Comments on the thermoelectric power of the f-electron metallic compounds
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Comments on the thermoelectric powerof the f-electron metallic compounds
A.E. Szukiel
Institute of Low Temperature and Structure Research,Polish Academy of Sciences, Wrocław, Poland (Dated: February 26, 2018)
Abstract
The anomalous temperature variation of the thermoelectric power in the f-electron metallic com-pounds, namely the sign reversal or the maxima, is sometimes interpreted as resulting from theconduction electrons scattering in Born approximation on the acoustic phonons and on the local-ized spins in the s–f exchange interaction. The experimenters rely on the results of some theoreticalworks where such thermoelectric power behavior was obtained within these simple models. In thepresent paper we prove that neither the electron–phonon scattering nor the magnetic s–f scatteringin the Born approximation (nor both of them) do lead to the effects mentioned above.
PACS numbers: 72.15.Jf, 72.10.-d, 72.10.Di, 75.30.-m . INTRODUCTION Over the last decades can be seen growing interest in the thermoelectric power of f-electron metallic compounds. The application studies concern mainly the strongly correlatedelectron systems (SCES), which seem to be promising as the thermoelectric materials.
The fundamental researches are focused both on the SCES, see e.g. Refs [3–11] as well ason the well-localized f-electron systems (WLS) – see Refs [12–19]. The thermoelectric power(TEP) of the SCES reaches typically the values one or two orders greater than those ofthe WLS, but in both material groups it exhibits the anomalies as the sign change and/orthe maxima (minima). In SCES these anomalies occur, in general, at the temperaturesmuch higher than in WLS. The main reasons of such TEP behavior in magnetic metals areassociated with phonons and the "magnetic" electrons of incomplete shell. If we disregardthe anomalies caused by the phonon-drag or the magnon-drag (see e.g. Ref [20,21]), we haveto deal with those due to the conduction electron scattering by phonons and those causedby the conduction electron and f-shell interaction (k-f interaction). In SCES the dominantrole plays the k-f interaction originating from the conduction and f-electron hybridization.Theoretical models that help to explain the great values and the high-temperature maxima(minima) with the possible sign change of TEP in SCES systems are founded on this typeof interaction, see e.g. Refs [22–26].In WLS even more important role than hybridization can play the Coulomb interaction .For 4f electron systems both type of interaction were described by Hirst (1978) within thesame form of generalized k-f interaction ( discussed briefly in Ref [27]). The contributionto the TEP anomalies from that interaction for rare-earths paramagnetic systems was con-sidered in Ref [28]. It has been shown that the anomaly arises in the third order of thescattering interaction when it fulfills some symmetry condition with respect to the symme-tries of the ground and the excited f-electron states in the crystalline field. In particularthe anomaly can be caused by the aspherical Coulomb interaction, whereas the isotropic s-fexchange interaction is excluded regardless the character of the crystal-field splitting. . Theanomaly manifests itself at the temperature corresponding to the half of the excited stateenergy as the maximum, or as the minimum combined with the sign reversal. Both thesecases were illustrated in Ref [29] for two examples of rare-earth intermetallics, and very goodagreement between theory and experiment was achieved. In the ordered f-electron systemsalso the isotropic s-f exchange scattering may contribute to the anomalies of thermoelectricpower. In ferromagnets the conduction-electron band splitting due to this interaction causesan asymmetry in the scattering intensities for the spin-up and spin-down electrons in theirscattering on the system of the f-electron localized moments. This leads to the maxima ofTEP below the Curie temperature, as it was shown in the Born approximation in Ref [30].The effect occurs both when the f-electron moments are described in the molecular fieldapproximation (MFA) and when the spin-wave approximation is applied.However the considerations in Ref [30] do not include the influence of the crystal fieldon the f-electron level (in the single-ion approximation) nor on the collective excitations in2-electron system. Thus the results of this work are applicable only for ferromagnets withoutthe crystal field splitting (see e.g. Ref [32]). Another possible cause of anomalous TEP atlow temperatures may be the conduction electron scattering on phonons, considered in thesecond Born approximation. This effect was demonstrated in Ref [33], and was called there"the phony phonon-drag", because it gives the TEP maxima of the similar magnitude andthe temperatures of occurrence as those attributed previously to phonon-drag.We have presented above some known mechanisms of the thermoelectric power anoma-lies possible for the f-electron metallic compounds. Among these referring to the low-temperature anomalies, Refs [28,30,33], the first two are related to WLS and the last mayconcern also SCES.It should be noted that all of these mechanisms (including those characteristic forSCES ) rely on the non-standard models: the second Born approximation for the scat-tering probability or the two-band model for the conduction electron experiencing the s-finteraction.In some experimental works (see e.g. Refs [4–19]) one can find the interpretations of TEPanomalies as caused by the isotropic s–f exchange scattering (in the one-band approach forthe conduction electrons) and/or the electron–phonon scattering – considered in the Bornapproximation. These interpretations are based on the results of Refs [34–36] where the TEPmaxima and the sign reversal were obtained within these oversimplified models.The authors of Refs [34–36] performed the numerical calculations of the electron–phononcontribution to the thermoelectric power, S ph ( T ) , according to the formula derived withinthe variational method of solving the Boltzmann equation. However, in their calculationsthey have not applied the scattering matrix elements presented in Refs [37–39], but other,received by themselves.
