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Journal of Physics Condensed Matter

Polarons on a one-dimensional non-linear lattice with two structural phases

Antonis D Mistriotis, Eleftherios N. Economou

Published Jan 10, 1994• 0 citations
Publisher

Abstract

The case of polaron and bipolaron formation due to electron-phonon coupling was studied on a lattice with soft non-linear intersite interactions. The ground state of a system that consists of one or two electrons coupled to a non-linear lattice was obtained numerically. The binding energy of the bipolaron was examined as a function of the degree of the non-linearity of the lattice. A model was investigated where the interatomic interactions in the lattice are described by a double-well potential. This type of non-linear interaction strongly influences the bipolaron formation and generally improves its stability and increases its binding energy by permitting a local structural change of the lattice.

DOI
10.1088/0953-8984/6/2/013

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arXiv:cond-mat/0211099•Physical review. B, Condensed matter•Jan 10, 2003•29 citations
The case of polaron and bipolaron formation due to electron-phonon coupling was studied on a lattice with soft non-linear intersite interactions. The ground state of a system that consists of one or two electrons coupled to a non-linear lattice was obtained numerically. The binding energy of the bipolaron was examined as a function of the degree of the non-linearity of the lattice. A model was investigated where the interatomic interactions in the lattice are described by a double-well potential. This type of non-linear interaction strongly influences the bipolaron formation and generally improves its stability and increases its binding energy by permitting a local structural change of the lattice.
In this paper we have studied the properties of a two-fermion system interacting with the phonon field in the framework of the Holstein Hubbard model in one, two and three dimensions. We have chosen the modified Lang-Firsov variational wave function with on-site and nearest-neighbour lattice distortion for the phonon subsystem to obtain an effective electronic Hamiltonian. This effective electronic Hamiltonian is then solved exactly to obtain the binding energy and effective mass of the bipolaron. We investigated the stability of the bipolaronic phase by comparing the ground-state energy of a bipolaron and two free polarons. We observed a critical repulsive on-site Hubbard interaction U c below which the bipolaronic phases are stable for a fixed electron-phonon coupling g . In the absence of on-site repulsion, bipolaronic phases are stable over the entire range of electron-phonon coupling for one dimension, whereas there is a critical electron-phonon coupling g c for formation of a stable bipolaron in two and three dimensions.
A large polaron is a quasiparticle that consists of a nearly free electron interacting with the phonons of a material, whose lattice parameters are much smaller than the polaron scale. The electron-phonon interaction also leads to an attractive interaction between electrons, which can allow two polarons to pair up and form a bipolaron. It has been shown that large bipolarons can form in theory due to strong one-electron--one-phonon coupling, but they have not been seen in real materials because the critical value of the required electron-phonon interaction is too large. Here, we investigate the effect of one-electron--two-phonon coupling on the large bipolaron problem. Starting from a generalization of the Fr\"ohlich Hamiltonian that includes both the standard one-electron--one-phonon interaction as well as an anharmonic one-electron--two-phonon interaction, we use the path-integral method to find a semianalytical upper bound for the bipolaron energy that is valid at all values of the Fr\"ohlich coupling strength $\ensuremath{\alpha}$. We find the bipolaron phase diagram and conditions for the bipolaron stability by comparing the bipolaron energy to the energy of two free polarons. The critical value of the Fr\"ohlich coupling strength ${\ensuremath{\alpha}}_{\text{crit}}$ is calculated as a function of the strength of the one-electron--two-phonon interaction. The results suggest that large bipolaron formation is more likely in materials with significant one-electron--two-phonon interaction as well as strong one-electron--one-phonon interaction, such as strontium titanate.
We present a diagrammatic Monte Carlo study of the properties of the Hubbard-Holstein bipolaron on a two-dimensional square lattice. With a small Coulomb repulsion UUU and with increasing electron-phonon interaction, and when reaching a value about two times smaller than the one corresponding to the transition of light polaron to heavy polaron, the system suffers a sharp transition from a state formed by two weakly bound light polarons to a heavy, strongly bound on-site bipolaron. Aside from this rather conventional bipolaron a new bipolaron state is found for large UUU at intermediate and large electron-phonon coupling, corresponding to two polarons bound on nearest-neighbor sites. We discuss both the properties of the different bipolaron states and the transition from one state to another. We present a phase diagram in parameter space defined by the electron-phonon coupling and UUU. Our numerical method does not use any artificial approximation and can be easily modified to other bipolaron models with longer range electron-phonon and∕or electron-electron interaction.
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.
The Holstein Hubbard and Holstein t-J models are studied for a wide range of phonon frequencies, electron-electron, and electron-phonon interaction strengths on finite lattices with up to ten sites by means of direct Lanczos diagonalization. Previously the necessary truncation of the phononic Hilbert space caused serious limitations to either very small systems (four or even two sites) or to weak electron-phonon coupling, in particular in the adiabatic regime. Using parallel computers we were able to investigate the transition from ``large'' to ``small'' polarons in detail. By resolving the low-lying eigenstates of the Hamiltonian and by calculating the spectral function, we can identify a polaron band in the strong-coupling case, whose dispersion deviates from the free-particle dispersion at low and intermediate phonon frequencies. For two electrons (holes) we establish the existence of bipolaronic states and discuss the formation of a bipolaron band. For the two-dimensional Holstein t-J model, we demonstrate that the formation of hole polarons is favored by strong Coulomb correlations. Analyzing hole-hole correlation functions, we find that hole binding is enhanced as a dynamical effect of the electron-phonon interaction. \textcopyright{} 1996 The American Physical Society.
The properties of the polaron and bipolaron are explored in the one-dimensional Jahn-Teller model with dynamical quantum phonons. The ground-state properties of the polaron and bipolaron are computed using a recently developed variational method. Dynamical properties of the ground state of a polaron are investigated by calculating the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega}).$ Our numerical results suggest that the Jahn-Teller and Holstein polarons are similar. However, in the strong-coupling regime qualitative differences in $\ensuremath{\sigma}(\ensuremath{\omega})$ between the two models are found and discussed. The influence of the electron-phonon coupling and the electrostatic repulsion on the bipolaron binding energy, bipolaron masses, and correlation functions is investigated.