Felix Hüning - Fundamentals of Electrical Engineering for Mechatronics

: Splitting of discrete energy levels into energy bands for increasing number of atoms.

metal (left); semiconductor (center); insulator (right).

Electrical properties can be distinguished depending on the presence and size of a band gap at the Fermi level.

Metals

The valence and conduction band overlap (the Fermi level lies within this overlapping band) or the Fermi level lies within the conduction band (not shown) and therefore the band is partly filled with electrons regardless of temperature. These electrons form a sea of practically free electrons moving in the background of the positively charged crystal structure ().

: The crystal structure of a metal: positively charged atomic cores surrounded by delocalized free electrons.

Insulators

In an conduction band is empty, resulting in no conductivity. At higher temperatures electrons can be excited to the conduction band due to thermal energy (leaving a hole in the valence band), but at reasonable temperatures the number of excited electrons is negligible and there is no conductivity.

Examples of insulators are glass or plastic materials, see .

Semiconductors

Like for insulators the Fermi level for pure semiconductors, (also known as intrinsic semiconductors), is within a band gap of width E, but this time the band gap is smaller. Semiconductors are isolators at absolute zero. Thanks to the smaller band gap, electrons can be excited to the conduction band more easily due to thermal energy. As the excitation of electrons is a thermal effect the number of intrinsic electrons in the conduction band ni is strongly temperature dependent:

Here kB=1.3810-23 J/K which is the Boltzmann constant.

When an electron is excited to the conduction band this electron is missing in the valence band. This missing electron is called an electron hole, or defect electron and also contributes to conductivity. The density of holes is the same as the density of excited electrons. The conductivity of semiconductors is higher than that of insulators, but still significantly lower than that of metals.

Fig. 1.4: The crystal structure of silicon.

As every excited electron generates a hole the density of electrons n0 and holes p0 is the same and corresponds to the intrinsic charge carrier density ni:

The opposite of the generation of electron-hole pairs is called recombination. The recombination rate depends on the carrier density of electrons and holes. The rate of generation and recombination of electron-hole pairs is temperature dependent. In thermal equilibrium both rates are the same and the number of free electron-hole pairs is constant at the given temperature. The equality of the two rates leads to the mass action law:

The product of the charge density of the free electrons and holes equals the square of the intrinsic charge carrier concentration. Mass action law also holds true for doped semiconductors.