Structures of Si and Ge nanowires in the sub-nanometer range

We report [1] {\it ab initio} and tight-binding calculations of several structures of pristine Si and Ge nanowires with diameters D between 0.5 and 5.0 nm. For nanowires with D $<$ 2 nm, the calculations are performed in the framework of Kohn-Sham density functional theory, within the generalized-gradient approximation. Total-energy differences are converged to within 10 meV/atom. For nanowires with D $>$ 2 nm, an order-N density-matrix tight-binding methodology (DMTB) is employed. We consider nanowires based on the diamond structure, high-density bulk structures, and fullerene-like structures. Our calculations indicate a transition from $sp^3$ geometries to structures based on denser bulk phases and fullerene-like structures, for diameters smaller than $\sim$1.2 nm. We show that a continuum model is able to reproduce quantitatively this transition. According to the model, the transition originates from the larger surface energy density of the $sp^3$ wires as compared to those of the denser wires. We also find that diamond-structure nanowires are unstable for diameters smaller than 1 nm, undergoing considerable structural transformations towards amorphous-like wires. For diameters between 0.8 nm and 1 nm, filled-fullerene wires are the most stable. For even smaller diameters ($\sim0.5~{\rm nm}$), we find that a simple hexagonal structure is particularly stable for both Si and Ge. [1] R. Kagimura, R. W. Nunes, and H. Chacham, Phys. Rev. Lett. 95, 115502 (2005)