7.3 Intrinsic Conduction
183
(a)
(b)
Fig. 7.3 a Band gap of silicon versus temperature. b Intrinsic carrier concentration of silicon versus temperature. Solid
line is (7.17) using E g = 1.204 eV − (2.73 × 10 −4 eV/K) T [564], symbols are experimental data from [565]
n i = p i =
N V N C exp
−
E g
2kT
(7.15)
= 2
kT
2π 2
3/2
(m e m h )
3/4 exp
−
E g
2kT
.
The mass-action law
n p = n i p i = n
2
i = p
2
i
(7.16)
will be essential also for light and moderately doped semiconductors. The intrinsic carrier concentration
is exponentially dependent on the band gap. Thus, in thermodynamic equilibrium intrinsic wide-gap
semiconductors have much smaller electron concentrations than intrinsic small-gap semiconductors
(see Table 7.1). The intrinsic carrier concentration of Si (in cm
−3 ) has been determined to be (within
1%, T in K)
n
Si
i = 1.640 × 10
15 T
1.706 exp
−
E g (T )
2kT
(7.17)
for temperatures between 77 and 400 K [562, 563] (Fig. 7.3).
As we will see later in Part II, many semiconductor devices rely on regions of low conductivity
(depletion layers) in which the carrier concentration is small. Since the carrier concentration cannot be
smaller than the intrinsic concentration (n + p ≥ 2n i ), an increase of temperature leads to increasing
ohmic conduction in the depletion layers and thus to a reduction or failure of device performance. The
small band gap of Ge leads to degradation of bipolar device performance already shortly above room
temperature. For silicon, intrinsic conduction limits operation typically to temperatures below about
300
◦ C. For higher temperatures, as required for devices in harsh environments, such as close to motors
or turbines, other semiconductors with wider band gaps need to be used, such as GaN, SiC or even
diamond.
From the neutrality condition for the intrinsic semiconductor (7.14) and (7.10) and (7.11), the Fermi
level of the intrinsic semiconductor can be determined as
E F = E i =
E V + E C
2
+
kT
2
ln
N V
N C
=
E V + E C
2
+
3
4
kT ln
m h
m e
.
(7.18)
Since the hole mass is perhaps a factor of ten larger than the electron mass, the second term has the
order of kT . Thus, for typical semiconductors where E g kT , the intrinsic Fermi level, denoted by
E i , is close to the middle of the band gap, i.e. E i ≈ (E C + E V )/2.
183
(a)
(b)
Fig. 7.3 a Band gap of silicon versus temperature. b Intrinsic carrier concentration of silicon versus temperature. Solid
line is (7.17) using E g = 1.204 eV − (2.73 × 10 −4 eV/K) T [564], symbols are experimental data from [565]
n i = p i =
N V N C exp
−
E g
2kT
(7.15)
= 2
kT
2π 2
3/2
(m e m h )
3/4 exp
−
E g
2kT
.
The mass-action law
n p = n i p i = n
2
i = p
2
i
(7.16)
will be essential also for light and moderately doped semiconductors. The intrinsic carrier concentration
is exponentially dependent on the band gap. Thus, in thermodynamic equilibrium intrinsic wide-gap
semiconductors have much smaller electron concentrations than intrinsic small-gap semiconductors
(see Table 7.1). The intrinsic carrier concentration of Si (in cm
−3 ) has been determined to be (within
1%, T in K)
n
Si
i = 1.640 × 10
15 T
1.706 exp
−
E g (T )
2kT
(7.17)
for temperatures between 77 and 400 K [562, 563] (Fig. 7.3).
As we will see later in Part II, many semiconductor devices rely on regions of low conductivity
(depletion layers) in which the carrier concentration is small. Since the carrier concentration cannot be
smaller than the intrinsic concentration (n + p ≥ 2n i ), an increase of temperature leads to increasing
ohmic conduction in the depletion layers and thus to a reduction or failure of device performance. The
small band gap of Ge leads to degradation of bipolar device performance already shortly above room
temperature. For silicon, intrinsic conduction limits operation typically to temperatures below about
300
◦ C. For higher temperatures, as required for devices in harsh environments, such as close to motors
or turbines, other semiconductors with wider band gaps need to be used, such as GaN, SiC or even
diamond.
From the neutrality condition for the intrinsic semiconductor (7.14) and (7.10) and (7.11), the Fermi
level of the intrinsic semiconductor can be determined as
E F = E i =
E V + E C
2
+
kT
2
ln
N V
N C
=
E V + E C
2
+
3
4
kT ln
m h
m e
.
(7.18)
Since the hole mass is perhaps a factor of ten larger than the electron mass, the second term has the
order of kT . Thus, for typical semiconductors where E g kT , the intrinsic Fermi level, denoted by
E i , is close to the middle of the band gap, i.e. E i ≈ (E C + E V )/2.