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[Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Thomas Scheike](https://orcid.org/0000-0002-9163-5524), [Hiroaki Sukegawa](https://orcid.org/0000-0002-4034-7848), [Yusuke Kozuka](https://orcid.org/0000-0001-7674-600X), [Seiji Mitani](https://orcid.org/0000-0002-1348-0774), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[Theory for tunnel magnetoresistance oscillation](https://mdr.nims.go.jp/datasets/1c2a7d7e-3f20-4eb1-b1a9-9c91dd9ab34f)

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Theory for tunnel magnetoresistance oscillationPHYSICAL REVIEW B 111, L220406 (2025)Letter Editors’ SuggestionTheory for tunnel magnetoresistance oscillationKeisuke Masuda ,1,* Thomas Scheike ,1 Hiroaki Sukegawa ,1 Yusuke Kozuka ,2Seiji Mitani,1,3 and Yoshio Miura 1,4,51Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan2Research Center for Materials Nanoarchitectonics, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan3Graduate School of Science and Technology, University of Tsukuba, Tsukuba 305-8577, Japan4Faculty of Electrical Engineering and Electronics, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan5Center for Spintronics Research Network, Graduate School of Engineering Science,Osaka University, Toyonaka, Osaka 560-8531, Japan(Received 17 June 2024; revised 25 March 2025; accepted 23 May 2025; published 9 June 2025)The universal oscillation of the tunnel magnetoresistance (TMR) ratio as a function of the insulating barrierthickness in crystalline magnetic tunnel junctions (MTJs) is a long-standing unsolved problem in condensedmatter physics. To explain this, we here introduce a superposition of wave functions with opposite spins anddifferent Fermi momenta, based on the fact that spin-flip scattering near the interface provides a hybridizationbetween majority- and minority-spin states. In a typical Fe/MgO/Fe MTJ, we solve the tunneling problem andshow that the TMR ratio oscillates with a period of ∼3Å by varying the MgO thickness, consistent with previousand present experimental observations.DOI: 10.1103/PhysRevB.111.L220406The tunneling effect is one of the most fundamen-tal phenomena in quantum mechanics originating from thewave nature of matter. In particular, quantum tunnelinghas played an important role for various topics in con-densed matter physics. For example, in the case of p-njunctions, electrons tunnel through the depletion layer un-der large electric field, giving rise to negative differentialresistance [1]. As another example, tunneling spectrum ina metal/insulator/superconductor junction provides a clearsignature of a gap structure in the density of states ofthe superconductor, validating the Bardeen-Cooper-Schrieffertheory [2]. Moreover, the scanning tunneling microscope uti-lizes tunneling electrons for imaging surfaces in the atomiclevel [3].The tunnel magnetoresistance (TMR) effect is anothertopic related to tunneling in the field of spintronics. This oc-curs in magnetic tunnel junctions (MTJs) consisting of an in-sulating barrier sandwiched between ferromagnetic electrodes[Fig. 1(a)]. The wave functions in different spin channelshave different transmission probabilities because of imbal-anced band structures, leading to finite magnetoresistance.One can estimate the magnitude of the magnetoresistanceby defining the TMR ratio as a ratio of resistances be-tween parallel and antiparallel magnetization states of thetwo ferromagnetic electrodes. In 2004, Parkin et al. [4] andYuasa et al. [5] reported significantly high TMR ratios in*Contact author: MASUDA.Keisuke@nims.go.jpPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.Fe(Co)/MgO/Fe(Co)(001) MTJs, which provided a basis forfurther fundamental studies of the TMR effect and their de-vice applications. However, there is a missing piece in themechanism of such a giant TMR effect; the universal oscilla-tion of the TMR ratio as a function of the insulating barrierthickness [5–8] has not been explained satisfactorily. In areport of the giant TMR effect in Fe/MgO/Fe(001) [5], Yuasaet al. observed an oscillation of the TMR ratio with a periodof ∼3 Å by varying the MgO thickness, referred to as theTMR oscillation. Subsequent experiments [6] clarified