Page 307 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
electrically disconnected diode is in a delicate dynamic equilibrium.
Here we can use the value of the Fermi energy predicted by (7.152a) only
as a reasonable starting value with which to iterate
2 q + -
(
∇ ψ = --- n – p + N – N ) (7.154)
ε d a
For reasonably doped (non-degenerate) diodes, i.e., where the Fermi
level is not closer than 3k T to either band edge, we can use Boltzmann
B
statistics, giving

2
2 q  E v – E + E g + - 
v
∇ E = ----- N Exp ------( ) – N Exp ----------------------) + N – N (7.155a)
(
v ε  v kT c kT d a 
Otherwise the doping is degenerate, and we have to use Fermi statistics,
so that (7.154) becomes
2
2 q  2 E v – E + E g + - 
v
(
∇ E = ----- N ---F ------( ) – N F ----------------------) + N – N a  (7.155b)
ε 
v
v
c 1
d
1
π --- kT
2 --- 2 kT
∞
⁄
where F x () = ( ( y) ( 1 + Exp y – x))) y is the Fermi integral of
d
(
⁄
12 0 ∫
order 12⁄ . We again have obtained an equation nonlinear in the energy,
but now it also is spatially coupled.
For 1D junctions (7.155a) can be solved numerically in a straightforward
manner, as shown in Figure 7.17. Many of the features of these plots,
however, can be obtained through assumptions that greatly simplify the
calculations. Two models are in use:
• If we assume that the doping profile is abrupt, and that the space-
charge has a piece-wise constant profile (often called a top-hat func-
-
tion). The value of the space charge density is N on the P side and
a
+
N on the N side of the junction. In this case the electric field in the
d
SCL is a piece-wise linear “hat” function, and the potential is piece-
wise quadratic.

304 Semiconductors for Micro and Nanosystem Technology