As the result, they have obtained S ph ( T ) exhibiting the signreversal and maxima. Calculating additionally the contribution from the s–f scatteringin Born approximation (but neglecting the conduction band splitting), and next the totalthermoelectric power S ( T ) with the use of the Matthiessen’s rule for the scattering matrixelements, they obtained the same effects. In consequence, they could reproduce the TEPmaxima occurring in some rare earths intermetallic systems like RE Al , Ref. [40].Notwithstanding, as we show in the present paper, neither the electron–phonon scatteringitself, nor in the combination with the isotropic s–f scattering (within the model of scatteringapplied in Refs [34]) do lead to the TEP sign reversal nor its maxima. We prove this bycalculating and analyzing the total thermoelectric power S ( T ) and the contributions fromthe s–f scattering, S mag ( T ) , and from the electron–phonon scattering, S ph ( T ) , with the useof the same formula for the TEP as in Refs [34–36]. We also discuss the variation of S ph ( T ) and S ( T ) in dependence on material parameters for the same parameter values as used inthe cited works. In conclusion we indicate the crucial points of calculations reported inRefs [34–36] that led to different results. 3 I. THE MODEL AND THE METHOD OF CALCULATION
We consider the thermoelectric power S ( T ) of cubic ferromagnetic metal with localizedspins, assuming, similarly as in Ref. [ 34], that it results from the free electron scattering inBorn approximation on acoustic phonons in the Debye model and on the localized spins inthe mean field approximation (MFA). For the calculation of S ( T ) and its particular phonon S ph ( T ) and magnetic S mag ( T ) contributions, we use the formula derived by Kohler, Refs [37,38], through the variational method of solving the linearized Boltzmann equation. Theessence of the method is the statement, known as the variational principle, that the solutionof Boltzmann equation (the wave vector k -dependent non-equilibrium electronic distributionfunction) should realize the maximum of the functional describing the production of theentropy induced by the scattering in the steady state (for transport process). In this way,solving Boltzmann equation becomes equivalent to the variational problem and has beensolved in Ref. [37] with the use of Ritz method. After expansion of the searched solutionwith respect to some base φ i ( k ) , i = 1 , ..., n , the coefficients of the expansion are found fromthe system of linear equations following from the variational principle. With the use of thefound coefficients, the electrical J and the thermal U currents can be represented as thecombinations of the trial currents J i , U i , J i = − e Z d k − ∂f k ∂ε k φ i ( k )( v k · u ) ,U i = − Z d k − ∂f k ∂ε k φ i ( k )( v k · u )( ε k − ζ ( T )) , (1)where f k = f ( ε k ) = (exp[( ε k − ζ ( T )) /k B T ] + 1) − is the equilibrium electron distributionfunction, u denotes the external field direction, v k – electron velocity, ε k = ( ~ k ) / m –electron energy, ζ ( T ) – the chemical potential.Substituting currents J , U to the Onsager transport equations one gets the transportcoefficients expressed by the products of the trial currents J i , U i and of the elements of the n -dimensional scattering matrix P ij P ij ( T ) = Vk B T Z d k Z d k ′ C ( k , k ′ ) f k (1 − f k ′ ) u ij ( k , k ′ ) u ij ( k , k ′ ) = [ φ i ( k ) − φ i ( k ′ )][ φ j ( k ) − φ j ( k ′ )] . (2)Here C ( k , k ′ ) is—for a given scattering process—the transition probability per unit time forthe free electron scattered from the state k to the state k ′ .The expressions for the electrical and the thermal conductivity, and for the thermoelectricpower of metals (for the highly degenerated electron gas) in the variational approximationof the n -th order were derived in Ref.[37] with the use of the base functions φ i ( k ) = ( k · u )( ε k − ζ ( T )) i − , i = 1 , . . . , n . (3)The lowest-order approximation for the thermoelectric power was obtained with φ ( k ) , φ ( k ) base functions, as it is seen from the formula in Eq. (27), Ref. [37]. The formula4known also as the Ziman’s formula) and its detailed derivation can also be find in Ziman’smonograph, see Eq. (9.12.13) therein. It can be written in the form S ( T ) = π k B e S ( T ) P ,S ( T ) = P − k B Tε F P − π k B Tε F P , (4)after using the reduced P ij = P ij / ( k B T ) i + j form of the scattering matrix elements andassuming that e > ( the electron charge is − e ).The lowest order variational approximations for the electrical ρ and the thermal W re-sistivity rely on the one-dimensional sets of the base functions – φ ( k ) in the first case and φ ( k ) in the second one ρ ( T ) = P J W ( T ) = T ( k B T ) P U , (5)where J , U are the trial currents (1), compare Eq. (19a) and the first of Eqs. (20a) inRef. [37].Considering the phonon S ph ( T ) and the magnetic S mag ( T ) contributions to the thermo-electric power, we will use the formula (4) defined correspondingly with the electron–phonon C ph ( k , k ′ ) and the s–f (magnetic) C mag ( k , k ′ ) scattering probability. The total thermoelec-tric power S ( T ) (4) is calculated with C ( k , k ′ ) = C ph ( k , k ′ ) + C mag ( k , k ′ ) , according toMatthiessen rule. The same rule concerns the scattering matrix elements (2) and, conse-quently, the electrical and the thermal resistivity (5) ρ ( T ) = ρ mag ( T ) + ρ ph ( T ) ,W ( T ) = W mag ( T ) + W ph ( T ) (6)From (4)–(6) the Kohlers rule follows S ( T ) = S ph ( T ) W ph ( T ) + S mag ( T ) W mag ( T ) W ( T ) , (7)which in its original form concerns the case when the total thermoelectric power was theresult of the electron–phonon and the electron–impurity scattering, see Eq. (8) in Ref. [38]. III. MAGNETIC CONTRIBUTION TO THE THERMOELECTRIC POWER:ELECTRON SCATTERING ON MAGNETIC IONS C mag ( k ′ , k ) , the transition probability for the conduction electron experiencing the s-fexchange interaction, can be expressed, after summation with respect to all possible changes5f the electron spin, by ℑ Tr χ ( q , ~ ω ) / (exp[ ~ ω/k B T ] − , for q = k ′ − k and ~ ω = ε ′ − ε ,where χ ( q , ~ ω ) is the susceptibility function for the system of localized f-electrons, and ε ′ = ( ~ k ′ ) / m , ε = ( ~ k ) / m , see e.g. Ref. [41]. Using the MFA susceptibilities for thecubic ferromagnet one gets C mag ( k ′ , k ) = 2 πj ( g − N ~ (cid:20)(cid:18) h J z i δ ( ε ′ − ε − ∆ )exp[ ∆ /k B T ] − h J z i δ ( ε ′ − ε + ∆ )1 − exp[ − ∆ /k B T ] (cid:19) + ( δJ z ) δ ( ε ′ − ε ) (cid:21) ( δJ z ) = h ( J z ) i − h J z i = J ( J + 1) − h J z i tanh( ∆ / (2 k B T )) − h J z i , (8)where j ex denotes the energy of the s–f exchange interaction, N is the number of ions perunit volume, h J z i is the MFA thermodynamical expectation value of the z -component of thef-electron total angular momentum; ∆ = 3 k B T c h J z i /J ( J +1) is the molecular field energy, T C the Curie temperature and J is the maximal eigenvalue of the operator J z .The components in round brackets refer to the inelastic scattering – the first describes thescattering with the energy absorption and the second with the emission. The last componentin square brackets refers to the elastic scattering.Since C mag ( k , k ′ ) is the even function with k and with k ′ C mag ( k ′ , k ) = C mag ( − k ′ , k ) = C mag ( k ′ , − k ) , (9)and the base functions φ i ( k ) (3) are odd, the only components of u ij ( k , k ′ ) giving contribu-tion to the scattering probabilities P mag ij ( T ) (2) are the products φ i ( k ) φ j ( k ) , φ i ( k ′ ) φ j ( k ′ ) .For this reason P mag ij ( T ) can be expressed in the form P mag ij ( T ) = 14 π Z d k (cid:18) − ∂f ( ε k ) ∂ε k (cid:19) τ ( ε k ) φ i ( k ) φ j ( k ) , (10)where τ ( ε k ) is the relaxation time depending only on the electron energy τ ( ε k ) = 14 π Z d k ′ C mag ( k , k ′ ) 1 − f ( ε k ′ )1 − f ( ε k ) , (11)and the identity f ( ε k )(1 − f ( ε k ′ )) ≡ k B T ( − ∂f ( ε k ) /∂ε k ) (1 − f ( ε k ′ )) / (1 − f ( ε k )) wasapplied.Performing the integration with respect to k and k ′ in the standard way (see e.g. Ref. (39)one gets P mag ij ( T ) = 2 m ~ ∞ Z d ε D ( ε ) ε ( ε − ε F ) i + j − (cid:18) − ∂f ( ε ) ∂ε (cid:19) τ ( ε ) , (12)where D ( ε ) = (2 m ) / ε / / (2 π ~ ) is the density of states for the free conduction electrons,6nd τ ( ε ) = πj N ~ D ( ε ) (cid:0) M ( y ( ε )) + ( δ J z ) (cid:1) , M ( y ( ε )) = h J z i (cid:18) − y ](exp[ z ] − − ( y + z )]) + 1 + exp[ − y ]1 − (exp[ − z ])(1 + exp[ − ( y − z )]) (cid:19) y = ( ε − ε F ) /k B T , z = ∆ /k B T . (13)Next we will calculate (12) with the use of the Sommerfeld expansion (A4) confiningourselves to the first non-vanishing term in the approximation of the strong degeneration ofthe electron gas. Noting, additionally, that M ( y ) is even, we get ∂ M ( ε ) /∂ε | ε = ε F = 0 , andin consequence P mag ij in the form P mag11 = P mag11 ( ε ) | ε = ε F = (2 m ) / π ~ ε / τ ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) ε = ε F P mag12 = π k B T ) ∂P mag11 ( ε ) ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ε = ε F = 2 π k B T ) ε F P mag11 P mag22 = π k B T ) P mag11 ( ε ) | ε = ε F . (14)The magnetic contribution to the thermoelectric power we get substituting in (4) P mag ij = P mag ij / ( k B T ) i + j − in place of P ij . After some algebra we can write it in the form S mag ( T ) = − π k B e k B Tε F . (15)This result is the same as that obtained from Mott formula when the solution ofBoltzmann equation is described by relaxation time depending on the electron energy as τ ( ε ) ∼ ε − / . The exemplification of that