Poole Ch.P., Jr. Handbook of Superconductivity

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286

Fig. 8.8.

Chapter 8:

Crystal Structures of High-Tc Superconducting Cuprates

O-'b' O

o,9

O ..... e

p4mm p2mm

9 9

"0,

9

p4mm c2mm

Rectangular and rhombal distortions of a primitive square cell (plane group

p4mm)

leading to a

primitive rectangular

(p2mm)

and a 2-fold centered rectangular

(c2mm)

cell, respectively.

tions of the 4-fold rotation axes and the mirror planes perpendicular to these (site

symmetry

4/mmm).

The tetragonal structure of BaTiO3 presents similar distor-

tions, the space group being, however,

P4mm,

because of additional deformations

leading to puckered layers.

When the atom layers described in the square planar group

p4mm

are added

to the 0101 structure, the symmetry remains tetragonal if the 4-fold rotation

points of the new layers coincide with the 4-fold rotation axes of the initial

structure. This is the case for an ideal basic structure, even if the cation sites in

neighboring layers are never stacked directly on top of each other, but are shifted

by a vector 89 i in the plane, that is, from one 4-fold rotation axis to another. The

space group of the resulting, undistorted structure will preserve the 4-fold rotation

axes parallel to the stacking direction, as well as the mirror planes perpendicular

to the cell axes [0 1 0] and [1 0 0], and to the diagonals of these.

As explained previously, the translation period of the resulting structure

must contain an even number of shifts by I 1, and therefore an even number of

cation-containing layers (see Section C,c). If the sum of the numbers in the four-

digit code (N = k + 1 + m + n) is even, the unit cell is primitive and the

translation period contains one stacking unit. If, on the other hand, it is odd,

the unit cell is body-centered and contains two stacking units. Thus, depending

on the number of layers in the stacking unit, the tetragonal Bravais lattice for an

ideal basic structure

AkBlCmDnOk+l+2m+2 ,

containing no AO' layers, will be

either primitive (P) or body-centered (I).

Each

AkBtOk+ t

and

CmOnO2m+2

slab has, in addition to the 4-fold rotation

axes and the mirror planes containing these, a symmetry plane (mirror plane m or

E. Symmetry 287

diagonal glide plane n), situated halfway between its external layers. When the

number of layers in the slab, (k + I) or (m + n), is even, the symmetry plane is

located between two cation-containing layers, that is, between two A O or two BO

(k = 0) layers for a AkBlOk+ l slab, and between two C layers, coinciding with an

inner O2 layer, for

a CmOnO2m+2

slab. Because of the shift by ) 1 of the cation

sites in consecutive layers, the plane is in this case a diagonal glide plane n. In

contrast, when the number of layers in the slab is odd, the symmetry plane

coincides with an atom layer. This may be an inner AO or BO (k = 0) layer for

the

AkBtOk+ t

slab and a C (n even) or an inner DO 2 (n odd) layer for the

CmDnOzm+2

slab. The plane is here a simple mirror plane m. Combining the two

kinds of slab, three cases may occur:

Both (k + I) and (m + n) are even: The combination of the two glide planes

n results in space group

P4/nmm

with one stacking unit in the translation

period. The 0222 structure, shown in Fig. 8.9 is an example of this

category.

Both (k + 1) and (m + n) are odd: The two mirror planes m give space

group

P4/mmm

with one stacking unit in the translation period. This case

is illustrated in Fig. 8.9 by the 1223 structure.

(k + I) and (m + n) are of different parity: One mirror plane m and one

diagonal glide plane n lead to space group

I4/mmm

with two stacking units

in the translation period. The 2201 structure is representative for this

category.

These cases are summarized in Table 8.1, which has been subdivided to consider

the different parity of the individual indices. It results from the stacking rules

described earlier that k and I cannot be odd simultaneously (if I = 1, k = 0). The

same is true for m and n, since m =

p(n-1).

The ideal basic structures can thus be described in one of the following

tetragonal space groups:

P4/mmm

(or cubic

Pm~3m

in the particular case of

idealized perovskite),

P4/nmm,

I4/mmm.

