- Email: [email protected]
Phonon thermal conductivity in single layered manganites
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 290–291 (2005) 937–939 www.elsevier.com/locate/jmmm
Phonon thermal conductivity in single layered manganites C. Baumanna,, C. Hessb, P. Reutlera,c, B. Bu¨chnera, A. Revcolevschic a Institute for Solid State Research IFW Dresden, P.O. Box 27 01 16, D-01171 Dresden, Germany De´partement de Physique de la Matie`re Condense´e, Universite´ de Gene`ve, 1211 Gene`ve, Switzerland c Laboratoire de Physico-Chimie des Solides, Universite´ de Paris-Sud, F-91405 Orsay Ce´dex, France
b
Available online 10 December 2004
Abstract We present measurements of the thermal transport in single crystals of single-layered manganites La1x Sr1þx MnO4 with 0pxp0:5: The data reveal a strong but non-monotonic suppression of the phonon thermal conductivity kph upon hole-doping. The suppression of kph originates from scattering of phonons by polaronic holes. The suppression is particularly strong when charge and orbital degrees of freedom are disordered and rather weak in the case of long-range charge and orbital ordering. Moreover, slight anomalies are found in the vicinity of antiferromagnetic phase transitions, probably due to scattering of phonons by magnetic fluctuations. r 2004 Elsevier B.V. All rights reserved. PACS: 63.20.Ls; 66.70.+f; 71.70.Ej; 44.10.+i Keywords: Magnetic oxides; Phonons; Thermal conductivity; Magnon–phonon interaction
Thermal transport measurements provide crucial information about the interplay between different degrees of freedom in transition metal oxides since the mean free path of excitations is governed by various scattering processes. In particular, there is a close relationship between the phonon thermal conductivity (kph ) and the lattice, spin, and charge degrees of freedom. For example, in La2x Srx NiO4 a clear correlation between kph and the so-called stripe order, i.e. the segregation of charge carriers and spins in a stripe-like pattern, was found. While static and longrange stripe correlations do not significantly affect kph ; it is strongly suppressed if the stripes become disordered or dynamic [1,2]. A similar interrelation was discussed to be relevant for the thermal transport in the cuprate superconductors La2x Srx CuO4 ; where the suppression of superconductivity in favour of static stripe order Corresponding author. Tel.: +49 351 4659 533; fax: +49 351 4659 313. E-mail address: [email protected] (C. Baumann).
upon rare earth doping is accompanied by an enhancement of kph [3,4]. In this paper, we focus on isostructural single-layered manganites La1x Sr1þx MnO4 ; which are isostructural to La2x Srx CuO4 : The physical properties of these compounds are governed by the content of Jahn–Teller active manganese-eg electrons, which can be changed by Sr-doping. In contrast to perovskite manganites, the layered materials show an insulating behaviour in the whole doping range. Moreover, the electronic transport is strongly anisotropic due to the layered structure [5]. In the undoped case (x ¼ 0), where no holes are present, an antiferromagnetic (AFM) spin order evolves below T N 125 K [6,7]. Moderate hole doping (x ¼ 0:125) reduces T N to 65 K [7], whereas long range AFM order disappears for 0:25pxp0:4 [7,8]. For this doping range a static, but rather disordered magnetic state was reported [5,6]. In the half-doped compound (x ¼ 0:5), charge and orbital order is established at T 230 K [1,5,7,9], and a CE-type AFM occurs below T N 110 K [6,7,9].
