Vacuum fluctuation forces between ultra-thin films

We have investigated the role of the quantum size effects in the evaluation of the force caused by electromagnetic vacuum fluctuations between ultra-thin films, using the dielectric tensor derived from the particle in a box model. Comparison with the results obtained by adopting a continuum dielectric model shows that, for film thicknesses of 1÷10 nm, the electron confinement causes changes in the force intensity with respect to the isotropic plasma model which range from 40% to few percent depending upon the film electron density and the film separation. The calculated force shows quantum size oscillations, which can be significant for film separation distances of several nanometers. The role of electron confinement in reducing the large distance Casimir force is discussed. Introduction. – Quantum mechanical forces, namely van der Waals and Casimir forces, between nano objects are expected to be important in the operation of micro and nano systems [1–3]. These forces depend significantly upon the optical properties of the interacting objects: for interacting films, variations in the experimental optical parameters, caused by film morphology, can determine a change in the force of the order of 10% [4, 5]. Since the basic work from Lifshitz and coworkers [6, 7] theoretical studies have been focused mainly on the determination of the forces between slabs, including semi-infinite slabs, on the basis of a continuum description of the material dielectric function [8–11]. This is a simplified description of a film, that neglects the modifications in the electronic structure related to the boundaries. It is expected to hold when the size of the film is large, so that the surfaces play a minor role in determining the dielectric response. For metallic films the calculated electronic distributions deviate significantly from the bulk behaviour when the size of the film is less than ten times the Fermi wavelength [12,13]. If the size of the film is of the order of few nanometers the continuum model does not provide an accurate description of the film properties and boundary effects cannot be neglected. Such effects arise as a consequence of the discretization of the energy bands due to the confinement potential, which produces the quantization of the electron energy levels in various sub-bands and affects the optical and electrical properties [14–21]. The quantum model. – The characteristics features induced by the quantization can be described by a model in which independent electrons of mass m are confined within a distance d in the direction of the surface normal (particle in a box model) [22, 23]. Assuming a jellium model and perfect planar surfaces the eigenvalue spectrum is simply given by: