Charge Carrier Dynamics in Interconnected Halide Perovskite Quantum Dot Networks

Charge Carrier Dynamics inInterconnected Halide PerovskiteQuantum Dot Networks David Otto Tiede UNIVERSITY OF SEVILLE DOCTORAL THESIS Charge Carrier Dynamics in Interconnected Halide Perovskite Quantum Dot Networks Author: David Otto TIEDE Supervisors: Prof. Hernán MÍGUEZ Dr. Juan GALISTEO-LÓPEZ A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Multifunctional Optical Materials Group Institute of Materials Science of Seville Spanish National Research Council (CSIC) May 27, 2024 iii Contents Popular Summary in English vii Resumen en Español ix 1 General Introduction 1 1.1 Charge Carrier Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 QuantumDots............................... 3 1.3 HalidePerovskites ............................ 6 1.4 Quantum-confined Perovskite Nanocrystals . . . . . . . . . . . . . . 8 1.5 ScopeofthisThesis............................ 10 1.6 Outline................................... 11 2 Conceptual Background 13 2.1 Quantum Dot Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.1 Theory of Quantum Confinement in low Dimensional Nanostructures.......................... 14 2.1.2 Colloidal Perovskite Quantum Dots . . . . . . . . . . . . . . 15 2.1.3 Fabrication of Ligand-free Perovksite Nanocrystals . . . . . 17 2.1.4 Structural Characterization of embedded Perovskite Nanocrystals ........................... 18 2.2 Charge Carrier Recombination in Perovskites . . . . . . . . . . . . . 22 2.2.1 Monomolecular Recombination . . . . . . . . . . . . . . . . . 24 2.2.2 Bimolecular Recombination . . . . . . . . . . . . . . . . . . . 26 2.2.3 Trimolecular Recombination . . . . . . . . . . . . . . . . . . 28 2.2.4 Recombination Constants in Excitonic Systems . . . . . . . . 30 2.3 Experimental Measurement Techniques . . . . . . . . . . . . . . . . 31 2.3.1 Time Correlated Single Photon Counting . . . . . . . . . . . 31 2.3.2 Transient Absorption Spectroscopy . . . . . . . . . . . . . . 33 iv 2.3.3 Fluence dependent Photoluminescence Quantum Yield . . . 36 2.4 Conclusion................................. 38 3 Extraction of Recombination Constants from Non-Canonical Systems 41 3.1 Introduction................................ 42 3.1.1 Limitations of the Shockley -Read- Hall model . . . . . . . . 42 3.2 Theoretical Description of the Photon Flux Representation . . . . . 45 3.2.1 Characteristics of Bimolecular Recombination . . . . . . . . 46 3.2.2 Characteristics of Monomolecular Recombination . . . . . . 48 3.3 Measurement and Analysis Protocol for the Photon Flux Representation.................................... 49 3.4 Experimental Application of the Photon Flux Model . . . . . . . . . 50 3.4.1 Application in a Canonical Material System . . . . . . . . . . 51 3.4.2 Application in a Non-Canonical Material System . . . . . . 53 3.5 Photon Flux Pattern of Isolated Quantum Dots . . . . . . . . . . . . 56 3.5.1 Theoretical Description in Isolated Systems . . . . . . . . . . 56 3.5.2 Experimental Validation in Isolated Systems . . . . . . . . . 59 3.6 Derivative Representations in Time-resolved Measurements . . . . 60 3.7 Conclusion................................. 62 4 Connectivity in embedded Quantum Dot Networks I: Effect on Carrier Transport and Recombination 65 4.1 Introduction................................ 66 4.2 Structural Properties and Linear Optical Response of Ligand-free Perovskite Quantum Dot Networks . . . . . . . . . . . . . . . . . . 67 4.3 Effect on Trap Filling Dynamics . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 Photon Flux Representation . . . . . . . . . . . . . . . . . . . 70 4.3.2 Emission Properties in Isolated QDs . . . . . . . . . . . . . . 73 4.4 Effect on Emission Efficiency . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Effect on Emission Linewidth . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Effect on Multiparticle Interaction . . . . . . . . . . . . . . . . . . . 79 4.7 Effect of Defect Landscape on Carrier Recombination . . . . . . . . 82 4.7.1 Defect Passivation in Isolated Emitters . . . . . . . . . . . . 82 4.7.2 Trap State-Limited Diffusion in Connected Quantum Dot Solids................................ 85 4.8 Conclusion................................. 87 v 5 Connectivity in embedded Quantum Dot Networks II: Effect on Ultrafast Carrier Dynamics 89 5.1 Introduction................................ 90 5.2 Slow Charge Carrier Cooling in Perovskite Quantum Dot Solids . . 92 5.2.1 Procedure for Charge Carrier Temperature Extraction . . . . 92 5.2.2 Interparticle-Spacing dependent Charge Carrier TemperatureEvolution........................... 93 5.3 Enhanced Auger Heating . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Reduced Phonon-assisted Charge Carrier Cooling . . . . . . . . . . 98 5.4.1 Ground State Bleach Formation Dynamics . . . . . . . . . . 102 5.5 Ultrafast Exciton Carrier-Coupling . . . . . . . . . . . . . . . . . . . 102 5.5.1 On the Macroscopic Origin of Exciton-Free Carrier-Phonon Couplings............................. 106 5.6 Conclusion................................. 107 6 Charge Carrier Propagation in Colloidal Networks: Excitonic Effects and Exciton-Polariton Condensation 109 6.1 Introduction................................ 110 6.2 Types of Dipole Interaction in Ligand-capped QDs . . . . . . . . . . 112 6.3 Charge Carrier Recombination in Coupled Colloidal QDs . . . . . . 114 6.3.1 Linear Optical Characterization of Coupled Colloidal QDs Solids................................ 115 6.3.2 Tuning Exciton-Exciton Interaction in Colloidal QD solids . 115 6.3.3 TA Fine Structure and Local Exciton-Exciton Interaction . . 118 6.4 Exciton-Polariton Condensation in Colloidal QD Solids . . . . . . . 121 6.4.1 Tuning of Strong Light-Matter Coupling . . . . . . . . . . . 121 6.4.2 Properties of Condensed Exciton-Polariton Emission . . . . 123 6.5 Conclusion................................. 126 7 General Conclusions and Perspective 129 7.1 SummarizedResults ........................... 130 7.2 Perspective................................. 131 A Data Curation 135 A.1 Extraction of Charge Carrier Density in TA Measurements . . . . . 135 A.2 Protocol to Extract Derivatives of (noisy) TRPL Decays . . . . . . . 136 vi Bibliography 139 Dissemination 161 Publications ................................... 161 Conference Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Acknowledgements 165 vii Popular Summary in English Semiconductor materials are a key element in current technology, where the number of devices that employ them is expected to grow exponentially in the near future. At the same time, increasing resource scarcity and the urgent need to optimize energy production and consumption is one of the main challenges for human society, such that an urgent need for more efficient optoelectronic devices arises. Understanding the charge carrier dynamics in semiconductors in optoelectronic devices is an essential tool to tackle this challenge. Precisely, perovskite quantum dot solids are emerging semiconductor materials that combine outstanding optoelectronic properties with the advantages of nanostructuring and, most importantly, quantum confinement effects, thus providing an exciting material platform for next generation devices. Interestingly, the intrinsic isolation required to obtain quantum confinement stands in contrast with the requirement for many optoelectronic devices to extract or inject electric charges. The analysis of the interplay between confinement and transport effects on carrier recombination in quantum dot lattices or networks pose significant experimental and fundamental challenges, which have so far hindered the realization of a systematic study. This thesis is an attempt to realize a comprehensive study on the charge carrier cooling, trapping and recombination dynamics along a wide temporal rage, comprising from the femtosecond to the millisecond regime, in perovskite quantum solids with varying connectivity. In this thesis, we establish a method based on macroscopic time resolved spectroscopy that allows to discriminate between local (isolated) and global (connected) recombination dynamics. We demonstrate how to generally apply density dependent recombination models on complex material systems, where time-dependent effects such as diffusion may blur out underlying processes. Based on this characterization, we have analysed the effects on connectivity between QDs on macroscopic optoelectronic properties such as emission yield or emission linewidth. These findings are discussed in terms of different charge carrier density regimes that correspond to different operational regimes for photovoltaic or emission applications. We observe that for different applications different degrees of connectivity will be required to obtain optimum efficiency and provide thus a roadmap that enables to screen material parameters for distinct optoelectronic devices. 2Chapter 1. General Introduction 1.1 Charge Carrier Dynamics "The only thing that is constant is change" - with these words, Heraclitus predicted with great precision the behaviour of charge carrier dynamics in semiconductors. This family of materials constitute a fundamental component of any optoelectronic device, as they open and close charge transport and act as active component, where light is transformed into electronic charge or vice versa. In semiconductors, upon photoexcitation the state of excited charge carriers is constantly evolving as they change their energy by interacting with their surroundings or other charge carriers. Understanding processes that dictate charge carrier recombination dynamics will largely contribute to both optimize semiconductor based devices and propose advanced designs with novel properties based on them . In the course of the last 120 years, the physical understanding of these phenomena has constantly evolved from simply describing the photoelectric effect, [2] which set the fundamentals, to explore light-matter interaction within the "diode era". In 1952, a large advancement in their theoretical description was made, introducing charge carrier recombinations as a series of possible recombination events, that can be described with material dependent probabilities (Figure 1.1a). Precisely, a scheme of defect assisted recombination was proposed within the Shockley-Read- Hall model [3, 4, 5] (Figure 1.1a). Soon after, this model was extended by multiparticle interactions described within the framework of Auger processes [6, 7] , adding further complexity to this model. Nowadays, seventy years later, the theoretical description is still based on the same fundamentals where, depending on the operational regime, scenarios with different complexity can be found [8, 9]. For instance, photovoltaic devices need to be described within a low charge carrier density scenario, whereas for emission applications a high charge carrier density needs to be considered. Furthermore, semiconductors are at the core of nowadays technology, where a clear trend towards nano-sized elements is evident [10, 11, 12]. All of this requires charge carrier recombination models to include complex phenomena such as trap filling, exciton trapping [13, 14] multiparticle processes [15] or diffusion processes [16, 17]. The profound understanding of charge carrier dynamics provides thus an essential tool towards the optimization of material and device structures [18], such that a major quest for the scientific community to find adequate models for the evolving material and device structures remains. 1.2. Quantum Dots 3 Many of the processes involved in the recombination of photo-excited carriers have different characteristic timescales albeit with overlapping intervals. These timescales span a broad time interval ranging from the femtosecond up to the millisecond. Experimental techniques able to provide a picture of the material at any given time after excitation are thus demanded. The most commonly employed method to study recombination is to utilize time-resolved spectroscopy techniques that provide time resolutions able to unveil the different mechanisms involving charge carriers. Here physical magnitudes such as light emission [19], and absorption [20], or changes in electronic properties, such as THz conductivity [21] or surface photovoltage (SPV) currents [22], are studied. The measured amplitude is then considered proportional to the instantaneous charge carrier density, such that a fluence and time-resolved dataset allows to globally simulate the recombination kinetics by tracking the evolution of the system as the weight of different processes changes. However, an intrinsic limitation of this analysis approach is the indirect nature of recombination rate extraction, which generally leads to a large range of reported rate constants combined, with a non-specified certainty, of multi-dependent constants, such that reliable values are often only limited to canonical material systems. This is in stark misalignment to the general trend towards working with a wide variety of compositions and more sophisticated sample architectures. In this thesis, a major focus is laid on presenting universally applicable analysis tools that enable a more direct access to recombination constants in complex material systems. We aim to provide extraction protocols that both circumvent and characterize artefacts induced by a more complex sample constitution. In particular we focus on the charge carrier dynamics of perovskite quantum dots in different ensemble structures, where the description of recombination has shown to be particularly challenging [8]. In what follows, we will first introduce the potential of quantum confinement technology before introducing the class of materials at the core of this Thesis, known as halide perovskites. 1.2 Quantum Dots "There is plenty of room at the bottom" - with this famous quote, Nobel laureate Richard Feynman introduced the concept of manipulation of matter at the nanoscale in a lecture in 1959 [23]. In his lecture, Feynman discussed a plethora 4Chapter 1. General Introduction FIGURE 1.1: (a) Decay processes that are described with probabilities have for a long time attracted interest in physics, as illustrated through a Galton Board. (b) In semiconductors, different recombination processes compete and can be described with recombination rates, here exemplified through cooling processes followed by a model of Shockley-Read-Hall Recombination. of possible applications of nanotechnology ranging from information storage and computation to biological applications and the possible achievement of new physical phenomena. Due to this encouraging vision he is nowadays seen as the father of nanotechnology for his inspiring perspective. Today, more than 60years later, research on nanotechnology is as extensive as never before and spans research areas ranging from biology, medicine, chemistry or physics to materials science, where nano-sized structures are both used to improve existing applications as well as a means to find new functionalities with large impact on society [24]. The control of light-matter interactions through nanostructures is appealing, as both the transport of light [25] as well as the electronic properties of optically active materials can be drastically modified when nano-sized structures are employed. The guidance of light can be engineered through a judicious design that structures matter in geometries in the order of the wavelength of light, where scattering and interference effects are exploited to redistribute the density of optical states [26]. Of particular relevance for this thesis, the electronic properties of semiconductors can be modified if their size approaches the de Broglie wavelength of the electron wavefunction [27]. If the lateral dimensions of a nanostructure reaches this limit, the electron wavefunction becomes spatially confined and leads to a quantization of the energy levels causing a drastic modification of the allowed density of electronic states (Figure 1.2). In this manner, the continuum 1.2. Quantum Dots 5 FIGURE 1.2: The potential of quantum dots can be exemplified through an adapted version of the logo of my research group. As the QD size is reduced, the edge states of the band structures becomes discretized and causes an increase in the band gap and thus in the emission energy. of electronic energy levels partially recover the discrete nature of the atomic constituents of the semiconductor. Since the first demonstration of particles sized small enough as to alter its optoelectronic properties in the 80ies [28, 29], an entire research field on the properties of the so called quantum dots (QDs) has emerged and has led to relevant applications that have been commercialized such as quantum dot-based displays [30, 31, 32]. The enormous potential of this effect was acknowledged last year, as the Nobel Prize of Chemistry was awarded to the development of QDs [1]. While a vast amount of knowledge on the electronic properties, synthetic strategies and additional functionalization have been gathered ever since the first reports of QDs, [33] a number of fundamental questions remain to be fully understood such as the presence of PL intermittency in these systems [34] that compromise their use in some applications. Nevertheless, QDs remain one of the materials with the bigger potential for applications in the near future ranging from quantum information [35] to biomedical imaging [36] or sensing and detection applications [37, 38]. Herein, a fundamental mechanism to consider is the charge extraction or injection into the QDs. This requires inevitably an interaction between the QDs, which will affect the charge dynamics within the QD ensembles. This thesis aims to provide a detailed description of the effects of these interactions on the macroscopic 6Chapter 1. General Introduction optoelectronic behaviour. To do so, we proceed to introduce the main material system studied in this work. The most widespread configuration of QDs is that comprising combinations of elements from the III-V and II-VI groups from the periodic table of elements, each of them with a limited optical bandgap range. Over the past two decades, new QD compositions with novel properties have emerged such as ternary QDs, silicon, carbon dots, QDs made from 2D-materials, etc. Very recently, an exciting versatile alternative has emerged in the shape of high quality halide perovskite-based quantum dots [39, 40]. We will next give a brief introduction to this new class of materials and why its combination with quantum confinement can provide an exciting route towards optoelectronic applications. 1.3 Halide Perovskites “One shouldn’t work on semiconductors, that is a filthy mess; who knows whether any semiconductors exist.” - When Wolfgang Pauli was making this statement in 1931, he was surely not thinking of halide perovskites. Among recent semiconductor material platforms, halide perovskite are often seen as the most messy ones. Nevertheless, they may show outstanding optoelectronic properties, which we will briefly address herein. Halide perovskites are a material system with the crystalline lattice structure ABX3, where A corresponds to a cation such as organic molecules (mostly methylammonium (MA) or formamidinium (FA)), Cs or Rb, B is a metal (typically Pb or Sn) and X is a halide anion (I, Br, Cl or mixtures of them). While halide perovskites have been known since the 70ies[41], their optoelectronic potential remained unstudied until the 90s [42, 43]. The true potential of this material was unveiled when the composition MAPbI3was first placed in a photovoltaic (PV) device by the group of Miyasaka in 2009 [44]. The exceptional optoelectronic properties of this material such as its defect tolerance [45], high charge carrier mobility [46] and the consequent large diffusion [47] length, the large absorption coefficient tunable throughout the visible [48], together with the possibility of manufacturing it by solution process methods soon led to the incorporation of this material in the research activity of many laboratories around the world. In less than a decade, the material parameter space was exploited extensively and compositional engineering further boosted the performance of devices based on these materials [49]. As halide perovskites are an easily processable family of materials, employed deposition techniques have been extended from lab-scale approaches such as spin coating to more industrial scale processes such 1.3. Halide Perovskites 7 FIGURE 1.3: The remarkable evolution of perovskite technology can be well appreciated in the efficiency chart of best performing solar cell devices certified by NREL. For simplicity, only the evolution of mass market Si technology and emerging perovskite devices are shown, evidencing an unprecedented rise in efficiency in photovoltaic technology. Taken from [56] as dip [50], blade [51] or spray coating [52] . Further, the possibility of processing this material by fabrication techniques already employed in the industry, such as vacuum evaporation [53], has been also eagerly explored. Tandem devices with bandgaps that allow to harvest different parts of the solar spectrum where designed [54, 55] and are nowadays among the most efficient PV devices. These combined efforts have led to an unprecedented rise in PV device efficiency for this material reaching over 26% in single junction perovskite devices and exceeding 33% in pervoskite-silicon tandem devices that benefit from the possibility of bandgap engineering through composition [56]. While for PV applications most record devices demand the use of thin film configurations, the evolution of perovskite technology is not only limited to thin films, but alternative morphologies have soon emerged as a means to boost specific properties of this material [57]. Single crystals have been used both to achieve a more fundamental understanding of material properties [58] as well as for particular applications such as radiation detection. For example, monocrystalline, centimeter-sized samples are usually employed for X-Ray detection applications [59]. Alternative approaches that focus on more specific properties such as the unique lattice softness to engineer nanofluidic memristors [60] that can be used to mimic neurons [61] have recently emerged and are expected to 8Chapter 1. General Introduction further pave the development of this material platform. Within the broad spectrum of perovskite configurations beyond conventional thin films, morphologies that facilitate carrier recombination such as two-dimensional perovskites [62] or perovskite QDs [39, 40] have attracted particular interest due to their enhanced emission and absorption properties, their structural stability as well as the presence of emerging collective phenomena such as superfluorescence [63]. In this thesis, we focus in particular on perovksite QD ensembles, as they have the potential to compete with their thin-flim counterpart, as evidenced for instance in PV [64] and light emission [65] , or even show complete new phenomena such as superlattices that exhibit superradiance [63, 66]. 