Symmetry properties of ZnO nanorods and nanotubes

The symmetry of ZnO hexagonal nanorods and nanotubes depends on the location of their axes with respect to the ZnO wurtzite lattice. When the axis coincides with a threefold rotation axis of the lattice (type I structure), the point and space symmetries are described by the C,, point group and the R 49 (p3ml) rod group, respectively. When the axis coincides with a 63 improper rotation axis (type II structure), the symmetries are described by the C-6v point group and the R 70 (p6(3)mc) rod group, respectively. The site symmetry of an atom in the structure is C-3v, C-s, or C-l, depending on its location. At any point of the one-dimensional Brillouin zone the direct optical transitions in the nanostructures obey the selection rules governed by the C-3v group (type I) or the C-6v one (type II). The latter group is that of the Gamma point of bulk ZnO. The rules for transitions involving bound states differ from those in the bulk since generally they are governed by smaller groups. The effect of a magnetic or/and an electric field on the symmetry properties of the nanostructures is established. The symmetry of heterostructures located along a single nanorod or nanotube is determined. The envelope function approximation is considered. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim