Polaronic effects in a Gaussian quantum dot

The problem of an electron interacting with longitudinal-optical (LO) phonons is investigated in an N -dimensional quantum dot with symmetric Gaussian confinement in all directions using the Rayleigh–Schrödinger perturbation theory, a variant of the canonical transformation method of Lee–Low–Pines, and the sophisticated apparatus of the Feynman–Haken path-integral technique for the entire range of the coupling parameters and the results for N = 2 and N = 3 are obtained as special cases. It is shown that the polaronic effects are quite significant for small dots with deep confining potential well and the parabolic potential is only a poor approximation of the Gaussian confinement. The Feynman–Haken path-integral technique in general gives a good upper bound to the ground state energy for all values of the system parameters and therefore is used as a benchmark for comparison between different methods. It is shown that the perturbation theory yields for the ground state polaron self-energy a simple closed-form analytic expression containing only Gamma functions and in the weak-coupling regime it provides the lowest energy because of an efficient partitioning of the Gaussian potential and the subsequent use of a mean-field kind of treatment. The polarization potential, the polaron radius and the number of virtual phonons in the polaron cloud are obtained using the Lee–Low–Pines–Huybrechts method and their variations with respect to different parameters of the system are discussed.