Editing Perovskite (structure)

{{Short description|Type of crystal structure}}
{{about|the synthetic compounds|the mineral|Perovskite}}
[[Image:Perovskite.jpg|thumb|Structure of a perovskite with general chemical formula ABX<sub>3</sub>. The red spheres are X atoms (usually oxygens), the blue spheres are B atoms (a smaller metal cation, such as Ti<sup>4+</sup>), and the green spheres are the A atoms (a larger metal cation, such as Ca<sup>2+</sup>). Pictured is the undistorted [[cubic crystal system|cubic]] structure; the symmetry is lowered to [[orthorhombic]], [[tetragonal]] or [[trigonal]] in many perovskites.<ref>{{cite journal| title = Energetics and Crystal Chemical Systematics among Ilmenite, Lithium Niobate, and Perovskite Structures| author = A. Navrotsky| journal = Chem. Mater.| date = 1998|volume = 10| issue = 10|page =2787| doi =10.1021/cm9801901}}</ref>]]
[[File:Perovskite mineral specimen.jpg|thumb|150px|[[Calcium titanate]]. The sample is black owing to impurities, typically Fe.]]
[[File:Perovskite ABO3.jpg|thumb|150px|Structure of oxide ABO<sub>3</sub>. The smaller B ion occupies the center of the "box" with eight A ions at its corners.]]

A '''perovskite''' is a crystalline material of formula ABX<sub>3</sub> with a [[crystal structure]] similar to that of [[Perovskite|the mineral perovskite]], this latter consisting of [[calcium titanium oxide]] (CaTiO<sub>3</sub>).<ref name="Min">{{cite book |title=Minerals: Their Constitution and Origin |first1=Hans-Rudolf |last1=Wenk |first2=Andrei |last2=Bulakh |publisher=Cambridge University Press |date=2004 |isbn=978-0-521-52958-7 |location=New York, NY}}</ref> The mineral was first discovered in the [[Ural Mountains|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 have wide-ranging properties and applications.<ref>{{Cite journal|last=Artini|first=Cristina|date=2017-02-01|title=Crystal chemistry, stability and properties of interlanthanide perovskites: A review|journal=Journal of the European Ceramic Society|language=en|volume=37|issue=2|pages=427–440|doi=10.1016/j.jeurceramsoc.2016.08.041|issn=0955-2219}}</ref>

==Structure==
Perovskite structures are adopted by many [[Chemical compound|compound]]s that have the chemical formula ABX<sub>3</sub>.  'A' and 'B' are positively charged [[ion]]s (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 crystal system|cubic structure]] has the B cation in 6-fold coordination, surrounded by an [[octahedron]] of anions, and the A cation in 12-fold [[cuboctahedron|cuboctahedral]] coordination. Additional perovskite forms may exist where both/either the A and B sites have a configuration of A1<sub>x-1</sub>A2<sub>x</sub> and/or B1<sub>y-1</sub>B2<sub>y</sub> and the X may deviate from the ideal coordination configuration as ions within the A and B sites undergo changes in their oxidation states.<ref>{{cite book| title = Mixed Ionic Electronic Conducting Perovskites for Advanced Energy Systems| editor= N. Orlovskaya, N. Browning| year=2003}}</ref> The idealized form is a cubic structure ([[space group]] Pm{{overline|3}}m, no. 221), which is rarely encountered. The [[orthorhombic]] (e.g. [[space group]] Pnma, no. 62, or Amm2, no. 38) 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 [[Calcium titanate|CaTiO<sub>3</sub>]], this mineral has a non-cubic structure. [[Strontium titanate|SrTiO<sub>3</sub>]] and CaRbF<sub>3</sub> 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.<ref>{{Cite book|doi=10.1002/9780470022184.hmm411|chapter=Crystallography and Chemistry of Perovskites|title=Handbook of Magnetism and Advanced Magnetic Materials|year=2007|last1=Johnsson|first1=Mats|last2=Lemmens|first2=Peter|arxiv=cond-mat/0506606|isbn=978-0470022177|s2cid=96807089}}</ref>

