Section 1.8.3.4. The Hall effect

Measurement of the Hall effect is simple and often useful. One takes a crystal and applies a magnetic field along the z axis. Then one imposes a current density along the x axis. One finds that the Lorentz force induces a voltage, or the equivalent electric field , in the y direction. The electric field is proportional to both the current and magnetic field. The ratio is the Hall constant . The inverse of is just the charge e and the speed of light c multiplied by the density of electrons : This provides a simple and accurate method of measuring the density of electrons. It works well when there is only one kind of current carrier and works well in semiconductors with a low density of carriers. A typical experiment for a semiconductor is to measure the conductivity and the Hall constant ; the mobility is then . If the conducting particles are holes in a semiconductor, the Hall constant has the opposite sign, which indicates positive charge carriers.

Measurement of the Hall effect does not work well if the semiconductor contains a mixture of different carriers, such as electrons and holes, or even electrons from different kinds of conduction bands. In these cases, the constant is not easily interpreted. Similarly, measuring the Hall effect is rarely useful in metals. It only works well in the alkali metals, which have all of the electrons in the first Brillouin zone on a spherical Fermi surface. In most metals, the Fermi surface extends over several Brillouin zones and has numerous pockets or regions of different curvatures. Regions of positive curvature act as electrons and give a negative Hall constant; regions of negative curvature act as holes and give a positive contribution to the Hall constant. Again, it is difficult to interpret the Hall constant when both contributions are present. In general, the Hall effect is most useful in semiconductors.