# Hall effect

The Hall effect [' hɔ ː l ], named after Edwin Hall, describes the occurrence of an electrical voltage in a current-carrying conductor, which is located in a stationary magnetic field. In this case, the voltage falls from both perpendicular to the current flow as well as to the magnetic field direction of the conductor, and is called the Hall voltage.

The size of the voltage can using the below equation derived

Of current, magnetic flux density, thickness of the sample ( parallel to ) and a material constant - the so-called Hall constant (also: Hall coefficient ) - are calculated.

## Explanation

The Hall effect occurs in a current-carrying electrical conductors, which is located in a magnetic field, wherein an electric field is built up, which is perpendicular to the current direction and magnetic field and compensate the load acting on the electrons Lorentz force.

By applying a voltage to the sample, a current flows. The charge carriers are electrons, in general, it may also prevail in hole conduction correspondingly doped semiconductors. The electrons move against the current technical direction with an average velocity ( drift velocity ) through the conductor. Because the magnetic field caused by the Lorentz force, the electron is deflected perpendicular to its direction of movement. This leads to the corresponding side of the conductor to an excess of electrons ( highlighted in blue), while it is on the opposite side in the same degree to electron-deficient (highlighted in red ). So you have to do it with a charge separation, comparable to that of a capacitor. Now the opposite negative and positive charge excess cause an electric field, which exerts a force on the electrons which is opposed to the Lorentz force. The gain of the charge separation comes to a halt when the two forces compensate straight. As with the capacitor, a voltage can be tapped, which is herein referred to as the Hall voltage. The Hall voltage follows current and magnetic field changes usually immediate. It increases linearly with the magnetic field and is inversely proportional to the ( signed ) charge carrier density, since a smaller number of charge carriers only at higher speed of the individual charges can lead to an unchanged current. Since the charge carrier density in semiconductors is much smaller than in metals, semiconductors to be used as Hall probes mainly.

The Hall voltage is in particular independent of the resistivity of the sample. The Hall effect to measure both the magnetic fields ( Hall- probe ) and to determine the type of charge carrier (electrons or holes) and the density used.

The specific characteristics of the line operation can be represented by the Hall coefficient.

## History

Edwin Hall described the effect later named after him in 1879 as part of his doctoral thesis. By his own admission he was a statement made by James Clerk Maxwell motivated to look for this effect, since this statement Maxwell seemed unnatural:

" It must be carefull fully remembered did the mechanical force Which urges a conductor carrying a current across the lines of magnetic force acts, not on the electric current, but on the conductor Which carries it. - The only force acts on electric currents Which is electromotive force ".

"It must be remembered mindful that the mechanical force that causes a conductor to carry a current through the lines of magnetic force does not act on the electric current, but on the conductor which carries it. - The only force acting on an electrical current, the electromotive force ".

Before Hall has a number of other physicists had such an effect sought (about Feilitzsch, Mach, Wiedemann and his doctor father Rowland ), but not until he reached a sufficient sensitivity. His doctoral work included measurements of the Hall effect in gold. At later measurements noticed Kelvin:

"The subject of the communication is by far the greatest discovery did HAS BEEN made in respect to the electrical properties of metals since the times of Faraday - a discovery Comparable with the greatest made by Faraday. "

" The content of the communication is by far the largest discovery in the field of electrical properties of metals since the time of Faraday - a discovery comparable to the greatest of Faraday. "

## Derivation

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

Charged particles in motion in a magnetic field experience the Lorentz force:

In the Hall effect, a compensating electric field is built up, which neutralizes the distracting effect of the magnetic field. Must therefore apply to the resultant force on the charge carriers:

For simplicity, the coordinate system is set so that the charge carriers in the x- direction will move and the magnetic field acts in the z- direction. So it is and. Thus the y-component of the above equation with respect to shortening to:

The current density in the conductor can be generally expressed by. Solving this relationship after and places them into the above equation, we obtain

About this relationship, the Hall coefficient is defined which characterizes the strength of the Hall effect.

