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[Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[Perpendicular magnetic anisotropy at the Fe/MgAl2O4 interface: Comparative first-principles study with Fe/MgO](https://mdr.nims.go.jp/datasets/0af4444d-67eb-4d5a-80a3-7fc2aaac44a3)

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Perpendicular magnetic anisotropy at the ${\rm Fe}/{\rm MgAl}_2{\rm O}_4$ interface: Comparative first-principles study with Fe/MgOPHYSICAL REVIEW B 98, 224421 (2018)Perpendicular magnetic anisotropy at the Fe/MgAl2O4 interface:Comparative first-principles study with Fe/MgOKeisuke Masuda1 and Yoshio Miura1,2,3,41Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS), 1-2-1 Sengen,Tsukuba 305-0047, Japan2Electrical Engineering and Electronics, Kyoto Institute of Technology, Kyoto 606-8585, Japan3Center for Materials Research by Information Integration, National Institute for Materials Science (NIMS), 1-2-1 Sengen,Tsukuba 305-0047, Japan4Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, Machikaneyama 1-3,Toyonaka, Osaka 560-8531, Japan(Received 2 April 2018; revised manuscript received 23 September 2018; published 26 December 2018)We present a theoretical study on interfacial magnetocrystalline anisotropy for Fe/MgAl2O4. This systemhas a very small lattice mismatch at the interface and therefore is suitable for realizing a fully coherentferromagnet/oxide interface for magnetic tunnel junctions. On the basis of density functional theory, we calculatethe interfacial anisotropy constant Ki and show that this system has interfacial perpendicular magnetic anisotropy(PMA) with Ki ≈ 1.2 mJ/m2, which is a little bit smaller than that of Fe/MgO (Ki ≈ 1.5–1.7 mJ/m2).Second-order perturbation analysis with respect to the spin-orbit interaction clarifies that the difference in Kibetween Fe/MgAl2O4 and Fe/MgO originates from the difference in contributions from spin-flip scatteringterms at the interface. We propose that the insertion of tungsten layers into the interface of Fe/MgAl2O4 is apromising way to obtain huge interfacial PMA with Ki � 3 mJ/m2.DOI: 10.1103/PhysRevB.98.224421I. INTRODUCTIONPerpendicular magnetic anisotropy (PMA) is an essentialproperty for ferromagnets (FMs) in magnetic tunnel junc-tions (MTJs) to realize nonvolatile magnetic random accessmemories (MRAMs) [1]. The PMA is beneficial for obtainingsufficiently high thermal stability and low critical currentin spin-transfer-torque MRAMs (STT-MRAMs), in whichcurrent-induced spin-transfer torque is used for magnetiza-tion switching [1]. Although large PMA has been observedin several FMs such as D022 Mn3Ga [2,3], D022 Mn3Ge[4,5], L10 MnGa [3], and L10 FePt [6], MTJs with theseFMs did not show sufficiently high tunnel magnetoresistance(TMR) ratios, which is another important requirement forMRAM applications. Therefore, interfacial PMA at interfacesbetween FMs and insulator barriers has attracted much atten-tion mainly in MTJs consisting of Fe-based FMs and MgObarriers.In addition to high TMR ratios [7–10], interfacial PMAhas also been obtained in the MgO-based MTJs. By usingthin CoFeB layers (∼1.3 nm), Ikeda et al. observed rela-tively large PMA at the interface of CoFeB/MgO/CoFeBMTJ [11]. In subsequent studies [12,13], Koo et al. demon-strated that Fe/MgO has a larger interfacial PMA than thatof CoFe(B)/MgO, in agreement with theoretical predictions[14,15]. Furthermore, interfacial PMA has also been observedin the heterostructure composed of the Heusler alloy Co2FeAland MgO [16,17]. The interfacial PMA is also advantageousfor voltage-torque MRAMs [18], because high interfacialPMA gives low write error rates in voltage-driven magneti-zation switching.The underlying mechanism of such interfacial PMA inFe-based FM/MgO heterostructures has been discussed inseveral theoretical studies. By analyzing the local density ofstates (LDOS) and band structure in Fe/MgO, Nakamura et al.