Nonvolatile electric field control of spin-valley-layer polarized anomalous Hall effect in a two-dimensional multiferroic semiconductor bilayer

The coupling of the lattice, charge, spin, and valley degrees of freedom is a fundamental scientific issue in spintronics and valleytronics. The nonvolatile electric field control of the spin and valley polarization has important potential applications for the development of next-generation ultrahigh speed processors and memories. However, the efficient electrical control of spin and valley degrees of freedom remains a challenge due to the weak coupling between them and external electric fields. Here we report the strong coupling between the spin, valley, and electric polarization in a two- dimensional multiferroic material ${\mathrm{Nb}}_3{\mathrm{I}}_8$ with a breathing Kagome lattice. The ferroelectric transition controls the local sublattice symmetry in the ferroelectric monolayer ${\mathrm{Nb}}_3{\mathrm{I}}_8$ and thus results in the sublattice-dependent Berry curvature switching in the momentum space, where the sign reversal of the Berry curvature can be controlled by reversing the ferroelectric polarization. More importantly, for the ferroelectric bilayer ${\mathrm{Nb}}_3{\mathrm{I}}_8$ with A-type antiferromagnetic coupling, the spin-valley-layer polarized anomalous Hall effect can be realized by coupling the electric polarization to spin, valley and layer degrees of freedom. So, the electrically driven the ferroelectric transition in a ${\mathrm{Nb}}_3{\mathrm{I}}_8$ bilayer can be applied to design nonvolatile memory and switch based on the spin, valley and layer dependent Berry curvature. Our findings open an avenue towards exploring the coupling between the ferroelectricity, ferromagnetism, and ferrovalley in the hidden local sublattice symmetry and demonstrate a nonvolatile electric-field controlled anomalous Hall effect in the atomically thin limit.

Figure 1

Schematic diagram of the structure of ${\mathrm{Nb}}_3{\mathrm{I}}_8$. (a) Side view of the bulk ${\mathrm{Nb}}_3{\mathrm{I}}_8$ crystal structure. (b) Top and side views of the breathing Kagome network of monolayers with opposite polarization directions. Nb atoms are in green, and I atoms are in purple. The direction of polarization is uniformly marked with yellow arrows. The Nb trimers are indicated by blue solid line triangles. (c), (d) 2D honeycomb lattice with two sublattices A and B. (e), (f) Berry curvature distribution in momentum space. $K_{+}$ and $K_{-}$ points are at opposite corners of the hexagonal Brillouin zone. (g), (h) Valley band and valley magnetic moment $\mathrm{m}\left(\mathrm{k}\right)$ near the $K_{+}$ and $K_{-}$ points.

Figure 2

Energy bands of the ${\mathrm{Nb}}_3{\mathrm{I}}_8$ monolayer with the SOC. Depending on the direction of the polarization and magnetization, four configurations can be distinguished: (a) $P\uparrow M\uparrow$, (b) $P\uparrow M\downarrow$, (c) $P\downarrow M\uparrow$, (d)$P\downarrow M\downarrow$. The red (blue) colors represent the bands of the spin projection in the positive (negative) directions of the z axis. (e)–(h) Berry curvatures calculated for (e) $P\uparrow M\uparrow$, (f) $P\uparrow M\downarrow$, (g) $P\downarrow M\uparrow$, (h) $P\downarrow M\downarrow$ configurations. (i)–(l) Schematic diagrams of anomalous Hall effects for (i) $P\uparrow M\uparrow$, (j) $P\uparrow M\downarrow$, (k) $P\downarrow M\uparrow$, (l) $P\downarrow M\downarrow$ configurations. The solid circles (hollow circles) indicate the holes in the $K_{+}\left(K_{-}\right)$ valley, and the upward red (downward blue) arrows represent the holes with spin-up (spin-down) states. The red (blue) curved arrows indicate the change of Berry curvature and hole transport processes under an applied external electric field $E$ (magnetic field $H$), while the green straight arrows denote the change under simultaneous action of $E$ and $H$.

Figure 3

Side and top views of the bilayer $\mathrm{AF}{\mathrm{E}}_{\mathrm{tt}}$ (a) and $\mathrm{AF}{\mathrm{E}}_{\mathrm{hh}}$ (b). The light green and light purple balls represent, respectively, the Nb and I atoms in the upper layer, while the dark green and dark purple balls denote, respectively, the Nb and I atoms in the lower layer. (c) Layer-resolved band structures of the $P_{\downarrow }^{\uparrow }{M}_{\uparrow }^{\downarrow }$ configuration with the SOC. The left side indicates the energy band contributions of the upper layer and the right side indicates the energy band contributions of the lower layer. (d) Under hole doping the layer-resolved spin Hall effect of the $P_{\downarrow }^{\uparrow }{M}_{\uparrow }^{\downarrow }$ configuration. (e) The schematic band structures under a positive or negative perpendicular electric field. The red and blue curves represent respectively the spin-up and spin-down bands.

Figure 4

The transition path of bilayer ${\mathrm{Nb}}_3{\mathrm{I}}_8$ from one FE state to the other opposite FE state.

Figure 5

(a)–(d) Energy bands of bilayer FE configurations with the SOC. Four configurations are classified according to different combinations of the electric polarization and magnetization directions, namely (a)$P_{\uparrow }^{\uparrow }{M}_{\downarrow }^{\uparrow }$, (b)$P_{\uparrow }^{\uparrow }{M}_{\uparrow }^{\downarrow }$, (c) $P_{\downarrow }^{\downarrow }{M}_{\downarrow }^{\uparrow }$, (d) $P_{\downarrow }^{\downarrow }{M}_{\uparrow }^{\downarrow }$. Layer-resolved bands and spin-resolved bands are shown respectively, where the orange circles and gray triangles represent respectively the contribution of the upper and lower layer, and the red and blue curves represent respectively the spin-up and spin-down channel. (e)–(h) Berry curvature calculated in the momentum space for (e) $P_{\uparrow }^{\uparrow }{M}_{\downarrow }^{\uparrow }$, (f) $P_{\uparrow }^{\uparrow }{M}_{\uparrow }^{\downarrow }$, (g) $P_{\downarrow }^{\downarrow }{M}_{\downarrow }^{\uparrow }$, (h) $P_{\downarrow }^{\downarrow }{M}_{\uparrow }^{\downarrow }$ configurations. (i)–(l) Schematic diagrams of the anomalous Hall effect for (i) $P_{\uparrow }^{\uparrow }{M}_{\downarrow }^{\uparrow }$, (j) $P_{\uparrow }^{\uparrow }{M}_{\uparrow }^{\downarrow }$, (k) $P_{\downarrow }^{\downarrow }{M}_{\downarrow }^{\uparrow }$, (l) $P_{\downarrow }^{\downarrow }{M}_{\uparrow }^{\downarrow }$ configurations.