Comparison of interband related optical transitions and excitons in ZnGeN$_2$ and GaN

Abstract

The optical dielectric function of ZnGeN$_2$ is calculated from the interband transitions using the energy bands calculated in the quasiparticle self-consistent (QS)$G\hat W$ method using two different levels of approximation: the independent particle approximation (IPA) and the Bethe-Salpeter Equation (BSE) approach. The first allows us to relate peaks in $\varepsilon_2(\omega)$ to specific bands and {\bf k}-points but does not include excitonic effects, while the latter does. The corresponding changes in the shape of $\varepsilon_2(\omega)$ are found to be similar to those in GaN. The screened Coulomb interaction $\hat W$ is here calculated including electron-hole interactions in the polarization function and gives a band structure already going beyond the random phase approximation. The static dielectric constants including only electronic screening, commonly referred to as $\varepsilon^\infty$, were calculated separately by extrapolating the wave vector dependent macroscopic $\varepsilon_M({\bf q},\omega=0)$ for ${\bf q}\rightarrow0$. Below the quasiparticle gap, we find three bound excitons optically active for different polarization. The convergence of these bound excitons with respect to the density of the {\bf k}-mesh used in the BSE is studied and found to require a fine mesh. It is also found that these bound excitons originate from only the lowest conduction band and the top three valence bands. To incorporate the lattice screening, we include a scaling factor $(\varepsilon^\infty/\varepsilon^0)^2$, which allows us to obtain exciton binding energies of the correct order of magnitude similar to those in GaN. The excitons are related to each of the three fold split valence bands and the splittings of the latter are also studied as function of strain. Finally, a relation between the anisotropic effective masses and the valence band splitting is pointed out and explained.