An Origin of Dzyaloshinskii-Moriya Interaction at Graphene-Ferromagnet Interfaces Due to the Intralayer RKKY/BR Interaction
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l An Origin of Dzyaloshinskii-Moriya Interaction at Graphene-Ferromagnet Interfaces Dueto the Intralayer RKKY/BR Interaction
Jin Yang, Jian Li, Liangzhong Lin, and Jia-Ji Zhu ∗ School of Science and Laboratory of Quantum Information Technology,Chongqing University of Posts and Telecommunications, Chongqing, 400065, China. School of Information Engineering, Zhongshan Polytechnic, Zhongshan, 528400, China
We present a theory of both the itinerant carrier-mediated RKKY interaction and the virtualexcitations-mediated Bloembergen-Rowland (BR) interaction between magnetic moments ingraphene induced by proximity effect with a ferromagnetic film. We show that the RKKY/BRinteraction consists of the Heisenberg, Ising, and Dzyaloshinskii-Moriya (DM) terms. In the case ofthe nearest distance, we estimate the DM term from the RKKY/BR interaction is about 0 .
13 meVfor the graphene/Co interface, which is consistent with the experimental result of DM interaction0 . ± .
05 meV. Our calculations indicate that the intralayer RKKY/BR interaction may be apossible physical origin of the DM interaction in the graphene-ferromagnet interface. This workprovides a new perspective to comprehend the DM interaction in graphene/ferromagnet systems.
PACS:
Introduction.—
Graphene has been the superstar for itsunique properties in condensed matter physics since 2004.Magnetism and magnetic phenomena in graphene are thekey issues in implementing spintronic devices. Sincethe intrinsic magnetism is absent in graphene, one mayinduce magnetism by some extrinsic strategies, for ex-ample, creating magnetic moments through introducingvacancies [1–3] , adding adatoms [4–6] , or doping magneticimpurities [7, 8] . However, the feasibility of these approachesis under debate, and the experimental realizations posetough challenges [9–11] . Therefore, the other extrinsic strate-gies such as edge engineering in zigzag nanoribbons [12–14] ,biased bilayer graphene [15, 16] , or magnetic proximity ef-fect borrowed from adjacent magnetic materials [17, 18] , re-ceive great attention. The magnetic proximity effect couldbring about the strong hybridization between the 2 p z or-bitals of the carbon atoms with the d states of the metallicsubstrate and the sublattice-symmetry breaking [19] . Theproximity effect may also promote the enhancement ofthe Rashba spin-orbit coupling (RSOC) [20] and anomalousHall effect in graphene-ferromagnetic films [17] . Betweenthe magnetic moments borrowed from the ferromagneticsubstrate, an indirect magnetic interaction emerges, whichis called the Ruderman-Kittel-Kasuya-Yosida (RKKY)interaction [21–23] .The RKKY interaction is an indirect magnetic in-teraction mediated by itinerant electrons or holes be-tween magnetic moments, which permits highly con-trollability thanks to the itinerant carriers. Therehave been intense researches on the RKKY interactionin graphene using various approaches like the Matsub-ara Green’s function technique [24–26] , the lattice Green’sfunction technique [27, 28] , and the exact diagonalizationapproach [29] . Some common conclusions have beenreached: i) The RKKY interaction is a usual isotropic Heisenberg-typed interaction in graphene; ii) The rangefunction decays as 1 /R with no oscillation in pristinegraphene and oscillates with 1 /R decay in doped graphene;iii) The RKKY interaction is ferromagnetic between themagnetic moments on the same sublattices and is antifer-romagnetic on the opposite sublattices. FIG. 1. The top-view of the crystal configurations of graphene-coated Ni or Co films in (a), and the (b) is the band structurefor the graphene-nickel system with λ R =50 meV, ∆=40 µ eV.The red (blue) lines represent conduction (valence) bands, and ↑ ( ↓ ) represents spin up (down). The progress of the spin-orbitronics permits highly effi-cient electrical control of chiral spin textures—skyrmions ordomain wall dynamics, and shows some exciting potentialapplications in spintronic memory and logic devices [30, 31] .These potential applications are based on interfacialDzyaloshinskii-Moriya (DM) interaction which has been ex-perimentally observed [32] . The DM interaction was firstproposed for explaining the weak ferromagnetism in ox-ide materials [33] and is related to spin-orbit coupling [34] .Typical DM interaction exists in noncentrosymmetric bulkmagnets [35, 36] and at interfaces between ferromagnets andmetals [37–40] . Yang et. al. showed that the physical originof interfacial DM interaction is the Rashba effect [41, 42] . In-deed, several kinds of research shows that there is a largeRashba spin-orbit splitting at the graphene-ferromagnet( e.g.
