The low temperature expansion of the free energy of atom/plane system is considered for general symmetric form of tensor conductivity of the plane. It is shown that the first correction is proportional to second order of the temperature ∼ T 2 and comes from TM mode. The agreement of the expansion and exact expressions for different models of conductivity is numerically demonstrated.

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 μ and mass gap parameter m. The key role plays the dependence of graphene conductivities on the μ and m. For simplicity, we made the manifest calculations for zero values of the Fermi velocity. For μ>m, the thermal correction ∼T2, and for μ<m, we confirm the recent result of Klimchitskaya and Mostepanenko, that the thermal correction ∼T5. In the case of exact equality μ=m, the correction ∼T. This point is unstable, and the system falls to the regime with μ>m or μ<m. The analytical calculations are illustrated by numerical evaluations for the Hydrogen atom/graphene system.

The Casimir energy is constructed for a system consisting of two semi-infinite slabs of anisotropic material. Each of them is characterized by bulk complex dielectric permittivity tensor and surface conductivity on the free boundary. We found general form of the scattering matrix and Fresnel coefficients for each part of the system by solving Maxwell equations in the anisotropic media.