PHYSICAL REVIEW B 1 APRIL 1999-IVOLUME 59, NUMBER 13

Sign of thermoelectric power in Co/Cu and Fe/Cr multilayers

E. Yu. Tsymbal and D. G. PettiforDepartment of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom

Jing ShiMotorola Phoenix Corporate Research Laboratories, Tempe, Arizona 85284

M. B. SalamonDepartment of Physics, University of Illinois, Urbana, Illinois 61801

~Received 9 November 1998!

We analyze the thermoelectric power~TEP! of Co/Cu and Fe/Cr multilayers in terms of the Mott formula,which relates TEP and energy dependent electric conductivity. The conductivity of the multilayers is treatedusing the model proposed by Tsymbal and Pettifor, Phys. Rev. B54, 15 314~1996!. The model is based on aquantum-mechanical formulation for the electron transport within a realisticspd tight-binding approximationfor the band structure and spin-independent disorder in the on-site atomic energy levels. We find that for adisorder, producing a realistic resistivity of the multilayers, the thermoelectric power is positive for Fe/Crmultilayers, but negative for Co/Cu multilayers, which is consistent with experiments. The difference in thesign of TEP is explained in terms of the difference in the electronic band structures for Co/Cu and Fe/Crmultilayers. We also show that the model accounts for the experimentally observed sign of the magnetother-moelectric power at room temperature, when the magnetizations of the consecutive ferromagnetic layerschange their alignment.@S0163-1829~99!10813-0#

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The discovery of giant magnetoresistance~GMR! in mag-netic multilayers1,2 has stimulated not only numerous expemental and theoretical studies of this phenomenon itself,also a number of physical properties related to GMR, onethem being magnetothermoelectric power~MTEP!.3–8 Theessence of this effect is a change in the thermoelectric po~TEP! of magnetic layered films when an applied magnefield changes the relative alignment of the magnetizationthe ferromagnetic layers. While MTEP, unlike GMR, is nvery promising from the point of view of practical applications, the importance of studying TEP and MTEP comfrom the fact that it helps to understand in more detailunderlying mechanisms responsible for spin-dependtransport in magnetic layered systems.8 By comparing resultsof experiments on TEP with theoretical predictions it bcomes possible to reveal strong and weak points of vartheoretical approaches and models for electron transpomagnetic multilayers.

Most magnetic multilayers, in which GMR is observe~see, e.g., Ref. 9!, show that the electric conductivity ihigher for the parallel configuration~high magnetic field!than for the antiparallel configuration~zero magnetic field!.Co/Cu and Fe/Cr multilayers are the well-known exampof such behavior.~We note, however, that there are exampof inverse GMR.10! On the other hand, TEP is a higher-ordtransport quantity11 and can be expressed through the deritive of the conductivity with respect to energy accordingthe Mott formula:12

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whereEF is the Fermi energy and the elemental chargee isassumed to be positive. As a consequence of this, bothand MTEP may have either sign. The TEP and MTEPtherefore expected to be very sensitive to the mechanwhich determines electron transport.

Experimental studies of thermoelectric power show tTEP is positive in Fe/Cr magnetic multilayers.3,5 The changein TEP associated with an applied magnetic field,DS5SP2SAP , is also positive at room temperature. HereSP andSAP are the values of TEP for the parallel and antiparaconfigurations, respectively. On the contrary, as was cfirmed by numerous experimental studies,6–8 both S andDSare negative in Co/Cu multilayers.

The MTEP was investigated theoretically for variomagnetic superlattices by Inoue, Itoh, and Maekawa.13 Theyproposed that the random scattering potentials caused byirregularities of atomic sites at the interfaces betweenlayers are the origin of GMR and MTEP, the approach beconsistent with an earlier idea.14 According to their modelthe s electrons carrying the electric current are scatteredthe magnetic atoms randomly distributed near the interfathrough s-d mixing. Because thed levels of the magneticatoms are spin dependent, spin dependent scattering octhat eventually leads to GMR and MTEP. The results ofcalculations based on this approach show thatDS5SP2SAP is positive for both Co/Cu and Fe/Cr multilayers. Athough the result for Fe/Cr is in agreement with experimeat room temperature,3,5 the result for Co/Cu is indisagreement.6–8