Indeed, the crystal structures of the

majority of the high-T c superconducting cuprates known up to now have been

refined in one of these groups. However, space groups of lower symmetry have

been used to describe and refine some crystal structures with significant

distortions. The group-maximal nonisomorphic subgroup relationships that

lead from one of the space groups listed previously to space groups used in

data sets given in Section H,b are indicated in Table 8.2. The lowering of

symmetry observed for the real structures may be caused by any of the following:

Displacements of the atoms from the ideal positions

An ordered arrangement of different cations within layers of the same kind

An ordered arrangement of vacancies

The insertion of extra atoms

Fig. 8.9.

Examples of cuprate structures with different symmetries. The symmetry planes (m, mirror; n, diagonal glide; b, axial glide plane) perpendicular to the stacking

direction ([0 0 1]) are indicated; the Pearson code and the space group are given.

Symmetry

289

Table 8.1.

Space groups for undistorted basic structures ofhigh-T c superconducting cuprates as derived from the

parity of the numbers k, l, m, and n in the four-digit code.

(k + !) even (k + 1) odd

k odd, 1 odd k even, I even k even, 1 odd k odd, 1 even

impossible (l = 2) (k --- 0, 1 -- 1) (1 = 2)

(m + n)

even

m odd, n odd

impossible

m even, n even

(p even)

X X X X

• P4/nmm 14/mmm 14/mmm

(m + n) m even, n odd • I4/mmm P4/mmm P4/mmm

odd m odd, n even • I4/mmm P4/mmm P4/mmm

(p odd)

Table 8.2.

Group-maximal nonisomorphic subgroup relationships a for basic structures of superconducting

cuprates. Standard settings according to the International Tables for Crystallography (Hahn, 1983)

are given within parentheses. Space group symbols within square brackets are not represented among

the data sets reported here but complete the transformation paths.

Space group of

ideal structure

Path to space group of

distorted basic structure

[Pm3m] I --+ P4/mmm

I ~ [R3m]

P4/mmm I --+ Pmmm

I ~ Cmmm

P4/nmm

I4/mmm

IIb --+ P4/mbm

I --+ Cmma

I --+ I4mm

I --~ Fmmm

IIa -+ [P42/mmc ]

IIb ~ R3c

IIa --+ Pman (Pmna)

IIb ~ Imam (Imma)

IIa --~ Bmab (Cmca)

IIa --~ Amaa (Cccm)

IIa --+ [Bbmb (Cccm)]

lib ~ P42/ncm

I --~ I2cm (Ima2)

I --+ A2aa (Ccc2)

IIa --+ Pnan (Pnna)

a I, translationengleiche (t) subgroup (all translations are retained but certain point symmetry

operations are lost); II, klassengleiche (k) subgroup (IIa, the unit cell is retained but the space

group is different; lib, the unit cell is larger owing to the loss of some of the integral translations and

the space group is different).

290

Chapter 8: Crystal Structures of High-Tr Superconducting Cuprates

These lead, in a majority of cases, to structures that are described in orthorhombic

space groups with cell vectors a + b, -a + b and c (d ~ b' ~ 5.4 A), where a, b,

and c are the cell vectors of the ideal structures. Note that the orthorhombic

structure of BaTiO3 (space group

Cm2m)

presents distortions resulting in a

similar cell. Owing to positional or occupational commensurate modulations,

superstructures with large unit cells are sometimes observed. Incommensurately

modulated structures are also known.

As stated previously, the symmetry considerations up to now do not apply

to structures containing AO' layers. The oxygen atoms in these layers center the

square edges only in one direction, breaking the 4-fold symmetry. Among the

basic structures of superconducting cuprates, there are two well-known examples,

Ba2YCu307 (1212, CuBa2YCu2OT) and Ba2YCu408 (2212, Cu2Ba2YCu208),

where the undistorted structures are orthorhombic. Both structures contain

additional

CuO layers, the copper atoms of which center mutually parallel

squares of oxygen atoms perpendicular to the layer. The replacement of a

single AO layer by a AO' layer, as in Ba2YCu3OT, does not alter the general

stacking scheme, but the 4-fold rotation axis is reduced to a 2-fold one. The

resulting orthorhombic structure, shown in Fig. 8.9, is described in space group

Pmmm,

which is a direct subgroup of

P4/mmm ((k + I)

odd, (m + n) odd). The

cell parameter b is slightly larger than a, because of the presence of oxygen atoms

along [0 1 0] in the AO' layer. The structure can be derived from the structure of

Ba2YCu306.26 with the same four-digit code (1212), space group

P4/mmm,

where one-third of the copper atoms form A layers, considering an ordered

arrangement of extra oxygen atoms in the

additional

layer.

