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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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[Band-folding-driven high tunnel magnetoresistance ratios in (111)-oriented junctions with <math>  <msub>    <mi>SrTiO</mi>    <mn>3</mn>  </msub></math> barriers](https://mdr.nims.go.jp/datasets/4aa4073c-1df7-43a5-9cf0-3e18b87e1ef2)

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Band-folding-driven high tunnel magnetoresistance ratios in (111)-oriented junctions with ${\rm SrTiO}_3$ barriersPHYSICAL REVIEW B 106, 134438 (2022)Band-folding-driven high tunnel magnetoresistance ratios in (111)-orientedjunctions with SrTiO3 barriersKeisuke Masuda ,1,* Hiroyoshi Itoh ,2,3 Yoshiaki Sonobe,4 Hiroaki Sukegawa ,1 Seiji Mitani,1,5 and Yoshio Miura 1,31Research 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, Japan4Research Organization for Nano & Life Innovation, Waseda University, Shinjuku 162-0041, Japan5Graduate School of Science and Technology, University of Tsukuba, Tsukuba 305-8577, Japan(Received 14 September 2022; accepted 18 October 2022; published 31 October 2022)We theoretically study the tunnel magnetoresistance (TMR) effect in (111)-oriented magnetic tunnel junctions(MTJs) with SrTiO3 barriers, Co/SrTiO3/Co(111) and Ni/SrTiO3/Ni(111). Our analysis combining the first-principles calculation and the Landauer formula shows that the Co-based MTJ has a high TMR ratio over500%, while the Ni-based MTJ has a smaller value (290%). Since the in-plane lattice periodicity of SrTiO3is about twice that of the primitive cell of fcc Co (Ni), the original bands of Co (Ni) are folded in the kx-ky planecorresponding to the ab plane of the MTJ supercell. We find that this band folding gives a half-metallic bandstructure in the �1 state of Co (Ni) and the coherent tunneling of such a half-metallic �1 state yields a high TMRratio. We also reveal that the difference in the TMR ratio between the Co- and Ni-based MTJs can be understoodby different s-orbital weights in the �1 band at the Fermi level.DOI: 10.1103/PhysRevB.106.134438I. INTRODUCTIONThe tunnel magnetoresistance (TMR) effect is essentialnot only for applications to magnetic sensors and memoriesbut also for deepening our understanding of spin-dependentelectron transport. A series of studies on Fe/MgO/Fe(001)magnetic tunnel junctions (MTJs) [1–4] has established theso-called coherent tunneling mechanism, which explains theTMR effect by bulk band structures of bcc Fe and MgO. Inthe � line of the Brillouin zone corresponding to the [001]direction, MgO has the slowest-decaying evanescent statewith �1 symmetry within the band gap, allowing the majorcontribution of the �1 wave function to the transmission.Since bcc Fe has a half-metallic band structure in the �1 state,the majority-spin �1 wave function can mainly tunnel throughMgO, leading to a giant TMR effect. Because of the successfulobservation of high TMR ratios [3,4], this mechanism hasbeen widely accepted and bcc(001)-oriented MTJs with MgObarriers have been mainly studied from both experimental andtheoretical points of view.In contrast, our recent studies [5,6] have focused on uncon-ventional fcc(111)-oriented MTJs with the stacking directionparallel to [111] directions of both the fcc ferromagnetic elec-trode and the insulator barrier. These MTJs are advantageousfor obtaining large perpendicular magnetic anisotropy (PMA),which is another requirement in addition to high TMR ra-tios for the application to magnetic random access memories.There are many fcc ferromagnetic materials with large mag-netic anisotropy along their [111] directions. Moreover, the(111) plane of the fcc structure is the closed-packed plane and*MASUDA.Keisuke@nims.go.jphas the lowest surface energy, indicating that (111)-orientedMTJs are compatible with fcc ferromagnetic electrodes. Thuswe have investigated