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Miniband-related 1.4–1.8 μ m Luminescence of Ge/Si Quantum Dot Superlattices 339
the same nanocluster. For a suffi cient concentration of non-equilibrium carriers this results in
the observed dependence J N
2
.
What benefi t for the PL intensity can be expected from the conversion to the type-I recom-
bination? In the type-I InAs/GaAs structure the electron and the hole are in the same QD, and
the overlap of their wavefunctions reaches 80% (direct exciton) [79] . In the Ge/Si structure this
value for the electron in Si and for the hole in Ge does not reach 15% (indirect exciton) [16] .
Therefore, the exciton oscillator strength due to the miniband should by more than one order
larger than without miniband. We have obtained a ratio J
/
J
1
10 for the periods number of
N 10 and J
/
J
1
12 for N 20, where J corresponds to the measured integrated intensity of
the QDSL band, J
1
– QDSL intensity, extrapolated for the linear dependence on N ( Fig. 10.16 ).
Thus the recombination transition becomes quasi-direct in real space, but probably remains
indirect in the momentum ( q ) space. Though for the strongly localized holes the relaxation of
q -conservation and, consequently, non-phonon recombination can be expected. The attempts of
an indirect–direct conversion in q -space in a Ge/Si structure have been taken for small Ge QDs
[48] , for thin Ge WLs [80] and for thin Si spacers [9] . Unfortunately, a hybrid situation, in which
the heavy holes are localized, but the electrons are not, can cause a reduced miniband transport.
10.4.4 Temporal profi le of QDSL PL
If one presumes an indirect–direct conversion of the exciton in our Ge/Si QDSLs, an increase in
oscillator strength and, consequently, a decrease in the exciton lifetime is expected. The typical
PL decay time for conventional Ge/Si QD structures is in the range 2–5 μ s [26, 81] . In this case,
the slow PL decay is due to the indirect character of transitions in real and momentum space.
The elimination of at least the fi rst factor could accelerate the transients by more than one order
of magnitude. This expectation is based on the following data. The PL of InAs QDs in an Si matrix
(type-II line-up, but q -direct exciton) is characterized by a decay time of about 440 ns [82] . The
TRPL from InAs QDs in a GaAs matrix (type-I line-up and q -direct exciton), which we measured
at the same wavelength (1.3 μ m), exhibited a time constant of 2 ns [83] .
Figure 10.17 presents the PL kinetics for a Ge/Si QDSL at 15 K. This temporal profi le was meas-
ured for an excitation with a fl ux density of 5 1 0
10
photons pulse
1
cm
2
. For the QD array
density of 1.1 1 0
10
c m
2
in the 20 layer structure it corresponds to an incomplete occupation
of 1 e states in Si QWs (see region of u 2.5 in Fig. 10.14 ). The PL decay profi le is well described
by one exponent with a time constant t
L
95 ns ( Fig. 10.17 ): I ( t ) I
0
exp[ ( t / t
L
)
s
]. The factor
of QD size dispersion s 1, i.e. all nanoclusters giving a contribution into the QDSL PL band
have similar sizes. Analysis of the full transient PL profi le is based on a three-level system, which
t
R
15 ns
t
L
95 ns
PL intensity (a.u.)
100 200 300
Time (ns)
Figure 10.17 PL decay profi le of the QDSL band at T 15 K for the p –(QDSL)– n -structure with 20 Ge/Si
periods. Time zero corresponds to a maximum of the laser pulse. Excitation – 5 1 0
10
photons pulse
1
cm
2
. The
solid line shows a fi t to the temporal evolution of PL, based on the three-level system (10.3).
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340 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
includes both radiative recombination with decay time t
L
and carrier relaxation into the 1 e states
with a PL rise time t
R
(fi tted line in Fig. 10.17 ):
It
t
tt
tt tt
L
LR
LR
( ) exp( ) exp( )∝
[]
(10.3)
Therefore, the observed decay time is assigned to the radiative lifetime of the excitons in the
QDSL. This value is much smaller than the time relevant for type-II recombination, where the
carriers are spatially separated. The 95 ns lifetime indicates the conversion to type-I recombina-
tion, which is likely to be caused by the creation of the electron miniband in the Ge/Si QDSL.
