# A Review: Thermal Stability of Methylammonium Lead Halide Based Perovskite Solar Cells
## 2 Structural Stability of Perovskite  
The structural stability of the perovskites is critical when used in photovoltaic devices, since they are expected to remain stable for more than 25 years [22]. Structural stability is defined as the ability for a crystalline phase to be stable over a wide range of external factors such as heat, pressure, moisture, etc. Perovskite compounds have a general chemical formula of ABX\({}_{3}\), where A and B are cations and X is an anion. Their crystal structure is similar to that of calcium titanium oxide. Usually, the A cations are larger than the B cations. Oxide perovskites have been studied extensively because of their multifunctional nature [23]. However, owing to their wide bandgap, oxide perovskites harvest only 8-20% of the solar spectrum, limiting their use in photovoltaic applications [23]. Instead, halide organic-inorganic perovskites were developed by replacing the oxygen anion of oxide perovskites with an inorganic halide (I\({}^{-}\), Cl\({}^{-}\), Br\({}^{-}\)) [23]. An organic or inorganic monovalent A\({}^{+}\) cation (e.g., Rb\({}^{+}\), Cs\({}^{+}\), CH\({}_{3}\)NH\({}_{3}\)\({}^{+}\), HC(NH\({}_{2}\))\({}_{2}\)\({}^{+}\)) and a divalent B\({}^{2+}\) metal cation (e.g., Pb\({}^{2+}\), Sn\({}^{2+}\)) were also implemented in the frame of the perovskite structures [23]. The halide perovskites obtain a desired crystal symmetry by maintaining an allowable tolerance factor. A tolerance factor developed by [PERSON] [24] determines the radii sizes associated with cubic symmetry, described by  
\[\mathrm{t}=\frac{\mathrm{R_{A}}+\mathrm{R_{X}}}{\sqrt{2}(\mathrm{R_{A}}+ \mathrm{R_{X}})} \tag{1}\]  
where \(\mathrm{R_{A}}\), \(\mathrm{R_{B}}\), \(\mathrm{R_{X}}\) are the ionic radii of A, B, X, respectively. The tolerance factor provides a rough estimate of the stability and distortion of crystal structures of a compound. In addition, it gives an idea of whether the phase is cubic (t = 1) or deviates into the tetragonal or orthorhombic phase [25]. In general, an established tolerance factor value for halide perovskites lies in the range of 0.85 \(<\) t \(<\) 1.11 [26]. Non-perovskite structures are formed when the tolerance factor is higher or lower. In an inorganic-organic hybrid perovskite, it is difficult to calculate the absolute tolerance factor as the organic cation has a non-spherical geometry [27]. However, it is possible to qualitatively analyze the transition of structure in these materials. For example, formamidinium lead iodide, HC(NH\({}_{2}\))\({}_{2}\)PbI\({}_{3}\) (FAPbI\({}_{3}\)), has a larger A cation than methylammonium lead iodide, CH\({}_{3}\)NH\({}_{3}\)PbI\({}_{3}\) (MAPbI\({}_{3}\)) and a larger cation would generally represent a higher tolerance factor [28]. Two phases can be obtained in solution-processed FAPbI\({}_{3}\) materials. One is the photoactive \(\alpha\)-phase (black phase) and the other is a non-photoactive phase (yellow phase or \(\delta\)-phase) [29]. Figure 1a,b shows the ideal case of cubic symmetry for perovskites. The cubic symmetry corresponds to a \(Pm3m\) space group with 12-fold coordination for the A cation, 6-fold coordination for the B cations, and BX\({}_{6}\) octahedra residing in the corners [23]. The deviation from cubic symmetry can be attributed to several factors including the atomic sizes of the constituents. The A cation does not directly affect the electronic properties [30; 31; 32]; however, the size of A cations can cause distortion of the B-X bonds, which undesirably affects the symmetry. The best electronic properties are obtained with cubic symmetry due to high ionic bonding. An octahedral tilting occurs when the tolerance factor exceeds the ideal range, and affects the electronic properties. In recent years, methylammonium lead trihalide (MAPbX\({}_{3}\)) has received increasing attention as an absorber material for perovskite solar cells. However, the stability of MAPbX\({}_{3}\) has been found to strongly vary depending on temperature, changing phase or crystal orientation.