21.2 Metal-Semiconductor Contacts
537
(a)
(b)
0
1 0
2 0
3 0
40
30
20
10
0
V (meV)
0
Au/
GaAs
V = T
0
k
e
0
0
k
e
Fig. 21.17 a Forward I –V characteristic of Au/GaAs diode for various temperatures. b Temperature dependence of the
voltage V 0 . The experimental data are fitted with T 0 = 45 ±8 K. Adapted from [1726]
Combined Theory
A combination of both theories [1729] considers both mechanisms to be in series. The current can then
be expressed as
j =
eN C v r
1 + v r /v D
exp
−
F Bn
kT
exp (βV ) − 1
(21.62a)
= A
∗∗ T
2
exp (βV ) − 1
(21.62b)
= j s
exp (βV ) − 1
.
Here v r = ¯
v/4 is a ‘recombination velocity’ [1730] at the top of the barrier according to j = v r (n −n 0 ),
n 0 being the equilibrium electron density at the top of the barrier and ¯
v is the average thermal velocity
in the semiconductor. v D is an effective diffusion velocity describing the transport of electrons from
the edge of the depletion layer (x = w) to the top of the barrier (x = x m ). It is defined as
v
−1
D =
w
x m
−e
μ n kT
exp
−
F Bn − E C (x)
kT
dx .
(21.63)
In [1729] μ n has been assumed to be independent of the electric field. This assumption is potentially
not realistic. If v D v r , thermionic theory applies and we obtain (21.48). The case v r v D ∼ μ n E m
relates to diffusion theory and we recover (21.61).
The constant A
∗∗ in (21.62b) is called the effective Richardson constant. Its calculated dependence
on the electric field is shown in Fig. 21.19 for Si. At room temperature for most Ge, Si and GaAs
Schottky diodes the thermionic emission of majority carriers is the dominating process. The effect of
lateral barrier height inhomogeneity in a diffusion model has been discussed in [1731].
537
(a)
(b)
0
1 0
2 0
3 0
40
30
20
10
0
V (meV)
0
Au/
GaAs
V = T
0
k
e
0
0
k
e
Fig. 21.17 a Forward I –V characteristic of Au/GaAs diode for various temperatures. b Temperature dependence of the
voltage V 0 . The experimental data are fitted with T 0 = 45 ±8 K. Adapted from [1726]
Combined Theory
A combination of both theories [1729] considers both mechanisms to be in series. The current can then
be expressed as
j =
eN C v r
1 + v r /v D
exp
−
F Bn
kT
exp (βV ) − 1
(21.62a)
= A
∗∗ T
2
exp (βV ) − 1
(21.62b)
= j s
exp (βV ) − 1
.
Here v r = ¯
v/4 is a ‘recombination velocity’ [1730] at the top of the barrier according to j = v r (n −n 0 ),
n 0 being the equilibrium electron density at the top of the barrier and ¯
v is the average thermal velocity
in the semiconductor. v D is an effective diffusion velocity describing the transport of electrons from
the edge of the depletion layer (x = w) to the top of the barrier (x = x m ). It is defined as
v
−1
D =
w
x m
−e
μ n kT
exp
−
F Bn − E C (x)
kT
dx .
(21.63)
In [1729] μ n has been assumed to be independent of the electric field. This assumption is potentially
not realistic. If v D v r , thermionic theory applies and we obtain (21.48). The case v r v D ∼ μ n E m
relates to diffusion theory and we recover (21.61).
The constant A
∗∗ in (21.62b) is called the effective Richardson constant. Its calculated dependence
on the electric field is shown in Fig. 21.19 for Si. At room temperature for most Ge, Si and GaAs
Schottky diodes the thermionic emission of majority carriers is the dominating process. The effect of
lateral barrier height inhomogeneity in a diffusion model has been discussed in [1731].