r = random # generator p = absorption probability

1. Novel Next-Generation Multijunction Quantum Dot Solar Panel Designs Using Monte Carlo-Based Modeling Valerie Ding r = random # generator p = absorption probability Theories and Model Basics Introduction Relevance The Sun provides a lot of power to Earth in the form of solar irradiance, photons with various energies. This provides a source of clean and unlimited renewable energy. At the current rate, 45 minutes of sunshine is enough energy to power the world for an entire year. Yet, in 2012, solar accounted for only 1% of world energy supply, which is still dominated by fossil fuels, contributing 30+ billion tons of CO2 emission each year – 100 times the weight of the entire human race! The key issue is the cost of solar energy generation. In the model, the number of photons was normalized to 10 million, distributed according to solar spectra as observed on Earth’s surface. Intensity I at photon energy E is fluxF (# photons striking surface) times E Flux changes as photons travel through solar cells. For one photon in one layer, change in flux can be expressed as a ceiling function: For a multijunction solar cell with 10 million incoming photons assumed, change in flux can be expressed as: The distribution of 10 million photons across the solar spectrum is described as following, assuming a Gaussian distribution of photon energies due to the quantum dot diameter distribution. The total solar power can be expressed as: With this model, power from a N-stack MJQDSC is related to the bandgap of quantum dots in each stack: The intrinsic spectral efficiency can be estimated as: Moreover, accounting for the thermodynamic effect due to the Carnot principle, with the temperatures of Earth’s surface and the Sun: These calculations were executed using Monte Carlo simulation and with programs written in the JAVA programming language. Raw solar spectra data from the NREL was converted from irradiance intensity as a function of wavelength l, I(l) toI(E), using calculus and computer programing: • This is a first numerical model to quantify MJQDSC efficiency. The best designs aided by the model are capable of delivering more than double the maximum efficiency of conventional solar cells. • This model provides an effective path to quickly design optimal MJQDSCs, reducing design and optimization time from years or months to days. • This method also enables direct correlation and calibration using experimental data from absorption when data is available, and is not limited to theoretical work. • With refinement, this method could be quickly adapted for other materials or a smaller-interval grid for quantum dot diameter, enabling finer optimization of MJQDSCs. • This model can quickly and drastically improve MJQDSC efficiency, enabling cost-competitiveness of clean and sustainable solar power. • The starting point of this project lies in obtaining absorption coefficients of quantum dots, which will be used in Monte Carlo modeling. • Schrödinger's equation dictates the electronic properties of each quantum dot: • h =Planck’s constant • m= particle’s mass • Ψ = particle’s wavefunction • E = eigenvalue of electron energy • Absorption coefficient of a photon with an energy E can be expressed as: • where • f represents the Fermi-Dirac distribution • δfunction is a step function = 1 when EC– EV– E = 0 and 0 otherwise. • With Quantum Dot Lab on NanoHUB.org, the eigenstates and wavefunctions for colloidal (spherical) can be calculated after materials (in this case, PbS) and diameter of quantum dots are defined. • For example, the following three graphs are eigenstates 1, 9, and 100, as calculated for colloidal PbS quantum dots with 5 nm diameter, relatively large quantum dots that absorb primarily infrared light energies. • Absorption spectra can be calculated for quantum dots with varying sizes. They are superimposed below. Sun to Earth: 1.74 X 1017 Watts Source: HoustonTomorrow.Org Accuracy and Other Factors TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD r = random # generator p = absorption probability • The model was normalized by maintaining a constant total thickness of QD layers at 18 microns. • Because the simulation involved a large number (10 million) of photons, the variation in estimated efficiencies from run to run, even using random number generation, was small. Results were stable to three significant figures. • This model accounts for fundamental spectral loss. Additional losses such as inefficient capture and transport need to be minimized to truly reap full benefits and achieve maximum efficiency. Source: International Energy Agency Cost and Efficiency of Solar Energy Solar energy costs are decreasing, but still two times those of fossil fuels. A key reason for this cost is that despite vast improvement over the years, conventional solar panels have an average efficiency of only about 15%: i.e., only 15% of solar irradiance is converted to usable electricity. Future Work • With the aid of more refined state-of-the-art quantum mechanical modeling software, a more refined grid size for quantum dots, and experimental validation of absorption coefficient data, this model is capable of accurately predicting the maximum efficiency of multijunction quantum dot solar cells using any materials and can identify the best designs. With additional programming, the vast majority of the design process could be fully automated. • With colloidal quantum dot solar cells progressing, the time has come for multijunction quantum dot solar cells to start shining. • This model can be used as a critical stepping stone in this endeavor. A key factor for the low efficiency is fundamental: the Shockley-Queisser Limit caps conventional solar cell efficiency at 34%, as most of the energy from solar irradiance is “unusable”.

