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Analysis of thermal conductivity data in heavy fermion compounds
Solid State Communications 135 (2005) 711–715 www.elsevier.com/locate/ssc
Analysis of thermal conductivity data in heavy fermion compounds Z. Kletowskia, B. Coqblinb,* a
W. Trzebiatowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, 50-950 Wrocław, Poland b Laboratoire de Physique des Solides (UMR8502), Baˆtiment 510, Universite´ Paris-Sud, 91405 Orsay, France Received 22 April 2005; received in revised form 2 June 2005; accepted 3 June 2005 by C. Lacroix Available online 16 June 2005
Abstract We present a revisiting analysis of thermal conductivity data in heavy fermion compounds. It is shown that the product WmagT, where Wmag is the magnetic part of the thermal resistivity taken as the difference between the inverse of the thermal conductivity of the Kondo compound and that of an equivalent non magnetic compound, is a good measurement of the strength of the Kondo interaction at sufficiently high temperatures with respect to the Kondo temperature and to the crystal-field splitting. In the considered temperature range, WmagT shows a log T dependence in a way similar to the magnetic part of the electrical resistivity. Such an analysis, which is analogous to the well known one for the electrical resistivity, has not been often used for the thermal conductivity. We present data for compounds with four anomalous Kondo rare earths Ce, Pr, Tm and Yb. q 2005 Elsevier Ltd. All rights reserved. PACS: 71.27.Ca; 72.15.Qm; 72.15.Kv Keywords: D. Heavy fermions; D. Kondo effects; D. Thermal conductivity
Many alloys and compounds containing cerium, ytterbium or other anomalous rare-earths exhibit a Kondo effect, characterized by a log T behavior of the electrical resistivity at high temperatures with respect to the Kondo temperature (TK) and by a T2 behavior of the electrical resistivity which is observed only at very low temperatures below TK. The magnetic and transport properties of many cerium or ytterbium compounds were accounted for by a model which takes into account both the crystalline field effect and the Kondo effect [1]. Typically, the magnetic resistivity of cerium compounds such as CeAl2 or CeAl3 presents a log T behavior at high temperatures, then undergoes a broad maximum at a temperature of order the overall crystal field splitting and finally decreases at low temperatures to either a magnetically ordered state or a nonmagnetic state exhibiting a heavy-fermion behavior.
* Corresponding author. Tel.: C33 1 69 15 60 94; fax: C33 1 69 15 60 86. E-mail address: [email protected] (B. Coqblin).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.06.005
Similarly the thermopower of such systems presents a broad maximum corresponding to a fraction of the crystal field spitting and eventually may exhibit a second maximum at lower temperatures corresponding roughly to the Kondo temperature [2,3]. On the other hand, the thermal conductivity has been also computed within the Coqblin–Schrieffer model including crystal-field effects [4], but in contrast to the electrical resistivity investigations, there are not many studies analyzing properties of the thermal conductivity in heavy fermion systems. Thus, the purpose of the present paper is to revisit the question of the thermal conductivity and to present an analysis of available experimental data in these systems. This analysis follows our recent results obtained for the thermal conductivity measurements in PrSn3 [5,6] and TmGa3 [7]. The analysis of the thermal conductivity measurements is rather difficult because there are several contributions, which enter either the thermal conductivity, if they were coming from different carriers, or the ‘thermal resistivity’ for a given type of carriers scattered by different scattering
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Z. Kletowski, B. Coqblin / Solid State Communications 135 (2005) 711–715
mechanisms. In fact, the phonon thermal conductivity cannot be in general neglected, except only in the case when the electronic contribution is much more important. In our present analysis, we make the simplifying assumption of neglecting the phonon contribution, which seems to be a reasonable assumption in the case of strong Kondo systems. Thus, we consider here only the electronic thermal conductivity and then, assuming that all the scattering mechanisms responsible for the electronic thermal resistivity are additive, according to the Matthiessen rule, the thermal conductivity k can be expressed as follows: kK1 Z W Z Wmag C We;ph
(1)
