Mathematical modeling and analysis of organic bulk heterojunction solar cells

Mathematical Modeling and Analysis of Organic Bulk Heterojunction Solar Cells Zhang Teng B.Eng.(Hons.), NUS A Thesis Submitted For the Degree of Doctor of Philosophy Deparment of Chemical and Biomolecular Engineering National University of Singapore May 2014 " .For actually the earth had no roads to begin with, but when many men pass one way, a road is made." - Lu Xun, excerpt from "Hometown", 1921. Typeset with AMS-LATEX. DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Zhang Teng 22 April 2014 Mathematical Modeling and Analysis of Organic Bulk Heterojunction Solar Cells Zhang Teng National University of Singapore Departrment of Chemical and Biomolecular Engineering Singapore 117576, Singapore Abstract Organic photovoltaics have become a promising alternative to today's silicon-based technologies with the potential of providing low-cost, large-area solar cells. However, organic solar cells still su er from signi cantly lower e ciencies than their silicon-based counterparts, and the understanding of their performance limiting factors is hampered by the convoluted morphologies of organic photoactive layers. In this context, this study seeks to establish a multiscale modelling framework to elucidate the structure-property relationships in organic bulk heterojunction (BHJ) solar cells. For this purpose, we have rst generalized the existing device models for organic bilayer and BHJ solar cells by reformulating the interfacial boundary conditions for charge carrier separation and recombination at the donor/acceptor interface. This generalized model could be reduced to either a bilayer or a BHJ model depending on the nature of the interface. The validity of this model has been assessed by calibrating and validating model predictions with experimental current-voltage measurements. In addition, we have utilized the model to investigate the morphology and loss mechanisms in the recently invented pseudo-bilayer solar cells. We employ a two-level modelling approach to investigate the structure-property relations in organic BHJ solar cells: First, we develop a three-dimensional (3D), two-phase device model that resolves the morphological details of representative photoactive layer morphologies; then, we volume-average the local structural details of the photoactive layer to derive a 1D, spatiallysmoothed model that reveals the e ects of inherent morphological characteristics on photovoltaic properties the solar cell in the form of mathematical relations. The spatially-smoothed model is consistent with the existing e ective-medium models, but it captures two essential morphological characteristics not found in existing models: the speci c interfacial area and the donor/acceptor volume fractions. In addition, we derive an analytical model for exciton transport that relates morphological characteristics explicitly with the charge carrier generation rate. This exciton transport model can be directly incorporated into the spatially-smoothed model, allowing it to capture the e ects of morphology and exciton transport on the performance of organic BHJ solar cells. ii With the spatially-smoothed device model derived and validated, we utilize it to investigate the optimum morphology of the recently demonstrated pillar-structured donor/acceptor organic solar cells. We illustrate that the e ective transport and recombination properties of the pillartype morphology are explicit functions of the speci c interfacial area and the donor/acceptor volume fractions. The cross-sectional shape of the pillars, on the other hand, has no major in uence on the performance of this type of solar cells. Based on these closed-form structure-property relations, we establish a fast computational framework to determine the optimal pillar-type morphologies. We further apply the spatially-smoothed device to study the e ect of morphology on the opencircuit voltage of organic solar cells. By solving the spatially-smoothed model analytically at open-circuit, we are able to derive a closed-form relation between the open-circuit voltage and the underlying donor/acceptor morphology. We nd that the in uence of morphology on the open-circuit voltage is attributed to a single morphological parameter: the ratio between the donor volume fraction and the speci c interfacial area. Our ndings are veri ed against detailed two-phase models with a range of randomly generated donor/acceptor morphologies. Finally, we highlight how the present work can be extended towards a hierarchical multiscale modelling framework to derive morphology-property relations in realistic, disordered BHJ morphologies. Keywords: organic solar cells; bulk-heterojunction; morphology; charge carrier transport; recombination; exciton; mathematical modeling; volume-averaging iii Preface This thesis presents topics on the mathematical modeling of organic solar cells with a focus on relating the solar cell performance with the microscopic morphology of the photoactive layer. Chapter introduces the background, motivation, and objectives of this work. In Chapter 2, the mathematical formulations for the existing bilayer and bulk-heterojunction (BHJ) device models are summarized and generalized in the form of a pseudo-bilayer device model. This generalized model could be reduced to either a bilayer or a BHJ model depending on the nature of the donor/acceptor interface. In addition, the implementation procedures for three-dimensional device models with