Henini M. Handbook of Self Assembled Semiconductor Nanostructures for Novel devices in Photonics and Electronics
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Effects of the Electron–Phonon Interaction in Semiconductor Quantum Dots 359
where
FCGj
j
j
ββ
β
βλλ
λ
λ
λ
ω
11
()
{}
()
{}
{}
ΩΩ
∞
∞
∑∑
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
(11.38)
Mmm
jjj
λββββ λββββ λββββ
ββ ββ
δδ
1234 1234 1111
12 34
() () ()
(11.39)
m
j
y
j
jjjj
j
λβ β β β
λ
λ
λ
βγβ βγβ
1234
2323
2
1
()
|
.
〈〉〈〉|||
()
†
(11.40)
The above restriction to one- and two-phonon contributions is justifi ed when the non-adiabatic
exciton–phonon interaction is weak:
||〈| |
′
〉
′
≠βγ β ω β β
λλ
2
for
. The functions
Q
ββ
′
()
()
1
Ω
and
Q
ββ
′
()
()
2
Ω
in Eq. (11.32), which describe contributions of one- and two-phonon processes to
non-diagonal matrix elements of the evolution operator, take the form
QGG
Gjm
j
ββ
β
βββ
βλ
λββββ
λβ
δ
ω
′
′′
′
∑
(1)
()
,
() () ()( )
()
ΩΩΩ
Ω
1
1
11
1
jj 1
∑
(11.41)
QGG
Gj
ββ
β
βββ
βλ
βββλλ
δ
ω
′
′′
∑
(2)
() () ()( )ΩΩΩ
Ω
1
11
12312
1
()
,,,
∑∑∑
{
jj
GjGjj
m
12
32212
31
1
1
212
1
,
()
βλβλλ
ββ
λββ
ωωω
δ
()( )ΩΩ
1123
1
21223
2
1123
1
2123
βββ
λββββ
λββββ
λββββ
′
′
⎡
⎣
⎢
() ()
() (
jj
jj
m
mm
22
22
2
1112
1
2233
1
1
)
() (
⎤
⎦
⎥
′
′
G
mm
jj
βββ
ββ
λββββ
λββββ
δ
δ
()( )
()
Ω
22
)
.
}
(11.42)
The absorption spectrum is thus expressed through the functions G
β
( Ω ), which in turn are deter-
mined by a closed set of equations (11.33) and (11.36) to (11.38).
In Figs 11.5, 11.6, and 11.7 the calculated optical absorption spectra are shown for a single
quantum dot, two stacked identical quantum dots, and two stacked dots with somewhat differ-
ent heights, respectively. The calculations were performed for low temperatures,
{
}
y
λ
1
,
when the absorption-line broadening due to the exciton–LO-phonon interaction is negligible. The
broadening shown in Figs 11.5 to 11.7 is introduced only to enhance visualization. From the
comparison of the spectra obtained in the adiabatic approximation with those resulting from the
non-adiabatic approach, the following effects of non-adiabaticity are revealed. First, the polaron
shift of the zero-phonon lines with respect to the bare-exciton levels is larger in the non-adiabatic
approach than in the adiabatic approximation. Second, there is a strong increase of the intensities
of the phonon satellites compared to those given by the adiabatic approximation. This increase can
be by more than two orders of magnitude. Third, in the optical absorption spectra found within
the non-adiabatic approach, there appear phonon satellites related to non-active bare-exciton states.
Fourth, the optical-absorption spectra demonstrate the crucial role of non-adiabatic mix-
ing of different exciton and phonon states in quantum dots. This results in a rich structure of
the absorption spectrum of the exciton–phonon system [14, 15, 17] . For the stacked quantum
dots, this effect is signifi cantly enhanced when the exciton-level splitting, caused by the coupling
between quantum dots, is comparable with an LO-phonon energy [see Figs 11.6b and 11.7b ].
Similar conclusions about the infl uence of the exciton–phonon interaction on the optical spec-
tra of quantum dots have been formulated in [72] for the “ strong coupling regime ” for excitons
and LO-phonons. Such a “ strong coupling regime ” is a particular case of the non-adiabatic
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360 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
mixing related to a resonance, which arises when the spacing between exciton levels is close to
the LO-phonon energy. As seen from Figs 11.6 and 11.7 , effects of non-adiabaticity lead to a
signifi cant increase of absorption peaks in the spectral ranges, characteristic for one- and two-
phonon satellites, even in the case when the exciton-level spacing is relatively far from satisfying
the aforementioned resonant conditions.
