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[Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), Hiroyoshi Itoh, Yoshiaki Sonobe, [Hiroaki Sukegawa](https://orcid.org/0000-0002-4034-7848), [Seiji Mitani](https://orcid.org/0000-0002-1348-0774), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[Interfacial giant tunnel magnetoresistance and bulk-induced large perpendicular magnetic anisotropy in (111)-oriented junctions with fcc ferromagnetic alloys: A first-principles study](https://mdr.nims.go.jp/datasets/0bfb2625-a374-43b1-bdb6-45b5218d0e91)

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Interfacial giant tunnel magnetoresistance and bulk-induced large perpendicular magnetic anisotropy in (111)-oriented junctions with fcc ferromagnetic alloys: A first-principles studyPHYSICAL REVIEW B 103, 064427 (2021)Interfacial giant tunnel magnetoresistance and bulk-induced large perpendicular magneticanisotropy in (111)-oriented junctions with fcc ferromagnetic alloys: A first-principles studyKeisuke Masuda ,1,* Hiroyoshi Itoh ,2,3 Yoshiaki Sonobe,4 Hiroaki Sukegawa ,1 Seiji Mitani,1,5,6 and Yoshio Miura1,3,51Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan2Department of Pure and Applied Physics, Kansai University, Suita 564-8680, Japan3Center for Spintronics Research Network, Osaka University, Toyonaka 560-8531, Japan4Samsung R &D Institute Japan, Yokohama 230-0027, Japan5Center for Materials Research by Information Integration, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan6Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8577, Japan(Received 2 July 2020; revised 7 February 2021; accepted 8 February 2021; published 22 February 2021)We study the tunnel magnetoresistance (TMR) effect and magnetocrystalline anisotropy in a series ofmagnetic tunnel junctions (MTJs) with L11-ordered fcc ferromagnetic alloys and MgO barrier along the [111]direction. Considering the (111)-oriented MTJs with different L11 alloys, we calculate their TMR ratios andmagnetocrystalline anisotropies on the basis of the first-principles calculations. The analysis shows that the MTJswith Co-based alloys (CoNi, CoPt, and CoPd) have high TMR ratios over 2000%. These MTJs have energeticallyfavored Co-O interfaces where interfacial antibonding between Co d and O p states is formed around the Fermilevel. We find that the resonant tunneling of the antibonding states, called the interface resonant tunneling, isthe origin of the obtained high TMR ratios. Such a mechanism is similar to that found in our recent work onthe simple Co/MgO/Co(111) MTJ [K. Masuda et al., Phys. Rev. B 101, 144404 (2020)]. In contrast, differentsystems have different spin channels where the interface resonant tunneling occurs; for example, the tunnelingmainly occurs in the majority-spin channel in the CoNi-based MTJ while it occurs in the minority-spin channelin the CoPt-based MTJ. This means that even though the mechanism is similar, different spin channels contributedominantly to the high TMR ratio in different systems. Such a difference is attributed to the different exchangesplittings in the particular Co d states contributing to the tunneling though the antibonding with O p states. Ourcalculation of the magnetocrystalline anisotropy shows that many L11 alloys have large perpendicular magneticanisotropy (PMA). In particular, CoPt has the largest value of anisotropy energy Ku ≈ 10 MJ/m3. We furtherconduct a perturbation analysis of the PMA with respect to the spin-orbit interaction and reveal that the largePMA in CoPt and CoNi mainly originates from spin-conserving perturbation processes around the Fermi level.DOI: 10.1103/PhysRevB.103.064427I. INTRODUCTIONMagnetic tunnel junctions (MTJs), in which an insulatingtunnel barrier is sandwiched between ferromagnetic elec-trodes, have attracted considerable attention not only fromthe viewpoint of fundamental physics, but also from theirpotential applications to various devices. In particular, forthe application to nonvolatile magnetic random access mem-ories (MRAMs), they need to have perpendicular magneticanisotropy (PMA) as well as high tunnel magnetoresistance(TMR) ratios. The PMA is more beneficial than in-planemagnetic anisotropy for achieving high thermal stability whendevice sizes are scaled down in ultrahigh-density MRAMs [1].The PMA is also preferred for the different types of mag-netization switching in MRAMs; the critical current for theswitching in spin-transfer-torque MRAMs (STT-MRAMs) [1]can be reduced and the write error rate in voltage-controlledMRAMs [2] can be decreased.To obtain both large PMA and high TMR ratios in MTJs,two types of approaches have been employed. One approach*MASUDA.Keisuke@nims.go.jpis to utilize ferromagnets with strong bulk magnetocrystallineanisotropy as electrodes of MTJs. The ordered alloys, L10FePt [3,4], D022 Mn3Ga(Ge) [5–8], and L10 MnGa [6], areferromagnets with such strong magnetic anisotropy along the[001] direction, by which one can achieve large PMA inthe (001)-oriented MTJs. However, unfortunately, these MTJsdid not show high TMR ratios even if one of the ferro-magnetic electrodes was replaced by CoFe(B) or Fe [9–13].The other approach is to combine the interface-inducedPMA and the established technology for high TMR ratiosin Fe(Co)/MgO/Fe(Co)(001) MTJs [14,15]. Actually, ex-periments on CoFeB/MgO/CoFeB MTJs [16] demonstratedrelatively large interfacial PMA (∼1.3 mJ/m2) and high TMRratios (>120% at room temperature). However, such an inter-facial PMA is sensitive to the interfacial oxidation condition[17,18] and the thickness of the ferromagnetic layers [16].Thus, large PMA due to bulk magnetocrystalline anisotropy isattractive for storage layers of MRAMs. It should also be re-marked that large bulk PMA is beneficial for the pinned layersin the synthetic antiferromagnetic structures in MRAM cells[19]. In this study, we theoretically demonstrate such largebulk-induced PMA and high TMR ratios in unconventionalMTJs and discuss their physical underlying mechanisms.2469-9950/2021/103(6)/064427(10) 064427-1 ©2021 American Physical Societyhttps://orcid.org/0000-0002-6884-6390https://orcid.org/0000-0001-6577-8313https://orcid.org/0000-0002-4034-7848http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.103.064427&domain=pdf&date_stamp=2021-02-22https://doi.org/10.1103/PhysRevB.101.144404https://doi.org/10.1103/PhysRevB.103.064427KEISUKE MASUDA et al. PHYSICAL REVIEW B 103, 064427 (2021)(a)(c)(b)CoNiMg OCo Ni MgOxyz xyzxyzFIG. 