﻿ Electron mobility

# Electron mobility

The mobility and mobility as a physical concept is defined by the constant ( fixed ) speed which reaches a body ( asymptotically ) when attacking a constant force at him.

In electrodynamics, the mobility is defined in a slightly different form, and thus with the other unit. The charge carrier mobility, usually referred to simply as mobility, referred to the relationship between the drift velocity of charge carriers and an applied electric field:

Basically, it is useful to introduce a mobility only in dissipative systems, ie where there is friction and thus inelastic scattering. Above a certain speed there is a balance between external force and friction force acting opposite, so that the motion is stationary (more generally, the mean velocity is stationary).

• 2.1 related to conductivity
• 2.2 Microscopic observation
• 2.3 Mobility in Solids
• 2.4 charge carrier mobility of some substances

## Mobility in the mechanics

A constant engaging force acts to a body as long as its acceleration, until the opposing force of friction (for example, air or friction ) is the same amount. Then, the steady state velocity is reached and the actual acceleration is zero. This is for example the reason why a falling body in the atmosphere is not as fast as desired. A cause of this law is the dependence of the friction on the speed of the body.

The mechanical motion is therefore defined as

In mechanics, the mobility thus has the unit s / kg. It is historically interesting that Aristotle has adopted this law as fundamental to his mechanics. The present mechanism, however, is based on the Newtonian axioms, which shows the law.

### Mobility in Stokes friction

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

The resultant force is made up of these two contributions:

At equilibrium, the net force and hence the acceleration is zero and the steady-state speed is reached:

The mobility is therefore

#### Mobility diameter

The mobility of a moving body in a liquid can be expressed also by the mobility- equivalent diameter and mobility diameter. This is the diameter of a sphere which has this mobility. Its value is in accordance with the Stokes' law, the Cunningham correction factor indicative of whether the fluid surrounding the body can be considered as a continuum, as molecular or free therebetween. The crucial factor is the mean free path of the fluid molecules and the mobility diameter of the body.

The constants and were determined empirically and are usually regarded as universally valid.

Applications of this size, especially in the aerosol technique, especially for ultrafine particles.

## Mobility in electrodynamics

In electrodynamics, the mobility is defined in a slightly different form. The charge carrier mobility (or simply mobility, especially for electrons: electron mobility ) is the relationship between an applied electric field and the drift velocity of charge carriers ( hard body: Defekt-/Elektronen, plasma: electrons / ions).

Wherein the unit has. Usually mobility in cm2 / (V · s ) is specified.

At low field strengths, however, is independent of the field strength, at high electric field strengths. The exact behavior is thereby significantly influenced by the material, so, for example, by whether an electric current flows through a solid body or a plasma. For very large field strengths, the average electron velocity no longer increases in solids and reaches the saturation velocity.

For the mobility of ions, see ion mobility.

### Related to conductivity

The electrical conductivity can be summed up with the mobility in combination. For conductive material is the material equation, which links the electric current density to the electric field applied on the electric conductivity:

The second equality holds, using the above definition of the mobility. In general, the current density is the charge density times velocity defined as ( = charge density is the charge density times the charge carriers ):

Thus one can obtain by equating the relationship between conductivity and mobility:

Where q is the electric charge present (not necessarily the elementary charge ) of a carrier (e.g., electron hole ion, charged molecule, etc.) and n is the charge carrier density. In metals, the charge carrier density changes with the temperature slightly, and the conductivity is determined by the temperature-dependent mobility

The conductivity of a semiconductor is composed of the electron density and their mobility and the hole density and their mobility

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

### Microscopic observation

Charge carriers move in a gas or solid state without an electric field in a rule at random, that is, the drift velocity is zero. In the presence of an electric field to move the charges on the other hand with the effective speed along the array, which is significantly lower than the average speed of the individual charges.

According to the Drude model, the drift velocity is equal to

From this one can read off directly the mobility:

Where: charge, mass, average collision time (time between two collisions ). The mean collision time can be written as the ratio of mean free path and average speed:

The average velocity is composed of medium thermal velocity and drift velocity. The drift velocity is much smaller than the thermal speed, so it can be neglected, they are not too large electric field strengths.

A quantum-mechanical consideration by Sommerfeld provides a similar result. There, however, the mass must pass through the effective mass ( may differ by several orders of magnitude of the electron mass ) are replaced. In addition, the mean collision time for the electrons with the Fermi energy must be used. For conductivity ( in degenerate systems, such as metals and highly doped semiconductors ) carry namely only the electrons with energy in the region around the Fermi energy at.

### Mobility in solids

In solids, the mobility strongly depends on the number of defects and the temperature so that it is difficult to specify values ​​. It should be noted that, in contrast to a single body, the velocity of the many existing carrier is randomly distributed. The necessary friction force, which prevents a constant acceleration is represented by the dispersion of voids in the crystal and by phonons. The mean free path is limited by the two spreading mechanisms. The electrons among themselves only rarely and at the lattice atoms scatter actually do not. Approximate way, the mobility as a combination of effects of lattice vibrations ( phonons) and from impurities by the following equation expressing ( Matthiessen's rule):

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

As the quantum mechanical view of summer field shows the mobility of the effective mass depends. It should be noted that the effective mass in the Generally, a tensor, that is direction-dependent. Thus, in single-crystal material, the mobility of the crystal orientation -dependent.

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 mass than holes and thus a higher mobility. If one of the two charge carriers dominated by 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) by foreign atoms suitable nature are deliberately introduced a certain amount of mobile charge carriers whose mobility is reduced, however, because the dopant atoms are impurities. Depending on the doping material created excess electrons ( n- doping) or electron defects (p- doping).

### Charge carrier mobility of some substances

Depending on the material structure, the mobility vary greatly. For example, they achieved the standard material of the electronics, the silicon (Si), only mean values ​​. In the gallium arsenide (GaAs ), however, it is significantly higher, with the result that this material is much higher operating frequencies from him created components permits than silicon, but also the higher material costs.

## Mobility in gas phase

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

Where - charge of the component - the collision frequency, - mass.

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

The diffusion constant, the mean free path length, the Boltzmann's constant and temperature, respectively.

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