Agility (physics)

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The mobility or mobility as a physical term is defined by the constant (stationary) speed which a body (asymptotically) reaches when a constant force acts on it.

In this context, one speaks of the drift speed .

In electrodynamics , mobility is defined in a slightly modified form and thus with a different unit. The charge carrier mobility describes the relationship between the drift speed of charge carriers and an applied electric field :

Basically, it only makes sense to introduce mobility in dissipative systems, i.e. where there is friction and thus inelastic dispersion. Above a certain speed there is a balance between external force and opposing frictional force, so that the movement is stationary (more generally: the mean speed is stationary).

Mobility in mechanics

A constant force acting on a body causes it to accelerate until the opposite frictional force (e.g. air or sliding friction ) has the same amount. Then the stationary speed is reached and the effective acceleration is zero. This is e.g. B. the reason why a body falling in the atmosphere does not become arbitrarily fast. One reason for this law is the dependence of friction on the speed of the body.

The mechanical mobility is therefore defined as

.

In mechanics, the mobility has the unit s / kg . What is historically interesting is that Aristotle adopted this law as fundamental to his mechanics. Today's mechanics, on the other hand, is based on Newton's axioms from which the law emerges.

Mobility with Stokes' Friction

A body is accelerated by an external force and braked by Stokes' friction . The Stokes' friction force is ; for the movement of a spherical particle in a fluid is considered wherein, the particle radius, the dynamic viscosity of the fluid and the Cunningham correction factor is.

The resulting force is made up of these two contributions:

In equilibrium, the resulting force and thus the acceleration is zero and the steady-state speed is reached:

So the mobility is

Mobility diameter

The mobility of a body moving in a liquid can also be expressed by the mobility-equivalent diameter or mobility diameter. This is the diameter of a sphere that has this mobility. Its value is according to Stokes' law , the Cunningham correction factor indicating whether the fluid surrounding the body can be regarded as a continuum, as free molecular or in between. The decisive factors are the mean free path of the fluid molecules and the mobility diameter of the body .

The constants , and were determined empirically and are i. d. Usually regarded as generally applicable.

This size is mainly used in aerosol technology, especially for ultra-fine particles .

Mobility in Electrodynamics

In electrodynamics , mobility is defined in a slightly modified form. The charge carrier mobility (or simply mobility , especially for electron: electron mobility ) denotes the relationship between an applied electric field and the drift velocity of charge carriers (Solids defect / electron , plasma: electron / ion ).

where the unit     has. The mobility is usually given in cm 2 / (V · s).

At low field strengths it is     independent of the field strength, but not at high field strengths. The exact behavior is significantly influenced by the material, e.g. B. by whether an electric current flows through a solid or a plasma. At very high field strengths, the mean electron speed in solids no longer increases and reaches the saturation speed .

For ion mobility , see Ion mobility .

Relation to conductivity

The electrical conductivity can be related to the mobility. For conductive materials, the material equation that links the electrical current density with the applied electrical field via electrical conductivity is :

The second equal sign is true using the definition of mobility above. In general, the current density is defined as charge density times speed (   is the charge density = charge times charge carrier density):

Thus by equating one comes to the connection between conductivity and mobility:

,

where the electrical charge (not necessarily the elementary charge) of a charge carrier (e.g. electron, hole, ion, charged molecule, etc.) and the charge carrier density represent. In metals, the charge carrier density changes little with temperature and the conductivity is determined by the temperature-dependent mobility

The conductivity of a semiconductor is made up of the electron density and their mobility as well as the hole density and their mobility

In semiconductors, the charge carrier density changes strongly (exponentially) with temperature, whereas the temperature dependence of mobility is small.

Microscopic observation

Charge carriers usually move randomly in a gas or solid body without an electric field, ie the drift speed is zero. In the presence of an electric field, on the other hand, the charges move at an effective speed along the field that is significantly lower than the average speed of the individual charges.

According to the Drude model , the drift speed is the same

From this you can read the mobility directly:

where charge, mass, mean impact time (time between two impacts). The mean peak time can be written as the quotient of the mean free path and mean speed:

The mean speed is made up of mean thermal speed and drift speed . If the electric field strengths are not too great, the drift speed is much smaller than the thermal speed, which is why it can be neglected.

