Rashba spin-splitting in ferroelectric oxides: from rationalizing to engineering

Ferroelectric Rashba semiconductors (FERSC), in which Rashba spin-splitting can be controlled and reversed by an electric field, have recently emerged as a new class of functional materials useful for spintronic applications. The development of concrete devices based on such materials is, however, still hampered by the lack of robust FERSC compounds. Here, we show that the coexistence of large spontaneous polarisation and sizeable spin-orbit coupling is not sufficient to have strong Rashba effects and clarify why simple ferroelectric oxide perovskites with transition metal at the B-site are typically not suitable FERSC candidates. By rationalizing how this limitation can be by-passed through band engineering of the electronic structure in layered perovskites, we identify the Bi$_2$WO$_6$ Aurivillius crystal as the first robust ferroelectric with large and reversible Rashba spin-splitting, that can even be substantially doped without losing its ferroelectric properties. Importantly, we highlight that a unidirectional spin-orbit field arises in layered Bi$_2$WO$_6$, resulting in a protection against spin-decoherence.We highlight moreover that a unidirectional spin-orbit field arises in Bi$_2$WO$_6$, in which the spin-texture is so protected against spin-decoherence.


I. INTRODUCTION
In non-magnetic solids, one can naively expect the energy bands of electrons of up and down spins to be degenerate in absence of magnetic fields. However, in systems that break spatial inversion symmetry, e.g. at surfaces and interfaces but also in non-centrosymmetric bulk crystals, spin-orbit coupling (SOC) can lift such spin band degeneracy through the so-called Rashba and Dresselhaus effects [1][2][3]. During the last decade, these phenomena have attracted increasing interests in various fields, including spintronics, quantum computing, topological matter and cold atom systems [4,5].
Recently, the concept of ferroelectric Rashba semiconductors (FERSC) has been introduced [6]. It defines a new class of functional materials combining ferroelectric and Rashba effects, in which the spin texture related to the Rashba Spin Splitting (RSS) can be electrically switched upon reversal of the ferroelectric polarisation. As such, FERSC offer exciting perspectives for spintronic applications. The Rashba spin precession of a current injected in such materials can be controlled in a non-volatile way by their reversible ferroelectric polarisation. Moreover, FERSC allow to envision new devices interconverting electron-and spin-currents based-on the Edelstein [7] and reverse-Edelstein [8] effects. In two-dimensional ferroelectric materials with inplane polarization and strong anisotropy in the electronic structure, the spin-orbit field (SOF) was proposed to have unidirectional out-of-plane alignment: Ω SOF ( k) = α( P × k) = αk yẑ , where α is a system-dependent coefficient [9]. In such a case, injected electrons with in-plane spins would therefore precess around the z axis, giving rise to a long-lived persistent spin helix (PSH), a concept originally proposed for quantum-wells of III-V semiconductors with fine-tuned Dresselhaus and Rashba coefficients [10][11][12][13][14][15] and very recently extended to a subclass of noncentrosymmetric bulk materials [16]. Independently, FERSC can also, in some cases, exhibit ferro-valley properties [17].
The basic idea of FERSC was first put forward theoretically in bulk GeTe [18] and then experimentally confirmed in GeTe thin films [18][19][20]. Unfortunately, GeTe does not appear as the best candidate for concrete applications, due to its very small bandgap and related large leakage currents that, in most cases, prevent polarization switching [6]. The identification of alternative robust FERSC is therefore mandatory to achieve full exploitation of the concept. Although different directions have been explored [21][22][23][24][25][26][27][28], no really convincing candidate has emerged yet.
Here, we rationalise by means of first-principles approaches (see Methods) the discovery of a promising FERSC in the family of oxide perovskite compounds. Focusing first on simple perovskites, we highlight that robust ferroelectricity and SOC are necessary but not sufficient conditions to get an efficient FERSC. Furthermore, we clarify why these materials are typically not suitable candidates. We then propose a strategy to by-pass their intrinsic limitation in layered perovskites and identify the Bi 2 WO 6 Aurivillius phase as the first robust ferroelectric with large and reversible Rashba spin-splitting at the bottom of the conduction band and unidirectional SOF. We finally show that a significant n-type doping does not lead to a loss of its ferroelectric properties, suggesting the possibility of creating a doped FERSC appropriate arXiv:1903.01241v3 [cond-mat.mtrl-sci] 21 Mar 2019 for practical applications.

