  By using first-principles electronic structure calculations, we have studied
the magnetic interactions in a proposed BaZn$_2$P$_2$-based diluted magnetic
semiconductor (DMS). For a typical compound
Ba(Zn$_{0.944}$Mn$_{0.056}$)$_2$P$_2$ with only spin doping, due to the
superexchange interaction between Mn atoms and the lack of itinerant carriers,
the short-range antiferromagnetic coupling dominates. Partially substituting K
atoms for Ba atoms, which introduces itinerant hole carriers into the $p$
orbitals of P atoms so as to link distant Mn moments with the spin-polarized
hole carriers via the $p$-$d$ hybridization between P and Mn atoms, is very
crucial for the appearance of ferromagnetism in the compound. Furthermore,
applying hydrostatic pressure first enhances and then decreases the
ferromagnetic coupling in
(Ba$_{0.75}$K$_{0.25}$)(Zn$_{0.944}$Mn$_{0.056}$)$_2$P$_2$ at a turning point
around 15 GPa, which results from the combined effects of the pressure-induced
variations of electron delocalization and $p$-$d$ hybridization. Compared with
the BaZn$_2$As$_2$-based DMS, the substitution of P for As can modulate the
magnetic coupling effectively. Both the results for BaZn$_2$P$_2$-based and
BaZn$_2$As$_2$-based DMSs demonstrate that the robust antiferromagnetic (AFM)
coupling between the nearest Mn-Mn pairs bridged by anions is harmful to
improving the performance of this II-II-V based DMS materials.
