532
21 Diodes
(a)
(b)
Fig. 21.13 Ideal diode I –V characteristics I = I s (exp(eV /kT ) − 1) (a) in linear plot and (b) semilogarithmic plot
with A
∗ being the Richardson constant given by
A
∗
=
4π e m
∗ k
2
B
h 3
=
e N C ¯
v
4 T 2 ,
(21.47)
where ¯
v is the average thermal velocity in the semiconductor. A
∗ for electrons in vacuum is
120 A cm
−2 K
−2 . A similar result is obtained for the thermionic emission of electrons from a metal
(overcoming the work function) into vacuum.
If the bias is changed, the current from the semiconductor to the metal increases in the forward
direction because the energy difference between the quasi-Fermi level and the top of the barrier is
reduced. The current is reduced for reverse bias. The barrier from the metal into the semiconductor
remains constant (except for the Schottky effect whose impact on the current-voltage characteristic is
discussed next). Therefore the current j m→s from the metal into the semiconductor is constant and can
be obtained from the condition j = j s→m + j m→s = 0 for zero bias. Therefore the current-voltage
characteristic in the thermionic-emission model is
j = A
∗ T
2 exp
−
F Bn
kT
exp (βV ) − 1
(21.48)
= j s
exp (βV ) − 1
.
The pre-factor
j s = A
∗ T
2 exp
−
F Bn
kT
(21.49)
is called the saturation current density. The saturation current increases with increasing temperature.
The dependence (21.49) represents the ideal diode characteristic and is plotted in Fig. 21.13.
The temperature dependence of the saturation current j s can be written as
ln
j s
T 2
= ln A
∗
−
F Bn
kT
(21.50)
by transforming (21.49). The plot of ln( j s /T
2
) to versus 1/T is called a Richardson plot and allows
the barrier height and the Richardson constant to be determined from a linear fit (Fig. 21.14b).
21 Diodes
(a)
(b)
Fig. 21.13 Ideal diode I –V characteristics I = I s (exp(eV /kT ) − 1) (a) in linear plot and (b) semilogarithmic plot
with A
∗ being the Richardson constant given by
A
∗
=
4π e m
∗ k
2
B
h 3
=
e N C ¯
v
4 T 2 ,
(21.47)
where ¯
v is the average thermal velocity in the semiconductor. A
∗ for electrons in vacuum is
120 A cm
−2 K
−2 . A similar result is obtained for the thermionic emission of electrons from a metal
(overcoming the work function) into vacuum.
If the bias is changed, the current from the semiconductor to the metal increases in the forward
direction because the energy difference between the quasi-Fermi level and the top of the barrier is
reduced. The current is reduced for reverse bias. The barrier from the metal into the semiconductor
remains constant (except for the Schottky effect whose impact on the current-voltage characteristic is
discussed next). Therefore the current j m→s from the metal into the semiconductor is constant and can
be obtained from the condition j = j s→m + j m→s = 0 for zero bias. Therefore the current-voltage
characteristic in the thermionic-emission model is
j = A
∗ T
2 exp
−
F Bn
kT
exp (βV ) − 1
(21.48)
= j s
exp (βV ) − 1
.
The pre-factor
j s = A
∗ T
2 exp
−
F Bn
kT
(21.49)
is called the saturation current density. The saturation current increases with increasing temperature.
The dependence (21.49) represents the ideal diode characteristic and is plotted in Fig. 21.13.
The temperature dependence of the saturation current j s can be written as
ln
j s
T 2
= ln A
∗
−
F Bn
kT
(21.50)
by transforming (21.49). The plot of ln( j s /T
2
) to versus 1/T is called a Richardson plot and allows
the barrier height and the Richardson constant to be determined from a linear fit (Fig. 21.14b).