Hall Effect

When an electric current flows through a conductor, electrons flow along its length in the opposite direction. When a magnetic field perpendicular to this current flow is introduced, a fascinating phenomenon emerges. Each electron experiences a force known as the Lorentz force, which deflects it sideways, perpendicular to both the direction of the current and the magnetic field lines.

This sideways deflection is crucial to understanding the Hall Effect. Imagine electrons navigating through a conductor under the influence of an electric field driving them along the conductor and a magnetic field exerting a perpendicular force. The Lorentz force compels these electrons to veer off their original trajectory, causing them to accumulate on one side of the conductor while creating an opposing charge buildup on the opposite side. This charge separation generates an electric field that counteracts further electron movement, resulting in an equilibrium state where the electric and magnetic forces balance each other. In addition, a voltage is generated inside the conductor, known as Hall voltage.

The following expression gives the Hall voltage:

Where

• V_H is the Hall voltage
• I_x is the current flowing through the conductor
• B_z is the applied magnetic field
• n is the number of charge carriers per unit volume
• e is the value of the electric charge
• t is the thickness of the conductor

The Hall coefficient is a fundamental parameter in understanding the behavior of charge carriers in materials subjected to a magnetic field. It is defined as the ratio of the Hall voltage to the product of the applied magnetic field, current density, and sample thickness. Mathematically, the Hall coefficient is given by

Where

• R_H is the Hall coefficient
• E_y is the electric field due to the Hall effect
• j_x is the current density

The Hall coefficient provides crucial insights into the material’s electronic properties. Its interpretation extends beyond a mere numerical value. It reflects the density and mobility of charge carriers within the material.

A. Magnetic field sensing and measurement:

• Utilizes the Hall Effect to measure magnetic fields accurately.
• Provides precise measurements across a wide range of magnetic field strengths
• Commonly employed in compasses, magnetometers, and magnetic field mapping devices

B. Current and voltage measurements:

• Hall Effect-based current sensors offer non-invasive and accurate measurement of electrical currents.
• Hall Effect voltage sensors enable precise measurement of voltages in various circuits and systems.
• Widely used in power electronics, automotive systems, and industrial applications

C. Semiconductor characterization:

• Hall Effect measurements provide valuable data for analyzing the electrical properties of semiconductors.
• Helps determine carrier concentration, mobility, and conductivity of semiconductor materials

D. Magneto-resistive sensors:

• Hall Effect-based magneto-resistive sensors detect changes in magnetic fields by measuring resistance.
• Used in applications such as automotive speed sensors, position sensors, and magnetic encoders

Problem 1: A thin rectangular metal plate of thickness 1 cm is subjected to a magnetic field of strength B = 0.5 T. When a current of I = 2 A flows through the plate, a Hall voltage of V_H = 0.1 V is measured. Calculate the charge carrier density in the metal.

Solution

The Hall voltage can be calculated using the formula:

V_H = I_xB_z/net

Rearranging the formula for n, we get:

n= I_xB_z/V_Het

Given data:

• I_x = 2 A
• B_z = 0.5 T
• V_H = 0.1 V
• t = 1 cm = 0.01 m (converted to meters)

Substituting the values, we get:

n = (2 A x 0.5 T)/(0.1 V x 1.6 x 10^-19 C x 0.01 m) = 6.25 x 10²¹  per m³