Hall Effect

The slab geometry and the Lorentz force

The textbook setup is a thin rectangular slab of conducting material — the Hall bar. Pick a coordinate system: current I flows along x with current density J_x; an external magnetic field B is applied along y, perpendicular to the slab's flat face; we measure voltage V_H along z, the third perpendicular direction. The slab has length L (along x), width w (along z), and thickness t (along y).

Inside the conductor, charge carriers drift with velocity v_d. For an n-type material the carriers are electrons with charge −e moving in the −x direction (since current flows in +x). Each electron feels the Lorentz force:

F = q (v × B) = (−e)(v_d ẑ' × B ŷ)

For our geometry (v_d in −x, B in +y), the cross product points in +z, but multiplied by the negative electron charge it becomes −z. So electrons drift toward the −z face of the slab. Negative charge accumulates there, leaving a positive deficit on the +z face. An electric field E_z builds up across the slab, pointing from +z toward −z. Equilibrium is reached almost instantly (within picoseconds for typical metals) when the Coulomb force on each new arriving electron exactly cancels its Lorentz deflection:

qE_z = q v_d B   ⟹   E_z = v_d B

The voltage across the slab in z is V_H = E_z × w = v_d B w. The drift velocity is related to current by I = nqv_d × (w t), so v_d = I/(nqwt). Substituting:

V_H = I B / (n q t)

This is the Hall voltage formula. Three observations: (1) it is independent of the slab length L; (2) it is inversely proportional to thickness t — thinner slabs give bigger signals; (3) it depends inversely on the carrier density n, which is why semiconductors with low n produce far larger Hall voltages than metals.

The Hall coefficient as a material property

To strip away geometry, define the Hall coefficient:

R_H = V_H t / (I B) = 1 / (n q)

R_H has units of m³/C and depends only on the material — its carrier density and the sign of its charge. Measuring V_H, I, B and t lets you extract R_H, and from R_H you read off both the carrier concentration n = 1/(R_H q) and the sign of the carriers from the sign of R_H itself.

Material	R_H (m³/C)	Carrier sign	n (m⁻³)
Copper	−5.5 × 10⁻¹¹	Electron (−)	1.13 × 10²⁹
Silver	−9.0 × 10⁻¹¹	Electron (−)	6.9 × 10²⁸
Gold	−7.2 × 10⁻¹¹	Electron (−)	8.6 × 10²⁸
Aluminium	+1.0 × 10⁻¹⁰	Hole (+)	6.2 × 10²⁸
Beryllium	+2.4 × 10⁻¹⁰	Hole (+)	2.6 × 10²⁸
Indium antimonide	−5.4 × 10⁻⁴	Electron (−)	1.2 × 10²²
Gallium arsenide	−1.0 × 10⁻³	Electron (−)	6.2 × 10²¹
n-type silicon (10¹⁵/cm³ doped)	−6.2 × 10⁻³	Electron (−)	10²¹
p-type silicon (10¹⁵/cm³ doped)	+6.2 × 10⁻³	Hole (+)	10²¹

Two surprises: aluminium and beryllium have positive Hall coefficients, even though they are metals. This is the signature of hole conduction — electrons in nearly-filled bands behave as if they were positive carriers, a result that classical Drude theory could not explain. Quantum band structure was needed.

The semiconductor R_H values are 10⁶ to 10⁷ times larger than for copper. This is why InSb is the workhorse material for Hall sensors — its low n (about 10²² vs 10²⁹ for copper) gives huge V_H per unit B.

Worked example: an InSb Hall sensor

Consider a commercial Hall sensor: a 100 μm × 50 μm thin film of indium antimonide, thickness t = 1 μm, biased at I = 1 mA. The carrier density in undoped InSb at 300 K is n ≈ 2 × 10²² m⁻³. We want the Hall voltage in a small magnetic field B = 10 mT (about 200× Earth's field):

V_H = I B / (n q t)
    = (1e-3 × 1e-2) / (2e22 × 1.602e-19 × 1e-6)
    = 1e-5 / 3.204e-3
    = 3.12e-3 V  ≈  3.1 mV

3 mV is easily readable with an op-amp. Compare with copper: at the same I, B and t, but n = 1.1 × 10²⁹:

V_H_copper = 1e-5 / (1.1e29 × 1.6e-19 × 1e-6) = 1e-5 / 1.76e4 = 5.7e-10 V = 0.57 nV

The semiconductor produces a signal 5 million times larger than the metal — and that is before tuning doping, geometry, or temperature. Real silicon-CMOS Hall sensors deliver about 30-50 mV/T at room temperature, with offset stability good to a few microtesla after chopper-stabilization. They cost about $0.50 each in volume.

