Rational Design of Hybrid Organic Solar Cells by Levi Lentz B.S., Mechanical Engineering, San Diego State University (2012) Submitted to the Department of Mechanical Engineering I MLr MASSACHUPETFE: 0OF TECC INSITTUTE 7 ' in partial fulfillment of the requirements for the degree Master of Science in Mechanical Engineering AUG 15 2014 at the MASSACHUSETTS INSTITUTE OF TECHNOLOGYJune 2014 @ Massachusetts Institute of Technology 2014. All rights reserved. Signature redacted Department of Mechanical Engineering May 9, 2014 Certified by.... Signature redacted ............... Alexie Kolpak Assistant Professor Thesis Supervisor Signature redacted Accepted by.... ......... .... low, David E. Hardt Chairman, Department Committee on Graduate Theses 2 Rational Design of Hybrid Organic Solar Cells by Levi Lentz Submitted to the Department of Mechanical Engineering on May 9, 2014, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Abstract In this thesis, we will present a novel design for a nano-structured organic-inorganic hybrid photovoltaic material that will address current challenges in bulk heterojunction (BHJ) organic-based solar cell materials. Utilizing first principles Density Functional Theory (DFT), we show that layered inorganic phosphates and tradition organic dyes can be combined to form a new class of bulk heterojunction photovoltaic with high electron and hole mobilities with low exciton recombination, potentially enabling very high efficiency with existing organic-based solar-cell molecules. We will discuss the physical origin of these properties and investigate several approaches for engineering the electronic structure of these materials. By using these methods, it will be possible to engineer the transport and optical properties of these materials, with potential applications beyond photovoltaics in areas from organic electronics to photoactuators. Thesis Supervisor: Alexie Kolpak Title: Assistant Professor 3 4 Acknowledgments Completion of a work such as a Master Thesis requires support from a wide variety of individuals, both personal and professional. In completion of this work, I have learned a great deal about computational physics and chemistry, something foreign to me at the start and more akin to a distant relative now. I would like to thank the people who have helped me immensely along that journey. Primarily, I would like to thank my adviser, Alexie Kolpak, for having the faith to bring me onto her lab. My background was not in the field of research that I now perform, yet she believed that I would be able to complete the work presented herein. Without her faith and, many times, patience, I would not have had the opportunity to excel at MIT the way I have. My bearded friend, Brian Kolb. Hired on as a Post-Doctoral Researcher in our lab, he was immesly helpful in guiding me along in the important aspects of computational physics. From providing resources, to writing up tutorials, to just being the person to bounce ideas off of, he created a lab environment inductive to groundbreaking research. I wish him luck in his future career work. My family, especially my brother Dillon, my father Kirk, and my step mother Judy. Living far away from home can be trying on the soul and a core support group was necessary to stymie the creeping thoughts of self-doubt. With out them, this work would have taken significantly longer. My MIT friends, too numerous to recount. They have been there for me to lament research ills, solve psets, study for the qualifying exams, and enjoy a lazy afternoon at the Boston Common. As we go forth from here to the corners of the world, I will not soon forget the times and friendships we shared here at MIT. The Show, and all my friends from undergrad. A personality is not singularly formed and a great deal of mine came from the friendships that I formed at all the sporting events attended with the wonderful people I love at San Diego State University. 5 6 Contents 17 1.1 Solar Energy. . . . . . . . . . . . . 18 1.1.1 Solar-Thermal . . . . . . . . 19 1.1.2 Photovoltaics . . . . . . . . 20 . . . Introduction 23 . . . . . . . . . . . . . . 26 Theory and Computational Principles 27 27 2.2 Density Functional Theory . . 29 2.3 Functionals 31 2.4 Pseudopotential Approximation 33 2.5 Electronic Properties from p[r] 34 2.6 DFT Bandgap Issue . . . . . 36 2.7 DFT Code . . . . . . . . . . . 38 2.8 DFT Scaling . . . . . . . . . . 38 2.9 Conclusion . . . . . . . . . . . 39 . . Computational Tools . . . . . . . . . . . . . . . . . . 41 3.1 Overall Design . . . . . . . . . . 41 3.2 M aterials . . . . . . . . . . . . . 43 3.2.1 Organic Semiconductor . . 43 3.2.2 Inorganic Phosphate Group 46 . . Rational Design of Nano-structured Hybrid Photovoltaics . 3 2.1 . 2 Objective . 1.3 . . . . . . . . . 1.2 *Bulk-Heterojunctions . 1 7 Exciton Separation . . 49 3.4 Mobility . . . . . . . . 52 3.5 Open Circuit Voltage . 52 3.6 Conclusion . . . . . . . 53 4 Atomic and Electronic Structure 55 . . . . 3.3 55 4.2 Pseudopotential Testing . . . . . 57 4.3 Computational Results . . . . . . 61 . . . Computational Details . . . . . . .. Bulk Zr(HPO 3 )2 4.3.2 Bulk a - Ti(HP04) 2 - H20 62 4.3.3 Titanium Doping of Zr(HPO.: 63 4.3.4 Layered Organic Zr(HPO 3 ) 2 67 4.3.5 Layered Zr(HPO 3 )2 with Org Photovoltaic . . . . . . . . . Scissor Operator 4.5 Organic MK2 Functionalization . 4.4 . 71 73 . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . 68 72 . Functionalized MK2 . 4.6 61 ... . 4.3.1 4.5.1 76 77 5 Engineered Band Alignment 77 Functional Group Band Shifts 5.2 Titanium Doping Band Alignment 79 5.3 Corrected Band Alignment . . . . 81 5.4 Conclusions . . . . . . . . .. . . . 84 . . . 5.1 87 Transport Properties 87 6.1.1 Governing Equations . . . 88 6.1.2 DFT Calculations . . . . . 90 . . Mobility Derivation . . . . . . . . . 6.1 Zr(HPO 3 ) 2 Mobility . . . . . . . 91 6.3 Conclusion . . . . . . . . . . . . . 97 . 6.2 . 6 . . . . . . . . . . . . . . . . . . . 4.1 8 7 Conclusions 99 7.1 Efficiency Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.2 Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A Molecule Figures 105 9 10 List of Figures 1-1 Solar spectrum at 1.5 air mass . . . . . . . . . . . . . . . . . . . . . . 18 1-2 Efficiency vs band gap . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1-3 Schematic of the working of a BHJ material . . . . . . . . . . . . . . 25 1-4 Schematic of types of BHJ materials. Left, low recombination. Right, high recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2-1 Relative computational cost of various methods. . . . . . . . . . . . . 28 2-2 Speedup associated with number of processors . . . . . . . . . . . . . 39 3-1 Schematic representing the proposed design. Alternating layers of organic and inorganic layers allow for efficient charge extraction . . . . 42 . . . . . . 43 3-2 Schematic showing ideal band alignment in bulk structure 3-3 The MK2 molecule (2-Cyano-3-[5-(9-ethyl-9H-carbazol-3-yl)-3,3,3,4-tetran-hexyl-[2,2,5,2,5,2]-quater thiophen-5-yl] acrylic) . . . . . . . . . . . 45 3-4 Different types of organic semiconductors studied . . . . . . . . . . . 45 ce-Zr(HPO 3 )2 3-5 Structure of 3-6 Structure of Zr(HPO 3 )2 layered with benzene rings . . . . . . . . . . 48 3-7 Schematic of dipole separation . . . . . . . . . . . . . . . . . . . . . . 51 3-8 Exciton separation in an ordered nano-structured material with an induced dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 51 3-9 VOC in BHJ materials. 4-1 K-point convergence of Bulk Zr(HPO 3 )2 . . . . . . . . . . . . . . . . 57 4-2 Atom-Resolved Density of States for a - Zr(HPO 3 )2 . . . . . . . . . . 62 . . . . . . . . . . . . . . . . . . . . . . . . . 11 54 4-3 Comparison between hydrated and unhydrated Ti(HPO 4 )2 . . . . . . 64 4-4 Titanium doping of Zr(HPO 3 )2 . . . . . . . . . . . . . . . . . . . . . 66 4-5 Atom resolved density of states, spatially separated into each layer . . 68 4-6 Computational structure of Zr(HPO 3 ) 2 bound to organic MK2 . . . . 69 4-7 Spatially resolved DOS of Zr(HPO 3 )2 -MK2. . . . . . . . . . . . . . . 70 4-8 Schematic showing energetic shift due to polarized MK2 . . . . . . . 70 4-9 Computational modifications of MK2 molecule . . . . . . . . . . . . . 73 . . . . . 74 4-11 Schematic showing functionalization of the organic MK2 molecule . . 75 . . . . 80 4-10 Windowed-averaged electrostatic potential of modified MK2 5-1 Spatial resolution of the Valence and Conduction Band Edges 5-2 Structural form of Titanium Doped Zirconium Phosphate . . . . . . 81 5-3 Titanium Doping of Zirconium Phosphate Layer . . . . . . . . . . . . 82 5-4 Scissor operator application to the bulk band structure . . . . . . . . 82 6-1 Directions of reciprocal space band structure . . . . . . . . . . . . . 93 6-2 Band Diagram of Zr(HPO 3 )2 along reciprocal axes . . . . . . . . . . 93 6-3 Plot of curve required to fit the elastic constant . . . . . . . . . . . . 94 6-4 Plot of curves required to calculate the deformation potential. .... 95 6-5 Effective Mass vs. Layers of Zr(HPO 3 ) 2 . . . . . . . . . . . . . . . . 96 7-1 Schematic representing the idea of the area fraction . . . . . . . . . . 100 7-2 Schematic for electron (hole) tunneling . . . . . . . . . . . . . . . . . 101 A-1 MK2 with Benzene interlinker . . . . . . . . . . . . . . . . . . . . . . 106 A-2 Chemical structures of MK2- Pyridine and Pyrimidine 107 . . . . . . . . A-3 Chemical structures of MK2- Pyridine and Pyrimidine functionalized with an OH group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A-4 Chemical structures of MK2- Pyridine and Pyrimidine functionalized with two OH groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 A-5 Chemical structures of Boron and Flourine functionalized MK2 . . . . 110 A-6 MK2 with both Fluorine and Boron . . . . . . . . . . . . . . . . . . . 111 12 A-7 BX shown with Benzene and Pyrimidine interlinkers . . . . . . . . . . 111 A-8 PX shown with Benzene and Pyrimidine interlinkers .... . . . . . . . 112 A-9 TPD shown with Benzene and Pyrimidine interlinkers . . . . . . . . . 112 13 14 3.2 Thiophene bandgap and chain length [5] 4.1 4.2 Pseudopoentials used in this study . 58 4.3 Bond distance comparison for STP gasses 58 4.4 Bond distances for carbon, sulfur, hydrogen, and nitrogen pseudopo- . Thiophene property modifications [30] . . 46 . 3.1 . List of Tables . 46 Zr(HPO 3 ) 2 experimental unit cell parameters 56 . . . . 59 4.5 Bulk zirconium lattice comparison . . . . 59 4.6 ZrO 2 unit cell computational error . . . . 59 4.7 Lattice constant error of Zr(HPO 3 ) 2 4.8 Experimental Structure of a - Ti(HPO 4 ) 2 - H 2 0 4.9 Similarities in Zr(HPO 3 ) 2 and Ti(HPO 4 )2 bond lengths in their bulk . . . tentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 63 65 4.10 Bangap Modification of Titanium doped Zirconium Phosphate . . . 4.11 Comparison between bulk structure and constitutive molecules. . . 69 4.12 Computational Optimization of MK2. Energies relative to Fermi Level 73 4.13 Voltage drop of functionalization . . . . . . . . . . . . . . . . . . . 75 5.1 Comparison between bulk structure and constitutive molecules . . . 78 5.2 Corrected bandgap of system by applying two scissor operators . . . 83 5.3 Corrected bandgap of system by applying two scissor operators . . . 84 6.1 Electron and hole transport properties of Zr(HPO 3 ) 2 . . . . . . . . 95 . . . . . . 65 . . structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.2 Comparison of mobilities between common semiconductors . . . . . . 97 7.1 Max efficiencies of select systems examined . . . . . . . . . . . . . . . 102 16 Chapter 1 Introduction Lower carbon emissions and a drive for clean and long-term sustainable energy generation is driving the development of renewable resources. Solar energy has the potential to replace the majority of conventionally generated electricity. In the United States alone, the National Renewable Resource Laboratory (NREL) estimates that the total potential in the United States is 155GW [35]. The foremost way this energy is captured is via silicon-based photovoltaic devices. Limited by the cost of manufacture, only recently have silicon photovoltaic materials become cost competitive with conventional grid-level coal or natural gas-based electricity generation. Organic-based photovoltaics are a promising alternative to silicon for the conversion of solar radiation into electrical power. These materials offer several advantages over traditional inorganic-based materials. In particular, organic photovoltaics can potentially be manufactured at scale using solution-based processing, which is significantly lower in cost [4] and more environmentally friendly than current semiconductor manufacturing techniques. In addition, organic-based materials could enable flexible photovoltaics, opening up possibilities for novel applications. In order to realize these possibilities at scale, however, a number of challenges must be overcome to improve the efficiency of organic-based photovoltaics. In the remainder of this chapter, we first discuss these challenges in the context of traditional, organic, and hybrid photovoltaics (PV). We then review the state-of-the-art in organic-based bulk heterojunction (BHJ) solar cells, and conclude with an outline of the main 17 Ground-Level Reference Spectra - ASTM G173-03 Reference Spectra - Blackbody at 5777K 15- 0.5-- 0 0 1000 2000 3000 4000 5000 6000 Wavelength [nm] Figure 1-1: Solar spectrum at 1.5 air mass objectives of this work, in which we use first-principles computations to design a novel class of nano-structured organic-based photovoltaic materials to overcome these challenges. 1.1 Solar Energy The solar spectrum illuminating the earth's surface provides an abundant amount of energy to supply nature's energetic needs with enough to spare to provide for the majority of humanity's endeavors. The energy that we receive from the sun comes in quantized packets called photons, following the Plank's distribution as if the sun emits as a black body. Because of the large distance between the earth and the sun, it can be estimated to emit at approximately 6000 C. NREL provides comprehensive measurements of the incident ground irradiation, shown in Figure 1-1. Overlaid is the exact solution to a black body with a solid angle of 6.87 x 10- steradians, the average solid angle of the sun-earth distance. In Figure 1-1, the solar spectrum was plotted at what is commonly referred to as 18 "air mass 1.5," or AM1.5. The definition of airmass is given as AM = , (1.1) Lo) where L is the length the light traveled through the atmosphere, and Lo is the length that light would travel, normal to the earth's surface at sea level. In this way, the length that light will have to travel through the atmosphere can be quantified. This is important because the longer that light is traveling through the atmosphere, the more scattering that it will experience, and hence the more the light will deviate from the perfect black body. AM1.5 is commonly used because it is the distance that light will have to travel in most populated areas. This is an industry standard and is used for the design of many solar devices. By integrating both curves represented in Figure 1-1, the black body spectrum yields a incident radiation of 1367 W/m2 while the ground-level spectrum yields a incident radiation of 872 W/m 2 . This loss in solar potential is due primarily to atmospheric interference that interacts with the solar spectrum. In this way, the sun deviates from being a perfect black body while viewed from the surface of the earth. By concentrating this spectrum, heat can be generated in a solar-thermal collection process to use for low-grade hot water generation or for high-grade steam generation. Alternatively, the light can be extracted directly from the photoelectric effect to directly provide energy in the form of electricity. Both of these processes will be described in the following sections. 1.1.1 Solar-Thermal One way to capture the sun's power is by capturing the light in the form of heat, which can be used for a variety of applications from rooftop water heaters to grid-level energy plants generating megawatts of electricity from high-level flux concentrators. This thesis will not be studying the applications of solar-thermal applications; however, knowledge of this area of solar applications warrants a brief overview. Solar thermal acutally provides the largest amount of energy generation from 19 solar in the world in the form of solar heat. This energy is traditionally harnesses in the form of solar water heaters, although it is rarely seen in the United States, accounting for only 0.4% of installed water heaters [21]. Israel, the world's larges producer of solar hot water per capita, produces many times amount, significantly offsetting the overall energy usage [21]. Because of this, solar thermal water heating could significantly offset the electrical/gas uses by homes around the US to warm their houses. The other form of solar thermal is obtaining high-grade heat by concentrating the solar flux and using it to generate steam. In general, solar concentrators are capable of generating a concentration factor of tens to hundresds of suns (equivalent to a 10100x increase in the solar flux [~-. 