Switchable in-plane anomalous Hall effect by magnetization orientation in monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$

In-plane anomalous Hall effect (IPAHE) is an unconventional anomalous Hall effect (AHE) where the Hall current flows in the plane spanned by the magnetization or magnetic field and the electric field. Here, we predict a stable two-dimensional ferromagnetic monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$ with collinear ordering of Mn moments in the basal plane. Moreover, we reveal that monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$ possesses a substantial periodic IPAHE due to the threefold rotational symmetry, which can be switched by changing the magnetization orientation by external magnetic fields. In addition, we briefly discuss the impacts of moderate strains on the electronic states and the AHE, which lead to a near quantized Hall conductivity. Our work provides a potential platform for realizing a sizable and controllable IPAHE that greatly facilitates the application of energy-efficient spintronic devices.

Figure 1

(a) The top and side views of monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$, where Mn, Si, Te-top, and Te-bottom are represented by pink, blue, green, and brown balls, respectively. The nattier blue face represents the distorted Te octahedral. The energy variation depends on (b) the polar angle of magnetization in the basal plane and (c) the plane perpendicular to the $a$ axis of monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$.

Figure 2

(a) Band structure of monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$ with and without SOC. ${\mathcal{C}}_n$ labels the degenerate points. (b) The distributions of degenerate points in the first BZ. (c) The dominant components of PDOS of monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$ without SOC. (d) The illustration of ${\mathcal{C}}_2$ and mirror $\mathcal{M}$ symmetry at $\varphi =\pi /6$. (e) The calculated AHC vs Fermi energy with $\varphi$ at $0{}^{\circ }, 20{}^{\circ }, 30{}^{\circ }, 40{}^{\circ }$, and $60{}^{\circ }$. (f) The AHC vs different in-plane magnetization with a given Fermi energy of 0.07 eV. The values represent the magnitudes of AHCs, and the blue and orange colors represent negative and positive AHCs, respectively.

Figure 3

Band-resolved Berry curvature distribution for some selected magnetization orientations in $0{}^{\circ }$ (a), $20{}^{\circ }$ (b), $30{}^{\circ }$ (c), and $40{}^{\circ }$ (d). In the direction of $\varphi ={30}^{\circ }$, the closure of the energy gaps in the $\mathrm{\Gamma }$-M path leads to AHC vanishing. As the direction changes from $30{}^{\circ }$ to $0{}^{\circ }$, the gaps increase gradually, and the AHC achieves its maximal value near $30{}^{\circ }$. For the directions $\varphi ={20}^{\circ }$ and $\varphi ={40}^{\circ }$, the Berry curvatures are opposite.

Figure 4

The calculated energy bands with magnetization orientations at $\varphi =0{}^{\circ }$ along the $\mathrm{\Gamma }$-M-K-$\mathrm{\Gamma }$ line and the AHE for selected in-plane strains (a) ${a/a}_0=0.98$, (b) ${a/a}_0=0.99$, (c) ${a/a}_0=1.01$, and (d) ${a/a}_0=1.02$. ${a/a}_0$ represents modification of the lattice constant relative to the initial one $a_0$.

Figure 5

The calculated phonon dispersion of monolayer ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$.

Figure 6

FM and three AFM configurations with the Néel vector in $x, y$, and $z$ directions corresponding to Table 3.

Figure 7

The calculated energy band of six representative orientations of the moments.

Figure 8

Band-resolved Berry curvature distribution for some selected magnetization orientations in $0{}^{\circ }$ (a), $20{}^{\circ }$ (b), $30{}^{\circ }$ (c), and $40{}^{\circ }$ (d).

Figure 9

The AHC at different compressible strains along the $c$ axis of bulk ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$, where $c/c_0$ represents the ratio between the strained lattice parameter $c$ and the initial value.

Figure 10

The AHC at different tensile strains along the $c$ axis of bulk ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$, where $c/c_0$ represents the ratio between the strained lattice parameter $c$ and the initial value.