QUANTUM DOTS
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft;
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties
in het openbaar te verdedigen op vrijdag 2 mei 2014 om 15.00 uur
door
Sybren TEN CATE
doctorandus geboren te Amsterdam
Copromotor: dr. J.M. Schins Samenstelling promotiecommissie:
Rector Magnificus voorzitter
prof. dr. L.D.A. Siebbeles Technische Universiteit Delft, promotor dr. J.M. Schins Technische Universiteit Delft, copromotor prof. dr. P. Dorenbos Technische Universiteit Delft
prof. dr. A. Schmidt-Ott Technische Universiteit Delft prof. dr. T. Gregorkiewicz Universiteit van Amsterdam prof. dr. ir. Z. Hens Universiteit Gent (Belgi¨e) prof. dr. A.M. Brouwer Universiteit van Amsterdam
This work is part of the Joint Solar Programme (JSP) of Hyet Solar and the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).
April 2014
1 Introduction 1
1.1 Quantum Dots . . . 2
1.2 Carrier Multiplication in Quantum Dots and other Nanocrys-tals . . . 3
1.3 Quantum Dots for Photovoltaics . . . 4
1.3.1 QD–Acceptor Heterojunctions . . . 4
1.3.2 Quantum-Dot Solids . . . 5
2 Experimental Methods 11 2.1 Sample Preparation . . . 11
2.1.1 Bulk Heterojunctions . . . 11
2.1.2 Non-Infilled PbSe QD Solids . . . 13
2.1.3 ALD-Infilled PbSe QD Solids . . . 14
2.2 Experimental Techniques . . . 15
2.2.1 Transient-Absorption Spectroscopy . . . 15
2.2.2 Time-Domain Terahertz Spectroscopy . . . 15
2.2.3 Time-Resolved Microwave Conductivity . . . 17
2.2.4 Photoluminescence . . . 20
2.2.5 Steady-State Optical Absorption . . . 21
3 Theoretical Analysis 23 3.1 Fraction of 800 nm Pump Photons Absorbed by PbS QDs in P3HT:PCBM Blend . . . 23
3.2 Absorption and Stimulated Emission of a Phonon by a Hot Charge . . . 25
3.3 Electron–hole pair dissociation energy . . . 27
3.4 Escape yield from double e–h pairs . . . 28
3.5 Charge Diffusion Length in Presence of Randomly Distrib-uted Deep Traps . . . 31
I Extracting Charges from Carrier Multiplication in Quantum Dots 33 4 Origin of Low Sensitizing Efficiency of Quantum Dots in Organic Solar Cells 35 4.1 Results and Discussion . . . 36
rier Multiplication in Quantum-Dot Films 47 5.1 Results . . . 48 5.2 Discussion . . . 55
II Carrier Multiplication and Diffusion in Infilled
PbSe Quantum Dot Solids 61
6 Activating Carrier Multiplication in PbSe Quantum Dot Solids by Infilling with Atomic Layer Deposition 63 6.1 Results and Discussion . . . 64 6.2 Conclusion . . . 71 7 Phonons Do Not Assist Carrier Multiplication in PbSe
Quantum Dot Solids 73
7.1 Results and Discussion . . . 74 7.2 Summary . . . 81 8 Micron Charge Diffusion Lengths in Quantum Dot Solids 83 8.1 Results . . . 85 8.2 Conclusion . . . 98 References 99 Summary 109 Samenvatting 113 Acknowledgements 117 Publication List 119 Curriculum Vitae 121
Introduction
(To appear in Acc. Chem. Res.)
I
n semiconductors, a photon of an energy equal to the band-gap energy can excite an electron from the top of the valence band to the bottom of the conduction band, yielding one electron–hole pair. If the photon energy exceeds the band-gap energy, absorption yields an electron of higher energy (hot electron) and/or a hole of higher energy (hot hole) than in case of band-gap photoexcitation. The hot electron and hot hole together possess an amount of excess energy equal to the difference between the photon energy and band-gap energy, and this excess energy can be converted in different ways.One way is to dissipate the excess energy in the form of heat. The hot electron does this (electron cooling) via interaction with nuclear-lattice vibrations (phonons), whereas the hot hole, on the other hand, can be occupied by valence electrons when these lower their energy through phonon emission (hole cooling).
Another possibility is that the hot electron and/or hot hole (par-tially) utilize their excess energy in exciting one or more additional electron–hole pair(s). This phenomenon—carrier multiplication (CM, also called ‘multiple exciton generation’ or ‘impact ionization’)— allows a single photon of sufficient energy to generate two or more electron–hole pairs. Besides attracting attention from a physics point of view, CM simultaneously holds the prospect of increased solar-cell efficiencies.
Although CM does occur in traditional solar-cell materials as bulk silicon, it has no practical benefits due to its insignificant oc-currence under solar illumination.1,2 The prediction in 20023,4 and experimental observation in 20045 that CM may be more efficient in nanometer-sized semiconductor crystals (nanocrystals) than in bulk caused a great interest in CM for application in solar cells. Currently, CM is a highly active field of research,6–8 with studies focusing not only on nanocrystals, but also on organic semiconductor molecules
Figure 1.1 | Demonstration of the relation between band-gap en-ergy and QD diameter (quantum size effect). Each vial contains QDs of a different diameter, and consequently, because of the quantum size ef-fect, luminesces a different colour of light under photoexcitation. Smaller QDs (larger band-gap energy) luminesce blue light, while larger QDs (smal-ler band-gap energy) luminesce red light. (Photograph credit: University of Utah)
such as pentacene9 and tetracene10 (where it is called ‘singlet
fis-sion’),11 and on carbon sheets (graphene)12 and tubes (single-walled
carbon nanotubes)12,13.
1.1 Quantum Dots
When reducing the size of a semiconductor in at least one dimension below a critical length scale, electron–hole pairs become spatially confined (quantum confinement). A well-known type of nanomaterial is the quantum dot (QD), which can be formed through solution-based synthesis. QDs are nanometer-sized ’spherical’ semiconductor crystals which confine electron–hole pairs in all three dimensions.
Quantum confinement affects some of the physical properties of semiconductors, most notably the band-gap energy. Smaller QDs have a larger band-gap energy as a consequence of stronger quantum confinement. This effect is clearly observed when illuminating differ-ently sized cadmium-selenide (CdSe) QDs with ultraviolet (UV) light (Figure 1.1). The UV phonons are of sufficient energy to excite (hot) electron–hole pairs, which can eventually recombine under the emis-sion of a photon (luminescence) of the band-gap energy. For CdSe QDs, these band-gap energies are such that the QDs luminesce light of visible colours: a unique colour for each QD size.
1.2 Carrier Multiplication in Quantum Dots and other Nanocrystals
That the size of a semiconductor can affect the CM efficiency was already apparent from the initial reports of increased CM efficiencies in nanocrystals versus bulk. Subsequent research has revealed that the CM efficiency also depends on other nanocrystal properties.14 Nanocrystal parent material (e.g. silicon,15–18 indium phosphide,19 and cadmium chalcogenides20)21 and architecture (e.g. core-shell or
alloyed,22 or more complex architectures23) were found to affect the efficiency and threshold energy of CM due to variations in the effect-ive masses of electrons and holes and in the strength of the electron-phonon interaction.24,25 Nanocrystal shape also affects the CM effi-ciency, the latter being reported to be higher in flattened nanocrystals (nanoplatelets) and elongated nanocrystals (nanorods)14,26–28than in
spherical nanocrystals (QDs).29
Of all nanocrystal shapes and compositions, lead chalcogenide (PbX) QDs are the most studied.6,7,22,30,31 An important reason for this is the range over which their band-gap energy can be tuned. Although the optimal band-gap energy of a semiconductor for har-vesting the solar spectrum is 1.34 eV (Shockley-Queisser limit), this optimum lowers when accounting for the occurrence of CM, being 0.7 eV in the limit of ideal CM.32–34Since PbX QDs display CM and
have a band-gap energy tunable over this energy range,35 they are an ideal candidate material for CM-enhanced solar cells.
Studies on the fundamental properties of isolated QDs showed that they posses qualities which are promising for application in solar cells. However, for such application, the isolation under which these qualities were discovered can impossibly be maintained. Aiming to harvest multiple charges from CM in QDs, the isolation of QDs must be broken and they must be connected (via intermediate molecules) to electrodes, while simultaneously maintaining the quantum con-finement and other beneficial properties of their isolated state to a maximum extent.
