Electron mobility

In solid-state physics, the electron mobility characterizes how quickly an electron can move through a metal or semiconductor when pushed or 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,. Then the electron mobility μ is defined as
Electron mobility is almost always specified in units of cm²/. This is different from the SI unit of mobility, m²/. They are related by 1 m²/ = 10⁴ cm²/.
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, 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

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. 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, or for a very short time. In these cases, drift velocity and mobility are not meaningful.

Definition and units

The electron mobility is defined by the equation:
where:

E is the magnitude of the electric field applied to a material,
v_d is the magnitude of the electron drift velocity caused by the electric field, and
μ_e is the electron mobility.

The hole mobility is defined by a similar equation:
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. When this is not true, 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 / = m²/. However, mobility is much more commonly expressed in cm²/ = 10⁻⁴ m²/.
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:
where:

a is the acceleration between collisions.
F is the electric force exerted by the electric field, and
is the effective mass of an electron.

Since the force on the electron is −eE:
This is the acceleration on the electron between collisions. The drift velocity is therefore:
where 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
where
Similarly, for holes we have
where
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, so that the total current density due to electrons is given by:
Using the expression for gives
A similar set of equations applies to the holes,. Therefore the current density due to holes is given by
where p is the hole concentration and the hole mobility.
The total current density is the sum of the electron and hole components:

Relation to conductivity

We have previously derived the relationship between electron mobility and current density
Now Ohm's law can be written in the form
where is defined as the conductivity. Therefore we can write down:
which can be factorised to

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:
where:

F is flux.
D_e is the diffusion coefficient or diffusivity
is the concentration gradient of electrons

The diffusion coefficient for a charge carrier is related to its mobility by the Einstein relation. For a classical system, it reads:
where:

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

For a metal, described by a Fermi gas, the quantum version of the Einstein relation should be used. Typically, temperature is much smaller than the Fermi Energy, in this case one should use the following formula:
where:

E_F is the Fermi energy
Examples

Typical electron mobility at room temperature in metals like gold, copper and silver is 30–50 cm²/. Carrier mobility in semiconductors is doping dependent. In silicon the electron mobility is of the order of 1,000, in germanium around 4,000, and in gallium arsenide up to 10,000 cm²/.
Hole mobilities are generally lower and range from around 100 cm²/ 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 , carbon nanotubes and freestanding graphene.
Organic semiconductors developed thus far have carrier mobilities below 50 cm²/, and typically below 1, with well performing materials measured below 10.

Electric field dependence and 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:
where ω_phonon is the optical-phonon angular frequency and m* the carrier effective mass in the direction of the electric field. The value of E_phonon 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.
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, 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 is scattering time, i.e. how long the carrier is ballistically accelerated by the electric field until it scatters 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. 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 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.