104- 10 5W/m2 ]). This allows for a higher amount of energy density per area, creating the possibility of high-grade heat necessary for electrical generation. Normally these systems have to be grid-scale and centrally located due to the fact that the energy then takes a 30-50% loss in the thermalelectric conversion process. In addition, large amounts of water and land are required to make even small thermal-electric power plants. 1.1.2 Photovoltaics Solar cells rely on the principle of the photovoltaic effect, described originally by Einstein in the Annus Mirabilis papers [10]. This effects relies on the particle-wave duality-of light, allowing energetic photons to impart their energy to electrons, generating charge carriers and providing a photocurrent that can be harvested as electricity. These devices offer a very unique way to generate electricity as they can be modular in nature, providing the opportunity to be small in form factor. Utilizing both the photoelectric effect as well as band theory, inorganic semiconductors can efficiently extract electricity from the solar spectrum. Conventional inorganic solar cells contain a p-n junction in the material, creating a built-in electric field that drives the charged carriers towards their respective electrodes. By utilizing the photoelectric effect, light creates a bound hole-electron pair, a quasiparticle is traditionally called an exciton, and the p-n junction is able to overcome coulumbic 20 interaction between the hole and electron pair, separating the charge and driving the photocurrent. This photoelectric effect can be observed in both metals as well as semiconductors. In the case of metals, the excited charges are simply used in the form of heat, whereas in semiconductors, this charge is in the form of useable electricity. This is because semiconductors have a built-in bandgap to the material, or a range of energies that electrons, or holes, cannot occupy. Bandgaps are composed of both a conduction band where excited electrons reside, and a valence band, where the holes reside. What this physically means is that a material that has a perfect absorptivity can absorb all photons with an energy greater than the bandgap of the material. When incident light has an energy greater than the bandgap, it photoexcites an electron to an energy state greater than the conduction band edge of the material. It then radiatively loses energy, in the form of heat, until it enters the conduction band of the material. Because of this radiative emission, there is actually a loss from the photon to electron that implies that there is an optimal bandgap associated with the material. Any photon that has a lower energy than the bandgap of the materials is not absorbed, a further loss in the system. For the purpose of photovoltaics, there exists two types of semiconductors: direct gap and indirect gap. This refers to the nature of the gap; a direct gap semi conductor has the conduction band minimum and valence band maximum at the same position in reciprocal space where as in a indirect gap semiconductor, there is a spatial separation of these two points. Direct gap semiconductors are materials traditionally used in lasers as well as photodiodes and have many uses as photovoltaics. This is desirable because the transition between the CBM and VBM is only energetic in nature. In an indirect gap semiconductor, momentum must be conserved, and in order for an electron to transition between the VBM and CBM, momentum must be imparted from the crystal into the electron, physically meaning that the material must be thicker to impart momentum to the system. While silicon is an indirect bandgap absorber, preventing it from being used as a thin film absorber, its abundance in nature as well as the industry know-how to 21 0.4 O0.301 0O.2 -1 0O.1- 0 0 0.5 I lI 1 I 1.5 I 2 2.5 3 EEgap gp(eV) Figure 1-2: Efficiency vs band gap process it from traditional electronic applications have propelled it to the forefront of the solar energy industry. Additionally, Silicon has a bandgap of ~1.1eV, occupying an almost ideal spot in the Shockley-Queisser limit, the efficiency limit of a singlejunction p-n junction [7]. The limit of efficiency vs bandgap is show in Figure 1-2, calculated following this limit. With all of these advantages in mind, silicon-based solar cells have become the main material used in solar energy capture. However, significant processing of the material is necessary as amorphous silicon suffers from severly degregaded efficiency over crystalline materials. Because of this, other alternatives that decrease the processing requirements as well as the cost per watt of electrical generation are of great interest. Even with these disadvantages, the solar-cell design offers many unique opportunities for solar-energy capture in the form of distributed electrical generation. In this way, modular units based off of photovoltaic cells can be used on rooftoops of homes or businesses to provide energy in a non-centralized way. This offers freedom 22 for both the individual as well as the corporation to create small-scale installations in residential areas or to create grid-scale solar capture plants based off of effecient solar cell devices. Consequently, a large amount of research is being done in both inorganic and organic materials. 1.2 Bulk-Heterojunctions One alternative to silicon and other inorganic photovoltaics is in the realm of organic semiconductors. Rather than being comprised of a single material doped for either por n-type carriers, two different materials are joined together, using energetics of the band alignments at their interfaces to drive exciton separation. This type of composite photovoltaic is known as a bulk-heterojunction (BHJ). BHJs are normally organic in nature, but hybrid composites comprised of both organic and inorganic materials also exist. BHJs, in general, have a lower overall efficiency compared to single materials, but their low cost of manufacture as well as low environmental impact drives research into this area of organic-based PV materials. The material considerations that limit the efficiencies of BHJs are primarily low carrier mobility and high exciton binding energy. The mobility is an indication of how fast charge carriers (electrons and holes) can move within the material. The lower the mobility, the slower charge carriers can move, thereby increasing the probability of recombination. The binding energy of the exciton is the the energy required to separate the bound hole-electron pair. This is a problem in all semiconductors, however in inorganic materials, the binding energy is of the order of the thermal energy at room temperature, while in organic materials, exciton binding energies much higher, on the order of 1.0eV. As a result, organic materials are observed to have much higher recombination of the electron and holes in the material. The exact nature of exciton binding energies will be described in Section 3.3. Recombination is simply when a bound hole-electron pair recombine or when a free charge carrier meets another charge carrier of the opposite charge. In both of these scenarios, that charge is then lost, leading to a decrease in the overall efficiency 23 of the system via loss of the photo-excited charge carrier. To overcome the issue of the large binding energy of the exciton, this design of the heterojunction was born. This design comes from the need to have an energetic difference large enough to separate the two bound charges. To this end, two organic materials are joined together that have a energetic difference between their conduction band edges that is large enough to encourage separation of the exciton. The main issue with this has to do with decreasing the effective bandgap of the material, thereby lowering the open circuit voltage of the material (Voc ). This is due to the fact that the efficiency in photovoltaics is defined as: FF x Vc x Isc E xAc ExA~ (1.2) where FF is the fill factor, an indication of how much power is utilized over ideal condictions; VOC is the open circuit voltage; ISc is the short circuit current, the max current that can go through the device; E x A, is simply the amount of energy per area time the total cross-sectional area (the total energy incident on the material). From this, it is apparent that controlling the VOC is extremely important to the efficiency of the material. To this end, the two disperate materials are carefully chosen in the BHJ to ensure that the exiton is properly separated without lowering the Voc significantly. This process is schematically shown in Figure 1-3. The other limiting factor of most bulk heterojunctions is the mobility issue. In these materials, the mobility is of the order 10- 4cm 2 /Vs [24], whereas traditional semiconductors have mobilities of 500 - 1000cm 2/Vs [49]. The extremely low mobility in organic materials contributes to the recombination of charge due to the long amount of time required for the charge carriers to travel to their respective electrodes. If the electron and holes could be guaranteed to never interact with each other, this would not be extremely detrimental. However, minimization of such interactions implies that the BHJ would have to be highly ordered. BHJs are generally solution-processed, leading to high degree of disorder in the material to maximize the surface area between the two materials, represented in Figure 1-4. In Figure 1-4a, a highly ordered BHJ material can be seen. This design takes 24 Exciton Separation ductio' -.. ncin Banj 1 .and -*-- -----.. rt OF =Ce alenceBn ..-------- Donor 9 Acceptor Figure 1-3: Schematic of the working of a BHJ material Anode Anode Cathode Cathode (a) Highly ordered BHJ schematic (b) Conventional BHJ schematic Figure 1-4: Schematic of types of BHJ materials. Left, low recombination. Right, high recombination 25 significant processing that leads to a lower amount of recombination. Conversely, Figure 1-4b shows how a real BHJ material would be made [46]. This design, while significantly easier to make, is more inducive to higher recombination. A material that could have the structure of Figure 1-4a while have the ease of processing as Figure 1-4b would combine the best of both material structures. Because of the intrinsic value of organic BHJ materials, a significant amount of work is being conducted to address the limiting factors preventing wide-scale acceptance of these materials. If these material issues can be addressed, cheap and environmentally friendly organic solar cells could become a reality. 1.3 Objective In this work, we will address the issues with traditional bulk heterojunction designs. These materials, because of their low cost of manufacture offer a unique opportunity for large scale solar capture the world over. By using first principle calculations, new materials will be explored, modified, and designed in such a way that the issues of recombination and low Voc can be alleviated or fixed. The remainder of this thesis is organized as follows: computational methods used in this work will be described from their mathematical foundations, the proposed material design that will address the issues with traditional BHJ will be described, the results of this study will be presented, concluded with discussion of the results as well as future design and modification of the proposed design. In this way, this work will provide a framework for highly efficient BHJ materials capable of producing cheap, and efficient, solar power. 26 Chapter 2 Theory and Computational Principles As we push our technology to higher and higher levels of sophistication, it becomes increasingly important to create materials that serve very specific applications, allowing novel technologies to exist in ways never thought possible. In this drive, experimental advancement have allowed for more accurate measurement of how materials behave in real applications. However, this drive has been matched by advancements in computing power, allowing theorist to contribute to the design and prediction of highly advanced materials. The realm of computational material design allows contribution to several material science areas including material property prediction, explanation of observed physical behavior, and effient prediction of new material properties. By accurately solving the many-body Schr6dinger equation, many material properties can be calculated accurately and quickly. 2.1 Computational Tools Computational modeling is a tool to understand how systems physically behave much like any number of physical tools are used to measure systems properties. Whereas in physical systems you have to balance cost of measurement with accuracy, in com27 Low cost/constituent, large length scales, long times Cn +-- ~Coarse- Continuum mdeling m lengt hw ies n luna Empiricalmehd potentials Tigh 0 Many ikexcoted Denitybinding functional theory Decreasing accuracy Figure 2-1: Relative computational cost of various methods. putational methods you have to balance the cost of computation with the accuracy of the modeling. In this way, there exists a hierarchy of the methods available. Some of these methods are shown in Figure 2-1. In this work, it is necessary to compute the atomic structure and electronic properties of materials. Numerous methods exist to model these properties, including molecular dynamics, density functional theory (DFT), Hartree Fock methods, and Beyond-DFT. Each of these has their own level of electronic accuracy possible to them. Material properties such as melting temperature, diffusion constants, and other molecular properties related to atomic motion can be modeled with MD methods. However, electronic properties such as band alignment and carrier transport require explicit quantum mechanical treatment of the electrons. As such, DFT and more advanced methods are required to model the electronic interactions. As computational resources have grown exponentially in recent years, methods for carrying out the solution to the exasperative Shr6dinger equation have been developed. One of the primary tools that enables solution of this equation is DFT. Al- though exact in principle, in practice DFT relies on several important approximations, which will be described in Section 2.2. More accurate methods have been developed 28 such as Beyond-DFT methods that do not rely on these assumptions. However, these methods are extremely computationally expensive, and would be near impossible to use on the systems considered in this work. Considering the time fram available to this work, traditional DFT has been employed to compute and measure the electronic properties of new BHJ designs. DFT can accurately model many different system properties. By solving for the electron density of the system, many system properties can be calculated. These properties include ground-state atomic structure, band-strcuture, local and atomresolved density of states, and other properties necessary to analyze a PV material. 