Hall Effect

Run a current through a semiconductor slab and apply a magnetic field at right angles to it, and a small transverse voltage appears across the slab's other pair of faces. That sideways voltage is the Hall effect, and pulling it apart tells you three things a plain resistance measurement never could: which charge carrier is doing the conducting, how many of them there are per unit volume, and how mobile they are.

The sideways push

Picture a rectangular semiconductor slab carrying a current I along its length, the x-axis, with a magnetic field B applied along the z-axis, straight through the slab's thickness. The carriers drift through the material with velocity v_d, and a moving charge in a magnetic field feels a sideways force, the Lorentz force:

F = −e(v_d × B)

For carriers drifting along x, that force points along y and pushes every one of them toward the same edge of the slab. Charge piles up there, leaving the opposite edge oppositely charged, and the two charged edges set up a transverse electric field, the Hall field E_H, that opposes any further pile-up. The geometry is easier to read than to describe:

Equilibrium, and where V_H comes from

Charge keeps arriving at the edge until the Hall field pushes back exactly as hard as the magnetic force pushes sideways:

eE_H = ev_dB  →  E_H = v_dB

With w the width of the sample across which that field is measured, it turns directly into a voltage:

V_H = E_H · w = v_dBw

From drift velocity to the Hall coefficient

v_d is awkward to measure directly, but it is tied to the current through the carrier concentration n. For a slab of width w and thickness t carrying current I:

I = ne(wt)v_d  →  v_d = I / (newt)

Substitute that back into V_H = v_dBw and the width cancels out entirely, leaving current, field and thickness:

V_H = IB / (net) = R_H · IB / t

Rearranged, that is the quantity the experiment actually measures:

R_H = V_Ht / (IB)

What the sign tells you

R_H carries a sign, and the sign is the whole point of doing the experiment this way rather than just measuring resistivity:

• R_H positive — the majority carriers are holes, and the sample is p-type.
• R_H negative — the majority carriers are electrons, and the sample is n-type.

Resistivity depends on how many carriers there are and how fast they move, but not on which way they are charged, so it can never tell p-type from n-type on its own. The Hall voltage's polarity can, which makes this the direct experimental route to identifying the majority carrier.

Carrier concentration and mobility

Once R_H is known, the carrier concentration follows immediately:

n = 1 / (R_He)

and combined with the sample's electrical conductivity σ, measured separately, R_H also gives the Hall mobility: how readily the carriers move under an applied field.

μ_H = R_H × σ

Checking it: V_H against B

Pass a constant current I through the sample, vary the magnetic field B, and record the Hall voltage at each step. Plotted against B, those readings should fall on a straight line through the origin — confirming V_H ∝ B — and the slope of that line, combined with the known I and t, is what gives R_H its graphical value.