thatthe TMR oscillation originates from resistance oscillationsin both parallel and antiparallel magnetization states. Here,electron tunneling through MgO occurs between the same(different) spin states of the two electrodes in the parallel(antiparallel) magnetization state. Recent experiments for aseries of MTJs with high crystallinity [9–11] found that theTMR oscillation with a period of ∼3 Å is universally observedand its amplitude is much larger than ever reported. Therefore,to elucidate the origin of the TMR oscillation will advanceour understanding not only on the TMR effect but also onthe quantum tunneling itself. This will also provide guidingprinciples for achieving even higher TMR ratios.Conventionally, high TMR ratios in Fe/MgO/Fe(001)have been explained by the �1 coherent tunneling mecha-nism; the half-metallic �1 band structure of Fe [Figs. 1(b) and1(c)] and the slowest decaying �1 evanescent state of MgOenable a selective tunneling of the perfectly spin-polarized �1state, leading to a high TMR ratio [12,13]. However, the TMRoscillation cannot be explained by this mechanism [12–14].Although additional effects, such as interference of evanes-cent states [12] and nonspecular tunneling [15], have beenconsidered, these provide a resistance oscillation only in theantiparallel magnetization state, qualitatively in disagreementwith the experimental results. Another study [16] proposed an2469-9950/2025/111(22)/L220406(6) L220406-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-6884-6390https://orcid.org/0000-0002-9163-5524https://orcid.org/0000-0002-4034-7848https://orcid.org/0000-0001-7674-600Xhttps://orcid.org/0000-0002-5605-5452https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/02956yf07https://ror.org/00965ax52https://ror.org/035t8zc32https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.111.L220406&domain=pdf&date_stamp=2025-06-09https://doi.org/10.1103/PhysRevB.111.L220406https://creativecommons.org/licenses/by/4.0/KEISUKE MASUDA et al. PHYSICAL REVIEW B 111, L220406 (2025)FIG. 1. (a) Schematic of the Fe/MgO/Fe MTJ. (b), (c) Majority(↑) -spin and minority (↓) -spin band structures of bcc Fe along the �line with k‖ = 0. (d) Illustration of our idea including a superpositionof wave functions with different Fermi momenta.oscillation of the TMR ratio due to the quantization in the fer-romagnetic layer, but this occurs when varying the thicknessof the ferromagnetic layer, inconsistent with the experimentalsituation.In this Letter, we show that the TMR oscillation can beexplained by taking into account a superposition of wavefunctions with opposite spins and different Fermi momentafor the tunneling problem. It is known that spin-flip scatter-ing occurs near interfaces of MTJs [17–19], indicating thatspin is not a good quantum number in this system. Thisprovides a hybridization between majority- and minority-spinstates with different Fermi momenta [Figs. 1(b) and 1(c)],which justifies our assumption on the superposition of wavefunctions. Focusing on Fe/MgO/Fe(001), we solve tunnel-ing problems assuming a superposition of majority-spin �1and minority-spin �2 wave functions with different Fermimomenta as a transmitted wave function [see Fig. 1(d)]. Weobtain transmittances in the parallel and antiparallel mag-netization states, from which the TMR ratio is calculated.It is found that the transmittances and the TMR ratio haveoscillatory behaviors with a period of ∼ 3 Å as a function ofthe MgO thickness, in agreement with previous and presentexperimental observations. We also show that the calculatedTMR ratio can reproduce our experimental results not onlyqualitatively but also quantitatively by tuning the parametersin our model. Although we focus on the TMR oscillation inthis Letter, the superposition of wave functions with differentFermi momenta is a general concept and would be helpful tounderstand transport properties in other tunnel junctions withsuperconductors, semiconductors, etc.To make the point of our approach clearer, we start by re-viewing the conventional analytical treatment of the tunnelingproblem in an MTJ. Let us consider the situation that the wavefunction in the left electrode propagates to the right electrodepassing through the insulating barrier, which