1. The unit cells of (a) L11 CoNi and (b) MgO, where the zaxes are set to their [111] directions of the original fcc cells. (c) Thesupercell of CoNi (7 ML)/MgO (7 ML)/CoNi (7 ML)(111).Let us here introduce unconventional (111)-oriented MTJs,where fcc ferromagnetic electrodes and the fcc tunnel barrierare stacked along their [111] directions [Fig. 1(c)]. It is nat-ural to consider such (111)-oriented MTJs for fcc materialssince the (111) plane is the close-packed plane of the fcclattice and has the lowest surface energy [20]. However, mostprevious studies addressed (001)-oriented MTJs with bcc ma-terials because of the initial success in Fe/MgO/Fe(001)[14,15,21,22]. Recently, three of the present authors theoreti-cally investigated the TMR effect in two simple (111)-orientedMTJs, Co/MgO/Co(111) and Ni/MgO/Ni(111), and ob-tained a high TMR ratio (∼2100%) in the Co-based MTJ [23].This result motivates us to study other (111)-oriented MTJsfor obtaining high TMR ratios.Another important merit of (111)-oriented MTJs is thatseveral magnetic superlattices and L11 alloys can be usedas ferromagnetic electrodes for large PMA [24–28]. For ex-ample, Seki et al. [24] recently observed large PMA withuniaxial magnetic anisotropic energy (Ku) of ∼0.5 MJ/m3 inepitaxial Co/Ni(111) multilayers, consistent with previous ex-periments [25]. In another experimental study [26], Sato et al.grew L11 CoPt films on an MgO(111) substrate and showedlarge PMA (Ku ∼ 3.7 MJ/m3). Furthermore, Yakushiji et al.[27] obtained PMA (Ku ∼ 0.5 MJ/m3) in Co/Pt(111) andCo/Pd(111) multilayers that have similar structures as L11films. All these studies indicate the potential of (111)-orientedMTJs with L11 alloys for large PMA; however, such MTJshave not been investigated both theoretically and experimen-tally in previous studies.In this work, we present a systematic theoretical study ofthe TMR effect and magnetocrystalline anisotropy in (111)-oriented MTJs with L11 alloys. We consider various possibleMTJs consisting of L11 alloys and the MgO tunnel bar-rier and calculate their TMR ratios and magnetocrystallineanisotropies by means of the first-principles calculations. It isshown that the MTJs with Co-based alloys (CoNi, CoPt, andCoPd) have high TMR ratios over 2000%. The detailed analy-sis of the electronic structures and conductances clarifies thatall the obtained high TMR ratios originate from the resonanttunneling of the interfacial d-p antibonding states called theinterface resonant tunneling [23], which is clearly differentfrom the conventional mechanism of the high TMR ratio inFe/MgO/Fe(001) [21,22]. The interface resonant tunnelingmainly occurs in the majority- and minority-spin channels inTABLE I. The optimized value of afcc in each L11 alloy and thecalculated TMR ratio in the corresponding (111)-oriented MTJ. Theanisotropy energy Ku calculated in each L11 alloy is also shown. Wecalculated TMR ratios using supercells with 7 ML of MgO. Onlythe TMR ratio in the bottom row was calculated for the thickerbarrier (MgO 13 ML). The values of Ku are given in units of MJ/m3(meV/cell). These Ku values were calculated using the unit cell ofeach L11 alloy [Fig. 1(a)] including 12 atoms.afcc (Å) TMR ratio (%) KuFePt 3.83 716 4.95 (5.21)CoPt 3.79 2534 9.86 (10.04)NiPt 3.78 650 −1.04 (−1.05)FePd 3.81 46 0.73 (0.76)CoPd 3.76 2172 1.88 (1.87)NiPd 3.76 585 0.45 (0.45)FeNi 3.56 484 0.67 (0.56)CoNi 3.51 3210 1.10 (0.89)CoNi 3.51 2361 (MgO 13 ML) 1.10 (0.89)the CoNi- and CoPt-based MTJs, respectively. Namely, thehigh TMR ratios in different systems come from the tunnelingin different spin channels. In the calculation of the magne-tocrystalline anisotropy, we obtain large PMA in many L11alloys. Among them, CoPt has the largest Ku of ≈10 MJ/m3.A second-order perturbation analysis of the PMA with respectto the spin-orbit interaction (SOI) clarifies that the large PMAin CoPt and CoNi originates from the spin-conserving pertur-bation processes around the Fermi level.II. MODEL AND METHODA. Structure optimizationSince the L11 phase can exist only in multilayer filmsowing to its metastable nature, it is hard to obtain the exper-imental lattice constants of the L11 alloys. This forces us toconduct the structure optimization to theoretically determinethe optimal lattice constants. In the present study, we consid-ered eight different L11 alloys (Table I) and prepared their unitcells with the z axis along the [111] direction of the originalfcc cell [Fig. 1(a)]. We optimized the value of afcc in each L11alloy by means of the density-functional theory (DFT) imple-mented in the Vienna ab initio simulation program (VASP)[29]. Here, we adopted the generalized gradient approxima-tion (GGA) [30] for the exchange-correlation energy and usedthe projected augmented wave (PAW) pseudopotential [31,32]to treat the effect of core electrons properly. A cutoff energy of337 eV was employed and the Brillouin-zone integration wasperformed with 23 × 13 × 5 k points. The convergence crite-ria for energy and force were set to 10−5 eV and 10−4 eV/Å,respectively. The obtained values of afcc are shown in Table I.By combining the unit cell of each L11 alloy [Fig. 1(a)] andthat of the (111)-oriented MgO [Fig. 1(b)], we built the super-cell of the corresponding (111)-oriented MTJ [Fig. 1(c)]. Thex- and y-axis