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Specific heat of CeAl2 in high magnetic fields
Journal of Magnetism and Magnetic Materials 7 (1978) 286-289 0 North-Holland Publishing Company
1
SPECIFIC HEAT OF CeA12 IN HIGH MAGNETIC FIELDS t C.D. BREDL and F. STEGLICH II. Physikalisches Institut der Universitiit zu Kdn, 0.5000 Kdn 41, W.-Germany Specific heat measurements on CeA12 show that in this compound the Kondo effect is of comparable strength (kBTK) to the Ce-Ce interactions which yield magnetic _order below TN 2 4 K. TK E 6 K is estimated from (a) the fact that the large specific heat coefficient y magnetic field up to 5 T, and (b) from the specific heats of (Ce,Lar_,)Al2 alloys (x < 0.7) in 5 T.
mining the strength of the magnetic Ce-Ce interactions. This may be inferred from specific heat results (in zero field) which show a distinct increase of the ordering temperature TN under application of a pressure of 7 kbar [12]. Thus, it appears to be of great importance with respect to the nature of this LTP to know the order of magnitude of the “characteristic temperature” TK. It is the purpose of this paper to estimate TK by analy-
The intermetallic CeA12 is often looked upon as an archetypical “Kondo compound”. This term is justified for two reasons: first, at sufficiently high temperatures (where Ce-Ce interactions are suppressed by thermal fluctuations [ 11) Kondo anomalies, modified by crystal field (CF) effects, are observed in several physical quantities [2-51, and secondly, high-field magnetization measurements display a loss of the 4f shell occupancy under applied pressure (a “nonmagnetic” or “mixed valent” state of the Ce ions is expected for 40 kbar) [6]. Such a pressure-induced demagnetization of the local 4f shell is well known for metallic Ce as well as for dilute Kondo alloys containing Ce impurities [7]. For a dilute Kondo alloy the energy kBTK, which measures the strength of the interaction between a single 4f shell and the conduction band states, can easily be determined from low temperature and zero field experiments. However, in the case of the “Kondo compound”, CeA12, this is prevented by the magnetic interactions between Ce ions: a phase transition at TN - 4 K has been established for CeA12 by several macroscopic properties [8]. The phase boundary in the (field, temperature) plane of its low temperature phase (LTP), as determined by magnetization [9] and magnetostriction [lo] measurements, is typical of a (not necessarily simple) antiferromagnet. Although CeA12 is magnetically ordered in its LTP, as proved by inelastic neutron scattering [ 51 and more directly by recent neutron diffraction experiments [ 111, the Kondo effect seems to play an important role in deter-
-1
CIT
Ce Al,
b&i-i 21
1.8
1.2
Cl6 al35 0
L
8
12
T2(K2)
Fig. 1. The specific heat of CeAl2 single crystals (per mole) in zero magnetic field and in ?.l and 5.0 T parallel to the [ 1001 and [ 1 lo] directions, plotted as C/T vs. T2.
