Advances in modeling the physics of disordered organic electronic devices

Advances in modeling the physics of disordered organic electronic devices

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Michael C. Heiber*,a, Alexander Wagenpfahl†,a, Carsten Deibel† *Center for Hierarchical Materials Design (CHiMaD), Northwestern University, Evanston, IL, United States, †Institut f€ur Physik, Technische Universit€at Chemnitz, Chemnitz, Germany

10.1

Introduction

Organic electronic devices have attracted significant attention over the last several decades as a potential low-cost, lightweight, flexible, semitransparent, and customizable solution for a wide variety of applications not well suited to traditional inorganic technologies. The most researched device applications have been organic photovoltaics (OPVs), organic light-emitting diodes (OLEDs), and organic field-effect transistors (OFETs). Creating high-performance organic electronic devices has required a detailed understanding of the mechanisms and processes that make them work, as well as those that lead to performance losses, so that materials and device architectures can be appropriately tuned. Due to the complexity of the materials’ structures and the resulting charge-carrier and exciton generation, transport, and recombination mechanisms, elucidating the physics of these devices to create predictive physical models has been a challenging task. Modeling techniques can span a large window of length and timescales, from very detailed simulations of individual molecules all the way to simpler analytical equations for device performance metrics. When building a model for a complex system, it is always important to determine how much detail is needed to capture the dominant physical phenomena that dictate the performance variable of primary interest. This becomes especially important for computational models, where calculation time constraints are the primary limitation on model complexity. Among the variety of modeling techniques that have been used, one could broadly classify them into two categories, primarily based on their level of detail: (1) microscopic techniques and (2) macroscopic techniques. In microscopic techniques, the goal is to build a mechanistic model that simulates the individual optoelectronic mechanisms that together determine the overall device response and performance. In macroscopic techniques, analytical effective medium expressions typically are used to capture overall behavior at the device level.

a

Both authors contributed equally to this work.

Handbook of Organic Materials for Electronic and Photonic Devices. https://doi.org/10.1016/B978-0-08-102284-9.00010-3 © 2019 Elsevier Ltd. All rights reserved.

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In this chapter, after a short discussion of the microscopic techniques, we will highlight the most popular one, kinetic Monte Carlo (KMC), describe its unique strengths, and explain how KMC simulations are constructed. We then briefly overview how KMC has been used to simulate the unique physics of charge carrier and exciton transport in organic semiconductors (OSCs), but we place the primary focus on a more detailed review that highlights how KMC simulations have been used to develop physical models for OPVs and OLEDs. Moving on to macroscopic simulations, we introduce the popular drift-diffusion technique and provide a brief tutorial on how the method is derived. We then highlight examples where drift-diffusion methods have been used to understand and optimize OPVs and OLEDs. In the end, we provide an outlook on the organic electronic device simulation and modeling field with recommendations for directions of future research.

10.2

Microscopic simulation and modeling

There are several microscopic modeling techniques where the goal is to simulate each of the individual optoelectronic mechanisms occurring in a device. These techniques are primarily used to understand the emergent device behavior and how experimentally observed phenomena can be traced back to the complex combination of fundamental mechanisms and materials structural features. Such techniques include primarily master equation and KMC methods, each of which can be performed at different length scales and with different levels of detail. In both cases, the individual mechanisms can be defined at a coarse-grained level, where the details of the individual molecules are disregarded; or at an ab initio level, where the positions of the molecules, their orientation, and the resulting electronic states and transitions are all derived from theory. While the use of multiscale ab initio KMC simulations is increasing, the more coarse-grained-lattice KMC implementation has been the most common technique for investigating many of the physical questions relevant for disordered organic electronic devices due to extreme computational cost of ab inito methods. In some cases, even lattice KMC is too computationally expensive, and researchers have preferred to use the much faster master equation methods. However, master equation methods are often inappropriate due to the challenges of correctly including particle-particle interactions that are particularly important for simulating electronic devices that contain charge carriers (Houili et al., 2006; Casalegno et al., 2013), and master equation methods can overestimate diffusion coefficients in disordered materials (Stehr et al., 2011). Here, we will focus our discussion mainly on lattice-KMC simulations, looking at (1) how KMC simulations are constructed, (2) how they are applied to simulate the unique transport physics in disordered OSCs, and (3) how they have been used in conjunction with experiments to elucidate the physics of OPVs and OLEDs.

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10.2.1 Introduction to KMC simulations Monte Carlo methods are a general class of algorithms that use random numbers to solve a wide range of problems. The first and most popular Monte Carlo algorithms were developed to determine equilibrium properties of systems. However, in the 1960s, researchers developed a new algorithm to describe the dynamic behavior of nonequilibrium systems (Voter, 2007), now most commonly called KMC. KMC simulation models have been applied to many different types of problems, but in the late 1970s and early 1980s, the method was adapted to describe the dispersive charge and exciton transport observed in disordered OSCs, through which B€assler and colleagues developed the well-known Gaussian disorder model (GDM) (B€assler, 1993). Expanding on this early work, numerous groups around the world have used KMC methods to simulate a wide variety of mechanisms and physical phenomena present in OSC devices. The KMC algorithm is used to simulate a physical system as it transitions from one microstate to the next over time. The method assumes that the dynamic evolution of the system has the properties of a Markov chain, whereby each transition is a stochastic process in which only knowledge of the present state is needed to determine the transition to the next state, and that the transition time is much shorter than the time between transitions (Voter, 2007). For organic electronic devices, the system’s state is defined by the positions of all charge carriers and excitons within the active semiconductor layer(s). Transitions occur whenever an additional exciton or charge is created or destroyed, or when any of them move. To then construct a KMC simulation, the transition rate constant must be known for all relevant optoelectronic mechanisms in the material of interest, and missing mechanisms always will be a potential source of systematic error. For each mechanism, one must explicitly define an analytical rate equation derived from theory or empirical relationships. The majority of KMC simulation studies have used a cubic, 3D lattice to represent a portion of the OSC materials in the device, whereby all charge carriers or excitons reside on discrete lattice sites. Without access to detailed knowledge about the positions of individual molecules or without the need for such detail, many important physical phenomena that result from the unique molecular nature of OSCs can be cast onto a cubic lattice with a lattice size resolution of around 1 nm, which is roughly the size of a single molecule or a short segment of a polymer. While using a lattice is not required, implementing the KMC algorithm with a lattice is significantly easier from a software design perspective, and the regularity of the lattice reduces computational complexity. Nevertheless, when one would like to capture the detailed impact of molecular structure and packing, precise positions of each molecule and its neighbors are needed, which requires a more detailed, off-lattice implementation. While lattice models have been used to capture positional disorder in molecularly doped materials, off-lattice models can simulate positional disorder effects more accurately (Oelerich et al., 2017). In most cases, there will be a number of possible transitions (events) at any given time. To determine which event will occur and when, most studies use the Gillespie

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first reaction method because it is very computationally efficient for systems with a large number of independent events (Gillespie, 1976). For each possible event, the event wait time is calculated, t¼

 ln X , k

(10.1)

where X is a uniform random number between 0 and 1 and k is the rate constant of the particular event. The wait time signifies the amount of time that must elapse before the particular event occurs. During the simulation, the wait time for all independent events is calculated and stored in a queue. Then the event with the shortest wait time is executed, and the wait time of the executed event becomes the time step for the iteration. Following the execution of each event, the queue is updated by adding or removing any newly enabled or disabled events, respectively, and then the wait time of the remaining events is decremented by the time step. Despite over three decades of using KMC simulations in the OSC field, there are still no widely accepted standard KMC software tools. However, in recent years, various software packages have become available namely, VOTCA-CTP for multiscale charge-transport simulations (www.votca.org) by the Andrienko group, MorphCT for multiscale charge-transport simulations (bitbucket.org/cmelab/morphct) by the Jankowski group, Bumblebee for OLED simulations (simbeyond.com/bumblebee) by Simbeyond, hophop for off-lattice charge-transport simulations (github.com/ janoliver/hophop) by Oelerich, and Excimontec for a variety of organic electronic devices (github.com/MikeHeiber/Excimontec) by Heiber. Of these, VOTCA-CTP, MorphCT, hophop, and Excimontec are freely available, open-source software tools under active development that may be of interest to readers looking to get started in this field. Nevertheless, most of the existing literature has been produced by research groups that maintain private codebases of varying complexity, efficiency, and reliability.

10.2.2 Fundamental transport modeling in OSCs 10.2.2.1 Charge transport In contrast to highly ordered inorganic semiconductors, charge carriers in disordered OSCs are relatively localized and form small polarons. As a result, charge-carrier motion proceeds via a thermally activated hopping mechanism. In addition, disordered semiconductors often exhibit dispersive charge transport, whereby some charge carriers move through the material much slower than others and the rate of transport becomes time dependent (Scher and Montroll, 1975). In most cases, this dispersion has been shown to be caused by a broadened distribution of hopping site energies (diagonal disorder) due to the fluctuations in the molecular orientations, conformation, and the surrounding dipolar environment, but positional disorder (off-diagonal disorder) due to fluctuations in the distance between molecules also can provide another type of disorder that affects charge transport (B€assler, 1993).

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To implement the broadened distribution of hopping site energies, the lattice sites can be randomly assigned values from the appropriate density of states (DOS) probability distribution. Most commonly, a Gaussian distribution is used to describe the energetic disorder, but in some cases, an exponential distribution or even more complex forms have been used (MacKenzie et al., 2012; Oelerich et al., 2012; Lange et al., 2013). In addition to the shape of the distribution, a number of studies have identified the presence of correlated energetic disorder in OSCs arising from statistical fluctuations in the molecular dipole orientation or molecular orientation correlations. As a result, there can be local clusters of high-energy or low-energy sites, and the presence and length scale of these site energy correlations can have a significant effect on charge transport (Novikov, 2003). Regardless of the DOS, two predominant models are used to describe the charge hopping transport mechanism: the Miller-Abrahams (Miller and Abrahams, 1960) and Marcus models (Marcus, 1993). In many cases, both models produce similar behavior. However, the details of these models, the various DOS models, and how they both affect the electric field, temperature, charge-carrier density, and time dependence of charge transport have been a heavily investigated and sometimes highly debated topic of fundamental solid-state physics research over the years, and several extensive review papers have been written on these subjects that are outside the scope of this chapter (Coehoorn and Bobbert, 2012; Baranovskii, 2014).

10.2.2.2 Exciton transport Due to the low dielectric constant of OSCs, excitons have a relatively large binding energy, are relatively localized, and are typically modeled as neutral quasiparticles. Two varieties of exciton states also can exist, as defined by their spin state, singlet or triplet. Singlet-exciton states are the primary product of optical excitation and typically emit a photon when relaxing back to the ground state. Conversely, when spinuncorrelated electrons and holes meet in an OSC, triplet-exciton formation usually prevails over singlet formation, and direct relaxation of triplet excitons to the ground state is a spin-forbidden transition that often results in significantly longer lifetimes and very low radiative efficiencies. In addition, exciton annihilation processes can occur whereby an exciton is quenched when encountering another exciton or a charge carrier. All together, there is a complex series of possible excitonic mechanisms, and for microscopic device simulations, it is important to make sure that all the relevant mechanisms are included for the device type and operating conditions of interest. While the details of exciton transport models have been somewhat less studied than charge-transport models, there are several good reviews on the physics of exciton transport and diffusion and the use of KMC simulation techniques in this area (Menke and Holmes, 2014; Bjorgaard and K€ ose, 2015; Mikhnenko et al., 2015). Like for charge carriers, KMC simulations of both singlet- and triplet-exciton transport have been shown to produce similar dispersive characteristics and can be similarly modeled using a broadened distribution of hopping site energies (Sch€onherr et al., 1980; Richert and B€assler, 1985). In this case, a Gaussian DOS is also most commonly used. As a result, following photoexcitation, the rate of exciton diffusion slows over

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time as the excitons hop to lower-energy sites in the tail of the DOS. Singlet-exciton transport is normally simulated using the F€ orster resonant energy transfer (FRET) model (F€ orster, 1948), and triplet-exciton hopping is usually simulated using the Dexter electron exchange model (Dexter, 1953). In general, singlet excitons can diffuse 5–20 nm from their origin before relaxing back to the ground state, but triplet excitons often can diffuse significantly farther due to their extended lifetimes (Mikhnenko et al., 2015).

10.2.3 Modeling organic photovoltaics In an OPV device, a complex series of optoelectronic mechanisms converts photon energy into electrical energy and dictates the final efficiency of the conversion process. A schematic illustration of these processes is presented in Fig. 10.1. First, light is absorbed by an active OSC layer to form singlet excitons. To overcome the excitonbinding energy and efficiently generate free charge carriers, a blend of OSCs that phase-separate to form a bulk heterojunction (BHJ) structure is used. In most cases, an electron-donating material (donor) and an electron-accepting material (acceptor) are utilized to create an energetic driving force for exciton dissociation into a

Fig. 10.1 Primary OPV current generation and loss mechanisms. (1) Photocurrent generation process showing an exciton (green) diffusing to the DA interface and dissociating into charge carriers that separate and are extracted at the proper electrodes, (2) exciton recombination, (3) geminate charge recombination where CT states are unable to separate, and (4) bimolecular charge recombination where free charge carriers meet and recombine.