where the two preceeding terms correspond to the so-called ‘magnetic’ term arising from collisions of the conduction electrons by Kondo impurities, Wmag and to the classical term describing scattering by phonons, We,ph. For the discussion of the thermal conductivity influenced by a strong Kondo effect, we make the same approximation as in the case of the electrical resistivity: we assume that the term We,ph in the Eq. (1) is equal to the thermal resistivity of an equivalent non magnetic compound. Such a situation occurs when the resulting thermal resistivity is much larger than that of the equivalent non magnetic compound. For example, it is the case of the thermal conductivities of PrSn3 and LaSn3, where the ‘magnetic’ contribution, Wmag, can be defined as the difference between the thermal resistivity of PrSn3 and that of LaSn3. More details on the separation procedure of the Wmag can be found in Refs. [5,6]. We can say finally that our present analysis, which is based on the same assumptions than the previous electrical resistivity one [1], is in fact more difficult to justify, which is the main explanation why the present analysis is much less obtained than the corresponding electrical resistivity analysis of Ref. [1]. Before discussing in detail experimental data, let us present now the theoretical analysis, which is performed for both the electrical resistivity [1] and the thermal conductivity [4] in the so-called ‘high temperature’ case, i.e. above the Kondo temperature TK. In this temperature regime, these two quantities are described by the well-known third-order perturbation treatment of the Coqblin–Schrieffer Hamiltonian including crystal-field effects. These theoretical calculations show that the product WmagT and the magnetic electrical resistivity rm exhibit the same temperature behaviour, i.e. they show a maximum at a temperature of order the crystalline-field splitting D and then they behave as log T in temperatures below and above D. Slopes of these two log T behaviours are dependent on the degeneracy a of the occupied levels, i.e. the ground state degeneracy for temperatures below D and the total degeneracy aZ2jC1 above D. The third-order parts of rm and WmagT yielding a log T behavior are given by: 2 a K1 3 J nðEF Þlog T rm Z 2R a
(2)
Wmag T Z 2
R L0
a2 K 1 3 J nðEF Þlog T a
(3)
where the constant R is given by: RZ
3m2 pn0 c e2 Z3 kF2
(4)
and L0Z(pkB)2/3e2 is the Sommerfeld value of the Lorenz number. Eqs. (2) and (3) allow us to calculate the exchange integral J, using values of the slopes for the log Tdependences for rm and WmagT obtained from experiment. The determination of the two calculated J values is clearly a good measurement of the Kondo problem and also a good way to see if the thermal conductivity measurements are a useful tool in investigating Kondo systems. Before the calculated J values will be discussed, we describe now available experimental data for temperature dependences of WmagT and rm in Kondo systems. We present in Fig. 1 the different experimental curves giving WmagT in a logarithmic plot for 8 different heavy fermion compounds, in order to see the log T behaviours at sufficiently high temperatures and the available shapes of the curves at lower temperatures. We have investigated thermal conductivity of monocrystalline samples of PrSn3 and its nonmagnetic reference compound LaSn3. PrSn3 is an antiferromagnetic system with the Neel temperature of 8.6 K. Previously the Kondo behavior was observed in the electrical resistivity and magnetoresistance of this compound [8,9] as well as in the Pr1KxLaxSn3 alloys [10]. In our experiment it was found that, for PrSn3, the quantity WmagT decreases as log T in a wide temperature range from 15 to 200 K [5,6]. Simultaneously, the electrical resistivity shows a wide range of the log T decrease in rm from 10 to 100 K [5,8]. In PrSn3, the Pr ions possess a 4f2 configuration and the overall crystal field splitting D is equal to 23 K only. Since D, TK and TN are very small, one observes the log T-dependence for both rm and WmagT for temperatures down to 10 K. Slopes of the log T dependences found for rm and WmagT are given in Table 1. Recently, the thermal conductivity of TmGa3 and LuGa3 single crystals has been measured. We observe a clear Kondo behaviour in the thermal conductivity of TmGa3, similarly as we previously found such a behaviour in the case of TmIn3 [11]. TmGa3 undergoes two first-order transitions very close to each other at TNZ4.26 K and TQZ 4.29 K, corresponding to antiferromagnetic and quadrupolar orderings [12]. The magnetic resistivity rm of TmGa3, taken as the difference between the resistivities of TmGa3 and LuGa3, exhibits a clear log T behavior between 50 and 250 K [13]. We have also found that, for TmGa3, the log T dependence of the WmagT factor exists in the range 20– 180 K [7], i.e. in a temperature range wider than that of the electrical resistivity. The compound TmIn3, which orders antiferromagnetically at 1.68 K presents a log T dependence
Z. Kletowski, B. Coqblin / Solid State Communications 135 (2005) 711–715
713
Fig. 1. Plot of WmagT (in mK2/W) versus log T for different heavy fermion compounds. The straight lines show the temperature ranges in which the linear dependence of WmagT versus log T occurs. Plots for the particular compounds are depicted by numbers: 1 for CeAl2, 2 for CeCu2, 3 for CeCu5, 4 for CeCu4Al, 5 for PrSn3, 6 for TmGa3, 7 for TmIn3 and 8 for YbCu4Ag.
