# Hall Effect in Physics

The Hall effect is a phenomenon in physics that occurs when a magnetic field is applied perpendicular to the flow of current in a conductor or semiconductor. This results in the development of a transverse voltage, known as the Hall voltage, across the material. Understanding the Hall effect is crucial because it provides insights into the behavior of charge carriers in materials. The effect has significant implications in various fields, from fundamental physics to practical applications in electronic devices.

### What Is the Hall Effect?

The Hall effect describes the generation of a voltage difference (Hall voltage) across an electrical conductor through which an electric current is flowing, when a magnetic field is applied perpendicular to the current. This effect was discovered by Edwin Hall in 1879 and is a critical tool in the study of electronic properties of materials.

### Importance of the Hall Effect

The Hall effect is important because it allows for the determination of the type (positive or negative), density, and mobility of charge carriers in a material. It is particularly useful in the characterization of semiconductors and in the design of various electronic components and sensors.

### History

American physicist Edwin Herbert Hall discovered the Hall effect in 1879 while working on his doctoral thesis at Johns Hopkins University. He observed that placing a current-carrying conductor in a perpendicular magnetic field generates a voltage across the conductor in a direction perpendicular to both the current and the magnetic field. This discovery provided a new way to probe the properties of conductors and semiconductors, significantly advancing the field of solid-state physics. Keep in mind, Hall’s observation occurred prior to the discovery of the electron.

### Theory Behind the Hall Effect

The Lorentz force explains the Hall effect. The Lorentz force acts on the moving charge carriers in the presence of a magnetic field. When a current flows through a conductor in the presence of a perpendicular magnetic field, the charge carriers (electrons or holes) experience a force that is perpendicular to both the current direction and the magnetic field. Positive and negative charges move in opposite directions. Charge carriers accumulate on one side of the conductor, creating a voltage difference across the conductor, known as the Hall voltage.

### Positive Charge Carriers

In situations involving metal wires, the charge carriers are electrons. In semiconductors, charge carriers can be holes (with electrons moving in the opposite direction). However, sometimes current involves positive charge carriers. Examples include:

• Ionic Conductors: In ionic conductors, the charge carriers are electrolytes, which are ions.
• Plasma: There are both electrons and free positive ions in plasma. Examples of environments involving plasma include stars, neon signs, and plasma televisions.
• Solid-State Materials: The primary charge carriers are ions with positive charges in certain ionic crystals or superconductors.

### Hall Voltage Formula

Calculate the Hall voltage (VH) using the following formula:

VH = B·I / n·q·d

where:

• B is the magnetic field strength (in Tesla, T),
• I is the current through the conductor (in Amperes, A),
• n is the charge carrier density (in carriers per cubic meter, m-3),
• q is the charge of the carriers (in Coulombs, C),
• d is the thickness of the conductor (in meters, m).

#### Example Problem

For example:

Problem:

A copper strip of thickness 0.01cm is in a magnetic field of 0.5T. A current of 3 A flows through the strip. Given that the charge carrier density in copper is approximately 8.5×1028 m−3 and the charge of an electron is 1.6×10−19 C, calculate the Hall voltage.

Solution:

1. Convert Thickness to Meters:
𝑑 = 0.01 cm = 0.01×10−2 m = 1×10−4 m
2. Identify Given Values:
𝐵 = 0.5 T
𝐼 = 3 A
𝑛 = 8.5×1028 m−3
𝑞 = 1.6×10−19 C
3. Plug the Values into the Hall Voltage Formula:
𝑉𝐻 = 𝐵⋅𝐼 / 𝑛⋅𝑞⋅𝑑 = 0.5 T ⋅ 3 A / 8.5×1028 m−3 ⋅ 1.6×10−19 C ⋅ 1×10−4 m
4. Calculate the Denominator:
𝑛⋅𝑞⋅𝑑 = 8.5×1028 ⋅ 1.6×10−19 ⋅ 1×10−4
𝑛⋅𝑞⋅𝑑 = 8.5×1.6×1028−19−4
𝑛⋅𝑞⋅𝑑=13.6 × 105
5. Calculate the Hall Voltage:
𝑉𝐻 = 0.5 ⋅ 3 / 13.6×105
𝑉𝐻 = 1.5 / 13.6×105
𝑉𝐻 = 1.10×10−6 V
VH ​= 1.10 μV

Result: The Hall voltage across the copper strip is 1.10 𝜇V.

### The Hall Coefficient

The Hall coefficient (RH) is a fundamental parameter that characterizes the Hall effect in a material. It is the ratio of the induced electric field (Hall voltage) to the product of the current density and the applied magnetic field. Mathematically, the Hall coefficient is:

RH = EH / J·B​​

where:

• EH​ is the Hall electric field,
• J is the current density,
• B is the magnetic field.

The Hall coefficient provides information about the type and density of charge carriers in the material. For a material with positive charge carriers (holes or cations), the Hall coefficient is positive, and for negative charge carriers (electrons or anions), it is negative.

### Using the Right-Hand Rule with the Hall Effect

The right-hand rule is a simple mnemonic for determining the direction of the Hall voltage. According to this rule, if you point the thumb of your right hand in the direction of the current and your index finger in the direction of the magnetic field, your middle finger (perpendicular to both the thumb and index finger) points in the direction of the induced Hall voltage.

### What Is a Hall Effect Sensor?

A Hall effect sensor is a transducer that varies its output voltage in response to changes in the magnetic field. These sensors are used to detect the presence, absence, or strength of a magnetic field. They are commonly used in applications such as:

• Automotive Ignition Systems: For determining the position of the crankshaft or camshaft.
• Proximity Sensors: For detecting the proximity of objects in automated systems.
• Speed Detection: In tachometers and anti-lock braking systems (ABS).
• Current Sensing: In power supplies and battery management systems.
• Nondestructive Testing: Sensors detect defects and irregularities, with applications in crack detection and thickness measurements.

### Hall Effect Controllers and Joysticks

Hall effect controllers and joysticks utilize Hall sensors for detecting the position of a control stick or lever. Unlike traditional potentiometers, Hall effect sensors do not suffer from mechanical wear and tear. Thus, they offer more reliable and longer-lasting performance. These controllers find use in various applications, including gaming, aviation, and industrial control systems.

### Applications of the Hall Effect

The Hall effect has numerous applications, including:

• Hall Effect Sensors: These devices use the Hall effect to measure magnetic field strength. They are popular in automotive, industrial, and consumer electronics for applications like current sensing, position sensing, and speed detection. The sensors also find use in blood flow meters and magnetically activated implants (e.g., cochlear implants) in medicine and biology.
• Magnetic Field Measurement: The Hall effect determines the strength and direction of magnetic fields in various scientific and industrial applications.
• Semiconductor Characterization: The Hall effect determines the carrier concentration, mobility, and type in semiconductors, which is essential for designing and optimizing electronic devices.
• Space Exploration: Hall effect thrusters are ion thrusters for spacecraft propulsion. The Hall effect also measures properties of plasma in space environments.

Additionally, the effect is important in fusion research and in understand star formation.

### Quantum Hall Effect

The quantum Hall effect occurs in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. A characteristic of the quantum Hall effect is the quantization of the Hall conductance in integer multiples of a fundamental constant. This discovery was made by Klaus von Klitzing in 1980.

### Spin Hall Effect

The spin Hall effect is a phenomenon where an electric current flowing through a material with strong spin-orbit coupling generates a transverse spin current, leading to spin accumulation on the material’s edges. This effect is essential for spintronics, a field of study that develops electronic devices based on electron spin rather than charge.

