A recently discovered magnetic semiconductor ${\mathrm{Ba}}_{1-\mathrm{x}}{\mathrm{K}}_{\mathrm{x}}{\left({\mathrm{Zn}}_{1-\mathrm{y}}{\mathrm{Mn}}_{\mathrm{y}}\right)}_2{\mathrm{As}}_2$, with its decoupled spin and charge doping, provides a unique opportunity to elucidate the microscopic origin of the magnetic interaction and ordering in dilute magnetic semiconductors (DMSs). We show that (i) the conventional density functional theory accurately describes this material, and (ii) the magnetic interaction emerges from the competition of the short-range superexchange and a longer-range interaction mediated by the itinerant As holes, coupled to Mn via the Schrieffer-Wolff $p\text{−}d$ interaction representing an effective Hund's rule coupling $J_H^{\text{eff}}$. The key difference between the classical double exchange and the actual interaction in DMSs is that an effective $J_H^{\text{eff}}$, as opposed to the standard Hund's coupling $J_H$, depends on the Mn $d$-band position with respect to the Fermi level, and thus allows tuning of the magnetic interactions. The physically transparent description of this material may also be applicable in more complicated DMS systems.

• Received 12 May 2014
• Revised 15 September 2014

DOI:https://doi.org/10.1103/PhysRevB.90.140403