Casimir–Polder Force on Atoms or Nanoparticles from Gapped and Doped Graphene: Asymptotic Behavior at Large Separations

Abstract

The Casimir–Polder force acting on atoms and nanoparticles spaced at large separations from real graphene sheets possessing some energy gaps and chemical potentials is investigated in the framework of the Lifshitz theory. The reflection coefficients expressed via the polarization tensor of graphene, found based on the first principles of thermal quantum field theory, are used. It is shown that for graphene the separation distances, starting from which the zero-frequency term of the Lifshitz formula contributes more than 99% of the total Casimir–Polder force, are less than the standard thermal length. According to our results, however, the classical limit for graphene, where the force becomes independent of the Planck constant, may be reached at much larger separations than the limit of the large separations determined by the zero-frequency term of the Lifshitz formula, depending on the values of the energy gap and chemical potential. The analytic asymptotic expressions for the zero-frequency term of the Lifshitz formula at large separations are derived. These asymptotic expressions agree up to 1% with the results of numerical computations starting from some separation distances that increase with increasing energy gaps and decrease with increasing chemical potentials. The possible applications of the obtained results are discussed.