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[Yoshio Miura](https://orcid.org/0000-0002-5605-5452), Jun Okabayashi

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Understanding magnetocrystalline anisotropy based on orbital and quadrupole moments Journal of Physics: Condensed MatterTOPICAL REVIEW • OPEN ACCESSUnderstanding magnetocrystalline anisotropybased on orbital and quadrupole momentsTo cite this article: Yoshio Miura and Jun Okabayashi 2022 J. Phys.: Condens. Matter 34 473001 View the article online for updates and enhancements.You may also likeLong-range chemical order effects uponthe magnetic anisotropy of FePt alloysfrom an ab initio electronic structure theoryJ B Staunton, S Ostanin, S S A Razee etal.-The effect of dynamical compressive andshear strain on magnetic anisotropy in alow symmetry ferromagnetic filmT L Linnik, V N Kats, J Jäger et al.-Evaluation of windowing techniques forintramuscular EMG-based diagnostic,rehabilitative and assistive devicesHassan Ashraf, Asim Waris, Syed OmerGilani et al.-This content was downloaded from IP address 144.213.253.16 on 05/10/2022 at 10:07https://doi.org/10.1088/1361-648X/ac943f/article/10.1088/0953-8984/16/48/019/article/10.1088/0953-8984/16/48/019/article/10.1088/0953-8984/16/48/019/article/10.1088/0953-8984/16/48/019/article/10.1088/0953-8984/16/48/019/article/10.1088/0953-8984/16/48/019/article/10.1088/1402-4896/aa6943/article/10.1088/1402-4896/aa6943/article/10.1088/1402-4896/aa6943/article/10.1088/1741-2552/abcc7f/article/10.1088/1741-2552/abcc7f/article/10.1088/1741-2552/abcc7fhttps://googleads.g.doubleclick.net/pcs/click?xai=AKAOjstN6Q_M4RyR4TCiFVuuM4uwhRt7dgjTVGZkChakI3xFmnRjcsdMmWkrmOWBwJmrERbLEtccgqxQ2WSxFXCgmt-9FrcTm8qVqRsY1SPYJAxRTBOcbSPwYO_-XT56xzgWEM-50gm5KQY2BwBbPfRLy2OEQ0hkZRG3S9dNjIVImyPtuRJydzrK-3z9rZQmLgczDePx92tc-E8NaSBhCu4FXJRH5fXJiLyW7xgdWdbLER2jMWhu8ZXseAnJ0p_5bLBRwfca2DJsESbvcMYwToPe0CO-rQR4hyIMVsos3YD-MfUtuQ&sai=AMfl-YQzTKkvUioUcvmqRRXaF50nWn4R9rkAVyQxkeo23fj7isaWwl_kJrOU7HyYKl6r9rIJChC-FlyqGjb8_2BkHQ&sig=Cg0ArKJSzMMNk4lG1tqO&fbs_aeid=[gw_fbsaeid]&adurl=http://iopscience.org/booksJournal of Physics: Condensed MatterJ. Phys.: Condens. Matter 34 (2022) 473001 (18pp) https://doi.org/10.1088/1361-648X/ac943fTopical ReviewUnderstanding magnetocrystallineanisotropy based on orbital andquadrupole momentsYoshio Miura1,2,∗ and Jun Okabayashi31 Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science(NIMS), Sengen 1-2-1, Tsukuba 305-0047, Japan2 Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, OsakaUniversity, Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan3 Research Center for Spectrochemistry, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, JapanE-mail: MIURA.Yoshio@nims.go.jpReceived 15 July 2022, revised 19 August 2022Accepted for publication 22 September 2022Published 5 October 2022AbstractUnderstanding magnetocrystalline anisotropy (MCA) is fundamentally important fordeveloping novel magnetic materials. Therefore, clarifying the relationship between MCA andlocal physical quantities observed by spectroscopic measurements, such as the orbital andquadrupole moments, is necessary. In this review, we discuss MCA and the distortion effects inmagnetic materials with transition metals (TMs) based on the orbital and quadrupole moments,which are related to the spin-conserving and spin-flip terms in the second-order perturbationcalculations, respectively. We revealed that orbital moment stabilized the spin moment in thedirection of the larger orbital moment, while the quadrupole moment stabilized the spin momentalong the longitudinal direction of the spin-density distribution. The MCA of the magneticmaterials with TMs and their interfaces can be determined from the competition between thesetwo contributions. We showed that the perpendicular MCA of the face-centered cubic Ni withtensile tetragonal distortion arose from the orbital moment anisotropy, whereas that of Mn-Gaalloys originated from the quadrupole moment of spin density. In contrast, in the Co/Pd(111)multilayer and Fe/MgO(001), both the orbital moment anisotropy and quadrupole moment ofspin density at the interfaces contributed to the perpendicular MCA. Understanding the MCA ofmagnetic materials and interfaces based on orbital and quadrupole moments is essential todesign MCA of novel magnetic applications.∗Author to whom any correspondence should be addressed.Original Content from this work may be used under theterms of the Creative Commons Attribution 4.0 licence. Anyfurther distribution of this work must maintain attribution to the author(s) andthe title of the work, journal citation and DOI.1361-648X/22/473001+18$33.00 Printed in the UK 1 © 2022 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/1361-648X/ac943fhttps://orcid.org/0000-0002-5605-5452https://orcid.org/0000-0002-9025-2783mailto:MIURA.Yoshio@nims.go.jphttp://crossmark.crossref.org/dialog/?doi=10.1088/1361-648X/ac943f&domain=pdf&date_stamp=2022-10-5https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewKeywords: magnetocrystalline anisotropy, orbital moment, quadrupole moment,first-principles calculation, spintronics, interface(Some figures may appear in colour only in the online journal)1. IntroductionMagnetocrystalline anisotropy (MCA), in which the internalenergy varies on the magnetization direction, is an import-ant physical property of magnetic materials [1]. Perpendicu-lar magnetization is required to enhance the thermal stabilityof magnetization directions in permanent magnets and spin-tronic materials, used in information storage media and mag-netoresistive random access memory (MRAM) [2, 3]. Con-trary to this, some studies suggest the importance of in-planeMCA of magnetic materials for applications such as a mag-netic under-layer of a perpendicular recording medium [4]and a spin torque oscillator for microwave-assisted magneticrecording [5, 6]. Furthermore, the development of soft mag-nets requires extremely small MCA under distortion to elimin-ate losses caused by magnetic hysteresis and circulating loopsof current [7]. Hence, understanding MCA is necessary forthe development of novel magnetic materials. Several experi-mental and theoretical studies have been conducted to determ-ine the microscopic origin of MCA.Theoretically, MCA originates from spin-orbit interaction(SOI), which is a one-body interaction between the electron’sspin and its orbital motion and is caused by the differencein the crystal symmetry because of the change in the mag-netization direction via SOI [8]. MCA of rare earth magnetsarises from strong SOI of f electrons, which has been ana-lyzed by the one-ion Hamiltonian, and understood as a com-petition between the crystalline electric field and SOI [9]. Onthe other hand, the MCA of magnetic materials with trans-ition metals (TMs) should be analyzed using electronic bandstructures [10–19].First-principles density functional theory (DFT)calculations [20–22] are important for quantitatively obtain-ing MCA energies and electronic structures. However, owingto the small SOI of TMs, understanding the origin of MCAis difficult by simply analyzing the electronic structures bychanging the magnetization directions. Therefore, a second-order perturbation analysis of SOI [23–31] will be useful indeducing the physical origin of MCA.In the second-order perturbation analysis of MCA, we canresolve MCA energies in terms of atoms, orbitals, and spins.By expanding the wavefunctions in DFT calculations withlocal atomic orbitals and calculating the second-order perturb-ation terms of SOI using the localized basis sets, the contribu-tion of specific atoms, orbitals, and spins to MCA can be cla-rified quantitatively [32, 33]. In addition, the spin-conservingand spin-flip terms in the second-order perturbation calcula-tions of SOI are related to the orbital and quadrupole momentsobserved in spectroscopy, respectively [23, 29, 30]. Brunotheoretically showed that the spin-conserving term betweenoccupied and unoccupied minority-spin states is related to theorbital moment and that MCA energies are proportional tothe difference in the orbital moments between the perpendic-ular and in-plane magnetization directions [23]. This formula,known as the Bruno relation, is widely used to estimate MCAenergies from orbital moments obtained by spectroscopicmeasurements based on x-ray magnetic circular dichroism(XMCD) and x-ray magnetic linear dichroism (XMLD), usingthe magneto-optical sum rule [34–37]. Furthermore, Wang,Wu, and Freeman showed that the spin-flip term betweenoccupied majority-spin and unoccupied minority-spin statesis related to the quadrupole moment of spin density [24]. Thequadrupolemoment of spin density is the same physical quant-ity with the magnetic dipole moment, which is extended toMCA analysis with XMCD and XMLD measurements by vander Laan and Stöhr [29, 30, 38].According to the [29], MCA energies (EMCA) in the form ofspin-conserving and spin-flip terms are expressed as the orbitalmoment anisotropy and the magnetic dipole moment:EMCA ≈ ξ4∆morb +212ξ2∆excmT, (1)where ∆morb and mT are the orbital moment anisotropy andmagnetic dipole moment, respectively, and ξ and ∆exc arethe SOI constant and exchange splitting of the correspond-ing materials, respectively. The orbital