Physics Letters A 303 (2002) 223–228

www.elsevier.com/locate/pla

Modeling of the thermopower of electron-doped manganites

Raymond Frésarda,∗, Sylvie Héberta, Antoine Maignana, Li Pi a, Jiri Hejtmanekb

a Laboratoire Crismat, UMR 6508 du Centre National de la Recherche Scientifique et de l’Institut des Sciences de la Matière etdu Rayonnement, 6, Bld. du Maréchal Juin, 14050 Caen Cedex, France

b Institute of Physics, Cukrovarnicka 10, 16200 Prague 6, Czech Republic

Received 19 August 2002; accepted 4 September 2002

Communicated by J. Flouquet

Abstract

The temperature dependence of the thermopower of electron-doped manganites is analyzed in terms of a microscopicHamiltonian. We show that the experimental data can be understood for a series of materials over a broad temperature rangeusing two parameters, namely the effective bandwidth for the quasiparticles, and the high temperature limit of the appropriatecorrelation function. These parameters are material-specific, and vary modestly over the series of considered materials. 2002 Elsevier Science B.V. All rights reserved.

PACS: 72.20.Pa; 71.27.+a; 71.30.+h; 74.62.Dh

1. Introduction

Materials with thermoelectrical applications keep on receiving a lot of attention. New developments in this areaare numerous, primarily following from the discovery of new skutterudites and oxides. In particular spectaculardevelopments have been achieved thanks to the discovery of large thermopower values in layered cobaltites [1] andlow electron-doped titanates [2] and manganites [3].

When it comes to interpret experimental data of thermopower out of a microscopic Hamiltonian, severe difficultytakes place, in particular for the above systems. Indeed the rather involved band structure, together with the strongcorrelation effects following from the large Coulomb and electron–phonon interactions look pretty discouragingwhen envisaging evaluating the correlation functions entering the expression for the thermopower (S). Neverthelessour understanding of the temperature dependence ofS improved in recent years, in particular in the context of thecuprates [4,5]. In the latter work it was shown for a large domain of density and temperature, that the ratio ofthe energy current–particle current correlation function to the particle current autocorrelation function, is mostlytemperature independent, implying that theT -dependence ofS results from the one of the chemical potential.More recently, Oudovenko and Kotliar evaluated this ratio in the high temperature regime for a two-band Hubbard

* Corresponding author.E-mail address: raymond.fresard@ismra.fr (R. Frésard).

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)01225-2

224 R. Frésard et al. / Physics Letters A 303 (2002) 223–228

model [6]. This gives us confidence that theT -dependence ofS for lightly electron-doped manganites can beanalyzed in terms of a simple model. Such an attempt is done here.

2. Model for the temperature dependence of the thermopower

We now attempt at interpreting the temperature dependence of the Seebeck coefficient data for lightly electron-doped manganites in simple terms. We resort to a microscopic effective Hamiltonian for oxides which takes intoaccount the bands which are crossing the Fermi level. For the manganites they result from the hybridization of thetwo eg levels on the manganese ions with the oxygen 2p levels. The bands resulting from the manganeset2g levelsare half-filled and are therefore not expected to play an important role, since the Hund’s rule coupling is strong inthese materials. They are left out in the following. At this stage the kinetic energy part of the HamiltonianT isgiven by:

(1)T =∑k,σ

Ψ†k,σ

εd 0 2itzx sin kxa2 2itzy sin kya2 2itzz sin kza2

0 εd 2itxx sin kxa2 2itxy sin kya2 0

−2itzx sin kxa2 −2itxx sin kxa2 εp 0 0

−2itzy sin kya2 −2itxy sin kya2 0 εp 0

−2itzz sin kza2 0 0 0 εp

Ψk,σ ,

with Ψ †k,σ = (d†

z,k,σ , d†x,k,σ ,p

†x,k,σ ,p

†y,k,σ ,p

†z,k,σ ). It is obtained by Fourier transformation ofd†

z,i,σ (d†x,i,σ ) which

is the creation operator for an electron in the 3z2 − r2 (x2 − y2) orbital on a manganese atom at sitei, andp

