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b Superconductivity Ql O o °<7> d) Charles P. Poole, Jr. Horacio A. Farach Richard /. Creswick Department of Physics and Astronomy University of South Carolina Columbia, South Carolina BEb 1 HVMiLjnoi.c; uU^y Academic Press San Diego London Boston New York Sydney Tokyo Toronto This book is printed on acid-free paper. Copyright © 1995 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Academic Press A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Poole, Charles P. Superconductivity / by Charles P. Poole, Jr., Horacio A. Farach, Richard J. Creswick. p. cm. Includes index. ISBN 0-12-561455-1. -- ISBN 0-12-561456-X (pbk.) 1. superconductivity. I. Farach, Horacio A. II. Creswick, Richard J. III. Title. QC611.92.P66 1995 537.6'23~dc20 94-31853 CIP. PRINTED IN THE UNITED STATES OF AMERICA 97 98 99 00 01 02 QW 9 8 7 6 5 4 3 2 Perovskite and Cuprate Crystallographic Structures I. INTRODUCTION Chapter 3 shows that the majority of single-element crystals have highly sym- metrical structures, generally fee or bec, in which their physical properties are the same along the three crystallographic di- rections x, y, and z. The NaCl-type and -415 compounds are also cubic. Some com- pounds do have lower symmetries, showing that superconductivity is compatible with many different types of crystallographic structure, but higher symmetries are cer- tainly more common. In this chapter we will describe the structures of the high- temperature superconductors, almost all of which are tetragonal or orthorhombic. In Chapter 3, we also gave some exam- ples of the role played by structure in determining the properties of supercon- ductors. The highest transition tempera- tures in alloys of transition metals are at the boundaries of instability between the bec and hep forms. The NaCl-type com- pounds have ordered vacancies on one or another lattice site. The magnetic and su- perconducting properties of the Chevrel phases depend on whether the large mag- netic cations (i.e., positive ions) occupy eightfold sites surrounded by chalcogenide ions or whether the small magnetic ions occupy octahedral sites surrounded by Mo ions. The structures described here are held together by electrons that form ionic or covalent bonds between the atoms. No ac- count is taken of the conduction electrons, which are delocalized over the copper ox- ide planes and form Cooper pairs respon- sible for the superconducting properties below T c . The following chapter will be devoted to explaining the role of these 173 774 7 PEROVSKITE AND CUPRATE CRYSTALLOGRAPHIC STRUCTURES conduction electrons within the frame- works of the Hubbard model and band theory. Whereas the present chapter de- scribes atom positions in coordinate space, the following chapter relies on a reciprocal lattice elucidation of these same materials. We begin with a description of per- ovskite and explain some reasons that per- ovskite undergoes various types of distor- tions. This prototype exhibits a number of characteristics that are common to the high-temperature superconducting cuprates (see Section V). We will empha- size the structural commonalities of these materials and make frequent comparisons between them. Our earlier work (Poole et al. y 1988) and the comprehensive review by Yvon and Frangois (1989) may be con- sulted for more structural detail on the atom positions, interatomic spacings, site and thallium high temperature supercon- ductors (Medvedeva et aL, 1993). We assume that all samples are well made and safely stored. Humidity can af- fect composition, and Garland (1988) found that storage of YBa 2 Cu 3 0 7 _ 5 in 98% humidity exponentially decreased the dia- magnetic susceptibility with a time con- stant of 22 days. II. PEROVSKITES Much has been written about the high-temperature superconductors being perovskite types, so we will begin by de- scribing the structure of perovskites. The prototype compound barium titanate, BaTi0 3 , exists in three crystallographic forms with the following lattice constants and unit cell volumes (Wyckoff, 1964): cubic: a - b - c = 4.0118 A K= 64.57 A 3 tetragonal: a = b = 3.9947, c = 4.0336 K= 64.37 A 3 (7-1) ort/iorhombic: a = 4.009\/2 A, b = 4.018i/2 A, c = 3.990 A K= 2(64.26) A 3 symmetries, etc., of these compounds. There have been reports of superconduc- tivity in certain other cuprate structures (e.g., Murphy et a/., 1987), but these will not be reported on in this chapter. There is a related series of layered compounds Bi 2 0 2 (M m _ 1 i? m 0 3m + 1 ) called Aurivillius (1950, 1951, 1952) phases, with the 12-coordinated M = Ca, Sr, Ba, Bi, Pb, Cd, La, Sm, Sc, etc., and the 6-coordinated transition metal R = Nb, Ti, Ta, W, Fe, etc. The m = 1 compound Bi 2 Nb0 6 be- longs to the same tetragonal space group lA/mmm, as the lanthanum, bismuth, For all three cases the crystallographic axes are mutually perpendicular. We will com- ment on each case in turn. A, Cubic Form Above 201°C barium titanate is cubic and the unit cell contains one formula unit BaTi0 3 with a titanium atom on each apex, a barium atom in the body center, and an oxygen atom on the center of each edge of the cube, as illustrated in Fig. 7.1. This corresponds to the barium atom, titanium atom, and three oxygen atoms being placed in positions with the following x, y, and z coordinates: Esite:Ti Fsite: O Csite: Ba (0,0,0) i Ti on apex (0,0, \); (0, £,0); ( ^,0,0) three oxygens centered on edges (l»i>i) Ba in center. (7.2) Figure 7.1 Barium titanate (BaTi0 3 ) perovskite cubic unit cell showing titanium (small black circles) at the vertices and oxygen (large white circles) at the edge-centered positions. Ba, not shown, is at the body center position (Poole et ai, 1988, p. 73). The barium in the center has 12 nearest- neighbor oxygens, so we say that it is 12- fold coordinated, while the titanium on each apex has 6-fold (octahedral) coordi- nation with the oxygens, as may be seen from the figure. (The notation E for edge, F for face, and C for center is adopted for reasons that will become clear in the dis- cussion which follows.) Throughout this chapter we will assume that the z-axis is oriented vertically, so that the x and y axes lie in the horizontal plane. Ordinarily, solid-state physics texts place the origin (0,0,0) of the perovskite unit cell at the barium site, with titanium in the center and the oxygens at the cen- ters of the cube faces. Our choice of origin facilitates comparison with the structures of the oxide superconductors. This structure is best understood in terms of the sizes of the atoms involved. The ionic radii of O 2 " (1.32 A) and Ba 2+ (1.34 A) are almost the same, as indicated in Table 7.1, and together they form a perfect fee lattice with the smaller Ti 4+ ions (0.68 A) located in octahedral holes surrounded entirely by oxygens. The octa- hedral holes of a close-packed oxygen lat- tice have a radius of 0.545 A; if these holes were empty the lattice constant would be o = 3.73 A, as noted in Fig. 7.2a. Each titanium pushes the surrounding oxygens outward, as shown in Fig. 7.2b, thereby increasing the lattice constant. When the titanium is replaced by a larger atom, the lattice constant expands further, as indi- cated by the data in the last column of Table 7.2. When Ba is replaced o by the smaller Ca (0.99 A) and Sr (1.12 A) ions, by contrast, there is a corresponding de- crease in the lattice constant, as indicated by the data in columns 3 and 4, respec- tively, of Table 7.2. All three alkaline earths, Ca, Sr, and Ba, appear prominently in the structures of 3 high-temperature superconductors. B. Tetragonal Form At room temperature barium titanate is tetragonal and the deviation from cubic, (c-a)/\{c+a\ is about 1%. All of the atoms have the same jc, y coordinates as in the cubic case, but are shifted along the z-axis relative to each other by ~ 0.1 A, producing the puckered arrangement shown in Fig. 7.3. The distortions from the ideal structure are exaggerated in this sketch. The puckering bends the Ti-O-Ti group so that the Ti-O distance increases while the Ti-Ti distance remains almost Table 7.1 Ionic Radii for Selected Elements 0 Small Cu 2+ 0.72 A Bi 5 + 0.74 A Small- Cu + 0.96 A Y 3 + 0.94 A Medium 0.95 A Bi 3+ 0.96 A Tl 3 + Ca 2+ 0.99 A Bi 3 + 0.96 A Nd 3+ 0.995 A Medium- Hg 2+ 1.10 A 1.14 A Large Sr 2+ 1.12 A La 3+ Pb 2+ 1.20 A Ag + 1.26 A Large . K + 1.33 A o 2 - 1.32 A Ba 2+ 1.34 A F" 1.33 A a See Table VI-2 of Poole et al. (1988) for a more extensive list. 176 7 PEROVSKITE AND CUPRATE CRYSTAL LOG RAP HIC STRUCTURES .73 A (t>) Figure 7.2 Cross section of the perovskite unit cell in the z = 0 plane showing (a) the size of the octahe- dral hole (shaded) between oxygens (large circles), and (b) oxygens pushed apart by the transition ions (small circles) in the hole sites. For each case the lattice constant is indicated on the right and the oxygen and hole sizes on the left (Poole et ai, 1988, p. 77). Figure 73 Perovskite tetragonal unit cell showing puckering of Ti-O layers that are perfectly flat in the cubic cell of Fig. 7.1. The notation of Fig. 7.1 is used (Poole et al. 9 1988, p. 75). the same. This has the effect of providing more room for the titanium atoms to fit in their lattice sites. We will see later that a similar puckering distortion occurs in the high-temperature superconductors as a way of providing space for the Cu atoms in the planes. C Orthorhombic Form There are two principal ways in which a tetragonal structure distorts to form an orthorhombic phase. The first, shown at Table 7.2 Dependence of Lattice Constants a of Selected Perovskites AM0 3 on Alkaline Earth A and Ionic Radius of Transition Metal Ion M + 4 ; the Alkaline Earth Ionic Radii are 0.99 A (Ca), 1.12 A (Sr), and 1.34 A (Ba) a Transitional metal Transitional metal o radius, A Lattice constant a, Ca Sr o A Ba Ti 0.68 3.84 3.91 4.01 Fe 3.87 4.01 Mo 0.70 3.98 4.04 Sn 0.71 3.92 4.03 4.12 Zr 0.79 4.02 4.10 4.19 Pb 0.84 4.27 Ce 0.94 3.85 4.27 4.40 Th 1.02 4.37 4.42 4.80 Data from Wyckoff (1964, pp. 39 Iff). the top of Fig. 7.4, is for the 6-axis to stretch relative to the a-axis, resulting in the formation of a rectangle. The second, shown at the bottom of the figure, is for one diagonal of the ab square to stretch and the other diagonal to compress, result- ing in the formation of a rhombus. The two diagonals are perpendicular, rotated by 45° relative to the original axes, and become the a\ b' dimensions of the new orthorhombic unit cell, as shown in Fig. 7.5. These a\ b r lattice constants are «' 4l times longer than the original constants, so that the volume of the unit cell roughly doubles; thus, it contains exactly twice as many atoms. (The same ^2 factor appears in Eq. 7.1 in our discussion of the lattice constants for the orthorhombic form of barium titanate.) When barium titanate is cooled below 5°C it undergoes a diagonal- or rhombal- type distortion. The atoms have the same z coordinates (z = 0 or \) as in the cubic phase, so the distortion occurs entirely in the *,>>-plane, with no puckering of the atoms. The deviation from tetragonality, as Figure 7.5 Rhombal expansion of monomolecular tetragonal unit cell (small squares, lower right) to bimolecular orthorhombic unit cell (large squares) with new axes 45° relative to the old axes. The atom positions are shown for the z = 0 and z = } layers (Poole et al., 1988, p. 76). Figure 7 A Rectangular- (top) and rhombal- (bot- tom) type distortions of a two-dimensional square unit cell of width a (Poole et aL, 1989). given by the percentage of anisotropy, 10016 ~a\ %ANIS= , —0.22%, (7.3) 2Kb +a) is less than that of most orthorhombic copper oxide superconductors. We see from Fig. 7.5 that in the cubic phase the oxygen atoms in the z = 0 plane are sepa- rated by 0.19 A. The rhombal distortion increases this O-O separation in one di- rection and decreases it in the other, in the manner indicated in Fig. 7.6a, to pro- duce the Ti nearest-neighbor configuration shown in Fig. 7.6b. This arrangement helps to fit the titanium into its lattice site. The transformation from tetragonal to orthorhombic is generally of the rhombal type for (La, _,Sr,) 2 Cu0 4 and of the recti- linear type for YBa 2 Cu 3 0 7 _ 5 . 