The short PL rise time of 15 ns ( Fig. 10.17 ) is explained by the fast capture of carriers from the
Si matrix into the miniband. The time constant of t
R
15 ns corresponds to the resonant tunnel-
ling time within the QDSL. The time constant of non-resonant tunnelling between Ge/Si QWs is
considerably longer (about 325 ns [84] ). The large difference between these time constants (more
than one order of magnitude) has been observed in GaAs QWs [85] and CdTe QWs [86] .
At a higher excitation level, corresponding to the region with a factor u 18 meV decade
1
( Fig. 10.14 ), the shape of the transient PL splits into two components – a fast one, not resolved
in our experiment, and a following slower one with a decay time of a few μ s. We suppose that
the slow component is typical for the lifetime of a spatially indirect exciton in Ge/Si heterostruc-
tures. It means that a high excitation power leads to a collapse of the miniband and a recovery of
the type-II recombination. The fast component appears to be due to high densities of non-
equilibrium carriers and most likely originates from such relaxation processes as Auger recombina-
tion and exciton–exciton scattering. The latter factor restricts the coherence length of the excitons
in the QDSL and makes the miniband unstable. The destruction of a miniband under high PL
excitation densities by exciton–exciton scattering was observed in InGaAs/GaAs QDSLs by [70] .
10.4.5 Temperature and power dependence of miniband
A description based on discrete levels in Si QWs ( Fig. 10.11 ) does not account for the following
experimental fi ndings: (i) multi-modal structure ( Figs 10.6 and 10.7 ); (ii) intensity rise ( Fig. 10.9 )
and (iii) rapid red shift ( Fig. 10.8 ) of QDSL PL as a function of temperature; (iv) superlinearity
for the integrated intensity ( Fig. 10.13 ); and (v) small blue shift ( Fig. 10.14 ) of the QDSL band in
the power dependence. However, these facts may be explained within the miniband model.
The electron subsystem of the studied Ge/Si QDSLs is made up from QWs with one confi ned
state. The possible transport regimes of such a superlattice under applied fi eld are a miniband
(MB), a Wannier–Stark (WS) ladder, and multi-QWs with spatially separated subbands and their
crossovers. Four main energy parameters play a role for a realistic description: the miniband
width ( Δ
MB
), the WS energy ( eFC ), the thermal energy ( kT ), and the structural and dynamic dis-
order broadening ( δ ). For a given QDSL, the Δ
MB
is determined by the overlap of the wavefunc-
tions of electrons in the adjacent identical QWs and the coherent length ( L
C
), in other words, by
the probability of resonant tunnelling. If the neighbouring QWs are not identical and/or an elec-
tric fi eld is applied the MB regime can be supported by temperature ( kT ) and dynamic broadening
(scattering). Simultaneously, the scattering processes (phonon, interface roughness, exciton–
exciton) shorten the coherence length in QDSL. Under an electric fi eld with eFC δ the miniband
breaks up into a series of WS states (WS ladder). The localization degree of an electron in the
WS state is determined also by the probability of tunnelling through the barrier. Depending
on this the electrons may be transported between WS states via hopping (moderate fi elds,
δ eFC Δ
MB
) or sequential non-resonant tunnelling (high fi elds, eFC Δ
MB
). In the last
extreme case, the wavefunctions are strongly localized in separate QWs. Such structures behave
like the uncoupled multi-QWs. At moderate fi elds the wavefunction of WS states can be well
delocalized throughout several periods and can give the additional optical transitions [87] .
