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A two-impurity model for the underscreened Kondo problem
ELSEVIER
Physica B 230-232 (1997)454-456
A two-impurity model for the underscreened Kondo problem K. Le Hur*, B. Coqblin Laboratoire de Physique des Solides, Universitb Paris-Sud, Brit. 510, 91405 Orsay, France
Abstract
The underscreened Kondo effect is studied within a model of two impurities with S = 1 interacting with the conduction 1 band and via a classical interimpurity coupling KS1 "$2. A completely asymmetric situation is obtained with one s = component strongly screened by the Kondo effect and the other one almost free to yield direct or indirect magnetism, which shows finally a possible coexistence between a strong local Kondo effect and the RKKY interaction between the two S = 1 spins, for some values of the K parameter. Keywords: Kondo compounds; Two-impurity model; Underscreened Kondo problem; Uranium compounds
K o n d o Cerium compounds have been extensively studied from both an experimental and theoretical point of view. In this case, the K o n d o effect is well described by either the s-f exchange Hamiltonian with a S f = ½ spin screened by only one conduction electron channel [1] or the so-called Coqblin-Schrieffer Hamiltonian [2] when orbital degeneracy and spin-orbit coupling are taken into account; in the two preceding cases, there is an equal number of 4f and conduction electrons, The ground state of the regular K o n d o effect is a nonmagnetic singlet state in the case of a single impurity [1] and the low-temperature properties are characterized by a Fermi-liquid behaviour. In the case of heavy-fermion compounds, there is a strong competition between the K o n d o effect and the magnetic ordering, which yields either nonmagnetic or magnetically ordered Cerium K o n d o compounds [3,4]. On the other hand, some Uranium compounds, such as UPt3, present also a heavy-fermion behaviour and are also superconducting. UPt3 has an outstanding behaviour, since it undergoes
* Corresponding author.
a transition to an antiferromagnetic ordering with a tiny ordered magnetic moment of 0.02 __+0.01~B below a Neel Temperature TN ~ 5 K [5] and becomes superconducting below Tc ~ 0.5 K [6]. A heavy-fermion behaviour characterized by a large electronic specific heat constant ,-~ 0.4 J/mol K 2 [7] and a T 2 term of the resistivity [8] is observed in UPt3 at low temperatures. A third characteristic temperature Ts = 17.6K, given by the maximum of the magnetic susceptibility, corresponds approximatively to the onset of spin fluctuations [8]. The heavy-fermion character decreases with pressure [7, 8], while the antiferromagnetic order disappears at roughly 5 kbar [9]. The purpose of the present paper is to present an explanation for the coexistence of the heavy-fermion character and tiny ordered magnetic moments in Uranium compounds such as UPt3. This explanation is based on the "underscreened K o n d o model" which appears to be appropriate to describe the 5f2 configuration of Uranium atoms. However, this explanation can be criticized since very small ordered magnetic moments have been also evidenced experimentally in Cerium compounds such as CeA13 [10].
0921-4526/97/$17.00 Copyright © 1997 Elsevier ScienceB.V. All rights reserved PII S092 1-4526(96)006 1 2-6
455
K. Le Hur, B. Coqblin / Physica B 230-232 (1997) 454-456
The underscreened K o n d o model corresponds to the case 2S > n, where S is the localized spin and n the number of screening channels coming from conduction electrons [11,12]. We study here the two-impurity K o n d o Hamiltonian in the underscreened model S = 1, n = 1. We use the notations of the paper by Affleck et al. [13] and the twoimpurity K o n d o Hamiltonian is written as
where each ~i corresponds to a ½-spin; we add the constraint that ~1 and ~z on one side and ~3 and z, on the other side are strongly ferromagnetically coupled. The next step is to refermionize again this problem of 4 sets of Pauli matrices ~i by the classical Jordan-Wigner transformation: r[ = d~ e i~2j . . . . . .
