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Exchange-mediated pairing in correlated systems
Physica C 153-155 (1988) 1267-1268 North-Holland, Amsterdam
EXCHANGE - MEDIATED PAIRING IN CORRELATED SYSTEMS
Jdzef SPALEK, Department of Solid State Physics, AGH, PL - 30-059 KrakSw, Poland Real space pairing among correlated electrons in both narrow-band and hybridized systems is discussed. In the former case the effective BCS Hamiltonian contains both two- and three-site terms. In the latter case a hybrid pairing coming from the Kondo-type interaction between hybridized electrons is introduced. It leads to an anisotropic BCS type of pairing among hybridized electrons. Implications for both heavy-fermion and high-To materials are stressed. 1. In this paper I discuss briefly two mechanisms of exchange-mediated pairing in real space: the d-d pairing due to virtual transitions,responsible also for antiferromagnetic (AF) kinetic exchange in strongly correlated metals [1], and the hybrid (interorbital) d-p or d-f pairing (in high-To and heavy-fermion materials, respectively) induced by virtual p-d (d-f) transitions which leads to the Kondo-type of interaction between localized (3d or 4f) and itinerant (2p or 5d) electrons [2]. The former mechanism has been proposed by Anderson [3] and elaborated by others [4] while the latter in its present form has not been (as far as I am aware of) obtained before. It should be realized that since the two mechanisms can be expressed equivalently either in terms of itinerant-spin variables Si = (ai+ai-,a+_ai+,(ni+ - ni_)/2, or in the terms of blj = (ai_a3+ - a4+aj_/v~, the magnetism (AF) should be treated on equal footing along with superconductivity. 2. In the case of d-d pairing in a narrow band [3-4], one notices that the effective Hamiltonian in the second order of the canonical perturbation [1] contains both two- and three-site terms as I have stressed in [4]. The latter term provides NN pair hopping ( a resonant behavior of the pair bond). In result, the effective Hamiltonian is H = E t l j b + a b j a -- E ij,~
i,j,k
2tijtJkb+bkj' U
(1)
where bier -- a i a ( 1 - n i - a ) , and the remaining terms are defined in [4]. Two further approximations are made in subsequent analysis of (1). First, the projected operators [1] bla are replaced by fermion operators ai~, and the parameters in (1) are renormalized by the band narrowing factor ~ = (1 -- n)/(1 -- n/2). Second, one makes the Hartree-Fock approximation of the BCS type. In effect, the quasiparticle energy in the superconducting phase is Ek =
-
+ (J01Akl ] ,
0921-4534/88/$03.50 ©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
where ek is the bare band energy, # is the chemical potential J0 = (2t2/U)z(1 - *)[1 + (z - 1)q)] the coupling constant, Ak = 27k(b), and 7k is the symmetry factor equal to ek/zt for extended s-wave pairing. The gap parameter A(O) = Jov~(b) = 1.5kBT¢ close to the metal-insulator (Mott) boundary. This is clearly an unphysical effect: It turns out that as either n --* 1 or U --~ 2W (for n = l ) then the antiferromagnetic phase is the stable phase [5]. Additionally, as n --* 1 the Hartree-Fock approximation may not be appropriate, since the exchange (pairing) part of (1) may be even larger than the single particle part. 3. Strictly speaking, we should start from hybridized model containing filled 2p shells or partially filled 5d-6s shells in the case of high- T¢ materials or heavyfermion materials, respectively. The proper starting point then is the periodic Anderson-Wolff Hamiltonian which reads in the atomic representation
H= E tm°c+ cn" +"ZNI° +OENi+Nii~t
Ill,n~
+
i
+
(3)
ilm~o"
If U is by far the largest parameter in (3) then we can decompose the interorbital mixing into the part weighted by (1 - Ni-~) and the part weigted by Ni-~, representing the virtual hopping onto localized (d or f) orbital in two different physical situations. The l a t t e r can be transformed out giving in the second order the coupling of the Kondo type between a and c electrons. Introducing further the hybrid-pairing operators 1
bim = ~ ( a i + C m - - ai-Cm+),
(4)
we obtain the effective Hamiltonian (3) with the last term in it replaced by
1268
J. SpMek / Exchange-mediated pairing
Vim(1 - Ni-a)(ai+aCma A- c+aa4a) i,m,ct
-- ~
2VmiVin
+
(5)
U-4-ef bimbin
i,m,l~