the potential of such MTJs in the TMReffect on the basis of the first-principles calculation. Wehave shown that several (111)-oriented MTJs with Co-basedferromagnetic electrodes and MgO barriers have high TMRratios [5,6], which originate from the interfacial resonant tun-neling, in contrast to the conventional coherent tunneling ofbulk electronic states in ferromagnetic electrodes.Although such a mechanism of high TMR ratios is phys-ically significant, the interfacial resonant tunneling mightbe sensitive to atomic configurations at interfaces of MTJs.Moreover, the application of bias voltages tends to sup-press the interfacial resonant tunneling, since the energylevel of the interfacial state is shifted oppositely in the twointerfaces. These motivate us to find other (111)-orientedMTJs with robustly high TMR ratios. In this work, weconsider (111)-oriented MTJs with SrTiO3 tunnel barriers.Historically, SrTiO3 has been recognized as an impor-tant material for tunnel barriers. Comparative experimentalstudies on Co/X/La0.7Sr0.3MnO3 (X = SrTiO3, Al2O3, andCe0.69La0.31O1.845) clarified that the spin polarization of ef-fective tunneling electrons but not that of ferromagnetsplays a crucial role in the TMR effect [7,8]. Moreover, ahigh TMR ratio was predicted theoretically in bcc(001)-oriented Co/SrTiO3/Co(001) [9]. Although experiments onsuch (001)-oriented MTJs have not succeeded in achievinghigh TMR ratios, unconventional (111)-oriented MTJs withSrTiO3 barriers may open a pathway for high TMR ratios. Wethus focus on fcc(111)-oriented MTJs, Co/SrTiO3/Co(111)and Ni/SrTiO3/Ni(111) (Fig. 1).Our first-principles-based transport calculation demon-strates that the Co- and Ni-based MTJs show relatively high2469-9950/2022/106(13)/134438(7) 134438-1 ©2022 American Physical Societyhttps://orcid.org/0000-0002-6884-6390https://orcid.org/0000-0001-6577-8313https://orcid.org/0000-0002-4034-7848https://orcid.org/0000-0002-5605-5452http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.106.134438&domain=pdf&date_stamp=2022-10-31https://doi.org/10.1103/PhysRevB.106.134438KEISUKE MASUDA et al. PHYSICAL REVIEW B 106, 134438 (2022)Sr TiCo or NiOabcbca(a))c()b(abcFIG. 1. Supercell of X/SrTiO3/X (111) (X = Co or Ni).(a) Three-dimensional view. (b) Side view from the a-axis and(c) top view from the c-axis directions.TMR ratios of 534 and 290%, respectively. We also reveal thatthe high TMR ratios can be explained by the coherent tun-neling of electronic states of bulk ferromagnets, meaning thatthe obtained TMR ratios are more robust against interfacialimperfections and bias voltage than those driven by the inter-facial resonant tunneling. The simple fcc Co and Ni given bytheir primitive unit cells have no �1 bands crossing the Fermilevel in the high-symmetry line � corresponding to the [111]direction. We show however that these ferromagnets have ahalf-metallic band structure in the �1 state when attached toSrTiO3, because the in-plane periodicity of SrTiO3 is abouttwice that of fcc Co (or Ni) and the original band structureof Co (or Ni) is folded in the kx-ky plane. This is a kindof “band-folding effect” found and studied in (001)-orientedMTJs with spinel-oxide tunnel barriers [10–14]. Finally, weaddress the difference in the TMR ratio between the Co-and Ni-based MTJs and clarify that this comes from differents-orbital weights in the �1 band at the Fermi level.II. MODEL AND METHODWe first considered supercells of Co/SrTiO3/Co(111) andNi/SrTiO3/Ni(111) (Fig. 1), in which fcc Co (or Ni) andSrTiO3 are stacked along their [111] directions. These su-percells have the hexagonal close-packed structure givenby the primitive translation vectors, a1 = a(1, 0, 0), a2 =a(−1/2,√3/2, 0), and a3 = (0, 0, c), where a = √2 afccwith afcc being the lattice constant of fcc Co (or Ni) and c isthe length of the supercell. We used afcc = 3.52 Å for both thesupercells and fitted SrTiO3 