UNIT CELL BY TAKING A CATION AS THE ORIGIN

Perovskite and Related Structure Systems

3.1.2. UNIT CELL BY TAKING A CATION AS THE ORIGIN

The perovskite structure has a simple cubic Bravais cell, in which the octahedra share comers and the origin of the Bravais cell is at the B cation, as given in Fig. 3.1a.

Alternatively, we can use the A cation as the origin and the unit cell is transformed into the form in Fig. 3.1c. In this configuration, the A cations are at the cubic vertices and the oxygen anions occupy the face centers of the six faces, forming an octahedron with the B cation at the center. In this arrangement the oxygen forms linear 0 2- _B4+ _02- triples parallel to the x, y, and z axes, with oxygen located at the centers of the squares formed by the A2+ cations. If an electric field is applied parallel to the z axis, for example, the 0 2- _B4+ _02- chain parallel to the z axis will be polarized, but the 0 2- _B4+ _02- chains parallel to the x and y axes may not be disturbed, resulting in a polarization of the crystal parallel to the z axis. This is the origin of ferroelectricity. Moreover, the displacement of the oxygen anions could distort the configuration of the A cations, resulting in a change in shape of the unit cell. This is the piezoelectric effect (Section

1.13.3).

3.1.3. OXYGEN CUBIC CLOSE PACKING

We now reexamine the AB0₃ perovskite structure from a different point of view.

We use the structure model of perovskite formed by the comer -sharing octahedra chains, e.g., the unit cell with B cations as the origin, as shown in Fig. 3.2a, in which the octahedron is sketched for clarity. From a topological point of view we may consider the perovskite structure as three similar oxygen cubes interpenetrating with each other, as shown in Fig. 3.2b. In the three types of cubes the B cations are located at the middle of the four edges, forming a plane parallel to one of the faces of each cube. The interpenetrating of three cubes forms the perovskitestructure (Fig. 3.2c). The A cations in the three types of cubes occupy the centers of the faces formed by oxygen parallel to the layers of the B cations. We may call the three types of oxygen cubes modules. A comparison of Fig. 3.2a and c can help us to understand the relationship between the three types of cubic modules with the perovskite structure.

Type 1 Type 2 Type 3

type 3

Figure 3.2. Basic structural modules in perovskite. (a) AB03 perovskite structure with B06 octahedra sharing corners. (b) Three ABO cubic modules used to assemble the perovskite structure, where the distribution of the A and B cations in each is different. (c) The interpenetrating of the ABO lattices forms the perovskite structure with B cation (left side) and A cation (right side) at the corners, respectively.

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By carefully observing the three modules, one can discover interesting clues to build the structure. Figure 3.3 shows the three modules and their stacking layers parallel to the (111) plane (Fig. 3.3a, b, c). For convenience the A cations located in different (111) stacking layers are labeled by different patterns. The triangles formed by oxygen are the same except for the position of the A cation. The A cations belonging to the same stacking layer are located at different sides of the triangles. The A cations belonging to the triangles labeled 1, 2, and 3 are the same lattice sites and they should coincide when constructing the unit cell of the perovskite structure. Accordingly, the arrangement relationship between the oxygen anions and A cations in the basic {Ill} stacking layers is shown in Fig. 3.3d, where an A cation is surrounded by six oxygen anions, forming a

Figure 3.3. (a)-(c) The three types of cubic modules and their stacking layers parallel to (111). (d) Assembling an A cation (111) layer in the perovskite structure using the stacking layers of the three modules shown in (a)-(c).

close-packed structure. Notice that three oxygen atoms form a triangle. Naturally the A cations also form a hexagonal array, but separated by the triangle close-packed oxygen anions. By the structure symmetry, a {Ill} stacking layer of the perovskite structure can be constructed (Fig. 3.4) where the chemical composition of this layer is A03. Since the charge of A is 2+, the valence charge of the layer is (A03)4-. To balance the local negative charge a layer with a charge of 4+ must be introduced. Therefore, a B⁴+ cation layer (without oxygen anions) should be the next stacking layer.

