Polaron lifetime and energy relaxation in semiconductor quantum dots

Since early attempts to evaluate the electron energy-loss rate in semiconductor quantum dots, a number of papers have proposed various reasons why the predicted longitudinal-optical ~LO! phonon bottleneck should be bypassed in actual dots ~for phonon-related mechanisms see, e.g., Refs. 3–5 and for Auger effects see, e.g., Refs. 6–8!. In particular, Li et al. have proposed that the finite LO phonon lifetime could alleviate the matching condition between the electron energy difference and the ~quasimonochromatic! LO phonon energy which results from Fermi golden rule. Recent experiments and calculations have pointed out that electrons and optical phonons in quantum dots are in a strongcoupling regime. This implies that their coupling can never be accounted for perturbatively and that, in fact, electrons and phonons form mixed modes. These mixed modes, the polarons, would be everlasting if both their constituents, the electron and the phonons, were stable elementary excitations. The electron lifetime ~limited by radiative decay in the case of photoexcitation of ideal dots! is long, typically 1 ns. Such is not the case of the LO phonons which are known in bulk semiconductors to disintegrate into two less energetic phonons due to the crystal anharmonicity ~see, e.g., Ref. 11!. The lifetime of LO phonons is 2 ps at room temperature in bulk GaAs. Hence, in a semiconductor quantum dot, the polaron decay will be triggered by the instability of its phonon component. We stress in this work that in contrast to bulk or quantum-well structures the energy relaxation in quantum dots is not due to the processes involving the emission of one LO phonon but of two phonons and that the relaxation mechanism is not associated to the sole electron–LO phonon coupling but rather to the phononphonon interaction on polaron states. This implies, in particular, that there is no need for the two electron states to differ by one LO phonon energy because the energy conservation in this relaxation path is that of the polaron, which may greatly differ from the electron one due to the strongcoupling regime between electrons and LO phonons in quantum dots. This relaxation mechanism in quantum dots is assessed in the following. We consider for simplicity the lower lying bound electron states in a nanometric quantum dot. To be specific we consider the case of InAs/GaAs dots. Their electron states can be approximately labeled by the z projection of the angular momentum of their envelope functions. Hereafter, they will be denoted uS& and uP6& with a typical energy distance of 50 meV. In quantum dots the electron–optical-phonon interac-