Poole Ch.P., Jr. Handbook of Superconductivity

Подождите немного. Документ загружается.

116 Chapter 6: Crystal Structures of Classical Superconductors

located. For alloys, optimal space filling is generally the dominating criterion and

the

coordination number,

that is, the number of nearest neighbors, increases with

the atomic radius. Small atoms in large interstices are often delocalized over

several off-centered positions.

Crystal structures rapidly become complex, and a description emphasizing

particular coordination polyhedra helps in visualizing and memorizing a structure

and finding relations between different structures. Examples of coordination

polyhedra that are commonly observed in crystal structures of inorganic

compounds are presented in Fig. 6.1. It may be noted that it is not always

easy to decide how many atoms belong to the first coordination sphere. There is

also a certain amount of subjectivity in the labeling of the polyhedron. For

instance, a distorted polyhedron built up by eight atoms can be considered as a

bicapped trigonal prism, but also as a square antiprism. In the same way, an

octahedron elongated or compressed along one of its 4-fold axes is still called an

octahedron, whereas if the deformation takes place along one of the 3-fold axes, it

is generally referred to as a trigonal antiprism. Straight trigonal prisms are

frequent in intermetallic compounds, such as borides, and silicides. In this case,

the first coordination sphere often includes atoms capping the rectangular faces,

sometimes located at a shorter distance from the central atom than the prism-

forming atoms. For a better visualization of the structure, the capping atoms are,

however, generally omitted in the description. The largest atom coordination

presented here, the 16-vertex Frank-Kasper polyhedron, is conveniently decom-

posed into a truncated tetrahedron (a polyhedron, obtained by cutting the 4

vertices of a tetrahedron, with 12 vertices and 4 hexagonal and 4 triangular faces)

and a tetrahedron formed by atoms capping the 4 hexagonal faces of this. Coor-

dination numbers exceeding 20 are found in some intermetallic compounds

combining atoms of very different sizes.

d. Definitions and Conventions

A crystallographic data set contains information about the cell parameters (the

edge lengths and the angles of the unit cell), the coordinates of the atoms in the

asymmetric unit (a representative triplet x y z for each atom site), and the space

group symbol. The

International Tables for Crystallography (Volume A)

(Hahn,

1983) defines how to choose the unit cell with respect to the symmetry operations

in each of the 230 possible space groups. By consulting the lists of equivalent

triplets in the

International Tables for Crystallography,

the coordinates of any

atom in the structure can be derived from this condensed information. Distinc-

tion is made between atom sites in general position (site symmetry 1) and atom

sites in special position, that is, situated on one or several symmetry elements.

Within each space group the general and the special positions are defined by the

so-called

Wyckoff letters,

which are lowercase letters starting from a for the

highest site symmetry. The Wyckoff letter is generally preceded by the corre-

Introduction 117

Fig. 6.1.

Coordination polyhedra commonly observed in structures of inorganic compounds.

118 Chapter 6: Crystal Structures of Classical Superconductors

sponding site multiplicity, the number of symmetry-related atoms in the unit cell

belonging to this site, to define together the Wyckoffposition. The multiplicity of

a special position is always a fraction of the multiplicity of the general

position. For atoms in special positions, all or part of the coordinates are fixed,

either to particular values (e.g., 0 1 0), or with respect to each other (e.g.,

x 2x z). As an example, the atom site given as Cu(2) in 4(c) with the triplet

0.175 88 0.061 in space group Pnma (No. 62 in the International Tables for

Crystallography) designates four atoms in the unit cell with the following

fractional coordinates: Cu(2)l at 0.175 ~ 0.061, Cu(2)2 at 0.325 3 0.561, Cu(2)3

at 0.825 43- 0.939, and Cu(2)4 at 0.675 88 0.439. The_ atom site given as W in 6(c)

with the triplet 0 0 0.176 in space group R3m (No. 166, hexagonal axes)

defines the positions of six atoms in the unit cell: W1 at 0 0 0.176, W2 at

0 0 0.824, W 3 at~ 89 0.509, W 4 at 2 89 0.157, W 5 at89 2 0.843,

W 6

at 89 2 0.491,

considering the translations characteristic of a rhombohedral lattice.

