Drift velocity

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The drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field. In general, an electron will rattle around in a conductor at the Fermi velocity randomly. An applied electric field will give this random motion a small net velocity in one direction

In a semiconductor, the two main carrier scattering mechanisms are ionized impurity scattering and lattice scattering.

Because current is proportional to drift velocity, which is, in turn, proportional to the magnitude of an external electric field, Ohm's law can be explained in terms of drift velocity.

Drift velocity is expressed in the following equations:

$J_{\it drift} = \sigma \cdot v_{\it avg}$ , where $J d r i f t$ is the current density , $σ$ is charge density in units C/m³, and v_avg is the average velocity of the carriers(drift velocity);
$v_{\it avg} = \mu \cdot E$ , where μ is the electron mobility in m²/(V·s) and E is the electric field in V/m.

[edit] Derivation

To find an equation for drift velocity, one can begin with the very definition of current:

$I = \frac{\Delta Q}{\Delta t}$

where

ΔQ is the small amount of charge that passes through an area in a small unit of time, Δt.

One can relate ΔQ to the motion of charged particles in a wire expecting a dependence on the number density of the charge carriers and using dimensional analysis:

$Δ Q$	$= \left( \mathrm{number \ of \ charged \ particles} \right) \times \left( \mathrm{charge \ per \ particle} \right)$
	$= \left( n A \Delta x \right) q$

where

n is the number of charge carriers per unit volume

A is the cross sectional area

Δx is a small length along the wire

q is the charge of the charge carriers

Now, normally particles move randomly, but under the influence of an electric field in the wire, the charge carriers gain an average velocity in a specific direction. This is what's called drift velocity, v_d. And since Δx = v_d Δt, we can plug it into the above equation.

$\Delta Q = \left( n A v_d \Delta t \right) q$