Lattice effects in high temperature superconductors

~

Progress in Materials Science Vol. 38, pp. 359-424, 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved. 0079-6425/94 $26.00

Pergamon

0079-6425(94)00009-3 L A T T I C E E F F E C T S IN H I G H T E M P E R A T U R E SUPERCONDUCTORS T. Egami* and S. J. L. Billinge~f, *Department of Materials Science and Engineering and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104-6272, U.S.A. tLos Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.

CONTENTS 1. INTRODUCTION 2. CRYSTALLOGRAPHICSTRUCTURE 2.1. Composition o f Superconducting Oxides 2.2. Structural Units 2.3. Crystallographic Structure o f Various H T S C Oxides 2.3.1. YBa2Cu307_ ~ 2.3.2. YBa2Cu40 s 2.3.3. La2 _ x (Sr, Ba )x CuO 4 2.3.4. N d 2_ x Cex Cu04 2.3.5. A , N2Mm_ICumO~+,+ 2 2.3.6. (Ba l xKx)Bi03 and Ba(Bi I_xPb~)03 2.3.7. Other H T S C oxides 2.4. Local Structure and Atomic Interaction 2.4. I. Nominal valence o f ions 2.4.2. Environment o f Cu ion 2.4.3. Environment o f 0 ion 2.5. Structural Modulation and Superlattice 3. LATTICE VIBRATION 3.1. Phonon Dispersion 3.2. Electron-Phonon Interaction and Superconductivity 3.3. Isotope Effect 4. LOCAL DEVIATIONS FROM CRYSTALLOGRAPHICSTRUCTURE 4.1. Debye-Waller Factor 4.2. Diffuse Scattering 4.3. Local Probes 4.3.1. E X A F S 4.3.2. Pulsed Neutron P D F analysis 4.3.3. Other evidence o f local displacements 4.4. Comparison with Crystallographic Analysis 4.4.1. YBCO-123 4.4.2. L S C 0 - 2 1 4 4.4.3. Tl-2212

360 362 362 363 364 364 364 364 365 366 367 367 368 368 370 371 371 372 372 373 376 377 378 380 382 382 384 388 389 389 389 390

~;Present address: Department of Physics and Astronomy, Michigan State University, East Lansing, MI 488241116, U.S.A. 359

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4.5. Modes o f Local Atomic Displacements 4.5.1. Apical oxygen 4.5.2. In-plane oxygen 4.5.3. Oxygen ions in other layers 4.5.4. Displacement o f other ions 4.5.5. Spatial range o f correlations 4.5.6. Composition dependence 5. LATTICEANOMALIESNEAR Tc 5.1. Lattice Dynamics 5.1.1. Phonons 5.1.2. Ion channeling 5.2. Structural Anomalies 5.2. I. EXAFS results 5.2.2. Neutron scattering 5.3. Other Lattice Anomalies 5.4. Nature o f the Lattice Anomalies 5.4.1. Magnitude o f the effect 5.4.2. Dynamics o f the anomalies 5.4.3. Coupling to electrons

6. STRUCTURALPHASETRANSITION 6.1. Structural Phases in La2_x(Sr, Ba)xCuO 4 6.2. H T T Phase 6.3. L T T Phase 6.4. Local Structure 7. IMPLICATIONSOFTHELATTICEANOMALIES 7.1. Lattice Anharmonicity and Double-well Potential 7.2. Lattice Polaron 7.3. Magnetic Mechanisms 7.4. Effect of Local Distortion on the Electronic Band Structure 7.5. Two-component Model 7.5.1. Lattice polaron scenario 7.5.2. Local pairing and superconductivity 7.6. Nature o f Electron-Lattice Interaction in HTSC Solids 7.6. I. Effect o f electron correlation 7.6.2. Bipolaron formation

8. CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

391 391 391 391 391 391 392 392 392 392 392 394 394 394 398 399 400 401 401 402 403 403 405 407 409 409 409 410 412 412 413 415 416 416 417 418 419 419

1. INTRODUCTION The discovery o f high-temperature superconductivity ( H T S C ) in 1986 o) was perhaps the m o s t significant achievement in solid state sciences in the late 20th century, with p r o f o u n d technological implications. Superconductivity until that time was a rather esoteric p h e n o m e n o n at very low temperatures with little or no real impact on technology. Since the late 70s there have been some attempts to use superconductors, or m o r e specifically Josephson junctions, for high-speed computers, but the efforts did n o t succeed due to technical difficulties. The newly discovered H T S C oxides, with the critical temperature, To, far exceeding 100 K, have opened up vast opportunities for real technological applications o f this amazing p h e n o m e n o n o f zero resistivity. The feverish drive for research in the years following the discovery has a g o o d reason. Partly because o f the unprecedented rate and intensity o f research in this field, there is a very large a m o u n t o f literature already published on the subject which is often confusing and contradictory. The purpose o f this article is to interpret m u c h o f this information, regarding

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the role of atomic structure on the HTSC phenomenon, within a framework of relatively simple chemical and physical ideas. We take as a starting point the crystal structures of these materials, but soon focus on the local deviations from the crystallographic structure. We will then discuss experimental observations concerning the electronic structure and doping, and relate them to observations of lattice effects, such as phase transformations and structural modulations, in terms of the bonding and crystal chemistry of these materials. Finally, we speculate on a very direct role of the lattice on the properties of these materials, taking into account the presence of electron correlations as well as the unique chemistry and structure of these oxides. This article has primarily been written with general readers, such as graduate students, in mind rather than the experts in the field. Thus, parts of the discussion are quite introductory and may be redundant for more advanced readers. However, we hope that it will still be interesting to a wider readership and stimulate further research in this intriguing field. It is not meant as an exhaustive survey of the literature. Consequently, references have been kept to a minimum and we apologize to authors whose work is omitted. High temperature superconducting (HTSC) oxides are fundamentally distinct from conventional metallic superconductors. Not only do they have surprisingly high superconducting transition temperatures, but they also show a number of totally unexpected properties in the normal, non-superconducting state such as a strongly temperature dependent Hall coefficient. In fact, many of their properties share similarities with those of the insulating oxides from which they are derived. It is striking that superconductivity with very high Tcs has only been found, to date, in the presence of specific chemical and structural motifs: sheets of covalently bonded copper and oxygen, separated by ionic intergrowth regions. These units are closely derived from perovskite, rocksalt and fluorite structures which are commonly found among covalently and ionically bonded solids, but are a far cry from close-packed, metallically bonded, conventional superconducting metals. We will discuss the importance of this bonding to the properties of these materials. In complex oxides the electronic structure, and therefore the properties, is delicately dependent on the atomic structure. In ionically bonded regions atoms are charged and strong electric field gradients exist. These are not fully screened because of low mobile carrier densities. Small variations in atom position will, thus, have a significant effect on the electron density distribution and the band structure. This is borne out by band structure calculations. In covalently bonded regions, the bonding is highly directional. In this case, small variations in atom position will intermix atomic orbitals differently in the bonding orbitals and will change the nature and symmetry of electronic wavefunctions. Thus, one might expect that changes in atomic structure--which may be long range, such as phase transitions, or local as we discuss--will have an effect on the electronic system and the properties. The lattice enters into conventional superconductivity since electrons become paired by sensing the polarization of the lattice due to phonons. ~2)Studies of lattice dynamics are, thus, also centrally important to understanding the properties of HTSC oxides. However, as we will discuss, many results are not easily interpreted in terms of linear electron-phonon coupling theories and suggest that the interaction is more complicated. There is also experimental evidence that suggests that there is an anomalous change in the lattice dynamics at the superconducting Tc which may suggest a more fundamental involvement of the lattice in the microscopic mechanism of high-To superconductivity. Related to this is evidence from local structural probes that the local structure deviates from the average crystal structure. The crystallographic structure describes only the periodic components of the actual structure,

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while the real, local structure apparently is slightly different from this, containing aperiodic deviations. The presence of such deviations, though not at present fully characterized, could be caused by non-linear, localized, electron-lattice interactions. These observations and ideas are discussed in Sections 3, 4 and 5. HTSC oxides undergo various structural transitions as temperature and composition are changed. In particular the La2_x(Ba, Sr)xCuO4 system shows complex phase behavior which was suggested by many researchers to be related to the appearance of superconductivity. If a loss of certain symmetry can be connected with the disappearance of superconductivity, it would be a major step forward in an attempt to understand the mechanism of superconductivity. However, the situation is far from clear, let alone the physical implication of the connection. In Section 6 we review some of the experimental results which are more pertinent in examining the effect of structural phase transition on superconductivity. A great deal of interest has been focused on the "strongly-correlated" nature of electrons in these systems. There is little doubt that the properties of the undoped parent compounds are dominated by the tendency of electrons to avoid each other in space-the electron correlations. A number of theories have been developed which attempt to explain the occurrence of superconductivity directly due to these correlations. In this article we discuss the correlations in light of the atomic structure of these materials. In particular the electron correlations significantly modify the response of electrons to the lattice. Lattice deformation affects electrons not only through the direct Coulombic electron-lattice interaction, but also indirectly by promoting carrier localization, or by promoting charge transfer between different electronic states. This is discussed in more detail in Section 7.

2. CRYSTALLOGRAPHIC S':I'RUCTURE 2.1. Composition of Superconducting Oxides In the few years following the discovery of the first HTSC oxide in the La-Ba-Cu-O system(~) a large number of superconducting oxide systems have been found. Many of them contain copper, and indeed all the HTSC oxides with Tc higher than 40 K are cuprates. As is well known, superconductivity in oxides usually occurs near the phase boundary that separates an insulator or semiconductor from a metal. Figure 1(3) shows the concentration dependence of the superconducting transition temperature, To, of La2_ xSrx CuO4. The carrier (hole) concentration, p, is assumed to be equal to x in this case due to an argument given later. Since replacing La 3+ with Sr 2+ changes the carrier concentration, Sr is called the "dopant". When p is zero (undoped) the solid, often referred to as the parent compound, is antiferromagnetic and insulating. When p is small (< 0.07) the solid shows semiconducting behavior judged from the temperature dependence and magnitude of the conductivity, and spin-glass behavior with Cu spins pointing in random directions. When p is sufficiently high (> 0.3) the solid behaves like a normal non-magnetic metal. Superconductivity is observed in the intermediate concentration range at low temperatures, and at temperatures above Tc the solid shows a mixture of semiconducting and metallic behaviors.

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FIG. 1. Schematicphase diagram for oxide superconductor. °) The composition with the maximum Tc is called "optimally doped". When p is more or less than this optimal value the solid is called "overdoped" or "underdoped", respectively. While some of the oxide superconductors are stoichiometric compounds such as YBa2 Cu3 07, it is useful to consider the composition variation in a similar way. Thus YBa2Cu307 is very nearly optimally doped, even though it is stoichiometric, while YBa2Cu307 ,~ with 0.03 < 5 < 0.6 are underdoped. Since the insulating phase adjacent to the superconducting phase shows antiferromagnetic spin correlations, magnetism is suspected to play some role in superconductivity. However, magnetism and superconductivity are usually exclusive of each other. Indeed in the La2_ ~SrxCuO4 system long-range antiferromagnetism is lost at a much lower concentration (p ~ 0.02) than at the onset of superconductivity. Also, even the local magnetic polarization quickly becomes small in magnitude as p is increased. The role of magnetic interaction and magnetic correlation in HTSC phenomena will be discussed later, but it is still poorly understood.

2.2. Structural Units The atomic structure of HTSC solids was studied by various crystallographic methods such as X-ray, neutron and electron diffraction, and the crystallographic structure of these compounds is well known. (*) They are all derived from the perovskite structure (Fig. 2). However, while the original cubic perovskite is made of three-dimensionally interpenetrating CuO2 planes (Fig. 3), most of the HTSC oxides, except for (Baj_,.Kx)BiO 3 and Ba(Bij xPbx)03, are two-dimensional, and are made of layers of CuO2 planes stacked on top of each other. It is widely believed that these CuO2 planes provide much of the electrical conduction and superconductivity. The rest of the structure is subsidiary, and its main role is to transfer charge carriers to the planes. Consequently various properties, such as electrical conductivity, are strongly anisotropic. This two-dimensional nature of the atomic structure is fundamentally important, and not only governs the physical properties but may well be crucial in raising the superconducting transition temperature, To, above 40 K.

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FIG. 2. Lattice structure of perovskite, ABO3. The body center site is called the A site, while the corner site is called the B site. FIG.3. Square planar CuO2plane. Closed circlesindicate Cu atoms and open circles O atoms. 2.3. Crystallographic Structure of Various HTSC Oxides 2.3.1. YBajCu307_a The most extensively studied HTSC oxide is YBa2Cu307_~ (YBCO-123, Tc = 93 K). This compound was discovered in late 1986, (5) and is the first compound for which T~ exceeded the liquid nitrogen temperature of 77 K. In that sense this compound is the epitome of the HTSC oxides. The structure of this solid (Fig. 4) (6) is relatively simple, and comprises of the two vital components of the HTSC oxides: CuO2 planes and a charge reservoir, in this case a CuO chain. There are two CuO2 planes in the unit cell which are connected by Y ions. Y ions are eight-coordinated by in-plane oxygen ions, 0 2 and 0 3 . The plane and the chain are sandwiching a BaO rocksalt plane. The oxygen ions in the BaO plane (04, (6) or O1 (7)) are on top of Cu ions and form pyramids with the in-plane oxygen ions. Thus, they are called the apical oxygen atoms. When the sample is reduced by annealing in an atmosphere with low oxygen pressure to remove oxygen, the chain oxygen atoms are the first to leave the structure. Oxygen vacancies show varying degrees of local ordering which affects properties including T~, as we will discuss later. 2.3.2. YBa2Cu408 YBa2Cu408 (YBCO-124, Tc = 80 K) is closely related to YBCO-123. It has two CuO chains instead of one for YBCO-123 (Fig. 5). (8) Unlike YBCO-123, this compound is stable with respect to oxygen stoichiometry. However, this solid has to be synthesized in high oxygen pressure at a high temperature to stabilize the structure. When the oxygen content is not stoichiometric the solid decomposes into YBCO-123 and CuO, or Y2 Baafu7 O15 (YBCO-247) and CuO. <9) 2.3.3. La2_x(Sr, Ba)xCu04 La2_xBaxCuO4 (LBCO-214) was the first HTSC oxide discovered by Bednorz and Miiller in 1986. (1) The confirmation of their result and identification of its structure by the group of

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FIG. 4. Structure of YBa2Cu307.

FIG. 5. Structure of YBa2Cu408.

Tanaka (~°'~l) started the gold-rush for the HTSC solids. Its unit structure (Fig. 6, K2NiF4 structure) comprises of a single CuO2 plane and two (La, Ba)O rocksalt planes (Fig. 7). The crystallographic unit cell, however, has two of these units forming a b.c.c, lattice pattern. The oxygen ions in the rocksalt plane are also the apical oxygens for Cu, and together with the in-plane oxygen ions form slightly distorted CuO6 octahedra. A variation of this structure is the overdoped La2CuO4+~. Because of the small concentration of the excess oxygen (& -~ 0.06 on average) and phase segregation the structure of this phase is not yet determined. (~2~ However, the structure of a similar compound, La2NiO4+~, has been determined313~ In this case excess oxygen ions form local peroxide dimers with the apical oxygen ions.

2.3.4. Nd2_ xCexCuO 4 While most other HTSC oxides are p-type conductors this solid is an n-type conductor. ~t4) This structure is another variation of the La2CuO4 structure (T-phase), and is called the T'-phase structure (Fig. 8). As in the T-phase structure, the structure is made of a single CuO2 plane and two (Nd, Ce)O planes, but the (Nd, Ce)O planes are shifted by a/2 in the x-direction, so that the oxygen ions in the (Nd, Ce)O planes are not on top of Cu ions, but are on top of oxygen ions in the CuO2 plane, forming a linear chain of oxygen ions parallel to the c-axis.

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FIG. 6. Structure of La2_x(Ba, Sr)xCuO4-

2.3.5. A,N2Mm_ I CumO2m+n+2 Examples of this class of structure are Bi2Sr2CaCu:O8 (BISCO-2212, Tc = 90K) ('5) and T12Ba2CaCu208 (TI-2212, Tc = 110 K). Cl6)They are made of three CuO2 planes glued by Ca, and two planes of BiO or T10 connected by SrO or BaO planes. Both the number of CuO2 planes and the number of BiO or TIO planes can be changed, resulting in a rich variety of structures (Fig. 9). The highest values of Tc in these series are Tc = 110 K for Bi2 Sr2 Ca2 Cu3 O10 and 127 K for well-annealed T12Ba2 Ca2 Cu30,0. A recently discovered member of this family, HgaSraCa2Cu3Ot0+~ showed the highest confirmed T~ of 137 K in ambient pressure °7~ and 164 K in high pressure. {J8)There has been a report of observing superconductivity near room temperature in this system,~19)but this result has not been reproduced.

o__ FIG. 7. Rocksalt block.

