hopping conduction and metal-insulator transition in isotopically enriched neutron-transmutation-doped 70Ge:Ga, Phys. Rev. Lett. 77, 4058-4061. 11. Watanabe, M. Ootuka, Y. Itoh, K.M. and Haller, E.E. (1998) Electrical properties of isotopically enriched neutron-transmutation-doped 70 Ge:Ga near the metal-insulator transition, Phys. Rev. B 58, 9851-9857. 12. Watanabe, M. Itoh, K.M. Ootuka, Y. and Haller, E.E. (2000) Localization length and impurity dielectric susceptibility in the critical regime of the metal-insulator transition in homogeneously doped p-type Ge, Phys. Rev. B 62, R2255-R2258.

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EXPERIMENTAL EVIDENCE FOR FERROELASTIC NANODOMAINS IN HTSC CUPRATES AND RELATED OXIDES

J. JUNG Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canada INTRODUCTION This paper reviews an experimental evidence for the presence of nanodomains in HTSC perovskite cuprates and in CMR perovskite manganites. Perovskite materials belong to a class of ferroelastic crystals [1]. A crystal is ferroelastic if it has two or more stable orientational states in the absence of a mechanical stress (and an electric field), and if it can be reproducibly transformed from one state to another of these states by the application of mechanical stress[2, 3]. Ferroelasticity is a structure-dependent property and is directly inferable from the crystal structure. In the ferroelastic state the crystal symmetry is reduced to a subgroup of a higher symmetry class by a small distortion which is a measure of the spontaneous strain. By the analogy to a ferroelectric case, as the spontaneous polarization is reoriented by application of an electric field, so in the ferroelastic case the spontaneous strain is reoriented by application of a mechanical stress. Reported maximum values of an atomic displacement in a variety of ferroelastic crystals range from 0.04 to 0.24 nm. Similarly to a ferroelectric crystal which exhibits a hysteresis of a polarization versus an electric field, a stress–strain hysteresis is manifested by a ferroelastic crystal. A transformation from one orientational state to the other in ferroelastic crystals is in general accompanied by the formation of ferroelastic domains which reflect concentration of strain due to static correlated displacements of atoms from their periodic lattice sites. For example, transmission electron microscopy (TEM) studies of a well know ferroelastic crystal of barium titanate (BaTiO3) revealed the presence of domain structures of the size down to 4-6 nanometers [4]. The onset of ferroelasticity, as a function of temperature or pressure is often accompanied by additional cooperative phenomena such as ferroelectric or magnetic ordering. Correlation between the ferroelastic and superconducting transition has been also observed in conventional superconductors such as V3Si and N b3Sn [3]. The occurrence of ferroelasticity in perovskite oxides has raised the question about the role of ferroelastic transitions in the origin of HTSC in cuprates and CMR in

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

311

manganites. This paper analyzes recent experimental evidence for ferroelastic transitions and nanodomains in HTSC and partly in CMR. The evidence for the ferroelastic transition in HTSC cuprates includes the measurements of stress-strain properties, helium ion channeling, and neutron and Raman scattering. The experimental evidence for ferroelastic nanodomains in HTSC cuprates has been provided by high resolution TEM, STM, inelastic neutron scattering, and magnetic studies of critical currents. FERROELASTICITY IN HTSC CUPRATES

One of the first experiments, that have indicated a ferroelastic nature of HTSC cuprates, were based on stress-strain measurements of Bi2Sr2CaCu2Ox whiskers of

Tc=75K by Tritt et al. [5]. 2212 whiskers are flexible ribbon-shaped single crystals with a c-axis perpendicular to the plane of the ribbon. The a-axis is the growth direction of the whiskers, perpendicular to the b axis Both a and b axes lie in the plane of the ribbon. The force (stress) was applied to the sample along the a-axis, and the displacement (strain) was measured capacitively. The results show a hysteresis in the stress-strain curves above 270K (Fig. 1), in addition to a maximum of Young’s modulus at 270K (Fig. 2). This suggests the existence of a structural phase transition. The data is characteristic of a displacive phase transformation and in particular of a ferroelastic transformation with the hysteresis which resembles that for a ferroelastic transition. The hysteresis and a temperature dependence of Young’s modulus

indicate a stress-related formation and dynamics (the hysteresis relaxation time is about 20 seconds) of ferroelastic domain walls.

Figure 1. Stress versus strain at various temperatures for a Bi2 Sr2 Ca Cu2 Ox whisker: (a) 50, 100 and 200K; (b) 255K; (c) 280K; (d) 290K; (e) 300K; (f) 315K. Hysteresis appears in the stress-strain relationship above 270K. The direction of the hysteresis loop is indicated by the arrows. The stress and strain relax towards the middle of the hysteresis loop with a characteristic time of approximately 20 seconds. The shape of the stress-strain curve at 270K gives a value for Young’s modulus of 20 GPa. The sample has the following dimensions L=0.65mm, A=12µm2 (From Tritt et al [5])

312

Figure 2. The normalized modulus, Y(T)/Y(270K), as a function of temperature for a sample of Bi2 Sr 2 Ca Cu 2 Ox (open circles) and NbSe3 sample (filled triangles). The value of Y at 270 K for Bi 2 Sr2 Ca Cu2 Ox and NbSe3 samples are Y=20 and 91 GPa, respectively (From Tritt et al [5]).

MeV helium ion channeling have been used by Sharma et al. [6] to probe lattice distortions (static or dynamic) in as a function of temperature and oxygen doping. Ion channeling provides a direct real space probe of extremely small (sub-picometer) displacement of atoms in single crystalline materials.

Sharma et al

measured the excess lattice distortion above the thermal background uex as a function of temperature between 30 and 300K (Fig. 3). The magnitude of uex was extracted from the measured FWHM (full-width-at-half-maximum) of the channeling angular scan. For an optimally doped YBCO (TC=92K) uex shows a drop from room temperature down to about 230K (T1) by about

. u ex exhibits a cusp (of the magnitude of about ) between 230K and about 140K (T2). Near Tc, uex drops further by about with a decreasing temperature. The authors have stated that the changes in uex are indicative of some kind of phase transition in the system, which enhances the fluctuation effects (static or dynamic). These changes could be attributed to ferroelastic phase transitions at temperatures and T3=Tc=90K. In the underdoped YBCO (TC=65K) a drop of uex between 300K and T1=240K and a cusp between 240K and T2=150K is less pronounced, except a sudden drop of uex at Tc by about Similar behavior has been observed in an underdoped YBCO of Tc=45K. Taking into account that oxygen deficiency (oxygen vacancies) in YBCO leads to nanoscopic disorder, the ferroelastic transitions are diffused since they are sensitive to a disorder.

Kaldis et al. [7] observed a displacive structural transformation in the copper-oxygen planes of YBa2Cu3Ox at the underdoped – overdoped phase separation line at x=6.95. They measured, as a function of x, the dimpling in the CuO2 planes using x-ray absorption fine-structure spectroscopy (EXAFS) at 25K and the oxygen O(2,3) in-phase mode Raman shifts. The data (Fig. 4) show for anomalously large static displacements of the Cu(2) atoms off the O(2,3) layer and a gap in the distribution of the O(2,3) in-phase Raman shifts. On doping from the underdoped side at x=6.805 up to x=6.885, the Cu(2) position was found to move along c-axis by about off the O(2,3) layer. From x=6.895 up to x=6.945 the dimpling of Cu(2) increases further by 313

Figure 3. Temperature dependence of excess atomic displacement above the thermal background uex in YBa 2 Cu3 O7–x. The top and middle figures: uex for an underdoped YBa 2 Cu3 O7–x with Tc of 45K and 65K, respectively. The bottom figure: u ex of an optimally doped YBa 2 Cu3 O7–x with TC=92.5K. (From Sharma et al [6]).

another to its maximum value of , almost entirely due to displacements of the Cu(2) atoms off the Y (yttrium) layer. At the onset of overdoping between x=6.970 and x=6.984 both the Cu(2) and O(2,3) layers shift off the Y-layer reducing the dimpling to . Earlier work by Conder et al. [8], based on standard refinements of neutron diffraction patterns (measured at 5K) has also shown a reduction in the dimpling of the CuO2 planes by about at x=6.974. The negative direction of this discontinuity has been attributed by Kaldis et al. to the structural transformation which develops first is

small domains of the crystal. Anomalous softening of the O(2,3) Raman shifts, which starts at x larger than 6.90, has been found to correlate with the anomalously large displacements of the Cu(2) atoms off the O(2,3) layer observed by EXAFS around x=6.95 (Fig. 5). The authors stated that the increase of the dimpling in the CuO2 planes softens the Cu(2) – O(2,3) bonds and thus may decrease the wave number of the O(2,3) in-phase vibrations. The drop of the Raman shift by –5 cm-1 within a narrow 314

Figure 4. Spacing between the Y-Cu(2) and Y-O(2,3) layers, as a function of oxygen concentration x in YBa2 Cu3 Ox. The spacing was determined from Y-EXAFS at 25K. Vertical arrows indicate the magnitude of dimpling in the Cu O2 planes. xopt marks the optimum oxygen concentration (From Kaldis et al [7]).

Figure 5. Raman shifts of the O(2,3) in-phase mode in YBa2 Cu3 Ox for oxygen concentrations between x=6.438 and 6.984. Dashed horizontal lines indicate the phase boundaries between coexisting phases, the drawn out horizontal lines (6.95) indicate the miscibility gap in the overdoped regime. The thick drawn boxes emphasize the sequence of phases occurring on doping (see Ref. 7 for more details) (From Kaldis et al [7]).

concentration range of gave an evidence that the deformation of the CuO2 planes is of the displacive type. On the other hand, an increase of the lattice constant in the c-axis direction is associated with a decrease of the a-axis lattice constant. The opposite behavior of these deformations suggest a martensitic (ferroelastic) nature of this phase transformation. Pulsed neutron diffraction experiments on YBa2 Cu3 O6+z (z=0.25, 0.45, 0.65, and 0.94) at 15K by Gutmann et al. [9] have revealed that the average copper Cu(2) - apical oxygen O(4) bond length changes from down to on going from z=0.1 up to z=0.94. The difference in these two bond lengths is . Displacements of 315

Cu(2) in directions out of the a-b plane are also consistent with the presence of diffuse scattering in the electron diffraction {Etheridge [10]} which originates from the ferroelastic distortions.

FERROELASTIC NANODOMAINS IN HTSC CUPRATES

One of the first studies that have addressed the presence of nanoscale structural perturbations (domains) in the copper-oxygen planes of cuprates, were the high resolution electron microscopy studies of YBCO by Etheridge [10]. They revealed a weak diffuse scattering in electron diffraction pattern of which arise from static displacements of atoms from their periodic lattice sites. The local atomic displacements partition the copper-oxygen planes into cells with dimensions comparable to a coherence length in the a-b planes (about 2 nm). These features were reported to be intrinsic to the orthorhombic form of and independent of the fabrication route. The intensity distribution of the diffuse streaks in the electron diffraction pattern is characteristic of scattering from atomic displacements and not of scattering from vacant chain-oxygen sites. It has been determined that there must be a static component to the atomic displacements, however the presence of a dynamic component to these displacements has not been found. In order to give the observed dark line contrast along

Figure 6.

A schematic illustrating the orientation of the irregular two-dimensional grid of walls of

nanoscopic cells (approx. 2nm in size) in the Cu O2 planes of the YBa 2 Cu 3 O6.95 (From Etheridge [10]).

316

axes perpendicular to c, the two-dimensional grids (cells) must be correlated along the c-axis (Fig. 6). From selected-area electron diffraction patterns, static atomic displacements have been identified with the geometry consistent with displacements of the planar copper and apical oxygen atoms that are nearest neighbors along the displacement directions, namely <2803> and <091>.

High resolution images have revealed that there are connected networks of oxygen-pyramidal planes (which link the ab planar oxygen atoms with the apical oxygen atom in the CuO5 pyramid) at which the charge density distribution is locally perturbed. These effectively partition each of the copper-oxygen planes into “cells” with dimensions of about 2 nm in the a-b plane. These studies have suggested that the a-b planes of buckle into the network of slightly misaligned cells in a struggle to relieve internal stresses. One source of internal stress is the mismatch between “natural” dimensions of the copper-oxygen planes and the chain-oxygen and barium-oxygen planes (by “natural” dimensions it is meant the dimensions that would minimize the energy of the plane if it were in isolation). The planar copper atom in the CuO5 pyramid is considered to be tightly bonded in the center of the oxygen pyramid so that the pyramid might be treated as a rigid unit (Fig. 7). As the structure distorts in a response to internal strain it might be expected to yield most easily at the “soft” apical oxygen site, possibly causing the CuO5 pyramid to tilt. These features suggest the effect of ferroelastic transition. Modulations along the copper-oxygen chains with a wavelength of approx. 1.5 nm have been inferred from scanning tunneling microscopy (STM) images of cleaved YBCO crystals by Edwards et al. [11]. This length scale is comparable with the cell dimensions which suggests that these two phenomena are coupled. This is not surprising, given that it is the apical oxygen that links the CuO chains to the CuO 2 planes, so the displacements of the planar copper atoms are coupled to the displacements of the apical

oxygen atoms two unit cell apart. In turn, the displacements of the apical oxygen atom distort locally the CuO chain, generating distortions in the chain with a spacing comparable to the cell structure.

Figure 7. The oxygen-pyramidal planes (delineated by the bold lines) in the average YBa 2 Cu3 O6.95 structure (From Etheridge [10]).

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The electrical transport and magnetic measurements of an optimally doped YBCO have revealed that the copper-oxygen planes behave like nanogranular superconducting systems (e.g. a conventional nanogranular niobium nitride (NbN) thin film). In a nanogranular superconductor with a coherence length comparable to the grain size, the temperature dependence of the critical current density Jc is governed by the Josephson tunnel junctions at low temperatures where the Ginzburg-Landau (GL) coherence length is smaller that the grain size. At higher temperatures (close to Tc) where the GLcoherence length is larger than the grain size, the temperature dependence of Jc is determined by the suppression of the order parameter in the grain. At the crossover

Figure 8. (a), (b). (c):Schematic representation of nanostructures in the a-b planes of

The

a- and b-axes are approx. 45 degrees relative to the domain walls, which are roughly along (110) directions (for details see Ref.10). An optimally doped superconductor could have nanostructures as in (a), where the critical current Ic at low temperatures is governed by the interdomain Josephson junctions. The temperature dependence of Ic is therefore that of Ambegaokar-Baratoff (AB) at low temperatures and that of Ginzburg-

Landau (GL) above approx. 0.85Tc as described by the Clem’s model (Ref.12) [(see solid triangles in (d)]. The dotted line in (d) shows Ic(T) of a single Josephson junction (pure AB dependence). An underdoped

superconductor is shown in (b) where Ic(T) is governed by the suppression of the order parameter in the nanodomain. Ic(T) is governed by the GL dependence see solid squares in (d). When a superconductor is a mixture of optimally doped and underdoped phases, its nanostructure is described schematically by “a disordered chessboard” shown in (c). In this case, Ic(T) is expressed by open circles shown in (e). The solid line in (e) is a superposition of two components; an optimally doped one with an AB-like Ic(T) (solid triangles) and an underdoped one with a GL-like Ic(T) (dashed straight line). in (d) and (e) is the Josephson coupling constant. See Reference 13 for more details.

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temperature T*, the GL coherence length equals the grain size (for NbN nanogranular

film T* 0.85Tc [12]). Consequently, at temperatures T T* , J c (T) is described by the Ambegaokar-Baratoff theory and at T>T* by the Ginzburg-Landau one [12] with J c (T) (Tc – T)3/2. It has been found that Jc(T) in YBCO exhibits this type of behavior [13] (Fig. 8). The measurements of Jc(T) in the a-b planes of YBCO have been done on c-axis oriented thin film ring-shaped samples. This geometry allows one to determine Jc(T) simply from the radial profile of the axial component of the self-field Bz(r) of a maximum (critical) persistent current Ic. The relationship between Ic and Bz(r) is provided by the Biot-Savart law. The magnitude of the Ginzburg-Landau coherence length at T* has been used to estimate the size of the grain (cell) in the a-b planes to be

approximately 3-4 nm. The cell size of 3-4 nm includes the width of the cell walls. This is in rough agreement with the TEM result of Etheridge [10].

The existence of large number of identical regions with diameters of about 3 nm (which have a relatively high density of low energy quasi-particle states) have been revealed in Bi 2 Sr 2 CaCu 2 (Tc=87K) by Hudson et al. [14] using low-temperature (4.2K) scanning tunneling spectroscopy (STS). The studies have been carried out with a high resolution scanning tunneling microscope (STM) which simultaneously could measure, with atomic resolution, both the surface topography and the local density of

states (LDOS) of a material. Following atomic resolution imaging of 130x130 nm2 area, the mapping of the differential conductance at zero-bias was performed on the same area. This kind of a map is a measure of the LDOS of low energy quasi-particles, and in a superconductor well below Tc, is expected to show a very low differential conductance. In contrast to this expectation, a typical zero-bias conductance map of BSCCO revealed a large number of localized features (domains), with a diameter about 3 nm, which have a relatively high zero-bias conductance (high LDOS near the Fermi energy) (Fig.9). These features appear to be randomly distributed. The authors suggested that these LDOS

Figure 9. A 130nm square zero-bias conductance map (from Hudson et al [14]). Quasi-particle scattering resonance (high LDOS) appear as bright regions approx. 3nm in diameter, owing to their higher zero-bias conductance.

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features are caused by quasi-particle scattering from atomic-scale defects or impurities, with oxygen inhomogeneities being most likely, due to the absence of the magnetic field effects. In view of the previously described evidence for ferroelastic nanodomains, the local high conductance features could originate at randomly distributed oxygen deficient

nanodomains. An indirect but an elegant experimental proof for the presence of static charged nanodomains has been provided by inelastic neutron scattering measurements of the temperature and composition dependencies of high energy Longitudinal Optical (LO) phonons in YBa 2 Cu 3 O 6+x {Petrov et al [15]}. The measurements of the composition dependence of the inelastic neutron scattering intensity from YBa2Cu3O6+x with x=0.20, 0.35, 0.60, and 0.93 at 10K was performed for the LO mode along the inplane Cu-O bond direction at energy transfers between 50-80 meV and momentum transfers Q along (100) direction from (3, 0, 0) to (3.5, 0, 0) in the unit of the reciprocal lattice vector (a*=1.63 ). This measurement detects only Longitudinal Optical (LO) phonons. According to Pintschovius et al. [16] the underdoped (x=0) sample has a dispersionless LO branch at 75 meV, while doping softens the zone-edge mode down to 55 meV, so one might expect a continuous softening at the zone-edge as doping level is increased. Surprisingly, the current experiments {Petrov et al. [15]} show that the LO branch is always split in two, the high energy branch around 75 meV and the low energy branch around 55 meV (Fig. 10). Instead of continuous softening the spectral weight is transferred from the high-energy branch to the low-energy one as doping is increased. The authors of this result, argued that this can be understood in terms of the two-phase picture if one associates the high-energy and the low-energy branches with the nanophases which have low and high charge densities, respectively. When the charges are segregated into nanoscopic domains, an increase of the doping level (x) does not change the local charge density in the domains, but instead increases their total volume fraction, which in turn causes the spectral weight transfer from the high-energy branch to the lowenergy one. The characteristic Q-dependence of these two LO branches indicates that the size of the charged domains is nanoscopic. The size of the charged domain roughly

Figure 10. Composition dependence of the inelastic neutron scattering intensity from YBa2 Cu3 Ox single crystals with x=0.2. 0.35, 0.60, and 0.93 at 10K. (From Petrov et al [15]).

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estimated from the shape of the dispersion-less portion of the phonon dispersion is about 2x1 nm {McQueeney et al. [17]}, which corresponds to the coherence length of the localized phonon in the a-b planes. According to Phillips [18] the flatness of the LO energy versus Q branch at 75 meV occurs because the phonons are localized in twodimensional nanodomains in the CuO2 planes.

