Quantum Wells

Abstract

We continue the development of the previous chapter to include the bandstructure modifications in quantum wells resulting from the quantum-confinement geometry. Section 6.1 shows how the envelope approximation method incorporates confinement effects into the k · p theory. The influence of quantum confinement on the valence band structure can be quite significant mixing especially the top two bulk semiconductor valence bands, i.e. the heavyhole and light-hole bands. We show in Sect. 6.2 how this mixing is treated in the context of the Luttinger Hamiltonian. Section 6.3 introduces the concept of elastically strained systems and shows how strain effects may be incorporated into the band-structure calculations. In order to compute gain and refractive index, we need the dipole matrix elements, which we derive in Sect. 6.4. Up to that point, the hole band-structure calculations are based on the bulk-material 4 x 4 Luttinger Hamiltonian, which ignores the effects of the additional split-off hole states with total angular momentum j = 1/2. Section 6.5 describes how these states can be included in the band-structure calculations. Reasons for doing so involves laser compounds based on phosphides and nitrides, where the spin-orbit energies are smaller than those of the arsenides. The nitride based compounds exist in the cubic and hexagonal crystal structures. Section 6.6 shows the modifications of the Luttinger Hamiltonian which are necessary in order to be applicable to the hexagonal geometry.