The low-temperature expansion of the Casimir-Polder free energy of an atom with graphene

TL;DR

This paper analyzes the low-temperature behavior of the Casimir-Polder free energy between an atom and graphene, revealing how chemical potential and mass gap influence thermal corrections with explicit calculations.

Contribution

It extends previous work by deriving the low-temperature expansion of the Casimir-Polder free energy considering different relations between chemical potential and mass gap in graphene.

Findings

Thermal correction scales as T^2 for μ > m.

Thermal correction scales as T^5 for μ < m.

At μ = m, the correction scales as T, but this state is unstable.

Abstract

We consider the low-temperature expansion of the Casimir-Polder free energy for an atom and graphene by using the Poisson representation of the free energy. We extend our previous analysis on the different relations between chemical potential $\mu$ and mass gap parameter $m$. The key role plays the dependence of graphene conductivities on the $\mu$ and $m$. For simplicity, we made the manifest calculations for zero values of the Fermi velocity. For $\mu >m$ the thermal correction $\sim T^2$ and for $\mu < m$ we confirm the recent result of Klimchitskaya and Mostepanenko, that the thermal correction $\sim T^5$. In the case of exact equality $\mu =m$ the correction $\sim T$. This point is unstable and the system falls to the regime with $\mu >m$ or $\mu <m$. The analytical calculations are illustrated by numerical evaluations for the Hydrogen atom/graphene system.