Perovskite structure (ABX₃)

The perovskite $AB{X}_3$ aristotype is the corner-sharing octahedral framework that organizes a huge slice of functional-materials physics: ferroelectricity, the manganite/cuprate oxides, and the halide photovoltaics. The ideal cell is the cubic prototype $Pm\bar{3}m$ ($O_h^1$, No. 221), a network of $B{X}_6$ octahedra sharing every corner, with the larger $A$ cation filling the 12-coordinate cuboctahedral cavity between them. Almost all the interesting physics lives in the distortions of this prototype — octahedral tilting, polar off-centering, Jahn–Teller elongation — and the recurring lattice-dynamics theme is that the high-symmetry cubic phase is often dynamically unstable at $0\mathrm{K}$, stabilized into existence only by anharmonicity and temperature.

Ideal cubic structure and stoichiometry

In the prototype the $B$ cation sits at the cell origin $(0,0,0)$ (Wyckoff $1a$), the $A$ cation at the body center $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ ($1b$), and the three $X$ anions at the face centers $(\frac{1}{2},0,0)$ and permutations ($3c$), midway along each $B$–$B$ edge. Each $B$ is octahedrally coordinated by six $X$; each octahedron shares all six corners with neighbors, so the framework is $\frac{1}{1}[B{X}_{6/2}]=B{X}_3$, and the $A$ cation occupies the resulting cuboctahedral $A{X}_{12}$ cavity. Equivalently the structure is a cubic-close-packed array of $A{X}_3$ layers with $B$ in one quarter of the octahedral holes (those coordinated entirely by $X$).

Charge neutrality $q_A+q_B+3q_X=0$ admits several families. For oxides ($X={\mathrm{O}}^{2-}$) the two common splittings are $A^{2+}{B}^{4+}{\mathrm{O}}_3$ (CaTiO₃, SrTiO₃, BaTiO₃) and $A^{3+}{B}^{3+}{\mathrm{O}}_3$ (LaMnO₃, LaAlO₃, the rare-earth orthoferrites); $A^{1+}{B}^{5+}{\mathrm{O}}_3$ (KNbO₃, NaTaO₃) also occurs. For halides ($X={\mathrm{F}}^{-},{\mathrm{Cl}}^{-},{\mathrm{Br}}^{-},{\mathrm{I}}^{-}$) neutrality forces $A^{1+}{B}^{2+}{X}_3$ — the optoelectronic lead/tin iodides CsPbI₃, CsSnI₃, MAPbI₃ all sit here. The cubic prototype has $Z=1$ formula unit per cell; the distorted variants below enlarge the cell ($Z=2$ for tetragonal, $Z=4$ for the orthorhombic $GdFe{O}_3$ structure).

Geometric stability: tolerance and octahedral factors

Whether a given $\{A,B,X\}$ triple adopts the perovskite topology, and with how much distortion, is captured to first order by two ionic-radius ratios. Goldschmidt’s tolerance factor measures how well the $A$–$X$ and $B$–$X$ bond lengths match the geometric constraint of the cubic cell — in the ideal structure the cube edge is $a=2(r_B+r_X)$ along the $B$–$X$–$B$ axis and the face diagonal is $a\sqrt{2}=2(r_A+r_X)$ along the $A$–$X$ contact, so equality of the two requires

$$t=\frac{r_A+r_X}{\sqrt{2}(r_B+r_X)}=1.$$

The octahedral factor $\mu =r_B/r_X$ asks separately whether the $B$ cation is large enough to sit stably in the octahedral $X_6$ cage; the rigid-sphere lower bound for octahedral coordination is $\mu \gtrsim 0.414$, and very large $B$ pushes toward higher coordination. The empirical regimes are:

• $t\approx 1$ ($\sim 0.9$–$1.0$): the cubic prototype is geometrically comfortable (SrTiO₃, the cubic halide perovskites at high $T$).
• $t<1$ ($\sim 0.71$–$0.9$): the $A$ cation is too small for the cavity, the framework relieves the underbonded $A$ by cooperative octahedral tilting, lowering the symmetry to tetragonal / orthorhombic / rhombohedral (CaTiO₃, LaMnO₃, GdFeO₃).
• $t>1$: the $A$ cation is too large (or $B$ too small); corner-sharing can no longer accommodate it and the structure goes to hexagonal face-sharing polytypes or abandons the perovskite topology altogether (e.g. BaNiO₃-type, or the non-perovskite $\delta$-phases of the cesium lead/tin halides).

