202
7 Electronic Defect States
With increasing concentration, the distance between impurities decreases and their wavefunctions
can overlap. Then, an impurity band develops (Fig. 7.25). A periodic arrangement of impurity atoms
would result in well-defined band edges as found in the Kronig-Penney model. Since the impurity atoms
are randomly distributed, the band edges exhibit tails. For high doping, the impurity band overlaps
with the conduction band. In the case of compensation, the impurity band is not completely filled
and contains (a new type of) holes. In this case, conduction can take place within the impurity band
even at low temperature, making the semiconductor a metal. This metal–insulator transition has been
discussed by Mott [619]. Examples for highly doped semiconductors are transparent conductive oxides
(Sect. 20), the contact layer for an ohmic contact (Sect. 21.2.6) or the active layers in a tunneling diode
(Sect. 21.5.9). The physics, properties and preparation of highly doped semiconductors are treated in
detail in [620].
The formation of the impurity band leads to a reduction of the impurity ionization energy as known
from (7.21). Typical results are shown in Fig. 7.26a for n-type Ge [594] and Fig. 7.26b for p-type
ZnTe [621]. At the critical doping concentration of N c = 1.5 × 10
17 , the activation energy for the
carrier concentration disappears. Similar effects have been observed for Si [622] and GaAs [623]. The
freeze-out of the carrier concentration (see Fig. 7.9) disappears as shown in Fig. 7.27. Critical doping
concentrations are listed in Table 7.6. The decrease of the ionization energy E
b (donor or acceptor)
follows the dependence [594, 622]
E
b
= E
b
0 − α N
1/3
i
= E
b
0
1 −
N i
N c
1/3
,
(7.52)
where N i is the concentration of ionized dopants. A refined theory, considering screening, shift and
tails of the conduction band and most importantly broadening of the donor level has been presented
in [624].
The critical density can be estimated from the Mott criterion when the distance of the impurities
becomes comparable to their Bohr radius (7.22)
2a D =
3
2π
N
1/3
c
.
(7.53)
The pre-factor 3/(2π) stems from the random distribution of impurities and disappears for a periodic
arrangement. The Mott criterion is (rewriting (7.53))
D(E)
E V
E D
E
(a)
(b)
(c)
E C
D(E)
E V
E C
E
D(E)
E V
E
Fig. 7.25 Principle of the formation of a (donor) impurity band. a Small doping concentration and sharply defined
impurity state at E D , b increasing doping and development of an impurity band that c widens further and eventually
overlaps with the conduction band for high impurity concentration. The shaded areas indicate populated states at T = 0 K
7 Electronic Defect States
With increasing concentration, the distance between impurities decreases and their wavefunctions
can overlap. Then, an impurity band develops (Fig. 7.25). A periodic arrangement of impurity atoms
would result in well-defined band edges as found in the Kronig-Penney model. Since the impurity atoms
are randomly distributed, the band edges exhibit tails. For high doping, the impurity band overlaps
with the conduction band. In the case of compensation, the impurity band is not completely filled
and contains (a new type of) holes. In this case, conduction can take place within the impurity band
even at low temperature, making the semiconductor a metal. This metal–insulator transition has been
discussed by Mott [619]. Examples for highly doped semiconductors are transparent conductive oxides
(Sect. 20), the contact layer for an ohmic contact (Sect. 21.2.6) or the active layers in a tunneling diode
(Sect. 21.5.9). The physics, properties and preparation of highly doped semiconductors are treated in
detail in [620].
The formation of the impurity band leads to a reduction of the impurity ionization energy as known
from (7.21). Typical results are shown in Fig. 7.26a for n-type Ge [594] and Fig. 7.26b for p-type
ZnTe [621]. At the critical doping concentration of N c = 1.5 × 10
17 , the activation energy for the
carrier concentration disappears. Similar effects have been observed for Si [622] and GaAs [623]. The
freeze-out of the carrier concentration (see Fig. 7.9) disappears as shown in Fig. 7.27. Critical doping
concentrations are listed in Table 7.6. The decrease of the ionization energy E
b (donor or acceptor)
follows the dependence [594, 622]
E
b
= E
b
0 − α N
1/3
i
= E
b
0
1 −
N i
N c
1/3
,
(7.52)
where N i is the concentration of ionized dopants. A refined theory, considering screening, shift and
tails of the conduction band and most importantly broadening of the donor level has been presented
in [624].
The critical density can be estimated from the Mott criterion when the distance of the impurities
becomes comparable to their Bohr radius (7.22)
2a D =
3
2π
N
1/3
c
.
(7.53)
The pre-factor 3/(2π) stems from the random distribution of impurities and disappears for a periodic
arrangement. The Mott criterion is (rewriting (7.53))
D(E)
E V
E D
E
(a)
(b)
(c)
E C
D(E)
E V
E C
E
D(E)
E V
E
Fig. 7.25 Principle of the formation of a (donor) impurity band. a Small doping concentration and sharply defined
impurity state at E D , b increasing doping and development of an impurity band that c widens further and eventually
overlaps with the conduction band for high impurity concentration. The shaded areas indicate populated states at T = 0 K