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[Ivan Kurniawan](https://orcid.org/0000-0001-5419-0047), [Keita Ito](https://orcid.org/0000-0002-3682-8614), [Takeshi Seki](https://orcid.org/0000-0003-3195-7051), [Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[Microscopic correlation between magnetostriction and magnetic damping](https://mdr.nims.go.jp/datasets/8384f47c-4abf-48bb-b8f8-3ed023a068a7)

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Microscopic correlation between magnetostriction and magnetic dampingIvan Kurniawan,1, ∗ Keita Ito,2 Takeshi Seki,2, 3 Keisuke Masuda,1 and Yoshio Miura1, 41Research Center for Magnetic and Spintronic Materials,National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan3Center for Science and Innovation in Spintronics, Tohoku University, Sendai 980-8577, Japan4Faculty of Electrical Engineering and Electronics,Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, 606-8585, JapanAlthough the relationship between magnetostriction and magnetic damping is often describedphenomenologically, their intrinsic connection remains unclear. In this study, we demonstrate thatthe magnitude of magnetic damping depends on the sign of magnetostriction in (Fe1–xCox )4Nand Ni1–yCoy alloys across various compositions, consistent with experimental observations. Thisbehavior is attributed to strain-induced changes in exchange splitting, which shift the minorityspin density of states near the Fermi level, thereby affecting both magnetostriction and dampingthrough spin-conserving transitions. Additionally, the presence of locally degenerate orbitals playsa crucial role in determining magnetostriction. These findings suggest that magnetization dynamicsand magnetostriction can be intrinsically controlled, facilitating the design of magnetic materialsfor applications such as flexible spintronics.The first observation of the spatial deformation of mag-netic materials under an applied magnetic field was re-ported by Joule for iron in 1842 [1], later termed mag-netostriction (λ), which is defined as a change in length(δl/l) between the demagnetized state and the magnet-ically saturated state along the magnetic field direction.At that time, Joule found that the elongation in soft ironbars is more prominent than in hardened steel [1], imply-ing that magnetostriction is inversely proportional to theelastic constants. It was later understood that contrac-tion (negative magnetostriction) is exhibited in nickel [2];hence, magnetostriction is not solely determined by theelastic constant but is also affected by the strain depen-dence of magnetic anisotropy energy (MAE) (which werefer to here simply as magnetoelasticity). When magne-toelasticity is nonzero, the preferred magnetic directionis changed by the strain; hence, the crystal structure willspontaneously deform in order to lower the total energy,which could result in elongation or contraction [3]. Mag-netostriction is expressed as the ratio of the magnetoe-lasticity to the elastic constant [3]. Since the elastic con-stant of typical magnetic materials is generally expectedto have a positive sign, magnetoelasticity predominantlydetermines the sign and magnitude of magnetostrictionover compositional change. Subsequently, first-principlescalculations were introduced by Wu et al. [4], providingintrinsic insights into the mechanisms of magnetostric-tion. This sparked significant interest, especially afterthe discovery of giant magnetostrictive materials like Ter-fenol [5].Around the same time as the establishment of first-principles studies on magnetostriction, a phenomenolog-ical model was proposed to explain the role of magne-tostriction in the magnetic damping constant (α) [6].