  We investigate the element-specific spin and orbital states and their roles
on 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 of
PMA in the low $x$ region ($x$ $<$ 1) can be mainly explained by the large
perpendicular orbital magnetic moments in the Co$^{2+}$ (3$d^7$) states
detected by X-ray magnetic circular and linear dichroisms (XMCD/XMLD). The XMLD
for a PMA film ($x=0.2$) with square hysteresis curve shows the oblate charge
distribution in the Co$^{2+}$ site, which is consistent with the change in
local nearest neighbor distance in Co detected by extended X-ray absorption
fine structure analysis. Our finding reveals that the microscopic origin of PMA
in Co-ferrite comes from the enhanced orbital magnetic moments along
out-of-plane [001] direction through in-plane charge distribution by tensile
strain, which adds the material functionalities in spinel ferrite thin films
from the viewpoint of strain and orbital magnetic moments.