This is a guide, not a law. $t$ predicts formability only probabilistically — a substantial fraction of $t\in (0.8,1.0)$ compounds are not perovskites, and some outside the window are — because it ignores covalency, the $B$-cation $d^0$/lone-pair electronic driving forces, and temperature. It is especially treacherous for hybrid halide perovskites, where the $A$ site is a molecular cation (methylammonium $\mathrm{C}{\mathrm{H}}_3\mathrm{N}{\mathrm{H}}_3^{+}$, formamidinium $\mathrm{CH}(\mathrm{N}{\mathrm{H}}_2{)}_2^{+}$) with no well-defined hard-sphere radius; one assigns an effective radius from the rotational envelope, and the classification degrades. The standard refinement is the data-driven Bartel tolerance factor $\tau$, a single dimensionless combination of $r_A$, $r_B$, $r_X$ and the $A$-cation oxidation state fitted to a large structure database, which separates perovskite from non-perovskite far more reliably than $t$ across both oxides and halides (Bartel et al., Sci. Adv. 5, eaav0693 (2019)).

Octahedral tilting, Glazer notation, and phase transitions

When $t<1$, the rigid octahedra rotate about pseudocubic axes to shorten $A$–$X$ contacts while preserving corner connectivity (the octahedra are far stiffer than the $A$–$X$ cage). Glazer notation classifies the resulting tilt systems by the rotation amplitude about each of the three cubic axes $⟨a⟩,⟨b⟩,⟨c⟩$ and the phase of successive layers: a superscript $+$ means adjacent octahedra along that axis rotate in phase, $-$ means out of phase, and $0$ means no tilt. Thus $a^0{a}^0{a}^0$ is the untilted cube; $a^0{a}^0{c}^{+}$ is a single in-phase tilt (tetragonal); $a^{-}{a}^{-}{c}^{+}$ (two equal out-of-phase tilts plus one in-phase) is the ubiquitous $GdFe{O}_3$ / orthorhombic $Pnma$ system of most distorted oxide and low-$T$ halide perovskites; $a^{-}{a}^{-}{a}^{-}$ gives rhombohedral $R\bar{3}c$. The 23 distinct Glazer systems and their space groups follow from group theory on the $R$- and $M$-point tilt modes (Glazer, Acta Cryst. B 28, 3384 (1972); Howard & Stokes, Acta Cryst. B 54, 782 (1998)).

A representative cooling sequence is cubic $\to$ tetragonal $\to$ orthorhombic, each step condensing an additional tilt (or polar) instability:

$$Pm\bar{3}m\stackrel{T_{c1}}{\to }P4/mbm\text{or}I4/mcm\stackrel{T_{c2}}{\to }Pnma.$$

The microscopic origin is a set of soft modes at the cubic zone boundary. The two tilt instabilities live at the $R$ point ${\mathbf{q}}_R=(\frac{1}{2},\frac{1}{2},\frac{1}{2})\frac{2\pi }{a}$ (out-of-phase tilts, $R_4^{+}$ irrep) and the $M$ point ${\mathbf{q}}_M=(\frac{1}{2},\frac{1}{2},0)\frac{2\pi }{a}$ (in-phase tilts, $M_3^{+}$); the ferroelectric/off-centering instability lives at $\Gamma$. Computing the dynamical matrix of the cubic phase at $0\mathrm{K}$ therefore generically returns imaginary frequencies ${\omega }^2({\mathbf{q}}_R)<0$, ${\omega }^2({\mathbf{q}}_M)<0$: the harmonic energy surface is a multi-well, the cubic structure a saddle point. Reading this correctly is the whole subject of imaginary modes: artifact vs. instability — a tilt mode at $R$/$M$ that deepens with $\mathbf{q}$-mesh and supercell convergence is a genuine instability, distinct from the spurious imaginary branches near $\Gamma$ that signal an under-converged calculation or a violated acoustic sum rule. Because the cubic phase is entropically stabilized — the free-energy minimum at finite $T$ sits at zero average tilt even though the $0\mathrm{K}$ enthalpy does not — its phonons must be computed anharmonically: the self-consistent harmonic approximation (anharmonic phonons & SSCHA) renormalizes the soft $R$/$M$ modes to real, positive frequencies above $T_c$ and locates the transitions as the temperature where the renormalized ${\omega }^2({\mathbf{q}}_{R/M})$ passes through zero.

Key examples

Inorganic halide perovskites — CsPbI₃, CsSnI₃. The flagship single-junction and tandem photovoltaic absorbers. Their defining complication is polymorphism / phase stability: the photoactive black perovskite phase ($\alpha$ cubic at high $T$, distorting to tetragonal $\beta$ and orthorhombic $\gamma$ on cooling) is only marginally stable against a yellow, photo-inactive non-perovskite $\delta$-phase (edge-sharing, $\mathrm{N}{\mathrm{H}}_4\mathrm{CdC}{\mathrm{l}}_3$-type for CsPbI₃) into which it readily converts at room temperature and humidity. CsPbI₃ has $t\approx 0.81$ — well inside the tilting regime and close to the perovskite/non-perovskite boundary — which is exactly why the black phase is so fragile and why A-site/B-site alloying (Br, FA, Sn) is used to push $t$ back toward 1. The electronic structure is dominated by the $n{s}^2$ lone pair of the $B^{2+}$ cation (Pb $6s$, Sn $5s$): the valence-band maximum is antibonding $Bns$–$Xnp$, giving the small effective masses and defect tolerance that make these materials good absorbers, while the conduction band is $Bnp$. Heavy-element spin–orbit coupling is not optional — for Pb it splits the conduction band by $\sim 1\mathrm{eV}$ and reorders the band edges, so any quantitative gap requires SOC. CsSnI₃ adds the Sn${}^{2+}$/Sn${}^{4+}$ oxidation problem (self-doping to a degenerate $p$-type metal).