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2.3.6. (Bal_:,Kx)Bi03 and Ba(Bil_xPbx)03 These oxides are not layered two-dimensional systems as are the preceding oxides, but their structure is the three-dimensional cubic perovskite itself (Fig. 2). ~2°)The A site is occupied by Ba or K, while the B site is occupied by Bi or Pb. Ba(Bi~_xPbx)O3 is one of the first oxide superconductors with Tc above 10 K. (21) T c of (Ba~ _xKx)BiO3 (BKB) is higher, but not more than 40 K. Since their Tc is relatively low and no magnetism is found in the neighboring compositions, they may not qualify as HTSC oxides. But in light of their low charge carrier density (in the order of 102~em -3) their Tc is anomalously high. Furthermore, the tendency of Bi +4 to split into Bi +3 and Bi +5 (charge disproportionation) suggests unusual electronic interaction, namely strong electron correlation effect, in this system. (22) Thus, understanding this system would undoubtedly greatly facilitate understanding of cuprates. 2.3.7. Other HTSC oxides Other HTSC oxides include the so-called infinite layer compound (m = or), (Ca, Sr)CuO2. While the stoichiometric compound Ca086Sr0.~4CuO2(23) is insulating, non-stoichiometric samples show superconductivity with the reported value of a Tc up to 110 K. ~24,25)However, the Meissner fraction is always small, and its true structure and the composition of the superconducting phase is yet to be determined. On the other hand, a doped infinite layer compound, (Ca, Nd)CuOz, shows real volume superconductivity, although its Tc is relatively low (,-~40 K). ~26)A synthetic HTSC oxide, BISCO with 8 layers of CuO2 was reported to show Tc approaching room temperature327) However, this observation has not been confirmed by a second group.

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FIG. 9. Structure o f TI 2 Ba 2C a C u : O 8 . The n u m b e r o f C u O 2 layers a n d TIO layers can be varied to form Ti, Ba2Ca,,_lCumO2,,+.+ 2 (n = 1, 2; m = 1, 2, 3).

2.4. Local Structure and Atomic Interaction 2.4.1. Nominal valence of ions As we mentioned before, the HTSC oxides are basically ionic/covalent solids derived from insulating parent oxides. Thus, it is useful to start the discussion by assigning the nominal valence for each ion, while the real valence often deviates from this nominal valence. For instance, in La 2CuO 4 the nominal valence of La is + 3, and that of O is - 2. Therefore the valence of Cu must be + 2. When L a is partially replaced by divalent Sr in La2_ x Srx CuO4 (LSCO-214), the valence of Cu has to be increased in order to maintain the charge neutrality.

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In other words, holes are transferred to Cu when La is replaced (doped) by Sr. F o r instance, when x = 0.15 the nominal valence of Cu is + 2.15. The excess positive charge carriers (holes) on Cu become mobile if their concentration is high enough, giving a metallic character to the solid. The solid therefore is a p - t y p e conductor. The nominal charge concentration in this case is equal to x. In a fully oxygenated (3 = 0) YBCO-123, since the valence of Y is + 3 and that of Ba is + 2, the average nominal valence of Cu is + 2.33. As we discuss later, m a n y of these excess charges are considered to be on the plane. When oxygen content is reduced to 6 = 0.5 (YBa2 Cu3 O65) the nominal valence of Cu is reduced to + 2. However, superconductivity is lost at a slightly larger value of fi, about 6 = 0.6, because some in-plane oxygen atoms also leave the solid during oxygen reduction treatment. (2s) In BISCO-2223, since the nominal valence of Bi is + 3 and those of Sr and Ca are + 2, the nominal valence of Cu is + 2. Thus, there seems to be no excess charge on Cu, which should make this solid insulating. The nominal valence consideration does not work well in this case, and we must consider more subtle means of providing charge carriers to the system, such as the internal charge transfer or defects329) The real valence of ions is different from the nominal valence since charges are shared by ions due to orbital hybridization, or in other words covalency. They can be best estimated by spectroscopic methods. For instance, X-ray absorption spectroscopy showed that the excess holes reside mostly on oxygen ions rather than copper ions33°) Thus, they reduce the valence of oxygen rather than increase the valence o f copper. The band structure calculations also provide good estimates, but they can sometimes be grossly in error. These points will be discussed later.

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FIG. 10. (a) Square planar, (b) pyramidal, (c) octahedral Cu environments.

FIG. 11. Oxygencoordination of chain (above) and plane (below) Cu in YBCO-123.

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2.4.2. Environment of Cu ion The most important structural element in the HTSC oxides is the CuO2 plane. In this plane a Cu ion has 4 in-plane oxygen ions and up to 2 apical oxygen ions as the nearest neighbors. Thus, there are three types of environment, namely square planar with Arc = 4, where Arc is the number of oxygen nearest neighbors of Cu, pyramidal with Arc = 5, and octahedral with Arc = 6 (Fig. 10). The square planar environment is most often found for divalent Cu ion, while the octahedral environment is characteristic of trivalent Cu. Therefore, Arc being more than 4 is a signature of the Cu valence being more than + 2. For instance, in YBCO-123 the chain Cu ion is four-coordinated while the plane Cu ion is five-coordinated (Fig. 11). Thus, obviously the chain Cu ion is likely to remain divalent, and the excess carriers are likely to be near the plane Cu ions. The nominal valence of plane Cu, however, is still closer to + 2 rather than + 3. Indeed the CuO6 octahedron is not perfect, but is elongated along the direction perpendicular to the plane. So that we may say Cu is basically square-planar coordinated with an additional apical oxygen as a second neighbor. This elongation of the CuO6 octahedron is understood in terms of the well-known Jahn-Teller effect. In fact the ionic/covalent bond between the plane Cu and the in-plane O is the strongest bond in the solid. Its bond strength is reflected to the frequency of the longitudinal optical phonon (about 20 T H z or 80 meV) ~31)in which Cu and O vibrate out-of-phase, stretching and compressing the C u ~ ) bond. This mode has the highest frequency among all the phonon modes. The relationship between the valence and the coordination of Cu ion immediately leads to the following insights:

Nd2_ x CGCuO4--The absence of the apical oxygen site in the T' structure is consistent with the valence of Cu being less than + 2, as an n-type conductor. The superconductivity of this solid is sensitive to oxygen content and local structure, and is observed only after an oxygen reducing annealing treatment. 04) A recent report suggests that the effect of annealing is to remove apical oxygen atoms which are accidentally present as-produced332) Then the apical oxygen should suppress superconductivity. This in fact makes perfect sense, since an apical oxygen ion will increase the Cu valence, and therefore should suppress n-type superconductivity.

Central layer of BISCO-2223--In BISCO-2223 there are three CuO2 planes or layers. Among them only the central layer has no apical oxygen associated with it. Thus, the central layer is expected to be in the divalent state and not to contribute to conduction. Indeed the Tc of BISCO-2223 (Tc = 110 K) is only slightly higher than that of the two layer compound BISCO-2212 (T~ = 90 K). Infinite layer compound--Since Tc of BISCO compounds goes up with the increasing number of layers, attempts have been made to create an artificial lattice with a larger number of CuO2 layers. However, as the number of layers is increased from three to four Tc was found to be reduced, and with five layers the solid became insulating. °3) This is perfectly understandable: since the CaCuO2 layer itself is neutral and the Cu ion in this layer does not have an apical oxygen, the Cu ions in the (CaCuO2), block except for those in the two surface layers should remain divalent, and the layer should be insulating. This cast serious doubt upon the validity of the reported high temperature superconductivity in the eight-layer BISCO ~27)which is yet to be confirmed by a second group. While it is still possible that the observation itself is correct, the actual composition and structure may well be different from what the authors

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of the report thought. The same applies to the infinite layer compound (Ca, Sr)CuO 2.(24) In this case it is generally agreed that the structure of the superconducting phase is different from the parent phase. ~34)It is possible that a similar compound, Sr2CuO3, is present as intergrowth layers. ~35)The high-pressure phase of this compound does indeed have apical oxygen atoms. 136~ 2.4.3. Environment of 0 ion Another key player in the HTSC oxides is oxygen ion. There are three types of environment for oxygen ions according to their site.

In-plane oxygen--An oxygen ion in the CuO 2 plane is covalently bonded to two Cu ions as shown in Fig. 3. However, it also touches four other in-plane oxygen ions, since the ionic radius of 0 2 is 1.40 A,~37)and the O-O distance in the plane is about 2.7 A. It also has two or four apical oxygen ions within the interacting distance (3.0-3.2 A). Furthermore, it has four alkali-earth or rare-earth ions as neighbors, although the covalency of this bond is weak. Thus oxygen ions have many interacting neighbors and a charge carrier on oxygen can hop to many neighbors, acquiring high mobility. Apical oxygen--Apical oxygen ions bridge the CuO2 plane and the charge reservoir, such as the chain in YBCO-123 and -124, and the rocksalt layers in other layered compounds. In YBCO-123 the apical oxygen ion has two Cu neighbors. It is, however, closer to the chain Cu than to the plane Cu. Indeed it is more logical to consider the apical oxygen forming a square-planar structure with the Cu~O chain (Fig. 1 I). It therefore has two chain oxygen ions as contacting neighbors at 2.7 A, and four in-plane oxygen ions as slightly more distant neighbors at 3.0-3.2 A. It also has four Ba neighbors, but the interaction with them is purely ionic. The role of apical oxygen will be discussed later in further detail. Oxygen ions in the charge reservoir-- Oxygen ions in the charge reservoir, such as the chain oxygen in YBCO-123, are considered to play only a minor role in the charge conduction and superconductivity of the HTSC oxides. They are usually a part of the rocksalt layers. 2.5. Structural Modulation and Superlattice While the structure of the HTSC oxides is basically made of the square-planar CuO2 planes and the reservoir rocksalt layers or chains, in many of the compounds there are secondary structural modulations which produce superstructure and make the real structure more complex. The main reason for such structural modulation is the steric effect of atomic size incompatibility. The structural modulation usually does not have significant effects on superconducting properties, while there are exceptions which will be discussed separately in Section 7. Here we will survey salient features alone.

La2 .... (Ba, Sr)xCuO4--Since the Cu-O bond in the CuO2 plane is the strongest bond as we discussed earlier, the dimension of the CuO2 plane is almost independent of the environment. The Cu-O distance is always about 1.9 ~, resulting in the lattice parameters in the directions in the plane of about 3.8 ]k in the tetragonal phase. In La2CuO4 this automatically sets the distance between the La site and the apical O site in the rocksalt plane to be 2.73 ~,. However, the ionic radius of La +3 is only 1.22/~,(37) which results in the equilibrium La-O distance of 2.62 ~. This mismatch between the intersite distance and the sum of the ionic radii produces room for structural modulation. An apical oxygen ion surrounded by four La +3 ions can lower the energy by breaking the symmetry and moving closer to one or two La ions. Since

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o 03

FIG. 12. Patterns of displacements in the T149 plane of T1-2212. Both patterns are possible and co-exist. Displacements are coherent over only a short range/43)

the displacement of an apical oxygen ion will push in-plane oxygen atoms out of the plane of the CuO2 plane, this results in tilting of the CuO6 octahedron, although the CuO6 octahedron is usually not rigidly tilted.(38) The tilting of the CuO6 octahedra produces modulation of the structure into various structures including the orthorhombic structure. Other structural modulations will be discussed separately. YBa 2Cu 30 7_ 6--When YBCO- 123 is reduced, oxygen vacancies initially form short range order, and after long annealing form a superlattice structure. For instance, when ~ = 0.5, instead of half of the chain oxygen sites being randomly occupied the vacancies line up in chains so that the system forms alternating full Cu-O and Cu-v chains, where v is the vacancy.(39)The ordering of vacancies has a small but clear effect on the magnitude of Tc.(4°) Bi and TI compounds--In BISCO compounds a significant mismatch exists between the ionic

size and the intersite distance. While the intersite Bi~O distance is 2.73/~, the sum of the ionic radii of O 2- and Bi3+ is only 2.43/81k.(37)This produces a long range incommensurate structural modulation. In BISCO-2212, for instance, the period of modulation is 4.7 unit cells in the (110) direction in the tetragonal index341) The modulation results in sharp superlattice peaks in the diffraction pattern. The periodicity of modulation depends upon concentration, but the dependence is different in the superconducting phase and non-superconducting phase, which may indicate a deeper role of the modulation in superconductivity342) In TI-2212 the size mismatch is even greater, since the sum of the ionic radii is only 2.4/~ (37) Instead of producing long range modulation, however, a short range structural modulation is formed, as shown in Fig. 12.t43) T1 or O vacancies are the likely cause for preventing the short range modulation from developing into a long range modulation. Similar modulations are seen also in BISCO, ~44)in addition to the long range modulation discussed above.

3. LATTICE VIBRATION 3.1. Phonon Dispersion Since the structure of HTSC oxides is not simple, containing many inequivalent sites, the phonon dispersion is also complex. The number of phonon branches is equal to three times

LATTICE

EFFECTS

IN

373

HTSC

O cO

O t.O

E

v

3~

O

O

0

0.1

0.2

0.3

0.4-

0.5

Fie. 13. Phonon dispersion in La2CuO4 calculated by a shell model.13~) the number of atoms in the unit cell. Thus, for instance, in La2CuO4 there are 14 atoms (two formula units because of the/-symmetry) in the unit cell, resulting in 42 phonon branches. Fortunately, details of phonon dispersion have been studied by neutron inelastic scattering from single crystals, mainly by the Karlsruhe group, for La2_ xSrx CuO4 and YBa2 Cu3 O7_ ~.(3~/ The phonon dispersion can be phenomenologically modeled by a "shell model". The shell model is essentially a ball-and-spring model including the electron shell elastically attached to a nucleus and the Coulomb interaction among the ions. Examples of the calculated dispersions are shown in Fig. 13. Dispersion is generally weak for most of the optical modes, particularly in the c-axis direction. As we mentioned earlier the highest energy modes are the in-plane longitudinal modes. 3.2. Electron-Phonon Interaction and Superconductivity According to the BCS theory the superconducting transition temperature, To, is given by: 1

knTc = 1.14hogt~e-~. = N(O)V,

(1)

where e~o is the Debye frequency, 2 is the electron-phonon coupling constant, N (0) is the electronic density of states at the Fermi level, and V is the average effective attractive

374

P R O G R E S S IN M A T E R I A L S S C I E N C E I

I

I

t

I

I

I

I

I

I

v~

Bi2Sr2CaCu208 11

10

+/,~

-

-

-

>

m

m

6 -FIG. 14. Schematic shape of the Fermi surface in HTSC oxides. Arrows indicate nesting. >

56 7 4

E

~

@

~

~

meV m

3

20

>18 E 16 14

_

5 meV

I

I

I

I

I

I

J

I

I

i

20 40 60 80 100 120 140 160 180 200

300 320

TEMPERATURE (K)

FIG. 15. Phonon energy width as a function of temperature for BISCO-2212353~

interaction between electrons at the Fermi level} ) The BCS equation, however, is applicable only when 2 is very small compared to unity and the dynamics of the phonons need not be considered explicitly. The BCS theory is therefore called the weak-coupling theory. Since, in HTSC oxides, kBTc/ha~o is as much as 1/3 or more, we cannot use the BCS theory. We have to use the strong-coupling theory of Eliashberg (45) which explicitly deals with the phonon dynamics and the retardation effect of coupling. However, in order to obtain quantitative results by the Eliashberg theory a very large number of coupled gap equations have to be solved simultaneously. This makes it impractical to use the strong-coupling theory to obtain physical insights. To circumvent this difficulty McMillan ~46)introduced a simplified equation based upon the solution of the Eliashberg equation for Nb. This widely used formula is given by: kBTc=\l.45

exp

2-#*-0.622#*

'

(2)

where #* describes the effect of Coulomb repulsion between the pair of electrons and is usually between 0 and 0.25. As the value of 2 is increased, however, Tc shows a maximum at about 2 = 2, and then decreases. As a result Tc cannot be more than 30 K for normal solids. This prediction was in the back of everyone's mind when HTSC oxides were discovered and the values of Tc over 100 K were observed. Consequently, many believed right away that the phonon mechanism could not provide such a high Tc and a totally different mechanism, such

LATTICE

EFFECTS

IN H T S C

375

as the one involving magnetism, had to be found to explain the result. However, the McMillan's limit appears to be merely spurious. Allen and Dynes (47) proposed instead an equation for large values of 2,

k~L ~ 0.18h ~

(~2),

(3)

where (~o 2) is the average of the square of the phonon frequencies, which predicts higher values of T~ to be achievable. According to these theories T¢ can be raised by increasing ogD, N(0) or V. Among these three variables N(0) is most sensitive to composition, and depending upon details of the Fermi surface it can be significantly larger than the value for free electrons. In particular, when the Fermi surface has a pseudo-one-dimensional quality by nesting, N (0) is strongly enhanced. Figure 14 shows a simplified Fermi surface of HTSC solids such as BISCO. Here the opposing fiat Fermi surface, which also has almost no dispersion in the c-direction, provides strong nesting as shown by the arrow in the figure. The problem is that the Fermi surface nesting at the same time leads to all kinds of effects which are harmful to superconductivity, such as magnetic ordering and phase transformation, and here lies the difficulty. There have been many attempts to estimate experimentally the value of 2 for HTSC oxides. However, because of various experimental difficulties, no definitive values of 2 for the cuprates have been established. But it appears from several different types of experiments (e.g. optical and photoemission spectroscopy, neutron scattering, specific heat, etc.) that electron phonon coupling in cuprates is at least modestly strong, with values of 2 roughly ranging between 1 and 3. (48) The strength of the electron-phonon interaction can be theoretically evaluated by the tirst-principle total energy calculation using the local density-functional approximation (LDA) within the adiabatic (Born-Oppenheimer) approximation. In this calculation the lattice is statically distorted to represent a certain phonon, in the so-called frozen phonon method. For Lal.85Sr0.15CuO 4 Krakauer eta/. (49/found an average value for 2 = 1.37, which from the Allen Dynes eq. (3) gives T¢ = 39 K, a value close to that actually measured. Because of the nesting of the Fermi surface (Fig. 14) there is a strong enhancement of the interaction at the momentum transfer of (0.5, 0.5, 0) or 0z, rr, 0) depending on whether the unit of reciprocal lattice vector is taken to be 2rc/a or 1/a. This Fermi surface nesting also promotes both magnetic and lattice susceptibility at these reciprocal lattice points. However, the largest contribution to this average 2 comes from the F-point (center of the Brillouin zone) axial oxygen breathing mode which has 2 = 11.7. In a similar vein, Zeyher and Zwicknagl ~5°~have calculated the shift in Raman-active phonon lines 151~in YBa2Cu307 caused by the appearance of superconductivity which will be discussed later. They found a reasonably good agreement with experimental results if they took 2 = 2.9. Cardona (521has discussed similar observations in some detail. In particular, he noted that analysis of phonon-frequency shifts and changes in linewidths upon cooling REBa2Cu307 (RE = rare earth) below T~, observed by Raman scattering, lead to an estimate of 2 = 0.8 if all phonons couple as B~g modes. However, as Cardona has emphasized, these q = 0 phonon effects depend strongly on sample quality, i.e. oxygen stoichiometry, purity, etc., and, therefore, one must be cautious in drawing conclusions about the strength of electron-phonon coupling. Other experimental evidence for (strong) electron-phonon coupling to q ¢ 0 modes comes from inelastic neutron scattering in Bi2Sr2CaCu2Os. (53~ Figure 15 gives plots of two high-energy oxygen-phonon linewidths as a function of temperature. Below T,. ~ 80 K the linewidths increase substantially, by between 1.5 and 2%. Such broadening is expected

