316
10 Recombination
(a)
(b)
Fig. 10.15 a Spectral width of the photoluminescence from CdS x Se 1−x alloys. Solid line is theory according to (10.30).
Adapted from [983]. (b) Spectral width of the bound exciton recombination in Al x Ga 1−x As with various Al content
within the direct-bandgap regime. Solid line is (10.30) with (10.28), dashed line with pre-factor 4π/3 instead of 10 π.
Adapted from [987]
p(N ) =
c c V
N
x
N
(1 − x)
c c V −N
.
(10.27)
The sampling volume for a luminescence event is the exciton volume (cf. (9.51)) that is given for the
free-exciton (in 1s hydrogen state) as [983, 986]
V ex = 10 π a
3
X = 10 π
m 0
m ∗
r
s a B
3
.
(10.28)
One should note that due to the variation of the involved material parameters V ex depends itself on
x. In GaAs there are about 1.2 × 10
6 cations in the exciton volume. In Al x Ga 1−x As, there are on
average xc c V ex Al atoms in the exciton volume. The fluctuation is given by the standard deviation of
the binomial distribution [986]
σ
2
x =
x (1 − x)
c c V ex
.
(10.29)
The corresponding energetic broadening (full width at half-maximum) of the spectral line is given by
E = 2.36 σ with
σ =
∂ E g
∂x
σ x =
∂ E g
∂x
x (1 − x)
c c V ex
.
(10.30)
We note that instead of the quantum mechanically correct factor 10 π [983, 986], often the factor 4π/3
[984] is used, resulting in larger theoretical broadening.
Experimental data for Cd x Se 1−x in Fig. 10.15a are consistent with (10.30). The theoretical dependence (10.30) is shown in Fig. 10.15b also for Al x Ga 1−x As together with experimental data and found
to disagree [987]. Since the exciton volume is much smaller (cf. Sect. 9.7.6) than in Al x Ga 1−x As, alloy
broadening in Mg x Zn 1−x O is much larger for a given for given x.
The spectral broadening due to alloy disorder masks the fine structure of recombination lines near the
band edge present for binary semiconductors. Often for all temperatures only a single recombination
line appears for alloys. Spectra for three different Mg x Zn 1−x O alloys are shown in Fig. 10.16a. The
increasing inhomogeneous broadening is obvious, causing a single peak for x > 0.03. The temperature
dependence of the peak positions is shown in Fig. 10.17 for the same samples. For x = 0.005 the bound
exciton (Al-donor) (D
0 ,X) and free exciton (X A ) recombination lines can still be resolved despite
10 Recombination
(a)
(b)
Fig. 10.15 a Spectral width of the photoluminescence from CdS x Se 1−x alloys. Solid line is theory according to (10.30).
Adapted from [983]. (b) Spectral width of the bound exciton recombination in Al x Ga 1−x As with various Al content
within the direct-bandgap regime. Solid line is (10.30) with (10.28), dashed line with pre-factor 4π/3 instead of 10 π.
Adapted from [987]
p(N ) =
c c V
N
x
N
(1 − x)
c c V −N
.
(10.27)
The sampling volume for a luminescence event is the exciton volume (cf. (9.51)) that is given for the
free-exciton (in 1s hydrogen state) as [983, 986]
V ex = 10 π a
3
X = 10 π
m 0
m ∗
r
s a B
3
.
(10.28)
One should note that due to the variation of the involved material parameters V ex depends itself on
x. In GaAs there are about 1.2 × 10
6 cations in the exciton volume. In Al x Ga 1−x As, there are on
average xc c V ex Al atoms in the exciton volume. The fluctuation is given by the standard deviation of
the binomial distribution [986]
σ
2
x =
x (1 − x)
c c V ex
.
(10.29)
The corresponding energetic broadening (full width at half-maximum) of the spectral line is given by
E = 2.36 σ with
σ =
∂ E g
∂x
σ x =
∂ E g
∂x
x (1 − x)
c c V ex
.
(10.30)
We note that instead of the quantum mechanically correct factor 10 π [983, 986], often the factor 4π/3
[984] is used, resulting in larger theoretical broadening.
Experimental data for Cd x Se 1−x in Fig. 10.15a are consistent with (10.30). The theoretical dependence (10.30) is shown in Fig. 10.15b also for Al x Ga 1−x As together with experimental data and found
to disagree [987]. Since the exciton volume is much smaller (cf. Sect. 9.7.6) than in Al x Ga 1−x As, alloy
broadening in Mg x Zn 1−x O is much larger for a given for given x.
The spectral broadening due to alloy disorder masks the fine structure of recombination lines near the
band edge present for binary semiconductors. Often for all temperatures only a single recombination
line appears for alloys. Spectra for three different Mg x Zn 1−x O alloys are shown in Fig. 10.16a. The
increasing inhomogeneous broadening is obvious, causing a single peak for x > 0.03. The temperature
dependence of the peak positions is shown in Fig. 10.17 for the same samples. For x = 0.005 the bound
exciton (Al-donor) (D
0 ,X) and free exciton (X A ) recombination lines can still be resolved despite