A layered heterostructure, comprising alternating layers of different semiconductors, wherein one of the atom species of one of the semiconductors has a faster diffusion rate along an oxidizing interface than an atom species of the other semiconductor at an oxidizing temperature, can be used to fabricate embedded nanostructures with arbitrary shape. The result of the oxidation will be an embedded nanostructure comprising the semiconductor having slower diffusing atom species surrounded by the semiconductor having the higher diffusing atom species. The method enables the fabrication of low- and multi-dimensional quantum-scale embedded nanostructures, such as quantum dots (QDs), toroids, and ellipsoids.