Modelling spin waves in noncollinear antiferromagnets: spin-flop states, spin spirals, skyrmions and antiskyrmions
Flaviano José dos Santos, Manuel dos Santos Dias, Samir Lounis
MModelling spin waves in noncollinear antiferromagnets: spin-flopstates, spin spirals, skyrmions and antiskyrmions
Flaviano Jos´e dos Santos, ∗ Manuel dos Santos Dias, and Samir Lounis Peter Gr¨unberg Institut and Institute for Advanced Simulation,Forschungszentrum J¨ulich and JARA, D-52425 J¨ulich, Germany (Dated: May 18, 2020)
Abstract
Spin waves in antiferromagnetic materials have great potential for next-generation magnonictechnologies. However, their properties and their dependence on the type of ground-state antiferro-magnetic structure are still open questions. Here, we investigate theoretically spin waves in one- andtwo-dimensional model systems with a focus on noncollinear antiferromagnetic textures such as spinspirals and skyrmions of opposite topological charges. We address in particular the nonreciprocalspin excitations recently measured in bulk antiferromagnet α –Cu V O utilizing inelastic neutronscattering experiments [Phys. Rev. Lett. , 047201 (2017)], where we help to characterize thenature of the detected spin-wave modes. Furthermore, we discuss how the Dzyaloshinskii-Moriyainteraction can lift the degeneracy of the spin-wave modes in antiferromagnets, resembling the elec-tronic Rashba splitting. We consider the spin-wave excitations in antiferromagnetic spin-spiral andskyrmion systems and discuss the features of their inelastic scattering spectra. We demonstratethat antiskyrmions can be obtained with an isotropic Dzyaloshinskii-Moriya interaction in certainantiferromagnets. ∗ [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y . INTRODUCTION The search for new technologies that are faster and more energy-efficient than present-dayelectronics stimulated the development of spintronics [1] and magnonics [2], which exploitmagnetic degrees of freedom instead of only mobile charges as in conventional electronics.The spin of the electron is central to spintronics, while magnonics builds upon spin wavesor magnons, which are collective motions of magnetic moments. Both traditionally involveferromagnetic materials, but recently antiferromagnets were recognized to have potentialadvantages, which led to the development of antiferromagnetic spintronics [3–6] while anti-ferromagnetic magnonics is still in its infancy [7, 8]. Skyrmionics can be seen as an interestingcrossover between these two fields [9–16]. Here, the key entity is the magnetic skyrmions [17–20] (topologically-quantized windings of the background magnetic structure), which can bevery robust against perturbations and also highly mobile under relatively small applied cur-rents. Skyrmions have been theoretically studied in antiferromagnetic materials [21–27], buthave not yet been experimentally discovered.Broadening the applications of antiferromagnetic materials, for instance, to advance an-tiferromagnetic magnonics, requires understanding the properties of their spin waves. Thebasic quantities are the spin-wave energy, how it relates to the wavevector, and what kindof precession of the magnetic moments takes place. The pioneering works of Kittel andKeffer [28, 29] explain antiferromagnetic resonance (zero wavevector) in collinear antiferro-magnets, and how it depends on the internal magnetic anisotropy and the external magneticfield. For larger wavevectors, the spin-wave energies are controlled by the magnetic exchangeinteraction and the Dzyaloshinskii-Moriya interaction (DMI) [30, 31]. The DMI is a chiralcoupling that arises from the relativistic spin-orbit interaction and was originally proposedto explain weak ferromagnetism in collinear antiferromagnetic materials. It can also lead tononcollinear antiferromagnetic structures, such as spin spirals and potentially antiferromag-netic skyrmions [32, 33]. In ferromagnetic materials, the DMI leads to the nonreciprocity ofthe spin-wave dispersion [34–37], so that the spin-wave energy is different for wavevectorsof equal length and opposite direction. A similar effect was recently observed in an anti-ferromagnetic material [38]. While spin waves are now well-understood in skyrmion-hostingferromagnetic materials [39], only some theoretical studies report on their antiferromagneticcounterparts [26, 27], as they remain to be discovered.2 complete experimental characterization of the spin-wave modes in an antiferromagneticmaterial is challenging, especially detecting what kind of precession is associated with eachmode. Antiferromagnetic resonance can be accessed by broadband spectroscopy [7, 40], whileinelastic neutron scattering can survey the spin-wave spectrum across the whole Brillouinzone, as shown for the antiferromagnetic spin-spiral system Ba CuGe O [41]. Polarizedneutron scattering experiments give access to the information about the precession but arevery difficult to perform. Resonant inelastic x-ray scattering is a new technique that holdspromise for detecting spin waves in bulk materials [42] and is applicable down to a singlelayer [43]. Recently, we presented a theoretical analysis of spin-resolved electron-energy-lossspectroscopy (SREELS) [44], which is a proposed extension of an existing surface-sensitiveexperimental method [45–48], and explained how the polarization analysis can be used tounderstand the nature of the spin-wave precession. Employing SREELS, one has accessto various spin-scattering channels, where the scattered electrons experience processes ofspin-flip nature or not. In contrast to collinear magnets, where only spin-flip processesare responsible for the emission of spin waves, non-spin-flip processes can generate spin-excitations in noncollinear materials. These general concepts are also applicable to otherspectroscopies.In this article, we study the inelastic scattering spectra of various noncollinear antifer-romagnetic spin structures in one and two dimensions using atomistic model systems. Inparticular, we address the nonreciprocity induced by the combined action of the DMI andan external magnetic field. We compare our results to inelastic neutron scattering measure-ments on a bulk antiferromagnet, α − Cu V O [38], providing a novel understanding of theexperimental data. In two dimensions, we explore antiferromagnetic systems with the C v and C v symmetries. We showed that DMI in these systems can induce a spin-wave Rashba-like effect characterized by a linear-angular-momenta locking. Furthermore, we demonstratethat antiskyrmions are natural skyrmionic occurrences in antiferromagnetic p(2 ×
