The study of relativistic effects in solids, i.e., the spin-orbit coupling (SOC), has attracted increasing interest in the field of spintronics since it allows for manipulation of electron’s spin degree of freedom without additional external magnetic field. Many intriguing SOCrelated phenomena were observed, including spin relaxation, spin Hall effect, spin galvanic effect, and spin ballistic transport. In a system lacking inversion symmetry, the SOC induces momentum (k)-dependent spin–orbit field (SOF), Ω( ~ ~k) ∝ E~ × ~p, where E~ is the electric field originated from the crystal inversion asymmetry and ~p is the momentum, that lifts Kramer’s spin degeneracy and leads to the nontrivial k-dependent spin textures of the spin-split bands through the so-called Rashba and Dresselhaus effects. In particular, the Rashba effect attracted much attention since it can be manipulated electrically to produce non-equilibrium spin polarization[9, 10], which plays an important role for spin-field effect transistors (SFET).
Practically, materials with strong Rashba effect are desirable in spintronic applications since a large Rashba SOC parameter is in favor of room temperature device operations. However, the strong Rashba SOC is also known to induce the undesired effect of causing spin decoherence, which plays a detrimental role for the spin lifetime. Due to the kdependent SOF, electron scatterings by defect and impurity randomize the spin in a diffusive transport regime, which induces the fast spin decoherence through the Dyakonov-Perel spin relaxation mechanism, and hence limits the performance of the spintronics functionality. This problem can further be resolved by designing the materials supporting unidirectional SOF. Under the unidirectional SOF, the spin textures are uniformly oriented in the k-space, resulting in the so-called persistent spin texture (PST), therefore enabling long-range spin transport without dissipation through persistent spin helix (PSH) mechanism.
Previously, GaAs/AlGaAs and InGaAs/InAlAs heterostructures have been reported to demonstrate the emergence of the PSH state arising from a balance between the strength of the Rashba and Dresselhaus SOCs; however, these artificial structures require atomic precision by tuning the quantum-well width and carefully controlled carrier densities through doping level and and applied external electric field.
Recently, a different method for achieving the PST has been proposed, implemented by enforcing the symmetry of the crystal. While it remains to be confirmed experimentally, this method allows us to design the PST without requiring the fine-tuning between the Rashba and Dresselhaus SOCs. For example, the PST was predicted in noncentrosymmetric materials having mirror symmetry (or glide mirror symmetry) operations as recently reported on bulk BiInO3, CsBiNb2O7, and several two-dimensional (2D) systems including WO2Cl2, GaXY (X=Se, Te; Y =Cl, Br, I) and various group-IV monochalcogenide monolayers (MLs). Moreover, the symmetry-protected PST with purely cubic spin splitting has also been recently reported in bulk materials crystallizing in the ¯6m2 and ¯6 point groups. In addition, the PST driven by the lower symmetry of the crystal has also been reported on wurtzite ZnO [10¯10] surface and several 2D transition metal dichalcogenides (TMDCs) with the line defect. Although the symmetry-enforced PSTs have been extensively studied, most of them are locally occured in the small part around the high symmetry k points or lines in the first Brilouin zone (FBZ). This implies that the spin deviation away from the uniform spin configurations appears for the larger k-space region in the FBZ, which significantly induces spin scattering, and hence limits the stability of the PSH states. Therefore, finding novel structures supporting the PST covering a substantially large region of the FBZ is highly desirable, which is important to produce a highly stable PSH state without controlling Fermi level to the specific FBZ region.
