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Physica Scripta

About the Problem of Existence of a Three-Dimensional Bipolaron

V. K. Mukhomorov

Published Jan 1, 2004• 5 citations
Publisher

Abstract

The range of existence of a translation invariant continuum three-dimensional bipolaron in space of two variables: of electron–phonon coupling constant and the dielectric parameters of polar medium is investigated. The critical values of the electron–phonon coupling constant are found. The possibility of existence of metastable bipolaron formations is scored. On the basis of definition of bipolaron existence the requirement of existence of a bound condition in a bipolaron potential well is necessary. It is shown, that use of a rigider requirement, than positivity of dissociation energy, namely presence of even one bound state narrows the range of existence of a bipolaron. The criteria for a bound state to arise are found by solving an integral equation, which corresponds to a Schrödinger equation describing internal states of a bipolaron.

DOI
10.1238/physica.regular.069a00139

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Physical review. B, Condensed matter•Feb 1, 1991•165 citations
The range of existence of a translation invariant continuum three-dimensional bipolaron in space of two variables: of electron–phonon coupling constant and the dielectric parameters of polar medium is investigated. The critical values of the electron–phonon coupling constant are found. The possibility of existence of metastable bipolaron formations is scored. On the basis of definition of bipolaron existence the requirement of existence of a bound condition in a bipolaron potential well is necessary. It is shown, that use of a rigider requirement, than positivity of dissociation energy, namely presence of even one bound state narrows the range of existence of a bipolaron. The criteria for a bound state to arise are found by solving an integral equation, which corresponds to a Schrödinger equation describing internal states of a bipolaron.
The energies of bipolaron states in three- and two-dimensional systems are calculated variationally, treating the electron-phonon interaction in the Fr\"ohlich approximation, and separating the relative motion from the center-of-mass motion. The bipolaron is bound if the electron-phonon coupling constant \ensuremath{\alpha} is larger than 6 in three dimensions and 2 in two dimensions, provided the ratio \ensuremath{\eta}=${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$/${\mathrm{\ensuremath{\epsilon}}}_{0}$ is smaller than a critical value which depends on \ensuremath{\alpha}. The theory yields the free-polaron energies as given in the Lee, Low, and Pines approximation for arbitrary values of the electron-phonon coupling constant \ensuremath{\alpha}. For \ensuremath{\alpha} larger than \ensuremath{\sim}10 and \ensuremath{\sim}4 in the three- and two-dimensional cases, respectively, the lowest free-polaron self-energies are obtained in the strong-coupling approach; it is shown that a trial harmonic envelope wave function, localized both in the center-of-mass and relative coordinates, with a coherent phonon state gives a bound localized bipolaron state of lower energy. It is interesting for application to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductivity that the bipolaron bounds more easily in two dimensions than in three, and that the mean value of the pair radius is a few angstroms. Furthermore, bipolaron states obtained when the linear total momentum is conserved have intrinsically high mobility, which is also an important condition to make the bipolaron mechanism consistent with high-temperature superconductivity.
The large bipolaron in two and three dimensions is investigated by combining the reasonable asymptotic relative wavefunctions and the Lee–Low–Pines–Huybrechts (LLP-H) variational method. The values of the critical coupling constant above which bipolarons may exist and the critical ratio of dielectric constants below which bipolarons may exist are evaluated, which are very close to the recent results obtained within totally different variational approaches. It is also found that low-dimensional materials are more favorable for the formation of bipolarons.
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.
Feynman path-integral techniques are used to study the large bipolaron system. A new trial action is introduced, which takes into account a nonzero average distance between the electrons. With this trial action, variational expressions are derived for the free energy for arbitrary electron-phonon interactions and spatial dimensions. For large interelectronic repulsions (twice) the Feynman upper bound for a single polaron is reobtained. Therefore it is possible to discuss the single-polaron--bipolaron transition within the same physical picture for both bipolarons and single polarons. Numerical results are presented for the case of LO-phonon interaction and a Coulombic repulsion between the electrons. For this system a scaling relation between the free energies in two (2D) and three dimensions (3D) is obtained. Bipolaron formation is only possible above a critical value for the coupling constant ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$, which is lower in two than in three dimensions (in 2D: ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$\ensuremath{\approxeq}2.9 and in 3D: ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$\ensuremath{\approxeq}6.8). This indicates more favorable conditions for bipolaron formation in two dimensions, which might be of relevance for the bipolaron model of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductivity. The single-polaron--bipolaron transition behaves much like a first-order phase transition.