Cosmic string

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https://en.wikipedia.org/wiki/Cosmic_string

Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry-breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking was not simply connected. Their existence was first contemplated by the theoretical physicist Tom Kibble in the 1970s.

The formation of cosmic strings is somewhat analogous to the imperfections that form between crystal grains in solidifying liquids, or the cracks that form when water freezes into ice. The phase transitions leading to the production of cosmic strings are likely to have occurred during the earliest moments of the universe's evolution, just after cosmological inflation, and are a fairly generic prediction in both quantum field theory and string theory models of the early universe.

Theories containing cosmic strings

In string theory, the role of cosmic strings can be played by the fundamental strings (or F-strings) themselves that define the theory perturbatively, by D-strings which are related to the F-strings by weak-strong or so called S-duality, or higher-dimensional D-, NS- or M-branes that are partially wrapped on compact cycles associated to extra spacetime dimensions so that only one non-compact dimension remains.

The prototypical example of a quantum field theory with cosmic strings is the Abelian Higgs model. The quantum field theory and string theory cosmic strings are expected to have many properties in common, but more research is needed to determine the precise distinguishing features. The F-strings for instance are fully quantum-mechanical and do not have a classical definition, whereas the field theory cosmic strings are almost exclusively treated classically.

Dimensions

Cosmic strings, if they exist, would be extremely thin with diameters of the same order of magnitude as that of a proton, i.e. ~ 1 fm, or smaller. Given that this scale is much smaller than any cosmological scale, these strings are often studied in the zero-width, or Nambu–Goto approximation. Under this assumption strings behave as one-dimensional objects and obey the Nambu–Goto action, which is classically equivalent to the Polyakov action that defines the bosonic sector of superstring theory.

In field theory, the string width is set by the scale of the symmetry breaking phase transition. In string theory, the string width is set (in the simplest cases) by the fundamental string scale, warp factors (associated to the spacetime curvature of an internal six-dimensional spacetime manifold) and/or the size of internal compact dimensions. (In string theory, the universe is either 10- or 11-dimensional, depending on the strength of interactions and the curvature of spacetime.)

Gravitation

A string is a geometrical deviation from Euclidean geometry in spacetime characterized by an angular deficit: a circle around the outside of a string would comprise a total angle less than 360°. From the general theory of relativity such a geometrical defect must be in tension, and would be manifested by mass. Even though cosmic strings are thought to be extremely thin, they would have immense density, and so would represent significant gravitational wave sources. A cosmic string about a kilometer in length may be more massive than the Earth.

However general relativity predicts that the gravitational potential of a straight string vanishes: there is no gravitational force on static surrounding matter. The only gravitational effect of a straight cosmic string is a relative deflection of matter (or light) passing the string on opposite sides (a purely topological effect). A closed cosmic string gravitates in a more conventional way.

During the expansion of the universe, cosmic strings would form a network of loops, and in the past it was thought that their gravity could have been responsible for the original clumping of matter into galactic superclusters. It is now calculated that their contribution to the structure formation in the universe is less than 10%.

Negative mass cosmic string

The standard model of a cosmic string is a geometrical structure with an angle deficit, which thus is in tension and hence has positive mass. In 1995, Visser et al. proposed that cosmic strings could theoretically also exist with angle excesses, and thus negative tension and hence negative mass. The stability of such exotic matter strings is problematic; however, they suggested that if a negative mass string were to be wrapped around a wormhole in the early universe, such a wormhole could be stabilized sufficiently to exist in the present day.

Super-critical cosmic string

The exterior geometry of a (straight) cosmic string can be visualized in an embedding diagram as follows: Focusing on the two-dimensional surface perpendicular to the string, its geometry is that of a cone which is obtained by cutting out a wedge of angle δ and gluing together the edges. The angular deficit δ is linearly related to the string tension (= mass per unit length), i.e. the larger the tension, the steeper the cone. Therefore, δ reaches 2π for a certain critical value of the tension, and the cone degenerates to a cylinder. (In visualizing this setup one has to think of a string with a finite thickness.) For even larger, "super-critical" values, δ exceeds 2π and the (two-dimensional) exterior geometry closes up (it becomes compact), ending in a conical singularity.

However, this static geometry is unstable in the super-critical case (unlike for sub-critical tensions): Small perturbations lead to a dynamical spacetime which expands in axial direction at a constant rate. The 2D exterior is still compact, but the conical singularity can be avoided, and the embedding picture is that of a growing cigar. For even larger tensions (exceeding the critical value by approximately a factor of 1.6), the string cannot be stabilized in radial direction anymore.

Realistic cosmic strings are expected to have tensions around 6 orders of magnitude below the critical value, and are thus always sub-critical. However, the inflating cosmic string solutions might be relevant in the context of brane cosmology, where the string is promoted to a 3-brane (corresponding to our universe) in a six-dimensional bulk.

Observational evidence

It was once thought that the gravitational influence of cosmic strings might contribute to the large-scale clumping of matter in the universe, but all that is known today through galaxy surveys and precision measurements of the cosmic microwave background (CMB) fits an evolution out of random, gaussian fluctuations. These precise observations therefore tend to rule out a significant role for cosmic strings and currently it is known that the contribution of cosmic strings to the CMB cannot be more than 10%.