The challenging nature of this task is illustrated as follows. Isol-ated QDs inherit thick insulating shells of protective molecules from their synthesis. While this protects the QDs, it also impedes charges transfer out of the QDs. For application of QDs in photovoltaic devices, it is thus considered beneficial to replace the long ligands by shorter surface-passivating molecules,36,37even single atoms38.
How-ever, doing so reduces the oxidative and photo-thermal stability of the QDs.39 In addition, changing the surface passivation may40 or may
not36 alter the properties determined in solution experiments, such as the CM efficiency. The remainder of this introduction outlines the recent progress in harvesting charges from CM in QDs.
1.3 Quantum Dots for Photovoltaics
Their relatively cheap production, tuneable band gap, and occur-rence of CM make colloidal QDs attractive building blocks for the absorber layers of low-cost, high-efficiency (i.e. ‘third-generation’) solar cells.6,32,41–43 In such application, the critical challenge is har-vesting multiple charges from CM in QDs and transporting them to their respective electrodes, i.e. electrons and holes must be able to both 1) move out of close proximity before recombination and 2) move freely toward electrodes.
1.3.1 QD–Acceptor Heterojunctions
One approach of harvesting charges is to contact QDs with acceptor molecules (a ‘binary heterojunction’ geometry). The acceptor mo-lecules accept electrons (or holes) from the QDs. This process relies on a staggered (type II) energy-level alignment across the interface. If an electron–hole pair reaches the interface between these two materi-als, the electron and hole are spatially separated and charge transport occurs via their respective host material.
Polymers and fullerenes are good candidates for such acceptor molecules. They have been extensively researched in the context of organic solar cells, are cheap and solution processable (like QDs), can form type-II energy-level alignments with QDs, and exhibit good charge transport.44 However, solar cells45 and photodiodes46 us-ing such binary heterojunctions suffer insignificant charge extraction from QDs.
Interestingly, these same photodiodes have shown efficient carrier separation and transport from QDs in a ternary heterojunction in which QDs were contacted with both electron- and hole-accepting material.46 This ternary heterojunction comprised a blend of PbS QDs, the polymer regio-regular poly(3-hexylthiophene) (P3HT), and the fullerene-derivative [6,6]-phenyl-C61-butyric acid methyl ester (PCBM), cf. Figure 1.2.
This finding induced systematic studies to the reason(s) underly-ing the dramatic difference in carrier-extraction efficiencies between
h+ e− PCBM P3HT
PbS
QD
O O S S S SFigure 1.2 | Illustration of the energetically favourable harvesting of the electron and hole from a PbS QD by means of the polymer P3HT and the fullerene-derivative PCBM.
binary and ternary heterojunctions. In the absence of the complic-ating effects of an electric field and electrodes, these studies found charge separation from the QDs to occur in the ternary blend,47–49 however, the resulting electrons and holes are subsequently Cou-lombically bound across the QD–acceptor interface (chapter 4).47 It was found that, on charge separation, a counter charge always remains localized in the QD. The strong Coulomb attraction that this induces, binds the electron and hole across the interface.47 This concept simultaneously explains the outcomes of the zero-field47–49
and the photodiode46 studies, because in the former case no elec-tric field is present to overcome the Coulomb attraction, while in the latter case a strong field is applied during operation. In general, the quantum confinement that causes the appealing features of nan-oparticles, can also localize charges to an extent where it enhances unwanted Coulombic forces.
1.3.2 Quantum-Dot Solids
To date, the most promising approach of harvesting charges from QDs is by forming a network of electronically coupled QDs (QD solid).37,45 In this process, the long molecules that passivate the QD
surface during synthesis are replaced by short linker molecules to im-prove carrier mobility. To avoid rapid degradation of QD solids in air (because the short linker molecules provide almost no protection against oxidation),39 a protective layer of metal oxides can be de-posited around the QDs by infilling the interstices of QD solids via atomic layer deposition.50 QD solids can simultaneously account for
light absorption, carrier multiplication,36,51 charge separation,37,52 and subsequent charge transport. The simultaneous occurrence of these processes was definitively proven in a QD solar cell (a solar cell utilizing a QD solid as the active material) where the number of collected electrons significantly exceeded the number of incident photons.43,53
Carrier Multiplication Efficiency
On forming a QD solid, the CM efficiency of isolated QDs may not ne-cessarily be maintained. Some report a conservation of CM efficiency on forming a QD solid from isolated QDs,36 while others report a
variation with the type of linker molecule and chemical treatment.40 In recent studies,40,54CM even appeared to be practically completely absent (already on picosecond timescales) in PbSe QDs coupled using 1,2-ethanedithiol (EDT) linker molecules. Interestingly, we observed that CM does occur in these QD solids when interstitially infilled with metal oxides (Fig. 1.3), and with a CM efficiency close to that of isol-ated PbSe QDs (chapter 6).51 Future studies must show whether the initial CM efficiency in QD solids does indeed depend on the choice of linkers and interstitial material, or that, instead, ultrafast recom-bination reduces an initially equal CM efficiency to a different extent depending on the particular QD solid.
In seeking higher CM efficiencies in QD solids, it is of benefit to consider the limitations to the CM efficiency of isolated QDs. The latter is determined through the competition of CM with hot-carrier cooling by phonon emission,14,55 i.e. hot-carrier cooling limits the time window in which CM can occur. Thus, the CM efficiency is affected by a change in the rate of the CM process and/or by a change in the rate of hot-carrier cooling.
Interestingly, both of these processes are in principle temperat-ure dependent (Fig. 1.4). At higher temperattemperat-ures, more phonons are available to compensate for energy mismatches in electronic trans-itions. Consequently, a hot single-exciton state has more energetic-ally allowed decay channels to different multiexciton states, and thus the CM rate is increased. As temperature increases, also the amp-litude of nuclear lattice vibrations increases, leading to a stronger electron-phonon coupling. Thus, the rate of hot-carrier cooling is also affected by temperature. In which way, however, is not known a priori, since phonon absorption and stimulated emission occur with different rates.
Carrier Multiplication
+
Infilled by ALD+
+
No Carrier Multiplication Non-InfilledFigure 1.3 | CM in (non-)infilled PbSe-EDT QD Solids. CM does occur in PbSe-EDT QD solids infilled with metal oxides via atomic layer deposition, whereas it was reported to be negligible or absent on (sub-)picosecond timescales in non-infilled PbSe-EDT QD solids. Future studies are needed to determine to what extent the initial CM yield and the rate of ultrafast exciton decay depend on the QD surface passivation and/or the interstitial material of the QD solid.
However, experiments have shown no significant effect of temper-ature on the CM efficiency.56 From this it was deduced that phonons do not assist CM in PbSe QD solids (chapter 7).56The main limiting
factor to the CM efficiency of the PbSe QDs was determined to be the (high) rate of spontaneous hot-carrier phonon emission which limits the time window in which CM can occur.56This rate does not
signi-ficantly depend on temperature, but varies with the properties of the QD parent material. This underlines the importance of the choice in QD material in reducing the cooling rate and thereby increasing the CM efficiency.14,25
Charge-Separation Efficiency
In photovoltaic devices, the generation of photocurrent requires that photogenerated electron–hole pairs are able to dissociate into mobile electrons and holes (charge separation). When multiple electron–hole pairs are generated in close proximity, charge separation competes with Auger recombination. Thus, the relevant quantity for devices is not the CM efficiency, but the multiple free charge generation ef-ficiency (MFCG). The latter is the number of free mobile electrons (and an equal number of free mobile holes) that result from the
ab-phonons
do not
assist CM
hot-carrier cooling CMEn
e
rg
y
Figure 1.4 | The cooling–CM competition versus temperature. Hot-carrier cooling and CM competitively determine the CM efficiency. In PbSe QD solids, both CM and cooling are shown to be independent of the presence of phonons.
sorption of a single photon.