1.4 Quantum-confined Perovskite Nanocrystals Confined perovskite structures with lateral dimensions of few to tens of nanometers were first reported some years after the emergence of halide perovskites for optoelectronic applications with high initial photoluminescence emission yields of ∼20% [67] that rocketed to values above ∼80% [68] in just one year. Nowadays, light emission devices with external emission efficiencies exceeding 23 % can be readily fabricated on a large scale [65] (Figure 1.4d). In commonly employed perovskite nanocrystals (NCs), the particle size is similar to the exciton Bohr radius and the electronic structure is not determined by the infinite periodic structure of the material that leads to a band structure but is rather defined by discrete, quantum confined states such that NCs can be considered as QDs. With respect to conventional III-V and II-VI QDs, pervoskite QDs offer unique advantages such as a continuously tunable bandgap throughout the visible spectrum by means of compositional engineering [69] (Figure 1.4a,b), enhanced optoelectronic properties due to the above mentioned defect tolerance as well as extremely narrow emission linewidths. The first reports presented QDs synthesized through a colloidal approach that lead to a large polydispersivity and shape heterogeneity, such that during the past years much effort has been placed on developing synthetic methods that allow a precise control on size and shape dispersion in perovskite NCs [39, 40]. This has led to the recent development of high quality, monodisperse pervoskite QDs [70] (Figure 1.4b) that enables strong excitonic transitions and which will be employed during certains chapters within this thesis. The QDs mentioned above are synthesized through a ligand assisted colloidal 1.4. Quantum-confined Perovskite Nanocrystals 9 FIGURE 1.4: Perovskite QDs allow a compositional tuning of the emission energy throughout the bandgap (a)-(b). (a) shows dispersions of QDs excited with UV light with the emission spectra displayed in (b) Adapted from [69] used under CC-BY license. (c) Low size dispersion QDs are synthesizable as illustrated by a HAADFSTEM image of CsPbBr3QDs. Reproduced with permission from [70]. (d) Bright FA0·9 GA0·1PbBr3QD LEDs exceeding 23% of external quantum efficiency (EQE) represent current state of the art light emission devices. Reproduced with permission from [65]. synthesis, which represent the most widespread approach within the field of QDs. Their properties will always be affected by some surface capping layer. Precisely, ligands play a crucial role at different levels: they control crystal growth, affect the optoelectronic properties of the QDs (i.e. by passivating surface defects), contribute to colloidal stability when kept in suspension and determine the electronic interaction among them when deposited in the shape of a QD film. The latter is a critical issue as films comprise the active element of any optoelectronic based on QDS, and a proper electronic coupling among them will determine the possibility of charge injection/extraction, which is at the heart of any optoelectronic application. As an alternative to this ligand-dominated system, mesoporous structures can be used as nano-reactors to confine perovskite crystal growth [71, 67, 72, 73, 74, 75]. In this approach, the void spaces in a host matrix of an insulating material is used to limit the size of individual crystals and prevent their aggregation. This method brings several advantages, such as the absence for additional purification steps during the synthesis, a large versatility in attaining different shapes as well as a strict control on the pore size distribution [76]. Importantly, the need to employ ligands to prevent aggregation is removed, enabling connectivity between QDs and hence electrical transport [77, 78]. As mentioned above, an efficient charge injection and/or extraction is crucial for device operation such that the removal of interparticle components is an important advantage in this approach. 10 Chapter 1. General Introduction The control of such connectivity and understanding how it influences the recombination of charge carriers within QD films will be addressed within this thesis. 1.5 Scope of this Thesis The main goal of this thesis deals with the last term of its title that has not been described explicitly so far in the introduction: Connectivity. As mentioned in the previous section, electronic coupling is crucial for the efficient use of QDs as active materials in optoelectronic devices as it enables injection and extraction of charge carriers. This may be at the expense of a loss of confinement, such that a compromise between connectivity and confinement needs to be established for every application. The characterization of connectivity in QD films is, however, nontrivial due to the involved charge propagation length scales that are much smaller than optical resolution. Further, charge propagation, in the case of colloidal QD networks, will inevitably be strongly influenced by the isolating effects of capping ligands. Systematic detailed studies on the effect of connectivity in quantum dot networks are missing. This thesis aims to fill this gap by providing a generalizable case study of the effects of connectivity and other types of coupling between excited states in QD networks. It employs, on the one hand, a ligand-free material system, where - in contrast to colloidal approaches - size confinement and interparticle distances can be widely tuned through the synthesis conditions and geometry of the host scaffold. On the other hand, colloidal QDs with excellent excitonic properties will be used to characterize the effect of dipole-dominated interaction processes. Precisely, the study is conceived to cover all relevant optoelectronic properties that need to be considered to implement QDs in a device, either photovoltaic or light emitting. To do so, macroscopic time-resolved spectroscopies are employed to study the effects of connectivity on the charge carrier dynamics. An analytical method, based on several time-resolved spectroscopy techniques, is developed to systemically identify contributions from connected or isolated QDs, characteristic data patterns being attained for the different degrees of connectivity analyzed. These patterns are then employed to study in detail the effect on other macroscopic optoelectronic properties. Especially, we aim at elucidating the role of the defect landscape in QD 1.6. Outline 11 networks, since controlling the impact of defects has been proven to be fundamental to achieve high quality perovskite-based optoelectronic devices. Further, in order to provide a complete picture comprising the entire time range from charge generation to its final recombination, we study the carrier thermalization and cooling dynamics and its dependence on electronic coupling through transient absorption spectroscopy. This will allow to provide an insight into the coupling dynamics between excitons and free carriers in this material system. We then turn to a truly exciton-dominated colloidal QD solid, where different types of dipole-assisted interactions between electronically isolated emitters are explored. Here, we analyse how dipole-dipole interactions as well as strong coupling of excitonic modes in perovskite QDs with optical modes of microcavities can be employed to mimic the role of electronic coupling. 1.6 Outline Chapter 2 provides a general description of the material systems under study in this thesis: ligand-free and colloidal halide perovskite QDs. Next, the relevant physical processes involved in charge carrier recombination in semiconductors are reviewed. Finally, a general overview of the experimental techniques employed throughout the thesis is given. Chapter 3 introduces a novel method to analyse time-resolved spectroscopic measurements that allows to discriminate between local and global charge carrier dynamics in canonical an non-canoncial material systems. Based on the Shockley-Read-Hall model for time resolved photoluminescence (TRPL) data sets, a theoretical description of different charge carrier recombination processes and their characteristic features in a derivative-amplitude data representation is provided. As a proof of concept, this derivative-amplitude representation is applied to experimental data of different systems. Finally, a generalization to other time-resolved measurements is given. Chapter 4 deals with the effect of connectivity on charge carrier transport, trapping by defect states and recombination dynamics in QD solids. The effect on macroscopic optoelectronic parameters such as emission efficiency, emission linewidth and multiparticle interactions is studied. Finally, the role of the defect 18 Chapter 2. Conceptual Background As mentioned in Section 1.4, QD synthesis within mesoporous matrices present a series of advantages with respect to colloidal synthetic approaches. In the course of the past seven years, the Multifunctional Optical Materials Group has specialized on this alternative synthesis method that leads to ligand-free perovskite QDs embedded in porous host matrices made out of inorganic nanospheres (namely SiO2and TiO2). Herein, the fabrication of this material system is briefly introduced (Figure 2.2). Preparation of SiO2Porous Scaffolds The porous matrices are prepared by means of dip-coating. In a first step, commercially available SiO2nano-sized spheres of 30nm diameter are dissolved in methanol with a concentration of 3% w/v. A low-fluorescence glass substrate was immersed into the solution with a 120mm/min withdrawal speed to dip-coat the SiO2spheres onto the substrate. This process was repeated 15 times to achieve a porous scaffold of roughly 1µm thickness, which was subsequently annealed at 150 °C for 30min to improve its mechanical stability and evaporate any organic component remaining within the matrix. Infiltration of FAPbBr3Quantum Dots The perovskite composition under consideration for these samples was formamidinium lead bromide (FAPbBr3) constituting a configuration that presents enhanced chemical stability and, when prepared in the shape of QDs, elevated emission quantum yield [84]. A perovskite solution precursor was prepared by dissolving FABr and PbBr2powders in a 1:1 molar ratio in dimethyl sulfoxide (DMSO). The precursor concentration with respect to DMSO was used to determine the amount of material to be grown within the SiO2host matrices and was varied between 5% and 30% w/w for the samples employed within this thesis. The precursor solution was deposited into the voids of the SiO2host matrix through spin-coating at 5000rpm for 60s, followed by an annealing process on a hot plate at 100°C. All perovskite deposition steps where performed in an inert N2environment within a glovebox. 2.1.4 Structural Characterization of embedded Perovskite Nanocrystals Part of this section is published in the supporting information of the journal article "Tiede, David O., et al. Effect of connectivity on the carrier transport and recombination 2.1. Quantum Dot Ensembles 19 FIGURE 2.2: Fabrication of perovskite quantum dots in porous matrices. (a) Sketch of the dip coating process employed to fabricate porous SiO2host matrices. (b) Sketch of the spin coating process that leads to crystalization of nanometer-sized FAPbBr3 crystals within the voids of the host matrix. Reproduced from [85] used under CC-BY license. dynamics of perovskite quantum dot networks", ACS Nano 18 (2024) 2325-2334. To study the photophysical properties of the ligand free QD solids as a function of the QD load and associate macroscopic properties with its microscopic origin, a structural characterization is crucial. Due the insulating nature of the SiO2scaffold and the presence of organic cations, typically present in lead halide pervoskites, imaging of the crystalline structure and distribution of ligand-free QDs via transmission electron microscopy is not trivial [86]. Further, in order to obtain information on the internal QD distribution, cross section images need to be taken. This, however, needs to be done with great care, as cutting the sample exposes the QDs to the environment and may cause its rapid degradation. While other members of my research group in cooperation with collaborators specialized in high resolution transmission electron microscopy (HRTEM) were able to record HRTEM images that evidenced the high crystalline quality of the synthesized QDs (Figure 2.3a), a statistical analysis of their size and spatial distribution is non-trivial. X-ray diffraction (XRD) measurements confirm no substantial changes in the crystallographic structure as the QD load is varied (Figure 2.3c) [86]. Based on this, the need for an alternative size and inter-particle distance estimation emerges. Since confined QDs are of particular interest because of the quantum-confinement enhanced optoelectronic properties they present, we focus on QDs small enough to observe absorption and emission properties characteristic of quantum-confinement, and employ optical characterization to estimate particle sizes. 20 Chapter 2. Conceptual Background FIGURE 2.3: Structural characterization of FAPbBr3QDs in porous scaffolds. (a) Sketch of FAPbBr3QD -SiO2film. (b) High-resolution high-angle annular darkfield scanning transmission electron microscopy (HAADF-STEM) micrograph of a FAPbBr3QD-SiO2film lamella with a particle size around 8nm. Inset shows the fast Fourier transform (FFT) image from the particle. (c) Normalized X-ray diffraction (XRD) diffractograms of FAPbBr3 QD@SiO2 films with different crystal sizes determined utilizing the Brus equation (8.7 olive, 10.2 light green, 11.0 yellow, 13.3nm size orange) and bulk film (red). (d) TEM micrograph from a lamella of a 1000 nm thick scaffold containing FAPbBr3QD within the pores. In this image, perovskite nanocrystals are distinguished as dark regions whereas 30 nm size SiO2nanoparticles appear in pale gray and stratified as thin layers. Reproduced from [86] used under CC-BY license. Average perovskite QD sizes for the different samples under study were estimated from photoluminescence emission energies by using the Brus equation [29], an approach that has proven to yield a good estimate in the past for similar samples [87]. We assume electron and hole effective mass to be identical, such that Equation (2.5) simplifies to : Eg,nano =Eg,bulk +h2 8m∗R2−1.786e2 4πϵ0ϵrR(2.6) where the values for effective mass (m∗) and dielectric constant (ϵr) were taken from reference [88] and the bulk bandgap energy Eg,bulk estimated from PL measurements of reference bulk samples. This equation assumes a fixed set of parameters for the effective mass of the electron m∗ e, hole m∗ has well as for the dielectric constant and assumes this set to remain unchanged under confinement. Further, the estimation of the bulk bandgap from the emission energy contains 2.1. Quantum Dot Ensembles 21 TABLE 2.1: Fill fraction, estimated average QD radius and interparticle spacing extracted from porosimetry and optical measurements Precursor concentration Cprec Fill fraction f f Estimated ⟨r⟩ # of QDs per m3dcenter dinterp 5% 0.0141 4.5nm 3.7 ×1022 m−337nm 28nm 10% 0.0405 5.7nm 5.2 ×1022 m−333nm 22nm 20% 0.092 6.26nm 9 ×1022 m−328nm 15nm 30% 0.145 6.67nm 8 ×1023 m−325nm 12nm uncertainties since the effect of an excitonic transition is not considered. In addition, possible effects from size distribution and charge funneling effects and modifications in the electronic band structure through e.g. strain are not reflected in Equation (2.6). Despite the uncertainties introduced by the mentioned effects, we employ the Brus equation to get an estimation of average QD sizes to obtain a figure of comparison between the different samples rather than to obtain an exact value for the actual size and its distribution. The estimated average sizes are shown in Table 2.1. To estimate the interparticle spacing between QDs, we assume the particles to be equally spaced within the host volume. In that case, in a volume Vfilled with N particles, the average volume vthat corresponds to one particle is given by v=V N(2.7) where, assuming spherical particles, the volume can be expressed as v=4 3πr3=V N(2.8) To obtain the average distance between particle centers, the previous equation can be rewritten as: dcenter =2r=23 r3V 4πN(2.9) , where the number of particles per volume unit present in the matrix can be expressed as: v=v V·f f (2.10) , where f f presents the perovskite filling fraction. The latter was estimated by 22 Chapter 2. Conceptual Background means of porosimetry measurements. The average void distance between particles can consequently be estimated as dinterp =d−2⟨r⟩(2.11) From this estimation, the values shown in Table 2.1 can be extracted. It becomes evident that the average particle size is only changing by a factor of 1.5, while the interparticle spacing is more than doubled. In an earlier work, we showed that nanocrystal formation in porous matrices includes both regimes where it is dominated by liquid-solid phase transitions, responsible for the creation of nucleation sites, as well as diffusion-controlled crystal growth, responsible for the overall crystal growth [89]. A variation of the precursor concentration Cprec will affect both growth regimes, such that a change in Cprec is expected to be reflected both in dinterp as well as in ⟨r⟩. 2.2 Charge Carrier Recombination in Perovskites Part of this section is published in the book chapter "Light emission of halide perovskites" in the book "Halide Perovskite Semiconductors: Structures, Characterization, Properties, and Phenomena". To understand the macroscopic behaviour of optically active materials, one needs to understand the dynamic processes that lead to light emission and those that hamper it. In a semiconductor, the emission of light is dictated by the way charge carriers recombine once they have been promoted to an excited state, mostly through optical or electrical excitation. In this section, the charge carrier recombination processes involving electrons and holes at the bottom (top) of the conduction (valence) band are examined. Recombination can take place not only by radiative means, where a photon is emitted, but also through processes involving trap-states associated with structural defects or interactions with other carriers. In order to study the recombination of charge carriers, one usually resorts to differential equations employing rate constants of all the processes involved. The most widespread model for charge carrier recombination is the Shockely - Read Hall model [4, 3], where the populations of electrons ne, holes nh 2.2. Charge Carrier Recombination in Perovskites 23 and trapped charges nTcan be described with the following model: dne dt =−Rpop(NT−nT)ne+RdetrapnT−Rehnenh(2.12) dnT dt = +Rpop(NT−nT)ne−RdetrapnT−RdepopnTnh(2.13) dnh dt =−RdepopnTnh−Rehnenh(2.14) In this model, where a total density of traps NTis considered, a charge trapping term Rpop(NT−nT)nedescribes the rate of trapping with the rate constant Rpop and the gradient between trapped/free carrier populations, RdetrapnTdescribes the detrapping probability that is proportional to the detrapping rate Rdetrap and the number of filled traps nT.Rehnenhaccounts for a bimolecular recombination Reh of electrons and holes that leads to the emission of a photon. Finally, RdepopnTnhaccounts for the nonradiative decay of a trapped charge with its counterparticle with the corresponding rate Rdepop. Note that this model only accounts for defect assisted and electron - hole recombination. Variations induced through multiparticle processes such as Auger recombination, charge redistribution effects like diffusion or non-equilibrium processes such as doping effects are not considered in this simplistic model. To be able to discuss the different types of recombination in detail, we can take a number of assumptions for the general case: primary photogenerated species are free carriers (as expected for perovskites taking into account the reported values for exciton binding energies [90]), the photogenerated carrier density is larger than the intrinsic one and the densities of photogenerated electrons and holes are similar. Under these assumptions, which are reasonable when studying halide perovskite, [91] the above rate equations can be reduced to the simplified expression commonly known as the ABC model: [92] dn dt =−An −Bn2−Cn3(2.15) The above expression, albeit highly simplified, is illustrative of the main recombination processes taking place in halide perovskites and can be used to highlight some of the peculiarities behind the photophysics of this material. The ABC model employs three terms, that summarize the probability of monomolecular (one-particle, Figure 2.4a), bimolecular (two-particle, Figure 2.4b) and trimolecular (three-particle, Figure 2.4c) processes to happen. In a general picture, electronhole recombination is a two-particle process including an electron and an electron 24 Chapter 2. Conceptual Background FIGURE 2.4: Illustration of recombination processes in the ABC model. (a) Defectinduced trapped charge carrier assisted recombination, where the trapping of one charge carrier results in an effective one-particle dependence. (b) Radiative bimolecular recombination, where an electron recombines with a hole and the excess energy is emitted as a photon. (c) Trimolecular Auger process. An electron and a hole recombine in a non-radiative way and the excess energy is transferred to excite a nearby charge carrier. hole. However, under certain conditions, this particle dependency can be varied, as elaborated in the following. We will first focus on bulk-like free carrier dominated systems and in a second step discuss the variations in confined, exciton dominated systems. 