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Hall Effect

Key idea: H3 Solid State Physics: Hall Effect — key ideas and exam-focused notes on bonding, crystal structures, conduction models, and band ideas.

  • Advanced Physics
On this page

Learning objectives

  • Connect energy bands, carrier response, Hall measurements, and semiconductor-device behaviour.
Before you start

Use this page to connect Hall-voltage measurements to carrier sign and carrier density. You should already be comfortable with magnetic force on a moving charge and drift current.

Use this page for:

  • deriving Hall voltage relations,
  • identifying majority carrier sign from measured voltage direction,
  • connecting microscopic drift variables to macroscopic measurements.

Fast start

  1. Magnetic force deflects moving carriers sideways.
  2. Charge separation builds an electric field until magnetic and electric forces balance.
  3. Hall voltage then encodes carrier sign and density information.

Imagine a conductor with rectangular cross-section d x t that is parallel to the y-z plane with (conventional) current passing in the positive x direction. With a magnetic field acting in the positive y direction, there is a Lorentz force on the charge carriers given by F = qv_dB. Fleming’s left hand rule suggests that both positive and negative charge carriers should experience a force to the side of the conductor.

A slab carries current I along +x in a magnetic field B along +y, with width d along z and thickness t along y. An end view shows the magnetic force pushing a moving carrier towards the +z face, an opposing electric force, charge separated across the width and a voltmeter reading the Hall voltage.
The magnetic force pushes carriers of either sign towards the same face. The charge that builds up creates a Hall field until the electric force balances the magnetic force; the charge signs shown are for electron carriers, and the Hall voltage is measured across the width d.

However, this build up of charge creates the Hall electric field that prevents the build-up of further charge. A steady state is reached where: F_B = F_E; Bqv_d = qE; Bv_d = V/d; B = V/v_dd; V = Bv_dd , where V is the hall voltage, vd is the drift velocity, d is the width of the conductor. Since I = Anv_dq, v_d = I/Anq Combining the equation for V and vd, V = Bd (I/Anq); V = BI/nq (d/A) Since the cross-sectional area A is equal to the product of the width of the sample d and its thickness t, (A = dt) V = BI/nqt The hall effect is used extensively to study conduction in materials, particularly in semiconductors. We have assumed so far that the mobile charge carriers within solids are electrons and this is in agreement with the sign of the Hall voltage for most materials. However, anomalous results can be obtained for metals such as aluminium or indium, and some semiconducting materials behave as if there were positive charge carriers in the materials. Back To Solid State Physics

Sign convention reminder

Keep conventional current direction, magnetic-field direction, and chosen voltage polarity explicit. Most Hall-effect mistakes come from hidden sign flips, not formula errors.

Next steps

Continue with the next resource in this course.

Course and syllabus information
Course
Advanced Physics
Edition
Advanced Physics