21.2 Metal-Semiconductor Contacts
527
Fig. 21.9 Electric-field
dependence of the image
charge lowering of the
Schottky barrier.
Dash-dotted line is for
vacuum dielectric constant,
dashed line is (21.26) for
r = 12. Adapted
from [1701]
x m =
q
16 π π s E
.
(21.25)
The Schottky barrier F B 0 is thus reduced by
if
B > 0 given by
if
B = −q
q E
4π π s
= 2 q E x m .
(21.26)
With the electrical field in the vicinity of the interface (x m w) given by E m from (21.16), the barrier
reduction is
6
if
B = e
e
3 N D
8 π 2 3
s
V bi − V ext − β
−1
1/4
.
(21.27)
For s = 0 (vacuum) and a field strength of 10
5 V/cm, the maximum position is at x m = 6 nm
and the barrier reduction is
if
B = 0.12 eV. For 10
7 V/cm, x m = 1 nm and
if
B = 1.2 eV. For
semiconductors with r ∼ 10 the effect is smaller (Fig. 21.9). The Schottky effect depends on the bias
voltage as visualized in Fig. 21.8b and therefore the barrier height depends on the applied bias voltage.
21.2.4 Capacitance
The total space charge Q (per unit area) in the semiconductor is (V = V ext )
Q(V ) = e N D w =
2 e N D s
V bi − V − β −1
.
(21.28)
and depends on the external voltage.
For measurement of the depletion layer capacitance first the external dc bias voltage V is set
which defines the extension of the depletion layer. The (differential) capacitance is probed by an ac
voltage with small amplitude δV V . First we assume that the ac frequency ω is small compared to
characteristic time constants of the electrically active impurities (quasi-static capacitance) and discuss
6 The term 3
s is technically s 2
d where d is the image-force dielectric constant. d is equal to s if the transit time of an
electron from the metal to the maximum of the potential energy is sufficiently long to build up the dielectric polarization
of the semiconductor [1681].
527
Fig. 21.9 Electric-field
dependence of the image
charge lowering of the
Schottky barrier.
Dash-dotted line is for
vacuum dielectric constant,
dashed line is (21.26) for
r = 12. Adapted
from [1701]
x m =
q
16 π π s E
.
(21.25)
The Schottky barrier F B 0 is thus reduced by
if
B > 0 given by
if
B = −q
q E
4π π s
= 2 q E x m .
(21.26)
With the electrical field in the vicinity of the interface (x m w) given by E m from (21.16), the barrier
reduction is
6
if
B = e
e
3 N D
8 π 2 3
s
V bi − V ext − β
−1
1/4
.
(21.27)
For s = 0 (vacuum) and a field strength of 10
5 V/cm, the maximum position is at x m = 6 nm
and the barrier reduction is
if
B = 0.12 eV. For 10
7 V/cm, x m = 1 nm and
if
B = 1.2 eV. For
semiconductors with r ∼ 10 the effect is smaller (Fig. 21.9). The Schottky effect depends on the bias
voltage as visualized in Fig. 21.8b and therefore the barrier height depends on the applied bias voltage.
21.2.4 Capacitance
The total space charge Q (per unit area) in the semiconductor is (V = V ext )
Q(V ) = e N D w =
2 e N D s
V bi − V − β −1
.
(21.28)
and depends on the external voltage.
For measurement of the depletion layer capacitance first the external dc bias voltage V is set
which defines the extension of the depletion layer. The (differential) capacitance is probed by an ac
voltage with small amplitude δV V . First we assume that the ac frequency ω is small compared to
characteristic time constants of the electrically active impurities (quasi-static capacitance) and discuss
6 The term 3
s is technically s 2
d where d is the image-force dielectric constant. d is equal to s if the transit time of an
electron from the metal to the maximum of the potential energy is sufficiently long to build up the dielectric polarization
of the semiconductor [1681].