6.12 Strain Effects on the Band Structure
173
Fig. 6.51 Bandgap of
GaAs 1−x N x , experimental
data from various sources
(symbols) and model
(curve) according to (6.62)
with V = V 0
√
x for
V 0 = 2.7 eV. Adapted
from [545]
Nitrogen concentration x (%)
1.5
1.4
1.3
1.2
1.1
1.0
0.9
GaAs N
1-x x
0
1
2
3
4
5
Energy (eV)
6.12.3 Interaction With a Localized Level
The normal dependence of the band gap on hydrostatic pressure is linear and given by (6.57). (Ga,In)As
containing nitrogen exhibits a remarkable deviation from this behavior as shown in Fig. 6.48a. This
is due to the interaction of the continuum states of the conduction band with the electron level of
the isoelectronic nitrogen impurity (Sect. 7.7.9) E N , being within the conduction band. For GaAs it
is 0.2 eV above the conduction band edge E C . This phenomenon has been investigated theoretically
within microscopic detail [544]. Within a simple ‘band anticrossing’ two-level model, the coupling
of the pressure-dependent conduction band edge E C and the nitrogen level can be obtained from the
Eigenwert equation
E − E C
V
V
E − E N
= 0 ,
(6.61)
V being the coupling constant. The determinant vanishes for
E ± =
1
2
E C + E N ±
(E C − E N ) 2 + 4V 2
.
(6.62)
Here the weak pressure dependence of E N is neglected for simplicity. This model can explain the
pressure dependence of the band gap of (Ga,In)As:N fairly well [538] (Fig. 6.48b). The coupling
parameter V is in the order of a few 0.1 eV for small nitrogen content. In photomodulated reflection
also the E + levels can be observed [545]. The anti-crossing model can also model the dependence of
the GaAs 1−x N x bandgap on the nitrogen concentration [545] (Fig. 6.51).
6.13 Density of States
6.13.1 General Band Structure
The dispersion relation yields how the energy of a (quasi-) particle depends on the k vector. Now we
want to know how many states are at a given energy. This quantity is called the density of states (DOS)
and is written as D(E). It is defined in an infinitesimal sense such that the number of states between
173
Fig. 6.51 Bandgap of
GaAs 1−x N x , experimental
data from various sources
(symbols) and model
(curve) according to (6.62)
with V = V 0
√
x for
V 0 = 2.7 eV. Adapted
from [545]
Nitrogen concentration x (%)
1.5
1.4
1.3
1.2
1.1
1.0
0.9
GaAs N
1-x x
0
1
2
3
4
5
Energy (eV)
6.12.3 Interaction With a Localized Level
The normal dependence of the band gap on hydrostatic pressure is linear and given by (6.57). (Ga,In)As
containing nitrogen exhibits a remarkable deviation from this behavior as shown in Fig. 6.48a. This
is due to the interaction of the continuum states of the conduction band with the electron level of
the isoelectronic nitrogen impurity (Sect. 7.7.9) E N , being within the conduction band. For GaAs it
is 0.2 eV above the conduction band edge E C . This phenomenon has been investigated theoretically
within microscopic detail [544]. Within a simple ‘band anticrossing’ two-level model, the coupling
of the pressure-dependent conduction band edge E C and the nitrogen level can be obtained from the
Eigenwert equation
E − E C
V
V
E − E N
= 0 ,
(6.61)
V being the coupling constant. The determinant vanishes for
E ± =
1
2
E C + E N ±
(E C − E N ) 2 + 4V 2
.
(6.62)
Here the weak pressure dependence of E N is neglected for simplicity. This model can explain the
pressure dependence of the band gap of (Ga,In)As:N fairly well [538] (Fig. 6.48b). The coupling
parameter V is in the order of a few 0.1 eV for small nitrogen content. In photomodulated reflection
also the E + levels can be observed [545]. The anti-crossing model can also model the dependence of
the GaAs 1−x N x bandgap on the nitrogen concentration [545] (Fig. 6.51).
6.13 Density of States
6.13.1 General Band Structure
The dispersion relation yields how the energy of a (quasi-) particle depends on the k vector. Now we
want to know how many states are at a given energy. This quantity is called the density of states (DOS)
and is written as D(E). It is defined in an infinitesimal sense such that the number of states between