Abstract
The recent demonstrations of electrical manipulation and detection of antiferromagnetic spins have opened up a new chapter in the story of spintronics. Here, we review the emerging research field that is exploring the links between antiferromagnetic spintronics and topological structures in real and momentum space. Active topics include proposals to realize Majorana fermions in antiferromagnetic topological superconductors, to control topological protection and Dirac points by manipulating antiferromagnetic order parameters, and to exploit the anomalous and topological Hall effects of zeronetmoment antiferromagnets. We explain the basic concepts behind these proposals, and discuss potential applications of topological antiferromagnetic spintronics.
Main
Topologically protected states of matter are unusually robust because they cannot be destroyed by small changes in system parameters. This feature of topological states has suggested an appealing strategy to achieve useful quantum computation^{1,2}. In spintronics, topological states provide for strong spinmomentum locking^{3}, high chargecurrent to spincurrent conversion efficiency^{4,5,6}, high electron mobility and long spin diffusion length^{7,8}, strong magnetoresistance^{8} and efficient spin filtering^{9}. Materials exhibiting topologically protected Dirac or Weyl quasiparticles in their momentumspace bands, and those exhibiting topologically nontrivial realspace spin textures^{6,10}, have both inspired new energyefficient spintronic concepts^{4,7,10,11,12,13}.
In a topological insulator (TI), timereversal symmetry enforces Dirac quasiparticle surface states with spinmomentum locking (see panel a in the figure in Box 1) and protection against backscattering^{3}. The much higher efficiency of magnetization switching by a currentinduced spin–orbit torque in a TI/magnetically doped TI (MTI) heterostructure^{4,14} than in a heavymetal/ferromagnet (FM) bilayer is thought to be associated with spinmomentum locking. This is a paradigmatic example of the potential for applications of topological materials in spintronics, although a full microscopic understanding of the underlying current–spin conversion mechanism is still absent^{15}. Progress in understanding and exploiting TIs in spintronics has so far been limited by unintentional bulk doping in TIs, and by the decreased stability of TI surface states at elevated temperatures^{15}. The practical utility of the topologically enhanced spin–orbit torque has also been limited by the cryogenic temperatures at which known MTIs order^{4,14}, although a recent report^{16} of interfacial ferromagnetism persisting to room temperature in an insulating FM (EuS)/TI heterostructure is promising in this respect.
A substantial rise in the critical temperature of an MTI (by a factor of 3 to 90 K) due to proximity coupling to an adjacent antiferromagnet (AF) has recently been demonstrated in a heterostructure consisting of the metallic AF CrSb sandwiched between two MTIs^{17}. Increased spin–orbit torque efficiency at heterojunctions between TIs and ferrimagnetic CoTb alloys containing antiferromagnetically coupled Co and Tb sublattices^{18} has also been reported. The later effect persists to room temperature, but with decreased efficiency enhancement^{15} at higher temperatures.
However, research on using antiferromagnetism to achieve a role for topological materials in spintronics is still at an early stage, and many ideas have so far only been addressed theoretically. For example, the practical advantages of TIs over heavymetal systems for spin–orbit torques are not yet established^{15}. The forms of magnetism so far incorporated in MTIs remain fragile because they are of interfacial^{16,17} or dilutemoment character^{14}. Other new ideas, beyond simply making TIs magnetic, are emerging at a rapid pace. In this article we review topological antiferromagnetic spintronics, the emerging field that is exploring the interplay between transport, topological properties in either momentum space or real space, and antiferromagnetic order.
Dirac quasiparticles in antiferromagnetic heterostructures
The roots of topological antiferromagnetic spintronics can be traced to studies of layered AFs of the SrMnBi_{2} type, which were reported to feature quasitwodimensional massive Dirac quasiparticles near the Fermi level^{19,20}. These were associated with the observation of enhanced mobilities, similar to those in graphene. Manipulation of the Dirac quasiparticle current and the quantum Hall effect in a EuMnBi_{2} AF by an applied strong magnetic field has been demonstrated, with the effect of the field mediated by Eu sublattices^{21}. Signatures of the twodimensional massless Dirac cones were found in the infrared spectra of the antiferromagnetic superconductor BaFe_{2}As_{2} (refs ^{22,23}).
