Mixed Weyl semimetals and low-dissipation magnetization control in insulators by spin orbit torques

Transcription

1 DOI:.38/s OPEN Mixed Weyl semimetls nd low-dissiption mgnetiztion control in insultors by spin orbit torques Jn-Philipp Hnke, Frnk Freimuth, Chengwng Niu, Stefn Blügel & Yuriy Mokrousov Relible nd energy-efficient mgnetiztion switching by electriclly induced spin orbit torques is of crucil technologicl relevnce for spintronic devices implementing memory nd logic functionlity. Here we predict tht the strength of spin orbit torques nd the Dzyloshinskii-Moriy interction in topologiclly nontrivil mgnetic insultors cn exceed by fr tht of conventionl metls. In nlogy to the quntum nomlous Hll effect, we explin this extrordinry response in the bsence of longitudinl currents s hllmrk of monopoles in the electronic structure of systems tht re interpreted most nturlly within the frmework of mixed Weyl semimetls. We thereby lunch the effect of spin orbit torque into the field of topology nd revel its crucil role in mediting the topologicl phse trnsitions rising from the complex interply between mgnetiztion direction nd momentum-spce topology. The presented concepts my be exploited to understnd nd utilize mgnetoelectric coupling phenomen in insulting ferromgnets nd ntiferromgnets. Peter Grünberg Institut nd Institute for Advnced Simultion, Forschungszentrum Jülich nd JARA, Jülich, Germny. Correspondence nd requests for mterils should be ddressed to J.-P.H. (emil: j.hnke@fz-juelich.de) NATURE COMMUNICATIONS 8: 479 DOI:.38/s

2 NATURE COMMUNICATIONS DOI:.38/s Progress in control nd mnipultion of the mgnetiztion in mgnetic mterils is pivotl for the innovtive design of future nonvoltile, high-speed, low-power, nd sclble spintronic devices. The effect of spin orbit torque (SOT) provides n efficient mens of mgnetiztion control by electricl currents in systems tht combine broken sptil inversion symmetry nd spin orbit interction 5. These current-induced torques re believed to ply key role in the prcticl implementtion of vrious spintronics concepts, since they were demonstrted to medite the switching of single ferromgnetic lyers 6,7 nd ntiferromgnets 8 vi the exchnge of spin ngulr momentum between the crystl lttice nd the (stggered) colliner mgnetiztion. Among the two different contributions to SOTs, the soclled ntidmping torques re of utter importnce owing to the robustness of their properties with respect to detils of disorder 5. Only recently, the reserch on electriclly controlled mgnetiztion switching strted to rech out to topologicl condensed mtter for exmple, very efficient mgnetiztion switching hs been chieved ltely in metllic systems incorporting topologicl insultors 9. Although in ltter cses strong torque cn be generted, the resulting electric-field response does not rely on the globl topologicl properties of these trivil systems. The discovery of quntized version of the nomlous Hll effect in mgnetic insultors with nontrivil topology in momentum spce 2 led to revolution in forging new spintronic device concepts tht utilize topology. On the other hnd, moving the field of mgnetiztion control by SOTs into the relm of topologicl spintronics would open bright venues in exploiting universl rguments of topology for designing mgnetoelectric coupling phenomen in mgnetic insultors. With this work, we firmly put the phenomenon of SOT on the topologicl ground. Employing theoreticl techniques we investigte the origin nd size of ntidmping SOTs nd Dzyloshinskii-Moriy interction (DMI) in prototypes of topologiclly nontrivil mgnetic insultors, demonstrte tht complex topologicl properties hve direct strong impct on the emergence nd mgnitude of SOT nd DMI in vrious clsses of mgnetic insultors, nd formulte intriguing perspectives for the electric-field control of mgnetiztion in the bsence of longitudinl chrge currents. Results Mixed Weyl semimetls nd SOT. In clen smple, the ntidmping SOT T cting on the mgnetiztion in liner response to the electric field E is medited by the so-clled torknce tensor τ, i.e., T = τe 3 (see Fig., b). The Berry phse nture of the ntidmping SOT mnifests in the fct tht the tensor elements τ ij re proportionl to the mixed Berry curvture Ω^mk ij ¼ ^e i 2Im P occ n ^m u kn j kj u kn of ll occupied sttes 3 5 which incorportes derivtives of lttice-periodic wve functions u kn with respect to both crystl momentum k nd mgnetiztion direction ^m. Here, ^e i denotes the ith Crtesin unit vector. Intimtely relted to the ntidmping SOT is the DMI 6,7 crucil for the emergence of chirl domin wlls nd chirl skyrmions 8 2 which cn be quntified by the so-clled spirliztion tensor D ˆm b z x y T = E z x y T ˆm θ = E d.2 9 c Energy (t ) Energy e.2 45 Mgn. direction Momentum k y Fig. Emergence of mixed Weyl points. The mgnetiztion ^m of topologiclly nontrivil insultor is subject to the ntidmping torque T if n electric field E is pplied. b The resulting reorienttion of the mgnetiztion by θ cn trigger topologicl phse trnsition to the trivil insultor. c Schemtic evolution of two energy bnds in the complex phse spce of crystl momentum nd mgnetiztion direction, where the colors of the bnds indicte different.ifk y is tuned, the electronic structure displys monopole, which is correlted with chnge in the mixed Chern number Z. Such crossing points re observed in d the model of mgneticlly doped grphene with hopping t, nd e the functionlizedbismuth film, where colors indicte the mgnetiztion direction ^m = (sin θ,, cos θ). The shown monopoles rise t θ = 9 nd k ¼ internl units for (e) :29 2π x ; :4 2π y for (d), nd θ = 43 nd k = (.48,.9) in 2 NATURE COMMUNICATIONS 8: 479 DOI:.38/s

