Nonlinear orbital and spin Edelstein effect in centrosymmetric metals

Nonlinear responses to external perturbations exhibit distinct symmetry properties compared to linear responses. As a result, novel types of nonlinear responses can emerge that are not allowed in the linear regime, such as shift current and unidirectional magnetoresistance^1,2,3,4,5. Unlike charge, spin possesses directional degrees of freedom, and spin-direction-related phenomena forbidden in the linear regime may arise in the nonlinear regime. Nonlinear spintronics, an emerging field that integrates nonlinear dynamics with spintronics, has gained attention in recent studies^6,7,8,9,10.

The orbital angular momentum (OAM) and spin densities can be generated from circular light^11,12,13 or static electric field^14,15. Especially, ref. ¹⁵ predicts that in the second-order response, an external electric field can generate spin density even in centrosymmetric bulk systems, which is forbidden in the linear regime. Such nonlinear spin generation may be called the nonlinear spin Edelstein effect (NSEE) (instead of the Rashba–Edelstein effect¹⁶) since it occurs even in centrosymmetric systems where the Rashba-type spin–momentum coupling is forbidden¹⁷. The findings discussed in ref. ¹⁵ offer fresh insights from a fundamental physics standpoint.

However, ref. ¹⁵ has a limitation. The Hamiltonian model used to elucidate the origin of NSEE requires inversion symmetry breaking, at least locally. Although the model has the global inversion symmetry (therefore, the described system has hidden Rashba coupling¹⁵), the inversion symmetry is broken locally, and the NSEE of the model vanishes when the local inversion symmetry is restored. This model does not provide a comprehensive explanation for the generic existence of NSEE, as the local inversion symmetry breaking is crucial. To go beyond this limitation and understand the NSEE in truly centrosymmetric systems (even at the local level), one needs to go beyond the spin degree of freedom, constructing a non-trivial Hamiltonian solely based on spin degrees of freedom is not feasible¹⁸.

Recently, there has been a growing recognition that spin dynamics are strongly influenced by orbital dynamics, which leads to the emerging field of orbitronics^{19,20,21,22,23,24,25,26,27}. Orbital dynamics play a crucial role in spin dynamics, particularly in time-reversal centrosymmetric systems where only the orbital exhibits non-trivial dynamics, and spin dynamics is induced by the orbital dynamics through spin–orbit coupling (SOC). Hence, to achieve a comprehensive understanding of spin dynamics in centrosymmetric systems, it is desired to grasp orbital dynamics¹⁸.

In this paper, we expand ref. ¹⁵ to include the orbital degree of freedom and investigate the NSEE and the nonlinear orbital Edelstein effect (NOEE) in centrosymmetric systems. First, we examine a centrosymmetric model system not only globally but also locally and demonstrate that it exhibits NOEE even without SOC. In contrast, the NSEE vanishes in the zero SOC limit and becomes finite only when the SOC is turned on. Our study indicates that the orbital degree of freedom is crucial for the NSEE in centrosymmetric systems; the nonlinear orbital Edelstein effect arises first, which in turn induces the NSEE through the SOC. This clarifies the NSEE’s mechanism and is our work’s first main result. Second, we demonstrate that the second-order orbital Edelstein effect can be understood intuitively. Suppose we investigate the E_iE_j response of OAM density generation. Then we can decompose this effect accordingly. First, when E_i is applied, it generates a momentum-dependent orbital moment, thereby converting a centrosymmetric system into an effective non-centrosymmetric system with the orbital Rashba effect. After that, E_j induces the orbital Edelstein effect in the effective orbital Rashba system. The reverse process (E_i ↔ E_j) also contributes to the total induced OAM density. Third, we perform density functional theory (DFT) calculations to establish the presence of NSEE and NOEE in a wide range of materials. To demonstrate this, we show that orthorhombic materials exhibit non-negligible NOEE as the NSEE. Furthermore, we illustrate that global and local centrosymmetric materials lacking NSEE in bulk due to C_3z and C_4z symmetries can have NSEE when sliced in specific orientations, which breaks those rotational symmetries. For instance, materials widely used in spintronics, such as fcc Pt, Rh, Ir, and Pd, can have non-negligible NSEE and NOEE if they are sliced along the (110) direction. Lastly, we remark that the polarization direction of spin/orbital current from the conventional spin/orbital Hall effect generated by the linear electric field and the polarization direction of the spin/orbital density derived from the NSEE generally exhibit disparities. This raises the possibility of harnessing this effect for the magnetic-field-free switching of the magnetization in ferromagnets¹.

