Spin-Orbit Engineering of Semiconductor Heterostructures

We present a systematic construction of the probability-current operator,based on a momentum power expansion of effective Hamiltonians. The result is valid in the presence of a Rashba term and when a D'yakonov--Perel contribution is included. We propose practical tools for spin-orbit engineering of semiconductor heterostructures. We apply this formalism to a paradigmatic system, the interface between two semi-infinite media, on one side a free-electron-like material and on the other side a barrier material with spin-orbit interaction. We show that the usual boundary conditions, namely the continuity of the envelope function and of a velocity at the interface, according to the BenDaniel-Duke approach, comply with the conservation of the probability current only when first- (Rashba-like) and second-order (free-electron-like) terms are taken into account in the Hamiltonian. We revisit the boundary conditions and we prove that the envelope function may be discontinuous at the interface.