376

PROGRESS

IN M A T E R I A L S

SCIENCE

because phonons having an energy greater than the superconducting energy gap A break Cooper pairs and, therefore, have additional decay channels. In contrast, the phonon mode at 5 meV has a linewidth that may decrease upon cooling below To, which is expected if the energy of this mode is less than the gap frequency, since in this case decay channels are removed by the opening of the gap. As with q = 0 phonons, changes in q > 0 phonon linewidths due to superconductivity are, in general, expected to be very small; therefore, the large increases found in the high-energy phonon linewidths of BISCO-2212 imply strong electron-phonon coupling for these modes. A further aspect of the data in Fig. 15 is the rapid decrease in linewidth as Tc is approached from above in the high-energy modes. Such a strong decrease is consistent ~53)with anharmonicity that also appears to be related to superconductivity, as evidenced by the abrupt arrest of linewidth narrowing upon approaching T¢. The measured phonon density-of-states in (Y, Pr)Ba 2(Cu, Zn)3 06 + ~, Bi2 Sr2 (Ca, Y ) C u 2 0 8 (54) and Lal.85Sr0.~5(Cu, Zn)O4¢55) suggests that an increase in T¢ is associated with the lattice softening, particularly of a decrease in the energy of the high frequency C u 4 ) modes. Such a softening has been observed for conventional superconductors and is further evidence of the importance of electron-phonon coupling for superconductivity356) These results indicate that the electron-phonon coupling in HTSC oxides is strong. However, it is not clear if such a strong coupling can be explained within a framework of the conventional electron-phonon interaction, nor if the electron-phonon coupling alone would be able to account for high T¢. For one thing, the conventional approximation includes a simplification (of neglecting the crossed Feynman diagrams) represented by the Migdal's theorem, (57) but this may fail for the HTSC solids because of the low carrier density which makes the Fermi energy comparable in magnitude to the phonon energies. It is, furthermore, possible that the electron-phonon coupling is greatly enhanced in the HTSC oxides beyond the level the LDA theory can calculate, because of the strong electron-electron interaction, as will be discussed later. 3.3. Isotope Effect One indication of the strength of electron-phonon interaction and the degree of phonon contribution to superconductivity is the isotope effect. When different isotopes are used in 0.6

I

,

r

i

I

i

J

i

J

0.7

i

i

i

i

I 0.10

I 0.15

t 0.20

I 0.25

Y l - x P r × B a 2 C u 3 O 6 92 0.5

i~

0.6

\ \ \

0,5

\

0.4

\ 0.4

o~ 0.3

*" 0.3

\ . ~~ "\\

•

0.2

0.2 0.1

o

o

• 0.1

0

Y B a . 2 ~' t

h 20

1-x i

x/3 6.94 t t L 40 60

i

~ 80

t 1O0

0.0 0.05

T R=0 (K)

F[o. 16. Exponent of the isotope effect, 0~, vs T c for (Y, Pr)Ba2Cu306.92 and YBa2(Cu, Zn)306.~. (58)

0.30

X

FIG. 17. Exponent of the isotope effect, ct, vs concentration for

La2_xSrxCuO4

.(61)

L A T T I C E E F F E C T S IN HTSC

377

the samples they affect the value of Tc through the change in the phonon frequency. Since the phonon frequency is related to the atomic mass M by:

oo (1 the change in T~ due to the change in the isotopic mass is given by: c~ lnTc - - = - ~ , O lnM

~=0.5.

(5)

In conventional superconductors such as A1 the value of a is close to 0.5, which is one of the strongest proofs of the BCS theory/2) In HTSC oxides the value of a due to oxygen is usually much smaller than 0.5, and depends upon composition. Values of a for alloyed YBCO-123 are shown against T¢ in Fig. 16J 58) There are two observations to note here: one is that in the same series ~ decreases with increasing To; the second is that the maximum Tc within a given series is associated with the smallest, or nearly so, value of a, which is less than 0.l. This result is usually interpreted as evidence that the lattice (phonons) do not play an important role in the HTSC phenomena, and other mechanisms, such as magnetic mechanism, have to be considered. However, small values of ~ may arise from very strong electron-phonon coupling, ~59)or lattice anharmonicity36°) Alternatively, however, it may be a signature of the enhancement of the electron-phonon interaction by electron correlation, as we will discuss later. Figure 17 shows the results of a systematic study of 0t vs x in the La2_xSrxCuO4 system. ~6~ In this case, a is a maximum for x = 0.12 and drops rapidly to a small value for x = 0.15--0.18, where Tc is maximum. At the composition (x = 0.125) where a is a maximum, T¢ exhibits a dip in the otherwise smooth variation of T¢ (x). The implication of this magic ratio (x = 1/8) and the relation between the structural phase transformation and the suppression of Tc will be discussed later. A recent report on the Cu isotope effect shows that the value of ~ for Cu is negative. (62)

4. LOCAL DEVIATIONS FROM CRYSTALLOGRAPHIC STRUCTURE As is well known, the crystallographic structure is determined with the presumption of perfect lattice periodicity, based upon the positions and intensities of the Bragg diffraction peaks in reciprocal space. The atomic structure of a real solid, on the other hand, does not have perfect periodicity. There may be lattice defects, including substitutional defects, and at any temperature atoms are moving because of thermal and quantum mechanical lattice vibrations. In addition, the HTSC oxides appear to have some intrinsic local deviations from perfect periodicity. Interestingly, there are indications that these deviations are intimately connected with the occurrence of superconductivity. This subject, however, is still controversial and actively debated, since conflicting experimental reports have been published. For instance, for YBCO-123 the crystallographic analysis ~63'64)gave a negative view with respect to the presence of anomalous behavior of the apical oxygen reported by the EXAFS measurement. ~65'66)In the following pages we will review some of the reported results and discuss their implications. As we will see, while some of the conclusions drawn from the EXAFS measurements may not be valid, the presence of significant local distortions appears unmistakable.

378

P R O G R E S S IN M A T E R I A L S S C I E N C E

4.1. Debye- Waller Factor The aperiodic deviations from the perfectly periodic lattice structure result in the decrease in the Bragg peak intensity through the Debye-Waller factor, and in diffuse scattering observed in the reciprocal space between the Bragg peaks. The intensity of X-rays or neutrons scattered by a sample is given by:

I (Q ) = IF(Q)12 1

F(Q ) = -v/ ~- ~ f ( Q )e'° R"

(6)

where Q is the momentum transfer in the scattering (=4~sin 0/2), f ( Q ) is the atomic scattering factor, <. . . . > denotes a compositional average, Ri is the position of the i-th atom, and N is the number of atoms in the system. (67) If the atoms are displaced from the crystallographic sites, R, = (R, > + ui

(7)

where u~ is the displacement and
F(Q ) - x / ~ ( f ( Q )) ~i f ( Q )e 'O'I+"'l _

(

1

,

~ f ( Q ) e iaI l + i Q

u~-~.[Q.ui)2+ ' ' '

).

(8)

By Fourier-transforming the displacement

ui=--

1

,,~q~

Uqe ,q.

(9)

the summation in eq. (8) becomes

Zf(Q

)ejQ " =

i

~.f(Q

)e~Q ( & ) q-

i 1 ~if(

Z f ( Q ) ~, iQ. i

o)2(Q.uq)(o.uq,)e

Uqe i(Q - q, <&>

q i(Q-q

q')..q_...

2 • q,q, A part of the third term with q = q' can be combined into the first term by: ~ f ( Q ) e ~Q 1 - ~ ( Q . u q ) 2 + i

....

e BQ2~f(Q)e'e,

q

(10)

(11)

i

where the factor e-~Q: is the Debye-Waller factor. B is called the thermal factor: 2 1> , B = ~1 = ~1 1 2 +
(12)

Here is the square of the thermal amplitude for phonons with the polarization parellel to Q, and 2 is the average squared amplitude of displacement due to strains. Thus the total scattered intensity becomes,

I(O ) = IB(Q ) + Io(Q) + " ,

(13)

e 2BQ2 IB(Q ) = N
(14)

where /

i,I

LATTICE

E F F E C T S I N I--ITSC

1

~ (O. Uq)2f(O)fjj(a) (eitQ ql.tR, R~I). Io(O ) __ N ( f (Q ) )2 ~J,q

379 (15)

The first term describes the Bragg diffraction peaks at the reciprocal lattice points while the second term describes the diffuse scattering intensity which is observed in-between the Bragg peaks. Remaining terms are the multiple-phonon scattering intensity which becomes important at large Q and at high temperatures. Thus by determining the Debye-Waller factor, the amplitude of deviations from the crystallographic structure can be known. The first indication of anomalous lattice distortion was seen early in 1987 in the Rietfeld analysis of the structure of La2 _xSrrCuO4 (LSCO-214) for which thermal factors were found to be larger than expected for phonons. ~6~)Since then many HTSC oxides have been found to have large thermal factors. ~4) By comparing with the expected phonon amplitude it is possible to estimate if significant non-phonon displacements exist or not. In the case of YBCO-123, the thermal factors are almost normal, except for the lateral displacements of the chain oxygen in the x - y plane. Since chains have no atoms in-between they are expected to be vibrating rather widely in the perpendicular direction. On the other hand, the thermal factors for T1-2212 are clearly anomalous. For instance, in T12Ba2CaCu20 s the apparent amplitude (U2) I/2 is so large that random displacement of oxygen in the T1-O plane by 0.4 A had to be introduced during the structural refinement: 69) Since the condition for melting by the Lindeman's rule corresponds to about 0.15 A in thermal amplitude, this apparent amplitude certainly cannot be ascribed to phonons. Thus, the anomaly in the Debye-Waller factor exists in some compounds. However, its magnitude sensitively depends upon composition, and a general statement cannot be made based upon the analysis of the Debye-Waller factor alone. Since the Debye-Waller factor is the volume average of the vibrational amplitude, if a small fraction of atoms have large amplitudes the total Debye-Waller factor is not likely to include their contributions. They can be observed only by more sensitive methods. Another problem is that large apparent thermal factors are often seen as a consequence of lattice defects, including the mixed ion effect. When two kinds of ions with different ionic radii are occupying the crystallographically equivalent sites, the atomic size difference produces local strain, and contributes to the thermal factor. Thus, thermal factors are small for stoichiometric compounds. Indeed, simple unmixed compounds such as YBa2Cu~O7 (YBCO-123) show rather small thermal factors. In addition, this compound is optimally doped as far as the charge carrier concentration is concerned, and represents a special case, while underdoped HTSC compounds are likely to show more pronounced anomalous lattice behavior, as will be discussed below. Often non-phonon displacements due to strain are rather small, and are difficult to differentiate from the thermal amplitude. In such a case more careful examinations beyond the Debye-Waller approximation are required. For instance the Debye-Waller approximation (e ~Quj) = e -2BQ~

(16)

assumes random Gaussian distribution of ui. However, if the local displacements are not random, such as in the case of a local double-well potential, this approximation breaks down. Instead, if ui is equal to ___u, assuming that u is parallel to Q, then ( e iO'ui) = cos (Qu),

(17)

which shows a slow oscillatory Q dependence. However, in the conventional crystallographic analysis the data are usually not taken up to high enough Q values to observe the oscillation. Then, for small values of Q (Qu << 1) eq. 17 is: JPMS 38:1 5---M

380

P R O G R E S S IN M A T E R I A L S S C I E N C E

cos(Qu) = 1

1

~

2

~(Qu) 2 +...,~e-I(Q",),

(18)

which cannot be differentiated from the Debye-Waller form unless the study is made in high Q ranges. For this reason, the effects of all the displacements including those in the double-well potential are usually swept into the Debye-Waller factor in the conventional analysis. Only in non-crystallographic methods such as EXAFS or PDF analysis is I(Q ) determined up to high enough Q values to differentiate local displacements from phonons.

4.2. Diffuse Scattering In eq. 15 the lattice sum requires K = Q - q, where K is the reciprocal lattice vector, for the observed intensity to be non-zero. Thus, the scattering caused by Uq is observed at the reciprocal space deviated from the Bragg point by q. Since the magnitude of Uqusually varies continuously with q, the scattered intensity is also a continuous function of q. For this reason it is called diffuse scattering. Diffuse scattering is caused by any deviation from perfect periodicity, including phonons (thermal diffuse scattering, TDS) and local strains due to defects (Huang scattering)37°) TDS is inelastic, since the phonon energy is exchanged in the scattering event. However, since the phonon energy (up to 80 meV for HTSC oxides) is much smaller than the conventional energy resolution for X-ray scattering (in the order of 10 eV), TDS appears elastic for X-ray scattering. The energy transfer can be resolved by a neutron scattering experiment with a triple-axis-spectrometer in which the resolution can be a fraction of a meV. An example of diffuse scattering is shown in Fig. 18, together with the Bragg intensity. Note that the scale is logarithmic, covering seven orders of magnitude. The peak described by a dashed line is the Bragg peak. Its width is limited by the Q resolution of the spectrometer with which the experiment was conducted. Diffuse scattering intensity is usually low, and it is not easy to differentiate diffuse scattering due to local strains from the TDS. An example of diffuse scattering is shown in Fig. 19.(71)The sample studied here is YBCO-123 doped with A1. The diffuse scattering is caused by the strain induced by the substituted A1 atom. By

8

Lal.sSro.2CuO 4 single crystal, T=10 K

6

~

2 0

-2

-0.25

f

i

-0,20

-0.15

i

-0.10 q [A -1]

i

-0,05

0.00

FIG. 18. X-ray scattering intensity in the vicinity of the Bragg peak (0, 0, 18) for Lal.sSro.2CuO 4. The dashed line describes the ideal Bragg peak shape limited only by instrumental resolution.

LATTICE

EFFECTS

381

IN HTSC

÷0.2

-0.2 40.2

--0.2 ~0.2

0

--0.2 ~.~

0

*0.2 --0,2

0

~012 --0.2

0

~.~

FIG. 19. Diffuse scattering from a single crystal of YBa2(Cu0.955A10.045)307divided by the scattering factor IF(Q)12around (a) (040), (b) (240) and (c) (220), and corresponding calculated Huang scattering (d f) and Huang scattering plus TDS (g-i) which compare well with the experimental results (a-c).tTt)

observing the difference in symmetry from the calculated T D S intensity the contribution due to A1 impurity can be isolated. Another example for La2_xSrxCuO4 is shown in Fig. 20. ¢72~Here the scan was made for the x = 0.2 sample along (3, 3, 18.2) in the tetragonal index (a* = 1.67 ~-~, c* = 0.47? ~ - ~ ) . The solid lines are the T D S intensities calculated using the shell model which was derived to reproduce the phonon dispersion determined by neutron inelastic experiments. Figure 20 shows that the diffuse scattering intensity at r o o m temperature can be accounted for as the T D S very well, but at low temperatures significant deviations appear and the TDS accounts for only a portion of the diffuse scattering. This extra non-phonon diffuse scattering (XDS) intensity depends both on composition and temperature, as shown in Fig. 21. Here the

382

P R O G R E S S IN M A T E R I A L S SCIENCE 350 []

T=300 K

14

[] Lal.sSro.2Cu04 ~ua

300 =

12

-~ --x=0ib] []

x=0.151

•

x=O.20j

La2_xSr,CuO 4

~o 250 10

(5

"B 200 "E

~ 15o 100

~ 6

(~

0 -0.2

, -0.1

8

I

I

0.0

0.1

0.2

FIG. 20. Diffuse scattering from a single crystal Lal.8 Sr0.2CuO4 along (~, ~, 18.2) at T = 300 K and 10 K.

Solid lines are calculated thermal diffuse scattering intensities.{ 72)

I

I

i

I

i

50

100

150

200

250

300

T [K] FIG. 21. Temperature dependence of the n0n-phonon diffuse scattering for Laz_xSrxCuO4(x =0.1,0.15, 0.2) represented by the integrated intensity of (~, ~, 18.15) from ~ = 0.02-0.1. {72)

non-phonon intensity is presented by integrating the intensity of (4, 4, 18 ___0.15) from = 0.02-0.1, after subtracting the TDS intensity from the data. The TDS intensity at T = 10 K is plotted in Fig. 22 as a function of Sr concentration, x. It is noteworthy that the intensity is minimum at x = 0.15 where Tc is maximum. Since the non-phonon intensity observed here depends upon temperature, and also on composition, in an unexpected way it is not due to static defects such as dislocations or small inclusions. It is most likely that they are induced by charge carriers in the solids, as we will further discuss later. The X D S intensity depends upon q, roughly as 1/Iql m with rn being between 3 and 4. Note that the TDS intensity and the intensity of diffuse scattering due to point defects are proportional to 1/Iq 12.~7°)Therefore, the X D S is not due to phonon softening nor to point-like inclusions. Comparison of intensities around different Bragg peaks suggests that the polarization of displacements, u, causing the X D S is primarily transverse, or u_Lq. The X D S intensity is not isotropic, but has a strong angular dependence. This strong angular dependence suggests that this scattering is not due to artifacts such as the finite resolution of the X-ray optics. The problem of resolution was also checked by a high resolution measurement using a crystal analyzer.