2.2 Density Functional Theory In general, the Schr6dinger eduation is a relationship describing the total energy of a quantum system. Originally pushed in his seminal paper "Quantisierung als Eigenwertproblem" [47], this equation has opened the realm of describing the subatomic meaning of the universe. In its most general form, the time-independent Schr6dinger equation takes the form of Equation 2.1. This equations allows relates the total energy operator to the total energy of the system. The total energy operator, called the Hamiltonian, can be formed to take the physics of the system into account. ET = HTW (2.1) Equation 2.1, however, is extremely trying to solve in a real system of greater complexity than the hydrogen atom. This is due to the fact that this equation involves 3N variables, the x-, y-, and z- components of the N electrons. This makes solving large system very intractable as N grows; DFT allows this system to be solved. DFT is based on the theory presented by Pierre Hohenberg and Walter Kohn in 1964 [19]. This theory provides that all electronic properties can be solved in terms of the charge density, p, alone. This turns a 3N problem into an Nth order computation. The largest power is that this provides a methodology to write the total energy in the following functional form: E0 = Eo[po]. Further, this threorem allowed for all other 29 material properties to be written as a function of the charge density alone, provided the functional relationship is known. Hohenberg and Kohn developed their equation in terms of an external potential V(r). By rewriting Equation 2.1 in dirac notation we can write it in the following form H IIF) = E I'). In the ground state, the ground-state energy can be calculated with the ground state wavefunction Eo = (TI H IT). By the variational principle, Kohn and Hohenberg were able to prove that for any external potential, all the ground state electronic properties were contained within the electron density. However, this raised a new set of problems as finding the ground state energy from the charge density is still not a trivial problem. Hodenberg and Kohn were able to show that for every trial charge density, ptrial (r) ;> 0 gives f Ptrial(r)dr= N, the number of electrons, and Eo <; Ev[ptria] where Ptrial is the ground state charge density for some other potential with wavefunction <trial. Kohn and Sham where then able to expand this into the method of Density Functional Theory (DFT) for finding po and for finding EO = Eo[po]. Kohn and Sham were able to fully expand this idea by writing expanding Equation 2.1 in DFT space with the foreknowledge of E = E[p], yielding [25]. E = I 2 2 T,[p] + Vext(r)p(r)dr + e - p(r)p(r') , drdr' + Exc[p], |r - r| (2.2) where T,[p] is the kinetic energy of a non-interacting, inhomogeneous electron gas, Vext(r)p(r)dr is the energy imparted by the external potential, the third term is the intergal form of the Hartee energy, and Exc[p] is the exchange-coorelation energy. The first three can be exactly solved if the charge density, p is known, where the Exchange-coorelation term is not exactly known, rather approximations have to be used to calculate it. Each of these terms were exactly solved by Kohn and Sham in terms of functionals with the charge density. For the kinetic energy, T,[p] that p(r) = E> #(r)#i (r). Where the #i = - J >j f q V20,(r)dr, such are the one-electron Kohn-Sham obritals. With these equations, it is possible to solve the energy of the whole system. Since 30 these are of little physical significance, solving can be instead be done by varying q(r) as shown in Equation 2.3. { 2V2 + Vext(r) + VH (r) + Vxc(r) 2mI Oi(r) = E6ji(r), where the only unknown is the then the exchange-coorelation. (2.3) This comprises four components: the kinetic correlation energy, the exchange energy, the correlation energy, and the self-interaction correction. 2.3 Functionals Since the expression for the exchange-correlation is not known, there are a variety of approximations that can be used for the exchange-correlation functional. These include the Local Density Approximation (LDA), the local spin-density approximation (LSDA), and the generalized gradient approximation (GGA). The most simple approximation to account for the exchange-coorelation is the LDA approximation. This approximation was developed by Kohn and Sham that follows the functional form of Equation 2.4, below. This functional was originally developed by Kohn and Sham in 1965 [25] and yields very good results despite its simplicity. Because of this simplicity, and low cost of computation, it is one of the most commonly used functionals to approximate the exchange-correlation. E CA j p(r)Exc[p(r)]dr (2.4) In this equation, EcX is the exchange correlation energy of a homogeneous electron gas with density p. As a simple way of explaining this, this exchange correlation is generated from the electron density at a given point, as if that point was surrounded by the same density electron gas. From this, it is a purely local approximation. Despite the simplicity of the approximation, this is one of the most widely used coorelations in part because it maintains the correct sum rule for the exchange-correlation hole. [22] In this work, however, the PBE functional has been used. This functional is in 31 the family of GGA functionals [31,32,39,42], and does not rely on empirical fitting. This was developed to enhance the issues associated with the LDA approximation when systems have significant variances in the density. In this way, PBE corrects LDA by including information about both the density at a position as well as the gradient of the density at a position. PBE was developed by John Perdew, Kieron Burke, and Matthias Ernzerhof in the late 1990s [40]. The functional form of PBE is shown below in Equation 2.5. Ex [P., Pd..n]= f (Pup(r),Pdown(r), I Vpup(r) 1, 1 pVdmn(r)|)dr (2.5) This functional was chosen for the exchange-coorelation because it is more apt at modeling the local charge variation in systems that have large changes in their charge density. In the systems that will be modeled, both organic and inorganic systems are combined, leading to significant charge fluctuations and localizations. It is computationally more expensive to model, however the gain in accuracy is necessary. Also important to accurate modeling in this work is the forces due to the van der Waals interactions. These interactions include forces not due to covalent bonds between materials or from hydrogen bonds, but are due to charge fluctuations and dipoles. In this work, the so-called London Force has been used that accounts for forces between induced dipoles. This allows forces from charge variation to be accounted for. By using the method developed by Stefan Grimme [16], the energy including the London Forces is now EDFT-D = EKS-DFT (2.6) + Eisp where this is now the existing Kohn-Sham energy combined with the dispersion energy given by Nat- Nat Edip = -86 i=1 j~i+1 6' ffdmp(Rij), 23 where Nat is the number of atoms in the system, C between atom i and j, 86 (2.7) is the dispersion coefficient is a global scaling factor that is dependent on the density 32 functional that is used, and Rj3 is the interatomic distance. fdmp is a dampening function used to avoid near-singularities given by 1 famp(Rij) = 1 + edU4/1). (2.8) In this equation R, is the sum of atomic vdW radii, derived from the radius of the O.O1a- 3 electron density contour from ROHF/TZV computations of the atoms in the ground state carried out by Grimme [15]. By including these interactions, a more accurate representation of the systems can be included in all modeling. 2.4 Pseudopotential Approximation The other main approximation used in DFT is the pseudopotential approximation, which relies on the idea that the core electrons do not significantly contribute to the system away from the nucleus. The reason for this approximation is that these electrons require many plane-wave to be represented accurately because of the sharp nature of the wavefunction within the core. To this end, a so-called pseudopotential is used so that the core electrons can be smoothed out, while the valence electrons are treated explicitly. This significantly decreases the computational cost with minimal loss of physical meaning. These pseudopoentials were developed and tested to accurately model a variety of systems. By testing these pseudopotentials in a wide variety of chemical environments, they can be shown to accurately describe the physics of the systems that they model. To develop a pseudopotential, forst and all-electron calculation is performed to get the all electron potential, VAE, as well as the all-electron wavefunction, 0$+ iAEI and the eigenvalues, O4 E. Further, the construction of this pseudopotential is subject to the following constraints: 33 f 1. EA =EAE 2. Of? 4 3. *>S P/(r)4ps(r)dr 4. J r > rc, where r, is the chosen cutoff radius = (r) IAE(r)dr ~PAE '90" r=rc "Ops -arm = r Ir=rc The above constraints imply that the pseudopotential must be made for each orbital. By using this information, a pseudopotential can be constructured for each element in the periodic table. To this end, many pseudopotentials have been developed and tested. This work will not develop unique pseudopoentials, rather developed pseudopotentials were tested to verify their accuracy, as described in Section 4.2. 2.5 Electronic Properties from p[r] The extent of this work is to examine the electronic properties and how they relate to the operation of a solar-cell material. To this end, once the ground-state system has been determined, several electronic properties are measured including the charge density, the density of states (DOS), and the electrostatic potential. All of these properties can be directly calculated from the DFT electron density. These properties directly indicate how the system will behave and act on a macroscopic scale; they are very important to calculate as accurately as possible soas to ensure accurate prediction of material properties. While these properties are generally built-in functions of the DFT code used, understanding how they relate to the charge density is important to undertand the base-line predictions that the code outputs. In this section, the mathematical relations that dictate the various properties are outlined. The number of states at a given energy level is indicative of several material properties including band-gap and mobility. The local density of state (LDOS) is a measure of the total number of states at a given energy level. This is defined in Equation 2.9. 34 N n(r, E) 1i(r)26(E - ci) = In the above euqation, Oi(r) are the Kohn-Sham orbitals. (2.9) To obtain the total density of states (DOS), integration over r is required as shown in Equation 2.10. n(c) n(r, c)dr = (2.10) Further, the DOS can be decomposed into the atomic orbitals, particularly useful to determine the chemical nature of bonds or the composition of band-edges. This is shown in Equation 2.11. fla(E) I (OPil = kta) 126(E _ 60) (2.11) Where a are the 1 and/or m quantum numbers. By integrating the LDOS over all c, the total charge density can be determined as in Equation 2.12, shown below. This can also be spatially resolved to show how the charge density varies throughout the system. p(r) n(r, c)dc = (2.12) The electrostatic potential will also be of use in calculating several material properties of the material including the induced electric field as well as isolating any effects that surface states may have. This is shown in Equation 2.13. V 2 (r) = 47rp(r) (2.13) All of the above equations will be exploited to determine the accurate electronic properties of all materials explored in this work. 35 2.6 DFT Bandgap Issue With all of the power of DFT, one issue that plagues it is the so-called "DFT band gap issue." This issue arises from the fact that DFT underestimates the bandgaps of a material by up to 50%. This has wide and reaching implications, especially with regard to lowband gap systems, where underestimating the bandgap can actually cause the system to become metallic. It can be shown [38], that for a N-electron system in an external potential, there arises a funamental bandgap given by (2.14) Egapud = I - A, where I is the ionization potential and A is the electron affinity given by I = Ev(N 1) - - E(N) (2.15) A = Ev(N ) - Ev(N - 1), which can also be written as a derivate for N electrons in the following form Egap deri = aN N6 9 ON N- 6 1 (2.16) In the case where the exchange coorelation is an explicit form of the charge density such as LDA or GGA, the above can be rewritten as deriv =KS gap gap KS = 6 LUMO KS ~ (2.17) EHOMO- where these values come from the KS equations. Cohen and Wang were able to show that DFT predicts too high of EHOMO and too low ELUMO. This is due to delocalization error associated with the exchange correlation that gives rise to a overal convex energy curve [38]. Even with this error in computing the bandgap, the generally accepted view is that trends in the bandgaps remain the same. In this way, even if the exact value 36 of a bandgap is not strictly right, any modification of the system will have the same relative error, if the same pseudopotentials and exchange correlation are used. To this end, several methods for fixing the bandgap have been proposed including the "scissor operator" in which the overall conduction band is shifted up by the delta-energy that is required to have the proper bandgap [11, p. 196]. This, in general, requires an exact knowledge of the experimental bandgap of the system, something that is not always known. This implies that the overall electronic character of the system will remain consistant, just shifted by an error that will remain consistant across similar systems. In this way, trends, rather than absolute bandgaps are studied in this work. There exists options to more precisely calculate the bandgap in systems, however these rely on an advanced type of DFT termed Time-Dependent DFT (TDDFT). One method is by using the GW approximation (GWA) that is an approximation that allows for an alternative way to calculate the self-energy of a many-body system of electrons. Lars Hedin proposed this method in his seminal paper, relies on approximation that the expansion of the self-energy into the single-particle Green's function (G) and the screened Coulomb interaction (W) [17]. In this way, Hartree Fock can be expanded to accurately describe the screening nature of the bandstructure. This approximation is very computationally expensive, as such, it was used sparingly in this work. However, this approximation has been shown to accurately calculate the bandgap of many materials [50]. In this way, the scissor operator can be defined formally as AEsciSor = EGap- ESGT. (2.18) This formalism allows for any interacting system to have the real bandgap be estimated as Eap = EDT + AESCissor. (2.19) This will become important in calculating the overall bandgap of the systems used throughout this work. 37 2.7 DFT Code In the preceding sections, the overall DFT theory was established from a mathematical approach. These principles are employed into several prepackaged codes that are optimized for highly parallel environments, allowing for computations of significant system size. Without the work of these groups, this work would not be possible within the time frame of a Master's Degree. The code used through out this work is an open source DFT code called Quantum Espresso, and formerly referred to as PWscf [14]. This code was chosen because of the "over-the-counter" nature of the open-source movement. Additionally, because of this open source nature, a wide variety of support is available as well as the ability to directly access the source for any modifications that may be necessary. 2.8 DFT Scaling Despite the power that DFT offers, it has one main limitation, the scalability of the system. As computational resources have grown, larger systems have been able to be modeled, however DFT codes do not scale linearly with the number of atoms in the system or the number of processors used. In this way, there are still systems that are not able to be calculated in a reasonable amount of time. One way of quantifying this is by looking how the code scales with the number of processors used for a fixed number of atoms. On the computational resources that were used in this work, the speed up with the number of processors used was calculated. This is shown in Figure 2-2. In this scalability study, a vanadium oxide supercell was used with 112 atoms represented. As it can be seen from this graph, even increasing the total number processors by 64 times only increases a speedup of approximately five times in total time. Similarly, the code scales poorly with the total number of atoms present. In this work, systems have atomic sizes of 32-300 atoms, all with varying chemical complexity. Scalability was a huge concern and some more advanced systems could not be exactly 38 6 II I I I I I I I I I I I I I I I I I I I I I I I I I I I I II 5- 2-0 20 40 60 80 100 120 Number of Processors Figure 2-2: Speedup associated with number of processors calculated because of the scaling issues associated with DFT. The computational limitations of DFT have put limits on the size of systems that can be modeled in this work; modifications to systems were performed to allow for advancement in the modeling and understanding of complex BHJ devices. In all cases the chemical character of the systems was preserved. 2.9 Conclusion Density functional theory is a powerful tool that affords several ways in which to calculate the properties of complex materials. By exploting the work of Hohenberg, Kohn, Sham, and others, advanced materials can be designed and qualified in an efficient computational environment. This work will exploit this theory extensively to determine the material properties of complex hybrid solar accepting materials. 