Hybrid organic–inorganic — MAPbI₃, FAPbI₃. Replacing Cs⁺ with the molecular methylammonium $\mathrm{C}{\mathrm{H}}_3\mathrm{N}{\mathrm{H}}_3^{+}$ on the $A$ site gives MAPbI₃, the compound that launched the perovskite-PV field (Kojima et al., J. Am. Chem. Soc. 131, 6050 (2009)). The dipolar MA⁺ cation rotates nearly freely in the cavity at room temperature; its reorientational dynamics couple to the $\mathrm{Pb}{\mathrm{I}}_6$ tilts and broaden the picture beyond a static lattice, but the band edges remain a lead-iodide framework property. MAPbI₃ runs the same cubic $\to$ tetragonal ($I4/mcm$, $\sim 330\mathrm{K}$) $\to$ orthorhombic ($Pnma$, $\sim 160\mathrm{K}$) tilt sequence. Formamidinium FAPbI₃ has a larger $A$ cation ($t$ closer to 1) and a more favorable $\sim 1.5\mathrm{eV}$ gap, making it the current absorber of choice, but it shares the black-$\alpha$/yellow-$\delta$ stability problem and is usually stabilized by Cs/MA/Br alloying.

Oxide perovskites. The classical functional oxides:

• CaTiO₃ — the namesake mineral (perovskite, after L. A. Perovski). Strongly tilted $Pnma$ ($a^{-}{a}^{-}{c}^{+}$, $t\approx 0.97$ but tilted); the prototype distorted oxide perovskite.
• SrTiO₃ — the canonical quantum paraelectric: an incipient ferroelectric whose polar $\Gamma$ mode softens on cooling but is arrested by quantum fluctuations before it condenses, so it never becomes ferroelectric at ambient pressure. It does undergo an antiferrodistortive $R$-point tilt transition at $\sim 105\mathrm{K}$ (cubic $Pm\bar{3}m\to$ tetragonal $I4/mcm$), a textbook soft-mode zone-boundary instability.
• BaTiO₃ — the prototypical ferroelectric: the $\Gamma$ polar mode condenses, taking the structure through cubic $\to$ tetragonal $\to$ orthorhombic $\to$ rhombohedral on cooling, each phase with the Ti off-centered along a progressively lower-symmetry direction (an order–disorder/displacive hybrid). The $d^0$ Ti${}^{4+}$ second-order Jahn–Teller off-centering is the driver.
• LaMnO₃ — $A^{3+}{B}^{3+}{\mathrm{O}}_3$ with high-spin Mn${}^{3+}$ ($d^4$, $e_g^1$): a cooperative Jahn–Teller elongation of the $\mathrm{Mn}{\mathrm{O}}_6$ octahedra, orbital ordering, and A-type antiferromagnetism — the parent of the colossal-magnetoresistive manganites.

Quantitative band gaps and the magnetic/orbital ordering of the late-$3d$ oxides (LaMnO₃ and relatives) are a stress test for the exchange–correlation functional: semilocal DFT badly underestimates the gap and can miss the insulating, orbitally ordered ground state because of self-interaction error, and the standard fixes — Hubbard $+U$ or hybrid functionals — are needed to recover the correct ground state. The same SIE caution, plus mandatory SOC, governs the halide-perovskite gaps.

Prerequisites

crystal symmetry & space groups · ionic radii, close packing & coordination polyhedra (foundational crystal chemistry)

Builds toward: perovskite variants — double & antiperovskites

Key references

• Tolerance factor (original) — V. M. Goldschmidt, Naturwissenschaften 14, 477 (1926).
• Octahedral tilt classification — A. M. Glazer, Acta Cryst. B 28, 3384 (1972); group-theoretical analysis: C. J. Howard & H. T. Stokes, Acta Cryst. B 54, 782 (1998).
• Revised (data-driven) tolerance factor — C. J. Bartel, C. Sutton, B. R. Goldsmith, R. Ouyang, C. B. Musgrave, L. M. Ghiringhelli & M. Scheffler, Sci. Adv. 5, eaav0693 (2019).
• Hybrid perovskite photovoltaics (first report) — A. Kojima, K. Teshima, Y. Shirai & T. Miyasaka, J. Am. Chem. Soc. 131, 6050 (2009).
• Halide-perovskite electronic structure & SOC — J. Even, L. Pedesseau, J.-M. Jancu & C. Katan, J. Phys. Chem. Lett. 4, 2999 (2013).