Abstract
Hidden Rashba and Dresselhaus spin splittings in centrosymmetric crystals with subunits/sectors having noncentrosymmetric symmetries (the R2 and D2 effects) have been predicted theoretically and then observed experimentally, but the microscopic mechanism remains unclear. Here we demonstrate that the spin splitting in the R2 effect is enforced by specific symmetries, such as nonsymmorphic symmetry in the present example, which ensures that the pertinent spin wavefunctions segregate spatially on just one of the two inversionpartner sectors and thus avoid compensation. We further show that the effective Hamiltonian for the conventional Rashba (R1) effect is also applicable for the R2 effect, but applying a symmetrybreaking electric field to a R2 compound produces a different spinsplitting pattern than applying a field to a trivial, nonR2, centrosymmetric compound. This finding establishes a common fundamental source for the R1 effect and the R2 effect, both originating from local sector symmetries rather than from the global crystal symmetry per se.
Introduction
Numerous physical effects and the technologies enabled by them are conditional on the presence of certain symmetries in the material that hosts such effects. Examples include effects predicated on the absence of inversion symmetry (noncentrosymmetric systems) such as the Dresselhaus effect^{1}, the Rashba effect^{2}, optical activity in nonchiral molecules^{3}, valley polarization and its derivative effects^{4}, and valley Hall effect in twodimensional (2D) layered structures^{5}. Although centrosymmetric systems are supposed to lack these effects, there is a large class of systems whose global crystal symmetry (GCS) is indeed centrosymmetric, but they consist of individual sectors with noncentrosymmetric local sector symmetry (LSS) (noncentrosymmetric site point groups). The term “hidden effect” refers to the general conditions where the said effect does exist even when the nominal GCS would disallow it. For example, the hidden Dresselhaus effect^{6} occurs in the diamondtype structure of Silicon, where each atom has a noncentrosymmetric LSS (the tetrahedral T_{d} point group) but the crystal as a whole has a centrosymmetric GCS (the octahedral O_{h} group). The theoretical prediction^{6} and subsequent experimental observations^{7,8,9,10,11,12,13} of “hidden spin polarization” in nonmagnetic centrosymmetric crystals triggered research on broader physical effects nominally disallowed under high GCS of systems, such as optical activity^{14}, intrinsic circular polarization^{15}, currentinduced spin polarization^{16,17}, superconductor^{18}, piezoelectric polarization^{6}, and orbital polarization^{19} in various centrosymmetric systems, as summarized in Table 1.
We use the designation “1” for cases where global inversion symmetry is absent (thus exhibiting the physical effects conditional on the absence of global inversion symmetry), as is the case of the conventional Rashba effect (R1) or Dresselhaus effect (D1). In parallel, we use the designation “2” for cases where the presence of global inversion symmetry hides the physical effects (conditional on the absence of symmetry), which is but revealed theoretically^{6} and observed experimentally^{7,8,9,10,11,12,13}. The latter is the case for the hidden Rashba effect (R2) or hidden Dresselhaus effect (D2)^{6}. It is noteworthy that in R2 or D2 nonmagnetic materials, even though the local spin polarization is nonzero, the net spin polarization remains zero (spin degeneracy), as imposed by the global inversion symmetry.
