Microwave Faraday Effect in Silicon and Germanium

The Faraday rotation and ellipticity in a system of quasifree carriers is discussed and applied to microwave measurements on semiconductors. The theoretical expressions for these effects are analyzed with a digital computer for various ranges of the magnetic field $B$, the mobility $\mu$, the conductivity $\sigma$, the frequency $\omega$, the collision time $\tau$ and the dielectric constant of the host material. It is possible to simplify these expressions in certain limiting cases. For $\mu B$ smaller than unity, the rotation and ellipticity are proportional to $B$. For $\mu B$ larger than both unity and $\omega \tau$, they decrease as $B^{-1\mathrm{}}$, and $B^{-3\mathrm{}}$, respectively. A maximum occurs near $\mu B=1\mathrm{}$ when $\omega \tau$ is small.

Rotation measurements on $n$- and $p$-type single crystals of silicon at room temperature, with resistivities from 0.5 to 40 ohm-cm, utilizing 9.6- and 35-kMc/sec radiation, are compared with the theory. Results for $n$-type germanium at 78°K, with $\mu B$ varied up to about 6, agree with the calculated low- and high-field behavior. Faraday ellipticity measurements on $n$-type germanium crystals at 78°K are in qualitative agreement with the theory. In the case of small losses, the sign of the ellipticity is determined by the sign of the quantity ($4\omega \tau -\frac{\sigma }{\omega {{\epsilon }_{\mathrm{st}}}^{\prime }}$).