21.2 Metal-Semiconductor Contacts
529
N D (w) is evaluated according to (21.31) using w = s /C from (21.29) [1703],
N D
w =
s
C
= −
2
e s
d
dV
1
C 2
−1
,
(21.33)
Using functional integration, the capacitance of a depletion layer can be expressed in terms of ϕ(0) =
−(V bi − V ) without explicit knowledge of the potential ϕ(x) [1697].
8 In the approximation −eϕ(0)
kT we obtain for homogeneous doping as refinement of (21.29)
C =
e N D s
2
V bi − V − β −1
n 0
N D
− ln
n 0
N D
,
(21.34)
where n 0 is the electron concentration in the neutral region. Within this general treatment, the validity
of (21.31) has been confirmed. Also, C does not diverge in the flatband case, for V → V bi , as for the
abrupt approximation but exhibits a maximum [1697].
At a given bias voltage, the charge (ionized donors or acceptors) at the boundary of the space-charge
region is tested by the capacitance measurement. However, this principle works only if the depth of the
space-charge region actually changes with the bias voltage. The method can therefore not be directly
applied to such systems like δ-doped layers or quantum wells.
Frequency and Temperature Dependence
The release of carriers from (and capture on) a donor occurs with a characteristic emission rate g c
(10.43) (capture rate r c ). This is true similarly for acceptors. Therefore, the capacitance depends on
the sampling frequency (Fig. 21.11a). If the capacitance is probed with a frequency much smaller than
the release rate, the system appears to be in equilibrium and has the (quasi-)static capacitance C 0 . If
the probe frequency is much higher, the system cannot follow and the donor does not contribute to the
capacitance. The characteristic frequency ˆ
f at the turning point of C( f ) is [1016, 1017]
2π ˆ
f = 2 g c ,
(21.35)
with corrections of the simple factor of 2 discussed in [1704]. Since the emission rate depends exponentially on temperature, for a given frequency the capacitance depends on temperature [1705]. This is
shown in Fig. 21.11b for ZnO which exhibits several donor levels. At low temperature shallow levels
release their carriers, at higher temperature the deeper levels start to contribute. The DX center in
AlGaAs (see Sect. 7.7.6) has been investigated in [1706] with this technique.
21.2.5 Current-Voltage Characteristic
The current transport through a metal-semiconductor junction is dominated by the majority charge
carriers, i.e. electrons (holes) in the case of an n-type (p-type) semiconductor, respectively.
In Fig. 21.12, the possible transport mechanisms are visualized for an n-type semiconductor.
Thermionic emission ‘above’ the barrier involves the hot electrons from the thermal distribution and
will be important at least at high temperatures. Tunneling ‘through’ the barrier will be important for
thin barriers, i.e. at high doping (w ∝ N
−1/2
D
, cf. (21.15)). ‘Pure’ tunneling for electrons close to the
(quasi-) Fermi level, also called field emission, and thermionic field emission, i.e. tunneling of electrons
8 This is valid as long as ϕ(x) is strictly monotonous.
529
N D (w) is evaluated according to (21.31) using w = s /C from (21.29) [1703],
N D
w =
s
C
= −
2
e s
d
dV
1
C 2
−1
,
(21.33)
Using functional integration, the capacitance of a depletion layer can be expressed in terms of ϕ(0) =
−(V bi − V ) without explicit knowledge of the potential ϕ(x) [1697].
8 In the approximation −eϕ(0)
kT we obtain for homogeneous doping as refinement of (21.29)
C =
e N D s
2
V bi − V − β −1
n 0
N D
− ln
n 0
N D
,
(21.34)
where n 0 is the electron concentration in the neutral region. Within this general treatment, the validity
of (21.31) has been confirmed. Also, C does not diverge in the flatband case, for V → V bi , as for the
abrupt approximation but exhibits a maximum [1697].
At a given bias voltage, the charge (ionized donors or acceptors) at the boundary of the space-charge
region is tested by the capacitance measurement. However, this principle works only if the depth of the
space-charge region actually changes with the bias voltage. The method can therefore not be directly
applied to such systems like δ-doped layers or quantum wells.
Frequency and Temperature Dependence
The release of carriers from (and capture on) a donor occurs with a characteristic emission rate g c
(10.43) (capture rate r c ). This is true similarly for acceptors. Therefore, the capacitance depends on
the sampling frequency (Fig. 21.11a). If the capacitance is probed with a frequency much smaller than
the release rate, the system appears to be in equilibrium and has the (quasi-)static capacitance C 0 . If
the probe frequency is much higher, the system cannot follow and the donor does not contribute to the
capacitance. The characteristic frequency ˆ
f at the turning point of C( f ) is [1016, 1017]
2π ˆ
f = 2 g c ,
(21.35)
with corrections of the simple factor of 2 discussed in [1704]. Since the emission rate depends exponentially on temperature, for a given frequency the capacitance depends on temperature [1705]. This is
shown in Fig. 21.11b for ZnO which exhibits several donor levels. At low temperature shallow levels
release their carriers, at higher temperature the deeper levels start to contribute. The DX center in
AlGaAs (see Sect. 7.7.6) has been investigated in [1706] with this technique.
21.2.5 Current-Voltage Characteristic
The current transport through a metal-semiconductor junction is dominated by the majority charge
carriers, i.e. electrons (holes) in the case of an n-type (p-type) semiconductor, respectively.
In Fig. 21.12, the possible transport mechanisms are visualized for an n-type semiconductor.
Thermionic emission ‘above’ the barrier involves the hot electrons from the thermal distribution and
will be important at least at high temperatures. Tunneling ‘through’ the barrier will be important for
thin barriers, i.e. at high doping (w ∝ N
−1/2
D
, cf. (21.15)). ‘Pure’ tunneling for electrons close to the
(quasi-) Fermi level, also called field emission, and thermionic field emission, i.e. tunneling of electrons
8 This is valid as long as ϕ(x) is strictly monotonous.