Nonlinear orbital Edelstein effect

We first consider the linear orbital Rashba–Edelstein effect (REE), which occurs in systems lacking inversion symmetry^{24,25,26,28,29}. The eigenstates of such systems can have nonzero expectation values of the orbital angular momentum (OAM) that vary with k. Although the total OAM vanishes in equilibrium due to the time-reversal symmetry, which enforces (f({varepsilon }_{n{{{bf{k}}}}})=f({varepsilon }_{bar{n}-{{{bf{k}}}}})) and (leftlangle {psi }_{n{{{bf{k}}}}}rightvert {{{bf{L}}}}leftvert {psi }_{n{{{bf{k}}}}}rightrangle =-leftlangle {psi }_{bar{n}-{{{bf{k}}}}}rightvert {{{bf{L}}}}leftvert {psi }_{bar{n}-{{{bf{k}}}}}rightrangle), where the state ((bar{n}-{{{bf{k}}}})) is the time-reversed state of the state (nk), the total OAM can acquire a finite value when an external electric field is applied since the field-induced Fermi surface shift is dissipative and breaks the time-reversal symmetry in the state occupation. Next, we consider our main systems where both inversion and time-reversal symmetry coexist. In such systems, the OAM expectation values vanish for their eigenstates. Consequently, the field-induced Fermi surface shift alone cannot generate the net OAM density. The situation changes in the second-order response since the extra order of an electric field can induce interband mixing, thereby distorting the wavefunction. For the interband mixing, the orbital texture plays an important role¹⁹. To illustrate this point, we note that energy eigenstates with an expectation value of zero for OAM, such as p_x and p_y orbitals, vary across k-space^19,30. When an electric field is applied, each eigenstate gets deformed ((leftvert {psi }_{n{{{bf{k}}}}}rightrangle to leftvert {tilde{psi }}_{n{{{bf{k}}}}}rightrangle)) due to the real orbital texture^19,30 and acquires k-dependent nonzero OAM expectation value (Fig. 1b). Roughly speaking, the acquired OAM L is proportional to E × k and thus generates the orbital Hall effect, where L, E, and v are perpendicular to each other. If we write this field-induced effect into a perturbation equation up to the first order, we obtain

which is derived in ref. ³¹. This can be interpreted as forming an electric field-induced orbital Rashba texture resulting from breaking inversion symmetry by the applied electric field (Fig. 1b and d). For instance, when an electric field is applied to a two-dimensional system along the x-direction, it gives rise to an L_z orbital current flowing along the y-direction, which can be expressed as α(k_yL_z)E_x. This phenomenon can be understood as forming an orbital Rashba texture induced by breaking inversion symmetry along the x-direction by the electric field. If an additional electric field is applied along the y-direction to the system where the inversion symmetry is effectively broken by an electric field along the x-direction [Eq. (1)], the y-direction field effectively breaks the time-reversal symmetry through the Fermi surface shift and gives rise to a finite total OAM (Fig. 1c). This intriguing observation corresponds to NOEE, which will be investigated in this paper. In the case of NSEE, it emerges through the conversion of NOEE by SOC. To achieve a giant NSEE, a crucial requirement is the presence of a substantial NOEE. Without such NOEE, achieving a giant NSEE becomes difficult. The reason lies in the role of SOC, which is crucial for generating NSEE. Considering that SOC is given as L ⋅ S in centrosymmetric systems, the effect of SOC is more pronounced when the L is large (in the case of a substantial NOEE), and conversely, if the L is small, the influence of SOC diminishes. Therefore, to make the giant NSEE, giant NOEE is primary.

a The schematic diagram for a real orbital texture. For the other figures b–e, the tangential (outer) band is only considered for a simple explanation. b The nonequilibrium orbital Rashba texture is initially induced by E_x. The transparent figure represents the equilibrium orbital texture. c Then, the nonequilibrium orbital Rashba texture is shifted by E_y, resulting in the formation of nonzero OAM. d The role of E_x and E_y can be exchanged so that the nonequilibrium orbital Rashba texture is generated by E_y (e) and the NOEE is generated by E_x The sign of two contributions to the NOEE (b and c vs. d and e) are opposite, so they are canceled by each other when C_3z or C_4z symmetries exist.