In the following, we build on our previous work ref. ^{6}, the idea of hidden spin polarization and the general conditions for its existence—global inversion symmetry and existence of inversionpartner sectors with polar site point group symmetries—were introduced. Here we focus on the microscopic mechanisms at play and how can they be translated into design principles for selecting highquality R2 materials for future experiments. We (i) show a common denominator for both R1 and R2 Rashba splitting, i.e., both effects originate from the symmetries of the local inversionpartner sectors rather than the global symmetries of the systems. (ii) As net polarization requires then that the doubly degenerate states on the different sectors will be prevented from mixing, we point out the mechanism of symmetryenforced wavefunction segregation, which prevents the doubly degenerate states on the different sectors from mixing. This is illustrated for the prototype compound in BaNiS_{2} where the requisite symmetry is nonsymmorphic operation. (iii) To clarify the difference between an R2 compound and a trivial centrosymmetric compound, we investigate the evolution of the R1 spin splitting from a R2 spin splitting (“R1 from R2”) by placing a tiny electric field on R2, which breaks the global inversion symmetry. We find that even for a tiny applied field the ensuing α_{R} for “R1 from R2” far exceeds the effect in the “R1 from trivial” case, highlighting that the observed R2 spin splitting is not due to inadvertent breaking of the inversion symmetry in an ordinary centrosymmetric compound as recently thought^{20}. This shows that angle resolved photoemission spectroscopy (ARPES) experiments can indeed probe band splitting genuinely coming from the hidden spin polarization and spin–orbit coupling (SOC), even if they are affected by surface sensitivity. This resolves another criticism raised by ref. ^{20} about potential difficulties in hidden spin polarization detection, namely the attribution of spin splitting to surface effects rather than to the bulk. This work sheds light on the view of the recent debate around the physical meaning and relevance of the “hidden spin polarization” concept and for the strong experimental and theoretical activity around it, motivated by the possibility to device materials with remarkable spin textures and technologically relevant properties. This finding offers clear experimental and computational frameworks to understand, tailor and use the R2/D2 effects.
Results
The evolution of R2 into R1 under an inversion symmetrybreaking electric field
One might naively think that the observed R2 spin splitting is due to inadvertent breaking of the inversion symmetry in an ordinary centrosymmetric compound.^{21} Indeed, a centrosymmetric R2 compound is distinct from a trivial centrosymmetric compound in that the former consist of individual polar sectors with noncentrosymmetric LSS (specifically, polar site point groups C_{1}, C_{2}, C_{3}, C_{4}, C_{6}, C_{1v}, C_{2v}, C_{3v}, C_{4v} and C_{6v}). A tiny electric field applied to a centrosymmetric trivial material such as cubic perovskites^{21} gives rise to a proportionally tiny spin splitting whose magnitude is proportional to the field. To clarify the difference between an R2 compound and a trivial centrosymmetric compound, which is often confused^{20}, we investigate the evolution of the R1 spin splitting from a R2 spin splitting (“R1 from R2”) by using the firstprinciples calculations on R2 compounds and placing on it a tiny electric field that breaks the global inversion symmetry.
An example of R2 compounds is BaNiS_{2}^{10}, which is a fivecoordinated Ni(II) structure consisting of puckered 2D layers of edgesharing square pyramidal polyhedral and crystalizes in the tetragonal system, space group P4/nmm. Conductivity and susceptibility measurements^{22,23} indicate that it is a metallic Pauli Paramagnet. Our DFT + U calculation (U = 3 eV, J = 0.95 eV) also predicts a lowtemperature antiferromagnetic phase with local Ni moments of ± 0.7 μ_{B} for bulk (± 0.6 μ_{B} for a monolayer) where the antiferromagnetic phase is slightly more stable than the nonmagnetic model by just 43 meV(f.u)^{−1} for bulk and 28 meV(f.u)^{−1} for monolayer. These DFT + U calculations had reported that BaNiS_{2} undergoes a phase transition from paramagnetic to antiferromagnetic as increasing the used Uvalue from 2 to 3 eV. Given the difficulty of estimating the proper Uvalue in the + U framework and experimental (conductivity and susceptibility) observation^{22,23} of metallic Pauli Paramagnet, in this work we nevertheless adopt a nonmagnetic phase for BaNiS_{2} to avoid the unnecessary complications from magnetic orders. Our relaxed lattice constants and interatomic distances in the nonmagnetic General Gradient Approximation (GGA) calculation agrees with the measured result within ~1%^{10,22}. In the nonmagnetic model, BaNiS_{2} possesses both inversion symmetry and timereversal symmetry; in the presence of SOC, each energy band is evenfold degenerate and thus has no R1 spin splitting.
