  In the search for novel materials for vacuum electron sources, multi-alkali
antimonides and in particular sodium-potassium-antimonides have been recently
regarded as especially promising due to their favorable electronic and optical
properties. In the framework of density-functional theory and many-body
perturbation theory, 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 a direct gap, which is on the order
of 1.5 eV in Na$_2$KSb and 1.0 eV in NaK$_2$Sb. In either system, 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 electron-hole interactions 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 are non-excitonic in
nature and that their binding energy is on the order of 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.
