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Transport coefficients of intermediate valent CeNix intermetallic compounds
536
Journal
TRANSPORT COEFFICIENTS COMPOUNDS U. GOTTWICK,
K. GLOOS,
of Magnetism
OF INTERMEDIATE
S. HORN,
and Magnetic
VALENT CeNi,
F. STEGLICH
Materials 47&48 (1985) 536-538 North-Holland, Amsterdam
INTERMETALLIC
and N. GREWE
Instiiut ftir Festkiirperphysik, Technische Hochschule Darmstadt, and SFB 65, D - 6100 Darmstadt, Fed. Rep. Germany
The resistivity and thermoelectric power of intermediate valent CeNi, intermetallic compounds and their La homologs were determined in the temperature ranges 1.5 to 350 K and 1000 K, respectively. The results are discussed in the frame of a theory for the quasiparticle density of states of intermediate valent systems.
From recent BIS experiments on various intermediate valent (IV) CeNi, intermetallic compounds a systematic increase of the 4f-conduction electron hybridization with increasing Ni concentration x was deduced [l]. The corresponding change in Ce valency was concluded to be rather small. A good emperical measure of the hybridization strength is provided by the temperature at which a large peak in the thermoelectric power (TEP) occurs [2]. For Ce-based IV systems this peak is always found to be positive [2,3]. In order to check these ideas we have performed TEP and resistivity measurements in the temperature range from 1.5 to 350 K and 1000 K, respectively, on Ce,Ni,, CeNi, CeNi,, CeNi,, Ce,Ni, and CeNi, and their La homologs. The results are analyzed using the phenomenological picture of quasiparticle bands developed by Hirst [4]. The most intriguing feature of fig. la is the giant S(T) maximum of = 55 pV/K near T= 110 K for CeNi. For CeNi, this peak decreases and shifts to T 2: 200 K. Clearly, this tendency continues with further increase in Ni concentration x, although at low temperatures a more complex behavior is apparent: the rise in S(T) for both CeNi, and Ce,Ni, seen at the high-T end indicates TEP peaks above T= 350 K. At low temperature the TEP of CeNi,, Ce,Ni, and CeNi, is determined by scattering from Ni-derived 3d states [6] and by the phonon-drag. For CeNi, with highest x, another maximum is expected at very high T [6], in accordance with the tendency seen in those compounds with lower x. (see: note added in proof) The Ce-richest system, Ce,Ni,, exhibits a doublepeak structure in S(T). Comparison with the result for the La homolog (fig. lb) suggests that the low-lying peak is due to the phonon drag. After subtraction of the 0304-8853/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V
phonon-drag contribution one is left with a negative diffusion TEP at low T being characteristic for a Kondo lattice [2,3,7]. The TEP behavior of the other five compounds puts them in the class of IV materials [2,3]. In order to see whether the anomalous scattering responsible for the large TEP peak can also be detected in the resistivity, we compare in figs. 2a and b p(T) results of the Ce compounds with those of their La homologs. Whereas at low temperatures, the Ce systems exhibit higher resistivities, they decrease below the La values at higher 7’. It is likely that in these cases the effect of the increased scattering is masked by an increased (5d6s) conduction-band density of states, due to an upward shift of the Fermi level with more delocalized 4f electrons. This agrees well with the expectation that the high peak positions of the measured TEP curves correspond to substantial hybridization. In fact, for
0
100
200
TIK)
Fig. 1. Thermopower as function of temperature for various CeNi, compounds (a) and La homologs (b). Note change in the vertical scale for CeNi in (a). To complete the data the result for LaNi was taken from ref. (51. A RENi, 0 RENi z, 0 RENi 3, w RE,Ni,, v RENi,, 0 RE,Ni,.