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<sup>+</sup>B<sup>2+</sup>X<sup>−</sup><sub>3</sub>, or 1:2 perovskites;<ref>{{Cite journal|last1=Becker|first1=Markus|last2=Klüner|first2=Thorsten|last3=Wark|first3=Michael|date=2017-03-14|title=Formation of hybrid ABX3 perovskite compounds for solar cell application: first-principles calculations of effective ionic radii and determination of tolerance factors|url=https://pubs.rsc.org/en/content/articlelanding/2017/dt/c6dt04796c|journal=Dalton Transactions|language=en|volume=46|issue=11|pages=3500–3509|doi=10.1039/C6DT04796C|pmid=28239731|issn=1477-9234|url-access=subscription}}</ref> A<sup>2+</sup>B<sup>4+</sup>X<sup>2−</sup><sub>3</sub>, or 2:4 perovskites; A<sup>3+</sup>B<sup>3+</sup>X<sup>2−</sup><sub>3</sub>, or 3:3 perovskites; and A<sup>+</sup>B<sup>5+</sup>X<sup>2−</sup><sub>3</sub>, 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<sub>6</sub> 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<sub>3</sub> 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==
[[image:Rhenium-trioxide-unit-cell-3D-balls-B.png|Rhenium trioxide is a simple example of a defect perovskite: the central atom found in classical perovskites is absent.|thumb|right|144px]]
Also common are the '''defect perovskites'''.  Instead of the ideal ABO<sub>3</sub> 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<sup>−</sup> occupies the O sites, and the large U<sup>3+</sup> 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<sub>2</sub>Cu<sub>3</sub>O<sub>7</sub>.  In this material Y<sup>3+</sup> and Ba<sup>2+</sup>, 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<sub>2</sub>Cu<sub>3</sub>O<sub>7</sub> is a superconductor.  The average oxidation state of copper is Cu<sup>(7/3)+</sup> since Y3+ and Ba2+ have fixed oxidation states.  When heated in the absence of O<sub>2</sub>, the solid loses its superconducting properties, relaxes to the stoichiometry YBa<sub>2</sub>Cu<sub>3</sub>O<sub>6.5</sub>, and all copper sites convert to Cu<sup>2+</sup>.  The material thus is an [[oxygen carrier]], shuttling between two defect perovskites:
:{{chem2|4 YBa2Cu3O7 <-> 4 YBa2Cu3O6.5 + O2}}

== Layered perovskites ==
[[File:Perovskite oxide thin film.jpg|thumb|Atomic resolution [[scanning transmission electron microscopy]] imaging of a perovskite oxide thin film system. Showing a cross section of a [[Lanthanum|La]]<sub>0.7</sub>[[Strontium|Sr]]<sub>0.3</sub>MnO<sub>3</sub> and LaFeO<sub>3</sub> bilayer grown on 111-SrTiO<sub>3</sub>. Overlay: A-cation (green), B-cation (grey) and oxygen (red).]]
Perovskites can be deposited as epitaxial thin films on top of other perovskites,<ref>{{cite journal |last1=Martin |first1=L.W. |last2=Chu |first2=Y.-H. |last3=Ramesh |first3=R. |title=Advances in the growth and characterization of magnetic, ferroelectric, and multiferroic oxide thin films |journal=Materials Science and Engineering: R: Reports |date=May 2010 |volume=68 |issue=4–6 |pages=89–133 |doi=10.1016/j.mser.2010.03.001|s2cid=53337720 |url=http://www.escholarship.org/uc/item/1gm2n89d}}</ref> 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.<ref>{{cite journal |last1=Yang |first1=G.Z |last2=Lu |first2=H.B |last3=Chen |first3=F |last4=Zhao |first4=T |last5=Chen |first5=Z.H |title=Laser molecular beam epitaxy and characterization of perovskite oxide thin films |journal=Journal of Crystal Growth |date=July 2001 |volume=227–228 |issue=1–4 |pages=929–935 |doi=10.1016/S0022-0248(01)00930-7|bibcode=2001JCrGr.227..929Y}}</ref>