∗ kurniawan.ivan@nims.go.jpThis model introduced interrelated equations of motionfor both strain and magnetization, resulting in tempo-ral relaxation expressed as magnetic damping, which isproportional to the square of magnetostriction. This in-terpretation was further supported by other phenomeno-logical models [7, 8]. Experimentally, Heusler alloys [9]demonstrated this proportional relationship. Meanwhile,variations in the composition of NixFe1–x revealed twodistinct regions: negative (positive) magnetostriction wasassociated with a large (small) damping constant, thoughthe underlying mechanism remains unknown [10]. It issignificant to note that both magnetoelasticity and mag-netic damping originate from spin-orbit interaction (SOI)[4, 11]. Moreover, the relationship between magnetostric-tion and magnetic damping is particularly valuable forspintronics applications, as it enables simultaneous op-timization of energy efficiency and multidirectional sens-ing, which are critical for flexible spintronics technologies[12].Since SOI links magnetostriction and damping, a con-ventional approach to control both would involve tuningSOI strength, ξ, as expressed in the spin-orbit Hamilto-nian HSO =∑I ξI L⃗ · S⃗, where L⃗ = (Lx, Ly, Lz) and S⃗ =(Sx, Sy, Sz) represent the orbital and spin angular mo-mentum operator, respectively. This has been studied byinclusion of heavy elements for damping in FePt1–xPdx[13] and magnetostriction in Tb-doped FeGa [14]. How-ever, practical usage of heavy elements is avoided consid-ering their scarcity, high cost, and environmental dam-age. We recently observed a strong intrinsic link be-tween magnetostriction and damping in (Fe1–xCox )4N,supported by theoretical calculations considering shiftedFermi level EF [15], as the calculated results for unshiftedEF shown in Fig. 1(a). The λ100 corresponds to themagnetostriction due to tetragonal distortion, which em-pirically means a change in length along the [001] direc-tion when the magnetization direction changes from theab plane to the c direction. In this Letter, we demon-2FIG. 1. Calculated magnetostriction (λ100) and damping con-stant (α) for (a) (Fe1–xCox )4N and (b) Ni1–yCoy . Crystalstructure of (a) cubic Fe4N and (b) fcc-Ni, showing sites ofFe1 (black), Fe2 (blue), Fe3 (green), Fe4 (red), N (grey), andNi (yellow) with the corresponding coordinate systemstrate that there exists the intrinsic link between mag-netostriction and magnetic damping not only for the(Fe1–xCox )4N but also for the Ni1–yCoy from their elec-tronic structure, facilitated by spin-conserving transi-tions, enabling simultaneous control of spatial and tem-poral responses under magnetic fields without significantSOI changes. These materials are also expected to un-dergo the sign change of magnetostriction by changing ofx (y) value, where a similar phenomenon is also found inthe NixFe1–x [10]. Interestingly, our calculation showsthat magnitude of damping seems affected by the sign ofmagnetostriction in [100] direction, where we may pro-pose a possible scenario. This finding may improve theunderstanding the relation behind magnetostriction andmagnetic damping.First-principles density functional calculations wereperformed using VASP [16] to study (Fe1–xCox )4N andNi1–yCoy , each of which based on the basic cubic struc-ture of Fe4N and fcc–Ni, respectively as illustrated inFig. 1. We employed the spin-polarized generalizedgradient approximation (GGA) [17] for the exchange-correlation energy and the projector augmented wave(PAW) [18] method to accurately account for core-electrons effects. Starting from the cubic prototype, weoptimized the lattice parameter by computing the to-tal energy over a range of lattice constants and per-forming a spline interpolation to identify the equilibriumlattice constant, c0, that minimizes the energy. Fromthis optimized cubic geometry, we constructed the un-strained unit cells: a single-formula-unit cubic cell for(Fe1–xCox )4N and a two-formula-unit tetragonal cell forNi1–yCoy (rotated 45◦ in-plane relative to the originalcubic cell). Strained structures were generated by vary-ing the out-of-plane lattice parameter c; for each strainvalue ε = (c−c0)/c0, we fully relaxed the in-plane latticeparameters a(= b), again using spline interpolation tolocate the minimum-energy configuration. The k-pointmeshes of 30×30×30 for (Fe1–xCox )4N and 25×25×17for Ni1–yCoy ensured convergence of the MAE, consis-tent with similar systems studied previously [19, 20].Without considering SOI, the face-centered