Anomalous Nernst and Hall effects in magnetized platinum and palladium
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Anomalous Nernst and Hall effects in magnetized platinum and palladium
G. Y. Guo,
1, 2, ∗ Q. Niu,
3, 4 and N. Nagaosa
5, 6 Department of Physics, National Taiwan University, Taipei 10617, Taiwan Graduate Institute of Applied Physics, National Chengchi University, Taipei 11605, Taiwan International Center for Quantum Materials and Collaborative InnovationCenter of Quantum Matter, Peking University, Beijing 100871, China Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: October 1, 2018)We study the anomalous Nernst effect (ANE) and anomalous Hall effect (AHE) in proximity-induced ferromagnetic palladium and platinum which is widely used in spintronics, within the Berryphase formalism based on the relativistic band structure calculations. We find that both the anoma-lous Hall ( σ Axy ) and Nernst ( α Axy ) conductivities can be related to the spin Hall conductivity ( σ Sxy ) andband exchange-splitting (∆ ex ) by relations σ Axy = ∆ ex e ~ σ Sxy ( E F ) ′ and α Axy = − π k B T ∆ ex ~ σ sxy ( µ ) ′′ ,respectively. In particular, these relations would predict that the σ Axy in the magnetized Pt (Pd)would be positive (negative) since the σ Sxy ( E F ) ′ is positive (negative). Furthermore, both σ Axy and α Axy are approximately proportional to the induced spin magnetic moment ( m s ) because the ∆ ex isa linear function of m s . Using the reported m s in the magnetized Pt and Pd, we predict that theintrinsic anomalous Nernst conductivity (ANC) in the magnetic platinum and palladium would begigantic, being up to ten times larger than, e.g., iron, while the intrinsic anomalous Hall conductivity(AHC) would also be significant. PACS numbers: 72.15.Gd, 72.15.Jf, 72.25.Ba, 75.76.+j
INTRODUCTION
Spin transport electronics (spintronics) has recently at-tracted enormous attention mainly because of its promis-ing applications in information storage and processingand other electronic technologies[1, 2]. Spin current gen-eration, detection and manipulation are three key issuesin the emerging spintronics. Large intrinsic spin Halleffect (SHE) in platinum has recently been predicted[3]and observed (see Refs. 4 and 5 and references therein).In the SHE, a transverse spin current is generated inresponse to an electric field in a metal with relativisticelectron interaction. The SHE enables us to generate andcontrol spin current without magnetic field or magneticmaterials, which would be an important step for spintron-ics. Furthermore, in the inverse spin Hall effect, a trans-verse voltage drop arises due to the spin current[6, 7],and this allows us to detect spin current by measuringthe Hall voltage. Therefore, platinum has been widelyused as a spin current generator and detector in recentspin current experiments, such as spin Seebeck effect[8],spin pumping[9] and spin Hall switching[10], and plays aunique role in recent developments in spintronics.Platinum is an enhanced paramagnet because its5 d -band is partially filled with a large density ofstates (DOS) at the Fermi level ( E F ) [ N ( E F ) = ∼ ∼ ∼ µ B /atom in Ni/Pt andFe/Pt multilayers[11, 12], respectively. In a ferromag-netic metal, a transverse charge current would be gen-erated in response to an electric field due to relativisticspin-orbit coupling (SOC), an effect known as the anoma-lous Hall effect (AHE)[16], discovered by Hall[17] longago. Since the AHE is another archetypal spin-relatedtransport phenomenon[16] and the SOC strength in Ptis large, it would be interesting to study the AHE inthe proximity-induced ferromagnetic platinum. Further-more, as pointed out in Ref. 18, the fact that the Hallvoltage could be generated by both the AHE and inverseSHE in the magnetized platinum, might complicate thedetection of the pure spin current and also related phe-nomena using platinum. Therefore, it is important tounderstand the transport and magnetic properties of themagentized platinum.In a ferromagnet, the Hall voltage could also arise whena thermal gradient instead of an electric field, is applied.This phenomenon, again due to the relativistic SOC, isrefered to as the anomalous Nernst effect (ANE)[19]. In-terestingly, the ANE could be used as a probe of thevortex phase in type II superconductors[20] and has beenreceiving considerable attention in recent years.