Ordered Hamiltonian and matching conditions for heterojunctions with wurtzite symmetry: ${\mathrm{G}\mathrm{a}\mathrm{N}/\mathrm{A}\mathrm{l}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ quantum wells

Using Burt’s exact envelope function theory together with symmetry arguments, we have derived an explicit form for the $6\times 6$ valence-band effective mass Hamiltonian for semiconductor heterojunctions with wurtzite symmetry $C_{6v}^4.$ The properly ordered Hamiltonian has a similar form to that of the bulk wurtzite effective Hamiltonian, and reduces to it when a homogeneous system is assumed. The wurtzite effective mass parameters are written in terms of dimensionless quantities that reveal the coupling with the various remote bands with ${\Gamma }_1,$ ${\Gamma }_5,$ and ${\Gamma }_6$ symmetries. Correct boundary conditions for heterojunctions with abrupt interfaces are presented. The widely used (although formally incorrect) symmetrized Hamiltonian and boundary conditions are shown to neglect the contribution of the d-like remote bands, specifically ${\Gamma }_6,$ which has character $\{d_{\mathrm{xz}}{,d}_{\mathrm{yz}}\}.$ Numerical calculations of the subband structure of ${\mathrm{G}\mathrm{a}\mathrm{N}/\mathrm{A}\mathrm{l}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ quantum wells display significant differences on the heavy-holes subband states. We find that the changes in dispersion are enhanced when the material parameters of the well and barrier differ considerably and/or for small widths of the quantum wells. Apart from quantitative energy shifts of the subbands, these results have experimental consequences, including level anticrossing and the corresponding changes to the transition matrix elements in quantum wells.