We believe that the multi-modal structure of the QDSL PL band refl ects the presence of a WS lad-
der in the QDSL at low temperatures ( Figs 10.6 and 10.7 ). The interval between PL neighbouring
components ( δ
m
10 and 20 meV) correlates well with the voltage drop per period ( Table 10.1 )
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Miniband-related 1.4–1.8 μ m Luminescence of Ge/Si Quantum Dot Superlattices 341
for p ( i ) n - and p
( i ) n
-structures: U
C
10.2 and 19 meV, respectively. In Fig. 10.18c we present
the sketched band line-up and optical transitions for a QDSL in the WS regime. In fact, the multi-
modal QDSL PL band refl ects the distribution of squared wavefunctions in the WS states of the
ladder. Thus the electron wavefunction is extended up to 10 periods in p ( i ) n -structures, i.e. a
coherence length L
C
70 nm. Due to the stronger built-in fi eld ( Table 10.1 ) only fi ve components
are distinguished in p
( i ) n
-structure ( L
C
43 nm). Two WS ladders with L
C
32.5 and 60 nm
were observed by the authors of [87, 88] in GaAs/AlGaAs superlattices. The periodicity of opti-
cal oscillations with F
1
-dependence was predicted in [89] . Hopping conduction was identifi ed
in the plane of QD arrays at low temperature in [71] for InGaAs and in [90, 91] for Ge QDs.
From this point of view the disappearance of the multi-modal structure in the QDSL PL spec-
trum can only be the result of miniband formation in the column due to temperature-induced
weakening of the built-in fi eld ( Table 10.1 ). The miniband develops gradually from separate WS
states, the electron wavefunctions extend throughout the QDSL ( Fig. 10.18a ) having an ave rage
thickness of 140–250 nm. The authors of [92] obtained a coherence length equal to 150 nm
for a miniband in a 20 period GaAs/AlGaAs superlattice. The miniband formation in the p ( i ) n -
structure occurs already at 10 K and in the p
( i ) n
-structure above 150 K due to differences in
the built-in fi eld strength in these structures ( Table 10.1 ). The p ( i ) n -fi eld at low temperatures
( F 14.6 kV c m
1
) can be considered a miniband breaking fi eld F δ / eC . In that case the
broadening energy of the electron level in the Ge/Si QDSL at low temperature should be around
10 meV. It is evident that such noticeable broadening will effectively support the MB regime in
QDSLs with narrow minibands and/or under strong electric fi elds, e.g. in p
( i ) n
-structures.
Acoustic-phonon scattering and scattering via interface roughness can give the disorder broad-
ening δ 10 meV at low temperatures. A short scattering time t / δ 70 fs, as well as some
modes in the Raman spectra ( Fig. 10.4 ) are an indication of both processes in Sb-doped QDSLs.
MB transport controlled acoustic-phonon scattering was found in [71] . The existence of true
miniband conduction via interface roughness scattering was highlighted by the authors of [93] .
(a)
(c)
(b)
Miniband
E
c
E
v
Ge
eFC
L
c
L
c
Si
WS states
E
c
E
c
Multi-QW states
L
c
MB
d
C
z
|
1hh
|
2
|
1e
|
2
3
2
1
1
2
Figure 10.18 Scheme of band edge profi le of Ge/Si QDSL in different regimes: (a) miniband (low electric fi eld,
L
C
C N – for not too low temperatures, low excitation densities); (b) multi-QW structure (low fi eld, L
C
C –
high excitation level); (c) WS ladder (moderate fi eld, L
C
5 C – low temperature, low excitation level), squared
wavefunctions and optical transitions [ 3; 2] are presented as well.
The miniband development in Ge/Si QDSLs results in spatially direct recombination. Due to
indirect–direct conversion the QDSL PL intensity increases as a function of temperature (Fig.
10.9). The maximum is reached at different temperature values depending on the original built-
in fi eld: 20 K for the p –(QDSL)– n -structure and 200 K for the p
–(QDSL)– n
. The magnitude of
this effect excludes its explanation by the carrier transfer from WL to the Ge QDs [80] . A further
intensity decrease is caused by the thermal activation of electrons from the miniband. A slope
with activation energy E
A
1
appears in the Arrhenius plot. Such behaviour was observed in all
QDSLs embedded into p
– n
-junctions. Typically, intensity increases by a factor of 3–4 in the
range of 5–200 K. As a rule, the QDSL intensity at room temperature is higher than at low tem-
peratures. For p ( i ) n -structures this ratio is always less than 1, namely from 0.1 to 0.01 depending
CH010-I046325.indd 341CH010-I046325.indd 341 6/27/2008 4:28:51 PM6/27/2008 4:28:51 PM
342 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
on the value of E
A
1
. The authors of [93] observed miniband transport between ~77 and 400 K in
GaAs/AlAs superlattices.
