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

TL;DR

This paper analyzes the asymptotic behavior of the Casimir-Polder force on atoms and nanoparticles near graphene with an energy gap and chemical potential, deriving analytical expressions and discussing the classical limit at large separations.

Contribution

It provides the first-principles-based asymptotic expressions for the Casimir-Polder force on graphene with energy gap and chemical potential at large distances.

Findings

Zero-frequency term dominates beyond certain separation distances.

Classical limit may occur at larger separations depending on energy gap and chemical potential.

Analytical asymptotic expressions match numerical results within 1%.

Abstract

The Casimir-Polder force acting on atoms and nanoparticles spaced at large separations from real graphene sheet possessing some energy gap and chemical potential 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 on the Planck constant, may be reached at much larger separations than the limit of 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 distance which increases with increasing energy gap and decreases with increasing chemical potential. Possible applications of the obtained results are discussed.