Integration of optically random media into optoelectronic devices

Integration of optically random media into optoelectronic devices José María Miranda Muñoz Integration of Optically Random Media into Optoelectronic Devices Doctoral thesis presented by José María Miranda Muñoz Supervised by Prof. Dr. Hernán Míguez García Dr. Gabriel Lozano Barbero Department of Inorganic Chemistry Faculty of Chemistry University of Seville Multifunctional Optical Materials Group Institute of Materials Science of Seville Spanish Research Council Seville, 2019 Contents 1 Introduction 1 1.1 Light scattering in optically random media . . . . . . . 2 1.2 Control of the optical response of optically random media............................... 7 1.3 Applications of optically random media . . . . . . . . . 11 1.3.1 Random lasing . . . . . . . . . . . . . . . . . . . 11 1.3.2 Imaging through optically random media . . . . 12 1.3.3 Light in- and out-coupling . . . . . . . . . . . . . 14 1.3.3.1 Dye-sensitized solar cells . . . . . . . . 16 1.3.3.2 Colourconvertersfor light-emitting devices.................... 21 1.4 Thisthesis .......................... 27 1.4.1 Objectives ...................... 27 1.4.2 Overview ...................... 28 2 Methods 47 2.1 Processing methods . . . . . . . . . . . . . . . . . . . . . 47 2.1.1 Nanoparticle synthesis . . . . . . . . . . . . . . . 47 2.1.1.1 Crystalline TiO2spheres . . . . . . . . 47 2.1.1.2 GdVO4-based nanophosphors . . . . . 48 2.1.2 Paste preparation . . . . . . . . . . . . . . . . . . 49 2.1.2.1 TiO2nanoparticle paste . . . . . . . . . 50 2.1.2.2 GdVO4-based nanophosphor paste . . 50 2.1.2.3 Integration of optical disorder . . . . . 51 2.1.3 Film preparation . . . . . . . . . . . . . . . . . . 52 2.1.3.1 High refractive index materials . . . . 52 2.1.3.2 Low refractive index materials . . . . . 56 2.1.4 Preparation of multifunctional films . . . . . . . 60 2.1.4.1 Sensitisation of films . . . . . . . . . . 60 2.1.4.2 Flexible films . . . . . . . . . . . . . . . 61 2.1.5 Solar cell fabrication process . . . . . . . . . . . 63 2.1.5.1 Electrode preparation . . . . . . . . . . 63 2.1.5.2 Cell assembly . . . . . . . . . . . . . . . 63 2.2 Characterisation methods . . . . . . . . . . . . . . . . . 64 2.2.1 Structural characterisation . . . . . . . . . . . . . 64 i 2.2.2 Optical characterisation . . . . . . . . . . . . . . 66 2.2.2.1 Reflectance and Transmittance . . . . . 66 2.2.2.2 Photoluminescence . . . . . . . . . . . 68 2.2.3 Optoelectrical characterisation of solar cells . . . 70 2.2.3.1 SpectralIncidentPhoton-to-CurrentEf- ficiency .................. 70 2.2.3.2 J-V characteristics . . . . . . . . . . . . 70 2.3 Theoretical methods . . . . . . . . . . . . . . . . . . . . 71 2.3.1 Response of an individual scattering particle . . 71 2.3.1.1 Mie formalism . . . . . . . . . . . . . . 72 2.3.1.2 Finite-Difference Time-Domain(FDTD) simulations................ 75 2.3.2 Response of a materialintegratingscatteringcentres.......................... 77 2.3.2.1 Scattering and transport mean free path 77 2.3.2.2 Determination of photon trajectory in layered media . . . . . . . . . . . . . . 80 2.3.3 DSSC modelling . . . . . . . . . . . . . . . . . . 82 2.3.3.1 Key performance parameters of DSSCs 83 2.3.3.2 ModellingofanidealDSSC.Schockley- Queisserlimitanddetermination of maximum short-circuit current density . . 85 2.3.3.3 Electron-generationfunctionandlight- harvesting efficiency . . . . . . . . . . . 87 3Mie glasses: fabrication and characterisation 95 3.1 Introduction......................... 95 3.2 Analytical study for different material combinations . . 96 3.3 TiO2scattering centres in a TiO2matrix . . . . . . . . . 101 3.3.1 Material fabrication for the experimental study . 101 3.3.2 Analysis of the optical response . . . . . . . . . 102 3.3.2.1 Characterisation of optical disorder . . 102 3.3.2.2 Finite size effects and photon statistics of light transport in diffusive random media...................108 3.3.2.3 Effects of optical disorder on the absorption of light . . . . . . . . . . . . . 117 3.4 TiO2scattering centres in a low refractive index matrix 119 ii 3.4.1 Ultralow refractive index films . . . . . . . . . . 120 3.4.2 Scatteringpropertiesofanopticallyrandommaterial based on a low refractive index film . . . . 124 3.4.3 Mie glasses as efficient light diffusers: comparison128 3.5 Conclusions .........................130 4Mie glasses for bifacial DSSCs with enhanced performance 135 4.1 Introduction.........................135 4.2 Theoretical study of the system . . . . . . . . . . . . . . 137 4.3 Counterelectrode optimisation and preparation . . . . . 142 4.4 Integrationofscattering centres in photoanodes:absorptance enhancement . . . . . . . . . . . . . . . . . . . . . 145 4.5 Electrical characterization of the cells . . . . . . . . . . . 148 4.6 Conclusions .........................156 5Mie glasses for light emission applications 165 5.1 Introduction.........................165 5.2 TiO2inclusions in a TiO2matrix for efficient colour conversion ............................166 5.3 TiO2scattering centres in a luminescent matrix . . . . . 173 5.3.1 Flexibleandtransparentluminescent mesoporous films .........................174 5.3.2 Optically random material based on a luminescentmatrix......................178 5.3.2.1 Scattering centre integration . . . . . . 178 5.3.2.2 Optical disorder as a route towards efficient light out-coupling in luminescentfilms.................181 5.4 Conclusions .........................188 General conclusions 193 Publications 195 iii 1 Introduction The work described in this thesis addresses the design, fabrication and characterisation of different optically random media consisting of porous transparent matrices integrating a random distribution of monodisperse spherical high refractive index nanoparticles. In these materials, the scattering of light by each centre is assumed to be independent from that of the other particles, and therefore correlation terms can be neglected. This approximation holds for a lowly concentrated and well dispersed distribution of particles in the host matrix. In the first place, optically random media consisting in different combinations of materials exhibiting this configuration are analysed, fabricated and optically characterised. In this thesis, the scattering of light by each centre is assumed to be independent from that of the other particles, and therefore correlation terms can be neglected. This approximation holds for a lowly concentrated and well dispersed distribution of particles in the host matrix. After sensitisation with absorbing or luminescent molecules, their integration into dye-sensitized solar cells (DSSCs), as well as their use for the fabrication of colour-converting materials aimed at implementation in lightemitting devices (LEDs), is assessed. The studies detailed throughoutthisthesisenableddisentanglingfundamentalaspectsoftheprop- agation of light in random media, as well as demonstrating the possibility of taking advantage of the diffuse propagation of light for the fabrication of optoelectronic devices with enhanced properties. This chapter provides a general overview on light scattering in optically random media. The magnitudes usually employed for the characterisation of light scattering as a function of the optical properties of the materials are defined. In order to justify the interest in the study of light propagation in these media, the regimes of weak and strong localisation are defined and some applications of optically random media are discussed. The progress in the control of the scattering properties of turbid media and the different strategies proposed for that purpose are briefly reviewed. Eventually, due to the interest of the integration of the materials proposed in this thesis as photoanodes into DSSCs and as a colour-converting element into LEDs, some 1 1. Introduction background and basic concepts on these optoelectronic devices are provided. 1.1 Light scattering in optically random media Photonics pursues the generation and harnessing of light, and as a consequence it offers numberless applications in a vast variety of technological fields1–3. Specifically, controlling light propagation has been demonstrated beneficial regarding the design and fabrication of optoelectronic devices4–6. Light manipulation usually involves the design and synthesis of materials with specific structural properties aimed at particular applications. As a consequence, ordered architectures structured in the wavelength range presenting a certain periodicity are widely employed7–11. Due to the ordered nature of these nanostructures, they generally rely on diffraction and interference effects for the modification of light transport. In contrast, an interesting strategy for moulding the flow of light is also found in random structures, where strong light scattering takes place12–19. Light scattering is generally understood as any deviation of light from rectilinear propagation caused by any wavelength-scale inhomogeneity, such as a particle of a certain refractive index, and it is generally an elastic event. An optically random medium commonly consists in a dielectric or semiconductor material in which its building blocks, or a fraction of these, are placed in a disordered manner, thus presenting a random modulation of the refractive index. The refractive index contrast between its components and the surrounding medium triggers light scattering at each of these building blocks, which act as scattering centres. Due to successive scattering events, light propagates in a diffusive manner, exiting the material at random directions and endowing it with a characteristic whitish opacity. This is the case of media such as clouds, sugar or colloidal suspensions, which despite their microscopic transparency appear white and opaque due to multiple scattering of the light. Because of the deviation of the light rays, the object appears blurry to the eye on the other side of the random medium, Fig. 1.1(a), and, for large enough thicknesses, the wave vector of the propagating beam loses memory of its initial direction and the source becomes invisible, as illustrated 2 1. Introduction tion. Besides, some of the most frequently employed materials, such as Ge, GaAs or Si, show strong absorption in the visible range of the spectrum, thus limiting the studies to be performed in the infrared region51–55, where they remain transparent. Absorption hinders the analysis of light propagation in random media and especially the observation of localisation phenomena. As for those random materials in liquid phase, i.e colloidal suspensions, the refractive index contrast between the scattering particles and the liquid medium does not provide sufficiently low values of 𝓁𝑠𝑐. Among the interesting optical phenomena derived from the propagation of light through highly scattering materials, the observation of Anderson localisation for electromagnetic waves has been long covetedandtherepeatedlyfailedattemptsatdetectingitin3-dimensional systems has made it become a holy grial in the study of optically random media. Above a certain degree of disorder, the diffusion of the light waves tends to zero and transport comes to a halt. This phenomenon is known as Anderson localisation and it derives from backscattered waves interfering constructively. In the limit of localisation, 𝓁𝑡→0, i.e transport is not possible, and diffusion theory fails at describing light propagation. Despite being proposed for electrons coherently scattered in semiconductor crystals presenting a certain degree of disorder20, due to the wave nature of the electron, Anderson localisation could be expected in any wave system, such as light21,22,61. Andersonlocalisationispredictedforstronglyscatteringmediawhere 𝑘𝓁𝑠𝑐 ≃1, according to the Ioffe-Regel criterion. In order to approach this condition, 𝓁𝑠𝑐 needs to be minimised, that is, 𝜎𝑠𝑐 should be maximised, according to Eq. 1.1, which is achieved for a particle size in the regime of Mie scattering. In the pursuit of experimental evidence of Anderson localisation of photons, scientists came across the observation of weak localisation of light, regarded as the precursor of strong or Anderson localisation13,23. Weak localisation is a consequence of time-reversal symmetry of the paths of light multiply scattered in a randommediumanditmanifestsitselfasacone of enhanced backscattered light. A light wave originated at a source entering a random material can undergo several scattering events and exit the material in the backward direction towards the source, accumulating a phase shift. Another wave can however follow the same path in the opposite direction due to time-reversal symmetry, developing the same phase 9 1. Introduction shift. The interference between these waves is constructive, resulting in an increase of intensity in the backward direction. Determination of the enhanced backscattering cone has been proved useful to understand the propagation of light in strongly scattering materials, since its width enables extraction of 𝓁𝑡57. Due to the interest in tuning and controlling optical disorder, efforts have particularly been focused on the fabrication processes of these materials and the subsequent analysis of the evolution of their optical response. In the first place, liquid suspensions employing PS spheres as scattering centres offered an easy manner to tune the amount of disorder through the size and concentration of PS spheres. Besides, the spherical shape of the centres ensured identical light scattering by all the inclusions, regardless of the incidence direction of the light. Indeed, the change of the features of the backscattering cone in the range of weak localisation, which are directly related to 𝓁𝑡, when varying the parameters dictating the conditions of the optical disorder, that is, either the size12,58,62 or the concentration12,13 of scattering particles, has been analysed in several ocassions. Wolf et al.62 also considered the effect of different degrees of absorption in the medium by adding a luminescent dye (Rhodamine 6G) in various concentrations. Later on, porous gallium phosphide (GaP) attracted interest as an optically random medium. It is a semiconductor material presenting a high bulk refractive index (n = 3.45 at 𝜆= 550 nm), transparent in part of the visible spectrum (𝜆> 550 nm) and, as a semiconductor, it offers the possibility to be manipulated in order to controllably introduce porosity by means of electrochemical etching of a single crystal. This allowed the fabrication of GaP chunks enabling control over the photonic strength through the amount of introduced porosity16,63,64. Further advances in the fabrication of turbid media with tunable scattering properties were made with the synthesis of semiconductor nanowires. Systems consisting in arrays of GaP, Si and InP nanowires have demonstrated exceptional photonic strength65, while providing control over their scattering through their diameter, volume filling fraction and alignment66. Apart from the study of the characteristics of light propagation in random media, these materials also attract interest owing to the possibility of their use in numberless applications, such as optical imaging in biological systems67, wavefront 10 1. Introduction shaping for sharp light focusing68,69, diffusing wave spectroscopy70 or implementation in optoelectronic devices40,65. 1.3 Applications of optically random media The characteristics of light propagation in optically disordered materials can find a variety of situations in which to be exploited with a specific purpose. In this section, some of the most common applications of these materials are briefly reviewed. 1.3.1 Random lasing Since the proposals of R. V. Ambartsumyan et al.71 and V. S. Letokhov et al.27, optically disordered materials for the generation of laser-like emission have been intensively studied. In traditional laser action, light scattering in the resonant cavity has always been considered a nuisance, since it removes photons from the lasing mode. However, under the right conditions, strong light scattering in a disordered medium can actually be fundamental to trigger laser action. In a random medium with optical gain, recurrent scattering can serve as a feedback mechanism for the amplification of light, analogous to the (a) Scattering medium with optical gain (b) Figure 1.6: (a) Illustration of the generation of closed loops due to strong light scattering in an amplifying medium giving rise to laser action. (b) Emission spectrum of ZnO powder displaying laser action at an excitation intensity of 875 kW cm−2 by a pulsed Nd:YAG laser at 𝜆= 355 nm (taken from Ref.[29]). 11 1. Introduction optical cavity in a conventional laser, as illustrated in Fig. 1.6(a). Laser action in a regular system is achieved above a threshold, where the total gain in the cavity surpasses the losses. In a strongly scattering medium, if 𝓁𝑠𝑐 becomes equal to or shorter than the wavelength, light canformclosed loop pathsthrough successive scattering,undergoing amplification. In the limit where such amplification exceeds losses, laser action occurs. In a diffusion process with amplification the total gain is defined by the volume, whereas losses are proportional to the total surface27. It is therefore possible to achieve a situation where the intensity diverges and laser action is triggered in a random medium. The emission spectrum of a random laser is typically broader in frequency than in a conventional laser, and narrow spikes are usually present. Furthermore, the statistics of photons emitted by a random laser resembles to a great extent that of regular laser emission. The frequencies and the direction of emission are dictated by the modes formed in the random structure. A random system for lasing action can be in principle easily fabricated by grinding laser crystals72,73, suspending high refractive index particles as individual scattering centres28,74 or forming scattering clusters75–77 in a dye solution, in semiconductor powder29, photonic glasses78 or in organic molecules employed as a gain medium79. 