Polarizability mechanisms and high temperature superconductivity

Polarizability mechanisms and high temperature superconductivity


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Physica C 153-155 (1988) 1325-1326 North-Holland, Amsterdam

POLARIZABILITY MECItANISMS AND HIGtt TEMPERATURE SUPERCONDUCTIVITY Annette BUSSMANN-HOLDER and Arndt SIMON Max-Planck-lnstitut f/Jr FestkSrperforschung, 7000 Stuttgart 80, FRG and Helmut BOTTNER Universit/it Bayreuth, FRG We describe a lattice dynamical model for high temperature superconductors which represents an extension of a polarizability model for displacive type ferroelectrics. The model is based on the instability of the oxygen ion 0 ~-, dynamically described by an on-site nonlinear electron-ion coupling, and on the valence instability of the copper ion, involving Cu(I), Cu(III), represented by a nonlinear electron-electron interaction which provides a pairwise coupling of the electrons. The transformation of the classical Hamiltonian to its quantum mechanical equivalent yields a combination of different Peierls-Hubbard-like Hamiltonians. 1. THE MODEL In ferroelectric perovskites and probably superconducting Cu-oxides the oxygen ion 02- plays a crucial role (1) which in ferroelectrics has been described by an on-site electron-two phonon coupling term (2),(3). The different oxidation states of the Cu-ion (Cu(I), Cu(III)) (4) induce a substantial change in the p-d hybridization of superconducting Cu-oxides (5) compared to ferroelectrics (6) which is simulated dynamically by extending the original shell model to a double shell model (7) where p and d electrons are coupled nonlinearly. The model ttamiltonian reads:

1 E,, m,~'. + 1_ ' ~(') -~ H + -2 2 E meivn


,.>, 1

(¢:>-,o+1) 2 +








(¢:)- u.) 4 1

2 ~°"+' - ¢:))' + 5 ~' ( ¢ : ) +








q2dqsin2qa + 2 f

(2) V¢ being the unit cell volume and g the self-consistent electron-phonon coupling (3). The transformation of equ.1 to its quantum mechanical equivalent is achieved by replacing the shell displacement coordinates v(~i) by electron-creation and annihilation operators, i.e.v(,1) = + n (2) = ~j,,, b+bj, = nj.d The Hamilto~j,~, aj~,aj~, = nj,v,, nian now reads:



2. RESULTS The dynamics of the coupled double-well problem are treated in the self-consistent-phonon approximation (SPA) from which two different phase transition temperatures can be derived. While ferroelectric transitions are attributed to the anharmonic core-shell potential (g2, g4) (3), a transition to a frozen-in shell-shell configuration is given by k2, k4 which yields for its corresponding transition temperature:

H =


Here ml, m,i, u,, v(ni)(i = 1,2) represent the core and shell-masses and their respective displacements, f', 1 and f are harmonic second nearest neighbor core-core, shell-shell and nearest neighbor core-shell couplings. g2 and k2 represent attractive core-shell and shell-shell force constants which are stabilized by fourth order repulsive terms g4, k4 respectively.



re,i,'. + f'(u.+,l - u.)' + f~4 + g,,


e'(a+.a~,. + b.c.) + ~ ~d(b+b~,. + ~.~.) iZ',~"






j pdpd


J pd


J j p p d d dp p pd d + 2njTnj~njtnj~ + njnjTnjl + njnjTnj~} -

E ~ { fu-a+,, ai~ + gu,a+,ai,, + 4g4u.n~tn~l } ..4jn



0921 4534/88/503.50 © ElsevierSciencePublishersB.V. (North-llollandPhysicsPublishingDivision)


A. Bussmann-Holder and A. Simon / Polarizability mechanisms

where eP, ed, U~,U~ are simple functions of the lattice dynamical model (equ.1) which e.g. for U] yields U] = 2k2 -t- 7k4 . It has to be pointed out that the physical origin of this equation is n ot equivalent to the usual Hubbard U-term. The transformation of equ.1 into equ.3 demonstrates the equivalence of a long known lattice dynamical model for ferroelectric soft modes and a one-dimensional Peierls-Hubbard Hamiltonian with on-site and intersite electron-phonon coupling (8). Two different models for superconductivity are combined in equ.3: a) the polarization induced palring mechanism (9) and b) an intramolecular vibration mechanism (10). The results of this combination (a,b) agree with the outcome of the original dynamical model. Intermolecular coupling enhances the tendency to dimerization of electrons (given by f in equ.2), simultaneously intramolecular vibrations (corresponding to g) reduce the superconducting Tc as now the electrons are dressed by phonon clouds (10). The reduction of this coupling (i.e. g ----, ~ ) leads to a delocalization of electron pairs into extended states and suppresses ferroelectric phase transitions. Thus the crucial interplay of inter-(f) and intramolecular vibrations (g~,g4) determines the magnitude of To. The electron pairing is suggested to arise from the electronic polarization of the oxygen ion p states hybridized with the Cu-ion d-states ( k2 / k4 ). 3. SUMMARY The above described model provides as well a dynamical description of the phase transition to a paried electronic state as a microscopic basis for a modified

Peierls-Hubbard Hamiltonian. ACKNOWLEDGEMENTS We thank P. Horsch for a critical reading of the manuscript and useful discussions. Support of the Deutsche Forschungsgemeinschaft is acknowledged. REFERENCES (1)

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