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A weak-coupling limit of the d - d excitation model for the high-Tc cuprates
PhysicaC 153-155(1988)1307-1308 North-Holland,Amsterdam
A WEAK-COUPLING HIGH-T~CUPRATES
LIMIT
OF THE
d - d EXCITATION
MODEL
FOR
THE
A.L. SHELANKOV*, X. ZOTOS** and W. W E B E R + *A.F. Ioffe Institute, Leningrad, 194021, USSR, and Inst. Theorie d. Kond. Mat. Univ. Karlsruhe, FRG; **Research Center of Crete, P.O. Box 1527, Heraklion, Crete, Greece, and Inst. Theorie d. Kond. Mat. Univ. Karlsruhe, FRG; +Kernforschungszentrum Karlsruhe, INFP, P.O. Box 3640, D-7500 Karlsruhe, FRG. An effective Hamiltonian is developed for the d - d excitation model in the atomic limit. In a weak coupling approximation it is shown that the interaction between itinerant oxygen p holes mediated by virtual d - d excitations of Cu ++ may lead to superconductivity of purely electronic origin.
1. I N T R O D U C T I O N The Cu - d - d excitation model for the pairing in the high T, copper oxide superconductors [1,2] allows many limiting cases. Using realistic parameters as estimated from energy band theory and/or spectroscopy, we are in the limit of strong coupling between oxygen p holes and copper d holes. This limit has been investigated for a finite size system elsewhere [3]. In this paper we study the atomic limit of the model, where oxygen and copper holes are only weakly coupled. This means that the band width tpp [4] of the oxygen hole band is large compared to p - d interactions which are of order t~d/U, t~a/Ep. The purpose of this paper is to demonstrate that in this limit d - d excitations can lead to superconductivity. We also discuss in a qualitative way the possible role of spin excitations. 2. T H E E F F E C T I V E H A M I L T O N I A N In the limit U, Ep ~ co, the system consists of two noninteracting parts, the Cu ++ ions with one d hole and a partially filled oxygen p hole band of width tpp. A d-hole on site i is specified by its spin cri; and a pseudospin Ii for the x - z orbital degree of freedom. For finite values of U and Ep, virtual hopping leads to an interaction between the Cu and oxygen subsystems described by an effective Hamiltonian H~yf = Ho + H d - d + Hspin. Here,
Ho=Et'~j,p+pj,.+EhI
i
(1)
( W ' d j ' I i ) p+pj'"
(2)
jj's
i
He-e : E ijj' 8
Hspin =
E
J.aft ~ , z , ~ aft ~Js'
ijj'af 0921--4534/88/$03.50 © ElsevierScience Publishers B.V.
(North-Holland Physics Publishing Division)
(3)
Ho is the renormalized single particle Hamiltonian of the Cu and O subsystems, tjj represents the on-site p orbital energy renormalized by the interactions Vz and Vz, and ~jj, describes the hopping tpp modified by indirect hopping via Cu. The pseudospin term is a matrix in z - z space with ( h I ) ~ = - ( h I ) , ~ = - - A , / 2 and (hI)~
tj~tj./Ep
= (hI),~ = t~, = - E
(4)
J where t j z , t j z are the standard p - x, p - z hopping terms. H d _ d describes the spin-independent part of the interaction between p holes and the Cu d - d excitations (pseudospins). We have
Ii) zz = V1 = A V / 2 + (t,j,tj,, - tjztj, z ) / D
( W i , j j, Ii )** = - ( W i , j j ,
(W~,jj, Ii)~z = ( W , j , j l i ) , ~ -- 112 = tjztj,~/2D
(5) with D -1 = ( U - E p ) -1 + E ~ -1. Finally, H~pi~ gives the exchange interaction of Cu and O spins. In standard spin notation, e.g. the z component of er~f is given by a~-/3, - a~-/3,, with a,/3 representing any z, z orbitals on the i-th site (a corresponding definition holds for the O spins S;j,) " Then J ~% 3 .3 , = t j ~ t j , f / 2 D with the orbitals a,/3 on site i. H e y f (Eqs. 1-3) neglects all processes higher than second order in p-d hopping, e.g. of superexchange type. The relative order of corrections to n e f f is t2~ vd / (\ U , E P ~ J and A Y / ( V , Ep).