for WmagT in the temperature range 15–55 K [11] and for rm in temperature range 30–70 K [14]. Although Kondo effect has been found in a number of compounds till now, data for the magnetic part of the thermal conductivity have been reported only in a limited number of Kondo systems. The thermal conductivity was measured in several cerium Kondo compounds. Measurements in CeCu2 and the reference compound YCu2 have
shown that WmagT and rm exhibit clearly two log T behaviors separated by a maximum at roughly 150 K [15]. A similar behavior was also observed in CeAl2 [16], where both WmagT and rm show log T dependences separated by a maximum around 80 K. For our analysis we use data of WmagT and rm taken from temperature ranges 180–300 and 150–300 K for CeCu2 and CeAl2, respectively. The thermal conductivity has been also measured in Ce(CuxAl1Kx)5
Table 1 The first two columns give the experimental values of the slopes versus log T of WmagT and rm Compound
Slopes versus log T of 2
CeAl2 CeCu2 CeCu5 CeCu4Al PrSn3 TmGa3 TmIn3 YbCu4Ag
WmagT (mK /W)
rm (mU cm)
8 [16] 11.5 [15] 3.3 [17,18] 8.6 [17,18] 2 [5,6] 0.9 [7] 0.6 [11] 13 [21]
17 [16] 25.5 [15] 1.9 [17,18] 12 [17,18,] 6 [5,6] 0.8 [13] 4.3 [14] 18 [22]
n (EF) (states/eV at.)
jJj calculated from WmagT (eV)
rm (eV)
2.3 2.3 2.3 2.3 2.3 0.6 1.6 1.8
0.092 0.103 0.085 0.118 0.055 0.143 0.046 0.156
0.087 0.100 0.053 0.098 0.059 0.102 0.066 0.130
The third column gives the density of states n (EF). The two last columns give the values of the exchange integrals J deduced from the calculations of WmagT and rm.
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Kondo compounds and we report values for WmagT found for CeCu5 and CeCu4Al [17–19]. The corresponding magnetic resistivities rm were also measured and we found that both WmagT and rm decrease as log T in a temperature range of 200–300 K for CeCu5 and a wider temperature range of 30–300 K for CeCu4Al. Deduced values of the log T slopes for WmagT and rm found for temperatures above the crystal field splitting are reported in Table 1. YbCu4Ag is another Kondo compound with a large electronic specific heat and its Kondo properties have been well fitted by the JZ7/2 Coqblin–Schrieffer model [20]. For this compound one observes also a decrease in WmagT and rm in log T from roughly 150 to 300 K [21,22]. Experimental curves of WmagT are plotted in Fig. 1 in a logarithmic plot and we see that the curves corresponding to three compounds with cerium and YbCu4Ag present a maximum, followed, with decreasing temperature, by a decrease and finally a small increase at low temperatures. Moreover, the values of the log T-slopes for WmagT and rm in the high temperature regime, i.e. for temperatures higher than D, TK and eventually TN, are given in Table 1. The deduced values of these slopes given in Table 1 correspond to a change of WmagT and rm in a temperature interval of 100 K. Then, we compute the corresponding values of J by use of formulae (2) and (3). For this computation, we use kFZ0.6 u.a., v0Z177 u.a. derived from the atomic volume, the density of states n(EF) derived from the experimental values of the electronic specific heat g of the equivalent non magnetic compound and the effective mass m* proportional to n(EF). Thus, we take n(EF)Z2.3 states/eV at. for one spin direction and m*Z3 in the case of the four cerium compounds and PrSn3, n(EF)Z1.8 states/eV at. for one spin direction and m*Z2.35 for YbCu4Ag, n(EF)Z 2.1 states/eV at. for one spin direction and m*Z2.7 for TmGa3, n(EF)Z1.6 states/eV at. for one spin direction and m*Z2.1 for TmIn3. We take also the degeneracy aZ2jC1 for each rare earth, equal to aZ6, 9, 13 and 8 for, respectively, Ce, Pr, Tm and Yb ions and