A. Simple perovskites
Ideal FERSC materials must meet a series of requirements. They should be non-magnetic ferroelectrics insulators with a sizable switchable polarisation and a reasonable bandgap. They should include heavy ions with large SOC exhibiting a significant RSS close to the valence or conduction band edge, which should be reversible with the polarization and, for applications based on spin/charge currents, should survive to appropriate doping.
Regarding ferroelectricity, it is natural to look to d 0 ABO 3 perovskites with a transition metal at the B-site [48], in which the bandgap is formally between O-2p and B-d states. As such, a large RSS around the bandgap would be more easily achieved by means of a heavy cation at the B-site while B-type ferroelectricity would likely favor an efficient polarization control of the RSS.
Tungsten oxide, WO 3 , is in line with previous requirements. It adopts the perovskite structure with an empty A-site and a heavy W atom on the B-site (see Fig. 1(a)).  (b) Evolution of the electronic band structure around the Fermi level when activating the polar distortion (P = 0 → P z s ) and SOC (λ = 0 → λ0). Projection on the t2g orbitals (dxy, dyz, dzx) of the reference structure (P = 0, λ = 0) are highlighted in colors. (c) Evolution of the splitting of the original t2g states at Γ-point for increasing polar distortion (P = 0 → P z s ) when including SOC (λ = λ0). The projection on the t2g orbitals are highlighted by mixing colors as in panel (c).
It is also an insulator with formal d 0 occupancy of the W 5d states. Although not intrinsically ferroelectricit adopts a nonpolar P 2 1 /c ground state [49] -, a recent study highlighted that it possesses low-energy metastable ferroelectric phases with large spontaneous polarizations (P s ≈ 50−70 µC·cm −2 ) arising from the opposite motion of W and O atoms (Supplemental Material I.A) [50]. Although never observed experimentally, these polar phases appear to be relevant prototypical states to investigate and rationalize the interplay between polarization and SOC in perovskite-like systems. Fig. 1(a) presents a sketch of the P 4mm ferroelectric phase of WO 3 , which exhibits a spontaneous polarization along the cartesian z-axis (P z s = 54 µC·cm −2 ). In Fig. 1(b), we show the calculated electronic band structure around the band gap of the cubic and tetragonal P 4mm phase of WO 3 with and without SOC. In the cubic phase (P z s = 0) without SOC (λ = 0), the bottom of the conduction band of WO 3 is at Γ and consists of triply-degenerate state of t 2g symmetry (pure d xy ,d yz and d zx orbitals). On the one hand, activating SOC (λ = λ 0 ) mixes the three t 2g states and produces a splitting ∆ SOC between a doubly-degenerate low-energy state of F 3/2,g symmetry (J = 3/2) and a higher-energy state of E 5/2,g symmetry (J = 1/2) [51]. On the other hand, the P 4mm phase (P = P z s ) without SOC has a splitting ∆ F E be-tween a low-energy state of B 2 symmetry (pure d xy orbital perpendicular to P z s at first perturbative order) and a higher-energy doubly-degenerate state of E symmetry (mixed d yz and d zx orbitals, partly hybridized with O 2p) [52]. In the presence of both SOC and ferroelectric polarization, three distinct levels of E 3/2 , E 1/2 and E 3/2 symmetry are present. For small amplitude of P z s , ∆ F E is small compared to ∆ SOC and all the three levels arise from a mixing of the three t 2g orbitals (see Fig.1(c)). As P z s and ∆ F E increase, the lowest E 3/2 acquires a dominant d xy character (like the B 2 state without SOC) while the higher-energy E 1/2 and E 3/2 levels are a mixing of d yz and d zx orbitals. This is supported by a simple tightbinding model (see Supplemental Material I.B).
