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Physics of the Solid State

Interaction of polarons in two and three dimensions

P. J. Baymatov, Pulat K. Khabibullaev

Published Mar 1, 1997• 0 citations
Publisher

Abstract

Two-dimensional and three-dimensional bipolarons of large radius, symmetrized with respect to the coordinate parts of the two-center wave functions, are investigated in the adiabatic approximation with allowance for dynamic interelectronic correlations. The adiabatic potential lines are plotted. It is shown that the quasimolecular configuration appearing in the Hartree-Fock approximation is unstable in both the two-dimensional and the three-dimensional cases. The ground state is a one-center configuration. Estimates are given for the binding energy and the heat of dissociation of a bipolaron.

DOI
10.1134/1.1129863

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Physics of the Solid State•Jan 1, 2008•6 citations
Two-dimensional and three-dimensional bipolarons of large radius, symmetrized with respect to the coordinate parts of the two-center wave functions, are investigated in the adiabatic approximation with allowance for dynamic interelectronic correlations. The adiabatic potential lines are plotted. It is shown that the quasimolecular configuration appearing in the Hartree-Fock approximation is unstable in both the two-dimensional and the three-dimensional cases. The ground state is a one-center configuration. Estimates are given for the binding energy and the heat of dissociation of a bipolaron.
The energy of a large bipolaron is calculated for various spacings between the centers of the polarization potential wells of the two polarons with allowance made for electron correlations (i.e., the explicit dependence of the wave function of the system on the distance between the electrons) and for permutation symmetry of the two-electron wave function. The lowest singlet and triplet 23 S states of the bipolaron are considered. The singlet polaron is shown to be stable over the range of ionic-bond parameter values η≤ηm≈0.143 (η=ɛ∞/ɛ0, where ɛ∞ and ɛ 0 are the high-frequency and static dielectric constants, respectively). There is a single energy minimum, corresponding to the single-center bipolaron configuration (similar to a helium atom). The binding energy of the bipolaron for η → 0 is J bp=−0.136512e 4 m */ℏ2ɛ ∞ 2 (e and m* are the charge and effective mass of a band electron), or 25.8% of the double polaron energy. The triplet bipolaron state (similar to an orthohelium atom) is energetically unfavorable in the system at hand. The single-center configuration of the triplet bipolaron corresponds to a sharp maximum in the distance dependence of the total energy J bp(R); therefore, a transition of the bipolaron to the orthostate (e.g., due to exchange scattering) will lead to decay of the bound two-particle state. The exchange interaction between polarons is antiferromagnetic (AFM) in character. If the conditions for the Wigner crystallization of a polaron gas are met, the AFM exchange interaction between polarons can lead to AFM ordering in the system of polarons.
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.
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.
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.
In this paper we consider the stability of a large bipolaron embedded in a polaron gas. The main conclusion is that an isolated nonstable bipolaron can be stabilized in the presence of a polaron gas exhibiting Fermi statistics. On the other hand, the exchange interaction tends to destabilize the bipolaron. This study has been performed both for bulk three-dimensional materials and for thin two-dimensional films using the Hartree-Fock approximation. The bipolaron has been described by an extension of the Feynman polaron model.
The conditions of the virial theorem imposed on the dependence of the total energy of a system consisting of two polarons on the distance between the centers of the polarization wells are analyzed for various trial two-electron functions in the limit of strong electron-phonon coupling. It is shown that the inclusion of the direct dependence of the wave function of the system on the interelectron distance (correlation effects) does not break the conditions of the virial theorem. This theorem holds for a single-center and a two-center configuration corresponding to a subsidiary minimum, which disappears when the electron correlations are included.