case is the elastic electron scattering on theionized impurities, see e.g. Ref. [21].For the magnetic part of the electrical ρ mag ( T ) and the thermal W mag ( T ) resistivities wesubstitute in (5) P mag ij with /τ ( ε F )1 τ ( ε F ) = G mk F Nπ ~ (cid:2) J ( J + 1) − h J z i − h J z i tanh( ∆ / k B T ) (cid:3) , (16)and J , U in the form J = e k F (3 π ~ ) , U = J π ( k B T ) e , (17)resulting from the integration in (1) with respect to k in the manner described above. Thefinal results are as follows ρ mag ( T ) = ρ mag0 (cid:2) J ( J + 1) − h J z i − h J z i tanh( ∆ / k B T ) (cid:3) W mag ( T ) = ρ mag ( T ) L T , (18)7here ρ mag0 = 3 π j mN/ (2 e ~ ε F ) , and L = π k B / (3 e ) is the Lorentz number.For T ≪ T C , after approximating h J z i MFA ∼ J − exp[ − T C / (( J +1) T )] , Ref.[31], there is ρ mag ∼ ρ mag0 J exp[ − T C / (( J + 1) T )] , W mag ∼ ρ mag0 JL T exp[3 T C / (( J + 1) T )] . (19)For T > T C there is h J z i MFA = 0 and hence ρ mag = ρ mag0 J ( J + 1) , W mag = ρ mag0 J ( J + 1) L T . (20)
IV. ELECTRON–PHONON SCATTERING CONTRIBUTIONTO THE THERMOELECTRIC POWER
We consider the phonon system in Debye approximation and the electron–phonon interac-tion in the deformation potential approximation. For the phonon system thermodynamicalequilibrium is assumed, so no phonon drag processes are considered. The transition proba-bility per unit time for the free electron normal (i.e. not
Umklapp ) scattering from the state k to the state k ′ = k + q , by phonon of the wave vector q and the energy ~ ω q , is: C ph ( k , k + q ) = 2 π ~ c ph ( q ) (cid:20) δ ( ε k + q − ε k − ~ ω q )exp[ ~ ω q /k B T ] − δ ( ε k + q − ε k + ~ ω q )1 − exp[ − ~ ω q /k B T ] (cid:21) , (21)see, e.g., Ref. [39], Eq. (9.5.6). For the considered model ~ ω q = ~ q v s , where v s denotes thesound velocity averaged over the directions in a crystal, and ≤ q ≤ q D , where q D is theDebye radius. The scattering amplitude c ph ( q ) has the form c ph ( q ) = 2 C q N q c , (22)where C = 2 ε F / is the interaction energy, N – the number of ions (under the assumptionof a one ion of the mass M per a primitive cell), q c = M v s / ~ . The first component of thesum (21) describes the scattering processes corresponding to the phonon absorption, andthe second one to its emission.For the electron–phonon scattering, unlike in the case of the electron-wave-vector inde-pendent magnetic scattering described in the previous Section, the relaxation-time solutionof the Boltzmann equation exists only in the temperatures much greater than the Debyetemperature, T D . Thus, for the calculation of the electron–phonon scattering contributionin the transport coefficients the variational method proved to be very useful.The thermoelectric power S ph ( T ) according to the scattering probability C ph ( k , k + q ) and the formula (4) was obtained by Kohler , and the same form of S ph ( T ) was derivedby Ziman. In the first subsection below (and in Appendix A) we perform the detailedcalculation of the scattering matrix elements P ph ij (5), applying slightly different methodthan Ziman, but obtaining the same results.8ur way of calculation is similar but simpler than that used by the authors of Ref. [36]. InAppendix B, we indicate the crucial points in their calculations which led them to the form ofthe electron–phonon scattering matrix elements differing from those of Kohler (Ziman) andours, and, in consequence, to the spectacular effects in the behavior of the thermoelectricpower. The same applies to the calculations of the scattering matrix elements in Refs [34,35],although they have been performed in a different way than in Ref. [36]. The results for S ph ( T ) in each of the papers Refs [34–36] are qualitatively the same as it is illustrated in Fig. 3 ofRef. [36]. For this reason, we refer only to the results of the last paper. A. The electron–phonon scattering matrix elements
Because of the form of C ph ( k , k + q ) it is convenient and natural to express and calculatethe scattering matrix elements P ph ij (2) as the integrals with respect to k and q ( q = k ′ − k ).Correspondingly, the functions u ij ( k , k ′ ) with the base (3) have the form u = ( q · u ) ,u = ( q · u ) ( ε k + q − ζ ( T )) + ( k · u )( q · u )( ε k + q − ε k ) ,u = ( q · u ) ( ε k + q − ζ ( T )) + ( k · u ) ( ε k + q − ε k ) + 2( k · u )( q · u )( ε k + q − ε k )( ε k + q − ζ ( T )) . (23)Then, as appropriate for a cubic symmetry, we average (23) with respect to all fielddirections, u , (cid:0) ( q · u ) → q / , ( k · u ) → k / , ( q · u )( k · u ) → ( k · q ) / (cid:1) , and substitute,by virtue of the energy conservation law, ε k + q − ε k = ± z q k · q = 12 q (cid:18) ± z q ε q − (cid:19) , where z q = ~ q v s , ε q = ~ q / m is the energy of the electron of the wave vector q ; theupper sign refers to