1.3.2 Imaging through optically random media When light coming from a source impinges on an optically disordered material, it is successively scattered and a fraction of it can reach the opposite interface of the material and exit as transmitted light. As previously mentioned, for a weakly scattering medium, a significant fraction of the light undergoes no scattering and it is ballistically transmitted. In this situation, the shape of the source can be retrieved. The image, however, appears somewhat blurry due to the light scattered and diffusively propagated. Here, image formation can be restricted to discerning between scattered and unscattered light and collect those photons corresponding to the latter. If the scattering in the material is too large, ballistic propagation is seriously hindered and information about the source becomes scrambled. As a result of this, the light transmitted by the material forms an apparently random speckle pattern.Fullcharacterisationof thescatteringsystem,which entails know- 12 1. Introduction ing the complete scattering matrix describing the disorder, would enable undoing the scattering and retrieving an image of the source24. Unfortunately this is not always feasible, since the usually countless elements forming the scattering matrix make it impracticable. An alternativemethod forunscramblingthisinformationreliesonthewave natureoflight,whichenablesinterferencebetweenthe scatteredwaves. Such interference carries phase information, which in principle enables compensating for the disorder regardless of the complexity and decode the information80. Biological tissue, such as the skin, generally behaves as a highly scattering medium. Unravelling the information hidden in the propagated light waves would enable imaging of objects underneath in a non-invasive manner. As a consequence, studies focused on the reconstruction of a source placed on the other side of a strongly scattering layer have been performed25,26,82. In particular, the procedure proposed by O. Katz et. al25 enables the reconstruction of an object between two scattering screens, as illustrated in Fig. 1.7. Light emitted by a source passes through a scattering screen and illuminates an object, which consists in a shape carved in a black screen. The object acts as an extended source and light becomes scrambled when passing through the second diffuser, which is afterwards detected by a Figure 1.7:Principle of reconstructionofan object placedbetweentwo scattering screens proposed by O. Katz et. al (taken from Ref.[81] and Ref.[25]). 13 1. Introduction camera. The recorded image consists in a virtually constant intensity distribution displaying some random fluctuations. Autocorrelation of such intensity distribution allows retrieving information on the shape of the object for its reconstruction. 1.3.3 Light in- and out-coupling In any application involving light harnessing, there are certain common issues, such as optical losses or parasitic absorptions, that are virtually inherent to any optical device and whose minimisation is generally desired when seeking for optimal performance. Since light management is performed through specific structures designed for a particular function, the incoming light must be in the first place coupled into the system. The presence of interfaces inevitably produces undesirable reflections reducing the fraction of light available for use. This is a well-known issue in the field of PV, where the high refractive index of the active materials hampers an efficient in-coupling of the incoming light, thus preventing maximal utilisation of the incident photons for photocurrent generation. In thin film technology, conceived to minimise material usage, insufficient optical thickness represents an additional drawback againstmaximal performance. On one hand, losses due to inefficient in-coupling are minimised through the implementation of anti-reflective coatings. On the other hand, texturised structures of a size in the wavelength range are integrated in order to cause a light-trapping effect due to successive light scattering and total internal reflection events and boost light absorption in the devices. Although such structures are commonly applied at the front transparent conductive oxide, they can also be found at deeper interfaces. In that sense, the integration of either inverted83 or upright pyramids4,83, as displayed in Fig. 1.8(a), or textures with randommorphologies is regarded as the standard procedure83–90. Two examples of disordered textures for enhanced light in-coupling are displayed in Fig. 1.8(b) and (c). Randomly textured structures present the advantage of allowing an in-coupling of the incident light into the absorber layer in a broad spectral range. They generate a Lambertian light distribution inside the cell due to diffuse light propagation. Periodic architectures have also been proposed91–93 and have demonstrated strong absorption enhancements89,94,95, yet their spectral reso- 14 1. Introduction Figure 1.8: Examples of structures for in- and out-coupling applications. (a) Micrograph of a periodically textured Si surface in the shape of upright pyramids (taken from Ref.[83]). (b) SEM image of a random texture on a Si (taken from Ref.[86]). (c) SEM image of the random surface on polycrystalline boron-doped ZnO film constituted by columnar grains (taken from Ref.[87]). (d) Photonic-crystal- based LED (upper picture) and top-view atomic force microscope of the periodic structure of the photonic crystal lattice (lower picture) (taken from Ref.[6]). (e) Illustration of the structure of an OLED integrating a disordered structure (scattering layer) for enhanced light extraction (taken from Ref.[37]). nances are typically sharper than in a disordered approach. Periodic structures provide a well-defined Fourier spectrum with strong, but scarce diffraction orders, while random architectures generate a large number of diffraction orders of weak intensity. There is, however, an optimal structure providing maximum light-trapping between these 15 1. Introduction two limit situations. With the purpose of reaching this optimum intermediate configuration, Martins et al.96 proposed a design based on the supercell concept, i.e. gratings of larger period containing smaller unit cells, allowing to tailor the Fourier spectra created by the structure through controlled design of its aperiodicity. Theoppositesituationalsoposesaproblem.Inphotoluminescence (PL) applications, the inefficient out-coupling of the generated light from the emitting material results detrimental for the eventual efficiency of the device. In the situation of light generated in a film of homogeneous interfaces, due to the different refractive index value of the emitting material to that of air and taking into account that light in a luminescent material is generally emitted in all directions, a fraction of the emitted light inevitably remains trapped inside. For light emitted at angles above the critical angle defined by the escape cone, which is dictated by the refractive index contrast between the film and the surrounding medium, total internal reflection occurs and eventually results in unusable light. Inefficient out-coupling has especially been addressed in the field of light-emission for lighting applications and approaches based on either ordered6,97 or disordered architectures, as shown in Fig. 1.8(d) and (e), respectively, have been proved to partly enhance light extraction34,37,38,98. Optical disorder can therefore serve as a tool to increase light incoupling in absorbing materials, while also be responsible for a more efficient light-extraction in photoluminescent materials. Scattering in a random medium is capable of enlarging the path length of the light and thus its residence time inside, which could result beneficial for boosting optical absorption and emission or to increase the PL of emitting films due to processes of improved out-coupling. The work presented in this thesis focuses specifically on the integration of materials presenting optical disorder as photoanodes into bifacial DSSCs and as conversion coatings into LEDs. 1.3.3.1 Dye-sensitized solar cells In the current picture of global warming, progressive depletion of fossil fuel reserves and the alarming expected population growth, aggravated by a constant increase in the energy demand by the world population, the task of meeting the human energetic needs while min- 16 1. Introduction imising the harmful impact on the environment appears as an urgent challenge to overcome. The still excessive dependence on fossil fuelbased resources for the performance of human activities is leading to serious environmental issues. In this regard, alternative energy resources represent a peerless option, since they are generally based on abundant natural resources, while offering a meaningful reduction of the consequences derived from the use of fossil fuels. Among the different alternative energy sources, solar energy is proving to be a powerful contender in order to meet the world’s energy demand without the environmental costs. In effect, the Earth’s atmosphere receives an average power density of 1367 W m−299, reaching the surface around 1000 W m−2 mostly due to absorption and scattering at the atmosphere. Most of the available PV techonology in use is based on silicon, commonly known as first generation PV, for which devices have reached efficiencies above 25%100. The development of a second generation of PV devices based on thin film technology overthrew silicon as the active material for solar cells, as modest efficiencies were demonstrated while reducing material usage. In spite of the benefits of such technologies, they however present considerable drawbacks derived from the high manufacturing costs (high purity of the active material, complex fabrication processes, high processing temperatures), toxicity, pollution and opacity hindering integration in buldings101,102. A third generation PV emerged in an attempt to overcome these disadvantages limiting the operation of the technologies belonging to the first two generations. Third-generation devices enable low-cost PV, since they generally encompass technologies requiring easy and inexpensive fabrication procedures and low material usage. This third generation comprises four main technologies and diverse variants of these, namely, organic solar cells, quantum dot solar cells, dye-sensitized solar cells and a novel and revolutionary technology based on the use of a material comprising a perovskite structure as active material. Perovskite-based solar cells are especially relevant due to their rapid and unprecedented development, placing them at a position to potentially compete against the current commercial and mature silicon-based technologies. In fact, perovskite-based PV started as a DSSC technology, which employed organometal halide perovskitecrystalsassensitisersofaTiO2photoanode103,104. The DSSC 17 1. Introduction appears as an attractive alternative to conventional PV devices due to their unique features, such as high transparency and lightweight105. They have been successfully developed on flexible substrates106 and they have demonstrated remarkable performance under indoor illumination107. Additionally, they offer the possibility of colour choice108, which results advantageous for architectural purposes, and they have demonstrated considerable stability109–111. Bifacial devices can be fabricated with them for harvesting of light impinging from all directions112,aswell as solar modules102,113, all of theabovepossiblethrough inexpensive fabrication procedures. DSSCs have seen a considerable wide number of applications due to different industrial sectors focused on their development. For instance, the Japanese company Fujikura has developed DSSCs for integration into indoor electronics, such as mobile phones. G24 Power started a nearly 5500-m2DSSC plant in 2007 while also achieving integration of these devices into flexible waterproof bags. Mini modules were fabricated by 3G Solar for charging electronic devices ranging from surveillance cameras to medical devices and Gunze Ltd. achieved the development of a wearable DSSC102. This thesis focuses on DSSCs as a target photovoltaic device for the implementation of optical disorder, mostly owing to their potential for industrialisation. Moreover, they offer the possibility to be manufactured at a large scale, since the materials are easily available, and fabrication by roll-to-roll processing is possible. This section introduces basic concepts of a DSSCs. In particular, the working principle of a DSSC is depicted and the limitations revealing the need for approaches in order to boost their performance addressed. Working principle of a DSSC: A DSSC is a photoelectrochemical system capable of generating a photocurrent from sunlight absorption. Ever since the first demonstration of a DSSC presenting sufficient absorption to become a competent PV technology114,115, a wide variety of approaches have been progressively developed and implemented in the cell to reach a maximum power conversion efficiency (PCE) close to 12%100,116. In these devices, the active material is an absorbing dye anchored to a wide-bandgap semiconductor. The typical architecture of a DSSC consists of a film of TiO2nanoparticles sen- 18 1. Introduction ange emission due to the Eu2+ and Mn2+ centres, respectively, have demonstrated white emission under UV irradiation137. Another disadvantage is the higher energy losses that are expected in the colourconversion processes as the excitation energy is increased. That in combination with losses owing to absorption of phosphor-emitted light by the other phosphors, places the strategy considering a blueemitting LED as the most commonly implemented. The use of PC-LED architectures entails drawbacks derived from employing inorganic phosphors as converting material. In particular, the high cost of the RE cations introduced as luminescence centres in most inorganic phosphors in combination with their processing, which usually involves high temperatures, high pressure or reducing conditions, limits their use. In that regard, other promising alternatives are being explored. Quantum dots (QDs), for instance, are attractive candidates. One reason for this is their high quantum yield (QY). In fact, QY values as high as 98% (for CdSe/CdS QDs)138 have been achieved as a consequence of the efforts focused on the maximisation of the QY of QDs139. Furthermore, they are susceptible to be easily dispersed in a polymeric matrix for integration into an LED device, in analogy to YAG:Ce phosphors. An intermediate approach involving inorganic phosphors and QDs was devised by H. S. Jang et al.140, who proposed the combination of red-emitting CdSe QDs with broad-emission greenish-yellow-emitting Sr3SiO5:Ce3+,Li+co-doped with Pr3+ so as to enhance the colour rendering of white-emitting LEDs. Another alternative currently under investigation consists in the use of organic phosphors for the processes of down-shifting, giving rise to the so-called white hybrid light-emitting diodes (WHLEDs). The group of organic phosphors encompasses a wide range of materials, such as organic dyes141,142, carbon nanodots143,144 or fluorescent proteins145,146. The emergence of this hybrid type of luminescent devices has attracted attention owing to their promising advantages. Whereas WHLEDs offer an easy design of the emission properties, high colour quality and a cost-effective and considerable environmentally friendly manufacture, their currently low thermal and chemical stability prevent them from becoming reliable and widely employed sources for lighting applications. 25 1. Introduction The possibility of employing nano-sized RE-doped phosphors has also been proposed as an alternative to common inorganic RE-based phosphors, which usually present a grain size of the order of the micrometer, owing to their properties, such as size tunability for improved optical quality, reduced light scattering or narrow emission band. The uncontrolled light scattering of YAG:Ce phosphor-based conversion layers hampers an optical design for a desired optical response of the material. In contrast, nano-sized phosphors offer the possibility of optical quality when employed for the fabrication of films, thus allowing control over the optical properties of the luminescent films through different strategies, such as nanostructuring or integration of particles. Current drawbacks and limitations of LED technology: Despite the benefits of the LED technology, there are still some issues to be overcome in pursuit of efficient operation and safety. Among those issues, two of them present special relevance. One is related to the strong blue component in the emission spectrum of white PC-LEDs based on a blue diode, Fig. 1.12(a). Figure 1.12(b) displays the emission spectrum of a standard white-light LED, which reveals an important contribution of the blue light emitted by the chip to the device emission spectrum. Whereas blue light has been proven to play a crucial role in mood147 and the correct regulation of circadian rhythms148, Figure 1.12: (a) Schematic of a light-emitting device in a PC-LED configuration. (b) Emission sprectrum of a typical white LED comprising a blue diode coated with yellow-emitting Ce:YAG phosphors as conversion material. 26 1. Introduction there are signs that it could in turn become a negative agent due to incessant exposure, especially for the eye, owing to the high transparency of the cornea and the lens to blue light149. Moreover, outdoor applications of white PC-LEDs could result in an intensification of light pollution due to the high scattering cross section of the molecules in the atmosphere at shorter wavelengths. Light pollution has been linked to a series of harms not only in humans, but especially wildlife, such as birds150–152, insects153 and trees154. Furthermore, as previously mentioned, the use of YAG:Ce as standard phosphors in combination with a blue-emitting chip yields poor colour rendering, as well as cold shades of white. The other main issue derives from the large refractive index of the materials required for the fabrication of the different components of LEDs. As a consequence, a significant fraction of the light produced in the active region can be trapped by means of total internal reflection in the different elements comprising the LED device, therefore not contributing to the usable light. In this regard, strategies for the enhancement of the extraction efficiency, such as surface roughening155 or nanostructures integration into the diode156, have been proposed. Hence the importance of developing approaches in order to achieve more efficient processes of light outcoupling from the different components of an LED device. 