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Deriving this phenomenon using the Moyal–Keldysh³² formula provides a formal expression that captures the physics described earlier, which is given by

where Γ is the energy level-broadening constant and X is either L or S. It allows us to quantitatively evaluate the interplay between the electric field-induced orbital Rashba texture [({Omega }_{nm{{{bf{k}}}}}^{i}) (Fig. 1b) and ({Omega }_{nm{{{bf{k}}}}}^{j}) (Fig. 1d)] and Fermi surface shift [(frac{partial f({varepsilon }_{n{{{bf{k}}}}})}{partial {k}_{j}}) (Fig. 1c) and (frac{partial f({varepsilon }_{n{{{bf{k}}}}})}{partial {k}_{i}}) (Fig. 1e)] to generate the resulting nonzero OAM density. A few crucial remarks follow. Firstly, it is worth noting that the OAM density derived by this represents the only term in the time-reversal and inversion symmetric metals¹⁵. Next, for Eq. (2b) to yield a nonzero result, both C_3z and C_4z symmetries must be broken since, otherwise, the two terms in Eq. (2b) cancel out each other. Hence, in order to achieve a substantial NOEE, we have to focus on materials that significantly break these symmetries. Moreover, the nonzero OAM density resulting from this is roughly proportional to the anisotropy of α(k) in k-space, which is linked to the anisotropy of the orbital Hall conductivity (OHC) in linear order induced by E_i or E_j. That is, the degree of anisotropy of the OHC governs the magnitude of the NOEE. We note that this concept aligns with the notion of anomalous spin polarizability, as introduced in ref. ¹⁵.

Tight-binding model illustration

To illustrate the physics described above, we introduce an intuitive tight-binding model. Our analysis considers a rectangular lattice with s, p_x, and p_y orbitals with nearest-neighbor hopping (Fig. 2b). The C_3z rotation symmetry is obviously absent, and the C_4z rotation symmetry is broken in the lattice spacing a and b along the x and y directions are different from each other. The hopping rule is schematically depicted in Fig. 2a. Here, η is the parameter for C_4z rotational symmetry breaking. Note, however, that the model system has the inversion symmetry both locally and globally.

a Hopping rules of the rectangular lattice with s, p_x, and p_y orbitals model (p_z orbitals are omitted). Since the lattice constants are different, the strength of sp hybridization is different, which is captured by the parameter η. b The band structure of Eq. (3) with η = 1 (dotted lines) and η = 0.9 (blue lines). Three bands correspond to three p-orbital character dominant bands, while the s-orbital character dominant band is omitted since it is far away from the p-orbital character bands. c Result in NOEE from tight-binding calculation with η = 0.9. Near the exact crossing point (around 0 and −1 eV), the OAM density shows significant growth. d The nonlinear OAM accumulation ({alpha }_{zxy}^{{rm {L}}}) with respect to the anisotropy η. e The nonlinear OAM ({alpha }_{zxy}^{rm {{L}}}) and spin ({alpha }_{zxy}^{{rm {S}}}) accumulation with respect to the SOC λ_SO. The inset graph: the ratio ({alpha }_{zxy}^{{rm {S}}}/{alpha }_{zxy}^{{rm {L}}}) between the nonlinear spin and OAM accumulation with respect to λ_SO.

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Figure 2 illustrates the band structure calculated at η = 1 and η = 0.9 and the results obtained by calculating Eq. (2) using the tight-binding model. In the case of a square lattice (a = b, η = 1), where C_4z symmetry is preserved, the NOEE caused by the three types of the quadratic perturbations to ({E}_{x}^{2}), ({E}_{y}^{2}) and E_xE_y fields are all zero. However, as the value of η deviates from 1, NOEE response to E_xE_y yields nonzero values, and the overall OAM density increases (Fig. 2c). This is due to the increasing anisotropy of the k-dependent orbital angular momentum derived from the linear order as η moves further away from 1. Figure 2d shows that the OAM density increases as the C_4z symmetry is broken. The NOEE due to ({E}_{x}^{2}) and ({E}_{y}^{2}) perturbation is still zero for this case due to the mirror M_x and M_y symmetry, respectively. From the comparison with the band structure (Fig. 2b), it becomes evident that the NOEE arises mainly near the band touching points. The conversion from the nonlinear OAM to the nonlinear spin density is proportional to the SOC strength λ_SO (Fig. 2e). Both the OAM and spin densities become smaller as λ_SO increases since the band gap, which generates the OAM and spin densities increase as λ_SO increases. An intriguing observation is that the NOEE and NSEE emerge even in this simplified model featuring distinct hopping parameters along the x-axis and y-axis. This suggests that the phenomenon may be prevalent in a wide range of multi-orbital systems exhibiting similar symmetry-breaking characteristics, providing insight into materials that can have large NOEE and NSEE.