Ni or Co) interface [20, 43] , and the RSOC could be en-hanced to a giant Rashba effect by the intercalation [44, 45] .However, the giant Rashba effect could also make somedifference to the RKKY interaction in graphene and mayresult in an anisotropic coupling which consists of the DMinteraction [46] . Vedmedenko et. al. showed that the in-terfacial DM interaction attributes to an interlayer DMterm from the RKKY interaction (L´evy-Fert model) andthe intralayer RKKY interaction is dominant a ferromag-netic Heisenberg term [47] . However, we will show that theRKKY interaction also contains an intralayer DM termwhich contributes to the interfacial DM interaction.In this paper, we present the RKKY interaction ingraphene between magnetic moments induced by proximityeffect of a ferromagnetic metal. The ferromagnet providesboth broken time-reversal symmetry and RSOC. Due to theRSOC resulting from the graphene-ferromagnet interface,we find that there are Heisenberg, Ising, and DM terms inthe RKKY interaction rather than the usual Heisenberg-typed interaction in graphene. This DM interaction couldalso induce magnetic chirality and weak ferromagnetism.We also consider the Bloembergen-Rowland (BR) interac-tion when the Fermi energy lies across the neutral point [48] ,and show that the BR interaction mainly consists of DMterm and Heisenberg term in the nearest distance. We es-timate the DM interaction arising from the RKKY/BR in-teraction is about 0 .
13 meV for the graphene-Co interface,which is consistent with the experimental result of DM in-teraction 0 . ± .
05 meV [41] . Our results indicate thatthe DM interaction may not be induced by the Rashbaeffect directly but may be induced by the Rashba effect in-directly via the BR/RKKY interaction. We also show thatthe RKKY interaction is nearly independent of Fermi en-ergy in the nearest distance, and almost the same as theBR interaction. The Fermi energy hardly influences ourestimate of the DM interaction.
Model.—
We consider two magnetic moments located ingraphene, which are borrowed from an adjacent ferromag-netic metal (Co or Ni films). The atomic distance of fcc
Ni(111) or hcp
Co(0001) planes is almost perfectly matchedwith the length of the basis vector of graphene, as showedin Fig .1(a). The total Hamiltonian can be written as H = H + H int , where H is the Hamiltonian of itinerantelectrons in graphene with RSOC and intrinsic spin-orbitcoupling (ISOC) [49, 50] , H = ~ v F ( τ x k x + τ y k y ) + λ R ( τ x σ y − τ y σ x ) + ∆ τ z σ z (1)where the λ R and ∆ are the RSOC and the ISOC constantrespectively, the v F denotes the Fermi velocity of graphene,Pauli matrices of pseudospin τ operate on A(B) sublattices,and σ are Pauli matrices for real electron spin.The exchange interaction between the local magnetic mo- ments, S and S , induced by proximity effect of a ferro-magnet with the itinerant electron spins σ can be givenby [49, 51, 52] H int = − J ( σ · S i ) δ ( r − R i ), where J denotesthe strength of the s - d exchange interaction.In the loop approximation we find the RKKY interactionbetween two magnetic moments in the form of [51, 52] H RKKYαβ = − J π Im Z ε F −∞ d ε Tr [( σ · S ) × G αβ ( R , ε ) ( σ · S ) G βα ( − R , ε ) i (2)where α, β are the sublattice indices, A or B, and R denotesthe lattice vector between the sublattice β and α . Here G αβ ( R ,ε ) is the unperturbed Green’s function in energy-coordinate representation and ε F is Fermi energy. Tr meansa partial trace over the spin degree of freedom of itinerantDirac electrons. The Green’s functions in real space can becalculated by integrating the corresponding Green’s func-tions in momentum space over the wave vector k in thevicinity of K valley [27, 49] G αβ ( ± R ,ε ) = 1Ω BZ Z d ke ± i k · R G αβ ( k , ε ) (3)The k dependent Green’s functions, G ( k ,ε ) =( ε + i η − H ) − , corresponding to the Hamiltonian (1), canbe rewritten as a 2 × H RKKYα,β = J αβH S · S + J αβDM ( S × S ) y + J αβz S z S z + J αβy S y S y (4)The effective anisotropic spin-spin interaction betweenlocal moments includes Heisenberg, Ising, and DM interac-tions. This behavior of RKKY interaction recurs in topo-logical