778 7 PEROVSKITE AND C U PRATE CKYSTALLOCRAPHIC STRUCTURES usefulness of this labeling scheme will be clarified in Section V. This completes our treatment of the structure of perovskite. We encountered many features that we will meet again in the analogous superconductor cases, and established notation that will be useful in describing the structure of the cuprates. However, before proceeding we present details about a cubic and a close-to-cubic perovskite superconductor in the following two sections. (b) Figure 7.6 Shift of the oxygens in the a, 6-plane around the titanium atom of perovskite from the room-temperature tetragonal (and cubic) configura- tion (a) to the rhombal configuration (b) of its low- temperature orthorhombic structure. D. Planar Representation Another way of picturing the structure of perovskite is to think of the atoms as forming horizontal planes. If we adopt the notation [E F C] to designate the occupa- tion of the E, F, and C sites, the sketches of perovskite presented in Figs. 7.1 and 7.3 follow the scheme z = 1 [Ti0 2 -] Ti at E, O at two F sites z = { [O-Ba] OatE,BaatC z = 0 [Ti0 2 -] Ti at E, O at two F sites. (7.4) The planes at the heights z = 0, \, and 1 can be labeled using this notation. The ML CUBIC BARIUM POTASSIUM BISMUTH OXIDE The compound Ba^K^BKV,, which forms for x > 0.25, crystallizes in the cubic pervoskite structure with a = 4.29 A (Cava et aL, 1988; Jin et aL, 1992; Mattheiss et aL, 1988). K + ions replace some of the Ba 2+ ions in the C site, and Bi ions occupy the E sites of Eq. (7.2) (Hinks et aL, 1988b; Kwei et aL, 1989; Pei et aL, 1990; Salem- Sugui et aL, 1991; Schneemeyer et aL, 1988). Some oxygen sites are vacant, as indicated by y. Hinks et aL (1989) and Pei et aL (1990) determined the structural phase diagram (cf. Kuentzler et aL, 1991; Zubkus et aL, 1991). We should note from Table 7.1 thaMhe potassium (1.33 A) and barium (1.32 A) ions are almost the same size, and that Bi 5+ (0.74 A) is close to Ti 4+ (0.68 A). Bismuth represents a mix- ture of the valence states Bi 3+ and Bi 5 + which share the Ti 4+ site in a proportion that depends on x and y. The larger size (0.96 A) of the Bi 3+ ion causes the lattice constant a to expand 7% beyond its cubic BaTi0 3 value. Oxygen vacancies help to compensate for the larger size of Bi 3+ . It is noteworthy that Ba 1 „ jr K jC Bi0 3 „ y becomes superconducting at a tempera- ture (^40 K for jc = 0.4) that is higher than the T c of all of the A15 compounds. This compound, which has no copper, has V. PEROVSKITE-TYPE SUPERCONDUCTING STRUCTURES been widely studied in the quest for clues that would elucidate the mechanism of high-temperature superconductivity. Fea- tures of Ba,_ x K x Bi0 3 _ > , ) such as the fact that it contains a variable valence state ion and utilizes oxygen vacancies to achieve charge compensation, reappear in the high-temperature superconducting com- pounds. IV. BARIUM LEAD BISMUTH OXIDE In 1983 Mattheiss and Hamann re- ferred to the 1975 "discovery by Sleight et aL of high-temperature superconductivity" in the compound BaPb 1 _^Bi jr 0 3 in the composition range 0.05 <x< 0.3 with T c up to 13 K. Many consider this system, which disproportionates 2Bi 4+ -» Bi 3+ + Bi 5+ in going from the metallic to the semiconducting state, as a predecessor to the LaSrCuO system. The metallic compound BaPb0 3 is a cubic perovskite with the relatively large lattice constant (Wyckoff, 1964; cf. Nitta et aL, 1965; Shannon and Bierstedt, 1970) listed in Table 7.3. At room temperature semiconducting BaBi0 3 is monoclinic {a~b~c/Jl, £ = 90.17°), but close to orthorhombic (Chaillout et aL, 1985; Cox and Sleight, 1976, 1979; cf. Federici et aL, 1990; Jeon et aL, 1990; Shen et aL, 1989). These two compounds form a solid solu- tion series BaPb l _^Bi x 0 3 involving cubic, tetragonal, orthorhombic, and monoclinic modifications. Superconductivity appears in the tetragonal phase, and the metal-to- insulator transition occurs at the tetrag- onal-to-orthorhombic phase boundary x ~ 0.35 (Gilbert et aL, 1978; Koyama and Ishimaru, 1992; Mattheiss, 1990; Mattheiss and Hamann, 1983; Sleight, 1987; cf. Ban- sil et aL, 1991; Ekino and Akimitsu, 1989a, b; Papaconstantopoulous et aL, 1989). The compound resembles B ai _,K,Bi0 3 _, with its variable Bi valence states, but it differs in not exhibiting superconductivity in the cubic phase. V. PEROVSKITE-TYPE SUPERCONDUCTING STRUCTURES In their first report on high-tempera- ture superconductors Bednorz and Miiller (1986) referred to their samples as "metallic, oxygen-deficient . . . perovskite- like mixed-valence copper compounds." Subsequent work has confirmed that the new superconductors do indeed possess these characteristics. In the oxide superconductors Cu 2+ re- places the Ti 4+ of perovskite, and in most cases the Ti0 2 -perbvskite layering is re- tained as a Cu0 2 layering with two oxy- gens per copper. Because of this feature of Cu0 2 layers, which is common to all of the high-temperature superconductors, such superconductors exhibit a uniform lattice size in the a, 6-plane, as the data in Table 7.3 demonstrate. The compound BaCu0 3 does not occur because the Cu 4+ ion does not form, but this valence constraint is overcome by replacement of Ba 2+ by a trivalent ion, such as La 3+ or Y 3 + , by a reduction in the oxygen content, or by both. The result is a set of "layers" con- taining only one oxygen per cation located between each pair of Cu0 2 layers, or none at all. Each high-temperature supercon- ductor has a unique sequence of layers. We saw from Eq. (7.2) that each atom in perovskite is located in one of three types of sites. In like manner, each atom at the height z in a high-temperature super- conductor occupies either an Edge (E) site on the edge (0,0, z), a Face (F) site on the midline of a face ((0,|,z) or ({,0, z) or both), or a Centered (C) site centered within the unit cell on the z-axis i\,\,z). The site occupancy notation [E F C] is used because many cuprates contain a suc- cession of [Cu 0 2 -] and [- 0 2 Cu] lay- ers in which the Cu atom switches between edge and centered sites, with the oxygens E B o U O o <N 0 WO A g V r- TJ "O § o o l t 2 a o wo to co co co ON 00 o o wo ( O (N o CN VO CN» ^ CN CO cn \/ h h oi v <n m n i—( v Q\ H H v t— < i— ( 3 u co CN o o o o o o o o o o T— f o o o o ON On 00 On r- wo oo CO VO wo IT) o\ 00 VO T-H vq CO ON wo CO CO r- wo CO <N 1 1 1 1 1 i 1 vd CO CO VO r-H i— 1 wo ON vd to On vd wo CN o CO wo CO t— 1 CO on r- o CO On 00 OO OO Xf 00 O o on <N CO CO CN T-H H Ov vq VO CN oq wo ON CO wo VO CO* Tfr XT CO CO 1— t 1—1 O r— CO On wo ON <n wo ON oi On CN t-H T— 1 T— 1 T— 1 CO CO CN CN CO T-H T-H cn 1— < ICS [cn [cN [cN CN wo CO CO VO wo CO o (N WO i—i On r- oo WO On 1— < VO o wo T— ( CO CO WO W0 vO VO VO O On o CN CO (N 00 On as CO oq oq oq oq oq OO oq oq CO XT CO CO CO CO CO CO CO CO CO CO CO* CO i * © c b 3 r-ti—t H 041—101 ^ H H O O (N N (N <N CN OA 1<N IcN IcN (cN t-H t-H r- I r-H T-H T-H T-H W0 WOt— <t-Ht-Ht-Ht-Ht-Ht-Ht-Ht— It— < < < << oo < < oo oo < < oo oooooooo<<!<<<<;<< UH OU H SUh OHO H OHHH H H H o E 00 T-H CO CO CN CO r-H CN o o t-H 1— 1 i— t CN o T— 1 cn cn CN CN CN CN CN CN o o o o CN CN CN CN CO t-h <N CO t-h CN CO OHM T3 C 3 O £ o U O o H H PQ PQ H a. «J CO PQ PQ PQ O s O ^ PQ O (N cd c3 aj PQ PQ *J O =3 » " ^ M /n ^ U M 3 3 3 ^ 3 [1 o o u SLQ y :on u u u ° 0 - ;ooo rn »o 3 3 3 P u It* — .^rorowv./ n n t h nj nj O => " =» U U U U ^ r'l r"? r°? CJ CJ U ^ea nj 00 ooPQCQPQrtrtcartrtpgpQpQ fij PQ <m <^ <n r^i fM PQ PQ PQ PQ PQ &i> ob u> »3 > > 180 V. PEROVSKITE-TYPE SUPERCONDUCTING STRUCTURES O 181 • Cu OO AB^BI, Ca ( Sr t Tt t Y Loytr Notation fEFq BASIC CELL SfTES «> O » V ( ) ( n CuO Layer OCu Layer [CuO,-] ^0 2 Cu| O r Y A -6 BaO Layer (Ba-OJ OBa Layer [O • Ba] Ca E - Layer [Ca--] Layers Ca C- Layer [--Ca] Image Layers Figure 7.7 Types of atom positions in the layers of a high-temperature superconductor structure, using the edge, face, center notation [E F C]. Typical site occupancies are given in the upper right (Poole et aL, 1989). remaining at their face positions. Similar alternations in position take place with Ba, O, and Ca layers, as illustrated in Fig. 7.7. Hauck et aL (1991) proposed a classi- fication of superconducting oxide struc- tures in terms of the sequence (1) super- conducting layers [Cu 0 2 -]and[-0 2 Cu], (2) insulating layers, such as [Y ] or [ Ca], and (3) hole-donating layers, such as [Cu O b -] or [Bi - O]. The high-temperature superconductor compounds have a horizontal reflection plane ( _L to z) called a h at the center of the unit cell and another a h reflection plane at the top (and bottom). This means that every plane of atoms in the lower half of the cell at the height z is duplicated in the upper half at the height 1-z. Such atoms, of course, appear twice in the unit cell, while atoms right on the symmetry planes only occur once since they cannot be reflected. Figure 7.8 shows a [Cu 0 2 -] plane at a height z reflected to the height 1-z. Note how the puckering preserves the reflection symmetry operation. Supercon- ductors that have this reflection plane, but lack end-centering and body-centering op- BaCuO* YCuO < BaCuO. [CuO [O-Ba] lal I ICuOj-] I--Y] [CuO,-] — i — 1-z [O-Ba] [CuO b -] Figure 7,8 Unit cell of YBa 2 Cu 3 0 7 showing the molecular groupings, reflection plane, and layer types. 