# Perovskite (structure)

A perovskite is a crystalline material of formula ABX$_3$ with a crystal structure similar to that of the mineral perovskite, this latter consisting of calcium titanium oxide (CaTiO$_3$). The mineral was first discovered in the Ural mountains of Russia by Gustav Rose in 1839 and named after Russian mineralogist L. A. Perovski (1792–1856). In addition to being one of the most abundant structural families, perovskites wide-ranging properties and applications.

## Structure

Perovskite structures are adopted by many compounds that have the chemical formula ABX$_3$. 'A' and 'B' are positively charged ions (i.e. cations), often of very different sizes, and X is a negatively charged ion (an anion, frequently oxide) that bonds to both cations. The 'A' atoms are generally larger than the 'B' atoms. The ideal cubic structure has the B cation in 6-fold coordination, surrounded by an octahedron of anions, and the A cation in 12-fold cuboctahedral coordination. Additional perovskite forms may exist where both/either the A and B sites have a configuration of A1$_{x-1}$A2$_x$ and/or B1$_{y-1}$B2$_y$ and the X may deviate from the ideal coordination configuration as ions within the A and B sites undergo changes in their oxidation states. The idealized form is a cubic structure (space group Pm3m, no. 221), which is rarely encountered. The orthorhombic (e.g. space group Pnma, no. 62, or Amm2, no. 68) and tetragonal (e.g. space group I4/mcm, no. 140, or P4mm, no. 99) structures are the most common non-cubic variants. Although the perovskite structure is named after CaTiO$_3$, this mineral has a non-cubic structure. SrTiO$_3$ and CaRbF$_3$ are examples of cubic perovskites. Barium titanate is an example of a perovskite which can take on the rhombohedral (space group R3m, no. 160), orthorhombic, tetragonal and cubic forms depending on temperature.
In the idealized cubic unit cell of such a compound, the type 'A' atom sits at cube corner position (0, 0, 0), the type 'B' atom sits at the body-center position (1/2, 1/2, 1/2) and X atoms (typically oxygen) sit at face centered positions (1/2, 1/2, 0), (1/2, 0, 1/2) and (0, 1/2, 1/2). The diagram to the right shows edges for an equivalent unit cell with A in the cube corner position, B at the body center, and X at face-centered positions.
Four general categories of cation-pairing are possible: A$^+$B$^{2+}$X$^−$$_3$, or 1:2 perovskites; A$^{2+}$B$^{4+}$X$^{2−}$$_3$, or 2:4 perovskites; A$^{3+}$B$^{3+}$X$^{2−}$$_3$, or 3:3 perovskites; and A$^+$B$^{5+}$X$^{2−}$$_3$, or 1:5 perovskites.
The relative ion size requirements for stability of the cubic structure are quite stringent, so slight buckling and distortion can produce several lower-symmetry distorted versions, in which the coordination numbers of A cations, B cations or both are reduced. Tilting of the BO$_6$ octahedra reduces the coordination of an undersized A cation from 12 to as low as 8. Conversely, off-centering of an undersized B cation within its octahedron allows it to attain a stable bonding pattern. The resulting electric dipole is responsible for the property of ferroelectricity and shown by perovskites such as BaTiO$_3$ that distort in this fashion.
Complex perovskite structures contain two different B-site cations. This results in the possibility of ordered and disordered variants.

