Using the density functional theory (DFT) with the hybrid nonlocal exchange correlation functional of Becke and Lee, Yang and Parr (B3LYP), we have calculated the optical gap and the oscillator strengths for several of the lowest, spin and symmetry allowed, electronic transitions of small Ge nanocrystals passivated by hydrogen. The largest nanoparticle has an approximate diameter of 2 nm. Our results show that the optical gap exhibits size dependence (due to quantum confinement) roughly similar to silicon nanoparticles. However, for this range of diameters, there is an indirect-to-direct transition in the spectra of Ge as the size of the nanocrystals decrease. The first allowed excitation (fundamental optical gap) of each germanium nanoparticle has relatively larger oscillator strengths compared to silicon. The diameter of the smallest Ge nanocrystal capable to emit in the visible region of the spectrum, is approximately 1.9 nm, compared to 2.2 nm for silicon nanocrystals.
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