182 7 PEROVSKITE AND CUPRATE CRYSTALLOGRAPHIC STRUCTURES erations (see Section VII), are called aligned because all of their copper atoms are of one type; either all on the edge (0,0,2) in E positions or all centered ({, \, z) at C sites. In other words, they all lie one above the other on the same verti- cal lines, as do the Cu ions in Fig. 7.8. VI. ALIGNED YBa 2 Cu 3 0 7 The compound YBa 2 Cu 3 0 7 , some- times called YBaCuO or the 123 com- pound, in its orthorhombic form is a superconductor below the transition tem- perature T c « 92 K. Figure 7.8 sketches the locations of the atoms, Fig. 7.9 shows the arrangement of the copper oxide planes, Fig. 7.10 provides more details on the unit cell, and Table 7.4 lists the atom positions and unit cell dimensions (Beno et at, 1987 Capponi et al., 1987; Hazen et al, 1987 Jorgensen et al, 1987; Le Page et ai, 1987 Siegrist et aL, 1987; Yan and Blanchin, 1991; see also Schuller et al., 1987). Con- sidered as a perovskite derivative, it can be looked upon as a stacking of three per- ovskite units BaCu0 3 , YCu0 2 , and BaCu0 2 , two of them with a missing oxy- gen, and this explains why c ~ 3a. It is, however, more useful to discuss the com- pound from the viewpoint of its planar structure. A, Copper Oxide Planes We see from Fig. 7.9 that three planes containing Cu and O are sandwiched be- tween two planes containing Ba and O and one plane containing Y. The layering scheme is given on the right side of Fig. 7.8, where the superscript b on O indicates that the oxygen lies along the 6-axis, as shown. The atoms are puckered in the two [Cu 0 2 -] planes that have the [ Y] plane between them. The third copper ox- ide plane [Cu O b -], often referred to as CuO 11.68* « Ba 2.16 * O 1.83 -CuO 0 Figure 7.9 Layering scheme of orthorhombic YBa 2 Cu 3 0 7 with the puckering indi- cated. The layers are perpendicular to the c-axis (Poole et aL, 1988, p. 101). Figure 7.10 Sketches of the superconducting orthorhombic (left) and nonsuperconducting tetragonal (right) YBaCuO unit cells Ther- mal vibration ellipsoids are shown for the atoms. In the tetragonal form the oxygen atoms are randomly dispersed over the basal plane sites (Jorgensen et at., 1987a, b; also see Schuller et al 1987) Table 7.4 Normalized Atom Positions in the YBa 2 Cu 3 0 7 Orthorhombic Unit Cell (dimensions a=3.83A, 6 = 3.88 A, and c= 11.68 A) Layer Atom [Cu O -] [O - Ba] [O - Ba] [Cu O -] Cu(l) 0 1 1 CK4) 0 0 0.8432 O(l) 0 i Ba I i 0.8146 Cu(2) 0 0 0.6445 ( [Cu0 2 -] CX3) 0 } 0.6219 °<2) T 0 0.6210 I,. [ Y] Y - I i !k 2 2 2 CK2) i o [Cu 0 2 -] CK3) 0 i 0.3790 10 0.3781 '"""I Cu(2) 0 0 0.3555 Ba 1 ? 0.1854 °< 4 > 0 0 0.1568 <X1) 0 } 0 Cu(l) o 0 n 7 PEROVSKITE AND CUPRATE CRYSTALLOGRAPHIC STRUCTURES 184 "the chains," consists of -Cu-O-Cu-O- chains along the b axis in lines that are perfectly straight because they are in a horizontal reflection plane a h \ where no puckering can occur. Note that, according to the figures, the copper atoms are all stacked one above the other on edge (E) sites, as expected for an aligned-type su- perconductor. Both the copper oxide planes and the chains contribute to the superconducting properties. B. Copper Coordination Now that we have described the planar structure of YBaCuO it will be instructive to examine the local environment of each copper ion. The chain copper ion Cu(l) is square planar-coordinated and the two coppers Cu(2) and Cu(3) in the plane ex- hibit fivefold pyramidal coordination, as indicated in Fig. 7.11. The ellipsoids at the atom positions of Fig. 7.10 provide a meas- ure of the thermal vibrational motion which the atoms experience, since the am- plitudes of the atomic vibrations are indi- cated by the relative size of each of the ellipsoids. C Stacking Rules The atoms arrange themselves in the various planes in such a way as to enable them to stack one above the other in an efficient manner, with very little interfer- ence from neighboring atoms. Steric ef- fects prevent large atoms such as Ba (1.34 A) and O (1.32 A) from overcrowding a layer or from aligning directly on top of each other in adjacent layers. In many cuprates stacking occurs in accordance with the following two empirical rules: 1. Metal ions occupy either edge or cen- tered sites, and in adjacent layers al- ternate between E and C sites. 2. Oxygens are found in any type of site, but they occupy only one type in a particular layer, and in adjacent layers they are on different types of sites. YBa 2 Cu 3 0 7 Figure 7.11 Stacking of pyramid, square-planar, and inverted pyramid groups along the c-axis of or- thorhombic YBa 2 Cu 3 0 7 (adapted from Poole et aL, 1988, p. 100). Minor adjustments to make more room can be brought about by puckering or by distorting from tetragonal to orthorhom- bic. D. Crystallographic Phases The YBa 2 Cu 3 0 7 _ 5 compound comes in tetragonal and orthorhombic varieties, as shown in Fig. 7.10, and it is the latter phase which is ordinarily superconducting. In the tetragonal phase the oxygen sites in the chain layer are about half occupied VI. ALIGNED YBa 2 Cu 3 0 7 in a random or disordered manner, and in the orthorhombic phase are ordered into -Cu-O- chains along the b direction. The oxygen vacancy along the a direction causes the unit cell to compress slightly so that a < b y and the resulting distortion is of the rectangular type shown in Fig. 7.4a. Increasing the oxygen content so that 8 < 0 causes oxygens to begin occupying the va- cant sites along a. Superlattice ordering of the chains is responsible for the phase that goes superconducting at 60 K. YBaCuO is prepared by heating in the 750-900°C range in the presence of vari- ous concentrations of oxygen. The com- pound is tetragonal at the highest temper- atures, increases its oxygen content through oxygen uptake and diffusion (Rothman et al, 1991) as the temperature is lowered, and undergoes a second-order phase transition of the order-disorder type at about 700°C to the low-temperature or- thorhombic phase, as indicated in Fig. 7.12 Temperature (K) "i i i i — i — I — i — r^~r 0 200 400 600 600 1000 Temperature (°C) Figure 7.12 Fractional occupancies of the (},0,0) (bottom) and (0,^,0) (top) sites (scale on left), and the oxygen content parameter 8 (center, scale on right) for quench temperatures of YBaCuO in the range 0-1000°C. The 8 parameter curve is the aver- age of the two site-occupancy curves (adapted from Jorgensen et a/., 1987a; also see Schuller et ai., 1987; see also Poole et aL, 1988). 185 (Jorgensen et al., 1987, 1990; Schuller et ai, 1987; cf. Beyers and Ahn, 1991; Metzger et ai, 1993; Fig. 8). Quenching by rapid cooling from a high temperature can produce at room temperature the tetrago- nal phase sketched on the right side of Fig. 7.10, and slow annealing favors the or- thorhombic phase on the left. Figure 7.12 shows the fractional site occupancy of the oxygens in the chain site (0, ±,0) as a func- tion of the temperature in an oxygen at- mosphere. A sample stored under sealed conditions exhibited no degradation in structure or change in T c four years later (Sequeira etai, 1992). Ultra-thin films tend to be tetragonal (Streiffer et ai, 1991). E. Charge Distribution Information on the charge distribu- tions around atoms in conductors can be obtained from knowledge of their energy bands (see description in Chapter 8). This is most easily accomplished by carrying out a Fourier-type mathematical transforma- tion between the reciprocal k x ,k y , A: 2 -space (cf. Chapter 8, Section II) in which the energy bands are plotted and the coordi- nate x ? y 9 z-space, where the charge is dis- tributed. We will present the results obtained for YBa 2 Cu 3 0 7 in the three ver- tical symmetry planes (x,z, y,z, and diago- nal), all containing the z-axis through the origin, shown shaded in the unit cell of Fig. 7.13. Contour plots of the charge density of the valence electrons in these planes are sketched in Fig. 7.14. The high density at the Y 3+ and Ba 2+ sites and the lack of contours around these sites together indi- cate that these atoms are almost com- pletely ionized, with charges of +3 and + 2, respectively. It also shows that these ions are decoupled from the planes above and below. This accounts for the magnetic isolation of the Y site whereby magnetic ions substituted for yttrium do not inter- fere with the superconducting properties. In contrast, the contours surrounding the Cu and O ions are not characteristic of an 186 Figure 7.13 Three vertical crystailographic planes (x, z-, y, z-, and diagonal) of a tetragonal unit cell of YBa 2 Cu 3 0 7 , and standard notation for the four crys- tailographic directions. o o v 7 PEROVSKITE AND CUPRATE CRYSTALLOCRAPHIC STRUCTURES ordinary ionic compound. The short Cu-0 bonds in the planes and chains (1.93-1.96 A) increase the charge overlap. The least overlap appears in the Cu(2)-0(4) vertical bridging bond, which is also fairly long (2.29 A). The Cu, O charge contours can be represented by a model that assigns charges of + 1.62 and - 1.69 to Cu and O, respectively, rather than the values of + 2.33 and -2.00 expected for a standard ionic model, where the charge +2.33 is an average of +2, +2, and +3 for the three copper ions. Thus the Cu-0 bonds are not completely ionic, but partly covalent. F. YBaCuO Formula In early work the formula YBa 2 Cu 3 0 9 _ 5 was used for YBaCuO because the proto- type triple pervoskite (YCu0 3 XBaCu0 3 ) 2 has nine oxygens. Then crystallographers showed that there are eight oxygen sites in the 14-atom YBaCuO unit cell, and the formula YBa 2 Cu 3 O g _ 5 came into widespread use. Finally, structure refine- ments demonstrated that one of the oxy- gen sites is systematically vacant in the chain layers, so the more appropriate ex- pression YBa 2 Cu 3 0 7 _ 5 was introduced. It would be preferable to make one more change and use the formula Ba 2 YCu 3 0 7 _ 5 to emphasize that Y is analogous to Ca in the bismuth and thallium compounds, but very few workers in the field do this, so we reluctantly adopt the usual "final" nota- tion. In the Bi-Tl compound notation of Section IX, B, Ba 2 YCu 3 0 7 _ s would be called a 0213 compound. We will follow the usual practice of referring to YBa 2 Cu 3 0 7 _ 5 as the 123 compound. <100> <110> <010> Figure 7.14 Charge density in the three symmetry planes of YBaCuO shown shaded in Fig. 7.13. The x, z, diagonal and the y, z planes are shown from left to right, labeled <100>, <110>, and <010>, respectively. These results are obtained from