Based on this analysis we can understand why the A^Z+B4+03 type of compound has the perovskite structure, what the stacking layers are, what the basic structure modules are, and how these layers are stacked. Katz and Ward (1964) used a different way to analyze the perovskite structure and obtained the same type of fundamental stacking layers. These types of stacking layers were also introduced by Wells (1984), and they have been used in analysis of structural similarities among perovskite-type compounds with different oxygen deficiencies (Anderson et al., 1993).

Based on our analysis of sodium chloride structure (or MO) and rutile structure (or MOz), the anions always form a close-packed layer, but for sodium chloride the cations also form a close-packed layer, an alternative stacking of the two forms of NaCl. In this MO structure, the radius ratio rM/rX < 1 and the anion octahedra share edges. For rutile the oxygen anions form a close-packed layer, but the cations cannot because the number of cations is half that of the oxygen anions and the valence charge of each cation is twice that of the oxygen anion. Therefore, the rutile structure exhibits octahedron chains with edge sharing, and the chains are connected by corner sharing (Fig. 2.6). The cations occupy only part of either voids 1, 3, and 5 positions or the 2, 4, and 6 positions as shown in Fig. 3.5, and this stacking structure makes rutile able to create stacking faults, in which

r:--- -,

I

e

^{A type Calion I}

I I

I Oxygen anion I

L.. _ _ _ _ _ _ _ _ .J

Figure 3.4. The fundamental (A03)4- (111) close-packing layer in the perovskite structure.

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the cations shift from voids 1, 3, or 5 positions to 2, 4, or 6 positions, forming the crystallographic shear plane. This is the reason that the rutile structure can have homologous phases Tin02n - 1• In NaCI and Ti02, the anion and cation layers are separated.

In perovskite structure, however, the mixture of anions and A cations forms close- packed layers. These layers have negative charge, equivalent to the pure anion layers in NaCI and rutile structures. These negatively charged layers and the positively charged B4+ layers stack alternately to build the perovskite structure. Based on this analysis, the A cations should have more influence on the oxygen anions than the B cations do, but the B cations should have valence variability to compensate the local charge. This is probably the origin of experimental feasibility for modifying the valence of B cations by some dopants of different valences.

3.1.4. ANION CLOSE PACKING AND FORMATION OF TETRAHEDRON AND OCTAHEDRON

As we discussed in describing the rutile structure, if the oxygen atoms are spheres, their packing is hexagonal, as illustrated in Fig. 3.5. In the sphere close-packed layer (we may say the A-type layer) there are six voids surround a sphere, denoted 1,2,3,4,5, and 6. When the two close-packed layers are stacked two cases can occur: the spheres in the second layer fall on the positions of the voids labeled (a) 1, 3, and 5, called the B-type layer, or (b) 2, 4, and 6, called the C-type layer. Ifthe stacking sequence is ABABAB ... , a hexagonal close packing is formed. In this packing configuration, we can see two types

Rutile-type comer- haring octahedra C

CACA ... sequence

Figure 3.5. Sphere close-packed (111) layers and the formation of the octahedron and tetrahedron interstices.

The first layer is the A layer. The B and C layers are stacked by translating the close-packing layer to the B and C void positions, respectively. The comer-sharing and the edge-sharing octahedra are shown at the right-hand side, as formed by different stacking sequences (see text).

of interstices: octahedron and tetrahedron (Fig. 3.5). The octahedron interstice is formed by six oxygen anions, while the tetrahedron is formed by four oxygens. The oxygen octahedra, formed by the two adjacent

AB

layers, may share edges (bottom left in Fig.

3.5) or comers in the following two cases: one is the comer-sharing octahedral chain formed by adjacent three layers in a

CAB

sequence (right side in Fig. 3.5), and the other is comer-sharing octahedral chain formed by adjacent three layers in a CAC sequence (lower right in Fig. 3.5), but the two chains are perpendicular. If the oxygen is ionic the positively charged cations must be at the center of the anion octahedron. The cation- anion-cation (or M - X - M) is the bonding bridge. The bonding length M - X and bonding angle M - X - M depend on the nature of the cation and the anion and the interaction between them.

If the stacking sequence of the oxygen close-packing layers is ABCABC ... , a face- centered cubic lattice is formed. If all of the interstices of oxygen octahedra are occupied by cations, the cations also form a face-centered cubic lattice, and the atomic ratio of cations to anions is 1: 1. The system has a composition of MX, such as N aCl. In this type of structure each cation is coordinated with six anions, so the MX6 octahedra share all of the faces. When half the centers of the oxygen octahedra are occupied by cations as in the rutile structure, the atom ratio of cations to anions is 1: 2, MX2, in which octahedra with edge sharing form chains and the two types of octahedra chains with different orientations share the comers, forming the rutile structure as discussed in Chapter 2.