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charge-transfer (CT) state at a donor-acceptor (DA) interface. As a result, when singlet excitons are created, they must diffuse to the interface to create a photocurrent. In high-performance devices, the CT state then efficiently separates to form free charge carriers. The free charge carriers then must travel though the complex BHJ morphology network, with electrons restricted to acceptor domains and holes to donor domains, before each carrier is extracted at its respective electrode. In opposition, there are a number of loss processes that reduce the photocurrent. Any excitons that are unable to reach the interface during their lifetime will relax back to the ground state. Additionally, if CT states are unable to form free charge carriers, the charges can recombine through a geminate recombination process. However, even if free charge carriers are generated, electrons and holes originating from separate excitons can meet and recombine through a bimolecular charge recombination process. For a more complete description of the physics of OPV devices, the reader is referred to several more detailed review papers that summarize the development of this field over the years (Clarke and Durrant, 2010; Deibel and Dyakonov, 2010; Proctor et al., 2013; Heeger, 2014). Groves (2013a) also has provided a prior overview of how KMC simulations in particular have contributed to advances in our understanding of OPV device physics. Here, we provide an updated overview and alternate perspective on this field. In addition, while a number of KMC studies have developed methods for simulating full current-voltage curves and have used them to make predictions about optimal devices, here, we primarily focus on work that elucidates the physics of the more detailed fundamental processes.

10.2.3.1 Bulk heterojunction morphology models The nanoscale morphology present in BHJ OPVs has been repeatedly shown to have a significant impact on device performance. As a result, retaining nanoscale detail in KMC simulations is particularly critical for developing structure-propertyperformance relationships. To model the BHJ morphology, Peumans et al. introduced an Ising-based method (Peumans et al., 2003), and this concept was later simplified and applied to KMC simulations by Watkins et al. (2005) to create the first BHJ OPV KMC simulations. Since these pioneering studies, Ising-based morphologies have been widely used in KMC simulations to elucidate OPV device physics. While this model may not capture all morphological features in BHJ blends, qualitatively, it produces a nanoscale bicontinuous morphology that is a reasonable and computationally accessible starting model. The ability to efficiently create Ising-based BHJ morphologies for KMC simulations and perform detailed 3D structural analysis is available through the open-source Ising_OPV software package developed by Heiber and Dhinojwala (2014) and Heiber (2016). Cahn-Hilliard model morphologies often have been used by the Groves group as a more complex alternative (Lyons et al., 2011, 2012). Most studies have used these BHJ morphology models as simplified models for a bicontinuous microstructure with pure donor and pure acceptor phases, and then they determined how simple structural features such as the interfacial-area-to-volume ratio and domain size affect a variety of different device behaviors. However, some

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efforts have been made to understand the impact of more detailed morphological features, such as mixed domains (McNeill et al., 2007; Lyons et al., 2012; Gagorik et al., 2013; Heiber and Dhinojwala, 2013; Jones et al., 2014) and anisotropic morphologies (Heiber et al., 2017), but these more complex morphological features that have been observed experimentally have not yet been rigorously probed using microscopic simulations.

10.2.3.2 Exciton diffusion and dissociation dynamics To produce an electrical current, photogenerated excitons must reach a DA interface before returning to the ground state, and as discussed in Section 10.2.2, most OSCs have a short exciton diffusion length, which places limitations on the maximum domain size of the BHJ structure. If domains are too large, too many excitons created in the interior of the domains are unable to reach an interface for dissociation. A number of KMC simulations have been used to probe this phenomenon and provide predictions about the optimal domain size when including exciton diffusion limitations with other domain size-dependent processes (Watkins et al., 2005; Yang and Forrest, 2008; Lyons et al., 2012). However, a number of studies have indicated that the domain-size restrictions due to the short exciton diffusion length may not always be so strict (Banerji, 2013; Caruso and Troisi, 2012). If excitons are delocalized over a larger volume, charge transfer can occur even when the exciton is centered on a molecule that is not at the interface. Another side effect of delocalization is a significant change in the exciton dissociation dynamics. When excitons are localized, the majority of the excitons must take time to diffuse to an interface for dissociation. However, when delocalized, a significant fraction of the excitons can dissociate immediately after creation. Guo et al. first experimentally quantified this behavior in P3HT:PCBM blends and determined that up to 50% of the excitons undergo immediate dissociation, and the rest require diffusion prior to dissociation (Guo et al., 2010); based on this finding, they concluded that P3HT has an exciton delocalization radius of about 4–7 nm. Taking this idea even further, Kaake et al. (2012) proposed that ultrafast dissociation of highly delocalized (>30 nm) excitons is a dominant mechanism occurring in many BHJ blends. In support of this conclusion, Banerji has highlighted a number of other experimental studies that support similar conclusions (Banerji, 2013). Probing this phenomenon further, Heiber and Dhinojwala used KMC simulations to model Guo’s experimental data and demonstrated the interplay among domain size, domain purity, and prompt dissociation. They estimated that excitons are likely to be significantly less delocalized than previously concluded from simpler analysis techniques (Heiber and Dhinojwala, 2013). They found that a significant amount of prompt exciton dissociation can be expected if domains are impure or if there is significant interfacial mixing and that the observation of prompt dissociation cannot be safely attributed to exciton delocalization without taking into account the details of the BHJ morphology. Overall, it remains unclear how much exciton delocalization increases the maximum domainsize limitation in state-of-the-art OPVs.

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10.2.3.3 Charge separation The efficient splitting of CT states into free charge carriers is a critical step in maximizing photocurrent generation. While a number of DA blends have demonstrated highly efficient charge separation, many others have not. Explaining how the donor and acceptor chemical structure, the resulting morphology at the interface and within the domains, and the energy levels of the relevant states each contribute to the resulting charge separation has been a hotly debated, fundamental physical question of major importance for the development of OPV materials. As a result, a number of recent reviews have been written that describe in greater detail the experimental and theoretical developments related to the charge-separation process (Dimitrov and Durrant, 2014; Gao and Ingan€as, 2014; Few et al., 2015). KMC simulations have played a key role in the development of new models and the understanding of the charge-separation process due to the ability to implement and then test new conceptual models. Immediately after exciton dissociation, traditionally the separation between the two charges in the CT state is assumed to be small enough that there should be an appreciable CT-state Coulomb binding energy, as depicted in Fig. 10.2A. In the traditional Onsager-Braun bound polaron-pair model, a thermally and/or field-activated separation process is required to overcome this binding energy and create free charges, and the overall separation yield depends on the CT state’s lifetime (Braun, 1984). When extending beyond the analytical Onsager-Braun model to investigate bound CT states at DA heterojunctions, KMC studies have shown that the overall chargeseparation yield and the electric-field dependence are both significantly affected by the presence of the heterojunction itself (Peumans and Forrest, 2004; Groves et al., 2008; Wojcik et al., 2010), and also the temperature (Offermans et al., 2005), electric field (Peumans and Forrest, 2004; Groves et al., 2008, 2010), mobility (Groves et al., 2008; Wojcik et al., 2010; Heiber and Dhinojwala, 2012), BHJ domain size (Marsh et al., 2007; Groves et al., 2008), and the energetic disorder of the materials (Offermans et al., 2005; Groves et al., 2010; Heiber and Dhinojwala, 2012). In addition, the separation yield remains critically dependent on the CT state’s lifetime (Offermans et al., 2005; Wojcik et al., 2010; Deibel et al., 2009a; Heiber and Dhinojwala, 2012). KMC simulations built on the bound-CT-state model largely predict relatively low charge-separation yields at device-relevant electric fields unless a very long CT-state lifetime is assumed (Offermans et al., 2005; Marsh et al., 2007; Heiber and Dhinojwala, 2012). In order to explain the highly efficient charge separation that occurs in many optimized BHJ blends, a number of theories have been proposed. Peumans and Forrest (2004) proposed and implemented a hot-CT-state dissociation model into KMC simulations, in which the initial charge-separation distance (thermalization radius) of the relaxed CT state simply correlates with the excess energy of exciton dissociation, without simulating the thermalization process itself. Building on this early work, the hot-CT-state model has evolved to its current form, where it is assumed that hot, delocalized CT states with a lower binding energy are formed, which makes it easier for charge carriers to separate, as depicted in

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Bound CT state model

(A)

Potential energy

Hot delocalized states

Hot CT state model

Potential energy

(B)

Charge delocalization model

Potential energy

(C)

(D)

Interfacial energy cascade model Interfacial disordered region

Charge separation distance

Fig. 10.2 Leading charge-separation models. Schematic diagram of the energetic landscape in leading charge-separation models: (A) Bound-CT-state model, (B) hot-CT-state model, (C) charge-delocalization model, (D) interfacial energy cascade model.

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Fig. 10.2B. Hot-CT states contain an electron and hole that are able to move much faster and diffuse away from each other. However, the benefit of the excess energy lasts only as long as the charge carriers retain the excess energy while occupying the higher vibrational and/or electronic states, typically lasting only around 0.5 ps (Jailaubekov et al., 2013). Heiber and Dhinojwala (2012) expanded the Peumans model by also simulating the thermalization process and including a parameterized excess energy thermalization rate and an enhanced mobility for hot charge carriers. However, they showed that a major mobility enhancement, an extremely slow excess energy relaxation rate, or both were needed to increase the charge-separation yield significantly. More recently, Jones et al. (2014) probed this problem further in model BHJ blends without simulating the thermalization process, and similarly concluded that an unexpectedly large thermalization radius is needed to reach high-separation yields. Nevertheless, the hot-CT-state model has been supported by a number of studies that find a correlation between the DA offset energy and the free charge-carrier generation yield (Dimitrov and Durrant, 2014) or show that charge separation through hot states is more efficient (Bakulin et al., 2012; Grancini et al., 2013; Dimitrov and Durrant, 2014). At the same time, in at least some blends, efficient charge separation has been observed from relaxed CT states (Vandewal et al., 2014) and in blends with a negligible energetic offset driving force (Menke et al., 2017). In such cases, even if excess energy relaxation is not needed to produce a high-charge separation yield, another related explanation for this phenomena proposed by many studies is that relaxation of the charge carriers into the DOS tail provides an additional longer-time-scale (100 ps) energetic relaxation process that promotes charge separation (Offermans et al., 2003, 2005; Groves et al., 2010; van Eersel et al., 2012; Howard et al., 2014). In this model, the charge-carrier mobility at short times is significantly enhanced until the charge carriers relax into the DOS tail. As a result, the separation yield depends not only on the width of the DOS, but also on the initial energetic position of the electrons and holes within their respective DOS distributions (Groves et al., 2010). Alternatively, Deibel et al. (2009a) proposed that the delocalization of relaxed hole polarons along a polymer chain could be a primary cause of efficient charge separation in polymer:fullerene blends and used KMC simulations to demonstrate a large increase in charge-separation yield when implementing delocalization. This idea has evolved to become the charge delocalization model, whereby following charge transfer, the electron and/or hole spreads out along a polymer chain or between adjacent molecules, causing the effective Coulomb attraction between the electron and hole to be greatly reduced, as depicted in Fig. 10.2C. The delocalization is assumed to be present even in the relaxed polaron states, and not a transient property associated with excess energy. Deibel et al. implemented hole delocalization by placing partial charges along a polymer-chain segment (Deibel et al., 2009a), but delocalization also can be implemented by treating the charge carriers as Gaussian spheres (Gagorik et al., 2015). In addition to charge delocalization, exciton delocalization has been proposed to have a significant impact. Guo et al. (2010) proposed that the magnitude of exciton delocalization controls the initial electron-hole separation distance following exciton

