Spin-Polarized Electronic Energy-Band Structure in EuS

The spin-polarized electronic energy bands for ${\mathrm{Eu}}^{++}$ ${\mathrm{S}}^{--}$ have been calculated by using the augmented-plane-wave (APW) method. The results show that this material is a ferromagnetic as well as a semiconducting material. The muffin-tin crystal potential energy for the ionic form of ${\mathrm{Eu}}^{++}$ ${\mathrm{S}}^{--}$, used to calculate the energy bands, is about 0.25 Ry higher than that for the case of neutral EuS. This influences the energy bands. The $f$ bands for up spin are well localized with width 0.002 Ry, and are located 0.515 Ry below the top of the valence band ${\Gamma }_{15}$. On the other hand, the $f$ bands for the down spin are conduction bands and are also localized, with a width of 0.031 Ry. These bands are located 0.115 Ry above the top of the valence band ${\Gamma }_{15}$. This quantity is also the direct band gap. At the zone center, the energy difference between these two $f$ bands is 0.630 Ry. The bottom of the conduction band is located at the zone edge $X$, and the indirect transition occurs from the valence $p$ band ${\Gamma }_{15}$ to the conduction $d$ band $X_3$ for up-spin electrons. The indirect energy gap obtained in this calculation is 0.111 Ry or 1.51 eV, compared with the experimental value 1.645 eV. This transition corresponds to the transition from the ground $3p$ shell in the ${\mathrm{S}}^{--}$ ion to the excited $5d$ shell in the ${\mathrm{Eu}}^{++}$ ion. The charge densities inside of the APW spheres have been analyzed, and are consistent with the energy-band picture. The constant-energy contours near the top of the valence band and near the bottom of the conduction band are shown.