Rapid Quantum Dot Solar Cell Optimization: Integrating Quantum Mechanical Modeling and Novel Solar Absorption Algorithm
By
Quantum dot (QD) solar cells are theoretically twice as efficient as conventional solar cells, but stateoftheart research QD cell efficiency is very low. Significant, rapid optimization is urgently needed.
A novel research method is proposed and demonstrated using PbSe QDs, combining quantum mechanical absorption coefficients and solar irradiance. Cloud computing and Python programming is used to implement the novel algorithm for each QD configuration. The model was validated against published experiments.
Results indicate that QD performance highly depends on shape and size, and optimal peak QD conditions are identified. This method will enable rapid optimization of QDs for solar applications.
Photovoltaics are the cornerstone of modernday renewable energy research. An emerging technology, quantum dot (QD) solar cells, is theoretically highly efficient. However, stateoftheart QD cells still are far behind in efficiency. Experts are predicting years, if not decades, before wide QD adoption. Rapid progress is urgently needed.
This study proposes and demonstrates a novel research method to rapidly improve QD cells. As a demonstration, PbSe QD absorption coefficients for various geometries and sizes were computed quantum mechanically with the help of cloud computing. A novel algorithm was developed to combine absorption coefficients and solar irradiance to calculate total solar energy absorption. A Python program was written to implement the algorithm for each configuration. A variety of configurations were compared to identify optimal condition for QD geometries.
Results indicate that QD performance is highly dependent on shape and size, and peak performances for various quantum shapes are optimized at different dimensions: for example, spheroids at 3.6 nm, cuboids at 3.0 nm, and spheroids at 4.0 nm.
This model was validated against previously published theoretical and experimental studies, which primarily focused on feasibility and material. This study is the first to combine quantum mechanical modeling with solar irradiance to optimize QD geometries specifically for solar absorption, enabling rapid optimization of QDs for solar applications much more quickly than current trialanderror experimentation does.
This model can be easily applied to other materials, and can be extended to include more aspects of QDs, such as multiple exciton generation, to further optimize QD cell efficiency.
“You now join Albert Einstein, Marie Curie, and Isaac Newton…”
These are my heroes. One month into my freshman year at the Catlin Gabel School in Portland, Oregon, I was at the 2011 Broadcom MASTERS National Finals, and MIT Lincoln Laboratory’s representative was announcing the naming of minor planets. I got mine—27392 Valerieding. That sealed it: I was tied to science, math, and engineering forever.
Eight months later, I was ogling a gigantic accelerator at CERN in Geneva, Switzerland. There, scientists were furiously searching for the Higgs boson, responsible for creating all mass in the universe… including 27392 Valerieding (and Valerie Ding herself). This trip was a result of an award from the 2012 Intel International Science & Engineering Fair for my whitelight LED research.
Amidst lectures on quantum, particle, and theoretical physics from CERN scientists, we visited the École Polytechnique Fédérale de Lausanne’s Photovoltaics Lab. I was inspired by discussions with their worldleading researchers on emerging photovoltaic technologies, especially quantum dot solar cells. An impulse crossed a synapse somewhere in my brain. Quantum dot solar cells, a perfect combination of my passions: quantum physics, math, programming, earth science, energy research. On my thunderstormdelayed flight home, a companion brainstorm raged and this project began.
Should I have the honor of also being selected by Google Science Fair, the path I’ve chosen thus far will truly be reaffirmed and validated. My next pursuit may well stem from stimulating discussion with a participating Google Science Fair scientist.
The key questions this study aims to address:
 How does quantum dot solar absorption vary with quantum dot shape and size?
 With that knowledge, can one determine what the optimal quantum dot geometric configuration is?
The hypotheses:
 Quantum dot solar absorption is strongly affected by quantum dot shape and size.
 Optimal quantum dot configurations can be determined by integrating the quantum mechanically calculated absorption coefficient spectra and the solar irradiance spectrum.
Lead selenide, PbSe, was chosen because it is more widely used in quantum dot solar cells.
Before investigation, it was expected that one could fully utilize known bulk materials of a chosen material, and that one could numerically compute the solutions to Schrodinger’s equation. Those solutions could be used to calculate absorption coefficient spectra as a function of energy. A literature search was to be conducted to determine potential quantum dot size range for investigation. Sizes wider than that range were to be tested for completeness. It was fully expected that computing software needed to be used for this timeconsuming computation. A data file source for the solar irradiance spectrum was to be found, and a computer program was to be written to implement integration of solar irradiance spectrum to absorption coefficient spectra for each quantum dot configuration to facilitate direct comparison.
