Form of kinetic energy in effective-mass Hamiltonians for heterostructures

We first prove that the class of effective-mass Hamiltonians H=${\mathit{T}}_{\mathrm{\alpha }}$+V with ${\mathit{T}}_{\mathrm{\alpha }}$=1/2${\mathit{m}}^{\mathrm{\alpha }}$ ${\mathit{pm}}^{\mathrm{-}1\mathrm{-}2\mathrm{\alpha }}$ ${\mathit{pm}}^{\mathrm{\alpha }}$, where m=m(x) is a twice differentiable (${\mathrm{C}}_2$) function and p=-id/dx can be mapped by an unitary transformation onto the constant-mass case with still a local potential that we give explicitly. The value of α is thus irrelevant. We then consider the case when m(x) is only piecewise ${\mathrm{C}}_2$, that is, the case of heterojunctions. We show that the general connection rule for the envelope function and its derivative at an unstrained heterojunction depends on a Hermitian 2×2 matrix which is intrinsic to the junction. Hence, the latter should be described by four parameters (three in the case of time-reversal invariant systems). The parameter values may be obtained from the scattering data for Bloch waves, which can be either measured or computed. Up to two Tamm states, localized at the interface, are allowed.