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- 1. Nano-photonic Light Trapping In Thin Film Solar Cells Thesis by Dennis M. Callahan Jr. In Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2015 (Defended October 1, 2014)
- 2. ii c 2015 Dennis M. Callahan Jr. All Rights Reserved
- 3. iii for Zhaoxia and little Dylan
- 4. iv Acknowledgements I would ﬁrst like to thank my thesis advisor, Harry Atwater, for taking a chance on a kid who took a chance on a school he never thought he could get into. Over the years, his mentorship and constant insight have been incredibly helpful and inspiring. I would like to thank him for creating an environment where an incoming student with limited scientiﬁc knowledge can quickly and eﬃciently learn and absorb so much about many diﬀerent areas of research. I would also like to thank Professors Oskar Painter, Julia Greer, Andrei Faraon and Bill Johnson for serving on my candidacy and thesis committees. I truly appreciate their time and it has been an honor to cross paths with them. I would like to acknowledge and thank all of the students and post-docs that I have worked with, learned from, and become friends with over the years. I would like to thank Dr. Jeremy Munday for helping me get started on the thin-ﬁlm GaAs project and for all the time we spent in front of the blackboard thinking about the light trapping limit. Jeremy was also a good friend and I am fortunate to have been assigned to work with him when I started full-time research in the group. I’d like to thank Dr. Jonathan Grandidier for working closely with me for a couple years on the nano sphere solar cell project and for letting me drive the Mustang to MRS in San Francisco. I’d like to thank Dr. Deirdre O’Carroll, the best “Good Cop, Bad Cop” partner one could ask for at MRS poster sessions. I would also like to thank Deirdre for humoring my ideas, trying to (unsuccessfully) make the world’s ﬁrst Ultrafast Deep Subwavelength Ultra-High Q-Switched Amplifying Nanorod Quantum
- 5. v Generating Hetero-Metamaterial Lasing Spasing Plasmonic FORABISER, and for the coﬀee. I’d like to thank Ragip Pala for working on the thin-ﬁlm GaAs project with me, helping with last-minute EQE measurements, and for always keeping a lighthearted atmosphere. I’d like to thank Dr. Marina Leite for sharing her expertise about semiconductor research with me. I’d like to thank Dr. Koray Aydin for humoring my ideas about metamaterials and for helping me get started with FDTD simulations. I’d like to thank Dr. Eyal Feigenbaum for teaching me about aspects of optics I had never thought about before. I’d like to thank Dr. Victor Brar for talking about physics, hanging out at the Rath, playing tennis, talking about basketball, and for the single-malt Scotch. I’d like to thank Raymond Weitekamp for being a good friend, inviting me to his shows, and participating in the short-lived generative art club. Jim Fakonas for being a good friend, lunchtime philosophy discussions and for knowing how to play all my favorite Rolling Stones songs. Ryan Briggs for teaching me about photonics and Purcell Factor. Stan Burgos for cigarettes and deep discussions among wandering homeless men on the streets of San Francisco. I’d like to thank everyone who came to meetings of the short-lived spontaneous emission club. I’d like to thank the spon- taneous emission group from my ﬁrst group retreat that made it so much fun and for trying to (unsuccessfully) get the term Wanteca into the nano-photonics vernacular (that’s a hybrid waveguide-antenna-cavity). I’d like to thank Colton Bukowsky for all the time and help he has recently given with electron beam lithography and reactive ion etching. For a period, I was working in collaboration with Albert Polman’s group at AMOLF in the Netherlands and I thank them for their contributions, particularly Piero Spinelli. I’d like to thank the talented undergraduate SURF students I had the privilege of mentoring over various summers including Reggie Wilcox, Clare Chen, Prateek Mantri and Kelsey Whitesell. Kelsey also returned to Caltech for a Masters degree
- 6. vi and I thank her for putting in so much time on the photonic crystal project and for being a good friend. I’d especially like to thank the Atwater group bluegrass band, Muskrat Water. This includes Matt “Bayou” Escarra, Dan “Moonshine” Turner Evans, Amanda “Thunderbird” Shing, Ana “Appalachia” Brown, and Cris “Huckleberry” Flowers. I reckon jamming with these folks was some of the most fun I had at this here school. I’d also like to thank Yulia Tolstova for introducing me to The Mountain Goats, The Tiger Lillies and for suggesting I watch The Double Life Of Vernonique by Krzysztof Kieslowski. I’d like to thank Raymond Wietekamp for introducing me to CocoRosie, Bibio and Flying Lotus. I’d like to thank Hal Emmer for introducing me to The Angus And Julia Stone, Carissa Eisler for suggesting I listen to glitch music, and all of Muskrat Water for introducing me to Old Crow Medicine Show. I’d like to thank everyone who contributed to the LMI-EFRC video contest in- cluding Raymond Weitekamp, Jonathan Grandidier, Emily Kosten, Eyal Feigenbaum, Stanley Burgos and Koray Aydin. I’d like to thank Professor Michael Roukes, Mary Sikora and everyone involved in TEDxCaltech, Feynman’s Vision: The Next 50 Years. I’d like to thank everyone involved in Caltech’s Art of Science program for creating such a great program and for exhibiting my submissions. I would like to thank all of my oﬃce mates over the years including Amir Safavi- Naeni, Lisa Mauger, Jeﬀ Hill, Alex Krause, Mike Deceglie, Victor Brar, Emily Kosten, and Matt Sheldon. I’d also like to thank the group’s administrative assistants April Neidholdt, Tiﬀany Kimoto, Jennifer Blankenship, and Lyra Haas. I would also like to deeply acknowledge Bob Dylan, Tom Waits, David Bowie, Brian Eno, Neko Case and Ryan Adams for providing the bulk of the soundtrack to the last 6 years. I sincerely think I could not have done it without them. I would like to thank my family for always believing in me and for letting me move 3,000 miles away from home without protest. Lastly and most importantly I would
- 7. vii like to thank my wife, Zhaoxia, for so much more than can be described with words at the end of a paragraph in a thesis. Closing out this chapter of my life, you are the only reason I’ve done any of this. Dennis M. Callahan Jr. September 2014 Los Angeles, CA
- 8. viii Abstract Over the last several decades there have been signiﬁcant advances in the study and understanding of light behavior in nanoscale geometries. Entire ﬁelds such as those based on photonic crystals, plasmonics and metamaterials have been developed, ac- celerating the growth of knowledge related to nanoscale light manipulation. Coupled with recent interest in cheap, reliable renewable energy, a new ﬁeld has blossomed, that of nanophotonic solar cells. In this thesis, we examine important properties of thin-ﬁlm solar cells from a nanophotonics perspective. We identify key diﬀerences between nanophotonic de- vices and traditional, thick solar cells. We propose a new way of understanding and describing limits to light trapping and show that certain nanophotonic solar cell de- signs can have light trapping limits above the so called ray-optic or ergodic limit. We propose that a necessary requisite to exceed the traditional light trapping limit is that the active region of the solar cell must possess a local density of optical states (LDOS) higher than that of the corresponding, bulk material. Additionally, we show that in addition to having an increased density of states, the absorber must have an appropriate incoupling mechanism to transfer light from free space into the opti- cal modes of the device. We outline a portfolio of new solar cell designs that have potential to exceed the traditional light trapping limit and numerically validate our predictions for select cases. We emphasize the importance of thinking about light trapping in terms of maxi- mizing the optical modes of the device and eﬃciently coupling light into them from
- 9. ix free space. To further explore these two concepts, we optimize patterns of superlat- tices of air holes in thin slabs of Si and show that by adding a roughened incoupling layer the total absorbed current can be increased synergistically. We suggest that addition of a random scattering surface to a periodic patterning can increase incou- pling by lifting the constraint of selective mode occupation associated with periodic systems. Lastly, through experiment and simulation, we investigate a potential high eﬃ- ciency solar cell architecture that can be improved with the nanophotonic light trap- ping concepts described in this thesis. Optically thin GaAs solar cells are prepared by the epitaxial liftoﬀ process by removal from their growth substrate and addition of a metallic back reﬂector. A process of depositing large area nano patterns on the surface of the cells is developed using nano imprint lithography and implemented on the thin GaAs cells.
