Quantum Dot Capacitors as Versatile Light Sources for Integrated
PhotonicsIntegrated Photonics Quantum Dot Capacitors as Versatile
Light Sources for
Academic year 2015-2016 Faculty of Engineering and
Architecture
Chair: Prof. dr. ir. Rik Van de Walle Department of Electronics and
Information Systems
Chair: Prof. dr. Isabel Van Driessche Vakgroep Anorganische en
Fysische Chemie
Master of Science in Engineering Physics Master's dissertation
submitted in order to obtain the academic degree of
Counsellors: Prof. dr. ir. Dries Van Thourhout, Suzanne Bisschop
Supervisors: Prof. dr. Zeger Hens, Prof. dr. ir. Kristiaan
Neyts
PREFACE ii
Preface A good friend of mine told me a few months ago: “quantum
dot capacitors and Francis,
that’s an excellent combination, since none of them is very bright!
:P”. We had a great
laugh at the time, yet from that day onwards, I have made it my
holy quest to prove to
the world that the opposite is true, by means of this very thesis.
Today, I can proudly
announce that my quantum dot capacitors indeed generate quite a bit
of light! Regarding
the other part of the statement, I leave this to the judgment of my
dear readers.
Time is shrinking but I would like to use the remaining 10 minutes
for thanking all
those who have made this thesis possible: in particular my two
brothers, my parents, grand
parents and ‘tante Rita’. I love them even more than they love me.
However, it might be
the case that I have neglected them slightly during the past few
months.
Furthermore, my thanks go to the colleges in my office, in
particular to Suzanne, who
has been a great mentor and to Kim, who was able to cheer me up
with a hug in my
darkest hours. Thanks also to Vignesh aka Vicky, who proposed to
proofread my thesis
but who was not available between 11:55 and 11:59 pm... Jorick,
Valeriia and Willem, of
course I did not forget about you!
Special thanks go to Michiel for depositing my metal contacts time
and again, you
did a great job! And to John for showing so much interest in my
work. Also thanks to
Woshun (did I get it right?) for keeping me company in the dark
dungeons underneath the
Great Tower in Zwijnaarde. And to Pieter Geiregat, who has a gift
for explaining the most
complicated things in a crystal clear way. Thank you: Hannes,
Pieter II, Kishu, Chen,
Dorian and especially Emile: I wish you the best of luck with ton
epouse Seraphine!
Last but certainly not least, I wish to thank my promotors: prof.
Neyts, prof. Van
Thourhout and Zeger in particular, for providing an incredible
support!
Francis Ryckaert, june 2016
Copyright Statement
The author gives permission to make this master dissertation
available for consultation
and to copy parts of this master dissertation for personal
use.
In the case of any other use, the limitations of the copyright have
to be respected, in
particular with regard to the obligation to state expressly the
source when quoting results
from this master dissertation.
Francis Ryckaert, june 2016
Master of Science in Engineering Physics
Academic Year 2015–2016
Promotors: prof. dr. ir. Z. Hens, prof. dr. dr. K. Neyts
Supervisors: prof. dr. ir. D. Van Thourhout, ir. S. Bisschop
Faculty of Engineering and Architecture
Ghent University
Department of Inorganic and Physical Chemistry
President: prof. dr. ir. I. Van Driessche
Summary
This work is aimed at designing a CMOS-compatible quantum dot-based
integrated light source, having a capacitor structure for
electrically exciting the quantum dots. On one hand, we fabricate
quantum dot capacitors with silicon nitride insulating layers, as
to characterize the actual mechanism for electroluminescence in
these devices. On the other hand, we determine the optimal
structure for extracting the generated light. We thereby
investigate two routes: the one of dielectric directivity
enhancement, where we optimize the waveguide material and
dimensions, and the one of plasmonic directivity enhancement, where
we additionally include nanoplasmonic structures.
Keywords
Quantum Dot Capacitors as Versatile Light Sources for Integrated
Photonics
Francis Ryckaert
Supervisor(s): prof. dr. ir. Zeger Hens, prof. dr. ir. Kristiaan
Neyts, prof. dr. ir. Dries van Thourhout, ir. Suzanne
Bisschop
Abstract— We aim at designing a quantum dot based integrated light
source, having a capacitor structure for electrically exciting the
quantum dots. On one hand, we fabricate quantum dot capacitors with
silicon ni- tride insulating layers, as to characterize the actual
mechanism for electro- luminescence in these devices. On the other
hand, we determine the op- timal structure for extracting the
generated light. We thereby investigate two routes: the one of
dielectric directivity enhancement, where we optimize the waveguide
material and dimensions, and the one of plasmonic directivity
enhancement, where we additionally include nanoplasmonic
structures.
Keywords—colloidal quantum dots, electroluminescence, integrated
pho- tonics, plasmonics
I. INTRODUCTION
IN integrated photonics as opposed to electronics, one uses light
as carrier of information, rather than electricity. The
field of photonics is considered to be crucial for developing the
next generation of devices in datacommunication, on-chip inter-
connects, sensing and biosensing and even in quantum comput- ing
[1]. One particular example is given by the electrical inter-
connects in between microprocessors, which are reaching their
limits in terms of both power consumption and bandwidth. On- chip
optical interconnects, compatible with the silicon-based CMOS
fabrication technology, could offer a viable solution to this
problem. However, the silicon on insulator and silicon nitride
material integrated photonics platforms have poor light emitting
and light modulating properties. As such, a cheap and efficient
integrated light source, compatible with the CMOS fab- rication
technology, is intensely sought after.
We aim at designing a quantum dot (QD) based electrically- driven
integrated light source, compatible with the CMOS fab- rication
technology. We thereby consider a quantum dot capac- itor device
architecture, with a layer of QDs sandwiched verti- cally between
two insulating layers with top and bottom electric contacts. The
great advantage of these structures is that their emission
wavelength can be altered simply by choosing another quantum dot
layer. In section II, we briefly discuss colloidal quantum dots and
explain their optoelectronic properties. In sec- tion III, we
present the quantum dot capacitor structures we fab- ricated, as
well as their electric and electroluminescent charac- terization,
where we rely on PSPICE simulations for interpreting our
measurements. In a next step, the devices have to be inte- grated
in a waveguide structure, where we want to maximize the coupling of
the generated light into a waveguide. In section IV, a maximal
coupling is obtained simply by altering the waveguide dimensions
and the waveguide material, both for single photon emitters and for
complete layers of quantum dots. We hence refer to this approach as
dielectric directivity enhancement. In section V, we follow the
route of plasmonic directivity enhance-
ment, where we include a nanoplasmonic antenna hybridized with the
dielectric waveguide. Optimization of the nanoantenna is performed
using the particle swarm optimization algorithm.
II. COLLOIDAL QUANTUM DOTS
A quantum dot (QD) is a nanometer-sized (2 -15 nm) piece of
semiconductor material. The QD and bulk optical properties
radically differ, due to the reduced size in all three dimensions —
the so-called quantum confinement effect. However, the crys- tal
structure and lattice constant of QDs in general closely re- semble
their bulk equivalents, hence the alternative appellation of
nanocrystal (NC). Figure 1(a) shows a Transmission Electron
Microscope (TEM) image [2] of a PbSe QD. Most importantly, the QD
emission wavelength can be tuned by altering the QD material and/or
the QD size. This way, emission of QD struc- tures covers the
entire visible and near infrared region.
(a) (c)(b)
Fig. 1. (a) A TEM image of a colloidal quantum dot, (b) Schematic
visualiza- tion of a core/shell structured QD, with a view on the
internal core structure (credit: Rusnano) and (c) TEM image of a
monolayer of colloidal CdSe/ZnS core shell QDs.
Quantum dots can be produced in large quantities via effi- cient
colloidal synthesis processes [3], which reduces their cost. We
make use of CdSe/CdS and PbS/CdS core/shell QDs emit- ting at 625
nm and 1550 nm respectively. Figure 1(b) gives a schematic
visualization of such a core/shell structured QD. An important
advantage of the core-shell structure in general is the improved
surface passivation of the inner core, which greatly in- creases
the quantum yield of these structures. For instance, the CdSe/ZnS
core/shell QDs sold by Aldrich Materials Science all exhibit room
temperature QYs surpassing 80 %. Colloidal QDs are easily deposited
in thin films, for example via the spin coat- ing technique or the
Langmuir Blodgett method. Figure 1(c) shows a TEM image [4] of a
monolayer of colloidal CdSe/ZnS core shell QDs.
III. QUANTUM DOT CAPACITOR
For electrically exciting a layer of QDs, we use the QD ca- pacitor
structure. A schematic view is given in figure 2(a). A layer of
CdSe/CdS QDs is sandwiched between top and bottom insulating
layers. All layers are deposited one by one on an ITO coated glass
substrate. For the insulating layers we employ PECVD Si3N4 layers.
Si3N4 is compatible with the CMOS fab- rication industry and can
also serve as waveguide core material, considering its refractive
index of about 1.98. In the end, a grid of top contacts of either
Au or Ag is deposited using e-beam PVD.
(b) QD capacitor samples
QDs
Si3N4
Si3N4
FIB induced Pt
e-beam induced Pt FIB damage to e-beam induced Pt
Ag top contact (~85 nm) top Si3N4 layer (~30 nm) CdSe/CdS QD layer
(~105 nm) bottom Si3N4 layer (~95 nm) ITO bottom contact (~30 nm)
glass substrate (~30 nm)
Fig. 2. (a) A schematic view of of the capacitor structure. (b)
Both a reference sample without QDs and a QD capacitor sample (c)
SEM cross sectional image of a QD capacitor. The Pt depositions on
top merely serve for SEM cross section imaging.
