280
9 Optical Properties
Table 9.6 Exciton energy (low temperature), LT splitting and exciton polariton oscillator strength for various semiconductors. Values for ZnO from [878], values for GaAs from [879], all other values from [880]
CdS A
CdS B
ZnO A
ZnO B
ZnSe
GaN A
GaN B
GaAs
ω T
(eV)
2.5528
2.5681
3.3776
3.3856
2.8019
3.4771
3.4816
1.5153
LT
(meV)
2.2
1.4
1.45
5
1.45
1.06
0.94
0.08
β
(10 −3 )
1.7
1.1
0.9
3.0
1.0
0.6
0.5
0.11
(ω) = b
1 +
β
1 − (ω 2 /ω 0 ) 2 + D k 2
.
(9.57)
The term k
2 with curvature D (for the exciton polariton D = /(M ω T )) plays a role in particular when
ω
2
T − ω
2
= 0. For k = 0 even a cubic material is anisotropic. The dimensionless curvature ˆ
D = Dk
2
should fulfill ˆ
D = /(Mc) 1 in order to make k
4 terms unimportant. For exciton polaritons
8
typically ˆ
D = ω T /(m c
2
) ≈ 2 × 10
−5 for ω T = 1 eV and m
∗
= 0.1.
From (9.56) together with (9.57) two solutions result:
2ω
2
= c
2 k
2
+ (1 + β + Dk
2
) ω
2
0
(9.58)
±
−4c
2 k
2
(1 + Dk
2
) ω
2
0 + (c
2 k
2
+ (1 + β + Dk
2
) ω
2
0 )
2
1/2 .
The two branches are shown schematically in Fig. 9.25a. Depending on the k value they have a photonic
(linear dispersion) or excitonic (quadratic dispersion) character. The anticrossing behavior at k
≈ ω T /c
(for ω T = 1 eV, k
≈ 0.5 × 10
−5 cm
−1 ) creates a bottleneck region in the lower polariton branch. This
name stems from the small emission rate of acoustic phonons (i.e. cooling) in that region, as predicted
in [876] and experimentally found, e.g. in CdS [877]. The polaritons decay into a photon when they hit
the surface. The effect of the oscillator strength of the dispersion is shown in Fig. 9.26 for two-exciton
resonance. In the case of several excitons (9.57) reads
(ω) = b
1 +
n
i=1
β i
1 − (ω 2 /ω 0,i ) 2 + D i k 2
.
(9.59)
For k = 0 either ω = 0 (lower polariton branch) or (ω L ) = 0. For the latter we find from (9.57)
ω L =
1 + β ω T .
(9.60)
Therefore, the energy splitting E LT , mostly denoted as LT , between the L- and T-exciton energy
given by
E LT = (ω L − ω T ) =
1 + β − 1
ω T ≈ β ω T /2
(9.61)
is proportional to the exciton oscillator strength (for experimental values see Table 9.6). We note that
if (D.9) is used for the dielectric function, β in (9.61) needs to be replaced by β/ b .
The effect of spatial dispersion on the reflection at the fundamental exciton resonance is depicted in
Fig. 9.25b. For non-normal incidence an additional feature due to the longitudinal wave is observed for
p-polarization [875]. For a detailed discussion additional effects due to anisotropy in wurtzite crystals,
8 The dependence of the optical-phonon energies on k is typically too small to make spatial dispersion effects important.
According to (5.19) ˆ
D = −(a 0 ω TO /4c) 2 ≈ 4 × 10 −11 for typical material parameters (lattice constant a 0 = 0.5 nm, TO
phonon frequency ω TO = 15 THz).
9 Optical Properties
Table 9.6 Exciton energy (low temperature), LT splitting and exciton polariton oscillator strength for various semiconductors. Values for ZnO from [878], values for GaAs from [879], all other values from [880]
CdS A
CdS B
ZnO A
ZnO B
ZnSe
GaN A
GaN B
GaAs
ω T
(eV)
2.5528
2.5681
3.3776
3.3856
2.8019
3.4771
3.4816
1.5153
LT
(meV)
2.2
1.4
1.45
5
1.45
1.06
0.94
0.08
β
(10 −3 )
1.7
1.1
0.9
3.0
1.0
0.6
0.5
0.11
(ω) = b
1 +
β
1 − (ω 2 /ω 0 ) 2 + D k 2
.
(9.57)
The term k
2 with curvature D (for the exciton polariton D = /(M ω T )) plays a role in particular when
ω
2
T − ω
2
= 0. For k = 0 even a cubic material is anisotropic. The dimensionless curvature ˆ
D = Dk
2
should fulfill ˆ
D = /(Mc) 1 in order to make k
4 terms unimportant. For exciton polaritons
8
typically ˆ
D = ω T /(m c
2
) ≈ 2 × 10
−5 for ω T = 1 eV and m
∗
= 0.1.
From (9.56) together with (9.57) two solutions result:
2ω
2
= c
2 k
2
+ (1 + β + Dk
2
) ω
2
0
(9.58)
±
−4c
2 k
2
(1 + Dk
2
) ω
2
0 + (c
2 k
2
+ (1 + β + Dk
2
) ω
2
0 )
2
1/2 .
The two branches are shown schematically in Fig. 9.25a. Depending on the k value they have a photonic
(linear dispersion) or excitonic (quadratic dispersion) character. The anticrossing behavior at k
≈ ω T /c
(for ω T = 1 eV, k
≈ 0.5 × 10
−5 cm
−1 ) creates a bottleneck region in the lower polariton branch. This
name stems from the small emission rate of acoustic phonons (i.e. cooling) in that region, as predicted
in [876] and experimentally found, e.g. in CdS [877]. The polaritons decay into a photon when they hit
the surface. The effect of the oscillator strength of the dispersion is shown in Fig. 9.26 for two-exciton
resonance. In the case of several excitons (9.57) reads
(ω) = b
1 +
n
i=1
β i
1 − (ω 2 /ω 0,i ) 2 + D i k 2
.
(9.59)
For k = 0 either ω = 0 (lower polariton branch) or (ω L ) = 0. For the latter we find from (9.57)
ω L =
1 + β ω T .
(9.60)
Therefore, the energy splitting E LT , mostly denoted as LT , between the L- and T-exciton energy
given by
E LT = (ω L − ω T ) =
1 + β − 1
ω T ≈ β ω T /2
(9.61)
is proportional to the exciton oscillator strength (for experimental values see Table 9.6). We note that
if (D.9) is used for the dielectric function, β in (9.61) needs to be replaced by β/ b .
The effect of spatial dispersion on the reflection at the fundamental exciton resonance is depicted in
Fig. 9.25b. For non-normal incidence an additional feature due to the longitudinal wave is observed for
p-polarization [875]. For a detailed discussion additional effects due to anisotropy in wurtzite crystals,
8 The dependence of the optical-phonon energies on k is typically too small to make spatial dispersion effects important.
According to (5.19) ˆ
D = −(a 0 ω TO /4c) 2 ≈ 4 × 10 −11 for typical material parameters (lattice constant a 0 = 0.5 nm, TO
phonon frequency ω TO = 15 THz).