Corrections to the continuum approximation of the Fröhlich polaron: The ground-state energy.

Abstract

Corrections to the continuum approximation in the formulation of the dielectric polaron are investigated. The starting point is the small-polaron Hamiltonian of Emin with an arbitrary electron-phonon interaction potential. The kq representation of Zak is then used. From this Hamiltonian and with an appropriate interaction potential, corrections to the effective-mass approximation in the Fr\"ohlich Hamiltonian are calculated from a perturbative expansion for a slowly varying electron-phonon interaction. This results in a renormalization of the interaction. A Debye cutoff is also imposed on the phonon wave vectors. The ground-state energy of the polaron is then calculated with use of the Fock approximation. It is found that in weak coupling and in strong coupling with small Debye cutoff the corrections to the continuum approximation are small, decrease the polaron self-energy, and are proportional to \ensuremath{\alpha}/L, where \ensuremath{\alpha} is the electron-phonon coupling constant and L is the wave vector at which the Debye cutoff is made. In strong coupling, for a large cutoff, the self-energy is also reduced but much more drastically and the strong-coupling behavior in ${\ensuremath{\alpha}}^{2}$ disappears. Polar crystals for which a large polaron is involved can all be classified in the first limit and the corrections to the continuum approximation are rather small, the largest one being of the order of 14% for LiF.