The actual samples are shown in figure 2(b). The top sample is a
reference sample, lacking the layer of QDs. A single sample
contains about a dozen individual QD capacitors. Throughout the
fabrication procedure, the left end of the substrate is cov- ered
with a high temperature resistant conductive tape. Upon removal of
the tape, the bottom ITO contact is accessible. Fig- ure 2(c) shows
a SEM cross section image of the device. The capacitor structure
has an overall thickness of order 100 nm.
From the capacitances of the reference samples on one hand and
samples containing QDs on the other hand, we determine a dielectric
constant of εSi3N4
= 7.6± 0.1 and εQD = 6.3± 0.3 for the Si3N4 and QD layers
respectively. For an electric character- ization of the QD
capacitors, we apply a 1 kHz sawtooth driving voltage over a series
circuit of the QD capacitor and a 21 k re- sistor. The measured
current response through the circuit is rep- resented by the dots
in figure 3. The device peak-to-peak voltage is mentioned for each
of the curves. At small driving voltages, the device behaves as an
ideal capacitor. For increasing volt- ages, the structure also
supports a resistive current, indicating a degradation of the Si3N4
insulating layers. For even higher voltages, an exponential
Shockley-like breakdown is noticed.
We set up a PSPICE model for the Si3N4 insulating layers and the QD
layer respectively, where both layers are characterized
independently by a capacitance in parallel with a resistor and
a
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
78 Vpp 62 Vpp 49 Vpp 23 Vpp
Fig. 3. The current response to a 1 kHz sawtooth voltage, of the QD
capacitor structure in series with a 21 k resistor. For each of the
curves, the device peak-to-peak voltage is indicated. The
measurements are added in dotted line, while the results of our
PSPICE simulations are added in solid line. For high peak-to-peak
voltages, the measurements have a non-physical offset, as to make
them symmetric. This way, they can be approximately described by a
symmetric diode pair, simplifying our model.
RQD
Is,SiN, nSiN Is,QD, nQD
Fig. 4. The model for our QD capacitor as employed in PSPICE
simulations. The different model parameters are defined, and the
voltages and currents are indicated.
forward and backward conducting diode pair. We refer to figure 4.
The Si3N4 resistor value is optimized for the reference sam- ples;
the Si3N4 resistance decreases when increasing voltages are
applied. This is an indication of degradation of the Si3N4
insulator material. The Si3N4 (Schockley) diode contribution, with
saturation current Is,Si3N4
and ideality factor nSi3N4 , only
depends on the electric field within the layer. Due to its asym-
metry, the device actually has a slightly asymmetric breakdown
characteristic. In order to keep our model as simple as possi- ble,
with identical forward and backward conducting diodes, we have
introduced a non-physical offset and try to optimize our model to
these curves. As such, the positive and negative peak currents on
the figure are equal in magnitude. The results are quite good for
small breakdown currents or, equivalently, for small device
peak-to-peak voltages, as can be seen on the figure in solid
line.
Via our PSPICE model, we are able to estimate the different current
contributions through the separate layers. The diode cur- rent
contribution through the QD layer is represented in figure 5.
40 30 20 10
I Q D
102 Vpp 146 Vpp
125 Vpp
Fig. 5. PSPICE results for the diode current contribution through
the QD layer, as a function of time. Starting form device voltages
of about 100 Vpp, the diode breakdown current through the QD layer
is significant.
30x103
25
20
15
10
5
0
122 Vpp
115 Vpp
Fig. 6. The electroluminescence of the QD capacitor when applying a
square wave signal of 10 kHz. Light is detected starting from
device voltages of about 100 Vpp.
Starting from device voltages of 102 Vpp, the diode breakdown
current through the QD layer is significant. For this device volt-
age, the voltage drop over the QD amounts to about 20 V, ac-
cording to our PSPICE simulations. This corresponds to a volt- age
drop per QD slightly higher than the QD excitonic band gap of 1.98
eV, where the QDs have a diameter of ∼ 10 nm.
The electroluminescence of the QD capacitor when applying a 10 kHz
square wave is given in figure 6. The device peak-to- peak voltage
is added for each of the curves. Light is detected starting from
device voltages of about 100 Vpp. This suggests that the diode-like
breakdown of the QD layer is indeed im- portant in electrically
exciting the quantum dots. We thereby think of electrons hopping
from one QD to the next. In those QDs where electrons and holes
come together, they can give rise to radiative decay. The device
voltage luminescent thresh- old is seen to be independent of the
frequency of the applied signal. However, high frequency signals
result in brighter emis- sion, which can be explained from the
higher repetition rate of current peaks passing through the QD
layer.
Our structures provide non-stop emission during about 20 minutes,
followed by either permanent breakdown of the Si3N4 material or
complete degradation of the top metal contacts. In this regard, the
use of silver instead of gold as top contact ma- terial results in
an improved stability. However, the electrolu- minescent mechanism
we observe is inherently unstable, with high currents flowing
through the Si3N4 insulating layers. These currents not only cause
power dissipation, but also degrade the
x
top contact
bottom contact
Fig. 7. Integrated source design. The QD layer is placed centrally
in the waveg- uide, where the waveguide material simultaneously
acts as insulating ma- terial for the QD capacitor structure. Top
and bottom electric contacts are provided.
Si3N4 material, especially when operating at higher device volt-
ages, superior to 120 Vpp. An alternative structure, with high-
quality insulating layers of SiO2 instead of Si3N4, and equally
compatible with the CMOS fabrication technology, could be
considered. However, the electroluminescent mechanism in these
alternative structures is yet to be studied. Secondly, in-
tegration of the device into a waveguide structure would not be
straightforward, due to the low refractive index of SiO2.
IV. DIELECTRIC DIRECTIVITY ENHANCEMENT
A next step is the integration of the QD capacitor structure into a
dielectric waveguide, as to design an integrated light source. Our
proposal, in which the QD capacitor is hybridized with a dielectric
strip waveguide, is shown in figure 7. In or- der to obtain a
maximal coupling efficiency into the waveguide structure, we simply
optimize the waveguide dimensions and materials. The results of our
Lumerical simulations are given in table I. In an optimized Si3N4
strip waveguide (400 × 200 nm) on glass, at λ0 = 625 nm, we obtain
a 29.7 % total (forward + backward) coupling for a central single
photon emitter and a 22.4 % total in-coupling for a complete
central QD layer. In our simulations, we did not include the effect
of top and bottom electric contacts, nor did we include the effect
of self-absorption within the QD layer. In a second step, we switch
to a-Si waveg- uides, which can be used in the near infrared
region. This mate- rial has a large refractive index of about 3.6 .
As such, the total in-coupling efficiency (at λ0 = 1550 nm) is
greatly improved, to 73.9 % and 62.4 % for single photon emitters
and layers of quantum dots respectively.
TABLE I THE TOTAL COUPLING FACTOR β
situation β [%] λ0 [nm] single QD in center of Si3N4 waveguide 29.7
625
single QD in center of a-Si waveguide 73.9 1550 QD slot in Si3N4
waveguide 22.4 625
QD slot a-Si waveguide 62.4 1550
We also managed to physically sandwich a layer of IR- emitting
PbS/CdS QDs in between two a-Si layers, where the QD layer did not
lose its photoluminescence. This is remark- able, since PbS-based
QDs are known to be very sensitive to elevated temperatures and the
PECVD a-Si deposition tempera- ture amounts to 180 °C.
8.0
7.8
7.6
7.4
7.2
y
z
Fig. 8. Nanoplasmonic Yagi-Uda antenna, hybridized with a Si3N4
dielectric waveguide. (a) Schematic view of a three-element
antenna, with reflector (R) feed (F) and director (D) element. (b)
The forward coupling trend during a particle swarm optimization
with 10 particles and 20 generations.
V. PLASMONIC DIRECTIVITY ENHANCEMENT
Yet another route for increasing the in-coupling efficiency of QDs
is by introducing a well-chosen geometry of nanoplas- monic
structures. We thereby think of a nanoplasmonic Yagi- Uda antenna
on top of the Si3N4 strip waveguide on glass, where the QD emitter
is located 10 nm below the feed element. Through Lumerical
simulations, we find an optimal interaction between the feed
element dipole resonance and the waveguide TE mode for waveguide
dimensions of 400 × 150 nm. The an- tenna geometry is shown in
figure 8(a). We employ the particle swarm optimization algorithm
with 10 particles and 20 genera- tions, within the Lumerical
simulation software, as to optimize LR, LD, dR and dD. All antenna
elements are silver bars with a cross section of 30 × 30 nm. The
feed element (LF = 59 nm) is chosen fixed and slightly off
resonance, such that its mode scattering factor is maximal. The
forward coupling trend for an x-oriented dipole — which is greatly
enhanced by the nanoplas- monic dipole resonance — is shown in
figure 8(b). The param- eters of the optimized antenna geometry are
given in table II.
TABLE II OPTIMIZED NANOPLASMONIC ANTENNA GEOMETRY.
LR 61.8 nm LD 53.4 nm dR 52.8 nm dD 97.2 nm
For determining the global forward in-coupling of a (non-
polarized) QD emitter, also y and z contributions are included.
Eventually, we obtain a 7.1 % forward coupling efficiency. This is
better than the coupling we obtain when only the feed ele- ment is
present (1.6 %), or when there is no plasmonic structure at all
(5.9 %), again with the QD located near the top facet of the strip
waveguide. As such, the nanoplasmonic antenna can indeed increase
the in-coupling of QD emission. However, as we have seen in the
previous section, a forward coupling fac- tor almost twice as large
is obtained when placing the QD in the center of the optimized
waveguide (400 × 200 nm), with- out nanoplasmonic structures on
top. The results are repeated in table III, together with the
optimal forward coupling results for single photon emitters of the
previous section.
