• • • In, the electronic band structure (or simply band structure) of a describes the range of that an within the solid may have (called energy bands, allowed bands, or simply bands) and ranges of energy that it may not have (called or forbidden bands). Band theory derives these bands and band gaps by examining the allowed quantum mechanical for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as and, and forms the foundation of the understanding of all (transistors, solar cells, etc.). Showing how electronic band structure comes about by the hypothetical example of a large number of carbon atoms being brought together to form a diamond crystal. The graph (right) shows the energy levels as a function of the spacing between atoms. When the atoms are far apart (right side of graph) each atom has valence atomic orbitals p and s which have the same energy.

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However when the atoms come closer together their orbitals begin to overlap. Due to the Pauli Exclusion Principle each atomic orbital splits into N molecular orbitals each with a different energy, where N is the number of atoms in the crystal. Since N is such a large number, adjacent orbitals are extremely close together in energy so the orbitals can be considered a continuous energy band. A is the atomic spacing in an actual crystal of diamond.

At that spacing the orbitals form two bands, called the valence and conduction bands, with a 5.5 eV band gap between them. Animation of band formation and how electrons fill them in a metal and an insulator The electrons of a single, isolated atom occupy each of which has a discrete. When two or more atoms join together to form into a, their atomic orbitals overlap. The dictates that no two electrons can have the same quantum numbers in a molecule.

So if two identical atoms combine to form a, each atomic orbital splits into two of different energy, allowing the electrons in the former atomic orbitals to occupy the new orbital structure without any having the same energy. Similarly if a large number N of identical atoms come together to form a solid, such as a, the atoms' atomic orbitals overlap. Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N~10 22) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10 −22 eV). The energy of adjacent levels is so close together that they can be considered as a continuum, an energy band. This formation of bands is mostly a feature of the outermost electrons () in the atom, which are the ones involved in chemical bonding. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow.

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Are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in the from which they arise.

Two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the bands associated with core orbitals (such as ) are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands. Higher bands involve comparatively larger orbitals with more overlap, becoming progressively wider at higher energies so that there are no band gaps at higher energies. Basic concepts [ ] Assumptions and limits of band structure theory [ ] Band theory is only an approximation to the quantum state of a solid, which applies to solids consisting of many identical atoms or molecules bonded together. These are the assumptions necessary for band theory to be valid: • Infinite-size system: For the bands to be continuous, the piece of material must consist of a large number of atoms.