Energy levels of Ge quantum wells embedded in Si: A tight-binding approach

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We calculate the energy levels of Ge quantum wells embedded in Si and grown on an arbitrary ${\mathrm{Si}}_{x}{\mathrm{Ge}}_{1\ensuremath{-}x}$ substrate. The calculations are carried out using the tight-binding-renormalization approach and reliable Slater-K\"oster parameters including interactions up to second neighbors. The substrate affects the positions of the Si and Ge atoms, and the Slater-K\"oster parameters are modified consistently using scaling laws beyond the ${d}^{\ensuremath{-}2}$ Harrison rule. Our results provide theoretical support to the observed photoluminescence lines within the Ge quantum wells. We study the effect of substrate alloy composition on the position of these lines and find that the two main energy transitions in the Ge quantum wells approach when the Ge concentration in the substrate increases.