lengths of the supercell were fixed to afcc/√2and√3 afcc/√2 in each supercell where the optimized afccof each alloy was used. The atomic positions along the zdirection in the supercells were relaxed using the DFT withthe aid of the VASP code. In these calculations for supercells,064427-2INTERFACIAL GIANT TUNNEL MAGNETORESISTANCE … PHYSICAL REVIEW B 103, 064427 (2021)23 × 13 × 1 k points were used, and the other calculationconditions were the same as the structure optimizations of theL11 alloys. More technical details of structure optimizationsof supercells are given in our previous work [33]. In eachsupercell, we compared energies for all interfacial atomicconfigurations and determined the energetically favored con-figuration. For example, in CoNi/MgO/CoNi(111), there arefour atomic configurations at the interface: Co-O, Ni-O, Co-Mg, and Ni-Mg. By comparing formation energies for thesecases, we found that the Co-O interface has the lowest energy.In Table I, each L11-ordered alloy is denoted as XY (X = Coand Y = Ni for CoNi). We confirmed that the X -O interfacewas energetically favored in each supercell. Such supercellswith energetically favored interfaces were used in the trans-port calculation explained below.B. Calculation method of TMR ratiosThe TMR ratio of each (111)-oriented MTJ was calcu-lated using the DFT and Landauer formula with the helpof the PWCOND code [34] in the QUANTUM ESPRESSOpackage [35]. We first constructed the quantum open systemby attaching the left and right semi-infinite electrodes of eachL11 alloy to the supercell. Here, the supercell is composedof seven monolayers (ML) of MgO sandwiched between7 ML of L11 alloys L11 (7 ML)/MgO (7 ML)/L11 (7 ML)[Fig. 1(c)] for the parallel configuration of the magnetiza-tion whose atomic positions are optimized following theprocedures mentioned in Sec. II A. The thickness of theMgO barrier layers is about 9.5 Å. In the case of the an-tiparallel magnetization, we need to use a supercell that istwice as long as that for the parallel magnetization to sat-isfy the translational invariance in the magnetization statealong the stacking direction; this is made by connecting twoL11 (7 ML)/MgO (7 ML)/L11 (7 ML) cells inverting one ofthem. The application of the DFT to the quantum open sys-tem provided the self-consistent potential, which was usedto derive the scattering equation mentioned below. In theDFT calculation, the exchange-correlation energy was treatedwithin the GGA, and the ultrasoft pseudopotentials wereused. The cutoff energies were set to 45 and 450 Ry forthe wave function and the charge density, respectively. Thenumber of k points was taken to be 23 × 13 × 1 and theconvergence criterion was set to 10−6 Ry. Since our systemhas translational symmetry in the xy-plane, the scatteringstates can be classified by an in-plane wave vector k‖ =(kx, ky). For each k‖ and spin index, we solved the scatteringequation derived under the condition that the wave functionand its derivative of the supercell are connected to those of theelectrodes [34,36]. These calculations gave the k‖-resolvedtransmittances from which the k‖-resolved conductances wereobtained through the Landauer formula: GP,↑(k‖), GP,↓(k‖),GAP,↑(k‖), and GAP,↓(k‖). Here, P (AP) refers to the par-allel (antiparallel) magnetization state of the electrodes and↑ (↓) indicates the majority-spin (minority-spin) channel.Note that in the antiparallel state, we defined the index σfor GAP,σ (k‖) as the channel for the left electrode; namely,GAP,↑(k‖) [GAP,↓(k‖)] corresponds to the electron tunnelingfrom the majority-spin (minority-spin) channel in the left elec-trode to the minority-spin (majority-spin) channel in the rightNumber of k|| point N (×104)TMR ratio (%)00.20.40.60.81.01.21.401.02.03.04500400035003000250020001500100050000 2 4 6 8 10GP (×10-2 e2 /h)GAP (×10-4 e2 /h)4.05.06.07.0GPGAP(a)(b)FIG. 2. The k‖-point number N dependencies of (a) GP(dashed line) and GAP (dotted line) and (b) the TMR ratio inCoNi/MgO (7 ML)/CoNi(111).electrode. We averaged each conductance over k‖ as, e.g.,GP,↑ = ∑k‖ GP,↑(k‖)/N , where N is the sampling numberof k‖ points. For each MTJ, we calculated the TMR ratiofollowing its optimistic definitionTMR ratio (%) = 100 × (GP − GAP)/GAP, (1)where GP(AP) = GP(AP),↑ + GP(AP),↓. In the present work, weneglect the SOI in the calculation of TMR ratios. This isbecause, as will be discussed in Sec. III B, it is expected thatthe SOI does not affect TMR ratios significantly, at least forthe present MTJs with Co-O interfaces that exhibit high TMRratios.We carefully considered k‖-point number N dependenciesof GP, GAP, and the TMR ratio. Figure 2 shows these quanti-ties in CoNi/MgO/CoNi(111) as a function of N , from whichwe found that N � 40 000 is required for the convergence ofthese quantities. In the following part of the paper, we showthe results calculated with N = 40 000.C. Estimation of magnetocrystalline anisotropyWe calculated the uniaxial magnetic anisotropy energy Kuof each L11 alloy on the basis of the DFT calculation includingthe SOI. We adopted the expression by the well-known forcetheorem [37,38]Ku = (E‖ − E⊥)/V, (2)where E‖ (E⊥) is the sum of the eigenvalues only over occu-pied states of the unit cell [Fig. 1(a)] with the magnetizationalong the x (z) direction, and V is the volume of the unit cell.Here, we used the optimized lattice constant mentioned abovefor each L11 alloy. From the definition in Eq. (2), a positive(negative) Ku indicates a tendency toward PMA (in-plane064427-3KEISUKE MASUDA et al. PHYSICAL REVIEW B 103, 064427 (2021)magnetic anisotropy). The VASP code was used for the DFTcalculation including the SOI, where we adopted the GGA forthe exchange-correlation energy, the PAW pseudopotential,and a cutoff energy of 337 eV. Since the energy scale of Kuis much smaller than that of the total energy of the system,the large number of k points are required to