t Work performed within the research program of the Sonderforschungsbereich 125 Aachen/Jiilich/Kbln. 286
CD. Bred4 F. Steglich / Specific heat of CeAl2 in high magnetic fields
zing specific heat data obtained from CeAIs and (Ce,La1_,)A12 alloys at low temperatures and in high magnetic fields. In fig. 1 the specific heat Cper mole is shown in a plot C/TVS. T2 for two CeA12 single crystals with [IOO] and [llO] axes parallel to the magnetic field (B = 0,3.1, and 5 T). In zero field a distinct anomaly is observed at Ti - 3.9 K. The dependence of TN on the magnetic field is in qualitative accordance with the boundary of the LTP as concluded from other measurements [9,10]. The remarkable anisotropy, visible in fig. I, confirms magnetization measurements [9] from which the [ 1 lo] direction was inferred to be magnetically easier than the [ 1001 direction. Surprisingly, the data in fig. 1 can be extrapolated at low temperatures to an intercept y applied field. A y value of this size is highly unusual for a magnetically ordered metal. On the other hand, the “Kondo compound”, CeAls , without magnetic order at low temperature [ 131 exhibits y = 1620 mJ/mole K2 [ 141. This huge value might be attributed to a narrow band (width kBTK ce*13) of quasiparticle states at the Fermi level due to strong electron-electron interactions in the “Kondo ground state” [15]. As was mentioned above, and is supported further by the large y value, the Kondo effect remains important in the LTP of CeA12 (though magnetically ordered). This may also be concluded from the observation [ 11, 131 that the effective magnetic moment per Ce ion, peff, in the LTP is evidently reduced compared to p7 corresponding to the I’7 CF ground state of the free ion. One could argue that in this LTP the majority of conduction electron states is governed by magnetic order, whereas a fraction still participates in the Kondo effect, thus giving rise to a narrow band (at the Fermi level). Its (quasiparticle) density of states would, of course, be considerably smaller than the one expected in the absence of magnetic order. In this context (and referring to Schotte and Schotte [ 15]), the field independence of y means a constant density of states in a field up to B = 5 T, providing a lower limit of the width, A, of this band, i.e. of the characteristic temperature TK = A/kn > Ig’lpBB/kB = 4.8 K. Here g’ = -1.43 is the Land6 factor for Ce3+ in the I’7 doublet state [ 161 (the other symbols have their usual meaning). In fig. 2 the specific heat C* per formula unit CeAla, divided by kg, is shown as a function of temperature (logarithmic scale) for the CeA12 crystal with its [ 1001
0.6‘.
Q
281
0.0T
+
3.1 T
l
5.0T
0.6 ..
0.1, ..
0.2 ..
0 i 0.3
1
3
Fig. 2. The specific heat, C*, per formula unit CeAla, divided by kg, versus temperature (logarithmic scale), obtained in magnetic fields of 0, 3.1, and 5.0 T parallel to the [ 1001 direction. The thick lines represent closely spaced data points, whereas the fine lines are merely interpolated between data points.
axis parallel to the field. It is found that application of 5 T not only depresses the ordering temperature TN, but also results in a broad specific heat anomaly at 6 K, which displays the Zeeman splitting of the I’7 doublet of the single Ce3+ ions. Since, unfortunately, no higher magnetic field was available to completely suppress antiferromagnetic order and, thus, completely resolve this “Schottky type” anomaly, we have mag netically diluted CeA12 with LaA12. In fig. 3 the specific heats of several (Ce,Lar_,)Ala alloys (1.5 a/o < x < 70 a/o), measured in B = 5 T, are also presented in the form C./k, vs. T (logarithmic scale). Besides a CF induced minimum above 5 K [ 161 and some residual influence of magnetic interaction effects at the low temperature end, these curves exhibit a “Schottky type” shape. The result for the most dilute alloy has previously [ 161 been compared by one of us to a theoretical model of Schotte and Schotte [ 151, who have calculated the thermodynamic properties of dilute Kondo alloys in high magnetic fields. It was shown [ 161 that a fit of this model to the zero field data yiel-
CD. Bredl, F. Steglich / Specific heat of CeAI2 in high magnetic fields
288
OX
CTk,
%(Steglich)
+
1.5
n
10 v.
[ 161 to CeA12, is caused by an increased lattice pressure due to a 1% decrease of the mean lattice spacing [ 18, 191. To conclude, analysis of low temperature and high field specific heat reveals for CeA12 a “characteristic temperature” TK z 6 K, which is of comparable size to the magnetic ordering temperature TN = 4 K. (Owing to the existence of short range order [ 171, kB TN presumably underestimates the actual magnetic interaction energy to which kBTK has to be compared). Apparently, the competition between magnetic order and Kondo effect can result in exciting low temperature behavior of a “Kondo compound”, e.g. a nonmagnetic “Kondo ground state” (CeAls) or weak antiferromagnetism (CeA12). The impact of the experimental data available leads one to expect a deeper understanding of these materials to develop within the near future [20].
