208
7 Electronic Defect States
Table 7.8 Binding energies (to conduction band) of double donor chalcogenide impurities in Si and Ge. All energies in
meV, data from [651, 652]
Host
State
S
Se
Te
Si
D 0
318
307
199
D +
612
589
411
Ge
D 0
280
268
93
D +
590
512
330
different position of the Fermi level, the trap (deep donor) energy is found to be E V + 0.375 eV as
indicated in Fig. 7.29b.
7.7.2 Double Donors
An impurity that has two extra electrons available after bonding in the matrix may give rise to a
double donor. Typical examples are substitutional chalcogenide atoms (S, Se or Te) in silicon [651]
and germanium [652], interstitial impurities such as Mg i in Si [653], or group-V atoms on a group-III
site in a III–V compound (antisite defect), such as P Ga in GaP [654] or As Ga in GaAs [655].
The double donor is electronically similar to a helium atom. Due to the repulsive Coulomb interaction
of the two electrons on the neutral double donor, the (single) ionization energy E 1 (also often labeled
E(0, 1) or E(0, +)) of D
0 is smaller than that of D
+ (E 2 , also labeled E(1, 2) or E(+, ++)). For He
and He
+ the ratio of ionization energies is 0.45; for chalcogenides in Si and Ge similar ratios have
mostly been found (Table 7.8).
The carrier statistics and the degeneracy factors for a double donor have been discussed in [585, 656].
Typically, the degeneracy factor for the ionization of the double donor D
0
→ D
+ is ˆ
g D = g 2 /g 1 = 1/2
and for the ionization D
+
→ D
++ is ˆ
g D = g 1 /g 0 = 2/1 = 2.
For the probabilities to find a neutral, single and double ionized donor we find following the treatment
in [656]
d
0
=
N
0
D
N D
=
exp
2 E F
kT
exp
E 1 +E 2
kT
+ exp
2 E F
kT
+ 2 exp
E 1 +E F
kT
(7.64a)
d
+
=
N
+
D
N D
=
exp
E 1 +E 2
kT
exp
E 1 +E 2
kT
+ exp
2 E F
kT
+ 2 exp
E 1 +E F
kT
(7.64b)
d
++
=
N
++
D
N D
=
2 exp
E 1 +E F
kT
exp
E 1 +E 2
kT
+ exp
2 E F
kT
+ 2 exp
E 1 +E F
kT
(7.64c)
The probabilities are depicted in Fig. 7.30a. The maximum of d
+ is at the energy (E 1 + E 2 )/2. Its
value is
d
+
E 1 + E 2
2
=
1
1 + exp
−
E 1 −E 2
2kT
(7.65)
and reaches a value close to one for (E 1 − E 2 )/kT 1. In Fig. 7.30b the number of electrons per donor
˜
n = (N
+
D + 2 N
++
D )/N D is shown as a function of the Fermi level; at (E 1 + E 2 )/2, exactly ˜
n = 1. In
Fig.7.31 the temperature dependent electron concentration in Si:Te is depicted. Up to 570 K the single
ionization is visible (other shallow impurities present in the sample in lower concentrations < 10
14 cm
−3
7 Electronic Defect States
Table 7.8 Binding energies (to conduction band) of double donor chalcogenide impurities in Si and Ge. All energies in
meV, data from [651, 652]
Host
State
S
Se
Te
Si
D 0
318
307
199
D +
612
589
411
Ge
D 0
280
268
93
D +
590
512
330
different position of the Fermi level, the trap (deep donor) energy is found to be E V + 0.375 eV as
indicated in Fig. 7.29b.
7.7.2 Double Donors
An impurity that has two extra electrons available after bonding in the matrix may give rise to a
double donor. Typical examples are substitutional chalcogenide atoms (S, Se or Te) in silicon [651]
and germanium [652], interstitial impurities such as Mg i in Si [653], or group-V atoms on a group-III
site in a III–V compound (antisite defect), such as P Ga in GaP [654] or As Ga in GaAs [655].
The double donor is electronically similar to a helium atom. Due to the repulsive Coulomb interaction
of the two electrons on the neutral double donor, the (single) ionization energy E 1 (also often labeled
E(0, 1) or E(0, +)) of D
0 is smaller than that of D
+ (E 2 , also labeled E(1, 2) or E(+, ++)). For He
and He
+ the ratio of ionization energies is 0.45; for chalcogenides in Si and Ge similar ratios have
mostly been found (Table 7.8).
The carrier statistics and the degeneracy factors for a double donor have been discussed in [585, 656].
Typically, the degeneracy factor for the ionization of the double donor D
0
→ D
+ is ˆ
g D = g 2 /g 1 = 1/2
and for the ionization D
+
→ D
++ is ˆ
g D = g 1 /g 0 = 2/1 = 2.
For the probabilities to find a neutral, single and double ionized donor we find following the treatment
in [656]
d
0
=
N
0
D
N D
=
exp
2 E F
kT
exp
E 1 +E 2
kT
+ exp
2 E F
kT
+ 2 exp
E 1 +E F
kT
(7.64a)
d
+
=
N
+
D
N D
=
exp
E 1 +E 2
kT
exp
E 1 +E 2
kT
+ exp
2 E F
kT
+ 2 exp
E 1 +E F
kT
(7.64b)
d
++
=
N
++
D
N D
=
2 exp
E 1 +E F
kT
exp
E 1 +E 2
kT
+ exp
2 E F
kT
+ 2 exp
E 1 +E F
kT
(7.64c)
The probabilities are depicted in Fig. 7.30a. The maximum of d
+ is at the energy (E 1 + E 2 )/2. Its
value is
d
+
E 1 + E 2
2
=
1
1 + exp
−
E 1 −E 2
2kT
(7.65)
and reaches a value close to one for (E 1 − E 2 )/kT 1. In Fig. 7.30b the number of electrons per donor
˜
n = (N
+
D + 2 N
++
D )/N D is shown as a function of the Fermi level; at (E 1 + E 2 )/2, exactly ˜
n = 1. In
Fig.7.31 the temperature dependent electron concentration in Si:Te is depicted. Up to 570 K the single
ionization is visible (other shallow impurities present in the sample in lower concentrations < 10
14 cm
−3