Page 310 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Inhomogeneities
(
bi Abrupt
x ( 2 ) = 2ε V ) N a - (7.164a)
------------------------------ --------------------------------
N abrupt
-
q + ( + N )
N N + a
d
d
2ε V ) N +
(
d
bi Abrupt
x ( 2 ) = ------------------------------ -------------------------------- (7.164b)
P abrupt - - +
q N N + N )
(
a a d
As an example we calculate parameters for the simple diode of
- 22 + 22
Figure 7.17, where N = 2.5×10 and N = 5.8×10 . For this case,
a d
the relations (7.159) and predict a built-in potential of V = 0.77 Volt
bi
and a depletion width of W = 0.14µm . The numerical prediction, based
on a self-consistent solution of (7.154), gives V = 0.8 and
bi
W = 0.4µm . The differing accuracy is reasonable, since the built-in
voltage depends on the amount of charge in the depletion region, whereas
the width of the depletion region is more sensitive to the exact distribu-
tion of charge, which we see from Figure 7.17 is more spread out than
the abrupt model assumes. More accuracy can be achieved by assuming a
linear variation of doping and hence of space charge across the junction,
which Figure 7.17 confirms to be closer to reality. In this case we do not
repeat the calculations, but just state the results
(
k T aW)
B
Linear
( V ) = ---------ln ------------------------ (7.165a)
bi Linear
q 2n i
( W) 3 Linear = 12ε ( (7.165b)
--------- V )
qa bi Linear
The two equations are nonlinearly interdependent, and can be solved
graphically, numerically, but most often the values are presented on con-
venient plots for a range of reasonable values of the doping.
Currents In thermal equilibrium the current through the diode is a delicate balance
of motion motivated by the electrostatic force on carriers qE , which we
term the drift current, and motion from a higher to a lower concentration
gradient and thus proportional to ∇– n and ∇– p , which we call the dif-

Semiconductors for Micro and Nanosystem Technology 307