Super-oscillations in the Interlayer Lattice Relaxation of Quantum Pb Films.*

Using first-principles total energy calculations, we study the interlayer relaxation in lead films and observe oscillations of the interlayer lattice relaxation with layer number (distance from vacuum) and with film thickness. By analyzing the charge distribution along the direction perpendicular to the film, we show that the former oscillations of lattice relaxation are induced by Friedel oscillations in charge density that decays slowly, following a 1/r law rather than a 1/r$^{2}$ law found in most metal films. The superoscillations are attributable to interference of the Friedel oscillations from the two boundaries of the film. Our results suggest that Friedel oscillations may be responsible for the quantum size effect observed in lead films.