band structure calcu- lations, as will be explained in the following chapter (Krakauer and Pickett, 1988). G. YBa 2 Cu 4 O a and Y 2 Ba 4 Cu 7 0 15 These two superconductors are some- times referred to as the 124 compound and the 247 compound, respectively. They have the property that for each atom at position (x,y,z) there is another identical atom at r. VII. BODY CENTERING 0(3) 0(3) Figure 7.15 Crystal structure of YBa 2 Cu 4 O g showing how, as a result of the side-centering symme- try operation, the atoms in adjacent Cu-O chains are staggered along the y direction, with Cu above O and O above Cu (Heyen et al y 1991; modified from Cam- puzano et al, 1990). position (x 9 y+ \,z+ \). In other words, the structure is side centered. This prop- erty prevents the stacking rules of Section C from applying. The chain layer of YBa 2 Cu 3 0 7 be- comes two adjacent chain layers in YBa 2 Cu 4 0 8 , with the Cu atoms of one chain located directly above or below the O atoms of the other, as shown in Fig. 7.15 (Campuzano et ai y 1990; Heyen et al, 1990a, 1991; Iqbal, 1992; Kaldis etaL, 1989; Marsh et ai, 1988; Morris et ai, 1989a). The transition temperature remains in the range from 40 K to 80 K when Y is re- placed by various rare earths (Morris et ai, 1989). The double chains do not exhibit the variable oxygen stoichiometry of the single ones. The other side-centered compound, Y 2 Ba 4 Cu 7 0 15 , may be considered accord- ing to Torardi, "as an ordered 1 : 1 inter- growth of the 123 and 124 compounds (YBa 2 Cu 3 0 7 + YBa 2 Cu 4 O g = Y 2 Ba 4 Cu 7 0 15 )» 187 (Bordel et al, 1988, Gupta and Gupta, 1993). The 123 single chains can vary in their oxygen content, and superconductiv- ity onsets up to 90 K have been observed. This compound has been synthesized with several rare earths substituted for Y (Morris et al, 1989b). VII. BODY CENTERING In Section V we discussed aligned-type superconductor structures that possess a horizontal plane of symmetry. Most high- temperature superconductor structures have, besides this a h plane, an additional symmetry operation called body centering whereby for every atom with coordinates (x,y, z) there is an identical atom with coordinates as determined from the follow- ing operation: x^x±\, x-*y±\ y z^z±\ (7.5) Starting with a plane at the height z this operation forms what is called an image plane at the height z + \ in which the edge atoms become centered, the centered atoms become edge types, and each face atom moves to another face site. In other words, the body-centering operation acting on a plane at the height z forms a body centered plane, also called an image plane, at the height z±\. The signs in these operations are selected so that the gener- ated points and planes remain within the unit cell. Thus if the initial value of z is greater than \, the minus sign must be selected, viz., z->z-|. Body centering causes half of the Cu-O planes to be [Cu 0 2 -], with the copper atoms at edge sites, and the other half to be [- 0 2 Cu], with the copper atoms at centered sites. Let us illustrate the symmetry features of a body-centered superconductor by con- sidering the example of Tl 2 Ba 2 CaCu 2 O g . This compound has an initial plane [Cu 0 2 -] with the copper and oxygen atoms at the vertical positions z = 0.0540 and 0.0531, respectively, as shown in Fig. 188 7 PEROVSKITE AND CUPRATE CRYSTALLOGRAPHIC STRUCTURES 1 -z 7 REFLECTED PLANE [Cu0 2 -] BODY CENTERED PLANE [0 2 Cu] REFLECTED AND BODY CENTERED l-0 2 Cu] INITIAL PLANE [CuQ 2 -] Figure 7.16 Body-centered tetragonal unit cell containing four puckered Cu0 2 groups showing how the initial group (bottom) is replicated by reflection in the horizontal reflection plane (z = }), by the body centering operation, and by both. 7.16. For illustrative purposes the figure is drawn for values of z closer to 0.1. We see from the figure that there is a reflected plane [Cu 0 2 -] at the height 1 - z, an image (i.e., body centered) plane [- 0 2 Cu] of the original plane at the height \ + z, and an image plane [- 0 2 Cu] of the reflected plane (i.e., a reflected and body centered plane) at the height \-z. Figure 7.16 illustrates this situation and indicates how the atoms of the initial plane can be transformed into particular atoms in other planes (see Problem 5). Figure 7.17 shows how the configurations of the Reflected Subcell II Body Centered Subcell III Reflected and Body Centered Subcell IV Basic Subcell I Figure 7.17 Body-centered unit cell divided into four regions by the reflection and body centering operations. atoms in one-quarter of the unit cell, called the basic subcell, or subcell I, determine their configurations in the other three subcells II, III, and IV through the sym- metry operations of reflection and body centering. VIII. BODY-CENTERED La 2 Cu0 4 AND Nd 2 Cu0 4 The body-centered compound M 2 Cu0 4 has three structural variations in the same crystallographic space group, namely the M = La and M = Nd types, and a third mixed variety (Xiao et aU 1989). Table 7.5 lists the atom positions of the first two types, and Fig. 7.18 presents sketches of the structures of all three. Each will be discussed in turn. A. Unit Cell Generation of La 2 Cu0 4 (T Phase) The structure of the more common La 2 Cu0 4 variety, often called the T phase, VIII. BODY-CENTERED La 2 Cu0 4 and Nd 2 Cu0 4 Table 7.5 Atom Positions in the La 2 Cu0 4 and Nd 2 Cu0 4 Structures 789 La 2 Cu0 4 structure Nd 2 Cu0 4 structure Layer Atom X y z Layer Atom X z i 2 a u i i 1 2 n u i i LCu u 2 -J cu n u i i ivu u 2 j n i i O(l) 0 1 2 1 r\{ -\ \ CXD I l La 1 2 2 U.OOZ I — NdJ i 2 2 a QAi no} n \j A VJ 0 SIR U.Olo net) n 1 2 4 i 2 1 2 U.OOZ 2 A U 3 4 [La - 0] I 3 0 u o tr U.DjO NH n o U.UJO l 2 u 2 \J\ij i 2 A u 1 2 [- 0 2 CuJ Cu 1 2 i 2 1 2 l~ U 2 CuJ cu 1 2 2 2 O(l) 0 1 y 1 2 0(1) 0 1 2 1 2 La u u [Nd J rt u rt U.30Z ILa - OJ 0(2) 1 2 2 0.318 h o 2 -] CK3) i 2 0 l 4 0(2) 0 0 0.182 (X3) 0 1 2 1 4 [O - La] La 1 2 2 0.138 [- - Nd] Nd 1 2 I 2 0.138 O(l) 0 1 2 0 CXI) 0 I 2 0 [Cu 0 2 -] Cu 0 0 0 [Cu 0 2 -] Cu 0 0 0 O(l) 2 0 0 CKl) 1 2 0 0 Figure 7.18 (a) Regular unit cell (T phase) associated with hole-type (La 1 _^Sr jr ) 2 Cu0 4 superconductors, (b) hybrid unit cell (T* phase) of the hole-type La 2 -.r-^>>Sr x Cu0 4 superconductors, and (c) alternate unit cell (T phase) associated with electron-type (Nd,_ x Ce Jt ) 2 Cu0 4 superconductors. The La atoms in the left structure become Nd atoms in the right structure. The upper part of the hybrid cell is T type, and the bottom is T'. The crystallographic space group is the same for all three unit cells (Xiao et a/., 1989; see also Oguchi, 1987; Ohbayashi et al. 9 1987; Poole et a/., 1988, p. 83; Tan et ai, 1990). 190 7 PEROVSKITE AND CUPRATE CRYSTALLOGWHIC STRUCTURES can be pictured as a stacking of Cu0 4 La 2 groups alternately with image (i.e., body centered) La 2 0 4 Cu groups along the c di- rection, as indicated on the left side of Fig. 7.19 (Cavaet et al, 1987; Kinoshita et al y 1992; Longo and Raccah, 1973; Ohbayashi et al, 1987; Onoda et aL, 1987; Zolliker et ai y 1990). Another way of visualizing the structure is by generating it from the group Cu.0 2 La, comprising the layers [O-La] and* {[Cu 0 2 -] in subcell I shown on the right side of Fig. 7.19 and also on the left side of Fig. 7.20. (The factor { appears because the [Cu 0 2 -] layer is shared by two subcells.) Subcell II is formed by re- flection from subcell I, and subcells III and IV are formed from I and II via the body-centering operation in the manner of Figs. 7.16 and 7.17. Therefore, subcells I and II together contain the group Cu0 4 La 2 > and subcells III and IV together contain its image (body centered) counter- part group La 2 0 4 Cu. The BiSrCaCuO and TIBaCaCuO structures to be discussed in Section IX can be generated in the same manner, but with much larger repeat units along the c direction. B. Layering Scheme The La 2 Cu0 4 layering scheme con- sists of equally-spaced, flat Cu0 2 layers with their oxygens stacked one above the other, the copper ions alternating between the (0,0,0) and (\,\,\) sites in adjacent layers, as shown in Fig. 7.21. These planes are body-centered images of each other, and are perfectly flat because they are vni. Formula Units Structure CuQ, La 2 O Sub Layers Cells [O-La] La 2 0 4 Cu (Image) UNIT CELL o [Cu0 2 -1 [O-La] [La-O] III r rt . . UNIT £.0,011 CELL Cu0 4 La 2 o- ILa-O] [O-La] [CuO a ] [O-U] IV Figure 7.19 Structure of La 2 Cu0 4 (center), showing the formula units (left) and the level labels and subcell types (right). Two choices of unit cell are indicated, the left-side type unit cell based on formula units, and the more common right-side type unit cell based on copper-oxide layers. Figur left) z tions ( in the at< VIII. BODY-CENTERED La 2 Cu0 4 and Nd 2 Cu0 4 La 2 Cu0 4 Sub Coll Nd, CuO, 2 «• [Cu0 2 -J [Cu0 2 -] [O-La] II [--Nd] [La - 0] III [-o 2 -j [Nd--] l-0 2 Cu] [-0 2 Cu] [La-0] IV [Nd--] [O-La] 1 [-0 2 -] [--Nd] [Cu0 2 -] [CuO,-] Figure 7.20 Layering schemes of the La 2 Cu0 4 (T, left) and Nd 2 Cu0 4 (T, right) structures. The loca- tions of the four subcells of the unit cell are indicated in the center column. 191 reflection planes. Half of the oxygens, O(l), are in the planes, and the other half, 0(2), between the planes. The copper is octahe- drally coordinated with oxygen, but the distance 1.9 A from Cu to (XD in the Cu0 2 planes is much less than the vertical distance of 2.4 A from Cu to the apical oxygen 0(2), as indicated in Fig. 7.22. The La is ninefold coordinated to four O(l) oxygens, to four 0(2) at (±, ±, z) sites, and to one 0(2) at a (0,0, z) site. C Charge Distribution Figure 7.23 shows contours of con- stant-valence charge density on a logarith- mic scale drawn on the back x,z-plane and on the diagonal plane of the unit cell sketched in Fig. 7.13. These contour plots are obtained from the band structure cal- culations described in Chapter 8, Section XIV. The high-charge density at the lan- thanum site and the low charge density around this site indicate an ionic state .1 -.. . . /+— 3.78 A — •>/ Figure 7.21 Cu0 2 layers of the La 2 Cu0 4 structure showing horizontal displacement of Cu atoms in alternate layers. The layers are perpendicular to the c-axis (Poole et al. y 1988, p. 87). La 3+ . The charge density changes in a fairly regular manner around the copper and oxygen atoms, both within the Cu0 2 planes and perpendicular to these planes, suggestive of covalency in the Cu-O bond- ing, as is the case with the YBa 2 Cu 3 0 7 compound. D. Superconducting Structures The compound La 2 Cu0 4 is itself an antiferromagnetic insulator and must be doped, generally with an alkaline earth, to exhibit pronounced superconducting prop- erties. The compounds (La,_^A/ Jt ) 2 Cu0 4 , with 3% to 15% of M = Sr or Ba replacing La, are orthorhombic at low temperatures and low M contents and are tetragonal otherwise; superconductivity has been found on both sides of this transition. The orthorhombic distortion can be of the rect- angular or of the rhombal type, both of which are sketched in Fig. 7.4. The phase diagram of Fig. 7.24 shows the tetragonal, orthorhombic, superconducting, and anti- ferromagnetically ordered regions for the lanthanum compound (Weber et aL, 1989; cf. Goodenough et al, 1993). We see that VIII. BODY-CENTERED La 2 Cu0 4 and Nd 2 Cu0 4 193 La2Cu0 4 <100> <110> Figure 7.23 Contour plots of the charge density of La 2 Cu0 4 obtained from band structure calculations. The x, 2-crystallographic plane labeled <100) is shown on the left and the diagonal plane labeled (110) on the right. The contour spacing is on a logarithmic scale (Pickett, 1989). 