Hall constant

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The Hall constant , also known as the Hall coefficient , is a (temperature-dependent) material constant that is specified in cubic meters per coulomb . In the measurement of the Hall effect, it determines as a proportionality factor , the Hall voltage in accordance with

when the examined layer has the thickness  . The Hall constant is through

given. If the Hall constant is calculated from the current  and the Hall voltage , the layer thickness  must be taken into account, which is not necessary if the electrical current density and the electrical field strength are used for this. The indices indicate the orientations of the respective quantities in a Cartesian coordinate system. The value of the Hall constant indicates how strong the electric field must be to compensate for the effects of the magnetic field on the moving charge carriers . The symbol is also used for the Hall constant , but it can be confused with the Hall resistance .   

Hall constant for free charge carriers

If the electrical conductivity of a material is only determined by one type of charge carrier, as in metals and heavily doped semiconductors , the Hall constant can be calculated from the reciprocal of the product of the charge carrier density and the charge of a charge carrier .

The type of charge carrier can be determined from the sign of the Hall constant. In the case of metals, these are electrons that carry a negative elementary charge ( ). In semiconductors, depending on the doping, both positive (predominantly hole conduction ) and negative (predominantly electron conduction ) values ​​occur for the Hall constant.

Since the type of charge carrier for a substance is usually known, the measurement of the Hall constant is mainly used to determine the charge carrier density. This is often temperature-dependent (very strongly in the case of semiconductors), which means that the Hall constant also changes with temperature.

If two different types of charge carriers contribute to the electrical conductivity, the formula becomes a little more complicated. This is the case in semiconductors, where electrons and positively charged holes occur. In this case, the Hall constant is calculated as follows

The index stands for holes or for electrons and for the respective mobility . It should be noted that even with intrinsic (undoped) semiconductors, the Hall constant can differ from zero due to different mobility.

Hall constant for quasi-free electrons

Metals can also have a positive Hall constant, such as B. aluminum , although here only electrons contribute to conductivity. This effect cannot be agreed with the acceptance of freely movable load carriers in the metal. Restrictions due to the band structure for allowed electron orbits play a decisive role here. Under certain conditions, conduction electrons can behave like a "hole", i. that is, they react to a magnetic field as if they had a positive charge.

history

Edwin Hall reported in his letter to the American Journal of Mathematics on Nov. 19, 1879 that the quotient is constant. He could not predict the constant itself, which was later named after him, since the electron and the elementary charge were still unknown in his time  .

Some typical values

copper −5.3 · 10 −11 m³ / C
silver −8.9 · 10 −11 m³ / C
aluminum + 9.9 · 10 −11 m³ / C
gold −7 · 10 −11 m³ / C
platinum −2 · 10 −11 m³ / C
zinc + 6.4 · 10 −11 m³ / C
Bismuth −5 · 10 −7 m³ / C
Indium antimonide ( semiconductor ) −2.4 · 10 −4 m³ / C

The specified values ​​of the Hall constant scatter widely. This depends on the one hand on the purity of the material and on the other hand on the temperature. For aluminum, for example, the value −3.4 · 10 −11 m³ / C is also given in publications. If the Hall constant has a positive value, it is a hole conductor , if it is negative it is a (predominantly) electron conductor .

literature

  • Gerthsen, Vogel: Physics . 17th edition. Springer, 1993
  • Charles Kittel: Introduction to Solid State Physics . 12th edition. Oldenbourg, 1999, ISBN 3-486-23843-4
  • Harald Ibach, Hans Lüth: Solid State Physics . 6th edition. Springer, 2002, ISBN 3-540-42738-4