We have investigated the growth condition, i.e. growth time and V/III ratio determining Ga-rich and N-rich conditions, influence on the formation of dislocation-free vertical GaN nano-rods grown on (1 1 1) Si substrate by molecular beam epitaxy. The hexagonal shape nano-rod with diameters ranging from < 10 to 350 nm is fully relaxed from lattice strain, having a very good crystal quality characterized by dislocation-free lattice images observed by transmission electron microscopy (TEM) and extremely strong photoluminescence excitonic lines near 3.47eV. The nano-rod starts to protrude after the formation of an approximately 0.4-mu m thick columnar film base, and its density and physical dimension, i.e. diameter and height, are strongly dependent on V/III ratio and growth time. We have found that the hexagonal nano-rod can be formed on Si substrate, not only in N-rich condition but also even in a Ga-rich condition, when it is formed without buffer layer at high growth temperatures. (c) 2005 Elsevier B.V. All rights reserved.