The Faraday effect in semiconductors

Abstract

Equations for the Faraday rotation and ellipticity in an anisotropic material are derived in terms of quantities related to the components of the high frequency magnetoconductivity tensor. The theory is valid for all orientations of H, (the magnetic field), and Eo, (the electric vector of the incident radiation), with respect to the crystal axes. The magnetoconductivity tensor is calculated for the [111] and [100] ellipsoid band models of a cubic semiconductor, assuming the scattering to be isotropic. The treatment is semi-classical, being based on the solution of the Boltzmann equation for a single valley which is an ellipsoid of revolution. It is convenient to consider axes such that H is along one axis. The tensors are therefore transformed bo systems in which H lies in a (100) or (110) plane, and Eo takes any orientation in the plane perpendicular to the field. The tensor for the isotropic model having spherical surfaces of constant energy is obtained as a special case. The series expansion for the current density inpowers of H is also developed to 0(H2), and the associated tensor coefficients evaluated for H in the (100) and (110) planes, as for the closed solution. Detailed calculations are made of the frequency, field and temperature dependence of the rotation and ellipticity in two specimens of n-type germanium, assuming the isotropic model. One is near intrinsic and accoustic phonon scattering is considered, while the other is moderately doped and ionized impurity scattering is assumed. Calculations are also made for the lattice scattering specimen only, using the [111] ellipsoid band model. An analysis is made of the anisotropy of the rotation and ellipticity as a function of the orientation of H and Eo, in addition to a consideration of the field and frequency dependence. Approximate expressions are derived and discussed in certain regions. Finally some calculations are presented for the same specimen, using the series expansion, and a comparison of the two treatments is made.