However, even when the symmetry elements are located in agreement with

the International Tables for Crystallography, there exists more than one way to

present the same crystal structure. In order to facilitate the recognition of isotypic

compounds, a standardization procedure was developed in Geneva (Parth6 and

Gelato, 1984, 1985; Parth6 et aL, 1993). This procedure applies a series of

criteria for the choice of the space group setting (e.g., setting Pnma is preferred to

Pmnb with interchanged axes), the cell parameters (important for monoclinic and

triclinic symmetry), the origin of the cell (considering space-group-permitted

origin shifts), and the representative triplets. All complete crystallographic data

sets given in Section J have been standardized.

One of the more common classification schemes applied to structures of

intermetallic compounds uses the so-called Pearson code. Two letters and a

number compose this code, for example, hP12. The lowercase letter indicates the

crystal system: a, anorthic (triclinic); m, monoclinic; o, orthorhombic; t, tetra-

gonal; h, hexagonal and trigonal; c, cubic. The uppercase letter stands for the

Bravais lattice: P, primitive (a lattice point at 0 0 0); S, side-face centered (for

instance, C-centered--0 0 0, 89 89 0); R, rhombohedral (0 0 023313'311232);

face-

centered (0 0 0, 0 1 89189 0 89 89 89 0); or I, body-centered (0 0 0 and 89 1 89 The

number corresponds to the number of atoms in the unit cell. For structures with

vacancies we have replaced this number by the sum of multiplicities of all partly

and fully occupied atom sites, when full occupation is not impeded by impossibly

short distances.

Since the atoms in isotypic compounds must occupy the same Wyckoff

positions, the so-called Wyckoffsequence, the sequence of Wyckoff letters of all

atom sites in the structure, becomes an important feature for the recognition of

isotypic structures. The two different layered structure types, 2H-MoS 2 and

2H-NbS2, for example, have similar unit cells and are described in the same

space group, P63/mmc. Structures crystallizing with one or the other modifica-

tion may be identified from the Wyckoff sequence, which isfc for the former type

and fb for the latter. It must be emphasized that Wyckoff sequences can only be

A. Introduction 119

compared for standardized data, since it is often possible to "move" an atom from

one Wyckoff position to another by shifting the whole structure by a space-group-

permitted origin shift (e.g., by 0 0 89 in many space groups). The letters in the

Wyckoff sequence are written in inverse alphabetic order. When several atom

sites occupy the same Wyckoff position, the corresponding letter is followed by

the number of times the particular position is present, written as a superscript

(e.g.,

g2db).

e. Strukturbericht Notation for Structure Types

Strukturberichte

is the German-speaking predecessor of the

Structure Reports,

which were published yearly for the

International Union of Crystallography

until

1990. The usefulness of grouping compounds crystallizing with similar atom

arrangements appeared early and a coding system for structure types was already

being used in the first volume of

Strukturberichte,

edited in 1931. The notation

starts with an uppercase letter, giving information about the ideal composition, or

the family of compounds. The letter A was assigned to element types, e.g., A 1 to

Cu, A2 to W, A3 to Mg. The number following the letter is here a simple ordering

number, assigned to chronology of discovery. Historical reasons are behind the

fact that the structure type Cr3Si, well-known among binary superconductors, is

referred to as A 15. The code was indeed originally assigned to the so-called r-

modification of elementary W, which was later shown to be an oxide. The letter B

was chosen for the ideal element ratio 1 : 1, C for the ratio 1:2, D for other binary

compounds, etc., whereas L was reserved for alloy structures. In the second

volume of

Strukturberichte

the notation was slightly modified and subscripts were

introduced to allow for a finer subdivision. A certain number of already-known

structure types changed codes, e.g., the code of cubic perovskite was modified

from G5 to E21, and that of CuaAu from L12 to L12. However, it rapidly became

clear that the number of existing structures and their complexity were far beyond

the limits of such a simple classification system. The coding was not continued

Structure Reports,

but a few authors went on extending the list of codes for

some time, in particular for intermetallic compounds. These more recent codes

may be recognized by subscripts containing lowercase letters. It may be noticed

that, because of the difficulty of recognizing isotypic structures described in

different settings, two or more codes have in some cases been assigned to the

same atom arrangement. After a closer comparison of the crystal structures, the

codes B31 (MnP) and

B d

(NiSi), for instance, turned out to be synonyms of B14,

defined on FeAs. In the literature, different substitution variants are often

grouped under the same notation and, for example, the binary Z phases are

often referred to as A12 (elementary ~-Mn). In a similar way, interstitial

compounds are often referred to by the Strukturbericht notation defined on the

parent type. A list of Strukturbericht notations defined on structure types treated

here is given in Table 6.2.

120 Chapter 6: Crystal Structures of Classical Superconductors

TABLE

6.2

Smd~rbericht notations for structure types.

