Casimir and Casimir-Polder Forces in Graphene Systems: Quantum Field Theoretical Description and Thermodynamics

Galina L Klimchitskaya,Vladimir M Mostepanenko

doi:10.3390/universe6090150

Abstract

We review recent results on the low-temperature behaviors of the Casimir-Polder and Casimir free energy an entropy for a polarizable atom interacting with a graphene sheet and for two graphene sheets, respectively. These results are discussed in the wide context of problems arising in the Lifshitz theory of van der Waals and Casimir forces when it is applied to metallic and dielectric bodies. After a brief treatment of different approaches to theoretical description of the electromagnetic response of graphene, we concentrate on the derivation of response function in the framework of thermal quantum field theory in the Matsubara formulation using the polarization tensor in (2 + 1)-dimensional space—time. The asymptotic expressions for the Casimir-Polder and Casimir free energy and entropy at low temperature, obtained with the polarization tensor, are presented for a pristine graphene as well as for graphene sheets possessing some nonzero energy gap Δ and chemical potential μ under different relationships between the values of Δ and μ. Along with reviewing the results obtained in the literature, we present some new findings concerning the case μ≠0, Δ=0. The conclusion is made that the Lifshitz theory of the Casimir and Casimir-Polder forces in graphene systems using the quantum field theoretical description of a pristine graphene, as well as real graphene sheets with Δ>2μ or Δ<2μ, is consistent with the requirements of thermodynamics. The case of graphene with Δ=2μ≠0 leads to an entropic anomaly, but is argued to be physically unrealistic. The way to a resolution of thermodynamic problems in the Lifshitz theory based on the results obtained for graphene is discussed.

Highlights

The attractive Casimir-Polder and Casimir forces act between an atom and an uncharged ideal metal plane and between two parallel ideal metal planes, respectively, in vacuum at zero temperature.These forces are entirely caused by the zero-point oscillations of quantized electromagnetic field and depend on the Planck constant h, speed of light c, atom–plane or plane–plane separation a, and on the static atomic polarizability αo ≡ α(0) [1,2] FCP ( a) = − 3α0 hc8π a4 where S is the plane area satisfying a condition a Universe 2020, 6, 150; doi:10.3390/universe6090150 FC ( a) = − √ π 2 hc S, 240 a4 (1) S
By and large we show that the Lifshitz theory of the Casimir-Polder and Casimir interaction in graphene systems using the polarization tensor is consistent with the requirements of thermodynamics
This is apparent from the fact that the Casimir-Polder and Casimir entropies go to zero with vanishing temperature in accordance with the Nernst heat theorem if the dielectric response of a pristine graphene is described on the basis of first principles of thermal quantum field theory by means of the polarization tensor

Summary

Introduction

The attractive Casimir-Polder and Casimir forces act between an atom and an uncharged ideal metal plane and between two parallel ideal metal planes, respectively, in vacuum at zero temperature.These forces are entirely caused by the zero-point oscillations of quantized electromagnetic field and depend on the Planck constant h, speed of light c, atom–plane or plane–plane separation a, and (in the case of the Casimir-Polder force) on the static atomic polarizability αo ≡ α(0) [1,2] FCP ( a) = − 3α0 hc8π a4 where S is the plane area satisfying a condition a Universe 2020, 6, 150; doi:10.3390/universe6090150 FC ( a) = − √ π 2 hc S, 240 a4 (1) S.

Methods

Discussion

Conclusion