The miniband formation is accompanied also by the transformation of a random electron dis-
tribution on the WS levels in separate Si QWs to the Fermi–Dirac distribution within a miniband.
The maximum of electron state density shifts towards the miniband bottom and deviates from
E
GeSi
( T ) ( Fig. 10.8 ). Further evolution of E
m
( T ) is related to the “ traffi c ” of Fermi–Dirac electron
gas within the miniband, namely to its middle. E
m
( T ) returns to E
GeSi
( T ). A similar, unusually
rapid temperature shift of the PL peak was observed for a planar InAlAs QD array [69] below
105 K, and for stacked InAs QDs [68] below 250 K. The phenomenon has been attributed to the
effective thermalization of carriers at strong coupling of InAs QDs and miniband formation.
The superlinear behaviour of PL and EL intensity ( [41, 42] and Fig. 10.13 ) has the same
nature as the temperature dependence – the conversion to the MB regime due to straightening
of a band bending. The lowering of the built-in electric fi eld in EL measurements is achieved by
a sequential growing of the forward bias. In PL intensity versus excitation power this is achieved
by an increase of the non-equilibrium carrier concentration (open-circuit regime).
A typical sign of type-I band alignment and of spatial direct exciton is a small u -factor of the
Δ E
m
(lg P )-dependence, which is caused by the well-known state fi lling effect. In contrast to this,
a large blue shift is typical for a type-II structure, in which electrons and holes are spatially sepa-
rated by the interface. A dipole layer is formed there due to the photo-excitation. A high Hartree
potential of this layer raises the confi nement energy of electrons and holes and causes a consider-
able blue shift in the PL spectrum [81, 94] . The value of that shift by one order exceeds the shift,
caused by the state fi lling effect. Therefore, the Δ E
m
(lg P )-dependence can serve as an indicator of
the band alignment type. The available data on Ge/Si QD structures deal with low temperatures
(4.5–10 K) and quite different power ranges within 0.6–400 Wcm
2
. For the sake of comparabil-
ity we introduced the u -factor and reduced all data to it. For the type-II Ge/Si heterostructures the
values of u ranged from 13 to 22 meV decade
1
[37, 80, 81, 94] . The II–I-type conversion has
been reported for the GeSi structures having strong intermixing in the annealed QDs [37] and in
WL [80] with u 1 meV decade
1
. Our QDSLs had u -factors less than 3 meV decade
1
for low
excitation densities ( u 2.5 meV decade
1
for P 6 W c m
2
in Fig. 10.14 ). This value agrees
very well with type-I recombination. At higher powers the u -factor reached 20 meV decade
1
. In
Fig. 10.14 the value u 18 meV decade
1
was observed for P 1 2 W c m
2
, where the fi lling of
the 2 e states occurred. By TRPL measurements we showed that a high excitation power leads to
the collapse of the miniband and the recovery of type-II recombination. At high density of non-
equilibrium carriers the exciton–exciton scattering restricts the coherence length L
C
in QDSLs. If
L
C
becomes extremely short ( L
C
W ), the miniband and WS ladder break up into loca lized states
of separate QWs. Sequential phonon-assisted tunnelling in such decoupled multi-QWs becomes
the dominant process ( Fig. 10.18b ).
