Perovskite structure (ABX₃): octahedra, tolerance factor and tilts

Every perovskite has the formula ABX₃. The smaller B cation sits at the centre of an octahedron of six X anions, and the octahedra share all six corners with their neighbours to make a three-dimensional framework. The larger A cation fills the cavity between eight octahedra, where it has twelve X neighbours (Bartel et al. 2019).

In the ideal cubic cell (space group Pm-3m, as in cubic BaTiO₃; Swaminathan et al. 2010) B sits at the body centre, the X anions at the face centres and A at the corners — or, choosing the origin differently, B at the corners and A at the centre. With one formula unit per cell, the B–X distance is half the cell edge and the A–X distance is a/√2 (geometry of the ideal cubic cell). In SrTiO₃, with a = 3.905 Å (Leroy et al. 2025), that gives Ti–O = 1.95 Å and Sr–O = 2.76 Å (calculated from a).

The X anion depends on the family: oxygen in oxide perovskites (ferroelectrics, catalysts, proton conductors), fluoride in some battery materials, and Cl, Br or I in the halide perovskites used in solar cells and light emitters (Bartel et al. 2019).

Whether a composition will form a perovskite has been estimated for over 90 years with the Goldschmidt tolerance factor, t, which needs only the chemical formula and the ionic radii of A, B and X (Bartel et al. 2019). It is a ratio that compares the A–X distance with the B–X distance, each taken as a sum of ionic radii. In the ideal cube A–X is √2 times B–X, so the ratio is scaled to give t = 1 when the ions fit the cubic cell exactly (from the ideal cubic geometry). A size mismatch between the ions distorts the cubic structure or leads to non-perovskite structures instead (Bartel et al. 2019).

Tested on 576 experimentally known ABX₃ solids, the range that best separates perovskites from non-perovskites is 0.825 < t < 1.059, but it is right only 74% of the time: 83% for oxides and fluorides, and much worse for chlorides (51%), bromides (56%) and iodides (33%) (Bartel et al. 2019).

The second classic test is the octahedral factor, μ = r_B / r_X, which asks whether B is large enough to hold six X ions in an octahedron; geometry gives 0.414 < μ < 0.732. Combining t and μ raises the accuracy to about 85%, and a newer one-number descriptor, τ, built from the same radii and oxidation states, reaches 92% across oxides and halides (Bartel et al. 2019).

Fewer than 10% of known perovskites are actually cubic. Size mismatch between the ions distorts the structure, and more than a dozen distinct structures, all built from corner-sharing BX₆ octahedra, count as perovskites. Even CaTiO₃, the mineral that gave the family its name, is orthorhombic (Bartel et al. 2019).

One common distortion is a cooperative tilt (rotation) of the octahedra. Tilt patterns are written in Glazer notation, one letter and superscript per cubic axis: cubic SrTiO₃, with no rotation, is a⁰a⁰a⁰, and below about 105 K its TiO₆ octahedra rotate about one axis, out of phase between neighbours, which is written a⁰a⁰c⁻; the structure becomes tetragonal and the primitive cell doubles (Leroy et al. 2025).

A second kind of distortion is a polar shift of the B cation off the centre of its octahedron, as in ferroelectric BaTiO₃, where titanium is displaced along the c axis in the tetragonal phase (Swaminathan et al. 2010).

• Oxide perovskites (ABO₃). X = O²⁻, so the charges of A and B add up to +6, for example Sr²⁺Ti⁴⁺O₃ or La³⁺Al³⁺O₃. They are used as ferroelectrics, electrocatalysts and proton conductors (Bartel et al. 2019), capacitor dielectrics (Dai et al. 2022) and substrates for oxide films (Kawasaki et al. 1994). The interface between two insulating oxide perovskites, LaAlO₃ and SrTiO₃, can even conduct (Ohtomo & Hwang 2004).
• Halide perovskites (ABX₃, X = Cl, Br, I). X is a halide ion, so A is +1 and B is +2: Cs⁺ or an organic cation such as methylammonium (CH₃NH₃⁺, MA) or formamidinium (HC(NH₂)₂⁺, FA) on A, and Pb²⁺ or Sn²⁺ on B. The tin and lead iodides are direct-gap semiconductors with gaps of 1.25–1.75 eV (Stoumpos et al. 2013). The tolerance factor predicts halide perovskites much less reliably than oxides (Bartel et al. 2019).