Perovskites may be structured in layers, with the {{chem|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:<ref>{{cite web|url=http://www.princeton.edu/~cavalab/tutorials/public/structures/perovskites.html|title=Cava Lab: Perovskites|first=Robert J.|last=Cava|publisher=Princeton University|access-date=13 November 2013}}</ref>
* [[Aurivillius phases|Aurivillius phase]]: the intruding layer is composed of a [{{chem|Bi|2|O|2}}]<sup>2+</sup> ion, occurring every ''n'' {{chem|ABO|3}} layers, leading to an overall chemical formula of [{{chem|Bi|2|O|2}}]-{{chem|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.<ref>{{Cite journal | last1 = Kendall | first1 = K. R. | last2 = Navas | first2 = C. | last3 = Thomas | first3 = J. K. | last4 = Zur Loye | first4 = H. C. | title = Recent Developments in Oxide Ion Conductors: Aurivillius Phases | doi = 10.1021/cm9503083 | journal = Chemistry of Materials | volume = 8 | issue = 3 | pages = 642–649 | year = 1996}}</ref>
* [[Dion−Jacobson phase]]: the intruding layer is composed of an alkali metal (M) every ''n'' {{chem|ABO|3}} layers, giving the overall formula as {{chem|M|+|A|(''n''−1)|B|''n''|O|(3''n''+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 {{chem|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<ref name="Structure 2006">{{cite journal|title=Structure, stability and electrical properties of the La<sub>(2−x)</sub>Sr<sub>x</sub>MnO<sub>4±δ</sub> solid solution series|journal=Solid State Ionics|date=15 October 2006|volume=177|issue=19–25|pages=1849–1853|doi=10.1016/j.ssi.2006.01.009|last1=Munnings|first1=C|last2=Skinner|first2=S|last3=Amow|first3=G|last4=Whitfield|first4=P|last5=Davidson|first5=I|url=https://nrc-publications.canada.ca/eng/view/accepted/?id=f2be87ec-d572-4c66-8e40-fbd22711297b}}</ref> or Sr<ref>{{cite journal|last1=Munnings|first1=Christopher N.|last2=Sayers|first2=Ruth|last3=Stuart|first3=Paul A.|last4=Skinner|first4=Stephen J.|title=Structural transformation and oxidation of Sr<sub>2</sub>MnO<sub>3.5+x</sub> determined by in-situ neutron powder diffraction|journal=Solid State Sciences|date=January 2012|volume=14|issue=1|pages=48–53|doi=10.1016/j.solidstatesciences.2011.10.015|bibcode=2012SSSci..14...48M|hdl=10044/1/15437|url=http://spiral.imperial.ac.uk/bitstream/10044/1/15437/2/Accepted%20vesrion%20SSC_OA.pdf|hdl-access=free}}</ref>) with the B ion being much smaller typically a transition metal (such as Mn,<ref name="Structure 2006"/> Co<ref>{{cite journal|last1=Amow|first1=G.|last2=Whitfield|first2=P. S.|last3=Davidson|first3=I. J.|last4=Hammond|first4=R. P.|last5=Munnings|first5=C. N.|last6=Skinner|first6=S. J.|title=Structural and sintering characteristics of the La<sub>2</sub>Ni<sub>1−x</sub>Co<sub>x</sub>O<sub>4+δ</sub> series|journal=Ceramics International|date=January 2004|volume=30|issue=7|pages=1635–1639|doi=10.1016/j.ceramint.2003.12.164|url=https://nrc-publications.canada.ca/eng/view/accepted/?id=feed36e4-af93-411c-a635-fbfee560f687}}</ref> or Ni<ref>{{cite journal|last1=Amow|first1=G.|last2=Whitfield|first2=P. S.|last3=Davidson|first3=J.|last4=Hammond|first4=R. P.|last5=Munnings|first5=C.|last6=Skinner|first6=S.|title=Structural and Physical Property Trends of the Hyperstoichiometric Series, La<sub>2</sub>Ni<sub>(1−''x'')</sub>Co<sub>x</sub>O<sub>4+δ</sub>|journal=MRS Proceedings|date=11 February 2011|volume=755|doi=10.1557/PROC-755-DD8.10}}</ref>).

== Complex perovskites ==
Although there is a large number of simple known ABX<sub>3</sub> perovskites, this number can be greatly expanded if the A and B sites are increasingly doubled / complex A{{prime|A}}B{{prime|B}}X<sub>6</sub>.<ref name=":2">{{Cite journal |last1=Vasala |first1=Sami |last2=Karppinen |first2=Maarit |date=2015-05-01 |title=A<sub>2</sub>{{prime|B}}B"O<sub>6</sub> perovskites: A review |url=https://www.sciencedirect.com/science/article/pii/S0079678614000338 |journal=Progress in Solid State Chemistry |language=en |volume=43 |issue=1 |pages=1–36 |doi=10.1016/j.progsolidstchem.2014.08.001 |issn=0079-6786|url-access=subscription }}</ref> Ordered [[Perovskite#Double perovskites|double perovskites]] are usually denoted as A<sub>2</sub>B{{prime|B}}O<sub>6</sub> where disordered are denoted as A(B{{prime|B}})O<sub>3</sub>. 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.<ref name=":2" /> The formation of rock-salt superstructures is dependent on the B-site cation ordering.<ref>{{Cite journal |last1=Serrate |first1=D |last2=Teresa |first2=J M De |last3=Ibarra |first3=M R |date=2007-01-17 |title=Double perovskites with ferromagnetism above room temperature |url=https://iopscience.iop.org/article/10.1088/0953-8984/19/2/023201 |journal=Journal of Physics: Condensed Matter |volume=19 |issue=2 |pages=023201 |doi=10.1088/0953-8984/19/2/023201 |s2cid=94885699 |issn=0953-8984|url-access=subscription }}</ref><ref>{{Cite journal |last1=Meneghini |first1=C. |last2=Ray |first2=Sugata |last3=Liscio |first3=F. |last4=Bardelli |first4=F. |last5=Mobilio |first5=S. |last6=Sarma |first6=D. D. |date=2009-07-22 |title=Nature of "Disorder" in the Ordered Double Perovskite Sr<sub>2</sub>FeMoO<sub>6</sub> |url=https://link.aps.org/doi/10.1103/PhysRevLett.103.046403 |journal=Physical Review Letters |volume=103 |issue=4 |pages=046403 |doi=10.1103/PhysRevLett.103.046403|pmid=19659376 |bibcode=2009PhRvL.103d6403M|url-access=subscription }}</ref> Octahedral tilting can occur in double perovskites, however [[Jahn–Teller effect|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<sub>3</sub>, 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 [[Octahedron|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 crystal system|orthorhombic]] or [[Tetragonal crystal system|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<sub>a</sub>, r<sub>b</sub> and r<sub>x</sub>, of the A, B, and X ions.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">Tolerance factor = <math>\frac{(r_a + r_x)}{\sqrt{2}(r_b + r_x)}</math></div>

