The van der Waals and Casimir energy of anisotropic atomically thin metallic films

We discuss the van der Waals (Casimir) free energies and pressures of thin metallic films, consisting from one to fifteen atomic layers, with regard to the anisotropy in their dielectric properties. Both free-standing films and films deposited on a dielectric substrate are considered. The computations are performed for a Au film and a sapphire substrate. According to our results, for free-standing Au films consisting of one and three atomic layers the respective relative error arising from the use of an isotropic (bulk) dielectric permittivity is equal to 73% and 37% for the van der Waals energy, and 70% and 35% for the pressure. We tabulate the energy and pressure van der Waals coefficients of thin Au films computed with account of their anisotropy. It is shown that the bulk permittivity of Au can be used for the films consisting of more than 30 atomic layers, i.e., more than approximately 7 nm thickness. The role of relativistic effects is also investigated and shown to be important even for the films consisting of two or three layers. The obtained results can find applications in the investigation of stability of thin films and development of novel nanoscale devices.