- 10. x Contents Acknowledgements iv Abstract viii 1 Introduction 5 1.1 Rise Of Nanophotonics . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Solar Cell Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Current State Of The Art Solar Cells . . . . . . . . . . . . . . . . . . 10 1.4 Why Thin Solar Cells? . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Light Trapping In Thin Film Solar Cells 13 2.1 Single Pass Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Modes Of A Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Accessing Trapped Modes . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 The Ray Optic Light Trapping Limit . . . . . . . . . . . . . . . . . . 20 3 Light Trapping Beyond The Ray Optic Limit 23 3.1 Condition for Exceeding The Ray Optic Limit . . . . . . . . . . . . . 25 3.2 Recovering The Ray Optic Limit . . . . . . . . . . . . . . . . . . . . 28 3.3 Nanophotonic Solar Cell Designs For Exceeding The Ray Optic Light Trapping Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 What Is The New Limit? . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Numerical Demonstration Of Exceeding The Ray Optic Limit . . . . 38
- 11. xi 3.6 Planar Waveguide Formulation for Exceeding The Ray Optic Limit . 41 4 Light Trapping With Dielectric Nanosphere Resonators 49 4.1 Dielectric Nanosphere Resonators for a-Si Absorption Enhancement . 50 4.2 Dielectric Nanosphere Resonators for GaAs Absorption Enhancement 58 5 Light Trapping In Ultrathin Film Si With Photonic Crystal Super- lattices 69 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Superlattices In A Square Photonic Crystal . . . . . . . . . . . . . . . 73 5.3 Superlattices In An Optimized Hexagonal Photonic Crystal . . . . . . 80 5.4 Addition Of A Randomly Textured Dielectric Incoupler . . . . . . . . 82 5.5 Thicker Absorber Layers . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6 Light Trapping In Thin Film GaAs 87 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 Epitaxial Liftoﬀ of Ultra-thin Film GaAs . . . . . . . . . . . . . . . . 90 6.3 Pattern Optimization and Electromagnetic Simulations . . . . . . . . 92 6.4 Light Trapping in Ultra-thin Film GaAs Using Nanoimprint Lithography 98 7 Summary And Outlook 106 7.1 Exceeding The Ergodic Limit In Si . . . . . . . . . . . . . . . . . . . 106 7.2 Purcell Enhanced Photovoltaics . . . . . . . . . . . . . . . . . . . . . 108 7.3 Photonic Molecules For Light Trapping And Angular Response Engi- neering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4 Light Absorption Enhancement in Thin Film GaAs . . . . . . . . . . 114 Appendices 116
- 12. xii A GaAs Epitaxial Liftoﬀ Procedure 116 B Mesa Etching And Contacting Procedure 120 Bibliography 123
- 13. 1 List of Figures 1.1 Current voltage curve for a p-n junction in the dark and under illumination. 8 1.2 Eﬀects of thinning a solar cell on Jsc, Voc, and eﬃciency. . . . . . . . . 12 2.1 Absorption depth for various photovoltaic materials . . . . . . . . . . . 14 2.2 Modes of a slab in 3 diﬀerent representations. . . . . . . . . . . . . . . 16 2.3 Coupling into a slab with a diﬀraction grating. . . . . . . . . . . . . . 19 2.4 Shcmatic showing modes of a slab accessed with planar and Lambertian surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 Traditional limits to photogenerated current. . . . . . . . . . . . . . . 24 3.2 Potential solar cell architectures. . . . . . . . . . . . . . . . . . . . . . 31 3.3 GaP and CdSe slot waveguide LDOS . . . . . . . . . . . . . . . . . . . 33 3.4 Photonic Crystal LDOS . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Ag nanoantenna LDOS . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 LDOS Enhancements needed to absorb the entire solar spectrum. . . . 37 3.7 Schematic of spectral reweighting needed to achieve density of states enhancements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.8 FDTD simulation of P3HT:PCBM absorber. . . . . . . . . . . . . . . . 39 3.9 Angular behavior of the P3HT:PCBM structure that exceeds the ergodic limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.10 FDTD simulation of P3HT:PCBM absorber with Ag resonators. . . . . 41 3.11 Waveguide dispersion relations. . . . . . . . . . . . . . . . . . . . . . . 44
- 14. 2 3.12 Plasmonic waveguides that beat the 4n2 limit. . . . . . . . . . . . . . . 46 3.13 Slot waveguides that beat the 4n2 limit. . . . . . . . . . . . . . . . . . 47 3.14 Absorption for a plasmonic waveguide cladded with a GaP top layer. . 48 4.1 Dielectric spheres on a-Si. . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Electric ﬁeld proﬁles for dielectric spheres on a-Si. . . . . . . . . . . . . 52 4.3 Angular behavior for dielectric spheres on a-Si. . . . . . . . . . . . . . 55 4.4 Sphere diameter vs. wavelength vs. absorption on a-Si. . . . . . . . . . 57 4.5 Current density of a ﬂat GaAs solar cell as a function of thickness. . . 59 4.6 Schematic of a GaAs solar cell with hexagonally close packed SiO2 nanospheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.7 Behavior of GaAs solar cell with SiO2 nanospheres. . . . . . . . . . . . 63 4.8 Behavior of GaAs solar cell with SiO2 nanospheres. . . . . . . . . . . . 64 4.9 Behavior of GaAs solar cell with SiO2 nanospheres. . . . . . . . . . . . 68 5.1 Conceptual diagram illustrating diﬀerent degrees of randomness and order. 69 5.2 Absorption spectra of a planar Si layer and photonic crystal. . . . . . . 71 5.3 Four diﬀerent superlattice geometries and their spectral absorption. . . 72 5.4 Absorbed current as a function of superlattice air hole diameter. . . . . 74 5.5 Field proﬁles for photonic crystal superlattices. . . . . . . . . . . . . . 75 5.6 Field proﬁles for photonic crystal superlattices. . . . . . . . . . . . . . 76 5.7 TE-TM averaged angular absorption spectra for photonic crystal super- lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.8 Integrated angular absorption spectra for photonic crystal superlattices. 78 5.9 Density of states maps of photonic crystal superlattices. . . . . . . . . 79 5.10 Absorption of Suzuki lattice photonic crystals. . . . . . . . . . . . . . . 81 5.11 Integrated absorbed current of Suzuki lattice photonic crystals. . . . . 82
- 15. 3 5.12 TE-TM averaged angular absorption spectra Suzuki lattice photonic crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.13 Schematic of superlattice photonic crystal with randomly textured in- coupler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.14 Thicker superlattice photonic crystals with randomly textured incouplers. 85 6.1 GaAs solar cell device architecture. . . . . . . . . . . . . . . . . . . . . 88 6.2 Images and J-V curve of GaAs device after processing. . . . . . . . . . 89 6.3 Sample to sample and cell to cell variation in thin-ﬁlm GaAs J-V curves made via epitaxial liftoﬀ. . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.4 Bilayer anti-reﬂection coating optimization for GaAs solar cell. . . . . . 91 6.5 Hexagonal TiO2 cylinder FDTD optimization. . . . . . . . . . . . . . . 92 6.6 RCWA simulations of angular response of planar cell and cell with opt- mized square lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.7 AM1.5G spectrally integrated absorbed current for planar anti-reﬂection coated cell and TiO2 nanopatterned cell. . . . . . . . . . . . . . . . . . 94 6.8 Electric ﬁeld intensities for several resonant modes observed in the an- gular absorption spectrum. . . . . . . . . . . . . . . . . . . . . . . . . 95 6.9 Quantiﬁcation of losses in each layer of the GaAs solar cell structure. . 96 6.10 Angular spectra of the GaAs solar cell with the rear contact removed with and without addition of an optmized nanopattern. . . . . . . . . 97 6.11 Integrated absorbed current for 4 diﬀerent GaAs solar cell conﬁgura- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.12 1.5 cm x 1.5 cm master and resulting PDMS stamp. . . . . . . . . . . 99 6.13 Optical image of a hexagonal pattern of TiO2 nanoparticles on Si. . . 100 6.14 SEM images of a hexagonal pattern of TiO2 nanoparticles on Si. . . . 101 6.15 GaAs solar cells after TiO2 nanopatterning and SiO2 deposition. . . . 102
- 16. 4 6.16 Current density-voltage curves for cells with and without a TiO2 nanopar- ticle coating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.17 External quantum eﬃciency measurements of diﬀerent positions of single cells with and without nanopatterning. . . . . . . . . . . . . . . . . . 103 6.18 Angle-dependent external quantum eﬃciency measurements of diﬀerent positions of single cells with and without nanopatterning. . . . . . . . 104 6.19 Short-circuit current as a funtion of angle as determined from EQE measurements for nanopatterned and AR coated areas of a single cell. . 105 7.1 Thicker superlattice photonic crystals with randomly textured incouplers.107 7.2 Maximum attainable open circuit voltage vs. Purcell Factor. . . . . . . 110 7.3 Potential photonic molecule conﬁgurations. . . . . . . . . . . . . . . . 112 7.4 Assembly of potential photonic molecules by colloidal self assembly. . . 113
- 17. 5 Chapter 1 Introduction World energy consumption is currently above 15 TW, and is expected to increase by 56% by 2040 [1]. The majority of this energy is extracted from a combination of coal, oil and natural gas. These non-renewable resources, while currently dominant, will eventually be depleted. Even before then, continued use of fossil fuels will only worsen the environmental impact of CO2 emissions associated with burning of these energy resources. The earth intercepts about 3.4 million EJ of radiation from the sun each year, about 7,500 times the worlds current energy consumption. This annual solar energy is also an order of magnitude greater than the energy contained in all the worlds estimated fossil fuel reserves and nuclear energy resources combined [2]. However, as of 2010, only about 0.1% of the worlds consumed energy is generated from photovoltaics [3]. No matter how one looks at it, it is obvious that solar energy utilization will have to grow to meet the worlds increasing energy needs. Given these numbers, it also seems inevitable that renewable energy resources like solar will eventually displace fossil fuels as the worlds dominant energy source. In fact, a recent study [2] predicts solar energy (photovoltaic and solar thermal) to provide 70% of world energy generation by 2100. A diﬀerent study predicts 11% of the worlds total energy needs to come from photovoltaics alone by 2050 [3]. These numbers are based on the sustained growth of the photovoltaics industry over the last decade. In 2012, the world surpassed 100 GW of globally installed PV, and has grown by about 30
- 18. 6 GW in each of the last 2 years [4]. To fuel a growing industry such as PV, research and development is needed to reduce the cost per watt of electricity generation. For photovoltaics, this translates to a need for research that reduces the usage of raw materials, reduces the cost of manufacturing and, equally as important, increases energy conversion eﬃciency. This thesis will explore concepts aimed at achieving these goals using new approaches to light harvesting and device design. 1.1 Rise Of Nanophotonics In parallel with the development of photovoltaic technology over recent years has been the development of the ﬁeld of nanophotonics. The unprecedented ability to manipulate and study light on a nano scale has been fueled by recent advances in technology such as electron beam lithography, near ﬁeld optics, as well as more ad- vanced computer processing power and computation techniques such as ﬁnite diﬀer- ence time domain (FDTD) and rigorous coupled wave analysis (RCWA). Entire ﬁelds such as those based on photonic crystals [5], plasmonics [6] and metamaterials [7] have been developed, accelerating the growth of knowledge related to nanoscale light manipulation. This has led to impressive strides in the development of ﬁelds such as quantum optics [8], nanoscale lasers [9], near ﬁeld imaging [10] and others. Despite the continual growth of the closely related ﬁelds of photovoltaics and nanophotonics, only recently have they started to merge, giving birth to a rich and interesting new research frontier, that of nanophotonic solar cells [11,12]. 1.2 Solar Cell Fundamentals The sun can be viewed as a blackbody at a temperature of about 5800 K that con- tinuously emits energy in the form of photons. The energy of these photons can be
- 19. 7 harvested in numerous ways, such as through heating of an absorbing material or ﬂuid (i.e. solar thermal), or by taking advantage of the photovoltaic eﬀect in a semicon- ductor and extracting excited charge carriers at a voltage to create electrical power. While this thesis focuses mainly on photovoltaic devices, the concepts regarding light trapping and absorption are applicable to both areas of solar energy harvesting. In a photovoltaic device, photons with energy above the material’s bandgap are absorbed, exciting electron-hole pairs that can be extracted as electrical current at an applied voltage to give electrical power. The amount of extracted power divided by the amount of incident power from the sun gives the energy conversion eﬃciency. In 1961, Shockley and Queisser calculated the theoretical maximum eﬃciency of a semiconductor photovoltaic device using the principle of detailed balance [13]. For this calculation, they ﬁrst considered a photovoltaic device in equilibrium with its surroundings at an ambient temperature, T. This ambient temperature will have an associated blackbody spectrum of ambient photons, some of which the device will absorb. Exactly how many it absorbs depends on the absorptivity, A(ω), of the device, a function of the complex dielectric function of the absorbing material and the geometry. Since the solar cell is in equilibrium, the photons that are absorbed have to be balanced by an outgoing ﬂux of emitted photons. The recombination causing this photon ﬂux constitutes the minimum dark current, Io, of the solar cell. This current ﬂowing across the junction will increase exponentially when an external voltage is applied, giving the diode equation: I(V ) ≈ Ioe qV kT (1.1) When the device is placed in a stronger illumination source, such as the sun, a much greater amount of electron hole pairs is excited and the device is no longer in equilibrium. These extra electron hole pairs ﬂow in the opposite direction across the junction and can be collected and extracted as electrical current. Deﬁning the total
- 20. 8 amount of photoexcited current IL, this gives a new I-V equation for an illuminated device: I(V ) ≈ Ioe qV kT − IL (1.2) The resulting dark current-voltage (I-V) curves for a diode in the dark and under illumination are shown in Figure 1. Figure'1.1:"(a)"Current"voltage"curve"for"a"p1n"junction"in"the"dark"and"under" illumination."The"important"solar"cell"parameters"Jsc,"Voc,"FF"and"ef?iciency"can" be"easily"extracted." Jsc Voc Pmax = Jmax * Vmax light dark Figure 1.1: Current voltage curve for a p-n junction in the dark and under illumina- tion. The important solar cell parameters Jsc, Voc, FF and eﬃciency can be easily extracted. Typically, the current is normalized by the area of the device receiving illumina- tion, and the curent density, J ( mA cm2 ) is used rather than I(A) when describing solar cell devices. This light J-V curve can be used to deﬁne a number of important parameters
- 21. 9 widely used to characterize a solar cell’s performance. 1. Jsc or short-circuit current: A measure of how well the solar cell absorbs light. This will depend on the thickness, geometry of the absorbing material as well as the dielectric function and the light-matter interaction. This is obtained by measuring the collected photocurrent a short-circuit when no voltage is applied to the device. 2. Voc or open-circuit voltage: A measure of how well the device separates charge. This will ﬁrst depend on the built-in voltage, a function of the bandgap of the material and the doping levels within the device. Voc will also depend on how much recombination occurs in the device, including radiative and non-radiative channels, which aﬀects the dark current, Jo. The Voc is deﬁned by solving the light J-V curve for V at J = 0 (i.e. the open-circuit condition): Voc = kT q ln( Jsc Jo ) (1.3) 3. FF: Corresponds to the squareness of the J-V curve and is deﬁned by FF = JmaxVmax JscVoc . This will depend mostly on series and shunt resistances in the device. These easily obtained parameters can be used to calculate the power conversion eﬃciency of the solar cell. η = Pmax Pinc = JscVocFF Pinc (1.4) where Pinc is the incident power density on the device.