## Defect perovskites

Also common are the defect perovskites. Instead of the ideal ABO$_3$ stoichiometry, defect perovskites are missing some or all of the A, B, or O atoms. One example is rhenium trioxide. It is missing the A atoms. Uranium trihydride is another example of a simple defect perovskite. Here, all B sites are vacant, H$^−$ occupies the O sites, and the large U$^{3+}$ ion occupies the A site.
Many high temperature superconductors, especially cuprate superconductor, adopt defect perovskite structures. The prime example is yttrium barium copper oxide (YBCO), which has the formula YBa$_2$Cu$_3$O$_7$. In this material Y$^{3+}$ and Ba$^{2+}$, which are relatively large, occupy all A sites. Cu occupies all B sites. Two O atoms per formula unit are absent, hence the term defect. The compound YBa$_2$Cu$_3$O$_7$ is a superconductor. The average oxidation state of copper is Cu$^{(7/3)+}$ since Y3+ and Ba2+ have fixed oxidation states. When heated in the absence of O$_2$, the solid loses its superconducting properties, relaxes to the stoichiometry YBa$_2$Cu$_3$O$_{6.5}$, and all copper sites convert to Cu$^{2+}$. The material thus is an oxygen carrier, shuttling between two defect perovskites:
    4 YBa$_2$Cu$_3$O$_7$ ⇌ 4 YBa$_2$Cu$_3$O$_{6.5}$ + O$_2$

## Layered perovskites

Perovskites can be deposited as epitaxial thin films on top of other perovskites, using techniques such as pulsed laser deposition and molecular-beam epitaxy. These films can be a couple of nanometres thick or as small as a single unit cell.
Perovskites may be structured in layers, with the ABO $_3$ structure separated by thin sheets of intrusive material. Based on the chemical makeup of their intrusions, these layered phases can be defined as follows:
* Aurivillius phase: the intruding layer is composed of a [Bi $_2$O $_2$]$^{2+}$ ion, occurring every n ABO $_3$ layers, leading to an overall chemical formula of [Bi $_2$O $_2$]-A $_{(n−1)}$B $_2$O $_7$. Their oxide ion-conducting properties were first discovered in the 1970s by Takahashi et al., and they have been used for this purpose ever since.
* Dion−Jacobson phase: the intruding layer is composed of an alkali metal (M) every n ABO $_3$ layers, giving the overall formula as M$^+$ A $_{(n−1)}$B $_n$O $_{(3n+1)}$
* Ruddlesden-Popper phase: the simplest of the phases, the intruding layer occurs between every one (n = 1) or multiple (n > 1) layers of the ABO $_3$ lattice. Ruddlesden−Popper phases have a similar relationship to perovskites in terms of atomic radii of elements with A typically being large (such as La or Sr) with the B ion being much smaller typically a transition metal (such as Mn, Co or Ni).

## Complex perovskites

Although there is a large number of simple known ABX$_3$ perovskites, this number can be greatly expanded if the A and B sites are increasingly doubled / complex AA′BB′X$_6$. Ordered double perovskites are usually denoted as A$_2$BB′O$_6$ where disordered are denoted as A(BB′)O$_3$. In ordered perovskites, three different types of ordering are possible: rock-salt, layered, and columnar. The most common ordering is rock-salt followed by the much more uncommon disordered and very distant columnar and layered. The formation of rock-salt superstructures is dependent on the B-site cation ordering. Octahedral tilting can occur in double perovskites, however Jahn–Teller distortions and alternative modes alter the B–O bond length.

## Antiperovskites

The lattice of an antiperovskites (or inverse perovskites) is the same as that of the perovskite structure, but the anion and cation positions are switched. The typical perovskite structure is represented by the general formula ABX$_3$, where A and B are cations and X is an anion. When the anion is the (divalent) oxide ion, A and B cations can have charges 1 and 5, respectively, 2 and 4, respectively, or 3 and 3, respectively. In antiperovskite compounds, the general formula is reversed, so that the X sites are occupied by an electropositive ion, i.e., cation (such as an alkali metal), while A and B sites are occupied by different types of anion. In the ideal cubic cell, the A anion is at the corners of the cube, the B anion at the octahedral center, and the X cation is at the faces of the cube. Thus the A anion has a coordination number of 12, while the B anion sits at the center of an octahedron with a coordination number of 6. Similar to the perovskite structure, most antiperovskite compounds are known to deviate from the ideal cubic structure, forming orthorhombic or tetragonal phases depending on temperature and pressure.
Whether a compound will form an antiperovskite structure depends not only on its chemical formula, but also the relative sizes of the ionic radii of the constituent atoms. This constraint is expressed in terms of the Goldschmidt tolerance factor, which is determined by the radii, r$_a$, r$_b$ and r$_x$, of the A, B, and X ions.
${\frac {(r_{a}+r_{x})}{{\sqrt {2}}(r_{b}+r_{x})}}$
For the antiperovskite structure to be structurally stable, the tolerance factor must be between 0.71 and 1. If between 0.71 and 0.9, the crystal will be orthorhombic or tetragonal. If between 0.9 and 1, it will be cubic. By mixing the B anions with another element of the same valence but different size, the tolerance factor can be altered. Different combinations of elements result in different compounds with different regions of thermodynamic stability for a given crystal symmetry..

