Nonequilibrium Casimir–Polder Interaction between Nanoparticles and Substrates Coated with Gapped Graphene

Abstract

The out-of-thermal-equilibrium Casimir–Polder force between nanoparticles and dielectric substrates coated with gapped graphene is considered in the framework of the Dirac model using the formalism of the polarization tensor. This is an example of physical phenomena violating the time-reversal symmetry. After presenting the main points of the used formalism, we calculate two contributions to the Casimir–Polder force acting on a nanoparticle on the source side of a fused silica glass substrate coated with gapped graphene, which is either cooler or hotter than the environment. The total nonequilibrium force magnitudes are computed as a function of separation for different values of the energy gap and compared with those from an uncoated plate and with the equilibrium force in the presence of graphene coating. According to our results, the presence of a substrate increases the magnitude of the nonequlibrium force. The force magnitude becomes larger with higher and smaller with lower temperature of the graphene-coated substrate as compared to the equilibrium force at the environmental temperature. It is shown that, with increasing energy gap, the magnitude of the nonequilibrium force becomes smaller, and the graphene coating makes a lesser impact on the force acting on a nanoparticle from the uncoated substrate. Possible applications of the obtained results are discussed.

Nonequilibrium Casimir–Polder Force between Nanoparticles and Graphene-Coated Silica Plate: Combined Effect of the Chemical Potential and Mass Gap

“…The analytic continuation of the polarization tensor to this region of the real frequency axis for a pristine and gapped graphene was performed in Refs. [67,70,71]. Here, we present the obtained results in a more convenient analytic form and generalize them for the case of gapped graphene with nonzero chemical potential.…”

“…[85] (see also Ref. [71] for the graphical representation of these data). Note that numerical computations of F SiO 2 neq are much more complicated than those for F SiO 2 eq made along the imaginary frequency axis owing to the quick variation of the integrands.…”

“…Among other things, the mass gap necessary arises in graphene sheets deposited on some substrates. Because of this, the obtained results were generalized for the case of a graphene-coated dielectric plate [71], as usually takes place in experimental situations. All these computations demanded the use of a supercomputer.…”

“…An impact of the plate material on the nonequilibrium Casimir-Polder force between nanoparticles and a silica plate coated with gapped graphene was considered in Ref. [71]. It was shown that the presence of a silica plate increases the magnitude of the nonequilibrium Casimir-Polder force.…”

“…It should be noted that computations of the nonequilibrium Casimir-Polder force acting on nanoparticles from real graphene sheet using the polarization tensor are rather cumbersome. They are connected with calculation of the multiple integrals of rapidly varying functions and were realized on a supercomputer [70,71].…”