Here we reexamine the arrangement of the cations in the rutile structure. Figure 3.6 shows an oxygen close-packing layer and voids 1,2,3,4,5, and 6. Voids 1,3, and 5 are the B-Iayer atom positions, and voids 2, 4, and 6 are the C-Iayer atom positions. In the rutile structure, the cations occupy 1 and 3 or 2 and 4 to satisfy the composition requirement, but the threefold symmetry gives three equivalent choices for defining the positions of (1, 3, 5) [or (2, 4, 6)] and each has equal probability. In the perfect rutile structure, however, only two choices are present. Therefore, this lower symmetry system is expected to have higher energy than the system with higher symmetry, resulting in the formation of a crystallographic shear plane in the rutile structure It is also well known that titanium dioxide without any defects is rarely found. If the three position choices are used equally, the octahedra form zigzag edge-sharing chains, such as the cx-Pb02

structure (Fig. 3.6 lower).

The analysis above is based only on a geometrical description. The actual structure of a compound depends strongly on the electronic band structure, structure type, bonding length, and angle. Octahedron distortion can also be significant. In titanium dioxide the octahedron is distorted to make two titanium-oxygen bonds slightly longer (0.198 nm) than the other four (0.195 nm) (Cromer and Herrington, 1955). At high pressure (40 kbar and at 450°C) titanium dioxide transforms to the cx-Pb02 structure, in which the two titanium-oxygen bonds are even longer (0.205 nm) than the other four (0.191 nm) (Simons and Dachille, 1967). If the distortion is large enough in a certain direction (say the c axis), the structure will be changed from rutile into, for example, a monoclinic one, such as V02 in which the cation distortion is along the c direction of the rutile. This transformation makes V02 metallic (Goodenough, 1971).

In Fig. 3.5 we showed that AC and AB stacking sequences can form tetrahedral and octahedral structural configurations based on the adjacent occupations. In Fig. 3.3 we analyzed the perovskite structure and obtained the fundamental stacking layer of (A03)4- , in which each A cation is surrounded by six oxygen anions and in each A cation triangle there is an oxygen anion triangle. In other words, an A cation hexagon contains

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six apex-sharing oxygen triangles. This character of the fundamental stacking layer (A03)4- detennines the features of the perovskite structure. Figure 3.7 shows the stacking layer of (A03)4- and the stacking sequence to fonn the perovskite structure. In Fig. 3.7a, three layers of A cations, represented by three types of shaded spheres, fonn the perovskite unit cell with A cations as the origin. The stacking sequence of the A cations is ABC in the A cation hexagonal, but in this A cation cube (the unit cell) there are only two triangles of oxygen anions for fonning an oxygen octahedron (Fig. 3.7b). In Figs.

3.7c and d, the B cation octahedron and the comer-sharing octahedra chains are shown, respectively, where the two arrowheads indicate the perpendicular relationship between the two chains in 3-D space. In Fig. 3.7e, the perovskite unit cell with B cation at the origin is shown. For the (A0_3)4- layer stacking, the A cation hexagon is the only choice to describe the stacking sequence as ABC, as shown in Fig. 3.8a, where the A position is at an A cation and the Band C positions are marked as ~ and y triangles, respectively. In this way we can clearly see that, in layers of (A03)4-, the basic mosaic is the rings containing an A cation surrounded by three y-and three ~ oxygen triangles which belong to the A layer. The 2-D Bravais cell is a hexagonal net which is the same as the A cation net. The ABC stacking of these layers builds a cubic unit cell. The ~ and y oxygen triangles have the same translation symmetry, and the stacking of the two types of oxygen triangles, which have an inversion center at their connected points, fonns the comer- sharing octahedra arrays in which the B cations are at the centers. If the (A03)4- layers

Rutile

Figure 3.6. The oxygen close-packed (111) layer and the formation of rutile and cx-Pb02 structures by choosing different cation locations. The oxygen octahedra are highlighted.

contain oxygen vacancies, there are three 2-D oxygen sublattices, as demonstrated in Fig.