In some cases (see, e.g., [11] ) the luminescence spectrum of a quantum dot can be easily
derived from its absorption spectrum. For example, under the assumption that the distribution
function of the states of an exciton coupled to the phonon fi eld, f ( Ω ), depends only on the energy
of a state, the luminescence intensity at low temperatures ( { y
λ
1 } ) can be represented as:
I
K
f
y
Kk
K
k
K
k
k
()
!
ΩΩ∝
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
∂
∂
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
∞
∑∑
∏
1
00
1
ω
λ
λ
⎟⎟
⎟
⎟
∑
→
λλ
α
λ
1
0
,,
{}
()
…
K
y
Ω .
(11.43)
Equation (11.43) is applicable, for instance, for the thermodynamic equilibrium photolumines-
cence. In this case the radiative lifetime of an exciton is much larger than the time characteristic
of radiationless relaxation between one-exciton states.
We have calculated the spectra of the thermodynamic equilibrium luminescence at T → 0 (not
shown here) for quantum dots with parameters indicated above. Both for single and for coupled
quantum dots, the intensities of phonon satellites in these spectra are signifi cantly smaller than
in the absorption spectrum. This is because in quantum dots under consideration the lowest one-
exciton energy level (with β 1) is less affected by the phonon-induced non-adiabatic mixing
of states than higher levels. The luminescence spectrum at zero temperature is determined by
Figure 11.5 Absorption spectra, calculated with the adiabatic approximation (panel (a)) and with the non-
adiabatic approach (panel (b)) for a single quantum dot. Optically active and non-active energy levels of a bare
exciton are shown as dotted and dashed lines, respectively. Ω
1
is the transition frequency for the lowest state of a bare
exciton. (Reprinted with permission from [20] . © 2004 by the American Physical Society.)
10
1
(a)
Absorption (arb. units)
3nm
8nm
8nm
10
1
10
2
10
3
10
4
10
5
1
Absorption (arb. units)
(b)
10
0
1
23
10
1
10
2
10
3
10
4
10
5
(ΩΩ
1
)/v
0
(InAs)
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Effects of the Electron–Phonon Interaction in Semiconductor Quantum Dots 361
transitions from the state with β 1, while the absorption spectrum contains appreciable contri-
butions due to transitions to higher exciton-phonon states.
11.5 Excitonic polarons in quantum dots: modifi cation of the optical spectra
The conclusion about the large enhancement of the two-phonon sidebands in the luminescence
spectra as compared to those predicted by the Huang-Rhys formula, which was explained in [11,
14] by non-adiabaticity of the exciton–phonon system in certain quantum dots, has been refor-
mulated in [72] in terms of the Fröhlich coupling between product states with different electron
and/or hole states.
A lens-shaped InAs/GaAs quantum dot is modelled in [72] as a truncated cone with basis
radius R and height h . The exciton states in this cylindrically symmetric system are classifi ed
in terms of the z -projection of the total angular momentum of the electron–hole pair L
z
. The
Coulomb term
H
e
C
eh
2
0
4πκε ||rr
(11.44)
with the dielectric constant κ mixes electron and hole pair states with the same L
z
(for example,
the states with L
z
0). In [72] , the case T 0 is analysed.
The exciton–phonon interaction Hamiltonian Eq. (11.3) with Eq. (11.20), leads to the direct
Fröhlich coupling between the factorized (adiabatic) exciton–phonon states, which differ by one
Figure 11.6 Absorption spectra, calculated with the adiabatic approximation (panel (a)) and with the non-
adiabatic approach (panel (b)) for two stacked identical quantum dots. Notations are the same as in Fig. 11.5 .
(Reprinted with permission from [20] . © 2004 by the American Physical Society.)
