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[Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Terumasa Tadano](https://orcid.org/0000-0002-8132-2161), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[Crucial role of interfacial <math>  <mrow>    <mi>s</mi>    <mtext>−</mtext>    <mi>d</mi>  </mrow></math> exchange interaction in the temperature dependence of tunnel magnetoresistance](https://mdr.nims.go.jp/datasets/debaf10e-4486-462c-b04a-7799130a0da3)

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Crucial role of interfacial $s\text{-}d$ exchange interaction in the temperature dependence of tunnel magnetoresistancePHYSICAL REVIEW B 104, L180403 (2021)LetterCrucial role of interfacial s-d exchange interaction in the temperaturedependence of tunnel magnetoresistanceKeisuke Masuda ,1,* Terumasa Tadano ,1 and Yoshio Miura 1,21Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan2Center for Spintronics Research Network, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan(Received 2 August 2021; revised 21 October 2021; accepted 25 October 2021; published 3 November 2021)The tunnel magnetoresistance (TMR) is one of the most important spintronic phenomena but its reduction atfinite temperature is a severe drawback for applications. Here, we reveal a crucial determinant of the drawback,that is, the s-d exchange interaction between conduction s and localized d electrons at interfacial ferromagneticlayers. By calculating the temperature dependence of the TMR ratio in Fe/MgO/Fe(001), we show that theobtained TMR ratio significantly decreases with increasing temperature owing to the spin-flip scattering in the�1 state induced by the s-d exchange interaction. The material dependence of the coupling constant Jsd is alsodiscussed on the basis of a nonempirical method.DOI: 10.1103/PhysRevB.104.L180403Understanding the physics of spin transport at finite tem-perature is of great importance not only from fundamentalbut also from application points of view. A particularlychallenging issue is the temperature decay of the tunnel mag-netoresistance (TMR) in magnetic tunnel junctions (MTJs)[Fig. 1(a)], which are used for various magnetic sensors andnonvolatile magnetic random access memories. Although agiant TMR ratio has been demonstrated at low temperature invarious MTJs [1–10], its significant reduction with increasingtemperature has also been observed [3–11]. This is a criticalproblem to be solved, since MTJs are usually used at roomtemperature.A clue to explain this phenomenon is conduction sp-electron states in ferromagnets; several experiments [7,10]have shown that sp-electron states with a smaller effectivemass than d-electron states provide dominant contributions totransport properties of MTJs. However, most previous theo-ries [12–19] have focused only on d-electron states and thed-d exchange interaction between d electrons on neighboringsites. This is because d-electron states have a large densityof states around the Fermi level and play the main role forstatic magnetic properties in bulk ferromagnets at finite tem-perature. For example, the Curie temperatures of 3d transitionmetals have been estimated by the Heisenberg model withthe d-d exchange interaction [12–15]. Moreover, the tem-perature dependencies of spin polarizations in Heusler alloyshave been understood by spin fluctuations in d-electron states[16–19]. In contrast, since transport properties in MTJs canbe dominated by sp-electron states as mentioned above, weneed to clarify how these states correlate with the temperaturedependence of the TMR ratio.In this Letter, we show that an intra-atomic s-d exchangeinteraction between conduction s and localized d electronsplays a significant role for the temperature decay of the TMR*MASUDA.Keisuke@nims.go.jpratio in Fe/MgO/Fe(001). While this interaction plays anessential role for