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It was recently reported that hydrogen-reduced PdCoO$_2$ films exhibit strongperpendicular magnetic anisotropy (PMA) with sign tunable anomalous Hall effect(AHE). Here, we provide extensive thickness-dependent study of this system, andshow that the electronic and magnetic properties are strongly dependent on thethickness and annealing conditions. Below a critical thickness of 25 nm, AHEshows clear PMA with hysteresis, and its sign changes from positive tonegative, and back to positive as the annealing temperature increases from 100$^\circ$C to 400 $^\circ$C. Beyond the critical thickness, both PMA and AHEhysteresis disappear and the AHE sign remains positive regardless of theannealing parameters. Our results show that PMA may have a large role on AHEsign-tunability and that below the critical thickness, competition betweendifferent AHE mechanisms drives this sign change.

Thickness-dependent, tunable anomalous Hall effect in hydrogen-reduced PdCoO$_2$ thin films