†α,i,σ which is the creation operator for an electron in theα (α = x, y, z) 2p orbital on a neighboring oxygen

atom pointing towards the corresponding Mn atom. In a cubic environment, the overlap integralstβα are given in

term of one single Slater and Koster parameter [7]pdσ = tzz astzx = tzy = 12tzz , andtxx = −txy =

√3

2 tzz . Once these

relationships are satisfied, cubic symmetry is obeyed. Typical values forpdσ for transition metal oxide-basedperovskites can be found in Ref. [8]. According to their calculation they are in the range of 1 eV, while the energydifference between the manganeseeg- and oxygen 2p-levels is about 2 eV [8]. Once it is recognized that oneof the five eigenvalues of the kinetic energy operator is given by∆ ≡ εp − εd , one can easily get the other foureigenvalues. They yield two sets of two bands, one dispersing aboveεd , and the other one belowεp . None of thebands disperse over the whole Brillouin zone in a way similar to what would be obtained from an effective simplecubic model. When considering the upper two bands, one disperses ask2 at the bottom of the band, while the otherone does not disperse in theΓX directions. The same behavior is found at the top of the band by symmetry, withboth bands interchanged. For each of the resulting bands, the bandwidth is given by:

(2)W = 1

2

(√∆2 + 24tz2

z −∆),

even though the bands are non-degenerate. Please note that the two parameters of the model,tzz and∆, enter theexpression for the bandwidth. In turn the density of states also depends on both parameters. It is seen in Fig. 1that its energy dependence is much stronger for large ratio oftzz /∆ than for smaller ones. Regardless of this ratiothe density of states remains finite at both band edges. This results from the particular dispersion relations foreg electrons. As a result, the chemical potential strongly depends on temperature, as shown in the inset of Fig. 1for several values oftzz . Since the density of states at the bottom of the band decreases with increasingtzz , thetemperature dependence ofµ is enhanced by increasingtzz .

On top of the above discussed kinetic energy term, one expects strong interaction terms in the Hamiltonianof the system. They primarily consist of Hubbard and Hund’s rule interactions between theeg and t2g electrons

R. Frésard et al. / Physics Letters A 303 (2002) 223–228 225

Fig. 1. Density of states for theeg electrons, for∆= 2 eV, tzz = 0.2 eV (full line) andtzz = 1 eV (dashed line). Inset: temperature dependenceof the chemical potential for 6% of electrons,∆= 2 eV, tzz = 0.5 eV (full line), tzz = 0.33 eV (dotted line),tzz = 0.25 eV (dashed line), andtzz = 0.2 eV (dashed-dotted-dotted line).

on the manganese ions. The additional interaction terms responsible for the Jahn–Teller effect, which are playingan important role in the Mn3+-rich side, are not expected to play a decisive role in the Mn4+-rich side. Theirinfluence is discussed below. The Hubbard and Hund’s rule interactions could be taken care of in the framework ofGutzwiller approximation [9] or in slave-boson mean-field approximation [10,11]. Nevertheless this would implydealing with a five band model, obviously a very involved task. Still, the expected output of such a calculation is,at the first place, that the one-particle bands split off into quasiparticle bands and lower and upper Hubbard bands.The quasiparticle bands are dispersing in a way that is similar to the original bands, but with a renormalizedtzz .Instead of determining it from a detailed calculation, which would require introducing additional parameters suchasU andJH , we rather determine the renormalization oftzz out of the experimental data.