[19] clarified that the Fe 3d3z2−r2 state is distributed awayfrom the Fermi level owing to its hybridization with the O2pz state, leading to interfacial PMA. Other studies [14,20]also indicated the importance of this hybridization using dif-ferent theoretical approaches. From a different point of view,the relation between PMA and orbital magnetic moment isanother significant issue. A second-order perturbation theoryby Bruno [21] revealed a proportional relation between mag-netic anisotropy and anisotropy of orbital magnetic moment,which is the so-called Bruno relation. Several theoreticalstudies have discussed the applicability of the Bruno relationto various Fe-based heterostructures [15,22,23]. Moreover,by means of x-ray magnetic circular dichroism (XMCD)measurements, Okabayashi et al. [24] showed that the in-terfacial PMA in Fe/MgO can be explained qualitatively bythe Bruno relation. This relation gives valuable informationfor understanding the interfacial PMA in Fe-based FM/MgOheterostructures.Although large interfacial PMA has been observed inFe/MgO heterostructures, the lattice mismatch between Feand MgO is rather large (∼4%), which is a drawback for prac-tical applications. On the other hand, spinel oxide MgAl2O4has a small lattice mismatch (<1%) with typical FMs suchas Fe, Co0.5Fe0.5, and Co2FeAl0.5Si0.5 [25]. Moreover, sincethe lattice constant of MgAl2O4 can be tuned by changingthe Mg/Al composition rate, one can achieve good lattice2469-9950/2018/98(22)/224421(9) 224421-1 ©2018 American Physical Societyhttp://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.98.224421&domain=pdf&date_stamp=2018-12-26https://doi.org/10.1103/PhysRevB.98.224421KEISUKE MASUDA AND YOSHIO MIURA PHYSICAL REVIEW B 98, 224421 (2018)xyzxyz(a)(b)Fe O Al MgFe O MgFIG. 1. Supercells of (a) Fe(5)/MgAl2O4(9) and (b)Fe(5)/MgO(5).matching with various FMs. Up to now, relatively highMR ratios have been observed in the MgAl2O4-based MTJs[25–28]. The interfacial PMA has also been obtained in someFM/MgAl2O4 heterostructures [29–31]. In particular, Kooet al. [29] reported that the Fe(0.7 nm)/MgAl2O4 heterostruc-ture has an interfacial PMA with interfacial anisotropy con-stant Ki of 0.9–1.6 mJ/m2, which is smaller than that ofthe Fe/MgO heterostructure (Ki ∼ 1.5–2.0 mJ/m2) with thesame Fe thickness [12]. Possible reasons for such a dif-ference in Ki should be clarified; however, no theoreticalstudy has addressed interfacial magnetocrystalline anisotropyin Fe/MgAl2O4.In this work, we study interfacial magnetocrystallineanisotropy in Fe/MgAl2O4 by means of first-principles cal-culations based on density functional theory. We find thatthis system has interfacial PMA with Ki ≈ 1.2 mJ/m2. Thisvalue of Ki is smaller than that calculated in Fe/MgO with asimilar barrier thickness (Ki ≈ 1.5–1.7 mJ/m2), in agreementwith the above-mentioned experimental results. To clarify theorigin of such a difference in Ki, second-order perturbationanalyses are carried out, which find that the smaller Ki inFe/MgAl2O4 is due to a smaller positive contribution in Kifrom spin-flip electron scattering. We show that these resultscan be naturally understood from the features of the LDOSsand band structures in these systems. We finally propose aninterfacial insertion of tungsten (W) layers into Fe/MgAl2O4as a possible way to achieve a larger Ki. It is shown that suchFe/W/MgAl2O4 systems with 4–5 layers of W have a largeKi of � 3 mJ/m2.II. CALCULATION METHODWe analyzed Fe/MgAl2O4(001) and Fe/MgO(001) bymeans of density functional theory (DFT) including the ef-fect of spin-orbit interactions, which is implemented in theVienna ab initio simulation program (VASP) [32]. We