1) ma-terials, even when the system has the C v symmetries. Finally, we calculate the inelasticscattering spectra of antiferromagnetic skyrmion and antiskyrmion lattices, identifying thespectral signature of its characteristic modes.3 I. THEORETICAL FRAMEWORK AND MODEL SYSTEMS
We employ the generalized Heisenberg model for spins S i (we take S = 1) on a lattice,and we measure lengths in units of the nearest-neighbor distance a = 1. The correspondingHamiltonian restricted to nearest-neighbor interactions reads H = (cid:88) (cid:104) i,j (cid:105) J ij S i · S j − (cid:88) (cid:104) i,j (cid:105) D ij · (cid:0) S i × S j (cid:1) − K (cid:88) i (cid:0) S zi (cid:1) − (cid:88) i B · S i . (1)Here the exchange interaction J ij is taken to be uniform and antiferromagnetic J > D ij = D ˆ n ij is taken to be uniform in magnitude D , with the direction for each bond given by ˆ n ij . Wealso include a uniaxial magnetic anisotropy with K > z , and theexternal uniform magnetic field B with magnitude B , also along z for most of the models.The brackets indicate a sum over the nearest-neighbor pairs.For the various investigated cases, we extract the most stable magnetic configurationeither by analytical means or by using atomistic-spin-dynamics simulations by solving theLandau–Lifshitz–Gilbert equation with the Spirit code [49]. For that, we used a supercellapproach with periodic boundary conditions. Once the ground state or a metastable state isfound, we compute the adiabatic spin-wave modes and the corresponding inelastic scatteringspectrum, based on time-dependent perturbation theory. The associated theoretical frame-work was presented in Refs. [44, 48]. Although we have access to several distinct scatteringchannels, in this work we always report results for the total inelastic scattering spectrum dueto spin waves (the sum over all scattering channels), as one would measure in an experimentwith an unpolarized scattering experiment.An important property of the spin-wave quantum is its angular momentum. In ferromag-nets, spin waves have quantized angular momenta oriented antiparallel to the spins formingthe background magnetization. Meanwhile, noncollinear systems can host spin-wave modeswith nonquantized angular momenta pointing in various directions, or even of vanishing mag-nitude [44]. Spin waves with finite angular momenta correspond to excitations with circularpolarization and are also called rotational or gyroscopic modes. In contrast, modes with van-ishing angular momenta are linearly polarized and can be seen as longitudinal excitations.Recently, we have shown that the angular momentum of a spin wave is connected to itshandedness (chirality), determining how the spin-wave properties respond to the DMI [37].4 a) (cid:45)(cid:54)(cid:0)(cid:0)(cid:9) yzx(b) (c) B = 0 B (cid:54) = 0FIG. 1. Inelastic scattering spectrum of a collinear antiferromagnetic spin chain. (a) The groundstate, where the spins align antiparallel among neighbors. (b) Inelastic scattering spectrum for theantiferromagnetic spin chain depicted in (a). The single spin-wave mode dispersing away from X is doubly-degenerate and the excitation gap is due to the magnetic anisotropy. (c) A magnetic fieldis applied along the z –axis, which breaks the degeneracy of the two modes. Model parameters: D = 0, K = 0 . J , B = 0 . J . Experimentally, this angular momentum can be measured via spin-resolved inelastic scat-tering spectroscopy. That is because the angular momentum determines in which scatteringchannels a given mode may be observed [44]. We used the theoretical SREELS spectra todetermine the spin-wave angular momenta in this work.
III. ANTIFERROMAGNETISM IN A SPIN CHAIN
To set the stage, we first study the case of one-dimensional antiferromagnetic spin chains.
A. Collinear antiferromagnetic chains: The effect of the DMI and the magneticfield on the spin waves
When B and D are zero, i.e., considering only the magnetic exchange interaction andthe magnetocrystalline anisotropy, the ground state of the system corresponds to a collinearalignment of the spins along the anisotropy easy-axis z . The nearest-neighbor spins areantiparallel to each other, as shown in Fig. 1(a), the inelastic scattering spectrum shows a5 a) (b) B = 0 B (cid:54) = 0FIG. 2. Inelastic scattering spectrum of a collinear antiferromagnetic spin chain with DMI. (a) TheDzyaloshinskii-Moriya vectors pointing along the easy-axis break the spin-wave mode degeneracy.The dispersion curves shift in opposite directions. (b) When a magnetic field is applied ( B = 0 . J ),the spectrum becomes nonreciprocal. Despite the DMI, the system has the same collinear groundstate shown in Fig. 1(a), stabilized by the magnetic anisotropy. Model parameters: D = 0 . J , K = 0 . J , B = 0 . J . The wavevector is given in units of π/a (we set the lattice spacing a = 1). single mode even though one could expect two, given that there are two magnetic sublattices(one for up and the other for down spins), see Fig. 1(b). This spin-wave mode is doubly-degenerate, having its lowest excitation energy at Γ (with vanishing scattering intensity)and at the Brillouin zone border, X = π , with an excitation gap opened by the magneticanisotropy. The degeneracy of the two modes is lifted by an external magnetic field parallelto the easy-axis. As the modes have opposite angular momenta (they can be measuredindividually with spin-resolved inelastic scattering spectroscopy [44]), the magnetic fieldraises the energy of one mode while lowers the energy of the other, as seen in Fig. 1(c).In the absence of an external magnetic field, the Dzyaloshinskii-Moriya interaction canalso lift the degeneracy. Introducing the DMI vectors collinear with the easy-axis but withmagnitude below the critical value D c = (cid:113)(cid:0) J + K (cid:1) K , see Appendix A, preserves thecollinear antiferromagnetic ground state. The DMI splits the modes apart shifting theirdispersion curves in opposite directions in the reciprocal space, as shown in Fig. 2(a). Thesespin-wave modes have their minimum excitation energies at X ± q , which were symmetricallyshifted away from the Brillouin zone boundary. This happens because the spiralizationinduced by the spin waves is energetically favored by the DMI, such that q = arctan( D/J ).Also, the energy gap