The occurrence of MFCG can be determined from photoconduct-ivity measurements: an enhanced photoconductphotoconduct-ivity at higher photon energies implies that CM occurred, followed by charge separation of multiple electron–hole pairs before their Auger recombination. Some instances of MFCG in QD solids were observed for specific treat-ments, e.g. PbSe-EDT QD solids treated with hydrazine,43 PbSe-EDT QD solids interstitially infilled with metal oxides,51 and 1,2-ethanediamine (EDA)-linked PbSe QD solids.36 However, the key
property that these treatments had in common remained unclear. We performed a systematic study in which the charge mobility was varied across PbSe QD solids of identical CM efficiencies.52 For each QD solid, the spectral dependence of the photoconductivity was measured, which allowed a determination of the MFCG efficiency. The latter was found to be tunable between zero and a value close to the CM efficiency in isolated solution-dispersed QDs by varying the sum of electron and hole mobility. For summed carrier mobilit-ies in excess of about 1 cm2 V−1 s−1, the near agreement between the CM efficiency in isolated QDs and the MFCG efficiency of the QD solid implies that the charge-separation efficiency saturates to unity (chapter 5, cf. figure 1.5).52The relation between mobility and separation efficiency is quantitatively understood by comparing the mobility-dependent charge hopping rate to the Auger recombination
Carrier
Multiplication
Multi-exciton
Dissociation
Figure 1.5 | Sketch of the photogeneration of multiple free charges in a QD solid. A photon of sufficient energy excites two or more electron-hole pairs via CM. When these charges have a combined mobility in excess of 1 cm2 V−1 s−1, the hopping rate much exceeds the recombination rate,
and charge separation occurs with unity efficiency.
rate.52
Charge Transport
Following CM and subsequent charge separation in QD solids for ap-plication in photovoltaic devices, the free mobile electrons and holes must be able to reach the electrodes within their lifetime to contribute to the current. Possible obstacles to their transport are charge decay via recombination or charge immobilization on a deep trap state. The largest distance that either the electron or hole needs to traverse to reach their electrode equals the thickness of the absorber layer.
One common architecture of QD photovoltaic devices uses a re-flective back contact (double pass absorption). Ideally, the absorber layer in such devices (using infrared-absorbing QDs, e.g. PbX QDs) needs to be 1 µm thick to nearly eliminate absorption losses due to incomplete light absorption.57∗In reality, however, the absorber layer must be kept thinner than this to allow photogenerated charges to reach the electrodes. The problem of simultaneously minimizing ab-sorption losses while maximizing collection of photogenerated charges leads to an absorptioncollection compromise.
This compromise can be broken in several ways. One is through ∗QD photovoltaics can thus employ significantly thinner absorber layers than
photovoltaics based on indirect band-gap materials such as silicon (the latter requires about 100 µm).57
novel device geometries: e.g. by means of a folded light path58(which increases absorption in a thin film via multiple reflections) or a de-pleted bulk heterojunction59 (where charges are collected at short intervals throughout a thick material). These solutions, however, are not ideal because they rely on an applied electric field to collect charges, which generally yields solar cells of reduced efficiencies.60
The ideal way to break the absorptioncollection compromise is by enlarging the charge diffusion length. We have recently realized this in PbSe-EDT QD solids infilled with Al2O3 or Al2O3/ZnO: large diffusion lengths of electron and hole were obtained of about 0.9 µm (chapter 8).61 This is about an order of magnitude larger than in
non-infilled PbSe QD solids, where 0.08 µm diffusion lengths were reported.62 Thus, infilled PbSe-EDT QD solids appear very suitable for future application in photovoltaics exhibiting fully diffusive charge collection (i.e. collection in the absence of an applied electric field) without absorption losses.
Experimental Methods
2.1 Sample Preparation
The preparation methods are described which were used for realizing the samples studied in this thesis. The bulk heterojunction films were prepared by me, the non-infilled QD solids were prepared by dr. C. S. Suchand Sandeep at the Delft University of Technology (the Netherlands), the ALD-infilled QD solids were prepared by dr. Yao Liu at the University of California, Irvine (United States).
2.1.1 Bulk Heterojunctions
Small PbS-oleate QDs were synthesized63,64∗ (cf. Fig. 2.1) with a mean diameter of 2.5 ± 0.1 nm (s.d.) as inferred from the central wavelength of the first absorption peak, using an empirical sizing re-lation65(the accuracy of which has been confirmed for small QDs66). The PbS:P3HT:PCBM bulk heterojunction films were prepared via drop casting as follows. Blend solutions were prepared by weigh-ing off PbS QDs, P3HT, and PCBM under ambient conditions (small PbS QDs are air stable66), and adding the solvent mixture (9:1 chlo-roform:chlorobenzene by volume) under inert gas to result in a QD concentration of 5 mg mL−1. The solutions were magnetically stirred overnight at room temperature.
One series of films was drop cast for measurements of optical ab-sorption spectra and photoluminescence under ambient conditions. A second film series was prepared for transient absorption and THz spectroscopy under N2 atmosphere to ensure sample stability. These solutions were desiccated in a glovebox to remove oxygen, and redis-persed in the solvent mixture and again stirred overnight. Films were ∗Chemicals used: lead(II) oxide (99.999%, metals basis), oleic acid (90%,
tech-nical grade), 1-octadecene (90%, techtech-nical grade), and toluene (99.8%, anhydrous) were purchased from Aldrich. Bis(trimethylsilyl)sulphide (98%) was purchased from Alfa Aesar. Acetone (99.9%, CHROMASOLV Plus for HPLC), chloroben-zene (99.8%, anhydrous), n-hexane (95%, anhydrous), and tetrachloroethylene (99%, anhydrous) were purchased from Sigma-Aldrich.
co lla p sa b le su p p o rt th e rmo me te r & re g u la to r magnetic stirrer vacuum line N2 @ 77 K N2 @ 2 9 5 K va cu u m p u mp Schlenk line heater N2 line valve valve co ld t ra p p re ssu re se n so r d isp la y syring e fixe d su p p o rt fume hood magnetic stir bar septa ice water Si lico n e o il b u b b le r
Figure 2.1 | Schematic of the synthesis equipment. The pictured equipment was used in Delft for QD synthesis via the hot-injection method. The QDs are grown in the three-neck flask, all other pictured equipment serves to realize and control this growth.
drop cast and continually kept under N2.
Quartz substrates and magnetic stirrers were ultrasonically cleaned in two ten-minute cycles: one with ethanol, followed by one with hexane. Drop casting was performed in a glovebox, and on a level surface. Per substrate, 100 µL of solution was deposited, and allowed to slowly dry. After the films have dried under atmospheric pressure, they are left under vacuum for one hour to ensure that all solvents are removed.
QDs in PCBM-containing BHJs required an oxygen-low atmo-sphere during laser excitation in order to avoid a blue shift of the QD band-gap absorption peak. We ascribe the blue shift to a oxidation of the outer layers of the QD by singlet oxygen, which is likely produced
by PCBM upon photo-excitation, since C60 is known to be a potent generator of singlet oxygen.67
2.1.2 Non-Infilled PbSe QD Solids
Lead oxide weighing 0.66 g was dissolved in a mixture of 30 ml oct-adecene and 2.2 ml oleic acid. The solution was degassed at 100 ◦C
under vacuum for 1 h. The solution was then heated to 180◦C in a
Schlenk line under nitrogen atmosphere. The synthesis was carried out by injecting 10.8 ml of 1M selenium (1M Se in trioctylphosphine mixed with 84 ml of diphenylphosphine) swiftly to the lead precursor. The reaction mixture was kept at 160◦C for 2 min after which it was
quickly cooled by a water bath. The QDs were precipitated with butanol and methanol and centrifuged at 5000 r.p.m. for 5 min, re-dissolved in hexane and washed with butanol/methanol mixture once again. The QDs were then dispersed in tetrachloroethylene (TCE) for absorption measurements and hexane for film preparation.
QD solids were formed via layer-by-layer dip-coating. To ensure good adhesion of the QDs to the quartz substrate, the substrate was first silanized with (3-aminopropyl)triethoxysilane (APTES). The PbSe QD films were prepared by layer-by-layer dip coating using a mechanical dip coater (DC Multi-8, Nima Technology) mounted in-side a nitrogen glove box. The silanized quartz substrate was dipped alternately into a 10 mM solution of PbSe QDs in hexane for 60 s, a 1M solution of the replacing ligands in methanol for 60 s and a rinsing solution of methanol for 30 s. The dipping procedure was repeated 15 times, resulting in homogenous thin films.
