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Physics of the Solid State

Spatial configuration of the bipolaron and the virial theorem

N. I. Kashirina, Victor Dmitrievich Lakhno

Published Jan 1, 2008• 6 citations
Publisher

Abstract

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.

DOI
10.1134/s1063783408010034

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Polarons and bipolarons in strongly interacting electron-phonon systems

Gerhard Wellein, Heinrich Röder, Holger Fehske

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.

arXiv:cond-mat/9512143•Physical review. B, Condensed matter•Apr 15, 1996•163 citations
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.
The work is devoted to issues related with implementation of the virial theorem for one-center bipolaron.The virial theorem expressions have been obtained for an electron system with Coulomb interactions in the phonon field.It is shown that for the bipolaron functional (one-center configuration) virial theorem holds for arbitrary electron-phonon coupling.As a specific example of the virial theorem for one-center bipolaron configuration, the author adduces numerical calculations of the energy of the ground state and the various contributions (kinetic energy, electron-phonon interaction, electron energy, phonon energy) into the energy of bipolaron, performed within the framework of Buimistrov-Pekar method.It is shown that the virial theorem is fulfilled with high accuracy for the two-electron systems with Coulomb interactions for an arbitrary value electron-phonon coupling.The necessary condition for formation of a bipolaron stable state is accounting electron correlations associated with the direct dependence of the trial electron wave function of the system from the interelectron distance.
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.
Bipolaron energies are calculated as a function of wave vector by a variational method of Gurari appropriate for weak or intermediate coupling strengths, for a model with electron-phonon interactions independent of phonon wave vector and a short-ranged Coulomb repulsion. It is assumed that the bare electrons have a constant effective mass. A two-parameter trial function is taken for the relative motion of the two electrons in the bipolaron. The energies of the bipolarons are compared with those of two single polarons as a function of wave vector for various parameter values. Results for the effective masses at the zone center are also obtained. Comparison is made with data of other authors for bipolarons in the Hubbard-Holstein model, which differs mainly from the present model in that it has a tight-binding band structure for the bare electrons.
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 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.