Semi-realistic tight-binding model for Dzyaloshinskii-Moriya interaction

Ahmed Hajr1,2, Abdulkarim Hariri1, Guilhem Manchon1, Sumit Ghosh1, and Aurelien Manchon1,3,4∗1King Abdullah University of Science and Technology (KAUST),

Physical Science and Engineering Division (PSE), Thuwal 23955-6900, Saudi Arabia.2King Fahd University of Petroleum and Minerals (KFUPM), Physics Department, Az Zahran, Saudi Arabia.

3King Abdullah University of Science and Technology (KAUST), Computer,Electrical and Mathematical Science and Engineering Division (CEMSE), Thuwal 23955-6900, Saudi Arabia. and

4Aix-Marseille Univ, CNRS, CINaM, Marseille, France

In this work, we discuss the nature of Dzyaloshinskii-Moriya interaction (DMI) in transition metalheterostructures. We first derive the expression of DMI in the small spatial gradient limit usingKeldysh formalism. This derivation provides us with a Green’s function formula that is well adaptedto tight-binding Hamiltonians. With this tool, we first uncover the role of orbital mixing: using botha toy model and a realistic multi-orbital Hamiltonian representing transition metal heterostructures,we show that symmetry breaking enables the onset of interfacial orbital momentum that is at theorigin of the DMI. We then investigate the contribution of the different layers to the DMI and revealthat it can expand over several nonmagnetic metal layers depending on the Fermi energy, therebyrevealing the complex orbital texture of the band structure. Finally, we examine the thicknessdependence of DMI on both ferromagnetic and nonmagnetic metal thicknesses and we find thatwhereas the former remains very weak, the latter can be substantial.

I. INTRODUCTION

Magnetic textures presenting a well-defined chiralityare of major interest due to their potential applications indata storage1, brain-inspired architectures2–4, and reser-voir computing5. Homochiral spin spirals6–8, quasi-onedimensional Neel walls9–11, magnetic skyrmions12–20 inperpendicularly magnetized systems, but also meronsin planar magnetic heterostructures21,22 are currentlythe object of intense theoretical and experimentalinvestigations as they display high current-velocitycharacteristics23,24. The key mechanism underlyingthese magnetic entities is the Dzyaloshinskii-Moriyainteraction25,26 (DMI), an antisymmetric magnetic ex-change that forces neighboring magnetic moments toalign perpendicular to each other.

In the atomistic limit, where the magnetic momentsare localized and well defined, the Dzyaloshinskii-Moriya(DM) energy reads

EDM =∑ij

Dij · (Si × Sj), (1)

where Si is the direction of the magnetic moment at sitei, Dij is the DM vector and the sum runs over all thepairs i, j of the system. In this general definition, DMIis not limited to nearest neighbors and from the sym-metry viewpoint, Dij is determined by Moriya’s rules26.In the micromagnetic limit, where the magnetic order isrepresented by a continuous vector field m with smoothspatial variation, DMI is rewritten

EDM =∑α

m · (Dα × ∂αm), (2)

where ∂α = ∂/∂α is the spatial gradient along the direc-tion eα and the DM vector Dα fulfills Neumann’s sym-metry principle. As discussed in this work, one can show

that Dα possesses the same tensorial form as the current-driven damping-like torque tensor27. From a theoreti-cal standpoint, DMI is usually studied within either theatomistic or the micromagnetic limit. Whereas the atom-istic form, Eq. (1), is certainly more general, the micro-magnetic form, Eq. (2), is often sufficient to describe thebehavior of magnetic soft modes such as smooth domainwalls and skyrmions. In contrast, the atomistic formis well adapted to study magnetic texture with strong,short-range canting like in weak ferromagnets and non-collinear antiferromagnets for instance.

The physical origin of this interaction at transitionmetal interfaces has been the object of numerous numer-ical investigations using density functional theory. Themost straightforward approach consists in computing theenergy of a spin cycloid or spiral in real space and deter-mining the energy difference between states of oppositechirality. In density functional theory, such a spin spiralcan be built by constraining the direction of the mag-netic moments by applying a penalty energy on each ofthem28.Upon varying the length of the spin spiral (i.e.,varying the size of the unit cell), the various DM vectorsfor nearest neighbors, next-nearest neighbors, etc. canbe extracted using Eq. (1). This approach has been usedto compute the DM vector in ferroelectric magnets suchas MgCr2O4

29 or Cu2OSeO330 and recently extended to

transition metal interfaces31. The ”constrained moment”method has the advantage of being applicable to materi-als with large spin-orbit coupling. However, it becomescomputationally prohibitive in the long-wavelength limit(typically when the spin spiral wavelength exceeds 10atomic sites) and is therefore more appropriate to com-pute the short-range DMI of insulating magnets than thelong-range DMI of magnetic metals.

