552
21 Diodes
(a)
C/Ci
(i)
(iii)
breakdown
+V
V
1.0
0.8
0.6
0.4
0.2
0
Ci
s=0
CFB
0
V min V T
(ii)
capacitance
s=
B
s=2
B
Ci
Cmin
Cmin '
(b)
C/Ci
1.0
0.8
0.6
0.4
0.2
0
0
2
4
-2
-4
Ci
Cmin '
V T
1.0
0.8
0.6
10Hz
10 Hz
2
10 Hz
3
10 Hz
4
10 Hz
5
Fig. 21.36 a Schematic dependence of the capacitance of a MIS diode on the bias for (i) low frequencies, (ii) high
frequencies and (iii) deep depletion. b High-frequency capacitance of a Si/SiO 2 diode. The inset shows the frequency
dependence. Adapted from [1758]
Between the charges −Q s and Q s the field strength E i in the insulator is
E i =
|Q s |
i
.
(21.94)
The voltage drop V i across the insulator is given by
V i = E i d =
|Q s |
C i
.
(21.95)
The total capacitance C of the MIS diode is given by the insulator capacitance in series with the
capacitance C d of the depletion layer
C =
C i C d
C i + C d
.
(21.96)
The capacitance of the space-charge region varies with the applied bias (Fig. 21.36). For forward bias
(accumulation), the capacitance of the space-charge region is high. Therefore, the total capacitance
of the MIS diode is given by the insulator capacitance C ≈ C i . When the voltage is reduced, the
capacitance of the space-charge region drops to C d = s /L D for the flat-band case ( s = 0). For a high
reverse voltage, the semiconductor is inverted at the surface and the space-charge region capacitance
is high again. In this case, the total capacitance is given by C ≈ C i again.
The previous consideration assumes that the charge density in the semiconductor can follow changes
of the bias sufficiently fast.
20 However, the inversion charge must be built up and then also disappear
via recombination that is limited by the recombination time constant τ . For frequencies around τ
−1
or faster, the charge in the inversion layer cannot follow and the capacitance of the semiconductor is
given by the value C d ∼ = s /w m ; w m (Fig. 21.37) is the maximum depletion-layer width present at the
beginning of inversion (cf. (21.10) and (21.88))
w m ∼ =
2 s
e N A
inv
s =
4 s kT
e 2 N A
ln
N A
n i
.
(21.97)
For further increased voltage (into the inversion regime), the electric field is screened by the inversion
charge and the width of the depletion layer remains constant. Therefore, the total capacitance in the
20 Typically, a dc bias voltage V is set and the capacitance is sampled with a small ac voltage of amplitude , with
δV V .
21 Diodes
(a)
C/Ci
(i)
(iii)
breakdown
+V
V
1.0
0.8
0.6
0.4
0.2
0
Ci
s=0
CFB
0
V min V T
(ii)
capacitance
s=
B
s=2
B
Ci
Cmin
Cmin '
(b)
C/Ci
1.0
0.8
0.6
0.4
0.2
0
0
2
4
-2
-4
Ci
Cmin '
V T
1.0
0.8
0.6
10Hz
10 Hz
2
10 Hz
3
10 Hz
4
10 Hz
5
Fig. 21.36 a Schematic dependence of the capacitance of a MIS diode on the bias for (i) low frequencies, (ii) high
frequencies and (iii) deep depletion. b High-frequency capacitance of a Si/SiO 2 diode. The inset shows the frequency
dependence. Adapted from [1758]
Between the charges −Q s and Q s the field strength E i in the insulator is
E i =
|Q s |
i
.
(21.94)
The voltage drop V i across the insulator is given by
V i = E i d =
|Q s |
C i
.
(21.95)
The total capacitance C of the MIS diode is given by the insulator capacitance in series with the
capacitance C d of the depletion layer
C =
C i C d
C i + C d
.
(21.96)
The capacitance of the space-charge region varies with the applied bias (Fig. 21.36). For forward bias
(accumulation), the capacitance of the space-charge region is high. Therefore, the total capacitance
of the MIS diode is given by the insulator capacitance C ≈ C i . When the voltage is reduced, the
capacitance of the space-charge region drops to C d = s /L D for the flat-band case ( s = 0). For a high
reverse voltage, the semiconductor is inverted at the surface and the space-charge region capacitance
is high again. In this case, the total capacitance is given by C ≈ C i again.
The previous consideration assumes that the charge density in the semiconductor can follow changes
of the bias sufficiently fast.
20 However, the inversion charge must be built up and then also disappear
via recombination that is limited by the recombination time constant τ . For frequencies around τ
−1
or faster, the charge in the inversion layer cannot follow and the capacitance of the semiconductor is
given by the value C d ∼ = s /w m ; w m (Fig. 21.37) is the maximum depletion-layer width present at the
beginning of inversion (cf. (21.10) and (21.88))
w m ∼ =
2 s
e N A
inv
s =
4 s kT
e 2 N A
ln
N A
n i
.
(21.97)
For further increased voltage (into the inversion regime), the electric field is screened by the inversion
charge and the width of the depletion layer remains constant. Therefore, the total capacitance in the
20 Typically, a dc bias voltage V is set and the capacitance is sampled with a small ac voltage of amplitude , with
δV V .