insulators [51] , topological semimetals [53, 54] , p -dopedtransition-metal dichalcogenides [55] etc. The three typeterms arise from the symmetry-breaking due to the RSOCand the ISOC.We list the results for two cases—the magnetic momentslocate on the same sublattices or the opposite sublattices,and one can see that the y direction Ising term and theDM term only depend on the RSOC. This means that theRashba effect plays a key role in realizing the DM interac-tion. The Ising term of the z direction results from boththe ISOC and the RSOC [56] . All the range functions J αβi of the RKKY interaction are listed in Table I. The RKKY/BR interaction.—
The well-preserved lineardispersion of graphene with the RSOC and ISOC is showedin Fig .1(b), where the splitting energy of subbands is about2 λ R . In our calculations we used the strength of s - d inter-action J about 1 eV [49] , the Fermi velocity 10 m/s [57] , andthe ISOC of graphene sheet ∆ = 40 µ eV [58] . TABLE I. The RKKY interaction range function including both of the same sublattices (AA) and the opposite sublattices (AB),where κ = π ~ v F Ω BZ , ζ ± = p ε − ∆ ± ε − ∆) λ R , J = − κ J π , the s ( s ′ ) = sgn( ε ± λ R ) respectively.AA AB J AAy = J Im R ε F −∞ d ε (cid:0) − (cid:1) J ABy = J Im R ε F −∞ d ε ( − Λ ) J AAz = J Im R ε F −∞ d ε (cid:0) − (cid:1) J ABz = J Im R ε F −∞ d ε (Λ − Λ ) J AADM = J Im R ε F −∞ d ε ( − Γ ) J ABDM = J Im R ε F −∞ d ε ( − (Λ + Λ ) J AAH = J Im R ε F −∞ d ε (cid:0) Γ − Γ + Γ (cid:1) J ABH = J Im R ε F −∞ d ε (cid:0) Λ Λ − Λ (cid:1) Γ = sζ + H (1)1 (cid:16) sRζ + ~ v F (cid:17) − s ′ ζ − H (1)1 (cid:16) s ′ Rζ − ~ v F (cid:17) Λ = sζ + H (1)1 (cid:16) sRζ + ~ v F (cid:17) + s ′ ζ − H (1)1 (cid:16) s ′ Rζ − ~ v F (cid:17) Γ = ( ε + λ R ) H (1)0 (cid:16) sRζ + ~ v F (cid:17) + ( ε − λ R ) H (1)0 (cid:16) s ′ Rζ − ~ v F (cid:17) Λ = − ( ε − ∆) H (1)0 (cid:16) sRζ + ~ v F (cid:17) + ( ε − ∆) H (1)0 (cid:16) s ′ Rζ − ~ v F (cid:17) Γ = (∆ + λ R ) H (1)0 (cid:16) sRζ + ~ v F (cid:17) + (∆ − λ R ) H (1)0 (cid:16) s ′ Rζ − ~ v F (cid:17) Λ = ( ε + ∆ + 2 λ R ) H (1)2 (cid:16) sRζ + ~ v F (cid:17) − ( ε + ∆ − λ R ) H (1)2 (cid:16) s ′ Rζ − ~ v F (cid:17) FIG. 2. All the RKKY interaction range functions of the samesublattice as a function of the distance R showed in (a) andFermi energy ε F showed in (b). (c) shows the BR interactionof the same sublattices depend on distance in the logarithmiccoordinate, and (d) shows the BR interaction of the same sub-lattices depending on RSOC in units of meV. We set ∆ = 40 µ eV for all cases, λ R = 50meV and ε F =200 meV for (a), λ R =50 meV and R =8 nm for (b), λ R = 50 meV and ε F =0 meV for(c), R =8 nm, ε F =0 meV for (d). We only need to discuss the case of the same sublatticeswhen we focus on the RKKY interaction in graphene onthe Ni(111) or Co(0001) films, because there one carbonatom of the graphene unit cell locates on top of the adja-cent Co(Ni) atom and another carbon atom locates abovethe hollow site, as shown in Fig. 1(a). Generally speaking,the asymptotic behavior of RKKY range functions is deter-mined by the dimensionality of the host materials [59] . In 2D electron gas or 2D materials, the range functions decayas 1 /R with distance R . We can see from Fig. 2(a) that allthe terms decay as 1 /R and conform to other 2D systems.To compare the different terms in the RKKY interaction,we plot all the four types of the RKKY range functionsdepending on the distance R or Fermi energy ε F in Fig.2(a) and (b) . We can see the Ising term of the z direc-tion is rather weak and relatively insensible to the increaseof the distance between moments, and this is because theIsing term J z relies mainly on the ISOC and the strengthof the ISOC is low. The other terms show decaying oscil-lations with increasing distance, as common RKKY rangefunctions do. Fig. 2(b) shows the dependence of the fourterms as functions of Fermi energy. Again the Ising term ofthe z direction shows a relatively flat curve with increasingFermi energy, and the other range functions show normallyenhanced oscillations with the increase of Fermi energy.While the Fermi energy is cross the neutral point andthe density of state