210
7 Electronic Defect States
Fig. 7.32 Inverse
(absolute) Hall coefficient
(cmp. Sect. 15.2.1) R
−1
H ,
i.e. charge concentration,
for three Ge:Zn samples
with different degree of
compensation with Sb
donors as labeled. The
dash-dotted lines indicate
typical slopes. The dashed
lines sketch the
Zn 0 → Zn − and the
Zn − → Zn −− processes.
Adapted from [659]
when the n-doping is sufficient to partially compensate the Zn and supply one electron for each Zn
atom but not two (2N Zn > N D > N Zn ). A similar situation has been observed for Zn in germanium,
exhibiting the levels E V + 0.03 eV and E V + 0.09 eV [659]. In Fig. 7.32 three different Ge:Zn samples
are compared. If the additional Sb donor concentration (N D ≈ 3.4 × 10
16 cm
−3 ) is larger than 2N Zn
(N Zn ≈ 1.2 × 10
16 cm
−3 ), the sample is n-type (upper curve). The slope is similar to the Ge:Sb donor
binding energy (Table 7.2). If compensation with donors is weak (N Zn > N D , middle curve) first the
shallow donor level with 0.03 eV activation energy is activated and subsequently the deeper one with
0.09 eV activation energy, creating p-conduction with a saturated hole density of p ≈ 2N A − N D > N Zn
(negative Hall coefficient). The two individual activation processes are sketched as dashed lines in
Fig. 7.32. If the Sb concentration is larger than N Zn but smaller than 2N Zn , the shallow acceptor
level is filled with electrons, leaving still the only partially filled deeper acceptor level available for
ionization (lower curve). In this case the sample is still p-type, but the saturation hole density is
p ≈ 2N A − N D < N Zn . The degeneracy factors for Zn in Si and Ge have been discussed in [601].
7.7.4 Jahn–Teller Effect
The lattice relaxation can reduce the symmetry of the defect. Many defects, such as a vacancy, a
tetrahedral interstitial or an impurity, occupy initially tetrahedral sites in the zincblende structure. The
lattice relaxation reduces the symmetry, e.g. to tetragonal or trigonal, and therefore causes initially
degenerate levels to split. Such splitting is called the static Jahn–Teller effect [639, 660]. The energy
change in terms of the atomic displacement Q can be denoted (using perturbation theory for the simplest,
nondegenerate case) as −I Q (I > 0). Including the elastic contribution with a force constant C, the
energy of a configuration Q is
E = −I Q +
1
2
C Q
2
.
(7.66)
The stable configuration Q min , for which the energy is minimal (E min ), is therefore given by
7 Electronic Defect States
Fig. 7.32 Inverse
(absolute) Hall coefficient
(cmp. Sect. 15.2.1) R
−1
H ,
i.e. charge concentration,
for three Ge:Zn samples
with different degree of
compensation with Sb
donors as labeled. The
dash-dotted lines indicate
typical slopes. The dashed
lines sketch the
Zn 0 → Zn − and the
Zn − → Zn −− processes.
Adapted from [659]
when the n-doping is sufficient to partially compensate the Zn and supply one electron for each Zn
atom but not two (2N Zn > N D > N Zn ). A similar situation has been observed for Zn in germanium,
exhibiting the levels E V + 0.03 eV and E V + 0.09 eV [659]. In Fig. 7.32 three different Ge:Zn samples
are compared. If the additional Sb donor concentration (N D ≈ 3.4 × 10
16 cm
−3 ) is larger than 2N Zn
(N Zn ≈ 1.2 × 10
16 cm
−3 ), the sample is n-type (upper curve). The slope is similar to the Ge:Sb donor
binding energy (Table 7.2). If compensation with donors is weak (N Zn > N D , middle curve) first the
shallow donor level with 0.03 eV activation energy is activated and subsequently the deeper one with
0.09 eV activation energy, creating p-conduction with a saturated hole density of p ≈ 2N A − N D > N Zn
(negative Hall coefficient). The two individual activation processes are sketched as dashed lines in
Fig. 7.32. If the Sb concentration is larger than N Zn but smaller than 2N Zn , the shallow acceptor
level is filled with electrons, leaving still the only partially filled deeper acceptor level available for
ionization (lower curve). In this case the sample is still p-type, but the saturation hole density is
p ≈ 2N A − N D < N Zn . The degeneracy factors for Zn in Si and Ge have been discussed in [601].
7.7.4 Jahn–Teller Effect
The lattice relaxation can reduce the symmetry of the defect. Many defects, such as a vacancy, a
tetrahedral interstitial or an impurity, occupy initially tetrahedral sites in the zincblende structure. The
lattice relaxation reduces the symmetry, e.g. to tetragonal or trigonal, and therefore causes initially
degenerate levels to split. Such splitting is called the static Jahn–Teller effect [639, 660]. The energy
change in terms of the atomic displacement Q can be denoted (using perturbation theory for the simplest,
nondegenerate case) as −I Q (I > 0). Including the elastic contribution with a force constant C, the
energy of a configuration Q is
E = −I Q +
1
2
C Q
2
.
(7.66)
The stable configuration Q min , for which the energy is minimal (E min ), is therefore given by