10.4 Phonon Replica
319
(a)
3.1
1LO
3.2
3.4
3.3
3.5
10
0
10
10
10
-1
-2
-3
GaN
5LO
2LO
4LO
3LO
92meV
(b)
Fig. 10.19 a Photoluminescence spectrum of GaN (grown on SiC substrate) at T = 50 K. In addition to emission from
free (FE) and bound (BE) excitons several phonon replica (labeled as 1LO–5LO) are observed. Vertical dashed lines
indicate energy positions of multiple LO-phonon energies ( LO = 92 meV) below the FE peak. Adapted from [993].
b Photoluminescence spectrum of 1 LO phonon-assisted recombination peak at T = 103 K (from the data of Fig. 10.5).
Data points (dots) and lineshape fit (solid line) according to (10.32) with the parameters L 1 = 0.9 and E 1 = 3.2955 eV
(and background)
Here, E ex represents the exciton kinetic energy. The factor w n (E ex ) accounts for the q-dependence of
the matrix element. It is typically expressed as
w n (E ex ) ∝ E
L n
ex .
(10.33)
Accordingly, as temperature dependent refinement of (10.31), the energy separation n of the energy
of the peak maximum of phonon replica from E 0 is given by
n = E n − E 0 = −n ph +
L n +
1
2
kT .
(10.34)
It is found theoretically that L 1 = 1 and L 2 =0 [994]. These relations are approximately fulfilled for
GaN [995]. A lineshape fit for the 1 LO phonon-assisted transition in ZnO is shown in Fig. 10.19b.
In Fig. 10.20a the ‘green band’ emission of ZnO is shown as presented in [996]. This band is mostly
attributed to a Cu impurity; recently, evidence has grown from isotope decay and annealing studies that
it is related to the zinc vacancy [997] (Fig. 10.20b). The zero phonon line is followed by many replica
with a maximum at about 6 LO phonons. The intensity I N of the N -th replica is given by [998, 999]
I N ∝ exp(−S)
S
N
N !
,
(10.35)
where S is the so-called Huang–Rhys parameter. In [997], a coupling parameter of S = 6.9 has been
determined.
Equation (10.35) is obtained from the consideration of transitions in the configuration diagram [998,
1000] (Fig. 10.21), using the Born–Oppenheimer approximation. Here the electronic wavefunctions
are separated from the vibrational wavefunctions, leading to the Franck–Condon principle, that optical
transitions occur with the positions of the nuclei fixed and thus vertical in the configuration diagram
Fig. 10.21. Assuming low temperatures, only the lowest state is (partially) occupied. The Huang–Rhys
parameter, the average number of phonons involved in the transition, is related to the displacement
δq = q 1 − q 0 of the two configurations
319
(a)
3.1
1LO
3.2
3.4
3.3
3.5
10
0
10
10
10
-1
-2
-3
GaN
5LO
2LO
4LO
3LO
92meV
(b)
Fig. 10.19 a Photoluminescence spectrum of GaN (grown on SiC substrate) at T = 50 K. In addition to emission from
free (FE) and bound (BE) excitons several phonon replica (labeled as 1LO–5LO) are observed. Vertical dashed lines
indicate energy positions of multiple LO-phonon energies ( LO = 92 meV) below the FE peak. Adapted from [993].
b Photoluminescence spectrum of 1 LO phonon-assisted recombination peak at T = 103 K (from the data of Fig. 10.5).
Data points (dots) and lineshape fit (solid line) according to (10.32) with the parameters L 1 = 0.9 and E 1 = 3.2955 eV
(and background)
Here, E ex represents the exciton kinetic energy. The factor w n (E ex ) accounts for the q-dependence of
the matrix element. It is typically expressed as
w n (E ex ) ∝ E
L n
ex .
(10.33)
Accordingly, as temperature dependent refinement of (10.31), the energy separation n of the energy
of the peak maximum of phonon replica from E 0 is given by
n = E n − E 0 = −n ph +
L n +
1
2
kT .
(10.34)
It is found theoretically that L 1 = 1 and L 2 =0 [994]. These relations are approximately fulfilled for
GaN [995]. A lineshape fit for the 1 LO phonon-assisted transition in ZnO is shown in Fig. 10.19b.
In Fig. 10.20a the ‘green band’ emission of ZnO is shown as presented in [996]. This band is mostly
attributed to a Cu impurity; recently, evidence has grown from isotope decay and annealing studies that
it is related to the zinc vacancy [997] (Fig. 10.20b). The zero phonon line is followed by many replica
with a maximum at about 6 LO phonons. The intensity I N of the N -th replica is given by [998, 999]
I N ∝ exp(−S)
S
N
N !
,
(10.35)
where S is the so-called Huang–Rhys parameter. In [997], a coupling parameter of S = 6.9 has been
determined.
Equation (10.35) is obtained from the consideration of transitions in the configuration diagram [998,
1000] (Fig. 10.21), using the Born–Oppenheimer approximation. Here the electronic wavefunctions
are separated from the vibrational wavefunctions, leading to the Franck–Condon principle, that optical
transitions occur with the positions of the nuclei fixed and thus vertical in the configuration diagram
Fig. 10.21. Assuming low temperatures, only the lowest state is (partially) occupied. The Huang–Rhys
parameter, the average number of phonons involved in the transition, is related to the displacement
δq = q 1 − q 0 of the two configurations