Where the Hall effect shows up

• Automotive position sensors. Hall ICs sense the passing of small magnets attached to crankshafts, camshafts, throttle bodies and ABS rings. A modern car has 20–40 Hall sensors. They cost $0.30–1.00 each, are immune to dust and oil, and survive the 125 °C engine bay environment that breaks optical sensors.
• Brushless DC motors. Three Hall sensors at 120° intervals around the rotor tell the motor controller which winding to energize next. Without them, BLDC motors need either back-EMF estimation or expensive optical encoders. Drone propellers, hard drive spindles, and electric vehicle traction motors all rely on Hall commutation.
• Current sensing in inverters and chargers. Closed-loop Hall current sensors put the Hall element inside a magnetic-core gap; the core's flux is proportional to the conductor current. They handle DC, isolate the measurement galvanically, and read currents from milliamps to thousands of amps. EV traction inverters use one per phase.
• Quantum resistance metrology. The integer quantum Hall resistance R_K = h/e² = 25812.807459 Ω is exact by definition since 2019. National metrology institutes maintain the international ohm by holding a 2D electron gas in a GaAs/AlGaAs heterostructure at 1.5 K and 10 T. Calibrations propagate to commercial standards with a few parts per billion uncertainty.
• Hall thrusters. Spacecraft propulsion that uses the Hall effect in plasma form. A radial magnetic field traps electrons in a closed E×B drift, ionizing xenon atoms; the ions are accelerated by an axial electric field to 15–25 km/s. Specific impulse 1500–3000 s, thrust 10–500 mN, efficiency 50–60%. Most modern geostationary satellites use Hall thrusters for stationkeeping.

The quantum Hall effect

In a two-dimensional electron gas at high magnetic field (5–20 T) and low temperature (1–4 K), the Hall resistance shows striking flat plateaus instead of the linear B-dependence of the classical formula:

R_xy = V_H / I = h / (ν e²) = 25812.807 Ω / ν

where ν is an integer (the integer quantum Hall effect, von Klitzing 1980) or a rational fraction like 1/3, 2/5 (the fractional quantum Hall effect, Tsui & Stormer 1982). Each plateau is flat to parts in 10⁹ over a range of magnetic field — accurate enough that the plateau resistance is now used as a primary standard. The von Klitzing constant R_K = h/e² is fixed by definition of h and e in the post-2019 SI system.

The flatness comes from topological protection. Electrons in 2D in a high field organize into Landau levels separated by gaps; impurities create localised states in the gaps that don't carry current; bulk states are localised; only edge channels carry current. The number of filled edge channels is the integer ν, and small disorder cannot change it without crossing an energy gap. The fractional version requires interactions between electrons — Laughlin's wavefunction predicted quasi-particles with charge e/3 that have since been measured directly by shot-noise experiments.

Variants and extensions

• Anomalous Hall effect. In ferromagnets V_H has an extra contribution proportional to magnetisation rather than external field: R_H = R_0 B + R_S μ₀ M. The extra term comes from spin-orbit coupling and Berry-phase contributions in the band structure. Used as a probe of magnetic ordering and spin polarization.
• Spin Hall effect. Even without a magnetic field, an electric current passing through a material with strong spin-orbit coupling separates electrons by spin: spin-up to one edge, spin-down to the other. No charge accumulation, but a transverse spin current. The basis of spin-orbit-torque magnetic memory.
• Quantum spin Hall effect. A topological insulator carries dissipationless edge currents protected by time-reversal symmetry. Predicted by Kane and Mele 2005, observed in HgTe quantum wells by Molenkamp 2007. Edge channels carry helical spin-momentum-locked electrons.
• Hall mobility measurement. Combining the Hall coefficient (giving 1/(nq)) with the resistivity (giving 1/(nqμ)) lets you extract carrier mobility μ = R_H / ρ separately from carrier density. Standard semiconductor characterisation method.
• Hall thruster. Plasma-physics inversion of the same effect — a closed E×B electron drift used to ionize and accelerate propellant. Operates in xenon or krypton at ~10⁻⁴ Torr; efficiency tracks the same Hall-parameter physics that classical electron Hall conduction obeys.

Common pitfalls

• Confusing the sign of charge with the direction of current. Conventional current flows opposite to electron drift. The sign of V_H depends on the carrier sign, not on which way the ammeter reads. Always work out the Lorentz force on the actual moving charges.
• Forgetting thickness. The Hall voltage scales as 1/t. A 10× thicker slab gives 10× smaller V_H for the same current. Designers exploit this by depositing nanometer-scale thin films to maximise sensitivity.
• Mixing Hall voltage with magnetoresistance. The Hall voltage is transverse (perpendicular to current). Magnetoresistance is the change in longitudinal resistance with field, a different measurement that requires four-terminal probes oriented along the current. Two-terminal voltage measurements blend both, ruining the analysis.
• Treating R_H as a single number when multi-band conduction is present. Aluminium has both electron and hole pockets; copper has only electron-like character. Materials with comparable carrier densities of opposite sign give R_H = (n_h μ_h² − n_e μ_e²)/(q(n_h μ_h + n_e μ_e)²) — sign and magnitude both depend on mobility, not just density.
• Ignoring temperature. Semiconductor n is strongly T-dependent. An InSb sensor calibrated at 25 °C drifts noticeably by 60 °C. Production sensors include temperature compensation either in CMOS analog blocks or as a digital calibration table.