1.4 This thesis 1.4.1 Objectives The work depicted in this thesis focuses on the study of the phenomena of light absorption and emission enhancement in optically random media, as well as their integration into light-emitting and thirdgeneration PV devices. The project proposes the fabrication of novel optically disordered media by randomly distributing scattering centresofcontrolledsizeandshapeinfilmsoftransparentmatrices,which enables a design of their scattering properties. A series of objectives are pursued from a fundamental point of view. In the first place, the project proposes the fabrication of optically disordered materials with the appropriate features to originate an enhancement of the optical absorption and emission. Gaining un- 27 1. Introduction derstanding of the effect of the phenomenon of multiple scattering in these materials on the processes of light absorption and emission by dye molecules embedded in their structure or by nanomaterials is sought for. At the same time, it is of interest to find if the features of these optically disordered materials, which display a photonic glass structure, can yield a resonant enhancement of the optical absorption and emission. Eventually, a complete characterisation of their optical response is required in order to analyse how light propagates in them, as well as to assess their suitability for integration into optoelectronic devices. From a technological perspective, a double objective is pursued. Firstly, an enhancement of the PCE of DSSCs by integrating different amounts of optical disorder into their sensitised electrode is proposed. The mechanism responsible for such improvement is based on an increase of the residence time of light inside the photoanode as a consequence of multiple scattering phenomena. A longer residence time of light would in principle boost the absorption probability and thus give rise to an increase of the number of generated photocarriers. The other side of this technological objective aims at increasing the efficiency of light-emitting devices integrating conversion layers comprising phosphors in combination with optical disorder. 1.4.2 Overview The research activity giving rise to the results depicted in this thesis was performed in the Institute of Materials Science of Seville under the supervision of Prof. Dr. Hernán Míguez and Dr. Gabriel Lozano. Realisation of this work was possible through funding received by the Spanish Ministry of Education and Vocational Training through an FPU programme under grant FPU14/00890. The results obtained during the development of this thesis project are depicted in the next chapters as follows: • Chapter 2 details the different experimental procedures employed for the fabrication of the diverse components of the Mie glasses, such as the synthesis of the scattering centres and nano-phosphors, as well asthepreparationoftransparentmatrices. The fabrication of DSSCs is also depicted. A second sec- 28 1. Introduction tion is devoted to the characterisation techniques used for the analysis of different aspects of light propagation and generation in the materials of study. Here, the approaches to perform a photoelectrical characterisation of DSSCs required for assessment of their operation are also described. The last section presents the theoretical methods required for the analysis of the systems at some point of the work, ranging from theoretical formalisms to numerical approaches employed for the simulation of the fabricated materials and their optical properties, as well as electrical characteristics of DSSCs. • Chapter 3 focuses on the analysis of light transport in the different fabricated Mie glasses and the characterisation of the optical disorder based on 𝓁𝑠𝑐 and 𝓁𝑡. In the first place, simulations of different combinations of materials for which strong scatteringisexpected are presentedandthesuitabilityofsuchcombinations for the aims of the work evaluated in terms of 𝜎𝑠𝑐 and 𝓁𝑠𝑐, thus justifying the combinations eventually selected. From these results, the optical properties of transparent matrices of TiO2and SiO2are analysed and the effect of the presence of disorder on their optical response is studied. The possibility of describing light propagation in these materials according to Mie formalism is investigated. • Chapter 4 is devoted to the integration of the TiO2-based Mie glass as photoanode into bifacial DSSCs. A theoretical study on the performance of the cell in relation to the features of the scattering centres is presented, which enables an optical design of the device. After that, an electrical characterisation of the fabricated cells allowing an evaluation of the cell operation with respect to the properties of the scattering centres is presented. Their performance as bifacial devices is also assessed. • Chapter 5 presents thestudyofemissionenhancement in those Mie glasses displayingluminescence.In thefirstplace,theemission improvement in the TiO2-based Mie glass when sensitised with molecules of an emitting dye as a function of the parameters of the integrated inclusions is evaluated. The last section 29 1. 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Eventually, the last section depicts the analytical and numerical tools employed for the modelling and design of the studied systems, providing the theoretical insight required for a thorough understanding of their properties. 2.1 Processing methods The highly experimental character of this thesis involves the use of a vast number of synthesis and fabrication methods. This section addresses experimental methods oriented toward the fabrication of random media for integration into either DSSCs or light-emission applications, giving details on the techniques and the implicated processes. 2.1.1 Nanoparticle synthesis This section provides references to the synthetic routes of nanoparticles employed in this research. Specifically, a procedure for the synthesis of spherical crystalline TiO2nanoparticles and the synthesis method of nano-sized phosphors are mentioned. 2.1.1.1 Crystalline TiO2spheres Spherical anatase particles in the sub-micrometre scale were used as scattering elements in order to endow transparent matrices with optical disorder. TiO2spheres were synthesised according to a modified 47 2. Methods 500 nm (a) 20 40 60 80 2 / degreesθ Intensity (b) Figure 2.1: (a) TEM image of the spherical structures resulting from the synthesis process before sintering. (b) X-ray diffraction diagram of the synthesised spheres before (upper half) and after (lower half) sintering. version of a procedure reported elsewhere1. This procedure consists in a sol-gel method combining a step of controlled hydrolysis and nucleation with a subsequent step of hydrothermal reaction. The spherical structures employed in this work correspond to those resulting from the former process, therefore omitting the step of hydrothermal reaction. The Transmission Electron Microscopy (TEM) image in Fig. 2.1(a)illustrates thesphericalshape of theresultingparticles afterthe synthesis process, which were in the amorphous phase, as evidenced by the upper X-ray diffraction (XRD) diagram in Fig. 2.1(b). A crystalline phase was displayed after the sintering process performed to achieve mechanical stability of the materials, as the lower XRD diagram in Fig. 2.1(b) demonstrates. The method allowed control over the size of the spherical structures through the hydrolisis conditions, more specifically, through the volume of H2O added to the initial solution. 2.1.1.2 GdVO4-based nanophosphors Due to the interest in the preparation of mesoporous luminescent matrices, near-UV excitable, Europium (Eu)- and Bismuth (Bi)-doped phosphors of nanometric size based on rare-earth (RE) vanadates, namely, GdVO4, were synthesised according a procedure already reported2. The synthesis is based on a precipitation process and it pro- 48 2. Methods (a) 200 nm 20 40 60 80 2 / degreesθ Intensity (b) Figure 2.2: (a) TEM image of the synthesised GdVO4:Bi3+,Eu3+ nanophosphors. (b) XRD diagram yielded by dry GdVO4:Bi3+,Eu3+ powder (upper half), along with the standard PDF 86-0996, ICDD 2014 reference (lower half) included for comparison. duced phosphors of an average size of (36 ±7) nm, as visible in the TEM image shown in Fig. 2.2(a). Such small size prevented visible light from being significantly scattered, thus yielding high transparency when used for film preparation. Co-doping with Bi3+ endowed the phosphor crystals with a cubic shape, as concluded after comparing with phosphors devoid of Bi3+ centres. The particles resulting from the synthesis presented an amorphous phase, which converted into crystalline after a thermal treatment, as confirmed by the XRD pattern shown in Fig. 2.2(b). Comparison with the standard PDF 86-0996, ICDD 2014 reference demonstrated crystallisation into the tetragonal GdVO4structure. A crystallite size of 39.5 nm was estimated from the Scherrer formula, in good agreement with that determined from the TEM image in Fig. 2.2(a). Alternatively, the Eu3+ cations could be replaced by Dy3+ cations so a to achieve a different emission spectrum by substituting the Eu-salt by the corresponding Dy-salt during the synthesis. 2.1.2 Paste preparation Most of the materials designed and analysed in this thesis were fabricated from precursors in the form of a viscous paste. These pastes generally consisted of a mixture of the nanoparticles and organic binders 49 2. Methods in a viscous solvent. Using pastes as precursors is revealed advantageous when aiming at large film thicknesses not feasible through other solution-processing methods. Besides, the deposition methods compatible with paste precursors can offer acceptable control over the thickness of the film. 2.1.2.1 TiO2nanoparticle paste Mesoporousanatasematriceswere prepared from a commercial paste (18-NRT, GreatCell Solar©) containing irregular TiO2particles with an average size of 20 nm. Films prepared employing this precursor displayed high optical transparency in the visible region. 2.1.2.2 GdVO4-based nanophosphor paste Transparent luminescent films comprising GdVO4-based nanophosphors doped with Bi3+and a RE cation were fabricated from a viscous paste. For the paste preparation, the total weight of the nanophosphors, mnph, in a methanol suspension resulting from the synthesis was determined in the first place. Further methanol was added to produce a 120-ml diluted suspension. Subsequent sonication was performed by means of tip sonication so as to minimise aggregation of the phosphors for the sake of optical quality of the resulting films. As organic binder, 0.3⋅mnph grams of ethyl cellulose (Sigma Aldrich, powder) were added to the suspension, followed by tip sonication. 4⋅mnph grams of 𝛼-terpineol (SAFC, ≥96%) were added as a solvent Figure 2.3: Description of the preparation process of a viscous paste of GdVO4:Bi3+,Eu3+ nanophosphors 50 2. Methods putable requirement towards this goal. Through combination with scattering centres of high refractive index, intense scattering is expected, since the scattering cross section of an inclusion, and thus the scattering strength, is highly dependent on the refractive index contrast between the inclusion itself and its surrounding medium. With this purpose, films consisting of a highly porous SiO2frame- work with a porosity close to 90% were fabricated from PS-sphere films via infiltration through CVD. The fabrication procedure consists of three steps: preparation of a scaffold consisting in a film of spherical PS particles randomly distributed, sequential infiltration of this scaffold through CVD for the formation of SiO2in the pores between the PS spheres and removal of the PS scaffold by means of a thermal treatment. In order to fabricate films consisting of disordered stacks of spherical PS particles, a suspension of PS spheres in a mixture of water and absolute ethanol as solvent was prepared. In particular, 20 ml of a 10 vol% dispersion of PS spheres of diameter d = (168 ±1) nm in water (IKERLAT Polymers S.L.©) were centrifuged at 24000 rpm for 40 minutes and the solvent removed. 9 ml Milli-Q water and 9 ml absolute ethanol were added to the solid part in order to prepare a 10:45:45 (v/v/v) dispersion. Eventually, the dispersion was sonicated for redispersion of the PS spheres. For the preparation of a film integrating TiO2spheres, the scattering centres were added from a dispersion in absolute ethanol to the PS-sphere dispersion. The added amount was dictated by the desired volume filling fraction, which can be calculated through Eq. 2.1 and 2.2, for which k = 1 has to be considered in this material. Figure 2.9: Deposition of a film from a liquid solution through spin coating. 57 2. Methods Figure 2.10: Process of deposition of a PS-sphere film via spin coating from a dispersion of PS spheres. The films were deposited via spin coating using this dispersion of spherical PS particles. Spin coating is one of the most common techniques for the deposition of thin films on a substrate. It relies on the centrifugal force originated by the high-speed rotation of a substrate for its coating from a small amount of a solution or a dispersion, usually in a volatile solvent. Spinning of the substrate causes spreading of the solution, while accelerating solvent evaporation, as illustrated in Fig. 2.9. After complete evaporation, often promoted through thermal annealing, a conformal layer results. Thickness of the eventual film depends exclusively on the rotation speed and the viscosity of the solution, offering an estimated precision around ±10 nm. 200 𝜇L of the suspension were casted on a 2.5 x 2.5 cm2glass substrate, spun at 2000 rpm for 40 s and left to dry for 10 minutes at 60𝑜C, see Fig. 2.10. A second deposition with the same conditions was performed. Eventually, the layers were dried overnight at 60𝑜C, yielding films constituted by spherical PS stacks of thickness between 1-2 𝜇m, as visible in Fig. 2.11. 500 nm Figure 2.11: SEM cross section image of a film constituted by PS spheres prepared by spin coating. 58 2. Methods For the fabrication of the material of interest, the PS scaffold was infiltrated through CVD. This deposition technique consists in the infiltration through the porous structure of a material serving as a scaffold with one or more volatile precursors in vapour phase. Reaction of silicon tetrachloride, SiCl4, and water in vapour phase was promoted inside the pores of the PS-sphere film through sequential infiltration. In a first stage, the film was infiltrated with SiCl4in vapour phase, followed by infiltration with water vapour, triggering the formation of a continuous SiO2layer in the pores of the PS structure, as well as some by-products. Specifically, reaction between SiCl4and H2O triggers the formation of silicic acid, Si(OH)4, and chloridric acid, HCl. This HCl catalyses in turn the formation of H2O and siloxane bonds from ≡Si-OH bonds3. For the deposition procedure, a homemade set-up was built. The setup consists of two lines like the one displayed in Fig. 2.12, for the circulation of the two precursors, which are stored in individual glass tubes. The gas flow is controlled by means of valves, which are manually opened and closed. Each line includes a N2(g) inlet, which is N2 Bubbler Reactor Basic solution Figure 2.12: Schematic of a gas line of a homemade setup in order to perform infiltration through Chemical Vapour Deposition (CVD) of a material in vapour phase. 59 2. Methods used to generate a gas flow throughout the line, bubble the precursors and enable their transport. The film is placed inside a tubular quartz reactor, carefully sealed in order to allow maintaining an inert atmosphere during the infiltration process. A first step of purging of the reactor by successive N2filling and emptying is required in order to remove any possible reactant inside the chamber. The line presents a gas outlet leading to a basic solution for neutralisation of the acidic by-products of the reaction. The first precursor, SiCl4(g), is bubbled by N2flow and transported throughout the line to the reactor, where infiltration of the film occurs. In order to trigger the reaction and subsequent formation of SiO2, a process of H2O(g) infiltration by N2 bubbling is subsequently performed. Posterior N2flow is advisable before sample removal in order to fully extract any acidic reactant left in the reactor chamber. For the formation of SiO2throughout the PS scaffold, 1-minute infiltration processes with each reactant were perfomed. Eventually, a thermal treatment consisting in a heating process up to 400𝑜C for two hours, where it plateaus for one hour, was performed in order to remove the PS scaffold, yielding an around 2 x 2-cm2film consisting in a SiO2network with porosities up to 90%. 2.1.4 Preparation of multifunctional films The porosity imposed upon the films enables infiltration with different fluorophores, chromophores or polymers in order to endow them with further functionality. According to such added functionality, the system can result suitable for specific applications. This part of the chapter briefly depicts the procedures of sensitisation of the materials with diverse molecules for particular purposes, as well as a procedure to transfer them into a flexible support. 2.1.4.1 Sensitisation of films With the purpose of endowing the material comprising a mesoporous TiO2matrix with functionality and analyse its suitability in particular applications, it was infiltrated with two types of dyes. 60 2. Methods Photoanode for DSSCs: Charge generation in DSSCs is enabled by anabsorbingdyeanchoredtotheinternalsurfaceof theiranatasephotoanode. For sensitisation of the material for DSSC integration, the films were immersed overnight in a 0.5 mM solution of N719 (Great- Cell Solar©), commonly employed in the field of DSSCs. After that, the infiltrated films were rinsed with absolute ethanol and dried with pressurised nitrogen. Colour conversion: Inorder todemonstratethepotentialoftheTiO2- based Mie glass as an efficient colour converter for solid-state lighting applications, it was infiltrated with fluorophores. Specifically, films of thicknesses around 7.5 𝜇m were sensitised through overnight immersion in a 0.075 wt% solution of a perylene dye, namely, Lumogen®F Red 305, BASF, in ethanol. 2.1.4.2 Flexible films The porous nature of the prepared films enabled infiltration with a polymer, which allowed endowing the material with flexibility4. The result was a self-standing and flexible version of the depicted materials. In this section, the procedure is depicted for the particular case of nanophosphor films, although application to the TiO2-based matrix is completely analogous. Inordertopreparea flexibleversion of a nanophosphor film, a sacrificial layer underneath the emitting material was required. Specifically, a dense SiO2layer was coated on the substrate from a SiO2 solution, which was prepared by diluting 18 ml tetraethyl orthosilicate (TEOS, Sigma Aldrich, 98%) in 136 ml absolute ethanol. 