10
1
10
1
10
2
10
3
10
4
10
5
(a)
Absorption (arb. units)
1
10
10
1
10
2
10
3
10
4
10
5
0123
(
ΩΩ
1
)/
v
0
(InAs)
Absorption (arb. units)
(b)
3nm
3nm
3nm
8nm
8nm
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362 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
phonon and for which either the electron on the hole quantum number changes:
〈
′′′
〉∝
〈
′
⋅
′
nnnH nnn
ni
e h int e h
nn
nn
e
hh
,,| |,,
|exp(
,
qq
q
qrδδ
q
1
,
eee
n
nh hh
nnin
ee
| ) |exp( | )〉〈
′
⋅〉
⎡
⎣
⎢
⎤
⎦
⎥
′
δ
,
.qr
(11.45)
The Hamiltonian of the exciton–phonon systems was diagonalized numerically in the basis,
which consisted of the zero-phonon states
| , , | , , ,| , , ,| , ,SS S P S P
eh eh e h e h
0, P 0 0 P 0〉〉〉 〉
;
the one-phonon continuum
| , , | , , ,| , , ,| , ,SS S P S P
eh eh e h e h
1, P P 1
qq q q
〉〉 〉
11
; and the
two-phonon continuum
|, , |, , , ,| , , , ,|SS S P S P
eh eh e h
1,1 , P 11 11
q
q
q
q
q
q
′′ ′
〉〉 〉
eeh
,,, P 11
q
q
′
〉
.
This diagonalization led to entangled exciton–phonon states, which are called “ excitonic polaron
states ” in [72] and which realize – in the aforementioned basis – a representation of the non-adi-
abatic exciton–phonon states introduced in [11] .
Their energies are shown in Fig. 11.8 as a function of the quantum dot size. The part of the
spectrum with L
z
±1 reveals a double anticrossing between
|, , | , ,SPS
eh e h
P 0 ,
〉〉0
a n d
|, ,S
eh
P 1
q
〉
. The magnitude of the anticrossings constitutes 7(10) meV for R 7(12.5) nm.
The symmetrical [73] L
z
0 states reveal anticrossings as large as 7 meV when a factorized
symmetrical zero-phonon state is symmetrical with a factorized one-phonon continuum. When a
factorized symmetrical zero-phonon state crosses a two-phonon polaron continuum, weaker anti-
crossings occur through the one-phonon continuum states
|, ,S
eh
P 1
q
〉
| and
|, ,PS
eh
1
q
〉
.
In the excitonic ground state, the electron and hole envelope functions are very similar.
Correspondingly, the excitonic polaron behaves as a neutral particle and manifests a weak polaronic
red shift by about 0.9 meV with respect to the unperturbed factorized state
|, ,SS
eh
0
〉
.
Figure 11.7 Absorption spectra, calculated with the adiabatic approximation (panel (a)) and with the
non-adiabatic approach (panel (b)) for two stacked quantum dots of different height. Notations are the same as in
Fig. 11.5 . (Reprinted with permission from [20] . © 2004 by the American Physical Society.)
10
1
(a)
Absorption (arb. units)
10
1
10
2
10
3
10
4
10
5
1
Absorption (arb. units)
0123
(ΩΩ
1
)/v
0
(InAs)
(b)
10
10
1
10
2
10
3
10
4
10
5
3.3 nm
3nm
3nm
8nm
8nm
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Effects of the Electron–Phonon Interaction in Semiconductor Quantum Dots 363
For absorption at T 0, the initial state is the vacuum with zero phonons. Therefore, the fi nal
state should contain an optically active zero-phonon component. As shown in Fig. 11.9 , the cal-
culated energies and intensities of interband absorption for excitonic polaron are signifi cantly dif-
ferent from the purely excitonic predictions. As in [11] , it is found that there are more absorption
lines within the excitonic polaron description than in the adiabatic exciton approximation. The
latter, in the range of quantum dot radii under consideration, gives two lines of comparable inten-
sity, which correspond to the L
z
0 states (corresponding to the
|,SS
eh
〉
and the symmetrical
combination of the
|,P
eh
P 〉
as shown in Fig. 11.8b ). The excitonic polaron absorption displays
up to fi ve lines of different strength. Except the ground-state transition, which mostly relates to
the
|, ,SS
eh
0〉
state, their energy positions and intensities vary substantially with radius. The
microscopic interpretation of the excitonic polaron absorption lines follows directly from Fig.
11.8b . In terms of the earlier works [11, 14] , those absorption lines are due to the non-adiabatic
transitions in the exciton–phonon system. The excitonic polaron reveals also an absorption
Figure 11.8 Excitonic polaron energies versus R ( h/R 0.1) in truncated cone-shaped InAs/GaAs quantum dots.
(a) L
z
1 states. (b) Symmetrical L
z
0 states. Only those states with more than 10% of zero phonon parts are
shown. The energy zero is taken at the energy of the factorized
|, , SS
eh
0〉
state. Solid lines: zero-phonon factorized
state. Dashed lines: one-phonon factorized states. Dotted lines: two-phonon factorized states. Symbols: excitonic
polaron energies. (Reprinted with permission after [72] . © 2002 by the American Physical Society.)
