The Spin-Orbit Effect on the Electronic Structures, Refractive Indices, and Birefringence of X₂PO₄I (X = Pb, Sn, Ba...

Introducing post-transition metal cations is an excellent strategy for enhancing optical properties. This paper focuses on four isomers, namely the X₂PO₄I (X = Pb, Sn, Ba, and Sr) series. For the first time, the paper's attention is paid to the changes in electronic structure, as well as refractive indices and birefringence, with and without the inclusion of spin-orbit effects in this series. The first-principles results show that spin-orbit effects of the 5p and 6p states found in these compounds lead to splitting of the bands, narrowing of the band gap, enhancement of the lone-pair stereochemistry, and enhancement of the refractive indices and birefringence. Moreover, a comparison of the lone-pair electron phosphates, X₂PO₄I (X = Pb and Sn), and the isomeric alkaline earth metal phosphates, X₂PO₄I (X = Ba and Sr), reveals that changes in the band structure have a greater effect on the enhancement of the birefringence than the slight enhancement of the lone-pair stereochemical activity. This study has important implications for a deeper understanding of the optical properties of crystals and the design of novel optical materials.