1308
A.L. Shelankov et aL / Weak-coupling limit of the d-d excitation model
3. W E A K C O U P L I N G LIMIT For realistic model parameters, H , yy (1-3) describes a strongly correlated system as tpp tZpd/(Ep, U). For simplicity, we assume tpp >> t2d/(E,, U). This corresponds to a weak coupling limit. In this case, and neglecting the t e r m / / s p i n for the moment, superconductivity can be studied in analogy to electron-phonon interaction (see also [5]). Diagonalizat i o n o f Ho leads to Op band states and to d levels split by A = ( A ~ + 4 ~ z ) l / 2 . The ground state is 5 = u z + v z with u 2 = A 2 / A 2, v 2 = 1 -- u 2. The interaction t e r m Hd-d leads to an effective attraction
where < > represents Fermi surface averages of the quantities V~ and V~. This attraction would lead to a BCS-like superconductivity with a transition temperature TCo ~ ~xexp ( - 1/(N(O)VeM)) (7) where N(0) stands for p hole density of states and the prefactor ~ represents the cut-off energy for attraction. As this expression includes only electronic parameters, one may expect Too to be quite large. Note that V~fy is very sensitive to the symmetry of the lattice. For a perfect Cu - 0 square plane, v 2 = 0, due to different rotational s y m m e t r y of the z and z orbitals. For the same reason < V2 > ~ 0 when Ep is close to the b o t t o m of the p band, so that < V~ >o¢ n, the p hole concentration. These restrictions do not hold for the Cu - 0 chains or for the orthorhombic Cu - 0 planes. Furthermore, any intersite x - z coupling, superexchange-like, would remove these s y m m e t r y restrictions. Let us now briefly discuss the influence of the Cu spins on To. The Cu spins can either form a coherent state of a Kondo-lattice type with a typical energy scale TK, a n d / o r short range spin ordering with a characteristic energy ('N6el t e m p e r a t u r e ' ) TN. In the simplest (and least probable) case, when To0 > TN, TK, Cu spins act as magnetic impurities causing p spin-flip with rates 1/~'~ ~ J2/tpp. Superconductivity will survive as long as l / r , < T~o. If the superconducting transition happens in the Kondo lattice regime; i.e. T~0 < TK, and spin correlations between Cu and O sites are the most i m p o r t a n t ones, we expect Tc ~ Ted" TK/~X, in accordance with arguments suggested for phononinduced superconductivity in heavy fermion systems [6].
If however, the Cu spin correlations are mainly controlled by superexchange interaction (not included in HEM), and TN > To0, we do not expect any BCS-like superconductivity to sustain the spin exchange. Estimates show that the repulsive interaction between the p holes m e d i a t e d by spin excitation can be as large as U. Yet superconductivity with a more complex order p a r a m e t e r is not excluded for this case. In conclusion we have developed a scheme for the description of the d - d excitation model in the atomic limit and have considered a weak coupling approximation of the model. We have shown that the interaction between itinerant holes mediated by a virtual valence conserving excitation of Cu ++ may lead to a superconductivity of purely electronic origin. ACKNOWLEDGMENTS We would like to thank Prof. A. Sehmid and all the members of the Institut fiir Theorie der Kondensierten Materie, Universit£t Karlsruhe, for their warm hospitality and the Deutsche Forschungsgemeinschaft for financial support.
REFERENCES [1] W. Weber, Z. Phys. B., to be published. [2] W. Weber, A.L. Shelankov and X. Zotos, this volulne.
[3] X. Zotos, A.L. Shelankov and W. Weber, this volume. [4] K n o t specified, notations follow Ref. 2. One particle energies are counted from the x level (Ed = 0). [5] J. E. Hirsch and D.J. Scalapino, Phys. Rev. B 32,
117 (1985). [6] H. Razafimandimby, F. Fulde and J. Keller, Z. Physik B 5 4 , 1 1 1 (1984).