to fcc Co (or Ni) in the ab plane.The length c was determined by the structure optimizationmentioned below. The supercell includes 13 monolayers (ML)of SrTiO3 and 7 ML of Co (or Ni). The thickness of SrTiO3layers is approximately 1.9 nm [15], which is a typical barrierthickness (1–2 nm) used in MTJs. Note here that there arefour possible candidates of the interfacial atomic configu-ration of the supercell as shown in Fig. 2. After preparingsupercells for all the cases, atomic positions in each super-cell were relaxed along the c direction and the formationenergy of each supercell was calculated. To determine the en-ergetically favored supercell, we compared formation energiesof Co/SrTiO3/Co(111) supercells with different interfacialatomic configurations [Figs. 2(a)–2(d)]. The formation energy(a) (b)(c) (d)c cc cab baababSrTiCoOOCoSrCoCoTiFIG. 2. Top view of different interfacial atomic configurations inCo/SrTiO3/Co(111). (a),(b) SrO-terminated interfaces with (a) Srand O on top of Co and (b) Sr and O on hollow sites. (c),(d) Ti-terminated interfaces with (c) Ti on top of Co and (d) Ti on hollowsites.for each supercell is expressed asEform = Etot −∑iNiμi, (1)where Etot is the total energy of the optimized supercell witheach interfacial atomic configuration, Ni is the number ofatoms of the element i and μi is its chemical potential. In thepresent work, we used μCo, μSr, μTi, and μO derived fromenergies of hcp Co, fcc Sr, hcp Ti, and O2 molecules. Table Ishows obtained formation energies. We find that the supercellwith the SrO-terminated interface with Sr and O on top of Co[Fig. 2(a)] has the lowest value of Eform/V . Therefore, thissupercell (Fig. 1) was selected for the calculation of the TMRratio. All the structural optimizations were performed usingthe first-principles calculation based on the density-functionaltheory (DFT) implemented in the Vienna ab initio simulationprogram (VASP) [16]. We adopted the generalized gradient ap-proximation (GGA) [17] for the exchange-correlation energyand used the projected augmented wave (PAW) pseudopoten-tial [18,19] to treat the effect of core electrons properly. Acutoff energy of 500 eV was employed and the Brillouin-zoneintegration was performed with 13 × 13 × 1 k points. Moredetails of the structural optimization are mentioned in ourprevious work [12].We calculated the TMR ratios on the basis of the bal-listic transport theory. The Landauer formula was used inTABLE I. Formation energy divided by the cell volume Eform/Vin each Co/SrTiO3/Co(111) supercell.Interfacial atomic configuration Eform/V (eV/Å3)Co-SrO (on top) −1.899 × 10−1Co-SrO (hollow) −1.878 × 10−1Co-Ti (hollow) −1.513 × 10−1Co-Ti (on top) −1.506 × 10−1134438-2BAND-FOLDING-DRIVEN HIGH TUNNEL … PHYSICAL REVIEW B 106, 134438 (2022)TABLE II. Conductances per unit areas and TMR ratios cal-culated using supercells with 13 ML of SrTiO3. The units are in�−1μm−2 and %, respectively. Here, A = 2.15 × 10−7 μm2 is thein-plane area of both the supercells.Co/SrTiO3/Co(111) Ni/SrTiO3/Ni(111)GP,↑/A 2.78 × 10−3 2.15 × 10−3GP,↓/A 1.51 × 10−1 3.56 × 10−2GAP,↑/A 1.20 × 10−2 4.85 × 10−3GAP,↓/A 1.22 × 10−2 4.84 × 10−3GP/A 1.53 × 10−1 3.78 × 10−2GAP/A 2.42 × 10−2 9.68 × 10−3TMR ratio 534 290ballistic transport calculations with the first-principles DFTmethod, which is implemented in the PWCOND code [20]in the QUANTUM ESPRESSO package [21]. We first con-structed the quantum open system by attaching the left andright semi-infinite electrodes of fcc Co (Ni) to the super-cell Co/SrTiO3/Co (Ni/SrTiO3/Ni). Then, the self-consistentpotential of the quantum open system was obtained by thefirst-principles calculation, where the GGA and the ultrasoftpseudopotentials [22] were used. The cutoff energies for thewave functions and the charge density were fixed to 58 and580 Ry, respectively, and 13 × 13 × 1 k points were usedfor the Brillouin-zone integration. Since the quantum