[21–28] Inthis context, it would be interesting to study the ANE inthe proximity-induced ferromagnetic platinum. On theother hand, spin Seebeck effect (another thermal phe-nomenon), which refers to the generation of a spin-motiveforce in a ferromagnet by a temperature gradient, has re-cently attracted considerable attention[8, 29, 30]. Again,this effect is usually measured as a transverse voltage ina nonmagnetic metal such as Pt in contact with the fer-romagnet via the inverse SHE[8]. Clearly, if the metalis magnetized due to the magnetic proximity effect, theANE would contribute to the measured Hall voltage too.In this connection, it is imperative to understand theANE in the magnetized platinum.Palladium is isoelectronic to platinum and thus hasan electronic structure similar to that of Pt except asmaller SOC strength (see, e.g., Refs. 3 and 31 andreferences therein). For example, like Pt, Pd also hasa large intrinsic spin Hall conductivity (SHC)[31] andis a highly enhanced paramagnetic metal with a large N ( E F ) = ∼ × − emu/mole among the nonmagnetic metals[32] andis usually considered to be nearly ferromagnetic. It couldbecome ferromagnetic when placed next to a ferromag-netic metal[33, 34] or fabricated as an atomic bilayer onsilver (001) surface[13] or a freestanding atomic chain[15].Recently, the AHE was observed in the Pd film on an yt-trium iron garnet (YIG).[35] Surprisingly, it was reportedthat the intrinsic anomalous Hall conductivity (AHC) inthe Pd film on the YIG layer has a sign opposite to thatfor the Pt/YIG bilayer.[35] This indicates that the AHCin a magnetized nonmagnetic metal does not simply scalewith the SOC strength. One would then ask what deter-mines the AHC in the magnetized metals.In this paper, therefore, we study the AHE and ANE inthe proximity-induced ferromagnetic platinum and palla-dium within the Berry phase formalism[36] based on first-principle relativistic band structure calculations. We alsoperform analytic calculations to identify possible rela-tions between the SHC in an nonmagnetic metal and theAHC in the corresponding magnetized metal. The restof this paper is organized as follows. In the next section,we briefly describe the Berry phase formalism for calcu-lating the AHC and ANC as well as the computationaldetails. In Sec. III, the calculated AHC and ANC willbe presented. Finally, the conclusions drawn from thiswork will be summarized in Sec. IV. THEORY AND COMPUTATIONAL DETAILS
The anomalous Hall conductivity and anomalousNernst conductivity (ANC) are calculated by using theBerry-phase formalism[36]. Within this Berry-phase for-malism, the AHC is simply given as a Brillouin zone (BZ)integration of the Berry curvature for all the occupied bands, σ Axy = − e ~ X n Z BZ d k (2 π ) f k n Ω zn ( k ) , Ω zn ( k ) = − X n ′ = n h k n | v x | k n ′ ih k n ′ | v y | k n i ]( ǫ k n − ǫ k n ′ ) , (1)where f k n and Ω zn are the Fermi distribution func-tion and the Berry curvature for the n th band at k ,respectively.[37] Similarly, the ANC can be written as α Axy = 1
T e ~ X n Z BZ d k (2 π ) f k n Ω zn ( k ) × [( ǫ k n − µ ) f k n + k B T ln(1 + e − β ( ǫ k n − µ ) )] , (2)where µ is the chemical potential and k B is the Boltz-mann constant.[22]The proximity-induced ferromagnetic platinum andpalladium are investigated by the constrained spin-density functional theory with the local density ap-proximation to the exchange-correlation potential.[38]Spin-polarized self-consistent scalar-relativistic electronicstructure calculations with the spin magnetic momentfixed to specified values, are performed. Using the resul-tant self-consistent charge densities, the fully relativisticband structures are then calculated for the AHC andANC calculations. The highly accurate all-electron full-potential linearized augmented plane wave (FLAPW)method, as implemented in the WIEN2K code[39], isused. The experimental lattice constants a = 3 .