Alternatively, one can build spin spirals in thereciprocal space32 employing the generalized Blochtheorem33,34. This approach, exact in the absence of

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spin-orbit coupling, permits the modeling of spin spiralsof arbitrary wavelength. DMI is then computed to thefirst order in spin-orbit coupling35,36. This method is lim-ited to materials with weak enough spin-orbit coupling.DMI introduces an additional dispersion that is odd inthe spin spiral momentum q and the DM vector is usu-ally evaluated taking the limit q → 0. This approach hasbeen used to compute the DM vector in a wide range oftransition metal interfaces6,37–43. It is particularly welladapted to identify the emergence of chiral ground states,such as homochiral spin spirals6,38.

In the magnetic multilayers where Neel walls and roomtemperature skyrmions are observed, these chiral mag-netic textures usually display smooth spatial gradientand long exchange length (typically 10 nm or more inperpendicularly magnetized materials). In this situation,the micromagnetic form, Eq. (2), seems more adaptedto describe the onset of magnetic chirality. Within themicromagnetic limit, the DM vector can be computedby expanding the magnetic energy to the first orderin magnetic gradient, an approach recently adopted byFreimuth et al.27,44,45 and Kikuchi et al.46. Within lin-ear response theory, it can be shown that the DM vec-tor is related to the Berry curvature in the mixed spin-momentum space. In Kikuchi’s theory, DMI is expressedas EDM = (~/2)

∫Ωm · [(Js ·∇)×m]d3r, where Js is the

equilibrium spin current that interacts with the magnetictexture. Mankovsky and Ebert47 have recently computedDMI using Freimuth’s theory implemented on fully rel-ativistic Korringa-Kohn-Rostoker Green’s function tech-nique.

Irrespective of the method employed, the theoreti-cal investigations of DMI at metallic interfaces havepointed out the importance of interfacial 3d-5d orbitalhybridization42. Since the magnetism is mostly localizedon 3d orbitals whereas spin-orbit coupling is mostly car-ried by 5d orbitals, a proper balance between 3d and 5dorbitals is required to obtain large DMI, a trend con-firmed experimentally48. The role of orbital hybridiza-tion has also been indirectly probed through the depen-dence of DMI on the magnetization induced in the non-magnetic metal49,50. While it is clear that DMI scaleswith 3d-5d hybridization51, the impact of inversion sym-metry breaking on the magnitude of DMI has remaineddifficult to established experimentally. Recently, Kim etal.52 demonstrated that DMI scales with the orbital as-phericity arising from interfacial symmetry breaking, afeature confirmed by density functional theory. This as-phericity is associated with the equilibrium orbital mag-netic moment, which was also suggested to play an im-portant role in the onset of DMI53. We also recentlyproposed to tune DMI through interfacial oxidation43,an effect confirmed experimentally54.

A question that remains scarcely addressed is the local-ized or delocalized nature of DMI. For instance, consid-ering magnetic transition metal chains deposited on topof nonmagnetic substrates, Kashid et al.37 have pointedout that DMI extends far beyond the nearest neighbor

interaction. Belabbes et al.42 showed that in W/Mn,DMI arises from the contribution of the first three Wmonolayers away from the interface. Experimentally,it is observed that DMI increases upon increasing thenonmagnetic metal thickness and saturates after a fewnanometers55, a scale that seems roughly comparable tothe spin relaxation length.

In the present work, we investigate the magnitude andsymmetry of DMI in a nonmagnetic metal/ferromagnetheterostructure using a multi-orbital tight-binding modelwithin the two-center Slater-Koster parameterization.We uncover the role of orbital mixing and show that DMIcan extend over several monolayers away from the inter-face. Correspondingly, we examine the thickness depen-dence of DMI and find that it can be substantial. ThisArticle is organized as follows: In Section II, we derive anexpression for DMI to the first order in spatial gradientusing Keldysh formalism. Then, Section III presents themulti-orbital tight-binding model of the transition metalheterostructure. The results are discussed in Section IVand confronted to the oversimplified Rashba model. Fi-nally, concluding remarks are given in Section V.

II. KELDYSH FORMALISM FOR DMI

As stated in the introduction, several methods havebeen proposed to compute DMI from first principles. Tothe best of our knowledge, the most popular approachesare the generalized Bloch theorem36 and the real-spacespin spiral methods56. In the present work, we aim to de-velop a Green’s function formula that is suitable to ournumerical platform. Such a Green’s function formula hasbeen derived by Freimuth et al.27 a few years ago by com-puting the energy of the system in the presence of a spinspiral and taking the long wave length limit. Here, wederive the DMI energy by computing the non-equilibriumresponse of the system in the presence of a gradient ofmagnetization within Keldysh formalism. As discussedbelow, in the limit of weak disorder and neglecting vertexcorrections, our results boil down to the formula derivedby Freimuth et al.27.