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dissociation. KMC simulations including exciton delocalization by Heiber and Dhinojwala (2012) have shown that the magnitude of the exciton delocalization radius can have a strong impact on the separation yield. However, like in the case of the hotCT-state model, a very large initial separation distance resulting from highly delocalized excitons is needed to reach the high-separation yields observed experimentally, and as discussed in the “Exciton diffusion and dissociation dynamics” subsection, there are still questions as to whether such highly delocalized excitons are present. Another prominent explanation is the interfacial energy cascade model. Experimental studies have shown that disorder in many OSCs increases the band gap due to a shift in the energy levels. Due to greater disorder near the DA heterojunction, McMahon et al. (2011) proposed that the energy of the transport states near the interface is higher than the domain interior, as depicted in Fig. 10.2D. In addition to disorder, complex electrostatic and induction effects can alter the energetic landscape near the interface and provide an energy gradient that favors charge separation (D’Avino et al., 2016). As a result, there can be an energetic driving force that promotes charge separation. Groves (2013b) has shown using KMC simulations with a simple interfacial energy cascade model that there is an overall increase in the charge-separation yield and a reduced electric-field dependence that depends on the magnitude of the cascade energy shift. However, he also demonstrated that if the interfacial region is too thick, the separation yield decreases again because the charge carriers are unable to reach the lower-energy regions before geminate recombination occurs. In addition, detailed morphological measurements have shown that the more disordered interfacial regions form diffuse/mixed interfaces (Collins et al., 2011). While the disordered mixed phase is still proposed to act as an energy-cascade interlayer so long as pure, more ordered domains are still present ( Jamieson et al., 2012), the complex interface can complicate the resulting behavior. Using KMC simulations, Lyons et al. (2012) demonstrated that charge-carrier percolation limitations in the mixed regions can increase the geminate recombination losses, and Burke and McGehee (2014) also showed a decreased separation yield when the interfacial region is mixed. In addition, increased disorder near the interface would likely be expected to reduce charge delocalization, thereby increasing the CT-state binding energy and reducing the charge-carrier mobility. Even with the interfacial energy cascade, Burke and McGehee (2014) showed that the separation yield is still strongly dependent on the mobility and the CT-state lifetime. In recent years an entropic driving force has been gaining traction to explain the charge-separation process. Early on, Clarke and Durrant (2010) proposed that entropic effects should be included in the conceptual explanation for the charge-separation process, and that a significant entropic driving force for charge separation exists at a DA interface. Instead of being simply dictated by the electrostatic potential as a function of the charge-carrier separation distance, they argued that the charge-separation process should be thought of in terms of the overall free energy of the system as the charges separate, which includes the gain in configurational entropy due to the large number of charge-separated states. Gregg provided further analysis of this concept by quantifying the entropy gain and total free-energy change as charges separate,

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depending on the dimensionality of the system (Gregg, 2011), and later, KMC simulations confirmed the impact of dimensionality (Giazitzidis et al., 2014; Nan et al., 2016). However, further theoretical work also has shown how energetic disorder limits the entropy gain due to fewer accessible states (Savoie et al., 2014; Hood and Kassal, 2016), and the entropic driving force can be lowered by reducing the BHJ domain size (Vithanage et al., 2013) and by having mixed interfaces (Lyons et al., 2012). In addition, the majority of studies have used equilibrium arguments to describe a definitively nonequilibrium process, and Giazitzidis et al. (2014) have calculated the free-energy curve under nonequilibrium conditions and found significant increases in the free energy of charge-separated states. Overall, while the entropic driving force certainly can play a role, it should not be argued to be the dominant reason for the unexpectedly high-charge separation yield. As Baranovskii et al. (2012) point out, Onsager theory already inherently includes entropy effects. In addition, we emphasize here that the numerous early KMC studies showing high-geminate recombination when implementing the bound-polaron-pair model already include entropy effects.

10.2.3.4 Charge transport in bulk heterojunction films While many studies have investigated the details of charge transport in neat OSC films, less is known about how the complex BHJ microstructure affects charge transport in OPVs. However, KMC simulation studies using model BHJ morphologies have revealed significantly lower mobilities than in a neat material due to the morphology (Frost et al., 2006; Groves et al., 2009), and showed that the presence of bottlenecks and dead ends can lead to charge buildup and increased recombination (Donets et al., 2013). On the other hand, morphologies containing domains with enhanced vertical alignment and direct charge-transport pathways give better charge-transport efficiency and less recombination (Donets et al., 2015). Using a master equation approach, Koster (2010) also probed charge transport in model BHJ blends and showed that the electric-field dependence of the mobility changes from positive to negative when comparing neat and BHJ films at low charge-carrier densities. Finally, Koster (2010) explained how the opposing domains in a BHJ morphology represent barriers to transport and that, when a barrier forces charge carriers to hop perpendicular to, or even against the electric field to bypass the barrier, these hops are driven by diffusion, and an increased electric field does not assist in their motion. In fact, an increased electric field can even hinder long-range transport if hopping against the field direction is needed to bypass a barrier. Expanding on this work further, Heiber et al. (2017) investigated how the quality of the charge-transport pathways in various BHJ morphologies, quantified by the average tortuosity, affects the transport behavior relative to a neat material system as a function of energetic disorder and temperature. KMC time-of-flight transport simulations showed that as tortuosity increases, the overall magnitude of the mobility is greatly reduced, and there is a dramatic decrease in electric-field dependence, as shown in Fig. 10.3. Interestingly, negative field dependence was shown not to be an inherent property of BHJ blends, but instead, the field dependence was found to depend on a combination of the tortuosity, energetic disorder, and temperature.

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σ/kBT = 3 Neat, t = 1 BHJ (d = 8 nm), t = 1.05 BHJ (d = 8 nm), t = 1.10 BHJ (d = 8 nm), t = 1.15 BHJ (d = 8 nm), t = 1.24

0.4

(A)

0.6

(B)

0.8

1.0

1.2

(F/F0)1/2

Fig. 10.3 Charge transport through a tortuous BHJ morphology. (A) Schematic showing how charge carriers must navigate through a complex, tortuous pathways. (Reproduced with permission from Heiber, M.C., Kister, K., Baumann, A., Dyakonov, V., Deibel, C., Nguyen, T.Q., 2017. Impact of tortuosity on charge-carrier transport in organic bulk heterojunction blends. Phys. Rev. Appl. 8, 054043.) (B) The impact of tortuosity (τ) on the electric-field dependence of the charge-carrier mobility. (Data from Heiber, M.C., Kister, K., Baumann, A., Dyakonov, V., Deibel, C., Nguyen, T.Q., 2017. Impact of tortuosity on charge-carrier transport in organic bulk heterojunction blends. Phys. Rev. Appl. 8, 054043.)

In addition to morphology effects, KMC modeling of experimental data has demonstrated that the charge-carrier mobility decreases over time as charge carriers relax into the DOS tail, and in some blends, this process is slow enough that quasiequilibrium mobilities are poor descriptors of the charge extraction process in at least some OPVs (Melianas et al., 2014, 2015). Not only does this phenomenon affect the chargeseparation yield, as we discussed in the “Charge separation” subsection, but it also potentially complicates the operation of BHJ OPVs and experimental device characterization. While it has not been shown how detailed morphological features combine with the DOS to affect mobility-relaxation behavior, morphological traps have been proposed to slow mobility relaxation further (Melianas et al., 2014).

10.2.3.5 Bimolecular charge recombination In most well-performing OPVs with efficient charge separation, bimolecular charge recombination is the dominant loss pathway and must be minimized in order to realize more efficient devices. We refer the reader to more comprehensive review papers discussing bimolecular charge recombination in OPVs for further details (Proctor et al., 2013; Lakhwani et al., 2014; G€ ohler et al., 2018). Bimolecular recombination is defined as a second-order process, Rrec ¼ 

dn ¼ kbr np, dt

(10.2)

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where kbr is the bimolecular recombination rate coefficient, n is the electron density, and p is the hole density. The starting model for bimolecular recombination in OPVs has been the Langevin model (Langevin, 1903), which assumes an encounter-limited reaction where the time it takes for an electron and hole to meet in the film is rate limiting. The result is that the recombination coefficient is primarily dictated by the mobility of the charge carriers. The Langevin model also assumes a spatially and energetically homogeneous and isotropic system, which is not strictly valid in BHJ OPVs. Typically, electrons are restricted to the acceptor phase and holes to the donor phase, and recombination can occur only at a DA interface. These spatial limitations on charge-carrier motion and recombination location are expected to alter the bimolecular recombination rate and cause deviations from the Langevin model. Perhaps not surprisingly, BHJ OPVs have frequently exhibited two major deviations. As recently summarized by Heiber et al. (2016), a literature survey shows first, that super-second-order recombination kinetics has been measured in many blend systems, and second, that the measured recombination rate sometimes is found to be up to several orders of magnitude lower than predicted by the Langevin model. As a result, assuming equal electron and hole densities, the recombination rate can be defined in greater detail as Rrec ¼ kbr ðnÞn2 ! Rrec ∝ n2 + δ ,

(10.3)

where the apparent higher recombination order is typically attributed to a carrier density-dependent recombination coefficient ðkbr ðnÞ∝nδ Þ. To quantify the so-called reduced recombination phenomenon, the magnitude of kbr is compared to the Langevin model, resulting in the characteristic reduction factor ðζ ¼ kbr =kL Þ. When highly reduced recombination was first observed, Pivrikas et al. (2005) proposed that a reduced recombination rate is an inherent property of BHJ blends due to the spatial segregation of the charge carriers, and a large number of subsequent studies have adopted this argument. However, in a number of other BHJ blends, recombination rates much closer to the Langevin model have been observed. Clarke et al. (2005) showed that even blends with very similar morphologies can have dramatically different bimolecular-recombination rates. While the measured effective reduction factor can be significantly smaller than the true value due to charge-carrier concentration gradients within the active layer (Deibel et al., 2009b), reduced recombination still has been observed in cases where there are not significant gradient effects. Understanding the various factors that ultimately determine kbr has been a critical fundamental endeavor. KMC simulations have shown that the domain size of model BHJ structures do not alter the recombination order, and when using a Gaussian DOS, second-order recombination kinetics are observed (Heiber et al., 2015; Coropceanu et al., 2017). However when using an exponential DOS, the recombination order is correlated with the characteristic energy of the distribution (Nelson, 2003; Coropceanu et al., 2017; G€ohler et al., 2018). In such a case, an exponential DOS leads to carrier density-dependent mobilities that then cause the carrier density dependence of the recombination

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coefficient. In addition, questions remain how to explain observations of blends where transport behavior appears to indicate a Gaussian DOS and recombination behavior indicates an exponential DOS (Gorenflot et al., 2014). As recently summarized by Street (2015), one leading explanation is that a small density of exponential trap states forms at the DA interface and recombination through these trap states accounts for the higher-order kinetics. In such a case, less traps may be present in the interior of the domains, and charge transport may be dictated by hopping within the primarily Gaussian DOS present in the interior of the domains (Gorenflot et al., 2014). Another idea is that regardless of the DOS shape, the charge-carrier mobilities can relax over time as the charge carriers relax into the DOS tail, as discussed in the OPV charge-transport subsection. Under these nonquasiequilibrium conditions, the charge-carrier mobility is primarily time-dependent. In such cases, the recombination coefficient can be timedependent as well, which causes super-second-order recombination kinetics that does not directly correlate with the DOS shape (Mozer et al., 2005; Clarke et al., 2012; Kurpiers and Neher, 2016). An important first step toward a more complete understanding of reduced recombination has been the testing of the fundamental effect of phase separation on kbr under encounter-limited conditions. In early theoretical work, Koster et al. (2006) developed the minimum mobility model, arguing that the bimolecular-recombination rate in a BHJ blend should be limited by the mobility of the slowest carrier. Groves and Greenham (2008) then used KMC simulations to show that the recombination rate lies somewhere between the Langevin model and the minimum mobility model with a weak dependence on domain size. However, other KMC simulations (Hamilton et al., 2010) have indicated that the domain size could have a much larger impact. Expanding on this work, Heiber et al. (2015) investigated the combined impact of the domain size and the mobilities on the recombination rate coefficient and the reduction factor, as shown in Fig. 10.4. Heiber et al. (2016) then further investigated the charge-carrier density dependence to develop the power mean model. The power mean model predicts a continuous deviation from the Langevin model as the domain-size increases that begins to approach the minimum mobility model when domains are very large. Most importantly, this work, as well as previous results by Groves and Greenham (2008) show that phase separation itself reduces the recombination coefficient by only about a factor of 2 when using domain sizes that are typical of optimized OPV devices and equal mobilities, and that highly reduced recombination is not an inherent property of BHJ blends (Heiber et al., 2015, 2016). Instead, the leading explanation for highly reduced recombination is that the recombination rate is not encounter-limited; rather, it is highly dependent on the lifetime of the CT state and the CT redissociation rate (Hilczer and Tachiya, 2010; Ferguson et al., 2011; Burke et al., 2015; Heiber et al., 2016; Coropceanu et al., 2017). Given the importance of the CT state’s lifetime and dissociation rate in the initial charge-separation process, correlations between efficient charge-separation behavior and reduced bimolecular recombination would seem likely, but the connections between the two phenomena have not been deeply investigated. Future work making connections between the two may help refine the range of competing models for each process and arrive at a more broadly applicable unifying model.

10 –8

d = 5 nm d = 10 nm d = 15 nm d = 20 nm

d = 25 nm d = 35 nm d = 45 nm d = 55 nm

10 –9

10 –10

0.1

me/m h = 1 me/m h = 10 me/m h = 100

kL (d = 5) kmin (d = 55) 10 –11

0.01 10–3

(A)

325

1

10 –7

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Recombination coefficient, kbr (cm3s–1)

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(m em h)

0

10–2 2

–1 –1

(cm V s )

(B)

20

10

30

40

50

60

Domain size (nm)

Fig. 10.4 Encounter-limited bimolecular recombination in a BHJ. The impact of domain size and charge-carrier mobilities on the bimolecular recombination coefficient (A) and reduction factor (B). (Data from Heiber, M.C., Baumbach, C., Dyakonov, V., Deibel, C., 2015. Encounter-limited charge-carrier recombination in phase-separated organic semiconductor blends. Phys. Rev. Lett. 114, 136602.)

