Feynman-Haken path-integral approach for polarons in parabolic quantum wires and dots

Abstract

Within the framework of the Feynman-Haken (FH) variational path-integral theory, the expression for the ground-state energy of the electron and longitudinal-optical phonon system in parabolic quantum wires and dots with arbitrary electron-phonon coupling constant and confining potential strength is derived in a unified way with the use of a general effective potential. For quantum dots, due to high symmetry, a simple closed-form analytical expression for the Feynman energy can be obtained, and the analytical results in the extended-state and localized-state limit can be further derived. It is shown both analytically and numerically that the present FH results could be better than those obtained by the second-order Rayleigh-Sch\"ordinger perturbation theory and the Landau-Pekar strong-coupling theory for all cases, which, therefore, shows the effectiveness of the present approach. In quantum wires, it is found in numerical calculations that the binding of polarons is monotonically stronger as the effective wire radius in the whole coupling regime. Interestingly, when the confining potential of quantum wire is extremely strengthened, even in the weak- and intermediate-coupling regime, this system could exhibit some strong-coupling features. More importantly, it is proven strictly that a very recent result in the literature that ``the binding can be weaker than in bulk case at weak coupling'' is not an intrinsic property of this system. In quantum dots, it is found numerically that the polaron binding energy increases with the decrease in size of the dot and is much more pronounced in two dimensions (2D) than in 3D, while the relative polaronic enhancement is stronger in 3D than in 2D for not too weak electron-phonon coupling.