In the case of Ba2YCu408 (2212), the two consecutive AO' layers are

shifted by 0 I with respect to each other if, by convention, the oxygen atoms in

the AO' layers are located along [0 1 0]. Such a shift corresponds to an axial

glide plane b, situated between the two layers, that is, halfway between the BO

layers delimiting the A2B204 slab. Combined with the mirror plane m coinciding

with the central C layer of the CD204 slab, this leads to space group

Ammm

(standard setting

Cmmm).

There are two stacking units in the translation period

along c and the cell parameter ratio

b/a

> 1 (Fig. 8.9).

c. Space Groups for Limiting Structures

Like the majority of the basic structures, the limiting structures with the general

formula

CpDO2p

are described in tetragonal space groups. When p is odd, the

translation period contains two mirror planes m. The first one coincides with the

DO 2 layer and the other one with the inner C layer. This combination yields

space group

P4/mmm

with one stacking unit in the translation period (0011,

(Sr, La)CuO2). When p is even, the mirror planes through the DO 2 layers alternate

with diagonal glide planes n, which coincide with the 02 layer situated between

E. Symmetry 291

the internal C layers. The resulting space group is I4/mmm with two stacking

units in the translation period (0021,

(Nd, Ce)zCuO3.92 ).

d. Space Groups for intergrowth Structures

As for the basic structures, the 4-fold rotation axes and the mirror planes

containing these are preserved in the intergrowth structures, as long as there

are no AO' layers. When the total number of cation-containing layers in all

stacking units forming an intergrowth structure, that is, the sum of the numbers in

all four-digit codes, is even, the tetragonal Bravais lattice is primitive; otherwise,

it is body-centered. Three cases may be distinguished:

The AkIBI Ok1+l 1 and Ak2Bt2Ok2+l 2 slabs are identical, that is,

k 1 -- k 2

and

I 1 = ! 2

The

CmlDnlOzml+ 2

and

CmzOnzOzm2+ 2

slabs are identical, that is, m I = m 2

and

n 1 = n 2

The corresponding slabs in the two stacking units are different

In cases (1) and (2), only the symmetry planes in the corresponding slabs

that are different will remain. As for the basic structures, the space group is either

P4/mmm, P4/nmm, or I4/mmm, depending in case (1) on the parity of (m I + nl)

and (m 2 -+-n2) and in case (2) on the parity of (k 1 + 11) and (k 2 +/2). This is

summarized in Table 8.3 and illustrated for case (2) by the idealized structure of

CBizSr4Cu201] (1201/2201) in Fig. 8.10.

Table 8.3.

Space groups for undistorted intergrowth structures of high-T c

superconducting cuprates with identical

AkB110kl+tl

and

Ak2Bt2Ok2+t 2

(or

CmDn O2m~+ 2

and

Cm2Dn202m2+2)

slabs as

derived from the parity of (m 1 4- n2) and (m 2 + n2) [or

(k I 4- il) and (k 2 4-/2)].

k 1 = k2, !1 = ! 2

[m l=m 2,nl=nz]

(m I + nl) even (m 1 4- nl) odd

[(k I 4- ll) even] [(kl + I1) odd]

(m 2 + n2) even

P4/nmm I4/mmm

[(k 2 4-12) even]

(m z + nz) odd

[(k 2 + 12) odd]

I4/mmm P4/mmm

292

Chapter 8: Crystal Structures of High-T~ Superconducting Cuprates

Fig. 8.10.

The 1201/2201 structure, an example of an intergrowth of two basic structures. The interfaces

and symmetry planes (m, mirror; n, diagonal glide plane) perpendicular to the stacking

direction ([0 0 1]) are indicated; the Pearson code and the space group are given. The real

structures of the examples chosen here have lower symmetry.