10.2.4 Modeling organic light-emitting diodes In a simple sense, OLEDs operate opposite to OPVs. In an OLED, charge carriers are injected at the opposing electrodes and ideally meet within the active emissive layer to form an exciton that then decays radiatively. As a result, understanding how charge carriers are injected into the OSC layers, transported through the material, and then eventually meet is important. And understanding exciton formation, diffusion, and radiative properties is needed as well. In addition, any nonradiative recombination mechanisms will lower the quantum efficiency of the device and must be minimized. All together, controlling each of these processes is critical for designing highefficiency OLEDs. We refer readers to more comprehensive reviews for details about the operation and design of OLEDs (Murawski et al., 2013; Reineke et al., 2013), and Kordt et al. (2015) have published an informative review on OLED modeling. Here, we highlight specific examples where KMC simulations have played a key role in understanding the fundamental device physics and processes in OLEDs. A variety of early KMC studies were done to model the injection of charges from a metal into a semiconductor layer using a hopping model, showing changes to the electric-field and temperature dependence of current injection due to charge hopping from the Fermi level of the metal into the tail of the DOS (Gartstein et al., 1996; Wolf et al., 1999). As a result of this phenomenon, a number of KMC and master equation studies have predicted the formation of current-density filaments in neat, disordered films (Yu et al., 2001; Tutisˇ et al., 2004; van der Holst et al., 2011). In a particularly thorough study, van der Holst et al. (2011) used KMC simulations of single-carrier diodes to show how the filaments change due to the injection barrier height (Δ)

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Correlated disorder s/kBT = 3, D = 0 eV

(B)

(A)

Correlated disorder s/kBT = 6, D = 1 eV

Uncorrelated disorder s/kBT = 6, D = 1 eV

(C)

(D)

Fig. 10.5 Filamentary charge transport in single-carrier diodes. Visualization of current-density hot spots from KMC simulations of charge transport in single-carrier diodes comparing cases of low energetic disorder and no injection barrier with an uncorrelated (A) and a correlated (B) Gaussian DOS and a high-injection barrier and high-energetic disorder with an uncorrelated (C) and a correlated (D) Gaussian DOS. (Adapted with permission from van der Holst, J.J.M., van Oost, F.W.A., Coehoorn, R., Bobbert, P.A., 2011. Monte Carlo study of charge transport in organic sandwich-type singlecarrier devices: effects of Coulomb interactions. Phys. Rev. B 83, 085206.)

and the energetic disorder (σ) with both an uncorrelated and correlated Gaussian DOS, as shown in Fig. 10.5. One of the primary drivers of this phenomenon is the chargeinjection hot spots due to fluctuations in the injection barrier resulting from the energetic disorder (Tutisˇ et al., 2004; van der Holst et al., 2011). As shown in Fig. 10.5A and B, with no injection barrier and low energetic disorder, injection hot spots are weak and the filaments largely disperse as they move away from the bottom injecting surface (van der Holst et al., 2011). However, the filaments can be enhanced further by the presence of correlated energetic disorder, as depicted in Fig. 10.5C and D (Yu et al., 2001; van der Holst et al., 2011). In general, the effect of the currentdensity filaments is particularly enhanced when the films are thin. Given the possible presence of significant current-density hot spots in the active layer, the primary question for OLEDs is how this affects the exciton-formation process. Most likely, injection hot spots for electrons and holes will not be correlated, so it may be more difficult for electrons and holes to encounter each other; and major questions include how large this effect is and how it can be minimized to create high-radiative-efficiency devices. In simple cases, recombination is expected to be described by the Langevin model, but KMC simulation studies have identified conditions where deviations from the Langevin model occur, due to transport anisotropy (Ries and B€assler, 1984; Gartstein et al., 1996; Groves and Greenham, 2008), energetic disorder (Richert et al., 1989; Gartstein et al., 1996), and the electric field (Albrecht and B€assler, 1995; Gartstein et al., 1996). While van der Holst et al. concluded that the Langevin model still works well in an isotropic system with the an uncorrelated Gaussian DOS at low electric fields so long as accurate mobility values

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are used (van der Holst et al., 2009), in cases where significant current-density filaments are expected, exciton formation and the resulting emission profiles can be highly inhomogeneous (Mesta et al., 2013; Shen and Giebink, 2015). Mesta et al. (2013) have shown that capturing these details is particularly important for accurately modeling and designing multilayer white OLEDs. In many of the more efficient phosphorescent OLEDs, in which photon emission comes from triplet excitons, so-called efficiency roll-off occurs where the quantum efficiency and luminous efficacy of the device decreases at higher current densities (Murawski et al., 2013). Due to the generally longer lifetime of triplet excitons, the nonradiative recombination mechanisms of triplet-triplet annihilation and tripletpolaron quenching can become significant due to the higher steady-state densities of triplet excitons and charge carriers in the device when operating at higher current densities or in current-density filament regions. To reduce this loss process and improve OLED performance, understanding the origins of efficiency roll-off is critical. A number of KMC studies have studied these phenomena to construct predictive models that relate electronic properties to expected losses so that device design can be optimized. In a particularly illuminating study, van Eersel et al. (2014) implemented a multilayer KMC model to fit experimental electrical and optical measurements and study the competing loss mechanisms and demonstrated the dominance of the triplet-polaron-quenching mechanism in two common host-guest OLED blends. Coehoorn et al. (2015) then showed that the triplet-polaron-quenching mechanism also can be linked to emitter molecule degradation. In these cases, triplet-polaronquenching causes degradation of OLED devices, and there is a direct relationship between roll-off behavior and device stability. Shen and Giebink (2015) further showed the molecular-degradation hot spots that can form due to the filamentary charge transport and the locally enhanced triplet-polaron quenching that occurs there. As new approaches are implemented to mitigate the undesirable roll-off and degradation processes, KMC simulation and modeling are sure to continue to be vital to understanding the complex kinetics and the impact of materials’ structural features.

10.3

Macroscopic simulation

When wanting to simulate full device behavior, especially at high charge-carrier densities, KMC simulation techniques can become prohibitively expensive, computationally speaking. Thus, alternative techniques are required. A major increase in calculation speed is obtained if the system is no longer calculated stepwise for each exciton or charge carrier, but rather by an analytic system of equations that solves the problem for all of them simultaneously. The drift-diffusion system of equations used for macroscopic simulations also offers the unique possibility to apply analytic theories without the requirement of numerical assumptions, such as the cubic lattice often used in KMC simulations. Here, we present a brief overview of how state-of-theart drift-diffusion simulations for application in OSC devices are built, and then highlight the important developments in the fields of OPV and OLED research using drift-diffusion simulations over the last decade.

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10.3.1 Introduction to drift-diffusion simulations The drift-diffusion system of equations is based on the Boltzmann-transport equation, which generally describes particle transport due to particle-density gradients and applied driving forces. For charge transport in semiconductor devices, this equation is often reduced, but not limited to, drift and diffusion in three dimensions, as follows: Jn ðxÞ ¼ qnðxÞμn FðxÞ qDn rnðxÞ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} + |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} , drift

(10.4)

diffusion

which is often denoted as the drift-diffusion equation. In this example for electrons, the electron current density (Jn) is determined by the elementary charge (q), the local electron density (n), the electron mobility (μn), the local electric field (F), the electron diffusion coefficient (Dn), and the local electron density gradient (rn). Most of these (bold) parameters can have a dependency on the spatial coordinate x. In the 1960s, efficient methods to discretize the drift-diffusion equation into a series of finite elements were developed. One of the main issues was that small changes in the electric field can lead to an exponential change of the charge-carrier density. This implies that the charge-carrier density does not scale linearly between two elements and that intermediate results might exceed the numerical range, which leads to severe numerical errors during the calculation. Scharfetter and Gummel developed a mathematical discretization scheme still used today for most drift-diffusion simulations. In its original and commonly used form, it requires the validity of the Einstein relation and ignores spatial variations in the charge-carrier mobility (Selberherr, 1984). The Einstein relation was demonstrated to be valid in OSCs, at least with nondispersive transport, where charge carriers are thermalized in the DOS (Wetzelaer et al., 2011). The charge-carrier mobilities can strongly depend on the charge-carrier density due to the energetic and spatial disorder that causes the broadened density of localized states. This implies that the current flow due to positional variations of the charge-carrier mobility must be smaller than drift or diffusive charge transport to produce valid simulation results. To simulate a real device, an analogous equation to Eq. (10.4) must be defined for holes as well as for all other particles of interest, such as excitons and CT states. Then, the temporal dependence of each particle type is determined using the continuity equation, X X ∂n 1 ¼ rJn ðxÞ  Ri ðxÞ + Gj ðx,tÞ, ∂t q i j

(10.5)

shown here for electrons. The continuity equation defines the current transport to and from each discretized spatial element, as well as the generation rate and recombination rate within each element. For each particle type, the generation rate is a sum of all possible generation mechanisms, and the recombination rate is a sum of all possible depletion mechanisms into other particles or the ground state. These rates interconnect

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all particle densities and result in the final population ratio between all considered particle densities by following detailed balance principles. Next, the electric field generated by the spatial distribution of charge carriers is calculated using the Poisson equation, rFðxÞ ¼ 

q ðn  p + CÞ, E0 Er

(10.6)

where Er is the relative dielectric constant, E0 is the vacuum permittivity constant, and C is the doping concentration. Spatial variations of Er are easily included in order to implement a stack of semiconducting layers. Finally, the system of equations created by Eqs. (10.4)–(10.6) is solved numerically. A variety of efficient approaches have been developed during the last decades. Which system is used mainly depends on the desired applied-voltage regime, the convergence accuracy of the selected algorithm, and the convergence speed. They all include a transformation of the physical variables into a numeric number space, a simultaneous or subsequent solution of all considered equations in an iterative loop, and then a retransformation into physical parameters. The discretization of the spatial dimensions is often disregarded in the implementation of drift-diffusion simulations. Mathematical requirements exist for the maximum distance in space (and in time) between two calculation steps to achieve valid simulation results (Selberherr, 1984; Mock, 1983). As an alternative to implementing and solving the drift-diffusion system of equations oneself, many open-source and commercial software packages are available to facilitate a first contact with drift-diffusion simulations. The most prominent examples are PC1D (sourceforge.net/projects/pc1d/), ASA (www.tudelft.nl), Setfos (www.fluxim.com/setfos/), Sentaurus (www.synopsys.com), and gpvdm (www. gpvdm.com).

10.3.2 Parameterization of disorder effects Drift-diffusion simulations require an analytic model for describing the drift and diffusion transport processes of each particle. This is particularly critical for charge-carrier transport, where a microscopic hopping model must be converted to a macroscopic analytic form. For disordered OSCs, the two most common models are the multiple-trapping-and-release (MTR) model and various empirical forms of the GDM.

10.3.2.1 Multiple-trapping-and-release model The MTR model considers the transport of an ensemble of localized charge carriers in a disorder broadened DOS (Shklovskii and Efros, 1984). The distribution is mostly assumed to be Gaussian or exponential, but one is not limited to one of these two examples. For hopping transport, charge carriers can either hop to an energetically lower state or they can be thermally excited to a state with higher energy.

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The restrictions are the limited thermal energy and availability of vacant states within the range of the hopping mechanism. The MTR model assumes that charge carriers at or above the so-called transport energy (Etr) can be interpreted as mobile, comparable to band transport, but charge carriers below Etr are interpreted as energetically trapped until they are again released to the transport energy. There are several ways of calculating the transport energy based on different hopping models (Oelerich et al., 2014), but the MTR model effectively discretizes the total population of charge carriers into two distinct groups. As a result, charge-carrier mobility always has to be considered as either the mobility of solely conductive charges (nc), or more commonly, as the lower average mobility of all charge carriers (n ¼ nc + nt). In the latter case, the effective mobility (μ) can be parameterized by the fraction (θ) of free charge carriers to the total charge-carrier density, and the corresponding mobility of the free charge carriers (μc), μ ¼ μc  θ ¼ μc

nc : nt + nc

(10.7)

The density of each subpopulation is calculated by integrating over all occupied states relative to the transport energy, nt ¼ nc ¼

ð Etr

DOSðEÞ  fFD ðE,EF , TÞdE

(10.8)

DOSðEÞ  fFD ðE, EF ,TÞdE

(10.9)

∞

ð∞ Etr

with the Fermi-Dirac function fFD(E, EF, T), where E is the state energy, EF is the Fermi energy, and T is the temperature. Whether the DOS is better described as exponential distribution, Gaussian distribution, or even a more complex function is still under consideration (Baranovskii, 2014). A Gaussian DOS distribution shows a fairly constant ratio of nc/nt for most relevant charge-carrier density regimes for OPV and OLED applications. An exponential DOS results in an exponentially decreasing share of conductive charges with decreasing total charge-carrier density (Mehraeen et al., 2013). The different behavior originates from the amount of available states above the Fermi energy, as shown in Fig. 10.6. A larger amount of available hopping sites allows for higher charge-carrier mobilities in Gaussian DOS distributions. Chargecarrier density-dependent mobilities with any distribution of the DOS can be directly calculated with drift-diffusion simulations (Baranovskii, 2014).

