Strain and crystallographic orientation effects on the valence subbands of wurtzite quantum wells

We present a derivation of the $6\times 6$ valence band Hamiltonian valid for general $\left[h0\mathrm{il}\right]$ oriented III-V wurtzite strained substrates. The derivation is performed within the exact envelope function theory approach in which the $\mathrm{k̂}$ operators are written in an ordered form when the Hamiltonian is applied to wurtzite superlattices. The choice of suitable basis functions for the $\mathrm{}\left(h0\mathrm{il}\right)\mathrm{}$ growth direction, written in terms of those for the typical (0001) case, allows us to define the heavy-, light-, and split-off-hole states in a transparent way and without ambiguities. The Hamiltonian for the $\left(101\mathrm{}¯0\right)$ oriented crystals is provided explicitly. The main differences between the bulk and superlattice $(\mathrm{k̂}$-ordered) Hamiltonians are contrasted. As an application, numerical calculations of the subband dispersion of ${\mathrm{G}\mathrm{a}\mathrm{N}/\mathrm{I}\mathrm{n}}_x{\mathrm{Al}}_{1-x}\mathrm{N}$ quantum wells for various directions are presented and discussed. We show that inclusion of strain effects and the properly $\mathrm{k̂}$-ordered Hamiltonian are essential in an accurate description of the valence subbands of quantum wells in these systems.