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We present a numerical model study of the zero-temperature infrared optical properties of (III, Mn) V diluted magnetic semiconductors. Our calculations demonstrate the importance of treating disorder and interaction effects simultaneously in modeling these materials. We find that the conductivity has no clear Drude peak, that it has a broadened inter-band peak near 220 meV, and that oscillator weight is shifted to higher frequencies by stronger disorder. These results are in good qualitative agreement with recent thin film absorption measurements. We use our numerical findings to discuss the use of f-sum rules evaluated by integrating optical absorption data for accurate carrier-density estimates.