Variational calculations of bipolaron binding energies

The energies of bipolaron states in three- and two-dimensional systems are calculated variationally, treating the electron-phonon interaction in the Fröhlich approximation, and separating the relative motion from the center-of-mass motion. The bipolaron is bound if the electron-phonon coupling constant α is larger than 6 in three dimensions and 2 in two dimensions, provided the ratio η=${\mathrm{\epsilon }}_{\mathrm{\infty }}$/${\mathrm{\epsilon }}_0$ is smaller than a critical value which depends on α. 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 α. For α larger than ∼10 and ∼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-${\mathit{T}}_{\mathit{c}}$ 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.