In this paper we apply the Mott formula~1! in conjunctionwith the model for conductivity proposed in Ref. 15 to stuthe thermoelectric power in Co/Cu and Fe/Cr multilayeThis model, which was originally developed to predict GM

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in magnetic multilayers and has been extended recentlspin valves,16 is motivated by experimental observations thlayered thin-film structures contain many grown-in defesuch as interfacial roughness, grain boundaries, stacfaults, dislocations, and lattice distortions. All of these dfects contribute to the electron scattering and determinedominant mechanism responsible for resistivity in the mnetic layered systems. Within this approach the influencethese defects on conductivity and GMR is modeled witthe Kubo-Greenwood formalism by assuming disorder inon-site atomic energy levels within a realistic tight-bindidescription of the electronic band structure. The defect stering is, thus, described by a single parameter which msures the variance in the on-site atomic energy level disbution and is assumed to be spin independent.appearance of GMR within this model is a result of the spolarization of the electronic structure. The spin-polarizband structure leads tospin-dependent scatteringas the re-sult of the spin asymmetry in the density of states~DOS! atthe Fermi level. It also determines the difference in the Fevelocities of the majority- and minority-spin electrons andthe Fermi surfaces. The application of this approachCo/Cu and Fe/Cr multilayers shows that if the single defescattering parameter is adjusted to provide realistic resistiof the multilayers at saturation, then the predicted valuesGMR are in quantitative agreement with those observedperimentally, supporting the validity of the theory. In thpresent paper, by applying this model to TEP and MTEPCo/Cu and Fe/Cr multilayers and comparing the results wexperiments, we are providing evidence supporting ourproach that spin-polarized band structure rather than spinpendent scattering potentials is the origin of spin dependtransport in magnetic layered systems.

According to the model,15 the conductivity per spin asfunction of electron energy is expressed by the formula

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~2p!3 Tr@Lm~k,E!Ln~k,E!#, ~2!

where the integration overk is to be carried out within thefirst Brillouin zone and where we have explicitly writtedown the dependence on electron energyE. The mean-free-path operatorLm(k,E) is defined by

Lm~k,E!5\ vm~k!Im„E2H0~k!2G~E!…21, ~3!

whereH0(k) is the tight-binding Hamiltonian of the perfecmultilayer andvm(k) is the m component of the velocityThe imaginary part of the self-energyG(E) is proportional tothe mean-square displacements of the on-site atomic eneg2 and to the partial density of statesnj a(E) related to layerj and orbitala, i.e.,

G j a, j a~E!5 ipg2nj a~E!, ~4!

the real part of the self-energy being neglected. The pareterg, which characterizes the degree of disorder withinmultilayer, is assumed to be spin independent. The spinpendence of the scattering rates within the model is demined explicitly by the spin dependence of the local den

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ties of states through Eq.~4!, i.e., t j a21(E)52 ImGja,ja /\

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The band structure of perfect fcc Co6/Cu6 and bccFe6/Cr6 multilayers was calculated for the parallel and anparallel alignment of magnetizations. The unit cell containg six atomic layers of each constituent for the parallel cofiguration was repeated to infinity in the stacking@001#direction, whereas the cell size was double for the antipalel configuration. The lattice parameter was set equal toof bulk bcc Cr for the Fe/Cr multilayer and of bulk fcc Cfor the Co/Cu multilayer, i.e., the lattice mismatch wasnored. The two-center, orthogonal, tight-binding parametfor the s, p, andd orbitals were taken to be the same asRef. 15. The conductivity of the multilayers was computedaccordance with formulas~2!–~4! by direct summation in thefirst Brillouin zone using meshes up to 72 000k points. Thenet conductivity was determined as a sum of the contritions from the two spin channels, i.e., spin-flip proceswere neglected.