opensystem has the translational symmetry in the ab plane, thescattering state can be classified by an in-plane wave vec-tor k‖ = (kx, ky), where the x axis was set to be parallel tothe a axis. Here, (x, y) and (kx, ky) are given in the Carte-sian coordinates. For each k‖ and spin index, we solvedthe scattering equation derived under the condition that thewave function and its derivative of the supercell are con-nected to those of the electrodes [20,23]. From the obtainedtransmittance, we calculated the conductance using the Lan-dauer formula. These calculations for both parallel (P) andantiparallel (AP) magnetization states of electrodes providethe following wave-vector-resolved conductances: GP,↑(k‖),GP,↓(k‖), GAP,↑(k‖), and GAP,↓(k‖), where ↑ (↓) indicatesthe up-spin (down-spin) channel. In this work, the up-spin(down-spin) channel is defined as the majority-spin (minority-spin) channel of the left electrode in both the parallel andantiparallel magnetization states. We calculated the averagedconductances as, e.g., GP,↑ = ∑k‖ GP,↑(k‖)/N , where N isthe sampling number of k‖ points. After confirming goodconvergence of the conductances and TMR ratio, N wasset to 150 × 150 = 2250. Using the averaged conductances,we calculated the TMR ratio given by the optimistic defi-nition, i.e., TMR ratio (%) = 100 × (GP − GAP)/GAP, whereGP(AP) = GP(AP),↑ + GP(AP),↓.III. RESULTS AND DISCUSSIONTable II shows the obtained conductances and TMR ra-tios in Co/SrTiO3/Co(111) and Ni/SrTiO3/Ni(111) MTJs.The Co-based MTJ exhibits a relatively high TMR ratio over500%, which is higher than that of the Ni-based MTJ (290%).Note that the down-spin conductance GP,↓ is much largerthan the up-spin conductance GP,↑ in both MTJs. This is asignificant feature for the present TMR effect and its origin isdiscussed below.In Fig. 3, we show the k‖-dependent conductances ofthe present MTJs, which provides key information to under-stand the mechanism of the TMR effect. Let us first focuson the Co/SrTiO3/Co(111) MTJ with a higher TMR ratio[Figs. 3(a)–3(c)]. In the down-spin conductance GP,↓(k‖)in Fig. 3(b), one can see a smooth peak centered atk‖ = (0, 0) = �. This reminds us of the similar peak inFe/MgO/Fe(001) [1,2], which was explained by the coherenttunneling of the �1 state at k‖ = �. In Fig. 3(a), the up-spinconductance GP,↑(k‖) has a spikelike structure distributedcircularly around the � point. Such a feature is often seen inother MTJs and is known to come from the interfacial resonanttunneling. In the antiparallel magnetization state [Fig. 3(c)],the k‖ dependence of the conductance is like a mixture ofGP,↑(k‖) [Fig. 3(a)] and GP,↓(k‖) [Fig. 3(b)] but the valueof the conductance is small in each k‖ point, because of themismatch of the conductive channels between the left andright electrodes. The k‖ dependences of conductances in theNi-based MTJ [Figs. 3(d)–3(f)] are almost similar to thoseof the Co-based MTJ. A minor difference is that the peakin the down-spin conductance [Fig. 3(e)] is not so smoothcompared to that of the Co-based MTJ. However, the sim-ilarity in the k‖-dependent conductances indicates that theTMR effects in these MTJs can be explained by the samemechanism.Let us discuss the mechanism of the TMR effect onthe basis of the electronic structures of the tunnel bar-rier and ferromagnetic electrodes. We mainly focus on theCo/SrTiO3/Co(111) MTJ, since the similar mechanism isexpected for both the MTJs. As mentioned above, the smoothpeak in the k‖-dependent conductance [Fig. 3(b)] reminds usof the well-known coherent tunneling mechanism, in whichbulk band structures of the tunnel barrier and ferromagneticelectrodes along the kz line at the � point can explain a highTMR ratio. In the conventional (001)-oriented MTJs, such ahigh symmetry line in the Brillouin zone is called the � line.On the other hand, in the present (111)-oriented MTJs, the �line corresponding