sites (Fe2,Fe3, Fe4) exhibit identical atomic environments [21]. Foroff-stoichiometric compositions between Fe-Co and Ni-Co, the virtual crystal approximation (VCA) [22] wasapplied equivalently to all Fe sites (both corner and face-centered) in Fe4N and Ni sites, respectively. This as-sumption for Fe4N is warranted by previous theoretical[23] and experimental results [24], indicating that Co hasno preferential site in Fe4N. In addition, our choice touse VCA is justified by good agreement with experimen-tal results for both of (Fe1–xCox )4N [15] and Ni1–yCoy[25]. In our implementation, each ”virtual” atom’s po-tential, formal valence, augmentation charges, and non-local pseudopotential were mixed according to the tar-get weighted composition, eliminating the need for largesupercells since compositional effects are fully capturedinformation within simple unit cell. Because the VCA isapplied uniformly at all equivalent sites, inversion sym-metry is preserved in this calculation.Magnetostriction was calculated from the derivative ofE001 and MAE with respect to ε using the following equa-tion [4, 26]:λ100 =23dMAE/dεd2E001/dε2(1)The values of E001 and MAE were obtained by includ-ing the SOI in a self-consistent calculation for each dif-ferent strained structure (total energy SCF method).We imposed a strict total-energy convergence criterionof 10−7 eV and employed the tetrahedron method forBrillouin-zone integration. Here, we define MAE asMAE = E100 −E001, where E100 and E001 represent theenergies of magnetic materials with magnetization orien-tations in [100] and [001] directions, respectively. Fur-3thermore, using the second-order perturbation analysiswith respect to the SOI [27, 28], we clarified the dominantatomic, orbital, and spin contributions to MAE hencealso the dMAE/dε. This perturbative approach is par-ticularly appropriate here because (1) it yields a nonzeroMAE already at second-order under strain (whereas forthe unstrained cubic lattice, nonzero contributions onlyappear at fourth order or higher), and (2) inversion sym-metry remains intact thanks to the uniform implemen-tation of the VCA, so no additional intraband termsare required (in contrast to low-symmetry cases [29, 30].We also verified that MAE values from the perturbativeanalysis closely agree with those obtained via both thetotal-energy SCF method and the force-theorem meth-ods, which can be used to understand the main contri-bution to the strain dependence of MAE.MAE = MAE(↑⇒↑) +MAE(↓⇒↓)+MAE(↑⇒↓) +MAE(↓⇒↑) (2)The magnetic damping was calculated using the torquecorrelation model [11, 31], which is derived from non-collinear magnetic calculations including the SOI.α =gπMs∑k∑nn′|Γ−nn′(k)|2× δ(EF − ϵkn′σ)2 + δ2δ(EF − ϵknσ)2 + δ2, (3)where the g is electron’s g-factor, Ms is the saturationmagnetization, and Γ−nn′(k) = ⟨kn′σ′|[S−, HSO]|knσ⟩,which is the matrix element of the spin-orbit torque op-erator [S−, HSO] =∑I ξI(S−Lz − SzL−) between eigen-states including the SOI, with energies ϵknσ labeled withwavevector k, band index n, and spin σ. These transi-tions occur near EF and vanish after interaction with lat-tice, which are phenomenologically parameterized withscattering rate δ. In this study, δ is estimated from theresidual resistance of typical magnetic alloys (0.01 eV)[32].The intrinsic relationship between magnetostrictionand damping in both (Fe1–xCox )4N and Ni1–yCoy isshown in Fig. 1. By changing the composition, we maysimultaneously tune the magnetostriction and damping.In addition, one may notice that there are two compo-sition regions: a region that has negative magnetostric-tion, which also exhibits large magnetic damping, andregions with positive magnetostriction that are associ-ated with small magnetic damping. Although one datapoint from Fe4N deviates from this general trend, mostdata points cluster within the described regions. To ex-plore this relationship further, we focus on (Fe1–xCox )4Nfirst. The second-order perturbation analysis reveals thesignificant contributions to dMAE/dε for (Fe1–xCox )4N[Figs. 2(a)-(c)]. As shown in Fig. 2(a), the gradient oftotal dMAE/dε corresponds to the sign and magnitude ofthe magnetostriction coefficient. From the second-orderperturbation analysis of MAE [27], it is found that theFIG. 