Hole subbands in strained GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Al}}_{\mathrm{x}}$As quantum wells: Exact solution of the effective-mass equation

Valence subbands of uniaxially stressed GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Al}}_{\mathrm{x}}$As quantum wells are found by solving exactly the multiband effective-mass equation for the envelope function; as in the particle in a box problem, we first solve the effective-mass equation in each bulk material, and then we impose boundary conditions on the linear combinations of bulk solutions. Discrete symmetries of the effective-mass Hamiltonian are used to decouple the spin-degenerate subbands; the energy levels are obtained as the zeros of an 8×8 determinant. The functional form of the wave functions is given analytically, and is used in order to discuss the heavy-hole–light-hole mixing at finite values of the in-plane vector ${\mathrm{k}}_{?}$; the mixing greatly increases when the applied stress reduces the energy separation at ${\mathrm{k}}_{?}$=0. Resonances are shown to arise and are due to the degeneracy of discrete levels with states of the continuum at different values of ${\mathrm{k}}_{?}$.