6.88 ml of Milli-Q water were added after vigorous stirring for some minutes. Subsequently, 0.32 ml of a 0.05 N hydrochloric acid (HCl, VWR, 37%) solution were added. After stirring for 24 hours, the solution was ready for use. The SiO2was coated on a glass slide by means of dip coating. Dip coating is a solution-processing technique for the deposition of thin films of precisely controlled thickness with an estimated precision around ±10 nm, Fig. 2.13. A dip coating process generally consists in the immersion of a substrate at a constant speed in a solution or particle suspension. After a dwell time for full wetting of the substrate, it is withdrawn at a controlled speed. The withdrawal 61 2. Methods Figure 2.13: Deposition of the sacrificial SiO2layer. speed is a key parameter allowing control over the thickness of the resulting layer, along with the concentration of the solution. Eventually, evaporation of the solvent leads to a thin film. In this work, the SiO2sacrificial layer was prepared by immersing a glass slide in the solution previously depicted and withdrawn at a speed of 150 mm min−1. The wet substrate was heated up to 500𝑜C for 30 minutes so as to remove any organic residue and achieve conformation of the film, yielding a transparent SiO2film. Poly(methyl methacrylate) (PMMA) was employed as support material and thus used for infiltration. The polymer was infiltrated through the pores of the nanophosphor film from a 15 wt% solution of PMMA (Alfa Aesar, powder) in anisole via spin coating at 2000 rpm for 40 seconds, followed by drying at 60𝑜C for at least one hour, Fig. 2.14(a). For etching of the SiO2sacrificial layer and release of the flexible film, the infiltrated layer was immersed in a 1% hydrofluoric acid (HF, Fluka, 48%) solution for 40-60 minutes, Fig. 2.14(b). After Figure 2.14: (a) Polymer infiltration of the nanophosphor film. (b) Immersion in HF solution and flexible film detachment. 62 2. Methods detachment, the flexible film was rinsed abundantly in water in order to wash away any HF trace. 2.1.5 Solar cell fabrication process The experimental procedure for the fabrication of the solar cells is depicted in this section. The preparation process of solar cells comprising a Mie glass-based photoanode encompasses the synthesis of a viscous paste for the subsequent deposition of the photoanode, the sensitization of the photoanode with an absorbing dye, the preparation of the counterelectrodes, cell assembly, electrolyte infiltration and sealing. 2.1.5.1 Electrode preparation Photoanode: The photoanodes were prepared and sensitised with N719 according to the procedure already described in Sec. 2.1.3.1 and 2.1.4.1. Counterelectrode: Clean FTO-coated glass substrates were drilled for posterior electrolyte injection and spin coated with a 6 mM chloroplatinic acid hexahydrate (H2PtCl6, Sigma-Aldrich, ≥37.50%) solution in 2-propanol. 10 𝜇l of the solution were deposited on the substrate and spin coated at 2000 rpm for 20 seconds after a first stage at 500 rpm for 5 seconds. Three depositions were performed with drying processes at 80𝑜C for 5 minutes in between. After that, the samples were heated up to 400𝑜C for 20 minutes so as to trigger thermal decomposition. 2.1.5.2 Cell assembly The cell was assembled by means of a thermoplastic polymer gasket (Surlyn-Low Temperature Thermoplastic Sealant, GreatCell Solar©), which melts at 110-130𝑜C. In that way, the photoanodeswere attached to the drilled counterelectrodes by applying heat and pressure, so that the polymer acted as a sealant after melting. The assembled system was afterwards infiltrated with an electrolyte consisting of an iodide/triiodide (I−/I− 3) redox couple in a mixture of acetonitrile and 63 2. Methods valeronitrile and some additives. Specifically, the electrolyte solution contained 0.7 M 1-butyl-3-methylimidazolium iodide (BMII, Sigma- Aldrich, 99%), I2(Sigma Aldrich, 99.999%), 0.1 M guanidinium-iso- thiocyanate (GdnSCN, Sigma-Aldrich, ≥97%), 0.5 M 4-tert-butylpyri- dine (Sigma-Aldrich, 99%) in acetonitrile (Sigma-Aldrich, ≥99.9%) and valeronitrile (Sigma-Aldrich, 99.5%) in a proportion 85:15 (v/v). The holes were eventually sealed with a cover glass small enough not to cover the active area in order to avoid additional reflections at operation under rear illumination using the thermoplastic polymer. 2.2 Characterisation methods Apartfromthesynthesisandfabrication,the materials weresubjected to diverse types of characterisation techniques in order to gain full understanding of the relation between their structural features and optical response, as well as to assess the operation of the devices into which they had been integrated. This section is devoted to the main techniques employed for the characterisation of different aspects of the fabricated materials, for which either commercial equipments or specific experimental setups were employed. 2.2.1 Structural characterisation The exploration of the structural properties of the fabricated materials was essential for an appropriate description of their configuration. It allowed, for instance, the verification of an adequate integration of the scattering centres into the transparent matrices or the determination of the size and shape of the different synthesised nanoparticles. For that purpose, the materials were subjected to examination by different techniques. This part of the section presents a brief description of each of the techniques employed for that purpose. Scanning Electron Microscopy (SEM): An SEM microscope relies on the interaction of an accelerated electron beam with a specimen to retrieve information on its structural properties. Most commonly, the properties of the secondary electrons generated from the interaction between the electron beam and the sample are analysed and used to 64 2. Methods produce an image. This technique mainly provides information on the topography and morphology of the specimen. In this work, an SEM microscope (S-4800, Hitachi®) allowed, for instance, the determination of film thickness through examination of cross section images or the observation of the stratified configuration of the scattering spheres when integrated into specific matrices. In some cases, a Focused Ion Beam combined with an SEM (FIB-SEM) was used for the preparation and exploration of the structures. In an FIB equipment, an accelerated beam of ions, usually gallium, is employed to produce controlled etching of a sample. When coupled to an SEM microscope, the equipment offers the possibility of sample examination. This technique appears as especially convenient for the exploration of fragile structures or when aiming at the observation of specific features susceptible to be easily ruined by mechanical action during the process of specimen preparation for microscope exploration. Cross section images obtained in this work through this technique were generated employing a dual-beam Auriga FIB-SEM (ZEISS®). Transmission Electron Microscopy (TEM): While similar to an SEM microscope inasmuch as the interaction between an accelerated electron beam and the specimen is the principle of image formation, theTEMtechniquecollectstheelectronstransmittedthroughthesample. As an example, images produced by a TEM microscope (CM200, Philips®) allowed determination of the size and shape of the synthesised nanophosphors. X-Ray Diffraction (XRD): XRD is a non-destructive technique providing information on the crystalline structure of any material. The technique is based on the analysis of the constructive interference of X-rays diffracted at specific lattices in a crystalline structure. The sample, which can rotate an angle 𝜃, defined by the incidence direction and the direction perpendicular to the surface of the sample, is irradiated with X-rays and the diffracted beam is collected by a detector at an angle 2𝜃. Whenever the geometry of the incident beam in relation to the structure of the sample verifies the Bragg equation for the diffraction, constructive interference occurs and a maximum in intensity is detected. 65 2. Methods 2.2.2 Optical characterisation A full study of the optical response of the fabricated random media is crucial in order to understand how the conditions of the included disorder affects the propagation of light. This characterisation is mostly based on spectroscopic techniques, such as reflectance and transmittance or photoluminescence (PL) in the case of emitting materials. 2.2.2.1 Reflectance and Transmittance In a large number of cases, spectroscopic reflectance and transmittancemeasurementswere performedonthematerialsinordertocharacterise their optical response. This type of characterisation was crucial for the determination of essential magnitudes studied in this thesis, such as 𝓁𝑠𝑐 and 𝓁𝑡. Specular and ballistic: The measurements of specular reflectance, Rs, and ballistic transmittance, Tb, were performed by means of a UV-Vis-NIR spectrophotometer (Cary 5000, Agilent Technologies©), which operates by collecting the light either transmitted or reflected by the sample and referring it to the intensity of the incoming beam, -90o 180o 90o 0o Tb Rs Td Rd Detector Sample Figure 2.15: Operation of the Universal Measurement Accessory, UMA (Agilent Technologies©), and different measurable magnitudes. 66 2. Methods Figure 2.18: Angular distribution of the light scattered by a 300-nm radius anatase sphere at 𝜆= 600 nm. Mie theory5,6. An interesting peculiarity of the Mie scattering for a sphere of a size comparable to the wavelength of the incident light is a largely forward-oriented distribution of the scattered light, see Fig. 2.18. Throughout this thesis, the propagation of light in random media was assessed through Mie theory considering spherical scattering particles embedded in an absorbing medium7, based on the standard parameters employed for the characterisation of turbid media, namely, 𝓁𝑠𝑐, and 𝓁𝑡. As mentioned in the previous chapter for Eq. 1.1, in the limit of independent scattering, 𝓁𝑠𝑐 is a function of the number particle density and the scattering cross section of an individual scattering centre. Taking into account the definition of particle number density and volume filling fraction, that is, 𝜌= 𝑁 𝑉;𝑓 = 𝑉𝑖𝑛𝑐𝑙 𝑉,(2.4) where N refers to the number of inclusions, V to the considered volume and Vincl to the total volume occupied by the scattering inclusions, which in this case correspond to spheres of radius r, Eq. 1.1 results 𝓁𝑠𝑐 =4 3𝜋𝑟3 𝑓 ⋅𝜎𝑠𝑐.(2.5) According to Mie theory, 𝜎𝑠𝑐for a spherical particle is a function of the radius of the sphere, the complex refractive index of both the particle, Np, and the surrounding medium, Nm, and the wavelength of the 73 2. Methods impinging light, 𝜆, and it is given by 𝜎𝑠𝑐 =2𝜋 𝑘2∞ ∑ 𝑛=1(2𝑛+1)(𝑎𝑛2+𝑏𝑛2), (2.6) where 𝑘= 2𝜋 𝜆𝑛𝑚refers to the modulus of the wave vector, in which nm corresponds to the real part of Nm. In this thesis, the model considered the possibility of a stratified sphere as scattering structure. This was necessary in order to account for the air shell surrounding the scattering particle present in some of the modelled materials, as well as for any possible coating of the particle. In that scenario, the Mie formalism provides the expression for the coefficients 𝑎𝑛and 𝑏𝑛as a function of the Riccati-Bessel functions. These coefficients were calculated by means of a script8,9 based on the formalism formulated by Arthur L. Aden and Milton Kerker10. In particular, 𝑎𝑛=𝜓𝑛(𝑦)[𝜓′ 𝑛(𝑚2𝑦)−𝐴𝑛𝜉′ 𝑛(𝑚2𝑦)]−𝑚2𝜓′ 𝑛(𝑦)[𝜓𝑛(𝑚2𝑦)−𝐴𝑛𝜉𝑛(𝑚2𝑦)] 𝐾𝑛(𝑦)[𝜓′ 𝑛(𝑚2𝑦)−𝐴𝑛𝜉′ 𝑛(𝑚2𝑦)]−𝑚2𝐾′ 𝑛(𝑦)[𝜓𝑛(𝑚2𝑦)−𝐴𝑛𝜉𝑛(𝑚2𝑦)],(2.7) 𝑏𝑛=𝑚2𝜓𝑛(𝑦)[𝜓′ 𝑛(𝑚2𝑦)−𝐵𝑛𝜉′ 𝑛(𝑚2𝑦)]−𝜓′ 𝑛(𝑦)[𝜓𝑛(𝑚2𝑦)−𝐵𝑛𝜉𝑛(𝑚2𝑦)] 𝑚2𝐾𝑛(𝑦)[𝜓′ 𝑛(𝑚2𝑦)−𝐵𝑛𝜉′ 𝑛(𝑚2𝑦)]−𝐾′ 𝑛(𝑦)[𝜓𝑛(𝑚2𝑦)−𝐵𝑛𝜉𝑛(𝑚2𝑦)],(2.8) being 𝐴𝑛=𝑚2𝜓𝑛(𝑚2𝑥)𝜓′ 𝑛(𝑚1𝑥)−𝑚1𝜓′ 𝑛(𝑚2𝑥)𝜓𝑛(𝑚1𝑥) 𝑚2𝜉𝑛(𝑚2𝑥)𝜓′ 𝑛(𝑚1𝑥)−𝑚1𝜉′ 𝑛(𝑚2𝑥)𝜓𝑛(𝑚1𝑥),(2.9) 𝐵𝑛=𝑚2𝜓𝑛(𝑚1𝑥)𝜓′ 𝑛(𝑚2𝑥)−𝑚1𝜓′ 𝑛(𝑚1𝑥)𝜓𝑛(𝑚2𝑥) 𝑚2𝜉′ 𝑛(𝑚2𝑥)𝜓𝑛(𝑚1𝑥)−𝑚1𝜉𝑛(𝑚2𝑥)𝜓′ 𝑛(𝑚1𝑥).(2.10) 𝜓𝑛(𝑧),𝜉𝑛(𝑧)and 𝐾𝑛(𝑧)refer to Riccati-Bessel functions of the form 𝜓𝑛(𝑧)=(𝜋𝑧 2)1/2𝐽(1) 𝑛+1 2(𝑧), (2.11) 𝜉𝑛(𝑧)=𝜓𝑛(𝑧)+𝑖𝑋𝑛(𝑧), (2.12) where 𝑋𝑛(𝑧)=−(𝜋𝑧 2)1/2𝑌(2) 𝑛+1 2(𝑧), (2.13) 74 2. Methods and 𝐾𝑛(𝑧)=(𝜋𝑧 2)1/2𝐻(1) 𝑛+1 2(𝑧), (2.14) where 𝐽(1) 𝑛+1 2(𝑧)corresponds to the semi-integer Bessel function of first kind, 𝑌(2) 𝑛+1 2(𝑧)to the semi-integer Bessel function of second kind and 𝐻(1) 𝑛+1 2(𝑧)is the semi-integer Hankel function of first kind. The optical characteristics of the different considered media play an essential role on the expressions above. Indeed, being Npand Ncthe complex refractive indices of the scattering particle and the coating material, respectively, the expressions for coefficients 𝑎𝑛and 𝑏𝑛depend on the refractive indices of the scattering sphere and its coating relative to the embedding medium, namely, 𝑚1=𝑁𝑝 𝑛𝑚and 𝑚2=𝑁𝑐 𝑛𝑚. Two size parameters are defined, 𝑥=𝑘𝑟𝑝for the scattering sphere and 𝑦=𝑘𝑟𝑐for the coating, where rpand rccorrespond to the radius of the particle and the coating thickness, respectively. Knowing 𝓁𝑠𝑐 allows the determination of 𝓁𝑡through the angular distribution of the light scattered by the particle predicted by Mie formalism, according to Eq. 1.2. 2.3.1.2 Finite-Difference Time-Domain (FDTD) simulations The calculations of the angular distribution of the scattered light and scattering efficiency of non-spherical particles were performed by means of 3-dimensional Finite-Difference-Time-Domain (FDTD) simulationswiththesoftware’FDTDSolutions’fromLumerical Solutions11. In order to account for shrinkage effects experimentally observed for the scattering particles, a hollow with the same shape as the considered particle between the inclusion and the embedding medium was included in the simulations. Besides, since the calculations were aimed at justifying the specific shape chosen for the scattering centres for the particular case of the integration into photoanodes of DSSCs, this hollow was considered to be filled with electrolyte. With these assumptions, the simulated system consisted of a scattering structure comprising a TiO2particle surrounded by an electrolyte-filled shell 75 2. Methods 500 600 700 Wavelength / nm 1.6 1.8 2.0 1.4 2.4 n 2.2 800400 Electrolyte TiO2(a) 800 Wavelength / nm 500 600 700 0 400 0.01 0.02 0.03 0.04 k Electrolyte TiO2 Dye (b) Figure 2.19: Optical constants of the materials used in the FDTD simulations of the light scattered by individual TiO2particles of diverse shapes: (a) Real part of the complex refractive index. (b) Imaginary part of the complex refractive index. of the same shape embedded in an external absorbing medium constituted by a sensitised mesoporous TiO2matrix infiltrated with electrolyte. The spectral complex refractive index of the materials considered in the simulations are displayed in Fig. 2.19. In order to numerically evaluate the absorption, scattering efficiency and angular distribution of the light scattered by individual particles,FDTDcalculationsconsideringa simulationbox sizeof 1.0× 1.0 ×1.0 𝜇m3and perfectly matching layer conditions on every boundary were performed. Four particle shapes were simulated, namely, spheres, cubes, truncated pyramids and distorted cubes. The volume of the simulated particles corresponds to that of a sphere of radius r = 150 nm, except for slight volume variations for irregular shapes. As excitation source, a plane wave of wavelength 𝜆= 650 nm was considered. As for the particles presenting irregular shapes, namely, truncated pyramids and distorted cubes, the resulting distributions and spectra derived from averaging over several angles and illumination sides. The mesh grid was established to 150 nm over the complete simulation volume, considering a refinement of 4 nm over a 0.72 × 0.72 ×0.72 𝜇m3volume spanning over and around the volume occupied by the scattering structure. These parameters provided convergent results for all the considered shapes. The scattered power was determined through the total-field scattered-field method by placing 76 2. Methods frequency-domain transmission monitors in the region of the scattered field. 2.3.2 Response of a material integrating scattering centres With the information of the scattering of light triggered by a single particle, the next step consisted in the simulation of light propagation in a disordered material including a random distribution of the previously modelled scattering centres. The first part of the section describes the mathematical formalism required for the determination of 𝓁𝑠𝑐 and 𝓁𝑡from experimental magnitudes. The second part provides the details of a model based on a Monte Carlo approach combined with Mie theory for the simulation of the photon trajectory throughout any multilayer system, thus enabling the modelling of light propagation. 