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.294
ARTICLE IN PRESS C. Baumann et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 937–939
938
We have grown a series of single crystals with x ¼ 0; 0.125, 0.25, and 0.5 using the traveling solvent floating zone method [7] and performed thermal and electrical transport measurements with standard four-probe techniques. The thermal conductivity measured parallel to the abplane (kab ), i.e. along the MnO2 layers of the different samples are shown in Fig. 1. For all compositions a lowtemperature (T 20 K) peak is found in kab : Such a peak is typical for phononic heat conductors, where it arises due to the different T-dependencies of the number of excited phonons and the phonon mean free path. At low temperatures, the mean free path is mainly determined by crystal boundaries and defects, and therefore almost independent of temperature. Since the number of phonons rapidly grows with rising T, kph increases strongly. When T is further increased the growing importance of umklapp-scattering reduces the phonon mean free path and over-compensates the effect of the growing phonon number, which results in the observed peak structure. Defects in the lattice periodicity normally lead to enhanced scattering of phonons and hence a reduction of the peak. Note, that additional electronic and magnetic contributions to kab are unlikely since the material is electrically insulating [5] and magnetic heat conduction, which at high temperatures has been observed in La2 CuO4 [10], is unlikely to significantly contribute at such low T. Fig. 1(a) demonstrates the intriguing doping dependence of the peak size: With increasing Sr content x the max in k is strongly suppressed, but, moreover, the 2
10
ab
40 20
κ
max
(W/mK)
60
x=0 x=1/2 x=1/8 x=1/4 κ ab(W/mK)
0
0
0.2
0.4
doping x
1
10
10
0
0
100
200
300
Temperature(K) Fig. 1. Temperature dependence of the in-plane thermal conductivity kab of La1x Sr1þx MnO4 for x ¼ 0; 0.125, 0.25 and 0.5. Inset: Maximal thermal conductivity kmax ab as a function of doping (). Dotted line: guide to the eyes.
doping dependence is non-monotonic. Apparently, the increase of the Sr-content suppresses the low temperature maximum kmax ph only up to xt0:25: As apparent from the figure, in the case of half doping kmax ph increases again and is largest of all compounds with xa0 not only at the peak but also in the whole measured temperature range. A gradual suppression of kph upon doping is a natural consequence of the successive substitution of different ions on the same lattice site, because these ions have different masses and sizes, which induces scattering of phonons. However, the scattering in binary systems is the lowest for parent compounds and increases upon substitution being maximal at the 1:1 ratio of the different ions [11]. Therefore, the scattering in La1x Sr1þx MnO4 is expected to decrease upon increasing doping x, just contrary to the observation (see inset of Fig. 1). Therefore, it is straightforward to attribute the strong suppression of kph for xa0 to additional scattering centers which arise due to introduction of holes with a strong electron–phonon coupling. This effect overcompensates the effect of lattice doping. All compounds are insulators, i.e. the holes in the MnO2 planes are localized [5]. The strong electron–phonon coupling then appears to be a natural consequence of the removal of Jahn–Teller active eg -electrons, which gives the holes a very strong polaronic character. This is supported by recent neutron scattering data which give evidence for short-range structural distortions due to the holes [8]. Therefore, we argue that the suppression of kph is due to phonon–polaron scattering. Note, that similar results have been reported recently by Zhou et al. for perovskite manganites, where the large phonon contribution of LaMnO3 was found to be strongly suppressed to a glass-like behaviour if holes are introduced by oxygen- or Sr-doping [12]. In contrast to the suppression of kmax ph for moderate x, the increase of kmax for x ¼ 0:5 cannot be attributed to a ph simple insertion of additional polarons. Since La1=2 Sr3=2 MnO4 is known to exhibit charge order (CO) below T CO 230 K [5,9], polaron dynamics and/ or polaronic order has to be considered. The CO has a strong impact on the electrical resistivity r (see Fig. 2(b)), where a pronounced step occurs at T CO : There is also a clear kink at T CO in kab of x ¼ 0:5 (see Fig. 2(a)). The data show that the in-plane thermal conductivity is significantly larger in the CO phase (ToT CO ) than at higher T. A similar observation has previously been reported for a polycrystalline sample [1]. At T CO ; the charge dynamics is strongly reduced, i.e. mobile and/or disordered holes become static as well as long-range ordered. Since for 0oxp0:25 phonon–polaron scattering was found to be a major scattering mechanism for kph ; it is straightforward to attribute the kink in kab to an enhancement of kph due to the long range crystallization of polaronic scattering centers.