Some QD solids, as prepared via layer-by-layer dip-coating, were studied by cross-sectional TEM. For the cross-sectional TEM meas-urements, 10 layers of PbSe QDs with 2DA ligands were dip-coated on a quartz substrate, on top of which another 10 layers of CdSe QDs were dip-coated. The CdSe QD layer serves as protective layer and helps to reduce the damage to the PbSe QD layer in the polishing of specimens for the cross-sectional TEM measurements. A glass sub-strate was then glued onto the film and a cross-section with a size of ∼ 1 mm was cut from the film using a diamond saw. This was then mechanically polished to a thickness of 10 mm and loaded on to a copper TEM grid. The sample was then further thinned down to electron transparency using a Gatan PIPS 691 ion mill, using Argon. A Tecnai F20ST/STEM (200 kV) was used for imaging of the sample.
2.1.3 ALD-Infilled PbSe QD Solids
PbSe QDs were synthesized†and purified using standard air-free
tech-niques. In a typical synthesis, a solution of 1.09 g PbO (4.9 mmol), 3.45 g oleic acid (12.2 mmol), and 13.5 g 1-octadecene was degassed in a three-neck flask and heated at 180◦C for 1 h to dissolve the PbO and
dry the solution. Fifteen milliliters of a 1 M solution of trioctylphos-phine selenide containing 0.14 g diphenylphostrioctylphos-phine (0.75 mmol) was then rapidly injected into this hot solution. The QDs were grown for 1 min, and the reaction was then quenched with a water bath and 20 mL of anhydrous hexane. The QDs were purified by three rounds of dispersion/precipitation in hexane/ethanol and stored in a glovebox as a powder. The particle size and size distribution were determined from transmission electron microscopy images of at least 100 QDs.
PbSe QD solids were prepared via layer-by-layer dip coating using a mechanical dip coater mounted inside of a glovebox (DC Multi-4, Nima Technology). Quartz substrates were cleaned by subsequently sonicating in acetone, rinsing with acetone and isopropanol and dry-ing under an N2 flow. Clean substrates were alternately dipped into a 2 mg mL−1 dispersion of QDs in dry hexane and then a 1 mM
solu-tion of EDT in dry acetonitrile. Resulting QD solids have a 1Sh1Se absorption peak at 1930 nm (0.64 eV; see Figure 6.1), and a thickness of ∼200–300 nm.
The interstitial space of a PbSe-EDT QD solid was infilled with amorphous Al2O3 (for some samples followed by crystalline ZnO) us-ing low-temperature ALD. This improves carrier mobility and envir-onmental stability of the QD solids. Al2O3 was deposited by ALD of trimethylaluminum and water, and ZnO was deposited by ALD of di-ethylzinc and water. A substrate temperature of 75◦C, pulse (purge)
duration of 20 ms (90–120 s), and operating pressure of ∼ 2×10−4 bar
were used. Synthesis of a 25 nm ALD film requires ∼ 9 h with these parameters. This procedure is described in more detail in 50.
†Chemicals used: lead oxide (PbO, 99.999%), selenium (99.99%), oleic acid
(tech. grade, 90%), diphenylphosphine (98%), trioctylphosphine (tech. grade, > 90%), 1-octadecene (90%), 1,2-ethanedithiol (> 98%), trimethylaluminum (97%), diethyl zinc and anhydrous solvents were purchased from Aldrich and used as received.
2.2 Experimental Techniques 2.2.1 Transient-Absorption Spectroscopy
Samples were photoexcited and probed using pulses from a Ti:Sapphire chirped-pulse amplified laser system (Libra-USP-HE, Co-herent Inc.) running at 1.5 kHz, delivering sub-50-fs, 3.5-mJ pulses with a central wavelength of 800 nm. The probe beam was imaged onto a silicon PIN photodiode (Vishay BPW34). The pump was spatially separated from the probe downstream of the sample, and orthogonal polarization of the beams allowed further separation by means of a polarizer (Fig. 2.2)
The normalized change in optical transmission due to photoexcit-ation of the sample is
σ ≡ −TTon− Toff offI0FA = − 1 I0FA " exp −LX i Ni∆σi ! − 1 # , (2.1) where Ton is the transmission through the sample in the excited state, Toff the transmission through the sample in the ground state, I0 is the incident fluence, FA is the fraction of incident light absorbed by the sample, L is the film thickness, Ni is the volume number density
of excited species i, ∆σi is the change in the absorption cross section
associated with excited species i. For small changes in absorption cross section, this is well described by the first-order approximation
σ = L I0FA X i Ni∆σi = X i φi∆σi, (2.2)
with the quantum yield of excited species i φi≡
NiL
I0FA
. (2.3)
Hence, ∆σi > 0 for photoinduced absorption of probe light and ∆σi <
0 for enhanced transmission of probe light due to stimulated emission and/or ground-state bleach.
2.2.2 Time-Domain Terahertz Spectroscopy
The ultrafast photoconductivity of samples was measured by time-domain terahertz photoconductivity. In this technique, the probe pulse is an electric field waveform of picosecond duration in the tera-hertz frequency range. Interaction with mobile charge carriers in the
ch o p p e r 0–1 ns delay 0–20 ns delay PC re g u la to r fs laser trigger N2 N2 polarization rotator sample p u mp p ro b e re fe re n ce a b so lu te ND filters p o la ri zo r p o la ri zo r
Figure 2.2 | Schematic of the Transient Absorption Spectro-scopy experiment. chopper 0–1 ns delay 0–20 ns delay 0–1 ns delay ZnTe PC re g u la to r ZnTe L/4 N2 @ 295 K external compressor fs laser tri g g e r THz THz ro ta me te r ND filters LWP LWP Wollaston prism p u mp g e n e -ra tio n d e te ct io n N2 valve N2 N2 sample
Figure 2.3 | Schematic of the Time-Domain Terahertz Spec-troscopy experiment. The chop-per modulates the pump beam, thus allowing measuring the response of the excited sample. By modulating the generation beam and blocking the pump beam, the probe waveform can be measured.
photoexcited sample leads to absorption of the probe to an extent that is determined by the conductivity. This experimental technique is detailed in more depth elsewhere.68,69 Samples were photoexcited with a < 50-fs laser pulse from the laser described above, and probed by a picosecond terahertz pulse. Terahertz pulses were generated by optical rectification in a ZnTe crystal. The excitation beam (dia-meter 5 mm) was aligned on the sample with the focused terahertz beam (diameter < 2 mm). The terahertz field transmitted through the samples was detected by means of the Pockels electro–optic effect in a 0.5-mm ZnTe crystal (Fig. 2.3).
The product of the quantum yield of charge-carrier photogenera-tion, φi, and the mobility, µi, summed over all charge carrier species
i, is related to the change in the transmission of electric field at the maximum of the terahertz waveform as68,70
µ =X i φiµi = − 2ncε0 e Eon− Eoff EoffI0FA , (2.4)
where n is the effective refractive index of the BHJ, c the speed of light in vacuum, ε0 the vacuum dielectric constant, e the elementary
charge, Eon is the amplitude of the THz field at the maximum of the THz wave form, as transmitted through the photoexcited sample, and Eoffis the amplitude of the THz field at the maximum of the THz wave form, as transmitted through the sample in the ground state. The effective terahertz refractive index is calculated by Maxwell–Garnett theory,71 where the volume-weighted average over the different con-stituent components is used for the medium refractive index, and the bulk optical refractive index is used for the QDs.
2.2.3 Time-Resolved Microwave Conductivity In the time-resolved microwave conductivity technique (TRMC),72,73
samples were mounted in an N2-filled microwave cavity (8.5 GHz resonance frequency) at the position of maximum electric field (∼ 100 V cm−1), cf. Fig. 2.4. This limits the time resolution to 16 ns due to the response time of the cavity. However, transients were decon-volved for the known cavity response function (see below), resulting in a time resolution limited by the 3 ns laser pulse. Samples were photoexcited via a grating in the copper end plate of the cell, which was covered and vacuum sealed with a quartz window. The other end of the cell was connected to a microwave source through a circulator detection scheme. Photoexcitation occurred at a 10 Hz repetition
p u mp th e rmo -me te r ns laser F a ra d a y ca g e N2 me ta l-g ra te d w in d o w p o w e r re si st o rs sa mp le oscilloscope PC PC control panel pneumatic system N D 2 .0 N2 @ 77 K N2 @ 295 K w a ve g u id e w a ve g u id e N D fil te rs sh u tt e r MW source MW detector ci rcu la to r offset regulator & amplifier
Figure 2.4 | Schematic of the Time-Resolved Microwave Conduct-ivity experiment.
rate using 3 ns pulses of tunable wavelength obtained by pumping an optical parametric oscillator with the third harmonic (355 nm) of a Q-switched Nd:YAG laser (Opotek Vibrant 355 II). The laser has a wavelength accuracy of ±5 nm (determined at 1000 nm).