- 22. 10 1.3 Current State Of The Art Solar Cells Using the method of Shockley and Quiesser, the maximum theoretical eﬃciency of a single junction solar cell of any given material can be calculated. All that is needed is an incident spectrum and the bandgap energy of the semiconductor. Two of the most important solar cell materials are Si and GaAs. The maximum eﬃciency of a single junction Si solar cell under the AM1.5G solar spectrum is about 30%. Currently, the highest achieved eﬃciency for a Si solar cell is 25% [14]. This eﬃciency for Si is currently limited by non-radiative recombination processes intrinsic to Si, such as Auger recombination. The thickness of typical record Si devices is usually above 50 microns. A Si record cell could beneﬁt from advanced light trapping by signiﬁcantly reducing the amount of active material needed, potentially down to below 5 microns, or more than an order of magnitude. The theoretical maximum eﬃciency for a GaAs solar cell is about 33%. The current maximum achieved eﬃciency is 28.8% [14], by Alta Devices [15] using the epitaxial liftoﬀ process of delaminating a thin ﬁlm of GaAs from its parent substrate and adding a back reﬂector. All three of the important solar cell parameters are just slightly below their theoretical maximum values. A GaAs cell could beneﬁt from advanced light trapping by reducing the amount of material needed to absorb the spectrum, potentially increasing the voltage by reducing the volume-dependent dark current. 1.4 Why Thin Solar Cells? Despite the best Si solar cells being many tens of microns thick, the ultimate solar cell is one that is as thin as possible. There are many advantages for thinning the active layer, with the obvious disadvantage that it decreases light absorption. If the exponential reduction in absorption while thinning the cell can be prevented, a solar
- 23. 11 cell could gain the following beneﬁts: • Reduced Cost The most obvious advantage of a thin solar cell is that is reduces the amount of raw material needed to make the device. As of 2012, about 60% of a Si module cost is due to the Si alone [16]. Reducing the thickness of a Si device by an order or magnitude or more could signiﬁcantly decrease this cost. If the solar cell is made from an epitaxial material like a III-V, reducing the active layer thickness will decrease both the amount of precursor gas needed for growth as well as reactor run time for each device. Though direct bandgap materials like GaAs are already relatively thin (∼2-5 microns) compared to Si, their thickness could still in theory be reduced to below a micron, as will be shown in the next section. • Flexibility While a crystalline wafer of a semiconductor such as Si or GaAs is extremely brittle and can crack or shatter if handled improperly, a thin ﬁlm of these materials can be extremely ﬂexible. This opens up many new applications for solar power that otherwise would not be feasible with wafer-based devices. For example, a ﬂexible module could be rolled out and used when needed as portable charging station, and rolled back up and stored conveniently when not needed. A ﬂexible device could be incorporated into clothing or baggage, wrapped around curved surfaces, or even integrated into the outside of an au- tomobile or aircraft. • Improved Voc and Eﬃciency Perhaps the most intriguing motivation for making a solar cell as thin as possible is an increased operating voltage and possibility of an overall increased eﬃciency
- 24. 12 compared to a thicker device. This is because the dark current density, Jo, is a function of the total recombination within the device. The more bulk material there is in the device, the more bulk recombination and the higher the dark current. This lowers the Voc as shown in Equation 1.3. Plotted in Figure 1.2a is a calculation of the Voc as a function of thickness for a GaAs solar cell with various amounts of bulk non-radiative recombination current. Also shown is how the Jsc decreases with thickness when there is only a single photonic pass. Clearly, in order to take advantage of the increased Voc for thinner cells, signiﬁcant light trapping is needed to maintain the Jsc of thicker cells. Shown in Figure 1.2b is the calculated eﬃciency of a GaAs cell with various amounts of light trapping. It can be seen that with increased light trapping, the maximum possible eﬃciency increases and occurs at thinner and thinner cells. Since both Voc and Jsc need to be maximized to increase eﬃciency, the ideal solar cell is one that is as thin as possible yet still absorbs the available light from the sun. 0" 0.05" 0.1" 0.15" 0.2" 0.25" 0.3" 0.35" 10" 100" 1000" 10000" Cell$Thickness$(nm)$ Jsc"Max"(Absorp4vity"="1)" Ergodic"Limit" 5"Passes" 2"Passes" 1"Pass" Efﬁciency Voc(V) a) b) Cell Thickness (μm) Cell Thickness (μm) Jsc(mA/cm2) Figure 1.2: a) Eﬀect of thinning a solar cell on Jsc and Voc with no light trapping. NR is the amount of non-radiative recombination. b) Eﬃciency as a function of cell thickness with various amounts of light trapping. The more light trapping, the higher the maximum possible eﬃciency.
- 25. 13 Chapter 2 Light Trapping In Thin Film Solar Cells 2.1 Single Pass Absorption Photons with energy above a semiconductor’s bandgap can be absorbed if given enough time to interact with the material. How quickly light is absorbed depends on the complex dielectric function of the material (i.e. refractive index and absorption co- eﬃcient), and the nature of the light-matter interaction. For light propagating freely as a plane wave in a homogeneous material, the amount absorbed, or intensity I, is exponentially dependent on how far it travels, given by the well known Beer-Lambert law: I(z) = Ioeαz (2.1) A commonly used metric for electromagnetic absorption is the absorption depth, 1 α deﬁned as or the depth at which the intensity of a wave decays to 1 e (about 35%) of its initial value. Plotted in Figure 2.1 is the absorption depth for some common semiconductors. A few things are clear from this plot. First, some materials absorb light much more strongly than others. For example, a direct bandgap material like GaAs has a
- 26. 14 Figure 2.1: Absorption depth for common photovoltaic materials used in this thesis. 1 10 100 1000 10000 100000 1000000 200 300 400 500 600 700 800 900 1000 1100 AbsorptionDepth(nm) Wavelength (nm) GaAs Si a-Si Figure 2.1: Absorption depth for common photovoltaic materials used in this thesis. much shorter absorption depth across the entire spectrum compared to an indirect semiconductor like Si. This will dictate typical thicknesses for each type of solar cell, with GaAs solar cell thicknesses being a few microns and Si thicknesses being many tens or even hundreds of microns. Second, for nearly all materials, the absorption depth is a strong increasing function of wavelength. This has strong implications for light trapping, as the wavelengths that need the most consideration will be the longer ones, and the bandwidth of interest will depend on the thickness of the material.
- 27. 15 2.2 Modes Of A Slab In order to design structures for light trapping, an understanding of the optical modes of a semiconductor slab is critical. Consider a dielectric slab of material with a refractive index, n2, surrounded by an upper dielectric material with index, n1, and a lower dielectric with index n3. Following Yariv [17], we ﬁnd that diﬀerent types of electromagnetic modes exist in the slab for a single frequency. Modes that have oscillating ﬁeld components both inside and outside the slab are known as radiation modes. Modes which are oscillating inside the slab but decay exponentially outside the slab are known as bound or trapped modes. Each of these will be described in more detail below. The nature of the modes of a slab depends on the relative surrounding refractive indices and a very important quantity known as the propagation constant, β, or in- plane wavevector, k||. It is useful to plot the relationship of wavevector and frequency in a dispersion diagram, as is shown in Fig. 2.2a,d. From this single diagram, we can describe and understand the modes that exist in most photonic structures. We start with the deﬁnition of wavevector for a plane wave propagating in a homogeneous medium: ω = ck n (2.2) where c is the speed of light and n is the refractive index of the material the photon is propagating in. If we deﬁne the in-plane wavevector k|| as the component of the wavevector in the plane of the slab, we can start to describe the photon with the dispersion diagram. For a photon propagating inside the incident material with index, n1, it will have a k|| of 0 when it is normal to the plane of the slab. As the incident angle in increased oﬀ-normal, it will gain a parallel component dependent on the angle and its frequency:
- 28. 16 0 10 20 30 1.2 1.4 1.6 1.8 2 300nm n=3.5 in Air k|| (um −1 ) PhotonEnergy(eV) TEo TM o TE 1 TM1 Air Lig Slab L TE2 Figure' 2.2:" (top)" Radiation" modes" of" a" slab" represented" (a)" on" a" dispersion" diagram" (b)" as" electric" 6ield" intensity" as" a" function" of" position" and" ©" schemcatically" in" terms" of" propagation" angles." (bottom)" radiation" modes" represented"in"the"same"3"ways." 0 10 20 30 1.2 1.4 1.6 1.8 2 300nm n=3.5 in Air k|| (um −1 ) PhotonEnergy(eV) TEo TMo TE 1 TM1 Air Lig Slab L TE2 PhotonEnergy(eV)PhotonEnergy(eV) k|| (um-1) k|| (um-1) airlightline airlightline radiation modes guided modes Normalized |E|2 a) b) c) d) e) f) Normalized |E|2 Figure 2.2: (top) Radiation modes of a 300 nm slab of index n = 3.5 in air represented (a) on a dispersion diagram (b) with ray optics and (c) as electric ﬁeld intensity as a function of position. (bottom) Guided modes represented in the same 3 ways. k|| = k sin(θ) (2.3) If the photon is propagating completely in the plane, all of its wavevector will be in the parallel component and it can be described by: k|| = nω c (2.4) This forms what is commonly known as the light line in the dispersion diagram as shown in Fig 2.2a. Here we can see that as the refractive index of the material of propagation in- creases, the wavevector also increases. This means that the dispersion of photons
- 29. 17 propagating in higher indices will appear towards the right on the dispersion dia- gram. A photon incident on a planar slab at any of these angles will refract at an angle according to Snell’s law, propagate through the slab at that angle, and exit the slab again at an angle according to Snell’s law (at each interface, of course, portions will not only be refracted but also reﬂected as dictated by the Fresnel equations). This is commonly known as a single pass through the material and would result in minimal absorption if the material were a solar absorber. Adding a back reﬂector will only add a second pass, with the photon exiting from the top instead of the bottom if it is not absorbed. The set of modes and corresponding in-plane wavevectors associated with this are known as radiation modes and are shown in three diﬀerent descriptions in Fig 2.2a-c. It is important to note that for a planar interface, these radiation modes are the only modes that can be accesed by photons incident from free space, which only results in 1-2 passes depending on if there is a back-reﬂector or not. Photons that gain extra in-plane momentum (k||) can access another set of modes that exist in the slab, called bound modes. These modes can be understood from a ray optics perspective as light trapped by total internal reﬂection. In the wave optics picture, these modes have oscillating ﬁelds within the slab and exponentionally decaying ﬁelds outside the slab. These modes will have in-plane wavevectors between the light line of the incident medium and the light line of the bulk medium of the slab, given by ω = ck n2 . They form discreet bands in the dispersion diagram and may have multiple solutions at a single frequency corresponding to diﬀerent mode orders. These modes are calculated and shown in Fig. 2.2d-f in 3 diﬀerent representations for a 300 nm slab with refractive index 3.5 surrounded by vacuum. A photon or wave in a trapped mode will stay trapped forever unless 1) something changes its momentum (k||) by scattering or diﬀracting it out of the waveguide or into a diﬀerent mode or 2) it is absorbed (if the material is absorbing at that wavelength).
