Growth of Multi-Dimensional Superlattices Using Step Array Templates: Evolution of the Terrace Size Distribution

Molecular beam epitaxy has revolutionized the fields of thin films by making possible the fabrication of new solid state materials and artificial superlattices, structures with grown-in periodicities along the growth direction. The wealth of new phenomena arising from the availability of these materials has stimulated the desire for two and three dimensional superlattices with periodicity in the in-plane direction. Unfortunately, the interesting region of the order of 100angstroms, where quantum phenomena at room temperature are expected, is beyond the resolution limit of current lithography schemes. However, one proposal, which might achieve the required dimensions, involves the use of a stepped surface as a template to pattern the in-plane periodicity. In this paper we report on a mathematical analysis of one aspect of this problem, a general and analytical description of the approach to uniformity of the terrace lengths in a step flow growth model.