The Hall effect

When a current $i$ flows in a flat conductor immersed in a magnetic field $\vec{B}$ perpendicular to the plane of the current, a transverse potential difference appears: this is the Hall effect (Edwin Hall, 1879).

Hall effect: the current in a transverse field accumulates charges on one edge, creating a voltage $\Delta V_H$ perpendicular to the current.

Physical explanation. The current is due to charge carriers $q$ (positive or negative depending on the material) with drift velocity $v_d$. On each of them the Lorentz force $\vec{F} = q\,\vec{v}_d\times\vec{B}$ acts, pushing them towards one edge of the conductor. The charges accumulated there create a transverse electric field $\vec{E}_H$ which, at steady state, balances the magnetic force:

$q\,E_H = q\,v_d\,B \quad\Longrightarrow\quad \ev{V_H = v_d\,B\,w}$

where $w$ is the width of the conductor. Substituting $v_d = i/(n\,e\,A)$, with $n$ the carrier density and $A = w\,t$ the cross-section ($t$ thickness):

$V_H = \frac{i\,B}{n\,e\,t}$

Key formula — Hall effect

$V_H = \frac{iB}{net}$ $R_H = 1/(ne)$: Hall coefficient, depends only on the material. The sign of $V_H$ reveals the sign of the carriers’ charge.

Principle — Consequences of the Hall effect

• It provides a direct measurement of $n$ (carrier density) and of the sign of the charge: in some semiconductors $V_H$ has the opposite sign to that predicted for electrons, indicating effective positive carriers (holes).
• Hall probes are used everywhere $\vec{B}$ needs to be measured electronically: brushless motors, automotive ABS sensors, magnetic pedals, guitar pickups.
• On the micro scale, the quantum Hall effect (von Klitzing, 1980; Nobel 1985) shows that at low temperatures and intense fields $V_H$ is quantised in integer or fractional multiples of $h/e^2$, defining the primary standard of resistance ($R_K = h/e^2 \approx 25\,812\;\Omega$).

Example — Laboratory Hall probe

An indium antimonide (InSb) sheet has thickness $t = 0.1\;\text{mm}$, width $w = 5\;\text{mm}$ and carrier density $n = 2.5\cdot 10^{22}\;\text{m}^{-3}$. With $i = 100\;\text{mA}$ in a field $B = 0.5\;\text{T}$: $V_H = \frac{iB}{net} = \frac{0.1\cdot 0.5}{2.5\cdot 10^{22}\cdot 1.6\cdot 10^{-19}\cdot 10^{-4}} \approx 0.125\;\text{V} = 125\;\text{mV}$ Easily measurable with a standard voltmeter. In a metal ($n \approx 10^{29}\;\text{m}^{-3}$) the same configuration would give $V_H \sim 30\;\mu\text{V}$: this is why Hall probes are made with semiconductors, not metals — few carriers, large voltage.

Summary

Hall: $V_H = iB/(net)$. It measures $B$, $n$ and the sign of the carriers. Hall probes are the most widespread magnetometer in the world.

Collegamenti

Argomenti: Magnetismo Concetti: Effetto Hall · Forza di Lorentz

Esercizi collegati: Worked exercise — charge in an outward field · Cyclotron frequency of a proton · Ranking the Lorentz force (four angles)