Following Keldysh formalism57,58, the lesser Green’sfunction reads

G< = (GR ⊗ Σ<)⊗ GA, (3)

where ⊗ ≈ 1 + i~2

(←−∂ p ·

−→∂ r −

←−∂ r ·−→∂ p

)is the Moyal

product expanded to the first order in spatial gradient.The retarded (advanced) Green’s function fulfills Dyson’sequation (

ε−H0 − ΣR(A))⊗ GR(A) = 1. (4)

Here, H0 is the system’s Hamiltonian in the absence of

disorder, and the symbol←−∂ i means that the derivative

applies to the left, while−→∂ i applies to the right. Let us

3

now derive the lesser Green’s function to the first orderin spatial gradient. We obtain

G< = GRΣ<GA − ~Im[∂pG

RΣ<∂rGA]

(5)

−~Im[∂pG

R∂rΣ<GA

]+ ~Im

[∂rG

R∂pΣ<GA].

In the limit of short range impurities, the self-energies

are local, i.e., Σα = niV20

∫d3k

(2π)3 Gα, with α = A,R,<.

Therefore, ∂pΣα = 0 and the last term in Eq. (5)

vanishes. Since the system is at equilibrium57, Σ< =(ΣA − ΣR

)f(ε), where f(ε) is Fermi-Dirac distribution.

Therefore, the first term in Eq. (5) reads

GRΣ<GA = GR(

ΣA − ΣR)GAf(ε). (6)

This term must also be expanded to the first order inspatial gradient. To do so, one uses Dyson’s equation forthe retarded Green’s function, Eq. (4), and expands theMoyal product. We obtain

GR = GR0 + GR0

(ΣR − ΣR0

)GRΣ<GA (7)

− i~2

(−GR0 ∂pH0∂rG

R + GR0 ∂rH0∂pGR)

− i~2GR0 ∂rΣ

R∂pGR.

Here ΣR0 is the self-energy at the zero-th order in spa-tial gradient, and we defined the unperturbed retarded

Green’s function GR0 =(ε−H0 − ΣR0

)−1

. The first or-

der perturbation of the retarded Green’s function, GR∇ =

GR − GR0 , reads

GR∇ = GR0 ΣR∇GR0 (8)

− i~2

(GR0 ∂rH0∂pG

R0 − GR0 ∂pH0∂rG

R0

),

where we defined ΣR∇ = ΣR − ΣR0 = niV20

∫d3k

(2π)3 GR∇.

After some algebra, and making use of

ΣA0 − ΣR0 =(GR0

)−1

−(GA0

)−1

, (9)

∂pGR0 = GR0 ∂pH0G

R0 , (10)

∂rGR0 = GR0 ∂r(H0 + ΣR0 )GR0 , (11)

we obtain the final expression for the first order perturba-

tion to the lesser Green’s function, G<∇ = Im[GR∇

]f(ε),

where

GR∇ = − i~2GR0

(∂r(H0 + ΣR0 )GR0 ∂pH0 (12)

−∂pH0GR0 (H0 + ΣR0 ) + 2iΣR∇

)GR0 .

One notices that Eq. (12) involves self-consistent treat-

ment of the disorder. In other words, GR∇ depends on

ΣR∇, which shows that the above expression includes ver-tex corrections, in the same spirit as Ref. 58. Now, wecan finally express the correction to the total energy

〈H0 − µ〉 = ~∫

dε

2πiTr[(H0 − µ)G<∇

]. (13)

By using the identity H0 − µ = ε − µ − ΣR0 −(GR0

)−1

and −(GR0

)2

= ∂εGR0 , we obtain the general expression

for the DM energy

〈H0 − µ〉 = A+B + C, (14)

with

A = −~Re

∫dε

2π(ε− µ)f(ε)Tr

[(∂r(H0 + ΣR0 )GR0 ∂pH0

−∂pH0GR0 ∂r(H0 + ΣR0 ) + 2iΣR∇

)∂εG

R0

], (15)

B = −~Re

∫dε

2πf(ε)Tr

[(∂r(H0 + ΣR0 )GR0 ∂pH0

−∂pH0GR0 ∂r(H0 + ΣR0 )

)GR0 ΣR0 G

R0

], (16)

C = 2~Im

∫dε

2πf(ε)Tr

[ΣR∇G

R0

(1 + ΣR0 G

R0

)]. (17)

Let us now simplify this formula. Neglecting the contri-bution of the self-energy, denoting vj = ∂pjH0 and recog-nizing that ∂rH0 = (m×∂rm) ·T , where T = m×∂mH0

is the torque operator, we obtain

〈H0 − µ〉 =∑ij

Dijei · (m× ∂jm) (18)

Dij = ~Re

∫dε

2π(ε− µ)f(ε)×

Tr[Ti(∂εG

R0 vjG

R0 − GR0 vj∂εGR0

)]. (19)

This expression is exactly the one obtained in Ref. 27(up to a ”-” sign). This is the expression we will use inthe next section to compute the DMI coefficient.