adoptedthe spin-polarized generalized gradient approximation (GGA)[33] for the exchange-correlation energy and used the projec-tor augmented wave (PAW) potential [34,35] to treat the effectof core electrons properly.Figures 1(a) and 1(b) show the supercells ofFe(5)/MgAl2O4(9) and Fe(5)/MgO(5) used in this study,where each number in parentheses represents each layernumber. Note that MgAl2O4(9) and MgO(5) have similarbarrier thicknesses, which are suitable for comparison ofinterfacial magnetic anisotropy. As mentioned in Sec. I, themost striking feature of Fe/MgAl2O4 is the significantly smalllattice mismatch between the electrode and the barrier; at theinterface, two unit cells of bcc Fe with 2aFe = 5.732 Å canbe well fitted to MgAl2O4 with aMgAl2O4/√2 = 5.72 Å. Thus,we fixed the in-plane lattice constant a of the Fe/MgAl2O4supercell to a = 2 aFe = 5.732 Å. On the other hand, thelattice mismatch is relatively large in Fe/MgO, for whichwe used two supercells with different in-plane latticeconstants a: one is a = aFe = 2.866 Å, and the other isa = aMgO/√2 = 2.98 Å. In all of these supercells, we carriedout structure relaxation, through which optimum atomicpositions and the interfacial distance between the electrodeand the barrier were determined. Here, we used the known factthat an interfacial atomic configuration where O atoms are ontop of Fe atoms [see Figs. 1(a) and 1(b)] is energeticallyfavored in both Fe/MgAl2O4 and Fe/MgO [36]. Thedetails of our structure relaxation are given in our previouspaper [40].In each optimized supercell, we calculated interfacial mag-netocrystalline anisotropy Ki using the well-known forcetheorem [41]Ki = (E[100] − E[001])/2S, (1)where E[100] (E[001]) is the sum of the eigenenergies of thesupercell with the magnetization parallel to the [100] ([001])direction, and S is the cross-sectional area of the supercell.Note that the factor 2 in the denominator reflects the factthat each supercell has two interfaces. In order to confirmwhether the force theorem gives reliable results for the presentsystems, we also calculated Ki in the self-consistent-field(SCF) manner using the total energies instead of the sumof eigenenergies in Eq. (1) [41]. In this paper, we representa set of k-point numbers used for the calculations as Nx ×Ny × Nz, where Nx, Ny , and Nz are the k-point numbersused for the x, y, and z directions of supercells, respectively.Figure 2(a) shows the values of Ki in Fe/MgAl2O4 as afunction of the number of in-plane k points N ≡ Nx = Nyobtained from the force theorem and the SCF total-energycalculation, where Nz is fixed to 1 or 3. We see that thevalue of Ki is saturated for N � 19 in all four of the casesshown in the figure and that the saturated values are almostthe same (Ki ∼ 1.2 mJ/m2). Similar saturations of Ki werealso obtained in two Fe/MgO systems with a = aMgO/√2 anda = aFe, as shown in Figs. 2(b) and 2(c), respectively. In bothof these systems, Ki is saturated to ∼1.6 mJ/m2 for N � 37.All these results indicate that the calculation using the forcetheorem and 19 × 19 × 1 (37 × 37 × 1) k points is sufficientto accurately estimate Ki in Fe/MgAl2O4 (Fe/MgO) [42];in the following, we use such calculation conditions. FromEq. (1), we can easily see that positive (negative) Ki indi-cates the tendency toward perpendicular (in-plane) magneticanisotropy. However, actual magnetic anisotropy is estimatedby Keff t = Ki + Edemagt , where t is the effective thicknessof the Fe electrode and Edemagt represents magnetic shapeanisotropy. The second term Edemagt always has a negativevalue, and therefore favors in-plane magnetic anisotropy. In224421-2PERPENDICULAR MAGNETIC ANISOTROPY AT THE … PHYSICAL REVIEW B 98, 224421 (2018) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 5  10  15  20  25  30 0 0.5 1.0 1.5 2.0 10  20  30  40  50  60 0 0.5 1.0 1.5 2.0 10  20  30  40  50  60Number of in-plane k points N (= Nx = Ny )Number of in-plane k points N (= Nx = Ny )Number of in-plane k points N (= Nx = Ny )Ki [mJ/m2 ]Ki [mJ/m2 ]Ki [mJ/m2 ](a)(b)(c)Nz=1, force theoremNz=1, SCFNz=3, force theoremNz=3, SCFNz=1, force theoremNz=1, SCFNz=3, force theoremNz=3, SCFNz=1, force theoremNz=1, SCFNz=3, force theoremNz=3, SCFFIG. 