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 (non-centrosymmetric systems) such as the Dresselhaus effect1, the Rashba effect2, optical activity in non-chiral molecules3, valley polarization and its derivative effects4, and valley Hall effect in two-dimensional (2D) layered structures5. 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 non-centrosymmetric local sector symmetry (LSS) (non-centrosymmetric 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 effect6 occurs in the diamond-type structure of Silicon, where each atom has a non-centrosymmetric LSS (the tetrahedral Td point group) but the crystal as a whole has a centrosymmetric GCS (the octahedral Oh group). The theoretical prediction6 and subsequent experimental observations7,8,9,10,11,12,13 of “hidden spin polarization” in non-magnetic centrosymmetric crystals triggered research on broader physical effects nominally disallowed under high GCS of systems, such as optical activity14, intrinsic circular polarization15, current-induced spin polarization16,17, superconductor18, piezoelectric polarization6, and orbital polarization19 in various centrosymmetric systems, as summarized in Table 1.

Table 1 Examples of reported hidden effects in centrosymmetric crystals
Full size table

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 (R-1) or Dresselhaus effect (D-1). 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 theoretically6 and observed experimentally7,8,9,10,11,12,13. The latter is the case for the hidden Rashba effect (R-2) or hidden Dresselhaus effect (D-2)6. It is noteworthy that in R-2 or D-2 non-magnetic 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 inversion-partner 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 high-quality R-2 materials for future experiments. We (i) show a common denominator for both R-1 and R-2 Rashba splitting, i.e., both effects originate from the symmetries of the local inversion-partner 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 symmetry-enforced wavefunction segregation, which prevents the doubly degenerate states on the different sectors from mixing. This is illustrated for the prototype compound in BaNiS2 where the requisite symmetry is non-symmorphic operation. (iii) To clarify the difference between an R-2 compound and a trivial centrosymmetric compound, we investigate the evolution of the R-1 spin splitting from a R-2 spin splitting (“R-1 from R-2”) by placing a tiny electric field on R-2, which breaks the global inversion symmetry. We find that even for a tiny applied field the ensuing αR for “R-1 from R-2” far exceeds the effect in the “R-1 from trivial” case, highlighting that the observed R-2 spin splitting is not due to inadvertent breaking of the inversion symmetry in an ordinary centrosymmetric compound as recently thought20. 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 R-2/D-2 effects.

Results

The evolution of R-2 into R-1 under an inversion symmetry-breaking electric field

One might naively think that the observed R-2 spin splitting is due to inadvertent breaking of the inversion symmetry in an ordinary centrosymmetric compound.21 Indeed, a centrosymmetric R-2 compound is distinct from a trivial centrosymmetric compound in that the former consist of individual polar sectors with non-centrosymmetric LSS (specifically, polar site point groups C1, C2, C3, C4, C6, C1v, C2v, C3v, C4v and C6v). A tiny electric field applied to a centrosymmetric trivial material such as cubic perovskites21 gives rise to a proportionally tiny spin splitting whose magnitude is proportional to the field. To clarify the difference between an R-2 compound and a trivial centrosymmetric compound, which is often confused20, we investigate the evolution of the R-1 spin splitting from a R-2 spin splitting (“R-1 from R-2”) by using the first-principles calculations on R-2 compounds and placing on it a tiny electric field that breaks the global inversion symmetry.

An example of R-2 compounds is BaNiS210, which is a five-coordinated Ni(II) structure consisting of puckered 2D layers of edge-sharing square pyramidal polyhedral and crystalizes in the tetragonal system, space group P4/nmm. Conductivity and susceptibility measurements22,23 indicate that it is a metallic Pauli Paramagnet. Our DFT + U calculation (U = 3 eV, J = 0.95 eV) also predicts a low-temperature anti-ferromagnetic phase with local Ni moments of ± 0.7 μB for bulk (± 0.6 μB for a monolayer) where the anti-ferromagnetic phase is slightly more stable than the non-magnetic 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 BaNiS2 undergoes a phase transition from paramagnetic to anti-ferromagnetic as increasing the used U-value from 2 to 3 eV. Given the difficulty of estimating the proper U-value in the + U framework and experimental (conductivity and susceptibility) observation22,23 of metallic Pauli Paramagnet, in this work we nevertheless adopt a non-magnetic phase for BaNiS2 to avoid the unnecessary complications from magnetic orders. Our relaxed lattice constants and interatomic distances in the non-magnetic General Gradient Approximation (GGA) calculation agrees with the measured result within ~1%10,22. In the non-magnetic model, BaNiS2 possesses both inversion symmetry and time-reversal symmetry; in the presence of SOC, each energy band is even-fold degenerate and thus has no R-1 spin splitting.

