Mesoscopic description of heterojunctions

Starting from the microscopic current associated with the single-electron wave functions and using Wannier functions suitably generalized for nonperiodic heterostructures we construct mesoscopic (either discrete or continuous) conserved currents associated with states having components only in the considered band. This provides a rigorous basis to the approximate envelope function approaches. As an application we analyze general one-dimensional heterojunctions, mapping explicitly the microscopic asymptotic states onto mesoscopic (one-band envelope) ones. This proves that, apart from the effective masses and the band offsets, the connection rules are characterized in general by three parameters, the values of which are unconstrained, confirming the results of phenomenological analyses.