is described by acoordinate system with the z axis along the stacking directionof the MTJ [Fig. 1(a)]. For simplicity, we focus on the wavefunctions with k‖ = (kx, ky) = (0, 0) providing the dominantcontribution to tunneling transport. When the Fermi momen-tum is given by kz = kL (kR) in the left (right) electrode, thewave function ψL (ψR) in the left (right) electrode and thewave function ψb in the insulating barrier are expressed asψL(z) = eikLz + R e−ikLz, (1)ψb(z) = A e−κz + B eκz, (2)ψR(z) = C eikRz, (3)where κ is the decaying wave number inside the insulatingbarrier. After determining R, A, B, and C from continuationconditions for the wave function and its derivative at z = 0 andd [20], we find the following expression for the transmittance:T = 16 k̃Lκ2k̃Re2κd[κ (k̃L + k̃R)(1 + e2κd )]2 + [(κ2 − k̃Lk̃R)(1 − e2κd )]2,(4)where d is the thickness of the insulating barrier,k̃L = (mb/mL) kL and k̃R = (mb/mR) kR. Here, mL(R) and mbare the effective masses in the left (right) electrode and theinsulating barrier, respectively [32]. We can obtain the con-ductance G by substituting Eq. (4) into the Landauer formulaG = (e2/h) T . The �1 coherent tunneling mechanism men-tioned above can be confirmed by employing this tunnelingtheory in combination with the first-principles calculation[34–36].However, this conventional tunneling theory cannot de-scribe the oscillation of the TMR ratio in the Fe/MgO/Fe(001) MTJ. Actually, the transmittance in the parallel mag-netization state is obtained by putting kL = kR = k1 (or k2)in Eq. (4), where k1 and k2 are the Fermi momenta of themajority-spin �1 and the minority-spin �2 bands of Fe, re-spectively [see Figs. 1(b) and 1(c)]. Note that the negativek2 value with a positive group velocity is chosen for theminority-spin state, since we focus on right-moving states.The transmittance in the antiparallel magnetization state issimilarly obtained by setting kL = k1 (or k2) and kR = k2(or k1) in Eq. (4). As seen from Eq. (4), both the paralleland antiparallel transmittances decrease exponentially withincreasing d in monotonic manner without any oscillations.To explain the oscillation in the transmittance, we in-troduce a superposition of wave functions between themajority-spin �1 and minority-spin �2 states for the transmit-ted wave in the tunneling problem [37]. Details on the choiceof these wave functions are discussed in the SupplementalMaterial [20]. Let us first calculate the parallel transmittanceTP. Based on the fact that the majority-spin �1 state pro-vides the dominant contribution to the TMR effect [12,13],TP can be calculated as TP = TP,↑ + TP,↓ ≈ TP,↑, where ↑(↓) indicates that tunneling electrons are in the majority-spinL220406-2THEORY FOR TUNNEL MAGNETORESISTANCE … PHYSICAL REVIEW B 111, L220406 (2025)(minority-spin) state in the left electrode. For the calculationof TP,↑, we consider a tunneling from the majority-spin �1state in the left electrode to the superposition state in the rightelectrode with the dominant contribution from the majority-spin �1 state, which is given byψL(z) = eik1z + R e−ik1z, (5)ψb(z) = A e−κz + B eκz, (6)ψR(z) = C (u eik1z + v eik2z ). (7)Here, u and v are matrix elements of the unitary matrix thatdiagonalizes the 2 × 2 Hamiltonian, including the effect of theinterfacial spin-flip scattering as off-diagonal elements. Anexplicit expression of the Hamiltonian and the derivation ofthe wave function in Eq. (7) are given in the SupplementalMaterial [20]. We impose |u| � |v| because of the dominanceof the majority-spin �1 state in TP,↑. The matrix elements alsosatisfy the normalization condition |u|2 + |v|2 = 1. By usingthe continuity of the wave function and its derivative at z = 0and d [38], we can derive the following expression for TP,↑:TP,↑ = k̃−11L [ũ2k̃1R + ṽ2k̃2R + ũṽ (k̃1R + k̃2R) cos ((k1 − k2)d − θ )]|C|2, (8)Denominator of |C|2 = (eκd − e−κd )2{κ4[1 + 2 ũṽ cos ((k1 − k2)d − θ )]− 2 κ2k̃1L[ũ2k̃1R + ṽ2k̃2R + ũṽ (k̃1R + k̃2R ) cos ((k1 − k2)d − θ )]+ k̃21L[ũ2k̃21R + ṽ2k̃22R + 2 ũṽ k̃1R k̃2R cos ((k1 − k2)d − θ )]}+ (eκd + e−κd )2{κ2k̃21L[1 + 2 ũṽ cos ((k1 − k2)d − θ )]+ 2 κ2k̃1L[ũ2k̃1R + ṽ2k̃2R + ũṽ (k̃1R + k̃2R ) cos ((k1 − k2)d − θ )]+ κ2[ũ2k̃21R + ṽ2k̃22R + 2 ũṽ k̃1Rk̃2R cos ((k1 − k2)d − θ )]}+ 2 (eκd + e−κd )(eκd − e−κd ) ũṽ(κ2 + k̃21L)κ (k̃1R − k̃2R) sin ((k1 − k2)d − θ ) (9)Numerator of |C|2 = 16 k̃21L κ2, (10)where k̃1L = (mb/mL) k1, k̃1R = (mb/mR) k1, and k̃2R =(mb/mR) k2. We put u = ũ and v = ṽ eiθ using positive realnumbers ũ and ṽ. The relation ũ2 + ṽ2 = 1 was used tosimplify the expression. Equations (8) and (9) include sev-eral terms with cos ((k1 − k2) d − θ ) or sin ((k1 − k2) d − θ ),leading to an oscillation of the transmittance as a functionof d . Physically speaking, this oscillation originates from theinterference of the majority-spin �1 and the minority-spin �2wave functions in the transmitted wave, which is seen fromthe analogy with the double-slit experiment in elementaryquantum mechanics [20]. The antiparallel transmittance TAP,↑is easily obtained by replacing ũ with ṽ and ṽ with −ũ inEqs. (8)–(10) [39]. Since ũ � ṽ, this replacement allows us toconsider the transmittance for the electron tunneling from themajority-spin �1 state in the left electrode to the superpositionstate in the right electrode with the dominant contributionfrom the minority-spin �2 state, which corresponds to TAP,↑.Using TAP,↑, the total antiparallel transmittance can be cal-culated as TAP = TAP,↑ + TAP,↓ ≈ 2 TAP,↑. We simply set ũ =0.95, θ = 0, mb/mL = 1.0, and mb/mR = 1.0 (–1.0) for thenumerical calculation of TP,↑ (TAP,↑); however, the period andoverall shape of the oscillation in the transmittance do notchange if we change ũ within the range of 1 > ũ � ṽ. Notehere that mb/mL and mb/mR need to have different signs inthe antiparallel state, which is discussed in the SupplementalMaterial [20].Figure 2 shows inverses of parallel and antiparallel trans-mittances, T −1P,↑ and T −1AP,↑, divided by the exponentiallyincreasing factor exp (2κd ). Here, we set κ = 0.2π/aMgO(aMgO = 4.217 Å: lattice constant of MgO), which is thedecaying wave number for the �1 complex band of MgOcalculated by the PWCOND code [40]. We also used k1 =1.0π/aFe and k2 = −0.9π/aFe (aFe = 2.866 Å: lattice con-stant of bcc Fe) obtained by calculating the band structureof bcc Fe [Figs. 1(b) and 1(c)] with the aid of QUANTUMESPRESSO [41]. In both T −1P,↑ and T −1AP,↑, we can see a clearoscillation with a period of 2π/(k1 − k2). From the valuesof k1 and k2 mentioned above, the period is estimated to be∼3 Å. These are consistent with the experimental fact thatresistances in both the parallel and antiparallel magnetizationstates have oscillatory barrier thickness dependencies withFIG. 2. Barrier thickness d dependencies of inverses of paralleland antiparallel transmittances, T −1P,↑ and T −1AP,↑, divided by exp (2κd ).L220406-3KEISUKE MASUDA et al. PHYSICAL REVIEW B 111, L220406 (2025)FIG. 3. TMR ratios as a function of the barrier thickness d fordifferent values of α. This was obtained by Eq. (11).periods of ∼3 Å [6,10,11]. Finally, we would like to point outthat either shape or phase in the oscillation needs to have adifference between T −1P,↑ and T −1AP,↑ for the occurrence of theTMR oscillation shown in Fig. 3 [42].By using TP,↑ and TAP,↑, we calculated the TMR ratiogiven by(TP − α TAP)/α TAP ≈ (TP,↑ − 2 α TAP,↑)/2 α TAP,↑. (11)Here, we introduced an electronic-structure parameter α, bywhich TAP is scaled relative to TP reflecting electronic struc-tures of Fe and MgO. The case with α = 1 corresponds tothe usual definition of the TMR ratio. However, as shownin Fig. 3, the TMR ratio takes negative values for α = 1,inconsistent with positive high TMR ratios observed in exper-iments. This is because the present analysis employs only thevalues of the Fermi momenta and the decaying wave numberand does not consider detailed electronic structures of Fe andMgO. If these electronic structures are taken into account byusing the first-principles calculation [12,13], values of TAP arearound one order of magnitude smaller than those of TP. Thus,we set α = 0.1 for a better comparison with experimentalvalues of the TMR ratio obtained at room temperature inFig. 