estimate Ku accu-rately. We thus used 51 × 27 × 11 k points after confirmingthe convergence of Ku with respect to the number of k points.In addition to these calculations, we also conducted asecond-order perturbation analysis of the magnetocrystallineanisotropy [39] to understand the origin of the PMA. Bytreating the SOI as a perturbation term, the second-order per-turbation energy is given byE (2) =∑kunocc∑n′σ ′occ∑nσ|〈kn′σ ′|HSOI|knσ 〉|2ε(0)knσ− ε(0)kn′σ ′, (3)HSOI =∑iξi Li · Si, (4)where ε(0)knσis the energy of an unperturbed state |knσ 〉with wave vector k, band index n, and spin σ . The index“occ” (“unocc”) on the summation in Eq. (3) means thatthe sum is over occupied (unoccupied) states of all atomsin the unit cell. In the Hamiltonian for the SOI HSOI, ξiis its coupling constant at an atomic site i, and Li (Si) isthe single-electron angular (spin) momentum operator. Wavefunctions and eigenenergies obtained in our DFT calculationswere used as unperturbed states and energies in Eq. (3). Themagnetocrystalline anisotropy energy within the second-orderperturbation was calculated as E (2)MCA = E (2)‖ − E (2)⊥ similar toEq. (2), where E (2)‖ (E (2)⊥ ) is the energy calculated by Eq. (3)for the magnetization along the x (z) direction of the unit cell.We can decompose E (2)MCA into four types of terms comingfrom different perturbation processes at each atomic site:E (2)MCA =∑iE iMCA, (5)EiMCA = �Ei↑⇒↑ + �Ei↓⇒↓ + �Ei↑⇒↓ + �Ei↓⇒↑. (6)Here, EiMCA is the magnetocrystalline anisotropy energy ateach atomic site i. The term �Ei↑⇒↑ (�Ei↓⇒↓) is the con-tribution from spin-conserving perturbation processes in themajority-spin (minority-spin) channel. The last two termsare the contributions from spin-flip perturbation processes:�Ei↑⇒↓ (�Ei↓⇒↑) comes from electron transition processesfrom majority- to minority-spin (minority- to majority-spin)channel. This decomposition provides us with information onthe origin of the PMA.III. RESULTS AND DISCUSSIONA. High TMR ratios and their possible originTable I shows the obtained TMR ratios in the (111)-oriented MTJs. The MTJs, including the Co-based alloys,have high TMR ratios over 2000%. In contrast, the Fe- andNi-based alloys give much lower TMR ratios (<1000%).To understand the origin of the high TMR ratios, the bulkband structures of the electrodes and the barrier were firstanalyzed because the high TMR ratio in the well-knownFe/MgO/Fe(001) MTJ [14,15] was explained by the bulkFIG. 3. Imaginary and real parts of kz calculated for the MgOunit cell [Fig. 1(b)]. (a) Imaginary part of kz as a function of realkx (ky = 0) at the Fermi level EF. (b) Imaginary and real parts of kzaround EF at kx = ky = 0.band structures of Fe and MgO on the basis of the coherenttunneling mechanism [21,22]. If a similar mechanism holdsfor the present MTJs, the bulk band structures along the �line in the Brillouin zone corresponding to the [111] directionshould explain the high TMR ratios.Figure 3(a) shows the imaginary part of kz, referred to asthe complex band, of the (111)-oriented MgO [Fig. 1(b)] asa function of kx. The smallest value of Im(kz ) is located at(kx, ky) = (0, 0) = �. This means that the � states, i.e., thewave function in the � line (0, 0, kz ), has the slowest decayand can provide the dominant contribution to the electrontransport. In Fig. 3(b), we show the complex and real bandsat the � line. We find that the smallest Im(kz ) at EF comesfrom the �1 state consisting of s and pz orbitals. Therefore,the �1 state decays most slowly in the barrier and the selec-tive transport of this state can occur. To study whether theL11 alloys have half-metallicity in the �1 state, bulk bandstructures of CoNi and CoPt, which provide the two highestTMR ratios, were analyzed. As shown in Figs. 4(a) and 4(b),both majority- and minority-spin bands from the d3z2−r2 state(belonging to the �1 state) cross the Fermi level in both alloys;namely, these alloys do not have half-metallicity in the �1state, which is in sharp contrast to the half-metallicity in the�1 state of Fe in Fe/MgO/Fe(001) [21,22]. All these resultsindicate that we cannot explain the present high TMR ratiosfrom the bulk band structures based on the coherent tunnelingmechanism as in Fe/MgO/Fe(001).Another possible way to understand the present high TMRratios is to focus on interfacial effects. In our previous study[23], we clarified that the interface resonant tunneling pro-vides a high TMR ratio in a simple (111)-oriented MTJ,Co/MgO/Co(111). To examine a similar possibility, we cal-culated the local density of states (LDOSs) at interfacial Coand O atoms of CoNi/MgO/CoNi(111) shown in Figs. 5(a)064427-4INTERFACIAL GIANT TUNNEL MAGNETORESISTANCE … PHYSICAL REVIEW B 103, 064427 (2021)FIG. 4. Band structures along the � line of (a) L11 CoNi and(b) L11 CoPt. In both panels, atomic orbitals contributing dominantlyto each band around EF are indicated, where d3z2−r2 and dx2−y2 areabbreviated as dz2 and dx2 , respectively.and 5(b). We can find a clear similarity in the energy de-pendence of the LDOS between the Co dzx (dyz) and O px(py) states in the majority-spin channel due to the forma-tion of the interfacial antibonding between these states. Atthe Fermi level, such O px and py states have large LDOSsand can provide interfacial resonant tunneling between theleft and right interfaces. Figure 5(c) shows the k‖-resolvedconductance GP,↑(k‖), which contributes dominantly to thehigh TMR ratio. The conductance has only a small value atk‖ = �, and their large values distribute around the � point,which is a characteristic in the conductance originating frominterfacial effects. We also analyzed the k‖-resolved LDOSsof the interfacial O px and py majority-spin states as shownin Figs. 5(d) and 5(e). The distribution of k‖ points with largeLDOS is similar to that with large conductance in Fig. 5(c),indicating that the interfacial O px and py states play