III. TIGHT-BINDING MODEL

A. Preliminaries

Before entering into the details of the multi-orbitalmodel proposed in this work, we introduce a simple min-imal model for DMI, inspired from Ref. 37. The modelis a diatomic chain along the x-direction, whose bot-tom non-magnetic atoms possess both pz and px orbitalswhile the top magnetic atoms possess pz orbitals only.The bottom atoms possess spin-orbit coupling, while thetop atoms carry magnetism. This toy model, depictedin Fig. 1(a), represents an oversimplified nonmagneticmetal/ferromagnet heterostructure. In the ptz,pbz,pbxbasis, where pην is the ν-th orbital of chain η, the Hamil-tonian of the system reads

4

Hchain =

εtk Vzz VzxV ∗zz εzk 0V ∗zx 0 εxk

. (20)

Here pην refers to the ν-th orbital of the top (η = t) orbottom chain (η = b), Vzz = (Vσ + Vπ) cos kxa/2 andVzx = −i(Vσ − Vπ) sin kxa/2. Vπ,σ are the Slater-Kosterhopping integrals59. In addition, we turn on spin-orbitcouplingHso on the bottom chain and magnetic exchangeHex on the top chain. Explicitly,

Hso = ξ

0 0 00 0 −iσy0 iσy 0

, (21)

and

Hex = ∆

σ ·m 0 00 0 00 0 0

. (22)

Let us now use Eq. (18) to obtain an explicit expressionof the DMI energy to the first order in exchange ∆ andspin-orbit coupling ξ. By doing so, we intend to revealthe orbital mixing due to symmetry breaking that is atthe origin of DMI. We first rewrite Eq. (19) as Dij =

~∫dε2π (ε− µ)f(ε)g(ε), with

g(ε) = ReTr[vjG

R0 [GR0 , Ti]GR0

]. (23)

The inner commutator can be extended to the first orderin both spin-orbit coupling and exchange,

[GR0 , Ti] ≈ [GR00HsoGR00, Ti], (24)

where

GR00 = (ε−H0 + i0+)−1, (25)

=∑n,s

|n〉 ⊗ |s〉〈s| ⊗ 〈n|ε− εn + i0+

, (26)

and |n〉⊗ |s〉 is the eigenstate of Hchain, i.e., evaluated inthe absence of spin-orbit coupling and exchange interac-tion. After some algebra, we obtain

g(ε) = −ξReIm [〈n|vj |m〉〈m|Ti|p〉〈p|Ll|n〉〈s|σk|s′〉〈s′|σl|s〉]

(ε− εm + i0+)(ε− εp + i0+)(ε− εn)2.

(27)

Summation over n,m, p and s, s′ is assumed for short-handedness. The diagonalization of Hamiltonian (20)gives us three eigenstates. In order to make our resultas simple as possible, we assume that εzk = εxk. Then, weend up with three bands with dispersion

ε0k = εzk, (28)

ε±k =εtk + εzk

2± 1

2γk, (29)

FIG. 1. (Color online) Schematics of the two tight-bindingmodels discussed in this work. (a) Two-orbital diatomicchain: The atoms of the bottom chain (gray) possess both px

and pz orbitals and spin-orbit coupling, while the atoms ofthe top chain (blue) has only pz orbitals and supports mag-netism. (b) Multi-orbital bilayer heterostructure: The het-erostructure is composed of two bcc monoatomic thin filmswhose elements possess all five d orbitals. The bottom layer(grey) is a nonmagnetic metal, whereas the top layer (blue)is magnetic. Both layers possess spin-orbit coupling.

with γk =√

(εtk − εzk)2 + 4(|Vzz|2 + |Vzx|2), correspond-ing to the eigenstates

|0〉 = −Vzx|pbz〉+ Vzz|pbx〉, (30)

|+〉 = cosχ|ptz〉+ sinχ(Vzz|pbz〉+ V ∗zx|pbx〉

), (31)

|−〉 = − sinχ|ptz〉+ cosχ(Vzz|pbz〉+ V ∗zx|pbx〉

),

(32)

where cos 2χ = (εtk − εzk)/γk and

Vzz =Vzz√

|Vzz|2 + |Vzx|2, Vzx =

Vzz√|Vzz|2 + |Vzx|2

.

After some algebra, we obtain

Dyx = ~s∆ξ(V 2σ − V 2

π )

∫dk

∆5k

sin kxa× (33)[v+x f(ε−)(ε− − µ)− v−x f(ε+)(ε+ − µ)

],

and all the other matrix elements are zero. We retrievein this simple expression all the key features of DMI atinterfaces. It is, to the lowest order, linear in both spin-orbit coupling and magnetic exchange and proportionalto the inversion symmetry breaking through Vzx. Thispotential characterizes the admixture between pbz and pbxorbitals, mediated by ptz orbitals. This admixture enablesthe onset of an orbital momentum along y, which results

5

in the emergence Dyx. One can extend this scenario tod orbitals: admixture between two orthogonal orbitals,mediated by a symmetry breaking coupling term, can re-sult in a non-vanishing orbital momentum, as illustratedin Fig. 2. The multi-orbital tight-binding model pre-sented below intends to encompass such admixtures atinterfaces.

FIG. 2. (Color online) Schematics of the spin momentuminduced by mixing d orbitals in the presence of spin-orbitcoupling.