Stmkturbericht Structure type S~bericht Structure type

A3'

A10

All

A12

A13

A15

B81

B82

B14

B15

B19

B27

B31

B33

Cll a

Cll b

Cu C14 MgZn z

W C15

MgCu z

Mg C16 0-CuAI z

~t-Nd C22 Fe2P (ZrNiAI)

13-Sn C32 AIB 2

C c

ot-ThSi z

a-Hg D02 CoAs 3 (LaFe4P12)

a-Ga D03 BiF 3 (MnCu2AI)

a-Mn (TisRe24) D011 Fe3C

fl-Mn (Mo4AI2C) D019 LT-Mg3Cd

Cr3Si

DO e

Ni3P

a-Pa (In) D13 BaA14 (CeAl2Ga2)

fl-U (CrFe) D21 CaB 6

NaCl D2 3 NaZnl3

CsCI

D2 b

ThMnl2

2H-ZnS

D2 d

CaCu 5 (CeCo3B2)

NiAs

D2f

UB12

Ni2In

D5c

Pu2C3

FeAs D7 3 ThaP 4

FeB a D8 6 Cul5Si4

[3'-AuCd D8 8 MnsSi 3 (HfsSnaCu)

FeB

D8 b

CrFe

MnP b D8 l

CrsB3

~t-TlI E21 CaTiO 3

~-(AgZn) E9 3 W3Fe3C

CrB ~ H 11 MgAI2 O4

WC L1 o

CuAu

p-FeS 2 L 12 Cu3Au

2H-MoS 2 L21 MnCu2AI

CaC 2 L6 0 SrPb 3

MoSi 2 L'3 e-FeNo. 5 (NiAs)

aSuperseded structure proposal; see B27.

blsotypic with FeAs, B 14.

r with a-TII, B33.

Elements

a. Close-Packed Element Structures

At ambient temperature and pressure the metal elements, which represent some

80% of all chemical elements, crystallize with a small number of different

B. Elements 121

structure types. The majority of them adopt structures that can be assimilated to

the packing of solid spheres. Balls on a flat surface will arrange themselves so

that six of these surround each ball. Most metal atoms do likewise, forming so-

called close-packed layers. One such layer is shown in the upper part of Fig.

6.2. To achieve an optimal space filling, identical layers are stacked in a way that

the atoms of one layer are placed over the triangular voids of the other

layer. Since there are twice as many voids as atoms, there are two possibilities

for placing the second layer. If we fix one atom from the first layer at 0 0 z 1

referring to a hexagonal unit cell, we can stack the next layer so that there is an

atom at

1 2 Z

2, or shift the layer slightly along the long diagonal of the cell so that

there is an atom at 2 89 z2" These three possibilities for placing a layer are

generally referred to as A, B, and C. Any stacking sequence of close-packed

layers is in principle possible, but only the simplest ones are found among the

element structures.

By shifting consecutive layers alternatively forwards and backwards along

the long diagonal of the hexagonal cell indicated at the top of Fig. 6.2, the atom

arrangement shown on the lower left, corresponding to the structure type defined

on Mg, is obtained. The overall symmetry is hexagonal and the unit cell contains

two atoms. The structure may be described by placing one atom at 0 0 0 and the

other at 89 2 89 however, in agreement with the

International Tables for Crystal-

lography,

space group

P63/mmc,

the origin of the cell is shifted so that the two

atoms have the coordinates 89 2 88 and 2 89 ~. The stacking mode where a layer is

sandwiched between two layers shifted in the same direction (e.g.,

ABA, CBC)

Fig. 6.2.

Close-packed layer, letters A, B, and C indicate the three possible stacking positions; atoms in

the (1 1 0) cross-section of the close-packed element structures Mg, a-Nd, and Cu (referring

to hexagonal setting).

122 Chapter 6: Crystal Structures of Classical Superconductors

called hexagonal, abbreviated h. The Mg type is often referred to as

hexagonal

close-packed,

h.c.p. Derived structures are sometimes preceded by the specifica-

tion 2H, where the digit indicates the number of layers in the translation unit and

H stands for hexagonal.

If, in contrast, consecutive layers are always shifted in the same direction

along the diagonal of the hexagonal cell, the translation unit in the stacking

direction will correspond to three layers, as sketched at the lower fight of Fig.

