Charge Dynamics in Organic Semiconductors: From Chemical Structures to Devices 9783110473636, 9783110473605

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Pascal Kordt Charge Dynamics in Organic Semiconductors

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Pascal Kordt

Charge Dynamics in Organic Semiconductors From Chemical Structures to Devices

D77 Dissertation submitted to Johannes Gutenberg University Mainz, Faculty of Physics, Mathematics and Computer Science on 16th September 2015. Conducted at Max Planck Institute for Polymer Research. Publications that were still in the review stage by the time the dissertation was submitted have been updated to their final versions in this edition. Supervisors: Prof. Dr. Kurt Kremer, Prof. Dr. Friederike Schmid Date of the oral examination: 13th November 2015

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Acknowledgements First, I would like to thank Kurt Kremer for the chance to conduct this work in his group, within a cooperative and inspiring atmosphere. Many thanks go to Friederike Schmid for examining my thesis as a second referee. I am highly grateful to Denis Andrienko, without whom this work would not have been possible. He contributed with expertise and excellent ideas, support for my publications, continuous discussions about my work and help wherever needed. It is my pleasure to thank for all contributions by external collaborators, namely by Thomas Speck, Sven Stodtmann, Alexander Badinski, Falk May, Christian Lennartz, Mustapha Al Helwi, Ole Stenzel, Volker Schmidt and Wolfgang Kowalsky. Within the group at the Max Planck Institute for Polymer Research I received support by many colleagues, especially I want to thank Björn Baumeier, Jeroen van der Holst, Carl Pölking, Jens Wehner and Anton Melnyk. I would like to express many thanks to Jens Wehner, Denis Andrienko, Patrick Gemünden and Christoph Scherer for carefully proofreading this thesis and their valuable comments. Many thanks to those who made working at the Max Planck Institute for Polymer Research an enjoyable time also with non-scientific activities. In particular I want to mention Gustav Waschatko, David Schäffel, Jens Wehner and Patrick Gemünden. Last but clearly not least, I want to thank my family and Vivien for their support beyond science throughout.

Abstract This thesis deals with simulations of charge transport in organic semiconductors. In particular, it contains strategies for linking the chemical composition and morphology of an organic layer to charge carrier mobility and current–voltage characteristics of a device. To build a bridge between chemical structure and device scale, different methods have to be combined, including density functional theory, molecular dynamics, kinetic Monte Carlo and drift–diffusion equations. The multiscale approach is still limited to several thousand atoms, leading to finite-size effects. The resulting error on the mobility and on the energy per charge carrier is quantified by analytic calculations and verified by simulations. A Monte Carlo algorithm for multiple charge carriers allows to investigate the influence of the charge density and its interplay with finite-size effects. Mobility values in the limit of an infinite bulk system are obtained after deriving a scaling relation of the mobility with the system size. Larger systems can be mastered by using coarse-graining techniques. The rates for charge transfer in this approach are obtained by stochastic modeling. A challenge lies in the accurate reproduction of spatially correlated site energies. This problem is addressed by devising an algorithm to generate energy values following the atomistic reference autocorrelation function. On the device scale, charge transport is usually modeled by a combination of analytic expressions for the mobility and drift–diffusion equations. The analytic expressions are parametrized from simulations in a limited regime of parameter values. Here, the limits of this approach are investigated and an alternative strategy is designed, which uses tabulated and interpolated mobility values, directly coupled to drift–diffusion equations. A further possibility is to use Monte Carlo simulations at the device scale. The combined methodology allows for the almost parameter-free prediction of current–voltage characteristics in organic films, that compares well to experimental measurements.

Contents Acknowledgements | V Abstract | VII Introduction | 1 1 1.1 1.2 1.3 1.4

Organic Semiconductor Devices | 9 Solar Cells | 9 Light Emitting Diodes | 12 Field Effect Transistors | 17 Lasers | 18

2 2.1 2.2 2.3

Experimental Techniques | 21 Mobility Measurements | 21 Charge Density Measurements | 27 Doping | 30

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Charge Dynamics at Different Scales | 34 Charge Transfer Reactions | 35 Band Theory vs. Hopping Transport | 40 Stochastic Description of Hopping Transport in Discrete Space | 41 Kinetic Monte Carlo | 45 Differential Equation Solver | 54 Gaussian Disorder Models | 55 Analytic Theories of Hopping Transport | 58 Continuous Space Charge Transport | 65 Heuristic Models | 67

4 4.1 4.2 4.3 4.4

Computational Methods | 70 Density Functional Theory | 71 Molecular Dynamics | 73 Electrostatic and Induction Calculations for Site Energies | 77 Coarse-Grained Morphology | 78

5 5.1 5.2 5.3 5.4

Energetics and Dispersive Transport | 83 Methods | 84 Scaling Relation | 85 Finite Carrier Density | 90 Conclusions | 93

X | Contents 6 6.1 6.2 6.3

Correlated Energetic Landscapes | 94 Atomistic Simulations | 95 The Algorithm | 98 Validation | 101

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Microscopic, Stochastic and Device Simulations | 104 Microscopic Modeling | 107 Coarse-Grained Models | 112 Lattice Models | 117 Charge Dynamics | 119 Device Simulations | 121 Experiment | 126 Materials | 127

8 8.1 8.2 8.3 8.4

Parametrization of Lattice Models | 131 Methodology | 132 Results | 135 Conclusions | 138 Supporting Information | 139

9 9.1 9.2 9.3

Drift–Diffusion with Microscopic Link | 143 Methods | 144 Validation | 151 Conclusions | 152

Conclusions and Outlook | 153 A

Molecule Abbreviations | 156

Bibliography | 157 Index | 189

Introduction Electric conductivity is a property naturally associated with metals. It can, however, also be observed in some organic compounds. The first observation of conductivity in an organic material dates back more than 150 years. In 1862 Letheby synthesized an organic polymer, probably polyaniline, that showed semiconducting properties [1]. Pochettino observed in 1906 that the conductivity of anthracene increased upon light absorption, an effect known as photoconductivity [2]. This effect received increasing attention in the 1970 in the context of photosensors, used in photocopying machines [3]. Pioneering work for organic semiconductors was done by Heeger, MacDiarmid and Shirakawa, who discovered in 1977 that oxidation of polyacetylene films made this material show 109 times higher conductivity than before [4]. In 2000 their work was honored by the Nobel Prize in Chemistry [1]. Their work directly received a lot of attention and the material was studied further, e.g., in the group of Wegner [5]. The real breakthrough in terms of applications, however, was the work by Tang published in 1986 and 1987. Tang had started working at Kodak on organic solar cells after finishing his dissertation, and his first patents in this field were granted in 1981. Since the internal clearance delayed the process [6], the publication on organic solar cells came out only in 1986 [7]. About the same time Tang discovered that he could obtain photoluminescence in the same device [8] – and an organic light emitting diode (OLED) was built for the first time. This publication has by now more than 12000 citations and stimulated both academic and industrial research in the field of organic electronics. Soon afterward, in 1987, the first organic field effect transistor was built [9].

π Orbitals What enables conduction in organic materials? Let us look at conjugated polymers as an example, i.e., macromolecules with a backbone chain that has alternating single and double bonds. Their ability to conduct charges lies in the sp2 hybridization of neighboring carbon atoms connected by double bonds [10]. Two out of the three p orbitals form a hybrid bond, the so called σ orbitals. The third orbital, p z , does not participate in the hybridization. p z orbitals of neighboring carbons stand next to each other and perpendicular to the linking bond, allowing them to overlap and form so called π orbitals, cf. Figure 1. These electrons are weaker bound and, hence, much more delocalized than electrons in σ orbitals. This, again, leads to an overlap with the single bonds, allowing to easily move electrons occupying π orbitals from one bond to the next and therefore resulting in one-dimensional semiconductors. While in polymers this intramolecular transport along the chain is the most important contribution, what counts in small molecule organic semiconductors is intermolecular transport. π orbitals of neighboring molecules can delocalize over different molecules, thus allowing electron hopping between molecules. In case of crystalline phases this can lead to

2 | Introduction pronounced anisotropic transport properties with high charge carrier mobility along the direction of π stacking [11].

Fig. 1. The p z orbitals of two neighboring molecules do not take part in the hybridization. Neighboring p z orbitals overlap and π electrons that are delocalized over the molecule.

Applications Potential applications of organic semiconductors are manifold. One of the reasons is that there is an infinite pool of molecules to choose from. With chemical intuition and nowadays with computational pre-screening it is possible to tailor a molecule fulfilling specific requirements – absorption and emission spectra, mechanical properties, etc. A unique feature of organic semiconductors compared to their inorganic counterparts is that they can be embedded in mechanically flexible films, thus allowing for new applications. Design studies and visions include rollable solar cells that can be taken to a camping site, semitransparent solar cells used as curtains that allow using solar energy without any complicated installation, or smart glass windows that allow less light to pass through upon illumination by sunlight or when a voltage is applied. OLED displays are already in use in many mobile phones. Compared to liquid crystal displays (LCDs) there is no backlight, but each pixel can be switched off. This allows for “darker black” in OLED displays, leading to contrast ratios by orders of magnitude higher than in LCD displays. If one assures that switched-off pixels actually have zero current flow, OLED displays, in principle, even allow for an infinite contrast ratio. Apart from that, with organic materials it is possible to make thinner and lighter displays. Samsung demonstrated a four inch display in 2008 that was only 50 μm thick [12] – for comparison: usual paper (80 g/m2 ) is about 100 μm thick . Also in the automotive sector OLED displays have a clear advantage. When driving at night, the bright background illumination of LCD displays does not allow the driver’s eyes to adopt to the darkness. OLED displays with their “dark black” do not have this disadvantage and thus allow for making cars safer. The automotive manufacturing company Continental has already presented a prototype OLED display for the automotive market [13]. OLEDs are also interesting for lighting applications. OSRAM currently develops suitable lamps [14] in cooperation with BMW and Audi. Again, one of the main ad-

Introduction

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vantages is that OLED lamps can be produced as thin layers. Therefore, designers do not have to reserve larger volumes for lamps in the coachwork. Instead, flat lighting panels become possible, which can also be used for other applications, such as futuristic lamp designs in buildings. Compared to inorganic (silicon-based) LEDs, OLEDs are still less efficient, with about 100 lumen/Watt compared to about 250 lumen/Watt for inorganic LEDs [6]. In any case this is still a lot better than the traditional incandescent light bulbs, yielding around 16 to 20 lumen/Watt. Considering that lighting makes up about 20 % of the world’s electricity consumption [15] this technology has a tremendous potential. It is also more efficient than compact fluorescent (CFL) lamps (also called “energy-saving light”), which yield around 50 to 70 lumen/Watt [16]. Compared to CFL lamps LEDs have a further advantage, namely they reach their maximum efficiency instantaneously, while CFL lamps need a couple of seconds or even minutes to reach their maximum brightness. The first inorganic LEDs were already constructed starting in the 1950s, with emission ranging from infrared to green. Blue LEDs, however, posed a major challenge to scientist which hindered the production of a white LED and thus their use in lighting. It took until the early 1990s for bright, blue LEDs to be produced. Akasaki, Amano and Nakamura, who made this breakthrough possible, were awarded the Nobel Prize in Physics of the year 2014 for their discovery [17]. OLEDs in current displays are used in an active-matrix scheme (AMOLED). Here each sub-pixel is addressed individually by a transistor. It is already possible to produce curved or semitransparent displays. Displays that are, to a certain extend, mechanically flexible have also been demonstrated in prototypes. With increased flexibility, this might allow rollable displays and thus large displays on very small, rollable mobile phones or, less futuristic, mobile phone screens that do not immediately break when the phone is dropped on the floor. The transistors to address pixels in the AMOLED are currently made of inorganic materials. However, ideal candidates would be organic field effect transistors (OFETs), and in prototypes this has already been realized. OFETs allow better compatibility to the organic LED material in the production process and also would allow to use the full potential of mechanical flexibility and, if wanted, transparency. An interesting topic in this area are light-emitting OFETs [18], which combine the functionality of the transistor and of an LED.

Production For the production of films of organic materials there are two classes of techniques available. The first one is vacuum deposition, where molecules are deposited one by one onto a solid surface. This allows controlled fabrication with high purity and the technique was used, for example, for the organic solar cell that currently holds the efficiency record [19] (January 2013). Maybe even more interesting is to print the layers. This requires soluble molecules. Solubility can be tuned by modifying the side chains. However, such modifications often also lead to changes in the absorption or emission spectrum or other material properties. Once suitable molecules that allow

4 | Introduction printing from solution have been found this bears the potential of cheap and largescale industrial production, which would make organic solar cells or other organic devices commercially interesting even if their efficiency is lower than the one of their inorganic counterparts.

Requirements Which properties make a good organic semiconductor? The answer depends on the particular application. In OLEDs, for example, the first criterion to be fulfilled is light emission at the desired wavelength. For organic solar cells, on the other hand, the aim is to absorb as much radiation energy as possible within the solar spectrum. Apart from looking for materials with broad absorption spectra one can also look for different materials with mostly non-overlapping absorption ranges and combine two layers to a tandem cell, as it is the case in the current record holder for organic solar cells [19]. With vacuum deposition, which allows very thin layers, it is even possible to have three absorption layers. Upon light absorption by a solar cell an exciton is created, which consists of a bound electron–hole pair. In organic solar cells the binding energy is high (Frenkel exciton) and both electron and hole are located on the same molecule. For generating a current this exciton has to be split into a free electron and a free hole, which works most efficiently at the interface of an electron donating layer and an electron accepting layer. To make sure that most of the excitons are split before they recombine, a large exciton diffusion length is of advantage in organic solar cells. To decrease the distance an exciton has to pass in order to reach the interface, usually a mixture of the two materials is used (bulk heterojunction) instead of using two separate films. This allows to use thicker films which absorb more light [20]. While in solar cells exciton recombination is undesirable, in OLEDs the opposite is the case. Here the radiative recombination leads to creation of a photon and light emission and an efficient recombination is a necessary condition for a good material in the respective layer of an OLED. In crystalline films this can be hindered by light scattering [21], which is why amorphous materials are preferable in OLEDs, while in solar cells crystalline films often show higher performance. The efficiency of a solar cells is proportional to the open circuit voltage, VOC , which is the voltage required to balance the current to zero. It can be tuned by suitable highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) levels of the electron donating and accepting material [20]. Similarly, matching energy levels are necessary for an efficient OLED. Due to the applied voltage, charges in an OLED move upward in energy, until they reach the emission layer and recombine. To keep the necessary voltage low it is, however, beneficial if the differences between energetic levels of the layers between the injecting electrode and the emission layer are small. Contrarily, to avoid a diffusion of charges from the emission layer to the collecting electrode instead of radiative recombination, an energy barrier

Introduction

| 5

in between the emission layer and the interlayer (charge-blocking layer) is advantageous. Apart from efficiency criteria there are other requirements. In solar cells, thermal stability is important, as they reach high temperatures under illumination by the sun. In OLEDs stability under application of a voltage is necessary. To allow for light-weight devices a low molecular mass is of advantage and for flexible devices the material has to be robust under a small bending radius. Furthermore, solubility is necessary if the device is fabricated by printing from solution.

Charge Mobility and Charge Density An important material property practically for all organic semiconductor devices is the speed of charge carrier motion, directly connected to the conductivity. It is characterized by the charge carrier mobility, i.e., the ratio of the average velocity in field direction and the absolute value of the applied field. Delocalized π orbitals and efficient stacking can be of advantage to achieve high mobilities. However, the mobility is not a material-specific constant only but a function of temperature, applied electric field and charge density. Since these conditions vary between different measurement techniques the extracted values can also differ. In particular, the dependence on charge density has not been fully studied in early work on organic semiconductors [22]. One of the objectives of this work is to understand this dependence in detail.

Current Challenges Throughout, a prime goal in the field of organic electronics has been the search for high-mobility materials, maximizing the efficiency of OLEDs, solar cells, organic lasers, or allowing the necessary operation speed in OFETs. Apart from that, the requirements for specific applications, as discussed above, have to be fulfilled. I have mentioned earlier that the Nobel Prize in Physics of the year 2014 was awarded for the work on realizing an efficient blue LED. Similarly, a major current challenge is to find suitable materials for efficient blue OLEDs. There are two mechanisms of light generation, fluorescence and phosphorescence. As discussed in Chapter 1, phosphorescence allows for significantly more efficient OLEDs. In materials for blue emission, where the excitation energy is highest as compared to red and green excitation energies, degradation of the material is a major problem. The lifetime of materials for blue phosphorescent emission is up to now too low, which is why commercial displays with phosphorescent blue emitters are not available yet (July 2015). A possible solution might be time-delayed fluorescence, as discussed later. Degradation is also an important issue in organic solar cells. Some research cells only have a lifetime of a few hours in air under illumination [2] and any new promising candidate material will only be suitable for industrial use if it has much longer lifetimes.

6 | Introduction The maximum efficiency of organic solar cells is currently 12 % [19] (January 2013), compared to a 46 % record in an inorganic cell [23] (December 2014). Even if organic solar cells never reach the efficiency of inorganic ones, their potential lies in costefficient materials and production. These requirements are among the most important challenges in order for organic solar cells to overcome the stage of expensive niche products.

Computational Modeling Computational approaches can assist chemical compound design by predicting material properties. To do this, various computational techniques, operating at different time- and length scales, have been developed. Quantum mechanical methods, in particular density functional theory (Section 4.1), can be used to calculate the ground state structure of a molecule, ground state energies and energetic changes upon charge transfer. With extensions, such as time-dependent density functional theory (TD-DFT) [24] or the GW method [25] (G stands for the Green function and W for the screened Coulomb potential), it is possible to treat also excited states. This, in turn, allows to determine the energetic levels corresponding to absorption and emission, and thus to predict the colors or spectra of these processes. Calculations of the morphology of amorphous materials on a nanometer scale can be performed using molecular mechanics methods (Section 4.2), where molecule interactions are modeled classically, i.e., governed by Newton’s equations of motion. For going to larger morphologies one can apply coarse-graining techniques (Section 4.4). With the morphology at hand, charge transport can be simulated by combining kinetic Monte Carlo schemes with analytical charger-transfer theories. In the past, such calculations have been done only based on lattice models instead of realistic morphologies. Nowadays, several software packages allow to couple material properties to charge transport simulations, one of them the VOTCA package [26]. In contrast to microscopic processes, simulations of devices are usually done using models continuous in space and the corresponding drift–diffusion equations.

Outline This theses starts with a general review of organic electronics in Chapters 1–4, followed by applications and results of my work in Chapters 5–9. In Chapter 1 different devices based on organic semiconductors, such as solar cells, organic light emitting diodes and field effect transistors, are explained. Chapter 2 reviews experimental techniques related to charge carrier mobility and density measurements. In Chapter 3 charge carrier dynamics is discussed. The chapter covers different scales, starting from charge transfer reactions between two molecules and the corresponding rates up to drift–diffusion equations for modeling in continuous space.

Introduction

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In amorphous materials, charge transport can be modeled by hopping between molecules. Solution methods for the corresponding master equation are kinetic Monte Carlo or numerical differential equations solvers. Also included in the chapter are model systems for hopping transport, namely Gaussian disorder models and analytic models. In order to go beyond model systems, a set of computational methods is required. These methods are reviewed in Chapter 4. They include density functional theory, molecular dynamics, electrostatic and induction calculations and coarse-graining methods. In Chapter 5 an investigation of the population of energy states in a Gaussian density of states is presented. By using a truncated Gaussian distribution it is possible to predict how finite-size effects affect the average charge carrier energy and thus their mobility. Scaling relations of the mobility as a function of temperature and system size allow extrapolations to the limit of infinite systems, which are applied in the following chapters. The model of a Gaussian density of states has been very successful for describing properties of organic semiconductors. The interaction of molecules, however, leads to strong spatial correlations of these energies and only by taking such correlations into account it is possible to explain, for example, the Poole–Frenkel field dependence of the mobility (cf. Section 3.9.6). In an extension of the Gaussian Disorder Model to include such correlations, a model of dipoles attached to each molecule is used, which fixes the spatial autocorrelation function to a 1/r dependence (r is the distance between two molecules) with only the lattice constant as a free parameter. The aim of Chapter 6 is to overcome this limitation. Calculations of the energies in realistic morphologies allow to deduce a correlation function. With the algorithm developed in this chapter it is then possible to reproduce this correlation function and obtain correctly distributed energies to use in lattice models or in coarse-grained stochastic systems. The combined microscopic approach for morphology, site energies and charge transport is presented in Chapter 7. Microscopic simulations are limited by computational resources, and typical system sizes are in the order of 104 molecules. As discussed in Chapter 5, this size is in many cases not sufficient to ensure system-size independent mobility values. Furthermore, it is also too small to investigate the effects of charge carrier interactions at relevant charge densities. Therefore, a second topic of this chapter are coarse-grained systems. Here the radial distribution function is reproduced by coarse-graining techniques, the electronic couplings entering the transfer rates by random sampling according to the microscopic reference and the correlated site energies by using the algorithm presented in the previous Chapter 6. A third topic of this chapter are Monte Carlo simulations at the length scales of devices, made possible by the coarse-grained model. Taking all these ingredients together it is possible to arrive at the current–voltage behavior of a material, starting from the chemical structure and without experimental input. The results are compared to current–voltage measurements for the OLED hole conducting material DPBIC, measured at BASF.

8 | Introduction If Monte Carlo simulations at the device scale are computationally too demanding, e.g., in a large system or in a multilayer stack, drift–diffusion continuum equations can be used. These equations incorporate the mobility as a function of charge density, electric field and temperature, μ(ρ, F, T). One possibility to obtain this function is discussed in Chapter 8, where a fixed functional form is parametrized from lattice simulations. The assumed functional form was derived using lattices with an energetic disorder up to about 0.15 eV at room temperature. Hence, one cannot expect the models to be valid in case of amorphous materials with higher energetic disorder, as it is the case in both of the materials examined here, DPBIC and DCV4T. Chapter 9 therefore presents a different approach. Here, the mobility is tabulated for a wide range of values for charge density, electric field and temperature, and then smoothed and interpolated. This allows a direct coupling to the drift–diffusion equations. Also presented in this chapter is a comparison of the tabulation approach and the fixed functional form.

1 Organic Semiconductor Devices Organic semiconductors are used in solar cells, light emitting diodes, field effect transistors or lasers, to name only the most important devices. There are different challenges in the search for suitable materials for these applications – in solar cells, for example, an efficient charge splitting is necessary, while in light emitting diodes materials that allow a conversion of all triplet states into light are favorable. In all cases there are different fabrication techniques: vacuum deposition allows for high purity and is used for smaller molecules [27], as they are usually found in light emitting diodes. Printing from solution, on the other hand, has the potential for cost-effective, large-scale fabrication, but needs soluble molecules. One property important to all devices is the speed with which electrons or holes move, characterized by the mobility, μ. It is closely related to the electron or hole density, which can differ by orders of magnitude between different devices and also between different operation conditions, such as high or low light input in a solar cell or a large or a small operation voltage of a light emitting diode. In this chapter I review the different devices mentioned and discuss the aspect of mobility and charge density in the different situations.

1.1 Solar Cells The first photovoltaic cell was built as early as 1839 by Becquerel. Einstein described the underlying photoelectric effect in 1905, for which he was awarded the Nobel prize in physics of the year 1921 [28]. The first observation of the effect in organic materials was much later, in the year 1958 [29]. Common inorganic solar cells have two semiconducting layers, a p-doped and an n-doped one. Upon illumination the incoming photon is absorbed by a molecule, generating a free electron–hole pair¹. The p-n junction leads to a built-in electric field across the sample, resulting in opposite forces on holes and electrons and thus in a drift towards the electrodes that prevents an electron–hole recombination. In organic photovoltaic (OPV) cells, in contrast, a strongly bound electron–hole pair, called a Frenkel exciton, is created [30]. Here electron and hole are located on the same molecule and are bound by Coulomb forces. Frenkel excitons have large binding energies in the order of 1 eV [31]. Since they recombine very easily, organic solar cells with a single semiconducting layer between the electrodes are inefficient (see

1 Strictly speaking it creates a Wannier–Mott exciton, i.e., an electron–hole pair that is still weakly bound by their Coulomb interaction. The latter, however, is very small, since in materials with a high relative permittivity ε there is a significant screening effect. This weakly bound exciton type is typical for inorganic photovoltaics.

10 | 1 Organic Semiconductor Devices Figure 1.1a). Therefore, usually a combination of an electron donating layer and an electron accepting layer is used (see Figure 1.1b). At the interface Frenkel excitons can be converted to charge transfer (CT) states, where hole and electron are still bound but located on different molecules [32]. Using two metallic electrodes with different Fermi levels, an internal electric field is generated, which allows the splitting of the CT state to a charge separated (CS) state, i.e., an unbound electron–hole pair where the charge carriers can drift to the respective electrode. Whether the semiconductor band model (mostly inorganic semiconductors) or the exciton picture (organic semiconductors) is an appropriate description depends on the strength of the electronic coupling: for weak coupling electron–hole pairs tend to be strongly bound, while for strong coupling the formation of free electrons and holes is more probable [33]. Details are discussed in Section 3.2. Another consequence of weak coupling is that organic semiconductors, in contrast to their inorganic counterparts, have well defined spin states, as it is also the case in isolated molecules [34]. One can therefore differentiate between singlet and triplet excitons The excitons generated by the incoming sunlight are short-lived singlets (≤ 300 ps at room temperature), but a conversion to triplet excitons, so-called inter system crossing (see Section 1.2 for details), is possible [34, 35]. Because of the spin-exchange interaction triplets usually have lower energies than singlets, longer lifetimes and thus a higher charge splitting efficiency [34]. Reported triplet densities are 1 % for MEH-PPV (poly(2-methoxy,5-(2’-ethylhexyloxy)p-phenylenevinylene)), 20 % for Alq3 (aluminum-tris(8-hydroxychinolin)) and 70 % for P3HT (poly(3-hexylethiophene)) [36, 37]. For light absorption the polymer film needs to have a thickness of typically around 100 nm, while the decay length of Frenkel excitons is around 6 nm [38]. As a result, in a double layer structure many excitons will decay before reaching the interface where they can be split to form CT and afterward CS states. To overcome this problem, often bulk heterojunctions are used, i.e., a blend of an electron conducting and an electron accepting material (fig. 1.1c). This increases the interface area and thus reduces exciton recombination. Effective splitting of excitons into free carriers requires favorably aligned energy levels: if the donor molecule has a low-lying LUMO and the acceptor a high-lying HOMO, the electron is transferred easily to the LUMO, while the hole remains in the HOMO of the donor molecule [39, 40]. The overall efficiency of an organic solar cell is determined by three factors: light absorption, exciton dissociation and charge collection. On a quantum mechanical level, it is measured as the external quantum efficiency (EQE), i.e., the number of generated electrons per present photon, or the internal quantum efficiency (IQE), where only absorbed photons are counted. The quantity on the device level is called power conversion efficiency η (PCE) and is defined as η=

Pout Isc Voc FF = , Pin Pin

(1.1)

1.1 Solar Cells

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Fig. 1.1. Schematic energy band diagram for different types of organic solar cells. (a) A single cell, where indium tin oxide (ITO) serves as a transparent anode (high work function) and Aluminum as a cathode (low work function). (b) A cell with an electron donor and an acceptor allows efficient charge separation at the interface. (c) In a bulk heterojunction cell the interface extends throughout the film, thus leading to better charge separation. Reprinted by permission from Macmillan Publishers Ltd: Nature Materials 5, 675, Copyright 2006 [38].

where Pin = E A, with E (W/m2 ) being the light input and A the surface area of the cell. Pout is the output power and Isc denotes the short-circuit current, which is the photocurrent measured under short-circuit conditions. Voc is the open-circuit voltage, i.e., the voltage required to balance the current to zero. Finally, FF = Im Vm /Isc Vsc is the fill factor, where Im and Vm are current and voltage that yield the maximal product. Organic solar cells currently reach maximum efficiencies of 12 % [19] (January 2013) in a tandem cell, while their inorganic counterparts go up to a record of 46 % [23] (December 2014) in a four-junction cell. Despite this difference, organic solar cells are interesting not only as a research topic but also to industry [10]. First, organic – or “plastic” – solar cell materials have the potential of cost-effective fabrication with low prices for the raw material, as opposed to, for example, inorganic gallium-arsenide cells. Second, it is possible to tune the desired properties by molecular design. An example of a desired property is solution processability which can be achieved by modifying the side chains of a molecule. These “printed” solar cells have the properties required for large-scale industrial production once their efficiencies and lifetimes are improved, which requires an investment in understanding the mechanisms driving their functionality. A third advantage of organic cells is the possibility to produce mechanically flexible cells, which allows for applications as built-in cells in bags, as a photovoltaic curtain or as a sunscreen foil that can be attached to a window without any further construction work. With respect to this application a fourth point is important, namely the possibility to fabricate semi-transparent cells and to tune their color [41].

12 | 1 Organic Semiconductor Devices The charge density in organic solar cells varies significantly depending on the illumination strength. Impedance spectroscopy measurements in a P3HT: PCBM blend [42] showed an electron density of 1018 cm−3 under at 1 sun (1000 W/m) illumination (at an open circuit voltage of Voc ≈ 0.6 V), while at 0.01 sun (Voc ≈ 0.2 V) it goes down to 1016 cm−3 . Transient absorption spectroscopy measurements of the same blend and in the same illumination range showed that densities under short circuit conditions are by about an order of magnitude smaller [43, 44]. For an operating solar cell outside a laboratory, the illumination can vary by more than three orders of magnitude between a sunny and a cloudy day, making it clear that a correct description of charge transport in solar cells is only possible by taking charge density account. Instead of dissociation into free carriers and diffusion to the electrodes, it is also possible that recombination or de-excitation takes place. In some cases this unfavored process goes along with light emission, in other cases the decay is non-radiative, e.g., due to conversion of the excitonic energy into thermal vibrations (“phonon-phonon non-radiative emission”). One distinguishes between exciton decay or intramolecular recombination, i.e., a bound Frenkel exciton decays, geminate recombination or bimolecular recombination, i.e., the intermolecular recombination of a CT state, and nongeminante recombination, i.e., the recombination of free charge carriers. The charge density also influences charge splitting and the loss of excitons. Specifically, higher hole densities have been observed to reduce the probability of populating CT states and to favor the dissociation into free carriers in expriments that compared doped and undoped samples of PCBM and PCPDTBT (see Appendix A for full names) [45]. On the downside, geminate recombination, which is usually in the millisecond regime, experiences an acceleration up to nanoseconds with increasing charge density [46].