2.2.1 Monomolecular Recombination The monomolecular rate constant A includes processes where the recombination only depends on electron or hole populations, which comprise trap assisted recombination where localized defect states interacting with either holes or electrons are considered (Figure 2.4a). These processes are particularly important at low excitation regimes commonly attained under operation conditions in photovoltaic devices. In the case of electron traps and considering a fast electron trapping rate ktrap such that the electron density equals the trapped electron density (ne≈NT(1−f), where NTis the total trap density and fthe probability of a trap state to be occupied), the recombination rate can be described as: R=ktrapneNT(1−f) = Ane(2.16) ,where neis the electron density, fis the probability of a trap state to be occupied and ktrap the capture coefficient. ktrapNT(1−f)can be considered as constant, if the transition from electrons to trap states is much faster than the recombination from trap states with holes. In this case, the recombination rate only 2.2. Charge Carrier Recombination in Perovskites 25 depends on the density of holes in the material, and the charge carrier density becomes particle-dependent and thus monomolecular. Such charge carrier trapping is found to happen mainly via a phonon-assisted transfer of free carriers to electronic trap states. The energetic position of the involved trap states determines the accessibility of those states for free charge carriers. The charge trapping constant can be described with ktrap =σtrapvthNT(2.17) where σtrap and vth represent the capture cross section and the thermal velocity of the free carriers. In a classical picture, σtrap is dependent on the allowed phonon modes that permit a transition between a free carrier and a trap state energy level. vth is correlated with the thermal energy of the system via Eth =1 2me f f v2 th =3 2kBT(2.18) vth =s3kBT me f f (2.19) where me f f represents the effective mass of the corresponding charge carrier. In thin film perovskites me f f has been reported to be in the range 0.2 −0.5me[90]. Finally, NTdescribes the ensemble of trap state energy levels, where the energetic position of the defects tates associated with individual traps within the bandgap affects the effective σtrap. A transfer to trap states via phonon emission is thus more likely and faster if the energy difference between the band edge and the trap state matches the available phonon modes in the lattice and thus depends both on available phonon modes and the energy levels of the trap states. Further, the energetic position of traps states determines their influence on recombination dynamics. Trap states lying deep inside the conduction or valence band will not affect the way charge carriers recombine. Trap states close to the conduction band minimum (CBM) or valence band maximum (VBM) with an energy difference to the CBM (VBM) smaller than the thermal energy kBTare easily ionized and act as shallow traps, that do not lead to nonradiative recombination but will induce doping. Trap states with an energy difference larger than kBTare labeled deep traps. Recombination from those sites occur only in a non-radiative way via vibrational decay paths where electron and electron hole recombine in the trap center. This non-radiative recombination via multiple phonon emission was first proposed by Shockley, Read and Hall in 1952 and is commonly referred 26 Chapter 2. Conceptual Background as Shockely-Read-Hall (SRH) Recombination [4, 3]. Perovskites are solution processed materials and possess a soft crystal structure in comparison with conventional inorganic semiconductors commonly used in optoelectronic devices such as Si or GaAs and this causes an intrinsically high defect density. In spite of this, halide perovskites can behave as good emitters as a result of the prevalence of shallow traps and only few deep traps within their bandgap. Point defects of different nature comprising interstitials, vacancies and antisites of standard components have been considered and surprisingly most of them have been identified as shallow traps with mainly halide related defects associated with deep traps (see for instance specific examples for MAPbI3and MAPbBr3in Figure 2.5) [93]. In particular, the role of halide interstitials and halide vacancies and their combined presence as Frenkel pairs have been pointed out theoretically to cause electron deep or shallow traps, depending on the exact employed halide and the charge state of the defect. The intra-band or shallow character of most of the remaining defects explains the high defect tolerance of halide perovskite semiconductors that turn them into a solution processed material with outstanding optoelectronic properties. We also have to note that another possible origin for the defect tolerance is the formation of charge polarons in halide perovskites [94], which has been proposed to provide a shielding mechanism against interaction with traps and other carriers [95]. 2.2.2 Bimolecular Recombination Bimolecular processes are those dominating charge carrier recombination at intermediate excited charge carrier densities. In halide perovskite materials, they are mainly attributed to direct band to band recombination (Figure 2.4b), between an electron at the bottom on the conduction band and a hole at the top of the valence band, and can be described by the recombination rate: R=Bnenh=Bn2if ne=nh(2.20) , where Bis the bimolecular recombination coefficient and ne(nh) the electron (hole) density. Upon excitation through electrical, optical or other means, the density of electrons and holes can be considered identical and the presence of intrinsic charges neglected, so that ne=nh. In a direct bandgap semiconductor such as lead halide perovskites, most of the recombination is expected to occur radiatively via 2.2. Charge Carrier Recombination in Perovskites 27 FIGURE 2.5: Density functional theory (DFT) calculations of different point defects in (a) MAPbI3 and (b) MAPbBr3. Energy levels that are close or in the conduction /valence band can be considered as shallow traps. Trap states in the band gap are considered as deep traps. Reproduced from [96] used under CC-BY license. emission of a photon. However, some thermal recombination via lattice vibrations can also contribute to the band to band charge carrier recombination, such that the coefficient Bhas to be extended to R=Bn2= (Brad +Bnon−rad)n2(2.21) In a direct semiconductor like halide perovskites, the radiative term can be expected to be much faster than the nonradiative recombination as the latter would involve several phonons thus reducing its probability. Nevertheless, in halide perovskites, the experimental recombination lifetimes are much longer than those expected from a pure direct bandgap semiconductor with a similar electronic gap, where the recombination lifetime can be estimated from the refractive index through the van Roosbroeck–Shockley relation [97]. Generally speaking, a fluorescence process can be slowed down, if the charges involved in the recombination are trapped or separated spatially, or its transition is forbidden momentum- or spinwise. Based on this, several mechanisms have been proposed to account for the prolonged lifetime, which are currently under debate (Figure 2.6). Beyond the 34 Chapter 2. Conceptual Background FIGURE 2.9: Transient Absorption Spectroscopy . Sketch of the setup used to perform transient absorption measurements. The light beam of an ultrafast laser is splitted in two, where one component is used to generate white light while the other one is used to generate a tunable excitation pulse. Both pulses are focused on a sample to excite (pump) and measure (probe) the sample. A delay line is employed to change the pump-probe delay such that differential absorption spectra can be obtained as a function of delay time. spectrum is recorded, such that a two-dimensional data set resolved in transmission wavelength and time delay with respect to the excitation pulse is created. The measurement amplitude is expressed as a change in absorption ∆A. As a change in the absorption requires an intense change in the charge carrier population, transient absorption spectroscopy cannot be used for low excitation density measurements. In this thesis, transient absorption measurements is thus mainly employed to measure multiparticle (bimolecular and trimolecular) recombination processes. To illustrate how these processes can be extracted from the ∆A, the main expected features from a TAS spectrum are shown in Figure 4.6. They consist of: 2.3. Experimental Measurement Techniques 35 FIGURE 2.10: Contributions to ∆Aspectra. (a) 2D False color map of ∆Aspectra as a function of delay time between pump and probe pulse of a FAPbBr3QD solid in a porous matrix. (b) ∆Aspectrum at 1ps. Shaded areas sketch the main contributions to the overall spectrum. • Ground state bleach (GSB). This feature can be associated to the population of photo-excited charge carriers residing at the band edge of the conduction and valence bands. As these excited carriers occupy electronic transitions that could otherwise serve as absorption events, the absorption is bleached at this energy. • Photobleach (PB). It has a similar origin than the GSB but originates from charge carriers with an energy higher than the band gap energy [111]. This feature corresponds to charge carriers that have not fully cooled to the ground state energy upon an excitation event. • Photoinduced absorption (PIA). This feature corresponds to a lower transmission caused by additional absorption pathways. It can have several origins such as interaction with already excited charge carriers that allow additional intraband transitions [112] or a change in refractive index [113]. Additional features such as band gap renormalization processes [114, 113] , additional signal due to stimulated emission [115] or excitation field induced processes such as the optical stark effect [116] can further contribute to the convoluted ∆A spectrum. In this thesis, we focus in particular on the evolution of the GSB signal, as it can be directly correlated to the evolution of the charge carrier density in the material. Also, we will analyse the evolution of the convoluted GSB and PB signal, as it provides information on the thermalization and cooling processes of excited "hot" charge carriers [114]. 36 Chapter 2. Conceptual Background FIGURE 2.11: Relation between PLQY and recombination constants. (a) Evolution of the charge carrier density extracted from fluence dependent TA measurement with a fit to the ABC model (grey lines, fit values indicated in the graph). The GSB amplitude is converted to the charge carrier density by normalizing with the initially injected charge carrier density calculated from excitation fluence and absorption cross section. (b) PLQY curve calculated from the recombinations constants in (a) (grey line) and measured with a photodiode (red points). Colored lines represent the recombination contribution of k1,k2and k3to the overall charge carrier recombination dn dt . Note that k2is dominating over k1and k2when the PLQY is highest. 2.3.3 Fluence dependent Photoluminescence Quantum Yield For optoelectronic applications, the share of radiative over the total recombination processes is a fundamental figure of merit to characterize a semiconductor and is described as the photoluminescence quantum yield (PLQY). Importantly, in semiconductors, the radiative and non-radiative recombination rates are charge carrier density dependent, such that a meaningful comparison of PLQY values of different samples can only be performed when being reported as a function of charge carrier density. In order to measure the fluence dependent PLQY throughout this thesis, the samples where excited with a pulsed, 190fs excitation source (Pharos from LightConversion) that allows to optically pump the sample in a large dynamic range without degrading it. The emission signal was collected with a fiber coupled spectrometer (Flame-T from OceanInsight) or a photodiode (PDM from MPD). Calibrated neutral density filters and variations in the integration time were used to adapt the signal amplitude to the dynamic range of the detector. The detected signal was corrected by the attenuation and divided by the excitation fluence to obtain the relative fluence dependent PLQY. The PLQY is directly correlated to the recombination constants as it comprises the 2.3. Experimental Measurement Techniques 37 ratio of radiative recombination Γrad over all recombination processes Γtot happening in the material system [117, 118]. It reads: PLQY(n) = Γrad Γrad +Γnonrad =Bn2 An +Bn2+Cn3(2.23) The sum of radiative and nonradiative recombination processes can be extracted from the charge carrier recombination dynamics and thus from the evolution of the GSB through the ABC-model introduced in Equation (2.15) (Figure 2.11a). The PLQY fluence dependence can thus be reconstructed from recombination constants extracted from TA measurements and be compared to directly measured values (Figure 2.11). It yields a bell-shaped dependence, where optimum PLQY is achieved at intermediate charge carrier densities when bimolecular processes (k2) are dominating. Note that a perfect overlap of the PLQY dependencies is not achieved, as the ABC model relies on a simplified description of charge carrier recombination processes and more complex contributions such as density dependent recombination constants kimight affect the actual measurement. Nonetheless, beyond extracting the same information in two independent measurement approaches, this representation also exemplifies the contributions of monomolecular, bimolecular and trimolecular recombination to the overall PLQY at a given charge carrier density. In a similar manner, the fluence dependent PLQY can also be defined for an excitonic system, where excitonic recombinations Aexnex can be assumed to be mainly radiative, while exciton-exciton annihilation Bexn2 ex is assumed to be nonradiative. The PLQY can thus be expressed as: PLQYex(n) = Γrad Γrad +Γnonrad =Aexnex Aexnex +Bexn2 ex (2.24) Here, it becomes evident that the PLQY will only decrease as the charge carrier density is increased, since the relative weight of radiative recombination is highest at low densities. Please note that within this description, Aexnex describes monomolecular recombination that can also possess a non-radiative, defect assisted contribution, such that the PLQY at low fluences will not yield 1. The relative evolution is, however, expected to be well approximated with . The PLQY fluence dependent representation, that will be employed throughout 38 Chapter 2. Conceptual Background FIGURE 2.12: Effects of recombination constants on PLQY fluence dependence. (a)- (b) Simulated PLQY and recombination rates within an ABC description. Rate constants are chosen similar to the expected ones for MAPbI3(a) Variation of the monomolecular contribution An by two orders of magnitude. A reduction on An causes a shift of the PLQY maximum towards lower fluences. (b) Variation of the Auger rate Cn3. A reduction of Cn3causes a PLQY shift towards higher fluences. (c) Simulation of PLQY within an excitonic AB description. A variation of the biexciton contribution Bexn2 ex changes the onset of the PLQY decrease. this thesis, can serve as a visual representation that yields information on the performance of a material in a given operational regime. To illustrate its potential, Figure 2.12 depicts simulated PLQY curves, where recombination constants are varied. In this thesis, we will explore material systems, where the amount of accessible defect states and thus the mono-molecular contribution will be varied. Figure 2.12a illustrates how a reduction in defect-assisted decay rates causes a shift of the PLQY curve towards lower fluences, such that the material will be more suited for low fluence applications, such as PV. On the other hand, a variation on the Auger rate C, which will also be treated within this thesis, causes a shift of the PLQY curves towards higher fluences (Figure 2.12b) and results thus in a system that is more suited for high intensity applications such as light emission. Finally, a decrease of the multiparticle effects in an exciton dominated scenario leads to a delayed onset of the PLQY reduction at elevated fluences (Figure 2.12, evidencing that exciton dominated systems work best at low excitation fluences. 2.4 Conclusion In this chapter, we have introduced the material systems that will be studied within this thesis and have emphasized on synthesis processes that can affect the final optoelectronic macroscopic behavior. We have given an insight into charge carrier recombination mechanisms in different sample configurations, that will 2.4. Conclusion 39 allow to better place into context the results of the following chapters. Finally we have presented the main aspects of the measurement techniques employed herein and have introduced principal analysis methods that will be relevant for this thesis. Here, the following chapters will further extend on this general vision of employed methods when necessary. This chapter provides, however, the fundamental basis to understand and contextualize the main findings of this thesis. 41 Chapter 3 Extraction of Recombination Constants from Non-Canonical Systems In this chapter, we theoretically and experimentally analyse charge carrier recombination processes of complex material systems that do not obey canonical charge carrier recombination models. We introduce a novel data representation method that allows to discriminate between global and local processes. Theoretical descriptions on charge carrier recombination processes of a canonical Shockley-Read-Hall (SRH) dominated material system are provided and confirmed experimentally in a MAPbI3thin film. In non-canonical systems, the introduced representation method allows to discriminate between global SRH dominated contributions and more complex, time dependent deviations. The method herein introduced allows to unambiguously extract recombination constants that cannot be extracted through global fittings and to identify contributions originating from local recombination processes. The method can be generalized to other time resolved spectroscopies as exemplary shown on a transient absorption measurement. 42 Chapter 3. Photon Flux Representation 3.1 Introduction Charge carrier dynamics play a crucial role in understanding the macroscopic behaviour of active optoelectronic materials, as they describe the different processes charge carriers can undergo upon photo-excitation. In a general picture, charges can diffuse [47, 119], interact with lattice vibrations [120], change their energy through cooling mechanisms [121], form quasiparticles such as polarons [94], excitons [90] or other multiparticles species [122, 123], get trapped at defect sites [124] or recombine with a particle of opposite charge. These processes can be strongly affected by the device architecture [18] in which the active material is employed and is further strongly morphology dependent [13, 17]. In particular, the recombination of charge carriers is of fundamental importance for device performance as this process ultimately determines whether the charge carriers can be employed for their intended use. A theoretical description of the recombination dynamics is often done with models that describe the change of charge carrier populations with differential rate equations that reproduce the different processes that can take place within a semiconductor. These differential rate equations are based on a description of the change in charge carrier populations dn dt and often a sophisticated modelling needs to be performed to fit these expressions to time resolved data. The analysis throughout this chapter focuses on a simple, but often overlooked, concept: Since the physical parameters of interest are given by the derivative of dn dt rather than by the absolute signal, a more direct analysis can be performed if the derivative of time resolved signals is analysed. Herein, we present a method where time resolved datasets are transformed into derivative datasets and provide theoretical descriptions of how to interpret these datasets. We apply this method in detail to TRPL data in different material configurations and show how non-canonical systems can be analysed with the presented model. Finally, an outlook is provided by applying the method to transient absorption measurements, demonstrating the large versatility of the proposed analysis scheme. 3.1.1 Limitations of the Shockley -Read- Hall model In the case of recombination dynamics in the ns to ms regime, the Shockley-Read- Hall (SRH) model [4, 3] can be employed to describe the variation in electron, hole and trapped carrier populations with the rate equations already mentioned 3.1. Introduction 43 in Chapter 2: dne dt =−Rpop(NT−nT)ne−Rehnenh(3.1) dnT dt = +Rpop(NT−nT)ne−RdepopnTnh(3.2) dnh dt =−RdepopnTnh−Rehnenh(3.3) , where Rpop(NT−nT)nedescribes the filling of trap states through electrons with the rate Rpop,Rehnenhdenotes the electron hole recombination with the rate Reh and RdepopnTnhrepresents the trap depopulation process with rate Rdepop where a trapped electron recombines with a hole. ne,nT,nhdescribe the densities of electrons, trapped charges and holes, whereas NTdenotes the global density of trap states present in the system. Note that in this description, electron traps are considered, but a description with hole traps would lead to a mirrored scenario with identical results. Please note that this description is a very simplistic one, where further additions to the model can be made to account for a more realistic scenario. Among others, intrinsic doping with different concentrations of electrons and holes[125], shallow trap cycling that leads to a slowed down PL [98] or the incorporation of several shallow traps with different energetic positions [126] have been considered. Further, additional effects such as polaron formation or photon recycling [91] can alter the recombination rates described in Equation (3.3) and cause an incomplete description of charge carrier recombination depending on the peculiarities of the sample under investigation. Finally it should be noted that additional processes with explicit time dependence dn dt (n,t), such as carrier diffusion, are not trivial to incorporate in the recombination models. The most commonly used experimental technique to study charge carrier recombination dynamics is time resolved PL (TRPL). Precisely, fluence dependent TRPL measurements can provide insights into population density dependent charge carrier recombination processes, as an increase in excitation fluence leads to faster dynamics as additional recombination paths are introduced. Figure 3.1a represents fluence dependent TRPL decay curves of a generic MAPbI3thin film, whose TRPL decay dynamics can be fitted within the model described in Equation (3.3) [124]. In Figure 3.1, we introduce an alternative way of plotting TRPL data, where the TRPL derivative is plotted against the TRPL amplitude. This kind of representation is inspired by an analysis scheme mainly introduced by the group of 50 Chapter 3. Photon Flux Representation FIGURE 3.3: Extraction protocol for the photon flux representation. (a) Initial PL of TRPL curves at different fluences of a MAPbI3thin film. A linear approximation in the log-log representation allows to scale the PL curves to charge carrier density. (b) Photon flux representation with scaled axis employing the correlation shown in (a) and adjusting the m=1.5 slope (black dashed line) to the higher amplitude tail of the dataset. The m=1 slope (grey dashed line) is consequently fitted to the lower amplitude tail of the dataset. (c) TRPL decay curves with the simulated decay curves extracted from (b). model is performed and compared with the experimental data Figure 3.3c, where only the parameter Rdepop is varied to achieve the best fit. Although no perfect simulation is achieved, the overlap of experimental data and the theoretical decay curves extracted from the photon flux representation is substantial. In fact, an attempt to model this data with a conventional SRH approach did not yield a satisfying fit result due to the possible deviations from the SRH model mentioned in Section 3.1. In such a sample that does not follow theoretical descriptions, the photon flux representation provides a unambiguous extraction method to characterize recombination constants. 3.4 Experimental Application of the Photon Flux Model To demonstrate the versatility of the photon flux model, the analysis technique is applied to several material systems that allows both to consolidate the established finding as well as to extend the analysis to material systems where a conventional fitting to a set of differential equations fails. Further, two approaches are employed to modify the defect landscape of the sample: changing the environmental conditions and thermally deactivating carrier traps. 