A WEAK-COUPLING HIGH-T~CUPRATES
LIMIT
OF THE
d - d EXCITATION
MODEL
FOR
THE
A.L. SHELANKOV*, X. ZOTOS** and W. W E B E R + *A.F. Ioffe Institute, Leningrad, 194021, USSR, and Inst. Theorie d. Kond. Mat. Univ. Karlsruhe, FRG; **Research Center of Crete, P.O. Box 1527, Heraklion, Crete, Greece, and Inst. Theorie d. Kond. Mat. Univ. Karlsruhe, FRG; +Kernforschungszentrum Karlsruhe, INFP, P.O. Box 3640, D-7500 Karlsruhe, FRG. An effective Hamiltonian is developed for the d - d excitation model in the atomic limit. In a weak coupling approximation it is shown that the interaction between itinerant oxygen p holes mediated by virtual d - d excitations of Cu ++ may lead to superconductivity of purely electronic origin.
1. I N T R O D U C T I O N The Cu - d - d excitation model for the pairing in the high T, copper oxide superconductors [1,2] allows many limiting cases. Using realistic parameters as estimated from energy band theory and/or spectroscopy, we are in the limit of strong coupling between oxygen p holes and copper d holes. This limit has been investigated for a finite size system elsewhere [3]. In this paper we study the atomic limit of the model, where oxygen and copper holes are only weakly coupled. This means that the band width tpp [4] of the oxygen hole band is large compared to p - d interactions which are of order t~d/U, t~a/Ep. The purpose of this paper is to demonstrate that in this limit d - d excitations can lead to superconductivity. We also discuss in a qualitative way the possible role of spin excitations. 2. T H E E F F E C T I V E H A M I L T O N I A N In the limit U, Ep ~ co, the system consists of two noninteracting parts, the Cu ++ ions with one d hole and a partially filled oxygen p hole band of width tpp. A d-hole on site i is specified by its spin cri; and a pseudospin Ii for the x - z orbital degree of freedom. For finite values of U and Ep, virtual hopping leads to an interaction between the Cu and oxygen subsystems described by an effective Hamiltonian H~yf = Ho + H d - d + Hspin. Here,
Ho=Et'~j,p+pj,.+EhI
i
(1)
( W ' d j ' I i ) p+pj'"
(2)
jj's
i
He-e : E ijj' 8
Hspin =
E
J.aft ~ , z , ~ aft ~Js'
ijj'af 0921--4534/88/$03.50 © ElsevierScience Publishers B.V.
(North-Holland Physics Publishing Division)
(3)
Ho is the renormalized single particle Hamiltonian of the Cu and O subsystems, tjj represents the on-site p orbital energy renormalized by the interactions Vz and Vz, and ~jj, describes the hopping tpp modified by indirect hopping via Cu. The pseudospin term is a matrix in z - z space with ( h I ) ~ = - ( h I ) , ~ = - - A , / 2 and (hI)~
tj~tj./Ep
= (hI),~ = t~, = - E
(4)
J where t j z , t j z are the standard p - x, p - z hopping terms. H d _ d describes the spin-independent part of the interaction between p holes and the Cu d - d excitations (pseudospins). We have
Ii) zz = V1 = A V / 2 + (t,j,tj,, - tjztj, z ) / D
( W i , j j, Ii )** = - ( W i , j j ,
(W~,jj, Ii)~z = ( W , j , j l i ) , ~ -- 112 = tjztj,~/2D
(5) with D -1 = ( U - E p ) -1 + E ~ -1. Finally, H~pi~ gives the exchange interaction of Cu and O spins. In standard spin notation, e.g. the z component of er~f is given by a~-/3, - a~-/3,, with a,/3 representing any z, z orbitals on the i-th site (a corresponding definition holds for the O spins S;j,) " Then J ~% 3 .3 , = t j ~ t j , f / 2 D with the orbitals a,/3 on site i. H e y f (Eqs. 1-3) neglects all processes higher than second order in p-d hopping, e.g. of superexchange type. The relative order of corrections to n e f f is t2~ vd / (\ U , E P ~ J and A Y / ( V , Ep).
