Million-Atom Pseudopotential Calculation of $\gamma$- $\mathit{X}$ Mixing in $\mathrm{GaAs}/\mathrm{AlAs}$ Superlattices and Quantum Dots

We have developed a “linear combination of bulk bands” method that permits atomistic, pseudopotential electronic structure calculations for $\sim {10}^6$ atom nanostructures. Application to $\left(\mathrm{GaAs}{)}_n/\right(\mathrm{AlAs}{)}_n$ (001) superlattices (SL's) reveals even-odd oscillations in the $\gamma -X$ coupling magnitude $V_{\gamma X}\left(n\right)$, which vanishes for $n=\mathrm{odd}$, even for abrupt and segregated SL's, respectively. Surprisingly, in contrast with recent expectations, 0D quantum dots are found here to have a smaller $\gamma -X$ coupling than equivalent 2D SL's. Our analysis shows that for large quantum dots this is largely due to the existence of level repulsion from many $X$ states.