TABLE III FORWARD COUPLING FACTOR β FOR SINGLE PHOTON
EMITTERS.
situation β [%] λ0 [nm] in center of Si3N4 waveguide 14.9 625
in center of a-Si:H waveguide 37.0 1550 non-central, without feed
element 5.9 625
non-central, with resonant feed element 1.6 625 non-central, with
optimized Yagi-Uda 7.1 625
More complex antenna geometries exist and more elaborate
optimization strategies might further improve the nanoantenna
performance. However, the major drawbacks of introducing
nanoplasmonic structures remain valid: first, the nanoplasmonic
dipole resonance in the visible range suffers from high absorp-
tion losses, primarily due to surface electron scattering. Second,
QDs usually have elevated quantum yields of about 80 %. The effect
of luminescence quenching will therefore dominate over the effect
of luminescence enhancement, when approaching the QD to the metal
nanoparticle. Third, our optimized Yagi-Uda antenna requires
fabrication technologies with a huge resolution of 1 nm, as we have
estimated from our Lumerical simulations.
Concerning the in-coupling efficiency of a complete layer of QDs,
we obtain a nanoantenna bandwidth of 65 nm, covering more or less
the with of the QD batch luminescence response. However, only those
QDs that are located near the nanoantenna feed element, with
separations of order 10 nm, show an im- proved in-coupling. QDs
that are located further away only ex- perience absorption losses
due to the parasitic elements, or are not affected at all. Hence,
this is not a viable approach for in- creasing the in-coupling
factor of a layer of QDs, even when using a grid of densely packed
nanoplasmonic antennae.
VI. CONCLUSION
In this work, we aimed at designing a CMOS-compatible quantum
dot-based integrated light source, having a capacitor structure for
electrically exciting the quantum dots. On one hand, we fabricated
quantum dot capacitors with silicon nitride insulating layers, and
characterized the actual mechanism for electroluminescence in these
devices. On the other hand, we de- termined the optimal structure
for extracting the generated light. We thereby investigated two
routes: the one of dielectric direc- tivity enhancement, where we
optimize the waveguide material and dimensions, and the one of
plasmonic directivity enhance- ment, where we additionally include
nanoplasmonic structures.
REFERENCES
[1] L. Pavesi and D. J. Lockwood, Silicon photonics, Vol. 1.
Springer Science & Business Media, 2004.
[2] P. Geiregat, Silicon compatible laser based on colloidal
quantum dots, 2015, http://www.photonics.intec.ugent.be/research/
topics.asp?ID=127. Accessed: 2016-04-23.
[3] M. Cirillo, et al. “Flash” Synthesis of CdSe/CdS Core-Shell
Quantum Dots, Chemistry of Materials 26.2 (2014): 1154-1160.
[4] P. P. Pompa, et al. Fluorescence enhancement in colloidal
semiconductor nanocrystals by metallic nanopatterns, Sensors and
Actuators B: Chemical 126.1 (2007): 187-192.
CONTENTS ix
1.2 Thesis report structure . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 2
2 Light and matter 3
2.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 3
2.2 Optical behavior of materials . . . . . . . . . . . . . . . . .
. . . . . . . . 4
2.2.1 Permittivity and electric susceptibility . . . . . . . . . .
. . . . . . 4
2.2.2 Refractive index and extinction coefficient . . . . . . . . .
. . . . . 5
2.3 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 6
2.3.2 Realistic dielectrics . . . . . . . . . . . . . . . . . . . .
. . . . . . . 7
2.4 Conductors . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 10
2.4.2 Indium tin oxide coated glass slides . . . . . . . . . . . .
. . . . . . 12
3 Colloidal quantum dots 14
3.1 What’s in a name? . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 14
3.2 Optoelectronic properties . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 16
3.2.2 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 16
3.2.4 Absorbance and luminescence . . . . . . . . . . . . . . . . .
. . . . 19
3.3 Dipolar emission . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 21
3.3.4 Quantum dots as dipoles . . . . . . . . . . . . . . . . . . .
. . . . . 25
3.4 Transition probabilities and rates . . . . . . . . . . . . . .
. . . . . . . . . 26
3.4.1 Photoexcitation . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 26
3.4.4 Quantum Yield . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 28
4.1 Processing techniques . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 30
4.1.2 Electron beam physical vapor deposition . . . . . . . . . . .
. . . . 31
4.1.3 Spin coating . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 31
4.2 Analysis techniques . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 32
5.1.3 Field-driven ionization . . . . . . . . . . . . . . . . . . .
. . . . . . 40
5.2 Device architecture . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 41
5.3 Electric characterization . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 42
5.3.2 Silicon nitride insulator material . . . . . . . . . . . . .
. . . . . . . 43
5.3.3 Quantum dot capacitors . . . . . . . . . . . . . . . . . . .
. . . . . 46
5.4 Device Electroluminescence . . . . . . . . . . . . . . . . . .
. . . . . . . . 50
5.5 Device degradation . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 50
6.1 Dielectric waveguides . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 52
6.1.1 Waveguide modes . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 52
6.1.3 The strip waveguide . . . . . . . . . . . . . . . . . . . . .
. . . . . 56
6.2 Coupling into a waveguide . . . . . . . . . . . . . . . . . . .
. . . . . . . . 57
6.2.1 Integrated source design . . . . . . . . . . . . . . . . . .
. . . . . . 57
6.2.2 The β-factor . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 58
6.4.1 PbS quantum dots in amorphous silicon structures . . . . . .
. . . 61
6.4.2 Amorphous silicon strip waveguide . . . . . . . . . . . . . .
. . . . 62
7 Plasmonic directivity enhancement 66
7.1 Metal-dielectric interface . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 66
7.2 Metal nanoparticles . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 68
7.3 Plasmonics hybridized with dielectric waveguides . . . . . . .
. . . . . . . 77
7.3.1 Boosting resonant lengths . . . . . . . . . . . . . . . . . .
. . . . . 77
7.3.2 Waveguide mode extinction . . . . . . . . . . . . . . . . . .
. . . . 78
7.3.3 Polarized quantum dot emitter . . . . . . . . . . . . . . . .
. . . . 82
7.4 Nanoplasmonic antenna . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 83
7.4.4 Forward coupling of single photon emitters . . . . . . . . .
. . . . . 88
7.4.5 Evaluation of the nanoplasmonic antenna . . . . . . . . . . .
. . . . 89
8 Conclusion 90
EL Electroluminescence
EM Electromagnetic
FD-QLED Field-driven QLED
HOMO Highest Occupied Molecular Orbital
IR Infrared
PL Photoluminescence
LED Light Emitting Device
SEM Scanning Electron Microscopy
TM Transverse Magnetic
QD Quantum Dot
QED Quantum Electrodynamics
1.1 Motivation of the research project
In integrated photonics as opposed to electronics, one uses light
as carrier of information, rather than electricity. The field of
photonics is considered to be crucial for developing the next
generation of devices in datacommunication, on-chip interconnects,
sensing and biosensing and even in quantum computing [1]. One
particular example is given by the electrical interconnects in
between microprocessors, which are reaching their limits in terms
of both power consumption and bandwidth. On-chip optical
interconnects, compatible with the silicon-based CMOS fabrication
technology, could offer a viable solution to this problem. However,
the silicon on insulator and silicon nitride material integrated
photonics platforms have poor light emitting and light modulating
properties. As such, a cheap and efficient integrated light source,
compatible with the CMOS fabrication technology, is intensely
sought after.
Colloidal quantum dots (QDs) or semiconductor nanocrystals can be
produced in large quantities, and the process of forming high
quality quantum dot layers from the colloidal solution is quite
straightforward, for instance using the spin coating technique.
Quantum dots have excellent luminescent properties, featuring high
quantum yields and color puri- ties. Additionally, their emission
wavelength can be tuned by altering either the quantum dot size or
the quantum dot material. As such, a broad range of wavelengths can
be covered, from the visible to the near infrared region.
In this thesis research, we aim at designing a QD-based
electrically-driven integrated light source, compatible with the
CMOS fabrication technology. Electrically-driven QD light emission
requires the formation of electron-hole pairs within the QDs,
followed by radiative decay. Electrons and holes can be directly
injected from outer contacts, although this demands a careful
choice of charge injection and transport layers. We consider a
quantum dot capacitor structure, with a layer of QDs sandwiched
vertically between two insulating layers with top and bottom
electric contacts. When applying an alternating voltage over the
device, the QDs are excited either through impact
excitation/ionization
1.2 Thesis report structure 2
by hot electrons stemming from the insulator interfaces or by field
driven ionization [2, 3]. The latter process generates free charges
within the quantum dot layer itself. In these quantum dot
capacitors, the choice of insulator material is not so critical and
any kind of QD layer can be inserted, which greatly extends the
versatility of our structures.
1.2 Thesis report structure
In chapter 2 we provide the basis for mathematically describing
electromagnetic waves and we explain the optical behavior of
materials. The refractive index and/or permittivity data is
presented for all materials that are used in the course of our
research. The same data is employed in our Lumerical
simulations.
In chapter 3 we introduce the concept of colloidal quantum dots,
which will make up the active medium of the light sources we
envisage. We elucidate the physics explaining their particularly
interesting optoelectronic properties and prove that QD emission
can be treated as dipolar emission. A brief quantum mechanical
description for photoexcitation and photon emission is provided,
and the Purcell effect is discussed. We also introduce the quantum
yield of an emitter.
In chapter 4 we outline the most important processing and analysis
techniques employed in the course of this thesis research.
In chapter 5 we present the mechanisms capable of providing QD
electroluminescence. In a second step, we fabricate our own quantum
dot capacitor structures and characterize their electroluminescent
behavior. We thereby partly rely on PSPICE simulations for
interpreting our results.