0.05 0 -i ► x Figure 7.24 Phase diagram for hole-type ^ a 2-jcSr J Cu0 4 _ >F indicating insulating (INS), antifer- romagnetic (AF), and superconducting (SC) regions. Figure VI-6 of Poole et al. (1988) shows experimental data along the orthorhombic-to-tetragonal transition line. Spin-density waves (SDW) are found in the AF region (Weber et al, 1989). the orthorhombic phase is insulating at high temperatures, metallic at low temper- atures, and superconducting at very low temperatures. Spin-density waves, to be discussed in Chapter 8, Section XIX, occur in the antiferromagnetic region. E. Nd 2 Cu0 4 Compound (T Phase) The rarer Nd 2 Cu0 4 structure (Skan- takumar et al, 1989; Sulewski et al, 1990; Tan et al, 1990) given on the right side of Fig. 7.18 and Table 7.5 has all of its atoms in the same positions as the standard La 2 Cu0 4 structure, except for the apical 0(2) oxygens in the [O-La] and [La-O] layers, which move to form a [- 0 2 -] layer between [- - La] and [La - -]. These oxygens, now called 0(3), have the same x, y coordinate positions as the O(l) oxygens, and are located exactly between the Cu0 2 planes with z = \ or f . We see from Fig. 7.18 that the Cu0 6 octahedra have now lost their apical oxygens, causing Cu to become square planar-coordinated Cu0 4 groups. The Nd is eightfold coordi- nated to four O(l) and four 0(3) atoms, but with slightly different Nd-O distances. The Cu0 2 planes, however,, are identical in the two structures. Superconductors with this Nd 2 Cu0 4 structure are of the electron type, in contrast to other high-temperature superconductors, in which the current car- riers are holes. In particular, the electron superconductor Nd 185 Ce 015 CuO 4 _ 5 with T c = 24 K has been widely studied (Fontcuberta and Fabrega, 1995, a review chapter; Allen 1990; Alp et al, 1989b; Bar- lingay et al, 1990; Ekino and Akimitsu, 1989a, b; Lederman et al, 1991; Luke et al, 1990; Lynn et al, 1990; Sugiyama et al, 1991; Tarason et al, 1989a). Other rare earths, such as Pr (Lee et al, 1990) and Sm (Almasan et al, 1992) have replaced Nd. The difference of structures associated with different signs attached to the current carriers may be understood in terms of the doping process that converts undoped ma- terial into a superconductor. Lanthanum and neodymium are both trivalent, and in the undoped compounds they each con- (7.6) (7.7) 794 tribute three electrons to the nearby oxygens, La->La 3+ + 3e - , Nd-»Nd 3+ + 3e', to produce O 2- . To form the superconduc- tors a small amount of La in La 2 Cu0 4 can be replaced with divalent Sr, and some Nd in Nd 2 Cu0 4 can be replaced with tetra- valent Ce, corresponding to Sr-^Sr 2+ + 2e" (in La 2 Cu0 4 ) Ce _ C e 4+ + 4e" (in Nd 2 Cu0 4 ) Thus, Sr doping decreases the number of electrons to produce hole-type carriers, while Ce doping increases the electron concentration and the conductivity is elec- tron type. There are also copper-oxide electron superconductors with different structures, such as Sr 1 _ x Nd ;c Cu0 2 (Smith et al, 1991) and TlCa^/^S^Cu.CVs, where R is a rare earth (Vijayaraghavan et al, 1989). Electron- and hole-type superconductivity in the cuprates has been compared (Katti and Risbud, 1992; Medina and Regueiro, 1990). F. La 2 _ x _ y /? x Sr ) ,Cu0 4 Compounds (T* Phase) We have described the T structure of La 2 Cu0 4 and the T structure of Kd 2 Cu0 4 . The former has 0(2) oxygens and the latter 0(3) oxygens, which changes the coordinations of the Cu atoms and that of the La and Nd atoms as well. There is a hybrid structure of hole-type supercon- ducting lanthanum cuprates called the T* structure, illustrated in Fig. 7.18b, in which the upper half of the unit cell is the T type with 0(2) oxygens and lower half the T' type with 0(3) oxygens. These two vari- eties of halfcells are stacked alternately along the tetragonal c-axis (Akimitsu et al, 1988; Cheong et al, 1989b; Kwei et al, 1990; Tan et al, 1990). Copper, located in the base of an oxygen pyramid, is fivefold-coordinated CuO s . There are two inequivalent rare earth sites; the ninefold- coordinated site in the T-type halfcell is PEROVSKITE AND CUPRATE CRYSTALLOCRAPHIC STRUCTURES preferentially occupied by the larger La and Sr ions, while the smaller rare earths R (i.e., Sm, Eu, Gd, or Tb) prefer the eightfold-coordinated site in the T' half- cell. Tan et al (1991) give a phase diagram for the concentration ranges over which the T and T* phases are predominant. IX. BODY-CENTERED BiSrCaCuO AND TlBaCaCuO Early in 1988 two new superconduct- ing systems with transition temperatures considerably above those attainable with YBaCuO, namely the bismuth- and thal- lium-based materials, were discovered. These compounds have about the same a and b lattice constants as the yttrium and lanthanum compounds, but with much larger unit cell dimensions along c. We will describe their body-centered struc- tures in terms of their layering schemes. In the late 1940s some related compounds were synthesized by the Swedish chemist Bengt Aurivillius (1950, 1951, 1952). A. Layering Scheme The Bi 2 Sr 2 Ca„Cu n+1 0 6+2n and Tl 2 Ba 2 Ca n Cu n + 1 0 6 + 2 n compounds, where n is an integer, have essentially the same structure and the same layering arrangement (Barry et al, 1989; Siegrist et al, 1988; Torardi et al, 1988a; Yvon and Francois, 1989), although there are some differences in the detailed atom positions. Here there are groupings of Cu0 2 layers, each separated from the next by Ca layers with no oxygen. The Cu0 2 groupings are bound together by interven- ing layers of BiO and SrO for the bismuth compound, and by intervening layers of TIO and BaO for the thallium compound. Figure 7.25 compares the layering scheme of the Tl 2 Ba 2 Ca n Cu n + 1 0 6+2 „ compounds with n = 0,1,2 with those of the lan- thanum and yttrium compounds. We also see from the figure that the groupings of [Cu 0 2 -] planes and [- 0 2 Cu] image IX. BODY-CENTERED BiSrCaCuO and TIBaCaCuO 795 Ba 2 YCu 3 0 7 r-[CuO-J- I-0 2 Cu] [O • Ba] [La-O] [Cu0 2 -] [O -Lai [-Y] (CuOH [Cu0 2 - ] [O-La] [O - Ba] [La-OJ -[CuO-J- l-0 2 Cu] Lapu0 4 [O-TI] [Ba - O] TI 2 Ba 2 Cu0 6 TI 2 Ba 2 CaCu 2 0 8 TI 2 Ba 2 Ca 2 Cu 3 O 10 Figure 7.25 Layering schemes of various high-temperature superconduc- tors. The Cu0 2 plane layers are enclosed in small inner boxes, and the layers that make up a formula unit are enclosed in larger boxes. The Bi-Sr compounds Bi 2 Sr 2 Ca n Cu n + 1 0 6 +2« have the same !averin g schemes as their Tl-Ba counterparts shown in this figure. (i.e., body centered) planes repeat along the c-axis. It is these copper-oxide layers that are responsible for the superconduct- ing properties. A close examination of this figure shows that the general stacking rules men- tioned in Section VI.C for the layering scheme are satisfied, namely metal ions in adjacent layers alternate between edge (E) and centered (C) sites, and adjacent layers never have oxygens on the same types of sites. The horizontal reflection symmetry at the central point of the cell is evident. It is also clear that YBa 2 Cu 3 0 7 is aligned and that the other four compounds are staggered. Figure 7.26 (Torardi et a/., 1988a) pre- sents a more graphical representation of the information in Fig. 7.25 by showing the positions of the atoms in their layers. The symmetry and body centering rules are also evident on this figure. Rao (1991) pro- vided sketches for the six compounds Tl m Ba 2 Ca n Cu rt + 1 0, similar to those in Fig. 7.26 with the compound containing one (m = 1) or two thallium layers (m = 2), where n =0,1,2, as in the Torardi et al figure. B. Nomenclature There are always two thalliums and two bariums in the basic formula for Tl 2 Ba 2 Ca n Cu n + 1 0 6+2n , together with n calciums and n + 1 coppers. The first three members of this series for n = 0, 1, and 2 are called the 2201, 2212, and 2223 com- pounds, respectively, and similarly for their 196 7 PEROVSKITE AND CUPRATE CRYSTALLOGRAPHIC STRUCTURES Tt? Jt? Jt? TI 2 Ba 2 Cu0 6 TI 2 Ba 2 CaCu 2 0 8 TI 2 Ba2Ca2Cu30 10 - a* - c« - Cu - St - TI - TI - 8* Cu Ca Cu Ca Cu B* TI TI Ba Cu Ca Cu Figure 7.26 Crystal structures of Tl 2 Ba 2 Ca rt Cu n + ,0 6 + 2rt superconducting compounds with « = 0,1,2 arranged to display the layering schemes. The Bi 2 Sr 2 Ca„Cu n + x 0 6+2n compounds have the same respective structures (Torardi et al, 1988a). BiSr analogues Bi 2 Sr 2 Ca n Cu rt + l O e + 2n - Since Y in YBa 2 Cu 3 0 7 is structurally analogous to Ca in the TI and Bi com- pounds, it would be more consistent to write Ba 2 YCu 3 0 7 for its formula, as noted in Section VLF. In this spirit Ba 2 YCu 3 0 7 _ 5 might be called the 0213 compound, and (La 1 _ x Af JC ) 2 Cu0 4 L 6 could be called 2001. C Bi-Sr Compounds Now that the overall structures and interrelationships of the BiSr and TIBa high-temperature superconductors have been made clear in Figs. 7.25 and 7.26 we will comment briefly about each com- pound. Table 7.3 summarizes the charac- teristics of these and related compounds. The first member of the BiSr series, the 2201 compound with n = 0, has octa- hedrally coordinated Cu and T c ~ 9 K (Torardi et ai, 1988b). The second mem- ber, Bi 2 (Sr,Ca) 3 Cu 2 0 8 + 5 , is a supercon- ductor with T c ~ 90 K (Subramanian et aL, 1988a; Tarascon et ai, 1988b). There are two [Cu 0 2 -] layers separated from each other by the [ Ca] layer. The spacing from [Cu 0 2 -] to [- - Ca] is 1.66 A, which is less than the corresponding spac- ing of 1.99 A between the levels [Cu 0 2 -] and [- - Y] of YBaCuO. In both cases the copper ions have a pyramidal oxygen coordination of the type shown in Fig. 7.11. Superlattice structures have been re- ported along a and b 7 which means that minor modifications of the unit cells re- peat approximately every