Bz5.01
0.3
0.2
0.1
0
0.3
1
3
10 T(K)
Fig. 3. Specific heat per formula unit CeA12, C*, in units of kB versus temperature (logarithmic scale), obtained from Ce,Laf__&2 samples (x = 0.015 1161, 0.1, 0.2,0.4,0.7) in an external magnetic field of 5 T.
ded a reasonable value of T~(0.3 K), whereas TK 0.75 K was obtained in 5 T. Since the “Schottky type” anomalies shift to higher temperatures upon increasing magnetic field, the observed increase of TK as a function of B might simply be caused by the “Korringa type” increase of the electronic scattering rate as a function of temperature. This effect is not accounted for in the Schotte model [ 1.51. A fit of the model to the specific heats of the (Ce,La1_,)A12 alloys in a field of 5 T, therefore, provides an upper limit of TK. The results of this fit, which will be explained in more detail elsewhere [ 171, are A/kn = 0.4, 1.05, 1.6, and 2.7 K (for x = 0.1,0.2,0.4, and 0.7, respectively). If these values of A/kn as a function of x are mean squared and extrapolated to x = 1, an upper limit of TK < 7.5 K is obtained for CeA12. The same magnitude is inferred from the low temperature (T > 4 K) width of the quasielastic line in the magnetic neutron spectra of CeA12 [S] . Furthermore, TK = (5 + 2) K is determined from the low temperature resistivity of CeA12 when dilutely dissolved in a nonmagnetic (La, Y)Ala host with the same mean lattice spacing [ 18,1]. The one order of magnitude increase of TK as observed, when going from dilute (Ce, L&U2
We wish to thank Dr. L. Kay Nicholson for critically reading this manuscript.
References [l] F. Steglich, Festkorperprobleme
(Advances in Solid State Physics), vol. 17, J. Treusch, ed. (Braunschweig, Vieweg, 1977) p. 319. (21 K.H.J. Buschow and H.J. van Daal, Phys. Rev. Lett. 23 (1969) 408; B. Cornut and B. Coqblin, Phys. Rev. B5 (1972) 4541. [ 31 W.M. Swift and W.E. Wallace, J. Phys. Chem. Solids 29 (1969) 2053. [4] C. Deenadas, A.W. Thompson, R.S. Craig and W.E. Wallace, J. Phys. Chem. Solids 32 (1971) 1853. [5] M. Loewenhaupt and F. Steglich, Physica 86-88B (1977) 187. [6] B. Barbara, H. Bartholin, D. Florence, M.F. Rossignol and E. Walker, Physica 86-88B (1977) 177. [7] M.B. Maple and D.K. Wohlleben, Magnetism and Magnetic Materials - 1973 (AIP, New York, 1974) p. 447. [8] E. Walker, H.G. Purwins, M. Landolt and F. Hullinger, J. Less Common Metals 33 (1973) 203; R.W. Hill and J.M. Machado da Silva, Phys. Lett. 30 (1969) 13. [9] B. Barbara, M.F. Rossignol, H.G. Purwins and E. Walker, Solid State Commun. 17 (1975) 1525. [Ia‘1 M. Croft, I. Zoric and R.D. Parks, Preprint. 111I B. Barbara, J.X. Boucherle, J.L. Buevoz, M.F. Rossignol and J. Schweizer, Proc. Int. Conf. on Rare Earths and Actinides, Durham (1977). !I A. Berton, J. Chaussy, G. Chouteau, B. Cornut, J. Peyrard 112 and R. Tournier, in: Valence Instabilities and Related Narrow Band Phenomena, R.D. Parks, ed. (Plenum Press, New York, 1977) p. 471.
CD. Bredl, F. Steglich / Specific heat of CeA12 in high magnetic fields [ 131 A. Benoit, J. Flouquet, M. Ribault and M. Chapellier, Preprint. [ 141 K. Andres, J.E. Graebner and H.R. Ott, Phys. Rev. Lett. 35 (1975) 1779. [ 151 K.D. Schotte and U. Schotte, Phys. Lett. 55A (1975) 38. [ 161 F. Steglich, Z. Phys. B23 (1976) 331. [17] CD. Bredl, F. Steglich and K.D. Schotte, to be published.
289
[ 181 F. Steglich, W. Franz, W. Seuken and M. Loewenhaupt, Physica 86-88B (1977) 503.
[ 191 F. Steglich and M. Loewenhaupt, in: Valence Instabilities and Related Narrow Band Phenomena, R.D. Parks, ed. (Plenum Press, New York, 1977) p. 467. [ 201 See recent work by Doniach et al., cf. R. Jullien, J. Fields and S. Doniach, Phys. Rev. Lett. 38 (1977) 1500, and refs. cited therein.