### Examples

Antiperovskites naturally occur in sulphohalite, galeite, schairerite, kogarkoite, nacaphite, arctite, polyphite, and hatrurite. It is also demonstrated in superconductive compounds such as CuNNi$_3$ and ZnNNi$_3$.
Discovered in 1930, metallic antiperovskites have the formula M$_3$AB where M represents a magnetic element, Mn, Ni, or Fe; A represents a transition or main group element, Ga, Cu, Sn, and Zn; and B represents N, C, or B. These materials exhibit superconductivity, giant magnetoresistance, and other unusual properties.
Antiperovskite manganese nitrides exhibit zero thermal expansion.

## Octahedral tilting

Beyond the most common perovskite symmetries (cubic, tetragonal, orthorhombic), a more precise determination leads to a total of 23 different structure types that can be found. These 23 structure can be categorized into 4 different so-called tilt systems that are denoted by their respective Glazer notation.
| Tilt System number | Tilt system symbol | Space group |
| Three-tilt systems |||
| 1 | a$^+$b$^+$c$^+$ | Immm (#71) |
| 2 | a$^+$b$^+$b$^+$ | Immm (#71) |
| 3 | a$^+$a$^+$a$^+$ | Im3 (#204) |
| 4 | a$^+$b$^+$c$^−$ | Pmmn (#59) |
| 5 | a$^+$a$^+$c$^−$ | Pmmn (#59) |
| 6 | a$^+$b$^+$b$^−$ | Pmmn (#59) |
| 7 | a$^+$a$^+$a$^−$ | Pmmn (#59) |
| 8 | a$^+$b$^−$c$^−$ | A2$_1$/m11 (#11) |
| 9 | a$^+$a$^−$c$^−$ | A2$_1$/m11 (#11) |
| 10 | a$^+$b$^−$b$^−$ | Pmnb (#62) |
| 11 | a$^+$a$^−$a$^−$ | Pmnb (#62) |
| 12 | a$^−$b$^−$c$^−$ | F1 (#2) |
| 13 | a$^−$b$^−$b$^−$ | I2/a (#15) |
| 14 | a$^−$a$^−$a$^−$ | R3c (#167) |
| Two-tilt systems |||
| 15 | a$^0$b$^+$c$^+$ | Immm (#71) |
| 16 | a$^0$b$^+$b$^+$ | I4/mmm (#139) |
| 17 | a$^0$b$^+$c$^−$ | Bmmb (#63) |
| 18 | a$^0$b$^+$b$^−$ | Bmmb (#63) |
| 19 | a$^0$b$^−$c$^−$ | F2/m11 (#12) |
| 29 | a$^0$b$^−$b$^−$ | Imcm (#74) |
| One-tilt systems |||
| 21 | a$^0$a$^0$c$^+$ | C4/mmb (#127) |
| 22 | a$^0$a$^0$c$^−$ | F4/mmc (#140) |
| Zero-tilt systems |||
| 23 | a$^0$a$^0$a$^0$ | Pm3m (#221) |
The notation consists of a letter a/b/c, which describes the rotation around a Cartesian axis and a superscript +/—/0 to denote the rotation with respect to the adjacent layer. A "+" denotes that the rotation of two adjacent layers points in the same direction, whereas a "—" denotes that adjacent layers are rotated in opposite directions. Common examples are a$^0$a$^0$a$^0$, a$^0$a$^0$a$^–$ and a$^0$a$^0$a$^+$ which are visualized here.