FERROELASTIC NANODOMAINS IN MANGANESE PEROVSKITES

Recent neutron scattering experiments {e.g. [19,20]} on manganites pointed out the presence of microscopic (nanoscopic) phase segregation. The small-angle neutronscattering (SANS) studies of La1–x CaxMnO3 (x=l/3), at temperatures larger than Tc, observed a short (weakly temperature dependent) ferromagnetic (FM) correlation length which has been attributed to magnetic clusters 1-2 nm in diameter [19]. High-resolution neutron diffraction and inelastic neutron scattering studies of Pr0.7Ca0.3MnO3 [20] revealed a two-phase segregation. One of the coexisting phases is a ferromagnetic (FM) metal, while the other is an insulator. This type of electronic segregation cannot become long-range, due to high Coulomb energy cost. Therefore a microscopically inhomogeneous state develops, whereby FM clusters, 1-2 nm in diameter, are interspersed into a charge localized “matrix”, the latter associated with much larger lattice distortions than the former. The studies point to intra-granular strain as the main driving force for phase segregation. These phase separation tendencies influence the properties of ferromagnetic region by increasing charge fluctuations [21]. REFERENCES 1.

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Abrahams, S.C. ( 1 9 7 1 ) Ferroelasticity, Mat. Res. Bull. 6, 881 -890.

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Bursill, L.A. and Lin, P.J. (1984) Microdomains observed al the ferroelectric / paraelectric phase transition of barium titanate, Nature 311, 550-552. 5. Tritt, T.M., Marone, M., Ehrlich, A.C., Skove, M.J., Gillespie, D.J., Jacobsen, R.L., Tessema, G.X., Franck, J.P., and Jung, J. (1992) Evidence in the elastic properties for a stress-related phase transition in the high Tc material: Bi 2 Sr 2 CaCu 2 O x , Phys. Rev. Lett. 68, 2531-2534. 6. Sharma, R.P., Ogale, S.B., Zhang, Z.H., Liu, J.R., Chu, W.K., Veal, B., Paulikas, A., Zheng, H. and Venkatesan, T. (2000) Phase transitions in the incoherent lattice fluctuations in YBa2 Cu3 Nature

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Kaldis, E., Röhler, J., Liarokapis, E., Poulakis, N., Conder, K and Loeffen, P.W. (1997) A displacive structural transformation in the CuO 2 planes of YBa 2 Cu3 Ox at the underdoped – overdoped phase

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separation line, Phys. Rev. Lett. 79, 4894-4897. Indications for a phase separation in YBa2 Cu3

in E. Sigmund and K.A. Müller (eds), Phase

Separation in Cuprate Superconductors, Springer – Verlag, Berlin, pp. 210-224. 9. Gutmann, M., Billinge, S.J.L., Brosha. E.L. and Kwei, G.H. (2000) Possible charge inhomogeneities in the CuO2 planes of YBa2 Cu3 O6+x (x= 0.25, 0.45, 0.65, 0.94) from pulsed neutron diffraction, Phys. Rev. B 61, 11762-11769. 10. Etheridge, J. (1996) Structural perturbations at intervals of the coherence length in YBa2 Cu 3

Philos. Mag. A 73, 643-668. 1 1 . Edwards, H.L., Barr, A.L., Markert, J.T. and deLozanne, A.L. (1994) Modulations in the CuO chain layer of YBa2 Cu 3 Charge density waves? Phys. Rev. Lett. 73, 1154-1157; Edwards, H.L., Derro, D.J., Barr, A.L.,Market, J.T. and de Lozanne, A.L. (1995) Spatially varying energy gap in the CuO chains of YBa 2 Cu 3 O 7 – x detected by scanning tunneling spectroscopy, Phys. Rev. Lett. 75, 1387-1390. 12. Clem, J.R., Bumble, B., Raider, S.I., Gallagher, W.J. and Shih, Y.C. (1987) Ambegaokar-BarratoffGinzburg-Landau crossover effects on the critical current density of granular superconductors, Phys. Rev B 35, 6637-6642.

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13. Jung, J., Yan, H., Darhmaoui, H., Abdelhadi, M., Boyce, B., Skinta, J., Lemberger, T. and Kwok, WK. (2000) Nanostructures in high temperature superconductors, Proc. of SPIE (in press); Darhmaoui, H. and Jung. J. (1996) Crossover effects in the temperature dependence of the critical current in

YBa2 Cu3 O7-δ, Phys. Rev. B 53, 14621-14630; (1998) Coherent Josephson nanostructures and the dissipation of the persistent current in the a-b planes of YBa 2 Cu3 thin films, Phys. Rev. B 57, 8009-8025; Yan, H., Jung, J., Darhmaoui, H., Ren, Z.F., Wang, J.H. and Kwok, W-K. (2000) Fast vortex motion and filamentary phase separation in high-T c thin films, Phys. Rev. B 61, 1 1 7 1 1 - 1 1 7 2 1 . 14. Hudson, E.W.. Pan, S.H., Gupta, A.K., Ng, K.W. and Davis, J.C. (1999) Atomic scale quasi-particle

scattering resonances in Bi 2 Sr 2 Ca Cu 2

Science 285, 88-91.

15. Petrov, Y., Egami, T., McQueeney, R.J., Yethiraj, M., Mook, H.A. and Dogan, F. (2000) Phonon signature of charge inhomogeneity in high temperature superconductors YBa2 Cu 3 O6+x, Cond.-Mat.

Preprint 0003414 / 25 Mar 2000. 16. Pintschovius, L., Pyka, N., Reichardt, W., Rumiantsev, A.Y., Mitrofanov, N.L, Ivanov, A.S., Collin, G. and Bourges, P. (1991) Lattice dynamical studies of HTSC materials, Physica C 185-189, 156-161. 17. McQueeney, R.J., Petrov, Y., Egami, T., Yethiraj, M., Shirane, G. and Endoh, Y. (1999) Anomalous

dispersion of LO phonons in La1.85 Sr0.15 Cu O4 at low temperatures, Phys. Rev. Lett. 82, 628-631. 18. Phillips, J.C. (2000) Zigzag filamentary theory of LO phonons in high temperature superconductors, preprint.

19. De Teresa, J.M., Ibarra, M.R., Algarabel, P.A., Ritter, C., Marquina, C., Blasco, J., Garcia, J., del Moral, Al. and Arnold, Z. (1997) Evidence for magnetic polarons in the magnetoresistive

perovskites. Nature 386, 256-259. 20. Radaelli, P.G., Ibberson, R.M., Argyriou, D.N., Casalta, H., Andersen, K.H., Cheong, S.W. and Mitchell, J.F. (2000) Mesoscopic and microscopic phase segregation in manganese perovskites,

Cond-Mat. Preprint 0006190 / 12 Jun 2000. 2 1 . Moreo, A., Yunoki, S. and Dagotto, E. (1999) Phase separation scenario for manganese oxides and

related materials, Science 283, 2034-2040.

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ROLE OF Sr DOPANTS IN THE INHOMOGENEOUS GROUND STATE OF La2–xSrxCuO4

D. HASKEL1, E. A. STERN2 and F. DOGAN3 1

Experimental Facilities Division, Advanced Photon Source Argonne National Laboratory, Argonne, IL 60439, USA

2

Department of Physics, University of Washington Seattle, WA 98195-1560, USA

3

Department of Materials Science and Engineering, University of Washington Seattle, WA 98195-2120, US A

INTRODUCTION The role of dopants in high Tc superconductors is widely seen as being limited to the introduction of hole carriers into the CuO2 planes of otherwise insulating parent compounds. This simplified assumption is partly driven by the lack of information on the local atomic and electronic structure around dopants. While experimental evidence favoring inhomogeneous charge distributions of the doped holes is still mounting, the role that dopants play in determining this inhomogeneous ground state, if any, is still unclear. Since high Tc superconductors manifest strong carrier-lattice interactions, as evidenced, e.g., in the presence of Jahn-Teller distorted CuO6 octahedra, structural techniques can, in principle, provide information on the spatial distribution of doped charges through the structural response. The x-ray absorption fine structure (XAFS) technique is particularly suited for elucidating such response around dopants, as it is element specific; i.e., by tuning the x-ray energy to a characteristic absorption threshold of a given dopant, it can determine the partial pair correlations between the dopant and its neighbors. This is of paramount importance since the dopants substitute at the crystal sites of the majority atoms and therefore techniques that sum over all pair correlations are dominated by the correlations involving the majority, host, atoms. In this paper we present evidence that the doped holes are spatially correlated to Sr dopants in La2–xSrxCuO4. This is manifested as a unique structural response of the Sr-O(2) distance across the x ~ 0.06 insulator-metal transition, while no such response is observed for the La-O(2) distance. This result by itself proves that the doped charge density is not uniformly distributed, and that the dopants play a role in determining the inhomogeneous charge state. In addition to local effects around dopants, we show evidence for the appearance, with Sr

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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doping, of an inhomogeneous structural ground state at This ground state is characterized by the presence of local domains with a larger orthorhombic distortion than the long-range averaged distortion determined by crystallography.

EXPERIMENTAL Samples of La2–xSrxCuO4 were prepared by (i) conventional solid-state reaction techniques starting from related oxides (Radaelli et al. [1], ) and (ii) by precipitation from ionic solution starting from related nitrates (Haskel et al. [2], ). Tc’s were determined by magnetic susceptibility measurements and average crystal structures were determined by neutron and x-ray powder diffraction (Refs. 1 and 2). The local environment around La/Sr atoms in La2–xSrx CuO4 is quite complex and includes nine oxygen neighbors at six different distances. In a powder XAFS experiment, an angular-average over all the relative orientations of the electric field vector and the La/Sr-O bond directions is performed. There is simply not enough information in a powder XAFS experiment to resolve all the different distances. By performing our experiments on c-axis magnetically aligned powders, we can exploit the angular dependence of the XAFS signal in the anisotropic (layered) cuprates to measure subsets of the local structure around the absorbing atom allowing a complete determination of the local structure. (For excitation of a 1s core electron, i.e., K-edges, this dependence is a one, with the angle between the electric field vector and the bond orientation). Since we are particularly interested in structural ground state properties, we present here results of measurements performed at T=10K using a closed cycle helium displex refrigerator. Orientation-dependent spectra were taken by rotating the oriented samples relative to the electric field vector of the synchrotron radiation. Experiments were performed at the National Synchrotron Light Source (NSLS, Brookhaven) and at the Advanced Photon Source (APS, Argonne). La K-edge measurements were performed in transmission geometry, which determines the energy-dependent absorption coefficient by measuring the attenuation of the x-ray intensity after passing through the sample thickness. Sr K-edge measurements were done in both transmission and fluorescence geometries, where the latter determines the absorption coefficient by measuring the secondary radiation that accompanies the de-excitation of the absorbing atom. The reader is referred to Refs. 1-3 for further

experimental details. RESULTS Figure 1 shows local Sr-O(2) and La-O(2) apical distances as determined from c-axis polarized XAFS experiments at Sr and La K-edges as a function of Sr content in La2–xSrxCuO4 at T=10K. The O(2) oxygens form the apices of CuO6 octahedra and the La/Sr-O(2) apical bonds nearly coincide with the crystallographic c-axis (slightly off c-axis due to tilts of CuO6 octahedra and correlated off-center displacements of La/Sr atoms in the La2O2 planes. See Radaelli et al.[1] for a detailed description of the crystal structure). We have previously reported evidence for the existence of a double site distribution for the apical O(2) oxygens only near Sr atoms, a distribution that can be described as two Sr-O(2) distances at 2.55 Å, respectively, with relative weights that strongly depend on x (Haskel et al. [4]). These previous results were obtained on samples prepared by solid-state reaction. The samples prepared by precipitation from solution did not show the broad (split) distribution in the Sr-O(2) distance, indicating a more homogeneous Sr environment in the latter. Since samples from both families have similar values of Tc, it is implied that the double site distribution observed previously must not be related to the mechanism of superconductivity. The single Sr-O(2) distance found in the samples precipitated from solution, however,

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Figure 1. Top: Local Sr-O(2) (centroids) and La-O(2) apical distances obtained from c-axis polarized Sr

and La XAFS at their K-edges at T=10K. Sr measurements were done on samples prepared by solid state reaction (circles) and by precipitation from solution (squares, crosses) and at different synchrotron facilities (APS, NSLS). Results from both transmission and fluorescence measurements are shown for comparison for x = 0.04. Crosses are displaced in x for clarity. Bottom: weighted averages of Sr-O(2) and La-O(2) distances as a function of x and their comparison with the results of crystallography.

agrees well with the weighted average of the two Sr-O(2) distances found previously. This is shown in Fig. 1 where the centroid of the Sr-O(2) apical distribution is plotted for samples made by the different methods (x = 0.07, 0.15). Good agreement is also obtained for Sr-O(2)

distances obtained by measuring XAFS at the Sr K-edge in transmission and fluorescence geometries (x = 0.04) as well as between measurements performed at different synchrotron facilities

Figure 1 shows that the local Sr-O(2) apical distance is significantly longer than the local La-O(2) distance. Their weighted average, ([La – O(2)] * (2 – x) + [Sr – O(2)] * x) / 2, agrees with the values obtained by crystallography, as expected (lower panel of Fig. 1). The longer Sr-O(2) local distance is readily explained by the stronger attraction of a negatively charged O(2) ion to a trivalent La+3 ion than to a divalent Sr+2. This has significant implications for the local electronic structure of La2–x Sr x CuO 4 , as discussed below. The local La-O(2) distance is nearly independent of x, and therefore the average expansion of the La/Sr-O(2)

distance determined by crystallography as the Sr content is increased is due to the increase in weight, with x, of the long Sr-O(2) distance. A more striking observation is the change in slope that is observed only in the Sr-O(2) distance at x ~ 0.06 but not in the La-O(2) distance. We recall that in La2–xSrx CuO4 an insulator-metal (I-M) transition takes place at The response of the Sr-O(2) distance to the delocalization of doped holes is indicative of a spatial correlation between these doped holes and the Sr dopants. This observed change in slope explains a similar (but opposite) change in slope measured by crystallography in the x dependence of the Cu-O(2) apical distance (Figure 2 and Ref. [1]). O(2) apicals bridge Cu and La/Sr atoms in a nearly collinear configuration. Since we know from Sr and La XAFS that the Sr-Cu and La-Cu distances along the c-axis are nearly identical (within 0.01 Å, Haskel et al. [5]) the Cu-O(2) apical distance measured by diffraction is a weighted average of a majority long Cu-O(2) distance (near La) and a minority short Cu-O(2) distance (near Sr), 325

Figure 2. Cu-O(2) apical distance measured by crystallography (Radaelli et al.[1]). The change in slope at the I-M transition reflects the similar (but opposite in sign) change in slope observed in the Sr-O(2) apical distance.

with the latter showing a reversal in slope at the I-M transition. We now turn to the experimental evidence showing the appearance of nanodomain structure in La2–xSrxCuO4 for The structural phase diagram of La2–xSrxCuO4 indicates a low temperature orthorhombic (LTO) ground state with correlated CuO6 octahedral tilts of magnitude ~ 3° about crystallographic axis. Crystallography finds that as the Sr content is increased the tilt angle of CuO6 octahedra gradually decreases, becoming zero at

x ~ 0.21 (at T=10K), at which point the second-order phase transition to a macroscopically tetragonal phase (HTT) is completed. Figure 3 shows the different tilt patterns of CuO6 octahedra encountered in La-cuprates. The Sr-doped system only exhibits the LTO and HTT phases, as determined by neutron diffraction. It is immediately obvious from Fig. 3 that by measuring Cu-O(1) and Cu-O(2) distances (i.e. the Cu atoms nearest neighbors’ distances) it is not possible to determine the direction nor the magnitude of the CuO6 octahedra tilts. This is because the tilts are nearly rigid and their magnitude too small to produce any measurable changes in the near-neighbors distances within the sensitivity of XAFS, ~ 0.01Å. The O(2) apical atoms lie in the La2O2 planes and different tilt directions and/or magnitudes result in significantly different La-O(2) planar radial distribution functions (Figure 3). Therefore by measuring in-plane polarized XAFS at the La site, we are extremely sensitive to tilt direction and magnitude. By measuring the local La-O(2) planar distribution of distances as a function of x we can follow the structural phase transition from the LTO phase (3 distances) to the HTT phase (a single distance). Figure 4 shows the results of such measurements. It is immediately obvious that although the local splitting (and therefore the local magnitude of tilt angle) initially decreases

up to x ~ 0.15, the local splitting deviates from the macroscopic value determined by crystallography above this concentration. In particular, whereas the averaged tilt angle goes to zero at the LTO HTT phase transition (x = 0.21), the local tilts do not vanish. DISCUSSION That the Sr-O(2) distance shows a large response to the delocalization of holes at the I-M transition but the La-O(2) distance does not (Figure 1) is direct evidence that a spatial correlation exists between the doped holes and the dopants that introduced them. This might not be surprising, as at low Sr concentrations the dopants’ potential is poorly screened and it is energetically favorable for a doped hole to remain in the vicinity of the Sr. At larger dopant concentrations screening becomes more efficient but remains poor at very short distances, so the doped holes, even if itinerant, are expected to have significant weight in the vicinity of

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Figure 3. Top: The different CuO6 octahedral tilt patterns encountered in La-cuprates together with the resultant La-O(2) planar distances caused by such tilts. Crystallography finds that only the LTO and HTT phases materialize in La 2–x Sr x CuO 4 While XAFS at the Cu sites is nearly insensitive to tilt direction and magnitude, XAFS at the La sites is very sensitive to both as seen in the very different radial distribution functions of La-O(2) distances. The splitting in La-O(2) distances is directly proportional to the magnitude of octahedral tilts.

