# Perovskite (structure)

The perovskite structure is a crystal structure of general formula ABX3 in which corner-sharing BX6 octahedra form a three-dimensional framework and the larger A cations occupy the twelve-coordinate cavities between them. The name comes from the mineral [calcium titanate](https://www.edgepedia.org/calcium-titanate) (CaTiO3), reported in 1839 by the German mineralogist Gustav Rose and named after the Russian mineralogist Count Lev Aleksevich von Perovski; any compound adopting the same structure type is called a perovskite.<sup>[1](https://cronfa.swan.ac.uk/Record/cronfa55557/Download/55557__18532__44566c04dfc74b4287634e09f6b6c143.pdf)</sup><sup> • </sup><sup>[2](https://time.com/87655/scientists-just-got-the-lead-out-of-solar-cells-with-tin/)</sup>

| Key fact | Detail |
|---|---|
| Ideal formula and symmetry | ABX3, cubic space group \( Pm\bar{3}m \); A cations on Wyckoff 1a, B cations on 1b, X anions on 3c<sup>[3](https://journals.iucr.org/a/issues/2019/02/00/kx5068/index.html)</sup> |
| Building block | Corner-linked BX6 octahedra (X anions at vertices, B cation at center), separated by A cations in cubo-octahedral voids<sup>[3](https://journals.iucr.org/a/issues/2019/02/00/kx5068/index.html)</sup> |
| Tolerance factor | \( t = (r_A + r_X)/(\sqrt{2}(r_B + r_X)) \), proposed by Goldschmidt in 1926; perovskite phases generally preferred for 0.8 < t < 1.0<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC6368436/)</sup><sup> • </sup><sup>[5](https://advanced.onlinelibrary.wiley.com/doi/full/10.1002/advs.202305799)</sup> |
| Octahedral factor | \( \mu = r_B/r_X \); halide perovskite formation reported for \( \mu > 0.442 \)<sup>[6](https://doi.org/10.1016/J.NANTOD.2015.04.009)</sup> |
| Tilt systems | Glazer (1972) classified octahedral tilts; Howard and Stokes (1998) confirmed 15 possible simple tilt systems by group theory<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC8256700/)</sup> |
| Layered variants | Ruddlesden–Popper \( A_{n+1}B_nO_{3n+1} \), Dion–Jacobson, and alternating-cation-in-interlayer (ACI) phases<sup>[8](https://www.sciencedirect.com/org/science/article/pii/S2052252523000489)</sup><sup> • </sup><sup>[9](https://onlinelibrary.wiley.com/doi/full/10.1002/aenm.202003877)</sup> |
| Double perovskites | \( A_2BB'X_6 \) with random, rock-salt, or layered B-cation ordering<sup>[10](https://www.sciencedirect.com/science/article/pii/007967869390004B)</sup> |

## Ideal structure

In the ideal cubic perovskite, the BX6 octahedra are connected by shared vertices in all three directions, and the A cations sit in the cubo-octahedral voids of this network. The A and B cation positions in the unit cell are symmetrically equivalent, connected by an origin shift of (½ ½ ½), so the structure can be described with either cation at the cell center.<sup>[3](https://journals.iucr.org/a/issues/2019/02/00/kx5068/index.html)</sup> The same framework can also be viewed as a ReO3-type network of corner-sharing octahedra into which the A cation is inserted.<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC6016328/)</sup>

The structure tolerates a vast range of elements on the A, B, and X sites, which is the basis of its wide functionality: oxides, halides, and hybrid organic–inorganic compositions all crystallize in this family.<sup>[12](https://www.annualreviews.org/content/journals/10.1146/annurev-matsci-092320-102133)</sup> In halide perovskites the B site is typically a divalent cation such as Pb2+, Sn2+, or Ge2+, and the three-dimensional framework accommodates only relatively small A-site cations, limited in practice to Cs+, methylammonium (MA), or formamidinium (FA).<sup>[13](https://pubs.rsc.org/en/content/articlehtml/2019/qi/c9qi00777f)</sup><sup> • </sup><sup>[8](https://www.sciencedirect.com/org/science/article/pii/S2052252523000489)</sup>

## Tolerance factor and structural stability

**Geometric prediction.** Victor Moritz Goldschmidt introduced the tolerance factor in 1926 as a dimensionless ratio of ionic radii that measures whether the A cation fits the cavity of the BX6 octahedral network:<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC6368436/)</sup>

\[ t = \frac{r_A + r_X}{\sqrt{2}\,(r_B + r_X)} \]