Arguments given above prove the presence of regimes typical for coupled superlattices (mini-
band and WS ladder) and uncoupled multi-QWs structures (sequential tunnelling) in Ge/Si
QDSLs. The crossovers between transport modes were investigated using two experimental
dependencies of PL, namely the temperature infl uence between 5 and 300 K at P 6 W c m
2
in the short-circuit regime and the power density dependence between 0.2 and 250 Wcm
2
at
T 15 K in the open-circuit regime, i.e. we actually have studied two cross-sections in the para-
metric space { T ; P } . Both parameters (each in its own regime) induced the change of built-in
electric fi eld without external bias. It demonstrates that the Ge/Si QDSL is a promising system
not only for Si-based optoelectronics, but also for other integrated optics.
10.5 Conclusions
Structural and luminescence properties of Sb-doped Ge/Si QDSLs grown by MBE were studied.
Highly strained columns and sharp Ge/Si interfaces were found to form a large conduction band
offset of 110 meV. The Si spacer QWs were found to have one single confi ned electron state only.
Kronig–Penney calculations showed that an electron miniband is formed in the QDSLs rather
than discrete electronic levels. The miniband width was found to be about 30 meV. The miniband
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Miniband-related 1.4–1.8 μ m Luminescence of Ge/Si Quantum Dot Superlattices 343
formation leads to a conversion of the band structure from indirect to quasi-direct, i.e. towards
type-I recombination. The proposed miniband model gives an adequate explanation of the exper-
imental results, i.e. the high effi ciency of photo- and electroluminescence (0.04%) in the 1.55 µm
region at room temperature; the square dependence of PL intensity on the number of periods;
the substantially decreased exciton lifetime of 95 ns; the temperature-induced growth of the pho-
toluminescence intensity and the superlinear dependence of luminescence on excitation power.
Acknowledgements
We gratefully acknowledge helpful discussions with V. Kveder, D. Kovalev, G. Abstreiter and
D. Grützmacher. We would like to thank A. Frommfeld for supporting the MBE growth and
S. Schwirzke-Schaaf for contributions to Raman measurements.
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Effects of the Electron–Phonon Interaction
in Semiconductor Quantum Dots
J.T. Devreese
,
1,2
V.M. Fomin
,
1
,2,
3
and V.N. Gladilin
1
,3
1
TFVS, Universiteit Antwerpen, Belgium ;
2
PSN, COBRA, TU Eindhoven, The Netherlands;
3
PMS, State University of Moldova, Chis¸ina˘u, Moldova
11.1 Introduction
The state-of-the-art nanodevices based on quantum dots have provided new insights in the
fundamental physical properties of confi ned semiconductors as well as advancement of novel
applications in microelectronics and optoelectronics and for manipulation of materials at the
nanoscale [1–3] .
The study of excitons in nanoscale systems has been triggered by the advancement of spectro-
scopic methods sensitive to the electronic structure of nanomaterials and dynamic processes of
charge carriers, including absorption, redistribution and emission of excitation energy. Features
of excitons in semiconductor quantum dots constitute a subject of an active cross-disciplinary
research [4] combining structural characterization, chemistry and physics. The quantum con-
fi nement effects on the exciton energy spectra in semiconductor quantum dots have been devoted
an extensive literature, see, e.g., monographs and reviews [5–9] and references therein.
Non-adiabaticity [10, 11] is an inherent property of exciton–phonon systems in various quan-
tum dot structures. The present authors and their co-authors found that non-adiabaticity can
drastically enhance the effi ciency of the exciton–phonon interaction . The effects of non-adiabaticity
are important to interpret the surprisingly high intensities of the phonon “ sidebands ” observed
in the optical absorption, the photoluminescence and the Raman spectra of quantum dots, in
particular, an enhancement of these intensities with decreasing the quantum dot size (see, e.g.,
[12, 13] ). Deviations of intensities of the phonon-peak sidebands, observed in some experimental
optical spectra, from the Franck-Condon progression, which is prescribed by the commonly used
adiabatic approximation, fi nd a natural explanation within our non-adiabatic approach [9–11,
14–18] .