ARTICLE IN PRESS C. Baumann et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 937–939 4 10
(a)
10
3.4
x=0.5
TN
2 x=0
1.5
-2
1
(b)
(a) 3 150
10
κc(Wm-1K-1)
3.6
0
6 200
250
300 150
200
250
x=0 x=1/8
300
Temperature(K)
Fig. 2. (a) kab at x ¼ 0:5 in the vicinity of T CO 230 K. (b) Comparison of the in-plane resistivity rab of La1x Sr1þx MnO4 at x ¼ 0:5 and x ¼ 0:25:
Finally, the coupling of phonons to the magnetic degree of freedom is discussed. Both, La1x Sr1þx MnO4 at x ¼ 0 and x ¼ 0:125 undergo a magnetic phase transition at T N 125 K and T N 65 K, respectively [7]. The onset of spin order is also visible by broad anomalies in the thermal conductivity at T N (Fig. 3). While at x ¼ 0; the anomaly is present in both kab and kc ; at x ¼ 0:125 only in kab it can be distinguished clearly from the phonon peak. (kc of x ¼ 0:125 is not shown.) In principle, magnetic excitations which can occur in the spin ordered phase may contribute to the thermal conductivity and, therefore, may account for the observed anomalies at T N : However, the magnetic dispersion of La1 Sr1 MnO4 is strongly anisotropic [8], and one would expect the magnetic contribution ksp to the thermal conductivity to be anisotropic. This is indeed observed for the strongly two-dimensional La2 CuO4 where a magnetic contribution to k is present only within the CuO2 layers [10]. Since La1 Sr1 MnO4 displays anomalies at T N in both, kab and kc ; the scenario of a significant ksp below T N is very unlikely. Consequently, additional scattering processes must account for the decrease of k at T N : One candidate for this process is enhanced phonon–magnon-scattering which may suppress kph close to T N [13]. One may also speculate whether the orbital degrees of freedom must be considered since thermal expansion data give evidence for a strong coupling between spin and orbital degrees of freedom in La1 Sr1 MnO4 [14]. Therefore, enhanced spin fluctuation may be connected to orbital fluctuations at T N ; and thus a perturbation of the ferro-orbital configuration in La1 Sr1 MnO4 [8] may result in an additional scattering channel. Although we cannot distinguish between phonon–magnon-scattering and scattering of phonons by orbital fluctuations, our results suggest enhanced scattering of phonons close to T N due to magneto-elastic coupling. In summary, we have measured the thermal conductivity of single-layered manganites. The thermal
κab(Wm-1K-1)
κab(W/mK)
x=0.25
3.2
2.5
2
TCO
ρab(Ωm)
x=0.5
3.8
TCO
939
(b) TN
5
4 TN 3
0
50
100 Temperature(K)
150
200
Fig. 3. (a) kc of La1x Sr1þx MnO4 at x ¼ 0 and (b) kab at x ¼ 0 and x ¼ 0:125 as a function of temperature. T N is indicated by arrows.
conductivity was found to be predominantly phononic. The suppression of the phonon heat transport in the compositional range 0oxp0:25 is assigned to the formation of polarons, which give rise to strong phonon–polaron scattering. The enhanced heat flow in the half-doped compound suggests, that dynamic polarons lead to a much stronger suppression of kph than static charges. Furthermore, the phonon thermal conductivity at the magnetic phase transitions is significantly reduced, which proves a magneto-elastic coupling.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
C. Hess, et al., Phys. Rev. B 59 (1999) 10397. C. Cassel, et al., J. Low Temp. Phys. 117 (1999) 1083. O. Barberski, et al., Europhys. Lett. 44 (1998) 335. C. Hess, et al., Phys. Rev. B 68 (2003) 184517. Y. Moritomo, et al., Phys. Rev. B 51 (1995) 3297. C. Baumann, et al., Physica B 326 (2003) 505. P. Reutler, et al., J. Crystal Growth 249/1–2 (2003) 222. S. Larochelle, et al., cond-mat. 0405424, 2004. B.J. Sternlieb, et al., Phys. Rev. Lett. 76 (1996) 2169. C. Hess, et al., Phys. Rev. Lett. 90 (2003) 197002. R. Berman, J. Brock, Proc. Roy. Soc. (London) A 289 (1965) 46. [12] J.-S. Zhou, J.B. Goodenough, Phys. Rev. B 64 (2001) 024421. [13] G.A. Slack, R. Newman, Phys. Rev. Lett. 1 (1958) 359; G.A. Slack, Phys. Rev. 122 (1961) 1451. [14] R. Klingeler, et al., J. Magn. Magn. Mater., in press, doi:10.1016/j.jmmm.2004.11.296.
