Hall effect, coefficient and mobility

Hall effect

When a conductor carrying current I is placed in a magnetic field B perpendicular to the current, a transverse electric field, and hence a potential difference (Hall voltage), appears across the conductor, perpendicular to both I and B.

Hall coefficient

Strip of width w, thickness t, carrying current I along x with field B along z. Carriers: charge q (magnitude e), density n, drift velocity v_d.

   strip: width w, thickness t
   current I along x, field B along z (thickness)
   Hall voltage appears across the width w

1. Magnetic force deflects carriers sideways, charges accumulate on the edges, and a Hall field E_H builds up until the forces balance: eE_H = e v_d B → E_H = v_d B
2. Current density J = n e v_d, so v_d = J/(ne)
3. E_H = J B/(n e)
4. The Hall coefficient is defined as R_H = E_H/(J B), so

R_H = 1/(n q)

(R_H is negative for electrons and positive for holes. R_H = −1/(ne) for electrons.) In terms of the Hall voltage, V_H = E_H w = R_H I B / t.

Mobility and conductivity

1. Conductivity: σ = J/E = n e v_d/E.
2. Mobility is the drift velocity per unit electric field: μ = v_d/E.
3. So σ = n e μ.
4. Multiply by R_H: |R_H| σ = (1/ne)(neμ)

μ = σ |R_H|

Therefore, measuring R_H (from Hall voltage) and σ (from resistance) gives n and the carrier mobility of the wire material.