1.2 Light Emitting Diodes An organic light emitting diode (OLED) in principle operates as an inverted photovoltaic cell, and indeed, as mentioned in the introduction, the first OLED was discovered in the course of OPV research. A schematic description of the working principles is shown in Figure (1.2). Electrons are injected in the reflective, low work function metal cathode, holes in the semi-transparent, high work function anode. Due to the external current they are transported upwards in energy to the emission layer. There an exciton is created, and subsequently decays radiatively, i.e., by emitting light. Between the cathode and the emission layer there is usually an electron transport and hole blocking layer. This can be a single layer of one material or two different layers, with the blocking layers adjacent to the emission layer and the transport layers adjacent to the cathode. Similarly, there is a hole transport and electron blocking layer between anode and emission layer. The aim of the layers is to allow efficient injection and transport to the emission layer by decreasing the energy gap between electrodes and emission layer, as well as to avoid electrons to diffuse to the anode and holes to

1.2 Light Emitting Diodes | 13

Fig. 1.2. Basic processes in an OLED. Electrons are injected at the cathode, holes at the anode. Due to the applied current they are transported towards the emission layer, where they recombine light emission. To avoid further transport into the opposite electrode blocking interlayers are used.

diffuse to the cathode instead of accumulating in the emission layer and recombining. An example of the full layer structure is shown in Chapter 7. A common material used both in the hole and in the electron conducting layer is Alq3 (aluminum-tris(8-hydroxychinolin)). More recently DPBIC has been used as a hole conductor and electron blocker. BCP (bath-ocuproine-4,7-diphenyl-2,9-dimethyl1,10-phenanthroline) is often used as an electron-conducting and hole-blocking layer [47, 48, 49, 50, 51]. The electron (hole) transport layer is sometimes an n-(p-)doped material; higher charge densities lead to larger mobilities and thus more efficient transport. The semi-transparent anode is most often made of indium tin oxide (ITO), a composite of SnO2 (10 % – 20 %) and In2 O3 (90 % – 80 %) [52] with a low ionization potential, allowing efficient hole injection from ITO to the organic material. Electrons are usually harder to inject than holes since the barrier at the interface between cathode and organic material is higher for most OLEDs. Low cathode work functions lead to better injection, but are often not air-stable. Stability is an inherent problem of activematrix OLED displays, since here a current flows continuously, contrary to liquid crystal displays [53]. To combine stability and a low work function, frequently a bi-layered structure is used [54], such as Al/LiF [51, 55] or Ag/MgAg [47, 49]. The emission layer usually consists of a host–guest system. The host should provide good stability and charge transport properties, while the role of the guest (emitter) molecules is to allow for efficient radiative exciton decay. The light emission is termed luminescence, and in this particular case, where exciton creation is due to an electric current, it is called electroluminescence. Photoluminescence, in contrast, describes the situation where incoming light creates excitons that subsequently decay radiatively. There is a further distinction by the different paths of exciton decay into phosphorescence, fluorescence and thermally-activated de-

14 | 1 Organic Semiconductor Devices

(a)

(b)

Fig. 1.3. Exciton harvesting mechanisms. (a) Purely organic molecules show fluorescence. Here inter system crossing (ISC) is small and the radiative S1 → S0 transition dominates. (b) In metalo-organic compounds phosphorescence is observed, where ISC is large and thus the radiative T1 → S0 transition dominates.

layed fluorescence (TADF). The different mechanisms influence the OLED efficiency, as further elaborated below. Let us first look at exciton formation and the different steps involved in this process. 1. It starts by trapping of a carrier (e.g., a hole) at an emitter molecule. High charge density enhances this process, thus it usually happens close to the interface [50, 56]. 2. This induces the formation of an oxidized complex and a molecular reorganization. 3. If a carrier of opposite charge (e.g., an electron) comes close enough to the trapped carrier it will be attracted. Due to the Coulomb interaction a bound electron–hole complex is formed. The binding energy depends on the separation and on the dielectric constant ϵ of the host material. 4. In contrast to inorganic semiconductors, excitons in organic semiconductors have well defined spin states, similar to the situation in isolated molecules, as a consequence of the weak bonding. The spin of the hole and the electron can be coupled quantum mechanically to four new, combined states – one singlet state (electron and hole spin are coupled such that the total spin angular momentum is 0) and three triplet states (total angular momentum is 1). All four states are formed with equal probability. As a result, triplets are formed three times more often. Energy splitting of singlet and triplet state can be disregarded at large electron–hole distances. 5. Due to Coulomb interaction, the electron moves further on the host molecules towards the hole. Once it reaches a host molecule that is a nearest neighbor of the emitter molecule on which the hole sits, their wave function overlap has to be taken into account. The resulting (short-range) interaction leads to an energysplitting of the singlet and triplet states. The lower excited states, S0 , T1 and S1 to

1.2 Light Emitting Diodes | 15

S n , are mostly confined to the guest (emitter) molecules, while higher excitations can extend to the host molecules [52]. In the ideal case all excitons recombine radiatively, creating photons. This process usually takes place on the guest (emitter) molecules, but under optical excitation it may also happen on the host molecules [52]. High charge densities enhance recombination and thus, due to space-charge formation, it occurs predominantly close to the interfaces, in an effective recombination zone of only about 5 to 10 nm thickness [52]. The two basic mechanisms of exciton decay are shown in Figure 1.3. In purely organic molecules the prevailing mechanism is fluorescence, as shown in Figure 1.3a Here singlets decay radiatively from the excited S1 state to the ground state S0 . A transition between singlet and triplet states, called inter system crossing (ISC), requires a spin flip that is energetically very unfavorable in organic molecules. Therefore, the T1 → S0 decay happens with a very low probability, meaning that triplets can not be harvested efficiently. The prevailing mechanism in organometallic compounds, which exist with emission in the whole visible and also in the infrared range, is called phosphorescence and is schematically depicted in Figure 1.3b. Here the effect of spin–orbit coupling is present, i.e., the spin angular momentum, s, of an electron or hole couples to its orbital angular momentum, l. This effect increases with increasing atomic mass of the metal atom, which is why heavy metals such as Rhenium (atomic mass: 186.2 u), Osmium (190.2 u), Iridium (192.2 u) or Platinum (195.1 u) are optimal (yet expensive) candidates in materials to allow for phosphorescence. Spin–orbit coupling allows easier spin flipping and, consequently, significantly increases the inter-system conversion. If the T1 → S0 decay is radiative, both singlets and triplets can decay in a radiative process. This is described by the term triplet harvesting and leads to an internal quantum efficiency close to one, which corresponds to an external quantum efficiency of about 20 %, limited by outcoupling losses [57, 58, 59]. Emission spectra and decay times of organometallic compounds are sensitive to environmental factors: oxygen, water, pH value, temperature or glucose concentration. This is an advantage for using them in sensors [60, 61, 62, 63, 64, 65]. For OLEDs, however, it is the most important disadvantage since it leads to material degradation and, consequently, a loss in efficiency, a problem especially prominent in blue phosphorescent emitters where the photon energy is highest. A third mechanism of exciton decay, known as thermally activated delayed fluorescence (TADF) [67], is present in some purely organic molecules and organometallic complexes with small spin–orbit coupling, such as copper-coordinated compounds. It results from a pronounced metal-to-ligand charge transfer. Since it yields 100 % quantum efficiency in the theoretical limit it is an interesting prospect to solve the problem of instability and to allow for triplet harvesting at the same time. Prerequisite for this mechanism is a small energy difference between the T1 and S1 levels, ∆E ST . Then T1 triplets can be activated thermally and can be lifted into the S1 level, a process known as reverse inter-system crossing. A small value of ∆E ST can be obtained by a large spa-

16 | 1 Organic Semiconductor Devices

Fig. 1.4. Study of the internal quantum efficiency, ηint , in purely organic molecules that show thermally activated delayed fluorescence. A small gap between the S1 and T1 level, ∆E ST , is favorable for a high efficiency ηint . This can be achieved by a high spatial separation of HOMO and LUMO. On the other hand, a large transition dipole moment, Q, is necessary for a good oscillator strength of fluorescence. It competes with the first property, and molecules with a large spatial separation of HOMO and LUMO but at the same time strong delocalization show highest quantum efficiencies. Between 1a and 2a Q is increased, while both have a low ∆E ST . Between 2a, 2b and 2c the delocalisation and thus Q is decreased. Calculated internal quantum efficiencies are 27 % (1a), 100 % (2a), 84 % (2b) and 72 % (2c). Optimised structures were calculated using TD-DFT. Reprinted by permission from Macmillan Publishers Ltd: Nature Materials 14, 330, Copyright 2015 [66].

tial separation of HOMO and LUMO [68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. On the other hand, a large transition dipole moment is necessary for a good oscillator strength of fluorescence, a property that competes with the HOMO–LUMO separation. Molecules that have a large spatial separation but at the same time a strong delocalization of HOMO and LUMO show the highest quantum efficiency [66], as depicted in Figure 1.4. Apart from the desired radiative decay, there are also non-radiative loss mechanisms for excited states. Triplet states of the host molecules must be higher in energy than the triplet states of the guest (emitter) molecules, otherwise triplets are transferred to the host and, usually, lost. Two triplets can annihilate with each other, as a result of interactions of adjacent excited guest (emitter) molecules, a process known as triplet–triplet quenching. Further possible losses are the radiationless transfer of triplets to impurities and vibronic coupling, i.e., an interaction between electronic and vibrational modes, which also leads to radiationless de-excitation and triplet–polaron quenching, i.e., energy transfer to polarons and the dissociation into free charge car-

1.3 Field Effect Transistors | 17

riers [59, 52]. There are contradicting results on whether the dominating process is triplet–triplet annihilation [78] or triplet–polaron quenching also plays an important role [59]. Experimentally, the triplet-triplet quenching rate can be probed by time resolved photoluminescence experiments [59]. The operation voltage of OLEDs is in the range of 5 to 10 V [52], leading to average charge densities of about about 1015 – 1017 cm−3 [44]. It varies, however, by several orders of magnitude within the layers [79], as a result of space charge formation. The spatial distribution of the charge density is discussed in Chapter 9. For an understanding of devices in simulations it is crucial to have a correct description of the dependence of electron and hole mobility on their density.

1.3 Field Effect Transistors The principle of field effect transistors (FETs) was already discovered as early as 1925 and first fabricated in 1960 [80, 81]. FETs have three electrodes: a current between source and drain electrode is regulated by changing the voltage applied to the gate electrode. In OLED displays FETs are a key component, allowing to control sub-pixels. When used in OLED displays, FET transistors have to be able to supply a high current density, such that the display brightness is sufficient for mobile applications, i.e., in the order of about 350 Cd m−2 [53]. This can be translated into a minimum requirement for the charge mobility in FETs, where estimates range from 1.5 [53] up to 5–10 cm V−1 s−1 [82]. Amorphous silicon does not meet this requirement, which is why most of the current OLED displays use FETs made of polycrystalline silicon [83]. The downside here is the compatibility of the materials. Organic field effect transistors (OFET, see references [53, 84, 85, 86] for reviews) would be the ideal candidates to be combined with OLEDs in displays, due to their good compatibility to flexible plastic and the possibility of vacuum deposition or solution processing. As a future perspective, this might even allow completely foldable displays, where bending to a very small radius (∼ 100 μm) occurs [87]. Due to the required high mobilities, OFETs usually use crystalline materials. Criteria for good performance in OFETs are (i) a HOMO/LUMO gap that is not too large, such that the necessary injection current is accessible, (ii) strong electronic couplings, (iii) high purity to avoid trapping and (iv) molecular alignment with efficient π–π stacking [88]. Materials with sufficiently high mobilities (above 10 cm V−1 s−1 ) have already been demonstrated [89, 90] and prototype OFET-driven OLED displays have been realized [91, 92, 93, 94], yet they are still in the development stage. An interesting perspective in this context is the combination of the functionality of a transistor and of a diode in a light-emitting OFET [18], which are not only excellent research systems to examine charge-carrier recombination and light emission in organic materials [95] but also pave the way for fully flexible displays [96]. Lightabsorbing OFETs with the reverse functionality have also been realized in prototypes

18 | 1 Organic Semiconductor Devices and are able to convert a light signal into an electrical current that is directly amplified in the transistor structure [18]. OFETs are most common as thin film transistors (TFT), i.e., a thin film of the organic material deposited on top of a dielectric (insulator) with underlying gate electrode and with source/drain electrode contacts on top [88], as shown in Figure (2.1). Other setups with, e.g., contacts at the bottom are also possible. The charge density in OFETs is usually very high, in the order of 1018 – 1019 cm−3 [44]. Since charge mobility increases with density, this facilitates high mobilities. To access materials for OFET applications experimentally, one should perform measurements in an OFET setup, such that the experimental charge density matches the one in the device. Details will be discussed in Section 2.1.

1.4 Lasers Since the first fabrication of a laser in 1960 [97], lasers have become an important tool in science, giving new insights into physics and chemistry, such as elemental analysis and gas sensing (spectrometry), detection of biological samples, studies of fluid dynamics, heat transfer and mass diffusion [98] up to the direct observation of vibrations of chemical bonds [99]. They are also used in medical applications, telecommunication, atomic clocks, space applications, industrial processing and many more [98]. Their high efficiency might also help them to become applicable, for example, in car lighting, superseding LED lighting technology at some point. Lasers generally consist of an optical gain medium and a resonator. The first can be a gas, a solution or a solid-state medium. An incident photon raises a ground state electron in the gain medium to an excited state. In a three-level laser that is from E0 to E2 . Subsequently, it relaxes to the first excited level E1 without emission of a photon. If it now relaxes further to the ground state level E0 a photon of energy E1 − E0 is emitted, a process called spontaneous emission. If such a photon hits another molecule with an excited electron in state E1 , it will absorb the photon and emit two photons of the same energy. This process is called stimulated emission and was first postulated by Einstein in 1916 [100]. For a laser to work, it is important to have more electrons in the excited state E1 than in the ground state E0 , a situation termed population inversion. A spontaneously emitted photon then leads to a chain reaction with more and more photons of the same energy, i.e., of the same wavelength, being created. Since only photons propagating perpendicular to the resonator areas contribute, the light is highly directional. To maintain population inversion, photons must continuously be “pumped” into the gain material. This is done either optically or electrically. Electric pumping provides high efficiency, since the conversion of electric current into photons happens directly in the gain material. A solid-state, electrically pumped laser is called a diode laser. Such lasers are cost-effective in their production, allow miniature size, offer high reliability and are comparably simple in their operation [98].

1.4 Lasers | 19

Fig. 1.5. In an organic laser both ground and the first excited singlet states have smeared out energies following a Gaussian distribution. This leads to an effective four level laser. The figure shows the processes in such a laser for some exemplary energy levels in the density of states.

Organic materials played a significant role in laser development already within a decade after the development of the first laser [101], with the first organic lasers built in 1967 [102], using both liquid and solid dyes. Similar to the use of organic materials for LEDs, they offer the potential of molecular tunability to certain wavelengths and efficient production using vacuum deposition or solution processing. Both small molecules [103] and conjugated polymers [104, 105, 106, 107, 108] have been used in organic lasers. Most state-of-the-art lasers work with a four-level system, with photon absorption leading to a E0 → E3 excitation, spontaneous, non-radiative decay to E2 , stimulated emission in the radiative E2 → E1 decay, and de-excitation from E1 to E0 . Four-level systems have the advantage that a population inversion between the levels with energies E2 and E1 is sufficient. Therefore the lasing threshold (minimum pumping power at which energy input and loss by absorption and emission are exactly equal and such the laser starts to work by stimulated emission) in such systems is comparably low. In organic materials, compared to their inorganic counterparts, there is a Gaussian distribution of site energies. This results in an effective operation as a four-level laser: upon excitation by a photon, a singlet exciton is created, with its energy following a Gaussian distribution (E0 → E3 ). This is followed by a fast relaxation into a singlet state with lower energy (E3 → E2 ). From the bottom of the singlet energy distribution the exciton decays radiatively to some ground state energy (E2 → E1 ), also called a

20 | 1 Organic Semiconductor Devices vibrationally excited ground state level. From there it relaxes to a lower lying ground state energy (E1 → E0 ) [101]. Figure (1.5) shows these processes schematically for four exemplary energy levels in the density of states. This has also been observed experimentally by a red shift (energy-lowering) of the emission within the first few picoseconds after excitation [101]. The mechanism increases the separation between absorption and fluorescence, which is helpful for lasing as it reduces the absorption at the lasing wavelength. Summarizing the last paragraphs, organic materials seem to be matching candidates for lasers, ideally for diode lasers. Optically pumped solid-state semiconducting polymer lasers have already been constructed in the 1990s [109]. In the year 2000 the report of an organic diode laser built at Bell Labs made it into the Science magazine [110]. However it turned out to be a fraud, a subsequent investigation at Bell led to the retraction of the article and the dismissal of the responsible scientist Hendrik Schön. The problem why OLEDs cannot simply be converted into organic diode lasers is the current density. In inorganic lasers it is about 105 times higher than in OLEDs (about 1000 compared to 0.01 A m−2 ). Such a current density in an organic material with much lower mobility would lead to an overheating of the material and destroy it. Apart from that, the short lifetime of singlet excitons makes a high pumping rate necessary in order to obtain population inversion which, however, leads to excitonexciton annihilation. Further problems are created by the electrode contacts, where losses are substantial and greatly increase the lasing threshold. A possible solution might, at some point, be light emitting OFETs, as discussed in the previous section. As opposed to OLEDs, research on organic lasers concentrates on fluorescent materials. Since phosphorescence is a forbidden transition, the available gain for triplets would be orders of magnitude lower than for singlets [101]. The challenge is to achieve high mobility, simple processing, high photoluminescence efficiency and efficient charge capture at the same time. A recent report [111] of an OLED with around 100 times higher luminosity (2.8 kA/cm2 ) than common OLEDs can be seen as a first step towards the goal of organic diode lasers.

2 Experimental Techniques The first part of this chapter gives an overview of experimental techniques to examine charge transport properties in organic semiconductors. Between different measurements the mobility for the same material can differ up to three orders of magnitude [112, 113, 114], which is primarily a result of different charge densities. In the second part of this chapter I therefore review techniques to extract the charge carrier density. Some techniques also allow a simultaneous extraction of charge mobility and density. The charge density can be increased dramatically by doping, which is the last topic of this chapter.

2.1 Mobility Measurements OFET and current–voltage measurements allow a quick extraction of charge transport properties by assuming certain models that relate the mobility to the current–voltage characteristics. A more direct technique to determine the charge carrier mobility is the time-of-flight (TOF) measurement and charge extraction by linearly increasing voltage (CELIV). Impedance spectroscopy also analyzes the current–voltage characteristics, but uses an alternating current and a model with the complex impedance. Not discussed is the Hall effect measurement [115, 116]. It is useful for band transport (cf. Section 3.2), however, for most organic semiconductors hopping transport prevails, such that the Drude model assumed in the derivation of the Hall mobility does not hold [117]. Other techniques, that are not discussed here, include xerographic discharge time-of-flight measurements [118, 119], transient electroluminescence [120, 121], pulse-radiolysis time-resolved microwave conductivity [122], Auston-switch photoconductivity [123], the conductivity/concentration method [124, 125], surface acustoelectric traveling wave [126], photoinduced transient Stark spectroscopy [127] and Seebeck effect measurements [128].

Field Effect Transistor (FET) In this method the organic semiconductor is incorporated into a field effect transistor, see also [129] for a review of the measurement technique. The main components of an OFET are shown schematically in Figure 2.1. A voltage VGS is applied (negative for hole conduction, positive for electron conduction) between gate and source electrode. Above a certain threshold voltage, VT , the FET is activated and a drain–source current, IDS , flows. The mobility can be analyzed from a plot of IDS vs. VGS , however, with the drain–source voltage VDS also entering the equations. There are two regimes above the threshold voltage. For a low drain–source voltage, VDS  VGS − VT for electrons or −VDS  −VGS + VT for holes, the FET is in the linear regime, IDS ∝ VGS , and acts as

22 | 2 Experimental Techniques a gate voltage-controlled resistor, with the current following the relation [129]   V2 w IDS = Ci μ (VGS − VT ) VDS − DS . l 2

(2.1)

Here l is the channel length, i.e., the distance between source and drain contact, and w is the channel width. Ci is the capacitance per unit area of the insulator between the gate electrode and the organic semiconductor, cf. Figure 2.1. The mobility can be calculated from the slope of this expression as μlin =

l 1 ∂IDS . w Ci ∂VGS

(2.2)

At higher voltages VDS the FET is in the saturation regime, where a depletion area forms at the drain contact and the dependence of the current on the gate–source voltage is 2 quadratic, IDS ∝ VGS , or more explicitly [129] IDS =

w Ci μ (VGS − VT )2 . l 2

The mobility in the saturation regime is calculated as p !2 l 2 ∂ ID μsat = . w Ci ∂VGS

(2.3)

(2.4)

An example of an FET measurement is shown in Figure 2.1. An analytic prediction of the mobility in the linear regime has been suggested by Vissenberg et al. [130], who derived an expression for the current from percolation theory and inserted it into Equation (2.2).

Fig. 2.1. The left-hand side shows a typical OFET structure, here in a top-contact bottom-gate setup. On the right is a schematic picture of an OFET measurements with the drain–source current, IDS , plotted against the drain–source voltage, VDS for a fixed gate–source voltage, VGS . A certain threshold voltage, VT , is necessary before any drain–source current flows.

2.1 Mobility Measurements | 23

Current–Voltage Measurement with Space-Charge-Limited Current (SCLC) Analysis Current–voltage measurements are performed by sandwiching the organic semiconductor between two electrodes, scanning different voltages, V, and recording the currents, I, through the semiconductor at these voltages. This technique serves as an indirect measurement of the mobility by assuming models that allows to extract the mobility from the current–voltage behavior. Characteristic for this setup is a strongly inhomogeneous charge density distribution throughout the device, with a high accumulation of charge carriers of the respective sign at the electrodes where they are injected. Due to this space charge formation the technique is also called space-charge-limited current (SCLC) measurement.

Fig. 2.2. Current–voltage measurement of an organic semiconductor with space charge limited current analysis. The plot schematically shows the three different regimes: (1) Ohmic transport at low voltages with I ∝ V, (2) the SCLC regime where, in an approximation, the Mott–Gurney relation, Equation (3.64), holds and I ∝ V 2 and (3) the SCLC trap-free regime for very high voltages. The second regime is used to extract the mobility after fitting to the Mott–Gurney relation or by more elaborate techniques, cf. Chapter 9.

The development of a theoretical description of SCLC started in 1911 with the work of Child [131] on ion dynamics and the transfer to electron dynamics by Langmuir [132], both for a vacuum diode, i.e., in a situation without scattering. The theory predicts 3 an I ∝ V 2 behavior of the current, known as Child’s law, see Section 3.9.1. It was extended in 1940 by Mott and Gurney [133] to the “Trap-free insulator model”, i.e., to the description of solids where scattering takes place. Helfrich and Mark first applied it to organic crystals in 1962 [134] and afterward it was used for many crystalline and amorphous organic semiconductors [135, 136, 137]. The Mott–Gurney law, cf. Section 3.9.2, predicts a I ∝ V 2 behavior. It is valid under the assumptions of (i) unipolar (hole-only or electron-only) transport (ii) no doping, (iii) constant mobility and relative permittivity, (iv) Ohmic contacts, i.e., no injection barriers and (v) one-dimensional current

24 | 2 Experimental Techniques flow. The dependency on the squared voltage is valid in the SCLC regime, i.e., starting from a certain applied voltage. At lower voltages (small electric field) transport is Ohmic and the current behavior is linear, I ∝ V. At very high voltages the charge carrier density increases up to a point where all traps are filled. At this trap-filling voltage, VTF , the current increases abruptly. The regime with V > VTF is called the SCLC trap-free regime. A schematic representation of the three regimes is shown in Figure 2.2. The mobility is often extracted from a fit to Equation (3.64) in the SCLC regime. The above mentioned conditions for the validity of the Mott–Gurney equation already suggest that it should be seen more as a model system than a reliable tool to extract the mobility. There have been attempts to resolve some of the limitations: In doped semiconductors there is a background charge, n0 , of additional free electrons (holes), which is significantly larger than the number injected charge carriers responsible for SCLC. In consequence, here an SCLC regime is not observed but the current behavior is Ohmic throughout, IOhmic (V) = qn0 μ Vd [138]. Another extension is to go from the one-dimensional to the two-dimensional case [139], where the sample is not sandwiched between the contacts but two contacts of distance d are placed on top of a sample of thickness h. The I ∝ V 2 of the Mott–Gurney law is still valid, but the 2 dependence on the sample thickness changes: I2D (V) = 2π ϵμ Vd2 . A transition from the one-dimensional to the two-dimensional is observed with increasing d/h ratio. In any case, the condition to have a constant mobility will never be fulfilled. In fact, the space-charge formation leads to a strongly inhomogeneous charge distribution. Since the mobility in organic semiconductors shows a strong dependence on the charge density, it will also be strongly inhomogeneous throughout the sample. In the following I describe a more elaborate way to analyze current–voltage measurement taking this fact into account. A study of the validity of the Mott–Gurney relation is presented in Chapter 9.

Current–Voltage Measurement with Gaussian Disorder (GDM) Drift–Diffusion Analysis Due to the above mentioned shortcomings of the Mott–Gurney model, the analysis of current voltage data has become more elaborate. Today a common procedure is to use one-dimensional drift–diffusion equations to describe charge transport, which will be discussed in detail in Chapter 9, see Equations (9.2) to (9.6). The mobility entering the drift–diffusion equations can be coupled to the functional form of lattice models, that describe the dependence on temperature, electric field and charge density. These models, called Gaussian disorder models (GDM) [22, 140, 114, 141] will be discussed in details in Section 3.6. A fit of the drift–diffusion equations coupled to the GDM, allows to extract the mobility from current–voltage measurements. The method is indirect, but contrary to the Mott–Gurney model it does not assume a constant mobility. Apart from the mobility, the method is also used to extract the energetic disorder, σ, and the average molecule spacing from the current–voltage data. In my work I evaluate in how

2.1 Mobility Measurements | 25

far this can be expected to yield microscopically adequate parameters, see Chapters 7 and 8.

Time-of-Flight (TOF) The time-of-flight technique was developed between 1957 and 1960 [142, 143, 144], and since then has routinely been used used for mobility measurements in organic semiconductors, for example in Alq3 [145]. Figure 2.3 explains the method schematically: The organic sample is sandwiched between two electrodes, one of them semitransparent (in this example ITO). A voltage, V, is applied to the sample and an oscilloscope connected to resistor R is used to monitor the current. A short laser pulse is used for photo-generation of free carriers at the transparent electrode. The arrival of the charge carriers at the opposite electrode at the transit time ttr can be seen in a characteristic response in the current vs. time plot, shown in the bottom right of Figure 2.3. The charge carrier mobility is then calculated using the equation μ = d2 /ttr V, where d is the sample thickness and thus the distance the carriers travel in the electric field direction [146].

Fig. 2.3. Setup and measured quantities of a time-of-flight measurement of the mobility.

Compared to current–voltage or FET measurements this technique provides a direct measurement of the mobility, without the need to assume certain models. This is, however, only true if the light pulse is short and weak enough to produce only a small amount of carriers, such that no space-charge formation occurs and the electron density is low enough to have a mobility that is density-independent (zero density limit). It is also only true in the above shown geometry with the sample sandwiched between two electrodes; a coplanar configuration, i.e., two contacts placed upon the sample, lead to a more complicated situation [147]. Due to the low charge density, TOF measurements usually yield lower mobilities than, for example, FET measurements.

26 | 2 Experimental Techniques Charge Extraction by Linearly Increasing Voltage (CELIV) The method of charge extraction by linearly increasing voltage (CELIV) [148] is based on the same experimental setup as the TOF method. The difference lies in the charge generation and in the applied voltage. The latter starts at zero and is increased linearly within a certain time tpulse . The current increases with the increase of the voltage, until carriers start reaching the opposite electrode, as shown in Figure 2.4. This time, tmax , allows to extract the charge mobility. A light pulse for the photo-generation of charges is not necessarily needed, as the applied voltage can provide a sufficient number of charges, making the CELIV measurement experimentally easier. However, using a light pulse to generate charge carriers (photo-CELIV) is also possible, and it is helpful for undoped, low conductivity semiconductors. While photo-generated carriers have to relax in energy and charge density, carriers generated in the bulk by the applied voltage are already in equilibrium, such that the conventional CELIV technique does not have the problem of “hot carriers” leading to artificially increased mobilities as it can be the case in photo-CELIV or TOF measurements.

Fig. 2.4. Voltage and current density as a function of time in a CELIV measurement.

For the extraction of the mobility one has to distinguish between three different cases, depending on the material’s conductivity, and to take into account a factor for photogeneration or charge generation in the bulk [146]. An advantage of the CELIV technique is that it allows a simultaneous determination of the charge density, cf. Section 2.2.

2.2 Charge Density Measurements | 27

Impedance Spectroscopy Impedance spectroscopy [149], also called dielectric spectroscopy, is a technique where an alternating current is applied to the sample and the complex impedance is measured at different frequencies. The extraction of the carrier mobility is done indirectly by assuming certain models, such as the Mott–Gurney model (Section 3.9.2) [150] or, more general, drift–diffusion equations coupled to the Poisson equation [151, 152]. In the latter case the frequency dependent drift current, Idrift (ω), is separated into a stationary dc component and a frequency-dependent ac part [151]: Iac (ω) + Idc

=

 
q ndc (x) + nac (x, ω) μdc Fdc (x) μac (ω)
× Fdc (x) + Fac (x, ω) + iωFac (x, ω),

(2.5)

with Idc/ac , μdc/ac , Fdc/ac and ndc/ac and denoting the dc/ac components of the current, the electric field and the charge density, respectively. A fit to the measured, frequency dependent conductance and capacitance allows to extract the mobility and charge density. Instead of directly using the complex impedance data one can also go via the dielectric constant, the dielectric loss [153] or the diffusion constant and the Einstein relation [154]. The method is less prone to dispersion effects in highly disordered materials, compared to, e.g., time-of-flight measurements [155]. A further advantage is that it serves for a simultaneous extraction of electron and hole mobilities [156].

2.2 Charge Density Measurements Experimental methods for determining the density of free charge carriers differ a lot in their approaches and in their applicability: certain techniques are based on currents during operation of a device while others aim to determine the density of free carriers after doping a material. As already mentioned in the previous section, CELIV measurements allow for a determination of the charge density. I will shortly discuss the density aspect in these measurements, referring to the previous section for details of the techniques. Current– voltage and impedance spectroscopy measurements yield a charge density distribution after fitting the IV characteristics to results from a drift-diffusion model, however this is only an indirect method. Additional methods discussed here are Kelvin probe force microscopy (KPFM), capacitance–voltage analysis and polaron-induced absorption spectroscopy, which are all suited to characterize free carriers in doped materials. The method of charge carrier extraction under illumination is helpful for characterizing solar cells. Not discussed here are time-resolved electroluminescence, which is used to study charge accumulation in the emission layer of an OLED [157] and the Suns-VOC method [158] for solar cells. Also not discussed is Raman spectroscopy, that

28 | 2 Experimental Techniques allows to probe the changing electron density in conjugated systems upon electron transfer [46, 159].

Charge Extraction by Linearly Increasing Voltage (CELIV) In a CELIV measurement the charge extracted at the electrode, Qe , can be evaluated by an integration of the current density over time, multiplied by the area of the electrode, S: tZpulse
Qe = S j(t) − j(0) dt. (2.6) 0

The extraction from the measurement is illustrated in Figure 2.4: The upper part of the figure shows the applied voltage that is increased linearly with time. In the resulting current response, the fraction j(0) is the current of the RC circuit (capacitive part) and ∆j(t) = j(t) − j(0) the current due to extracted charge carriers. Figure 2.4 shows the different quantities graphically.

Charge Extraction under Illumination Organic semiconductors that are used in OPV can be characterized by a charge extraction technique [160, 161]. In such a measurement the solar cell is held at a fixed point of the current–voltage curve under illumination. It is then switched to short circuit and the illumination is turned off. Switching to short circuit conditions creates a discharge current. The integral over time of this current is equal to the free charge in the cell in the current–voltage point before switching, provided that recombination is small enough to be neglected. By varying the illumination intensity one can derive the relation between charge carrier density and short-circuit current [44].

Kelvin Probe Measurement Kelvin probe force microscopy (KPFM) [162] is a variation of atomic force microscopy (AFM). While a cantilever, i.e, a microscopically small “needle” is used in AFM, KPFM makes use of a conducting tip that does not touch the sample, the Kelvin probe. A voltage applied between tip and sample allows to determine the surface work function, i.e., the energy needed to remove a surface electron from the sample. The work function, again, is directly connected to the density of free charge carriers. Hamwi et al. [163] showed for the example of BCP (bath- ocuproine-4,7-diphenyl-2,9-dimethyl-1,10phenanthroline) p-doped with MoO3 , how the charge density can be extracted from the variation of the surface work function throughout the sample, which is a result of space charge formation. Their technique requires a gradual deposition of doped BCP on an electrode (here ITO), KPFM measurements for different thicknesses, and a subsequent extraction of the charge density from the space charge model. The grad-

2.2 Charge Density Measurements | 29

ual deposition can, however, not exactly model the situation in a thicker bulk sample [164]. The Fermi level change coming along with the increased charge density after doping can also be quantified by KPFM [165].

Capacitance–Voltage Analysis In the capacitance–voltage (C–V) technique [166] the organic semiconductor is connected to a metal electrode. When a voltage is applied, a depletion region forms in the semiconductor close to the electrode, i.e., a region without free charge carriers. This depletion region acts as a capacitor. Measurements of the capacitance under a variation of the gate voltage, V, which goes along with a varying size of the depletion region, allow insights into properties of the semiconductor. The charge density can be extracted from the Schottky–Mott analysis [81] 2 (V − V ) 1 = 2 bi , C2 A qε0 ε r n

(2.7)

where C is the capacitance, Vbi the built-in potential, A the semiconductor–electrode contact area, q = ±e the charge per carrier, ε0 the vacuum permittivity, ε r the relative permittivity and n the free charge carrier density. After plotting 1/C2 against the voltage, the charge density can be extracted from the slope without previous knowledge of the built-in potential. The technique, first applied to inorganic materials, has routinely been used for a variety of organic semiconductors [167, 168] and dopants [169].

Polaron Induced Optical Absorption Spectroscopy In absorption spectroscopy a sample is placed between a radiation source and a spectrometer. The energy spectrum after absorption by the sample is compared to the original spectrum. If the sample consists not of atoms only but of molecules, molecule specific excitations, such as rotational or vibrational excitations are observed in addition to electronic excitations, leading to spectral bands (as opposed to sharp spectral lines). For polaron induced optical absorption spectroscopy, two measurements are performed. The first one is a measurement of the undoped sample to determine the polaron absorption cross section [170], that includes an analysis assuming the space charge limited current model and an additional measurement of the electron or hole mobility as an input parameter. A second measurement is performed to determine the material’s absorption. The free charge carrier density is then calculated by using the ratio of the material’s absorption to the polaron absorption cross section at the maximum of the latter [168].

30 | 2 Experimental Techniques

2.3 Doping Doping is the process of introducing electron donating or accepting impurities into a material. This leads to higher hole or electron densities and is of vital importance for many semiconductor devices. In organic semiconductors doping is equally important, but faces different challenges, as reviewed, for example, by Gregg et al. [171] or Walzer et al. [128]. The main difference is that organic semiconductors are bound by weak intermolecular forces (van der Waals bonds), and not by strong ionic bonds. As a result, the introduction of dopant molecules often leads to bending or breaking of bonds, hindering the formation of free charges [171].

(a)

(b)

Fig. 2.5. Doping of organic semiconductors. (a) Hole density and (b) conductivity as a function of pdoping density in ZnPc doped with F4 TCNQ. Reprinted with permission from Maennig et al., Phys. Rev. B 64, 195208 (2001) [172], Copyright 2001 by the American Physical Society.

p-type doping, which increases the hole density, requires a dopant material (acceptor molecule) with a LUMO level near the HOMO of the host material. This condition is comparably easy to fulfill, making p-type doping technologically easier. A suitable dopant (depending on the host material) can, for example, be F4 -TCNQ (2,3,5,6tetrafluoro-7,7,8,8-tetracyanoquinodimethane), which is a very strong electron acceptor. As an example, Figure 2.5 shows how doping ZnPc (zinc phthalocyanine) with F4 -TCNQ influences hole density and conductivity: both increase up to three orders of magnitude with increasing doping ratio. The effect is similar both for a polycrystalline and for an amorphous phase of ZnPc. Compared to the undoped sample, the conductivity increases by six orders of magnitude with 1 % molar doping concentration [128]. A superlinear increase of the conductivity with dopant concentration has been observed independent of morphology, material or type of doping [171, 128].

