Effective exponent for the size dependence of luminescence in semiconductor nanocrystallites

The size $\mathrm{}\left(d\right)\mathrm{}$ dependence of photoluminescence from nanocrystalline semiconductors is examined. The overall luminescence is determined by two distinct physical mechanisms: (i) the variation of the semiconductor gap with size $d$ (typically $\sim {1/d}^{\alpha },$ $\alpha >1),$ and (ii) the variation of the oscillator strength $f_{\mathrm{osc}}$ with size (typically ${1/d}^{\beta },$ $5<~\beta <~6).$ We present an analytical framework to understand the luminescence line shape based on the above two mechanisms, taking no recourse to computational simulations. We show that the peak energy varies with the mean particle size $d_0$ as $d_0^{-\beta },$ where $\beta$ is an effective exponent determined by the disorder in the system. Our results can explain conflicting experimental observations on the luminescence from silicon nanocrystallites.