c the concentration of rare-earth atoms. Finally, experimental values of the log T-slopes of the electrical resistivity and the thermal conductivity yield, respectively, two values for the exchange integral J which are given in the two last columns of Table 1. An extension of the data in the very low temperature regime for the compounds presenting a maximum at high temperatures would be very instructive in order to check the relationship directly connected with the degeneracy a of the occupied levels between the low and high temperature slopes of WmagT. To summarize this rapid analysis of the thermal conductivity results for a few Kondo compounds, we can conclude that we have found a log T decrease in the quantity WmagT in several compounds with cerium, ytterbium, praseodymium and thulium. A log T decrease in the magnetic resistivity rm has been also observed in these compounds. Two points are very interesting to be noticed: first, a Kondo behavior has been observed in the thermal
conductivity as well as in the electrical resistivity and second the Kondo effect has been observed not only in cerium and ytterbium compounds, but also in praseodymium and thulium compounds, which is obviously relatively rare. Exchange integrals J were calculated using the thirdorder perturbation method performed within the Coqblin– Schrieffer Hamiltonian including crystal-field effects in the case of thermal conductivity and electrical resistivity data. Although the J values vary from a compound to another one, the two deduced values of J for a given compound are reasonable and very close to each other. This means clearly that our analysis of the thermal conductivity data in terms of the Kondo effect is correct and that our theoretical analysis of the thermal conductivity gives evidence for a Kondo effect in compounds with Ce, Yb, Pr and Tm. Our results indicate that a precise measurement of the thermal conductivity is a useful tool for studying the Kondo effect in anomalous rare-earth systems at high and certainly also at low temperatures. Since the thermal conductivity of heavy fermion compounds is relatively small, it can contribute to increase the figure of merit in such systems. Thermal conductivity measurements in heavy fermion systems are clearly much rarer than other transport measurements, probably due to their experimental difficulty. However from a theoretical point of view it would be very interesting to get further experimental data of the thermal conductivity, not only in the high temperature regime, but also in the low temperature range below the Kondo temperature.
References [1] B. Cornut, B. Coqblin, Phys. Rev. B5 (1972) 4541. [2] A.K. Bhattacharjee, B. Coqblin, Phys. Rev. B13 (1976) 3441. [3] V. Zlatic, B. Horvatic, I. Milat, B. Coqblin, G. Czycholl, C. Grenzebach, Phys. Rev. B68 (2003) 104432. [4] A.K. Bhattacharjee, B. Coqblin, Phys. Rev. B38 (1988) 338. [5] Z. Kletowski, J. Mucha, H. Misiorek, Y. Onuki, B. Coqblin, J. Magn. Magn. Mater. 272–276 (2004) e97 (presented at ICM, Roma, July 2003). [6] Z. Kletowski, J. Mucha, H. Misiorek, Y. Onuki, B. Coqblin, J. Alloys Compd. 383 (1–2) (2004) 173 (presented at SCTE, Linz, July 2003). [7] Z. Kletowski, J. Mucha, H. Misiorek, in press. [8] P. Lethuillier, O. Haen, Phys. Rev. Lett. 35 (1975) 1391. [9] R. Settai, K. Sugiyama, A. Yamaguchi, S. Araki, K. Miyake, T. Takeuchi, K. Kindo, Y. Onuki, Z. Kletowski, J. Phys. Soc. Jpn. 69 (2000) 3983. [10] A.I. Abou Aly, S. Bakanowski, N.F. Berk, J.E. Crow, T. Mihalisin, Phys. Rev. Lett. 35 (1975) 1387. [11] J. Mucha, A. Jezowski, A. Czopnik, H. Misiorek, Z. Kletowski, J. Alloys Compd. 199 (1993) 145. [12] A. Czopnik, H. Ma¨dge, R. Pott, Phys. Status Solidi B 163 (1991) 473.
