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We investigate the element-specific spin and orbital states and their roleson magnetic anisotropy in the Co-ferrite (Co$_x$Fe$_{3-x}$O$_{4+\delta}$ (001))thin films which exhibit perpendicular magnetic anisotropy (PMA). The origin ofPMA in the low $x$ region ($x$ $<$ 1) can be mainly explained by the largeperpendicular orbital magnetic moments in the Co$^{2+}$ (3$d^7$) statesdetected by X-ray magnetic circular and linear dichroisms (XMCD/XMLD). The XMLDfor a PMA film ($x=0.2$) with square hysteresis curve shows the oblate chargedistribution in the Co$^{2+}$ site, which is consistent with the change inlocal nearest neighbor distance in Co detected by extended X-ray absorptionfine structure analysis. Our finding reveals that the microscopic origin of PMAin Co-ferrite comes from the enhanced orbital magnetic moments alongout-of-plane [001] direction through in-plane charge distribution by tensilestrain, which adds the material functionalities in spinel ferrite thin filmsfrom the viewpoint of strain and orbital magnetic moments.

Origin of Perpendicular Magnetic Anisotropy in Co$_x$Fe$_{3-x}$O$_{4+\delta}$ Thin Films Studied by X-ray Magnetic Circular and Linear Dichroisms