2. The interfacial anisotropy constant Ki as a function ofthe number of in-plane k points N (= Nx = Ny) in (a) Fe/MgAl2O4,(b) Fe/MgO (a = aMgO/√2), and (c) Fe/MgO (a = aFe).the present work, we calculated Edemagt by summing upthe magnetostatic dipole-dipole interaction between atomicmagnetic moments [41] with the use of the Ewald-summationtechnique [43].In addition to these calculations, we further carried outa detailed second-order perturbation analysis to understandmagnetocrystalline anisotropy in Fe/MgAl2O4 and Fe/MgOmore deeply. By treating the spin-orbit interaction HSO asa perturbation term, the second-order perturbation energy isexpressed asE(2) =∑kunocc∑n′σ ′occ∑nσ|〈kn′σ ′|HSO|knσ 〉|2ε(0)knσ − ε(0)kn′σ ′, (2)HSO =∑iξi Li · Si , (3)where ε(0)knσ is the energy of an unperturbed state |knσ 〉 withwave vector k, band index n, and spin σ . The index “occ”(“unocc”) on the summation means that the sum is overoccupied (unoccupied) states of all atoms in the supercell[44,45]. Note here that the state |knσ 〉 can be expanded as|knσ 〉 = ∑iμ ckniμσ |iμσ 〉, where μ is an atomic orbital at sitei and ckniμσ = 〈iμσ |knσ 〉 [44]. In the spin-orbit interactionHSO, ξi is its coupling constant at site i, and Li (Si) is thesingle-electron angular (spin) momentum operator. As the val-ues of ξi , we used ξFe = 54.3 meV, ξMg = 47.5 meV, ξAl =10.8 meV, and ξO = 24.3 meV for Fe, Mg, Al, and O atoms,respectively. The Wigner-Seitz radius of each atom was set torFe = 1.302 Å, rMg = 1.524 Å, rAl = 1.402 Å, rO = 0.820 Å.All these values of spin-orbit coupling constants and Wigner-Seitz radii are those listed in the pseudopotential files inVASP. We used wave functions and eigenenergies obtainedin our DFT calculations as unperturbed states and energiesin Eq. (2). The magnetocrystalline anisotropy energy withinthe second-order perturbation E(2)MCA (∝ Ki) was calculated asE(2)MCA = E(2)[100] − E(2)[001], where E(2)[100] (E(2)[001]) is the energy forthe magnetization along the [100] ([001]) direction obtainedby Eq. (2). In the process of such an analysis, we can decom-pose E(2)MCA = ∑i EiMCA into four types of terms coming fromdifferent electron scattering around the Fermi level:E(2)MCA=∑i(�Ei↑⇒↑ + �Ei↓⇒↓ + �Ei↑⇒↓ + �Ei↓⇒↑). (4)Here, �Ei↑⇒↑ (�Ei↓⇒↓) originates from spin-conserving elec-tron scattering between occupied and unoccupied majority-spin (minority-spin) states. On the other hand, �Ei↑⇒↓(�Ei↓⇒↑) corresponds to spin-flip electron scattering fromoccupied majority-spin (minority-spin) states to unoccupiedminority-spin (majority-spin) states. The details of these cal-culations are given in a previous paper [44]. As we show inthe next section, differences in magnetocrystalline anisotropybetween different systems can be explained naturally by thesesecond-order perturbation analyses.III. RESULTS AND DISCUSSIONTable I shows the values of Ki, Edemagt, Keff t, �Morb,i,and Mspin,i for Fe/MgAl2O4 and Fe/MgO obtained in thisstudy. Here, �Morb,i is the anisotropy of the interfacial Feorbital magnetic moment and Mspin,i is the spin magneticmoment at interfacial Fe atoms. We see that Fe/MgAl2O4has a positive Ki of 1.192 mJ/m2. Since this value exceedsthe negative shape anisotropy (Edemagt = −0.895 mJ/m2),this system has interfacial PMA (Keff t = 0.296 mJ/m2 > 0).224421-3KEISUKE MASUDA AND YOSHIO MIURA PHYSICAL REVIEW B 98, 224421 (2018)TABLE I. List of Ki, Edemagt, Keff