92 and3.89 (˚A) are used, respectively, for Pt and Pd. In bothcases, the muffin-tin sphere radius ( R mt ) of 2.5 a.u. isadopted. The wave function, charge density, and poten-tial are expanded in terms of the spherical harmonicsinside the muffin-tin spheres and the cutoff angular mo-ment ( L max ) used is 10, 6 and 6, respectively. The wavefunction outside the muffin-tin sphere is expanded interms of the augmented plane waves (APWs) and a largenumber of APWs (about 70 APWs per atom, i. e., themaximum size of the crystal momentum K max = 9 /R mt )are included in the present calculations. The tetrahedronmethod is used for the BZ integration[40]. To obtain ac-curate ground state properties, a fine 21 × ×
21 grid of11616 k -points in the first BZ is used. For the AHC andANC calculations, a very find grid of 258156 k -points onthe magnetic irreducible wedge (1/16 BZ) in the BZ isused. This is equivalent to a large number of k -points of ∼ X line into 70 intervals. Comparison with testcalculations with a denser grid of 381915 k -points (80 di-visions of the Γ X line) indicates that the calculated AHCand ANC converge to within a few %. RESULTS AND DISCUSSION
The relativistic band structure and also AHC ( σ Axy ) asa function of the Fermi energy ( E F ) for the magnetizedplatinum and palladium with the spin magnetic moment m s = 0 . µ B /atom are displayed in Figs. 1 and 2, re-spectively. All Kramer-degenerate bands in nonmagneticplatinum (see Fig. 1 in [3]) and palladium (see Fig. 1in [31]) are now exchange-split due to the induced mag-netization in the magnetized Pt and Pd. This is clearlyvisible for the d -dominated bands [i.e., energy bands be-low 1.0 eV in Fig. 1(a) or 0.5 eV in Fig. 2(a)] since theferromagnetism is mainly caused by the exchange inter-action among the d -electrons. The band spin-splittingsare largest in the flat bands of almost pure d charactersuch as the bands around 0.5 eV (0.3 eV) in the vicinityof the W-point in Fig. 1(a) [Fig. 2(a)]. -5-4-3-2-1012 E n e r gy ( e V ) -200 0 200 σ xy (S/cm) fcc Pt W Z Γ W L Γ X m s = 0.1 µ B A (a) (b) FIG. 1: (Color online) (a) Relativistic band structure and(b) anomalous Hall conductivity (AHC) of the magnetizedplatinum with a spin magnetic moment of 0.1 µ B /atom. Thehorizontal dotted line at the zero energy indicates the Fermilevel. Anomalous Hall effect
Figures 3 and 4 show the calculated AHC and ANC( α Axy ) as well as the exchange splitting (∆ ex ) as a functionof the induced spin magnetic moment ( m s ) in platinumand palladium, respectivelty. ∆ ex refers to the splittingof the spin-up and spin-down bands, and we calculate∆ ex as the spin splitting of the scalar-relativistic bandsabove the Fermi level at the W-point [Figs. 1(a) and2(a)]. First of all, it is clear from Figs. 3 and 4 that thecalculated σ Axy and ∆ ex increase monotonically with m s .In fact, ∆ ex is almost perfectly proportional to m s , whilethe amplitude of the σ Axy increases linearly with m s for -5-4-3-2-1012 E n e r gy ( e V ) -400 0 400 σ xy (S/cm) fcc Pd W Z Γ W L Γ X m s = 0.1 µ B A (a) (b) FIG. 2: (Color online) (a) Relativistic band structure and(b) anomalous Hall conductivity of the magnetized palladiumwith a spin magnetic moment of 0.1 µ B /atom. The horizontaldotted line at the zero energy indicates the Fermi level. small m s values up to 0.30 and 0.25 µ B /atom for Pt andPd, respectively.Secondly, the AHC is large. In particular, the mag-nitude of the AHC per µ B ( σ Axy /m s ) for m s ≤ . µ B /atom in Pt and Pd is, respectively, ∼