is now smaller than in the absence of DMI, and it closes completely forthe critical DMI magnitude, D c . This phenomenon is analogous to the Rashba effect [50],6here electrons acquire a finite group velocity at zero wavevector due to the spin-orbitcoupling. Furthermore, in the electronic Rashba effect, electrons propagating to oppositedirections with the same wavelength have opposite spins. Similarly, the two spin-wavemodes in Fig. 2 have opposite angular momenta along the z –axis, thus perpendicular to thepropagation direction y .Finally, we can turn on the external magnetic field parallel to the easy-axis in the presenceof the DMI. Again, one of the curves is raised in energy, while the other is lowered. Thiscauses a break of the symmetry in the reciprocal space, which is a phenomenon known asnonreciprocity, as observed in Fig. 2(b). Note that the isolated action of either the magneticfield or the DMI preserves the reciprocity of the spectrum. B. Antiferromagnetic spin spirals
In the current situation, we have an external magnetic field which is parallel to half of thespins and antiparallel to the other half. Therefore, increasing the magnetic field does notaffect the total energy of the collinear AFM state shown in Fig. 1(a). However, when themagnetic field is large enough, the system undergoes a spin-flop transition, where the spinsare mostly perpendicular to the easy-axis but with a small component parallel to the field.Furthermore, an antiferromagnetic spin spiral can form with a rotational axis parallel to theeasy-axis such as to gain energy from the DMI, see Fig. 3(a). This new state is unfavored bythe magnetic anisotropy, but the loss is compensated by the Zeeman and the DMI energygains. The pitch of the antiferromagnetic spin spiral is given by q = arctan( D/J ) just as forferromagnetic systems, as shown in Appendix A. The inelastic scattering spectrum of thisnew state becomes reciprocal again. Despite the stronger magnetic field, which previouslywas causing the nonreciprocity in combination with the DMI, the inelastic scattering spec-trum of this new state is reciprocal. The signal is formed by a central feature surroundedby two side modes of lower intensity, whose energy minima occurs at ± Q = X ± q .In Ref. [38], Gitgeatpong et al. measured for the first time the nonreciprocity of spinwaves in antiferromagnets by performing inelastic neutron scattering experiments on thebulk antiferromagnet α –Cu V O . The spin-wave physics of this compound is analogous tothe model just described. The nonreciprocity is observed by applying an external magneticfield as shown in Fig. 4(a), which resembles Fig. 3(a). Our results establish a perfect parallel7 Q + Q (a) (b) (cid:45)(cid:54)(cid:0)(cid:0)(cid:9) xyz FIG. 3. Inelastic scattering spectrum of an antiferromagnetic spiral spin chain. (a) The groundstate generated by a spin-flop transition induced by an external magnetic field along z . The spinslie in the plane perpendicular to the applied field forming an antiferromagnetic spin spiral. Asmall component of each spin still points along z what makes it a conical spin spiral. (b) Spin-wave scattering spectrum of the antiferromagnetic spin spiral depicted in (a). The spectrum issymmetric and formed by an intense gapped mode centered at X, enclosed by two faint gaplessmodes dispersing away from ± Q = X ± q , where q is the wavevector of the spiral. Model parameters: D = 0 . J , K = 0 . J , B = 0 . J . with their measurements.We now present our understanding of these experimental results. A spin spiral hoststhree universal spin-wave modes [44], instead of two for collinear two-sublattice antiferro-magnets. Two of the spin-wave modes of the spiral have dispersion curves with minimain the wavevector of the spiral ± Q and are rotational modes with equal and opposite netangular momenta. The third mode is symmetric around a high-symmetry point (Γ or X),and it has no net angular momentum but corresponds to a longitudinal oscillation of thenet magnetization, which is generated perpendicularly to the plane of rotation of the spiral(see a related discussion in Ref. [44]). This longitudinal (linear polarized) mode is the oneresponsible for the high-intensity feature in the inelastic scattering spectrum, as obtainedtheoretically in Fig. 3(b), and measured in Fig. 4(b). Furthermore, our theoretical calcu-8 a) (b)FIG. 4. Inelastic neutron scattering spectra of bulk α –Cu V O . (a) Nonreciprocal scatteringspectrum due to an external magnetic field, B = 6 T. The system is collinear antiferromag-netic. (b) For a high field, B = 10 T, the system undergoes a spin-flop transition into a spinspiral and the spectrum becomes symmetric. The arrows denote the magnetic Bragg peaks. Fig-ure reprinted with permission from Gitgeatpong, G. et al. , Phys. Rev. Lett. , 047201 (2017)(https://doi.org/10.1103/PhysRevLett.119.047201), Ref. [38]. Copyright (2020) by the AmericanPhysical Society. lation enlightens the existence of two weaker features in the inelastic scattering spectrum,see Fig. 3(b). These two modes have energy minima at the magnetic Bragg peaks of theantiferromagnetic spin spiral, which are given by the spiral wavevector ± Q . Therefore, theshifts of the minima out of the high-symmetry point are only related to the DMI indirectlythrough Q . Although signatures of these two feeble branches also appear in the experimentaldata shown in Fig. 3(b), the insufficient counts and lack of theoretical support probably ledthe authors of Ref. [38] to leave them unremarked.To close this section, we remark that the nonreciprocity of spin waves in noncollinearsystems is discussed at length in Ref. [51], where we present a general theory of how todetect asymmetries in the inelastic scattering spectrum due to the DMI. IV. TWO-DIMENSIONAL ANTIFERROMAGNETS
In the previous section, we have considered a one-dimensional antiferromagnetic spinchain, which allowed us to study the spin-wave Rashba effect due to the Dzyaloshinskii-9oriya interaction as well as the reciprocal-symmetry breaking in response to an appliedmagnetic field. Now, we place our focus on the properties of spin waves in two-dimensionalantiferromagnetic systems, which have the minimal dimension to host antiferromagneticskyrmionic structures as discussed, for example, in Refs. [21, 27]. First, we consider theformation of antiferromagnetic spin spirals, and subsequently the occurrence of antiferro-magnetic skyrmions and antiskyrmions.