U. Gottwick et al. / Transport coefficients of IV- CeNi, compoundr
CeNi with weakest hybridization Ap = pCeNi - pLaNi stays positive up to T= 670 K. A peak in Ap(T) is found near T = 140 K (inset of fig. 2) i.e. not too far from the position of the TEP peak for this compound. As a first step towards a theoretical description one can try to use a relaxation time T(E), which is derived from the density pc40 (E) of 4f-states in the phenomenological picture of Hirst, for a calculation of transport integrals. A connection of the form [8) h/~~ = -2
Im Tkk = V’~(~f)(c,),
(1)
where Tkkg is the transition matrix and V a hybridization matrix element, is known to hold within Anderson’s impurity model. Although in a perfect lattice at T = 0 one expects rk = cc for zk = eF this relation is supposed to become valid at higher temperatures, where coherence is destroyed by thermal fluctuations. Due to general arguments [9] the effect of coherence on the electrical resistivity should be larger than on the TEP. Therefore, the use of eq. (1) might be reasonable for a calculation of the TEP, but is poor for the low-temperature behavior of the resistivity, where a better knowledge of the T-matrix is needed [lo]. Hirst proposes a Lorentzian form of the 4f-density of states with width F = (Q++ Q-)A, where A = 7N,V2 and the sum rule strengths Q+= 6, Q-= 1 in case of Ce. In agreement with recent results, pc4” should have an integrated weight of Q’z + Q-(1 -z), determined by the valency Y = 3 + z [ll]. Although it is known [12] that the shape of pc40 near and below the Fermi energy deviates from a Lorentzian, this form is used for simand the plicity. The position c,, of the maximum Anderson width A are treated as fit parameters. Within
0
500
T(K)
1000
0
500
T(K)
531
Boltzmann theory and relaxation TEP for an isotropic system reads
time
approach
the
An explicit result follows with the above assumptions: AT
S=-
co - CF)
Az2(
B2 + T*
’
lel
’
A similar expression holds for the conductivity [9]. In the following we analyze the two systems CeNi and CeNi 2 showing the large IV-derived TEP maximum along with two prototypical IV compounds, CeSn, and CeBe,, [13]. By plotting T/S against T* in fig. 3, A and B, i.e. co -eF and F, can be determined. It is remarkable that the analytic form of eq. (3) is well obeyed over the whole measured temperature range, which contains the complete TEP peak, except at low T, where an upturn in T/S indicates a stronger decrease of the TEP, in accord with the expected influence of coherence. As a further consistency check the valency at T = 0 can be determined from the condition l-z=
‘F dc pc4’)( c) / -rn
(4)
and the linear specific heat coefficient yc4u from the resulting pc40(cF). Table 1 shows a comparison of yc4n
1000
Fig. 2. Resitivity as function of temperature for various CeNi, compounds (a) and La homologs (b). Inset shows difference AP = PCeNi - pLaN, VS. T. A RENi, 0 RENi,, 0 RENi,, n RE,Ni,, v RENi,, 0 RE,Ni,.
0
200*
300’ T2( K21
Fig. 3. Thermopower S(T) of CeNi and CeNi, in a plot T/S vs. T’. Straight line is consistent with S(T) = AT/(B’ + T2), see eq. (3). Literature results on CeSn, and CeBe,, 1131 are included in the inset.
U. Gorrwick et al. / Trcrnspor~ coefficienls 01 I V-CeNl I: compounds
538
Table 1 Analysis of the thermopower results shown in fig. 3. F and eF - c0 are the half width and position density
of states.
Y and
yc40 are the Ce valence
and 4f-derived
linear
specific-heat
coefficient
of the peak in the 4f-one particle estimated from TEP. yexp is the
specific-heat coefficient directly measured Compound
CeNi
CeNi,
CeSn ,
CeBer,
7;,,,(K) co - r,(meV) I‘(meV) f;14/) (mJ/molK2)
110 5.8 15.9 853.21
200 4.2 33.9 403.17
220 6.0 30.7 443.18
120 1.2 17.3 113.22
ycxP b/m01
65 ’
K2)
24.9 b
53&10
d
115*10
‘I
a ref. [13], ’ ref. [14], ’ ref. [15].
with
the experimental
value
together
with
cc, - eF and
A qualitative agreement in the relation between experimental and theoretical specific heat data is apparent. Also an increased hybridization in CeNi, relative to CeNi can be recognized in accord with the spectroscopic data and the shift of the TEP peak. The valency in all four IV compounds apparently shows little variation and always stays close to 3.2. The low and nearly constant value of the Ce-valence agrees well with recent spectroscopic results [l], which also support the above result connecting the shifted TEP peak with an increased hybridization.