Z. Kletowski, B. Coqblin / Solid State Communications 135 (2005) 711–715 [13] Z. Kletowski, P.J. Markowski, A. Czopnik, Solid State Commun. 65 (1988) 593. [14] A. Czopnik, private communication. [15] E. Gratz, E. Bauer, B. Barbara, S. Zemirli, F. Steglich, C.D. Bredl, W. Lieke, J. Phys. F: Met. Phys. 15 (1985) 1975. [16] E. Bauer, E. Gratz, G. Adam, J. Phys. F: Met. Phys. 16 (1986) 493. [17] E. Bauer, E. Gratz, G. Hutflesz, H. Mu¨ller, J. Phys.: Condens. Matter 3 (1991) 7641.
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[18] E. Bauer, Adv. Phys. 40 (1991) 417. [19] E. Bauer, E. Gratz, G. Hutflesz, A.K Bhattacharjee, B. Coqblin, J. Magn. Magn. Mater. 108 (1992) 159. [20] M.J. Besnus, P. Haen, N. Hamdaoui, A. Herrr, A. Meyer, Physica B163 (1990) 571. [21] E. Bauer, E. Gratz, G. Hutflesz, A.K. Bhattacharjee, B. Coqblin, Physica B186–188 (1993) 494. [22] E. Bauer, R. Hauser, E. Gratz, K. Payer, G. Oomi, T. Kagayama, Phys. Rev. B48 (1993) 1583.
Analysis of thermal conductivity data in heavy fermion compounds Z. Kletowskia, B. Coqblinb,* a
W. Trzebiatowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, 50-950 Wrocław, Poland b Laboratoire de Physique des Solides (UMR8502), Baˆtiment 510, Universite´ Paris-Sud, 91405 Orsay, France Received 22 April 2005; received in revised form 2 June 2005; accepted 3 June 2005 by C. Lacroix Available online 16 June 2005
Abstract We present a revisiting analysis of thermal conductivity data in heavy fermion compounds. It is shown that the product WmagT, where Wmag is the magnetic part of the thermal resistivity taken as the difference between the inverse of the thermal conductivity of the Kondo compound and that of an equivalent non magnetic compound, is a good measurement of the strength of the Kondo interaction at sufficiently high temperatures with respect to the Kondo temperature and to the crystal-field splitting. In the considered temperature range, WmagT shows a log T dependence in a way similar to the magnetic part of the electrical resistivity. Such an analysis, which is analogous to the well known one for the electrical resistivity, has not been often used for the thermal conductivity. We present data for compounds with four anomalous Kondo rare earths Ce, Pr, Tm and Yb. q 2005 Elsevier Ltd. All rights reserved. PACS: 71.27.Ca; 72.15.Qm; 72.15.Kv Keywords: D. Heavy fermions; D. Kondo effects; D. Thermal conductivity
Many alloys and compounds containing cerium, ytterbium or other anomalous rare-earths exhibit a Kondo effect, characterized by a log T behavior of the electrical resistivity at high temperatures with respect to the Kondo temperature (TK) and by a T2 behavior of the electrical resistivity which is observed only at very low temperatures below TK. The magnetic and transport properties of many cerium or ytterbium compounds were accounted for by a model which takes into account both the crystalline field effect and the Kondo effect [1]. Typically, the magnetic resistivity of cerium compounds such as CeAl2 or CeAl3 presents a log T behavior at high temperatures, then undergoes a broad maximum at a temperature of order the overall crystal field splitting and finally decreases at low temperatures to either a magnetically ordered state or a nonmagnetic state exhibiting a heavy-fermion behavior.