o~ = Ho + Hk + Hi, where
H o = fd3ke(k)~t}P~k , Hk = f d3kl fd3k2T~'a~T~k~.[V(k~)*V(k2)Sl + V( - kl)*V( - kz)Sz],
(2)
Hi = KS1S2, where $1, $2 are two S = 1 impurity spins, V(k) is proportional to the k-f hybridization term and to e ik~, if the two impurities are taken symmetrically at + R/2 and - R / 2 . The second term Hk is the K o n d o two-impurity Hamiltonian but, as usual, we add the two-impurity direct interaction Hi. Then, we use several transformations on the Hamiltonian equation (1) in order to show finally the coexistence between the K o n d o effect and magnetism, indirect or direct according to the K values. First of all, projecting the K o n d o problem on isoenergy surfaces, using the particle-hole symmetry of the new Hamiltonian and taking odd and even combinations for the conduction electron operators transforms Hk in the sum of two terms proportional to ($1 + $2) and ($1 - $ 2 ) , as previously done for the two s =½ impurity case [14, 15]. Then, we bosonize the resulting one-dimensional Hamiltonian Ho + Hk by using the standard one-dimensional relations between Bose and Fermi fields. However, in contrast to the s = ½ case, we have to decompose the two S = 1 spins into 81 : "gl + ~'2, 8 2 = T 3 + "c4,
(with nj = 0, 1).
(4)
(1)
(3)
Finally, we obtain a resulting symmetric Hamiltonian given as a function of the operators d i for the f-electrons and ~u for the conduction electrons; in fact, in order to diagonalize the Hamiltonian, we have used a particular mean-field approximation [15] which leads us to neglect all the terms composed of more than four operators. We then discuss the results as a function of the strength and sign of the interimpurity coupling K. The main results can be summarized as follows: When K = 0, within the present mean-field approximation, the two initial S = 1 spins are decoupled and we have obtained the following picture: the two components z2 and z4 are completely decoupled from the conduction band, while the two components zl and z3 give two independent Kondo effects. Thus, when K = 0, one s = ½ component of each S = 1 spin is completely screened by the Kondo effect and the other s = ½ component keeps its magnetism and is completely decoupled from the Fermi-liquid component. - W h e n K < 0, the ferromagnetic coupling between the two S = 1 spins is direct and consequently the system goes from two disconnected K o n d o effects for K = 0 to a K o n d o effect in the configuration S = 2 and n = 2 for K ~ - ~ . For K < 0 the direct coupling between the two S = 1 spins does not disturb the K o n d o effect. - W h e n the interimpurity coupling is antiferromagnetic and moderate, there is no quantum critical point in the phase diagram, which appears to be different from the case of the regular screened s = ½ K o n d o model, where the existence of a critical point appears to be a controversial subject [13]. In fact, the low-energy physics appears clearly in our treatment of the Hamiltonian: the two
456
K. Le Hur, B. Coqblin / Physica B 230-232 (1997) 454-456
components zl and z3 are still strongly screened by the Kondo effect; the resulting Kondo coupling decreases with K due to the two residual moments z2 and z4, which are now weakly ferromagnetically coupled to the conduction electrons (due to the Pauli principle) and can give a magnetic indirect interaction of RKKY type. This antiferromagnetic and indirect K coupling disturbs weakly the strong and local Kondo effect but a good coexistence between these two processes is expected. When K --, + oo, the two S = 1 impurities prefer to form a spin-singlet and to annihilate the Kondo effect. Detailed calculations will be published elsewhere [16]. The same completely asymmetric situation is also obtained for the underscreened Kondolattice with S = 1 decomposed into two s = ½components [16], within a mean-field treatment describing both the Kondo effect and long-range indirect correlations. Thus, the "underscreened Kondo model" with a moderate antiferromagnetic intersite interaction can account for the complex situation of compounds such as UPt3 which has a heavy fermion behaviour and presents a weak magnetic order at very low temperatures. In our model, the tiny ordered magnetic moment is in fact induced by the long-range scale of the RKKY interaction.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10]
