Imagine a crystal with a random doping profile in one dimension
Figure 14.1 - A Crystal With One Dimensional Random Doping
In thermal equilibrium EF is a constant throughout. The variation of the band edge energy is V(x) and so everywhere we can write the net electron current as
J=-eDe
¶ n
¶ x
+eµen.E=0
[J=(diffusion)+(drift)]
however the Boltzmann distribution say approximately that
n=(constant).e
-
eV(x)
kT
Þ
¶ n
¶ x
=-n
e
kT
.
¶ V(x)
¶ x
where ¶ V(x)/¶ x is the built-in electric field. So the expression for J is
J=
neF.De
kT
-|e|µenEº 0
this must be true everywhere and at all temperatures.
Þ
De
µe
=
kT
e
this is the Einstein Relationship. This is exactly the same for the holes too
Dh
µh
=
kT
e
14.3 Carrier Recombination
In thermal equilibrium np=ni2 so an increase in np above this (eg. by photo-excitation)
Figure 14.2 - Photo-Excitation
This will recombine in a time trecombine until thermal equilibrium is restored. This implies if, for example, light is generating carriers at a rate G ( s-1 ) in the steady state, there will be an everage of D N=Gtrecombine extra electrons in the solid.