A. Antiferromagnetic spin spirals and Rashba spin locking
When the magnetic exchange interaction between neighboring atoms is dominant andantiferromagnetic, the formation of collinear structures of antiparallel spins takes place.Considering a two-dimensional square lattice, the most common ones can be labelled ac-cording to the notation for surface reconstructions as the c(2 ×
2) and p(2 × ×
2) structure, a spin moment is antialigned to all its nearestneighbors. In the p(2 × ×
2) structure as the ground state, see Figs. 5(a) and (c).In an isotropic medium, beyond nearest-neighbor interactions are required to stabilize thep(2 ×
1) structure [52]. To stick to only nearest-neighbor interactions, we circumvent thisobstacle with a spatially anisotropic exchange interaction in Model II, such that J ij = − J along the y and J ij = + J along the x –directions, as illustrated in Fig. 5(b). Its ground state,the p(2 ×
1) structure, is shown in Fig. 5(d). The respective low-energy spin excitations havewavevectors around the M–point for the c(2 ×
2) structure and around the X–point for thep(2 ×
1) structure. The M–point ( π, π ) corresponds to a precession where a given spin is de-phased by π with respect to all its nearest-neighbors. For the c(2 ×
2) structure, this meansthat nearest-neighbor spins are kept perfectly antiparallel throughout the whole precessionrevolution. Such an excitation, therefore, costs no energy and corresponds to the Goldstonemode. Similarly, the X–point ( π,
0) guarantees a precession phase of π between spins alonghorizontal lines, and no phase shift along the vertical ones, minimizing the excitation energyfor the p(2 ×
1) structure. 10 a) (b)(c) (d)(e) (f) (cid:45)(cid:54)(cid:2)(cid:2)(cid:14) xyzFIG. 5. Two-dimensional antiferromagnetic structures on a square lattice. (a) and (b) representModel I and II, respectively. The gray atoms interact antiferromagnetically with the nearest-neighbor atoms in red, and ferromagnetically with the ones in blue. Both models also include aneasy-axis magnetic anisotropy along z (normal to the lattice plane). (c) and (d) show the c(2 × ×
1) phases, which are the ground states of Model I and II without DMI, respectively.(e) and (f) show spin spirals formed due to Dzyaloshinskii-Moriya interactions in Models I and II,respectively. The energy per atom of the configuration in (c) and (d) is E = − . J , and in (e)and (f) E = − . J . Model parameters: D = 0 . J , K = 0 . J and B = 0. Since we want to study whether chiral skyrmions are supported by these antiferromagneticsystems, we need to consider the effects of the Dzyaloshinskii-Moriya interactions. Thus,we add to both models in-plane isotropic nearest-neighbor Dzyaloshinskii-Moriya vectors,which circulate counterclockwise perpendicularly to the bonds. Furthermore, both modelsinclude an easy-axis magnetic anisotropy favoring the spins to align along z . The DMI favorsa noncollinear alignment among the spins, while the anisotropy defines a preferred directionfor the spin to point along. The competition between the DMI, the magnetic anisotropy, andthe exchange interaction determines the characteristics of the possible noncollinear states,11uch as spin spirals and skyrmions.The ground states for Models I and II with DMI are shown in Fig. 5(e) and (f), respec-tively, which we shall call c(2 ×
2) and p(2 × × × × × × × q = q ˆ x with magnitude q = π/ × q . Similarly, Fig. 6(b) displays the total spec-trum around the X–point for Model II in the p(2 × ± q , while the central and moreintense one is the longitudinal mode gapped by the magnetic anisotropy. The energy scaleof the spin waves of the p(2 × q is roughly half of that for the c(2 × x –axis have polarization along y , while the ones dis-persing along y are polarized along x . These observations uncover a locking between thelinear and angular momenta of the spin waves, which is another characteristic of the Rashbaeffect discussed in Sec. III A.On the paths perpendicular to the spin-spiral wavevector (panels on the right-hand sidein Fig. 6), we initially observe two modes whose energy minima are shifted from the high-symmetry point. Interestingly, the location in the reciprocal space of these minima is notrelated to the spin-spiral wavevector but directly to the strength of the Dzyaloshinskii-Moriyainteraction in a linear manner. We demonstrate this in Fig. 7(a) and (b), where we increasedthe DMI strength from D = 0 . J to D = 0 . J without relaxing the spin structure, whichresulted in a change of the minima from M ± .
048 to M ± .
072 ( π ). Furthermore, the largersplitting induced by the enhanced DMI reveals a third mode that was indistinguishable be-fore, see also Fig. 6(a) and (b) (right-hand-side panels). Finally, Fig. 7(c) demonstrates that12 a)(b) (cid:113)(cid:113) (cid:113)(cid:113) XMYΓFIG. 6. Inelastic scattering spectra for 2D antiferromagnetic spin spirals. (a) and (b) show thedispersion curves for the c(2 ×
2) and p(2 × q are shown in the left-hand side and paths perpendicular to it on the right-hand side.The inset in (b) depicts the high symmetry points of the Brillouin zone for the underlying squarelattice (the scattering process unfolds the spectra). Both the c(2 ×
2) and p(2 × D = 0 . J , K = 0 . J . the two DMI-shifted modes are susceptible to external magnetic fields. An applied fieldalong the x –axis, therefore parallel to the polarization of these modes, breaks the reciprocalsymmetry around the band center, increasing the energy of one mode while decreasing theenergy of the other one. The inelastic scattering spectra of ferromagnetic and antiferro-magnetic spin spirals have some similarities, such as the characteristic three helimagnonbranches. However, only the antiferromagnetic case displays DMI-shifted branches in allreciprocal space directions. 13 a) (b) (c)FIG. 7. Inelastic scattering spectra along the X-M-X–path for the Model I on the c(2 × D = 0 . J to (b) D = 0 . J without relaxing the spin structure ( B = 0). In (a) the energy minima are locatedat M ± . ± . π . Thus thescaling on D is linear. Due to the further splitting, a third mode can be distinguished, which iscentered at M. (c) Next, we apply an external magnetic field B = 0 . J along q to the case in (b).The spectrum becomes nonreciprocal, with the magnetic field raising the excitation energies of onemode and lowering those of the other. Parameter: K = 0 . B. Antiferromagnetic skyrmions
In the previous section, we considered two model Hamiltonians with c(2 ×
2) and p(2 × J changes sign for different directions. For a given set of DMI, the antiferromag-netic alignment reverses the chirality of the spin spirals in comparison to the chirality ofa ferromagnetic arrangement, see Fig. 8(c). As the p(2 ×
