Orbitally Dominated Rashba-Edelstein Effect in Noncentrosymmetric Antiferromagnets
Leandro Salemi, Marco Berritta, Ashis K. Nandy, Peter M. Oppeneer
OOrbitally Dominated Rashba-Edelstein Effect in NoncentrosymmetricAntiferromagnets
Leandro Salemi, ∗ Marco Berritta, Ashis K. Nandy, † and Peter M. Oppeneer Department of Physics and Astronomy, Uppsala University, P. O. Box 516, S-751 20 Uppsala, Sweden (Dated: May 22, 2019)Efficient manipulation of magnetic order with electric current pulses is desirable for achieving fastspintronic devices. The Rashba-Edelstein effect, wherein a spin polarization is electrically inducedin noncentrosymmetric systems, provides a mean to achieve current-induced staggered spin-orbittorques. Initially predicted for spin, the orbital counterpart of this effect has been disregarded up tonow. Here, we present a generalized Rashba-Edelstein effect, which generates not only spin polar-ization but also orbital polarization, which we find to be far from being negligible and could play acrucial role in the magnetization dynamics. We show that the orbital Rashba-Edelstein effect doesnot require spin-orbit coupling to exist. We present first-principles calculations of the frequency-dependent spin and orbital Rashba-Edelstein susceptibility tensors for the noncentrosymmetric an-tiferromagnets CuMnAs and Mn Au. We show that the electrically induced local magnetization hasboth staggered in-plane components and non-staggered out-of-plane components, and can exhibitRashba-like or Dresselhaus-like symmetries, depending on the magnetic configuration. Furthermore,there is an induced local magnetization on the nonmagnetic atoms as well, that is smaller in Mn Authan in CuMnAs. We compute sizable induced magnetizations at optical frequencies, which suggestthat electric-field driven switching could be achieved at much higher frequencies.
I. INTRODUCTION
The efficient manipulation of the magnetization of ma-terials remains a crucial challenge in the field of spin-tronics [1–3]. Although previously disregarded, antifer-romagnets have recently emerged as appealing candidatematerials for information storage devices since they of-fer various advantages [4–6]. Specifically, antiferromag-nets are robust against external magnetic field pertur-bations, they are available as insulators, semiconductorsand metals, allowing thus for versatile environment in-tegration, and they often have a high N´eel temperature[7–9], suitable for room-temperature operation of the de-vices. Moreover, their intrinsic spin dynamics is ultrafast,in the THz domain [10–12] (compared to GHz dynamicsreported for ferromagnets [13, 14]).Achieving efficient control over the magnetization inantiferromagnets is however an entirely different issue.The spin Hall effect has proven to generate a spin-polarized current [15, 16] and thereby create a spin-orbit torque that can act efficiently on the magnetiza-tion of a ferromagnetic layer [17–20]. A different effect,the Rashba-Edelstein effect (REE) has been proposed asa method to induce a nonequilibrium spin polarizationthrough an electrical current in solids lacking inversionsymmetry [21]. This effect (also called inverse spin gal-vanic effect) was initially predicted in 1990 by Edelstein[21] using a Rashba spin-orbit coupling (SOC) [22]; itwas experimentally observed in GaAs heterostructures[23–25]. ∗ [email protected] † Current address: School of Physical Sciences, National Instituteof Science Education and Research, HBNI, Jatni 752050, India.