In chapter 6 we discuss the concept of dielectric waveguides and
waveguide modes. We provide a simple integrated source design and
estimate the optimized coupling of both a single photon emitter and
a layer of quantum dots into the waveguide structure, based upon
Lumerical simulation software. In a second step, we switch to
amorphous silicon strip waveguides, which show a greatly increased
in-coupling factor. We also prove that it is possible to integrate
PbS/CdS core/shell QDs, emitting in the near-IR, into the amorphous
silicon waveguide material.
In chapter 7 we investigate nanoplasmonic particles. Using the
particle swarm algorithm within the Lumerical simulation software,
we prove that a well-chosen geometry of these structures — a
so-called nanoplasmonic antenna — can increase the in-coupling of a
single photon emitter into a waveguide. However, we also indicate
that very precise and expensive fabrication technologies are
required. We ultimately show that these structures are not capable
of increasing the in-coupling of emission stemming from a complete
layer of QDs.
LIGHT AND MATTER 3
Light and matter
In this chapter, we first introduce the Maxwell equations for
describing electromagnetic radiation. They provide a mathematical
‘light’ description in terms of vector fields that are mutually
coupled through a set of partial differential equations. The
materials of interest are thereby described either by their
permittivity or by their refractive index. We include some
simplified models that explain the permittivity of dielectrics and
conductors. Ulti- mately, these insights are employed in
understanding the optical behavior of all materials that will be
used during the course of this project.
2.1 The Maxwell equations
What is commonly referred to as ‘light’ — being the visible and
occasionally including the infrared up to ultraviolet region — is
only a small part of a much broader spectrum of electromagnetic
radiation. All electromagnetic radiation satisfies the Maxwell
equations. For instance, in the Minkowski formulation [4], these
are given by:
∇× E = −∂B
∂t , (2.1)
∇×H = ∂D
∂t + J, (2.2)
∇ ·D = ρ, (2.3)
∇ ·B = 0, (2.4)
in which E represents the electric field, H the magnetic field, D =
εE the displacement field, B = µH the magnetic induction, J = σE
the current density and ρ the charge density. Throughout this
thesis, we will always assume all materials to be isotropic,
implying a scalar (and not a tensorial) nature of the permittivity
ε, the permeability µ and the conductivity σ. In addition we
suppose all materials to be linear.
Often it is more convenient to switch to the frequency domain. We
employ the engi- neering formalism when introducing a harmonic time
dependency ejωt for all fields. As
2.2 Optical behavior of materials 4
such, the explicit variables change from (r, t) to (r, ω):
∇× E = −jωB, (2.5)
∇ ·D = ρ, (2.7)
∇ ·B = 0. (2.8)
We will further assume all materials to be electrically neutral or,
equivalently, ρ = 0 C/m3. For dielectrics, σ = 0 S/m, resulting in
J = 0 A/m2. More generally, the current density can be incorporated
into the displacement field by passing to a new permittivity ε = ε
+ σ/jω, resulting in:
∇× E = −jωB, (2.9)
∇×H = jωD, (2.10)
∇ ·D = 0, (2.11)
∇ ·B = 0. (2.12)
2.2 Optical behavior of materials
Electromagnetic radiation interacts with matter since matter
contains electric charges. Locally, ‘matter’ is fully characterized
by its permittivity and permeability — both de- pending on the
field pulsation ω. From now on we only consider non-magnetic
materials, thus µ = µ0 = 4π×10−7 H/m. Usually the relative
permittivity εr of a material is introduced, where ε = εrε0, ε0 =
8.854× 10−12 F/m being the vacuum permittivity.[5]
2.2.1 Permittivity and electric susceptibility
Each material that is subjected to an electric field, is affected
on the microscopic level: electron clouds deform, electric dipoles
change orientation, mobile charges acquire a di- rected motion, ...
Assuming the field amplitudes are small and excluding
ferro-electricity, the material’s response is linear. As a result,
all these effects can be incorporated into a generalized
polarization density P:
P(ω) = ε0χ(ω)E(ω), (2.13)
where χ(ω) is the electric susceptibility. This polarization
density contributes to the dis- placement field through D = ε0E + P
= εrε0E, connecting the electric susceptibility to the relative
permittivity of the material:
εr = 1 + χ. (2.14)
The total displacement field D that emerges when imposing an
electric field E, contains both the response of free space
(contribution ‘1’) and the response of the material itself
2.2 Optical behavior of materials 5
(contribution χ). Alternatively, we can separate the real and
imaginary parts of χ = χ′ + jχ′′ and εr = ε′r + jε′′r ,
obtaining:
ε′r = 1 + χ′, (2.15)
ε′′r = χ′′. (2.16)
Due to causality, the real and imaginary part of the electric
susceptibility are related to each other according to the
Kramers-Kronig relations:
χ′(ω) = 2
χ′′(ω) = 2
ω2 − ω′2dω ′, (2.18)
in which P indicates the Cauchy principal value of the integral.
Using the Kramers-Kronig relations, one can deduce the imaginary
(real) part of χ knowing its real (imaginary) part over the entire
frequency range. Therefore, as will become clear later on,
dispersive materials will always show some absorption and visa
versa.
2.2.2 Refractive index and extinction coefficient
For homogeneous, infinitely extended media, (2.9) through (2.12)
can be combined into Helmholtz equations for both the electric and
magnetic field:
E(r) + ω2εrε0µ0E(r) = 0, (2.19)
H(r) + ω2εrε0µ0H(r) = 0. (2.20)
It is common to introduce the vacuum wavenumber k0 = ω √ ε0µ0 = ω/c
and the complex
refractive index n = n − iκ, for which εr = n2. n is called the
refractive index and κ the extinction coefficient of the material
in question. This way, e.g. (2.19) can be replaced by E(r) +
n2k2
0E(r) = 0. Solutions are found that only depend on a single
coordinate: the so-called plane waves. In general a plane wave,
having an arbitrary propagation direction u, (u = 1) is represented
by:
E(r, t) = E0 exp [j(ωt− nk0 · r)] (2.21)
= E0 exp [j(ωt− nk0 · r)] exp(−κk0 · r), (2.22)
where we have introduced the free space wavevector k0 = k0u and the
electric field am- plitude E0. With λ0 = 2π/k0 the vacuum
wavelength, n determines the wavelength λ in the material through λ
= λ0/n. n, if its wavelength dependence is significant, will also
cause dispersion. The extinction coefficient κ governs attenuation
and gain. Passive mate- rials have a positive extinction
coefficient, causing an exponential decay of all fields in the
direction of propagation, whereas active materials have a negative
extinction coefficient, resulting in gain.
2.3 Dielectrics 6
The relation between (n, κ) on one hand and the relative
permittivity εr = ε′r + jε′′r on the other hand is immediately
clear:
n2 − κ2 = 1 + ε′r, (2.23)
−2nκ = ε′′r . (2.24)
The refractive index and extinction coefficient might provide more
insight into the behavior of light propagating through a material.
The refractive index n of materials is typically positive, although
metamaterials with a zero [6] or negative [7] effective refractive
index over an extended wavelength domain have been reported.
Assuming n > 0, the material can only be lossy (κ > 0) if
ε′′r < 0. The concept of ‘effective index’ will be put forward
in section 6.1.
2.3 Dielectrics
A dielectric is a material that does not contain any mobile
charges, yet it becomes polarized when applying an electric field.
This material polarization density P, as introduced in section
2.2.1, can be obtained either through reorientation of permanent
dipoles or through material excitations engendering an induced
dipole moment. [4] In sinusoidal regime, P oscillates with the
frequency of the field. For optical fields this frequency takes a
value between 430 and 770 THz.
2.3.1 Damped oscillator model
In the damped-oscillator model [5], the dielectric is replaced by a
volume density N of 1D oscillators with mass m, charge e, force
constant k and damping ratio ζ. As to ease the notation, the
natural frequency ω0 =
√ k/m is introduced. Going through Newton’s second
law (d2x dt
+ 2ζω0 dx dt
+ ω2 0x = eE
m ejωt) and passing to the frequency domain, an expression is
obtained for the electric susceptibility of such a
dielectric:
χ = Ne2
. (2.25)
It should be noted that this model does not take into account the
electric field created by the neighboring oscillators. The real and
imaginary parts of the permittivity εr = ε′r + jε′′r are readily
found:
ε′r = 1 + Ne2
1
Figure 2.1: The permittivity εr = ε′r + jε′′r as a function of the
frequency ω, for a resonant dielectric according to the damped
oscillator model.
Figure 2.1 displays the characteristic behavior of ε′r and ε′′r in
the neighborhood of the
natural frequency ω0. In the low frequency limit, εr → 1 +
Ne2
mε0/ω2 0. The polarization is in
phase with the electric field and losses are low: ε′′r → 0. Second,
for very high frequencies, εr → 1. The polarization can no longer
follow the electric field. χ now introduces a 180 ° phase lag and P
= 0 C/m2. Consequently, the electromagnetic field does not feel the
material: it is as if the EM waves were propagating through free
space. That is why, in general, all materials tend to be
transparent (well) above their natural frequency ω0. Once again,
losses are low (ε′′r → 0). Since the system is significantly
underdamped (ζ < 1/
√ 2),
one will find a resonant frequency ωr = ω0
√ 1− 2ζ2 ≈ ω0 in between these two extreme
cases. In the neighborhood of ω0 the imaginary part ε′′r becomes
strongly negative, giving rise to huge losses.
