The inclusion of multiple reflections in the theory of the Faraday effect in semiconductors

The theory of the isotropic free-carrier Faraday effect in semiconductors is reformulated, taking into account multiple reflections in the specimen. Exact expressions are given for the Faraday rotation Θ and ellipticity Δ, and also for the amplitude ratio E and phase shift ψ of the transmitted wave. Approximations are developed for small and large values of the specimen thickness z; in general these are useful only for moderately low frequencies and weak magnetic fields. The conventional expressions are obtained in the limit of high frequency and large z.

The dependence of Θ, Δ, E and ψ upon frequency and thickness is illustrated by numerical calculations based on experimental data for n-type germanium. In general behaviour E resembles Θ and ψ resembles Δ. Compared with the conventional treatment, the frequency dependence of all four quantities exhibits substantial changes in the microwave region and oscillatory behaviour when the wavelength in the specimen is comparable to the thickness. At sufficiently high frequencies, dependent upon the thickness, the effect of reflections becomes negligible. The thickness dependence is examined in the range 10 μm - 1 cm at microwave frequencies, and shows that the small z and non-reflection approximations represent limiting behaviour, with the large z approximation providing a useful intermediate description under suitable conditions.