the well-known Kondo effect [20], its impor-tance in the TMR effect has yet to be suggested. We calculatethe temperature dependence of the TMR ratio by employingthe tight-binding model with the s-d exchange interaction.As shown in Fig. 1(b), increasing the temperature enhancesspin fluctuations in d-electron states, as suggested in previousstudies [16–19]. We find that such spin fluctuations propagatefrom d- to s-electron states through the s-d exchange inter-action. As a result, spin-flip scattering occurs in s-electronstates, leading to a significant reduction of the TMR ratio[21]. These findings indicate that the s-d not d-d exchangeinteraction is the main origin of the TMR reduction, since theTMR ratio never drops significantly for a small s-d exchangeinteraction even if d spins fluctuate. We also find that thes-d exchange interaction at interfacial ferromagnetic layers……T = 0 T > 0Jsdd spins electronFe O Mg(a)(b)xyzJddFIG. 1. (a) An Fe/MgO/Fe(001) MTJ. Spin fluctuations in theshaded interfacial layers provide a significant reduction of the TMRratio with increasing temperature. (b) Illustrations of our idea. Whenthe temperature increases, the spins of s electrons fluctuate throughthe exchange coupling with d-electron spins, which reduces the TMRratio significantly.2469-9950/2021/104(18)/L180403(6) L180403-1 ©2021 American Physical Societyhttps://orcid.org/0000-0002-6884-6390https://orcid.org/0000-0002-8132-2161https://orcid.org/0000-0002-5605-5452http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.104.L180403&domain=pdf&date_stamp=2021-11-03https://doi.org/10.1103/PhysRevB.104.L180403MASUDA, TADANO, AND MIURA PHYSICAL REVIEW B 104, L180403 (2021)contributes dominantly to the TMR reduction. We will finallyestimate the coupling constant of the s-d exchange interac-tion using a nonempirical method. We show that the materialdependence of the TMR reduction can be explained by theestimated coupling constants. Our results would be quite im-portant for future materials design for a smaller temperaturedependence of the TMR ratio.Our calculation is based on the tight-binding model,H0 =∑i j∑μνσtμνi j c†iμσ c jνσ +∑iμσεμiσ niμσ , (1)where c†iμσ creates an electron with spin σ in orbital μ at site i,tμνi j is the hopping integral of electrons, εμiσ is the on-site poten-tial measured from the Fermi level EF, and niμσ = c†iμσ ciμσ .We constructed this type of Hamiltonian for bcc Fe and MgO[23]. In addition to the one-body terms H0, we considered thes-d exchange interaction at each Fe site,Hsd = −2Jsd∑isi · Si, (2)where si ≡ 12∑σσ ′ c†isσ τσσ ′cisσ ′ is the spin operator for s elec-trons with τσσ ′ being the Pauli matrices and Si is the operatorfor the localized spin in the d orbitals, which is assumedto have S = 2 because of six 3d valence electrons in Fe.Since the p-orbital states are far from the Fermi level anddo not affect our results, we neglected the p-d exchangeinteraction between the p and d electrons. When the tem-perature increases, the localized spin Si has fluctuations inthe longitudinal (Siz) and transverse (Six and Siy) directions,leading to spin-flip scattering in the s-electron states throughHsd . We treat this spin-flip scattering at finite temperature bymixing the up-spin and down-spin s states within the coher-ent potential approximation (CPA) [26,27]. By introducingthe orbital-diagonal coherent potentials �sσ (σ =↑,↓) in thes orbital, the Hamiltonian H = H0 + Hsd of Fe is rewrit-ten as H = K + V , with K = H0 + ∑iσ �sσ c†isσ cisσ and V =∑iσσ ′ c†isσ (−Jsd τσσ ′ · Si − �sσ δσσ ′ )cisσ ′ ≡ ∑i vi. Using viand the unperturbed Green’s function P ≡ 1/(ω − K ), thescattering operator ti is defined as ti = vi(1 − Pvi )−1, whereω is the energy relative to the Fermi level and is set to 0. Wedetermined the values of �sσ at each temperature from theCPA condition 〈ti〉 = 0. Technical details for