The 30-band k ⋅ p theory of valley splitting in silicon thin layers

The valley splitting of the conduction-band states in a thin silicon-on-insulator layer is investigated using the 30-band k ⋅ p theory. The system composed of a few nm thick $\text{Si}$ layer embedded within thick SiO₂ layers is analyzed. The valley split states are found to cross periodically with increasing quantum well width, and therefore the energy splitting is an oscillatory function of the quantum well width, with period determined by the wave vector K ₀ of the conduction band minimum. Because the valley split states are classified by parity, the optical transition between the ground hole state and one of those valley split conduction band states is forbidden. The oscillations in the valley splitting energy decrease with electric field and with smoothing of the composition profile between the well and the barrier by diffusion of oxygen from the SiO₂ layers to the Si quantum well. Such a smoothing also leads to a decrease of the interband transition matrix elements. The obtained results are well parametrized by the effective two-valley model, but are found to disagree from previous 30-band calculations. This discrepancy could be traced back to the fact that the basis for the numerical solution of the eigenproblem must be restricted to the first Brillouin zone in order to obtain quantitatively correct results for the valley splitting. (paper)