to the [111] direction plays the key role forthe coherent tunneling.Figure 4(a) shows real and complex band structures ofSrTiO3 along the � line. Here, the Fermi level EF is set tothat of SrTiO3 attached to Co, which was estimated by theCo/SrTiO3/Co(111) supercell. Around E = EF, the real bandhas an insulating gap of ∼1.04 eV, which is smaller than thetypical theoretical value in SrTiO3 (∼1.9 eV) estimated bysimilar first-principles calculations [24,25]. This is becausethe in-plane lattice constant of SrTiO3 is shrunk so as to fitthat of fcc Co and the tensile strain (∼23%) is applied alongthe [111] direction. Because of such a small band gap inSrTiO3, the present MTJs have small values of resistance-areaproduct (RA), which are beneficial for realizing read sensorsof high-density hard disk drives and Gbit-class magnetic ran-dom access memories. By calculating the inverse of GP/A inTable II, we obtained RA of 6.52 and 26.46 �μm2 in theCo- and Ni-based MTJs, respectively. These values are muchsmaller than that in the typical Fe/MgO/Fe(001) MTJ witha similar barrier thickness (∼103 �μm2) [see Fig. 4(b) ofRef. [26]].134438-3KEISUKE MASUDA et al. PHYSICAL REVIEW B 106, 134438 (2022)GP,↑(k||) [e2/h] GP,↓(k||) [e2/h] GAP,↑(k||) [e2/h](a) )c()b(k a/2πk a/2πk a/2 kπ a/2πk a/2π k a/2π(d)k a/2πGP,↑(k||) [e2/h](e)k a/2πGP,↓(k||) [e2/h]k a/2πk a/2πGAP,↑(k||) [e2/h](f)k a/2πk a/2πCo/SrTiO3 (13 ML)/Co(111)Ni/SrTiO3 (13 ML)/Ni(111)FIG. 3. k‖-dependent conductances in Co/SrTiO3 (13 ML)/Co(111) [(a)–(c)] and Ni/SrTiO3 (13 ML)/Ni(111) [(d)–(f)], where k‖ =(kx, ky ) is given in the Cartesian coordinates. (a),(d) Up-spin conductances GP,↑(k‖) and (b),(e) down-spin conductances GP,↓(k‖) in theparallel magnetization configurations. (c),(f) Up-spin conductances GAP,↑(k‖) in the antiparallel magnetization configurations.-3-2-1 0 1 2 3 0π/cE-E F [eV]π/cRe(kz)Im(kz)Λ3-3-2-1 0 1 2 3E-E F [eV]Λ (s, p d )Λ (s, p d ) 0 π/c 0 π/ckz kz(b) (c)-3-2-1 0 1 2 3E-E F [eV](d) (e) 0 π/c 0 π/ckz kzSrTiO3fcc Co (4 atoms/plane) fcc Co (1 atom/plane)Up Down Up Downabcacbacb(a)Λ1Λ3Λ3Λ (d )Λ (d d )Λ (p p )Λ (p )Λ (p p )Λ (p p )Λ (d d )FIG. 4. (a) Real and complex band structures along the � line of SrTiO3. (b) Up-spin and (c) down-spin band structures along the � lineof fcc Co calculated for the unit cell with four atoms in each plane. In (a)–(c), the irreducible representation and atomic orbitals contributingdominantly to each band are indicated, where d3z2−r2 and dx2−y2 are abbreviated as dz2 and dx2 , respectively. (d),(e) The same as (b),(c) but forthe unit cell with one atom in each plane. The unit cells used in the calculations are also shown.134438-4BAND-FOLDING-DRIVEN HIGH TUNNEL … PHYSICAL REVIEW B 106, 134438 (2022)In Fig. 4(a), three complex bands cross E = EF, where oneof them has �1 symmetry (red curve) and the others have�3 symmetry (black curves). The s, pz, and d3z2−r2 orbitalsrotationally symmetric along the [111] direction belong tothe �1 state and the other p and d orbitals belong to the �3state. Note here that Im(kz ) provides a decay rate of the wavefunction in the barrier layer. Since all three complex bandshave similar values of Im(kz ) at E = EF, both �1 and �3 wavefunctions of the electrode are expected to decay with a similarlength scale in the SrTiO3 barrier.We next calculated the band structure of fcc Co along the� line as shown in Figs. 4(b) and 4(c), using the unit cell ex-tracted from the Co/SrTiO3/Co(111) supercell. One can findthe half-metallic nature in the �1 state (red curves); namely,the �1 band in the down-spin state crosses EF, while that inthe up-spin state does not cross EF. The relatively high TMRratio in this system is attributed to the coherent tunneling ofthe half-metallic �1 state. However, the TMR ratio (534%)is lower than that of the conventional Fe/MgO/Fe(001) MTJ(>1000%) with a half-metallic �1 state in Fe [1,2]. As shownin Figs. 