3 NATURE COMMUNICATIONS DOI:.38/s b Energy (t ) t so, λ nl.4.4 Γ 2 Γ..2 k y (2π / y ) t λ c σ xy (e 2 /h) d τ yx (e) e D yx (mt / uc) Energy (t ) reflecting the chnge of the free energy F due to chirl perturbtions j^m ccording to F ¼ P ij D ij^e i ^m 3 j^m. Optimizing the efficiency of mgnetiztion switching in spintronic devices by current-induced SOTs relies crucilly on the knowledge of the microscopic origin of most prominent contributions to the electric-field response. To promote the understnding, it is rewrding to drw n nlogy between the mk ntidmping SOT s given by Ω ^ ij nd the intrinsic nomlous Hll effect s determined by the Berry curvture Ω kk ij ¼ 2Im P occ n ki u kn j kj u 22 kn. Both Ω kk nd Ω mk ^ re components of generl curvture tensor Ω in the composite (k, ^m) phse spce combining crystl momentum nd mgnetiztion direction 23,24. Bnd crossings, lso referred to s mgnetic monopoles in k-spce, re known 25 to ct s importnt sources or sinks of Ω kk. When trnsferring this concept to currentinduced torques, crossing points in the composite phse spce cn be nticipted to give rise to lrge mixed Berry curvture Ω^mk, which in turn yields the dominnt microscopic contribution to torknce nd spirliztion. Thus, mterils providing such monopoles close to the Fermi energy cn be expected to exhibit notbly strong SOTs nd DMI. In the field of topologicl condensed mtter 26,27 the recent dvnces in the reliztion of quntum nomlous Hll or Chern insultors hve been striking,2. These mgnetic mterils re chrcterized by quntized vlue of the nomlous Hll conductivity nd n integer nonzero vlue of the Chern number in k-spce, C¼=ð2πÞ R Ω kk xy ddk y. On the other hnd, topologicl semimetls hve recently ttrcted gret ttention due to their exceptionl properties stemming from monopoles in momentum spce. Among these ltter systems, mgnetic Weyl semimetls host gpless low-energy excittions with liner dispersion in the vicinity of nondegenerte bnd crossings t 2 3 Fig. 2 Model of mgneticlly doped grphene. Sketch of the tight-binding model. b Bnd structure with out-of-plne mgnetiztion nd t so =.3 t, λ =. t, nd λ nl =.4 t. In ddition, we show the vlence bnd mximum (red circles) nd conduction bnd minimum (blue squres) for given vlues of k y, where the depicted gp closing occurs for θ = 27 nd k ¼ :7 2π 2π ; :9. The bold integers denote the mixed Chern number x y Z in the insulting regions, nd y = 3/2. c e Energy dependence of the nomlous Hll conductivity σ xy ¼Ce 2 =h ¼ e 2 =ð2πhþ R Ω kk xy ddk y, the torknce τ yx, nd the spirliztion D yx, respectively, for n out-of-plne mgnetiztion. Insets show the corresponding momentum-spce distributions summed over ll occupied sttes in the vicinity of the -point generic k-points 28 3, which re sources of Ω kk. Also referred to s Weyl fermions, these qusiprticles re conventionlly described by the Hmiltonin H w ¼ P i v ik i σ i, where σ = (σ x, σ y, σ z ) is the vector of Puli mtrices. Besides the on-going intensive efforts in discovering new type-i nd type-ii Weyl semimetls 29,3,32, scrutinizing the stbility nd symmetry protection of the Weyl points nd uncovering exotic trnsport properties of the Weyl phse re hot topics of ever-growing reserch interest 33. Here, we introduce the concept of mixed Weyl semimetl by formlly replcing one of the momentum vribles with the mgnetiztion direction (specified by n ngle θ) in the usul Weyl Hmiltonin. This results in the low-energy description of the system in the combined phse spce of k = (, k y ) nd θ by H mw = v x σ x + v y k y σ y + v θ θσ z, where θ is the ngle tht the mgnetiztion ^m = (sin θ,, cos θ) mkes with the z-xis. By introducing the concept of mixed Weyl semimetl, we endevour to generlize the notion of Weyl point to the cse of entngled crystl momentum nd mgnetiztion direction vribles. A distinct property of mixed Weyl semimetl is tht while it is n insultor for generl vlue of θ, it exhibits bnd crossings t the Fermi energy for certin distinct vlues of the mgnetiztion direction, determined by the symmetry of the system. In other words, s illustrted in Fig. c, mixed Weyl semimetls s described by the Hmiltonin H mw feture monopoles in the composite phse spce of k nd θ. These monopoles serve s the sources of the generl curvture tensor Ω. In nlogy to conventionl Weyl semimetls 28, we cn chrcterize the topology nd detect mgnetic monopoles by