1308
A.L. Shelankov et aL / Weak-coupling limit of the d-d excitation model
3. W E A K C O U P L I N G LIMIT For realistic model parameters, H , yy (1-3) describes a strongly correlated system as tpp tZpd/(Ep, U). For simplicity, we assume tpp >> t2d/(E,, U). This corresponds to a weak coupling limit. In this case, and neglecting the t e r m / / s p i n for the moment, superconductivity can be studied in analogy to electron-phonon interaction (see also [5]). Diagonalizat i o n o f Ho leads to Op band states and to d levels split by A = ( A ~ + 4 ~ z ) l / 2 . The ground state is 5 = u z + v z with u 2 = A 2 / A 2, v 2 = 1 -- u 2. The interaction t e r m Hd-d leads to an effective attraction
where < > represents Fermi surface averages of the quantities V~ and V~. This attraction would lead to a BCS-like superconductivity with a transition temperature TCo ~ ~xexp ( - 1/(N(O)VeM)) (7) where N(0) stands for p hole density of states and the prefactor ~ represents the cut-off energy for attraction. As this expression includes only electronic parameters, one may expect Too to be quite large. Note that V~fy is very sensitive to the symmetry of the lattice. For a perfect Cu - 0 square plane, v 2 = 0, due to different rotational s y m m e t r y of the z and z orbitals. For the same reason < V2 > ~ 0 when Ep is close to the b o t t o m of the p band, so that < V~ >o¢ n, the p hole concentration. These restrictions do not hold for the Cu - 0 chains or for the orthorhombic Cu - 0 planes. Furthermore, any intersite x - z coupling, superexchange-like, would remove these s y m m e t r y restrictions. Let us now briefly discuss the influence of the Cu spins on To. The Cu spins can either form a coherent state of a Kondo-lattice type with a typical energy scale TK, a n d / o r short range spin ordering with a characteristic energy ('N6el t e m p e r a t u r e ' ) TN. In the simplest (and least probable) case, when To0 > TN, TK, Cu spins act as magnetic impurities causing p spin-flip with rates 1/~'~ ~ J2/tpp. Superconductivity will survive as long as l / r , < T~o. If the superconducting transition happens in the Kondo lattice regime; i.e. T~0 < TK, and spin correlations between Cu and O sites are the most i m p o r t a n t ones, we expect Tc ~ Ted" TK/~X, in accordance with arguments suggested for phononinduced superconductivity in heavy fermion systems [6].
If however, the Cu spin correlations are mainly controlled by superexchange interaction (not included in HEM), and TN > To0, we do not expect any BCS-like superconductivity to sustain the spin exchange. Estimates show that the repulsive interaction between the p holes m e d i a t e d by spin excitation can be as large as U. Yet superconductivity with a more complex order p a r a m e t e r is not excluded for this case. In conclusion we have developed a scheme for the description of the d - d excitation model in the atomic limit and have considered a weak coupling approximation of the model. We have shown that the interaction between itinerant holes mediated by a virtual valence conserving excitation of Cu ++ may lead to a superconductivity of purely electronic origin. ACKNOWLEDGMENTS We would like to thank Prof. A. Sehmid and all the members of the Institut fiir Theorie der Kondensierten Materie, Universit£t Karlsruhe, for their warm hospitality and the Deutsche Forschungsgemeinschaft for financial support.
REFERENCES [1] W. Weber, Z. Phys. B., to be published. [2] W. Weber, A.L. Shelankov and X. Zotos, this volulne.
[3] X. Zotos, A.L. Shelankov and W. Weber, this volume. [4] K n o t specified, notations follow Ref. 2. One particle energies are counted from the x level (Ed = 0). [5] J. E. Hirsch and D.J. Scalapino, Phys. Rev. B 32,
117 (1985). [6] H. Razafimandimby, F. Fulde and J. Keller, Z. Physik B 5 4 , 1 1 1 (1984).