Coupled spin-valley, Rashba effect and hidden persistent spin polarization in WSi$_2$N$_4$ family

The new two-dimensional materials, MoSi$_2$N$_4$ and WSi$_2$N$_4$ are experimentally synthesized successfully and various similar structures are predicted theoretically. Here, we report the electronic properties with a special focus on the band splitting in WA$_2$Z$_4$ (A=Si, Ge; Z=N, P, As), using state-of-the-art density functional theory and many-body perturbation theory (within the framework of G$_0$W$_0$ and BSE). Due to the broken inversion symmetry and strong spin-orbit coupling effects, we detect coupled spin-valley effects at the corners of the first Brillouin zone (BZ). Additionally, we observe cubically and linearly split bands around the $\Gamma$ and M points, respectively. Interestingly, the in-plane mirror symmetry ($\sigma_h$) and the reduced symmetry of arbitrary $k$-point, enforce the persistent spin textures (PST) to occur in full BZ. We induce the Rashba splitting by breaking the $\sigma_h$ through an out-of-plane external electric field (EEF). The inversion asymmetric site point group of the W atom introduces the hidden spin polarization in centrosymmetric layered bulk counterparts. Therefore, the spin-layer locking effect, namely, energy degenerate opposite spins spatially segregated in the top and bottom W layers, has been identified. Our low energy $k.p$ model demonstrates that the PST along the M-K line is robust to EEF and layer thickness, making them suitable for applications in spintronics and valleytronics.