1
SPECIFIC HEAT OF CeA12 IN HIGH MAGNETIC FIELDS t C.D. BREDL and F. STEGLICH II. Physikalisches Institut der Universitiit zu Kdn, 0.5000 Kdn 41, W.-Germany Specific heat measurements on CeA12 show that in this compound the Kondo effect is of comparable strength (kBTK) to the Ce-Ce interactions which yield magnetic _order below TN 2 4 K. TK E 6 K is estimated from (a) the fact that the large specific heat coefficient y magnetic field up to 5 T, and (b) from the specific heats of (Ce,Lar_,)Al2 alloys (x < 0.7) in 5 T.
mining the strength of the magnetic Ce-Ce interactions. This may be inferred from specific heat results (in zero field) which show a distinct increase of the ordering temperature TN under application of a pressure of 7 kbar [12]. Thus, it appears to be of great importance with respect to the nature of this LTP to know the order of magnitude of the “characteristic temperature” TK. It is the purpose of this paper to estimate TK by analy-
The intermetallic CeA12 is often looked upon as an archetypical “Kondo compound”. This term is justified for two reasons: first, at sufficiently high temperatures (where Ce-Ce interactions are suppressed by thermal fluctuations [ 11) Kondo anomalies, modified by crystal field (CF) effects, are observed in several physical quantities [2-51, and secondly, high-field magnetization measurements display a loss of the 4f shell occupancy under applied pressure (a “nonmagnetic” or “mixed valent” state of the Ce ions is expected for 40 kbar) [6]. Such a pressure-induced demagnetization of the local 4f shell is well known for metallic Ce as well as for dilute Kondo alloys containing Ce impurities [7]. For a dilute Kondo alloy the energy kBTK, which measures the strength of the interaction between a single 4f shell and the conduction band states, can easily be determined from low temperature and zero field experiments. However, in the case of the “Kondo compound”, CeA12, this is prevented by the magnetic interactions between Ce ions: a phase transition at TN - 4 K has been established for CeA12 by several macroscopic properties [8]. The phase boundary in the (field, temperature) plane of its low temperature phase (LTP), as determined by magnetization [9] and magnetostriction [lo] measurements, is typical of a (not necessarily simple) antiferromagnet. Although CeA12 is magnetically ordered in its LTP, as proved by inelastic neutron scattering [ 51 and more directly by recent neutron diffraction experiments [ 111, the Kondo effect seems to play an important role in deter-
-1
CIT
Ce Al,
b&i-i 21
1.8
1.2
Cl6 al35 0
L
8
12
T2(K2)
Fig. 1. The specific heat of CeAl2 single crystals (per mole) in zero magnetic field and in ?.l and 5.0 T parallel to the [ 1001 and [ 1 lo] directions, plotted as C/T vs. T2.
t Work performed within the research program of the Sonderforschungsbereich 125 Aachen/Jiilich/Kbln. 286
CD. Bred4 F. Steglich / Specific heat of CeAl2 in high magnetic fields
zing specific heat data obtained from CeAIs and (Ce,La1_,)A12 alloys at low temperatures and in high magnetic fields. In fig. 1 the specific heat Cper mole is shown in a plot C/TVS. T2 for two CeA12 single crystals with [IOO] and [llO] axes parallel to the magnetic field (B = 0,3.1, and 5 T). In zero field a distinct anomaly is observed at Ti - 3.9 K. The dependence of TN on the magnetic field is in qualitative accordance with the boundary of the LTP as concluded from other measurements [9,10]. The remarkable anisotropy, visible in fig. I, confirms magnetization measurements [9] from which the [ 1 lo] direction was inferred to be magnetically easier than the [ 1001 direction. Surprisingly, the data in fig. 1 can be extrapolated at low temperatures to an intercept y applied field. A y value of this size is highly unusual for a magnetically ordered metal. On the other hand, the “Kondo compound”, CeAls , without magnetic order at low temperature [ 131 exhibits y = 1620 mJ/mole K2 [ 141. This huge value might be attributed to a narrow band (width kBTK ce*13) of quasiparticle states at the Fermi level due to strong electron-electron interactions in the “Kondo ground state” [15]. As was mentioned above, and is supported further by the large y value, the Kondo effect remains important in the LTP of CeA12 (though magnetically ordered). This may also be concluded from the observation [ 11, 131 that the effective magnetic moment per Ce ion, peff, in the LTP is evidently reduced compared to p7 corresponding to the I’7 CF ground state of the free ion. One could argue that in this LTP the majority of conduction electron states is governed by magnetic order, whereas a fraction still participates in the Kondo effect, thus giving rise to a narrow band (at the Fermi level). Its (quasiparticle) density of states would, of course, be considerably smaller than the one expected in the absence of magnetic order. In this context (and referring to Schotte and Schotte [ 15]), the field independence of y means a constant density of states in a field up to B = 5 T, providing a lower limit of the width, A, of this band, i.e. of the characteristic temperature TK = A/kn > Ig’lpBB/kB = 4.8 K. Here g’ = -1.43 is the Land6 factor for Ce3+ in the I’7 doublet state [ 161 (the other symbols have their usual meaning). In fig. 2 the specific heat C* per formula unit CeAla, divided by kg, is shown as a function of temperature (logarithmic scale) for the CeA12 crystal with its [ 1001
0.6‘.
