526
21 Diodes
21.2.3 Schottky Effect
The barrier height is reduced by the image-charge effect that has been neglected so far. This effect
has been worked out for metals by Schottky [1698] and has been adapted to semiconductors [1699,
1700]. An electron (charge q = −e) at position x in the semiconductor is facing a metal surface at
x = 0 (Fig. 21.8a). The potential distribution of the free charge is modified since the metal surface is
an equipotential surface. The potential distribution outside the metal is identical to that if an image
charge −q was located at −x. This image charge exerts a force (image force F if ) on the electron
F if = −
q
2
16 π π s x 2 ,
(21.22)
where s is again the relative dielectric constant of the semiconductor. In order to bring an electron
from infinity to x the work E if ,
E if =
x
∞
F if dx = −
q
2
16 π π s x
,
(21.23)
is needed. This image potential energy is shown in Fig. 21.8a. The total energy E tot (solid line in
Fig. 21.8a) of the electron in the presence of an electric field E is given by (F B 0 is the relabeled
Schottky barrier (21.3) without image charge effect, q = −e < 0 and E < 0)
E tot = F B 0 − q E x −
q
2
16 π π s x
.
(21.24)
We note that the divergence at x = 0 stems from the continuum idealization of the problem. It can
be removed by shifting the position of the image charge slightly against the interface or by using an
extended charge [1188].
The maximum of this function (dE tot /dx = 0) is at x m ,
(a)
w
E C
eV bi
F B
x m
E F
E F
if
F B0
F Bn
(b)
0
1
2
-1
0
1
V=-V bi
V=0
V=V bi /2
0
)
B0
Fig. 21.8 a Energy of a particle with respect to the metal surface (dashed line), conduction band in semiconductor
depletion layer (dash-dotted line) and combined effect (solid line). The image charge energy lowers the potential barrier
F B0 by the amount if
B to F Bn . b Conduction band on the semiconductor side of a metal-semiconductor junction at
various bias voltages (V = 0, +V bi /2, and −V bi as labeled) taking into account the Schottky effect. The width of the
depletion layer is indicated with a short vertical dashed line. The barrier height without Schottky effect is F B0 . The
dashed line is the situation without Schottky effect for zero bias
21 Diodes
21.2.3 Schottky Effect
The barrier height is reduced by the image-charge effect that has been neglected so far. This effect
has been worked out for metals by Schottky [1698] and has been adapted to semiconductors [1699,
1700]. An electron (charge q = −e) at position x in the semiconductor is facing a metal surface at
x = 0 (Fig. 21.8a). The potential distribution of the free charge is modified since the metal surface is
an equipotential surface. The potential distribution outside the metal is identical to that if an image
charge −q was located at −x. This image charge exerts a force (image force F if ) on the electron
F if = −
q
2
16 π π s x 2 ,
(21.22)
where s is again the relative dielectric constant of the semiconductor. In order to bring an electron
from infinity to x the work E if ,
E if =
x
∞
F if dx = −
q
2
16 π π s x
,
(21.23)
is needed. This image potential energy is shown in Fig. 21.8a. The total energy E tot (solid line in
Fig. 21.8a) of the electron in the presence of an electric field E is given by (F B 0 is the relabeled
Schottky barrier (21.3) without image charge effect, q = −e < 0 and E < 0)
E tot = F B 0 − q E x −
q
2
16 π π s x
.
(21.24)
We note that the divergence at x = 0 stems from the continuum idealization of the problem. It can
be removed by shifting the position of the image charge slightly against the interface or by using an
extended charge [1188].
The maximum of this function (dE tot /dx = 0) is at x m ,
(a)
w
E C
eV bi
F B
x m
E F
E F
if
F B0
F Bn
(b)
0
1
2
-1
0
1
V=-V bi
V=0
V=V bi /2
0
)
B0
Fig. 21.8 a Energy of a particle with respect to the metal surface (dashed line), conduction band in semiconductor
depletion layer (dash-dotted line) and combined effect (solid line). The image charge energy lowers the potential barrier
F B0 by the amount if
B to F Bn . b Conduction band on the semiconductor side of a metal-semiconductor junction at
various bias voltages (V = 0, +V bi /2, and −V bi as labeled) taking into account the Schottky effect. The width of the
depletion layer is indicated with a short vertical dashed line. The barrier height without Schottky effect is F B0 . The
dashed line is the situation without Schottky effect for zero bias