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I know that the complex conductivity ($\sigma = \sigma_1+i\sigma_2$) is related to the dielectric function ($\epsilon = \epsilon_1+i\epsilon_2$) by: $$ \epsilon_1 = 1 - \frac{4\pi\sigma_2}{\omega} \\\epsilon_2 = \frac{4\pi\sigma_1}{\omega} $$ and I also know that, for semiconductors, the following relation exists: $$\sigma=\sigma_0e^{-\frac{E_{gap}}{2k_BT}}$$ where $E_{gap}$ is the band gap. My question is, can either of these quantities be obtained from the band structure of other types of materials, like superconductors/insulators/insulators? What other information might be required to do so?

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