the perovskite structure
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The perovskite structure, named after the mineral perovskite (calcium titanate, CaTiO3), is the great workhorse of electroceramics, and its general formula ABO3 packs in two different-sized cations plus oxygen. Picture a cage: small B cations sit at the corners of a cube, oxygens sit at the middle of every cube edge, and a single big A cation floats in the large cavity at the centre.
Read another way, each B cation is caged by six oxygens in an octahedron (coordination 6), those BO6 octahedra share corners to build a three-dimensional framework, and the roomy A cation nestles among twelve oxygens (coordination 12). Charge balance is flexible because only the sum A + B must equal 6: it works for 2+ and 4+ (barium titanate, BaTiO3), for 1+ and 5+ (potassium niobate), or for 3+ and 3+ (lanthanum aluminate). A tolerance factor, t = (r_A + r_O) / (sqrt(2) times (r_B + r_O)), measures how snugly the ions fit; t near 1 gives an ideal cube, and departures tilt and distort the octahedra.
That slight distortion is where the magic lives. In barium titanate, cooling below the Curie point (about 120 degrees C) lets the small Ti4+ ion shift off-centre within its oxygen cage, like a marble resting in one of two dimples, giving the crystal a permanent, switchable electric polarisation, this is ferroelectricity, the basis of multilayer ceramic capacitors and piezoelectric sensors and actuators (PZT is also a perovskite). Honest caveat: the ideal cubic perovskite is the exception, not the rule; almost every useful perovskite is a slightly tilted or distorted version, and it is precisely that broken symmetry that switches on its electrical properties.
Barium titanate, BaTiO3, is the textbook perovskite: a framework of corner-sharing TiO6 octahedra with big barium ions in the 12-coordinate cavities. Below its Curie point the titanium shifts off-centre, and that tiny shift is the ferroelectric charge storage inside billions of capacitors.
Corner-sharing BO6 octahedra with a large A cation in the 12-fold cavity: ABO3.
A perovskite is not held together by one cation size but two: a large 12-coordinate A cation and a small 6-coordinate B cation. Whether the structure is cubic or distorted is set by how well those two sizes match, captured by the tolerance factor.