monitoring the flux of the mixed Berry curvture through plnes of constnt k y s given by the integer mixed Chern number Z¼ =ð2πþ R mk Ω ^ yx dθd (Fig. c). Tking the generl viewpoint of mgnetiztion dynmics in topologiclly nontrivil mterils, here we demontrte the existence of mixed Weyl semimetls nd focus on the implictions of the corresponding monopoles for mgnetoelectric properties, leving the nlysis of symmetry requirements which gurntee their emergence for future studies. In the following, we show tht significnt electric-field response ner monopoles in mixed Weyl semimetls is invluble in pving the rod towrds low-dissiption mgnetiztion control by SOTs 34. Mgneticlly doped grphene. We begin with tight-binding model of mgneticlly doped grphene 35 : H ¼ t P c y iα c P jα þ it so ^e z ðσ d ij Þc y iα c jβ hijiα hijiαβ þλ P ð^m σþc y iα c P jβ λ nl ð^m σþc y iα c jβ; iαβ hijiαβ which is sketched in Fig. 2. Here, c y iα (c iα) denotes the cretion (nnihiltion) of n electron with spin α t site i, restricts the sums to nerest neighbors, nd the unit vector d ij points from j to i. Besides the usul hopping with mplitude t, the first line in Eq. contins the Rshb spin orbit coupling of strength t so originting in the surfce potentil grdient of the substrte. The remining terms in Eq. re the exchnge energy due to the locl (λ) nd nonlocl (λ nl ) exchnge interction between spin nd mgnetiztion. Depending on ^m, the nonlocl exchnge describes hopping process during which the spin cn flip. The Methods section provides further detils on the tight-binding model nd its numericl solution. First, by monitoring the evolution of the mixed Chern number Z we demonstrte tht the bove model hosts mixed Weyl semimetl stte. Indeed, s shown in Fig. 2b, the topologicl index Z chnges from 2 to t certin vlue of k y, indicting thus ðþ NATURE COMMUNICATIONS 8: 479 DOI:.38/s

4 NATURE COMMUNICATIONS DOI:.38/s Bi H c θ k y = kk Ω xy e σ xy (e 2 /h) 2 b.2 Γ M Γ.2.5 k y (2π/) d θ = k y = = = = ˆ Ω yx mk = = f τ yx (e) g D yx (mev/uc) Fig. 3 Mgnetoelectric properties of mixed Weyl semimetl. Crystl structure of the semi-hydrogented Bi() bilyer. b First-principles bnd structure for n out-of-plne mgnetiztion, where the region of the topologiclly complex bnd gp nd the trivil one bove re highlighted. In ddition, the evolution of vlence bnd mximum (red circles) nd conduction bnd minimum (blue squres) with k y is shown. Bold integers denote the mixed Chern number Z, nd is the in-plne lttice constnt. c, d Monopole-like field of momentum spce nd mixed Berry curvtures ner one of the mixed Weyl points. Arrows indicte the direction of the curvturefield ð Ω^mk yy ; Ω^mk yx ; Ωky Þ, nd logrithmic color scle is used to disply two of its components, where drk red (drk blue) denotes lrge positive (negtive) vlues. e, g Energy dependence of σ xy, τ yx, nd D yx for mgnetiztion perpendiculr to the film plne. Insets show the microscopic distributions in momentum spce ner nd the presence of bnd crossing in composite phse spce tht crries topologicl chrge of +2. One of these monopoles ppers ner the -point off ny high-symmetry line if the mgnetiztion is oriented in-plne long the x-direction (see Fig. d). The emergence of the quntum nomlous Hll effect 35 (Fig. 2c), over wide rnge of mgnetiztion directions cn be understood s direct consequence of the mgnetic monopoles cting s sources of the curvture Ω kk. Correspondingly, for ^m out of the plne, the system is quntum nomlous Hll insultor. Moreover, lrge vlues of the mixed curvture Ω mk ^ in the vicinity of the monopole re visible in the momentum-spce distributions of torknce nd spirliztion in the insets of Fig. 2d, e, respectively. For n out-of-plne mgnetiztion, the primry microscopic contribution to the effects rises from n voided crossing long Γ residue of the Weyl point in (k, θ)-spce. Since the expression for the mixed Berry curvture relies only on the derivtive of the wvefunction with respect to one of the components of the Bloch vector, the symmetry between nd k y in the distributions of torknce nd spirliztion is broken nturlly (see Methods). As consequence of the monopole-driven momentum-spce distribution, the energy dependence of the torknce τ yx (Fig. 2d) displys decent mgnitude of. e in the insulting region (with being the intertomic distnce), nd stys constnt throughout the bnd gp. In contrst to the Chern numbers C nd Z, the torknce τ yx is, however, not gurnteed to be quntized to robust vlue, i.e., the height of the torknce plteu in Fig. 2d is sensitive to fine detils of the electronic structure such s mgnetiztion direction nd model prmeters. Becuse of their intimte reltion in the Berry phse theory 3,36,37, the plteu in torknce implies liner behvior of the spirliztion D yx within the gp, chnging from 8 mt/uc to 6mt/uc s shown in Fig. 2e, where uc refers to the in-plne unit cell contining two toms. To provide relistic mnifesttion of the model considertions bove, we study from b initio systems of grphene decorted by trnsition-metl