t, �Morb,i, and Mspin,i obtained in this study.System Ki (mJ/m2) Edemagt (mJ/m2) Keff t (mJ/m2) �Morb,i (μB/atom) Mspin,i (μB/atom)Fe/MgAl2O4 1.192 −0.895 0.296 0.026 2.81Fe/MgO (a = aMgO/√2) 1.617 −0.828 0.788 0.030 2.73Fe/MgO (a = aFe) 1.552 −0.908 0.643 0.020 2.78Note that O layer is the termination layer of MgAl2O4,as mentioned in Sec. II. Thus, the interfacial hybridizationbetween Fe 3d3z2−r2 and O 2pz states plays a key role forthe interfacial PMA of Fe/MgAl2O4 in the same way asFe/MgO.The values of Ki and Keff t in Fe/MgAl2O4 are smaller thanthose in Fe/MgO. As mentioned in Sec. I, the relationshipbetween magnetocrystalline anisotropy and anisotropy of theorbital magnetic moment provides important information onPMA in these systems. From Table I, we find that both Kiand �Morb,i of Fe/MgO with a = aMgO/√2 are larger thanthose of Fe/MgAl2O4, which indicates that the Bruno relation(Ki ∝ �Morb,i) holds for these two systems. On the otherhand, it seems that this relation is not applicable to Fe/MgOwith a = aFe, because this system has a larger Ki but a smaller�Morb,i than Fe/MgAl2O4. Therefore, the following second-order perturbation analysis is required to deeply understandinterfacial PMA in all of these systems.In Fig. 3(a), we show the results of the second-orderperturbation analysis for the magnetocrystalline anisotropyin Fe/MgAl2O4. We see that the interfacial Fe layer has thelargest positive EiMCA, which provides the dominant contri-bution to the positive Ki in this system. This indicates thatFe/MgAl2O4 has interfacial PMA. At the interfacial Fe layer(Fe1), the anisotropy due to minority-spin scattering (�Ei↓⇒↓)provides the largest contribution. In order to understand thisfeature, we utilize the following simplified expressions for thelocal magnetocrystalline anisotropy [46]:EiMCA ≈ �Ei↓⇒↓ + �Ei↑⇒↓, (5)�Ei↓⇒↓ = ξ 2i∑u↓,o↓∣∣〈u↓|Liz|o↓〉∣∣2 − ∣∣〈u↓|Lix |o↓〉∣∣2εu↓ − εo↓, (6)�Ei↑⇒↓ = ξ 2i∑u↓,o↑∣∣〈u↓|Lix |o↑〉∣∣2 − ∣∣〈u↓|Liz|o↑〉∣∣2εu↓ − εo↑, (7)where the meanings of �Ei↓⇒↓ and �Ei↑⇒↓ are the same asthose in Eq. (4). Here, ξi is the spin-orbit coupling constant,Liα (α = x, z) is the local angular momentum operator at sitei, and |oσ 〉 (|uσ 〉) is a local occupied (unoccupied) state withspin σ and energy εoσ(εuσ). To derive these expressions, itis assumed that the occupied d states in the majority-spinchannel are located deep below the Fermi level. Therefore,we neglected �Ei↑⇒↑ and �Ei↓⇒↑ in Eq. (5) [compare withEq. (4)]. Let us now focus on the local density of states(LDOS) of interfacial Fe atoms in Fe/MgAl2O4 shown inFig. 3(b). From the inset of the figure, we find that themajority-spin states have quite small LDOS around the Fermilevel. Thus, we can expect that the term �Ei↓⇒↓ provides thedominant contribution in Eq. (5) in the case of Fe/MgAl2O4,which is consistent with our results shown in Fig. 3(a). Inthe previous paragraph, we mentioned that the Bruno rela-tion holds in this system, which is reasonable because only�Ei↓⇒↓ is taken into account in the derivation of the Brunorelation [47].FIG. 3. (a) Results of second-order perturbation analysison the interfacial PMA in Fe/MgAl2O4. The verticalgreen, yellow, blue, red, and pink bars show the values of�Ei↑⇒↑, �Ei↓⇒↓, �Ei↑⇒↓, �Ei↓⇒↑, and the local anisotropyenergy EiMCA, respectively, at each Fe layer. [See Eq. (4) and thecorresponding text for details.] (b) Projected LDOSs for Fe 3d statesat the interface of Fe/MgAl2O4. In panel (b), positive and negativevalues indicate the majority- and minority-spin projected LDOSs,respectively. The inset of panel (b) shows a magnified view near theFermi level.224421-4PERPENDICULAR MAGNETIC ANISOTROPY AT THE … PHYSICAL REVIEW B 98, 224421 (2018)(a)(b)(c)FIG. 4. The wave-vector-resolved information on the interfacialPMA in Fe/MgAl2O4. (a) The