• SrTiO₃ (strontium titanate). The textbook cubic perovskite at room temperature (a = 3.905 Å), with an antiferrodistortive transition near 105 K and quantum paraelectric behaviour at low temperature (Leroy et al. 2025). Its atomically flat, TiO₂-terminated surfaces make a well-defined substrate for oxide films (Kawasaki et al. 1994), and it hosts a high-mobility electron gas at its interface with LaAlO₃ (Ohtomo & Hwang 2004).
• BaTiO₃ (barium titanate). The classic ferroelectric: cubic above a Curie point near 120 °C, then tetragonal, orthorhombic and rhombohedral on cooling (Tan et al. 2015; Zhang et al. 2006). It is the main raw material of multilayer ceramic capacitors (Dai et al. 2022).
• CsPbBr₃ (caesium lead bromide). An all-inorganic halide perovskite. Colloidal CsPbX₃ nanocrystals emit across 410–700 nm with narrow lines and quantum yields of up to 90% (Protesescu et al. 2015).
• MAPbI₃ (methylammonium lead iodide). The hybrid perovskite that started perovskite photovoltaics: 3.8% efficiency as a dye-cell sensitizer in 2009 (Kojima et al. 2009) and 10.9% in a solid-state cell in 2012, with a 1.55 eV band gap (Lee et al. 2012).
• FAPbI₃ (formamidinium lead iodide). A narrower band gap than MAPbI₃ but a less stable perovskite phase; alloying with MAPbBr₃ stabilized it and gave solar cells above 18% (Jeon et al. 2015).
• 2D (layered) perovskites. Slabs of corner-sharing octahedra separated by layers of large organic cations, such as (PEA)₂(MA)₂Pb₃I₁₀, which is more moisture resistant than MAPbI₃ (Smith et al. 2014).

In layered (2D) hybrid perovskites the three-dimensional framework is cut into slabs of corner-sharing octahedra separated by layers of large organic cations. These lower-dimensional organic–inorganic perovskites, known since the late 19th century, offer great chemical flexibility (Saparov & Mitzi 2016). In solar cells, (PEA)₂(MA)₂Pb₃I₁₀ (PEA = C₆H₅(CH₂)₂NH₃⁺) gave 4.73% efficiency with a larger band gap than MAPbI₃ and films that could be made under ambient humidity (Smith et al. 2014).

• Capacitors. BaTiO₃ is the main raw material of multilayer ceramic capacitors (Dai et al. 2022).
• Piezoelectrics. BaTiO₃-based ceramics as lead-free piezoelectrics (Tan et al. 2015; Liu & Ren 2009).
• Solar cells. Lead halide perovskite absorbers, from 3.8% in 2009 to more than 18% by 2015 (Kojima et al. 2009; Jeon et al. 2015).
• Light emission. CsPbX₃ nanocrystals with quantum yields up to 90% and colour covering up to 140% of the NTSC gamut (Protesescu et al. 2015).
• Photocatalysis. Al-doped SrTiO₃ splits water with an external quantum efficiency of up to 96% at 350–360 nm (Takata et al. 2020).
• Oxide electronics. Conducting interfaces between insulating perovskites (Ohtomo & Hwang 2004).
• Catalysis, proton conduction and batteries. Oxide and fluoride perovskites (Bartel et al. 2019).

For a paper, slide or poster, the clearest picture is the cubic cell drawn as polyhedra: one BX₆ octahedron at the centre, or eight octahedra sharing corners with the A cation in the cavity between them. Show the A cation as a large sphere and X at the octahedron corners; to show tilts, rotate neighbouring octahedra in opposite senses (a⁰a⁰c⁻). In LabCanvas you can draw the cell in the figure editor, show the coordination polyhedra, repeat the cell to show the corner-sharing network, recolour the A, B and X sites for an oxide or a halide perovskite, and export a high-resolution image.