Recent advances in tunable spin–orbit coupling using ferroelectricity

Spintronic devices, which utilize and control the spin degree of freedom of electrons, have additional functionalities that the electric and magnetic signals can be interconverted with attractive advantages, such as non-volatile control, high-speed procession, high-density, and low power consumption.^1–5 Considering the orbital degree of freedom, the coupling between spin, charge, and orbit has become the key ingredient in this rapidly emerging field.⁶ These inter-couplings result in many novel physical phenomena, among which the spin–charge interconversions^7–11 and spin-torque effects^12–16 draw special attention because of the controllable electric and magnetic properties. Ferroelectric materials become the choice of materials since the inherent coupling between charge, lattice, orbit, and spin provides a natural way to control the spin properties. Its ferroelectric polarization induced intense electric fields can tune the carrier density, band structure, and spin–orbit coupling (SOC) of the adjacent materials with multi-functional properties. In addition, the energy consumption for polarization switching is ultralow, which is about a thousand times smaller than those needed to switch ferromagnets.^4,17 These make ferroelectricity attractive in spintronics for both fundamental science and industry applications with multifunctional properties.^11,17,18

In this Research Update, we first introduce the concept of SOC and its related novel physical effects, which are essential for understanding the topic. We then discuss the strength of SOC of different materials and heterostructures. The ferroelectrics with spin-dependent band splitting are highlighted in this part. Then, recent breakthroughs on ferroelectricity tuned spin–charge interconversion and spin transfer torque are introduced. A short summary and prospects on spin–orbit coupling via ferroelectricity for future device applications are discussed.

Figure 1 illustrates the basic concepts of SOC. The spin–orbit coupling is the interaction of spin angular momentum and orbit angular momentum of an electron. It is a relativistic interaction,^11,19 where an electric field (⁠$E⇀$⁠) acting on the moving electrons equals to an effective magnetic field (⁠$H⇀SO=E⇀×v⇀$⁠, where $v⇀$ is the velocity of moving electrons). As an example shown in Fig. 1(a), a moving electron around a nucleus experiences a momentum dependent magnetic field from the nucleus, and its spin state thus changes via precession, damping, and torque. Since the electric field from the nucleus is proportional to the atomic number (Z), the strength of SOC increases with Z as calculated from different methods shown in Fig. 1(b):²⁰ Z⁴ dependence is derived from the simple hydrogenic model and Z² scaling is obtained from Landau–Lifshitz’s estimation of the outermost electrons, while the color lines are Hartree–Fock method²¹ and the colored circles and the shaded area are calculated for outermost electrons by Shanavas et al.²⁰ 

For a nonzero net spin–orbital magnetic field, it is necessary to have an asymmetric electric field in the system, as well as unbalanced moving electrons. It was first reported by Dresselhaus²² in non-centrosymmetric zinc blende structured semiconductors, such as InSb and GaAs,²³ and the electronic band shows spin- and momentum-(k) dependent splitting. This is known as Dresselhaus SOC, which is attributed to the bulk inversion asymmetry (BIA) of the lattice and its crystal field with band splitting proportional to k³.^24,25 With strain inside the semiconductor, the additional band splitting from strain is reported to be linear with the k.^26,27 Meanwhile, Rashba²⁸ reported that in wurtzite semiconductors with the polar axis, the band shows spin-dependent splitting linear with the k, the same as the linear Dresselhaus SOC, but the effect on spin orientations is different according to their Hamiltonian [Figs. 1(d) and 1(e)].^25,29 Recently, ferroelectric materials with electric polarization become attractive because of the inherent coupling between charge, lattice, orbit, and spin, providing a natural way to control the spin properties with electric fields.^30,31 Besides the crystal field in bulk, an electric field normal to the surface or the interface could also contribute to SOC, which is named as Rashba-SOC. It is originated from the structural inversion asymmetry (SIA) and has been widely reported in quantum wells,^32–34 2D electron gas at the oxides interface,^35–37 and metal interfaces^8,38,39 with confined dimension asymmetry. Figure 2 presents the SOC-based effects with spin–charge interconversions and spin-torque effects, which can be further tuned by the ferroelectricity.