[11] [12] [13] [14] [15] [16]
K.G. Wilson, Rev. Mod. Phys. 47 (1975) 773. B. Coqblin and J.R. Schrieffer, Phys. Rev. 185 (1969) 847. S. Doniach, Physica B 91 (1977) 231. B. Coqblin, J. Arispe, J.R. Iglesias, C. Lacroix and K. Le Hur, J. Phys. Soc. Japan 65 (1996) Suppl. B, 64. G. Aeppli, E. Bucher, C. Broholm, J.K. Kjems, J. Baumann and J. Hufnagl, Phys. Rev. Lett 60 (1988) 615. G.R. Stewart, Z. Fisk, J.O. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679; L. Taillefer, K. Behnia, K. Hasselbach, J. Flouquet, S.M. Hayden and C. VeRier, J. Magn. Magn. Mater 90-91 (1990) 623. G.E. Brodale, R.A. Fisher, N.E. Phillips, G.R. Stewart and A.L. Giorgi, Phys. Rev. Lett. 57 (1986) 234. J.O. Willis, J.D. Thomson, Z. Fisk, A. de Visser, J.J.M. Franse and A. Menovsky, Phys. Rev. B 31 (1985) 1654. S.M. Hayden, L. Taillefer, C. Vettier and J. Flouquet, Phys. Rev. B 46 (1992) 8675. A. de Visser, J. Flouquet, J.J.M. Franse, P. Haen, K. Hasselbach, A. Lacerda and L. Taillefer, Physica B 171 (1991) 190. D.M. Cragg and P. Lloyd, J. Phys. C 12 (1979) L215; P. Nozi~res and A. Blandin, J. Physique 41 (1980) 193. P. Coleman and J. Gan, Physica B 171 (1991) 3; J. Gan, P. Coleman and N. Andrei, Phys. Rev. Lett. 68 (1992) 3476. I. Affleck, A.W.W. Ludwig and B.A. Jones, Phys. Rev. B 52 (1995) 9528. B.A. Jones and C.M. Varma, Phys. Rev. Lett. 58 (1987) 843. Phys. Rev. B 40 (1989) 324. C. Sire, C.M. Varma and H.R. Krishnamurthy, Phys. Rev. B 48 (1993) 13 833. K. Le Hur and B. Coqblin, to be published.
Physica B 230-232 (1997)454-456
A two-impurity model for the underscreened Kondo problem K. Le Hur*, B. Coqblin Laboratoire de Physique des Solides, Universitb Paris-Sud, Brit. 510, 91405 Orsay, France
Abstract
The underscreened Kondo effect is studied within a model of two impurities with S = 1 interacting with the conduction 1 band and via a classical interimpurity coupling KS1 "$2. A completely asymmetric situation is obtained with one s = component strongly screened by the Kondo effect and the other one almost free to yield direct or indirect magnetism, which shows finally a possible coexistence between a strong local Kondo effect and the RKKY interaction between the two S = 1 spins, for some values of the K parameter. Keywords: Kondo compounds; Two-impurity model; Underscreened Kondo problem; Uranium compounds
K o n d o Cerium compounds have been extensively studied from both an experimental and theoretical point of view. In this case, the K o n d o effect is well described by either the s-f exchange Hamiltonian with a S f = ½ spin screened by only one conduction electron channel [1] or the so-called Coqblin-Schrieffer Hamiltonian [2] when orbital degeneracy and spin-orbit coupling are taken into account; in the two preceding cases, there is an equal number of 4f and conduction electrons, The ground state of the regular K o n d o effect is a nonmagnetic singlet state in the case of a single impurity [1] and the low-temperature properties are characterized by a Fermi-liquid behaviour. In the case of heavy-fermion compounds, there is a strong competition between the K o n d o effect and the magnetic ordering, which yields either nonmagnetic or magnetically ordered Cerium K o n d o compounds [3,4]. On the other hand, some Uranium compounds, such as UPt3, present also a heavy-fermion behaviour and are also superconducting. UPt3 has an outstanding behaviour, since it undergoes
* Corresponding author.