Finally, a rating system was to be developed to compare each quantum dot configuration. The results were to be validated against prior experiments. Upon model validation, optimal QD configurations were to be determined.
One of the newest research topics in renewable energy has been quantum dot (QD) solar cells. QD cells are theoretically highly efficient—66%, which is double the theoretical siliconbased solar cell maximum efficiency of 33%. However, as seen in the chart (edited NREL compilation), the best demonstrated research QD cell efficiencies remain at only 7% in March 2013.
Quantum dots are semiconductor nanoparticles with tunable properties for various applications. Below is a design for QD intermediate band solar cell, courtesy Yoshitaka Okada.
Researchers are working hard on improving QD cells, exemplified by latest studies:

The most recent QD cell work, published in Nano Letters in February 2013 by University of Toronto and Tsinghua University researchers, uses colloidal QDs and gold to prove that QD cells can be made capable of absorbing infrared light, which is something that siliconbased cells cannot capture due to the nature of their electronic bandgap. However, the authors of the paper write that it is a “proof of principle” and “more work needs to be done.”
 Some experimental studies investigate the effect of QD size on efficiency. The consensus is that for spheroidlike PbSe QDs, diameters < 2 nm do not work well and large diameters are not efficient. However, optimum size is not yet defined.
 However, TEM scans (below, by Gao et. al.) that the studies do not have solid control over QD size or geometry.
Evidently, there has been a lot of work in advancing QD cells. However, ultimately the record QD cell efficiency is still 7%. I was curious as to how it could possibly be that a multitude of researchers collaborating at institutions around the world (such as EPFL) could not raise the QD cell efficiency to meet that of other technologies. I went through an extensive scientific literature and article search to determine stateoftheart techniques and methods in QD experimentation. After consolidating my thoughts on what I had read, I realized that a major reason for this uncharacteristic lag in QD cell efficiency was simply that there is currently no effective way to rapidly improve QD cells without conducting massive amounts of trialanderror experimentation (as has been the norm).
But quantum physics has been wellestablished as a technique to investigate materials, including those used in QD cells. In fact, quantum physics can help determine the absorption coefficient of a material, which is directly related to how a material will interact with light. So why isn't quantum mechanical modeling widely used in QD cell research? While examining the solar irradiance spectrum on Earth's surface (from PV Education), I realized that quantum mechanics had to be used in conjunction with solar spectrum analysis in order to obtain any meaningful results.
Thus, I determined that my goal was to combine quantum mechanically obtained absorption coefficient spectra with analysis of the solar irradiance spectrum into a more inclusive algorithm.
As it turns out, this is the first time such an approach has been utilized.
Research Process
 PbSe bulk properties such as lattice constants, atomic mass, and effective electron mass were obtained.
 Extensive internet research was done to determine a viable software; eventually, cloudcomputing Quantum Dot Lab on NanoHUB was chosen.
 Thirty QD configurations were identified, based on criteria such as synthesis feasibility, viability in solar technology, and quantum mechanical complexity.
 Absorption coefficient spectra were computed, using Eq. 1,2, for each of the thirty QD configurations.
 The solar irradiance spectrum I(E), published and maintained by the National Renewable Energy Laboratory, was obtained.
 An algorithm for total solar absorption Eq. 8 that combined the absorption coefficient spectra and solar irradiance spectrum was developed, by mathematically integrating the product of absorption coefficient and solar irradiance over the entire solar energy spectrum.
 A Python program was written to implement the total solar absorption algorithm. This entailed approximating the integral from the total solar absorption algorithm with Riemann sums Eq. 9, and additionally using interpolation to align absorption coefficient data values to solar irradiance data values.
 Total solar absorption was calculated, through the written Python program (embedded below), for each of the thirty QD configurations.
 A rating system (which normalized absorptions to the highestperforming, highestabsorption QD configuration) Eq. 10 was developed to directly compare performance of individual QD configurations.
 Model results were checked against each and every known published experimental study on PbSe QDs, especially regarding QD size (see "Research"), to ensure that this model was consistent with experimental data.
 By comparing ratings, optimal QD cell conditions were determined.
Variables
Because this is primarily a theoretical work, there are no experimental variables. However, the input parameters for modeling are:
 Chosen QD material (PbSe) and material properties such as bulk properties, atomic mass, etc.
The independent variables are:
 QD shape (cuboid, cylinder, dome, spheroid, pyramidal)
 QD size (final range: 2.7 to 6 nm)
And the dependent variables are:
 Absorption coefficient spectra
 Total solar energy absorption (in the form of integration of absorption coefficient spectra with solar irradiance spectrum)
 Ratings for each QD configuration (through normalization of total solar energy absorption)
Resources Used
 Work was performed on a personal laptop computer at home, school, and libraries.