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..<ref>Xia W, Zhao Y, Zhao F, et al. Antiperovskite Electrolytes for Solid-State Batteries. Chem Rev. 2022;122(3):3763-3819. </ref>

===Examples===
Antiperovskites naturally occur in sulphohalite, galeite, schairerite, [[kogarkoite]], nacaphite, [[arctite]], polyphite, and hatrurite.<ref name=":0">{{Cite journal|last=Krivovichev|first=Sergey|date=2008-01-01|title=Minerals with antiperovskite structure: A review|url=https://www.researchgate.net/publication/244749034|journal=Zeitschrift für Kristallographie |volume=223|issue=1–02|pages=109–113|doi=10.1524/zkri.2008.0008|bibcode=2008ZK....223..109K|s2cid=94097089}}</ref>  It is also demonstrated in [[superconductivity|superconductive]] compounds such as CuNNi<sub>3</sub> and ZnNNi<sub>3</sub>.

Discovered in 1930, metallic antiperovskites have the formula M<sub>3</sub>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]].<ref>{{Cite news |last=Stanier |first=Carol |date=27 Sep 2011 |title=A material for all weathers (with zero thermal expansion) found in anti&shy;perovskite manganese nitrides |url=http://ceramics.org/ceramic-tech-today/a-material-for-all-weathers-with-zero-thermal-expansion-found-in-antiperovskite-manganese-nitrides |access-date=9 May 2011 |work=Ceramic Tech Today |publisher=[[American Ceramic Society]] |location=Westerville, OH}}</ref><ref>{{Cite journal |last1=Takenaka |first1=K. |last2=Takagi |first2=H. |date=2009-03-30 |title=Zero thermal expansion in a pure-form antiperovskite manganese nitride |url=https://pubs.aip.org/apl/article/94/13/131904/118320/Zero-thermal-expansion-in-a-pure-form |journal=Applied Physics Letters |language=en |volume=94 |issue=13 |doi=10.1063/1.3110046 |bibcode=2009ApPhL..94m1904T |issn=0003-6951|url-access=subscription }}</ref>

