Tunable valley-spin splitting in a Janus $XM{\mathrm{SiN}}_2$ monolayer $(X=\mathrm{S}, \mathrm{Se}; M=\mathrm{Mo}, \mathrm{Cr})$ and giant valley polarization via vanadium doping

Exploring spin-valley coupling in two-dimensional (2D) materials with strong spin-orbit coupling (SOC) is of great significance for fundamental physics and practical applications. Using first-principles calculations, we investigate the valley-related properties of Janus $XM{\mathrm{SiN}}_2$ $(X=\mathrm{S}, \mathrm{Se}; M=\mathrm{Mo}, \mathrm{Cr})$ monolayer. The Janus $XM{\mathrm{SiN}}_2$ monolayer forms a pair of nonequivalent valleys, and the conduction and valence bands are degenerated at the valleys. The inversion symmetry breaking and the SOC effect induce remarkable valley spin splitting and Rashba spin splitting. Our calculations indicate that not only valley-contrasting transport properties but also optical selection rules with spin-valley coupling result in the coexistence of spin and valley Hall effects in the Janus $XM{\mathrm{SiN}}_2$ monolayer. Moreover, we demonstrate that the valley-spin physics of the Janus $XM{\mathrm{SiN}}_2$ monolayer can be modulated by in-plane biaxial strains, allowing its extraordinary potential for spintronics and valleytronic applications. We also show that V-doped ${\mathrm{SMoSiN}}_2$ monolayer can exhibit giant valley polarization of 89.51 meV $(-24.48 \mathrm{meV})$ for the valence (conduction) band. These findings could be helpful for the valleytronic applications of the Janus $XM{\mathrm{SiN}}_2$ monolayer.

Figure 1

Top and side views of the atomic structures of the $XM{\mathrm{SiN}}_2$ monolayers after structural optimization. The red, lavender, blue, yellow, green, and gray spheres representing Cr, Mo, Si, S, Se, and N atoms, respectively.

Figure 2

The calculated phonon spectra for the (a) ${\mathrm{SMoSiN}}_2$, (b) ${\mathrm{SCrSiN}}_2$, (c) ${\mathrm{SeMoSiN}}_2$, and (d) ${\mathrm{SeCrSiN}}_2$ monolayers, respectively.

Figure 3

The orbital-projected band structure of the ${\mathrm{SMoSiN}}_2$ monolayer (a) without SOC and (b) with SOC. The Brillouin zone with some high-symmetry points is shown as a inset in (b). (c) Representation of valley spin splitting at the $K/K^{\prime }$ valleys and optical transitions, where the red and blue parabolas represent the spin-up and the spin-down bands, respectively. The red (${\omega }_u$) and blue (${\omega }_d$) arrows represent the minimal energies for spin-up and spin-down optical transitions at the $K$ valley. The orbital-projected band structures of the ${\mathrm{SCrSiN}}_2$ monolayer (d) without SOC and (e) with SOC. The (f) projected density of states (PDOS) of the ${\mathrm{SCrSiN}}_2$ monolayer. The Fermi level $E_F$ is set to 0.

Figure 4

The band structures of (a) the monolayered ${\mathrm{SeMoSiN}}_2$ and (b) the monolayered ${\mathrm{SeCrSiN}}_2$. The top and bottom figures refer to the calculated electronic structures without and with SOC, respectively.

Figure 5

The spin projected band structures of (a) the ${\mathrm{SMoSiN}}_2$ and the (b) ${\mathrm{SCrSiN}}_2$ monolayers with SOC. The color bars represent the weights of the spin component $S_Z$, and red (blue) is spin-up (spin-down) component.

Figure 6

The magnified valence bands calculated by (a) PBE and (b) $\mathrm{PBE}+\mathrm{SOC}$. The red and black lines represent the energy bands of the ${\mathrm{SeMoSiN}}_2$ and the ${\mathrm{SMoSiN}}_2$ monolayers, respectively.