4(b) and 4(c), the �3 band crosses EF in both up-and down-spin states. Since the �3 state has a similar decayrate as the �1 state [Fig. 4(a)], these up- and down-spin �3bands enhance the conductance GAP and thus decrease theTMR ratio. We also analyzed the band structure of fcc Ni inNi/SrTiO3/Ni(111) and found a similar half-metallic naturein the �1 state (not shown). Thus the TMR effect in theNi-based MTJ can also be understood by the �1 coherenttunneling, even though the TMR ratio is not so high comparedto that of the Co-based MTJ. The origin of such a differencewill be discussed later.Note here that the half-metallic band structure in the �1state [Figs. 4(b) and 4(c)] can be interpreted as a result ofthe band folding in the kx-ky plane. To discuss this, let usconsider the primitive unit cell of (111)-oriented fcc Co shownon top of Figs. 4(d) and 4(e), which has one Co atom ineach ab-plane layer and fits a simpler tunnel barrier witha smaller in-plane area like MgO. Using this unit cell, wecalculated up- and down-spin band structures along the �line shown in Figs. 4(d) and 4(e). It is seen that no bandcrosses EF in both spin states, i.e., there is no �1 state atEF. In the case of the larger unit cell that fits SrTiO3 shownon top of Figs. 4(b) and 4(c), the a- and b-axis lengths aretwice as long as those of the primitive cell and each ab-planelayer has four Co atoms. Therefore, the band structures ofthis extended cell are identical to those obtained by foldingthe band structures of the primitive cell in the kx-ky plane.Actually, by comparing Figs. 4(b), 4(c), 4(d), and 4(e), wesee that the band folding provides additional bands cross-ing EF, leading to the half metallicity in the �1 state. Weemphasize that this is in sharp contrast to the band-foldingeffect in Fe/MgAl2O4/Fe(001) [10–14]; the band foldinggives an additional minority-spin band in the �1 state ofFe, which breaks the �1 half metallicity and lowers a TMRratio. In our previous study [5], we studied the TMR effect inCo/MgO/Co(111). In this case of MgO(111), as mentionedabove, the bulk band structure of Co has no �1 state at EF andcannot contribute to a high TMR ratio. Instead, in this system,we showed that the characteristic interfacial state gives a highTMR ratio through the resonant tunneling [27]. However, suchTABLE III. Conductances per unit areas and TMR ratios calcu-lated for Co/SrTiO3(n ML)/Co(111) (n = 7, 13). The units are in�−1μm−2 and %, respectively.SrTiO3 thickness 7 ML (11 Å) 13 ML (19 Å)GP/A 7.46 1.53 × 10−1GAP/A 2.20 2.42 × 10−2TMR ratio 240 534a high TMR ratio might be fragile against interfacial defectsor impurities as mentioned in Sec. I. In contrast, the presentlyobtained high TMR ratio is owing to the �1 half metallicityin the bulk electronic state and is expected to be more robustagainst interfacial imperfections than the interface-driven highTMR ratio.The above mentioned coherent tunneling scenario of the�1 state is also supported by the SrTiO3 thickness dependenceof conductances and the TMR ratio. We additionally calcu-lated these quantities in the Co/SrTiO3(7 ML)/Co(111) MTJand compared them with those in the original Co/SrTiO3(13ML)/Co(111) MTJ as shown in Table III. We see that theTMR ratio increases with increasing the SrTiO3 thickness.This is because the selective transport of the �1 state becomesmore prominent as the barrier thickness increases. A similarbehavior is also seen in Fe/MgO/Fe(001) [1,2], where theTMR effect originates from the selective transport of the �1state. The decay of the parallel conductance GP/A can beroughly estimated from the complex band shown in Fig. 4(a).When we use κ = Im(kz ) = 0.6 π/c as the complex wavevector for the �1 state, the decay factor of the conductanceis calculated as exp(−2κd ) ≈ 1.86 × 10−2, where we usedd = 8 Å as the increment in the SrTiO3 thickness (7 → 13ML) and c = 7.57 Å as the c-axis length of the