The violent oscillations of cosmic strings generically lead to the formation of cusps and kinks. These in turn cause parts of the string to pinch off into isolated loops. These loops have a finite lifespan and decay (primarily) via gravitational radiation. This radiation which leads to the strongest signal from cosmic strings may in turn be detectable in gravitational wave observatories. An important open question is to what extent do the pinched off loops backreact or change the initial state of the emitting cosmic string—such backreaction effects are almost always neglected in computations and are known to be important, even for order of magnitude estimates.

Gravitational lensing of a galaxy by a straight section of a cosmic string would produce two identical, undistorted images of the galaxy. In 2003 a group led by Mikhail Sazhin reported the accidental discovery of two seemingly identical galaxies very close together in the sky, leading to speculation that a cosmic string had been found. However, observations by the Hubble Space Telescope in January 2005 showed them to be a pair of similar galaxies, not two images of the same galaxy. A cosmic string would produce a similar duplicate image of fluctuations in the cosmic microwave background, which it was thought might have been detectable by the Planck Surveyor mission. However, a 2013 analysis of data from the Planck mission failed to find any evidence of cosmic strings.

A piece of evidence supporting cosmic string theory is a phenomenon noticed in observations of the "double quasar" called Q0957+561A,B. Originally discovered by Dennis Walsh, Bob Carswell, and Ray Weymann in 1979, the double image of this quasar is caused by a galaxy positioned between it and the Earth. The gravitational lens effect of this intermediate galaxy bends the quasar's light so that it follows two paths of different lengths to Earth. The result is that we see two images of the same quasar, one arriving a short time after the other (about 417.1 days later). However, a team of astronomers at the Harvard-Smithsonian Center for Astrophysics led by Rudolph Schild studied the quasar and found that during the period between September 1994 and July 1995 the two images appeared to have no time delay; changes in the brightness of the two images occurred simultaneously on four separate occasions. Schild and his team believe that the only explanation for this observation is that a cosmic string passed between the Earth and the quasar during that time period traveling at very high speed and oscillating with a period of about 100 days.

Currently the most sensitive bounds on cosmic string parameters come from the non-detection of gravitational waves by pulsar timing array data. The earthbound Laser Interferometer Gravitational-Wave Observatory (LIGO) and especially the space-based gravitational wave detector Laser Interferometer Space Antenna (LISA) will search for gravitational waves and are likely to be sensitive enough to detect signals from cosmic strings, provided the relevant cosmic string tensions are not too small.

String theory and cosmic strings

During the early days of string theory both string theorists and cosmic string theorists believed that there was no direct connection between superstrings and cosmic strings (the names were chosen independently by analogy with ordinary string). The possibility of cosmic strings being produced in the early universe was first envisioned by quantum field theorist Tom Kibble in 1976, and this sprouted the first flurry of interest in the field. In 1985, during the first superstring revolution, Edward Witten contemplated on the possibility of fundamental superstrings having been produced in the early universe and stretched to macroscopic scales, in which case (following the nomenclature of Tom Kibble) they would then be referred to as cosmic superstrings. He concluded that had they been produced they would have either disintegrated into smaller strings before ever reaching macroscopic scales (in the case of Type I superstring theory), they would always appear as boundaries of domain walls whose tension would force the strings to collapse rather than grow to cosmic scales (in the context of heterotic superstring theory), or having a characteristic energy scale close to the Planck energy they would be produced before cosmological inflation and hence be diluted away with the expansion of the universe and not be observable.

Much has changed since these early days, primarily due to the second superstring revolution. It is now known that string theory in addition to the fundamental strings which define the theory perturbatively also contains other one-dimensional objects, such as D-strings, and higher-dimensional objects such as D-branes, NS-branes and M-branes partially wrapped on compact internal spacetime dimensions, while being spatially extended in one non-compact dimension. The possibility of large compact dimensions and large warp factors allows strings with tension much lower than the Planck scale. Furthermore, various dualities that have been discovered point to the conclusion that actually all these apparently different types of string are just the same object as it appears in different regions of parameter space. These new developments have largely revived interest in cosmic strings, starting in the early 2000s.

In 2002, Henry Tye and collaborators predicted the production of cosmic superstrings during the last stages of brane inflation, a string theory construction of the early universe that gives leads to an expanding universe and cosmological inflation. It was subsequently realized by string theorist Joseph Polchinski that the expanding Universe could have stretched a "fundamental" string (the sort which superstring theory considers) until it was of intergalactic size. Such a stretched string would exhibit many of the properties of the old "cosmic" string variety, making the older calculations useful again. As theorist Tom Kibble remarks, "string theory cosmologists have discovered cosmic strings lurking everywhere in the undergrowth". Older proposals for detecting cosmic strings could now be used to investigate superstring theory.

Superstrings, D-strings or the other stringy objects mentioned above stretched to intergalactic scales would radiate gravitational waves, which could be detected using experiments like LIGO and especially the space-based gravitational wave experiment LISA. They might also cause slight irregularities in the cosmic microwave background, too subtle to have been detected yet but possibly within the realm of future observability.