39 40 Chapter 3 Rational Design of Nano-structured Hybrid Photovoltaics The main objective of this work is to employ first principle computational tools to rationally design a novel class of nano-structured hybrid materials that overcome the issues inherent to traditional bulk heterojunction photovoltaic materials. This chapter will outline the the general approach, focusing on engineering materials to enhance exciton separation and decrease recombination in a hybrid organic-inorganic nanostructured photovoltaic material. By designing to address these issue, an efficient BHJ hybrid material can be realized. 3.1 Overall Design Addressing the issues with traditional BHJ materials requires a layered structure; by alternating layers of organic and inorganic regions, a piecewise material can be designed in which photons are absorbed in the organic material, generating excitons which are separated into electrons and holes by an intrinsic dipole field. The electrons and holes are driven by an intrinsic electric field to the inorganic layers at lower and higher potentials, respectively, as illustrated schematically in Figure 3-1. 41 3A Anode S18.5 A Figure 3-1: Schematic representing the proposed design. Alternating layers of organic and inorganic layers allow for efficient charge extraction The design shown in the figure has the ability to effectively separate bound excitons. In the organic regions, a local molecular dipole is engineered to provide sufficient energetic potential to separate excitons into free holes and electrons and drive the free charge carriers away from each other, decreasing the overall probability of recombination. Once the free carriers reach the high-mobility inorganic regions, they are rapidly advected out of the material to the electrodes. As Figure 3-1 shows, the dipole changes direction in alternating organic regions suck that the separated electrons and holes are at different energy levels; thus, each inorganic region contains carriers of a single type. Furthermore, once carriers are in an inorganic region, there is no thermodynamic driving force for recombination with carriers of opposite charge in adjacent inorganic layers. Together, these properties dramatically decrease recombination. It is also worth noting that the alternating dipoles prevent any macroscopic polarization, or electric field, from being generated. In an ideal system, such as that shown schematically in Figure 3-2, there will be no energetic losses at the organic-inorganic interfaces. Bound electron-hole pairs would be generated in the organic region, then separated by the intrinsic local dipole into 42 Figure 3-2: Schematic showing ideal band alignment in bulk structure free carriers, which would be effectively removed into the inorganic regions before they could recombine. Thus, this design would increase efficiency in two ways: First, due to the presence of the local dipole in the organic material and the nanoscale distance between inorganic regions, charge carriers will only remain in the organic region for a short period before being driven into high-mobility inorganic layers and transported to the electrodes, thereby minimizing recobination. Second, since no energetic losses are required to separate excitons, the Voc can be maximized to obtain optimal power output per absorbed photon. 3.2 Materials The nature of the design proposed in Section 3.1 requires two different materials to be joined together: an organic semiconductor and a layered inorganic material. In this section, we discuss the types of materials we have explored for each component in the system. Of utmost interest are polar organic materials and inorganic materials that readily form layered structures with organic materials and also posses a higher mobility than typical organic semiconductor. 3.2.1 Organic Semiconductor The organic semiconductors used throughout this study are derived from structures that are experimentally used and studied. These materials are traditionally molecules 43 from a specific subset of BHJ materials called dye sensitized solar cells (DSSC). These materials normally are liquid-suspended, with the other material used in the heterojunction being liquid in nature, where a redox reaction sustains the charge in the system. Because these materials are extensively studied, they were chosen as a baseline type of organic semiconductor to examine. In this way, we are able to readily compare our computational results to experimentally available data. A group of these DSSC semiconductors that is widely studied is based on polythiophene. Thiophene is a sulfer-based molecules that readily polymerizes; the degree of polymerization create the ability to tune the energetic properties of the BHJ material. In addition, thiophene can be functionalized with other orgnaic molecules to tune the bandgap of the semiconductor. As a result, several experimentally available compounds are created with thiophene acting as the charge generator. An exam- ple that was explored heavily in this work is 2-Cyano-3-[5-(9-ethyl-9H-carbazol-3-yl)3,3,3,4-tetra-n-hexyl-[2,2,5,2,5,2]-quater thiophen-5-yl] acrylic, commonly referred to as MK2. This was chosen because the organic structure is both linear, compact, and polar in nature. MK2 is one of several organic molecules based on polythiophene. This includes MK1-MK6, each manufactured in such a way as to have better absorptance cooeffiencients, or to have an optimized bandgap. Organic MK2 is shown in Figure 3-3. From experiment, MK2 has a bandgap of 1.85eV, a peak absorptance at 480nm, and a incident photon conversion effiency of 5.01% while bound to titanium dioxide [51]. From the Shottkey-Quassier limit, MK2 itself is predicted to have a max efficiency of 22.56%, relying on the assumption tha there is no recombination; the low efficiency in the experimental results is likely due to the inherent non-directionality of the TiO 2 - MK2 heterojunction design, as well as the relatively small amount of MK2 covering the surface. By combining this material with a high-mobility inorganic material in a directional fashion as the proposed design dictates, overall recombination should be limited. In addition to organic thiophene, several other semiconductors were mixed with thiophene to change the optoelectical properties of a single thiophene molecule. They 44 \ CN HOOC N Figure 3-3: The MK2 molecule (2-Cyano-3-[5-(9-ethyl-9H-carbazol-3-yl)-3,3,3,4-tetran-hexyl-[2,2,5,2,5,2]-quater thiophen-5-yl] acrylic) S NNN N I Thiophene N N 0 N NN 5-methyl-4H-thieno[3,4-cpyrrole-4,6(5)-dione benzo(c(1,2,51oxadlazole [1.2.5]oxadiazolo-(3,4-cpyridine Figure 3-4: Different types of organic semiconductors studied were also chosen to tune the energy levels within the inorganic layers. These include BX (benzo[c][1,2,5]oxadiazole), PX ([1,2,5]oxadiazolo-[3,4-c]pyridine), and TPD (5-methyl-4H-thieno[3,4-c]pyrrole-4,6(5H)-dione), which are shown in Figure 3-4. Specifically, each of these different types of materials are chosen because of their bandgaps, and their electron affinity when bound to thiophene. The electron affinity is an indication of the electron-acceptor strength of the material, or how easily it it can transfer electrons out of the material. Previous work has used TDDFT to evaluate the total band energies of these molecules bound between a thiophene group in the following structure: thiophene-A-thiophene, where A is either BX, PX, TPD or thiophene [30]. The results of this study are shown in Table 3.1. 45 Table 3.1: Thiophene property modifications [30] Molecule Tri-Thiophene Thiophene-TPD-Thiophene Thiophene-BX-Thiophene Thiophene-PX-Thiophene Bandgap 3.27 3.15 2.56 2.44 Electron Affinity -1.69 -2.19 -2.7 -3.01 Table 3.2: Thiophene bandgap and chain length [5] 00 Computational Gap [eV] 5.51 3.65 2.93 1.71 Experimental Gap [eV] 5.37 4.12 3.52 - Thiophene Chain Length 1 2 3 The large bandgap of the tri-thiophene is to be expected. Both experimentally and computationlly, the bandgap of polythiophene depends on the number of thiophene units, decreasing significantly from over 5eV for a thiophene molecule to less than 2eV for above six linked thiophene units. In another computational work that accurately calculated the band gap of thiophene chains [5], the trend in bandgap decrease can be observed, shown in Table 3.2. In this way, the bandgap of these materials can be controlled by changing the total number of thiophene repeats present in the organic photovoltaic. This will become important in Chapter 4 when this nature will be exploited to make this work more computationally streamlined. 3.2.2 Inorganic Phosphate Group Transitional metal phosphates are a group of materials that readily form bulk layered strucutres. These structures form in a number of different chemical compositions with different phosphate moieties, providing a range of electronic and chemical properties. Similar to the more familiar graphene and transition metal dichalocogenides that are currently of interest for numerous applications, these phosphates are composed of stacks of weakly interacting two-dimensional sheets held together mainly via van 46 der Waals interactions; a mixture of strong covalent/ionic bonding occurs within the plane of each 2D sheet. Even more interestingly, it has been experimentally demonstrated that these materials can be chemically modified to insert ordered arrays of covalently boud organic molecules between the 2D sheets. These properties make the 2D transition metal phosphate materials extremely useful to rectify several of the issues of traditional BHJ materials. These materials have the following two different categories of chemical formulation: M(HPO 3 ) 2 as well as XH 2 P 3 0 10 where M can be materials in +4 such as zirconium, hafnium, and titanium while X is materials in +3 such as Al, Ga, Fe, Mn, V, and Cr [44]. All of these structures have been experimentally explored for various electronic properties. A wide variety of materials is beneficial to qualify if there will be a material that will display the properties desired: high mobility and proper band alignment with organic material. The primary inorganic material studied in this work is Zr(HPO 3 )2 . This material has been studied primarily for the ionic exchange properties that it posses [8]. It has been used in a variety of fields from nanocomposites to nuclear applications, yet its electrical and optical properties have not been well-studied, likely because of its large bandgap. Originally chosen because of the experimental evidence that it forms layered structures with organic material, we wil show that Zr(HPO 3 ) 2 also has excellent optoelectronic properties necessary for the proposed novel BHJ material design. This structure forms both a and y phases, and has been experimentally shown to' form both in Zr(HPO 3 ) 2 and Zr(HPO 4 ) 2 structures [1,36]. For both of these studies, the unit cell parameter did not vary significantly as the additional oxygen was terminated normal to the layers. The simpler Zr(HPO 3 ) 2 structure was used throughout this work, shown in Figure 3-5. This material also forms layers with organic materials [1, 9], such as the structure shown in Figure 3-6. This was explored experimentally in order to form structures with controlled porosity for ionic transport applications, but few experiments have been performed to explore its electronic or transport properties. 47 oH p 0 @0 Zr OZr Figure 3-5: Structure of a-Zr(HPO 3 )2 op x@0 L e r Figure 3-6: Structure of Zr(HP03)2 layered with benzene rings 48 The nature of forming layered structures with organic materials means that the main issue will be determining if this material will behave in such a manner as to increase the optoelectronic properties of the organic MK2 molecule. This will be discussed in Chapter 4. 3.3 Exciton Separation The idea of an exciton was originally proposed by Frenkel in his governing paper that described the interaction of absorption of light into heat in crystals [13]. In this, he described how light can generate a bound hole and electron pair caused by the interaction of light with the material. The generated charge would leave behind in its space a positive charge normally called a hole. These two particles would form what is classically called an exciton. The interaction between these two is given by the coulomb potential energy e2 Ec = Er , (3.1) where e is the relative permittivity (dielectric constant) in the material, e is the fundamental charge of an electron, and r is the distance between the charges. In this way, both the size of the exciton as well as the material dielectric constant affects the exciton binding energy. These exictons are classically termed Frenkel Excitons [33]. Other types of excitons exist such as Wannier-Mott excitons, however, Frenkel excitons will be considered primarily in this work because organic materials generally have low dielectric constants. For example, benzene (C6 H6 ) has a dielectric constant of ~3 [37] whereas silicon has a relative permittivity of 11.4 (ab initio) or 12.7 (exp) [3]. Further types of excitons can be observed, especially when there are systems with free charge, that can screen the Coulomb potential, yielding Escreened = -- e 2r exp ( (3.2) where A is the screening length. Organic materials do not have a significant amount of free charge to effectively screen the charge. In this way, in typical organic materials, 49 the binding energy of the exciton can range between 0.1-1.5eV. As discussed in Chapter 1, exciton separation in traditional BHJ materials is accomplished via conduction band-edge alignment; two materials are joined together because their conduction bands have an energetic difference AECB greater than the energetic binding energy of the exiton. However, exitonic binding energies are normally large, requiring a large AECB that decreases the Voc by the same amount. Inaccurate control of the band alignment leads to low open circuit voltages or low probability of exciton separation because of processing requirements, material properties, and the dual nature of a traditional BHJ design. Additionally, this method relies on electron/hole diffusion with no built-in potential driving the separation; the lack of directionality increases the probability of free carriers recombining, rather than leave the organic region at the electrodes. In contrast to traditional BHJ operation, this work proposes a design in which a local dipole is responsible for the separation of the exciton. With such a design, it is possible to addresses the two afformentioned issues with BHJ materials: First the dipole allows for a maximized Voc, as the excitonic sepatation is not due to an energy loss, but rather to an electric field. Due to the dipole-induced field in the organic region, this design allows for the exciton separation without the need of the Voc to decrease to achieve this separation. Second, the dipole can provide a driving force to separate the charge carriers, decreasing the chances of recombination in the material before extraction. Referring to Figure 1-3, the lowing of the Voc can be observed in the material. Rather, in Figure 3-7, if a molecule has an inheirant material dipole, a directionality can be given to the exiton. In Figure 3-8, the exitonic separation in the proposed system is shown; a dipole in the organic region, as indicated in the change in the vacuum level (Xorganic), and the energetic alignments of the organic and inorganic region are only necessary to extract the exciton from the organic region. In this way, a near perfect band alignment between can be realized as very small energetic differences are necessary for the transfer of charge between the organic and inorganic region. This idea of exitonic separation relies on an intrinsic dipole that can generate an 50 N Dipole of Molecule Figure 3-7: Schematic of dipole separation * 4 MU Inorganic XOrganic -Inorganic EC I ~m. NEMENE :ONEENNE U Ev 0000 1 i Figure 3-8: Exciton separation in an ordered nano-structured material with an induced dipole 51 electrostatic potential greater than that of the exciton binding energy. Further, in order for this to effectively separate the exciton with little recombination, an ordered structure is required; a dipole alone cannot create the separation of the exiton, a layered structure is required. 