In other words, the Kondo-type interaction combined with three-site terms leads to the hybrid pairing. One should emphasise that here both a and c electrons are itinerant; the itineracy of a electrons is provided by the residual hybridization in (5) which cannot be transformed out, since we assume that the bare atomic and conduction band states overlap energetically. The last term in (5) represents the correction to the current models [6] of heavy fermions which consider mainly infinite U limit only. Using the ansatz of Rice and Ueda [6] and making the BCS-type approximation we obtain the effective Hamiltonian of pairing in the lower hybridized band in the form
H
(6)
=
kcr
k - -k+J + h ' c with A = ( - l / N ) ~ - ~ V q ( b q ) , and 7k being complicated function of k (almost constant if the hybridization is intraatomic). In that case one can derive an approximate formula for T¢ which is
kBTc =
.2v~
. . u + ~f ~/(~)2 [~T~fJexp[ 2v~¢p(~)V4
+ 4v2]
where Ae is a distance of the atomic level from the middle point of the band and p(p) is the density of hybridized states per site per spin at the Fermi energy. If we take U=5 eV, A e = - 2 e V , • = 0.04 (i.e. nf = 0.98), p(#) = 2O0states/eV/spin (corresponding to the value of 3' = 1J/moleK 2) and IVI=O.6 eV, then Tc=1.76 K. Hence, we obtain a reasonable value of T¢ for heavyfermion materials taking typical values of the parameters. Additionally, the value of Tc grows by more than two orders of magnitude with IVI growing from 0.5 to 1 eV. Moreover, it is crucial to note that (6) is of the same form as the effective Hamiltonian for exchange mediated superconductivity in strongly correlated narrow band [4]. The difference is in the k dependence in the two cases of both starting band energies, in the form of 7k and in that that in the present case the pairing vanishes as nf --~ 1. One should also note that in the fourth order in V a d-d (or f-f) pairing appears and corresponds directly to that considered in the previous
Section. This type of pairing is of higher order with respect to that coming from the second order. 4. One may ask what happens in the situation when the bare atomic level ef is placed deeply below the Fermi level of the band electrons. In that case one can apply to (3) a Schrieffer-Wolff transformation in the real space and instead of (5) we have now the secondorder part in the form 2VmiVlnU b+ b
where Tij = Em(VimVmj /el)(1 --~m,,). Hence, the periodic Anderson-Wolff Hamiltonian reduces to the two band model with a hybrid pairing. The separate question is whether the occupancy of the narrow can be noninteger in this limit. Summarizing, I have provided a microscopic Hamiltoulan with exchange mediated pairing in both narrow band and in the hybridized band of correlated electrons. The two-site part of pairing provides the binding energy whereas the three-site part supplies a hopping of pairs. The antiferromagnetic (Kondo type) interaction leads to a superconducting phase when the coherent normal Fermi liquid state is achieved for the Kondo lattice. N o t e added: A similar type of pairing has been proposed by Newns (Phys. Rev. B36 (1987) 5595). He uses the slave boson representation for the periodic Anderson model whereas here we obtain the effective Hamiltonian with hybrid pairing directly via canonical transformation. It would be interesting to apply the slave boson to our effective Hamiltonian ( one boson is needed only). Also, a similar transformation has been used by Zaanen and Ole~ using the method devised in [1] and [2]. They concentrate on d-d (fourth-order) pairing only and suggest that p-d (second-order) interaction suppresses pairing (see this proceedings). It may be due to the fact that they do not use representation (4) introduced above for the first time. It would be of interest to combine the p-d and d-d pairings into one formalism.
[1] K.A. Chao, J. Spalek and A.M. OleO, J.Phys.C10 (1977)L271; Phys. Rev. B18 (1978) 3453. [2] J. Spa~ek, A.M. Oleff and K.A. Chao, phys. stat. sol. (5)87 (1978) 625; Phys. Rev. B18 (1978) 3748. [3] P.W. Anderson, Science 235 (1987) 1196. [4] M. Cyrot, Sol. St. Comm., 62 (1987) 821; A.E. Ruckenstein et al, Phys. Rev. B36 (1987) 857; J. Spa~ek B37 (1988) No.1. [5] W. Wdjcik and J. Spatek, to be published. [6] T.M. Rice and K. Ueda, Phys. Rev. B34 (1986) 6420. [7] O.V. Dolgov et al, Z. Phys. B67 (1987) 63 and references therein.