10.3.2.2 Gaussian disorder models Another approach to modeling the charge-carrier mobility is to fit results of KMC transport simulations using empirical analytic expressions. Such an approach was successfully introduced by B€assler (1993) to parameterize the mobility in a Gaussian DOS, known as the GDM. The model captured how the mobility depends on the

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e

e

Transport energy

DOS

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Transport energy

DOS

s2 kT

eF

eF Carrier density Carrier density

Fig. 10.6 Influence of the DOS on mobility. Quasiequilibrium density of occupied states and transport energies of an exponentially distributed density of states (left) and Gaussiandistributed density of states (right). (Reproduced with permission from Baranovskii, S.D., 2014. Theoretical description of charge transport in disordered organic semiconductors. Phys. Status Solidi B 251, 487–525.)

energetic disorder (σ) and temperature (T) in the low charge-carrier density regime, and includes a Poole-Frenkel dependence on the electric field (F), 

2σ μ∝ μ0 exp  3kB T

2 !

pffiffiffi  exp Aðσ, TÞ F :

(10.10)

Additional fit parameters are summarized by A. Due to shortcomings with the original GDM, several extensions have been derived to more accurately capture additional physical phenomena. The correlated disorder model (CDM) introduces spatial correlations between site energies due to randomly oriented permanent dipoles, which extends the Poole-Frenkel electric-field dependence of the mobility to low field strengths (Novikov et al., 1998). Later, the extended GDM was developed to capture the charge-carrier density dependence that appears at higher carrier densities, but again without considering energetic correlation. Eventually, both extensions were merged to form the extended correlated disorder model (ECDM) (Bouhassoune et al., 2009). However, these models were all derived from lattice KMC simulations, and recent work has shown how the electric-field dependence of the mobility changes when including positional disorder in off-lattice KMC simulations and has argued for the use of an effective temperature model in drift-diffusion simulations (Oelerich et al., 2017). The GDM-based models limit simulations to scenarios with a Gaussian DOS, but effective temperature models also exist for an exponential DOS. Regardless, any of these transport models can be inserted directly into the drift-diffusion system of equations, so long as the current created by the spatially dependent charge-carrier mobilities remains small enough.

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10.3.3 Modeling organic photovoltaics 10.3.3.1 BHJ morphology effects Simulating OPVs is uniquely challenging due to the complex, 3D BHJ morphology that is required to produce efficient devices. In many cases, the effective medium approach is used in macroscopic simulations, where electrons are transported on the lowest unoccupied molecular orbital (LUMO) of the acceptor phase and holes are transported in the highest occupied molecular orbital (HOMO) of the donor phase. While both levels are separated in energy, no spatial constraints for electrons and holes exist in an effective medium model. Morphology effects on recombination and charge transport, among others, have to be included parametrically. In addition, band-gap differences between the donor and acceptor materials are not treated explicitly, and an effective band gap is used instead. As discussed in the “Microscopic simulation and modeling” section, the details of the BHJ morphologies are often important, so this is one of the main reasons to apply 2D or 3D drift-diffusion models instead of effective medium models. Among the first, Buxton and Clarke (2006) investigated the concentration profiles of excitons, electrons, and holes in 2D morphologies formed by diblock copolymers. Other work considered the optimum ratio between well-mixed BHJs and regions with neat donor and acceptor phases. It turned out that the average domain size in the well-mixed regions is important to the device efficiency. If one compares the influence of well-blended BHJ morphologies containing neat domains of donor and acceptor molecules in a simulated 3D device, the question of whether a complex 3D morphology can be calculated by a 1D drift-diffusion simulation can be answered. It was found that the total current from a solar cell can be approximated by the sum of two 1D simulations: one accounting for currents originating from the mixed phase, and one for charge carriers from an interface region between the mixed-phase and neat (acceptor) domains. Because the current generated from the blended phase always exceeds the contribution of the neat phases, 1D simulations are often sufficient to predict the current-voltage characteristics (Bartesaghi and Koster, 2015). In summary, this means that in contrast to KMC simulations, effects of percolation are often not explicitly addressed in macroscopic simulations. However, effects of a stack of electrically active materials typical for devices can be addressed only in macroscopic simulations, which show variations in the spatial charge-carrier density distributions (to cite one example). This effect makes recombination spatially dependent—a fact neglected by simple recombination models. This is one reason for observed lower-than-expected recombination rates in OSCs by averaging charge-extraction experiments.

10.3.3.2 Charge recombination There are a variety of charge-recombination mechanisms and models, including the Langevin, Onsager-Braun, Shockley-Read-Hall (SRH), Auger, and other models (Wagenpfahl, 2017). Any of these can be directly implemented into a drift-diffusion simulation to probe their influence on the device performance.

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In traditional solar cells, the dominant charge-carrier recombination mechanism can be estimated from current-voltage characteristics. The J-V curve is often treated as an ideal diode described by the Shockley equation. For a perfect pn-junction, the recombination mechanism can be approximated by the ideality factor,  nid ¼

kB T ∂ ln J q ∂V

1

:

(10.11)

Values of unity indicate direct Langevin recombination, whereas values of 2 and 2/3 indicate SRH and Auger recombination, respectively (G€ohler et al., 2018). Most experimental measurements on OPVs do not show distinct values, but rather more complex dependencies, such as charge-carrier density (illumination) or temperature. Drift-diffusion simulations can help to understand this by modeling a more realistic scenario than captured by the Shockley equation. Under the assumption of exponential tail states, ideality factors between unity and 1.5 can be obtained by additional recombination pathways through tail-state traps. Also, the spatial distribution of charge carriers in the device can decrease the ideality factor (Kirchartz and Nelson, 2012). However, ideality factors above 1 are not necessarily a sign of trap-assisted recombination. In contrast to the recombination of geminate pairs such as excitons or CT states, charge-carrier losses mediated by tail states seem to be most relevant to reproduce current-voltage curves with ideality factors not equal to 1 (Soldera et al., 2012). Due to the weak charge-carrier density dependence of mobilities in a system with a Gaussian DOS, exponential tail states or deep traps might be necessary to model current-voltage characteristics with higher ideality factors. Also, numerical fits to current-voltage characteristics point to the relevance of trap-assisted recombination (Liu and Li, 2011). However, the dominant mechanism can depend on illumination conditions. At low illumination intensities, recombination tends to be dominated by trap-assisted recombination, whereas direct Langevin recombination becomes more prominent with higher light intensities around one sun (Tress et al., 2013). With the help of macroscopic simulations, it was shown that current-voltage characteristics of OPVs often cannot be described by the Shockley equation. Instead, the low charge-carrier mobilities typical for OSCs lead to a current-transport-limited current-voltage curve, in which charge-carrier accumulations hinder charge extraction. It is, therefore, much more precise to determine ideality factors from Voc-Jsc pairs with changing temperature or illumination instead of from dark current-voltage curves (Tvingstedt and Deibel, 2016). The ideality factor is not only sensitive to recombination in the bulk of the film but also to surface recombination. In cases where electrons and holes are mutually transferred into a metal electrode, ideality factors less than or equal to 1 are reported. If one charge-carrier species is not efficiently transported across an interface, S-shaped current-voltage curves are found, which no longer follow the Shockley equation and show a plateau of the current increase around the open-circuit voltage. Besides surface recombination, charge-carrier mobility mismatch between two adjacent semiconductor layers, or blocking layers, also have been found to create such S-shaped

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current-voltage curves. The common origin in all cases is the formation of spacecharge regions under certain voltage regimes. Thus, S-shaped curves indicate a current transport limitation during extraction or an injection barrier into the adjacent layer. Corresponding simulations have helped to explain such a charge extraction limitation, thereby enabling researchers to pinpoint the back contact as the bottleneck in one case (H€ ubler et al., 2011).

10.3.3.3 Optimizing material properties and device design A key strength of macroscopic simulations is the ability to simulate current-voltage characteristics and predict how the device performance depends on variations in material properties, such as charge-carrier mobility. Parameters typically used to quantify a solar cell are the open-circuit voltage (Voc), short-circuit current (Jsc), fill factor (FF), and power conversion efficiency (PCE). All of these values are a function of the parameters used to simulate a solar cell. Since the early applications of macroscopic simulations in OPVs, it has become clear that the PCE does not automatically increase with a higher charge-carrier mobility. The reason for this is that, while high mobility leads to a high Jsc and low remaining charge-carrier densities in the device, this is accompanied by a larger Langevin recombination rate and a lower Voc. For low charge-carrier mobilities, Voc is high with a lower recombination rate, but slow charge exaction limits Jsc, which thereby reduces the PCE. It was also demonstrated that balanced (equal) electron and hole mobilities are required to gain a high PCE, as shown in Fig. 10.7 (Wagenpfahl et al., 2010). Rules for designing efficient OPVs have been

Fig. 10.7 Power conversion efficiency as function of balanced mobilities. Driftdiffusion simulations predict optimal mobility to maximize PCE in OPVs for a standard device (solid black line), a device with surface passivation (dashed blue line), and for limited transfer velocities of electrons and holes over both metalsemiconductor interfaces (dotted red line). (Data from Wagenpfahl, A., Deibel, C., Dyakonov, V., 2010. Organic solar cell efficiencies under the aspect of reduced surface recombination velocities. IEEE J. Sel. Top. Quantum Electron. 16 (6), 1759–1763.)

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derived from these findings. Also, the energetic disorder has to be recognized as a relevant factor for the photocurrent. In addition, the question of how to design a device with optimum PCE using a provided set of materials can be answered. The first question may be whether a planar bilayer heterojunction or a BHJ architecture is better. As it turns out, both device structures differ in the charge-carrier distribution as a function of the metal electrode work function and the specific properties of the semiconductor layers, such as the exciton diffusion length (Foertig et al., 2012). Regardless of the heterojunction architecture used, interference effects in thin films must be considered to simulate the exciton generation process accurately. Optical interference has been shown to influence the photocurrent of organic solar cells significantly, as shown in Fig. 10.8. Capturing this phenomenon requires knowledge about the complex refractive index and the thicknesses of all layers to generate accurate optical absorption profiles that determine the overall exciton generation rate and the resulting photocurrent. Especially in complex structures such as tandem solar cells, optical simulations are required to tune the thickness of the layers optimally so that each subcell contributes the same current density and the recombination is minimized. All together, drift-diffusion simulations can combine all optical and electronic effects to predict the optimal heterojunction architecture and layer thicknesses needed to reach the highest PCE (H€ausermann et al., 2009).

–10

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Fig. 10.8 Optical interference effects in P3HT:PCBM solar cells. Transfer-matrix simulations are used to predict optical interference effects in P3HT:PCBM BHJ OPVs, leading to several peaks in the photocurrent as a function of the layer thickness that are reduced by recombination losses. (Adapted with permission from H€ausermann, R., Knapp, E., Moos, M., Reinke, N.A., Flatz, T., Ruhstaller, B., 2009. Coupled optoelectronic simulation of organic bulk-heterojunction solar cells: parameter extraction and sensitivity analysis. J. Appl. Phys. 106 (10), 104507.)

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10.3.3.4 Transient simulations A number of important experiments in OPV research are based on the transient response of devices, which reveals the charge-carrier generation and recombination kinetics. With these techniques, a short illumination pulse generally is used to create charge carriers, and then one measures the transient current response of the device under an applied bias. Transient drift-diffusion simulations allow one to model such experiments, but it is challenging due to issues with the stability of the differential equation solver (Mock, 1983), as well as the limitation of most macroscopic simulations to the transport of thermalized charge carriers. Recent developments show that the limitations of dispersive transport can be overcome (Felekidis et al., 2016). Using transient macroscopic simulations, transient photocurrent curves can be reproduced, and one can conclude that the initial increase of the photocurrent is a function of the free charge-carrier formation, charge-transport, and charge-carrier recombination. Fits to photocurrent transients have shown the influence of charge-carrier trapping and detrapping, as well as the importance of trap-assisted recombination. In some cases, an exponential distribution of tail-state traps seems to be relevant for modeling the experimental data (MacKenzie et al., 2012). Different slopes in the transient photocurrent decay were attributed to changing from a nondispersive regime to a dispersive charge-transport regime within an exponential DOS (Christ et al., 2013). Another typical application of transient experiments is the estimation of chargecarrier mobilities. Depending on the average extraction time after the excitation pulse and the device thickness, the charge-carrier mobility can be calculated. The photo-charge extraction by linear increasing voltage (photo-CELIV) technique applies a linearly increasing voltage ramp to extract the charge carriers and can be used to estimate the charge-carrier mobilities. Because analytic approximations of the photo-CELIV response are often less accurate, transient simulations can provide more accurate parameter extraction (Neukom et al., 2011). These benefits also apply to other experiments, including the standard transient photocurrent measurements.