dtoms (Fig. 4). These systems, which exhibit complex spin orbit medited hybridiztion of grphene p sttes with d sttes of the trnsition metl, hve by now become one of the prototypicl mteril clsses for reliztion of the quntum nomlous Hll effect Detils on the first-principles clcultions re provided in the Methods section. In the Chern insultor phse of these mterils with mgnetiztion perpendiculr to the grphene plne, depending on the trnsition-metl dtom, both torknce nd spirliztion cn rech colossl mgnitudes tht originte from mixed Weyl points. In the cse of W in 4 4-geometry on grphene, for exmple, the torknce mounts to very lrge vlue of τ yx = 2.9 e (with being Bohr s rdius), nd the spirliztion D yx rnges from 5 mev /uc to 6 mev /uc (Fig. 4b e), surpssing thoroughly the vlues obtined in metllic mgnetic heterostructures 5,3 nd non-centrosymmetric bulk mgnets 2. Since the detils of the electronic structure cn influence the vlue of the torknce in the gp, upon replcing W with other trnsition metls, the mgnitude of SOT nd DMI cn be tilored in the gpped regions of corresponding mterils ccording to our clcultions. Functionlized bismuth film. Aiming t reveling pronounced mgnetoelectric coupling effects in mgnetic insultors with lrger bnd gps s compred to the bove exmples, we turn to semi-hydrogented Bi() bilyer (Fig. 3), which is prominent exmple of functionlized insultors relizing nontrivil topologicl phses 42. As we show, semi-hydrogented Bi() bilyer is mixed Weyl semimetl. For n out-of-plne mgnetiztion, the system is vlley-polrized quntum nomlous Hll insultor 43 with mgnetic moment of. μ B per unit cell, nd it exhibits lrge bnd gp of.8 ev t the Fermi energy s well s distinct symmetry between the vlleys nd (Fig. 3b). Anlyzing the evolution of the mixed Chern number Z s function of k y in Fig. 3b, we detect two mgnetic monopoles of 4 NATURE COMMUNICATIONS 8: 479 DOI:.38/s

5 NATURE COMMUNICATIONS DOI:.38/s c σ xy (e 2 /h) 2 9 f τ yx (e ).2 ev. ev.3 ev..4 ev g C W d.2 G Bi b.5.5 Γ M Γ τ yx (e ) e D yx (mev /uc) h θ i. ev.8 ev M Γ M Γ M Fig. 4 Monopole-driven spin orbit torques in mixed Weyl semimetls. Crystl structure of grphene decorted by W dtoms in 4 4 geometry. b Firstprinciples bnd structure for n out-of-plne mgnetiztion. The topologiclly nontrivil gp round the Fermi level nd the trivil gp bove re highlighted. c e Energy dependence of nomlous Hll conductivity σ xy, torknce τ yx, nd spirliztion D yx, respectively. Insets show the microscopic distributions in momentum spce ner the -point. f Energy dependence of the torknce τ yx in GBi film upon pplying n exchngefield B = B (sin θ,, cos θ) perpendiculr to thefilm plne, i.e., θ =. Numbers denote the vlue of B. g Crystl structure of the system. h Evolution of vlence bnd mximum (squres) nd conduction bnd minimum (tringles) with θ for B =. ev (dshed blck) nd B =.8 ev (solid red). i Bnd structures for θ = nd two different vlues of B, where colors encode the spin polriztion perpendiculr to the film opposite chrge tht emerge t the trnsition points between the topologiclly distinct phses with Z¼ nd Z¼. Alterntively, these crossing points nd the monopole chrges in the composite phse spce could be identified by monitoring the vrition of the momentum-spce Chern number C with mgnetiztion direction. These monopoles occur t generic points ner the vlley for θ = 43 (see Fig. e) nd in the vicinity of the -point for θ = 37, respectively. The presence of such mixed Weyl points in the electronic structure drsticlly modifies the behvior of the generl curvture Ω in their vicinity, s visible from the three-dimensionl representtion of Ω displyed in Fig. 3c, d. Reveling chrcteristic sign chnges when pssing through monopoles in composite phse spce, the singulr behvior of the Berry curvture underlines the role of the mixed Weyl points s sources or sinks of Ω. For n out-of-plne mgnetiztion, the complex nture of the electronic structure in momentum spce mnifests in the quntiztion of C to + (Fig. 3e), which is primrily due to the pronounced positive contributions ner, where the bnds come closest to ech other. Clcultions of the energy dependence of the torknce nd spirliztion in the system, shown in Fig. 3f, g, revel the extrordinry mgnitudes of these phenomen of the order of. e for τ yx nd 5 mev /uc for D yx, exceeding by fr the typicl mgnitudes of these effects in mgnetic metllic mterils 5,3,2. At this point, we would like to comment on the role of the mgnetic nisotropy energy for the effects tht we study. Our clcultions show tht the mgnetic stte of the considered systems is stbilized by nisotropy energies on the order of mev, which is comprble to the vlues obtined in metllic heterostructures such s Co/Pt. As consequence of the mgnetic nisotropy energy, the mgnetiztion is subject to n dditionl torque if ^m is not ligned with the esy xis (or lies outside of the esy plne). This mgnetic nisotropy torque is qulittively very distinct from the electriclly controlled ntidmping SOT s it is not determined by the Berry