Journal of Magnetism and Magnetic Materials 290–291 (2005) 937–939 www.elsevier.com/locate/jmmm
Phonon thermal conductivity in single layered manganites C. Baumanna,, C. Hessb, P. Reutlera,c, B. Bu¨chnera, A. Revcolevschic a Institute for Solid State Research IFW Dresden, P.O. Box 27 01 16, D-01171 Dresden, Germany De´partement de Physique de la Matie`re Condense´e, Universite´ de Gene`ve, 1211 Gene`ve, Switzerland c Laboratoire de Physico-Chimie des Solides, Universite´ de Paris-Sud, F-91405 Orsay Ce´dex, France
b
Available online 10 December 2004
Abstract We present measurements of the thermal transport in single crystals of single-layered manganites La1x Sr1þx MnO4 with 0pxp0:5: The data reveal a strong but non-monotonic suppression of the phonon thermal conductivity kph upon hole-doping. The suppression of kph originates from scattering of phonons by polaronic holes. The suppression is particularly strong when charge and orbital degrees of freedom are disordered and rather weak in the case of long-range charge and orbital ordering. Moreover, slight anomalies are found in the vicinity of antiferromagnetic phase transitions, probably due to scattering of phonons by magnetic fluctuations. r 2004 Elsevier B.V. All rights reserved. PACS: 63.20.Ls; 66.70.+f; 71.70.Ej; 44.10.+i Keywords: Magnetic oxides; Phonons; Thermal conductivity; Magnon–phonon interaction
Thermal transport measurements provide crucial information about the interplay between different degrees of freedom in transition metal oxides since the mean free path of excitations is governed by various scattering processes. In particular, there is a close relationship between the phonon thermal conductivity (kph ) and the lattice, spin, and charge degrees of freedom. For example, in La2x Srx NiO4 a clear correlation between kph and the so-called stripe order, i.e. the segregation of charge carriers and spins in a stripe-like pattern, was found. While static and longrange stripe correlations do not significantly affect kph ; it is strongly suppressed if the stripes become disordered or dynamic [1,2]. A similar interrelation was discussed to be relevant for the thermal transport in the cuprate superconductors La2x Srx CuO4 ; where the suppression of superconductivity in favour of static stripe order Corresponding author. Tel.: +49 351 4659 533; fax: +49 351 4659 313. E-mail address: [email protected] (C. Baumann).
upon rare earth doping is accompanied by an enhancement of kph [3,4]. In this paper, we focus on isostructural single-layered manganites La1x Sr1þx MnO4 ; which are isostructural to La2x Srx CuO4 : The physical properties of these compounds are governed by the content of Jahn–Teller active manganese-eg electrons, which can be changed by Sr-doping. In contrast to perovskite manganites, the layered materials show an insulating behaviour in the whole doping range. Moreover, the electronic transport is strongly anisotropic due to the layered structure [5]. In the undoped case (x ¼ 0), where no holes are present, an antiferromagnetic (AFM) spin order evolves below T N 125 K [6,7]. Moderate hole doping (x ¼ 0:125) reduces T N to 65 K [7], whereas long range AFM order disappears for 0:25pxp0:4 [7,8]. For this doping range a static, but rather disordered magnetic state was reported [5,6]. In the half-doped compound (x ¼ 0:5), charge and orbital order is established at T 230 K [1,5,7,9], and a CE-type AFM occurs below T N 110 K [6,7,9].
