Phonon-mediated electron-electron interaction in confined media: low-dimensional bipolarons

We study the criterion for the formation of confined large bipolarons and their stability. In order to deal with this specific subject of polaron theory, it is required to adopt some particular approximation methods, because the polaronic systems do not admit exact analytic solutions in general. Those approximation techniques, which are applied to the low-dimensional one-polaron problems, are presented to some extent to form a working basis for our main theme, bipolarons. As the model of confined bipolaron, the electrons are treated as bounded within an external potential while being coupled to one another via the Fröhlich interaction Hamiltonian. Within the framework of the bulk-phonon approximation, the model that we use consists of a pair of electrons immersed in a reservoir of bulk LO phonons and confined within an anisotropic parabolic potential box, the barrier slopes of which can be tuned arbitrarily from zero to infinity. Thus, encompassing the bulk and all low-dimensional geometric configurations of general interest, we obtain an explicit tracking of the critical values of material parameters for the bipolarons to exist in confined media. First, in the limit of strong electron-phonon coupling, we present a unified insight into the stability criterion by applying the Landau-Pekar strong coupling approximation. This crude approximation provides us the condition on the ratio of dielectric constants (η = epsilon substcript infinity/epsilon substrcript 0) for large values of electron-phonon coupling constant α. For more reliable results, we consider the path-integral formulation of the problem adopting the Feynman-polaron model to derive variational results over a wide range of the Coulomb interaction and phonon coupling strengths. It is shown that the critical values of α and η exhibit some non-trivial features as the effective dimensionality is varied, and the path integral results conform to those of strong coupling approximation in the limit of large α.