DFT calculation

In this section, we calculate both NOEE and NSEE in real materials with the aforementioned symmetries broken. In centrosymmetric orthorhombic metals, such as IrW, MoRh, MoIr, and TiPt^33,34, we find that when the electric field E = 10⁵ V/m ^35,36 is applied, the NOEE-induced OAM and NSEE-induced spin density are given by δL_z and δS_z ~ 10⁻⁷−10⁻⁸ μ_B/(nm)³. We also find that even the materials that do not exhibit the NOEE and the NSEE in bulk can have the NOEE and the NSEE when they are sliced into films in the proper direction. Materials widely used in spintronics, such as fcc Pt, Rh, Pd, Ir, and bcc V^{37,38,39,40,41}, can have non-negligible NOEE and NSEE if it is sliced along the (110) direction since they only have C_2z rotational symmetry. Leaving the calculation details in the supplementary, we evaluate those materials’ NOEE-induced OAM and NSEE-induced spin densities and summarize them in Table 1. When the electric field E = 10⁵ V/m ^35,36 is applied, the NOEE-induced OAM and NSEE-induced spin density are given, δL_z and δS_z ~ 10⁻⁷−10⁻⁸ μ_B/(nm)³ at the true Fermi energy in room temperature with thickness ≈ 3 nm. We also find that with doping, NSEE values of Pd and Ir can be enlarged. (Please see the supplementary information for details.)

In this section, we explore potential applications of the NSEE, particularly its use in magnetic-field-free switching. We propose the feasibility of magnetic-field-free switching utilizing the NSEE generated by centrosymmetric metals. In a typical experimental setup for switching, an electric field is applied to the FM/heavy metal (HM) structure, where the spin current from the spin Hall effect in the HM exerts an influence on the FM’s magnetization (Fig. 3a). Traditionally, deterministic switching requires an external magnetic field^42,43,44,45 or mirror symmetry breaking of the sample^{46,47,48,49,50}. However, with the materials we have proposed, an alternative scheme to realize the magnetic-field-free deterministic switching is possible since the second-order effect of the electric field generates spin density with a polarization distinct from that originating from the spin Hall effect. This spin density can diffuse into FM and can serve a similar function, potentially replacing the need for an external magnetic field (Fig. 3a). To validate the scenario described above, the spin density generated in the NM material must be transferred to the FM. To investigate this, we truncate the NM material to a finite size to observe if the NSEE was induced at the sample’s edge. The results (Please see the supplementary information for details.) showed that the NSEE phenomenon occurred not only within the bulk of the material but also at the sample’s edge with the comparable size of the bulk value. This indirectly suggests that the spin density generated at the NM sample’s edge due to the NSEE in the NM/FM structure can effectively diffuse into the FM material. This implies that the NSEE-induced spin/NOEE-induced OAM density at the edges plays an equally important role at the edges as the spin/OAM density induced by conventional linear REE.

a The microscopic procedure for the magnetic-field-free switching in a nonmagnet/ferromagnet bilayer. b The schematic diagram for the magnetic-field-free switching measurement.

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Now, we provide a more detailed explanation of an experimental setting for magnetic-field-free switching. Figure 3b illustrates the experimental setup designed to capture the characteristics mentioned above for the magnetic-field-free switching effect. We consider a situation where the magnetism is switched in the opposite direction when it is out-of-plane in equilibrium (+z or −z direction). First, the magnetization is reoriented to lie along the in-plane direction by the spin/orbital torque generated by the spin/orbital Hall effect induced by the linear order of the electric field⁴⁶. Without the NSEE, the probability that the magnetization is returned to its initial direction or flopped is equal. However, with the NSEE, δS_z is generated (Fig. 3a), which makes deterministic switching. Furthermore, using the property that the sign of the spin induced by NSEE depends on the current path (I₁ vs. I₂ in Fig. 3b), it becomes possible to achieve the magnetic-field-free switching in both directions (+ z to −z and −z to +z). Since the NSEE and NOEE have (sin (2phi )) dependence, the out-of-plane spin/orbital density induced by NSEE/NOEE is opposite for I₁ and I₂. Specifically, when an electric field is applied along the I₁ direction and induces an out-of-plane density of +z, the equilibrium magnetization can only be switched when it is initially along the −z direction. On the other hand, if the magnetization is initially oriented in the +z direction, it will return to its original direction. In the I₂ direction, the opposite is true. This current path-dependent magnetic-field-free switching property serves as a characteristic that can experimentally verify the presence of NSEE.