Estimate of the RSS strength in the P 4mm phase through the effective Rashba parameter α R = 2E R /k R [53,54] gives a sizable value α R ≈ 0.7 eV·Å for the upper bands linked to E 3/2 and E 1/2 states. However, α R ≈ 0 for the band linked to the lowest E 1/2 state with strongly dominant d xy character (d xy is perpendicular to P z s ). The same conclusions apply to the ferroelectric Amm2 phase of WO 3 (see Fig. 2 and Supplemental I.C) where the polarization is along the xy pseudo-cubic direction (x in a reference axis rotated by 45 o around z with respect to x) and with a calculated P x s = 69 µC cm −2 . In this orthorhombic phase, the reference t 2g states are split in three levels of E 1/2 symmetry. The lowest state has a strongly dominant d y z character (d y z is perpendicular to P x s ) and does not show any significant RSS. These results are generic to ABO 3 perovskites and remain valid in presence of a ("non-empty") A-cation, as in KTaO 3 (see Ref. [55] and Supplemental II): the first unoccupied d-band does not show RSS in the presence of ferroelectric polarization.
A natural question at this stage is why the lowest t 2g state does not show RSS. As highlighted from a simple tight-binding model restricted to the t 2g subspace (see Supporting Information I. B), all the three levels are allowed to show RSS but α ∝ ∆ SOC /∆ F E and should vanish for all states in the limit of large ∆ F E . The question is then rather why the upper t 2g states show significant RSS. A plausible explanation is their interactions with the 2p states of bridging oxygen atoms. Combining an extended tight-binding model and first-principles calculations, we instead demonstrate that the dominant effect comes actually from their hybridization with the e g states.
This rationalize that significant RSS can appear in the t 2g conduction states of d 0 ABO 3 perovskites with heavy B-site atoms. However, RSS is restricted to the upper t 2g levels showing significant hybridization with the e g states. Consequently, achieving a large α R at the conduction band bottom of perovskites would require to get rid of the lowest energy state associated with the d ⊥ orbital perpendicular to P s . As we now show, this can be achieved if one confines the ferroelectric material in the direction perpendicular to P s , which is naturally realized for WO 3 in the Bi 2 W n O 3n+3 Aurivillius series, a family of single-phase layered compounds alternating WO 3 perovskite blocks with Bi 2 O 2 fluorite-like layers.

B. Layered perovskites
Bi 2 WO 6 is the n = 1 member of the Bi 2 W n O 3n+3 series. It is a strong ferroelectric with large polarization (P s ≈ 50 µC cm −2 ) and high Curie temperature (T c = 950 K). It has a measured experimental gap of 2.7 -2.8 eV [56,57] defined between the O 2p and W 5d states of the perovskite block (see Supplemental Material III.B). Furthermore, Bi 2 WO 6 is prone to n-type doping. [58,59] Bi 2 WO 6 exhibits a polar orthorhombic P 2 1 ab phase up to 670 o C, at which it undergoes a phase transition to another polar orthorhombic phase of B2cb symmetry, stable up to 950 o C [60,61]. As discussed in Ref. [61], the polar B2cb and P 2 1 ab phases are small distortions of the same reference I4/mmm high-symmetry structure and arise from the consecutive condensation of independent atomic motions: (i) a polar distortion along the x -axis (Γ − 5 symmetry) lowering the symmetry from I4/mmm to F mm2, (ii) tilts of the oxygen octahedra along the x -axis (X + 3 symmetry) lowering further the symmetry to B2cb and (iii) rotations of the oxygen octahedra around the z-axis (X + 2 symmetry) bringing the system in its P 2 1 ab ground state.
The polar F mm2 phase of Bi 2 WO 6 is comparable to the Amm2 phase of bulk WO 3 (Fig. 2) with a spontaneous polarization P x s in the xy pseudo-cubic directions and oriented in plane (i.e. perpendicular to the stacking direction). In Fig. 2 we compare the electronic band structure of Amm2 WO 3 and F mm2 Bi 2 WO 6 in the presence of SOC. In both cases, the t 2g states at Γ are split into 3 distinct E 1/2 levels. However, in Bi 2 WO 6 due to the asymmetry imposed by the Bi 2 O 2 layers along the z-axis, the states associated to the W d x z and d y z orbitals are pushed to much higher energy than the d x y . Consequently, the E 1/2 level at the conduction band bottom is now the one with dominant d x y character and it exhibits a large α R of 1.28 eV·Å.