References
r.
]21
]3] I41 ]51 ]61
]‘I
Note added in proof The expected high-T peak in S(T) of CeNi, has recently in fact been observed at T,,, = 650 K (R.V. Lutsiv, M.D. Koterlin, 0.1. Babich and 0.1. Bodak, Sov. Phys. Solid State 26 (1984) 7161.
]81 ]91 I101 ]Jll ]I21 (131 ]I41 ]I51
F.U. Hillebrecht, J.C. Fuggle, G.A. Sawatzky, M. Campagna, 0. Gunnarsson and K. Schonhammer, Phys. Rev. B34 (1984) 1977. D. Jaccard and J. Sierro, in: Valence Instabilities, eds. P. Wachter and H. Boppart (North-Holland, Amsterdam, 1982) p. 409. S. Horn, W. Klamke and F. Steghch, ibid., p. 459. L.L. Hirst, Phys. Rev. B15 (1977) 1. E. Gratz. J. Magn. Magn. Mat. 24 (1981) 1. S. Cabus, K. Gloos, U. Gottwick, S. Horn, M. Klemm, J. Kubler, F. Steglich and R.D. Parks, Solid State Commun. 51 (1984) 909. A.E. Sovestnov, V.A. Shaburov. I.A. Markova, EM. Savitskii. O.D. Chistyakov and T.M. Shkatova, Sov. Phys. Solid State 23 (1983) 1652. D.C. Langreth, Phys. Rev. 150 (1966) 516. K. Gloos, Diploma Thesis, TH Darmstadt (1984) (unpublished). N. Grewe, work in progress. Y. Kuramoto, Proc. NATO-AS1 Summer School, Pearson College, Vancouver Island, Canada (1983) forthcoming. N. Grewe, Z. Phys. B53 (1983) 271. J.R. Cooper, C. Rizzuto and G. Olcese, J. de Phys. 32X1 (1971) 1136. J.M. Machado da Silva and R.W. Hill, J. Phys. C 5 (1972) 1584. D. Gignoux, F. Givord, R. Lemaire and F. Tasset, J. Less-Common Metals 94 (1983) 165.
Journal
TRANSPORT COEFFICIENTS COMPOUNDS U. GOTTWICK,
K. GLOOS,
of Magnetism
OF INTERMEDIATE
S. HORN,
and Magnetic
VALENT CeNi,
F. STEGLICH
Materials 47&48 (1985) 536-538 North-Holland, Amsterdam
INTERMETALLIC
and N. GREWE
Instiiut ftir Festkiirperphysik, Technische Hochschule Darmstadt, and SFB 65, D - 6100 Darmstadt, Fed. Rep. Germany
The resistivity and thermoelectric power of intermediate valent CeNi, intermetallic compounds and their La homologs were determined in the temperature ranges 1.5 to 350 K and 1000 K, respectively. The results are discussed in the frame of a theory for the quasiparticle density of states of intermediate valent systems.
