Rationalizing and engineering Rashba spin-splitting in ferroelectric oxides

Abstract

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 polarization 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 a 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.

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 polarization. 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 polarization. Moreover, FERSC allow to envision new devices interconverting electron- and spin-currents based on the Edelstein⁷ and reverse-Edelstein⁸ effects. In two-dimensional ferroelectric materials with in-plane polarization and strong anisotropy in the electronic structure, the spin–orbit field (SOF) was also proposed to have unidirectional out-of-plane alignment: ${\vec{\Omega }}_{{\mathrm{SOF}}}(\vec{k})=\alpha (\vec{P}\times \vec{k})=\alpha {k}_{y}\hat{z}$, where $\alpha$ is a system-dependent coefficient.⁹ In such a case, injected electrons with in-plane spins would therefore precess around the $\vec{z}$ 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 non-centrosymmetric 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 GeTe¹⁸ 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 rationalize by means of first-principles approaches 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 for practical applications.

Results

Simple perovskites

Ideal FERSC materials must meet a series of requirements. They should be non-magnetic ferroelectrics insulators with a sizable switchable polarization 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 at ${d}^{0}$$AB$O${}_{3}$ perovskites with a transition metal at the B-site,²⁹ 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. 1a). It is also an insulator with formal ${d}^{0}$ occupancy of the W 5$d$ states. Although not intrinsically ferroelectric—it adopts a nonpolar $P{2}_{1}/c$ ground state³⁰—a recent study highlighted that it possesses low-energy metastable ferroelectric phases with large spontaneous polarizations (${P}_{s}\approx 50-70\ \upmu$C$\cdot$cm⁻²) arising from the opposite motion of W and O atoms (Supplemental Material II.A).³¹ 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

Electronic dispersion curves in the P4mm phase of WO${}_{3}$. a Sketch of the P4mm phase of WO${}_{3}$, with ${P}_{s}$ along the $z$-axis. b Evolution of the electronic band structure around the Fermi level when activating the polar distortion ($P=0\to {P}_{s}^{z}$) and SOC ($\lambda =0\to {\lambda }_{0}$). Projection on the ${t}_{2g}$ orbitals (${d}_{xy}$, ${d}_{yz}$, ${d}_{zx}$) of the reference structure ($P=0$, $\lambda =0$) are highlighted in colors. c Evolution of the splitting of the original ${t}_{2g}$ states at $\Gamma$-point for increasing polar distortion ($P=0\to {P}_{s}^{z}$) when including SOC ($\lambda ={\lambda }_{0}$). The projection on the ${t}_{2g}$ orbitals are highlighted by mixing colors as in panel (b). d Sketch of the level splittings at $\Gamma$ produced by SOC (Δ_SOC) and ferroelectric distortion (Δ_FE)

Figure 1a presents a sketch of the P4mm ferroelectric phase of WO${}_{3}$, which exhibits a spontaneous polarization along the cartesian $z$-axis (${P}_{s}^{z}=54$ µC$\cdot$cm⁻²). In Fig. 1b, we show the calculated electronic band structure around the bandgap of the cubic and tetragonal P4mm phase of WO${}_{3}$ with and without SOC. In the cubic phase (${P}_{s}^{z}=0$) without SOC ($\lambda =0$), the bottom of the conduction band of WO${}_{3}$ is at $\Gamma$ 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 ($\lambda ={\lambda }_{0}$) mixes the three ${t}_{2g}$ states and produces a splitting Δ_SOC (Fig. 1d) 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 ($J=1/2$).³² On the other hand, the P4mm phase ($P={P}_{s}^{z}$) without SOC has a splitting Δ_FE (Fig. 1d) between a low-energy state of ${B}_{2}$ symmetry (pure ${d}_{xy}$ orbital perpendicular to ${P}_{s}^{z}$ 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 $2p$).³³ 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 ${P}_{s}^{z}$, Δ_FE is small compared with Δ_SOC and all the three levels arise from a mixing of the three ${t}_{2g}$ orbitals (see Fig. 1c). As ${P}_{s}^{z}$ and Δ_FE 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 tight-binding model (see Supplemental Material II.B).