five lattice spac- ings, as explained in Sect. IX.E. The third member of the series, Bi 2 Sr 2 Ca 2 Cu 3 O 10 , has three Cu0 2 layers separated from each other by [ Ca] planes and a higher transition temperature, 110 K, when doped with Pb. The two Cu ions have pyramidal coordination, while the third is square planar. )X. BODY-CENTERED BiSrCaCuO and TlBaCaCuO Charge-density plots of Bi 2 Sr 2 CaCu 2 0 8 indicate the same type of covalency in the Cu-O bonding as with the YBa 2 Cu 3 0 7 and La 2 Cu0 4 compounds. They also indi- cate very little bonding between the adja- cent [Bi - O] and [O - Bi] layers. D. Tl-Ba Compounds The TIBa compounds Tl 2 Ba 2 Ca n Cu n + 1 0 6+ 2n have higher transition temperatures than their bismuth counterparts (Iqbal et ai, 1989; Subramanian et al y 1988b; Torardi et ai, 1988a). The first member of the series, namely Tl 2 Ba 2 Cu0 6 with n = 0, has no [ Ca] layer and a relatively low transition temperature of ~ 85 K. The second member (n = 1), Tl 2 Ba 2 CaCu 2 0 8> called the 2212 compound, with T c = 110 K 197 has the same layering scheme as its Bi counterpart, detailed in Figs. 7.25 and 7.26. The [Cu 0 2 -] layers are thicker and closer together than the corresponding lay- ers of the bismuth compound (Toby et aL, 1990). The third member of the series, Tl 2 Ba 2 Ca 2 Cu 3 O 10 , has three [Cu 0 2 -] layers separated from each other by [ Ca] planes, and the highest transi- tion temperature, 125 K, of this series of thallium compounds. It has the same cop- per coordination as its BiSr counterpart. The 2212 and 2223 compounds are tetrag- onal and belong to the same crystallo- graphic space group as La 2 Cu0 4 . We see from the charge-density plot of Tl 2 Ba 2 Cu0 6 shown in Fig. 7.27 that Ba 2+ is ionic, Cu exhibits strong covalency, espe- cially in the Cu-O plane, and Tl also ap- pears to have a pronounced covalency. The bonding between the [Tl - O] and [O - Tl] planes is stronger than that between the [Bi - O] and [O - Bi] planes of Bi-Sr. [100] [100] [110] DISTANCE (a.u.) Figure 7.27 Contours of constant charge density on a logarithmic scale in two high-symmetry crystallographic planes of Tl 2 Ba 2 Cu0 6 . Oxygen atoms O(l), (X2), and CK3) are denoted 1, 2, and 3, respectively. The planar Cu-Ol binding is strongest (Hamann and Mattheiss, 1988; see Pickett, 1989). 198 E. Modulated Structures The x-ray and neutron-diffraction pat- terns obtained during crystal structure de- terminations of the bismuth cuprates Bi 2 Sr 2 Ca„Cu„ +l 0 6+ 2n exhibit weak satel- lite lines with spacings that do not arise from an integral multiple of the unit cell dimensions. These satellites o have modula- tion periods of 21 A, 19.6 A, and 20.8 A, respectively, for the n = 0, 1, and 2 com- pounds (Li et aL, 1989). Since the lattice constant a = 5.41 A (b = 5.43 A) for all three compounds, this corresponds to a superlattice with unit cell of dimensions ~ 3.8a, b, c, with the repeat unit along the a direction equal to ~ 3.8a for all three compounds. A modulation of 4Jb has also been reported (Kulik et aL, 1990). This structural modulation is called incommen- surate because the repeat unit is not an integral multiple of a. Substitutions dramatically change this modulation. The compound Bi^Ca^XO^O, 7 PEROVSKITE AND CUPRATE CRYSTALLOCRAPH1C STRUCTURES Tsukada, 1989a, b) to calculate the quasi- particle spectrum of superlattices. has a period that decreases from about 4.86 for x = 0 to the commensurate value 4.0b for x = 1 (Inoue et aL, 1989; Tamegai et aL, 1989). Replacing Cu by a transition metal (Fe, Mn, or Co) produces nonsuper- conducting compounds with a structural modulation that is commensurate with the lattice spacing (Tarascon et aL, 1989b). A modulation-free bismuth-lead cuprate su- perconductor has been prepared (Mani- vannan et aL, 1991). Kistenmacher (1989) examined substitution-induced superstruc- tures in YBa^Cu^M^O?. Superlattices with modulation wavelengths as short as 24 A have been prepared by employing ultra-thin deposition techniques to inter- pose insulating planes of PrBa 2 Cu 3 0 7 be- tween superconducting Cu-O layers of YBa 2 Cu 3 0 7 (Jakob et aL, 1991; Lowndes et aL, 1990; Pennycook et aL, 1991; Ra- jagopal and Mahanti, 1991; Triscone et aL 1990). Tanaka and Tsukada (1991) used the Kronig-Penney model (Tanaka and F. Aligned Tl-Ba Compounds A series of aligned thallium-based su- perconducting compounds that have the general formula TlBajCa^u^ i0 5 + 2fI with n varying from 0 to 5 has been re- ported (Ihara et aL, 1988; Rona, 1990). These constitute a series from 1201 to 1245. They have superconducting transi- tion temperatures almost as high as the Tl 2 Ba 2 Ca n Cu n + 1 0 6+2n compounds. Data on these compounds are listed in Table 7.3. G. Lead Doping In recent years a great deal , of effort has been expended in synthesizing lead- doped superconducting cuprate structures (Itoh and Uchikawa, 1989). Examples in- volve substituting Pb for Bi (Dou et aL, 1989; Zhengping et aL, 1990), for Tl (Barry et aL, 1989; Mingzhu et aL, 1990), or for both Bi and Tl (Iqbal et aL, 1990). Differ- ent kinds of Pb, Y-containing super- conductors have also been prepared (cf. Mattheiss and Hamann, 1989; Ohta and Maekawa, 1990; Tang et aL, 1991; Tokiwa et aL, 1990, 1991). X. ALIGNED HgBaCaCuO The series of compounds HgBa 2 Ca„Cu n + 1 0 2rt + 4 , where n is an integer, are prototypes for the Hg family of superconductors. The first three members of the family, with n = 0,1,2, are often referred to as Hg-1201, Hg-1212, and Hg-1223, respectively. They have the structures sketched in Fig. 7.28 (Tokiwa- Yamamoto et aL, 1993; see also Martin et aL, 1994; Putilin et al^ 1991). The lattice constants are a = 3.86 A for all of them, and c = 9.5, 12.6, and 15.7 A for n = 0, 1,2, respectively. The atom positions of the m — 1 compound are listed in Table 7.6 (Hur et aL, 1994). The figure is drawn Table 7.6 Normalized Atom Positions in the Tetragonal Unit Cell of HgBa 2 Ca 086 Sr 0 14 Cu 2 0 6+ / Layer Atom y z Hg 0 0 1 [Hg - -] 0(3) i 2 i 2 1 0(2) 0 0 0.843 [O - Ba] Ba 1 2 ] 2 0.778 Cu 0 0 0.621 [Cu 0 2 -] OU) 0 t 2 0.627 OU) 1 2 0 0.627 [- - Ca] Ca, Sr 1 2 1 2 2 OU) 1 2 0 0.373 [Cu 0 2 -] OU) 0 I 2 0.373 Cu 0 0 0379 Ba 1 2 1 2 0.222 [O - Ba] CK2) 0 0 0.157 CK3) 2 1 2 0 [Hg - -] Hg 0 0 0 0 Unit cell dimensions a = 3.8584 A and c = 12.6646 A, space group is P4/mmm y D\ h . The Hg site is 91% occupied and the 0(3) site is 11% occupied (6 = 0.11). The data are from Hur et al (1994). 200 7 PEROVSKITE AND CU PRATE CRYSTALLOGRAPHIC STRUCTURES © Hg # Ca • Cu O o (Partial Occupancy) Figure 7.29 Schematic structure of the HgBa 2 CaCu 2 0 6 + 5 compound which is also called Hg-1212 (Meng et ai, 1993a). with mercury located in the middle layer of the unit cell, while the table puts Hg at the origin (000) and Ca in the middle (~ \ \). Figure 7.29 presents the unit cell for the n = 1 compound HgBa 2 CaCu 2 0 6+5 drawn with Ca in the middle (Meng et ai, 1993a). The symbol 8 represents a small excess of oxygen located in the center of the top and bottom layers, at positions ^ 0 and \ \ 1 which are labeled "partial occupancy" in the figure. If this oxygen were included the level symbol would be [Hg - O] instead of [Hg ]. These Hg compound structures are similar to those of the series TlBa 2 Ca„Cu n + ^2^ + 4 mentioned above in Section IX.F. We see from Fig. 7.28 that the copper atom of Hg-1201 is in the center of a stretched octahedron with the planar oxy- gens O(l) at a distance of 1.94 A, and the apical oxygens 0(2) of the [O - Ba] layer much further away (2.78 A). For n = 1 each copper atom is in the center of the [— [H9 " ] — | [O-Ba] [CuQ 2 -l [O - Ba] -[Hg--]— J I— [Hg--]— , [O - Ba] [Cu0 2 -] [--Ca] [Cu0 2 ] [0 - Ba] ' — [Hg - -] - [Hg--]-n [O - Ba] [Cu0 2 -] [ -Ca] [CuO*-] [-Ca] [CuQ 2 -] HgBa 2 CuQ 4 HgBa 2 CaCu 2 0 6 [O - Ba] I— [Hg--]- HgBa 2 Ca 2 Cu 3 O e Figure 7.30 Layering schemes of three HgBa 2 Ca n Cu n+ x0 2n + A compounds, using the nota- tion of Fig. 7.25. base of a tetragonal pyramid, and for n = 2 the additional Cu0 2 layer has Cu atoms which are square planar coordinated. The layering scheme stacking rules of Section VI.C are obeyed by the Hg series of com- pounds, with metal ions in adjacent layers alternating between edge (E) and centered (C) sites, and oxygen in adjacent layers always at different sites. We see from Table 7.6 that the [O - Ba] layer is strongly puckered and the [Cu 0 2 -] layer is only slightly puckered. The relationships between the layering scheme of the HgBa 2 Ca n Cu n + 1 0 2rt + 4 series of compounds and those of the other cuprates may be seen by comparing the sketch of Fig. 7.30 with that of Fig. 7.25. We see that the n = 1 compound HgBa 2 CaCu 2 0 6 is quite similar in struc- ture to YBa 2 Cu 3 0 7 with Ca replacing Y in the center and Hg replacing the chains [Cu O -]. More surprising is the similarity between the arrangement of the atoms in the unit cell of each HgBa 2 Ca rt Cu n + 1 0 2 n + 4 compound and the arrangement of the atoms in the semi-unit cell of the corre- sponding Tl 2 Ba 2 Ca /1 Cu n + 1 0 2n + 6 XL BUCKMINSTERFULIERENES compound. They are the same except for the replacement of the [Tl - O] layer by [Hg ], and the fact that the thallium compounds are body centered and the Hg ones are aligned. Supercells involving polytypes with ordered stacking sequences of different phases, such as Hg-1212 and Hg-1223, along the c direction have been reported. The stoichiometry is often Hg 2 Ba 4 Ca 3 Cu 5 0, corresponding to equal numbers of the Hg-1212 and Hg-1223 phases (Phillips, 1993; Schilling et al, 1993, 1994). Detailed structural data have already been reported on various Hg family com- pounds such as HgBa 2 Cu0 4 + 6 (Putlin et al, 1993) and the n = 1 compound with partial Eu substitution for Ca (Putlin et al, 1991). The compound Pb 0 7 Hg 0 3 Sr 2 Nd 0 3 Ca 0 7 Cu 3 0 7 has Hg in the position (0.065,0,0), slightly displaced from the origin of the unit cell (Martin et al, 1994). Several researchers have reported synthesis and pretreatment procedures (Adachi et al, 1993; Itoh et al, 1993; Isawa 1994a; Meng, 1993b; Paran- thaman, 1994; Paranthaman et al, 1993). Lead doping for Hg has been used to improve the superconducting properties (Iqbal et al, 1994; Isawa et al, 1993; Martin et al, 1994). XI. BUCKMINSTERFULLERENES The compound C 60 , called buckmin- sterfullerene, or fullerene for short, con- 201 sists of 60 carbon atoms at the vertices of the dotriacontohedron (32-sided figure) that is sketched in Fig. 3.35 and discussed in Chapter 3, Section XVI. The term fullerene is used here for a wider class of compounds C rt with n carbon atoms, each of whose carbon atoms is bonded to three other carbons to form a closed surface, with the system conjugated such that for every resonant structure each carbon has two single bonds and one double bond. The smallest possible compound of this type is tetrahedral C 4 , which has the three resonant structures shown in Fig. 7.31. Cu- bic C 8 is a fullerene, and we show in Problem 17 that it has nine resonant struc- tures. Icosahedral C 12 is also a fullerene, but octahedral C 6 and dodecahedral C 20 are not because their carbons are bonded to more than three neighbors. These hypo- thetical smaller C„ compounds have never been synthesized, but the larger ones, such as Cw, C 70 , C 76 , C 78 , and C 82 , have been made and characterized. Some of them have several forms, with different arrange- ments of polygons. Clusters of buckmin- sterfullerenes, such as icosahedral (C 60 ) 13 , have also been studied (T. P. Martin et al, 1993). There are several interesting geomet- rical characteristics of fullerenes (Chung and Sternberg, 1993). Since each carbon (vertex) joins three bonds (edges) and each edge has two vertices, the number of edges £ in a structure C n is 50% greater than the number of vertices V. There is a gen- eral theorem in topology, called Euler's Theorem, that the number of faces F of a (a) (b) (c) Figure 731 The three resonant structures of the (hypothetical) tetrahedral compound C 4 . 