4.4: Application - electron in a crystalline solid

The above is the 1D statement of Bloch’s Theorem, the basis of study of electrons in solids. If we imagine applying an electric field $(\mathcal{E})$ in the $x$-direction, then the rate at which work is done is:

$$−e\mathcal{E} v_g = \frac{dE}{dt} = \frac{dE}{dk}\frac{dk}{dt} \nonumber$$

Using the expression for $v_g$ we find that the rate of change of $\hbar k$ is proportional to the external force, rather like Newton’s second law.

$$−e\mathcal{E} = F = \hbar \frac{dk}{dt} \nonumber$$

If we now consider acceleration:

$$a = \frac{dv_g}{dt} = \frac{dv_g}{dk} \frac{dk}{dt} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2} F \nonumber$$

we find a quantity $\hbar^2 / \frac{d^2E}{dk^2}$ which is known as the effective mass, relating external force to acceleration in a solid, and allowing us to avoid further consideration of the effect of the lattice.