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Chemical and magnetic structures of substituent Co ions in ZnO:Co dilute magnetic oxides are investigated by using first-principles density functional calculations of isolated Co, clustered Co-O-Con (n>0) and chained -Co-O-Co-O-. It demonstrates that the substitutional Co ions in ZnO tend to form -Co-O-Co-O- chains or Co-O-Con clusters rather than stay isolated. The most stable structure was identified to be antiferromagnetic -Co-O-Co-O- chains via O on ab plane rather than Co-O-Con clusters. Furthermore, substituent Cu codoping in ZnO is disclosed to suppress the Co's tendency towards chain-like structures or clusters, thus enhancing its homogeneous distribution.