2.3 Doping

| 31

105

luminance (cd/m²)

104 Al (100 nm) 103

LiF (1 nm) BPhen:Li (30 nm)

102

BPhen (10 nm)

p-i-n OLED without n-doping

Alq3 (20 nm)

101

TPD (5 nm) m-MTDATA:F4TCNQ (100 nm)

100

ITO substrate

2

3

4

5

6

7

8

voltage (V)

Fig. 2.6. Luminance profile of an OLED, once with an n-doped transport layer (BPhen:Li), as shown in the stack structure, and once without n-doping of BPhen. Reprinted with permission from Walzer et al., Chem. Rev. 107, 1233 [128]. Copyright 2007 American Chemical Society.

n-type doping to increase the electron density requires a dopant material with a HOMO level near or above the LUMO of the host material. This is more challenging than ptype doping since it is difficult to find air-stable materials with such a high HOMO level. For OPV applications an electron affinity of around 4.0 eV is needed, for typical OLED materials it has to be around 3.0 eV (note that a low electron affinity is energetically unfavorable). With decreasing electron affinity the materials become more and more unstable in air. An estimated threshold for air-stability is 4.3 eV. For this reason it is especially hard in OLED applications to find suitable n-type dopants [128]. One category of n-type dopants are small molecules (such as O2 , Br2 , I2 ). The problem in this case is the high diffusion coefficient: because of the weak intermolecular forces such molecules can diffuse and therefore do not allow a doping profile that is stable in space and time [171]. In particular, p-n junctions will be destroyed by such an unwanted diffusion. A second category is formed by alkali metals, specifically lithium (Li, Li2 O, LiF) and caesium. Lithium can be used for bulk doping, i.e., by coevaporation together with the organic material. The most common use, however, is in the form of an LiF layer between the electron transport layer and the cathode (mostly aluminum), i.e., the very common combination Alq3 |LiF|Al [173, 174, 175]. In my cooperation project with BASF a DPBIC|LiF|Al setup was used, see Chapter 7. X-ray photoelectron spectroscopy (XPS) [176, 177, 178] and secondary ion mass spectroscopy (SIMS) measurements [179, 180] have been conducted to clarify whether or not lithium atoms diffuse into the electron transport layer, thus doping it. Results were contradictory, observing no diffusion or diffusion up to 80 nm into the organic layer. Walzer et al. suggested this might be a result of different experimental setups: only if the metal electrode is deposited, a process done at very high temperatures, the temperature-dependent lithium diffusion takes place [128]. While lithium atoms are wanted in the electron transport layer to increase the mobility, they are not wanted in the emission layer; here they

32 | 2 Experimental Techniques lead to quenching and decrease the OLED lifetime [181]. Therefore a possible alternative dopant might be Caesium, which has a lower diffusion constant [128]. A third category are large aromatic molecules, which can donate their π electrons, such as BEDT-TTF (bis(ethylenedithio)-tetrathiafulvalene) [182]), TTN (tetrathianaphthacene) [183], CoCp2 (bis(cyclopentadienyl)-cobalt(II) or cobaltocene) [184], or the transition metal complex [Ru(terpy)2 ]0 (bis(terpyridine)ruthenium) [185]. The last category are cationic dyes, that have been successful dopants in OPV applications [186, 187], while for OLEDs lower oxidation potentials are necessary. Two layers of differently doped materials, a p-doped and an n-doped one, are called a p-n junction. The charge gradient leads to an internal electric field, which, for example, helps to split bound carriers of opposite charge in photovoltaic cells. While these junctions are usually stable when using inorganic semiconductors, the weak intermolecular forces in organic semiconductors can allow ion diffusion instead of charge diffusion, destroying the p-n junction. This is especially a problem when using small doping molecules, such as O2 , Br2 or I2 , but also for the above mentioned lithium, while molecular doping gives better stability. If the host material of the pdoped and the n-doped layer is the same, the junction is called a homojunction, otherwise it is called a heterojunction. Apart from being an interesting model system for research, homojunctions also have the advantage of a high open-circuit voltage, and are thus useful for efficient photovoltaic cells. It is, however, a big challenge to find a working combination of a host, a p-dopant and an n-dopant molecule, where the prerequisites of a p-dopant LUMO near the host HOMO and an n-dopant HOMO near or above the host LUMO are fulfilled simultaneously. The first successful realization of a stable organic homojunction was reported in 2005 [188], using ZnPc as host, F4 -TCNQ as a p-dopant and [Ru(terpy)2 ]0 as an n-dopant material. It has also proven useful to use an intrinsic, undoped layer (p-i-n junction) to allow for an efficient formation of a space-charge region [189]. In organic solar cells doped layers fulfill various purposes. One of them is as optical spacers, i.e., transparent interlayers that separate the active material from the electrodes, such that the active material is in the region with the highest optical field, leading to efficient absorption [190, 191, 192, 193, 194, 195]. While undoped materials can be used for such layers with thicknesses of about 10 – 30 nm [196, 197], in doped layers the mobility is high enough to use more than 100 nm thick layers without significant losses in carrier transport [198]. Doped layers are also important in monolithically integrated tandem cells, i.e., tandem cells which are not simply two mechanically separated cells stacked on top of each other but which consist of sub-cells stacked together and connected in a series by a charge recombination zone, where photo-generated electrons from one cell recombine with holes arriving from the opposite site. This leads to a continuous current throughout the device, with the voltage being the sum of the voltages of the sub-cells [198]. For the cell efficiency thick recombination interlayers are advantageous when optimizing the absorption. High mobilities of doped materials allow to

2.3 Doping

| 33

use thick layers [190, 199]; up to 300 nm thick recombination layers have been used in a tandem cell without decreasing open circuit voltage or fill factor significantly [195]. In OLEDs, n-doping is used to increase the mobility in the electron transport layer and p-doping to increase the mobility in the hole transport layer. The resulting increase in conductivity lowers the required driving voltages. A special type of n-doping is the above mentioned interlayer between the (aluminum) cathode and the electron transport layer; diffusion of, e.g., lithium ions into the electron transport layer leads to a minimization of the energy barrier between cathode and electron transport layer, thus giving (approximately) Ohmic contacts. Figure 2.6 shows the luminance of two OLEDs, one where the BPhen electron transport layer is doped with lithium and one where it is undoped. The luminance at a given voltage is increased by about one order of magnitude by this n-doping. Note that both cells also have an LiF interlayer (n-doping) and a p-doped hole transport layer. The energetic effect of doping, namely a shift of the Fermi level, can be extracted by ultraviolet photoemission spectroscopy (UPS) [180, 200]. Infrared spectroscopy allows to determine to which degree the charge is transferred from the dopant molecule to the host [172]. Prerequisite for good charge transfer is a favorable energy level alignment. For a theoretic description of the conductivity as a function of the doping ratio a percolation model [172] and a Miller–Abrahams hopping transport model [201, 202] have been proposed.

3 Charge Dynamics at Different Scales Electron or hole transfer between two molecules in an organic semiconductor, i.e., an oxidation/reduction reaction, takes place at the molecular (nanometer) scale. Contrariwise, experimental current–voltage characterizations of OLEDs, organic solar cells or OFETs are measured in devices which are several micrometers thick. Consequently, there are several orders of magnitude difference in different descriptions and the question is how one can link the device performance to elementary processes occurring at the nanometer scale processes. transport mechanism (3.2) hopping transport described band transport by the master eqution (3.3) solution methods: KMC (3.4) numerical solver (3.5)

continuous space models (3.8)

models: GDM (3.6) analytic (3.7)

heuristic models (3.9)

1 nm and less

ca. 10 nm

100 nm and more

molecular scale

mesoscopic scale

device scale

charge transfer reactions (3.1)

Fig. 3.1. Overview of different scales of charge dynamics and where in this chapter they are discussed.

In this chapter I start from the smallest scale in this picture and review different models of charge transfer processes between two molecules. I then discuss how charge dynamics can be described by band transport in inorganic semiconductors and crystalline organic semiconductors. In amorphous semiconductors charge transfer takes place via a series of charge transfer events, and can be described as hopping transport. This transport can mathematically be described in terms of a Poisson process and the master equation. For systems with multiple charge carriers a mean field approximation of the master equation allows solving it for the necessary system sizes. A different approach is kinetic Monte Carlo (KMC), which can incorporate various additional effects (recombination, Coulomb interaction etc.), but is computationally more demanding. I discuss both methods and show how they allow to obtain a dependence of the charge carrier mobility, μ, on external parameters, namely temperature, external electric field and the charge carrier density, but also on specifics of the material, such as its energetic landscape, the strength of charge transfer reactions and the intermolecular spacing. Different functional forms of the mobility have been developed in order to simplify the description as a function of external and material-specific parameters. They can

3.1 Charge Transfer Reactions | 35

be separated into two main classes, namely functions derived heuristically from KMC simulations and attempts to solve the problem analytically. Finally, with the charge carrier mobility and its dependencies at hand, one can tackle the problem at the macroscopic scale, where it is described by the drift– diffusion equation. I discuss properties of the equation and how it can be linked to the smaller scales. The chapter concludes with a discussion of heuristic models developed between 1911 and 1951, which are still still widely used for experimental analysis of current–voltage measurements. The applicability of such models will be assessed later in Chapter 9.

3.1 Charge Transfer Reactions In this section I recapitulate charge transfer theories, compare respective charge transfer rates, and summarize the underlying assumptions. For details on charge transfer, including the topics of intramolecular charge transfer, delocalization over different molecules, bridging molecules and derivations of the transfer rates, the books by Weiss [203] and by May and Kühn [204] are recommended. The dynamics of a molecular system made of nuclei and electrons can be described by a Hamiltonian that includes the kinetic energy of all nuclei and electrons and all electrostatic interaction potential operators (nucleus–nucleus, electron– electron, electron–nucleus). The state of the system is then described by a many-body wave function, which is the solution to the Schrödinger equation with appropriate boundary conditions (neglecting relativistic effects). For macroscopic systems, where the number of atoms is in the order of 1023 , even just storing the wave function a hopeless task, not talking about the necessary processor power for computing it. Efficient approximations are therefore indispensable for addressing the problem. A common approximation applied to the Hamiltonian is the Born–Oppenheimer approximation or adiabatic approximation. It is based on the assumption that electron movement is fast compared to nucleus movement. Thus one can decouple the corresponding degrees of freedom and treat nucleus positions parametrically when describing electron dynamics. This also allows to define a so-called adiabatic basis of electronic wave functions, in which the problem is solved conveniently in case of sufficiently delocalized electronic states. In many organic materials, however, electronic states are rather localized on a single molecule or parts of it. The coupling between such states is small and the rearrangement of nucleus positions can lead to transitions between electronic states, which is called nonadiabatic charge transfer. The electronic states are easier to describe in a basis adjusted to this problem, called the diabatic basis. The Marcus and Weiss–Dorsey rate expressions discussed here fall into this category.

36 | 3 Charge Dynamics at Different Scales I will use the notation ω ij for the charge transfer rate from molecule i to molecule j, as it is commonly done in chemistry and physics. Most publications from the field of mathematics follow the opposite convention.

3.1.1 Miller–Abrahams Rates The Miller–Abrahams rate expression [205] neglects the effect of nuclear reorganization. It has two contributions. The first one is an electronic coupling factor,
exp −2r ij /rloc , that describes the likelihood of electronic tunneling. Here r ij is the distance between the two molecules and rloc is the localization length of the charge. The second contribution is a Boltzmann factor with the energy difference between the two diabatic states:      ∆E ∆E ij > 0 2r ij exp − kTij , (3.1) ω ij = ω0 exp − rloc 1, ∆E ≤ 0 ij

The factor ω0 is the attempt frequency. Miller–Abrahams rates are the simplest rate expression and are often used in lattice model systems such as Gaussian disorder models, see Section 3.6.

3.1.2 Marcus Theory The development of Marcus theory started in 1956 [206]. R. A. Marcus was awarded the Nobel Prize in Chemistry of the year 1992 for his theory. Marcus rates [207, 208] were the first to take the effect of the environment into account. the D Into E rate expres ˆ el ˆ el = sion enter transfer integrals for electronic coupling, J ij = ψ i H ψ j , where H

Tˆ el + Vˆ el−el + Vˆ nuc−el is the electronic Hamiltonian (kinetic energy of electrons, interaction between electrons and interaction between electrons and nuclei) and |ψ i i are the electronic wave functions in the diabatic basis. A second, additional parameter is the reorganization energy, λ ij , that describes the energetic change due to the relaxation of the nuclei after the charge transfer. The rate reads¹: "
2 # J 2ij ∆E ij − λ ij 2π p ω ij = exp − . ~ 4λ ij kB T 4πλ ij kB T

(3.2)

The Marcus rate can formally be derived by using linear response theory to describe a coupling to the environment, also called the “heat bath”. At sufficiently high tem-

1 Note that this rate expression assumes ∆E ij = E i − E j . Sometimes in the literature the definition
2 ∆E ij = E j − E i is used, which changes the expression in brackets to ∆E ij + λ ij .

3.1 Charge Transfer Reactions | 37

peratures the electron has to tunnel from reactant to product state. This requires nuclear vibrations (“bath fluctuations”) to bring reactant and product energy levels into resonance. The prefactor in the rate expression describes the probability of electronic tunneling once the levels are in resonance. The second term is a Boltzmann factor that takes the reorganization energy into account, in other words the activation energy required for favorable bath fluctuations. Marcus rates are valid for small electronic coupling, when the reaction is nonadiabatic. The calculation of the necessary ingredients for the rates, J ij , λ ij and ∆E ij , will be discussed in Chapter 7.

3.1.3 Weiss–Dorsey Rates Weiss–Dorsey rates [209, 210, 211, 203] are a generalization of the Marcus rate expression to a wider temperature regime. They are derived in the spin-boson model, also called the dissipative two level system. The name stems from the fact that it describes a system of two states coupled to a surrounding that can be imagined as a “heat bath” of bosons. The two levels are often regarded as the spin-up and spin-down state of a spin- 21 system. In the context of electron transfer the two states describe the donor and acceptor state. The corresponding Hamiltonian (cf. [203] Chapter 20) consists of a term for tunneling between the two states, a fluctuating polarization energy acting on the dipole moment of the states and the “bath”, a Gaussian reservoir. The rates derived from this Hamiltonian, in the general form, read [209] ω ij =

J 2ij ~2

Z∞

−∞

dt exp



 ∆E ij it − Q (t) , ~

with t being a complex time and the dimensionless function Q defined as      ~ν ~ Z∞ 4 j(ν) cosh 2kB T − cosh ν 2kB T − it   Q (t) = . dν 2 π ~ν ν sinh 2k T B 0

(3.3)

(3.4)

j (ν) is the spectral density  ofa heat bath and is often modeled by the Ohmic spectral π density j (ν) = 2 αν exp − ννc with a characteristic frequency νc . The general expression from Equation (3.3) can be simplified to two limiting cases:

High temperature limit ~νc /kB T  1. In this case the expressions can be simplified to yield the Marcus rate expression, Equation (3.2).

38 | 3 Charge Dynamics at Different Scales Low temperature limit ~νc /kB T  1. For low temperatures, the Weiss–Dorsey rate can be simplified to the following expression:   2 1−2α Γ α + i ∆E ij  J 2ij 2πkB T ~νc (3.5) ω ij (ϵ) = Γ (2α) ~2 νc 2πkB T !   ∆E ij ∆E ij × exp exp − , 2kB T ~νc where νc is the characteristic frequency, which is here the largest frequency in the bath. This frequency is related to the reorganization energy by λ = 2α~νc [212], and Γ (z) denotes the Gamma function. The Kondo parameter, α, is an additional parameter compared to the Marcus rate expression, that describes the coupling strength between the charge and the bath. The review by Legget et al. [213] gives details about this parameter. For the case of regio-regular poly(3-hexylethiophene) (rr-P3HT), Asadi et al. [212] calculated it from a fit to IV curves at different temperatures and obtained a value of α = 2.8.

3.1.4 Comparison and Validity Regimes of Rates A direct comparison of Miller–Abrahams to Marcus rates in a system with microscopic input parameters faces the difficulty of choosing an attempt frequency ω0 . As a simple test I took the Marcus rates for a DPBIC system of 4000 molecules (see Section 7) and chose ω0 such that the mean over all rates was the same for both theories. Results are shown in Figure 3.2, which reveals that the rates have little in common. Note that a different choice of ω0 would only shift the rates in (a) upwards or downwards and shift the scatter plot in (b) to the left or to the right, not leading to any possible improvement in the correlation of the rates. A special observation is that there are some extremely low Marcus rates were Miller–Abrahams rates are much higher. This can be attributed to the inverted regime of Marcus rates, which does not exist in the Miller– Abrahams picture. It predicts that for an energy difference larger than the reorganization energy, including an external electric field, the rates decrease with increasing external field. This inverted regime was predicted in Marcus’ original work [206] in 1956 and experimentally proven almost thirty years later in 1984 [214]. In the inverted regimes tunneling processes at low temperatures become particularly important. Figure 3.2a also shows the low temperature limit of Weiss–Dorsey rates for comparison. Two different values for the unknown Kondo parameter, α, are displayed, with the characteristic frequency calculated from the relation λ = 2α~νc and λ obtained from microscopic simulations. With such apparent differences between the rate expressions naturally the question arises which one to use for a correct description of charge transfer. Cottar et al.

3.1 Charge Transfer Reactions | 39

1030

1020

α=

rate (s−1 )

1010

1015

1

α=

100

α=

0.5

0.25

10−10 10−20 10−30 10−40

Miller–Abrahams Marcus Weiss–Dorsey LT -0.6 -0.4 -0.2

0

Marcus rate (s−1 )

1020

1010 105 100 10−5

10−10

0.2

0.4

0.6

energy difference ∆ǫij (eV)

(a) Rates as a function of energy difference.

10−15 10−1610−1410−1210−10 10−8 10−6 10−4 10−2 100 Miller-Abrahams rate (s−1 )

(b) Comparative rate distribution. The dashed black line indicates the case of matching rates.

Fig. 3.2. Comparison of Miller–Abrahams and Marcus rates. Marcus rates are results of a microscopically simulated DPBIC system of 4000 molecules. Miller–Abrahams rates are obtained by using the microscopic energy values and choosing ω0 such that the mean rate is the same for both models. Parameters are J = 0.0099 eV, λ = 0.067 eV for the Marcus rate and ω0 = 1.37 s−1 for the Miller–Abrahams rate. In (a) additionally a comparison with the low temperature limit of Weiss– Dorsey rates is given. The unknown parameter α is set to the arbitrary values 0.25, 0.5 and 1, ν c is calculated from the relation λ = 2α~νc . All rates are at T = 300 K.

[215] compared the mobility in a percolation calculation (Section 3.7.4) for both rate expressions and found different temperature dependencies of the zero field mobility:

μ0 (T) ∝ exp −C σˆ 2 for Miller–Abrahams rates and μ0 (T) ∝ T γ exp − 12 σˆ 2 − a σˆ with γ = λ − 32 for Marcus hopping. Here σˆ is defined by σˆ = σ/kB T. The density dependence was unaffected by the choice between these two rate expressions. Hoffmann et al. investigated the question for which temperature the tunneling described by Miller–Abrahams rates prevails and for which temperatures the “hightemperature” approximation of Marcus rates is better by comparing to experimental results [216]. They concluded that for charges in time-of-flight experiments the turning point is around 250 K. However, they remarked that this is only an estimate and that the turning point depends on energetic disorder, minimum hopping time, phonon energy and thermal energy. Fishchuk et al. compared the mobilities in an effective medium approximation (Section 3.7.3) for both rate expressions [217]. In particular, they investigated the dependency of the mobility on the charge carrier density. At high densities the mobility decreased abruptly when using Miller–Abrahams rates. The decrease was also seen when using Marcus rate, however, it was significantly weaker. Recently the interest in Weiss–Dorsey rates increased. Asadi et al. [212] found good agreement with experimental field-effect transistor measurements in a temperature

40 | 3 Charge Dynamics at Different Scales regime from 18 K to 300 K when using the low temperature approximation of Weiss– Dorsey rates, Equation (3.5).

3.2 Band Theory vs. Hopping Transport In inorganic, crystalline semiconductors band theory [218, 219] provides an effective tool to describe charge transport. In a crystal electrons move in a periodic lattice and the solution of the single electron Schrödinger equation are Bloch waves, i.e., wave functions of the form ψ n,k (r) = e ik·r u n,k (r) , (3.6) where r is the position and k is the wave vector. The function u is a periodic function and describes the periodicity of the crystal by fulfilling the relation u n,k (r) = u n,k (r + R) for all lattice vectors R. The integer n enumerates the different solutions and is called the band index. For all vectors K of the reciprocal lattice the relation ψ n,k+K (r) = ψ n,k (r)

(3.7)

for the wave function is fulfilled. Furthermore the eigenvalues of the Hamiltonian fulfill the relation ϵ n,k+K = ϵ n,k . (3.8) The energy eigenvalue is a smooth function of the wave vector k and the interval of possible values that ϵ n,k can take is called the nth band. Within such a band electrons are strongly delocalized, leading to efficient transport. Some highly crystalline organic material also show band transport. For most amorphous organic materials, however, hopping transport prevails. Since molecules in amorphous semiconductors are randomly oriented and usually also larger than in inorganic semiconductors, their energy can differ a lot between different molecular sites and no energy band is formed. Instead, the overlap of molecular orbitals is small, electrons are highly localized on one molecule each and electron transport can be described as a sequence of hops between those sites. In the picture of the Marcus or Weiss–Dorsey rate expression, it is the electronic coupling, J ij , that determines which transport mechanism prevails. Large electronic coupling elements lead to band transport while small ones lead to hopping transport [33]. However, it is not always straightforward to decide which transport model is a correct description [220], and especially for organic single crystals, that show high mobilities, this poses a problem for the correct understanding of charge transport [221]. An intermediate description is the multiple trapping and release model discussed briefly in Section 3.7, that assumes a conduction band with additional trap states. This model also motivated the transport energy concept discussed in Section 3.7.5, that aims to find an energy similar to the band energy but in hopping transport. Some materials also show band-like transport at very low temperature and hopping transport at higher

3.3 Stochastic Description of Hopping Transport in Discrete Space |

41

temperatures, which has stimulated the search for a theory that combines both mechanisms and reflects this transition [222, 223]. In this work the focus lies on amorphous materials at not too low temperatures, where transport can be described as a sequence of hops. In the following I will discuss the stochastic background for this description and how to solve the underlying master equation

3.3 Stochastic Description of Hopping Transport in Discrete Space Hopping in a system of sites with positions ri and with transfer rates ω ij can be described in terms of a stochastic process. The memory-less nature of this process – the next hop of a charge does not depend on previous hops but only on the current state – makes it a Markov process. Hopping transport also fulfills the properties of the subclass of Poisson processes and thus shows exponentially distributed waiting times. The time evolution of such a process is described by the master equation.

3.3.1 Markov Processes A stochastic process is a collection of random variables that represent the evolution of random values over time. If the time steps are discrete, it is called a discrete time process, otherwise it is a continuous time process. If such a stochastic process with a continuous time dependent random variable X(t) and arbitrary times ... < t l < t m < t n fulfills the relation

P X(t n ) = x|X(t m ) = y, X(t l ) = z, . . . = P X(t n ) = x

(3.9)

it is called a (continuous time) Markov process, named after the Russian mathematician Andrey Markov. Here P(A) denotes the probability of an event A. Equation (3.9) means that the probability of a certain event to occur is independent of the history of the system but only depends on the current state of the system. Markov chains fulfill the Chapman-Kolmogorov equation

=


P X(t n ) = x|X(t l ) = z Z∞

P X(t n ) = x|X(t m ) = y P X(t m ) = y|X(t l ) = z dy.

−∞

(3.10)

42 | 3 Charge Dynamics at Different Scales 3.3.2 Poisson Processes The basic form of a homogeneous Poisson process is the continuous time counting process that counts the number N(t) of events occurring up to a certain time t. It is defined by its properties 1. N(0) = 0. 2. Independent increments, i.e., the number of events in disjoint time intervals are independent from each other. 3. Stationary increments, i.e., the probability distribution of events within a certain time interval only depends on the length of this interval. 4. No simultaneous events. Such a process by definition is a Markov process. This basic form of a Poisson process, the continuous time counting process, is often just called “the” Poisson process. From these properties one can derive the probability density (see, e.g., p. 147 in [224]) for such a process, given by the exponential distribution f (t) = λe−λt I[0,∞) (t)

(3.11)

where λ > 0 is called the intensity and I[0,∞) is a step function that is 1 for t ∈ [0, ∞) and 0 otherwise. The probability of k events to occur in an interval (t, t + τ] is given by
e−λτ (λτ)k . (3.12) P N(t + τ) − N(t) = k = k! If we consider the system of a single charge carrier hopping between different sites, this can be described by such a process. Then the probability for a single hop (k = 1) to occur in a time span τ is given by

P N(t + τ) − N(t) = 1 = P N(τ) = 1 =

Zτ

f (t)dt = λτe−λτ ,

(3.13)

−∞

i.e., the time between two consecutive carrier hops is exponentially distributed with parameter λ. This distribution also implies an expectation value of 1λ for the average time between two events. For hopping of multiple, independent charge carriers, one can now define the escape rate ω i as X ωi = ω ij . (3.14) j

It describes the average number of carriers per time escaping from site i. The inverse escape rate must be equal to the intensity, λ = ω−1 i , i.e., the time that a carrier will spend on site i is given by an exponential distribution with that parameter tesc ∼ Exp (λ) , λ = ω−1 i . i

(3.15)

3.3 Stochastic Description of Hopping Transport in Discrete Space |

43

Apart from such a homogeneous counting process there is also a more general case of a Poisson process, the inhomogeneous one, which allows λ to be time dependent, λ = λ(t). Such a process is still a Markov process. For charge hopping this situation occurs, for example, if Coulomb interaction is taken into account, such that the rates have to be updated in each time step.

3.3.3 Master Equation The dynamics of a physical system with discrete states is described by the master equation X  d P α (t) = T αβ P α (t) − T βα P β (t) , (3.16) dt β

where P α is the probability for the system to be in state α and T αβ a transition rate from state α to β. P α (t) is interpreted as the probability density of a Markov chain. The integrated version of the master equation is equal to the Chapman-Kolmogorov Equation (3.10). If a Markov chain fulfills the detailed balance principle, T αβ P α (t) = T βα P β (t),

(3.17)

it is called a reversible Markov chain. For physical systems in equilibrium this condition should be fulfilled. It was already formulated in 1867 by Maxwell [225] to describe gas kinetics. When a system with one charge carrier is simulated, a state of the system can be characterized by the (only) occupied site of that system. That implies that Equation (3.16) can equivalently be written in terms of the occupation probabilities of the sites, p i , and the charge transfer rates between sites, ω ij , i.e., the average number of carriers per time that are transferred from site i to site j: X  d p i (t) = ω ij p i (t) − ω ji p j (t) . dt

(3.18)

j

In the context of amorphous organic semiconductors, each site corresponds to a molecule. For convenience one can choose the positions entering the charge transfer rates as the center of masses of these molecules. The rates ω ij and molecule positions have to be calculated first. After that one can determine the unknown occupation probabilities, p i . In order to do so, one has to solve a system of coupled, linear, first order, ordinary differential equations (ODE). An overview of different approaches for solving the master equation can be found, e.g., in [226] and comprises analytic methods for certain special cases as well as direct numerical ODE solvers. In the case of multiple charges a state α of the system is no longer equivalent to an occupied site i but to a list of occupied sites. As a consequence, the master equation for the site occupations has a much more complicated form. Directly tackling it with

44 | 3 Charge Dynamics at Different Scales an ODE solver is only possible after making a mean field approximation and rewriting it in terms of occupation probabilities and electron transfer rates: X

 d p i (t) = ω ij p i (t) 1 − p j (t) − ω ji p j (t) 1 − p i (t) . (3.19) dt j

with the constraint of the total probability being the number of charge carriers: X n= pi . (3.20) i

The differential equations are no longer linear, and thus more complicated algorithms are needed when solving the system of ODEs. Furthermore, all correlations between the occupation probabilities of different sites are neglected (mean field approximation) [227]. Here the advantage of using a Monte Carlo algorithm comes into play, which does not rely on this approximation.

3.3.4 Mobility and Diffusion Constant The mobility is the key property of interest in charge transport simulations. If there is an external field it is defined as the average velocity in field direction divided by this field.

1-Dimensional Case Here the definition reads

D

∂x ∂t

E

hvi = , (3.21) F F where F is the applied electric field. The mathematical definition of the diffusion constant is [228]

2  x (t) − hx (t)i2 . (3.22) D = lim 2t t→∞ For practical purposes the limit is omitted and an average over a sufficiently large time is taken. For pure diffusion, where the external field, F, is zero, the position averaged over many different runs, hx (t)i, is zero. In this case the diffusion constant can be evaluated as

2 ∆x D= , (3.23) 2τ where ∆x is the distance that a particle moved in the time step τ.

μ=

3-Dimensional Case If the material shows no anisotropic effects the one-dimensional approach can also be applied to a three-dimensional system by considering the movement x(t) in the direc-

3.4 Kinetic Monte Carlo | 45

tion of the applied electric field. For the general case with anisotropic effects included the mobility becomes a second order tensor  

 μ11 μ12 μ13 ∂x i t × Fj ∇r · F   =  μ21 μ22 μ23  , (3.24) μ= with μ ij = ∂t 2 . 2 τF τF μ31 μ32 μ33 F is a (constant) external electric field and Ft denotes the transposed vector. The diffusion tensor is given by [229]  



 D11 D12 D13 t ∆r · ∆r  ∆r i ∆r j  D= with D ij = , (3.25)  D21 D22 D23  , 2τ 2τ D31 D32 D33

where ∆r i (t) = r i (t + τ) − r i (τ) and τ is the time step. The Einstein relation, or Einstein–Smoluchowski relation, describes the relationship between mobility and diffusion tensor/constant. For the case of a single charge carrier in thermal equilibrium in the limit of zero applied external field the occupation probabilities are given by Boltzmann statistics and the Einstein relation reads [230]: qD = μkB T.

(3.26)

3.4 Kinetic Monte Carlo A possible strategy to solve the master equation is the Kinetic Monte Carlo (KMC) method. It produces a statistically correct trajectory and provides an estimator of occupation probabilities which solve the master equation. The method with some variations has been referred to under a variety of different names during its development – Gillespie algorithm, residence-time algorithm, n-fold way and Bortz-Kalos-Lebowitz algorithm. The development started in 1942 with Doob’s work on Markov chains [231, 232, 233, 234]. The time update of the KMC presented here (Equation (3.30)) was first used by Young and Elcock [235], the same algorithm was independently developed and applied to solve the Ising model by Bortz, Kalos and Lebowitz [236]. Gillespie derived the method using a physics approach and applied it to surface reactions [233, 234]. Fichthorn advanced his algorithm [237] and Jansen developed the so called variable step size method (VSSM), making it computationally more efficient [238]. For an overview of the method see the review by Jansen [239].

3.4.1 Multiple Charge Carriers and Forbidden Events The VSSM algorithm can treat reactions at different levels. This section describes a two-level VSSM algorithm for the treatment of charge transport with multiple charge

46 | 3 Charge Dynamics at Different Scales carriers. Charge movement is assumed to occur between sites i = 1 . . . N located at positions ri . In the context of charge hopping between localized states the position is the center of mass of a molecule. Single atoms are not taken into account in this description. The rate for a hop from a site i to a site j is given by ω ij . Possible rate expressions have been discussed in Section 3.1. In the system of N sites there are now n < N charge carriers (electrons or holes). In the described version of the algorithm unipolar transport is assumed, i.e., charge transport is dominated either by electrons or by holes and the other carriers, as well as recombination of carriers, are neglected. This assumption is valid, e.g., in the holeor electron conducting layers of an OLED, but not in situations where recombination plays an important role. Relevant are now all sites i at which a carrier is located. Such a carrier is connected to a certain number, m, of neighbors; this number is called the coordination number and it can vary between different sites and depends on the cutoff chosen for connections in a neighbor list. For each site i that is occupied by a charge carrier the escape rate is defined as the sum of all the outgoing rates to neighbors: ωi =

m X

ω ij .

(3.27)

j=1

The probability that a carrier from a site i will hop to some other site is given by the P escape rate divided by the sum of all escape rates of occupied sites, ω i / k occupied ω k . Selecting a site corresponding to a carrier which will hop is the first level of the VSSM algorithm and is performed randomly according to this probability. On the second level the destination of the hopping reaction is chosen. All the rates ω ij , j = 1 . . . m denote a possible hop, and each rate is proportional to the probability of that hop. A destination is chosen randomly with probability ω ij /ω i . A complexity in the algorithm arises from Coulomb repulsion between charge carriers of the same type. A rigorous approach would be to take Coulomb interaction directly into account. This, however, means updating the transfer rates, ω ij , in every Monte Carlo step and is therefore computationally very demanding. How it can be done will be explained in subsection 3.4.2. An alternative approach is to take only the largest contribution into account: two electrons occupying the same site would experience a very strong Coulomb repulsion, which makes double occupation virtually impossible. Hence, double occupancy is forbidden in the Monte Carlo algorithm. This is referred to as an exclusion principle. Formally, it leads to a situation equivalent to the Pauli principle, where each state can only be occupied once, despite the fact the the Pauli principle is a theory for spin particles and we do not take the electron spin into account here. Still, the exclusion principle results in Fermi–Dirac occupation statistics of the sites, as it will be shown in simulations later, cf. Chapter 7. In the algorithm the exclusion principle leads to the situation that some events are forbidden: if a carrier is located at a site i and the neighboring site j is already occupied this hop is forbidden and the rate ω ij should in principle be set to zero when calculating the escape rate, Equation (3.27), before the selection of an event. This is, however,

3.4 Kinetic Monte Carlo |

47

Fig. 3.3. Overview of the two-level Kinetic Monte Carlo algorithm for multiple charge carriers following an exclusion principle. The time step here is not fixed but drawn from an exponential distribution with parameter 1/ω i , as described in the bottom part of the figure.

48 | 3 Charge Dynamics at Different Scales computationally inefficient, since it requires checking the occupational status of all neighboring sites before every hop. To improve the efficiency, it is therefore useful to take another approach [240]: Once a hopping destination is chosen, the occupation of that site is checked. If it is unoccupied the charge hops, otherwise a different hopping destination is chosen and the occupied site is added to a list of forbidden destinations. In case all surrounding sites are occupied, the escape node is added to a list of forbidden escape nodes and a different escape node is chosen. The time is updated on the second level (selection of the escape node), regardless of whether the charge escaped or not. This approach yields correct statistics, which can be seen as follows: Assume the charge is located on site i and one of its neighboring sites j is already occupied. Now P if the rate ω ij is not removed from the sum in the escape rate ω i = k ω ik , then the probability per time for the charge to try hopping to any unoccupied neighboring site is given by
P(1) (t) = ω i − ω ij e−ω i t . (3.28)

The probability that it first attempts a hop to site j and then to another, unoccupied neighboring site is given by P(2) (t)

=

Zt 0

=


′ ′ dt′ ω ij e−ω i t · ω i − ω ij e−(ω i −ω ij )(t−t )

ω i − ω ij

i
h −(ω i −ω ij )t e − e ωi t .

Now, the probability of the charge to either hop directly to an unoccupied site or attempt to hop to an occupied site and afterward hop to an unoccupied site is given by
P(1) (t) + P(2) (t) = ω i − ω ij e−(ω i −ω ij )t .

(3.29)

That is the same expression that one would get when checking the occupation of all neighboring sites and removing the rate ω ij beforehand from the list of events. Analogously, the same is true for multiple occupied neighboring sites and not just one, which shows that both approaches lead to the same statistics. In the description of the Poisson process, Section 3.3.2, I showed that the escape ∼ Exp (λ) , with parameter λ = ω−1 times are exponentially distributed, tesc i . The last i step of the VSSM algorithm consists of a time update following this distribution. In practice, this is achieved by choosing a random number u from a uniform distribution in the interval (0, 1] and setting the time step to
1 ∆t = − P ln (u) , u ∼ U (0, 1] . i ωi

(3.30)

A summary of the algorithm in the presented form is given in Figure 3.3. To conclude the section let me discuss how to select events. Figure 3.4 shows the process schematically: in the first level all escape rates for occupied sites are divided

3.4 Kinetic Monte Carlo |

49

1 choose site of the carrier that hops here: carrier on site 2 hops

2 choose hopping destination here: carrier hops to site 3

Fig. 3.4. Choosing events with probability proportional to their rates in the two-level KMC algorithm. In the example there are four possible sites from which a carrier can hop. The corresponding rates ω1 , ω2 , ω3 , ω4 are assigned sub intervals of the interval (0, 1] such that each sub-interval has a length proportional to the rate. A random number X1 is chosen from a uniform distribution in the interval (0, 1], depicted by the arrow. It determines the chosen rate, in this example ω2 . In the second step there are three possible hopping destinations with rates ω21 , ω22 , ω23 . The procedure is repeated and the third option is chosen so that, overall, the hopping event 2 → 3 with rate ω23 is chosen in this example.

by the sum of all these rates, giving the probability for a certain escape events. Now the interval (0, 1] is divided into sub-intervals with the length corresponding to the probP abilities ω i / j occupied ω j . A random number is drawn from a uniform distribution in
this interval, X1 ∼ U (0, 1] , depicted by the arrow in the figure. Now, the probability that this arrow points on a certain escape rate is the probability of this escape event to occur. The procedure is repeated for the selection of the hopping destination.

3.4.2 Explicit Molecular Coulomb Interaction Going beyond the exclusion principle described in the previous section, it is also possible to take Coulomb interaction into account explicitly. There are two different aspects which can be treated at different levels of accuracy. One is the decision between using a single charge on a molecule or using partial charges on atoms, the other one is the treatment of the long-range part. Both aspects will be discussed in this section. First let us look at the question how Coulomb interaction affects charge transfer. Figure 3.5 shows three molecular sites i, j, k, where i and k both are occupied by a charge carrier, j is unoccupied, and the question is how the charge transfer rate, ω ij , for the i → j transition is affected by the interaction with the charge located on site k.

50 | 3 Charge Dynamics at Different Scales

j

i

j

i

k 1 before hopping

k 2 after hopping

Fig. 3.5. Hopping of a charge carrier from site i to site j under the influence of Coulomb interaction with a carrier located on site k. Both initial and final state are affected by the Coulomb interaction. The energy of site i is modified by the Coulomb contribution, E i → E i + ECoul ik , as well as the energy 0 . The initial energy difference ∆E = E − E therefore changes to ∆E ij = of site j, E j → E j + ECoul i j ij jk − ECoul ∆E0ij + ECoul jk . ik

In the initial configuration the Coulomb interaction energy reads ECoul = ik

1 q2 , 4πε0 |ri − rk |

(3.31)

where q = ±e and ε0 is the vacuum permittivity. After hopping, a Coulomb interaction with energy ECoul acts between the sites occupied in the final configuration. In jk consequence, the energy difference between sites i and j, ∆E0ij , changes to² ∆E ij = ∆E0ij + ECoul − ECoul ik jk . For several occupied sites k the equation can be generalized to  X  Coul ∆E ij = ∆E0ij + E ik − ECoul . jk

(3.32)

(3.33)

= k6 i k occupied

So far the assumption was that a site is either charged or uncharged, with the charge located at the center of mass of the molecule. In lattices it is the only available model. In a real morphology, however, where each molecule consists of different atoms, a more accurate description is possible, namely using partial charges located on the atoms of the molecule, as depicted in Figure 3.6. The idea is that a molecule k has a certain charge distribution. This distribution of is different between the charged and the neutral state. The molecule consists of different atoms a k . For an efficient computational treatment the charge distribution can be represented by fractional charge numbers assigned to the atoms, q a k , that are chosen such that the potential created

2 Note that sometimes in the literature the definition ∆E0ij = E j − E i is used, which changes the sign
2 within the brackets to ∆E ij + λ ij and ∆E ij to∆E ij = ∆E0ij − ECoul + ECoul ik jk .

3.4 Kinetic Monte Carlo | 51

i i k

k 1 site Coulomb interaction

2 partial charges Coulomb interaction

Fig. 3.6. (1) In a simple picture Coulomb interaction is the interaction of two charge carriers on sites i and k. (2) Each molecule k is composed of different atoms a k . If the charge distribution of the molecule is represented by partial charges on these atoms, the molecular Coulomb interaction can be modeled by the resulting Coulomb interaction of all different atoms, as given in Equation (3.34).

partial charges Coulomb energy (eV)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.4

0.6

0.8

1

1.2

1.4

1.6

site Coulomb energy (eV)

Fig. 3.7. Comparison of site Coulomb interaction, Equation (3.31), with the partial charges Coulomb interaction, Equation (3.34), as obtained in a microscopically simulated DPBIC morphology of 4000 molecules for an exemplary site and all possible interacting sites. Image charges are taken into account by using Ewald summation.

by the charge distribution is reproduced as accurately as possible. The Coulomb interaction in Equation (3.31) can then be refined to the interactions of all involved partial charges of molecule i with molecule k: ECoul = ik

1 X X qna i qca k . 4πε0 |ra i − ra k | ai

(3.34)

ak

Equations (3.33) and (3.34) give a recipe on how to modify the energy difference in the transfer rates to take Coulomb interaction into account. A comparison of the obtained values for ECoul ik is shown in Figure (3.7) for microscopic calculations in an amorphous morphology. Note that the Coulomb contributions, ECoul ik , can be pre-computed for each pair. In a Monte Carlo step it is therefore not necessary to compute these expressions but

52 | 3 Charge Dynamics at Different Scales they can be taken from a look-up table. What is, however, necessary is an update of the involved rates in each step. For a higher accuracy of reproducing the electric field formed by the charge distribution one can use dipoles or multipoles of higher order instead of partial charges.

Fig. 3.8. Coulomb interaction with image charges. A charge carrier i interacts with a charge carrier k, as well as with the images of i itself and of k.

Apart from the question to what detail to compute the Coulomb interaction there is another question, namely up to what maximal distance to compute it. The simplest approach is the exclusion principle, which would correspond to an interaction range of 0; only if charges attempt to hop on the same molecule they repel each other, making this hop impossible. The next possibility is to compute all Coulomb interactions in a sphere of radius rcut around the molecule of interest. The slow 1/r decay of Coulomb interaction, however, leads to a significant contribution also from charges that are further away. A special complication is created by periodic boundary conditions: a charge at site i interacts not only with a charge at site k, but also with all periodic images of this charge, as shown in Figure 3.8. Additionally, it interacts with the periodic images of itself. Coulomb interaction can be computed approximately using a cutoff or exactly by using Ewald summation [241, 242]. In the latter case a short-range part is solved in real space and a long-range part in the inverted space, thus allowing an exact evaluation of the infinite Coulomb sum. For amorphous Alq3 it was found that the cutoff-based calculation incorrectly shifts the density of states by about 1 eV [243]. An overview of all the possibilities to model Coulomb interaction discussed in this and the previous section is given in Figure (3.9). From the figure the following possible treatments of Coulomb interaction can be seen:

3.4 Kinetic Monte Carlo | 53

A double-occupancy exclusion principle. Short-range interaction of site Coulomb interaction, i.e., using Equations (3.31) and (3.33) and evaluating the Coulomb interaction within a certain radius, rcut . 3. Long range site Coulomb interaction, i.e., the same equations as in (2) but taking periodic images into account. 4. Short-range partial charges Coulomb interaction, i.e., using Equations (3.34) and (3.33) and evaluating the Coulomb interaction within a certain radius, rcut . 5. Long-range partial charges Coulomb interaction, i.e., the same Equations as in (4) but taking periodic images into account. 6. Long-range interaction of partial charges as in (5), with additional inclusion of (short-range) polarization.

Interaction Detailedness

accuracy

1. 2.

Interaction Range infinite long range part evaluated with Ewald summation

partial charges Coulomb interaction i

additional long range part with imited number of images

k site Coulomb interaction short range cut-off

i k

exclusion principle

Fig. 3.9. Overview of different levels of accuracy with increasing complexity for treating Coulomb interaction. The simplest form is an exclusion principle. Above the dashed lines different combinations are possible, for treating the detailedness and the range of the interaction. Not shown is the single carrier simulation, which completely neglects interaction.

54 | 3 Charge Dynamics at Different Scales The different levels of accuracy come along with increasing computational demands. For option (1) there exists an implementation on a lattice [114]. I implemented an offlattice version for realistic morphologies [244], cf. Chapter 8. Options (2) and (3) have been implemented on a lattice [245, 246]. For option (2) it was shown that Coulomb interaction starts to play a role at relative densities of 0.01 carriers per site and above [245]. In collaboration with Jeroen van der Holst option (5) was realized [247], see Chapter 7. Option (6) is in practice impossible, since the polarization contribution cannot be pre-computed, making a complete electrostatic calculation in every Monte-Carlo step necessary. Such a calculation for 4000 molecules, for example, takes a few days on several hundred processors, while 107 –108 Monte Carlo steps are necessary.

3.5 Differential Equation Solver The kinetic Monte Carlo approach can easily be extended (at least conceptually) to include effects such as Coulomb interaction or charge recombination. The downside of the technique is the high computational demand. An alternative method is to use numerical differential equation solvers. For one charge carrier the master equation for charge transport, Equation (3.18), is an ordinary, linear differential equation, which is relatively simple to solve. For many charges only the mean-field version, Equation (3.19), can be treated, meaning that all correlations between the occupation probabilities of different sites are neglected [227]. Additionally, the equation is no longer linear in the occupation probability, p i , making the numerical solution more complicated. Possible methods are Implicit Runge-Kutta or Newton’s method. Details of an implementation of the master equation solver have been given by Klein [248]. The approach of a numerical differential equation solver has been applied, e.g., for the parametrization of the EGDM (see Section 3.6) [114]. In a benchmark of the single charge version against KMC the differential equation solver was four orders of magnitude faster [249]. The solution of the master equation are occupation probabilities, p i . The mobility cannot be calculated directly from the trajectory as in KMC simulations, but is evaluated from the relation μ=

P

i,j

ω ij p i (1 − p j )∆r ij P , F i pi

(3.35)

where ∆r ij is the positional difference of sites i and j in the direction of the electric field of magnitude F.

3.6 Gaussian Disorder Models

| 55

3.6 Gaussian Disorder Models The Gaussian disorder model (GDM) was discussed in detail in Bässler’s pioneering work [22] and has become today’s most frequently used model of organic semiconductors. In this model molecules (sites) are arranged on a regular, simple cubic (sc) lattice. The site energies are assumed to follow a Gaussian density of states  2 1 (ϵ − m) exp − fDOS (ϵ) = √ (3.36) 2σ2 2πσ2 with mean m and width σ. In the context of the GDM the width of the Gaussian is called the energetic disorder, which is a key quantity in the model. The reasoning behind this energy dependence is the assumption that molecules in an amorphous solid are randomly oriented and that the interaction of their dipole moments leads to Gaussian distributed site energies. While this assumption could not be tested at the time, today’s microscopic simulations show it is indeed often possible to fit a Gaussian distribution to the site energies, see for example the results in Chapter 7. There are, however, also cases where the distribution is not perfectly described by a Gaussian [250]. For charge transfer the GDM assumes Miller–Abrahams rates, Equation (3.1), and nearest neighbors are connected with periodic boundary conditions at the sides of the box. Other parameters of the model are the lattice constant, a, and the mobility in the limit of zero field, μ0 . Bässler performed Kinetic Monte Carlo simulations of charge carrier hopping for this model and derived an empirical formula for the mobility as a function of temperature and electric field. The site energies in the original model are assumed to be uncorrelated.

Fig. 3.10. Schematic picture of the Gaussian Disorder Model (GDM). Molecules are represented as lattice points and have Gaussian distributed energies. The parameters of the model are the lattice constant, a, and the width of the Gaussian, σ, called the energetic disorder. A third parameter that is not shown is the mobility at zero field and zero density, μ0 .

56 | 3 Charge Dynamics at Different Scales The first modification of the model was to introduce spatial correlations. Since site energies are not independent quantities but a result of interaction with surrounding molecules, they show correlation [251]. It is quantified by the spatial autocorrelation function 


E ( ϵ (r i ) − m ) ϵ r j − m , (3.37) κ ( r ) = κ ri − rj = σ2 where r = ri − rj is the distance between two molecules, E [·] denotes the expectation

value, m is the mean of the energy distribution and σ its width, the energetic disorder. The autocorrelation function takes values between 0 (no correlation) and 1 (complete correlation). Novikov et al. [140] modeled the energetic correlation by representing molecules by dipoles, pj , which have a fixed absolute value but a random orientation and which interact with a test charge. The site energy of a molecule i is then given by the equation
X qpj rj − ri (3.38) Ei = − 3 , ε r j − r i =j6 i

where q = ±e is the charge, ri the molecule (center-of-mass) position and ε the material’s relative permittivity. The sum runs over all molecules in the lattice (no restriction to nearest neighbors or a cutoff sphere) and is evaluated using the Ewald summation technique [252, 242]. The energetic disorder resulting from this procedure has a width of σ = 2.35 q p/ε a2 [253, 140]. For a simple cubic lattice, the energy correlation function can be approximated by [254] κ(r) = 0.74 a r−1 . Thus, in this model the only free parameter in the autocorrelation function is the lattice constant a. In Chapters 6, 7 and 8 I investigate whether or not this can be confirmed by microscopic simulations. A success of the correlated disorder model (CDM) was the correct  √  description of the field dependence of the mobility at small fields, μ ∝ exp α F , that is known as the Poole–Frenkel dependence, cf. Section 3.9.6. Another extension of the model was to include finite charge carrier density. This is important because the charge carrier mobility strongly depends on the density. Pasveer et al. [114] used a differential equation solver (cf. Section 3.5) for the meanfield version of the master equation for multiple charge carriers, Equation (3.19). Their simulation results allowed for an empirical parametrization of the mobility as a function of temperature, T, spatial charge carrier density (carriers per volume), ρ, and electric field, F. This model is often referred to as the extended Gaussian disorder model (EGDM): μ(T, ρ, F) = μ0 (T)g(T, ρ)f (T, F). (3.39) The functions μ0 , g and f are defined by h i μ0 (T) = 1.8 × 10−9 μ0 exp −C σˆ 2 , C = 0.42, g(T, ρ) = exp

   δ  1 2 σˆ − σˆ 2ρa3 , 2

3.6 Gaussian Disorder Models

"



f (T, F) = exp 0.44 σˆ

3/2

| 57

#  q 2 −1 , 1 + 0.8Fred − 2.2


ln σˆ 2 − σˆ − ln(ln 4) δ=2 . σˆ 2 μ0 is a material specific property, Fred = eaF/σ, and σˆ = σ/kB T. Bouhassoune et al. [141] combined the correlated disorder of the dipolar model with the effects of multiple charge carriers. The combined model is usually referred to as the extended correlated Gaussian disorder model (ECDM) and postulates the following mobility dependence μ(T, ρ, F) = μ0 (T)g(T, ρ)f (T, F, ρ),

(3.40)

where h i μ0 (T) = 1.0 × 10−9 μ0 exp −C σˆ 2 , C = 0.29,

h 

i 2 3 δ  , exp 0.25σˆ + 0.7σˆ 2ρa     ρa3 < 0.025 h i g(T, ρ) =
 exp 0.25σˆ 2 + 0.7σˆ (2 × 0.025)δ      ρa3 ≥ 0.025, "

   r(σ) ˆ f (T, Fred , ρ) = exp h(Fred ) 1.05 − 1.2 ρa3 

× σˆ

3/2

−2

 p

1 + 2Fred − 1

# 

,

eaF * ˆ = 0.7σˆ −0.7 , Fred = r(σ) , Fred = 0.16, σ  F*  Fred < 2red  43 FFred * , red 2   h(Fred ) =  * Fred *  −1 ,  1 − 43 FFred * 2 ≤ F red ≤ F red , red


ln 0.5σˆ 2 + 1.4σˆ − ln(ln 4) δ = 2.3 . σˆ 2 Gaussian Disorder models are widely used for the analysis of experimental data. Due to their popularity there has also been a lot of effort in finding analytic solutions rather than empirical equations based on numerical simulations. These analytic approaches are discussed in Section 3.7. To conclude, I remark that the Gaussian disorder models assume Miller–Abrahams rates for charge transfer. Furthermore, their parametrization from simulations was

58 | 3 Charge Dynamics at Different Scales done in a regime up to an energetic disorder of about 0.15 eV (at room temperature, i.e., 300 K). It is not immediately clear (1) whether or not these models are applicable to disordered amorphous semiconductors with a higher energetic disorder, σ > 0.15 eV, and (2) whether or not the model parameters, namely the lattice constant, a, the energetic disorder, σ, and the mobility in the limit of zero field and zero charge density, μ0 , have a physical meaning or are merely fitting parameters. This question is addressed in Chapter 8 by a comparison of microscopic simulations to the EGDM and ECDM, parametrized from the microscopically obtained mobilities.

3.7 Analytic Theories of Hopping Transport Hopping transport on a lattice with a Gaussian density of states has not only been a subject of simulations but also of analytic calculations. For the simpler case of a onedimensional chain it can be solved without any approximations [255]. Provided that rates are the same for all events it can also be solved without approximations in a cubic lattice, and is known as the continuous time random walk (CTRW). Strategies of an analytic solution of the full GDM are percolation theory and the effective medium approximation. All approaches will be discussed in detail in the following. Another model, which I mention here only briefly, is the multiple trapping and release model [256, 257, 258, 259]. It does not assume localized quantum states and hopping transport, but is based on a conduction band, i.e., delocalized electronic states. This conduction band is complemented by trap states with low-lying energies. From this model, drift–diffusion equations with an additional trapping term are derived. The density of traps in the model is taken to be either uniform or Gaussian distributed. The multiple trapping and release model was the motivation for the concept of transport energy, an “effective transport band” in case of hopping transport, also discussed in this section. Some of the approaches do not assume a Gaussian DOS but an exponential DOS. From microscopic calculations, cf. Chapter 7, it is clear that a Gaussian DOS is a suitable model. Also mathematically the exponential DOS leads to problems, namely to infinite diving in energy in the limiting case of zero charge density [260]. The reason why it is still often used is that it allows simplifications in many analytic expressions. In a limited range it can be used as an approximation of the Gaussian DOS, but only with precaution.

3.7.1 One-Dimensional Hopping Hopping transport in a periodic, one-dimensional chain can be treated analytically. Even though the one-dimensional case does not correspond to a realistic physical system, its study is instructive due to the fact that the dependencies of the mobility on

3.7 Analytic Theories of Hopping Transport | 59

temperature and on the system size are often similar in one and in three dimensions. This is discussed in detail in Chapter 5. Derrida studied the problem of hopping in a chain from a general point of view, i.e., for the one-dimensional master equation with general transition probabilities [228]. He derived the time evolution for the mean displacement, hx (t)i, the mean square



 
displacement, x2 (t) , as well as the diffusion constant, D = 21 limt→∞ dtd x2 (t) − hx (t)i2 . Cordes et al. applied the expression to charge hopping in one-dimensional, disordered organic semiconductors [255]. They started from the general form derived by Derrida, assumed a Gaussian density of states for the hopping sites and Miller– Abrahams rates, Equation (3.1), for hopping reactions. From these assumptions they derived an analytic expression for the one dimensional GDM transport. Assuming an uncorrelated energetic landscape and for sufficiently low fields (i.e., in the limit F → 0) their expression reads 2 !      eaF σ ω0 a exp −1 . (3.41) exp − μ (F, T ) = 2F kB T kB T I refer to the original publication [255] for the lengthy general expression and for the case of including energetic correlation. Around the same time Seki and Tachiya also presented an analytic solution of the one-dimensional hopping problem, however, using Marcus rates, Equation (3.2), for charge transfer. Under the same assumptions, i.e., for an uncorrelated Gaussian density of states, low electric fields (F → 0) and with the additional assumption of a large reorganization energy, λ, the mobility has the form  2 ! 2πJ ea2 1 λ 3 σ exp − − μ= p . (3.42) 4 kB T 4 kB T ~ 4πλkB T kB T The resulting temperature dependence is   3 μ ∝ T − 2 exp −AT −1 − BT −2

(3.43)

for suitable constants A, B > 0. This relation has been used successfully for an interpolation from high temperature transport to obtain non-dispersive mobilities at room temperature [261] and will further be treated in Chapter 5. Seki and Tachiya also solved the problem in a correlated energetic landscape with Marcus rates, for arbitrary values of the energetic disorder and for larger electric fields. Note that all one-dimensional models have the intrinsic drawback of being limited to a single charge. They do therefore not allow insight into effects of charge carrier density.

3.7.2 Continuous Time Random Walk (CTRW) The continuous time random walk (CTRW) was first described by Montroll and Weiss [262]. They considered nearest-neighbor hopping in a lattice (simple cubic, face cen-

60 | 3 Charge Dynamics at Different Scales tered cubic) with periodic boundary conditions in which they aim to calculate the mean first passage time by the method of Green functions. In the simplest form there is no external field and all hopping rates are equal. Montroll and co-workers extended the model, amongst others, by introducing an external bias and traps [263, 264]. Scher and Montroll also considered the off-lattice case of randomly distributed step widths [265, 266]. The CTRW can be seen as a limiting case of no energetic disorder and no external field, compared to the more general GDM.

3.7.3 Effective Medium Approximation (EMA) Green Function Approach Movaghar and co-workers developed a Green function based formalism [267] in order to find an analytic solution of hopping transport. Their starting point is the master equation for occupation probabilities in the mean field approximation form, Equation (3.19). The equation is further approximated by replacing the probabilities in brackets, (1 − p j (t)), by their mean value, namely by a Fermi distribution: (1 − fFD (ϵ j )). By intro˜ ij = ω ij (1 − fFD (ϵ i )), this allows to rewrite the ducing effective hopping probabilities, ω mean field version of the master equation into a standard master equation for hopping of a single particle:  dp i (t) X  ˜ ij p i (t) − ω ˜ ji p j (t) . = ω (3.44) dt j

To solve this equation they used a Green function formalism and applied the effective medium approximation (EMA) to it. The EMA is based on the initial ideas by Bruggemann [268] and was adjusted to the problem of conductivity by Kirkpatrick [269]. Its idea is to model the percolation problem as a random resistor network and find an “effective medium”, i.e., an effective conductivity or an effective rate, such that the the net field inside the network is equal to the external field. Movaghar and co-workers initially assumed a linearly increasing or an exponential density of states [270, 271]. The analytic expressions were simplified for some special cases, such as a linear chain or a Bethe lattice (also known as a Caley tree) [272]. Later they also examined the case of a Gaussian density of states with Miller– Abrahams hopping rates [273, 274]. Within their theoretical framework they derived a 1/T 2 proportionality of the mobility at high temperatures, a 1/T proportionality at low temperatures [273], and the equilibrium energy for a single carrier (not for multiple carriers), given by ϵ∞ = −σ2 /kB T. The approach is seldom used for the analysis of experimental data, probably due to the lack of a closed expression for the mobility that can easily be evaluated, as it is the case in the GDM parametrizations, Section 3.6.

3.7 Analytic Theories of Hopping Transport |

61

Rate Averaging The above mentioned effective medium approximation by Kirkpatrick [269] is also the basis of the work of Fishchuk and co-workers, who aimed to derive an analytic theory for the GDM, i.e., for charge hopping in a Gaussian density of states. They reformulated Kirkpatrick’s conductivity condition in terms of a condition for the charge transfer rate in an effective medium:   ω12 − ωe = 0, (3.45) ω12 +ω21 + 2ωe 2

where the angular brackets denote the configuration averaging over the distribution of site energies and distance between two sites with hopping rates ω12 and ω21 . The rate ωe is called the effective hopping rate which is searched for. From this rate the effective steady state mobility can be calculated. The approach was first used for a single carrier only [275, 276, 277, 278, 279] and then extended to the case of multiple carriers with either Miller–Abrahams or (linearized) Marcus hopping rates [217, 280]. The resulting mobility in case of Marcus rates reads  r   r 2 (3.46) μ = μ0 k0 T exp −2 T exp (−xx α ) a b exp(− 12 t2 +xt)  R xT  dt 1+exp(x(t−x 1 −∞ F )) × exp − xxT R , exp(− 21 t2 + 12 xt) xT 2 dt 1+exp(x(t−xF )) −∞

where constant transfer integrals, J ij = J, and reorganization energies, λ ij = λ, are as
√ sumed, μ0 = ea2 2πJ 2 / σ~ 4πλkB T , x α = λ/σ, xF = ϵF /σ , xT = ϵT /σ. Here ϵT denotes the effective transport energy, further discussed in Section 3.7.5, that has to be calculated from an integral equation. Similarly, the hopping distance at and below this transport energy, rT = r(ϵT ), has to be calculated from an integral equation. The interested reader is referred to the original publication (Equations (44) and (45) in [217]).

3.7.4 Percolation Theory Percolation theory was introduced in the field of Mathematics by Broadbent and Hammersley [281] in 1957 for the description of liquid flow through a random maze. For a review of the (earlier) applications to disordered semiconductors see also the article by Shklovskii et al. [282]. It started with a description of transport in amorphous inorganic semiconductors by Ambegaokar et al. [283] in 1971. In the model the solid is represented as a resistor network, i.e., a voltage is applied to the lattice and the conductivity is calculated by applying Kirchhoff’s laws. The overall conductivity in this approach is determined by the conductivity, Gcrit , of a single critical bond. It is defined  as the bond with the largest conductivity value such that the subset G ij |G ij > Gcrit still contains a connected path from one side of the system to the other. In other words, it is the lowest conductivity in the dominant path of conduction. The overall conductivity of the system is assumed to be proportional to the critical conductivity, Gcrit . .

62 | 3 Charge Dynamics at Different Scales

Fig. 3.11. Illustration of percolation theory for charge transport on a two-dimensional square lattice with Marcus rates. The line thickness corresponds to the current. At high temperatures (a) the current distribution is rather homogeneous while at low temperatures (c) only a single percolation path remains. The arrow in (c) points to the lowest conductance, Gcrit , of the path at the single critical bond. Reprinted with permission from Cottaar et al., Phys. Rev. Lett. 107, 136601 [215], Copyright 2011 by the American Physical Society.

Ambegaokar et al. use Miller–Abrahams rates for hopping and a uniform distribution of site energies between zero and a maximum value. Vissenberg and Matters [130] and Baranovskii [284] applied the model to a system with an exponential density of states, connected by Miller–Abrahams rates. Cottaar et al. [215] published results comparing both Marcus and Miller–Abrahams rates in a Gaussian density of states. They extended the method to the concept of “fat percolation” by Dyre and Schrøder [285], thus achieving quantitative agreement with numerical simulations. The idea here is that it is not a single critical bond that determines the hole systems but that there exist other possible bonds which are still important enough to influence the overall conductivity. Consequently, in addition to a single conductivity, Gcrit , a distribution of critical conductivities, f (Gcrit ), is used, where both Gcrit and f (Gcrit ) are derived from the rate expressions. An illustration is given in Figure 3.11. Here transport was simulated in a two-dimensional lattice with an applied voltage and Marcus hopping rates. (a) shows the highest temperature, in (b) it is decreased up to the lowest temperature in (c). In this low temperature case the standard percolation approach works well, because transport is limited by a single path and the critical bond inside. With increasing temperature, however, “fat percolation” becomes increasingly superior for a quantitative description. With their approach Cottaar et al. obtained an expression for the mobility in the Gaussian disorder model given by  2 ! 1 EF (T, ρ) 1 σ + , (3.47) μ (T, ρ) = μ0 (T ) exp ρ kB T 2 kB T

3.7 Analytic Theories of Hopping Transport |

63

with EF denoting the Fermi energy. μ0 (T ) is given by μ0 (T ) = B

eω0 2

Nt3 σ



σ kB T

1−γ

1 exp − 2



σ kB T

2

E − crit kB T

!

.

(3.48)

Here Nt is the site density. The critical current is   e2 ω0 EF (T, ρ) − Ecrit exp Gcrit = kB T kB T and f (Gcrit ) by

B kB T . (3.49) A σ If Miller–Abrahams rates are used the rate prefactor ω0 is the one entering the rate expression, Equation (3.1), while in case of Marcus rates it is given by ω0 = p 2πJ/~ 4πλ ij kB T. They compared these expressions to results from lattice calculations, applying a differential equation solver on the master equation (cf. Section 3.5), and thus obtaining numerical values for the free parameters. For Miller–Abrahams rates in the simple cubic lattice, i.e., for the standard Gaussian disorder model, they obtained A = 2.0, B = 0.47, γ = 0.97 and Ecrit = −0.491σ. In case of Marcus rates they obtained A = 1.8 and γ = 0.85. The parameters B and Ecrit show a dependence on the energetic disorder, λ, that enters the Marcus rate expression. For λ = σ they obtained B = 0.51, Ecrit = 0.620σ, in the limit of λ → ∞ the values are B = 0.66, Ecrit = 0.766σ. Interesting results of a comparison of the percolation approach using the two different rate expressions are that the dependence of the charge mobility on the charge density is the same for both rates. The temperature dependence, however, differs. Contrarily, Fishchuk et al. also found differences in the dependence on the charge carrier densit,y depending on the reorganization energy (they call it the polaron activation energy) using an effective medium approach [286]. In summary, the percolation approach is an alternative to the EGDM [114], which is currently the most used formalism for a parametrization of the mobility, μ (ρ, F, T ). It does not allow a purely analytic quantitative prediction, since parameters are obtained from lattice simulations. Gcrit f (Gcrit ) =

3.7.5 Transport Energy The transport energy is a concept originally introduced by Monroe [287], that was subsequently used by many researchers trying to describe hopping transport intuitively. Its derivation is based on the Miller–Abrahams rate expression, Equation (3.1). For carriers in high energy states it is favorable to hop to a state down in energy. If, however, a carrier is already relatively deep down in the density of states, it becomes harder

64 | 3 Charge Dynamics at Different Scales to find lower lying energy states and hopping upwards in energy due to thermal activation becomes more probable. Monroe defined the transport energy as the energy at which thermal activation starts to predominate over hopping downwards in energy. In other words, hopping from energies ϵ < ϵt occurs mainly upwards to energies close to ϵt , hopping from energies ϵ > ϵt occurs mainly downwards in energy [260]. The motivation for defining such an energy stems from the picture of band transport in the multiple trapping and release model [256, 257, 258, 259] mentioned in the introduction, and the idea to find a similar energy fitting this picture. Initially the transport energy was derived using an exponential density of states [287, 288] and the limit of zero charge density [287, 288]. It was later on modified to describe the Gaussian density of states in organic semiconductors [289] and extended to finite charge densities, using the Fermi–Dirac distribution instead of a Boltzmann distribution [290]. Some derivations use percolation theory [291, 292, 293, 294, 295, 290] while others do not [289, 288], some also use a different definition of the transport energy [201, 202]. A result for the transport energy by Baranovskii and co-workers derived using percolation theory and Miller–Abrahams rates reads [260, 290]   ϵ σ (3.50) , F , ϵt = σx Na3 , kB T kB T where x is the solution of the equation  =

1 + exp



ϵF − σx kB T

 √ − 13 9 2πB−1 Na3





x √

Z

2

2

exp −t  1 + exp ϵF − −∞  2 kB T x exp − σ 2  

dt




 43 

 √ 2σt kB T

(3.51)

with B = 2.7 ± 0.1 being a result of percolation theory. N is the number of sites per volume and a the charge localization length. Arkhipov commented that the transport energy after Monroe and Baranovskii is the one that maximizes the probability for upwards jumps but is not the energy equivalent to a transport band. He therefore defined an effective transport energy [292] that takes backward hops into account and lies slightly below the transport energy after Monroe and Baranovskii. The initial derivation of this energy was based on Miller–Abrahams rates and the limit of zero charge density. It was later extended to finite charge densities [296] and to to finite charge densities combined with the Marcus rate expression [217], however in a linearized version valid only if the energetic disorder is small compared to the reorganization energy, σ  λ. This assumption does not hold for many organic semiconductors. Different approaches yield different results and their comparison lacks an experimental validation. An experimental approach suggested by Cleve [297] was to trace the most frequently visited sites. This has also been done in numerical studies [298,

3.8 Continuous Space Charge Transport |

65

299, 300]. However, oscillations between trap states produce highly visited sites that are unimportant for transport, making the approaches unsuitable [299, 260]. The lack of consensus in the community for what definition to use, missing experimental validation and the missing full (i.e., not linearized) Marcus rate integration are obstacles in using the transport energy concept, yet it is popular since its analogy to band theory allows an intuitive interpretation.

3.8 Continuous Space Charge Transport A continuous space approach using the drift–diffusion (also: convection–diffusion) equation allows to model charge dynamics at the scale of semiconductor films or devices. The drift–diffusion equation describes the evolution of the charge density with time. It can be derived from two different approaches. One is an heuristic approach using physical arguments. It is based on Fick’s laws, which were postulated already in 1855. The second option, which is presented here, is a derivation from probability theory. Here the probability for particles to be found in a certain volume and not the density is described. Starting from the master equation in continuous space the Fokker–Planck equation can be derived. The drift–diffusion equation is a specific case of the Fokker–Planck equation.

3.8.1 Fokker–Planck Equation Instead of the master equation (3.16) with discrete states P α (t) (still with continuous time) it is also possible to write down a master equation for states continuous in space, p (x, t), and with transition rates ω (∆x, t). For the one-dimensional case and electrons it reads Z Z ∂ p(x, t) = dr [ω (x − r, r) p (x − r, t)] − p (x, t) drω (x, −r) . (3.52) ∂t A Taylor expansion of the first integral, called the Kramers–Moyal expansion, leads to the one-dimensional Fokker–Planck equation ∂ ∂ 1 ∂2 p (x, t) = − [A (x, t) p (x, t)] + [B (x, t) p (x, t)] ∂t ∂x 2 ∂x2

(3.53)

with a drift expression A (x, t) and a diffusion expression B (x, t). If A and B are constant in time and space, it can be written in the simpler form q

∂ ∂ ∂2 p (x, t) = qμF (x) p (x, t) + qD 2 p (x, t) ∂t ∂x ∂x

(3.54)

with mobility μ = A/qF and diffusion constant qD = 21 B. This form of the Fokker– Planck equation is also called the convection–diffusion equation or drift–diffusion

66 | 3 Charge Dynamics at Different Scales equation. The diffusion constant in this notation is given by the expression [301] 2

(∆x) , ∆t,∆x→0 2∆t

D = lim

(3.55)

where ∆x and ∆t denote the infinitesimally small steps in space and time. Equation (3.53) can be rewritten to the form of a probability conservation law: q

∂ ∂ p (x, t) = − J(x, t) ∂t ∂x

(3.56)

where J (x, t)

=

Jdrift (x, t)

=

J diff (x, t)

=

Jdrift (x, t) + Jdiff (x, t) ∂ μp (x, t) U(x) ∂x ∂ −qD p (x, t) . ∂x

(3.57)

Here J = limA→0 I/A is the current density, i.e., the electric current I per unit area A. In the drift current expression it has been assumed that the electric field, F, is due to an external potential, U, such that the replacement F (x) = −∂/∂x U (x) can be made. In case of hole transport the sign of the drift current is opposite.

3.8.2 Three-Dimensional Anisotropic Drift–Diffusion Equation All equations can also be generalized to the three-dimensional case. The drift– diffusion form of the Fokker–Planck then reads q


∂ p(r, t) = ∇ · p(r, t) q μ · F ± qD(r)∇p(r, t) . ∂t

(3.58)

A three-dimensional diffusion current in an anisotropic medium is described as Jdiff (r, t)

D(r)∇p(r, t)

(3.59)

Jdrift (r, t) = q p(r, t)μ ∇U(r)

(3.60)

=

The drift current is generalized to

with ∇U(r) = qF(r) if F is the local net electric field. It has been pointed out both for the one-dimensional [302] and the three-dimensional [230] case that this expression is not valid for inhomogeneous temperature distributions. The general Equation (3.58) describes the non-equilibrium dynamics of a single charge. If we are interested in the steady state the total probability flux ∂/∂t p(r, t) = 0 is zero, simplifying the equation to 0 = μ q p(r)∇U(r) ± qD(r)∇p(r). (3.61)

3.9 Heuristic Models |

67

3.8.3 Poisson Equation The Poisson equation is a second order partial differential equation. The version that describes electrostatic interaction of a potential, U(r), with a charge density, ρ(r), reads ρ . (3.62) ∆U = − ε0 ε r and is derived from Maxwell’s equations. The Poisson equation allows a self-consistent solution of the drift–diffusion equation for charge density and current. This is done by iteratively solving Equations (3.56) and (3.62). An external field qF = ∇U enters as a constant term on the left-hand side of Equation (3.62). The probability, p, in Equation (3.56) is replaced by the charge number density, n, i.e., the number of charges per volume, with ρ = qn. A constant number density, n0 , serves as a starting condition.