Photoluminescence excitation (PLE) experiments for self-organized InAs/GaAs quantum dots
[19] revealed size-dependent excitation resonances. The observed PLE resonances are identifi ed
based on their energy, relative intensity, and sensitivity to size variations in comparison to results
of eight-band k
·
p calculations of exciton transitions for pyramidal InAs/GaAs QDs. Band mix-
ing, strain, and the particular geometry of the three-dimensional confi nement lead to a rich fi ne
structure with a variety of “ forbidden ” excitonic transitions. A good agreement between experi-
ment and theory is found for large QDs (with a ground-state transition energy
E 11.eV
) ,
whereas the agreement becomes worse for smaller QDs because of the uncertainties in the size- and
CHAPTER 11
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Effects of the Electron–Phonon Interaction in Semiconductor Quantum Dots 347
growth-dependent variations of the QD shape and composition as well as Coulomb-induced
localized wetting layer states.
We overview in this chapter the present understanding of the effects of non-adiabaticity of the
exciton–phonon systems (section 11.2), which allow for the interpretation of the surprisingly
high intensities of the phonon sidebands observed in the optical absorption, the photolumines-
cence, the PLE and the Raman spectra of some types of quantum dots. We start with the theory
of non-adiabatic processes in spherical quantum dots (section 11.3). Next we analyse stacked
quantum dots, modelling their shape by parallelepipeds (section 11.4) and discuss the notion of
excitonic polarons in quantum dots and their effect on the optical properties of quantum dots
(section 11.5). Finally, we consider some recent studies of the effects of the electron–phonon
interaction in semiconductor quantum dots (section 11.6).
11.2 Non-adiabaticity of the exciton–phonon systems in spherical quantum dots
In the last two decades, there has been an increasing experimental interest in multiphonon pho-
toluminescence [12, 21–24] and Raman scattering [25–30] of nanosize quantum dots.
Photoluminescence peaks due to phonon-assisted processes were fi rst observed in colloidal
quantum dots (nanocrystals) under size-selective excitation [21, 22] . Observation of multipho-
non photoluminescence of quantum dots embedded in glass is complicated because of a quick
trapping of holes onto the local surface levels [31] . Overlapping of photoluminescence bands
related to recombination of the electron–hole pairs, which are present in different states local-
ized at the surface, smears the features of the spectrum due to the phonon-assisted processes
[12, 32] . Distinct phonon-line progressions, which are caused by recombination of the elec-
tron–hole pairs in “ interior ” states of a quantum dot (i.e. the states spatially quantized due to
the confi nement potential), were observed in the fast components of photoluminescence of CdSe
quantum dots embedded in glass [12, 23] using time-resolved spectroscopy.
Existing attempts to interpret [12, 24, 26, 33, 34] the aforementioned experiments on the
basis of the adiabatic theory of multiphonon transitions in deep centres by Pekar [35] and Huang-
Rhys [36] meet considerable diffi culties. In the framework of the adiabatic theory, the values of
the Huang–Rhys parameter were obtained in [33, 34] using a spherical model Hamiltonian for
the electron–hole pair in a quantum dot [37–39] and taking into account the valence-band mix-
ing. The values of the Huang-Rhys parameter obtained in this way are signifi cantly (by one–two
orders of magnitude) smaller as compared with those derived from experiment. This discrepancy
is due to the fact that under a strong confi nement the charge density of the electron–hole pair in
the ground state is small everywhere in the quantum dot. In order to achieve agreement with the
experimental data, additional mechanisms have been intuitively introduced, which ensure sepa-
ration of the electron and hole charges in space: (i) additional built-in charge [26, 33, 40] ; (ii)
traps which would localize holes [22] ; and (iii) different boundary conditions for electrons and
holes [25] .
The main idea of the adiabatic approach consists in the assumption that a stationary state of
charge carriers is formed for each instantaneous position of ions (i.e. charge carriers follow the
ion motion adiabatically). In the framework of this approach, the exciton–phonon interaction
leads only to a modifi cation of the exciton wave function, but does not give rise to transitions
between different exciton states.
However, two circumstances are to be mentioned, which imply that the exciton–phonon sys-
tems in quantum dots are essentially non-adiabatic.