curvture, nd thus not responsive to the presence of mixed Weyl points. Performing explicit clcultions, we estimte tht for the exmple of the functionlized bismuthfilm with θ = 3 the mgnitude of the ntidmping SOT exceeds the mgnetic nisotropy torque if the pplied inplne electric field strength is lrger thn reltively smll vlue of 5 mv/å. The exct reltion between the mgnetic nisotropy torque nd the ntidmping SOT cn thus be influenced either by tuning the mgnitude of the pplied in-plne electric field, or by tuning the strength of the mgnetic nisotropy brrier vi the ppliction of n out-of-plne electric field 39. Proof of monopole-driven SOT enhncement. An importnt question to sk is whether the colossl mgnitude of the SOT in the insultors considered bove cn be unmbiguously identified with the mixed Weyl semimetllic stte. In the following, we nswer this question by explicitly demonstrting the utter importnce of the emergent mixed monopoles for driving pronounced mgnetoelectric response. First, by removing the mixed Weyl points from the electronic structure of the model () vi, e.g., including n intrinsic spin orbit coupling term, we confirm tht the electric-field response is strongly suppressed, which promotes the monopoles s unique origin of lrge SOT nd DMI. Second, to verify this sttement from the first-principles clcultions, we nlyze the electric-field response throughout the topologiclly trivil gps bove the Fermi level tht re highlighted in Figs. 3b nd 4b. Since these gps do not exhibit the mixed Weyl points, we obtin gretly diminished mgnitude of the torknce τ yx within these energy regions s pprent from Figs. 3f nd 4d. Finlly, we clerly demonstrte the key role of these specil points by studying n illustrtive exmple: thin film of GBi with tringulr lttice structure (Fig. 4g). The initil system is nonmgnetic trivil insultor, on top of which we rtificilly pply n exchngefield B = B (sin θ,, cos θ), with the purpose of NATURE COMMUNICATIONS 8: 479 DOI:.38/s

6 NATURE COMMUNICATIONS DOI:.38/s triggering topologicl phse trnsition s function of the exchnge field strength, see Supplementry Note. When tuning the exchnge field strength B we crefully monitor the evolution of the system from trivil mgnetic insultor for B.2 ev to mixed Weyl semimetl s indicted by the emergence of mgnetic monopoles in the electronic structure. The ltter phse is ccompnied by the quntum nomlous Hll effect prominent for finite rnge of directions θ, for instnce, if B is perpendiculr to thefilm plne (Fig. 4h, i). Compring in Fig. 4f the electric-field response for these two distinct phses, we uniquely identify drstic chnges in sign nd mgnitude of the torknce τ yx with the trnsition from the trivil insultor to the mixed Weyl semimetl hosting monopoles ner the Γ-point. This proves the crucil relevnce of emergent monopoles in driving mgnetoelectric coupling effects in topologiclly nontrivil mgnetic insultors. Discussion Remrkbly, the mgnetiztion switching vi ntidmping torques in mixed Weyl semimetls cn be utilized to induce topologicl phse trnsitions from Chern insultor to trivil mgnetic insultor medited by the complex interply between mgnetiztion direction nd momentum-spce topology in these systems s illustrted in Fig., b. In the cse of the functionlized bismuth film, for instnce, the mteril is trivil mgnetic insultor with bnd gp of.25 ev if the mgnetiztion is oriented prllel to the film plne. Nevertheless, the resulting ntidmping torknce in this trivil stte is still very lrge, nd the DMI exhibits strong vrition within the gp, see Supplementry Note 2 nd Supplementry Figs. nd 2. We therefore motivte experimentl serch nd reliztion of lrge mgnetoelectric response nd topologicl phse trnsitions in quntum nomlous Hll systems fbricted to dte 2, Overll, mixed Weyl semimetls tht combine exceptionl electric-field response with lrge bnd gp (such s, e.g., functionlized bismuthfilms) ly out extremely promising vists in room-temperture pplictions of mgnetoelectric coupling phenomen for lowdissiption mgnetiztion control subject which is currently under extensive scrutiny (see, e.g., refs. 34, 47, 48 ). In contrst to the ntidmping SOT in mgnetic metllic bilyers (such s Co/ Pt) for which lrge spin orbit interction in the nonmgnetic substrte is necessry for generting lrge spin Hll effect nd lrge vlues of SOT 4, the mgnitude of the SOT in insulting phses of mixed Weyl semimetl is driven by the presence of the mixed monopole rther thn the spin orbit strength itself. This opens perspectives in exploiting strong mgnetoelectric response of wek spin orbit mterils. In the exmples tht we considered here, the nontrivil topology of mixed Weyl semimetls leds to DMI chnges over wide rnge of vlues throughout the bulk bnd gp, implying tht proper electronic structure engineering enbles us to tilor both strength nd sign of the DMI in given system, for instnce, by doping or pplying strin. Such verstility could be prticulrly vluble for the stbiliztion of chirl mgnetic structures such s skyrmions