3.9 Heuristic Models This section gives an overview of some rather old models for the description of current–voltage characteristics (Child’s Law, Mott–Gurney model, Shockley diode equation), as well as for a dependence of the mobility on temperature (Arrhenius law, Mott’s T 1/4 law) and on the electric field (Poole–Frenkel behavior). These models should not be considered as a fully correct description in all cases; however, due to their simple relationships and partly also for historic reasons they are frequently used as a reference when drawing conclusions from measured current–voltage characteristics or mobilities. The models are not applied in this work but reviewed here briefly due to their frequent usage. In Chapter 9 the Mott–Gurney law is compared to results from microscopic calculations.

3.9.1 Child’s Law In 1911 Child proposed a theory to describe the current–voltage behavior for spacecharge limited current (SCLC) in a vacuum diode [131]. It was originally designed for ion transport and then applied to electron currents in 1913 by Langmuir [132]. The theory it is also sometimes referred to as the Child–Langmuir law or as the “threehalves power” law. It is only valid under the assumption of vacuum electron transport, meaning that no scattering events occur and predicts the relationship r 3 4ε0 2e V 2 I(V) = , (3.63) 9 me d2 where e is the electron charge and me its mass, V is the anode voltage and d the distance between anode and cathode. ε0 is the vacuum permittivity. In metals electrons

68 | 3 Charge Dynamics at Different Scales can move relatively free, making the model of vacuum electron transport a good approximation.

3.9.2 Mott–Gurney The Mott–Gurney model [303], also called the trap-free insulator model, was published in 1940. In contrast to Child’s law it does not assume vacuum transport and is thus applicable to insulators and semiconductors. The current–voltage characteristics predicted by the model reads J(V) =

9 V2 , εμ 8 d3

(3.64)

where J is the current density, d the thickness of the sample and ε the material’s relative permittivity. Assumptions leading to this relation are (i) hole-only or electron-only transport, (ii) no doping, (iii) constant mobility and relative permittivity and (iv) no injection barriers. The model also gives expressions for the electric potential and hole density throughout the sample: Vint (x) = V

n(x) =



d−x d

 32

,

(3.65)

3 ϵV 1 √ , 4 qd 32 d − x

(3.66)

where 0 ≤ x ≤ d denotes the position. In Chapter 9 the Mott–Gurney model is compare to to results from lattice models and to microscopic simulations coupled to the drift–diffusion equation. Despite the striking deviations, the model is still frequently used for the experimental analysis of current–voltage curves, cf. Section 2.1.

3.9.3 Shockley Diode Equation The Shockley diode equation, or ideal diode equation, was proposed by Shockley in 1949 [304]. He was awarded the Nobel prize in Physics of the year 1956 for his work on semiconductors and the discovery of the transistor effect. The Shockley diode equation describes the current–voltage behavior in a p-n junction with diode voltage V:     qV −1 I = I0 exp ηkB T

(3.67)

Here I0 is the saturation current (or scale current), which is the current that flows for a small, negative voltage V in the order of a few kB T/q. η is called the ideality factor,

3.9 Heuristic Models |

69

that should be one for an ideal realization of the theory. Trap-assisted recombination events, however, lead to a larger ideality factor [305]. Typically it is the range between 1 and 2 [306].

3.9.4 Arrhenius Temperature Dependence Arrhenius proposed an equation to describe the rate constant in chemical reactions in 1889 [307, 307]. In the context of organic semiconductors, however, Arrhenius law typically refers to a temperature dependence of the mobility of the form [308] ln μ ∝

1 . T

(3.68)

3.9.5 Mott’s T 1/4 law Mott proposed a temperature dependence of the dc conductivity, σ, or the mobility, μ, of the form [309, 310, 311]   14 T0 (3.69) ln σ ∝ − T for steady state transport at sufficiently low temperatures, derived from variable range hopping theory. Shklovskii and Éfros proposed a modification leading to an exponent of 21 instead of 14 [312, 313] after the inclusion of Coulomb effects. Aharony et al. then introduced a generalized version, that has the limiting cases of the Mott behavior for high temperatures and of the Éfros–Shklovskii behavior for low temperatures, agreeing with experiments that showed this crossover [314].

3.9.6 Poole–Frenkel Field Dependence Poole–Frenkel field dependence refers to the relationship of the conductivity, σ, or the mobility, μ, to the applied electric field of the form ln σ ∝

√

F,

(3.70)

derived by Frenkel [315] in 1938. In his work he briefly mentions a relation called √ Poole’s law, that predicts a proportionality to F instead of F, thus the name Poole– Frenkel. The model is based on band transport with additional trap states that capture a large part of the charges, similar to the multiple trap and release model developed later, cf. Section 3.7.

4 Computational Methods Schrödinger’s groundbreaking publication [316] from the year 1926 defines a full nonrelativistic¹ description of a solid, consisting of nuclei and electrons. The Hamiltonian reads ˆ 1 , ..., RN ; r1 , ..., rn ) H(R

=

2 N n 2 X X pˆ i Pˆ i + 2M i 2m i=1

N

+

1X 1 2 4πε0 i,j=1

i6=j

−

(4.1)

i=1

N X n X i=1 j=1

n

Z Z e2 1X 1 e2 i j + Ri − Rj 2 4πε0 ri − rj i,j=1 i6=j

2

1 Ze i . 4πε0 Ri − rj

Here N denotes the number of nuclei, n the number of electrons, M i the mass of the ith nucleus, m ≈ 9.109×10−31 kg the electron mass, e ≈ 1.602×10−19 C the absolute value of the electron charge, ε0 ≈ 8.854×10−12 AsV−1 m−1 the vacuum permittivity, Ri and ri the nucleus and electron positions, Pi and pi the nucleus and electron momenta, and Z i the atomic numbers. The terms in this equation are the kinetic energies of nuclei and electrons, the Coulomb interaction of nuclei with each other, of electrons with each other and of nuclei with electrons. The corresponding stationary Schrödinger equation for the wave function Ψ(R1 , ..., RN ; r1 , ..., rn ) of all nuclei and electrons is given by ˆ HΨ(R (4.2) 1 , ..., RN ; r1 , ..., rn ) = EΨ(R1 , ..., RN ; r1 , ..., rn ). While the equation, in principle, completely describes the material, it remains a problem to solve it. Even for a molecule as simple as H+2 , consisting of only two protons (the nuclei) and one electron, there is no analytic solution. Furthermore, it is impossible simply to store the wave function without approximations for a macroscopic solid, where the number of atoms is in the order of 1023 . Ab initio methods aim to solve the Schrödinger equation approximately. The Hartree–Fock method and density functional theory (DFT) are methods that fall into this category. For large, disordered systems, these methods are too expensive. Here, instead of solving the Schrödinger equation, one can describe materials on a classical level with Newtons equations of motion, neglecting quantum mechanic effects. Molecular mechanics (MM) methods can be used to solve the classical description, namely molecular dynamics (MD) and Monte Carlo methods. A combination of both, quantum mechanical and classical descriptions, is commonly referred to as QM/MM. 1 Just a year later, Dirac published a description that takes special relativity into account [317]. Relativistic effects are important in materials with heavy elements, where spin–orbit coupling is strong [318]. Here the discussion is limited to the non-relativistic Schrödinger equation.

4.1 Density Functional Theory | 71

In the first section of this chapter ab initio methods are discussed, with an introduction to the DFT method. The classical treatment using MD is the subject of the second section. In the third section classical calculations of Coulomb interaction and induction are explained. The last section covers coarse-graining techniques.

4.1 Density Functional Theory Density functional theory allows to evaluate the electronic ground state of a molecule. It also allows to calculate energy changes upon electron removal or addition and, thus, a calculation of the ionization potential (IP) and the electron affinity (EA) of a molecule in vacuum. The vacuum levels serve as a first approximation for molecular energies in the solid state, which can be refined using a perturbative scheme that includes electrostatic and induction effects (Section 4.3). DFT also allows to calculate transfer integrals, J ij , that enter the rates for charge transfer, see Section 3.1. A third application here is to calculate reorganization energies, λ ij , which enter the charge transfer rates.

Born–Oppenheimer Approximation The Born–Oppenheimer or adiabatic approximation [319] allows a simplification of the Schrödinger equation of a solid by assuming that the complete wave function can be written as a product of electron and nucleus functions, where the nuclei positions are fixed in the electrons’ reference system. This assumption is reasonable due to the fact that electron motion is by orders of magnitude faster than nucleus motion.

Hartree–Fock Method A first solution method for the electronic wave function was proposed by Hartree. Here the idea is to treat electron–electron interactions in a mean field approximation and factorize the many-electron wave function into a product of single electron wave functions. The Hartree method was improved by Fock and Slater, such that the Pauli principle is obeyed. This is taken into account by requiring an anti-symmetric many electron wave function, written as a Slater determinant.

Hohenberg–Kohn Theorem Hohenberg and Kohn proved in 1964 that for a system of interacting electrons (with non-degenerate ground state) in an external potential Vext (r) there is a unique electron density n(r) [320]. For any ground state property, the energy can be expressed as E = R drn(r)Vext (r) + F[n(r)]. The density that minimizes this energy is the ground state density, the corresponding energy the ground state energy. A generalized proof of this theorem was given by Levi in 1982 [321].

72 | 4 Computational Methods Kohn–Sham Equations The Kohn–Sham equations [322] cast the many-body problem into a set of effective one-electron equations which form the basis of the DFT approach. Kohn and Sham divided the energy functional, E[n(r)], into several contributions: Z E[n] = T[n] + V H [n] + drn(r)Vext (r) + Exc [n]. (4.3) The first term, T[n], is the kinetic energy of non-interacting electrons: T[n] =

n Z X

drψ*i (r)

i=1



 ~2 ∆ ψ i (r). − 2m

(4.4)

Here the electron density, n(r), is expressed in terms of the single electron wave functions X ψ i (r) 2 . n(r) = (4.5) i

The second term is the Hartree energy, which describes the Coulomb interaction between electrons: Z Z 1 e2 n(r)n(r′ ) . (4.6) V H [n] = drdr′ r − r′ 4πε0 2

The third term describes the interaction of electrons with an external potential. The last term, Exc [n], includes exchange and correlation effects between electrons. Its functional form is unknown. Various approximations are used, leading to different DFT functionals, which is the most important systematic limitation of DFT. A variation of the total energy with respect to n leads to the following expression for the effective potential: Z e2 δExc n(r′ ) dr′ + Veff (r) = Vext (r) + . (4.7) r − r′ 4πε0 δn(r) The single-electron wave functions are defined by the Kohn–Sham equations  2  ~ − ∆ + Veff (r) ψ i (r) = ϵ i ψ i (r). 2m

(4.8)

These equations have to be solved self-consistently.

Exchange–Correlation Energy Functionals DFT relies on an approximation of the exchange–correlation energy. The simplest one is the local density approximation (LDA) Z Exc [n] = n(r)ϵxc [n(r)]dr. (4.9)

4.2 Molecular Dynamics | 73

Here ϵxc [n] is the exchange–correlation energy per electron in a homogeneous electron gas that has a constant density n. ϵxc [n] is calculated once, afterward Exc [n] can be calculated easily from the above equation. The method is exact for a homogeneous electron gas. For solid state systems it often yields good results, even if their electron density is (globally) strongly inhomogeneous. An extension of LDA is the generalized gradient approximation (GGA) that includes also a gradient term of the density. Hybrid functionals try to combine the computational efficiency of DFT with the exact exchange, as it is used in the Hartree–Fock method. This exact exchange is expressed in terms of Kohn–Sham orbitals and in terms of the electron density. One of the most frequently used hybrid functionals is B3LYP (Becke, 3-parameter, Lee-Yang-Parr) [323, 324], that has portions of LDA, GGA and exact exchange. The fixed parameters for the different portions were obtained from fits of relevant properties by comparison to Hartree–Fock calculations.

4.2 Molecular Dynamics Molecular dynamics (MD) simulations help to understand morphological properties, i.e., the structure of assemblies of molecules and their microscopic interactions. Together with Monte Carlo simulations of the morphology it falls into the category of molecular mechanics (MM) calculations. MD simulations also allow to study the time development of the system and to extract quantities such as the diffusion constant of molecules from it. This section gives a brief overview of MD. For details I refer to exemplary reviews and books [325, 326], out of many treatments of the broad topic. In contrast to quantum mechanical techniques, such as DFT or the Hartree–Fock method, MD is a classical simulation method. Its basis are Newton’s equations of motion applied to the atoms of a molecule, with masses m i and (center-of-mass) positions ri . Newton’s second law is then written in the form m i r¨i = fi ,

fi = −∇ri U (r1 , . . . , rN ) ,

(4.10)

where the force fi acting on atom i is expressed as the gradient of a potential U(r1 , . . . , rN ). This potential depends on all atom positions and describes all inter-atomic interactions. The interactions are split into a non-bonded, intermolecular part and a bonded, intramolecular part. The non-bonded part is split into 1-body, 2-body, 3-body terms etc.: X XX
Unonbonded (r1 , . . . , rN ) = u (ri ) + v rr , rj + . . . . (4.11) i

i

j>i

Here the first term describes an external potential and is not needed for bulk systems with periodic boundary conditions. The second term is the pair potential. Higher order

74 | 4 Computational Methods potentials are often neglected. The most commonly used pair potential is the Lennard–Jones potential "  # 12  6 d d − . vLJ (r) = 4ε r r

(4.12)

Here ε is called the depth of the potential well, d is the distance at which the interparticle potential vanishes and r = |ri − rj |. The parameters are usually empirically determined. An additional term is the interaction of the molecular charges. In principle, it is a double integral over the charge densities of the two molecules, which is, however, computationally too expensive to calculate in practice. The simplest approximation to handle this problem are partial charges, i.e., fractional charges assigned to the atoms such that the outer electrostatic potential of a molecule is reproduced well. The Coulomb interaction is then modeled as the interaction of the partial charges of atoms i and j: qi qj vCoulomb (r) = . (4.13) 4πϵ0 r Bonded (intramolecular) interactions consists of three contributions: stretching or contraction of bonds, bending around certain angles and torsion. A common functional form of these interactions reads
2 1 X r Ubonded (r1 , . . . , rN ) = k ij r ij − req (4.14) 2 bonds
2 1 X ϑ + k ijk ϑ ijk − ϑeq 2 bending angles

+



2 1 X φ,m k ijkl 1 + cos mφ ijkl − γ m . 2 torsion angles

Bond stretching/contraction depends on two atomic coordinates that define the bond. Bending angles are defined by three atomic coordinates and torsion angles by four coordinates (three bonds). The torsion interactions can be expanded in periodic functions of order m = 1, 2, . . .. Here the bending term is of a quadratic form but periodic potentials are also used. The presented form of Ubonded is one of the simplest versions. Sometimes also cross terms of these interactions are included. The total interaction potential U that is used is called the force field, and is a crucial part of an MD simulation. Depending on the aim of the simulation, force fields of different complexity have to be used – simple models can be sufficient to understand basic processes, while sometimes the model has to be as accurate as possible for, e.g., a reliable structure prediction. While for biomolecules sometimes force-fields can be used “out of the box”, this is normally not the case for organic semiconductors with their extended π-conjugated systems. Here, force fields usually have to be parametrized from DFT calculations, where the

4.2 Molecular Dynamics | 75

different degrees of freedom of a single molecule are scanned and the parameters are adjusted such that the energy change of the force field due to bending and torsion agrees well with the DFT results for this geometry. An example of such a force field parametrization is given in Chapter 7. Note also that the vibrations due to stretching and contraction of bonds usually have a very high frequency. In fact, they should be treated quantum mechanically. Therefore in MD they are usually constrained to a fixed bond length. It is also common to impose other constraints, for example to keep a molecule reasonably flat. The identification of important degrees of freedom has to be done “by hand” and is not easy to automatize, however, there are efforts into this direction [327, 328]. MD simulations usually employ periodic boundary conditions, i.e., a molecule that leaves the box from one side automatically enters at the opposite site. This also leads to interaction between molecules and their images. In MD, usually the minimum image convention is used, i.e., only the interaction with the closest image is taken into account. In case of long-range correlations in charged systems or systems with high molecular dipole moments this can lead to problems and a more accurate evaluation becomes necessary. The time span of MD simulations here is typically in the nanoseconds regime (10−9 s). In other applications simulations can go up to microseconds (10−6 s). Typical time steps are in the femtoseconds (10−15 s) regime. Software packages, both commercial and freely available, for MD simulations include GROMACS, ESPResSo, ESPResSo++, LAMMPS, CHARMM, and AMBER. The latter two, CHARMM and AMBER, are at the same time the names of the force fields used by these packages. There are also software suites for computational chemistry that include DFT, MD and other calculations, for example CP2K, Material Studio or Scigress. These examples are not intended to give a comprehensive list. Other important force fields include MM3, MM4, AMBER, MMFF, GROMOS or OPLS.

4.2.1 Ensembles A solution of Newton’s equation of motion leads to simulations at a fixed number of particles, N, in a fixed volume, V, and without energy exchange with the environment, i.e., at a fixed total energy, E. This NVE ensemble is also called the microcanonical ensemble in statistical physics. Another ensemble is the canonical ensemble with constant NVT. Most experiments are carried out at constant pressure, i.e., in the isobaric-isothermal ensemble, with constant NPT. In both cases the average kinetic energy per particle fulfills the relation D E 1 (4.15) v2α = kB T, 2

76 | 4 Computational Methods where v α with α = x, y, z are the components of the velocity vector. However, during the time evolution statistical fluctuations of the kinetic energy of particles occur. To simulate at constant T a thermostat is necessary, modeling the energy exchange of the system with the environment, or a “heat bath”. An extensive overview of different available thermostats has been given by Hünenberger [329]. The simplest idea would be a temperature re-scaling of all molecule velocities in each MD step, such that the temperature given by Equation (4.15) is obtained. This approach, however, suppresses the statistical temperature fluctuations of the canonical ensemble. A weaker formulation is the Berendsen thermostat [330] (“temperature coupling”), that re-scales the velocities in each step, such that the change of temperature is proportional to the difference between the current and the desired temperature. The Berendsen thermostat suppresses fluctuations of the kinetic energy and therefore does not generate correct trajectories. For systems in the order of hundreds or thousands of particles, however, this approximation usually yields good results. A different approach is the Andersen thermostat [331]. Here, occasionally random particles are selected and their velocity is updated to a random value drawn from Boltzmann statistics for the desired temperature, T. This can be understood as an energy transfer between heat bath and particle. The strength of the coupling (frequency of updating particles) can be adjusted. Similar to the Berendsen thermostat, the Andersen thermostat also does not yield correct trajectories. The random collisions lead to discontinuities in the velocity trajectory in disagreement with the real nature of the system. If physically correct trajectories are needed, the conceptually more complicated and computationally more expensive Nosé–Hoover thermostat [332, 333] allows deterministic MD simulations at constant temperature. Sometimes systems are first equilibrated using the Andersen or Berendsen thermostat, which are computationally more efficient, and a subsequent calculation using the Nosé–Hoover thermostat is used to examine the correct dynamics. In the NPT ensemble an additional barostat is used to control the motion of the box volume [325] to simulate at fixed pressure, P.

4.2.2 Polarizable Force Fields Initially, the interaction between charge distributions was modeled by the interaction of static, partial charges, usually centered on the atoms. However, changes in the charge distribution of the environment cause a re-arrangement of the charge distribution of a molecule, an effect known as polarization or induction. It can affect both the structural properties and the energetic landscape. By adjusting partial charges and van der Waals interaction, it is possible to take polarization into account. In most force fields the parametrization of effective charges is performed using experimental input for thermodynamic properties [334]. Such an

4.3 Electrostatic and Induction Calculations for Site Energies | 77

approach limits the applicability of the force field, since the parametrization depends on the compound or the class of compounds, as well as on external parameters. A more general but computationally expensive approach is to model polarization by including higher multipoles. This approach allows for a parametrization by ab initio methods. Therefore, in contrast to experimentally parametrized force fields, it is possible to use the technique for systematic pre-screening of materials solely based on simulations. As a compromise between computational efficiency and accuracy, structural properties are often calculated using standard force fields, complemented by polarizable force fields for energy calculations [334].

4.3 Electrostatic and Induction Calculations for Site Energies The delocalized π orbital is the highest occupied molecular orbital (HOMO), while its antibonding counterpart π* is the lowest unoccupied molecular orbital (LUMO) [335]. In a crude approximation, i.e., neglecting the effects of the environment and of molecular rearrangement, electron transport takes place in the LUMO level. A more appropriate description of the energy levels can be achieved using the ionization potential (IP) and the electron affinity (EA). They can be calculated as the sum of a quantum mechanical part, Eint , for a molecule in the gas phase, and classically pol el evaluated electrostatic and induction (polarization) contributions, E i = Eint i +E i +E i . The quantum mechanical part is evaluated using DFT calculations. Both ionization and electron acceptance lead to a rearrangement of atomic positions. Denoting the energy of a charged molecule in its charged geometry with EcC and the energy of a neutral molecule in its neutral geometry with EnN , the ionization potential and elech tron affinity can be expressed as Eint IP = E cC − E nN , where the index h (hole) stands e for the anionic state, and Eint EA = E cC − E nN , with the index e (electron) denoting the cationic state. The index i of the molecule has been omitted here. It is computationally too expensive to evaluate the electrostatic and induction contributions as an integral over two charge distributions. Hence, the electrostatic part is calculated as the sum of multipoles of all molecules in the system, either using a cutoff for the maximal interaction range or Ewald summation to evaluate the complete sum. One method to include induction effects is the Thole model.

Thole Model The Thole model [336] is based on a point dipole model, where each atom has a polarizability which is calculated from a dipole field tensor. Based on these polarizabilities, a molecular polarizability is calculated. A problem in this approach is that for atoms closer than a certain limit the interaction energy diverges, which is why the Thole model modifies the dipole–dipole interaction using a damping term. The model relies

78 | 4 Computational Methods on fitting parameters for the atomic polarizabilities. Initially these were parametrized based on experimental input, while nowadays one can use ab initio calculations.

Ewald Summation The interaction energy of partial charges is often computed using a distance cutoff. However, the 1/r decay of the Coulomb potential leads to significant long-range effects. Part of this contribution is the interaction of partial charges with their own images and images of other charges, leading to an infinite sum when using periodic boundary conditions. Instead of using a cutoff it is also possible to evaluate this sum exactly, including the long range contribution. This is done by splitting the potential into a short-range and a long-range part: ϕ (r) = ϕsr (r) + ϕlr (r) .

(4.16)

The short-range part is evaluated in real space. The long-range part, on the contrary, is solved in Fourier (reciprocal) space, where it converges quickly. This method is known as Ewald summation [242]. An efficient implementation is particle mesh Ewald [241] (PME). Here, a fast Fourier transform (FFT) on a discrete lattice (mesh) is used. This
allows a computational efficiency of O (N log N ) compared to O N 2 for the direct method. The method can be generalized to include dipoles or higher multipoles. Regarding induction effects the long range contribution is less important, such that it is usually sufficient to use a (large enough) cutoff.

4.4 Coarse-Grained Morphology In small simulation boxes the charge carrier mobility is usually overestimated, as discussed in Chapter 5. A strategy to obtain larger morphologies without a restriction to Gaussian disorder models is a stochastic reproduction of all quantities entering charge transfer rates from microscopic simulations, as discussed in Chapter 7. In this section different methdos for a reproduction of the center-of-mass distribution of molecules are discussed, which are useful in cases where system sizes are not accessible by MM methods. A measure of this distribution is the radial distribution function (RDF), g(r). Its product with the averaged molecular density, ρg(r), describes the density of molecules at a distance r from a reference molecule. There are different strategies to reproduce this target function – some are mathematically motivated and others use physical considerations. They also differ in computational demands and in their flexibility to describe also more complicated systems, such as polymer chains or multi-component systems, i.e., systems with different types of molecules. A conceptually simple and mathematically motivated strategy is thinning of a Poisson process [337, 244]. It is suitable for small-molecule, single-component systems and can yield reasonably good approximations of g(r) in the short range regime. The al-

4.4 Coarse-Grained Morphology

| 79

gorithm starts by producing points in space that are independently, uniformly distributed in a sub-volume w, with N ∼ Poi (ρw) being the number of points within the sub-volume and ρ the point density. These points can have very short distances, that would correspond to the unphysical situation of overlapping molecules. Hence, in a next step, each point i is assigned a random radius R i . Afterward, all points i whose sphere is contained in in the sphere of another point j are deleted. Now the density of the obtained morphology will be smaller than the density aimed for. Hence, in a next step, a new set of potential points is created and, again, all the points whose spheres are contained in the spheres of previous realizations are deleted. The procedure is repeated iteratively until the desired point density ρ is reached. Thinning of a Poisson process is a fast and approximate scheme that does not exactly reproduce the radial distribution function. The method has been applied to dicyanovinyl-substituted quaterthiophene (DCV4T) in Chapter 8. If one needs a more accurate description of g(r) or if a more complicated system has to be described, coarse-graining is a more suitable method. The wide subject cannot be treated here in detail, therefore I refer to reviews [338, 339, 340, 341, 342] for details. Coarse-graining methods aim to find potentials that reproduce structural properties of the underlying atomistic system. These structural properties are, in general, not limited to the RDF, g(r), but can also include conformational degrees of freedom or distances between groups of atoms. Accordingly, interaction potentials are usually divided into two groups, bonded and non-bonded interactions. Including bonded interaction is important for larger molecules, where the molecule is subdivided into different fragments and degrees of freedom such as the distance between fragments or their torsional or dihedral angle to each other are considered. Including intramolecular degrees of freedom, consequently, requires molecule-specific method development. Here I limit the discussion to amorphous, small-molecule semiconductors, where intermolecular degrees of freedom, and in particular the RDF, g(r), are sufficient and the methods are generally applicable without molecule-specific adjustment. Strategies to determine coarse-grained potentials fall into two principal categories. One is to use an empirical form of the potential and fit its parameters such that the RDF of the atomistic simulation is reproduced well [343, 344]. A simple example of a parametrized potential is the Lennard–Jones potential, see Section 4.2. The second category comprises methods that use numerically derived, tabulated potentials of arbitrary shape. I will focus here on the latter, more general category and describe in the following three important structure-based methods: Boltzmann inversion [345], iterative Boltzmann inversion (IBI)[346] and inverse Monte Carlo (IMC) [347, 348, 349]. In principle, the solution to the problem of finding a pair potential reproducing a given RDF is unique up to an additive constant, as stated be the Henderson theorem [350]. Different numerical solutions for the pair potential, however, may reproduce the same RDF without a noticeable error, which is why some methods allow applying additional constraints on thermodynamic properties, e.g., pressure and compressibility.

80 | 4 Computational Methods Boltzmann inversion This technique makes use of the fact that the probability of a certain configuration in the canonical ensemble is Boltzmann distributed with respect to a degree of freedom q, i.e.,
exp −βU(q) P(q) = , (4.17) Z

where U(q) is a potential and β = 1/kB T. The partition function is given by Z =
R exp −βU(q) dq. If the probability distribution P is known, e.g., from atomistic reference calculations, one can invert this equation to obtain the potential
U(q) = −kB T ln P(q) .

(4.18)

The term ln (Z ) has been omitted since it will only shift the potential by a constant factor and therefore is of no importance. In case of a one-component small molecule system, the intermolecular distance is the only coarse-grained degree of freedom, q = r, and the probability distribution is given by the RDF, P(q) = g(r). The same technique, however, also works for potentials that depend on different degrees of freedom, U(q1 , q2 , . . . , q N ), provided that they are independent. If that is the case the probability distribution factorizes, P(q1 , q2 , . . . , q N ) = P(q1 )P(q2 ) . . . P(q N ), and, as a consequence, the potential U can be written as a sum of the contributions of different degrees of freedom. A point to take care of when implementing the method is to normalize the distributions obtained from histograms, such that P(q i ) is volume normalized, see [346, 26] for details. Also important to keep in mind is to check that coarse-grained degrees of freedom were chosen such that they are uncorrelated, because otherwise the probabilities do not factorize. Since the reference potential obtained from atomistic simulations is tabulated for discrete values, a smooth interpolation, e.g., using splines, is needed before inversion. A single Boltzmann inversion is exact only in the limit of infinitely dilute systems [346], i.e., with molecular density ρ = 0. It is mostly used for bonded potentials.

Iterative Boltzmann inversion An important extension of the method is iterative Boltzmann inversion (IBI). The idea is to consider the difference between the potential U i obtained from the ith Boltzmann inversion cycle and the reference potential, and express it in form of probability distributions by making use of Equation (4.18):  i  P (q) . (4.19) ∆U i = U i − U ref = −kB T ln Pref (q) Now the next potential in the iteration is calculated as U i+1 := U i + ∆U i .

(4.20)

4.4 Coarse-Grained Morphology

| 81

A potential that reproduces the probability function is a fixed point of the iteration, which, in principle, guarantees convergence of the algorithm. In practice, the correction term ∆U i tends to over-correct the potential of the current iteration. It is therefore often of advantage to rescale it by a factor between 0 and 1 to make the algorithm numerically stable. In case of non-bonded interactions an important parameter influencing the speed of convergence is the distance cutoff of the potential, which therefore should not be chosen unnecessarily large [346]. An example is given in Chapter 7, where I apply IBI to DPBIC, building up a stochastic description for hole transport up to the device scale.

Inverse Monte Carlo Another method to obtain coarse-grained potentials is inverse Monte Carlo (IMC). The name originates from the fact that the method was in the first place used in combination with Monte Carlo sampling. However, it can also be used in combination with MD or stochastic dynamics, making its name somewhat misleading. I limit the discussion here to the compact version of non-bonded interactions, as discussed in [349]. First, the Hamiltonian of the full system is expressed as the sum of all pair-interaction potentials, U(r ij ), which are assumed to depend only on the intermolecular distance r ij : H=

X

U(r ij ).

(4.21)

i,j

The potential is then evaluated at certain grid points within the interval [0, rcut ], with r α = ∆r α, where α = 1, . . . , M and rcut is the chosen maximal interaction range to be included. With U α := U(r α ) the Hamiltonian can be written as X H= Uα Sα , (4.22) α

where S α denotes the number of pairs that have an intermolecular distance within the bin corresponding to r α . Now, the average number of pairs at a certain distance, hS α i, is equal to the RDF times a normalization constant,

N (N − 1) 4πr2α ∆r g(r α ). (4.23) 2 V Here N is the number of molecules in the system and, consequently, N(N − 1)/2 the number of pairs, V is the total volume of the system and 4πr2α ∆r is the volume of the considered α slice. On the other hand, the average hS α i depends on the interaction potential and can be written as a Taylor expansion:   X ∂ hS α i ∆ hS α i = ∆U γ + O ∆U 2 . (4.24) ∂U γ hS α i =

γ

The derivatives

∂ hS α i ∂ = ∂U γ ∂U γ

R


P dqS α (q) exp −β λ U λ S λ (q)
R . P dq exp −β λ U λ S λ (q)

(4.25)

82 | 4 Computational Methods αi can be rewritten by applying the quotient rule [347]. With the definition A αγ := ∂hS ∂U γ they read β β A αγ = 2 (hS α S γ i − hS α i hS γ i) = 2 Cov [Sα , Sγ ]. (4.26) Z Z Now, one can write down a set of linear equations that has to be solved in order to calculate the correction potential ∆U γ :

hS α i − Sref α = A αγ ∆U γ .

(4.27)

i+1 := U i + ∆U i , Here Sref α is the reference value. The potential is updated iteratively, U until the algorithm converges. As in the case of IBI, it can be helpful to scale the potential update ∆U i by a factor between 0 and 1 to make the algorithm numerically stable.

Force Matching Another method for obtaining coarse-graining potentials is force matching. In contrast to the structure-based techniques discussed above, here not the RDF is reproduced but the aim is to match the forces in the coarse-grained description as good as possible with respect to the atomistic reference. For details about the technique and its implementation see [26, 351, 352, 353].

5 Energetics and Dispersive Transport Reprinted with permission from: Pascal Kordt, Thomas Speck, Denis Andrienko Finite-size scaling of charge carrier mobility in disordered organic semiconductors Physical Review B 94, 014208 (2016) Copyrighted by the American Physical Society. DOI: 10.1103/PhysRevB.94.014208

Simulations of charge transport in amorphous semiconductors are often performed in microscopically sized systems. As a result, charge carrier mobilities become systemsize dependent. We propose a simple method for extrapolating a macroscopic, nondispersive mobility from the system-size dependence of a microscopic one. The method is validated against a temperature-based extrapolation [261]. In addition, we provide an analytic estimate of system sizes required to perform nondispersive charge transport simulations in systems with finite charge carrier density, derived from a truncated Gaussian distribution. This estimate is not limited to lattice models or specific rate expressions.

Introduction Charge carrier mobility is the key characteristic of organic semiconductors. Experimentally, it can be extracted from time-of-flight measurements [145, 146], current– voltage characteristics in a diode [135, 136] or field effect transistor [354, 355], pulseradiolysis time-resolved microwave conductivity measurements [122], or other techniques [118, 120, 123, 126, 127, 148, 156, 356, 357, 358]. In amorphous organic materials the energetic landscape sampled by a charge carrier can be rather rough, with the width of the density of states as large as 0.2 eV. As a result, charge transport in thin films becomes dispersive; that is, the extracted mobility varies with the film thickness [359, 360, 361]. Consequently, the intrinsic value of mobility is difficult to measure: for example, the film thickness has to be large enough in time-of-flight experiments, imposing stringent requirements on the accuracy of measurements of transient currents. A similar situation is encountered in computer simulations of charge transport in organic semiconductors. Here, both lattice and off-lattice models employ system sizes which are usually much smaller than those used in experimental setups. This leads to an artificial increase in the average charge carrier energy and, as a result, to overestimated values of the charge mobility [244, 247, 261]. In fact, there are two reasons for finite-size effects. The first one is the dependence of the percolation threshold on the system size: in small systems the mobility fluctuations can become comparable

84 | 5 Energetics and Dispersive Transport to the mobility itself and are system-size dependent [362]. Second, the average charge carrier energy increases with decreasing system size, leading to overestimated values of the charge carrier mobility [261]. Here, we will mostly deal with the second effect, which seems to be the dominant contribution to the finite-size scaling. To overcome the limitations imposed by small system sizes and the resulting energy increase, a method based on a temperature-extrapolation procedure has recently been proposed [261]. Its main idea is to simulate charge transport at a range of elevated temperatures. At high temperatures transport becomes nondispersive, and one can then extract the nondispersive mobility at relevant (lower) temperatures from the mobility-temperature dependence μ(T). This method relies on the analytical dependence of the mobility on the temperature derived for a one-dimensional system [363] with Gaussian-distributed energies and Marcus rates for charge transfer (see Section 5.1), given by      a 2 b μ . (5.1) μ(T) = 03 exp − − T T T2

Note that it is also possible to derive a similar relation using percolation theory¹. [215] In three dimensions μ0 , a, and b are treated as fitting parameters instead of the parameters derived analytically in one dimension. Hence, for three-dimensional transport, Equation (5.1) has to be validated for every particular system. It would therefore be useful to have an alternative approach, which does not rely on the ad hoc μ(T) function. This is the first target of our paper: solving the one-dimensional stochastic transport, we derive the system-size scaling of the mobility and benchmark it against the temperature-based extrapolation. Due to the filling of the energy levels in the tail of the density of states the charge density is known to have a strong impact on the mobility [113, 114, 286, 364, 365, 366, 367]. With increasing density the energy per carrier is increased, leading to higher mobilities. On the other hand, in small systems mobilities are artificially increased. One can therefore presume that the error induced by finite-size effects will decrease with increasing charge carrier density. Hence, the second task of this paper is to provide a criterion for the system size required for nondispersive transport simulations at finite charge concentration.

