Discontinuous envelope function in semiconductor heterostructures

Based on a proper definition of the current operators for non-quadratic Hamiltonians, we derive the expression for the transport current which involves the derivative of the imaginary part of the free-electron current, highlighting peculiarities of the extra terms. The expression of the probability current, when Spin-Orbit Interaction (SOI) is taken into account, requires a reformulation of the boudary conditions. This is especially important for tunnel heterojunctions made of non-centrosymmetric semiconductors. Therefore, we consider a model case: tunneling of conduction electrons through a [110]-oriented GaAs barrier. The new boundary conditions are reduced to two set of equations: the first one expresses the discontinuity of the envelope function at the interface while the other one expresses the discontinuity of the derivative of the envelope function.