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The Physics of Metals and Metallography

Electron-phonon interaction in a spatially disordered system with a strong interelectron correlation

M. Kh. Kharrasov, И. Р. Кызыргулов, А. Т. Хусаинов

Published Feb 1, 2011• 0 citations

Abstract

The Hubbard model augmented by the Holstein interaction of localized electrons and optical dispersionless phonons is reduced (with the help of a canonical transformation) to a model, in which there occur jumps of polarons and scattering on “impurity” scatterers, which are accompanied by jumps and scatterings of an unlimited number of phonons. This circumstance and the fact that there is a strong on-site interaction of electrons inherent in the Hubbard model, lead to the need for the formulation of a new theory of the phenomenon and the use of a diagrammatic technique. A theory of perturbations in disorder for electrons moving in the random field of “impurity” scatterers has been developed. A collective mode of the phonon cloud surrounding a polaron has been found and the influence of the processes of scattering of this mode on the energy spectrum of both localized and delocalized polarons has been established. The motion of the delocalized polarons related to their scattering is described in the approximation of chain diagrams, in which the manyparticle irreducible Green’s functions of the polaron are ignored. The electron-energy spectrum obtained consists of four subbands.

DOI: 10.1134/s0031918x11010182

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The Hubbard model augmented by the Holstein interaction of localized electrons and optical dispersionless phonons is reduced (with the help of a canonical transformation) to a model, in which there occur jumps of polarons and scattering on “impurity” scatterers, which are accompanied by jumps and scatterings of an unlimited number of phonons. This circumstance and the fact that there is a strong on-site interaction of electrons inherent in the Hubbard model, lead to the need for the formulation of a new theory of the phenomenon and the use of a diagrammatic technique. A theory of perturbations in disorder for electrons moving in the random field of “impurity” scatterers has been developed. A collective mode of the phonon cloud surrounding a polaron has been found and the influence of the processes of scattering of this mode on the energy spectrum of both localized and delocalized polarons has been established. The motion of the delocalized polarons related to their scattering is described in the approximation of chain diagrams, in which the manyparticle irreducible Green’s functions of the polaron are ignored. The electron-energy spectrum obtained consists of four subbands.

We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov. A new diagram technique is used in order to handle the strong Coulomb repulsion of the electrons, which leads to a closed set of integral equations for the irreducible on-site many-particle Green's functions. The coupling to the phonons leads to an essential renormalization of the one- and two-particle irreducible Green's functions. It also influences the parameters which control the metal-insulator or superconducting phase transition. In addition we find a collective mode of phonon clouds which surround the polarons. The physics of the emission and absorption of this mode by the polarons has been investigated.

We investigate the Anderson-Holstein model with strong repulsion of impurity electrons using a strongcoupling approximation. The strong electron-phonon interaction determines the formation of polarons with heavy clouds of phonons surrounding the impurity electrons. We describe the collective mode of phonon clouds and the relation between electron propagators and correlation functions. We obtain the dependence of the collective mode energy on the hybridization parameter in the case T = 0 and prove that it is suppressed by increasing hybridization.

The interaction of strongly correlated electrons with phonons in the framework of the Hubbard-Holstein model is investigated. The electron-phonon interaction is considered to be strong and is an important parameter of the model, in addition to the Coulomb repulsion of electrons and the band filling. This interaction with nondispersive optical phonons is transformed to the problem of mobile polarons using the canonical transformation of Lang and Firsov. We discuss the case where the on-site Coulomb repulsion is exactly canceled by the phonon-mediated attractive interaction. It is suggested that polarons exchanging phonon clouds can lead to polaron pairing and superconductivity. The fact that the frequency of the collective mode of phonon clouds is larger than the bare frequency then determines the superconducting transition temperature.

We develop a strong-coupling approach to investigate the Anderson-Holstein model with strong repulsion on impurity centers. We derive the relation between electron propagators and correlation functions and prove that for the impurity electrons the latter is identical to the mass operator of the conduction electrons. Strong electron-phonon interaction determines formation of polarons with heavy clouds of phonons surrounding impurity electrons. We demonstrate the existence of a collective excitation mode of these clouds and obtain the dependence of its energy on the hybridization of the impurity with band states. Hybridization is shown to cause softening of the collective mode and its total suppression at sufficiently large values of the hybridization parameter.

The three-band $p\text{\ensuremath{-}}d$ model of strongly correlated electrons interacting with optical phonons via diagonal and off-diagonal electron-phonon interactions is considered within the cluster perturbation theory. In the beginning, the exact diagonalization of the Hamiltonian of a

{\mathrm{CuO}}_{4}

cluster results in the construction of local polaronic eigenstates $\left|p\right\ensuremath{\rangle}$ with hole numbers

{n}_{h}=0,1,2

per unit cell. The intercluster hoppings and interactions are exactly written in terms of Hubbard operators ${X}_{f}^{pq}=\left|p\right\ensuremath{\rangle}\ensuremath{\langle}q|$ determined via the multielectron polaronic eigenstates $\left|p\right\ensuremath{\rangle}$ at site

\mathbf{f}

. The Fermi-type single-electron quasiparticle dispersion and spectral weight are calculated for the undoped antiferromagnetic parent insulator like

{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}

. The quasiparticle dispersion of Hubbard polarons is determined by a hybridization of the Hubbard fermion subbands with local Franck-Condon resonances so the main polaronic effect of the quasiparticle band structure is a splitting of the Hubbard bands on the number of Hubbard polaron subbands. Increasing of the EPI constant results in an increase of splitting, decrease of the subband width, transfer of the spectral weight to high-energy multiphonon excitations, and subsequent localization of the charge carriers. Herewith, the effect of such renormalization for the conduction band and the valence one differs depending on the ratio of the diagonal and off-diagonal EPI. In the framework of the GTB method, the Franck-Condon broadening of the spectral function of polaronic excitations is also reproduced for strongly correlated systems with strong electron-phonon interaction.