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Physical review. B, Condensed matter

Electron localization with and without barrier formation

V. V. Kabanov, O. Yu. Mashtakov

Published Mar 1, 1993• 147 citations
Publisher

Abstract

The possibility of polaron formation within a model of local electron-phonon interactions is considered. It is shown that, unlike the continuous limit, the solution to the Schro\ifmmode\ddot\else\textasciidieresis\fi{}dinger equation on a two-dimensional lattice in the adiabatic approximation provides evidence for the presence of an energy barrier that separates a small-size polaron and delocalized electronic state. The polaron stability and metastability regions are determined. The possibility of polaron formation in high-${\mathit{T}}_{\mathit{c}}$ superconductors is discussed.

DOI
10.1103/physrevb.47.6060

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arXiv:0806.2483•Physical Review B•May 11, 2009•30 citations
The possibility of polaron formation within a model of local electron-phonon interactions is considered. It is shown that, unlike the continuous limit, the solution to the Schro\ifmmode\ddot\else\textasciidieresis\fi{}dinger equation on a two-dimensional lattice in the adiabatic approximation provides evidence for the presence of an energy barrier that separates a small-size polaron and delocalized electronic state. The polaron stability and metastability regions are determined. The possibility of polaron formation in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors is discussed.
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.
We review a series of exact, analytical and numerical results obtained on the adiabatic Holstein–Hubbard model, many of which are new and non-trivial. We study next the role of the quantum lattice fluctuations that were initially neglected. The possibility of having high- T c bipolaronic superconductors is analysed on the basis of these results. We suggest that models that involve only electron–phonon coupling are very unlikely to produce bipolaronic superconductivity, which is prevented by the spatial ordering of the bipolarons associated with a lattice instability. This is due to a very large effective mass of the bipolarons that can be related to the large Peierls–Nabarro energy barrier required to move these bipolarons through the lattice in the adiabatic limit. We conjecture that high- T c superconductivity originates specifically from the exceptionally well-balanced competition between electron–phonon coupling and electron–electron repulsion. In the restricted region in which, within the mean field approach, the energy of a bipolaron is close to those of two polarons, a new type of electron pairing occurs by formation of pairs of polarons in the spin singlet state. Such a polaron pair, called a spin resonant bipolaron (SRB), is not the standard bipolaron (both could exist in the adiabatic case). Its Peierls–Nabarro energy barrier can be sharply depressed, almost to zero. As a result of quantum lattice fluctuations, its tunnelling energy is sharply enhanced. Then, superconductivity could persist for a relatively large electron–phonon coupling, with an unusually large critical temperature as a Bose condensate of SRBs before becoming, at larger coupling and in any case, an insulating spin–Peierls polaronic phase.
We calculate the real space and time formation of a polaron quasiparticle from a bare electron. The time-dependent Schr\"odinger equation for the Holstein model of electron-phonon coupling is numerically integrated in a large Hilbert space to obtain the time evolution of the electron and phonon densities and the electron-phonon (el-ph) correlation functions. The quantum dynamical nature of the phonons is preserved. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. A potential barrier between the quasifree and heavy polaron states exists in dimensions $D\ensuremath{\geqslant}2$, consistent with earlier adiabatic theory. We compare to recent experiments.
The formation of a polaron quasiparticle from a bare electron is studied in the framework of the Holstein model of electron-phonon coupling. Using Schrödinger's formalism, we calculate the time evolution of the distribution of the electron and phonon density, lattice deformation, and the electron-phonon (el-ph) correlation functions in real space. The quantum dynamical nature of the phonons is preserved. The polaron formation time is related to the dephasing time of the continuum of unbound phonon excited states in the spectral function, which depends on the phonon frequency and the el-ph coupling strength. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. In the adiabatic regime, we find that a potential barrier between the quasi-free and heavy polaron states exists in both 2D and 3D, which is crucial for polaron formation dynamics. We compare to recent experiments.
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 solve the problem of polaron localization on an attractive impurity by means of direct-space diagrammatic Monte Carlo implemented for the system in the thermodynamic limit. In particular we determine the ground-state phase diagram in dependence on the electron-phonon coupling and impurity potential strength for the whole phonon frequency range. Including the quantum phonon effects we find and characterize a phase where self-trapped polarons are not localized at shallow impurities, which is missing in the adiabatic approximation. We show that near the localization transition a region with a mixture of weak- and strong-coupling spectral responses is realized.