More recently, the Rashba-Edelstein effect has beenproposed as a method to create a current-induced stag-gered spin polarization and spin-orbit torque in the non-centrosymmetric antiferromagnets CuMnAs and Mn Au,causing the antiferromagnetic magnetic moments to flipto a perpendicular direction [26–30]. These recent ex-periments have shown that current driven switching ofthe N´eel vector is possible, however, the operation ofthe spin-orbit torque and the switching path are not un-derstood yet. Microscopic investigations indicate that acomplex switching process with both domain wall mo-tion and domain flips may occur [31, 32]. It is moreovera question how large the induced staggered moments are.So far, linear-response tight-binding calculations withRashba SOC [33, 34] and an ab initio calculation [26] ofthe current induced magnetic fields have been performed,that however differed considerably. Also a semiclassicalmodel based on the Boltzmann equation has been em-ployed to compute the induced magnetization in a Weylsemimetal [35]. These investigations concentrate more-over on the induced spin polarization and neglect anypossible contribution stemming from an induced orbitalmagnetization.In the work, we investigate computationally the fullmagnetic polarization induced by an applied electric fieldin the noncentrosymmetric antiferromagnets CuMnAsand Mn Au. To this end we employ an accurate full-potential, all-electron code WIEN2k [36] to compute boththe spin REE (SREE) and orbital REE (OREE) tensorswithin the density-functional theory (DFT) framework.These tensors are computed over a wide frequency spec-trum, i.e. we do not restrict the calculation to the caseof static electric fields. Our calculations bring new in-sights into the Rashba-Edelstein effect in these antiferro-magnets. We find that the dominant contribution to theinduced polarizations stems from the orbital REE. The a r X i v : . [ c ond - m a t . m t r l - s c i ] M a y OREE tensor can have a symmetry different from thatof the SREE tensor (e.g., Rashba-type vs. Dresselhaus-type of symmetry). Due to the pronounced Rashba sym-metry of the OREE tensor, a strong orthogonal orbital-momentum locking is obtained for in-plane electric fields.We find furthermore that quite sizable moments can beelectrically induced on the nonmagnetic atoms. Investi-gating the origin of the large induced orbital polariza-tions, we show that these are present even without spin-orbit interaction, whereas the spin REE tensor is propor-tional to the SOC and vanishes without SOC, signifyingthat the latter are induced through the relativistic SOC,whereas the former have a nonrelativistic origin.
II. THEORETICAL FRAMEWORK
We use linear-response theory to evaluate the magneticresponse to a time-dependent electric field E ( t ). Theinduced magnetic polarization δ M = µ B δ ( L + 2 S ) inthe frequency domain reads δM i ( ω ) = (cid:88) j (cid:16) χ Lij ( ω ) + 2 χ Sij ( ω ) (cid:17) E j ( ω ) , (1)where χ Lij ( ω ) and χ Sij ( ω ) ( i = x, y, z ) are the orbital andspin Rashba-Edelstein susceptibility tensor, respectively(in units of µ B nm/V). These tensors can be derived byconsidering the influence of the time-varying electric fieldas a perturbation ˆ V ( t ) = − e E ( t ) · ˆ r within a periodiccrystal potential, with e the electron charge and ˆ r theposition operator, to the unperturbed time-independentKohn-Sham Hamiltonian ˆ H . Employing linear-responsetheory the tensors can be expressed in terms of the solu-tions of the unperturbed Kohn-Sham Hamiltonian as χ B αβ ( ω ) = iem e (cid:90) Ω d k Ω (cid:88) n (cid:54) = m f m k − f n k (cid:126) ω nm k B αmn k p βnm k ω − ω nm k + iτ − − iem e (cid:90) Ω d k Ω (cid:88) n ∂f n k ∂(cid:15) B αnn k p βnn k ω + iτ − , (2)where f n k is the occupation of Kohn-Sham state | n k (cid:105) with energy (cid:15) n k at wavevector k , Ω the Brillouin zonevolume, p nm k are the momentum-operator matrix ele-ments and B nm k are the matrix elements of the spin ( ˆ S )or orbital angular momentum ( ˆ L ) operator, respectively,and (cid:126) ω nm k = (cid:15) n k − (cid:15) m k .The REE tensor (2) contains two distinct contribu-tions, interband ( n (cid:54) = m ) and intraband ( n = m ) con-tributions. The former describe transitions between thevalence and conductions states, the latter describes theresponse of electrons around the Fermi energy. τ − is a broadening parameter that accounts for the finiteelectron-state lifetime; it can be different for intrabandand interband transitions. Here, for sake of simplicity, weuse the same value of τ for the interband and intrabandcontributions (see Appendix), because our aim is to un-derstand the role and symmetry of the spin and orbital Rashba-Edelstein effect and not their precise value. Wefurther note that our interband formulation is differentfrom another recent investigation [33, 34], in which theexpression f m k − f n k ω nm k + i/τ is used, but the analytic dependencein Eq. (2) guarantees that the response is causal andthat Kramers-Kronig relations between real and imagi-nary part are fulfilled, which is necessary for the REE atnonzero frequencies. III. RESULTS
To evaluate the frequency-dependent SREE andOREE tensors, we adopt the DFT formalism as imple-mented in the full-potential linearized augmented plane-wave (FLAPW) all-electron code WIEN2k [36]. De-tails of the computational approach are given in the Ap-pendix. In the following we apply this framework tononcentrosymmetric CuMnAs and Mn Au that have re-cently drawn attention for antiferromagnetic spintronics[4, 6, 26, 28, 29, 37].
