22.6 The Hall Effect

The Hall effect occurs when a magnetic field deflects charge carriers inside a current-carrying conductor, producing a measurable voltage across the conductor's width. This voltage tells you what type of charge carriers a material has, how dense they are, and how strong the surrounding magnetic field is.

The Hall Effect

How the Hall Voltage Forms

Picture a flat conductor carrying current while a magnetic field points perpendicular to the conductor's surface. The moving charge carriers experience a transverse Lorentz force:

$\vec{F} = q\vec{v} \times \vec{B}$

where $q$ is the carrier charge, $\vec{v}$ is the drift velocity, and $\vec{B}$ is the magnetic field.

This force pushes carriers toward one side of the conductor. As they pile up, a charge imbalance develops: one edge becomes more negative, the other more positive. That imbalance creates a transverse electric field ($\vec{E}_H$) inside the conductor that pushes back against the Lorentz force.

Carriers stop accumulating once the electric force exactly balances the magnetic force. At that equilibrium, the potential difference across the conductor's width is the Hall voltage ($V_H$).

Calculating the Hall Voltage

The Hall voltage is given by:

$V_H = Blv$

where $B$ is the magnetic field strength (T), $l$ is the width of the conductor (m), and $v$ is the drift velocity of the charge carriers (m/s).

You can express drift velocity in terms of more easily measured quantities:

$v = \frac{I}{nqA}$

• $I$ = current (A)
• $n$ = charge carrier density (carriers per m$^{-3}$)
• $q$ = charge per carrier (C)
• $A$ = cross-sectional area of the conductor (m$^2$)

Substituting this into the Hall voltage equation gives a form that connects $V_H$ directly to current, magnetic field, and material properties.

The sign of $V_H$ tells you the carrier type. If positive carriers (holes) dominate, the voltage is positive on the side predicted by the right-hand rule. If negative carriers (electrons) dominate, the voltage flips.

Practical Applications

Identifying charge carrier type. The sign of the Hall voltage reveals whether a material's majority carriers are holes or electrons. A positive $V_H$ points to hole conduction (as in p-type semiconductors), while a negative $V_H$ points to electron conduction (as in n-type semiconductors). This was historically important because it confirmed that some materials conduct via positive-like carriers, not just electrons.

Measuring magnetic fields. Rearranging the Hall voltage formula gives $B = V_H / (lv)$. If you know the conductor's properties and measure $V_H$, you can determine the external field strength.

Hall effect sensors. These small, inexpensive devices show up in many technologies:

• Automotive systems (wheel speed sensors, anti-lock brakes)
• Industrial controls (proximity switches, current sensing)
• Consumer electronics (smartphone compasses, laptop lid detection)

Characterizing materials. By measuring the Hall voltage alongside current and field strength, researchers can extract carrier density and mobility, which is especially useful when developing new semiconductors like gallium arsenide.

Additional Hall Effect Parameters

• Hall coefficient ($R_H$): Relates the Hall electric field to the product of current density and magnetic field. Its sign indicates carrier type, and its magnitude gives carrier density.
• Carrier mobility ($\mu$): Describes how fast carriers move through the material per unit electric field. Higher mobility means better conductivity for a given carrier density.
• Hall resistivity ($\rho_H$): The transverse resistivity caused by the Hall effect, equal to the product of the Hall coefficient and the magnetic field.
• Magnetoresistance: The change in a material's electrical resistance when an external magnetic field is applied. It's a separate but related effect often measured in the same experiments as the Hall voltage.