1) state has a ferromagnetic14 a) (b) (cid:45)(cid:54) xy (c) (cid:45)(cid:54) yz (cid:45)(cid:54) xzFIG. 8. Skyrmionic structures in antiferromagnetic backgrounds. (a) Antiferromagnetic skyrmionthat is formed when the exchange interaction with all nearest neighbors is negative, Model I. Theskyrmion lives in a c(2 ×
2) antiferromagnetic background. (b) Antiferromagnetic antiskyrmion,which results from spatially anisotropic exchange interactions. J is negative along- x and positivealong- y , Model II. The antiskyrmion lies in a p(2 ×
1) antiferromagnetic background. (c) Cross-sections along the x and y directions of the antiferromagnetic antiskyrmion in (b). They correspondto spin spirals with different chirality despite the same DMI, whose orientation is represented bythe black circles. Model parameters: D = 0 . J , K = 0 . J and B = 0. The total energy of bothspin configuration is E = − . J . cross-section along y and an antiferromagnetic cross-section along x , the antiskyrmion isthe natural occurrence for this type of antiferromagnets. We also confirmed this result byemploying a next-nearest-neighbor isotropic Hamiltonian with C v symmetries that favorsthe p(2 ×
1) state through exchange frustration, therefore, without invoking a spatiallyanisotropic magnetic exchange interaction.Next, we investigate the inelastic scattering spectra related to these skyrmionic structures.Fig. 9(a) shows the spectra for the antiferromagnetic skyrmion, which is related to the15 a)(b) (cid:113)(cid:113) (cid:113)(cid:113)
XMYΓFIG. 9. Inelastic scattering spectra for antiferromagnetic skyrmionic lattices. (a) Spectrum forthe antiferromagnetic skyrmion in the c(2 ×
2) background. (b) Spectrum for the antiferromagneticantiskyrmion in the p(2 ×
1) background. In contrast to the spin spirals, the intense features forthe skyrmion lattices are much more broadened. The inset in (b) depicts the high symmetry pointsof the crystal Brillouin zone. Model parameters: D = 0 . J , K = 0 . J and B = 0. c(2 ×
2) antiferromagnetic structure, while Fig. 9(b) corresponds to the antiferromagneticantiskyrmions, whose background is a p(2 ×
1) antiferromagnet. Overall, both spectra havea lot more structure in comparison to those of the parent antiferromagnetic spin spirals, withmany faint modes almost forming a continuum of spin-wave excitations. Yet, it is possibleto clearly resolve distinct intense modes throughout most of the reciprocal path.For small excitation energies around M or X, we can observe that the modes dispersemostly linearly with the changing wavevector. This is in contrast to the quadratic behaviorof the low-energy spin-wave modes for ferromagnetic skyrmion lattices [44]. In Fig. 8(a),we singled out a few points in the spectrum with features common to all four panels. Thearrow labeled A indicates a continuous dispersing mode which is circularly polarized withnet angular momentum along + y on the left half of the Y − M − Y path. The same feature on16he right-hand side has angular momentum along − y . The polarization of this mode rotatesto the x direction in the X − M − X path, for example. The B arrow indicates a hotspotseen in all four panels with nonvanishing energy and localized at the high symmetry points.It corresponds to a mode with linear polarization along z . The intensity marked by theC arrow is due to two degenerate rotational modes with opposite angular moment along z ,therefore, reproducing the polarization of the native modes of the collinear antiferromagneticbackground. The modes dispersing out of the Bragg peaks, such as the one indicated byD, are linearly polarized along z . The precession nature and relative energy alignmentof the excitations due to points B and C are in accordance with the ones reported forantiferromagnetic skyrmions confined in nanodiscs [26].Notice that all the spectra are symmetric with respect to the high-symmetry points.As the net magnetization is zero and no magnetic field has been applied, the spin-waveenergies must be reciprocal. Nevertheless, hidden nonreciprocity of individual spin-wavemodes induced by the DMI could be observed as an asymmetry in the inelastic scatteringspectra if spin-polarized spectroscopies are to be used [37]. V. CONCLUSIONS
We studied simple models of magnetic materials whose magnetic exchange interaction ispredominately antiferromagnetic, intending to contribute to the development of antiferro-magnetic spintronics and magnonics. We first considered one-dimensional antiferromagneticspin chains. We observed that the DMI can lift the degeneracy of the two spin-wave modes inthe collinear antiferromagnetic structure, resulting in a mode splitting similar to the Rashbaeffect for electronic bands. Because these two spin-wave modes have opposite angular mo-menta, a magnetic field can induce a nonreciprocity of the spectrum. For even higher fields,the collinear state gives way to a spin-flop state where a spin spiral is formed. The new statehas a reciprocal spin-wave spectrum formed of the three universal helimagnon modes [39].Our calculations compare well with the recent experimental results obtain with inelasticneutron scattering for the bulk material α –Cu V O [38], and explain some unremarkedfeatures in the measurements.We also investigated noncollinear spin textures in two dimensions. In particular, we com-puted the inelastic scattering spectra for two model systems with spin-spiral ground states,17ne with spatially-isotropic and another with spatially-anisotropic magnetic exchange inter-actions sharing the same set of isotropic DMI. The isotropic interactions favor the c(2 × ×
1) row-wiseantiferromagnetic structure in the anisotropic case, and the DMI leads to spiral structuresbased on those reference collinear states. The spin-wave spectra have some similarities withthose for ferromagnetic spin spirals studied in Ref. [44], but are centered at high-symmetrypoints at the edges of the Brillouin zone instead of at the zone center. The different modeswere characterized in terms of their precessional character, which is tied to the chosen pathin reciprocal space in a way that is once again reminiscent of the Rashba spin-momentumlocking. With the same models, we could also stabilize antiferromagnetic skyrmion lattices.We demonstrated that antiskyrmions are the natural skyrmionic occurrence in p(2 × ACKNOWLEDGMENTS
This work is supported by the Brazilian funding agency CAPES under Project No.13703/13-7 and the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (ERC-consolidator Grant No. 681405-DYNASORE). We gratefully acknowledge the computing time granted by JARA-HPC onthe supercomputer JURECA at Forschungszentrum J¨ulich and by RWTH Aachen Univer-sity.