10.3.3.5 Extracting accurate fit parameters A frequently raised question is whether macroscopic simulations can be used to extract fit parameters from a set of measurements and use them to predict the device behavior accurately. Fits to steady-state current-voltage characteristics are generally possible, but not unambiguous. Parametrical fits often are used to reproduce certain measurements, but it was recently demonstrated that most fit parameters show a correlation to each other, and therefore, they cannot be easily separated. An accurate and unique parameter set can be extracted only from a set of steady-state and transient measurements (Neukom et al., 2012).

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10.3.4 Modeling organic light-emitting diodes Applications of the drift-diffusion model for OLEDs are less numerous in the recent research literature. Due to the progress of the OLED technology, it can be expected that significant work has been conducted in companies without revealing the results to the general public. There should be no doubt that drift-diffusion simulations play a very important role in the design of state-of-the-art OLED devices and displays. However, current transport and calculating the origin of the generated photons is just one step in the development of OLED devices. The propagation and out coupling of the generated light is comparably important. In terms of charge-carrier transport, the current-voltage characteristics of OLEDs can be fit by macroscopic simulations. Tunneling currents from metals into OSCs seem to be occasionally necessary for that purpose (Martin et al., 2005; Altazin et al., 2016). Analogous to OPVs, the influence of charge-carrier injection layers, layer thicknesses, or charge-carrier mobilities can be systematically studied to maximize the radiative recombination of charge carriers. The recombination zone can be engineered by choosing semiconductor layers (or Dopants) that accumulate electrons and holes at a mutual position (Erickson and Holmes, 2013). Charge-carrier gradients and charge-carrier mobilities limit the overall recombination efficiency (Kasparek et al., 2018). Charge-carrier mobilities and activation energies often are measured by charge extraction and impedance spectroscopy experiments. However, it was shown that Arrhenius plots, which are used to examine the temperature dependence of charge-carrier mobilities, are of limited use in certain OLEDs because chargecarrier gradients and nonlinear electric-field distributions inside the device structure lead to systematic errors (Z€ ufle et al., 2017). The integration of single OLEDs to more complex devices such as displays can be seen as the next step. For these applications, the switching (on/off/pulse) behavior of OLEDs and the corresponding light emission must be optimized (Pflumm et al., 2008). Aging effects need to be understood such as the appearance of additional hole traps in aged devices made of the often-used semiconductor known as “super yellow” PPV (Niu et al., 2016). With the knowledge of the radiative recombination efficiency, complex multilayer OLEDs with several recombination centers with different wavelengths can be simulated. In combination with light-propagation calculations, the angular-dependent emission of white-light LEDs can be simulated (Perucco et al., 2012). Even the most complex systems, such as AMOLED displays, can be simulated at large scales, even though additional assumptions and calculations might be required (Diethelm et al., 2018). Overall, drift-diffusion simulations are a powerful tool to model and predict the electrical behavior of OPV and OLED devices in steady-state and transient conditions. Avoiding prohibitive costs of precise microscopic transport models tracking each individual particle, drift-diffusion simulations investigate the influence of materials’ properties and device architectures using a faster analytic framework. Drift-diffusion simulations represent a link between theoretical models and real-life applications, where other models can be applied only in certain regions of a more complex structure. Macroscopic simulations allow this kind of analysis, and therefore, they are an important tool to understand the physics of OPVs, OLEDs, and other devices.

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Conclusions and outlook

As with all computational methods, there are pros and cons to using KMC, driftdiffusion, or other methods. There are physical phenomena uniquely captured by KMC simulations and cases where drift-diffusion modeling is more appropriate. To choose the most appropriate technique, researchers should carefully consider their specific scientific question and the resources that are available. While coarse-grained KMC simulations are often much less computationally intensive than ab initio methods, most device-simulation problems require the use of high-performance computing clusters. Alternatively, less computationally expensive techniques, like driftdiffusion and master equation methods, can be reasonably executed on a personal computer. Despite these significant costs, for problems where one is interested in the impact of materials’ microstructure on device performance or on nonequilibrium phenomena, KMC is usually the most appropriate tool. However, due to the major computational cost of explicit electrostatic interaction calculations and simulating charge injection with Ohmic contacts, fitting KMC simulations to experimental device data is often prohibitively expensive. When wanting to perform extensive experimental device data-fitting, drift-diffusion techniques are much more appropriate. Looking to the future, there are many avenues that will lead to further progress. From an application standpoint, it is still not possible to predict a priori the performance of a particular material or blend in a specific device application. This limitation stems largely from shortcomings in our physical models on many length scales. At the molecular level, there has recently been an increasing effort to simulate the arrangement of the OSC molecules and calculate the resulting electronic properties of the material. However, validation of these predictions is often difficult due to measurement limitations, large computational cost, and difficulty verifying with experiments. Nevertheless, as molecular-simulation methods and computational power continue to improve, it will be important to utilize this information in device simulations to help us understand which molecular details dictate the device physics. Even on a larger scale, materials’ microstructure at the nanoscale to microscale due to crystalline-amorphous domains and DA domains can have a major impact on device performance. Microscopic modeling techniques could be greatly improved by integrating more information from state-of-the-art morphology characterization techniques to construct more accurate structure-property relationships. Further, for predicting device performance using macroscopic device modeling, the accuracy could be improved significantly if analytical expressions were explicitly derived from microscopic simulations. Fitting drift-diffusion models to device data also could be improved significantly if fit uniqueness was enhanced by fitting to higher-dimensionality datasets, including a series of measurements with various temperatures, illumination intensities, layer thicknesses, and other data, or to multiple types of measurement techniques. To make these broad multiscale-modeling efforts possible, there is a distinct need in the organic electronics community to create high-quality, open, and interoperable simulation, modeling, and analysis software tools.

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Acknowledgments Michael C. Heiber is supported by financial assistance award No. 70NANB14H012 from the US Department of Commerce, National Institute of Standards and Technology, as part of the Center for Hierarchical Materials Design (CHiMaD).

References Albrecht, U., B€assler, H., 1995. Langevin-type charge carrier recombination in a disordered hopping system. Phys. Status Solidi B 191, 455–459. Altazin, S., Z€ufle, S., Knapp, E., Kirsch, C., Schmidt, T., J€ager, L., Noguchi, Y., Br€ utting, W., Ruhstaller, B., 2016. Simulation of OLEDs with a polar electron transport layer. Org. Electron. 39, 244–249. Bakulin, A.A., Rao, A., Pavelyev, V.G., van Loosdrecht, P.H.M., Pshenichnikov, M.S., Niedzialek, D., Cornil, J., Beljonne, D., Friend, R., 2012. The role of driving energy and delocalized states for charge separation in organic semiconductors. Science 335, 1340–1344. Banerji, N., 2013. Sub-picosecond delocalization in the excited state of conjugated homopolymers and donor-acceptor copolymers. J. Mater. Chem. C 1, 3052–3066. Baranovskii, S.D., 2014. Theoretical description of charge transport in disordered organic semiconductors. Phys. Status Solidi B 251, 487–525. Baranovskii, S.D., Wiemer, M., Nenashev, A.V., Jansson, F., Gebhard, F., 2012. Calculating the efficiency of exciton dissociation at the interface between a conjugated polymer and an electron acceptor. J. Phys. Chem. Lett. 3, 1214–1221. Bartesaghi, D., Koster, L.J.A., 2015. The effect of large compositional inhomogeneities on the performance of organic solar cells: a numerical study. Adv. Funct. Mater. 25 (13), 2013–2023. B€assler, H., 1993. Charge transport in disordered organic photoconductors. Phys. Status Solidi B 175, 15–56. Bjorgaard, J.A., K€ose, M.E., 2015. Simulations of singlet exciton diffusion in organic semiconductors: a review. RSC Adv. 5, 8432–8445. Bouhassoune, M., van Mensfoort, S., Bobbert, P., Coehoorn, R., 2009. Carrier-density and fielddependent charge-carrier mobility in organic semiconductors with correlated Gaussian disorder. Org. Electron. 10 (3), 437–445. Braun, C.L., 1984. Electric field assisted dissociation of charge transfer states as a mechanism of photocarrier production. J. Chem. Phys. 80, 4157–4161. Burke, T.M., McGehee, M.D., 2014. How high local charge carrier mobility and an energy cascade in a three-phase bulk heterojunction enable >90% quantum efficiency. Adv. Mater. 26, 1923–1928. Burke, T.M., Sweetnam, S., Vandewal, K., McGehee, M.D., 2015. Beyond Langevin recombination: how equilibrium between free carriers and charge transfer states determines the open-circuit voltage of organic solar cells. Adv. Energy Mater. 5, 1500123. Buxton, G.A., Clarke, N., 2006. Predicting structure and property relations in polymeric photovoltaic devices. Phys. Rev. B 74 (8), 085207. Caruso, D., Troisi, A., 2012. Long-range exciton dissociation in organic solar cells. Proc. Natl. Acad. Sci. 109, 13498–13502.

340

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Casalegno, M., Bernardi, A., Raos, G., 2013. Numerical simulation of photocurrent generation in bilayer organic solar cells: comparison of master equation and kinetic Monte Carlo approaches. J. Chem. Phys. 139, 024706. Christ, N., Kettlitz, S., Mescher, J., Valouch, S., Lemmer, U., 2013. Dispersive transport in the temperature dependent transient photoresponse of organic photodiodes and solar cells. J. Appl. Phys. 113 (23), 234503. Clarke, T.M., Durrant, J.R., 2010. Charge photogeneration in organic solar cells. Chem. Rev. 110, 6736–6767. Clarke, T.M., Rodovsky, D.B., Herzing, A.A., Peet, J., Dennler, G., DeLongchamp, D., Lungenschmied, C., Mozer, A.J., 2005. Significantly reduced bimolecular recombination in a novel silole-based polymer:fullerene blend. Adv. Energy Mater. 1, 1062–1067. Clarke, T.M., Peet, J., Denk, P., Dennler, G., Lungenschmied, C., Mozer, A.J., 2012. NonLangevin bimolecular recombination in a silole-based polymer: PCBM solar cell measured by time-resolved charge extraction and resistance-dependent time-of-flight techniques. Energy Environ. Sci. 5, 5241–5245. Coehoorn, R., Bobbert, P.A., 2012. Effects of Gaussian disorder on charge carrier transport and recombination in organic semiconductors. Phys. Status Solidi A 209, 2354–2377. Coehoorn, R., van Eersel, H., Bobbert, P.A., Janssen, R.A.J., 2015. Kinetic Monte Carlo study of the sensitivity of OLED efficiency and lifetime to materials parameters. Adv. Funct. Mater. 25, 2024–2037. Collins, B.A., Tumbleston, J.R., Ade, H., 2011. Miscibility, crystallinity, and phase development in P3HT/PCBM solar cells: toward an enlightened understanding of device morphology and stability. J. Phys. Chem. Lett. 2, 3135–3145. Coropceanu, V., Bredas, J.L., Mehraeen, S., 2017. Impact of active layer morphology on bimolecular recombination dynamics in organic solar cells. J. Phys. Chem. C 121, 24954–24961. D’Avino, G., Muccioli, L., Castet, F., Poelking, C., Andrienko, D., Soos, Z.G., Cornil, J., Beljonne, D., 2016. Electrostatic phenomena in organic semiconductors: fundamentals and implications for photovoltaics. J. Phys. Condens. Matter. 28, 433002. Deibel, C., Dyakonov, V., 2010. Polymer-fullerene bulk heterojunction solar cells. Rep. Prog. Phys. 73, 096401. Deibel, C., Strobel, T., Dyakonov, V., 2009. Origin of the efficient polaron-pair dissociation in polymer-fullerene blends. Phys. Rev. Lett. 103, 036402. Deibel, C., Wagenpfahl, A., Dyakonov, V., 2009. Origin of reduced polaron recombination in organic semiconductor devices. Phys. Rev. B 80, 075203. Dexter, D.L., 1953. A theory of sensitized luminescence in solids. J. Chem. Phys. 21, 836–859. Diethelm, M., Penninck, L., Altazin, S., Hiestand, R., Kirsch, C., Ruhstaller, B., 2018. Quantitative analysis of pixel crosstalk in AMOLED displays. J. Inf. Disp. 316, 1–9. Dimitrov, S.D., Durrant, J.R., 2014. Materials design considerations for charge generation in organic solar cells. Chem. Mater. 26, 616–630. Donets, S., Pershin, A., Christlmaier, M.J.A., Baeurle, S.A., 2013. A multiscale modeling study of loss processes in block-copolymer-based solar cell nanodevices. J. Chem. Phys. 138, 094901. Donets, S., Pershin, A., Baeurle, S.A., 2015. Optimizing the fabrication process and interplay of device components of polymer solar cells using a field-based multiscale solar-cell algorithm. J. Chem. Phys. 142, 184902. Erickson, N.C., Holmes, R.J., 2013. Investigating the role of emissive layer architecture on the exciton recombination zone in organic light-emitting devices. Adv. Funct. Mater. 23 (41), 5190–5198.

