Generalized effective mass theory of sub-surface scanning tunneling microscopy: application to cleaved quantum dots

An effective mass theory of sub-surface STM is developed. Sub-surface structures such as quantum dots embedded into a semiconductor slab are considered. States localised around sub-surface structures match on to a tail that decays into the vacuum above the surface. It is shown that the lateral variation of this tail may be found from a surface envelope function provided that the effects of the slab surfaces and the sub-surface structure decouple approximately. The surface envelope function is given by a weighted integral of a bulk envelope function that satisfies boundary conditions appropriate to the slab. The weight function decays into the slab inversely with distance and this slow decay explains the sub-surface sensitivity of STM. These results enable STM images to be computed simply and economically from the bulk envelope function. The method is used to compute wave function images of cleaved quantum dots and the computed images agree very well with experiment.