* Corresponding author. Tel.: C33 1 69 15 60 94; fax: C33 1 69 15 60 86. E-mail address: [email protected] (B. Coqblin).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.06.005
Similarly the thermopower of such systems presents a broad maximum corresponding to a fraction of the crystal field spitting and eventually may exhibit a second maximum at lower temperatures corresponding roughly to the Kondo temperature [2,3]. On the other hand, the thermal conductivity has been also computed within the Coqblin–Schrieffer model including crystal-field effects [4], but in contrast to the electrical resistivity investigations, there are not many studies analyzing properties of the thermal conductivity in heavy fermion systems. Thus, the purpose of the present paper is to revisit the question of the thermal conductivity and to present an analysis of available experimental data in these systems. This analysis follows our recent results obtained for the thermal conductivity measurements in PrSn3 [5,6] and TmGa3 [7]. The analysis of the thermal conductivity measurements is rather difficult because there are several contributions, which enter either the thermal conductivity, if they were coming from different carriers, or the ‘thermal resistivity’ for a given type of carriers scattered by different scattering
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Z. Kletowski, B. Coqblin / Solid State Communications 135 (2005) 711–715
mechanisms. In fact, the phonon thermal conductivity cannot be in general neglected, except only in the case when the electronic contribution is much more important. In our present analysis, we make the simplifying assumption of neglecting the phonon contribution, which seems to be a reasonable assumption in the case of strong Kondo systems. Thus, we consider here only the electronic thermal conductivity and then, assuming that all the scattering mechanisms responsible for the electronic thermal resistivity are additive, according to the Matthiessen rule, the thermal conductivity k can be expressed as follows: kK1 Z W Z Wmag C We;ph
(1)
where the two preceeding terms correspond to the so-called ‘magnetic’ term arising from collisions of the conduction electrons by Kondo impurities, Wmag and to the classical term describing scattering by phonons, We,ph. For the discussion of the thermal conductivity influenced by a strong Kondo effect, we make the same approximation as in the case of the electrical resistivity: we assume that the term We,ph in the Eq. (1) is equal to the thermal resistivity of an equivalent non magnetic compound. Such a situation occurs when the resulting thermal resistivity is much larger than that of the equivalent non magnetic compound. For example, it is the case of the thermal conductivities of PrSn3 and LaSn3, where the ‘magnetic’ contribution, Wmag, can be defined as the difference between the thermal resistivity of PrSn3 and that of LaSn3. More details on the separation procedure of the Wmag can be found in Refs. [5,6]. We can say finally that our present analysis, which is based on the same assumptions than the previous electrical resistivity one [1], is in fact more difficult to justify, which is the main explanation why the present analysis is much less obtained than the corresponding electrical resistivity analysis of Ref. [1]. Before discussing in detail experimental data, let us present now the theoretical analysis, which is performed for both the electrical resistivity [1] and the thermal conductivity [4] in the so-called ‘high temperature’ case, i.e. above the Kondo temperature TK. In this temperature regime, these two quantities are described by the well-known third-order perturbation treatment of the Coqblin–Schrieffer Hamiltonian including crystal-field effects. These theoretical calculations show that the product WmagT and the magnetic electrical resistivity rm exhibit the same temperature behaviour, i.e. they show a maximum at a temperature of order the crystalline-field splitting D and then they behave as log T in temperatures below and above D. Slopes of these two log T behaviours are dependent on the degeneracy a of the occupied levels, i.e. the ground state degeneracy for temperatures below D and the total degeneracy aZ2jC1 above D. The third-order parts of rm and WmagT yielding a log T behavior are given by: 2 a K1 3 J nðEF Þlog T rm Z 2R a
(2)
Wmag T Z 2
R L0
a2 K 1 3 J nðEF Þlog T a
(3)
where the constant R is given by: RZ
3m2 pn0 c e2 Z3 kF2
(4)
