Hall Effect

Every semiconductor datasheet you have ever read uses a value measured by the Hall effect. It is the standard lab technique for finding carrier type, concentration, and mobility all from one experiment.

Core Concept

When a current-carrying conductor is placed in a magnetic field perpendicular to the current, a voltage develops across the third axis. This is the Hall effect, discovered by Edwin Hall in 1879. In semiconductors, it reveals not just the magnitude of carrier density but also the type (N or P) from the polarity of the Hall voltage.

Consider an N-type silicon bar with current Ix flowing in the +x direction. This means electrons move in the -x direction. If a magnetic field Bz points into the page (-z direction), the Lorentz force F = q(v × B) pushes electrons in the -y direction. Electrons accumulate on the bottom face, making it negative and the top face positive. This built-up voltage opposes further accumulation until equilibrium is reached. The voltage across the bar is the Hall voltage VH.

For P-type material, holes move in the +x direction with current. The same Bz pushes holes in the -y direction too. Holes accumulate at the bottom, making it positive. So the polarity of VH is opposite for P-type. This sign flip is the definitive test for carrier type. Hall effect measurements are used routinely to characterise silicon wafers before fabricating ICs.

Key Equations

Hall voltage:

VH = (Ix * Bz) / (q * n * d) where n is carrier concentration in cm^-3, d is the sample thickness in cm

Hall coefficient:

RH = VH * d / (Ix * Bz) = 1 / (q * n) for N-type: RH is negative. For P-type: RH = +1/(q*p) is positive.

Hall mobility:

μH = |RH| * σ = |RH| / ρ where σ is conductivity and ρ is resistivity in Ω·cm

Carrier concentration from Hall measurement:

n = 1 / (|RH| * q)

Given:
  N-type silicon bar
  Current Ix = 2 mA = 2 × 10^-3 A
  Magnetic field Bz = 0.5 T
  Measured Hall voltage VH = 3.125 mV = 3.125 × 10^-3 V
  Sample thickness d = 2 mm = 0.2 cm
  q = 1.6 × 10^-19 C

Why this formula:
  Hall voltage VH depends on carrier concentration n and sample dimensions.
  Rearranging gives n directly.

Formula:
  n = (Ix * Bz) / (q * VH * d)

Substitution:
  n = (2 × 10^-3 × 0.5) / (1.6 × 10^-19 × 3.125 × 10^-3 × 0.2)

Calculation:
  Numerator = 2 × 10^-3 × 0.5 = 10^-3
  Denominator = 1.6 × 10^-19 × 3.125 × 10^-3 × 0.2
  Denominator = 1.6 × 10^-19 × 6.25 × 10^-4
  Denominator = 1.0 × 10^-22
  n = 10^-3 / 10^-22 = 10^19 cm^-3

Final Answer:
  Electron concentration n = 10^19 cm^-3
  Since VH polarity indicates N-type, majority carriers are electrons.

Exam Tip: GATE problems on the Hall effect ask either for carrier concentration from VH, or for the sign of RH and type of semiconductor. For N-type, RH = -1/(nq) is negative. For P-type, RH = +1/(pq) is positive. Students frequently get the sign wrong because they forget that electrons moving in -x direction produce the same conventional current Ix as holes moving in +x direction, but the Lorentz force pushes them to opposite faces. Always draw the force diagram to determine VH polarity.

Key Properties

• Hall coefficient RH = -1/(nq) for N-type (negative) and +1/(pq) for P-type (positive). The sign identifies carrier type.
• Hall voltage VH = Ix * Bz / (q * n * d). Larger VH means lower carrier concentration.
• Hall mobility μH = |RH| * σ gives a direct measurement of carrier mobility without separate conductivity and concentration measurements.
• In compensated semiconductors (both donors and acceptors present), n = ND - NA and the Hall coefficient reflects the net carrier type.
• Hall sensors are used in brushless DC motor controllers, current sensors, and position encoders. The SS49E is a popular linear Hall sensor IC.
• Hall effect is the standard method for characterising semiconductor wafers in IC fabrication, measuring doping uniformity and mobility.

Quick Revision

• Hall effect: magnetic field deflects moving carriers, producing a transverse voltage.
• VH = Ix * Bz / (q * n * d).
• RH = -1/(nq) for N-type (negative). RH = +1/(pq) for P-type (positive).
• Carrier concentration: n or p = 1 / (|RH| * q).
• Hall mobility: μH = |RH| / ρ where ρ is resistivity.
• Thinner samples produce larger VH for the same current and field.
• Used in practice to measure doping level and identify carrier type in silicon wafers.
• Exam trap: Students forget that RH is negative for N-type. If a GATE question asks which sign of RH corresponds to N-type, the answer is negative, not positive, because electrons carry negative charge.