Since the F mm2 phase is not observed experimentally, we now analyze how oxygen octahedra rotations (X + 3 and X + 2 ) present in the B2cb and P 2 1 ab phase on top of the polar distortions (Γ − 5 ) affect the RSS. In order to clarify the independent role of X + 3 and X + 2 distortions, we compare, in Table I ). It appears that the RSS is dominantly produced by the polar Γ − 5 distortion, while oxygen rotations play a detrimental but much minor role (see Supporting Information III.C): the X + 3 distortion tends to decrease α R , while the X + 2 distortion has no direct effect. In fact k R stays almost unchanged in all the phases, while E R is more affected. Overall, the amplitude of α R in the P 2 1 ab ground state is slightly reduced but remains comparable to that of the F mm2 phase. Fig. 3(a) shows the electronic dispersion curves of the P 2 1 ab phase, highlighting the significant spin splitting at the conduction band bottom. We notice an additional band splitting due to the presence of the oxygen tilts (X + 3 distortion) that double the unit cell in the y z-plane. Constant energy maps are also shown for an energy of 2.0 eV, along with the corresponding spin texture. The relative orientation of the coupled k and S components is determined by the symmetry of the system; in our case, the four polar phases belong to the C 2v point group that contains a C 2x two-fold rotation around the polar x -axis and two mirror planes, m ⊥y and m ⊥z . The electronic structure has the shape of two partially overlapping revolution paraboloids with revolution axes symmetrically shifted in opposite directions with respect to k y = 0. These two paraboloids are associated to electrons with opposite S z spin component and an additional S y contribution en-  Fig. 3(b)). It can also be noted that the spin splitting vanishes along the Γ → X path, corresponding to the polarization direction. As such, all the symmetry constraints and design criteria proposed in Ref. [9,16] in order to have a unidirectional SOF are met. We therefore conjecture the spin-lifetime in Bi 2 WO 6 to be long, due to reduction of spin decoherence mechanisms (the latter being related in Bi 2 WO 6 only to higher order momentum k-cube term in SOF). In addition, we expect a long-lived and nanometersized PSH, which could be of high relevance for future spintronic applications.

C. Doping
So far, we have shown Bi 2 WO 6 to be a robust switchable ferroelectric with large reversible RSS at the conduction band bottom. To be also of practical utility for spintronic applications based on charge/spin currents, it should additionally be possible to dope it with electrons, which contrary to some other Aurivillius, appears to be naturally the case [58,59]. Moreover, it should keep its FERSC properties when n-doped. This is far from obvious, since adding conduction electrons is expected to suppress ferroelectricity (and related RSS). Nevertheless, recent studies have shown that prototypical ferroelctrics like BaTiO 3 can preserve their ferroelectric distortion under n-doping concentrations up to 0.1e/u.c. [41,62].
In Fig. 3, we report the evolution of structural and electronic properties of the P 2 1 ab phase of Bi 2 WO 6 un- der electron doping (see Methods). In line with the electronic structure of the pristine material, doping electrons occupy the W 5d states around the conduction band bottom. Due to the dominant d x y character of these states, these electrons form a two-dimensional electron gas (2DEG) confined in the perovskite layer ( Fig.  3(c)). Amazingly, symmetry-adapted mode analysis of the atomic distortion of the doped structure with respect to the I4/mmm reference structure indicates that the global Γ − 5 polar distortion remains constant under electron doping (Figure 3e), rather than being suppressed. Further insights are given by the projection of this distortion on the phonon eigendisplacement vectors of the I4/mmm reference (in Fig. 3(f)). The global Γ − 5 polar distortion arises in fact from the condensation of two distinct phonon modes: a "W-mode" confined in the WO 3 layer and related to the off-centering of W in its O octahedron cage and a "RL-mode" (i.e. rigid-layer mode [61,63]), related to a nearly rigid motion of the Bi 2 O 2 layer with respect to the perovskite block. Although the global polar distortion remains constant under n-doping, the contribution of the W-mode is progressively suppressed when increasing the population of the W 5d xy states, while that of the RL-mode is amplified.
