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We experimentally investigate second-harmonic transverse voltage response toac electrical current for a magnetic nodal-line semimetal Fe$_3$GeTe$_2$. Forzero magnetic field, the observed response depends as a square of thelongitudinal current, as it should be expected for non-linear Hall effect. Themagnetic field behavior is found to be sophisticated: while the first-harmonicresponse shows the known anomalous Hall hysteresis in FGT, the second-harmonicHall voltage is characterized by the pronounced high-field hysteresis and flat($B$-independent) region with curves touching at low fields. The high-fieldhysteresis strongly depends on the magnetic field sweep rate, so it reflectssome slow relaxation process. For the lowest rates, it is also accomplished bymultiple crossing points. Similar shape of the second-harmonic hysteresis isknown for skyrmion spin textures in non-linear optics. Since skyrmions havebeen demonstrated for FGT by direct visualization techniques, we can connectthe observed high-field relaxation with deformation of the skyrmion lattice.Thus, the second-harmonic Hall voltage response can be regarded as a tool todetect spin textures in transport experiments.

Second-harmonic response as a tool to detect spin textures in magnetic nodal-line semimetal Fe$_3$GeTe$_2$