4.3. Local Probes 4.3.1. E X A F S

The X-ray absorption coefficient has long been known to show some oscillatory structure as a function of the photon energy, or wavelength, just above the absorption edge. Stern e t al. (73) gave a simple explanation of this effect in terms of the local structure of atoms, and succeeded in deriving useful structural information from these oscillations. This method is now called the extended X-ray absorption fine structure (EXAFS or XAFS) method. In this

LATTICE

EFFECTS

240

35

La2-xSrxCuO4

o

A 4 term

T=10 K

•

A o term

200

IS

E'~" 160 --~ >, 120

,0

i

,

B ~

,

,

9, k./

,

g N 8o

,0 K

A \

5

40

0

383

IN HTSC

0

J 05

io

V

i 15

20

25

3.0

105 K > 35

4o

45

50

R (A)

i

I

i

0.10

0.15

0.20

FIG. 23. Pair d i s t r i b u t i o n function from Cu a l o n g the c-axis in Y B C O - 1 2 3 as a function of t e m p e r a t u r e o b t a i n e d by E X A F S . (66)

X

FIG. 22. C o m p o s i t i o n d e p e n d e n c e o f the n o n - p h o n o n diffuse scattering at T = 10 K for L a 2 x S r x C u O 4 evaluated at (0.05, 0, 18.15)/72)

method the structural information is contained in the quantum interference of the outgoing and incoming wavefunctions of the photo-excited electrons which modifies the X-ray absorption coefficient. A classical analogy is the back-scattering diffraction of the photo-excited electrons. The EXAFS method has a particular advantage in that it is capable of describing the distribution of atomic distances (pair distribution function, PDF) from a specific element, for instance Cu, as shown in Fig. 2 3 . (66) It determines the structure factor up to large values of Q ( = 2k ), typically 30 A - ~. On the other hand, an important part of the structure factor, up to k about 3 A-1 or Q = 6 A-~, cannot be observed by EXAFS because of the interference by the band effects (X-ray absorption near edge structure, XANES), and is left out of the analysis. This often renders the coordination number and sometimes even the near neighbor distances determined by the EXAFS method less than reliable. To overcome this difficulty the analysis is often done not by using the P D F but by simulating the energy dependence of the absorption coefficient using a model. Using single crystals or highly textured grain-oriented samples and polarized X-rays a directional P D F can be determined by EXAFS. By this method, for instance, the Cu O distances along the c-axis can be separated from the Cu~O distances in the plane. An example of the P D F shown in Fig. 23 along the c-axis for YBCO-123 was obtained in this manner. The distances are shortened compared to the real distance because of the phase shift that occurs during the scattering of an electron by an atom. The peak at 1.4/~ corresponds to the distance between the chain copper, Cul, and the apical oxygen, 0 4 (1.84 A), and the smaller peak at 1.8/~ is the in-plane copper, Cu2, to apical oxygen distance (2.28/~). By back-Fourier transforming the model P D F they found that the Cu2-O4 distance is bifurcated into two sub-peaks, indicating the double-well interatomic potential. However, this widely known result has been very controversial. As we mentioned, an X-ray single crystal study of YBa 2Cu 306,88 (63) and a neutron single crystal study of YBa 2Cu 306.98 (64) did not detect any indication of such bifurcation. A recent independent EXAFS study observed splitting of the Cu2-O4 distance only for oxygen reduced YBCO-123. (74) Thus, the initial report of large splitting of the apical oxygen site (66) is likely to be in error. In general, results of EXAFS experiments carried out with oriented powder samples such as the one in

384

PROGRESS

IN M A T E R I A L S

SCIENCE

discussion need to be viewed with some caution. In order to separate the in-plane CuO distance from the out-of-plane distances, grain-oriented powder samples were used in these measurements. But, if each powder particulate contains more than one crystal grain, these particles will be misoriented during the orientation process by the applied magnetic field, and the in-plane and out-of-plane signals become mixed up. This can produce a false double-well pattern. Furthermore, data analysis of EXAFS results is delicate. The results of the EXAFS study are excellent indicators of the presence of anomaly, but they are not necessarily quantitatively reliable. Since the initial impact of the first EXAFS study was so significant this controversy has caused an unfortunate backlash of doubting the whole concept involving the double-well instability. However, fully oxygenated YBCO-123 may be an exception as we discuss later, and anomalous atomic displacements have been found in many other materials, for instance for BISCO-2212 (75) and T1-2212.{76) 4.3.2. Pulsed neutron PDF analysis In conventional crystallographic structural analysis the periodicity of the structure is assumed a priori. Thus, only the Bragg peaks are considered for the analysis, and diffuse scattering intensities are discarded as a part of the background. Consequently any deviation from periodicity can be handled only within the Debye-Waller approximation as we discussed above. However, real materials are hardly perfect and there can be numerous deviations from perfect periodicity. These deviations can be much more directly studied by the method of atomic pair-distribution function (PDF) analysis in which no assumption of periodicity is made. For this reason PDF analysis has been widely used in the study of liquids and g l a s s e s . (67'77) However, it has rarely been used for the study of crystalline solids, since high energy probes are needed to implement this technique successfully for crystalline materials. With the advent of synchrotron base sources, such as pulsed neutron or synchrotron radiation sources, the method can now be widely applied to various crystalline solids. The PDF, p (r), is obtained by the Fourier-transformation of the normalized scattering intensity, or the total structure factor, S (Q) = I ( Q )/2: p (r) = Po + ~

,f

[S (Q) - 1]sin (Wr)QdQ,

(19)

where P0 is the average number density of atoms and describes the probability of two atoms being separated by a distance r. (67'77) It is a one-dimensional quantity averaged over all solid angles. In obtaining the PDF not only the Bragg diffraction intensity but also diffuse scattering intensities are included, which makes this method powerful in studying deviations from periodicity. For a long time the experimental limitation on the range of Q has made it impractical to apply this method to well-ordered materials. In glasses and liquids S (Q) attenuates quickly to unity beyond Q = 20-25/~ -1 because the structural disorder makes the scattering incoherent, therefore featureless, at large Q. In crystals, on the other hand, the structure in S (Q) persists up to larger values of Q. Fortunately even in a perfect crystal, thermal and quantum mechanical positional disorder due to lattice vibrations renders the structure factor S (Q) incoherent for very large values of Q, as represented by the Debye-Waller factor discussed above. Usually when S (Q) is determined up to Q of 35-45/~-I the Fourier-transformation can be calculated without serious termination errors. {78) With the advent of synchrotron base sources, neutrons and X-rays of sufficiently short wavelengths are now available, so that S (Q) can be determined up to these high ranges of momentum transfer.

385

L A T T I C E EFFECTS IN HTSC

The main advantage of this technique is that correlation in the atomic displacements can be directly studied. For instance, if two atoms are oscillating in-phase, the Debye-Waller factor records the amplitude of each atom. However, the P D F shows no oscillation at all because the interatomic distance does not change with time. On the other hand, if they are oscillating out-of-phase, the P D F shows an amplitude twice as large as the Debye-Waller factor does. Thus, by properly modeling the P D F correlation among the atomic displacements can be determined. Furthermore, by the virtue of collecting data up to high values of Q, anharmonic displacements such as the formation of the double-well states can be directly detected. One of the problems of the P D F method is that in spite of the large Q ranges over which the data are collected some spurious oscillations still remain. They are due to statistical as well as systematic errors. Methods to evaluate the statistical errors have been developed, (7s) and are used in the modeling process so that real effects can be separated from spurious effects. Another problem is that unless isotopic substitution is used we can obtain only the total PDF, not resolved for correlations among the elements. However, in most cases the three-dimensional structure can be identified by a careful modeling process. Yet another difficulty often mentioned is the uniqueness of the model. This problem, however, is common to any structural study, conventional or otherwise. This problem can be overcome only by careful execution of the study and confirmation by other direct or indirect observations. The P D F is a one-dimensional correlation function averaged over all solid angles. In order to deduce a three-dimensional structure, modeling or structural refinement has to be carried out. This can be done by using the agreement (A) factor defined by: [p~xp(r)-- rCca~c(r)]2dr/

A = 1

02 dr

,

(20)

*)rl

where Pexp(r) is the experimentally determined PDF, Pcalc(r) is the calculated PDF, and rl and r2 define the range over which the A-factor is calculated. The A-factor is a real-space version of the crystallographic R-factor. In the modeling process the initial structural model is modified so that the A-factor is minimized, for instance starting with the crystallographic structure. We call this procedure a real-space structural refinement, as opposed to the usual 0.3

0.2

D

0.0 2

3

4

,

,

,

,

5

6

7

8

9

r [A]

FIG. 24. Atomic pair-distribution function of YBa2Cu40 s determined by pulsed neutron scattering at T = 10 K. O2) The dotted line is the model PDF calculated for the crystallographic structure.

386

P R O G R E S S IN M A T E R I A L S S C I E N C E

FIG. 25. Pattern of displacements for the chain oxygen atoms in YBCO-124392)

process of reciprocal space refinement in which the difference between an experimental S (Q) and a theoretical S (Q) is minimized. The modeling can be done by the so-called simulated annealing, using the Monte-Carlo method and the generic algorithm. ~79,8°) We can also minimize the A-factor with respect to a small number of controlled variables when the choices of these variables are clear. ¢8t) Since 1987 we have been studying the local structure and its temperature dependence by using PDF analysis for a number of HTSC o x i d e s . (43'79'81-9~)All the HTSC oxides we examined showed in one way or another marked deviations as seen in the PDF from the crystallographic structure. However, the magnitude and mode of displacement are not identical, and vary from one system to another. In some cases the displacements are merely the consequence of static or quasi-static disorder unrelated to superconductivity, such as in the case of TI~O plane rearrangement in T1 (2212). "3) Also, in some cases they are merely the consequence of mixed ion effect or defects. Exceptions are fully oxygenated YBCO-123 and YBCO-124, since they are nearly stoichiometric.

YBCO-123 and 124---The PDFs (solid lines) of YBa2Cu408 (124) and YBa2Cu307 (123) are compared with the PDFs calculated for the crystallographic structure (dotted lines) in Fig. 24. ~91) The calculated PDFs were convoluted by a Gaussian function to represent thermal vibrations. The amplitude of thermal vibration assumed ((u~)~/z=0.048.A,) is consistent with the phonon dispersion in these solids. The measured PDFs agree reasonably well with the calculated PDFs up to about 4 ~, but beyond 4 ~ they show significant departures from the calculated PDFs. The amplitude of oscillations in the measured PDFs is less than that in the calculated PDFs beyond 4/~, suggesting that larger values of (u~) have to be assumed to account for the part of the PDF beyond 4 ~. This implies that there are collective atomic displacements with the correlation length of about 4 ,~. Similar amounts of deviations are seen for (124) and for (123). The degree of deviation can be assessed using the A-factor. The A-factor evaluated for the range of 2.17-9 A, is 0.0866 for (124) and 0.0887 for (123). The real space modeling process suggested that in (124) the chain oxygen (04) atoms are significantly displaced from the atomic position in the average structure392) At low temperatures (below 100 K) the 0 4 atoms are displaced by about 0.1 .~ in the x-direction, which is perpendicular both to the chain and to the c-axis. The displacements are not random, but are highly correlated as shown in Fig. 25, within a domain of about 7 x 15 A, with the long dimension perpendicular to the chain. Since Cu atoms in the chain are not much displaced these collective displacements of oxygen atoms break the mirror symmetry and create a local electrical polarization, as in ferroelectric solids. At temperatures above 190 K, however, the correlations in the displacements are lost.

387

L A T T I C E EFFECTS IN HTSC

A more direct indication of the size of the domain is obtained in the following way. A c factor is defined by: + Ar/2

Ir [pexp(r,)_ P~l~(r ,)]2r ,2dr , [AG(r)]2 = d r - ~tr/2

~

(21 )

r + Ar[2 P 2 r ' z d r ' -- Ar/2

As Pexp(r), the P D F at 190 K was chosen. As p~alc(r), the P D F at 10 K was processed to simulate the P D F at 190 K in the normal condition. Namely, the r scale was adjusted to account for thermal expansion (0.1%), and then the P D F was convoluted with the Gaussian function with a = 0.075 ~ so that the peak width at T = 10 K (a = 0.069/~) is increased to the width expected at 190 K (~ = 0.102 ~ ) in order to simulate the phonon amplitude at 190 K. If the evolution in the structure is merely thermal expansion and the increase in the phonon population, this corrective procedure should make the P D F at 10 K look very much like the P D F at 190 K, except for statistical noise. The difference evaluated by A~ factor as a function of r with Ar = 5/~-I is shown in Fig. 26. The solid line in the figure represents the difference expected for statistical noise. Clearly there is systematic deviation below 20 &, indicating that the structure in this range changes between 10 K and 190 K; this cannot be explained in terms of phonons. This analysis shows that in YBCO-124 ferroelectric micro-domains, with a size about 20/~,, are formed at low temperatures, but they disappear above 190 K. Implications of such results will be discussed later. T l - 2 2 1 2 - - E a r l i e r studies carried out for T12 Ba2 CaCu2 08 first uncovered local displacements in the T1-O layer as shown in Fig. 12. (43) The displacements are correlated over short range, and are dynamic down to low temperatures, freezing probably around 20 K. These large displacements are the consequence of severe ionic size mismatch, discussed above. Smaller displacements were observed for oxygen ions around in-plane Cu. The apical oxygen site is split into two, with Cu~O distances of 2.4 and 2.8/~. The displacements are 0.3

@

o

0.2

@®0

#

@

000

@@°®O°oooOo@oo 0.1

0,0

o

~o 0 uuo-

oo®Q

I

I

I

I

I

I

I

s

lo

15

2o

2~

so

s5

4o

r [A] FIG. 26. A factor defined by eq. 22 for the PDFs of YBCO-124 at T = 10 K and at 190 K .

(92)

388

P R O G R E S S IN M A T E R I A L S S C I E N C E

dynamic and seen only by instantaneous PDF, as will be discussed in more detail later. At the same time, the in-plane oxygen atoms are displaced parallel to the c-axis, and the change in correlation between the apical oxygen and the in-plane oxygen produces the anomaly(79) discussed later.

LSC0-214 The most pronounced ionic displacement other than those associated with the tilting of CuO6 octahedra in La2_xSrxCuO4 are the displacements of apical and in-plane oxygen along the c-axis. Collective displacements of in-plane oxygen ions result in not only the tilting of the plane in the direction other than that specified by crystallographic structure (as we discuss in Section 6), but local buckling,e3) Nd-214 Significant displacements were observed for the in-plane oxygen ions in Ndl.s35Ce0.165CuO4 .fsl) One mode of displacement is in the x - y plane, and rotates the CuO4 square collectively, while the other one is out-of-plane and leads to buckling of the C u 4 ) plane (Figs 27 and 28). In particular, the second mode results in the 0-49 distance modulation of the oxygen chain parallel to the c-axis. This reduction in some 0 4 9 distances may produce holes (p-type carrier) in the system, in addition to the n-type carrier introduced chemically by Ce doping. This two-carrier picture is consistent with the transport properties of this solid.(94)

B K B - - I n the insulating mother compound of (Ba0.6K0.4)BiO3, BaBiO3, the BiO 6 octahedra are rotated and oxygen ions are displaced along the cube edge in the breathing mode to result in two types of Bi environment, indicative of the charge disproportionation, with a large BiO6 octahedra for Bi3+ and a small BiO6 octahedra for BiS+.eS) When alloyed with K, the breathing mode is all but wiped out, but the tilting modes remain, so that the PDF of (Ba0.6K0.4)BiO3 is far from that for the ideal perovskite structure. ~88) 4.3.3. Other evidence of local displacements The crystallographic structures of HTSC oxides have inversion symmetry and, therefore, Raman active and infrared (IR) active optical phonon modes are mutually exclusive. Using polarized Raman spectroscopy, Sugai(96) found Raman active modes at 145, 282, 367 and

• Copper

FIG. 27. Pattern of oxygen displacements in the plane of Ndl.835Ce0.ttsCuO4} TM

CuO 2

FIG.28. Tiltingof in-plane oxygenin Ndl.835Ce0.165CuO 4 and resultingmodulation(dimerization)in the oxygen chain.

