Hall Effect - Electronic Devices - Electronics and Communication Engineering

Table of Contents
1. Introduction
2. Qualitative explanation
3. Theory and formulae
4. Measurement of carrier concentration and mobility
5. Example calculation
6. Uses and applications
7. Types of currents in semiconductors
8. Practical notes and sign convention
9. Summary
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Introduction

The Hall effect was discovered by Edwin Hall in 1879. When a current-carrying conductor or semiconductor is placed in a magnetic field that is perpendicular to the direction of current, an electric field is induced in the conductor in the direction mutually perpendicular to both the current and the magnetic field. This induced transverse electric potential difference is called the Hall voltage and the phenomenon is known as the Hall effect.

Qualitative explanation

Consider a rectangular slab of conducting material with current flowing along one axis and a magnetic field applied perpendicular to the current. Charge carriers moving through the magnetic field experience a magnetic force which deflects them toward one side of the slab. As charges accumulate on the side, an electric field develops that opposes further charge separation. At steady state, the electric force due to this field balances the magnetic force; the resulting transverse potential difference is the Hall voltage.

As electrons (or holes) move through the conductor in the presence of a magnetic field B, the magnetic component of the Lorentz force causes the carriers to be displaced to one side. This creates one side negatively charged and the opposite side positively charged, producing the Hall voltage.

The separation of charge produces an electric field (the Hall electric field) and a measurable voltage between the two faces. The voltage increases until the electric force equals the magnetic force on the moving charges; this equilibrium condition defines the Hall effect.

Notation used in later derivations:

Where q denotes the charge of a carrier (for electrons q = -e),

= the magnetic field

= the drift velocity of carriers

= the Hall electric field

d = distance between the faces across which the Hall voltage is measured (thickness of the sample in the appropriate direction).

This transverse voltage is the Hall EMF (Hall voltage).

Theory and formulae

The force on a charge q moving with velocity v in electric field E and magnetic field B is given by the Lorentz force:

F = q(E + v × B).

For charges drifting through a conductor with drift velocity v_d in a magnetic field perpendicular to their motion, the magnetic force q(v_d × B) causes lateral charge separation. At steady state the magnetic force is balanced by the electric force due to the Hall field E_H.

Derivation of the Hall field and Hall voltage:

Balance of forces gives qE_H = q(v_d × B).

Therefore E_H = v_d B (magnitude).

Drift velocity v_d is related to current I, carrier concentration n and cross-sectional area A by v_d = I/(n q A).

Substitute v_d into E_H to get E_H = (I B)/(n q A).

For a rectangular slab with cross-section A = t × w (t = thickness, w = width), the Hall voltage V_H measured across the width is V_H = E_H · w.

Substituting E_H gives V_H = (I B w)/(n q t w) = (I B)/(n q t).

Thus V_H = (I B)/(n q t), where

• V_H is the Hall voltage (measured across the sample),
• I is the current through the sample,
• B is the magnetic flux density (perpendicular to current),
• n is the number density of charge carriers,
• q is the carrier charge (±e),
• t is the sample thickness in the direction of the magnetic field).

The Hall field per unit current density J and magnetic field B gives the Hall coefficient R_H:

R_H = E_H/(J B) = 1/(n q).

For materials where electrons are the majority carriers, q = -e and therefore R_H = -1/(n e). The sign of R_H (and thus the measured Hall voltage polarity) indicates the sign of the majority carriers: negative for electrons, positive for holes.

Relation with conductivity and mobility:

The electrical conductivity σ of a material with carrier concentration n and mobility μ is σ = n q μ.

Therefore the carrier mobility can be expressed using Hall coefficient as μ = |R_H| σ.

Measurement of carrier concentration and mobility

From the measured Hall voltage V_H, known current I, applied magnetic field B and sample thickness t, the carrier concentration n is obtained as

n = (I B)/(q t V_H).

Once n and conductivity σ are known (σ can be measured from the sample resistance and geometry), the mobility μ is given by μ = σ |R_H|.

Example calculation

Consider a sample of thickness t carrying a current I with an applied magnetic field B and a measured Hall voltage V_H. To find n proceed as follows.

From the Hall voltage expression V_H = (I B)/(n q t), rearrange to get n = (I B)/(q t V_H).

Insert the numerical values for I, B, q = e (1.602 × 10⁻¹⁹ C) and t in SI units to calculate n in m⁻³.

After finding n, measure the sample resistance R to determine conductivity σ = (l)/(R A) for a bar of length l and cross-section A. Then compute μ = σ |R_H|.

Uses and applications

• The Hall effect is used to determine the type of majority carriers (sign of charge carriers) in a semiconductor.
• It is used to measure the carrier concentration n and the mobility μ of charge carriers in semiconductors.
• Hall probes (sensors) are widely used to measure magnetic flux density B in laboratories and instruments.
• Hall sensors are used in position sensing, speed and tachometry of motors, and in brushless DC motor commutation.
• Hall-effect current sensors measure current without direct electrical contact by sensing the magnetic field produced by the current.
• Hall devices assist in characterising materials as conductor, semiconductor or insulator and in quality control of semiconductor fabrication.

Types of currents in semiconductors

Two important types of current in semiconductors are diffusion current and drift current.

• Diffusion current: This current is caused by the motion of charge carriers from regions of higher concentration to regions of lower concentration. When an N-type region and a P-type region are brought into contact, electrons diffuse from the N-side to the P-side and holes diffuse from the P-side to the N-side. The region where charge carriers are depleted is called the depletion region, and the current due to concentration gradient in absence of an external electric field is the diffusion current.

• Drift current: This current arises from the motion of charge carriers under the influence of an external electric field. When an electric field is applied across a semiconductor, carriers accelerate and attain a drift velocity; the flow of charges due to this drift constitutes the drift current. Drift current determines device behaviour under applied bias and is important for active circuit operation.

Practical notes and sign convention

The observed Hall voltage polarity depends on the sign of dominant carriers. In practice, proper wiring and polarity checks are essential when using Hall probes. Temperature affects both carrier concentration and mobility, so measurements are often temperature-calibrated. Thin samples give larger Hall voltages for the same current and field, because V_H ∝ 1/t.

Summary

The Hall effect provides a direct method to probe the nature and quantity of charge carriers in conductors and semiconductors and to measure magnetic fields. Key relations are V_H = (I B)/(n q t) and R_H = 1/(n q). Measurement of Hall voltage together with conductivity yields carrier concentration and mobility - essential parameters in electronic device characterisation.