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Journal of Physics Condensed Matter

The polaron - bipolaron transition for acoustical three-dimensional polarons

Wilson B. da Costa, François Maria Peeters

Published Mar 25, 1996• 10 citations
Publisher

Abstract

The ground-state energy of an acoustical bipolaron in three dimensions (3D) is studied within the Feynman path integral method. The acoustical polaron exhibits a transition from the quasi-free state to the self-trapped state. Depending on the value of the Debye cut-off in phonon wavevector space this transition can be continuous or discontinuous. If two electrons are present a second transition becomes possible: the transition to the bipolaron state in which the two electrons are self-trapped around the same position. The corresponding phase diagram is calculated as function of the cut-off , the electron - phonon coupling strength , and the repulsion (U) between the electrons. We found that: (i) the bipolaron transition is a first-order transition; (ii) it is also possible between two polarons which are in the quasi-free state, i.e. are not self-trapped; and (iii) the effective particle mass increases substantially at the transition point.

DOI
10.1088/0953-8984/8/13/009

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Physica Scripta•Jan 1, 1991•18 citations
The ground-state energy of an acoustical bipolaron in three dimensions (3D) is studied within the Feynman path integral method. The acoustical polaron exhibits a transition from the quasi-free state to the self-trapped state. Depending on the value of the Debye cut-off in phonon wavevector space this transition can be continuous or discontinuous. If two electrons are present a second transition becomes possible: the transition to the bipolaron state in which the two electrons are self-trapped around the same position. The corresponding phase diagram is calculated as function of the cut-off , the electron - phonon coupling strength , and the repulsion (U) between the electrons. We found that: (i) the bipolaron transition is a first-order transition; (ii) it is also possible between two polarons which are in the quasi-free state, i.e. are not self-trapped; and (iii) the effective particle mass increases substantially at the transition point.
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.
We calculate the low-temperature properties of an acoustic polaron in three dimensions in thermal equilibrium at a given temperature using a specialized path-integral Monte Carlo method. In particular we find numerical evidence that the chosen Hamiltonian for the acoustic polaron describes a phase transition from a localized state to an unlocalized state for the electron as the phonon-electron coupling constant decreases. The phase transition manifests itself with a jump discontinuity in the potential energy as a function of the coupling constant. In the weak coupling regime the electron is in an extended state whereas in the strong coupling regime it is found in a self-trapped state.
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 bound states of two electrons in the adiabatic Holstein-Hubbard model are studied numerically in one and two dimensions from the anticontinuous limit. This model involves a competition between a local electron-phonon coupling (with a classical lattice) which tends to form pairs of electrons and the repulsive Hubbard interaction U≥0U \geq 0U≥0 which tends to break them. In 1D, the ground-state always consists in a pair of localized polarons in a singlet state. They are located at the same site for U=0. Increasing U, there is a first order transition at which the bipolaron becomes a spin singlet pair of two polarons bounded by a magnetic interaction. The pinning mode of the bipolaron soften in the vicinity of this transition leading to a higher mobility of the bipolaron which is tested numerically. In 2D, and for any UUU, the electron-phonon coupling needs to be large enough in order to form small polarons or bipolarons instead of extended electrons. We calculate the phase diagram of the bipolaron involving first order transitions lines with a triple point. A pair of polarons can form three types of bipolarons: a) on a single site at small UUU, b) a spin singlet state on two nearest neighbor sites for larger UUU as in 1D and c) a new intermediate state obtained as the resonant combination of four 2-sites singlet states sharing a central site, called quadrisinglet. The breathing and pinning internal modes of bipolarons in 2D generally only weakly soften and thus, they are practically not mobile. On the opposite, in the vicinity of the triple point involving the quadrisinglet, both modes exhibit a significant softening. However, it was not sufficient for allowing the existence of a classical mobile bipolaron (at least in that model).
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.
The bipolaron ground state binding energy and the effective masses are calculated self-consistently in a scheme where the electron-phonon interaction is described by the Fröhlich interaction. We explicitly use the total linear momentum conservation and both two-and three-dimensional systems are considered. We review results for binding energies and show that the bipolaron effective mass increases with the electron-phonon coupling constant α more rapidly than two free polaron masses. As expected, the increase is greater in two than in three dimensions. We estimate the screening effects due to an electronic or hole density n in a range of values such that n R 2 b ≪ 1 (here R b is the bipolaron radius). We find that the bipolaron binding energy decreases with n and eventually becomes positive indicating the existence of a metastable bipolaron state. Finally we discuss the possible connections between our results and high T c superconductivity.