Q
281
0.0T
+
3.1 T
l
5.0T
0.6 ..
0.1, ..
0.2 ..
0 i 0.3
1
3
Fig. 2. The specific heat, C*, per formula unit CeAla, divided by kg, versus temperature (logarithmic scale), obtained in magnetic fields of 0, 3.1, and 5.0 T parallel to the [ 1001 direction. The thick lines represent closely spaced data points, whereas the fine lines are merely interpolated between data points.
axis parallel to the field. It is found that application of 5 T not only depresses the ordering temperature TN, but also results in a broad specific heat anomaly at 6 K, which displays the Zeeman splitting of the I’7 doublet of the single Ce3+ ions. Since, unfortunately, no higher magnetic field was available to completely suppress antiferromagnetic order and, thus, completely resolve this “Schottky type” anomaly, we have mag netically diluted CeA12 with LaA12. In fig. 3 the specific heats of several (Ce,Lar_,)Ala alloys (1.5 a/o < x < 70 a/o), measured in B = 5 T, are also presented in the form C./k, vs. T (logarithmic scale). Besides a CF induced minimum above 5 K [ 161 and some residual influence of magnetic interaction effects at the low temperature end, these curves exhibit a “Schottky type” shape. The result for the most dilute alloy has previously [ 161 been compared by one of us to a theoretical model of Schotte and Schotte [ 151, who have calculated the thermodynamic properties of dilute Kondo alloys in high magnetic fields. It was shown [ 161 that a fit of this model to the zero field data yiel-
CD. Bredl, F. Steglich / Specific heat of CeAI2 in high magnetic fields
288
OX
CTk,
%(Steglich)
+
1.5
n
10 v.
[ 161 to CeA12, is caused by an increased lattice pressure due to a 1% decrease of the mean lattice spacing [ 18, 191. To conclude, analysis of low temperature and high field specific heat reveals for CeA12 a “characteristic temperature” TK z 6 K, which is of comparable size to the magnetic ordering temperature TN = 4 K. (Owing to the existence of short range order [ 171, kB TN presumably underestimates the actual magnetic interaction energy to which kBTK has to be compared). Apparently, the competition between magnetic order and Kondo effect can result in exciting low temperature behavior of a “Kondo compound”, e.g. a nonmagnetic “Kondo ground state” (CeAls) or weak antiferromagnetism (CeA12). The impact of the experimental data available leads one to expect a deeper understanding of these materials to develop within the near future [20].