3.4. Experimental Application of the Photon Flux Model 51 3.4.1 Application in a Canonical Material System As a testbed for a proof of principle study, fluence dependent TRPL decay curves of a conventional MAPbI3thin film were measured at room temperature in vacuum, in humid air and in vacuum at 77K (Figure 3.4). Figure 3.4a shows the photon flux representation of the TRPL data of a MAPbI3thin film in vacuum following the analysis presented in Section 3.3. A clear transition of a monomolecular to a bimolecular emission regime is observed. The extracted electron-hole recombination constant yields Reh =3×10−10 cm3/s. The fit of the monomolecular recombination slope yields RpopNT=15µs−1. Both values are aligned with reported literature values [102, 129, 103, 124] and reproduce to a satisfying degree the reconstructed time resolved decays (Figure 3.4b). Note that the overlap is not perfect, as photoinduced processes and environmental contamination as the sample was transferred to the sample chamber might affect the emission properties, where photo-activation and -bleaching mechanisms can affect the recombination properties during the measurement [130, 91]. Nonetheless, the unambiguous extraction of recombination parameters in Figure 3.4a highlights even more the potential of the photon flux extraction method, as even varying measurement conditions can be compensated. This is particularly relevant for materials that may undergo variations due to measurement conditions such as lead halide perovskites. Next, the sample was exposed to ambient air and stabilized in dark conditions for an extended time period before repeating the fluence dependent TRPL measurement series (Figure 3.4b). A decrease in the amplitude of the slope associated to the monomolecular contributions can be appreciated without the need of any modelling. Figure 3.4b shows an identical Reh, whereas the monomolecular contribution RpopNTdecreases by one order of magnitude. It is well established, that ambient O2is expected to passivate halide defects, such that the reduction in aRpop can be associated to a reduction in the defect concentration NTwith respect to a measurement in vacuum [131, 132]. Again, the parameters from the photon flux representation can be used to qualitatively reproduce the TRPL decay curves (Figure 3.4e). Finally, the sample was measured at 77K in vacuum (Figure 3.4c). Here, a clear reduction in the monomolecular contribution RpopNTand an increase in the bimolecular contribution Reh of roughly one order of magnitude with respect to the 52 Chapter 3. Photon Flux Representation FIGURE 3.4: Photon flux representation in a MAPbI3thin film. (a)-(c) Photon flux representation of the MAPbI3thin film in (a) vacuum, (b) air and (c) vacuum at 77K. (d)-(f) TRPL decay curves of the datasets shown in (a)-(c) with the simulated SRH decays extracted from the photon flux analysis (grey lines). sample in vacuum at room temperature is observed. The change in the trap population contribution can be associated to temperature activated defect sites originating from ionized impurities [133]. In particular, halide related defects have been shown to originate from halide Frenkel pairs [134, 135], whose activation energies were shown both experimentally [136, 137, 138] and theoretically [139, 140] to be well suited to show significantly varying defect densities at different sample temperatures. The increase in the bimolecular recombination constant Reh can be associated to a reduced charge carrier mobility at reduced temperature [133] or a reduction in trap recycling processes [98] due to the reduction of thermally activated shallow defects. While a detailed description of the contribution of each of the mentioned mechanisms is beyond the scope of this chapter, the extracted values coincide with previous reports and a good agreement between simulated and experimentally measured TRPL curves is observed (Figure 3.4d-f). Overall, the photon flux representation provides an analysis route that allows to extract recombination constants without the need to employ sophisticated fitting procedures of differential equations. As a proof of concept, the variation of recombination constants in different environmental and thermal conditions has been tested in a conventional MAPbI3thin film. 3.4. Experimental Application of the Photon Flux Model 53 3.4.2 Application in a Non-Canonical Material System To exploit the full potential of the photon flux representation, we next consider a material system that can not be easily described within a SRH formulation. A QD solid comprising formamidinium lead bromide (FAPbBr3) QDs embedded in a silica porous matrix (see Section 2.1.3) were employed as a testbed system that will be proven in this thesis to show slow charge carrier diffusion. Figure 3.5a-c shows the fluence dependent TRPL decay curves of a FAPbBr3QD sample with 30% precursor concentration - a magnitude that is proportional to the QD load in the porous scaffold. The QDs have an average size of 6.7nm (see Section 2.1.3) that causes weak quantum confinement, such that similar recombination dynamics with respect to a bulk sample are expected. It is anticipated that these samples show limited connectivity inducing a reduced charge carrier diffusion as demonstrated in the analysis in Chapter 4. The fluence dependent TRPL decay curves do not follow the trend observed for the MAPbI3 film with faster dynamics as the fluence is increased. As mentioned in Section 2.1, now TRPL decays seem to become slower and then faster as the fluence is icnreased. Such behaviour cannot be described within the standard SRH model described in Equation (3.3). Nevertheless, important information on the recombination dynamics can be extracted from the photon flux analysis presented in Figure 3.5d-f. The fluence dependent dPL dt are substantially different from those observed for the canonical case presented in Section 3.4.1, as early time deviations (corresponding to the high carrier part of each fluence) are observed for the different fluences. Nonetheless, after a certain decrease in PL intensity, the dPL dt do converge towards a global decay dynamics, that is very similar to what is observed in Section 3.4.1. In this regime, a similar analysis on the recombination regimes as presented in Section 3.4.1 can be performed. The dPL dt can thus be separated in two distinct regimes. (1) An initial one, in which the dynamics depend on the initial excitation density. Such a scenario implies an explicit time ("history") dependence of the charge carrier density evolution dn dt (n,t), which can be explained by a locally varying charge carrier density. (2) As the dPL dt signals evolve, a global trend where the effects induced by the employed initial excitation density have vanished is observed. In this regime no explicit time dependence is observed (dn dt (n,C t)). We first focus on the interpretation of the regime without explicit time dependence before discussing in detail the latter regime. 54 Chapter 3. Photon Flux Representation FIGURE 3.5: Photon flux representation of FAPbBr3QDs with Cprec =30% in a porous matrix. (a)-(c) Fluence dependent TRPL decay curves of the FAPbBr3QD solid in (a) vacuum, (b) air and (c) vacuum at 77K. (d)-(f) Photon flux representation of the datasets shown in (a)-(c) that cannot be adjusted to SRH decays. As the time dependence of TRPL signals is removed in the photon flux representation, it allows to analyse the recombination dynamics at a given PL amplitude without being affected from history dependent effects. Consequently, similar analysis as that proposed in Section 3.4.1 can be performed. In a vacuum environment (Figure 3.5d), both the monomolecular and bimolecular recombination regimes are identified and the recombination constants are extracted. The values for Reh and RpopNTyield values similar to what is expected for bulk FAPbBr3[16]. As the sample is exposed to ambient air, the monomolecular contribution is reduced by one order of magnitude (Figure 3.5e). In analogy to the MAPbI3bulk sample, this observation can be associated to a reduction in the defect density NT as molecular oxygen has been proposed to passivate halide defect sites [131, 132]. If the sample is further cooled down to 77K, in addition to the reduction of thermally activated defects, a reduction in Reh is observed. This reduction can again be associated to a reduction in charge carrier mobility [133] or a reduction in trap recycling processes [98]. Without entering in a detailed discussion of the underlying mechanisms of the observed variations in recombination, a strikingly similar phenomenology is observed for the canonical and noncanonical material system, confirming the validity of the presented analysis. Thus the photon flux representation allows to extract charge carrier recombination constants in systems that 3.4. Experimental Application of the Photon Flux Model 55 cannot be modelled easily with differential equations. Next, the time dependent contributions are discussed in greater detail. These contributions are of different magnitude depending on the initial charge carrier excitation density, but their initial amplitude follows a first order power law (indicated by a dashed line in Figure 3.5a-c). In addition, this feature vanishes at higher charge carrier densities, when bimolecular recombination is dominating (Intersection with the 2√Reh slope). A striking similarity of these qualitative trends can be found with drift diffusion measurements, where interfacial recombination of the minority carrier at charge transporting layers act as charge carrier sinks [141]. In particular, the simulation results in Figure 7 of Ref [141] demonstrate an identical behaviour than the one observed in Figure 3.5. The observed behaviour can be generalized as a non-equilibrium process, where charge carriers are extracted from the material system to a carrier sink. In the case of QDs solids in porous matrices, this carrier sink is associated to neighbouring QDs. We rationalize this as follows: Upon charge carrier excitation, a heterogeneous excitation of the QD ensemble is expected, where a distribution of excited carriers among identical QDs is expected to follow Poisson statistics. In systems with a certain heterogeneity, a even larger variance in QD charge carrier population can be expected, as variations in the absorption cross section may be present. As time evolves, the excited charge carriers will redistribute from QDs with high charge carrier density to QDs with lower density and cause a rapid decrease of the mean charge carrier density, resulting in an effective sink-like dynamic. Once this redistribution process has finished, charge carrier recombination will follow global dynamics, as evidenced by the converging curves at large time instants shown in Figure 3.5. As this evolution depends on the ease of redistribution, these time-dependent features are expected to be more pronounced with lower charge carrier mobility between adjacent QDs. In other words, the stronger the time-dependent contribution, the higher is the degree of QD isolation, such that the presence of these features can be employed as a way to sense the QD inter-dot connectivity. Please note that these features show a slope m>1.5, which prevents an assignment to local recombination, as discussed in the following section. 56 Chapter 3. Photon Flux Representation 3.5 Photon Flux Pattern of Isolated Quantum Dots So far, we have only considered systems that follow a global charge carrier population density. In many nano-structured systems, a complete isolation of QDs is achieved, such that the charge carrier recombination will not depend on the global charge carrier density any more. Charge carriers in isolated QDs are not affected by the density in neighbouring QDs, such that a macroscopic description of charge carrier populations with densities per volume unit loses its meaning. Instead, a description based on probabilities can be employed, where one excitation per QD is assumed. To understand the implications of such a material system on the photon flux representation, we first perform a theoretical, qualitative analysis before comparing it to experimental data. This understanding will allow to tackle a system with an unknown degree of connectivity and systemically discriminate between local and global dynamics in the photon flux representation. Please note that this section does not aim to provide an exact image of recombination dynamics in QDs, since this would require more sophisticated analysis such as single dot spectroscopy and time-tagged time-resolved PL (TTTRPL) measurements (e.g. [142]), but wants to give an overview of decay shapes that may be expected in the photon flux representation. 3.5.1 Theoretical Description in Isolated Systems In an isolated emitter, each excited electron has a hole associated to it, that cannot be regarded as independent populations, as the recombination of the electron inevitably also anneals the corresponding hole. Thus, differential equations that describe the recombination cannot contain terms that include various populations, such as electrons and holes. Instead, we change this description to equations that consider excitations nex, where recombination rates are only expressed as rates per time unit. Note that this description is similar, but not equal to a fully excitonic picture, where bound electron-hole states are considered. Here, electrons and holes are directly correlated, but their movement is not necessary restricted by a Coulombic binding to the opposite charge, but rather limited by the finite size of the QD, such that the present description includes both excitons and localized charges in small particles. The decay processes can then be described as 3.5. Photon Flux Pattern of Isolated Quantum Dots 57 illustrated in Figure 3.6a and expressed below: dnex dt =−(krad +knonrad)nex (3.13) , where krad represents radiative and knonrad nonradiative decay pathways. Note that the exact nature of nonradiative PL quenching mechanisms in QDs is an ongoing discussion and will not be addressed in detail here [33]. Such a description leads to a single exponential TRPL decay, regardless of the relative contribution of radiative and nonradiative decay components. However, many QD ensembles do show multiexponential decays as reported in literature and will be evidenced in different chapters within this thesis. In particular, long decay dynamics than cannot be explained simply by an increase in krad or knonrad, will be addressed. In what follows we will show that many QD ensembles show a decay shape, that is more complex than a monoexponential decay, but corresponds to a decay rate of up to m=1.5 in the photon flux representation. To assess such an enlarged decay component, we consider different decay paths in QDs ensembles which may present a dispersion in size and shape but also a distribution of defects. These defects may not only act as additional decay paths that lead to a faster dnex dt through an additional non-radiative decay channel, but may also introduce an additional trap recycling process as illustrated in Figure 3.6b. The rate equations displayed in Figure 3.6b present a generic version of possible defect assisted recombination pathways, that may consist of excitation trapping and detrapping, as well as of nonradiative depopulation of the trap state. Such a description is well aligned with commonly observed QD ensemble emission kinetics, such as QD PL blinking [34, 143]. Without further specifying which of these processes are actually taking place in different isolated QDs, a simulation of such trap assisted paths leads to a biexponential behaviour with a fast trapping component and a slow detrapping contribution Figure 3.6c,d. In an ensemble measurement, it can be expected that different QDs may present different recombination pathways, such that the global recombination can be described as a sum of exponential decays with different prefactors: PLtot(t) = ∑ i aie−kit(3.14) 58 Chapter 3. Photon Flux Representation FIGURE 3.6: Simulation of isolated QD recombination dynamics within the photon flux representation. (a) Photon flux representation of a simulated generic dataset for an isolated emitter. (b) Contributions of the different recombination processes as displayed in (e) (c) Evolution of the probability of electrons, holes and trapped charges to be present in the QDs. (d) Sketch of an ensemble of isolated identical QDs with independent recombination dynamics. (e) Proposed recombination processes with corresponding rate equations. Such a description will lead to a multiexponential decay scenario, that is simulated in Figure 3.6c,d in a simplistic scenario with only two different QD configurations (depicted in Figure 3.6a,b) with varying defect-delayed weight on the global recombination. In a scenario with little defect-delayed contribution (Figure 3.6c), a single exponential decay is observed that leads to a slope of m=1 in the photon flux representation (Figure 3.6e). As the number of recombination pathways in the QD ensemble is increased, a multiexponential decay behaviour is 3.5. Photon Flux Pattern of Isolated Quantum Dots 59 observed (Figure 3.6d), that causes a slope of m=1.5 in the photon flux representation (Figure 3.6f). At final stages, when depopulation dynamics are dominating, a deviation from this m=1.5 slope towards a rapidly decaying dPL dt might be observed (Figure 3.6f). Without further specifying of which decay components a QD ensemble is constituted, we can generalize that for a mono-disperse QD ensemble comprising QDs with identical recombination pathways a slope of m=1 is expected, while for an ensemble of QDs with varying decay contributions a slope of up to m=1.5 can be observed. 3.5.2 Experimental Validation in Isolated Systems Having theoretically described the characteristics of isolated QDs, we now turn to a comparison with experimental data. As test-bed, a dispersion of mono-disperse CsPbBr3QDs with an average diameter of 8.2nm fabricated by the group of Prof. M. Kovalenko at ETH Zürich is employed. Note that these QDs have been measured after ∼2 years of storage, such that some degradation might have occurred. Further, the loss of ligands over such long periods may introduce non-raditive decay paths. Fluence dependent TRPL measurements ranging over four orders of magnitude of excitation density were performed and are shown in Figure 3.7a. A fluence independent TRPL decay is observed, where a fast decay that persists over three orders of magnitude is followed by a long, low intensity decay that lasts several µs, such that two recombination regimes are expected. Figure 3.7b shows the photon flux representation of the dataset shown in Figure 3.7a, where identical, but amplitude-shifted dPL dt curves are detected. Each curve comprises (1) a higher amplitude component with a slope of m=1.5 followed by (2) a strong decrease of dPL dt with barely changing amplitude in PL. Following the description of Section 3.5.1, the first feature can be assigned to direct emission from free excitations whereas the second feature stems from an emission dominated by trap-recycled excitations. Note that the second feature is only present at very low PL intensities and that it depends on the feasibility of trap recycling and on the overall emission yield of the QD, such that this feature might not be visible in some TRPL measurements of QDs. To account for the amplitude-shifted curves, we consider a varying share of excited QDs. Figure 3.7c shows several replica of the same simulation of a QD following the model in Section 3.5.1 multiplied by different exponential factors. A comparison between theory and experiment (Figure 3.7b and c) yields a satisfactory reproduction of experimentally observed trends, where a variation in 66 Chapter 4. Connectivity: Transport and Recombination 4.1 Introduction The content of this chapter is part of two publications. Section 4.2-Section 4.6 are covered within an article in ACS Nano under reference "Tiede, David O., et al. Effect of Connectivity on the Carrier Transport and Recombination Dynamics of Perovskite Quantum-Dot Networks." ACS Nano 18 (2024) 2325-2334. Section 4.7 is part of a publication under preparation. Many of the outstanding optoelectronic properties of QDs such as an enhanced quantum yield [147], an enhanced optical absorption [148], size tunable binding energies [70] or single photon emission [149] are obtained through the confinement provided by the particle size. The latter is achieved by limiting the crystal boundaries, most commonly employing growth limiting organic spacers [150], or host structures that limit the growth of crystals [76]. This methodology also affects other material properties such as the local surroundings, where the dielectric environment can strongly affect the electronic properties of the oscillator transition within the material [151, 152] or the surface termination of the crystals. It has been shown that when organic spacing ligands are employed, the macroscopic properties of the colloidal film can be affected dramatically [153, 11]. On the other hand, if the QD surface is not ligand passivated, the presence of traps can alter the defect landscape. Further, the larger surface to volume ratio in QD films can make these materials more sensitive to ambient conditions such as the surrounding atmosphere [154]. Finally, as any optoelectronic application (from photovoltaics to light emission or detection) relies on the efficient injection and extraction of carriers, the possibility of achieving electronic coupling among the QDs is crucial. Many of the above mentioned variations from a bulk system occur, however, on a length scale of a few nanometers and are difficult to measure and characterize directly, as most direct imaging techniques employ light and are thus diffraction limited. As an alternative approach with increased resolution, carrier transport between adjacent QDs was measured by combining optical measurements with high resolution techniques such as atomic force microscopy [155]. In this chapter, we aim to provide a general understanding of the spectroscopic response of material systems with varying charge carrier transport without directly applying spatially resolved techniques. Time-resolved, macroscopic techniques are employed 4.2. Structural Properties and Linear Optical Response of Ligand-free Perovskite Quantum Dot Networks 67 to analyse the effect of connectivity on charge carrier transport and recombination mechanisms and different regimes are identified. We employ ligand-free perovskite QDs embedded within the pores of metal-oxide nanoporous films as a test bench to understand the different collective phenomena leading to charge carrier recombination in a QD solid. By tuning the average inter-QD separation we are able to modulate the interaction between them in a controlled manner and move from a scenario of isolated QDs to one of interconnected QDs where collective phenomena dictate the optoelectronic properties of the network. Time-resolved spectroscopic techniques are employed to study charge carrier recombination and unveil the gradually increasing influence of interconnectivity as the average interparticle distance is reduced. Further, we explore different operational regimes from those comprising low carrier densities, where recombination is limited by defect-assisted processes, relevant for photovoltaic applications, to those comprising large carrier densities where many-particle processes come into play, a scenario characteristic of light emitting devices. Results reveal that dot-to-dot transport and diffusion result in the exposure of the carriers to a larger defect landscape, which determines the efficiency of the different recombination processes occurring in the ensemble. 