moment anisotropy isdefined by ∆morb = mperp.orb −minp.orb , where mperp.orb and minp.orb arethe orbital moment with the perpendicular and in-plane mag-netization, respectively.Equation (1) connects MCA energies to orbital and quadru-pole moments measured by spectroscopy and is important forthe microscopic understanding of MCA in realistic materials.According to equation (1), the positive and negative magneticdipole moments (mT) contribute to a perpendicular MCA andan in-plane MCA, respectively. However, there is a lack ofintuitive understanding on the relationship between the mag-netic dipole moments and MCA, i.e. why the positive (neg-ative) mT contributes to the perpendicular (in-plane) MCA.Because equation (1) is based on several approximations andassumptions, separately verifying this equation for varioussystems with XMCD and XMLD measurements is necessary.In addition, examining the conformity of theoretical results bythe second-order perturbation calculations based on DFT cal-culations with experimental results is important.In this review, in order to obtain intuitive understandingsof MCA, we investigate the dependence of MCA in magneticmaterials with TMs on orbital and quadrupole moments basedon first-principle DFT and the second-order perturbation cal-culations, together with experimental verifications. First, weshow the detailed derivations of the relation between MCAenergies and spectroscopic quantities through second-orderperturbation calculations of SOI. The studies of [23, 29] havebeen intended not to give the final answers of MCA, but to2J. Phys.: Condens. Matter 34 (2022) 473001 Topical Reviewgive the important direction to understand MCA energy. Togive a further important step in the right direction to under-stand MCA, intuitive understandings and classifications ofMCA for various system depending on the interface chemicalbonding will be necessary. Thereafter, we review collaborat-ive work involving theoretical calculations and XMCD andXMLD measurements for understanding MCA of TM films.We review also strain dependent MCA of bulk and interfaces,because they give rise to the precise determination of MCAcharacteristics.First, we discuss the MCA of face-centered cubic (fcc) Niwith in-plane lattice distortion by the piezoelectricity of theunderlayer BaTiO3 [39]. We find that the anisotropy of theorbital moments could describe the MCA of fcc Ni with tetra-gonal distortion. We then discuss the MCA of Mn-Ga alloys,which shows a strong perpendicularMCAwithout a heavy ele-ment. We clarify that the perpendicular MCA ofMn-Ga alloysis attributed to the positive magnetic dipole ofMn owing to thecigar-type distribution of the spin density [40]. Next, we dis-cuss the MCA of the Co(t)/Pd(111) multilayer showing per-pendicular MCA when the thickness of the Co layer t is lessthan 0.6 nm [41]. We reveal that both the interfacial orbitalmoments anisotropy in Co and magnetic dipole moment in Pdare crucial for the perpendicular MCA. Thereafter, we discussthe perpendicular MCA of Fe/MgO(001) [42, 43], frequentlyinvestigated as an interface MCA of magnetic tunnel junctions(MTJs) used in MRAM applications. We find that tetragonaldistortions in MTJs, as well as in interface structures, signi-ficantly affect the contribution of the orbital and quadrupolemoments to MCA. This dependence of MCA on tetragonaldistortions may prove useful for the voltage control of MCAin future spintronic devices [44–46]. We will give an system-atic discussion on MCA of bulk and interfaces and intuitiveunderstandings of MCA based on the orbital and quadrupolemoments.2. Second-order perturbation calculations of SOI2.1. MCA energiesMCA energies are analyzed using second-order perturbationof SOI. The spin-orbit (SO) Hamiltonian is given byĤSO =∑IξI⃗̂L · ⃗̂S,where ⃗̂L= (L̂x, L̂y, L̂z) and ⃗̂S= (Ŝx, Ŝy, Ŝz) are the angularmomentum and spin angular momentum operators, measuredin units of Dirac’s constant ℏ, respectively; I is the atomic pos-ition index, and ξI is the spin–orbit coupling strength at I. ξIhas a localized character and is given byξI ≡ ξI(r) =ℏ22m2c21rdVI(r)dr, (2)where c is the speed of light, m is the electron mass, r isthe distance from the atomic center, and VI(r) is the potentialbetween the electron and atomic nucleus [47]. We assume thatH(0) is the one-electron Hamiltonian without SOI, satisfyingthe non-perturbative Schrödinger equationH(0)∣∣∣⃗knσ〉= ϵ⃗knσ∣∣∣⃗knσ〉 ,where |⃗knσ⟩ is an unperturbed state of energy ϵ⃗knσ with indicesk⃗−point k⃗, band n, and spin σ. Using the eigenstate and eigen-value, the variation in the total energy due to the second-orderperturbation of SOI is given byE(2) =−∑k⃗unocc∑n ′σ ′occ∑nσ∣∣∣〈⃗kn ′σ ′∣∣∣ĤSO∣∣∣ k⃗nσ〉∣∣∣2ϵ⃗kn ′σ ′ − ϵ⃗knσ. (3)|⃗knσ⟩ can be expanded with an orthogonal basis of atomicorbitals labeled µ (or λ):∣∣∣⃗knσ〉=∑jµck⃗njµσ |µσ⟩ei⃗k·R⃗j , (4)where R⃗j is the atomic position at site j in the unit cell. Wecan obtain the second-order contribution of ĤSO to the totalenergy as a sum over terms depending on the spin transitionprocesses, atomic orbitals, and atomic sites:E(2) =−∑σσ ′∑II ′ξIξ′I∑λλ ′µ ′µ〈λσ|⃗L̂ · ⃗̂S|λ ′σ ′〉〈µ ′σ ′ |⃗L̂ · ⃗̂S|µσ〉×Gσσ ′II ′ (λλ ′;µ ′µ), (5)where Gσσ ′II ′ (λλ ′;µ ′µ) is an integral of joint local density ofstates (LDOSs) given byGσσ ′II ′ (λλ ′;µ ′µ) =∑k⃗occ∑nunocc∑n ′ck⃗n∗I ′λσck⃗n ′I ′λ ′σ ′ck⃗n′∗Iµ ′σ ′ck⃗nIµσϵ⃗kn ′σ ′ − ϵ⃗knσ. (6)To derive the equation (5), we assume that SOI acts only at thesame atomic site and use the relation∑jj ′ ei⃗k·(R⃗j−R⃗j ′ )ξ(|R⃗j−R⃗I|) = ξI. Although SOI is effective within the same atomicsite, the joint LDOS Gσσ ′II ′ (λλ ′;µ ′µ) includes the electronicstates for different atomic sites in the unit cell (I and I′)because of the hybridization of atomic orbitals in the crys-tal. The joint LDOS Gσσ ′II ′ (λλ ′;µ ′µ) can be calculated usingfirst-principles DFT calculations, and the coefficients ck⃗njµσ areobtained from the projections of unperturbed eigenstate oneach localized atomic orbital. We calculated Gσσ ′II ′ (λλ ′;µ ′µ)using the Vienna ab initio simulation package (VASP) code[48–50]. Furthermore, we set the spin–orbit coupling con-stants defined by equation (2), which are used within VASPcode, as follows: ξMn(3d) = 41.5 meV , ξFe(3d) = 54.3 meV,ξCo(3d) = 69.4 meV, ξNi(3d) = 87.2 meV, ξO(2p) = 24.3 meV,ξMg(2p) = 47.5 meV, ξGa(3p) = 35.4 meV, ξPd(4d) = 187 meV.To obtain MCA energies, its dependence on the magnetiz-ation direction must be considered. Noncollinear magnetiza-tion can be introduced by expressing the spin-quantum axis of3J. Phys.: Condens. Matter 34 (2022) 473001 Topical Reviewlocal basis sets |σ⟩ (σ =↑ or ↓) with a spin-1/2 rotation matrixaccording to Kübler’s formulation [51, 52]| ↑⟩= eiϕ2 cosθ2∣∣∣↑̃〉+ e−iϕ2 sinθ2∣∣∣↓̃〉 , (7)| ↓⟩=−eiϕ2 sinθ2∣∣∣↑̃〉+ e−iϕ2 cosθ2∣∣∣↓̃〉 , (8)where θ and ϕ are the polar and azimuthal angles of magnetiz-ationwith respect to the z and x-axes of the system, and z axis isperpendicular to the plane of the tetragonal or hexagonal sys-tems. |σ̃⟩ indicates a spin state along the global spin-quantumaxis, fixed to the z axis of the system.By using equations (7) and (8) in the matrix element⟨µ ′σ ′ |⃗L̂ · ⃗̂S|µσ⟩, we obtain the following expressions depend-ing on the magnetization direction:⟨µ′ ↑ |⃗L̂ · ⃗̂S|µ ↑⟩= 12[sinθ cosϕ⟨µ′|L̂x|µ⟩−sinθ sinϕ⟨µ′|L̂y|µ⟩+ cosθ⟨µ′|L̂z|µ⟩],⟨µ′ ↓ |⃗L̂ · ⃗̂S|µ ↓⟩=−12[sinθ cosϕ⟨µ′|L̂x|µ⟩−sinθ sinϕ⟨µ′|L̂y|µ⟩+ cosθ⟨µ′|L̂z|µ⟩],⟨µ′ ↑ |⃗L̂ · ⃗̂S|µ ↓⟩= 12[(cosθ cosϕ− isinϕ)⟨µ′|L̂x|µ⟩− (cosθ sinϕ+ icosϕ)⟨µ′|L̂y|µ⟩− sinθ⟨µ′|L̂z|µ⟩],⟨µ′ ↓ |⃗L̂ · ⃗̂S|µ ↑⟩= 12[(cosθ cosϕ+ isinϕ)⟨µ′|L̂x|µ⟩− (cosθ sinϕ− icosϕ)⟨µ′|L̂y|µ⟩− sinθ⟨µ′|L̂z|µ⟩],where we have used the eigenstate and eigenvalue relationbetween ⃗̂S= (Ŝx, Ŝy, Ŝz) and |σ̃⟩ and the orthonormal propertyof the eigenstate ⟨σ̃|σ̃ ′⟩= δσ̃σ̃ ′ . Because we consider the uni-axial MCA energies, the energy difference between the per-pendicular magnetization (θ= 0 and ϕ= 0) and in-plane mag-netization (θ = π2 and ϕ= 0) should be calculated. In this case,the matrix elements of SOI for each magnetization directionare given by⟨µ ′ ↑ |⃗L̂ · ⃗̂S|µ ↑⟩θ=0,ϕ=0 =−〈µ ′ ↓ |⃗L̂ · ⃗̂S|µ ↓〉θ=0,ϕ=0=12〈µ ′|L̂z|µ〉, (9)⟨µ ′ ↑ |⃗L̂ · ⃗̂S|µ ↑⟩θ=π2 ,ϕ=0 =−〈µ ′ ↓∣∣∣⃗̂L · ⃗̂S∣∣∣µ ↓〉θ=π2 ,ϕ=0=12〈µ ′∣∣∣L̂x∣∣∣µ〉 , (10)〈µ ′ ↑∣∣∣⃗̂L · ⃗̂S∣∣∣µ ↓〉θ=0,ϕ=0=〈µ ′ ↓∣∣∣⃗̂L · ⃗̂S∣∣∣µ ↑〉∗θ=0,ϕ=0=12〈µ ′∣∣∣L̂x− iL̂y∣∣∣µ〉 , (11)Table 1. Nonzero matrix elements of the angular momentumoperator ⃗̂L= (L̂x, L̂y, L̂z) for the d orbitals in Cartesian coordinates.