Nonlinear Orbital and Spin Edelstein Effect in Centrosymmetric Metals

Nonlinear spintronics combines nonlinear dynamics with spintronics, opening up new possibilities beyond linear responses. A recent theoretical work [Xiao et al., Phys. Rev. Lett. 130, 166302 (2023)] predicts the nonlinear generation of spin density [nonlinear spin Edelstein effect (NSEE)] in centrosymmetric metals based on symmetry analysis combined with first principle calculation. However, its microscopic mechanism is limited to a specific set of materials with local inversion symmetry breaking and is not applicable to general materials. This paper focuses on the fundamental role of orbital degrees of freedom for the nonlinear generation in centrosymmetric systems. Using a combination of tight-binding model and density functional theory calculations, we demonstrate that nonlinear orbital density can arise independently of spin-orbit coupling. In contrast, spin density follows through spin-orbit coupling. We further elucidate the microscopic mechanism responsible for this phenomenon, which involves the NSEE induced by electric-field-induced orbital Rashba texture. In addition, we also explore the potential applications of the nonlinear orbital and spin Edelstein effect for field-free switching of magnetization.


Introduction
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 effect [16]) since it occurs even in centrosymmetric systems where the Rashba-type spin-momentum coupling is forbidden [17].The findings discussed in Ref. [15] offer fresh insights from a fundamental physics standpoint.
However, Ref. [15] has a limitation.The model Hamiltonian 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 coupling [15]), 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 * hwl@postech.ac.kr to go beyond the spin degree of freedom, constructing a non-trivial Hamiltonian solely based on spin degrees of freedom is not feasible [18].
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 dynamics [18].
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 the 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; one order of an electric field generates a momentum-dependent orbital moment, thereby converting a centrosymmetric system into an effective noncentrosymmetric system with the orbital Rashba effect.The other order of an electric field then induces the orbital Edelstein effect in the effective orbital Rashba system.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 globally and locally centrosymmetric materials lacking NSEE in bulk due to C 3z and C 4z symmetries can have NSEE when sliced in specific orientations, which break 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 field-free switching of the magnetization in ferromagnets [1].