B. Transition metal heterostructure

We now move on to the description of the tight-bindingmodel of our transition metal heterostructure. Since thismethod has been described in Ref. 60, we summarize itsmain features below and refer the reader to Ref. 60 formore details. The structure is depicted on Fig. 1(b) andconsists of two adjacent transition metal layers with bcccrystal structure and equal lattice parameter. The modelis constituted of monolayers stacked on top of each otheralong the (001) direction. The individual Hamiltonian ofa monolayer reads

H0 = Hmono ⊗ σ0 +Hex +Hsoc. (34)

The first term is the 10×10 Hamiltonian of the mono-layer without magnetic exchange. Hmono is written inthe basis dxy,dyz,dzx,dx2−y2 ,dz2 and its matrix el-ements are written assuming two-center Slater-Kosterparameterization59. The second term is the exchangeinteraction between the itinerant spins and the mag-netic order, and the third term is the spin-orbit couplingHamiltonian written in Russel-Saunders scheme,

Hsoc = ξso

0 iσy −iσx 0 2iσz−iσy 0 iσz −i

√3σx −iσx

iσx −iσz 0 i√

3σy −iσy0 i

√3σx −i

√3σy 0 0

−2iσz iσx iσy 0 0

.

(35)Each monolayer is connected to its top first and second-

nearest neighbor through off-diagonal matrices, T1 and

T2, respectively. The Hamiltonian of one bcc layer isthen

Hlayer =

H0 T1 T2 0

T †1 H0 T1 T2. . .

T †2 T †1 H0 T1. . .

0 T †2 T †1 H0. . .

. . .. . .

. . .. . .

(36)

We adopt the parameters computed byPapaconstantopoulos61 for bulk bcc Fe and bcc W(see Ref.60 for details). With these parameters, wedetermine the Hamiltonian for the nonmagnetic andferromagnetic layers, HNM and HF. Finally, the het-erostructure is obtained by stitching two individual slabstogether, yielding the total Hamiltonian

H =

(HF T FN

T FN,† HNM

). (37)

The hopping matrix T FN is simply given by T1 and T2

adopting the parameters of Table I in Ref.62. At zerotemperature, the chemical potential equals the Fermi en-ergy, µ = EF = 14 eV. The density of state of the struc-ture can be obtained by computing − 1

π Im[GR], where

GR = (ε−H + iΓ) is the retarded Green’s function andΓ is the homogeneous broadening, as shown in Fig. 3.Our minimal multi-orbital model serves as a platform toour investigation on DMI.

FIG. 3. (Color online) The spin-resolved density of states ofFM(5)/NM(7) bilayer with EF = 14 eV. The blue shaded areacorresponds to the contribution of the nonmagnetic metal,while the red shaded area corresponds to the ferromagneticmetal contribution. The vertical dotted line indicates EF =12.6 eV.

We would like to emphasize that because magnetismarises from both spin and orbital moments, DMI also pos-sesses both orbital and spin contributions, as discussedin the case of La2CuO4 by Ref. 53. In certain systems,

6

FIG. 4. (Color online) Orbital-resolved band structure of FM(5)/NM(7) bilayer. The band structure is projected on (a) dxy,(b) dyz, (c) dzx, (d) dz2 and (e) dx2−y2 . The scale spans from 0 (dark blue) to 1 (red). Close to Fermi level (white dashed line),one can see that dxy, dz2 and dx2−y2 contributions are isotropic in momentum, while dyz and dzx contributions are anisotropic.

such as correlated oxides, the orbital contribution to theoverall magnetism is important and therefore can sub-stantially contribute to DMI53. The theory presented insection II does in principle account for the orbital contri-bution. Nonetheless, in transition metal multilayers theorbital moment is usually quenched due to the high sym-metry of the bulk metal and slightly increases close tothe interface due to symmetry lowering63. Although weacknowledge that the influence of this orbital moment de-serves further study, we neglect this orbital contributionin the present work.

Before closing this brief presentation, let us inspect theband structure of the heterostructure along the X−Γ−Y

path, projected on the various d orbitals, as displayed inFig. 4. As mentioned in the previous section, the ad-mixture of two orthogonal such orbitals favors the onsetof DMI (see Fig. 2), and it is therefore instructive toidentify the momentum-dependent orbital texture closeto Fermi level. From Fig. 4, we see that dxy, dx2−y2 anddz2 are isotropic in momentum [Figs. 4(a), (d), and (e)],dz2 being dominant [light blue in Fig. 4(d)] over dxy anddx2−y2 [blue in Figs. 4(a,e)] at Fermi level. In contrast,dyz and dzx are weaker [dark blue in Figs. 4(b,c)] and dis-play an anisotropic texture: their magnitude along alongthe Γ − Y path is different from their magnitude alongΓ− X path.