2. Strain dependence of (a) total, (b) ↓⇒↓, and (c)↑⇒↓ contribution to the MAE calculated from second-orderperturbation analysis for (Fe1–xCox )4N. (d) Spin-resolveddensity of states of (Fe1–xCox )4NFIG. 3. Strain dependence of the ↓⇒↓ normalized orbitalcontribution to the MAE of the (Fe1–xCox )4N calculated fromsecond-order perturbation analysis for (a) dxz - dyz of Fe2, (b)dxz - dyz of Fe3 (Fe4) , (c) dx2−y2 - dxy of Fe2, and (d) dx2−y2- dxy of Fe3 (Fe4)spin-conserving ↓⇒↓ [Fig. 2(b)] and spin-flip ↑⇒↓ [Fig.2(c)] contributions dominate the matrix elements nearEF. In order to understand the role of each contribution,let us focus on three compositional regions. First, atlow Co contents (x < 0.3), the spin-conserving process ismore dominant than the competing spin-flip term, whichcontrols the behavior of the total MAE. At moderatelevel of Co content (0.3 ≤ x ≤ 0.5), dMAE(↓⇒↓)/dε be-come smaller compared to the positive dMAE(↑⇒↓)/dεleading to a sign change in λ100. When Co content isfurther increased (x > 0.5), both of dMAE(↓⇒↓)/dεand dMAE(↑⇒↓)/dε are positive. The negligible con-tributions of spin-conserving ↑⇒↑ and spin-flip ↓⇒↑ arenot shown. This behavior can be understood from theelectronic structure of (Fe1–xCox )4N with more thanhalf-filled transition metal elements, where the majority-spin states are fully occupied and the electronic states4around EF mainly come from the minority-spin states[Fig. 2(d)].The relationship between magnetic damping and mag-netostriction can be explained as follows. An intrinsicmagnetic damping is described using the torque correla-tion model proposed by Kambersky [11], as expressed inEq. (3). In general, the precession motion of magneti-zation is damped by interactions with conduction elec-trons through the SOI. This interaction is mediated byspin-orbit torque, which contains operators for spin-flip(S−Lz) and spin-conserving (SzL−) contributions. Notethat |knσ⟩ = Σjµcknjµσ|µσ⟩eikRj , where the Bloch states|knσ⟩ are expanded with an orthogonal basis of atomicorbitals labeled µ (or ν) for site j with atomic position Rjin the unit cell and the coefficient of the expansion cknjµσfrom projected state on each localized atomic orbitals.The precession is annihilated and resulting in the creationof the electron-hole pairs which may occupy the same(intraband) or different (interband) band indices n, n′.This process concludes with scattering between electron-hole pairs and the lattice, parameterized by the electron-lattice scattering rate δ into an equilibrium state. It isknown that intraband terms are proportional to the den-sity of states at EF [33]. Assuming a pure spin state atEF, spin-flip contributions may be neglected in the in-traband mechanism, leaving the spin-conserving matrixelements:Γ−nn′(k) =∑IξI∑µµ′ckn′∗Iµ′↓cknIµ↓⟨µ′|L−|µ⟩ (4)Meanwhile, the spin-conserving term in MAE can also beexpressed as follow:MAE(↓⇒↓) =∑II′ξIξI′∑νν′µµ′[⟨ν|Lz|ν′⟩⟨µ′|Lz|µ⟩−⟨ν|Lx|ν′⟩⟨µ′|Lx|µ⟩]G↓↓II′(νν′;µµ′) (5)where theG↓↓II′(νν′;µµ′) =∑kocc∑nunocc∑n′ckn∗I′ν↓ckn′I′ν′↓ckn′∗Iµ′↓cknIµ↓ϵkn′σ′ − ϵknσ(6)Thus, spin-polarized magnetic materials with dominantminority-spin states exhibit nonzero cknIµ↓ values, giv-ing nonzero spin-conserving transition matrix elementsof the angular momentum operator responsible for bothdamping (⟨µ′|L−|µ⟩) and magnetoelasticity (⟨µ′|Lz|µ⟩,⟨µ′|Lx|µ⟩). This connection facilitates the strong cor-relation between magnetostriction and magnetic damp-ing. Additionally, one can propose an intuitive picturewhere negative magnetostriction is associated with rela-tively large magnetic damping. In all the systems, theincrease of the strain from negative to positive increasesthe magnetization as shown in Fig. S1 in Supplemen-tal Materials [34]. This means that the increase of thestrain increases the exchange splitting of the system. Inother words, the increase of the strain shifts the majority-spin and minority-spin states to the lower and higher en-ergy sides, respectively. These