1. The states of an exciton in a quantum dot including the ground state are, generally speak-
ing, degenerate . In this connection, it is worthwhile recalling [41–46] that the electron–
vibrational interaction in the impurity centres with a degenerate electron level may cause
dynamic Jahn–Teller effect [47] , or, equivalently, so-called internal non-adiabaticity . Namely,
if the electron interaction with some vibrational modes is described by non-commutative
matrices calculated on the basis of a degenerate electron level (those vibrational modes
are usually called the non-adiabatic or Jahn–Teller modes), there is a non-adiabatic mixing
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348 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
of electron states belonging to this level. In a direct analogy to this scenario of the proper
Jahn–Teller effect for the impurity centres, transitions do occur between different states of a
degenerate exciton level in a quantum dot, provided that the exciton–phonon interaction in
the basis of this degenerate level is characterized by non-commutative matrices.
2. Differences between energy levels of an exciton in a quantum dot can become compara-
ble with the optical phonon energy in the experiments [12, 21–29] . Therefore, the result-
ing effects of external non-adiabaticity , or so-called pseudo Jahn–Teller effect [45] , in the
phonon-assisted optical transitions are of crucial importance. The term “ external non-
adiabaticity ” is used in order to make a distinction between this class of non-adiabatic phe-
nomena and the above-described proper Jahn–Teller effect relevant to a degenerate exciton
level.
11.3 Photoluminescence of spherical quantum dots
11.3.1 The light absorption by quantum dots
Quantum dots are considered to be embedded in a medium with a weak dispersion of refractive
index. In this section, the quantum dots are supposed to be identical. On the basis of the Kubo
formula [48] , the linear coeffi cient of absorption by an ensemble of quantum dots in the electric
dipole approximation [49] can be represented as:
α()
()
()Ree ,
()
Ω
Ω
Ω
Ω
Ω
8
3
1
π ᏺ
cn
dt d d
k
B
T
ii
it
Ht Ht
eee
i
〈
†
〉→→
∞
∫
0
0
,
(11.1)
where Ω is the light frequency, ᏺ is the concentration of quantum dots, n ( Ω ) is the refractive
index of the medium, d is the dipole moment operator, and … denotes the averaging over the sta-
tistical ensemble of the exciton–phonon systems. The total Hamiltonian of the exciton–phonon
system:
HH H H
ex ph int
(11.2)
contains the exciton Hamiltonian H
ex
, the phonon Hamiltonian H
ph
, and the exciton–phonon
interaction Hamiltonian:
Haa
int
()γγ
λλ
λλ
λ
††
∑
(11.3)
where
a
λ
†
and a
λ
are the creation and annihilation operators for the phonons of the λ th vibra-
tional mode, the interaction operators γ
λ
will be specifi ed below.
Note that in the experiments [12, 21–24] , the exciton energy is much larger than both the pho-
non energy and the value k
B
T . This means, in particular, that the probability of thermal generation
of an exciton is vanishing. For optical transitions leading to generation of an exciton and start-
ing from the exciton vacuum
|0〉
whose energy is chosen as zero, the initial states of the exciton–
phonon system are described by the wave functions
||0〉〉n
, where
|n〉
are eigenstates of the phonon
Hamiltonian H
ph
. Further, the fi nal states of the exciton–phonon system are not occupied, so that in
Eq. (11.1), one can omit the term which is proportional to exp (
–
h Ω /k
B
T ) and describes a correction
due to stimulated emission of light with frequency Ω . Finally, the frequency of the exciting radiation
is much larger than the phonon frequency; hence, the interaction of this radiation with free pho-
nons can be neglected, i.e.
dn nd||〉〉
. Hence, the transformation of the average follows:
ed d n d d n
ii i i
Ht Ht
n
Ht Ht
n
††
e|e|e|0|
ph
ρ 〈〈0 〉〉
∑
〈〉〈 〉
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
′
〉〈
′
∑
0〈|| | |e e | | ||
ph
dnnd
n
Ht
Ht
n
i
i
†
ββ ρ β β
00〉
(11.4)
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