Figure 7

The spin textures of the (a) second highest occupied state and (b) highest occupied state for the monolayered ${\mathrm{SMoSiN}}_2$. The spin textures of the (c) second highest and (d) highest occupied states for the ${\mathrm{SeMoSiN}}_2$ monolayer. The arrow represents the in-plane component of spin polarization, and the color represents the out-of-plane component of spin polarization, i.e., red (blue) is spin-up (spin-down).

Figure 8

The band structures of the ${\mathrm{SMoSiN}}_2$ monolayer under different strains and SOC is included in the calculation.

Figure 9

The valley-related spin splitting, which is calculated by Eq. (2), as a function of in-plane strains for (a) the ${\mathrm{SMoSiN}}_2$ and (b) the ${\mathrm{SCrSiN}}_2$ monolayers. The optical transition energies (${\omega }_u$ and ${\omega }_d$), as schematically shown in Fig. 3, of the (c) ${\mathrm{SMoSiN}}_2$ and the (d) ${\mathrm{SCrSiN}}_2$ monolayers varied by strains.

Figure 10

The spin projected band structures of [(a) and (b)] ${\mathrm{SMoSiN}}_2$ and [(c) and (d)] ${\mathrm{SCrSiN}}_2$ monolayers under certain tensile strains with SOC. The color bar represents the weight of the spin component $S_z$, and red (blue) is spin-up (spin-down).

Figure 11

Optical absorption of (a) the ${\mathrm{SMoSiN}}_2$, ${\mathrm{SCrSiN}}_2$, and ${\mathrm{MoSi}}_2{\mathrm{N}}_4$ monolayers. Optical absorption of (b) the ${\mathrm{SMoSiN}}_2$ monolayer under different in-plane strains.

Figure 12

The Berry curvature distribution for (a) the ${\mathrm{SMoSiN}}_2$ and (b) the ${\mathrm{SCrSiN}}_2$ monolayer, respectively. The contributions from all valence bands are considered. The circular polarizations of (c) the ${\mathrm{SMoSiN}}_2$ and (d) the ${\mathrm{SCrSiN}}_2$ monolayers, respectively. (e) The spin and valley Hall effect diagram of hole-doped ${\mathrm{SMoSiN}}_2$ and ${\mathrm{SCrSiN}}_2$ monolayers. The blue/red sphere represents the hole in the $K/K^{\prime }$ valley, respectively, and the up/down arrow represents spin-up/down, respectively. (f) The spin and valley Hall effect diagram of the ${\mathrm{SMoSiN}}_2$ and the ${\mathrm{SCrSiN}}_2$ monolayers under the illumination of circularly polarized light at a specific frequency. The $\pm$ symbol in the sphere indicates hole/electron carrier.

Figure 13

The Berry curvature ${\mathrm{\Omega }}_Z$ distributions for the (a) ${\mathrm{SeMoSiN}}_2$ and (b) ${\mathrm{SeCrSiN}}_2$ monolayers with all valence bands are included, respectively. The circular polarization of the monolayered (c) ${\mathrm{SeMoSiN}}_2$ and (d) ${\mathrm{SeCrSiN}}_2$, respectively.

Figure 14

The distribution of the Berry curvature ${\mathrm{\Omega }}_Z$ in the whole BZ for the strained ${\mathrm{SMoSiN}}_2$ monolayer with (a) 6% and (b) $-6$% in-plane biaxial strains, respectively. The distribution of circular polarization of ${\mathrm{SMoSiN}}_2$ monolayer in the whole BZ under in-plane strains of (c) 6% and (d) $-6$%.

Figure 15

(a) The V-doped ${\mathrm{SMoSiN}}_2$ monolayer. The yellow and green isosurfaces represent spin up and spin down charges, respectively. The isosurface value is set to $0.01\mathrm{e}/{\AA }^3$. (b) The splitting of $d$ orbitals for the V atom owing to the trigonal prismatic crystal field. (c) The band structure and (d) orbital-projected density of states of the V-doped ${\mathrm{SMoSiN}}_2$ monolayers without SOC. The red and blue lines represent spin-up and spin-down components, respectively. (e) The band structure of the V-doped ${\mathrm{SMoSiN}}_2$ monolayer with SOC. The inset shows the amplified details of the conduction (left) and valence (right) bands at the $K/K^{\prime }$ valleys, respectively.