5.1 Methods To perform mobility simulations, we use the Gaussian disorder model [22, 114, 140, 141]; that is, we assume that molecular sites are arranged on a cubic lattice and that

1 The resulting equation has the same temperature dependence as Equation (5.1), however, with a different exponent in the power law (0.15 instead of 3/2 for a simple cubic lattice with Marcus rates). In our simulations the 1D expression agrees better with simulation results.

5.2 Scaling Relation | 85

√
site energies ϵ i follow a Gaussian distribution, f (ϵ) = 1/σ 2π × exp −ϵ2 /2σ2 , where σ denotes the energetic disorder. We assume a mean of zero throughout. We use the Marcus expression for charge transfer rates [207, 208, 277], "
2 # J 2ij ∆ϵ ij + qF · rij + λ ij 2π p exp − ω ij = (5.2) ~ 4λ ij kB T 4πλ ij kB T

for transitions j → i, where ∆ϵ ij = ϵ i − ϵ j is the site energy difference, q is the charge, F is an external field, rij = ri − rj is the distance between two sites, λ ij is the reorganization energy and J ij denotes the electronic coupling. As a simplification, we assume a constant reorganization energy, λ ij = λ; a constant transfer integral, J ij = J; and a lattice spacing of |rij | = a. Further, T is the temperature, and kB is the Boltzmann constant. The Marcus rates allow us to link the mobility to the chemical composition of organic semiconductors [250, 244, 368, 369, 370, 371]. Charge–charge interactions are modeled by an exclusion principle²; that is, each site can be occupied by only one charge carrier at a time. As a result, the equilibrium site occupation is given by Fermi–Dirac statistics [373]    −1 ϵ − ϵF p(ϵ) = exp +1 , (5.3) kB T where the Fermi energy ϵF is implicitly determined by the number of charges in the system through Z∞ p(ϵ)f (ϵ)dϵ = n. (5.4) −∞

Here, n is the charge carrier density, i.e., the number of charges divided by the number of sites. The average energy per charge carrier, ϵc , is then given by R∞ ϵ p(ϵ)f (ϵ)dϵ ϵc = R−∞ . (5.5) ∞ p(ϵ)f (ϵ)dϵ −∞ Note that in the limit of zero charge carrier density or for high temperatures the Fermi–Dirac distribution can be approximated by the Boltzmann distribution, pB (ϵ) =
exp −ϵ/kB T , which yields ϵc = −σ2 /kB T.

5.2 Scaling Relation 5.2.1 Derivation We now derive and test the system-size dependence of the charge carrier mobility μ(N) in the limit of zero charge carrier density. The derivation is based on the model of a 2 A more elaborate model of Coulomb interaction than the exclusion principle would lead to small deviations from Fermi–Dirac statistics [372] but is not taken into account here.

86 | 5 Energetics and Dispersive Transport 10−7

10−7

(b)

10−8

10−9 10−10 10−11 10−12 10−13

10−10 10−11 10−12

600 K 300 K 200 K

0

10−9

0.02 0.04 0.06 0.08 0.1 0.12 0.14

10−13

100

1/L

non-dispersive regime

mobility µ (m2 V−1 s−1 )

mobility µ (m2 V−1 s−1 )

10−8

dispersive regime

(a)

1000

600 K 300 K 200 K

10000

100000

temperature T (K)

Fig. 5.1. (a) System-size extrapolation for energetic disorder of σ = 0.1 eV, external field of F = 106 V/m, lattice spacing of 1 nm, transfer integral of J = 10−3 eV, and reorganization energy of λ = 0.3 eV. (b) Validation using the temperature extrapolation. Large symbols denote the extrapolated mobilities μ∞ , summarized in Table 5.1.

one-dimensional chain of length N with Gaussian-distributed, uncorrelated energies, and hopping taking place only between adjacent sites according to the Marcus rates, Equation (9.1). An electric field of strength F = |F| is applied in the direction of the chain. We will require the mean velocity D E N−1 v N (F) = (N − 1) τ−1 , ≈ N hτ N i

(5.6)

R where h·i denotes the average over the energetic disorder, i.e., hg (ϵ)i ≡ dϵf (ϵ) g (ϵ). Here, we have approximated the mean rate by the inverse mean first passage time hτ N i, which can be calculated more readily. For completeness, we now present a detailed derivation of hτ N i. At steady-state conditions for a given realization of the disorder, the mean first-passage time to traverse the chain starting at i = 1 reads [374, 375, 376, 377] τN =

i N−1 X X i=1 k=1

ω 1 ω i−1,i · · · k,k+1 . ω i+1,i ω i,i−1 ω k+1,k

(5.7)

   The rates fulfill a detailed balance, ω ij /ω ji = exp −β ϵ i − ϵ j − f (i − j) , which leads to   i−1 i−1  X  Y ω j,j+1 = e−βf (i−k) exp β (ϵ j+1 − ϵ j ) ω j+1,j   j=k

j=k

=e

−βf (i−k)+β(ϵ i −ϵ k )

,

(5.8)

5.2 Scaling Relation | 87

where f = qFa and β = 1/kB T. After shifting i − k 7→ k, this can be rewritten as τN =

N−1 X i=1

1 ω i+1,i

i−1 X

e−βfk+β(ϵ i −ϵ i−k ) .

(5.9)

k=0

For Gaussian-distributed energies the exponential e ϵ i is log-normal distributed with

 2 mean e σ /2 . Since, furthermore, energies are uncorrelated, ϵ i ϵ j = σ2 δ ij , for k > 0 we have E D 2 1 (5.10) e−βϵ i−k = e 2 (βσ) , leading to

" # N−1  βϵ i  i−1 X 2 X 1 e (βσ) −βfk hτ N i = 1 + e2 e ω i+1,i i=1

=

N−1  X i=1

k=1

βϵ i

e ω i+1,i



1+e

2 1 2 (βσ)

z − zi 1−z



(5.11)

(geometric series with z = e−βf < 1). We can split the average because the first term only involves i and i + 1 and the second term involves sites < i. The first term becomes  βϵ  1 D βϵ i +β/(4λ)(ϵ i+1 −ϵ i +λ′ )2 E e i = I(f ) = e , (5.12) ω i+1,i ω0 p where ω0 = 2πJ 2 /~ 4πλkB T is the prefactor of the rates equation (9.1) and λ′ = λ − f is the shifted reorganization energy. It is convenient to transform site energies as ϵ i = ϵ¯ − The brackets above become I(f ) =

1 δ, 2

ϵ i+1 = ϵ¯ +

1 δ. 2

 1 2 1 δ dϵ¯ dδ exp − 2 ϵ¯ 2 − σ 4σ2  β 1 + β ϵ¯ − βδ + (δ + λ′ )2 2 4λ

1 2πσ2 w0

(5.13)

Z

(5.14)

since integrals over other sites reduce to unity and the variable change has unity Jacobian. The integral over ϵ¯ can be evaluated straightforwardly, leading to  Z 2 1 1 1 1 dδ exp − 2 δ2 − βδ I(f ) = √ e 4 (βσ) 2 4σ 2 πσω0  β + (δ + λ′ )2 . (5.15) 4λ Evaluating the second integral, we have  σ˜ 1 1 f2 ˜ 2 2 I(f ) = exp (βσ)2 + (β σ) w0 σ 4 2 λ  2  1 f + βλ 1 − 4 λ

(5.16)

88 | 5 Energetics and Dispersive Transport with

β 1 1 = − , σ˜ 2 σ2 λ

σ˜ 2 =

λσ2 σ2 = . λ − βσ2 1 − βσ2 /λ

(5.17)

Note that this result implies, in principle, an upper bound (βσ)2 < βλ for the disorder σ beyond which the mean waiting time diverges and, consequently, the approximation in Equation (5.6) breaks down. The mean waiting time finally reads hτ N i = I(f )

 N−1  X 2 z − zi 1 1 + e 2 (βσ) , 1−z

(5.18)

i=1

which constitutes our first central result.

5.2.2 Mobility We first consider the limit z → 1 corresponding to a vanishing field, F → 0, before evaluating the sum. We thus obtain hτ N i = I(0)

N−1 h X

2

1

1 + e 2 (βσ) (i − 1)

i=1

i

  2 N 1 = I(0)(N − 1) 1 + e 2 (βσ) ( − 1) 2

(5.19)

z − zi = i − 1. 1−z

(5.20)

using lim z→1

For large N,

N (N − 1) (5.21) 2 and the velocity decays as v N (0) ∼ 1/N. On the other hand, evaluating the sum first, we obtain   2 (N − 1)z(1 − z) − (z − z N ) 1 , (5.22) hτ N i = I(f ) (N − 1) + e 2 (βσ) (1 − z)2 1

2

hτ N i ≈ I(0)e 2 (βσ)

which reduces to the same result as Equation (5.21) in the limit z → 1 applying L’Hôpital’s rule twice. For a finite field, by putting Equation (5.22) into Equation (5.6) and taking the limit N → ∞, we get  2 2 1 1 z  z 1 = I(f ) 1 + e 2 (βσ) ≈ Ie 2 (βσ) . (5.23) v∞ 1−z 1−z With a few simplifications for sufficiently large N, we can thus write   1 1 1 1 ≈ 1− . v N v∞ N 1−z

(5.24)

5.2 Scaling Relation | 89

Table 5.1. Extrapolated mobilities for the thermodynamic limit μ∞ from the temperature extrapolation method and the system-size extrapolation method (in m2 V−1 s−1 ). T extrapolation 200 K 300 K 600 K

−13

3.2 × 10 4.9 × 10−10 3.5 × 10−8

N extrapolation 2.2 × 10−13 2.8 × 10−10 3.3 × 10−8

To leading order this results in v N ≈ v∞



1 1 1+ N 1−z



> v∞ .

Specifically, for the zero-field mobility we obtain   βw0 σ ∂v∞ 1 3 2 μ∞ = exp − (βσ) − βλ , = ∂f 4 4 σ˜

(5.25)

(5.26)

f =0

which is the same result that Seki and Tachiya [363] obtained in their Equation (6.7) [see Equation (5.1)]. Going beyond their result and including the leading-order correction, we find for the mobility of the one-dimensional chain of N sites  c μ N = μ∞ 1 + (5.27) N

with c = 32 . For large N this result can be further simplified to ln μN ≈ ln μ∞ +

c N.

5.2.3 Numerical test Like for the temperature-based extrapolation, we now assume that Equation (5.27) also holds in three dimensions, but with a different constant c. The dependence on the number of sites is replaced by a dependence on the box length, L = N 1/3 , in three dimensions. To test this assumption and to extrapolate the mobility value, we performed kinetic Monte Carlo simulations in cubic lattices from 8 × 8 × 8 to 50 × 50 × 50 sites, with Gaussian distributed energies and Marcus rates as described in Section 5.1. The master equation for occupation probabilities is solved using the variable-step-size Monte Carlo algorithm [237, 238, 247]. The charge mobility, μ = d/τF, is evaluated using the charge trajectory, where d is the distance traveled by the charge along the field F during time τ. Results are shown in Figure 5.1(a). One can see that the mobility (and its logarithm) scales indeed linearly with the inverse box length. The extrapolated mobilities, μ∞ = μ(N → ∞), also agree well with the temperaturebased extrapolation, which is shown in Figure 5.1(b). Both methods are compared in more detail in Table 5.1.

90 | 5 Energetics and Dispersive Transport 0 -0.05

n = 10−1 n = 10−3 n = 10−5 n=0

carrier energy (eV)

-0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -σ 2 /kB T 100 101 102 103 104 105 106 107 108 109 box size N Fig. 5.2. Convergence of the carrier energy with system size N for different carrier densities n. Symbols are the results of random-number experiments, Equation (5.28), while solid lines are the predictions of our analytic model, Equation (5.35). The dashed lines are the exact values for ϵc as N → ∞ obtained from Equation (5.5).

5.3 Finite Carrier Density We now turn to the estimation of the error introduced by finite-size effects in systems with finite charge carrier density. A generalization of the approach of Section 5.2 to multiple interacting carriers is not straightforward since we are now faced with an exclusion process in the presence of disorder, for which an analytical result for the mean first-passage time is not available. Instead, we use the average energy per carrier ϵc (N) as a figure of merit. This also makes the error estimate independent of the rate expression and the positional order of sites. Hence, it is also applicable to realistic morphologies and models with different charge transfer rates. We first perform a direct evaluation of ϵc (N) by drawing N random energies ϵ i from a Gaussian distribution of width σ and calculating the ensemble average as PN ϵ i p(ϵ i ) . ϵc (N) = Pi=1 N i=1 p(ϵ i )

(5.28)

Results for σ = 0.1 eV and T = 300 K are shown in Figure 5.2 (symbols) for different charge densities. This demonstrates that in finite systems there is a significant deviation from ϵ c , especially at low charge carrier densities. In order to obtain a closed-form expression for the finite-size error, we first note that the probability to draw an energy smaller than ϵ0 from a Gaussian distribution

5.3 Finite Carrier Density | 91

Table 5.2. Necessary system size (order of magnitude) for different values of energetic disorder σ and charge carrier density n to ensure that the relative error on the energy per charge carrier, δϵc /ϵc , is smaller than 5% and 0.1%, assuming a temperature of 300 K. Errors are calculated using the difference of the analytic estimate, Equation (5.35), to the exact value in an infinite system, Equations (5.5).

n 0 10−7 10−6 10−5 10−4 10−3 10−2 10−1

σ=

δϵc / |ϵc | ≤ 5% 0.001 eV 0.01 eV 0.1 eV

δϵc / |ϵc | ≤ 0.1% 0.001 eV 0.01 eV 0.1 eV

103 103 103 103 103 103 103 103

105 105 105 105 105 105 105 105

103 103 103 103 103 103 103 103

107 107 106 106 105 104 103 103

f (x) reads P (ϵ ≤ ϵ0 ) =

Zϵ0

f (x)dx = F(ϵ0 ),

105 105 105 105 105 105 105 105

> 1010 1010 109 108 107 106 105 103

(5.29)

−∞

where F(x) is the cumulative distribution function   1 1 x F(x) = + erf √ . 2 2 2σ2

(5.30)

The probability to draw an energy larger than ϵ0 is then given by P (ϵ > ϵ0 ) = 1− F(ϵ0 ). If we draw N independent energies, the probability that none of them will be smaller than ϵ0 reads  N P (ϵ i > ϵ0 , i = 1 . . . N ) = 1 − F(ϵ0 ) . (5.31) The probability to find one value ϵ ≤ ϵ0 is then given by

P (∃ i : ϵ i ≤ ϵ 0 ) = 1 − P ( ϵ i > ϵ 0 , i = 1 . . . N )  N = 1 − 1 − F(ϵ0 ) ,

(5.32)

which is the cumulative distribution function for ϵ0 . The respective probability distribution for the minimum sampled value (MSV) function is obtained by differentiation and reads fMSV (x) = −N [F (x)]N−1 f (x). (5.33) We now assume that in a sample of finite size the site energy distribution is given by a truncated Gaussian distribution. The lower cutoff ϵmin is the expectation value for the minimum energy, obtained when drawing N energies ϵmin =

Z∞

−∞

x fMSV (x) dx.

(5.34)

92 | 5 Energetics and Dispersive Transport 4.5

occupation probability

4 3.5 3

ǫF (ρ)

2.5 2 DOS

1.5 1 0.5 0

ODOS -0.3 -0.2 -0.1 ǫmin (N )

0

0.1

energy ǫ (eV)

0.2

0.3

ǫmax (N )

Fig. 5.3. The occupational density of states (ODOS) is the product of the Gaussian density of states (DOS) and the Fermi–Dirac distribution (red). For finite systems the DOS is approximated by a truncated Gaussian distribution. The light blue part is the missing contribution to the ODOS in a finite system, leading to increased energy values. An increase of the Fermi energy (larger density) or a decrease of ϵmin (N) (larger box size) reduces this part and thus the finite-size error.

The expectation value for the maximum sampled energy is given by ϵmax = −ϵmin . With this model distribution function we obtain an estimate for the size-dependent average energy R ϵmax

ϵ p(ϵ)f (ϵ)dϵ ϵc (N) ' Rϵmin , ϵmax p(ϵ)f (ϵ)dϵ ϵmin

(5.35)

which constitutes our second central result. This estimate is also shown in Figure 5.2 (solid lines) and is in good agreement with the values simulated directly. Figure 5.3 illustrates how the cutoff leads to a finite-size error. Hence, given the error, δϵc (n, σ, N) = ϵc (n, σ, N) − ϵc (n, σ, N → ∞) , we can estimate the necessary system size N for our simulations. Such estimates are shown in Table 5.2. As we have anticipated, large energetic disorder requires large system sizes, while for large charge densities one can use smaller systems. As of today, atomistically resolved simulations can handle systems of approximately 5000 molecules. With coarse-grained models one can increase this number to about 106 molecules [247]. In lattice models system sizes are up to 3 × 106 sites [227]. Comparing this to Table 5.2 shows that for low carrier densities and for large values of energetic disorder, the necessary system size is computationally still inaccessible, be it a microscopic, stochastic, or lattice model, and the extrapolation schemes from the previous section have to be used. For a sufficiently large charge carrier density

5.4 Conclusions | 93

or small energetic disorder, however, the error is within an acceptable range even for simulations in smaller systems.

5.4 Conclusions To conclude, we have derived a system-size dependence of charge carrier mobility and provided a simple way of correcting for finite-size effects in computer simulations of charge transport in disordered organic semiconductors. We have also estimated the system sizes required for simulating charge transport of carriers with a given energetic disorder in the density of states. Our results are general and are applicable to different rate expressions as well as off-lattice morphologies.

Acknowledgements This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) under the Priority Program “Elementary Processes of Organic Photovoltaics” (SPP 1355) and the program IRTG 1404, by BMBF grants MESOMERIE (FKZ 13N10723) and MEDOS (FKZ 03EK3503B) and by the European Research Council (ERC) NMP-20-2014 – “ Widening materials models ” program under Grant Agreement No. 646259 (MOSTOPHOS). We are grateful to J. Wehner and C. Scherer for a critical reading of the manuscript.

6 Correlated Energetic Landscapes Reprinted with permission from: Pascal Kordt, Denis Andrienko Modeling of Spatially Correlated Energetic Disorder in Organic Semiconductors Journal of Chemical Theory and Computation 12, 36–40 (2016) Copyright 2016 American Chemical Society. DOI: 10.1021/acs.jctc.5b00764

Mesoscale modeling of organic semiconductors relies on solving an appropriately parametrized master equation. Essential ingredients of the parametrization are site energies, or driving forces, which enter the charge transfer rate between pairs of neighboring molecules. Site energies are often Gaussian-distributed and are spatially correlated. Here we propose an algorithm which generates these energies with a given Gaussian distribution and spatial correlation function. The method is tested on an amorphous organic semiconductor, DPBIC, illustrating that the accurate description of correlations is essential for the quantitative modeling of charge transport in amorphous mesophases.

Introduction Early models of charge transport in organic semiconductors, such as the Gaussian disorder model (GDM), employed simple lattices in order to describe material morphology, postulated a Gaussian distribution of uncorrelated site energies driving charge transfer reactions, and used the thermally-activated tunneling (Miller– Abrahams) [205] rate expression to compute charge transfer rates [22]. Material properties were thus represented by a small number of essential parameters: the width of the density of states, the lattice spacing, and the hopping attempt frequency. In spite of its simplicity, the GDM could already explain the non-trivial dependencies of the charge carrier mobility on the external electric field, temperature, and the charge carrier density [22, 141, 260, 378]. The GDM has been gradually refined in order to improve the agreement with experiment. First, it has been shown that the long-range electrostatic interactions of a charge with molecular dipoles [140, 253] as well as a local alignment of conjugated segments in polymeric systems [379, 380] can lead to spatial correlations of site energies. By accounting for these correlations, the (extended) correlated Gaussian disorder models (ECDM, CDM) [140, 141], could explain the experimentally observed Poole– √ Frenkel dependence, ln μ ∝ F, also for small external fields [140, 251]. Second, a more direct link to the underlying chemical composition of the material could be established by combining quantum-chemical and classical force-field-based methods,

6.1 Atomistic Simulations | 95

which have been used to predict material morphologies and evaluate charge transfer rates [368, 369, 370, 381, 382]. The atomic-scale modeling also stimulated the development of off-lattice stochastic models [337, 247]. In these models, molecular positions and rates are generated according to the distribution functions evaluated in a small atomistic morphology. Stochastic models allow to simulate charge transport in micrometer-thick layers while at the same time retaining the link to the chemical composition. Both stochastic and atomistic models have helped to pinpoint drawbacks of GDM and CDM. In particular, the CDM assumes a charge interacting with randomly oriented, constant magnitude dipoles of surrounding molecules [253, 254], which leads to an energy correlation function of the form c(d) ∼ 1/d. The underlying assumption is, however, not well justified: Figure 6.1 shows that the distribution of magnitudes of molecular dipole moments in an amorphous mesophase can be rather broad. A similar situation has been observed in other systems, e.g., typical dye molecules used as donors in organic solar cells [244]. As a consequence, the spatial correlations of site energies can have decays of the correlation function different from 1/d dependence [244, 369]. For accurate parametrizations of stochastic models it is therefore desirable to have an algorithm which can generate site energies with a predefined correlation function. The goal of this work is to devise such an algorithm and to test it on a typical amorphous semiconductor, DPBIC, the chemical structure of which is shown in the inset of Figure 6.1.

6.1 Atomistic Simulations To evaluate the reference spatial correlation function, we first use atomistic molecular dynamics to generate an amorphous morphology of 4000 DPBIC molecules. We then employ quantum-chemical calculations and polarizable force-fields to evaluate electronic couplings, site and reorganization energies. To validate the algorithm and to illustrate the vital role of correlations, we finally study charge dynamics by solving the master equation with the help of the kinetic Monte Carlo algorithm. Charge hopping rates entering the master equation are evaluated within the high-temperature limit of the non-adiabatic charge transfer theory. For morphology simulations we use an OPLS-based [384, 385] force-field with missing bonded interactions parametrized using the potential energy surface scans obtained using the density functional theory (B3LYP functional and 6-311g(d,p) basis set), as described elsewhere [247]. Partial charges are fitted using the Merz–Kollman scheme [386, 387]. To obtain the amorphous morphology, the box of 4000 molecules is first simulated at 700 K, which is above the glass transition temperature, and then quenched to 300 K. The size of the final box is about 16 nm. All calculations are performed in the NPT ensemble with the Berendsen barostat and thermostat [330].

96 | 6 Correlated Energetic Landscapes 12 10

number of molecules

Ir N

3

8

N

6 4 2 0 0

2

4

6

8

10

12

14

16

dipole moment (Debye) Fig. 6.1. Distribution of molecular dipoles in an atomistic morphology of 4000 DPBIC molecules. The inset shows the chemical structure of DPBIC (Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-ylidene)1,2-phenylene]Ir), which is used as a hole-conducting and an electron-blocking layer in modern OLEDs [383, 250].

2.5

relative frequency

2

1.5

1

0.5

0 4.6

4.8

5

5.2

5.4

5.6

5.8

6

energy (eV) Fig. 6.2. Site energy distribution evaluated in the atomistic morphology and the fit to a Gaussian distribution with σ = 0.176 eV.

6.1 Atomistic Simulations | 97

atomistic generated ECDM, a=1.06 nm a=0.44 nm

spatial corelation, c(d)

0.6

0.4

0.2

0 0

1

2

3

4

5

6

7

8

center-of-mass distance, d (nm) Fig. 6.3. Spatial site energy correlation evaluated in an atomistic morphology (symbols) and its reconstruction using the numerical scheme described in the text (dashed line). Thin lines show correlation functions of the ECDM model for two different lattice spacings.

The hole site energies are evaluated using a perturbative approach, in which electrostatic and induction energies are added to the gas-phase ionization potential of a A A A A molecule A, i.e., E A = Eint + Eel + Epol , where Eint is the ionization potential in vacuum, A A Eel is the electrostatic interaction energy of partial charges, and Epol is the polarization contribution. Since rates depend only on site energy differences [205, 207, 208], the ionization potential does not affect rates or the spatial correlation function of a A mono-component system. The electrostatic contribution to the site energies, Eel , is calculated using the Ewald summation technique adapted for charged systems [242, 388]. All induction effects are incorporated in this scheme using the Thole model [336, 389] with a cutoff of 3 nm. Details of this approach are described elsewhere [390]. The distribution of site energies, shown in Figure 6.2, is, to a good accuracy, Gaussian with a standard deviation of σ = 0.176 eV. The spatial correlation function, c(d), which describes the correlation of site energies of molecules at a (center-of-mass) separation d is calculated as h i c(d) = E E A E B /σ2 , (6.1)

where E A and E B are energies of molecules A and B separated by distance d and E[·] is the expectation value. The spatial correlation cref (d), evaluated in the atomistic system, is shown in Figure 6.3 (symbols). Since we are dealing with amorphous materials, we assume thermally-activated type of transport and use the semi-classical charge transfer rate expression [207, 208]

98 | 6 Correlated Energetic Landscapes

ω AB =

 

J 2AB

2π  p exp − ~ 4πλkB T

∆E AB − λ 4λkB T

2 

 .

(6.2)

Here the reorganization energy, λ, reflects the effect of molecular rearrangement in response to the change of the charge state. J AB describes the strength of the electronic coupling between two localized (diabatic) states, and ∆E AB = E A − E B is the aforementioned driving force. For DPBIC the calculated (B3LYP, 6-311g(d,p)) internal reorganization energy for hole transport is λ = 0.13 eV. Electronic coupling elements, J AB , between the initial and the final states of a molecular dimer are evaluated by approximating its diabatic sates with the highest occupied molecular orbitals of monomers (frozen core approximation) and using a projection method [391, 392, 393]. To model charge dynamics, we solve the corresponding master equation using the kinetic Monte Carlo (KMC) algorithm [233, 234]. The mobility is calculated from the KMC trajectory as [247] D E N ~v i · ~F X 1 , (6.3) μ= N F2 i=1

where ~v i is the velocity of the i-th carrier, h· · · i denotes the average over the simulation time, and the sum is performed over N carriers.

6.2 The Algorithm Given the spatial correlation function, cref (d), and the variance, σ2 , of the Gaussian site energy distribution in the atomistic system, our goal is to generate site energies for a large stochastic morphology, with the same distribution and spatial correlation. The idea is to add to a site energy of a molecule A appropriately weighted energies of neighboring sites. To do this, we first assign N + 1 independent random numbers, X jA (0 ≤ j ≤ N), to every molecule (site) A. These random numbers are Gaussian-distributed with mean 0 and variance 1, X jA ∼ N(0, 1). A nonzero mean value of m can be achieved by replacing E A by E A + m. We now express the site energy of a molecule A as s N √ A X bj X A E = aX0 + (6.4) X jB . A j=1

`j

B∈S(r j ,A)

Here S(r j , A) is a sphere of radius r j centered at a molecule A, the sum runs over the sites B 6= A located within this sphere, `Aj = 34 πr3j ρ is the number of sites (molecules) in this sphere, and ρ is the site number density (molecules per volume), which we

6.2 The Algorithm | 99

Fig. 6.4. Mixing-in of site energies: the site energy correlation of two molecules A and B is determined by the (weighted) number of molecules, `AB j (red), within the overlap of spheres of radii r j centered at A and B.

assume to be constant. This is shown schematically in Figure 6.4. The first term in Equation (6.4) is used to adjust the width of the final distribution (σ), while the second term, which mixes in the energies of the neighbors, is used to adjust the decay of the correlation function. The choice of energies in the form provided by Equation (6.4) allows to obtain a simple analytic expression for the correlation function and the variance of the resulting Gaussian distribution. Following the definition of site energies, Equation (6.4), we can rewrite the expectation value as s N X N h i X X bi bj X A B δ CD δ ij E E E = A B `i `i

i=1 j=1

=

N X i=1

=

N X i=1

C∈S(r i ,A) D∈S(r j ,B)

X

bi

q

B `A i `i

`AB

bi q i

1

C∈S(r i ,A)∩S(r i ,B)

B `A i `i

=

  N X d ξ bi . 2r i

(6.5)

i=1

Here we used the fact that the random numbers X iA are independent, hence E[X iA X jB ] = δ ij δ AB . `AB = ρV AB is the number of molecules in the intersection of two spheres i at a separation d, S(r i , A) ∩ S(r i , B), V AB is the volume of this intersection, V AB = 2 1 12 π (4r i + d ) (2r i − d ) for d ≤ 2r i and zero otherwise (see Figure 6.4). The standard deviation of X A can be evaluated in a similar fashion, yielding 2

σ =E



E

A

2 

N h i2 X − E EA = a + bi . i=1

(6.6)

100 | 6 Correlated Energetic Landscapes Inserting these expressions into the definition of the correlation function, Equation (6.1), we obtain the spatial correlation function for molecules A and B at separation d c(d) =

  N 1 X d ξ bj , 2r j σ2

(6.7)

j=1

where ξ (x) is a finite-support function, ( 1 − 23 x + 12 x3 , ξ (x) = 0,

x≤1 x > 1,

(6.8)

and σ2 is the variance of site energies, σ2 = a +

N X

bj .

(6.9)

j=1

An intuitive geometric interpretation of Equation (6.7) is that ξ (d/2r) provides a (normalized) overlap of two spheres of radii r, with their centers separated by a distance d. This overlap determines the spatial correlation function, since sites belonging to the overlap contribute equally to site energies of two molecules located in the centers of the spheres (see Figure 6.4). = cref (d i ), i = By providing the reference correlation function in N points, cref i 1, 2, . . . , N, we obtain N linear equations cref i =

N 1 X ξ ij b j , σ2

(6.10)

j=1

where ξ ij = ξ (d i /2r j ). The solution of Equation (6.10) provides the weighting coefficients b j and thus a unique interpolation of the reference correlation function in terms of finite support functions ξ , i.e, piecewise-defined cubic polynomials. Once the coefficients b i are known, a can be determined from Equation (6.9). In practice, however, p some coefficients b i can be negative, leading to imaginary values of b i in Equation (6.4) and thus unphysical energies. In this case a recursive scheme, which provides an approximate solution to Equation (6.10), becomes more practical. To devise the recursive scheme, we note that there is still a certain degree of flexibility in choosing the grid points d i and r j . We now choose the second grid such that r i = d i+1 /2, where d i are the points in which the reference function is evaluated, i = 1 . . . N − 1, and r N = (d N + ∆)/2, with ∆ = d N − d N−1 . Note that the generated correlation function will be zero in the last point, c(2r N ) = 0. Given this choice of grid points, ξ ij = 0 for i < j, i.e., ξ ij becomes a triangular matrix. Equation (6.10) then simplifies to σ2 cref i = b i ξ ii +

N X

j=i+1

ξ ij b j ,

(6.11)

6.3 Validation |

101

and can now be solved for b i recursively, starting from b N . The final recursive algorithm reads 2 1. Evaluate b N = ξσNN cref N . 2. Starting from N − 1 evaluate recursively for N − 1 ≥ i ≥ 1   N 1  2 ref X bi = σ ci − ξ ij b j  (6.12) ξ ii j=i+1

If b i becomes negative, rescale b i+1 by a factor 0 < η < 1 and recalculate b i . If b i+1 < δ, set it to zero. 3. Evaluate a according to Equation (6.9). 4. For M sites and N values of the reference function cref i generate M× ( N + 1) random A variables X i ∼ N(0, 1), 1 ≤ A ≤ M, 0 ≤ i ≤ N. 5. Evaluate site energies of all molecules according to Equation (6.4). Note that the second step includes an ad hoc way of enforcing all b i coefficients to be positive.

6.3 Validation We now apply the developed scheme and study hole transport in an amorphous morphology of DPBIC. The reference spatial correlation function has been evaluated in the atomistic system of 4000 molecules as described in sec. 6.1. To improve the numerical stability of the algorithm, we first smoothen the reference data which is evaluated in N = 24 points and fit a stretched exponential, α exp(−βd γ ), to the atomistic correlation (α = 2.83, β = −1.73, γ = 0.56). The fitted function is then used as an input for the algorithm. Figure 6.3 shows an excellent agreement between correlations functions evaluated in the atomistic reference and in the generated stochastic system of 40000 sites. We have used the value of δ = 10−5 and the scaling factor η in the range between 0.8 and 0.99, which yields sufficient accuracy and fast convergence of the recursive algorithm. For comparison we also show the correlation function of the ECDM, cECDM (d) = 0.74a/d, for two different values of the lattice constant a. The lattice constant of 1.06 nm corresponds to the average intermolecular distance of the nearest neighbors and leads to a significant overestimation of the correlation. a = 0.44 nm is obtained by fitting the charge mobility of an atomistic reference to the analytical expression provided by the ECDM [247]. The rationale behind a much smaller fitted lattice constant is now apparent: the only adjustable parameter of the ECDM which enters the correlation function is the lattice spacing a. Since the field dependence of the charge carrier mobility is very sensitive to spatial correlations of site energies, the ECDM tries to provide the best approximation to the atomistic correlation function – by reducing the lattice

102 | 6 Correlated Energetic Landscapes -9.5

log(µ/m2 V−1 s−1 )

-10

-10.5

-11

-11.5

uncorrelated generated from ECDM generated from atomistic

-12 2000

4000

p

6000

8000

10000

F/ V m−1

Fig. 6.5. Poole–Frenkel plot for charge transport simulated in a box with identical positions and distributions of site energies but different spatial correlations. A lattice constant of 1.06 nm has been used in the ECDM.

spacing to an unphysical value. equation Finally, we shall ask ourselves whether an accurate reproduction of the spatial correlations is important. To answer this question we have performed charge transport simulations in systems of 40000 sites, with the positions of molecules obtained using the iterative Boltzmann inversion method, which reproduces the radial distribution function of the atomistic reference [26, 247]. To this end, we have simulated charge transport in three systems, all with Gaussiandistributed (σ = 0.176 eV) site energies. The site energies were (i) uncorrelated (ii) correlated with the correlation function of the atomistic reference (iii) correlated with the ECDM correlation function and lattice constant a = 1.06 nm. Since the EGDM site energy correlation is very long-ranged, the (non-uniform) grid in the algorithm was extended to 15 nm. Charge transport simulations were performed at a concentration of 10−4 carriers per site (four charges), which leads to finite size effects smaller than 0.5 % [394]. The Poole–Frenkel plots, i.e., the logarithm of the mobility versus the square root of the external field, are shown in Figure 8.2 for all three cases. It can be seen that wrong spatial correlations result in ca. two orders of magnitude difference in the mobility values. Furthermore, the slopes also differ, which can be rationalized in terms of the “effective” energetic disorder, i.e., the energy variation of those sites which are most frequently occupied by charge carriers [395]. We conclude that the spatial correlation of site energies should be reproduced as accurate as possible in order to achieve quantitative modeling of transport in amorphous organic materials.

