FRIEDRICH-ALEXANDER-UNIVERSITÄT ERLANGEN-NÜRNBERG TECHNISCHE FAKULTÄT • DEPARTMENT INFORMATIK Lehrstuhl für Informatik 10 (Systemsimulation) Optical Simulation of Upconversion Nanoparticles for Solar Cells Constantin Vogel Master Thesis Optical Simulation of Upconversion Nanoparticles for Solar Cells Constantin Vogel Master Thesis Aufgabensteller: Prof. Dr. Ch. Pflaum Betreuer: M. Sc. J. Hornich, Dr. K. Forberich Bearbeitungszeitraum: 01.07.2015–18.01.2015 Erklärung: Ich versichere, dass ich die Arbeit ohne fremde Hilfe und ohne Benutzung anderer als der angegebenen Quellen angefertigt habe und dass die Arbeit in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegen hat und von dieser als Teil einer Prüfungsleistung angenommen wurde. Alle Ausführungen, die wörtlich oder sinngemäß übernommen wurden, sind als solche gekennzeichnet. Der Universität Erlangen-Nürnberg, vertreten durch den Lehrstuhl für Systemsimulation (Informatik 10), wird für Zwecke der Forschung und Lehre ein einfaches, kostenloses, zeitlich und örtlich unbeschränktes Nutzungsrecht an den Arbeitsergebnissen der Master Thesis einschließlich etwaiger Schutzrechte und Urheberrechte eingeräumt. Erlangen, den 18. Januar 2016 . Acknowledgements I want to thank Christoph Pflaum and Julian Hornich (LSS), Karen Forberich (iMEET), Robyn Klupp Taylor and Fabrizio-Zagros Sadafi (LFG) for their productive collaboration. 4 Abstract Energy is a resource that experiences shortage due to climate change and consequences mankind has drawn. Thus, research tries to boost efficiency of solar cells to help society to focus on renewable energy sources. This thesis is dedicated to make progress at this point. By implementing optical simulation of upconversion nanoparticles for solar cells, spatial propagation of light and temporal evolution of energy level population densities is investigated. For this task, properties of used materials such as semiconductors, crystalline matter and upconversion lanthanides are gathered along with physical background about radiometric quantities for solar illumination and solid-state physics for surface plasmon polariton local field enhancement. Time harmonic inverse iteration method is performed for propagation simulation which properly handles negative permittivity of silver. A performed coverage study concludes that absorption exhibits a maximum for 50 % of the surface that is covered by silver patches. Furthermore, rate equation analysis confirms red and green upconversion lines to be of nonlinear second order, while the blue upconversion line shows nonlinear fourth order dependency to the incident electric field. 5 Contents Glossary 7 Symbols 8 1 Introduction 11 2 Solar cells 12 2.1 Radiometric quantities . 13 2.2 Semiconductor materials . 16 2.3 Shockley-Queisser limit . 17 3 Upconversion nanoparticles 19 3.1 Hexagonal prism . 19 3.2 Sodium yttrium fluoride . 21 3.3 Extrinsic Euler rotation . 22 3.4 Einstein transitions . 23 3.5 Energy transfer upconversion . 24 3.6 Silver patches . 26 3.7 Surface plasmon polariton . 26 4 Optical simulation 28 4.1 Finite-difference time-domain method . 28 4.2 Courant-Friedrichs-Lewy condition . 29 4.3 Boundary conditions . 30 4.4 Finite integration technique . 31 4.5 Time harmonic inverse iteration method . 31 4.6 Particle layouts . 32 4.7 Rate equations . 33 5 Results 37 5.1 Coverage study . 37 5.2 Absorption behavior . 38 5.3 Upconversion efficiency . 39 5.4 Accuracy analysis . 40 6 Conclusion 42 Bibliography 43 Figures 45 Tables 46 Appendix I 6 Glossary Term Acronym Description Carbon dioxide CO2 Courant-Friedrichs-Lewy condition CFL Energy transfer upconversion ETU phonon-assisted nonradiative energy transfer Erbium Er3+ lanthanide ion with atomic number 68, dopant for sodium yttrium fluoride Excited state absorption ESA stimulated absorption from system already in an excited state Finite integration technique FIT refinement strategy for curved hyperplanes in a rectangular grid Finite-difference time-domain method FDTD iterative algorithm to solve Maxwell’s equations Gallium arsenide GaAs direct semiconductor Ground state absorption GSA stimulated absorption from system in ground state Hydrogenated amorphous silicon a-Si:H non-crystalline silicon Partial differential equation PDE Perfectly matched layer PML absorbing boundary condition Power conversion efficiency PCE percentage of power transformed from optical to electrical domain Silicon Si indirect semiconductor, standard material for solar cells Silver Ag Sodium yttrium fluoride NaYF4 material of upconversion nanoparticle Time harmonic inverse iteration method THIIM stable finite-difference time-domain method iteration algorithm for negative permittivity Upconversion nanoparticle UCNP nano structure, adding energy transfer upconversion to solar cells Ytterbium Yb3+ lanthanide ion with atomic number 70, dopant for sodium yttrium fluoride 7 Symbols Quantity Symbol Value/Unit Description Angle θ rad Angular frequency ! rad=s frequency of the phase rotation Area A m2 Attenuation coefficient α =m coefficient of Lambert-Beer’s law, describing exponential decay in lossy medium 23 