Persistent spin texture (PST), a property of solid-state materials preserving a uniform spin configuration in the k-space, offers a route to deliver the necessary long carrier spin lifetimes through persistent spin helix (PSH) mechanism. However, most of the discovered PST are locally observed in the small part around certain high symmetry k-points or lines in the first Brilouin zone (FBZ), thus limiting the stability of the PSH state. Herein, by using first-principles calculations supplemented with ~k · ~p analysis, we report the emergence of full-zone PST in the monolayers of group IV-V A2B2 (A = Si, Sn, Ge; B = Bi, Sb) compounds. Due to in-plane mirror symmetry operation of the crystal and reduced symmetry of the arbitrary k-point in the FBZ, a fully out-ofplane spin texture preserves in the whole region of the FBZ. Importantly, we observed giant spin splitting in which the PST sustains, supporting large SOC parameters and small wavelengths of the PSH states. Our ~k · ~p analysis also demonstrated that the PST is robust for the k point along the specific high symmetry line under the external electric field, thus offering a promising platform for future spintronic applications.
FIG. 1. (a) Atomic structures of A2B2 MLs corresponding to the first Brillouin zone (FBZ) (b) are shown. The symmetry operations in the crystal including in-plane mirror symmetry Mxy, out-ofplane mirror symmetry Mxz, and three-fold rotation C3z are shown by the dashed red lines. The unit cell of the crystal is indicated by the blue lines. The A atoms represent the group IV elements (Si, Sn, Ge), while the B atoms show the group V elements (Bi, Sb). The FBZ is characterized by high symmetry ~k points as indicated by the Γ, X, Y , and M points. (c) Phonon dispersion spectra and (d) AIMD simulation calculated for the optimized structures of Si2Bi2 ML are shown.
FIG. 4. (a) Spin-resolved projected to the bands along Γ − M − K − Γ symmetry line is shown. The color bars represent the expected values of the Sx, Sy, and Sz spin components. (b) Schematic view of the spatially periodic mode of the spin polarization in the persistent spin helix (PSH) state is presented. The wavelength of the PSH, LPSH, is indicated by blue arrow. (c) Spin-resolved projected to the bands along the Γ − M − K lines calculated for the CBM is presented. (d) Spin-resolved projected to to the Fermi line calculated at constant energy cut of 0.1 eV above the degenerated state at the CBM around the M point is shown. The color bars represent the expected values of the Sz spin components. The expectation values of the in-plane spin components (Sx,Sy) for Figs. 3(c) and 3(d) are not shown here since their magnitude is zero.
FIG. 5. (a) Schematic view of A2B2 MLs under out-of-plane external electric field, Ez, where the broken of the in-plane mirror symmetry Mxy is indicated. (b) Spin-resolved projected to the bands of Si2Bi2 ML under Ez = 0:2 V/˚ A is shown along Γ − M − K − Γ symmetry line. The color bars represent the expected values of the Sx, Sy, and Sz spin components. (c)-(d) Spin-resolved projected to the bands of Si2Bi2 ML calculated along the Γ − M − K lines at the CBM with and without Ez is shown, respectively.
In summary, based on first-principles DFT calculations supported by ~k · ~p model, we systematically investigated the SOC-related properties of the group IV-V A2B2 (A = Si, Sn, Ge; B = Bi, Sb) ML compounds. We show that the existence of the in-plane mirror symmetry Mxy operation of the crystal together with the reduced symmetry of arbitrary k point in the FBZ lead to the full-zone PST, in which the unidirectional out-of-plane spin textures maintain in the whole region of the FBZ. Importantly, this PST exhibits giant spin splitting, which is particularly visible in the proximity of the CBM, supporting large SOC parameters and small wavelengths of the PSH states. Our ~k · ~p analysis based on the symmetry argument demonstrated that the PST is robust for the k point along the specific high symmetry line under the external electric field. Considering the fact that the present system demonstrated the full-zone PST with large spin splitting, this work opens a new avenue to achieve highly efficient spintronic devices operating at room temperatures.
Since the full-zone PST found in the present study are solely enforced by the in-plane mirror symmetry Mxy operation of the crystal, it is expected that this PST can also be achieved on other 2D materials having similar crystal symmetry. Therefore, our prediction is expected to trigger further theoretical and experimental studies in order to find novel 2D systems supporting the PST, which is useful for future spintronic applications.