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Figure 4. La-O(2) planar and La-La planar distances as a function of x determined by in-plane (electric field parallel to La2O2 planes) polarized La K-edge XAFS in La2–xSrxCuO4 at T=10K. Solid lines show the results of crystallography (Radaelli et al.[1]), with the macroscopically averaged splitting of La-O(2) and La-La planar distances going to zero at x = 0.21 (the splitting is proportional to the tilt angle of CuO6 octahedra). The XAFS results show that, although decreasing up to x = 0.15, the splitting remains present in the local structure (as do the tilts) even in the nominal HTT phase.

the dopants. The structural response to the change in localization of the holes also shows that

there is a strong local interaction between the doped hole and the lattice. We believe this is the first direct experimental evidence for a spatial correlation between dopants and doped holes, although a similar conclusion was derived from the interpretation of NQR data by Hammel et al.[6]. The local distortion in the Sr-O(2) apical distance has an important role in determining the local electronic structure of La-cuprates. The electronic orbital character of doped holes was determined by polarization-dependent x-ray absorption near edge structure (XANES) measurements, which, at the absorption threshold, determine the density of unoccupied states (holes) at the Fermi level, projected into the angular momentum states allowed by dipole selection rules. O K-edge (1s) measurements showed that doped holes acquire more out-ofplane, O 2pz orbital character, compared to their in-plane, O 2px,y character, as doping is increased (Chen et al. [7]). This phenomenon is hard to explain by band structure calculations using the periodic average structure of crystallography and a rigid band model to account for the changes in chemical potential with doping. However, the introduction of Sr+2 dopants results in a long, local, Sr-O(2) distance which raises the local energy of O 2pz orbitals towards the Fermi level, allowing them to become more populated by the doped holes. The

raise in orbital energy is due to both a smaller Madelung energy contribution of the Sr+2 ions compared to La+3 together with an increased overlap of O 2pz orbitals with ones that results from the closer O(2)-Cu distance. The XANES measurement cannot determine which oxygens in the structure are being populated by the doped holes, as it averages over all oxygens. Our measurements indicate that the out-of-plane holes are being introduced in O(2) apicals neighboring the Sr dopant atoms. 328

This result is crucial in understanding the c-axis transport properties of La2–xSrxCuO4, as O 2pz orbitals provide connectivity along the c-axis. For example, the c-axis normal state conductivity is insulating/semiconducting throughout the superconducting region of the phase diagram, due to the small overlap of O 2pz orbitals (the normal state ab-plane conductivity is metallic). As the Sr content increases, this overlap increases with a concomitant increase in c-axis conductivity, to result in c-axis metallic conduction at x ~ 0.25. (At this concentration of dopants, ca. 12% of La sites, a random distribution of Sr atoms results in at least two Sr atoms per unit cell). That the structure of La2–xSrxCuO4 is composed of nanodomains for is readily seen from Figure 4. For these concentrations the local structure is different from the average structure; specifically the local tilt angle of CuO6 octahedra (and the related splitting of La-O(2) and La-La planar distances) is larger than its macroscopically averaged value. In particular, while the long range averaged tilt becomes zero at x ~ 0.21, the local tilt remains, with a magnitude of comparable size to the one at x = 0.15. This can be explained by the presence of structural disorder in the form of nanodomains. XAFS obtains local structural information within a length scale that is determined by the photoelectron mean free path, ~ 10Å. Diffraction techniques average over much longer length scales. In order for the macroscopically averaged tilt angle to be smaller than the local tilt the latter has to become disordered over the length scale measured by diffraction. The tilts, therefore, are locally ordered within domains whose size is determined by the correlation length of this ordering. These domains are at least as big as the XAFS length scale (~ 10 Å) as no evidence for local disorder is found in the XAFS measurements. We do not see the domain boundaries in our measurements but we can put an upper limit, on the order of 50 Å, to the domain size. Larger LTO domains would be visible in the diffraction measurements, but those were not observed. The presence of nanodomains with size l, 50 Å, for is a direct consequence of our measurements. Most theories aiming at describing the transport properties of La 2–x Sr x CuO 4 limit the Sr dopants’ involvement to their introduction of hole carriers into the CuO2 planes. A much more active role for the dopant sites is postulated by J. C. Phillips in the context of his zigzag filamentary theory of high Tc superconductors (Phillips [8]). In this theory dopants are resonant tunneling centers that serve the role of providing interlayer connectivity by creating a percolative current path. A large carrier-lattice coupling at the dopant sites aids in establishing this connectivity. The observation, in our measurements, of a structural response in the Sr environment (and not in the La one) to hole-delocalization at the I-M transition is a signature of the inhomogeneity of the wave function of the doped holes, which is peaked in the vicinity of the dopant sites. This provides some support for the ideas in Ref. [8] which also depend on the inhomogeneity of the carriers’ wave function and on a large hole-lattice coupling at the dopant sites.

The appearance of structural disorder at could be related to the decrease of Tc in the overdoped regime; however, it has also been argued that the presence of local, short-ranged orthorhombic domains with orthorhombic distortion larger than the macroscopic orthorhombic distortion measured by crystallography favors superconductivity (Phillips [8]). The role played by disorder introduced with dopants is currently being debated, particularly how it influences the topology of inhomogeneous charge distributions in high Tc cuprates (Hasselmann et al.[9]).

Summary Our experiments in La2–xSrxCuO4 provide evidence for a spatial correlation between the doped holes and the Sr dopants, in addition to the presence of a large hole-lattice coupling only in the vicinity of the Sr dopants. The local structure deviates from the macroscopic, average, structure at indicating the appearance of structural disorder. This disorder is in

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the form of nanodomains, within which the local tilts are ordered, but tilts become disordered relative to each other in going from one domain to the other. These results indicate that Sr dopants might have a much more significant role in determining normal and superconducting

properties of La2–xSrxCuO4 than is typically assumed. Acknowledgments It is a pleasure to thank V. Polinger, J. C. Phillips, Y. Yacoby, A. R. Moodenbaugh, D. G. Hinks, A. W. Mitchell, J. D. Jorgensen, F. Perez and M. Suenaga for valuable discussions. This work supported by DOE grant no. DE-FG03-98ER45681 and DOE contract no. W-31-109-Eng-38. REFERENCES 1.

2. 3.

4.

5. 6. 7.

8.

Radaelli, P.G., Hinks, D. G., Mitchell, A.W., Hunter, B.A., Wagner, J.L., Dabrowski, B., Vandervoort, K.G., Viswanathan, H.K. and Jorgensen, J.D. (1994) Structural and superconducting properties of La2–xCuO4 as a function of Sr content, Phys. Rev. B 49, 4163-4175. Haskel, D., Stern, E.A., Dogan, F. and Moodenbaugh, A.R. (2000) XAFS study of the low-temperature tetragonal phase of La2–x Ba x CuO 4 : Disorder, stripes, and Tc suppression at x = 0.125, Phys. Rev. B 61, 7055-7076. Haskel, D., Stern, E.A., Hinks, D.G., Mitchell, A.W., Jorgensen, J.D. and Budnick, J.I. (1996) Dopant

and Temperature induced structural phase transitions in La 2–xSrx CuO4, Phys. Rev. Lett. 76, 439-442. Haskel, D., Polinger, V and Stern, E.A. (1999) Where do the doped holes go in La2–xSrxCuO4? A close look by XAFS High Temperature Superconductivity, AIP Conference Proceedings 483, 241-246. Haskel, D., Stern, E.A., Hinks, D.G., Mitchell, A.W. and Jorgensen, J.D. (1997) Altered Sr environment in La2–xSrx CuO4, Phys. Rev. B. 56, R521-524. Hammel, P.C., Statt, B.W., Martin, R.L., Chou, F.C., Johnston, D.C. and Cheong, S.W. (1998) Localized holes in superconducting lanthanum cuprate, Phys. Rev. B. 57, R712-715. Chen, C.T., Tjeng, L.H., Kwo, J., Kao, H.L., Rudolf, P., Sette, F. and Fleming, R.M. (1992) Out of plane orbital character of intrinsic and doped holes in La2–xSrxCuO4, Phys. Rev. Lett. 68, 2543-2547. Phillips, J.C. (1999) Dopant sites and structure in high Tc layered cuprates, Philos. Mag. B. 79, 14771498; Phillips, J.C. (1997) Filamentary microstructure and linear temperature dependence of normal state transport in optimized high temperature superconductors, Proc. Natl. Acad. Sci. USA 94, 12771-12775;

Phillips, J.C. (1999) Is there an ideal phase diagram for high Tc superconductors?, Philos. Mag. B. 79, 9.

527-536 Hasselmann, N., Castro Neto, A.H., Morais Smith, C. and Dimashko, Y (1999) Striped Phase in the Presence of Disorder and Lattice Potentials, Phys. Rev. Lett. 82, 2135-2138.

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Universal Phase Diagrams and “Ideal” High Temperature Superconductors:

J. L. Wagner1, T. M. Clemens1, D. C. Mathew1, O. Chmaissem2, B. Dabrowski2, J.D. Jorgensen3 and D.G. Hinks3 1

Physics Department, University of North Dakota, Grand Forks, ND 58201

2

Physics Department, Northern Illinois University, Dekalb, IL 60115

3

Science and Technology Center for Superconductivity, and Material Science Division, Argonne National Laboratory, Argonne, IL 60439

INTRODUCTION The series of compounds have the highest superconducting temperatures of any known class of materials with Tc’s of 98, 128, and 135 K for the n= 1, 2, and 3 compounds, respectively. The first member of this series, (Hg1201)1, has been the most extensively studied of the Hgcuprates and possesses the simplest structure with Tc determined by the variable oxygen defect concentration Many studies have investigated the structure and superconductivity over limited doping regions, 2 - 6 It has been reported to span the entire range of superconducting behavior from a Tc ~ 0 K in the underdoped regime to Tc ~0 K in the Figure 1. Crystal structure of The O3 overdoped regime. Therefore, site is a partially occupied defect site (see text). Hg-1201 exhibits a remarkable range of superconducting properties spanning nearly 200 K, making it an ideal compound to

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investigate the interplay between doping, atomic structure and physical properties. The structure of Hg-1201 is show in Fig.l and consists of [Cu-O2] planes separated by a blocking layer composed of planes. Defects within the layer determine the structural and superconducting properties of this compound. The exact nature of all defects in the is still controversial. All studies have reported the O3 oxygen atom, shown in Fig. 1, as the predominant defect in the Hg-1201 structure, having the greatest variation and effect on physical properties. Different groups have also reported evidence of small amounts of Cu/CO3 substitution on the Hg-site1, 7-9 as well as a second type of oxygen defect, O4, in the plane.2, 5 However, the concentrations of these latter defects are small with little variation, making it difficult to establish their role on

physical properties. In this paper, only the O3 defect is discussed. Even when considering only the O3 defect, significant differences in the functional and quantitative dependence of Tc on exist in the literature. Simple parabolic doping dependence consistent with the universal doping behavior proposed by Presland et al.10 have been reported.3, 4, 6 Other studies have shown more complex doping dependence typified by trapezoidal-shaped curve exhibiting-a plateau across the region of optimal Tc.5 In this work, the doping curve consists of three linear regions corresponding to the underdoped, optimally doped and overdoped regions. A wide variation in the optimal doping level necessary for maximum Tc, also exists. Reported values of range from 0.06 to 0.18 (refs. 1 and 3, respectively), spanning nearly the entire range of doping concentration for Hg-1201. These quantitative disagreements may be due to uncertainties

in values of obtained through structural refinements, however, the functional differences in reported doping trends are not easily reconciled. In the work by Chmaissem et al.,5 a trapezoidal doping curve was found in which the maximum Tc persisted over a range of oxygen contents from to 0.16. Anomalies

in lattice parameters, cell volume and charge transfer were found upon crossing this plateau. Possible electronic origins for these anomalies were proposed. In particular, observed anomalies were considered to be a structural response to the proximity of the

Fermi level to features in the electronic structure as the doping is varied near maximum Tc. These structural distortions may involve charge ordering within the CuO2 planes (i.e., stripe formation), a change in the defect chemistry, or possible oxygen ordering. Regardless of origin, the ordering/distortions would then lead to a suppression in Tc, giving rise to a plateau rather than the expected parabolic behavior. If this picture is correct, then the maximum Tc for Hg-1201 could be considerably higher if these distortions could be suppressed. An alternative theoretical view has been proposed by Phillips11,12 that explains this novel behavior based upon the formation of filamentary, nanodomain structures in cuprate materials. In this model, under appropriate conditions the oxygen defect atoms selforganize to form percolative, metallic nanofilaments embedded in an insulating matrix. Within these nanofilaments, the dopant-electron/phonon interaction is ~25 times larger than that of a normal Fermi liquid, thus giving rise to maximum Tc. This filamentary structure persists over a range of doping concentrations, resulting in the observed plateau. This model also predicts a continuos phase transition associated with the formation of nano-filamentary structures at low defect concentrations, and a discontinuous phase transition associated with an abrupt disappearance of this filamentary structure and superconductivity at high concentrations. Both models predict deviations from simple parabolic doping behavior though arising from different origins. If present, electronically driven structural distortions will lead to a suppression of Tc from that possible in the undistorted structure. The filamentary theory of Phillips is an enhancement in Tc arising from the formation the filamentary structure. To test these theories, it is essential to probe the resulting change in Tc arising from samples of differing degrees of disorder and/or structural distortion.

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In this paper, results of neutron and x-ray diffraction structural experiments are presented on two series of samples comprised of 34 individual samples, spanning the complete range of superconducting properties of Hg-1201. Different annealing conditions were used for these series to achieve underdoped and optimally doped samples. High pressure, high temperature processing was used to fully overdoped samples to Tc = 0 K. Distinct differences in both the structure and superconductivity were found for samples in the underdoped and optimally doped region. Analysis of these differences show that total oxygen content alone is not sufficient to determine the superconducting and structural properties of Hg-1201. Between samples of identical oxygen content, variations in Tc as large as 30 K were found. EXPERIMENTAL

Samples for this work were from two, large master synthesis (~12 g each) that were then processed after initial synthesis at low temperatures to adjust the doping level Two series of samples, denoted A and B, were obtained and are comprised of 34 individual samples spanning the entire underdoped and overdoped regime. Samples within each series were from a single master synthesis. Results of powder neutron diffraction studies on series A have been previously published.5 Both master samples were synthesized in sealed quartz tubes as described previously. 2 Two different methods of crossing the intermediate underdoped to optimally doped region (50 K < Tc < 98 K) were used. For series A, doping was varied by anneals at a fixed temperature of 400 °C in varying oxygen partial pressures, 10-7 < P(O2) < 1 atm. For series B, samples were annealed in flowing Ar gas (P(O2) < 10-8 atm) at annealing temperatures from 150 to 400 °C. Both series of samples were cooled to room temperature over time intervals of ~30 min. For strongly underdoped samples (Tc < 50 K), samples of series B were annealed in Ar gas at temperatures from 400 to 650 °C. Moderately overdoped samples (Tc > 50 K) for both series were obtained by annealing samples in 10 to 150 atm of oxygen at temperatures under 180 °C for 100 hrs. Strongly overdoped samples for series B were obtained from high-pressure anneals up to 6 Gpa at 800 °C for 1 hr. No structural differences were found for overdoped samples prepared using the moderate-pressure/long-time anneals with those made at high-pressure. Details of the high-pressure work will be described elsewhere.14 Fully overdoped samples (Tc = 0 K ) could only be obtained by high pressure techniques. Neutron powder-diffraction data were collected on samples of series A on the Special Environment Powder Diffractometer (SEPD) at Argonne’s Intense Pulsed Neutron Source (IPNS).14 X-ray diffraction data were taken for samples of series B on a Philip’s PW1400 diffractometer with neutron-diffraction data also collected for selected samples in series B. Structural parameters were refined using the General Structural Analysis Software (GSAS) suite of programs.15 Tc’s were determined from ac susceptibility measurements. RESULTS AND DISCUSSION

Variation in lattice parameters and cell volume with Tc

Annealing experiments were conducted on two series of samples, A and B, utilizing different conditions to produce moderately underdoped to optimally doped samples. Series A samples were annealed at a fixed temperature with varying P(O2). Series B samples were annealed at a fixed P(O2) with varying temperatures. Similar annealing conditions

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were used to produce strongly underdoped and overdoped samples in both series. All samples were found to be single-phase by neutron and x-ray diffraction after anneals. Figure 2 (a) shows the a lattice parameter vs. Tc for series A and B. Both series show a general decrease in a with increasing doping. Similar values of a are found for strongly underdoped samples (Tc < 50 K) and overdoped samples, consistent with similar annealing conditions used to obtain these samples in both series. However, differences in a are found in the underdoped region as Tc increases and are most pronounced in the region of optimal Tc. Across the region of maximum Tc, values of a are seen to be discontinuous in series A, indicating a range of values corresponding to the same maximum Tc. That is, for the slowcooled samples in series A the maximum value of Tc is nominally constant over a range of doping levels, producing the plateau behavior in the doping curve as discussed earlier.5 For samples in series B, this discontinuity is not found. In both series, doping trends in the underdoped and overdoped regions are distinctly different. For underdoped samples, a kink in the doping curve appears near 70 K, with this feature being more pronounced in series A. For overdoped samples, a is found to smoothly decrease with increased doping. Figure 2 (b) shows the c lattice parameter vs. Tc. Values of c are found to be different for both series throughout the entire doping range, though general trends observed in these series are similar. For series A, a discontinuity in c is observed at maximum Tc, like that observed in the a lattice parameter. No discontinuity is seen for series B. The differences in doping trends between underdoped and overdoped samples are similar to that found in a-

lattice parameters. The origin for the different c lattice parameters throughout the entire doping range is not understood, but likely arise from small variations in other structural defects. This illustrates the importance of using a single, large master batch in which to conduct these doping experiments. The unit cell volume vs. Tc for series A and B are plotted in Figure 2 (c). Differences in cell volume are seen in the underdoped region with those of series A being on the order of 0.2% less than those for series B. To within our experimental uncertainty, no difference in cell volume is detected for overdoped samples. The dependence of lattice parameters and unit cell volume vs Tc show distinct differences between the two series of samples, with the largest variations occurring at optimal Tc.

Tc and volume dependence on

Determination of the excess oxygen is crucial to understand the physical properties of Hg-1201. Neutron diffraction data were collected for samples in series A and previously reported.5 Rietveld analysis of the neutron data allowed determination of the O3 occupancy with an estimated uncertainty of less than 0.01 oxygen atoms/unit cell for all samples in series A. For samples in series B, neutron diffraction data were collected on four samples and x-ray diffraction data were collected for all samples. Using values of obtained from neutron data, values of were calculated for the remaining samples of seried B using a linear interpolation of vs. a lattice parameters across the entire doping range. The Tc vs. dependence for the 34 samples in series A and B are shown in Fig. 3. For series A, the curve exhibits a plateau where Tc is nominally constant and also corresponds to maximum Tc's for this series of samples.

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Figure 2. (a) a-lattice parameter, (b) c-lattice parameter and (c) unit cell volume V vs. Tc for series and Estimated standard deviations for all quantities are less than symbol size. Lines are drawn as guide to the eye. An oval is drawn indicating the samples near maximum Tc.

For series B, the maximum Tc occurs at a value of Series B also exhibits a shoulder in the doping curve for to 0.10 over which Tc gradually increases from 70 K to 80 K. It is of interest to note that the onset of the shoulder feature in series B coincides with the beginning of the plateau feature for series A. Both series exhibit regions where Tc is nominally constant over a range of doping starting at The onset of these features corresponds to the kinks seen in curves of lattice parameters and cell volume vs. Tc, in which these structural parameters change rapidly over a narrow range of Tc. In terms of possible structural distortions occurring as doping increases, it is useful to plot the unit cell volume vs. as shown in Fig. 4. For samples in series B, the kink present in the volume vs. Tc curve is absent and the volume is found to monotonically decrease with increasing This suggests that the nature of the oxygen defect remains unchanged and structural distortions are absent as this series is doped throughout its entire range. For series A, however, this is not the case.

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Figure 3. Tc vs. oxygen content for series

and

Lines drawn are as guides to the eye. Error bars are shown for series A only.

For samples in series A, the volume vs. curve has three distinct regions, corresponding to the underdoped, optimally doped and overdoped regions. In the underdoped region, volume for samples in series A are less than those found in series B for

a given The decrease in volume with increasing is more rapid and continues until these samples achieve optimal Tc at For overdoped samples, the volume dependence on is the same in both series. For optimally doped samples (i.e. those that lie on the plateau of the doping curve in Fig. 3) the volume is found to expand as is

increased across the region of maximum Tc. From the doping curves for series A and B, it is clear that alone is not sufficient to uniquely determine Tc nor the structure of Hg-1201. For example, samples with are found to have differences in Tc of 30 K between series A and B. Unlike series A, Tc and volume are found to be well defined for a given value of series B. It is therefore of interest to look at the amount of charge transfer present in the samples of series A.16 The structural variables most sensitive to charge transfer are the z coordinates of the Ba and apical oxygen O2 atoms. These atoms move in opposite directions along the c-axis as charge is transferred from the [Hg-Od] layer to the CuO2 plane. Therefore the parameter most sensitive to the charge transfer on the CuO2 plane is the structural quantity [z(Ba) – z(O2)]. Fig. 5 (a) shows Tc plotted as a function of [z(Ba) – z(O2)] for slow-cooled samples of series A. In this plot the four samples of series A spanning the plateau in fig. 3 with 0.06 to 0.16 cluster about a single point. This indicates that all samples having the maximum Tc in series A have the same charge transfer, regardless of different oxygen

content.