4(b) [43]. Figure 3 shows barrier thickness d dependen-cies of the TMR ratio for different values of α. For all valuesof α, the TMR ratio shows an oscillation with a period of2π/(k1 − k2) ∼ 3 Å, similarly to T −1P,↑ and T −1AP,↑.Let us directly compare our calculation results with anexperimentally observed TMR oscillation. To this aim, wefabricated an MTJ structure and measured magnetotransportproperties. The experimental method is explained in the Sup-plemental Material [20]. Figure 4(a) shows barrier thicknessd dependencies of the resistance-area product (RA) in theparallel and antiparallel magnetization states, where RA is aproduct of the resistance and the cross-sectional area of theMTJ. We show values of the RA divided by exp (a · d + b),where a and b were determined to be 5.48 (5.73) nm−1 and−2.36 (−1.35) in the parallel (antiparallel) magnetizationstate, respectively, from the fits using the exponential function.As seen in Fig. 4(a), the RA has an oscillatory d dependencewith a period of 3.1 Å in both the parallel and antiparallelmagnetization states [44]. We also find that both the shapeand phase of the RA oscillation are slightly different betweenFIG. 4. Experimental results of (a) RA values and (b) TMR ratiosin Fe/Mg4AlOx/Fe(001) at room temperature. Panel (b) also showstheoretical values of TMR ratios for different values of β calculatedby Eq. (12) with α = 0.1 [47].the parallel and antiparallel magnetization states [45], leadingto the TMR oscillation in Fig. 4(b). For a direct comparisonbetween theoretical and experimental results, we recalculatedthe TMR ratio using[(TP + β e−2κd ) − α (TAP + β e−2κd )]/α (TAP + β e−2κd ),(12)where α is fixed to 0.1. Note here that another parameterβ is introduced for transmittances at k‖ �= 0 [46]. In thederivation of Eqs. (8)–(10), we considered only the elec-tronic states at k‖ = 0 because of its dominant contributionto the TMR effect. However, in actual experiments, elec-tronic states at k‖ �= 0 can also contribute to the transmission,which is expressed by the β-related terms in Eq. (12). Sincetransmittances at different k‖ should have different periodsof oscillations, these oscillations are mixed and cancel eachother. Thus, we treated the β-related terms as simple exponen-tial functions. In Fig. 4(b), calculated TMR ratios for differentvalues of β are compared with experimental results. We findthat a saw-tooth-like shape of the TMR oscillation is quitesimilar between theoretical and experimental results [48]. Inaddition, the TMR ratios calculated for β = 15 are found toquantitatively agree with experimental values. Therefore, weconclude that the TMR ratio calculated by Eq. (12) can repro-duce the experimental results not only qualitatively but alsoL220406-4THEORY FOR TUNNEL MAGNETORESISTANCE … PHYSICAL REVIEW B 111, L220406 (2025)quantitatively. To further validate our theory, we have con-ducted additional experiments and discussed the relation withprevious experiments using Heusler alloys, which are ex-plained in the Supplemental Material [20].In summary, we proposed a theory for explaining the uni-versal oscillation of the TMR ratio called the TMR oscillation.Based on the fact that spin-flip scattering occurs near inter-faces of MTJs, we took into account the superposition ofthe majority-spin �1 and minority-spin �2 wave functionswith different Fermi momenta for the tunneling problem inFe/MgO/Fe(001). We analytically calculated transmittancesin the parallel and antiparallel magnetization states, fromwhich the TMR ratio was obtained. It was found that the trans-mittances and the TMR ratio have oscillatory barrier thicknessdependencies with a period of ∼3 Å, consistent with the ex-perimental observations. According to our theory, the periodof the TMR oscillation is determined by the difference ofthe Fermi momenta between the majority- and minority-spinstates in the ferromagnetic electrode. Therefore, the periodof ∼3 Å is specific to bcc Fe used as electrodes. If MTJswith other ferromagnetic electrodes are successfully made,TMR oscillations with periods different from 3 Å would beobserved. We expect future experimental studies using a widerrange of materials will provide further information for theTMR oscillation.The authors are grateful to S. Yuasa and H. Imamurafor fruitful discussions. The authors also thank S. Kasaifor magnetotransport measurements and H. Ikeda for hertechnical support on device microfabrication. This workwas supported by Grants-in-Aid for Scientific Research(Grants No. 22H04966, No. 23K03933, and No. 24H00408)and MEXT Program: Data Creation and Utilization-TypeMaterial Research and Development Project (Grant No. JP-MXP1122715503). 