Vladimir Kaganer1 David van Treeck1 Gabriele Calabrese1 Johannes Zettler1 Sergio Fernández-Garrido1 Oleg Konovalov2

1, Paul Drude Institute for Solid State Electronics, Berlin, , Germany
2, ESRF, Grenoble, , France

Self-assembled GaN nanowires (NWs) prepared by molecular beam epitaxy typically possess lengths of up to several microns and diameters of 20-100 nm. These NWs elongate along the surface normal with a spread in orientations of 2-3°. The aim of the present work is to study the NW diameter distributions by small-angle X-ray scattering (SAXS). We analyze three different kinds of GaN NW ensembles which exhibit significantly different NW densities and diameters: (i) NW ensembles on Si with densities over 1010 cm-2 and diameters of 20-100 nm, (ii) NW ensembles on Si thinned down to 6 nm in diameter using a post-growth thermal decomposition process, and (iii) NW ensembles on sputtered Ti films that possess densities of the order of 109 cm-2 and diameters of 20-50 nm.
The small-angle scattering measurements are performed with the incident x-ray beam directed normal to the NW axis, i.e., at a grazing angle to the substrate. We avoid the complications of the grazing incidence scattering by choosing the incidence angle notably larger than the critical angle. The analysis of the X-ray data requires development of new approaches since, unlike to the common case of small-angle scattering, NWs are aligned. The range of orientations is, however, notably broader than one meets in grazing incidence X-ray scattering studies of epitaxial islands, so that the characteristic features typical for scattering from these structures (like the facet truncation rods) are not observed.
We separate the scattered intensity from NWs from all kinds of parasitic scattering caused by surface roughness, 3D islands, etc. by taking into account that an angular spread of orientations of long NWs gives rise to a respective cone of intensity in the reciprocal space. The sharp facets of NWs should result in the Porod law for the intensity I(q) at large wave vectors, I(q)~q-4. The plot of I(q)*q4 vs. q possesses a maximum that allows to determine the mean NW diameter directly from the measured intensity, even without a model fit. The fit provides the whole diameter distribution, which we compare with the scanning electron microscopy analysis, for the samples where the latter is possible.
Deviations of the measured intensity from the Porod law are however observed at large wave vectors q, despite the sharp NW facets. We find two different effects causing the Porod law violation in the respective samples. First, a combination of the finite NW lengths (albeit large compared with their diameters) with a finite range of their orientations gives rise to an increased intensity at large wave vectors. Secondly, NW bundling causes an interference between scattering from individual NWs in the bundle, which provides additional intensity in another range of wave vectors. Summarizing, we state that grazing incidence SAXS directly provides the GaN NW diameter distributions. Violation of the Porod law at larger wave vectors brings to light further features of the NW ensembles.