Spin-dependent Dirac electrons and valley polarization in the ferromagnetic stanene/$\mathrm{Cr}{\mathrm{I}}_3$ van der Waals heterostructure

Realization and regulation of valley polarization is a core issue in the valleytronics fields. Here, through first-principles calculations, we find that the valley polarization (up to 39.6 meV) can be realized in stanene, by stacking it on a monolayer ferromagnetic insulator $\mathrm{Cr}{\mathrm{I}}_3$, forming the ferromagnetic stanene/$\mathrm{Cr}{\mathrm{I}}_3$ van der Waals heterostructure. The heterostructure possesses a magneto band-structure effect, whereby the spin orientation of Cr atoms can significantly influence valley polarization. When the spin orientation of Cr atoms is parallel (perpendicular) to the $c$ axis, the largest (smallest) valley polarization is achieved with the value of 71.7 meV (1.7 meV). Additionally, the spin-polarized Dirac electrons and hole doping are induced in stanene due to the magnetic proximity effect, and the Dirac cone of stanene is opened with the value of 187.6 meV. These results indicate that the stanene/$\mathrm{Cr}{\mathrm{I}}_3$ heterostructure is very promising to be applied in future spintronics and valleytronics.

Figure 1

The (a) top and (b) side views of stanene/$\mathrm{Cr}{\mathrm{I}}_3$ heterostructure. The green and red arrows in (a) represent the shift directions, and the gray area in (b) represents equilibrium interlayer distance. (c) The binding energy as a function of the shift distance along the [100] and $\left[1\overline{1}0\right]$ directions in stanene/$\mathrm{Cr}{\mathrm{I}}_3$ heterostructure. (d) The first Brillouin zone and high symmetry points of the system.

Figure 2

(a) The spin-dependent PBS of stanene/$\mathrm{Cr}{\mathrm{I}}_3$ FM vdWH without SOC effect in (left) spin-up and (right) spin-down channels. The sizes of the red and green colors represent the projected weight of stanene and $\mathrm{Cr}{\mathrm{I}}_3$ components, respectively. (b) The PBS of stanene/$\mathrm{Cr}{\mathrm{I}}_3$ FM vdWH with SOC effect. The sizes of the black and green colors represent the projected weight of stanene and $\mathrm{Cr}{\mathrm{I}}_3$ components, respectively. The left is the PBS of $\mathrm{Cr}{\mathrm{I}}_3$ and the right is the PBS of stanene. The red value insert in the right band is the value of valley polarization ($\mathrm{\Delta }{E}_{K{K}^{\prime }}$). The Fermi level is set as zero.

Figure 3

The electrostatic potential of (a) stanene, (b) $\mathrm{Cr}{\mathrm{I}}_3$ monolayer and (c) stanene/$\mathrm{Cr}{\mathrm{I}}_3$ FM vdWH. (d) The transfer direction of electrons and the movement of the Fermi level in the process of forming heterostructure. The 3D surface of the charge density difference is also displayed in (c). The yellow and cyan areas represent electron accumulation and depletion, respectively, and the black arrows represent the direction of built-in electric field.

Figure 4

(a) The side view of structure model of the stanene/$\mathrm{Cr}{\mathrm{I}}_3$ FM vdWH. The right coordinate system represents the angle between the spin orientation of Cr atoms and the plane of heterostructure. (b) The values of valley polarization as a function of the angles (θ) between the spin direction of Cr atoms and the plane of heterostructure. (c) A schematic diagram of the characteristics of valley polarization at θ > 0° and θ < 0°.

Figure 5

The projected band structures of stanene with the effect of spin direction of the Cr atoms. The green and black colors denote the components of $\mathrm{Cr}{\mathrm{I}}_3$ and stanene, respectively. The values in the blue boxes represent the angles between the spin direction of Cr atoms and the plane of heterostructure and the values in red boxes are the valley polarization values.