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[Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051), [Ryo Iguchi](https://orcid.org/0000-0002-8112-4608), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[First-principles study of the anisotropic magneto-Peltier effect](https://mdr.nims.go.jp/datasets/973dd595-b331-4a8c-ac41-601540594642)

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First-principles study of the anisotropic magneto-Peltier effectPHYSICAL REVIEW B 99, 104406 (2019)First-principles study of the anisotropic magneto-Peltier effectKeisuke Masuda,1 Ken-ichi Uchida,1,2,3,4 Ryo Iguchi,1 and Yoshio Miura1,2,51Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan2Center for Materials Research by Information Integration, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan3Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan5Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531, Japan(Received 18 June 2018; revised manuscript received 18 February 2019; published 6 March 2019)We study theoretically the anisotropic magneto-Peltier effect, which was recently demonstrated experimen-tally. A first-principles-based Boltzmann transport approach including the spin-orbit interaction shows that Nihas a larger anisotropy of the Peltier coefficient (��) than Fe, consistent with experiments. It is clarified thatspin-flip electron transitions due to the spin-orbit interaction are the key in the mechanism of the large anisotropicmagneto-Peltier effect. Using our method, we further predict several ferromagnetic metals with much larger ��than that of Ni.DOI: 10.1103/PhysRevB.99.104406I. INTRODUCTIONThe spin-orbit interaction (SOI) plays a key role inmechanisms of various spintronic phenomena, such as thespin Hall effect [1], the Rashba–Edelstein effect [2,3], mag-netic anisotropies [4], and the anisotropic magnetoresistance(AMR) [5–7]. Among them, a transport phenomenon stronglycoupled with the magnetization is the AMR in ferromagnets,where the electrical resistivity depends on the relative anglebetween the charge current and the magnetization owing tothe SOI acting on spin-polarized charge carriers.Similar to the AMR, thermoelectric coefficients also de-pend on the direction of the magnetization. The Seebeckcoefficient S in a ferromagnet is dependent on the relativeangle between the directions of the thermal gradient ∇Tand the magnetization M [see Fig. 1(a)], which is calledthe anisotropic magneto-Seebeck effect (AMSE) [8–19]. Thereciprocal effect of the AMSE called the anisotropic magneto-Peltier effect (AMPE), in which the Peltier coefficient �depends on the relative angle between the charge current Jcand M, has also been investigated recently [20–22]. Uchidaet al. [21] directly observed temperature change due to thedifference in the Peltier coefficient �� = �‖ − �⊥, where�‖ (�⊥) is the Peltier coefficient for M ‖ Jc (M ⊥ Jc) inferromagnetic metal slabs [see Fig. 1(b)]. Uchida et al. [21]found that Ni, Ni95Pt5, and Ni95Pd5 exhibit clear AMPE. Onthe other hand, in their experiments, Fe did not show clearAMPE and Ni45Fe55 showed only small signals. They alsoobserved the similar tendency in the AMSE measurements;the AMSE signal of Ni was clear but that of Fe was negligiblysmall [21]. Although both the AMPE (AMSE) and AMRare associated with the SOI, we can find a clear differencein the material dependence between these phenomena; whileFe exhibits the small but finite AMR [7], no clear AMPEsignal was obtained in Fe in contrast with the case of Niwith a large AMPE coefficient [21]. The origin of such astrong material dependence of the AMPE and AMSE shouldbe clarified; however, no theoretical study has addressed thematerial dependence of these phenomena so far.In this study, we investigate theoretically the intrinsicmechanism and the material dependence of the AMPE byanalyzing the anisotropy of the Peltier coefficient �� onthe basis of the first-principles-based Boltzmann transportapproach including the SOI. We show that �� in Ni ismuch larger than that in Fe, in agreement with experimentalobservations, and that such a difference in �� comes fromthe presence of the spin-flip electron transition around theFermi level in Ni. Using this calculation method, we revealthat |��| of several ferromagnetic metal alloys containing Pt[23] is much larger than that of Ni. Although we focus onlyon the AMPE in this study, the