Densities of states

Key point: The density of states is not uniform across a band: in most cases, the states are densest close to the centre of the band. The number of energy levels in an energy range divided by the width of the range is called the density of states, (Fig. 3.68). The density of states is not uniform across a band because the energy levels are packed together more closely at some energies than at others. This variation is apparent even in one dimension, for—compared with its edges—the centre of the band is relatively sparse in orbitals (as can be seen in Fig. 3.63). In three dimensions, the variation of density of states is more like that shown in Fig. 3.69, with the greatest density of states near the centre of the band and the lowest density at the edges. The reason for this behaviour can be traced to the number of ways of producing a particular linear com bination of atomic orbitals. There is only one way of forming a fully bonding molecular orbital (the lower edge of the band) and only one way of forming a fully antibonding orbit al (the upper edge). However, there are many ways (in a three-dimensional array of atoms) of forming a molecular orbital with an energy corresponding to the interior of a band.

The density of states is zero in the band gap itself—there is no energy level in the gap. In certain special cases, however, a full band and an empty band might coincide in energy but with a zero density of states at their conjunction (Fig. 3.70). Solids with this band structure are called semimetals. One important example is graphite, which is a semimetal in directions parallel to the sheets of carbon atoms.

Fig. 3.68 The density of states is the number of energy levels in an infinitesimal range of energies between E and E+dE.

Fig. 3.69 Typical densities of states for two bands in a three-dimensional metal.

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مواضيع ذات صلة

The Fermi level

Band formation by orbital overlap

The conductivities of inorganic solids

Nonstoichiometry

Intrinsic point defects

Solubility

Thermal stabilities of ionic solids

The Born–Mayer equation

Structure maps

The radius ratio

Ionic radii

Characteristic structures of ionic solids