[A value](https://www.edgepedia.org/a-value) of t = 1 corresponds to a perfectly fitting, cubic structure. Values slightly below 1 predict that the octahedral network distorts, typically by tilting, to accommodate a smaller A cation, and the perovskite structure is generally preferred for 0.8 < t < 1.0.<sup>[5](https://advanced.onlinelibrary.wiley.com/doi/full/10.1002/advs.202305799)</sup><sup> • </sup><sup>[14](https://pubs.rsc.org/en/content/articlehtml/2024/mh/d4mh00689e)</sup> A second descriptor, the octahedral factor \( \mu = r_B/r_X \), assesses whether the B cation fits the octahedral hole; for halides, formation has been reported for \( \mu > 0.442 \).<sup>[6](https://doi.org/10.1016/J.NANTOD.2015.04.009)</sup>

**Limits of the factor.** The tolerance factor is an empirical descriptor of average geometric compatibility, not a deterministic predictor. Its hard-sphere model assumes fixed ionic radii and idealized coordination, and it does not account for bond covalency, hydrogen bonding, temperature, or synthesis history; a given t does not necessarily correspond to a unique stable phase.<sup>[15](https://link.springer.com/article/10.1007/s44503-026-00017-x)</sup> For inorganic iodides, a tolerance factor within a range calculated from Shannon radii is a necessary condition for perovskite formation but not a sufficient one.<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC6016328/)</sup> Newer descriptors improve on it: a tolerance factor \( \tau \) developed with the [SISSO](https://www.edgepedia.org/sisso) machine-learning method correctly classified 92% of an experimental set of 576 ABX3 compounds as perovskite or non-perovskite, and generalized with 91% accuracy to 1,034 realized single and double perovskites.<sup>[16](https://arxiv.org/abs/1801.07700)</sup>

## Octahedral tilting

The most common distortion of the ideal structure is tilting of the BX6 octahedra, driven by a size mismatch between the A and B cations as measured by the tolerance factor.<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC8256700/)</sup> A. M. Glazer of the [University of Oxford](https://www.edgepedia.org/university-of-oxford) classified the simple tilt patterns in 1972 using a three-letter notation in which each letter gives the relative tilt magnitude along a principal axis and a superscript +, −, or 0 marks in-phase tilting, out-of-phase tilting, or no tilt; for example, a<sup>0</sup>a<sup>0</sup>c<sup>−</sup> denotes out-of-phase rotation about the c axis only.<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC8256700/)</sup><sup> • </sup><sup>[17](https://journals.iucr.org/b/issues/2022/01/00/ra5104/index.html)</sup> Howard and Stokes later confirmed by group-theoretical analysis that 15 simple tilt systems are possible, and established the group–subgroup hierarchy descending from the cubic \( Pm\bar{3}m \) aristotype.<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC8256700/)</sup><sup> • </sup><sup>[17](https://journals.iucr.org/b/issues/2022/01/00/ra5104/index.html)</sup> Tilting lowers the symmetry to tetragonal, orthorhombic, rhombohedral, and other phases, and in halide perovskites it correlates with the bandgap and carrier dynamics.<sup>[18](https://www.nature.com/articles/s41570-025-00687-6)</sup>

## Layered and double perovskites

**Ruddlesden–Popper phases.** Slicing the 3D structure into perovskite slabs and restacking them with extra cation layers gives the layered Ruddlesden–Popper (RP) phases, with oxide formula \( A_{n+1}B_nO_{3n+1} \) and hybrid halide formula \( (A')_2(A)_{n-1}B_nX_{3n+1} \), where n counts the connected octahedral layers and n = ∞ recovers the 3D structure.<sup>[8](https://www.sciencedirect.com/org/science/article/pii/S2052252523000489)</sup><sup> • </sup><sup>[19](https://pubs.rsc.org/en/content/getauthorversionpdf/c9cs00254e)</sup> In RP phases successive layers are offset by a/2 + b/2, so octahedra in one layer sit over the cavities of the next; in the related Dion–Jacobson (DJ) family, built on diammonium spacers, the layers stack directly on top of one another. A third family, the alternating-cation-in-the-interlayer (ACI) phases, has formula \( A'A_nB_nX_{3n+1} \).<sup>[8](https://www.sciencedirect.com/org/science/article/pii/S2052252523000489)</sup><sup> • </sup><sup>[9](https://onlinelibrary.wiley.com/doi/full/10.1002/aenm.202003877)</sup>

**Double perovskites.** Ordering two different cations on the B site gives double perovskites of formula \( A_2BB'X_6 \). A survey of more than 300 such compounds found three distinct B-cation arrangements: random, rock-salt, and layered.<sup>[10](https://www.sciencedirect.com/science/article/pii/007967869390004B)</sup>

