We report the physical properties and electronic structure calculations of a layered chromium oxypnictide, Sr$_2$Cr$_3$As$_2$O$_2$, which crystallizes in a Sr$_2$Mn$_3$As$_2$O$_2$-type structure containing both CrO$_2$ planes and Cr$_2$As$_2$ layers. The newly synthesized material exhibits a metallic conduction with a dominant electron-magnon scattering. Magnetic and specific-heat measurements indicate at least two intrinsic magnetic transitions below room temperature. One is an antiferromagnetic transition at 291 K, probably associated with a spin ordering in the Cr$_2$As$_2$ layers. Another transition is broad, occurring at around 38 K, and possibly due to a short-range spin order in the CrO$_2$ planes. Our first-principles calculations indicate predominant two-dimensional antiferromagnetic exchange couplings, and suggest a KG-type (i.e. K$_2$NiF$_4$ type for CrO$_2$ planes and G type for Cr$_2$As$_2$ layers) magnetic structure, with reduced moments for both Cr sublattices. The corresponding electronic states near the Fermi energy are mostly contributed from Cr-3$d$ orbitals which weakly (modestly) hybridize with the O-2$p$ (As-4$p$) orbitals in the CrO$_2$ (Cr$_2$As$_2$) layers. The bare bandstructure density of states at the Fermi level is only $sim$1/4 of the experimental value derived from the low-temperature specific-heat data, consistent with the remarkable electron-magnon coupling. The title compound is argued to be a possible candidate to host superconductivity.
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