Conduction band of Si-Ge_{x}Si_{1-x} superlattices using the envelope-function approximation

We have calculated the structure of the conduction band in Si-${\mathrm{Ge}}_{\mathrm{x}}$ ${\mathrm{Si}}_{1\mathrm{-}\mathrm{x}}$ superlattices using the envelope-function approximation, neglecting the transverse directions. Our description of the bulk materials is valid throughout the Brillouin zone along one direction and consequently takes the multiple minima of the conduction band into account. We find that in contrast to those for superlattices made from direct-band-gap semiconductors, the energy bands for Si-${\mathrm{Ge}}_{\mathrm{x}}$ ${\mathrm{Si}}_{1\mathrm{-}\mathrm{x}}$ superlattices exhibit a splitting that we assign to the interaction between the two conduction-band minima along the superlattice direction. It is shown that in the limit of infinitely deep wells, the splitting depends sinusoidally on the well width.