60
50
40
30
S
e
, S
h
, 1
q
S
e
, P
h
, 0
P
e
, S
h
, 0
Energy (meV)
20
10
80
(a)
90 100 110 120 130 140 150 160
R (Å)
≈
≈
S
e
, S
h
, 1
q
1
q
´
S
e
, P
h
, 1
q
P
e
, S
h
, 1
q
P
e
, P
h
, 0
(b)
100
90
80
70
60
50
80 110 120 130 140
R (Å)
90 100
40
Energy (meV)
≈
≈
®
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364 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
line at
–
h ω
LO
above the ground-state transition, but for parameters used in [72] it is too weak to be
shown in Fig. 11.9 .
The excitonic polaron model of [72] has a restricted basis and therefore could not explain the
totality of the absorption lines in the energy range of the expected excitonic transitions observed
in [74] .
The key difference between luminescence and absorption in the excitonic polaron picture is
that even at T 0 there exists a possibility of recombining to the vacuum state with 0,1,2 … pho-
nons. This gives rise to 1-, 2- … phonon satellite lines, which are red shifted from the zero-phonon
line by
–
h ω
LO
, 2
–
h ω
LO
… . The calculation in [72] showed that, e.g., for a quantum dot with R 8 n m
a two-photon satellite has intensity, which is about 100 times larger than that predicted within
the adiabatic Pekar–Huang-Rhys approach. Thus the prediction of large enhancement of the
phonon satellites in the luminescence spectra of exciton–phonon systems obtained within the
non-adiabatic approach [11] has been rederived using the Fröhlich coupling between the adi-
abatic product states [72] .
11.6 Recent studies
Due to non-adiabaticity, multiple absorption peaks appear in spectral ranges characteristic for
phonon satellites. From the states which correspond to these peaks, the system can rapidly relax
to the lowest emitting state. Therefore, in the photoluminescence excitation (PLE) spectra of
quantum dots, pronounced peaks can be expected in spectral ranges characteristic for phonon
satellites. Experimental evidence of the enhanced phonon-assisted absorption due to effects of
non-adiabaticity has been provided by PLE measurements on single self-assembled InAs/GaAs
[75] and InGaAs/GaAs [76] quantum dots. The polaron concept was also invoked for the expla-
nation of the PLE measurements on self-organized In
x
Ga
1
x
As/GaAs [77] and CdSe/ZnSe [78]
quantum dots. Recently [79] , the infl uence of quantum dot occupancy and incident light polari-
zation on the carrier–lattice interactions in electron-doped InAs self-assembled quantum dots
grown on GaAs has been investigated up to 28 T by far-infrared magnetotransmission spectros-
copy. An enhancement of the effi ciency of the electron–phonon coupling with increased electron
population from one to two electrons per dot is found, in good agreement with the predicted
2
Figure 11.9 Calculated energies and intensities of interband optical transitions in truncated cone-shaped InAs/
GaAs quantum dots versus R ( h/R 0.1). Solid lines: excitonic picture. Symbols: excitonic polaron description. The
radius of each symbol is proportional to the corresponding oscillator strength. Only symbols with radii larger than
5% of the ground transition one are displayed. The energy zero is taken at the ground excitonic transition. (Reprinted
with permission after [72] . © 2002 by the American Physical Society.)
100
90
80
70
60
50
90 100 110 120 130 140
R (Å)
80
0
1
2
Energy (meV)
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Effects of the Electron–Phonon Interaction in Semiconductor Quantum Dots 365
increase due to the antisymmetrization of the electron wave function in the two-electron quan-
tum dot. The authors of [80] have reported the direct observation of polarons in InAs/GaAs
self-assembled quantum dots populated by a few electrons. The degree of coupling is varied in a
systematic way with a set of samples having electron intersublevel spacing changing from larger
to smaller than the longitudinal-optical-phonon energy. The signature of polarons is manifested
clearly by the observation of a large (12–20 meV) anticrossing for both InAs and GaAs-like QD
phonons using resonant Raman spectroscopy.
Preparation of the CdS quantum dots with HgS quantum wells (QDQWs) was described in
[81] . Such structures possess greater photochemical stability as compared to ordinary quan-
tum dots. Excitons, localized in the HgS quantum well, are separated from the localized surface
levels by the CdS shell and, as a result, have longer lifetime and improved quantum yield/pho-
toresponse. It has been revealed that QDQWs are preferentially tetrahedral particles [82] with
zinc-blende crystal lattice. An insight into the nature of the optical response of CdS/HgS/CdS
QDQWs has been provided by a simultaneous consideration of the tetrahedral shape of a QDQW,
interface optical phonons, and non-adiabatic phonon-assisted transitions [18, 83] . The exciton
states and the photoluminescence spectra appear to be very sensitive to the shape of the QDQWs.