the absorption and the lower one to the emission. After representing k = q ( η/ε q + ζ ( T ) /ε q ) and denoting η = ε k − ζ ( T ) we obtain u ± = 13 q ,u ± = 13 q (cid:20) η ± z q z q (2 ε q ) (cid:21) ,u ± = 13 q (cid:20) η + 2 z q ηε q ± z q η + z q ζ ( T ) ε q ± z q ε q (cid:21) . (24)The scattering matrix elements P ph ij can be then written in the way P ph ij = P ph ij + + P ph ij − ,P ph ij ± = Vk B T Z d q q c ph ( q ) H ± ij ( q, z q ) , (25)9here H ± ( q, z q ) = F ± ,H ± ( q, z q ) = F ± ± z q F ± + z q ε q F ± ,H ± ( q, z q ) = F ± ± z q F ± + 2 z q ε q F ± + ζ ( T ) z q ε q F ± ± z q ε q F ± , (26)and for n = 0 , , : F ± n ≡ F ± n ( z q , T ) = ± Z d k ( ε k − ζ ( T )) n δ ( ε k + q − ε k ∓ z q ) f ( ε k )(1 − f ( ε k ± z q ))exp[ ± z q /k B T ] − . (27)After the transformations made in Appendix A with the use of the approximation of theelectron gas strong degeneration the functions (27) have the form F ± n ( z q , T ) = ςq θ ( q max − q ) θ ( q ) ( k B T ) n +1 I n ( z q ) , (28)where ς = 2 π m / ( ~ ) , θ ( x ) is the Heaviside step function, and I ± n ( z q ) ≡ ± ∞ Z −∞ d y y n f ± ( y, z q ) ,f ± ( y, z q ) = (exp[ ± z q /k B T ] − − (exp[ y ] + 1)(1 + exp[ − ( y ± z q /k B T )]) . (29)As the immediate consequence of the symmetry property f + ( y, z q ) = f − ( − y, z q ) of theintegrands in I ± n ( z q ) , we obtain the following symmetry properties for F ± n ( z q , T ) F +0 = F − ,F +1 = − F − ,F +2 = F − . (30)Accounting (30) in (26) we reach H + ij ( q, z q ) = H − ij ( q, z q ) and from (25) the conclusionthat (i) the contribution to the scattering from the absorption and from the emission must beequal P ph ij + = P ph ij − . (31)After substitution F ± n ( z q , T ) (28) with I ± n ( z q ) (A5) to (26) we get H ± ( q, z q ) = F ± ,H ± ( q, z q ) = z q ε q F ± H ± ( q, z q ) = (cid:20) π k B T ) + (cid:18) ε F ε q − (cid:19) z q (cid:21) F ± ,, (32)10nd we can reach the subsequent conclusion that (ii) the powers n of q n in all the components in the integrand in P ph ij (25) originatingfrom (32) must have the same parity.The final form of P ph ij = 2 P ph ij + , (25), results from the trivial integration with respect todirections of q and using the Debye integrals x Z d z z n sinh ( z/ ≡ J n ( x ) . After changing the integral variable z = ~ qv s /k B T , and taking q D = k B T D / ~ v s , one gets P ph11 = 2 P ph0 (cid:18) TT D (cid:19) J (cid:18) T D T (cid:19) ,P ph12 = ε s P ph11 ,P ph22 = 2 P ph0 ( k B T ) (cid:18) TT D (cid:19) (cid:20)(cid:18) π ε s ε F ( k B T ) (cid:19) J (cid:18) T D T (cid:19) − J (cid:18) T D T (cid:19)(cid:21) , (33)where , and ε s = 2 mv s is the energy of the electron of the wave vector q s = 2 mv s / ~ .The above result for P ph ij is the same as P ij derived in Ch. IX of Ref. [39]. It also corre-sponds to the result presented in Ref. [37], Eq.(18), after using the equivalence k B Tε F k B Tε s ≡ (cid:18) TT D (cid:19) n s , (34)taken for n s = (1 / / . The parameter n s is related to the electron gas density n a , n s =( n a / / , and for metals n s ≥ (1 / / .Notice that the n of J n ( u ) , occurring in all P ph ij (33), has the same parity according tothe conclusion (ii) above. B. Electron–phonon scattering contribution to the thermoelectric power
The electron–phonon contribution to the thermoelectric power S ph ( T ) results from sub-stitution in (4) P ph ij in place of P ij . Considering it as the function of the reduced temperature t = T /T D and applying the identity (34) we get after some algebra S ph ( t ) = − π k B e k B T D ε F t R ph ( t ) R ph ( t ) = π t − J t / (4 J ) + n s π t / − J t / (6 J ) + n s , (35)where J n ≡ J n (1 /t ) .Calculating S ph ( t ) (35), which depends on n s , the Fermi energy ε F and the Debye temper-ature T D , we fixed the last two parameters, similarly as in Ref. [36], assuming the same values11 ABLE I: The parameter n s = ( n a / / , related to the electron gas density n a , the sound velocity v s and the Debye radius q D , corresponding to values of ε s . ε s [K] n s v s [m/s] q D [nm − ]1.5 0.652 4770 5.493.0 1.30 6740 3.886.0 2.61 9540 2.7512.0 5.22 13500 1.9424.0 10.43 19100 1.3748.0 20.87 27000 0.97 ε F = 1 . eV and T D = 200 K, as were fixed therein. The values of n s corresponding to thevalues of ε s borrowed from Ref. [36], and the other material constants, are presented in TA-BLE I, providing the corrected (with respect to the values in the Table I ) correspondencebetween ε s , v s and q D .Irrespective of the numerical analysis of S ph ( t ) , which we present as the graphs in Fig. 1,some general conclusions can be obtained from the very analysis of the above formula. Thebehavior of S ph ( t ) , illustrated in Fig. 1, is determined by the properties of the function R ph ( t ) :(I) < R ph ( t ) < ;(II) lim t → R ph ( t ) = 1 , lim t →∞ R ph ( t ) = 3 ;(III) ∂R ph ( t, n s ) ∂n s < .