3.4 Mobility The second major bottleneck in tradition BHJs is the mobility issue. The mobility of a material is a measure of the speed at which an electron, or hole, can move through a material in the presence of an electric field. This can be written as Vd = where Vd [cm 2 /s] is the drift velocity, /. (3.3) pLE, [cm 2 /(V . s)] is the mobility, and E [V/cm] In this way, a material with a high mobility will be able to is the electric field. move charge faster with the same applied electric field. This is of interest because when the material, such as organic materials, has a high exciton binding energy, the longer it is in the material, the higher the chances are for recombination. As an example, the mobility in [6-6]-phenyl C71 butyric acid methyl ester (PC6oBM) has a mobility of 1.5 x 10- 2 cm2 V-IS-1 [24] where traditional silicon has a mobility of 102 - 10 cm 2 /(V . s) [49]. This very small mobility in the organic material, high recombination of charge can be observed, and low photon conversion efficiency (PCE) can be observed. Calculation of the mobility will be described in Section 6.1. 3.5 Open Circuit Voltage Besides recombination, the open circuit voltage Voc in BHJ materials can also be observed to be quite low compared to traditional inorganic solar cells. 52 The open circuit voltage is determined in the following way Voc = AkbT 1n e JC+1), Js (3.4) where Jsc is the current generated in the short circuit, Js is the saturation current (reverse bias), and A is the diode quality factor for a p-n junction [23]. While this was originally developed for p-n junction solar cells, it has been used to describe the nature of BHJ organic materials. In an idealized state, the VOC is the total bandgap of the material. While this cannot be realized in a true system because of atomic defects in a real crystal. The total bandgap of a material is used as a rubric for judging the overall material. BHJ devices rely on decreasing the overall bandgap of the system to separate the exciton, shown schematically in Figure 3-9, decreasing the overall Voc. However, the bandgap that interacts with the solar spectrum is still the large bandgap of the accepting material in the BHJ. In this way, both the effeciency of the device is decreased by the nature absorption nature of the optical bandgap (described by the Shottkey-Quisser limit) as well as taking a decrease in the operational effeciency described by FF x Vc x Isc ExA= E x A. (3.5) This implies that the effiency of the overall material can also be increased simply by increasing the Voc. BHJ materials do not exploit this effectively because of the requirement of separating the exiton via band energy differences. 3.6 Conclusion In this chapter the existing issues with traditional BHJ designs was discussed and a new design was proposed. By combining a layered structure of organic and inorganic materials it may be possible to create a piece-wise system that overcomes the issues with traditional BHJ designs. This design may also make using existing DSSC materials possible because of the inherently directional nature to the 2D structure. 53 4 Egap-1 -Photon r j Egap-2 -I Figure 3-9: VOC in BHJ materials. In the following chapters, the system is quantified from a calculation standpoint to determine the viability of such a design as well as any improvements that can be made to this system. 54 Chapter 4 Atomic and Electronic Structure In previous chapters, the framework for the ideal BHJ material design has been proposed. The organic DSSE material was explained as well as a proposed framework for the using Zr(HPO 3 ) 2 as an inorganic inter-linker to convect charge away. In this section, the results of studying this bulk structure will be discussed. This will provide a framework for discussing how to effectively modify the structure in ways that further optimize the optoelectronic properties of the bulk structure. 4.1 Computational Details In DFT, like most computational paradigms, certain computational values must be converged. In the case of DFT, the plane-wave basis set must be fully converged in order to accurately compute the electronic properties of the system. In order to properly converge this basis set, two key values must be converged, firstly the energy cutoff (Ece,) of the plane waves, and secondly the k-point mesh. This energetic convergence will be described in this section. The convergence test of the energies and k-points were done on the smallest system in this study, represented by the bulk structure of Zr(HPO 3 ) 2 . The experimental unit-cell parameters of this system are shown in Table 4.1. This system was chosen precisely because smaller unit cells have the most strict convergence criteria principally because these plane-waves are evaluated in k-space, because of the Fourier 55 Table 4.1: Zr(HPO 3 ) 2 experimental unit cell parameters Value Parameter a b c a 5.42000 A 5.42000 A 5.57950 A 900 900 1200 nature of the calculations. The reciprocal space system is defined as a= bxc 27r a.(bxc) b* = 2 cxa a.(bxc) (4.1) axb C* = 27r a.(bxc) where a, b, c are the prinicple lattice vectors, and a*, b*, c* are the reciprocal lattice vectors. It is possible to see that the smaller the overall volume of the real-space unit cell, the larger the reciprocal lattice will be. Organic molecules generally have very large unit cells and superstructures that will be studied also have very large lattice constants. Bulk Zr(HPO 3 ) 2 has the smallest overall volume; any system converged in this structure will be converged in larger unit cells. First the cutoff energy was converged to within 20 meV. Increments of 5Ry (~68.028eV) were used in calculating the convergence values. The convergence criteria is a reverse looking criteria in that AEc,,, = E, - E,_ 1 < 20meV. In a similar fashion, the k-point mesh was also converged. This mesh is defined as Na, Nb, N, where N is the number of k-points along the a*, b*, and c* lattice vectors. This was converged after the total energy of the structure was converged and was also found to be converged in a reverse looking criteria within 20meV. This is plotted in 4-1. From this criteria, it was found to be converged at a 3,3,3 kpoint mesh. To this end, all other structures were ran at minimum energy cutoff of 50Ry = 680.28eV, and a maximum k-point mesh of 3, 3, 3. As unit cell lengths grew in real 56 -4484.4 I I I I I I -4484.6- & -4484.8 -4485S 0 1 I I I I I 2 3 4 5 6 7 I I 8 9 10 K-point Mesh (Na ,Nb ,Nc) Figure 4-1: K-point convergence of Bulk Zr(HPO 3 ) 2 space, the k-point mesh along that direction could be relaxed, however the dense k-point mesh was used throughout this work. 4.2 Pseudopotential Testing Quantum Espresso comes with a variety of prepackaged pseudopotentials as well as a plefora of other pseudopotentials developed by other research groups. In general, the pseudopotentials that are employed by the Quantum Espresso library are as simple as need be, without the addition of more complex approximations that hybrid functionals use. The pseudopotentials used throughout this work are shown in Table 4.2. All pseudopotentials were used from the Quantum Espresso pseudopoential library, generated from PBE exchange-correlation [40,41,43]. A general pseudopotential without these added complexities was desired to eliminate the addition of artifacts from the hybrid nature of other pseudopotentials. Where possible, the pseudopoetentials were tested to guarantee that they returned sensical answers. For the standard temperature and pressure (STP) gasses (02, N 2 , H 2 ), the pseudopotentials were tested to guarantee that they calculated the correct bond length of 57 Table 4.2: Pseudopoentials used in this study Species C 0 N H S P Zr Ti F B Pseudopotential Config 2s2 2p2 3d-2 2s2 2p4 2s2 2p3 1si 3s 3p 3s 3p 4s 4p 4d 5s 5p 3s 3p 4s 3d 2s 2p 2s 2p 3d Table 4.3: Bond distance comparison for STP gasses Bond Gas 02 O O N2 H2 N H N H Experimtal Distance [A] Computational Distance [A] 1.2075 1.0977 0.7414 1.2374 1.1109 0.7521 Error [%] 2.48 1.20 1.44 the material. These pseudopotentials have been extensively used and tested in other works, so in this way, this was the primary test used to test these pseudopotentials. Oxygen, Nitrogen, and Hydrogen gas experimentally form bond length structures of 1.2075A, 1.0977A, and 0.7414A, respectively [20]. The computational results of isolated molecules are shown below in Table 4.3 and shows superb agreement with experiment. These pseudopotentials will also be compared to how they interact with carbon to further verify their validity. Carbon and sulfer form a wide variety of materials in nature, so to quantify these pseudopotentials, the carbon-carbon distance in benzene and the carbon-sulfur distance in thiophene was used as a benchmark for the pseudopotential's viability. In benzene, the carbon-carbon distance is 1.3970A and in thiophene, the carbon-sulfur distance is 1.714A; additionally, the carbon-hydrogen distance is 1.079A [18]. Further, in pyridine, the experimental N-C distance was found to be 1.340A. Once again, the computational results can be found in Table 4.4; near perfect bonding distances can 58 Table 4.4: Bond distances for carbon, sulfur, hydrogen, and nitrogen pseudopotentials Bond C S C N C C H C Experimtal Distance [A] 1.397 1.714 1.079 1.34 Computational Distance [A] 1.393 1.72 1.089 1.338 Error [%] 0.29 0.35 0.93 0.15 Table 4.5: Bulk zirconium lattice comparison Experimental Computational % error a [A] c[A] a/c 3.225 3.230 0.169% 5.134 5.17 0.695% 1.592 1.60 0.521% be observed. Bulk zirconium forms a hexagonal close-pack structure with lattice constants of a = 3.225A, c = 5.134A, and c/a = 1.592 [12]. The comparison of this with compu- tational detail is shown in Table 4.5. It can be seen to have great agreeance with the experimental values found in literature. The pseudopotentials were also tested against a common material in zirconium dioxide (ZrO 2 ). This structure forms a monoclinic unit cell with paramaters shown in Table 4.6 [52] along with the computational error of this unit cell. The total error was found to be very small with the used pseudopotentials. As an additional test of the pseudopotential accuracy, the enthalpy of formation was calculated for ZrO 2 . The enthalpy of formation is simply the energy required to Table 4.6: ZrO 2 unit cell computational error Lattice Parameter a b c a Experimental 5.1501A 5.2077A 5.3171A 90.000 0 7 99.2240 90.000 59 Computational 5.1200A 5.216A 5.281A 90.000 99.010 90.000 Error [%] 0.58 0.16 0.68 0.00 0.22 0.00 Table 4.7: Lattice constant error of Zr(HPO 3 ) 2 Lattice Parameter a b c Experimental Value 5.418A 5.418A 5.579A 900 Computational Value 5.517A 5.517A 5.660A 900 0 900 1200 900 1200 -y Error [%] 1.83 1.83 1.45 0.00 0.00 0.00 form 1 mole of a compound from the constitutive elements defined as AH; = Z(vH; )roducts - (4.2) Z(vH)reactants, where v is the number of moles for each element in the reaction and HY is the enthalpy of elements. In this way, the formation of ZrO 2 is given by + 0 2 -+ ZrO (4.3) 2 . Zr From a computational standpoint, the enthalpies are taken as the total energy of the consititutive elements. In this way, Equation 4.2 is simplified to be A Ef = Z(vEf),roducts - Z(vEf) (4.4) reactants- Experimentally, the enthalpy of formation has been found to be -1097.46kJ/mol [6]. In this way, the total enthalpy of formation can be calculated from Ezr02 Ezr = -11.57eV = -1116.6lkJ/mol. E02- This gives an error of only 1.744%, an excellent agreement with experiment. These pseudopotentials were also tested against the bulk phase of Zr(HPO 3 ) 2 from experimental values [36]. The results of this calculation are shown in Table 4.7. Other pseudopotentials were likewise tested to verify that they matched experimental properties. They were also chosen from the well-tested quantum espresso pseudopotential library and have been used in many other computational works. 60 The results of this section show that the chosen pseudopotentials have great agreement with the existing experimental systems. Moving forward, these pseudopotentials will predict accurate optoelectronic properties of the tested systems and will be used over a wide variety of systems to verify the design proposed in the previous chapter. 4.3 Computational Results In this section, we present the computational results of this work. The overall structures used will be described as well as their ground-state electronic properties. These properties will be used to iterate, on the design of this BHJ. The backbone of this design will rely on Zr(HPO 3 )2 as well as the organic MK2 molecule. 4.3.1 Bulk Zr(HPO 3 )2 In the realm of layered phosphates, one of the most studied is a-Zr(HPO 3 )2 . It is primarily used as an ionic transport material in experimentally realized structures. Of utmost interest is the fact that this structure forms layered structures with organic materials, allowing for solution processing of the bulk structure, limiting the processing cost associated with creating the photovoltaic structure. Additionally, layered phosphate compounds form with zirconium, titanium, and halfnium with almost identical unit cell parameters, to be discussed later. This structure would allow for a highly ordered layer structure with a piecewise mobility increase, addressing both of the primary issues with traditional BHJ designs. The optimized DFT lattice parameters of Zr(HPO 3 )2 are shown in Table 4.7. By calculating the atom-projected density of states (pDOS), both the energetic properties of the band edges as well as their character can be examined. The pDOS of Zr(HPO 3 )2 is shown in Figure 4-2. Although this figure shows that Zr(HPO 3 )2 is a large bandgap semiconductor with a DFT-gap of 4.41eV, it is clear from the pDOS that the bonding is not fully ionic, but has significant covalent nature to the bonds. In particular, the valence band is dominated by oxygen p-states but there is significant hybridization (overlap) of Zr d-, H s-, and P p-states at lower energies away from 61 - IIIZ - 4 H-s - -- P-p o-p 2,- Q -4 -2 0 2 4 E - Ef [eV] Figure 4-2: Atom-Resolved Density of States for a - Zr(HPO 3 ) 2 the band edges. The empty conduction band states also display this same trend of hybridization with the conduction band being made up of hybridized Zr d-, 0 p-, and P p-states. The nature of these hybridizations contribute to overall system properties such as the mobility of the system, to be discussed in subsequent chapters. 4.3.2 Bulk a - Ti(HPO 4 ) 2 . H 2 0 In order to understand the nature of, and potentially modifying, the bandgap of Zr(HPO 3 ) 2 , other transition metal phosphates were explored. a - Ti(HPO 4 ) 2 - H 2 0 forms a similar reduced unit cell as Zr(HPO 3 ) 2 [45] with titanium offering different electrical properties compared to zirconium. The experimentally realized structure is hydrated, with water molecules occupying the space between the independent layers. These water molecules generally do not serve to change the electrical properties of a material, but further processing is generally required to fully remove them from the structure. The 'hydrated' unit cell parameters of a - Ti(HPO 4 ) 2 - H 2 0 are shown below in Table 4.8. A hydrated structure simply means that there are unbound water 62 Table 4.8: Experimental Structure of a - Ti(HPO 4 ) 2 - H 2 0 Lattice Parameter a b C a 0 7 Experimental Value 8.611A 4.99330A 16.15070A 900 110.2060 900 molecules within the super structure. This is normally due to the processing or other experimental creation of the material. The DOS was calculated for hydrated titanium phosphate and is shown in Figure 4-3a. The character of the band edges are now comprised of titanium states, where before the states were made up of zirconium states. It has a DFT-bandgap of 2.88eV, significantly less than the bandgap of the Zr(HPO 3 )2 structure potentially providing a way to control the bandgap of Zr(HPO 3 ) 2 as well as controlling the band alignment in the bulk structure, to be described later. This titanium phosphate structure can be represented as an unhydrated layer, yielding the chemical formula a - Ti(HPO 4 ) 2 , where water molecules are not suspended inside the layer. This now has a much closer lattice constant to the Zr(HPO 3 ) 2 lattice that was experimentally shown. In this unhydrated layer, the lattice constants become a=5.054A, b=4.984A, c = 6.O1A, a = 0 = 90.0', -y = 60.00. These lattice constant are almost identical to the values found in Table 4.7. The DOS was again plotted, and this is shown in Figure 4-3b. In this way, the overall DFT-gap can be seen to be 2.77eV, significantly less than the overall DFT-gap of Zr(HPO 3 ) 2 (4.41eV). 4.3.3 Titanium Doping of Zr(HPO 3 ) 2 The nature of the lower bandgap in both hydrated and unhydrated Ti(HPO 4 ) 2 , the possiblity of doping the Zr(HPO 3 )2 1ayer with titanium atoms becomes an interesting way to control the bandgap of Zr(HPO 3 )2 . In particular, they both share close lattice constants and have similar bond-lengths in the their bulk form as shown in Table 4.9. 