- 30. 18 Similarly, as stated before, a photon starting out in the lower index incident medium cannot enter one of these trapped modes unless something changes its momentum. Clearly, a photon trapped in a bound mode (having many eﬀective passes) has a much greater chance of getting absorbed than a photon in a radiation mode (only 1- 2 passes). Getting photons from free space with low in-plane momentum into trapped modes where they can be most eﬃciently absorbed is the central goal of solar cell light trapping. 2.3 Accessing Trapped Modes In order for a photon incident from free space to access the guided modes of a slab, its in-plane momentum must be changed so that it matches that of the trapped modes. A photon can elastically scatter oﬀ of a particle that has dielectric contrast with its surroundings if the particle’s size is comparable to the wavelength. This will alter the momentum of the photon and may allow it to couple to nearby optical modes with various momenta. In the wave picture, this occurs because the photon is localized during the scattering event, introducing new, larger k-vectors (modes in k- space are inversely proportional to modes in real space) in the vicinity of the scattering particle [18]. If the particle is close enough to an optical mode (e.g. a guided mode of a slab), and the ﬁelds overlap, the energy of the incident wave can be transferred to the optical mode. In the quantum picture, scattering events are described by Fermi’s golden rule, which states that a scattering event will send the scattered particle (i.e. a photon) into a new state depending on the nearby density of states. As we will see in the next section, a dielectric slab with a higher index than the incident medium has a higher density of optical states than its surroundings, and will preferentially accept photons scattered at the interface between the two media. A single isolated scatterer or a set of random, uncoupled scatterers will ideally redistribute the incident light in
- 31. 19 many diﬀerent directions, or angles from its incident angle. Figure' 2.2:" (top)" Radiation" modes" of" a" slab" represented" (a)" on" a" dispersion" diagram" (b)" as" electric" 6ield" intensity" as" a" function" of" position" and" ©" schemcatically" in" terms" of" propagation" angles." (bottom)" radiation" modes" represented"in"the"same"3"ways." 0 10 20 30 1.2 1.4 1.6 1.8 2 300nm n=3.5 in Air k|| (um −1 ) PhotonEnergy(eV) TEo TMo TE 1 TM1 Air Lig Slab L TE2 PhotonEnergy(eV) k|| (um-1) b) Δk = 2π Λ Λ a) 0 10 20 30 1.2 1.4 1.6 1.8 2 300nm n=3.5 in Air k|| (um −1 ) PhotonEnergy(eV) TEo TM o TE 1 TM 1 Air Lig Slab L TE 2 PhotonEnergy(eV) k|| (um-1) d) Δk = 2π Λ Λ c) k|| = 0 kz k k|| = k sin(θ) kz k θ Figure 2.3: Coupling into a slab with a diﬀraction grating. The interscting lines in (b) and (d) occur at wavelengths where light can be coupled into guided modes at a given angle. (top) Normal incidence. (bottom) Oﬀ-normal incidence. Another way to access the guided modes of a slab is through diﬀraction. Diﬀrac- tion can be viewed as a series of scattering events with the scatterers arranged in an ordered, periodic pattern. In the wave picture, certain angles of propagation are allowed due to constructive interference between the scattered waves, and other an- gles disallowed due to destructive interference. An ordered array of scatterers with period Λ, will add and subtract in-plane momentum from an incident plane wave with values of 2πn Λ , where n is an integer. These new values of in-plane momentum
- 32. 20 can also correspond to new angles of propagation given by the diﬀraction equation: mλ = Λsin(θm) (2.5) where λ is the wavelength of light and m is an integer. This is schematically shown in Fig. 2.3a using the dispersion diagram of Fig. 2.2. Light can be coupled into the slab at the locations where the diﬀraction lines intersect the bands corresponding to the trapped modes of the slab. If the incident angle is changed, the starting value of k|| will change, but the values of added momentum will not. This will only change the resonant wavelengths which couple into the slab as shown in Figure 2.3b. It is important to note that at a given angle, there is only a subset of available modes that can be accessed with a diﬀraction grating (i.e. the intersecting lines in Fig. 2.3), depending on the period of the grating and the incident angle. Alternately, scattering from single point sources or random surfaces as previously described can potentially couple light into all of the available modes of the slab. If the scattering surface is ideal, it will also do this at all angles. For this reason, random scattering surfaces are typically more useful for broadband, varied angle incoupling for applications such as solar cells. 2.4 The Ray Optic Light Trapping Limit In 1982, Yablonovitch [19] published a theory describing a fundamental limit to how much light trapping could be achieved in a thin slab of semiconductor with ideal scattering surfaces. This limit is known as the ray optic light trapping limit, also often called the ergodic limit. Yablonovitch arrives at this limit by considering a homogeneous slab of material with a refractive index, n, and immersing it in a vacuum (with refractive index 1 by deﬁnition) that contains blackbody radiation at a temperature, T. When the
- 33. 21 radiation inside the slab reaches equilibrium with its surroundings, we can compare energy densities inside and outside the slab: Uin(r, ω) = n3 ω2 π2c3 · 1 e ¯hω kT − 1 · ¯hω (2.6) Uout(r, ω) = ω2 π2c3 · 1 e ¯hω kT − 1 · ¯hω (2.7) They clearly only diﬀer by a factor of n3 . Absorption is directly related to optical intensity, so by converting the energy density to an intensity by multiplying by the group velocity vg = dω dk and using the well known relation ω = ck n , the intensities become: Iin(r, ω) = n2 π2c2 · ¯hω3 e ¯hω kT − 1 (2.8) Iout(r, ω) = 1 π2c2 · ¯hω3 e ¯hω kT − 1 (2.9) Which diﬀer by only n2 . If a simple back reﬂector is added to add a second pass for any light that escapes the rear of the slab, this intensity enhancement increases to 2n2 . After averaging over all angles using a ray optics approach, Yablonovitch comes to an overall absorption, or path length, enchancement of 4n2 . Figure 2.4: (a) modes of a planar slab accessed (grey regions) at all incident angles, (b) modes of a Lambertian textured slab accessed at all incident angles. Ω 2π 2π 2π (4π w/BR) a) b) Figure 2.4: (a) Modes of a planar slab accessed at all incident angles, (b) Modes of a Lambertian textured slab accessed at all incident angles.
- 34. 22 An intuitive way to understand where the 4n2 path length enhancement comes from is by considering the escape cone for light inside the slab. As mentioned pre- viously, a subset of modes known as radiation modes exist inside the slab and can be accessed from free space. If the angle of incident light were swept through all 2π steradians outside the slab as shown in Fig. 2.4, the refracted rays inside the slab would sweep out a cone known as the escape cone. The solid angle, Ω, of this cone is less than the 2π steradians outside because of Snell’s law of refraction and the fact that the index of the slab is higher than free space. Any angles outside of this cone correspond to trapped modes of the slab and cannot be accessed from free space with- out a momentum changing event. Now, consider a slab with perfectly Lambertian textured surfaces. These surfaces will scatter the light ideally into all 4π steradians that exist inside the slab. This ratio of internal angles, or modes accessed, gives the 4n2 light trapping limit. 4π Ω ≈ 4n2 (2.10) by series expansion and trigonometric identity where Ω = 2π(1 − cos θ) and θ = sin−1 (1 n ). In this sense, reaching the light trapping limit can be understood as when incident light is able to access all of the modes inside the slab (both radiative and bound).
