Hall Effect: Charges Pile Up Sideways

A magnetic force with nowhere to go but sideways

Send a current through a flat conducting strip and apply a magnetic field perpendicular to it. Every moving charge carrier inside the strip feels the Lorentz force, F = qv × B, pushing it sideways, perpendicular to both its own velocity and the field. In an ordinary wire that sideways push has nowhere to send the charge except toward the strip's edge — the geometry of the sample turns a magnetic force into an accumulation of charge on one side and a depletion on the other.

That charge does not pile up forever. As it accumulates, it builds an internal electric field pointing across the strip, opposing further sideways drift. Equilibrium is reached the instant that new electric force exactly cancels the magnetic force on the current-carrying charges, and the strip is left with a small, steady, measurable voltage across its width — the Hall voltage — discovered by Edwin Hall in 1879, well before the electron itself was identified.

V_H = I * B / (n * q * t)
      I  current through the strip
      B  magnetic field, perpendicular to the strip
      n  carrier density (carriers per unit volume)
      q  carrier charge
      t  strip thickness

Telling electrons and holes apart

The single most useful thing the Hall effect reveals is not just the magnitude of the effect but its sign. A conventional current flowing one direction can, physically, be either negative electrons drifting the opposite way or positive holes drifting the same way — and those two pictures curve to opposite edges of the strip under the same magnetic field, because the Lorentz force depends on the sign of the charge. Measuring which edge builds up positive voltage tells you directly which carrier type dominates conduction in a given material — the reason p-type and n-type semiconductors, which are built to have holes or electrons as their majority carrier, can be distinguished with a simple Hall measurement rather than any more exotic technique.

Why semiconductors show a much bigger effect than metals

Notice that the Hall voltage formula has carrier density n in the denominator. A typical metal has an enormous free-electron density — the same modest current is carried by an enormous number of electrons, each contributing only a tiny share of the sideways deflection, so the resulting voltage is small and hard to measure precisely. A doped semiconductor has a carrier density many orders of magnitude lower, so the same current is carried by far fewer carriers, each of which is deflected more strongly on average — producing a Hall voltage large enough to be measured cheaply and precisely. This is exactly why practical Hall sensors are built from semiconductor material rather than ordinary metal.

Hall sensors: measuring current and position without contact

Because the Hall voltage is directly proportional to the magnetic field for a fixed current, a small semiconductor Hall element makes an inexpensive magnetic-field sensor. Wrap a Hall sensor around (but not touching) a current-carrying wire and it reports the field generated by that current, giving a non-contact ammeter that never needs to be spliced into the circuit. Mount a small magnet on a rotating shaft and place a Hall sensor nearby, and every pass of the magnet produces a voltage pulse — the basis of the position and speed sensors in car engines, brushless motors, and countless keyboard and joystick mechanisms.

The quantum Hall effect

Cool a two-dimensional electron system to near absolute zero and expose it to an intense magnetic field, and the classical picture above stops being the whole story. The electrons' allowed energies collapse into discrete Landau levels, and the Hall resistance no longer rises smoothly with field — it settles onto a staircase of plateaus, each precisely quantized in units of Planck's constant divided by the square of the electron charge, independent of the sample's exact geometry or purity. This quantum Hall effect, discovered by Klaus von Klitzing in 1980, is measured so precisely and so reproducibly that it now defines the international standard for electrical resistance.

Frequently asked questions

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