2.3.2 Realistic dielectrics
In general several mechanisms can contribute to the material
polarization. A typical ex- ample of the relative permittivity of a
realistic dielectric is shown in figure 2.2. At low frequencies,
dipole relaxation effects play an important role. In an oscillating
electric field, permanent electric dipoles constantly relax into
their newly defined equilibrium orienta- tion. Due to the viscosity
of the medium, there is some energy dissipation. For increasing
frequencies, the permittivity shows multiple resonances that arise
from vibrational and electronic excitations. Each resonance is
characterized by a peak in absorption, indicated by a sudden
strongly negative value of ε′′r . The real permittivity part, ε′r,
is seen to steadily rise in frequency — and hence drop in
wavelength — in between every two subsequent res- onances. As a
result, the refractive index n of most dielectrics will decrease
with increasing wavelengths in between resonances. We refer to
figure 2.3 for some examples. The general trend of ε′r is
downwards, with a significant drop near each resonance, eventually
reaching 1 in the high frequency limit. It is not hard to imagine
that multiple resonances can coincide,
2.3 Dielectrics 8
Frequency [Hz]
Figure 2.2: The permittivity εr = ε′r + jε′′r as a function of
frequency f = 2πω, for a typical dielectric with multiple
resonances.[8]
giving rise to a more complicated behavior. At optical frequencies
(430 to 770 THz), the electronic excitations predominantly dictate
the optical behavior of materials.
2.3.3 Overview of dielectrics used
As indicated in section 2.2.2, the refractive index n, together
with the extinction coefficient κ of a material, offer a good
understanding of how this material interacts with light. The
dielectrics used throughout this project are (hydrogenated)
amorphous silicon, aluminum oxide, and silicon nitride. Their (n,
κ) data is shown in figure 2.3.
Amorphous silicon Hydrogenated amorphous silicon (a-Si:H or a-Si)
is a non-crystalline form of silicon, to which hydrogen was added
in order to maximally passivate the Si dangling bonds. It can be
deposited in thin films, e.g. using PECVD [9]. We will use this
technique (see section 4.1.1) when fabricating our devices, in the
clean room of the Ghent University Photonics Research Group. The
a-Si material is deposited at temperatures of 180 to 200 °C. It has
a band gap between 1.71 and 1.92 eV, and surface roughnesses of
2.15 nm can be obtained [10]. Figure 2.3(a) shows the a-Si (n, κ)
data as a function of the wavelength. Clearly, the absorption
losses at optical wavelengths (390 to 700 nm) are high. However, it
can perfectly serve as an optical material in IR; even more so
because of its high refractive index, which is about 3.54 in this
region. At the right hand side, a detailed view is given near 1550
nm, which is an important wavelength in fiber communication.
2.3 Dielectrics 9
a-Si:H
n
5
4
3
2
1
0
n
κ (b) Al2O3
MF MF
Figure 2.3: Refractive index and extinction coefficient of (a)
a-Si, (b) Al2O3 and (c) Si3N4 (MF: mixed frequency, HF: high
frequency), as function of wavelength. The left hand side displays
all data available for λ < 2000 nm, while the right hand side
provides a detailed view on two regions of interest: around 625 nm
(emission wavelength of CdSe/CdS Flash QDs [11]) and/or 1550 nm
(fiber optics).
2.4 Conductors 10
Aluminum oxide Aluminum oxide or alumina (Al2O3) is a large band
gap material (Eg ≈ 8.4 eV). It can be deposited at Ghent University
using ALD, resulting in thin layers of high-quality insulator
material. The Al2O3 (n, κ) data [12] is shown in figure 2.3(b). It
has low absorption losses across the entire VIS-IR range, and a
refractive index of about 1.75 .
Silicon nitride Silicon nitride (Si3N4) is an amorphous material
with a band gap of about 5 eV. Si3N4
waveguide structures and devices are compatible with the
silicon-on-insulator microelectronics fabrication technology, hence
the great interest of silicon photonics. The material is de-
posited using PECVD (see section 4.1.1). We will use HTHF (high
temperature high frequency) and HTMF (high temperature mixed
frequency) Si3N4; both are deposited at a 270 °C temperature. HF
Si3N4 generally results in a somewhat denser layer, while MF Si2N4
is better suited for deposition onto a (non-flat) quantum dot
layer, where its HF counterpart has been found to show cracks. In
terms of refractive index and extinction coefficient, there is only
a small difference between HF and MF Si3N4. We refer to figure
2.3(c). Si3N4 can be used both in the visible and near-infrared
range.
2.4 Conductors
In metals, electromagnetic radiation will predominantly interact
with mobile charge car- riers: the free electrons. Light impinging
on a metal gives rise to microscopic currents, which are accounted
for by the generalized polarization density P introduced above. The
complex permittivity εrε0 should thereby include the effects of
conduction.
2.4.1 The Drude model for metals
The Drude-model explains the basic electrodynamic properties of
metals. Thereby, the free electrons are thought of as a gas of
particles, merely interacting through collisions. It can be seen as
a special case of the damped oscillator model for a resonant
dielectric. The latter includes a restoring force with force
constant k, whereas the free electrons are not bound at all: k → 0.
Newton’s second law is slightly rewritten as d2x
dt + 2γ dx
dt = eE
m ejωt.
Damping is now governed by γ, a parameter related to the mean free
time of the electrons in the Drude model. Going through the same
procedures as we did for dielectrics, we obtain [5] an expression
for the electric susceptibility of metals:
χ = ω2 p
−ω2 + 2jγω . (2.28)
2.4 Conductors 11
-4 -2 0
700600500 -20 -10
0
700600500
Figure 2.4: Left panel: the ε′r data for gold and silver, as a
function of wavelength. The Au data are shifted over −100 (ε′Au −
100) for clarity. Right panel: the ε′′r data for gold and silver.
The Au data are shifted over −5 (ε′′Au− 5) for clarity. Drude model
fits are added in dashed lines. The inset of the figure zooms in on
the 500 to 700 nm region — leaving out the offset for the Au
data.
The permittivity εr = ε′r + jε′′r of a metal becomes:
ε′r = 1− ω2 p
ω2 + 4γ2 , (2.29)
ε′′r = −2γ
ω2 + 4γ2 . (2.30)
The left and right panels of figure 2.4 respectively show the ε′r
and ε′′r behavior of both gold and silver, as a function of
wavelength. The experimental data [12] is ac- companied by a Drude
fit, according to formulae 2.29 and 2.30. The fitting parameters
(2πωp,Au = 1.93 PHz, 2πγAu = 10.6 THz, 2πωp,Ag = 2.10 PHz, 2πγAg =
9.98 THz) are calcu- lated using matlab and closely resemble the
values reported in literature [13],[14],[15],[16]. At large
wavelengths, the strongly negative real part prevails, indicating
that the free elec- trons efficiently shield electromagnetic
fields. For shorter wavelengths, interband effects occur that
should be described using the Lorentz-Drude model, by introducing
Lorentz contributions [17].
One would expect the free electrons in a metal to interact
significantly with each other, resulting in a poor overall
performance of the Drude model. Indeed, an assumption of non-
interacting particles/oscillators was made, both in the Drude model
and in the damped oscillator model. However, due to electric field
screening, the electrons in a solid are dressed
2.4 Conductors 12
with a cloud of particle-hole excitations. The resulting electron
quasiparticles behave as if they were independent particles
[18].
2.4.2 Indium tin oxide coated glass slides
5
4
3
2
1
0
n
2.00 1.98 1.96
κ (b) SiO2
Figure 2.5: The refractive index and extinction coefficient of (a)
ITO [19] and (b) glass [12], as a function of the wavelength. At
the right hand side, some enlarged views for wavelengths of
interest are given.
Indium tin oxide or ITO (In2O3/SnO2) cannot be seen as a metal, nor
as a dielectric. It is a degenerate, large band gap (Eg ≈ 4 eV)
semiconductor, where SnO2 serves as a ‘dopant’ of (high)
concentration in the In2O3 material (typically at 10 %wgt).
Amorphous ITO films can be deposited on all kinds of substrates,
for instance using PVD [20] (see section 4.1.1). The resulting thin
amorphous layers are not only conductive, but also transparent in
the visible region, with a refractive index of about 1.98 .
Although ITO contains mobile charges
2.4 Conductors 13
at high concentrations, optical transparency is not jeopardized due
to the unusually low plasma frequency, in the near IR. Free carrier
absorption is therefore pushed into the IR region; the UV is
characterized by band-to-band absorption [21].
We use ITO coated glass slides (25 × 25 × 1.1 mm), provided by
Sigma-Aldrich. These can serve both as substrate and as transparent
electric contact, with a square resistance of 30 to 60 /. The ITO
layer thickness varies from 20 to 30 nm [22]. Glass (SiO2) shows
almost no absorption at optical frequencies and has a refractive
index of about 1.44 .
COLLOIDAL QUANTUM DOTS 14
Colloidal quantum dots
Colloidal quantum dots (QDs) make up the active medium of the light
source we envisage, and as such they deserve a chapter of their
own. QDs were discovered [23, 24] in the early 80s and have been a
popular and promising field of science ever since. We first
introduce the concept of colloidal QDs. Secondly, we elucidate the
physics explaining their particularly interesting optoelectronic
properties. We also prove that QD emission is dipolar emission, an
insight that will be exploited in our simulations. In a last
section, the quantum mechanics related to optoelectronic
transitions within QDs is briefly discussed. Throughout this
chapter, the particular strengths of QDs will become apparent,
being their high luminescent efficiency, narrow emission spectra,
tunable emission wavelength and the possibility of (cheap) QD
synthesization through colloidal methods.
3.1 What’s in a name?
A quantum dot (QD) is a nanometer-sized (2 -15 nm) piece of
semiconductor material. The QD and bulk optical properties
radically differ, due to the reduced size in all three dimensions —
the so-called quantum confinement effect. However, the crystal
structure and lattice constant of QDs in general closely resemble
their bulk equivalents, hence the alternative appellation of
nanocrystal (NC). Figure 3.1(a) shows a Transmission Electron
Microscope (TEM) image of a PbSe QD. Indeed, the rock-salt bulk
crystalline ordering of atoms becomes apparent.