6.3 Validation |

103

Discussion and Conclusions To summarize, the proposed algorithm allows to generate charge transport networks with millions of molecules, which helps to reduce finite-size effects and to study systems of small (but physically relevant) charge densities. One can, therefore, simulate current-voltage characteristics of realistic devices by first tabulating mobility values as a function of temperature, charge density and external electric field and then solving continuous drift–diffusion equations. This approach yields results which are in an excellent agreement with experimental measurements [396]. It is also possible to extend the scheme to two or more molecule types. In this case one has to take into account not only the autocorrelation function but also the correlation between two different types of molecules and add an additional index for the molecule type to the b coefficients and random numbers. The described algorithm also improves the moving-average scheme proposed earlier [337]. It can reproduce arbitrary correlation functions, is easy to automatize, and does not need iterative generation of site energies for trial correlation functions. To conclude, we have developed a method which can generate Gaussian-distributed energies with a predefined spatial correlation function. We have illustrated that the correlated and extended correlated Gaussian disorder models are unable to reproduce spatial correlations of atomistic systems, or use effective (and unphysical) values of the lattice constant to do this. The developed method can be used to either refine the family of Gaussian disorder models or to construct accurate stochastic off-lattice models.

Acknowledgements This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) under the Priority Program “Elementary Processes of Organic Photovoltaics” (SPP 1355), BMBF grant MESOMERIE (FKZ 13N10723) and MEDOS (FKZ 03EK3503B), and DFG program IRTG 1404. The project has received funding from the NMP-20-2014 – “Widening materials models” program under grant agreement No. 646259 (MOSTOPHOS). We are grateful to Jens Wehner, Anton Melnyk, Carl Poelking, Christoph Scherer, and Tristan Bereau for a critical reading of the manuscript.

7 Microscopic, Stochastic and Device Simulations Reproduced with permission from: Pascal Kordt, Jeroen J. M. van der Holst, Mustapha Al Helwi, Wolfgang Kowalsky, Falk May, Alexander Badinski, Christian Lennartz, Denis Andrienko Modeling of organic light emitting diodes: from molecular to device properties Advanced Functional Materials 25, 1955–1971 (2015). DOI: 10.1002/adfm.201403004

We review the progress in modeling of charge transport in disordered organic semiconductors on various length-scales, from atomistic to macroscopic. This includes evaluation of charge transfer rates from first principles, parametrization of coarse-grained lattice and off-lattice models, and solving the master and drift-diffusion equations. Special attention is paid to linking the length-scales and improving the efficiency of the methods. All techniques are illustrated on an amorphous organic semiconductor, DPBIC, a hole conductor and electron blocker used in state of the art organic light emitting diodes (OLEDs). The outlined multiscale scheme can be used to predict OLED properties without fitting parameters, starting from chemical structures of compounds.

Introduction The discovery of electrical conductivity in conjugated polymers [397] has marked the birth of the field of organic semiconductors, providing a new class of materials suitable for manufacturing of field effect transistors [398], light emitting diodes [399], and photovoltaic cells [7]. Cost-efficiency, enhanced processability, and chemical tunability have immediately been recognized as distinct properties of organic semiconductors [400, 401, 402, 403]. Since then, organic light emitting diodes (OLEDs) have been incorporated into active-matrix displays and nowadays are produced on an industrial scale [404, 399]. At the same time, white OLEDs are considered as promising candidates for lighting applications [404, 405]. In spite of this progress, there is still a need to improve lifetimes and efficiencies, especially of blue phosphorescent OLEDs, a task which requires designing new materials and optimizing device architectures. This is, of course, impossible without understanding elementary processes occurring in a device. In a phosphorescent OLED, schematically shown in Figure 7.1, electrons and holes are injected into the transport layers, which ensures their balanced delivery to the emission layer (EML). To allow for triplet-harvesting mediated by spin-orbit coupling, the EML consists of an organic semiconductor (host) doped by a transition-metal coordinated compound, the emitter (guest). The excitation of the emitter can be achieved either by an energy transfer pro-

105

7 Microscopic, Stochastic and Device Simulations |

DPBIC

BTDF

1.98

Si

Si

Ir N

3 O

N

TBFMI

BCP N

N

O Ir N

3 N

Fig. 7.1. Multilayered structure of an OLED with corresponding energy levels and chemical structures. Electrons/holes move upwards/downwards in energy due to an applied voltage. Higher electron/hole energy levels correspond to smaller electron affinities/ionization potentials. In this OLED, DPBIC (Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-phenylene]Ir) is used in the hole-conducting layer; TBFMI (Tris[(1,2-dibenzofurane-4-ylene)(3-methyl-1/1-imidazole-1-yl-2(3/1)ylidene)]Ir(III)) is the emitter (guest); BTDF (2,8-bis(triphenylsilyl)dibenzofurane) is the host material and 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline (bathocuproine, BCP) is the electron conductor.

cess, i.e., by the formation of an exciton on a host molecule with subsequent energy transfer to the dopant, or by a direct charge transfer process. In the latter case, one of the charge carriers is trapped on the emitter and attracts a charge of opposite sign, forming a neutral on-site exciton. Hence, an interplay between the hole, electron, and exciton mobilities, relative energy offsets, energetic disorder, and recombination rates determines the charge/exciton profiles in a device and therefore its current–voltage– luminescence characteristics. In principle, it is possible to reconstruct these profiles by using charge-transport and exciton-transport models and fitting the experimental current–voltage curves [113, 406, 407]. This approach, however, does not provide a link to the chemical composition or morphological order and hence does not allow to prescreen organic compounds prior to their synthesis. In this situation, computer simulations can help by linking device efficiency to the morphological order and molecular structures. A frequently used ansatz incorporates simulations of atomistic morphologies using classical force-fields, evaluation of charge/exciton transfer rates with first principles methods, and a subsequent analysis of the solution of the master equation [11, 408, 250, 368, 369, 395, 409, 410, 411, 412]. This microscopic model is, however, computationally demanding: a layer of a few hundred nanometers thickness requires simulation boxes of ∼ 107 atoms, even if periodic boundary conditions are used in the directions perpendicular to the applied field. To remedy the situation, extensions to this scheme have been proposed, as depicted in Figure 7.2. The most straightforward solution is to parametrize a lattice model by matching certain macroscopic properties of the atomistic and lattice models. For example, if charge carrier mobility is the property of interest, one can use the parametric dependencies provided by the family of Gaussian disorder models (GDM) [22,

106 | 7 Microscopic, Stochastic and Device Simulations

Lattice models Energetic disorder Lattice spacing

Drift-diffusion equations First principles

Microscopic models

Morphologies Site energies Electronic couplings Rates

s

~

le nm u 0 lec ~31 mo 10

Long-range Coulomb interactions Poisson equation

Device properties

Ground state geometries Distributed multipoles and polarizabilities Force-field parameters Reorganization energies Ionization potentials Electron affinities

Current-voltage characteristics Charge density distributions Current density distributions

Electrodes Injection barriers Mirror images Injection rates

Off-lattice models ~1 nm

Coarse-grained morphologies Models for site energies Models for electronic couplings

Kinetic Monte Carlo -

+

Forbidden events Binary tree search Aggregate Monte Carlo

les

m u 00 n lec ~ 1 106 mo 5 – ~10

Fig. 7.2. Possible workflows of parameter-free OLED simulations: polarizable force-fields and electronic properties of isolated molecules obtained from first principles are used to generate amorphous morphologies and evaluate charge transfer rates in small systems (microscopic models). Coarse-grained models are parametrized either by matching macroscopic observables, e.g., charge mobility, of the microscopic and coarse-grained (lattice) models. The resulting analytical expressions for mobility are then used to solve drift-diffusion equations for the entire device, after incorporating long-range electrostatic effects and electrodes. Alternatively, off-lattice models can be developed by matching distributions and correlations of site energies, electronic couplings, and positions of molecules. The master equations for this model can be solved using the kinetic Monte Carlo algorithm, yielding macroscopic characteristics of a device.

114, 140, 141, 413]. Note that a similar approach is often employed to interpret experimental data, i.e., to fit the measured mobility as a function of applied field, charge carrier density, and temperature. In our case simulated instead of experimentally measured mobility values are used. This strategy, however, has two significant drawbacks. First, it relies on approximations made to construct the underlying models, which may not hold for the system of interest. Second, the model parametrization is usually performed in stationary conditions. It is therefore not immediately obvious that the same model can be used to describe non-equilibrium (transient) properties of the system. An alternative approach is to construct an off-lattice model by matching mesoscopic system properties, such as distributions of molecular positions and orientations, electronic couplings, and site energies [244, 337]. Once parametrized, this

7.1 Microscopic Modeling |

107

model will not require explicit simulations of atomistic morphologies and quantumchemical evaluations of rates and can therefore be used to simulate large systems. In this review we illustrate both approaches on an amorphous phase of DPBIC, the chemical structure of which is shown in Figure 7.1, a compound used in holeconducting and electron-blocking layers in blue phosphorescent OLEDs [250, 383]. In particular, we show how material properties such as density, radial distribution functions, ionization potentials and electron affinities, energetic disorder, charge mobility and eventually current–voltage characteristics can be extracted from simulations. The paper is organized as follows: In Section 7.1 we recapitulate the methods required to compute charge transfer rates in (microscopic) morphologies. Section 7.2 deals with the construction and validation of coarse-grained models. Section 7.3 discusses how lattice models can be parametrized from an atomistic reference. In Section 7.4 we review efficient kinetic Monte Carlo algorithms needed to simulate charge transport in systems with large energetic disorder. We then present device simulations and a comparison to experiments in Section 7.5. Finally, in Section 7.7 we review computer simulation predicted material properties for a number of OLED materials, including BTDF, TBFMI, BCP, Alq3, and α-NPD.

7.1 Microscopic Modeling We will begin with the computationally most demanding approach, which is later used as a reference for constructing coarser charge transport models. For relatively small systems of circa 104 molecules it is possible to parametrize the master equation for charge/exciton transport by combining first-principles and classical force field based methods [368, 369, 408, 409]. First, for every molecule type, the potential energy surface, reorganization energy, ionization potential and electron affinity, as well as distributed multipoles and polarizabilities are evaluated using first-principles calculations. These are used to parametrize the classical (polarizable) force field and to simulate amorphous morphologies. The polarizable force field is also employed to evaluate the electrostatic and induction interactions of a localized charge with the environment in a perturbative fashion [414]. Finally, electronic coupling elements and rates are computed and used to formulate the master equation. The solution of the master equation yields charge carrier mobility and occupation probabilities in the system. This scheme has been applied to study transport in a number of organic semiconducting materials, such as tris(8-hydroxyquinoline)aluminium (Alq3) [261, 369, 415, 416], fullerene (C60) and its derivatives [417, 418], dicyanovinyl-substituted oligothiophenes [11, 395, 419], poly(3-hexylthiophene) (P3HT) [411], and poly(bithiophene-altthienothiophene) (PBTTT) [412], to name a few. In the next sections we briefly recapitulate the methods used to evaluate the ingredients of charge transfer rates. These will later be used to parametrize coarse-grained models of an organic semiconductor.

108 | 7 Microscopic, Stochastic and Device Simulations

(b)

N

energy, kJ/mol

3

force field DFT

150

10

Ir N

(c)

force field DFT

energy, kJ/mol

(a)

5

100

50

0

0 -20

-15

-10

-5 γ, deg

0

5

10

0

90

180

270

360

β, deg

Fig. 7.3. Examples of the potential energy scans of the dihedral angles γ (b) and β (c), shown in (a). DFT calculations (B3LYP/6-311g(d,p)) and the force field cross-sections (after adjusting its parameters) of the potential energy surfaces are shown in (b) and (c).

7.1.1 Morphology In general, predicting realistic molecular arrangements of organic semiconductors is a challenging task: organic materials can be highly crystalline, form partially ordered liquid crystalline, smectic, or lamellar mesophases, or be completely amorphous. Many are polymorphs and practically all have a fairly high density of defects, which has a large impact on their conductive properties [11, 395]. X-ray scattering, tunneling electron microscopy (TEM), and solid-state nuclear magnetic resonance (NMR) spectroscopy are common experimental techniques which are used to probe molecular ordering of organic materials [420, 421, 422]. Organic semiconductors employed in OLEDs are usually amorphous, which simplifies modeling of their morphologies. In order to simulate large-scale morphologies, atomistic and coarse-grained models are often employed. Both rely on force fields, i.e., parametrizations of the potential energy surface of inter-molecular (non-bonded) and intra-molecular (bonded) interactions. While for bio-molecules a number of force fields exist and can be used “out of the box”, this is normally not the case for organic semiconductors, which have extended π-conjugated systems. Such compounds require – at least partial – reparametrizations of existing force fields. For bonded interactions this can be performed by scanning the cross-sections of the potential energy surface (e.g., dihedral potentials) using first-principles methods [416, 423, 424], as illustrated for DPBIC in Figure 7.3, where two potential energy surfaces are shown, one of the bending of a benzine ring out of the ligand plane (γ) and the other one of its rotation out of this plane (β). The situation with non-bonded interactions is more complex: Coulomb and Lennard-Jones potentials can also be parametrized by using first-principles calculations [414]. This, however, requires distributed multipoles and polarizabilities, as well as many-body van der Waals interactions, making force-evaluation computationally demanding. As a compromise, most force fields account for the induction contribution

7.1 Microscopic Modeling |

109

by adjusting partial charges and van der Waals parameters. These parametrizations are, however, state-point dependent and should be validated against experimental data. With the force field at hand, an amorphous morphology of an OLED layer can be simulated by using Monte Carlo or molecular dynamics (MD) techniques [369, 408, 425, 426]. Several protocols were suggested for this purpose. One can, for example, anneal the system above the glass transition temperature, and then quench to room temperature using the NPT ensemble [395]. It is also possible to use Monte Carlo [415, 425] or molecular dynamics simulations to gradually deposit the material [427]. Due to large deposition rates in molecular dynamics, simulated morphologies can be very far from equilibrium, while the Monte Carlo technique does not control the kinetics of the deposition process. As an illustration, an amorphous morphology of DPBIC, simulated using the annealing protocol, is shown in Figure 9.1. Here, the Berendsen barostat and thermostat [330] were used and the system was first equilibrated at 700 K for 1 ns, then quenched to 300 K and equilibrated for another 1.3 ns The final density of the system was 1.29 g/cm3 , which corresponds to a number density of 8.50 × 1020 cm−3 .

7.1.2 Charge Transfer Charge transport in organic materials is intimately linked to the underlying material morphology: in crystals and at low temperatures transport is coherent, charges are delocalized, and consequently a band-like transport prevails [428, 429, 430]. At higher temperatures intramolecular and intermolecular vibrations destroy the coherence, charges become localized by the dynamic disorder and transport becomes diffusive [431, 432, 433, 434]. In disordered materials with weak electronic couplings between molecules, static disorder localizes charged states already at low temperatures and a thermally-activated hopping transport dominates [435]. Descriptions of transport in polymeric systems are the most challenging: along chains it can be diffusionlimited by dynamic disorder while between the chains it is in the hopping limit [436, 437]. Since we are dealing with amorphous materials, we assume thermally-activated type of transport (this can be verified experimentally by studying temperature- and field-dependencies of mobility). The simplest expression for a charge transfer rate which takes into account energetic landscape, polaronic effects, and electronic coupling elements, is the semi-classical Marcus rate [207, 208] "
2 # J 2ij ∆E ij − λ ij 2π p exp − ω ij = . (7.1) ~ 4λ ij kB T 4πλ ij kB T Here the reorganization energy, λ ij , reflects the effect of molecular rearrangement in response to the change of the charge state, the electronic coupling element, J ij , de-

110 | 7 Microscopic, Stochastic and Device Simulations scribes the strength of the electronic coupling between two localized (diabatic) states, and E ij = E i − E j is the driving force, where E i is the site energy of molecule i [204]. Note that the semiclassical Marcus rate is the high temperature limit of a more general, quantum-classical rate [438] and as such has a limited range of applicability. There exist also other rate expressions, which go beyond the harmonic approximation for the thermal bath [209, 210, 211] and have been reported to describe experimental data in a wide temperature and field range [212]. The distribution of rates for DPBIC, evaluated using the semi-classical rate expression, Equation (8.1), is shown in Figure 7.5(e). This distribution is very broad, with rates varying by several orders of magnitude, which is typical for amorphous systems. Large variations are a result of pronounced energetic disorder (see Section 7.1.3), as well as of a broad distribution of electronic couplings (see Section 7.1.5).

7.1.3 Site Energies Site energy differences, or driving forces, which enter the charge transfer reaction rate, Equation (8.1), can be evaluated perturbatively: first the ionization potential (IP, hole transfer) or electron affinity (EA, electron transfer) of an isolated molecule is calculated and then the interaction with the environment (electrostatic and polarization) is accounted for, pol el E i = Eint i + E i + E i + qF · r i .

(7.2)

Here F is the external electric field, r i is the center of mass of a molecule, and q is its charge. The IP and EA can be evaluated using first principles methods as the energy difference of a charged molecule (in its charged geometry) and a neutral molecule (in its neutral geometry), i.e., Eint = UcC − UnN , where the first (lowercase) index denotes the charge state and the second (uppercase) index denotes the geometry. The interaction with the environment is often computed on a classical level, using distributed multipoles [439] and polarizabilities [336, 389, 440]. For each molecule i the electrostatic interaction, Eel i , is evaluated for all molecules in a sphere of a certain radius (cutoff). Note that, even in amorphous materials, an ample cutoff is required to converge the electrostatic energy. For molecular assemblies with a long-range orientational or positional order it is often not feasible to achieve the convergence by increasing the cutoff radius [441] and the long-range Coulomb interaction energy should be evaluated using the Ewald summation technique [242, 388, 440, 441]. The induction interaction energy, Epol i , can be evaluated using the Thole model [336, 389]. For organic molecules the Thole parameters must be adjusted to reproduce the molecular polarizability tensor. Normally, the induction contribution converges for much smaller cutoffs than the electrostatic one, though special care should be taken

7.1 Microscopic Modeling | 111

(a)

(b)

(c) 0.6 5.4

5.4

5.3

5.3 5.2

5.2

autocorrelation, κ(r)

site energy, eV

5.5

0.4

4.7 0.2

5 5.3 5.6 site energy, eV

5.9

16 12

0

8

4 x , nm

8

4

12 16

0

y, nm

0 1

2

3

4

5

6

r , nm

Fig. 7.4. (a) Amorphous morphology of DPBIC with a highlighted one-molecular-layer thick crosssection for which the energetic landscape is shown in (b). (c) The spatial autocorrelation function of site energies, κ(r). The inset shows the distribution of site energies, which is approximately Gaussian with a mean of 5.28 eV and energetic disorder of 0.176 eV.

in order to not have artificially induced dipoles at the surface of the cutoff sphere (e.g., by choosing a larger electrostatic cutoff radius than the polarization one). In order to evaluate site energies (their differences drive charge transfer reactions) for an amorphous morphology of DPBIC, we have used an adapted version of the Ewald summation method for long-range electrostatic interactions [336, 388, 242, 440, 441] (i.e., the electrostatic contribution is calculated without a cutoff) and the Thole model with a cutoff of 3 nm for the induction interaction. Atomic charges, fitted to reproduce the molecular electrostatic potential (shown in Figure 9.1), were used for electrostatic calculations, while Thole parameters were adjusted to reproduce the molecular polarizability tensor of DPBIC. The energetic landscape, shown in Figure 7.4(b), is spatially correlated, which is due to a long-range dipolar electrostatic contribution [140, 254]. This can be seen qualitatively in the cross-section of the potential energy landscape, Figure 7.4(b), and quantified by the the autocorrelation function, κ(r), shown in Figure 7.4(c). The distribution of site energies is approximately Gaussian with a width of σ = 0.176 eV. Note that the Gaussian shape of the distribution is only expected if the change in the molecular polarizability upon charging is small [250], which is indeed the case for DPBIC: one third of the trace of the polarizability tensor is 108.5 3 for a neutral molecule and 127.1 3 for a cation (B3LYP/TZVP).

7.1.4 Reorganization Energy The reorganization energy, λ ij , quantifies the energetic response of a molecule due to its geometric rearrangement upon charging or discharging. It has two contributions: the internal (intramolecular) part, i.e., the energy difference due to the internal molecular degrees of freedom, and the outersphere (intermolecular) part, which is due to the relaxation of the environment [204]. The internal part is normally evaluated as the energy difference of the involved charged/neutral states in a charged/neutral geom-

112 | 7 Microscopic, Stochastic and Device Simulations etry using density functional calculations in vacuum [382]. The outersphere contribution can be computed using the dieletric response function [369], polarizable force fields [442], or hybdrid quantum mechanics/molecular mechanics methods [443]. For DPBIC the calculated (B3LYP, 6-311g(d,p)) internal reorganization energies for hole transport are 0.068 eV (discharging) and 0.067 eV (charging). Here we neglect the outersphere contribution, which becomes important in strongly polarizable environments.

7.1.5 Electronic Couplings Evaluation of electronic coupling elements, J ij , between the initial and the final states of a charge transfer complex (molecular dimer) requires the knowledge of diabatic states and the Hamiltonian of a dimer. Diabatic states are often approximated by the highest occupied molecular orbital (HOMO) of monomers in case of hole transport and the lowest unoccupied molecular orbital (LUMO) for electron transport [391, 392, 393] (“frozen core” approximation) or constructed using the constrained density functional approach [444]. In some cases the explicit evaluation of the dimer Hamiltonian can be avoided by employing semi-empirical methods [382, 445, 446, 447]. Electronic couplings are intimately connected to the overlap of electronic orbitals participating in a charge transfer reaction and are therefore very sensitive to relative molecular positions and orientations. Together with the energetic landscape, they constitute one of the main factors which influence charge transport in organic semiconductors [11, 395, 448, 449]. Depending on the first-principles method and approximations used, computed electronic coupling elements can easily vary by a factor of ten [450, 451]. For the amorphous morphology of DPBIC, the distribution of electronic couplings of molecular pairs at a certain distance is approximately Gaussian with a mean decaying exponentially for intermolecular separations larger than 1.1 nm, see Figure 7.5(d). Below this distance electronic couplings are constant (on average) and are of the order of 10−3 −10−2 eV. Such a behavior has also been observed for amorphous mesophases of Alq3 and DCV4T [244, 337].

7.2 Coarse-Grained Models The microscopic model outlined in Section 7.1, in combination with the charge dynamics simulations (see Section 7.4), provides a direct link between the molecular structure and macroscopic observables, e.g., charge mobility. It is, however, limited to relatively small systems of ca. 104 molecules, which is far from sizes required for studying, e.g., rare events during material degradation. Furthermore, one can have pronounced finite size effects, especially in materials with large energetic disorder [244, 261].

7.2 Coarse-Grained Models | 113

In view of this, various coarse-grained models have been developed [244, 452, 453]. The idea behind these models is to reproduce the key morphological and electronic properties of the underlying materials without explicit simulations of atomistic morphologies, evaluations of electronic couplings and site energies. The link to the chemical composition is retained by an appropriate parametrization of mesoscopicscale quantities, e.g., distributions and correlations of site energies, electronic couplings, and pair distribution functions of molecular positions, evaluated as described in Section 7.1. Coarse-grained models allow to study charge dynamics in systems of more than 106 molecules. In what follows we describe how such models can be parametrized.

7.2.1 Morphology The first step in constructing a coarse-grained model is the prediction of molecular positions. In the general case, the algorithm should reproduce all (including many-body) correlation functions of molecular positions in the reference morphology. For amorphous molecular arrangements, however, the pair distribution function is sufficient to adequately describe the molecular ordering. Thus, one has to develop an algorithm that generates point positions of a specified density and reproduces the radial distribution function, g(r), of the microscopic model. To do this, one can use a Poisson process [244, 337]. This process first positions N uniformly distributed points in a volume w, N ∼ Poi (ρw), where ρ is the point density. Unphysically close contacts are removed by assigning a random radius R n to every point n and by deleting the points whose sphere is contained in the sphere of another point m. The point density is then adjusted by introducing more points in the voids. This procedure is repeated iteratively until the desired density, ρ, is reached. The algorithm yields reasonably good approximations of g(r) [244, 337]. An exact match of the radial distribution function can be achieved by using techniques employed in soft condensed matter simulations, namely the iterative Boltzmann inversion (IBI) [345, 346] or inverse Monte Carlo [347, 348, 349] methods. Both rely on the Henderson theorem [350], which states that for every g(r) there is a unique (up to an additive constant) interaction potential, U(r), such that sampling the system in the canonical ensemble reproduces this g(r). Hence, it suffices to construct an algorithm capable of finding U(r) from a given g(r). In IBI, this is done by simulating a coarse-grained system with a potential which is iteratively refined at the i-th iteration as   i g (r) i+1 i . (7.3) U (r) = U (r) − kB T ln ref g (r) A potential that reproduces the reference radial distribution function is a fixed point of this relation. In practice, the second term is rescaled by a factor between 0 and 1 to make the algorithm numerically stable. Inverse Monte Carlo follows the same idea,

114 | 7 Microscopic, Stochastic and Device Simulations (a)

0.5

0.5 0.4 0.3 4.7

0.2

5 5.3 5.6 energy, eV

5.9

0.1 0

0 0

1

2

3

4

0 1.1

5

1.2

1.3

(d)

(e) atomistic coarse-grained

-3

1.4

1.5

1.6

1.7

1

2

r, nm

r, nm

10−6

atomistic coarse-grained

10−7

µ, m2 /Vs

distribution

-7

4

5

6

(f)

0.2

-5 -6

3 r, nm

-4

m(r), σ(r)

atomistic coarse-grained

0.6

correlation, κ(r)

1

0.7

atomistic coarse-grained

1

atomistic coarse-grained connection probability

radial distribution function, g(r)

(c)

(b) 2

0.1

-8

atomistic coarse-grained 9 × 107 V/m 7 × 107 V/m 5 × 107 V/m 3 × 107 V/m

10−8

10−9

-9 0

-10 0.8

1

1.2 r, nm

1.4

-5

0

5 log10 (ωij · s)

10

15

10−10

1024

1025 ρ, m−3

Fig. 7.5. Comparison of the atomistic (17 × 17 × 17 nm3 ) and coarse-grained (50 × 50 × 120 nm3 ) models. (a) Radial distribution function, g(r). (b) Probability of two sites to be connected (added to the neighbor list) as a function of their separation. (c) Spatial site energy autocorrelation function, κ(r); Inset: Site energy distribution. (d) Mean m and width σ of a distribution of the logarithm of electronic couplings, log10 (J 2 / eV2 ), for molecules at a fixed separation r. (e) Rate distributions. (f) Mobility as a function of hole density, plotted for four different electric fields.

except that the correction to the potential is rigorously derived (in a linear approximation) and contains cross-correlations of the number of particles at different separations r [347, 348, 349]. Both methods are implemented in the VOTCA software package [26]. Figure 7.5(a) shows the radial distribution functions of a small atomistic (reference) systems of 4000 DPBIC molecules and of a much larger system of 255083 interacting points generated using the potential inverted with the IBI procedure. Note that the generation of the coarse-grained morphology is very fast, since there are significantly less degrees of freedom involved as well as the smooth coarse-grained interaction potential leads to much faster (molecular) dynamics and hence shorter equilibration times. In total, 250000 steps (1 ns) were used to equilibrate the system of 255083 interacting points in the NPT ensemble, starting from a lattice.

7.2 Coarse-Grained Models | 115

7.2.2 Site Energies Early models for charge transport simulations assumed a Gaussian distribution of width σ of spatially uncorrelated [22] or correlated [140] site energies, where σ is referred to as the energetic disorder of the system. In the spatially correlated model, all molecules are assumed to have randomly oriented dipole moments, p, of fixed magnitude. The site energies, E i , are then evaluated as
X qp j r j − r i (7.4) Ei = − 3 , =j6 i ε r j − r i

where q is the charge, r i is the position of molecule i and ε is the material’s relative permittivity. This sum, evaluated using the Ewald summation method [242], has a spatial

 correlation function κ(r) ≡ corr[E i , E j ] which decays approximately as 1/r, where the average is taken over all molecules with center-of-mass distance r j − r i = r. Since the spatial correlation function can deviate from the 1/r asymptotics [244] (e.g., when the molecular dipole also varies [395, 244]), an algorithm that can reproduce any (physically relevant) spatial correlation function is required. For Gaussiandistributed site energies this can be achieved by mixing in the energy values of the neighbors: for each site A, N independent Gaussian distributed random variables X iA ∼ N(0, 1), 0 ≤ i ≤ N, are drawn. The energy of the molecule is then calculated as s N √ A X bi X A X iB , (7.5) E = aX0 + A i=1

`i

B∈S(r i ,A)

where S(r i , A) denotes a sphere of radius r i around molecule A and `Ai is the number of molecules within this sphere. The first term in Equation (7.5) corresponds to an uncorrelated energetic landscape. By adjusting a, one can change the energetic disorder, σ, without affecting the spatial correlation function κ(r). The sum over molecules within a certain cutoff r i adds spatial correlations. By adjusting the weighting coefficients, b i , one can reproduce the reference correlation function. It is also possible to devise an iterative scheme which converges to a set of required weights. For DPBIC, the autocorrelation functions of the reference (evaluated in the atomistic system) and the coarse-grained (constructed by iteratively refining the weighting coefficients) systems are shown in Figure 7.5(c). The autocorrelation function is perfectly reproduced and, in fact, is far from the 1/r dependence assumed in lattice models.

7.2.3 Neighbor List Since electronic coupling elements decrease exponentially with intermolecular separation, charge transfer reactions normally happen only between nearest molecular

116 | 7 Microscopic, Stochastic and Device Simulations neighbors. This physical argument is frequently used to reduce the size of the list of molecular pairs (neighbor list) for which charge transfer rates are evaluated. In atomistic simulations the neighbor list is constructed based on the cutoff for the shortest distance between π-conjugated molecular fragments [369]. In a coarse-grained model, where the molecular information is not available, the neighbor list can be constructed by analyzing the probability in the reference system of two molecules at a certain separation to be connected. This distribution is shown in Figure 7.5(b). One can see that all molecules separated by less than a certain minimal distance, rmin , are connected while there are no connected pairs if separation exceeds rmax . In the coarse-grained model each pair of sites is then connected based on this probability distribution. For DPBIC, this approach reproduces well both the connectivity of the reference system (see Figure 7.5(b)) and the coordination number (11.60 for the atomistic and 11.44 for the coarse-grained model).

7.2.4 Electronic Couplings In case of electronic coupling elements, one can analyze the distance-dependent distribution P[J ij (r)], i.e., the distribution of electronic couplings at a given separation r. For amorphous systems it has been found that the logarithm of the coupling element, log(J 2ij (r)/eV2 ), is Gaussian distributed with a mean m(r) and standard deviation σ(r) which depend on the molecular separation [244, 337]. The exponential decay of J ij (r) with intermolecular separation results in a linear decrease of the mean of the distribution for r ≥ 1.1 nm, as shown in Figure 7.5(d). To reproduce this behavior in a coarse-grained model, the values for the logarithm of the coupling are drawn from
the normal distribution, log(J ij (r)2 /eV2 ) ∼ N m(r), σ2 (r) , where m(r) and σ(r) are chosen according to the atomistic reference.

7.2.5 Charge Mobility and Model Validation The coarse-grained model can be validated by comparing the charge transfer rates, as well as the mobility–field and mobility–charge density dependencies of both models. The rate distributions are shown in Figure 7.5(e) and are in good agreement with each other. The mobility versus charge density is shown in Figure 7.5(f) for different electric fields and also agree with each other, validating the coarse-grained model. Small deviations are due to statistical fluctuations in relatively small systems studied (4000 molecules).

7.3 Lattice Models | 117

7.3 Lattice Models In Section 7.2 we have described how to parameterize a coarse-grained model based on a microscopic reference and use it to simulate relatively large (mesoscopic) systems. Alternatively, one can adopt an approach often employed to analyze experimental data, where analytical expressions for charge mobility are parametrized by fitting the IV characteristics of, e.g., a simple diode. The expressions themselves are derived with the help of lattice-based models, such as the Gaussian disorder model [22], the extended Gaussian disorder model (EGDM), which includes the effect of electron density [114], the correlated Gaussian disorder model (CDM), which takes into account spatial correlations of site energies [140], or the extended correlated Gaussian disorder model (EGDM) [141]. T, K

(a)

10−5 10−6 10−7

10−7 10−10

0.001

10−8

10−11

10−12 10−9

10−8 0

10−9

V/m V/m V/m V/m

dispersive regime

non-dispersive regime

T 3/2 µ(T ), K3/2 m2 /Vs

10−2

10−4

300 9 × 107 7 × 107 5 × 107 3 × 107

10−1

10−3

(b)

1200

µ ,m2 /Vs

50000

0.002 1/T , K−1

0.003

10−13

EGDM ECDM 0 1

1024

1025 ρ, m−3

Fig. 7.6. (a) Extrapolation of high-temperature mobility simulations to room temperatures. For a system of 4000 molecules transport simulations with one carrier are non-dispersive only above 1200 K. Hence, mobilities in the temperature range between 1200K and 50000 K (circles) are used for extrapolations to non-dispersive mobility values at 300 K (crosses). (b) A fit to EGDM (solid lines) and ECDM (dashed lines) models. All four dependencies are fitted simultaneously.