LATTICE EFFECTS IN HTSC

389

670 cm ~only for superconducting LSCO-214 samples with x = 0.12 and 0.20 (but with Tcs of only 10 K); these are not seen in non-superconducting samples with x = 0 and 0.34. The modes at 145, 367 and 670 cm-~ coincide with infrared-active transverse optical modes which have polarization vectors parallel to the Cu~O plane. This clearly indicates that inversion symmetry is lost on a local scale, contrary to the predictions from the crystallographic structure. Further, the Raman spectra for the x = 0.34 sample show local orthorhombic symmetry despite the sample having the crystallographically tetragonal HTT structure. Further evidence for deviations of the local structure from the crystallographic structure has been found by Hammel et al. (97) from Cu nuclear quadrupole resonance (NQR) measurements on a single crystal of superoxygenated La2 CuO4 ÷~with 6 = 0.03. This material phase separates into a nearly stoichiometric antiferromagnetic phase (6 .,~0.01) and an oxygen-rich metallic superconducting phase (6 ~ 0.06) at T = 265 K. Peaks in the Cu NQR spectrum of the superconducting phase indicate at least two distinct copper environments in contrast to the single site expected from crystallography. Further, for equivalent hole doping levels, the NQR frequency of these peaks agrees (to within 2%) with similar peaks observed in LSCO-214, although this material also should have only a single copper site. Therefore, this effect is independent of the type of dopant (interstitial O vs substitutional Sr) and indicates a change in local structure that results from the crystal response to the presence of doped holes. 4.4. Comparison with Crystallographic Analysis Since these deviations have been reported using relatively new methods of structural study such as EXAFS and pulsed neutron atomic pair-distribution analysis, there has been strong skepticism over the validity of these results. As we mentioned earlier, conventional methods of structural analysis tended to contradict the reports of local distortions, although they did confirm some of these results. In the following we discuss how the two approaches compare for each case. 4.4.1. YBCO-123 As we have previously discussed, the model of a split apical oxygen site proposed by EXAFS analysis is probably in error, at least for fully oxygenated YBCO-123. As shown in Fig. 29, (63) the atomic density map determined by a single crystal X-ray diffraction does not show large displacements of apical oxygen 04. This result is reasonable since the distance between the apical oxygen 04 and the chain copper Cul (= 1.84/~) is smaller than the expected C u ~ ) distance (1.9/~). If splitting of 04 occurs this distance has to be further reduced, which is unlikely. On the other hand, crystallographic analyses indicate in-plane lateral displacements of apical oxygen and the chain oxygen.(63'64'98)These results are consistent with the pulsed neutron observation on a similar compound, YBCO-124.(92)Figure 29 also shows the thermal ellipse of in-plane Cu2 elongated along the c-axis, suggesting some vertical displacement of Cu2. It is still possible that the anomaly of the C u ~ ) bond occurs not by the displacement of 04 but by the displacement of Cu2. 4.4.2. LSC0-214 A recent single crystal neutron diffraction study(99) showed that the thermal factors for La2 CuO4 are normal. For La2_ xSrx CuOa the thermal factors are much smaller than reported earlier. However, thermal factors increase rapidly with concentration, perhaps more than

390

P R O G R E S S IN M A T E R I A L S S C I E N C E

f~.

-"~ . . . . . . . . .

~..

Cu(2)

Z

O(1)

1A

Cu(l)

FIG. 29. Atomic density map of YBa2Cu306.s7 along the c-axis determined by single crystal X-ray diffraction363)

expected for size effects. Thus, crystallographic results are not in conflict with minor local displacements in LSCO-214. 4.4.3. T/-2212 The short range reconstruction of the T1-O planer reported by pulsed neutron PDF analysis (43) is consistent with the large thermal factor found in crystallographic (Rietveld) analysis. (69) Similar displacements exist in the Bi-O plane of BISCO-2212. ('1°°'~°~)

LATTICE

EFFECTS IN HTSC

391

4.5. Modes of Local Atomic Displacements We will now summarize the common features observed in the pattern of atomic displacements in HTSC oxides. 4.5.1. Apical oxygen The e-axis displacement of the apical oxygen was first reported by EXAFS measurement for YBCO-123, but as discussed above--at least for fully oxygenated YBCO-123--this result is probably incorrect. Indeed, this displacement has not been observed by the pulsed neutron PDF analysis of YBCO-123, except when it is oxygen reduced. On the other hand, the splitting of the Cu-apical-O distance was observed by PDF analysis in T1-2212~79) and in LSCO-214393) In YBCO-124 the apical oxygen moves in the b-direction, perpendicular to the chain and the c-axis. Thus, even though the presence of the anomaly in the Cu to apical-oxygen distance in YBCO-123 is in doubt, such a splitting may exist. But it certainly is not universal, and its general importance to the mechanism is somewhat questionable. 4.5.2. In-plane oxygen A change in the Cu~O bond length in the Cu~5) plane, for instance, induced by the breathing mode of lattice vibration or the in-plane longitudinal optical (LO) phonon mode, couples directly to the local charge densitY on Cu or O. Initially the electron-phonon coupling through this breathing mode was suspected to be the primary contributor to superconductivity31°2) Since the frequency of this LO mode is high (80 meV), Tc can also be high when a strong coupling to this mode produces superconductivity. However, there is no strong indication that it is indeed the case. No anomalous change in the in-plane Cu~O distance is seen by PDF analysis. Longitudinal displacements beyond the phonon amplitude are either absent or very small, less than 0.02/~. However, there are indications of dynamic anomalies, as we mention later. On the other hand, in-plane oxygen ions were often found to be displaced in the direction perpendicular to the plane. The magnitude of displacement is about 0.03 ~, for YBCO-124 and more for LSCO-214. These displacements bend the C u ~ ) - C u bond, and thus bring about the mixing of the O-p, state to the C u 4 ) cr bond. The consequence of this mixing on the electronic structure will be discussed later. 4.5.3. Oxygen ions in other layers Oxygen ions in the charge reservoir layers have been seen to be displaced in the plane as shown in Fig. 26. However, in the case of T1-2212 and BISCO-2212 this is the consequence of the ionic size mismatch, and is inconsequential to superconductivity. 4.5.4. Displacement of other ions Cu ions have been observed to be displaced in the direction opposite to the displacement of the nearest oxygen ion. However, the magnitude of displacement is usually a half or less of that of oxygen ions. Heavy ions such as Ba or Bi do not appear to be displaced much from the crystallographic position, and thus are not participating in the electron-lattice interaction. 4.5.5. Spatial range of correlations It is important to note that at least at low temperatures the observed displacements are not random, but locally well-correlated and coherent, often resulting in new, well-defined peaks in the PDF. Also, not all the atoms are displaced. Nearly half of the atoms are located

392

P R O G R E S S IN M A T E R I A L S SCIENCE

at the average crystallographic sites. At low temperatures well-defined coherent displacements occur collectively in a small domain of 10-20/~ size. Thus, the total system appears to be made of micro-domains of two phases: with and without local atomic displacement. The dynamics of the displacements change near or above To, as we will discuss in more detail later. Particularly, the oxygen-oxygen correlation seems to behave anomalously with temperature. We first have to emphasize that the displacements observed are not expected due to ionic size effects. Even though we observe only the nuclear positions by neutron scattering, these observations strongly suggest that charges must be involved in producing these displacements, as we will discuss later. 4.5.6. Composition dependence While there are very few systematic studies of the effect of composition on local displacements, available data suggest that composition is likely to be an important factor. For instance deviations from the crystallographic structure are small in fully oxygenated YBCO-123, but they increase as oxygen is reduced. A part of this may be simply due to disorder introduced by oxygen vacancies, but it is also possible that the deviations are smaller when T~ is higher. This possibility is most clearly demonstrated in Fig. 22. Here disorder due to the Sr doping should increase linearly with the Sr concentration, but the local strain is minimum when T~ is highest. While more data have to be compiled before a conclusion can be derived, this offers a very interesting possibility with regard to the role of lattice distortion in the HTSC phenomena, as we will discuss later. 5. LATTICE ANOMALIES NEAR T c

5.1. Lattice Dynamics 5.1.1. Phonons In the conventional superconductors the onset of superconductivity alters the phonon frequency and the phonon lifetime. Some of the observations in HTSC oxides can be interpreted as the normal consequence of the electron-phonon interaction, as we discussed above. However, there are some anomalies as well. For instance, in YBCO-123 the phonon mode at 340 cm- l (= 43 meV) corresponding to the buckling of the C u ~ ) plane is strongly affected by the onset of superconductivity.0°3) If the energy of this phonon corresponds to the superconducting gap, then the ratio 2A/kTc is more than 5, much larger than expected for the BCS superconductor (=3.5). One technique that measures the total dynamic response of ions in the lattice, including localized and extended phonons, is neutron resonance absorption spectroscopy (NRAS). In this technique, the Doppler broadening of resonant absorption peaks gives the total kinetic energy associated with a particular ion. By studying aligned samples it is also possible to extract directional information. The kinetic energy of Cu in a sample of BISCO-2212 °°4) showed normal behavior in the case of the c-axis vibrations, but a distinct and large (20%) softening at Tc for the in-plane vibrations.

5.1.2. 1on channeling A beam of inert ions, such as He ions, directed into a crystal is scattered by colliding with atoms (Rutherford scattering). When the direction of the ion beam is parallel to an atomic plane in the crystal the ions can slide into a crystal by channeling, reducing the back-scattering

LATTICE EFFECTS IN HTSC

393

I

<3

0.03 h_ C~

a_ 0.025 ~- 0.02

vO

° 0.015

T <~ 0.01 0

100 200 Temperoture, K

300

FIG. 30. Temperature dependence of the PDF peak height at 3.4 A for T12Ba2CaCu20s. The solid line describes the expected temperature dependence calculated from the phonon density of states. The arrow indicates superconducting transition temperature/79) intensity. If, however, the direction of the ion beam is not completely parallel to the plane of the crystal but at a small angle to it, the back-scattering intensity is only partially reduced. Thus, as a function of angle, the back-scattered intensity exhibits a dip. The width of this dip, measured as the angular full width at half m a x i m u m ( F W H M ) depends upon the lattice dynamics and the defects in the atomic plane. The angle deviation can be interpreted in terms of the apparent phonon amplitude. Sharma e t al. °°5) found that the F W H M value of channeling changes abruptly at To, and from this result concluded that the apparent phonon amplitude changes significantly at Tc for YBCO-123. Y a m a y a e t al. (1°6) found that the change occurs primarily for Cu and O. The results by Sharma e t al. suggest that, below To, displacement amplitudes perpendicular to the c-axis (either static or dynamic) of ions in the Cu(1)-O(4)-Cu(2) columns are getting smaller, or more highly correlated, or both. This conclusion of lattice hardening below Tc is apparently at odds with the conclusion derived from the N R A S experiment which suggests lattice softening. Furthermore, the sense of the change found by Y a m a y a e t al. is opposite

1.0

-l-

a) o_

0.5

r

~

0

50

i

r

i

4

1oo 150 200 250 Temperature [K] FIG. 31. Temperature dependence of the PDF peak height at 4.3 ,~ (A), 3.88/~ (Fq), 5.38/~ (O), for YBa2Cu40s compared to the expected temperature dependence calculated from the phonon density of states (solid line).(92)

394

P R O G R E S S IN M A T E R I A L S S C I E N C E

to that found by Sharma et al. On the other hand, the measurements of macroscopic elastic constant and phonon frequencies suggest only small changes. This conflict points to the need for further careful experimental studies and the danger of interpreting these results in terms of usual harmonic phonons. The phenomena are subtle and are likely to be local and inhomogeneous. Different experiments which are sensitive to quantities volume-averaged in different ways can give different answers. 5.2. Structural Anomalies 5.2.1. E X A F S results The first indication of structural anomaly near Tc was observed by an EXAFS experiment on YBCO-123/65) The most widely known case is the anomaly in the dynamics of the apical oxygen of YBCO-123. (66)The results were interpreted in terms of tunneling between two sides of the double-well potential and the change of the tunneling barrier height at To. As the origin of the double-well potential, Mustre-de Leon et al. proposed a many-body mechanism involving non-adiabatic electron-lattice interaction and strong electron-electron correlation, rather than lattice anharmonicity.°°7) Similar anomalous double-well behavior was seen also in TI-2201. (76) 5.2.2. Neutron scattering Anomalous temperature dependence of the PDF peak height was observed for a number of compounds. Among them the observation made for one peak of TI-2212 (79) is the best and most unambiguous case. As shown in Fig. 30, the height of the peak at 3.4/~ changes anomalously near the superconducting transition temperature of 110 K, shown by an arrow. Here the solid line is the normal temperature dependence calculated from the phonon density of states determined by a neutron inelastic measurement. The deviation of the data points from the expected normal behavior near T¢ can be concluded with the 99.999% statistical 6000

5000

•

4000

O•00

o~

"-'- 3000

2000

1000

0

100

200

300

400

500

Temperature[KI FIG. 32. Temperature dependence of N M R spin-lattice relaxation rate,

I/T~, for YBa2Cu40 s.(l°s)

LATTICE

EFFECTS

IN HTSC

395

confidence. If one attempts to interpret this change in terms of the change in the phonon frequencies, the magnitude of the change in the P D F corresponds to the change in the phonon frequency by as much as 20% or so. While some changes in lattice dynamics are expected at To, the changes usually are much smaller, amounting only to about 2%, as we discussed above. Since the change in the P D F peak height cannot be explained in terms of the change in the phonon frequency, it is appropriate to call this change an anomaly. In other compounds the anomaly near Tc is often found, but it tends to be weaker than in the case of T1-2212. In the case of YBa2CuaO 8 (124) the anomalous change was observed at temperatures significantly above Tc .e2~ As shown in Fig. 31, while the peak at 3.88 A, which corresponds to the a and b translation, and the peak at 5.38/~ show normal temperature dependence, the peak at 4.3/~ showed a rapid decrease in the peak height, followed by saturation at temperatures above 200 K. It is very interesting to compare the observed temperature dependence with the temperature dependencies of the Cu N M R spin-lattice relaxation time, 7"1, shown in Fig. 32, ~1°8~the Cu Knight shift, and the resistivity divided by T3 mgl Below around 160 K (To = 80 K), 1/TI rapidly decreases with lowering temperature. This decrease in 1/T~ corresponds closely to the change in the height of some of the peaks of the P D F of YBCO-124, including the peak at 4.3 A. Since the nuclear spin relaxation rate, I/T~, is proportional to the imaginary part of the spin susceptibility of the system, X "(q = 0, 09 --+ 0), this behavior is usually interpreted to indicate an opening-up of the spin gap. Indeed, the neutron inelastic scattering measurement detects a decrease in X "(q, 09) below this temperature in La2_ ~SrxCuO4 and YBa2Cu307_ a.° ~0) This opening of the spin gap is a consequence of the increase in the Cu spin-spin correlation, which, in the spin fluctuation theory, I'~H'2~ is considered to provide the mechanism for high temperature superconductivity. However, the P D F results suggest that the spin gap may be, at least in part, structural in origin and that this phenomenon is also a part of the lattice effect, as discussed below. The anomalies in the P D F are dynamic in nature. If the neutron scattering experiment is conducted such that the transfers of momentum and energy are resolved, for instance, by using a triple-axis spectrometer, the neutron scattering cross section is given by: k

21

c~coOf~ - k

'

N (b )2S (Q, oo),

(22)

where N is the number of atoms, ( b ) is the compositionally-averaged neutron scattering length, k is the momentum of incident neutron, k' is the momentum of the scattered neutron, Q = k ' - k , ho~ is the energy transfer; S(Q, ~o) is the dynamical structure factor:

S (Q, co) - N (b)~ ~ bibj e x p ( - ie)t )((exp{iQ. [ri(0) - rj(t )]}))dt,

,

ij

f

(23)

d

where (( . . . . )) denotes an ensemble (quantum and thermal) average, b~ is the neutron scattering length of the i-th atom, and ri(t) is the position of the i-th atom at time t. ~'3~ In the pulsed neutron powder scattering experiments, discussed above, the energy transfer is not resolved. Thus, the measured neutron scattering intensity is given by:

am (Qo) = ~S (Q (~),

(.o ) d(.o .

(24)

Here Q weakly depends upon ~o, and Q ( 0 ) = Q0 = 2k sin O. This ~o dependence was first discussed by Placzek. m4) If for the moment we neglect this co dependence and integrate eq. 24 from - oo to + oo, then integrating eq. 23 leaves only t = 0. Thus, the total structure factor describes the instantaneous correlation:

396

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Stotal(a ) = ~

MATERIALS

SCIENCE

E b, bj((exp{ ia . [r;(0) - rj(0)]})).

(25) ij On the other hand, if only the ~o = 0 component is measured, ri and rj are temporarily decoupled, so that: 1

S (Q, 0) = N ( b ) 2 i.~, bibj((exp[iQ, r~]))((exp[- iQ rj])),

(26)

which describes the correlation among the average atomic positions. However, in the pulsed neutron time-of-flight diffraction experiment Q is dependent on ~o. The effect on Sm(Q) can be approximated by:

Sm(Q ) =

f0

s (Q, o))&o, (27) ~0 where 090 is an effective window of acceptance within which Q (09) is not significantly different from Q (0). The width of the window depends upon the diffraction angle 0. The window is wider at lower diffraction angles. Its Fourier transform by eq. 19 thus describes the average distances for the pairs of atoms oscillating faster than ~o0, and the instantaneous distances for the pairs of atoms oscillating slower than 090. The width of the window is in the order of 10 meV for the pulsed neutron diffractometer (Special Environment Powder Diffractometer (SEPD) o f Intense Pulsed Neutron Source (IPNS) at Argonne National Laboratory) used in these measurements, but further analysis is underway. It was found that the magnitude of the apparent anomaly depends upon the diffraction angle 0. The result given in Fig. 30 (79) is obtained mainly by using the scattering data of the detector bank at 0 = 45 °. When the data from the detector bank at 0 = 75 ° is used (83) the anomaly is smaller. Furthermore, if the inelastic component is eliminated by using a triple-axis spectrometer, the result is much different. Such a measurement was performed for T1-2212 using the high temperature neutron source of Institut-Laue-Langevin. Figure 33 shows the elastic structure factor, S(Q, 0), given by eq. 27. (86) Its Fourier transform, 0,3-,[,..,

I . . . . . . . .

j ....

4 0.2

i ............ 5

10

15

20

i

.