Exciton energies and wave functions have been found numerically, within a fi nite-difference
scheme [18] , as eigenstates of the Hamiltonian:
ˆ
[
ˆ
()] [
ˆ
()] ( )HHV HV V
eehh ehex sa sa int
,
rrrr
(11.46)
where
ˆ
H
e
is the two-band electron Hamiltonian and
ˆ
H
h
is the six-band hole Hamiltonian
derived from a general eight-band Hamiltonian for heterostructures [85] . The dielectric mis-
match at the boundaries of QDQW’s shells leads to the appearance of the electron and hole
self-interaction potentials V
s a
and signifi cantly changes the potential of the electron–hole inter-
action V
int
.
While only a few lowest electron states lie below the CdS bulk conduction band edge and are
localized near the HgS shell, many hole states are trapped in the HgS shell and lie above the CdS
bulk valence band edge. As shown in [18] , the electron in the ground state is practically uni-
formly distributed in the HgS shell, while the hole in the ground state is localized near the edges
of the tetrahedral HgS shell. Unlike ground state, the hole in the fi rst excited state substantially
penetrates the region of facets of the HgS shell, where the electron density in the ground state is
high. The electron states are widely separated by energy and the Coulomb mixing between the
electron–hole pair states with different electron states can be ignored. As distinct from that, the
hole states are closely spaced and the Coulomb mixing between the pair states with different hole
states is taken into account [18] .
In the QDQW’s absorption spectrum (Eq. 11.11), the intensity of a zero-phonon transition to
the exciton level β is
||d
β
2
(cp. Eq. 11.12). Figure 11.10 shows that the intensity of the second
Figure 11.10 Normalized intensities of the lowest exciton levels in the absorption spectrum of the tetrahedral
QDQW (the inset gives the intensities for the spherical QDQW considered in [86] ). Spectral lines are broadened
by a Gaussian to enhance visualization. (Reprinted with permission after [18] . © 2004 by the American Physical
Society.)
1
0.8
0.8
1
0.6
0.4
Ground state
Dark
excitons
Dark
excitons
Spherical
QDQW
Tetrahedral
QDQW
0.2
Intensity
0
Ground state
1940 1960
1980 2000 2020
Exciton energy (meV)
0.6
0.4
0.2
0
1860 1880 1900 1920
Exciton energy (meV)
Intensity
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366 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
exciton level is 8.5 times larger than that of the ground state level. Since the exciton ground
state is luminescent and the second exciton state is absorptive, a Stokes shift should be observed
between the excitation and the zero-phonon line in the equilibrium photoluminescence spec-
trum. In addition to the fact that the calculated exciton ground state energy of 1857 meV in [18]
is in excellent agreement with experimental data of [82, 84] , the obtained energy difference of
18 meV between the second and ground states of the exciton is very close to the value of 19 meV
of the Stokes shift found in the fl uorescence line narrowing spectrum [82] . The inset to Fig.
11.10 shows the result of an analogous fi nite-difference calculation of the exciton states for a
spherical QDQW [86] with the same thickness of the HgS shell as that for the tetrahedral QDQW.
The exciton ground state is eight-fold degenerate for both spherical and tetrahedral QDQWs. The
remarkable effect of the QDQW’s shape is that the exciton in the ground state is bright for the tet-
rahedral QDQW shape, while it is dark for the spherical QDQW.
The bulk-like and interface phonon modes and the amplitudes of the electron–phonon interac-
tion for the tetrahedral QDQW are theoretically derived in [18, 83] using the same fi nite-difference
scheme as for the excitons.
Figure 11.11 represents a comparison of the photoluminescence spectrum calculated using
the non-adiabatic theory of [11] for a single tetrahedral QDQW with two experimental spectra
measured at different excitation wavelengths and intensities [84, 87] . Spectral diffusion, which
has been explained recently by a reorganization of local electric fi elds in or around the particles
[88] , is most likely responsible for rather broad lines in the photoluminescence spectra of a sin-
gle QDQW. Experimentally defi ned average phonon frequency is 35.3 0.6 meV and the Huang-
Rhys parameter is 0.25 0.05 [84] . Because of the localization of an exciton in the HgS well, the
amplitudes of bulk-like phonon modes are much smaller than those of interface-phonon modes.
