Classical limit of the Casimir interaction for thin films with applications to graphene

The Casimir interaction between two thin material films, between a film and a thick plate, and between two films deposited on substrates is considered at large separations (high temperatures) which correspond to the classical limit. It is shown that the free energy of the classical Casimir interaction between two insulating films with no free charge carriers and between an insulating film and a material plate depends on film thicknesses and decreases with separation more rapidly than the classical limit for two thick plates. The free energy of thin films characterized by the metallic-type dielectric permittivity decreases as the second power of separation, i.e., demonstrates the standard classical limit. The obtained results shed light on the possibility to describe dispersion interaction between two graphene sheets and between a graphene sheet and a material plate by modeling graphene as a thin film possessing some dielectric permittivity. It is argued that the most reliable results are obtained by describing the reflection properties on graphene by means of the polarization tensor in (2 + 1)-dimensional space-time.

Figure 1

Stratified medium consisting of three layers of finite thickness $d_{-},a$, and $d_{+}$ with dielectric permittivities ${\varepsilon }^{(-1)}\left(\omega \right)$, ${\varepsilon }^{\left(0\right)}\left(\omega \right)$, and ${\varepsilon }^{\left(1\right)}\left(\omega \right)$, enclosed between two semispaces $z\le -a/2-d_{-}$ and $z\ge a/2+d_{+}$ with dielectric permittivities ${\varepsilon }^{(-2)}\left(\omega \right)$ and ${\varepsilon }^{\left(2\right)}\left(\omega \right)$, respectively.