790 and3500 S/(cm · µ B ), being much larger than that of ∼ · µ B ) in iron[37]. Interestingly, the ratio σ Axy /m s for Pt is smaller than that for Pd, indicating that theAHC in a proximity-induced ferromagnetic metal is notnecessarily correlated with the SOC strength. Thirdly,the sign of the AHC in Pt is opposite to that in Pd, be-ing in good agreement with the recent experiments onthe Pt/YIG and Pd/YIG bilayers[35]. Correlation between anomalous and spin Hallconductivities
In order to gain insight into the key factors that deter-mine the AHC in a magnetized nonmagnetic metal, letus consider the two-current model to connect the con-ductivities for the different sorts of Hall effects. Withinthe two-current model approximation, σ Axy and σ Sxy canbe written as[41–43] σ Axy ( E ) = σ ↑ xy ( E ) + σ ↓ xy ( E ) and − ~ e σ Sxy ( E ) = σ ↑ xy ( E ) − σ ↓ xy ( E ), where σ ↑ xy and σ ↓ xy arethe spin-up and spin-down Hall conductivities, respec-tively. In a non-magnetic metal, the spin magnetic mo-ment m s = 0 and thus, σ Axy = 0. In the magnetizedmetal, σ Axy ( E ) = σ ↑ xy ( E − ∆ ex ) + σ ↓ xy ( E + ∆ ex ) ≈ σ ↑ xy ( E ) − ∆ ex σ ↑ xy ( E ) ′ + σ ↓ xy ( E ) + ∆ ex σ ↓ xy ( E ) ′ , where∆ ex is the exchange splitting and is proportional to m s ,as shown in the inset in Fig. 3(a) and Fig. 4(a). There- σ xy A ( S / c m ) Eq. (1)Eq. (3)0 0.1 0.2 0.3 0.4 0.5spin magnetic moment ( µ B /atom)048 α xy A / T ( - A / m K ) T = 300 KT = 100 KEq. (4)0 0.1 0.2 0.3 0.4 0.500.10.20.3 ∆ e x ( e V ) fcc Ptfcc Pt (a)(b) FIG. 3: (Color online) (a) Anomalous Hall conductivity ( σ Axy )and (b) anomalous Nernst conductivity ( α Axy ) as a function ofthe induced spin magnetic moment ( m s ) in platinum. Ex-change splitting (∆ ex ) is displayed as a function of m s in theinset in (a). In (b), T denotes temperature. fore, we find σ Axy ( E F ) ≈ ∆ ex e ~ σ Sxy ( E F ) ′ . (3)Equation (3) tells us that the AHC is proportional tothe energy derivative of the spin Hall conductivity [ σ Sxy ]as well as the exchange splitting (∆ ex ). Interestingly,the SOC strength does not appear explicitly in Eq. (3),in contrary to conventional wisdom. We notice thatplatinum and palladium have similar AHC-versus-energy[ σ Axy ( E )] curves which have a prominent peak near the E F (see Fig. 1 in both [3] and [31]). However, the E F falls on the up-hill side of the peak in Pt[3] but on thedown-hill side of the peak in Pd[31], resulting in the posi-tive σ Sxy ( E F ) ′ for Pt and negative σ Sxy ( E F ) ′ for Pd. This,together with Eq. (3), naturally explains why both thecalculated and observed AHCs in Pt and Pd have oppo-site signs.To examine quantitatively the validity of Eq. (3), herewe repeat the calculations of the SHC for Pt[3] and Pd[31]but using the more accurate FLAPW method with thesame computational details as described already in Sec.II. The calculated SHC for Pt and Pd as a function ofenergy is displayed in Fig. 5(a). The σ Sxy at the E F is 2200 ( ~ /e)S/cm for Pt and 1242 ( ~ /e)S/cm for Pd,being in good agreement with the corresponding resultscalculated previously using the linear muffin-tin orbitalmethod with the atomic sphere approximation.