Concomitantly with the suppression of the W-mode distortion, α R (Fig. 3(b)) is progressively reduced under doping, highlighting that large polar distortion is not enough to lead to large α R ; rather, the polar distortion pattern must occur around the W atom responsible for the RSS, as in the W-mode. Although progressively re-duced, α R keeps nevertheless a sizable value up to large n-doping: at a doping level of 0.5 e − /u.c. (≈ 10 21 cm −3 ), α R is still as large as 0.3 eV·Å. Fig.3(g) shows the related electronic dispersion curves and spin texture.

IV. CONCLUSION
Combining first-principles calculations, symmetry analysis and tight-binding models, we have first rationalized step by step the RSS in the important family of ABO 3 perovskites with a transition metal at the B-site, demonstrating why they typically do not show significant RSS at the conduction band bottom. Relying on the concept of band-structure engineering in layered structures, we have then identified the Aurivillius Bi 2 WO 6 compound to be the first known ferroelectric oxide to show a large Rasba-like spin splitting at the conduction band bottom that can be reversed upon application of an external electrical field. Beyond being a practical ferroelectric, Bi 2 WO 6 offers additional and appealing peculiarities with respect to previously proposed FERSC candidates: i) a unidirectional spin-orbit field (arising from the combined presence of in-plane polarization, strong layering-induced anisotropy in the electronic structure and related symmetry properties) that protects the spintexture from spin dephasing; ii) the persistence of desired properties (such as robust ferroelectricity, large Rashba spin splitting and unidirectional spin-orbit field) upon sizable n-doping.
A similar behavior can a priori be found in other fer-  In TABLE S II, we report the main features of the low energy metastable P 4mm and Amm2 polar phases of WO 3 (more information about the other phases of WO 3 and their internal energy can be found in Ref. 50). We notice that the theoretical band gap E g is much larger in the Amm2 phase than in the P 4mm. This behavior, observed in several perovskites, is explained in term of B-cation off-centering displacements that increases the anti-bonding character of the orbital at the CBM [67].
B. The origin of Rashba splitting In this section, combining minimal and extended models with first-principles calculations, we shed light on the origin of Rashba splitting in simple ferroelectric perovskites.

Minimal model
Since the lowest conduction bands of the cubic (undistorted) phase consist in triply degenerate t 2g states, which are split from higher-energy doubly degenerate e g states due to the octahedral crystal field ∆ o , we will consider at first an effective model for t 2g ={yz, zx, xy} electrons only. In the high-symmetry phase, hopping to neighboring transition-metal ions is mediated by bridging oxygen p states, being strongly direction-dependent and resulting in substantially decoupled bonding networks for the three t 2g bands. Using Slater-Koster parametrization to keep track of the angular dependence of hopping interactions [68][69][70], the unperturbed Hamiltonian H 0 is diagonal in the t 2g manifold with eigenvalues: where α, β = x, y, z and t 0 =t 2 pd /∆ pd is taken as the energy reference, being t pd and ∆ pd the hopping amplitude and the splitting between O-p and metal-d orbital states, respectively.
