Nanotechnology-Enabled Sensors

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Chapter 6: Inorganic Nanotechnology Enabled Sensors
equation are used to overcome this problem. It is also assumed that the
temperature is high enough to ionise all donors (concentration of ionised
donors equals the bulk electron density nb), and all the electrons in the depletion
layer are captured on surface levels.
For obtaining the second relation, two different cases will be considered:
32
Case 1. Large grains/crystallites (d>>λD comparable to the Debye
length)
In this case, only the surface, and not the bulk region, is affected. It is
possible to treat the situation for large grains as dealing with a planar or
quasi-infinite object where qVS is the band bending, z0 is the depth of the
depleted region and A the covered area. In this case, the electroneutrality
and the Poisson equations for energy (E) can be written as (Qss is the surface
charge): 32
αθ [ S ] A A = Q , (6.39)
t
= nb
z0
0
SS
2
2
d E(
z)
q . nb
= . 2
(6.40)
dz εε
The boundary conditions for the Poisson equation are:
It can be shown that:
dE(
z)
dz
z=z
0
E =
= 0 , (6.41)
( z)
E
z=
z C . (6.42)
0
q . nb
E(
z)
= EC
+ .( z − z
2.
ε.
ε
As V = E/q, the potential is found to be:
q n
( = z
2εε
2
V z)
b .( z −
and the surface band bending is:
V
S
2
0
0
0
0
)
2
0
)
2
. (6.43)
, (6.44)
2
q nb
2
= . z0
. (6.45)
2εε