effects couple the strainand the magnetic damping through the density of states.When EF is located slightly to the left of a peak inthe minority spin states (for small x and y values in(Fe1–xCox )4N and Ni1–yCoy cases, respectively), it re-sults in large damping due to the proximity to the peak ofthe minority spin states. At the same time, there is a neg-ative strain dependence of the density of states becausestrain pushes the peak farther from EF, thereby reduc-ing the states at EF. This leads to negative magnetoe-lasticity and magnetostriction, because of the reductionof the spin-conserving term in the MAE with increasingthe strain. Conversely, when EF is situated in the val-ley of the minority spin states (for large x and y valuesin (Fe1–xCox )4N and Ni1–yCoy cases, respectively), thesystem exhibits small damping. In this scenario, strainenhances the states at EF by bringing them closer to thenearest peak of occupied minority spins, leading to posi-tive magnetostriction.In addition to the relationship between magnetostric-tion and damping, the substitution of Fe by Co resultsin a sign change in magnetostriction in (Fe1–xCox )4N,as shown in Fig. 1(a). We now examine the role ofatomic sites and orbitals in this sign change of the ↓⇒↓magnetoelasticity of (Fe1–xCox )4N, as shown in Fig.3. It was observed that the face-centered atoms (Fe2,Fe3, Fe4) significantly influence the strain dependenceof the MAE, compared to the corner site (Fe1) (notshown). Notably, the dominant orbital contribution in(Fe1–xCox )4N varies depending on the symmetry of theatomic sites when the magnetization direction is changed.For Fe2 (aligned along the c-axis), the MAE contributionfrom the transition between dxz and dyz strongly corre-lates with the compositional dependence of the magne-toelasticity [Fig. 3(a)]. This substantial contributionto MAE persists because both orbitals remain degen-erate even under tetragonal distortion [see Fig. S2(b)in Supplemental Materials [34]]. Furthermore, for Fe3(aligned along the b-axis) and Fe4 (aligned along thea-axis), the dx2−y2 and dxy orbitals dictate the magne-toelastic behavior [Fig. 3(d)]. These dx2−y2 and dxyorbitals do not remain degenerate under tetragonal dis-tortion at the same atom, but their similar behavior inFe3-dx2−y2 (Fe4-dx2−y2) and Fe4-dxy (Fe3-dxy) can beexplained by the symmetry invariance of the dx2−y2 anddxy orbitals between Fe3 and Fe4 sites [see Fig. S2(c)-(d) in Supplemental Materials [34]]. It is important tonote that similar contributions were not observed for dxzand dyz in Fe3 or Fe4 [Fig. 3(b)], nor for dx2−y2 anddxy in Fe2 [Fig. 3(c)], nor any other orbitals [see Fig.S3 in Supplemental Materials [34]], highlighting the roleof the symmetry in the orbital contribution to magne-toelasticity. Note that the orbital contribution is de-fined as the sum of the |⟨µ′|Lx(z)|µ⟩|2G↓↓II′(µ′µ′;µµ) +|⟨µ|Lx(z)|µ′⟩|2G↓↓II′(µµ;µ′µ′), normalized so that the zerovalue for the unstrained structure (ε = 0) was obtained.In the case of Ni1–yCoy , the main contribution to thesign change of magnetostriction is also expected to bethe same as that of (Fe1–xCox )4N [Figs. 4(a)-(c)]. The5FIG. 4. Strain dependence of (a) total, (b) ↓⇒↓, and (c)↑⇒↓ contribution to the MAE calculated from second-orderperturbation analysis for Ni1–yCoy . (d) Spin-resolved densityof states of fcc–Ni1–yCoyFIG. 5. Strain dependence of the ↓⇒↓ normalized orbitalcontribution to the MAE calculated from second-order per-turbation analysis for (a) dx2−y2 - dxy and (b) dyz - dz2 ofNi1–yCoytotal dMAE/dε [Fig. 4(a)] is still dominated by the spin-conserving term dMAE(↓⇒↓)/dε [Fig. 4(b)]. The onlynotable difference with (Fe1–xCox )4N is that, with theinclusion of Co (y > 0.6), the spin-flip term dMAE(↑⇒↓)/dε slightly shifts from positive to a small negative value[Fig. 4(c)], while the spin-flip term dMAE(↑⇒↓)/dε of(Fe1–xCox )4N is always positive [Fig. 2(c)]. Since thetotal density of states at EF in this system is mainlyfrom minority-spin state [Fig. 4(d)], the same mechanismof spin-conserving transitions mediated by minority-spingoverns the MAE and its strain dependence, while