Colloidal synthesis, in particular the Hot Injection Method [25] is
a very efficient method for synthesizing QDs. It can be classified
as a bottom-up approach, implying the assembly of QDs starting from
precursor molecules or monomers. First, a precursor solution is
injected into a hot solvent. Nuclei are thereby formed and grow
further. We obtain a quantum dot colloid : a solution of inorganic
QDs, capped with an organic ligand shell. The ligands, e.g. oleic
acid, ensure steric stabilization of the colloid. The ultimate QD
size can be controlled through the growth temperature and the
concentration of this stabilizing ligand. Third, the resulting
colloidal quantum dots are brought into an appropriate organic
solvent, e.g.
3.1 What’s in a name? 15
(a) (c)(b)
Figure 3.1: (a) TEM image of a colloidal PbSe QD [30]. (b)
Schematic visualization of a core/shell structured QD, with a view
on the internal core structure (credit: Rus- nano). (c) TEM image
of a monolayer of colloidal CdSe/ZnS core-shell QDs [31].
toluene (C6H5CH3) or tetrachloroethylene (C2Cl4). Via colloidal
synthesis, high quality QD batches of small size dispersion (<
10 %) can be synthesized [26]. This synthesis method is deemed most
promising for producing large quantities of QDs for commercial
applications.
The most common colloidal QD materials include all kinds of II-VI,
IV-VI and III-V compound semiconductors, in particular the
sulfides, selenides and tellurides of Zn, Cd, Hg and Pb. It should
be noted that toxic materials are often involved. For instance, The
International Agency for Research on Cancer classified Cd as
carcinogenic to humans [27].
Provided that their lattice mismatch is not too large, an
additional shell can be grown epitaxially around the core material,
obtaining a core/shell quantum dot [28, 29]. One has recently
proposed a so-called ‘flash’ synthesis method [11] for producing
high qual- ity CdSe/CdS core/shell QDs in only a matter of minutes.
An important advantage of the core-shell structure in general is
the improved surface passivation of the inner core. Additionally,
it allows for ‘bandstructure engineering’, with a far-reaching
impact on the wavefunctions of electrons and holes within the QD.
Figure 3.1(b) gives a schematic visu- alization of a colloidal
core-shell structured QD. The stabilizing ligands (e.g. oleic acid)
are clearly visible; these ensure the stability of the QD colloid
through steric stabilization. Figure 3.1(c) shows a monolayer of
colloidal CdSe/ZnS core-shell QDs. Layers of this kind can be
produced using the Langmuir-Blodgett method. For depositing thicker
QD layers, spincoating can be employed.
3.2 Optoelectronic properties 16
3.2.1 From single atoms to bulk material
Single atoms have well-defined, discrete electronic states and
excitation energies. A gas of barely interacting helium atoms
serves as a nice example. Indeed, its light absorption and emission
spectra are line spectra. Thereby, the absorption or emission
photon energy equals the separation of the electronic energy levels
involved.
When bringing together a moderate number of N atoms into a bound
state, the in- dividual atomic orbitals (AOs) overlap to form
delocalized molecular orbitals (MOs). In this ‘molecule’, the MOs
maintain their discrete nature and the fundamental excitation
energy is simply the separation of the lowest unoccupied molecular
orbital (LUMO) and the highest occupied molecular orbital (HOMO),
quite similar to the case of the single atom.
With increasing N however, the number of MOs equally increases and
their energy levels become more densely packed. In the limit of N
→∞, the energy levels form quasi- continuous energy bands with
intraband spacings no longer exceeding the thermal energy kBT . In
particular, one can distinguish the completely filled valence band
and the empty (or only partially filled) conduction band. In the
specific case of a semiconductor, the HOMO (top of valence band)
and LUMO (bottom of conduction band) energy levels are
well-separated by a band gap Eg of order 1 eV kBT . This is the
fundamental excitation energy of the bulk semiconductor electron
system.
For bulk semiconductors, an incoming photon can in this view only
be absorbed if its energy exceeds the band gap energy Eg. Thereby,
a valence band electron is promoted to the conduction band, leaving
behind a hole. The remaining energy is put into the kinetic energy
of this electron and hole. We assume a direct band gap material for
simplicity. Because of energy conservation, the photon energy ~ω
should equal the energy of the electron-hole (e-h) excitation: ~ω =
Eg + ~2k2/2µ, where k is the electron/hole wavenumber and µ =
1/m∗−1
e +m∗−1 h the reduced mass. m∗e andm∗h are the electron and hole
effective masses
respectively. The absorption spectrum will show a √ ~ω − Eg
behavior, proportional to
the joint DOS of electrons and holes [32].
3.2.2 Excitons
In general, the electron and hole in a bulk semiconductor follow
their own separate path through the crystal. However, bound e-h
excitations also exist. These short-lived neutral quasiparticles
are called excitons and move through the crystal as a whole. The
most simple way of describing an exciton is using a hydrogenic
Hamiltonian:
H = pe
3.2 Optoelectronic properties 17
where electron and hole are held together by a Coulombic
attraction, against a background medium of permittivity εrε0 — the
semiconductor permittivity. Employing the same tech- niques as
those used for solving the hydrogen system, we obtain the energy
Eex,bulk(n) that is required for creating an exciton in a bulk
semiconductor:
Eex,bulk(n) = Eg − µ
n2 , (3.2)
where ERy = 13.6 eV is the Rydberg energy and m0 the free electron
mass. The second term in (3.2) is the exciton binding energy.
Theoretically we expect to find discrete exciton absorption peaks
at ~ω = Eex,bulk(n), thus below the fundamental absorption offset
Eg. In practice however, bulk excitons can only be observed in
semiconductors of high purity and at low temperatures.
The hydrogen model for bulk excitons also predicts the most
probable spatial separation of the electron and hole, the so-called
exciton Bohr radius a∗B:
a∗B = εr m0
µ aB, (3.3)
in which aB = 0.53 A is the conventional Bohr radius. Because of
the relatively good screening — εr ranges from 5 to 12 for
inorganic semiconductors — and the low effective masses of
electrons and holes, its value typically amounts to several
nanometers [32].
As mentioned earlier, quantum dots lie somewhere in between the two
extremes of atoms on one hand and bulk material on the other hand.
The number N of atoms involved typically ranges from 102 to 105 .
Photon absorption leads to electron-hole excitations, quite similar
to the case of a bulk semiconductor. However, as the QD diameter is
com- parable to the exciton Bohr radius a∗B, electrons and holes
will inevitably be subjected to Coulombic attraction. Therefore, we
can call them excitons, as we did for the bulk case. However, such
a confinement of charge carriers into a small piece of matter
influences the energy levels considerably. The so-called quantum
confinement effect will turn out to be the dominant energy
contribution for excitons in QDs.
3.2.3 The quantum confinement effect
The qualitative effect of confining a particle into a small amount
of space can be best illustrated using the model of a particle in a
1D square infinite box. The Heisenberg uncertainty principle
dictates that xpx ≥ ~/2, where x can be approximated by the width
of the box L and px =
√ p2 x − px2. As the particle is unable to leave the box
p = 0. Its average kinetic energy thus becomes: T = p2/2m ≥ ~/8mL2.
Clearly, when limiting the movement of a particle (L→ 0), its
kinetic energy drastically increases. We will further refer to this
(kinetic) energy as the confinement energy.
In case the exciton resides in a quantum dot as opposed to bulk
material, the Hamilto- nian of (3.1) should be modified
correspondingly in order to account for this confinement
3.2 Optoelectronic properties 18
effect. We assume the conservation of the bulk lattice structure
and thereby recycle the concept of (bulk) effective mass, although
this was shown not to be very accurate [33]. The QD itself is
replaced by an infinite spherical potential well of radius R.
Similar to the above, its material is taken homogeneous, with
permittivity εrε0. As such, the Schrodinger equation describing the
exciton in the QD reads: pe
2
Φ(re, rh) = EΦ(re, rh). (3.4)
The potential energy V clearly consists out of two parts: the first
contribution represents a Coulombic attraction while the second
contribution forbids the charges to leave the (infinite) potential
well, with U(re, rh) rocketing to infinity whenever re or rh
exceeds R [34].
As proposed in [35], two limiting situations can now be discerned,
depending on which energy contribution dominates the system. On one
hand, in the regime of weak confine- ment (R a∗B), the Coulombic
term prevails: the exciton is primarily bound through a mutual
Coulombic attraction of electron and hole, quite similar to the
exciton in a bulk semiconductor. Confinement will only have a
moderate effect on the exciton energy levels, and can be included
in perturbation. On the other hand, in the regime of strong
confine- ment (R a∗B), the electron and hole can in first instance
be described independently, since the Hamiltonian of (3.4) is
dominated by the kinetic/confinement energy one-particle operators.
Afterwards, their Coulombic interaction can be included as a
perturbation, correlating the electron and hole.
We are primarily interested in the latter case of strong
confinement. According to the particle-in-a-box model, a basis of
wavefunctions for an electron in an infinite spherical potential
well is given by the spherical Bessel functions. We will only
retain the s-like wavefunctions Ψn with energies En:
Ψn(re) = cn r
sin nπre
R , (3.5)
En = ~2π2n2
2meR2 . (3.6)
The most simple unperturbed (and uncorrelated) exciton wavefunction
is obtained by sim- ply taking the fundamental particle-in-a-box
state for both electron and hole: Φ(re, rh) = Ψ1(re)Ψ1(rh). In a
second step, their Coulombic interaction can be incorporated using
first order perturbation theory. Numerically one obtains [36] the
following energy correc- tion: Φ|VCoulomb|Φ = −1.786 e2/4πεrε0R. We
finally have the energy Eex,QD required for creating an exciton in
a quantum dot — the so-called excitonic band gap:
Eex,QD = Eg + ~2π2
3.2 Optoelectronic properties 19
From this expression we see that the confinement energy and the
Coulombic interaction scale differently with the size of the QD.