thedominant role in the high TMR ratio through the interfacialresonant tunneling.As shown in the bottom of Table I, we also calculatedthe TMR ratio in the thicker MgO case (∼19 Å) using asupercell CoNi/MgO (13 ML)/CoNi. A high TMR ratio over2000% was obtained for this thicker barrier, although thevalue is lower than the thinner barrier case. When the MgObarrier becomes thicker, the interfacial resonant tunnelingis weakened depending on the value of the smallest Im(kz )[Fig. 3(a)] at the k‖ points with interfacial resonance states.Actually, as shown in Table II, the total conductance GP inthe parallel magnetization state decreases to 2.64 × 10−6 e2/hby increasing the number of MgO layers. However, since theconductance GAP in the antiparallel magnetization state alsolargely diminishes, the TMR ratio has a high value (>2000%)as mentioned above. Although not shown here, we confirmedthat the k‖ dependence of the conductance GP,↑(k‖) in thethicker MgO case is almost the same as that in the thinnerMgO case [Fig. 5(c)], indicating that the interfacial resonantFIG. 5. The electronic structures and transport properties ofCoNi/MgO (7 ML)/CoNi(111). (a, b) Projected LDOSs at interfa-cial Co and O atoms, where d3z2−r2 and dx2−y2 are abbreviated asdz2 and dx2 , respectively. (c) The k‖ dependence of the majority-spinconductance in the parallel configuration of magnetizations. (d, e)The k‖-resolved LDOSs at E = EF in the majority-spin channelprojected onto the px and py states of interfacial O atoms.tunneling is still active for the thicker MgO. The decay ofthe parallel conductance GP with increasing the MgO thick-ness (Table II) can be roughly estimated from the complexband shown in Fig. 3. When we use complex wave vectorκ = Im(kz ) = 0.94 π/c for simplicity, the decay factor forthe conductance is calculated as exp(−2κd ) ≈ 9.81 × 10−5,where we used d = 9.5 Å as the increment in the MgO thick-ness (7 → 13 ML) and c = √3 × 3.51 Å as the c-axis lengthTABLE II. The conductances GP = GP,↑ + GP,↓ and GAP =GAP,↑ + GAP,↓ obtained using CoNi/MgO (n ML)/CoNi (n = 7, 13)supercells.MgO thickness 7 ML (9.5 Å) 13 ML (19 Å)GP (e2/h) 1.19 × 10−2 2.64 × 10−6GAP (e2/h) 3.24 × 10−4 1.07 × 10−7TMR ratio (%) 3561 2361064427-5KEISUKE MASUDA et al. PHYSICAL REVIEW B 103, 064427 (2021)FIG. 6. The same as Fig. 5, but forCoPt/MgO (7 ML)/CoPt(111). Note that the conductance andLDOSs in the minority-spin channel are shown in panels (c) to (e).of the MgO cell [Fig. 1(b)]. Using this factor and GP for 7 MLMgO, the GP for 13 ML MgO is approximately estimated asGP(7 ML MgO) × exp(−2κd ) ≈ 1.17 × 10−6 e2/h, which isclose to 2.64 × 10−6 e2/h (Table II) obtained in the actualtransport calculation. Note that this type of comparison makessense for the order of the conductance in the present case.This is because the present TMR comes from the interfacialresonance effect, in which many k‖ points on the rings around� [Fig. 5(c)] provide a large contribution to GP and each k‖point has a different value of κ .We also studied the interfacial LDOSs and k‖-resolvedconductance of CoPt/MgO/CoPt(111) with the second high-est TMR ratio [Figs. 6(a) to 6(e)]. In this case, the interfacialantibonding related to the high TMR ratio is formed in theminority-spin state, not the majority-spin state. As shown inFig. 6(b), O px and py minority-spin states have large LDOSsat the Fermi level owing to the antibonding with Co dzx anddyz states. These interfacial states provide a high TMR ratiothrough the interface resonant tunneling. Actually, the con-ductance with the largest contribution to the high TMR ratiois that in the minority-spin state GP,↓(k‖) [Fig. 6(c)], whosek‖ dependence can be reproduced by that of the LDOSs inthe interfacial O px and py minority-spin states [Figs. 6(d)and 6(e)].Such a difference in the spin channel contributing to thehigh TMR ratio between the CoNi- and CoPt-based MTJscomes from different exchange splittings in the interfacialCo dzx and dyz states. By comparing Figs. 5(a) and 6(a), wecan easily see that the exchange splitting in the dzx and dyzstates in the CoNi-based MTJ is clearly smaller than that inthe CoPt-based MTJ. In fact, the magnetic moment projectedonto each d orbital in the interfacial Co atom was estimated inboth MTJs. We obtained 0.96 μB in the dzx and dyz orbitalsfor the CoNi-based MTJ and 1.10 μB for the CoPt-basedMTJ. In the other d orbitals, the difference in the projectedmagnetic moment was found to be quite small. Therefore, inthe CoNi-based MTJ, the dzx and dyz majority-spin states havefinite majority-spin LDOSs at the Fermi level, leading to thelarge O px and py majority-spin LDOSs through the interfacialantibonding [Fig. 5(b)]. In contrast, the CoPt-based MTJ hasnegligibly small dzx and dyz majority-spin LDOSs at the Fermilevel owing to the larger exchange splitting [Fig. 6(a)], whichprovides the dominance of the minority-spin LDOSs in theinterfacial O p states [Fig. 6(b)].Although not shown here, we confirmed that the high TMRratio in the CoPd-based MTJ (2172%) can also be explainedby the interface resonant tunneling of the interfacial O pxand py minority-spin states. Our present study revealed thatnot only the Co/MgO/Co(111) MTJ [23] but also several(111)-oriented MTJs with Co-based L11 alloys exhibit highTMR ratios due to the interface resonant tunneling. This factallows us to expect that such a mechanism may be universalfor high TMR ratios in (111)-oriented MTJs.B. Effect of the SOI on TMR ratiosTo discuss the effect of the SOI on the TMR effect, wetried the calculation of conductances considering the SOIin CoNi/MgO/CoNi(111) and CoPt/MgO/CoPt(111), whichwere shown to have quite high TMR ratios for the case withoutthe SOI. Figures 7(e) and 7(f) show the k‖-resolved con-ductances for the parallel configuration of magnetization inCoNi/MgO/CoNi(111) and CoPt/MgO/CoPt(111), respec-tively. We see that the feature of the k‖ dependence can benaturally understood by combining those of the majority-and minority-spin conductances in the absence of the SOI[Figs. 7(a) to 7(d)]. In addition, the difference in the k‖-averaged conductance between the cases with and without theSOI is not so large; for example, we obtained GP = 4.98 ×10−3 e2/h in CoPt/MgO/CoPt(111) with the SOI, which is25% smaller than GP = GP,↑ + GP,↓ = 6.61 × 10−3 e2/h inthe case without the SOI. On the other hand, in the antipar-allel configuration of magnetization, the self-consistent-fieldcalculation for the supercell did not converge in both systemswithin a realistic calculation time. Thus, we could not estimateTMR ratios directly from such calculations.However, we can approximately estimate the effect ofthe SOI on TMR ratios only from the obtained par-allel conductance GP. We here consider the case ofCoPt/MgO/CoPt(111), where GP is decreased to 75% ofthe original value by including the SOI as shown above.Let us here introduce an assumption that the conductance is064427-6INTERFACIAL GIANT TUNNEL MAGNETORESISTANCE … PHYSICAL REVIEW B 103, 064427 (2021)FIG. 7. The comparison of k‖-resolved conductances betweenthe cases with and without SOI in CoNi/MgO (7 ML)/CoNi(111)(a), (c), (e) and CoPt/MgO (7 ML)/CoPt(111) (b), (d), (f) with theparallel configuration of magnetization. (a, b) Majority-spin conduc-tances GP,↑(k‖) and (c, d) Minority-spin conductances GP,↓(k‖) forthe case without SOI. (e, f) Conductances GP(k‖) for the case withSOI. The unit of the color bar is e2/h in all panels.proportional to the product of LDOSs at the left and rightinterfaces. This is the similar assumption used in the deriva-tion of the well-known Julliere formula [40]; however, thebulk DOSs of electrodes in the original assumption are re-placed by the interfacial LDOSs in the present case sincethe interfacial resonant tunneling is found to be the originof the present TMR. Based on this picture, the parallel con-ductance is given by GP ∝ DL↑DR↑ + DL↓DR↓ ≈ DL↓DR↓,where DL↑(↓) and DR↑(↓) are the majority-spin (minority-spin)LDOSs at EF in the left and right interfaces, respectively.Here, we assumed that the minority-spin LDOSs DL(R)↓ aresufficiently large compared to the majority-spin ones DL(R)↑as seen from Figs. 6(a) and 6(b) [41]. From GP ≈ DL↓DR↓and the fact that the SOI decreases GP to 75% of the originalvalue, it is estimated that each of DL↓ and DR↓ becomes√0.75 times smaller than the original value. Since the an-tiparallel conductance is given by GAP ∝ DL↑DR↓ + DL↓DR↑,TMR ratio (%) ≈ 100 × (GP/GAP) becomes 0.75/√0.75 =√0.75 times smaller than the original value [42]. Therefore,by using the original TMR ratio in Table I, it is concludedthat the TMR ratio of CoPt/MgO/CoPt(111) decreases from2534% to 2194% by including the SOI. Although this is arough estimation, we can expect that the SOI does not affectthe TMR ratio significantly. Note here that such a small effectof the SOI would be related to the fact that the interface ismade by Co and O atoms (not containing Pt atoms with alarge SOI). As shown in Sec. II A, such a Co-O interfaceis energetically stable from the theoretical point of view.However, another interface such as Pt-O may occur in actualexperiments, where the effect of the SOI might be large. Thus,the systematic analysis of TMR ratios fully including the SOIshould be addressed in future studies.C. Large PMA and its correlation with perturbation processesWe listed the obtained values of Ku in Table I. All the alloysexcept NiPt have positive Ku indicating a tendency towardPMA. Among them, CoPt possesses the largest value closeto 10 MJ/m3. In this section, we discuss the origin of Ku inCoNi and CoPt as representatives based on the second-orderperturbation analysis of the magnetocrystalline anisotropy.Here, we used ξCo = 69.4 meV, ξNi = 87.2 meV, and ξPt =523.8 meV as the coupling constants of the SOI ξi. We alsoset the Wigner-Seitz radius of each atom to rCo = 1.302 Å,rNi = 1.286 Å, and rPt = 1.455 Å for obtaining projectedwave functions used in the calculation. All these values arethose in the pseudopotential files in the VASP code.Figure 8(a) shows the results of the second-order pertur-bation analysis of Ku in CoNi. We see that Ni has a muchlarger positive EiMCA than Co and contributes dominantly tothe PMA. In the Co atom, �Ei↓⇒↓ and �Ei↑⇒↓ have largevalues but with opposite signs, leading to a small �EiMCA. Incontrast, in the Ni atom, the spin-conserving term �Ei↓⇒↓ inthe minority-spin channel is positive and much larger than theother terms, giving a large positive EiMCA. This is consistentwith the LDOSs of Ni shown in Fig. 8(c), where the minority-spin state has large values around EF, while the majority-spinstate has only small values. It is known that the expression of�Ei↓⇒↓ within the second-order perturbation theory is analyt-ically given by�Ei↓⇒↓ = ξ 2i∑u↓,o↓|〈u↓∣∣Liz∣∣o↓〉|2 − |〈u↓∣∣Lix∣∣o↓〉|2εu↓ − εo↓, (7)where Liα (α = x, z) is the local angular momentum operatorat an atomic site i, and |oσ 〉 (|uσ 〉) is a local occupied (unoccu-pied) state with spin σ and energy εoσ(εuσ) [43]. This expres-sion indicates that the matrix element of Liz gives a positivecontribution to �Ei↓⇒↓ while that of Lix gives a negative con-tribution. Actually, we confirmed that 〈dx2−y2 ,↓ |Liz|dxy,↓〉and 〈dxy,↓ |Liz|dx2−y2 ,↓〉 have large values in our perturba-tion calculation, which is consistent with large minority-spinLDOSs in the dx2−y2 and dxy states shown in Fig. 8(c).Figure 9 presents the results for CoPt. From the pertur-bation analysis [Fig. 9(a)], we find that in all spin-transitionprocesses Pt has much larger anisotropy energy than Co,meaning that the PMA in CoPt mainly comes from theanisotropy in Pt. In the Pt atom, a large positive anisotropy�Ei↑⇒↓ is found in the ↑⇒↓ spin-flip process, but this iscanceled out by �Ei↓⇒↑ in the other spin-flip process. Thus,the dominant contribution to the large positive anisotropy inPt is given by �Ei↑⇒↑ in the ↑⇒↑ spin-conserving process.Similar to Eq. (7), the analytical expression of �Ei↑⇒↑ is given064427-7KEISUKE MASUDA et al. PHYSICAL REVIEW B 103, 064427 (2021)FIG. 8. (a) Results of second-order perturbation analysis on thePMA in L11 CoNi. (b, c) Projected LDOS for Co and Ni atoms inL11 CoNi, where d3z2−r2 and dx2−y2 are abbreviated as dz2 and dx2 ,respectively.as follows [43]:�Ei↑⇒↑ = ξ 2i∑u↑,o↑|〈u↑∣∣Liz∣∣o↑〉|2 − |〈u↑∣∣Lix∣∣o↑〉|2εu↑ − εo↑, (8)from which the matrix element of Lz is found to give a positivecontribution to �Ei↑⇒↑. As clearly seen in Fig. 9(c), the dx2−y2and dxy states have much larger LDOSs than the other d statesaround EF in the majority-spin channel. Such LDOSs yieldlarge values of 〈dx2−y2 ,↑ |Liz|dxy,↑〉 and 〈dxy,↑ |Liz|dx2−y2 ,↑〉,leading to a large positive �Ei↑⇒↑. The importance of theFIG. 