On photoexcitation, the change in microwave power reflected from the cell was measured. For small photo-induced changes in the con-ductance of the sample, ∆G(t), the relative change in microwave probe power P can be written as
Pon(t) − Poff Poff
= −K∆G(t), (2.5)
with pump–probe delay time t, frequency-dependent sensitivity factor K,74, and subscript on (off) signifying the photoexcited
(ground-state) film. The photoconductance ∆G(t) can be expressed as ∆G(t) = eβI0FA
φe(t)µe+ φh(t)µh
with elementary charge e, aspect ratio of the rectangular waveguide β = 2.08, fluence incident on the sample I0, fraction of photons
ab-sorbed by the sample FA, the transient quantum yield φe (or φh) is the number of mobile electrons (or holes) per absorbed photon, and µe(or µh) the corresponding mobility.
Thus, the photoinduced change in microwave probe power P relates to the sum of quantum yields φi(t) for photogenerated
elec-trons and holes weighted by their mobilities µi, as
φe(t)µe+ φh(t)µh = − 1 eKβI0FA Pon(t) − Poff Poff , (2.7)
The yield per incident photon η is related to the yield per absorbed photon φ as η = φFA.‡
The temperature of the films was regulated with an average pre-cision of ±0.1 K from 90–295 K using liquid nitrogen and power resistors. The excitation density was determined as the product of the reflection-reduced pump laser fluence and QD absorption cross section, i.e. Ii(1 − FR)σ, with Ii the incident excitation fluence, FR the fraction of incident light reflected by the film surface, and σ the absorption cross section35 of a PbSe QD in dispersion.§
The most prominent source of uncertainty in our TRMC meas-urements is in the determination of incident photon flux. This varied between 3-8% depending on the wavelength of excitation. Another source of error in the calculated photoconductivity is from the meas-urement of fraction of absorbed light, which varied between 0.1-3.0%, again dependent on the wavelength.
Deconvolution of Transients
Transients can be deconvolved with the microwave cavity response function exp(−t/τR), with response time τR. The latter has the value τR = (π∆)−1, with ∆ the full width at half maximum of ‡The MFCG efficiency is determined from the increase of the quantum yield φ
with photon energy. Hence a constant fraction of charge trapping does not lead to misinterpretation of the efficiency, which is not the case75for ultrafast transient
absorption experiments employing the peak–tail method.
§This is the value of the excitation density at the interface of the sample where
the pump light first enters. It is therefore a maximum value, and the excitation density will be lower in all other parts of the sample due to a Beer–Lambert absorption profile. Trapped charge carriers recombine between successive laser pulses (10 Hz repetition rate) since laser pump fluence did not affect the photo-normalized signal.
2 0 1 as measured 300 200 100 0 Time (ns) as deconvolved (cm 2 V −1 s −1 )
φ
e(
t)
µ
e+
φ
h(
t)
µ
hFigure 2.5 | Example of a deconvolved transient. Microwave con-ductivity transients (as measured and as deconvolved) of an Al2O3-infilled
PbSe-EDT QD solid excited at low fluence. The vertical axis displays the product of the mobility µ and yield of mobile charge carriers per photoexcit-ation φ, summed for photogenerated electrons and holes: φe(t)µe+ φh(t)µh.
a Lorentzian reduction of the cavity reflection spectrum determ-ined at five temperatures in the range of 90–295 K. The response time has a value of 15.9 ns (17.2 ns) at 295 K (90 K) and for in-termediate temperatures T describes the second-order polynomial τR/ns = C0+C1(T /K)+C2(T /K)2with C0 = 16.37, C1 = 1.37×10−2, and C2= −5.16 × 10−5.
Figure 2.5 shows the charge mobility transients of an Al2O3 -infilled PbSe-EDT QD solid excited at low fluence: as measured (grey curve) and as deconvolved (black curve). The sample was excited at a photon energy of 1.87Eg (λ = 1032 nm) such that CM is not pos-sible. The maximum deconvolved signal is 2 cm2 V−1 s−1. Since
the quantum yield is unity at most, this gives the lower limit for the mobility
(µe+ µh) = 2 cm2 V−1 s−1.
2.2.4 Transient and Steady-State Photoluminescence Transient and steady-state photoluminescence was measured with a LifeSpec-ps spectrometer. The excitation source is a diode laser at 405 nm. Specular reflection is removed using a 475-nm longpass filter. Detection is integrated over a 16-nm range, centered at 820 nm, which is at the maximum in the photoluminescence spectrum, see inset
in Fig. 4.2. A repetition frequency of 100 kHz is used, which is sufficiently low so that the sample relaxes back to the ground state between successive excitation pulses.
2.2.5 Steady-State Optical Absorption
Absorption spectra were measured with a Perkin–Elmer Lambda 900 spectrometer equipped with integrating sphere. ALD-infilled QD solids were measured in air inside the integrating sphere. Baseline spectra obtained from the quartz substrate, recorded in the integ-rating sphere under identical conditions were subtracted from the sample spectra to obtain the final absorption spectra. Air-sensitive films could be kept under N2 atmosphere using air-tight sample hold-ers. The absorption spectrum of these films was obtained from the measured transmission and reflection spectra. The band-gap energy of a QD solid, Eg, is taken as that photon energy at which the lowest-energy absorption peak shows a maximum. The QD band gap and the optical absorption cross section at the photoexcitation energies used in the experiments have negligible temperature dependence over 90–295 K.76
Theoretical Analysis
3.1 Fraction of 800 nm Pump Photons Absorbed by PbS QDs in P3HT:PCBM Blend
Photoexcitation of a PbS:P3HT:PCBM blend film at 800 nm involves absorption of photons by PbS quantum dots with fraction X and absorption by the organic material component (P3HT:PCBM) with fraction 1 − X. This appendix comprises a calculation of the fraction X from experimental data.
The base-e absorption coefficient of films was determined from the Lambert–Beer law as
α = − ln IT I0 L = − ln IT II− IR L = − ln T 1 − R L, (3.1)
where ITis the intensity of light transmitted through the film of thick-ness L, I0= II− IR is the initial intensity of light upon entering the film where IIis the intensity of incident light and IRis the intensity of reflected light, T ≡ IT/IIis the transmitted fraction of incident light, and R ≡ IR/II is the reflected fraction of incident light. Transmis-sion and reflection at 800 nm were measured using a Perkin–Elmer Lambda900 spectrometer, and the thickness was measured using a Dektak profilometer.
According to the Lambert–Beer law, the change in light intensity (dI) per unit path length (dz) equals
dI dz = − h αQD(WQD) + αORG(WORG) i I. (3.2)
Here the subscripts QD and ORG represent the PbS QDs and the organic blend P3HT:PCBM, respectively, and the inverse optical at-tenuation length due to material component i is
αi(Wi) = σini(Wi),
where σi is the absorption cross section of a material unit that is
present with number density ni(Wi), which depends on Wi, the weight
fraction of component i in the BHJ. The fraction of photons absorbed by PbS QDs is thus X = αQD(WQD) αQD(WQD) + αORG(WORG) . (3.3) Furthermore, ni(Wi) = ρi(Wi) Mi , and ρi(Wi) = ρWi,
with ρ the mass density of the film and Mi the weight of a material
unit of component i. Thus, the inverse attenuation length in the blend film is
αi(Wi) = ρWi
αi(Wi = 1)
ρi(Wi = 1)
,
with αi(Wi = 1) and ρi(Wi = 1) the values for films consisting of
only PbS QDs or only P3HT:PCBM. Using this relation it is found that X = WQD αQD(WQD= 1) ρQD(WQD= 1) WQD αQD(WQD= 1) ρQD(WQD= 1) + WORG αORG(WORG = 1) ρORG(WORG = 1) , (3.4) with WQD+ WORG = 1.
For the PbS:P3HT:PCBM BHJ studied in this work the weight fraction of PbS QDs is 1/3, and the remaining weight fraction of P3HT:PCBM is 2/3, thus WQD= 1 3, WORG = 2 3.