Appendix A: Antiferromagnetic spin spirals1. Spin-spiral model
Let us suppose that we have the magnetic properties of a given system. That is, wehave the spin moment of each site and the magnetic interactions, both the exchange andthe Dzyaloshinskii-Moriya, from each pair of atoms in a Bravais lattice. We want to de-18ermine whether a spin spiral can be an energetically more favorable state than a collinearantiferromagnetic phase. However, there can be many types of spin spirals, with differentorientations, wavevectors, etc. Thus, we restrict our search among spirals given by thefollowing equation: S i = cos φ i sin θ n + sin φ i sin θ n + cos θ n , (A1)where n is a unity vector defining the axis around which the spins rotate, and that formsan orthonormal basis set for the three dimension space together with n and n . θ is theconical angle between the spins and n . φ i = ( q AF + q ) · R i , where q AF accounts for acollinear antiferromagnetic phase, q is the spiral wavevector, and R i the position vector ofthe i -th spin.
2. Spin-spiral energy
Regarding the hamiltonian of Eq. (1), the classical energy (per atom) of such a spin spiralcan be decomposed in four terms.The exchange one: ε J ( q ) = 1 N (cid:88) ij J ij S i · S j = 1 N (cid:88) ij J ij (cid:2) cos (cid:0) ( q AF + q ) · R ij (cid:1) sin θ + cos θ (cid:3) . (A2)The Dzyaloshinskii-Moriya term: ε D ( q ) = − N (cid:88) ij D ij · S i × S j = − N (cid:88) ij sin θD ij sin (cid:0) ( q AF + q ) · R ij (cid:1) , (A3)where we used the symmetry property of the Dzyaloshinskii-Moriya interaction that imposes D ij = − D ji .And the uniaxial magneto-crystalline anisotropy contribution: ε K ( q ) = − KN (cid:88) i ( ˆ K · S i ) = − K (cid:20)
12 sin θ (cid:16) ( K ) ( δ q , { , q AF } + 1) + ( K ) (1 − δ q , { , q AF } ) (cid:17) + sin 2 θ (cid:0) K K δ q , q AF (cid:1) + cos θ ( K ) (cid:3) , (A4)19here we considered that q is restricted to the first Brillouin zone. The unit vector thatrepresent the magnetic anisotropy axis decomposes as ˆ K = K n + K n + K n with (cid:80) α ( K α ) = 1. Thus, we can see that the anisotropy energy is a constant for every wavevec-tor different of zero: ε K ( q (cid:54) = 0) = − K (cid:26)
12 sin θ (cid:2) ( K ) + ( K ) (cid:3) + cos θ ( K ) (cid:27) , (A5)and it contains a singularity for the antiferromagnetic state: ε K ( q = 0) = − K (cid:2) sin θ ( K ) + cos θ ( K ) (cid:3) . (A6)The component K does not appear in this last result because for q = 0 in the definition ofthe spin spiral, Eq. (A1), the circular components of the spins point along n only.The magnetic field term is given by: ε B ( q ) = − (cid:88) i B · S i = − (cid:88) i B cos θ n , (A7)where B α = B · n α .
3. One-dimensional model
We now consider the one-dimensional model introduced in Sec. III, with D = D ˆ z , K = ˆ z ,and B = B ˆ z . Energy of the antiferromagnetic state with spin parallel to the anisotropy easy-axis reads ε AF = − J − K . (A8)Meanwhile, the energy of the antiferromagnetic spin-spiral state with rotational axis along z is ε spiral ( q ) = − J (cid:2) cos (cid:0) aq (cid:1) sin θ − cos θ (cid:3) − D sin (cid:0) aq (cid:1) sin θ − K cos θ − B cos θ . (A9)The spiral wavevector that minimizes the total energy is given by ∂ε spiral ( q min ) ∂q =2 a (cid:16) J sin (cid:0) aq min (cid:1) − D cos (cid:0) aq min (cid:1)(cid:17) sin θ = 0= ⇒ tan (cid:0) aq min (cid:1) = DJ . (A10)20hus, the mininum energy is achieved when the spin-spiral wavevector satisfy tan( aq min ) = D/J . Replacing this result in the total energy, we have that the minimum energy is: ε spiral ( q min ) = − √ J + D sin θ − ( − J + K ) cos θ − B cos θ , (A11)where we used the transformation sin( aq ) = D/ √ J + D and cos( aq ) = J/ √ J + D .The spin spiral becomes more favorable when its energy is lower than the antiferromag-netic state energy thus satisfying: ε AF − ε spiral ( q min ) > ε = (cid:0) √ J + D − K + 2 J (cid:1) sin θ − J + B cos θ > . (A12)The first term is larger than zero when K < √ J + D + 2 J , where it is maximized for θ = π/
2. The term that depends on the magnetic field is maximized for θ = 0. For B = 0and θ = π/
2, the condition is satisfied when
D > (cid:115)(cid:18) J + K (cid:19) K . (A13)Let us find the global maximum of the energy difference as a function of θ :∆ ε ( θ ) = (cid:0) √ J + D − K + 2 J (cid:1) sin θ − J + B cos θ∂∂θ ∆ ε ( θ ) = (cid:0) √ J + D − K + 2 J (cid:1) θ cos θ − B sin θ ,∂ ∂θ ∆ ε ( θ ) = (cid:0) √ J + D − K + 2 J (cid:1) (cid:0) θ − (cid:1) − B cos θ . (A14)The critical points are given by (we only need to check for 0 ≤ θ ≤ π ): θ = 0 , cos θ = B (cid:0) √ J + D − K + 2 J (cid:1) . (A15)Analyzing the concavity of these points, we have: ∂ ∂θ ∆ ε ( θ ) = (cid:0) − cos θ (cid:1) B cos θ > ,∂ ∂θ ∆ ε ( θ ) = (cid:0) cos θ − (cid:1) B cos θ < , (A16)because 0 ≤ cos θ ≤
