Simulation of electron and hole states in Si nanocrystals in a SiO$_{2}$ matrix: choice of parameters of the empirical tight-binding method

The problem of the optimal choice of parameters of the empirical tight-binding method to simulate the quantum-confined levels of Si nanocrystals embedded into an amorphous SiO$_{2}$ matrix is studied. To account for tunneling from nanocrystals to SiO$_{2}$, the amorphous matrix is considered as a virtual crystal with a band structure similar to that of SiO$_{2}$ $\beta$-cristobalite and with a lattice constant matched to the lattice constant of bulk Si. The electron density distributions in $\mathbf{k}$ space for electrons and holes quantum-confined in a Si nanocrystal in SiO$_{2}$ are calculated in a wide energy region, which provides a means to see clearly the possibility of the existence of efficient direct optical transitions for hot electrons at the upper quantum-confined levels.