⟨dx2−y2 |L̂x|dyz⟩= 1 ⟨dxy|L̂x|dzx⟩= 1 ⟨d3z2−r2 |L̂x|dyz⟩=√3⟨dx2−y2 |L̂y|dzx⟩= 1 ⟨dyz|L̂y|dxy⟩= 1 ⟨d3z2−r2 |L̂y|dzx⟩=√3⟨dx2−y2 |L̂z|dxy⟩= 2 ⟨dzx|L̂z|dyz⟩= 1〈µ ′ ↑∣∣∣⃗̂L · ⃗̂S∣∣∣µ ↓〉θ=π2 ,ϕ=0=〈µ ′ ↓∣∣∣⃗̂L · ⃗̂S∣∣∣µ ↑〉∗θ=π2 ,ϕ=0=−12〈µ ′∣∣∣L̂z+ iL̂y∣∣∣µ〉 . (12)Since we define the MCA energy as positive for perpendicu-lar magnetization (θ= 0 and ϕ= 0), the MCA energy in thesecond-order perturbation calculations is given byE(2)MCA ≡ E(2)(θ =π2,ϕ= 0)−E(2)(θ = 0,ϕ= 0)= E↑↑MCA +E↓↓MCA +E↑↓MCA +E↓↑MCA (13)=∑II ′ξIξI ′4∑λλ ′µµ ′[⟨λ|L̂z|λ ′⟩⟨µ ′|L̂z|µ⟩− ⟨λ|L̂x|λ ′⟩⟨µ ′|L̂x|µ⟩]×G↑↑II ′(λλ′;µ ′µ)+∑II ′ξIξI ′4∑λλ ′µµ ′[⟨λ|L̂z|λ ′⟩⟨µ ′|L̂z|µ⟩− ⟨λ|L̂x|λ ′⟩⟨µ ′|L̂x|µ⟩]G↓↓II ′(λλ′;µ ′µ)−∑II ′ξIξI ′4∑λλ ′µµ ′[⟨λ|L̂z|λ ′⟩⟨µ ′|L̂z|µ⟩− ⟨λ|L̂x|λ ′⟩⟨µ ′|L̂x|µ⟩]×G↑↓II ′(λλ′;µ ′µ)−∑II ′ξIξI ′4∑λλ ′µµ ′[⟨λ|L̂z|λ ′⟩⟨µ ′|L̂z|µ⟩− ⟨λ|L̂x|λ ′⟩⟨µ ′|L̂x|µ⟩]G↓↑II ′(λλ′;µ ′µ). (14)In E(2)MCA, ⟨L̂y⟩ terms in equations (11) and (12) are automat-ically canceled out. Eσσ ′MCA is the spin-resolved MCA energy,where E↑↑MCA and E↓↓MCA are spin-conserving terms, and E↑↓MCAand E↓↑MCA are spin-flip terms. It is important to note that thesign of the matrix elements of L̂z and L̂x are different forthe spin-conserving (positive) and spin-flip (negative) terms.This means that the local electronic structures around theFermi level have an opposite contribution to MCA in the spin-conserving and spin-flip processes. The opposite sign betweenthe spin-conserving and the spin-flip terms arises from theeigenvalue of Ŝz for the spin eigenstate |σ⟩, i.e. Ŝz| ↑⟩=+ 12 | ↑⟩and Ŝz| ↓⟩=− 12 | ↑⟩. The spin-conserving term includes twoeigenvalues with the same sign, wheres the spin-flip term hastwo eigenvalues with the different sign, leading the oppositecontribution to MCA.In table 1, we show the nonzero matrix elements of theangular momentum operator ⃗̂L= (L̂x, L̂y, L̂z) for the d orbit-als in Cartesian coordinates. The matrix elements of L̂z arenonzero for d orbitals with the same magnetic quantum num-bers for occupied and unoccupied states, such as dxy− dx2−y2and dyz− dzx, contributing to the perpendicular MCA in thespin-conserving terms (E↑↑MCA and E↓↓MCA) and in-plane MCAin the spin-flip terms (E↑↓MCA and E↓↑MCA). On the other hand,4J. Phys.: Condens. Matter 34 (2022) 473001 Topical Reviewthe matrix elements of L̂x are nonzero for d orbitals when thedifference in magnetic quantum numbers between the occu-pied and unoccupied states is ±1, contributing to the in-planeMCA in E↑↑MCA and E↓↓MCA terms and the perpendicular MCA inE↑↓MCA and E↓↑MCA terms. These analyses enable us understandthe relationship between the local electronic structures aroundthe Fermi level and their MCA contribution at each atomicsite; This may be useful in designing new ferromagneticmaterials with a strong perpendicular MCA for spintronicsapplications.2.2. Orbital momentsThe first-order correction of wavefunctions for perturbation ofSOI is given by|⃗knσ⟩(1) = |⃗knσ⟩+unocc∑n ′σ ′occ∑nσ〈⃗kn ′σ ′∣∣∣ĤSO∣∣∣ k⃗nσ〉ϵ⃗kn ′σ ′ − ϵ⃗knσ∣∣∣⃗kn ′σ ′〉.(15)The orbital moment ⟨L̂ζ⟩, which is the component of ⃗̂L par-allel to the direction of the spin quantum axis ζ = (θ,ϕ), isgiven by as the expectation value of the angular momentumoperator in equation (15). Because the expectation value ofL̂ζ for the unperturbed states is zero, the orbital moment isgiven by⟨L̂ζ⟩ ≃ 2∑k⃗unocc∑n′σ′occ∑nσ⟨⃗knσ|L̂ζ |⃗kn′σ′⟩ ⟨⃗kn′σ′|ĤSO |⃗knσ⟩ϵ⃗kn′σ′ − ϵ⃗knσ,where the factor of 2 comes from the Hermitian conjug-ate of the SOI Hamiltonian. We neglect the squared termof ĤSO.By expanding |⃗knσ⟩ with the orthogonal basis given byequation (4), we obtained the orbital moment at atomic siteI using the following equation:〈L̂ζ〉I=−2ξI∑σσ′∑I′∑λλ′µ′µ〈λσ|L̂ζ |λ′σ′〉〈µ′σ′∣∣∣⃗̂L · ⃗̂S∣∣∣µσ〉×Gσσ′II′ (λλ′;µ′µ).We note that ⟨L̂ζ⟩=∑I⟨L̂ζ⟩I. The sign of the orbital momentis determined by the term ⟨λσ|L̂ζ |λ ′σ ′⟩, which is equivalent tothe ⃗̂L component parallel to the direction of the spin quantumaxis, and the following relation holds [23]:⟨λσ|L̂ζ |λ ′σ ′⟩= 2sgn(σ)δσσ ′⟨λσ|⃗L̂ · ⃗̂S|λ ′σ ′⟩, (16)where sgn(↑) = +1 and sgn(↑) =−1. Due to δσσ ′ inequation (16), the spin-flip terms do not contribute to orbitalmoment. Thus, the orbital moment can be written only by thespin-conserving terms as follow:⟨L̂ζ⟩I =−4ξI∑σsgn(σ)∑I′∑λλ′µ′µ⟨λσ|⃗̂L · ⃗̂S|λ′σ⟩⟨µ′σ|⃗̂L · ⃗̂S|µσ⟩×GσσII′ (λλ′;µ′µ).Finally, the orbital moment with the spin moment along thez axis (θ= 0 and ϕ= 0) corresponding to mperp.orb and the spinmoment along the x axis (θ = π2 and ϕ= 0) corresponding tominp.orb are given by⟨L̂z⟩I = ξI∑I′∑λλ′µ′µ⟨λ|L̂z|λ′⟩⟨µ′|L̂z|µ⟩×[G↓↓II′ (λλ′;µ′µ)−G↑↑II′ (λλ′;µ′µ)],⟨L̂x⟩I = ξI∑I′∑λλ′µ′µ⟨λ|L̂x|λ′⟩⟨µ′|L̂x|µ⟩×[G↓↓II′ (λλ′;µ′µ)−G↑↑II′ (λλ′;µ′µ)].To obtain these equations, we used the relation in equations (9)and (10), and the eigenstates and eigenvalues of the spin angu-lar momentum operator.If we assume that the spin-flip terms E↑↓MCA and E↓↑MCA andthe spin-conserving term E↑↑MCA in equation (14) are negligiblecompared to the spin-conserving term E↓↓MCA, the MCA energycan be expressed as the sum of the orbital moment anisotropyat each atomic site∆mIorb:E(2)MCA ≈ 14∑IξI[〈L̂z〉I−〈L̂x〉I]=14∑IξI∆mIorb. (17)This is the Bruno relation, commonly used to connect MCAenergies and the observed orbital moments by XMCD.Because the ferromagnetic materials with more than halfelements as Fe, Co, and Ni have fully occupied majority-spinstates, the unoccupied majority-spin states around the Fermilevel are negligible. In this case, neglecting E↑↑MCA and E↓↑MCAis reasonable for MCA energies. However, there are casesin which the spin-flip-term through the unoccupied minority-spin states E↑↓MCA is not too small to be ignored compared tothe spin-conserving term E↓↓MCA, and the Bruno relation failsto describe MCA energies, both quantitatively and qualitat-ively. If the energy depth of the occupied majority-spin statesis far from the Fermi level owing to strong exchange split-ting, we can neglect the spin-flip term E↑↓MCA because of thelarge denominator in equation (6), and the Bruno relation welldescribes MCA energies.2.3. Quadrupole momentsWang,Wu, and Freeman proposed that the spin-flip termE↑↓MCAcan be related to quadrupole moments [24]. We start the for-mulation of MCA energies in equation (3) in the second-order perturbation. We then expand the unperturbed unoc-cupied eigenstates |⃗kn ′ ↓⟩ using the local atomic orbitals ofequation (4) and retain the occupied eigenstates |⃗kn ↑⟩5J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewE↑↓MCA =−14∑k⃗unocc∑n ′occ∑n∑Iξ2I×[∑λ ′µ ′ ck⃗n′∗Iλ ′↓ck⃗n ′Iµ ′↓⟨λ ′ ↓ |L̂z |⃗kn ↑⟩⟨⃗kn ↑ |L̂z|µ ′ ↓⟩ϵ⃗kn ′↓ − ϵ⃗kn↑−∑λ ′µ ′ ck⃗n′∗Iλ ′↓ck⃗n ′Iµ ′↓⟨λ ′ ↓ |L̂x |⃗kn ↑⟩⟨⃗kn ↑ |L̂x|µ ′ ↓⟩ϵ⃗kn ′↓ − ϵ⃗kn↑].(18)The factor of 14 comes from the eigenvalues of two spin angu-lar momentum operators, and we leaves brakets of spin states|σ⟩ to clearly specify the spin state in the second-order per-turbation term. Here, we express a spin state by σ for theglobal spin-quantum axis along z-axis. Wang, Wu, and Free-man introduced two approximations to relate the spin-flip termand quadrupole moment. First, the replacement of the dif-ference in eigenvalues between the unoccupied and occupiedstates in the denominator with ∆exc is considered:ϵ⃗kn ′↓ − ϵ⃗kn↑ ≈∆exc, (19)where ∆exc is the exchange splitting of ferromagnetic mater-ials, corresponding to the range of eigenvalues between themajority-spin occupied and minority-spin unoccupied states.Second, the use of the completeness relation of brakets onoccupied majority-spin states:occ∑n|n ↑⟩⟨n ↑ | ≈ 1. (20)This relation holds if we consider ferromagneticmaterials withmore than half elements as Fe, Co, and Ni, where the majority-spin states are fully occupied. In this case, the sum of all occu-pied states is equivalent to the sum of all states.By using equations (19) and (20), the square of the expect-ation of the angular momentum operator can be expressedas the expectation of the square of the angular momentumoperator:E↑↓MCA ≈−14∑k⃗unocc∑n′∑Iξ2I×∑λ′µ′ ck⃗n′∗Iλ′↓ck⃗n′Iµ′↓[⟨λ′ ↓ |L̂2z |µ′ ↓⟩− ⟨λ′ ↓ |L̂2x |µ′ ↓⟩]∆exc.Furthermore, if we assume the tetragonal system which hasthe same lattice structure for the in-plane x and y axes, thefollowing relation holds.⟨L̂2x ⟩= ⟨L̂2y ⟩=12⟨L̂2x + L̂2y ⟩=12⟨L̂2 − L̂2z ⟩.Thus, for tetragonal systems, we haveE↑↓MCA ≈−18∑Iξ2I∑k⃗unocc∑n′∑µ′ |ck⃗n′Iµ′↓|2 ⟨µ′ ↓ |3L̂2z − L̂2|µ′ ↓⟩∆exc.Here, the matrix of ⟨L̂2 ⟩ and ⟨L̂2z ⟩ are diagonal foratomic-orbital basis set, and we use the orthonormal prop-erty, ⟨λ ′|µ ′⟩= δλ ′µ ′ . The operator 3L̂2z − L̂2 has the sameform as the z-component of intra-atomic quadrupole momentoperator:Q̂zz ≡221(3L̂2z − L̂2). (21)Finally, we can express the spin-flip term of MCA ener-gies related to the quadrupole moments of the unoccupiedminority-spin electron densitiesE↑↓MCA ≈−2116∑Iξ2I∑k⃗unocc∑n ′∑µ ′ |ck⃗n′Iµ ′↓|2 ⟨µ ′ ↓ |Q̂zz|µ ′ ↓⟩∆exc.