In case (3) all symmetry planes perpendicular to the 4-fold rotation axes

present in the basic structures are lost and the space group is either P4mm or

I4mm, depending on the sum of the numbers in the two four-digit codes.

Note that if the site occupations in the two slabs with identical indices are

different, the symmetry plane perpendicular to the stacking direction is lost. As

stated previously, structures where the

AkBllOkl+l ~

and

Ak2Bl2

Ok2+/2

slabs and/or

the

CmlOnlO2m1+ 2

and

CmzDn202m2+ 2

slabs differ only in the chemical composi-

tion are here considered as superstructures resulting from cation ordering (e.g.,

1201, (C,T1,Pb)Sr2CuOs) and not as intergrowth structures.

Ideal and Real Cation Coordinations

293

Table 8.4.

Group-maximal nonisomorphic subgroup relationships for inter-

growth structures of superconducting cuprates (for notations, see

Table 8.2).

Space group of

ideal structure

Path to space group of

distorted intergrowth structure

[P4/mmm] I ---+ [Cmmm]

IIa --+

Pbmm (Pmma)

I4/mmm I ~ [Fmmm]

IIa ~

Abmm (Cmma)

The majority of the intergrowth structures known up to now are described

in subgroups of the space groups derived for the ideal structures. Group-maximal

nonisomorphic subgroup relationships are presented in Table 8.4.

Intergrowth of Ba2YCu307 (1212) and BazYCu408 (2212), crystallizing in

the orthorhombic space groups

Pmmm

and

Ammm,

respectively, is known for the

compound Ba4YzCu7014.94 (1212/2212, CuBazYCuzO6.94/Cu2BazYCu208).

The total number of layers in the four slabs is odd, that is, there are two stacking

units from each parent structure in the translation period. Since the basic

structures contain identical

CD204 (YCu204) slabs,

the axial glide planes b,

situated between the two consecutive AO' (CuO) layers in the

A2B204

slab from

2212, and the mirror planes m, coinciding with the AO' layers in the AB203 slab

from 1212, remain from the basic structures and alternate along [0 0 1]. The

space group of the resulting structure is

Ammm

and the cell parameter ratio

b/a

> 1 (oxygen atoms in the AO' layers located along [0 1 0]).

Ideal and Real Cation Coordinations

The ideal coordination of the cation derives from the stacking of the layers, the

cation in each layer being surrounded by oxygen atoms from the same and/or the

directly neighboring layers (see Fig. 8.3). The real coordinations are often

distorted to better accommodate a particular cation. The strongest covalent

bonding is generally observed within the DO 2 (CuO2) layers and within the

coordination polyhedra of the A cations.

a. Coordination Polyhedra

Figure 8.11 shows ideal coordination polyhedra represented in high-T c super-

conducting cuprates. The coordination number of the D (Cu) atoms can be 6, 5,

or 4. Six oxygen neighbors form an octahedron (6o), as in perovskite. This

coordination of the D atoms is observed only in cuprates k/01 with a single

294

Fig. 8.11.

Chapter 8:

Crystal Structures of High-T~ Superconducting Cuprates

A (21) A (31) A (4s) A (5y)

A (60) A (60) A (8aplc) A (12co)

B (8ap) B(8aplc) B (8ap2) B(12co)

C (8p) D (4s) D (5y) D (60)

Commonly observed coordination polyhedra built up of oxygen atoms around the A, B, C, and

D cations: 21, two collinear atoms; 31, triangle; 4s, square; 5y, square pyramid; 6o,

octahedron; 8ap, square antiprism; 8aplc, monocapped square antiprism; 8ap2, square

antiprism with two additional atoms; 12co, cuboctahedron; and 8p, tetragonal prism.

conducting

DO 2 layer in the stacking unit (e.g., 1201, T10.92(Ba,La)2CuO4.86).

