268
9 Optical Properties
Table 9.3 Functional dependence of the joint density of states for critical points in 3, 2 and 1 dimensions. E 0 denotes
the energy (band separation) at the critical point, C stands for a constant value. The type of critical point is given (min.:
minimum, saddle: saddle point, max.: maximum)
Dim.
Label
Type
D j for E < E 0
D j for E > E 0
3D
M 0
M 1
M 2
M 3
min.
saddle
saddle
max.
0
C −
√
E 0 − E
C
√
E 0 − E
√
E − E 0
C
C −
√
E − E 0
0
2D
M 0
M 1
M 2
min.
saddle
max.
0
− ln(E 0 − E)
C
C
− ln(E − E 0 )
0
1D
M 0
M 1
min.
max.
0
√
E 0 − E
√
E − E 0
0
D j (E cv ) = 2
S( ˜
E)
d
2 S
(2π/L) 3
1
|∇ k E cv |
,
(9.42)
where E cv is an abbreviation for E c (k) − E v (k) and d
2 S is a surface element of the constant energy
surface with ˜
E = E cv . The spin is assumed to generate doubly degenerate bands and accounts for the
pre-factor 2. Singularities of the JDOS (van-Hove singularities or critical points) appear where ∇ k E cv
vanishes. This occurs when the gradient for both bands is zero or when both bands are parallel. The
latter generates particularly large JDOS because the condition is valid at many points in k-space.
Generally, the (three-dimensional) energy dispersion E(k) around a three-dimensional critical point
(here developed at k = 0) can be written as
E(k) = E(0) +
2 k
2
x
2m x
+
2 k
2
y
2m y
+
2 k
2
z
2m z
.
(9.43)
The singularities are classified as M 0 , M 1 , M 2 and M 3 with the index being the number of masses m i
in (9.43) that are negative. M 0 (M 3 ) describes a minimum (maximum) of the band separation. M 1 and
M 2 are saddle points. For a two-dimensional k-space there exist M 0 , M 1 and M 2 points (minimum,
saddle point and maximum, respectively). For a one-dimensional k-space, there exist M 0 and M 1 points
(minimum and maximum, respectively). The functional dependence of the JDOS at the critical points
is summarized in Table 9.3. The resulting shape of the dielectric function is visualized in Fig. 9.8.
9.7.2 Direct Transitions
Transitions between states at the band edges at the point are possible (Fig. 9.9). The k conservation
requires (almost) vertical transitions in the E(k) diagram because the length of the light k vector,
k = 2π/λ, is much smaller than the size of the Brillouin zone |k| ≤ π/a 0 . The ratio of the lengths of
the k vectors is of the order a 0 /λ and typically about 10
−3 for NIR wavelengths.
For isotropic parabolic bands the band-band transition energy versus wavevector relation is
E cv (k) = E g +
2
2
1
m ∗
e
+
1
m
∗
h
k
2
.
(9.44)
9 Optical Properties
Table 9.3 Functional dependence of the joint density of states for critical points in 3, 2 and 1 dimensions. E 0 denotes
the energy (band separation) at the critical point, C stands for a constant value. The type of critical point is given (min.:
minimum, saddle: saddle point, max.: maximum)
Dim.
Label
Type
D j for E < E 0
D j for E > E 0
3D
M 0
M 1
M 2
M 3
min.
saddle
saddle
max.
0
C −
√
E 0 − E
C
√
E 0 − E
√
E − E 0
C
C −
√
E − E 0
0
2D
M 0
M 1
M 2
min.
saddle
max.
0
− ln(E 0 − E)
C
C
− ln(E − E 0 )
0
1D
M 0
M 1
min.
max.
0
√
E 0 − E
√
E − E 0
0
D j (E cv ) = 2
S( ˜
E)
d
2 S
(2π/L) 3
1
|∇ k E cv |
,
(9.42)
where E cv is an abbreviation for E c (k) − E v (k) and d
2 S is a surface element of the constant energy
surface with ˜
E = E cv . The spin is assumed to generate doubly degenerate bands and accounts for the
pre-factor 2. Singularities of the JDOS (van-Hove singularities or critical points) appear where ∇ k E cv
vanishes. This occurs when the gradient for both bands is zero or when both bands are parallel. The
latter generates particularly large JDOS because the condition is valid at many points in k-space.
Generally, the (three-dimensional) energy dispersion E(k) around a three-dimensional critical point
(here developed at k = 0) can be written as
E(k) = E(0) +
2 k
2
x
2m x
+
2 k
2
y
2m y
+
2 k
2
z
2m z
.
(9.43)
The singularities are classified as M 0 , M 1 , M 2 and M 3 with the index being the number of masses m i
in (9.43) that are negative. M 0 (M 3 ) describes a minimum (maximum) of the band separation. M 1 and
M 2 are saddle points. For a two-dimensional k-space there exist M 0 , M 1 and M 2 points (minimum,
saddle point and maximum, respectively). For a one-dimensional k-space, there exist M 0 and M 1 points
(minimum and maximum, respectively). The functional dependence of the JDOS at the critical points
is summarized in Table 9.3. The resulting shape of the dielectric function is visualized in Fig. 9.8.
9.7.2 Direct Transitions
Transitions between states at the band edges at the point are possible (Fig. 9.9). The k conservation
requires (almost) vertical transitions in the E(k) diagram because the length of the light k vector,
k = 2π/λ, is much smaller than the size of the Brillouin zone |k| ≤ π/a 0 . The ratio of the lengths of
the k vectors is of the order a 0 /λ and typically about 10
−3 for NIR wavelengths.
For isotropic parabolic bands the band-band transition energy versus wavevector relation is
E cv (k) = E g +
2
2
1
m ∗
e
+
1
m
∗
h
k
2
.
(9.44)