Estimate of the RSS strength in the P4mm phase through the effective Rashba parameter ${\alpha }_{R}=2{E}_{R}/{k}_{R}$^34,35 (see Supplemental Material I) gives a sizable value ${\alpha }_{R}\approx 0.7$ eV$\cdot$Å for the upper bands linked to ${E}_{3/2}$ and ${E}_{1/2}$ states. However, ${\alpha }_{R}\approx 0$ for the band linked to the lowest ${E}_{1/2}$ state with strongly dominant ${d}_{xy}$ character (${d}_{xy}$ is perpendicular to ${P}_{s}^{z}$).

The same conclusions apply to the ferroelectric $Amm2$ phase of WO${}_{3}$ (see Fig. 2 and Supplemental Material II.C) where the polarization is along the $xy$ pseudo-cubic direction ($x^{\prime}$ in a reference axis rotated by 45° around $z$ with respect to $x$) and with a calculated ${P}_{s}^{x^{\prime} }=69$ 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 ${d}_{y^{\prime} z}$ character (${d}_{y^{\prime} z}$ is perpendicular to ${P}_{s}^{x^{\prime} }$) and does not show any significant RSS.

Fig. 2

Comparison of WO${}_{3}$ and Bi${}_{2}$WO${}_{6}$. Sketch of the atomic structure and electronic dispersion along the $\Gamma$–Y direction focusing on the lowest conduction states, ${E}_{F}=0$) of the $Amm2$ phase of WO${}_{3}$ (left) and $Fmm2$ phase of Bi${}_{2}$WO${}_{6}$ (right). Contribution of the ${t}_{2g}$ orbitals (${d}_{xy}$, ${d}_{yz}$, ${d}_{zx}$) to the different electronic states are highlighted in colors

These results are in fact generic to ABO${}_{3}$ perovskites and remain valid in presence of a (“non-empty”) A-cation, as in KTaO${}_{3}$ (see ref. ³⁶ and Supplemental Material III): the first unoccupied $d$-band is related to the orbital perpendicular to ${P}_{s}$ (${d}_{\perp }$) and does not show significant 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 Supplemental Material II.B.1), all the three levels are allowed to show RSS but $\alpha_{R} \propto {\Delta }_{{\mathrm{SOC}}}/{\Delta }_{{\mathrm{FE}}}$ and should vanish for all states in the limit of large ${\Delta }_{{\mathrm{FE}}}$. 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 identify that the dominant effect comes actually from their hybridization with the ${e}_{g}$ states (see Supplemental Material II.B.2-3).

This rationalizes that significant RSS can appear in the ${t}_{2g}$ conduction states of ${d}^{0}$$AB$O${}_{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 ${\alpha }_{R}$ at the conduction band bottom of perovskites would require to get rid of the lowest energy state associated with the ${d}_{\perp }$ 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.

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 (${P}_{s}\approx 50$ µC cm^−2 37, see Supplemental Material IV.A), and high Curie temperature (${T}_{c}=95{0}\,^{\circ }{\rm{C}}$³⁸). It has a measured experimental gap of 2.7–2.8 eV ^39,40 defined between the O $2p$ and W $5d$ states of the perovskite block (see Supplemental Material IV.B). Furthermore, Bi${}_{2}$WO${}_{6}$ is prone to n-type doping.^37,41

Bi${}_{2}$WO${}_{6}$ exhibits a polar orthorhombic $P{2}_{1}ab$ phase up to 670 °C, at which it undergoes a phase transition to another polar orthorhombic phase of $B2cb$ symmetry, stable up to 950 °C.^38,42 As discussed in ref. ⁴², 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 $x^{\prime}$-axis (${\Gamma }_{5}^{-}$ symmetry) lowering the symmetry from $I4/mmm$ to $Fmm2$, (ii) tilts of the oxygen octahedra along the $x^{\prime}$-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 $Fmm2$ phase of Bi${}_{2}$WO${}_{6}$ is comparable to the $Amm2$ phase of bulk WO${}_{3}$ (Fig. 2) with a spontaneous polarization ${P}_{s}^{x^{\prime} }$ 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 $Fmm2$ Bi${}_{2}$WO${}_{6}$ in the presence of SOC. In both cases, the ${t}_{2g}$ states at $\Gamma$ 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 ${d}_{x^{\prime} z}$ and ${d}_{y^{\prime} z}$ orbitals are pushed to much higher energy than the ${d}_{x^{\prime} y^{\prime} }$. Consequently, the ${E}_{1/2}$ level at the conduction band bottom is now the one with dominant ${d}_{x^{\prime} y^{\prime} }$ character and it exhibits a large ${\alpha }_{R}$ of 1.28 eV$\cdot$Å.