For the P3HT:PCBM blend, the inverse optical attenuation length was measured as described above, and the mass density was calculated as the content-weighted average. The resulting values are
αORG(WORG = 1) = 2.0 × 104 m−1, ρORG(WORG = 1) = 1.3 × 103 kg m−3,
and
αORG(WORG = 1) ρORG(WORG = 1)
= 15 m2 kg−1.
For PbS QDs, the absorption cross section σQD was obtained by scaling the absorption cross section measured at 400 nm from liter-ature65 with the relative magnitude of the optical absorption at 800 nm. The mass was obtained as the mass of a PbS sphere (with dia-meter 2.5 nm and mass density77 7.6 × 103 kg m−3 of PbS) and the mass of the oleic acid surface ligands (4.7 × 10−25 kg per ligand mo-lecule) with density of (3.0 ± 0.4) ligands per nm2of PbS QD surface
area.78 The resulting values are
σQD = 1.6 × 10−20 m2, MQD = 9.0 × 10−23 kg, and αQD(WQD= 1) ρQD(WQD= 1) = σQD MQD = 1.8 × 10 2 m2 kg−1.
Using the weight fractions and inverse optical attenuation lengths for the PbS:P3HT:PCBM blend, Eq. 3.4 yields that the fraction of photons absorbed by PbS QDs is equal to
X = 0.86.
3.2 Absorption and Stimulated Emission of a Phonon by a Hot Charge
The energy rate of change for a hot charge of kinetic energy E (equa-tion 7.4) can be written as
dE dt = C [X(T ) − Y ] , with X(T ) = n(T ) " sinh−1r E ¯ hω − sinh −1r E ¯ hω − 1 # , Y = sinh−1r E ¯ hω − 1. Defining x ≡ E/(¯hω) we can write
Since in the present work x ≫ 1, because the charge energy much exceeds the phonon energy, the second of the two square roots can be expanded via the binomial series
√ x − 1 =√x r 1 − x1 =√x − 1 2√x + . . . , ≡√x + ∆ ⇔ ∆ ≡√x − 1 −√x.
Note that ∆ is nearly zero because x ≫ 1. Substituting the above expression into equation (3.5) and multiplying by ∆/∆ yields
X(T ) = −n(T )∆ sinh −1(√ x + ∆) − sinh−1√x ∆ . In the limit that ∆ → 0, this expression simplifies as
lim ∆→0X(T ) = ∆→0lim −n(T )∆ d sinh−1√x d√x = lim ∆→0−n(T ) ∆ √ x + 1 = lim ∆→0n(T ) √ x −√ √x − 1 x + 1 .
Because in the present case ∆ is close to zero, we may use the above expression as a good approximation to X(T ). Expanding the square roots via the binomial series and neglecting terms of second order and higher (since x ≫ 1), yields
X(T ) ≈ n(T ) ¯hω 2E + ¯hω
≈ n(T )2E¯hω. (3.6)
Equation (3.6) becomes exactly identical to equation (3.5) in the limit when E/(¯hω) tends to infinity, and is very nearly identical for E ≫ ¯hω. At present, E/(¯hω) ≥ 39 and equation (3.6) is an excellent approximation to equation (3.5).
The expression for Y may be simplified using the definition sinh−1(z) ≡ lnhz +p1 + z2i
and by noting that E/(¯hω) ≫ 1 in the present work, to yield Y ≈ ln 2r E ¯ hω ! = 1 2ln 4E ¯ hω . Thus for hot charges, equation 7.4 simplifies to
dE dt = C 2 n(T )¯hω E − ln 4E ¯ hω . (3.7)
3.3 Electron–hole pair dissociation energy
To estimate the dissociation energy of an e–h pair residing in one QD we first consider the energy E of a well-separated electron–hole pair where the electron and hole occupy different QDs and do not interact. With respect to the unexcited QD film the energy is:
E = E1e− E1h+ E pol
e + E
pol
h . (3.8)
Here E1e and E1h are the site energies of the electron and hole, re-spectively and Epol
e and E pol
h are the electron and hole self energies. When the electron and hole are brought together on a single QD, the energy becomes: Ee–h= E1e− E1h+ E pol e + E pol h + E dir e,h+ E pol e,h. (3.9) Edir
e,h is the direct Coulomb interaction between electron and hole and Epol
e,h is the interaction of the electron with the polarization induced by the hole, and vice versa. The e–h pair dissociation energy Edis,eh is obtained as the difference between E and Ee–h and contains the direct Coulomb interaction Edir
e,h between electron and hole, as well as the cross-polarization energy Epol
e,h. Expressions for these contri-butions have been derived by Delerue resulting in the following final expression for the dissociation energy:79
Edis,eh= 1.79e2 4πε0εina + e 2 4πε0a εin− εout εinεout , (3.10)
with vacuum permittivity ε0, QD radius a, dielectric constant of PbSe
QDs εin, and dielectric constant outside of the QDs εout.
Equation 3.10 shows that the dissociation energy and the dielec-tric constants are directly related. A complexity comes from the fact
that the dielectric constant is frequency dependent. In bulk semicon-ductors, the exciton binding can be determined experimentally, and the proper value of the dielectric constant can be determined.80With the magnitude of the exciton binding energy the angular frequency of the electron and hole changes. If this frequency is higher than the frequency of the optical phonons the use of the optical dielec-tric constant is appropriate, if the frequency is well below the optical phonon frequencies, the static dielectric constant applies.80 We
as-sume that for QDs the use of the optical dielectric constant is always more appropriate. This results from the fact that electrons and holes always have a high kinetic energy as a result of quantum confinement and the associated frequency will be higher than the optical phonon frequencies.81
εout is determined by the capping molecules, in this case 1,2-ethanediamine, and neighboring QDs. We consider it to be the ef-fective dielectric constant of the film and obtain an estimate of its value by applying the Bruggeman effective medium theory:82
f εin− hεi εin+ κhεi
= (f − 1)εεm− hεi m+ κhεi
, (3.11)
with QD fill factor f , effective dielectric function of the film hεi (i.e. hεi = εout), and dielectric function of the capping material εm. For QDs with a radius of 3.0 nm, a capping layer of 0.2 nm and a total packing density of 0.7, the fill factor is 0.58. With the optical dielec-tric constant of PbSe (23.9) and 2DA (2.11) this results in εout= 10.3 and an Edis,eh= 58 meV.
3.4 Escape yield from double e–h pairs
Note that we define a difference between yield, an overall fraction of charges indicated by the symbol φ0 and efficiency, the slope of a
charge yield versus energy, indicated by the symbol η.
The overall yield of free charge carrier generation is determined by the initial CM efficiency ηCM and the escape yield of multiple e–h pairs from recombination. The latter process is shown schematically in Figure 3.1. It involves the dissociation of double e–h pairs into positive or negative trions (Γdis,1), the further dissociation of those trions into single e–h pairs (Γdis,2a) or doubly charged QDs (Γdis,2b) and finally into single charges per QD (Γdis,3a and Γdis,3b). The overall efficiency of generation of free charge carriers ηMFCGis thus given by: ηMFCG= ηCMφesc= ηCMφdis,2ehφdis,Tφdis,eh (3.12)
Aug,2eh Γ Γdis,1 dis,2a Γ dis,2b Γ Aug,T Γ dis,3b Γ dis,3a Γ rec Γ dis,eh Γ a
+
b+
++
+
+
+
+
+
++
+
++
Figure 3.1 | Dissociation of e–h pairs. Schematic of the dissociation of a, a single e–h pair and b, multiple e–h pairs into free charges in a QD film. The dissociation of single e–h pairs occurs in competition with recombination to the ground state, whereas the dissociation of multiple e–h pairs occurs in competition with Auger recombination. The dissociation of a double e–h pair involves multiple charge transfer steps. The first step is the rate-limiting step.
with φesc the yield of escape from Auger recombination, φdis,2eh the yield of dissociation of a double e–h pair into a trion and a single charge, φdis,T the yield with which a trion decays into a single e–h pair and a free charge, and φdis,eh the yield of dissociation of a single e–h pair into free charges.
As explained in chapter 5 the term φdis,ehis always near unity and can thus be dropped from Eq. 3.12. The rate of dissociation of single e–h pairs depends on the hopping rate of electrons and holes and the
dissociation energy of an e–h pair: Γdis,eh= X Γhope − Edis kBT = kBT NP µ e∆2 . (3.13)
Here P Γhop is the sum of the hopping rates for electrons and holes, Edis is the dissociation energy of an e–h pair, N is the number of nearest QD neighbors, P µ is the sum of the electron and hole mo-bility and ∆ is the interparticle hopping distance.