3.8b using different eyeball patterns. The oxygen vacancy can be formed at one or two sublattices, but it cannot have three or more vacancies, otherwise the lattice will be destroyed by Coulomb repulsion between cations. If one oxygen sublattice is vacant, the layer composition is (A02)2- (Fig. 3.8c). If two oxygen sublattices are vacated, the layer composition is AO, so the layers are neutral, provided the valence of cation A is 2+ and

A

ABO) unit cell

Comer- haring octahedra chain r- - - - -

I Close-packing layer at Aile:

I I~ @ Close-packing layer at Bite I

L ~_ Clo~!a~~ ~y~ ~ ~~:

ABO) perovskite structure fonned by comer- hariog octahedra chain

Figure 3.7. The perovskite structure is formed by an alternate stacking of the (A03)4- and B cation layers parallel to (Ill). (a) The A cation stacking sequence, (b) the unit cell of perovskite, (c) the formation of the B06

octahedron, and (d) the connection between B06 octahedron chains, where the arrowheads indicate two primary perpendicular directions in perovskite structure in 3-D space. (e) Perovskite structure with B cation as the origin, where the oxygens behind the octahedra are not drawn for clarity.

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(b) (A<>J)4- layer

(d) (A03_x)(4-2x)- layer

(c) (AOV2-layer

e

type I oxygen ublaltice

Q

^type 2 oxygen ublattice

8

type 3 oxygen subJanice oxygen vacancy in lype I oxygen ublanice oxygen vacancy in type 2 oxygen sublaltice

oxygen vacancy in lype 3 oxygen sublallice

Figure 3.8. (a) The (A03)4- layer can be visualized as the ~-and y-oxygen triangle rings with A cations at the centers. The unit cell of perovskite structure is outlined. (b-d) Three two-dimensional oxygen sublattices in the (A0_3)4- layers without vacancy [(A03)4- j, with one vacancy [(A02)2-j, with x vacancy [(A0_{3_x})(4-2x)-j,

respectively, where the little eyeballs indicate the types of the anions and vacancies belonging to different oxygen sublattices in Fig. 3.3.

no electrostatic force attracts the B4+ cation layers. Thus, the perovskite structure cannot be formed. In a general case the oxygen vacancy can easily occur partially in the local unit of the (A03)4- layer, but it is rare to create three oxygen vacancies in one sublattice.

If oxygen vacancies are partially formed in the (A03)4- layer, the close-packing layer should be (A03_J4-2x)- (Fig. 3.8d). Therefore, there are two choices to have oxygen vacancies in the three oxygen sublattices. If one oxygen vacancy is formed in the oxygen sublattice, it can be at different layers and/or different sub lattices of oxygens. The symmetry of oxygen triangles makes it possible to have vacancies at different corners of triangles belonging to alternate stacking layers. This creates the five- and four- coordinated oxygen polyhedra (Figs. 3.9 and 3.10). In Figs. 3.9 and 3.10 the total number of B cations does not change in the structure, but their valence states may change to balance the local charge. The A cations in the (A03)4- layer may change their valence states and/or their total number in the layer (e.g., stoichiometry). On the other hand, to maintain stability of the (A03)4- layer (even it can be corrugated) too many A cations cannot be lost. Therefore, replacing the A cations by an element with a different valence may be the optimal choice to induce oxygen deficiency and/or a mixed valence state of

Figure 3.9. The local oxygen vacancy arrangements and stacking of (A03)4--type layers (with vacancies). (a) Single- and double-oxygen vacancies in a B06 octahedron. (b) The connections of octahedra without, with one, and with two oxygen vacancies. VI, V2, and V3 indicate oxygen vacancies in the first, second, and third layers, respectively.

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the B cations. This process is known as tailoring the characteristics of the perovskite structure.

The differences between the rutile-type and the perovskite-type structures, as described from the oxygen close-packing point of view, are summarized as follows.

1. The fundamental structures of the stacking layers are different: pure oxygen close-packing layers for rutile, layers of (A03)4- for perovskite.

