Uncovering and tailoring hidden Rashba spin–orbit splitting in centrosymmetric crystals

Hidden Rashba and Dresselhaus spin splittings in centrosymmetric crystals with subunits/sectors having non-centrosymmetric symmetries (the R-2 and D-2 effects) have been predicted theoretically and then observed experimentally, but the microscopic mechanism remains unclear. Here we demonstrate that the spin splitting in the R-2 effect is enforced by specific symmetries, such as non-symmorphic symmetry in the present example, which ensures that the pertinent spin wavefunctions segregate spatially on just one of the two inversion-partner sectors and thus avoid compensation. We further show that the effective Hamiltonian for the conventional Rashba (R-1) effect is also applicable for the R-2 effect, but applying a symmetry-breaking electric field to a R-2 compound produces a different spin-splitting pattern than applying a field to a trivial, non-R-2, centrosymmetric compound. This finding establishes a common fundamental source for the R-1 effect and the R-2 effect, both originating from local sector symmetries rather than from the global crystal symmetry per se.


Supplementary Note 1: Identification of the two inversion-partner sectors in ferroelectric SnTe and illustration of how the R-1 spin-splitting originates from the local symmetries of the sectors rather than being a consequence of the global inversion symmetry breaking.
At room temperature, α-SnTe has the centrosymmetric rocksalt (space group Fm-3m) structure 1 (Supplementary Figure 1a) and, as temperature is lowered, 2 it undergoes a (ferroelectric) phase transition to the non-centrosymmetric rhombohedral (space group R3m) R-1 structure, 3,4 where the Te atom is spontaneously displaced along the [111] direction relative to Sn atom. We focus on the bands These effects give the R-1 compound α-SnTe a residual dipole field felt by band states and thus give rise to a finite Rashba spin splitting according to the unified model described by Eq. (3), similar to that of R-2 spin splitting in BaNiS 2 . This illustrates the identification of the two inversion-partner sectors in non-layered SnTe and shows how the R-1 spin-splitting originates from the local symmetries of the sectors rather than being a consequence of the global inversion symmetry breaking.

Supplementary Note 2: Non-symmorphic screw axis symmetry enforces the four-fold degeneracy of the energy bands at point of the tetragonal P4/nmm crystal
We show in Figure 2d that a four-fold degeneracy is present at X point of the tetragonal P4/nmm crystal. In non-magnetic material, such four-fold degenerate at time-reversal-invariant (TRI) k points can be enforced by a Hermitian symmetry operator that anti-commutes with the spatial inversion symmetry. 5 In the following, we will illustrate that, it is the non-symmorphic screw symmetry Considering two-fold symmetries, which have two eigen-energies, are the most common Hermitian symmetry operators, we choose as one of the two-fold symmetries and obtain = − G . After inserting it into Supplementary Eq. (2), it is straightforward to have which reveals the anti-commutation between and , Therefore, a Hermitian symmetry operator , which anti-commutates with spatial inversion symmetry, It is easy to verify that { , P} = 0 taking = 45 | 2 , 0, 0 , and, therefore, the screw axis symmetry enforces the energy bands to be four-fold degenerate at TRI X point. Above, the nonsymmorphic symmetry is essential for the anti-commutation relation and the four-fold degeneracy at certain TRI points (on Brillouin Zone boundary).

Supplementary Note 3: Restore the spin-split sector-segregated bands of monolayer BaNiS 2 from effective Hamiltonian
Based on the theory of invariants 6 Where is the lattice vector and "(#) = ±( /4, /4, yn s(t) ) ( is the in-plane lattice parameter of the monolayer BaNiS 2 ) is the relative position of ( ) Ni atom within the primary cell for the origin setting in the middle of two Ni atoms.
The transformation property of the basis is ready to obtain upon application of the symmetry operations given in Supplementary Eq. (5). Taking the glide reflection operator 5 | 2 , 0, 0 as an example, we find that wavefunctions of the bands at the X -point segregated on Ni α and Ni β , respectively, have opposite parity under symmetry operation: { 5 |( /2,0,0)} Ni "(#) , , X = ± Ni "(#) , , X and thus, their interaction is strictly forbidden by symmetry. The energy equivalence but interaction forbidden of these two states causes an additional degeneracy besides the Kramers' degeneracy for bands at the TRI X-point, which is consistent with the observed four-fold degenerate bands at the Xpoint in the first-principles calculations. The same transformation rule could also apply to states along X − M direction, obtaining forbidden interaction between degenerate states segregated on two inversion-partner sectors and leading to the two-fold degeneracy of the energy bands in agreement with the first-principles calculation, as shown in Figure 1c.
The  Table 2 for the case without considering SOC.
According to the theory of invariant 6 , the effective Hamiltonian must be invariant under all eight symmetry operations of the D 2h point group and, from Supplementary Table 2, we learn that all possible invariants up to second order in wavevector k at the X-point are ( , 5 4 , e 4 ) ‚ and 5 " . Neglecting the second order parabolic term, which will not contribute to spin splitting, the effective Hamiltonian of the spin splitting bands is therefore Here, t is a parameter that characterizes the strength of the interaction between two atomic d states located on two Ni atoms, respectively, within the primary cell. A diagonalization of Supplementary Eq.
(11) yields the energy dispersion relation We, therefore, find that these two bands (without spin) arising from the atomic d orbitals of two Ni atoms, respectively, are degenerate when 5 is zero (i.e., along X − M direction), but spits when 5 is non-zero (i.e., along X − Γ direction). Such energy dispersion relation of the effective Hamiltonian explains well the results obtained from the first-principles calculations for the case without SOC, as shown in Figure 1c.
It is straightforward to learn that the four energy bands (including spin) along both X − M and X − Γ directions spits into two doubly degenerate branches (branch A and branch B), which agrees with the results (shown in Figure 1d) obtained from the first-principles calculations for the case with SOC. It worth noting that since the off-diagonal elements are zero when 5 = 0 in the effective Hamiltonian given by Supplementary Eq. (13), the interaction between two atomic d states originating from two inversion-partner sectors are forbidden by the symmetry along X − M direction ( 5 = 0). The nonzero off-diagonal elements when 5 = 0 implies allowed mixture between two atomic d states originating from two inversion-partner sectors along X − Γ. These results justify our findings of the wavefunction segregation and fully delocalization on inversion-partner sectors along X − M and X − Γ directions, respectively.
Upon applying a small bias potential perpendicular to the monolayer BaNiS 2 , the potential dropping on two inversion-partner sectors lifts the degeneracy of both branch A and branch B, giving a justification for the observation of the first-principles calculation as shown in Figure 1e. The effect of the applied field can be described by ‚ • added to Supplementary Eq. (13). Take it as a small perturbation, to first order in energy, immediately, the electric field induced splitting is ~2 .