Figure 1a shows the structure of a monolayer of this centrosymmetric crystal, which has two separated crystallographic sectors–Sα and its inversion partner Sβ (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 C4v, having its local internal dipole field10 (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 first-principles calculations, one finds along high-symmetry path \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) a single, fourfold degenerate band whose degeneracy is imposed by the non-symmorphic screw-axis symmetry {C2x|(a/2, 0, 0)}; {C2y|(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 out-of-plane electric field external electric field generates asymmetric potential on the two inversion-partner sectors and thus breaks the global inversion symmetry, but conserves the time-reversal symmetry.

Fig. 1
figure 1

The crystal structure and energy bands of the monolayer BaNiS2. a The crystal structure of a centrosymmetric monolayer of BaNiS2 taken from the bulk with P4/nmm space group, showing its two inversion-partner sectors Sα and Sβ. b Energy band dispersion of the monolayer in an extended zone. The Rashba bands of interest are highlighted in red square. Insert shows schematically the 2D Brillouin zone of the monolayer. c–e Zoom-in the energy dispersion of the lowest four conduction bands near the X point along \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) and \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) directions when SOC is turned off (c) and turned on (d, e). Relative to the result shown in d, in case shown in e we apply a small electric field of 1 mV Å−1 to the monolayer along the z-direction, as schematic digram shown in a, to break the inversion symmetry. The inversion symmetry-breaking electric field lifts the degeneracy of both branches A and B into the Sα-Rashba band and the Sβ-Rashba band, with an energy separation at the X point denote as Δαβ. The band with its wavefunction segregated on the sector Sα is represented by red and on the on sector Sβ by blue. Arrows are used to illustrate the spin orientation

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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 Eext, as shown in Fig. 1e. This splitting, denoted Δαβ, occurs at the time-reversal invariant (TRI) \({\bar{\mathrm X}}\) point and is dependent linearly on Eext (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 spin-down component of the high-energy branch A and the spin-up component of the low-energy branch B have wavefunctions confined in sector Sα, and thus pair as one orbital band (hereafter, termed Sα-Rashba band). The spin-up component of the branch A and the spin-down component of the branch B possess wavefunctions confined in sector Sβ (hereafter, termed Sβ-Rashba band). We therefore identify the splitting δEAB(k) as a consequence of the R-2 effect quantified by a Rashba parameter αR(R2) = 0.24 VÅ. The applied electric field further adds/subtracts the R-1 spin splitting to/from the R-2 splitting δEAB(k) of the Sα- and Sβ-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 Eext: αR of the Sα-Rashba band increases and the Sβ-Rashba band decreases at rates of the same magnitude but opposite sign as increasing Eext. The extrapolations of these two αR functions cross at Eext = 0, giving rise to αR = 0.24 VÅ, a value being the same as the (zero field) R-2 spin splitting αR(R-2).

