We present an analytic model that explains the self-ordering of quantum nanostructures grown on nonplanar surfaces. Self-limiting growth in these structures results from the interplay among growthrate anisotropy, curvature-induced capillarity, and, for alloys, entropy of mixing effects. Experimental results on self-limiting organometallic chemical vapor deposition on corrugated surfaces are in quantitative agreement with the model. The implications of the self-limiting growth characteristics on the self-ordering of quantum wells, wires, and dots are discussed. [S0031-9007(98)07220-2] PACS numbers: 68.65. + g, 81.10.Bk, 82.65.Dp Two-or three-dimensionally quantum-confined semiconductor structures have attracted much attention because of their interesting physical properties and potential device applications [1]. To overcome limitations in size and interface quality related to traditional lithography techniques, many efforts have been devoted to study their formation during the epitaxial process [2]. This can be accomplished if a suitable driving force is introduced to yield the desired lateral heterostructure patterning. A widely used approach in this direction is to exploit self-ordering processes on planar surfaces, as for strained-induced Stranski-Krastanow growth of quantum dots (QDs) [3,4]. Such techniques have the advantage that self-ordering is achieved without any surface patterning prior to growth; however, they suffer from a limited control on uniformity and deposition site due to the intrinsic random nature of the nucleation process.Self-ordering of nanostructures on nonplanar surfaces has the potential for solving these problems, as the corrugated surface can provide a template for the nucleation sites. In fact, organometallic chemical vapor deposition (OMCVD) and molecular beam epitaxy (MBE) on substrates patterned with corrugations (see Fig. 1) [5,6] or with pyramidal patterns [7] have been successfully employed to fabricate uniform arrays of quantum wires (QWRs) and QDs. Despite the accurate structural control demonstrated with this approach, the understanding of the self-limiting growth mechanism on such corrugated surfaces has been essentially phenomenological [8]. Existing models can, in fact, predict only constant growth rates of thick layers on mm-sized facets, depending on their orientation and environment, as a result of gas-phase and surface diffusion [9,10]. The growth behavior of facets in the 10-nm scale, relevant to the self-ordering of quantum nanostructures, cannot be explained with such models, since facet-size dependent surface diffusion fluxes should be invoked to account for the self-limiting growth [11].In this Letter we address the self-limiting growth of a corrugated surface, and establish a model that quantitatively describes the self-ordering of quantum wells (QWs), QWRs, and QDs on such patterned substrates.The formation of surface patterns during growth relies on lateral gradients in the surface chemical potential m. Considering, for simplicity, variations in only one dim...