In order to make the equation a bit handy, you can the conductor in which even a charge separation has taken place, conceived as a plate capacitor. For this the relation

In addition, the current density can be expressed as in the present case. Putting these two spellings, we obtain for the Hall voltage a depending only on simple measurable terms apply:

This equation is correct also for conductors with different types of charge carriers, however, the Hall constant can then no longer be calculated. From the equation, the so-called Hall resistance can specify:

Characterized the Hall resistance, a Hall element, however, has nothing to do with the measured electrical resistance of a Hall element. He gives the ratio transverse voltage to power a Hall element on at a certain magnetic flux density:

## Application

In the electronics of the Hall effect in the so-called Hall probe is used to measure the magnetic flux density. A current flows through the conductor, it can be calculated by measuring the Hall voltage generated by the above formula. Materials with large Hall coefficient are distinguished with high sensitivity. For this reason, the semiconductor materials are most commonly used. The mass production for wide use in industry was made possible by the integration of Hall plates in CMOS technology. Only then can temperature dependencies and other effects compensated and the Hall voltage are evaluated accordingly and processed digitally. Today, there are more and more complex Hall sensors in CMOS-based applications for angle, position, velocity and current measurement.

Another field of application is the determination of charge carrier densities, by measuring the Hall constant. By an additional measurement of the electrical conductivity (or resistivity ), it is also possible to determine the mobility of the charge carriers in the material. A convenient method for determining the resistivity and the Hall constants of thin layers is the van der Pauw method of measurement.

An electronic compass can be built with Hall probes.

Practical applications can also be found in aerospace, at Hall ion thrusters.

## Quantum Hall effect

→ Main article: Quantum Hall effect

Already around 1930 Landau had the thought that at very low temperatures, strong magnetic fields and two-dimensional conductors, quantum effects should occur. The quantum Hall effect causes in two-dimensional systems with very strong magnetic fields and low temperatures (a few Kelvin), the Hall voltage divided by the current can not vary arbitrarily when the magnetic field strength is varied, but that the ratio varies in steps. For example at interfaces or surfaces under the specified conditions always an integer fraction of the von Klitzing constant

( in units of ohms = V / A; h is the Planck's constant, e is the elementary charge ). The specified scale values for the ratio are thus:, and so on. The accuracy with which these plateaus can be reproduced is so good that has been established by international treaties as a standard for electrical resistance. The quantum Hall effect is largely understood. Klaus von Klitzing received the Nobel Prize for this discovery in 1985.

## Generalization: Spin Hall effect

Electrons have except the elementary electric charge nor any other " fundamental property ", namely the so-called "spin", a kind of magnetic alignment, which assumes only the two values, the reduced Planck constant is. The direction of the spin is continuous as vectors and can be influenced by magnetic fields.

So if the ordinary currents are (precisely " cargo flows " ) replaced by " spin currents ", is obtained instead of the " ordinary Hall effect, " the so-called spin Hall effect, which is currently, 2010, in physics very timely.

More specifically, the spin Hall effect also, as well as the ordinary Hall-effect, proportional to the longitudinal component of the electric field, ie to the common current value, and the spin current generated is also, similar to the ordinary Hall voltage UH, " transverse" directed. However, there is the following major difference from the ordinary Hall effect: there is no magnetic field is required, but the spin Hall effect results from extrinsic mechanisms (eg impurities ) or by intrinsic effects (eg by the so-called spin-orbit coupling).

## The planar Hall effect

The so-called planar Hall effect is a magnetoresistive effect in ferromagnetic materials, which has nothing to do despite the name with the ordinary Hall effect. The main difference from the ordinary Hall effect - and also reason for naming - is that in this case the magnetic field is not perpendicular to the sample, but in the sample (ie, " planar" ) runs, but " transverse" to the longitudinal current, also extrinsic and intrinsic effects can be distinguished. Insofar as the spin Hall effect is rather similar to the planar Hall effect as the ordinary Hall effect.