As pointed out in ref. ^{24}, TI phases are possible in AFs even though timereversal symmetry \({\mathscr{T}}\) is broken, and are protected instead by \({T}_{\frac{1}{2}}{\mathscr{T}}\) where \({T}_{\frac{1}{2}}\) is a halfmagneticunitcell translation operation, as we illustrate in panel a in the figure in Box 1. The proposed lowtemperature AF candidate, GdPtBi, has not yet been confirmed as a TI by angleresolved photoemission spectroscopy (ARPES), presumably due to the imperfect crystalmomentum resolution of the measurement^{25}. A path of research related to topological superconductivity has demonstrated signatures of the coexistence of a twodimensional TI, that is, the quantum spin Hall effect (see Fig. 1a), and a superconducting state in holedoped and electrondoped antiferromagnetic monolayers of FeSe (ref. ^{26}). FeSe is the metallic building block of the ironbased highT_{C} superconductors, and the combined effect of substrate strain, spin–orbit coupling and electronic correlations was shown to induce band inversion and quantum spin Hall effect edge states, as shown in Fig. 1a,b (ref. ^{26}). Separately, quantum spin Hall effect states in an AF have also been predicted in honeycomb lattice systems^{27}.41586_2018_136_Tab1_ESM.jpg
The fortunate lattice constant match between the TI (Bi,Sb)_{2}Te_{3} and the hightemperature AF CrSb has been exploited to grow epitaxial interfaces between these materials^{17,28}. CrSb/TI (Bi,Sb)_{2}Te_{3}/AF CrSb trilayers exhibit cusps in the magnetoresistance, that presumably correspond to a topological phase transition of Dirac quasiparticles at the interfaces^{28}. The phase transition from the quantum anomalous Hall state (QAHE, see panel c of the figure in Box 1 and Fig. 1c,e) to what is presumed to be an axion insulator (quantized electric polarization induced by magnetism^{24}, Fig. 1d,e) was observed in an MTI/TI/MTI trilayer by reoerienting the exchange fields of the MTIs from ferromagnetic to antiferromagnetic^{29}. The control of the QAHE and axion insulator states by the external magnetic field and electric gating yields very large magneto/electroresistance changes h/e^{2} ~ 25.8 kΩ and ~GΩ, albeit at millikelvin temperatures^{29,30}.
Threedimensional topological semimetal AFs
Topological semimetal states arise when conduction and valence bands touch at discrete points, lines or planes in a bulk Brillouin zone at energies close to the Fermi level. The lowenergy physics of topological semimetals can be governed by effective Dirac or Weyl equations^{11,13,35}. Threedimensional (3D) Dirac and Weyl quasiparticles in nonmagnetic bulk systems have attracted attention because of reports of suppressed backscattering, measurements of exotic topological surface states and interest in unique topological responses such as lowdissipation axial currents^{8,11,36}. These properties are thought to be responsible for experimental observations of chiral magnetotransport^{12,37}, and strong magnetoresistance^{38,39}, although the topological origin of these phenomena is not yet firmly established^{8}. For instance, the strong magnetoresistance in WTe_{2} semimetals was originally explained on the basis on the carrier compensation in the tiny electron–hole pockets at the Fermi level^{40}, and only later linked to the presence of Weyl fermions^{39}.
Topological metal–insulator transitions in 3D Dirac semimetal AFs
In a system with time reversal \({\mathscr{T}}\) and spatial inversion \({\mathscr{P}}\) symmetries, the electronic bands are doubly degenerate, resulting in a lowenergy Dirac Hamiltonian, \({{\mathscr{H}}}_{{\rm{D}}}({\bf{k}})\). In its simplest form^{11,13,35},
where σ is the vector of Pauli matrices, v_{F} is the Fermi velocity, k = q − q_{0} is the crystal momentum measured from the Dirac point at q_{0} and m is the mass (in units of energy). The corresponding energy dispersion is E(k) = \(\pm \hslash {v}_{{\rm{F}}}\sqrt{{k}_{x}^{2}+{k}_{y}^{2}+{k}_{z}^{2}+{\left(\frac{m}{\hslash {v}_{{\rm{F}}}}\right)}^{2}}\). The mass can be absent because of a crystalline symmetry, and in this case \({{\mathscr{H}}}_{{\rm{D}}}({\bf{k}})\) describes the fourfold degenerate band touching^{11,13} of a 3D Dirac semimetal illustrated schematically in panel b in the figure in Box 1. In a 3D Dirac semimetal, the topological invariants and nontrivial surface states can be linked to the crystalline symmetry protecting the degeneracy^{41,42}.