This feature promotes the inverse orbital galvaniceffect, i.e., the generation of non-equilibrium orbitalmomentum64, illustrated on Fig. 5. From Fig. 5(a,b), wesee that the Lx and Ly components are antisymmetric inmomentum k along Γ− Y and Γ− X paths, respectively.In contrast, the Lz component is isotropic and even inmomentum. In other words, L ∝ z×k. As a consequence,based on Fig. 2 and Fig. 4, we can propose the follow-ing scenario: around Fermi level, the admixture dxy-dzx(dxy-dyz) produces a non-equilibrium orbital momentumLy (Lx) along the Γ− X (Γ− Y) path. In the presence ofmagnetization gradient along x, this orbital momentumpromotes the onset of a DM vector along y, i.e., Dyx.Symmetrically, the admixture dxy-dyz promotes the on-set of a DM vector along x for a magnetization gradientalong y, i.e., Dxy.

IV. RESULTS

A. Symmetry analysis

As explicitly demonstrated in Ref. 27, the DMI coeffi-cient Dij possesses the same symmetries as the damping-like torque coefficient, tij , defined as T iDL = tijEj , Ejbeing the j-th component of the electric field. At an in-terface with the highest symmetry C∞, the damping-liketorque reads65

TDL ∝m× [m× (z×E)], (38)

or, equivalently,

tC∞ ∝

mxmy m2z +m2

y 0−m2

z −m2y −mxmy 0

mymz −mxmz 0

≡ DC∞ . (39)

By identifying the matrix elements of DC∞ to that of tC∞ ,we obtain the DMI energy

EDM = Dm · [(z×∇)×m], (40)

7

FIG. 5. (Color online) Band structure of FM(5)/NM(7) bilayer projected on the three components of the orbital momentum.Lx and Ly are antisymmetric along Y − Γ − Y and X − Γ − X, respectively, whereas Lz remains even in momentum.

−6 −4 −2 0 2

−6

−3

0

3

6

·10−3

EF − 14 (eV)

Dij

(eV

/Å)

Dxx

Dxy

Dyy

Dyx

FIG. 6. (Color online) The different coefficients of theDMI tensor as a function of the Fermi energy, computed inFM(5)/NM(7) bilayer using our multi-orbital tight-bindingapproach. In this calculation, the broadening is set to Γ = 50meV and EF = 14 eV.

as expected at such interfaces. The coefficient D can beobtained by solving Eq. (18) for the magnetic RashbaHamiltonian,

H =p2

2m+ αR(z× σ) · k + ∆σ ·m. (41)

In the constant relaxation time approximation, the re-tarded Green’s function reads GR = 1

2

∑s

1+sσ·nε−εk,s+iΓ ,

where

εk,s =~2k2

2m+ sλk, n = −(αR/λk)z× k, (42)

λk =√

∆2 + α2R − 2∆αRk sin θ sin(ϕ− ϕm). (43)

Using Eq. (18), we obtain

EDM = −~∆Re

∫dε

2π(ε− εF )f(ε)

∫d2k

4π2× (44)

Tr[vjG

R(σ · ∂jm)∂εGR − vj∂εGR(σ · ∂jm)GR

].

= αR∆∑s

∫d2k

2π2

s(εk,s − EF )f(εk,s)

(εk,s − εk,−s)2[n× (z×∇)] ·m

(45)

After solving the integral, we get the final expression

EDM =αREF

2π

m

~2m · [(z×∇)×m] , (46)

from which we can see that

Dxy = −Dyx =αREF

2π

m

~2, (47)

Dxx = Dyy = 0 (48)

In this expression, the magnetic exchange ∆ does notappear explicitly due to an accidental cancellation withthe denominator ∝ εk,s − εk,−s, a feature specific to theideal case of the free electron Rashba gas.

In systems that deviate from the ideal Rashba case,such as transition metal multilayers, the damping-liketorque display higher order behavior that feature torquecomponents of unusual symmetry66,67, beyond that ofEq. (38). Since we did observe these additional featuresin the tight-binding model presented here62, one couldreasonably expect that the associated DMI might alsodisplay higher order contributions. The DMI coefficientsfor the transition metal heterostructure are reported onFig. 6 as a function of the Fermi energy. The multi-orbital tight-binding model, in spite of its much highercomplexity than the Rashba Hamiltonian, Eq. (41), alsodisplays Dxx = Dyy = 0 and Dxy = −Dyx 6= 0, incontrast with the damping-like torque discussed above.

8

We conclude this discussion by computing the Dxy co-efficient as a function of the disorder strength Γ. Asexposed in Eqs. (33) and (46), DMI is an intrinsic mech-anism in the sense that when disorder vanishes, it con-verges to a finite value27, similarly to the damping-liketorque in this respect68. However, as discussed in SectionII, in the presence of short-range disorder the self-energy

is non-vanishing and reads ΣR(A) = niV20

∫d3k

(2π)3 GR(A).

