Environmental Engineering Reference
In-Depth Information
1
=
2
W ¼½
2
ee
o
ðV
B
VÞ=
eN
A
:
ð
3
:
63a
Þ
In the depletion region, on the N-side, for example, the charge density is
r¼
þN
D
e
, from which, using the Poisson equation for the electric potential
V
,is
2
r
ðVÞ¼r=ee
o
¼þN
D
e
=ee
o
:
ð
3
:
64
Þ
2
V
/
q
x
2
¼þN
D
e
/
ee
o
, so that the electric
eld
E¼
q
V
/
q
x¼þN
D
ex
/
ee
o
þ c
, with
c
an arbitrary constant. So the electric
eld
increases linearly from zero, at the outer edges of the depletion layer, reaches a
peak at the junction center, where
E
(
x
) must be continuous from right to left. The
potential
V
(
x
), sketched in Figure 3.16c, is therefore quadratic in
x
in the depletion
region, with an in
ection point at the center of the junction.
We now seek to understand the outline of the PN junction in Figure 3.16c. This is
the most important semiconductor device! How does it work?
It is important to understand the role of
minority carriers
,
n
p
electrons in P-region
and
p
n
holes in N-region. (As we will see, these concentrations set the reverse current
density.) From Equation 3.53, we see that these concentrations are small relative to
the majority concentrations, which are essentially set by the levels of dopants,
N
D
on
the N-side and
N
A
on the P-side. The numbers of minority carriers are set by the
equilibrium equations (3.54
-
3.58), but the minority carrier concentration can also be
understood as a balance between a generation rate (a thermal bond-breaking rate) and
a
minority carrier lifetime t
. An electron on the upper left of Figure 3.16c lives only a
short-lifetime
t
before it encounters one of the many holes and recombines
(annihilates) either by emitting a light photon or by giving off other forms of energy.
The minority electrons in diffusion length
L
n
(Figure 3.16c, upper left) can reach the
junction electron field and cross to the other side in the minority carrier lifetime
t
,so
the sum of these two diffusion current densities, comprising the reverse current
density, is given as
For one dimension, this becomes
q
J
rev
¼ eðL
n
n
p
=t
n
ÞþeðL
p
p
n
=t
p
Þ:
ð
3
:
65
Þ
Since the diffusion lengths generally are larger than
W
, the minority carriers are in
a
field-free region and diffuse randomly. Equation 3.64 can be expressed in a slightly
different formbymaking use of theminority carrier diffusion constant
D
, unitsm
2
/s.
D
is closely related to the minority carrier mobility
m
, through the relation
m¼ eD
/
kT
,
with units expressed as (m
2
/Vs). The diffusion length is
L¼
(
Dt
)
1/2
, the distance
traversed in a random walk, so the reverse current can be expressed also as
1
=
2
1
=
2
J
rev
¼ en
p
ðD
n
=t
n
Þ
þep
n
ðD
p
=t
p
Þ
ð
3
:
66
Þ
and a further variation is possible using
D¼ k
B
Tm
/
e
. By using the product rule (3.58),
and setting
p
p
N
A
, and so on, the approximate form (3.67) can be reached.
D
n
N
A
L
n
þ
D
h
N
D
L
h
J
rev
¼ J
o
¼ e n
i
ð
3
:
67
Þ
