Correction to the Casimir force due to the anomalous skin effect

Abstract

The surface impedance approach is discussed in connection with the precise calculation of the Casimir force between metallic plates. It allows to take into account the nonlocal connection between the current density and electric field inside of metals. In general, a material has to be described by two impedances $Z_{s}(\omega,q)$ and $Z_{p}(\omega,q)$ corresponding to two different polarization states. In contrast with the approximate Leontovich impedance they depend not only on frequency $\omega$ but also on the wave vector along the plate $q$. In this paper only the nonlocal effects happening at frequencies $\omega<\omega_{p}$ (plasma frequency) are analyzed. We refer to all of them as the anomalous skin effect. The impedances are calculated for the propagating and evanescent fields in the Boltzmann approximation. It is found that $Z_p$ significantly deviates from the local impedance as a result of the Thomas-Fermi screening. The nonlocal correction to the Casimir force is calculated at zero temperature. This correction is small but observable at small separations between bodies. The same theory can be used to find more significant nonlocal contribution at $\omega\sim\omega_p$ due to the plasmon excitation.Comment: 29 pages. To appear in Phys. Rev.

The Casimir free energy in high- and low-temperature limits

“…For the anomalous skin effect these impedances were found in [30] (see also [27]). When q → 0 (propagating field) both impedances coincide and reproduce the Leontovich impedance for the anomalous skin effect [33].…”

“…We stressed [25,27] that to analyse the low-temperature behaviour one has to take into consideration the anomalous skin effect. Description of nonlocal metals can be done with two nonlocal dielectric functions [26] or equivalently with two impedances [30], which are known in the theory of metals. It was demonstrated that the Leontovich impedance, which is used for the description of propagating fields, cannot be applied for the Casimir problem because evanescent fields are important in this case.…”

“…On the other hand, the denominator in equation (14) is small and the condition v 1 will be fulfilled at a sufficiently low frequency (temperature). In this limit the dielectric functions behave as [30] …”

The Casimir effect: medium and geometry

“…The Lifshitz result for two parallel semi-infinite dielectrics separated by a vacuum slit is obtained from (25) when Fresnel coefficients r T M (iω, k), r T E (iω, k) are substituted into (25) [6,22]. The limit ε(iω) → +∞ corresponds to reflection coefficients of a perfectly conducting metal plane: r T M = +1, r T E = −1.…”

“…Contribution of T M part is obtained in full analogy. Free energy has the form (25) ω n = 2πnT are Matsubara frequencies, prime means n = 0 term is taken with the coefficient 1/2, r…”