1 202 polyhedron is given by the formula F = E-V+2. (7.8) In a fullerene C„ where n = V three edges meet at each vertex, so we have £ = 3K/2, 7 PEROVSKITE AND CU PRATE CRYSTALLOGRAPHY STRUCTURES V It is shown in Problem 16 that s (7.9) (7.10) (7.11a) (7.11b) where F s is the number of faces with s sides, and of course, F= £ Fj . (7.12) S Combining Eqs. (7.10)-(7.12) gives the fullerene face formula £(6-5)^ = 12. (7.13) This expression does not place any restric- tions on the number of hexagons (F s ), but it does severely limit the number of other polyhedra. The two smallest hypothetical fullerenes, the tetrahedron and the cube, have no hexagons, and the larger ones consist of 12 pentagons (F s ), from Eq. (7.13), and numerous hexagons. For exam- ple, the molecule with V=60 has 12 pentagons and 20 hexagons. Table 7.7 gives the geometric characteristics of the five Platonic solids, the solids generated by truncating all of their vertices, and several other regular polygons, most of which are fullerenes. The fullerenes of current inter- est are and larger molecules consisting of 12 pentagons and numerous hexagons, such as C 70 , C 76 , C 78 , and C 82 . Some have several varieties, such as the isomers of C 78 with the symmetries C 2u , D 3 , and D 3h (Diederich and Whetten, 1992). The outer diameter of the C 60 molecule is 7.10 A and its van der Waals separation is 2.9 A, so that the nearest- neighbor distance (effective diameter) in a Characteristics of Several Regular Solids" Figure Tetrahedron Octahedron* Cube Icosahedron 6 Dodecahedron (pentagonal) Hexadecahedron Truncated tetrahedron Truncated octahedron Truncated cube Dotriacontohedron (truncated icosahedron) Truncated dodecahedron Heptatriacontohedron Tetracontahedron Hentetracontohedron Dotetracontohedron Large Fullerene Vertices Edges Faces 4 6 4 6 12 8 8 12 6 12 30 20 20 30 12 28 42 16 12 18 8 24 36 14 24 36 14 60 90 32 60 90 32 70 105 37 76 114 40 78 116 41 84 126 44 n in \n + 2 Face (polygon) type all equilateral triangles all equilateral triangles all squares alt equilateral triangles all regular pentagons 12 pentagons, 4 hexagons 4 equilateral triangles, 4 hexagons 6 squares, 8 hexagons 8 equilateral triangles, 6 octagons 12 regular pentagons, 20 hexagons 20 equilateral triangles, 12 decagons 12 pentagons (2 regular), 25 hexagons 12 pentagons, 28 hexagons 12 pentagons, 29 hexagons 12 pentagons, 32 hexagons 12 pentagons, \n - 10 hexagons "The first five solids are the Platonic solids, and the seventh to eleventh are truncations of the Platonic Sh^E ca rbons occupy the vertices all correspond to fullerenes except the octahedron and the S3J^ which 3K* 2E. The smallest compounds in this table have never been synthesized. h Not a fullerene because the vertices have more than three edges. XI S< fi a: a< 1c st w ta 1! a! in la h< 01 (C XI th pi te th be oi T< al sit CO XIK SYMMETRIES solid is 10.0 A. The bonds shared by a five-membered and a six-membered ring are 1.45 A long, while those between two adjacent six-membered rings are 1.40 A long. Above 260 K these molecules form a face centered cubic lattice with lattice con- stant 14.2 A; below 260 K it is simple cubic with a = 7.10 A (Fischer et al, 1991; Kasa- tani et al, 1993; Troullier and Martins, 1992). When C 60 is doped with alkali met- als to form a superconductor it crystallizes into a face centered cubic lattice with larger octahedral and smaller tetrahedral holes for the alkalis. The ions are orientationally disordered in the lattice (Gupta and Gupta, 1993). XII. SYMMETRIES Earlier in this chapter we mentioned the significance of the horizontal reflection plane a h characteristic of the high- temperature superconductors, and noted that most of these superconductors are body centered. In this section we will point out additional symmetries that are present. Table VI-14 of our earlier work (Poole et a/., 1988) lists the point symmetries at the sites of the atoms in a number of these compounds. 203 In the notation of group theory the tetragonal structure belongs to the point group 4/mmm (this is the newer interna- tional notation for what in the older Schonflies notation was written D 4h ). The unit cell possesses the inversion operation at the center, so when there is an atom at position (x,y,x\ there will be another identical atom at position (-*,-)>, -z). The international symbol 4/mmm indi- cates the presence of a fourfold axis of symmetry C 4 and three mutually perpen- dicular mirror planes m. The Schonflies notation D Ah also specifies the fourfold axis, h signifying a horizontal mirror plane a h and D indicating a dihedral group with vertical mirror planes. We see from Fig. 732 that the z-axis is a fourfold (90°) symmetry axis called C 4 , and that perpendicular to it are twofold (180°) symmetry axes along the x and y directions, called C 2 , and also along the diagonal directions (Q) in the midplane. There are two vertical mirror planes cr tJ , two diagonal mirror planes a d which are also vertical, and a horizontal mirror plane <j h . Additional symmetry operations that are not shown are a around the z axis, 180° rotation C\ C| ~~ C4C4 , (7.14) \ v Figure 7 32 Symmetry operations of the tetragonal unit cell showing a fourfold rotation axis C 4 , three twofold axes C 2 , and reflection planes of the vertical a t = a v , horizontal a xy = <r„; and diagonal a d types. 204 (a) (b) Z271 Figure 7.33 Rotational symmetry operations of an orthorhombic unit cell (a) with rectangular distortion, and (b) with rhombal distortion from an originally tetragonal cell. and the improper fourfold rotation S z around z that corresponds to C\ followed by, or preceded by, a h > S z 4 = C z * h = a h Cl (7.15) where C\ and a h commute. The orthorhombic structure has mmm, D 2h symmetry. We see from Fig. 7.33 that both the rectangular and rhombal unit cells, which correspond to Figs. 7.4a and 7.4b, respectively, have three mutually per- pendicular twofold axes, and that they also have three mutually perpendicular mirror planes a, which are not shown. The two cases differ in having their horizontal axes and vertical planes oriented at 45° to each other. Cubic structures, being much higher in symmetry, have additional symmetry oper- ations, such as fourfold axes Q, C$, and d along each coordinate direction, three- fold axes C 3 alortg each body diagonal, and numerous other mirror planes. These can be easily seen from an examination of Fig. 7.1. Buckyballs belong to the icosohedral group, which has twofold (C 2 ), fivefold (C 5 ), and sixfold (C 6 ) rotation axes, hori- zontal reflection planes, inversion symme- try, and sixfold (S 6 ) and tenfold (5 10 ) improper rotations, for a total of 120 indi- vidual symmetry operations in all (Cotton, 1963). 7 PEROVSKITE AND CUPRATE CRYSTALLOGRAPHIC STRUCTURES XIII. CRYSTAL CHEMISTRY In Chapter 3 we briefly described the structures of some classical superconduc- tors, and in this chapter we provided a more detailed discussion of the structures of the cuprate superconductors. The ques- tion arises of how structure is related to the presence of metallic and superconduct- ing properties. Villars and Phillips (1988; Phillips, 1989a) proposed to explain the combina- tions of elements in compounds that are favorable for superconductivity at rela- tively high temperatures by assigning three metallic coordinates to each atom, namely an electron number N e , a size r, and an electronegativity X. The electron numbers are given in Table 3.1 for most of the elements, with N e = 3 for all of the rare earths and actinides; several correlations of N e with T c have already been given in Chapter 3. The sizes and electronegativi- ties were determined empirically from a study of some 3,000 binary intermetallic compounds of types AB, AB 2 , AB 3 , and A 2 By The resulting values for each atom are listed in Fig. 7.34 together with their electron numbers. These values, although arrived at empirically on the basis of the constraint of self-consistency, do have a spectroscopic basis, and thus are called, respectively, spectroscopic radii and spec- troscopic electronegativities. The metallic coordinates of the atoms can be employed to calculate the three Villars-Phillips (VP) coordinates for each compound, namely (a) average number of valence electrons N, = <N e > av , (b) spectro- scopic electronegativity difference A AT, and (c) spectroscopic radius difference A/?, where we are using the VP notation. For example, for the compound NbN, with T c = 17.3 K, we have, using the data from Fig. 7.34, N v = i(4 + 5) = 4.5, AR = 2.76 - 0.54 = 2.22, (7 ' 16) A*= 2.03 -2.85 = -0.82. XIII. CRYSTAL CHEMISTRY H i 2.10* 1.25* Lit 0.90 1.61 Nal 0.89 2.65 K 1 0.80 3.69 Rbl 0 80 4.10 B*2 1.45 1.06 M g 2 1.31 2.03 C.2 1.17 3.00 C$1 0.77 4.31 Sr2 1.13 3.21 B*2 1.06 3.402 Frl |fU2 0.70* J0.90* 4.37* I 3.53* Sc3 1.50 2.75 Y3 1.41 2.94 U3 1.35 3.08 Ac 3 1.10* xir Ti4 1.86 2.58 Zr4 1.70 2.825 Hf4 1.73 2.91 V5 2.22 2.43 Nb5 2.03 2 76 T»5 1.94 2.79 Cr6 2.00 2.44 Mo 6 1.94 2.72 W6 1.79 2.735 Mn7 2.04 2.22 Tc7 1IH 2.65 Re 7 2.06 2.6M 1.67 2.11 1.97 2.605 I. HA 2.65 Co 9 1.72 2.02 Rh9 I 99 Ir9 1.87 V628 Nt 10 1.76 2.16 PdtO 2.08 2.45 I*i 10 1.91 2.70 Cu II 1.08 2.04 A* II 1.07 2.375 Au 1 1 1 19 2.66 Zn 12 1.44 1.88 Cd 12 1.40 2.215 H«12 1.49 2.41 83 1 90 0 795 AI3 164 1.675 G«3 1.70 1.695 In 3 1 6.1 1.0$ T13 1.69 2.235 C4 2.37 064 S.4 1.98 1.42 04 1.99 I 56 Sn4 1.88 1.88 Pb4 1.92 2.09 S5 2.85 0.54 P5 2.32 1.24 A* 5 2.27 1 415 So 5 2.14 1.765 B. o 2.14 1.997 06 332 0.465 S6 2.65 1.10 S*6 2.54 1.285 Te6 2.38 1.67 Po6 2.40 1 90 F7 X78 0.406 CI 7 2.98 1.01 Be 7 2.63 1.20 I 7 2.76 1.585 At 7 2.64 1.83 Ctr3 Pr3 Nd3 I*m3 Sm.t Ku .1 (Id 3 Th3 o>. 