Fig. 3

a Real-space 3D view of the cubic crystal structure of AMX₃ (A = Cs, CH₃NH₃, M = Pb, Sn, and X = I, Br, Cl) with the Pm3m space group. CH₃NH₃ ⁺ cation (red ball) is located at the center of the cube. b Reciprocal-space 3D view showing the first BZ of the Pm3m space group. Points of high symmetry in the cubic BZ: Γ denotes the origin of the BZ; X is the center of a square face at the BZ boundary; M is a center of a cube edge; and Rs are vertices of the cube. c Electronic band structure for the high-temperature cubic Pm3m phase of MAPbI₃ with spin–orbit coupling (SOC) at the local density approximation (LDA) level of theory. An upward energy shift of 1.4 eV has been applied to match the experimental bandgap value at R. Irreducible representations obtained from a Pm3m double group analysis are given at R and M points for the electronic states close to the band gap. Vertical arrows show various possible optical transitions close to the band gap energy. Optical transitions along the line between the M and R points generate carriers that easily relax toward the R point. Reprinted with permission from 23