This self-energy should be computed self-consistently inorder to account for all scattering orders. Since this pro-cedure is highly computationally demanding, it is conven-tional to reduce the self-energy to a constant broadeningΣR(A) = ∓iΓ, equivalent to the constant relaxation timeapproximation in Boltzmann transport equation. We re-port the disorder dependence of the Dxy coefficient fortwo different Fermi energies in Fig. 7. Due to numericallimitations, we could not test the limit of vanishing disor-der. Nonetheless, the DMI coefficient displays a smoothdecay as a function of disorder [∝ 1/(η2 + Γ2)], smallerthan the one expected for an extrinsic effect (∝ 1/Γ).This observation is important because it emphasizes themajor impact of disorder on the DMI coefficient, despiteits ”intrinsic” origin. In the ideally clean limit, the in-trinsic origin of the DMI reveals itself through the im-portance of the ”band anticrossing”, resulting in sharppeaks when spanning across the band structure69,70. Inreal materials though, thermally activated phonons anddefects induce a finite broadening Γ, which washes outthese singularities. From the transport calculations per-formed in Ref. 60, we estimate that this broadening isabout 20 meV, corresponding to a conductivity of ∼107

Ω−1·m−1. In other words, in realistic systems the valueof DMI is unlikely to be equal to the one obtained inthe clean limit and should be substantially smaller. Anestimation solely based on the clean limit systematicallyoverestimates DMI.

B. Orbital decomposition of DMI

As we have seen in Section III, DMI arises from theorbital momentum stemming from the admixture of theatomic orbitals induced by symmetry breaking. Whereasthe toy model of Section III was based solely on px and pzorbitals, giving rise to Ly orbital momentum, our multi-orbital model for the transition metal heterostructure in-volves all the 10 d-orbitals. To understand which or-bitals are involved in the emergence of interfacial DMI,one can contemplate the chart provided in Fig. 2. Thisfigure schematically represents the spin momentum di-rection induced by the atomic spin-orbit coupling uponthe mixing of two d atomic orbitals.

In order to stabilize a perpendicular Neel spin spiralpropagating along x, the orbital momentum must bealigned along y, which can be obtained by the followingadmixture: dzx-dz2 , dxy-dyz and dzx-dx2−y2 . Similarly,in order to induce a perpendicular Neel spin spiral alongy, the orbital momentum must be aligned along x, which

30 60 90 120 150 180 210−8

−6

−4

−2

0·10−3

Γ (meV)

Dxy

(eV

/Å)

EF = 14eV

EF = 12.6eV

FIG. 7. (Color online) The DMI coefficient Dxy as a functionof the disorder broadening in FM(5)/NM(7) bilayer, for twovalues of the Fermi energy.

can be obtained by mixing: dyz-dz2 , dzx-dxy and dyz-dx2−y2 . In Fig. 8, the DMI coefficient Dxy is calculatedby only turning on the spin-orbit coupling coefficient thatmixes two specific orbitals. In this figure, the spin-orbitcoupling of the ferromagnetic layer is set to zero, forsimplicity. We see that the dominant contributions toDMI come from dyz-dz2 (red), dzx-dxy (cyan) and dyz-dx2−y2 (orange), all orbital combinations giving an or-bital momentum along Lx. We also notice the reducedcontribution from dxy-dx2−y2 (blue), which produces anorbital momentum along Lz. This orbital-resolved dia-gram demonstrates that the scenario discussed in SectionIII remains mostly valid in our multi-orbital system. No-tice that the specific orbital contributions are stronglyenergy dependent, which reflects the fact that the elec-tronic band structure displays strong orbital texture [Fig.4]. Finally, we emphasize that performing the same anal-ysis on the DMI coefficient Dyx gives orbitals combina-tions that yield an orbital momentum Ly.

Figure 9 displays the same orbital-resolved DMI whenspin-orbit coupling is present in both metals. Whereasthe DMI orbital decomposition remains mostly unaf-fected for low (EF < 12 eV) and high energies (EF > 15eV), we notice that the contribution from dxy-dx2−y2(blue) increases substantially, reflecting the importantrole of interfacial orbital mixing between the magneticand nonmagnetic orbitals.

C. Thickness dependence

An important question that remained to be addressedis whether DMI is localized at the interface or whetherit extends away from it. As mentioned in the intro-duction, it has been experimentally observed that in

9

−6 −4 −2 0 2

−6

−3

0

3

6

·10−3

EF − 14 (eV)

Dxy

(eV

/Å)

dxy-dzxdyz-dx2y2

dyz-dz2

FIG. 8. (Color online) Dxy as a function of the Fermi energywhen turning on only specific coefficients of the spin-orbitcoupling matrix, in FM(5)/NM(7) bilayer. Here the spin-orbit coupling of the ferromagnetic layer is turned off. Thebroadening is set to Γ = 50 meV and EF = 14 eV.

−6 −4 −2 0 2

−6

−3

0

3

6

·10−3

EF − 14 (eV)

Dxy

(eV

/Å)

dxy-dzxdyz-dx2y2

dyz-dz2

FIG. 9. (Color online) Dxy as a function of the Fermi energywhen turning on only specific coefficients of the spin-orbitcoupling matrix, in FM(5)/NM(7) bilayer. The spin-orbitcoupling of both the ferromagnetic and nonmagnetic layers isturned on. The broadening is set to Γ = 50 meV and EF = 14eV.