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Figure 4. Unit cell volume vs. oxygen content Error bars are shown for series A only.

for series

and

It has been proposed that as oxygen is added in crossing the plateau for series A, the charge remains constant as a result of a structural distortion occurring over this range of oxygen compositions. This was inferred from the discontinuity in unit cell volume shown in fig. 5 (b). This indicates that while Tc and charge transfer remains constant over the plateau, the volume increases as increases. This volume increase can be considered a distortion, and is thought to be a structural response to the proximity of the Fermi level to features in the electronic density of states. This distortion then produces the observed plateau in the doping curve rather than the expected parabolic behavior. Whether this distortion involves charge inhomogeneities within the CuO2 planes (i.e., charge stripe formation), different defect oxygen sites, oxygen ordering, or is purely of electronic origin is not addressed. Whatever the cause, it was argued that higher Tc’s would be possible if this distortion were suppressed. The new data from series B provides a test of this hypothesis, since no such structural anomalies are observed for these samples as is varied between 0.00 and 0.25. For overdoped samples in both series, the volume dependence is the same. However, differences exist for all underdoped and optimally doped samples between these series. This suggests that all underdoped and optimally doped samples of series A are structurally different from those of series B despite identical oxygen contents. In terms of possible distortions present, samples of series B appear to possess less distortion than those of series A based upon the volume vs. dependence shown in figure 4. However, Tc values for series B samples are less than those found for series A. That is, whatever “distortions” are present across the maximum-Tc, plateau region in series A, these distortions actually enhance rather than suppress Tc.

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Figure 5. (a) Tc and (b) unit cell volume vs. the structural parameter [z(Ba) – z(O2)], a measure of the charge transfer, for samples in series A.

Therefore, it appears that the trapezoidal doping dependence found for series A is not

readily explained by a structural distortion that competes with superconductivity in the region of maximum Tc. Whatever ordering is present, it must account for the constant charge transfer and optimal Tc observed for those samples across the plateau region. The filamentary model of Philips11,12 readily accounts for these observations, as well as providing a mechanism capable of producing the high Tc’s found in cuprate materials. CONCLUSIONS The structural and superconducting properties of have been investigated over an extended range of doping Two series of samples were measured in which oxygen content was varied using different annealing conditions. Both series were

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found to have non-parabolic doping dependence as well as unusual structural phenomena in the vicinity of maximum Tc. Samples of identical oxygen content were found to have Tc variations as large as 30 K. These differences in superconducting and structural properties

are evidence of oxygen ordering present in This oxygen ordering was observed samples processed at different temperatures, and was found to enhance superconductivity, consistent with the nano-filamentary models for superconductivity recently proposed by Phillips. Work is in progress to investigate the nature of the oxygen ordering present in and means by which to tune this ordering. ACKNOWLEDGEMENTS This work was funded by the Office of Naval Research through grant #N000 14-99-1-0574 (JLW, TMC, DCM), the US Department of Energy, Division of Basic Energy Science – Materials Science, contract No. W-31-109-ENG-38 (JDJ, DGH), and the National Science Foundations, Office of Science and Technology Centers grant No. DMR 91-20000 (OC, BD).

REFERENCES 1. S.N. Putlin, E.V. Antipov, O. Chmaissem and M. Marezio, Nature (London) 346, 226 (1993) 2. J.L. Wagner, P.O. Radealli, D.G. Hinks, J.D. Jorgensen, J.F. Mitchell, B. Dabrowski, G.S. Knapp and M.A. Beno, Physica C 210,447 (1993).

3. Q. Huang, J.W. Lynn, Q Xiong and C.W. Chu, Phys Rev. B52, 462 (1995). 4. A. Fukuoka, A Tokowa-Yamamoto, M. Itoh, R. Usami, S. Adachi, H. Yamauchi and K. Tanabe, Physica C265, 13 (1996). 5. O. Chmaissem, J.D. Jorgensen, D.G. Hinks, J.L. Wagner, B. Dabrowski and J.F. Mitchell, Physics B 241-243, 805 (1998).

6. A.M. Balagurov, D.V. Sheptyakov, V.L. Aksenov, E.V. Antipov, S.N. Putlin, P.G. Radaelli and M. Marezio, Phys. Rev. B 59, 7209 (1999).

7. D. Pelloquin, V. Hardy, A. Maignan and B. Raveau, Physica C 273, 205 (1997). 8.

P. Bordet, F. Duc, S. LeFloch, JJ. Capponi, E. Alexandre, M. Rosa-Nunes, S. Putlin and E.V. Antipov, Physica C 271, 189(1996).

9. S.M. Loureiro, E.T. Alexandre, E.V. Antipov, J.J. Capponi, S. de Brion, B. Souletie, J.L. Tholence, M. Marezio, Q Huang and A. Santoro, Physica C 243, 1 (1995), 10. M.R. Presland, J.L. Tallon, R.G. Buckley, R.S. Liu and N.E. Flower, Physica C 176, 95 (1991). 11. J.C. Phillips, Phil. Mag. B 79, 527 (1999). 12. J.C. Phillips, “Filamentary theory of cuprate superconductivity phase diagram and giant electron-phonon interactions”, Superconducting and Related Oxides: Physics and Nanoengineering III, SPIE Proc., 3481, 87, 1998. 13. J.L. Wagner, J. D. Jorgensen, D.G. Hinks, T.M. Clemens and D.C. Mathew, in preparation. 14. J.D. Jorgensen, J. Faber, Jr., J.M. Carpenter, R.K. Crawford, J.R. Haumann, R.L. Hitterman, R. Kleb, G.E. Ostrowski, F.J. Rotella and T.G. Worlton, J. Appl. Crystallogr. 22, 321 (1989).

15. A.C. Larson and R.B. VonDreele, General Structure Analysis System, LAUR 86-748 (1986-1990). 16. J.D. Jorgensen, O. Chmaissem, D.G. Hinks, A. Knizhnik, Y. Eckstein, H. Shaked, J.L. Wagner, B. Dabrowski, S. Short J.F. Mitchell and J.P. Hodges, “Novel Structural Phenomena at the Maximum Tc in 123 and Superconductors: Evidence for a Structural Response That Competes With Superconductivity”, Chemistry and Technology of High-Temperature Superconductors and Related Advanced Materials, Kluwer Acad. Publ. B.V., Moscow, 1998.

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COEXISTENCE OF SUPERCONDUCTIVITY AND WEAK FERROMAGNETISM IN Eu1.5Ce0.5RuSr2Cu2O10

I. FELNER The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel 91904

INTRODUCTION The general antagonism nature between superconductivity (SC) and long range magnetic order is one of the fundamental problems of condensed matter physics and has been studied experimentally and theoretically for almost four decades. In conventional swave superconductors, local magnetic moments break up the spin singlet Cooper pairs and hence strongly suppress SC, an effect known as pair-breaking. Therefore, a level of

magnetic impurity of only 1 %, can result in a complete loss of SC. In a limited class of intermetallic systems, SC occurs even though magnetic ions with a local moment occupy all of one specific crystallographic site, which is well isolated and de-coupled from the conduction path. The study of this class of magnetic-superconductors was initiated by the discovery of RRh 4 B 4 and RMo6S8 compounds (R=rare-earth), and has been recently revitalized by the discovery of the RNi2B2C system. In all three systems, both SC and antiferromagnetic (AFM) order states coexist. The onset of SC takes place at Tc ~ 2-15 K, while AFM order appears at lower temperatures (except for DyNi2B2C), thus, the ratio TN/TC is ~ 0.1-0.5. Many of the high Tc superconducting systems (HTSC) contain magnetic R ions as structural constituents, and are AFM ordered at low temperatures, e.g. in GdBa2Cu3O7 (Tc = 92 K), and TN(Gd)=2.2 K. The R sublattice is electronically isolated from the Cu-O planes, and has no adverse effect upon the superconducting state. Much attention has been focused on a phase resembling the RBa2Cu3O7 materials, having the composition R1.5Ce0.5MSr2Cu2O10 (M-2122, M= Nb, Ru or Ta) [1]. The tetragonal M-2122 structure (space group I4/mmm) evolves from the RBa2Cu3O7 structure by inserting a fluorite type R1.5Ce0.5O2 layer instead of the R layer in RBa2Cu3O7, thus shifting alternate perovskite blocks by (a+b)/2. The M ions reside in the Cu(l) site and only one distinct Cu site (corresponding to Cu(2) in RBa2Cu3O7) with fivefold pyramidal coordination, exists. The hole doping of the Cu-O planes, which results in metallic behavior and SC, can be optimized with appropriate variation of the R/Ce ratio[2]. SC occurs for Ce contents of 0.5-0.7, where the highest Tc was obtained for Ce=0.5, the concentration which has been studied here. The Nb-2122 and Ta-2122 materials are SC with TC~28-30 K.

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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Coexisting of weak-ferromagnetism (W-FM) and superconductivity (SC) was discovered about three years ago in R1.5Ce0.5RuSr2Cu2O10 (R=Eu and Gd, Ru-2122)

layered cuprate systems [3-6], and more recently [7] in GdSr2RuCu2O8. In both systems, the magnetic order does not vanish when SC sets in at Tc, and remains unchanged and coexist with the SC state. The Ru-2122 materials (for R=Eu) display a magnetic transition at TM= 125-180 K and bulk SC below TC = 32-50 K (TM >TC) 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
Figure 9. The temperature dependence of the coercive field (HC) for the ASP and the sample annealed at 75 atm. The inset shows the M(H) curve up to 50 kOe for the ASP material.

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for this material are similar to the 75 atm. sample (Fig. 5b), but exhibit an increase of the relative abundance of the large gaps. Mossbauer effect studies (ME) on 57Fe doped samples has been proved to be a

powerful tool in the determination of the magnetic nature of the Fe site location. When the ions of this site become magnetically ordered, they produce an exchange field on the Fe

ions residing in this site. The Fe nuclei experience a magnetic hyperfine field leading to a sextet in the observed ME spectra. As the temperature is raised, the magnetic splitting decreases and disappears at TM.

Figure 10. Mossbauer spectra of 0.5 %57Fe doped in Ru-2122 (R=Gd) below and above T M (Ru).

The main effect to be seen in Fig. 10 is that the ME spectra of Ru-2122 (R=Gd), consist of one site only, below and above TM(Ru)=175 K. A least square fit to the

spectrum at 180 K yields an isomer shift (IS) of 0.30(1) mm/s (relative to Fe metal) with a line width=0.35 mm/s, and a quadrupole splitting of 1.00(1) mm/s. We attribute this doublet to paramagnetic Fe ions in the Ru site. This interpretation is consistent with: (i) the similarity of the chemical properties of Fe and Ru (Ru resides below Fe in the periodic table), (ii) and with the fact that in most HTSC materials the Fe atoms are found to occupy predominantly the Cu(l) sites which is equivalent to the Ru site in the Ru-2122. At low temperatures, all spectra display magnetic hyperfine splitting,

which is a clear evidence for long-range magnetic ordering. The fitting parameters of the single sextet obtained at 4.1 K are: IS= 0.40(l)mm/s, Heff(0)=467(3) kOe and an effective quadrupole splitting value of Using the relation: we obtained for the Ru site a hyperfine field orientation, relative

to the tetragonal symmetry c axis. As the temperature is raised, Heff decreases and disappears completely at T M (Ru)= 1875(5) K. Heff values obtained at 110, 130, 150 K and 160 K are: 399(3), 358(5), 312(2) and 279(5) kOe. The variation of the normalized Heff(T)/Heff(0) values, as a function of the reduced temperature is exhibited in Fig. 11. It was shown theoretically [3], that for all Fe-doped YBa2Cu3O7 (as well as M-2122)

materials, when Fe reflects the magnetic behavior of Cu(2) sites, the normalized Heff values 349

Figure 11. The temperature dependence of (I) the normalized hyperfine field acting on Fe in the Ru sites for R=Gd (left scale) and (II) the reduced magnetic moment of Ru deduced from magnetic measurements for R=Eu (right scale). The dashed line is the universal theoretical curve for Fe-Cu exchange strength. Note the deviation of the experimental data from the universal curve.

fall on one universal curve, regardless of Fe or oxygen concentration and whether Y is replaced by Pr. The model assumes that the temperature dependence of magnetization of Cu(2) and Fe3+ as a probe, behaves like a spin 1/2 and spin 5/2 systems and that the Fe-Cu exchange is only 26% of the Cu-Cu exchange strength. The dashed line in Fig. 11 is the universal theoretical curve calculated in this way. The deviation of the experimental data from this universal curve is our supporting evidence that the magnetic sextet in Fig. 10 is due to Fe in the Ru site. Moreover, the fact that both the reduced magnetic moment obtained directly from the M(H) curves for R=Eu , and the data obtained from ME for R=Gd, lie on the same curve, indicates clearly that the Ru-Fe and Ru-Ru exchange strength are quite similar.

(II) The effect of hydrogen on the SC and magnetic behavior of Ru-2122 We have demonstrated [5] that Ru-2122H0.35, the effect of hydrogen is to suppress SC and to enhance the W-FM properties of the Ru sublattice (TM increased to 225 K). The hydrogen atoms reside in interstitial sites, and their effect is reversible. Namely, by depletion of hydrogen, SC is restored and TM drops back to 122 K. This is in contrast to the behavior observed in all other HTSC materials, in which charging and/or depletion of hydrogen is irreversible and destructive. Two scenarios that could lead to this phenomenon are: (a) in addition to the change of p in the Cu-O2 planes, there is a transfer of electrons from hydrogen to the Ru 4d sub-bands, resulting in an increase in the Ru moments, and hence to enhance the magnetic parameters; (b) the enhancement arises from a change of the anti-symmetric exchange coupling of the DM type between the adjacent Ru moments, discussed above, which causes the spins to cant out of their original direction to a larger angle. The question arises whether TC decreases monotonically with hydrogen concentration until SC is suppressed completely, or whether small hydrogen quantities

induce phase separation. In this case, TC may remain unchanged (32 K), but the fraction of the SC phase reduces with increasing the hydrogen concentration, thus smearing the transition, until SC is globally suppressed. To address this question, the magnetic measurements of the hydrogen charged Ru2122HX materials are shown in Figs 12-14. Fig. 12 shows ZFC and FC branches of Ru-

2122H0.07 (H=0.07at %), measured at 50 Oe. and for the sake of clarity, we display again 350

Figure 12. ZFC and FC susceptibility curves for ASP and Ru-2122H0.07.

the data of the ASP sample. Here again, the magnetic properties due to the Ru are all enhanced, as compared to the ASP material. The ZFC and FC values are much higher, and both Tirr and TM (Ru) are shifted to 167(2) and 225(2) K respectively. Similar Tirr and TM values have been also obtained for samples with H>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
ZFC and FC magnetic measurements for the (Ru,Nb)-2122 sample annealed under 50 atm. of oxygen, were performed over a broad range of applied magnetic fields, and typical M/H vs T curves measured at 10 Oe., and the isotherm magnetization measured at various temperatures, are shown in Figs.16-18. The onset of the SC transition is shifted to TC =41 K (confirmed also by four point resistivity measurements), and the ZFC and FC branches merge at Tirr=142 K. All M (H) curves below TM, are strongly dependent on the field until a common slope is reached (Fig. 17). At 5 K, a value which is much smaller than obtained for the ASP sample, Ms decreases with T, and becomes zero at TM(Ru)=160(2). Thus, reduction of the Ru content does not change the typical coexistence of SC and W-FM found in the Ru-2122 system. (The enhancement of TC and TM is consistent with Fig. 3 and related to the extra oxygen content in this material). The clear peak at TC in both the ZFC and FC branches in (Ru,Nb)-2122 (Fig. 16), and the negative signal in the ZFC curve below TC are quite evident. On the other hand, in contrast to Figs. 6-7, the typical expulsion of magnetic flux lines at TC (the MS) in the FC curve, as well as the flatness at low temperatures, are readily observed. The small positive contribution of the W-FM Ru sublattice to the total magnetic moment decreases the internal field, and as a

Figure 16. ZFC and FC susceptibility curves measured at 50 Oe. for Ru,Nb -2122. Note the flux expulsion at TC.

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Figure 17. Isothermal magnetization curves of (Ru,Nb) -2122.

result the SVP is weaker. Therefore, the typical MS is clearly observed, although the M/H

values in the FC branch are positive. Therefore, we may conclude that the absence of the MS in Ru-2122 is caused either (or both) by the significantly high Ru moment contribution to the overall moment, and by the enhanced internal fields which lead to the SVP. The hysteresis loop at 5 K, at low applied fields is shown in Fig. 18. Note (a) the

increase of negative moments up to 200 Oe. in the virgin curve, and (b) the particular hysteresis loop obtained at low applied fields. In contrast to the typical FM hysteresis loop obtained for Eu-2122 shown in Figs. 6 (inset) and 13) this loop is a superposition of SC and W-FM properties of the sample. Here again, the SC properties are not masked by the reduced W-FM features.

Figure 18. A superposition of SC and W-FM properties in the hysteresis loop at 5 K for (Ru,Nb) -2122.

DISCUSSION We suggest two scenarios that could lead to the observed phenomena. A central assumption is that Ru in Ru-2122 orders magnetically at elevated temperatures, and bulk SC is confined to the CuO2 planes. Both sublattices are practically decoupled, and thus the present system is the first magnetic-superconducting system in the HTSC based materials. Supporting evidence for this interpretation is (1) the high SF obtained for R=Eu in the SC 354

state, and (2) the overlapping of the two normalized curves exhibited in Fig. 11. The second interpretation invokes analogy to inhomogeneous materials, e.g., the reason for the two physical phenomena are grains with different oxygen concentrations, part of them are SC and the rest magnetic. Moreover, one may argue that the magnetic anomalies exhibited in Figs. 6-7 are due to SrRuO3 impurity phase which is ferromagnetically ordered at 165 K(l 1). In order to reconcile these arguments we have prepared pure and Fe doped SrRuO3 samples, and measured their magnetic and ME properties. The measured curve (at 10 Oe) is a typical one obtained for a ferromagnetic-like sample and in the ME spectra at T>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.

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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<7) have shown that the energy spectra can be interpreted as a superposition of three stable states, corresponding to three different types of clusters in the CuO2 planes, two metallic and one semiconducting with weight depending on oxygen concentration, with superconductivity resulting from the formation of a two-dimensional network. The localization range is estimated to be a few unit cells and the concentration of the twoplateau structure of Tc predicted by means of a bond percolation model.. From neutron powder diffraction data and pair distribution analysis, local tilted directions of the CuO6 octahedra have been inferred, with local displacements ordered over

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lengths of the order of 10A, this local structure being the same as the one of antiferromagnetic spin correlations and of the superconducting coherence [21]. Coherence lengths for magnetic fluctuations of the order 10-20A and the existence of locally ordered mesoscopic domains are obtained in the La2–xSrxCuO4 system with 0.02 < x < 0.08 through the measurements of the 139 La spin-lattice relaxation rates vs. temperature [22]. 1/f noise measurements on copper oxide superconductors [23,] have evidenced that in the normal state the noise is higher, typically 7-10 orders of magnitude, than that of a conventional material. However, there are cases, like Tl2Ba2CaCu2O8 thin films, were the magnitude of the 1/f noise spectral density is much lower than in other cuprates [24]. The important role played by morphology has been discussed by the authors just mentioned. They make the assumption that the noise is due to tunneling transport across domain boundaries of percolation grains. From the fitting of experimental data with tunneling transport models these domains are found to have dimension of 100A and a wall thickness 10A, i.e much smaller than the one of domains and grains existing in the oxides even in the single crystal form, which have spacing of order 1µ or of defects, like twins, which have spacing of the order 1000A. These studies, together with the resistivity studies, indicate that the inhomogeneites are intrinsic to the microscopic structure and that superconductivity can occur in microscopically small regions of space. In the following Table 5 a resume of parameters for the model is given.

Support to the idea that superconductivity can occur in reduced space comes from the huge experimental evidence in various systems which exemplify a number of physical situations of reduced space. Metal clusters and metallic particles are examples. The superconducting properties of metal clusters have been of interest for many years. Zeller and Giaver [25] studied isolated, superconducting Sn clusters embedded in an oxide layer.

