4.4: Application - electron in a crystalline solid

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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.