A. CuMnAs
Our DFT calculations give that CuMnAs has an an-tiferromagnetic ground state with Mn atoms carrying amagnetic moment of ∼ . µ B , in agreement with re-cent experiments [38, 39]. The tetragonal cell of CuM-nAs (space group P /nmm ), shown in Fig. 1(a), consistsof six inequivalent atoms, two of each chemical species.Both the Mn and As atoms have the 4 mm point groupwhereas the Cu atoms possess the − m { } Mn planes are antiferromagnetically coupled whileMn atoms laying in the same plane are ferromagneticallyordered. The As atoms are also found to carry a smallmagnetic moment of ∼ .
33 10 − µ B . Their orientation issuch that { } As planes are ferromagnetically coupledto the closest { } Mn plane. The Cu atoms are foundto be non-magnetic.The AFM moments can orient along different N´eel vec-tor axes and this direction of the AFM moments de-pends sensitively, for thin films, on the interplay of intrin-sic magneto-crystalline anisotropy and shape anisotropy.Experimentally, an orientation of the spins in the ab -plane has been observed for thin films [39].The REE tensors depend on the orientation of the mo-ments. We compute them here for different orientationsof the moments, and in addition, we compute the atom-resolved tensors’ spectra, using a specific labeling of theatoms as shown in Fig. 1(a).We first consider the case where magnetic moments areoriented along the c -axis and the applied field is along the a -axis (Fig. 1(a)). This magnetic configuration does notbreak the 4-fold rotational symmetry about the c -axis(hard magnetization axis). The nonzero components of FIG. 1. Magnetization induced by the Rashba-Edelstein effect in antiferromagnetic CuMnAs. (a) Sketch of the tetragonal unitcell of CuMnAs with the magnetic moments constrained along the c -axis. The red arrows on the manganese atoms representthe initial magnetic moments. Applying an electric field E along the (100) direction (grey arrow) induces a non-equilibriummagnetization mainly along the (010) direction (green arrows). (b) Symmetry of the induced spin magnetization as a functionof the static electric field direction for Mn . (c) Symmetry of the induced orbital magnetization as a function of the staticelectric field direction for Mn . (d) Real part of the frequency-dependent spin and orbital Rashba-Edelstein susceptibility. Onlythe real part of the nonzero tensor components is displayed. the atom-resolved spin and orbital Rashba-Edelstein ten-sors are displayed in Fig. 1(d). Several remarkable ob-servations can now be made. First, there are frequency-dependent induced moments not only on the Mn atoms,but also on the Cu and As atoms. Second, the orbitalcontribution that was thus far disregarded, is not negli-gible. In fact the staggered orbital part χ Lxy is the domi-nating part of the response and is ∼
60 times larger thanits spin counter part at ω = 0. In the near-infrared re-gion ( (cid:126) ω = 0 . χ Lxy dominates even more, sincethe spin response χ Sxy is almost zero. Third, apart fromthe staggered components, that are such that antiferro-magnetic Mn1 and Mn2 atoms experience an oppositeresponse (off-diagonal xy and yx ), there are also homo-geneous induced components that act in the same direc-tion for a given atomic species (see diagonal diagonal xx , yy and zz tensors elements in Fig. 1(d)). These non-staggered induced magnetizations can alter the atomictorques and influence eventual spin switching. Lastly,we note that SREE and OREE tensors of the individ-ual elements obey different symmetries, specific to the atomic site’s point group. For the Mn atoms we observe χ S,Lxy = − χ S,Lyx , and χ S,Lxx = χ S,Lyy . The same symmetry ofthe tensors is obtained for the As atoms, but for the Cuatoms χ S,Lxy = χ S,Lyx and χ S,Lxx = χ S,Lyy .The calculated orientation of the induced momentsas a function of the direction of an in-plane applied static electric field is displayed in Figs. 1(b) and 1(c)for the spin and orbital part, respectively. We observea nearly Rashba-like behavior for the spin part withnonorthogonal spin-momentum locking, whereas the or-bital part possesses a perfect Rashba symmetry with or-thogonal orbital-momentum locking (for a definition, seee.g. [40, 41]). These plots are obtained by computing thetensors at ω = 0 while varying the direction of E . It isimportant to note that the induced spin and orbital mo-ments depend on the frequency ω . In addition, the factthat the spin and orbital polarization are induced