1. Theferore, θ should be the global maximum with energy differencegiven by ∆ ε ( θ ) = B (cid:18) θ + cos θ (cid:19) − J . (A17)21
1] Igor ˇZuti´c, Jaroslav Fabian, and S. Das Sarma, “Spintronics: Fundamentals and applications,”Rev. Mod. Phys. , 323–410 (2004).[2] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, “Magnon spintronics,”Nature Physics , 453–461 (2015).[3] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, “Antiferromagnetic spintronics,”Nature Nanotechnology , 231 (2016).[4] P. Wadley, B. Howells, J. ˇZelezn´y, C. Andrews, V. Hills, R. P. Campion, V. Nov´ak,K. Olejn´ık, F. Maccherozzi, S. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth,Y. Mokrousov, J. Kuneˇs, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds,B. L. Gallagher, and T. Jungwirth, “Electrical switching of an antiferromagnet,” Science ,587–590 (2016), https://science.sciencemag.org/content/351/6273/587.full.pdf.[5] Kamil Olejn´ık, Tom Seifert, Zdenˇek Kaˇspar, V´ıt Nov´ak, Peter Wadley, Richard P. Campion,Manuel Baumgartner, Pietro Gambardella, Petr Nˇemec, Joerg Wunderlich, Jairo Sinova, PetrKuˇzel, Melanie M¨uller, Tobias Kampfrath, and Tomas Jungwirth, “Terahertz electrical writ-ing speed in an antiferromagnetic memory,” Science Advances , eaar3566 (2018).[6] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, “Antiferromag-netic spintronics,” Rev. Mod. Phys. , 015005 (2018).[7] Tobias Kampfrath, Alexander Sell, Gregor Klatt, Alexej Pashkin, Sebastian M¨ahrlein, ThomasDekorsy, Martin Wolf, Manfred Fiebig, Alfred Leitenstorfer, and Rupert Huber, “Coherentterahertz control of antiferromagnetic spin waves,” Nature Photonics , 31–34 (2011).[8] Matthew W. Daniels, Ran Cheng, Weichao Yu, Jiang Xiao, and Di Xiao, “Nonabelianmagnonics in antiferromagnets,” Phys. Rev. B , 134450 (2018).[9] Albert Fert, Vincent Cros, and Jo˜ao Sampaio, “Skyrmions on the track,” Nature Nanotech-nology , 152–156 (2013).[10] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, “Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures,” Nature Nanotechnology , 839–844 (2013).[11] Dax M. Crum, Mohammed Bouhassoune, Juba Bouaziz, Benedikt Schweflinghaus, StefanBl¨ugel, and Samir Lounis, “Perpendicular reading of single confined magnetic skyrmions,” ature Communications , 1–8 (2015), number: 1 Publisher: Nature Publishing Group.[12] Wanjun Jiang, Pramey Upadhyaya, Wei Zhang, Guoqiang Yu, M. Benjamin Jungfleisch,Frank Y. Fradin, John E. Pearson, Yaroslav Tserkovnyak, Kang L. Wang, Olle Heinonen,Suzanne G. E. te Velthuis, and Axel Hoffmann, “Blowing magnetic skyrmion bubbles,” Sci-ence , 283–286 (2015), https://science.sciencemag.org/content/349/6245/283.full.pdf.[13] Xichao Zhang, Motohiko Ezawa, and Yan Zhou, “Magnetic skyrmion logic gates: conversion,duplication and merging of skyrmions,” Scientific Reports , 1–8 (2015), number: 1 Publisher:Nature Publishing Group.[14] F. Garcia-Sanchez, J. Sampaio, N. Reyren, V. Cros, and J.-V. Kim, “A skyrmion-based spin-torque nano-oscillator,” New Journal of Physics , 075011 (2016), publisher: IOP Publishing.[15] Albert Fert, Nicolas Reyren, and Vincent Cros, “Magnetic skyrmions: advances in physicsand potential applications,” Nature Reviews Materials , natrevmats201731 (2017).[16] Christian H Back, Vincent Cros, Hubert Ebert, Karin Everschor-Sitte, Albert Fert, MarkusGarst, Tainping Ma, Sergiy Mankovsky, Theodore Monchesky, Maxim V Mostovoy, Naoto Na-gaosa, Stuart Parkin, Christian Pfleiderer, Nicolas Reyren, Achim Rosch, Yasujiro Taguchi,Yoshinori Tokura, Kirsten von Bergmann, and Jiadong Zang, “The 2020 skyrmionicsroadmap,” Journal of Physics D: Applied Physics (2020).[17] AN Bogdanov and DA Yablonskii, “Thermodynamically stable “vortices” in magneticallyordered crystals. The mixed state of magnets,” , 178 (1989).[18] A. Bogdanov and A. Hubert, “Thermodynamically stable magnetic vortex states in magneticcrystals,” Journal of Magnetism and Magnetic Materials , 255–269 (1994).[19] UK R¨oßler, AN Bogdanov, and C Pfleiderer, “Spontaneous skyrmion ground states in mag-netic metals,” Nature , 797–801 (2006).[20] Naoto Nagaosa and Yoshinori Tokura, “Topological properties and dynamics of magneticskyrmions,” Nature Nanotechnology , 899–911 (2013).[21] Joseph Barker and Oleg A. Tretiakov, “Static and Dynamical Properties of AntiferromagneticSkyrmions in the Presence of Applied Current and Temperature,” Physical Review Letters , 147203 (2016).[22] H Velkov, O Gomonay, M Beens, G Schwiete, A Brataas, J Sinova, and R A Duine, “Phe-nomenology of current-induced skyrmion motion in antiferromagnets,” New Journal of Physics , 075016 (2016).