The strength of SOC can be theoretically calculated either by the tight-binding-based model Hamiltonian approaches²⁰ or by first-principle calculations of band structures.⁴⁰ Experimentally, the band structure can be directly observed by spin- and angle-resolved photoemission spectroscopy (ARPES).^40,41 SOC-related physical effects (Fig. 2) with spin–charge interconversion, such as spin Hall effect (SHE),¹⁰ inverse spin Hall effect (ISHE),⁹ spin galvanic effect (SGE, also called inverse Rashba–Edelstein effect, IREE),^7,42 and inverse spin galvanic effect (ISGE, sometimes called Edelstein effect),^43–45 provide more convenient ways to characterize the strength of SOC. These charge–spin interconversions are generally interpreted as the spin- and direction-dependent scattering. The H_SO acts on an electron spin with an additional magnetic energy (E_SO = μ_BH_SO, where μ_B is the Bohr magneton¹⁹), which shifts the electron band up (parallel to the spin) or down (antiparallel to the spin). This would lead to the spin-dependent scattering with higher scattering probability for the electrons with spin parallel aligned with the H_SO than those antiparallel aligned. While the scattering is isotropic, its total contribution to H_SO is zero. An electric current can break the balance, where the moving electrons with up and down spins would be deflected differently and form transverse spin imbalance because of the H_SO, i.e., the SHE for bulk and ISGE at the interface, and vice versa, a spin current can generate charge accumulation known as ISHE and SGE for the bulk and interface, respectively.

The spin–charge interconversion has been extensively investigated in the past decade. Based on charge-to-spin conversion, Kato et al.²⁴ observed SHE by a magneto-optical Kerr microscope, where spin polarization with opposite signs accumulated at the two edges of strained InGaAs semiconductors, as shown in Fig. 3(a). The spin accumulation can be absorbed by ferromagnets with changed magnetism via spin–orbit torque (SOT),⁴⁶ and thus, the SOC can be characterized magnetically by Faraday rotation²⁶ and Kerr rotation⁴⁷ techniques, as well as the magnetism-related electric transport properties, such as anomalous Hall effect^48,49 or anisotropic magnetoresistance.^23,50,51 Using the spin accumulation at the edges, a p-n junction based light-emitting diode [LED, Fig. 3(b)]^52–54 has been designed and fabricated, in which the circular polarization of the emission light is controlled by the direction of current [Fig. 3(c)]. The SHE can also be detected electrically, as a nonlocal device shown in Fig. 3(d).⁸ The current in Al injected from a ferroelectric electrode (FM1) is spin polarized with unequal spin up and spin down electrons, which are deflected in opposite direction and thus accumulated at the two edges with a measurable voltage. For pure spin current, the conversion of spin-to-charge can be detected electrically by ISHE voltage. The pure spin current can be generated magnetically by spin pumping using ferromagnetic resonance [FMR, Fig. 3(e)],^55–57 optically by circular polarized light,⁵⁸ or thermally by the spin Seebeck effect.⁵⁹ The flow of spin current in non-magnetic materials experiences spin-orbital magnetic field and scattered differently to generate charge accumulation, i.e., ISHE voltage as an example shown in Fig. 3(f).^37,60 These studies have greatly advanced the development of spin-orbitronics, although the efficiency of spin–charge interconversion still needs to be improved for future device applications.

The spin–charge interconversion efficiency in bulk is characterized by the spin Hall angle (SHA, θ_SHA), per se it is the tangent of an angle defined as the ratio of spin density and charge density. Figure 4(a) lists the SHA and the spin flipping length (l_sf) of some commonly used SOC-metals.¹⁰ Besides the low SHA value, the data present the trade-offs of SHA and the spin flipping length. Apparently, the conversion of spin to charge depletes the spin quantum states and thus limits the length of spin diffusion. Thence, λ = θ_SHAl_sf, generally called the inverse Rashba–Edelstein length for interface induced SOC, is used to characterize the efficiency of spin–charge interconversions. Figure 4(b) shows the values of λ for different systems. With confined dimensions, the metal interface (e.g., Bi–Ag)⁵⁵ and oxide interface with two-dimensional electron gas (2DEGs, e.g., LAO/STO interface)³⁷ show larger λ than bulk materials, which can be further tuned by a gate voltage.⁶¹ This is attributed to the Rashba-SOC, where the reduced crystal symmetry generates an electrical field normal to the interface. Recently, the van der Waals heterostructure with both strong SOC materials and weak SOC materials has been developed to achieve the coexistence of strong SOC and long spin flipping length. Much higher λ values have been reported for WS₂-,⁴³ WTe₂-,⁶² and MoTe₂-⁶³based heterostructure devices.