in insulting ferromgnets. In the ltter cse, very lrge vlues of the ntidmping SOT rising in these systems would open exciting perspectives in mnipultion nd dynmicl properties of chirl objects ssocited with miniml energy consumption by mgnetoelectric coupling effects. Generlly, we would like to remrk tht mgnetic monopoles in the composite phse spce, which we discuss here, do not only govern the electric-field response in insulting mgnets but re lso relevnt in metls, where they pper on the bckground of metllic bnds. Ultimtely, in nlogy to the (nonquntized) nomlous Hll effect in metls, this mkes the nlysis of SOT nd DMI in metllic systems very complex owing to competing contributions to these effects from vrious bnds present t the Fermi energy. In ddition, the electric-field strength in metls is typiclly much smller, limiting thus the rechble mgnitude of response phenomen s compred to insultors. At the end, we revel the relevnce of the physics discussed here for ntiferromgnets (AFMs) tht stisfy the combined symmetry of time reversl nd sptil inversion. SOTs in such AFMs re intimtely linked with the physics of Dirc fermions, which re doubly degenerte elementry excittions with liner dispersion 49,5. In these systems, the relible switching of the stggered mgnetiztion by mens of current-induced torques hs been demonstrted very recently 8. In nlogy to the concept of mixed Weyl semimetls presented here, we expect tht the notion of mixed Dirc semimetls in combined phse spce of crystl momentum nd direction of the stggered mgnetiztion vector will prove fruitful in understnding the microscopic origin of SOTs in insulting AFMs. Following the very sme interprettion tht we formulted here for ferromgnets, monopoles in the electronic structure of AFMs cn be nticipted to constitute prominent sources or sinks of the corresponding generl non- Abelin Berry curvture, whose mixed bnd digonl components correspond to the sublttice-dependent ntidmping SOT, in nlogy to the spin Berry curvture for quntum spin Hll insultors nd Dirc semimetls Correspondingly, exploiting the principles of electronic structure engineering for topologicl properties depending on the stggered mgnetiztion could result in n dvnced understnding nd utiliztion of pronounced mgnetoelectric response in insulting AFMs. Methods Berry phse expressions for torknce nd spirliztion. In order to chrcterize the ntidmping SOTs, we evlute within liner response the torknce 3 τ ij ¼ 2e i Xocc ^m Im ^m u kn j kj u kn ; ð2þ N k^e kn where N k is the number of k-points, nd e > denotes the elementry positive chrge. Similrly, the spirliztion 3 is obtined s D ij ¼ ^e i N k V Xocc ^m Im ^m u kn jh kn j kj u kn ; ð3þ kn where h kn = H k + ε kn 2ε F, H k is the lttice-periodic Hmiltonin with eigenenergies ε kn, ε F is the Fermi level, nd V is the unit cell volume. Tight-binding clcultions. To rrive t the model Hmiltonin (), the model in ref. 35 hs been generlized to ccount for rbitrry mgnetiztion directions ^m nd the nonlocl exchnge interction. We obtined 4 4 mtrix representtion of the resulting Hmiltonin on the biprtite lttice of grphene by introducing four orthonorml bsis sttes Nα tht describe electrons with spin α = {, } on the sublttice N = {A, B}. Using Fourier trnsformtions, we trnsformed this mtrix to representtion H(k) in momentum spce, which ws subsequently digonlized t every k-point to ccess the electronic nd topologicl properties. The model prmeters t so =.3 t, λ =. t, nd λ nl =.4 t were employed in this work. We chose the mgnetiztion direction s ^m = (sin θ,, cos θ) for direct comprison between the model nd the first-principles clcultions. First-principles electronic structure clcultions. Using the full-potentil linerized ugmented plne-wve code FLEUR (see we performed selfconsistent density functionl theory clcultions of the electronic structure of () grphene decorted with W dtoms in 4 4 geometry, nd (2) semihydrogented Bi() bilyer. The structurl prmeters of refs. 39, 43 were ssumed in the respective cses. While we used here the PBE (Perdew Burke- Ernzerhof) exchnge nd correltion functionl, other choices led to the sme mixed Weyl points but t slightly different positions. The effect of spin orbit coupling ws treted within the perturbtive second-vrition scheme. Strting from the converged chrge density, the ohn Shm equtions were solved on n equidistnt mesh of 8 8 k-points (6 6 in cse ()) for 8 different mgnetiztion directions ^m = (sin θ,, cos θ), where the ngle θ covers the unit circle once. Bsed on the resulting wve-function informtion in the composite phse spce, we constructed single set of higher-dimensionl Wnnier functions 54 (HDWFs) for ech of the systems by employing our extension of the wnnier9 code 55. In cse (), we generted 274 HDWFs from 36 bnds with the 6 NATURE COMMUNICATIONS 8: 479 DOI:.38/s