6.2. The overall symmetry of this structure type, defined on Cu, is cubic. The

conventional unit cell is face-centered with atoms located at the vertices of the

cube and the centers of the cube faces. Figure 6.3 shows a portion of such an

atom arrangement corresponding to several unit cells. One of the comers of

the large cube has been cut to evidence the close-packed layers perpendicular to

[1 1 1]. Note that, in agreement with the cubic symmetry, similar layers are found

also perpendicular to the other body diagonals i.e., [-1 1 1], [1 -1 1], and

[1 1 -1]. The structure type is also referred to as

face-centered cubic

(f.c.c.),

cubic close-packed

(c.c.p.), or 3C. A close-packed layer sandwiched between

two layers shifted in opposite directions (e.g.,

ABC, BAC)

is considered to be c-

stacked. It may be noted that in the more general case, where the interatomic

distances within and between the layers are not identical, the overall symmetry is

trigonal with an R (rhombohedral) lattice (atoms at 0 0 0, 2 1 1, and 1 2 2 in the

triple hexagonal cell). The overall stacking is then referred to as 3R.

A third variant of stacking of close-packed layers, shown at the lower center

position of Fig. 6.2, is labeled after t~-Nd. This stacking is a combination of the

Fig. 6.3.

Cubic close-packed, i.e., f.c.c., atom arrangement (Cu type). One comer of a large cube is cut

to emphasize the close-packed layers perpendicular to the body diagonal.

B. Elements 123

two modes described previously, since consecutive layers are alternately shifted in

the same and in opposite directions (ABAC). The overall stacking is consequently

called hc. The resulting structure is hexagonal and the translation unit in the

stacking direction corresponds to four layers. The type is sometimes referred to

as d.h.c.p. (double hexagonal close-packed).

Since the stacking is defined by the relative shift along the diagonal of the

hexagonal cell, structures with close-packed layers are conventionally represented

by the atoms in the cross-section (1 1 0). Cross-sections of the three structures

just described are presented in Fig. 6.2. Note that a projection of all atoms in the

structure onto the same plane would contain twice as many circles. Indepen-

dently of the stacking sequence, each atom has, in addition to six atoms in the

close-packed layer, three near neighbors in each of the adjacent layers. In the

case of a c-stacked layer, the twelve atoms form the cuboctahedron sketched in

Fig. 6.1. For an h-stacked layer the top and bottom triangles have the same

orientation and the polyhedron is called an anticuboctahedron, also sketched in

Fig. 6.1.

The highest superconducting transition temperature observed for an element

in close-packed arrangement is reported for technetium, which crystallizes with a

Mg-type structure. The transition temperature was earlier reported as 11.2 K, but

is now generally accepted to be 7.9 K, the actual value depending strongly on the

purity. Isotypic Re and c~-T1 are superconducting at lower temperatures. Rela-

tively high critical temperatures for Cu-type stacking are reached by Pb (7.2 K)

and/%La (6.1 K). Superconductivity is also observed for another modification of

the latter element, c~-La with an c~-Nd-type structure (4.9 K).

b. Other Element Structures

Another very simple structure type, defined in W (also called c~-Fe) is adopted by

all alkaline metal elements and a certain number of transition elements. The

cubic, body-centered unit cell contains two atoms, located at the origin and the

center (0 0 0, 1 89 89 corresponding to Fig. 6.4a with a 1 - a 2 - a 3. Eight atoms

forming a cube surround each atom. The type is generally referred to as b.c.c.,

but it should be noted that the term body-centered cubic may also be applied to

any crystal structure that has a body-centered cubic translation lattice: for

example, La3Rh4Snl3 , the cubic unit cell of which contains 320 atoms. The

same remark is also valid for the term f.c.c., designating the Cu type. Figure 6.4

illustrates the close relationship that exists between a cubic body-centered (left)

and a cubic close-packed (fight) atom arrangement. The projections along one of

the cell edges show identical square-mesh patterns; however, the ratios of the

translation units perpendicular to and in the plane of projection are different. It is

thus possible to go from the b.c.c. W type to the f.c.c. Cu type by extending the

structure by a factor of v/2 along one of the 4-fold axes.

124 Chapter 6: Crystal Structures of Classical Superconductors

Fig. 6.4.

Structures ofNb (W type, b.c.c.) (a) and fl-La (Cu type, f.c.c.) (b) in projections along [0 0 1].

Dark shading: z - 1; light shading: z - 1. For both structures a 1

a 2.

