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The next step in understanding the operation of semiconductors is the
concept of diffusion and drift of the charge carriers.
Note that the nature, amount and regional location of these charge carriers
can be be manipulated according to the intentions of the chip designers.
Diffusion
Whenever there is a concentration gradient, material will move in the direction
of the highest rate of decrease of the gradient.
(As if to remove the concentration difference  see figure 26.)
Figure 26:
Movement of material down a concentration gradient.

Fick's Law
The flux of particles is proportional to the gradient of particles.
(Flux is the number of particles /second /area.)
In the case of charge carriers, the flux is the current density with the appropriate sign.
For example, in the case of electrons as charge carriers :

(106) 
where is the number of charge carriers (now per unit volume)
and is the diffusion coefficient with units :

(107) 
Drift
When an electric field is applied to a semiconductor, the carriers will move at a velocity
that is proportional to the magnitude of the field. This velocity
is called the drift velocity .

(108) 
where is called the mobility, with units

(109) 
and is the electric field.
This constitutes a drift current

(110) 
At equilibrium, the drift and diffusion currents are equal.

(111) 
so

(112) 
Figure 27:
Equilibrium condition for drift and diffusion currents.

We may write the electric field as the gradient of the potential

(113) 
Therefore

(114) 
so

(115) 
This differential equation has the solution

(116) 
for the electron distribution per unit volume.
But because this nonuniform electron distribution has to be
maintained against the potential using thermal energy, we must also have

(117) 
We therefore find the Einstein relations
That is, we can express the diffusion coefficient as

(119) 
a result which we will soon use.
Note
Carrier mobility is very important. It is affected by temperature, doping concentration and the
magnitude of the applied field. It also depends on the effective mass.
Carriers with small effective masses have large mobilities.
As a result, holes are significantly less mobile than electrons.
Next: Junctions, depletion regions, band
Up: From Semiconductivity to Microelectronics
Previous: Fermi statistics, charge carrier
Simon Connell
20041004