..

i r,

FIG. 33. Elastic structure factor, S(Q,0), for T12Ba2CaCu208 at T = 70 K determined by neutron FIG.34. Elastic (averaged)PDF (solid line) and instanscattering with a triple-axis spectrometer and a high taneous PDF (dashed line) for T12B%CaCu208 at temperature neutron source. T = 120 K (top), 90 K (middle) and 70 K (bottom).(86)

LATTICE EFFECTS IN HTSC

397

1

Pet(r)

=

P0

2~2---"~.f[S (Q, O) - e-2Be2]sin (Qr )QdQ,

+

(28)

describes the distances between the atomic positions regardless of time correlation,

jp,(r )p,(r + r')dr',

1 ('

pel(r) = ~

(29)

where p,(r) is the single atom density function. Comparison of the PDF obtained by the powder diffractometer at IPNS and the elastic PDF shown in Fig. 34 shows significant differences at 3.2 # distance, where the anomaly was observed. Also, the temperature dependence of the peak height shown in Fig. 35~86)indicates that while the peak height of the instantaneous PDF shows anomaly at Tc, the peak height of the elastic PDF does not. Thus, it is clear that the anomaly in PDF is produced by the change in the dynamic displacement of oxygen atoms. A further piece of evidence for a dynamic lattice anomaly is suggested by the more direct measurement of the dynamical structure factor S (Q, ~o) by the inelastic neutron scattering conducted by Arai et al. "15) Using the neutron chopper (mechanical monochromator) and time-of-flight method at the ISIS pulsed neutron source at the Rutherford-Appleton Laboratory, these authors saw a change in S (Q, co) above and below Tc in fully oxygenated YBCO-123. The change is small but is larger than the experimental errors and was reproduced in two runs carried out with different incident neutron energies. From an analysis of the Fourier transform of S (Q, co), the authors interpret the changes as coming from short-range vibrations whose dynamical correlation length diverges right at To. The importance of this interpretation is that it appears to be a local mode which couples strongly to the electrons, presumably associated with the local lattice distortions which have already been discussed. Localized lattice distortions allow for the possibility that a very strong and non-linear electron-lattice coupling can exist without inducing a macroscopic phase transition. A similar study made recently on YBCO-124 shows further evidence of dynamic anomaly. As shown in Fig. 36 the Q-integrated structure factor,

S (oa ) = s (o, oa)da,

(30)

shows changes with temperature. As shown in the inset, the portion at 50 meV changes appreciably near 80 K (= To), while the portion abound 70 meV shows a change at 160 K. ° 16) '

I

'

*

'

L

I

'

'

'

'

• ILL

I I

a IPNS

CO O

c4

O,2 O O

i

0

r

i

i

i00

200

300

Temperature, K FIG. 35. Temperature dependence of the PDF peak height at 3.4/~ for T12Ba2CaCu208 for instantaneous (open square) and averaged (closed circle) PDFs. ~s6)

398

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I

I

I

I

I

40

80

120

160

200

¢9

•

T=40 K

--T=80

K

"--~-

T=120 K

"®"

T=200 K

I

I

I

I

I

40

50

60

70

80

90

(o (meV)

FIG. 36. S(~o) integrated over Q from 1 to 20/~ -~ for YBazCu4Os. ~tl6) The inset is the intensity at 50 meV.

Also, the comparison of the elastic PDF obtained by eq. 28 and the instantaneous PDF obtained by Fourier-transforming eq. 25 are given in Fig. 37. (116~Differences are larger than the statistical errors, and suggest the presence of correlated dynamic local distortions in this solid.

5.3. Other Lattice Anomalies Clear evidence for coupling of a lattice instability to the electronic system has come from elastic constant measurements on single crystal Lal.86Sro~4CuO4 by Nohara et al. <~7~using a high-resolution ultrasound technique. The evolution of the CH-CI2 mode (in orthorhombic notation) with temperature is shown in Fig. 3 8 .
LATTICE EFFECTS IN HTSC

399

0.2

YBa2Cu408

T=40 K --

TOTAL

0.0

2

3

4 r [A]

5

6

FIG. 37. Elastic (averaged) PDF (dashed line) and instantaneous PDF (solid line) for YBa2Cu40 ~ at T = 10 K. ° 161

fields. With H parallel to the c-axis Tc is shifted to lower temperature and the lattice hardening follows as can be seen in Fig. 38(c). None of the other modes that were studied showed this anomalous softening and hardening. This is a very clear demonstration of the lattice coupling to the superconductivity. It draws to mind very similar observations in the A-15 class of compounds. 0~8) Testardi and Bateman showed that in V3Si the softening of a shear mode, which ultimately leads to a martensitic transformation, is arrested by the appearance of superconductivity. Similar behavior was also found in Nb 3Sn and ultimately in all of the ,4-15s which had high Tcs, whereas similar behavior was not observed in the low Tc A- 15s. tits1 This led to the belief that phonon softening associated with the instability was instrumental in T¢ enhancement. It is interesting, however, that a lattice softening strong enough to produce a structural transition in these materials enhances T¢ by as much as 17 K. While the lattice softening alone may not be able to explain the high T~ of LaLs6Sr0.14CuO4, the ultrasound results provide an important demonstration of a strong electron-lattice coupling in these materials. Two recent reports appear to be consistent with the local lattice distortion described above. One is the observation by scanning tunneling microscope (STM) of inhomogeneous charge distribution on the surface of YBCO (123) which appears like local charge density wave ( C D W ) . 019) Usually CDW or polarons are mobile, but at the surface they could be pinned by surface irregularities. The other is an observation of temperature dependent diffuse scattering by electron diffraction on T1 (2212)312°) In addition, a recent high-resolution imaging study by transmission electron microscopy (HRTEM) shows some very weak but unmistakable contrast of micro-domains which could be due to charge inhomogeneitiesJ ~2L>

5.4. Nature of the Lattice Anomalies The evidence of unusual or anomalous temperature dependence of some structural characteristics near Tc is widespread and unmistakable. The evidence gives arguments in favor of a strong electron-lattice interaction, of some nature, contributing to superconductivity in these materials. However, details of the results project an unsatisfactory and confusing picture. Before we proceed any further, let us examine whether it is justified to call these phenomena anomalies or not. Furthermore, if it is indeed anomalous, we must differentiate quantitative and qualitative anomalies. In other words, in one case just the magnitude of the

400

PROGRESS 98.9

IN MATERIALS

SCIENCE

F Single Crystal

98.8

(a) H // 11101 98.7

98.6

L

(b)

;;; 98.8

2

H/I [liO]

9T

,

8

1

I

98.7

C? I

5

98.6

(c) H // [OOl]

98.8

98.7 t

98.6

98.5

I

0

20

40

I

60

D

T (W FIG. 38. Temperature dependence of the elastic constant (C,, - C,,)/2 for

La,,,,Sr,,,,CuO,measured

in various magnetic fields shown in the figure. The arrows indicate Te.(“‘)

effect is large and anomalous, while in the other the phenomenon itself is unusual. Finally, we have to judge whether these effects are merely an interesting but unimportant consequence of unconventional superconductivity, or whether the lattice effect is directly, and essentially, involved in the mechanism of superconductivity in some unconventional sense. 5.4.1. Magnitude of the e@ct If some of these observations are interpreted in terms of harmonic lattice vibrations, the observed phenomena correspond to changes in the elastic constant by 20% or more, while changes of only a few % are expected for conventional mechanisms. Thus, the effects are at

LATTICE EFFECTS IN HTSC

401

least quantitatively anomalous. Furthermore, the sense of the change is utterly confusing. As we mentioned earlier, the NRAS experiment suggests that the lattice softens below Tc, while the ion channeling result suggests the opposite. Therefore, the effect cannot be interpreted in terms of changes in the elastic constant. 5.4.2. Dynamics of the anomalies The anomalies near Tc themselves are subtle, and each method has to be pushed almost to its limit of detection to observe them. Consequently, while the presence of anomaly itself appears to be firmly established, reliable information regarding their dynamics is scant. For instance, the dynamics of the double-well potential were deduced from EXAFS data, ~66)but the result depends upon many assumptions, and was contradicted by crystallographic measurements. However, neutron experiments described above provide good confidence that the phenomena are dynamic and anharmonic. The energy range of Eg ~ 50 meV determined from the S(Q, e~) measurement on YBCO-123 ~j~5) and YBCO-124 ~6) gives a good measure of the energy scale at which the electron-lattice coupling is very strong. Strong electron-phonon interaction is expected at the phonon frequency corresponding to 2A, where A is the superconducting energy gap. However, the ratio Eg/kTc = 6.24 for Eg = 50 meV is far larger than the value expected for weak coupling BCS theory, if Eg corresponds to the superconducting energy gap. Interestingly, various examples of gap-like behavior have been observed at the energy 6-8 times kTc, as listed below: 1. IR pseudo gap starts at about 8 kTc .022) 2. Tunneling gap is also often 6-8 kTc .~23) 3. Strong phonon frequency renormalization observed by Raman scattering is also in the range of 5-6 kTc.~J°3) 4. The superconducting energy gap in BISCO-2212 observed by photoemission is strongly anisotropic, but in the direction in which the gap is maximum it is about 50 meV ~ 6 kTc .~124) These results strongly suggest that the lattice dynamics in the range of 50 meV are strongly coupled to superconductivity. This point will be discussed later. 5.4.3. Coupling to electrons The observed patterns of ionic displacements involved in the anomalous behavior suggest that they occurred as a consequence of interaction with charge carriers. There are three types of displacements: 1. Displacements that produce local polarization, such as the case of chain oxygen displacements in YBCO- 123 and - 124. 2. Apical oxygen displacements along the c-axis. 3. Displacement of in-plane oxygen in the e-direction. The first type clearly indicate the existence of an electric field that induced the polarization. Thus, it is natural to assume that these oxygen ions are responding to the charge motion which is slow enough to cause local polarization. The second t y p e of displacement would induce changes in the local Cu charge, and thus couple to the charge fluctuation. The third type involves a decrease in the O - O distance as a result. In YBCO, LSCO, and T1 or Bi compounds, the distance between the apical oxygen and the in-plane oxygen is reduced. In N d - C e - C u - O the distance between the in-plane oxygen and the oxygen in the rocksalt layer is reduced. In each case, since oxygen ions are negatively charged it is difficult to understand

402

P R O G R E S S IN M A T E R I A L S S C I E N C E

Laz_xBaxCu04 I

I

I

I

I

I

400

300

Tetragonal (HTr)

200 0rthorhombic

\

(LT0) \

100 T- . T~4-/(,. ~_

/f I 0

~ i

i

0

0.05

~?

,,{

Tetragonal (LTT) " I

I

I

0.I0 0.15 0.20 Composition [x]

[

0.25

FIG. 39. Structuralphase diagram of La2_xBaxCuO496)

why they come close, unless we assume that they trap a hole in-between. In other words, the electronic levels associated with these oxygen ions cross the Fermi level and become unoccupied as a result of the deformation. Thus, the nature of the atom displacements themselves tell the story: they are a result of electron-lattice interaction, and reflect the fact that the charge dynamics in the system are slow enough for the lattice to follow. Then the crucial question is whether such slow charge dynamics are the direct consequence of the interaction with the lattice, or occur for a completely different reason, and the lattice is merely following them.

6. STRUCTURAL PHASE TRANSITION One way of acquiring indirect but important information regarding the mechanism of superconductivity is to learn what suppresses it in an unexpected way. In La2 x(Sr, Ba)xCuO4 two kinds of structural phase transformation have been suggested to suppress superconductivity. One is the transformation from orthorhombic (LTO) phase to tetragonal (HTT) phase, and the other is the transformation from LTO to low-temperature tetragonal (LTT) phase. In both cases the situation is not as simple as was initially thought, and even after numerous studies there is still no clear evidence that these transformations are directly linked to the disappearance of superconductivity. Nevertheless, there are ample indications that these phase changes do affect superconductivity in some manner. We will first review what has been learnt so far, and then discuss the implications.

LATTICE

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403

6.1. Structural Phases in Laz_ x(Sr, Ba)xCuO 4 The structural phase diagram for the Ba-doped material is shown in Fig. 39. At high temperature and/or high Ba concentration, the sample is in the high-temperature tetragonal (HTT, space group I 4 / m m m ) phase ~° and the C u - O planes are crystallographically flat. On cooling, it undergoes a second order transition into the LTO phase. °25) Finally, over a restricted composition range the sample undergoes a further transition to the low-temperature tetragonal (LTT, space group P4z/ncm) phase. 026'127~These transitions have been understood as arising primarily from nearly rigid tilts of the CuO6 octahedra accompanied by small strains which distort the octahedra. ~128)In the LTO phase the octahedra tilt about [t 10] axes in the tetragonal indices, referred to the Abma space group. At the L T O - L T T transition the average octahedral tilts reorient through 45 °, the tilt axes now being about [010] directions.

6.2. H T T Phase There has been speculation that superconductivity may not occur in the H T T phase, since the superconducting and structural phase boundaries seem to be in the same region at high doping (compare Figs 1 and 39 in the region x ,~ 0.2). The existence of superconductivity is well established in the LTO phase. Some superconductivity can also be seen beyond the phase boundary on the H T T side; however, less than 100% superconducting fractions typically are reported and low Tc s are observed. One view is that the primary reason for the disappearance of superconductivity at large values of x is overdoping, and the presence of structural phase boundary at the same region is a mere coincidence. The other view is that the deviation from tetragonal symmetry is vital for superconductivity to occur. While the conventional view is the first one, the second view has some reasonable grounds as well. For instance, in highly doped La-214 samples N M R and magnetic susceptibility measurements report observation of a spin pseudo-gap, considered to be characteristic of an "underdoped" HTS materialf129"13°)This suggests that these materials are not overdoped, yet Tc is falling to zero. Takagi et al. ~la° gave support to this view by studying samples which were annealed at high temperature in oxygen for I month. The authors found that in their well-annealed samples the superconducting fraction (as measured from the Meissner signal in a 10 Oe field) was significantly lower in the H T T phase than in the LTO phase. The residual superconductivity in the H T T phase samples could be explained, even after these long anneals, if the samples were not fully homogenized or a second phase had precipitated. The authors argued that in principle, an ideal anneal, which would be impractically long, would homogenize the samples completely and no superconductivity would be seen in the H T T phase. However, this interpretation has been questioned and evidence now exists that demonstrates bulk superconductivity in the H T T phase. The most convincing work is by Nagano et al. °321 These authors prepared samples using a spray-dry technique which produces an intimate mixture of reagents on a microscopic scale. These samples were reacted and also given a post-anneal to enhance homogeneity. Careful susceptibility measurements were carried out in extremely low fields (0.20e). These authors demonstrated a full shielding fraction ( > 100%) for all samples up to x = 0.25, well beyond the LTO H T T transition. By introducing P r 3+ into Sr doped La-214 samples, Schfifer et a l . 033) w e r e able to shift the structural phase boundary to higher Sr concentrations. However, the superconducting boundary fell to T = 0 K at the same Sr (and therefore hole) doping level as in the material with no Pr. The superconducting transition temperature appears to be a function of hole concentration alone and independent of the structural phase transition.

404

PROGRESS IN MATERIALS SCIENCE

These results clearly demonstrate that superconductivity can occur in the tetragonal structure, and the phase transformation itself does not suppress superconductivity. However, it is important to note that the relationship established here is between the crystallographic symmetry and superconductivity. Even when the long-range order is tetragonal, locally the symmetry around ions can be lower. As we discussed earlier, the important structural condition is not necessarily the symmetry of the long-range crystallographic structure, but the structure within the superconducting correlation length. In this regard it is important that the PDF study finds lower symmetry even in the tetragonal phase, just beyond the phase boundary. (89) In other words, the tetragonal structure just beyond the phase boundary may well be made of randomly oriented orthorhombic micro-domains. As long as the size of the micro-domains is large enough to sustain superconducting correlation this deviation from tetragonality may well be important still to superconductivity. Thus the question still remains open. It is interesting to note that there is a report of a correlation between a feature in the normal state c-axis resistivity in Sr-doped La-214 single crystals and the H T T - L T O transition. 034,135) A distinct upturn in Pc is observed at a temperature which seems to correlate with the H T T - L T O transition temperature. No such effect is seen in the in-plane resistivity. While this correlation has not been established for wide enough ranges of composition, it is interesting that a metal-insulator (MI) transition is seen in the c-axis conductivity in the region of doping where samples superconduct. The M I transition temperature decreases with increasing doping until at high doping, apparently, it merges with the superconducting transition temperature where superconductivity disappears. This interesting result appears to suggest that quasi-two-dimensional nature is necessary for superconductivity to occur. In the superconducting region of the phase diagram the anisotropy of conductivity, Pc/P~b, is highly temperature dependent. (134) Nakamura et al. (]34) argue that this demonstrates that the scattering mechanisms for c-axis transport are different from the in-plane scattering mechanisms and conductivity is two-dimensional. In the overdoped region the anisotropy parameter becomes temperature independent. Metallic conductivity is observed along both ab and c directions, indicating the dominance of the same La2-xBaxCu04 i

t

i

I :~.

A i ~1

100

/'\.

? 00

f

A

i i i A/

60

I

i

A~ ~ ~~ ] -~ i ~ i

25 20

/

..... iA t i

o

>

ira9

40 I

!

I

i <---/--- ~ i

20 o

0

i

A

~ ~i ~i

10

A

A~:~

'"".A ,

x o

I

I I

o

0.05

I

I

0.10 0.15 0.20 Composition Ix]

~

I

0.25

FIG. 40. T¢ and the volume fraction of the LTT phase vs x for La2_,BaxCuO496)

405

L A T T I C E EFFECTS IN HTSC

scattering mechanism parallel and perpendicular to the C u ~ planes and a crossover to three-dimensional behavior. At the same time, superconductivity is suppressed. This tends to suggest that the two-dimensional nature of the crystal structure is a key component in the properties. However, we should point out that in fully doped YBCO-123 (with Tc = 90 K), the c-axis conductivity is metallic. °36) Thus, this conclusion is not universal.