In order to evaluate the Seebeck coefficient, we use the standard expression [12]

(3)S = 1

eT

L12

L11

whereL11 is the particle current autocorrelation function, andL12 the heat current–particle current correlationfunction. Writing the heat current operator asjQ = jE −µj , Eq. (3) can be cast into two terms as

(4)S = kB

|e|β(µ−E0)

whereE0 represents the contribution following from the energy current operatorjE . Our analysis assumes that thetemperature dependence ofE0 can be neglected. Thus an expression for it can be derived from a particular modelusing, e.g., a high temperature expansion. This has been done in particular for a degenerate two-band Hubbardmodel [6]. Here we keepE0 as a parameter accounting for the combined result of disorder and correlation effects.In this way one may gain some insight in the origin of particular structures in theT -dependence ofS. ObviouslyE0should go toµ (T = 0) in the lowT limit. This cannot happen within the above assumption, since we determineE0from its highT limit. This implicitly sets a low temperature bound to the validity of our approach.

3. Numerical application

We now turn to the analysis of experimental data. They consists of both the A site-, and B site-, substitutedmanganite CaMnO3, in the low doping regime. In this regime, the Jahn–Teller effect, which is primarily connected

226 R. Frésard et al. / Physics Letters A 303 (2002) 223–228

Fig. 2. Thermopower of CaMn0.94Ru0.06O3 (circles), CaMn0.94Re0.06O3 (triangles), and CaMn0.96Mo0.04O3 (diamonds). The original dataare in µV/K. Here the thermopower for the Ru-substituted sample is divided by three, while 25 µV/K are subtracted from the ones for theRe-substituted ones, for clarity. Theoretical calculation (rescaled as above) for (tzz = 0.42 eV andE0 = −16 meV) (full line), (tzz = 0.58 eVandE0 = −27 meV) (dashed line), and (tzz = 0.60 eV andE0 = −23 meV) (dashed-dotted line).

to Mn3+ ions, is not expected to play any dominant role. Let us start with the B site-substituted manganites. Hereone brings in charge carriers in this band insulator by substituting Mn4+ ions by either 6+ ions, such as Mo [13], or5+ ions such as Ru, and Re [14]. We apply our analysis to CaMn0.94Ru0.06O3, CaMn0.94Re0.06O3 (correspondingto a density of 6% of electrons in theeg bands), and CaMn0.96Mo0.04O3 (corresponding to a density of 8% ofelectrons in theeg bands). The result of this analysis is shown in Fig. 2. Manifestly the calculatedS reproducesthe experimental data over a wideT -range. This range extends from room temperature down to 100 K, or evenbelow. Rather remarkably there is a fairly small scattering of the value oftzz needed to describe this set of data.It ranges from 0.42 eV for Mo, to 0.60 eV for Re, thus a factor two smaller than the predictions of the bandstructure calculation [8]. The theoretical curves are not equally sensitive to all parameters. In thisT -range,∆ hasvery little influence onS, once large enough, namely as long as the lower bands are not thermally excited. On thecontrary, the slope ofS at room temperature is very sensitive to the value oftzz , and a 2% change of the latter quiteseverely affects the quality of the agreement. Therefore one can make use of theS data to accurately determinethe bandwidth of a particular compound. This also represents a clear signature ofeg electrons. Indeed, carryingout the same analysis using free electron dispersion (or equivalently, for this purpose,t2g electrons) would yield avery different result since in this case the chemical potential has a far weaker temperature dependence. Generically,the thermopower has a fairly linear temperature dependence around room temperature. Here we see that even thedownturn is reproduced by our calculation as well. It is therefore not related to any particular physical process, butrather to the temperature at whichE0 becomes the dominant contribution toS. Here we see that theT -dependenceof E0 becomes relevant forT < Tlow, with Tlow = 80 K for the Ru-substituted sample, 90 K (Re), and 110 K (Mo).ThusTlow increases with decreasing values of|E0|. Remarkably, the magnetic transition taking place at 110 K inthe Ru-substituted sample only has a small influence onS, since only a small departure from the calculated curvecan be observed below 110 K. On the contrary,Tlow corresponds to the magnetic transition temperatures in Re andMo.