The 3D Dirac semimetal state is not possible in FMs because \({\mathscr{T}}\)symmetry breaking prevents the double band degeneracy. On the other hand, a topological crystalline 3D Dirac semimetal was predicted in an AF, namely in the orthorhombic phase of CuMnAs^{43,44}. Here \({\mathscr{P}}\) and \({\mathscr{T}}\) symmetries are absent separately, but the combined \({\mathscr{P}}{\mathscr{T}}\) symmetry ensures double band degeneracy over the whole Brillouin zone, as illustrated in Fig. 2b–d. In this case, the Dirac point is protected by \({\mathscr{P}}{\mathscr{T}}\) symmetry together with an additional crystalline nonsymmorphic symmetry, as we explain in Fig. 2e. The orthorhombic CuMnAs AF is an attractive minimal case for magnetic Dirac semimetals induced by band inversion, since only a single pair of Dirac points appears near the Fermi level of the ab initio band structure. Electronfillingenforced semimetals with a single Dirac cone are also a possibility, as indicated theoretically in twodimensional model AFs^{45}.
Novel effects have been predicted in topological Dirac semimetal AFs that are based on the possibility of controlling topological states by controlling only Néel vector orientation, not the presence or absence of antiferromagnetic order, and this can be accomplished using currentinduced spin–orbit torques. The latter effect, discussed in detail by Železný et al.^{46} in this Focus issue, has been experimentally demonstrated in CuMnAs^{47}. The coexistence of Dirac fermions and spin–orbit torques in CuMnAs implies a new phase transition mechanism, referred to as the topological metal–insulator transition^{44}. The origin of the effect is in Fermi surface topology, which is sensitive to the changes in the magnetic symmetry upon reorienting the Néel vector, as explained in Fig. 2e,f. The transport counterpart of the topological metal–insulator transition is topological anisotropic magnetoresistance, which in principle can reach extremely large values^{44}. The topological anisotropic magnetoresistance can be understood as a limiting case of crystalline anisotropic magnetoresistance. The effect is different in origin and presumably more favorable for spintronics than the metal–insulator transition observed in the pyrochlore iridate family, which is driven by combined correlation and external field effects^{48}, or the extreme magnetoresistance observed in the AF topological metal candidate NdSb^{49}.
An antiferromagnetic Dirac nodal line semimetal has also been proposed^{44}. Since the nodal lines were observed several electronvolts deep in the ab initio Fermi sea of tetragonal CuMnAs, the search is still on for more favorable candidate AF materials featuring nodal lines closer to the Fermi level.
Weyl fermions in AFs
When \({\mathscr{P}}\) or \({\mathscr{T}}\) symmetry, or both, is broken and the double band degeneracy is lifted, the touching points of two nondegenerate bands can form a 3D Weyl semimetal (panel d in the figure in Box 1). Fermi states in a Weyl semimetal are described by the Weyl Hamiltonian^{11,13}:
Weyl points act as monopole sources of Berry curvature flux and generate a topological charge defined by
Here δS is a small sphere surrounding the Weyl point at k_{z,W} with the surface normal vector n, \({\mathscr{C}}\) (equation (1)) refers to the plane slightly below and above the Weyl point, k_{z,W} ± δ. The difference in integration area between the first and second forms of equation (6) is justified by Gauss’s theorem. In the vicinity of the Weyl point the Berry curvature takes the monopole form, b(k) = ±k/(2k^{3}). Weyl points always come in pairs with opposite topological charges and in general do not rely on any specific symmetry protection. The only way to remove them is to annihilate two Weyl points with opposite topological charges. The 3D nature of the Weyl point is crucial here since the corresponding Weyl equation uses all three Pauli matrices. Consequently, any small perturbation that is expressed as a linear combination of Pauli matrices that form the basis of the 2 × 2 Hilbert space just shifts—and does not gap—the Weyl point. For example, for a perturbation of the form mσ_{ z }, the dispersion is renormalized as E(k) = \(\pm \hslash {v}_{{\rm{F}}}\sqrt{{k}_{x}^{2}+{k}_{y}^{2}+{\left({k}_{z}+\frac{m}{\hslash {v}_{{\rm{F}}}}\right)}^{2}}\).