9. The same as Fig. 8, but for L11 CoPt.↑⇒↑ term is also found in Pt of L10 FePt with large PMA[44] and is a feature in ordered alloys with Pt atoms.Conventionally, PMA has been explained with the helpof the Bruno theory [45], which states that PMA mainlycomes from the anisotropy of the orbital magnetic moment,namely, the spin-conserving term �Ei↓⇒↓ in Eq. (6). This the-ory is applicable to typical ferromagnets with large exchangesplittings, since such ferromagnets have almost occupiedmajority-spin states, and only minority-spin states are locatedclose to the Fermi level. In contrast, many recent studies onPMA [24,44,46–50] focused on its unconventional mecha-nism due to the spin-flip terms �Ei↑⇒↓ and �Ei↓⇒↑ in Eq. (6).These terms can be interpreted in terms of the quadrupole064427-8INTERFACIAL GIANT TUNNEL MAGNETORESISTANCE … PHYSICAL REVIEW B 103, 064427 (2021)moment and provide novel physical insight into PMA. Upto now, it has been shown that the spin-flip terms playa significant role for PMA in various systems includingferromagnet/MgO interfaces and ferromagnetic multilayers[24,44,46–50]. In the present study, we obtained large valuesof spin-flip terms in L11 CoNi and CoPt. However, as men-tioned above, �Ei↑⇒↓ is canceled by �Ei↓⇒↓ in CoNi and twotypes of spin-flip terms are canceled with each other in CoPt.Therefore, the unconventional physical picture is not suitableto explain PMA in the present CoNi and CoPt. A similar can-cellation of the spin-flip terms has also been reported recentlyin an FeIr/MgO system [51].IV. SUMMARYWe theoretically investigated the TMR effect and mag-netocrystalline anisotropy in (111)-oriented MTJs with L11alloys based on the first-principles calculations. Our trans-port calculation showed that the MTJs with Co-based alloys(CoNi, CoPt, and CoPd) have high TMR ratios over 2000%,which are attributed to the interface resonant tunneling. Wealso found that the tunneling mainly occurs in the majority-spin channel in the CoNi-based MTJ while it occurs in theminority-spin channel in the CoPt-based MTJ, meaning thatdifferent spin channels provide dominant contributions to thehigh TMR ratios in different systems. This can be understoodfrom the different exchange splittings in the dzx and dyz statesof interfacial Co atoms contributing to the TMR effect throughantibonding with O px and py states. The analysis of the mag-netocrystalline anisotropy revealed that many L11 alloys havelarge PMA and CoPt has the largest value of Ku ≈ 10 MJ/m3.Through a detailed second-order perturbation calculation, weclarified that the large PMA in CoPt and CoNi is attributed tothe spin-conserving perturbation processes around the Fermilevel. All these findings would be useful for understandingexperimental results in (111)-oriented MTJs, which will beobtained in future studies.ACKNOWLEDGMENTSThe authors are grateful to S. Takahashi and K. Nawafor helpful discussions and critical comments. This workwas partly supported by Samsung Electronics, Grant-in-Aids for Scientific Research (S) (Grand No. JP16H06332and No. JP17H06152), Scientific Research (B) (Grand No.JP20H02190), and for Early-Career Scientists (Grant No.JP20K14782) from the Ministry of Education, Culture,Sports, Science and Technology, Japan, and NIMS MI2I. Thecrystal structures were visualized using VESTA [52].[1] B. Dieny, R. B. Goldfarb, and K. J. Lee, Introduction to Mag-netic Random-Access Memory (Wiley, Hoboken, NJ, 2016).[2] Y. Shiota, T. Nozaki, S. Tamaru, K. Yakushiji, H. Kubota, A.Fukushima, S. Yuasa, and Y. Suzuki, Appl. Phys. Express 9,013001 (2016).[3] T. Klemmer, D. Hoydick, H. Okumura, B. Zhang, and W. A.Soffa, Scr. Metall. Mater. 33, 1793 (1995).[4] S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y.Shimada, and K. Fukamichi, Phys. Rev. B 66, 024413 (2002).[5] F. Wu, S. Mizukami, D. Watanabe, H. Naganuma, M. Oogane,Y. Ando, and T. Miyazaki, Appl. Phys. Lett. 94, 122503 (2009).[6] S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T.Kubota, X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T.Miyazaki, Phys. Rev. Lett. 106, 117201 (2011).[7] H. Kurt, N. Baadji, K. Rode, M. Venkatesan, P. S. Stamenov,S. Sanvito, and J. M. D. Coey, Appl. Phys. Lett. 101, 132410(2012).[8] S. Mizukami, A. Sakuma, A. Sugihara, T. Kubota, Y. Kondo,H. Tsuchiura, and T. Miyazaki, Appl. Phys. Express 6, 123002(2013).[9] M. Yoshikawa, E. Kitagawa, T. Nagase, T. Daibou, M.Nagamine, K. Nishiyama, T. Kishi, and H. Yoda, IEEE Trans.Magn. 44, 2573 (2008).[10] T. Kubota, Y. Miura, D. Watanabe, S. Mizukami, F. Wu, H.Naganuma, X. Zhang, M. Oogane, M. Shirai, Y. Ando, and T.Miyazaki, Appl. Phys. Express 4, 043002 (2011).[11] Q. Ma, T. Kubota, S. Mizukami, X. Zhang, H. Naganuma,M. Oogane, Y. Ando, and T. Miyazaki, Appl. Phys. Lett. 101,032402 (2012).[12] T. Kubota, Q. L. Ma, S. Mizukami, X. M. Zhang, H. Naganuma,M. Oogane, Y. Ando, and T. Miyazaki, J. Phys. D 46, 155001(2013).[13] H. Lee, H. Sukegawa, J. Liu, Z. Wen, S. Mitani, and K. Hono,IEEE Trans. Magn. 52, 4400204 (2016).