Since the $Fmm2$ 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 (${\Gamma }_{5}^{-}$) affect the RSS. In order to clarify the independent role of ${X}_{3}^{+}$ and ${X}_{2}^{+}$ distortions, we compare, in Table 1, ${\alpha }_{R}$ in distinct fully relaxed ferroelectric phases: $Fmm2$ (${\Gamma }_{5}^{-}$), $B2cb$ (${\Gamma }_{5}^{-}$+${X}_{3}^{+}$), $B2cm$ (${\Gamma }_{5}^{-}$+${X}_{2}^{+}$), and $P{2}_{1}ab$ (${\Gamma }_{5}^{-}$+${X}_{3}^{+}$+${X}_{2}^{+}$). It appears that the RSS is dominantly produced by the polar ${\Gamma }_{5}^{-}$ distortion, while oxygen rotations play a detrimental but much minor role (see Supplemental Material IV.C): the ${X}_{3}^{+}$ distortion tends to decrease ${\alpha }_{R}$, 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 ${\alpha }_{R}$ in the $P{2}_{1}ab$ ground state is slightly reduced but remains comparable to that of the $Fmm2$ phase.

Table 1 Computed spontaneous polarization (${P}_{s}$), $k$-vector splitting (${k}_{R}$), energy splitting (${E}_{R}$), Rashba parameter (${\alpha }_{R}$), and energy gap (E${}_{g}$) for distinct ferroelectric phases of Bi${}_{2}$WO${}_{6}$. Values in few selected reference systems are reported for comparison

Fig. 3

Pristine and n-doped $P{2}_{1}ab$ phase of Bi${}_{2}$WO${}_{6}$. a, b ${S}_{z}$ spin-projected band dispersion around the Fermi level (bottom) and spin-texture into ${k}_{x}{k}_{y}$ plane at E = 2.0 eV (top) of pristine Bi${}_{2}$WO${}_{6}$ in its $P{2}_{1}ab$ phase for a up and b down polarization directions. A reversible spin-splitting is observed at the conduction band bottom. c Charge-density of the conduction bands in the $P{2}_{1}ab$ phase of Bi${}_{2}$WO${}_{6}$ with a doping of 0.5 e/u.c. d–f Evolution with the electronic doping concentration of, respectively, d the Rashba parameter, e the ${\Gamma }_{5}^{-}$, $X^{+}_{2}$, and $X^{+}_{3}$ distortion amplitudes, and f the respective W-mode and RL-mode contributions to ${\Gamma }_{5}^{-}$ distortion in the $P{2}_{1}ab$ phase of Bi${}_{2}$WO${}_{6}$. g${S}_{z}$ spin-projected bands (bottom) and spin-texture into ${k}_{x}{k}_{y}$ plane at E${}_{F}$ = 0 eV (top) in the $P{2}_{1}ab$ phase of Bi${}_{2}$WO${}_{6}$ with a doping of 0.5 e/u.c. A characteristic 2DEG-like behavior is observed from the parabolic band shape

Figure 3a 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 doubles the unit cell in the $y^{\prime} z$-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 ${C}_{2x}$ twofold rotation around the polar $x^{\prime}$-axis and two mirror planes, ${m}_{\perp y}$ and ${m}_{\perp z}$. 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 ensuring rotation of S in the region where the ellipsoids cross. No ${S}_{x}$ component is observed, consistently with the Rashba-like effect. The RSS is proportional to the polarization and is reversed under polarization switching (Fig. 3b).

It can also be noted that the spin-splitting vanishes along the $\Gamma$$\to$X path, corresponding to the polarization direction. As such, all the symmetry constraints and design criteria proposed in refs. ^9,16 in order to have a unidirectional SOF are met. Specifically, the crystal symmetries impose that such unidirectional SOF is preserved even when including higher-order spin-momentum coupling terms, thus ruling out spin-decoherence effects based on the Dyakonov-Perel relaxation mechanism.^11,12 We therefore conjecture the spin-lifetime in Bi${}_{2}$WO${}_{6}$ to be long, possibly leading to a long-lived and nanometer-sized persistent spin helix,^{10,11,12,13,14,15} which could be of high relevance for future spintronic applications.