Results
Nonlinear orbital Edelstein effect.We first consider the linear orbital Rashba-Edelstein effect (REE), which occurs in systems lacking inversion symmetry [24-26, 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 timereversal symmetry, which enforces f 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 fieldinduced 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 role [19].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 (|ψ nk → | ψnk ) due to the real orbital texture [19,30] which is derived in [31].This can be interpreted as forming an electric field-induced orbital Rashba texture resulting from breaking inversion symmetry by the applied electric field [Figs. 1 (b) 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 y L 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 ydirection 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 timereversal symmetry through the Fermi surface shift and gives rise to a finite total OAM [Fig.1(c)].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 gi-ant 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.Deriving this phenomenon using the Moyal-Keldysh [32] 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 [Ω i nmk (Fig. 1(b)) and Ω j nmk (Fig. 1(d))] and Fermi surface shift [ ∂f (ε nk ) ∂kj (Fig. 1(c)) and ∂f (ε nk ) ∂ki (Fig. 1(e))] 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 metals [15].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 tightbinding model.Our analysis considers a rectangular lattice with s, p x , and p y orbitals with nearest-neighbor hopping [Fig.2(b)].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. 2(a).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.
Figure 2 illustrates the band structure calculated at η = 1 and η = 0.9 and 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 2 x , E 2 y and E x E y fields are all zero.However, as the value of η deviates from 1, NOEE response to E x E y yields nonzero values, and the overall OAM density increases [Fig.2(c)].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 2(d) shows that the OAM density increases as the C 4z symmetry is broken.The NOEE due to E 2 x and E 2 y perturbation is still zero for this case due to the mirror M x and M y symmetry, respectively.From the comparison with the band TABLE I.The NOEE-induced OAM density δLz and spin density δSz with unit 10 −7 µB/(nm) 3 at the true Fermi energy for 11-layered films from DFT calculation at T = 300 K.The external electric field is set as E = 10 5 V/m [35,36] 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 = 10 5 V/m [35,36] is applied, the NOEE-induced OAM and NSEEinduced 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 into Table I.When the electric field E = 10 5 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 [42].Discussion In this section, we explore potential applications of the NSEE, particularly its use in field-free switching.We propose the feasibility of 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.3(a)].Traditionally, deterministic switching requires an external magnetic field [43][44][45][46] or mirror symmetry breaking of the sample [47][48][49][50][51].However, with the materials we have proposed, an alternative scheme to realize the fieldfree deterministic switching is possible since the secondorder 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.3(a)].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 [42] 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.Now, we provide a more detailed explanation of an experimental setting for field-free switching.Figure 3(b) illustrates the experimental setup designed to capture the characteristics mentioned above for the field-free switching effect.We consider a situation where the magnetism is switched in the opposite direction when it is out-ofplane in equilibrium (+z or −z direction).First, the magnetization is reoriented to lie along in-plane direction by the spin/orbital torque generated by the spin/orbital Hall effect induced by the linear order of the electric field [47].Without the NSEE, the probability that the mag-netization is returned to its initial direction or flopped is equal.However, with the NSEE, δS z is generated [Fig.3(a)], which makes deterministic switching.Furthermore, using the property that the sign of the spin induced by NSEE depends on the current path [I 1 vs.I 2 in Fig. 3(b)], it becomes possible to achieve the fieldfree switching in both directions (+z to −z and −z to +z).Since the NSEE and NOEE have sin(2φ) dependence, the out-of-plane spin/orbital density induced by NSEE/NOEE is opposite for I 1 and I 2 .Specifically, when an electric field is applied along the I 1 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 I 2 direction, the opposite is true.This current-path-dependent field-free switching property serves as a characteristic that can experimentally verify the presence of NSEE.

Conclusion
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 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.

Methods
Details of the Tight-binding Hamiltonian.The tight-binding Hamiltonian in the (s, p x , p y , p z ) basis is given by where 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 ), interp-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).
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 linearization augmented wave method [52] with code FLEUR [53].We use Perdew-Burke-Ernzerhof exchange-correlation functional within the generalized gradient approximation [54].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, respectively [55].The muffin-tin radii of each atom and the plane wave cutoffs were set to 2.3 a 0 and 5.0 a −1 0 (where a 0 is Bohr radius) for all structures.
Next, we obtain the maximally localized Wannier functions (MLWFs) from the Bloch states with WANNIER90 code [56].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 uniformly distributed 240 × 240 × 240 k-mesh grid for bulks and 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 eV to 2 eV with 0.01 eV step, where the true Fermi energy is set to ε F = 0.

FIG. 1 .
FIG. 1.(a) The schematic diagram for real orbital texture.For 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 Ex.The transparent figure represents the equilibrium orbital texture.(c) Then, the nonequilibrium orbital Rashba texture is shifted by Ey resulting the formation of nonzero OAM.The role of Ex and Ey can be exchanged so that the nonequilibrium orbital Rashba texture is generated by Ey (d), and the NOEE is generated by Ex (e).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 C3z or C4z symmetries exist.
and acquires k-dependent nonzero OAM expectation value [Fig.1(b)].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

FIG. 2 .
FIG. 2. (a) Hopping rules of rectangular lattice with s, px, and py orbitals model (pz 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's far away from p-orbital character bands.(c) Result in NOEE from tight-binding calculation with η = 0.9.Near the exact crossing point (around 0 eV and −1 eV), the OAM density shows significant growth.(d) The nonlinear OAM accumulation α L zxy with respect to the anisotropy η.(e) The nonlinear OAM α L zxy and spin α S zxy

FIG. 3 .
FIG. 3. (a) The microscopic procedure for the fieldfree switching in a nonmagnet/ferromagnet bilayer.(b) The schematic diagram for the field-free switching measurement.