Figure 1a shows the structure of a monolayer of this centrosymmetric crystal, which has two separated crystallographic sectors–S_{α} and its inversion partner S_{β} (shown in Fig. 1a as red and blue planes, respectively); each sector contains a single B atom (here, B = Ni, Pd, or Pt) with a polar site group C_{4v}, having its local internal dipole field^{10} (calculated and shown below). We focus our attention on the lowest four conduction bands (including spin) around the \({\bar{\mathrm X}}\) point (highlighted with a red square in Fig. 1b). Figure 1c shows that when SOC is turned off in the firstprinciples calculations, one finds along highsymmetry path \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) a single, fourfold degenerate band whose degeneracy is imposed by the nonsymmorphic screwaxis symmetry {C_{2x}(a/2, 0, 0)}; {C_{2y}(0, a/2, 0)} (explained in Supplementary Note 2 and 3). When SOC is turned on, the fourfold degenerate band splits into two branches A and B (Fig. 1d) and each branch is doubly degenerate and has two orthogonal spin components. The applied outofplane electric field external electric field generates asymmetric potential on the two inversionpartner sectors and thus breaks the global inversion symmetry, but conserves the timereversal symmetry.
The spin degeneracy of both branches A and B along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) and at the \({\bar{\mathrm X}}\) point is lifted upon application of an external electric field E_{ext}, as shown in Fig. 1e. This splitting, denoted Δ_{αβ}, occurs at the timereversal invariant (TRI) \({\bar{\mathrm X}}\) point and is dependent linearly on E_{ext} (see below). The finite splitting at the TRI point rules out the Rashba effect as the origin of the splitting of the two spin components of branch A (and branch B) along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\). Figure 2a indeed shows that the spindown component of the highenergy branch A and the spinup component of the lowenergy branch B have wavefunctions confined in sector S_{α}, and thus pair as one orbital band (hereafter, termed S_{α}Rashba band). The spinup component of the branch A and the spindown component of the branch B possess wavefunctions confined in sector S_{β} (hereafter, termed S_{β}Rashba band). We therefore identify the splitting δE_{AB}(k) as a consequence of the R2 effect quantified by a Rashba parameter α_{R}(R2) = 0.24 VÅ. The applied electric field further adds/subtracts the R1 spin splitting to/from the R2 splitting δE_{AB}(k) of the S_{α} and S_{β}Rashba bands, respectively, along the \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction. Figure 3a shows the corresponding Rashba parameters \(\alpha _{\mathrm{R}} = {\mathrm{\delta }}E_{{\mathrm{AB}}}\left( {{\mathbf{k}}  {\bar{\mathrm X}}} \right)/2\left( {{\mathbf{k}}  {\bar{\mathrm X}}} \right)\), which exhibits a linear response to E_{ext}: α_{R} of the S_{α}Rashba band increases and the S_{β}Rashba band decreases at rates of the same magnitude but opposite sign as increasing E_{ext}. The extrapolations of these two α_{R} functions cross at E_{ext} = 0, giving rise to α_{R} = 0.24 VÅ, a value being the same as the (zero field) R2 spin splitting α_{R}(R2).
The magnitude of the R2 spin splitting can be determined unambiguously by placing on a candidate R2 compound an electric field, then extrapolating to the zero field to uncover a finite, zerofield (R2) Rashba parameter. The significant magnitude illustrated above of the ensuing α_{R} for “R1 from R2” relative to the “R1 from trivial” scenario highlights the fact that the R1 spin splitting is inherited from the R2 effect in bulk Rashba systems, i.e., from the local asymmetric dipole fields of the individual sectors. This finding obviates the concern of Li and Appelbaum^{20} who suggested that the Rashba surface spin splitting detected experimentally (e.g., via ARPES) might originate from the unavoidable inversion symmetrybroken surface, as this contribution is indistinguishable from bulk R2 effect.