2.3.2.1 Scattering and transport mean free path The spectral 𝓁𝑠𝑐 of each material is determined by fitting measurements of Tband Rsof the material to Lambert-Beer’s law for the diffusepropagationoflightinrandommediainavalid wavelengthrange. If the material is described as a homogeneous slab of thickness L, the intensity attenuation of a beam of light impinging on its surface due to the scattering of light, see Fig. 2.20, is given by 𝐼𝑏(𝑧)=𝐼0𝑒−𝑧/𝓁𝑠𝑐,(2.15) z Iinc RsTb Td L L Intensity Figure 2.20: Illustration of the intensity attenuation of a beam through a slab of thickness L in a turbid medium according to Lamber-Beer’s law. 77 2. Methods being Ib(z)the intensity of the beam at any point inside the slab along the direction of incidence, z, and I0the value of the intensity of the incoming beam at the entrance plane inside the slab. The presence of interfaces causes reflection losses at the entrance and end of the slab, which are accounted for through a decrease in the intensity measured at z = 0 inside the slab and at z = L outside of it, that is, 𝐼0=𝐼𝑖𝑛𝑐−𝐼𝑅, where Iinc corresponds to the intensity of the beam impinging on the slab and IRthe value of the total intensity lost due to reflections at the interfaces. The intensity measured outside of the slab along the propagation direction of the incident beam is therefore determined by the expression 𝐼𝑏(𝐿)=(𝐼𝑖𝑛𝑐 −𝐼𝑅)𝑒−𝐿/𝓁𝑠𝑐,(2.16) and dividing by Iinc results 𝑇𝑏=(1−𝑅𝑠)𝑒−𝐿/𝓁𝑠𝑐,(2.17) Finally: 𝑇𝑏 1−𝑅𝑠=𝑒−𝐿/𝓁𝑠𝑐,(2.18) where Tbis obtained from the measurement of the emerging intensity at any point after the slab along the direction of the incident beam and Rsincludes interference effects due to reflections at the entrance and exit of the slab. Eq. 2.18 is used in this work for the extraction of the spectral 𝓁𝑠𝑐 of the different proposed systems from experimental data. Light transport throughout a turbid media is considered diffusive if verifying the condition 𝜆≪𝓁𝑡≪ 𝐿. In the weak-scattering limit, where the density of scattering centres is low and/or 𝜎𝑠𝑐is small enough to yield 𝑘𝓁𝑠𝑐 ≫1, while having enough thickness to achieve randomisation of the propagated light, its transport can be described by the diffuse equation, 𝑆( 𝑟,𝑡)=(𝜕 𝜕𝑡−𝐷⋅∇2+𝑣𝑒 𝓁𝑖)𝐼( 𝑟,𝑡), (2.19) where S( 𝑟,t) represents a light source, D is the diffusion coefficient, vethe energy velocity and 𝓁𝒾corresponds to the inelastic absorption length, i.e the average distance into the slab at which the intensity of the light, I( 𝑟,t), has been attenuated by a factor e. This approximation is not valid in conditions of localisation, that is, 𝑘𝓁𝑠𝑐 ≃1, where 78 2. Methods the scattering is excessively strong and light propagation is seriously hampered. The stationary solution of the monodimensional diffusion equation relates the total transmittance through the slab, T(L,𝜆), to 𝓁𝑡(𝜆)and the slab thickness: 𝑇(𝐿,𝜆)= 1 𝛼𝑧𝑒𝑠𝑖𝑛ℎ[𝛼(𝑧𝑒+𝑧𝑝)]𝑠𝑖𝑛ℎ[𝛼𝑧𝑒] 𝑠𝑖𝑛ℎ[𝛼(𝐿+2𝑧𝑒)] (2.20) Here, 𝛼corresponds to the reciprocal absorption length and 𝑧𝑒and 𝑧𝑝are extrapolation and penetration lengths, respectively, which can be calculated according to12 𝑧𝑝=𝑧𝑒=1 2𝛼𝑙𝑛(1+𝛼𝑧0 1−𝛼𝑧0), (2.21) and 𝑧0=2 3𝓁𝑡(1+𝑅 1−𝑅), (2.22) being R the polarisation-averaged Fresnel internal reflectivity at an imaginary boundary between the disordered medium and the embedding material13. In line with the derivation of Eq. 2.18, Eq. 2.20 can be modified in order to account for reflections at the interfaces of the material: 𝑇(𝐿,𝜆) 1−𝑅𝑠(𝜆) =1 𝛼𝑧𝑒𝑠𝑖𝑛ℎ[𝛼(𝑧𝑒+𝑧𝑝)]𝑠𝑖𝑛ℎ[𝛼𝑧𝑒] 𝑠𝑖𝑛ℎ[𝛼(𝐿+2𝑧𝑒)] (2.23) Equation 2.23 is often referred to as the photonic Ohm’s law. The total transmittance through the slab is directly proportional to 𝓁𝑡and inversely proportional to L, 𝑇(𝐿,𝜆) (1−𝑅𝑠(𝐿)) ∼𝓁𝑡(𝜆)/𝐿. The expression in Eq. 2.23 is employed in this thesis for the extraction of the spectral 𝓁𝑡values of the material comprising a random distribution of TiO2spheres in a mesoporous TiO2matrix from experiments. Another way to determine 𝓁𝑡from experimental magnitudes entails measurements of the enhanced backscattering cone14,15. Whereas a sample presenting a large thickness is required for determination of 𝓁𝑡through the latter, the former relies on the measurement of total transmission of several samples of different thicknesses. In this work, estimation of 𝓁𝑡is performed through fitting Eq. 2.23 to experimental data of T(L,𝜆) for slabs of diverse thicknesses. 79 2. Methods 2.3.2.2 Determination of photon trajectory in layered media An optical model able to predict light propagation is a key tool in order to perform a full description of the interaction of light with the optically random materials studied in this thesis, since it enables designing the optical response of the system, as well as provides a theoretical support for the empirical observations. For this purpose, a Monte Carlo approach combined with Mie theory was employed in order to simulate the trajectory of individual photons propagating throughout a multilayer system16–19. The model consists in a raytracing method capable of fully accounting for the trajectory of each photon through any multilayer structure. It provides full information on light interaction with the system, specifically, specular and total reflectance, ballistic and total transmittance, their corresponding diffuse magnitudes, the absorption profile at each of the layers, the angular distribution of light emerging from the system and photon scattering event statistics, allowing an angular representation of the photons undergoing a specific number of scattering events. The system is represented as a 3-dimensional multilayer structure comprising any number of layers, each characterised by their thickness, L𝑖, and the optical constants of the material constituting it, n𝑗𝑖, Fig. 2.21(a). Additionally, each layer can include any number of components, such as a homogenous distribution of a different phase or scattering centres, for which further input parameters are required. Figure 2.21: (a) Illustration of a general multilayer structure considered in the model. (b) Different possible fates for a photon contemplated in the model: I. Transmitted photon, II. Reflected photon, III. Transmitted and scattered photon, IV. Transmitted and absorbed photon. 80 2. Methods In this description, the incident beam is depicted as a collection of individual photons. A photon is randomly generated outside the structure and impinges on the first interface of the system at a given angle. The distance travelled by the photon before undergoing an scattering or absorption event, 𝓁, is given by 𝓁=−𝑙𝑛[𝑟]/𝛼𝑒𝑥𝑡,(2.24) being r a randomly generated number in the range 0 < r ≤1 and 𝛼𝑒𝑥𝑡 an extinction coefficient accounting for different phenomena depending on the components included in that layer. Extinction is herein regarded as any phenomenon substracting photons from the incident beam, such as absorption or scattering, thus resulting 𝛼𝑒𝑥𝑡 =𝛼𝐴+𝛼𝑠𝑐,(2.25) where 𝛼𝐴and 𝛼𝑠𝑐 refer to the absorption and scattering coefficient, respectively. The scattering coefficient is given by 𝛼𝑠𝑐 = 𝜌𝜎𝑠𝑐, being 𝜌 the number density of the scattering inclusions and 𝜎𝑠𝑐 their scattering cross section, according to the Mie formalism. If 𝓁> Li, the photon reaches the following interface without deviation from the trajectory of the incident beam. At this point, there is a certain likelihood that the photon results either transmitted or back reflected, Fig. 2.21(b) I and II, respectively, as defined by the Fresnel coefficients. If the photon is transmitted, the process is repeated at the next layer. In the case the photon undergoes an absorption or scattering event, then 𝓁< Li. Here, a new random number r’ is generated and a casuistry for the fate of the photon depending on its value unfolds. Specifically, if r’ ≤ 𝛼𝑠𝑐/𝛼𝑒𝑥𝑡, the photon is scattered, Fig. 2.21(b) III, while r’ ≥𝛼𝑠𝑐/𝛼𝑒𝑥𝑡 results in a photon absorbed at the layer, Fig. 2.21(b) IV. This latter condition can be expanded for a layer comprising more than one single component in order to account for the relative contribution of each of the components to the absorption at the layer. In the case of scattering of the photon, the new propagation direction, (𝜙,𝜃), is determined by the angular distribution of the light scattered by a particle yielded by Mie theory. The absorption by the external medium is included when evaluating this. The process is extrapolated to a larger number of photons and their trajectories are traced. Collection of the photons emerging from the structure at the incoming and outgoing media enables re- 81 2. Methods 90o 180o 270o 0o Figure 2.22: Angular distribution of the light scattered by a disordered system simulated according to the depicted model. The conditions of the material correspond to a 7-𝜇m thick mesoporous anatase slab integrating nanocrystalline TiO2spheres of size r = 225 nm in a 12% concentration at 𝜆= 600 nm. trieving information on the reflectance and transmittance, while distinguishing between diffuse light and Tbor Rsthrough the photon exit angle. In a similar way, the model provides information on the spatial absorption of photons at each layer, therefore enabling the generation of spatial absorption profiles by each component comprising thelayers.Besides, information regardingthedirectionofeach emerging photon in both the incoming and outgoing medium enables the determination of the pattern of angular distribution of the scattered light, which illustrates the number of photons exiting the system at every angle. An example is presented in Fig.2.22, where 0𝑜and 180𝑜 refer, respectively, to Tband Rs. 2.3.3 DSSC modelling The numerical simulation method depicted above in Sec. 2.3.2.2 was applied to a DSSC. Considering the cell as a multilayer system, the model enabled extraction of magnitudes relevant for the characterisation of the operation of the device, as well as allowing comparison with the empirical values. At this point, it is convenient to define the principal magnitudes involved in the optoelectrical characterisation of a photovoltaic device. 82 2. Methods numerical approach depicted in Sec. 2.3.2.2 allows the calculation of the absorption profile by each material at any layer of the structure, it is therefore possible to distinguish between productive and parasitic absorptance, that is, between light absorbed, respectively, by the dye molecules and light absorbed by any other material in the cell, such as the FTO film, the catalytic Pt or the electrolyte, thus allowing an estimation of 𝜂𝐿𝐻(𝜆). Incident-Current-to-Photon Efficiency (𝜂𝐼𝑃𝐶𝐸(𝜆)): Eq. 2.26 indicates that 𝜂𝐼𝑃𝐶𝐸(𝜆)can be expressed as a product of efficiencies of different processes involved in the generation of the photocurrent from the absorption of photons. For the calculation of 𝜂𝐼𝑃𝐶𝐸(𝜆),𝜂𝐼𝑁𝐽(𝜆)= 𝜂𝑅𝐸𝐺(𝜆)=1was assumed, that is, lossless electron injection and dye regeneration processes, thus 𝜂𝐼𝑃𝐶𝐸(𝜆)=𝜂𝐿𝐻(𝜆)⋅𝜂𝐶𝑂𝐿(𝜆). The integration of optical disorder mainly modifies the absorption of light along the electrode and thus 𝜂𝐿𝐻)(𝜆). However, due to the dependence of 𝜂𝐶𝑂𝐿(𝜆)on g(z,𝜆), the former is expected to be affected as well. For the calculation of 𝜂𝐼𝑃𝐶𝐸(𝜆),𝜂𝐶𝑂𝐿(𝜆)was computed according to 𝜂𝐶𝑂𝐿(𝜆)= ∫𝐿 0𝑔(𝑧,𝜆)⋅𝑐𝑜𝑠ℎ(𝐿−𝑧 𝐿𝑒) 𝑐𝑜𝑠ℎ(𝐿 𝐿𝑒)𝑑𝑧 ∫𝐿 0𝑔(𝑧,𝜆)𝑑𝑧 ,(2.34) where L refers to the thickness of the photoanode and Lecorresponds to an ’effective’ electron diffusion length, which accounts for the average distance covered by an electron before undergoing recombination. 89 Bibliography [1] Y. J. Kim, M. H. Lee, H. J. 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Snaith, “Estimating the maximum attainable efficiency in dye-sensitizedsolarcells,”Advanced Functional Materials,vol. 20, pp. 13–19, 2010. 93 3Mie glasses: fabrication and characterisation 3.1 Introduction Liquid media comprising a suspension of spherical particles have drawn special attention due to their interesting optical properties. As long as the concentration of the scattering centres is kept low enough soas toallow neglectingcorrelation effects,thescattering phenomena and optical properties of these suspensions are well described by Mie theory, that is, taking into account single-particle considerations1,2. Such liquid suspensions are especially advantageous for several reasons. In the first place, their generally low refractive index promotes a high dielectric constant contrast between the dispersed phase and the liquid medium, ensuring considerable scattering strength. Furthermore, they accept chromophores, so that light scattering triggered by the presence of particles can result in an enhancement of their absorption, thus also emission, intensity. These features were exploited in the decade of 1990, which led to the development of random lasers3. In the case of solids, however, pre-designing the optical response of the disordered medium by means of analytical models has exclusively met with success for photonic glasses, which consist in disordered packings of identical unit blocks4–6. As for the remainder, the spectralproperties of solid randommedia generallyremain unknown prior to fabrication. Attempts to design the scattering properties of diffusive media for subsequent fabrication have been recently made, such as the case of V. Y. F. Leung et al.7, who analysed light transport in YAG:Ce plates, the most commonly employed material for colour conversion in LEDs, which are known to present significant light scattering. This chapter introduces a procedure for the fabrication in an uncomplicated and cost-effective manner of films of materials of an optically random medium in a novel configuration, which can be regarded as a solid version of a liquid dilute suspension of spheres. Such configuration features the advantage of enabling a design of its scattering response prior to fabrication according to Mie formalism and was accordingly coined Mie glass. In the first place, different combinations of materials in a Mie glass configuration are tested in 95 3. Mie glasses: fabrication and characterisation order to assess their performance as highly scattering media. The results of these calculations enable choosing the most suitable combinations for intense scattering strength, thus defining the materials to be developed and analysed throughout this thesis. After that, the fabrication procedure of each material is depicted, along with a integral characterisation of their optical response, including that of the base transparent matrix when required. 3.2 Analytical study for different material combinations The presence of disorder in a non-absorbing medium typically causes successive elastic scattering of the light, therefore producing diffuse propagation of the waves. In this thesis, maximisation of the diffusion strength in a controlled manner was sought for, due to its potential regarding the fabrication of optoelectronic devices with enhanced properties. 𝓁𝑠𝑐 and 𝓁𝑡serve as a measure of the diffusion ability of a slab of a random material in relation to its thickness. As previously mentioned, in the approximation of independent scattering, Sec. 1.1 in Ch. 1, 𝓁𝑠𝑐 exclusively depends on 𝜎𝑠𝑐 of an individual inclusion embedded in a support material (a porous transparent matrix in this case) and the volume concentration of particles, 𝜌, embedded in it: 𝓁𝑠𝑐 =1 𝜌⋅𝜎𝑠𝑐.(3.1) At the same time, and as described by Eq. 2.6, 𝜎𝑠𝑐 is a function of the size of the inclusions, their refractive index, that of the surrounding medium and, more specifically, their constrast, as well as of the wavelength of the incident light. The work presented in this thesis proposes the fabrication of solid films of controlled thickness integrating optical disorder in a controlled manner named Mie glasses. Specifically, the proposed configuration consists in porous films displaying transparency in the visible range of the spectrum, such as TiO2or SiO2, including a random distribution of monodisperse TiO2spheres of size around a few hundred nanometres, that is, of the order of the wavelength of visible light, as 96 3. Mie glasses: fabrication and characterisation Transparent matrix Scattering centres Figure 3.1: Illustration of the structure proposed for the optically random media studied in this thesis. illustrated in Fig. 3.1. In the first place, identical scattering by each inclusion is ensured due to the monodispersity in size of the inclusions, as well as their well-defined shape. Indeed, the disorder can be easily tuned through the size and volume filling fraction of particles dispersed in the matrix, which enables control over the photonic strength of the resulting film. A spherical shape was preferred for the inclusions, since Mie formalism provides an analytical expression for 𝜎𝑠𝑐, which is essential for the calculation of the optical response of the material. Moreover, a spherical shape can be easily synthesised through standard experimental procedures. However, it is important to highlight that any other shape would have been equally valid for the intended study, as long as the shape would remain identical for all inclusions. In this case, the determination of 𝜎𝑠𝑐would require numerical computational methods, such as Finite-difference time-domain (FDTD) calculations. In the limit of high dillution, correlation terms among inclusions regarding scattering can be neglected and the scattering of light in the material can be described taking into account exclusively single-particle considerations. In fact, for high inclusion density values Eq. 3.1 is not valid, and dependent scattering causes an increase of 𝓁𝑠𝑐8, that is, a reduction of the diffusion strength. A size of the inclusions comparable to the wavelength of the incident light ensures scattering in the Mie scattering range, where 𝜎𝑠𝑐 is maximal. This in combination with sufficient refractive index contrast between them and the surrounding medium allows minimisation of 𝓁𝑠𝑐 for high scattering strength. Furthermore, porosity is imposed in the transparent matrices, which not only helps reducing their refrac- 97 3. Mie glasses: fabrication and characterisation tive index to increase the contrast with the scattering centres, but also provides them with versatility, since they would accept infiltration with fluorophores, chromophores or polymers that could endow them with further funcionality. A noteworthy advantage of the disordered materials herein proposed is that their fabrication process depends to a great extent on solution-processing methods, which are known to be cost-effective and easily implemented. Based on the premise that light propagation in the media herein studied could be properly described by Mie formalism, it was employed in order to explore the scattering properties of different combination of materials in the configuration of interest. With this purpose, a series of different available materials susceptible to be synthesised as a colloid in the shape of spherical particles, such as titanium dioxide (TiO2), silicon dioxide (SiO2) or void scatterers resulting from the etching of organic spheres embedded in an inorganic matrix, were considered. As for the matrices, only low density media for which a wide number of synthetic routes had been developed, such as mesostructured TiO2and SiO2, were taken into account. The value of 𝜎𝑠𝑐 of the scattering inclusions, necessary for the calculation of 𝓁𝑠𝑐, were semianalitically obtained through Mie theory. A 50% porosity was assumed for the matrices in the calculations. Figure 3.2 displays the spectral optical constants of the different materials employed in these simulations. Figure 3.3(a) and (b) compare 𝜎𝑠𝑐 and 𝓁𝑠𝑐 values in 2.8 2.4 2.0 1.6 1.4 0.8 400 800 1200 1600 2000 Wavelength / nm n (a) TiO2 Porous TiO matrix 2 Air 0.003 0.002 0.001 0 400 800 1200 1600 2000 Wavelength / nm k (b) TiO2 Porous TiO matrix 2 Air Figure 3.2: Optical constants of the materials considered in the simulations. (a) Real part of the complex refractive index. (b) Imaginary part of the complex refractive index. 