- 35. 23 Chapter 3 Light Trapping Beyond The Ray Optic Limit To utilize many of the beneﬁts of ultrathin solar cells, the active layers for most ma- terials have to be reduced to thicknesses where traditional light trapping is no longer eﬀective. At some point, even light trapping that reaches the ray optic, ergodic limit is not enough to generate reasonable short circuit currents, particularly in the wave- length range near the semiconductor bandgap. For any given thickness, there is a range of wavelengths that is near fully absorbed upon 4n2 optical passes and another range that is not fully absorbed due to the exponential nature of the absorption pro- cess and the fact that the absorption coeﬃcient approaches zero at the semiconductor band edge (Fig. 2.1). Therefore we can divide our conceptual approach to the light trapping limit between the spectral region where enhanced absorption is needed and where it is not. The transition region between these two regimes depends on the material properties, thickness, structure, etc. There is previous evidence that solar cells with non-slablike geometries that op- erate in the wave optic rather than ray optic limit may exceed this limit [20–24], and recent theoretical work has successfully explained the phenomena in selected cases [25,26]. The theory of Yu et al. establishes a relationship between the modes supported in a slablike solar cell and the maximum absorption attainable but is not easily extensible to the broad portfolio of light trapping schemes currently being ex-
- 36. 24 Figure 3.1: Traditional limits to photogenerated current. (a) Current density resulting from AM1.5 solar spectrum absorption for various thicknesses of a bulk slab of Si (inset) assuming either exponential absorption from a single pass or 4n2 passes at the ergodic light trapping limit. (b) Spectrally resolved absorbed current for a 100 nm slab of Si with light trapping at the ergodic limit. There is a signiﬁcant region near the band edge where increased light trapping is needed beyond the ergodic limit. (c) A thin slab of semiconductor where the LDOS is affected by a lower index surrounding environment and (d) a higher index surrounding environment.. Figure 3.1: Traditional limits to photogenerated current. (a) Current density resulting from AM1.5 solar spectrum absorption for various thicknesses of a bulk slab of Si (inset) assuming either exponential absorption from a single pass or 4n2 passes at the ergodic light trapping limit. (b) Spectrally resolved absorbed current for a 100 nm slab of Si with light trapping at the ergodic limit. There is a signiﬁcant region near the band edge where increased light trapping is needed beyond the ergodic limit. (c) A thin slab of semiconductor where the LDOS is aﬀected by a lower index surrounding environment and (d) a higher index surrounding environment. plored in the literature [11,21,27]. We ﬁnd that a common deﬁning feature of light trapping structures that can ex- ceed the ray optic limit is that the electromagnetic local density of optical states (LDOS) integrated over the active absorber region must exceed that of the corre- sponding homogeneous, bulk semiconductor. That the density of optical states plays an important role in deﬁning the light trapping limit has been alluded to in previous work [25, 28, 29], and we use it here as a deﬁning approach for understanding light trapping. We develop a uniﬁed framework for thin ﬁlm absorption of solar radiation and show that an enhanced integrated LDOS is the fundamental criterion that de-
- 37. 25 termines whether light trapping structures can exceed the ray optic light trapping limit, leading to a strategy for design of solar absorber layers with wavelength-scale characteristic dimensions that exhibit optimal light trapping. 3.1 Condition for Exceeding The Ray Optic Limit As in the original ray optics formulation [19], the electromagnetic energy density serves as a starting point for examining the partitioning of energy between the solar cell and the surrounding environment. For any given geometry, the electromagnetic energy density is a function of position. One way to express the electromagnetic energy density is in terms of the local density of optical states (LDOS), ρ(r, ω), the modal occupation number ν , and the energy of each mode, ¯hω: U(r, ω) = ρ(r, ω) · ν · ¯hω (3.1) The modal occupation number is summed over all modes of the structure. In a homogeneous environment with refractive index, n, and with purely thermal Bose- Einstein occupation, the electromagnetic energy density takes the familiar form: U(r, ω) = n3 ω2 π2c3 · 1 e ¯hω kT − 1 · ¯hω (3.2) By examining the ratio of energy densities inside and outside the solar cell at thermal equilibrium, we can determine the maximum light intensity enhancement and thus the maximum light trapping that is possible [19]. We do this by taking the ratio of electromagnetic energy densities in two distinct cases of interest. This gives an important ratio, that includes the LDOS ratio and the modal occupation number ratio:
- 38. 26 U(r, ω)case1 U(r, ω)case2 = ρ(r, ω)case1 ρ(r, ω)case2 · ν case1 ν case2 (3.3) This same ratio can be obtained by starting with an expression for the local absorption rate: A(r, ω) = 1 2 ω · (r, ω) · E(r, ω)2 (3.4) The absorption can be related to the energy density by rewriting the energy density in terms of the ambient dielectric function and the electric ﬁeld intensity. In limit of a lossless, dispersionless medium: U(r, ω) = 1 2 (r, ω) · E(r, ω)2 + B(r, ω)2 2µ(r, ω) = (r, ω) · E(r, ω)2 (3.5) Equating this with equation 3.1 and rearranging we have for the electric ﬁeld intensity: E(r, ω)2 = 1 (r, ω) · ρ(r, ω) · ν · ¯hω (3.6) We can insert this into our expression for local absorption rate to determine how the LDOS and modal occupation number aﬀect absorption: A(r, ω) = 1 2 (r, ω) · · ρ(r, ω) · ν · ¯hω2 (3.7) Because equations 1 and 4 assume lossless materials, we can only introduce a small amount of absorption before these concepts become ill-deﬁned. It is neverthe- less common to use expressions like this in the limit of low absorption to describe light trapping phenomena, as has been done both in the original derivation [19] and subsequent treatments [25,26]. We will ﬁnd next, however, that Eq. 3.7 is not neces- sary but rather has been used to verify the importance of energy density ratios. As
- 39. 27 with the expressions for energy density, we can now take the ratio of local absorption rate for 2 distinct cases of interest and again obtain Eq. 3.3. We can use this ratio to examine numerous cases of interest. In the limit of large thicknesses, we will recover the ray optic light trapping limit for homogeneous mate- rials in Section 3.2. First, we compare a homogeneous absorber with an arbitrarily designed absorber that may be capable of exceeding the ray optic light trapping limit: ρ(r, ω)case1 ρ(r, ω)case2 · ν case1 ν case2 = ρ(r, ω)cell ρ(r, ω)bulk · ν cell ν bulk (3.8) where we have assumed integration over all modes of the structure. When an incoupling mechanism fully populates the modes of each structure, the light trapping enhancement is simply the LDOS ratio between the cell structure and a homogeneous bulk structure which achieves the ergodic limit. This implies that if the LDOS of the cell structure is larger than that of a homogeneous medium with the same refractive index, the ergodic light trapping limit can be exceeded. Inputting the expression for LDOS of a homogeneous medium with refractive index n, we reach the condition for exceeding the ergodic limit: π2 c3 n3ω2 ρ(r, ω)cell > 1 (3.9) assuming that all modes of the structure are fully occupied via an appropriately designed light incoupler, which is often taken to be a Lambertian scattering surface. We now brieﬂy discuss the importance of fully populating the modes of the arbi- trarily designed solar cell structure. As the ratio of equation 3.8 indicates, a structure with an elevated LDOS is necessary but not suﬃcient to exceed the light trapping limit, which also depends on the modal occupation numbers. There are numerous examples of solar cell structures that very likely have an elevated LDOS; however, whether or not they have exceeded the traditional light trapping limit has remained
- 40. 28 uncertain [20,23]. Structures that use diﬀractive elements such as gratings and pho- tonic crystals often couple into only a small subset of the available modes, dictated by momentum conservation. This severely limits the number of modes that are oc- cupied, much like the above-mentioned case of a planar interface. If these structures were integrated with a scattering layer, the full modal spectrum of the device could in principle be excited, and much more light could be absorbed within the solar cell. This would also resolve the relevant problem of limited angular and polarization response associated with many designs. We also note that full modal occupation is not necessary to surpass the ergodic limit, as will be shown below using FDTD calculations. 3.2 Recovering The Ray Optic Limit Considering ﬁrst a bulk, homogeneous slab of semiconductor with a planar interface, we ﬁnd that light incident from all 2 steradians gets coupled into a small subset of the modes within the slab given by the escape cone deﬁned by Snell’s law. We call these radiation modes. We assume that all incident light is transmitted to the solar cell, thus each of these radiation modes is fully occupied with the maximum occupation number deﬁned through the modiﬁed radiance theorem [30,31]. Each incident mode contains an occupation number, νinc, and is mapped into a single radiation mode within the semiconductor deﬁned by Snell’s Law with maximum occupation number νmax = νinc [30, 31]. No more light can be coupled into these radiation modes within the slab as they are fully occupied to the maximum extent dictated by thermodynamics. There are of course 4π - Ω modes inside the semiconductor left unoccupied in this case, and we call these trapped or evanescent modes [30]. These trapped modes are inaccessible from free space without a momentum changing event such as scattering or diﬀraction. We count the number of excitations within the slab by multiplying the
- 41. 29 number of modes accessed by their occupation number. We now look at the same slab of semiconductor with a lambertian scattering surface. We assume that this surface does not alter the density of states within the slab as is often assumed [19,30]. This means that when we take the ratio of equation 3.8, the LDOS terms will cancel, and we will be left considering only the diﬀerence in modal occupation numbers. For an incident mode with occupation number νinc, the scattering event splits the energy in this mode between all equal modes within the semiconductor. Thus, each of the internal modes, both radiation and trapped, now have a fraction of this energy, let us say νinc 4π . This means that each of the internal modes can be fed more energy as their occupation numbers are not at their maximum value. We can continue to couple light into the semiconductor until each of these 4π internal modes has an occupation number νmax = νinc. We again count the number of excitations by multiplying the number of modes by their occupation numbers. We can now take the ratio in equation 3.8 for each of these two cases, ρ(r, ω)lambertian ρ(r, ω)planar · ν lambertian ν planar = ν lambertian ν planar = 4π ν inc θc ν inc = 4π θc ≈ 4n2 (3.10) where we have assumed integration over all modes of the structure. This gives the traditional, or ergodic, light trapping limit. 3.3 Nanophotonic Solar Cell Designs For Exceed- ing The Ray Optic Light Trapping Limit We now outline a portfolio of wavelength-scale solar absorber layer designs that ex- hibit an LDOS modiﬁed relative to that of a homogeneous slab of the same material. We begin by examining ultrathin, planar solar cells for which we calculate the LDOS