== Octahedral tilting ==
Beyond the most common perovskite symmetries ([[Cubic crystal system|cubic]], [[Tetragonal crystal system|tetragonal]], [[Orthorhombic crystal system|orthorhombic]]), a more precise determination leads to a total of 23 different structure types that can be found.<ref>{{Cite journal |last=Woodward |first=P. M. |date=1997-02-01 |title=Octahedral Tilting in Perovskites. I. Geometrical Considerations |url=https://scripts.iucr.org/cgi-bin/paper?s0108768196010713 |journal=Acta Crystallographica Section B: Structural Science |language=en |volume=53 |issue=1 |pages=32–43 |doi=10.1107/S0108768196010713 |bibcode=1997AcCrB..53...32W |issn=0108-7681|url-access=subscription }}</ref> These 23 structure can be categorized into 4 different so-called tilt systems that are denoted by their respective Glazer notation.<ref>{{Cite journal |last=Glazer |first=A. M. |date=1972-11-15 |title=The classification of tilted octahedra in perovskites |url=https://scripts.iucr.org/cgi-bin/paper?S0567740872007976 |journal=Acta Crystallographica Section B: Structural Crystallography and Crystal Chemistry |language=en |volume=28 |issue=11 |pages=3384–3392 |doi=10.1107/S0567740872007976 |bibcode=1972AcCrB..28.3384G |issn=0567-7408|url-access=subscription }}</ref>
{| class="wikitable"
|+
!Tilt System
number
!Tilt system symbol
!Space group
|-
| colspan="3" |Three-tilt systems
|-
|1
|a<sup>+</sup>b<sup>+</sup>c<sup>+</sup>
|''Immm'' (#71)
|-
|2
|a<sup>+</sup>b<sup>+</sup>b<sup>+</sup>
|''Immm'' (#71)
|-
|3
|a<sup>+</sup>a<sup>+</sup>a<sup>+</sup>
|''Im''{{overline|3}} (#204)
|-
|4
|a<sup>+</sup>b<sup>+</sup>c<sup>−</sup>
|''Pmmn'' (#59)
|-
|5
|a<sup>+</sup>a<sup>+</sup>c<sup>−</sup>
|''Pmmn'' (#59)
|-
|6
|a<sup>+</sup>b<sup>+</sup>b<sup>−</sup>
|''Pmmn'' (#59)
|-
|7
|a<sup>+</sup>a<sup>+</sup>a<sup>−</sup>
|''Pmmn'' (#59)
|-
|8
|a<sup>+</sup>b<sup>−</sup>c<sup>−</sup>
|''A''2<sub>1</sub>/''m''11 (#11)
|-
|9
|a<sup>+</sup>a<sup>−</sup>c<sup>−</sup>
|''A''2<sub>1</sub>/''m''11 (#11)
|-
|10
|a<sup>+</sup>b<sup>−</sup>b<sup>−</sup>
|''Pmnb'' (#62)
|-
|11
|a<sup>+</sup>a<sup>−</sup>a<sup>−</sup>
|''Pmnb'' (#62)
|-
|12
|a<sup>−</sup>b<sup>−</sup>c<sup>−</sup>
|''F''{{overline|1}} (#2)
|-
|13
|a<sup>−</sup>b<sup>−</sup>b<sup>−</sup>
|''I''2/''a'' (#15)
|-
|14
|a<sup>−</sup>a<sup>−</sup>a<sup>−</sup>
|''R''{{overline|3}}''c'' (#167)
|-
| colspan="3" |Two-tilt systems
|-
|15
|a<sup>0</sup>b<sup>+</sup>c<sup>+</sup>
|''Immm'' (#71)
|-
|16
|a<sup>0</sup>b<sup>+</sup>b<sup>+</sup>
|''I''4/''mmm'' (#139)
|-
|17
|a<sup>0</sup>b<sup>+</sup>c<sup>−</sup>
|''Bmmb'' (#63)
|-
|18
|a<sup>0</sup>b<sup>+</sup>b<sup>−</sup>
|''Bmmb'' (#63)
|-
|19
|a<sup>0</sup>b<sup>−</sup>c<sup>−</sup>
|''F''2/''m''11 (#12)
|-
|29
|a<sup>0</sup>b<sup>−</sup>b<sup>−</sup>
|''Imcm'' (#74)
|-
| colspan="3" |One-tilt systems
|-
|21
|a<sup>0</sup>a<sup>0</sup>c<sup>+</sup>
|''C''4/''mmb'' (#127)
|-
|22
|a<sup>0</sup>a<sup>0</sup>c<sup>−</sup>
|''F''4/''mmc'' (#140)
|-
| colspan="3" |Zero-tilt systems
|-
|23
|a<sup>0</sup>a<sup>0</sup>a<sup>0</sup>
|''Pm''{{overline|3}}''m'' (#221)
|}
[[File:Tilt systems.png|thumb|One-tilt and zero-tilt systems in perovskites]]
The notation consists of a letter a/b/c, which describes the rotation around a [[Cartesian coordinate system|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<sup>0</sup>a<sup>0</sup>a<sup>0</sup>, a<sup>0</sup>a<sup>0</sup>a<sup>–</sup> and a<sup>0</sup>a<sup>0</sup>a<sup>+</sup> which are visualized here.

==Examples==

=== Minerals ===
Aside from [[perovskite]] itself, some perovskite minerals include [[loparite]] and [[bridgmanite]].<ref name="Min"/><ref name=Mindat>[http://www.mindat.org/min-45900.html Bridgemanite] on [[Mindat.org]]</ref>  [[Bridgmanite]] is a silicate with the chemical formula {{chem2|(Mg,Fe)SiO3}}.  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.<ref name="Min" />

At the high pressure conditions of the Earth's [[mantle (geology)|lower mantle]], the [[pyroxene]] [[enstatite]], MgSiO<sub>3</sub>, which otherwise has tetrahedral Si sites, transforms into a denser perovskite-structured [[polymorphism (materials science)|polymorph]]; this phase may be the most common mineral in the Earth.<ref>{{Cite book|title=QI: The Book of General Ignorance|last=John Lloyd|author-link=John Lloyd (producer)|author2=John Mitchinson|author2-link=John Mitchinson (researcher) |chapter=What's the commonest material in the world|isbn=978-0-571-23368-7|publisher=Faber & Faber|title-link=QI|year=2006}}</ref> This phase has the orthorhombically distorted perovskite structure (GdFeO<sub>3</sub>-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, [[silicate perovskite|MgSiO<sub>3</sub> 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<sub>3</sub> are well known. A large family of metallic perovskite compounds can be represented by RT<sub>3</sub>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<sub>3</sub>B, RRh<sub>3</sub>B and CeRu<sub>3</sub>C are examples. MgCNi<sub>3</sub> 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<sub>3</sub>AuO, which contain large alkali cations in the traditional "anion" sites, bonded to O<sup>2−</sup> and Au<sup>−</sup> anions.<ref>{{cite journal |doi=10.1002/anie.199310491 |title=Cs<sub>3</sub>AUO, the First Ternary Oxide with Anionic Gold |date=1993 |last1=Feldmann |first1=Claus |last2=Jansen |first2=Martin |journal=Angewandte Chemie International Edition in English |volume=32 |issue=7 |pages=1049–1050 }}</ref>

