Direct observation of local Rashba spin polarization and spin-layer locking in centrosymmetric monolayer PtSe$_2$

The generally accepted view that spin polarization is induced by the asymmetry of the global crystal space group has limited the search for spintronics [1] materials to non-centrosymmetric materials. Recently it has been suggested that spin polarization originates fundamentally from local atomic site asymmetries [2], and therefore centrosymmetric materials may exhibit previously overlooked spin polarizations. Here by using spin- and angle-resolved photoemission spectroscopy (spin-ARPES), we report helical spin texture induced by local Rashba effect (R-2) in centrosymmetric monolayer PtSe$_2$ film. First-principles calculations and effective analytical model support the spin-layer locking picture: in contrast to the spin splitting in conventional Rashba effect (R-1), the opposite spin polarizations induced by R-2 are degenerate in energy while spatially separated in the top and bottom Se layers. These results not only enrich our understanding of spin polarization physics, but also may find applications in electrically tunable spintronics.

The new insight that spin polarization in nonmagnetic materials originates from relativistic spin-orbit coupling (SOC) from the local asymmetry (atomic site group) [2][3][4][5] rather than the global asymmetry (bulk space group) has revolutionized our understanding of spin polarization physics. This has led to two forms of hidden spin polarization in centrosymmetric materials, R-2 (by site dipole field) and D-2 (by site inversion asymmetry) to be distinguished from the conventional Rashba (R-1, by dipole field) [6][7][8][9][10][11][12][13] and Dresselhaus (D-1, by bulk inversion asymmetry) [14] effects previously discovered in non-centrosymmetric materials. Different from the R-1 effect with a net spin polarization, the R-2 effect is characterized by compensated spin polarizations of opposite signs that are spatially segregated into two real space sectors forming the inversion partners, e.g. top and bottom layers (see schematic cartoon in Fig. 1). Compared to the R-1 effect with a large internal electric field which is difficult to be reversed by an external field, the spins induced by R-2 might have advantages for electrically tunable spintronics devices due to the easy manipulation via the application of an external electric field [15,16]. Layered transition metal dichalcogenides (TMDs) [17][18][19][20] are good candidates for realizing the R-2 effect due to the large SOC and large site dipole field. Although hidden spin polarization has been recently reported in the bulk crystal of WSe 2 [4] and predicted in other layered materials, e.g. LaOBiS 2 [15,16] and (LaO) 2 (SbSe 2 ) 2 [21] films, so far stable semiconducting thin films with large spin polarization induced by the R-2 effect still remain to be realized experimentally.
Monolayer PtSe 2 contains one Pt layer sandwiched between two Se layers, forming trigonal structure when projected onto the (001) plane ( Fig. 2(a)). It has centrosymmetric space group P3m1 for bulk structure, polar point group C 3 for both Pt and Se sites, and it is semiconducting [22]. These properties make it a promising candidate for realizing electrically tunable spintronics by R-2 effect [2]. First-principles calculations ( Fig. 2(b)) show that the top three valence bands (labeled by α, β and γ) are mostly contributed by the p orbitals of Se, and the fourth valence band (labeled by δ) is mainly contributed by the d orbitals of Pt. From the calculations, all the bands should be doubly degenerate without any net spin polarization since monolayer PtSe 2 has both inversion and time-reversal symmetries.
Here, by combining a full three-dimensional spin analysis using spin-ARPES and theoretical calculations, we report the hidden helical spin texture and spin-layer locking in monolayer PtSe 2 induced by R-2 effect.
Figure 2(c) shows the low energy electron diffraction (LEED) pattern of the high quality monolayer PtSe 2 thin film grown on Pt(111) substrate [22]. The semiconducting property of monolayer PtSe 2 has been reported [22]. ARPES data measured along two high symmetry directions M-Γ-M ( Fig. 2(d)) and K-Γ-K ( Fig. 2(e)) show similar dispersions, suggesting that the electronic structure is overall rather isotropic. Correspondingly, circular shape is observed in the constant energy maps ( Fig. 2(g)) for all the bands, and hexagonal warping is observed at relatively high binding energy. Analysis from energy distribution curves (EDCs) shows that the measured dispersions are in good agreement with first-principles calculations ( Fig. 2(b)) and the four valence bands are separated from each other at the Γ point. Figure 3 shows the three dimensional spin analysis for data measured along the M-Γ-M direction. A large spin contrast is observed along the tangential direction (θ) for β, γ and δ bands at emission angle of 7.5°(dashed line in Fig. 3(a)) with up to 50% polarization ( Fig. 3(c)), while negligible spin contrast is observed along the radial (r) and out-of-plane (⊥) directions ( Figs. 3(b,d)). The spin directions are illustrated by blue crosses (into the plane) and red dots (out of the plane) in Fig. 3(a). We extend the in-plane tangential spin analysis to other momenta along the M-Γ-M direction. polarization along the in-plane tangential direction with opposite signs on the two sides of the Γ point suggests that these bands may exhibit helical spin texture. Figure 4 shows the spin analysis for data measured along the K-Γ-K direction. A large spin polarization is also observed for the tangential component (Figs. 4(b, c)), in agreement with helical spin texture. The α band shows negligible spin polarization near the Γ point.
