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On nonlinear electron-lattice interaction and anharmonicity in HTSC
Physica B 165&166 (1990) 1079-1080 North- Holland
ON NONLINEAR ELECTRON-LATTICE INTERACTION AND ANHARMONICITY IN HTSC
Jiirgen SCHREIBER, Peter HARTWICH Technische Vniversitiit Dresden, Sektion Physik, Mommsenstr. 13, Dresden, DDR-8027.
A model of electron-phonon coupling quadratic in the lattice displacements is applied to the Cu-O and Bi-O based superconductors. T e, the isotope effect and the renormalization of phonon frequencies are discussed.
1
Introduction
Let us start with the idea that there is a unique microscopic mechanism for high-Te superconductivity in the Cu-O and Bi-O based systems. As in the Bi-O systems electronelectron correlation is rather weak, magnetic or pure electroninc mechanisms are not very likely. Therefore we suggest that the electron-lattice interaction produces the pairing of electrons where the observed anomalies in the HTSC can be connected with the nonlinearity of the interaction [1] and the anharmonicity in the lattice dynamics [2]. The aim of this paper is to investigate a model describing electron-phonon coupling quadratic in the lattice displacements in the Cu(Bi)-O planes. In the framework of an extended Eliashberg theory, T e and the isotope effect are discussed.
2
Model
Applying the LCAO-Xa method of Seifert et al. [3], cluster calculations can be done to determine the electron-phonon coupling in Cu(Bi)-O clusters, (see fig. 1). Using a minimum basis set of Cu 3d-,4s-,4p-, Bi 6s-,6pand 0 2s-,2p-, orbitals, it can be shown that the highest occupied state is the nondegenerated, relatively isolated, strongly covalent anti bonding state with energy f. The dependence of I' on the displacements rO (a = x, y, z) of the characteristic atom in the cluster (cf. fig.l) has the form I' = 1'(0) + Pa(ro)2 (where pz(Cu)=5.8eV/ 1\2, Px,y(Cu)~O, py(0)=18.5eV/ A2, px,z(0)=2.1eV/A2). A small linear contribution to I' for the Cu(z) vibration in the [CuOs]-cluster can be neglected. The calculations for the cubic Bi-O systems show qualitatively the same picture. The parameters of the electron-lattice coupling are slightly smaller than for Cu-O clusters. Hence let us consider the following simplified Hamiltonian for describing the band of anti bonding states:
+ Lt;jcicj c7c; + Hlattice
H = L
+ LPv(xi)
2
The index i labels the antibonding state. The hopping elements t;j are also influenced by atomic displacements. For 1990 - Elsevier Science Publishers B.V. (North-Holland)
these parameters the linear coupling term dominates, the order of magnitude of which is by far smaller than the corresponding values for f. Therefore their contribution is neglected.
3
T c and isotope effect
The usual Eliashberg equations can be derived, where instead of the one-phonon GF, the two-phonon GF enters in the case of quadratic interaction. For the estimation of T e we use an approximated solution of the Eliashberg equations derived by Krezin [4],
Using harmonic dispersionless optical phonons within our model, the parameters >. and 0 take the form
(;11
02
4p~ <
Xv
>2 +p~ < u~ >
>.-1 LN(EF )2 < u~ x4 {p~
< Xv >2
+p~
>T=Ow v
< u~ >},
Xv
=<
Xv
> +uv
With these formulae we can give numerical values for>' and T e. II labels the different phonon modes. The data f~r < u~ > and the phonon frequencies are available from neutron scattering data [5]. The values of the density of states at the Fermi energy N(O) are in the region generally accepted now for the discussed systems (N(0)~0.5/eVCuat). The results of the calculations for Cu systems are as follows: T e(LaSrCuO)=38.5K, T e(YBaCuO)=89.7K. Even in the harmonic approximation, but including the electron-lattice coupling quadratic in the displacements, the relation of T e values for LaSrCuO and YBaCuO systems agrees with the experimental ones. This is due to the strong change of the Cu-contribution to the electron-phonon interaction. Iell alters from 0.33 eV/ A for LaSrCuO to 2.6 eV/ A in YBaCuO. Considering Bi-O systems, T e is correspondingly smaller also due to the reduced N(O). In the framework of the present
J. Schreiber, P. Hiirtwich
1080
theory we have analysed the parameter of the isotope effect, a = &In Tel &In M. The results show that harmonic phonons alone cannot explain the real situation in HTSC, quadratic coupling enhances a in comparison with the BSC value 0.5, At least anharmonic lattice effects have to be incorporated. Assuming local anharmonic 4>4- potentials for the O(z) and Cu(z) motion the possible change of a is estimated within the self consistent phonon approximation. Indeed, anharmonicity leads to correction in a towards the experimental values. To illustrate the influence of an unstable mode, which causes a structural phase transition (SPT) detailed investigations within the 4>4- model have been performed. In such a case T e has a maximum if it is equal to the SPT temperature. Independence on the parameters of the model, a wide range of a-values is possible, even zero or negative values are obtained in [6]. As it was discussed earlier, the anharmonicity has a complex influence on T e working in a second order theory. Especially in the case where long time correlations are present [7], T e will be considerably enhanced.