===Organic perovskites ===
[[File:Rectangular perovskite crystal.jpg|thumb|150px|[[Methylammonium lead halide|MAPbBr<sub>3</sub>]] crystal]]
[[File:CH3NH3PbI3 structure.png|thumb|Crystal structure of CH<sub>3</sub>NH<sub>3</sub>PbX<sub>3</sub> perovskites (X=I, Br and/or Cl). The methylammonium cation (CH<sub>3</sub>NH<sub>3</sub><sup>+</sup>) is surrounded by PbX<sub>6</sub> octahedra.<ref name=r5>{{cite journal|doi=10.1038/ncomms8497|pmid=26105623|pmc=4491179|title=Ionic transport in hybrid lead iodide perovskite solar cells|journal=Nature Communications|volume=6|page=7497|year=2015|last1=Eames|first1=Christopher|last2=Frost|first2=Jarvist M.|last3=Barnes|first3=Piers R. F.|last4=o'Regan|first4=Brian C.|last5=Walsh|first5=Aron|last6=Islam|first6=M. Saiful|bibcode = 2015NatCo...6.7497E}}</ref>]]
Of interest in the context of solar energy are materials of the type {{chem2|[R4N]+[MX3]-}}. Thus, the [[quat cation]] occupies the B site and the metals occupy the A sites.These materials are the basis of [[perovskite solar cell]]s. These materials have high [[charge carrier]] [[Electron mobility|mobility]] and charge carrier [[Carrier lifetime|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.<ref>{{cite journal |doi=10.1021/acs.chemrev.8b00477 |title=Tuning the Luminescence of Layered Halide Perovskites |date=2019 |last1=Smith |first1=Matthew D. |last2=Connor |first2=Bridget A. |last3=Karunadasa |first3=Hemamala I. |journal=Chemical Reviews |volume=119 |issue=5 |pages=3104–3139 |osti=1528780 }}</ref><ref>{{cite journal |doi=10.1021/acs.chemrev.9b00107 |title=Perovskites for Next-Generation Optical Sources |date=2019 |last1=Quan |first1=Li Na |last2=Rand |first2=Barry P. |last3=Friend |first3=Richard H. |last4=Mhaisalkar |first4=Subodh Gautam |last5=Lee |first5=Tae-Woo |last6=Sargent |first6=Edward H. |journal=Chemical Reviews |volume=119 |issue=12 |pages=7444–7477 |hdl=10356/140305 |hdl-access=free }}</ref><ref>{{cite journal |doi=10.1021/acs.chemrev.3c00214 |title=Synergy of 3D and 2D Perovskites for Durable, Efficient Solar Cells and Beyond |date=2023 |last1=Metcalf |first1=Isaac |last2=Sidhik |first2=Siraj |last3=Zhang |first3=Hao |last4=Agrawal |first4=Ayush |last5=Persaud |first5=Jessica |last6=Hou |first6=Jin |last7=Even |first7=Jacky |last8=Mohite |first8=Aditya D. |journal=Chemical Reviews |volume=123 |issue=15 |pages=9565–9652 |pmid=37428563 |url=https://hal.science/hal-04160913 }}</ref>

