Hall resistance and its quantization

Hall effect

When a current-carrying conductor is placed in a magnetic field perpendicular to the current, a voltage appears across the conductor, perpendicular to both the current and the field. This is the Hall effect.

        B (into page)
   +-----------------------+
   |  ->  ->  ->  I  ->    |  top edge:    + (or -)
   |                       |
   +-----------------------+  bottom edge: - (or +)
        V_H across width

Moving carriers feel the force F = qv_d × B, so they pile up on one edge. This builds a transverse field E_H until it cancels the magnetic force.

Hall resistance

1. Balance of forces: eE_H = e v_d B, so E_H = v_d B.
2. Hall voltage across width w: V_H = E_H w = v_d B w.
3. Drift speed from I = n e v_d (w t): v_d = I/(n e w t), where t is the thickness.
4. So V_H = IB/(n e t).
5. Hall resistance: R_H(res) = V_H/I = B/(n e t) = B/(e n_s), where n_s = nt is the number of electrons per unit area (sheet density).

It rises linearly with B in an ordinary conductor.

Quantization of Hall resistance

In a 2D electron gas at very low temperature (~1 K) and strong field (~10 T), the electron energies form Landau levels. Each level holds n_s = eB/h electrons per unit area. If ν levels are completely filled, n_s = ν eB/h. Putting this in step 5:

V_H/I = h/(ν e²), with ν = 1, 2, 3, …

So the Hall resistance does not vary smoothly with B. It stays on flat plateaus at h/e², h/2e², h/3e², … with h/e² = 25 812.8 Ω. The values depend only on the fundamental constants h and e, not on the sample, and this is used as the resistance standard (integer quantum Hall effect).