[3, 31] Wethen evaluate numerically the energy derivative of the σ xy A ( S / c m ) Eq. (1)Eq. (3)0 0.1 0.2 0.3spin magnetic moment ( µ B /atom)-5-4-3-2-10 α xy A / T ( - A / m K ) T = 300 KT = 100 KEq. (4)0 0.1 0.2 0.300.10.2 ∆ e x ( e V ) fcc Pdfcc Pd (a)(b) FIG. 4: (color online) (a) Anomalous Hall conductivity ( σ Axy )and (b) anomalous Nernst conductivity ( α Axy ) as a function ofthe induced spin magnetic moment ( m s ) in palladium. Ex-change splitting (∆ ex ) is displayed as a function of m s in theinset in (a). In (b), T denotes temperature. SHC using the σ Sxy ( E ) displayed in Fig. 5. We obtain σ Sxy ( E F ) ′ = 1081 and -4245 ( ~ /e)S/cm-eV for Pt andPd, respectively. Figures 3(a) and 4(a) also show the σ Axy evaluated using Eq. (3) together with the calculated σ Sxy ( E F ) ′ and ∆ ex . It is clear that Eq. (3) holds verywell for small m s up to ∼ µ B /atom for Pt and Pd[Figs. 3(a) and 4(a)].We have also calculated the SHC in the magnetized Ptand Pd metals. The calculated SHC for Pt and Pd isshown as a function of the spin magnetic moment in Fig.5(b). In both Pt and Pd, the SHC initially increases with m s up to ∼ µ B /atom and then decreases slowly as m s further increases [Fig. 5(b)]. Nevertheless, the SHC forboth Pt and Pd remains in the same order of magnitudeall the way up to m s = 0.5 µ B /atom.The validity of Eq. (3) may be understood at the mi-croscopic level. The two-current model can be derivedfrom an approximation in which the spin-flipping partof the SOC is ignored. The spin-conserving part of theSOC can still lead to nontrivial results on the transversetransport coefficients. This non-flip approximation canbe justified for crystals with inversion symmetry and inthe limit of zero magnetization. This is because thatKramer’s theorem implies a two-fold degeneracy of theband structure at general k -points even in the presence ofthe SOC. The SOC term in the Hamiltonian, being sym-metric under spatial inversion and time reversal, mustbehave as a constant within the degenerate space. There- -0.6 -0.4 -0.2 0 0.2 0.4 0.6Energy (eV)05001000150020002500 σ xy [( h / e ) S / c m ] PtPd0 0.1 0.2 0.3 0.4 0.5spin magnetic moment ( µ B /atom)05001000150020002500 σ xy [( h / e ) S / c m ] PtPd S -- S (a)(b) FIG. 5: (Color online) (a) Spin Hall conductivity ( σ Sxy ) as afunction of energy in nonmagnetic Pt and Pd metals. Thevertical dotted line at 0 eV indicates the Fermi energy ( E F ). σ Sxy ( E F ) = 2200 and 1242 ( ~ /e)S/cm for Pt and Pd, respec-tively. σ Sxy ( E F ) ′ = 1081 and -4245 ( ~ /e)S/cm-eV for Pt andPd, respectively. (b) Spin Hall conductivity as a function ofthe induced spin magnetic moment ( m s ) in magnetized Ptand Pd metals. fore, it must also commute with the representation ofthe spin operator within the two-fold degenerate space.In the presence of a small magnetization, the degeneratebands are split to first order in the Zeeman energy ac-cording to the representation of the spin operator withineach of the original degenerate space. Not being able tomix these split levels directly, the spin-flip part of theSOC term in the Hamiltonian can be safely discarded,because its residual effect must be of second order (in aprocess going out and back to the degenerate space). Anomalous Nernst effect