They found an increase of the superconting transition temperature with decreasing cluster size, exceeding the one of bulk Sn, for clusters smaller than 40A.. Granular films composed

of Bi clusters have also been found superconducting by [26] Wei, while the bulk metal Bi is not.. They find that superconductivity disappears for cluster size above 200A. Although in principle cluster superconductivity in such systems could be explained as surface superconductivity, normal coupling with phonons within the clusters can also lead to results in agreeement with the observations. Thus, this is an interesting evidence for the appearance of superconductivity in small systems. In general the superconductivity of small metallic particles is a well established phenomenon and the interest in the corresponding materials has always been high. Although there is an intrinsic lower l i m i t of size for the superconductivity to occur, due to the strong fluctuations expected for the order parameter, such limit can be estimated around

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20-25A, so for larger sizes of order 100A or so there is a concrete possibility for superconductivity to develop. Further experimental and theoretical studies support the viability of a percolative nature of the charge carriers in cuprates. Experimental evidence of granular superconductivity has been reported in YBaCuO system by Cai et al. [27] , through the measurement of anomalous transient voltage excursions; the conclusions of these studies are that granular superconductivity in this material can exist with Tc as high as 160K for grains of the order 1000A.. Percolation models based on the existence of an intragrain superconducting transition and a coherence transition induced by the Josephson coupling between neighboring grains have been advanced, for example, to explain of the whole resitivity curves obtained in YBaCuO ceramics [28].

A percolation model also has been introduced by Hizhnyakov et al.[29] to calculate the critical magnetic cluster size prior to the disappearance of the antiferromagnetic long

range order in La2–x(Ba,Sr)xCuO4 and YBa2Cu3O6+x and to describe the transition from the antiferromagnetic to the metallic or superconducting state. This work interestingly shows that percolative models can be extended to the region of the phase diagram where antiferromagnetism and its fluctuations exists. The limiting form of reduced space is the unit cell and several experimental groups have found superconducting behaviour in one-unit cell thick YBa2Cu3O7 films with Tc around 20K-30K [30]. Models based on tunneling coupling of the Josephson type of two

adjacent CuO2 planes have been used to explain the data [30] Phase separation of charge can also be considered as a percolation effect . In this respect there are a number of experimental evidences of phase electronic separation in cuprates, i.e in La2–xSrxCuO4 and By measuring the infrared

activity of photoinduced carriers, Kim et al. [31] found evidence of phase separation of the photocarriers into metallic domains in the CuO2 planes and of their condensation in a superconducting state with Tc = 40K. It is experimentally now well established from Mossbauer spectroscopy, microwave absorption, neutron scattering and EPR [32] that phase separation occurs through domains of conducting and superconducting type. Using experimental data obtained from electrical resistivity, susceptibility and magnetization measurements, the phase separation in

and La2–xSrxCuO4 has been demonstrated

to be of electronic and percolative type [32,33]. The glassy behaviour exhibited by the high Tc oxides can be taken as evidence for

intragrain Josephson junctions arising from well identified metallic-superconductor grains separated by thin insulating barriers [34]. The interesting aspect is that glassy behaviour is exhibited in single phase samples and even in single crystals. It has been suggested that

granular behaviour can result from the short coherence length of cuprates and the suggested order of magnitude of these domains is 100-1000A with barriers of the order of 10A [34]. HOPPING CONDUCTIVITY WITHOUT INTERACTION

In the case of phonon-assisted hopping [35] first gave and expression for the conductivity. Their expression derived by the golden rule, essentially contains a coupling parameter which is assumed a deformation potential term, an overlap integral of the

localized states and the phonon population factor. For the purpouse of the present paper, their expression can be reelaborated to arrive at a simpler formula. When the overlap between wave functions is suffciently small, i.e the localized states are sufficiently distant apart with respect to the radius of the localized states, taking account that the number of

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carriers participating in the hopping process is form:

one arrives at an expression of the

(1) for the conductivity of a two-site process, where E is the coupling parameter, NF the density

of states at the Fermi energy, R the separation of the centres of localized orbitals,

the

energy difference between states that has to be supplied by phonons, the localization radius and nq the phonon population at wavevector q . There are two useful limiting forms of this equation for low and high temperatures. We obtain (2) (3) Eq. (1) relating to a two-site process, has to be integrated over the whole process of subsequent hops.

Miller and Abrahams showed that the current can be reduced to a problem of equivalent resistors connecting sites, and the conductivity is related to an integral over all conducting paths, a problem connected with a kind of percolation problem in which the paths carrying the current can be evaluated as a critical percolative problem. A procedure, alternative to the integration over paths has been suggested by Mott. It is a saddle-point method which amounts to resolve the integration on selecting those paths

where the transition probability per unit time has a maximum value. From this one can deduce simple rules for calculating the conductivity under more complex situations. Mott has argued that a simple geometric argument can be established to obtain a relationship between the parameter states within a hopping sphere

and R. It amounts to require that the total number of This gives

in three

dimensions. Similar reasoning can be applied to lower dimensionality. Then the saddle point maximation procedure leads to the results:

(4)

(5) where

and is a factor of order one. Mott’s argument holds for a constant density of states at the Fermi level. There have been various attempts to remove such an assumption, thereby improving on the Mott’s law, on assuming more general dependencies of the density of states at the Fermi level on energy and also avoiding the geometrical argument. These modifications usually change the preexponential of the law, while leaving the exponential factor unchanged. From eqs. (2-3) one finds that as a result of the phonon population at high and low temperatures, the conductivity at low T turns out to be higher than the one at high T by the factor This result is common to other models (see i.e granular metals) and has relevance in connection with the pseudogap behaviour of the conductivity.

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The functional form of variable range hopping may be modified close to a metal-insulator transition, the fractal nature of the wavefunction leading to a possible modification of the pre-exponential. We shall discuss this problem in connection with the granular metal models, which share close resemblance with the localization problem

HOPPING CONDUCTIVITY WITH INTERACTION. COULOMB GAP

Mott’s law in its various forms is due for neutral centres. The effects of the Coulomb repulsions have been considered by Efros and Shklovskii [36]. Their results for the conductivity can be deduced by noting that in such a case the activation energy between two centres is given by the expression where R is the separation of sites and a suitable static dielectric constant. On using the maximation procedure similar to the Mott’s law one gets the result (6) (7) with

and The factor A(T) may include corrections due to fractal nature of wavefunctions. In fact the same caution for the pre-exponential has to be used in the case of vicinity of the MIT like in Mott’s law. The results above are shown to arise from a modification of the density of states from the coulomb repulsions, leading to a density of states vanishing at the Fermi level, the so-called coulomb gap. Efros and Shklovskii have shown that the density of states follows the law The constant A can be deduced on imposing that i.e the density of states goes over continuosly to its umperturbed value for eenrgies greater than the gap. One has for instance in three dimensions. An equation which generalizes Mott’s argument is for an arbitrary dependence of the density of states on energy is

This is due for the

particular case of 3D. In lower dimensions suitable modifications of the volume element in this equation has to be carried out. For the case of a constant density of states one finds back Mott’s result for and in thecase of the coulomb gap one ends with the expression of the coulomb activation energy. In general, a cross-over occurs from the Mott’s law to the Efros-Shklovskii law on varying the temperature, since at higher T the activation energy for the neutral case becomes smaller than the coulomb activation energy and thus the ½ law is confined to smaller T. Aharony et al. [37]. have discusseed the cross-over on combining the two activation energies for the charged and neutral cases. It follows from these treatments a form of universality or scaling in the cross-over region. If the combination of the two activation energy is written as Aharony et al. [37] find that such a crossover occurs at the temperature with the conductivity behaving as: In

i.e a universal function of x = T / T x . It is shown by

Aharony et al. that the Efros-Shklovskii and Mott’s results are limiting cases of this with and

expression when T << Tx and T >> 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.

369

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
=2/(m*rs2), and it is given by: (3)

Uemura et al. and Keller et al. [14-16] have measured

from the London

penetration depth in different cuprate perovskites at a fixed doping, showing the linear

relation for Tc versus

376

Within the BCS approximations the critical temperature increases by increasing the electron-electron attraction, and by decreasing the size of the pairs, i. e., the coherence length of the superconducting phase. However the BCS approximations breaks down in the strong coupling regime where the electron-electron attraction is larger than the Fermi energy. In this extreme limit all electrons form localized pairs (LP). These local pairs are formed below the high temperature Tp however the superconducting critical temperature Tc occurs at low temperature where the local pairs Bose condense and in the strong coupling the factor f>>1. Therefore the critical temperature Tc reaches a maximum in a optimum intermediate coupling (OIC) regime where [17].

THE METALLIC HETEROGENEOUS PHASE The heterogeneous structure of a Cu2+ cuprate perovskite is shown in Fig. 1. The CuO2 layers form a fcc layer of a tetragonal structure with crystallographic axis at =bt=3.94 Å. The Cu ion form a square pyramid or bipyramid with planar Cu-O distance R=1.97 Å and axial Cu-O(A) distance 2.4-2.6 Å due to cooperative Jahn Teller effect for the Cu2+ 3d9 ion that removes the degeneracy of the and orbital. The bcc CuO2 layers are intercalated between insulating rocksalt fcc AO layers, This second material fit in the heterostructure by rotating its orthorhombic axis a0=b0 by 450 and

for the distance The electronic structure of the CuO2 plane is a charge transfer insulator with a half filled valence band. The covalency of the Cu-O bond is very high and the single hole per

Figure 1. The heterostructure of a Cu2+ cuprate perovskite and the molecular orbitals forming the electronic

structure of the CuO2 plane. 377

Cu ion is both in the

orbital and/or in the molecular orbital combination of the

oxygen 2p orbitals of local bi symmetry

There is a strong

local Coulomb repulsion for two holes in the same Cu 3d orbital, Udd~6 eV, that gives a Mott-Hubbard gap for the charge transfer where indicates a hole in the orbital.

The gap for the electron transfer of a hole from the Cu(3d9), or to the oxygen orbital is smaller than the Hubbard gap. This charge transfer gap is given by where in the final state configuration there is a Coulomb repulsion UdL between a hole on Cu and a hole on the nearest oxygen, and is the energy separation between O(2p) and orbital. The relevant local inter-atomic Coulomb repulsion UdL~2 eV has been determined by joint x-ray photoemission and x-ray absorption and the optical gap for the insulating compound since [18]. The metallic phase in the CuO 2 plane is obtained by two separate steps in the design of the material: first, chemical dopants that play the role of acceptors and pump electrons from the CuO2 plane, are introduced in the charge reservoir blocks; second, multiple substitutions of metallic ions A (A=Ba,Sr,La,Nd,Ca,Y ) in the rocksalt layers are made in such a way to change the average ionic radius of the rocksalt layers. In cuprates with multiple CuO2 layers the rocksalt layers between the copper planes loose completely their oxygen ions. Doping introduces holes in the O 2p orbital and a single hole remains in the Cu site [19]. However the symmetry of the molecular orbital for the added hole have a mixed symmetry with a component with local a 1 symmetry mixed with These results have shown that the symmetry of the doped holes is not the pure m"=2 symmetry of the antiferromagnetic insulator at half filling. Therefore the doped holes in the metallic phase are associated with a local lattice distortions (LLD) mixing states with different orbital momentum. These LLD distortions are expected for a pseudo Jahn Teller electron lattice interaction of the doped holes [23]. Here the key point has been to show that the electronic correlation lower the Jahn-Teller energy separation between the states with and symmetry from about 1.5 eV to about 0.5 eV since the Coulomb repulsion UdL for the configuration is much smaller than for The 2D electron gas in the CuO2 plane of cuprate perovskites is therefore a strongly correlated electron gas described by the Hubbard Hamiltonian. Moreover there is a relevant electron lattice interaction of the type of cooperative pseudo Jahn-Teller coupling of charges with Q2-type local modes. This can be described by the Holstein Hamiltonian with a next-near neighbour hopping integral t’. Therefore its metallic phase is described by the Hamiltonian:

(4)

The first two terms describe the itinerant charges in a 2D square lattice simulating the

CuO2 plane where t is the electron transfer integral between nearest-neighbor sites and t` is the electron transfer integral between next-nearest-neighbor sites <>, is the local electron density, denotes the electron creator operator at site i.

378

The third term is the Hubbard Hamiltonian describing the electronic correlation in the CuO2 plane. The Hubbard term induces a mass renormalization of a factor of the order of 5 giving an effective mass in the direction, m*/m0~2. The coupling of the charges with local lattice distortions (LLD) of the CuO4 unit can be described by the Holstein Hamiltonian (Hph +HI). The position of the lattice site is

indicated by Ri and represents the creation operator for phonon with wavevector q, is the frequency of the optical local phonon mode and g indicates the coupling of the charge with this local lattice mode. This term describes the weak electron-phonon interaction while for g sufficiently large the charge is coupled with local lattice distortions. The local lattice distortion Q follows the equation

therefore the electron-lattice interaction provides a force that induces a displacement of the equilibrium position. The local lattice distortion (LLD) appears when becomes larger than zero energy vibration amplitude. In the present square lattice it is possible to identify the electron lattice coupling constants where d=2 for a 2D electron gas. We have found that in the cuprate perovskites, while the first coupling constant is in the weak coupling limit, the second one is in the intermediate-strong coupling limit and it is expected to give local lattice distortions. In this situation we are in an intermediate

regime where charges trapped into LLD coexist with itinerant charges. This situation is expected to occur in special cases in the intermediate electron-lattice coupling regime.

The experimental evidence for LLD due to pseudo JT electron-lattice interaction (JTLLD) was provided by the presence of two different types of doped holes in the oxygen orbital [20-23]: of partial a1 symmetry, mixed with (orbital angular momentum and of b1 symmetry mixed with (orbital angular momentum The pseudo JT-LLD should be associated with the doped holes since the Q2 type lattice distortion forms molecular orbital of mixed and angular momentum. The search for these local lattice distortions motivated the Rome group to solve the

incommensurate structural modulation of the CuO2 plane in Bi2Sr2CaCu2O8.2 (Bi2212) by joint single crystal x-ray diffraction and EXAFS. We have found in 1992 that the pseudo JT-LLD get self organized in linear arrays, i.e., stripes [24-26]. The co-existing itinerant particles form rivers of charges and at the Erice workshop in 1992 [25] the scenario of superconducting stripes, where “the free charges move mainly in one direction, like the water running in the grooves of a corrugated iron foil ” , was introduced for the first time in the field of high Tc superconductors. A heterogeneous phase of the matter is a generic phenomenon following doping of a high correlated antiferromagnetic insulating electronic system. The formation of a microscopic electronic phase separation with the formation of metallic droplets in a antiferromagnetic background was first shown in doped magnetic semiconductors [27]. Experimental evidence that at very low doping in the cuprates the doped holes segregate into strings of charges that play the role of domain walls between anti ferromagnetic domains, forming a glassy phase of strings, has been reported [28]. At higher doping if the counterions are mobile the system is unstable toward a macroscopic phase separation between macroscopic metallic domains and insulating antiferromagnetic domains [29,30]. 379

The high Tc superconductors are a special case of heterogeneous doped magnetic superconductors since in the metallic droplets we have the coexistence of doped charges in the weak coupling limit, phase A, with doped charges associated with local lattice distortions associated with pseudo Jahn Teller electron lattice interaction phase B. There is now clear experimental evidence that there are two types of doped charges in the cuprates [31]. The ordering of charges trapped by the pseudo Jahn-Teller LLD with an associated modulation of the orbital angular momentum gives stripes and orbital density waves. The phase diagram of the metallic phase of high Tc superconductors is usually given as a function of hole doping, measuring the distance from the antiferromagnetic (AF) insulating Mott Hubbard phase. In the two components 2D electron fluid it is necessary to measure the actual density of the itinerant component by using the electron density parameter rs measured by the Hall effect at low temperature. The phase diagram of La2-x SrxCuO4 as a function of electron density parameter rs is shown in Fig. 2 (lower panel).

Figure 2. Phase diagram of the La2-xSrxCuO4 (lower) and

(upper) as a function of electron density

parameter rs of the itinerant 2D electron gas measured by Hall effect. Usual notations are used to denote different regimes of the phase diagram. The structural phase transition boundary between the orthorhombic

and tetragonal phase is shown.

The system La2-xSrxCuO4 shows a complex phase diagram typical of a glassy system due to the random distribution of countercharges (Sr ions) in the block layers. The AF phase appears in the range rs>37, and a spin glass phase appears for 37>rs>15. The metallic phase, 15>rs>5, shows a typical glassy phase with several crossover temperatures that depend on the measuring time of each experimental probe.

380

To understand the basic physics of the metallic phase of cuprate superconductors we need to study a simple crystalline system. This is provided by (and where the itinerant holes in the CuO2 plane are compensated by the negative charges carried by the mobile interstitial oxygen in the LaO layers (and in the BiO charge reservoir layer) that can get ordered. There is no frustrated phase separation regime in due to mobile counterions, therefore it does not show the spin glass phase of the doped Mott insulator in the range 37>rs>15, where it shows the expected phase separation below about 300K between an insulating doped AF lattice (rs~37) and a metallic phase (rs1~12). This first superconducting phase with Tc=32 K shows the universal 1D dynamical spin fluctuations below Tsdw~60K and the 1D stripes [32], CDW and/or ODW, indicated by the ordering of local lattice distortions of the CuO6 octahedra (tilting) below a critical temperature Tc0=190 K [33], as shown in Fig. 2 (upper panel). A second stable phase with the highest critical temperature appears at 5
Figure 3. Temperature dependence of the order parameter for the stripe formation with wavevector in La2CuO4.1 . The fit shows a critical temperature Tco=190 K for charge ordering.

In the second harmonic peaks we can well separate the charge ordering from the 3D oxygen ordering peaks. The square root of the intensity plotted in Fig. 3 gives the direct measure of the density of charge ordered in the CuO2 plane with wavevector q, that is the order parameter for the charge ordered phase. The solid line is a fit to the 381

experimental intensities with an expression

which represents a typical second

order phase transition with Tco~188 K. This effect is clearly due to formation of charge stripes in the CuO2 plane since the oxygen mobility is frozen below 200K. In fact the 3D oxygen ordering has already been established at higher temperature (in the range 270-230

K) in the system as evidenced by temperature evolution of the resolution-limited diffraction peaks. The second harmonic of the charge modulation at 0.416b* has the same wavevector of the nesting vector at 2kF ~0.4b* or observed by Saini et al. [18] in the Fermi surface of Bi2212 that induces the suppression of the spectral weight at selected spots in the k space and gives a broken Fermi surface. The 3D ordering of dopants stabilizes the orthorhombic phase, as shown in Fig. 1, and the symmetry of the CuO2 plane is broken. The ordering of stripes in the b direction gives a superconducting phase in a broken symmetry, with a broken Fermi surface. The Fermi surface is therefore formed not by closed circles but by segments and the "mini-gaps" in the density of states due to the 1D superlattice of stripes [19-20] give origin to the "pseudo-gap" scenario. In this stripe scenario the amplification of the superconducting critical temperature is realized by tuning

the Fermi level at a "shape resonance" of the superlattice [19-20]. The pairing mechanism is mediated by charge fluctuations in a metal with the anomalous dielectric constant typical of an anomalous Fermi fluid at rs>4 [17].

Figure 4. The critical temperature of different cuprate perovskites at fixed doping at (open circles) and (open squares) as function of the ratio ρ/m*, where is the condensate density and m* the effective mass. The dashed line shows the calculated Tc using formula (5) for the coherence length of the order of the distance between particles =2rsaB.