6.3. L T T Phase

As shown in Fig. 40, in the Ba-doped La-214 material at a hole doping concentration, p, of exactly p = 1/8 (x = 0.125), superconductivity is suppressed; (137)whereas, at compositions on either side of this magic fraction, samples show almost optimal bulk superconductivity 2500

2000

0K i~ 1500

1500

"~ 1000

60

I

50O

0

1.0

1.330

-

i

1.335 1.340 d-space (A)

-

-

-

.

-

-'B-"-~ ~ ~

-

-

,

-

-

-

1.345

0

1.335 1 . 3 4 0 d-space (A)

1.345

0

(e)

oO0~: (d)

o 0.8

0

0.6

-0.02

0.2 ~I~_~

0.4

0

1.330

•

20

.

40 60 80 Temperature (K)

10O

~ -0.04 -0.06 -0.08 -0.10

20 30 Temperature (K)

10

40

FIG. 41. (4,0,0) and (0,4,0) Bragg diffraction peaks by high resolution X-ray scattering, for La2_xBaxCuO 4 (a) x =0.125, (b) x =0.15. The fraction of the LTT phase determined from these patterns is plotted as a function of temperature in (c), for x = 0.125 (open square), 0.15 (filled circle), compared to magnetic susceptibility (d). While the superconducting fraction (d) is different between these two samples, LTT fraction is virtually the same, suggesting that the appearance of the LTT phase does not affect superconductivity. "45)

406

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with T~s of ~27 K. This observation has been correlated with the appearance of the LTT phase which occurs in the same region of the phase diagram. (u6'127) The LTO-LTT transition is first order and samples do not transform fully into the LTT phase. Axe et al. 026) postulated that the LTT phase is not superconducting. In this picture, at compositions on either side of x = 0.125, the superconductivity comes from residual LTO material; whereas, exactly at x = 0.125 the sample fully transforms and superconductivity is annihilated. Further credibility was lent to this view by measurements of the normal state properties. °27'm) An electronic transition was evident in x = 0.125 samples at a temperature of approximately 60 K, very close to the LTO-LTT structural transition temperature: thermopower changes from positive to negative, resistivity shows an abrupt upturn and Hall resistance a sharp drop at this temperature. Much theoretical effort has been applied to explain why the LTT structure does not superconduct whereas the LTO structure d o e s ; (]39-143) indeed, some theories postulate that the LTT structure itself is stabilized by the electronic transition. "39~ However, subsequent experiments°44']45) show beyond doubt that the electronic and structural transitions are not uniquely coupled. First, it was demonstrated in closely related doubly doped (La, Sr, Nd)-214 materials that the LTT structure does support superconductivityY44J46) In these materials the average dopant ion size (which controls the structural phase transitions "46)) can be varied independently of the hole concentration (which controls the electronic transition)047) by controlled doping of, for example, strontium and neodymium instead of Sr or Ba alone. Superconductivity is observed in the LTT phase in these materials. (144'146)Furthermore, in pure La2_~BaxCuO4 it has also been shown recently that the LTT phase can superconduct perfectly well. This has been demonstrated quantitatively by comparing the phase

' LT+-Ri'etvild 6 0 0 "- - - L T O - R i e t v e l d

'(c)

'(a)

,

'i

'(b)

II

400 200 0 -200 -400

V

-600

I

IV I

I

x=0.125,10K 600 "---x=0.125,80K

I

v-v I

I

~ I

"V'" I

/X._

""-

V I

I

.,,~¢'X A - .

-v-"VV

I

I

I

I

(f)

(e)

(d)

400 200 0 -200 -400 v

O"V

-600 |

0.20

0.30 r(nm)

0.40

0.60

0.70 r(nm)

,

|

I

0.60

1.00

I

!

1.10 r(nm)

I

I

1.20

F]G. 42. Pulsed neutron P D F s from LTO and L T T structures. (a-c) show model P D F s calculated for ideal structures. (d-f) show the actual P D F s for the LTO phase (dashed line) and the L T T phase (solid line) for different distance ranges. Differences predicted by the models are not observed in the experimental data. 049)

LATTICE

EFFECTS IN HTSC

407

composition of a bulk superconducting sample with x = 0.15 with that of a non-superconducting x =0.125 sample. The phase composition was determined by curve fitting to high-resolution X-ray diffraction data collected at a synchrotron source, shown in Fig. 41(a) and (b). Within the errors of the measurement, both samples are 95% transformed into the L T T phase at low temperature although the samples had very different superconducting properties (Fig. 41(c) and (d)). ~45) The L T T structure supports bulk superconductivity with a T~ of 27 K in this system. Finally, Nagano et a l Y 2) recently found a sharp superconducting anomaly, similar to that observed in (La, Ba)-214, was induced in microscopically homogenized Sr-doped samples with x = 1/8. No structural transition to the LTT phase has been seen in this system suggesting that the electronic transition can occur in the LTO phase. These experimental results together indicate that the structural and electronic transitions are not coupled directly. This understanding is placed in a new light by studies of the local structure of (La, Ba)-214 which are described below. The superconducting anomaly at p = 1/8 in the La-214 materials still begs for a satisfactory explanation. It is interesting that FtSR measurements indicate that the Cu spins develop static (on a microsecond time scale) antiferromagnetic order at low temperature at this hole concentration, ~H8~though no evidence of this has been seen in neutron diffraction experiments. 6.4. Local Structure The local structure of the LTO and L T T phases has been studied using P D F analysis of neutron powder diffraction data. Two Ba-doped samples were studied with x = 0.125 and x = 0.15. These samples were used in the synchrotron X-ray experiment to establish phase composition, as described above. ~145)Part of the X-ray diffraction patterns from the x = 0.125 sample at 80 K and 10 K can be seen in the Fig. 41(a). The phase transition is very clear in the diffraction data. In the average crystal structure model, the tilt axes of the CuO6 octahedra are rotated through 45 ~at this L T O - L T T transition. This results in a significant rearrangement of atoms within the unit cell, especially of the apical 0 2 ion, and should be clearly visible in the PDF. This is illustrated in the upper panels (a-c) of Fig. 42049~ where PDFs calculated from the crystal structure models of the LTO and L T T structures are compared to each other. The difference curves in Fig. 42(a-c) indicate that large changes are expected for certain P D F peaks. For example, the peak at 2.6 ~ , which originates from a La-O2 near-neighbor pair correlation, is very sensitive to the local tilt orientation. The data are shown in the lower panels (Fig. 42(d-f)). It is clear that the changes predicted by the crystal structure models do not occur. In particular, there is no change, within the errors, to the indicator peak at 2.6 A, although the crystal structure models predict changes to this peak many times larger than the experimental errors. It is clear that the octahedra do not change their local tilt direction at this transition. Small changes are observed to certain peaks at higher r values, as can be demonstrated by considering the variation of peak heights through the L T O - L T T transitionJ ~ag~This suggests that a change in the intermediate-range pattern of tilt ordering occurs at this transition. In any case, the P D F demonstrates dramatically that, on the length scale of the superconducting coherence length, changes to the structure at this transition are much smaller than implied by the crystal structure models. The fact that the direction of local tilting does not change at this transition implies that the crystal structure models do not describe correctly the direction of local tilting in one of

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the two phases. The local tilts were modeled by refining structure models to the PDFs using a least-squares regression code. The fully converged refinements are shown in Fig. 4 3 . (1491 The data are from the x = 0.125 sample measured at 80 K (LTO phase). In the low-r region, the L T T model fits best, indicating that, even in the LTO phase, the tilts are in the [110] (LTT) directions. However, the best fit to the intermediate-range data comes from a model in which two local L T T variants, rotated by 90 ° with respect to each other, are superimposed. This model retains the local L T T tilt directions, consistent with the P D F results, but recovers the LTO tilt pattern on the average, thus reconciling the observed average crystal structure. This model is not unique, but in unbiased refinements it gives the best agreement for PDFs from samples in the LTO phase. The model also predicts the L T O - H T T phase transition as being a continuous disordering of correlated tilts. (~491 The existence of an inhomogeneous local tilting pattern in La-214 has been suggested from early P D F results (821 and is consistent with the observation of a very fine mesoscale strain distribution observed in convergent beam electron diffraction experiments. (15°) This result indicates that, in the LTO phase, a structural correlation length exists which is in the order of the superconducting coherence, the electronic mean-free-path and antiferromagnetic correlation lengths. The local tilts are always in the L T T directions in this material. Below the structural coherence length the tilts are ordered in the L T T sense. However, the locally ordered domains are interleaved with domains whose tilts are rotated by 90 ° . The tilt magnitudes and directions obtained from crystal structure analysis are the average values only. These P D F results suggest that we may have to modify the way we understand the structures and transitions in the La-214 materials. Resonant ultrasound measurements (]5u521 at the H T T - L T O transition also indicate that the canonical soft-mode description (1281may not give the complete picture. However, much more work is required to establish the generality and wider importance of these results.

600 -I~t i-t

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r (nm) FIG. 43. P D F o f Lal.a75Ba0125CuO 4 at 80 K (crosses), compared to the P D F calculated for the L T T model with local tilt (above) and the P D F calculated for the LTO model. While the long range structure has the L T O symmetry, the P D F compares better with the L T T model, indicating that locally the structure is close to the L T T structure. (t491

L A T T I C E EFFECTS IN HTSC

409

7. IMPLICATIONS OF THE LATTICE ANOMALIES 7.1. Lattice Anharmonicity and Double-well Potential As discussed above, structural phase transition and lattice instability are often associated with the HTSC phase or the phase close to the HTSC phase. This situation is similar to the case of A 15 compounds, in which lattice softening provides stronger electron-lattice coupling.(56,~18~While there is no clear evidence that these phase transitions are directly related to the HTSC phenomenon, propensity for phase transition implies strong anharmonicity. Electronically as well, the HTSC phenomenon appears in the vicinity of the phase transition from the Mott insulator to a metal. Thus electronic phase bifurcation is likely, "53'~54~further promoting lattice anharmonicity. With this background the idea of the split apical oxygen site of YBCO-123 and the double-well potential associated with it looked eminently plausible when it was proposed. It prompted much theoretical, as well as experimental, work. When the experimental evidence against the split apical oxygen site in fully oxygenated YBCO-123 started to mount, however, an unfortunate backlash against the whole concept of the double-well potential occurred. But it is completely premature to dismiss the idea, particularly since many other anharmonic displacements have been observed, including the displacements of the apical oxygen and the in-plane oxygen toward each other. The mechanism proposed for apical oxygen could well work in another circumstance.

7.2. Lattice Polaron As we discussed in Section 4.5, the local lattice distortions in the HTSC are not random, but are coherent over 10-20~, as shown in Fig. 26 and also proven in the study of Nd-Ce-Cu-O. t8° The density of such domains is comparable to the charge carrier density, so that if they are associated with the charge carriers one or two carriers would be present for each domain. This suggests that the domains of local distortion may represent polarons o r bi-polarons. (79,86,9°,9H55A56) When the electron-lattice coupling is strong, electrons are heavily dressed by phonons and their dynamics are slow. If the coupling is so strong that the velocity of the dressed electron (quasi-particle) becomes comparable to the sound velocity, the electron becomes self-trapped to form a polaron. If a pair of polarons form a bound state by sharing the strain field they form a bi-polaron. Since bi-polarons are bosons, they Bose-condense to a superfluid state, and can show superconductivity. Actually the idea of bi-polaronic superconductivity proposed by Schafroth057) predates the BCS theory. The coherence length in conventional BCS superconductors is usually 1000 A or more, so that the wavefunctions of many Cooper pairs overlap each other in real space. Cooper pairs are formed by coupling the electrons in the states (k, a) and ( - k , - a ) , in the k-space, through the exchange of phonons. This spatial spread is important in order to avoid the cost of Coulomb repulsion between the electrons. In contrast, bi-polarons are real space pairs. They do not overlap in real space, but since they are represented by wave-packets made by summing the states over the whole or a part of the Brillouin zone, they overlap in the k-space. The problems with the real space pairs are the Coulomb repulsion within the pair because they are close to each other, and a low Bose-Einstein condensation temperature because of their heavy mass. The Bose-Einstein condensation temperature, TBE, is proportional to (n/~om) 1/2where n is the density of particles and Eois the low frequency dielectric constant. ~581

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While TBE is in the order of 1000 K for free electrons, if the mass is as large as that of an oxygen atom it is only 10 K or less. Consequently, as the strength of the electron-lattice coupling is increased Tc initially increases, but decreases again as the carriers become polaronic and heavy. In order to avoid this predicament Emin and Holstein ~159)proposed the concept of large polarons. Large polarons extend over several lattice spacings, so that the magnitude of displacement of each atom is small, making the effective mass of the large polaron much smaller than the ionic mass. However, it is difficult to stabilize large polarons with the delicate balance of short and long coulomb interactions. It is more likely that large polarons are formed because of a particular feature of the Fermi surface which accidentally prefers polarons of a certain size as we discuss later, or due to the lattice modulation as discussed in detail by Bianconi for the case of BISCO. ~75~4t) Polarons have been detected in underdoped LSCO samples. ~6°)Also, photo-excited carriers in the underdoped YBCO-123 have a long life-time characteristic of polarons/6~ Since the photo-excited carriers show superconductivity, a polaronic state most likely persists in the superconducting phase. 062-164)

7.3. M a g n e t i c M e c h a n i s m s As discussed in Section 5, the lattice anomalies observed by various experiments indicate an intimate relation between charge carriers and the lattice. It therefore implies that the dynamics of at least some of the carriers are slow enough for phonons to respond. One possible origin of the slow charge dynamics is the magnetic interaction. Since the energy scale of magnetic interaction, the exchange constant J, is the order of 0.1 eV, carriers are slowed down when they are dragged by spins. A number of models have been proposed to account for the behavior of doped holes when strong magnetic exchange interactions are present. The undoped parent compounds such as La2 CuO4 are insulating and their Cu spins are antiferromagnetically ordered. The long range antiferromagnetic order inhibits conduction by creating the spin polarization gap, therefore for the carriers to move in the background of anti ferromagnetic order spins nearby need to be reoriented. Incidentally, the LDA band structure calculations predict the parent compounds to be metallic and non-magnetic.~65) The insulating nature and the antiferromagnetic ordering is a consequence of strong electron-electron repulsion in the d-states of Cu, which is not properly included in the LDA calculations. Such a strong electron repulsion is better described by the Hubbard Hamiltonian:~66) H = -- t ~ (ci.~ci+~.~ + + ci+~,~c,~) + + U~ni,Tni,+, i,6,a

(30)

i

where c:~ t,ff is an operator to create an electron with spin a (T or l) on the local orbital of the i-th atom, 6 specifies the nearest neighbor, and ni,o = c + ci.~.

(31)

This Hamiltonian predicts the system of one electron per atom (half-filled band) to become insulating when U is larger than the band width. Such an insulator was first predicted by Mott, ¢16v) and is called the Mott insulator. One model of dealing with the Hubbard Hamiltonian in the k-space resulting in the magnetic capturing of the carrier, or the spin polaron, is the spin-bag model by Schrieffer3 ~6s)

LATTICE

EFFECTS

IN

411

HTSC

In this approach the insulating state is created by the formation of spin-density wave, and it is essentially BCS-like, relying upon the Hartree-Fock approximation. This theory has been criticized since the magnitude of U (about 10 eV) is too large to be treated by a perturbative approach; the effect of U will be more violent. Instead, an approach starting from the infinitely large U, the t - J Hamiltonian: H = - t ~. (ci+~c,+a,~+ c++a,~c,.,) + J ~ Si. Si+j, i,fi,a

(32)

i,6

has been proposed, 069)and has been studied by a very large number of researchers. A radically different scheme for the infinitely large U case is the resonating valence bond (RVB) model of P. W. AndersonP 7°) In this theory the ground state is made of local pairs in the spin-singlet state. Based upon the solution for a one-dimensional system(tT~) the spin degrees freedom (spinons) are supposed to be separate and independent from the charge degrees of freedom (holons). ~72) However, in either the t - J model or the RVB model superconductivity has not been shown to take place within reasonable ranges of parameters. Another more phenomenological theory is the spin-fluctuation theory of Moriya"~) and Pines. ~1~2~ In this theory carriers interact with local spins through the interaction Hamiltonian: 1

Hint = -~ ~ g (q)s (q)S ( - q),

(33)

q

where s (q) and S (q) describe the spins of free carriers and the localized Cu, respectively. In the Pines theory the Cu spins are assumed to be correlated, such that the susceptibility is given by: ZQ Z (q,e)) = 1 + ~ 2(q _ Q )2

-- io.)/(Os F '

(34)

with Q = (~/a, rc/a). The parameters are determined to fit the NMR data. This susceptibility is sharply peaked around the X (rt/a, rt/a) point of the Brillouin zone corresponding to the antiferromagnetic order of the Cu spins. This theory has been successful in describing various experimental observations. In particular it predicts the d-symmetry for the superconducting wavefunction, while most other theories predict s-symmetry. Recent experiments are indeed beginning to show some evidence of d-symmetry for YBCO-123. (~751 However, this subject of the symmetry of wavefunction is highly controversial, both experimentally and theoretically. Moreover, just as in the case of the apical oxygen double-well potential problem it is very dangerous to generalize the result for YBCO-123 to other compounds. The jury is still out on this subject. Also, this theory is a phenomenological theory, without a microscopic basis, and the spin correlation length determined from the NMR measurements is much longer than that deduced from the neutron scattering measurements. Furthermore, the functional form of the spin susceptibility Equation 34 is not consistent with the findings of neutron scattering. In any case, since the magnitude of the Cu spin exchange interaction J, which is about 0.1 eV, is comparable to the phonon energies, the dynamics of the charge, which is strongly interacting with spins, will be slow enough for the lattice to react. Thus, as we mentioned above, the spin polaron is most likely to have a lattice component. Whether the lattice or the spin correlation is more important in localizing the carrier, therefore, is a quantitative question rather than the choice of one excluding the other. JPMS 38/1-5--~N

