Semiclassical (band theory) conductivity

The following properties from band theory will be plugged into Drude model for conductivity

$\sigma = \frac{ne^2\tau}{m^*}$

to provide a semiclassical estimation of conductivity.

Group velocity

$v_\text{g} = \frac{1}{\hbar}\left(\frac{\partial E(k)}{\partial k}\right)$

Effective mass $m^*$

$m^*= \hbar^2\left(\frac{\partial^2E(k)}{\partial k^2}\right)^{-1}$

Or simply can be thought of as a second-order expansion of Hamiltonian ( $H = \hbar^2k^2/2m + V(r)$ ) near $k=0$ .