Advanced Solid State Physics

## Free-electron model

In the free-electron model, the electron dispersion relation is,

$$E(\vec{k})= \frac{\hbar^2 k^2}{2m^*}.$$

Here $m^*$ is the effective mass. At low temperatures, all of the $k$ states below the Fermi wave vector $k_F$ are filled and all the states above $k_F$ are empty. The electron density is the volume of the Fermi sphere, times two for spin, divided by the $k$-space volume per state,

$$n =\frac{2}{(2\pi)^3}\frac{ 4\pi k_F^3}{3}=\frac{k_F^3}{3\pi^2}.$$

Many equilibrium thermodynamic properties of the free electron model depend only on the electron density and the effective mass. The transport properties of the free electron model depend on the electron density $n$, the effective mass $m^*$ and the relaxation time $\tau$.