(22)This is an approximate expression of the spin-flip term E↑↓MCArelated to the quadrupole moment. Based on the definitionof quadrupole moment in equation (21), oblate distributionsof unoccupied minority-spin electrons along z axis (cigar-like quadrupole), e.g. d3z2−r2 orbital, gives a negative quadru-pole moment, yielding perpendicular (positive) MCA throughequation (22). In contrast, prolate distributions of unoccupiedminority-spin electrons along xy plane (pancake-like quadru-pole), e.g. dx2−y2 and dxy orbitals, produce a positive quadru-pole moment, yielding an in-plane (negative) MCA.In equation (22), the spin-flip term of MCA energies E↑↓MCAis described by the quadrupole moments of the unoccupiedminority-spin electrons. In addition, E↑↓MCA can be expressedby the quadrupole moments of occupied minority-spinelectrons:E↑↓MCA ≈+2116∑Iξ2I∑k⃗occ∑n∑µ |ck⃗nIµ↓|2 ⟨µ ↓ |Q̂zz|µ ↓⟩∆exc,(23)where∑unoccn ′ =∑alln −∑occn and the quadrupole moment⟨Q̂zz⟩↓I including all states (occupied and unoccupied states)is zero, and ⟨Q̂zz⟩↓I =∑k⃗∑n∑µ |ck⃗nIµ↓|2 ⟨µ ↓ |Q̂zz|µ ↓⟩.Equations (22) and (23) shows that the contribution ofquadrupole moments toMCA energies is opposite between theoccupied and unoccupied states in the minority-spin electrons.Then, we considered the quadrupole moments of spin densitycorresponding to the magnetic dipole moment mT. Again, weassumed that magnetic materials with more than half elementsas Fe, Co, and Ni have fully occupied majority-spin states.This provides the following relationship:∑k⃗occ∑n∑µ|ck⃗nIµ↑|2 ⟨µ ↑ |Q̂zz|µ ↑⟩≈∑k⃗all∑n∑µ|ck⃗nIµ↑|2 ⟨µ ↑ |Q̂zz|µ ↑⟩= 0.6J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewThus, we can write equation (23) using a intra-atomic mag-netic dipole moment:E↑↓MCA ≈−2116∑Iξ2I∆exc∑k⃗occ∑n∑µ[∣∣∣ck⃗nIµ↑∣∣∣2 ⟨µ ↑ |Q̂zz|µ ↑⟩−∣∣∣ck⃗nIµ↓∣∣∣2〈µ ↓ |Q̂zz|µ ↓〉]=−218∑Iξ2I∆exc∑k⃗occ∑n∑µ∑σ∣∣∣ck⃗nIµσ∣∣∣2〈µσ|Q̂zzŜz|µσ〉=+218∑Iξ2I∆excmIT (24)where the magnetic dipole moment and quadrupole momentof the spin density at each atomic site is related by followingequation:mIT ≡−⟨Q̂zzŜz⟩I =−12[⟨Q̂zz⟩↑I −⟨Q̂zz⟩↓I ]. (25)Furthermore, we can rewrite the intra-atomic magnetic dipolemoment using the spin moment projected to each atomic orbit-als mµspin as follow:mIT =− 121(mpxspin +mpyspin − 2mpzspin + 6mdx2−y2spin + 6mdxyspin−3mdyzspin − 3mdzxspin − 6md3z2−r2spin). (26)The intra-atomic magnetic dipole momentmIT can be observedby XMCD and XMLD measurements [38], and a positive(negative) mIT indicates the contribution of the spin-flip termto the perpendicular (in-plane) MCA. Because the mIT term isdirectly related to MCA energies, we do not need to considerthe anisotropy ofmIT to discuss its contribution toMCA, unlikethe orbital moments in equation (17). Furthermore, becausethe mIT term in equation (24) is obtained using non-perturbedeigenstates, the mIT term is irrelevant to SOI despite beingderived from the second-order perturbation of SOI. In fact, wecan obtain the mT using equations (25) and (26) without SOI.This is because the approximation in equation (20), which cor-responds to neglecting the quantum uncertainty in the angularmomentum operators,√⟨Lζ⟩2 −⟨L2ζ ⟩ ≈ 0.The above treatment of angular momentum operators res-ults in a classical picture of magnetic anisotropy, namely,the shape magnetic anisotropy (SMA) due to magnetostaticdipole-dipole interactions. Stöhr pointed out that there is arelationship between the magnetic dipole moment derivedfrom the spin-flip term of MCA energies and magnetostaticdipole-dipole interaction (see appendix B in [30]). Further-more, because equation (20) requires the occupied majority-spin and the unoccupied miniroty-spin states in Eσσ ′MCA, themagnetic dipole moment (quadrupole moment of spin dens-ity) in equations (22) and (23) are related to the spin-flip termE↑↓MCA of MCA energies.2.4. Intuitive understanding of MCA based on orbital andquadrupole momentsBased on discussions above, MCA energies can be presen-ted by using the orbital moment anisotropy and the mag-netic dipole moment (quadrupole moment of spin density) asfollows:EMCA ≈∑I14ξI∆mIorb +218∑Iξ2I∆excmIT. (27)A difference in the coefficient of the mIT term by a factor of 14is noticed in this equation compared to equation (1). The dif-ference comes from the eigenvalues of the spin operator ⃗̂S asmentioned in equation (18), because Wang et al described theSOI Hamiltonian by HSO = ξ⃗̂L · ⃗̂σ [24], where ⃗̂σ is the Paulispin matrices (twice of ⃗̂S) and a factor 12 is included in ξ. Ourformulations of the spin-flip term of MCA energies related tothe magnetic dipole moments (quadrupole moments of spin-density) in equations (18)–(24) using HSO = ξ⃗̂L · ⃗̂S (thus afactor of 12 is not included in ξ) are consistent with the for-mulation in Stöhr’s paper (see equation (27) in [30]).Furthermore, it is important to notice that Bruno term(orbital moment) and van der Laan term (quadrupole momentof spin density) [53] are not sufficient to describe the MCA,because they ignore perturbation terms involving unoccupiedmajority-spin states [54]. In spite of the approximation, theequation (27) is still important to connect the MCA energywith the local physical quantities observed by the spectro-scopic experiments. Following, we add an intuitive pictureon MCA to the Bruno term and the van der Laan term.Then, we would like to comment on the difference betweenequation (27) of the present paper and equation (28) of [29].The equation (27) is the MCA energy formulated throughthe second order perturbation of SOI under the approxima-tion of equations (19) and (20). On the other hand, the lat-ter is the energy due to the second-order perturbation derivedfrom equation (9) of [29], and is not the MCA energy. Thus,to obtain the MCA energy, the energy difference betweenthe perpendicular and in-plane magnetization should be con-sidered. In taking the energy difference, the ELS term inequation (28) of [29] will be canceled out between the perpen-dicular and in-plane magnetization directions, because the ELSis independent of magnetization direction. Thus, we ignore theELS term in equation (27) of the present paper.Figure 1 shows schematic images of the orbital and quad-rupole moments’ contributions to MCA. The contribution ofthe orbital moment to MCA can be understood from theSOI Hamiltonian in second-order perturbation. Because of theinternal product term ⃗̂L · ⃗̂S in equation (3), an orbital momentparallel to the spin moment is more favorable to stabilize themagnetization direction. This implies that the spin momentaligns with a larger orbital moment. Thus, the orbital momentanisotropy ∆morb is proportional to the MCA energy. Theoblate (pancake-like) distributions of minority-spin electronsin xy plane cause the perpendicular orbital moment (alongz axis), contributing to the perpendicular MCA, whereas the7J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewFigure 1. Schematic image on effects of the orbital moment anisotropy and quadrupole moment of spin density on MCA. (a) Spin momentcan be stabilized along the direction of the larger orbital moment due to SOI, described as the Bruno term in equation (17). (b) Spin momenttend to be parallel to the longitudinal direction of the spin-density distribution, characterized by the quadrupole moment (intra-atomicmagnetic dipole moment) and described as van der Laan term in equation (24).prolate (cigar-like) distributions along z axis cause the in-plane orbital moment, contributing to the in-plane MCA (inxy plane). In contrast, the contribution of the quadrupolemoments of spin density (magnetic dipole moments) to MCAimplies that the shape of the spin-density distribution directlyaffects the MCA .If the quadrupole moments (Qzz) of the spin densitydefined by equation (21) is zero, the spin-density distribu-tion is spherical and does not contribute to MCA. However,if Qzz is non-zero, the spin moment tends to orient in thelongitudinal direction of the spin-density distribution, and theprolate (cigar-type) distribution contributes to the perpendicu-lar MCA, whereas the oblate (pancake-type) distribution con-tributes to the in-plane MCA. Therefore, the contributions toMCA from the two shapes of the spin-density distributionare opposite for the orbital moment and quadrupole moment(magnetic dipole moment). This is consistent with the oppos-ite signs of the matrix elements of L̂z and L̂x for the spin-conserving term (orbital moment anisotropy) and the spin-flipterm (the quadrupole moment of spin density). Therefore, theMCA of magnetic materials with TMs can be determined fromthe competition between the orbital moment anisotropy andthe anisotropy of the spin-density distribution (the quadrupolemoment of spin-density).Recently, Suzuki and Miwa [55] formulated the expecta-tion of the magnetic dipole operator by using wavefunctionswith first-order perturbative corrections of SOI, and directlydescribe the spin-flip term of perturbative MCA energies withthe measurable physical parameters related to the magneticdipole moment. The formulation does not require the approx-imation and assumption such as equations (19) and (20), andcan apply magnetic materials with small exchange splitting.To confirm the relationship between the spin-flip term of theMCA and the correction of the magnetic dipole term due to theSOI perturbation in [55], based on first-principles DFT calcu-lations and spectroscopic experiments, will be future work.3. Fcc Ni with in-plane distortionsThe coupling between MCA and lattice distortion is a funda-mental issue in magnetism. The lattice distortion changes thesymmetry of a crystal and produces different electronic struc-tures affecting magnetization directions. Therefore, magneticproperties and electronic structures strongly couple with lat-tice structure and symmetry, known as the inverse magneto-striction (magneto-elastic) effect. For example, fcc Ni doesnot have MCA in the cubic structure. However, the tensileand compressive tetragonal lattice distortions with respect tothe (001) plane cause the perpendicular and in-plane MCAs,respectively.Recently, the magnetic properties of Ni/Cu multilayers onferroelectric BaTiO3 substrate were controlled by the mech-anical strain through the piezo electric effect . This strain isinduced by an applied electric field E to BaTiO3, switchingthe magnetization from the perpendicular to in-plane easy axisby tuning the lattice distortion with E [39]. In this experi-ment, without E, the Ni layer exhibited a tensile strain of 2%through the sandwiched Cu layers, and the Ni layer showed a8J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewTable 2. The orbital magnetic moment of Ni in Ni/Cu multilayer on BaTiO3 measured by XMCD for Ni L edges in the normal incident (NI)and the grazing incident (GI) geometries, with and without electric field (E) . The calculated orbital magnetic moments of fcc Ni for theperpendicular (θ= 0) and in-plane (θ = π/2) magnetization configurations with and without tensile distortion (2%), corresponding to withand without electric field (E) in the experiment. Reproduced from [39]. CC BY 4.0.Experiment E= 0 kV cm−1 E= 8 kV cm−1morb (NI) 0.06 µB 0.04 µBmorb (GI) 0.04 µB 0.05 µBCalculation 2% tensile strain without strainmorb (θ= 0) 0.0542 µB 0.0506 µBmorb (θ = π/2) 0.0505 µB 0.0504 µBFigure 2. (a) DFT calculation results of MCA energies and the orbital moment anisotropies of fcc Ni as a function of the in-plane latticeconstant a∥ (tetragonal distortion is defined as (a∥ − a0)/a0 × 100), where a0 = 3.524 Å. The crystal structure of fcc Ni for the conventionalunit cell and the unit cell with 45◦ in-plane rotation is also shown in the inset. (b) Spin-resolved MCA energies in the second-orderperturbation of SOI from equation (13) for fcc Ni as a function of a∥. Reproduced from [39]. CC BY 4.0.perpendicular MCA. With E, the change in the domain struc-tures reduces the lattice constant of BaTiO3 and releases thetensile strain in the Ni layers, resulting in magnetization alongthe in-plane easy axis. In addition, a strain-induced change inthe orbital magnetic moments of Ni was observed by XAS andXMCD measurements. The spectra for NiCu/BaTiO3 weremeasured for Ni L edges in the normal incident (NI) and graz-ing incident (GI) geometries, with and without E (8 kV cm−1).Table 2 shows the observed orbital moments of Ni in theNi/Cu multilayers on BaTiO3 by XMCD measurement. Theorbital moments in the NI geometry correspond to those withthe magnetization normal to the (001) plane, while the orbitalmoments in the GI geometry include half of the in-plane com-ponents owing to the grazing angle (60◦) from the sample sur-face normal (cos60◦ = 1/2). The values of the orbital momentanisotropy of Ni between the NI and GI geometries wereestimated to be 0.02 µB (for E= 0 and tensile strain= 2%) and0.01 µB (for E= 8 kV cm−1 and tensile strain= 0). These res-ults indicate that an applied E modulates the orbital momentsand their anisotropies, resulting in changes in MCA owing tolattice distortion.To examine the relationship between MCA and lattice dis-tortion in detail, we performed second-order perturbation cal-culations for the MCA of fcc Ni with tetragonal distortionin the (001) plane. Details of DFT calculations are presentedin [39] . Figure 2(a) shows the MCA energy and orbitalmoment anisotropy of fcc Ni as a function of the distor-tion ratio (a∥ − a0)/a0 × 100, where a∥ is the in-plane latticeparameter, and a0 = 3.524 Å is the optimized lattice con-stant of the cubic fcc Ni in DFT calculations. The inset offigure 2(a) shows the crystal structure and unit cell of distortedfcc Ni. The out-of-plane lattice parameter a⊥ is fully relaxedfor each a∥.As shown in figure 2(a), the MCA energy increases withan increasing distortion ratio (i.e. tetragonal tensile strain),which is consistent with the experimental results. Becausethe orbital moment anisotropy∆morb monotonically increaseswith an increasing distortion ratio, the change in the perpen-dicular MCA can be attributed to the change in the orbitalmoment anisotropy given by equation (17). Figure 2(b) showsthe dependence of the spin-resolved MCA energies in second-order perturbation from equation (13) on the tetragonal strain.The spin-conserving term E↓↓MCA has the largest dependence onthe distortion ratio and is the main contributing factor toMCA.In contrast, for the spin-flip term E↑↓MCA, the magnitude is smal-ler than the spin-conserving term, and E↑↓MCA has the oppositedependence on the distortion ratio compared toMCA energies.To validate equation (27), we plotted the Bruno term fromequation (17) in figure 3(a) and van der Laan term fromequation (24) as a function of the distortion ratio, where∆Eexc9https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewFigure 3. (a) Bruno term (orbital moment anisotropy) expressed by equation (17) and van der Laan term (magnetic dipole moment)expressed by equation (24) of fcc Ni as a function of a∥. (b) The d orbital contribution to the MCA energies of tetragonally distorted fcc-Nias a function of the in-plane lattice constant a∥, where G↓↓Ni is the joint local density of states expressed by equation (6). Reproduced from[39]. CC BY 4.0.was set to 1 eV. By comparing figures 2(b) and 3(a), we foundthat the strain dependence of Bruno and van der Laan terms(on the distortion ratio) agrees with that of the spin-conservingterm E↓↓MCA and the spin-flip term E↑↓MCA, respectively, indic-ating that the orbital moment anisotropy and the magneticdipole (the quadrupole moment of spin density) successfullydescribe the MCA of distorted fcc Ni. In this case, the orbitalmoment anisotropy is more responsible for the MCA of fcc Niwith tetragonal distortion than the anisotropy of the quadru-pole moment (the magnetic dipole term).To further understand the MCA of fcc Ni, in figure 3(b),we show the square of the matrix elements of the spin-conserving terms corresponding to Lz and Lx for each d orbitalas a function of the distortion ratio. The matrix elements|⟨dxy|Lz|dx2−y2⟩|2G↓↓Ni in equation (14) show the main contri-bution to the perpendicular MCA, and their lattice distortiondependence is similar to that of EMCA, i.e. it increases withincreasing distortion ratio, where the positive (negative) mat-rix elements indicate the contribution to the perpendicular(in-plane) MCA. We confirm that the number of minority-spin electrons in dx2−y2 increases owing to tensile distortion,whereas that in dxy changes negligibly. This change in elec-tron distribution in dx2−y2 under tensile distortion enhances thematrix element |⟨dxy|Lz|dx2−y2⟩|2G↓↓Ni and the orbital momentalong the perpendicular direction, leading to the origin of theperpendicular MCA of fcc Ni under tensile distortion.4. Mn-Ga alloysIn section 3, we introduced a bulk system with a perpendic-ular MCA originating from the orbital moment anisotropy.We now discuss a perpendicular MCA arising from the quad-rupole moment of the spin density. Recently, XMCD andXMLD measurements were performed to detect the orbitaland quadrupole moments in Mn-Ga binary alloys to clarifythe origin of the perpendicular MCA of Mn3−δGa alloys [40].Mn3−δGa is one of the candidates that could overcome someof the problems in spintronics devices by reducing the energyconsumption during magnetization reversal and enhancing thethermal stability due to their strong perpendicular MCA withthe ferrimagnetic property.Schematics of the crystal structures of L10-MnGa(δ= 2)and D022-Mn3Ga(δ = 0) are shown in figures 4(a) and (b),respectively. In Mn3Ga, the MnI is located at the same (001)plane with Ga having negative spin moment, while MnII islocated between the two (001) MnI-Ga planes having posit-ive spin moment. Detecting the element specific quadrupoletensor Qzz is possible with the XLMD measurements and thesum rule in the Mn L edges. According to the XMCD andXMLDmeasurements, MnGa and Mn3Ga had relatively largemagnetic dipole moments mT of Mn (of the order of 0.01 µB)while the orbital moment anisotropies ∆morb were negligiblysmall (less than 0.01) despite the perpendicularMCAofMnGaand Mn3Ga.Table 3 shows the calculated spin moments, orbital momentanisotropies ∆morb, and magnetic dipole moments mT of Mnin L10-MnGa and D022-Mn3Ga with the experimental latticeconstant shown in [56, 57], respectively. The details of DFTcalculations are described in [40]. The calculated magneticdipole moments are larger than the orbital moment aniso-tropies both in MnGa and Mn3Ga, demonstrating consistencywith the experimental results. This indicates that the magneticdipole moment related to the cigar-type quadrupole momentof the spin density plays an important role in the perpendicu-lar MCA of these alloys. Figures 4(c) and (d) show the spin-resolved MCA energies in the second-order perturbation ofthe SOI for MnGa and Mn3Ga. We found that the spin-flipterm (E↑↓MCA) of Mn in MnGa and MnII in Mn3Ga are themain contributors to MCA energies. Since Mn3Ga is a fer-rimagnet, MnI and MnII have mutually opposite spin direc-tions, where the contribution of MnI to MCA is smaller thanthat of MnII. As discussed in section 2, the spin-flip termE↑↓MCA of the MCA energy is described by the magnetic dipolemoment, indicating that the origin of the perpendicular MCAcan be attributed to the cigar-type quadrupole moment of thespin density rather than the orbital moment anisotropies. Infigures 4(e)–(g), we show the local density of states (LDOS)10https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewFigure 4. Schematics of crystal structures and spin configurations of (a) L10-MnGa and (b) D022-Mn3Ga. Spin-resolved MCA energies inthe second-order perturbation of the SOI for (c) Mn in L10-MnGa and (b) MnI and MnII in D022-Mn3Ga. Local density of states (LDOS) ofeach d orbital as a function of energy relative to the Fermi energy EF for (e) Mn in L10-MnGa, (f) MnI and (g) MnII in D022-Mn3Ga.Reproduced