The log derivative of the conductivity, which enters fomula ~1! for TEP, was determined numerically from the coductivity computed as a function of electron energy. Sinthe local densities of states are energy dependent and coa fine structure, thes(E) displays also a fine structurewhich makes the evaluation of the derivative difficult. Awas noticed in Ref. 13, however, the fine structure ofdensity of states is meaningless within the energy scalethe order of ImG. We therefore averaged the local DOS ovthe energy interval of 0.1 eV, which is a typical value of tscattering induced broadening that provides realistic resisity of the multilayers. This procedure results in a numericerror in the log derivative, which was estimated as 0.4 eV21

for the Co/Cu and 0.2 eV21 for the Fe/Cr.The top panel of Fig. 1~a! shows the DOS of the paralle

aligned Co6/Cu6 multilayer. As seen, the DOS is asymmetrbetween the majority and minority spins, which is typical fferromagnetic metals. The important feature of the DOSthe majority-spin electrons is that the Fermi level lies withthespbandabovethe top of thed band. Since the dispersivsp electrons are characterized by a high velocity and adensity of states, the conductivity of the majority-spin eletrons is high. This is due to the fact that the mean free patproportional to the velocity and inversely proportional to tdensity of states~3!. On the contrary, as seen from the botom panel of Fig. 1~a!, the Fermi level of the Co/Cumultilayer lies within thed band for the minority-spin electrons. Thed band is localized in a relatively narrow energwindow, i.e., the velocity of thed electrons is low and theDOS is high. That leads to a short mean free path and aconductivity associated with thed electrons. We note that thcontribution of thesp electrons within thed band is sup-pressed by the strongsp-d hybridization. These argumentsuggest, therefore, that the conductivity of the paraaligned Co/Cu multilayer is dominated by the majority-spelectrons.

Since we are interested in the sign of thermoelecpower, we should understand now how the conductivitythe majority-spin electrons changes with electron energy.obvious from the top panel of Fig. 1~a!, the majority DOSdoes not change significantly in the vicinity of the Ferm

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level. Therefore the conductivity change is mainly detmined by the change in the electron velocity at the Felevel. As was demonstrated in Ref. 15, the velocity aconsequently, the conductivity of the Co/Cu multilayeraffected strongly by the hybridization betweenspandd elec-trons. The hybridization of the dispersivespbands with thedbands results in a ‘‘slowing down’’ of the current-carryingspelectrons, thereby decreasing the conductivity. The strenof this effect depends on the position of thed band withrespect to the Fermi energy. If thed bands cross the Fermenergy, the effect of the hybridization on the conductivitymost pronounced. However, the effect is still sizeable formajority-spin channel, where thed band lies below the Fermlevel. Increasing the electron energy in this case resultincreasing the electron velocity, because thespelectrons be-come less affected by thesp-d hybridization. We can therefore conclude that the conductivity of the Co/Cu multilayincreaseswith energy and that, according to the Mott fomula, implies anegativesign of the thermoelectric power.

These qualitative conclusions are supported by quantive results. Fig. 1~b! shows the conductivity of the Co6/Cu6multilayer as a function of electron energy in the vicinitythe Fermi energy. In this calculation the parameterg charac-terizing the disorder within the multilayer was taken as 0

FIG. 1. Calculated results for the Co6 /Cu6 multilayer as a func-tion of electron energy:~a! density of states at saturation fomajority-spin ~top! and minority-spin~bottom! electrons;~b! con-ductivity for the parallel~circles! and antiparallel~squares! align-ment of magnetizations;~c! the log derivative of the conductivitywith error bars. The Fermi energy lies at zero.

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eV which provides a realistic room-temperature resistivitysaturation, i.e., about 22mV cm. As is evident from Fig.1~b!, the conductivity increases with increasing electron eergy. This leads to positive values of] ln s/]E which aredisplayed with error bars in Fig. 1~c!. The value of] ln s/]Eat the Fermi energy is 1.860.4 eV21. Assuming thatT5293 K, we obtain from formula~1! that SP521363 mV/K, which is close to the experimentally observevalues of TEP at room temperature, namely220 mV/K ~Ref.7! and226 mV/K.8

As seen from square symbols in Fig. 1~b!, the conductiv-ity of the antiparallel-aligned Co/Cu multilayers also icreases with energy. We found, however, that the valuethe log derivative are smaller in this case, which impliesnegativeDS @squares in Fig. 1~c!#. Our calculations showthat ] ln s/]E51.360.4 eV21 at the Fermi energy. TheMTEP defined byDS/SAP is therefore of the order of 40%which is in agreement with experiments,7,8 where DS/SAP547% andDS/SAP536% were found, respectively.

