Full-band-structure calculation of second-harmonic generation in odd-period strained (Si)n/(Ge)nsuperlattices

We report a full-band-structure calculation of frequency-dependent second-harmonic generation in odd-period strained [(Si${)}_{\mathit{n}}$/(Ge${)}_{\mathit{n}}$]/Si (001) superlattices for n=1, 3, and 5. We derive sum rules to show that for materials with filled valence bands and empty conduction bands, independent of crystal symmetry, the expression for X${\to }^{\mathrm{}\left(2\right)\mathrm{}}$(-2ω;ω,ω) does not diverge in the zero-frequency limit. We use a minimal-basis orthogonalized linear combination of Gaussian orbitals technique, in conjunction with a linearized sampling method, to obtain χ${\to }^{\mathrm{}\left(2\right)\mathrm{}}$(-2ω;ω,ω). We then use these results and a simple model to estimate second-harmonic generation at Si/Ge interfaces.