Avogadro constant NA 6:022 · 10 =mol amount of atoms/molecules in one mole −23 Boltzmann constant kB 1:3806505 · 10 J=K proportional constant of the law for ideal gases Boundary condition Γ behavior of propagation phenomenon at boundary of simulation domain Ωs 3 Charge density ρe C=m sources of the electric field Complex refractive index n¯ material constant for the optical density Computational complexity C Coverage ratio p percentage of hexprism’s surface area that is covered with silver patches Cross-section σ cm2 cross-section of stimulated radiant transition between energy levels Ei and Ej Current density ~| A=m2 moving charges Curvature a second derivative of a function Decay rate γ =s rate of spontaneous radiant transition from energy level Ei to level Ej Earth e Edge length `h m edge length of hexprism’s regular hexagonal face Electric conductivity σe F=m=s damping factor for the electric field E in finite-difference time-domain method Electric field E V=m Electric potential Ψ V 8 Quantity Symbol Value/Unit Description Electron e− Energy E J Energy level E =cm Energy transfer upconversion coefficient κ cm3=s coefficient of phonon-assisted nonradiative energy transfer Extinction coefficient ke lossyness of the medium, imaginary part of complex refractive index Height hh m height of hexprism Hole h+ 2 Irradiance Ee W=m incident radiant flux per area Lebesque norm L2 Lebesque norm 2 Magnetic conductivity σm N=A =s damping factor for the magnetic field H in finite-difference time-domain method Magnetic field H A=m 3 Mass density ρm g=cm relation between mass and volume Molar mass Mm g=mol mass of one mole of a certain substance 3 Molar volume Vm cm =mol volume of one mole of a certain substance Nonradiative decay rate γ~ =s rate of spontaneous nonradiative transition from energy level Ei to level Ej Normal vector n vector perpendicular on an area Numeric accuracy A Permeability µ 1:2566370614·10−6 N=A2 magnetic field constant Permittivity 8:854187817 · 10−12 F=m electric field constant 3 Photon density Φd =cm amount of photons per volume Planck constant h 6:6260693 · 10−34 Js constant that maps frequency of a photon to its energy Population density N =cm3 amount of ions per volume that occupies energy level Ei Poynting vector S W=m2 energy transport direction of an electro-magnetic wave Projection P projects a vector from R3 onto a hyperplane 2 Radiance Le;Ω W=m =sr radiant flux that is emitted by a light source per area dA per solid angle dΩ 9 Quantity Symbol Value/Unit Description 2 Radiant exitance Me W=m emitted radiant flux per area for specific angle Radiant flux Φe W energy that is transferred to a defined volume per time. also called optical power Radius r m Reduced Planck constant ~ 1:05457168 · 10−34 Js constant that maps a photon’s angular frequency to its energy Refractive index nr real part of complex refractive index Rotation R rotates a vector in R3 Simulation domain Ωs 3 subset of three-dimensional space, i.e. Ωs ⊂ R Solid angle Ω sr surface area of a sphere inside a cone related to the sphere’s radius 2 Spectral exitance Me,λ W=m =nm emitted radiant flux per area per wavelength for specific angle Spectral flux Φe,λ W=nm spectral power, i.e. energy that is transferred to a defined volume per time per wavelength range dλ 2 Spectral irradiance Ee,λ W=m =nm incident radiant flux per area per wavelength 2 Spectral radiance Le;Ω,λ W=m =sr=nm spectral flux that is emitted by a light source per area per solid angle per wavelength range Speed of light c 2:99792458 · 108 m=s propagation speed of lightwaves/photons in empty space Stefan-Boltzmann constant σ 5:67 · 10−8 W=m2=K4 emitter constant of black bodies Sun s Temperature T K Time t s Upconversion efficiency ηuc Wavelength λ µm distance between two adjacent wave fronts of the same phase X direction x Y direction y Z direction z 10 Chapter 1 Introduction Climate change poses a threat to human life [14]. Coastline erosion and land loss intensify with rising sea level due to increasing temperatures and melting ice caps [33], causing necessary measures like displacing hundreds of million of people in developing countries [14]. Furthermore, the probability of natural disasters rises due to higher total entropy in the atmosphere [36]. To counter this threat, governments take efforts to minimize carbon dioxide emission by shutting down coal-fired power plants [23] and setting limits to exhaust gases [18]. Such situations create demand for alternative, renewable energy sources. In 2011, this demand drastically increased after the nuclear disaster of Fukushima Daiichi. After the temporal nuclear memorial, the German government decided to stop using nuclear power after 2022 [8]. In order to compensate the retirement of both coal-fired and nuclear power plants, the renewable energy sector is boosted to bring up new technologies for alternative and at the same time efficient energy harvesting. Alpine height differences offer great potential for hydropower plants, which is already heav- ily exploited: By 2009, Austria harvested 68.1 % of its annual hydropower potential ranging at 56.1 TWh [48].