EFFECTIVE-MASS THEORY FOR ABRUPT HETEROJUNCTIONS

For an abrupt heterojunction between two otherwise homogeneous semiconductors in one-dimension we study the effective-mass Hamiltonian H = -HBAR2/2 [m(z)alpha(d/dz)m(z)beta(d/dz)m(z)alpha] + E(c)(z) with 2alpha + beta = -1, where m(z) is the position-dependent effective-mass and E(c)(z) the position-dependent conduction band edge. Through exact model calculations on heterostructures, we find when the effective-mass theory is valid the parameters alpha and beta to be 0 and -1, respectively. This condition is obtained in the asymptotic limit of the energy near the band edge and holds for equal lattice parameter. Results are also compared to other theories.