A polar distortion along, say, the z direction has two major effects on the band structure of a cubic perovskite: first, it lifts the degeneracy within the t 2g manifold, inducing a splitting ∆ F E between a lower energy b 2 = xy state and higher-energy doubly degenerate e = {yz, zx} states [52,71]; second, it opens new covalency channels in the metaloxygen network due to orbital/lattice polarization effects, i.e., a polarization of the atomic-like orbital states and a change of the metal-oxygen bonding angle affecting the angular dependence of the two-center hopping integrals [69,70]. Focusing only on the band-structure properties in the plane perpendicular to the polar axis, the effect of the polar distortion can be modeled by the following perturbative term: where we followed the notation used in Ref. [70] for the orbital/lattice polarization coupling term  The polar-activated new hybridization channels are responsible of spin-splitting effects once spin-orbit coupling (SOC) is included. The atomic-like spin-orbit coupling for the t 2g manifold in the {d yz , d zx , d xy } ⊗ (↑, ↓) basis reads: where λ is the SOC coupling constant. The effect of such atomic-like interaction is to split the degenerate t 2g bands in two-fold degenerate states with total momentum j = 1/2 and local energy 2λ and in four-fold degenerate states with total momentum j = 3/2 and energy −λ, producing a splitting ∆ SOC = 3λ. The band structure around Γ point can be analyzed by diagonalizing the full Hamiltonian H = H 0 + H F E + H soc at k = 0 and including linear terms in k as subdominant contributions. The eigenvalues and eigenvectors of H at Γ point are given by: where s = ± and tan θ = 2 √ 2∆ F E /(∆ F E + 9λ), while the eigenvectors are expressed in the basis of |j, j z states: Clearly, ∆ F E couples only states with j z = ±1/2, since the 3 2 , s 3 2 state comprises only yz, zx orbital states. In the limit of vanishing ∆ F E , one recovers the SOC-split states, where E 1 = E 2 → −λ belonging to the j = 3/2 manifold while E 3 → 2λ. On the other hand, the ferroelectric crystal field ∆ F E affects the SOC-induced mixing of the t 2g states, reducing the mixed character of the relativistic eigenstates between |ψ xy,s and |ψ yz(zx),s which is linked to the Rashba-like spin-splitting effects. The xy character of |Ψ 1s and |Ψ 3s as a function of ∆ F E can be exactly evaluated, being: highlighting the fact that the lowest (highest) band acquires a rapidly increasing (decreasing) pure d xy character, as also shown in FIG. S 5.
In the rotated basis the k dependence of the Hamiltonian up to linear order in crystal momentum reads: where σ are Pauli matrices and σ 0 is the 2×2 identity matrix. Since the three manifolds are well separated in energy, and all off-diagonal terms are already linear in k, the leading term of the spin-momentum coupling is that parametrized by the diagonal Rashba parameters α ii , while the effect of the off-diagonal terms could be included using the standard Löwdin partitioning [72], resulting in cubic spin-momentum coupling terms. The Rashba coefficient for the lowest-energy state is given by: where the last result is valid in the limit of ∆ SOC /∆ F E 1. Therefore, the Rashba coupling of the lowest-energy state is controlled by the same parameter ∆ SOC /∆ F E which measures its pure xy character. In the limit of very large ∆ F E ∆ SOC , the Rashba splitting vanishes, simply because there is no effective coupling between opposite-spin manifolds.
On the one hand, the minimal model considered so far allows to explain why the lowest-energy state shows a negligibly small spin-splitting in the presence of a sizeable polar distortion. In fact, the Rashba coupling constant appears to be directly proportional to the activated new covalency channel (modeled by the γ 3 parameter), at the same time being inversely proportional to the energy splitting ∆ F E ; the competing and compensating effects of these two physical ingredients lead to a subtstantial suppression of spin splitting in the E 1 manifold. On the other hand, it fails in reproducing the large spin splittings observed in the E 2 , E 3 manifolds. In fact, no linear spin-momentum coupling appears in the E 2 manifold, while α 33 = −α 11 for the E 3 manifold, implying its strong suppression as a function of the polar crystal field.

Extended model
As previously noticed in Ref. [73], the main reason for this is that the atomic t 2g SOC is not an accurate enough description, and k-dependent corrections to the purely atomic H soc need to be included. Two such corrections, both activated by orbital/lattice polarization effects, can be envisaged: i) including the effect of SOC onto bridging oxygen ions mediating the d-d hopping interactions or ii) including contributions from e g states. The former results in spin-flip hopping terms between yz/zx ↑ and yz/zx ↓ states [73], giving Such Hamiltonian provides the following corrections to the diagonal Rashba coupling terms A Rashba-like spin-momentum coupling emerges in the E 2 manifold with a coupling constant which is directly proportional to the inversion asymmetry correction of the O-mediated hopping interactions. It is worth to notice that while the Rashba coupling in the lowest-energy state is still found to vanish as ∆ SOC /∆ F E 1, a finite coupling term survives in the E 3 manifold.