Avoided compensation of the R2 spin polarization in BaNiS_{2} enforced by nonsymmorphic symmetry
We next clarify under what circumstances the hidden R2 effect can be large or small. This physics can be gleaned by looking at a single nonmagnetic centrosymmetric R2 ABX_{2} system in two different directions in the Brillouin zone (BZ). Figure 1 shows that these R2 bands along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) and \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) directions exhibit two different types of spinsplitting behaviors associated with the distinct transformation properties of the wavefunction under nonsymmorphic glide reflection symmetry (see Supplementary Note 3 for details). This realization then would help us establish the distinguishing features of R1 vs. R2 materials.
Wavefunction segregation causes sizable R2 spin splitting along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction
To quantify the degree of wavefunction segregation (DWS) of the wavefunction, we introduce a measure D(φ_{k}) for states φ_{k} at the wavevector k, where
and
\(P_{\varphi _{\mathbf{k}}}(S_\alpha )\) is the component of the wavefunction φ_{k} localized on the sector S_{α}. The DWS explicitly quantifies the locality of wavefunction, in contrast to the implicit measure^{10} by means of the integral of the local spin density operator restricted on a given sector.
It is evident that D(φ_{k}) = 0 for a wholly delocalized wavefunction over two inversionpartner sectors, whereas, D(φ_{k}) = 100% indicates that the wavefunction is entirely confined either on sector S_{α} or sector S_{β}. One expects, in general, that any linear combination of two degenerate states should still be an eigenstate and prevent us from obtaining a unique DWS for the energydegenerate bands.^{20} However, we demonstrated in Supplementary Note 3 that, in R2 compounds, the symmetry of the wavevectors along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction prohibits the mixing of two degenerate states arising from two inversionpartner sectors (S_{α} and S_{β}), respectively, as a result of the glide reflection symmetry, and hence dissociates any linear combinations of the degenerate states for tracing back to the symmetryenforced segregated states. SantosCottin et al.^{10} had shown the localization of wavefunction in BaNiS_{2} to provide the basis to decouple two effective Rashba Hamiltonians associated with each sector. Our calculations also (Fig. 2a) show segregated wavefunctions (localized either on sector S_{α} or S_{β}) and D(φ_{k}) = 88% (k = (0.025, 0.5, 0)(2π/a), here a is the lattice constant, for both spin components of doubly degenerate branches A and B along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction. This fact obviates the concern of validity of hidden spinsplitting theory due to the possible lack of gauge invariance, raised by Li and Appelbaum^{20}.
The relation between wavefunction segregation and the R2 effect can be appreciated as follows: in 2D quantum wells or heterojunctions, one obtains the Rashba parameter α_{R} due to the R1 effect as^{24}
where r_{R,i} is a materialspecific Rashba coefficient of the ithband, the electric field E(r) = (1/e)∇V is the local gradient of the crystal potential V, and angular brackets indicate an average of the local Rashba parameter r_{R,i}E(r) of the well and barrier materials weighted by the wavefunction amplitude. In a crystal without external fields, the electric field originates from the local dipole and is termed E_{dp}(r), which does not have to vanish at all atomic sites even in centrosymmetric systems. Figure 2e shows the x–y planaraveraged internal local dipole fields E_{dp}(z) in the monolayer BaNiS_{2}. It exhibits that E_{dp}(z) varies rapidly within a single sector and is inversion through a point located on the sulfur atom (or point reflection). The internal dipole fields are finite (and in fact atomically large) within a single sector, whereas the sum over both inversionpartner sectors is zero as expected. The segregation of wavefunctions on a single sector with D(φ_{k}) = 88% for states along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction indicates that this band experiences a net effective field of the internal dipole fields within a single sector (as illustrated in Fig. 2f) and is immune to full compensation from the opposite dipole fields within its inversionpartner sector. According to Eq. (3), a finite Rashba parameter α_{R} is thus obtained for R2 bands along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction. Thus, the large R2 effect along this BZ direction originates from wavefunction segregation on each of the two inversionpartner sectors, avoiding mutual compensation of local dipolar electric fields.