When moving to a large emission angle along the Γ-K direction, the α band is separated from other bands and we can resolve its spin polarization more easily here. In Fig. 4(d), spin polarization is observed in the α band and its direction is determined to be the same as β band. Considering that the electronic structure is rather isotropic, the observed large spin polarization for the tangential direction along both Γ-K and Γ-M directions suggests that the spin polarization has an overall helical texture. Different from the Γ-M direction, a small yet detectable spin polarization of ≈ 5% is observed in the radial component for drawing in Fig. 4(h).
The observed helical spin texture is fundamentally different from that of the R-1 effect.
For the conventional R-1 Rashba effect, two spin-splitting bands possess opposite spin helicities and their eigen-energies become degenerate at the Γ point (k = 0) due to TR symmetry, as shown in Fig. 1(a). In contrast, all these four bands are well split at the Γ point, as clearly shown in Fig. 2(f). In additional, the α and β bands share the same spin helicities, completely different from the R-1 effect. Compared to the first-principles calculations in Fig. 2(b), it seems confusing that although energy dispersion agrees well between theory and experiments, spin texture does not appear in the calculations due to double degeneracy 8 of two spin bands. This discrepancy can be understood by recognizing the fact that the light for ARPES measurements has a limited penetration depth and thus the top Se layer contributes more significantly to the observed spin polarization. To test this idea, we project spin polarization into the top and bottom Se layers and helical spin textures indeed emerge in each Se layer, as shown in Fig. 5(a). Strikingly, the obtained spin textures on the top Se layer in Fig. 5(a) reproduce qualitative features observed in spin-ARPES measurements for all four bands. For example, the α and β bands indeed have the same spin polarization while those of γ and δ bands are opposite. For the α band, the spin polarization is negligible compared to that of the β band close to the Γ point. The consistency between the calculations and experiments suggests that layer-dependent spin texture (Fig. 5(c)), which was also theoretically proposed in other layered materials including LaOBiS 2 [2,15] and More theoretical understanding can be obtained by analyzing the orbital natures of these four bands at the Γ point (the atomic limit), as shown in Fig. 5(b). It is found that the p orbitals of Se atoms dominates the bands near the Fermi energy and the d orbitals of Pt atoms are fully occupied. The strong anisotropy introduce a strong energy splitting between the p x,y orbitals and the p z orbital of Se atoms, pushing Se p z orbitals to lower energy. As a result, the conduction and valence bands around the band gap mainly consist of the p x,y orbitals of Se atoms. The hybridization of the Se p x,y orbitals between the top and bottom Se layers can be mediated by the central Pt layer and leads to a band gap opening between the conduction and valence bands, which are formed by the bonding and anti-bonding states of Se p x,y orbitals, respectively. After taking into account SOC, we find that the α and β bands correspond to the bonding states of Se p x,y orbitals with total z-direction angular momentum ± 3 2 and ± 1 2 , respectively. We explicitly show the atomic orbital form of the basis wave function and construct the corresponding low energy effective Hamiltonian for the α and β bands based on the symmetry principle in the Supplementary Information.
The effective Hamiltonian clearly shows layer dependent spin textures for the β bands. In addition, it is found that the vanishing spin texture of the α band at a small momentum results from the ± 3 2 angular momentum, in contrast to the ± 1 2 angular momentum of the β 9 band.
In summary, for the first time, we report the experimental realization of unconventional R-2 Rashba effect in a centrosymmetric monolayer PtSe 2 thin film. Considering that the monolayer PtSe 2 sample is very stable, only monolayer thick and semiconducting, growth or transfer of such thin film onto insulating substrates may provide exciting opportunities to realize electrically controllable spintronics devices.

Methods
Experiments. The PtSe 2 thin film was grown by direct selenization of Pt(111) [22]. The Calculations. The first-principles calculations are performed using the density functional theory as implemented in the Vienna ab initio simulation package [24] with the projector augmented-wave method. Perdew-Burke-Ernzerhof parametrization of the generalized gradient approximation is used for the exchange-correlation potential [25]. We adopt a default plane-wave energy cutoff, and the Brillouin zone is sampled by a Γ-centered 6 × 6 × 1 k-point mesh. The monolayer structure of PtSe 2 is modeled with a vacuum region more than 15 A thick to eliminate the spurious interaction between neighboring layers. A perpendicular electric field of 0.1 eV/Å was applied to simulate the effect of the substrate. Spin-orbit coupling (SOC) is included in all electronic structure calculations.