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.

Contents 1 Conceptual overview 2 Examples 3 Relation to conductivity 4 Saturated velocity 5 See also 6 References 7 External links

[edit] 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):

.

[edit] 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).^[1]

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)^[2], carbon nanotubes (100,000 cm²/ V·s at room temperature) ^[3] and more recently, graphene (200,000 cm²/ V·s at low temperature)^[4]. Organic semiconductors (polymer, oligomer) developed thus far have carrier mobilities below 10 cm²/(V·s), and usually much lower.

[edit] 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 σ satisfies:^[5]

σ = neμ

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^[5]

σ =	∑	niμi | qi |
	i

where the ith species has number density n_i, charge q_i, and mobility μ_i.

[edit] Saturated velocity

In a semiconductor, carrier velocity can not be indefinitely increased with applied field. Carriers speed up in response to a stronger field until velocity saturation occurs, where higher fields do not result in any increase beyond the saturation drift velocity. This velocity is a characteristic of the material and a strong funtion of doping or impurity levels and temeprature. It is one of the key material and semiconductor device properties that determine a device such as a transistor's ultimate limit of speed of response and frequency.

[edit] See also

• Speed of electricity
• Semiconductor carrier mobility

[edit] References

1. ^ http://www.ioffe.rssi.ru/SVA/NSM/Semicond/Si/electric.html
2. ^ Harris et al., J. Applied Physics 61, no.3 , pp.1219, 1987. doi:10.1063/1.338174
3. ^ Durkop et al., Nano Letters 4, no. 1, pp.35, 2004 doi:10.1021/nl034841q
4. ^ Bolotin et al., http://arxiv.org/abs/0802.2389
5. ^ ^{a b} Online textbook by B. Van Zeghbroeck

[edit] External links

• semiconductor glossary entry for electron mobility
• Bart Van Zeghbroeck's Online Book, Principles of Semiconductor Devices
• Resistivity and Mobility Calculator from the BYU Cleanroom