Wavefunction delocalization leading to vanishing R2 spin splitting along the \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) direction
In sharp contrast to the \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction, Fig. 1c shows that along \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) direction these four bands already split into two doublets even in the absence of SOC and the magnitude of their splitting is barely changed after turning on the SOC. We attribute such band splitting to symmetry allowed interaction between states stemming from two inversionpartner sectors S_{α} and S_{β} (see Supplementary Note 3). Thereby, we denote two spin components of the branch A by \(S_{\alpha /\beta }^ \downarrow \left( {{\mathrm{A}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right)\) and \(S_{\alpha /\beta }^ \uparrow \left( {{\mathrm{A}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right)\), respectively, whereas, for branch B we use \(S_{\alpha /\beta }^ \downarrow \left( {{\mathrm{B}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right)\) and \(S_{\alpha /\beta }^ \uparrow \left( {{\mathrm{B}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right).\) The wavefunction of the spindown component of the branch A is 49% confined, and that of branch B is 51% confined in sector S_{α}, respectively, so as Fig. 2b shows DWS is \(D\left( {S_{\alpha /\beta }^ \downarrow \left( {{\mathrm{A}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right)} \right) = D\left( {S_{\alpha /\beta }^ \downarrow \left( {{\mathrm{B}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right)} \right) = 2\%\) for spindown components of both A and B branches. Similarly, the wavefunction of the spinup component of the branch A is 43% confined, and that of branch B is 57% confined in sector S_{α} so DWS \(D\left( {S_{\alpha /\beta }^ \uparrow \left( {{\mathrm{A}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right)} \right) = D\left( {S_{\alpha /\beta }^ \uparrow \left( {{\mathrm{B}},{\mathbf{k}}_{{\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}}} \right)} \right)\) is 14% for spinup components. Thus, the wavefunctions of the \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) bands are essentially delocalized over both inversionpartner sectors S_{α} and S_{β}. Such wavefunction delocalization naturally leads to a complete compensation of the undergoing local internal dipole fields within S_{α} by that within S_{β}, when each local dipole weighted by its wavefunction amplitudes gives rise to zero average Rashba parameter α_{R} according to Eq. (3).
Unification of R1 and R2 into a single theoretical framework
The smooth “R1 from R2” evolution (Fig. 3a) suggests that when applying an external electric field E_{ext} to an R2 system, the electric field E(r) acting on electrons is a superposition of E_{ext} and the internal local dipole (dp) electric fields E_{dp}(r),
Thus, both R1 and R2 spin splitting have a common fundamental source being the dipole electric fields of the local sectors rather than from the global crystal asymmetry per se. Such local dipole electric field “lives” within individual local sectors. The fundamental difference between R1 and R2 effects is that in R2 the spin splitting is hidden by the overlapping energy bands arising from two inversionpartner sectors, whereas in the R1 case such overlap is forbidden by the global inversion asymmetry.
Figure 1e also shows that the applied electric field lifts the spin degeneracy of the bands along \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) direction and raises α_{R} linearly from zero at E_{ext} = 0 to saturation at E_{ext} = 10 mV Å^{−1} at an odd large rate. This behavior is in striking contrast to the linear field dependence of the bands along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction (see Fig. 3a). Such unusual field dependence of α_{R} confirms again that the R2 spin splitting evolves smoothly to the R1 spin splitting upon the breaking of the global inversion symmetry, regarding the bands along \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) direction have vanishing R2 spin splitting with α_{R}(R2) = 0 in the absence of an external field. Upon application of electric field, the delocalized wavefunctions of the \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) bands become gradually segregated on one of two inversionpartner sectors as a result of Stark effect^{25}. Subsequently, Fig. 3c shows that the applied field amplifies substantially the DWS (Eq. (1)) of the spinup component of both branches from 14% to > 80% as the magnitude of E_{ext} increases from 0 to 50 mV Å^{−1}. However, D(φ_{k}) is barely changed once E_{ext} > 50 mV Å^{−1} (saturation field). It is noteworthy that DWS of the corresponding spindown components is not shown but has a similar response to the applied electric field. It is straightforward to learn that the internal electric dipole fields acting on these bands become uncompensated as their wavefunctions change into segregation on a single sector, evoking the R2 effect with its strength highly related to D(φ_{k}) according to Eq. (3). The rapid amplification of D(φ_{k}) by the applied electric field explains that the (unusual) rapid rise of α_{R} for those bands along \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) direction is mainly due to the enhancement of the wavefunction segregation rather than to the increase of the total electric dipole field.