In this regime a generalized BCS scheme remains valid and the dynamical pairing described by the BCS is still possible up to the point where the size of the pairs of quasi-

particles is of the order of the wavelength of the electrons at the Fermi level

In this

regime the coexistence of local pairs (bosons) and fermions in the normal phase is expected. The proximity to Wigner localization gives a metal with a negative dielectric

permitivity with

382

predicted by Dolgov and Ginsburg, needed for high Tc The highest critical temperature is reached for

close to the average

distance between two electrons =2rsaB in a 2D electron gas. This is the highest critical temperature possible in an extended BCS scheme in the intermediate coupling: (5)

where is measured in Å-2, and m* is the effective mass. In this limit the critical temperature depends only on the ratio and using the phenomenological value, f=2, from tunneling data at optimum doping, the critical temperature has been calculated as a function of in ref. 17. The predicted temperature is plotted in Fig. 4 and it is in very good agreement with the experimental data. The expected density of charge carriers is constant but the condensate density is different in different cuprate perovskites. Therefore there is a term in the Hamiltonian of the metallic phase of the CuO2 layers, that changes the electronic structure and drives the critical temperature and the condensate density. The object of this work is the identification of the term in the Hamiltonian that drives the CuO2 plane to the optimum intermediate coupling regime giving high Tc superconductivity. THE

PHASE DIAGRAM OF BI2212

After the discovery of HTcS the physics of cuprates was described by a generic phase diagram where the critical temperature is plotted as a function of doping i. e., a measure of the charge density and the distance from the Mott Hubbard insulator at By increasing the system goes through quite different states. At low doping the doped holes form a disordered electron glass. At very high doping a normal metallic phase appears. The high Tc superconducting phase appears between these two phases. In 1993 we presented the phase diagram for Bi2212, shown in Fig. 5. The doping was measured in units of

Figure 5. The phase diagram

for Bi2212. The charge density is measured in units of the critical doping

for the expected commensurate polaron crystal (CPC) also if in this family by changing the doping we do not cross the CPC phase. We observe 1) an insulating phase A of an electron gas in the localization limit (kF1<1) for 2) a homogeneous metallic phase B at and 3) a region of co-existence of the two different phases, a 1D-polaronic incommensurate charge density waves and the superconducting phase. 383

the critical density for the insulating commensurate JT charge ordered crystal at in La1-xBaxCuO4 This electronic crystal is in competition with superconductivity and it is at the origin of the huge suppression of the superconducting critical temperature in La1xBaxCuO4 at x=l/8. The long range Coulomb interaction between charge trapped in pseudo JT-LLD is expected to play a key role in the formation of the ordered phase of localized charges. Therefore we call this phase a generalized Wigner commensurate JT polaron crystal (CPC). The CPC was not observed at in the phase diagram of Bi2212 shown in Fig. 5 where only a weak minimum of Tc appears at By decreasing the temperature, below about T*=TCO=130 K, the system form an inhomogeneous phase where a onedimensional (1D) incommensurate polaronic charge density wave (ICDW) or polaronic stripes coexist with free carriers. In the underdoped phase, the charge density of free carriers is smaller than that of JT polarons in the ICDW. In the high doping phase,

the charge density of free carriers is larger than that of JT polarons in the ICDW.

This particular ICDW does not suppress but promotes the pairing of the free carriers below Tc. In fact in this unexpected metallic phase of condensed matter, a superlattice of quantum mesoscopic stripes of width L, the chemical potential is tuned to a "shape resonance". The "shape resonance" occurs when the de Broglie wavelength of electrons at the Fermi level The measure of the stripe width L in 1993 has established the presence of the "shape resonance" at optimum doping. At the shape resonance the chemical potential is tuned to the bottom of a superlattice subband and therefore to a narrow peak in the

density of states (DOS). At optimum doping a BCS-like superconductivity is observed and the highest Tc is reached where the chemical potential is tuned to this narrow DOS peak via the calculated "shape resonance" effect on the superconducting gap. A patent has been granted with priority date 7 Dec 1993 for a method of Tc amplification via the "shape resonance" effect in new materials formed by a superlattice of quantum wires. A very good agreement with experimental data has been found for the calculated critical temperature plotted as a solid line in Fig. 5 assuming the pairing mechanism mediated by charge fluctuations in a superlattice of quantum wires at the "shape resonance" [17].

Figure 6. The average bond length measured by Cu K-edge EXAFS as a function of the average radius of metal ions in the rocksalt layers.

384

THE MICRO STRAIN QUANTUM CRITICAL POINT

The electron-lattice interaction g of the pseudo JT type in the cuprates is controlled by the static distortions of the CuO4 square plane and the Cu-O(apical) distance. In fact where Q is the conformational parameter for the distortions of the CuO4 square, like the rhombic distortion of CuO4 square; is the dimpling angle given by the displacement of the Cu ion from the plane of oxygen ions; and is the JT energy splitting that is controlled by the Cu-O(apical) bond length.

Figure 7. The critical temperature for charge ordering function of the micro strain at optimum doping

and the superconducting critical temperature Tc as

There is an external field acting on the CuO2 plane of the cuprates that controls via the micro-strain of the CuO2 lattice due to the compressive stress generated by the lattice mismatch between the metallic bcc CuO2 layers and the insulating rocksalt fcc AO layers [40,41]. The bond-length mismatch across a block-layer interface is given by the Goldschmidt tolerance factor where [r(A-O)] and [r(Cu-O)]=d0 are the respective equilibrium bond lengths in homogeneous isolated parent materials A-O and CuO2 [42]. The hole doped cuprate perovskite heterostructure is stable in the range 0
controlled by the oxygen doping in the charge reservoir blocks. The stress due to the mismatch or the chemical pressure acting on the CuO2 plane is controlled by the average ionic radius in the rocksalt layers. The stress increases going from Ba to Sr to La. We have measured the average Cu-O bond lengths by Cu K-edge EXAFS, a local structural probe, and shown in Fig. 6 as a function of average ionic radius of metallic ions in the rocksalt layers . Decreasing is equivalent to a anisotropic chemical pressure acting on the CuO2 plane. We define the local or micro strain of the CuO in plane bond length /d0), where d0=1.97 Å is the equilibrium Cu-O distance at doping in many different systems.

385

In the cuprate perovskites the micro-strain e drives the system to a quantum critical point for the formation of a superlattice of quantum stripes. The stripes of local lattice distortions are detected by x-ray diffraction above a critical micro-strain We have plotted in Fig. 7 the variation of the critical temperature for charge ordering Tco and the superconducting critical temperature Tc as function of the micro-strain at optimum doping

Figure 8. The superconducting critical temperature Tc plotted in a half-tone scale (from TC=0K, black, to Tc ~ 135K, white) as a function of the micro-strain and doping and

The maximum Tc occurs at the critical point

In Fig. 8 we report the critical temperature Tc in a color plot (the critical temperature increases from black, Tc=0K, to white, the maximum Tc ~ 135K) as a function of the microstrain and doping for all superconducting cuprate families. The figure shows that the maximum Tc occurs at the critical point From these data we can derive a qualitative phase diagram for the normal metallic phase of all cuprate perovskites that give high Tc superconductivity that is shown in Fig. 9. This phase diagram solves the long standing puzzle of the phase diagram of the normal phase of the cuprates. There was a hidden physical parameter, the micro-strain, that triggers the electron-lattice interaction at a critical value for the onset of charges trapped into pseudo JT-LLD. The doping of the strained antiferromagnetic lattice forms both free carriers and charges trapped into the JT-LLD above the critical micro-strain For as it was discussed for the case of oxygen doped Bi2212 and La 124, the systems show a quasi

first order phase transition as a function of doping. The quantum critical point QCP is well defined at constant finite doping as a function of the micro-strain as it is shown in Fig. 8. Direct experimental evidence for quantum critical local lattice fluctuations has been obtained by measuring the dynamical fluctuations of the Cu-O bond at a high temperature TH>TCO in all families of cuprates (TH~200K). In conclusion we have deduced a phase diagram for the superconducting phases where Tc depends from both doping and micro-strain. The anomalous normal phase of cuprate superconductors is determined by an inhomogeneous phases with co-existing polaronic stripes and itinerant carriers that appears for an electron lattice interaction larger than a critical value. Fluctuations of lattice-charge stripes appear in this critical regime. The micro-strain drives the electron lattice interaction to a QCP of a quantum phase transition [43]. Near this QCP the stripes get self organized in a superlattice of quantum wires of charges trapped into JT-LLD that co-exist with free carriers. This superlattice forms an 386

array of superstripes where the chemical potential is tuned to a shape resonance. The plot reaches the highest temperature at the critical point

Figure 9. The phase diagram of the normal phase of doped cuprate perovskites as a function of micro-strain

on the Cu-O(planar) bond and doping. The high Tc superconductivity occurs in the region of quantum fluctuations around the micro-strain quantum critical point QCP.

This research has been supported by INFM, by MURST - Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale, and by "Progetto 5% Superconduttività del CNR.

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ELECTRON STRINGS IN OXIDES

F.V.Kusmartsev School of MAP, Loughborough University, LE11 3TU, and Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 0EH, UK, Landau Institute, Moscow, Russia

INTRODUCTION The electronic phase separation scenario originally introduced by Nagaev into the physics of magnetic semiconductors[1] is very popular now as a possible scenario for a creation of a colossal magnetoresistance phenomenon[2-7]. The idea of string formation also belongs to this stream flow[8,9]. The phenomenon of the colossal magneto-resistance observed in hole doped manganites [10-14]. (see, also references inside) may have a natural explanation in such a novel framework. The strings are arising in solids with narrow bands[8,9] due to an effect of a self-traping(ST). For a single particle the self-trapping phenomenon has been intensively studied in the past (see, for example, in Ref.[15-22]. Since the mass of electrons is a lot smaller than the atomic masses the electrons move much faster than the atoms. Therefore, conventionally, an adiabatic approximation for the motion of the atomic lattice is assumed. If we are interested in stationary points of the adiabatic potential the kinetic energy of the atoms may be neglected. The state of a charged particle together with the surrounding static deformation associated with a stationary point of the adiabatic potential was originally called a ”deformon” and was later conventionally termed a self- trapped (ST) state which is analogous to a conventional polarons[18-22]. The effective radius of this deformation cloud is the radius of the self-trapped state. Such an effect is strongly enhanced for low dimensional systems or for systems with strong electron-phonon interactions [17]. In 3D crystals with wide bands the self-trapping process is associated with a penetration of the electron-phonon system through a self-trapped barrier[23-25]. while in 1D system the self- trapped barrier is absent. In 2D systems the selftrapping arises only if the electron-phonon coupling is greater than some threshold value[17]. The origin of ST barrier in 3D systems and the ST threshold are due to a competition (or a balance) between the kinetic and potential energies of the electrons(holes)[23]. The most interesting self-trapped phenomena to date include self-trapped excitons ob-

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

389

served in rare gas solids (see, Ref.[17-26] and references therein). The creation of such states gives rise to an electronic mechanism of the defect creation in solids[24,27]. That is the creation of self-trapped excitons by light causes structural defects in solids. The self-trapping terminates the tails in the electronic density of states in disordered semiconductors[25,28]. Finally the self-trapping gives rise to a deformation arising in vortex cores[29] observed recently in superconducting Nb by polarized neutron scattering[30]. In the case of the many-particle self-trapping (a string’s formation) the situation changes drastically since the potential deformation or polarization energy as well as a Coulomb energy are dominant. The self-trapping process is barrierless in such a state the deformation energy

takes maximal possible value that causes a localization of many particles in a potential of a linear shape. Here, our main result is that the phenomenon of electronic string-defect formation - a multi-particle self-trapping[8,9] may also arise in manginites due to an interaction of electrons(holes) with acoustical and Jahn-Teller phonons. Note that this phenomenon is arising only in solids with narrow bands where the kinetic energy of electrons is limited by the small width of the band. Therefore, due to the small kinetic energy of electrons (holes) in a single deformation potential well many charged particles may be self-trapped. In this case the ST deformation well is self-consistently created by all the particles trapped by this well. Although the shape of this well may be arbitrary, it is primarily dictated by a Coulomb repulsion between the particles. The contribution of the electron kinetic energy may be important for the formation of conducting string associated with long-range electron-phonon interaction with polar or longitudinal optical phonons[9]. Qualitatively, if the deformation potential well traps M electrons(holes), the lattice deformation Q is proportional to M and consequently the elastic energy of the lattice is proportional to Q2 ~ M2. Hence, the electron kinetic energy of the self-trapped particles and, therefore, the lattice adiabatic potential of the ST state decreases as ~ –E P M 2 , where Ep is a polaron shift. However, this decrease is opposed by the Coulomb repulsion between the particles trapped by the string well, which energy has an additional log M multiplier[8,9] so that the Coulomb energy is approximately equal to ~ VM2 log M, where V is a constant of the intersite inter-electron Coulomb repulsion. A balance between these energies which is determined by a minimisation of the total energy gives a stationary many particles ST string state where the string length N ~ M ~ exp (Ep/V – 1/2), (see, for example, and compare with Ref.[8]. The precise dependence of the Coulomb energy of the trapped particles depends strictly on the shape of the self- trapped spot. When all particles well separated, of course, the Coulomb energy is minimal. But in this case the deformation energy and the adiabatic potential are not of the lowest one. For the multi-particle self-trapping spot the lowest Coulomb energy corresponds to a linear chain when all the self-trapped particles are located along a single line. We call such a many-body self-trapping object as a self-trapped string. We describe here an electron(hole) string consisting of M trapped particles and with the length Na created due to electron-phonon interaction with Jahn-Teller and acoustical phonons in a hypercubic d – dimensional lattice where a is a characteristic distance between nearestneighboring atoms. In a doped antiferromagnet such a string may have a lower energy than M spatially separated self-trapped particles. Such ST strings in a hypercubic lattice may take arbitrary nonlinear configurations, which correspond to different highest values of the adiabatic potential. Indeed, recent neutron scattering and electron microscopy experiments

[12-14] indicate on electronic phase separation and stripe or string ordering in semiconducting La1–xCaxMnO3 and analogous compounds which may be related to the effect of string formation as described in the present paper. With the increase of the doping the forming insulating electron strings may be ordered into the parallel stripes[12-14] If the material consists of layers with ordered magnetic moments like LaMnO3 then the threshold of a bond or a site percolation in the 2D plane (a single layer) is equal to xc = 0.5. Since the strings are charged objects then it is very probable that at such a percolation threshold the percolating charged strings will be ordered. As a result

390

the system of charged parallel stripes as it was observed in the recent experiments[12] will be formed. However for a proper comparison with existing experimental data[10-14] an investigation of the possible microscopic forms of electronic phase separation (like, the described string formation) in the framework of the double exchange model of spinful fermions having orbital degrees of freedom and interacting with Jahn-Teller phonons is urgently needed. The

present work may serve as an attempt in this direction. HAMILTONIAN

We consider a Hamiltonian of charged spinful fermions having orbital degrees of freedom and interacting with acoustical and Jahn-Teller phonons on a d – dimensional hypercubic lattice:

(1)

where is a matrix of the electron hopping-integrals, the operator creates (destroys) a fermion at a lattice site i, ni, is a density operator which is expressed via the occupation number operator as and the operator is an operator of the creation (destruction) of an –branch of phonons. We consider Jahn-Teller and acoustical phonons. The summations in eq.(l) extend over the lattice sites i and -as indicated by < i, j > -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.

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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)

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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”

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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

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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

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(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.

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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].

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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

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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.

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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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TAILORING THE PROPERTIES OF HIGH-TC AND RELATED OXIDES: From Fundamentals To Gap Nanoengineering

DAVOR PAVUNA Department of Physics - IPA Ecole Polytechnique Federale de Lausanne CH - 1015 Lausanne EPFL, Switzerland Email: [email protected]

In a direct analogy with a successful (Al)GaAs band-gap engineering, I discuss an equivalent nanotechnology: the advanced heteroepitaxy of layered oxides enables us to ‘nano-engineer’ desired superconducting gap and/or insulating barrier. However, in most oxides, the progress is somewhat hindered by intrinsic materials problems: growth induced disorder, difficult local doping and non-homogeneous oxygen distribution. As textbook understanding of the fundamentals is obviously needed, I also discuss electronic properties across the electronic phase diagram: the ‘pseudogap’ controversy, metal-insulator transition and anomalous transport. As we gradually solve remaining obstacles we will be able to fully exploit the potential of layered cuprates (and related oxides) and tailor their electronic (and/or magnetic) properties at will. New concepts and applications will emerge from such an integrated nano-engineering technology in the 21st century, yet for real ambient technology we still require colossal superconductors with INTRODUCTION: TAILORING THE PROPERTIES OF LAYERED OXIDES Some 65,000 papers after the striking discovery of high-Tc superconductivity in cuprates [1], we are still trying to fully understand these versatile solids and control their properties. Complete understanding of fundamentals of a given class of electronic materials often results in a successful new technology. That may require huge interdisciplinary effort of the whole generation of scientists and engineers. As the main topic of this workshop is the role of self-organisation in complex electronic materials, it is useful to begin with applications of high-Tc oxides and compare them to prominent electronic materials, like Si or GaAs. Pure Si is an electronically clean solid that, thanks to the skillful use of doping and SiO2 insulating oxide, can be functionally nanoengineered (down to about 3nm) and integrated into large scale multi-transistor chips, GaAs is optically clean material in which the direct band-gap and desired AlGaAs heterostructure properties can be modified at will

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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(and calculated in advance) by a suitable MBE hetero-epitaxy [2]. As a result, the bandgap engineering photonic technology produces its own original, ‘archetype’ device - the laser of a desired wavelength. Although many of the ‘leaders’ of the semiconductor industry or analysts from the Wall Street consider the field of high-Tc superconductivity an investment risk, the applications are advancing successfully and may even dominate some technologies of the 21st century. Note that the successful Si-based technology has so far accumulated more than 107 menyears of know-how, III-V (GaAs) photonic technology 106 men-years, while all superconductivity hasn’t even reached the 105 men-years. We clearly need at least another decade of intensive R & D before giving any definite conclusions to the global media. Especially so, as superconducting oxides have their own ‘archetype device’ (ultrafast Josephson switch, and/or RSFQ-logic) while the magnetic oxides (for example, manganites) provide some of the most versatile magnetic memories [4]. Moreover, there is no doubt that the in-depth understanding of the fundamentals [3-7] of this field (superfluidity included) will be relevant to many branches of advanced science and technology in the 3rd millennium. To illustrate the concept of high-Tc oxide ‘nano-engineering’ I now use a direct analogy with the bandgap engineering of (Al)GaAs lasers [2]: the direct bandgap, and consequently the emiting wavelength, is altered by varying the Al content in an optically

clean, epitaxially grown AlGaAs heterostructure. An equivalent conceptual approach to layered high-Tc (and related) oxides requires, at the very least, the following: i)

Atomically flat heteroepitaxy of electronically clean constituent blocks (sub-layers) and ‘clean’ interfaces. Here, considerable progress hase been made [4,8].

ii)

Very precise control of the stoichiometry and local carrier doping, across the whole electronic (magnetic) and crystallographic phase diagram on nano-, meso- and macroscopic scale. Controlled local doping still poses formidable challenge in some oxides. Several reports in this workshop clearly illustrate the difficulties.

iii) Reproducible variation of the superconducting gap (or the pseudogap in some cases) and/or of the thin insulating barrier for Josephson junctions [4,7]. Again, as shown by other contributors in this workshop, this is often a non-trivial task, given the growth-induced disorder and the self-organisation in some oxides. iv) Suitable integration [8] into novel devices and systems that range from array of Josephson junctions, RSFQ-logic [2] to magnetic storage and nanomagnetism. In photonic technology an engineer designs the characteristics of the functional AlGaAs heteroepitaxy, say, a laser with wavelength of 1.5µm, the computational physicist solves the apropriate Schroedinger equation for a predetermined model-structure and the crystal-grower delivers it by using a computer-controlled molecular beam epitaxy [2]. One can consider this whole technology as an ‘applied quantum mechanics’. There is effectively a one-to-one correspondence between the calculated and the fabricated structure and the corresponding, designed and obtained laser properties. At present, this is already possible to achieve with some layered oxides. the ‘nanoengineered’ superconductivity in BSCCO cuprates and ‘nanoengineered’ magnetism in manganites [4,8]. However, complete variation of the doping of an YBCO-123 film is often hindred by oxygen mobility and by incipient, possibly intrinsic growth-induced disorder [28]. Namely, even as insulators, most of the technologically interesting transition metal oxides are not even electronically clean (like pure Si), let alone optically clean as GaAs waveguides. Furthermore, upon doping, these artificially obtained ionic metals, show the distribution of oxide vacancies which leads to localized states and/or to (nano-)phase 432

separation [9]. That in turn poses a challenge to our understanding of the true electrodynamic response as well as correct intepretation of the microscopic origin of the socalled ‘pseudogap’ [3,10], the exact role of point defects, twins and grain boundaries, the symmetry of the gap-parameter [3] and ultimately the understanding of the puzzling difference between hole- and electron doped cuprates [6]. As oxide nano-engineering progress is discussed by leading resarchers at length elsewhere [4], I will now discuss some of the ‘anomalies’ in the electronic properties of high-Tc cuprates. ANOMALIES IN ELECTRONIC PROPERTIES OF HIGH-TC OXIDES

To fully understand all difficulties associated with the proposed heterostructure nanoengineering of layered oxides one should realize that they are i) usually truly thermodynamically stable only as insulators, ii) as illustrated in Table 1, highly anisotropic ternary and quaternary solids (except for the isotropic Ba1–xKxBiO3), and iii) artificial ionic metals (and superconductors below Tc) obtained by doping of the parent insulating compound. Their properties are complex as compared to usual normal metals and alloys [318]. For example, there is well documented existence of the so-called ‘pseudogap’ [10,14], distinct from the true superconducting, condensation gap [11]. Furthermore, in the underdoped regime and under very high magnetic fields some of these oxide superconductors exhibit phase transition directly into the insulating state [12], although this could be due to the granularity of the sample. As illustrated in Figure 1 the structure of a high-Tc oxide can be represented as a layered structure that consists of two CuO2 planes separated by a spacer. Between these bilayers are interlayer regions which, in the case of YBa2Cu 3O7, correspond to the CuO chains [6]. In general, most cuprate superconductors can be discussed in terms of the

block-reservoir and the doped CuO2 plane and evidently they can be artificially ‘constructed’ by block-by-block epitaxial growth. Recent advent of combinatorial

chemistry enables a fast search for new promising phases [8].