4.2 Structural Properties and Linear Optical Response of Ligand-free Perovskite Quantum Dot Networks Ligand free FAPbBr3QDs were synthesized within nanoporous SiO2matrices by spin coating a solution of perovskite precursors in DMSO into it, following the procedure thoroughly described in Section 2.1.3. Upon annealing at 100◦C, perovskite QDs form in the voids of the SiO2matrix, as illustrated in Figure 4.1a, where the precursor concentration Cprec defines the overall amount of material that is injected into the host system. Figure 4.1b shows a transmission electron microscopy (TEM) micrograph of an intermediate concentration, which illustrates the dispersion of crystallites attained in the matrix and their characteristic size distribution. The average crystal size, the number of particles formed per unit volume of the pores (or filling fraction, ff), and the average interparticle distance are affected by Cprec (see Table 2.1), allowing to tune the connectivity without affecting their chemical composition, morphology or surface properties. All samples show clear signatures of quantum confinement effects, as evidenced by the shift towards 68 Chapter 4. Connectivity: Transport and Recombination FIGURE 4.1: Structural and linear optical characterization of ligand-free perovskite QDs embedded in porous matrices. (a) Sketch of the analyzed materials system with an optical photograph of the thin film. (b) TEM cross section of Cprec =20% at different magnifications. (c) Linear absorption normalized at 2.8 eV. (d) Normalized PL of bulk and QD samples. The color code is the same as in (c). (e) PLQY of the different samples shown in (c) and (d) higher energies of both the absorption edge (Figure 4.1c) and emission spectrum (Figure 4.1d) as the average QD radius, 〈r〉,is reduced [76], although contributions to this shift from the dielectric environment cannot be discarded [156, 157]. A reduction in QD size also leads to an enhancement in the continuous wave photoluminescence quantum yield (cw-PLQY) (Figure 4.1e). While this is the expected trend for an increasing level of quantum confinement, the key factor determining emission yield will be dictated by the average interparticle distance, as discussed below. It should be noticed that the excitonic peak at the absorption edge, clearly evident for the bulk case, is absent for the QDs under study. Such behavior has been found in QD solids in the past and associated with the presence of inter-dot electronic coupling and/or size polydispersity [158, 159]. The key structural parameters, 〈r〉and ff, are obtained by fitting the spectral position of the PL maximum to the Brus equation [29] and performing porosimetry measurements, respectively. The Brus equation, while bearing a number of uncertainties due to the assumptions made, (Section 2.1.4) has been proven to yield a good estimate of 〈r〉for similar samples [87]. Table 2.1 shows the estimated average size, which 4.3. Effect on Trap Filling Dynamics 69 lies always below the Bohr radius for this material (8nm) [88]. Both 〈r〉and ff values are used to estimate the average spacing between the embedded QDs, dQD, as summarized in Table 2.1. Please note that the infiltrated system we are dealing with shows heterogeneities as a result of a tendency of QDs to cluster at certain regions of the matrix, as it is evident from the TEM images shown in Figure 4.1b. Hence, interconnectivity can only be reached when these clusters coalesce, allowing charge percolation, which occurs above a certain QD load. In this context, the mean QD separation dQD should be taken only as a quantitative label that is inversely proportional to the degree of connectivity, and dQD =12nm must not be taken as the actual value at which dot-to-dot transport occurs. The data in Table 2.1 clearly show that while changing Cprec modifies the QD size, the main effect is on the filling fraction and therefore on the average separation between QDs, dQD, thus allowing control on inter-dot connectivity. 4.3 Effect on Trap Filling Dynamics To understand the role of QD interconnectivity on the charge carrier recombination we start by studying the dynamics of the PL measured from the set of samples described above, which show a gradually varying interparticle distance dQD. Results are presented in Figure 4.2a-e. The fluence conditions used in this experiment can be considered to be on the lower end, with an average of 10−3−10−1 excitations per QD. For the case of a reference FAPbBr3bulk sample (Figure 4.2a) we obtain long PL lifetimes on the order of µs, which are strongly fluence dependent. The Shockley-Read-Hall model [4, 3] can be applied to this scenario, where the recombination dynamics depend on the global population density of charge carriers that can freely move and distribute within the excited area, which is limited by the density of carriers trapped at defects. Here, the experimentally measured TRPL dynamics depend on the macroscopic density of electrons ne, electron holes nhand trapped carriers nTand their recombination dynamics can be described by coupled rate equations (see Section 2.2), which account for the processes schematized in Figure 4.2f. This model is well established for perovskite bulk samples [125, 124, 8] and yields a perfect fit of the TRPL dynamics of the bulk reference (gray solid lines in Figure 4.2a), which provide defect densities and recombination rates in order with those reported in the literature (Rpop = 3.39 ×10−8cm3s−1,Rdetrap =5.74 ×107s−1,Rdepop =2.99 ×10−9cm3s−1,Reh = 70 Chapter 4. Connectivity: Transport and Recombination 3.6 ×10−11 cm3s−1,NT=1.8 ×1015 cm3s−1). This evidences that charge recombination under these conditions is dominated by trap filling and de-trapping dynamics. Interestingly, the fluence-dependence of the TRPL in the QD films becomes less pronounced as the filling fraction in the porous films decreases, and is eventually lost for the samples with the lowest amount of perovskite in the pore (Cprec =5%, i.e., those with the largest dQD). In what follows, we analyze these dynamics in more detail. For the cases of sparsely separated QDs ( Cprec =30 −10 % ↔dQD =12 −22nm, Figure 4.2b-d), the TRPL dynamics retain a fluence dependent behavior, albeit with markedly distinct trends with respect to the bulk sample. The PL decays now present two components: a fast (tens of ns) and intense initial decay that can be associated to redistribution effects followed by a much slower (several µslong decay) due to trap assisted recombination dynamics. This assignment will become more evident in the analysis of the results presented in Figure 4.2i-m. As the fluence increases, the second component becomes more rapid and dominant in amplitude. Notably, the SRH model that has been successful in the bulk case fails to yield a good fit to these PL dynamics, indicating a more complex scenario than that depicted in Figure 4.2f. For the extreme case of the largely separated QDs (Cprec =5% ↔dQD =28nm), the TRPL dynamics turn fluence independent (Figure 4.2e). This can be interpreted in terms of isolated QDs where excitations are now well localized and dot-to-dot charge transport is not allowed. In this isolated QD scenario, the TRPL lifetime is determined by the ratio between non-radiative and radiative recombination pathways, characterized by rate constants knr and kr, respectively (Figure 4.2h). In what follows, we first focus on the interpretation of the sparsely separated QDs (Section 4.3.1) before analysizing in greater detail the pecularities of the isolated system (Section 4.3.2). 4.3.1 Photon Flux Representation The effect of QD connectivity can be directly visualized and further analyzed by plotting the derivative of the emitted photon flux (PL in counts per second) as a function of the emitted photon flux (Figure 4.2i-m) as described in Chapter 3. This derivative can be interpreted as proportional to an instantaneous effective carrier lifetime τe f f (n), similar to the differential lifetime introduced by Kirchartz in Ref. [127] , which bears information on the predominant recombination mechanism at a given time or carrier density (see below). Such a representation in the 4.3. Effect on Trap Filling Dynamics 71 log-log scale is shown in Figure 4.2i-m for all samples at all excitation densities employed. In order to interpret these graphs, it should be considered that, in a quasi-equilibrium condition and assuming the photophysical model depicted in Figure 4.2f, the first derivate of the photoluminescence can be written as (see Section 3.2): dPL dt =Reh(Rehnenh·(ne+nh)) + RdepopnTnh·ne(4.1) Considering that the emitted photon flux is proportional to the product of the free electron and hole densities (nenh), Equation (4.1) implies that the time variation of the photon flux is directly dependent on the total photon flux and hence the curves for all different excitation conditions must all line up in the log-log representation. Also, a monomolecular decay should result in a linear dependence of the dPL/dt vs. PL curve with a slope close to 1, while a bimolecular one would yield a slope of around 1.5 (see discussion in Section 3.2). With this in mind, we can straightforwardly conclude that the PL decay dynamics of the bulk perovskite, shown in Figure 4.2i, is mainly dominated by trap-limited monomolecular recombination (slope m=1.2), with an effective lifetime, τe f f (n), that depends only on the initial excitation density. On the other hand, a similar analysis of the curves plotted in Figure 4.2j-l reveals that the emitted photon flux does not follow a monotonic trend with changing excitation density. The effective lifetime is now a time-dependent quantity, τe f f (n,t), pointing at an out-of-equilibrium process that appears at early times in all measurements. This effect may be considered a direct consequence of the presence of interconnected QDs in the samples under analysis. Connectivity allows dot-to-dot charge transport and thus diffusion of carriers through the QD network to take place. So, when the excitation pulse gives rise to a high charge carrier population (as depicted in Figure 4.2g), diffusion-mediated redistribution effects lead to a time-dependent reduction of the local average charge carrier density. This redistribution of charge carriers to neighboring QDs causes carriers to access additional defect states in adjacent sites and will thus increase the fraction of trapped carriers, corresponding to the fast initial component observed in the TRPL (Figure 4.2b-d). As fluence increases, more defects will be filled, this process becomes less relevant and the weight of the fast component is reduced. At the same time, the larger sampling of the defect landscape accounts for the reduced PLQY observed for these Cprec, as shown in Figure 4.1e. This 72 Chapter 4. Connectivity: Transport and Recombination FIGURE 4.2: TRPL measurements and corresponding charge carrier recombination models (a)-(e) Fluence dependent TRPL measurements for bulk and QD samples with different Cprec. Maximum and minimum fluences are indicated as carrier densities (or average excitations per QD) for the bulk (or QD) cases. Gray solid lines in (a) indicate fits to the SRH model, while the gray line in (e) corresponds to the fit to a log-normal distribution of decay rates. Insets in (b) and (e) illustrate the expected crystal size and spacing (not to scale). (f) Model of the electron transition processes considered within the SRH model. (g) Drawing exemplifying the reduced diffusivity caused by the limited interconnection of QDs. (h) Model of an isolated emitter excitation and recombination. (i)-(m) Derivative of emitted photon flux against photon flux for the same samples in (a)-(e). initial fast TRPL decay component is similar to that reported for some bulk samples [160, 146] and associated with diffusion and fast charge trapping [161]. In the most extreme case of isolated QDs (Figure 4.2m) a set of parallel curves are observed, indicating a recombination dynamics that does not depend on the initial excitation density. In this case, all decays are multiexponential as faster (slower) 4.3. Effect on Trap Filling Dynamics 73 emitters in an ensemble of QDs will contribute more at early (late) time scales. 4.3.2 Emission Properties in Isolated QDs Isolated QD samples show a distinct behavior as compared to interconnected ones both in static and dynamic PL measurements. For a homogenous distribution of monodisperse QDs, a single exponential decay is the expected characteristic of excitonic monomolecular recombination [70] (see also Section 3.5). However, in our case (Figure 4.2e) the TRPL presents a clear multi-exponential behavior, which is best fitted considering a lognormal distribution of decay rates(Figure 4.3c-e) [162]. The lognormal distribution can be associated to different ratios of radiative and nonradiative contributions in isolated QDs, caused by a size dispersion or different rates induced by strain or variations in the defect density. The PL spectrum for the Cprec =5% (Figure 4.3a) is broader than the rest of the samples under consideration where a certain degree of interconnection is expected. Further, TRPL measurements show a spectral dependence, which is expected for a sample comprising a distribution of independent QDs. The PL decay of the high energy components of the PL spectrum (green band in Figure 4.3a) is much faster than for the low energy components (orange band). The latter further matches the overall integrated decay (Figure 4.3b). Upon fitting the PL decays to a lognormal distribution of decay rates, a lifetime distribution can be retrieved for each spectral range (Figure 4.3f-h) where the presence of a distribution of isolated QDs with different emitting properties becomes evident. The comparison with samples that present a strong connectivity provides further insight into the importance of funneling mechanisms in the charge carrier lifetimes. While we expect a similar size, energy and defect dispersion in all samples, the dispersion effects in the TRPL dynamics shown in Figure 4.3a-h vanish in the case of connected samples (Figure 4.3i-j). In this scenario, the TRPL dynamics are dictated by the recombination centers where charge carriers reside after diffusion processes have redistributed the charge carrier density in the system. 74 Chapter 4. Connectivity: Transport and Recombination FIGURE 4.3: Spectrally resolved effective lifetime distributions of isolated QDs (Cprec =5%). (a) PL emission spectrum of Cprec =5% at RT. Shaded green (orange) area indicates high (low) energetic emission ranges considered in the following panels. (b) Comparison of TRPL decays detected without spectral detection filter (yellow dots), detected with energies below 2.34eV (orange dots) and above 2.53 eV (green dots). Color code used for the different detection ranges is identical for all panels. (c)-(e) Fit (grey lines) of TRPL decays to a log normal distribution of effective lifetimes. Bottom panel shows residual values of fit. (f)-(h)Emission weight of the effective lifetimes extracted from (c)-(e). (i)-(j) PL emission spectrum of Cprec =20% at RT. (j) Corresponding TRPL decays (excitation density 0.4⟨N⟩) from different spectral regions as indicated by color coding in (i). 4.4. Effect on Emission Efficiency 75 4.4 Effect on Emission Efficiency To further understand the impact of the interconnectivity on the macroscopic behavior of the perovskite QD networks under study, we next consider PLQY values as a function of fluence and temperature. This type of measurements allows identifying the prevalent recombination mechanism as a function of the injected carrier density, as well as the role played by defect trapping or lattice vibrations. We start the discussion with the bulk scenario where recombination dynamics depend on the charge carrier population density as described within the SRH model. By extending this model to higher-order recombination mechanisms such as Auger processes, and considering some assumptions [91], the recombination dynamics can then be described in a simplified way within the ABC model [8], where A, B, and C are nonradiative monomolecular, radiative bimolecular and nonradiative multiparticle recombination constants respectively. Assuming only the bimolecular processes to be radiative, the effective emission quantum yield can be calculated as: PLQY(n) = Bn2 An +Bn2+Cn3(4.2) The PLQY fluence dependence of the bulk material (red dots in Figure 4.4a) follows the theoretically expected bell shape (gray line), estimated considering the recombination coefficients extracted from the fitting of transient absorption spectroscopy (TAS) measurements using the ABC model. These experiments were carried out exploring recombination dynamics over a broad range of carrier densities (see Section 2.3.3). The observed bell shape is indicative of increased dominance of radiative bimolecular recombination due to the filling of charge traps at increasing carrier population [163]. Above a certain threshold, non-radiative (trimolecular, Auger) recombination processes take over and the PLQY consequently decreases. Films with higher concentrations of QDs (Cprec =30% ↔dQD =12nm, Cprec = 20% ↔dQD =15nmCprec =20, Figure 4.4b-c) in which inter-dot connection is more likely, show a similar PLQY bell-shaped curve (for the higher temperatures this may be more clearly seen in the normalized PLQY curves shown in Figure 4.4f-i). In these examples (see Figure 4.4b-c) it can be readily observed that for very low excitation densities, ⟨N⟩ ≪ 1 (where ⟨N⟩is the average number of excitations per QD), trap state assisted recombination gives rise to a largely reduced PLQY and that for ⟨N⟩ ≫ 1 non-radiative multiparticle processes are dominating, also causing a reduction of the PLQY. The optimum PLQY of interconnected 82 Chapter 4. Connectivity: Transport and Recombination 4.7 Effect of Defect Landscape on Carrier Recombination We have demonstrated in Section 4.3 that the trap filling dynamics strongly depends on the connectivity between QDs as, depending on the charge carrier diffusion, injected charge carrier can access a different portion of the surrounding defect landscape. On the other hand, trap states can also act as diffusion limiters, as charges can get trapped at surrounding defect sites that hamper further diffusion. A powerful way of controlling the defect density in particular in systems with high surface to volume ratio, is the use of passivating agents such as reactive environmental species (O2and H2O) [171, 130] as well as surface passivating strategies (e.g. PMMA) [86]. In this section, we study and discuss the role of crystalline defects in the charge carrier recombination dynamics in ligand-free FAPbBr3QD films. As a test-bed, we employ two samples, one with isolated QDs (Cprec =5 % ↔ dQD =28 nm) and another one with some degree of connectivity (Cprec =20% ↔ dQD =15 nm). The first one is used to understand and illustrate the isolated effect of passivation on a non-interacting system. With this knowledge, the effects of defect passivation and the consequences on charge carrier transport can be discriminated for the latter. Again, TRPL is used as tool to identify connectivity and to interpret defect passivation mechanisms. 4.7.1 Defect Passivation in Isolated Emitters Halide perovskites have been shown to be highly sensitive to reactive environmental gases such as O2or H2O, both of which can increase emission yield through defect passivation and decrease emission yield through sample degradation at extended exposure and/or in combination with illumination [172, 173, 174]. This so called "light soaking" is assigned to the passivation or creation of point defect states [131, 175]. In perovskites, halide-related defects have been shown to be the most abundant point defects due to their low formation energy [96] and it was demonstrated that they can both act as deep trap states in the bulk [96] as well as on the surface [176]. The low migration energy associated with this ionic species causes halide-related defects to be highly mobile under external stimuli such as applied bias [136]. Similarly they are also highly responsive to atmospheric gases, which can easily help to passivate these defects but also cause sample degradation [171, 130]. In particular, reports have pointed 4.7. Effect of Defect Landscape on Carrier Recombination 83 FIGURE 4.7: TRPL of isolated QD solids in different atmospheres. (a)-(c) Fluence dependent TRPL measurement of a Cprec =5% sample in (a) air, (b) O2and (c) passivated with PMMA. Grey curves indicate a reference decay in N2atmosphere. Excitation fluences range from ⟨N⟩=0.008 to ⟨N⟩=0.8 (d) TRPL measurment of the same sample at constant fluence in different environments.(e) Photon flux representation of the measurements displayed in (d). (f) Sketch illustrating the molelcular character of the studied system. out the action of environmental agents on both surface [132] and bulk [131, 175] defect states. While an extensive analysis of environmental effects exceed the scope of this thesis, the presence of environmental gases will be used as a mean to passivate defect states on both surface and bulk sites. In the case of QDs, the effect of surface effects is expected to be larger than in bulk samples due to the higher surface to volume ratio. Figure 4.7 shows the fluence dependent TRPL dynamics of the isolated QD sample (Cprec =5%) in different atmospheres in comparison with inert N2atmosphere (grey curve). As evidenced in Figure 4.2e, the TRPL decay dynamics are fluence independent within all atmospheric conditions, indicating isolated emitters (Figure 4.7a-c). Nonetheless, significant changes can be observed when comparing TRPL decay dynamics in different atmospheres. Upon change from inert N2at- mosphere to pure O2slightly larger lifetime is observed (Figure 4.7a), indicating passivation of nonradiative recombination pathways (see also Section 2.2.1). As the environmental conditions are changed from O2to air and finally to a film with a PMMA-passivated surface, this effect becomes even more pronounced. Since the TRPL dynamics are fluence independent, a direct comparison of decay curves 84 Chapter 4. Connectivity: Transport and Recombination can be carried out (Figure 4.7d). To interpret this comparison, the photon flux representation (Chapter 3) is employed. In Figure 4.7e, it becomes evident that the effective TRPL decay rate dPL dt is decreasing continuously, as a more passivating atmosphere is employed. dPL dt is proportional to the decay rate of charge carriers dn dt . Since the TRPL decay rates are fluence independent in all atmospheres, a molecular description of the decay dynamics can be employed (Figure 4.7f) where the decay rate is independent on the global charge carrier density (dn dt (Z n)): dn dt =−(krad +knonrad)n(4.3) that is likely to present a distribution of decay rates, as introduced in Section 3.5 of Chapter 3. Since the rate krad describes a bimolecular, radiative recombination event that is bandstructure dependent and thus an intrinsic material property [177] (see also Section 2.2.2), it is not expected to change with a changing surrounding. The reduction of dPL dt observed in Figure 4.7e is thus assigned to a reduction in the nonradiative recombination rate knonrad. To explain the dependencies of this ratio, we recall Equation (2.17): ktrap =knonrad =σtrapvth NT(4.4) Here, knonrad is directly affected by the number of trap sites NTpresent in the system and by the capture cross section of the trap states σtrap, while the thermal velocity of the free charge carriers vth is expected to remain unchanged. The reduction of the number of trap sites can be explained as halide related defects are passivated (e.g.: through defect site filling by environmental gases), while a change in σtrap can be assigned to a change in the energy level of a trap state. The latter was proposed in Ref. [131] to occur if an interstitial iodide oxidizes with ambient oxygen changing from a deep trap to a shallow trap. To shed more light into the exact origin of the reduction in knonrad, the distribution of recombination rates is analysed in the following. As mentioned in Section 3.5 of Chapter 3, the fluence independent decay rates do not follow a monoexponential decay rate, but rather a multiexponential decay, that can be well described with a lognormal distribution of decay rates (Figure 4.8a, see also Figure 4.3). Following Equation (4.3), the distribution of decay rates can be associated to both the non-radiative knonrad and radiative krad components, where non-radiative components are expected to show both larger decay 4.7. Effect of Defect Landscape on Carrier Recombination 85 FIGURE 4.8: Lifetime distribution in isolated emitters. (a) TRPL for a sample low connectivity sample (Cprec =5%) infiltrated with PMMA. Grey (black) curve represents experimental data (fit to a lognormal distribution of decay rates). Bottom panel shows the residual of the fit. (b) Average decay rate of the distributions of the Cprec =5% sample with different passivating agents. (c) Lognormal distribution of decay rates from fit to experimental data to bare samples (black curve) and samples infiltrated with PMMA (red curve). times and a broader distribution of decay times. As passivating agents (atmosphere and PMMA) are acting on the sample, a reduction of the decay rate distribution is observed (Figure 4.8b). This distribution can be described with the mean decay rate kmand the width of the distribution at a height of 1/e[162], where a continuous decrease of both parameters is observed as surface defects become passivated through a passivating agent (Figure 4.8c). Since both parameters decrease in the same monotonic way, the global reduction in decay rates can be solely assigned to the reduction in the number of trap states. A significant change in the distribution of defect states would involve a more drastic change in the width ∆k, rather than in the mean rate km. Since this is not the case, a global reduction of the dominant defects rather than the reduction of just one specific defect can be assumed. Further, an assignment of the broad distribution to a size and energy dispersion due to the heterogeneity of the crystals is discarded, as in this case, km and ∆kwould not follow identical dynamics. 