The electronic band structures of ferromagnetic Cd${\mathrm{Cr}}_2$${\mathrm{S}}_4$ and Cd${\mathrm{Cr}}_2$${\mathrm{Se}}_4$ are self-consistently calculated by using the discrete variational $X\alpha$ method. The general features of the band structures are quite similar between sulfide and selenide; each structure consists of relatively narrow valence bands, fairly wide conduction bands, and very narrow $d$ bands. The $3d\epsilon$ and $3d\gamma$ bands for up spin lie in the energy region near the top of the valence bands and around the bottom of the lowest conduction band, respectively, and both $d$ bands for down spin fall in the conduction bands. The maximum point of the valence bands has ${\Sigma }_4$ symmetry for both compounds, and the minimum point of the conduction band has ${\Lambda }_1$ for sulfide and ${\Gamma }_1$ for selenide. The fundamental energy gap at the $\Gamma$ point is 2.6 eV for sulfide and 2.3 eV for selenide. The spin polarization of the $3d$ orbitals of Cr is about 3.5, in which 0.5 comes from the $3d\gamma$ components mixed with the valence bands, while the spin polarization of the outermost $p$ orbitals of chalcogen ion has the opposite sign, the magnitude of which is about 0.3.

• Received 29 January 1980

DOI:https://doi.org/10.1103/PhysRevB.22.872