VOLUME 63, NUMBER 4 PHYSICAL REVIEW LETTERS 24 JULY 1989 Diffraction Determination of the Size Distribution of Noncrystalline Regions on a Crystalline Substrate D. W. Kruger, D. E. Savage, and M. G. Lagally ' Department of Materials Science and Engineering, University of Wisconsin, Madison, Wisconsin 53706 (Received 2 February 1989) It is demonstrated that analysis of the shape of diAracted beams in LEED can be used to determine the size and separation distribution functions of structurally incoherent regions growing on a crystalline substrate. The method has been used to investigate the growth of In clusters on cleaved GaAs(110). An exponent of x =0.5 ~ 0. 15 is obtained for the change in mean island size with dose. The data imply that coalescence is important in the growth process. PACS numbers: 68. 55.Gi, 61. 14.Hg, 61. 50.Cj Although most overlayers or films deposited on crys- talline substrates initially form in islands (they do when- ever there is a net attractive adatom interaction), they usually do not conveniently form a simple epitaxial lat- tice or even a superlattice with a rational relationship to the substrate. There are many examples: amorphous patches, such as oxides; liquidlike layers; films that form with a three-dimensional liquid or solid morphology; translationally incommensurate layers; 2D layers that have no preferred rotational arrangement with respect to the substrate; and so forth. This class of systems in- cludes many examples of technological importance and also several (e.g. , the growth of diamond) in which there is current high scientific interest. It is frequently desirable to know the spatial and size distributions of such regions on the surface, and to follow their evolution with, for example, time, temperature, quantity of material deposited, or deposition rate. We demonstrate here that it is possible to do so using low- energy electron diH'raction (LEED), using as a model system the deposition of In on GaAs(110). From the analysis we are able to determine the growth law of In clusters on GaAs(110) as a function of dose. The use of diffraction to investigate surface disorder is predicated on the existence of extended defects, or boun- daries between ordered regions, that introduce new Fourier components into diffracted-beam profiles, caus- ing them to broaden in specific ways. ' For epitaxial layers or layers that form a superstructure, relatively straightforward procedures have been developed for ex- tracting the size (and in some cases separation) distribu- tion function of the ordered islands by analysis of diffracted-beam profiles' from the overlayers. Over- layers with superlattices [e. g. , (n x m) ] form new diffracted beams whose shape can be directly interpreted in terms of an overlayer size distribution. The shapes of the fundamental refiections (those that occur also for the clean substrate) are modified by the existence of an over- layer made up of islands. For epitaxial overlayers it has been shown' that one can extract the size and separa- tion distribution functions of the overlayers from the be- havior of these shapes with energy, coverage, time, etc. The essence of the analysis is that the system (even though it may contain many partially filled layers) can be treated in diA'raction as a two-layer problem: Islands of overlayer at one height are separated by regions of substrate at the lower level. The amplitudes of radiation scattered from the two layers destructively interfere at appropriate diffraction conditions (the "out-of-phase" conditions) producing profile broadening that reAects the density of up-down boundaries and hence the sizes and separations of the islands. The systems we wish to consider here, e.g. , amorphous patches, liquid droplets, or a crystalline 3D cluster, gen- erally do not have a integral relationship to the substrate, and hence the simple physical picture of interference of amplitudes scattered from atoms at two heights but hav- ing an epitaxial relationship would seem to be inap- propriate. Nevertheless we can demonstrate that, in fact, the same analysis can be used. One can think of the problem in the following manner. If, in a scattering volume, matter is removed in discrete regions, these holes must modify the scattering. Babinet's principle from optics states that a configuration of holes in a volume produces the same diffraction pattern as would the equivalent configuration of matter at the positions of the holes. Thus one can conceive of the possibility of measuring the size and separation distribution functions of the holes by analysis of the line shapes of beams dif- fracted from the volume. Here we make the extension to surfaces. If one removes scattering regions from the sub- strate, the order in the substrate is disturbed. These "holes" in the substrate can be created by an overlayer material that prefers to form in islands or clusters. By following the line-shape changes of diffracted beams as- sociated with the clean crystalline substrate surface as a function of dose of overlayer material, time, or tempera- ture, the size distribution of the "overlayer" patches can be determined. The analysis can be performed in terms of a simple two-layer 3-on-3 model extended to B on A. Consider monolayer-height islands of material A adsorbed epitaxi- ally on A (Fig. 1, top panel). This system can be con- sidered as "up" islands and "down" islands (substrate) 402 1989 The American Physical Society