Disorder and size effects in the envelope-function approximation

We investigate the validity and limitations of the envelope-function approximation (EFA), widely accepted for the description of the electronic states of semiconductor heterostructures. We consider narrow quantum wells of GaAs confined by ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ barriers. Calculations performed within the tight-binding approximation using ensembles of supercells are compared to the EFA results. Results for miniband widths in superlattices obtained in different approximations are also discussed. The main source of discrepancy for narrow wells is the treatment of alloy disorder within the virtual crystal approximation. We also test the two key assumptions of the EFA: (a) that the electronic wave functions have Bloch symmetry with well-defined $k\to$ in the alloy region; (b) that the periodic parts of the Bloch functions are the same throughout the heterostructure. We show that inaccuracies are mainly due to the former assumption.