Impurity and free-carrier effects on the far-infrared dispersion spectrum of silicon

We investigate a generalized Cauchy power-series expansion for the index of refraction of an n-type elemental semiconductor in the region of IR transparency. A plot of index versus photon energy squared should be very nearly linear if all absorptions lie above the transparent region. However, free carriers produce far-IR absorption, and the dispersive signature of this is a deviation from linearity in the mid- to far-IR. By retaining terms with negative exponents in the index expansion, we find a substantially improved fit to index measurements. Moreover, the free-carrier density may be determined from the coefficients in the regression fit. The method has been used to evaluate several extensive index measurements for doped silicon found in the literature.