(I)–(III) can be derived directly from the formula for R ph ( t ) (35) if one takes into accountthe metallic values of n s ( n s ≥ (1 / / ) and uses the estimation J (1 /t ) / J (1 /t ) < (1 /t ) for t > / / π and the approximation J (1 /t ) / J (1 /t ) ≃ ζ (7) / ζ (5) for t ≤ / / π .The last follows from J n ( x ≫ ≃ n ! ζ ( n ) – the approximation of the Debye integrals bythe zeta-Riemann function ζ ( n ) for x ≫ ( ζ (7) /ζ (5) ≃ ).With the use of (I)–(III) we get the conclusion that S ph ( t ) is bounded by the linearfunctions − π k B e k B T D ε F t < S ph ( t ) < − π k B e k B T D ε F t , (36)being its asymptotes, correspondingly for t → ∞ and for t → .The subsequent conclusion, which we get from (36) and (15) is that for every t > / < S mag ( t ) S ph ( t ) < , (37)where S mag ( t ) = − π k B T D t/ (3 ε F ) .The limitations of S ph ( t ) and its asymptotic behavior is seen from the graphs in Fig. 1.12 S ph ( t ) ,[ k B / e ] t=T/T D n s ph (t 0)S ph (t ∞ ) FIG. 1: The electron–phonon scattering contribution to the thermoelectric power S ph ( t ) (35) labeledby the parameter n s , TABLE I. Additionally, on the basis of (III) we can conclude that S ph ( t, n s ) (35) increases its valuesfor the increasing values of n s . This conclusion is also illustrated by the graphs in Fig. 1.Our results can be compared with those presented in Ref. [36], which depend on ε s (seeFig. 2 therein), if one takes into account the relation between n s and ε s , see TABLE I.Note at the end of this section, that after substituting P ph11 , P ph22 (33) and J , U (17) into(5), one gets the standard results for the electron–phonon part of the thermal W ph ( t ) andthe electrical ρ ph ( t ) resistivity, Refs [37–39]. W ph ( t ) = ρ ph ( t ) L T D t (cid:20) π n s t − π J (1 /t ) J (1 /t ) (cid:21) ,ρ ph ( t ) = ρ ph0 t J (1 /t ) , (38)where ρ ph0 = 3 πm C q D / (16 e N M v s k F ) .For t ≪ , because of J (1 /t ) = 5! ζ (5) , there is ρ ph ∼ ρ ph0 t , W ph ∼ ρ ph L T D π n s t . (39)13or t ≫ one can approximate J (1 /t ) ≃ / (4 t ) and hence ρ ph ∼ ρ ph0 t, W ph ∼ ρ ph0 L T D . (40) V. THE TOTAL THERMOELECTRIC POWER FROM THE ELECTRON––PHONON AND THE ELECTRON–LOCALIZED SPIN SCATTERING
In order to examine the dependence of the total thermoelectric power S ( t ) on the reducedtemperature t = T /T D we use Kohler rule (7). Some general conclusions about S ( t ) can bedrawn directly from this rule when we account the results of Sections III and IV.B. Writing(7) in the form S ( t ) = S ph ( t ) 1 + [ S mag ( t ) / S ph ( t )][ W mag ( t ) /W ph ( t )]1 + W mag ( t ) /W ph ( t ) , (41)and applying (36)–(37) one can easily find that S ph ( t ) < S ( t ) . Similarly, by the mutual ex-change of indices ‘ph’ and ‘mag’ in (41), it can be shown that S ( t ) < S mag ( t ) . Summarizing,we get for S ( t ) − π k B e k B T D ε F t < S ( t ) < − π k B e k B T D ε F t (42)the same limiting conditions, as these for S ph ( t ) , (36). Additionally, for t ≪ there is W mag /W ph ∼ t − exp[ − T C / ( T D ( J + 1)] (see (19), (39)) and for t ≫ correspondingly W mag /W ph ∼ γ J ( J + 1) /t , where γ = ρ mag0 /ρ ph0 (see (20) and (40)). Applying this in (41)and including (37), we obtain that S ( t ) shows the same asymptotic behavior in low and hightemperatures as S ph ( t ) .The properties of S ( t ) discussed above can be seen from the results of the numerical cal-culations presented as the graphs in Fig. 2. We have obtained them for the set of parametersused in Ref. [34] as corresponding to GdAl : T D = 289 K, T C = 180 K, k B T D /ε F = 0 . , γ = 0 . , n s = 4 . , J = 3 . (see Table 2 therein), and additionally for various values of n s , J and γ .The dependence of S ( t, p ) on the free parameter p = n s , γ, J can be examined in ananalytical way, similarly as for S ph ( t ) , after writing the formula (41) in the form S ( t ) = − π k B e k B T D ε F t R ( t ) R ( t ) = 3 + (3 /π ) n s /t − (3 / π ) J / J + ρ mag ( t ) /ρ ph ( t )1 + (3 /π ) n s /t − (1 / π ) J / J + ρ mag ( t ) /ρ ph ( t ) , (43)resulting from the substitution in (41) W α ( t ) (18), (38) and S α ( t ) (15), (35), ( α = mag, ph).It can be shown, by repeating the reasoning conducted previously in relation to R ph ( t, n s ) that ∂R ( t, p ) /∂p < for p = n s , γ and t > ( ρ mag ( t )) /ρ ph ( t ) is linear with respect to γ ). Thesimilar justification for ∂R ( t, J ) /∂J < can be performed under the condition t > T C /T D ,where ρ mag ( t ) (20) depends sufficiently simply on J .14 S ( t ) [ k B / e ] t=T/T D n s J γ ∞ ) FIG. 2: The total thermoelectric power S ( t ) (43) for different values of the parameters n s , J and γ = ρ mag0 /ρ ph0 and for the fixed T D = 289 K, T C = 180 K, k B T D /ε F = 0 . . The continuous, thickline corresponds to the data for GdAl according to Table 2 in Ref. (34). We can conclude that S ( t, p ) (43) is the increasing function of p = n s , γ, J in the appro-priate range of temperature (in the case of J for t > T C /T D ). This property is illustrated inFig.2. VI. SUMMARY AND CONCLUSIONS