solving 〈ti〉 = 0are presented in the Supplemental Material [28]. To solve this,we assumed a typical temperature dependence of 〈Siz〉, 〈Siz〉 =S√1 − (T/TC)2, where TC is the Curie temperature of Fe(TC = 1040 K). As the temperature increases, 〈Siz〉 decreasesfollowing this equation. Such a decrease in 〈Siz〉, i.e., theenhancement of d-spin fluctuation, propagates to s-electronstates through the s-d exchange interaction [Eq. (2)], leadingto the spin-flip s-electron scattering. The real part of �sσ givesan exchange splitting and the imaginary part gives a finitelifetime, namely, the occurrence of spin-flip scattering from|s, σ 〉 to |s, σ̄ 〉. Since the effect of the exchange splitting isalready included in the on-site potential in H0, we consideredonly the imaginary part of the coherent potential Im(�sσ ) inour transport calculations.We calculated the electronic states of Fe/MgO/Fe(001)by using the recursive Green’s function method [29,30]in combination with the above-mentioned parameters (tμνi j ,εμiσ , and �sσ ). This method allows us to calculate theGreen’s function at the (n + 1)th layer gn+1 from that atthe nth layer gn: gn+1 = (ω − ε − t†gnt)−1. Here, ε is theon-site potential matrix including εμiσ and �sσ and t is thehopping-integral matrix composed of tμνi j . In addition to thehopping integrals of bulk Fe and MgO, we also need thoseat the interface, which were approximately determined byapplying Harrison’s method [31] to the hopping integralsof bulk Fe. Starting from the left (right) surface Green’sfunction of Fe, we obtained the Green’s function at eachlayer from Fe to MgO by using the above recursive equa-tion, leading to the Green’s functions of the left (right)semi-infinite system [30]. From these we can obtain theGreen’s functions of the entire system Fe/MgO/Fe(001)[32]. By applying the Kubo-Greenwood formula [32,33]to the obtained Green’s functions, temperature-dependentconductances were calculated. Since our system has trans-lational symmetry in the xy plane, the electronic statesare labeled by the in-plane wave vector k‖ = (kx, ky). Theconductances GP(k‖) = GP,↑(k‖) + GP,↓(k‖) and GAP(k‖) =GAP,↑(k‖) + GAP,↓(k‖) for parallel and antiparallel magneti-zation configurations were calculated for each k‖ and wereaveraged as GP = ∑k‖ GP(k‖)/N . Here, the sampling num-ber N of k‖ points was set to 100 × 100 for ensuring goodconvergence of the conductances. The TMR ratio was es-timated using the optimistic definition, TMR ratio (%) =100 × (GP − GAP)/GAP.Figure 2(a) shows the temperature dependencies of theTMR ratio for different values of Jsd . We focused on negativevalues of Jsd because they are reasonable as discussed later.The TMR ratio decreases with increasing the temperature forall the values of Jsd . When the temperature increases, theimaginary part of the coherent potential |Im(�sσ )| increasesas shown in Figs. 2(b) and 2(c), which means an enhancementof spin-flip scattering and leads to the reduction of the TMRratio. A larger |Jsd | gives a faster decrease in the TMR ratiobecause of a faster increase in |Im(�sσ )| with increasing thetemperature. More detailed behaviors of |Im(�sσ )| shown inFigs. 2(b) and 2(c) can be understood as follows. Note herethat the s-d exchange interaction [Eq. (2)] can be rewrit-ten as Hsd = −2Jsd∑i [ 12 (si+Si− + si−Si+) + sizSiz], wheresi± = six ± isiy and Si± = Six ± iSiy. At T = 0, the localizedd spin has the largest Siz of Siz = 2. Thus, the term si+Si−in the s-d exchange interaction provides a decrease in Siz ofthe localized d spin and an increase in siz of conduction selectrons, namely, down-to-up spin-flip s-electron scatteringrepresented by Im(�s↓). This is the reason for the rela-tion |Im(�s↓)|   |Im(�s↑)| at low temperature (T < 100 K).When the temperature is increased over 100 K, the localizedd spin comes to have a smaller Siz, which enhances the