### Experimental Setup for Measuring the Hall Effect

Measuring the Hall effect involves a set-up for observing the transverse voltage generated across a material when an electric current flows through it in the presence of a perpendicular magnetic field. Here’s a detailed description of the experimental setup and procedure:

#### Experimental Setup

1. Hall Bar Sample: A rectangular strip of the material (conductor or semiconductor). The dimensions of the sample should be well-defined, especially the thickness (𝑑).
2. Constant Current Source: A DC power supply that provides a stable and known current (𝐼) through the Hall bar sample.
3. Magnetic Field Source: Typically an electromagnet or a permanent magnet capable of generating a uniform magnetic field (𝐵) perpendicular to the current flow in the sample. Make sure the strength of the magnetic field is adjustable and measurable.
4. Voltmeter: Use a sensitive voltmeter or a digital multimeter for measuring the Hall voltage (𝑉𝐻) across the sample. Connect the voltmeter to the sides of the Hall bar where the voltage develops.
5. Positioning Apparatus: Use clamps or other equipment that secures the Hall bar sample in place and ensures that the magnetic field is perpendicular to the current flow.
6. Temperature Control (Optional): If the experiment requires measurements at different temperatures, include a temperature-controlled environment or a heating/cooling apparatus.

#### Procedure

1. Prepare the Hall Bar Sample:
• Cut the sample into a rectangular strip with well-defined dimensions.
• Attach electrical contacts to the ends of the sample for the current and to the sides for measuring the Hall voltage.
2. Setup the Circuit:
• Connect the sample to the constant current source, ensuring that the current flows uniformly through the length of the Hall bar.
• Connect the voltmeter across the sides of the sample.
3. Apply the Magnetic Field:
• Place the sample in the magnetic field generated by the electromagnet or permanent magnet.
• Ensure that the magnetic field is perpendicular to the plane of the Hall bar sample and the direction of the current.
• Adjust and measure the strength of the magnetic field using a gaussmeter or a similar device.
4. Measure the Hall Voltage:
• Turn on the current source and set the desired current (𝐼) through the Hall bar.
• With the magnetic field applied, read the Hall voltage (𝑉𝐻) from the voltmeter. This voltage is the potential difference across the width of the sample due to the Hall effect.
5. Repeat for Different Values:
• Repeat the measurements for different values of the magnetic field strength (𝐵) and current (𝐼) to study the dependence of the Hall voltage on these parameters.
• Optionally, repeat the measurements at different temperatures if temperature control is available.

#### Data Analysis

1. Calculate the Hall Coefficient:
• Using the measured Hall voltage (𝑉𝐻​), current (𝐼), magnetic field strength (𝐵), and thickness of the sample (𝑑), calculate the Hall coefficient (𝑅𝐻​) using the formula: 𝑅𝐻 = 𝑉𝐻⋅𝑑 / 𝐼⋅𝐵​
2. Determine Carrier Density and Type:
• The sign of the Hall coefficient indicates the type of charge carriers: positive (𝑅𝐻 > 0) for holes and negative (𝑅𝐻 < 0) for electrons.
• Use the magnitude of 𝑅𝐻 for calculating the charge carrier density (𝑛) using the relationship: 𝑛 = 1 / 𝑅𝐻⋅𝑞​ where 𝑞 is the charge of the carrier (e.g., 1.6×10−19 C for electrons).
3. Analyze Mobility:
• Calculate the mobility (𝜇) of the charge carriers if the conductivity (𝜎) of the material is known: 𝜇 = 𝜎 / 𝑛⋅𝑞​

### References

• Braiding, C. R.; Wardle, M. (2012). “The Hall effect in star formation”. Monthly Notices of the Royal Astronomical Society. 422 (1): 261. doi:10.1111/j.1365-2966.2012.20601.x
• Hall, Edwin (1879). “On a New Action of the Magnet on Electric Currents”. American Journal of Mathematics. 2 (3): 287–92. doi:10.2307/2369245
• Karplus, R.; Luttinger, J. M. (1954). “Hall Effect in Ferromagnetics”. Phys. Rev. 95 (5): 1154–1160. doi:10.1103/PhysRev.95.1154
• Ohgaki, Takeshi; Ohashi, Naoki; et al. (2008). “Positive Hall coefficients obtained from contact misplacement on evident n-type ZnO films and crystals”. Journal of Materials Research. 23 (9): 2293. doi:10.1557/JMR.2008.0300
• Ramsden, Edward (2011). Hall-Effect Sensors: Theory and Application. Elsevier. ISBN 978-0-08-052374-3.