562
21 Diodes
Fig. 21.46 Carrier
densities (a, c) and current
densities (b, d) (linear
scales) in a pn diode under
(a, b) forward bias and
(c, d) reverse bias
(a)
(b)
(c)
(d)
p,n
j n
p,n
j ,j
p n
j ,j
p n
j n
j p
j n
j p
p n
-x p
n p
p n
x n
-x p x n
-x p
x n
-x p
x n
p n0
n p
p n0
n p0
n p0
x
x
x
x
From the continuity equation and the boundary condition that far away from the depletion layer the
hole density is p n 0 , the hole density on the n-side is given by
δp n (x) = p n (x) − p n 0 = p n 0
exp (βV ) − 1
exp
−
x − x n
L p
,
(21.128)
where L p =
D p τ p is the hole (minority carrier) diffusion length. In [1763] it is argued from analytical
and numerical analysis that the majority carrier density follows δn n (x) = δp n (x).
The hole current density at the boundary of the depletion layer on the n-side is
j p (x n ) = −e D p
∂ p n
∂ x
| x n =
e D p p n 0
L p
exp (βV ) − 1
.
(21.129)
Similarly, the electron current (again δp p (x) = δn p (x) in the neutral region [1763]) in the depletion
layer is
j n (−x p ) =
e D n n p 0
L n
exp (βV ) − 1
.
(21.130)
The total current due to diffusion is
j d = j p (x n ) + j n (−x p ) = j s
exp (βV ) − 1
,
(21.131)
with the saturation current given by
j
d
s =
e D p p n 0
L p
+
e D n n p 0
L n
.
(21.132)
This dependence represents the ideal diode characteristic and the famous result from Shockley. Here,
only the minority diffusion currents have been considered; a more in-depth discussion is delivered
in [1763] where also the majority carrier density and majority carrier drift and diffusion currents are
analyzed (minority drift currents can be neglected altogether). However, the analysis leads to the same
result (21.131) and (21.132).