The superconducting transition temperature of the W-type element Nb,

9.3 K, is the highest observed for an element at ambient pressure. Isotypic V and

Ta, with the same number of valence electrons, become superconducting around

5 K. fl-Zr may be stabilized by the presence of, for example, Rh, and then

undergoes a transition to a superconducting state at 6.5 K. A comparison of

critical temperatures reported for different elements early revealed a correlation

between T c and the valence electron concentration. The critical temperatures of

the transition elements and their solid solutions as a function of the number of

valence electrons show well-defined maxima around 4.65 and 6.5 electrons per

atom. It has been pointed out that the same values characterize the borders of the

stability domain of the W-type atom arrangement, in competition with the

hexagonal Mg type on both sides.

The nonmetals tend to complete their electron octet by forming covalent

bonds. As expected, no superconductivity is observed for nonmetals at ambient

pressure. Near the zigzag Zintl line, which draws a rough border between metals

and nonmetals in the periodic table, structures related to the metal element

structures, but with a tendency toward partly localized bonding, are formed.

Mercury occupies a particular position in the history of superconductivity, since

the phenomenon was discovered during low-temperature studies of this element,

for which the superconducting transition takes place at 4.2 K. The structure of o~-

Itg may be considered as a deformation variant of the cubic close-packed Cu

type. The rhombohedral unit cell contains one atom, but it is usual to represent

the structure referred to its triple hexagonal cell (c-axis along the body diagonal

of the rhombohedral cell). The

c/a

ratio of the hexagonal cell of a-Hg is 1.93,

i.e., significantly lower than ~ = 2.45, corresponding to the Cu type described

in a similar hexagonal cell. As a consequence, the distances between atoms in

B. Elements 125

neighboring layers are shorter than the interatomic distances within the

layers. The 12 atoms of the original cuboctahedron in the Cu type are subdivided

into 6 atoms at a shorter distance, forming a trigonal antiprism, and 6 atoms at a

longer distance.

The tetragonal structure of In represents a different deformation variant of

the Cu type. The originally close-packed structure is here extended along one of

the 4-fold axes. Described in a tetragonal face-centered cell similar to the cubic

cell of Cu, the

c/a

ratio is 1.08. However, following crystallographic conven-

tions, tetragonal face-centered cells are reduced to body-centered cells and the

structure is correctly described in a tetragonal cell with one atom at the origin and

one atom at the center. The

c/a

ratio of the body-centered cell, 1.52, should here

be compared with x/2 - 1.41 to estimate the deformation. The 12 atoms of the

original cuboctahedron are subdivided into 4 atoms at a shorter distance, forming

a square, and 8 atoms at a longer distance. Indium itself becomes superconduct-

ing at 3.4 K, but the highest T c for this structure type is observed for a second

modification of mercury. The cell parameter ratio of fl-Hg is significantly lower

than unity and the coordination polyhedron is reduced to a compressed tetragonal

prism, with two additional atoms capping the square faces. This structure is

better seen as a deformation derivative of W and may be classified under the

corresponding branch of the structure type, defined on a-Pa.

The semiconducting, gray a-modification of tin crystallizes with the cubic

diamond structure, where each atom is covalently bonded to four others situated

at the comers of a tetrahedron. White [3-Sn has metallic character and becomes

superconducting at 3.7 K. This structure is tetragonal, with four atoms in the unit

cell. The coordination number is 6 and the six-vertex polyhedron can be

described as a distorted octahedron formed by a heavily compressed tetrahedron

and two additional atoms at a slightly longer distance. The elements Si and Ge

from the same group IVB crystallize with this structure type at high pressure.

The quenched samples show superconducting transition temperatures of up to

7.1 K at 13.6 GPa for Si. Equiatomic mixtures of elements from the surrounding

columns IIIB and VB of the periodic table, like GaSb and InSb, adopt the same

structure type within certain ranges of pressure and become superconducting at

4.2 and 2.1 K, respectively.

Gallium presents several superconducting structural modifications. The

structure of the orthorhombic ambient-pressure modification ~-Ga contains

strongly puckered close-packed layers. Each atom has seven nearest neighbors,

one in either the over- or underlying layer and six within the same layer. The

distance to the atom in the neighboring layer is shorter (2.48 ~) than that to the

other atoms, so that Ga-Ga pairs are sometimes considered. This structure

modification becomes superconducting at a modest temperature of 1.1 K, whereas

critical temperatures increasing from 6.2 to 7.9 K are observed for the metastable

modifications r, 7, and ~. In monoclinic [~-Ga the atoms are arranged in infinite

zigzag chains with interatomic distances of 2.68/~. Six additional atoms are

located around each atom at distances ranging from 2.77 to 2.92A. The