The Fe/Cr system behaves differently from the Co/system. As is evident from Fig. 2~a!, the Fermi level in theFe/Cr multilayer lies within thed band for both spin orien-tations. However, the DOS exhibits a pronounced valleythe minority spins, with the Fermi level lying almost at th

FIG. 2. Calculated results for the Fe6 /Cr6 multilayer as a func-tion of electron energy:~a! density of states at saturation fomajority-spin ~top! and minority-spin~bottom! electrons;~b! con-ductivity for the parallel~circles! and antiparallel~squares! align-ment of magnetizations;~c! the log derivative of the conductivitywith error bars. The Fermi energy lies at zero.

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bottom of this valley. This feature in the DOS is a consquence of both the bcc structure of the Fe/Cr multilayer athe similarity in the minority-spin electronic structure of buFe and bulk Cr. In the region of this valley electrons amore dispersive and consequently the conductivity ofFe/Cr multilayer displays an enhancement at thenergies.15 We note that, contrary to the Co/Cu system, tconductivity of the parallel-aligned Fe/Cr multilayer is domnated by the minority-spin electrons. With increasing eltron energy we are moving away from the valley to thegion with less-dispersive bands and a higher DOS. Duethis, the conductivitydecreaseswith increasing energy andthat, according to the Mott formula, implies apositiveTEP.

These arguments are supported by accurate calculatthe results of which are displayed in Fig. 2~b!. The value ofthe disorder parameter was taken in these calculations tthe same as for the Co/Cu multilayer, i.e.,g50.7 eV. Theresistivity of the saturated Fe6/Cr6 multilayer in this case isabout 39mV cm, which is a representative value charactizing experiments at room temperature. As is evident frFig. 2~b!, the conductivity decreases with increasing electenergy. This leads to negative values of] ln s/]E, which aredisplayed with error bars in Fig. 2~c!. The value of] ln s/]Eat the Fermi energy is20.960.2 eV21. Assuming thatT5293 K, we arrive atSP5762 mV/K, which is close to theexperimentally observed value 11mV/K at T5250 K.3

As seen from the square symbols in Fig. 2~b!, the conduc-tivity of the antiparallel-aligned Fe/Cr multilayers also dcreases with energy. We found, however, that the absovalues of the log derivative are smaller in this case@comparesquares and circles in Fig. 2~c!#. This implies apositiveDS.Our calculations show that] ln s/]E520.760.2 eV21 at the

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Fermi energy. The MTEP is therefore about 30% whichconsistent with the experimental resultDS/SAP516% atT5250 K.3

We note that similar to GMR, the magnitude of TEP aMTEP depends on the degree of disorder in the multilayand is different for various values of the parameterg. How-ever, unlike GMR, which decreases monotonically withcreasing defect scattering,15 the behavior of TEP and MTEPmay be more complex.

For some multilayers we are able to predict the signTEP by analyzing their band structures without calculatthe conductivity. For example, applying similar argumenwhich have been used for predicting the sign of TEP inCo/Cu and Fe/Cr systems, we can conclude that the TEnegative in Co/Au, Co/Ag, Ni/Cu, Ni/Au, Ni/Ag, Fe/Cu, FeAu, and Fe/Ag multilayers, whereas the TEP is positiveCr/Cu, Cr/Au, and Cr/Ag multilayers. This prediction is cosistent with the experiments on Fe/Cu~Ref. 5! and Fe/Au,7

whereas experiments on the other multilayers remain toperformed.

In conclusion, we have shown that the Mott formulaconjunction with a model for electric conductivity, whicconsidersspin dependent disorderas the dominant mechanism of electron scattering within an accuratespin-polarizedband structure, accounts for the experimentally observesign of the thermoelectric power and magnetothermoelecpower in Co/Cu and Fe/Cr multilayers.

E.Yu.T. is grateful to Hewlett-Packard LaboratoriesPalo Alto for financial support. The computations were pformed in the Materials Modelling Laboratory at the Depament of Materials, University of Oxford, on an HP ExemplV-class computer jointly funded by Hewlett-Packard aHEFCE through the JREI scheme.

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