in dif-ferent directions, and can even be antiparallel (see below)has an important consequence. The resultant torque fieldthat acts on the atomic moments in a Landau-Lifshitz-Gilbert spin-dynamics formulation can then not be rep- FIG. 2. Rashba-Edelstein effect in CuMnAs with magnetic moments along the a -axis. (a) Sketch of the tetragonal unit cell ofCuMnAs. The red arrows on the Mn atoms represent the initial magnetic moments. Applying an electric field E along the (100)direction (grey arrow) induces a nonequilibrium magnetization mainly along the (010) direction (green arrows). (b) Symmetryof the induced spin magnetization as a function of the static electric field direction for Mn . (c) Symmetry of the inducedorbital magnetization as a function of the static electric field direction for Mn . (d) Real part of the frequency-dependent spinand orbital Rashba-Edelstein susceptibility tensors. Only the real part of the nonzero components is displayed. resented in the form of a single atomic Zeeman field, cor-responding to an interaction ( µ B / (cid:126) )( ˆ L +2 ˆ S ) · H , with H the applied atomic Zeeman magnetic field, as this wouldlead to a proportional induced spin and orbital atomicmoment.We now consider the case of CuMnAs with an in-plane magnetization along the (100) direction which cor-responds to the magnetic structure realized in experi-ments. As shown in Fig. 2(a), applying a static elec-tric field (grey arrow) along the magnetization direction(red arrows) induces magnetic moments (green arrows)mainly on the Mn atoms. Those magnetic moments aremainly staggered, i.e., they are practically antiparallel toeach other for AFM coupled Mn atoms. However, a smallparallel out-of-plane contribution is also present. Thisnon-staggered feature of the magnetic response can berecognized by looking at the SREE and OREE tensors,shown in Fig. 2(d) (the nonzero χ zx tensor components).Notably, the by far dominant part of the induced mag-netic polarization is again contained in the staggered xy and yx components of the orbital response.Another important point to be noticed is that the nonzero homogeneous tensor components have changedwith the changed direction of the N´eel vector. The non-staggered components for CuMnAs with magnetizationalong (001) were the diagonal xx , yy and zz componentswhile for the magnetization along (100), these are the xz and zx components. As can be seen in Figs. 1(d) and2(d), the computed SREE spectra are very similar, withan inverted sign ( χ Szz −→ χ Sxz , and χ Sxx −→ − χ Szx ). Thisis a direct demonstration that the electrically inducedmagnetization depends on the underlying magnetizationdirection itself. This can be understood as an influence ofthe magnetization direction on the eigenstates which af-fects the induced magnetization [34]. This effect has alsobeen observed experimentally in (Ga,Mn)As [42]. Com-puting the symmetry of the momentum-dependent in-duced spin and orbital polarizations for an in-plane elec-tric field, we find that the spin-resolved part exhibits aDresselhaus-like symmetry whereas the orbital-resolvedpart exhibits Rashba symmetry (Figs. 2(b) and 2(c)).Here, it can be recognized that the induced spin andorbital polarizations cooperate for a static in-plane elec-tric field along the a axis ([100]) and exert thus spin and FIG. 3. Theory of Rashba-Edelstein effect in Mn Au with magnetic moments along the c -axis. (a) The unit cell of Mn Au, withred arrows on the Mn atoms depicting the initial magnetic moments. Applying an electric field E along the (100) direction (greyarrow) induces a non-equilibrium magnetization tilted in between the (010) and (100) direction (green arrows). (b) Symmetryof the induced spin magnetization as a function of the static electric field direction for Mn . (c) Symmetry of the inducedorbital magnetization as a function of the static electric field direction for Mn . (d) Real part of the frequency-dependent spinand orbital Rashba-Edelstein susceptibility. Only the real part of the nonzero components is displayed. orbital torques in the same direction. However, for an in-plane field along the b -axis ([010]) the induced spin andorbital polarizations are opposite and thus act againsteach other. We further note that the symmetries ofthe REE tensor are now such that χ Sxy (cid:54) = − χ Syx , but χ Lxy = − χ Lyx for the Mn and As atoms. The latter tensorelements are the largest, which illustrates the dominanceof the orbital REE.