3 Ho 3 Er3 Tm3 Yb3 Lu3 11' 1. 1' 12* 1.15- 1.2* 1.1V 1.1- 1.2* 1.15' 1.2* 1.2* 1 1.2* 1.1* 1.2* 4.50* 4.48* 3.99* 3.99* 4.14* 3.94* 3.91* :i.H9* 3.67* 3.65* 3.63* |3.60* 359* 3.37* Tha Pa 3 U.I Np3 Pu3 Am 3 i.r 1.5* 1.7* I..T 1.3' 1.3- 496* 4.96* 4.72* 4.ai* 4.91* 4.H9* Figure 734 Periodic table listing metallic valences (upper right), sizes (center), and electronegativities (bottom) in the box of each element, according to the Villars-Phillips model (Phillips, 1989a, p. 321). 205 The VP coordinates for the ^415 com- pound Ge 3 Nb with T c = 23.2 K are calcu- lated as follows: AT V = |(4 + 3x5) = 4.75, A/? = |(1.56 - 2.76) = -0.60, (7.17) AX = |(1.99 -2.03)= -0.02. The text by Phillips (1989a) tabulates the VP coordinates for more than 60 super- conductors with T c > 10 K and for about 600 additional superconductors with tran- sition temperatures in the range 1 < T c < 10 K. When the points for the 600 com- pounds with lower transition temperatures are plotted on a three-dimensional coordi- nate system with axes N v , AX, and AR, they scatter over a large range of values, but when the points for compounds with T c > 10 K are plotted, they are found to cluster in three regions, called islands, as shown in Fig. 7.35. Island A contains the A15 compounds plus some complex inter- metallics, island B consists mainly of the NbN family plus some borides and car- bides, and island C has closely clustered Chevrel phases, with the high-jT c cuprates on the left. When ternary ferroelectric ox- ides with Curie temperatures that exceed 500°C are plotted in the same diagram as the superconductors they cluster between the Chevrel group and the cuprates. These ferroelectric oxides are not superconduc- tors, though Phillips (1989a) suggested that doping them with Cu and alkaline earths could produce superconductors with high transition temperatures. Thus we see that the high transition temperatures of classical superconductors are favored by particular structures and by particular combinations of metallic coordi- nates for each of these structures. The Villars-Phillips approach provides both structural and atomic criteria for the pres- ence of high T c . We have discussed the Phillips ap- proach to a crystal chemistry explanation of the superconductivity of the cuprates. Other researchers have offered alternate, in some cases somewhat related, ap- proaches to understanding the commonali- 206 7 PEROVSK1TE AND CU PRATE CRYSTALLOCRAPHIC STRUCTURES 6 STRUCTURE SYMBOLS * cFB Had . oP16 BzURu • cI40 C3PU2 . hP2 CW ♦ hRt5 MoePbSe » cP5 CoOjTi x tl* (U,Sf>2Co04 t cF56 AlzM^O^ -tl" (BuPbfeSrzCoCuzOs . oP14 B02CU3O7Y o cPB Cr 3 Si « cI58 Mn v tlt2 Ai 2 Cu * tPSOOfe o cP4 AuCuj • tP18 B4CCC04 F/eure 7 35 Regions in the Villars-Phillips configuration space where superconduc- tivity occurs at relatively high temperatures (Phillips, 1989a, p. 324; Villars and Phillips, 1988). ties of the various high-temperature and classical superconductors (Adrian, 1992; Schneider, 1992; Tajima and Kitazawa, 1990; Whangbo and Torardi, 1991; Tor- race, 1992; Yakhmi and Iyer, 1992; Zhang and Sato, 1993). XIV. COMPARISON WITH CLASSICAL SUPERCONDUCTOR STRUCTURES Many elements such as copper and lead are face centered cubic, while many other elements, such as niobium, are body o centered cubic, with a = 3.30 A for Nb. The A15 compounds, such as Nb 3 Se, are (simple) cubic with lattice constant a ~ 3.63a/2 and have parallel chains of Nb atoms 5.14 A apart. Other types of classi- cal superconductors, such as the Laves and Chevrel phases, are cubic or close to cubic. The new oxide superconductors are tetrag- onal or orthorhombic close to tetragonal, and they all have a « b « 3.85 A, which is somewhat greater than the value for the A15 compounds. The third lattice constant c varies with the compound, with the values 13.2 A for LaSrCuO, o 11.7 A for YBaCuO, and = 23 to 36 A for the BiSrCaCuO and TIBaCaCuO compounds. These differences occur because the num- ber of copper-oxygen and other planes per unit cell, as well as the spacings between them, vary from compound to compound due to the diverse arrangements of atoms between the layers. Thus relatively high- symmetry crystal structures are character- istic of many superconductors. XV. CONCLUSIONS Almost all the high-temperature oxide superconductors have point symmetry D Ah (a = b) or symmetry close to D 4h (a ~ b). These superconductors consist of horizon- tal layers, each of which contains one posi- tive ion and either zero, one, or two oxy- gens. The copper ions may be coordinated square planar, pyramidal, or octahedral, with some additional distortion. Copper oxide layers are never adjacent to each other, and equivalent layers are never ad- jacent. The cations alternate sites verti- cally, as do the oxygens. The copper oxide layers are either flat or slightly puckered, in contrast to the other metal oxide layers, which are generally far from planar. The highest T c compounds have metal layers PROBLEMS (e.g., Ca) with no oxygens between the copper oxide planes. FURTHER READING The Wyckoff series, Crystal Structures (1963, Vol. 1; 1964, Vol. 2; 1965, Vol. 3; 1968, Vol. 4) provides a comprehensive tabulation of crystal structures, but many important classical superconductors such as the A15 compounds are not included. The International Tables for X-Ray Crystallography (Henry and Lons- dale, 1965, Vol. 1) provide the atom positions and symmetries for all of the crystallographic space groups. The Strukturbericht notation, e.g., A\5 for Nb 3 Ge, is explained in Pearson's compilation (1958). Details of cuprate crystallographic structures are given by Beyers and Shaw (1989; YBa 2 Cu 3 0 7 ), Burns and Glazer (1990), Hazen (1990), Poole et al. (1988, Chapter 6), Santoro (1990), and Yvon and Francois (1989). Phillips (1989a) provides an extensive discus- sion of the crystal chemistry of the cuprates. Our earlier work (Poole et al., 1988, p. 107) lists the site symmetries in perovskite and cuprate structures. Billinge et al (1994) reviewed lattice effects in high temperature superconductors, and Zhu (1994) re- viewed structural defects in YBa 2 Cu 3 0 7 _ 5 . The microstructure of high temperature super- conductors studied by electron microscopy are re- viewed by Chen (1990), Gai and Thomas (1992), Gross and Koelle (1994), and Shekhtman (1993). Oxygen stoichiometry in HTSCs is reviewed by Chan- drashekhar et al, (1994), Green and Bagley (1990) and by Routbert and Rothman (1995). Electron-doped superconductors are reviewed by Almasan and Maple (1991) and by Fontcuberta and Fabrega (1995). The March 1992 special issue of Accounts of Chemical Research (Vol. 25, No. 3) is devoted to reviews of buckminsterfullerenes. Two recent books are edited by Billups and Ciofolini (1993) and by Kroto and Walton (1993), and the review by Dressel- haus et al. (1994) are devoted to fullerenes. The thallium compounds were reviewed by Hermann and Yakhimi (1993) and the mercury superconductors by Chu (1995). PROBLEMS 1. Show that the radius of the octahedral hole in an fee close-packed lattice of atoms of radius r 0 is equal to [yfl - l]r 0 . What is the radius of the hole if the lattice is formed from oxygen ions? 2. Show that the radius of the tetrahedral hole in an fee close-packed lattice of atoms of radius r 0 is equal to [(3/2) I/2 - l)r 0 . What is the radius of the hole if the lattice is formed from oxygen ions? 3. The "image perovskite" unit cell is generated from the unit cell of Fig. 7.1 by shifting the origin from the point (0,0,0) to the point (i,~,^). Sketch this "image" cell. Show that the planes of atoms in this cell are the image planes related by the body centering operation to those of the original per- ovskite. This image cell is the one that usually appears to represent perovskite in solid-state physics texts. 4. Calculate the distance between the yt- trium atom and its nearest-neighbor Ba, Cu, and O atoms in the supercon- ductor YBa 2 Cu 3 0 7 . 5. Write down the x,y,z coordinates for the five numbered atoms in the initial plane of Fig. 7.16. Give the explicit symmetry operations, with the proper choice of sign in Eq. (7.5) for each case, that transform these five atoms to their indicated new positions on the other three planes. 6. Explain how the international and Schonflies symbols, mmm and D 2h re- spectively, are appropriate for desig- nating the point group for the or- thorhombic superconductors. 7. What are the symmetry operations of the ,415 unit cell of Fig. 3.19? 8. The D 2h point group consists of eight symmetry operations that leave an or- thorhombic cell unchanged, namely an identity operation E that produces no change, three twofold rotations C\ along i=x 9 y y z, three mirror reflec- tion planes a ijy and an inversion /. Examples of these symmetry opera- tions are E x ->jc y ^y z ->z X ~> JC y •y Z -> —2 X -*Jt y z -> -z i x -» — x y y z -> — Z A group has the property that succes- sive application of two symmetry oper- 208 7 PEROVSKITE AND CUPRATE CRYSTAL LOG RAPH1C STRUCTURES ations produces a third. Thus, we have, for example, CWxy = a zx ^2^2 = ^2 iC{ = <r zx Vzx a yz = C 2' These results have been entered into the following multiplication table for the D lh group. Fill in the remainder of the table. Hint: each element of a group appears in each row and each column of the multiplication table once and only once. E C x 2 C{ Cf / a xy a yz a zx E C\ &ZX C{ u 2 i °~zx °yi °zx c 2 9. Construct the multiplication table for the D Ah point group which contains the 16 symmetry elements that leave a tetragonal unit cell unchanged. Which pairs of symmetry elements A and B do not commute, i.e., such that AB ^ BAl Hint: follow the procedures used in Problem 8. 10. Draw diagrams analogous to those in Fig. 7.25 for the first two mem- bers of the aligned series TlBa 2 Ca„Cu n + 1 0 5 + 2n> where n =0, 1. 11. Draw the analogue of Fig. 7.19 for the Nd 2 Cu0 4 compound, showing the lo- cation of all of the Cu and O atoms. How do Figs. 7.21 and 7.22 differ for Nd 2 Cu0 4 ? 12. Calculate the Villars-Phillips coordi- nates for the three superconductors MoP 3 , V 3 Sn, and NbTi. 13. Select one of the compounds (Tl 2 Ba 2 Cu0 6 , Bi 2 Sr 2 CaCu 2 0 8 , Bi 2 Sr 2 Ca 2 Cu 3 O 10 , n 2 Ba 2 Ca 2 Cu 3 0 6 ) and construct a table for it patterned after Tables 7.5 or 7.6. 14. 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