EXCHANGE - MEDIATED PAIRING IN CORRELATED SYSTEMS
Jdzef SPALEK, Department of Solid State Physics, AGH, PL - 30-059 KrakSw, Poland Real space pairing among correlated electrons in both narrow-band and hybridized systems is discussed. In the former case the effective BCS Hamiltonian contains both two- and three-site terms. In the latter case a hybrid pairing coming from the Kondo-type interaction between hybridized electrons is introduced. It leads to an anisotropic BCS type of pairing among hybridized electrons. Implications for both heavy-fermion and high-To materials are stressed. 1. In this paper I discuss briefly two mechanisms of exchange-mediated pairing in real space: the d-d pairing due to virtual transitions,responsible also for antiferromagnetic (AF) kinetic exchange in strongly correlated metals [1], and the hybrid (interorbital) d-p or d-f pairing (in high-To and heavy-fermion materials, respectively) induced by virtual p-d (d-f) transitions which leads to the Kondo-type of interaction between localized (3d or 4f) and itinerant (2p or 5d) electrons [2]. The former mechanism has been proposed by Anderson [3] and elaborated by others [4] while the latter in its present form has not been (as far as I am aware of) obtained before. It should be realized that since the two mechanisms can be expressed equivalently either in terms of itinerant-spin variables Si = (ai+ai-,a+_ai+,(ni+ - ni_)/2, or in the terms of blj = (ai_a3+ - a4+aj_/v~, the magnetism (AF) should be treated on equal footing along with superconductivity. 2. In the case of d-d pairing in a narrow band [3-4], one notices that the effective Hamiltonian in the second order of the canonical perturbation [1] contains both two- and three-site terms as I have stressed in [4]. The latter term provides NN pair hopping ( a resonant behavior of the pair bond). In result, the effective Hamiltonian is H = E t l j b + a b j a -- E ij,~
i,j,k
2tijtJkb+bkj' U
(1)
where bier -- a i a ( 1 - n i - a ) , and the remaining terms are defined in [4]. Two further approximations are made in subsequent analysis of (1). First, the projected operators [1] bla are replaced by fermion operators ai~, and the parameters in (1) are renormalized by the band narrowing factor ~ = (1 -- n)/(1 -- n/2). Second, one makes the Hartree-Fock approximation of the BCS type. In effect, the quasiparticle energy in the superconducting phase is Ek =
-
+ (J01Akl ] ,
0921-4534/88/$03.50 ©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
where ek is the bare band energy, # is the chemical potential J0 = (2t2/U)z(1 - *)[1 + (z - 1)q)] the coupling constant, Ak = 27k(b), and 7k is the symmetry factor equal to ek/zt for extended s-wave pairing. The gap parameter A(O) = Jov~(b) = 1.5kBT¢ close to the metal-insulator (Mott) boundary. This is clearly an unphysical effect: It turns out that as either n --* 1 or U --~ 2W (for n = l ) then the antiferromagnetic phase is the stable phase [5]. Additionally, as n --* 1 the Hartree-Fock approximation may not be appropriate, since the exchange (pairing) part of (1) may be even larger than the single particle part. 3. Strictly speaking, we should start from hybridized model containing filled 2p shells or partially filled 5d-6s shells in the case of high- T¢ materials or heavyfermion materials, respectively. The proper starting point then is the periodic Anderson-Wolff Hamiltonian which reads in the atomic representation
H= E tm°c+ cn" +"ZNI° +OENi+Nii~t
Ill,n~
+
i
+
(3)
ilm~o"
If U is by far the largest parameter in (3) then we can decompose the interorbital mixing into the part weighted by (1 - Ni-~) and the part weigted by Ni-~, representing the virtual hopping onto localized (d or f) orbital in two different physical situations. The l a t t e r can be transformed out giving in the second order the coupling of the Kondo type between a and c electrons. Introducing further the hybrid-pairing operators 1
bim = ~ ( a i + C m - - ai-Cm+),
(4)
we obtain the effective Hamiltonian (3) with the last term in it replaced by
1268
J. SpMek / Exchange-mediated pairing
Vim(1 - Ni-a)(ai+aCma A- c+aa4a) i,m,ct
-- ~
2VmiVin
+
(5)
U-4-ef bimbin
i,m,l~
