The Hall effect is an elementary occurrence in solids, but it had a large impact on the development of solid state physics. It was first studied in 1878 by E. H. Hall, who was attempting to test whether a magnetic field acts only on current-carrying electrons in a wire or on the entire wire. Experiments showed that the magnetic field acted only upon the moving charges. However, experiments on a variety of different materials had some inexplicable results, which were later explained by a quantum mechanical treatment of solid state physics. The Hall effect is currently used to quickly characterize solids (e.g. when one wants to determine whether a semiconductor is n-type or p-type).

The Hall experiment

The idea by the Hall effect is to place a current-carrying slab in a magnetic field. Magnetic fields act on moving charges through the Lorentz force in a direction perpendicular to both the magnetic field vector and the direction of current flow. Consider a solid with current flowing along the positive y-axis and a magnetic field pointing along the positive z-axis. See the diagram below.

              ------------     z
            /|           /|    ^
           / |          / |    |
          /  |         /  |    |
         |------------|   |    |
 Current |   |        |   |     ------>y
 ------->|   ---------|--/    /
 in      | /          | /    /
         |/           |/    x

             field H

The Lorentz force acting on a current-carrying charge due to the applied magnetic field is -qvH/c, where q is the charge (positive or negative) of a current carrier, v is the speed of the carrier along the positive y-axis, and c is the speed of light. Assuming the current-carrying charges are negatively-charged electrons, and given that the current flows in the positive-y direction, the electrons must have negative velocities (i.e. they move in the negative-y direction). Their deflection in the magnetic field (being careful with negative signs) is in the negative-x direction.

The deflection of electrons in the negative-x direction creates an electric field. The solid is positively-charged in the positive-x direction and negatively-charged in the negative-x direction. The electric field corresponds to a measurable voltage, called the Hall voltage. The Hall voltage is used to characterize the material being tested.

For convenience, a Hall coefficient R is defined to be E/jH, where E is the electric field corresponding to the Hall voltage and j is the current density. First-order classical analysis, using the so-called Drude model, suggests that R = 1/nqc, where n is the concentration of electrons.

Results of the experiment

Noble metals and Alkali metals have Hall coefficients that are close to those predicted by the Drude-model analysis. However, some materials have drastically different Hall coefficients. Most importantly, in some materials (e.g. beryllium, aluminum, and magnesium) the Hall coefficient is negative. The only explanation for the negative Hall coefficients is that positively-charged carriers, not electrons, are the main current carriers in those materials. These positively-charged carriers, which we now call holes, were explained by the application of quantum mechanics to solids.

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