10.3 Exciton Recombination
315
Fig. 10.14 (Temperature
dependent PL intensity of
(D 0 ,X) in GaN and (A0,X)
in AlN:Mg recombination.
Solid lines are fits with
(10.26). Data from [960,
978]
The luminescence intensity I (T ) of bound exciton lines is quenched with increasing temperature
due to ionization of the excitons from the impurities. The temperature dependence can be modeled
using the relation [976]
I (T )
I (T = 0)
=
1
1 + C exp(−E A /kT )
,
(10.26)
E A being the thermal activation energy and C a pre-factor. Often the activation energy is found equal
to the localization energy, E A = Q (Fig. 10.14, cmp. Table 10.2). If several processes contribute,
additional exponential terms can be added with further activation energies. For acceptor-bound excitons
in GaAs two processes are found to contribute, the ionization from the impurity into a free exciton
(E
1
A ≈ Q) and into an electron-hole pair (E
2
A ≈ Q + E
b
X ) [976]. In [977] the model is refined by
considering the temperature dependence of the parameter C due to the ionization of the impurity itself.
So far single excitons bound to a center have been discussed. Also bound exciton complexes [979]
containing up to six excitons have been observed at sufficient excitation density, e.g. for substitutional
boron [980] or phosphorus [981] and interstitial Li [982] in silicon. In a multi-valley semiconductor
several electrons are available to form bound excitons which follow approximately a shell model and
exhibit further fine structure.
10.3.3 Alloy Broadening
The bound-exciton recombination peak in a binary compound is spectrally fairly sharp (Sect. 10.3.2),
even in the presence of isotope disorder (Fig. 10.11). In an alloy (see Sect. 3.7), the random distribution
of atoms (with different atomic order number Z ) causes a significant broadening effect of the luminescence (and absorption) line, the so-called alloy broadening [983, 984]. As an example, we treat
Al x Ga 1−x As. The exciton samples, at different positions of the lattice, different coordinations of Ga
and Al atoms. If the experiment averages over these configurations, an inhomogeneously broadened
line is observed.
The cation concentration c c for the zincblende lattice is given as c c = 4/a
3
0 , for the wurtzite lattice
as c c = 4/(
√
3 a
2 c). For example, c c = 2.2 × 10
22 cm
−3 for Al x Ga 1−x As in the entire composition
range 0 ≤ x ≤ 1 since the lattice constant does not vary significantly, and c c = 4.2 × 10
22 cm
−3 for
wurtzite Mg x Zn 1−x O [985]. In a random alloy, the probability p(N ) to find exactly N Ga atoms in a
given volume V (with a total of c c V cations) is given by the binomial distribution
315
Fig. 10.14 (Temperature
dependent PL intensity of
(D 0 ,X) in GaN and (A0,X)
in AlN:Mg recombination.
Solid lines are fits with
(10.26). Data from [960,
978]
The luminescence intensity I (T ) of bound exciton lines is quenched with increasing temperature
due to ionization of the excitons from the impurities. The temperature dependence can be modeled
using the relation [976]
I (T )
I (T = 0)
=
1
1 + C exp(−E A /kT )
,
(10.26)
E A being the thermal activation energy and C a pre-factor. Often the activation energy is found equal
to the localization energy, E A = Q (Fig. 10.14, cmp. Table 10.2). If several processes contribute,
additional exponential terms can be added with further activation energies. For acceptor-bound excitons
in GaAs two processes are found to contribute, the ionization from the impurity into a free exciton
(E
1
A ≈ Q) and into an electron-hole pair (E
2
A ≈ Q + E
b
X ) [976]. In [977] the model is refined by
considering the temperature dependence of the parameter C due to the ionization of the impurity itself.
So far single excitons bound to a center have been discussed. Also bound exciton complexes [979]
containing up to six excitons have been observed at sufficient excitation density, e.g. for substitutional
boron [980] or phosphorus [981] and interstitial Li [982] in silicon. In a multi-valley semiconductor
several electrons are available to form bound excitons which follow approximately a shell model and
exhibit further fine structure.
10.3.3 Alloy Broadening
The bound-exciton recombination peak in a binary compound is spectrally fairly sharp (Sect. 10.3.2),
even in the presence of isotope disorder (Fig. 10.11). In an alloy (see Sect. 3.7), the random distribution
of atoms (with different atomic order number Z ) causes a significant broadening effect of the luminescence (and absorption) line, the so-called alloy broadening [983, 984]. As an example, we treat
Al x Ga 1−x As. The exciton samples, at different positions of the lattice, different coordinations of Ga
and Al atoms. If the experiment averages over these configurations, an inhomogeneously broadened
line is observed.
The cation concentration c c for the zincblende lattice is given as c c = 4/a
3
0 , for the wurtzite lattice
as c c = 4/(
√
3 a
2 c). For example, c c = 2.2 × 10
22 cm
−3 for Al x Ga 1−x As in the entire composition
range 0 ≤ x ≤ 1 since the lattice constant does not vary significantly, and c c = 4.2 × 10
22 cm
−3 for
wurtzite Mg x Zn 1−x O [985]. In a random alloy, the probability p(N ) to find exactly N Ga atoms in a
given volume V (with a total of c c V cations) is given by the binomial distribution