Effective-mass approximation in semiconductor heterostructures: One-dimensional analysis

The effective-mass method (EMM) has been widely used for the description of layered semiconductor structures with sharp boundaries (quantum wells, superlattices, etc.), although it does not work for rapidly varying potentials. Here, I analyze the EMM in a simple case of one-dimensional heterostructures. Being able to obtain the exact solutions (using the transfer-matrix method), I can extract from them the envelope functions within the one-band or multiband EMM. I can then derive the boundary conditions (BC's) for these envelopes at the interfaces. These BC's turn out to be very different from what has been adopted so far. They involve microscopic parameters of the bulk materials forming the layered structures; these parameters cannot be expressed by the effective masses or other band-structure characteristics. In the multiband case the BC's are energy dependent. For the band edges of different types on two sides of the interface (as in GaAs/AlAs or Si/Ge), the BC's are unusual and allow for the existence of intrinsic interface states. I also point out some artifacts of the EMM; unphysical solutions in the bands and in the gaps which should be rejected.