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7.4. Effect o f Local Distortion on the Electronic Band Structure If the lattice is mainly responsible for determining the dynamics of the charge carriers there must be a signature of strong electron-lattice interaction in the electronic band structure. While, generally speaking, the band structure observed by the photo-emission experiments shows well defined dispersion and appears to be well accounted for by the LDA band structure calculations, the observed band structure is different from the LDA band structure in detailed but important ways. In particular, the saddle point at the X-point in the Cu-O plane (M-point in some structures, but always in the direction of the Cu-O bond) is supposed to be more than 50 meV below the Fermi level according to the LDA calculation,(174)but in all of the HTSC solids other than Nd-Ce~Cu-O it is within 20 meV of the Fermi level, and is very much more extended than in calculation,°24) as shown in Fig. 44 and schematically in Fig. 14. In order to explain this phenomenon Andersen et al. (175)pointed out that the displacements of either apical oxygen or in-plane oxygen in the c-direction could produce the extended saddle point. These displacements are indeed observed by pulsed neutron scattering as we discussed above. Also they reduce the oxygen-oxygen distance. According to Andersen et al., these displacements mix the oxygen rc band into the C u ~ ) a* band. Here the Cu-O a* band is the anti-bonding band made of the Cu-d(x 2 - y2) orbital and the O-p (x) orbital. (165~The rc band, O-p (z) band, is a non-bonding band and is fiat. By intermixing with this band the flat dispersion is introduced into the a* band. 7.5. Two-component Model The Fermi surface depicted in Fig. 14 clearly shows two kinds of carriers: heavy carriers in the extended saddle points, and light carriers on the more strongly dispersed parts of the Fermi surface. The dispersed regions (straight lines connecting the extended saddle points)

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of the Fermi surface produce nesting indicated by the arrows in the figure because of the near square shape of the Fermi surface. The Fermi surface nesting is known to produce short range Cu spin correlation in L S C O (176) and in Y B C O . (177) It could also lead to charge density wave (CDW) formation,(68) but so far there is no evidence of CDW except possibly on the surface,tll9) as we mentioned earlier. We have searched the evidence of CDW near the (n, n) point of LSCO by X-ray scattering, but were not able to find any diffuse scattering intensity which could be indicative of a CDW. There have been reports of observing CDW by electron diffraction, but that appears to be a radiation induced effect. Therefore, the fast dispersing carriers must be delocalized and nearly free. Thus, the system should have two kinds of carriers with rather different characters: heavy carriers which are nearly localized and light carriers which are nearly free. A phenomenological two component model was extensively discussed to explain the IR conductivity. As shown in Fig. 45, ~178) a typical IR absorption conductivity has a low frequency part which looks just as the free carrier Drude behavior, and a mid-IR part which is very broad. 079J8°) It is natural to associate the free carriers with the light carriers in the strongly dispersing section of the Fermi surface and the mid-IR carriers with the heavy carriers at the extended saddle point. 7.5.1. Lattice polaron scenario Carriers in the fiat saddle point, as shown in Figs 14 and 44, can easily become localized by a small disturbance, such as randomness (Anderson localization), by lattice distortion forming a polaron, or contained by magnetic correlation forming a spin polaron. <~Sj)Even when the mechanism of localization is Anderson localization or spin polaron formation rather than the lattice polaron self-trapping, once the carriers become localized the lattice around them would deform, leading to polaronic state anyway. The size of the extended saddle point over which the energy is nearly constant should be inversely related to the size of the polaron. This happens to be about 10 x 20 ~, quite consistent with the size of the micro-domains suggested by the pulsed neutron PDF study. Note that the electron dispersion indicates that the energy quickly increases moving away from the extended saddle point. This steep increase keeps the large polaron from collapsing

414

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into a small polaron, since a small polaron is formed by summing up all the states in the Brillouin zone, which costs a large energy. Thus, only the states in a portion of the k-space form large polarons which co-exist with the free carriers in the rest of the k-space. The two-component model also explains the Hall effect of LSCO-214. As shown in Fig. 46, the Hall coefficient varies strongly with temperature and composition. °s2) They can be scaled into one master temperature dependence, with the characteristic temperature T* being strongly composition dependent, as shown in Fig. 47. o80) Note that the negative temperature dependence of RH is a signature of thermally activated carriers as in semiconductors. If the 1.2

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04

L A T T I C E E F F E C T S IN HTSC

415

binding energy of the localized carriers is not large they could become delocalized at high temperature. Indeed the energy level of the extended saddle point is only 20 meV ( = 230 K) or so below the Fermi level('24~ which is comparable to T*, so that they can be thermally excited. It is interesting to note that the temperature and composition dependence of the non-phonon diffuse scattering shown in Fig. 21 is similar to that of the Hall effect. For instance, the non-phonon diffuse scattering intensity for x = 0.2 sample depends strongly upon temperature. (72~ In other words, the local lattice distortion which gives rise to the non-phonon diffuse scattering may be related to the localization of the carriers. It may not be the lattice distortion associated with a single polaron, but it could well be the lattice distortion induced by collective fluctuation of polarons. The possibility of the observed electronic transport properties being consistent with the idea of polarons has been extensively discussed by Emin, (183~ and more recently by Alexandrov et al. in terms of Anderson localization.( ~8~ 7.5.2. Local pairing and superconductivity If, for some reasons, the localized carriers have a tendency to form pairs, or bi-polarons, and the light and heavy carriers are mixed, for instance because of local distortion, it could lead to ideal conditions for the two-component superconductivity. (~8~s6~ In the scenario for two-component superconductivity the localized carries form negative-U centers, or bi-polarons, and provide the pairing force. The light carriers provide conductivity, and by hybridizing with the localized bi-polarons produce superconductivity. In this juncture it is further appealing that the superconducting gap is maximum in the direction of the saddle point, suggesting that the driving force for pair formation indeed comes from the carriers on the extended saddle point which are presumably localized. In this picture, at low concentrations of carriers the bi-polarons are strongly localized and only weakly interacting through hybridization with free carriers. Thus, the bi-polaron mass is large, making the Bose condensation temperature, T~E, lOW, while the bi-polaron formation temperature or the spin-gap temperature, Tg, is high. Upon increasing the carrier concentration, the bi-polaron binding energy becomes reduced due to increased screening. Consequently the bi-polaron formation temperature is reduced, while the Bose condensation temperature is increased because of the reduced bi-polaron mass. When TBEand Tg cross each other Tc becomes maximum. To summarize, to the left of the optimum concentration at which T~ is maximum, or in the underdoped state, bi-polarons are formed above Tc which is the bi-polaron Bose condensation temperature. To the right of the optimum concentration, or

-I"

X FIG. 48. Schematic diagram of the carrier density (x) dependence of T~, TBE and TsG.

416

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in the overdoped state, bi-polarons overlap each other, and the system becomes a normal phonon mediated strong coupling superconductor (Fig. 48). Thus, the lattice displacements are large and relatively easily seen in the underdoped state, but in the overdoped state the atomic displacements associated with the Cooper pairs are essentially phonon-like, and difficult to differentiate from thermal phonons. Our observations of the lattice distortion are largely consistent with this picture. Larger atomic displacements are seen in YBCO(124) and oxygen reduced YBCO(123) which are underdoped, than in fully oxygenated YBCO(123). Also the non-phonon diffuse scattering in LSCO-214 is minimum at the concentration at which Tc is maximum. However, more thorough studies of concentration dependence need to be carried out. It is interesting to note that a recent photoemission study of overdoped BISCO showed a temperature dependence of the superconducting gap strongly suggestive of a two-component model.(~sT)In the F-M direction (parallel to the Cu-O bond) the gap is maximum with the value of 2A/kTc = 4.6 at T<< To, while in the F-X direction (45 ° away) the gap is smaller but non-zero at low temperatures and decrease quickly with increasing temperature. The result contradicts both the pure d(x 2 - y2) wave scenario, and the single anisotropic s-wave scenario. However, it is consistent with the two-component picture where a large gap forms at the saddle point (M point); the induced superconductivity along the fast dispersing Fermi surface, including the X point, will have a much smaller gap. When the carrier concentration is reduced, the gap in the X-direction is expected to become smaller because of the reduced hybridization, and the gap anisotropy would increase, consistent with various reports of d-like anisotropy of the gap. 024Jss) It is, furthermore, possible that the d-wave is actually involved. For instance, as for the symmetry of the wavefunction of the bi-polarons, the BCS-like phonon mean-field would result in the s-wave, but if the coupling is local and mediated by a specific mode of local lattice distortion, it may be possible that the d-wave state occurs even in the lattice mechanism. 7.6. Nature of Electron-Lattice Interaction in HTSC Solids Experimental evidence discussed at length in the present article points to strong, and possibly unusual, electron-lattice interaction. Once the slow charge dynamics are established this could account for various anomalous lattice behaviors observed by different experimental methods. However, there is no clear evidence that the lattice is directly involved in superconductivity by providing forces to pair carriers. In this section we will speculate on the possible lattice mechanism of superconductivity, if the lattice were to be directly involved. 7.6.1. Effect of electron correlation The effects of strong electron correlation on the magnetic properties of transition metal oxides are widely known and well studied as we briefly touched upon above, but relatively less attention has been given to its effects on the lattice. In fact, the electron-lattice interaction can be significantly enhanced because of electron correlation. It has been shown that, at the crossover point between the Mott state and the ionic state, the electron-lattice interaction is strongly enhanced by the electron correlation effect in a one-dimensional system.089) Let us consider two atoms, A and B, with the electron energy level of EA and EB (EA -- EB = A). If the electron repulsion energy for two electrons to occupy each state is U, the system can have three ground state configurations. When A is strongly negative and IAI > U, two electrons will occupy the A atom level. If A < U, one electron will be on each atom. When A > U, both electrons will be on the B atom level. The situation with IA[ < U

LATTICE

EFFECTS

IN

417

HTSC

can be described as the Mott state, in the sense that the repulsion energy U decides the electron configuration. In the limit of A = 0 this state is the usual M o t t state. A = U signals the crossover condition. If we introduce a hopping term in-between the A and B levels, the effect of modulating the hopping integral is m a x i m u m at the crossover point. This effect of Peierls distortion (dimerization) on the two-band Peierls-Hubbard model was calculated by the exact diagonalization in one-dimension. The Hamiltonian is given by: H

+ aCB,]+a-'~-CB,],aCA,i,a)"t-AECAd,aCA,i+a-'[+ + : t ~ (CA,i, i,],a

UAEnA,i, TnA,i,~'-[- UB~nBj.rnBj,~.

i,a

i

(35)

j

The energy gain, AE, due to Peierls distortion which makes the hopping integral modulated to t + fit was calculated for a ring of eight atoms (four A and four B atoms) as shown in Fig. 49. Here the energy gain, normalized to the value at Un = 0, shows a strong m a x i m u m at the crossover point from the band insulator (left of the maximum) to the Mott insulator (right of the maximum). Near the crossover point the spins at the nearest neighbors are strongly correlated antiferromagnetically if the transfer integral t between the pairs is increased by deformation (by the distance becoming shorter) to to + fit, and are weakly correlated if t is reduced to to - fit. When U is large, the on-site energy is reduced by spin polarization, but this usually increases the kinetic energy. In the local resonant state the atoms are dynamically spin-polarized, but are in the local singlet state, thus preventing the kinetic energy from becoming too large. Over a fairly wide range of the values of Un the energy gain is significant, showing that the effect of lattice distortion is enhanced by U. It is important to note that this enhancement occurs only at a specific k value of deformation, in this case with the wavelength equal to 2a. For deformation modes with different wavelengths than 2a the electron correlation exerts no effect. 7.6.2. Bipolaron formation Preliminary calculations show that even when the charge density is away from the half-filling, similar response to the lattice distortion near the crossover point occurs. In that case, however, the wavelength of the distortion to which the system responds strongly, 2, is not 2a, but 2a/n, where n is the charge density. Thus, by this distortion charge carriers are divided into groups of two, forming bi-polarons. Note that the wavevector of distortion, 2rt/2, corresponds to the usual Fermi surface nesting condition. In the regular band case the Fermi I

i

.--,a---

~"

i

i

i

I

i

,

I

i

I

J

J

I 4

At=0.025

----o-- At=0.05

j/

"~

.<

0

L 2

~

~

~

,

I 3

~

~

J

L

UB (eV)

Dependence of the energy gain due to Peierls distortion for the two-band Hubbard model. The energy gain is maximum at the crossover point from the band insulator (UB< UB,¢~t)to the Mott insulator ( U B > UB.~n~).(189) FIG. 49.

418

P R O G R E S S IN M A T E R I A L S S C I E N C E

surface nesting produces lattice distortion and CDW which creates the superlattice and new Brillouin zones, so that the electronic states are partitioned in the new Brillouin zone, two electrons per unit cell. In the case where electron repulsion is dominant, the CDW is accompanied by the bond order wave (BOW). The BOW can bc seen as the local resonating valence bond (RVB) as well. In this case partitioning occurs in the real space, into pairs of electrons. Thus strong electron correlation promotes bi-polaron formation, at least in one-dimension. In two-dimensions if the bipolarons could be formed by local RVB the wavefunction can have a d-symmetry. It is important to note that this enhancement occurs due to the charge transfer between A and B atoms induced by lattice distortion. Thus a large part of the energy gain is carried by the electron-electron interaction energy. Therefore, this effect cannot be described by the conventional deformation potential picture, but requires the description of the heavy dressing of phonons by electrons due to electron-electron interaction. Near the crossover point a small lattice deformation can induce a large electronic effect duc to this enhancement. If this crossover point corresponds to the charge concentration to produce maximum To, at that point the magnitude of the lattice deformation necessary to produce polarons is minimum. As the charge density is reduced and the condition departs from the crossover point a larger lattice deformation becomes necessary. This agrees with the pulsed neutron PDF observation that for the optimally doped YBa2Cu307 the lattice distortion is small, whereas in underdoped (124) and reduced (123) the distortions are more easily observable. This, furthermore, is consistent with the isotope effect which is minimum when Tc is maximum, and the diffuse scattering study for which the non-phonon intensity is again minimum when Tc is maximum as shown in Fig. 22. At the crossover point the polarons are almost electron-polarons, since most of the screening is achieved by transferred charges induced by small lattice deformation. 8. CONCLUSIONS

The long range technological implications of high temperature superconductivity are staggering. In one estimate the global market size for HTSC related technology in the year 2020 could reach $150-200 billion. °9°) Furthermore, a vast amount of experimental observations already made on this fascinating class of materials revealed that the HTSC oxides are drastically different from the conventional metallic superconductors in many aspects, including the mechanism of superconductivity. Consequently an unprecedented amount of research effort has been focused on the HTSC oxides. Measurements were made with accuracy never attempted before. Thus, various phenomena hitherto unknown have been uncovered. Numerous lattice anomalies associated with the onset of superconductivity are examples of these phenomena. Anomalous atomic displacements were observed in HTSC, some of which were associated with Tc or spin-gap temperature, while conventional crystallographic structural measurements did not record much of these anomalous displacements. It is not so surprising, however, that the crystallographic methods failed to observe the structural anomalies, since the anomalies occur at a very local level, while crystallographic methods are sensitive only to long range order. Even though there are many controversies regarding the details of these anomalies, it appears that the presence of some notable lattice anomalies is unmistakable. Neither is it so surprising that they should be observed given the close structural relationship of the cuprates to ferroelectric perovskites in which lattice instabilities and strong electron-lattice coupling is endemic.

L A T T I C E E F F E C T S IN H T S C

419

However, the physical significance of these anomalies is not clear. One possibility is that they are merely a structural response to some unconventional superconductivity. Even in this case careful studies of this effect might reveal relevant information regarding the mechanism. A more interesting possibility is that they are indicative of positive involvement of the lattice in the high temperature superconductivity. One scenario of such involvement calls for the formation of lattice polarons for carriers in the extended saddle point near the Brillouin zone boundary. If these polarons become bi-polarons, for example, through the electron-lattice interaction enhanced by electron correlation effect, and hybridize with light carriers in the fast dispersing part of the Fermi surface, two-component superconductivity may be observed. A huge number of papers, totaling 35 000 in one estimate,(~9~)have been published on HTSC materials and related subjects. In spite of such an extensive effort it appears that surprisingly little real understanding of these materials has been attained. The primary reason is that they are fundamentally different from conventional materials, having dual faces of metallic conductor and inorganic insulator. Atomic bonding in these materials is partially covalent and partially ionic, and electron correlation plays a major role in determining the electronic properties. Thus, the limitation of the methodology of metal physics is apparent. On the other hand, HTSC oxides represent only a small fraction of materials classified as conducting ceramics. New knowledge acquired by studying cuprates could be very effectively used in tapping the vast and largely underexplored field of electronic ceramics and could lead to new opportunities for applications. The lattice effects reviewed in this article represent a group of new phenomena the study of which could not only provide information regarding the HTSC mechanism but also the basic nature of conducting ceramics.

ACKNOWLEDGEMENTS

This work was supported by the National Science Foundation through DMR93-00728 and DMR91-20668. Assistance in generating figures by W. Dmowski is deeply appreciated. Discussions with O. K. Andersen, M. Arai, Y. Bar-Yam, A. Bianconi, A. R. Bishop, S. D. Conradson, A. L. de Lozanne, W. Dmowski, J. Etheridge, J. Goodenough, C. Humphreys, E. Kaldis, G. H. Kwei, E. Mele, D. E. Moncton, M. Onellion, M. Salamon, S. K. Sinha, G. Shirane, E. A. Stern, J. D. Thompson and K. Yamada are gratefully acknowledged.

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