Large bipolarons in two and three dimensions

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.