The object of the present invention is to provide a fabrication method of fine structures which have few carrier trap centers and light absorption levels and find applications in quantum wires and quantum boxes having arbitrary configurations at least within a two-dimensional plane. The fabrication method comprises the steps of having a sharp tip held in close proximity to the surface of a substrate 1 and having a metal constituting the tip evaporated thereof or having a metal contained in ambient vapor or solution decomposed by a tunnel current or the like for deposition of said metal locally on the substrate surface and then having a finely structured crystal grown on said locally deposited region by means of a vapor phase-liquid phase-solid phase reaction.
