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We show that the magnetic and vacancy orders in the 122$(A_{1-y}Fe_{2-x}Se_2)$ iron-chalcogenides can be naturally derived from the$J_1-J_2-J_3$ model with $J_1$ being the ferromagnetic (FM) nearest neighborexchange coupling and $J_{2}, J_3$ being the antiferromagnetic (AFM) next andthird nearest neighbor ones respectively, previously proposed to describe themagnetism in the 11(FeTe/Se) systems. In the 11 systems, the magnetic exchangecouplings are extremely frustrated in the ordered bi-collinearantiferromagnetic state so that the magnetic transition temperature is low. Inthe 122 systems, the formation of iron vacancy order reduces the magneticfrustration and significantly increases the magnetic transition temperature andthe ordered magnetic moment. The pattern of the 245 iron-vacancy order($\sqrt{5}\times \sqrt{5}$) observed in experiments is correlated to themaximum reduction of magnetic frustration. The nature of the iron-vacancyordering may hence be electronically driven. We explore other possible vacancypatterns and magnetic orders associated with them. We also calculate the spinwave excitations and their novel features to test our model.

Magnetic Frustration and Iron-Vacancy Ordering in Iron-Chalcogenide