When considering the virtual processes between t 2g and e g manifolds, the additional hopping interactions γ 1 and γ 2 , arising from orbital/lattice polarization effects between the {yz/zx} and z 2 and x 2 −y 2 orbital states, respectively [70], must be included alongside the atomic SOC interacting terms; the resulting k−dependent SOC Hamiltonian reads: where This interacting term also produces Rashba-like spin-momentum coupling in all three manifolds, whose coupling constants can be expressed as: In this case, the Rashba-like splitting in the E 2 (E 3 ) manifold would raise mostly from virtual processes with the z 2 (x 2 − y 2 ) orbital state, while the splitting in the lowest-energy state is again found to vanish as the crystal-field splitting ∆ F E increases. Given the several energy scales entering in the proposed model, identifying the most relevant mechanism responsible for the observed spin splitting is not a trivial task. It is worth to notice, however, that the spin-splitting of the band connected to the lowest-energy state is always found to be strongly suppressed as its character becomes strongly xy-type, i.e., perpendicular to the polar axis.

First-principles results
In order to analyse the origin of Rashba splitting more quantatively, we then carry out first-principles calculations. First of all, we artificially switch on/off the partial SOC matrix elements to directly invistigate the orbital dependence of the Rashba splitting in the P 4mm phase of WO 3 . When we compare in FIG. S 6 panel (a) with (h) and panel (b) with (g), we observe that they are exactly the same, indicating that oxygens produce no significant Rashba splitting. Although the band structures without SOC from W-p orbitals (FIG. S 6(d)) look quite similar to the one with full SOC (FIG. S 6(a) According to our previous analytical model, such an effect can be closely related to the hybridization between t 2g and e g states, which is confirmed by our density-of-states calculations. As clearly shown in FIG. S 7, for the lowest E 1 band, there is no W-e g components, resulting in the absence of Rashba splitting. However, the minor occupied e g states hybridize with the dominant t 2g orbitals for the higher E 2 and E 3 bands, and then trigger their large Rashbatype splitting. Especially, the hybridization for the E 2 band is mainly between t 2g and W-d z 2 states, in accordance to the EQ (16).
To directly assess the influence of e g states on Rashba splitting, we then use orbital selective external potential (OSEP) method [74,75]. This approach can introduce a special external potential on any selected orbitals, which is analogous to the DFT + U method [76].Within the frame of OSEP, the specifically assigned atomic orbit |inlmσ feels the potential V ext , making the system Hamiltonian become as follows: where i denotes the atomic site, and n, l, m, σ are the principle, orbital , magnetic and spin quantum numbers, respectively. H 0 KS is the unperturbed Kohn-Sham Hamiltonian. Since the strength of overlap between orbitals is strongly dependent on their energy difference, we can modify the orbital interaction between states by applying an artificial field to shift the energy levels. As a representative, in FIG. S 8, we shift the energy level of the W-d z 2 orbital in the P 4mm phase of WO 3 to investigate its influence on Rashba-like splitting of the E 2 band.
It is clealry shown in FIG. S 8(a) that the artificial field indeed shifts the energy level of the W-d z 2 orbital and greatly affects the DOS. When the potential energy V d z 2 ext changes from negative to positive, the position of z 2 states moves upward. Their density of states is progressively reduced during the process, indicating that the hybridization between t 2g and z 2 states becomes smaller. When an artificial field is applied to up-shift z 2 orbital by 2 eV, due to its weaker hybridization with the t 2g states, the Rashba-type splitting in the E 2 band is suppressed. Such a large external potential will also bring the Fermi level upward, therefore the E 2 band is equivalently shifted down as shown in FIG. S 8(b). With the reduction of the hybridization between t 2g and z 2 orbitals by enlarging the positive V ext to 5 eV, the E 2 band moves downward continuously. Meanwhile, its Rashba splitting further decreases. Inversely, if we introduce a negative field to the P 4mm phase of WO 3 ( V d z 2 ext = −2 eV), the energy level of z 2 orbital is shifted down. As expected, stronger orbital interaction between t 2g and z 2 states results in an enlarged Rashba splitting in electronic dispersion for the E 2 band. In order to summarize the influence of applied potential on the Rashba-type splitting, we compare the splitting energy in FIG. S 8(c). It is clear that with the enhancement of the external field applying to the W-d z 2 orbital, the splitting energy of the E 2 band is gradually decreased. We therefore can conclude that the orbital interaction between t 2g and z 2 states can is the main feature responsible for the Rashba-type splitting, which is consistent with our analytical model and gives a strong evidence to confirm the critical role of e g states on Rashba splitting. the lowest band acquires a dominant d y z character, while the higher-energy levels mix d x y and d x z orbitals.