23] Rick Keesman, Mark Raaijmakers, A. E. Baerends, G. T. Barkema, and R. A. Duine,“Skyrmions in square-lattice antiferromagnets,” Phys. Rev. B , 054402 (2016).[24] Xichao Zhang, Yan Zhou, and Motohiko Ezawa, “Antiferromagnetic skyrmion: stability,creation and manipulation,” Scientific reports , 24795 (2016).[25] B¨orge G¨obel, Alexander Mook, J¨urgen Henk, and Ingrid Mertig, “Antiferromagnetic skyrmioncrystals: Generation, topological hall, and topological spin hall effect,” Phys. Rev. B ,060406 (2017).[26] Volodymyr P. Kravchuk, Olena Gomonay, Denis D. Sheka, Davi R. Rodrigues, KarinEverschor-Sitte, Jairo Sinova, Jeroen van den Brink, and Yuri Gaididei, “Spin eigenexci-tations of an antiferromagnetic skyrmion,” Phys. Rev. B , 184429 (2019).[27] Sebasti´an A. D´ıaz, Jelena Klinovaja, and Daniel Loss, “Topological Magnons and Edge Statesin Antiferromagnetic Skyrmion Crystals,” Physical Review Letters , 187203 (2019).[28] C. Kittel, “Theory of antiferromagnetic resonance,” Phys. Rev. , 565–565 (1951).[29] F. Keffer and C. Kittel, “Theory of antiferromagnetic resonance,” Phys. Rev. , 329–337(1952).[30] I. Dzyaloshinsky, “A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics,”Journal of Physics and Chemistry of Solids , 241–255 (1958).[31] Tˆoru Moriya, “Anisotropic Superexchange Interaction and Weak Ferromagnetism,” PhysicalReview , 91–98 (1960).[32] A. N. Bogdanov, U. K. R¨oßler, M. Wolf, and K.-H. M¨uller, “Magnetic structures and re-orientation transitions in noncentrosymmetric uniaxial antiferromagnets,” Phys. Rev. B ,214410 (2002).[33] Randy S. Fishman, Toomas R˜o om, and Rog´erio de Sousa, “Normal modes of a spin cycloidor helix,” Phys. Rev. B , 064414 (2019).[34] L. Udvardi and L. Szunyogh, “Chiral Asymmetry of the Spin-Wave Spectra in UltrathinMagnetic Films,” Physical Review Letters , 207204 (2009).[35] A. T. Costa, R. B. Muniz, S. Lounis, A. B. Klautau, and D. L. Mills, “Spin-orbit couplingand spin waves in ultrathin ferromagnets: The spin-wave Rashba effect,” Physical Review B , 014428 (2010).[36] Kh. Zakeri, Y. Zhang, J. Prokop, T.-H. Chuang, N. Sakr, W. X. Tang, and J. Kirschner,“Asymmetric Spin-Wave Dispersion on Fe(110): Direct Evidence of the Dzyaloshinskii-Moriya nteraction,” Physical Review Letters , 137203 (2010).[37] Flaviano Jos´e dos Santos, Manuel dos Santos Dias, and Samir Lounis, “Nonreciproc-ity of spin waves in noncollinear magnets due to the Dzyaloshinskii-Moriya interaction,”arXiv:2003.11649 [cond-mat] (2020), arXiv: 2003.11649.[38] G. Gitgeatpong, Y. Zhao, P. Piyawongwatthana, Y. Qiu, L. W. Harriger, N. P. Butch, T. J.Sato, and K. Matan, “Nonreciprocal Magnons and Symmetry-Breaking in the Noncentrosym-metric Antiferromagnet,” Physical Review Letters , 047201 (2017).[39] Markus Garst, Johannes Waizner, and Dirk Grundler, “Collective spin excitations of he-lices and magnetic skyrmions: review and perspectives of magnonics in non-centrosymmetricmagnets,” Journal of Physics D: Applied Physics , 293002 (2017).[40] P. Nˇemec, M. Fiebig, T. Kampfrath, and A. V. Kimel, “Antiferromagnetic opto-spintronics,”Nature Physics , 229–241 (2018).[41] A. Zheludev, S. Maslov, G. Shirane, I. Tsukada, T. Masuda, K. Uchinokura, I. Zaliznyak,R. Erwin, and L. P. Regnault, “Magnetic anisotropy and low-energy spin waves in thedzyaloshinskii-moriya spiral magnet ba cuge o ,” Phys. Rev. B , 11432–11444 (1999).[42] Jungho Kim, D. Casa, M. H. Upton, T. Gog, Young-June Kim, J. F. Mitchell, M. van Vee-nendaal, M. Daghofer, J. van den Brink, G. Khaliullin, and B. J. Kim, “Magnetic excitationspectra of sr iro probed by resonant inelastic x-ray scattering: Establishing links to cupratesuperconductors,” Phys. Rev. Lett. , 177003 (2012).[43] MPM Dean, RS Springell, C Monney, KJ Zhou, J Pereiro, I Boˇzovi´c, Bastien Dalla Piazza,HM Rønnow, E Morenzoni, J Van Den Brink, et al. , “Spin excitations in a single la2cuo4layer,” Nature Materials , 850–854 (2012).[44] Flaviano Jos´e dos Santos, Manuel dos Santos Dias, Filipe Souza Mendes Guimar˜aes, JubaBouaziz, and Samir Lounis, “Spin-resolved inelastic electron scattering by spin waves innoncollinear magnets,” Physical Review B , 024431 (2018).[45] M. Plihal, D. L. Mills, and J. Kirschner, “Spin Wave Signature in the Spin Polarized ElectronEnergy Loss Spectrum of Ultrathin Fe Films: Theory and Experiment,” Physical ReviewLetters , 2579–2582 (1999).[46] R. Vollmer, M. Etzkorn, P. S. Anil Kumar, H. Ibach, and J. Kirschner, “Spin-Polarized Elec-tron Energy Loss Spectroscopy of High Energy, Large Wave Vector Spin Waves in Ultrathinfcc Co Films on Cu(001),” Physical Review Letters , 147201 (2003).
47] E. Michel, H. Ibach, and C. M. Schneider, “Spin waves in ultrathin hexagonal cobalt filmson W(110), Cu(111), and Au(111) surfaces,” Physical Review B , 024407 (2015).[48] Flaviano Jos´e dos Santos, Manuel dos Santos Dias, and Samir Lounis, “First-principles inves-tigation of spin-wave dispersions in surface-reconstructed Co thin films on W(110),” PhysicalReview B , 134408 (2017).[49] Gideon P. M¨uller, Markus Hoffmann, Constantin Dißelkamp, Daniel Sch¨urhoff, StefanosMavros, Moritz Sallermann, Nikolai S. Kiselev, Hannes J´onsson, and Stefan Bl¨ugel, “Spirit:Multifunctional framework for atomistic spin simulations,” Physical Review B , 224414(2019).[50] Yu. A. Bychkov and ´E. I. Rashba, “Properties of a 2D electron gas with lifted spectral degen-eracy,” Soviet Journal of Experimental and Theoretical Physics Letters , 78 (1984).[51] Flaviano Jos´e dos Santos, Manuel dos Santos Dias, and Samir Lounis, “Nonreciprocity ofspin waves in noncollinear magnets due to the dzyaloshinskii-moriya interaction,” (2020),arXiv:2003.11649 [cond-mat.mes-hall].[52] P. Ferriani, I. Turek, S. Heinze, G. Bihlmayer, and S. Bl¨ugel, “Magnetic Phase Control inMonolayer Films by Substrate Tuning,” Physical Review Letters , 187203 (2007)., 187203 (2007).