results can be applied to theAMSE simply by dividing �� by the temperature T on thebasis of the Onsager reciprocal relations.II. CALCULATION METHODIn the present analysis, we focus on the intrinsic mech-anism of the AMPE by applying the first-principles-basedBoltzmann transport approach to bulk ferromagnets. This isbecause the AMPE (AMSE) discussed in this study occursin bulk ferromagnets and does not require any interfaces,unlike other phenomena; e.g., the spin-dependent Seebeckand Peltier effects in magnetic nanostructures [24–26] andthe magneto-Seebeck and Peltier effects in magnetic tunneljunctions [27–29].The electronic structure of each system was calculatedby using the full-potential linearized augmented plane-wave(FLAPW) method including the SOI implemented in theWIEN2K program [30]. We employed conventional unit cellsfor bcc Fe and fcc Ni [31], where we set M along the [001]direction [see insets of Fig. 2(a)]. By applying Boltzmanntransport theory [32] to the obtained electronic structures, we2469-9950/2019/99(10)/104406(6) 104406-1 ©2019 American Physical Societyhttp://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.99.104406&domain=pdf&date_stamp=2019-03-06https://doi.org/10.1103/PhysRevB.99.104406MASUDA, UCHIDA, IGUCHI, AND MIURA PHYSICAL REVIEW B 99, 104406 (2019)VTS|| MJcMferromagnetic metal||VAnisotropic magneto-Peltier effect (AMPE)Anisotropic magneto-Seebeck effect (AMSE)T(a)(b)Sferromagnetic metalMheat releasedheat absorbedFIG. 1. Schematic illustrations of (a) the AMSE and (b) theAMPE.calculated the Peltier coefficient �α (α =⊥, ‖) given by�α = −1e∫σα (ε)(ε − μ)(− ∂ f∂ε)dε∫σα (ε)(− ∂ f∂ε)dε, (1)where μ is the chemical potential, f (ε) ={exp [(ε − μ)/kBT ] + 1}−1 is the Fermi distribution function,and σα (ε) = e2τN∑i,k vα (i, k)vα (i, k)δ(ε − εi,k ) is theenergy-dependent conductivity. Here, εi,k is the eigenenergywith wave vector k in band i, v‖(i, k) [v⊥(i, k)] is the groupvelocity along the direction parallel (perpendicular) to M, τis the relaxation time assumed to be constant, and N is thenumber of k points used in the summation. The temperatureT in the Fermi function was fixed to 300 K in our analysisto compare with experiments performed at room temperature[21]. We estimated the AMPE from the anisotropy of thePeltier coefficient �� ≡ �‖ − �⊥.We also analyzed the AMR given by the anisotropy of theelectrical resistivity. The AMR ratio is defined as �ρ/ρav =(ρ‖ − ρ⊥)/( 13ρ‖ + 23ρ⊥), where ρ‖ (ρ⊥) is the electrical resis-tivity when the electric current is parallel (perpendicular) tothe magnetization. Note here that ρα (α =⊥, ‖) is the inverseof the electrical conductivity σα and is formulated as followsin Boltzmann transport approach:ρα = 1σα= 1∫σα (ε)(− ∂ f∂ε)dε. (2)By comparing Eqs. (1) and (2), we see that the numeratorof Eq. (1) gives the difference in the material dependencebetween the AMPE and AMR, which will be discussed inmore detail in the next section.-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15-0.2 -0.15 -0.1 -0.05  0  0.05  0.1  0.15  0.2NiFe(a)MFeMNi-1.0-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8(b)NiFeAMPEAMR 0.3 0.35 0.4 0.45 0.5 0.55 0.6-0.2 -0.15 -0.1 -0.05  0  0.05  0.1  0.15  0.2NiFeDOS [states/eV/atom](c)DOSFIG. 2. Calculated μ dependencies of (a) the anisotropy of thePeltier coefficient �� = �‖ − �⊥ and (b) the AMR ratio �ρ/ρavfor Fe and Ni. (c) The total DOS of Fe and Ni.III. RESULTS AND DISCUSSIONIn Fig. 2(a), we show the calculated anisotropy of thePeltier coefficient �� as a function of the chemical potentialμ for Fe and Ni. Here, μ = 0 corresponds to the Fermi levelεF. At μ = 0, Ni has a �� of 0.114 mV, which is muchlarger than that of Fe [33]. These values of �� are consis-tent with the experimental ones [21], not only qualitativelybut also quantitatively. We also calculated the μ dependen-cies of AMR ratios for Fe and Ni, as shown in Fig. 2(b),where the AMR ratio of Fe is larger than that of Ni aroundμ= 0. Such opposite material dependencies of the AMPE andAMR clearly indicate that these phenomena follow differentphysical pictures, consistent with the findings of previoustheoretical studies [34,35].To explain the fundamental physical properties of thePeltier coefficient, one can utilize the following approximate104406-2FIRST-PRINCIPLES STUDY OF THE ANISOTROPIC … PHYSICAL REVIEW B 99, 104406 (2019)expression of Eq. (1), called the Mott formula [36]:� ≈ −π2(kBT )23e1D(εF)[dD(ε)dε]ε=εF, (3)where � is the Peltier coefficient and D(ε) is the density ofstates (DOS). This expression suggests that a smaller D(εF)and a larger [dD(ε)/dε]ε=εF are better for obtaining a larger� and its anisotropy. As shown in Fig. 2(c), the DOS of Nisatisfies such conditions.Our results for the AMR ratios can be understood fromthe relation that the electrical conductivity is approximatelyproportional to the DOS at the Fermi level D(εF), which isderived in the same manner as Eq. (3) [36]. This relationholds for the present case because we confirmed that thecalculated σ⊥ (σ‖) of Fe is larger than that of Ni, as expectedfrom the DOS shown in Fig. 2(c). Owing to such a materialdependence in the conductivity, the anisotropy σ⊥ − σ‖ is alsolarger in Fe than in Ni, leading to the AMR ratios shownin Fig. 2(b). Note here that our calculations for the AMRratios took into account only the intrinsic contribution fromthe band structures of Fe and Ni. On the other hand, in actualexperiments, s-d scattering due to impurities provides non-negligible contributions to the conductivity [37], leading to theexperimental behavior that the AMR ratio of Ni is larger thanthat of Fe [7]. However, in the case of the AMPE, not onlythe DOS itself but also its derivative give a large contributionto the Peltier coefficient. Therefore, in ferromagnets havinga large [dD(ε)/dε]ε=εF and a small D(εF), the effect of thes-d scattering is relatively weakened. This is a possible reasonwhy our calculated value of �� in Ni agrees quantitativelywith the experimental results, even though the effect of the s-dscattering is disregarded and the relaxation time is assumed tobe constant in the present study.To obtain further insight into the large �� in Ni, we focuson spectral weights A(k, ε) = ∑i δ(ε − εi,k ) [38] of Fe andNi. Figures 3(a)–3(c) show the two-dimensional wave vector(k⊥, k‖) dependencies of A(k, εF) in Fe at kxa/π = 0, 0.5,and 1, respectively. Here, k‖(⊥) represents the wave-vectorcoordinate parallel (perpendicular) to M; we set k‖ = kz andk⊥ = ky in the present case. Owing to the presence of theSOI, fourfold symmetry of A(k, ε) in the (k⊥, k‖) plane isslightly broken for all kx. This gives the anisotropy of A(k, ε)between k⊥ and k‖ directions, �A(k, εF) ≡ A(kx, ky, kz, εF) −A(kx, kz, ky, εF), as shown in Figs. 3(d)–3(f). From Figs. 3(d)and 3(f), we see that non-negligible values of �A(k, εF) occuraround the k⊥ and k‖ lines through (0,0), which can yielda finite value of ��. Figures 3(j)–3(l) show the anisotropy�A(k, εF) in Ni calculated from A(k, εF) shown in Figs. 3(g)–3(i). We find that Ni has large values of �A(k, εF) especiallyat kxa/π = 0 [Fig. 3(j)], which distribute broadly around thek⊥ and k‖ lines through (0,0). Such large and distributed�A(k, εF) can be the origin of the large �� in Ni.To clarify the reason for the difference in �A(k, εF)between Fe and Ni, we next analyzed the band structuresalong the k⊥ and k‖ lines through � = (0, 0, 0), as shown inFigs. 4(a)–4(d). Note that these band structures are calculatedfor conventional unit cells introduced in Sec. II. This is whythe band structures in Figs. 4(a)–4(d) are seemingly differentfrom the well-known ones calculated for the primitive unitFIG. 3. (a)–(c) The in-plane wave vector (k⊥, k‖) dependenciesof the spectral weight A(k, εF ) in Fe at (a) kxa/π = 0, (b) kxa/π =0.5, and (c) kxa/π = 1. (d)–(f) The in-plane wave vector (k⊥, k‖)dependencies of the anisotropy of the spectral weight �A(k, εF ) inFe at (d) kxa/π = 0, (e) kxa/π = 0.5, and (f) kxa/π = 1. (g)–(i) Thesame as (a)–(c) for Ni. (j)–(l) The same as (d)–(f) for Ni.cells [39,40]. The band structures for the conventional cellsare identical to those obtained by folding the band structuresfor the primitive cells. If the SOI is absent, we can identifyboth majority- and minority-spin bands, which have identicaldispersions in the k⊥ and k‖ lines [Figs. 4(a) and 4(c)]. Whenthe SOI is taken into account, the majority- and minority-spinbands are mixed, and the k⊥ and k‖ lines have different bandstructures [Figs. 4(b) and 4(d)].The SOI ξL · S = ξ [ 12 (L+S− + L−S+) + LzSz] has two ef-fects on band structures, where L and S are, respectively,the orbital and spin angular-momentum operators. First, theterm ξ2 (L+S− + L−S+) gives the spin-flip electron transitionbetween majority- and minority-spin bands, leading to theband splitting