BOUNDARY-CONDITIONS FOR HETEROSTRUCTURES IN THE MULTIBAND EFFECTIVE-MASS THEORY

We consider the Luttinger Hamiltonian for the degenerate Gamma(8) vale nce bands. Based on the bulk Hamiltonian we develop a Hermitian Hamilt onian for heterostructures, where the Luttinger parameters are spatial ly varying. The correct boundary conditions are derived from the minim al mathematical restrictions on the resulting coupled set of second or der differential equations. The equations can be generally written to k(alpha)D(jj')(alpha beta),k(beta)F(j') + E(v)F(j) = epsilon F-j' with the effective mass tensor fulfilling the symmetry property (D-jj'(alp ha beta))(dagger) = D-j'j(beta alpha). This requires F-j' and D-jj'(al pha beta),k(beta)F(j') continuous in the effective mass equation. Thes e boundary conditions automatically imply the conservation of current in the sample. This strict mathematical treatment of the boundary cond itions for the multiband effective mass equation for heterostructures can reconcile the controversy on this subject. In an Appendix we comme nt on the new envelope function method for heterostructures, developed by Burt.