63 5 1 1 II- 4- 3 - 4-0 P P-P Ti-d H-s 2- -4 -2 0 2 4 E-Ef [eV] (a) Atom-resolved DOS of a - Ti(HPO 4 )2 H 2 0 4- Ti-d -- O-p H-s - 2 0 0 -4 -2 2 0 4 E - Ef [eV] (b) Atom-resolved DOS of Ti(HPO 4 )2 Figure 4-3: Comparison between hydrated and unhydrated Ti(HPO 4 ) 2 64 Table 4.9: Similarities in Zr(HPO 3 ) 2 and Ti(HPQ 4 ) 2 bond lengths in their bulk struc- tures Length [A] 1.961 2.068 1.586 1.513 Bond Ti0 Zr 0 Ti P Zr P The primary reason to dope the Zr(HPO 3 ) 2 layer would be to change the energies and character of the band endges. Additionally, the titanium may control other bulk system properties such as mobility To explore these system properties, doping of the zirconium structure with a range of titanium atoms was performed. These dopant levels go as Ti:Zr, and were performed for the following ratios: 1:1, 1:3, and 1:5. This limit was reached because after a 1:5 ratio, computational time became excessive, and a trend could be established. The variation of the system DOS with the ratio of titanium doping levels are in Figure 4-4. It is apparent that the doping level does not significantly change the character of the band edges with respect to eachother. The band gap remains the same to the third significant figure, and can seen to be 0.7eV, significantly less than the bandgap observed in zirconium phosphate of 6.9eV. This is mainly because of the gap state, or the titanium states present between the valence band and the higher energy conduction band, lower the effective bandgap the material will project. These energetic states slowly decrease in width as Ti:Zr ratio decreases and the material becomes more Zr(HPO 3 ) 2 like. This can be seen in the Table 4.10. Table 4.10: Bangap Modification of Titanium doped Zirconium Phosphate Dopant Level (Ti:Zr) 1:1 1:3 1:5 Bandgap [eV] 1.67 1.7 1.7 Titanium Gap-State Width [eV] 0.8 0.7 0.6 From Table 4.10, it is apparent that the width of the titanium gap states decreases 65 Z,-d CA 2- 0E - 0-4 -2 2 0 4 E-E, [eV] (a) 1:1 doping of Ti:Zr O-p -4 0 2 4 E-Ef [eV] (b) 1:3 doping of Ti:Zr 2- 4 -Ti -d ZT-d -O-p Pp 0 2- 0 V" 4 -2 2 0 4 E-Ef [eV] (c) 1:5 doping of Ti:Zr Figure 4-4: Titanium doping of Zr(HPO 3 ) 2 66 in a predictable way, yet the bandgap remains constant. In this way, the bandstructure of the zirconium phosphate can be controlled as long as the titanium gap states remain present. As the level of titanium doping decreases, the overall structure behaves more like Zr(HPO 3 ) 2 , while still having a modified bandgap. This presents a unique tool that can help with controlling any band alignment issues in the bulk structure. 4.3.4 Layered Organic Zr(HPO 3 ) 2 As was mentioned previously, Zr(HPO 3 )2 readily forms benzene-ring interlinkers between layers, primarily used for ionic transport. The large bandgap of Zr(HPO 3 )2 makes it impractical for typical photovoltaic materials and the optoelectronic properties of Zr(HPO 3 ) 2 have not been studied in depth. It is a small jump to imagine a structure where organic solar materials are used as the interlinking material instead of simple benzene rings. In this way, a layered structure with optically-optimized organic interlinking molecules can be imagined and realized. In order to design this structure, it is first worth exploring the experimental structure, comprised only of benzene rings. This structure was shown in Figure 3-6. Complete testing of this structure was performed, and an atom-projected DOS was plotted as described before and is shown in Figure 4-5. Due to symmetry, the unit cell had two layers of zirconium phosphate as well as two layers of dibenzene to account for the directionality that the zirconium layer has. Because of this, this calculation has the following chemical formula: (Zr(P0 3 ) 2 ) 2 Hl6 C 24 , representing a rather large structure in DFT-space. Figure 4-5 shows that there is no significant optical properties in the experimentally realized layered Zr(HPO 3)2 structure. Inside of the benzene region, the DFT-band gap is 2.58eV, making this region inefficient at capturing solar radiation. Further, any exitonic generation will quickly recombine without leaving the organic region due to the fact that there is no favorable energetic band alignment between the two materials making up this heterojunction. Additionally, dibenzene has no directional dipole, preventing the separation of the exciton. 67 As such, this cannot Zr-d P-p o-p -C-p C6# -4 -2 0 2 4 E-Ef [eV] Figure 4-5: Atom resolved density of states, spatially separated into each layer be considered a solar material. However, very interesting material properties manifest when the polar molecules that were discussed in Chapter 3 are used in lieu of dibenzene. This will be discussed in detail in the following section. 4.3.5 Layered Zr(HPO 3 )2 with Organic Photovoltaic To design a more efficient BHJ material, the dibenzene layers were replaced with organic materials used as dyes in dye-sensitized solar cells. The inorganic zirconium phosphate structures are able to control the directonality of the organic interlacers by acting as the backbone of the material. This system is shown in figure 4-6. By replacing the dibenzene layers with truly organic photoacceptors, the bulk structure will be shown to have optoelectrical properties necessary for efficient light capture. As a first computation, the unfunctionalized MK2 molecule was inserted between the inorganic phosphate layers. The DOS for the structure shown in Figure 4-6 is shown in Figure 4-7. Comparing this result to the results in Figure 4-5 shows a large shift in the valence band edges are shifted by approximately 0.82eV. This band shift 68 C 0 N S P H Zr Figure 4-6: Computational structure of Zr(HPO 3 ) 2 bound to organic MK2 Table 4.11: Comparison between bulk structure and constitutive molecules. MK2-Layer CB [eV] 0.11 3.36 0.35 Shift [eV] N/A N/A 0.24 VB [eV] -0.71 -0.95 -0.47 Shift [eV] N/A N/A 0.24 DFT-Gap [eV] 0.82 4.31 0.82 (Zr(HPO 3 ) 2 )Top 2.02 -1.34 -1.87 -0.92 3.89 (Zr(HPO 3 ) 2 )Bottom 1.28 -2.08 -2.71 -1.76 3.99 Structure MK2-Molecule (Zr(HPO 3 ) 2 )ulk is due to the total dipole, or electric field, inherent to the polar nature of organic MK2. Table 4.11 shows the energetic shifts in the bulk structure when bound with unfunctionalized MK2, this trend is shown schematically in Figure 4-8. The band shift observed is indicative of two trends: first it shows that the MK2 molecule's polarity can drive effects in the inorganic region (namely the band shifts), second this dipole is an indication of how much energy the dipole is inducing into the exciton. If this shift can be pushed higher, band alignment between the band edges can be perfected. If a perfect band alignment can be achieved, where perfect means there is no energetic gap or barrier at the interface of the organic region and the inorganic layer, the Voc of the material can be optimized as the effective band gap of the material is not hindered under a perfect band-alignment scenario. 69 o-p - N-p S-p - 0 C-p Zr-d P-p 0 0 0 -4 -3 -2 -1 1 0 2 3 4 E-Ef [eV] Figure 4-7: Spatially resolved DOS of Zr(HPO 3 ) 2 -MK2. ECB Eg, Zr(HPO 3)2 EF E Eg, Zr(HPO 3)2 Eg, Zr(HPO 3)2 E9, Org E., Org E9, EVB Un-polarized dibenzene system Zr(HPO 3)2 Polar MK2 system Figure 4-8: Schematic showing energetic shift due to polarized MK2 70 4.4 Scissor Operator As described in Section 2.6, the scissor operator is a way in which to correct the overall bandgap of a system. In this system, there are two scissor operators present, the scissor operator for the inorganic layer, and the scissor operator of the organic region. This was chosen in such a way as to describe the physics of both the organic and inorganic system better. The bandgap of Zr(HPO 3 ) 2 was not readily available in literature, however this system is small enough to calculate the exact bandgap with accurate GW methods as described in Section 2.6. From performing a fully converged GW calculation with the TDDFT functionality of Vasp [26-29], an exact band gap of Zr(HPO 3 ) 2 was able to be calculated. By performing this calculation, a bandgap of 6.9eV was found for Zr(HPO 3 ) 2 . In this way, the scissor operator defined in Equation 2.18 is found to be AEscissor-Inorg = 2.49eV. (4.5) For the organic MK2 molecule, the experimental bandgap was known to be 1.85eV [51]. A "molecule in a box" calculation was performed on an isolated MK2 molecule converged in such a way that the periodic boundary did not allow the adjacent MK2 molecules to interact with one another. The bandgap of this computation yielded a DFT-bandgap of 0.663eV. The scissor operator is then found to be JAEscissororg = 1.187eV. (4.6) All bandgaps repersented herein are underestimated by AEsciSSO. Since the effect of the scissor operator is to merely move the conduction band up in energy by AEsci, 8 r, conceptually and fundamentally all band caracteristics and shifts will remain constant with the main effect being on the overall bandgap of the system. By using these scissor operators, the exact band gap of the system can be calculated. 71 4.5 Organic MK2 Functionalization One of the goals of this design is to maximize the force that will separate the exciton in the organic region. To this end, organic MK2 has been functionalized with electron activating and deactivating groups. The directionality of the molecule allows a coherent dipole to be created across molecule, providing the driving force that can separate the bound hole and exciton pair. After establishing the ability to modify the energy levels of the inorganic region, it becomes necessary to verify that this induces a potential change sufficient to separate the exiton out of the organic region. The organic MK2 molecule is extremely large and represents a computational limit on the study of the bulk structures. The majority of these atoms are used to change the bandgap of the material, and modify the electronic properties of the system. However, this study will be further modifying the structure to see how functional modification will change the electronic character of the molecule, implying that the majority of these functional atoms may be unnecessary. This is possible because Thiophene is the main contributor to the exciton generation; the other atoms are there to facilitate charge transfer. Further, polythiophene serves mainly to decrease the bandgap of the material. To increase the computational efficiency, the MK2 molecule was decreased in sized to the bare minimum amount of atoms to represent the electronic character of the molecule. This is shown in figure 4-9. What we are most interested in is how the band gap, and edges, change with the number of total molecules and the total number of thiophene groups. The results are shown below in Table 4.12. It is apparent that the electronic properties are not significantly changed as the number of atoms are decreased. The main take-away is that the molecule does behave as is to be expected, as the number of thiophene-units decreased the bandgap also decreases. By limiting the number of atoms present in MK2 system, the overall computational time can be decreased without significant loss in overall electronic character of the system. 72 HN N HN S ~~8 s CN S ON CN -9 144 Atoms 66 Atoms 44 Atoms Increasing Computational Time Figure 4-9: Computational modifications of MK2 molecule Table 4.12: Computational Optimization of MK2. Energies relative to Fermi Level Structure Experimental MK2 MK2 without alkyl chains MK2 with one # Thiophene Groups CBM [eV] Band Gap [eV] 4 -0.35 0.38 0.73 4 -0.69 0.15 0.84 1 -1.04 0.43 1.47 thiophene group 4.5.1 VBM [eV] Functionalized MK2 Functionalization of the organic region is the driving force behind the energetic shifts observed in the Zr(HPO 3 ) 2 layers. Additionally, these functionalizing groups were chosen to induce a dipole across the organic layer, creating a potential field strong enough to separate the exiton. Ideally, functional groups that created this energetic shift in the phosphate layers will also create the dipole necessary to separate the exciton. To calculate the induced electric field, and dipoles of the material, the same unit cell as the bulk layered zirconium phosphate structure was used. However, only the 73 0 - -0.2 -0.4-0.6- - -0.8 0 20 40 60 80 Length along z-direction [Bohr] Figure 4-10: Windowed-averaged electrostatic potential of modified MK2 relaxed organic molecules occupied the unit cell. This allows the electric field across the organic molecule to be observed directly. In order to calculate the voltage drop across the material, a dipole correction was induced into the calculation. This was applied far in the vacuum and is required to guarantee no field from the periodic calculation. A windowed-average, with window size of 2.5Bohr (the C-C distance), of the electrostatic potential was then taken [2]. This average is shown in Figure 4-10. The difference of the discontinuity that is seen in the graph is the change in vacuum level, representing the total voltage drop as an electron travels across the material. This calculation was carried out over a wide variety of functional groups. In this way, we can verify that the molecule has enough potential to separate the exciton generated by the thiophene groups (0.18eV [30]). Through functionalization as shown in Figure 4-11, both the voltage drop across the molecule as well as the band shifts in the inorganic region can be controlled. The results of these calculations are shown in Table 4.13. It is apparent that several of these have a dipole-potential large enough to overcome this binding energy in the material. It is worth noting that MK2 without 74 0 S \ / OH NH2 N H Functional Group I Functional Group 2 Figure 4-11: Schematic showing functionalization of the organic MK2 molecule Table 4.13: Voltage drop of functionalization Molecule MK2 MK2 MK2 MK2 MK2 BX-Thiophene BX-Thiophene PX-MK2 PX-MK2 TPD-MK2 TPD-MK2 Functional Group 1 Pyrimidine Boron Flourine Flourine Benzene Pyrimidine Benzene Pyrimidine Benzene Pyrimidine Functional Group 2 Boron - Total Voltage Drop [V] 0.102 0.194 0.418 0.160 0.115 0.582 0.119 0.00 0.123 0.223 0.173 functionalization is unable to separate the exciton by itself, so functionalization is necessary. By functionalizing the organic region, the voltage drop across this region can be increased over the baseline voltage drop of MK2. It appears that the combination of a pyridine ring actually lowers the overall voltage drop, while combining the material with atoms with a variety of electronegativity shows an overall trend in the MK2 voltage drop. As an example, boron seemed to have the largest voltage drop, but also of interest is how simply adding molecules such as BX, PX, or TPD can change the voltage drop over a single thiophene. This functionalization is a way to control how strongly the exciton is separated. Functionalization with simple atoms with varying 75 electron negativity can increase the overall potential drop, effectively increasing the overall energetic efficiency of the material. 4.6 Conclusion In this chapter several results were laid out. The pseudopotential testing results were presented with these pseudopotentials being used throughout this work. The experimental layered organic Zr(HPO 3 ) 2 structure was examined, but no optical properties were seen. However, by using organic MK2 and the inter-linker between the Zr(HPO 3 ) 2 layers, it is easy to band shifts within the inorganic region become evident and imply that this structure could be further modified to create a truly efficient organic heterojunction. By examining the total vacuum level change across the organic molecules, organic MK2 needed to be functionalized in such a way to create a potential drop necessary to separate the exciton. The electronic properties of Zr(HPO 3 ) 2 were also modified by doping the structure with titanium. All of this knowledge will be used to iterate on our design in the next chapter to alleviate these issues. 76 Chapter 5 Engineered Band Alignment In Chapters 4 and 6, the material properties of a layered Zr(HPO 3 )2 solar material was proposed and the electronic properties of the system were measured. Even though minor band shifts were observed, the energetic barrier in the MK2-Zr(HPO 3 )2 structure are too large for efficient exciton separation., In this chapter, the band alignment will be engineered through chemical modifications to the superstructure. 