Faraday Effect in Germanium at Room Temperature

Abstract

When a plane polarized electromagnetic wave passes through a semiconductor and a static magnetic field is applied along the direction of propagation, there occurs a rotation of the plane of polarization and the transmitted radiation is found to be elliptically polarized. This effect is due to the influence of the free charge carriers in the semiconductor and has been analyzed using the Drude-Zener model. For small losses, weak magnetic fields, and small values of $\ensuremath{\omega}\ensuremath{\tau}$ (assuming the relaxation time $\ensuremath{\tau}$ to be energy independent) the angle of rotation of the plane of polarization can be expressed to a first order of approximation by the simple formula: $\ensuremath{\theta}=\frac{1}{2}{(\frac{{\ensuremath{\mu}}_{0}}{{\ensuremath{\epsilon}}_{0}})}^{\frac{1}{2}}(\frac{{\ensuremath{\sigma}}_{0}\ensuremath{\mu}B}{\ensuremath{\surd}{K}^{\ensuremath{'}}})t,$ where $\ensuremath{\mu}$ is the Hall mobility, ${\ensuremath{\sigma}}_{0}$ is the dc conductivity, $B$ is the magnetic field, $t$ is the thickness of sample traversed, ${K}^{\ensuremath{'}}$ is the dielectric constant of the material at the frequency employed in the experiments, and ${\ensuremath{\epsilon}}_{0}$ and ${\ensuremath{\mu}}_{0}$ are the dielectric constant and the permeability of free space respectively. For spherical energy surfaces, the degree of ellipticity, which is a second-order effect, can be expressed by the relation $\mathcal{E}={(\frac{{\ensuremath{\mu}}_{0}}{{\ensuremath{\epsilon}}_{0}})}^{\frac{1}{2}}[\frac{{\ensuremath{\sigma}}_{0}(\ensuremath{\mu}B)(\ensuremath{\omega}\ensuremath{\tau})}{\ensuremath{\surd}{K}^{\ensuremath{'}}}]t$ where $\frac{\ensuremath{\omega}}{2\ensuremath{\pi}}$ is the frequency and $\ensuremath{\tau}$ is the relaxation time. Thus, for small losses the ellipticity is proportional to the latter quantity. For the case of low frequencies, the effect can be explained by the introduction of a Hall-effect type field into Maxwell's equations. In general, the angle of rotation and the ellipticity may depend on the direction of the applied magnetic field because of the nonspherical nature of the energy surfaces.Room-temperature measurements of the angle of rotation at microwave frequencies on both $n$- and $p$-type samples of germanium gave values of 3780 ${\mathrm{cm}}^{2}$/volt-sec and 3300 ${\mathrm{cm}}^{2}$/volt-sec for the electron and hole mobilities respectively. The method should be applicable to the determination of mobilities in powdered samples without using electrodes, if the field inside the powdered particles is determined by a Clausius-Mosotti type approximation.