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
------------
^^^^^^^^^^^
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Magnetic
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.