in-plane wave-vector (k‖) dependenceof �E(k‖) ≡ E[100](k‖) − E[001](k‖). (b) and (c) The band structureof the supercell in the majority- and minority-spin states, respec-tively. In panels (b) and (c), orbital components of each band areindicated by colors.In order to obtain further information on the PMAin Fe/MgAl2O4, we analyzed wave-vector-resolved magne-tocrystalline anisotropy and band structures of the supercell.Previous studies using this type of analysis on other ferro-magnetic systems have shown that localized d states aroundthe Fermi level provide the dominant contribution to themagnetocrystalline anisotropy [19,48–50]. Figure 4(a) showsthe in-plane wave-vector (k‖) dependence of �E(k‖) ≡E[100](k‖) − E[001](k‖) [51]. Here, we plotted the case of kz =0 because kz dependence is very weak owing to the long c-axisconstant of the supercell. Note that Ki is proportional to thesum of �E(k‖) over all k‖ in the two-dimensional Brillouinzone. We see that large positive anisotropy is obtained aroundthe � point, which provides the dominant contribution to theinterfacial PMA in this system. We can naturally understandFIG. 5. The same as Fig. 3, but for Fe/MgO with a = aMgO/√2.this behavior from the band structures of the supercell shownin Figs. 4(b) and 4(c). Actually, as seen from Fig. 4(c), theminority-spin bands around the � point have dzx- and dyz-orbital components near the Fermi level, leading to finitevalues of 〈dyz,↓ |Liz|dzx,↓〉 and 〈dzx,↓ |Liz|dyz,↓〉 includedin the first term of the numerator of Eq. (6). Such a bandstructure is consistent with sharp peaks in the dyz- and dzx-orbital LDOSs in the minority-spin states around the Fermilevel [see Fig. 3(b)].We next discuss PMA in Fe/MgO to understand the dif-ference from the case of Fe/MgAl2O4. Figure 5(a) showsthe results of the second-order perturbation calculations inFe/MgO with a = aMgO/√2 = 2.98 Å. In this case, we ob-tained similar results with Fe/MgAl2O4; anisotropy energydue to minority-spin scattering (�Ei↓⇒↓) at the interfaceprovides the dominant contribution to the PMA. The structureof the LDOS at the interfacial Fe atoms is also similar tothat of Fe/MgAl2O4 [see Fig. 5(b)]; the majority-spin statehas quite small LDOS around the Fermi level. In Fig. 5(a),a non-negligible difference with Fe/MgAl2O4 is that the224421-5KEISUKE MASUDA AND YOSHIO MIURA PHYSICAL REVIEW B 98, 224421 (2018)-1-0.5 0 0.5 1-1-0.5 0 0.5 1-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5ΓMX(a)(b)(c)E-E F [eV]Γ X M ΓE-E F [eV]Majority-spinMinority-spinFIG. 6. The same as Fig. 4, but for Fe/MgO with a = aMgO/√2.Fe/MgO has a larger positive spin-flip component �Ei↑⇒↓ atthe interfacial Fe atoms. To clarify the reason of this behavior,we show the k‖ dependence of �E(k‖) in Fig. 6(a). We findthat positive anisotropy occurs mainly around the X point.Similarly to the case of Fe/MgAl2O4, dzx and dyz states inthe minority-spin bands give finite values of 〈Liz〉 as shownin Fig. 6(c), leading to positive �Ei↓⇒↓. In addition, themajority-spin occupied dxy band and minority-spin unoccu-pied dzx band yield finite values of 〈dzx,↓ |Lix |dxy,↑〉, asseen from Figs. 6(b) and 6(c). This gives positive �Ei↑⇒↓following Eq. (7), which is the reason why this system hasthe non-negligible contribution from spin-flip scattering in theinterfacial PMA.