Halide perovskites have emerged as a highly versatile class of materials due to their unique crystal structures and exceptional optoelectronic properties. The general formula for these perovskites is ABX3, where A is a monovalent organic (methylammonium, MA; formamidinium, FA) or inorganic (cesium, Cs) cation, B is a divalent metal (lead, Pb; tin, Sn), and X is a halide anion (iodide, I; bromide, Br; chloride, Cl). The structural flexibility of these materials allows them to adopt multiple phases—cubic, tetragonal, and orthorhombic—depending on temperature and composition, with each phase exhibiting distinct optoelectronic characteristics.

The cubic phase is the highest symmetry structure, belonging to the Pm-3m space group. In this arrangement, the B-site cation sits at the center of an octahedron formed by six X anions, while the A-site cation occupies the cuboctahedral voids. The cubic phase is typically stable at high temperatures and is characterized by isotropic properties. For example, MAPbI3 transitions to the cubic phase above 330 K. The cubic structure is ideal for optoelectronic applications due to its high symmetry, which promotes efficient charge carrier transport. However, it is often metastable at room temperature for many compositions.

Upon cooling, the cubic phase undergoes symmetry-lowering transitions to tetragonal and orthorhombic phases. The tetragonal phase, with the I4/mcm or P4/mbm space group, arises due to slight tilting of the BX6 octahedra, reducing the symmetry while maintaining long-range order. For MAPbI3, the tetragonal phase is stable between 160 K and 330 K. The octahedral tilting in this phase introduces anisotropy in electronic properties, slightly modifying the band structure compared to the cubic phase. Further cooling leads to the orthorhombic phase (Pnma space group), where more pronounced octahedral distortions occur. This phase is typically stable below 160 K for MAPbI3 and exhibits the lowest symmetry among the three, resulting in further alterations to charge transport and optical absorption properties.

The stability and formation of these phases are governed by the Goldschmidt tolerance factor (t), a geometric parameter defined as t = (rA + rX) / [√2 (rB + rX)], where rA, rB, and rX are the ionic radii of the respective ions. For an ideal perovskite structure, t should lie between 0.8 and 1.0. Values outside this range lead to non-perovskite structures or distorted phases. For instance, MAPbI3 has a tolerance factor of approximately 0.91, favoring the perovskite structure, whereas FAPbI3, with a larger FA cation, has t ≈ 1.03, leading to a tendency to form a non-perovskite hexagonal phase at room temperature unless stabilized by additives or strain. The tolerance factor also influences phase transition temperatures; compositions with t closer to 1 exhibit higher cubic phase stability over a broader temperature range.

Structural variations significantly impact the optoelectronic properties of halide perovskites. The bandgap, a critical parameter for light absorption and emission, is highly sensitive to the crystal phase and octahedral distortions. The cubic phase generally has the lowest bandgap due to its high symmetry, while the orthorhombic phase exhibits a larger bandgap because of reduced orbital overlap from octahedral tilting. For example, MAPbI3 has a bandgap of ~1.6 eV in the cubic phase, which increases slightly in the tetragonal phase and further in the orthorhombic phase. Halide substitution also plays a key role; replacing I with Br or Cl increases the bandgap due to higher electronegativity and reduced orbital mixing.

Charge carrier mobility is another property influenced by crystal structure. The cubic phase typically exhibits the highest mobility due to minimal octahedral distortions and isotropic charge transport. In contrast, the tetragonal and orthorhombic phases show anisotropic mobility due to preferential directions of orbital overlap. For instance, in MAPbI3, the mobility along the [100] direction in the tetragonal phase is higher than along [001] because of the specific octahedral tilting pattern. Structural defects, such as vacancies or grain boundaries, further influence mobility by introducing scattering centers.

The role of the A-site cation is crucial in determining structural stability and optoelectronic behavior. Larger cations like FA increase lattice parameters but can destabilize the perovskite phase if the tolerance factor exceeds 1. Smaller cations like Cs favor cubic symmetry but may introduce strain if mismatched with the BX6 framework. Mixed-cation approaches (e.g., MA/FA or MA/Cs) are often employed to optimize phase stability and optoelectronic performance. Similarly, mixed halides (e.g., I/Br or Br/Cl) allow fine-tuning of bandgaps while maintaining structural integrity.

Temperature-induced phase transitions also affect material properties. Thermal expansion and contraction alter bond lengths and angles, modifying electronic structure and carrier dynamics. For example, heating MAPbI3 from orthorhombic to tetragonal phase reduces the bandgap and improves charge transport, but excessive heating can lead to decomposition. Understanding these transitions is critical for applications requiring thermal stability.

In summary, halide perovskites exhibit rich structural chemistry with cubic, tetragonal, and orthorhombic phases dictated by composition and temperature. The Goldschmidt tolerance factor serves as a key predictor of stability, while octahedral distortions govern optoelectronic properties. Bandgap tuning and charge carrier mobility are directly linked to structural variations, making precise control of crystal phases essential for optimizing material performance. Future research into strain engineering, defect passivation, and advanced characterization will further elucidate the structure-property relationships in these materials.