’charge reservoir’ CuO2 plane spacer CuO2 plane

’charge reservoir’

Figure 1. A schematic model unit of double CuO2 layer cuprate.

Before discussing the electronic phase diagram of cuprates I strongly emphasize that even the very best single crystals of HTSC oxides often contain various defects and imperfections like oxygen vacancies, twins, impurities ... These imperfections are not only relevant to their physical properties but possibly even essential to their basic thermodynamic stability. It may well turn out that various imperfections found in HTSC

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crystals are intrinsic to these materials. That may be due to the fact that even their parent insulators can never be made optically clean. Therefore it is more difficult to achieve true one-to-one correspondence between the ‘computational solid’ (as used by most mean-field theories) and the actual sample, made and measured by experimentalists. This gets worse upon doping. Note that to this day some of the highest Tc phases, like Hg-cuprates, were synthesized only as a small single crystals <1mm in diameter and majority of reported measurements were performed mainly on LSCO, YBCO and BSCCO crystals.

Table 1. Most representative HTSC compounds. The index n refers to the number of CuO2 superconducting layers within a given crystallographic structure. m refers to the number of ’chains’ in the structure; m=1.5 corresponds to the case of alternating ’chains’.

It is indeed important to understand that the materials science of HTSC oxides is a non-trivial pursuit and that the understanding of phase diagrams, crystal chemistry, of the preparation and stability of layered oxides requires an in-depth study and often hands-on experience in the laboratory. The advancement of our understanding of physics, chemistry

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and appearance of applications therefore depends very much on the advancements in materials research [4,8] that is still very much in progress.

Figure 2. Schematic phase diagram of cuprate superconductors. Various authors attach different name or significance to various regions. Note that the intermediate phase, left of the maximum of Tc (and below T*) is a highly controversial subject [9-19].

True textbook understanding [6] of measured properties of high-Tc oxides and their electronic phase diagram (Figure 2) still presents a major challenge, despite of a remarkable progress in both, sample preparation and advanced experimental techniques [3-19]. One of the reasons for this is that as we dope these layered oxides, we do not encounter only rather complex electronic phases; the underlying crystallographic (structural) and ’metallurgical’ phase diagrams of these quaternary solids are often even more complex and the disorder and (possibly intrinsic) inhomogeneities clearly play an important role. Consequently, it is difficult to determine which properties are ‘intrinsic’ to these solids and which are induced by disorder as one varies the doping in real samples. Most reports in the literature have

initially ignored inhomogeneity effects in samples yet recent results are clear: these effects are real. That partly explains why simple mean field theories cannot describe phenomena observed in these solids and why experimentalists and especially chemists often ignore most advanced theories. At present, most experiments seem to indicate that at the left hand side of the phase diagram we have 2D antiferromagnetic insulator while, at the right hand side, highly overdoped 3D perovskites tend to exhibit more Fermi-liquid-like properties. There is an agreement on the existence of the Fermi surface in the optimally doped and overdoped samples (see Figure 3) but no definite agreement on the fine features [16]. Still, while many fine details remain to be clarified, especially in the underdoped and overdoped samples, there is now well established evidence, shown by most experiments, of the existence the 'pseudogap', T* . The 'pseudogap' was originally introduced to cuprates already in 1987 by 435

Figure 3. Experimentally determined Fermi surface of Bi-2212: filled squares – experimental Fermi surface locations; open squares – Fermi surface obtained by symmetry operations; closed circles – superlattice band crossings along the M-Z and Y high symmetry directions ; open circles – locations in the Brillouin zone where the ARPES spectra were taken [15].

Phillips [10] and subsequently by Friedel [13] in a different context from presently dominant idea of a pre-formed pairs in the normal state [7,9,14], Some authors consider that ‘direct’ observation of the ‘pseudogap’ can be made by ARPES, yet this can be misleading [16] and possibly wrong due to the disorder effects [15,16,27]. Deutscher approached the ‘pseudogap’ problem somewhat differently [11]. He has compared gap energies, measured by different experimental techniques, and has shown that these reveal the existence of two distinct energy scales: p and c. The first, determined either by angleresolved photoemission spectroscopy or by tunnelling, is the single-particle excitation energy - the energy (per particle) required to split the paired charge-carriers that are required for superconductivity. The second energy scale is determined by Andreev reflection experiments, and Deutscher associates it with the coherence energy range of the superconducting state: the macroscopic quantum condensate of the paired charges. In the overdoped regime, p and c converge to approximately the same value, as would be the case for a BCS superconductor where pairs form and condense simultaneously. In the underdoped regime, where the pseudogap is measured, the two values diverge and p (T*) is larger than c (Tc) [11]. Phenomenologically, this indeed corresponds to the schematic phase diagram given in Figure 2. However, very different explanations of Tc and T* are given by theories that consider the phonon mechanism of superconductivity [10,17] as compared to more unconventional approaches [18-20]. Metal-insulator transition transition is not understood even in simpler solids like Si:P so it is not surprising that it remains highly disputed topic in high-Tc oxides [3,7,9,10], where anomalous behavior was reported by Boebinger et.al [12]: deep in the underdoped regime LSCO and BSCCO samples directly transit from superconducting into the insulating state under very high magnetic field. This, however, could be due to the ‘granularity’ of the sample so more work is needed until the experimental situation is clarified. There is also evidence for formation of stripes [5] in some underdoped perovskites yet the exact role of stripes in the overall HTSC scenario is still unclear. While some authors consider them crucial for the pairing mechanism, other consider them only as a stabilizing feature within a low-dimensional system. What seems clear though is that the appearance of stripes is a consequence of self-organisation.

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In the superconducting state the agreement exists that the charge q = 2e and that in most phases holes are the carriers (see Table 2), that the pairing is singlet and that the symmetry of the gap is not a conventional, isotropic s-wave. In highly doped samples, measured properties often appear as BCS-like [6], yet this is also highly disputed by some authors. Namely cuprates are highly anisotropic, the coherence length is very short, and there are only Cooper pairs per coherence volume; much less than in the conventional BCS model [6]. In the underdoped regime the disagreement among researchers is complete and more careful research is needed to clarify the controversies. In the underdoped to optimally doped samples, majority of experiments indicate a dominant dwave symmetry [3]. There seem to be, however, some notable exceptions [3]: there is no evidence for d-wave component in the electron doped cuprates (Maryland Center), Sharvin experiments on LSCO give a finite minimum gap (Deutscher et al [3]) and electronic Raman experiment on Hg-2201 compound (Sacuto et al [3]) is not compatible with d-wave but rather with extended s-wave (with nodes). Only few experiments were performed systematically in the overdoped regime [15,16]. While the main characteristics of the (anomalous) transport in cuprates seem to be well established [18,21], several recent results [22-24] pose a new challenge to our understanding. Slightly underdoped, disordered Sr2RuO4–y, superconducting (TC=0.9K) perovskite without Cu, exhibits linear resistivity over three decades of temperatures [22,25], up to 1050K, yet the temperature dependence of the Hall coefficient is similar to what was measured in cuprates [22]. This suggests that the linear temperature dependence of resistivity is not an exclusive signature of the anomalous normal state of high-Tc cuprates but rather of layered oxides in general, especially single layer perovskites, possibly independently of the magnitude of the superconducting temperature. What is really striking is that very clean Sr2RuO4 crystals, grown by zone melting, show ‘usual’ T2 term at low temperatures [26] implying that the linear resistivity can be “introduced” by the growthinduced disorder. Note that deliberately introduced disorder (by electron bombardment of samples [15,16,27]) introduces the ‘pseudogap’ into ARPES spectra of optimally doped BSCCO-2212 samples which show no ARPES pseudogap before bombardment [27]. While in-depth analysis requires more experimentation, obviously the disorder plays some role in the appearance of the pseudogap [16]. The ongoing pioneering photoemission experiments on in-situ grown films (other than BSCCO-2212 compound) may provide better insight to such controversies [28]. Let us also note that in single crystals of T1-2212 [Tc =111K exhibits ‘usual’ linear behavior and follows generally metallic-like, positive slope. However, there is a clear crossover of to semiconductor-like behavior close to Tc and, for the first time, above 500K. Under high pressures (<15 kbar) the magnitude of strongly decreases, yet slope does not change [23]. That suggests pressure independent out-of-plane mechanism like in resonant tunneling in quasi-one-dimensional organic conductors, proposed by Weger [24]. Above 500K the hopping is activated hence the measured crossover in [23]. In general, the out-of-plane transport in layered oxides is not settled, partly due to very different theoretical approaches [18-26]. CONCLUDING REMARKS

As can be seen from other contributions in this workshop, new superconducting phases are still being discovered [29] so it is premature to propose very definite conclusions in this field. Even more so as there is now some new evidence for the co-existence of superconductivity and magnetism in oxides. This may provide further possibilities for creative nanoengineering of new functional heterostructures. As there are many puzzling

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results and open questions related to understanding of high-Tc and related oxides, we still need many more systematic experiments on very carefully prepared and characterised samples of both, under- and over- doped, films and crystals. Moreover, to achieve mature oxide nano-engineering technology we have to demonstrate all milestones discussed in the introduction, above all – the complete control of the local doping and a full mastery of all length scales in the material. In short we have to be able to alter at will the electronic dispersion relation, E(k), within oxide heterostructure. Such work is currently in progress worldwide [4] and there are oxide electronics companies [8] that are developing promising oxide heterostructures and new electronic devices. “Future History” of Superconductivity: New Millennium Perspective

Figure 4. ‘Future history’ of high-Tc superconductivity: Note that superconducting materials above the

boiling temperature of water, colossal superconductors with superconductor technology at ambient temperatures.

are required for a genuinely useful

However, at a dawn of the new millennium, it is probably important to emphasize that, whatever the progress in understanding of physics, we still need new, colossal superconductors with Tc 450K, rather than 300K as generally assumed. Tc 450K is needed as most devices usually operate at temperatures roughly 2/3 of Tc [6]. For commercial ambient technology we do require such colossal superconductors. As we progress in quantum-nanoengineering of layered oxides and the controlled layer-by-layer epitaxial growth of new phases we will eventually create new artificially layered superconducting oxides (ALSO), and ultimately, in an analogy with manganites (that are very similar to cuprates) - first the giant and ultimately the colossal superconductors (see Figure 4). Although it may seem optimistic, let us note that are no known fundamental physics limits to the appearance of superconductivity up to, at least, Debye temperature range i.e. ~500K. Therefore, the opportunity and the means to realize such remarkable materials and technology are within reach. 438

ACKNOWLEDGEMENTS I gratefully acknowledge important contributions of my numerous friends and colleagues, especially of all co-authors listed in references 27 and 28. I also acknowledge the support by the EPFL and the Swiss National Science Foundation. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

J. G. Bednorz and K.A. Müller, Z Phys. B 64, 189 (1986). B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics , J. Wiley (1991). J. Bok, G. Deutscher, D. Pavuna, S.A. Wolf (eds.) The Gap Symmetry and Fluctuations in High-Tc Superconductors, Proc. NATO-ASI B371 Kluwer (1998). D. Pavuna and I. Bozovic (eds.), Superconducting and Related Oxides : Physics and Nano-engineering I, II, III, IV SPIE volumes 2058, 2697, 3481, 4058 SPIE, Bellingham (1994, 1996, 1998, 2000). A. Bianconi (ed.), Stripes I & II, special issue of J. of Superconductivity (1997, 1999). M. Cyrot and D. Pavuna, Introduction to Superconductivity and High-Tc Materials, World Scientific, London, Singapore, New Jersey (1992). S. Barnes et. al. (eds.) High Temperature Superconductivity (eds..) CP483 American Institute of Physics (1999). Ivan Bozovic, Oxxel GmbH (Bremen), private communication, see www.oxxel.de for further details on atomic layer-by-layer MBE growth and promissing devices. See the Proceedings of the Klosters’ (April 1-10) Millenium Superconductivity Symposium, J. of Superconductivity incl. Novel Magnetism vol. 13 (5) (2000). J. C. Phillips, Phys. Rev. Lett. 59 1856 (1987) and the paper in this volume. G. Deutscher, Nature 397, 410 (1999) G. S. Boebinger et al, Phys. Rev. Lett. 77 5417 (1996). J. Friedel, Physica C 153-155, 1610(1988). B. Batlogg and C. Varma, Physics World 13 (2), 33 (2000) and refs therein. I. Vobornik, ‘Investigation of the Electronic Properties and Correlation Effects in the Cuprates and in Related Transition Metal Oxides’, D.Sci. Thesis, EPFL (1999). D. Pavuna, I. Vobornik, G. Margaritondo, J. of Superconductivity 13 (5) 749 (2000). J. Bok. and J. Bouvier in ref. 2 and ref. 9. P.W. Anderson, Theory of High-Tc Superconductivity in Cuprates, Princeton (1997). D. Ariosa and H. Beck, Int. J. Mod. Phys. B 13 3472 (1999). B. Normand, D.F. Agterberg, T.M. Rice, Phys. Rev. Lett. 82 (1999). N.P. Ong, Science 273, 321 (1996) and references therein. H. Berger, L. Forró and D. Pavuna, Europhysics Letters 41 (5), 531 (1998). J.P. Salvetat et. al., unpublished (2000); L. Forró, Int. J. Mod. Phys. B 8, 829 (1994). M. Weger, J. Phys. Colloq. C 6, 1456 (1978). D. Pavuna, L. Forró, H. Berger, p. 412 in ref. 7. A.P. Mackenzie et.al. Phys. Rev. Lett. 76 3786 (1996). I. Vobornik, H. Berger, D. Pavuna, M. Onellion, G. Margaritondo, F. Rullier-Albenque, L. Forró, M. Grioni, Phys Rev. Lett. 82, 3128 (1999). M. Abrecht, T. Schmauder, D. Ariosa, O. Touzelet, S. Rast, M. Onellion and D. Pavuna, in print in August issue of Surface Review and Letters (2000). See articles by I. Felner et. al., and by J. Dow et. al. in this NATO-Kluwer volume.

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DESIGNING PROTEIN STRUCTURES

HAO LI1, CHAO TANG2, and NED S. WINGREEN2 1

Department of Biochemistry and Biophysics, University of California, San Francisco, 513 Parnassus Avenue, Box 0448, San Francisco, CA 94143-0444

2

NEC Research Institute, 4 Independence Way, Princeton, NJ 08540

Recent work on lattice models of protein folding has identified a designability principle for structures [1,2]. It is of course well known that some sequences make better behaved

proteins than others. The designability principle suggests that some structures make for better behaved proteins as well. A goal of current research is to apply this principle to the design of novel protein folds. Here we briefly discuss the designability principle and its application to the design problem. Essential characteristics of natural, water-soluble proteins are thermodynamic stability of the ground state, stability against mutations of the amino-acid sequence, and relatively fast, unassisted folding into the ground state. It is known that only a small fraction of all possible polypeptide sequences have all these properties [3]. Natural protein sequences have clearly been carefully selected. There is considerable evidence as well that natural protein structures have been selected from among all possible folds of a polypeptide chain. There are estimates that out of the huge number of possible folds, only 2000 are in use by living organisms [4]. Are some structures more apt to give protein-like behavior than others? In our attempt to answer this question, we have focused on the concept of “designability”. Specifically, the designability of a structure is the number of sequences which have that structure as their unique ground state. Designability as a concept can therefore be applied to real polypeptide chains as well as to lattice models of proteins. In lattice-model studies we always find a small group of structures with designabilities much higher than the average. Furthermore, these highly designable structures are the ground states of the sequences with the most protein-like properties, and the structures themselves have striking geometrical regularities reminiscent of real proteins. It is this connection between the highly designable structures and the most protein-like sequences which we refer to as the principle of designability. Since it is very difficult to determine the designability of real polypeptide structures, we have concentrated instead on simplified lattice models. In these models, a “protein” is a

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

441

chain of monomers, restricted to fall on the sites of a 2D or 3D square lattice (see Fig. 1). A structure is the path a chain takes through the sites of the lattice. For many of our studies, we considered only the compact structures for chains of a given length. For example, the compact structures of a chain of 27 monomers in three dimensions are all paths filling a 3x3x3 cube (Fig. l(a)). Instead of the twenty amino acids appearing in nature, our model sequences consist of only two types of monomers: H for hydrophobic, and P for polar. A sequence is therefore simply a string of H’s and P’s, e. g., HHPHHHPPPHPPPHH..., and

models of this type are called HP models [5].

Figure 1. Structures with the largest designability in 3D 3x3x3 (A) and 2D 6x6 (B) lattice models. The black balls represent H monomers and the gray balls represent P monomers for particular HP sequences which fold into the structures.

The thermodynamic and dynamic properties of a lattice-model protein depend on the choice of an energy function. That is, an energy must be defined for each sequence in each possible structure. We focus here on a theoretically convenient choice which reproduces the behavior of more realistic energy functions. Specifically, the sites of each compact structure are divided into two types, surface and core. For example, in Fig. l(b), the innermost square of 16 sites is defined to be the core. The energy of a sequence folded into a given compact structure is taken to be -1 times the number of hydrophobic (H) monomers falling on core sites. This definition allows a particularly convenient geometrical formulation of the designability of structures.