In other words, the Kondo-type interaction combined with three-site terms leads to the hybrid pairing. One should emphasise that here both a and c electrons are itinerant; the itineracy of a electrons is provided by the residual hybridization in (5) which cannot be transformed out, since we assume that the bare atomic and conduction band states overlap energetically. The last term in (5) represents the correction to the current models [6] of heavy fermions which consider mainly infinite U limit only. Using the ansatz of Rice and Ueda [6] and making the BCS-type approximation we obtain the effective Hamiltonian of pairing in the lower hybridized band in the form
H
(6)
=
kcr
k - -k+J + h ' c with A = ( - l / N ) ~ - ~ V q ( b q ) , and 7k being complicated function of k (almost constant if the hybridization is intraatomic). In that case one can derive an approximate formula for T¢ which is
kBTc =
.2v~
. . u + ~f ~/(~)2 [~T~fJexp[ 2v~¢p(~)V4
+ 4v2]
where Ae is a distance of the atomic level from the middle point of the band and p(p) is the density of hybridized states per site per spin at the Fermi energy. If we take U=5 eV, A e = - 2 e V , • = 0.04 (i.e. nf = 0.98), p(#) = 2O0states/eV/spin (corresponding to the value of 3' = 1J/moleK 2) and IVI=O.6 eV, then Tc=1.76 K. Hence, we obtain a reasonable value of T¢ for heavyfermion materials taking typical values of the parameters. Additionally, the value of Tc grows by more than two orders of magnitude with IVI growing from 0.5 to 1 eV. Moreover, it is crucial to note that (6) is of the same form as the effective Hamiltonian for exchange mediated superconductivity in strongly correlated narrow band [4]. The difference is in the k dependence in the two cases of both starting band energies, in the form of 7k and in that that in the present case the pairing vanishes as nf --~ 1. One should also note that in the fourth order in V a d-d (or f-f) pairing appears and corresponds directly to that considered in the previous
Section. This type of pairing is of higher order with respect to that coming from the second order. 4. One may ask what happens in the situation when the bare atomic level ef is placed deeply below the Fermi level of the band electrons. In that case one can apply to (3) a Schrieffer-Wolff transformation in the real space and instead of (5) we have now the secondorder part in the form 2VmiVlnU b+ b
where Tij = Em(VimVmj /el)(1 --~m,,). Hence, the periodic Anderson-Wolff Hamiltonian reduces to the two band model with a hybrid pairing. The separate question is whether the occupancy of the narrow can be noninteger in this limit. Summarizing, I have provided a microscopic Hamiltoulan with exchange mediated pairing in both narrow band and in the hybridized band of correlated electrons. The two-site part of pairing provides the binding energy whereas the three-site part supplies a hopping of pairs. The antiferromagnetic (Kondo type) interaction leads to a superconducting phase when the coherent normal Fermi liquid state is achieved for the Kondo lattice. N o t e added: A similar type of pairing has been proposed by Newns (Phys. Rev. B36 (1987) 5595). He uses the slave boson representation for the periodic Anderson model whereas here we obtain the effective Hamiltonian with hybrid pairing directly via canonical transformation. It would be interesting to apply the slave boson to our effective Hamiltonian ( one boson is needed only). Also, a similar transformation has been used by Zaanen and Ole~ using the method devised in [1] and [2]. They concentrate on d-d (fourth-order) pairing only and suggest that p-d (second-order) interaction suppresses pairing (see this proceedings). It may be due to the fact that they do not use representation (4) introduced above for the first time. It would be of interest to combine the p-d and d-d pairings into one formalism.
[1] K.A. Chao, J. Spalek and A.M. OleO, J.Phys.C10 (1977)L271; Phys. Rev. B18 (1978) 3453. [2] J. Spa~ek, A.M. Oleff and K.A. Chao, phys. stat. sol. (5)87 (1978) 625; Phys. Rev. B18 (1978) 3748. [3] P.W. Anderson, Science 235 (1987) 1196. [4] M. Cyrot, Sol. St. Comm., 62 (1987) 821; A.E. Ruckenstein et al, Phys. Rev. B36 (1987) 857; J. Spa~ek B37 (1988) No.1. [5] W. Wdjcik and J. Spatek, to be published. [6] T.M. Rice and K. Ueda, Phys. Rev. B34 (1986) 6420. [7] O.V. Dolgov et al, Z. Phys. B67 (1987) 63 and references therein.