Induced Charge-Density Oscillations at Metal Surfaces
  Induced charge-density (ICD) oscillations at the Cu(111) surface caused by an external impurity are studied within linear response theory. The calculation takes into account such properties of the Cu(111) surface electronic structure as an energy gap for three-dimensional (3D) bulk electrons and a $s-p_z$ surface state that forms two-dimensional (2D) electron system. It is demonstrated that the coexistence of these 2D and 3D electron systems has profound impact on the ICD in the surface region. In the case of a static impurity the characteristic ICD oscillations with the $1/\rho^2$ decay as a function of lateral distance, $\rho$, are established in both electron systems. For the impurity with a periodically time-varying potential, the novel dominant ICD oscillations which fall off like $\sim1/\rho$ are predicted. 
V. M. Silkin, I. A. Nechaev, E. V. Chulkov, and P. M. Echenique