Band structure and symmetry analysis of coherently grown ${\mathrm{Si}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Ge}}_{\mathit{x}}$ alloys on oriented substrates

A semiempirical tight-binding method was used to calculate the band structures of ${\mathrm{Si}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Ge}}_{\mathit{x}}$ alloys coherently grown on (001)-, (111)-, and (110)-oriented ${\mathrm{Si}}_{1\mathrm{-}\mathit{y}}$ ${\mathrm{Ge}}_{\mathit{y}}$ substrates. The distorted lattice and Brillouin zone, as well as the band-edge splittings and shifts which vary with the Ge content of both the ${\mathrm{Si}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Ge}}_{\mathit{x}}$ strained layer and the substrate, are given. The band structures and symmetry properties of the coherently strained ${\mathrm{Si}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Ge}}_{\mathit{x}}$ alloys along high-symmetry lines of the distorted Brillouin zone are analyzed. The calculation results show that for the [001] and [111] growth cases, the conduction-band minima appear in the growth direction when the epilayer is under tensile strain. When the ${\mathrm{Si}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Ge}}_{\mathit{x}}$ alloys are grown on a (111)- or (110)-oriented substrate, the four-degree degenerate state ${\mathit{X}}_5$ in the [001] direction of an unstrained diamond structure splits into two bands with even and odd parities, respectively. This splitting results in a nonlinear effect, which increases rapidly with increasing strain and results in the downward bending of ${\mathit{E}}_{\mathit{c}}^{\mathrm{\Delta }}$(6) and ${\mathit{E}}_{\mathit{c}}^{\mathrm{\Delta }}$(2) for Si-rich alloys grown on (111) and (110) Ge substrates, respectively. This effect deviates from the band-edge variation trend given by linear deformation-potential theory, which does not explicitly include the nonlinear effect. Corresponding to the reduced symmetry of the distorted diamond structures, irreducible representations of the space groups are obtained and used to denote the calculated energy bands. Relations among the irreducible representations of the energy bands, such as compatibility, the relation between energy bands of unstrained and strained diamond structures, and additional degeneracies due to time-reversal symmetry, are obtained and shown. Finally, selection rules for direct optical transitions are obtained within the framework of the electric-dipole approximation, and the effects of the polarization of incident light are discussed.