Quantum size effects in Friedel oscillations inside films

An analytical formula of density oscillations is found for metallic films of finite thickness. The result shows that the quantum size effect on density oscillations is surprisingly more evident in the middle of the film. As a result, the density oscillations in a finite film cannot be regarded as a simple addition of the two sets of Friedel oscillations for half-infinite metal no matter how thick the film is. This analytical result is confirmed by our numerical jellium-model computation. Such quantum size effect should exist in all the electron-mediated interactions that are driven by the Friedel oscillations. As an example, we indeed find it also exists in Ruderman–Kittel–Kasuya–Yosida interactions inside films.