Advances in modeling the physics of disordered organic electronic devices

341

Felekidis, N., Melianas, A., Kemerink, M., 2016. Nonequilibrium drift-diffusion model for organic semiconductor devices. Phys. Rev. B 94 (3), 1–7. Ferguson, A.J., Kopidakis, N., Shaheen, S.E., Rumbles, G., 2011. Dark carriers, trapping, and activation control of carrier recombination in neat P3HT and P3HT:PCBM blends. J. Phys. Chem. C 115, 23134–23148. Few, S., Frost, J.M., Nelson, J., 2015. Models of charge pair generation in organic solar cells. Phys. Chem. Chem. Phys. 17, 2311–2325. Foertig, A., Wagenpfahl, A., Gerbich, T., Cheyns, D., Dyakonov, V., Deibel, C., 2012. Nongeminate recombination in planar and bulk heterojunction organic solar cells. Adv. Energy Mater. 2 (12), 1483–1489. F€ orster, T., 1948. Zwischenmolekulare energiewanderung und fluoreszenz. Ann. Phys. 437, 55–75. Frost, J.M., Cheynis, F., Tuladhar, S.M., Nelson, J., 2006. Influence of polymer-blend morphology on charge transport and photocurrent generation in donor-acceptor polymer blends. Nano Lett. 6, 1674–1681. Gagorik, A.G., Mohin, J.W., Kowalewski, T., Hutchison, G.R., 2013. Monte Carlo simulations of charge transport in 2D organic photovoltaics. J. Phys. Chem. Lett. 4, 36–42. Gagorik, A.G., Mohin, J.W., Kowalewski, T., Hutchison, G.R., 2015. Effects of delocalized charge carriers in organic solar cells: predicting nanoscale device performance from morphology. Adv. Funct. Mater. 25, 1996–2003. Gao, F., Ingan€as, O., 2014. Charge generation in polymer-fullerene bulk-heterojunction solar cells. Phys. Chem. Chem. Phys. 16, 20291. Gartstein, Y.N., Conwell, E.M., Rice, M.J., 1996. Electron-hole collision cross section in discrete hopping systems. Chem. Phys. Lett. 249, 451–458. Giazitzidis, P., Argyrakis, P., Bisquert, J., Vikhrenko, V.S., 2014. Charge separation in organic photovoltaic cells. Org. Electron. 15, 1043–1049. Gillespie, D.T., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phys. 22, 403–434. G€ ohler, C., Wagenpfahl, A., Deibel, C., 2018. Nongeminate recombination in organic solar cells. Adv. Energy Mater. 1 (1), 1–2. Gorenflot, J., Heiber, M.C., Baumann, A., Lorrmann, J., Gunz, M., K€ampgen, A., Dyakonov, V., Deibel, C., 2014. Nongeminate recombination in neat P3HT and P3HT: PCBM blend films. J. Appl. Phys. 115, 144502. Grancini, G., Maiuri, M., Fazzi, D., Petrozza, A., Egelhaaf, H.J., Brida, D., Cerullo, G., Lanzani, G., 2013. Hot exciton dissociation in polymer solar cells. Nat. Mater. 12, 29–33. Gregg, B.A., 2011. Entropy of charge separation in organic photovoltaic cells: the benefit of higher dimensionality. J. Phys. Chem. Lett. 2, 3013–3015. Groves, C., 2013. Developing understanding of organic photovoltaic devices: Monte Carlo models of geminate and non-geminate recombination, charge transport, and charge extraction. Energy Environ. Sci. 6, 3202–3217. Groves, C., 2013. Suppression of geminate charge recombination in organic photovoltaic devices with a cascaded energy heterojunction. Energy Environ. Sci. 6, 1546–1551. Groves, C., Greenham, N.C., 2008. Bimolecular recombination in polymer electronic devices. Phys. Rev. B 78, 155205. Groves, C., Marsh, R.A., Greenham, N.C., 2008. Monte Carlo modeling of geminate recombination in polymer-polymer photovoltaic devices. J. Chem. Phys. 129, 114903. Groves, C., Koster, L.J.A., Greenham, N.C., 2009. The effect of morphology upon mobility: implications for bulk heterojunction solar cells with nonuniform blend morphology. J. Appl. Phys. 105, 094510.

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Groves, C., Kimber, R.G.E., Walker, A.B., 2010. Simulation of loss mechanisms in organic solar cells: a description of the mesoscopic Monte Carlo technique and an evaluation of the first reaction method. J. Chem. Phys. 133, 144110. Guo, J., Ohkita, H., Benten, H., Ito, S., 2010. Charge generation and recombination dynamics in poly(3-hexylthiophene)/fullerene blend films with different regioregularities and morphologies. J. Am. Chem. Soc. 132, 6154–6164. Hamilton, R., Shuttle, C.G., O’Regan, B., Hammant, T.C., Nelson, J., Durrant, J.R., 2010. Recombination in annealed and nonannealed polythiophene/fullerene solar cells: transient photovoltage studies versus numerical modeling. J. Phys. Chem. Lett. 1, 1432–1436. H€ausermann, R., Knapp, E., Moos, M., Reinke, N.A., Flatz, T., Ruhstaller, B., 2009. Coupled optoelectronic simulation of organic bulk-heterojunction solar cells: parameter extraction and sensitivity analysis. J. Appl. Phys. 106 (10), 104507. Heeger, A.J., 2014. 25th anniversary article: bulk heterojunction solar cells: understanding the mechanism of operation. Adv. Mater. 26, 10–28. Heiber, M.C., 2016. Ising_OPV v3.0. https://doi.org/10.5281/zenodo.60505. Heiber, M.C., Dhinojwala, A., 2012. Dynamic Monte Carlo modeling of exciton dissociation in organic donor-acceptor solar cells. J. Chem. Phys. 137, 014903. Heiber, M.C., Dhinojwala, A., 2013. Estimating the magnitude of exciton delocalization in regioregular P3HT. J. Phys. Chem. C 117, 21627–21634. Heiber, M.C., Dhinojwala, A., 2014. Efficient generation of model bulk heterojunction morphologies for organic photovoltaic device modeling. Phys. Rev. Appl. 2, 014008. Heiber, M.C., Baumbach, C., Dyakonov, V., Deibel, C., 2015. Encounter-limited charge-carrier recombination in phase-separated organic semiconductor blends. Phys. Rev. Lett. 114, 136602. Heiber, M.C., Nguyen, T.Q., Deibel, C., 2016. Charge carrier concentration dependence of encounter-limited bimolecular recombination in phase-separated organic semiconductor blends. Phys. Rev. B 93, 205204. Heiber, M.C., Kister, K., Baumann, A., Dyakonov, V., Deibel, C., Nguyen, T.Q., 2017. Impact of tortuosity on charge-carrier transport in organic bulk heterojunction blends. Phys. Rev. Appl. 8, 054043. Hilczer, M., Tachiya, M., 2010. Unified theory of geminate and bulk electron-hole recombination in organic solar cells. J. Phys. Chem. C 114, 6808–6813. Hood, S.N., Kassal, I., 2016. Entropy and disorder enable charge separation in organic solar cells. J. Phys. Chem. Lett. 7, 4495–4500. Houili, H., Tutisˇ, E., Batistic, I., Zuppiroli, L., 2006. Investigation of the charge transport through disordered organic molecular heterojunctions. J. Appl. Phys. 100, 033702. Howard, I.A., Etzold, F., Laquai, F., Kemerink, M., 2014. Nonequilibrium charge dynamics in organic solar cells. Adv. Energy Mater. 4, 1301743. H€ ubler, A., Trnovec, B., Zillger, T., Ali, M., Wetzold, N., Mingebach, M., Wagenpfahl, A., Deibel, C., Dyakonov, V., 2011. Printed paper photovoltaic cells. Adv. Energy Mater. 1 (6), 1018–1022. Jailaubekov, A.E., Willard, A.P., Tritsch, J.R., Chan, W.L., Sai, N., Gearba, R., Kaake, L.G., Williams, K.J., Leung, K., Rossky, P.J., Zhu, X.Y., 2013. Hot charge-transfer excitons set the time limit for charge separation at donor/acceptor interfaces in organic photovoltaics. Nat. Mater. 12, 66–73. Jamieson, F.C., Domingo, E.B., McCarthy-Ward, T., Heeney, M., Stingelin, N., Durrant, J.R., 2012. Fullerene crystallisation as a key driver of charge separation in polymer/fullerene bulk heterojunction solar cells. Chem. Sci. 3, 485–492.

Advances in modeling the physics of disordered organic electronic devices

343

Jones, M.L., Dyer, R., Clarke, N., Groves, C., 2014. Are hot charge transfer states the primary cause of efficient free-charge generation in polymer:fullerene organic photovoltaic devices? A kinetic Monte Carlo study. Phys. Chem. Chem. Phys. 16, 20310–20320. Kaake, L.G., Jasieniak, J.J., Ronald, C., Bakus, I., Welch, G.C., Moses, D., Bazan, G.C., Heeger, A.J., 2012. Photoinduced charge generation in a molecular bulk heterojunction material. J. Am. Chem. Soc. 134, 19828–19838. Kasparek, C., R€orich, I., Blom, P.W., Wetzelaer, G.J.A., 2018. Efficiency of solution-processed multilayer polymer light-emitting diodes using charge blocking layers. J. Appl. Phys. 123 (2), 024504. Kirchartz, T., Nelson, J., 2012. Meaning of reaction orders in polymer:fullerene solar cells. Phys. Rev. B 86, 165201. Kordt, P., van der Holst, J.J.M., Helwi, M.A., Kowalsky, W., May, F., Badinksi, A., Lennartz, C., Andrienko, D., 2015. Modeling of organic light emitting diodes: from molecular to device properties. Adv. Funct. Mater. 5, 1955–1971. Koster, L.J.A., 2010. Charge carrier mobility in disordered organic blends for photovoltaics. Phys. Rev. B 81, 205318. Koster, L.J.A., Mihailetchi, V.D., Blom, P.W.M., 2006. Bimolecular recombination in polymer/ fullerene bulk heterojunction solar cells. Appl. Phys. Lett. 88, 052104. Kurpiers, J., Neher, D., 2016. Dispersive non-geminate recombination in an amorphous polymer:fullerene blend. Sci. Rep. 6, 26832. Lakhwani, G., Rao, A., Friend, R.H., 2014. Bimolecular recombination in organic photovoltaics. Annu. Rev. Phys. Chem. 65, 557–581. Lange, I., Kniepert, J., Pingel, P., Dumsch, I., Allard, S., Janietz, S., Scherf, U., Neher, D., 2013. Correlation between the open circuit voltage and the energetics of organic bulk heterojunction solar cells. J. Phys. Chem. Lett. 4, 3865–3871. Langevin, P., 1903. Recombinaison et mobilites des ions dans les gaz. Ann. Chim. Phys. 28, 433. Liu, L., Li, G., 2011. Investigation of recombination loss in organic solar cells by simulating intensity-dependent current-voltage measurements. Sol. Energy Mat. Sol. C 95 (9), 2557–2563. Lyons, B.P., Clarke, N., Groves, C., 2011. The quantitative effect of surface wetting layers on the performance of organic bulk heterojunction photovoltaic devices. J. Phys. Chem. C 115, 22572–22577. Lyons, B.P., Clarke, N., Groves, C., 2012. The relative importance of domain size, domain purity and domain interfaces to the performance of bulk-heterojunction organic photovoltaics. Energy Environ. Sci. 5, 7657–7663. MacKenzie, R.C.I., Shuttle, C.G., Chabinyc, M.L., Nelson, J., 2012. Extracting microscopic device parameters from transient photocurrent measurements of P3HT:PCBM solar cells. Adv. Energy Mater. 2, 662–669. Marcus, R.A., 1993. Electron transfer reactions in chemistry. Theory and experiment. Rev. Mod. Phys. 65, 599–610. Marsh, R.A., Groves, C., Greenham, N.C., 2007. A microscopic model for the behavior of nanostructured organic photovoltaic devices. J. Appl. Phys. 101, 083509. Martin, S.J., Walker, A.B., Campbell, A.J., Bradley, D.D.C., 2005. Electrical transport characteristics of single-layer organic devices from theory and experiment. J. Appl. Phys. 98 (6), 063709. McMahon, D.P., Cheung, D.L., Troisi, A., 2011. Why holes and electrons separate so well in polymer/fullerene photovoltaic cells. J. Phys. Chem. Lett. 2, 2737–2741.

