Electronic structure and optical properties of Na$_2$KSb and NaK$_2$Sb from first-principles many-body theory

In the search for novel materials for vacuum electron sources, multi-alkali antimonides and in particular sodium-potassium-antimonides have been regarded as especially promising due to their favorable electronic and optical properties. In the framework of non-relativistic density-functional theory and many-body perturbation theory ($GW$ approximation and solution of the Bethe-Salpeter equation) we investigate the electronic structure and the dielectric response of two representative members of this family, namely Na$_2$KSb and NaK$_2$Sb. We find that both materials have direct gaps of the order of 1.5 eV and that their valence and conduction bands are dominated by Sb states with $p$- and $s$-character, respectively. The imaginary part of the dielectric function, computed upon explicit inclusion of excitonic effects to characterize the optical response of the materials, exhibits maxima starting from the near infrared region, extending up to the visible and the ultraviolet band. With our analysis, we clarify that the lowest-energy excitations correspond to weakly correlated interband transitions that can be dissociated into free carriers with an energy penalty of about 100 meV. Our results confirm the potential of Na$_2$KSb and NaK$_2$Sb as photoemissive materials for vacuum electron sources, photomultipliers, and imaging devices.