Fig. 2
figure 2

Wavefunction segregation and local dipole fields in BaNiS2 monolayer. a, b Charge density of the lowest four conduction bands at \({\mathbf{k}}_{{\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}} = \left( {0,0.475,0} \right)(2{\mathrm{\pi }}/a)\) and \({\mathbf{k}}_{{\bar{\mathrm X}} - {\bar{\mathrm M}}} = \left( {0.025,0.5,0} \right)(2{\mathrm{\pi }}/a)\), respectively. The isosurface of charge density is represented by purple. The Ni, S, and Ba atoms are represented by green, yellow, and gray balls, respectively. The degree of wavefunction segregation and the percentage of the charge density localized on the sectors Sα and Sβ are also listed for each state. c The crystal structure of the monolayer BaNiS2, a view perpendicular to the (11̄0) plane. d Planar-averaged crystal potential of the monolayer BaNiS2. e The z-component of the internal local dipole fields \({\mathbf{E}}_{{\mathrm{dp}}}(z) = \left( {1/e} \right)\partial \bar V\left( z \right)/\partial z\) along the z-direction. Red arrows indicate the dipole fields within the sector Sα and blue arrows for the dipole fields within the sector Sβ. f, g Charge density of the \(S_\alpha ^ \uparrow (B,{\mathbf{k}}_{{\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}})\) and \(S_\alpha ^ \uparrow (B,{\mathbf{k}}_{{\bar{\mathrm X}} - {\bar{\mathrm M}}})\) states of the monolayer BaNiS2 in the absence of external fields

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Fig. 3
figure 3

The evolution of Rashba physics under electric field in monolayer BaNiS2. a The Rashba parameters of the spin-splitting bands segregated on the sector Sα (red empty squares or circles) and sector Sβ (blue solid squares or circles), respectively, along \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) (square) and \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) (circle) directions as a function of applied electric field. b Electric field induced energy separation (Δαβ) between the Sα-Rashba band and the Sβ-Rashba band at X point. c Degree of wavefunction segregation of branch A (upper panel) and branch B (lower panel) along \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) and \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) directions, respectively, as functions of the applied electric field. It is noteworthy that the Rashba parameter shown in a is a sum of dipole fields weighted by corresponding wavefunction amplitudes and are not necessary to display a simple linear correlation with the DWS (shown in c), a degree of wavefunction segregation defined in Eqs. (1) and (2)

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The magnitude of the R-2 spin splitting can be determined unambiguously by placing on a candidate R-2 compound an electric field, then extrapolating to the zero field to uncover a finite, zero-field (R-2) Rashba parameter. The significant magnitude illustrated above of the ensuing αR for “R-1 from R-2” relative to the “R-1 from trivial” scenario highlights the fact that the R-1 spin splitting is inherited from the R-2 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 Appelbaum20 who suggested that the Rashba surface spin splitting detected experimentally (e.g., via ARPES) might originate from the unavoidable inversion symmetry-broken surface, as this contribution is indistinguishable from bulk R-2 effect.

Avoided compensation of the R-2 spin polarization in BaNiS2 enforced by non-symmorphic symmetry

We next clarify under what circumstances the hidden R-2 effect can be large or small. This physics can be gleaned by looking at a single non-magnetic centrosymmetric R-2 ABX2 system in two different directions in the Brillouin zone (BZ). Figure 1 shows that these R-2 bands along \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) and \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) directions exhibit two different types of spin-splitting behaviors associated with the distinct transformation properties of the wavefunction under non-symmorphic glide reflection symmetry (see Supplementary Note 3 for details). This realization then would help us establish the distinguishing features of R-1 vs. R-2 materials.

Wavefunction segregation causes sizable R-2 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(φk) for states φk at the wavevector k, where

$$D(\varphi _{\boldsymbol{k}}) = \left| {\frac{{P_{\varphi _{\mathbf{k}}}\left( {S_\alpha } \right) - P_{\varphi _{\mathbf{k}}}\left( {S_\beta } \right)}}{{P_{\varphi _{\mathbf{k}}}\left( {S_\alpha } \right) + P_{\varphi _{\mathbf{k}}}\left( {S_\beta } \right)}}} \right|,$$
(1)

and

$$P_{\varphi _{\mathbf{k}}}(S_{\alpha ,\beta }) = \int_{{\mathrm{\Omega }}\, \in \,S_{\alpha ,\beta }}^{} \left| {\varphi _{\mathbf{k}}\left( {\mathbf{r}} \right)} \right|^2d^3{\mathbf{r}}.$$
(2)

\(P_{\varphi _{\mathbf{k}}}(S_\alpha )\) is the component of the wavefunction φk localized on the sector Sα. The DWS explicitly quantifies the locality of wavefunction, in contrast to the implicit measure10 by means of the integral of the local spin density operator restricted on a given sector.