344

Handbook of Organic Materials for Electronic and Photonic Devices

McNeill, C.R., Westenhoff, S., Groves, C., Friend, R.H., Greenham, N.C., 2007. Influence of nanoscale phase separation on the charge generation dynamics and photovoltaic performance of conjugated polymer blends: balancing charge generation and separation. J. Phys. Chem. C 111, 19153–19160. Mehraeen, S., Coropceanu, V., Bredas, J.L., 2013. Role of band states and trap states in the electrical properties of organic semiconductors: hopping versus mobility edge model. Phys. Rev. B 87 (19), 195209. Melianas, A., Pranculis, V., Devizˇis, A., Glubinas, V., Ingan€as, O., Kemerink, M., 2014. Dispersion-dominated photocurrent in polymer:fullerene solar cells. Adv. Funct. Mater. 24, 4507–4514. Melianas, A., Etzold, F., Savenije, T.J., Laquai, F., Ingan€as, O., Kemerink, M., 2015. Photogenerated carriers lose energy during extraction from polymer-fullerene solar cells. Nat. Commun. 6, 8778. Menke, S.M., Holmes, R.J., 2014. Exciton diffusion in organic photovoltaic cells. Energy Environ. Sci. 7, 499–512. Menke, S.M., Ran, N.A., Bazan, G.C., Friend, R.H., 2017. Understanding energy loss in organic solar cells: toward a new efficiency regime. Joule 2, 1–11. Mesta, M., Carvelli, M., de Vries, R.J., van Eersel, H., van der Holst, J.J.M., Schober, M., Furno, M., L€ussem, B., Leo, K., Loebl, P., Coehoorn, R., Bobbert, P.A., 2013. Molecular-scale simulation of electroluminescence in a multilayer white organic light-emitting diode. Nat. Mater. 12, 652–658. Mikhnenko, O.V., Blom, P.W.M., Nguyen, T.Q., 2015. Exciton diffusion in organic semiconductors. Energy Environ. Sci. 8, 1867–1888. Miller, A., Abrahams, E., 1960. Impurity conduction at low concentrations. Phys. Rev. 120, 745–755. Mock, M.S., 1983. Analysis of Mathematical Models of Semiconductor Devices. Boole Press Limited, Dublin. € Mozer, A.J., Dennler, G., Sariciftci, N.S., Westerling, M., Pivrikas, A., Osterbacka, R., Jusˇka, G., 2005. Time-dependent mobility and recombination of the photoinduced charge carriers in conjugated polymer/fullerene bulk heterojunction solar cells. Phys. Rev. B 72, 035217. Murawski, C., Leo, K., Gather, M.C., 2013. Efficiency roll-off in organic light emitting-diodes. Adv. Mater. 25, 6801–6827. Nan, G., Zhang, X., Lu, G., 2016. The lowest-energy charge-transfer state and its role in charge separation in organic photovoltaics. Phys. Chem. Chem. Phys. 18, 17546–17556. Nelson, J., 2003. Diffusion-limited recombination in polymer-fullerene blends and its influence on photocurrent collection. Phys. Rev. B 67, 155209. Neukom, M., Reinke, N., Ruhstaller, B., 2011. Charge extraction with linearly increasing voltage: a numerical model for parameter extraction. Sol. Energy 85 (6), 1250–1256. Neukom, M., Z€ufle, S., Ruhstaller, B., 2012. Reliable extraction of organic solar cell parameters by combining steady-state and transient techniques. Org. Electron. 13 (12), 2910–2916. Niu, Q., Wetzelaer, G.J.A.H., Blom, P.W.M., Cr€aciun, N.I., 2016. Modeling of electrical characteristics of degraded polymer light-emitting diodes. Adv. Electron. Mat. 2 (8), 1600103. Novikov, S.V., 2003. Charge-carrier transport in disordered polymers. J. Polym. Sci. Pol. Phys. 41, 2584–2594. Novikov, S.V., Dunlap, D.H., Kenkre, V.M., Parris, P.E., Vannikov, A.V., 1998. Essential role of correlations in governing charge transport in disordered organic materials. Phys. Rev. Lett. 81, 4472–4475.

Advances in modeling the physics of disordered organic electronic devices

345

Oelerich, J.O., Huemmer, D., Baranovskii, S.D., 2012. How to find out the density of states in disordered organic semiconductors. Phys. Rev. Lett. 108, 226403. Oelerich, J.O., Jansson, F., Nenashev, A.V., Gebhard, F., Baranovskii, S.D., 2014. Energy position of the transport path in disordered organic semiconductors. J. Phys. Condens. Matter. 26 (25), 255801. Oelerich, J.O., Nenashev, A.V., Dvurechenskii, A.V., Gebhard, F., Baranovskii, S.D., 2017. Field dependence of hopping mobility: lattice models against spatial disorder. Phys. Rev. B 96, 195208. Offermans, T., Meskers, S.C.J., Janssen, R.A.J., 2003. Charge recombination in a poly (para-phenylene vinyl)-fullerene derivative composite film studied by transient, nonresonant, hole-burning spectroscopy. J. Chem. Phys. 119, 10924–10929. Offermans, T., Meskers, S.C.J., Janssen, R.A.J., 2005. Monte-Carlo simulations of geminate electron-hole pair dissociation in a molecular heterojunction: a two-step dissociation mechanism. Chem. Phys. 308, 125–133. Perucco, B., Reinke, N.A., Rezzonico, D., Knapp, E., Harkema, S., Ruhstaller, B., 2012. On the exciton profile in OLEDs-seamless optical and electrical modeling. Org. Electron. 13 (10), 1827–1835. Peumans, P., Forrest, S.R., 2004. Separation of geminate charge-pairs at donor-acceptor interfaces in disordered solids. Chem. Phys. Lett. 398, 27–31. Peumans, P., Uchida, S., Forrest, S.R., 2003. Efficient bulk heterojunction photovoltaic cells using small-molecular-weight organic thin films. Nature 425, 158–162. Pflumm, C., Gartner, C., Lemmer, U., 2008. A numerical scheme to model current and voltage excitation of organic light-emitting diodes. IEEE J. Quantum Electron. 44 (7–8), 790–798. Pivrikas, A., Jusˇka, G., Mozer, A.J., Scharber, M., Arlauskas, K., Sariciftci, N.S., Stubb, H., € Osterbacka, R., 2005. Bimolecular recombination coefficient as a sensitive testing parameter for low-mobility solar-cell materials. Phys. Rev. Lett. 94, 176806. Proctor, C.M., Kuik, M., Nguyen, T.Q., 2013. Charge carrier recombination in organic solar cells. Prog. Polym. Sci. 38, 1941–1960. Reineke, S., Thomschke, M., L€ussem, B., Leo, K., 2013. White organic light-emitting diodes: status and perspective. Rev. Mod. Phys. 85, 1245–1293. Richert, R., B€assler, H., 1985. Energetic relaxation of triplet excitations in vitreous benzophenone. Chem. Phys. Lett. 118, 235–239. Richert, R., Pautmeier, L., B€assler, H., 1989. Diffusion and drift of charge carriers in a random potential: deviation from Einstein’s law. Phys. Rev. Lett. 63, 547–550. Ries, B., B€assler, H., 1984. Monte Carlo study of bimolecular charge-carrier recombination in anisotropic lattices. Chem. Phys. Lett. 108, 71–75. Savoie, B.M., Rao, A., Bakulin, A.A., Gelinas, S., Movaghar, B., Friend, R.H., Marks, T.J., Ratner, M.A., 2014. Unequal partnership: asymmetric roles of polymeric donor and fullerene acceptor in generating free charge. J. Am. Chem. Soc. 136, 2876–2884. Scher, H., Montroll, E.W., 1975. Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455–2477. Sch€onherr, G., Eiermann, R., B€assler, H., Silver, M., 1980. Dispersive exciton transport in a hopping system with Gaussian energy distribution. Chem. Phys. 52, 287–298. Selberherr, S., 1984. Analysis and Simulation of Semiconductor Devices. Springer-Verlag, Wien, New York. Shen, Y., Giebink, N.C., 2015. Monte Carlo simulations of nanoscale electrical inhomogeneity in organic light-emitting diodes and its impact on their efficiency and lifetime. Phys. Rev. Appl. 4, 054017.

346

Handbook of Organic Materials for Electronic and Photonic Devices

Shklovskii, B.I., Efros, A.L., 1984. Electronic Properties of Doped Semiconductors. Springer Series in Solid-State Sciences, vol. 45. Springer, Berlin, Heidelberg, p. 388. Soldera, M., Taretto, K., Kirchartz, T., 2012. Comparison of device models for organic solar cells: band-to-band vs. tail states recombination. Phys. Status Solidi A 209 (1), 207–215. Stehr, V., Pfister, J., Fink, R.F., Engels, B., Deibel, C., 2011. First-principles calculations of anisotropic charge-carrier mobilities in organic semiconductor crystals. Phys. Rev. B 83, 155208. Street, R.A., 2015. Electronic structure and properties of organic bulk-heterojunction interfaces. Adv. Mater. 28, 3814–3830. Tress, W., Leo, K., Riede, M., 2013. Dominating recombination mechanisms in organic solar cells based on ZnPc and C60. Appl. Phys. Lett. 102 (16), 163901. Tutisˇ, E., Batistic, I., Berner, D., 2004. Injection and strong current channeling in organic disordered media. Phys. Rev. B 70, 161202. Tvingstedt, K., Deibel, C., 2016. Temperature dependence of ideality factors in organic solar cells and the relation to radiative efficiency. Adv. Energy Mater. 6 (9), 1502230. van der Holst, J.J.M., van Oost, F.W.A., Coehoorn, R., Bobbert, P.A., 2009. Electron-hole recombination in disordered semiconductors: validity of the Langevin formula. Phys. Rev. B 80, 235202. van der Holst, J.J.M., van Oost, F.W.A., Coehoorn, R., Bobbert, P.A., 2011. Monte Carlo study of charge transport in organic sandwich-type single-carrier devices: effects of Coulomb interactions. Phys. Rev. B 83, 085206. van Eersel, H., Janssen, R.A.J., Kemerink, M., 2012. Mechanism for efficient photoinduced charge separation at disordered organic heterojunctions. Adv. Funct. Mater. 22, 2700–2708. van Eersel, H., Bobbert, P.A., Janssen, R.A.J., Coehoorn, R., 2014. Monte Carlo study of efficiency roll-off of phosphorescent organic light-emitting diodes: evidence for dominant role of triplet-polaron quenching. Appl. Phys. Lett. 105, 143303. Vandewal, K., Albrecht, S., Hoke, E.T., Graham, K.R., Widmer, J., Douglas, J.D., Schubert, M., Mateker, W.R., Bloking, J.T., Burkhard, G.F., Sellinger, A., Frechet, J.M.J., Amassian, A., Riede, M.K., McGehee, M.D., Neher, D., Salleo, A., 2014. Efficient charge generation by relaxed charge-transfer states at organic interfaces. Nat. Mater. 13, 63–68. Vithanage, D.A., Devizˇis, A., Abramavicˇius, V., Infahsaeng, Y., Abramavicˇius, D., MacKenzie, R.C.I., Keivanidis, P.E., Yartsev, A., Hertel, D., Nelson, J., Sundstr€ om, V., Gulbinas, V., 2013. Visualizing charge separation in bulk heterojunction organic solar cells. Nat. Commun. 4, 2334. Voter, A.F., 2007. Introduction to the kinetic Monte Carlo method. In: Sickafus, K.E., Kotomin, E.A., Uberuaga, B.P. (Eds.), Radiation Effects in Solids. In: vol. 1. Springer, Dordrecht, pp. 1–23. Wagenpfahl, A., 2017. Mobility dependent recombination models for organic solar cells. J. Phys. Condens. Matter. 29 (37), 373001. Wagenpfahl, A., Deibel, C., Dyakonov, V., 2010. Organic solar cell efficiencies under the aspect of reduced surface recombination velocities. IEEE J. Sel. Top. Quantum Electron. 16 (6), 1759–1763. Watkins, P.K., Walker, A.B., Verschoor, G.L.B., 2005. Dynamical Monte Carlo modelling of organic solar cells: the dependence of internal quantum efficiency on morphology. Nano Lett. 5, 1814–1818. Wetzelaer, G.A.H., Koster, L.J.A., Blom, P.W.M., 2011. Validity of the Einstein relation in disordered organic semiconductors. Phys. Rev. Lett. 107, 066605.

Advances in modeling the physics of disordered organic electronic devices

347

Wojcik, M., Michalak, P., Tachiya, M., 2010. Geminate electron-hole recombination in organic solids in presence of a donor-acceptor heterojunction. Appl. Phys. Lett. 96, 162102. Wolf, U., Arkhipov, I., B€assler, H., 1999. Current injection from a metal to a disordered hopping system. I. Monte Carlo simulation. Phys. Rev. B 59, 7507–7513. Yang, F., Forrest, S.R., 2008. Photocurrent generation in nanostructured organic solar cells. ACS Nano 2, 1022–1032. Yu, Z.G., Smith, D.L., Saxena, A., Martin, R.L., Bishop, A.R., 2001. Molecular geometry fluctuations and field-dependent mobility in conjugated polymers. Phys. Rev. B 63, 085202. Z€ ufle, S., Altazin, S., Hofmann, A., J€ager, L., Neukom, M.T., Br€ utting, W., Ruhstaller, B., 2017. Determination of charge transport activation energy and injection barrier in organic semiconductor devices. J. Appl. Phys. 122 (11), 115502.