Tunable spin polarization and electronic structure of bottom-up synthesized ${\mathrm{MoSi}}_2{\mathrm{N}}_4$ materials

Manipulation of spin-polarized electronic states of two-dimensional (2D) materials under ambient conditions is necessary for developing new quantum devices with small physical dimensions. Here, we explore spin-dependent electronic structures of ultra-thin films of recently introduced 2D synthetic materials $M{\mathrm{Si}}_2{Z}_4$ $(M=\mathrm{Mo} \mathrm{or} \mathrm{W} \mathrm{and} Z=\mathrm{N} \mathrm{or} \mathrm{As})$ using first-principles modeling. Stacking of $M{\mathrm{Si}}_2{Z}_4$ monolayers is found to generate dynamically stable bilayer and bulk materials with thickness-dependent properties. When spin-orbit coupling (SOC) is included in the computations, $M{\mathrm{Si}}_2{\mathrm{N}}_4$ monolayers display indirect band gaps and large spin-split states at the $K$ and $K^{\prime }$ symmetry points at the corners of the Brillouin zone with nearly 100% spin polarization. The spins are locked in opposite directions along an out-of-the-plane direction at $K$ and $K^{\prime }$, leading to spin-valley coupling effects. As expected, spin polarization is absent in the pristine bilayers due to the presence of inversion symmetry, but it can be induced via an external out-of-plane electric field much like the case of $\mathrm{Mo}\left(\mathrm{W}\right){\mathrm{S}}_2$ bilayers. A transition from an indirect to a direct band gap can be driven by replacing N by As in $M{\mathrm{Si}}_2{(\mathrm{N},\mathrm{As})}_4$ monolayers. Our study indicates that the $M{\mathrm{Si}}_2{Z}_4$ materials can provide a viable alternative to the ${\mathrm{MoS}}_2$ class of 2D materials for valleytronics and optoelectronics applications.

Figure 1

Atomic arrangement of (a) four and (b) two layers of ${\mathrm{MoSi}}_2{\mathrm{N}}_4$ with AB stacking. The dashed box identifies the bulk unit cell of 2H-${\mathrm{MoSi}}_2{\mathrm{N}}_4$. The red dot in the middle of the van der Waals gap in (b) marks the spatial center of inversion, which is absent in the monolayer. (c) Top view of monolayer ${\mathrm{MoSi}}_2{\mathrm{N}}_4$. (d) Mo-N trigonal and (e) Si-N tetrahedral local coordination structures in ${\mathrm{MoSi}}_2{\mathrm{N}}_4$ monolayers. The calculated phonon dispersion of (f) monolayer (1ML), (g) bilayer (2ML), and (h) bulk ${\mathrm{MoSi}}_2{\mathrm{N}}_4$.

Figure 2

Orbitally resolved band structure of monolayer ${\mathrm{MoSi}}_2{\mathrm{N}}_4$ (a) without and (b) with spin-orbit coupling (SOC). Spin-resolved bands around $K$ along the $\mathrm{\Gamma }\text{−}K$ direction for (c) $E_z$ = 0 eV/Å and (d) $E_z$ = 0.03 eV/Å with SOC. The color bar in (d) denotes the degree (in percent) of spin polarization. (e) Spin-polarization decay profile of the states at the top of the valence band around the $K$ point. Large spin polarization ($>99.9\%$) persists over a wide momentum range along the $\mathrm{\Gamma }\text{−}K$ direction. (f) Schematic representation of spin-valley locking in monolayer ${\mathrm{MoSi}}_2{\mathrm{N}}_4$. Red (blue) color represents spin pointing out of (into) the plane.

Figure 3

(a) Band structure of bilayer ${\mathrm{MoSi}}_2{\mathrm{N}}_4$ in the absence of external electric field ($E_z$ = 0). (b) Same as (a) but for $E_z=0.03$ eV/Å. Spin splitting in the band structure is evident. (c) Evolution of the top four valence bands around the $K$ point with varying external electric field strength. The color scale gives the degree (in percent) of spin-polarization of the bands. Markings 1 and 2 identify the doublets associated with the first and second layers of the bilayer. (d) Degree of spin-splitting at the $K$ point as a function of $E_z$. Blue (red) markers show the intralayer (interlayer) ${\mathrm{\Delta }}_{\mathrm{intra}}$ (${\mathrm{\Delta }}_{\mathrm{inter}}$) components of the spin splitting. (e) A schematic of the electric-field effect on the bilayer band structure.

Figure 4

(a) Orbitally resolved band structure of bulk ${\mathrm{MoSi}}_2{\mathrm{N}}_4$ in the bulk hexagonal Brillouin zone. (b) Band gap and (c) average spin-polarization of the top layer as a function of the number of layers.

Figure 5

Calculated phonon spectrum of (a) monolayer (1ML), (b) bilayer (2ML), and (c) bulk ${\mathrm{MoSi}}_2{\mathrm{As}}_4$. Orbitally decomposed band structure of (d) monolayer, (e) bilayer, and (f) bulk ${\mathrm{MoSi}}_2{\mathrm{As}}_4$.