In summary, we have identified the microscopic origin of the NSEE and dealt with the technical issue of the energy level broadening by which the NSEE is influenced severely. For the microscopic origin, we have elucidated that the orbital degree of freedom fundamentally underpins the origin of the NSEE. For the energy level-broadening issue, we have applied the energy level broadening to the calculations and found that the NSEE is sizable only at low temperatures. The substantial NSEE that we have discovered is anticipated to give rise to significant dynamics and pave the way for diverse phenomena like magnetic-field-free switching. This research stands as a bridge between nonlinear dynamics and nonlinear spin/orbitronics, opening up exciting prospects for both fundamental understanding and practical applications.

Details of the tight-binding Hamiltonian

The tight-binding Hamiltonian without the spin–orbit coupling in the (s, p_x, p_y, p_z) basis is given by

where ({H}_{ss}={E}_{s}+2{t}_{s}[cos ({k}_{x}a)+eta cos ({k}_{y}a)]), ({H}_{s{p}_{x}}=2i{gamma }_{sp}sin (a{k}_{x})), ({H}_{s{p}_{y}}=2ieta {gamma }_{sp}sin (b{k}_{y})), ({H}_{{p}_{x}{p}_{x}}={E}_{p}+2{t}_{ppsigma }cos ({k}_{x}a)+2eta {t}_{pppi }cos ({k}_{y}b)), ({H}_{{p}_{y}{p}_{y}}={E}_{p}+2{t}_{pppi }cos ({k}_{x}a)+2eta {t}_{ppsigma }cos ({k}_{y}b)), ({H}_{{p}_{z}{p}_{z}}={E}_{p}+2{t}_{pppi }[cos ({k}_{x}a)+2eta cos ({k}_{y}b)]). We choose E_s = 3.0, E_p = t_s = γ_sp = t_ppσ = −0.5, t_ppπ = 0.2 in eV. Note that the magnitude of the sp hybridization (γ_sp), inter-p-orbital hopping parameters (t_ppσ, and t_ppπ) varies between x and y directions by the parameter η. Thus, the difference η−1 may be regarded as the strength of the C_4z rotation symmetry breaking. We consider two η values (η = 1 and 0.9).

The tight-binding Hamiltonian of the spin–orbit coupling in the (s, p_x, p_y, p_z) basis is given by ({{{mathcal{{H}}}_{{{{rm{SO}}}}}}}=lambda {{{bf{L}}}}cdot {{{bf{S}}}})^19,20 where λ is the spin–orbit coupling strength, S is the spin operator represented as Pauli matrices, and the components of the OAM operator L are

Details of the first-principle calculation

We proceed with the full-potential DFT calculation as follows. First, the self-consistent electronic structures are obtained using the full-potential linearized augmented wave method⁵¹ with code FLEUR (http://www.flapw.de). We use Perdew–Burke–Ernzerhof exchange-correlation functional within the generalized gradient approximation⁵². The Brillouin zone is sampled using the 16 × 16 × 16 Monkhorst–Pack k-point mesh for bulks and the 32 × 32 × 1 Monkhorst–Pack k-point mesh for films, respectively⁵³. The muffin-tin radii of each atom and the plane wave cutoffs were set to 2.3a₀ and (5.0,{a}_{0}^{-1}) (where a₀ is Bohr radius) for all structures.

Next, we obtain the maximally localized Wannier functions (MLWFs) from the Bloch states with WANNIER90 code⁵⁴. The Brillouin zones are sampled with the equidistant 8 × 8 × 8 k-mesh for bulks and with the equidistant 8 × 8 × 1 k-mesh for films respectively, including the Γ point. Bloch states are initially projected into (s,,{p}_{x},,{p}_{y},,{p}_{z},,{d}_{xy},,{d}_{yz},,{d}_{xz},,{d}_{{x}^{2}-{y}^{2}}), and ({d}_{{z}^{2}}) states. We obtained 18 MLWFs out of 36 bands for each atomic site. The frozen windows are set to include a region 5 eV higher than the true Fermi energy. We employ the Kubo Formula within quadratic response theory to evaluate the induced orbital and spin magnetization derived in the previous section. The k integration in Eq. (2) is calculated using a uniformly distributed 240 × 240 × 240 k-mesh grid for bulks and a uniformly distributed 480 × 480 k-mesh grid for films, respectively. The temperature is set to T = 300 K. The nonlinear OAM and spin accumulation are calculated by increasing the Fermi energy ε_F from −2 to 2 eV with 0.01 eV step, where the true Fermi energy is set to ε_F = 0.