This suggests that the approximation is indeed valid provided that
R is taken sufficiently small. The fundamental excitation energy of
the QD system depends both on the material choice (through Eg) and
on the dimensionality of the QD (through R). This will allows the
tuning of optical properties just by picking QDs of an appropriate
material/size combination. In particular, the quantum confinement
effect is reflected in the luminescence and absorbance spectra of a
QD colloid.
3.2.4 Absorbance and luminescence
First of all, an electronic system can be characterized by studying
its absorption. Imagine [37] an electromagnetic wave of intensity
Ii traveling through a sample containing QDs. The outgoing wave has
an intensity If < Ii as a result of absorption. Encouraged by
the exponential decay of section 2.2.2, the absorbance A of the
sample is defined as:
A = log10
( Ii
If
) . (3.8)
Quite naturally, one introduces the absorption coefficient α, which
is independent of the sample thickness t. It is related to both the
absorbance of the sample and the effective extinction coefficient
κeff of the sample:
α = ln(10)A
λ . (3.9)
The absorbance in absolute value can be used to determine the QD
concentration. However, we will consistently normalize all
absorbance data to the first exciton peak.
The photoluminescence (PL) spectrum of a QD colloid is measured by
exciting the sample well above its excitonic band gap. The created
excitons thermalize into the lowest energy state, possibly followed
by radiative exciton decay. The PL spectrum shows the number of
detected photons as a function of the emission wavelength,
normalized to the maximum of this emission peak.
The absorbance and PL spectra for various colloids of PbS (core)
QDs in C2Cl4 are shown in figure 3.2(a). The absorbance
measurements are performed with a Perkin Elmer Lambda 950
spectrometer. The PL spectra are taken using an Edinburgh
Instruments FLSP920 UV-vis-NIR spectrofluorimeter with a 450 W
xenon lamp excitation source and a Hamamatsu near-IR photodetector.
In each sample, we note a distinct absorption peak corresponding to
the excitonic band gap. It is indicative of the creation of the
fundamental QD exciton state. From its position we calculate the
average QD size using the empirical sizing formula proposed by
[38]. The PbS bulk band gap at 300 K is 0.41 eV (or 3350 nm). Due
to quantum confinement however, the excitonic band gap is
significantly higher, and further blueshifts with decreasing QD
diameter. Our excitonic band gap predictions using (3.7) are marked
by black arrows. For the electron and hole effective masses we
use
3.2 Optoelectronic properties 20
2.90/ — nm
Figure 3.2: (a) Absorbance (solid line) and PL (shaded area) for
some PbS core QD col- loids of various QD sizes; black arrows mark
a prediction of the excitonic band gap using (3.7). (b) A series of
CdSe-based photoluminescent QD colloids (credit: Iwan Moreels). (c)
Absorbance (solid line) and PL (shaded area) for some ‘flash’
CdSe/CdS core/shell QD colloids, including the absorbance of the
naked 2.90 nm core.
m∗e ≈ m∗h = 0.085m0. For the permittivity we take εPbS = 17.2 .
Clearly this prediction is not capable of reproducing the measured
position of the absorption peak. In fact, the effective mass
approximation breaks down for small R, as the band edge is then no
longer parabolic, due to electron-hole correlation effects. This is
true for both PbS [39] and CdS [40] QDs with particle diameters
below 10 nm. As a result, the particle-in-a-box model largely
overestimates the excitonic band gap energy. Semi-empirical
tight-binding methods [41, 42] give much better predictions.
The PL spectra are slightly redshifted with respect to the first
exciton absorption peak. This is commonly referred to as the Stokes
shift and is due to band edge relaxation of excitons via acoustic
phonon emission. The PL emission peak is in general quite narrow.
By altering the QD size, its position can be chosen freely
throughout the entire near-
3.3 Dipolar emission 21
IR region. In particular, PbS core (and PbS/CdS core/shell) QDs
that emit at about 1300 nm and 1550 nm can be synthesized. Both
wavelengths are of high interest for fiber communication.
Figure 3.2(b) shows a series of CdSe-based photoluminescent QD
colloids of varying size. With some effort, the QDs can be chosen
to emit anywhere in the visible range. Figure 3.2(c) shows the
absorbance and PL spectra of various colloids containing CdSe/CdS
core/shell ‘flash’ QDs in toluene. The absorbance spectra are
collected using a Perkin Elmer UV/vis Lambda 2 spectrometer and the
photoluminescence is registered by a Hamamatsu R928P PMT detector.
The size of the core/shell structures is determined empirically
using an appropriate sizing formula for the position of the first
exciton absorption peak. In addition, the absorbance spectrum of
the CdSe cores alone is given. Their size of about 2.90 nm is
derived from a TEM image. Quite expectedly, the shell material
weakens quantum confinement, causing a redshift. We also note a
second exciton absorption peak. The PL of CdSe/CdS ‘flash’ QDs
having a 3 nm core size can be tuned within the range of 580 to 640
nm.
3.3 Dipolar emission
It appears that small emitters of electromagnetic radiation behave
predominantly as electric dipole emitters. As will be shown, to a
certain extent this is true for QDs as well, but this concept will
also return when treating nanoplasmonic scatterers.
3.3.1 The elementary dipole
In electrostatics, a classical dipole p = qd in a dielectric medium
of permittivity εdε0, is constituted of a negative and a positive
charge ±q, separated by distance vector d. The electrostatic
potential of this charge distribution is easily calculated using
Coulomb’s law. An elementary dipole, is the idealized version of a
classical dipole, taking the limit of d→ 0 while q → ∞, carried out
in such a way that p is preserved. By also taking this limit of the
classical dipole potential, one obtains the electrostatic potential
V of an elementary dipole [4]:
V = 1
3.3.2 The Hertzian dipole
Now consider the (dynamic) system of a Hertzian dipole, being a
harmonically oscillating elementary dipole p ejωt. Equivalently,
one can imagine a current I = I0 ejωt, periodically flowing between
two points in space which are separated by distance vector d. These
points thereby acquire harmonically oscillating charges ±q = ±I0
ejωt /ω. Starting from this configuration, one realizes a Hertzian
dipole by letting d→ 0 while I0 →∞, and such
3.3 Dipolar emission 22
0
45
90
135
180
225
270
315
φ [°] =
D [dBi]
Figure 3.3: Radiation pattern of a Hertzian dipole along z in free
space, with directivity D(θ, φ) expressed in dBi — taking the
isotropic emitter as a reference. The left panel shows the
directivity D(90 °, φ) in the orthogonal plane, while the right
panel depicts the directivity in any plane containing the dipole,
e.g. an xz cut D(θ, 0 °).
that p remains constant. For a Hertzian dipole along z, the nonzero
far field components of E and H are given by [43]:
lim kr→∞
Eθ = jωµ0Id
µ0/εdε0 the characteristic impedance of the medium.
The directivity D(θ, φ) of an emitter (or receiver) of EM power is
the ratio of the power radiated in solid angle sin(θ)dθdφ to some
reference value. For a Hertzian dipole along z and taking the
isotropic emitter as a reference, we obtain:
D = 3
2 sin2 θ. (3.13)
The directivity of a source is mostly depicted in a so-called
radiation pattern, which is a polar plot showing its base 10
logarithm for a given plane of intersection. As an example, the
radiation pattern of a z-oriented Hertzian dipole in the orthogonal
xy plane and in some plane containing the dipole z axis are given
in figure 3.3. Apparently, a vertical Hertzian
3.3 Dipolar emission 23
dipole emits maximally and omnidirectionally in the horizontal
(orthogonal) plane, while there is no emission along its (vertical)
axis. The directional behavior of an emitter can be expressed by
its half power beamwidth, being the angular range θ3 dB over which
the power emission is at least half of its maximal value. For a
Hertzian dipole, θ3 dB = 90 °. As such, one can hardly consider the
Hertzian dipole as a directive source.
3.3.3 The dipole approximation
One may wonder how to model a quantum dot when for instance
simulating a QD-based structure. This is where we call upon the
dipole approximation [44], which states that ‘small’ electronic
systems predominantly behave as Hertzian dipole emitters and
absorbers. We will justify this statement for a system of one
electron captured in a potential well U . Throughout this section,
typically belonging to the ‘physics domain’, we will employ the
physics formalism, with factor e−iωt for harmonically oscillating
fields. We first write down the Hamiltonian Htot for the combined
system of the bound electron state and the electromagnetic
field:
Htot = Hwell +Hrad +Hcoupl, (3.14)
where:
. (3.17)
Hwell represents the isolated electronic subsystem of an electron
in a static potential well U . p denotes the electron momentum and
m0 the electron mass. The eigenstates |ξ, as well as the
corresponding energy levels Eξ of this subsystem are assumed to be
well known. ξ is a (group of) quantum numbers describing the
individual electron states. The total Hamiltonian Htot is obtained
by minimal substitution (p→ p−eA) into Hwell, and addition of Hrad.
The latter is associated to the EM fields solely, in absence of
matter. Its expression is the result of quantifying the classical
electromagnetic field using the Coulomb gauge. It contains a sum of
energy contributions ~ω over all eigenmodes |kεi with wavevector k
and polarization (unit) vector εi, i = 1, 2. Additionally, a zero
point energy equal to 1
2 ~ω must
be included. a†i (k) and ai(k) respectively are the creation and
annihilation operators of mode |kεi. A denotes the vector potential
and fully describes the electromagnetic field.
