Fig. 4: Emergence of topological Hall effect in H_xSrRuO₃.

a Ex-situ measured proton distribution profile in a sample (~32 nm) with topological Hall effect. b P-polarized SHG intensity as a function of the polarization direction of the incident light (0 corresponds to s-polarization) for both pristine SrRuO₃ and gated H_xSrRuO₃ films. The weaker SHG in the pristine film is due to surface contributions, while the enhanced SHG intensity of the H_xSrRuO₃ state suggests the breaking of inversion symmetry in bulk. c Magnetic field dependent Hall resistivity for H_xSrRuO₃ gated with V_G = 1.8 V at different temperatures. The blue and red arrows denote the field sweeping direction. Ordinary Hall term is subtracted through the linear fitting of R_HB at higher magnetic fields. An offset is applied per curve for clarity, while the dotted lines denote the center of the hysteresis loops. The estimated topological Hall resistivity with different signs is marked with different colors. d Color map of estimated topological hall resistivity \(\left( {\rho _{YX}^{\mathrm{T}}} \right)\) and characteristic fields (H_C and H_P) obtained at H_xSrRuO₃ gated with V_G = 1.8 V. H_C (black filled symbol) represents the coercive field and the H_P (white open diamond) denotes the field where the topological Hall resistivity reaches its maximum.