alsocontrolling the intraband transitions in the minority-spinstates for the magnetic damping, due to the absence ofmajority-spin d states around EF.In Ni1–yCoy , two significant transitions dx2−y2-dxyand dyz-dz2 determine the compositional dependence ofdMAE/dε [Figs. 5(a)-(b)]. Both orbital contributionsshow the decrease of MAE with increasing the strain infcc-Ni, consistent with the negative magnetostriction offcc-Ni observed in many previous experiments. The sub-stitution of Ni by Co significantly affects these two transi-tions, leading to the increase of MAE with increasing thestrain. It is worth noting that Co doping does not signifi-cantly affect other orbital contributions to the MAE andits strain dependence [see Fig. S4 in the SupplementalMaterials [34]]. Unlike the Fe sites in (Fe1–xCox )4N, theNi sites in Ni1–yCoy have the same atomic environmentas each other and equally contribute to the magnetoe-lasticity. These results emphasize the important role ofcrystal symmetry and the role of light elements (here,Nitrogen atom) in understanding the effects of atomicsubstitution on magnetoelasticity.In summary, we theoretically investigated the corre-lation between magnetostriction and intrinsic magneticdamping as a function of composition in (Fe1–xCox )4Nand Ni1–yCoy alloys. We found that damping is largerwhen the magnetostriction constant is negative andsmaller when the magnetostriction constant is positive.This is because the magnitude of magnetization changes(exchange splitting changes) when strain is applied, andthe change in exchange splitting shifts the density ofstates near the Fermi level, affecting both magnetostric-tion and magnetic damping. We also found that the pres-ence of locally degenerate orbitals is important with re-spect to the magnetostriction constant and that the pres-ence of degenerate orbitals in the same atom or atoms ofthe same symmetry dominantly contributes to the straindependence of the MAE. These results suggest the pos-sibility of controlling magnetization dynamics by the ap-plication of strain to magnetic materials and provide im-portant guidelines for future experiments.ACKNOWLEDGMENTSThe authors are grateful to T. Tadano and G. Xing ofNIMS for their valuable discussions on this paper. Thisresearch was partially supported by Grants-in-Aid forScientific Research (Grant No. JP22H04966) from theJapan Society for the Promotion of Science and the JapanScience and Technology Agency (JST) CREST (GrantNo. JPMJCR21O1). The calculations were performedon the Numerical Materials Simulator at NIMS.[1] J. P. Joule, On a new class of magnetic forces, AnnalsElectric. Magn. Chem. 8, 219 (1842).[2] S. Bidwell, On the effect of tension upon magneticchanges of length in wires of iron, nickel, and cobalt,Proc. Roy. Soc. 47, 469 (1890).[3] C. Kittel, Physical theory of ferromagnetic domains, Rev.Mod. Phys. 21, 541 (1949).[4] R. Wu, L. Chen, and A. Freeman, First principles deter-mination of magnetostriction in bulk transition metalsand thin films, J. Magn. Magn. Mater. 170, 103 (1997).[5] A. E. Clark, and H. S. Belson, Giant room-temperaturemagnetostrictions in TbFe2 and DyFe2, Phys. Rev. B 5,3642 (1972).[6] H. Suhl, Theory of the magnetic damping constant, IEEE6Trans. Magn. 34, 1834 (1998).[7] C. Vittoria, S. D. Yoon, and A. Widom, Relaxationmechanism for ordered magnetic materials, Phys. Rev.B 81, 014412 (2010).[8] E. Rossi, O. G. Heinonen, and A. H. MacDonald, Dy-namics of magnetization coupled to a thermal bath ofelastic modes, Phys. Rev. B 72, 174412 (2005).[9] O. M. Chumak, A. Pacewicz, A. Lynnyk, B. Salski, T.Yamamoto, T. Seki, J. Z. Domagala, H. Glowiński, K.Takanashi, L. T. Baczewski, H. Szymczak, and A. Nabi-alek, Magnetoelastic interactions and magnetic dampingin Co2Fe0.4Mn0.6Si and Co2FeGa0.5Ge0.5 Heusler alloysthin films for spintronic applications, Sci. Rep. 11, 7608(2021).[10] Y. Endo, O. Mori, Y. Shimada, S. Yabukami, S. Sato, andR. Utsumi, Study on measurement technique for magne-tization dynamics of thin films, Appl. Phys. Lett. 112,252403 (2018).[11] V. Kamberský, On ferromagnetic resonance damping inmetals, Czech. J. Phys. 26, 1366 (1976).