from [40]. CC BY 4.0.Table 3. Calculated MCA energy EMCA, the spin moment mspin, theorbital moment anisotropy ∆morb, and the magnetic dipole momentmT of L10-MnGa and D022-Mn3Ga with the experimental latticeconstant [56, 57].Calculation L10-MnGa D022-Mn3GaEMCA 1.463 MJm−3 1.856 MJm−3mspin 2.667 µB −3.177 µB (MnI) 2.546 µB (MnII)∆morb 0.0014 µB −0.003 µB (MnI) 0.0064 µB (MnII)mT 0.0493 µB 0.018 µB (MnI) 0.0801 µB (MnII)of L10-MnGa and D022-Mn3Ga by the DFT calculation. In theLDOS of the MnI and MnII sites, all orbital states were splitthrough exchange interaction. However, exchange splittingwas incomplete where complete spin splitting was required,in the Bruno formula, which enabled the transitions by spinmixing between occupied spin-up and unoccupied spin-downstates.To further understand how atomic orbitals contribute to theperpendicularMCA in L10 MnGa andD022-Mn3Ga,we showsin figure 5 the square of the matrix elements of Lz and Lx foreach d orbital ofMn in the spin-conserving and spin-flip terms.As shown in figure 5(a), the matrix elements that contributepositively to the MCA energy of L10-MnGa (perpendicularMCA) are |⟨dyz|Lx|d3z2−r2⟩|2G↑↓Mn and |⟨d3z2−r2 |Lx|dyz⟩|2G↑↓Mn,both of which are spin-flip matrix elements. Although the mat-rix element |⟨dxy|Lz|dx2−y2⟩|2G↓↓Mn shows a large positive value,its spin-flip term |⟨dxy|Lz|dx2−y2⟩|2G↑↓Mn shows a large negativevalue, leading to a small contribution of dxy and dx2−y2 orbitalsto the perpendicular MCA because each term is canceled out.For D022-Mn3Ga, the matrix elements of MnI are much smal-ler than those of MnII. Furthermore, the spin-flip matrix ele-ments of Lx between d3z2−r2 and dyz of MnII had large positivevalues, indicating the origin of the perpendicularMCA. There-fore, the perpendicular MCA of L10 MnGa and D022-Mn3Gaoriginates from the cigar-type distribution of spin density,especially because of d3z2−r2 and dyz orbitals, where the spinmoment of Mn tends to be oriented in the longitudinal (per-pendicular) direction of the spin-density distributions.Because the dependence of MCA on lattice distortion isan important physical aspect, we investigated the change inMCA energies with tetragonal distortion. Figure 6(a) showsMCA energies and MCA contributions of each spin transitionprocess in the second-order perturbation for Mn atom in L10-MnGa as a function of the in-plane distortion ratio. We findthat the spin-flip term E↑↓MCA has the largest contribution to theperpendicular MCA. Whereas its dependence on the in-planedistortion ratio is weaker than that on the total MCA energy.In contrast, the spin-conserving term E↓↓MCA exhibits a strongerdependence on in-plane distortion ratio and connects MCAenergies to lattice distortion, which is consistent with the res-ults for fcc Ni in figure 3.The above behavior is also confirmed in figure 6(b), whereBruno term in equation (17) and van der Laan term inequation (24) for Mn in L10-MnGa are plotted as a function ofthe distortion ratio. Here, we used∆Eexc = 2 eV. These resultsindicate that the orbital moment anisotropy is more sensitiveto lattice distortion than the magnetic dipole moment (quad-rupole moment of spin density), which can be understood interms of an orbital-striction or an orbital-elastic effect as isdiscussed in [39].11https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewFigure 5. Matrix elements of the second order perturbation of SOI between d-orbitals of Mn atoms in L10 MnGa and D022-Mn3Ga for thespin-conserving (↓↓) and the spin-flip (↑↓) processes, where G↓↓Mn is the joint local density of states expressed by equation (6). Positive andnegative values indicate the contribution of the matrix elements to the perpendicular and the in-plane MCA, respectively. Reproduced from[40]. CC BY 4.0.Figure 6. (a) Spin-resolved MCA energies in the second-order perturbation of SOI for L10-MnGa shown in equation (13) as a function ofa∥. (b) Bruno term shown in equation (17) and van der Laan term shown in equation (24) of L10-MnGa as a function of a∥.5. Co/Pd(111) multilayersSo far, we have discussed MCA in bulk materials. In thissection, we discuss MCA for interface systems. Among vari-ous interface combinations, the interaction between 3d TMsand 4d or 5d TMs is considered for a study of the interface-driven MCA because of the large spin moment of 3d TMs andthe strong SOI in 4d and 5d TMs.Ultrathin Co/Pd(111) multilayers are typical artificialnanosystems exhibiting an interface perpendicular MCA. Thedevelopment of synthesized thin films with perpendicular-magnetization has led researchers to expect ultrahigh dens-ity recording media. Furthermore, using Co/Pd interfaces andmultilayers, researchers have demonstrated the photo-inducedprecession of magnetization [58, 59], creation of skyrmionsusing the interfacial Dzyaloshinskii-Moriya interaction [60],and magnetization reversal using the spin-orbit torque [61].Despite much interest in Co/Pd interfaces, the mechanism ofthe perpendicular MCA and the role of Co and Pd sites werenot fully understood. To clarify the origin of the perpendicularMCA of Co/Pd(111), element-specific XMCD measurementsTable 4. The spin moment mspin, the orbital moment anisotropy∆morb, and the magnetic dipole mT of interface Co and Pd inCo(4ML)/Pd(8ML)(111) multilayer obtained from the XMCDmeasurement and DFT calculations. Reproduced from [41].CC BY 4.0.Co PdµB XMCD DFT XMCD DFTmspin 1.82 1.87 0.25 0.31∆morb 0.03 0.032 0.00 −0.001mT 0.0014 −0.0137 0.0014 0.0106for both Co and Pd were performed on ultrathin films ofCo/Pd(111) multilayers [41].In the experiments, ultra-thin multilayered samples of [Co(4 ML)/Pd (8 ML)(111)]5 with an out-of-plane easy axis werefabricated. A thicker Co monolayer (ML) case exhibit in-plane MCA because of large SMA. Then, the spectra of thesesamples were measured for the Pd M2,3 and Co L2,3 edges inthe NI and GI geometries. Table 4 presents the experimental12https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewFigure 7. The dependence of the MCA energy (EMCA), the SMA energy (ESMA), and the effective magnetic anisotropy energy(Eeff = EMCA −ESMA) on the number of Co layer (nCo) for Co(nCo)/Pd(8 ML)(111) multilayer. (b) Spin-resolved MCA energies in thesecond-order perturbation of SOI from equation (13) for interface Co and (c) Pd of Co(4 ML)/Pd(111). Reproduced from [41]. CC BY 4.0.values of mspin, ∆morb, and mT for the sample having out-of-plane magnetization together with those estimated from DFTcalculations.From table 4, a positive ∆morb for Co sites and its negli-gible value for Pd sites were observed in XMCD, indicatingthat the Bruno term (17) in the Co sites is the origin of the per-pendicular MCA of Co/Pd(111). In contrast, a finite mT wasobserved at approximately 0.01 µB for both Co and Pd sites.Because the SOI constant of Pd atoms ξPd is three times thatof Co atoms ξCo, the contribution of van der Laan term (24)to the perpendicular MCA for Pd atoms is three times that ofCo atoms. Furthermore, the observed ∆morb of Co was neg-ligibly small for the sample with thicker Co layers having in-plane magnetization, indicating that∆morb of the interface Cois responsible for the perpendicular MCA.Figure 7(a) shows EMCA, the SMA energy (ESMA =12µ0M2s ), and the effective magnetic anisotropy energy Eeff =EMCA −ESMA of Co(nCo ML)/Pd(8ML)(111), whereMs is thetotal magnetization of Co/Pd(111), and µ0 is the permeabil-ity in vacuum. The in-plane lattice constants and out-of-planeatomic distances of the Co(nCo)/Pd(111) supercells were fullyoptimized for each the number of Co layers (nCo) by DFT cal-culations. Other details of DFT calculations are presented in[41]. EMCA (red line points) decreases with an increase in nCo,indicating that interface Co has the main contribution toMCA.Because the SMA energy of Co/Pd(111) (blue line points)increases with increasing nCo, Co/Pd(111) has a negative Eeffaround nCo = 6∼ 7 and prefers in-plane magnetization, whichis consistent with the experimental result.To confirm these results, we performed the second-orderperturbation analyses of Co/Pd(111) multilayers. Figures 7(b)and (c) show the spin-resolved MCA energies in thesecond-order perturbation of the SOI for the interface Coand Pd atoms in Co(4ML)/Pd(111). We find that for theinterface Co, the spin-conserving term E↓↓MCA is the maincontributor to the perpendicular MCA. Whereas for the inter-face Pd, the spin-flip term E↑↓MCA shows the main contribution.Because E↓↓MCA and E↑↓MCA are related to ∆morb and mT,these results are also consistent with the experimental resultsin table 4.To examine the layer-by-layer contributions to MCA ener-gies of Co/Pd(111), we show in figure 8 Bruno and van derLaan terms estimated from ∆morb and mT at each atomic siteof Co(4ML)/Pd(8ML)(111). To obtain van der Laan term, weuse ∆exc = 4 eV. It is noted that the value of exchange split-ting ∆exc = 4 eV is too large for nonmagnetic element Pd.However, the ∆exc is defined by equation (19) indicating theenergy range of eigenstate ϵ⃗k,n. Because the valence statesof Pd in Co/Pd(111) shows broad energy range correspond-ing to 4∼ 5 eV due to delocalized character of 4d element(see figure 5(d) of [41]), we use 4 eV for ∆exc. As shownin figure 8, the Bruno term increases only at the interface Cosites, whereas