Zero-point motion and temperature effects on the band gap of semiconductor nanoclusters

We calculate the band gap renormalization of semiconductor nanoclusters, avoiding the large computational costs associated with the calculations of the self-energy (Fan) and the Debye-Waller terms. This approach allows us to address clusters with a few hundred atoms. For Silicon nanoclusters, we obtain a band gap reduction of hundreds of of meV associated with the quantum zero point motion. This reduction rapidly increases with decreasing cluster size. Based on the Bose-Einstein distribution, we further study the temperature dependence of the band gap in semiconductor nanoclusters and find a band gap shift of -580~meV and -270~meV when going from T=0 to room temperature for silicon clusters with radius of 9.6 and 11.9~\AA, respectively. Furthermore, we find that the band gap renormalization of semiconductor nanoclusters is dominated by the optic-like vibrational modes with $\Gamma_4$ point group symmetry.