## Applications

The structure family underpins several device technologies. Three-dimensional halide perovskites such as MAPbI3 are prominent photovoltaic absorbers, but their instability to moisture, light, and heat limits commercialization; the layered 2D RP-type halide perovskites formed by bulky organic ammonium spacers show much greater environmental stability and behave as natural quantum wells with much larger exciton binding energies than their 3D analogs.<sup>[20](https://pubmed.ncbi.nlm.nih.gov/29028138/)</sup> Layered perovskites reach exciton binding energies up to hundreds of meV, against roughly 3 meV for low-bandgap tin materials and tens of meV for MAPbI3 in 3D structures, so excitons rather than free carriers dominate at room temperature in many layered compositions.<sup>[9](https://onlinelibrary.wiley.com/doi/full/10.1002/aenm.202003877)</sup> Mixed-dimensional films that combine 3D and 2D phases are used to raise both efficiency and stability in solar cells.<sup>[21](https://www.nature.com/articles/s41578-018-0065-0)</sup> Oxide perovskites of the same structure type serve as functional ceramics, with properties including ferroelectricity, magnetism, and catalysis arising from the same octahedral framework and its distortions.<sup>[12](https://www.annualreviews.org/content/journals/10.1146/annurev-matsci-092320-102133)</sup>

## References

1. [Perovskite solar cells: introduction and background, University of Bath thesis](https://cronfa.swan.ac.uk/Record/cronfa55557/Download/55557__18532__44566c04dfc74b4287634e09f6b6c143.pdf)
2. [Scientists Just Got the Lead Out of Solar Cells — with Tin, Time](https://time.com/87655/scientists-just-got-the-lead-out-of-solar-cells-with-tin/)
3. [Group-theoretical analysis of 1:3 A-site-ordered perovskite formation, Acta Crystallographica A](https://journals.iucr.org/a/issues/2019/02/00/kx5068/index.html)
4. [New tolerance factor to predict the stability of perovskite oxides and halides, Science Advances (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC6368436/)
5. [Understanding defects in perovskite solar cells through computation, Advanced Science](https://advanced.onlinelibrary.wiley.com/doi/full/10.1002/advs.202305799)
6. [Under the spotlight: the organic–inorganic hybrid halide perovskite, Nano Today](https://doi.org/10.1016/J.NANTOD.2015.04.009)
7. [Structural chemistry of layered lead halide perovskites containing single octahedral layers, IUCrJ (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC8256700/)
8. [Understanding structural distortions in hybrid layered perovskites with the n = 1 Ruddlesden–Popper structure, IUCrJ](https://www.sciencedirect.com/org/science/article/pii/S2052252523000489)
9. [Layered perovskites in solar cells: structure, optoelectronic properties, and device design, Advanced Energy Materials](https://onlinelibrary.wiley.com/doi/full/10.1002/aenm.202003877)
10. [B-cation arrangements in double perovskites, Progress in Solid State Chemistry](https://www.sciencedirect.com/science/article/pii/007967869390004B)
11. [On the application of the tolerance factor to inorganic and hybrid halide perovskites: a revised system, CrystEngComm (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC6016328/)
12. [Layered double perovskites, Annual Review of Materials Research](https://www.annualreviews.org/content/journals/10.1146/annurev-matsci-092320-102133)
13. [Luminescent perovskites: recent advances in theory and experiments, Inorganic Chemistry Frontiers](https://pubs.rsc.org/en/content/articlehtml/2019/qi/c9qi00777f)
14. [Rethinking tolerance factor analysis for chalcogenide perovskites, Materials Horizons](https://pubs.rsc.org/en/content/articlehtml/2024/mh/d4mh00689e)
15. [Learning from oxide perovskites for defect engineering in halide perovskite photovoltaics, Springer](https://link.springer.com/article/10.1007/s44503-026-00017-x)
16. [New tolerance factor to predict the stability of perovskite oxides and halides, arXiv](https://arxiv.org/abs/1801.07700)
17. [Phase transitions and (p–T–X) behaviour of centrosymmetric perovskites, Acta Crystallographica B](https://journals.iucr.org/b/issues/2022/01/00/ra5104/index.html)
18. [Octahedral units in halide perovskites, Nature Reviews Chemistry](https://www.nature.com/articles/s41570-025-00687-6)
19. [Two-dimensional Ruddlesden–Popper perovskite solar cells, Chemical Society Reviews](https://pubs.rsc.org/en/content/getauthorversionpdf/c9cs00254e)
20. [2D Ruddlesden–Popper perovskites for optoelectronics, Advanced Materials (PubMed)](https://pubmed.ncbi.nlm.nih.gov/29028138/)
21. [Dimensional tailoring of hybrid perovskites for photovoltaics, Nature Reviews Materials](https://www.nature.com/articles/s41578-018-0065-0)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Condensed matter physics › Crystal and structural condensed matter › Crystal lattices and symmetry › Crystal structure types*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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