From recent BIS experiments on various intermediate valent (IV) CeNi, intermetallic compounds a systematic increase of the 4f-conduction electron hybridization with increasing Ni concentration x was deduced [l]. The corresponding change in Ce valency was concluded to be rather small. A good emperical measure of the hybridization strength is provided by the temperature at which a large peak in the thermoelectric power (TEP) occurs [2]. For Ce-based IV systems this peak is always found to be positive [2,3]. In order to check these ideas we have performed TEP and resistivity measurements in the temperature range from 1.5 to 350 K and 1000 K, respectively, on Ce,Ni,, CeNi, CeNi,, CeNi,, Ce,Ni, and CeNi, and their La homologs. The results are analyzed using the phenomenological picture of quasiparticle bands developed by Hirst [4]. The most intriguing feature of fig. la is the giant S(T) maximum of = 55 pV/K near T= 110 K for CeNi. For CeNi, this peak decreases and shifts to T 2: 200 K. Clearly, this tendency continues with further increase in Ni concentration x, although at low temperatures a more complex behavior is apparent: the rise in S(T) for both CeNi, and Ce,Ni, seen at the high-T end indicates TEP peaks above T= 350 K. At low temperature the TEP of CeNi,, Ce,Ni, and CeNi, is determined by scattering from Ni-derived 3d states [6] and by the phonon-drag. For CeNi, with highest x, another maximum is expected at very high T [6], in accordance with the tendency seen in those compounds with lower x. (see: note added in proof) The Ce-richest system, Ce,Ni,, exhibits a doublepeak structure in S(T). Comparison with the result for the La homolog (fig. lb) suggests that the low-lying peak is due to the phonon drag. After subtraction of the 0304-8853/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V
phonon-drag contribution one is left with a negative diffusion TEP at low T being characteristic for a Kondo lattice [2,3,7]. The TEP behavior of the other five compounds puts them in the class of IV materials [2,3]. In order to see whether the anomalous scattering responsible for the large TEP peak can also be detected in the resistivity, we compare in figs. 2a and b p(T) results of the Ce compounds with those of their La homologs. Whereas at low temperatures, the Ce systems exhibit higher resistivities, they decrease below the La values at higher 7’. It is likely that in these cases the effect of the increased scattering is masked by an increased (5d6s) conduction-band density of states, due to an upward shift of the Fermi level with more delocalized 4f electrons. This agrees well with the expectation that the high peak positions of the measured TEP curves correspond to substantial hybridization. In fact, for
0
100
200
TIK)
Fig. 1. Thermopower as function of temperature for various CeNi, compounds (a) and La homologs (b). Note change in the vertical scale for CeNi in (a). To complete the data the result for LaNi was taken from ref. (51. A RENi, 0 RENi z, 0 RENi 3, w RE,Ni,, v RENi,, 0 RE,Ni,.
U. Gottwick et al. / Transport coefficients of IV- CeNi, compoundr
CeNi with weakest hybridization Ap = pCeNi - pLaNi stays positive up to T= 670 K. A peak in Ap(T) is found near T = 140 K (inset of fig. 2) i.e. not too far from the position of the TEP peak for this compound. As a first step towards a theoretical description one can try to use a relaxation time T(E), which is derived from the density pc40 (E) of 4f-states in the phenomenological picture of Hirst, for a calculation of transport integrals. A connection of the form [8) h/~~ = -2
Im Tkk = V’~(~f)(c,),
(1)
where Tkkg is the transition matrix and V a hybridization matrix element, is known to hold within Anderson’s impurity model. Although in a perfect lattice at T = 0 one expects rk = cc for zk = eF this relation is supposed to become valid at higher temperatures, where coherence is destroyed by thermal fluctuations. Due to general arguments [9] the effect of coherence on the electrical resistivity should be larger than on the TEP. Therefore, the use of eq. (1) might be reasonable for a calculation of the TEP, but is poor for the low-temperature behavior of the resistivity, where a better knowledge of the T-matrix is needed [lo]. Hirst proposes a Lorentzian form of the 4f-density of states with width F = (Q++ Q-)A, where A = 7N,V2 and the sum rule strengths Q+= 6, Q-= 1 in case of Ce. In agreement with recent results, pc4” should have an integrated weight of Q’z + Q-(1 -z), determined by the valency Y = 3 + z [ll]. Although it is known [12] that the shape of pc40 near and below the Fermi energy deviates from a Lorentzian, this form is used for simand the plicity. The position c,, of the maximum Anderson width A are treated as fit parameters. Within
0
500
T(K)
1000
0
500
T(K)
531
Boltzmann theory and relaxation TEP for an isotropic system reads
time
approach
the
An explicit result follows with the above assumptions: AT
S=-
co - CF)
Az2(
B2 + T*
’
lel
’
A similar expression holds for the conductivity [9]. In the following we analyze the two systems CeNi and CeNi 2 showing the large IV-derived TEP maximum along with two prototypical IV compounds, CeSn, and CeBe,, [13]. By plotting T/S against T* in fig. 3, A and B, i.e. co -eF and F, can be determined. It is remarkable that the analytic form of eq. (3) is well obeyed over the whole measured temperature range, which contains the complete TEP peak, except at low T, where an upturn in T/S indicates a stronger decrease of the TEP, in accord with the expected influence of coherence. As a further consistency check the valency at T = 0 can be determined from the condition l-z=
‘F dc pc4’)( c) / -rn
(4)
and the linear specific heat coefficient yc4u from the resulting pc40(cF). Table 1 shows a comparison of yc4n
1000
Fig. 2. Resitivity as function of temperature for various CeNi, compounds (a) and La homologs (b). Inset shows difference AP = PCeNi - pLaN, VS. T. A RENi, 0 RENi,, 0 RENi,, n RE,Ni,, v RENi,, 0 RE,Ni,.
