21.4 Bipolar Diodes
557
Fig. 21.42 Built-in
voltage as a function of
doping for one-sided Si and
GaAs pn diodes
∂
2 V
∂ x 2 = −
e N D
s
, 0 ≤ x ≤ x n
(21.103a)
∂
2 V
∂ x 2 =
e N A
s
, −x p ≤ x ≤ 0 .
(21.103b)
One integration yields (together with the boundary conditions that the field is zero at the boundaries
of the depletion layer) the electric field in the two regions
E(x) =
e
s
N D (x − x n ), 0 ≤ x ≤ x n
(21.104a)
E(x) = −
e
s
N A (x + x p ), −x p ≤ x ≤ 0 .
(21.104b)
The maximum field strength E m is present at x = 0 and is given by
E m = −
e N D x n
s
= −
e N A x p
s
.
(21.105)
The continuity of the field at x = 0 is equivalent to the overall charge neutrality
N D x n = N A x p .
(21.106)
Another integration yields the potential (setting V (x = 0) = 0)
V (x) = −E m
x −
x
2
2x n
, 0 ≤ x ≤ x n
(21.107a)
V (x) = −E m
x +
x
2
2x p
, −x p ≤ x ≤ 0 .
(21.107b)
The built-in potential V bi = V (x n ) − V (−x p ) > 0 is related to the maximum field via
V bi = −
E m w
2
,
(21.108)