Properties of a polaron confined in a spherical quantum dot

A Frohlich Hamiltonian describing the electron-phonon interaction in a spherical quantum dot embedded in another polar material is derived, taking into account interactions with both bulk longitudinal optical and surface optical phonons. The Hamiltonian is appropriate to the general case of a finite confining potential originating from a bandgap mismatch between the materials of the dot and the surrounding matrix. This Hamiltonian is then applied to treat the electron-phonon interaction in the adiabatic approximation for various quantum dot systems. It was found that, as the radius of the dot decreases, the magnitude of the electron-phonon interaction energy first increases, passes through a maximum, and then gradually decreases to the value appropriate to the situation where the electron is weakly localized inside the dot. For most dot radii the polaron properties are described well by a model assuming perfect electron confinement. Based on this result, the problem of the bound polaron confined perfectly in the quantum dot was investigated within the adiabatic and all-coupling variational approaches. The polaron properties have been studied performing both analytical and numerical calculations for various radii of the quantum dot and for different impurity positions inside the dot. Within the adiabatic approximation, it was found that the magnitude of the electron-phonon interaction increases as the radius decreases for any impurity position. It was also shown that the input from the electron-surface-phonon interaction to the total polaron energy is much larger than was found earlier for the free polaron confined in the dot. As a function of the impurity position, the electron-surface-phonon interaction energy increases as the impurity is shifted towards the surface, reaches its maximum when the impurity is positioned inside the dot and then decreases as the impurity moved close to surface. The all-coupling approach gave rise to the following results: for any value of coupling and for small binding strength the total electron-phonon interaction energy depends weakly on the impurity position and has a maximum when the impurity is at the center of the dot. As the binding strength increases, the maximum of the electron-phonon interaction energy shifts inside the dot. For small values of the coupling the magnitude of the electron-surface-phonon interaction has a maximum on the surface of the dot, unlike the results of the adiabatic limit.