van der Laan term at the Pd sites. This indicatesthat an interface-driven perpendicular MCA originates fromboth the orbital moment anisotropy and cigar-type quadrupolemoment of spin density (corresponding to positive magneticdipole moment). In Co/Pd(111), owing to the hybridizationbetween the Co 3d and Pd 4d orbitals, there is large proximitythat induces SOI in the Co states and spin polarization in thePd states, which leads to a large induced spin moment in Pd ofapproximately 0.3 µB.Moreover, the strong proximity effect in the Co/Pd(111)multilayer leads to the orbital moment anisotropy in Co site∆morb and the magnetic dipole moment in Pd site mT corres-ponding to the quadrupole moment of spin density. From theLDOS of interfacial Co in the Co/Pd(111) multilayer, the Co3dxy, 3dx2−y2 , 3dyz, and 3dzx orbitals contribute to the perpen-dicular MCA because of being dominant at the Fermi level inthe minority-spin states [41]. In addition, these states providelarge spin-conserving matrix elements of the interface Co,such as |⟨dx2−y2 |Lz|dxy⟩|2G↓↓Co and |⟨dyz|Lz|dzx⟩|2G↓↓Co, enhan-cing the perpendicular components through the Bruno term.Furthermore, we found large spin-flip matrix elements of theinterface Pd, such as |⟨dyz|Lx|dx2−y2⟩|2G↑↓Pd , contributing to thecigar-type distribution of spin density and the perpendicularMCA through van der Laan term.13https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewFigure 8. Bruno term shown in equation (17) and van der Laan term shown in equation (24) at each atomic site of Co and Pd inCo(4ML)/Pd(8ML)(111) multilayer. The atomic structure of Co(4ML)/Pd(8ML)(111) is shown below the graph, where each atomicposition along z axis corresponds to the points in the graph. Reproduced from [41]. CC BY 4.0.6. Fe/MgO(001) interfaceFinally, we discuss the MCA of Fe/MgO(001) interfacesin MTJs, which are important for spintronics applications.Fe/MgO-based MTJs exhibit large tunneling magnetoresist-ive (TMR) ratios of over 400% at room temperature becauseof coherent tunneling of the highly spin-polarized∆1 evanes-cent states through the MgO barrier [62–64]. To enhance thethermal stability of magnetization directions at a finite temper-ature, MTJs with perpendicular magnetization are required.However, because body-centered cubic (bcc) Fe and FeCoalloys have cubic structures in bulk, we cannot expect a largeperpendicular MCA in the bulk electrode regions of MTJs.Thus, the perpendicular MCA of the interface plays an import-ant role in stabilizing the magnetization directions while pre-serving the large TMR ratios.Maruyama et al reported perpendicular magnetization of aFe/MgO(001) interface and a large voltage-controlled MCA(VCMA) change in a few atomic layers of Fe [44]. Ikeda et alshowed both a high TMR ratio of over 120% at room tem-perature and the perpendicular magnetization of CoFeB/MgO/CoFeB(001) MTJs when the CoFeB layer was approxim-ately 1.3 nm thick [65]. Furthremore, a large perpendicularMCA of 1.4MJm−3 in Fe/MgO(001) was observed for 0.7 nmthick Fe layer with adsorbate-induced surface reconstruction[66]. XMCD measurements were also performed for ultrathinFe/MgO(001), and the orbital moment anisotropy was domin-ant at the Fe/MgO interface perpendicular to MCA; the con-tribution of quadrupole moments was small but finite at thelattice distorted interfaces [43].Several studies have discussed the origin of the perpendic-ular MCA of Fe/MgO(001) and the VCMA effect in termsof the orbital moment anisotropy and hybridization of the Fe3d3z2−r2 orbital with the O 2pz orbital [67–71] from a the-oretical perspective. However, the effects of the quadrupolemoments (anisotropy of the spin-density distribution) on theperpendicular MCA and the correlation between the latticedistortion and MCA have not been thoroughly discussed forFe/MgO(001) [42, 72].We calculated theMCA energies of theFe(nFe)/MgO(5 ML)(001) interface for various layer Fe layerthickness nFe as a function of the in-plane lattice constant a∥,where a∥ was changed from aFe = 2.8309 Å to aMgO/√2=3.0043 Å with an interval of approximately 0.03 Å. The val-ues of aFe and aMgO were obtained using DFT calculations forbulk bcc Fe and rock-salt MgO. In addition, we changed thenumber of Fe-layer nFe from 5 ML to 11 ML in Fe/MgO(001)supercells. The out-of-plane atomic distances of the supercellswith each a∥ were fully optimized by DFT calculations. Otherdetails of DFT calculations were the same as those in [42].Figure 9(a) shows the MCA energy of Fe/MgO(001) inter-face as a function of a∥. First, we found that the MCA ener-gies of Fe/MgO(001) exhibited similar a∥ dependences, irre-spective of nFe, although there were some differences. Weconfirmed that the Fe/MgO(001) supercells with nFe ≦ 4 MLshowed totally different a∥ dependence compared to that withnFe ≧ 5 ML in figure 9(a). This indicates that nFe ≧ 5 MLis necessary to correctly describe the characteristics of theFe/MgO(001) interface. The MCA energy of Fe/MgO(001)in figure 9(a) exhibits a non-monotonic behavior with respectto the change of the in-plane lattice constant; it graduallyincreases and reaches a maximum around a∥ = 2.92 Å andthen gradually decreases with increasing a∥. This result is con-sistent with the previous calculation results [72]. Figure 9(b)shows the out-of-plane interlayer distance as a function of a∥for nFe = 11 ML. The result shows a monotonic decrease withincreasing a∥, indicating that the non-monotonic behavior ofMCA energies with tetragonal distortion do not originate fromthe structural properties of Fe/MgO(001).To clarify the tetragonal distortion effect on MCA inFe/MgO(001), we calculated the spin-resolved MCA energiesin second-order perturbation of the SOI EMCAσσ ′ for the inter-face Fe atom as a function of the in-plane lattice constant14https://creativecommons.org/licenses/by/4.0/J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewFigure 9. (a) The tetragonal distortion dependence of MCA energies of Fe(nFe)/MgO(5 ML)(001) for nFe = 5, 7, 9 and 11 ML. (b) Theoptimized (001) interlayer distance of Fe(11 ML)/MgO(5 ML)(001) interface, bulk MgO, and bulk Fe as a function of the in-plane latticeconstant a∥. The atomic structure of Fe/MgO(001) interface is shown in the inset.Figure 10. (a) Spin-resolved MCA energies in the second-order perturbation of SOI for Fe(11 ML)/MgO(5 ML)(001) interface shown inequation (13) as a function of a∥. (b) Bruno term in equation (17) and van der Laan term in equation (24) of Fe/MgO(001) interface as afunction of a∥.a∥. Figure 10(a) shows that the spin-conserving term EMCA↓↓and spin-flip term EMCA↑↓ mainly contribute to the distortiondependence of MCA. The spin-conserving term EMCA↓↓ exhib-its non-monotonic behavior for a∥, whereas the spin-flip termEMCA↑↓ monotonically increases with an increase in a∥. Thespin-conserving term EMCA↓↓ is slightly larger than the spin-flipterm EMCA↑↓ in Fe/MgO(001) interface.To confirm the relationship between the MCA energy andthe orbital and quadrupole moments, we show in figure 10(b)Bruno and van der Laan terms of Fe interface as a functionof a∥. To obtain van der Laan term, we use ∆exc = 4 eV.Figure 10(b) shows that the tetragonal distortion dependenceof Bruno and van der Laan terms are consistent with those ofthe spin-conserving term EMCA↓↓ and the spin-flip term EMCA↑↓ ,respectively. This means that the orbital moment anisotropyand magnetic dipole moment effectively describe the MCA ofFe/MgO(001) with tetragonal distortion.The increase in van der Laan term (the spin-flip term)with increasing a∥ indicates that Fe/MgO(001) interfacehas a cigar-type distribution of spin density with tensiledistortion, and the perpendicular MCA can be stabilizedby orienting the spin moments along the longitudinal dir-ection of the spin-density distributions. Because the Brunoterm (the orbital moment anisotropy) of Fe/MgO(001) fora∥ = aFe has approximately the same value as that for a∥ =aMgO, the increase in the perpendicular MCA owing to thetensile tetragonal distortion of Fe/MgO(001) is mainly causedby the additional contribution of the quadrupole momentof spin density around the Fe interface. The XMCD andXMLD measurements for Fe/MgO(001) reported that theorbital moment anisotropy was dominant in Fe/MgO(001),with a finite contribution of the quadrupole moment of spindensity.Our calculation results suggest that the contribution of theanisotropy of quadrupole moment is enhanced when the in-plane lattice constant is close to that of MgO. The contribu-tion of the spin-flip term in EMCA is expected to be sensitivenot only to the in-plane lattice constant, but also to the appliedelectric field. As a next step, we intend to clarify the relation-ship between the MCA and lattice distortion, together with theVCMA effects at various magnetic interfaces.15J. Phys.: Condens. Matter 34 (2022) 473001 Topical Review7. DiscussionsConsidering above case studies, we discuss the modulationof orbital moments by local strain. The orbital moments arestrongly affected to the anisotropic local environments in thenearest sites through the hybridization. In the case of highlysymmetric cubic structure, orbital moments are completelyquenched. When the stress can be applied along some direc-tion, the orbital hybridization along this