5.1 Functional Group Band Shifts Calculation similar to those described in Chapter 4 were carried out for the molecules with functional groups added as electron activating/deactivating groups. This was implemented to explore the effects that the overall dipole induced by the molecule dipole. Additionally, several molecules were sustituted as an alternative to thiophene. These molecules were chosen for their differences in acceptor strengths (correlated with the LUMO energy); specifically these were BX, PX, and TPD. In general, by combining these structures with simple functional groups, it is possible to significantly alter the relative energy levels of the zirconium phosphate band edges in the two distinct inorganic layers. By enabling this control over band alignment, it may be possible to design a nanostructured photovoltaic material with optimal band alignment, thereby minimizing recombination and maximizing both carrier extraction and Voc , leading to significant improvement in power efficiency 77 Table 5.1: Comparison between bulk structure and constitutive molecules Base Molecule MK2 MK2 MK2 MK2 MK2 MK2 MK2 MK2 MK2 MK2 BX BX PX PX TPD TPD R1 R2 - - Pyrimidine Pyridine Pyridine Pyridine Pyrimidine Pyrimidine Flourine Flourine Boron Benzene Pyrimidine Benzene Pyrimidine Benzene Pyrimidine - OH 20H OH 20H - Boron - Hole Barrier 1.13 0.32 0.37 0.34 0.18 0.37 0.53 0.25 0.42 1.35 0.96 0.37 1.92 0.42 0.57 0.37 Electron Barrier 0.86 0.64 0.37 0.44 1.95 0.50 0.18 0.62 0.86 0.59 1.97 2.37 2.11 2.09 1.28 1.63 relative to traditional BHJ photovoltaics. In Table 5.1, the band shifts of the systems as well as their energetic barriers are shown. The name of each molecule corresponds to a molecule represented in Appendix A. Of most importance is the difference between the band edges between the organic molecule and the phosphate layer. In Table 5.1 the hole barrier and electron barrier represents the energetic barrier between the the organic layer and the inorganic layer for the holes and electrons. This barrier prevents charges from leaving the organic region, leading to recombination. The selection of the functional groups were chosen to explore the effects that electronegativity has on shifting the bands of the zirconium phosphate layer. In this way, the dipole of the molecule can be controlled as well as separating the exiton. The combination of these functional groups still present a large energetic barrier at both the VB and the CB, yet it is apparent that by combining these functional groups 78 the energetic barrier can be controlled. Interactions with these functional groups and the the Zr(HPO 3 ) 2 layers can also be contributing to these energetic barriers. These barrier are still excessive and will prevent efficient carrier extraction, to be discussed in the next chapter. The spatial resolution of the conduction and valence conductions bands is indicative of how the exciton will be separated. The spatial resolution of these bands are plotted below in Figure 5-1. The location of the conduction band and valence band edges are indicative of how the charge is being separated in the material. This separation means that the exciton is immediately separated into locations that are not spatially near, encouraging the dissociation of the exciton. The energetic barriers described in Table 4.11 still prevent the charge from fully leaving the organic region, however, with other modifications to the structure, these barriers can be reduced and eliminated. Once this is eliminated, the dipole across the molecule will allow the the charge to freely leave the organic region, as indicated by the spatial separation of the band edges. 5.2 Titanium Doping Band Alignment The large bandgap of the zirconium phosphate layer makes band-alignment with the small band-gap organic semiconductor very difficult. As described in Section 4.3.2, titanium phosphate also forms in a very similar structure as zirconium phosphate. However, this structure has not been experimentally explored to form the same layered structure as Zr(HPO 3 )2 with organic structures. To this end, the bulk Zr(HPO 3 )2 structure was doped with titanium with the following chemcial formula in the phosphate layer TiZr 3 (PO 3 )s. This level of doping (1:3) is extremely high and was done because of the size of the computational structure. This dopant level was chosen to examine the interaction of the titanium and zirconium without needing an excessive amount of computational resources. The bi-layer nature of the computation allowed the other layer to remain as undoped zirconium layer. This allows the effect of the disperate layers to be observed. 79 0r (a) Spatially resolved Valence Band (b) Conduction Band Figure 5-1: Spatial resolution of the Valence and Conduction Band Edges 80 Figure 5-2: Structural form of Titanium Doped Zirconium Phosphate Between the layers, MK2 functionalized with pyrimidine is used. This structure is shown in Figure 5-2. This allows direct comparison to the values in Table 4.11. The DOS of this calculation is shown below in Figure 5-3. From this figure, the titanium doping significantly modifies the conduction band character of the doped zirconium layer; addressing the band alignment through chemical modification of the phosphate layer. This is particularly useful because of the myraid of chemical species that form the same phosphate layer. In this structure, the doped layer has a DFT-bandgap of 2.63eV. Additionally, while this layer addresses the conduction band alignment issue, there still represents a energetic barrier of 0.61eV in the valence band. It was not explored in this study, but by combining other functional groups, the energetic barrier at the valence band can be addressed allowing the titanium doping to correct the conduction band alignment. 5.3 Corrected Band Alignment As was mentioned in Section 4.4, both the organic region and the inorganic region have bandgaps that are not the true gaps that would be observed in experiment. In this way, by applying a scissor operator to both region's conduction band energies, the correct band gap can be estimated. The exact nature of this shift can be seen schematically in Figure 5-4. 81 Ti-d Zr-d P-p - -- O-p S-p - C-p r- 0n -4 -2 0 E - E [eV] 2 4 Figure 5-3: Titanium Doping of Zirconium Phosphate Layer EDFT+Scissornor( EDFT+Scissor EDFT EDFT+Scissorg "EDFT. EDFT Figure 5-4: Scissor operator application to the bulk band structure 82 Table 5.2: Corrected bandgap of system by applying two scissor operators Structure Accurate Bandgap AEscss[eV] Zr(HPO 3 ) 2 6.9 4.77 1.85 2.56 2.44 3.15 2.49 1.97 1.187 1.83 1.33 2.07 Ti(HPO 4 ) 2 MK2 BX PX TPD In the inorganic region, the scissor operator was used as a constant 2.49eV (Equation 4.6). In the organic region, the scissor operator varied based on the character of the light-absorbing molecule. For MK2, the scissor operator was used as 1.187eV(Equation 4.5). For BX, PX, and TPD, the scissor operator was determined from the computational work performed by K6se [30], these operators are given as Eso, = 1.83eV ESPLSO, = 1.33eV ESTP, (5.1) = 2.07eV This was done for select systems that had large voltage drops across the system. The DFT gap, the experimental (or GW band gap), and the scissor operator are shown in Table 5.2. These shifts are used to accurately calculate the bandgap of the system as well as the energetic barriers between the organic and inorganic layers. Even with these shifts, there are still energetic barriers between the organic and inorganic layers. These are represented in Table 5.3. Since the overall band alignment and structure is now exactly known, the exact barriers can be examined. The differences in the scissor operators actually make the energetic barriers greater than with the underestimated DFT bandgaps. The energetic barriers at both of the inorganic interfaces means that the total bandgap is in the organic region. This bandgap will be used to estimate the overall efficiency of the material, to be discussed later. This information combined with the voltage drops represented in Table 4.13, can be used to further estimate the efficiency of the 83 Table 5.3: Corrected bandgap of system by applying two scissor operators Organic Molecule MK2 MK2 MK2 BX BX PX PX TPD TPD MK2 FG1 CB-Difference [eV] 1.95 Pyrimidine 1.89 Flourine 1.92 Benzene 2.63 Pyrimidine 2.61 Benzene 3.31 Pyrimidine 3.35 Benzene 1.78 Pyrimidine 1.95 Titanium Doped Pyrimidine 0.37 Bandgap [eV] 2.78 2.54 2.99 2.84 2.82 2.60 1.95 3.77 2.78 2.74 material. Ideally, there would be no energetic barrier at the interface between the organic and inorganic interfaces. These gaps imply that the energetic shift in the inorganic region is not due entirely to the voltage drop across the organic region. Other reasons for these barriers would be due to an interface dipole or charge sharing between the atoms at the surfaces. Further exploration would be required to develop a more accurate model of the driving effects of these barriers. 5.4 Conclusions In this chapter, a way to energetically shift the conduction band and valence bands of zirconium phosphate has been discovered. By combining functional groups with the organic molecule, significant band shifts in the inorganic region can be observed. These functional groups were only able to shift the energetic levels in Zr(HPO 3 )2 to a certain extent, and other methods for achieving perfect band alignment were explored. By doping the Zr(HPO 3 )2 layer with titanium, the band alignment of the conduction band was able to be addressed, and the barrier for electrons was removed. However, the hole barrier was still present. Further exploration of functional groups is necessary 84 to fully remove these barriers as they represent the largest obstacle to an efficient material. 85 86 Chapter 6 Transport Properties Key to an efficient solar cell material is the overall mobility of the structure. Given the low mobility of organic photovoltaics, the designed BHJ presented in Chapters 4 and 5 addressed the issue of band alignment in the composite material. In this section the mobility of the Zr(HPO 3 )2 layer will be calculated and quantified. Further, a piecewise model for the system mobility will be presented. 6.1 Mobility Derivation The mobility of charge carriers is an excited-state material property and thus is time-dependent, a property that cannot be captured dirctly with DFT because it solves the time-independent Schr6dinger equation. Nevertheless, mobility is directly related to the band structure of the material and can be well approximated in the limit of a perfect crystal by determining the electronic properties as a function of deformation. Perfect crystals have the highest mobility possible as experimental systems have crystallographic defects that serve to lower the overall mobility by acting as scattering centers in the material. Due to the relatively low concentratioi of defects in real materials, as well as the large number of defect combinations within the material, realistic inclusion of defects in DFT requires both large system sizes and numerous calculations, defects are normally computationally intractable. The calculation described in this section are an upper limit to the mobility of any system. 87 6.1.1 Governing Equations In this section, the derivation of the mobility calculation formulated Zhigang Shuai, et al, [48, p.67-88] will be described. By following this derivation, the mobility can be calculated using very little computational time as the final form of the mobility relies on energetic properties. By starting with the Boltzmann's tranport equation, all properties of the mobility can be determined. We start by assuming the distribution function f(f, k, t) describes how a particle in phase space evolves in time. Taking the time derivative of this gives df dt Of t Of dr Of dk = -- + + Or dt Ok dt .(6.1) this can be simplified using the knowledge that df/dt = v is the packet velocity and the external force acting on the particle is h , yielding: (6.2) ( df af Of Of F(r) . + _v(k) + = dt Ot Or Ok This is now treated as the scattering time of the system, if it is assumed that the external force is purely mechanical in nature. This equation is still complicated to deal with because of the time-dependent nature of the system. This can be further simplified by rewriting the above equation if the distribution function is a FermiDirac distribution at equilibrium, fo = 1/{exp{[E(k) - EF]/kBT} + 1}. For charge transport, and group velocity, this equation becomes Of a;: at sct -eoEv(k) Ofo 09E , (6.3) where E is the electric field, and afo/E represents the derivative of the Fermi-Dirac distribution with respect to the energetic bands, E. In order to get this to a tractable equation, several approximations must be made: the relaxation time approximation and the deformation potential. Using the relaxation time approximation this derivative is converted into a probability distribution relating the probability to transition between states k and k'. By making this approxi88 mation, the final mobility equation can be derived as performed by Shuai [48, p.67-88]. This is shown below in Equation 6.4. f r(i, k)v'(i, k)exp[cFc(k)/kBT]dk E e(h) - eo iE(CB)(VB) kT a E f exp[-ej(k)/kBT]dk iE(CB)(VB) This equation holds for both the hole and electronic mobility of the material, where the required properties that need to determined are the relaxation time, T(k), the group velocity va(i, k), and the band energy, ei(k) (the minimum or maximum in the band energies for the CB and VB, respectively). The work by Long [34] made calculation of the group velocity trivial from the band energies. To this end, once the band structure is known, both the band energy and the group velocity is known from va(i, k) = V" (k) (6.5) h and Shuai was able to show that the relaxation time could then be approximated by 1 v(k') eEl~ 27rkBTE2 - c(k )] - Vkk T(k)1 (6.6) This introduces two new constants, E1 and cq, the deformation potential and the electric constant. However, this form of the mobility and the relaxation time requires an extremely dense kpoint mesh. As an example, Long used a 64x64 kpoint mesh in the plane of a graphene sheet. This quickly becomes computationally difficult for complex systems that would required many more kpoints to get the physics of the system correct. To alleviate the reliance on a dense k-point mesh, one last approximation must be made. The effective mass approximation allows the mobility to be solved by only knowing the band diagram of a system. Using a parabolic band argument, the dispersion relation can be written as (k - ko) 2 c(k) = co + ( 89 (6.7) where the given approximation comes from m* = h2 (d2 E/dk2 )-', giving an "effective" mass of the electrons and holes. As the equation indicates, these masses come from the curvature of the conduction and valence bands, respectively. This allows us to write v(k) = hk/m*, simplifying the relaxation time to 1 r(k) 2E2kBTm* kh cq 1 3(6.8) Due to the anisotropic nature of the hybrid systems studied in this work, we compute the 2D mobility fo these systems (specifically the mobility within the plane of the phosphate structures). The final equation for the 2D mobility is then given by. __2eohocq 1-2D - 2EOkB3 q'(6.9) 2 2kBTm* 2E Now that a concise equation has been developed to describe the mobility, only three constants need to be calculated, the deformation potential (E2), the elastic constant (cq) and the effective mass (m*). 6.1.2 DFT Calculations Within the effective mass approximation, the three required constants come from the band structure alone. In this section, we will describe the physical meaning of these constants as well as how they are computed. The deformation potential is given as El = AV. , Al/10 (6.10) where this fitting comes from straining the unit cell. AV is the change in energy of the conduction band (electrons) or valence band (holes) and Al/lo is simply the strain along one of the lattice directions. By straining the unit cell at various different increments (in general, at 1% increments from +3%), a curve fit to this strain can give the deformation potential. 90 The elastic constant is given as _cQI'Ll'\ - - = VO 2 lo (6.11) , ' AE where V is the original unit cell volume, lo is the lattice constant along the a direction, AE is the change in total energy for the strained unit cell, and Al/10 is the strain along the a direction. It is worth noting that V takes a different character for either 1D, 2D, or 3D mobility. In this work the 2D mobility was calculated, so V is actually the cross sectional area perpendicular to the required axis. The effective mass is defined as m* = h2( d 2 -. dk (6.12) The only term of computational interest in this equation is (d 2 E/dk?)-1, which simply is the inverse of the curvature of the conduction band (electrons) or the valence band (holes). In this way, all three can be calculated by running approximately seven calculations at different levels of strain. All of these can be obtained provided that the whole structure is energetically relaxed to the ground state, both for atomic positions as well as lattice constant(s). 