up-to-down s-electron scattering through the term si−Si+. This alsoprovides a saturation of the down-to-up s-electron scattering,since the effect of the term si+Si− is relatively weakened.These are characterized by an increase in |Im(�s↑)| [Fig. 2(b)]and a saturation of |Im(�s↓)| [Fig. 2(c)], respectively. In thiswork, we neglected the p-d exchange interaction as mentionedabove, since the energy levels of p states in Fe are much higherthan EF (E − EF ≈ 1 eV) and the p-d exchange interactionhas little effect on our results. We confirmed this point byL180403-2CRUCIAL ROLE OF INTERFACIAL s-d EXCHANGE … PHYSICAL REVIEW B 104, L180403 (2021)Jsd = −2.0 eVJsd = −1.5 eVJsd = −1.0 eVJsd = −0.5 eVTMR ratio (%)Temperature (K)(a)(b)(c) 100 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1.0 1.5 2.0 0  50  100  150  200  250  300FIG. 2. Temperature dependencies of (a) TMR ratios,(b) |Im(�s↑)|, and (c) |Im(�s↓)| for different values of theexchange interaction Jsd .similar calculations including the p-d exchange interaction(see the Supplemental Material [28]).Let us further discuss the reduction of the TMR ratio fromthe viewpoint of electronic structures. Figure 3(a) shows theconductances and TMR ratio as a function of temperature atJsd = −1.5 eV. As is shown later, this value of Jsd is closeto the one estimated by a nonempirical method. When thetemperature increases, the antiparallel conductance GAP =GAP,↑ + GAP,↓ largely increases (GAP at 300 K is almosttwice as large as that at 10 K) while the parallel conduc-tance GP = GP,↑ + GP,↓ hardly changes [35], leading to thesignificant reduction of the TMR ratio. Such a dominance ofGAP in the temperature dependence of the TMR ratio is con-sistent with experimental results in various MTJs [5,11,36].To deeply understand this behavior, we next focus on thek‖-resolved conductances shown in Figs. 3(d)–(l). At lowtemperature (T = 10 K), the well-known features of the �1coherent tunneling [37,38] are seen; the up-spin conductanceGP,↑(k‖) in the parallel magnetization state [Fig. 3(d)] has abroad peak centered at k‖ = (0, 0) =   while the down-spinone GP,↓(k‖) [Fig. 3(g)], which has only a small value at the  point, instead has relatively large values in a ring-shapedregion surrounding the   point. Such a clear difference inthe conductance can be naturally explained by the half metal-licity in the �1 state of Fe [Figs. 3(b) and 3(c)], as shownby pioneering theoretical studies [37,38]. In the antiparallelmagnetization state [Fig. 3(j)], the conductance has only avery small value at the   point owing to the absence ofthe �1 down-spin band crossing E = EF [Fig. 3(c)]. Whenwe increase the temperature to 50 K, the effect of the spinmixing clearly appears: The down-spin conductance GP,↓(k‖)[Fig. 3(h)] has large values around the   point and the valuejust at the   point is not so small, which comes from thefeature of the up-spin conductance GP,↑(k‖). On the otherhand, the values of GP,↑(k‖) hardly change [Fig. 3(e)], sincethe spin-mixing effect from GP,↓(k‖) is quite small owingto the relation GP,↓(k‖) � GP,↑(k‖). In the antiparallel state[Fig. 3(k)], the conductance at the   point largely increasescompared to Fig. 3(j) due to the spin-mixing effect in the �1state, which is the reason why GAP,↑ (= GAP,↓) significantlyincreases as shown in Fig. 3(a). In other words, �1 electronsscattered from the up-spin to down-spin state contribute domi-nantly to the enhancement of GAP and thereby the reduction ofthe TMR ratio. When we increase the temperature to 100 K,GP,↓(k‖) and GAP,↑(k‖) [Figs. 3(i) and 3(l)] increase furtherwhile GP,↑(k‖) [Fig. 3(f)] hardly changes, leading to a furtherreduction of the TMR ratio.To see the effect of spin mixing at different regions offerromagnetic layers, we calculated the TMR ratio for twoadditional cases: (i) The s-d exchange interaction was con-sidered