[12] S. Ota, A. Ando, and D. Chiba, A flexible giant magne-toresistive device for sensing strain direction, Nat. Elec-tron. 1, 124 (2018).[13] P. He, X. Ma, J. W. Zhang, H. B. Zhao, G. Lüpke, Z. Shi,and S. M. Zhou, Quadratic scaling of intrinsic Gilbertdamping with spin-orbital coupling in L10 FePdPt films:experiments and ab initio calculation, Phys. Rev. Lett.110, 077203 (2013).[14] J. Niu, K. Yan, Y. Xu, Y. Wu, E. Liu, Z. Fu, J. Li,X. Gao, X. Mu, B. Liu, X. Wang, Y. Li, J. Wang, andC. Jiang, Understanding the intrinsic mechanism of thegiant magnetostriction in binary and alloyed FeGa solidsolutions, Phys. Rev. B 109, 014417 (2024).[15] K. Ito, I. Kurniawan, Y. Shimada, Y. Miura, Y. Endo,and T. Seki, Giant tunability of magnetoelasticity inFe4N system as a platform to unveil correlation be-tween magnetostriction and magnetic damping, Com-mun. Mater. 6, 53 (2025).[16] G. Kresse and J. Furthmüller , Efficient iterative schemesfor ab initio total-energy calculations using a plane-wavebasis set, Phys. Rev. B 54, 11169 (1996).[17] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalizedgradient approximation made simple, Phys. Rev. Lett.77, 3865 (1996).[18] P. E. Blöchl, Projector augmented-wave method, Phys.Rev. B 50, 17953 (1994).[19] S. Isogami, K. Masuda, and Y. Miura, Contributions ofmagnetic structure and nitrogen to perpendicular magne-tocrystalline anisotropy in antiperovskite ε-Mn4N, Phys.Rev. Mater 4, 014406 (2020).[20] J. Okabayashi, Y. Miura, and T. Taniyama, Strain-induced reversible manipulation of orbital magnetic mo-ments in Ni/Cu multilayers on ferroelectric BaTiO3, npjQuantum Mater. 4, 21 (2019).[21] E. L. P. y. Blancá, J. Desimoni, N. E. Christensen, H.Emmerich, and S. Cottenier, The magnetization of γ′-Fe4N: theory vs. experiment, Phys. Status Solidi B 246,909 (2009).[22] L. Bellaiche and D. Vanderbilt, Virtual crystal approx-imation revisited: Application to dielectric and piezo-electric properties of perovskites, Phys. Rev. B 61, 7877(2000).[23] P. Monachesi, T. Björkman, T. Gasche, and O. Eriks-son, Electronic structure and magnetic properties of Mn,Co, and Ni substitution of Fe in Fe4N, Phys. Rev. B 88,054420 (2013).[24] K. Ito, T. Sanai, Y. Yasutomi, T. Gushi, K. Toko, H.Yanagihara, M. Tsunoda, E. Kita, and T. Suemasu,Mössbauer study on epitaxial CoxFe4–xN films grownby molecular beam epitaxy, J. Appl. Phys. 117, 17B717(2015).[25] M. Yamamoto, T. Nakamichi, Magnetostriction con-stants of nickel-copper and nickel-cobalt alloys, J. Phys.Soc. Japan 13, 228 (1958).[26] The strain dependence of E001 and MAE is fitted usinginterpolation.[27] Y. Miura and J. Okabayashi, Understanding magne-tocrystalline anisotropy based on orbital and quadrupolemoments, J. Phys.: Condens. Matter 34, 473001 (2022).[28] The values of ξ used for Fe, Co, Ni, and N in the second-order perturbation calculation are 54.3, 69.4, 87.2, and 20meV, respectively. The coefficient of the expansion fromprojected state on each localized atomic orbitals wereobtained by simple projection with option LORBIT=2in VASP.[29] M. Cinal, Magnetic anisotropy and orbital magnetic mo-ment in Co films and Co/X bilayers (X=Pd and Pt),Phys. Rev. B 105, 104403 (2022)[30] M. Cinal, Intraband terms of magnetocrystallineanisotropy energy in layered systems without inversionsymmetry, Phys. Rev. B 109, 024424, (2024).[31] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Identifica-tion of the dominant precession-damping mechanism inFe, Co, and Ni by first-principles calculations, Phys. Rev.Lett. 99, 027204 (2007).[32] R. Hiramatsu, D. Miura, and A. Sakuma, First-principlescalculations for Gilbert damping constant at finite tem-perature, Appl. Phys. Express 15, 013003 (2022).[33] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Spin-orbitprecession damping in transition metal ferromagnets, J.Appl. Phys. 103, 07D303 (2008).[34] See Supplemental Material at [URL will be insertedby publisher] for strain dependence of magnetizationand orbital contribution to MAE of (Fe1–xCox )4N andNi1–yCoy , and atomic-resolved partial density of statesof tetragonally distorted Fe4N.