Ferroelectric materials have long-range dipolar order with static electric fields allowing the possibility for SOC.⁶⁴ Spin-dependent band splitting has been theoretically predicted in ferroelectric ordered bulk systems, such as polar semiconductors (e.g., BiTeI⁶⁵ and GeTe³¹), complex oxides (e.g., PbTiO₃,⁶⁶ BiAlO₃,²⁵ and Bi₂WO₆⁶⁷), perovskites (e.g., CsPbBr₃⁶⁸ and FASnI₃⁶⁹), and mono-layer ferroelectric materials.^70,71 Experimentally, the band splitting has been observed for GeTe by ARPES.^30,72–74 The band splitting parameters, known as Rashba parameters (α_R),^25,67,71 are summarized in Fig. 5. Comparing the surface-induced band splitting, ferroelectric materials have much higher α_R because of the polarization field. It is noteworthy that the heavy element in materials (for instance, GeS compared with GeTe and SnS), the specific crystalline symmetry (e.g., Bi₂WO₆ with different structures⁶⁷), and the bandgap affect the strength of the ferroelectricity-based Rashba-SOC.

Ferroelectric materials provide a knob to tune SOC at the interface via the density of electrons, the shift of Fermi levels,^75–77 the changes of spin polarization,^78–83 and the Rashba-band splitting. The ferroelectricity controlled SOC with tunable spin-torque effects and spin–charge interconversion efficiency will be introduced in this part.

The in-plane current induced perpendicular magnetization switch has been reported because of the spin-torque effects.^13,84,85 Figure 6 shows an example of the current-induced perpendicular magnetization switching¹³ and its ferroelectric control. A current (I_y) in Pt generates spin accumulation with moment along the x axis (σ_x) at the Pt/Co interface [Fig. 6(a)] because of the SHE, followed by the absorption of spin current by the Co layer with perpendicular magnetization (m_z). Two torques, the spin transfer torque from the accumulated spins (⁠$τ⇀ST=m⇀×σ⇀×m⇀$⁠, the direction is along the spin direction) and the external magnetic field (along the y axis) induced torque (⁠$τ⇀field=−m⇀×B⇀$⁠), act on the moment of electrons and change the switch of the magnetic alignments. As the sketched map shown in Fig. 6(b), an external magnetic field along the y axis tilts the perpendicular magnetic moment differently. Thence, for m_z > 0 and m_z < 0, the magnetization switched by the spin transfer torque is either clockwise or anticlockwise. This chiral magnetization switching can be detected by anomalous Hall resistance (along the x axis), as shown in Figs. 6(c) and 6(d) for B_y > 0 and B_y < 0, respectively. It was then reported that even without the external magnetic field, the chiral current-induced perpendicular magnetization switching can be observed in a hybrid ferromagnetic/ferroelectric heterostructure, Pt (4 nm)/CoNiCo (0.4/0.2/0.4 nm)/Pt (2 nm) multilayer on the ferroelectric PMN-PT substrate, as the device structure shown in Fig. 6(e).⁸⁶ By electrically changing the in-plane ferroelectric polarization of the substrate, the current-induced magnetization switching can be switched differently, as the chiral hysteresis loops shown in Figs. 6(f) and 6(g). This in-plane electric field control is also observed in a lateral symmetry-breaking Ta/CoFeB/TaO_x(wedge) device,⁸⁴ where the non-uniform oxygen content along the interface contributes to an in-plane electric field and switches the perpendicular magnetization. The ferroelectric controlled magnetization switching was further developed for logic device applications with all-electric programmable functions.^87,88