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.294
ARTICLE IN PRESS C. Baumann et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 937–939
938
We have grown a series of single crystals with x ¼ 0; 0.125, 0.25, and 0.5 using the traveling solvent floating zone method [7] and performed thermal and electrical transport measurements with standard four-probe techniques. The thermal conductivity measured parallel to the abplane (kab ), i.e. along the MnO2 layers of the different samples are shown in Fig. 1. For all compositions a lowtemperature (T 20 K) peak is found in kab : Such a peak is typical for phononic heat conductors, where it arises due to the different T-dependencies of the number of excited phonons and the phonon mean free path. At low temperatures, the mean free path is mainly determined by crystal boundaries and defects, and therefore almost independent of temperature. Since the number of phonons rapidly grows with rising T, kph increases strongly. When T is further increased the growing importance of umklapp-scattering reduces the phonon mean free path and over-compensates the effect of the growing phonon number, which results in the observed peak structure. Defects in the lattice periodicity normally lead to enhanced scattering of phonons and hence a reduction of the peak. Note, that additional electronic and magnetic contributions to kab are unlikely since the material is electrically insulating [5] and magnetic heat conduction, which at high temperatures has been observed in La2 CuO4 [10], is unlikely to significantly contribute at such low T. Fig. 1(a) demonstrates the intriguing doping dependence of the peak size: With increasing Sr content x the max in k is strongly suppressed, but, moreover, the 2
10
ab
40 20
κ
max
(W/mK)
60
x=0 x=1/2 x=1/8 x=1/4 κ ab(W/mK)
0
0
0.2
0.4
doping x
1
10
10
0
0
100
200
300
Temperature(K) Fig. 1. Temperature dependence of the in-plane thermal conductivity kab of La1x Sr1þx MnO4 for x ¼ 0; 0.125, 0.25 and 0.5. Inset: Maximal thermal conductivity kmax ab as a function of doping (). Dotted line: guide to the eyes.
doping dependence is non-monotonic. Apparently, the increase of the Sr-content suppresses the low temperature maximum kmax ph only up to xt0:25: As apparent from the figure, in the case of half doping kmax ph increases again and is largest of all compounds with xa0 not only at the peak but also in the whole measured temperature range. A gradual suppression of kph upon doping is a natural consequence of the successive substitution of different ions on the same lattice site, because these ions have different masses and sizes, which induces scattering of phonons. However, the scattering in binary systems is the lowest for parent compounds and increases upon substitution being maximal at the 1:1 ratio of the different ions [11]. Therefore, the scattering in La1x Sr1þx MnO4 is expected to decrease upon increasing doping x, just contrary to the observation (see inset of Fig. 1). Therefore, it is straightforward to attribute the strong suppression of kph for xa0 to additional scattering centers which arise due to introduction of holes with a strong electron–phonon coupling. This effect overcompensates the effect of lattice doping. All compounds are insulators, i.e. the holes in the MnO2 planes are localized [5]. The strong electron–phonon coupling then appears to be a natural consequence of the removal of Jahn–Teller active eg -electrons, which gives the holes a very strong polaronic character. This is supported by recent neutron scattering data which give evidence for short-range structural distortions due to the holes [8]. Therefore, we argue that the suppression of kph is due to phonon–polaron scattering. Note, that similar results have been reported recently by Zhou et al. for perovskite manganites, where the large phonon contribution of LaMnO3 was found to be strongly suppressed to a glass-like behaviour if holes are introduced by oxygen- or Sr-doping [12]. In contrast to the suppression of kmax ph for moderate x, the increase of kmax for x ¼ 0:5 cannot be attributed to a ph simple insertion of additional polarons. Since La1=2 Sr3=2 MnO4 is known to exhibit charge order (CO) below T CO 230 K [5,9], polaron dynamics and/ or polaronic order has to be considered. The CO has a strong impact on the electrical resistivity r (see Fig. 2(b)), where a pronounced step occurs at T CO : There is also a clear kink at T CO in kab of x ¼ 0:5 (see Fig. 2(a)). The data show that the in-plane thermal conductivity is significantly larger in the CO phase (ToT CO ) than at higher T. A similar observation has previously been reported for a polycrystalline sample [1]. At T CO ; the charge dynamics is strongly reduced, i.e. mobile and/or disordered holes become static as well as long-range ordered. Since for 0oxp0:25 phonon–polaron scattering was found to be a major scattering mechanism for kph ; it is straightforward to attribute the kink in kab to an enhancement of kph due to the long range crystallization of polaronic scattering centers.