When E_{ext} reaches ~25 mV Å^{−1}, α_{R} of both high and lowenergy doublets become linear fielddependent but in rates of opposite signs, which is in a similar field dependence as that along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction. Figure 3c shows that the response of D(φ_{k}) of the \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) bands to E_{ext} is, however, barely modified by the external field, indicating those states remain fully localized on one of two inversionpartner sectors. The linear change of α_{R} along \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction as shown in Fig. 3a thus arises entirely from the external field induced asymmetry, i.e., in Eq. (3) the change α_{R} is solely arising from the electric field. The calculated Rashba parameter of the R2 spin splitting can be explained regarding the model of the R1 spin splitting (Eq. (3)), indicating a unified theoretical view for both R1 and R2 effects in bulk systems. Specifically, the effective electric field that promotes either R1 and/or R2 Rashba effects is a superposition of the applied external electric field plus the internal local electric fields originating from the dipoles of the individual local sectors, weighted by the wavefunction amplitude on the corresponding sectors.
We also apply this unifying theoretical framework to a nonlayered R1 example, the αSnTe^{6} or similarly the αGeTe (a standard ferroelectric bulk R1 compound predicted in 2013^{26} and experimentally confirmed in 2016^{27,28}), where one can identify two inversionpartner sectors and the corresponding wavefunction becomes segregated due to the lack of inversion symmetry in the rhombohedral phase (details see Supplementary Note 1). According to the unified model described by Eq. (3), such wavefunction segregation gives a residual dipole field felt by band states and thus give rise to a finite Rashba spin splitting, similar to that of R2 spin splitting in BaNiS_{2}. As displacing the Te atom from Sn along [111] direction, the αGeTe will undergo a phase transition from noncentrosymmetric rhombohedral phase to centrosymmetric rocksalt phase. We demonstrate that in the centrosymmetric rocksalt phase wavefunctions are evenly distributed among two inversionpartner sectors, leading to a perfect compensation of the local dipole fields and thus vanishing Rashba effect in the centrosymmetric rocksalt phase according to Eq. (3).
Design principles for increasing the strength of the R2 effect
R2 materials^{6} are defined by having global inversion symmetry and two recognizable inversionpartner sectors with polar site point group symmetries. Designing R2 materials possessing large hidden spin splitting and hence strong local spin polarization can benefit from two additional design principles:

(i)
Minimizing the mixing and entanglement of the wavefunction on the different inversion partners sectors. Here we point to a nontrivial mechanism of symmetryenforced wavefunction segregation, keeping the doubly degenerate states on the different sectors from mixing (in contrast to the trivial physical separation of the two inversionpartner sectors). It is noteworthy that R1 compounds do not have to maintain segregationinducing symmetries to have Rashba effect, because its inversion asymmetry alone ensures the avoidance of wavefunction entanglement by lifting the degeneracy of states from the two partner sectors, as illustrated in Supplementary Note 1 for rhombohedral SnTe. The wavefunction segregation enforcing symmetry illustrated here is the nonsymmorphic symmetry along the \({\bar{\mathrm X}}  {\bar{\mathrm M}}\) direction in the BaNiS_{2} BZ. Other segregation enforcing symmetry operations may exist in general cases, but they have not been discovered yet.