disordered donor/acceptor morphologies are outlined. In Chapter 3, the pseudo-bilayer device model is utilized to study the morphology and loss mechanisms in the recently discovered pseudo-bilayer organic solar cell which contains partially intermixed photoactive layer. Chapter 4-5 present the derivation and veri cation of the spatially-smoothed device model, which represent a key contribution of this thesis. In particular, Chapter demonstrates the development of spatially-smoothed forms of the Poisson and charge carrier continuity equations based on volume-averaging the local charge carrier generation, transport and recombination properties. The spatially-smoothed models not only features the simplicity of existing e ective-medium models, but also captures the e ects of inherent morphological characteristics on the photovoltaic properties the solar cell. In Chapter 5, the spatially-smoothed model is further extended to include the e ect of exciton transport and morphology on the interfacial charge carrier generation rate. With the spatially-smoothed model secured, Chapter 6-7 illustrate two applications of this new modeling framework in elucidating structure-property relationships. In Chapter 6, the spatially-smoothed model is utilized to study the recently demonstrated pillar-structured donor/acceptor organic solar cells. Closed-form expressions are derived for the e ective charge carrier transport and recombination properties of this type of device, and the optimum characteristics of the pillar-type structures are derived. In Chapter 7, the spatially-smoothed model is solved analytically at open-circuit to derive an analytical expression of the open-circuit voltage of organic BHJ solar cells as a function of the underlying morphology. A single morphological parameter is identi ed to govern the open-circuit voltage at leading order. The analytical results in Chapter 6-7 are veri ed with detailed two-phase device modeling with randomly generated donor/acceptor morphologies. Finally, Chapter summarizes the main ndings of this thesis, and discusses possible extensions of the current work. iv The following journal publications are based on research carried out for this doctoral thesis: 1. T. Zhang, E. Birgersson, K. Ananthanarayanan, C. H. Yong, L. N. S. A. Thummalakunta, and J. Luther, Analysis of a device model for organic pseudo-bilayer solar cells, J. Appl. Phys. 112, 084511 (2012). 2. T. Zhang, E. Birgersson, and J. Luther, A spatially-smoothed device model for organic bulk heterojunction solar cells, J. Appl. Phys. 113, 174505 (2013). 3. T. Zhang, E. Birgersson, and J. Luther, Closed-form expression correlating exciton transport and interfacial charge carrier generation with the donor/acceptor morphology in organic bulk heterojunction solar cells. Physica B 456 267 (2015) 4. T. Zhang, E. Birgersson, and J. Luther, Modelling the structure-property relations in pillarstructured donor/acceptor solar cells, Organ. Electron. 15 2742 (2014). 5. T. Zhang, E. Birgersson, and J. Luther, Relating morphological characteristics with the open circuit-voltage of organic bulk-heterojunction solar cells, accepted in Appl. Phys. Express, 2014. v Acknowledgements I owe my deepest gratitude to my PhD advisor, Dr Erik Birgersson, who is the most enthusiastic teacher and one of the smartest people I know. Erik has supported me throughout my PhD journey with thoughtful guidance on my research work, invaluable advise on my career and personal development, whilst allowing me the room to learn and work in my own way. I could not have imagined having a better advisor and mentor for my PhD study. I would like to express my sincere gratitude to Professor Joachim Luther for his insightful comments on my research papers and many enlightening discussions on the physics of solar cells. I would like to thank Associate Professor Peter Ho for the constructive feedback to my manuscripts. I am also in debt to my colleagues - Yong Chian Haw and L. N. S. A, Thummalakunta for the help in conducting experiments for model validations, and Dr Krishnamoorthy Ananthanarayanan, Set Ying Ting, and To Thin Tran for many fruitful discussions and helpful suggestions. I cannot nd words to express my gratitude to my parents who always provide their unconditional love and care. I dedicate the thesis to my ancee , Tang Pan, for her continued love, support, and encouragement for me to pursue an academic career. Finally, I gratefully acknowledged the nancial support from the Solar Energy Research In- stitute of Singapore and the Singapore Economic Development Board for the research work conducted for this thesis. Contents Introduction 1.1 Organic solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Device physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Photoexcitaton and exciton transport . . . . . . . . . . . . . . . . . . . . 1.2.2 Charge carrier separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Charge carrier transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Summary of the device operation . . . . . . . . . . . . . . . . . . . . . . . photoactive layer morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The bulk-heterojunction morphology . . . . . . . . . . . . . . . . . . . . . 1.3.2 Morphology control and characterization . . . . . . . . . . . . . . . . . . . 11 Device models for organic solar cells . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 E ective-medium models . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.2 Two-phase models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Objectives and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 1.4 1.5 Mathematical formulation 2.1 21 Mathematical formulation for the pseudo-bilayer device model . . . . . . . . . . . 22 2.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.3 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 E ective-medium and two-phase models . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Three-dimensional two-phase models . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 Morphology generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Device physics of organic pseudo-bilayer solar cells 35 3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 The device model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 vii Bibliography 113 [22] M. 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(a) the typical structure of an organic bulk heterojunction solar cell, and (b) length scales for the donor/acceptor blend in an averaging volume 4.2 44 49 Simulated (lines) and measured (symbols) current densities for organic BHJ solar cells under AM1.5 solar spectrum: the training set with an illumination intensity of 1 sun (N), and test sets with intensities of 0.55 sun ( ) and 0.75 sun ( ) 4.3... carriers, which are bound in the form of excitons in organic 1.1 Organic solar cells 3 Figure 1.2: Schematic of a typical organic solar cell semiconductors A major milestone in the development of organic photovoltaic devices is the concept of bilayer heterojunction solar cells introduced by Tang et al in 1986 [17] The idea behind this concept is to form an interface between two organic materials with di erent... smaller optical bandgaps is therefore an e ective route to enhance the PCE of organic solar cells, as it allows absorption into the near-infrared portion of the solar spectrum Some examples of low-bandgap polymers include PCPDTBT synthesized by Brabec et al [29] with a bandgap of ~1:5 eV; PDPP3T reported by Janssen et al [30] with a low bandgap of ~1:3 eV; the 1:6 eV bandgap polymers PTB7 [31] and PBDT-TT-CF... characterizing organic solar cells can be found in [74, 101] 1.4 Device models for organic solar cells The interplay between physics and morphology hampers the interpretation of experimental ndings and the understanding of the inherent photovoltaic processes { especially since it is di cult 1.4 Device models for organic solar cells 13 to achieve precise control, as well as accurate quanti cation, of the active... photo-degradation of active materials [10, 11, 12, 13], further hamper their commercialization In order to further enhance the performance and stability of organic solar cells, further research and development are needed in the theoretical understanding of device physics, the design and synthesis of new materials, the development of new device architectures, as well as the characterization and control of the... of organic solar cells [80] 1.3.2 Morphology control and characterization Due to the signi cant in uence of D/A morphology on the solar cell operation, morphology engineering is an e ective route to improving the performance of organic solar cells The solution processing of D/A blends, however, allows for limited control over the resulting nanoscale morphology The the domain size and connectivity of. .. List of Figures 1.1 Research-cell e ciencies of emerging PV technologies over the past decade 2 1.2 Schematic of a typical organic solar cell 3 1.3 Molecular structures and band diagrams for the donor material P3HT and the acceptor material PC60 BM 5 1.4 Photovoltaic processes in an organic solar cell 8 1.5 Schematics of the organic bulk- heterojunction. .. rst generation of organic solar cells have a homojunction structure, in which a single layer of organic semiconductor is swandwiched between two metal electrodes of di erent work functions [15, 16] This type of solar cells usually have e ciencies less than 0.1%, because the electric eld generated by the asymmetrical work functions of the electrodes is insu cient to drive the separation of charge carriers,... low-temperature and high-throughput 1 2 1.1 Organic solar cells Figure 1.1: Research-cell e ciencies of emerging PV technologies over the past decade The data is extracted from Ref 4 roll-to{roll printing process [7, 8, 9] In addition, the synthetic variability of organic materials provides ample opportunities to continue enhancing and optimizing the optoelectronic properties of organic solar cells, as well... reduce the cost of active materials Despite of these advantages, organic solar cells still exhibit much lower photo-conversion e ciencies as compared to their wafer-based counterpart, primarily due to the low charge carrier mobility, the strong exciton binding energy, and the relatively narrow absorption spectrum of most organic semiconductors [5] Stability issues of organic solar cells, arising partly . Mathematical Modeling and Analysis of Organic Bulk Heterojunction Solar Cells Zhang Teng B.Eng.(Hons.), NUS A Thesis Submitted For the Degree of Doctor of Philosophy Deparment of Chemical and. 22 April 2014 Mathematical Modeling and Analysis of Organic Bulk Heterojunction Solar Cells Zhang Teng National University of Singapore Departrment of Chemical and Biomolecular Engineering Singapore. volume-averaging iii Preface This thesis presents topics on the mathematical modeling of organic solar cells with a focus on relating the solar cell performance with the microscopic morphology of the photoactive layer. Chapter