Note that most of these proposals depend, however, on the appropriate cosmological fundamentals (strings, branes, etc.), and no convincing experimental verification of these has been confirmed to date. Cosmic strings nevertheless provide a window into string theory. If cosmic strings are observed which is a real possibility for a wide range of cosmological string models this would provide the first experimental evidence of a string theory model underlying the structure of spacetime.

Electron mobility

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electron_mobility

In solid-state physics, the electron mobility characterises how quickly an electron can move through a metal or semiconductor when pulled by an electric field. There is an analogous quantity for holes, called hole mobility. The term carrier mobility refers in general to both electron and hole mobility.

Electron and hole mobility are special cases of electrical mobility of charged particles in a fluid under an applied electric field.

When an electric field E is applied across a piece of material, the electrons respond by moving with an average velocity called the drift velocity, ${\displaystyle v_d}$. Then the electron mobility μ is defined as

$${\displaystyle v_d=\mu E.}$$

Electron mobility is almost always specified in units of cm²/(V⋅s). This is different from the SI unit of mobility, m²/(V⋅s). They are related by 1 m²/(V⋅s) = 10⁴ cm²/(V⋅s).

Conductivity is proportional to the product of mobility and carrier concentration. For example, the same conductivity could come from a small number of electrons with high mobility for each, or a large number of electrons with a small mobility for each. For semiconductors, the behavior of transistors and other devices can be very different depending on whether there are many electrons with low mobility or few electrons with high mobility. Therefore mobility is a very important parameter for semiconductor materials. Almost always, higher mobility leads to better device performance, with other things equal.

Semiconductor mobility depends on the impurity concentrations (including donor and acceptor concentrations), defect concentration, temperature, and electron and hole concentrations. It also depends on the electric field, particularly at high fields when velocity saturation occurs. It can be determined by the Hall effect, or inferred from transistor behavior.

Introduction

Drift velocity in an electric field

Main article: Drift velocity

Without any applied electric field, in a solid, electrons and holes move around randomly. Therefore, on average there will be no overall motion of charge carriers in any particular direction over time.

However, when an electric field is applied, each electron or hole is accelerated by the electric field. If the electron were in a vacuum, it would be accelerated to ever-increasing velocity (called ballistic transport). However, in a solid, the electron repeatedly scatters off crystal defects, phonons, impurities, etc., so that it loses some energy and changes direction. The final result is that the electron moves with a finite average velocity, called the drift velocity. This net electron motion is usually much slower than the normally occurring random motion.

The two charge carriers, electrons and holes, will typically have different drift velocities for the same electric field.

Quasi-ballistic transport is possible in solids if the electrons are accelerated across a very small distance (as small as the mean free path), or for a very short time (as short as the mean free time). In these cases, drift velocity and mobility are not meaningful.

Definition and units

See also: Electrical mobility

The electron mobility is defined by the equation:

$${\displaystyle v_d={\mu }_{e}E.}$$

where:

• E is the magnitude of the electric field applied to a material,
• v_d is the magnitude of the electron drift velocity (in other words, the electron drift speed) caused by the electric field, and
• µ_e is the electron mobility.

The hole mobility is defined by a similar equation:

$${\displaystyle v_d={\mu }_{h}E.}$$

Both electron and hole mobilities are positive by definition.

Usually, the electron drift velocity in a material is directly proportional to the electric field, which means that the electron mobility is a constant (independent of the electric field). When this is not true (for example, in very large electric fields), mobility depends on the electric field.

The SI unit of velocity is m/s, and the SI unit of electric field is V/m. Therefore the SI unit of mobility is (m/s)/(V/m) = m²/(V⋅s). However, mobility is much more commonly expressed in cm²/(V⋅s) = 10⁻⁴ m²/(V⋅s).

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 material.

Derivation

Starting with Newton's Second Law:

$${\displaystyle a=F/m_e^{\ast }}$$

where:

• a is the acceleration between collisions.
• F is the electric force exerted by the electric field, and
• ${\displaystyle m_e^{\ast }}$ is the effective mass of an electron.

Since the force on the electron is −eE:

$${\displaystyle a=-\frac{eE}{m_e^{\ast }}}$$

This is the acceleration on the electron between collisions. The drift velocity is therefore:

$${\displaystyle v_d=a{\tau }_c=-\frac{e{\tau }_c}{m_e^{\ast }}E,}$$

where ${\displaystyle {\tau }_c}$ is the mean free time

Since we only care about how the drift velocity changes with the electric field, we lump the loose terms together to get

$${\displaystyle v_d=-{\mu }_{e}E,}$$

where ${\displaystyle {\mu }_e=\frac{e{\tau }_c}{m_e^{\ast }}}$

Similarly, for holes we have

$${\displaystyle v_d={\mu }_{h}E,}$$

where ${\displaystyle {\mu }_h=\frac{e{\tau }_c}{m_h^{\ast }}}$ Note that both electron mobility and hole mobility are positive. A minus sign is added for electron drift velocity to account for the minus charge.