It is evident that D(φk) = 0 for a wholly delocalized wavefunction over two inversion-partner sectors, whereas, D(φk) = 100% indicates that the wavefunction is entirely confined either on sector Sα or sector Sβ. 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 energy-degenerate bands.20 However, we demonstrated in Supplementary Note 3 that, in R-2 compounds, the symmetry of the wavevectors along \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) direction prohibits the mixing of two degenerate states arising from two inversion-partner sectors (Sα and Sβ), respectively, as a result of the glide reflection symmetry, and hence dissociates any linear combinations of the degenerate states for tracing back to the symmetry-enforced segregated states. Santos-Cottin et al.10 had shown the localization of wavefunction in BaNiS2 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 Sα or Sβ) and D(φk) = 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 spin-splitting theory due to the possible lack of gauge invariance, raised by Li and Appelbaum20.

The relation between wavefunction segregation and the R-2 effect can be appreciated as follows: in 2D quantum wells or heterojunctions, one obtains the Rashba parameter αR due to the R-1 effect as24

$$\alpha _{{\mathrm{R}},{\mathrm{i}}} = \left\langle {r_{{\mathrm{R}},i} \cdot {\mathbf{E}}({\mathbf{r}})} \right\rangle$$
(3)

where rR,i is a material-specific Rashba coefficient of the ith-band, 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 rR,iE(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 Edp(r), which does not have to vanish at all atomic sites even in centrosymmetric systems. Figure 2e shows the x–y planar-averaged internal local dipole fields Edp(z) in the monolayer BaNiS2. It exhibits that Edp(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 inversion-partner sectors is zero as expected. The segregation of wavefunctions on a single sector with D(φk) = 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 inversion-partner sector. According to Eq. (3), a finite Rashba parameter αR is thus obtained for R-2 bands along \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) direction. Thus, the large R-2 effect along this BZ direction originates from wavefunction segregation on each of the two inversion-partner sectors, avoiding mutual compensation of local dipolar electric fields.

Wavefunction delocalization leading to vanishing R-2 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 inversion-partner sectors Sα and Sβ (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 spin-down component of the branch A is 49% confined, and that of branch B is 51% confined in sector Sα, 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 spin-down components of both A and B branches. Similarly, the wavefunction of the spin-up component of the branch A is 43% confined, and that of branch B is 57% confined in sector Sα 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 spin-up components. Thus, the wavefunctions of the \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) bands are essentially delocalized over both inversion-partner sectors Sα and Sβ. Such wavefunction delocalization naturally leads to a complete compensation of the undergoing local internal dipole fields within Sα by that within Sβ, when each local dipole weighted by its wavefunction amplitudes gives rise to zero average Rashba parameter αR according to Eq. (3).

Unification of R-1 and R-2 into a single theoretical framework

The smooth “R-1 from R-2” evolution (Fig. 3a) suggests that when applying an external electric field Eext to an R-2 system, the electric field E(r) acting on electrons is a superposition of Eext and the internal local dipole (dp) electric fields Edp(r),

$${\mathbf{E}}\left( {\mathbf{r}} \right) = {\mathbf{E}}_{{\mathrm{dp}}}\left( {\mathbf{r}} \right) + {\mathbf{E}}_{{\mathrm{ext}}}$$
(4)