Magnetic Weyl semimetals have remained experimentally elusive for a long time, despite several promising antiferromagnetic candidates including pyrochlore irridates^{50} such as Eu_{2}Ir_{2}O_{7}^{51}, or the YbMnBi_{2} AF, which was controversially suggested^{35} to be either a Weyl^{52,53} or a Dirac semimetal^{54,55}. Recently, magnetic Weyl fermions were predicted^{56}, and reported^{57} in Mn_{3}Sn (see Fig. 3a), a noncollinear AF from the Heusler family that is potentially more relevant for metallic spintronics, due for example to the relatively high Néel temperature, 420 K (ref. ^{58}). In Fig. 3b we show the measured ARPES overlaid with the ab initio calculated band structure in Mn_{3}Sn. In Fig. 3b is the measured positive magnetoconductance, which is believed to be a signature of the chiral anomaly and Weyl fermions in condensed matter^{11,12,35}. The chiral anomaly refers to a nonconservation of left and righthanded Weyl quasiparticles in parallel electric and magnetic fields^{12}.
a, The noncollinear antiferromagnetic structure of Mn_{3}Sn. b, The angleresolved photoemission spectrum overlaid with the ab initio band structure of Mn_{3}Sn. c, Measured positive magnetoconductance for parallel electric and magnetic fields in Mn_{3}Sn. d, Tilted Weyl points^{39} and Fermi arc band dispersion in Mn_{3}Sn as predicted by the surfaceweighted densityofstates ab initio calculation. The Fermi arcs can be of a rather complex shape; however, their qualitative behaviour corresponds to the simple picture presented in panel d in the figure in Box 1. LDOS, local density of states. e, The calculation of the surface Fermi arcs connecting the tilted Weyl points^{39} in Mn_{3}Ge. The Weyl points marked by + and − are distributed antisymmetrically around the Γ point. f, The largest contribution to the intrinsic AHE in Mn_{3}Ge originates from avoidedcrossing lines, where the kspace resolved fictitious magnetic field, that is the Berry curvature \({b}_{y}\left({k}_{x},{k}_{y}\right) \sim {\int }_{{k}_{z}}{\rm{d}}{k}_{z}f({\bf{k}}){b}_{y}({\bf{k}})\), takes the largest value. Adapted from ref. ^{57}, Macmillan Publishers Ltd (b,c); and ref. ^{56}, IOP (d,e).
Figure 3d,e illustrate the surfaceweighted density of states predicted by ab initio calculations, which exhibits the typical Fermi arc features (see also panel d in the figure in Box 1) together with trivial Fermi surface pockets^{56}. Weyl semimetal states can also be realized in the paramagnetic and AF phase of GdPtBi^{12,37} by applying a magnetic field and, in contrast to Dirac semimetals, Weyl semimetals can in principle also exist in FMs^{59}. The Mn_{3}(Ge/Sn) AF was shown to host a large anomalous Hall effect (AHE), whose origin is discussed in the next section.