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5  0  0.5  1  1.5  2-0.6-0.4-0.2 0 0.2-0.4 -0.2  0  0.2  0.4-0.100.10.20.30.40.5-0.2Projected LDOS [states/eV/orbital]E-EF [eV](a)(b)Fe1 Fe2 Fe3xzFe1OMgFe2Fe3FIG. 7. The same as Fig. 3, but for Fe/MgO with a = aFe.We also carried out the same perturbation analysis on theother Fe/MgO with a = aFe = 2.866 Å, which provides valu-able insight as explained below. Figure 7(a) shows the resultsof the calculations, in which we find a clear difference fromthose of Fe/MgAl2O4 and also from those of Fe/MgO witha = aMgO/√2. Namely, at the interfacial Fe atoms, the spin-flip component �Ei↑⇒↓ is quite large and has a similar valueto that of the spin-preserving component �Ei↓⇒↓. This is thereason why the Bruno relation does not hold in this systemas mentioned above. We can naturally understand the originof this behavior by analyzing the LDOSs of interfacial Featoms shown in Fig. 7(b). As seen from the inset of the figure,the majority-spin state has a finite d3z2−r2 LDOS around theFermi level. Since these d3z2−r2 states give finite values of〈dyz,↓ |Lix |d3z2−r2 ,↑〉, the large positive �Ei↑⇒↓ can occurfollowing Eq. (7) [52]. This is the reason for the magnituderelation �Ei↓⇒↓ ≈ �Ei↑⇒↓ in this system. This feature canalso be confirmed by the k‖ dependence of �E(k‖) andband structures of the supercell shown in Figs. 8(a)–8(c). Themajor difference from the case of a = aMgO/√2 is that the224421-6PERPENDICULAR MAGNETIC ANISOTROPY AT THE … PHYSICAL REVIEW B 98, 224421 (2018)-1-0.5 0 0.5 1-1-0.5 0 0.5 1Γ X M ΓE-E F [eV]E -E F [eV]Majority-spinMinority-spin-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5ΓMX(a)(b)(c)FIG. 8. The same as Fig. 4, but for Fe/MgO with a = aFe.majority-spin d3z2−r2 band crosses the Fermi level around the� point as shown in Fig. 8(b). Thus, the occupied d3z2−r2states in this band and the unoccupied dyz states in theminority-spin bands give finite values of 〈dyz,↓ |Lix |d3z2−r2 ,↑〉 [see Figs. 8(b) and 8(c)], by which �Ei↑⇒↓ becomes largercompared to the case of a = aMgO/√2. An experimentallyrealized Fe/MgO heterostructure is expected to have an in-termediate in-plane lattice constant between aFe = 2.866 Åand aMgO/√2 = 2.98 Å. Although the anisotropy energy fromspin-flip scattering �Ei↑⇒↓ is sensitive to the in-plane latticeconstant, we can conclude from our results that Fe/MgO hasa larger positive �Ei↑⇒↓ than Fe/MgAl2O4. This is a possibleexplanation for the fact that the experimentally observed Ki inFe/MgO is larger than that in Fe/MgAl2O4.-1.5-1-0.500.511.522.533.544.5xyz(a)(b)Ki [mJ/m2 ]W(3)Fe W OAl MgW(4)W(5)W(0)FIG. 9. (a) The supercell of Fe(5)/W(3)/MgAl2O4(9) usedin our calculation. (b) Values of Ki obtained for Fe(5)/W(3–5)/MgAl2O4(9) and two types of Fe(5)/W(3–5)/MgO(5) withdifferent in-plane lattice constants. The data indicated by W(0) arethose when W layers are not inserted (the values of Ki were alreadyshown in Table I).IV. A WAY TO OBTAIN LARGER PMAIn order to obtain larger interfacial PMA for MTJs with theMgAl2O4 barrier, we propose an insertion of thin W layersbetween MgAl2O4 and the Fe electrode as shown in Fig. 9(a).The insertion of W layers at the interface of Fe/MgAl2O4is based on the theoretical prediction of huge PMA in theFe/W(001) multilayer with the in-plane lattice constant ofbulk bcc Fe [53]. Experimentally, Matsumoto and co-workersconfirmed the large change of magnetic anisotropy fromnegative to positive by reducing the W-layer thickness inthe Fe/W(001) multilayer, where the in-plane lattice constantof W approaches from that of bulk W to that of bulk Fe[53,54]. Motivated by these theoretical and experimental re-sults, we examined the possibility that the insertion of thinW layers into the interface of Fe/MgAl2O4 with the in-planelattice constant of bcc Fe can enhance the interfacial PMAin this junction. In Fig. 9(b), we show the calculated valuesof Ki in Fe/W(n)/MgAl2O4(001) and Fe/W(n)/MgO(001).As in-plane lattice constants, we adopted a = 2 aFe for theMgAl2O4-based junction and a = aFe and a = aMgO/√2for the MgO-based junction. As can be seen in Fig. 9(b),Fe/W(3–5)/MgAl2O4 and Fe/W(4–5)/MgO with a = aFehave large positive Ki, indicating that the insertion of W layerssignificantly enhances the PMA in these junctions. Note