One-phonon relaxation of localized electronic states in anharmonic nanoparticles

There is currently great interest in the dynamics of electrons and phonons in low-dimensional systems, where the effects of quantum confinement cause a dramatic difference in their behaviour as compared with bulk systems. In this paper we consider a localized electronic impurity state (an electronic two-level system) linearly coupled to the vibrational modes of an isolated nanometre-scale insulating crystal, and study the phonon emission rate at frequencies less than that of the lowest internal vibrational mode, i.e., i n the acoustic 'gap'. We show that, at finite temperature, electronic energ y relaxation below the acoustic gap can occur as a result of anharmonic bro adening of vibrational modes, and we calculate the frequency and temperatur e dependence of the relaxation provided by this mechanism.