== 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.<ref>{{cite journal|title= Laser action in LaAlO<sub>3</sub>:Nd<sup>3+</sup> single crystal |journal=Journal of Applied Physics|volume=103|issue=4|pages=043102–043102–8|doi=10.1063/1.2842399|year=2008|last1=Dereń|first1=P. J.|last2=Bednarkiewicz|first2=A.|last3=Goldner|first3=Ph.|last4=Guillot-Noël|first4=O.|bibcode = 2008JAP...103d3102D}}</ref><ref>Wallace, John (28 March 2014) [http://www.laserfocusworld.com/articles/2014/03/high-efficiency-perovskite-photovoltaic-material-also-lases.html High-efficiency perovskite photovoltaic material also lases]. ''LaserFocusWorld''</ref><ref name=k1403>{{cite web |url=http://www.rdmag.com/news/2014/03/study-perovskite-solar-cells-can-double-lasers |title=Study: Perovskite solar cells can double as lasers |publisher=Rdmag.com |date=2014-03-28 |access-date=2014-08-24}}</ref> 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 [[Electronics|electronic]] factors ([[Jahn–Teller effect|Jahn-Teller distortions]]).<ref name=Lufaso>{{cite journal|doi=10.1107/S0108768103026661|pmid=14734840|title=Jahn–Teller distortions, cation ordering and octahedral tilting in perovskites|date=2004|last1=Lufaso|first1=Michael W.|last2=Woodward|first2=Patrick M.|journal=Acta Crystallographica Section B|volume=60|issue=Pt 1|pages=10–20|bibcode=2004AcCrB..60...10L |url=https://digitalcommons.unf.edu/cgi/viewcontent.cgi?article=1002&context=achm_facpub|url-access=subscription}}</ref> The financially biggest application of perovskites is in [[ceramic capacitor]]s, in which BaTiO<sub>3</sub> is used because of its high dielectric constant.<ref>{{Cite web |title=Capacitor Market Size, Share, Scope, Trends, Opportunities & Forecast |url=https://www.verifiedmarketresearch.com/product/capacitor-market/ |access-date=2022-12-15 |website=Verified Market Research |language=en-US}}</ref><ref>{{Cite journal |last=Merz |first=Walter J. |date=1949-10-15 |title=<nowiki>The Electric and Optical Behavior of BaTi${\mathrm{O}}_{3}$ Single-Domain Crystals</nowiki> |url=https://link.aps.org/doi/10.1103/PhysRev.76.1221 |journal=Physical Review |volume=76 |issue=8 |pages=1221–1225 |doi=10.1103/PhysRev.76.1221|url-access=subscription }}</ref> [[Light-emitting diodes]] exploit the high [[photoluminescence]] [[quantum efficiency|quantum efficiencies]] of perovskites.<ref>{{cite journal |last1=Wang |first1=Heyong |last2=Kosasih |first2=Felix Utama |last3=Yu |first3=Hongling |last4=Zheng |first4=Guanhaojie |last5=Zhang |first5=Jiangbin |last6=Pozina |first6=Galia |last7=Liu |first7=Yang |last8=Bao |first8=Chunxiong |last9=Hu |first9=Zhangjun |last10=Liu |first10=Xianjie |last11=Kobera |first11=Libor |last12=Abbrent |first12=Sabina |last13=Brus |first13=Jiri |last14=Jin |first14=Yizheng |last15=Fahlman |first15=Mats |last16=Friend |first16=Richard H. |last17=Ducati |first17=Caterina |last18=Liu |first18=Xiao-Ke |last19=Gao |first19=Feng |title=Perovskite-molecule composite thin films for efficient and stable light-emitting diodes |journal=Nature Communications |date=December 2020 |volume=11 |issue=1 |pages=891 |doi=10.1038/s41467-020-14747-6|pmid=32060279 |pmc=7021679 |bibcode=2020NatCo..11..891W |doi-access=free}}</ref><ref>{{cite journal |last1=Andaji-Garmaroudi |first1=Zahra |last2=Abdi-Jalebi |first2=Mojtaba |last3=Kosasih |first3=Felix U. |last4=Doherty |first4=Tiarnan |last5=Macpherson |first5=Stuart |last6=Bowman |first6=Alan R. |last7=Man |first7=Gabriel J. |last8=Cappel |first8=Ute B. |last9=Rensmo |first9=Håkan |last10=Ducati |first10=Caterina |last11=Friend |first11=Richard H. |last12=Stranks |first12=Samuel D. |title=Elucidating and Mitigating Degradation Processes in Perovskite Light-Emitting Diodes |journal=Advanced Energy Materials |date=December 2020 |volume=10 |issue=48 |pages=2002676 |doi=10.1002/aenm.202002676|bibcode=2020AdEnM..1002676A |s2cid=228806435 |url=https://www.repository.cam.ac.uk/handle/1810/312192}}</ref> In the area of photoelectrolysis, water electrolysis at 12.3% efficiency can use perovskite photovoltaics.<ref name=Science-Luo>{{cite journal|author1=Jingshan Luo|display-authors=etal|title=Water photolysis at 12.3% efficiency via perovskite photovoltaics and Earth-abundant catalysts|journal=Science|date=26 September 2014|volume=345|issue=6204|pages=1593–1596|doi=10.1126/science.1258307|pmid=25258076|bibcode = 2014Sci...345.1593L |s2cid=24613846}}<!--|access-date=26 September 2014--></ref><ref name=PO-harvest>{{cite news|title=Harvesting hydrogen fuel from the Sun using Earth-abundant materials|url=http://phys.org/news/2014-09-harvesting-hydrogen-fuel-sun-earth-abundant.html|access-date=26 September 2014|publisher=Phys.org|date=Sep 25, 2014}}</ref> Scintillators based on cerium-doped lutetium aluminum perovskite (LuAP:Ce) single crystals were reported.<ref name="QChen2018">{{cite journal|last1=Chen|first1=Quishui|title=All-inorganic perovskite nanocrystal scintillators|journal=Nature|date=27 August 2018|volume=561|issue=7721|pages=88–93|doi=10.1038/s41586-018-0451-1|pmid=30150772|bibcode=2018Natur.561...88C|s2cid=52096794}}</ref> 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.<ref name=":0:">{{Cite journal|last1=Xie|first1=Aozhen|last2=Maddalena|first2=Francesco|last3=Witkowski|first3=Marcin E.|last4=Makowski|first4=Michal|last5=Mahler|first5=Benoit|last6=Drozdowski|first6=Winicjusz|last7=Springham|first7=Stuart Victor|last8=Coquet|first8=Philippe|last9=Dujardin|first9=Christophe|last10=Birowosuto|first10=Muhammad Danang|last11=Dang|first11=Cuong|date=2020-10-13|title=Library of Two-Dimensional Hybrid Lead Halide Perovskite Scintillator Crystals|url=https://doi.org/10.1021/acs.chemmater.0c02789|journal=Chemistry of Materials|volume=32|issue=19|pages=8530–8539|doi=10.1021/acs.chemmater.0c02789|s2cid=224916409|issn=0897-4756|url-access=subscription}}</ref><ref name=":1">{{Cite journal|last1=Maddalena|first1=Francesco|last2=Xie|first2=Aozhen|last3=Arramel|last4=Witkowski|first4=Marcin E.|last5=Makowski|first5=Michal|last6=Mahler|first6=Benoit|last7=Drozdowski|first7=Winicjusz|last8=Mariyappan|first8=Thambidurai|last9=Springham|first9=Stuart Victor|last10=Coquet|first10=Philippe|last11=Dujardin|first11=Christophe|date=2021-03-01|title=Effect of commensurate lithium doping on the scintillation of two-dimensional perovskite crystals|url=https://pubs.rsc.org/en/content/articlelanding/2021/tc/d0tc05647b|journal=Journal of Materials Chemistry C|language=en|volume=9|issue=7|pages=2504–2512|doi=10.1039/D0TC05647B|s2cid=233789445|issn=2050-7534|url-access=subscription}}</ref> In addition this class of materials have shown capability for wide-range particle detection, including [[alpha particle]]s and thermal [[neutron]]s.<ref>{{Cite journal|last1=Xie|first1=Aozhen|last2=Hettiarachchi|first2=Chathuranga|last3=Maddalena|first3=Francesco|last4=Witkowski|first4=Marcin E.|last5=Makowski|first5=Michał|last6=Drozdowski|first6=Winicjusz|last7=Arramel|first7=Arramel|last8=Wee|first8=Andrew T. S.|last9=Springham|first9=Stuart Victor|last10=Vuong|first10=Phan Quoc|last11=Kim|first11=Hong Joo|date=2020-06-24|title=Lithium-doped two-dimensional perovskite scintillator for wide-range radiation detection|journal=Communications Materials|language=en|volume=1|issue=1|page=37|doi=10.1038/s43246-020-0038-x|bibcode=2020CoMat...1...37X|issn=2662-4443|doi-access=free|hdl=10356/164062|hdl-access=free}}</ref>