- 42. 30 as a function of position within the semiconductor using the Greens dyadic for a lay- ered planar system [10, 18, 32] and then integrate over position to get the spatially averaged LDOS. For the case of a ﬁnite thickness planar absorber, in Fig 3.1c-d, the LDOS modiﬁcation can be viewed heuristically in terms a slab of thickness t with a homogeneous LDOS (LDOSo) which is modiﬁed from its surface to some depth tm by the LDOS of the neighboring material. In Fig. 3.2 we plot this averaged LDOS en- hancement for various absorber structures as a function of thickness and wavelength. When t >> 2tm, the absorber LDOS modiﬁcation by the surrounding materials is negligible, but when t ≈ 2tm the change in the absorber layer LDOS due to the in- ﬂuence of surrounding materials becomes signiﬁcant. First, in Fig. 3.2a we note that the integrated LDOS for a thin semiconductor slab surrounded by free space or lower index materials is always less that that of its corresponding bulk material as the slab thickness is reduced, implying that the light trapping limit in this structure is actually lower than that of the same material in bulk form. This has been pointed out before using a diﬀerent methodology [30]. However, when the surrounding material includes a metal back reﬂector as shown in Fig. 3.2b, the integrated LDOS can exceed that of its corresponding bulk material in ﬁnite thickness layers, particularly for thicknesses below 200-300 nm. Further, the metal-insulator-metal (MIM) structure shown in Fig. 3.2c exhibits an integrated LDOS enhancement relative to its corresponding bulk ab- sorber material that is very signiﬁcant for all thicknesses below 500nm, exceeding 50 for all relevant wavelengths with ultrathin layers below 20 nm. For a two-dimensional planar structure, the modal density of states can be ob- tained from the optical dispersion relations. In this case, the density of guided modes per unit area per unit frequency for the mth mode can be written as [30]: ρm = km 2π dω dkm −1 (3.11) where km is the propagation constant. In order to increase the density of modes
- 43. 31 Figure 3.2: Energy(eV) k(nm-1) (μm) Thickness (μm) 1 0.4 400 1.1 1"0"0.4 1.2 1.2 3.0 1000 1.1 13 1.7 absorber absorber metal metal metal absorber low index high index high index Bulk Si thin Si slab (large & slow vg)Bulk Si SPP mode (slow vg) Bulk n=2 Slot waveguide Bulk Si (large & slow vg) Thickness (μm) 1"0" Thickness (μm) 1"0" Thickness (μm) 1"0" 228 1 Thickness (μm) 0.10.02 550 1 x position (nm) 1000 6.8 0.05 A B C DGaAs A B C D "0""""""""""""""""""""""""""""2" " LDOS"Enhancement" (a) (b) (c) (d) (e) (f) 1.01.52.02.53.0 0.0 0.5 1.0 1.5 2.0 Wavelengthnm⇥ Averaged LDOS Enhancement GaAsPhCSolarCellLDOSEnhancement k(nm-1) k(nm-1) k(nm-1) Figure 3.2: Potential solar cell architectures. (a) Planar slab of Si in air in which the LDOS is always below the bulk value. Structures with the potential to beat the ergodic limit: (b) planar slab of Si on an Ag back reﬂector, (c) Ag/Si/Ag planar structure, (d) high/low/high index structure with refractive indices 3/1.5/3, (e) a photonic crystal with index 3.7, and (f) a split dipole antenna made of Ag with a Si gap. The second row shows the LDOS enhancement (on a log color scale) over the bulk material for each structure for various cell thicknesses and wavelengths. Values of LDOS enhancement > 1 correspond to beating the traditional absorption limit. For (a - d), the bottom row shows examples of the 2D modal dispersion curves for each structure. The bottom of column e shows that the integrated LDOS enhancement > 1 for most wavelengths, and the bottom of column f shows the wavelength dependent LDOS enhancement > 1 for radial positions between the split dipole. above the homogeneous value, either km > k0 or vm = dω dkm < vo , where k0 and v0 are the homogeneous values of the propagation constant and group velocity, respectively. Figure 3.2 also shows for each planar structure examples of the modal dispersion relations, showing regions where the density of states is increased due to either slow (low group velocity) modes or large in-plane propagation constants. Note that these dispersion plots do not fully depict the total LDOS, due to the existence of multiple modes; however, many of the key features can be obtained from the dispersion rela- tions. A full 2D waveguide formulation requires a slightly diﬀerent analysis, which
- 44. 32 will be the subject of section 3.4. For the metallodielectric structures, the plasmonic modes that contribute to the LDOS enhancement exhibit parasitic Ohmic metallic losses, and so exceeding the light trapping limit in any structure incorporating metals, requires the useful absorption within the solar cell semiconductor absorber layer to dominate over parasitic metallic losses. This represents a challenge for successful im- plementation of certain cell designs incorporating metals, but we demonstrate below that it is possible to exceed the light trapping limit in metallodielectric structures with proper design. A analogous planar structure without metallic losses is a dielectric slot waveguide [33] in which a low index absorber material is sandwiched between two nonabsorbing cladding layers of higher index, shown in Fig. 3.2d for a core of dielectric constant 1.5 and a cladding of dielectric constant 3. The LDOS within the low index slot will be increased due to the presence of the high index cladding layers, which have an LDOS signiﬁcantly higher than that for a bulk slab of the low index material. Over the entire range of the solar spectrum there is a signiﬁcant integrated LDOS enhancement in the dielectric slot (Fig. 3.2d). This eﬀect can also be observed for common inorganic semiconductor materials such as GaP and CdSe. We perform a similar calculation to Fig. 3.2d using the dielectric functions of these realistic, all inorganic materials. For the cladding layer we use the weakly absorbing indirect semiconductor GaP and for the core we use CdSe. These materials ﬁt the criterion at all wavelengths of interest that the refractive index of the core is lower than that of the cladding. In Fig. 3.3 we plot the spatially averaged LDOS enhancement within the CdSe slot for various slot thicknesses. It can be seen that over all wavelengths of interest there is an increased LDOS due to the slot waveguide eﬀect, becoming more pronounced as the slot thickness is reduced. Dielectric slot waveguides have considerable potential as structures to achieve absorption enhancement in polymer and low index solar cells as there is no metallic
- 45. 33 1100 1000 900 800 700 600 500 400 Figure 3.3: (a) Schematic of slot waveguide structure using realistic materials GaP and CdSe. (b) Averaged LDOS enhancement within the CdSe slot as a function of slot thickness with a constant total device thickness of 1um. 1 um total 3.8 1.08 Log LDOS Enhancement vary t GaP CdSe GaP Wavelength(nm) 0 20 40 60 80 100 Slot Thickness, t (nm) Figure 3.3: (a) Schematic of slot waveguide structure using realistic materials GaP and CdSe. (b) Averaged LDOS enhancement within the CdSe slot as a function of slot thickness with a constant total device thickness of 1um. loss mechanism. This phenomenon for similar structures has also been recognized in two recent works using diﬀerent methodologies [25,26]. Next we examine other structures with potential to exceed the light trapping limit. Photonic crystals [5,34] are interesting candidates as they are known to possess elevated LDOS around ﬂat bands in their dispersive bandstructures. There are a least two ways that photonic crystals can be taken advantage of for solar cells. A photonic crystal could be carved out of a bulk, planar solar cell, or an unstructured, planar solar cell could be integrated with a photonic crystal. The former is certainly intriguing as large LDOS enhancements would be expected but has issues regarding increased surface recombination associated with nanostructuring the active region. Still, utilizing a liquid junction [21] or appropriate passivation of a solid state surface [35] could solve this problem. Here we consider integrating a non-absorbing planar photonic crystal with an unstructured, planar solar cell. This could be easily realized by making the photonic crystal out of an anti-reﬂection coating material such as SiNx, ZnS or TiO2, or by patterning a window layer such as AlGaAs or InGaP on a III-V solar cell such as GaAs. Here, we take a material of index 3.7, common
- 46. 34 for III-V materials like GaAs, InGaP, and AlGaAs, and create a hexagonal lattice of air holes in it with a radius of 205 nm and lattice constant of 470 nm. These parameters are designed so that there is a slow mode with high density of states at the GaAs bandedge at 870 nm. We put this photonic crystal on a thin 10 nm planar layer of the same material and calculate the LDOS as a function of position in the middle of the 10 nm plane. In Fig 3.4a it can be seen that there are signiﬁcant LDOS enhancements at various positions throughout the structure. In Fig 3.4b we also show the spatial LDOS proﬁles for select wavelengths. It can be seen that at the bandedge and within the visible part of the spectrum there are signiﬁcant LDOS enhancements and a suppression of the LDOS within the photonic bandgap starting at 870 nm. In Fig. 2e we show the LDOS averaged over position for this structure. It exhibits an integrated LDOS enhancement relative to a GaAs ﬁlm over the wavelengths of interest above the semiconductor bandgap. Lastly, we point out that integrating solar cell absorber layers into an array of subwavelength scale optical cavities also oﬀers a prospect of large LDOS and absorp- tion enhancement beyond the ray optic limit. As an example, we calculate the LDOS within the gap of a metallic split dipole nanoantenna, shown in Fig. 3.2f. The LDOS enhancement is large and relatively broadband due to the hybridization of each of the monomer resonances and the low quality factor of the antenna. This structure allows for a large degree of tunability of the resonance wavelength, the LDOS peak enhancement and bandwidth by tuning the antenna length/diameter ratio as shown via FDTD simulations in Fig. 3.5. Of course, it is also possible to design all-dielectric cavities of this type which would eliminate any metallic losses.
- 47. 35 Figure 3.4: (a) Schematic of photonic crystal solar cell (b) LDOS map as a function of wavelength and position along the white dotted like shown in (a). (c) Spatial maps of the LDOS enhancement within the center of the 10nm planar layer of dielectric at various wavelengths. a) b) c) Figure 3.4: (a) LDOS map as a function of wavelength and position along the white dotted like shown within the schematic inset. (b) Spatial maps of the LDOS enhance- ment within the center of the 10nm planar layer of dielectric at various wavelengths. 3.4 What Is The New Limit? Figure 3.6 shows the density of states enhancement needed to absorb 99.9% of the solar spectrum for various thicknesses of Si and GaAs. For ultrathin layers we ﬁnd that the density of states enhancement needed is very reasonable, especially in the context of the above results. It is thus possible to imagine absorbing nearly the entire solar spectrum with ultrathin layers of semiconductors as thin as 10-100 nm. The most important criterion that gives an upper bound to the absorbance of a
- 48. 36 Figure' 3.5:" 3D" FDTD" calculated" absorp3on" spectra" of" a" 10nm" layer" of" P3HT:PCBM" clad" with" various" arrays" of" plasmonic" Ag" resonators." Depending" on" the" resonator" diameter"(D),"height"(H)"and"laIce"spacing,"the"ergodic"limit"can"be"exceeded"over" diﬀerent"por3ons"of"the"spectrum." "" FractionAbsorbed 0.6 0.5 0.4 0.3 0.2 0.1 0.0 550 600 650 700 Wavelength (nm) Figure 3.5: 3D FDTD calculated absorption spectra of a 10nm layer of P3HT:PCBM clad with various arrays of plasmonic Ag resonators. Depending on the resonator diameter (D), height (H) and lattice spacing, the ergodic limit can be exceeded over diﬀerent portions of the spectrum. solar cell absorber layer is the extent to which the density of states can be raised in the wavelength range where absorption is required, which is determined by the density of states sum rules [36,37] which dictate that as the density of states is increased in a region of the spectrum, then it must decrease in another region of the spectrum so as to satisfy: ∞ 0 ρcell − ρvac ω2 dω = 0 (3.12) We show in Fig. 3.7 how this spectral reweighting could occur to allow a density of states enhancement in the desired region of the spectrum. Spectral reweighting occurs naturally whenever a change in the density of states occurs, but can potentially be artiﬁcially engineered at will with clever tuning of dispersion using photonic crystals or metamaterials, for example. Thus the density of states could potentially be enhanced to a reasonably large value which is limited by the bandwidth of the spectral reweighting region as we will show.
