Jahn-Teller coupling in spinel-type crystals doped with transition metal ions

In this article, we report the theoretical evaluation of the vibronic constants for the spinel-type crystals ZnAl₂S₄, ZnAl₂Se₄, CdAl₂Se₄, and CdIn₂S₄ doped with the transition metal ions Cr³⁺, Cr⁴⁺, Ti²⁺, and Ti³⁺. In these crystals the ions Cr³⁺, Ti³⁺ and Cr⁴⁺, Ti²⁺ occupy positions with octahedral and tetrahedral surrounding correspondingly. In the case of a tetrahedral complex the interaction with full symmetric A₁, tetragonal E and two trigonal vibrations T₂⁽¹⁾ and T₂⁽²⁾ is considered, while for the octahedral cluster along with the A₁ and E vibrations only one trigonal vibrational mode is taken into account. The examined semiconductor systems are mainly covalent and application of the crystal field point charge model loses proper exactity. For this reason, we employ the exchange charge model for the crystal field that accounts for the covalence effects and provides relatively simple expressions for the crystal field and vibronic parameters keeping at the same time a reasonable level of accuracy. The vibronic coupling constants are numerically calculated and for each case an appropriate multi-mode Jahn-Teller vibronic problem is found out.