We now turn to A site-substituted manganites, taking as examples SmxCa1−xMnO3, with x = 0.05 andx = 0.10. The results are presented in Fig. 3. Forx = 0.05, the numerical result and the experimental dataagree above the Curie temperature of 110 K, where an abrupt change in theT -dependence ofS takes place.For x = 0.10, the curves start to depart smoothly at 120 K, reflecting the contribution of the large ferromagneticfluctuations toE0. We note that the experimental data forx = 0.05 correspond totzz = 0.45 eV, and the ones forx = 0.10 totzz = 0.51 eV. Therefore increasing the charge carrier density corresponds to an increase of the effectivebandwidth. This appears to be in contradiction to what one expects when modeling the series SmxCa1−xMnO3 with

R. Frésard et al. / Physics Letters A 303 (2002) 223–228 227

Fig. 3. Thermopower of Sm0.05Ca0.95MnO3 (diamonds), and Sm0.10Ca0.90MnO3 (circles) compared to Eq. (4) with (tzz = 0.496 eV,E0 = −22 meV andρ = 0.1) (full line), tzz = 0.45 eV,E0 = −10 meV, andρ = 0.05) (dashed line).

an extended Hubbard model with fixed parameterstβα , U andJH , where an increase in the charge carrier density

leads to a decrease of the quasiparticle bandwidth. However, one should keep in mind that increasing the Sm contentalso induces an increase of the Mn–O–Mn angle, and therefore of the relevant overlap integrals. This effects seemsto overcompensate the reduction of the quasiparticle bandwidth due to the electron–electron interaction.

The values ofE0 in the above analysis are ranging from 10 to 25 meV. They appear to be too small to beexplained with the result of Oudovenko and Kotliar [6]. Indeed, usingU � 3 eV, and (in their notation)n � 4%,one obtainsE0 � 120 meV. The difference, together with the scattering of the results, should be attributed todisorder effects, which may hardly be ignored in these ceramic materials.

4. Summary

In summary we have presented a model calculation of the temperature dependence of the thermopower for bothA site-, and B site-, substituted CaMnO3, in the low electron-doped regime. Based on a microscopic Hamiltonian,this calculation proved to accurately reproduce a series of experimental data over a broad temperature range. Theparameters entering the calculation are seen to be modestly varying over this series of materials. Calculations forother systems involvingt2g electrons are in progress.

Acknowledgements

We are grateful to Ch. Simon, Ch. Martin, and P. Prelovšek for helpful discussions.

References

[1] I. Terasaki, Y. Sasago, K. Uchinokura, Phys. Rev. B 56 (1997) R12 685.[2] T. Okuda, K. Nakanishi, S. Miyasaka, Y. Tokura, Phys. Rev. B 63 (2001) 113104.[3] A. Maignan, C. Martin, F. Damay, B. Raveau, J. Hejtmanek, Phys. Rev. B 58 (1998) 2758.[4] G. Hildebrand, T.J. Hagenaars, W. Hanke, S. Grabowski, J. Schmalian, Phys. Rev. B 56 (1997) R4317.[5] J. Jaklic, P. Prelovšek, Adv. Phys. 49 (2000) 1.[6] V.S. Oudovenko, G. Kotliar, Phys. Rev. B 65 (2002) 075102.

228 R. Frésard et al. / Physics Letters A 303 (2002) 223–228

[7] J.C. Slater, G.F. Koster, Phys. Rev. 94 (1954) 1498.[8] P. Mahadevan, N. Santhi, D.D. Sarma, Phys. Rev. B 54 (1996) 11 199.[9] J. Bünemann, W. Weber, F. Gebhard, Phys. Rev. B 57 (1998) 6896.

[10] H. Hasegawa, J. Phys. Soc. Jpn. 66 (1997) 1391.[11] R. Frésard, G. Kotliar, Phys. Rev. B 56 (1997) 12 909.[12] G.D. Mahan, Many Particle Physics, 3rd Edition, Kluwer Academic/Plenum Publishers, New York, 2000.[13] L. Pi, et al., in preparation.[14] B. Raveau, A. Maignan, C. Martin, M. Hervieu, Mater. Res. Bull. 35 (2000) 1579.