A quantum mechanical analysis according to Sommerfeld provides a similar result. There, however, the mass must be replaced by the effective mass (which can differ from the electron mass by several orders of magnitude). In addition, the mean peak time for the electrons with the Fermi energy must be used. This is because only electrons with energy in the range around the Fermi energy contribute to conductivity (in degenerate systems such as metals and highly doped semiconductors) .

Mobility in solids

In the case of solids, the mobility depends strongly on the number of defects and the temperature, so that it is difficult to give values. It should be noted that in contrast to a single body, the speed of the many charge carriers present is statistically distributed. The necessary frictional force, which prevents constant acceleration, is given by the scattering at defects in the crystal and at phonons . The mean free path is limited by these two scattering mechanisms. The electrons only rarely scatter among each other and actually not at all on the lattice atoms. The mobility can be roughly expressed as a combination of the effects of lattice vibrations (phonons) and impurities using the following equation ( Matthiessensche rule ):

.

The mobility depends on the material, the density of impurities, the temperature and the field strength. At low temperatures, the electrons scatter mainly with impurities, at higher temperatures they scatter more often with phonons (the higher the temperature, the more phonons are excited).

As the quantum mechanical analysis according to Sommerfeld shows, mobility depends on the effective mass . It should be noted that the effective mass is generally a tensor, i.e. it is direction-dependent. In the case of monocrystalline materials, the mobility is therefore dependent on the crystal orientation.

In semiconductors, the mobility is also different for electrons in the conduction band and holes (= holes) in the valence band . Electrons usually have smaller effective masses than holes and thus a higher mobility. If one of the two charge carriers dominates due to doping, the conductivity of the semiconductor is proportional to the mobility of the majority charge carriers . By doping a high-purity semiconductor material (typically silicon) with foreign atoms of a suitable nature, a certain amount of mobile charge carriers are introduced in a targeted manner, but their mobility is reduced because the doping atoms are impurities. Depending on the doping material, excess electrons (n-doping) or electron defects (p-doping) are created.

Charge carrier mobility of some substances

The mobility can vary greatly depending on the material structure. For example, in the standard electronics material, silicon (Si), it only achieves average values. In gallium arsenide (GaAs), on the other hand, it is much higher, with the result that this material allows components made from it to be operated at far higher frequencies than silicon, but at a likewise higher material cost.

Electron and hole mobility of different materials in cm 2 · V −1 · s −1 at 300 K.
material Electrons Holes Remarks
organic semiconductors ≤ 10
Rubren 40 highest mobility among organic semiconductors
common metals ≈ 50
Silicon (crystalline, undoped) 1,400 450
Germanium 3,900 1,900
Gallium arsenide 9,200 400
Indium antimonide 77,000
Carbon nanotubes 100,000
Graph 10,000 on SiO 2 carrier
Graph 350,000 at 1.6 K; previous maximum value
Two-dimensional electron gas 35,000,000 close to absolute zero

Mobility in the gas phase

Mobility is defined individually for each component of the gas phase. This is of particular interest in plasma physics . The definition is:

where - charge of the component, - impact frequency, - mass.

The relationship between mobility and diffusion coefficient is known as the Einstein equation :

where denotes the diffusion constant, the mean free path, the Boltzmann constant and the temperature.

See also

Web links

Individual evidence

  1. Luca Banszerus, Michael Schmitz, Stephan Engels, Jan Dauber, Martin Oellers, Federica Haupt, Kenji Watanabe, Takashi Taniguchi, Bernd Beschoten and Christoph Stampfer: Ultrahigh-mobility graphene devices from chemical vapor deposition on reusable copper . In: Science Advances . No. 6 , 2015, doi : 10.1126 / sciadv.1500222 .
  2. V. Umansky, M. Heiblum, Y. Levinson, J. Smet, J. Nübler, M. Dolev: MBE growth of ultra-low disorder 2DEG with mobility exceeding 35 × 10 6 cm 2  / V s . In: Journal of Crystal Growth . No. 311 , 2009, p. 1658–1661 , doi : 10.1016 / j.jcrysgro.2008.09.151 .