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 making it already a promising candidate for FERSC optical devices. 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.^37,41 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 0.1 e/u.c.^43,44

In Fig. 3, we report the evolution of structural and electronic properties of the $P{2}_{1}ab$ phase of Bi${}_{2}$WO${}_{6}$ under 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 ${d}_{x^{\prime} y^{\prime} }$ character of these states, these electrons form a two-dimensional electron gas (2DEG) confined in the perovskite layer (Fig. 3c). 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 ${\Gamma }_{5}^{-}$ polar distortion remains constant under electron doping (Fig. 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. 3f). The global ${\Gamma }_{5}^{-}$ 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^42,45), 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 5${d}_{xy}$ states, while that of the RL-mode is amplified.

Concomitantly with the suppression of the W-mode distortion, ${\alpha }_{R}$ (Fig. 3b) is progressively reduced under doping, highlighting that large polar distortion is not enough to lead to large ${\alpha }_{R}$; rather, the polar distortion pattern must occur around the W atom responsible for the RSS, as in the W-mode. Although progressively reduced, ${\alpha }_{R}$ keeps nevertheless a sizable value up to large n-doping: as illustrated in Fig. 3d, g, at a doping level of 0.5 e⁻/u.c. ($\approx \hskip -4pt 10^{21}$ cm⁻³), ${\alpha }_{R}$ is still as large as 0.3 eV$\cdot$Å. Spin textures at intermediate carrier concentrations are reported in Supplemental Material IV.D. Such a survival of the polar distortion and related sizable RSS under significant doping is a necessary requirement for FERSC but, of course, it does not guarantee that it is still possible to achieve electric switching of the polarization, which ask for further experimental confirmation.

Discussion

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 under polarization switching. 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 spin transport from spin dephasing; (ii) the persistence of desired properties (such as robust polar distortion, large Rashba spin-splitting and unidirectional spin–orbit field) upon sizable n-doping.

A similar behavior can a priori be found in other ferroelectric Aurivillius phases, like SrBi${}_{2}$Ta${}_{2}$O${}_{9}$, or even Bi${}_{4}$Ti${}_{3}$O${}_{12}$. However, the RSS depends on the strength of the $B$-cation SOC that increases with the oxidation state⁴⁶ and ${P}_{s}$, which are maximized in W-based compounds (see Supplemental Material V). Within the Bi${}_{2}$W${}_{n}$O${}_{3n+3}$ series, Bi${}_{2}$W${}_{2}$O${}_{9}$ and Bi${}_{2}$W${}_{3}$O${}_{12}$ show a large ${\alpha }_{R}$ as well (Supplemental Material V). However, Bi${}_{2}$W${}_{2}$O${}_{9}$ is not ferroelectric⁴⁷ and Bi${}_{2}$W${}_{3}$O${}_{12}$ has not been synthesized yet. Therefore, Bi${}_{2}$WO${}_{6}$ emerges as the best candidate so far for large RSS and unidirectional SOF in the whole family of perovskite-based oxides, calling for experimental confirmations of our theoretical predictions. Our work also motivates and rationalizes the search of alternative candidates in other families of naturally layered perovskites like Ruddlesden-Popper and Dion-Jacobson series.⁴⁸

Methods

First-principles calculations relied on density functional theory (DFT)^49,50 within the projected augmented waves (PAW) method⁵¹ as implemented in the Vienna Ab initio Simulation Package (VASP).^52,53 Many results were also checked using ABINIT^54,55,56 with norm-conserving pseudopotentials. The exchange-correlation effects were estimated within the generalized gradient approximation (GGA) using the PBEsol parameterization.⁵⁷ The following electrons were treated as valence states: Bi(5${d}^{10}$6${s}^{2}$6${p}^{3}$), W(5${p}^{6}$6${s}^{2}$5${d}^{4}$), and O(2${s}^{2}$2${p}^{4}$). Convergence was reached using a Monkhorst-Pack⁵⁸ 8$\times$8$\times$2 k-point mesh for Bi${}_{2}$WO${}_{6}$ (8$\times$8$\times$8 for WO${}_{3}$) and a 600 eV energy cutoff. Structural relaxations were converged until forces are <1 meV$\cdot$Å⁻¹. The spin–orbit coupling was included into the calculations as in ref. ⁵⁹. Electron doping was performed by adding electrons to the total electronic density and introducing a neutralizing homogeneous background charge to compensate the additional electrons, as previously done in several works;^44,60,61,62 electronic and atomic relaxations were carried out at fixed volume.⁶³ The spin-texture was analyzed using the script PyProcar,⁶⁴ the structural distortion was analyzed with AMPLIMODE⁶⁵ and the figures of atomic structures were elaborated with VESTA.⁶⁶