Problems in the theory of thermal Casimir force between dielectrics and semiconductors

The application of the Lifshitz theory to describe the thermal Casimir force between dielectrics and semiconductors is considered. It is shown that for all true dielectrics (i.e., for all materials having zero conductivity at zero temperature) the inclusion of a nonzero conductivity arising at nonzero temperature into the model of dielectric response leads to the violation of the Nernst heat theorem. This result refers equally to simple insulators, intrinsic semiconductors, Mott-Hubbard dielectrics and doped semiconductors with doping concentration below a critical value. We demonstrate that in the insulator-metal transition the Casimir free energy changes abruptly irrespective of whether the conductivity changes continuously or discontinuously. The application of the Lifshitz formula to polar dielectrics results in large thermal correction that is linear in temperature. A rule is formulated on how to apply the Lifshitz theory to real materials in agreement with thermodynamics and experiment.