Hall effect

Hall effect

1. What Hall Effect Is

Hall effect = when current-carrying conductor/semiconductor is placed perpendicular to magnetic field $B$, a transverse voltage (Hall voltage $V_H$) develops perpendicular to both current and $B$ due to Lorentz deflection of carriers.

Discovered by E.H. Hall (1879); measures carrier type and concentration.

2. Derivation (geometry)

Consider slab thickness $t$ (along $B$?), width $w$, length $l$, current $I$ along $x$, $B$ along $z$, carriers charge $q$ (±e). Lorentz $F = q(v_d×B)$ pushes carriers along $y$ until transverse Hall electric field $E_H$ balances:

Hall voltage

with sign depending on $q$.

For electrons ($q=-e$): $V_H$ negative on one side; for holes ($q=+e$): opposite sign ⇒ sign tells n vs p.

General:

where Hall coefficient

• Magnitude: $|R_H| =1/(n|q|)$ ⇒ large for semiconductors (small $n$) vs metal ($n$ huge ⇒ $V_H$ µV).
• Mobility: from conductivity $σ = n e μ$ and $R_H$, get

3. Uses

• Carrier type: sign of $V_H$.
• Carrier concentration:

or

• Mobility as above.
• Magnetic field sensor (Hall probe) and current sensor (clamp).

4. Worked Examples

(a) n-type Si Hall bar t=0.1 mm, I=1 mA, B=0.1 T: V_H= -1e-3·0.1/(1e22·1.6e-19·1e-4)= -62.5 µV? Compute: n=1e22, denominator 1.6e-19·1e-4·1e22=1.6e-1? Actually t=1e-4, n q t=1e22·1.6e-19·1e-4=0.16 ⇒ V=1e-3·0.1/0.16=0.000625=625 µV? Check sign.

(b) Need to find n: measure V_H=10 mV, I=10 mA, B=1 T, t=1 µm: n= I B/(e V_H t)=1e-2·1/(1.6e-19·1e-2·1e-6)=6.25e21 m⁻³

(c) Metal Cu n=8.5e28 ⇒ R_H≈-7.35e-11 m³/C ⇒ V_H tiny.

5. Exam Points

• State Lorentz balance, derive $V_H=IB/(n q t)$ and $R_H=1/nq$ — 7 marks, very frequent (common derivation).
• Explain sign tells n vs p — 5 marks, very frequent.
• Show how to get $n$ and $μ$ — 5 marks, frequent.
• Sketch slab with $I, B, V_H$ directions — 5 marks, frequent.