7 NATURE COMMUNICATIONS DOI:.38/s frozen window up to 4 ev bove the Fermi level, nd in the cse (2), we extrcted from 28 bnds 4 HDWFs for frozen window tht extends to 2 ev bove the Fermi energy. We used the Wnnier interpoltion 56,57 tht we generlized to tret crystl momentum nd mgnetiztion direction on n equl footing 54 in order to evlute the Berry curvtures Ω kk nd Ω mk ^. Tking into ccount the bove prmetriztion of the mgnetiztion direction by θ, we were thereby ble to ccess efficiently the nomlous Hll conductivity σ ij, the torknce τ yj, nd the spirliztion D yj. Convergence of these quntities ws chieved using k-points in the Brillouin zone. We obtined the mixed Chern number Zðk y Þ¼ =ð2πþ R 2Im P occ n h θ u kn j kx u kn idθd by integrting the mixed Berry curvture on uniform mesh of 24 -vlues nd 52 ngles θ in [, 2π]. Dt vilbility. The tight-binding code nd the dt tht support thefindings of this study re vilble from the corresponding uthors on request. Received: 24 April 27 Accepted: 2 August 27 References. Chernyshov, A. et l. Evidence for reversible control of mgnetiztion in ferromgnetic mteril by mens of spin-orbit mgnetic field. Nt. Phys. 5, (29). 2. Miron, I. M. et l. Current-driven spin torque induced by the Rshb effect in ferromgnetic metl lyer. Nt. Mter. 9, (2). 3. Miron, I. M. et l. Fst current-induced domin-wll motion controlled by the Rshb effect. Nt. Mter., (2). 4. Grello,. et l. Symmetry nd mgnitude of spin-orbit torques in ferromgnetic heterostructures. Nt. Nnotech 8, (23). 5. Freimuth, F., Blügel, S. & Mokrousov, Y. Spin-orbit torques in Co/Pt () nd Mn/W () mgnetic bilyers from first principles. Phys. Rev. B 9, (24). 6. Miron, I. M. et l. Perpendiculr switching of single ferromgnetic lyer induced by in-plne current injection. Nture 476, (2). 7. Liu, L., Lee, O., Gudmundsen, T., Rlph, D. & Buhrmn, R. Current-induced switching of perpendiculrly mgnetized mgnetic lyers using spin torque from the spin Hll effect. Phys. Rev. Lett. 9, 9662 (22). 8. Wdley, P. et l. Electricl switching of n ntiferromgnet. Science 35, (26). 9. Mellnik, A. R. et l. Spin-trnsfer torque generted by topologicl insultor. Nture 5, (24).. Hldne, F. D. M. Model for quntum Hll effect without lndu levels: Condensed-mtter reliztion of the prity nomly. Phys. Rev. Lett. 6, (988).. Yu, R. et l. Quntized nomlous Hll effect in mgnetic topologicl insultors. Science 329, 6 64 (2). 2. Chng, C.-Z. et l. Experimentl observtion of the quntum nomlous Hll effect in mgnetic topologicl insultor. Science 34, 67 7 (23). 3. Freimuth, F., Blügel, S. & Mokrousov, Y. Berry phse theory of Dzyloshinskii- Moriy interction nd spin-orbit torques. J. Phys. Condens. Mtter 26, 422 (24). 4. urebyshi, H. et l. An ntidmping spin-orbit torque originting from the Berry curvture. Nt. Nnotech. 9, 2 27 (24). 5. Lee, i-seung. et l. Angulr dependence of spin-orbit spin-trnsfer torques. Phys. Rev. B 9, 444 (25). 6. Dzyloshinsky, I. A thermodynmic theory of wek ferromgnetism of ntiferromgnetics. J. Phys. Chem. Solids 4, (958). 7. Moriy, T. Anisotropic superexchnge interction nd wek ferromgnetism. Phys. Rev. 2, 9 (96). 8. Neubuer, A. et l. Topologicl Hll effect in the phse of MnSi. Phys. Rev. Lett. 2, 8662 (29). 9. nzw, N. et l. Lrge topologicl Hll effect in short-period helimgnet MnGe. Phys. Rev. Lett. 6, 5663 (2). 2. Frnz, C. et l. Rel-spce nd reciprocl-spce Berry phses in the Hll effect of Mn x Fe x Si. Phys. Rev. Lett. 2, 866 (24). 2. Gyles, J. et l. Dzyloshinskii-Moriy interction nd Hll effects in the skyrmion phse of Mn x Fe x Ge. Phys. Rev. Lett. 5, 3662 (25). 22. Ngos, N., Sinov, J., Onod, S., McDonld, A. H. & Ong, N. P. Anomlous Hll effect. Rev. Mod. Phys. 82, (2). 23. Xio, D., Shi, J. & Niu, Q. Berry phse correction to electron density of sttes in solids. Phys. Rev. Lett. 95, 3724 (25). 24. Freimuth, F., Bmler, R., Mokrousov, Y. & Rosch, A. Phse-spce Berry phses in chirl mgnets: Dzyloshinskii-Moriy interction nd the chrge of skyrmions. Phys. Rev. B 88, 2449 (23). 25. Fng, Z. et l. The nomlous Hll effect nd mgnetic monopoles in momentum spce. Science 32, (23). 26. Hsn, M. Z. & ne, C. L. Colloquium: topologicl insultors. Rev. Mod. Phys. 82, (2). 27. Qi, X.-L. & Zhng, S.-C. Topologicl insultors nd superconductors. Rev. Mod. Phys. 83, 57 (2). 28. Xu, G., Weng, H., Wng, Z., Di, X. & Fng, Z. Chern semimetl nd the quntized nomlous Hll effect in HgCr 2 Se 4. Phys. Rev. Lett. 7, 8686 (2). 29. Xu, S.-Y. et l. Discovery of Weyl fermion semimetl nd topologicl Fermi rcs. Science 349, (25). 3. Lv, B. Q. et l. Experimentl discovery of Weyl semimetl TAs. Phys. Rev. 5, 33 (25). 3. Gosálbez-Mrtnez, D., Souz, I. & Vnderbilt, D. Chirl degenercies nd Fermi-surfce Chern numbers in bcc Fe. Phys. Rev. B 92, 8538 (25). 32. Soluynov, A. A. et l. Type-II Weyl semimetls. Nture 527, (25). 33. Ji, S., Xu, S.-Y. & Hsn, M. Z. Weyl semimetls, Fermi rcs nd chirl nomlies. Nt. Mter. 5, 4 44 (26). 34. Avci, C. O. et l. Current-induced switching in mgnetic insultor. Nt. Mter. 6, (27). 35. Qio, Z. et l. Quntum nomlous Hll effect in grphene from Rshb nd exchnge effects. Phys. Rev. B 82, 644(R) (2). 36. Thonhuser, T. Theory of orbitl mgnetiztion in solids. Int. J. Mod. Phys. B 25, (2). 37. Hnke, J.