4.7.2 Trap State-Limited Diffusion in Connected Quantum Dot Solids Having characterized the effect of passivating agents on isolated QDs, we now turn to the case of connected QDs. Figure 4.9 shows the TRPL decay dynamics for a sample that presents intermediate connectivity (Cprec =20%). In inert N2the sample shows a fluence independent dynamics (Figure 4.9a) indicating that no radiative emission is detected from diffused charge carriers. However, as the environment is changed to defect passivating O2or surface is passivated via PMMA, 86 Chapter 4. Connectivity: Transport and Recombination FIGURE 4.9: TRPL of connected QD solids in different atmospheres. (a)-(c) Fluence dependent TRPL measurement of a Cprec =20% sample in (a) N2, (b) O2and (c) passivated with PMMA. Excitation fluences range from ⟨N⟩=0.01 to ⟨N⟩=1. (d) Photon flux representation of the data shown in (a)-(c). the TRPL dynamics start to become fluence dependent (Figure 4.9b-c). To quantify these changes, the photon flux analysis described in Chapter 3 can be employed and results are shown in Figure 4.9d-f. It becomes evident that the samples in N2 show an isolated behaviour, as parallel, disconnected decay dynamics for different fluences are observed. As passivating O2is introduced into the system, the final decay dynamics are converging, indicating a global density dependence (dn dt (n)) at final stages. This global density dependence is associated to charge carriers that are mobile enough to feel the effect of filled adjacent defect sites, such that a larger charge carrier density leads to larger fraction of filled trap states and thus larger emission rate. In other words, charge carrier diffusion length is enhanced sufficiently to be affected by neighboring trap filling events. This effect becomes more pronounced in the case of PMMA-passivated samples, as final dynamics start to converge strongly. The above described observations can be interpreted with the sketch shown in Figure 4.10. The Cprec =20% samples present limited connectivity, where a charge diffusion from a QD to an adjacent one is a process of limited probability. In the 4.8. Conclusion 87 FIGURE 4.10: Proposed picture for carrier recombination and the role of deep traps. (Top) for bare QD films carriers generated in a given QD are rapidly recombine nonradiatively at a trap within the same or an adjacent QD. (Bottom) Upon passivation of the surface traps, carriers can diffuse across the QD film before recombining. case of a N2inert atmosphere, the presence of a large global trap state density (Section 4.7.1) hampers charge transport as all charge carriers that access adjacent QDs are expected to be trapped at defect sites therein (Figure 4.10 top). In contrast, in samples with reduced trap state density (ambient or PMMA passivated), diffusing charge carriers can travel easily across the QD network without being trapped at surface defect states (Figure 4.10 bottom) and thus recombine with other charge carriers present in the material system. These freely moving charge carriers behave as in a bulk sample, where recombination depends on the global charge carrier density present in the system. 4.8 Conclusion In summary, the effects of connectivity on relevant optoelectronic properties of ligand-free QD solids have been analysed in detail. Trap filling dynamics dramatically depend on how far charge carriers can propagate within the QD network, as a larger propagation distance allows to access a larger defect landscape. Local, time dependent and global, time independent recombination dynamics are discriminated through the use of the photon flux representation. A drastic change is observed as a system with fully isolated QDs is analysed. The ease of access of the defect landscape further reflects in the fluence dependent emission efficiencies, where different optimum emission regimes with respect to excitation fluence and temperature are identified. Isolated samples have shown to best perform at lowest excitation densities as multiparticle effects quench emission at higher excitations, whereas connected samples reach the highest emission yield when the optimum between trap filling saturation and the multiparticle onset is obtained. 88 Chapter 4. Connectivity: Transport and Recombination Connectivity has been further shown to affect the emission linewidth, as a certain degree of connectivity allows charge funneling processes to the lowest energetic sites that reduce the emission linewidth. To demonstrate the versatility of the photon flux model, the analysis technique is applied to several material systems that allows both to consolidate the well established finding as well as to extend the analysis to material systems where a conventional fitting to a set of differential equations fails. As the role of the defect landscape and the level of accessibility of trap states is shown to be central to recombination dynamics, the effects of a varying defect landscape were analysed in greater detail by employing environmental (O2and H2O) as well as surface passivation (PMMA) strategies. In isolated emitters, it is found that by passivating trap states, the nonradiative contribution to the recombination rates can be reduced. In the case of connected samples, this reduction of nonradiative decay paths leads to an enhanced charge carrier propagation in the material system, as carriers are not trapped by adjacent defect sites. 89 Chapter 5 Connectivity in embedded Quantum Dot Networks II: Effect on Ultrafast Carrier Dynamics In this chapter, we focus on processes happening on a time scale faster than a nanosecond in ligand-free QD solids. These initial processes dictate the energy distribution of excited charge carriers and determine whether they will bind to excitons. We unveil that charge carrier cooling dynamics are strongly dependent on the QD separation both in the low and high excitation density regimes. We show how Auger heating causes extended cooling dynamics in isolated system and analyse how phonon decay pathways are affected by interparticle separation. Finally, we employ 2D electronic spectroscopy to analyse exciton formation and rationalize about the origin of weak excitonic features in ligand free perovskite QDs. 90 Chapter 5. Connectivity: Ultrafast Dynamics 5.1 Introduction In the previous chapter, the effect of connectivity in perovskite quantum dot (QD) solids on their macroscopic optoelectronic properties has been analysed in detail. Most of the presented phenomena such as recombination dynamics or charge diffusion processes occur within the ns- to µs regime. However, many relevant photophysical mechanisms such as thermalization, cooling and exciton formation occur on a much shorter time scale within the fs- to ns-range. These processes are highly relevant both for optoelectronic applications such as hot carrier solar cells or lasing applications as well as for a fundamental understanding of electronic processes in a semiconductor. For solar cells, losses due to charge carrier cooling represent a mayor limitation in energy harvesting, where the maximum conversion efficiency in a single junction solar cell is given by the Shockely-Queisser limit and only reaches 33.7% [178, 179]. Harvesting excited charge carriers before their cooling has completed is an intensively investigated optimization strategy to improve the performance of solar cells [180], an approach also sought for in the field of halide perovskites [169, 181]. On the other hand, for emission applications, a fast cooling is desired as emission from the same energy level is needed for a monochromatic emission [182]. In particular, for collective emission such as amplified spontaneous emission or emission from collective states such as polariton condensates, a fast cooling process is of utmost importance [183, 184]. Thermalization and cooling processes describe the relaxation of excited charge carriers with an excess of energy to the bottom (top) of the conduction (valence) band. Here, immediately upon (photo-)excitation, charge carriers are distributed in a thermodynamically non-equilibrium population, as their energy matches exactly the one of the absorbed light pulse. Within 100s of fs, this charrier population redistributes into an equilibrium distribution of carriers with elevated energy through carrier-carrier scattering and carrier-phonon scattering events, commonly referred to as thermalization [185]. The ensemble of charge carriers in the system can then be described with a statistical distribution of charges with different energies, where a Fermi-Dirac distribution can be employed to represent the excited ensemble [186]. A characteristic magnitude of this distribution is the carrier temperature TC, whose evolution towards lower amplitudes happen via carrier-phonon scattering events and is labeled charge carrier cooling. The latter will be analysed in great detail within this chapter. Precisely, this TCcan be 5.1. Introduction 91 estimated through the analysis of time resolved PL [187, 188] or absorption measurements [114, 185]. A commonly employed approach to modify the cooling dynamics in semiconductors is nanostructuring [186]. On the one hand, discretization of electronic energy levels through quantum confinement has been extensively explored as a way to impose multi-phonon participation in the thermalization process, thus resulting in a much slower rate than in bulk semiconductors [189, 190, 191]. Notably, most of these investigations have been carried out on colloidal quantum structures with capping organic ligands, which inhibit any inter-particle interactions. On the other hand, the phonon landscape can also be strongly altered in quantum confined structures [192, 193]. As a consequence, it has been observed that the carrier temperature is significantly different in solids made up of compact assemblies of quantum dots in comparison to solutions of isolated dots. In terms of connected QD structures, it was suggested that carrier-phonon coupling strength is modified in the presence of Coulombic coupling from neighboring quantum dots, resulting in a faster decay rate in compact quantum dot assemblies [194]. Alternatively, the presence of excitonic effects is also suggested as the origin of the variance in the carrier temperatures observed in quantum dot solids [195]. An in-depth understanding of the role of inter-particle coupling in cooling dynamics is however still missing. This is particularly relevant given that solid-state assemblies of quantum dots are the only viable material architectures that are technologically compatible [196]. The main material system under study in this thesis, namely ligand free FAPbBr3 QD solids, provides a unique testbed that allows to study the cooling dynamics in interconnected quantum dot solids. This chapter is dedicated to understand the photophysical mechanisms behind carrier cooling dynamics and how they are affected by inter-dot coupling. To do so, we study two charge carrier density regimes. When high charge carrier densities are considered, we show how Auger reheating processes are dominating and strongly dependent on the interparticle spacing. In a low carrier concentration regime, where electron-phonon coupling plays a key role, we show how connectivity affects the phonon-assisted cooling mechanisms. In this sense, we obtain valuable information that points at different material designs depending on whether the application under consideration demands high or low carrier densities. Finally, as the ultimate process relevant prior to recombination processes, we analyse how excitons form and interact in 98 Chapter 5. Connectivity: Ultrafast Dynamics shown in Figure 5.3(d). With limited access to neighboring QDs for smaller material loads, there is a reduced inter-dot Auger transfer, leading to the re-heating process illustrated in Figure 5.3(e) which results in slow cooling of the carriers. Importantly, the Auger rate depends both on the density of excited carriers present in the system ⟨N⟩exc as well as on the degree of localization of the excited carriers (indicated by arrows in Figure 5.3f). The effect of localization can be considered as an effective increase in the density of states, as charge carriers are more likely to encounter themselves if their mobility is reduced as the QDs are more isolated. The latter highlights that Auger processes in QD solids cannot be seen as only size dependent, but also emphasizes the role of inter-dot coupling in these processes. 5.4 Reduced Phonon-assisted Charge Carrier Cooling From the above description, it becomes evident that the long-lasting Tcin QD solids is a result of the competition between phonon-assisted cooling processes and Auger reheating. The other observation that has not been considered yet pertains to the differences in the initial carrier temperature across samples, which increases with reduced inter-dot distance. Moreover, the dynamics shown in Figure 5.3 deviate from the cubic dependence at initial times of up to ∼2 ps. We hypothesize that these initial dynamics and elevated carrier temperatures are related to the role of phonon dynamics in the cooling mechanism. To elaborate on this, we consider the low fluence dynamics at an excitation density of approximately 0.5⟨N⟩, where the Auger-heating process is minimized. It can be seen in Figure 5.4(a) that at this excitation density, a higher initial temperature is observed for lower QD load samples. To describe the evolution of Tc, a multiple temperature model can be employed [198], considering the effective temperatures of photo-generated carriers, as well as those of the optical and acoustic phonons. Note that the specific energetics and nature of the phonon are not explicitly considered, but a generalized bath is considered instead. Importantly, each of the considered populations can be assigned to a specific heat capacity ci, illustrated with "baths" in Figure 5.4b. The interaction between the populations can be quantified through the respective coupling parameters: an electron-optical phonon coupling coefficient αe−ph, and an optical to acoustic phonon decay rate αLO−ac. The time dynamics of the associated temperatures of carriers (Te), optical phonons (Tph) and acoustic phonons (Tac) can be described through [199]: 5.4. Reduced Phonon-assisted Charge Carrier Cooling 99 ce dTe dt =−αe−ph(Te−TLO)(5.4) cph dTph dt =αe−ph(Te−TLO)−αLO−ac(TLO −Tac)(5.5) cac dTac dt =αLO−ac(TLO −Tac)(5.6) When changing the QD load in the material system, variations in the heat capacity cph and the phonon-phonon coupling constant αLO−ac can be expected, where three possible mechanisms can affect the cooling dynamics. • On a first look, the distinct density of vibrational modes at the sample surface can alter the phonon landscape and anharmonicity and thus cph [200]. As the QD size changes with QD load, a change in the surface-to-volume ratio (S/V) can be expected. However, since S/V=3r−1and considering the average QDs sizes (see Section 2.1.3), a change of less than 20% is expected as the QD load is tuned, such that a dominating effect of the S/Vvariation is unlikely. • Second, apart from the QD size, one key aspect of our samples is the dielectric matrix surrounding the QDs based on SiO2, which can have a significant contribution to the overall heat capacity of the system. Here, a reduction of the QD load causes an increase of the ratio of SiO2to QD material. Since SiO2has a larger lattice stiffness than the perovskite, one could expect that the overall material system presents a larger effective heat capacity and thus a faster lattice cooling with reduced QD load. However, the observed trend is opposite, with a reduced cooling as the QD load is decreased, such that an influence of phonon modes of the SiO2matrix can also be discarded. This can likely be associated with the fact that the contact between perovskite material and SiO2does not change as we modify the QD load, such that the contribution of the host SiO2matrix remains unchanged. This assumption is further sustained by a possible lattice mismatch between the perovskite QD and the SiO2host matrix, which may reduce the heat transport between QD and host matrix. • Third and more important, the local environment of QDs is altered through the presence of adjacent QDs, where the void space in the host matrix gets 100 Chapter 5. Connectivity: Ultrafast Dynamics FIGURE 5.4: Phonon-assisted carrier T cooling. (a) Extracted charge carrier cooling of for samples with different QD load and bulk at an excitation density of ∼0.5⟨N⟩. (b) Bath representation of the expected phonon-assisted decay channels. (c)-(d) Simulation of the phonon decay channels described in Equation (5.6) with parameters ce=1, cph =3, cac =1000, αe−ph =αLO−ac =7·1012 with parameter variation of cph (c) and αLO−ac (d).Blue, orange and green lines represent Tc,TLO and Tac, respectively. Note the parameter values are of qualitative character and arbitrarily scaled. To highlight this, absolute units are omitted as they do not represent scaled values. filled with perovskite material as the precursor load is increased. We consider that this proximity effect will have significant impact on the effective phonon landscape and lattice vibrational structure. Here, the highly heterogeneous spatial distribution of QDs observed in previous chapters (e.g Section 4.3.2) must be highlighted such that even in a sample with a small material load adjacent QDs with connected surfaces are expected. A variation of QD load will thus change the amount of such connected QD networks present in the sample. We hypothesize that the changes in the Tcevolution is predominantly related to such variations in the local environment with changing connectivity between adjacent QDs. To qualitatively describe the effect of varying connectivity, we associate cph to the density of states of LO-phonons and αLO−ac to LO-phonon to acoustic phonon decay rate and vary them to simulate the cooling dynamics with Equation (5.6). Note that at initial timescales hot carriers have not yet thermalized and cannot be 5.4. Reduced Phonon-assisted Charge Carrier Cooling 101 described within Fermi-Dirac statistics, such that the extraction described in Section 5.2.1 does not yield reliable values for Tc. Accordingly, we only consider the carrier dynamics starting from a pump-probe delay of ∼0.5ps, which are shown in Figure 5.4. We recall that those dynamics are characterized by (1) a higher initial temperature and (2) a faster decrease in Tcfor smaller QD loads, where a strong difference is observed between bulk and QD samples. The solid lines in Figure 5.4a are only guides to the eye, as we do not seek to directly fit the experimental data with our model. Instead, we compare the TCdynamics with independent simulations shown in Figure 5.4c-d after 0.5ps (dashed line). We can see how increasing either cph or αLO−ac leads to a larger simulated initial Tcat 0.5ps and a faster decay from then on. At this point, due to the limited resolution of the measurement points (restricted by the used 190fs excitation and probe pulse) and the uncertainty resulting from the carrier extraction method that relies on fully thermalized charge carrier populations (refer to Section 5.2.1), we cannot unambiguously determine whether a change in αLO−ac or in cph is responsible for the observed variations. We can, however, hypothesise on the expected changes as the inter-dot spacing is varied. For perovskite thin films it has been shown that charge carrier cooling presents large spatial heterogeneities [201, 202], suggesting local sites that favor the LO-acoustic phonon decay as prominent shallow traps where found at grain boundaries that enhance cooling dynamics. In the system under study here, reducing interparticle QD spacing increases the defect landscape of charge carriers (see Chapter 4) and thus the probability of hot carriers to access the above-mentioned shallow traps, increasing the available phonon modes reflected in cph and thus experience a faster cooling. On the other hand, a change in the LO- phonon to acoustic phonon decay αLO−ac can also be rationalized, as Coulombic interactions of the highly polar surfaces of neighbouring perovskite QDs might affect the degree of freedom of vibrational modes at the QD surface [194]. This inter-dot spacing dependent Coulombic force can potentially affect the density of states of phonon modes and consequently vary the LO-phonon to acoustic phonon decay. Thus interparticle spacing in QD solids will strongly determine the cooling dynamics in the early stages (and low fluence regime) as it will determine the interaction between carriers and phonons and among the latter. 