A phenomenological model of electronic band structure in ferroelectric Pb(In_1/2Nb_1/2)O₃-Pb(Mg_1/3Nb _2/3)O₃-PbTiO₃ single crystals around the morphotropic phase boundary determined by temperature-dependent transmittance spectra

The optical properties of ferroelectric Pb(In_1/2Nb _1/2)O₃-Pb(Mg_1/3Nb_2/3)O ₃-PbTiO₃ (PIN-PMN-PT) single crystals around the morphotropic phase boundary (MPB) have been investigated using ultraviolet-infrared transmittance spectra in the temperature range of 8-300 K. Based on the temperature-dependent spectral measurement of the band gap, we propose a phenomenological model of band structure vs. temperature to explain both the negative and positive band narrowing coefficient dE_gd/dT in ferroelectric PIN-PMN-PT crystals around the MPB where multiple phases coexist. The peculiar positive coefficient only exists in the fragile multiphase region of the MPB, while the negative coefficient, caused by thermal expansion of the lattice and renormalization of the band structure by electron-phonon interaction, exists in the rhombohedral or tetragonal single-phase region as well as in the stationary multiphase region of the MPB. The origin of the positive coefficient is a long-range increasing fraction of coexistence from the monoclinic phase with small band gap to rhombohedral phase with large band gap at elevated temperature. In agreement with optical transmittance results of PMN-PT/PIN-PMN-PT, the model predicts that these unusual positive band narrowing coefficients may exist for all ferroelectrics around the MPB where the coexistence of phases lacks thermodynamic stability.