Solids are divided into two major classes: metals and insulators. A metal – or a conductor – is a solid in which an electric current flows under the application of electric field. By contrast, application of an electric field produces no electric current in an insulator. There is a simple criterion for distinguishing between the two classes on the basis of the band structure. If the valence electrons exactly fill one or more bands, leaving others empty, the crystal will be an insulator. An external electric field will not cause current flow in an insulator. Provided that a filled band is separated by an energy gap from the next higher band, there is no continuous way to change the total momentum of the electrons if every accessible state is filled. Nothing changes when the field is applied.
Fig. 1 Occupied states and band structures giving (a) an insulator, (b) a metal or a semimetal because of band overlap, and (c) a metal because of electron concentration. In (b) the overlap need not occur along the same directions in the Brillouin zone. If the overlap is small, with relatively few states involved, we speak of a semimetal.
On the contrary if the valence band is not completely filled the solid is a metal. In a metal there are empty states available above the Fermi level like in a free electron gas. An application of an external electric field results in the current flow.
It is possible to determine whether a solid is a metal of an insulator by considering the number of valence electrons. A crystal can be an insulator only if the number of valence electrons in a primitive cell of the crystal is an even integer. This is because each band can accommodate only two electrons per primitive cell. For example, diamond has two atoms of valence four, so that there are eight valence electrons per primitive cell. The band gap in diamond is 7eV and this crystal is a good insulator
However, if a crystal has an even number of valence electrons per primitive cell, it is not necessarily an insulator. It may happen that the bands overlap in energy. If the bands overlap in energy, then instead of one filled band giving an insulator, we can have two partly filled bands giving a metal (Fig.1b). For example, the divalent metals, such as Mg or Zn, have two valence electrons per cell. However, they are metals, although a poor ones – their conductivity is small.
If this overlap is very small, we deal with semimetals. The best known example of a semimetal is bismuth (Bi).
If the number of valence electrons per cell is odd the solid is a metal. For example, the alkali metals and the noble metals have one valence electron per primitive cell, so that they have to be metals.
The alkaline earth metals have two valence electrons per primitive cell; they could be insulators, but the bands overlap in energy to give metals, but not very good metals. Diamond, silicon, and germanium each have two atoms of valence four, so that there are eight valence electrons per primitive cell; the bands do not overlap, and the pure crystals are insulators at absolute zero.
There are substances, which fall in an intermediate position between metals and insulators. If the gap between the valence band and the band immediately above it is small, then electrons are readily excitable thermally from the former to the latter band. Both bands become only partially filled and both contribute to the electric condition. Such a substance is known as a semiconductor. Examples are Si and Ge, in which the gaps are about 1 and 0.7 eV, respectively. Roughly speaking, a substance behaves as a semiconductor at room temperature whenever the gap is less than 2 eV. The conductivity of a typical semiconductor is very small compared to that of a metal, but it is still many orders of magnitude larger than that of an insulator. It is justifiable, therefore, to classify semiconductors as a new class of substance, although they are, strictly speaking, insulators at very low temperatures.