B. Mn Au Mn Au crystallizes in the tetragonal structure shownin Fig. 3(a) ( I /mmm space group). The ground state ofMn Au is computed to be antiferromagnetic with mag-netic moments of 3 . µ B only on the manganese atoms.Experimentally, the magnetization of Mn Au films isfound to lay in { } (basal) planes, with ∼ µ B mo-ments on Mn [9]. The unit cell consists of two equivalentAu atoms and two pairs of inequivalent Mn atoms, la-beled Mn and Mn in Fig. 3(a). The four Mn atomshave the 4 mm (polar) point group symmetry and the two Au atoms have the 4 /mmm (centrosymmetric) pointgroup symmetry.Figure 3(d) shows the nonzero SREE and OREE ten-sor elements, computed for Mn moments along the c axis.The calculated tensors exemplify that the REE of Mn Auis in several aspects different from that in CuMnAs. Incontrast to CuMnAs, the spin and orbital responses forboth the xy and yx components are staggered in Mn Auand the homogeneous part of the response is in the diago-nal xx , yy and zz components, similar to CuMnAs. Also,while the orbital part of the response is far from beingnegligible, however, it is not as dominant as in the caseof CuMnAs. The largest orbital in the off-diagonal ele-ments is almost 3 times larger than the spin contributionfor ω = 0. We can furthermore observe that the non-magnetic Au atoms do not display any finite staggered response, consistent with the centrosymmetric nature ofits point group 4 /mmm .The directional dependence of the current-induced mo-ments on Mn atoms as a function of the direction of anin-plane applied static electric field is shown in Figs. 3(b)and 3(c) for the spin and orbital response, respectively. FIG. 4. Rashba-Edelstein effect in Mn Au with magnetic moments along the a -axis. (a) The unit cell of Mn Au. Thered arrows on the Mn atoms represent the initial magnetic moments. Applying an electric field E along the (100) direction(grey arrow) induces a nonequilibrium magnetization mainly along the (010) direction (green arrows). (b) Symmetry of theinduced spin magnetization as a function of the static electric field direction for Mn . (c) Symmetry of the induced orbitalmagnetization as a function of the static electric field direction for Mn . (d) Real part of the frequency-dependent spin andorbital Rashba-Edelstein susceptibility. Only the real part of the nonzero components is displayed. The spin response exhibits a Rashba-like behavior andthe orbital counterpart possesses a Rashba symmetry,too, but notably opposite to that of the spin response.Hence, for any applied in-plane field, the current-inducedspin and orbital moments will exert torques in antiparal-lel directions during a switching process.We now consider Mn Au with moments laying in the ab -plane along the (001) direction, see Fig. 4(a). As forCuMnAs, the magnetic moments have been experimen-tally found to lay in the ab -plane [9]. Here, the calcu-lated nonzero REE tensor elements, shown in Fig. 4(d),are the xy , yx , xz and zx components. In this configu-ration, the staggered responses for both spin and orbitalcontributions are present in the xy and yx componentswhereas the non-staggered responses are present in the zx and xz components, that however give smaller contri-butions. The mainly staggered response corroborates theinvestigation of ˇZelzen´y et al . [33], who predicted stag-gered spin-orbit fields on the two Mn sublattices. Thesymmetries of the main staggered tensor elements are χ Sxy (cid:54) = − χ Syx but χ Lxy = − χ Lyx , as we also obtained forCuMnAs with N´eel vector along the a -axis. Considering the symmetry of the induced polariza-tions for an in-plane field in Figs. 4(b) and 4(c), we findthat the SREE exhibits a Rashba-type behavior and theOREE exhibits a pure Rashba symmetry. Again, the pos-sible non-cooperativity of the OREE and SREE whenexerting a torque can be fully recognized. When boththe static moments and electric field are along the a -axis ([100]) the induced spin and orbital magnetizationsare antiparallel and the torques will partially compen-sate each other. For an in-plane electric field E alongthe b -axis the orbital and spin magnetizations do alsocounteract each other (see also Fig. 4(d)), but this con-figuration only leads to an induced moment along thestatic AFM moments that does not exert a local torqueon the atomic moment. This exemplifies that devisingoptimal switching conditions can be quite intricate. FIG. 5. Rashba-Edelstein effect as a function of spin-orbitcoupling strength computed for Mn1 atom of antiferromag-netic CuMnAs with moments along the c axis. Shown arethe two components χ Sxx and χ Sxy of the SREE susceptibil-ity tensor (top panels) and χ Lxx and χ Lxy of the OREE tensor(bottom panels), computed for scaled values of the SOC, asgiven in the panels.