Thus, both R-1 and R-2 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 R-1 and R-2 effects is that in R-2 the spin splitting is hidden by the overlapping energy bands arising from two inversion-partner sectors, whereas in the R-1 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 αR linearly from zero at Eext = 0 to saturation at |Eext| = 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 αR confirms again that the R-2 spin splitting evolves smoothly to the R-1 spin splitting upon the breaking of the global inversion symmetry, regarding the bands along \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) direction have vanishing R-2 spin splitting with αR(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 inversion-partner sectors as a result of Stark effect25. Subsequently, Fig. 3c shows that the applied field amplifies substantially the DWS (Eq. (1)) of the spin-up component of both branches from 14% to > 80% as the magnitude of Eext increases from 0 to 50 mV Å−1. However, D(φk) is barely changed once Eext > 50 mV Å−1 (saturation field). It is noteworthy that DWS of the corresponding spin-down 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 R-2 effect with its strength highly related to D(φk) according to Eq. (3). The rapid amplification of D(φk) by the applied electric field explains that the (unusual) rapid rise of αR 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 |Eext| reaches ~25 mV Å−1, αR of both high- and low-energy doublets become linear field-dependent 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(φk) of the \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) bands to Eext is, however, barely modified by the external field, indicating those states remain fully localized on one of two inversion-partner sectors. The linear change of αR 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 αR is solely arising from the electric field. The calculated Rashba parameter of the R-2 spin splitting can be explained regarding the model of the R-1 spin splitting (Eq. (3)), indicating a unified theoretical view for both R-1 and R-2 effects in bulk systems. Specifically, the effective electric field that promotes either R-1 and/or R-2 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 non-layered R-1 example, the α-SnTe6 or similarly the α-GeTe (a standard ferroelectric bulk R-1 compound predicted in 201326 and experimentally confirmed in 201627,28), where one can identify two inversion-partner 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 R-2 spin splitting in BaNiS2. As displacing the Te atom from Sn along [111] direction, the α-GeTe will undergo a phase transition from non-centrosymmetric rhombohedral phase to centrosymmetric rocksalt phase. We demonstrate that in the centrosymmetric rocksalt phase wavefunctions are evenly distributed among two inversion-partner 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 R-2 effect

R-2 materials6 are defined by having global inversion symmetry and two recognizable inversion-partner sectors with polar site point group symmetries. Designing R-2 materials possessing large hidden spin splitting and hence strong local spin polarization can benefit from two additional design principles:

  1. (i)

    Minimizing the mixing and entanglement of the wavefunction on the different inversion partners sectors. Here we point to a nontrivial mechanism of symmetry-enforced wavefunction segregation, keeping the doubly degenerate states on the different sectors from mixing (in contrast to the trivial physical separation of the two inversion-partner sectors). It is noteworthy that R-1 compounds do not have to maintain segregation-inducing 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 non-symmorphic symmetry along the \({\bar{\mathrm X}} - {\bar{\mathrm M}}\) direction in the BaNiS2 BZ. Other segregation enforcing symmetry operations may exist in general cases, but they have not been discovered yet.

  2. (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 materials2,24 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 R-2 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 BaNiS2. 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 αR vs. field for bands along \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) direction (Fig. 3a), demonstrating tailoring of the R-2 effect. For instance, Otani and colleagues29 have recently found a strong correlation between the charge density distribution and the strength of the Rashba effect at non-magnetic metal/Bi2O3 interfaces. Furthermore, the unexpected rapid rise of αR vs. field for bands along \({\bar{\mathrm X}} - {\bar{\mathrm \Gamma }}\) direction (Fig. 3a) implies that one might effectively tune the strength of R-2 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

First-principles band structure calculation

Electronic structures are calculated using density functional theory (DFT)30,31,32-based first-principles methods within the GGA33 implemented in the Vienna Ab initio simulation package (VASP)34. A plane-wave expansion up to 400 eV is applied and a Г-centered 16 × 16 × 1 Monkhorst-Pack35 k-mesh is used for the BZ sampling. The lattice constants used in the first-principles calculations are taken directly from the experimental data. The monolayer slab of BaNiS2 are separated by a 17.8 Å vacuum layer. We adopt the GGA + U method36 to account the on-site Coulomb interaction of localized Ni-3d orbitals. We follow the approach proposed by Neugebauer and Scheffler37 to apply a uniform electric field to monolayer BaNiS2 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 self-consistently such that the electrostatic field-induced 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 x-direction) 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.