206
7 Electronic Defect States
Centers that can capture electrons and holes lead to nonradiative recombination of electrons through
the deep level into the valence band (see also Chap. 10). This can be useful for the fabrication of
semi-insulating layers with low carrier concentration and fast time response of, e.g., switches and
photodetectors.
While the electronic properties of deep levels can be readily characterized, the microscopic origin
is not immediately apparent. Next to theoretical modeling of defects and correlation with experimental
results, paramagnetic hyperfine interactions have proven useful to identify the microscopic nature of
various defects [643].
7.7.1 Charge States
The deep level can have different charge states depending on the occupancy with electrons. The energy
position within the gap varies with the charge state due to the Coulomb interaction. Also, the lattice
relaxation around the defect depends on the charge state and modifies the energy level.
The localized charge q d at the defect is the integral over the change ρ of the charge density
compared to the perfect crystal over a sufficiently large volume V ∞ around the defect
q d =
V ∞
d
3 r =
n e
r
.
(7.59)
In semiconductors, the charge q d r is an integer multiple of the elementary charge. The defect is said to
be in the nth charge state. Each charge state has a certain stable atomic configuration R n . Each charge
state has a ground state and excited states that can each have different stable atomic configurations.
Now, we discuss how the concentration of the various charge states depends on the position of the
Fermi level. The overall constraint of global charge neutrality determines the chemical potential of the
electron, i.e. the Fermi level in Fermi–Dirac statistics. We use the approximation that the concentration
of defects is so small that the mutual interaction of defects becomes negligible.
As an example, we treat the possible reaction V
0
V
+
+ e
− , where V
0 denotes a neutral vacancy
and V
+ is a positively charged vacancy, created by the ionization of an electron from the vacancy into
the conduction band. The free energy G depends on the numbers n 0 of neutral and n + of positively
charged vacancies. The minimum condition is met by
dG =
∂G
∂n 0
dn 0 +
∂G
∂n +
dn + = 0 .
(7.60)
The neutrality constraint is dn 0 + dn + = 0 and therefore the minimum condition reads
∂G
∂n 0
=
∂G
∂n +
.
(7.61)
For noninteracting defects and using (4.9) we write
∂G
∂n 0
=G
f
(V
0
) + kT ln
n 0
N 0
(7.62a)
∂G
∂n +
=
∂G(V
+
)
∂n +
+
∂G(e
−
)
∂n +
= G
f
V + + kT ln
n +
N +
+ μ e − ,
(7.62b)
7 Electronic Defect States
Centers that can capture electrons and holes lead to nonradiative recombination of electrons through
the deep level into the valence band (see also Chap. 10). This can be useful for the fabrication of
semi-insulating layers with low carrier concentration and fast time response of, e.g., switches and
photodetectors.
While the electronic properties of deep levels can be readily characterized, the microscopic origin
is not immediately apparent. Next to theoretical modeling of defects and correlation with experimental
results, paramagnetic hyperfine interactions have proven useful to identify the microscopic nature of
various defects [643].
7.7.1 Charge States
The deep level can have different charge states depending on the occupancy with electrons. The energy
position within the gap varies with the charge state due to the Coulomb interaction. Also, the lattice
relaxation around the defect depends on the charge state and modifies the energy level.
The localized charge q d at the defect is the integral over the change ρ of the charge density
compared to the perfect crystal over a sufficiently large volume V ∞ around the defect
q d =
V ∞
d
3 r =
n e
r
.
(7.59)
In semiconductors, the charge q d r is an integer multiple of the elementary charge. The defect is said to
be in the nth charge state. Each charge state has a certain stable atomic configuration R n . Each charge
state has a ground state and excited states that can each have different stable atomic configurations.
Now, we discuss how the concentration of the various charge states depends on the position of the
Fermi level. The overall constraint of global charge neutrality determines the chemical potential of the
electron, i.e. the Fermi level in Fermi–Dirac statistics. We use the approximation that the concentration
of defects is so small that the mutual interaction of defects becomes negligible.
As an example, we treat the possible reaction V
0
V
+
+ e
− , where V
0 denotes a neutral vacancy
and V
+ is a positively charged vacancy, created by the ionization of an electron from the vacancy into
the conduction band. The free energy G depends on the numbers n 0 of neutral and n + of positively
charged vacancies. The minimum condition is met by
dG =
∂G
∂n 0
dn 0 +
∂G
∂n +
dn + = 0 .
(7.60)
The neutrality constraint is dn 0 + dn + = 0 and therefore the minimum condition reads
∂G
∂n 0
=
∂G
∂n +
.
(7.61)
For noninteracting defects and using (4.9) we write
∂G
∂n 0
=G
f
(V
0
) + kT ln
n 0
N 0
(7.62a)
∂G
∂n +
=
∂G(V
+
)
∂n +
+
∂G(e
−
)
∂n +
= G
f
V + + kT ln
n +
N +
+ μ e − ,
(7.62b)