Temperature correction to Casimir-Lifshitz free energy at low temperatures: Semiconductors

The Casimir force and free energy at low temperatures have been the subject of focus for some time. We calculate the temperature correction to the Casimir-Lifshitz free energy between two parallel plates made of dielectric material possessing a constant conductivity at low temperatures, described through a Drude-type dielectric function. For the transverse magnetic (TM) mode such a calculation is made. A further calculation for the case of the TE mode is thereafter presented which extends and generalizes previous work for metals. A numerical study is undertaken to verify the correctness of the analytic results.

Figure 1

$F^{\mathrm{TM}}$ and its approximation with (a) $\overline{\epsilon }=11.67$ and (b) $\overline{\epsilon }=1$. Correction curves, calculated from Eq. (3.31), are shifted to match the numerical calculations at $T=0$ in each graph.

Figure 2

The quantity $R$ defined in Eq. (3.38) plotted for the TM result (3.31) and numerical calculations.

Figure 3

Temperature dependence of the free energy for TE at $1\text{micron}$ separation. The solid line is an exact numerical calculation including all terms of Eq. (1.1) with $\overline{\epsilon }=11.66$ and ${\omega }_0=8.0\times {10}^{15}{\mathrm{s}}^{-1}$, the dashed line is the parabolic temperature correction (4.3). The term $\propto T^{5∕2}$ is too small to be visible in the graph.

Figure 4

The difference between the numerically calculated free energy and the quadratic $T$ term of Eq. (4.18) (equal to the difference between the graphs in Fig. 3) plotted against the absolute value of the $T^3$ term of Eq. (4.18).