Envelope wave functions and subband energies in superlattices with complex bases: Analytical solutions and numerical examples

We study envelope states in superlattices with complex bases including a general coupled-well basis and a general step-well basis. Analytical solutions for the subband energies and the electronic wave functions for all values of the wave number are obtained. Numerical examples are also given. It is shown that the wave functions of the first subband are, in general, localized at the wide well of a coupled-well basis and at the lowest well of a step-well basis, whereas the envelope states of the second subband basically extend to the whole complex basis. The calculation results also show that the wave functions corresponding to band edges have definite parities for superlattices with inversion symmetry. This work can be regarded as an extension of the formalism of Pan and Feng [Phys. Rev. B 44, 5668 (1991)].