Topological transport in AFs
Until recently the AHE was viewed as a combined consequence of timereversal symmetry breaking in an FM and spin–orbit coupling^{60}. In the case of collinear AFs, either \({T}_{\frac{1}{2}}{\mathscr{T}}\) symmetry or \({\mathscr{P}}{\mathscr{T}}\) symmetry forces the Hall conductivity to vanish. Recent ab initio calculations^{61,62} inspired by earlier theoretical works^{63,64,65,66} have shown that timereversal symmetry breaking by AF order can yield a finite Hall response in some noncollinear AFs, even those with zero net magnetization and even in the absence of spin–orbit coupling. The timereversal symmetry breaking is manifested by a nonzero Berry curvature, as we show in Figs. 3f and 4a–d. The intrinsic contribution to the Hall conductivity, \({\sigma }_{{\rm{H}}}=\frac{1}{2}\left({\sigma }_{xy}{\sigma }_{yx}\right)\), depends only on the band structure of the perfect crystal and can be calculated from linear response theory^{60}:
a, In the simplest Weyl semimetal, the net emergent magnetic field b (Berry curvature) is parallel to the line connecting the two Weyl points with opposite chiralities and the AHE occurs in the perpendicular plane. b, The nonzero b(k) and AHE in Mn_{3}Ge are due to the breaking of the combinined timereversal and mirrorplane symmetries by the noncollinear antiferromagnetic order on the kagome lattice. j, current; V_{H}, Hall voltage. c, The emergent magnetic field b_{ i } at position r can also be generated by the realspace spin chirality (proportional to the solid angle subtended by the three noncoplanar spins S_{A}, S_{B} and S_{C}) and yields a topological Hall effect. For the sake of simplicity, the noncoplanar spins of the antiferromagnetic quantum topological Hall effect candidate K_{0.5}RhO_{2} are translated to a common origin. The effect survives when spin–orbit interactions are turned off. d, Emergent magnetic field (sum of the contribution from each tetrahedron face) can also be found in the magnetic order on a distorted facecentred cubic or pyrochlore lattice (we show a fragment of a pyrochlore lattice^{77}), in phase with a cubic symmetry broken by strain^{64} or a magnetic field parallel to the [111] direction^{77}. e, The measured AHE resistivity in the Mn_{3}Ge flips its sign on applying a weak magnetic field \(\pm {\bf{B}}\parallel [100]\). f, Temperature dependence of the AHE conductivity in Mn_{3}Ge and its magnitude dependence on the antiferromagnetic texture controlled by a weak magnetic field^{67,68}. g, The measured temperature dependence of the topological Hall conductivity in the chiral spin liquid without longrange dipolar order on the pyrochlore lattice in Pr_{2}Ir_{2}O_{7}. FC, field cooled; ZFC, zero field cooled. Adapted from ref. ^{68}, APS (e,f); and ref. ^{77}, Macmillan Publishers Ltd (g).
where \({b}_{z}^{n}({\bf{k}})\) is the zcomponent of the Berry curvature (equation (2)), f is the Fermi–Dirac distribution function and n is the band index.
The AHE in noncollinear AFs
The AHE was recently observed in the hexagonal noncollinear AFs Mn_{3}Sn and Mn_{3}Ge^{67,68,69}, which have Weyl points close to the Fermi level. However, ab initio calculations of the intrinsic AHE in Mn_{3}Ge, which predict a magnitude consistent with experiment, reveal that the dominant contribution to the AHE originates instead from avoided crossings in the band structure^{58}, as shown in Fig. 3f. In Fig. 4e,f we show the observed AHE. For instance, σ_{ xz } ~ 380 Ω^{−1} cm^{−1} and corresponds to a large effective emergent magnetic field \(\left{{\bf{b}}}_{[010]}\right\) ~ 200 T (refs ^{67,68}). We also note that a large AHE was achieved in the collinear AF GdPtBi by canting the staggered order^{70}. The discovery of a large AHE in Mn_{3}Sn and Mn_{3}Ge, which are metals but have a relatively small density of states at the Fermi level, inspires a search for the quantized and dissipationless limits of anomalous transport in topological semiconducting/insulating AFs, for instance the QAHE^{32,71}, or dynamical axion fields^{72}.
Topological Hall effect in AFs
Realspace order parameter textures can be induced in AFs, and their presence can be detected by the socalled topological Hall effect. In this phenomenon, the role of the spin–orbit coupling is substituted by the chirality of the spin texture (see Fig. 4c). The effect of the corresponding fictitious magnetic field, \(\hat{{\bf{m}}}\) ⋅ (∂_{ x }\(\hat{{\bf{m}}}\) × ∂_{ y }\(\hat{{\bf{m}}}\)), on the Bloch electrons generates a Hall response. The topological Hall effect can be experimentally distinguished from the AHE by, for example, analysing the disorder dependence^{73}. However, making the distinction might be difficult in heterojunction systems, as was pointed out in studies of monolayer Fe deposited on an Ir(001) surface^{74}. Noncoplanar AFs can have also a pronounced topological orbital moment due to the scalar spin chirality. For example, textures in Fe/Ir(001), Mn/Cu(111) or the γphase of FeMn alloy^{74,75,76}, and their control by spin torques, might yield novel material functionalities.