98 3. Mie glasses: fabrication and characterisation 2 4 6 8 10 ln(T /(1-R )) b s (a) Thickness / µm 0 -2 -4 12 2 4 6 8 10 ln(T /(1-R )) b s (b) Thickness / µm 0 -2 -4 12 -6 Figure 3.7: Values of ln( 𝑇𝑏 1−𝑅𝑠) in relation to slab thickness and their corresponding lineal fittings, when possible, for the Mie glass under consideration including scattering centres with the conditions r = (95 ±20) nm - f = 5% (blue dots), r = (225 ±20) nm - f = 5% (red dots), r = (225 ±20) nm - f = 10% (orange dots) and r = (225 ±20) nm - f = 15% (green dots) at (a) 𝜆= 800 nm and (b) 𝜆= 500 nm. and slab thickness is not linear for every wavelength, as illustrated in Fig. 3.7(b). Here, whereas a linear behaviour was revealed for the cases r = (95 ±20) nm - f = 5% and r = (225 ±20) nm - f = 5%, it was no longer possible to obtain a reasonable fitting assuming a linear dependence for the rest of conditions. In order to standardise the procedure, only linear fits yielding a correlation coefficient verifying r2> 0.95 were considered. Else, extraction of a faithful value of 𝓁𝑠𝑐from the experimental data was not possible. The slope of the linear fit allowed extraction of 𝓁𝑠𝑐 at the considered wavelength, Fig. 3.8(a). As the size or the concentration of the inclusions increased, so did the slope of the resulting linear fit, which was translated into shorter 𝓁𝑠𝑐 values, in accordance with the theoretical predictions, Fig. 3.8(b). The fair agreement between the curves obtained from the experiments and those resulting from the calculations provided evidence so as to assert that the behaviour of light in this material can be described to a good extent by means of Mie theory, i.e. assuming single-particle considerations for the scattering of the light, hence corroborating its behaviour as a Mie glass. The possible reasons for the deviation from a linear dependence depicted in Fig. 3.7(b) hindering the determina- 105 3. Mie glasses: fabrication and characterisation tion of a reliable value of 𝓁𝑠𝑐 can be classified according to the wavelength range of the deviation: • Range of long wavelength values: The expected 𝓁𝑠𝑐 values significantly exceeded the maximum thickness of the films. • Range of short wavelength values: Due to the wavelength of the light in comparison with the size of the inclusions and theirmeanseparationdistance,scatteringcorrelationsbetween them can become relevant enough, so that the single-particle approximation would no longer be accurate at describing the diffusive behaviour of the material. The angular distribution of the scattered light plays a crucial role on the randomisation of light propagation. In order to gain a better insight into this feature, 𝓁𝑡was studied. Measurements of T, and Rs allowed the determination of 𝓁𝑡, according to the photonic Ohm’s law, Eq. 2.23. In this expression, zpand zedepend on the polarisationaveraged Fresnel reflectivity, R, see Eq. 2.21 and 2.22, calculated at an imaginary boundary between the disordered medium and the embedding material, which for a mesoporous TiO2matrix integrating spherical TiO2scattering centres yielded R = 0.07 for f = 5%, R = 0.16 400 800 1200 1600 lsc / m µ (a) Wavelength / nm 20 0 2000 10 30 40 400 800 1200 1600 lsc / m µ (b) Wavelength / nm 20 0 2000 10 30 40 Figure 3.8: Spectral 𝓁𝑠𝑐 curves obtained for the Mie glass under consideration from (a) experimental data and (b) according to Mie formalism. The considered conditions correspond to r = (95 ±20) nm - f = 5% (blue), r = (225 ±20) nm - f = 5% (red), r = (225 ±20) nm - f = 10% (orange) and r = (225 ±20) nm - f = 15% (green). 106 3. Mie glasses: fabrication and characterisation 2 4 6 8 10 Thickness / µm 12 T/(1-R ) s 1.0 0.8 0.6 0.4 Figure 3.9: Values of 𝑇 1−𝑅𝑠in relation to the slab thickness and their corresponding fittings employing Eq. 2.23 for the Mie glass under consideration including scattering centres with the conditions r = (95 ± 20) nm - f = 5% (blue dots), r = (225 ±20) nm - f = 5% (red dots), r = (225 ±20) nm - f = 10% (orange dots) and r = (225 ±20) nm - f = 15% (green dots) at 𝜆= 800 nm. for f = 10% and R = 0.27 for f = 15%. Along the same lines of the 𝓁𝑠𝑐 analysis, Fig. 3.9 displays the fittings of the experimental data to Eq. 2.23 at 𝜆= 800 nm. In this case, Rsand T were measured em- 400 500 600 700 lt/ m µ (a) Wavelength / nm 20 0 800 10 30 40 lt/ m µ (b) Wavelength / nm 20 0 10 30 40 400 500 600 700 800 Figure 3.10: Spectral 𝓁𝑡curves obtained for the Mie glass under consideration integrating inclusions with different conditions from (a) experimental data and (b) Mie formalism. The considered conditions correspond to r = (95 ±20) nm - f = 5% (blue), r = (225 ±20) nm - f = 5% (red), r = (225 ±20) nm - f = 10% (orange) and r = (225 ±20) nm - f = 15% (green). 107 3. Mie glasses: fabrication and characterisation ploying an integrating sphere coupled to a spectrophotometer capable of detecting intensity up to 𝜆= 850 nm. Figure 3.10(a) displays the 𝓁𝑡curves resulting from these fittings. For the case r = (95 ±20) nm - f = 5%, 𝓁𝑡yielded larger values than the largest thickness of the fabricated films, L ∼12 𝜇m, for 𝜆> 575 nm, therefore not fulfilling the condition L ≫𝓁𝑡. In this situation, Eq. 2.23 is unable to describe the propagation of light for 𝜆> 575 nm and these values have been accordingly not included in the graph. Shorter values of 𝓁𝑡were observed for larger inclusions or concentrations, holding 𝓁𝑡>𝓁𝑠𝑐, as expected for particles yielding non-isotropic light scattering, see Eq. 1.2. Fair agreement was observed when comparing to the spectra derived from Mie formalism, Fig. 3.10(b), in spite of the discrepancies for the case of the material yielding the weakest scattering strength and the unability to reproduce resonant features. The latter was justified by the experimental polydispersity in size of the TiO2inclusions. Mie formalism predicts resonances sustained by scattering particles of a well-defined size at specific wavelength values. Nevertheless, when polydispersity is introduced, the size distribution is not sharp enough and the resonant features easily smooth out. Additionally, while each scattering centre was considered to share its geometrical centre with its corresponding surrounding air shell in the model, this was not a faithful reproduction of the real system. As a consequence, discrepancies in the shape of the angular distribution of the scattered light arose, hindering an accurate reproduction of the expected 𝓁𝑡values. Discrepancies were more pronounced in the determination of 𝓁𝑡, as it is a function of 𝓁𝑠𝑐, which depends on both, 𝜎𝑠𝑐 and the angular distribution of the scattered light, which is apparently highly sensitive to shape deviations of the inclusions. Notwithstanding, the reasonable agreement of the results proves that the optically random medium herein proposed behaves as a solid version of a dilute dispersion of spherical particles, as well as demonstrating the validity of the Mie formalism for the description its angular scattering properties. 3.3.2.2 Finite size effects and photon statistics of light transport in diffusive random media This section presents a general study based on a combination of experiments with calculations on how the thickness of a finite slab of the 108 3. Mie glasses: fabrication and characterisation optically random medium under consideration affects the degree of randomisation of light transport, from a regime where ballistic transmittance prevails to the limit where full randomisation is achieved. An analysis on photon statistics in terms of the number of scattering events these photons underwent is also performed. Analysis of the spectral and angular optical response: In order to study finite size effects on light propagation through this Mie glass, the spectral and angular response of three slabs of thickness L = 9 𝜇m characterised by different 𝓁𝑠𝑐 values were determined. The value of 𝓁𝑠𝑐 was controlled through the size and concentration of scattering centres. Specifically, the scattering centre conditions of the materials under consideration were r = (95 ±20) nm - f = 5%, r = (225 ± 20) nm - f = 5% and r = (225 ± 20) nm - f = 10%, which according to Mie formalism yielded 𝓁𝑠𝑐 of 6.6 𝜇m, 1.7 𝜇m and 1.1 𝜇m, respectively, at 𝜆= 600 nm. Negligible contributions due to absorption by the material were expected at this wavelength. The Tbspectra, as displayed in Fig. 3.11(a), revealed that shorter 𝓁𝑠𝑐 values caused a decrease of the fraction of light ballistically transmitted, as expected. In particular, a decrease of Tbfrom 29% to 6% was observed at 𝜆= 600 nm for a change of 𝓁𝑠𝑐 from 6.6 𝜇m to 1.7 𝜇m, and a 3% Tbwas yielded 400 600 800 Tb (a) Wavelength / nm 1.0 0.8 0.6 0.4 0.2 0 500 700 400 600 800 Td (b) Wavelength / nm 1.0 0.8 0.6 0.4 0.2 0 500 700 Figure 3.11: (a) Ballistic and (b) diffuse transmittance spectra of 9-𝜇m thick films of the Mie glass under consideration for the conditions r = (95 ±20) nm - f = 5% (blue), r = (225 ±20) nm - f = 10% (green) and r = (225 ±20) nm - f = 15% (orange). The spectra corresponding to the simulated systems have been included as thin lines. 109 3. Mie glasses: fabrication and characterisation 0.3% 0.2% 0.1% (a) 90o 270o 0o 180o L = 9 mµ (b) 90o 270o 0o 180o 0.3% 0.2% 0.1% r = (95 20) nm - f = 5%± Figure 3.12: Evolution of theangulardistributionofthelightscattered by films of different characteristics. (a) Variable 𝓁𝑠𝑐, namely, 6.6 𝜇m (blue), 1.7 𝜇m (green) and 1.1 𝜇m (orange), respectively corresponding to the conditions r = (95 ±20) nm - f = 5%, r = (225 ±20) nm - f = 5% and r = (225 ±20) nm - f = 10%, for a fixed thickness of L = 9𝜇m at 𝜆= 600 nm. (b) Fixed 𝓁𝑠𝑐 = 6.6 𝜇m, corresponding to the condition r = (95 ±20) nm - f = 5%, and variable slab thickness, namely, (1.84 ±0.07) 𝜇m (violet), (6.59 ±0.18) 𝜇m (black) and (9.3 ±0.3) 𝜇m (red), at 𝜆= 600 nm. The angular distributions corresponding to the simulated systems were included as thin lines. Calculations consider the conditions r = 115 nm - f = 8%, r = 225 nm - f = 7.5% and r = 225 nm - f = 12%. for a further reduction of 𝓁𝑠𝑐 to 1.1 𝜇m. Shorter 𝓁𝑠𝑐 implies an increase of the scattering strength of the material, resulting in an increase of Td, Fig. 3.11(b). The size dispersion of the spheres (around 20 nm) was considered in the calculations by averaging the spectra yielded by spheres of sizes within the experimental dispersion range. The angular distribution of the scattered light at 𝜆= 600 nm was measured, Fig. 3.12(a). A shortening of𝓁𝑠𝑐modifiesthe patternoftheangular distributions by causing an angular broadening of both, the transmitted and reflected intensity. A larger number of scattering events led to a higher fraction of light diffusively propagated and, as a consequence, theangularpatternapproachedthedistributionof a Lambertian emitter as the diffusive strength of the material gradually increased. The effect of a variation of L was inspected for the material with scattering 110 3. Mie glasses: fabrication and characterisation centre conditions of r = (95 ±20) - f = 5%, corresponding to 𝓁𝑠𝑐 = 6.6 𝜇m, an intermediate value in relation to the available thickness range. The angular distributions in Fig. 3.12(b) displayed a similar trend in terms of the evolution of the angular patterns, evidencing that an increase of the scattering strength was equivalent to an increase of the slab thickness, as the average number of scattering events escalated in both cases. The corresponding theoretical spectra and angular distributions, calculated according to the approach detailed in Sec. 2.3.2.2 ofCh. 2, were included in Fig. 3.11 and3.12 forthe sakeof comparison. Size and filling fraction values of the simulated systems were chosen within the experimental uncertainty regarding the nominal experimental values. The simulated systems included a 100-nm thick film ofintermediate refractive index between thatoftheairandtheporous TiO2, that is, n = 1.35, on top of the Mie glass in order to account for the slightly inhomogeneous nature of the interface, which would result in an increased in-coupling of the light into the system. The spectra and angular distributions calculated employing the model confirmed the trends experimentally observed, thus demonstrating its reliability and proving its ability to predict light propagation in the considered material without accounting for phase correlation effects, that is, the total scattering was described as a series of successive individual scattering events as dictated by Mie theory. Based on this agreement, the model was hereinafter applied in order to inspect the origin of the angular distribution of light yielded by slabs of the Mie glass depending of their thickness in relation to their scattering strength. Scattered photon statistics: This section presents an analysis of the shape of the angular distribution of the light after interacting with slabs of different thicknesses of the Mie glass under study in terms of the distribution of scattering events that light undergoes. The structural parameters considered in the simulated film correspond to r = 225 nm - f = 12%, for which 𝓁𝑠𝑐 = 0.7 𝜇m and 𝓁𝑡= 3.0 𝜇m at 𝜆= 600 nm according to Mie formalism. The value of 𝓁𝑡was also extracted from applying Eq. 2.23 to the results of T and Rsfor different slab thicknesses yielded by theoretical experiments employing the model. This procedureprovideda value of 𝓁𝑡= 3.7 𝜇m, whichroughlyagreedwith the 𝓁𝑡= 3.0 𝜇m derived from Mie theory. Light is expected to reach 111 3. Mie glasses: fabrication and characterisation full diffusive propagation for L ≫𝓁𝑡. For that reason, the values of L for the simulated films ranged from L = 𝓁𝑡up to L = 7𝓁𝑡(corresponding to 30𝓁𝑠𝑐), which covered thickness values from a point where the optical response should be strongly determined by the finite size of the slab to displaying the typical features of a fully developed random ’bulk’ material. If thickness was short, e.g. L = 𝓁𝑡, light only underwent a few scattering events during propagation and a small fraction of light deviated from the incidence direction, thus resulting in an angular distribution displaying a strong ballistic component. In effect, in this case, around 24% of the incident light was transmitted ballistically, as shown in Fig. 3.13(a). As Fig. 3.13(b) revealed, in a (b) 90o 270o 0o 180o 4% 2% L / = 3.5lt 0.5% 1.0% (c) 90o 270o 0o 180o 4% 2% L / = 7lt 0.3% 0.5% (e) 90o 270o 0o 180o 0.2% 0.1% x3 x5 x20 (f) 90o 270o 0o 180o 0.2% 0.1% x10 x25 x30 (a) 90o 270o 0o 180o 20% 10% L / = 1lt 2% 4% 6% (d) 90o 270o 0o 180o 4% 2% x25 x300 x1500 Figure 3.13: Evolution with thickness of the angular distribution of the light scattered by a simulated slab with the conditions r = 225 nm - f = 12%. (a)-(c) Total angular distribution for slabs of thickness L = 𝓁𝑡, L = 3.5𝓁𝑡and L = 7𝓁𝑡, respectively. The insets show a zoom-in of the respective angular distributions for a better visualisation of the shape. (d)-(f) Angular distribution of photons undergoing s = 1 (blue), s = 3 (orange), s = 5 (violet) and s = 10 (pink) scattering events in slabs of thickness L = 𝓁𝑡, L = 3.5𝓁𝑡and L = 7𝓁𝑡, respectively. The calculations were performed at 𝜆= 600 nm. 112 3. Mie glasses: fabrication and characterisation thicker slab, L = 3.5𝓁𝑡(equivalent to L = 15𝓁𝑠𝑐), light was given the possibility to be scattered a larger number of times, thus resulting in a higher fraction of diffuse light at the expense of Tb, which was reduced to an exiguous 1.2%. As a consequence of the pronunciated decreaseof total transmittance,the film shoulddisplay a whitish opacity typical of an optically random material. Moreover, in contrast to the slab of thickness L = 𝓁𝑡, the angular pattern of the light yielded by a film of thickness L = 3.5𝓁𝑡presented a fairly isotropic shape, resembling that of a Lambertian diffuser, due to the important fraction of light propagating diffusively. Eventually, for film thicknesses significantly larger than 𝓁𝑡, for instance L = 7𝓁𝑡(corresponding to 30𝓁𝑠𝑐), light underwent numberless scattering events, which hampered transmittance and, as a consequence, most of the incident light was diffusively reflected, resulting in high opacity of the film, as illustrated in Fig. 3.13(c). Furthermore, the perfectly Lambertian profile of the transmitted intensity indicates full randomisation of the light. While Tb is largely affected by the irregular variations of the refractive index throughout the material, under the assumptions considered herein and in good agreement with the measurements, the value of Rsis dictated to a great extent by the refractive index contrast between that of the Mie glass and the surrounding medium, air in this case. This became evident when inspecting the evolution of Rsfor increasing thickness of the slab, namely, 5.5%, 5.3% and 5.3% for L = 𝓁𝑡,L= 3.5𝓁𝑡and L = 7𝓁𝑡, respectively, as shown in Fig. 3.14. For short enough 0.0 7.00 Rs L / lt 0.08 0.07 0.06 0.05 3.501.75 5.25 Figure 3.14: Evolution of Rsfor simulated slabs of the system under consideration for increasing value of L/𝓁𝑡. 113 3. Mie glasses: fabrication and characterisation values of L, where the probability of photons undergoing a scattering event is low, a fraction of the incident photons can be reflected at the second interface of the slab, travel back throughout the material and escape in the backward direction without undergoing any further scattering, thus contributing to Rs. As thickness is increased, the probability of a photon crossing the material slab twice avoiding scattering by an inclusion is reduced, therefore causing a decrease of Rs up to a critical thickness, above which no contribution from unscattered photons reflected at the back interface is possible. At this point, the value of Rsstabilised at 5.3%, which was independent on L. This point is discussed again below when inspecting the angular distributions described by the fraction of light undergoing a specific number of scattering events. The angular distributions in Fig. 3.13(a)-(c) can be understood as a sum of the individual angular patterns described by each fraction of light undergoing a specific number of scattering events. Indeed, Fig. 3.13(d)-(f) illustrate particular cases for the three slab thicknesses under consideration. Even though by definition any phenomenon removing photons from the incident beam would be regarded as a scattering event, photons specularly reflected at any of the slab interfaces were considered as non-scattered light. Indeed, in the model only modifications of the trajectory of the photons originated from interaction with the spherical inclusions were considered scattering events. Comparison between Fig. 3.13(d) and (a) evinced that the total angular distribution for a slab of L = 𝓁𝑡was mainly defined by those photons scattered once. Photons scattered once preserved to a great extent the forward-oriented angular distribution typical of Mie scattering. In contrast, owing to the low probability of undergoing a higher number of scattering events across such thickness, the contribution of higher order distributions played a small part in the total intensity. This was evidenced by the high magnifications required in Fig. 3.13(d) for the visualisation of distributions of photons scattered above once. For a thicker slab, e.g. L = 3.5𝓁𝑡the contribution of light undergoing more than one scattering event to the intensity of the total angular distribution started becoming relevant, as pointed out by the lower magnifications required in Fig. 3.13(e) in comparison to those previously observed for the case L = 𝓁𝑡. For such thickness, not only was there substantial probability that photons were scattered 3 and 5 times, but the slab was still thin enough for these photons to reach 114 3. Mie glasses: fabrication and characterisation increase of the number of infiltration cycles, on the average refractive index of the eventual structure. The main aim of this analysis was not only the determination of the spectral refractive index of the fabricated films, but also inspect the possibility of experimentally tuning their value during the preparation process. The results corresponding to this study are shown in Fig. 3.20. Figure 3.20(a) displays the measured Rsfor the prepared films, along with that for a glass substrate for the sake of comparison. The spectra revealed a reduction of Rsfor increasing number of infiltration cycles. Such decrease was attributed to a higher amount of SiO2constituting the film, a consequence of increasing the number of infiltration cycles, which would reasonably result in higher refractive index values. At first glance, Fig. 3.20(a) clearly revealed an anti-reflective character of the fabricated films when comparing to the reference Rsspectrum of a glass, being more pronounced for increasing number of infiltration cycles within the range under consideration.As a matter of fact, calculations Figure 3.18: Fabrication process of a ultralow refractive index film. (a) Sequential infiltration of the PS scaffold through CVD and partial filling of the voids with SiO2. (b) Removal of the PS scaffold and formation of a film consisting in a porous SiO2network. The coating in red represents the SiO2. 