21.2 Metal-Semiconductor Contacts
525
w 0 =
2 s
e N D
V bi − β −1
.
(21.14)
When a potential difference V ext is applied externally to the diode, (21.14) is modified by the change
in the interface boundary condition, ϕ(0) = −V bi + V ext . The band diagram is shown schematically
in Fig. 21.4a for a forward bias and in Fig. 21.4c for a reverse bias. Therefore, we obtain for the
depletion-layer width (within the abrupt approximation)
w(V ext ) =
2 s
e N D
V bi − V ext − β −1
.
(21.15)
Now we can also give explicitly the value of the maximum electric field E m
(at x = 0)
E m = −
2 e N D
s
V bi − V ext − β −1
(21.16)
= −
2
w
V bi − V ext − β
−1
.
We note that so far the barrier height is independent of the applied bias voltage. In the next section, it
is shown that this is actually not the case.
An alternative approach to solve the Poisson equation is to consider the field as a function of the
potential, E(φ). Then,
dE
2
dφ
= 2 E
dE
dφ
,
(21.17)
and
dE
dφ
=
dE
dx
dx
dφ
= −
dE
dx
1
E
= −
d
2
φ
dx 2
1
E
.
(21.18)
Combining these two equations and using (21.7) and (21.11) we find
dE
2
dφ
= −
2 e N D
s
1 − exp(β φ)
.
(21.19)
The integration of this equation yields with the boundary condition E(φ = 0) = 0,
E
2
(φ) = −
2 e N D
s
φ −
exp(β φ)
β
+
1
β
.
(21.20)
At the interface φ(x = 0) = −(V bi − V ), and for sufficient voltage drop in the semiconductor
exp β φ 1, and therefore (21.16) for E m is recovered without using the abrupt approximation or its
correction (21.13).
5 Without approximation,
E m = −
2 e N D
s
V bi − V ext −
1 − exp[−β(V bi − V ext )]
β
.
(21.21)
5 The functional integration method is limited to bijective potentials φ(x), i.e. strictly monotonously falling or rising
potentials [1697] and thus covers monotonously varying doping density within the depletion layer.