at their crossing point. In this case, the magneticquantum numbers m in these bands need to differ with eachother by 1 [41]. Second, the term ξLzSz gives the spin-conserving electron transition between bands with the samespin and the same m, also leading to the band splitting [42].Since the SOI ξL · S mainly affects the band structure alongthe k‖ line, this interaction gives anisotropic band structures inbetween k‖ and k⊥ lines, leading to finite �A(k, εF).Let us discuss each band structure of Fe and Ni in moredetail. In the band structure of Fe in the absence of the SOI[Fig. 4(a)], we have a crossing point between minority-spinbands close to εF [see dashed arrow in Fig. 4(a)]. However,the band splitting due to the SOI is rather weak at this point[Fig. 4(b)]. We can also find other crossing points between themajority- and minority-spin bands, but they are not close to104406-3MASUDA, UCHIDA, IGUCHI, AND MIURA PHYSICAL REVIEW B 99, 104406 (2019)FIG. 4. Band structures along the k⊥ and k‖ lines through � forFe in the (a) absence and (b) presence of the SOI. (c), (d) The sameas (a) and (b) but for Ni. In panels (a) and (c), d orbitals contributingto each band are indicated on the curve, where d3z2−r2 and dx2−y2 are,respectively, abbreviated as dz2 and dx2 for simplicity.εF; the closest crossing point from εF is at ε − εF ≈ −0.09 eV[see solid arrow in Fig. 4(a)]. Moreover, at this crossing point,since both majority- and minority-spin bands originate fromthe same dyz state (m = ±1), the spin-flip electron transition,i.e., the band splitting, does not occur when the SOI is takeninto account [Fig. 4(b)]. This is why Fe has small �A(k, εF).On the other hand, Ni has a favorable band structure forlarge �A(k, εF). First, near the center of the k line, threeminority-spin bands cross nearly at a single point around εF[see the dashed arrow in Fig. 4(c)]. At this point, since oneband includes the dxy component and the other two bands havethe dx2−y2 component, the spin-conserving transition occursthrough Lz, leading to the band splittings. Moreover, we canalso find two crossing points between majority- and minority-spin bands at ε − εF ≈ 0.02 eV and ≈−0.01 eV, sufficientlyclose to εF [see the solid arrows in Fig. 4(c)]. In addition,at one of them with ε − εF ≈ 0.02 eV, the majority-spinband comes from the d3z2−r2 (m = 0) and dx2−y2 (m = ±2)components and the minority-spin band includes the dxz (m =±1) component. Thus, at this point, the spin-flip transitionFIG. 5. Experimental results on the AMPE and AMSE in thesingle-crystalline Ni. (a) Lock-in amplitude Aeven and phase φevenimages. (b) Schematic of the setup in the AMPE experiments. (c),(d) LIT frequency f dependencies of Aeven and φeven at the corners Land R of the U-shaped Ni sample.occurs between the majority-spin d3z2−r2(x2−y2 ) state and theminority-spin dxz state through the operator L−(+), leading tosignificant band splittings. We can conclude that, in the caseof Ni, both the spin-conserving and spin-flip transitions occuraround εF, which is in sharp contrast with the case of Fewith only a weak spin-conserving transition around εF. Thesetransitions give rise to a large anisotropy of the band structurebetween the k⊥ and k‖ lines [Fig. 4(d)], which is the origin ofthe large �A(k, εF) in Ni shown in Fig. 3(j). It is well knownthat the exchange splitting of Ni is smaller than that of Fe [43].This is the reason why the spin-flip transition is more effectivein Ni than in Fe.As mentioned above, our calculated value of �� in Niagrees with the experimental results in Ref. [21]. However,we assumed single-crystalline Ni in our calculations, al-though polycrystalline samples were used in the previousstudy. Thus, we carried out the AMPE experiments using asingle-crystalline Ni for direct comparison of our theory withexperiments. Using the lock-in thermography (LIT) [44–51],we observed the distribution of the temperature modulationinduced by the AMPE on the surface of a U-shaped single-crystalline Ni slab with M along the [001] direction. Duringthe LIT measurements, we applied a magnetic field H withmagnitude H = ±10.0 kOe and a rectangularly modulatedac charge current with the amplitude of Jc = 1.0 A and thefrequency of f = 25.0 Hz, and zero dc offset to the slab[Fig. 5(b)]. Since the AMPE exhibits an even dependenceon the M direction [21], we extract the H-even componentfrom the raw LIT images [Fig. 5(a)], where the LIT amplitude104406-4FIRST-PRINCIPLES STUDY OF THE ANISOTROPIC … PHYSICAL REVIEW B 99, 104406 (2019)Ni FePtCoPtNi3Pt-0.6-0.4-0.200.20.40.60.811.21.41.6NiPtFeCoPt (L10)NiPt (L10)Ni3Pt (L12)CoPtNiPtNiPtMMMFIG. 