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

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.

Low-temperature expansion of the Casimir–Polder free energy for an atom interacting with a conductive plane

“…The Casimir-Polder force is usually referred to as the van der Waals force on large distances between micro-particles and macro-objects when the retardation of interaction is taken into account. The Casimir-Polder force essentially depends on the material of the macro-objects, its dimension, shapes, conductivity, and temperature [3,4], and it is important for the interaction of graphene with micro-particles [5][6][7][8][9]. The Casimir-Polder force and torque for anisotropic molecules have been the subject of investigations in the recent years [10][11][12][13][14].…”

“…[15][16][17][18][19][20][21]. In the last few years, much attention is given to the low-temperature expansion of the Casimir-Polder free energy for the atom/graphene system [7,[19][20][21]. In the case of an atom/ideal plane, the low-temperature correction to the Casimir-Polder free energy is proportional to the fourth degree of temperature ∼ T 4 .…”

The Low-Temperature Expansion of the Casimir-Polder Free Energy of an Atom with Graphene

“…At the same time, in Refs. [7,24], the positive value of correction was observed. The disagreement is connected with that in which we use distance a = 100 nm for numerical evaluations, which is out of distances considered in Refs.…”

“…The Casimir-Polder force is usually referred to as the van der Waals force on large distances between micro-particles and macro-objects when the retardation of interaction is taken into account. The Casimir-Polder force essentially depends on the material of the macro-objects, its dimension, shapes, conductivity, and temperature [3,4], and it is important for the interaction of graphene with micro-particles [5][6][7][8][9]. The Casimir-Polder force and torque for anisotropic molecules have been the subject of investigations in the recent years from the theoretical point of view [10][11][12][13][14], as well as experimentally [15,16].…”

“…[19][20][21][22][23][24][25]. In the last few years, much attention has been given to the low-temperature expansion of the Casimir-Polder free energy for the atom/graphene system [7,[23][24][25]. In the case of an atom/ideal plane, the low-temperature correction to the Casimir-Polder free energy is proportional to the fourth degree of temperature ∼ T 4 .…”