534
21 Diodes
assuming all donors are ionized (otherwise N D shall be replaced by the electron density n, not to be
confused here with the ideality factor). Accordingly, the saturation current density (21.49) is rewritten as
j s,f = A
∗ T
2 exp
−
F
f
B
kT
(21.55)
Laterally Inhomogeneous Barrier
The increase of the ideality factor due to a spatially inhomogeneous barrier height has been proposed
early on [1715–1719]. The barrier height F Bn (y, z) across the contact area is typically assumed to have
a Gaussian probability distribution [1720] p(F Bn ) with a mean value ¯
F Bn and a standard deviation σ F .
It turns out that the barrier height F
C
Bn responsible for the capacitance, and thus the diffusion voltage
determined by C–V spectroscopy, is given by the spatial average, i.e. F
C
Bn = ¯
F Bn . The barrier height
F
j
Bn determining the current-voltage characteristics (cf. (21.48)) via
j = A
∗ T
2
exp (βV ) − 1
exp
−
F Bn
kT
p(F Bn ) dF Bn
(21.56)
= A
∗ T
2 exp
−
F
j
Bn
kT
exp (βV ) − 1
is given by
F
j
Bn = ¯
F Bn −
σ
2
F
2 kT
.
(21.57)
Thus, the barrier height determined from the current-voltage characteristic underestimates the spatial
average of the barrier height.
10 The Richardson plot (21.50) is now modified (and is nonlinear in
1/T ) to
ln
j s
T 2
= ln A
∗
−
F Bn
kT
+
σ
2
F
2 k 2 T 2 .
(21.58)
In [1721] the barrier nonuniformity due to the random distribution and discreteness of impurity charges
in the depletion region is evaluated. This mechanism, yielding increasing barrier inhomogeneity for
larger doping, represent a fundamental limit to the ideality of a Schottky diode.
If the distribution of barrier heights is discrete, at low voltages the current will first flow at certain
spots with low barrier (‘hot spots’). This leads to kinks in the I –V characteristics as found for SiC
[1722] or ZnO [1723] (Fig. 21.15).
Temperature Dependence
Figure 21.16a shows the temperature-dependent I –V characteristics of a Pd/ZnO Schottky diode.
A straightforward evaluation according to (21.48) results in a barrier height of about 700 meV and a
Richardson constant that is orders of magnitude smaller than the theoretical value of 32 A K
−2 cm
−2 (for
m
∗
e = 0.27). A fit of the temperature-dependent data with (21.57), as shown in Fig. 21.16b, results in
¯
F Bn = 1.1 eV, in agreement with the (temperature-independent) value obtained from CV spectroscopy,
and σ F = 0.13 eV [1724].
The temperature dependence of the ideality factor n is given by [1725]
10 This phenomenon is similar to the red-shift of luminescence lines (Stokes shift) due to thermalization in the presence
of disorder, see Sect. 12.4.
21 Diodes
assuming all donors are ionized (otherwise N D shall be replaced by the electron density n, not to be
confused here with the ideality factor). Accordingly, the saturation current density (21.49) is rewritten as
j s,f = A
∗ T
2 exp
−
F
f
B
kT
(21.55)
Laterally Inhomogeneous Barrier
The increase of the ideality factor due to a spatially inhomogeneous barrier height has been proposed
early on [1715–1719]. The barrier height F Bn (y, z) across the contact area is typically assumed to have
a Gaussian probability distribution [1720] p(F Bn ) with a mean value ¯
F Bn and a standard deviation σ F .
It turns out that the barrier height F
C
Bn responsible for the capacitance, and thus the diffusion voltage
determined by C–V spectroscopy, is given by the spatial average, i.e. F
C
Bn = ¯
F Bn . The barrier height
F
j
Bn determining the current-voltage characteristics (cf. (21.48)) via
j = A
∗ T
2
exp (βV ) − 1
exp
−
F Bn
kT
p(F Bn ) dF Bn
(21.56)
= A
∗ T
2 exp
−
F
j
Bn
kT
exp (βV ) − 1
is given by
F
j
Bn = ¯
F Bn −
σ
2
F
2 kT
.
(21.57)
Thus, the barrier height determined from the current-voltage characteristic underestimates the spatial
average of the barrier height.
10 The Richardson plot (21.50) is now modified (and is nonlinear in
1/T ) to
ln
j s
T 2
= ln A
∗
−
F Bn
kT
+
σ
2
F
2 k 2 T 2 .
(21.58)
In [1721] the barrier nonuniformity due to the random distribution and discreteness of impurity charges
in the depletion region is evaluated. This mechanism, yielding increasing barrier inhomogeneity for
larger doping, represent a fundamental limit to the ideality of a Schottky diode.
If the distribution of barrier heights is discrete, at low voltages the current will first flow at certain
spots with low barrier (‘hot spots’). This leads to kinks in the I –V characteristics as found for SiC
[1722] or ZnO [1723] (Fig. 21.15).
Temperature Dependence
Figure 21.16a shows the temperature-dependent I –V characteristics of a Pd/ZnO Schottky diode.
A straightforward evaluation according to (21.48) results in a barrier height of about 700 meV and a
Richardson constant that is orders of magnitude smaller than the theoretical value of 32 A K
−2 cm
−2 (for
m
∗
e = 0.27). A fit of the temperature-dependent data with (21.57), as shown in Fig. 21.16b, results in
¯
F Bn = 1.1 eV, in agreement with the (temperature-independent) value obtained from CV spectroscopy,
and σ F = 0.13 eV [1724].
The temperature dependence of the ideality factor n is given by [1725]
10 This phenomenon is similar to the red-shift of luminescence lines (Stokes shift) due to thermalization in the presence
of disorder, see Sect. 12.4.