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Motivated by the recently observed unconventional Hall effect in ultra-thinfilms of ferromagnetic SrRuO$_3$ (SRO) we investigate the effect ofstrain-induced oxygen octahedral distortion in the electronic structure andanomalous Hall response of the SRO ultra-thin films by virtue of densityfunctional theory calculations. Our findings reveal that the ferromagnetic SROfilms grown on SrTiO$_3$ (in-plane strain of $-$0.47$\%$) have an orthorhombic(both tilting and rotation) distorted structure and with an increasing amountof substrate-induced compressive strain the octahedral tilting angle is foundto be suppressed gradually, with SRO films grown on NdGaO$_3$ (in-plane strainof $-$1.7$\%$) stabilized in the tetragonal distorted structure (with zerotilting). Our Berry curvature calculations predict a positive value of theanomalous Hall conductivity of $+$76\,S/cm at $-$1.7$\%$ strain, whereas it isfound to be negative ($-$156\,S/cm) at $-$0.47$\%$ strain. We attribute thefound behavior of the anomalous Hall effect to the nodal point dynamics in theelectronic structure arising in response to tailoring the oxygen octahedraldistortion driven by the substrate-induced strain. We also calculatestrain-mediated anomalous Hall conductivity as a function of reducedmagnetization obtained by scaling down the magnitude of the exchange fieldinside Ru atoms finding good qualitative agreement with experimentalobservations, which indicates a strong impact of longitudinal thermalfluctuations of Ru spin moments on the anomalous Hall effect in this system.

Tailoring the anomalous Hall effect of SrRuO$_3$ thin films by strain: a first principles study