190
7 Electronic Defect States
E F = E C − E
b
D + kT ln
⎛
⎜
⎝
1 + 4 ˆ
g D
N D
N C
exp
E
b
D
kT
1/2 − 1
2 ˆ
g D
⎞
⎟
⎠ .
(7.30)
For T → 0 the Fermi level is, as expected, in the center between the populated and unpopulated states,
i.e. at E F = E C − E
b
D /2. In Fig. 7.9a the position of the Fermi is shown for a donor with 45 meV binding
energy in Si. For low temperatures the solution can be approximated as (dashed curve in Fig. 7.9b)
E F ∼ = E C −
1
2
E
b
D +
1
2
kT ln
N D
ˆ
g D N C
.
(7.31)
The freeze-out of carriers in n-type silicon has been discussed in detail in [587], taking into account
the effects of the fine structure of the donor states. We note that the fairly high donor binding energy
in silicon leads to freeze-out of carriers at about 40 K and is thus limiting for the low-temperature
performance of devices. Ge has smaller donor ionization energies and subsequently a lower freeze-out
temperature of 20 K. For n-type GaAs, conductivity is preserved down to even lower temperatures.
We note that the freeze-out of carriers involves the recombination of free electrons with the ionized
donors. This aspect is considered in Sect. 10.9. Microscopically this process is equal to the emission
of a (far infrared) photon [588, 589]. Similarly the release of an electron from the donor is due to the
absorption of a photon.
For higher temperatures, when the electron density saturates towards N D , the approximate solution
is (dash-dotted curve in Fig. 7.9a)
E F ∼ = E C + kT ln
N D
N C
.
(7.32)
The electron density n is given (still in the Boltzmann approximation) by
n = N C exp
−
E
b
D
kT
1 + 4 ˆ
g D
N D
N C
exp
E
b
D
kT
1/2 − 1
2 ˆ
g D
(7.33)
=
2 N D
1 +
1 + 4 ˆ
g D
N D
N C
exp
E
b
D
kT
1/2 .
The theoretical electron density as a function of temperature is shown in Fig. 7.9b. It fits very well to
experimental data for arsenic doped germanium [594] as shown in Fig. 7.10 (Arrhenius plot, ln n vs.
1/T ).
For low temperatures, the solution (7.34) is close to
n ∼ =
N D N C
ˆ
g D
exp
−
E
b
D
2kT
=
n 1 N D .
(7.34)
For high temperatures, n ∼ = N D . This regime is called exhaustion or saturation since all possible
electrons have been ionized from their donors. We note that even in this case np = n i p i holds,
however, n p.
While the characteristic energy for the ionization of electrons from donors is E
b
D , at high enough
temperatures electrons are transferred also from the valence band into the conduction band. Thus, in
order to make the above consideration valid for all temperatures, the intrinsic conduction also has to
7 Electronic Defect States
E F = E C − E
b
D + kT ln
⎛
⎜
⎝
1 + 4 ˆ
g D
N D
N C
exp
E
b
D
kT
1/2 − 1
2 ˆ
g D
⎞
⎟
⎠ .
(7.30)
For T → 0 the Fermi level is, as expected, in the center between the populated and unpopulated states,
i.e. at E F = E C − E
b
D /2. In Fig. 7.9a the position of the Fermi is shown for a donor with 45 meV binding
energy in Si. For low temperatures the solution can be approximated as (dashed curve in Fig. 7.9b)
E F ∼ = E C −
1
2
E
b
D +
1
2
kT ln
N D
ˆ
g D N C
.
(7.31)
The freeze-out of carriers in n-type silicon has been discussed in detail in [587], taking into account
the effects of the fine structure of the donor states. We note that the fairly high donor binding energy
in silicon leads to freeze-out of carriers at about 40 K and is thus limiting for the low-temperature
performance of devices. Ge has smaller donor ionization energies and subsequently a lower freeze-out
temperature of 20 K. For n-type GaAs, conductivity is preserved down to even lower temperatures.
We note that the freeze-out of carriers involves the recombination of free electrons with the ionized
donors. This aspect is considered in Sect. 10.9. Microscopically this process is equal to the emission
of a (far infrared) photon [588, 589]. Similarly the release of an electron from the donor is due to the
absorption of a photon.
For higher temperatures, when the electron density saturates towards N D , the approximate solution
is (dash-dotted curve in Fig. 7.9a)
E F ∼ = E C + kT ln
N D
N C
.
(7.32)
The electron density n is given (still in the Boltzmann approximation) by
n = N C exp
−
E
b
D
kT
1 + 4 ˆ
g D
N D
N C
exp
E
b
D
kT
1/2 − 1
2 ˆ
g D
(7.33)
=
2 N D
1 +
1 + 4 ˆ
g D
N D
N C
exp
E
b
D
kT
1/2 .
The theoretical electron density as a function of temperature is shown in Fig. 7.9b. It fits very well to
experimental data for arsenic doped germanium [594] as shown in Fig. 7.10 (Arrhenius plot, ln n vs.
1/T ).
For low temperatures, the solution (7.34) is close to
n ∼ =
N D N C
ˆ
g D
exp
−
E
b
D
2kT
=
n 1 N D .
(7.34)
For high temperatures, n ∼ = N D . This regime is called exhaustion or saturation since all possible
electrons have been ionized from their donors. We note that even in this case np = n i p i holds,
however, n p.
While the characteristic energy for the ionization of electrons from donors is E
b
D , at high enough
temperatures electrons are transferred also from the valence band into the conduction band. Thus, in
order to make the above consideration valid for all temperatures, the intrinsic conduction also has to