Efficient band-structure calculations of strained quantum wells

The effective-mass equations using the Luttinger-Kohn Hamiltonian taking into account the strain effects are solved exactly by making a unitary transformation. Using this method, we need to solve two 2\ifmmode\times\else\texttimes\fi{}2 matrices instead of the original 4\ifmmode\times\else\texttimes\fi{}4 matrix. The eigenvalues and eigenvectors for the heavy hole and the light hole can be expressed analytically. When applied to heterojunctions such as quantum wells, the reduction in the number of unknowns makes the method a more efficient approach to the calculations of valence-band structures, which takes into account the valence-band mixing and the strain effects with proper boundary conditions. Detailed numerical results and significant features for strained ${\mathrm{Ga}}_{\mathit{x}}$${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As grown on an ${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$${\mathrm{As}}_{\mathit{y}}$${\mathrm{P}}_{1\mathrm{\ensuremath{-}}\mathit{y}}$ lattice matched to InP are presented. For a Ga mole fraction x less than 0.468 (compression), the top subband is heavy-hole-like. For x greater than 0.468 (tension strain), the top band can be either a heavy-hole or a light-hole subband, depending on the well width and the shear deformation potential. Interesting subband structures, which show a negative effective mass in the top subband, are discussed.