![]() Two adjacent bands may simply not be wide enough to fully cover the range of energy. The bands have different widths, with the widths depending upon the degree of overlap in the atomic orbitals from which they arise. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow.īand gaps are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. This formation of bands is mostly a feature of the outermost electrons ( valence electrons) in the atom, which are the ones involved in chemical bonding and electrical conductivity. The energy of the adjacent levels is so close together that they can be considered as a continuum, an energy band. 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). Each discrete energy level splits into N levels, each with a different energy. Similarly, if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap with the nearby orbitals. This tunneling splits ( hybridizes) the atomic orbitals into molecular orbitals with different energies. If two atoms come close enough so that their atomic orbitals overlap, the electrons can tunnel between the atoms. The electrons of a single, isolated atom occupy atomic orbitals with discrete energy levels. The second model starts from the opposite limit, in which the electrons are tightly bound to individual atoms. This model explains the origin of the electronic dispersion relation, but the explanation for the band gaps is subtle in this model. In this model, the electronic states resemble free electron plane waves, and are only slightly perturbed by the crystal lattice. : 161 The first one is the nearly free electron model, in which the electrons are assumed to move almost freely within the material. The formation of electronic bands and band gaps can be illustrated with two complementary models for electrons in solids. Animation of band formation and how electrons fill them in a metal and an insulator At the actual diamond crystal cell size denoted by a, two bands are formed, separated by a 5.5 eV band gap. The Pauli exclusion principle limits the number of electrons in a single orbital to two, and the bands are filled beginning with the lowest energy. Since there are many atoms, the orbitals are very close in energy, and form continuous bands. When the atoms come closer (left side), the orbitals hybridize into molecular orbitals with different energies. When the atoms are far apart (right side of graph) the eigenstates are the atomic orbitals of carbon. The right graph shows the energy levels as a function of the spacing between atoms. Why bands and band gaps occur A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal.
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