Editing Electron mobility (section)

==Introduction==
{{Unreferenced section|date=March 2021}}

===Drift velocity in an electric field===
{{Main|Drift velocity}}
Without any applied electric field, in a solid, [[electron]]s  and [[electron hole|holes]] [[Brownian motion|move around randomly]].{{What|reason=Electrons are bound by metallic bond in silicon... how could they move randomly?|date=May 2025}} 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 [[Crystallographic defect|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:
<math display="block">v_d = \mu_e E.</math>
where:
*''E'' is the [[Euclidean vector|magnitude]] of the [[electric field]] applied to a material,
*''v<sub>d</sub>'' is the [[Euclidean vector|magnitude]] of the electron drift velocity (in other words, the electron drift [[speed]]) caused by the electric field, and
*''μ''<sub>e</sub> is the electron mobility.
The hole mobility is defined by a similar equation: 
<math display="block">v_d = \mu_h E.</math>
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 [[Metre per second|m/s]], and the SI unit of electric field is [[volt|V]]/[[metre|m]]. Therefore the SI unit of mobility is (m/s)/(V/m) = [[square metre|m<sup>2</sup>]]/([[volt|V]]⋅[[second|s]]). However, mobility is much more commonly expressed in cm<sup>2</sup>/(V⋅s) = 10<sup>−4</sup> m<sup>2</sup>/(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]]:
<math display="block">a = F/m_e^* </math>
where:
*''a'' is the acceleration between collisions.
*''F'' is the electric force exerted by the electric field, and
*<math>m_e^* </math> is the [[Effective mass (solid-state physics)|effective mass]] of an electron.

Since the force on the electron is −''eE'':
<math display="block">a = -\frac{eE}{m_e^*} </math>

This is the acceleration on the electron between collisions. The drift velocity is therefore:
<math display="block">v_d = a \tau_c = -\frac{e\tau_c}{m_e^*}E,</math> where <math>\tau_c</math> 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 
<math display="block">v_d = -\mu_e E,</math> where <math>\mu_e = \frac{e\tau_c}{m_e^*}</math>

Similarly, for holes we have 
<math display="block">v_d = \mu_h E,</math> where <math>\mu_h = \frac{e\tau_c}{m_h^*}</math>
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 <math>-e v_d</math>, so that the total current density due to electrons is given by:
<math display="block">J_e=\frac{I_n}{A} = - e n v_d</math>
Using the expression for <math>v_d</math> gives
<math display="block">J_e = e n\mu_e E</math>
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
<math display="block">J_h =e p \mu_h E</math>
where p is the hole concentration and <math>\mu_h</math> the hole mobility.

The total current density is the sum of the electron and hole components:
<math display="block">J=J_e+J_h=(en\mu_e+ep\mu_h)E</math>

===Relation to conductivity===
We have previously derived the relationship between electron mobility and current density 
<math display="block">J=J_e+J_h=(en\mu_e+ep\mu_h)E</math>
Now [[Ohm's law]] can be written in the form
<math display="block">J=\sigma E</math>
where <math>\sigma</math> is defined as the conductivity. Therefore we can write down:
<math display="block">\sigma=en\mu_e+ep\mu_h</math>
which can be factorised to
<math display="block">\sigma=e(n\mu_e+p\mu_h)</math>

===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]]:
<math display="block">F=-D_\text{e}\nabla n</math>
where:
*''F'' is flux.
*''D''<sub>e</sub> is the [[diffusion coefficient]] or diffusivity
*<math>\nabla n</math> is the concentration gradient of electrons

The diffusion coefficient for a charge carrier is related to its mobility by the [[Einstein relation (kinetic theory)|Einstein relation]]. For a classical system (e.g. Boltzmann gas), it reads:
<math display="block">D_\text{e} = \frac{\mu_\text{e} k_\mathrm{B} T}{e}</math>
where:
*''k''<sub>B</sub> 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 (Fermi liquid), 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:
<math display="block">D_\text{e} = \frac{\mu_\text{e} E_F}{e}</math>
where:
*''E''<sub>F</sub> is the Fermi energy