  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.