102 Chapter 5. Connectivity: Ultrafast Dynamics 5.4.1 Ground State Bleach Formation Dynamics Another way to further characterize the cooling dynamics is to track the formation of the ground state bleach (GSB), as it characterizes how long charges require to reach their ground state energy (i.e. the band edges). Figure 5.5a and b show the GSB formation of the studied samples at room temperature and at 77K for low excitation fluences like those used in the previous section. A slowed down cooling dynamic is observed at reduced temperature, validating the assignment of cooling processes to thermally activated phonon-processes. The dynamics at 77K shown in Figure 5.5b further confirm a reduced cooling in samples with smaller QD load, as expected from the modified phonon landscape. While a more detailed analysis of the formation dynamics is limited by the temporal solution of the laser system, a striking observation is present when comparing QD samples with its bulk thin film counterpart: A fast, roughly 0.5ps long intense initial feature is observed, that is strongly enhanced at low temperature with respect to room temperature. It can be assigned to a pump-polarization induced process: As an intense light pulse passes the sample, the induced polarization can cause a shift of the energy of excitonic transitions, that will reflect as an derivative-like feature in TA spectra. This derivative-like behaviour is confirmed, when comparing to the normalized PIA absorption plotted in Figure 5.5b, that follows an identical dynamic [116]. Such a behaviour is an unambiguous confirmation of the presence of a strong excitonic oscillation. Intriguingly, this feature is not observed in the QD samples, indicating a much reduced excitonic contribution with respect to the bulk counterpart. This assumption is supported by other measurements, such as the absence of a well defined excitonic peak in the (temperature resolved) absorption spectra. This observation is striking and on a first perspective counter-intuitive, as one would expect an enlargened exciton contribution for confined QDs. To provide insight into this phenomenon, we next switch to a different spectroscopic technique, that allows to analyse cross-correlation between absorption and emission features. 5.5 Ultrafast Exciton Carrier-Coupling The measurements presented in this section were performed during a one week stay at the lab of Prof. Silva-Acuña. at Georgia Tech, Atlanta, United States. The measurements where scheduled solely to meet with the thesis objective and I participated in the presented 5.5. Ultrafast Exciton Carrier-Coupling 103 FIGURE 5.5: Ground state bleach formation. Normalized minima of TA measurements with similar excitation conditions as shown in Figure 5.2 at (a) 300K and (b) 77K. In (b), the normalized evolution of the lower energetic photo-induced absorption is further shown. measurements. However, due to the limited time of the stay at Georgia Tech, mayor parts of the analysis where performed by Esteban Rojas-Gatjens. It let to the publication: Rojas- Gatjens, E., Tiede, D. O., Koch, K. A., Romero-Perez, C., Galisteo-López, J. F., Calvo, M. E., ... & Kandada, A. R. S. (2024). Exciton-carrier coupling in a metal halide perovskite nanocrystal assembly probed by two-dimensional coherent spectroscopy. Journal of Physics: Materials, 7(2), 025002. Part of the data shown herein is published in this article. A striking observation of the afore presented cooling dynamics is the remarkable difference in the excitonic nonlinear response between bulk and quantum dot samples. To analyse the interplay between excitonic and free charge carrier processes, we next employ 2 Dimensional Electronic Spectroscopy (2DES), performed in a COLBERT setup [203]. We focus on a sample with 30% QD load, which is representative of a well connected QD network. While a detailed description of this technique is beyond the scope of this thesis, a main finding to complement the picture we draw on charge carrier dynamics in QD solids stems from 2DES. We thus ask the reader interested in the technical and experimental details to refer to reference works in particular from the Silva lab such as [204, 205]. Further, we only focus here on the main finding of the performed studies and ask the reader to refer to Ref. [206] for a detailed description. In brief, 2DES consists of measuring the correlation between absorptive and emissive electronic states by measuring the electric field induced by charge carrier populations. To do so, a train of three ultrashort pulses (∼15fs) are sent on a sample in a BoxCAR geometry. Such a coherent excitation sequence generates a third-order nonlinear polarization in the material, which emits coherent radiation 104 Chapter 5. Connectivity: Ultrafast Dynamics FIGURE 5.6: Fano lineshape evidences free carrier exciton interaction. (a)-(d) Absolute (top) and real (bottom) components of the 2D coherent rephasing spectrum after a time delay t1=10 fs at 10K and 110K, respectively. (e)-(h) Simulation of the rephasing response at 10 fs via a discrete and a continuum state separeted by (e,g) ∆E=40meV and (f,h) ∆E=20meV. (e)-(f) show the absolute and (g)-(h) the real components. (i) Simplyfied sketch of the modeled Fano-resonance. For simplicity, the coupling constant qwas assumed to be 1 in the simulations. in the phase-matching direction. A fourth pulse that serves as a local oscillator enables the measurement of the amplitude and phase of the emitted field. The first pulse generates a coherence that evolves for a time (t1), the second pulse further drives the system to a population state which relaxes within a population time (t2). Finally, the third pulse establishes another coherence which emits the electric field. We measure the emitted electric field from the ensemble of photo-excited species through spectral interferometry whose energy corresponds with the emission axis of the two-dimensional spectrum, labeled ¯hω3here. By scanning t1and performing a discrete Fourier transform the excitation axis can be recovered and is labeled ¯hω1. Herein, we specifically focus on the nonlinear response with a phasematching condition,  ksig =− ka+ kb+ kc, referred to as the rephasing pathway. The resulting data is a complex 2D map of correlated changes in the electric field of absorption (¯hω3) and emission (¯hω3). Figure 5.6 shows the absolute and real components of the 2D coherent spectrum at a time delay t1of 10fs at 10K and 110K. Features on the diagonal axis (dotted line) can be associated to absorption and emission events that happen at the same energy, e.g. processes that do not include any transfer processes. In contrast, negative off-diagonal elements correspond to additional excited state absorptions (ESA) that crosslink higher and 5.5. Ultrafast Exciton Carrier-Coupling 105 FIGURE 5.7: Evolution of the rephasing 2D coherent spectra as a function of population time. Rephasing dynamcis were measured at 10K. The panels (a-e) correspond to the absolute components and (f-j) correspond to the real component. lower energetic states. Such features are labeled A and B in Figure 5.6. Further a positive off diagonal streak is identified at higher energies, which is labeled C. Such an correlation picture has recently be described by Finkelstein-Shapiro et. al. through a Fano-like interference between a discrete exciton state and a continuum of free carrier states [207, 208]. Based on this formalism, we simulated a Fano-like interference between an exciton state at 20meV and 40meV below a continuous free charge carrier band edge at 2.31eV. The simulation reproduces strikingly well the experimental data at different temperatures, that correspond of a different degree of energetic separation between exciton and free carrier state that can be explained through thermal broadening effects. This simulation suggests thus strongly, that there is a strong coupling between excitonic and free carrier states, that reduced the excitonic character of this size-confinded material system. The fact, that a Fano-like coupling is present despite an energy separation between the coupled states indicate that broadening effects must be present, that will be discussed later. To further characterize this coupling, a time evolution of the absolute and real components of the rephasing 2D coherent spectra at 10K is displayed in Figure 5.7. Here, an increase in the relative weight of feature A with respect to feature B is observed which is accompanied by the appearance of a new feature D with similar properties as C. The change in relative weight in A and B can be assigned to the creation of a population asymmetry with an increasing weight in the continuum 106 Chapter 5. Connectivity: Ultrafast Dynamics states. This assignment is further validated by the additional continuum feature D. As time elapses, a dephasing of excitons to free carrier continuum states is thus observed, indicating a strong exciton-free carrier coupling and rationalizing the weak excitonic contribution in nonlinear (such as the weak optical stark effect) and linear (such as non resolved excitonic peaks in the absorption spectra) optoelectronic properties. We now proceed to combine the observed observations to provide a general picture on the exciton-free carrier-phonon coupling dynamics. 5.5.1 On the Macroscopic Origin of Exciton-Free Carrier-Phonon Couplings The overlap A strong exciton carrier coupling through a Fano-resonance requires an energetic overlap of the interfering features due to spectral broadening. An energy broadening due to size dispersions is considered unlikely, since the Fanoresonance occurs locally within a QD or its intermediate surrounding. Instead, a local broadening of spectral features is more plausible. Here, the excitonic transition in an QD can be screened through interaction within phonon modes of the lattice through Fröhlich interaction (FI), which depends on the electrostatic potential of the phonon mode. In contrast to thin film FAPbBr3, the phonon modes may be spatially confined in QDs. In other material system, this confinement has successfully been modelled with a hydrodynamic model [209, 210, 211], resulting a a quantum confined electrostatic potential with discrecte modes φlmn, where l,m,nrepresent quantum numbers. Note that for such a description, QDs with hard surfaces, e. g. without capping ligands that can act as phonon sinks, where considered. While in a bulk system, phonons can freely propagate and possess extended wavefunctions, in QDs both exciton and phonon wavefunctions are confined within the QD, such that an enhanced FI can be expected. The FI interaction in turn can spectrally broaden the excitonic transition and cause the enhanced Fano resonance. This effect is to be particularly strong in ligand free QDs, as the absence of ligands at the QD surface prevents a propagation of phonon modes outside of the QD and a stronger confinement is expected. Here, it has been show in a similar system, that electron-phonon interaction enhances in ligand free QDs whose size approach the Bohr radius aR[212]. A reduced excitonic contribution to the isolated QDs can thus be rationalized through an enhanced FI of the electrostatic potential of the phonon mode with excitonic modes. While this hypothesis seems reasonable after consideration of all the presented experimental evidences, 5.6. Conclusion 107 further studies will be required to confirm the presented hypothesis. At this point, a study with QDs with varying surface stiffness is suggested, but is beyond the scope of this thesis. 5.6 Conclusion In summary, in this chapter, we have carried out a detailed analysis of carrier cooling in QD solids as a function of interdot spacing. In the first part, we have demonstrated a clear dependence of the charge carrier cooling dynamics on the inter-dot spacing, where a phonon bottleneck behaviour is observed for large excitation intensities ⟨N⟩>1. In stark contrast to the FAPbBr3bulk semiconductor, the bottleneck behaviour is not associated to a delayed carrier decay due to an accumulation of phonons, but is rather caused by a competition between phononassisted and Auger re-heating process. Here, as connectivity is increased, the possibility of carriers to access nearby QDs lowers the probability of Auger interaction with other excited carriers and thus hampers Auger re-heating and thus lowers carrier cooling dynamics. For the case of low carrier densities and early cooling times, the interaction between carriers and LO-phonons and the latter with acoustic phonons determine the dynamics of hot carrier cooling. Upon introducing a three-temperature model where all these interactions are considered, we postulate that connectivity plays a role in this regime by giving access to the carriers to an extended defect landscape that provides additional cooling pathways in the shape of shallow traps. Hence we have unveiled the role of connectivity on the carrier cooling mechanisms of QD solids involved in two regimes of relevance for PV and emission applications comprising low and high charge carried densities respectively. This chapter contributes to complete the picture of how electronic coupling determines the fate of photo-generated carriers in this type of system and thus determines its performance in an optoelectronic device. In the second part, we have employed 2DES to analyse the interaction of excitons with the continuum of free charge carrier states, where a Fano-like resonance was detected. A time evolution of 2DES spectra indicate a transition from excitonic to continuum free charge carrier states. This is rationalized with an expected enhanced FI-mediated enhanced electron phonon coupling causes by confined phonon modes. While the latter needs to be further confirmed, it points towards 114 Chapter 6. Excitonic Effects a two level system with transition energies of the exciton ¯hωex and photonic mode ¯hωex can be considered and was introduced by Jaynes and Cummings [238]. The hybridization of the involved oscillators causes an energy splitting of ¯hΩRbetween the hybridized states, where ΩRis the so called Rabi splitting and the hybrid states are labeled upper (UP) and lower polaritons (LP), respectively (Figure 6.1e). If many oscillators are coupled collectively to the same light mode, the Rabi splitting increases and scales with the number of oscillators as √N[239]. The additional oscillators further cause a creation of N−1 states that cannot be accessed directly from the ground state, labelled dark states (DS). Intriguingly, the properties of these uncoupled DS do follow a delocalized character that spans over the entire cavity [240]. In what follows, we will study how this strong-light matter coupling affects the recombination dynamics. 6.3 Charge Carrier Recombination in Coupled Colloidal QDs This section treats a material system that was prepared and mainly studied by Clara Bujalance, a PhD candidate of my research group. She developed the device structure and performed all structural and linear optical characterization shown herein. The QDs employed were synthesized by the group of Prof. Kovalenko at ETH Zürich. The time-resolved emission and absorption measurements were performed between Clara Bujalance and me. A part of the data shown herein as well as a detailed study of the Strong Light-Matter Coupling are published in Bujalance, C., Caliò, L., Dirin, D. N., Tiede, D. O., Galisteo-López, J. F., Feist, J., ... & Míguez, H. (2024). Strong Light–Matter Coupling in Lead Halide Perovskite Quantum Dot Solids. ACS nano. 18 (2024) 4922-4931 In this section, we focus on charge carrier recombination dynamics of QDs in different environments and how they are affected by the transfer of energy mediated by exciton-exciton interactions. As in previous chapters, conclussions are drawn from results obtained from time-resolved spectroscopy. However, in order to introduce the material system, we first briefly discuss the linear optical characterization performed by my coworker Clara Bujalance. 6.3. Charge Carrier Recombination in Coupled Colloidal QDs 115 6.3.1 Linear Optical Characterization of Coupled Colloidal QDs Solids Highly mono-disperse CsPbBr3QDs having 6.6nm diameter were fabricated by the group of Prof. Kovalenko following a synthesis approach reported elsewhere [70]. Figure 6.2a shows a TEM micrograph of isolated CsPbBr3QDs dissolved in toluene. Both emission and absorption spectra show strong excitonic features that correspond to the 1p-1p and 1s-1s excitonic transitions [70] (Figure 6.2d). To obtain a solid sample in the shape of a QD film, the dispersion was mixed with a stabilizing agent (polystyrene, 15wt% with respect to the weight of CsPbBr3-QDs) before depositing it through spin-coating on a glass substrate. As shown in the SEM micrograph in Figure 6.2b, the QDs are now more closely packed, while the strong excitonic features in linear absorption and emission are maintained (Figure 6.2e). Finally, the CsPbBr3QD solid was deposited in between two silver mirrors forming an optical cavity in order to couple the excitonic modes with the photonic mode of the cavity. To achieve this coupling, the optical quality of the CsPbBr3 QD solid as well as the device structure were carefully optimized as discussed in Ref. [83]. This causes a strong modification of the absorption spectrum shown in Figure 6.2f, where a strong angular dependence of the absorption is observed as the cavity mode is shifted in energy and couples to the two excitonic modes of the CsPbBr3, leading to the formation of the above mentioned hybridized polaritonic modes. Note that in the present system, lower (LP), middle (MP) and upper polariton (UP) modes are formed, as a result of the coupling to two excitonic modes. While an extensive study on this device configuration is reported in Ref. [83], we will only focus here on the emission and absorption properties of the three configurations. 6.3.2 Tuning Exciton-Exciton Interaction in Colloidal QD solids To shed light into the recombination dynamics of colloidal QD solids, we again employ fluence dependent TRPL measurements. For the case of CsPbBr3QDs dispersed in toluene, we observe no changes in the PL decay dynamics over several orders of magnitude of varying fluence (Figure 6.3a). This results in a set of parallel curves in the photon flux representation shown in Figure 6.3d). Following the argumentation in Chapter 3, this can be associated to a localized recombination dynamics, that does not depend on the overall charge carrier density in the system, but is only determined by the specific recombination pathways present in an individual QD. Intriguingly, the slope in the photon flux representation now 116 Chapter 6. Excitonic Effects FIGURE 6.2: Electron microscopy and linear optical absorption. (a) TEM image of CsPbBr3-QDs dispersed on a grid. Inset shows the schematic structure of a 6.6nm lecithin-capped spheroidal CsPbBr3nanocrystal. (b) HRTEM image of a spin-casted CsPbBr3-QD film cross section. Inset shows a photograph of such film. (c) SEM image of a cross-section of a CsPbBr3-QD metallic optical cavity. Inset shows a photograph of such cavity. (d) Absorbance and photoluminescence (black and red curve, respectively) spectra of a 0.73mg/mL CsPbBr3QD dispersion in toluene. (e) Absorptance and PL (black and blue curve, respectively) spectra of a CsPbBr3-QD film. (f) Absorptance spectra of the cavity containing a QD solid at angles of incidence 26◦,36◦and 46◦(purple, red and blue, respectively) attained with unpolarized light. The dotted colored lines represent the position of the reflectance minima of the third order optical cavity mode at each angle (same color code). The absorptance of the bare film at 0° (gray dashed line) is plotted for the sake of comparison. is almost m=1, indicating very low broadening contributions through blinking or defect distributions among the QDs (see Section 3.5.1). Such dynamics can be explained within an excitonic description dnex dt =−Anex −Bn2 ex (6.1) where Adescribes the exciton recombination rate comprising radiative and non radiative contributions within an isolated QD. At elevated excitation densities, biexcitonic annihilation with quadratic scaling can also be expected described 6.3. Charge Carrier Recombination in Coupled Colloidal QDs 117 FIGURE 6.3: TRPL of coupled colloidal QD networks. (a)-(c) Fluence dependent TRPL decay curves for CsPbBr3QDs in (a) a dispersion of toluene (b) deposited as a QD solid (c) embedded into a planar cavity. Insets illustrate the respective sample configuration. (e)-(f) Photon flux representation of the data shown in (a)-(c).Insets illustrate the connectivity between the QDs. through the biexciton annihilation rate B[241]. However, since biexciton annihilation in isolated QDs is happening after exciting twice the same QD, the two excitons involved in the recombination are closely confined within the volume of the QD, which causes a fast recombination that is reported to happen in the timescale of few ps [14, 123, 106]. Our time resolved PL (TRPL) measurement setup has a minimum resolution of 50ps such that fast biexcitonic annihilation in isolated QDs cannot be detected. If the QDs are deposited in a thin film, the TRPL decays become fluence dependent (Figure 6.3b). As theoretically described in Chapter 3, this causes the dPL dt curves in the photon flux representation to overlap (Figure 6.3e) and can be associated to aglobal dependency of the charge carrier recombination, where recombination is dependent on the population of neighbouring QDs. In fact, at sufficiently high PL amplitudes, the slope of dPL dt turns from close to m=1 to m=1.5, the latter corresponds to a quadratic dependency of the charge carrier recombination. Following Equation (6.1), such a dependency can be associated to biexcitonic annihilation events, that show however at much larger times than those expected for isolated QDs [14, 123, 106]. Such behaviour can only be explained with exciton propagation through the QD solid. It has recently been shown that exciton transport can 118 Chapter 6. Excitonic Effects happen via FRET over long distances in CsPbBr3QD solids [219, 222]. As excitons propagate through QD solids, for high excitation fluences they can interact with other excitons through biexcitonic annihilation. Likewise, two excitons generated in the same QD can also dissipate to neighbouring QDs before recombining and thus reducing the rate of biexcitonic annihilation. Since this mechanism is now diffusion assisted, its characteristic time constant is much slower than an annihilation caused by two excitons within one QD and depends strongly on the overall exciton density in the QD solid. This highlights again the importance of interdot connectivity on the overall recombination dynamics, as the biexcitonic annihilation rate Bbecomes interdot spacing-dependent. Finally, to fully couple excitons with the macroscopic surroundings, we have placed the QD solid into a planar cavity, such that excitons can couple with the photonic mode of the embedding cavity to form exciton-polaritons. The TRPL decay turn again fully fluence independent (Figure 6.3c), that reflects in parallel dPL dt slopes in the photon flux representation (Figure 6.3f). This indicates that the above observed inter-particle interactions are again heavily suppressed. The coupling of excitons with photonic modes lead to the formation of different polariton modes and excitons that are not coupled to the photonic mode (dark states). Due to symmetry arguments, these dark states are not coupled to the electromagnetic field such that they cannot be accessed from the ground state and thus do not emit photons [236]. This causes exciton-exciton interactions to be largely reduced, such that the polariton emission lifetime does not depend on the number of excitons present in the system. These results thus evidence how exciton-exciton interactions can be further controlled by manipulation of the surrounding electromagnetic field. 