- 49. 37 Figure 3.6: Enhancements needed to absorb the entire solar spectrum. (a) Absorption enhancement over 4n2 (proportional to LDOS enhancement over the homogeneous value) for various thicknesses of Si and GaAs needed to fully absorb the AM1.5 solar spectrum. (b) Needed density of states proﬁles for two thicknesses of Si needed to fully absorb the spectrum. Enhancementover4n 2 Energy (eV) 10 nm GaAs! 100 nm Si! 1000 nm Si! Densityofstates(Msm -3 ) Energy (eV) 100 nm Si! 1000 nm Si! Homogeneous Si! a) b) Figure 3.6: LDOS Enhancements needed to absorb the entire solar spectrum. (a) Absorption enhancement over 4n2 (proportional to LDOS enhancement over the ho- mogeneous value) for various thicknesses of Si and GaAs needed to fully absorb the AM1.5 solar spectrum. (b) Needed density of states proﬁles for two thicknesses of Si needed to fully absorb the spectrum. For a solar of material index n, absorption coeﬃcient α, and thickness x, the density of states need to absorb all of the incident radiation is given by: ρmax = ρbulk 1 − e−α(4n2)x (3.13) where ρbulk = n3ω2 π2c3 is the homogeneous density of density of states of the bulk material. If the cell is to absorb all of the incident light over the spectral range from ω0 to ω1, e.g. over the solar spectrum, then over this range, ρcell → ρmax. By the sum rule, over some other spectral range, ω2 to ω3, the density of states of the cell must be reduced. If n and alpha are approximately constant over the range ω0 to ω1 and n is constant over ω2 to ω3, then the minimum bandwidth of the suppression region, ω3 − ω2, corresponds to: (ω3 − ω2) = (ω1 − ω0) e−α(4n2)x − 1 (3.14) Since the bandwidth of needed enhancement is limited (i.e. the solar spectrum)
- 50. 38 E(!2)" E(!3)"E(!0)" E(!1)" #" $0" $" Energy (eV)" Densityofstates(sm-3)" c! Homogeneous Si" Solar spectrum" Figure 3.7: Schematic of spectral reweighting needed to achieve density of states enhancements. Figure 3.7: Schematic of spectral reweighting needed to achieve density of states enhancements. and the region from which we can aﬀord to decrease the density of states is nearly unlimited (i.e. regions outside of the solar spectrum), we expect other physical and practical limits to be relevant before a limit is reached for increasing the density of states, such as saturated absorption in the active layer or device and processing issues with extremely thin devices. 3.5 Numerical Demonstration Of Exceeding The Ray Optic Limit As a simple proof-of-principle, we perform three-dimensional electromagnetic sim- ulations using ﬁnite diﬀerence time domain methods for two test structures using the measured optical properties of real materials that demonstrate these concepts. Figure 3.8 shows the fraction of incident illumination absorbed by a 10 nm layer of P3HT:PCBM, a common polymeric photovoltaic material for both a dielectric slot waveguide (Fig. 3.8a) and a hybrid plasmonic structure (Fig. 3.8b). For the slot waveguide, the P3HT:PCBM layer is cladded on both sides by 500 nm of roughened GaP, a high index material with little to no loss in the wavelength range of interest owing to its indirect bandgap at 549 nm. In this structure, the elevated
- 51. 39 Figure 3.8: Full$wave)simula-ons)of)a)P3HT:PCBM)structure)cladded)with:)a,)roughened)GaP)or) b,)plasmonic)resonators)with)absorp-on)enhancement)exceeding)the)ergodic)limit.)The) absorp-on)greatly)exceeds)the)LDOS)limit)for)the)slab)of)P3HT:PCBM)in)air)(dashed)orange)line)) and)approach)the)LDOS)limit)for)the)P3HT:PCBM)slot)(dashed)blue)line))with)reﬁned)incoupling.) LDOS limit! FDTD simulation! Ergodic limit! Slab in air! LDOS limit! FDTD simulation! Ergodic limit! Slab in air! !"#$%!&'() *+!) *+!) a! b! !"#$%!&'() ,-../.) 01) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 560 600 640 680 Wavelength (nm) 560 600 640 680 Wavelength (nm) FractionAbsorbed 0.8 0.6 0.4 0.2 0.0 FractionAbsorbed a) b) Figure 3.8: Full-wave simulations of a P3HT:PCBM structure cladded with roughened GaP (a) without and (b) with a back reﬂector that exceeds the ergodic limit. The absorption greatly exceeds the LDOS limit for the slab of P3HT:PCBM in air (dashed orange line) and approach the LDOS limit for the P3HT:PCBM slot (dashed blue line) with reﬁned incoupling. LDOS leads to an enhancement of the absorption that exceeds the ray optic light trapping limit even without full population of all of the modes of the structure at all wavelengths where the GaP cladding is transparent. This structure exceeds both the 4n2 and 2n2 limits for absorption enhancement, with and without a back reﬂector, respectively. Ideally, Lambertian scattering surfaces could be designed to allow more nearly equal incoupling to all of the modes, which will further enhance the absorption over the entire range of the spectrum to the new limit shown by the blue dashed line in Fig. 3.8a. We also calculate the LDOS limit for this slab in air (orange dashed line), which is exceeded by our simulation for all wavelengths. The original ergodic limit calculation assumed isotropic illumination. Thus to truly exceed this limit a given design should have higher absorption when integrated over all angles. We thus calculate the angular response of the structure in Fig. 3.8 at two wavelengths to ensure the design truly exceeds the ergodic limit. Fig. 3.9 shows fraction of light absorbed at 550 and 600 nm as a function of angle of incidence, averaged over both polarizations. The ergodic limits are also shown for these two
- 52. 40 wavelengths. The limit is exceeded for most angles for each of the wavelengths, falling oﬀ only at very steep angles as expected. The angle averaged absorbances for 550 nm and 600 nm light are 0.48 and 0.34, respectively, indicating that the ergodic limit is surpassed with this structure even with isotropic illumination. Figure 3.9: Angular behavior of the P3HT:PCBM structure that exceeds the ergodic limit. Similarly, Fig. 3.10 indicates that plasmonic resonators situated above and below the absorber layer also enable absorption enhancements that surpass the ergodic limit. These two examples demonstrate that with proper design of LDOS and incoupling, the ergodic limit can be exceeded in several diﬀerent ways. We have identiﬁed the local density of optical states as a key criterion for exceed- ing the ray optic light trapping limit in solar cells. To exceed the ray optic limit, the absorber must exhibit a local density of optical states that exceeds that of the bulk, homogeneous material within the active region of the cell. In addition to this, the modal spectrum of the device must be appreciably populated by an appropriate incoupling mechanism. We have outlined a portfolio of solar absorber designs that can meet the LDOS criterion implemented utilizing plasmonic materials, dielectric waveguides, photonic crystals, and resonant optical antennas, and examples are given to show that incouplers can be designed that provide appreciable mode population.
- 53. 41 Figure 3.9: Full$wave)simula-ons)of)a)P3HT:PCBM)structure)cladded)with)plasmonic)resonators) that)exceeds)the)ergodic)limit.) Slab in air! LDOS limit! FDTD simulation! Ergodic limit! Slab in air! b! !"#$%!&'() ,-../.) 01) 560 600 640 680 Wavelength (nm) 0.8 0.6 0.4 0.2 0.0 FractionAbsorbed Figure 3.10: Full-wave simulations of a P3HT:PCBM structure cladded with plas- monic resonators with absorption enhancement exceeding the ergodic limit. These concepts and design criteria open the door to design of a new generation of nanophotonic solar cells that have potential to achieve unprecedented light trapping enhancements and eﬃciencies. 3.6 Planar Waveguide Formulation for Exceeding The Ray Optic Limit If we limit the scope of the previous section and consider only thin absorbing slabs (thickness, h ∼ λ) that contain propagating modes, we can describe light trapping limits by extending a method initially developed by Stuart and Hall [30]. By consid- ering the reduction in the number of photonic modes in an absorbing slab as the slabs thickness is reduced, they concluded that the maximum absorption enhancement is reduced below 4n2 . Despite the early work suggesting that the absorption enhancement in thin slabs is less than 4n2 , recent work has shown that the enhancement may in fact exceed this limit through the use of photonic crystals [38], plasmonic waveguides [39], high
- 54. 42 index claddings [25, 26], or an elevation of the local density of optical states [40]. Here we revisit the case of a thin waveguide and determine that the 4n2 limit can be exceeded for a number of structures that support large propagation constants and/or slow modal group velocities. We follow a formalism similar to Refs [30,39] and consider the thermal occupation of electromagnetic modes in a waveguide surrounded by vacuum. The total density of modes per unit volume and frequency is given by [30]: ρtot = ρrad + ρm (3.15) where ρrad = 1 − 1 − 1 n2 sc n3 scω2 π2c3 (3.16) is the density of radiation modes not trapped by total internal reﬂection within a material of index nsc, and ρm = β 2πhvm g (3.17) is the density of waveguide modes with modal propagation constants βm and group velocities vm g . Assuming that the modes of the structure are equally occupied from an appropriate incoupler, the total fraction of incident light absorbed by the ﬁlm is [39]: Ftot = ρrad ρtot frad + m ρm ρtot fm (3.18) where frad = α α + 1 4 ρtotvrad g ρ0v0 g h (3.19)
- 55. 43 and frad = αΓm αΓm + 1 4 ρtotvrad g ρ0v0 g h (3.20) α is the attenuation coeﬃcient, Γm is modal conﬁnement factor, ρ0 = n3 scω2 π2c3 is the bulk density of states in vacuum, and v0 g = c is the speed of light in the surrounding vacuum. We note that the ratios of the density of states gives the fraction of light that enters each mode and frad and fm give the rate of absorption divided by the total rate of energy loss (absorption and out coupling). We now consider the case of a thin waveguide that only supports one guided mode. For this case ρrad << ρm, and the total fraction of light absorbed can be approximated by: f1−m = αΓm αΓm + 1 4 ρtotvrad g ρ0v0 g h = αΓm αΓm + 1 4 βmπc3 2ω2hvm g nsc h (3.21) which can be compared to the fraction of light absorbed in the traditional 4n2 limit [41]: F4n2 = α α − 1 4n2 sch (3.22) In order to exceed the 4n2 limit, F1−m must be greater than F4n2 . With the condition F1−m > F4n2 , we obtain our central result: βm k0 vrad g vm g Γm > 4nsc h λ (3.23) Thus in order to surpass the 4n2 limit, we want well-conﬁned modes (Γm ∼ 1) that have large propagation constants (βm > k0), small group velocities (vm g < vrad g ), and waveguide thicknesses that are small compared to the wavelength of the incident
- 56. 44 light. By comparing Eq. 3.20 and 3.21, we note that the path length enhancement lpath is: lpath = c vm g βm k0 h λ Γm (3.24) Note that the actual path length is lpathh for the single mode waveguide case compared to 4n2 h for the traditional limit. For a given waveguide structure, the dispersion relation is calculation to determine the modal parameters needed for Eq. 3.23. Figure 3.11 shows the dispersion relation for several structures based on a thin ﬁlm of P3HT:PCBM, a common polymer solar cell blend. The propagation constants are obtained from the x-axis, and the slopes of the curves are proportional to the group velocities. Traditional photonic modes for the slab are conﬁned to the region between the vacuum light line and the absorber light line (light blue region). In this region, βm < k0, vm g > vrad g , and Γm << 1 when the waveguide is thin. Thus, these photonic slab modes cannot surpass the 4n2 limit. Figure 3.11: Waveguide dispersion relations. Plasmonic (solid red), Metal-Insulator- Metal (dashed red), and high-low-high index slot (green dashed) waveguides have dispersion relations that lie to the right of the bulk absorber light line (solid blue), which have large propagation constants, slow group velocities, and are well-conﬁned.