The yield of dissociation of a double e–h pair into a trion and a free charge is given by:
φdis,2eh = Γdis,1 Γdis,1+ ΓAug,2eh ≈ 2Γdis,eh 2Γdis,eh+ ΓAug,2eh , (3.14)
as there are twice as many charge carriers in a double e–h pair as in a single e–h pair, and the energy of dissociating the former into a trion is in first approximation identical to the energy required to dissociate a single e–h pair (in both cases the initial state is neutral and the final state involves two QDs with a single net charge), it is found that Γdis,1 ≈ 2Γdis,eh. However, since the Auger recombination rate is ∼ 150 times higher than the non-radiative recombination rate of single e–h pairs,83 the yield of double e–h pair dissociation can be
much lower than the yield of single e–h pair dissociation.
The dissociation of trions can occur via two paths, 2a and 2b in Figure 3.1, and the overall yield depends on the sum of the rates of these paths:
φdis,T =
Γdis,2a+ Γdis,2b Γdis,2a+ Γdis,2b+ ΓAug,T ≈ 10Γ 10Γdis,eh+ 0.005Γdis,eh
dis,eh+ 0.005Γdis,eh+ ΓAug,2eh/4
. (3.15)
Trion dissociation occurs in competition with Auger recombination of trions, which is four times slower than Auger recombination of double e–h pairs.84 Path 2a does not involve an increase in the
Cou-lomb energy of the system, and hence the rate of this process is ∼ exp[Edis/(kBT )] times higher than the rate of single e–h pair dis-sociation. For Edis = 58 meV, as determined above, this amounts to
a factor 10. Path 2b involves an increase of the Coulomb energy of ∼ 2Edis. In addition, only a single carrier is involved in this step, versus two carriers in the single e–h pair dissociation. Hence the overall rate of path 2b is ∼ 200 times lower than Γdis,ehand it can be disregarded.
Upon comparing Eqs. 3.14 and 3.15, it is clear that the yield of dissociation of a double e–h pair into a trion and a free charge is limiting the overall escape yield. In the limit of low escape yield the trion dissociation yield is 20 times higher than the double e–h pair dissociation yield. Hence the escape yield of double e–h pairs is well-approximated by the yield of their decay into a trion and a free charge.
3.5 Charge Diffusion Length in Presence of Randomly Distributed Deep Traps
In the limit that t approaches infinity, the fraction of surviving charge carriers s(t) becomes zero and all charge carriers are trapped. At this final instance, each charge carrier i has been trapped a distance ri
from its initial position. The diffusion length is defined as the square root of the mean squared displacement L ≡ phr2i, with the latter
given as hr2i = N1 N X i=1 ri2,
with total number of charge carriers N . Alternatively, this can be written in terms of a probability of obtaining a given displacement, as hr2i = Z all space r2f (r) dV, (3.16)
with f (r) the probability density function (PDF), i.e. the probability of finding a trapped charge carrier in the infinitesimal volume dV , divided by the volume dV . Note: charge-carrier diffusion in the QD solid is assumed to be isotropic, hence the PDF only depends on the displacement r and not on the direction of displacement. The function f (r) is determined as follows.
At any given pump–probe delay time t, the fraction of charge carriers that has been trapped is 1 − s(t). Thus, the fraction of charge carriers that reaches a trap state per unit time is −ds(t)/dt. In the present model of deep traps, once charge carriers are trapped
they are not reemitted but recombine rapidly. Thus between t and t + dt, a fraction −ds(t) of charge carriers are trapped after having experienced trap-free diffusion for the duration of t. Their spatial distribution at the moment of trapping t is therefore described by the PDF for trap-free particle diffusion f0(r, t). Weighting this PDF
by the fraction of charges for which it holds and integrating over all time yields an expression for f (r), as
f (r) = − ∞ Z t=0 f0(r, t) ds(t) = 1 Z s=0 f0(r, t) ds (3.17)
Substituting this into Eq. 3.16, and noting that the mean squared displacement of trap-free diffusion is hr2
0i = 2dDt, yields hr2i = 1 Z 0 Z all space r2f0(r, t) dV ds = 2dD 1 Z 0 t ds.
Note that t is expressed in terms of s by means of the inverse function of s, i.e. t(s) = s−1(t) = τ∗[ln(1/s)](d+2)/d. Evaluating the integral
yields the mean squared displacement in a system of randomly dis-tributed deep traps
hr2i = Γ 2 +2 d 2dDτ∗, (3.18)
with Γ the gamma function. The diffusion length is defined as the square root of the mean squared displacement, thus
L = s Γ 2 + 2 d 2dDτ∗.
The carrier lifetime (the average time a carrier takes to reach a deep trap) is calculated as τ ≡ hti = R1
s=0t ds, where evaluating as above
yields τ = Γ 2 + 2 d τ∗. (3.19)
Thus the diffusion length in the presence of randomly distributed deep traps, specified in terms of the lifetime τ , is
Extracting Charges from
Carrier Multiplication in
Quantum Dots
Origin of Low Sensitizing Efficiency of Quantum Dots in Organic Solar Cells
(ACS Nano, 2012, 6, 8983–8988)
C
urrently, there is a great interest in the development of cheap,solution-processable organic materials for application in photo-voltaic devices.85,86 In this context, conjugated polymers as light absorber with fullerene derivatives as electron acceptor attract a great deal of attention.87,88 Unfortunately, the near-infrared (NIR) portion of the solar spectrum remains largely unabsorbed by these materials. The spectral sensitivity can be extended by including lead-chalcogenide quantum dots (QDs) as sensitizers, which have a high absorption coefficient with an onset that can be tuned through-out the NIR by variation of their size.89 Indeed, it was recentlyconcluded that electrons and holes can be extracted from excited PbS-oleate QDs in a BHJ with the fullerene derivative [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) and the conjugated poly-mer regio-regular poly(3-hexylthiophene) (P3HT).48,49 However, the NIR power-conversion efficiency of QD-sensitized bulk-heterojunction (BHJ) photovoltaics remains orders of magnitude lower than that of either fully organic or fully QD-based devices.45 The poor perform-ance of a BHJ containing spherical QDs may be due to inefficient charge separation. The latter has been associated with the nano-particle shape, and it was demonstrated that the charge separation efficiency increases when going from spherical QDs to rods or tet-rapods.90 Another way of achieving efficient charge separation is by creating a layer of linked spherical QDs.37,45,91
The question remains why BHJ photovoltaics comprising lead-chalcogenide QDs, a conjugated polymer and/or PCBM, have a poor NIR power-conversion efficiency. In this paper we aim to ex-plain the poor performance of lead-chalcogenide QDs as NIR sens-itizers in BHJ photovoltaics. This is achieved by quantifying the efficiency of charge transfer by means of ultrafast transient
tion spectroscopy measurements (TA), and by measuring the mobil-ity of the charges using ultrafast time-resolved terahertz spectroscopy (THz),68,69 see chapter 2.2.1 and 2.2.2. Combined use of these two experimental techniques on a series of samples of different compos-ition enables a direct characterization of charge carrier behaviour. Charge transfer from photoexcited PbS QDs to the organic material components was found to occur, however, the charges produced in this way have a negligible mobility compared to those produced in a fully organic BHJ. We conclude this to be due to coulomb attraction between the charge transferred from a QD, and the remaining oppos-ite charge that is localized on a single QD. Thus we reach a general finding which explains the data presented here, and other literature data.
4.1 Results and Discussion
The PbS QDs studied in this work have an average diameter of 2.5 nm and contain oleic acid as passivating surface ligand, similar to earlier device and spectroscopic studies.46,48,49 The first optical absorption peak is at 780 nm. It is inferred from the long photoluminescence lifetime that charge trapping at surface defects is negligible (Fig. 4.2). The small PbS QD size ensures a favourable driving force for electron transfer92 from a photoexcited PbS QD to PCBM and hole transfer to P3HT in π-stacked domains (cf. Fig. 4.1). A pure QD film, and BHJs of PbS QDs with P3HT and/or PCBM, and of P3HT with PCBM were prepared by drop-casting from solution with equal weight fraction of each material. The optical absorption spectra of the BHJs exhibit a clearly discernible optical absorption peak due to PbS QDs with the band gap being retained upon blending (Fig. 4.3).