ARTICLE IN PRESS C. Baumann et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 937–939 4 10
(a)
10
3.4
x=0.5
TN
2 x=0
1.5
-2
1
(b)
(a) 3 150
10
κc(Wm-1K-1)
3.6
0
6 200
250
300 150
200
250
x=0 x=1/8
300
Temperature(K)
Fig. 2. (a) kab at x ¼ 0:5 in the vicinity of T CO 230 K. (b) Comparison of the in-plane resistivity rab of La1x Sr1þx MnO4 at x ¼ 0:5 and x ¼ 0:25:
Finally, the coupling of phonons to the magnetic degree of freedom is discussed. Both, La1x Sr1þx MnO4 at x ¼ 0 and x ¼ 0:125 undergo a magnetic phase transition at T N 125 K and T N 65 K, respectively [7]. The onset of spin order is also visible by broad anomalies in the thermal conductivity at T N (Fig. 3). While at x ¼ 0; the anomaly is present in both kab and kc ; at x ¼ 0:125 only in kab it can be distinguished clearly from the phonon peak. (kc of x ¼ 0:125 is not shown.) In principle, magnetic excitations which can occur in the spin ordered phase may contribute to the thermal conductivity and, therefore, may account for the observed anomalies at T N : However, the magnetic dispersion of La1 Sr1 MnO4 is strongly anisotropic [8], and one would expect the magnetic contribution ksp to the thermal conductivity to be anisotropic. This is indeed observed for the strongly two-dimensional La2 CuO4 where a magnetic contribution to k is present only within the CuO2 layers [10]. Since La1 Sr1 MnO4 displays anomalies at T N in both, kab and kc ; the scenario of a significant ksp below T N is very unlikely. Consequently, additional scattering processes must account for the decrease of k at T N : One candidate for this process is enhanced phonon–magnon-scattering which may suppress kph close to T N [13]. One may also speculate whether the orbital degrees of freedom must be considered since thermal expansion data give evidence for a strong coupling between spin and orbital degrees of freedom in La1 Sr1 MnO4 [14]. Therefore, enhanced spin fluctuation may be connected to orbital fluctuations at T N ; and thus a perturbation of the ferro-orbital configuration in La1 Sr1 MnO4 [8] may result in an additional scattering channel. Although we cannot distinguish between phonon–magnon-scattering and scattering of phonons by orbital fluctuations, our results suggest enhanced scattering of phonons close to T N due to magneto-elastic coupling. In summary, we have measured the thermal conductivity of single-layered manganites. The thermal
κab(Wm-1K-1)
κab(W/mK)
x=0.25
3.2
2.5
2
TCO
ρab(Ωm)
x=0.5
3.8
TCO
939
(b) TN
5
4 TN 3
0
50
100 Temperature(K)
150
200
Fig. 3. (a) kc of La1x Sr1þx MnO4 at x ¼ 0 and (b) kab at x ¼ 0 and x ¼ 0:125 as a function of temperature. T N is indicated by arrows.
conductivity was found to be predominantly phononic. The suppression of the phonon heat transport in the compositional range 0oxp0:25 is assigned to the formation of polarons, which give rise to strong phonon–polaron scattering. The enhanced heat flow in the half-doped compound suggests, that dynamic polarons lead to a much stronger suppression of kph than static charges. Furthermore, the phonon thermal conductivity at the magnetic phase transitions is significantly reduced, which proves a magneto-elastic coupling.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
C. Hess, et al., Phys. Rev. B 59 (1999) 10397. C. Cassel, et al., J. Low Temp. Phys. 117 (1999) 1083. O. Barberski, et al., Europhys. Lett. 44 (1998) 335. C. Hess, et al., Phys. Rev. B 68 (2003) 184517. Y. Moritomo, et al., Phys. Rev. B 51 (1995) 3297. C. Baumann, et al., Physica B 326 (2003) 505. P. Reutler, et al., J. Crystal Growth 249/1–2 (2003) 222. S. Larochelle, et al., cond-mat. 0405424, 2004. B.J. Sternlieb, et al., Phys. Rev. Lett. 76 (1996) 2169. C. Hess, et al., Phys. Rev. Lett. 90 (2003) 197002. R. Berman, J. Brock, Proc. Roy. Soc. (London) A 289 (1965) 46. [12] J.-S. Zhou, J.B. Goodenough, Phys. Rev. B 64 (2001) 024421. [13] G.A. Slack, R. Newman, Phys. Rev. Lett. 1 (1958) 359; G.A. Slack, Phys. Rev. 122 (1961) 1451. [14] R. Klingeler, et al., J. Magn. Magn. Mater., in press, doi:10.1016/j.jmmm.2004.11.296.