442

Since a structure consists of only two types of sites, each structure can be represented as a string of 1s and 0s, 1 for a core site and 0 for a surface site. Similarly, each sequence can be represented as a string of 1’s and 0’s, 1 for a hydrophobic (H) monomer and 0 for a polar (P) monomer. The energy of a sequence folded into a structure is therefore given by

This energy can be rewritten as

Since the last two terms are effectively constants is constant for a given sequence and is constant for compact structures, which all have the same number of core sites - only the first term determines which structure will be the ground state of a given sequence. The important observation is that the first term in the energy, is a Hamming distance in the space of strings of 1s and 0s. Each vertex in this string space corresponds to a possible sequence, and a subset of the vertices correspond to the allowed compact structures. The energy of a particular sequence folded into a particular structure is just the distance from

the sequence's vertex to the vertex for the structure. The designability of a structure, i. e. the number of sequences with that structure as a unique ground state, is then just the number of vertices which lie closer to that structure than to any other. A schematic depiction of this geometrical construction of designability is shown in Fig. 2. In the space of strings, some regions have a dense population of structures, while some regions are very scarcely populated with structures. It is precisely in the scarcely populated regions of string space that one finds the highly designable structures. It is natural that in regions with very few structures, the volume lying closest to each structure will be large, and, by the geometrical construction, the number of vertices in each volume is just the designability of the structure. Within this framework it is easy to see how designability is related to mutational and thermodynamic stability. By definition a sequence which folds into a given structure is “stable” under mutation if the mutated sequence also folds uniquely into the same structure. Since the designability of a structure is just the total number of associated sequences, it is obvious that high designability corresponds to high mutational stability. The additional insight provided by the geometrical construction is that since the sequences associated with a given structure occupy a connected volume in the space of strings, they are generally all connected by point mutations. The thermodynamic stability of sequences associated with highly designable structures follows from the small density of competing structures. For the sequences associated with any one structure in string space, the nearby structures are low lying excited states. Since the highly designable structures fall in regions with a small density of neighbors, their associated sequences have a small density of low-lying excited states. This is exactly the requirement for thermodynamic stability of the sequences. Fast folding, in turn, correlates well with thermodynamic stability. Finally, what produces the geometrical regularities observed in highly designable structures? A recent study [6] has demonstrated that for 6x6 structures, as in Fig. l(b), designability is strongly correlated with mirror symmetry. A plausible explanation is that designable structures are made from repeated designable substructures. Current work aims at pinning down this connection. The discussion so far has centered on lattice models of proteins. How could the principle of designability be tested, and perhaps applied, in the context of real polypeptide chains? a sequence specifically for this structure, and demonstrate that the sequence folds into the structure in the laboratory. Currently, attempts to design novel protein structures by other schemes have not been successful. The designability principle helps clarify why design has proven so difficult. In 443

Figure 2. Schematic plot of string space. The shaded volume contain the sequences which lie closer to the

central structure than to any other, and hence have that structure as their unique ground state. Degenerate structures have the same strings and hence cannot be the unique ground state of any sequence.

the lattice models, the vast majority of structures fall in densely occupied regions of structure space. The sequences associated with these structures are always thermodynamically unstable because of their high density of low-lying excited states. While real polypeptide chains cannot simply be mapped onto a space of strings, one can still argue that a space of structures exists, with some regions densely occupied and some regions sparsely occupied. Previous attempts to design novel structures have, unfortunately, focused on optimizing the sequence for an arbitrary chosen target structure. But an arbitrarily chosen structure is generally unstable, even for the optimal sequence, because it is surrounded by a high density of competing structures. The principle of designability makes it clear that one must first select a highly designable structure as a target. Current work is aimed at theoretically generating novel protein structures of high designability. There are several foreseeable difficulties. First, real polypeptide chains have a large number of degrees of freedom. A typical globular protein consists of 100-200 amino acid residues, and each residue has two free bond angles. Second, the energy function for real polypeptide chains is not well known. Indeed, even the approximate functions used in simulations are very complicated, involving van der Waals interactions, Coulomb interactions, hydrogen bonds, and hydrophobic interactions. Nevertheless, we believe a balance can be achieved between fidelity to nature and computational practicality. Specifically, the degrees of freedom of the chain can be greatly restricted while still allowing realistic backbone configurations. Furthermore, the designability 444

of structures is in general not very sensitive to the choice of energy function. In the event, we anticipate being able to generate short, i. e. 20-30 monomer, highly designable protein segments in the near future. The generation of highly designable large structures remains challenging, but will likely prove possible as well.

REFERENCES 1.

2.

Li, H., Helling, R., Tang, C., and Wingreen, N. (1996) Emergence of preferred structures in a simple model of protein folding, Science 273, 666-669. Li, H., Tang, C., and Wingreen, N.S. (1998) Are protein folds atypical?, Proc. Natl. Acad. Sci. USA 95,

4987-4990. 3.

4. 5.

6.

Davidson, A.R. and Sauer, R.T. (1994) Folded proteins occur frequently in libraries of random amino acid sequences, Proc. Natl. Acad. Sci. USA 91, 2146-2150. Chothia, C. (1992) Proteins. One thousand families for the molecular biologist, Nature 357, 543-544; Govindarajan, S., Recabarren, R., and Goldstein, R.A. (1999) Estimating the total number of protein folds, Proteins 35, 408-414. Dill, K.A. (1985) Theory for the folding and stability of globular proteins, Biochemistry 24, 1501-1509; Lau, K.F. and Dill, K.A. (1989) A lattice statistical mechanics model of the conformational and sequence spaces of proteins, Macromolecules 22, 3986-3997. Wang, T., Miller, J., Wingreen, N.S., Tang, C., Dill, K.A., unpublished.

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PARTICIPANTS

J. Acrivos Department of Chemistry San Jose State University San Jose, CA 95192-0101 USA [email protected]

A.J. Coleman Dept. of Mathematics and Statistics Queen’s University Kingston, ON K7L 3N6 CANADA [email protected]

J. Banavar Department of Physics Pennsylvania State University University Park, PA 16802 USA [email protected]

V. Dallacasa Dept. of Science and Technology (Mat. Analysis Lab.) University of Verona Strada Le Grazie I-37134 Verona ITALY

A. Bianconi Unità INFM Dipartimento di Fisica Università di Roma La Sapienza 00185 Roma ITALY [email protected]

[email protected]

Punit Boolchand University of Cincinnati Electrical & Computer Engineering Dept. PO Box 210030 Cincinnati, OH 45221-0030 USA [email protected] T.G. Castner Department of Physics University of Massachusetts, Lowell 1 University Ave. Lowell, MA 01854 USA [email protected]

Martin Dove Department of Earth Sciences Cambridge University Downing Street Cambridge CB2 3EQ ENGLAND [email protected] John Dow 6031 East Cholla Lane Scottsdale, AZ 85253 USA [email protected] J. Dyre IMFUFA Roskilde University POB 260 DK-4000 Roskilde DENMARK [email protected]

447

M. Dzugutov Department of Numerical Analysis Royal Institute of Technology 100 44 Stockholm SWEDEN

[email protected]

D. Haskel Argonne National Laboratory Exp. Facilities Div., Adv. Photon Source APS, Bldg. 401 Argonne, IL 60439 USA

[email protected] Hellmut Eckert Institut für Physikalische Chemie Westfälische Wilhelms Universität Münster Schlossplatz 7 D-48142 Munster GERMANY [email protected] A.L. Efros

Department of Physics University of Utah 201 JFB Salt Lake City, UT 84112 USA [email protected]

Jan A. Jung Department of Physics University of Alberta Edmonton, AB T6G 2J1 CANADA [email protected] Richard Kerner Lab. de Gravitation et Cosmologie Relativistes Univ. P. & M. Curie – BC 142. 4 P1. Jussieu 75252 Paris 05 FRANCE [email protected]

1. Felner Racah Institute Physics The Hebrew University Jerusalem 91904 ISRAEL [email protected]

F.V. Kusmartsev

D. Georgiev Dept. of ECECS University of Cincinnati Cincinnati, OH 45221-0030 USA [email protected]

G. Lucovsky Department of Physics North Carolina State University Raleigh, NC 27695-8202 USA [email protected]

G.N. Greaves Department of Physics The University of Wales Aberystwyth Ceredigion SY23 3BZ UNITED KINGDOM [email protected]

M. Micoulaut Lab. Physique Theor. Liquides Univ. P. & M. Curie, Boite 121 4 Pl. Jussieu 75252 Paris Cedex 05 FRANCE mmi@ccr .jussieu.fr

J. Haase 2. Physikalisches Institut Universität Stuttgart Pfaffenwaldring 57 70569 Stuttgart GERMANY [email protected]

G.G. Naumis Instituto de Fisica UNAM A.P. 20-364 0-1000 Mexico D.F. MEXICO [email protected]

448

School of MAP Loughborough University Loughborough, Leics LE11 3TU UNITED KINGDOM [email protected]

D. Pavuna Department of Physics - IPA Ecole Polytechnique Federale de Lausanne CH – 1015 Lausanne SWITZERLAND [email protected] J.C. Phillips Lucent Technologies Bell Labs Innovations

600 Mountain Ave. Murray Hill, NJ 07974-0636 USA [email protected] J.-L. Pichard Dept. L’Etat Condense, CEA Cent. Etud. Saclay F-91191 Gif-s.-Yvette FRANCE [email protected]

John Wagner Physics Department University of North Dakota Grand Forks, ND 58201 USA [email protected]

Y. Wang Department of Physics Osaka University 1-1 Machikaneyama, Toyonaka Osaka 560-0043 JAPAN [email protected] M. Watanabe Nanostructure Physics, KTH Lindstedtsvagen 24 SE-100 44 Stockholm SWEDEN [email protected]

E.A. Stern Department of Physics

N. Wingreen NEC Research Institute 4 Independence Way

University of Washington,

Princeton, NJ 08540

Box 351560 Seattle, WA 98195 USA

USA

[email protected]

R. Zallen Department of Physics Virginia Tech

M.F. Thorpe Physics and Astronomy Dept. Michigan State University East Lansing, MI 48824 USA [email protected]

[email protected] .nec.com

Blacksburg, VA 24061-0435 USA [email protected]

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INDEX

Activation energy, 102, 106 Adiabatic approximation, 391 Agglomeration model, 172 Agglomeration theory, 144, 157, 171 Alternative gate dielectrics, 204 Amino acids, 442 Amorphisation, 226, 233 Amorphous metals, 38 Amorphous solid, 161 Anderson localization, 265 Anderson transition, 263 Anharmonic, 105 Antiferromagnetic correlations, 397 Antiferromagnetic spin fluctuations, 413 Applied stress backbone, 57 Atomic scaffolding, 225 Average cluster size, 178 Average coordination, 123 Avrami relation, 232 Berlinite, 237 Binary glass, 65 Binodal points, 179 Boltzmann factors, 174 Bond bending, 47, 70 Bond bending networks, 50 Bond ionicity, 190, 200 Bond stretching, 47 Bragg diffraction, 209 Bragg peaks, 209 Broken symmetry, 3 Busbar geometry, 56 Calorimetry, 72, 143 Central Force Network, 46, 49 Ceramics, 232 Chalcogenide glasses, 65–66, 123, 161, 172 Channel openings, 226 Charge domains, 362 Charge transfer, 332 Chemical ordering, 135 Chemical phase separation, 197

Chemical threshold, 124 Clathrasil, 238 Clausius-Mossotti equation, 265 Collapse, 236 Colossal magnetoresistivity, 209 Compressibility, 235 Connectivity, 60, 65, 181 Conductivity threshold, 40 Connectivity percolation, 1, 44 Constraint theory, 162 Constraints, 4, 46 Continuous random networks, 43, 192, 206 Continuous transition, 2 Correlated electrons, 247 Correlation length, 291 Correlation matrix, 23, 27 Corrugated, 379 Coulomb gap, 247, 250, 264, 366 Coulomb glass, 248 Coulomb interactions, 264, 267, 444 Covalent bonded atoms, 162 Covalent glasses, 43 Covalent network, 163 Critical exponents, 292 Critical point, 286 Critical temperature, 376, 414 Critical volume fraction, 38 Crossover from strong to weak disorder, 248 Crystalline silicon, 190 Crystallization, 143 Cuprate perovskites, 375 Debye behavior, 102 Debye relaxation, 102 Defects, 332 Degenerate structures, 444 Density matrices, 23 Devices, 432 Dielectric constant enhancement, 200 Dielectrics, 190 Diffuse scattering, 213 Diffusion coefficient, 117

451

Disordered systems, 1

Displacive phase transformation, 312

Displacive structural transformation, 313 Distortions, 332

Dopants, 323 Doped antiferromagnet, 390, 398 Doped semiconductors, 291 Dynamical ergodicity, 112

Heteroepitaxy, 431 High-temperature superconductivity, 403, 413 Hopping conductivity, 364, 366 Hydrogen, 350 Hydrogen bonds, 444 Hydrophobic interactions, 444 Hyperfine coupling, 414 Ice I, 237

Edwards-Anderson order parameter, 249

Impurities, 433

Effective medium approximation, 1 Einstein relations, 277, 282 Elastic energy, 106 Elastic properties, 55 Electron gas, 383 Electron glass, 248 Electron strings, 389

Impurity bands, 1 Impurity dielectric susceptibility, 292 Inelastic neutron scattering, 362 Inelastic scattering cross-section, 212

Energy barriers, 116

Inhomogeneity, 292 Insulator-metal transition, 247, 323 Integral operator, 23 Interface properties, 189 Interface, 200 Interfacial fixed charge, 200 Interfacial limitations, 204 Intermediate phase, 1, 5, 43, 50, 52, 54, 65, 74 Ion channeling, 313

Enthalpic rigidity, 66

Ionic solids, 398

Electronegativity, 190

Electron-electron interaction, 247, 291 Electronic properties, 433 Electron-phonon system, 389 Elementary building blocks, 177

Epitaxial growth, 433 Ergodic diffusion, 114

Ergodicity breaking, 119

Irreversibility, 238 Isothermal magnetization, 347 Isotopes, 293

Exotic superconducting phase, 375

Expansion coefficient, 235

Jahn-Teller, 323, 392 Josephson junctions, 432

Fermat’s Last Theorem, 1–2 Fermi energy, 319

Fermi liquid, 2 Fermi surface, 12, 32, 382 Ferroelastic nanodomains, 311, 316, 321 Ferroelasticity, 311–312 Ferroelectricity, 209

Filamentary metals, 10 Filamentary, 1

Finite-temperature scaling, 293 First-order transition, 2 Fixed positive charge, 199 Floppy, 85 Floppy modes, 44, 89, 161–162, 168, 242 Floppy regions, 88 Floppy units, 88 Gate dielectric materials, 205

Generic networks, 45 Geometrical regularities, 441 Germanium, 291 Glass preparation, 86 Glass transition, 71, 101, 254 Glass transition temperature, 65, 123, 143, 175 Glassy state, 1 1 1

Global chaotic connectivity, 113 Granular metals, 367 Hall number, 272 Hartree-exchange, 281

452

Kauzmann paradox, 111 Knight shifts, 414

Lagrangian, 44 Lagrangian bonding constraints, 66 Landau-Ginzburg theories, 2 Lattice instabilities, 33 Lattice models, 442 Layered oxides, 431 Liquid-glass transition, 111, 143 Local bonding, 189

Local distortion, 328 Local structure, 209–210 Localization, 358, 390

Localization length, 248, 291 Localized states, 11 London penetration length, 375 Loopless networks, 58 Low temperatures, 295 Luminescent ions, 135

Magic angle spinning, 124 Magnetic behavior, 350 Magnetic moments, 390 Magnetic vortices, 18 Manganese perovskites, 321 Manganites, 311 Maximal homogeneity, 173 Mean coordination, 56

Mean-field, 66 Mean-field constraint theory, 85 Mechanical rigidity, 242 Medium range order, 127, 157 Meissner state, 353 Melting, 233, 235

Polaronic charge density wave, 384 Polarons, 389 Polymeric networks, 65, 135

Polypeptide chains, 441 Porosity, 225 Protein structures, 441

Metal-insulator transition, 263, 291

Pseudogap, 431

Metallic heterogeneous phase, 377 Minimal local fluctuations, 174 Mode-coupling theory, 112 Modulus, 313 Molecular dynamics, 117 Molecular orbitals, 377

Pseudoground states, 254

Mössbauer spectra, 138 Mössbauer spectroscopy, 344 Mott law, 248 Muon spin polarization, 409 Muon spin rotation relaxation, 409

Quadrupole interaction, 414

Quantum computers, 1, 3 Quantum critical point, 385 Quantum mesoscopic stripes, 384 Quantum percolation, 12, 357 Quantum phase transition, 291 Quantum wires, 375 Quartz, 237 Quasi-particles, 319

Raman scattering, 75, 85–86 Raman shifts, 314

Nanodomains, 329 Nanofilaments, 332 Network, 65 Network connectivity, 71, 73, 143 Network constraints, 193 Network glasses, 1, 4, 43, 146 Network stiffening, 123 Network stress, 73 Neutron diffraction experiments, 315 Neutron scattering, 321, 390, 426

Raman spectroscopy, 156 Random close packed ionic structures, 192 Random networks, 66, 174 Rayleigh-Ritz, 25 Re-assembly, 243 Reduced density matrix, 23 Relaxation processes, 111 Relaxation times, 103

Neutron-transmutation doping, 292

Reversibility, 99, 238

Non-Arrhenius behavior, 104

Rigidity, 85, 168, 181, 225 Rigid units, 88

Non-crystalline solids, 65 Non-Debye relaxations, 106 Non-ergodic diffusion, 115

Non-ergodic dynamics, 111 Nuclear magnetic resonance, 124, 414 Nucleation, 232

Rigidity percolation, 44, 51 Rigidity threshold, 67, 243 Rigidity transition, 66, 71, 163, 171 Ring distribution, 181

Scaling exponent, 268 Orbital shift, 414 Oxygen, 344 Oxygen ordering, 332 Oxygen vacancies, 433 Pair-breakers, 403

Pairing mechanism, 413

Pauling electronegativity, 190 Percolating charged strings, 390

Percolation, 39, 358, 390, 398 Percolation processes, 39

Percolation theory, 123 Percolation threshold, 399 Percolation topology, 40 Perovskite, 311 Perovskite oxides, 311 Phase diagram, 293, 380, Phase separation, 33 Phase stability, 189 Phase transitions, 19, 235 Phonons, 398 Photomelting, 79

Scaling relationship, 115 Scaling theory, 8 Scattering measurements, 219 Screening, 247

Second quantization, 24 Self-organization, 43, 51, 157, 375, 436 Self-organize, 65 Self-organized networks, 55, 71 Self-trapped excitons, 389 Self-trapped state, 389 Shear modulus, 233 Short-range order, 125 Silicate alloys, 197 Silicate glasses, 172 Slow relaxation, 254

Solidity, 101 Solidity length, 104 Specific heat, 9

Spectral diffusion, 255 Spin correlations, 363 Spin pseudogap, 18 Spin shifts, 414

453

Spontaneous vortex phase, 353

Stability, 62 Stiffness transition, 85, 90 Stressed rigid phase, 76

Transition temperature, 161 Transport properties, 14, 263, 359 Twins, 433 Two-dimensional lattices, 38

Stress-free, 65 Stress-free oxide network, 66 Strings in manganites, 400

Universality class, 292

Stripe formation, 381 Stripes, 414 Structural relaxation, 87

Vacancy-ordering, 362 van der Waals interactions, 444

Superconducting gap, 432 Superconducting networks, 61 Superconducting phase, 382 Superconductive transition temperatures, 2 Superconductivity, 1, 33, 209, 331, 341, 362, 369, 375,

403, 408, 413, 436

Variable range hopping, 248, 253, 292 Vibrational properties, 88 Viscosity, 233 Viscous liquids, 101, 103

Weak ferromagnetism, 341

Supercooled liquids, 111 Superstripes, 375 Swiss cheese, 40

Wigner localization, 382 Window glass, 171 Wrong bond, 99

Target structure, 444

X-ray absorption fine structure, 216 X-ray diffraction, 243, 333

Ternary systems, 135 The Pebble Game, 48 Thermal relaxation, 85 Thermodynamic entropy, 115 Thermodynamic fluctuations, 254

Young’s modulus, 7, 65

Yukawa potential, 265

Thermodynamic stability, 443 Threshold, 389

Zeeman splitting, 414

Time-reversal symmetry, 292

Zeolite collapse, 242

Transition metal silicates and aluminates, 190 Transition region, 85

Zeolites, 225 Zero-frequency modes, 48

454