SrTiO3 cell.Using this factor and GP/A for 7 ML SrTiO3, GP/A for 13 MLSrTiO3 is approximately estimated as GP/A (7 ML SrTiO3) ×exp(−2κd ) ≈ 1.39 × 10−1 �−1 μm−2, which is close to1.53 × 10−6 �−1 μm−2 (Table III) obtained in the actualtransport calculation. All these results on the SrTiO3 thick-ness dependence indicate that the coherent tunneling of the�1 state driven by the band folding provides the dominantcontribution to the TMR effect in the present systems.We finally address the difference in the TMR ratio betweenthe Co- and Ni-based MTJs. As mentioned above, since bothCo and Ni in the present MTJs have the �1 half metallic-ity, a more detailed comparison of the electronic structureis required. Here, we focus on the weight of the s-orbitalcomponent in the �1 band, since the s-orbital state contributesdominantly to transport properties including the TMR effectowing to its small effective mass. Such a significance of thes-orbital state on the TMR effect has been reported in manyprevious studies [14,28–30]. Figures 5(a) and 5(b) show thedown-spin band structures of Co and Ni, respectively, wherethe relative weight of the s-orbital component is indicatedas the linewidth of each band using color. We find that themain �1 band crossing EF in Co has more s-orbital weightaround EF than in Ni. This larger s-orbital component at EFcan provide a larger conductance; in fact, as shown in Table II,the down-spin conductance GP,↓ in the Co-based MTJ is more134438-5KEISUKE MASUDA et al. PHYSICAL REVIEW B 106, 134438 (2022)Down Downfcc Co fcc NiΛ1 Λ1(a) (b)E-E F [eV] 0 π/ckz  0 π/ckz-2-1 0 1 2FIG. 5. Down-spin band structure along the � line of (a) fcc Coand (b) fcc Ni with s-orbital projection. The relative weight of thes-orbital component is indicated as the linewidth of each band usingcolor.than four times larger than that in the Ni-based MTJ, whilethe up-spin conductance GP,↑ is almost similar for both MTJs.These results indicate that a higher TMR ratio in the Co-basedMTJ is attributed to a larger s-orbital component at EF in the�1 band of Co.IV. SUMMARYWe investigated the TMR effect in unconventional (111)-oriented MTJs with SrTiO3 tunnel barriers by means of thefirst-principles calculation and the Landauer formula. Weobtained relatively high TMR ratios of 534 and 290% inCo/SrTiO3/Co(111) and Ni/SrTiO3/Ni(111), respectively.The analysis of the bulk band structure in the electrode and thebarrier regions of the MTJ clarified that the TMR effect in thepresent MTJs can be explained by the coherent tunneling ofelectronic states of bulk ferromagnets; actually, we found thatfcc Co and Ni in the MTJs have half-metallic band structuresin the �1 state and these half-metallic states transmit throughSrTiO3 with an evanescent �1 state, leading to relatively highTMR ratios. A usual primitive cell of fcc Co (Ni) has no�1 state at EF. However, since the in-plane lattice constantof SrTiO3 is about twice as long as that of fcc Co (Ni), the2 × 2 in-plane cell of fcc Co (Ni) fit the unit cell of SrTiO3.This yields a band folding in fcc Co (Ni) in the kx-ky planeand the folded bands give a half metallicity in the �1 state.Therefore, we can conclude that the band folding is the keyfor the �1 half metallicity and resultant high TMR ratios. Wealso discussed the difference in the TMR ratio between theCo- and Ni-based MTJs and found that this is attributed tothe different weights of the s-orbital component in the �1band at the Fermi level. Unfortunately, the TMR ratios ofthe present MTJs are not so high compared to that of theconventional Fe/MgO/Fe(001). This is because SrTiO3 hasa slow-decaying evanescent state with �3 symmetry, as wellas that with �1 symmetry. Since fcc Co (Ni) has both up-and down-spin �3 bands at EF, these bands degrade a TMRratio by increasing the conductance in the antiparallel magne-tization state. 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