 Cloudcomputing software Quantum Dot Lab on NanoHUB was used to facilitate heavy quantum mechanical computation.
 The Python programming language was used to implement the developed total solar absorption algorithm.
Validation of Model
To ensure that my methodology was sound, I compared notes with what I had learned from EPFL and discussed my idea and approach with Dr. Bjoern Seipel, a former physics professor at Portland State University who has investigated quantum dots and who currently works as a materials scientist for SolarWorld, a leading solar manufacturer. He validated the theoretical accuracy and viability of this model on absorption coefficients, and the solar irradiance spectrum data is wellestablished and trustworthy.
My model's results were 100% consistent with the previously published experimental studies cited in the "Research" section.
Python Code Written
Quantum dot geometries, clockwise from top left: cuboid, cylinder, dome, spheroid, pyramidal.
For any QD geometry, solutions to Equation 1 can be obtained. In lieu of showing 3,000 images, results for PbSe pyramidal QD with 4.5 nm dimensions are shown here as an example, with wavefunction visualizations for lowest three out of 100 energy levels (first three graphs) and the first 100 energy states with zoomin on first four (last graph). Visualizations were assisted by Quantum Dot Lab on NanoHUB.
Absorption coefficient plots (with normal solar incidence) were obtained using Equation 2 for each QD configuration through computing about 5,000 possible photon/electron interaction scenarios between lowest 100 energy states, integrated over reciprocal space. This computation was conducted for 30 different QD configurations using NanoHUB Quantum Dot Lab. Below are two examples of absorption coefficient spectra for spheroid QD at 4.2 nm dimensions (first graph) and pyramidal QD at 4.5 nm dimensions (second graph).
Using interpolation, I wrote a Python program to use Riemann sums for integrating the solar spectrum on Earth’s surface and absorption coefficient to obtain total solar absorption for each of the 30 QD configurations. Using obtained total absorption values, I developed a normalized rating system (Equation 10).
Following are normalized ratings data for all 30 QD geometries studied, with the highest absorption (spheroid QD at 3.6 nm dimensions) normalized to a rating of 1.0. The first graph plots solar absorption rating as a function of linear QD dimension, while the second plots rating as a function of QD volume.
The graphs show that neither linear dimension nor volume is the dominant factor, though both are relevant. For any geometry, there is a different optimal point, above which absorption decreases due to excessive red shift associated with larger quantum dots driving towards the infrared region. Below the optimal point, the quantum dots become small and the geometry becomes less regular, which leads to model divergence. Moreover, there is a blue shift away from infrared, which negates the benefit of using quantum dots.
Absorption ratings are highly dependent on QD shape and size, and each shape has its unique optimal size. For example, a 10% shift in linear QD dimension can cause absorption to increase or decrease by 30%, and at the same linear dimension, QD shape can cause absorption to differ by up to a factor of five. This illustrates how crucial it is to optimize QD absorption through an integrated approach such as this model. Based on the level of observed variation for QD size and shape in stateoftheart QD cell research TEM/SEM images, QD cell efficiency may be doubled with the help of my model.
As seen below, rating peaks at 3 nm for cuboid PbSe QDs, 3.2 nm for cylinders, 3.7 nm for domes, 4.7 nm for pyramids, and 3.6 nm for spheroids.
Highestperforming QD geometries are spheroid and cuboid. Spheroid PbSe QD at 3.6 nm gives highest absorption of thirty QD configurations analyzed.
Results Summary
The combination model of quantum mechanics and solar irradiance produced solid predictions. QD absorption rating heavily depends on QD shape and size. For example, 10% linear dimension shift could cause up to 30% absorption rating change. Moreover, QDs of the same linear dimension but different shape can vary in absorption by a factor of up to five. Each QD geometry has its unique optimal size. Out of all thirty PbSe QD configurations, the bestperforming QD geometries are spheroids followed by cuboids, and the top configuration is spheroid at 3.6 nm.
Hypotheses Confirmed
Both hypotheses are confirmed. The results fully demonstrate that QD absorption is strongly dependent on QD size and shape, as described above. This model was capable of quantitatively predicting QD performance, as the normalized rating system effectively compared QD configurations encompassing different shapes and sizes. This model is 100% consistent with and thus fully validated by several recent experimental studies (outlined in the "Research" section) that cover aspects of spheroidlike PbSe QDs.
Limitations
My model is focused on QD absorption of solar light, so it does not cover certain aspects of QDs such as efficiency of multiple exciton generation or electron trapping after photoelectron generation. Also, this model becomes less accurate when QD size is too small, as the theoretical shape begins to break down. More experimentation is needed to fully confirm this model's with actual QD cells, so model predictions can be fully utilized in QD production for solar technology.