6. Calculated values of �� at μ = 0 for Fe, Ni, FePt, CoPt,NiPt, and Ni3Pt. White and black stars indicate experimental valuesestimated by the AMPE and AMSE measurements, respectively.Note that Fe does not exhibit clear AMPE in both the calculationsand experiments. Crystal structures of CoPt, NiPt, and Ni3Pt are alsoshown.and phase of the H-even component are denoted by Aeven andφeven, respectively. As seen in the top two panels of Fig. 5(a),the signal is generated around the corners of the U-shapedstructure and the sign of the temperature modulation at thecorner L is opposite to that at the corner R, which is thebehavior expected for the AMPE. In Figs. 5(c) and 5(d), weshow the f dependencies of Aeven and φeven. By combiningthese results with numerical calculations based on finite ele-ment method [21], we obtained �� = 0.11 mV for single-crystalline Ni, which is almost the same as that for polycrys-talline Ni. In Fig. 6, we compare the �� values obtained byour calculations with those by the experiments. Here, whiteand black stars indicate experimental values estimated fromthe AMPE and AMSE measurements, respectively. We seethat our theoretical values agree well with all the experimentalvalues.The above calculations remind us of the importance of theSOI in the mechanism of the AMPE. On the basis of thisknowledge, we predict promising systems for obtaining largeAMPE. We considered L10-ordered FePt, CoPt, and NiPt, andL12-ordered Ni3Pt [52], since Pt has a strong SOI (Fig. 6).Here, since the [001] direction is special for the L10 structure,we set M along the [100] direction and estimated �� =�‖ − �⊥ = �[100] − �[010]. We found that NiPt has a huge�� of 1.31 mV, which is more than ten times larger than thatof Ni. It was also found that CoPt and Ni3Pt have relativelylarge |��|, which are about four times larger than that of Ni(note that CoPt has a negative ��). On the other hand, FePthas a small �� of about half the value in Ni, although FePtis a well-known ferromagnetic metal with strong SOI. Sucha nontrivial material dependence of �� clearly indicates thatnot only a strong SOI but also a small exchange splitting isrequired for obtaining a large ��. Note that, although thelargest �� was obtained, NiPt might have a low Curie tem-perature (TC ∼ 200 K) as shown in a previous experimentalstudy [23]. Thus, to realize huge AMPE at room tempera-ture, CoPt (TC � 800 K) or Ni3Pt (TC � 300 K) would behopeful.IV. SUMMARYWe give a microscopic physical picture of the AMPE bycalculating the anisotropy of the Peltier coefficient �� onthe basis of the first-principles-based Boltzmann transportapproach including the SOI. We showed that Ni has a muchlarger �� than Fe, consistent with recently reported observa-tions on polycrystalline Fe and Ni. By carrying out additionalAMPE experiments using single-crystalline Ni, we confirmedthat our calculated �� also agrees with the experimental oneestimated in single-crystalline Ni, which can emphasize theconsistency between our theory and experiments. Analysisof the band structures clarified that the spin-flip electrontransition due to the small exchange splitting is the key for thelarge �� in Ni. Such an insight is important not only for ad-vancing the understanding of the AMPE and AMSE but alsofor developing research for other spin-caloritronic phenomenawith interconversion between charge and heat currents due tothe SOI. We further calculated �� in some ordered alloysincluding Pt. It was found that L10-ordered CoPt and NiPtand L12-ordered Ni3Pt can have huge |��|, which are aboutseveral to ten times larger than that of Ni. Our first-principlesanalysis clarified the microscopic mechanism of the AMPEand predicted hopeful materials to obtain larger AMPE, whichis beneficial for developing nanoscale thermal managementtechnologies using electronic and spintronic devices. Furtherexperiments on the temperature dependence of the AMPEwould provide more detailed information on the relationshipbetween �� and band structures, which will be addressed infuture works.ACKNOWLEDGMENTSThe authors are grateful to S. Mitani, M. Tsujikawa, K.Nawa, and S. 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