We investigated the magnetic properties of (La$_{1-x}$Ba$_{x}$)(Zn$_{1-x}$Mn$_{x}$)AsO with $x$ varying from 0.005 to 0.05 at an external magnetic field of 1000 Oe. For doping levels of $x$ $leq$ 0.01, the system remains paramagnetic down to the lowest measurable temperature of 2 K. Only when the doping level increases to $x$ = 0.02 does the ferromagnetic ordering appear. Our analysis indicates that antiferromagnetic exchange interactions dominate for $x$ $leq$ 0.01, as shown by the negative Weiss temperature fitted from the magnetization data. The Weiss temperature becomes positive, i.e., ferromagnetic coupling starts to dominate, for $x$ $geq$ 0.02. The Mn-Mn spin interaction parameter $mid$$2J/k_B$$mid$ is estimated to be in the order of 10 K for both $x$ $leq$ 0.01 (antiferromagnetic ordered state) and $x$ $geq$ 0.02 (ferromagnetic ordered state). Our results unequivocally demonstrate the competition between ferromagnetic and antiferromagnetic exchange interactions in carrier-mediated ferromagnetic systems.
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