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Abe, and M. Shirai, First-principlesstudy of tunneling magnetoresistance in Fe/MgAl2O4/Fe(001)magnetic tunnel junctions, Phys. Rev. B 86, 024426 (2012).[35] K. Masuda and Y. Miura, First-principles study on magnetictunneling junctions with semiconducting CuInSe2 and CuGaSe2barriers, Jpn. J. Appl. Phys. 56, 020306 (2017).[36] K. Masuda and Y. Miura, Bias voltage effects on tunneling mag-netoresistance in Fe/MgAl2O4/Fe(001) junctions: Comparativestudy with Fe/MgO/Fe(001) junctions, Phys. Rev. B 96, 054428(2017).[37] We also considered a similar superposition of wave functionsin the incident and reflection waves. This additional analysisclarified that these do not affect the oscillatory behavior ofthe transmittance given by the superposition in the transmittedwave. Thus, we omit these effects for simplicity.[38] We used the same continuation conditions as mentioned in theSupplemental Material.[39] This is because TAP,↑ can be calculated by using ψR(z) =C(−v∗eik1z + ueik2z ) instead of Eq. (7).[40] A. Smogunov, A. D. Corso, and E. Tosatti, Ballistic con-ductance of magnetic Co and Ni nanowires with ultrasoftpseudopotentials, Phys. Rev. B 70, 045417 (2004).[41] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C.Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I.Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi,R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M.Lazzeri, L. Martin-Samos et al., Quantum ESPRESSO: Amodular and open-source software project for quantum sim-ulations of materials, J. Phys.: Condens. Matter 21, 395502(2009).[42] In our results, differences in both the shape and phase betweenthe T −1P,↑ and T −1AP,↑ oscillations provide the TMR oscillation.Although the oscillation shapes of T −1P,↑ and T −1AP,↑ look quitesimilar, these have a slight difference owing to the differencein the effective mass and the replacement of the coefficients (uand v) in the expression of the transmittance.[43] TMR ratios at low temperature are more than twice as high asthose at room temperature and can be reproduced by using asmaller value of α.[44] The amplitude of the RA oscillation hardly changes as the bar-rier thickness d increases, consistent with the results in Fig. 2.In our theory, the amplitude of the transmittance oscillation isdetermined by the values of u and v in Eq. (7). Since u and v areassumed to be constant, the amplitude of the oscillation hardlychanges with increasing d . This assumption is reasonable, sincethe interfacial exchange interaction determining u and v is in-dependent on d .[45] Note that the phase difference in RA is smaller than thatin the inverse of the transmittance in Fig. 2; however, thisdoes not mean that our theory is inconsistent with experi-mental observations [5,6,8–11]. It is experimentally knownthat the phase difference highly depends on the quality ofthe sample and is not a universal feature for the TMRoscillation.[46] We introduced nonoscillatory terms βe−2κd decreasing expo-nentially as the barrier thickness increases.[47] For a better comparison with experimental results, we applied ashift in d by 0.8 Å for Eq. (12).[48] Experimentally, the shape of the TMR oscillation is different fordifferent samples. Actually, Mg4AlOx-based MTJs have saw-tooth-like shapes [10] similarly to our present results, whileMgO-based MTJs have sinelike shapes [9,11]. These differ-ent shapes can be reproduced by tuning the parameters in ourmodel.L220406-6https://doi.org/10.1103/PhysRev.152.683https://doi.org/10.1103/PhysRev.149.491https://doi.org/10.1016/0031-8914(47)90013-Xhttps://doi.org/10.1103/PhysRev.94.1111http://kkr.issp.u-tokyo.ac.jphttps://doi.org/10.1103/PhysRevLett.77.3865https://doi.org/10.1103/PhysRev.156.809https://doi.org/10.1143/JPSJ.64.2152https://doi.org/10.1103/PhysRevB.66.174429https://doi.org/10.1063/1.2931023https://doi.org/10.1103/PhysRevB.56.11827https://doi.org/10.1103/PhysRevB.86.024426https://doi.org/10.7567/JJAP.56.020306https://doi.org/10.1103/PhysRevB.96.054428https://doi.org/10.1103/PhysRevB.70.045417https://doi.org/10.1088/0953-8984/21/39/395502