Hwell and Hrad commute as they are acting in two completely
different spaces. They give rise to a common set of eigenstates of
the form |φ = |ξ ⊗ |kεj = |ξ; kεj. The combination of these two
non-interacting subsystems Hwell + Hrad will be regarded as the
unperturbed system, whereas the coupling term Hcoupl will be
treated in perturbation. The
3.3 Dipolar emission 24
(small) interaction between matter and radiation is thus completely
governed by Hcoupl, given by (3.17). Further investigation learns
that, in first order, its first term has to do with interactions
involving two photons, while the second term governs single photon
absorption and emission. We only wish to study the latter and
denote the second term as V for convenience. We consider the
transition of an excited state |ξ′ towards a relaxed state |ξ, with
the emission of a photon in the otherwise empty mode |kεj. The
corresponding interaction matrix element φi|V |φf that relates the
initial and final states |φi and |φf reads:
φf |V |φi = ξ; kεj| − eA · p m0
|ξ′; 0, (3.18)
where |0 represents the empty photon mode. Using the equality p =
m0
i~ [r, Hwell] and since we know the action of Hrad on the
electronic states, for instance Hrad|ξ = Eξ|ξ, we can further
rewrite (3.18):
φf |V |φi = ie
~ (Eξ′ − Eξ)ξ; kεj|A · r|ξ′; 0. (3.19)
We now introduce the expression for the vector potential A as
obtained through quan- tification of the classical electromagnetic
field:
A(r, t) = ∑ k′
′, t)eik ′·r + a†i (k
′, t)e−ik ′·r ) εi, (3.20)
where ai(k, t) = e−iωt ai(k) and a†i (k, t) = eiωt a†i (k). In
developing the theory, one has introduced a (non-physical) box with
volume v0 and periodic boundary conditions. As such, v0 is but a
theoretical construct and should not appear in the final result.
There is only one term in the expression for A that contributes to
the right hand side of (3.19), namely the one that contains the
creation operator a†i (k
′, t) with k′ = k and i = j. The interaction matrix element
becomes:
φf |V |φi = ie
√ ~
2v0ε0ω ξ| e−i(k·r−ωt) r · εj|ξ′kεj|a†j(k)|0. (3.21)
Using kεj|a†j(k)|0 = kεj|kεj = 1, we finally obtain:
φf |V |φi = ie√
2v0ε0~ω (Eξ′ − Eξ) eiωtξ| e−ik·r r|ξ′ · εj. (3.22)
In a next step, we want to evaluate the matrix element ξ| e−ik·r
r|ξ′. We will thereby only consider photons of a wavelength much
longer than the typical spatial variations of the electron
orbitals. The factor e−ik·r now varies slowly with respect to the
electron orbitals r|ξ, r|ξ′ and only its zeroth order Taylor
contribution needs to be retained: e−ik·r ≈ 1. In doing so, we
obtain the dipole approximation for the matrix element in
question:
φf |V |φi = ie√
2v0ε0~ω (Eξ′ − Eξ) eiωtξ|r|ξ′ · εj. (3.23)
3.3 Dipolar emission 25
where ξ|r|ξ′ will lead to ‘selection rules’, which express
restrictions on the orbital quantum numbers ξ and ξ′. For reasons
of symmetry, (3.23) governs both photon emission and absorption, in
absence of spectator photons and within the dipole
approximation.
Classically one expects that a ‘dipole’ interaction can be written
as Vdip = −e r · Eloc, being the interaction of the dipole e r
formed by the electron in the potential well with the (homogeneous)
local electric field Eloc. Within QED, this electric field can be
expanded into contributions belonging to the different modes:
Eloc(r, t) = i ∑ k′
′·r ) εi. (3.24)
Using this quantification for the electric field, we indeed
find:
φf | − e r · Eloc|φi = ie
√ ~ω
2v0ε0 eiωtξ|r|ξ′ · εj, (3.25)
which is the same expression as (3.23), provided that one only
considers transitions in which energy is conserved: Eξ′ − Eξ = ~ω.
As a result, small electronic systems that interact with EM fields
can be regarded as dipole absorbers and/or emitters. Thereby, the
interaction to be considered is of the dipole form −e r ·Eloc. Note
that this approach only holds in case the field varies slowly with
respect to the spatial variations of the electronic system, which
is always true for systems that are sufficiently small. In practice
one should compare the spatial extension of the orbitals to the
wavelength of EM radiation.
3.3.4 Quantum dots as dipoles
In general, the interaction of QDs with EM fields can be described
semiclassically as a multipolar expansion. For interactions with
far fields, the higher order transitions are often too weak and
cannot be observed. In particular, the (second order) magnetic
dipole and electric quadrupole contributions can be neglected and
only the (first order) electric dipole term should be retained. For
instance the 10.2 nm diameter CdSe/CdS flash QDs of figure 3.2(c),
emitting at about 625 nm, can at first instance be treated as
(isotropic) dipole emitters and absorbers.
Near fields on the other hand, show stronger spatial variations.
Higher order multipolar interactions now also have to be taken into
account. For example, it was demonstrated [45] that, when
approaching a QD with a 10 nm diameter laser illuminated gold tip,
the electric dipole and quadrupole QD absorption rates are
comparable in size.
3.4 Transition probabilities and rates 26
3.4 Transition probabilities and rates
Quantum mechanics dictates that the probability for a certain
transition to take place, is given by the absolute square of the
transition matrix element involved. In particular we will use the
transition rate Γ, being the transition probability per unit of
time. We will focus on the processes of absorption and emission,
thereby approximating the QD as an exciton two-level system.
Additionally, non-radiative decay is treated.
3.4.1 Photoexcitation
Consider the case of an electron, initially occupying the relaxed
state |ξ and absorbing a photon of mode |k, ε, with mode occupation
number n(k, ε). In doing so, the electron is promoted to the final
(excited) state |ξ′. Applying the dipole approximation, we obtain
the differential absorption rate through Fermi’s Golden Rule:
∂Γ
1
~ δ
( Eξ′
~ − Eξ
~ − ω
) , (3.26)
where the initial and final states of the global system are
respectively given by |φi = |ξ; kε⊗n(k,ε) and |φf = |ξ′;
kε⊗[n(k,ε)−1]. The Dirac function ensures energy conservation.
Introducing expression (3.24), and integrating over the photon
energy ~ω′, we get the absorption rate of the QD:
Γ(k, ε) = πe2
~εdε0 n(k, ε)
|fLF|2|ξ′|r · ε|ξ|2. (3.27)
The relative permittivity εd is included as to account for the
surrounding (dielectric) medium. %(ω) represents the spectral
density of the radiation field, being the energy density per volume
of radiation. An expression is provided by Plank’s law. The local
field factor fLF has to be incorporated in case of QD absorption:
it is a proportionality factor that relates the field inside the
(spherical) particle to the external driving field Eloc. For QDs in
a low-index environment, the local field is reduced as a result of
screening. An expression will be derived within the
dipole/electrostatic approximation, in the context of nanoplas-
monic spheres; see (7.13). As an example, for a layer of CdSe/CdS
QDs (εCdSe/CdS ≈ 10) surrounded by ligands (εlig ≈ 2.2), we have
fLF = 0.45. In the particular case of metal nanospheres however,
fLF turns resonant at certain frequencies, thence greatly enhancing
the local field.
3.4.2 Photon emission
Once the QD is excited, the exciton can recombine radiatively, i.e.
accompanied by photon emission. The excited state can be achieved
through various processes. One example is absorption, in which case
the emission is classified as photoluminescence. Another is
direct
3.4 Transition probabilities and rates 27
injection of electrons and holes, in which case we refer to the
emission as electrolumines- cence.
We consider the spontaneous emissive transition from an excited
electron-hole state |ξ′ to the ground state |ξ. The corresponding
transition rate is again found using Fermi’s Golden Rule. Very
similar to the case of absorption one obtains the spontaneous
emission rate:
Γ(k, ε) = 2π
~ e2~ω√εd
2v0ε0 ρ(ω)|fLF|2|ξ′|r · ε|ξ|2, (3.28)
in which ρ(ω) = v0ω 2/π2c3 is the free space photon density of
states. As such, v0 cancels
out and the decay rate is indeed independent of the cavity
volume.
From (3.28) we see that the exact form of the QD orbitals r|ξ and
r|ξ′ will only influ- ence the polarization ε and not the direction
uk of the emitted photons. As a consequence, QDs will act as
isotropic emitters of dipole radiation. We additionally assume that
QDs are spherically symmetric such that their orbitals do not have
any preferred orientation. The photon polarization ε will in this
case be at random in the plane orthogonal to uk. When averaging
over all possible relative orientations of r, ε and k, we
obtain:
Γ = 2π
. (3.29)
In our simulations however, we will make use of Hertzian dipole
antennas with a prede- fined dipole orientation. We introduce a
Cartesian x, y, z coordinate system with z along uk, the direction
of photon emission. As we have seen, dipole emitters do not emit
along their axis and indeed, with ε ⊥ uk//uz, the corresponding
matrix element ξ′|zuz · ε|ξ is zero. Considering r2 = x2 + y2 + z2,
the emissive decay rate of (3.29) can now be de- composed into
three parts: Γ = Γx + Γy + Γz, associated with x, y and z-oriented
dipole emission respectively. For example:
Γx = 2π
. (3.30)
We can now replace the QD by three dipolar antennas, along the
three principal axes of our coordinate system. They contribute
equally, each for one third, with radiative power Pi = ~ωΓi, i = x,
y, z. The total power emitted by these antennae should equal the
actual QD power emission: P = Px + Py + Pz = ~ωΓ.
Starting from (3.29) and assuming that the QD orbitals |ξ, |ξ′ can
be described using products of Bloch functions and some envelope,
one predicts a QD spontaneous emission decay rate proportional to
the field pulsation ω. Experimentally, a slightly deviant
supralinear decay rate is measured however, rather than the linear
decay rate one would expect for the ideal two-level exciton system
[46].
3.4 Transition probabilities and rates 28
3.4.3 The Purcell effect
Purcell noted that, in case the emitter is placed in a resonant
cavity, the density of final states is no longer given by the free
space photon density of states ρ(ω)