We have calculated the contributions to the thermoelectric power of the f-electron metalsfrom the s–f scattering, S mag ( T ) , and from the electron–phonon scattering, S ph ( T ) , applyingthe same standard approach as the authors of Refs [34–36] (Born approximation for the scat-tering and Kohler (Ziman) variational formula for the thermoelectric power). For the totalthermoelectric power S ( T ) we have used Kohler rule, equivalently to using the variationalformula and the Matthiessen rule for the scattering probabilities in Ref. [34].In the case of the s–f scattering, basing on the symmetry of the scattering probability inthe molecular field approximation for the f-electron system, we have shown the equivalenceof the variational formula and the Mott formula (with the relaxation time dependent onthe electron energy as τ ∼ ε − / ). This gave us the linear dependence of S mag ( T ) with the15egative sign.For the electron–phonon scattering we performed the detailed calculation of the scatteringmatrix elements by an equivalent method to that applied in Ref. [36] and, unlike them,we have got the results, obtained previously by Kohler and Ziman. We have shown thatthese matrix elements satisfy some symmetry conditions, which we have formulated as theconclusions (i)–(ii) in Section IV.A.Analyzing the variational formula for S ph ( T ) with these scattering matrix elements, wewere able to show that S ph ( T ) has the constant and negative sign, being bounded by itshigh-temperature and low-temperature asymptotes. The same concerns S ( T ) , as we haveshown with the use of Kohler rule.From the results of our numerical calculations, presented in Fig. 1, the weak nonlinearityin the intermediate temperatures depends, for the constant Fermi energy and the Debyetemperature, on the electron gas density characterized by the parameter n s . Moreover, aswe could state analyzing the formula for S ph ( T ) , it is the increasing function of n s , what isalso seen from Fig. 1.The same nearly linear behavior in low and high temperatures and weakly nonlinear inthe intermediate temperatures is exhibited by the graphs of the total thermoelectric power S ( T ) in Fig. 2, obtained similarly as S ph ( T ) for the fixed values of the Fermi energy andthe Debye temperature, and, additionally, for the constant Curie temperature. The graphsin Fig. 2 illustrate also the property of S ( T ) , which we have derived analytically, that it isthe increasing function of the parameters: n s , the quantum number J and the parameter γ , characterizing the relative contribution of the s–f scattering. For the rising values of thelast two parameters S ( T ) , as it is seen from Fig. 2, approaches S mag ( T ) .Analyzing the way of calculation of the electron–phonon scattering matrix elements inRef. [36], we have found that some terms were omitted in the integrands, and also the contri-bution to the scattering from the emission processes was completely omitted. In consequenceof these inaccuracies, the scattering matrix elements derived in Ref. [36] do not fulfill thesymmetry conditions which should be fulfilled. This can be seen by comparing (B2) and(B3) with (i)–(ii) in Section IV.A.The same concerns the scattering matrix elements presented in Refs [34,35], as we couldstate performing calculations by the way applied there – the slightly different one than thatapplied in Ref. [36]. We can conclude that the sign reversal and the maxima of S ph ( T ) obtained in Refs [34–36] and these of S ( T ) in Ref. [34] have their origin in the flaws in thescattering matrix elements calculations, which we have described above. The direct meaningof this conclusion is that the interpretation of the anomalous behavior of TEP can not bebased on these simple models which have been applied in Refs [34–36]. However, it alsorestores the question about an adequate models for explanation of the thermoelectric poweranomalies in the metallic f-electron systems. This concerns particularly the anomalies inlow temperatures which are not explained by dynamically developing theories of stronglycorrelated electron systems. In the Introduction we gave only a partial answer to thisquestion. We have payed attention there on the models described in Refs [28,30], which can16xplain the anomalies in paramagnetic systems with crystal-field (CF) splitting or in theferromagnets - not taking this splitting into account.There are examples of the f-electron systems with CF splitting, the thermoelectric powerof which exhibits the anomalies in the magnetically ordered phase, see Ref [9–12] or examplesin Ref [42]. Finding the appropriate models is therefore an important issue for furtherresearch. In particular, an interesting problem could be the extension of considerations ofRefs [28,30] for the cases of the anomaly in the ordered phase of the f-electron systems withthe CF splitting. Appendix A