## Examples
### Minerals

Aside from perovskite itself, some perovskite minerals include loparite and bridgmanite. Bridgmanite is a silicate with the chemical formula (Mg,Fe)SiO$_3$. It is the most common mineral in the Earth's mantle. At high pressures associated with the deeper mantel, the Si sites feature octahedral units.
At the high pressure conditions of the Earth's lower mantle, the pyroxene enstatite, MgSiO$_3$, which otherwise has tetrahedral Si sites, transforms into a denser perovskite-structured polymorph; this phase may be the most common mineral in the Earth. This phase has the orthorhombically distorted perovskite structure (GdFeO$_3$-type structure) that is stable at pressures from ~24 GPa to ~110 GPa. However, it cannot be transported from depths of several hundred km to the Earth's surface without transforming back into less dense materials. At higher pressures, MgSiO$_3$ perovskite, commonly known as silicate perovskite, transforms to post-perovskite.

### Inorganic perovskites lacking oxygen

Although the most common perovskite compounds contain oxygen, there are a few perovskite compounds that form without oxygen. Fluoride perovskites such as NaMgF$_3$ are well known. A large family of metallic perovskite compounds can be represented by RT$_3$M (R: rare-earth or other relatively large ion, T: transition metal ion and M: light metalloids). The metalloids occupy the octahedrally coordinated "B" sites in these compounds. RPd$_3$B, RRh$_3$B and CeRu$_3$C are examples. MgCNi$_3$ is a metallic perovskite compound and has received lot of attention because of its superconducting properties. An even more exotic type of perovskite is represented by the mixed oxide-aurides of Cs and Rb, such as Cs$_3$AuO, which contain large alkali cations in the traditional "anion" sites, bonded to O$^{2−}$ and Au$^−$ anions.

### Organic perovskites

Of interest in the context of solar energy are materials of the type [R$_4$N]$^+$[MX$_3$]$^−$. Thus, the quat cation occupies the B site and the metals occupy the A sites.These materials are the basis of perovskite solar cells. These materials have high charge carrier mobility and charge carrier lifetime that allow light-generated electrons and holes to move far enough to be extracted as current, instead of losing their energy as heat within the cell.

## Applications, real and aspirational

Probably the dominant applications of perovskites are in microelectronics and telecommunications, which exploit the ferroelectric properties of barium titanate, lithium niobate, lead zirconium titanate and others.
Physical properties of interest to materials science among perovskites They are applicable to lasers. They are also some interests for scintillator as they have a large light yield for radiation conversion. Because of the flexibility of bond angles inherent in the perovskite structure there are many different types of distortions that can occur from the ideal structure. These include tilting of the octahedra, displacements of the cations out of the centers of their coordination polyhedra, and distortions of the octahedra driven by electronic factors (Jahn-Teller distortions). The financially biggest application of perovskites is in ceramic capacitors, in which BaTiO$_3$ is used because of its high dielectric constant. Light-emitting diodes exploit the high photoluminescence quantum efficiencies of perovskites. In the area of photoelectrolysis, water electrolysis at 12.3% efficiency can use perovskite photovoltaics. Scintillators based on cerium-doped lutetium aluminum perovskite (LuAP:Ce) single crystals were reported. Layered Ruddlesden-Popper perovskites have shown potential as fast novel scintillators with room temperature light yields up to 40,000 photons/MeV, fast decay times below 5 ns and negligible afterglow. In addition this class of materials have shown capability for wide-range particle detection, including alpha particles and thermal neutrons.