drops to zero, there are no carriersin the graphene sheet. The RKKY interaction seems tovanish due to the absence of the mediated carriers. How-ever, the virtual excitations could also offer mediated car-riers, and the RKKY interaction would turn to the BRinteraction [48, 51] . The BR interaction, in general, expo-nentially decays with increasing the distance between mo-ments. We can see from Fig. 2(c) the perfect exponentialdecaying behavior of the BR interaction. The Heisenbergterm falls more rapidly than the other three terms. The in-tensity of the Heisenberg interaction will be lower than thatof the DM interaction in the distance larger than 5 . z direction will be higher than that of the DM interactionin the distance larger than 12 . The DM interaction from the intralayer RKKY/BRinteraction.—
Since the adjacent Co(Ni) atoms locate un-der the same sublattice of graphene with a distance of about0.25 nm [60, 61] , we turn our attention to the case of R = 0 . λ R = 265 meV and for Ni film λ R = 136meV [20] , we can calculate the corresponding DM interac-tion J DM = 0 .
133 meV for graphene/Co and J DM = 0 . J DM = 0 . ± .
05 meV of the graphene/Co film [41] , and the theoretical estimate is perfectly consistent withthe experimental result. This coincidence indicates that theDM interaction may not be induced by the Rashba effectdirectly but may be induced by the Rashba effect indirectlyvia the BR interaction.
FIG. 3. (a) The range functions of the BR interaction of thesame sublattices depending on RSOC in units of meV withFermi energy ε F =0 meV. (b) The range functions of the RKKYinteraction of the same sublattices depending on Fermi energywith RSOC λ R =50 meV. ∆=40 µ eV, R =0.25 nm are both for(a) and (b). Fig. 3(b) shows the RKKY interaction is nearly inde-pendent of Fermi energy. The RKKY interaction is almostthe same as the BR interaction even if the Fermi energydrastically increases. The corresponding DM terms fromthe RKKY interaction ( ε F = 200 meV) can be extractedas J DM = 0 .
134 meV for graphene/Co and J DM = 0 . J DM and the ferromagnetic Heisenberg term J H . If we could extract the J DM and J H from asymmet-ric spin-wave dispersion by measuring the highly resolvedspin-polarized electron energy loss spectra [62] , we may ver-ify our conclusion by comparing the theoretical value andthe experimental result of the J DM /J H . Conclusion.—
We have studied the long-range RKKY in-teraction (the short-range BR interaction) mediated byitinerant carriers (virtual excitations) in graphene betweenmagnetic moments induced by proximity effect with a ferro-magnetic film. Thanks to the giant RSOC of the graphene-ferromagnet interface, the RKKY/BR interaction consistsof the Heisenberg, DM, and Ising terms. Since the adja-cent atoms from a ferromagnetic substrate locate under thesame sublattice of graphene, we focus on the RKKY/BRinteraction on the same sublattice and find that the DMterm and the Heisenberg term make the main contribu-tion. While in the case of the nearest distance, we showthe RKKY interaction nearly independent of the Fermi en-ergy and is almost the same as the BR interaction. TheHeisenberg and the Ising terms are also insensible to theRSOC except for the DM term. We estimate the DM termfrom the BR/RKKY interaction is about 0 .
13 meV for thegraphene/Co interface, which is consistent with the exper-imental result of DM interaction 0 . ± .
05 meV. Thisresult indicates that the DM interaction may not be in-duced by the Rashba effect directly but may be induced bythe Rashba effect indirectly via the intralayer BR/RKKYinteraction.This work was supported by the NSFC (Grants No.11404043, 1160041160), by the key technology innova-tions project to industries of Chongqing (cstc2016zdcy-ztzx0067) and by Graduate Research Innovation Projectof Chongqing (Grants No. CYS18253). ∗ [email protected][1] Yazyev O V and Helm L 2007 Phys. Rev. B Nature Phys. Phys. Rev. Lett.
Phys. Rev. Lett.
Nat. Commun. , 2010[6] Gonz´alez-Herrero H et al 2016 Science
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Nature. Nanotech.
152 [32] Cho J et al 2015
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