Even–Odd Layer Oscillatory Behavior of Electronic and Phononic Specific Heat in an Ultra-Thin Metal Film

Both electronic and phononic statistical and thermal properties, modulated by the quantum size effect, are suggested in a thin metal film. In order to show the quantum size effect of specific heat, the densities of the electron and phonon states of an ultra-thin film are treated within the framework of quantum statistics. It was found that strong and weak “even–odd layer oscillatory behavior” was exhibited by the ultra-thin metal film in electronic and lattice specific heat, respectively. Such a behavior, which depends on film thickness, results from the quantum confinement of electrons and phonons in the vertical (thickness) direction of the film, where both electrons and phonons form their respective quantum well standing wave modes. If, for example, the thickness of the ultra-thin metal film is exactly an integer multiple of a half wavelength of the standing wave of electrons in the thickness direction, the corresponding density of states would become maximized, and the electronic specific heat would take its maximum. In the literature, less attention has been paid to the size-dependent electron Fermi wavelength for quantum size effects, i.e., the Fermi wavelength in ultra-thin metal films has always been identified as a constant. We shall show how the Fermi wavelength varies with the size of a nanofilm, including an explicit analytic formulation for the thickness dependence of the electron Fermi wavelength. Size-dependent resonantly oscillatory behavior, depending on the ultra-thin or nanoscale film thickness, would have possible significance for researching some fundamental physical characteristics (e.g., low-dimensional quantum statistics) and may find potential applications in new thermodynamic device design.