Bz5.01
0.3
0.2
0.1
0
0.3
1
3
10 T(K)
Fig. 3. Specific heat per formula unit CeA12, C*, in units of kB versus temperature (logarithmic scale), obtained from Ce,Laf__&2 samples (x = 0.015 1161, 0.1, 0.2,0.4,0.7) in an external magnetic field of 5 T.
ded a reasonable value of T~(0.3 K), whereas TK 0.75 K was obtained in 5 T. Since the “Schottky type” anomalies shift to higher temperatures upon increasing magnetic field, the observed increase of TK as a function of B might simply be caused by the “Korringa type” increase of the electronic scattering rate as a function of temperature. This effect is not accounted for in the Schotte model [ 1.51. A fit of the model to the specific heats of the (Ce,La1_,)A12 alloys in a field of 5 T, therefore, provides an upper limit of TK. The results of this fit, which will be explained in more detail elsewhere [ 171, are A/kn = 0.4, 1.05, 1.6, and 2.7 K (for x = 0.1,0.2,0.4, and 0.7, respectively). If these values of A/kn as a function of x are mean squared and extrapolated to x = 1, an upper limit of TK < 7.5 K is obtained for CeA12. The same magnitude is inferred from the low temperature (T > 4 K) width of the quasielastic line in the magnetic neutron spectra of CeA12 [S] . Furthermore, TK = (5 + 2) K is determined from the low temperature resistivity of CeA12 when dilutely dissolved in a nonmagnetic (La, Y)Ala host with the same mean lattice spacing [ 18,1]. The one order of magnitude increase of TK as observed, when going from dilute (Ce, L&U2
We wish to thank Dr. L. Kay Nicholson for critically reading this manuscript.
References [l] F. Steglich, Festkorperprobleme
(Advances in Solid State Physics), vol. 17, J. Treusch, ed. (Braunschweig, Vieweg, 1977) p. 319. (21 K.H.J. Buschow and H.J. van Daal, Phys. Rev. Lett. 23 (1969) 408; B. Cornut and B. Coqblin, Phys. Rev. B5 (1972) 4541. [ 31 W.M. Swift and W.E. Wallace, J. Phys. Chem. Solids 29 (1969) 2053. [4] C. Deenadas, A.W. Thompson, R.S. Craig and W.E. Wallace, J. Phys. Chem. Solids 32 (1971) 1853. [5] M. Loewenhaupt and F. Steglich, Physica 86-88B (1977) 187. [6] B. Barbara, H. Bartholin, D. Florence, M.F. Rossignol and E. Walker, Physica 86-88B (1977) 177. [7] M.B. Maple and D.K. Wohlleben, Magnetism and Magnetic Materials - 1973 (AIP, New York, 1974) p. 447. [8] E. Walker, H.G. Purwins, M. Landolt and F. Hullinger, J. Less Common Metals 33 (1973) 203; R.W. Hill and J.M. Machado da Silva, Phys. Lett. 30 (1969) 13. [9] B. Barbara, M.F. Rossignol, H.G. Purwins and E. Walker, Solid State Commun. 17 (1975) 1525. [Ia‘1 M. Croft, I. Zoric and R.D. Parks, Preprint. 111I B. Barbara, J.X. Boucherle, J.L. Buevoz, M.F. Rossignol and J. Schweizer, Proc. Int. Conf. on Rare Earths and Actinides, Durham (1977). !I A. Berton, J. Chaussy, G. Chouteau, B. Cornut, J. Peyrard 112 and R. Tournier, in: Valence Instabilities and Related Narrow Band Phenomena, R.D. Parks, ed. (Plenum Press, New York, 1977) p. 471.
CD. Bredl, F. Steglich / Specific heat of CeA12 in high magnetic fields [ 131 A. Benoit, J. Flouquet, M. Ribault and M. Chapellier, Preprint. [ 141 K. Andres, J.E. Graebner and H.R. Ott, Phys. Rev. Lett. 35 (1975) 1779. [ 151 K.D. Schotte and U. Schotte, Phys. Lett. 55A (1975) 38. [ 161 F. Steglich, Z. Phys. B23 (1976) 331. [17] CD. Bredl, F. Steglich and K.D. Schotte, to be published.
289
[ 181 F. Steglich, W. Franz, W. Seuken and M. Loewenhaupt, Physica 86-88B (1977) 503.
[ 191 F. Steglich and M. Loewenhaupt, in: Valence Instabilities and Related Narrow Band Phenomena, R.D. Parks, ed. (Plenum Press, New York, 1977) p. 467. [ 201 See recent work by Doniach et al., cf. R. Jullien, J. Fields and S. Doniach, Phys. Rev. Lett. 38 (1977) 1500, and refs. cited therein.