Boundary conditions for heterostructures in the multiband effective mass theory

Abstract

We consider the Luttinger Hamiltonian for the degenerate Γ8 valence bands. Based on the bulk Hamiltonian we develop a Hermitian Hamiltonian for heterostructures, where the Luttinger parameters are spatially varying. The correct boundary conditions are derived from the minimal mathematical restrictions on the resulting coupled set of second order differential equations. The equations can be generally written to kα Dαβjj' kβFj' + EνFj = εFj' with the effective mass tensor fulfilling the symmetry property (Dαβjj')† = Dβαj'j. This requires Fj' and Dαβjj' kβFj' continuous in the effective mass equation. These boundary conditions automatically imply the conservation of current in the sample. This strict mathematical treatment of the boundary conditions for the multiband effective mass equation for heterostructures can reconcile the controversy on this subject. In an Appendix we comment on the new envelope function method for heterostructures, developed by Burt.