10.2 Band–Band Recombination
305
(a)
1.35
1.40
1.45
1.50
1.55
0
1
2
3
4
5
GaAs
E=
46meV
E =
1.423eV
g
(b)
Fig. 10.2 a Photoluminescence spectrum of an undoped LPE-grown epitaxial GaAs layer at room temperature and low
cw (λ = 647 nm) excitation density (10 W/cm 2 ). The solid line is a lineshape fit with (10.3) and E g =1.423 eV and
T = 293 K. (b) Room temperature, direct (e –h ) recombination from heavily n-doped (10 19 cm −3 ) germanium (1 µm
thick Ge layer on silicon (001)) with biaxial (thermal) tensile strain. The strain-split valence band edge (Fig. 6.50) causes
the e–hh and e–lh transitions (individual contributions with lineshape according to (10.3) shown as dashed lines) to occur
at different energies. Adapted from [939]
Fig. 10.3 Carrier
temperature T C in GaAs as
a function of excitation
density at a lattice
temperature of 1.6 K. The
dashed line is guide to the
eye, the solid line
corresponds to an
activation energy of
33 meV, similar to the
GaAs optical phonon
energy. Adapted
from [940]
15
20
40
100
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
10
-4
10
-3
10
-2
10
-1
1
Excitation density (arb. units)
1/T (K )
C
-1
T (K)
C
n-type
p-type
GaAs
The lineshape of the band–band recombination with k-conservation
2 is proportional to the joint
density of states (9.42) and the Fermi distribution function. At small excitation and at low doping it
can be approximated by the Boltzmann distribution function and the lineshape is given as
I (E) ∝
E − E g exp
−
E
kT
.
(10.3)
An experimental spectrum is shown in Fig. 10.2 together with a fit according to (10.3). The expected
FWHM of the peak is 1.7954 kT , which is about 46 meV at T = 300 K. At low sample temperature,
the temperature of the carrier gas is typically higher than the lattice temperature, depending on the
cooling mechanisms (carrier–carrier scattering, optical phonon emission, acoustic phonon emission,
recombination, . . .) and the excitation rate. The carrier temperature in GaAs, determined from the
Boltzmann tail of spontaneous emission (photoluminescence) is depicted in Fig. 10.3 as a function of
excitation density; clearly it increases with increasing excitation.
The recombination rate in indirect semiconductors is small since the transition is phonon-assisted.
For silicon, an internal quantum efficiency in the 10
−6 -range has been reported [941]. For germanium,
2 Excitonic effects are neglected here, e.g. for temperatures kT E b
X . Such effects are discussed in Sect. 10.3.
305
(a)
1.35
1.40
1.45
1.50
1.55
0
1
2
3
4
5
GaAs
E=
46meV
E =
1.423eV
g
(b)
Fig. 10.2 a Photoluminescence spectrum of an undoped LPE-grown epitaxial GaAs layer at room temperature and low
cw (λ = 647 nm) excitation density (10 W/cm 2 ). The solid line is a lineshape fit with (10.3) and E g =1.423 eV and
T = 293 K. (b) Room temperature, direct (e –h ) recombination from heavily n-doped (10 19 cm −3 ) germanium (1 µm
thick Ge layer on silicon (001)) with biaxial (thermal) tensile strain. The strain-split valence band edge (Fig. 6.50) causes
the e–hh and e–lh transitions (individual contributions with lineshape according to (10.3) shown as dashed lines) to occur
at different energies. Adapted from [939]
Fig. 10.3 Carrier
temperature T C in GaAs as
a function of excitation
density at a lattice
temperature of 1.6 K. The
dashed line is guide to the
eye, the solid line
corresponds to an
activation energy of
33 meV, similar to the
GaAs optical phonon
energy. Adapted
from [940]
15
20
40
100
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
10
-4
10
-3
10
-2
10
-1
1
Excitation density (arb. units)
1/T (K )
C
-1
T (K)
C
n-type
p-type
GaAs
The lineshape of the band–band recombination with k-conservation
2 is proportional to the joint
density of states (9.42) and the Fermi distribution function. At small excitation and at low doping it
can be approximated by the Boltzmann distribution function and the lineshape is given as
I (E) ∝
E − E g exp
−
E
kT
.
(10.3)
An experimental spectrum is shown in Fig. 10.2 together with a fit according to (10.3). The expected
FWHM of the peak is 1.7954 kT , which is about 46 meV at T = 300 K. At low sample temperature,
the temperature of the carrier gas is typically higher than the lattice temperature, depending on the
cooling mechanisms (carrier–carrier scattering, optical phonon emission, acoustic phonon emission,
recombination, . . .) and the excitation rate. The carrier temperature in GaAs, determined from the
Boltzmann tail of spontaneous emission (photoluminescence) is depicted in Fig. 10.3 as a function of
excitation density; clearly it increases with increasing excitation.
The recombination rate in indirect semiconductors is small since the transition is phonon-assisted.
For silicon, an internal quantum efficiency in the 10
−6 -range has been reported [941]. For germanium,
2 Excitonic effects are neglected here, e.g. for temperatures kT E b
X . Such effects are discussed in Sect. 10.3.