121 3. Mie glasses: fabrication and characterisation (a) 400 nm (b) 200 nm (c) Figure 3.19: (a) SEM cross section image of the film prepared as described, consisting in a highly porous SiO2network. (b) Zoom-in of the film, displaying the highly porous nature of the material. (c) Picture of the prepared film on a glass substrate displaying high transparency. demonstrated a reduction of the average Rsof the film before reaching a minimum value as its average refractive index grew, as displayed in Fig. 3.21. Specifically, the calculations considered in a first approximation a wavelength-independent refractive index for the materials and the values of Rscorrespond to spectra averaged within the wavelength range 400 - 800 nm. The considered films were simulated and their spectral Rsdetermined, Fig. 3.20(b), employing for that purpose the refractive index spectra extracted from experimental measurements, Fig. 3.20(c). For the extraction of the refractive index, measurements of Rsand Tbfor three different incidence angles and both polarisation states, sand p, were fitted to those calculated by means of the transfer-matrix method employing a genetic algorithm, designed to find the required complex refractive index providing the best fitting according to a designated model. Here, the Cauchy Absorbent model was considered. The trend experimentally observed for the Rs spectra was faithfully reproduced by the calculations, thus displaying good agreement between measurements and theory. As anticipated, Fig. 3.20(c) demonstrated that the refractive index of the resulting films was directly affected by the number of infiltration cycles performed via CVD. Indeed, the value of the refractive index increased for a higher number of infiltration cycles, as expected. Specifically, a nominalrefractiveindexaslowasn= 1.03 at 𝜆= 600 nmwasachieved for the film subjected to one infiltration cycle. This analysis demonstrated that, not only the proposed experimental procedure enabled 122 3. Mie glasses: fabrication and characterisation (a) 0.06 0.05 Rs 0.07 0.08 0.09 0.10 Wavelength / nm 400 500 600 700 800 Wavelength / nm Rs (b) 0.06 0.05 0.07 0.08 0.09 0.10 400 500 600 700 800 (c) 1.05 1.00 n 1.10 1.15 1.20 Wavelength / nm 400 500 600 700 800 0o 90o 270o 1.0 0.8 0.6 0.4 0.2 0.0 (d) Figure 3.20: Optical characterisation of the fabricated films. (a) Experimental and (b) calculated Rsspectra for a glass substrate (gray line) and a glass substrated coated with a highly porous SiO2film fabricated through one (dark-yellow line), two (green line) and three (orange line) CVD infiltration cycles. (c) Spectral refractive indices extracted from Rsand Tbmeasurements for the films fabricated with one (dark-yellow line), two (green line) and three (orange line) CVD infiltration cycles. (d) Angular distribution of the light transmitted through a glass substrate (gray line) and a glass substratecoated with a highly porous SiO2film prepared through one (dark-yellow line), two (green line) and three (orange line) CVD infiltration cycles at 𝜆 = 575 nm. the preparation of wide-area and highly transparent films of ultralow refractive index, but it also allowed control over its optical properties. A porosity of p = 0.9 could be estimated for the film subjected to one infiltration cycle according to the Lorentz-Lorenz expression considering an average refractive index of n1= 1.05 for the SiO2-network 123 3. Mie glasses: fabrication and characterisation 1.0 1.2 1.4 Refractive index 0.6 0.7 0.8 0.9 Average R (%) Figure 3.21: Calculated average Rsas a function of the refractive index of the SiO2film. film and taking n2= 1.425 for SiO2: 𝑝=1−𝑛2 1−1 𝑛2 2−1.(3.3) From the high transparency observed in Fig. 3.19(b), the scattering strength of the films was expected to be negligible, which was confirmed by measurement of the angular patterns of the light scattered by the films, Fig. 3.20(d). The largely forward-oriented profiles of the angular distributions indicated that most of the transmitted light propagated ballistically, revealing a virtually inexistent diffuse propagation as a consequence of a lack of significant scattering. Indeed, the angular distributions of the films were highly similar to that for a bare glass substrate, which demonstrated a high degree of transparency. 3.4.2 Scattering properties of an optically random material based on a low refractive index film After the development of ultralow refractive index films, the next step in the fabrication of an optically random medium of high diffusive strength consisted in the inclusion of high refractive index scattering particles, namely, monodisperse nanocrystalline TiO2spheres. The fabrication procedure allowed the inclusion of scattering particles in the eventual film by dispersing them in the initial suspension of PS spheres, calculated according to Eq. 2.1. The amount of TiO2in relation to the volume of PS spheres approximately defined the volume 124 3. Mie glasses: fabrication and characterisation 1 µm (a) (b) Figure 3.22: (a) SEM cross section image of the fabricated optically disordered medium consisting of a highly porous SiO2film integrating a random distribution of crystalline TiO2spheres of size r = (225 ±20) nm in a 5% concentration. (b) Picture of the material exhibiting high opacity. filling fraction of scattering centres in the eventual film. Due to experimental limitations, a volume filling fraction f = 5% was considered in the study herein performed. Suspensions prepared for f > 5% were not suitable for the fabrication of films with good optical quality due to inhomogeneities arising during the deposition process. The image shown in Fig. 3.22(a) reveals the desired structural properties of the resulting material, namely, a random distribution of monodisperse spherical TiO2particles in a SiO2matrix of high porosity forming a film. Due to the presence of TiO2particles, intense light scattering was triggered in the film, therefore adopting a white appearance and high opacity typical of turbid media, as visible in Fig. 3.22(b). The opacity of the resulting material was evidenced by the determination of the Tbspectra, Fig. 3.23, which revealed a considerable reduction when compared to the Tbspectrum of the initial low refractive index SiO2film. Following the same procedure previously employed in Sec. 3.3.2.1 for the characterisation of the disorder, the spectral 𝓁𝑠𝑐 were determined from Rsand Tbmeasurements of films integrating optical disorder with different conditions, namely, different sizes of the scattering centres in a 5% concentration, as visible in Fig. 3.24(a), and the corresponding spectra calculated through Mie theory, Fig. 3.24(b). A porosity p = 0.9 was considered for the SiO2 matrix of the simulated films according to Eq. 3.3. Smaller inclusions yielded shorter values of 𝓁𝑠𝑐, as opposed to the trend exhibited by the Mie glass based on a mesoporous TiO2matrix previously discussed in 125 3. Mie glasses: fabrication and characterisation Sec. 3.3.2.1, for which bigger scattering centres were required to minimise 𝓁𝑠𝑐. In order to understand this, it was necessary to analyse 𝓁𝑠𝑐 Wavelength / nm 400 500 600 700 800 1.0 0.8 0.6 0.4 0.2 Tb 0.0 Figure 3.23: Tbspectra for a Mie glass consisting of a highly porous SiO2film integrating nanocrystalline spherical TiO2particles of size r = (146 ±15) nm (blue line) and r = (250 ±30) nm (orange line) in a 5% concentration. The spectrum corresponding to a ultralow refractive index SiO2-based film before the inclusion of scattering centres have been included (gray line) for comparison. (a) 0 lsc / µm 1 2 3 4 Wavelength / nm 400 500 600 700 800 0 lsc / µm 1 2 3 4 Wavelength / nm (b) 400 500 600 700 800 Figure 3.24: 𝓁𝑠𝑐 spectra extracted from (a) measurements and (b) calculations for a Mie glass comprising a low refractive index SiO2film including a random distribution of nanocrystalline TiO2spheres of size r = (146 ±15) nm (blue line), r = (188 ± 15) nm (dark yellow line), r = (225 ± 20) nm (green line) and r = (250 ± 30) nm (orange line) in a f = 5% concentration. The dashed lines in (b) correspond to a hypothetical system consisting in spherical scattering centres of the same size suspended in air in a 5% concentration. 126 3. Mie glasses: fabrication and characterisation r / nm 220 260 Wavelength / nm 400 500 600 700 800 Qsc/ / nmr -1 0.00 0.02 0.04 0.06 180 140 Figure 3.25: Calculation of the ratio Q𝑠𝑐/r for an optically random medium consisting of a low refractive index SiO2film integrating nanocrystalline TiO2spheres in a 5% concentration. The dashed lines highlight the corresponding sizes considered in the analysis. in terms of the ratio of the scattering efficiency, Qsc, to the size of the particle, i.e Qsc/r. Considering a spherical shape, 𝓁𝑠𝑐 (Eq. 2.5), can be expressed in terms of the volume filling fraction, the scattering efficiency and the size of the particles: 𝓁𝑠𝑐 =4 3𝑓𝑄𝑠𝑐 𝑟,(3.4) where 𝑄𝑠𝑐 =𝜎𝑠𝑐/𝜋𝑟2, was taken into account. In light of Eq. 3.4, 𝓁𝑠𝑐 was expected to vary inversely with Qsc/r. With that purpose, the modification of Qsc/r as a function of the size of the inclusions was determined, Fig. 3.25, which justified the trends observed for 𝓁𝑠𝑐 in Fig. 3.24. 𝓁𝑠𝑐 values as short as 436 nm at 𝜆= 565 nm were attained for a particle size r = (146 ±15) nm due to the high refractive index contrast between the TiO2inclusions and the porous SiO2matrix, thus justifying the quest for a ultralow refractive index support matrix for the scattering inclusions. The spectra calculated through Mie formalism, Fig. 3.24(b), reproduced the trends of the spectra extracted from experiments. Such agreement demonstrated the validity of Mie formalism for the description of light transport in this material, proving it to behave as a Mie glass. The observation that the spectra extracted from experiments were generally shorter than those yielded 127 3. Mie glasses: fabrication and characterisation by the calculations was attributed to a likely somewhat larger filling fraction of scattering centres in the materials fabricated, since this parameter does not experimentally offer a fine control over its value, as previously noticed for the Mie glass based on a porous anatase matrix, Sec. 3.3.2.2. The 𝓁𝑠𝑐 spectra corresponding to a hypothetical situation where the TiO2scattering centres are randomly suspended in air were also included in Fig. 3.24(b). The great similarities between these spectra and those corresponding to the Mie glass evinced an airlike character of the porous SiO2matrix in terms of optical properties. This confirmed the ultralow refractive index of the porous SiO2ma- trix, demonstrating its behaviour as a film of air to a great extent, yet capable of supporting particles serving as scattering centres. In this regard, if a SiO2film exhibiting this property were to be infiltrated with any photoluminescent molecule, an efficient out-coupling of the light could be expected. Owing to the small refractive index contrast between the film and the surrounding air, the escape cone would result wide enough to ensure small losses due to light trapped by total internal reflection, guided along the film and lost through the edges. 3.4.3 Mie glasses as efficient light diffusers: comparison In order to assess the relevance of the photonic strength of this optically random material within the framework of strongly diffusive Mie glasses, a comparison of its 𝓁𝑠𝑐 spectrum with that for the TiO2- based Mie glass previously analysed in Sec. 3.3 with the same conditions of disorder was established. In particular, the conditions of the integrated disorder consisted in scattering centres of size r = (225 ±20) nm and concentration 5%. The comparison is illustrated in Fig. 3.26(a).Inbothcases,𝓁𝑠𝑐displayedconsiderablyshortvalues,between 1𝜇m and 3 𝜇m. The relative scattering strength between both materials was, however, revealed to be importantly dependent on the considered wavelength range. Specifically, 𝓁𝑠𝑐 showed shorter values in the wavelength range 400 - 635 nm for the TiO2-based Mie glass, becoming the trend reversed thenceforth, where the strongest diffusion strength was expected for the Mie glass based on a low refractive index material. This trend verified the relationship between Q𝑠𝑐/r and 𝓁𝑠𝑐 previously inspected in Fig. 3.24 and 3.25 according to Eq. 3.4, as evidenced through the determination of Qsc/r for both materials, Fig. 128 3. Mie glasses: fabrication and characterisation lsc / µm (a) 2 1 3 400 500 600 700 Wavelength / nm 800 0o 90o 270o 180o (c) 20% 40% 80% 60% λ = 500 nm λ = 700 nm 0.2% 0.1% 400 500 600 700 Q /r / nm sc -1 (b) Wavelength / nm 0.02 800 0.01 0o 90o 270o 180o (d) 20% 40% 80% 60% 0.2% 0.1% Figure 3.26: (a) Calculated 𝓁𝑠𝑐 spectra for a Mie glass comprising a mesoporous TiO2(dark red line) or a highly porous SiO2(green line) matrix integrating nanocrystalline TiO2spheres of size r = (225 ±20) nm in a 5% concentration. (b) Spectral values of the ratio Q𝑠𝑐/r for the TiO2-based (dark red line) and the SiO2-based (green line) Mie glasses under consideration. (c) Calculated angular distributions of the light emerging from a 2-𝜇m slab of the Mie glasses based either on a TiO2(dark red line) or a highly porous SiO2(green line) matrix under consideration at 𝜆= 500 nm and (d) 𝜆= 700 nm. The insets in (c) and (d) display a zoom-in of the corresponding angular distributions. 3.26(b). The choice of the material for its use as an efficient diffuser would therefore depend on the targeted wavelength, which dictates thepredominanceoftheintensityof thescattering strength ofone material over the other. Indeed, at 𝜆= 500 nm, where the TiO2-based Mie glass was expected to yield a stronger photonic strength than its SiO2 counterpart, the angular distribution of the scattered light displayed 129 3. Mie glasses: fabrication and characterisation a weaker ballistic component as a consequence of an increased scattering intensity, Fig. 3.26(c). Higher diffusion strength removes photons from the ballistic direction, forcing them to emerge from the film at greater angles, thus justifying the broadening of its angular distribution shown in the inset. The opposite behaviour was observed at 𝜆= 700 nm, Fig. 3.26(d), where a more diffusive character was expected for the SiO2-based Mie glass. 3.5 Conclusions This chapter focuses on the design and fabrication of new optically random media in the configuration of a Mie glass. The Mie glass was defined as an optically disordered material consisting of a transparent matrix integrating a random distribution of spherical scattering centres in a controlled manner with concentration values within the range of high dillution for a description of light propagation according to Mie formalism for the scattering of individual particles. First of all, different combinations of materials in this configuration were theoretically tested with the purpose of finding specific combinations maximising the photonic strength. From this study, two particular materials demonstrated the desired properties: a mesoporous TiO2matrix and a ultralow refractive index porous SiO2matrix, both integrating nanocrystalline spherical TiO2scattering centres. In the case of the TiO2-based material, films integrating disorder with different conditions of the scattering centres were fabricated and a characterisation of the optical disorder through the determination of 𝓁𝑠𝑐 and 𝓁𝑡from experiments was performed. Comparison with the results of the corresponding calculations performed by means of Mie theory proved that in specific wavelength ranges, light propagation could be predicted in terms of Mie scattering from the scattering of light by an individual particle. Furthermore, the effects of finite size regarding light propagation were studied for this material. Specifically, the effect of shortening 𝓁𝑠𝑐 through the conditions of the optical disorder included for a fixed size of a slab of this material was demonstrated to affect light transport in a similar way to increasing the thickness of the slab for a particular 𝓁𝑠𝑐 value. A theoretical model was employed for the analysis of the modification of the angular distribu- 130 4. Mie glasses for bifacial DSSCs with enhanced performance the effects of diffuse light propagation in cells including submicron spherical particles in their photoanode. In the study presented herein, the proposed system is modelled and the performance of the device is analysed in relation to the conditions of the scattering particles. The validity of the approach for absorption enhancement regardless of the incidence side of the light is demonstrated and, since bifacial operation of the devices was sought for, a process of optimisation of the transmittance of the counterelectrodeisdescribed.Also, an electricalcharacterisationofthe fabricated cells is presented, through which the relationship between the conditions of the introduced disorder and the performance of the cell previously inspected during the modelling is confirmed. 