Contribution of Drifting Carriers to the Casimir-Lifshitz and Casimir-Polder Interactions With Semiconductor Materials

We develop a theory for Casimir-Lifshitz and Casimir-Polder interactions with semiconductor or insulator surfaces that takes into account charge drift in the bulk material through use of the classical Boltzmann equation. We derive frequency-dependent dispersion relations that give the usual Lifshitz results for dielectrics as a limiting case and, in the quasistatic limit, coincide with those recently computed to account for Debye screening in the thermal Lifshitz force with conducting surfaces with small density of carriers.

Figure 1

Behavior of the functions $g_{\mathbf{k}}^p(i\xi )$ used to compute the Casimir-Lifshitz free energy and entropy for semiconductor materials with account of drifting carriers. The reflections coefficients are given by (6, 7), parameters are for intrinsic Ge (see text), and the distance is set to $d=1\mu \mathrm{m}$. The variation with temperature (in the range $T=0–300\mathrm{K}$) of the TE function is not perceptible on the scale of the figure. The corresponding functions without account of Debye screening and carrier drift correspond to the $T=0\mathrm{K}$ plots in this figure.

Figure 2

Casimir-Lifshitz free energy at $T=300\mathrm{K}$ for intrinsic semiconductor parallel plates for different conductivity models as follows: Carrier drift with Debye-Hückle screening, and a dc conductivity term $4\pi {\sigma }_0/\xi$ added to the bare permittivity. The free energy is normalized to the free energy computed with the standard Lifshitz theory using the bare permittivity only. Parameters are as follows: For Ge, the bare permittivity is quoted in the text, the Debye length is $R_D=0.68\mu \mathrm{m}$ and the dc conductivity is ${\sigma }_0=1/(43\Omega \mathrm{cm}$). For Si, ${\overline{ϵ}}_0=11.87$, ${\overline{ϵ}}_{\infty }=1.035$, ${\omega }_0=6.6\times {10}^{15}\mathrm{rad}/\sec $, $R_D=24\mu \mathrm{m}$, and ${\sigma }_0=1/(2.3\times {10}^5\Omega \mathrm{cm}$).