==See also==
* [[Antiperovskite (structure)|Antiperovskite]]
* [[Aurivillius phases]]
* [[Diamond anvil]]
* [[Goldschmidt tolerance factor]]
* [[Ruddlesden-Popper phase]]
* [[Spinel]]

==References==
{{Reflist}}

==Further reading==
* {{Cite book|last = Tejuca|first = Luis G|title = Properties and applications of perovskite-type oxides|publisher = Dekker|date = 1993|location = New York|pages = 382|isbn = 978-0-8247-8786-8}}
* {{Cite book|last = Mitchell|first = Roger H|title = Perovskites modern and ancient|publisher = Almaz Press|date = 2002|location = Thunder Bay, Ontario|pages = 318|isbn = 978-0-9689411-0-2}}
* [http://pubs.acs.org/doi/abs/10.1021/ja305709z Superionic Conductivity in Lithium-Rich Anti-Perovskites]
* [http://www.hindawi.com/journals/acmp/2013/214120/ Lattice and Magnetic and Electronic Transport Properties in Antiperovskite Compounds]

==External links==
* {{cite web | url=http://cst-www.nrl.navy.mil/lattice/struk/e2_1.html | title=Cubic Perovskite Structure | work=Center for Computational Materials Science | publisher=[[United States Naval Research Laboratory|U.S. Naval Research Laboratory]] | url-status=dead | archive-url=https://web.archive.org/web/20081008092209/http://cst-www.nrl.navy.mil/lattice/struk/e2_1.html | archive-date=2008-10-08}} (includes a [https://web.archive.org/web/20080412093413/http://cst-www.nrl.navy.mil/lattice/struk.jmol/e2_1.html Java applet] with which the structure can be interactively rotated)
* [https://catalogmineralov.ru/mineral/perovskite.html Перовскит в Каталоге Минералов]

{{Titanium minerals}}
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{{DEFAULTSORT:Perovskite (Structure)}}
[[Category:Mineralogy]]
[[Category:Solar power]]
[[Category:Perovskites|*]]
[[Category:Crystal structure types]]
[[Category:Crystallography]]
[[Category:Materials science]]


[[de:Perowskit#Kristallstruktur]]