a transition to an antiferromagnetic ordering with a tiny ordered magnetic moment of 0.02 __+0.01~B below a Neel Temperature TN ~ 5 K [5] and becomes superconducting below Tc ~ 0.5 K [6]. A heavy-fermion behaviour characterized by a large electronic specific heat constant ,-~ 0.4 J/mol K 2 [7] and a T 2 term of the resistivity [8] is observed in UPt3 at low temperatures. A third characteristic temperature Ts = 17.6K, given by the maximum of the magnetic susceptibility, corresponds approximatively to the onset of spin fluctuations [8]. The heavy-fermion character decreases with pressure [7, 8], while the antiferromagnetic order disappears at roughly 5 kbar [9]. The purpose of the present paper is to present an explanation for the coexistence of the heavy-fermion character and tiny ordered magnetic moments in Uranium compounds such as UPt3. This explanation is based on the "underscreened K o n d o model" which appears to be appropriate to describe the 5f2 configuration of Uranium atoms. However, this explanation can be criticized since very small ordered magnetic moments have been also evidenced experimentally in Cerium compounds such as CeA13 [10].
0921-4526/97/$17.00 Copyright © 1997 Elsevier ScienceB.V. All rights reserved PII S092 1-4526(96)006 1 2-6
455
K. Le Hur, B. Coqblin / Physica B 230-232 (1997) 454-456
The underscreened K o n d o model corresponds to the case 2S > n, where S is the localized spin and n the number of screening channels coming from conduction electrons [11,12]. We study here the two-impurity K o n d o Hamiltonian in the underscreened model S = 1, n = 1. We use the notations of the paper by Affleck et al. [13] and the twoimpurity K o n d o Hamiltonian is written as
where each ~i corresponds to a ½-spin; we add the constraint that ~1 and ~z on one side and ~3 and z, on the other side are strongly ferromagnetically coupled. The next step is to refermionize again this problem of 4 sets of Pauli matrices ~i by the classical Jordan-Wigner transformation: r[ = d~ e i~2j . . . . . .
o~ = Ho + Hk + Hi, where
H o = fd3ke(k)~t}P~k , Hk = f d3kl fd3k2T~'a~T~k~.[V(k~)*V(k2)Sl + V( - kl)*V( - kz)Sz],
(2)
Hi = KS1S2, where $1, $2 are two S = 1 impurity spins, V(k) is proportional to the k-f hybridization term and to e ik~, if the two impurities are taken symmetrically at + R/2 and - R / 2 . The second term Hk is the K o n d o two-impurity Hamiltonian but, as usual, we add the two-impurity direct interaction Hi. Then, we use several transformations on the Hamiltonian equation (1) in order to show finally the coexistence between the K o n d o effect and magnetism, indirect or direct according to the K values. First of all, projecting the K o n d o problem on isoenergy surfaces, using the particle-hole symmetry of the new Hamiltonian and taking odd and even combinations for the conduction electron operators transforms Hk in the sum of two terms proportional to ($1 + $2) and ($1 - $ 2 ) , as previously done for the two s =½ impurity case [14, 15]. Then, we bosonize the resulting one-dimensional Hamiltonian Ho + Hk by using the standard one-dimensional relations between Bose and Fermi fields. However, in contrast to the s = ½ case, we have to decompose the two S = 1 spins into 81 : "gl + ~'2, 8 2 = T 3 + "c4,
(with nj = 0, 1).