Figures 3(b) and 4(b) indicate firstly that the anoma-lous Nernst conductivity α Axy increases monotonicallywith the spin moment m s for m s up to at least 0.5 µ B /atom in Pt and for m s up to 0.25 µ B /atom in Pd.Like σ Axy , α Axy is approximately proportional to m s for m s ≤∼ . µ B /atom in both Pt and Pd. Secondly, thecalculated α Axy is large, especially in Pt [Fig. 3(b)]. Infact, α Axy for Pt at m s ≥ . µ B /atom could be ten times larger than the intrinsic α Axy [ α Axy /T = 0 . × − A/(m-K ) at T = 293 K] of iron[27]. The magnitude of α Axy for Pd for m s ≥ . µ B /atom is also several timeslarger than that of iron[27].At low temperatures, Eq. (2) can be simplified as theMott relation, α Axy = − π k B Te σ
Axy ( µ ) ′ , (4)which relates the ANC to the AHC. Therefore, it is notsurprising that the magnetized platinum has a very large α Axy since the E F is located on the steep slope of σ Axy ( E )[Fig. 1(b)], resulting in a large energy derivative of σ Axy ( E ) at E F . In Figs. 1(b) and 2(b), the α Axy calcu-lated using the Mott relation [Eq. (4)] is displayed as afunction of the induced m s . Clearly, α Axy calculated di-rectly [Eq. (2)] and using the Mott relation at T = 100K are in good agreement with each other for Pt [see Fig.3(b)] and also for m s ≤ . µ B /atom for Pd [see Fig.4(b)]. On the other hand, α Axy at T = 300 K calculateddirectly differs noticeably from that from the Mott rela-tion, indicating that T = 300 K cannot be considered asa low temperature in this context.Differentiating Eq. (3) and substituting the result intoEq. (4), we find α Axy /T = − π k B ∆ ex ~ σ sxy ( µ ) ′′ . (5)Eqs. (3) and (4) indicate that for small m s , boththe AHC and ANC are proportional to the exchange-splitting. As mentioned above, the exchange-splitting isalmost a perfect linear function of m s , and hence thisexplains why both the σ Axy and α Axy are approximatelyproportional to m s . Furthermore, this suggests that the σ Axy and α Axy are proportional to each other for small m s ,as shown in Fig. 3 and Fig. 4. Therefore, we can rewriteEq. (3) as σ Axy ( E F ) ≈ [ e ~ ∆ ex ( m s ) m s σ Sxy ( E F ) ′ ] m s = βm s , (6)where constant β can be determined solely by first-principle calculations for a certain spin moment m s . Inthe present work, we find that β = 788 S/cm/ µ B forPt and β = − µ B for Pd. Similarly, we canrewrite Eq. (4) as α Axy /T ≈ − [ π k B ∆ ex ( m s ) ~ m s σ Sxy ( E F ) ′′ ] m s = γm s . (7)And using the results of the first-principle calculations,we obtain that constant γ = 0 .
034 A/(m-K µ B ) for Ptand γ = − .
027 A/(m-K µ B ) for Pd. CLOSING REMARKS
Recently, the possible magnetic proximity-inducedspin moment in Pt films in the Pt/YIG bilayerswas measured by magnetic x-ray circular dichroismexperiments[44], and m s was found to be 0.054 µ B /atomat 300 K and 0.076 µ B /atom at 20 K. Using m s = 0 . µ B /atom together with Eqs. (6) and (7), we can esti-mate the intrinsic AHC and ANC for the Pt film to be σ Axy = 40 S/cm and α Axy = 0.51 A/(m-K ) ( T = 300K). The anomalous Seebeck coefficient E y / ( − ∂ x T ) = ρ xx ( α xy − Sσ xy ) where S = α xx /σ xx is the ordinary See-beck coefficient. At T = 300 K, ρ xx = 10 . µ Ωcm and S = − . µ V/K (see Refs. 45 and 46). Resultantly, E y / ( − ∂ x T ) = 0 . µ V/K. Using the sample sizes andthe temperature gradient in the Pt/YIG bilayers[47, 48],one would obtain the Hall voltage due to the ANE inthe order of ∼ µ V, being comparable with the Hallvoltage ( ∼ µ V in Au/YIG and ∼ µ V in Pt/YIG)produced by the spin Seebeck effect via the inverse spinHall effect.
Acknowledgments
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