Relation to current density

The drift current density resulting from an electric field can be calculated from the drift velocity. Consider a sample with cross-sectional area A, length l and an electron concentration of n. The current carried by each electron must be ${\displaystyle -e{v}_d}$, so that the total current density due to electrons is given by:

$${\displaystyle J_e=\frac{I_n}{A}=-en{v}_d}$$

Using the expression for ${\displaystyle v_d}$ gives

$${\displaystyle J_e=en{\mu }_{e}E}$$

A similar set of equations applies to the holes, (noting that the charge on a hole is positive). Therefore the current density due to holes is given by

$${\displaystyle J_h=ep{\mu }_{h}E}$$

where p is the hole concentration and ${\displaystyle {\mu }_h}$ the hole mobility.

The total current density is the sum of the electron and hole components:

$${\displaystyle J=J_e+J_h=(en{\mu }_e+ep{\mu }_h)E}$$

Relation to conductivity

We have previously derived the relationship between electron mobility and current density

$${\displaystyle J=J_e+J_h=(en{\mu }_e+ep{\mu }_h)E}$$

Now Ohm's Law can be written in the form

$${\displaystyle J=\sigma E}$$

where ${\displaystyle \sigma }$ is defined as the conductivity. Therefore we can write down:

$${\displaystyle \sigma =en{\mu }_e+ep{\mu }_h}$$

which can be factorised to

$${\displaystyle \sigma =e(n{\mu }_e+p{\mu }_h)}$$

Relation to electron diffusion

In a region where n and p vary with distance, a diffusion current is superimposed on that due to conductivity. This diffusion current is governed by Fick's Law:

$${\displaystyle F=-D_{\text{e}}\mathrm{\nabla }n}$$

where:

• F is flux.
• D_e is the diffusion coefficient or diffusivity
• ${\displaystyle \mathrm{\nabla }n}$ is the concentration gradient of electrons

The diffusion coefficient for a charge carrier is related to its mobility by the Einstein relation:

$${\displaystyle D_{\text{e}}=\frac{{\mu }_{\text{e}}{k}_{\mathrm{B}}T}{e}}$$

where:

• k_B is the Boltzmann constant
• T is the absolute temperature
• e is the electric charge of an electron

Examples

Typical electron mobility at room temperature (300 K) in metals like gold, copper and silver is 30–50 cm²/ (V⋅s). Carrier mobility in semiconductors is doping dependent. In silicon (Si) the electron mobility is of the order of 1,000, in germanium around 4,000, and in gallium arsenide up to 10,000 cm²/ (V⋅s). Hole mobilities are generally lower and range from around 100 cm²/ (V⋅s) in gallium arsenide, to 450 in silicon, and 2,000 in germanium.

Very high mobility has been found in several ultrapure low-dimensional systems, such as two-dimensional electron gases (2DEG) (35,000,000 cm²/(V⋅s) at low temperature), carbon nanotubes (100,000 cm²/(V⋅s) at room temperature) and freestanding graphene (200,000 cm²/ V⋅s at low temperature). Organic semiconductors (polymer, oligomer) developed thus far have carrier mobilities below 50 cm²/(V⋅s), and typically below 1, with well performing materials measured below 10.

List of highest measured mobilities [cm²/ (V⋅s)]

Material	Electron mobility	Hole mobility
AlGaAs/GaAs heterostructures	35,000,000
Freestanding Graphene	200,000
Carbon nanotubes	79,000
Cubic boron arsenide (c-BAs)	1,600
Crystalline silicon	1,400	450
Polycrystalline silicon	100
Metals (Al, Au, Cu, Ag)	10-50
2D Material (MoS2)	10-50
Organics	8.6	43
Amorphous silicon	~1

Electric field dependence and velocity saturation

Main article: Velocity saturation

At low fields, the drift velocity v_d is proportional to the electric field E, so mobility μ is constant. This value of μ is called the low-field mobility.

As the electric field is increased, however, the carrier velocity increases sublinearly and asymptotically towards a maximum possible value, called the saturation velocity v_sat. For example, the value of v_sat is on the order of 1×10⁷ cm/s for both electrons and holes in Si. It is on the order of 6×10⁶ cm/s for Ge. This velocity is a characteristic of the material and a strong function of doping or impurity levels and temperature. 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.

This velocity saturation phenomenon results from a process called optical phonon scattering. At high fields, carriers are accelerated enough to gain sufficient kinetic energy between collisions to emit an optical phonon, and they do so very quickly, before being accelerated once again. The velocity that the electron reaches before emitting a phonon is:

$${\displaystyle \frac{m^{\ast }{v}_{\text{emit}}^2}{2}\approx \hslash {\omega }_{\text{phonon (opt.)}}}$$

where ω_phonon(opt.) is the optical-phonon angular frequency and m* the carrier effective mass in the direction of the electric field. The value of E_{phonon (opt.)} is 0.063 eV for Si and 0.034 eV for GaAs and Ge. The saturation velocity is only one-half of v_emit, because the electron starts at zero velocity and accelerates up to v_emit in each cycle. (This is a somewhat oversimplified description.)

Velocity saturation is not the only possible high-field behavior. Another is the Gunn effect, where a sufficiently high electric field can cause intervalley electron transfer, which reduces drift velocity. This is unusual; increasing the electric field almost always increases the drift velocity, or else leaves it unchanged. The result is negative differential resistance.