The dominant contribution to the intensity of the one-phonon lines in the calculated photolumi-
nescence spectrum is due to p -like interface-phonon modes. Two highest one-phonon peaks corre-
spond to the phonon modes with frequencies 34.3 and 36.7 meV. The average frequency, weighted
with the heights of the two peaks, is 35.5 meV, which is in a good agreement with the experimen-
tally defi ned value. The intensity of the calculated two-phonon lines can hardly be compared
with experimental spectra because of the high background level in this spectral region. The inset
to Fig. 11.11 shows one- and two-phonon bands in the photoluminescence spectrum calculated
within the adiabatic approximation. Since the exciton ground state β
0
is eight-fold degenerate and
its intensity
||d
β
0
2
is small compared to the intensities of the higher lying exciton states (see Fig.
11.10 ), the heights of phonon peaks in the adiabatic approximation are much lower than those
calculated using the non-adiabatic theory. Only the simultaneous consideration of the tetrahedral
1
0.8
0.6
0.4
0.2
0
1780
2-phonon lines
2ω
LO
HgS
ω
LO
HgS
1-phonon lines
0-phonon lines
2ω
LO
CdS
ω
LO
CdS
0
4000
40
Adiabatic approximation
5 10
5
5 10
3
1800
1820
1840 1860
Energy (meV)
PL intensity
Figure 11.11 Photoluminescence spectrum of a tetrahedral QDQW normalized to the zero-phonon line’s height.
Calculated spectral lines (black curve) are broadened by a Gaussian to enhance visualization. Experimental spectra
are measured at T 10 K with excitation wavelength 633 nm and intensity 15 kW/cm
2
(light grey curve) [84] and
with excitation wavelength 442 nm and intensity 5 kW/cm
2
(dark grey curve) [87] . The insert gives one- and two-
phonon bands (magnifi ed by factors 40 and 4000, respectively) as calculated within the adiabatic approximation.
(Reprinted with permission after [18] . © 2004 by the American Physical Society.)
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Effects of the Electron–Phonon Interaction in Semiconductor Quantum Dots 367
shape of a QDQW, interface optical phonons, and non-adiabatic phonon-assisted transitions
allows for a quantitative interpretation of the observed photoluminescence spectra of a CdS/HgS/
CdS QDQW [18, 83] .
11.7 Conclusions
Due to the non-adiabaticity of the exciton–phonon system in several types of quantum dots,
different new channels of the phonon–assisted optical transitions open. As distinct from the adi-
abatic approximation, which describes only phonon transitions through intermediate exciton
states coinciding either with its initial or its fi nal state, the non-adiabatic approach also takes
into consideration additional phonon transitions: (i) between different exciton states belonging to
the same degenerate level (proper Jahn–Teller effect), as well as (ii) between exciton states of dif-
ferent energy (pseudo Jahn–Teller effect). In contrast to the works based on the theory of Pekar
and Huang-Rhys, which adequately account for the adiabatic phonons only, the non-adiabatic
approach enables describing correctly the transitions involving all types of phonons, including
Jahn–Teller phonons.
The effect of the channels leads to a considerable enhancement of the phonon-assisted transi-
tion probabilities in the photoluminescence of quantum dots even with relatively weak electron–
phonon coupling strength. The resulting multiphonon optical spectra are considerably different
from the Franck–Condon progression. In order to obtain agreement between theory and experi-
ment, in particular regarding the optical properties of self-assembled quantum dots, it is neces-
sary to take into account these new channels of the phonon-assisted optical transitions.
Acknowledgements
We acknowledge discussions with Evghenii Pokatilov, Serghei Klimin, Vladimir Fonoberov,
Serghei Balaban, Fritz Henneberger, Jean-Pierre Leburton, Dieter Bimberg, the late Robert Heitz,
Andrei Schliwa, Marius Grundmann, Mario Capizzi, Paul Koenraad, Andrei Silov, Jos Haverkort,
Alberto García-Cristóbal, Joachim Wolter, Maurice Skolnick, Laurence Eaves, Mohamed Henini,
Jörg Kotthaus, Victor Moshchalkov, Manus Hayne, Frank Wise, Thomas Basché, and Isaac
Bersuker. This work was supported by the FWO-V project G.0435.03, the WOG WO.035.04N
(Belgium) and the European Commission SANDiE Network of Excellence, contract No.
NMP4-CT-2004-500101.
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