Strain and band-edge alignment in single and multiple layers of self-assembled Ge/Si and GeSi/Si islands

Strain in coherently embedded Ge/Si islands significantly modifies the band-edge alignment in and around the nanostructures. Our calculations on embedded flat truncated Ge pyramids show that tensile strain in the surrounding Si causes a splitting of the sixfold-degenerate $\Delta$ valleys into the fourfold-degenerate $\Delta \mathrm{}\left(4\right)\mathrm{}$ and twofold-degenerate $\Delta \mathrm{}\left(2\right)\mathrm{}$ valleys. This strain-induced splitting energy can be larger than 400 meV in stacked Ge/Si islands. The $\Delta \mathrm{}\left(2\right)\mathrm{}$ valleys in the Si constitute the conduction-band minimum, and the heavy hole in the island constitutes the valence-band maximum. The band gap in the Si above and below the Ge island is smaller than for bulk Si, in perfect agreement with recent experiments. Relevant energies are worked out as a function of Si interlayer thickness, number of islands, and Ge concentration in the islands. We compare our calculations of the band-gap energy with photoluminescence experiments on embedded Ge islands, yielding an average Ge fraction in the nanostructures of $60\%.$