IV. DISCUSSIONA. Importance of spin-orbit interaction
In his original work, Edelstein predicted an electricallyinduced out-of-equilibrium spin magnetization generatedby Rashba SOC [21]. Here, without assuming any specificshape for the SOC, we find that, depending on the mag-netic configuration, the symmetry of the induced magne-tization can adopt Rashba-like or Dresselhaus-like behav-iors. Remarkably, we find that the previously neglectedorbital polarization can in fact be much larger than theinduced spin polarization.The importance of SOC on the magnetoelectric sus-ceptibilities can be accessed by reducing or switching offSOC in the calculations. Doing so, we find that the spinRashba-Edelstein effect computed without SOC com-pletely vanishes; therefore, consistent with Edelstein’swork [21], this is an intrinsic effect which occurs due tothe broken inversion symmetry in the presence of SOC.Surprisingly, however, for the orbital component our cal-culations without SOC give an unchanged, non-vanishingOREE response for the dominant off-diagonal tensor ele-ments, as shown in Fig. 5, bottom-right panel. In Fig. 5(top panels) we show the computed SOC dependence ofthe xx and xy tensor elements of the SREE susceptibil-ity of Mn1 in CuMnAs with antiferromagnetic momentsalong the c axis. These elements decrease linearly with adecreasing SOC. For the OREE, in Fig. 5 (bottom-rightpanel), we find that the staggered components χ Lxy (and χ Lyx , not shown) are present even without SOC, and arenot even changed by SOC strength which suggests thatthe leading off-diagonal term is independent of SOC. In contrast, without SOC the non-staggered OREE compo-nents χ Lxx and also χ Lyy and χ Lzz (not shown) vanish, andthese can consequently be identified as intrinsic SOC-related quantities. This observation is quite crucial andunexpected, since the staggered SREE components aregenerally believed to be at the origin of switching, ine.g. CuMnAs, and to be SOC related. We find that thedominant nonrelativistic contribution is in the staggeredOREE components while smaller staggered spin compo-nents and non-staggered orbital components are gener-ated by SOC.As yet we know little about the influence of the OREEfor a magnetization switching event, but a cautioningremark is warranted. Although the OREE can be large,to act on the spin moments present in an AFM, it needsto couple to these through spin-orbit interaction. Then,the overall torque on the antiferromagnetic spin momentswill eventually be proportional to the SOC.To analyze the origin of the induced orbital polariza-tions, we observe that due to the staggered nature of theinduced moments in Fig. 5, the sum of the induced or-bital moments on all atoms in the unit cell cancels, butthe contributions on individual atoms do not. There isthus an atomic orbital polarization present even withoutSOC. The orbital angular momentum dynamics inducedby the applied potential ˆ V ( t ) = − e E ( t ) · ˆ r can be evalu-ated from the Heisenberg expression in a single-electronpicture as d ˆ L ind dt = 1 i (cid:126) (cid:104) ˆ L ind , ˆ V ( t ) (cid:105) = ˆ r × e E ( t ) , (3)which is the quantum mechanical counter-part of theclassical equation of motion for angular momentum, d L dt = r × F , where F is an externally applied force.In this picture, the electric field acts as a torque on thecenter of mass of the electrons on an atom. This doesnot require the interplay of SOC as the field couples di-rectly to the position of the electrons and thereby affectsthe orbital momentum. Therefore, the OREE does notarises only from the small relativistic SOC, and sizeableeffects might thus even be observed in systems with smallSOC.As a further remarkable aspect, we point out that thehere-observed appearance of an orbital polarization inthe unit cell in the absence of SOC is distinct from otherrecent theoretical predictions of nonzero orbital textures[43–45]. Hanke et al. showed that a nonzero static orbitalmoment can arise in the noncoplanar antiferromagnet γ -FeMn without SOC due to spin chirality [44]. Here, inthe absence of spin chirality, we predict nonzero orbitalmoments that are present without SOC when an appliedelectric field is present. Yoda et al. proposed that in a chi-ral crystal the solenoidal electron hopping motion couldlead to an orbital magnetization in systems with time-reversal symmetry, in the sense of an orbital Edelsteineffect [45]. Our here-computed induced orbital polariza-tion is distinctly different, as it does not require a chiralcrystal symmetry. B. Frequency and magnetization dependence