[14] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando,Nat. Mater. 3, 868 (2004).[15] S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes,M. Samant, and S.-H. Yang, Nat. Mater. 3, 862 (2004).[16] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan,M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno,Nat. Mater. 9, 721 (2010).[17] H. X. Yang, M. Chshiev, B. Dieny, J. H. Lee, A. Manchon, andK. H. Shin, Phys. Rev. B 84, 054401 (2011).[18] A. Hallal, H. X. Yang, B. Dieny, and M. Chshiev, Phys. Rev. B88, 184423 (2013).[19] K. Yakushiji, A. Sugihara, A. Fukushima, H. Kubota, and S.Yuasa, Appl. Phys. Lett. 110, 092406 (2017).[20] N. Ting, Y. Qingliang, and Y. Yiying, Surf. Sci. 206, L857(1988).[21] W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M.MacLaren, Phys. Rev. B 63, 054416 (2001).[22] J. Mathon and A. Umerski, Phys. Rev. B 63, 220403(R) (2001).[23] K. Masuda, H. Itoh, and Y. Miura, Phys. Rev. B 101, 144404(2020).[24] T. Seki, J. Shimada, S. Iihama, M. Tsujikawa, T. Koganezawa,A. Shioda, T. Tashiro, W. Zhou, S. Mizukami, M. Shirai, and K.Takanashi, J. Phys. Soc. Jpn. 86, 074710 (2017).[25] M. T. Johnson, J. J. de Vries, N. W. E. McGee, and J. aan deStegge, F. J. A. den Broeder, Phys. Rev. Lett. 69, 3575 (1992).[26] H. Sato, T. Shimatsu, Y. Okazaki, H. Muraoka, H. Aoi, S.Okamoto, and O. Kitakami, J. Appl. Phys. 103, 07E114 (2008).[27] K. Yakushiji, T. Saruya, H. Kubota, A. Fukushima, T.Nagahama, S. Yuasa, and K. Ando, Appl. Phys. Lett. 97,232508 (2010).064427-9https://doi.org/10.7567/APEX.9.013001https://doi.org/10.1016/0956-716X(95)00413-Phttps://doi.org/10.1103/PhysRevB.66.024413https://doi.org/10.1063/1.3108085https://doi.org/10.1103/PhysRevLett.106.117201https://doi.org/10.1063/1.4754123https://doi.org/10.7567/APEX.6.123002https://doi.org/10.1109/TMAG.2008.2003059https://doi.org/10.1143/APEX.4.043002https://doi.org/10.1063/1.4737000https://doi.org/10.1088/0022-3727/46/15/155001https://doi.org/10.1109/TMAG.2016.2519283https://doi.org/10.1038/nmat1257https://doi.org/10.1038/nmat1256https://doi.org/10.1038/nmat2804https://doi.org/10.1103/PhysRevB.84.054401https://doi.org/10.1103/PhysRevB.88.184423https://doi.org/10.1063/1.4977565https://doi.org/10.1016/0039-6028(88)90008-8https://doi.org/10.1103/PhysRevB.63.054416https://doi.org/10.1103/PhysRevB.63.220403https://doi.org/10.1103/PhysRevB.101.144404https://doi.org/10.7566/JPSJ.86.074710https://doi.org/10.1103/PhysRevLett.69.3575https://doi.org/10.1063/1.2830097https://doi.org/10.1063/1.3524230KEISUKE MASUDA et al. PHYSICAL REVIEW B 103, 064427 (2021)[28] K. Mizunuma, S. Ikeda, J. H. Park, H. Yamamoto, H. Gan, K.Miura, H. Hasegawa, J. Hayakawa, F. Matsukura, and H. Ohno,Appl. Phys. Lett. 95, 232516 (2009).[29] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).[30] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,3865 (1996).[31] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).[32] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).[33] K. Masuda and Y. Miura, Phys. Rev. B 96, 054428 (2017).[34] A. Smogunov, A. Dal Corso, and E. Tosatti, Phys. Rev. B 70,045417 (2004).[35] S. Baroni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi, http://www.pwscf.org.[36] H. J. Choi and J. Ihm, Phys. Rev. B 59, 2267 (1999).[37] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, Phys.Rev. B 41, 11919 (1990).[38] M. Weinert, R. E. Watson, and J. W. Davenport, Phys. Rev. B32, 2115 (1985).[39] Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa, K. Abe, andM. Shirai, J. Phys. Condens. Matter 25, 106005 (2013).[40] M. Julliere, Phys. Lett. A 54, 225 (1975).[41] Note that, in addition to |DL(R),↑| � |DL(R),↓|, the effect of theSOI on DL(R),↑ is considered to be much smaller than that onDL(R),↓ since the energy dependence of DL(R),↑ around EF ismuch flatter than that of DL(R),↓ as seen in Figs. 6(a) and 6(b).[42] We approximately neglected the change in DL(R),↑ due to theSOI using the fact mentioned in Ref. [41].[43] D. S. Wang, R. Wu, and A. J. Freeman, Phys. Rev. B 47, 14932(1993).[44] S. Ueda, M. Mizuguchi, Y. Miura, J. G. Kang, M. Shirai, and K.Takanashi, Appl. Phys. Lett. 109, 042404 (2016).[45] P. Bruno, Phys. Rev. B 39, 865(R) (1989).[46] S. Miwa, M. Suzuki, M. Tsujikawa, K. Matsuda, T. Nozaki,K. Tanaka, T. Tsukahara, K. Nawaoka, M. Goto, Y. Kotani,T. Ohkubo, F. Bonell, E. Tamura, K. Hono, T. Nakamura,M. Shirai, S. Yuasa, and Y. Suzuki, Nat. Commun. 8, 15848(2017).[47] K. Masuda and Y. Miura, Phys. Rev. B 98, 224421(2018).[48] J. Okabayashi, Y. Iida, Q. Xiang, H. Sukegawa, and S. Mitani,Appl. Phys. Lett. 115, 252402 (2019).[49] J. Okabayashi, Y. Miura, and H. Munekata, Sci. Rep. 8, 8303(2018).[50] J. Okabayashi, Y. Miura, Y. Kota, K. Z. Suzuki, A. Sakuma, andS. Mizukami, Sci. Rep. 10, 9744 (2020).[51] S. Miwa, T. Nozaki, M. Tsujikawa, M. Suzuki, T. Tsukahara,T. Kawabe, Y. Kotani, K. Toyoki, M. Goto, T. Nakamura, M.Shirai, S. Yuasa, and Y. Suzuki, Phys. Rev. B 99, 184421(2019).[52] K. Momma and F. Izumi, J. Appl. Cryst. 44, 1272 (2011).064427-10https://doi.org/10.1063/1.3265740https://doi.org/10.1103/PhysRevB.54.11169https://doi.org/10.1103/PhysRevLett.77.3865https://doi.org/10.1103/PhysRevB.50.17953https://doi.org/10.1103/PhysRevB.59.1758https://doi.org/10.1103/PhysRevB.96.054428https://doi.org/10.1103/PhysRevB.70.045417http://www.pwscf.orghttps://doi.org/10.1103/PhysRevB.59.2267https://doi.org/10.1103/PhysRevB.41.11919https://doi.org/10.1103/PhysRevB.32.2115https://doi.org/10.1088/0953-8984/25/10/106005https://doi.org/10.1016/0375-9601(75)90174-7https://doi.org/10.1103/PhysRevB.47.14932https://doi.org/10.1063/1.4959957https://doi.org/10.1103/PhysRevB.39.865https://doi.org/10.1038/ncomms15848https://doi.org/10.1103/PhysRevB.98.224421https://doi.org/10.1063/1.5127665https://doi.org/10.1038/s41598-018-26195-whttps://doi.org/10.1038/s41598-020-66432-9https://doi.org/10.1103/PhysRevB.99.184421https://doi.org/10.1107/S0021889811038970