4
Renormalization of phonon frequenCIes
Finally, let us point at the possibility of a qualitatively new influence of quadratic electron-lattice coupling on the renormalization of phonon frequencies WI" In this case, w~ contains an electronic contribution of the form p < nl >, which is absent for linear coupling. For the one band case the averaged electron number < n > does not change crossing T e • However, if the plane band and the chain band overlap < n > can be varied if a gap is opened in the plane band. Consequently, electrons from the chain band go over to the plane band, enhancing < n > and lowering the frequency of phonons with atomic displacements in the planes, only if p < O. The order of magnitude of the < n >-change is given by ~~N(O), (~-gap energy). Concerning ego the 0(2)-0(3) antiphase mode, a reduction of the frequency is expected to be about 1% which could explain the experimental findings
[8].
z
8FIGURE 1 10[CuO ] - and [Cu2 0 7 ] -clusters for S the simulation of Cu- and O-vibrations
REFERENCES [1] N. Kumar. Phys. Rev. B9 (1974) 4993. [2] N.M. Plakida, V.L. Aksenov, S.L. Drechsler. Europhys. Lett. 4 (1987) 1309. [3] K. Bieger, G. Seifert, H.Eschrig. Z Phys.Chem 266 (1985) 751: Z Phys. Chern. 267 (1986) 529,. [4] V.Z. Krezin. Phys. Lett. A122 (1987) 434. [5] J.J. Capponi, C. Chaillout, A.W. Hewatet al. Europhys. Lett. 3 (1987) 1301. [6] S. Flach, P. Hartwich, J. Schreiber, submitted to Solid State Comm. [7] J. Schreiber, S. Flach, Physica C 153-155 (1988) 237. [8] L.Genzel, A: Wittlin et al. Phys. Rev. B40 (1989) 2170.
ON NONLINEAR ELECTRON-LATTICE INTERACTION AND ANHARMONICITY IN HTSC
Jiirgen SCHREIBER, Peter HARTWICH Technische Vniversitiit Dresden, Sektion Physik, Mommsenstr. 13, Dresden, DDR-8027.
A model of electron-phonon coupling quadratic in the lattice displacements is applied to the Cu-O and Bi-O based superconductors. T e, the isotope effect and the renormalization of phonon frequencies are discussed.
1
Introduction
Let us start with the idea that there is a unique microscopic mechanism for high-Te superconductivity in the Cu-O and Bi-O based systems. As in the Bi-O systems electronelectron correlation is rather weak, magnetic or pure electroninc mechanisms are not very likely. Therefore we suggest that the electron-lattice interaction produces the pairing of electrons where the observed anomalies in the HTSC can be connected with the nonlinearity of the interaction [1] and the anharmonicity in the lattice dynamics [2]. The aim of this paper is to investigate a model describing electron-phonon coupling quadratic in the lattice displacements in the Cu(Bi)-O planes. In the framework of an extended Eliashberg theory, T e and the isotope effect are discussed.
2
Model
Applying the LCAO-Xa method of Seifert et al. [3], cluster calculations can be done to determine the electron-phonon coupling in Cu(Bi)-O clusters, (see fig. 1). Using a minimum basis set of Cu 3d-,4s-,4p-, Bi 6s-,6pand 0 2s-,2p-, orbitals, it can be shown that the highest occupied state is the nondegenerated, relatively isolated, strongly covalent anti bonding state with energy f. The dependence of I' on the displacements rO (a = x, y, z) of the characteristic atom in the cluster (cf. fig.l) has the form I' = 1'(0) + Pa(ro)2 (where pz(Cu)=5.8eV/ 1\2, Px,y(Cu)~O, py(0)=18.5eV/ A2, px,z(0)=2.1eV/A2). A small linear contribution to I' for the Cu(z) vibration in the [CuOs]-cluster can be neglected. The calculations for the cubic Bi-O systems show qualitatively the same picture. The parameters of the electron-lattice coupling are slightly smaller than for Cu-O clusters. Hence let us consider the following simplified Hamiltonian for describing the band of anti bonding states:
+ Lt;jcicj c7c; + Hlattice
H = L
+ LPv(xi)
2
The index i labels the antibonding state. The hopping elements t;j are also influenced by atomic displacements. For 1990 - Elsevier Science Publishers B.V. (North-Holland)
these parameters the linear coupling term dominates, the order of magnitude of which is by far smaller than the corresponding values for f. Therefore their contribution is neglected.