and L0Z(pkB)2/3e2 is the Sommerfeld value of the Lorenz number. Eqs. (2) and (3) allow us to calculate the exchange integral J, using values of the slopes for the log Tdependences for rm and WmagT obtained from experiment. The determination of the two calculated J values is clearly a good measurement of the Kondo problem and also a good way to see if the thermal conductivity measurements are a useful tool in investigating Kondo systems. Before the calculated J values will be discussed, we describe now available experimental data for temperature dependences of WmagT and rm in Kondo systems. We present in Fig. 1 the different experimental curves giving WmagT in a logarithmic plot for 8 different heavy fermion compounds, in order to see the log T behaviours at sufficiently high temperatures and the available shapes of the curves at lower temperatures. We have investigated thermal conductivity of monocrystalline samples of PrSn3 and its nonmagnetic reference compound LaSn3. PrSn3 is an antiferromagnetic system with the Neel temperature of 8.6 K. Previously the Kondo behavior was observed in the electrical resistivity and magnetoresistance of this compound [8,9] as well as in the Pr1KxLaxSn3 alloys [10]. In our experiment it was found that, for PrSn3, the quantity WmagT decreases as log T in a wide temperature range from 15 to 200 K [5,6]. Simultaneously, the electrical resistivity shows a wide range of the log T decrease in rm from 10 to 100 K [5,8]. In PrSn3, the Pr ions possess a 4f2 configuration and the overall crystal field splitting D is equal to 23 K only. Since D, TK and TN are very small, one observes the log T-dependence for both rm and WmagT for temperatures down to 10 K. Slopes of the log T dependences found for rm and WmagT are given in Table 1. Recently, the thermal conductivity of TmGa3 and LuGa3 single crystals has been measured. We observe a clear Kondo behaviour in the thermal conductivity of TmGa3, similarly as we previously found such a behaviour in the case of TmIn3 [11]. TmGa3 undergoes two first-order transitions very close to each other at TNZ4.26 K and TQZ 4.29 K, corresponding to antiferromagnetic and quadrupolar orderings [12]. The magnetic resistivity rm of TmGa3, taken as the difference between the resistivities of TmGa3 and LuGa3, exhibits a clear log T behavior between 50 and 250 K [13]. We have also found that, for TmGa3, the log T dependence of the WmagT factor exists in the range 20– 180 K [7], i.e. in a temperature range wider than that of the electrical resistivity. The compound TmIn3, which orders antiferromagnetically at 1.68 K presents a log T dependence
Z. Kletowski, B. Coqblin / Solid State Communications 135 (2005) 711–715
713
Fig. 1. Plot of WmagT (in mK2/W) versus log T for different heavy fermion compounds. The straight lines show the temperature ranges in which the linear dependence of WmagT versus log T occurs. Plots for the particular compounds are depicted by numbers: 1 for CeAl2, 2 for CeCu2, 3 for CeCu5, 4 for CeCu4Al, 5 for PrSn3, 6 for TmGa3, 7 for TmIn3 and 8 for YbCu4Ag.
for WmagT in the temperature range 15–55 K [11] and for rm in temperature range 30–70 K [14]. Although Kondo effect has been found in a number of compounds till now, data for the magnetic part of the thermal conductivity have been reported only in a limited number of Kondo systems. The thermal conductivity was measured in several cerium Kondo compounds. Measurements in CeCu2 and the reference compound YCu2 have
shown that WmagT and rm exhibit clearly two log T behaviors separated by a maximum at roughly 150 K [15]. A similar behavior was also observed in CeAl2 [16], where both WmagT and rm show log T dependences separated by a maximum around 80 K. For our analysis we use data of WmagT and rm taken from temperature ranges 180–300 and 150–300 K for CeCu2 and CeAl2, respectively. The thermal conductivity has been also measured in Ce(CuxAl1Kx)5
Table 1 The first two columns give the experimental values of the slopes versus log T of WmagT and rm Compound
Slopes versus log T of 2
CeAl2 CeCu2 CeCu5 CeCu4Al PrSn3 TmGa3 TmIn3 YbCu4Ag
WmagT (mK /W)
rm (mU cm)
8 [16] 11.5 [15] 3.3 [17,18] 8.6 [17,18] 2 [5,6] 0.9 [7] 0.6 [11] 13 [21]
17 [16] 25.5 [15] 1.9 [17,18] 12 [17,18,] 6 [5,6] 0.8 [13] 4.3 [14] 18 [22]
n (EF) (states/eV at.)
jJj calculated from WmagT (eV)
rm (eV)
2.3 2.3 2.3 2.3 2.3 0.6 1.6 1.8
0.092 0.103 0.085 0.118 0.055 0.143 0.046 0.156
0.087 0.100 0.053 0.098 0.059 0.102 0.066 0.130
The third column gives the density of states n (EF). The two last columns give the values of the exchange integrals J deduced from the calculations of WmagT and rm.