4.2 Theoretical study of the system According to the approach proposed in this thesis for the fabrication of optically disordered media, the features of the scattering centres employed to introduce optical disorder the material are deliberately selected among the different possible options in such a manner that light propagation can be reasonably well predicted by means of calculations, as previously illustrated in Ch. 3. A meaningful consequence derived from this particularity is the possibility of tailoring the disorder and designing the material properties in order to meet a specific application. Based on this concept, this section describes a theoretical analysis justifying the shape of the particles employed as scattering centres, as well as the results from calculations related to the maximum achievable Jsc values depending on the conditions of the considered inclusions. The way light is scattered when impinging on a particle of a size comparable to its wavelength is not generally intuitive and it is highly dependent on its structural, as well as its material properties. The shape of the particle inevitably defines the pattern described by the electromagnetic field posterior to the scattering event, provided that its size is comparable to the wavelength of the incoming radiation. Calculations of the angular distribution of the light scattered by TiO2 particles of diverse shapes were performed according to the methods described in Sec. 2.3.1.2 in Ch. 2. Due to the the presence of an air shell 137 4. Mie glasses for bifacial DSSCs with enhanced performance surrounding the TiO2sphere, as addressed in Ch. 3, an electrolytefilled layer surrounding the particle and with the same shape of this was included in the calculations, as detailed in Ch. 2. The patterns resulting from the calculations are displayed in Fig. 4.1(a) for the shapesindicatedasschematicsintheinset,specifically,spheres,cubes, truncated pyramids and distorted cubes. The angular distributions presenting the narrowest forward-oriented patterns corresponded to the spherical and the cubic shapes, for which they happened to be virtually identical. As a matter of fact, spherical and cubic TiO2par- Figure 4.1: (a) FDTD calculated angular distribution of the light scattered by TiO2nanoparticles of diverse shapes and surrounded by an electrolyte shell of thickness 0.5r, where r is the radius of a spherical particle, and same shape as the particle, embedded in a mesoporous TiO2matrix, at 𝜆= 650 nm. The same volume was considered for the four scattering structures. The distributions are the result of averaging several distributions yielded for different orientations of the particles with respect to the incident light. The angular patterns correspond to (i) a sphere of size r = 150 nm (blue solid line), (ii) a cube of side L = 242 nm (orange dashed line), (iii) a truncated pyramid of hexagonal base (green dash-dotted line) and (iv) a modified cube (dark red dotted line). (b) FDTD calculated scattering efficiency of a TiO2sphere of size r = 150 nm (blue solid line) and a cube of the same volume (orange dashed line). The scattering efficiency for a sphere of size r = 150 nm obtained through analytical calculations by means of Mie theory was included for the sake of comparison (pink dotted line). 138 4. Mie glasses for bifacial DSSCs with enhanced performance Figure 4.2: Illustration of the different components constituting the bifacial solar cell under consideration. ticles displayed similar scattering efficiencies, as Fig.4.1(b) demonstrates. Thus, cubic- and spherical-shaped scattering centres direct light more efficiently towards deeper regions of the photoanode, effectively reducing reflection losses due to backward-scattered light at the entrance of the cell. In the same line of thought, the broader angular distributions of the light scattered by more irregular shapes, e.g. truncated pyramid and distorted cube, prove them as less preferable options for the aim pursued herein. Despite resulting both the cube and the sphere optimal for the integration into photoanodes, only the spherical shape was considered throughout the rest of the study due to the ease of experimental fabrication. 400 500 600 700 800 Wavelength / nm 0 1 3 2 Scattering efficiency Figure 4.3: Scattering efficiency spectra of crystalline TiO2spheres of size r = 80 nm (blue line), r = 100 nm (purple line), r = 112 nm (red line) and r = 135 nm (orange line) surrounded by an electrolyte-filled shell of thickness sh = 0.5r embedded in a mesoporous TiO2matrix. The spectra were obtained from Mie formalism. 139 4. Mie glasses for bifacial DSSCs with enhanced performance Thephotovoltaicsystemconsideredinthis studycomprised a photoanode based on a dye-sensitised TiO2Mie glass incorporating spherical TiO2scattering centres and sandwiched between front and rear contacts consisting in an FTO-coated glass substrate, being the latter coated with platinum for catalytic action, Fig. 4.2. Not only the concentration of the scattering centres increases the diffusive propagation of light throughout the photoanode, as pointed out by Eq. 1.1, but also their size, as evidenced by the calculated scattering efficiency spectra shown in Fig. 4.3. In order to theoretically assess the effect of the variation of these two parameters on the absorptance and further performance of the PV system herein proposed, calculations of the maximum expected Jsc values with respect to both the volume filling fraction, f, and the size of scattering centres, r, were carried out for a cell under front, Fig. 4.4(a), and rear, Fig. 4.4(b), illumination. As Eq. 2.27 reveals, the value of Jsc is directly related to the efficiency of the cell. The optimum configuration is met within scattering centre filling fraction up to 20% and size up to r = 300 nm. In effect, Jsc = 15.62 mA cm−2 and Jsc = 10.90 mA cm−2 are yielded by the calculations 100 200 300 0 5 10 15 20 Filling fraction (%) Scatterer radius / nm (a) J / mA cm sc -2 Scatterer radius / nm 100 0 5 10 15 200 300 Filling fraction (%) 20 6.70 16.0011.36 (b) Figure 4.4: Simulated upper limit of Jsc for a cell under (a) front and (b) rear illumination in relation to the volume filling fraction value and the size of the scattering inclusions. The black dashed line in (a) corresponds to the value Jsc = 14.20 mA cm−2 yielded by a device including a backscattering layer. The value Jsc = 3.87 mA cm−2 corresponding to the same configuration under rear illumination has not been included in (b), since it is shorter than that for a reference cell. 140 4. Mie glasses for bifacial DSSCs with enhanced performance as the highest attainable Jsc values for r = 180 nm and f = 19% under front and rear illumination, respectively. Such values correspond to a 35% efficiency enhancement under front illumination and a 40% efficiency enhancement under rear illumination when compared to a bare cell, that is, f = 0%. Even though the colour maps displayed a trend of increasing Jsc for bigger inclusions and larger concentrations, the parameter map was restricted to a 20% concentration and size r = 300 nm due to limitations of the model. This Monte Carlo approach fails at predicting light transport in the considered materials outside of the high-dillution regime, since it describes light scattering according to single-particle considerations and does not account for correlation terms between inclusions. For high enough particle concentration or size, the event of light scattering by an inclusion is not anymore independent on the light scattered by the surrounding centres. As previously indicated, the most common approach for the increase of optical absorption in the photoanode of DSSCs consists in the inclusion of a scattering layer made out of larger TiO2particles than those of the electrode between the photoanode and the Pt-coated back contact. Considering this scattering layer in a reference cell, the calculations yielded a maximum Jsc = 14.20 mA cm−2 (front illumination), included in Fig. 4.4(a), and a Jsc = 3.87 mA cm−2 (rear illumination). This shows that the integration of a scattering layer, although resulting beneficial for operation under front illumination, hinders the performance of the device when illuminated from the rear side. It is noteworthy that the addition of these two values is close to that considering those corresponding to a reference cell, specifically, Jsc = 11.69 mA cm−2 and Jsc = 7.83 mA cm−2 under front and rear illumination, respectively. The implications of the shape of the scattering particles and its relationship with the anisotropy of the yielded scattered light pattern, earlier explored in Fig. 4.1(a), on the attainable efficiency enhancement in the device was evidenced when analysing a hypothetical situation, in which an isotropic angular distribution of the light scattered by the inclusions was assumed. Figure 4.5 presents the results of the maximum attainable Jsc values for front and rear illumination, in analogy to Fig. 4.4, when considering equal probability at all angles of the scattered light and identical scattering intensity than a spherical inclusion surrounded by an electrolyte-filled shell. In this nonfac- 141 4. Mie glasses for bifacial DSSCs with enhanced performance 100 200 300 0 5 10 15 20 Filling fraction (%) Scatterer radius / nm (a) J / mA cm sc -2 Scatterer radius / nm 100 0 5 10 15 200 300 Filling fraction (%) 20 (b) 6.70 16.0011.36 Figure 4.5: Simulated upper limit of Jsc for a cell under (a) front and (b) rear illumination integrating imaginary scattering inclusions yielding perfectly isotropic angular distributions of the scattered light. tual situation, only a maximum Jsc = 13.2 mA cm−2 and Jsc = 8.97 mA cm−2 can be at best achieved under front and rear illumination, respectively, which means a scant 13% and 14% enhancement, in comparison to the 35% and 40% improvement demonstrated in Fig. 4.4 for spherical scattering centres. Such contrast revealed the relevance of employing inclusions yielding a highly narrow forward-oriented light scattering on the final performance of the device, resulting otherwise in optical losses due to backscattered light. This observation justifies the seek for particle shapes triggering highly anisotropic scattering initially contemplated at the beginning of the section. 4.3 Counterelectrode optimisation and preparation The back part of the cell, or counterelectrode, comprises a conductive substrate coated with Pt. This coating is in fact essential for proper operation of the device, since it acts as a catalyst for the reaction of regeneration of the dye molecules through the electrolyte in which the entire system is immersed. In a bifacial configuration of the cell, the high absorption by the Pt coating results detrimental for the performance of the device when operating under rear illumination, as it would entail a lower light intensity reaching the photoanode for ex- 142 4. Mie glasses for bifacial DSSCs with enhanced performance citation of the dye molecules. For this reason, trade-off between high transparency and maintaining enough catalytic action must be met. A study of the performance of cells depending on the amount of Pt coating the counterelectrode was performed. Two concentrations of the H2PtCl6solution, namely, 6 mM and 7 mM, and up to three depositions were tested. Coating of the substrates by casting a drop of the solution was also tested for the sake of comparison. The transmittance of the different counterelectrodes was measured, Fig. 4.6(a) 500 600 700 800 Wavelength / nm 0 0.2 0.4 0.6 0.8 1.0 (a) Transmittance 500 600 700 800 Wavelength / nm 0 0.2 0.4 0.6 0.8 1.0 (b) Transmittance 500 600 700 800 Wavelength / nm 0 2 4 6 8 10 (c) J / mA cm-2 500 600 700 800 Wavelength / nm (d) 0 2 4 6 8 10 J / mA cm-2 Figure 4.6: Transmittance of glass/FTO/Pt-coated substrates prepared with different deposition number through spin coating of a (a) 6-mM and (b) 7-mM concentration H2PtCl6solution. (c) J-V curves determined for solar cells fabricated employing the as-prepared counterelectrodes employing a 6-mM and a (d) 7-mM concentration H2PtCl6solution. The spectra correspond to counterelectrodes coated with one (blue line), two (green line) and three (orange line) depositions of a H2PtCl6solution. Data for a system prepared through drop casting were also included (gray line). 143 4. Mie glasses for bifacial DSSCs with enhanced performance Table 4.1: Electrical parameters measured for devices integrating a counterelectrode prepared with a H2PtCl6solution of different concentration. 6-mM H2PtCl6solution concentration Cell Jsc exp. / mA cm−2 Voc / mV FF (%) PCE (%) Drop casting 7.91 799 69.6 4.4 1 deposition 7.87 784 70.0 4.3 2 depositions 8.29 781 69.4 4.5 3 depositions 8.83 782 68.2 4.7 7-mM H2PtCl6solution concentration Drop casting 8.22 762 70.0 4.4 1 deposition 8.37 785 65.2 4.3 2 depositions 8.73 759 66.6 4.4 3 depositions 8.54 774 67.8 4.5 and (b). From these spectra, it was straightforward to conclude that the method of drop casting yielded lower transmittance values than spin coating. As a consequence, it was preferable to rely on the latter for the coating process. After that, these counterelectrodes were employed for the fabrication of standard cells comprising an 8-𝜇m thick photoanode. An electrical characterisation of the devices allowed the determination of the J-V curves for the evaluation of the influence of the amount of Pt coating the counterelectrode on the performance of the cell. Although no significant influence of the solution concentration on the transmittance spectra on was revealed, the J-V curves, Fig. 4.6(c) and (d), displayed a clear dependence. Whereas the efficiency of the devices including counterelectrodes prepared by using a 7-mM Pt solution remained virtually unvariable, see Table 4.1, the efficiency yielded by those devices including a counterelectrode prepared with a 6-mM solution displayed a clear increasing trend with the number ofdepositions, showing amaximum 4.7% PCE for3depositions. Such value corresponded to the maximum observed for all of the considered cases. In light of these results, a 6-mM concentration solution 144 4. Mie glasses for bifacial DSSCs with enhanced performance and 3 depositions by spin coating were established as the most suitable conditions for the preparation of the counterelectrodes. 4.4 Integration of scattering centres in photoanodes: absorptance enhancement The entire fabrication process of photoanodes integrating optical disorder, from the synthesis of the spherical inclusions to the final solid film, is based on solution-processing methods, which are known to be inexpensive. Plus, the lack of order in the included structures reduces the need for sophisticated integration methods, therefore demanding lower complexity during the fabrication process. Altogether, these characteristics demonstrate the suitability of the proposed approach when aiming at an efficient and low-cost solar technology, typical features of third-generation photovoltaics. The integration of optical disorder in the sensitised mesoporous TiO2films typically employed as photoanode in DSSCs was previously proved in Ch. 3 to induce an effective increase of the optical absorption of light by the dye molecules. Hence the potential of this material for the fabrication of solar cells with improved performance. Essential for the fabrication of bifacial solar cells is the ability of the device to efficiently operate regardless of the incidence side and angle of the incoming light, which is ensured by the random nature of the approach herein proposed20–22,24. For the demonstration of this point, measurements of the absorptance were performed under front and rear illumination for two conditions of the inclusions, Fig. 4.7, revealing spectra that remained virtually unaffected by the illumination side. Such effect was also promoted by the fully symmetric shape of the scattering particles, which ensured identical scattering regardless of the incidence angle. Although the absorption enhancement yielded by a system integrating a backscattering layer was revealed significant under front illumination, the absorptance dramatically decreased under illumination from the rear side due to light lost through efficient diffuse reflection that could not be coupled into the system. Were this electrode configuration to be part of a device, one should take into account its unsuitability for bifacial operation. Let us remember that the integration of a backscattering layer is a widely 145 4. Mie glasses for bifacial DSSCs with enhanced performance Figure 4.7: Demonstration of the invariability of the absorptance spectra of a system glass/FTO/sensitised electrode including scattering centres of size r = (100 ±15) nm and concentration f = 10% (green line) and f = 20% (red line) under (a) front and (b) rear illumination. The absorptance spectra for a system glass/FTO/sensitised bare electrode/backscattering layer have been included as a black dashed line in (a) for front illumination and as a pink dashed line in (b) for rear illumination in order to illustrate the strong dependence of the light-harvesting capability of this system on the illumination side. extended strategy for the fabrication of high-performance DSSCs. In fact, the certified record PCE of 11.9% for a DSSC up to date was yielded by a device including a 5-𝜇m backscattering layer29. On the whole, such analysis proved one of the major points of this approach, that is, its appropriateness for the fabrication of photoanodes for DS- SCs to operate under bifacial conditions. Determination of the diffuse properties of the base material would allow an assessment of its diffusive strength, generally regarded as a way to evaluate its scattering and light-trapping ability. The haze is in fact a straightforward manner to easily assess the diffusive character of a material and it accounts for the fraction of light scattered by a material and it is defined as the ratio of Tdto T of the light emerging from the material at the exit interface. High haze values have been reported to yield high 𝜂𝐼𝑃𝐶𝐸34. However, it should not be regarded as a unequivocal magnitude providing the light-trapping ability of a material, since it lacks information about essential factors of the diffuse propagation of light. The haze only considers the fraction of light being transmitted. Besides, it cannot account for the 146