6.2 Zr(HPO 3 )2 Mobility In the previous section, the mobility was derived in such a way that it could be calculated with DFT. In this section, the mobility of Zr(HPO 3 )2 will be analytically calculated and will be shown to be much higher than that of traditional organic heterojunction materials; to increase the overall efficiency of the material, the phosphate simply needs to have a mobility greater than 10 4 cm 2 /Vs, the order of charge mo- bility in organic semiconductors. A literature review did not yield any information about the mobility, or other electronic properties, of the transition metal phosphate compounds studied in this work. This is principally due to the fact that these ma91 terials have such large bandgaps; thus this is the first study aiming to quantify the charge carrier mobility within these materials. In Equation 6.9, several values have to be calculated. Namely cq, the electric constant, E1 , the deformation potential, and m*, the effective mass. Each of these values have a value for both the holes and electrons. All of these constants come from the band diagram of the structure. For zirconium phosphate, this was calculated and was found to be highly directionally dependent. The bandstructure was calculated along, the reciprocal a*, b*, and c* directions. Along a* and b*, the band diagram was plotted from F to M. Along the c* axis the band diagram was plotted from F to A and was found to be completely flat. These points are the high symmetry points inside the bruillion zone, and are generally accepted to be points of special interest. These points were chosen because of their directionality relative to the real-space directions as we primarily are interested in the mobility normal to the phosphate sheet as well as along the sheet. These direction are shown in Figure 6-1. The band diagram was then calculated along these directions and is shown in Figure 6-2. The mobility of the structure is then only calculated in the a* and b* directions, however, due to symmetry, these results are identical. As Figure 6-2 shows, the curvature of the bands is zero along the c* direction (the direction normal to the Zr(HPO 3 )2 layers, where van der Waals interactions dominate), leading to an infinite effective mass and a mobility of exactly zero along this direction. In contrast, the bands in the plane of the 2D sheets have a finite curvature, and therefore a meaningful mobility along this direction. By fitting a curve to this raw data, a curvature can be found by taking the second-derivative of the fitting function. By using Equation 6.12, this curvature then leads to the the effective mass. To calculate the elastic constant and the deformation potential and eleastic constant, the unit cell is now strained along the a* direction (only required along this direction because of symmetry as well as the fact that the mobility along the c* direction is identically zero). Increments of 1% are induced along this unit cell. In this way, two curves are obtained, as dictated by Equations 6.10 and 6.11. For 92 k M b* iA j .b r C* -A -M (b) Reciprocal space showing the direction (a) Reciprocal space showing the direction along c* along a* Figure 6-1: Directions of reciprocal space band structure 6 6 4 4 2 0 -2- rI2 -2 -4 -4- -6 -6 -M -1 M K [1/A] (a) Band Diagram from F to A K [1/A] (b) Band Diagram from F to M Figure 6-2: Band Diagram of Zr(HPO 3 ) 2 along reciprocal axes 93 A 0.015 I ' U - Calculated Points Fit(12.316x 0.0065x-3e-5) 0.010 -- W 0.005- -0.04 -0.02 0 0.02 0.04 Al/ 0 Figure 6-3: Plot of curve required to fit the elastic constant the elastic constnat, only one curve needs to be calculated, this is shown in Figure It is worth noting that this is acutally plotting 2AE/Vo = (Al/1b) 6-3. 2 , to make curve fitting easier. By fitting the curve with a parabola, the elastic constant can be calculated as the coefficient to the parabolic curve that fits this data of the form y = Ejx 2 +C 1 x+Cox0 . The additional constants are indicative of the error associated with using the parabolic band approximation. In the same way, the deformation potential can be calculated. This, however, requires knowledge of both the conduction band and valence band. Equation 6.10 dictates that the conduction band minimum is required for electrons and the valence band maximum for holes. These curves are plotted in 6-4. By fitting a linear curve to these curves, the deformation potential for both the holes and electrons can be calculated. The absolute value of this fitting will be used to guarantee a positive mobility value. Now knowing the effective mass, the elastic constant, and the deformation potential, the mobility can be calculated using Equation 6.9. Each of these constants were then calculated for various "layer thicknesses." This simply means that the bulk structure properties were calculated as well as a structure with a vaccuum (20A) on either side of the slab. 94 A combination of bulk layers 0.3 N - 0 - 0.2- - BM VBM CBM Fit (-0.9691x+0.0022) VBM Fit (6.03x+0.0072) 0.1- - 0 -0.1- -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 A1I1 Figure 6-4: Plot of curves required to calculate the deformation potential. Table 6.1: Electron and hole transport properties of Zr(HPO3 )2 # Layers cq [eV/A2] 1 2 3 4 5 Bulk 13.18 12.32 12.32 12.35 13.06 12.87 E1 (e) [eV] -2.92 -0.97 -0.79 1.11 0.52 3.32 E1 (h) 3.71 6.03 6.25 8.17 5.8 10.1251 m*' [mo] 0.95 0.81 0.77 0.78 0.83 0.75 m*' [mo] 1.43 1.49 1.05 1.45 1.34 1.45 were then combined, from a monolayer until five total repeats. The results of these calculations are shown in Table 6.1. The inverse-square relation of the effective mass of the electrons/holes are of the most important property to the electronic property to dictate the mobility of the holes and electrons in the material. The effective mass for each layer is plotted in Figure 6-5, showing that the effective mass does not change significantly from a bulk structure to a monolayer. Between a monolayer and a bulk-structure, the hole-mass grew by 27.5% while the electron mass only shrunk by 1.78% between a bulk and mololayer. This implies that a monolayer can be used without sacrificing a significant 95 1.6 1 - Y-V Electron Mass -V Hole Mass 1.2- - I 0.81 2 4 3 5 Bulk # Layers Figure 6-5: Effective Mass vs. Layers of Zr(HPO 3 )2 amount of performance over the bulk structure properties. By combining the knowledge of Table 6.1 and Figure 6-5, it is then immediately possible to use Equation 6.9 to calculate the electron and hole mobility in the plane of the zirconium phosphate. Carrying this out it is found to be 107cm 2 /Vs for holes and 388cm 2 /Vs for electrons, a significant improvement over the total mobility of organic semiconductors. The comparison between common semiconductors are shown in Table 6.2. It is important to note that the comparison should be between Zr(HPO 3 ) 2 and traditional organic semiconductors. While traditional inorganic semiconductors such as silicon and gallium-arsenide have much higher mobilities, they serve a different purpose than is desired of Zr(HPO 3 )2 . By having a mobility several orders of magnitude greater than the organic semiconductors, the overall mobility of a traditional BHJ material can be increased, allowing for an overall higher efficiency in the material. 96 Table 6.2: Comparison of mobilities between common semiconductors Material Electron Mobility (cm 2 /Vs) Hole Mobility (cm 2 /Vs) Monolayer Zr(HPO 3 )2 Silicon Ga-As P3HT-PEDOT 388 1400 8500 107 450 400 10 4 10-4 6.3 Conclusion In this chapter, calculation of the mobility with existing DFT tools was derived. By using this derivation, it was found that Zr(HPO 3 )2 has a significantly higher mobility than organic semiconductors. By using this material with organic semiconductors, an overall increase in the material mobility can be realized. This presents a key finding to hybrid organic-inorganic BHJ operations and opens the door for higher efficiency in such materials. 97 98 Chapter 7 Conclusions In this thesis a model for a highly efficient solar material was presented. The functional groups that were explored were shown to both contribute to the overall voltage drop across the organic molecule as well as shift the energetic levels of both the CB and VB inside of Zr(HPO 3 )2 . Further, the high mobility in the Zr(HPO 3 )2 layer was shown to be significantly higher than that of the organic layer. 7.1 Efficiency Estimate The energetic barriers due to the band alignment will severely limit the overall efficiency of the material. Elimination of these barriers would make the system work in an ideal way, but it would be prudent to estimate the overall efficiency of the devices as they currently stand. There are three main ways in which the system will be limited in terms of the overall efficiency: the Shockley-Queisser (SQ) limit, the area fraction, and the efficiency associated with the energetic barrier. The SQ barrier is the max efficiency associated with the bandgap. This was described in Section 1.1.2. This is a measure of the max efficiency that the material can achieve with a given bandgap. In organic photovoltaics such as the materials observed in this study, recombination would have to be taken into account to describe the actual efficiency. The SQ limit was originally developed for inorganic material, and hence no recombination is included in the use of it. In this way, the SQ limit is 99 Absorbs Light Absorbs Light Figure 7-1: Schematic representing the idea of the area fraction an upper limit. The area fraction needs to be taken into account because the phosphate materials do not interact with the solar spectrum. AF = Area interacting with solar spectrum Total Area This is shown schematically in Figure 7-1. (7.1) Because of the length of the organic semiconductors used in this material, the area fraction will be quite high (~ 90%). The largest efficiency drop will be due to the energetic barriers at both the conduction and valence bands. In classical physics, energy barriers would prevent particles from moving over this, however, electrons and holes behave quantum-mechanically. One of the benefits of this is that the electrons can tunnel through an energetic barrier that it does not have the energy to overcome. This is a well known effect and can be found in any introductory physics book. Schematically, this tunneling is shown in Figure 7-2. If an electron can tunnel through this barrier, it would maintain the same energy that was incident on the barrier, but the wavelength of its wave-function would be decreased. In this way, the transmission coefficient, T, describes the probability that the electron can tunnel through the barrier at either interface. T can be described functionally as T= e- 2 k2L where k 2 = f2m(Uo - E)/h and L is the length of the barrier. In k 2 , (7.2) m is the rest mass of an electron, Uo is the energetic barrier, and E is the energy the electron (hole) posses. The transmission coefficient is then used as a proxy to the efficiency 100 U Barrier E Electron EElectron with probability T j Energy X=0 x=L Figure 7-2: Schematic for electron (hole) tunneling of an electron leaving the organic region. The energy (E) of the electron is taken to be and electron in the electric field created by the organic region and the length of the barrier L is taken to be the length of a benzene ring, a characteristic length in the system. This length was chosen because it is a good characteristic length of the whole system. Because of the fact that there exists two energetic barriers in the systems studies (a barrier for holes and a barrier for electrons), the larger of the two barriers is taken as a Uo. By using this model, the total efficiency of the system will be - WTotal ~~ TSQ * T7AF * 77Tunneling (7-3) Each of these values are calculated for the the materials listed is Table 5.3. The results of this efficiency study is shown in Table 7.1. From this table is is apparent that these devices are not the highly efficient materials that are to be desired. It can 101 Table 7.1: Max efficiencies of select systems examined Molecule MK2 MK2 MK2 BX BX PX PX TPD TPD FG1 - Pyrimidine Flourine Benzene Pyrimidine Benzene Pyrimidine Benzene Pyrimidine Max-Barrier 2.83 1.89 1.92 3.27 3.25 3.45 3.49 2.66 2.83 Bandgap [eV] 1.90 2.54 2.99 2.20 2.17 2.65 1.80 2.89 1.90 77SQ 12.39% 11.59% 4.52% 18.51% 19.03% 22.35% 27.09% 6.06% 25.03% Wunneling 7lTotal 1.81% 2.40% 2.25% 0.92% 0.20% 0.25% 0.09% 0.15% 0.92% 0.50% 0.53% 1.15% 0.95% 0.11% 0.10% 0.13% 0.06% 0.21% be seen that this is due to the extremely low tunneling coefficient due to the large energetic gap between the organic and inorganic material. 7.2 Future Outlook Even though the materials that have been calculated have been shown to have rather low efficiencies, shown in Table 7.1, many positive outcomes can be gained from the completion of this work. Primarily, the ability to control the energetic levels in the phosphate group creates the ability to accurately control these levels, first computationally observed in this work. The phosphate groups themselves have much higher mobilities than the organic materials, a property that has not been studied thus far. The ability of Zr(HPO 3 ) 2 to form layered structures with organic molecules implies that it can be used as a hybrid organic-inorganic BHJ material. The directionality Zr(HPO 3 ) 2 creates for the organic molecules creates the ability the ability for traditional semiconducting organic materials to become highly ordered, utilizing their intrinsic dipole to effectively separate the exciton. Several materials were shown to have intrinsic dipoles great enough to effectively separate the generated exciton. The main limiter in the estimated efficiencies of these materials is the energetic barrier at each interface. The exact nature of this barrier is not precisely known, but it could be due to charge transfer at the interface between the two materials, 102 or due to an interface dipole at this interface. However, the ability to control the energetic properties of these phosphate layers via organic modification has many implications to the organic BHJ designs as well other novel material designs. Further, the order layered phosphate groups create a way to make highly ordered BHJ designs, allowing for the organic molecules to use a designed dipole to overcome the exciton binding energy. Further optoelectrical properties can be imagines from such functional materials as optomechanical actuators as well as tunable-bandgap materials. Utilizing this design and the results contained within this work, it will be possible to engineer band alignment between the organic and inorganic material. The ability to control the properties of the layered inorganic phosphates through organic functionalization could have wide implications. Fixing the energetic barriers present in the material is the primary issue that needs to be addressed. Elimination of these barriers would create a material that would be highly efficient photovoltaic material. This would realize a cheap, solution-processable, organic photovoltaic material that would open the world to abundant solar-electricity. 103 104 Appendix A Molecule Figures Because of the large number of combination that are present in this work, common names were used extensively used throughout this work. In this appendix the chemical structure of each molecule is represented in concise form. These figures represent the organic materials that were used throughout this work to modify the bands of the Zr(HPO 3 ) 2 layers. These molecules were placed between layers of Zr(HPO 3 ) 2 as well as titanium-doped Zr(HPO 3 )2 and their electronic properties were measured. 105 HN H 2N 0 0 Figure A-i: MK2 with Benzene interlinker 106 HN HN s S H 2N H 2N o 0 0 0 N N N (b) MK2-Pyrimidine (a) MK2-Pyridine Figure A-2: Chemical structures of MK2- Pyridine and Pyrimidine 107 OH OH HN HN H2N NH 2 S S H 2N O 0 NN 0 N (a) MK2-Pyridine-OH (b) MK2-Pyrimidine-OH Figure A-3: Chemical structures of MK2- Pyridine and Pyrimidine functionalized with an OH group 108 OH HO OH HO HN HN S S NH 2 0 NH 2 0 0 N N (a) MK2-Pyridine-(OH) 2 0 N (b) MK2-Pyrimidine-(OH) 2 Figure A-4: Chemical structures of MK2- Pyridine and Pyrimidine functionalized with two OH groups 109 HN HN S NH 2 0 H2B NH 2 0 o 8H 2 F (a) MK2 with Boron 0 F (b) MK2 with Flourine Figure A-5: Chemical structures of Boron and Flourine functionalized MK2 110 H 2B BH2 HN NH 2 0 0 F F Figure A-6: MK2 with both Fluorine and Boron HN HN N N \I N 0 N N -~N (b) BX with Pyrimidine interlinker (a) BX with Benzene interlinker Figure A-7: BX shown with Benzene and Pyrimidine interlinkers 111 HN HN N)N NMNN NM. 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