Quantum Yields of Charge Transfer
Ultrafast optical pump-probe spectroscopy was used to determine the yields of charge transfer from photoexcited PbS QDs to PCBM and P3HT (see chapter 2.2.1). Both pump and probe laser pulses were centred at 800 nm in order to photoexcite and probe excitons or excess charges at the band gap of our PbS QDs. Fig. 4.4a shows the transient change of the optical absorption cross section per absorbed pump photon
σ =X
i
−6 −5 −4 −3 P3HT PbS QD PCBM En er g y (e V) −3.5 −5.1 −3.7 −5.9 −2.7 −4.7 P3HT PCBM −3.7 −5.9 −2.7 −4.7 QD excitation CT excitation
Figure 4.1 | Energy diagram showing HOMO and LUMO levels of PbS QDs89
of the size as studied, and P3HT and PCBM.93
Electron transfer to PCBM and hole transfer to P3HT (dashed arrows) is energetically feasible from a photoexcited PbS QD. Excitation of the charge-transfer (CT) state, in which an electron is charge-transferred from the P3HT HOMO to the PCBM LUMO is also possible at our excitation wavelength. The P3HT levels depend on P3HT crystallinity,94 which may differ in the
proximity of QDs.
(see chapter 2.2.1), with φi the quantum yield for photogeneration
of species i and ∆σi the change in absorption cross section due to
this species. An incident pump fluence of 4.4 × 1013 photons cm−2 was found to be sufficiently low for higher-order interactions between excitons or charges to be insignificant. The change in absorption cross section at the band gap of the PbS QDs due to the presence of excitons or excess charges was obtained from a TA measurement on a neat film of PbS QDs (solid black curve, Fig. 4.4a). An exciton in a PbS QD results in a reduced ground state absorption due to the presence of a hole and stimulated emission from the excited electron. The bleach at the band gap is determined by state-filling effects, within the approximation that transition strengths are not affected by a spectator charge. In this case, the cross section for bleach, ∆σPbS, of electron and hole are equal. This is found to agree with
experiment.95 Therefore, the initial bleach is given by
1 10 100 P h o to lu m in e sce n ce ( a rb . u .) 900 600 300 0 Time (ns) 900 800 PL (a rb . u .) O D (a rb . u .) Wavelength (nm) 700
Figure 4.2 | Optical data of PbS QDs in tetrachloroethene solu-tion. The long photoluminescence lifetime of 1.5 µs exceeds the expected lifetime for this size of QDs,65 which implies that defect states do not
sig-nificantly affect exciton decay. The maximum in the photoluminescence spectrum is Stokes-shifted to 820 nm (inset: black curve) from the first peak in the optical absorption spectrum at 780 nm (inset: grey curve).
0.4 0.3 0.2 0.1 0.0 950 900 850 800 750 700 650 Wavelength (nm) P3HT:PCBM PbS PbS:PCBM PbS:P3HT PbS:P3HT:PCBM Fraction absorbed of incident light
FA
Figure 4.3 | Absorption spectra of drop-cast blend films. The first absorption peak due to PbS QDs is retained in the blends, which implies conservation of quantum confinement.
0.4 0.3 0.2 0.1 0.0 Transient THz photoconductivity 0.5 µ ( cm 2 V − 1 s − 1 ) 100 101 102 103 104 Time (ps)
b
100 101 102 103 104 Time (ps)a
4 3 2 1 0 −1 PbS PbS:PCBM PbS:P3HT:PCBM ( Model) PbS:P3HT P3HT:PCBMTransient optical absorption
(
Å
2 )
−2 5
Figure 4.4 | Time-resolved spectroscopic data of four BHJs of dif-ferent composition, and a pure PbS QD film. a, Transient change of optical absorption per absorbed excitation photon. The vertical axis dis-plays the weighted mean of the changes in cross section associated with the photogenerated species: σ ≡ P
iφi∆σi (see chapter 2.2.1). b,
Time-resolved terahertz conductivity per absorbed excitation photon. The ver-tical axis displays the weighted mean of the charge mobilities: µ≡P
iφiµi
(see chapter 2.2.2). Colour coding as in panel ‘a’. a & b, From comparing results (dashed) of a simple model (see text) with the ternary blend data (yellow) we conclude that charges transferred from QDs have negligible mo-bility.
The amplitude of the TA transient of the PbS QD film yields ∆σPbS= −0.51 ± 0.02 ˚A2
(standard error of mean, s.e.m.).
The optical bleach exhibits no decay on the timescale of 20 ns, in agreement with the long exciton lifetime of 1.5 µs in solution (Fig. 4.2), from which we infer an absence of exciton decay at de-fects. Interestingly, the bleach of the PbS QD film appears to slightly increase during time, which is attributed to energetic relaxation of ex-citons by diffusion to larger QDs with smaller band gap and larger cross section for optical bleach.
Photoexcitation of the charge-transfer state (CT) in the P3HT:PCBM BHJ at 800 nm leads to promotion of a valence electron from the highest occupied molecular orbital (HOMO) of P3HT to the lowest unoccupied molecular orbital (LUMO) of PCBM.96 The TA
probed at 800 nm (green curve, Fig. 4.4) is due the P3HT+ cation only, since at this wavelength the PCBM– anion does not give a change in optical transmission.97–99The initial magnitude of the TA
is equal to the cross section for absorption by P3HT+; i.e.
σP3HT:PCBM = ∆σP3HT+ (4.3)
= 3.9 ± 0.1 ˚A2 (s.e.m.).
The decay of the TA after 100 ps is due to geminate electron–hole recombination, while the remaining absorption on times exceeding a few nanoseconds is due to free charges that decay on a millisecond timescale.97
The optical cross sections, as determined above, are used below to obtain the quantum yields of charge transfer from photoexcited PbS QDs to PCBM and P3HT from TA measurements on BHJs.
The TA for the PbS:PCBM BHJ (blue curve, Fig. 4.4a) is less negative than that for the PbS QD film, which is due to electron transfer from photoexcited PbS QDs to PCBM. The amplitude of the bleach for the PbS:PCBM BHJ is determined by excitons in PbS QDs that have not undergone charge transfer and holes remaining in a PbS QD after electron transfer to PCBM with quantum yield φe;
i.e.
σPbS:PCBM = (2 − φe)∆σPbS (4.4)
With the value of ∆σPbS determined above, this results in φe= 0.41 ± 0.06 (s.e.m.).
The step-like kinetic behaviour implies that electron transfer takes place within 100 fs, while charge recombination occurs on times ex-ceeding 20 ns. The high rate for electron transfer from a photoexcited PbS QD to PCBM can be understood on basis of the classical Marcus rate for electron transfer,100 given by
k = 2πJ 2 ¯ h√4πλkBT exp −(∆G + λ) 2 4λkBT , (4.5)
with J the electronic coupling matrix element, ¯h the reduced Planck constant, λ the reorganization energy, ∆G the driving force, kB the
Boltzmann constant and T the temperature. Figure 4.1 shows that ∆G = −0.2 eV, which is close to the negative of the reorganization energy involved in electron transfer to a PCBM molecule,101 so that ∆G+λ is negligible and the Marcus rate is close to optimal. To obtain a transfer rate exceeding (100 fs)−1 for λ = 0.2 eV the electronic coupling must be
J > 16 meV,
which is not particularly high for charge transfer between organic molecules.102Hence, the high rate for electron transfer in the present system stems from (near) cancellation of ∆G and λ.
The amplitude of the TA change for the PbS:P3HT BHJ (red curve, Fig. 4.4a) is less negative than for the PbS QD film, which is attributed to reduced bleach due to hole transfer from photoexcited PbS QDs to P3HT and induced absorption by P3HT+. The step-like kinetic behaviour implies that hole transfer occurs promptly within 100 fs, while charge recombination plays no role on the 20-ns time scale. The absence of decay implies that charge recombination is slower than 20 ns. The amplitude of the transient of the PbS:P3HT BHJ is given by
σPbS:P3HT= (2 − φh)∆σPbS+ φh∆σP3HT+, (4.6) with φh the quantum yield for hole transfer. Using the cross sections
for bleach due to a charge in a PbS QD and absorption by P3HT+, it is found that