thatsuch a large enhancement was not obtained in very thin Wcases (n = 1–2). Thus, at least 3 layers of W are required224421-7KEISUKE MASUDA AND YOSHIO MIURA PHYSICAL REVIEW B 98, 224421 (2018)(a) (b)(c) (d)-0.20-0.15-0.10-0.05 0 0.05 0.10 0.15 0.20 0.25Projected LDOS [states/eV/atom]-0.20-0.15-0.10-0.05 0 0.05 0.10 0.15 0.20 0.25Projected LDOS [states/eV/atom]E-EF [eV]-2.0 -1.5 -1.0 -0.5  0  0.5  1.0  1.5  2.0 -2.0 -1.5 -1.0 -0.5  0  0.5  1.0  1.5  2.0E-EF [eV]FIG. 10. (a)–(d) Projected LDOSs for 3d states at themiddle layer of the W atom in Fe(5)/W(3)/MgAl2O4(9),Fe(5)/W(3)/MgO(5) (a = aFe), and Fe(5)/W(3)/MgO(5) (a =aMgO/√2), where positive and negative values indicate the majority-and minority-spin projected LDOSs, respectively.for enhancing PMA. On the other hand, Fe/W(n)/MgO witha = aMgO/√2 shows a negative or small Ki for any layernumber of W up to n = 5, which means that the insertion of Wlayers degrades the interfacial PMA in this junction becauseof the lattice mismatch between Fe and MgO. These resultsare consistent with those of Fe/W multilayers in Ref. [53].We can conclude that the insertion of W layers into theinterface of Fe/MgAl2O4 is a promising way to obtain hugePMA owing to the good lattice matching between Fe andMgAl2O4, indicating the advantage of MgAl2O4 as comparedwith MgO. Perhaps it might be even better to use W layers asunderlayers of Fe/MgAl2O4, because the interfacial insertionof nonmagnetic metals between FMs and insulator barrierstends to decrease TMR ratios of MTJs. However, furtheranalysis for the PMA in such a system is beyond the scopeof this study and will be addressed in our future work.From the second-order perturbation analysis, we foundthat the large PMA in the Fe/W/MgAl2O4 multilayer ismainly attributed to perturbation processes through unoccu-pied majority-spin states (�Ei↓⇒↑ and �Ei↑⇒↑) of the middle-layer W atoms. Figures 10(a)–10(d) show the projectedLDOSs of middle-layer W atoms in Fe(5)/W(3)/MgAl2O4(9)and Fe(5)/W(3)/MgO(5) (a = aFe and a = aMgO/√2). Ascan be seen in Figs. 10(a) and 10(c), there are large unoc-cupied dxy and dyz (dzx) states just above the Fermi level,when the in-plane lattice constant corresponds to that of bccFe (a = aFe). Because W is a transition metal element withless than half d electrons, it has unoccupied majority-spind states. These unoccupied majority-spin states provide aconsiderable contribution to the PMA through the second-order perturbation of the spin-orbit interaction between un-occupied majority-spin states and occupied states in boththe spin channels. In the present case, the matrix elements〈dxy,↑ |Liz|dx2−y2 ,↑〉 and 〈dyz,↑ |Lix |d3z2−r2 ,↓〉 show posi-tive contributions to the PMA of Fe(5)/W(3)/MgAl2O4(9)and Fe(5)/W(3)/MgO(5) (a = aFe) in the perturbation pro-cesses.V. SUMMARYWe theoretically investigated interfacial magnetocrys-talline anisotropy in Fe/MgAl2O4, which has a potentialapplicability to spintronic devices because of its quite smalllattice mismatch at the interface. By means of density func-tional theory, we calculated interfacial anisotropy constantKi of this system and compared it with those of twoFe/MgO systems with different in-plane lattice constants.We found that Fe/MgAl2O4 has interfacial perpendicularmagnetic anisotropy (PMA) with Ki ≈ 1.2 mJ/m2, which isslightly smaller than that of Fe/MgO systems. By carryingout second-order perturbation calculations on the PMA incombination with detailed analyses of the LDOSs and bandstructures, we clarified that the smaller Ki in Fe/MgAl2O4 isdue to the smaller positive anisotropy energy from spin-flipelectron scattering. We finally proposed insertion of tungsten(W) into the interface of Fe/MgAl2O4 as a possible way toobtain larger interfacial PMA. 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