In the regime of velocity saturation (or other high-field effects), mobility is a strong function of electric field. This means that mobility is a somewhat less useful concept, compared to simply discussing drift velocity directly.

Relation between scattering and mobility

Recall that by definition, mobility is dependent on the drift velocity. The main factor determining drift velocity (other than effective mass) is scattering time, i.e. how long the carrier is ballistically accelerated by the electric field until it scatters (collides) with something that changes its direction and/or energy. The most important sources of scattering in typical semiconductor materials, discussed below, are ionized impurity scattering and acoustic phonon scattering (also called lattice scattering). In some cases other sources of scattering may be important, such as neutral impurity scattering, optical phonon scattering, surface scattering, and defect scattering.

Elastic scattering means that energy is (almost) conserved during the scattering event. Some elastic scattering processes are scattering from acoustic phonons, impurity scattering, piezoelectric scattering, etc. In acoustic phonon scattering, electrons scatter from state k to k', while emitting or absorbing a phonon of wave vector q. This phenomenon is usually modeled by assuming that lattice vibrations cause small shifts in energy bands. The additional potential causing the scattering process is generated by the deviations of bands due to these small transitions from frozen lattice positions.

Ionized impurity scattering

Semiconductors are doped with donors and/or acceptors, which are typically ionized, and are thus charged. The Coulombic forces will deflect an electron or hole approaching the ionized impurity. This is known as ionized impurity scattering. The amount of deflection depends on the speed of the carrier and its proximity to the ion. The more heavily a material is doped, the higher the probability that a carrier will collide with an ion in a given time, and the smaller the mean free time between collisions, and the smaller the mobility. When determining the strength of these interactions due to the long-range nature of the Coulomb potential, other impurities and free carriers cause the range of interaction with the carriers to reduce significantly compared to bare Coulomb interaction.

If these scatterers are near the interface, the complexity of the problem increases due to the existence of crystal defects and disorders. Charge trapping centers that scatter free carriers form in many cases due to defects associated with dangling bonds. Scattering happens because after trapping a charge, the defect becomes charged and therefore starts interacting with free carriers. If scattered carriers are in the inversion layer at the interface, the reduced dimensionality of the carriers makes the case differ from the case of bulk impurity scattering as carriers move only in two dimensions. Interfacial roughness also causes short-range scattering limiting the mobility of quasi-two-dimensional electrons at the interface.

Lattice (phonon) scattering

At any temperature above absolute zero, the vibrating atoms create pressure (acoustic) waves in the crystal, which are termed phonons. Like electrons, phonons can be considered to be particles. A phonon can interact (collide) with an electron (or hole) and scatter it. At higher temperature, there are more phonons, and thus increased electron scattering, which tends to reduce mobility.

Piezoelectric scattering

Piezoelectric effect can occur only in compound semiconductor due to their polar nature. It is small in most semiconductors but may lead to local electric fields that cause scattering of carriers by deflecting them, this effect is important mainly at low temperatures where other scattering mechanisms are weak. These electric fields arise from the distortion of the basic unit cell as strain is applied in certain directions in the lattice.

Surface roughness scattering

Surface roughness scattering caused by interfacial disorder is short range scattering limiting the mobility of quasi-two-dimensional electrons at the interface. From high-resolution transmission electron micrographs, it has been determined that the interface is not abrupt on the atomic level, but actual position of the interfacial plane varies one or two atomic layers along the surface. These variations are random and cause fluctuations of the energy levels at the interface, which then causes scattering.

Alloy scattering

In compound (alloy) semiconductors, which many thermoelectric materials are, scattering caused by the perturbation of crystal potential due to the random positioning of substituting atom species in a relevant sublattice is known as alloy scattering. This can only happen in ternary or higher alloys as their crystal structure forms by randomly replacing some atoms in one of the sublattices (sublattice) of the crystal structure. Generally, this phenomenon is quite weak but in certain materials or circumstances, it can become dominant effect limiting conductivity. In bulk materials, interface scattering is usually ignored.

Inelastic scattering

During inelastic scattering processes, significant energy exchange happens. As with elastic phonon scattering also in the inelastic case, the potential arises from energy band deformations caused by atomic vibrations. Optical phonons causing inelastic scattering usually have the energy in the range 30-50 meV, for comparison energies of acoustic phonon are typically less than 1 meV but some might have energy in order of 10 meV. There is significant change in carrier energy during the scattering process. Optical or high-energy acoustic phonons can also cause intervalley or interband scattering, which means that scattering is not limited within single valley.

Electron–electron scattering

Due to the Pauli exclusion principle, electrons can be considered as non-interacting if their density does not exceed the value 10¹⁶~10¹⁷ cm⁻³ or electric field value 10³ V/cm. However, significantly above these limits electron–electron scattering starts to dominate. Long range and nonlinearity of the Coulomb potential governing interactions between electrons make these interactions difficult to deal with.

Relation between mobility and scattering time

A simple model gives the approximate relation between scattering time (average time between scattering events) and mobility. It is assumed that after each scattering event, the carrier's motion is randomized, so it has zero average velocity. After that, it accelerates uniformly in the electric field, until it scatters again. The resulting average drift mobility is:

$${\displaystyle \mu =\frac{q}{m^{\ast }}\overline{\tau }}$$

where q is the elementary charge, m* is the carrier effective mass, and τ is the average scattering time.