only in the interfacial Fe layers (H interfacesd in Fig. 4);and (ii) the s-d exchange interaction was considered only inthe bulk regions of Fe except the interfacial Fe layers (Hbulksdin Fig. 4). Figure 4 shows the temperature dependencies ofthe TMR ratio for the three cases, which clarifies that thespin mixing at interfacial Fe layers provides the dominantcontribution to the sharp reduction of the TMR ratio. Thisindicates the importance of selecting ferromagnets with small|Jsd | at the interface for preventing the temperature decay ofthe TMR ratio.Finally, we estimate the coupling constant Jsd using anonempirical method. A pioneering theory by Schrieffer andWolff [39] has shown that the s-d exchange interaction[Eq. (2)] can be derived by applying a canonical transforma-tion to the Anderson Hamiltonian and the coupling constantJsd can be expressed asJsd ≈ |Vsd |2 Uεd (εd + U ), (3)where εd is the energy level of a d orbital with respect to EF,U is the Coulomb interaction in the d orbital, and Vsd is thehybridization between the d and s states. This expression canbe understood from virtual electron transitions in the second-order perturbation processes [40]. Here, we apply Eq. (3)to bcc Fe1−xCox (0�x�1), since this series of materials istypically used for MTJs with an MgO tunnel barrier and isknown to give high TMR ratios [1,2,41].The values of εd , U , and Vsd can be estimated on the basisof the maximally localized Wannier function (MLWF) methodimplemented in the RESPACK code [42]. We first conducteddensity functional theory (DFT) calculations of bcc Fe1−xCoxusing the QUANTUM ESPRESSO code [43]. We employed thePerdew–Burke–Ernzerhof exchange-correlation potential [44]L180403-3MASUDA, TADANO, AND MIURA PHYSICAL REVIEW B 104, L180403 (2021)FIG. 3. (a) Temperature dependencies of the conductances and TMR ratio for Jsd = −1.5 eV. (b), (c) Up- and down-spin bands of Fealong the � line contributing dominantly to the TMR effect. (d)–(l) The k‖-resolved conductances calculated for Jsd = −1.5 eV [34]. (d)–(f)GP,↑(k‖) at T = 10, 50, and 100 K, respectively. (g)–(i) The same as (d)–(f) but for GP,↓(k‖). (j)–(l) The same as (g)–(i) but for GAP,↑(k‖). Theunit of the color bars in (d)–(l) is e2/h.and the optimized norm-conserving Vanderbilt (ONCV) pseu-dopotentials from PseudoDojo [45]. The primitive bcc unitcell with a lattice parameter of 2.866 Å was used for all x.The alloys with 0 < x < 1 were treated by the virtual crystalapproximation. We used 10 × 10 × 10 k-point grids and anenergy cutoff of 108 Ry for the wave functions and 432 Ryfor the electron charge densities. We next constructed theMLWFs [46,47] using the RESPACK code [42]. By adoptingatomic s, p, and d orbitals as initial projection functions, weobtained nine MLWFs that reproduce the original DFT banddispersion around the Fermi level. Here, the inner and outerTemperature (K)TMR ratio (%) 200 300 400 500 600 700 800 0  50  100  150  200  250  300FIG. 4. Temperature dependencies of the TMR ratio calculatedwith Jsd = −1.5 eV for three different cases: The s-d exchange in-teraction was considered in the whole region of the electrodes (Hsd ),only in the interface region (H interfacesd ), and only in the bulk region(H bulksd ).energy windows were set to [1 eV, 30 eV] and [0 eV, 55 eV],respectively, for all x, where the Fermi level was located at17.86 eV for x = 0 (Fe) and 17.17 eV for x = 1 (Co). Theobtained MLWFs are not the same as the initial atomic orbitalsbut sufficiently maintain their features. We also obtained thehopping integrals and the on-site energy of each MLWF. Weused the nearest-neighbor hopping integrals between s and dorbitals as Vsd and on-site energies of d orbitals as εd . TheCoulomb interaction parameters were also calculated usingthe RESPACK code. Here, we adopted the usual random phaseapproximation for calculating the dielectric