2. MJQDSC: High Efficiency at Low Cost Results and Discussion Conclusions • Multijunction solar cells, with multilayer structures and each layer fine-tuned to absorb and convert specific energy bands of sunlight, have been demonstrated to bypass the Shockley-Queisser Limit. • It is generally accepted that multijunction solar cells are the key to improving efficiency. However, constructing effective multijunction solar cells integrating many different materials can be prohibitively expensive. • An idea has been proposed that the same materials, with the help of varying sizes of quantum dots, can be used for different stacks. This is the idea of multijunction quantum dot solar cells (MJQDSCs). • Following is a schematic showing a MJQDSC. Photons of various energies (depicted as red, green, and blue rays) are absorbed by a particular layer of specifically-sized quantum dots, with their energy converted to electricity, depicted as electrons (e-). This ensures improved efficiency in solar cells. • Using a grid of 0.5 nm for PbS quantum dot diameter and Monte Carlo modeling as discussed above, various MJQDSCs were designed and evaluated. Theirefficiencies were calculated and compared. • Following is the progression of energy spectra for best designs of 2, 3, 5 and 9-layer MJQDSCs. With a fixed total thickness, the spectral change at each quantum dot layer were tracked and plotted. The highest-efficiency PbS MJQDSC designs identified with this model are listed below. Within these MJQDSC designs, a detailed breakdown of how each layer contributes was obtained through this model. As expected, since the total thickness is held constant for comparison, increasing the number of quantum dot layers leads to higher efficiency. Nearly 80% intrinsic efficiency is achievable with 9 QD stacks. Of course, the incremental improvement decreases as the number increases; diminishing return is observed. Utilizing excellent traceability at the individual photon level in this model, statistical analysis was performed to assess the effectiveness of all aspects of these solar cells. For example, for the best 9-stack MJQDSC design, a set of detailed indicators were calculated and used to access the impact of each factor on total efficiency. Accounting for thermodynamic effect due to the Carnot principle, the maximum efficiencies for optimized designs aided by the model were calculated. • Maximum 50.0%, 57.5%, 66.1% and 75.0%-efficiency 2,3, 5, and 9-junction lead sulfide (PbS) quantum dot solar cells, respectively, were designed using Monte Carlo simulation, significantly surpassing the conventional solar cell maximum efficiency of 33.7%. • By combining quantum mechanical predictions and Monte Carlo simulation, the first ever novel model to design and optimize multijunction quantum dot solar cells was developed and tested to quickly design and optimize multijunction quantum dot solar cells, cutting design and testing time from months or even years to merely days or hours of computation. • This model has an open architecture capable of utilizing absorption properties either obtained theoretically or experimentally, enabling rapid calibration of data and refinement of predictive abilities in the future. e- e- e- e- e- e- e- e- e- References Altermatt, P. 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TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD AREA – THIS GUIDE WILL BE REMOVED BEFORE PRINTING – TRIFOLD Objective of Project Multijunction quantum dot solar cell is a relatively new concept proposed in the past few years. No commercial products exist. Researchers are in exploration stage focusing on understanding fundamental properties of quantum dots to utilize its properties. This project attempts to go one step beyond to achieve breakthroughs. Colloidal PbS quantum dots, the best experimentally studied low cost system is utilized. Absorption properties using quantum mechanical modeling are integrated using Monte Carlo simulation to predict photon and quantum dot interactions, which in turn is used to calculate the intrinsic solar cell efficiency to identify optimized MJQDSC configurations.