If the effective mass is anisotropic (direction-dependent), m* is the effective mass in the direction of the electric field.

Matthiessen's rule

Normally, more than one source of scattering is present, for example both impurities and lattice phonons. It is normally a very good approximation to combine their influences using "Matthiessen's Rule" (developed from work by Augustus Matthiessen in 1864):

$${\displaystyle \frac{1}{\mu }=\frac{1}{{\mu }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\mu }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}.}$$

where µ is the actual mobility, ${\displaystyle {\mu }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}$ is the mobility that the material would have if there was impurity scattering but no other source of scattering, and ${\displaystyle {\mu }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}$ is the mobility that the material would have if there was lattice phonon scattering but no other source of scattering. Other terms may be added for other scattering sources, for example

$${\displaystyle \frac{1}{\mu }=\frac{1}{{\mu }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\mu }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}+\frac{1}{{\mu }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{s}}}+\cdots .}$$

Matthiessen's rule can also be stated in terms of the scattering time:

$${\displaystyle \frac{1}{\tau }=\frac{1}{{\tau }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}}}+\frac{1}{{\tau }_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}+\frac{1}{{\tau }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{s}}}+\cdots .}$$

where τ is the true average scattering time and τ_impurities is the scattering time if there was impurity scattering but no other source of scattering, etc.

Matthiessen's rule is an approximation and is not universally valid. This rule is not valid if the factors affecting the mobility depend on each other, because individual scattering probabilities cannot be summed unless they are independent of each other. The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability. For example, lattice scattering alters the average electron velocity (in the electric-field direction), which in turn alters the tendency to scatter off impurities. There are more complicated formulas that attempt to take these effects into account.

Temperature dependence of mobility

Typical temperature dependence of mobility

	Si	Ge	GaAs
Electrons	∝T −2.4	∝T −1.7	∝T −1.0
Holes	∝T −2.2	∝T −2.3	∝T −2.1

With increasing temperature, phonon concentration increases and causes increased scattering. Thus lattice scattering lowers the carrier mobility more and more at higher temperature. Theoretical calculations reveal that the mobility in non-polar semiconductors, such as silicon and germanium, is dominated by acoustic phonon interaction. The resulting mobility is expected to be proportional to T ^−3/2, while the mobility due to optical phonon scattering only is expected to be proportional to T ^−1/2. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.

As $\frac{1}{\tau }\propto ⟨v⟩\mathrm{\Sigma }$, where ${\displaystyle \mathrm{\Sigma }}$ is the scattering cross section for electrons and holes at a scattering center and ${\displaystyle ⟨v⟩}$ is a thermal average (Boltzmann statistics) over all electron or hole velocities in the lower conduction band or upper valence band, temperature dependence of the mobility can be determined. In here, the following definition for the scattering cross section is used: number of particles scattered into solid angle dΩ per unit time divided by number of particles per area per time (incident intensity), which comes from classical mechanics. As Boltzmann statistics are valid for semiconductors ${\displaystyle ⟨v⟩\sim \sqrt{T}}$.

For scattering from acoustic phonons, for temperatures well above Debye temperature, the estimated cross section Σ_ph is determined from the square of the average vibrational amplitude of a phonon to be proportional to T. The scattering from charged defects (ionized donors or acceptors) leads to the cross section ${\displaystyle {\mathrm{\Sigma }}_{\text{def}}\propto {⟨v⟩}^{-4}}$. This formula is the scattering cross section for "Rutherford scattering", where a point charge (carrier) moves past another point charge (defect) experiencing Coulomb interaction.

The temperature dependencies of these two scattering mechanism in semiconductors can be determined by combining formulas for τ, Σ and ${\displaystyle ⟨v⟩}$, to be for scattering from acoustic phonons ${\displaystyle {\mu }_{ph}\sim T^{-3/2}}$ and from charged defects ${\displaystyle {\mu }_{\text{def}}\sim T^{3/2}}$.

The effect of ionized impurity scattering, however, decreases with increasing temperature because the average thermal speeds of the carriers are increased. Thus, the carriers spend less time near an ionized impurity as they pass and the scattering effect of the ions is thus reduced.

These two effects operate simultaneously on the carriers through Matthiessen's rule. At lower temperatures, ionized impurity scattering dominates, while at higher temperatures, phonon scattering dominates, and the actual mobility reaches a maximum at an intermediate temperature.

Disordered Semiconductors

Density of states of a solid possessing a mobility edge, ${\displaystyle E_C}$.

While in crystalline materials electrons can be described by wavefunctions extended over the entire solid, this is not the case in systems with appreciable structural disorder, such as polycrystalline or amorphous semiconductors. Anderson suggested that beyond a critical value of structural disorder, electron states would be localized. Localized states are described as being confined to finite region of real space, normalizable, and not contributing to transport. Extended states are spread over the extent of the material, not normalizable, and contribute to transport. Unlike crystalline semiconductors, mobility generally increases with temperature in disordered semiconductors.

