Faraday-rotation spectra of semimagnetic semiconductors

An analytical expression of the Verdet constant of semimagnetic semiconductors is derived from a microscopic analysis of the transverse susceptibility responsible for the Faraday effect. This expression is obtained by integrating, in k space, the wave-vector-dependent transverse polarizability. The latter is obtained within the density-matrix formalism using a Lorentzian shape for the wave-vector dependence of the exchange interaction. Using only two parameters, this model reproduces very well the experimental Faraday-rotation spectra available for both ${\mathrm{Cd}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Mn}}_{\mathit{x}}$Te and ${\mathrm{Zn}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Mn}}_{\mathit{x}}$Te for various manganese concentrations and sample temperatures. The normalization parameter is independent of the manganese concentration and sample temperature for a given type of semimagnetic semiconductor, and the same exchange parameter is used for all manganese concentrations at a given temperature. Using the experimental data obtained in ${\mathrm{Cd}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Mn}}_{\mathit{x}}$Te samples of precisely known manganese concentrations, we are able to extract the value q=0.022 at 77 K and q=0.018 at 290 K for the exchange interaction parameter with a relative precision of 5%, while for ${\mathrm{Zn}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Mn}}_{\mathit{x}}$Te a smaller value (q=0.003 at room temperature) is obtained from previous studies.