Orbital Magneto-Nonlinear Anomalous Hall Effect in Kagome Magnet ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$

It has been theoretically predicted that perturbation of the Berry curvature by electromagnetic fields gives rise to intrinsic nonlinear anomalous Hall effects that are independent of scattering. Two types of nonlinear anomalous Hall effects are expected. The electric nonlinear Hall effect has recently begun to receive attention, while very few studies are concerned with the magneto-nonlinear Hall effect. Here, we combine experiment and first-principles calculations to show that the kagome ferromagnet ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ displays such a magneto-nonlinear Hall effect. By systematic field angular and temperature-dependent transport measurements, we unambiguously identify a large anomalous Hall current that is linear in both applied in-plane electric and magnetic fields, utilizing a unique in-plane configuration. We clarify its dominant orbital origin and connect it to the magneto-nonlinear Hall effect. The effect is governed by the intrinsic quantum geometric properties of Bloch electrons. Our results demonstrate the significance of the quantum geometry of electron wave functions from the orbital degree of freedom and open up a new direction in Hall transport effects.

Figure 1

Crystal structure and basic transport characterization of ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$. (a) Crystal structure. The crystal includes a honeycomb Sn layer sandwiched between kagome ${\mathrm{Fe}}_3\mathrm{Sn}$ bilayers (green slices). The left illustration shows the breathing kagome ${\mathrm{Fe}}_3\mathrm{Sn}$ layer. The gray dashed line indicates the mirror plane perpendicular to the $a$ axis. (b) Temperature-dependent longitudinal resistivity along the $a$ axis. The inset depicts the current direction. The $x$, $y$, and $z$ axes correspond to the $a$, $a_{\perp }$, and $c$ axes, respectively. (c),(d) Magnetoresistance [$\mathrm{MR}=\{[{\rho }_{xx}(H)-{\rho }_{xx}(0)]/{\rho }_{xx}(0)\}$] and Hall resistivity under an out-of-plane field at various temperatures, respectively.

Figure 2

In-plane Hall effect and its angular dependence. (a) Hall resistivity for the magnetic field rotating in the $zx$ plane at $T=50\mathrm{K}$. (b) Hall resistivity for the magnetic field rotating in the $yz$ plane at $T=50\mathrm{K}$. The top-left insets in (a) and (b) depict the current and magnetic field directions. The top-right inset in (b) shows ${\rho }_{\mathrm{IAHE}}$ from 0 to 7 T. (c) $H$-antisymmetric Hall resistivity ${\rho }_{xy}^{H\text{−}\mathrm{Asy}}$ for magnetic field rotated in the $xy$ plane at ${\mu }_{0}H=6\mathrm{T}$ and $T=5\mathrm{K}$. ${\rho }_{xy}^{H\text{−}\mathrm{Asy}}$ consists of two sinusoidal contributions with periods of $2\pi$ and $2/3\pi$. The $2\pi$ oscillation is contributed by the out-of-plane Hall component due to a misalignment sketched in the inset. (d) Angular dependence of ${\rho }_{\mathrm{IAHE}}^0$ (blue, left axis) and ${\rho }_{\mathrm{IAHE}}^H/{\mu }_{0}H$ (red, right axis) at $T=5\mathrm{K}$. The inset depicts the measurement configuration and mirrors (green dashed lines) of the crystal.

Figure 3

Temperature dependence of the in-plane Hall response. (a) Magnetic field dependence of the Hall resistivity for $H$ along the $y$ axis at various temperatures. The in-plane Hall signal consists of a jump around zero field and a $H$-linear dependence after that. (b) Hall conductivity ${\sigma }_{\mathrm{IAHE}}^0$ (blue, left axis) and the slope ${\sigma }_{\mathrm{IAHE}}^H/{\mu }_{0}H$ (red, right axis) for $H$ along the $y$ axis as functions of temperature. ${\sigma }_{\mathrm{IAHE}}^H/{\mu }_{0}H$ from the experiment and the first-principles calculations are compared in the inset.

Figure 4

Calculated AOP of ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ with magnetization in the $y$ direction. (a) The band projection of $yy$ component of AOP in units of ${\mu }_{\mathrm{B}}\AA /\mathrm{eV}$, which is concentrated in small-gap regions. (b) Calculated band structures around two Weyl points appearing within an energy window of 1 meV about the Fermi surface of ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$. The two Weyl points are located in the $k_x=0.793$ plane of the Brillouin zone. (c) Distribution of the $k$-resolved antisymmetrized AOP dipole on the Fermi surface, i.e., $f_0^{\prime }(v_y{F}_{yx}^{\mathrm{O}}-v_x{F}_{yy}^{\mathrm{O}})$, which is the integrand of Eq. (2) obtained via an integration by parts, in the $k_x=0.793$ plane of the momentum space. Here, $f_0^{\prime }=\partial f_0/\partial ϵ$. The black points show a pair of Weyl points in this plane, which are the hot spots of AOP dipole. The coordinate of M point is (0.793, 0, 0).