Adsorption of Cu on the polar GaN(0001) surface in a 2×2 geometry is studied by the generalized gradient approximation (GGA) with ultrasoft pseudopotential within the density functional theory (DFT). Different adsorption sites and the effects of different coverages are considered. It is found that copper atoms preferentially adsorb at the H3 sites compared with doing at the T4 site, while the copper adsorption on top of a gallium atom (atop position) is totally unfavorable in each coverage considered. Both GaN(0001) surface and its adsorption have the semi-metallic behaviour and remain to be of n-type conduction character.