Magnetoelectric Coupling in Multiferroic Bilayer ${\mathrm{VS}}_2$

Based on the first-principles prediction, we report the magnetoelectric coupling effect in two-dimensional multiferroic bilayer ${\mathrm{VS}}_2$. The ground-state $3R$-type stacking breaks space inversion symmetry, therefore introducing a spontaneous polarization perpendicular to the layer plane. We further reveal that the out-of-plane ferroelectric polarization of bilayer ${\mathrm{VS}}_2$ can be reversed upon interlayer sliding of an in-plane translation. Each ${\mathrm{VS}}_2$ layer has a ferromagnetic state with an opposite magnetic moment between two antiferromagnetically ordered layers. We found that ferroelectricity and antiferromagnetism can be coupled together by a ferrovalley in bilayer ${\mathrm{VS}}_2$ to realize electronic control of magnetism. Remarkably, a net magnetic moment is generated by reducing the interlayer distance, and an electric field is able to achieve linear and second-order nonlinear magnetoelectric coupling in bilayer ${\mathrm{VS}}_2$.

Figure 1

(a) and (b) are the side views of the bilayer ${\mathrm{VS}}_2$ crystal structures in which the ferroelectric polarizations are antiparallel and parallel to positive $z$ axes, respectively. The gray and red spheres indicate the top layer and bottom layer V atoms, respectively; the dark yellow and bright yellow spheres are for the top layer and bottom layer S atoms, respectively. The band structure with SOC of ferroelectric antiferromagnetic configurations $P\downarrow M\uparrow \downarrow$ and $P\uparrow M\downarrow \uparrow$ is shown in (c), and the Berry curvature distribution in the first Brillouin zone is shown in (d). The band structure with SOC of configurations $P\downarrow M\downarrow \uparrow$ and $P\uparrow M\uparrow \downarrow$ is shown in (e), and the Berry curvature is shown in (f). In (c) and (e), the red (blue) colors represent the bands of the spin projection in the positive (negative) directions of the $z$ axis (spin up or spin down).

Figure 2

Multistate control of two-dimensional multiferroic bilayer ${\mathrm{VS}}_2$. The red and gray arrows represent the spin directions of the bottom layer and top layer ${\mathrm{VS}}_2$, respectively. The blue arrows represent the ferroelectric polarization direction of the system. At the two energy valleys $K_{-}$ and $K_{+}$, red and blue curves represent bands where the spin projection is positive (spin-up) and negative (spin-down) along the $z$ axis, respectively.

Figure 3

Total magnetic moments (without SOC) on bilayer ${\mathrm{VS}}_2$ as a function of the interlayer distance variation and external electric field. The definition of the bilayer ${\mathrm{VS}}_2$ interlayer distance $d$ is shown in (a), which is the distance between two V atomic layers. The interlayer distance of the equilibrium state obtained through structural optimization is defined as $d_0$, and ${\delta }_d$ is the decrease of the interlayer distance $d$ as shown in (b). The net magnetic moment of the bilayer ${\mathrm{VS}}_2$ system increases with the increase of ${\delta }_d$. (c) and (d) show the relationship between the net magnetic moments and the external electric field when ${\delta }_d=0.6$ and 0.8 Å, respectively. The red and blue data points and lines are the calculated and fitting results.