Valence-band ordering and magneto-optic exciton fine structure in ZnO

Abstract

Using first-principles linear muffin-tin orbital density functional band structure calculations, the ordering of the states in the wurtzite ZnO valence-band maximum, split by crystal-field and spin-orbit coupling effects, is found to be ${\ensuremath{\Gamma}}_{7(5)}>{\ensuremath{\Gamma}}_{9(5)}>{\ensuremath{\Gamma}}_{7(1)},$ in which the number in parentheses indicates the parent state without spin-orbit coupling. This results from the negative spin-orbit splitting, which in turn is due to the participation of the Zn $3d$ band. The result is found to be robust even when effects beyond the local density approximation on the Zn $3d$ band position are included. Using a Kohn-Luttinger model parametrized by our first-principles calculations, it is furthermore shown that the binding energies of the excitons primarily derived from each valence band differ by less than the valence-band splittings even when interband coupling effects are included. The binding energies of $n=2$ and $n=1$ excitons, however, are not in a simple $1/4$ ratio. Our results are shown to be in good agreement with the recent magneto-optical experimental data by Reynolds et al. [Phys. Rev. B 60, 2340 (1999)], in spite of the fact that on the basis of these data these authors claimed that the valence-band maximum would have ${\ensuremath{\Gamma}}_{9}$ symmetry. The differences between our and Reynolds' analysis of the data are discussed and arise from the sign of the Land\'e g factor for holes, which is here found to be negative for the upper ${\ensuremath{\Gamma}}_{7}$ band.