The Hall effect associated with the spin chirality was reported initially in the chiral spin liquids of the pyrochlore iridates (see Fig. 4d,g)^{77} and later in MnSi chiral antiferromagnetic alloys^{78,79}. We note that the term ‘topological’ used to label the effect does not imply in this case a correspondence to a topological invariant. In contrast, a ferromagnetic skyrmion spin texture (next section) carries an integer topological charge, which is accompanied by a topological Hall effect (\(\left{\bf{b}}\right\) ~ −13 T; ref. ^{80}). In this case, the term topological refers to the association of the Hall response with a topological invariant. A distinct example of such a correspondence is the quantized topological Hall effect of a noncollinear magnet defined by the nonzero Chern number in kspace as proposed for the noncoplanar AF K_{0.5}RhO_{2} (ref. ^{81}) and shown schematically in Fig. 4c.
Spin currents and torques in AFs
While the AHE arises from the Berry curvature in momentum space, other important spintronic phenomena can be associated with Berry curvatures in different parameter spaces. For instance, the spin–orbit torkance tensor τ_{ ij } (ref. ^{82}) is defined by the linear response relation T_{ i } = τ_{ ij }E_{ j }, where \({\bf{T}}=\frac{{\rm{d}}{\bf{m}}}{{\rm{d}}t}\) is the spin–orbit torque exerted on the magnetization m in a magnet subject to an applied electric field E. The intrinsic part of the spin–orbit torque can be rewritten in terms of a mixed Berry curvature, \({b}_{ij}^{\hat{{\bf{m}}}{\bf{k}}}={\hat{{\bf{e}}}}_{i}\cdot 2{\rm{Im}}{\sum }_{n}\left\langle {\partial }_{\hat{{\bf{m}}}}{u}_{{\bf{k}}n}\left{\partial }_{{k}_{j}}{u}_{{\bf{k}}n}\right.\right\rangle\), where e_{ i } denotes the ith Cartesian unit vector and m is a unit vector in the direction of magnetization^{82}. A large spin–orbit torque in a topologically nontrivial insulating FM has been associated with the existence of monopoles with mixed Berry curvature. These are termed mixed Weyl points, as they correspond formally to a Weyl Hamiltonian \({\mathscr{H}}({\bf{k}},\hat{{\bf{m}}})\) = \(\hslash {v}_{{\rm{F}}}\left({k}_{x}{\sigma }_{x}+{k}_{y}{\sigma }_{y}\right)+{v}_{\theta }\theta {\sigma }_{z}\) in the mixed momentum–magnetization space^{82}. (Here θ is the azimuthal angle of the magnetization.) The recent discovery of the spin–orbit torque and the prediction of a Dirac semimetal state in antiferromagnetic CuMnAs motivates a search for analogous dissipationless pronounced spin–orbit torques in insulating AFs. Additionaly, large spin Hall angles were also predicted for the Weyl semimetals^{83}, and it was theoretically proposed that the spin Hall effect can also occur due to the breaking of the spin rotational symmetry in noncollinear AFs without the need for either spin–orbit coupling or spin chirality^{84}.
Finally, a topological spin Hall effect was predicted for skyrmions^{85}, where the spin Hall response occurs even in the absence of spin–orbit coupling, in analogy with the above topological (charge) Hall effect. Antiferromagnetic skyrmionic crystals were predicted to have nonzero topological spin Hall effect, but vanishing topological Hall effect^{86}.
Antiferromagnetic skyrmions
Magnetic skyrmions are noncollinear magnetization textures in which the spin quantization axis changes continuously over length scales that vary from a few nanometres to a few micrometres. For twodimensional systems, the winding number,
of a magnetization texture measures the number of times the sphere of magnetization directions is covered upon integrating over space and must take integer values^{10,87}. Here \(\hat{{\bf{m}}}\) = m(x, y, z) is the normalized magnetization field in real space and \(\hat{{\bf{m}}}\) ⋅ (∂_{ x }\(\hat{{\bf{m}}}\) × ∂_{ y }\(\hat{{\bf{m}}}\)) is the fictitious emergent magnetic field. Antiferromagnetic skyrmions can be visualized as two interpenetrating ferromagnetic skyrmions, where the index (j) = (A, B) labels the two antiferromagnetic sublattices, as shown in Fig. 5a. Microscopically, the skyrmionic magnetization modulation is caused by the Dzyaloshinskii–Moriya interaction of noncentrosymmetric crystals or due to the inversion asymmetry at the interfaces^{6}. Remarkably, the Dzyaloshinskii–Moriya interaction in the bulk is more abundant in AFs than FMs^{88}. By comparing to equations (1) and (6), we see that Q topologically protects skyrmion textures in real space, just as Weyl points and the QAHE state are protected in momentum space. The calculated energy barrier for skyrmion annihilation in discrete magnetic skyrmions is of the order of 0.1 eV in Fe/Co(111)^{89}. Because this barrier is finite, the stability of skyrmions in experimental systems relies in part on other physical limitations, for example a combined effect of spin rotation and skyrmion diameter shrinking, rather than on topological protection itself^{89}.
