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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 

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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 

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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 



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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 ( 


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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. Locate a twofold (C 2 ), fivefold (C 5 ), 
and sixfold (C 6 ) rotation axis, and also 
a reflection plane a h in the buckyball 
sketch of Fig. 3.35. How many of each 
type of operation are there? 

15. We can see by examining Fig. 3.35 that 
a buckyball has inversion symmetry. 
Identify a sixfold (5 6 ) and tenfold (5 10 ) 
improper rotation axis, where an im- 
proper rotation is understood to in- 
volve a sequential inversion and a 
proper rotation. How many 5 6 and how 
many 5 10 axes are there? 

16. Show that the total number of edges E 
in a fullerene is given by 

s 

and the number of vortices is 

s 

where F s is the number of faces with s 
sides. 

17. Show that the cubic fullerene com- 
pound C 8 has nine resonant struc- 
tures. 



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