Quantum Dot Capacitors as Versatile Light Sources for Integrated Photonics

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 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—colloidal quantum dots, electroluminescence, integrated photonics, 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 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.
We aim at designing a quantum dot (QD) based electrically- driven integrated light source, compatible with the CMOS fabrication technology. We thereby consider a quantum dot capacitor device architecture, with a layer of QDs sandwiched vertically 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 section III, we present the quantum dot capacitor structures we fabricated, as well as their electric and electroluminescent characterization, where we rely on PSPICE simulations for interpreting our measurements. In a next step, the devices have to be integrated 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 crystal structure and lattice constant of QDs in general closely resemble 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 structures covers the entire visible and near infrared region.
(a) (c)(b)
Fig. 1. (a) A TEM image of a colloidal quantum dot, (b) Schematic visualization 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 efficient colloidal synthesis processes [3], which reduces their cost. We make use of CdSe/CdS and PbS/CdS core/shell QDs emitting 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 increases 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 coating 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 capacitor 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 fabrication 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 covered 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 characterization of the QD capacitors, we apply a 1 kHz sawtooth driving voltage over a series circuit of the QD capacitor and a 21 k resistor. The measured current response through the circuit is represented 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 voltages, 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 samples; 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 possible, 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 current 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 voltage, the voltage drop over the QD amounts to about 20 V, according to our PSPICE simulations. This corresponds to a voltage 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 important 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 emission, 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 material results in an improved stability. However, the electroluminescent 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 waveguide, where the waveguide material simultaneously acts as insulating material for the QD capacitor structure. Top and bottom electric contacts are provided.
Si3N4 material, especially when operating at higher device voltages, 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, integration 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 order 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 waveguides, which can be used in the near infrared region. This material 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 temperature 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 nanoplasmonic 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 antenna geometry is shown in figure 8(a). We employ the particle swarm optimization algorithm with 10 particles and 20 generations, 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 nanoplasmonic dipole resonance — is shown in figure 8(b). The parameters 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 element 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 factor almost twice as large is obtained when placing the QD in the center of the optimized waveguide (400 × 200 nm), without 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 absorption 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 improved 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 increasing 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 determined the optimal structure for extracting the generated light. We thereby investigated 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.
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 engineering 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 depending 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 displacement 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 materials 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 example 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 orientation. 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 resonances. 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 deposited 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 carriers: 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 accompanied 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 calculated 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 electrons 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 quality 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 visualization 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 individual 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
where electron and hole are held together by a Coulombic attraction, against a background medium of permittivity εrε0 — the semiconductor permittivity. Employing the same techniques 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 comparable 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
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 confinement (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 confinement (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 simply 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
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
2.90/ — nm
Figure 3.2: (a) Absorbance (solid line) and PL (shaded area) for some PbS core QD colloids 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
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
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
(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 quantification 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)
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 nanoplasmonic 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
injection of electrons and holes, in which case we refer to the emission as electroluminescence.
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].
3.4.3 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)

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