- 57. 45 Figure 3.11 also shows the dispersion relation for three structures that have large βm, slow vm g , and large conﬁnement factors. These structures include a surface plasmon polariton waveguide, a high-low-high index dielectric slot waveguide, and a metal-insulator-metal (MIM) waveguide. These dispersion curves represent the lowest order TM modes for the structures; however, higher order modes may exist depending on the thickness of the slab. While Eq. 3.23 is strictly valid only for a single, highly conﬁned mode, the behavior of the fundamental mode typically dictates the absorption characteristics, as we shall see below. A thin surface plasmon polariton waveguide (Fig. 3.12) satisﬁes the necessary conditions to surpass the 4n2 limit for a variety of wavelengths and slab thicknesses. Using Eq. 3.23, we ﬁnd that the path length enhancement should exceed 4n2 for absorber thicknesses between 50-105 nm. However, the condition of only one well- conﬁned mode in the structure is not met for thicknesses larger than 70 nm, because a second mode exists for larger slab thicknesses. Further, below 50 nm, the conﬁnement factor is reduced. Figure 3.12a shows the actual absorption enhancement beyond 4n2 for λ=580 nm using Eq. 3.21, which takes into account both propagating and radiation modes. This structure surpasses the 4n2 limit for the entire range. The wavelength resolved absorption for a 50 nm thick slab of P3HT:PCBM on Ag is shown in Fig. 3.11b. A high-low-high index dielectric slot waveguide is also found to exceed the 4n2 limit. This structure consists of a 10 nm P3HT:PCBM layer cladded on both sides by a 45 nm layer of GaP, a wide bandgap semiconductor with an indirect energy gap whose absorption is negligible over the wavelength region of consideration and whose real part of the refractive index exceeds that of P3HT:PCBM. This structure supports multiple modes; however we can use our expression for the path length enhancement to predict whether or not the structure could surpass the 4n2 limit if only one mode were present. Figure 3.13a shows that the lowest order TE and TM modes for =580
- 58. 46 Figure 3.12: Plasmonic waveguides that beat the 4n2 limit. (a) Calculated absorp- tion enhancement surpasses the 4n2 limit for a large range of thicknesses. Shaded region corresponds to the region where the limit is surpassed if only one mode were present. The appearance of a second mode (vertical dashed line) allows the absorp- tion to surpass the 4n2 limit beyond the region predicted by the single mode model. (b) Plasmonic waveguide absorption surpasses the 4n2 limit throughout the useful absorbing spectrum of the polymer. Inset: Schematic of the plasmonic waveguide and the modal proﬁle. nm would both have path length enhancements in excess of 4n2 if this were a single mode waveguide. Figure 3.13b shows the total absorption in the slot waveguide and the fraction of absorption coming from each of the modes. For wavelengths below 620 nm, there are three propagating modes; however, for longer wavelengths, there are only two modes since the second order TE mode is cut oﬀ. Because the cladding layer is thin, the radiation modes were assumed to be incident from vacuum and resulted in < 1% of the total absorption. If the radiation modes were allowed to occupy the full 4π steradians, ρrad would still contribute to less than 3% of the total absorption. Thus the main contribution to the absorption still comes from the propagating waveguide modes. Finally, we explore a hybrid waveguide which combines the beneﬁts of both a high index cladding and a plasmonic back reﬂector to achieve over 90% absorption in a 10 nm P3HT layerwell in excess of the 4n2 limit for bulk P3HT. The high index
- 59. 47 Figure 3.13: Slot waveguides that beat the 4n2 limit. (a) Path length enhancement if only one mode were present. Both the lowest TM and TE modes surpass 4n2 . Insets: Mode proﬁles. (b) Modal decomposition of the absorbed fraction of the incident light. The total absorption surpasses the 4n2 limit. Inset: Schematic of the high-low-high index slot waveguide. GaP cladding further conﬁnes the mode and reduces its group velocity. Figure 3.14 shows the total waveguide absorption, the absorption expected for 4n2 passes, and the absorption obtained by a 10 nm thick P3HT ﬁlm surrounded by vacuum. Note that the 10 nm thick P3HT ﬁlm in vacuum has a signiﬁcantly lower absorption than would be expected by the 4n2 limit. This is due to the fact that the photonic modes are not well conﬁned, have higher group velocities than do modes in the bulk, and have small propagation constants, which are in agreement with the conclusions of Ref. [30]. In summary, we have described a general method for predicting whether or not thin waveguides can have absorption enhancements in excess of the 4n2 limit based on their dispersion relations. We gave several examples of such structures by including plasmonic back reﬂectors and/or high index claddings in order to manipulate the modal propagation constant, group velocity, and conﬁnement factor. Our results provide both a qualitative and quantitative design scheme for improving the light trapping in thin ﬁlms, which is highly desired for solar cell applications.
- 60. 48 Figure 3.14: Absorption for a plasmonic waveguide cladded with a high index (GaP) top layer. Waveguide absorption (circles) is well in excess of the 4n2 limit (solid line) and the calculated absorption for a planar slab in vacuum (dashed line).
- 61. 49 Chapter 4 Light Trapping With Dielectric Nanosphere Resonators We have seen how various nanophotonic elements can act as useful tools to enhance optical absorption for photovoltaic devices. Here, we will examine in greater detail the behavior of a simple low cost, low loss scheme for absorption enhancement based on dielectric nanosphere resonators. The resonators can act as both scattering centers or as diﬀractive elements that change the momentum of incident photons and enables access of trapped modes in the absorbing layer. The periodic arrangement of the resonators allows for great ﬂexibility for optical dispersion engineering and tuning of optical absorption for diﬀerent materials applications. Here we show how this simple scheme can be applied to both traditional a-Si solar cell designs, as well as current thin-ﬁlm GaAs solar cells. While neither of the two cases discusses here exceed the ergodic light tapping limit, further investigation with higher index resonators could potentially achieve this.
- 62. 50 4.1 Dielectric Nanosphere Resonators for a-Si Ab- sorption Enhancement We propose here a light trapping conﬁguration for a-Si solar cells based on coupling from a periodic arrangement of resonant dielectric nanospheres. It is shown that photonic modes in the spheres can be coupled into particular modes of the solar cell and signiﬁcantly enhance its eﬃciency. We numerically demonstrate this enhancement using full ﬁeld ﬁnite diﬀerence time domain (FDTD) simulations of a nanosphere array above a typical thin ﬁlm amorphous silicon (a-Si) solar cell structure. The incoupling element in this design is advantageous over other schemes as it is composed of a lossless material and its spherical symmetry naturally accepts large angles of incidence. Also, the array can be fabricated using simple, well developed methods of self assembly and is easily scalable without the need for lithography or patterning. This concept can be easily extended to many other thin ﬁlm solar cell materials to enhance photocurrent and angular sensitivity. Thin ﬁlm photovoltaics oﬀer the potential for a signiﬁcant cost reduction [11] compared to traditional, or ﬁrst generation photovoltaics usually at the expense of high eﬃciency. This is achieved mainly by the use of amorphous or polycrystalline optoelectronic materials for the active region of the device, for example, a-Si. The resulting carrier collection eﬃciencies, operating voltages, and ﬁll factors are typically lower than for single crystal cells, which reduce the overall cell eﬃciency. There is thus great interest in using thinner active layers combined with advanced light trapping schemes to minimize these problems and maximize eﬃciency. Wavelength-scale dielectric spheres [42] are interesting photonic elements because they can diﬀractively couple light from free space and also support conﬁned resonant modes. Moreover, the periodic arrangement of nanospheres can lead to coupling between the spheres, resulting in mode splitting and rich bandstructure [43,44]. The
- 63. 51 Figure'4.1:"(a)"3D"schema-c"of"the"dielectric"nanosphere"solar"cell."(b)"Current"density"calculated" in"the"amorphous"silicon"layer"with"and"without"the"presence"of"nanospheres."The"a>Si"thickness" is"t=100"nm"and"the"sphere"diameter"is"D=600"nm."(c)"Cross"sec-on"of"a"silica"nanosphere"on"an" amorphous"silicon"layer"with"an"AZO"and"silver"back"contact"layer."(d)"Integrated"electric"ﬁeld" and"absorp-on"calculated"from"ﬁg."2(a)"for"normal"incidence"of"a"cross"sec-on"at"the"center"of" the"sphere"at"the"resonant"frequency"λ=665"nm."" ! a) b) c) d) Figure 4.1: (a) 3D schematic of the dielectric nanosphere solar cell. (b) Current density calculated in the amorphous silicon layer with and without the presence of nanospheres. The a-Si thickness is t=100 nm and the sphere diameter is D=600 nm. (c) Cross section of a silica nanosphere on an amorphous silicon layer with an AZO and silver back contact layer. (d) Integrated electric ﬁeld and absorption calculated from ﬁg. 2(a) for normal incidence of a cross section at the center of the sphere at the resonant frequency λ =665 nm. coupling originates from whispering gallery modes (WGM) inside the spheres [44, 45]. When resonant dielectric spheres are in proximity to a high index photovoltaic absorber layer, incident light can be coupled into the high index material and can increase light absorption. Another important beneﬁt of this structure for photovoltaic application is its spherical geometry that naturally accepts light from large angles of incidence. Figure 4.1 (a) depicts a solar cell where close packed dielectric resonant nanospheres stand atop a typical a-Si solar cell structure. A cross section is represented in Fig. 4.1
- 64. 52 (c). We use a silver back contact and, in order to avoid diﬀusion between the silver layer and the a-Si layer, a 130 nm aluminium-doped zinc oxide (AZO) layer is placed between the silver and the a-Si layer [46]. An 80 nm indium tin oxide (ITO) layer is used as a transparent conducting front contact and also acts as an anti-reﬂection coating. 600 nm diameter silica nanospheres with a refractive index of n=1.46 are directly placed on the ITO as a hexagonally close packed monolayer array. Figure'4.2:"(a)"Electric"ﬁeld"intensity"for"a"cross"sec4on"at"the"middle"of"a"sphere"in"the"(x,z)"plane" at"λ=665"nm."Below"is"a"zoom"of"the"aBSi"layer."The"direc4on"of"the"electric"ﬁeld"of"the"incident" plane"wave"is"also"represented."(b,c)"Ey"component"of"the"electric"ﬁeld"from"a"cross"sec4on"in" the"(x,y)"plane"in"the"middle"of"the"aBSi"layer."(d,e,f)"Same"as"(a,b,c)"for"λ="747"nm"which" corresponds"to"a"frequency"oﬀ"resonance."" ! a) b) c) d) e) f) Figure 4.2: (a) Electric ﬁeld intensity for a cross section at the middle of a sphere in the (x,z) plane at λ=665 nm. Below is a zoom of the a-Si layer. The direction of the electric ﬁeld of the incident plane wave is also represented. (b,c) Ey component of the electric ﬁeld from a cross section in the (x,y) plane in the middle of the a-Si layer. (d,e,f) Same as (a,b,c) for λ= 747 nm which corresponds to a frequency oﬀ resonance. In order to study the response of the system, we perform three-dimensional full