Past exam of the condensed matter physics course of the University of Oxford / 2014 / 3 / b

An electron hole is a missing electron in an otherwise almost filled valence band. A completely filled band has no net transport current: its velocities cancel over the Brillouin zone. Describing a few missing states as mobile excitations is therefore simpler than following almost every occupied electron state.

Near an isotropic valence maximum, write the electron dispersion as $E_v(\mathbf{k})=E_v(0)-{\hslash }^2{k}^2/(2m_h)$ with $m_h>0$. The electron curvature mass is negative there. Removing that electron creates an excitation with momentum $\hslash {\mathbf{k}}_h=-\hslash \mathbf{k}$ and energy measured upward from the filled reference,ThusThe sign refers to the positive excitation energy of a missing electron near a maximum, not to negative vacuum mass. A general maximum gives the corresponding positive hole-curvature tensor.^[ref]

Removing an electron of charge $-e$ leaves charge $+e$ relative to the filled-band reference. This also reproduces the current. If the missing electron state had velocity ${\mathbf{v}}_e(\mathbf{k})$, subtracting its contribution from the zero current of a filled band givesThe hole dispersion with ${\mathbf{k}}_h=-\mathbf{k}$ gives ${\mathbf{v}}_h({\mathbf{k}}_h)={\mathbf{v}}_e(\mathbf{k})$, soLikewise $\hslash \dot{\mathbf{k}}=-e\mathbf{E}$ becomes $\hslash {\dot{\mathbf{k}}}_h=+e\mathbf{E}$. Positive hole charge and positive hole effective mass consistently describe drift along the electric field, while the valence electrons themselves drift through an almost full band. A hole is a quasiparticle, not a proton moving through the crystal.