9.6 Electron–Photon Interaction
267
30
20
10
0
|p | (eV)
cv
2
A
A
L M
H K
60
40
20
0
A
A
L M
H K
E||c
E c
C
B
A
GaN
Fig. 9.7 Theoretical momentum matrix elements | p cv | 2 along high-symmetry directions in the Brillouin zone (see
Fig. 3.38d) for transitions between valence and conduction bands in GaN and light polarized perpendicular (left panel)
and parallel (right panel) to the c-axis. The transitions are A: 9 (A)→ 7c , B: 7 (B)→ 7c , C: 7 (C)→ 7c (see
Fig. 6.44). Adapted from [840]
given by
=
1
4ππ 0
2π e
m ω
2
|p cv |
2
k
δ (E c (k) − E v (k) − ω) d
3 k
(9.39a)
= 1 +
k
e
2
0 m ω 2
cv
2 |p cv |
2
m cv
1
1 − ω 2 /ω 2
cv
d
3 k ,
(9.39b)
with cv = E c (k) − E v (k). Equation (9.39b) has been obtained via the Kramers–Kronig relations
4
(see Appendix C).
Comparison with (D.7) yields that the oscillator strength of the band–band absorption is given by
f =
e
2
0 m ω 2
cv
2 |p cv |
2
m cv
=
e
2
0 m ω 2
cv
N cv ,
(9.40)
with the classical ’number’ of oscillators with the frequency ω cv ,
N cv =
2 |p cv |
2
m cv
.
(9.41)
9.7 Band–Band Transitions
9.7.1 Joint Density of States
The strength of an allowed optical transitions between valence and conduction bands is proportional
to the joint density of states (JDOS) D j (E cv ) (cf. (6.63), (6.64) and (9.39a))
4 The real and imaginary parts of the dielectric function are generally related to each other via the Kramers–Kronig
relations.