Multiple trapping and release

Mott later developed the concept of a mobility edge. This is an energy ${\displaystyle E_C}$, above which electrons undergo a transition from localized to delocalized states. In this description, termed multiple trapping and release, electrons are only able to travel when in extended states, and are constantly being trapped in, and re-released from, the lower energy localized states. Because the probability of an electron being released from a trap depends on its thermal energy, mobility can be described by an Arrhenius relationship in such a system:

Energy band diagram depicting electron transport under multiple trapping and release.

$${\displaystyle \mu ={\mu }_0\exp \left(-\frac{E_{\text{A}}}{k_{\text{B}}T}\right)}$$

where ${\displaystyle {\mu }_0}$ is a mobility prefactor, ${\displaystyle E_{\text{A}}}$ is activation energy, ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, and ${\displaystyle T}$ is temperature. The activation energy is typically evaluated by measuring mobility as a function of temperature. The Urbach Energy can be used as a proxy for activation energy in some systems.

Variable Range Hopping

At low temperature, or in system with a large degree of structural disorder (such as fully amorphous systems), electrons cannot access delocalized states. In such a system, electrons can only travel by tunnelling for one site to another, in a process called variable range hopping. In the original theory of variable range hopping, as developed by Mott and Davis, the probability ${\displaystyle P_{ij}}$, of an electron hopping from one site ${\displaystyle i}$, to another site ${\displaystyle j}$, depends on their separation in space ${\displaystyle r_{ij}}$, and their separation in energy ${\displaystyle \mathrm{\Delta }{E}_{ij}}$.

$${\displaystyle P_{ij}=P_0\exp \left(-2\alpha r_{ij}-\frac{\mathrm{\Delta }{E}_{ij}}{k_{B}T}\right)}$$

Here ${\displaystyle P_0}$ is a prefactor associated with the phonon frequency in the material, and ${\displaystyle \alpha }$ is the wavefunction overlap parameter. The mobility in a system governed by variable range hopping can be shown to be:

$${\displaystyle \mu ={\mu }_0\exp \left(-{\left[\frac{T_0}{T}\right]}^{-1/(d+1)}\right)}$$

where ${\displaystyle {\mu }_0}$ is a mobility prefactor, ${\displaystyle T_0}$ is a parameter (with dimensions of temperature) that quantifies the width of localized states, and ${\displaystyle d}$ is the dimensionality of the system.

Measurement of semiconductor mobility

Hall mobility

Main article: Hall effect

 

Hall effect measurement setup for holes

 

Hall effect measurement setup for electrons

Carrier mobility is most commonly measured using the Hall effect. The result of the measurement is called the "Hall mobility" (meaning "mobility inferred from a Hall-effect measurement").

Consider a semiconductor sample with a rectangular cross section as shown in the figures, a current is flowing in the x-direction and a magnetic field is applied in the z-direction. The resulting Lorentz force will accelerate the electrons (n-type materials) or holes (p-type materials) in the (−y) direction, according to the right hand rule and set up an electric field ξ_y. As a result there is a voltage across the sample, which can be measured with a high-impedance voltmeter. This voltage, V_H, is called the Hall voltage. V_H is negative for n-type material and positive for p-type material.

Mathematically, the Lorentz force acting on a charge q is given by

For electrons:

$${\displaystyle {\overrightarrow{F}}_{Hn}=-q({\overrightarrow{v}}_n\times {\overrightarrow{B}}_z)}$$

For holes:

$${\displaystyle {\overrightarrow{F}}_{Hp}=+q({\overrightarrow{v}}_p\times {\overrightarrow{B}}_z)}$$

In steady state this force is balanced by the force set up by the Hall voltage, so that there is no net force on the carriers in the y direction. For electrons,

$${\displaystyle {\overrightarrow{F}}_y=(-q){\overrightarrow{\xi }}_y+(-q)[{\overrightarrow{v}}_n\times {\overrightarrow{B}}_z]=0}$$

$${\displaystyle \Rightarrow -q{\xi }_y+q{v}_x{B}_z=0}$$

$${\displaystyle {\xi }_y=v_x{B}_z}$$

For electrons, the field points in the −y direction, and for holes, it points in the +y direction.

The electron current I is given by ${\displaystyle I=-qn{v}_{x}tW}$. Sub v_x into the expression for ξ_y,

$${\displaystyle {\xi }_y=-\frac{IB}{nqtW}=+\frac{R_{Hn}IB}{tW}}$$

where R_Hn is the Hall coefficient for electron, and is defined as

$${\displaystyle R_{Hn}=-\frac{1}{nq}}$$

Since ${\displaystyle {\xi }_y=\frac{V_H}{W}}$

$${\displaystyle R_{Hn}=-\frac{1}{nq}=\frac{V_{Hn}t}{IB}}$$

Similarly, for holes

$${\displaystyle R_{Hp}=\frac{1}{pq}=\frac{V_{Hp}t}{IB}}$$

From the Hall coefficient, we can obtain the carrier mobility as follows:

$${\displaystyle \begin{array}{rl}{\mu }_n& =\left(-nq\right){\mu }_n\left(-\frac{1}{nq}\right)\\ & =-{\sigma }_n{R}_{Hn}\\ & =-\frac{{\sigma }_n{V}_{Hn}t}{IB}\end{array}}$$

Similarly,

$${\displaystyle {\mu }_p=\frac{{\sigma }_p{V}_{Hp}t}{IB}}$$

Here the value of V_Hp (Hall voltage), t (sample thickness), I (current) and B (magnetic field) can be measured directly, and the conductivities σ_n or σ_p are either known or can be obtained from measuring the resistivity.

