Self-assembled Ge quantum dots were formed by in-situ thermal annealing of a thin amorphous Ge layer deposited by molecular beam epitaxy on a thin porous TiO_2 layer grown on SiO_2 on Si(001). For samples with dot diameters ranging from 10 to 35 nm, the dot photoluminescence (PL) appeared primarily as a wide near-infrared band peaked near 800 meV. The peak energy of the PL band reflects the average dot size and its shape depends on the dot size distribution. Using a tight binding bandgap model, we have analyzed the PL spectrum in terms of the dot size distribution. The observed size distribution determined from transmission electron and atomic force microscopy allowed the determination of the nonlinear increase in the PL efficiency with decreasing dot diameter. Knowing this generic PL efficiency, we show that it is possible to evaluate the size distribution of Ge dots from their PL energy dependence.
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