function [42]. Theenergy cutoff for the dielectric function was set to 60 Ryfor ensuring good convergence of the Coulomb interactionparameters. The polarization function was calculated using 60bands. The obtained intraorbital screened Coulomb interac-tion U in each d orbital was used to estimate Jsd given byEq. (3). The screened Hund exchange interaction between sand d electrons was also obtained for each d orbital.Figure 5 shows x dependencies of Jsd , εd , U , and |Vsd |2averaged over the d orbitals. First of all, Jsd is negative (i.e.,antiferromagnetic coupling) for all the values of x, since εdand εd + U have different signs. As x is increased from 0(Fe) to 1 (Co), the d level εd gets deeper, which is naturalsince Co has more valence electrons than Fe. This also de-creases the hybridization |Vsd | between the d state and thes state near EF. These changes in εd and |Vsd | lead to adecrease in |Jsd | with increasing x. We summarized the valuesof Jsd and �TMR in Table I, indicating that the reductionin the TMR ratio monotonously decreases with increasing x.This tendency is consistent with previous experimental resultson Fe/MgO/Fe (�TMR ∼ −500%) [11] and Co/MgO/Co(�TMR � −100%) [41]. Note that there exists another con-tribution to Jsd different from Eq. (3). It is the Hund exchangeL180403-4CRUCIAL ROLE OF INTERFACIAL s-d EXCHANGE … PHYSICAL REVIEW B 104, L180403 (2021)J sd, ε d, U, ε d+U (eV)|Vsd|2 (eV2)xεdεd+UU|Vsd|2oCeF−2−1.5−1−0.5 0 0.5 1 1.5 2 0  0.2  0.4  0.6  0.8  1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4JsdFIG. 5. The x dependencies of Jsd , εd , U , εd + U , and |Vsd |2 inbcc Fe1−xCox .interaction [49], more generally called the direct exchangeinteraction [40], between s and d electrons. Using the MLWFmethod, we estimated its contribution to Jsd and obtainedsmall values of 0.25−0.28 eV for all the values of x inFe1−xCox. Therefore, we can conclude that the s-d exchangeinteraction due to the second-order perturbation [Eq. (3)] pro-vides the dominant contribution to Jsd . The total value of Jsdincluding both the contributions is estimated to be ∼ − 1.5 eVfor x = 0 (Fe), which justifies our choice of Jsd = −1.5 eV inFigs. 3 and 4.In summary, we theoretically investigated the temperaturedependence of the TMR effect in Fe/MgO/Fe(001). We clar-ified a crucial importance of the s-d exchange interaction inthe degradation of the TMR at finite temperature: The s-dexchange interaction at the interfacial ferromagnetic layersTABLE I. The x dependencies of Jsd and the reduction of theTMR ratio at room temperature, �TMR = TMR ratio (300 K) −TMR ratio (0 K), in bcc Fe1−xCox [48].x 0 0.2 0.4 0.6 0.8 1.0Jsd (eV) –1.75 –1.52 –1.35 –1.22 –1.11 –1.03�TMR (%) –610 –560 –510 –460 –420 –380provides spin-flip scattering in the �1 states, leading to asignificant reduction of the TMR ratio. To the best of ourknowledge, most of the previous theories on the TMR effectmight have missed this fact, since they have focused onlyon the d-d exchange interaction in bulk ferromagnets. Ourfindings are also supported by the experimental fact that con-duction sp-electron states provide dominant contributions tothe transport properties in MTJs. We finally estimated the cou-pling constant Jsd of the s-d exchange interaction on the basisof a nonempirical method. By using the present approach, onecan predict the material dependence of the TMR reduction atroom temperature for a wide range of ferromagnets, whichwould be quite useful for designing MTJs with a weak tem-perature dependence of the TMR ratio.The authors are grateful to H. Itoh, S. Honda, M. Mat-sumoto, and H. Sukegawa for helpful discussions. This workwas partly supported by TDK Corporation and Grant-in-Aids for Scientific Research (Grants No. JP16H06332, No.JP17H06152, No. JP20H02190, and No. JP20K14782). 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