7.7 Deep Levels
211
Fig. 7.33 Charge states of
the vacancy in silicon. Left:
level scheme without
lattice relaxation, right:
level scheme including the
Jahn–Teller effect. For a
Fermi level below (above)
E(0, 2) the charge state
V ++ (V 0 ) is dominant
V
V
V
0
V
V
0
E (0,1)
0
E(1,2)
E (1,2)
0
E(0,1)
E(0,2)
Q min =
I
C
(7.67a)
E min = −
I
2
2C
.
(7.67b)
Several equivalent lattice relaxations may exist, e.g. a 3-fold minimum for remaining C 3v symmetry.
The energy barrier between them has a finite height. Therefore, e.g. at sufficient temperature, the defect
can switch between different configurations and eventually again becomes isotropic (dynamic Jahn–
Teller effect). The experimental observation depends on the relation between the characteristic time of
the experiment and the reorientation time constant of the defect.
7.7.5 Negative-U Center
We explain the principle of a so-called negative-U center [661] for the Si vacancy [662] (cf. Fig. 4.2).
It was first proposed by Anderson to explain the properties of amorphous chalcogenide glasses [663].
Many defects in semiconductors exhibit negative-U behavior, e.g. also the boron interstitial in Si
[662, 664]. Coulomb energy and the Jahn–Teller effect compete for the position of the occupancy
level for different charge states. U refers to the additional energy upon charging of the defect with
an additional electron. The Coulomb repulsion of electrons leads to an increase of the energy, i.e.
positive U , which has been calculated to be 0.25 eV for the Si vacancy [665] for all charge states.
The occupation level (cf. Sect. 4.2.2) E 0 (1, 2) (the index 0 indicates effects only due to many-electron
Coulomb interaction), separating the domination of V
++ and V
+ (Fig. 7.33) is 0.32eV above the
valance-band edge. Therefore, the occupation level E 0 (0, 1) is expected to lie at about 0.57 eV about E V .
The Jahn–Teller effect may lead to a splitting of the otherwise 4-fold degenerate states of the vacancy.
A detailed experimental study using hyperfine interactions can be found in [666]. The schematic level
diagram for the Jahn–Teller splitting is shown in Fig. 7.34. The V
++ state ( A 1 is always populated
V
T 2
A 1
V
V
0
V
Fig. 7.34 Jahn–Teller splitting for different charge states of the vacancy. A 1 and T 2 refer to irreducible representations
of the T d point symmetry group. A 1 is nondegenerate and therefore does not exhibit a Jahn–Teller effect. T 2 is triply
degenerate. The arrows represent electrons and their spin orientation
211
Fig. 7.33 Charge states of
the vacancy in silicon. Left:
level scheme without
lattice relaxation, right:
level scheme including the
Jahn–Teller effect. For a
Fermi level below (above)
E(0, 2) the charge state
V ++ (V 0 ) is dominant
V
V
V
0
V
V
0
E (0,1)
0
E(1,2)
E (1,2)
0
E(0,1)
E(0,2)
Q min =
I
C
(7.67a)
E min = −
I
2
2C
.
(7.67b)
Several equivalent lattice relaxations may exist, e.g. a 3-fold minimum for remaining C 3v symmetry.
The energy barrier between them has a finite height. Therefore, e.g. at sufficient temperature, the defect
can switch between different configurations and eventually again becomes isotropic (dynamic Jahn–
Teller effect). The experimental observation depends on the relation between the characteristic time of
the experiment and the reorientation time constant of the defect.
7.7.5 Negative-U Center
We explain the principle of a so-called negative-U center [661] for the Si vacancy [662] (cf. Fig. 4.2).
It was first proposed by Anderson to explain the properties of amorphous chalcogenide glasses [663].
Many defects in semiconductors exhibit negative-U behavior, e.g. also the boron interstitial in Si
[662, 664]. Coulomb energy and the Jahn–Teller effect compete for the position of the occupancy
level for different charge states. U refers to the additional energy upon charging of the defect with
an additional electron. The Coulomb repulsion of electrons leads to an increase of the energy, i.e.
positive U , which has been calculated to be 0.25 eV for the Si vacancy [665] for all charge states.
The occupation level (cf. Sect. 4.2.2) E 0 (1, 2) (the index 0 indicates effects only due to many-electron
Coulomb interaction), separating the domination of V
++ and V
+ (Fig. 7.33) is 0.32eV above the
valance-band edge. Therefore, the occupation level E 0 (0, 1) is expected to lie at about 0.57 eV about E V .
The Jahn–Teller effect may lead to a splitting of the otherwise 4-fold degenerate states of the vacancy.
A detailed experimental study using hyperfine interactions can be found in [666]. The schematic level
diagram for the Jahn–Teller splitting is shown in Fig. 7.34. The V
++ state ( A 1 is always populated
V
T 2
A 1
V
V
0
V
Fig. 7.34 Jahn–Teller splitting for different charge states of the vacancy. A 1 and T 2 refer to irreducible representations
of the T d point symmetry group. A 1 is nondegenerate and therefore does not exhibit a Jahn–Teller effect. T 2 is triply
degenerate. The arrows represent electrons and their spin orientation