2. The cations occupy only half positions of the A, B, and C voids and the oxygen hexagonal network is the basis of stacking in the rutile structure. In the perovskite structure, the basis of stacking is the A cation hexagon, and the

Figure 3.10. A total of 14 perovskite-type structural modules with oxygen vacancies, where the oxygen anions are represented by different patterns to distinguish their stacking layers, V stands for vacancy. The combination of these modules can reproduce the crystal structures of many compounds (see text).

oxygen close-packing net penetrates the A cation net, fonning an oxygen close- packing array that separates the A cations located in the same plane.

3. In rutile the stacking sequence is ABAB ... and only two out of three choices of stacking positions are occupied, resulting in the movability of cations and creating crystallographic shear planes. In perovskite the stacking sequence is ABCABC ... and the B cations always have close packing, but the fundamental structure of the (A03

t-

layer can be changed so that oxygen vacancies can be created due to the variation in either the size of the A cation or its valence, as well as the change in coordination number of the B cation from 6 to 5 or 4.

Based on these comparisons, we may figure out the relationship between the A or B cations with the oxygen anions in the perovskite structure. It seems that the A cations are tightly related to the oxygen anions and contain more ionic bonding; the radius and the

Figure 3.10. (continued)

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valence of the A cation strongly affect the oxygen anions. The B cations may be regarded as the dependent subordinations that rely on the structure of (A03)4-. The bonding of the B cations with the surrounding oxygens may be ionic, covalent, or partially covalent. If the (A03)4- layer has defects, such as vacancies, the stacking of these layers will form distorted or deficient oxygen octahedra. Thus, the B cation is required to have the flexibility (or multivalences) to modify its valence state for balancing the local charge. In other words, if we intend to modify the valence of the B cation it is better to change the surrounding structure of the A cations first via doping or other techniques. This is very important clue for modifying the structure and property of perovskite.

3.2.

POSSIBLE TYPES OF ANION-DEFICIENT PEROVSKITE STRUCTURES 3.2.1. THE 14 FuNDAMENTAL STRUCTURE UNITS

Perovskite-type structures are the basis of many oxides, in which oxygen-deficient perovskites are a key group of materials that possess functionality. The anion deficiency can change the coordination number of the B cation octahedra, and the size and type of Bravais unit cell of the compounds. Therefore, the band structure and the property can also be modified. It is important to know the possible types of anion-deficient perovskite structures for enhancing the insight in fabricating new materials by controlling the stacking of the (A03)4- layers.

Figure 3.9a shows an AB stacking of the (A03)4- layers, in which the anions in the B layer are positioned at the y-type position of the oxygen close-packing triangles, equivalent to shifting the A position to the B position (the A, B, and

C

positions are indicated on the left of the figure). The octahedra are formed by the superimposed oxygen triangles belonging to the A and B layers. If an oxygen vacancy is created in the B layer, as indicated by V 2, the five-coordinated polyhedron is formed. By the same token, oxygen vacancies can be created in the octahedra formed by the superposition of the B and C layers (Figs. 3.9a, b), where the numbers 1,2, and 3 represent the anions in A, B, and C layers, respectively. As discussed before, any (A03)4- stacking layer can have oxygen vacancies. A five-coordinated polyhedron is formed if only one oxygen vacancy is created in the B layer while the A layer is perfect. If two oxygen vacancies (V 1 and V 2)

are created in the two adjacent layers within the same octahedron, a four-coordinated square is formed (Fig. 3.9a). It is impossible to have more than two vacancies in one octahedron because four apices are the minimum to form a 3-D connection for maintaining the perovskite-type of frame. Figures 3.10a and b show the BOs and B0⁴ structures in an octahedron sheet, respectively. Missing one oxygen at a vertex forms a five-coordinated unit (Fig. 3.lOa), while missing two oxygens located at two opposite vertices forms a four-coordinated square (Fig. 3. lOb). Missing three oxygens in one octahedron forces the unit to break down its connections with other octahedra and thus, is unlikely to occur. If we refer to the three cubic oxygen modules in Fig. 3.2, the four- coordinated oxygen square is due to the missing of two adjacent oxygens in the same module but belonging to different oxygen stacking layers. Figure 3. Wc shows a cluster constructed with consideration of the symmetric distribution of two oxygen octahedra, resulting in four five-coordinated polyhedra and two four-coordinated squares. It has a composition of ABS030 and it demonstrates how the different coordinated polyhedra may be connect with each other in three dimensions.