strain direction is pref-erentially modulated. Within the orbital sum rule in XMCD,the electron occupation in 3d states which is modulated bySOI can be controlled by external strain, resulting in the mod-ulation of orbital moments [73]. Therefore, the relationshipbetween strain and orbital moments can be systematically for-mulated. Our studies in this review generalize the orbital con-trol by strain for some cases.Until now, magneto-striction or magneto-elastic effect hasbeen recognized as strain effect in magnetism as a phe-nomenologically macroscopic understanding for some mater-ials which are quite essential for applications such as motor,actuator, and mobile devices [74]. However, the microscopicunderstanding considering the electronic structures and orbitalstates has not been clarified yet. Recent studies using ultra-thin films strongly require more detailed analysis for strictiveeffect. Now, we develop novel concept of orbital-strictive ororbital-elastic effect using the results of previous sections.In the case of Ni/Cu multilayers in section 3, the orbitalmoments are formulated as a linear relation with strain, whichcan be understood as orbital-elastic effect. A linear relationoriginated from the enhancement of orbital moments in thespin-conserving electron motion within the in-plane directionat the interfaces, which is categorized as the case of figure 1(a).As shown in figure 2(a), orbital moments are related to in-plane strain, resulting in the perpendicular MCA energy. Inthis case, the direction of strain and enhanced orbital momentsare orthogonal in principle. Similar scenario can be adoptedto the perpendicular MCA in Co-ferrite CoFe2O4 as orbital-elastic effect in Co2+ site with large orbital moment [75].Second type can be understood as quadrupole cases; the dir-ection of strain becomes an easy axis. In this case, the contri-bution from orbital moments is small and the charge distribu-tion along elongated direction stabilizes the MA. The relationbetween strain andMCA does not exhibit a linear and spin-flipcontribution is dominant for the MCA, which is typical in thecase of Mn3−δGa because of the small contribution of orbitalmoment in Mn compounds. These two-types of orbital-elasticeffects can be categorized as a microscopic origin of strictivephenomena from the viewpoints of the electronic structures.Other cases for MCA can be also proposed using asym-metric Rashba-type SOI [76]. For example, Au(111) surfacehas a strong Rashba-type SOI, which induces the MCA onthe deposited ultra-thin Fe layer. Since the potential profilebetween Fe/Au is different, the interfacial electric gradientpromotes the change of charge distributions, which is alsodetected by XMCD and other spectroscopic probes [77, 78].Therefore, the MCA from asymmetric momentum space canbe also developed as other orbital-strictive phenomena includ-ing a topological physics.As discussed in [79, 80], the of MCA energy obtained bythe first-principles calculations often requires some scalingfactors to compare the experimental results. This can be attrib-uted to many factors related to problems both theoretical andexperimental points of view. In the calculations, the Brunoterm requires to neglect the majority-spin spin-conservingterm and the spin-flip terms, which leads to the requirementof the scaling factors. In addition, the first-principles calcula-tions do not include the second Hund’s law, which leads to theunderestimation of the orbital moment [32]. Furthermore, inthe relationship between the spin-flip MCA term and the mag-netic dipole moment, the exchange splitting ∆exc acts as thescaling factor. On the other hand, in the experimental points ofview, the MCA energy tends to be smaller than the theoreticalpredictions due to problems of crystallinity, interface rough-ness and degree of order of magnetic thin films, leading to thescaling factors in the Bruno and van der Laan terms.8. SummaryIn this review, we discussed the perpendicular MCA of mag-netic materials and their interfaces based on the orbital andquadrupole moments. First, we reviewed the relationshipamong the orbital moments, quadrupole moments, and MCAbased on the detailed formulation of the second-order per-turbation of the SOI. We argued that the orbital momentstabilizes the spin moment parallel to the direction with a lar-ger orbital moment, whereas the quadrupole moment stabil-izes the spin moment along the longitudinal direction of thespin-density distribution. These effects are expressed as thespin-conserving Bruno term (related to the orbital momentanisotropy) and the spin-flip van der Laan term (related to theanisotropy of the quadrupole moment and the magnetic dipolemoment). We demonstrated that the contributions of d orbitalsto these two effects are mutually opposite. In the Bruno term,in-plane d orbitals, such as dx2−y2 and dxy, provide a perpen-dicular MCA, whereas out-of-plane d orbitals, such as d3z2−r2 ,prefer an in-plane MCA. In contrast, in van der Laan term,the out-of-plane and in-plane d orbitals provide a perpendic-ular MCA and an in-plane MCA, respectively. The MCA ofmagnetic materials and interfaces with TMs can be determ-ined from the competition between these two contributions.We then applied our formulations of MCA to various mag-netic systems by comparing the theoretical results with theXMCD and XMLD measurements. We showed that fcc Niwith tensile tetragonal distortion shows perpendicular MCAarising from the Bruno term (the orbital moment anisotropy),while MnGa alloys such as L10-MnGa and D022-Mn3Ga canbe attributed to van der Laan term (the quadrupole moment).Furthermore, the MCA of magnetic systems with interfaceswas discussed. We found that the perpendicular MCA of theCo/Pd(111) multilayer originates from the orbital momentanisotropy of the interface Co and the anisotropy of the quad-rupole moment at the interface Pd. Finally, we examined thetetragonal distortion dependence of MCA for Fe/MgO(001)interfaces, which are important systems as the perpendicularlymagnetized MTJs with high TMR ratios. The perpendicular16J. Phys.: Condens. Matter 34 (2022) 473001 Topical ReviewMCA of Fe/MgO(001) exhibited a non-monotonic behaviorwith respect to the in-plane lattice constant, which can beattributed to both Bruno term (the orbital moment anisotropy)and van der Laan term (the quadrupole moment of spin dens-ity). These fundamental understandings ofMCAwill be essen-tial in the theoretical design of novel magnetic materials andinterfaces, as well as for the control of MCA through latticedistortion and applied bias voltage.Data availability statementThe data generated and/or analysed during the currentstudy are not publicly available for legal/ethical reasons butare available from the corresponding author on reasonablerequest.AcknowledgmentsWe are grateful to S Mitani, H Sukegawa, and K Masudaat NIMS, H Yanagihara at the University of Tsukuba,Y Kota at Fukushima College, M Shirai and A Sakumaat Tohoku University for their valuable discussions on ourwork. For sections 3–5, we acknowledge several collaborators,H Munekata at Tokyo Institute of Technology, T Taniyamaat Nagoya University, S Mizukami and K Z Suzuki atTohoku University who prepared excellent samples andprovided faithful comments. Y M sincerely thanks Y Suzukiat Osaka University for showing calculation notes regardingdetailed derivations of the equations in [55]. This work waspartly supported by the Grants-in-Aid for Scientific Research(Grant Nos. JP16H06332, JP20H00299, JP20H02190 andJP22H04966) from the Japan Society for the Promotionof Science(JSPS), Center for Spintronics Research Network(CSRN) of Osaka University, and Cooperative Research Pro-ject Program of the Research Institute of Electrical Commu-nication (RIEC), Tohoku University.ORCID iDsYoshio Miura https://orcid.org/0000-0002-5605-5452Jun Okabayashi https://orcid.org/0000-0002-9025-2783References[1] Sander D 2004 J. Phys.: Condens. Matter 16 R603–36[2] Dieny B and Chshiev M 2017 Rev. Mod. 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Phys.: Condens. Matter 34 (2022) 473001 Topical Review[53] The relationship between the spin-flip term of the second orderperturbation of SOI and the quadrupole moment of spindensity is first proposed by Wang et al [24]. However, sincevan der Laan’s works [28, 29], which were later developedinto the relationship with XMCD and XMLDmeasurements, have been more widely known, it would beeasier for the reader to call it van der Laan term. Therefore,we refer to the quadrupole terms as van der Laan terms inthis paper as well[54] In the derivation of Bruno term and van der Laan term, weassume that the magnetic materials with more-than-halfelements such as Fe, Co, and Ni have fully occupiedmajority-spin states. In the calculation of the orbital and themagnetic dipole moment, however, we consider theunoccupied majority-spin states. 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Introduction 2. Second-order perturbation calculations of SOI 2.1. MCA energies 2.2. Orbital moments 2.3. Quadrupole moments 2.4. Intuitive understanding of MCA based on orbital and quadrupole moments 3. Fcc Ni with in-plane distortions 4. Mn-Ga alloys 5. Co/Pd(111) multilayers 6. Fe/MgO(001) interface 7. Discussions 8. Summary References