6.3.3 TA Fine Structure and Local Exciton-Exciton Interaction Since we expect exciton-exciton interaction in isolated QD to occur on a timescale of a few ps, we next turn to TA measurements with a time resolution of <200fs. Upon non-resonant excitation (λpump =420nm, RR =1kHz), photobleach (PB) signals appeared at the maxima of the absorption spectra presented in Figure 6.2d- f, which correspond to the population of exciton and polariton states, respectively. In the case of the dispersed QDs a strong population is observed at the first exciton position (PB1) and a weak population at the second exciton position (PB2) 6.3. Charge Carrier Recombination in Coupled Colloidal QDs 119 (Figure 6.4a). The fact that no continuously broadened PB towards higher energies is observed further confirms the discrete nature of confined states in the QDs [242]. Both features follow identical population dynamics, indicating cooling mechanisms that homogeneously populate the excitonic states (Figure 6.4d) in times of ca. 2ps. Fluence dependent TA kinetics of the PB1 are analysed in Figure 6.4g and evidence no fluence dependence at large time scales as expected from TRPL measurements. In contrast, during the initial stages, a fast decay component is observed (see inset in Figure 6.4g), that can be associated to excitonexciton annihilation processes [14, 123, 106]. As mentioned above, these annihilation processes are occurring in twofold excited isolated QDs, that cause the rapid interaction among excitons. with a time constant of ∼26ps. Next, we consider the CsPbBr3QD film, where very similar TA maps and spectral features are observed (Figure 6.4b,e). However, the fluence dependent dynamics now show a clear increase in the recombination dynamics for higher exciton densities. In contrast, initial dynamics are less pronounced. This further corroborates the proposed mechanism for exciton-exciton interactions: As connectivity is enhanced through close packing of the QDs, excitons can diffuse and reduce exciton-exciton annihilation of twofold excited QDs. However, as they propagate through the QD solid, diffusing excitons can coincide in the same QD and cause recombination, causing a density dependence at longer times than for the isolated case. Finally, the TA dynamics of strongly coupled excitons in microcavities are studied. PB features now appear at the position of the polaritonic modes (Figure 6.4c). However, their formation dynamics differ from the QD samples without photonic structure as evidenced in Figure 6.4f. Here PB1 shows a fast formation (<1ps) followed by a strong decrease. PB2 has a slower formation and continues to increase while PB1 is decreasing, both signals reaching a stationary value for similar (ca. 2ps) times. Such a behaviour is characteristic of a process where energy is transferred from the lower energy PB1 to higher energy PB2. To contextualize this, we have to take into account the interplay of exciton-polaritons with the optically dark states. Even though not directly accessible from the ground state, dark states can be filled both from the lower polariton state and from higher energy levels as hot electrons cool down [240]. From these dark states, carrier exchange can happen to both polaritonic modes PB1 and PB2 as a significant energetic overlap between the dark and polaritonic states is inferred from the moderate separation 120 Chapter 6. Excitonic Effects FIGURE 6.4: TA of coupled colloidal QD networks. (a)-(c) False color maps of ∆A versus λand time for the CsPbBr3in (a) solution, (b) bare film and (c) embedded in a planar cavity, in which the main signals are labelled. Fluences were varied in the range 12µJ/cm−2-118µJ/cm−2according to the absorption at 420nm to ensure an excitation of ⟨N⟩<1. Grey lines represent the corresponding linear absorption spectra, evidencing the correspondence of TA features with the absorption maxima. (d)-(f) Evolution of the spectral features indicated in (a)-(c). (g)-(i) Fluence dependent time evolution of the GSB extracted from the lowest energy bleach in (a)-(c) (indicated by the dark blue arrow). Fluences range from 4µJ/cm−2-540µJ/cm−2. Insets in (g) and (i) show a zoom into the the initial dynamics. between LP and MP polariton [243, 244]. The interaction dynamics in the few ps time scale in the cavity scenario can thus be associated to polariton redistribution effects that depend on the overlap between polaritons and dark states. An increase in the excitation fluence causes an increase in this overlap, since a larger population causes a reduction in the Rabi splitting [83]. Consequently, a more pronounced transfer process is observed for higher excitation fluences (inset of Figure 6.4i). Furthermore, the reduction in the Rabi splitting causes the contribution of nonradiative decay into the dark states to increase, such that a variation of the long time dynamics is observed (Figure 6.4i). Hence, for QDs embedded in optical cavities, interaction between individual excitations are mediated through the 6.4. Exciton-Polariton Condensation in Colloidal QD Solids 121 photonic coupling, where a modification of this coupling can be achieved through a variation in the Rabi splitting. 6.4 Exciton-Polariton Condensation in Colloidal QD Solids The results of this section where obtained within a research stay at IBM Research, Zürich in the group of Dr. Rainer Mahrt and Dr. Thilo Stöferle. I prepared and optimized the CsPbBr3QD solids following a procedure developed by C. Bujalance and L. Caliò in my host group, based on the controlled deposition of colloidal suspensions prepared by the group of Prof. Kovalenko at ETH Zürich. All optical measurements shown herein where done at IBM in close collaboration with Ioannis Georgakilas, with whom I will share co-first authorship of a to be submitted manuscript. Since we have seen that the interaction between excitations in QDs embedded in microcavities is governed by the hybridization with photonic modes, we now turn to the possibility of exploiting such interaction. The bosonic character of the polaritonic modes can be potentially employed to achieve a macroscopic, coherent state of excitation through Bose-Einstein Condensation (BEC), which represents a macroscopically occupied quantum state [230]. Such a condensate has a vast technological interest, with possible applications that range from polariton lasers [245] to quantum simulators [246] or ultrafast polaritonic logics [247] and polariton transistors [248]. If the polariton density is increased such that the particle separation approaches the thermal de Broglie wavelength, BEC is expected to occur [249]. To achieve the necessary polariton density for BEC we now turn towards a sample configuration that allows to strongly confine polaritons spatially . 6.4.1 Tuning of Strong Light-Matter Coupling A thin film of highly monodisperse colloidal perovskite CsPbBr3QDs blended with a small quantity of stabilizing and homogenizing polystyrene was placed inside a tunable patterned cavity consisting of two pairs of Distributed Bragg Reflectors (DBRs) as displayed in Figure 6.5a. The employed cavity structure is comprised of two halves, separately placed on two nanopositioning stages for 122 Chapter 6. Excitonic Effects FIGURE 6.5: Strong Coupling in a tunable Gaussian defect confined cavity. (a) Sketch of the tunable cavity consisting of DBR layer stacks with Gaussian defect and 245nm thick perovskite QD film as emitter. (b) White light transmission spectra as a function of cavity length. Red dots represent the extracted peaks for the planar cavity mode and the LG00 Gaussian mode. Red solid lines represent the fits of the upper and lower polariton for the planar cavity mode and the lower polariton for the LG00 mode. The dark yellow dashed line indicates the fitted exciton energy of 2.484eV, which matches well the exciton peak of the PLE measurement. The two dashed bright yellow lines are the photonic dispersions of the planar cavity mode and of the LG00 mode adjusting their relative position and the distance between them. The lower half consists of a DBR which is deposited on a glass substrate, and on top of it 235nm of CsPbBr3QDs are deposited. A patterned DBR, with a Gaussian shaped deformation of ∼1µm diameter and ∼50nm depth, is used for the upper half to complete the cavity. The Gaussian defect acts as a potential well for the polaritons, inducing lateral confinement of their wavefunction and therefore supporting a set of discrete energy states in the form of Laguerre-Gaussian modes (LG) [250]. For optimal reflectivity, the bottom DBR consists of 9.5 pairs of SiO2/Ta2O5, whereas the top DBR has a slightly reduced reflectivity with only 6.5 pairs of SiO2/Ta2O5to enable an optical coupling to the cavity. This coupling was performed with an optical microscope having a large (100x) magnification objective that allows for both transmission and reflection measurements. To demonstrate strong coupling in this configuration, transmission measurements as a function of the vertical separation between top and bottom DBR were performed (Figure 6.5b). The transmission spectrum of this sample configuration is composed of the spatially confined LG modes as well as of a mode originating from the planar cavity. 6.4. Exciton-Polariton Condensation in Colloidal QD Solids 123 By tuning the photonic mode through the exciton resonance, a characteristic anticrossing behaviour is observed when entering the strong coupling regime. The resulting cavity length-dependent polaritonic modes can be modelled via a Tavis- Cumming oscillator [251, 252]. It reads: ¯hωγ1 2¯hΩR 1 2¯hΩR¯hωs! Cγ Cs!=¯hωLP,UP Cγ Cs!(6.2) , where ωγ,ωs,ΩRrepresent the oscillator frequency of the photonic mode, the first exciton transition 1s−1sand the Rabi oscillator frequency, respectively. Cγ and Csare the Hopfield coefficients, that are proportional to the photon and exciton contribution to the polaritonic mode respectively. ¯hωLP,UP are the energy dispersions of the two identified polariton branches. Modelling of the data in Figure 6.5 with two different photonic modes for planar and first Gaussian mode LG00 yield ΩR=49.4 meV and ΩR=55.3 meV, respectively, evidencing a clear strong coupling dispersion. The obtained extracted polariton dispersion is then compared to a Transfer Matrix Model (TMM) simulation to determine the cavity length. 6.4.2 Properties of Condensed Exciton-Polariton Emission To achieve polariton condensation, a focused λ=400 nm, 150fs short excitation pulse train with a 1kHz repetition rate was employed to induce a high local polariton density at the Gaussian defect without degrading the sample. An imaging spectrograph in Fourier space configuration was employed to analyse the angleresolved emission below and above condensation threshold as displayed in Figure 6.6. Below threshold, polaritons populate both the ground state LG00 of the Gaussian potential as well as the first excited state LG01 (Figure 6.6a). As the system is driven above the condensation threshold, a predominant population of the ground state of the Gaussian defect is observed, while the emission of the first excited state LG01 remains unchanged, as discussed below. In addition, the dispersion of LG00 now presents a significantly reduced linewidth accompanied with a blueshift with respected to the uncondensated state (Figure 6.6b). The emission profile in real space further shows a Gaussian shape mode profile with a lateral extension of ∼2.5 µm, consistent with the patterned emission expected from the deformation potential (Figure 6.6c). 130 Chapter 7. General Conclusions and Perspective 7.1 Summarized Results • The derivative-like representation of fluence dependent datasets obtained with time resolved spectroscopies allows to discriminate processes that are dependent and independent on the global charge carrier population density. For canonical systems amenable to be described within the Shockley-Read- Hall (SRH) model, this approach allows to determine the main recombination mechanisms as well as its characteristic rate without the need for cumbersome global fitting procedures. In complex systems such as diffusionlimited QD networks, hidden recombination dynamics associated with trapping and diffusion can be extracted. • Trap filling dynamics in QD solids are strongly determined by the diffusionlimited accessibility of defect states. The interparticle spacing determines whether recombination dynamics follow an isolated, molecule-like or a connected, density dependent recombination pattern. This behaviour causes the emission yield of isolated QDs to be most effective in the low carrier concentration regime where photovoltaic devices operate, whereas connected QDs show higher yields for higher carrier concentration where light emission devices operate. Energy dispersion effects are most prominent in isolated QD ensembles, as charge funneling effects are prevented, an effect that is reflected in the emission linewidth. • The charge carrier cooling dynamics of heterogeneous samples can be characterized through a convolution of the energy dispersion with a carrier temperature evolution. In the low carrier concentration regime, phonon-assisted processes dominate the cooling of hot carriers in QD solids, similar to the case of reference bulk samples. In the high carrier concentration regime, delayed cooling times are observed as a consequence of an Auger reheating process that becomes more efficient in isolated QDs as a result of an effectively reduced density of states. Upon charge carrier relaxation, exciton species that strongly interact with a continuum of unbound states are formed in ligand free QD solids. The latter rationalizes the free carrier-like dynamics in size confined systems. • In excitonic QD solids formed by well separated QDs, energy propagation 7.2. Perspective 131 mechanism are found to be mediated by the different ways dipole interact. At nanometric separation length scales imposed by the organic capping ligands surrounding the QDs, resonant dipole-dipole interactions enable exciton propagation. This strongly affects Auger-like exciton-exciton annihilation processes, where fast intradot Auger processes are reduced and slow diffusion assisted interdot Auger processes are enhanced. At macroscopic length scales, exciton interaction can be mediated through coupling with confined optical modes within resonators and form exciton-polaritons. Strong light-matter coupling in defect-confined microcavities was employed to attain exciton polariton condensation, leading to a macroscopic quantum state. 7.2 Perspective The findings presented within this thesis, while applied to a technologically relevant material such as lead halide perovskites, should be considered in the broader context of a toolbox to analyse the effects of connectivity in any type of semiconductor light emitting system that shows limited connectivity. In this context, a logical progression would be to extend the methods described herein to QDs solids formed by nanocrystals capped with ligands of gradually varying length. In doing so, a precise control on the coupling among QDs over different regimes, from dipole-dipole interactions to the overlap of the wavefunctions of charge carriers leading to electronic transport, can be achieved. The presented methods provide a toolbox to screen QD solids with different degrees of electronic coupling among their constituent units mapping their optoelectronic properties and optimize their connectivity before their implementation in any optoelectronic device. As demonstrated throughout this thesis, depending on the envisaged operational regime (PV or light emission), different degrees of connectivity are expected to yield optimal optoelectronic behaviour. To confirm and exploit this prevision, an exciting perspective is to extend the established analysis protocol to optoelectronic devices (both PV and light emitting) that employ low dimensional perovskite structures. With respect to connectivity, a special interest lies in studying the propagation of charge carriers in layers stacks that contain charge transporting layers. We believe that representing the derivative of time resolved spectroscopic data against its amplitude is a simple, but powerful approach to extract carrier 132 Chapter 7. General Conclusions and Perspective concentration dependent recombination and interaction mechanisms. In many cases, the magnitude time in time resolved measurements is actually a misleading magnitude, as no direct dependency is stated in the corresponding rate equations. Instead, we suggest that in many cases, an analysis of derivative/amplitude datasets can actually provide more direct insight than a global modelling of amplitude/time dynamics. This allows to directly extract dominating processes in the distinct charge carrier density regimes , where power-law fittings allow to directly identify number of particles involved (e.g. mono-, bi- or trimolecular behaviour). Besides the convenient aspect of removing global fitting procedures, it provides information on the relative significance of each process as a function of charge carrier concentration. We thus expect that this analysis approach can and should be extended both to different material systems as well as to other time-resolved spectroscopies, such as transient absorption spectroscopy, terahertz spectroscopy, microwave conductance or surface photovoltage spectroscopy. From a materials perspective, there are still open questions to be addressed regarding the studied systems. In the case of the ligand free QD solids, while founded with very solid arguments, no direct evidence for charge carrier diffusion is provided in this thesis. Here, spatially and time resolved PL has been employed to map propagation of charge carriers. We are currently exploring options for a collaboration with an expert in energy diffusion in semiconductor materials. Beyond the direct visualisation and quantification of the expected diffusion in the material, answering the question whether the diffusivity is exciton or free carrier dominated is highly interesting. In fact, while addressed in Section 5.5, elucidating whether this material system is exciton and free carrier dominated and how the observed confinement and simultaneous free carrier characteristics can be brought together, are probably among the most challenging remaining scientific question within this system. Again, we can expect that answering this question will not only describe the present material system, but most likely provide insights of general interest on the behaviour of ligand free QD systems. In colloidal QDs, dipole-dipole interactions have led to exciton transport. Since dipole-dipole interaction depends strongly on the dipole orientation, optimizing the respective dipole arrangement will allow to enhance these interactions. Here, effects such as superradiance are anticipated and were already observed in this system. Further, the demonstration of strong coupling and exciton condensation in perovskite quantum dots opens up a whole new field of study, both to design 7.2. Perspective 133 polaritonic optoelectronic devices and further exploit the possibilities of optical confinement structures as well as to study the fundamental properties of exciton polariton condensates in perovskite QD solids. The optical confinement potential and its consequent high density of polaritonic states provides an exciting platform to study polariton-polariton interactions. The room temperature exciton polariton condensation strongly calls for further implementations of prototype devices for polaritonic logics. 135 Appendix A Data Curation A.1 Extraction of Charge Carrier Density in TA Measurements In this thesis, the ground state bleach (GSB) signal of TA measurements is converted in several chapters into a charge carrier density. In order to convert amplitudes of ∆Ainto charge carrier density, fluence dependent TA measurements are employed. The maximum value of the GSB is assumed to be proportional to the injected charge carrier density through the excitation pulse. The averaged charge carrier density is estimated assuming a Lambert-Beer absorption in the material [188]: navg 0=f l ·(1−exp(−αd)) dEph (A.1) , where f l is the excitation fluence estimated from the laser power, repetition rate and measurement spot size, alpha the absorption coefficient at the excitation wavelength extracted from a linear absorption spectrum, dthe sample thickness and Eph the energy of an exciting photon. In the case of porous samples of QDs inside non-absorbing SiO2,dis corrected to an effective thickness employing the fill fraction f f estimated through porosimetry de f f =f f ·d(A.2) 136 Appendix A. Data Curation FIGURE A.1: Calibration curve for conversion of ∆Ato n. Values for excitation densities and corresponding maximum ∆Avalues of a fluence dependent measurement on a canonical MAPbI3thin film. Grey line represents a second order polynominal fit. The values for initial excitation densities are plotted against the maximum negative value of the ∆Amaps. Figure A.1 shows the extracted data points with an empirical polynominal fit: n(∆A) = a·∆A+b·∆A2(A.3) This calibration function is then applied to every data point of the GSB decay curves to obtain an evolution of the charge carrier density. This approach accounts thus for slight nonlinearities of the ∆A-nrelation at higher excitation densities. Note that slight uncertainties in this method are induced due to charge carrier relaxation effects before carriers reach their ground state energy and spectral broadening of the GSB. In some cases, the data was further treated by extracting the derivative of the charge carrier evolution with a similar approach as elaborated in Appendix A.2. A.2 Protocol to Extract Derivatives of (noisy) TRPL Decays While a derivative of time resolved data can provide valuable information, the extraction of the derivative of noisy data can also increase significantly the uncertainty of the measurement points as the derivative is strongly affected from point to point variations. To account for this, a protocol to extract the derivative of exponentially decaying evolutions is implemented. Measurements of TRPL data were carried out in a calibrated system, where both A.2. Protocol to Extract Derivatives of (noisy) TRPL Decays 137 FIGURE A.2: Calibration curve for conversion of ∆Ato n. Values for excitation densities and corresponding maximum ∆Avalues of a fluence dependent measurement on a canonical MAPbI3thin film. Grey line represents a second order polynominal fit. excitation and detection paths were filtered with different calibrated neutral density filters to vary both the excitation fluence as well as the detection efficiency to maintain the signal within the single photon limit of the detector. The signal was corrected by the measurement time and employed filters. Also, while care was taken to have minimum background illumination, a dark count measurements was performed to subtract the averaged dark counts normalized to the measurement time. At large times the signal was cropped at a fixed threshold value of the smoothed decay curve. Resulting normalized decay curves are shown in Figure A.2a. To extract the derivative, both x-axis (time) and y-axis(PL amplitude) where transformed to log-log scale (Figure A.2b). This allows to equally weight initial, high intensity and final, low intensity signals. 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