# Explain the Hall effect

The **Hall effect** ['hɔːl-], discovered by Edwin Hall in 1879, is the appearance of an electrical voltage in a current-carrying conductor that is in a stationary magnetic field. The voltage drops perpendicular to the direction of the current flow and the direction of the magnetic field on the conductor and becomes *Hall voltage**U*_{H} called.

The magnitude of the stress can be calculated using the formula derived below

from amperage *I.*, magnetic flux density*B.*, Thickness of the sample *d* (parallel to *B.*) and a material constant - the so-called *Hall constants* (also: Hall coefficient) *R.*_{H} - be calculated.

The reverb effect bears its name in honor of its discoverer and has nothing to do with reverberation in acoustics.

### Explanation

The Hall effect occurs in an electrical conductor through which current flows, which is located in a magnetic field, an electrical field being built up that is perpendicular to the direction of the current and the magnetic field and which compensates for the Lorentz force acting on the electrons.

A current flows when a voltage is applied to the sample. The charge carriers are generally electrons, but hole conduction can also predominate in appropriately doped semiconductors. The electrons move against the technical direction of current at a medium speed *v* (Drift speed) through the conductor. Because of the Lorentz force caused by the magnetic field, the electron is deflected perpendicular to its direction of movement. This leads to an excess of electrons on the corresponding side of the conductor (highlighted in blue), while on the opposite side there is to the same extent an electron deficiency (highlighted in red). So you are dealing with a charge separation comparable to that of a capacitor. The opposing negative and positive excess charges cause an electric field that exerts a force on the electrons that is opposite to the Lorentz force. The intensification of the charge separation comes to a standstill when both forces are just compensating each other. As with the capacitor, a voltage can be tapped, here as *Hall voltage* referred to as. The Hall voltage usually follows changes in current and magnetic field directly. It increases linearly with the magnetic field and is inversely proportional to the (signed) charge carrier density. In particular, the Hall voltage is independent of the specific resistance of the sample. The Hall effect is used both for measuring magnetic fields (with *Hall sample*) as well as to determine the type of charge carrier (electrons or holes) and their density.

The specific properties of the conduction process are determined by the Hall constant *R.*_{H} reproduced.

### Derivation

*Basic knowledge of vector calculus and electrodynamics is helpful in understanding this section.*

At this point a short derivation of the formula for the Hall voltage should be sketched. The validity of the derivation is limited to electrical conductors with only one type of charge carrier, such as metals (electrons) or heavily doped semiconductors (either predominantly holes or electrons).

Moving charge carriers in a magnetic field experience the Lorentz force:

With the Hall effect, a compensating electric field is built up, which neutralizes the deflecting effect of the magnetic field. The following must therefore apply to the resulting force on the load carriers:

For the sake of simplicity, the coordinate system is placed in such a way that the charge carriers move in the x direction and the magnetic field acts in the z direction. So it is other . This becomes the y component of the above equation after shortening *q* to:

The current density one can generally get through in the conductor express. One dissolves this relationship *v*_{x} and put it in the above equation, one obtains

This relationship becomes the Hall constant *R.*_{H} defines which characterizes the strength of the Hall effect.

To make the equation a little more manageable, the conductor in which charge separation has taken place can be viewed as a plate capacitor. For this the relationship applies

- .

In addition, the current density *j*_{x} in the present case *j*_{x} = *I.* / *b**d* be expressed. If you use these two notations, you get the hall voltage *U*_{H} an expression that only depends on easily measurable quantities:

- .

This formula is also correct for conductors with different types of charge carriers, but the Hall constant can no longer be passed through *R.*_{H} = 1 / *n**q* to calculate.

The so-called **Hall resistance** specify:

The Hall resistance characterizes a Hall element, but has nothing to do with the electrical resistance measured on a Hall element. It indicates the ratio of transverse voltage to current of a Hall element at a certain magnetic flux density:

### application

In electronics, the Hall effect is used in so-called Hall probes to measure the magnetic flux density. If a current flows through the conductor, the generated Hall voltage can be measured according to the above formula *B.* be calculated. Materials with a large hall constant are characterized by their high sensitivity. For this reason, semiconductor materials are mostly used. Mass production for widespread use in industry was only possible through the integration of Hall plates in CMOS technology. Only then can temperature dependencies and other effects be compensated and the Hall voltage evaluated accordingly and digitally processed. Today there are increasingly complex Hall sensors based on CMOS in applications for angle, position, speed and current measurement.

Another area of application is the determination of charge carrier densities by measuring the Hall constants. By additionally measuring the electrical conductivity (or the specific resistance) it is also possible to determine the mobility of the charge carriers in the material.

### Quantum Hall Effect

The quantum Hall effect causes the Hall voltage to be increased in the case of strong magnetic fields and low temperatures of around a few Kelvin *U*_{H} divided by the stream *I.* cannot vary arbitrarily if the magnetic field strength is varied, but always an integral fraction of the von Klitzing constant

(in units of kOhm, *H* is Planck's constant, *e* the elementary charge) is, therefore *R.*_{K}, *R.*_{K} / 2, *R.*_{K} / 3 and so on. The accuracy with which these plateaus can be reproduced is so extremely good that *R.*_{K} has been established as the standard for electrical resistance by international treaties. The quantum Hall effect is largely understood. Klaus von Klitzing received the Nobel Prize in 1985 for this discovery.

### So see

Categories: Electrodynamics | magnetism

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