Valley spin polarization in two-dimensional h−MN (M=Nb,Ta) monolayers: Merger of valleytronics with spintronics

The search for new two-dimensional (2D) semiconductors with strong spin-orbit coupling, merging Rashba effect with valley physics, is essential for advancing the emerging fields of spintronics and valleytronics. $h\text{\ensuremath{-}}M\mathrm{N}$ $(M=\mathrm{Nb},\mathrm{Ta})$ monolayers are found to host valley physics together with Rashba effect. $h$-NbN (TaN) monolayers show Zeeman-type valley spin splitting of 32 (112) meV and 130 (406) meV at valence-band and conduction-band edges, respectively, based on density-functional theory calculations. One of the three degenerate valleys around the $K/{K}^{\ensuremath{'}}$ point in the first Brillouin zone (BZ) lies fully enclosed within the first BZ. Berry curvature $\ensuremath{\sim}50(73)\phantom{\rule{0.16em}{0ex}}{\AA{}}^{2}$ shown by h-NbN (TaN) monolayers is much higher than that in ${\mathrm{MoS}}_{2}$ monolayer $(\ensuremath{\sim}11\phantom{\rule{0.16em}{0ex}}{\AA{}}^{2})$, owing to the lower band gap and wave vector magnitude in $h$-NbN (TaN) monolayers. The Rashba energy splitting and Rashba constant induced by strong spin-orbit coupling (SOC) in $h$-NbN (TaN) monolayer is found to be 52 (74) meV and 2.9 (4.23) eV \AA{} respectively, which are amongst the giant Rashba spin-splitting parameters observed so far in 2D materials. The degree of Zeeman-type valley spin splitting and Rashba-type spin splitting is substantially tunable via in-plane biaxial strain and out of-plane electric field. Higher SOC in $h$-TaN monolayer relative to $h$-NbN monolayer caused by the heavier Ta atoms gives rise to a higher Berry curvature, valley-, and Rashba spin splitting in the former as compared to the latter. Valleytronic and spintronic properties in the studied monolayers are found to be superior to that in $h\text{\ensuremath{-}}{\mathrm{MoS}}_{2}$ and Janus MoSSe monolayers and are therefore proposed for an effective coupling of spin and valley physics.