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Energy Bands

Continuous ranges of allowed energy levels for electrons in a solid, arising from the periodic potential and quantum mechanical effects.
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The statement of the theorem

The allowed energy spectrum EE for an electron in a crystal lattice is defined by the function E(k)E(\mathbf{k}), where k\mathbf{k} is the crystal momentum restricted to the first Brillouin zone (FBZ). The existence of these continuous bands is a direct consequence of the periodicity of the potential V(r)V(\mathbf{r}) and the Bloch theorem. Mathematically, the allowed energies EE are the eigenvalues of the Hamiltonian HH for which the secular equation has a solution for k∈FBZ\mathbf{k} \in \text{FBZ}. The bands are characterized by the dispersion relation En(k)E_n(\mathbf{k}), where nn is the band index.