Magnetically controllable nonlinear valley Hall effect in centrosymmetric ferromagnets

Abstract

Valley Hall effect is fundamental to valleytronics and provides a promising avenue for advancing information technology. While conventional valley Hall effect requires the inversion symmetry breaking, the recently proposed nonlinear valley Hall (NVH) effect removes the symmetry constraint, and broaden material choices. However, existing studies are limited to nonmagnetic materials without spin involvement and rely on external strain to break rotational symmetry. Here, to address these limitations, we design a magnetically controllable NVH effect in centrosymmetric ferromagnets, by the tight-binding model and first-principles calculations. The model calculations demonstrate nonvanishing NVH conductivities can emerge in pristine hexagonal lattice without external strain, with the magnitude, sign, and spin polarization of the conductivities being all dependent on the magnetization orientation. The effect thus generates various spin-polarized valley Hall currents, characterized by distinct combinations of current direction and spin polarization. First-principle results on a ferromagnetic VSi$_2$N$_4$ bilayer confirm considerable NVH conductivities and their dependence on the magnetization. The magnetically controllable NVH effect unlocks the potential of centrosymmetric magnets for valleytronics, and offer opportunities for novel spintronic and valleytronic devices.

Figures (4)

• Figure 1: Electronic band structure and NVH conductivities of the tight-binding model of a centrosymmetric ferromagnet, with the magnetization along the $x$-axis. (a) Electronic bands with dominating spin components being along the $x$-axis. (b) Evolutions of $\chi^{\text{NVH}}_{xyy}$ and $\chi^{\text{NVH}}_{yxx}$ as functions of the chemical potential $\mu$, where the energy window is around the conduction band edge. In (a) and (b), the conduction band minima are set to zero energy. (c) and (d) Momentum-resolved $\Lambda_{xyy}$ in small regions centered at $K_{+}$ and $K_{-}$, respectively, where $\mu=15$ meV that is near the positive peak of $\chi^{\text{NVH}}_{xyy}$ in (b). The locations of $K_\pm$ points are denoted by black dots. $q_x$ and $q_y$ are wave vectors with respect to $K_\pm$ points.
• Figure 2: Magnetically controllable NVH conductivities. (a) Evolutions of $\chi^{\text{NVH}}_{xyy}$ and $\chi^{\text{NVH}}_{yxx}$ with the angles $\phi$ and $\theta$ at $\mu=15$ meV, respectively. $\theta$ and $\phi$ are set to zero in (a) and (b), respectively. (c)-(e) Schematic depictions of the NVH effect with the magnetization along the $x$, $-x$, and $y$ directions, respectively. Red and blue balls stand for carriers from $K_+$ and $K_-$ valleys, respectively, with arrows representing spin directions. Cyan and purple arrows denote directions of the NVH current and the spin polarization, respectively.
• Figure 3: Crystal and electronic band structures of the AA$^\prime$-stacked VSi$_2$N$_4$ bilayer. (a) Top view and (b) side view of the bilayer. Red, blue and white balls stand for V, Si and N atoms, respectively. (c) Band structure of the bilayer, with the magnetization along the $x$-axis. (d) Enlarged band structures around the conduction and valence band edges. In (c) and (d), the valence band maxima are set to zero energy.
• Figure 4: NVH conductivities in the VSi$_2$N$_4$ bilayer. (a) Evolutions of $\chi^{\text{NVH}}_{xyy}$ and $\chi^{\text{NVH}}_{yxx}$ as functions of the chemical potential $\mu$, with the valence band maxima set to zero energy. (b) Evolutions of $\chi^{\text{NVH}}_{xyy}$ and $\chi^{\text{NVH}}_{yxx}$ as functions of the angle $\phi$, with $\theta=0$. (c) Evolutions of $\chi^{\text{NVH}}_{xyy}$ and $\chi^{\text{NVH}}_{yxx}$ as functions of the angle $\theta$, with $\phi=0$. $\mu$ is chosen to be 415 meV in (b) and (c).