Our calculations predict sizable induced polarizationsat finite frequencies, which raises the question whetherelectric field driven magnetic moment switching couldbe achieved at high frequencies. It is well known thattime-dependent magnetic fields cannot drive fast spindynamics of ferromagnets in the optical regime becausethe magnetic permeability µ ( ω ) decays quickly as ω in-creases to the infrared region [46]. The situation is how-ever entirely different for the SREE. The magnetic per-meability is due to a magnetic field H that acts on thespin through the Zeeman interaction in the Hamiltonian, µ B ˆ σ · H ( t ), whereas for the SREE the electric field cou-ples to the charge, − e ˆ r · E ( t ). The electric charges canindeed follow the rapidly changing E -field, implying thatan equally fast magnetic response can be anticipated.Due to their electrical origin, the induced magnetizationscan be driven at petahertz frequencies, thus opening forpotential routes to achieve petahertz spintronics.In the DC limit, ω = 0, the real part of the REE isnonzero and its imaginary part vanishes exactly. At fi-nite frequencies, both the real and imaginary parts ofthe tensor components can be nonzero. The nonzeroimaginary REE susceptibility has a specific influenceon the evolving magnetization dynamics. For a givendriving electric field E ( t ), the induced spin polarization δ M S ( t ) can be retrieved from a Fourier transform of δ M S ( ω ) = χ S ( ω ) E ( ω ). The induced spin polarizationfollows the driving field, but it has a phase difference dueto the imaginary SREE susceptibility. An equivalent re-lation holds for the orbital polarization. The induced spinand orbital polarizations at a frequency ω will thus stillprovide staggered torques on the existing static moments,but these torques will alternate with time. A major ques-tion is then how fast the switching of the static momentscan proceed, whether this can be pushed to the PHzregime. Recent experiments demonstrated that switchingof CuMnAs is possible at THz frequencies [37]. Poten-tially, on account of the above, the switching could thusbe even faster in antiferromagnets, in particular whenthe torques could be enhanced, but the boundaries onthe switching speed are as yet unexplored.To verify whether the SREE and/or OREE can be atthe origin of ultrafast switching, and what the intrinsicfrequency limit is, one should perform atomistic spin-dynamics simulations. The inclusion of both inducedspin and orbital magnetic moments would notably be re-quired to achieve the full picture. This in turn would ne-cessitate the handling of two coupled moments per atom,as was done recently for the dynamics of the 4 f and 5 d moments on gadolinium [47]. Such spin-dynamics simu-lations could clarify as well the role of the non-staggered,homogeneous components for the switching and the in-fluence of Joule heating, inherently present in all exper-iments. It was shown recently that Joule heating playsan essential role as it drastically decreases the requiredswitching field and enhances the spin-orbit torque effi- ciency [48]. Also for Mn Au it was lately concluded thatJoule heating can provide a sufficient thermal activationfor switching processes [29]. Lastly, it should be empha-sized that the switching dynamics of an antiferromagnetis distinct from that of a ferromagnet, since the mag-netization dynamics of an antiferromagnet is describedby a second-order differential equation, which contains amagnetic inertia term for the spins [49–51]. This antifer-romagnetic inertia can provide an important stimulus forthe switching, because, even after the pulse is switchedoff, the already induced torques will act for a longer timeas drivers of the dynamics.