a, An antiferromagnetic skyrmion can be viewed as consisting of two antiferromagnetically coupled ferromagnetic skyrmions. For the sake of clarity, we draw the two opposite magnetic moments in the antiferromagnetic unit cell as coinciding and their moments as perfectly compensated. Note that the structure of the skyrmion is analogous to the momentumspace Berry curvature, and we illustrate here the Néel type of the skyrmion. b, A synthetic antiferromagnetic skyrmion of the Bloch type in an Fe–Cu–Fe trilayer. c, Micromagnetic simulation of ferromagnetic (upper panel) and antiferromagnetic (lower panel) skyrmion motion driven by a spin–orbit torque. The ferromagnetic skyrmion is deflected by the Magnus force, while the antiferromagnetic skyrmion can move in a straight line due to mutual compensation between Magnus forces from the two magnetic sublattices, as schematically shown in a. Adapted from ref. ^{96}, AIP (b); and ref. ^{93}, Wiley (c).
Spintronics aspects of antiferromagnetic skyrmions^{90,91}, namely their manipulation by an electrical current, have been discussed only recently, and only theoretically^{88,92,93,94}. Micromagnetic simulations show that antiferromagnetic skyrmions move faster than ferromagnetic skyrmions, can be driven with lower current densities (j_{crit.} ~ 10^{6}–10^{7} A cm^{−2}) and, most importantly, move in straight lines, as explained in Fig. 5a,c^{88,92,93}. Antiferromagnetic skyrmions were also recently studied in detail in synthetic AFs (for example in an Fe–Cu–Fe trilayer) in which skyrmions in the two ferromagnetic layers are coupled antiferromagnetically^{95,96}, as shown in Fig. 5b. The topological spin Hall effect was suggested as a probe to monitor the antiferromagnetic skyrmions, as well as to generate a spin current^{86,96}. While ferrimagnetic skyrmions were observed recently in GdFeCo^{97}, antiferromagnetic skyrmions remain to be discovered.
Perspectives
Potential advantages of nearly and perfectly compensated antiferromagnetic materials for spintronics are discussed throughout this Focus issue. In this brief article, we have focused on possible cooperation between antiferromagnetism and topological properties in both momentum and real spaces. In many cases, novel phases that combine antiferromagnetism and topology have been discussed only very recently and remain experimentally elusive, for example antiferromagnetic TIs, antiferromagnetic Dirac semimetals, new QAHE systems and new systems that support skyrmions. As we have explained, these topological antiferromagnetic states can, once realized at room temperature, enable more stable nanospintronic devices that dissipate less energy and have new functionalities related to unique AF symmetries, or to the possibility of tuning by coupling to the antiferromagnetic order. Recently, signatures of a correlated magnetic Weyl semimetal were observed in the AF Mn_{3}Sn (ref. ^{57}).
Fast topological memories, in which states are written by the topological spin–orbit torque in an antiferromagnetic TI or an antiferromagnetic Dirac semimetal^{44,82,98}, and read out via the large magnetoresistance effects associated with bandgap tuning^{44,99,100}, seem particularly attractive. Another possibility exploits the phase transitions of topological phases, as we have explained in the context of the CuMnAs Dirac AF in Fig. 2. Here one can foresee the possibility of a topological transistor operating at high frequencies and low current densities (see Fig. 6a)^{101}. Another scenario starts by forming a p–n junction in a single layer of FeSe by gating to obtain a superconducting region and a region with a quantum spin Hall effect^{102}. Coupling this system to one ferromagnetic electrode from each side would then localize Majorana modes at the interfaces as illustrated in Fig. 6b, providing an alternative realization of the Fu–Kane Majorana fermion proposal^{103} that could survive at higher temperatures^{26}. The associated twolevel states can function as quantum bits that encode information nonlocally and are therefore robust against decoherence^{2}.