(ii)
Instilling strong local dipole fields, i.e., designing individual sectors with maximal asymmetry of the local potential within the sector. Thus, whereas the creation and enhancing Rashba effect in conventional (e.g., interfacial) Rashba materials^{2,24} entails, by tradition, breaking inversion symmetry, here our design principles for Rashba effect in centrosymmetric compounds focuses on using other symmetry operations that enhance segregation and avoid mixing.
Applying the design principles (i) and (ii) one could design strong R2 materials via selecting compounds where the wavefunctions are concentrated in real space locations that have a larger magnitude of local dipole fields. An example illustrated here is BaNiS_{2}. Such wavefunction segregation can be tailored through application of an external electric field, strain, atom mutation, or modifications of the polar cation ordering.^{24} This is illustrated by the rapid rise of α_{R} vs. field for bands along \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) direction (Fig. 3a), demonstrating tailoring of the R2 effect. For instance, Otani and colleagues^{29} have recently found a strong correlation between the charge density distribution and the strength of the Rashba effect at nonmagnetic metal/Bi_{2}O_{3} interfaces. Furthermore, the unexpected rapid rise of α_{R} vs. field for bands along \({\bar{\mathrm X}}  {\bar{\mathrm \Gamma }}\) direction (Fig. 3a) implies that one might effectively tune the strength of R2 effect. We thus present an alternative mechanism for boosting the strength of the Rashba effect, which is commonly achieved by enhancing the breaking of inversion symmetry.
Methods
Firstprinciples band structure calculation
Electronic structures are calculated using density functional theory (DFT)^{30,31,32}based firstprinciples methods within the GGA^{33} implemented in the Vienna Ab initio simulation package (VASP)^{34}. A planewave expansion up to 400 eV is applied and a Гcentered 16 × 16 × 1 MonkhorstPack^{35} kmesh is used for the BZ sampling. The lattice constants used in the firstprinciples calculations are taken directly from the experimental data. The monolayer slab of BaNiS_{2} are separated by a 17.8 Å vacuum layer. We adopt the GGA + U method^{36} to account the onsite Coulomb interaction of localized Ni3d orbitals. We follow the approach proposed by Neugebauer and Scheffler^{37} to apply a uniform electric field to monolayer BaNiS_{2} slab in the calculations. This approach treats the artificial periodicity of the slab by adding a planar dipole sheet in the middle of the vacuum region. The strength of the dipole is calculated selfconsistently such that the electrostatic fieldinduced dipole is compensated for. For the calculations including the spin–orbit interaction, the spin quantization axis set to the default (0 +, 0, 1) (the notation 0 + implies an infinitesimal small positive number in the xdirection) with zero atomic magnetic moments. The VASP configuration files and related codes that support the findings of this study are available from the corresponding authors upon reasonable request.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
J.L. and S.L. were supported by the National Natural Science Foundation of China (NSFC) under Grant Number 61888102. J.L. was also supported by the National Young 1000 Talents Plan. Work of A.Z. and Q.L at CU Boulder was supported by the National Science foundation NSF Grant NSFDMRCMMT Number DMR1724791. Q.L. was also supported by the NSFC under Grant Number 11874195. X.Z. was supported by the NSFC under Grant Number 11774239.
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L.Y. performed the electronic structure calculations, prepared the figures, and developed the tightbinding models with the help of Q.L. J.L. proposed the research project. J.L. and A.Z. established the project direction and conducted the analysis, discussion, and writing of the paper with input from L.Y., Q.L., and X.Z. S.L. provided the project infrastructure and supervised L.Y.’s study.
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Yuan, L., Liu, Q., Zhang, X. et al. Uncovering and tailoring hidden Rashba spin–orbit splitting in centrosymmetric crystals. Nat Commun 10, 906 (2019). https://doi.org/10.1038/s41467019088364
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DOI: https://doi.org/10.1038/s41467019088364
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