Importance of Structural Instability to High-Temperature Superconductivity

The orthorhombic-tetragonal structural phase transition of Lap — Sr Cu04 is quantitatively analyzed as a function of composition x within an anharmonic electron-phonon interaction model. The correct temperature dependence of the soft mode and the elastic constant c66 is obtained. The double-well potential in the electron-phonon interaction is derived self-consistently and found to vary strongly with x. In the vicinity of the superconducting transition temperature T, electron-two-phonon interactions dom-inate the harmonic ones which may explain the high T, 's observed.

H =g4+[q;n; (q;+n; )+q; n;tn;t], where the p; and q; are the phonon momentum and dis-The origin of the pairing mechanism in high-temperature superconductors [1] is still an open problem. Including the perovskites Bai,K,Bi03 [2] and BaPbi -"Bi,03 [3] in the class of "high-temperature" superconductors that show BCS-like properties would suggest that a phonon-mediated mechanism may also be responsible for the layered superconducting compounds with much higher T, 's. In this work we derive an electron-phonon interaction model that quantitatively describes the phononmediated structural phase transition observed in Laq -"-Sr Cu04. This transition is analyzed on the basis of a local p potential in the electron-phonon interaction (not nonlinear phonon-phonon interaction) [4] which induces the substantial structural instability observed in all high-T, compounds [5]. It will be shown that this electronphonon-induced structural instability requires an extension of the Migdal theorem [6], leading to a BCS-type superconducting state, where the harmonic electron-phonon interaction is enhanced by electron-density-two-phonon couplings. (For example, in YBa2Cu307, the high dielectric constant and high pyroelectric coefticients point to an incipient dipolar instability. ) The model Hamiltonian we start with represents a combination of two diITerent electron-phonon models [7,8] extended by higher-order interaction terms H H=g[p;/2M+ -, ' gqqP+ 2 g4q; + -, ' K(q;iq;) ] placement coordinates, c;1 and c; are electron creation and annihilation operators with n; =g c;t c;, and k and k represent on-site and intersite couplings of phonon coordinates with the electron density. The fourth-order term in the phonon coordinates q; is a consequence of the nonlinear electron-phonon interaction potential [9,10].
Higher-order interactions in the electron density have been omitted. The potential in the q; is equivalent to those used by Hardy and Flocken, Plakida and coworkers, and various other groups [11]. However, in contrast to those models, and an important consequence of this electron-phonon interaction potential, are the terms appearing in Eq.
(2) describing electron-density-twophonon interactions. The Migdal theorem, where higher-order perturbation corrections [12] in the electronphonon interaction are considered on the basis of the Frohlich Hamiltonian [8], which resembles (1) with neglect of H, is not applicable to our Hamiltonian as the higher-order interactions do not result from perturbation methods but from the p potential in the electronphonon interaction. This means that X&g4. Microscopically the Hamiltonian (1) has its origin in the instability of the oxygen 2p configuration [13],which, due to small phonon displacements, may easily change its character from bound to unbound thus inducing a polarizability catastrophe. The electronic configuration 2p of 0 is only stable in a crystal where Coulomb interactions with the surrounding ions provide the ionic stability. In an isotropic environment the p,p~,p, orbitals are also isotropic. Anisotropy and covalency strongly favor the tendency of an elliptic ground state. For example, in YBa2-Cu307 the p, orbitals of the apical oxygen ion O(4) are strongly delocalized towards the Cu02 planes which then provide a mechanism for the pairwise attraction of holes in the planes [10,14]. picture of bipolaron formation [15) (and especially large bipolaron formation [16]) drastically because the higherorder electron-phonon interaction terms of 8 are usually neglected. Furthermore, early calculations of Hui and Allen [17], using a Hamiltonian with g4 terms in the phonon coordinate q; only, did not lead to an enhancement of the electron-phonon coupling. The calculations by Hardy and Flocken, Plakida, and others [11] start from a double-well potential in the phonon coordinates q;, and an enhancement was found, yet not large enough to explain high T,. Our analysis is based on the classical equivalent of Hamiltonian (1), and the procedures of Enz [18] and Pytte and Feder [19]  shell displacement coordinates and V~represents the ionic interaction potential with neighboring cells. Within the framework of the self-consistent phonon approximation (SPA), which corresponds to an expansion in the first cumulant of the relative electron-ion displace-ment~;, the structural phase transition of La~85-Sro t5Cu04 is quantitatively described. A comparison of inelastic-neutron-scattering data with model calculations is shown in Fig. 1. The softening of the acoustic branch at the zone boundary with decreasing temperature is self-consistently calculated (Fig. 1). Together with the softening of the zone-boundary frequency the elastic constant c66 softens [21] and shows perfect Curie-Weiss behavior, which also results in quantitative agreement with the model.
To find the temperature dependence of the soft mode related to the structural transition for other compositions and thus other T"resonant ultrasound spectroscopy (RUS) [22] was used to confirm a universal behavior of the softening of c66 and thus of the related soft mode.
Because previous RUS studies [21] of a well-characterized single crystal of La] 86Sro]4Cu04 established that only c66 softened at T"we could, for this work, use an unoriented flake from a very small single crystal of Cu-0 fiux-grown La~9oSro~oCu04 [23] in the RUS system.
Any observed temperature dependence could then be taken to be that of c66. Again a Curie-Weiss-like temperature dependence of c66 was obtained. Thus the power law (exponent of unity) is correctly determined. The confirmation of a universal temperature dependence of the soft mode as depicted in Fig. 2 (where the new results on c66 are inserted) enabled the self-consistent determination of the relevant electron-phonon couplings g2, g4 as a function of T, and x. Note that both quantities are nonlinearly dependent on x and T, (Fig. 3). Also shown in Fig. 2 is the predicted temperature dependence of c66, derived from dru/dq for q 0. It displays the same softening for other T, .
The electron-phonon interaction potential derived self-