3
T c and isotope effect
The usual Eliashberg equations can be derived, where instead of the one-phonon GF, the two-phonon GF enters in the case of quadratic interaction. For the estimation of T e we use an approximated solution of the Eliashberg equations derived by Krezin [4],
Using harmonic dispersionless optical phonons within our model, the parameters >. and 0 take the form
(;11
02
4p~ <
Xv
>2 +p~ < u~ >
>.-1 LN(EF )2 < u~ x4 {p~
< Xv >2
+p~
>T=Ow v
< u~ >},
Xv
=<
Xv
> +uv
With these formulae we can give numerical values for>' and T e. II labels the different phonon modes. The data f~r < u~ > and the phonon frequencies are available from neutron scattering data [5]. The values of the density of states at the Fermi energy N(O) are in the region generally accepted now for the discussed systems (N(0)~0.5/eVCuat). The results of the calculations for Cu systems are as follows: T e(LaSrCuO)=38.5K, T e(YBaCuO)=89.7K. Even in the harmonic approximation, but including the electron-lattice coupling quadratic in the displacements, the relation of T e values for LaSrCuO and YBaCuO systems agrees with the experimental ones. This is due to the strong change of the Cu-contribution to the electron-phonon interaction. Iell alters from 0.33 eV/ A for LaSrCuO to 2.6 eV/ A in YBaCuO. Considering Bi-O systems, T e is correspondingly smaller also due to the reduced N(O). In the framework of the present
J. Schreiber, P. Hiirtwich
1080
theory we have analysed the parameter of the isotope effect, a = &In Tel &In M. The results show that harmonic phonons alone cannot explain the real situation in HTSC, quadratic coupling enhances a in comparison with the BSC value 0.5, At least anharmonic lattice effects have to be incorporated. Assuming local anharmonic 4>4- potentials for the O(z) and Cu(z) motion the possible change of a is estimated within the self consistent phonon approximation. Indeed, anharmonicity leads to correction in a towards the experimental values. To illustrate the influence of an unstable mode, which causes a structural phase transition (SPT) detailed investigations within the 4>4- model have been performed. In such a case T e has a maximum if it is equal to the SPT temperature. Independence on the parameters of the model, a wide range of a-values is possible, even zero or negative values are obtained in [6]. As it was discussed earlier, the anharmonicity has a complex influence on T e working in a second order theory. Especially in the case where long time correlations are present [7], T e will be considerably enhanced.
4
Renormalization of phonon frequenCIes
Finally, let us point at the possibility of a qualitatively new influence of quadratic electron-lattice coupling on the renormalization of phonon frequencies WI" In this case, w~ contains an electronic contribution of the form p < nl >, which is absent for linear coupling. For the one band case the averaged electron number < n > does not change crossing T e • However, if the plane band and the chain band overlap < n > can be varied if a gap is opened in the plane band. Consequently, electrons from the chain band go over to the plane band, enhancing < n > and lowering the frequency of phonons with atomic displacements in the planes, only if p < O. The order of magnitude of the < n >-change is given by ~~N(O), (~-gap energy). Concerning ego the 0(2)-0(3) antiphase mode, a reduction of the frequency is expected to be about 1% which could explain the experimental findings
[8].
z
8FIGURE 1 10[CuO ] - and [Cu2 0 7 ] -clusters for S the simulation of Cu- and O-vibrations
REFERENCES [1] N. Kumar. Phys. Rev. B9 (1974) 4993. [2] N.M. Plakida, V.L. Aksenov, S.L. Drechsler. Europhys. Lett. 4 (1987) 1309. [3] K. Bieger, G. Seifert, H.Eschrig. Z Phys.Chem 266 (1985) 751: Z Phys. Chern. 267 (1986) 529,. [4] V.Z. Krezin. Phys. Lett. A122 (1987) 434. [5] J.J. Capponi, C. Chaillout, A.W. Hewatet al. Europhys. Lett. 3 (1987) 1301. [6] S. Flach, P. Hartwich, J. Schreiber, submitted to Solid State Comm. [7] J. Schreiber, S. Flach, Physica C 153-155 (1988) 237. [8] L.Genzel, A: Wittlin et al. Phys. Rev. B40 (1989) 2170.