212
7 Electronic Defect States
with two electrons) is resonant with the valence band. The T 2 state lies in the band gap. When the
Jahn–Teller effect (now on the T 2 state) is included, the energies of the different charge states depend
on the configuration coordinate (a mostly tetragonal distortion in the case of the Si vacancy).
E V 0 = E(0, Q) =E(0, Q = 0) − 2I Q +
1
2
C Q
2
(7.68a)
E V + = E(1, Q) =E(1, Q = 0) − I Q +
1
2
C Q
2
(7.68b)
E V ++ = E(2, Q) =E(2, Q = 0) +
1
2
C Q
2
.
(7.68c)
For the n = 2 state the T 2 gap state is empty and thus no degeneracy and Jahn–Teller term arises.
For n = 1 there is a linear Jahn–Teller term. The occupation with two electrons (V
0 ) causes an
approximately twice as large Jahn–Teller splitting for the n = 0 state. The force constant is assumed
to be independent of the charge state. The energies for the minimum configurations Q
n
min are therefore
E(0, Q
0
min ) =E(0, Q = 0) − 4
I
2
2C
(7.69a)
E(1, Q
1
min ) =E(1, Q = 0) −
I
2
2C
(7.69b)
E(2, Q
2
min ) =E(2, Q = 0) .
(7.69c)
The Jahn–Teller energy E JT = I
2
/2C lowers the position of the occupancy levels E 0 calculated with
Coulomb terms only. The occupancy levels including the Jahn–Teller contribution are therefore given as
E(1, 2) = E 0 (1, 2) − E JT
(7.70a)
E(0, 1) = E 0 (0, 1) − 3 E JT .
(7.70b)
For the vacancy in silicon the Jahn–Teller energy E JT is about 0.19 eV. Thus the E(1, 2) level is
lowered from 0.32 eV to 0.13 eV. The E(0,1) occupancy level, however, is reduced from 0.57 eV to
0.05 eV [662, 667] (see Fig. 7.33). The occupancy level E(0, 2) is in the middle between E(0, 1) and
E(1, 2) (E(0, 2) = (E(0, 1)+ E(1, 2))/2) and indicated in Fig. 7.35a. At this energy, c(V
0
) = c(V
++
)
and the value of c(V
+
) is small (≈ exp
E 1 −E 2
2kT
) since E(0, 1) < E(1, 2) (cmp. (7.65)).
The relative concentrations of the three charge states are determined by (7.63) (degeneracy and
entropy terms have been neglected)
c(V
++
)
c(V + )
= exp
E(1, 2) − E F
kT
(7.71a)
c(V
+
)
c(V 0 )
= exp
E(0, 1) − E F
kT
.
(7.71b)
They are depicted in Fig. 7.35a in a plot related to Fig. 7.30a. Therefore, V
++ dominates if E F < E(0, 1)
and V
0 dominates for E F > E(1, 2). In the intermediate range E(0, 1) < E F < E(1, 2) we know
from (7.71a, b) that V
+ is dominated by V
0 and V
++ . However, at this point it is not clear whether
V
++ or V
0 dominates overall. The ratio of the concentrations of V
++ and V
0 is given by
7 Electronic Defect States
with two electrons) is resonant with the valence band. The T 2 state lies in the band gap. When the
Jahn–Teller effect (now on the T 2 state) is included, the energies of the different charge states depend
on the configuration coordinate (a mostly tetragonal distortion in the case of the Si vacancy).
E V 0 = E(0, Q) =E(0, Q = 0) − 2I Q +
1
2
C Q
2
(7.68a)
E V + = E(1, Q) =E(1, Q = 0) − I Q +
1
2
C Q
2
(7.68b)
E V ++ = E(2, Q) =E(2, Q = 0) +
1
2
C Q
2
.
(7.68c)
For the n = 2 state the T 2 gap state is empty and thus no degeneracy and Jahn–Teller term arises.
For n = 1 there is a linear Jahn–Teller term. The occupation with two electrons (V
0 ) causes an
approximately twice as large Jahn–Teller splitting for the n = 0 state. The force constant is assumed
to be independent of the charge state. The energies for the minimum configurations Q
n
min are therefore
E(0, Q
0
min ) =E(0, Q = 0) − 4
I
2
2C
(7.69a)
E(1, Q
1
min ) =E(1, Q = 0) −
I
2
2C
(7.69b)
E(2, Q
2
min ) =E(2, Q = 0) .
(7.69c)
The Jahn–Teller energy E JT = I
2
/2C lowers the position of the occupancy levels E 0 calculated with
Coulomb terms only. The occupancy levels including the Jahn–Teller contribution are therefore given as
E(1, 2) = E 0 (1, 2) − E JT
(7.70a)
E(0, 1) = E 0 (0, 1) − 3 E JT .
(7.70b)
For the vacancy in silicon the Jahn–Teller energy E JT is about 0.19 eV. Thus the E(1, 2) level is
lowered from 0.32 eV to 0.13 eV. The E(0,1) occupancy level, however, is reduced from 0.57 eV to
0.05 eV [662, 667] (see Fig. 7.33). The occupancy level E(0, 2) is in the middle between E(0, 1) and
E(1, 2) (E(0, 2) = (E(0, 1)+ E(1, 2))/2) and indicated in Fig. 7.35a. At this energy, c(V
0
) = c(V
++
)
and the value of c(V
+
) is small (≈ exp
E 1 −E 2
2kT
) since E(0, 1) < E(1, 2) (cmp. (7.65)).
The relative concentrations of the three charge states are determined by (7.63) (degeneracy and
entropy terms have been neglected)
c(V
++
)
c(V + )
= exp
E(1, 2) − E F
kT
(7.71a)
c(V
+
)
c(V 0 )
= exp
E(0, 1) − E F
kT
.
(7.71b)
They are depicted in Fig. 7.35a in a plot related to Fig. 7.30a. Therefore, V
++ dominates if E F < E(0, 1)
and V
0 dominates for E F > E(1, 2). In the intermediate range E(0, 1) < E F < E(1, 2) we know
from (7.71a, b) that V
+ is dominated by V
0 and V
++ . However, at this point it is not clear whether
V
++ or V
0 dominates overall. The ratio of the concentrations of V
++ and V
0 is given by