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Kondo compounds and we report values for WmagT found for CeCu5 and CeCu4Al [17–19]. The corresponding magnetic resistivities rm were also measured and we found that both WmagT and rm decrease as log T in a temperature range of 200–300 K for CeCu5 and a wider temperature range of 30–300 K for CeCu4Al. Deduced values of the log T slopes for WmagT and rm found for temperatures above the crystal field splitting are reported in Table 1. YbCu4Ag is another Kondo compound with a large electronic specific heat and its Kondo properties have been well fitted by the JZ7/2 Coqblin–Schrieffer model [20]. For this compound one observes also a decrease in WmagT and rm in log T from roughly 150 to 300 K [21,22]. Experimental curves of WmagT are plotted in Fig. 1 in a logarithmic plot and we see that the curves corresponding to three compounds with cerium and YbCu4Ag present a maximum, followed, with decreasing temperature, by a decrease and finally a small increase at low temperatures. Moreover, the values of the log T-slopes for WmagT and rm in the high temperature regime, i.e. for temperatures higher than D, TK and eventually TN, are given in Table 1. The deduced values of these slopes given in Table 1 correspond to a change of WmagT and rm in a temperature interval of 100 K. Then, we compute the corresponding values of J by use of formulae (2) and (3). For this computation, we use kFZ0.6 u.a., v0Z177 u.a. derived from the atomic volume, the density of states n(EF) derived from the experimental values of the electronic specific heat g of the equivalent non magnetic compound and the effective mass m* proportional to n(EF). Thus, we take n(EF)Z2.3 states/eV at. for one spin direction and m*Z3 in the case of the four cerium compounds and PrSn3, n(EF)Z1.8 states/eV at. for one spin direction and m*Z2.35 for YbCu4Ag, n(EF)Z 2.1 states/eV at. for one spin direction and m*Z2.7 for TmGa3, n(EF)Z1.6 states/eV at. for one spin direction and m*Z2.1 for TmIn3. We take also the degeneracy aZ2jC1 for each rare earth, equal to aZ6, 9, 13 and 8 for, respectively, Ce, Pr, Tm and Yb ions and c the concentration of rare-earth atoms. Finally, experimental values of the log T-slopes of the electrical resistivity and the thermal conductivity yield, respectively, two values for the exchange integral J which are given in the two last columns of Table 1. An extension of the data in the very low temperature regime for the compounds presenting a maximum at high temperatures would be very instructive in order to check the relationship directly connected with the degeneracy a of the occupied levels between the low and high temperature slopes of WmagT. To summarize this rapid analysis of the thermal conductivity results for a few Kondo compounds, we can conclude that we have found a log T decrease in the quantity WmagT in several compounds with cerium, ytterbium, praseodymium and thulium. A log T decrease in the magnetic resistivity rm has been also observed in these compounds. Two points are very interesting to be noticed: first, a Kondo behavior has been observed in the thermal
conductivity as well as in the electrical resistivity and second the Kondo effect has been observed not only in cerium and ytterbium compounds, but also in praseodymium and thulium compounds, which is obviously relatively rare. Exchange integrals J were calculated using the thirdorder perturbation method performed within the Coqblin– Schrieffer Hamiltonian including crystal-field effects in the case of thermal conductivity and electrical resistivity data. Although the J values vary from a compound to another one, the two deduced values of J for a given compound are reasonable and very close to each other. This means clearly that our analysis of the thermal conductivity data in terms of the Kondo effect is correct and that our theoretical analysis of the thermal conductivity gives evidence for a Kondo effect in compounds with Ce, Yb, Pr and Tm. Our results indicate that a precise measurement of the thermal conductivity is a useful tool for studying the Kondo effect in anomalous rare-earth systems at high and certainly also at low temperatures. Since the thermal conductivity of heavy fermion compounds is relatively small, it can contribute to increase the figure of merit in such systems. Thermal conductivity measurements in heavy fermion systems are clearly much rarer than other transport measurements, probably due to their experimental difficulty. However from a theoretical point of view it would be very interesting to get further experimental data of the thermal conductivity, not only in the high temperature regime, but also in the low temperature range below the Kondo temperature.
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