Impact
This is the first time that anyone has actively combined the physics of quantum mechanical modeling and the earth science of the solar irradiance spectrum on Earth's surface to develop a novel research method for QDs in solar cell applications. Experimentalists can highly benefit from this work, in that they can now utilize my model to preoptimize QD configurations for best results prior to timeconsuming and costly experimentation. For example, based on the level of observed variation for QD size and shape in stateoftheart QD cell research cross sections and TEM / SEM images, QD cell efficiency may be doubled with the help of my model, which quantitatively predicts the effects of QD shape and size. Also, this methodology can be easily applied to any quantum dot material besides PbSe.
By accelerating the research process for QD cells, this model addresses the bottleneck facing wide QD cell adoption globally, which would be a significant boost to renewable energies  thus, ultimately a more sustainable world.
Conclusion
This study proposes, implements, and demonstrates, assisted by cloud computing, a novel research methodology and rating system for quantum dot solar cells. This model was fully validated by recently published experimental studies, and it was used in this study to investigate thirty PbSe QD configurations, revealing best results for spheroids at 3.6 nm diameter among other leading options. This modeling and algorithmbased rating system can rapidly optimize quantum dots for solar cells, enabling rapid breakthroughs in critical QD solar cell efficiency and thus potentially widespread adoption.
Acknowledgements
 Dr. Bjoern Seipel, a materials scientist at SolarWorld (leading solar manufacturer located in Oregon) and former physics professor at Portland State University who has also studied quantum dots. Dr. Seipel educated me on emerging solar technologies that were being investigated commercially, elaborated on the role of quantum physical and materials modeling in solar research, and provided many helpful tips on my project.
 Mr. Andrew Merrill, computer science and math teacher at my school. Mr. Merrill has been my computer science teacher and one of my science research advisors for the past two years, and has continually been available to help me during the investigation process. I was also able to implement my model using the programming techniques I learned from his class.
 Ms. Veronica Ledoux, chemistry and science research teacher at my school. Ms. Ledoux critiqued various drafts of my research paper, and has provided a lot of guidance, feedback, and encouragement over the past two years.
 CERN, Society for Science & the Public, and the Intel International Science & Engineering Fair for generously providing me with an award to visit CERN and EPFL in the summer of 2012. Many thanks to Ms. Diane Rashid and Dr. Wolfgang Von Rueden, our chaperones on the trip.
 Ecole Polytechnique Federale de Lausanne (EPFL), especially Dr. Christophe Ballif and his team, for intriguing discussion on solar technologies and various designs for solar cells, especially quantum dot solar cells. This event was instrumental in sparking my interest in quantum dot technologies and its applications in solar cells.
 Catlin Gabel School, for providing me with a nurturing educational environment, encouragement to pursue science research, and rigorous and highly enjoyable science and math curriculum.
 The Caroline D. Bradley Scholarship, for generously providing me with a fulltuition scholarship to Catlin Gabel; especially Ms. Bonnie Raskin of the Institute for Educational Advancement for her support over the past four years.
 The Julian C. Stanley Study of Exceptional Talent (SET) and Johns Hopkins University's Center for Talented Youth (CTY), for support and encouragement of my academic pursuits.
 The Davidson Institute, for support through the Davidson Young Scholar program.
 Intel Northwest Science Expo (NWSE), for encouraging my early science fair participation and nominating me for the Intel ISEF and Broadcom MASTERS.
 The Society for Science & the Public, the Broadcom MASTERS, and MIT Lincoln Laboratory, for fueling my passion for research early on.
 Mr. Manny Norse, Oregon ARML (American Regions Mathematics Leagues) coach, for encouragement of my scientific and mathematical endeavors.
 Professor Marek Perkowski, director of the Intelligent Robotics Lab at Portland State University. I have visited Professor Perkowski at Portland State University many times, attended various courses and seminars taught by Professor Perkowski, as well as private sessions with him, on quantum computing, and had stimulating discussions with him about the role of quantum mechanics in various devices.
 Professor Erik Sanchez, principal investigator at the Sanchez NanoDevelopment Lab at Portland State University. I visited the Sanchez Lab and had very good discussions with Professor Sanchez regarding the influence on nanostructures on band structures and optical properties, among other things.
 Quantum Dot Lab on NanoHUB, a free online cloudcomputing modeling resource maintained by researchers at Purdue University, which I was able to access after creating an account.
 My parents, for their indispensable encouragement and support.
 The Google Science Fair, for motivating me to further my research.
 Google Science Fair judges, for their time and consideration.
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