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Electron mobility
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In physics, electron mobility (or simply, mobility), is a quantity relating the drift velocity of electrons to the applied electric field across a material, according to the formula:
where
- is the drift velocity in m/s (SI units) or cm/s (cgs units).
- is the applied electric field in V/m (SI) or statvolt/cm (cgs).
- is the mobility in m2/(V·s), in SI units, or cm2/(statvolt·s), in cgs units.
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Encyclopedia
In physics, electron mobility (or simply, mobility), is a quantity relating the drift velocity of electrons to the applied electric field across a material, according to the formula:
where
- is the drift velocity in m/s (SI units) or cm/s (cgs units).
- is the applied electric field in V/m (SI) or statvolt/cm (cgs).
- is the mobility in m2/(V·s), in SI units, or cm2/(statvolt·s), in cgs units. A mixed mobility unit of 1 cm2/(V·s) = 0.0001 m2/(V·s) is also often used.
It is the application for electrons of the more general phenomenon of electrical mobility of charged particles in a fluid under an applied electric field.
In semiconductors, mobility can apply to electrons as well as to holes.
Conceptual overview
In a solid, electrons (and, in the case of semiconductors, both electrons and holes) will move around randomly in the absence of an applied electric field. Therefore, if one averages the movement over time there will be no overall motion of charge carriers in any particular direction. However upon applying an electric field, electrons will be accelerated in an opposite direction to the electric field. The summation of the time between acceleration of electrons due to electric field and deceleration of electrons due to collisions and lattice scattering events (caused by phonons, crystal defects, impurities, etc.) over the mean free path between scattering events results in the electrons having an average drift velocity. This net electron motion must be orders of magnitude less than the normally occurring random motion, otherwise the mobility equation is not valid (i.e., typical drift speeds in copper being of the order of 10-4 m·s-1 compared to the speed of random electron motion of 105 m·s-1).
In a semiconductor the two charge carriers, electrons and holes, will typically have different drift velocities for the same electric field.
In a plasma there is analogous behavior with ions and free electrons.
In a vacuum, electrons will accelerate continuously in an electric field according to Newton's second law of motion (until they reach a relativistic speed). This is known as "ballistic transport". A steady-state drift velocity is never achieved, and thus electron mobility is undefined for electronic movement in a vacuum.
In a solid, if the electrons must move only a very short distance (distance comparable with the Brownian motion length scale), quasi-ballistic transport is possible.
Since mobility is usually a strong function of material impurities and temperature, and is determined empirically, mobility values are typically presented in table or chart form. Mobility is also different for electrons and holes in a given semiconductor.
An approximation of the mobility function can be written as a combination of influences from lattice vibrations (phonons) and from impurities by the Matthiessen's Rule (developed from work by Augustus Matthiessen in 1864):
.
Mobility in gas phase
Mobility is defined for any species in the gas phase, encountered mostly in plasma physics and is defined as:
where
is the charge of the species,
is the momentum transfer collision frequency, and
is the mass.
Mobility is related to the species' diffusion coefficient through an exact (thermodynamically required) equation known as the Einstein relation:
,
where
is the Boltzmann constant,
is the gas temperature, and
is a measured quantity that can be estimated. If one defines the mean free path in terms of momentum transfer, then one gets:
.
But both the momentum transfer mean free path and the momentum transfer collision frequency are difficult to calculate. Many other mean free paths can be defined. In the gas phase, is often defined as the diffusional mean free path, by assuming a simple approximate relation is exact:
,
when is the root mean square speed of the gas molecules:
where
is the mass of the diffusing species. This approximate equation becomes exact when used to define the diffusional mean free path.
Examples
Typical electron mobility for Si at room temperature (300 K) is 1400 cm2/ (V·s) and the hole mobility is around 450 cm2/ (V·s).
Very high mobility has been found in several low-dimensional systems, such as two-dimensional electron gases (2DEG) (3,000,000 cm2/ V·s at low temperature), carbon nanotubes (100,000 cm2/ V·s at room temperature) and more recently, graphene (200,000cm2/ V·s at low temperature).
Organic semiconductors (polymer, oligomer) developed thus far have carrier mobilities below 10 cm2/(V·s), and usually much lower.
Relation to conductivity
There is a simple relation between mobility and electrical conductivity. Let n be the number density of electrons, and let µ be their mobility. In the electric field E, each of these electrons will move with the velocity vector -µE, for a total current density of neµE (where e is the elementary charge). Therefore, the electrical conductivity s satisfies:
When there is more than one species (e.g., a plasma with electrons and ions, or a semiconductor with electrons and holes), the total conductivity is
where the ith species has number density ni, charge qi, and mobility µi.
See also
External links
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