Field-effect mobility

See also: MOSFET

Not to be confused with Wien effect.

The mobility can also be measured using a field-effect transistor (FET). The result of the measurement is called the "field-effect mobility" (meaning "mobility inferred from a field-effect measurement").

The measurement can work in two ways: From saturation-mode measurements, or linear-region measurements. (See MOSFET for a description of the different modes or regions of operation.)

Using saturation mode

In this technique, for each fixed gate voltage V_GS, the drain-source voltage V_DS is increased until the current I_D saturates. Next, the square root of this saturated current is plotted against the gate voltage, and the slope m_sat is measured. Then the mobility is:

$${\displaystyle \mu =m_{\text{sat}}^2\frac{2L}{W}\frac{1}{C_i}}$$

where L and W are the length and width of the channel and C_i is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode:

$${\displaystyle I_D=\frac{\mu C_i}{2}\frac{W}{L}(V_{GS}-V_{th}{)}^2.}$$

where V_th is the threshold voltage. This approximation ignores the Early effect (channel length modulation), among other things. In practice, this technique may underestimate the true mobility.

Using the linear region

In this technique, the transistor is operated in the linear region (or "ohmic mode"), where V_DS is small and ${\displaystyle I_D\propto V_{GS}}$ with slope m_lin. Then the mobility is:

$${\displaystyle \mu =m_{\text{lin}}\frac{L}{W}\frac{1}{V_{DS}}\frac{1}{C_i}.}$$

This equation comes from the approximate equation for a MOSFET in the linear region:

$${\displaystyle I_D=\mu C_i\frac{W}{L}\left((V_{GS}-V_{th})V_{DS}-\frac{V_{DS}^2}{2}\right)}$$

In practice, this technique may overestimate the true mobility, because if V_DS is not small enough and V_G is not large enough, the MOSFET may not stay in the linear region.

Optical mobility

Electron mobility may be determined from non-contact laser photo-reflectance technique measurements. A series of photo-reflectance measurements are made as the sample is stepped through focus. The electron diffusion length and recombination time are determined by a regressive fit to the data. Then the Einstein relation is used to calculate the mobility.

Terahertz mobility

Electron mobility can be calculated from time-resolved terahertz probe measurement. Femtosecond laser pulses excite the semiconductor and the resulting photoconductivity is measured using a terahertz probe, which detects changes in the terahertz electric field.

Time resolved microwave conductivity (TRMC)

Main article: Time resolved microwave conductivity

A proxy for charge carrier mobility can be evaluated using time-resolved microwave conductivity (TRMC). A pulsed optical laser is used to create electrons and holes in a semiconductor, which are then detected as an increase in photoconductance. With knowledge of the sample absorbance, dimensions, and incident laser fluence, the parameter ${\displaystyle \varphi \mathrm{\Sigma }\mu =\varphi ({\mu }_e+{\mu }_h)}$ can be evaluated, where ${\displaystyle \varphi }$ is the carrier generation yield (between 0 and 1), ${\displaystyle {\mu }_e}$ is the electron mobility and ${\displaystyle {\mu }_h}$ is the hole mobility. ${\displaystyle \varphi \mathrm{\Sigma }\mu }$ has the same dimensions as mobility, but carrier type (electron or hole) is obscured.

Doping concentration dependence in heavily-doped silicon

The charge carriers in semiconductors are electrons and holes. Their numbers are controlled by the concentrations of impurity elements, i.e. doping concentration. Thus doping concentration has great influence on carrier mobility.

While there is considerable scatter in the experimental data, for noncompensated material (no counter doping) for heavily doped substrates (i.e. ${\displaystyle {10}^{18}{\mathrm{c}\mathrm{m}}^{-3}}$ and up), the mobility in silicon is often characterized by the empirical relationship:

$${\displaystyle \mu ={\mu }_o+\frac{{\mu }_1}{1+{\left(\frac{N}{N_{\text{ref}}}\right)}^{\alpha }}}$$

where N is the doping concentration (either N_D or N_A), and N_ref and α are fitting parameters. At room temperature, the above equation becomes:

Majority carriers:

$${\displaystyle {\mu }_n(N_D)=65+\frac{1265}{1+{\left(\frac{N_D}{8.5\times {10}^{16}}\right)}^{0.72}}}$$

$${\displaystyle {\mu }_p(N_A)=48+\frac{447}{1+{\left(\frac{N_A}{6.3\times {10}^{16}}\right)}^{0.76}}}$$

Minority carriers:

$${\displaystyle {\mu }_n(N_A)=232+\frac{1180}{1+{\left(\frac{N_A}{8\times {10}^{16}}\right)}^{0.9}}}$$

$${\displaystyle {\mu }_p(N_D)=130+\frac{370}{1+{\left(\frac{N_D}{8\times {10}^{17}}\right)}^{1.25}}}$$

These equations apply only to silicon, and only under low field.