Integration in F ± n ( z q , T ) (27) with respect to the angle Θ = ∠ ( k , q ) can be done by theintegration with respect to k in the spherical coordinate system, where q is parallel to thepolar axis. With the use of the delta-Dirac function properties and changing the variable y = ( ε k − ζ ( T )) /k B T in the integration with respect to k we have F n ( q, z q ) ± = ( k B T ) n +1 m ~ q ± ∞ Z Ξ ( q, ∓ z q ,T ) /k B T d y y n (exp[ ± z q /k B T ] − − (exp[ y ] + 1)(1 + exp[ − ( y ± z q /k B T )]) , (A1)where Ξ ( q, ∓ z q , T ) = ( ~ / m ) | z q m/ ~ q ∓ q/ | − ζ ( T ) .In the approximation of the strong degeneration of the electron gas and for the experi-mentally accessible temperature range there is − ζ ( T ) k B T ≃ − ε F k B T ≃ −∞ , and Ξ ( q, ∓ z q , T ) can be considered as temperature-independent for that temperature range.Thus, for q fulfilling the inequality Ξ ( q, ∓ z q ) < ( z q = ~ q v s ) and the equivalent one (cid:12)(cid:12)(cid:12) q ∓ q s (cid:12)(cid:12)(cid:12) − k F < , (A2)where q s = mv s / ~ , the lower limit of the integral can be replaced by −∞ , for the sufficientlylow T . Comparing the typical for metals value of the sound velocity, v s = ~ q s /m < · m/s,with the value of the Fermi velocity, v F = ~ k F /m ≃ · m/s, one gets q s /k F ≃ − .Because for metals there is q D ≤ / k F the range ÷ q D can be accepted as the solution of theinequality (25), both for the phonon absorption and emission. It justifies the approximation Ξ ( q, ∓ z q , T ) /k B T ≃ −∞ in (A1) which now can be represented in the form (28)–(29).The integrals I ± n ( z q ) (29) can be calculated with the aid of the formula ∞ Z −∞ d y F ( y )(exp[ y ] + 1)(1 + exp[ − ( y + z )]) = ∞ Z −∞ d y (cid:20) G ( y ) − G ( y − z )(1 − exp[ − z ]) (cid:21) (cid:18) − ∂f ( y ) ∂y (cid:19) , G ( y ) = y Z d y ′ F ( y ′ ) , (A3)17nd next with the use of the Sommerfeld expansion ∞ Z −∞ d y (cid:18) − ∂f ( y ) ∂y (cid:19) H ( y ) = H (0) + π ∂ H ( y ) ∂y (cid:12)(cid:12)(cid:12)(cid:12) y = 0 + · · · . (A4)When F ( y ) in the integrand (A3) has the form of polynomial, like in I ± n ( z q ) , the firstterm of the above expansion is the exact value of the integral, what gives I ± ( z q ) = 4 z q /k B T sinh [ z q / k B T ] − ,I ± ( z q ) = ± z q I ± ( z q )2 k B T ,I ± ( z q ) = (cid:20) π z q k B T ) (cid:21) I ± ( z q ) . (A5) Appendix B
Comparing u ij ( k , k + q ) (23) with the equation (3.8) of Ref. 36, one can see that thecomponent ( q · u )( k · u ) has been omitted there. Since the authors of Ref. 36 had usedthis incomplete form f u ij ( k , k + q ) as non-averaged over field directions, they had to takeinto account the space directions of the vectors q and k when calculating the integralsin the scattering matrix elements. This made the integration more complex than in ourcalculations presented in Section IV.A. Despite this, our method of calculation, althoughsimpler, is equivalent to theirs.Performing on f u ij the same transformations which led from (23) to (24) one gets f u ± = 13 q , f u ± = 13 q ( η ± z q ) , f u ± = 13 q (cid:20) η ± z q η + z q + z q ηε q + z q ζ ( T ) ε q (cid:21) . (B1)Repeating with (B1) in place of (24) all the calculations made in Section IV.A andAppendix A one obtains the form of (32) which we denote as g H ± ij ( q, z q ) g H ± ( q, z q ) = F ± , g H ± ( q, z q ) = ± z q F ± g H ± ( q, z q ) = (cid:20) π k B T ) + (cid:18) ε F ε q + 13 (cid:19) z q ∓ z q ε q (cid:21) F ± . (B2)Substitution (B2) in (25), in place of (26), leads to the non-equivalence of the contri-butions to the scattering matrix elements from the absorption and the emission processes18 P ph + ij = ] P ph − ij . However the authors of Ref. ( 36) have omitted the emission processes intheir calculations. It can be easily verified by the substitution g H + ij ( q, z q ) (B2) in P ph+ ij (25)in place of H + ij ( q, z q ) . Performing next all the transformations described in Section IV.A,which led to the final form of P ph ij (33) one gets the scattering matrix elements, the averaged P DA ij = P DA ij / ( P ph0 ( k B T ) i + j − t ) form of which P DA = J (1 /t ) ,P DA = 12 J (1 /t ) ,P DA = (cid:20)(cid:18) π ε s ε F ( k B T ) (cid:19) J (1 /t ) − ε s k B T J (1 /t ) + 13 J (1 /t ) (cid:21) , (B3)is the same as those in the equations (4.2a)–(4.2c) in Ref.( 36), corrected by the Erratum . S.Paschen, Thermoelectric aspects of strongly correlated electron systems, in:
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