Effective-mass Hamiltonian for strained superlattices

For an abrupt heterojunction between two otherwise homogeneous semiconductors in one dimension, the effective-mass Hamiltonian

is examined with 2α + β = −1, where m(z), a(z), and E_c(z), are the position dependent effective mass, the local lattice parameter, and the local conduction band edge, respectively. The exact Schrödinger equation is compared with that for the effective-mass equation in the case of an analytically solvable model. In the asymptotic limit of the energy close to the band edge, the conclusions obtained are shown to be dependent on the parity of the bands.