V. CONCLUSIONS
Switching in antiferromagnets is believed to be dueto locally staggered spin-orbit fields that drive oppositedynamics of moments on the two AFM sublattices [26–31, 33, 37]. Our investigation strongly supports that theRashba-Edelstein effect is an excellent candidate to ex-plain the microscopic origin of such staggered fields. Be-yond this, we report several surprising discoveries: first,there exists a significant orbital Rashba-Edelstein effectthat can be much larger than the spin Rashba-Edelsteineffect. Second, we find that there exists not only stag-gered but also non-staggered components to the REEtensors. In both CuMnAs and Mn Au, we find that thestaggered response is strongest. This causes a locking ofthe orbital momentum perpendicular to the applied field.Computing the symmetry of the induced polarizationswith respect to an in-plane electric field, we find thatthese can have a Rashba-like or a Dresselhaus-like charac-ter and that these characters can in general be distinct forthe induced spin and orbital polarizations; for example,a Dresselhaus-like symmetry for the SREE and a Rashbasymmetry for the OREE of CuMnAs with in-plane AFMmoments. As a consequence, the spin and orbital fieldscan enhance each other or cancel each other, i.e., act ina cooperative or a non-cooperative way for switching ofthe sublattice magnetizations.The most surprising part of this work is undoubtedlythe strong induced orbital polarization, which can bemuch larger than the induced spin dipole magnetization.The nonequilibrium orbital polarization is notably evenpresent in the absence of spin-orbit interaction. Thisimplies that it does not arise from a small relativisticeffect, but has a more fundamental, nonrelativistic ori-gin. While our focus here has been on the two antifer-romagnets that are of current interest for antiferromag-netic spintronics, the large dominant orbital fields couldgain importance in the emerging field of spinorbitron-ics [52]. As the induced spin and orbital polarizationsoriginate from the coupling of the electric field to theelectron charges, these induced polarizations can more-over be driven at high frequencies, opening prospects forachieving spintronics at petahertz frequencies.Lastly, on a more general note, the here-developed gen-eral ab initio framework can be employed for the studyof nonequilibrium electric-field induced polarizations ina wide range of materials, as e.g. bulk compounds andmetal/ferromagnet or metal/antiferromagnet interfaces.While bulk materials can already display rich spin-orbit-related physics, interfaces of a heavy metal with a mag-netic layer, where the SOC is increased at the interface,can feature an enhanced Rashba-Edelstein effect that cane.g. be utilized to control the spin orientation in the mag-netic layer [40]. Our ab initio theory framework can pro-vide a materials’ specific understanding of the mecha-nisms behind electrical spin control and lead to the designof suitable interfaces for future spintronics applications.
VI. ACKNOWLEDGMENT
This work has been supported by the Swedish Re-search Council (VR), the K. and A. Wallenberg Foun-dation (grant No. 2015.0060), the European Union’sHorizon2020 Research and Innovation Programme un-der grant agreement No. 737709 (FEMTOTERABYTE),and the Swedish National Infrastructure for Computing(SNIC). The calculations were performed at the PDCCenter for High Performance Computing and the Up-psala Multidisciplinary Center for Advanced Computa-tional Science (UPPMAX).
VII. APPENDIX
To evaluate the frequency-dependent SREE andOREE tensors, we adopt the DFT formalism as imple- mented in the full-potential linearized augmented plane-wave (FLAPW) all-electron code WIEN2k [36]. Themomentum matrix elements are computed through theWIEN2k package [53] while we use our own implemen-tation for the spin and orbital momentum matrix ele-ments. In all of our calculations, we use the PBE-GGAexchange correlation potential [54]. The broadening (cid:126) τ − is set to 0 .
41 eV for both intra- and inter-band transitionswhich was found to give realistic results for the (spin-orbit related) magneto-optical properties of metallic sys-tems [55]. The spin and orbital responses to the electricfield are computed over a whole range of frequency, i.e.,we do not restrict our formalism to static electric fields( ω = 0). Both the real and imaginary parts of the SREEand OREE susceptibility tensors are computed. Our rel-ativistic DFT calculations include spin-orbit interactionconsistent with Dirac theory, without resorting to anyspecific form of SOC such as Rashba or Dresselhaus.For CuMnAs, the product between the smallest muffin-tin radius R MT and the largest reciprocal vector K max was R MT × K max = 7 .
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