Frequency-Dependent Hall Effect in Normal and Superconducting Metals

The Hall current flow occurring in a normal and a superconducting metal when both a static magnetic field ($H_0$) and an electromagnetic wave are applied on the metal is calculated. The entire frequency range of the electromagnetic wave is discussed although the emphasis is on the microwave range. The nonlocal, transverse Hall current in a normal metal is calculated by solving the Boltzmann equation. It is shown that the microwave Kerr rotation in a circular cylindrical cavity provides a good test for the nonlocal Hall current in a normal metal. The relation between a longitudinal and a transverse Hall current in a superconductor is briefly discussed. A detailed theory of the transverse Hall current in a superconductor based on the Bardeen-Cooper-Schrieffer model and including the effect of collective excitations is presented. The field $H_0$ is assumed constant in space and a general result for the Hall current in $Q$ space is derived. When the electric field is constant in space ($Q\to 0$), it is shown that the Hall current is proportional to the microscopic analog of the fraction of normal electrons of a two-fluid model.