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Physical Review B

Elasticity driven self-organization of polarons

Panagiotis Maniadis, Turab Lookman, A. R. Bishop

Published Oct 17, 2008• 7 citations
Publisher

Abstract

We use a strain description to couple long-range elastic fields adiabatically to electronic density to describe the behavior of a quantum particle in an elastic medium. We show that in this generalization of the Holstein polaron problem, a bound polaronic state results with strong long-range angular dependence in the elastic fields, but a localized electronic core. The deformation of the elastic fields creates an anisotropic, indirect interaction between polarons extending to large distances. For a given density of polarons, this interaction favors the formation of strings of polarons in preferred directions.

DOI
10.1103/physrevb.78.134304

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We use a strain description to couple long-range elastic fields adiabatically to electronic density to describe the behavior of a quantum particle in an elastic medium. We show that in this generalization of the Holstein polaron problem, a bound polaronic state results with strong long-range angular dependence in the elastic fields, but a localized electronic core. The deformation of the elastic fields creates an anisotropic, indirect interaction between polarons extending to large distances. For a given density of polarons, this interaction favors the formation of strings of polarons in preferred directions.
We present results for the time evolution of a one-dimensional system consisting of an electron, described by a tight-binding Hamiltonian and a harmonic latttice, coupled by a deformation-type potential. We solve numerically the nonlinear system of equations of motion for this model in order to study the effects of varying the electronic effective mass for several initial conditions and coupling strengths. Different types of localized and extended states are formed with features that are absent from the traditional polaronic states and depend very strongly on the initial electronic configuration and effective mass in a very often unexpected manner. We find that, in general, an increase of the initial electronic energy decreases the ability of the system to form localized states. However, a large effective mass favors localized polaron formation for initially localized electrons, but this is not always the case for initially extended electronic states. In the latter case, increasing the effective mass of an electron initially close to the bottom of the band makes localization more difficult, while for an initially highly excited electron, localized polaron formation is possible only when the electronic effective mass and the atomic masses of the lattice become of the same order. Finally, for a small parameter range, we find an impressive recurrence, a periodic and a complete exchange between the electronic and vibrational degrees of freedom of a small part of the initial electronic energy.
We investigate polaron formation in a many-electron system in the presence of a local repulsion sufficiently strong to prevent local-bipolaron formation. Specifically, we consider a Hubbard-Holstein model of interacting electrons coupled to dispersionless phonons of frequency ω 0 . Numerically solving the model in a small one-dimensional cluster, we find that in the nearly adiabatic case ω 0 < t , the necessary and sufficient condition for the polaronic regime to occur is that the energy gain in the atomic ( i.e. , extremely localized) regime ε pol overcomes the energy of the purely electronic system ε el . In the antiadiabatic case, ω 0 > t , polaron formation is instead driven by the condition of a large ionic displacement g / ω 0 < 1 ( g being the electron-phonon coupling). Dynamical properties of the model in the weak and moderately strong coupling regimes are also analyzed.
The polaron features for long-range electron–phonon interaction are investigated by extending a variational approach previously proposed for the study of systems with local coupling. The ground state spectral weight, the average kinetic energy, the mean number of phonons, and the electron–lattice correlation function are discussed for a wide range of model parameters focusing on the adiabatic regime and comparing the results with the short-range case (Holstein model). A strong mixing of electronic and phononic degrees of freedom for small values of the electron–phonon coupling constant is found in the adiabatic case due to the long-range interaction. Finally a polaron ‘phase diagram’ is proposed.
The polaron formation is investigated in the intermediate regime of the Holstein model by using an exact diagonalization technique for the one-dimensional infinite lattice. The numerical results for the electron and phonon propagators are compared with the nonadiabatic weak- and strong-coupling perturbation theories, as well as with the harmonic adiabatic approximation. A qualitative explanation of the crossover regime between the self-trapped and free-particle-like behaviors, not well-understood previously, is proposed. It is shown that a fine balance of nonadiabatic and adiabatic contributions determines the motion of small polarons, making them light. A comprehensive analysis of spatially and temporally resolved low-frequency lattice correlations that characterize the translationally invariant polaron states is derived. Various behaviors of the polaronic deformation field, ranging from classical adiabatic for strong couplings to quantum nonadiabatic for weak couplings, are discussed.
The polaron formation is investigated in the intermediate regime of the Holstein model by using an exact diagonalization technique for the one-dimensional infinite lattice. The numerical results for the electron and phonon propagators are compared to the nonadiabatic weak- and strong-coupling perturbation theories, as well as with the harmonic adiabatic approximation. A qualitative explanation of the crossover regime between the self-trapped and free-particle-like behaviors, not well understood previously, is proposed. It is shown that a fine balance of nonadiabatic and adiabatic contributions determines the motion of small polarons, making them light. A comprehensive analysis of spatially and temporally resolved low-frequency lattice correlations that characterize the translationally invariant polaron states is derived. Various behaviors of the polaronic deformation field, ranging from classical adiabatic for strong couplings to quantum nonadiabatic for weak couplings, are discussed.