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Physical review. B, Condensed matter

Large bipolarons in two and three dimensions

Guy L. M. M. Verbist, François Maria Peeters, J. T. Devreese

Published Feb 1, 1991• 165 citations
Publisher

Abstract

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-${\mathit{T}}_{\mathit{c}}$ superconductivity. The single-polaron--bipolaron transition behaves much like a first-order phase transition.

DOI
10.1103/physrevb.43.2712

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Two-dimensional Hubbard-Holstein bipolaron

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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 $U$ 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 $U$ 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 $U$. 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.

arXiv:cond-mat/0401305•Physical Review B•Jun 23, 2004•110 citations
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 \ensuremathαc{\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}\ensuremathαc​, which is lower in two than in three dimensions (in 2D: \ensuremathαc{\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}\ensuremathαc​\ensuremath{\approxeq}2.9 and in 3D: \ensuremathαc{\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}\ensuremathαc​\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.
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 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.
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 variational calculation is presented for large bipolarons in the strong-coupling limit. By means of the phonon coherent state, two coupled self-consistent integrodifferential equations for the center-of-mass and relative motions have been derived. Compared with previous works, lower binding energies for large bipolarons in three dimensions (3D) and two dimensions (2D) have been calculated. As a result, the values of lower bounds to critical ratios between dielectric constants below which bipolarons can exist are obtained as ${\mathrm{\ensuremath{\eta}}}_{\mathit{c}}$=1.23 in 3D and ${\mathrm{\ensuremath{\eta}}}_{\mathit{c}}$=1.64 in 2D.
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.