  The inverse spinel ferrimagnetic NiCo$_2$O$_4$ possesses high magnetic Curie
temperature $T_C$, high spin polarization, and strain-tunable magnetic
anisotropy. Understanding the thickness scaling limit of these intriguing
magnetic properties in NiCo$_2$O$_4$ thin films is critical for their
implementation in nanoscale spintronic applications. In this work, we report
the unconventional magnetotransport properties of epitaxial (001) NiCo$_2$O$_4$
films on MgAl$_2$O$_4$ substrates in the ultrathin limit. Anomalous Hall effect
measurements reveal strong perpendicular magnetic anisotropy for films down to
1.5 unit cell (1.2 nm), while $T_C$ for 3 unit cell and thicker films remains
above 300 K. The sign change in the anomalous Hall conductivity ($\sigma_{xy}$)
and its scaling relation with the longitudinal conductivity ($\sigma_{xx}$) can
be attributed to the competing effects between impurity scattering and band
intrinsic Berry curvature, with the latter vanishing upon the thickness driven
metal-insulator transition. Our study reveals the critical role of film
thickness in tuning the relative strength of charge correlation, Berry phase
effect, spin orbit interaction, and impurity scattering, providing important
material information for designing scalable epitaxial magnetic tunnel junctions
and sensing devices using NiCo$_2$O$_4$.