A recent report showed that in the LSMO/PZT/Pt device, the interface spin–orbit coupling can be tuned by the ferroelectric polarization of the PZT layer, as shown in Fig. 7.⁸⁹ The spin polarized current tunneling from the LSMO electrode is determined by the inverse spin Hall effect in Pt, which has reversible signs by the polarization of the PZT [Figs. 7(a)–7(c)]. This ferroelectric dependence is vanished either by inserting a Cu layer (LSMO/PZT/Cu/Pt device) or by increasing the thickness of Pt, ruling out the possibility of changes at the LSMO/PZT interface. The spin–charge conversion efficiency shows Pt thickness dependence, which is further fitted using interface SHA (θ_in) and bulk SHA (θ₀), as shown in Fig. 7(d). It was found that θ_in is dozens of times higher than θ₀ with the sign reversed by the PZT polarizations. The possible mechanisms are proposed as follows: (1) the atomic lattice displacement and the ferroelectric field altered electronic structure of the Pt layer at the interface; (2) the imperfect components such as interdiffusion, mixing, or redox would change the electronic structure at the interface; and (3) the Rashaba spin–orbit coupling at the PZT/Pt interface, as supported by density functional theory (DFT) calculations [Figs. 7(e) and 7(f)]. It is noteworthy that the band structures for upward PZT polarization show Rashba spin-splitting bands with α_R = −0.152 eV Å. The spin hall angle (θ) can be estimated from the inverse Rashba–Edelstein effect length (λ), which is the function of α_R and the spin flipping time (τ): $θ=2λdin=2αRτℏdin$⁠, where d_in is the thickness of the interface. The SHA estimated from band splitting is −0.18, which is in agreement with the interface spin Hall angle [−0.2, Fig. 7(d)] from the fitting of experiment data. This study provides experimental support on determining the mechanism of reversible TMR in magnetic ferroelectric tunnel junctions and highlights the significance of spin manipulation by ferroelectricity at the interface.

The ferroelectric control of spin–orbit coupling has also been observed in a SrTiO₃ Rashba system,⁴ where the local electric field at the interface promotes ferroelectric-polarization-direction-dependent Rashba-SOC and thus spin–charge conversions.

The magnetoelectric coupling describes the changes in electric polarization with an applied magnetic field, or conversely the changes in magnetization with an applied electric field, in multiferroic crystals^91–92 or at the ferroelectric/ferromagnetic epitaxial interface.^94–95 Combining this with SOC, a magnetoelectric spin–orbit (MESO)^17,96 device is proposed for charge-based logic computing. Figure 8 shows a MESO device proposed by Intel Corporations. It has two inverters: the first node converts the input current [I_C(input)] into the magnetization of the ferromagnet [Fig. 8(b)], and the second node converts the spin polarized current (I_s) injected from the ferromagnet into charge accumulation via SOC with an output current [I_C(output), Fig. 8(c)]. The I_C(output) can be used as the input current for the next device, and thus, a cascaded logic device is proposed with non-volatility and energy efficiency.

The observation of SOC induced novel physical effects has stimulated the rapid growth of spin-orbitronics over the past decade. The challenges lie in improving the efficiency of spin–charge interconversion, the trade-off of SOC and spin-flipping length, and active control of properties. Ferroelectricity-based SOC has advantages such as the polarization field induced strong Rashba-SOC, the external electric field tunable local control, the nonvolatility, and the low power consumption required for switching, which may extend the Moore’s law scaling via the development of new materials and the structure design of devices. The ferroelectricity controlled SOC, however, is still in a preliminary stage. Recent prototype devices demonstrate the intercoupling of spin–charge–orbit–lattice with ferroelectricity tunability, indicating possible applications in spin-based logic devices and memories. Future efforts may focus on developing devices that meet the requirements of integration, lower power, higher speed/density, and cost-effectiveness to promote the development of information technology. In particular, the interface controlled by atomic, magnetic, and electronic processes should be focused to produce emergent properties and functions. Last but not least, the precise control of ferroelectricity on each process (such as the spin–charge interconversion, the spin/charge transfer, and interface hybridization) should be carried forward to reveal the underground physics, which is extremely challenging.

This work was supported by the National Key Research and Development Program of China (Grant No. 2016YFA0300702), the National Natural Science Foundation of China (Grant Nos. 12074071 and 11504055), the Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ2480), the Program of Shanghai Academic Research Leader (Grant No. 18XD1400600), and the Shanghai Municipal Natural Science Foundation (Grant No. 18JC1411400).