CoFeB/Pt heterostructure DMI increases upon increas-ing the nonmagnetic metal thickness and saturates af-ter a few nanometers55, on a scale that seems roughlycomparable to the spin relaxation length. From thetheoretical viewpoint, Yang et al.31 computed DMI inCo/Pt(111) and found that is it dominated by the up-permost Pt layer, while Belabbes et al.42 computed DMIin Mn/W(001) and found that the first three W layers

contribute to the total DMI. Although these two calcula-tions are performed using different methods (real-spacespin spiral versus momentum-space spin spiral), they in-dicate that different materials may display quite differentbehaviors.

In Fig. 10, we report the energy-dependent DMI coef-ficient when turning on the spin-orbit coupling param-eter of a given monolayer away from the interface inFM(3)/NM(10) while turning off the spin-orbit couplingof the other layers. This procedure is only valid in thelimit of weak spin-orbit coupling, but does provide a qual-itative picture of the delocalized nature of DMI as longas the overall band structure remains weakly modifiedby the spin-orbit coupling of individual layers. Figure 10shows that whereas DMI is often dominated by the up-permost nonmagnetic metal monolayer (thick blue line),the contribution of the sub-monolayers is very sensitive tothe energy (thin colored lines). At Fermi energy, DMI isentirely dominated by the uppermost nonmagnetic metallayer. However, around 13.5 eV contributions from thesecond and third monolayers become significant (verti-cal dotted line in Fig. 10), indicating that the Blochstates participating to DMI have a delocalized character.At 12.6 eV, only the second and third layers contributewhereas the first layer close to the interface does not (ver-tical dashed line in Fig. 10). This complex behavior re-flects again the high sensitivity of the orbital compositionof the band structure as a function of the energy. It alsoindicates that the nature of DMI, localized close to theinterface or delocalized away from it, is material sensi-tive. This suggests that such a feature could be tuned bydoping the nonmagnetic metal and modifying the Fermilevel.

FIG. 10. (Color online) Dxy as a function of the Fermi energywhen turning on the spin-orbit coupling only on specific lay-ers. The vertical dashed and dotted lines indicate EF = 12.6eV and EF = 13.5 eV, respectively.

To conclude this study, let us now turn our attention

10

towards the thickness dependence of DMI. Upon varyingthe thickness of the nonmagnetic metal, DMI shows alarge modulation as reported in Fig. 11(a). At 14 eV(blue symbols), this oscillation only extends over a fewmonolayers (typically 1 nm), which is understood fromour previous discussion: at 14 eV, the DMI is dominatedby the first nonmagnetic metal layer, resulting in oscilla-tions confined close to the interface. In contrast, at 12.6eV (red symbols), DMI does not saturate before about 20monolayers, corresponding to about 2.3 nm, revealing thedelocalized nature of DMI at this energy. Figure 11(b)shows the dependence of DMI when varying the thicknessof the ferromagnetic layer. At both Fermi energies, 14 eVand 12.6 eV, DMI saturates after a few monolayers only(∼ 8 monolayers, corresponding to less than 1 nm). Thisfast saturation is attributed to the spin dephasing, i.e., tothe alignment of the spin of the delocalized electrons onthe local magnetization of the ferromagnet. Due to thelarge exchange, any spin misalignment due to the mag-netic texture is absorbed close to the interface, resultingin an interfacial behavior.

0 4 8 12 16 20 24−8

−6

−4

−2

0

Dxy×10

3(

eV/Å

)

(a)

0 4 8 12 16 20 24−8

−6

−4

−2

0

Number of layers

(b)

FIG. 11. (Color online) Thickness dependence of Dxy when(a) varying the nonmagnetic thickness and setting the ferro-magnetic layer to 3 monolayers, and (b) varying the ferromag-netic thickness and setting the nonmagnetic layer to 5 mono-layers. The calculations have been performed for EF = 14 eV(blue) and 12.6 eV (red).

V. CONCLUSION

In this work, we discussed the nature of DMI in tran-sition metal heterostructures. We first derived the ex-pression of DMI in the weak spatial gradient limit withinKeldysh formalism. This derivation provides us with aGreen’s function formula that is well adapted to tight-binding Hamiltonians. With this tool, we first uncoverthe role of orbital mixing and show that symmetry break-ing enables the onset of interfacial orbital momentumthat is at the origin of the DMI. We finally investigatethe different layers to the DMI and reveal that it can ex-pand over several nonmagnetic metal layers depending onthe Fermi energy, thereby revealing the complex orbitaltexture of the band structure. Finally, we examine thethickness dependence of DMI on both ferromagnetic andnonmagnetic metal thicknesses and we find that whereasthe former remains very weak, the thickness dependenceof DMI as a function of the nonmagnetic metal thicknesscan be substantial.

ACKNOWLEDGMENTS

This work was supported by the King Abdullah Uni-versity of Science and Technology (KAUST) throughthe Office of Sponsored Research (OSR) [Grant NumberOSR-2017-CRG6-3390].

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