(4)
(1)
(3)
Finally, we obtain a resulting symmetric Hamiltonian given as a function of the operators d i for the f-electrons and ~u for the conduction electrons; in fact, in order to diagonalize the Hamiltonian, we have used a particular mean-field approximation [15] which leads us to neglect all the terms composed of more than four operators. We then discuss the results as a function of the strength and sign of the interimpurity coupling K. The main results can be summarized as follows: When K = 0, within the present mean-field approximation, the two initial S = 1 spins are decoupled and we have obtained the following picture: the two components z2 and z4 are completely decoupled from the conduction band, while the two components zl and z3 give two independent Kondo effects. Thus, when K = 0, one s = ½ component of each S = 1 spin is completely screened by the Kondo effect and the other s = ½ component keeps its magnetism and is completely decoupled from the Fermi-liquid component. - W h e n K < 0, the ferromagnetic coupling between the two S = 1 spins is direct and consequently the system goes from two disconnected K o n d o effects for K = 0 to a K o n d o effect in the configuration S = 2 and n = 2 for K ~ - ~ . For K < 0 the direct coupling between the two S = 1 spins does not disturb the K o n d o effect. - W h e n the interimpurity coupling is antiferromagnetic and moderate, there is no quantum critical point in the phase diagram, which appears to be different from the case of the regular screened s = ½ K o n d o model, where the existence of a critical point appears to be a controversial subject [13]. In fact, the low-energy physics appears clearly in our treatment of the Hamiltonian: the two
456
K. Le Hur, B. Coqblin / Physica B 230-232 (1997) 454-456
components zl and z3 are still strongly screened by the Kondo effect; the resulting Kondo coupling decreases with K due to the two residual moments z2 and z4, which are now weakly ferromagnetically coupled to the conduction electrons (due to the Pauli principle) and can give a magnetic indirect interaction of RKKY type. This antiferromagnetic and indirect K coupling disturbs weakly the strong and local Kondo effect but a good coexistence between these two processes is expected. When K --, + oo, the two S = 1 impurities prefer to form a spin-singlet and to annihilate the Kondo effect. Detailed calculations will be published elsewhere [16]. The same completely asymmetric situation is also obtained for the underscreened Kondolattice with S = 1 decomposed into two s = ½components [16], within a mean-field treatment describing both the Kondo effect and long-range indirect correlations. Thus, the "underscreened Kondo model" with a moderate antiferromagnetic intersite interaction can account for the complex situation of compounds such as UPt3 which has a heavy fermion behaviour and presents a weak magnetic order at very low temperatures. In our model, the tiny ordered magnetic moment is in fact induced by the long-range scale of the RKKY interaction.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10]
[11] [12] [13] [14] [15] [16]
K.G. Wilson, Rev. Mod. Phys. 47 (1975) 773. B. Coqblin and J.R. Schrieffer, Phys. Rev. 185 (1969) 847. S. Doniach, Physica B 91 (1977) 231. B. Coqblin, J. Arispe, J.R. Iglesias, C. Lacroix and K. Le Hur, J. Phys. Soc. Japan 65 (1996) Suppl. B, 64. G. Aeppli, E. Bucher, C. Broholm, J.K. Kjems, J. Baumann and J. Hufnagl, Phys. Rev. Lett 60 (1988) 615. G.R. Stewart, Z. Fisk, J.O. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679; L. Taillefer, K. Behnia, K. Hasselbach, J. Flouquet, S.M. Hayden and C. VeRier, J. Magn. Magn. Mater 90-91 (1990) 623. G.E. Brodale, R.A. Fisher, N.E. Phillips, G.R. Stewart and A.L. Giorgi, Phys. Rev. Lett. 57 (1986) 234. J.O. Willis, J.D. Thomson, Z. Fisk, A. de Visser, J.J.M. Franse and A. Menovsky, Phys. Rev. B 31 (1985) 1654. S.M. Hayden, L. Taillefer, C. Vettier and J. Flouquet, Phys. Rev. B 46 (1992) 8675. A. de Visser, J. Flouquet, J.J.M. Franse, P. Haen, K. Hasselbach, A. Lacerda and L. Taillefer, Physica B 171 (1991) 190. D.M. Cragg and P. Lloyd, J. Phys. C 12 (1979) L215; P. Nozi~res and A. Blandin, J. Physique 41 (1980) 193. P. Coleman and J. Gan, Physica B 171 (1991) 3; J. Gan, P. Coleman and N. Andrei, Phys. Rev. Lett. 68 (1992) 3476. I. Affleck, A.W.W. Ludwig and B.A. Jones, Phys. Rev. B 52 (1995) 9528. B.A. Jones and C.M. Varma, Phys. Rev. Lett. 58 (1987) 843. Phys. Rev. B 40 (1989) 324. C. Sire, C.M. Varma and H.R. Krishnamurthy, Phys. Rev. B 48 (1993) 13 833. K. Le Hur and B. Coqblin, to be published.