a, A concept for a topological antiferromagnetic transistor. In contrast to the related proposal in crystalline TIs^{109}, one can achieve a highly mobile bulk Dirac quasiparticle current (present in the ‘ON’ state) that is controllable by gating or by ultrafast spin–orbit torque. b, A Majorana qubit in an antiferromagnetic FeSe monolayer. QSHI, quantum spin Hall insulator. c, The skyrmion racetrack memory concept. Magnetic information is stored in a topological charge Q instead of the magnetization of the magnetic domain. ‘1’ encodes the skyrmion with Q = 1, while ‘0’ corresponds to a uniform domain with Q = 0. Spintransfer torque shifts the register. Bits are read by the topological Hall effect and written by nucleating or deleting the skyrmion. Adapted from ref. ^{110}, Macmillan Publishers Ltd (c).
Many of the novel effects we have discussed follow directly from AF symmetries and cannot be realized in FMs, for instance (1) magnetism combined with the quantum spin Hall effect, and superconductivity, and (2) magnetism combined with Dirac semimetal phases. The conditions for a good Dirac quasiparticle in AF spintronics have recently been carefully specified^{13}. Further afield, strong electronic correlations introduce additional challenges in finding and realizing topological phases, but they generate even richer phase diagrams and their interplay with topology represents a very active area of research^{30,48,50}.
An example of a system in which the interesting effects are mostly established is the noncollinear AHE AFs Mn_{3}Sn and Mn_{3}Ge. The sign and magnitude of their AHE depends on the noncollinear spin texture orientation. This, together with the demonstration of the possibility of manipulating the noncollinear spin texture by a spin torque^{104}, can allow for memory devices in noncollinear AFs, with electrical readout via the AHE, as illustrated in Fig. 4e. Moreover, optical and thermal counterparts of the d.c. AHE should be present in noncollinear AFs^{105–108}, opening the prospect of antiferromagnetic topological optospintronic and spincaloritronic devices.
The skyrmion might represent the smallest micromagnetic object that can store information, short of truly quantum atomic or molecular spins^{6}. For instance, in the skyrmionic racetrack memory shown in Fig. 6c, the magnetic information is stored in skyrmions instead of magnetic domains separated by domain walls^{10}. Antiferromagnetic skyrmions can be driven by the spin–orbit torque at lower current densities, and thanks to their stability have advantages over domain walls, especially in the curved parts of the race track.
To conclude, beyond providing an interesting new context in which to identify and understand the physical consequences of topological properties of momentumspace bands or realspace textures, topological antiferromagnetic spintronics has the tantalizing possibility of converting important fundamental advances into truly valuable new applications of quantum materials.
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Change history
30 May 2018
In the version of this Review Article originally published, three of the citations corresponded to the wrong references. Ref. 16 should have corresponded to Nature 533, 513–516 (2016), ref. 17 to Nat. Mater. 16, 94–100 (2016), and ref. 18 to Phys. Rev. Appl. 6, 054001 (2016).
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Acknowledgements
L.Š. acknowledges support from the Grant Agency of Charles University, no. 280815, and EU FET Open RIA Grant 766566. We acknowledge support from the Ministry of Education of the Czech Republic Grants LM2015087 and LNSMLNSpin, and the Grant Agency of the Czech Republic Grant 1437427G. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the programme ‘Projects of Large Research, Development, and Innovations Infrastructures’ (CESNET LM2015042) is greatly appreciated. Y.M. acknowledges funding from the German Research Foundation (Deutsche Forschungsgemeinschaft, Grant MO 1731/51). B.Y. acknowledges the support of the Ruth and Herman Albert Scholars Program for New Scientists at Weizmann Institute of Science, Israel. A.H.M. was supported by SHINES, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award SC0012670, Army Research Office (ARO) under Contract No. W911NF1510561:P00001, and by Welch Foundation Grant TBF1473.
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Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
 Libor Šmejkal
Institut fur Physik, Johannes Gutenberg Universitat Mainz, Mainz, Germany
 Libor Šmejkal
Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic
 Libor Šmejkal
Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, Jülich, Germany
 Yuriy Mokrousov
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel
 Binghai Yan
Department of Physics, University of Texas at Austin, Austin, TX, USA
 Allan H. MacDonald
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