0
200*
300’ T2( K21
Fig. 3. Thermopower S(T) of CeNi and CeNi, in a plot T/S vs. T’. Straight line is consistent with S(T) = AT/(B’ + T2), see eq. (3). Literature results on CeSn, and CeBe,, 1131 are included in the inset.
U. Gorrwick et al. / Trcrnspor~ coefficienls 01 I V-CeNl I: compounds
538
Table 1 Analysis of the thermopower results shown in fig. 3. F and eF - c0 are the half width and position density
of states.
Y and
yc40 are the Ce valence
and 4f-derived
linear
specific-heat
coefficient
of the peak in the 4f-one particle estimated from TEP. yexp is the
specific-heat coefficient directly measured Compound
CeNi
CeNi,
CeSn ,
CeBer,
7;,,,(K) co - r,(meV) I‘(meV) f;14/) (mJ/molK2)
110 5.8 15.9 853.21
200 4.2 33.9 403.17
220 6.0 30.7 443.18
120 1.2 17.3 113.22
ycxP b/m01
65 ’
K2)
24.9 b
53&10
d
115*10
‘I
a ref. [13], ’ ref. [14], ’ ref. [15].
with
the experimental
value
together
with
cc, - eF and
A qualitative agreement in the relation between experimental and theoretical specific heat data is apparent. Also an increased hybridization in CeNi, relative to CeNi can be recognized in accord with the spectroscopic data and the shift of the TEP peak. The valency in all four IV compounds apparently shows little variation and always stays close to 3.2. The low and nearly constant value of the Ce-valence agrees well with recent spectroscopic results [l], which also support the above result connecting the shifted TEP peak with an increased hybridization.
References
r.
]21
]3] I41 ]51 ]61
]‘I
Note added in proof The expected high-T peak in S(T) of CeNi, has recently in fact been observed at T,,, = 650 K (R.V. Lutsiv, M.D. Koterlin, 0.1. Babich and 0.1. Bodak, Sov. Phys. Solid State 26 (1984) 7161.
]81 ]91 I101 ]Jll ]I21 (131 ]I41 ]I51
F.U. Hillebrecht, J.C. Fuggle, G.A. Sawatzky, M. Campagna, 0. Gunnarsson and K. Schonhammer, Phys. Rev. B34 (1984) 1977. D. Jaccard and J. Sierro, in: Valence Instabilities, eds. P. Wachter and H. Boppart (North-Holland, Amsterdam, 1982) p. 409. S. Horn, W. Klamke and F. Steghch, ibid., p. 459. L.L. Hirst, Phys. Rev. B15 (1977) 1. E. Gratz. J. Magn. Magn. Mat. 24 (1981) 1. S. Cabus, K. Gloos, U. Gottwick, S. Horn, M. Klemm, J. Kubler, F. Steglich and R.D. Parks, Solid State Commun. 51 (1984) 909. A.E. Sovestnov, V.A. Shaburov. I.A. Markova, EM. Savitskii. O.D. Chistyakov and T.M. Shkatova, Sov. Phys. Solid State 23 (1983) 1652. D.C. Langreth, Phys. Rev. 150 (1966) 516. K. Gloos, Diploma Thesis, TH Darmstadt (1984) (unpublished). N. Grewe, work in progress. Y. Kuramoto, Proc. NATO-AS1 Summer School, Pearson College, Vancouver Island, Canada (1983) forthcoming. N. Grewe, Z. Phys. B53 (1983) 271. J.R. Cooper, C. Rizzuto and G. Olcese, J. de Phys. 32X1 (1971) 1136. J.M. Machado da Silva and R.W. Hill, J. Phys. C 5 (1972) 1584. D. Gignoux, F. Givord, R. Lemaire and F. Tasset, J. Less-Common Metals 94 (1983) 165.