-P. et l. Role of Berry phse theory for describing orbitl mgnetism: from mgnetic heterostructures to topologicl orbitl ferromgnets. Phys. Rev. B 94, 24(R) (26). 38. Ding, J., Qio, Z., Feng, W., Yo, Y. & Niu, Q. Engineering quntum nomlous/vlley Hll sttes in grphene vi metl-tom dsorption: n binitio study. Phys. Rev. B 84, (2). 39. Zhng, H., Lzo, C., Blügel, S., Heinze, S. & Mokrousov, Y. Electriclly tunble quntum nomlous Hll effect in grphene decorted by 5d trnsition-metl dtoms. Phys. Rev. Lett. 8, 5682 (22). 4. Acost, C. M., Lim, M. P., Miw, R. H., d Silv, A. J. R. & Fzzio, A. Topologicl phses in tringulr lttices of Ru dsorbed on grphene: b initio clcultions. Phys. Rev. B 89, (24). 4. Hu, J., Zhu, Z. & Wu, R. Chern hlf metls: new clss of topologicl mterils to relize the quntum nomlous Hll effect. Nno Lett. 5, (25). 42. Ren, Y., Qio, Z. & Niu, Q. Topologicl phses in two-dimensionl mterils: review. Rep. Prog. Phys. 79, 665 (26). 43. Niu, C. et l. Functionlized bismuth films: gint gp quntum spin Hll nd vlley-polrized quntum nomlous Hll sttes. Phys. Rev. B 9, 433 (25). 44. Checkelsky, J. G. et l. Trjectory of the nomlous Hll effect towrds the quntized stte in ferromgnetic topologicl insultor. Nt. Phys, (24). 45. ou, X. et l. Scle-invrint quntum nomlous Hll effect in mgnetic topologicl insultors beyond the two-dimensionl limit. Phys. Rev. Lett. 3, 372 (24). 46. Chng, C.-Z. et l. High-precision reliztion of robust quntum nomlous Hll stte in hrd ferromgnetic topologicl insultor. Nt. Mter. 4, (25). 47. Chu, Y.-H. et l. Electric-field control of locl ferromgnetism using mgnetoelectric multiferroic. Nt. Mter. 7, (28). 48. Chib, D. et l. Mgnetiztion vector mnipultion by electric fields. Nture 455, (28). 49. Tng, P., Zhou, Q., Xu, G. & Zhng, S.-C. Dirc fermions in n ntiferromgnetic semimetl. Nt. Phys. 2, 4 (26). 5. Šmejkl, L. S.,Železný, J. Z., Sinov, J. & Jungwirth, T. Electric control of dirc qusiprticles by spin-orbit torque in n ntiferromgnet. Phys. Rev. Lett. 8, 642 (27). 5. Murkmi, S. Phse trnsition between the quntum spin Hll nd insultor phses in 3d: emergence of topologicl gpless phse. New J. Phys. 9, 356 (27). 52. Murkmi, S. & ug, S.-i Universl phse digrms for the quntum spin Hll systems. Phys. Rev. B 78, 6533 (28). 53. Yng, B.-J. & Ngos, N. Clssifiction of stble three-dimensionl Dirc semimetls with nontrivil topology. Nt. Commun. 5, 4898 (24). 54. Hnke, J.-P., Freimuth, F., Blügel, S. & Mokrousov, Y. Higher-dimensionl Wnnier functions of multiprmeter Hmiltonins. Phys. Rev. B 9, 8443 (25). 55. Mostofi, A. A. et l. An updted version of wnnier9: tool for obtining mximlly-loclised Wnnier functions. Comput. Phys. Commun. 85, (24). 56. Wng, X., Ytes, J. R., Souz, I. & Vnderbilt, D. Ab initio clcultion of the nomlous Hll conductivity by Wnnier interpoltion. Phys. Rev. B 74, 958 (26). NATURE COMMUNICATIONS 8: 479 DOI:.38/s

8 NATURE COMMUNICATIONS DOI:.38/s Ytes, J. R., Wng, X., Vnderbilt, D. & Souz, I. Spectrl nd Fermi surfce properties from Wnnier interpoltion. Phys. Rev. B 75, 952 (27). Acknowledgements We grtefully cknowledge computing time on the supercomputers JUQUEEN nd JURECA t Jülich Supercomputing Center s well s t the JARA-HPC cluster of RWTH Achen, nd funding from the Germn Reserch Foundtion (Deutsche Forschungsgemeinschft) under Grnt No. MO 73/5- nd SPP 666. We further cknowledge funding from the Europen Unions Horizon 22 reserch nd innovtion progrmme under grnt greement number (FET-Open project MAGicSky). This work hs been lso supported by the Deutsche Forschungsgemeinschft (DFG) through the Collbortive Reserch Center SFB 238. Author contributions J.-P.H. uncovered the mixed Weyl points s origin of lrge mgnetoelectric coupling effects through model considertions nd first-principles clcultions. J.-P.H. nd Y.M. wrote the mnuscript. All uthors discussed the results nd reviewed the mnuscript. Additionl informtion Supplementry Informtion ccompnies this pper t doi:.38/s Competing interests: The uthors declre no competing finncil interests. Reprints nd permission informtion is vilble online t reprintsndpermissions/ Publisher's note: Springer Nture remins neutrl with regrd to jurisdictionl clims in published mps nd institutionl ffilitions. Open Access This rticle is licensed under Cretive Commons Attribution 4. Interntionl License, which permits use, shring, dpttion, distribution nd reproduction in ny medium or formt, s long s you give pproprite credit to the originl uthor(s) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde. The imges or other third prty mteril in this rticle re included in the rticle s Cretive Commons license, unless indicted otherwise in credit line to the mteril. If mteril is not included in the rticle s Cretive Commons license nd your intended use is not permitted by sttutory regultion or exceeds the permitted use, you will need to obtin permission directly from the copyright holder. To view copy of this license, visit licenses/by/4./. The Author(s) 27 8 NATURE COMMUNICATIONS 8: 479 DOI:.38/s