Coordination number five, a square pyramid (5y), is present when there are two or

more DO 2 layers (1212, T1Baz(Ca,T1)Cu207). In the latter case the 5-fold

coordinated D atoms are located in the external DO 2 layers, whereas the D

atoms in the internal layers are 4-fold coordinated, the oxygen atoms forming a

square (4s) (1223, (T1,Pb)Sr2(Ca, T1,Pb)2Cu309) (see also Fig. 8.5). In all cases

four oxygen atoms are situated in the DO 2 layer, the apical oxygen atom, when

present, belonging to the neighboring BO layer.

In all structures of superconducting cuprates, only one type of coordination

polyhedron is observed for the C atoms, that is, a tetragonal prism (8p). The

E Ideal and Real Cation Coordinations 295

oxygen atoms forming the tetragonal prism may belong to two neighboring DO 2

layers (0212, (La, Sr, Ca)z(Ca,La)Cu2Os.91), one DO 2 and one O2 layer (0222,

(]N~d, Sr)2(Nd, Ce)zCuzO7.84), or two O 2

layers (0232, Srz(Ce,Y)3(Cu,Fe)2Olo) (see

also Fig. 8.6).

The coordination number of the B atoms in perovskite and cuprates

containing one

bridging BO

layer in the stacking unit is 12, the surrounding

oxygen atoms forming a cuboctahedron (12co) (0112, BaY(Cu,Fe)2Os). Four

oxygen atoms are located in the BO layer, and the other eight in the two

neighboring DO 2 layers. When a second BO layer is added, the coordination

number of the B atoms is reduced to 9, the nine oxygen atoms forming a

monocapped square antiprism (8aplc) (0212). In the real structures the BO layers

are often puckered so that the B atoms are located closer to the center of the

polyhedron. This coordination is also observed when an AO layer with oxygen

atoms centering all squares of the cation square mesh is present (1212,

T1Baz(Ca, T1)Cu207) (see also Fig. 8.4). Depending on the number and the

arrangement of the oxygen atoms in the neighboring

additional

layer, the

coordination number of the B cation varies: 8 (square antiprism, 8ap) for an A

layer (1212,

CuBa2YCu206.26),

10 (two capping atoms on one side, 8ap2) for an

AO' layer (1212, CuBazYCu207), and 12 (cuboctahedron) for an AO 2 layer

(4212, (Ti,Gd, Ca)4Baz(Gd, Ca)Cu2Ol2). The B sites are occupied by Ba, Sr, or

rare-earth elements in ionized states, with little localized bonding and conse-

quently few requirements on the shape of the coordination polyhedron.

Depending on the exact type of

additional

layers present (A, AO, AO', AO',

or AO2) , a wide range of different coordinations are found. In contrast to the B

metal atoms, the elements found on the A sites, generally nonmetals, have

characteristic coordination polyhedra, with a large proportion of covalent bond-

ing. Five types of ideal coordination polyhedra are observed: two collinear atoms

(21), a triangle (31), a square (4s), a square pyramid (5y), and an octahedron (6o).

Linear coordination is typical for Hg or Cu. In this case the

additional

layer

contains no oxygen atoms (type A), and the neighboring layers are either BO

(1201, HgBazCuO4.18) or AO (3212, (Pb,Cu)3SrzYCu208). Trigonal coordination

is characteristic of carbon atoms (1201, C(Ba, Sr)zCuOs.05), the structures

containing planar carbonate units, perpendicular to the AO" layers. Square

coordination is observed for copper atoms in A O' layers, the two neighboring

layers being AO and/or BO (1212, CuBazYCu207). To achieve square pyramidal

coordination, sometimes found for lead atoms (3212, (Pb,Cu)3SrzYCu208), one

of the surrounding layers must contain no oxygen atoms (type A). Bi, Pb, and T1

preferentially form AO layers, which would lead to 6-fold, octahedral coordina-

tion. The real coordination polyhedra are, however, often strongly distorted

because of the presence of lone-electron pairs, and the cations displaced from

the centers of the original octahedra (2212, (Bi,Pb)zSrz(Y, Ca)Cu2Os). The

presence of AO 2 layers will give rise to higher coordination numbers, suit-

able for metal atoms. When the structure contains four

additional

layers in

the stacking sequence

-AOz-AO-AO-AO2- ,

for instance, the cations of the