9.7 Band–Band Transitions
279
(a)
(b)
Fig. 9.24 a Absorption spectra of GaN bulk (0.38 µm thick epilayer on sapphire) for various temperatures T = 100,
200, 300, 350, 400, 450, and 475 K. Adapted from [868] b Homogeneous broadening as a function of temperature,
symbols are experimental data, solid lines are fits, rf. Table 9.5
Table 9.5 Phonon broadening parameters (FWHM) of various bulk semiconductors. Values from fits with (9.54) to
experimental data for GaAs [871], ZnSe [869], GaN [868], ZnO [872] (phonon energy fitted) as shown in Fig. 9.25b
Material
LO (meV)
0 (meV)
γ AC (µeV/K)
γ LO (meV)
GaAs
36.8
0
4 ± 2
16.8 ± 2
ZnSe
30.5
1.9
0 ± 7
84 ± 8
GaN
92
10
15 ± 4
408 ± 30
ZnO
33
1.2
32 ± 26
96 ± 24
9.7.8 Exciton Polariton
Electrons and holes are particles with spin 1/2. Thus, the exciton can form states with total spin S = 0
(para-exciton, singlet) and S = 1 (ortho-exciton, triplet). The exchange interaction leads to a splitting
of these states, the singlet being the energetically higher. The singlet state splits into the longitudinal
and transverse exciton with respect to the orientation of the polarization carried by the Bloch functions
and the center-of-mass motion K of the exciton. Dipole transitions are only possible for singlet excitons
(bright excitons). The triplet excitons couple only weakly to the electromagnetic field and are thus also
called dark excitons.
The coupling of these states to the electromagnetic field creates new quasi-particles, the exciton
polaritons [873, 874]. The dielectric function of the exciton (with background dielectric constant b ) is
= b
1 +
β
1 − (ω 2 /ω X ) 2
∼ = b
1 +
β
1 − (ω 2 /ω T ) 2 + K 2 /(M ω T )
,
(9.55)
where β is the oscillator strength and the energy is X = T +
2 K
2
/2M. T is the energy of the
transverse exciton at K = 0. With this dispersion the wave dispersion must be fulfilled, i.e.
c
2 k
2
= ω
2
,
(9.56)
where k is the k-vector of the light that needs to be k = K due to momentum conservation. The
dependence of the dielectric function on k is called spatial dispersion [875]. Generally, up to terms in
k
2 it is written as
279
(a)
(b)
Fig. 9.24 a Absorption spectra of GaN bulk (0.38 µm thick epilayer on sapphire) for various temperatures T = 100,
200, 300, 350, 400, 450, and 475 K. Adapted from [868] b Homogeneous broadening as a function of temperature,
symbols are experimental data, solid lines are fits, rf. Table 9.5
Table 9.5 Phonon broadening parameters (FWHM) of various bulk semiconductors. Values from fits with (9.54) to
experimental data for GaAs [871], ZnSe [869], GaN [868], ZnO [872] (phonon energy fitted) as shown in Fig. 9.25b
Material
LO (meV)
0 (meV)
γ AC (µeV/K)
γ LO (meV)
GaAs
36.8
0
4 ± 2
16.8 ± 2
ZnSe
30.5
1.9
0 ± 7
84 ± 8
GaN
92
10
15 ± 4
408 ± 30
ZnO
33
1.2
32 ± 26
96 ± 24
9.7.8 Exciton Polariton
Electrons and holes are particles with spin 1/2. Thus, the exciton can form states with total spin S = 0
(para-exciton, singlet) and S = 1 (ortho-exciton, triplet). The exchange interaction leads to a splitting
of these states, the singlet being the energetically higher. The singlet state splits into the longitudinal
and transverse exciton with respect to the orientation of the polarization carried by the Bloch functions
and the center-of-mass motion K of the exciton. Dipole transitions are only possible for singlet excitons
(bright excitons). The triplet excitons couple only weakly to the electromagnetic field and are thus also
called dark excitons.
The coupling of these states to the electromagnetic field creates new quasi-particles, the exciton
polaritons [873, 874]. The dielectric function of the exciton (with background dielectric constant b ) is
= b
1 +
β
1 − (ω 2 /ω X ) 2
∼ = b
1 +
β
1 − (ω 2 /ω T ) 2 + K 2 /(M ω T )
,
(9.55)
where β is the oscillator strength and the energy is X = T +
2 K
2
/2M. T is the energy of the
transverse exciton at K = 0. With this dispersion the wave dispersion must be fulfilled, i.e.
c
2 k
2
= ω
2
,
(9.56)
where k is the k-vector of the light that needs to be k = K due to momentum conservation. The
dependence of the dielectric function on k is called spatial dispersion [875]. Generally, up to terms in
k
2 it is written as