Lifshitz-Matsubara sum formula for the Casimir pressure between magnetic metallic mirrors

We examine the conditions of validity for the Lifshitz-Matsubara sum formula for the Casimir pressure between magnetic metallic plane mirrors. As in the previously studied case of nonmagnetic materials [Guérout et al., Phys. Rev. E 90, 042125 (2014)], we recover the usual expression for the lossy model of optical response, but not for the lossless plasma model. We also show that the modes associated with the Foucault currents play a crucial role in the limit of vanishing losses, in contrast to expectations.

Figure 1

Schematic representation of the poles and zeros of the function $p_{\mathbf{k}}^{\text{TE}}$ in the vicinity of the origin for the Au-Ni system (mixed magnetic-nonmagnetic mirrors). Zeros are indicated by crosses. Dashed intervals represent an infinite number or poles and zeros. Here we are considering $\gamma <{\omega }_m$ to study the limit $\gamma \to 0$.

Figure 2

Schematic representation of the poles and zeros of the function $p_{\mathbf{k}}^{\text{TE}}$ in the vicinity of the origin for the Ni-Ni system (identical magnetic mirrors). Zeros are indicated by crosses together with their multiplicities, poles by a dot. Dashed intervals represent an infinite number or poles and zeros.

Figure 3

Contribution to the Casimir pressure from the Foucault modes as a function of the wave vector $k$. From top to bottom, for the Au-Au, Au-Ni, and Ni-Ni systems. Positive (red) parts correspond to repulsive contribution and negative (green) parts to attractive ones.

Figure 4

Calculated Casimir pressures $P_{\gamma }$ at $T=300$ K for the Au-Au, Au-Ni, and Ni-Ni systems of plane mirrors separated by a distance $L=300$ nm (from bottom to top) as a function of the relaxation rate $\gamma$. The squares are the calculations using relaxation rates given in the text. The triangles are $P_0^{\mathcal{LM}}$ using the Lifshitz-Matsubara sum formula. The circles are $P_0^{\mathcal{L}}$ using the Lifshitz formula, which naturally takes into account the Foucault currents.

Figure 5

Contribution from the Foucault modes to the total Casimir pressure as a function of the distance $L$ between the plates. From top to bottom for the Au-Au, Au-Ni, and Ni-Ni systems. The inset shows the same data, in absolute value, on a log-log scale.