II. The case of KTaO3
A. Relaxed polar structures KTaO 3 is known to be an incipient ferroelectric, which can however be made ferroelectric under strain engineering [77]. Under suficcient tensile epitaxial biaxial strain, it adopts a ferroelectric Amm2 ground state with polarization in-plane (along the xy-direction) ; under sufficient compressive epitaxial biaxial strain, it adopts a ferroelectric P 4mm ground state with polarization out-of-plane (along the z-direction). TABLE S III report the main characteristics of KTaO 3 in its reference P m3m cubic phase and in the P 4mm and Amm2 polar phases obtained under −3% and +3% epitaxial biaxial strain. In FIG. S10, we report the evolution electronic band-structure of KTaO 3 in terms of SOC and polar distortion when going (i) from the P 4/mmm paraelectric reference to the P 4mm polar ground state under −3% epitaxial biaxial strain and (ii) from the P 4/mmm paraelectric reference to the Amm2 polar ground state under +3% epitaxial biaxial strain. We identify the same behavior as for WO 3   In TABLE S IV, we report the cell parameters and the amplitude of the atomic distortions with respect to the paraelectric I4/mmm reference (expressed in terms of symmetry-adapted mode amplitudes as obtained with AM-PLIMODE [79]) for the four ferroelectric metastable phases. We report amplitudes of primary distortions (Γ − 5 , X + 2 and X + 3 ) as well as of secondary modes (X − 1 , X − 3 , M + 5 and M − 3 ). The relaxed structures of the B2cb andP 2 1 ab phases are in good agreement with the experiment (disciption typically better in PBEsol than in LDA). We see no significant effect of the SOC on the atomic structure.

B. Electronic band structure
The atomically projected bands for the different phases of Bi 2 WO 6 are presented in FIG. S11. As expected from the rather ionic character of the compound, the lowest conduction bands are dominated by the W 5d states while the highest valence states have a dominant O 2p character. The Bi-6p bands lay at much higher energy.

C. Role of individual distortions on the RSS
In a complementary way to Table I comparing the RSS in the relaxed metastable phases of Bi 2 WO 6 , in TABLE S V, we report the evolution of the Rashba parameter α R and theoretical bandgap when condensing individually or together the 3 primary distortions connecting the I4/mmm reference to the P 2 1 ab ground state. As deduced already from Table I, the RSS is essentially produced by the Γ − 5 polar distortion, the c-axis rotation X + 2 does not have any effect on it and the a-axis tilt X + 3 tends to reduce it strength. Here the volume is fixed to the paraelectric I4/mmm reference to avoid any strain effect (this justifies why the values in TABLE S V and in TABLE I are slightly different). . S IV. Symmetry-adapted mode amplitudes (Å) respect the I4/mmm paraelectric reference and lattice parameters (Å) of distinct polar phases of Bi2WO6 fully relaxed in GGA (PBEsol) with and without SOC. Mode decomposition was performed using the AMPLIMODE software [79]. Modes amplitudes are in Å. Comparison with experimental data [? ] is provided for the B2cb intermediate phase and P 21ab ground state. LDA relaxation are performed with ABINIT package [80].

IV. Other Aurivillius materials
In order to generalize our findings, we computed the band structure and Rashba spin splitting for Bi 2 W 2 O 9 , Bi 2 W 3 O 12 , SrBi 2 Ta 2 O 9 , and Bi 4 Ti 3 O 12 Aurivillius compounds. The values of α R are given in TABLE S VI. In FIG. S9, we show also the band structure of Bi 2 W 3 O 12 : it is interesting to note that the splitting at the conduction band bottom is due to d x y orbitals of W 1 and W 3 , the interface tungsten atoms that are linked to the Bi 2 O 2 layers, while . S V. evolution of the Rashba parameter αR (eV·Å) and theoretical bandgap (eV) when condensing individually or together the 3 primary distortions (Γ − 5 , X + 2 and X + 3 ) connecting the I4/mmm reference to the P 21ab ground state. The cell parameters are fixed to those of the I4/mmm phase. the d x y orbital of the central W 2 atom is located at higher energy.