Electronic Band Structure - Why Bands Occur in Materials

Why Bands Occur in Materials

The electrons of a single isolated atom occupy atomic orbitals, which form a discrete set of energy levels. If several atoms are brought together into a molecule, their atomic orbitals split into separate molecular orbitals each with a different energy. This is due to the Pauli exclusion principle, which says that electrons that are close together must have different sets of quantum numbers (energy). This produces a number of molecular orbitals proportional to the number of valence electrons. When a large number of atoms (of order ×1020 or more) are brought together to form a solid, the number of orbitals becomes exceedingly large. Consequently, the difference in energy between them becomes very small. Thus, in solids the levels form continuous bands of energy rather than the discrete energy levels of the atoms in isolation. However, some intervals of energy contain no orbitals, no matter how many atoms are aggregated, forming band gaps.

Within an energy band, energy levels can be regarded as a near continuum for two reasons. First, the separation between energy levels in a solid is comparable with the energy that electrons constantly exchange with phonons (atomic vibrations). Second, this separation is comparable with the energy uncertainty due to the Heisenberg uncertainty principle, for reasonably long intervals of time. As a result, the separation between energy levels is of no consequence.

Several approaches to finding band structure are discussed below.