High perpendicular magnetic anisotropy (PMA), a property needed for nanoscale spintronic applications, is rare in oxide conductors. We report the observation of a PMA up to 0.23 MJ/m 3 in modestly strained (-0.3%) epitaxial NiCo2O4 (NCO) films which are room-temperature ferrimagnetic conductors. Spin-lattice coupling manifested as magnetoelastic effect was found as the origin of the PMA. The in-plane x 2 -y 2 states of Co on tetrahedral sites play crucial role in the magnetic anisotropy and spin-lattice coupling with an energy scale of 1 meV/f.u. The elucidation of the microscopic origin paves a way for engineering oxide conductors for PMA using metal/oxygen hybridizations. sin( ) cos( ) = [sin 2 ( ) − sin( ) cos( )], with
127, the spherical angles of magnetization, and the angle between magnetization and anisotropy axis. As 128 the anisotropy scale K increases, the magnetization direction of minimum energy tilts further from the field 129 direction and more towards the anisotropy axis.
131A consequence of the relative orientations of the laser incidence and magnetization is that a large enough 132 misalignment of the directions leads to a flipping of the sign of the magneto-optic effect. Explicitly,
133̂′ ⋅̂= 0 when the angle of incidence satisfies tan( ) = , and on either side of this value, the effect of 134 the magnetization switches signs. This is displayed visually in the supplementary Fig S3(a)-(d). The severe 135 misalignment of the laser k'-vector and magnetization direction seen in the NCO/MAO(111) measurements 136 leads directly to the emergence of inverted hysteresis loops, and the three-fold rotational symmetry of the 137 angle-dependent MOKE measurement.
139We can further analyze the effect of magnetization direction relative to incidence direction modeling the 140 Magneto-optic Kerr Effect in terms of the Lorentz force experienced by charges in the magnetized 141 material. In our case of s-polarized light (i.e. polarization perpendicular to the plane of reflection), the 142 electric field forces an oscillation of the charges, which then experience a Lorentz force. With light 143 polarization along the x-axis (i.e. s-polarized light), charges will be forced to oscillate along this same 144 axis. Given the most general magnetization direction ̂= ( , , ), the force experienced will be