In principle, some parameters of these models can be obtained by analyzing the microscopic model directly: the energetic disorder, σ, for example, can be calculated from the site energy distribution, as described in Section 7.1.3. The lattice constant, a, can be estimated by matching the densities of the atomistic and lattice models, i.e.,
1/3 a = V /N , where V is the volume and N the number of molecules. In practice, however, it is better to obtain the model parameters by fitting the mobility to results of atomistic or stochastic simulations. The reason for this is that the analytical expressions provided by the lattice models rely on certain approximations, which might

118 | 7 Microscopic, Stochastic and Device Simulations not hold for the microscopic model. For example, spatial correlations of site energies, present in the microscopic model, can be accounted for by an smaller energetic disorder in the GDM model. By comparing the two approaches one can then assess whether or not a particular lattice model is capable of reproducing material properties [244] or can be used in drift-diffusion simulations of devices [246]. Table 7.1. Lattice spacing a, energetic disorder σ, and mobility at zero field and in the limit of zero charge density for the microscopic, EGDM, and ECDM models. The last row corresponds to the experimentally measured IV-characteristics fitted to the ECDM model.

microscopic EGDM, simulation ECDM, simulation ECDM, experiment

a [nm]

σ [eV]

μ0 (300 K) [m2 /Vs]

1.06 1.67 0.44 0.74

0.176 0.134 0.211 0.121

3.4 × 10−12 2.1 × 10−11 1.8 × 10−13 1.2 × 10−10

As an example, a mobility versus charge density fit has been performed for an amorphous DPBIC morphology. To correct for finite-size effects at small charge densities, an extrapolation procedure was used to obtain non-dispersive mobilities [261] via a simulation of charge transport at high temperatures and a subsequent extraction of the room-temperature mobility from the known mobility-temperature dependence, as shown in Figure 7.6(a). The extrapolated values, together with mobilities at charge densities typical for an operating OLED, were used to fit the EGDM and ECDM models, as shown in Figure 7.6(b). Notably, the model with energetic correlations yields a worse fit than the uncorrelated one, even though spatial site energy correlations are present in the system. This has been observed before in simulations of amorphous dicyanovinyl-substituted quaterthiophene [244]. The reason is that the spatial correlation function of site energies does not decay as 1/r at large r, as correlated disorder models predict. This is also reflected in the fitting parameters, which are summarized in Table 7.1: for both EGDM and ECDM models the estimated energetic disorder differs from the microscopic value by about 0.04 eV. The EGDM underestimates the disorder, while the ECDM overestimates it. Similarly, the microscopic zero-field mobility, μ0 , is in between the two fitted values. The absence of correlations in the EGDM is compensated by an effectively increased lattice constant, while in the ECDM the lattice constant is much smaller than the density-based estimate. Note that both models were originally developed for energetic disorder up to 0.15 eV, while here the atomistic and ECDM values are higher than this.

7.4 Charge Dynamics | 119

7.4 Charge Dynamics The outcome of both coarse-grained and atomistic models are center-of-mass positions of molecules and charge transfer rates between them, i.e., a directed graph. To model charge dynamics in this network, one needs to solve the corresponding master equation. While the efficiency of the solver is not important for relatively small systems, it becomes essential for modeling transport in large coarse-grained systems of millions of sites. Here we summarize several approaches which help to make these solvers computationally efficient.

7.4.1 Master Equation Our target is the solution of the master equation, which describes the time evolution of the system  dP i (t) X  = T ij P i (t) − T ji P j (t) , (7.6) dt j

where P i is the (unknown) probability of state i and T ij is the transition probability from state i to j. For one charge carrier, the state of the system is given by the index of the charged molecule and the rates are equal to charge transfer rates, i.e., P i = p i , where p i is the occupation probability of a site i, and T ij = ω ij . This equation can be solved analytically in some special cases. In general, however, direct numerical differential equation solvers are used [226]. If more than one charge carrier is present in the system, the number of system states increases rapidly (it is proportional to the binomial coefficient) and ordinary differential equation (ODE) solvers quickly become impractical unless additional approximations are employed. One of them is the mean-field approximation, which allows to rewrite the master Equation (7.6) in terms of site-occupation probabilities, p i , 
dp i X  (7.7) = ω ij p i 1 − p j − ω ji p j (1 − p i ) . dt j

One can see that the differential equation becomes nonlinear, with the 1 − p term accounting for the prohibited double-occupancy of a site in a mean-field way. An alternative approach to solve the master equation is to use kinetic Monte Carlo (KMC) algorithms [233, 234], i.e., to construct a continuous-time Markov process, where the probability of a certain event to occur depends only on its current state. We will discuss particular implementations of KMC in the next sections. Since KMC does not rely on the mean-field approximation, it is possible to evaluate the error introduced by neglecting higher-order correlations. It turns out that for an energetic disorder of σ/kB T = 2 the correction due to pair correlations is smaller than 3 % [227]. In more disordered systems this error increases, as well as it is not clear how higher order correlations might affect this result.

120 | 7 Microscopic, Stochastic and Device Simulations Once the equation for p i is solved, the mobility tensor, μ, can be evaluated as
1 X ω ij p i 1 − p j r ij,α F β , μ αβ = (7.8) ρF 2 V ij

where ρ is the charge density, F is the external electric field, V is the box volume, and r ij = r i − r j . It is also possible to compute the mobility tensor from the KMC trajectory μ αβ =

 N

1 X v i,α F β , N F2

(7.9)

i=1

where hv i i is the velocity of the ith carrier, h· · · i denotes the average over the simulation time, and the sum is performed over N carriers. The mobility at zero external field can also be calculated from the diffusion tensor using the generalized Einstein relation [454] (generalization is required due to the Fermi-Dirac and not Boltzmann statistics of the site-occupation probability) qD

∂ρ = μρ. ∂η

(7.10)



Here D αβ = ∆r α ∆r β /2τ, where τ is the total simulated time, ρ the charge density, and η the chemical potential. Note that the Einstein relation holds only for a steady state that fulfills the detailed balance condition [455] and for small fields [456]; it cannot be applied to inhomogeneous temperature distributions [302, 230] or unrelaxed (“hot”) carriers.

7.4.2 Kinetic Monte Carlo The variable step size method (VSSM) [237, 238] is one of the kinetic Monte Carlo solvers of the master equation. It is a hierarchical method, i.e., events can be grouped and VSSM can be used for both the groups and events in the group. In case of charge transport, for example, it is convenient to have two levels: first a charge is selected P based on the escape rate, ω i = j ω ij , and then the hopping destination is chosen. VSSM allows an efficient treatment of forbidden events [238], without the need to recalculate escape rates at every step. In charge transport simulations, forbidden events are due to the site single-occupancy constraint which leads to Fermi-Dirac statistics for the equilibrium distribution of site occupations, as shown in Figure 7.7(a) for a DPBIC morphology of 4000 molecules. As expected, both the Fermi level and occupation probability increase with the increase of charge density. This is explained in Figure 7.7(b), where the Gaussian density of states is filled up to the Fermi energy, and the product of the Fermi-Dirac distribution and Gaussian density of states, integrated up to the Fermi energy, gives the charge density in the system. Note that the standard implementation of the VSSM algorithm would require about 108 KMC steps to reach the steady state for charge flux in DPBIC, since carriers

7.5 Device Simulations | 121

(a)

(b)

n = 0.001 n = 0.004 n = 0.016

1 0.8

2 probability

probability

Fermi Dirac distribution DOS (fit) product

2.5

0.6 0.4

1.5 n = 0.1 1

0.2 0.5 0 0 4.7

4.8

4.9 energy, eV

5

5.1

4.8

EF

hEi 5.4

5.6

5.8

energy, eV

Fig. 7.7. (a) Site-occupation probability as a function of site energy plotted for three charge densities. Points are Monte Carlo simulations in a DPBIC morphology of 4000 molecules. Solid lines are fits to the Fermi-Dirac distribution. The Fermi level (dashed lines) increases with increasing relative charge density n (charges per site). (b) Filling of the Gaussian density of states (DOS) for a number density of n = 0.1. The product of the Fermi-Dirac distribution and the Gaussian DOS yields the occupation distribution (red curve). The area under this curve, evaluated up to the Fermi level is equal to the charge density. The width of the DOS matches the energetic disorder of DPBIC (0.176 eV) and is centered at 5.28 eV.

can be trapped in aggregates of low-energy sites. This can be circumvented using an aggregate Monte Carlo (AMC) algorithm [452], i.e., by coarsening the state space [457, 458, 459, 460, 461] into “super-states”. Efficient coarsening methods that retain physically sensible trajectories are the stochastic watershed algorithm [457, 462, 463], which is an improved version of the watershed algorithm [464, 465], and a graphtheoretic decomposition [466]. AMC has been tested on the amorphous phase of Alq3, where it leads to about two orders of magnitude faster convergence [452]. KMC solvers with explicit long-range Coulomb interactions require rate updates at (practically) every KMC step. Efficiency of such algorithms can be improved by using a binary tree search and update algorithm [467]. It updates only those rates which are affected by a moving carrier and has a logarithmic scaling with the number of events, whereas a linear search scales linearly. The binary tree algorithm becomes more and more beneficial with increasing number of charge carriers: while for 1 % carrier density (in 4000 molecules) it is slower than a primitive search, at 10 % it is already faster by about a factor of 10.

7.5 Device Simulations In order to simulate IV characteristics of an entire device, two additional ingredients must be incorporated in the model: the effect of metallic electrodes, which inject and

122 | 7 Microscopic, Stochastic and Device Simulations collect charges, and the strongly inhomogeneous distribution of charge density in the device. This can be done on a macroscopic level, by complementing the drift-diffusion equations with boundary conditions and Gauss’s law for charge density, as described in Section 7.5.1. An alternative approach is to extend the Monte Carlo scheme by including the electrodes and the Poisson equation solver for long-range electrostatics. This extension has already been implemented for lattices [246], see also Section 7.5.2, and is discussed for off-lattice models in Section 7.5.3. To disentangle the contributions of different functional layers of an OLED device here we will simulate a simpler single-carrier , which is shown schematically in Figure 7.9(a). The experimental measurements (Section 7.6) are also performed using this setup.

7.5.1 Drift-Diffusion Modeling The drift-diffusion model assumes that local charge density, ρ(x), charge mobility, μ(x; ρ, F, T), field strength, F(x), and diffusion constant, D(x; ρ, F, T), are changing continuously. In one dimension, the corresponding drift-diffusion equation then reads J = qρμF + qD

∂ρ . ∂x

(7.11)

Here the first (drift) term describes transport of holes/electrons in/against the direction of the electric field. The second (diffusion) term is due to transport of charges against the gradient of charge density. The drift-diffusion equation is complemented by Gauss’s law, ∂F(x)/∂x = ρ(x)/ε, where ε is the relative permittivity. The dependence of mobility on charge density, electric field, and temperature can be obtained from Monte Carlo simulations as described in Section 7.3. μ and D are related via the generalized Einstein equation [468]. Boundary conditions fix the charge carrier density at the electrodes, ρel , to the value given by a thermal equilibrium between the electrode and the organic layer, ρel =

Z∞

−∞

g(E) h i dE, 1 + exp E+∆ kB T

(7.12)

where g(E) is the density of states. To correct for the presence of the image potential, an equilibrium between the Fermi level of the injecting electrode and the maximum of the sum of the applied driving bias, the local potential as obtained from the Poisson equation and the image potential is assumed. The lowering of the injection barrier, ∆, between the Fermi level of the electrode and of the organic site is then described by [469] r qFc ∆′ = ∆ − q . (7.13) 4πε

7.5 Device Simulations | 123

(a)

104

(b)

102 current density, A/m2

current density, A/m2

102 100 10−2 10−4

0.0 0.2 0.4 0.6 0.8 1.0

10−6 T = 300 K 10−8

103

0

2

4

6

voltage, V

eV eV eV eV eV eV 8

10

101 100 10−1 10−2

KMC experiment

10−3

T = 313 K 273 K 233 K

10−4

0

2

4

6

8

10

voltage, V

Fig. 7.8. (a) Simulated current–voltage characteristics of a single-carrier device with electrodes. Squares show the results of Monte Carlo simulations in a cubic lattice, which are compared to driftdiffusion simulations (solid lines) using the EGDM mobility parametrized on a microscopic system. Different colors correspond to different values of the injection barrier ∆. (b) Current–voltage characteristics of the off-lattice (coarse-grained) model obtained by Monte Carlo simulations (squares) and measured experimentally (circles) for three different temperatures.

The field at the electrode, Fc , and the lowering of the injection barrier, ∆′ , are determined self-consistently via an iterative procedure [470]. When the field at the electrode is directed into the electrode itself, i.e., Fc < 0, the injection barrier is not lowered by the image potential. Before solving the drift-diffusion equation we have parametrized the field and temperature dependencies of the mobility, as described in Section 7.3, i.e., fitted the results of Monte Carlo simulations performed in an amorphous morphology of 4000 DPBIC molecules to the EGDM. Lattice spacing, energetic disorder, and zero-field mobility were used as fit parameters and are summarized in Table 7.1. The drift-diffusion equations were then solved for several values of the injection barrier, ∆, and are shown in Figure 7.8(a). Up to ∆ = 0.2 eV the IV curves are practically on top of each other. Only from 0.4 eV the injection barrier starts to affect the IV characteristics. One can also see that for larger values of ∆ (less effective injection) the current density decreases by several orders of magnitude. Since in organic materials the transport of charge carriers is percolative and the current density in the device has a filamentary structure [471, 472, 473], it is not immediately clear how reliable drift-diffusion device models are [474]. This is addressed in the next section by comparing the drift-diffusion solution to the results of lattice simulations. Drift-diffusion equations, combined with the EGDM and ECDM models, are often used for fitting to experimentally measured IV characteristics, in order to characterize a particular material. The results of such a fit, are summarized in Table 7.1. Here we performed a simultaneous fit to 9 different IV curves, measured using three layer thick-

124 | 7 Microscopic, Stochastic and Device Simulations nesses (108 nm, 162 nm, 219 nm) and three temperatures (233 K, 293 K, 313 K). The model includes a fitted injection barrier (0.05 eV) and doped injection layers which provide the (almost) Ohmic injection, similar to the experimental setup shown in Figure 7.9(a)).

7.5.2 Lattice Monte Carlo To be able to use the Monte Carlo procedure for devices, additional sites, which play the role of electrodes, are added. The injection/collection of a charge carrier to/from the electrode is modeled as a hopping of this charge carrier between the electrode and the lattice site to/from which the charge carrier is injected/collected [474, 246]. Additional contributions to site energies (due to electrodes) include a linear drop in the applied potential, eV(1 − x/L). Here the Fermi energy of the electron injecting electrode (x = L) is taken as a reference, while the applied potential at the collecting electrode is set to V. An injection barrier, ∆, is introduced between the Fermi level of the electrode and the one of the sites of organic material, such that the energy difference to the electrode is raised by ∆, Eel,i = −Eel + E i + ∆, while for the charge transfer into the electrode E i,el = Eel − E i − ∆. These contributions are depicted in Figure 7.9(d). Coulomb interactions are split into a short-range and a long-range contribution. The short-range part is evaluated explicitly for all charges (and their periodic images) within separations smaller than rc . Between one and ten images provide sufficient accuracy, such that a computationally demanding Ewald summation can be avoided. The long-range contribution is evaluated by first calculating the local charge carrier density, ρ(x), by averaging the number of charges in two-dimensional (yz) slabs oriented perpendicular to the electric field direction (along the x axis). Then the electrostatic potential, ϕ(x), is reconstructed using the Poisson equation, ϕ′′ (x) = −ρ(x)/ε. Note that the layer-averaged potential also takes into account the short-range interactions and one needs to correct for this double counting. The current density throughout the device is evaluated as P q k ∆r k · F , (7.14) J= AτF where ∆r k · F/F is the change in position of a charge along the applied field F, A is the surface area of the electrodes, and τ is the total simulated time. Apart from device simulations, the lattice Monte Carlo method can be used to benchmark the accuracy of the continuum drift-diffusion solver, discussed in Section 7.5.1. Figure 7.8(a) shows IV curves obtained by Monte Carlo simulations in a cubic lattice with lattice spacing a = 1.67 nm and Gaussian-distributed site energies of width σ = 0.134 eV, i.e., the same parameters as in the EGDM mobility parametrization, see Section 7.3. One can see that lattice Monte Carlo results agree reasonably well with the EGDM model combined with the drift-diffusion solver. Note that at low volt-

electrode mirror images

mirror images

electrode

7.5 Device Simulations | 125

Fig. 7.9. (a) Experimental setup for single-carrier-type measurements, here of the DPBIC holeconduction layer. (b) Simulation setup mimicking the device. (c) Evaluation of the long- and shortrange electrostatic contributions in a device geometry. (d) Additional contributions to site energies due to electrodes: a linear drop in the applied potential and an injection barrier ∆.

ages (small drift currents) the KMC algorithm converges very slowly, which is why we have included KMC results only for voltages higher than 2 V.

7.5.3 Off-lattice Monte Carlo The Monte Carlo scheme on a lattice can be generalized to off-lattice models ¹, thus helping to avoid parametrizations of mobility as well as assumptions incorporated in lattice models. Again, the effects that need to be taken into account are Coulomb interactions and the injection from the electrodes into the organic layer. To model the injection process, an electronic coupling of the form Jinj = exp(−2αr ij ) is assumed between the electrodes and the organic material, where α is the inverse wave function decay length and r ij is the distance between the electrode and the site to or from which the charge carrier is injected or collected. This introduces two parameters, α and d, where the latter is the distance between the electrode and the organic site nearest to that electrode. Close to the electrodes the image charge effect leads to an effective lowering of the energetic disorder [475]. This is accounted for by a factor γ

1 Details will be explained in a more technical publication that is currently in preparation.

126 | 7 Microscopic, Stochastic and Device Simulations which rescales the site energy, E i , of injection/collection sites to E′i = γ(E i − hE i i)+ hE i i, where hE i i is the average site energy. Coulomb interactions are split into a short-range and a long range contribution. The short-range part is evaluated in a similar way as for the lattice, except that ten to hundred images are included to improve accuracy. For the long-range part the total simulation box is split into three-dimensional slabs of equal thickness, as shown in Figure 7.9(b). All charges in a slab are summed up to obtain the slab-averaged charge carrier density, ρ i . The slab-averaged potential, ϕ(x), is then evaluated by solving the Poisson equation, where the slab-averaged charge carrier densities are used as a source term. Again, similar to the lattice Monte Carlo, the layer-averaged potential already takes into account short-range interactions. Thus double-counting should be corrected for, as shown in Figure 7.9(c). The computationally expensive update of the averaged long-range potential and the re-evaluation of the hopping rates is done every 100 – 1000 hops, which yields sufficient accuracy of long-range effects. To perform charge transport simulations, we used the coarse-grained off-lattice model of DPBIC, parametrized as discussed in Section 7.2. An amorphous morphology of 255083 molecules was generated in a box of 123 nm × 50 nm × 50 nm to match the experimentally used layer thickness (see Section 7.6 for details). Periodic boundary conditions were employed in the directions perpendicular to the electric field. The simulated IV curves, together with the experimental data points are shown in Figure 7.8(b) for α = 13 nm−1 , d = 0.5 nm, ∆ = 0.5 eV, and γ = 0.55, and are in good agreement with each other.

7.6 Experiment In order to decouple different processes in OLEDs, experiments are often performed in single-layer hole-only or electron-only devices [476]. Here we used a setup shown in Figure 7.9(a) with 5 nm molybdenum trioxide (MoO3 ) layers on both sides of a 123 nm DPBIC layer. The function of the MoO3 layers is to allow band bending [477], resulting in an (almost) Ohmic injection to the DPBIC layer [478]. The devices were fabricated by vacuum thermal evaporation of the organic material on a glass substrate patterned with a 140 nm thick indium tin oxide (ITO) layer as an anode. Prior to deposition, substrates were cleaned in an ultrasonic bath using subsequently detergents and de-ionized water. The samples were dried and treated in an ozone oven to additionally clean them and modify the ITO workfunction. A 100 nm film of thermally evaporated Aluminium (Al) is used as a cathode. In order to exactly determine the thickness of the DPBIC layer, we simultaneously deposited the same amount of DPBIC on a Si-Wafer mounted adjacent to the substrate and measured its thickness using optical ellipsometry.

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| 127

The fabricated samples were placed into the oil reservoir of a cryostat, where the temperature was varied between 233 K and 313 K. The experimentally measured IV curves are shown in Figure 7.8 for three different temperatures, together with simulation results described in Section 7.5.

7.7 Materials DPBIC

BTDF

Ir Si

N

Si

3 N O

TBFMI BCP N

N

O Ir N

3 N

Alq3

-NPD O Al

N

N

N

3

Fig. 7.10. Chemical structures of different materials used in OLEDs.

To this end, we have simulated current–voltage characteristics of a diode with a hole-conducting DPBIC used as organic layer. While this example is so far the only exhaustive illustration of multiscale simulations techniques which can be used to simulate charge transport in disordered organic semiconductors, a number of similar methods have already been used to evaluate material properties of such materials [368, 369]. Here we will provide a (far from complete) review of such case studies. In particular, we will focus on charge transport in amorphous mesophases of DPBIC, BTDF, TBFMI, BCP, α-NPD, and Alq3, chemical structures of which are shown in Figure 7.10.

128 | 7 Microscopic, Stochastic and Device Simulations DPBIC Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-pheny-lene]Ir is used in OLEDs as a hole-transporting and electron-blocking material [479, 480, 481, 482]. Apart from IV dependencies discussed here, simulations of amorphous DPBIC predict a density value of 1.29 g/cm3 and an energetic disorder of 0.176 eV, which compares reasonably well to the experimentally determined value of 0.121 eV, that relies on the ECDM [141]. In amorphous morphology the perturbative approach yields an ionization potential (IP) of 5.28 eV, in a perfect agreement with the Conductor-like screening model (COSMO) [483] giving the same value of 5.28 eV. Both the Poole-Frenkel dependence and the absolute value of mobility are in agreement with experimental measurements: For small fields the simulated hole mobility varies from 6.6 × 10−10 m2 /Vs (ρ = 4.25 × 1023 m−3 ) to 3.6 × 10−8 m2 /Vs (ρ = 1.36 × 1025 m−3 ), while the experimental value is 9 × 10−9 m2 /Vs.

BTDF 8-bis(triphenylsilyl)-dibenzofuran (BTDF) is an electron-conducting and hole-blocking material [479] which, in combination with a deep blue emitter can lead to quantum efficiencies above 17 % [410]. Its glass transition temperature is at 107 C◦ [410]. Molecular dynamics simulations predict an amorphous density of 1.37 g/cm3 [410]). Calculations based on polarizable force-fields predict an energetic disorder of σ = 0.085 eV (electrons) and σ = 0.078 eV (holes). Simulated in the zero-charge density limit mobilities, 1.0 × 10−9 m2 /Vs for holes and 3.2 × 10−8 m2 /Vs for electrons, are comparable to the experimental values obtained using admittance spectroscopy, 2.5×10−10 m2 /Vs for holes and 1.0×10−8 m2 /Vs for electrons, measured at an electric field of 9×106 V/m (holes) 3 × 107 V/m (electrons).

TBFMI Tris[(1,2-dibenzofurane-4-ylene)(3-methyl-1/1-imidazole-1-yl-2(3/1)-ylidene)]Ir(III)) (TBFMI) has been used as an emitter molecule in various host-guest mixtures, for example, with BTDF. An important quantity in such mixtures is the relative alignment of densities of states of the host and guest materials. For example, in TBFMI:BTDF the energy offset between the host and the guest is very small, which is mostly due to environmental effects. In particular, the change in molecular polarizability upon charging plays an important role [250].

BCP 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline (bathocuproine, BCP) is a typical electron conducting and hole blocking material which is used in phosphorescent OLEDs [48, 484, 485, 486, 487]. Simulations of charge transport in a crystalline phase of this

7.7 Materials

| 129

material [487], using a model based on quantum mechanic calculations of a sample dimer, show mobilities of 1.8 × 10−6 m2 /Vs in a reasonable agreement with a reported experimental value of 1.1 × 10−7 m2 /Vs [488].

α-NPD N,N’-bis(1-naphthyl)-N,N’-diphenyl-1,1’-biphenyl-4,4’-diamine (α-NPD or NPB) can be used in OLEDs as a hole-transporting, electron-blocking [48, 489, 490, 491, 492, 493], or blue-emitting layer [494, 495], and as a host material [496, 497]. It has a relatively high glass transition temperature of 95 C◦ , favorable for the material’s stability. In the amorphous phase, a density of 1.16 g/cm3 has been reported from X-ray reflectometry measurements [498]. IV curves in an α-NPD based diode were simulated by solving the drift-diffusion equations using both extended and correlated Gaussian disorder models. Both models provide a consistent description of the IV characteristics at different temperatures. However, the lattice constant obtained from the ECDM fit is closer to the experimentally known intersite distance, i.e., spatial correlations of site energies in this material are important. The amorphous morphology of α-NPD has been studied by using the Monte Carlo algorithm that mimics the vacuum deposition process [425] as well as by MD simulations [499]. The latter, combined with electrostatic and induction calculations, gave energetic disorder values of σ = 0.09 − 0.11 eV (electrons) and σ = 0.14 − 0.15 eV (holes). Simulated mobilities, obtained by applying an analytic solution of the master equation to the transport network, varied from 4.2 × 10−10 m2 /Vs to 1.3 × 10−9 m2 /Vs (holes) and from 1.1 × 10−6 m2 /Vs to 4.1 × 10−6 m2 /Vs (electrons), in a good agreement with the experimentally measured value of 7.8 × 10−10 m2 /Vs for holes [500].

Alq3 Mer-tris(8-hydroxyquinolinato)aluminium (Alq3) was used in the first organic diode already in 1987 [8] and ever since has been studied intensively. The amorphous morphology of Alq3, obtained by annealing above the glass transition temperature and subsequent cool-down [416] to room temperature, resulted in density of 1.18 g/cm3 (Williams 99 force field) and 1.29 g/cm3 (OPLS force field), while Monte Carlo-based vacuum-deposition algorithm with the local relaxation after deposition led to 1.22 g/cm3 (OPLS force field) [425]. Experimentally reported values range from 1.3 g/cm3 [501] to 1.5 g/cm3 [502]. Simulations predict energetic disorder of 0.21 eV for holes [369]. Charge transport simulations have shown that spatial correlation of site energies is responsible for the Poole-Frenkel behavior of the charge mobility [369] and that finite-size effects should be carefully accounted for in systems with large energetic disorder [261].

130 | 7 Microscopic, Stochastic and Device Simulations

Conclusions and Outlook We have reviewed a set of simulation techniques used to understand charge transport in amorphous organic materials. The distinct feature of these techniques is that they all start with chemical structures and predict, by building a hierarchical set of models, macroscopic properties of devices, in our case charge carrier mobility and current-voltage characteristics of an organic light emitting diode. The advantages and shortcomings of all techniques have been illustrated by simulating (and validating experimentally) charge dynamics in a unipolar device with a hole conducting material used in state of the art diodes. We have also identified areas where substantial method development is still required in order to achieve a parameter-free modeling of realistic devices. In particular, still missing are (i) first-principles evaluations of charge injection rates (from metal to organic material), (ii) explicit treatment of the induction interaction when solving the master equation (re-evaluation of rates at every Monte Carlo step), (iii) quantitative treatment of excited states embedded in a heterogeneous polarizable molecular environment, (iv) descriptions of charge-exciton and exciton-exciton interactions. Advancements in all these directions are absolutely vital for devising accurate structureproperty relationships for organic semiconductors.

Acknowledgements This work was partly supported by the DFG program IRTG 1404, DFG grant SPP 1355, and BMBF grants MEDOS (FKZ 03EK3503B) and MESOMERIE (FKZ 13N10723). We are grateful to Carl Poelking, Anton Melnyk, Jens Wehner, and Kurt Kremer for a critical reading of this manuscript.

8 Parametrization of Lattice Models Reprinted with permission from: Pascal Kordt, Ole Stenzel, Björn Baumeier, Volker Schmidt, Denis Andrienko Parametrization of Extended Gaussian Disorder Models from Microscopic Charge Transport Simulations Journal of Chemical Theory and Computation 10, 2508–2513 (2014) Copyright 2014 American Chemical Society. DOI: 10.1021/ct500269r

Gaussian Disorder ModelSimulations of organic semiconducting devices using driftdiffusion equations are vital for the understanding of their functionality as well as for the optimization of their performance. Input parameters for these equations are usually determined from experiments and do not provide a direct link to the chemical structures and material morphology. Here we demonstrate how such a parametrization can be performed by using atomic-scale (microscopic) simulations. To do this, a stochastic network model, parametrized on atomistic simulations, is used to tabulate charge mobility in a wide density range. After accounting for finite-size effects at small charge densities, the data is fitted to the uncorrelated and correlated extended Gaussian disorder models. Surprisingly, the uncorrelated model reproduces the results of microscopic simulations better than the correlated one, compensating for spatial correlations present in a microscopic system by a large lattice constant. The proposed method retains the link to the material morphology and the underlying chemistry and can be used to formulate structure-property relationships or optimize devices prior to compound synthesis.

Introduction The optimization of organic photovoltaic cells [7, 503], light emitting diodes [406], and field effect transistors [398] requires the improvement of the charge-carrier mobility, μ, of an organic semiconductor, which depends on charge-carrier density, ρ, external electric field, F, and temperature, T [217, 114, 364, 366, 367, 504]. Different experimental setups have been proposed to measure these dependencies, e.g. time-of-flight [361, 505], field effect transistor [506, 507, 508], diode [406], microwave conductivity [509], or charge extraction by linearly increasing voltage [148] measurements. They each operate at different charge densities and hence should be accompanied by an appropriate model in order to recover the full dependence [113]. Analytical expressions have therefore been proposed by analyzing various model systems [22, 114, 140, 141, 413] and are routinely used to interpret experiments [510, 511, 512, 513, 514] as well as to parametrize charge transport models. While being useful for the fine-tuning of the de-

132 | 8 Parametrization of Lattice Models vice performance for a specific material combination, these parametrizations do not directly relate the chemical structure or material morphology to charge mobility and hence cannot be applied to compound screening. Computer simulations can help to retain the link to the chemical structure and should ideally provide the mobility μ as a function of ρ, T, F over the ranges relevant for use in device simulations. At an atomistic level of detail, which is required for describing morphologies and charge transfer processes without fitting parameters [368, 369, 408, 515], it is possible to study only relatively small systems comprising up to several thousands of molecules and concomitantly at high carrier densities. Larger systems and thereby lower carrier densities can be simulated by employing a stochastic model [337, 516] parametrized on the distribution and correlation functions of atomistic simulations, both for morphologies and transport parameters. With the help of stochastic models, the dependence of mobility on charge density and field can, in principle, be tabulated and used to solve the macroscopic drift-diffusion equations. In practice, the range of charge densities accessible to stochastic simulations is still limited and pronounced finite-size effects occur [261], especially at low densities. In this work we illustrate how to overcome these limitations and parametrize the analytical expressions resulting from the Gaussian disorder models. To do this, we tabulate the mobility as a function of charge density using atomistic (microscopic and stochastic) approaches, eliminate finite-size effects at small carrier densities, and then fit the data to the analytical forms provided by the extended and extended correlated Gaussian disorder models. The approach is illustrated on an amorphous phase of dicyanovinyl-substituted quaterthiophene (DCV4T), whose chemical structure is shown in Figure 8.1(a). DCV4T is a thermally stable dye [517, 518] with a small optical band gap, which renders it as an excellent donor for bulk heterojunction solar cells [32, 519, 520, 521, 522, 523]. When mixed with C60, DCV4T retains (at least partially) its crystallinity [11, 395, 419]. Here, however, we will study pure amorphous (glassy) systems, where morphological disorder results in a large energetic disorder and hence pronounced finite size effects, making these systems ideal for illustrating and testing the method.

8.1 Methodology To perform molecular dynamics simulations, the reparametrized [11, 395] version of the OPLS [384, 385] force field and the GROMACS package were used [524]. An amorphous phase of DCV4T has been generated using the isothermal-isobaric (NPT) ensemble with the Berendsen barostat and thermostat [330] by equilibrating 4096 molecules at T = 800 K for 10 ns, quenching the system to T = 300 K, and equilibrating again for 10 ns. The final cubic box of 13.7 × 13.7 × 13.7 nm3 is shown in Figure 8.1(d).

8.1 Methodology | 133

(a)

N

(d) N

HC

HC

CH

CH

S HC

S

S

CH

S HC

CH

HC

CH

N N

(b)

(c)

Fig. 8.1. (a) chemical structure, (b) electrostatic potential (at ±0.7 V ), (c) highest occupied molecular orbital, and (d) amorphous morphology of 4096 molecules of dicyanovinyl-substituted quaterthiophene (DCV4T).

Charge transport simulations were carried out using the VOTCA package [369]. The rates of a charge transfer reaction were evaluated using the high temperature limit of the Marcus theory [207, 208] "
2 # J 2ij ∆E ij − λ ij 2π p ω ij = exp − , ~ 4λ ij k B T 4πλ ij kB T

(8.1)

where ∆E ij = E i − E j is the energy difference between localized states, λ ij is the reorganization energy, and J ij is the electronic coupling. For each molecule i with center-ofpol el mass coordinate ri the site energy was calculated as E i = Eint i + E i + E i +qF · ri , int where E i is the internal molecular energy, i.e., the adiabatic ionization potential of the molecule. Eel i is the electrostatic energy due to variations of the local electric field, evaluated using atomic partial charges. The corresponding electrostatic potential, which demonstrates the acceptor-donor-acceptor character of the molecule, is shown in Figure 8.1(b). Epol i is the induction energy, evaluated using the Thole model [336, 389], qF · ri is the energy due to the interaction of charged molecules with an external electric field. The corresponding parameters can be found in the supporting information of Refs. [11, 395]. Note that the acceptor-donor-acceptor architecture of DCV4T, in combination with non-planar molecular geometries in the amorphous morphology, leads to large molecular dipolar moments and pronounced spatially correlated energetic disorder of σ = 0.253 eV. Electronic couplings were evaluated for all molecule pairs in the neighbor list using the semi-empirical ZINDO method [445, 447]. The frozen core approximation was used, i.e., the HOMO orbital, which is shown in Figure 8.1(c), provides the main con-

134 | 8 Parametrization of Lattice Models tribution to the diabatic states of the dimer. The neighbor list was constructed using a cutoff of 0.7 nm between rigid fragments (thiophenes and dicyanovinyl groups). The variable-step-size implementation of the kinetic Monte Carlo algorithm was used to solve the master equation [369] and simulate charge dynamics in the system. Coulomb interaction between charges was accounted for approximately by excluding double occupancy of a site. For 4096 molecules one to sixteen charges per simulation box were used, corresponding to hole densities of 0 (no interactions for a single charge carrier) to 6.22×1018 cm−3 , which covers typical values for OFET measurements [113]. Simulation times varied between 0.1 s and 0.0001 s, depending on the number of carriers (systems with many carriers undergo faster relaxation). The mobility μ was evalP uated by averaging over several trajectories and all carriers, μ = NF1 2 Nk=1 hvk · Fi, where vk is the velocity of carrier k, and h· · · i denotes the average over all runs. High computational costs limit the accessible system size in atomistic simulations. Therefore a stochastic model [337, 516] was used to reproduce distribution functions for center of mass positions, site energies, pair connection and transfer integrals. Positions were parametrized using the thinning of a Poisson process: we first generated a purely random pattern of points (candidates for site positions) such that the number M of points in a given volume V is Poisson distributed, M ∼ Poi(ρsite V), with ρsite > 1.59 denoting the density of points per unit volume. In a second step points that are too close to each other (i.e., the unphysical situation of overlapping molecules) were deleted by assigning a radius R n = r h + X n to each point and deleting those points for which a sphere of radius R n is contained in the sphere of another point with radius R m > R n . Here r h = 0.1 nm is the minimum separation observed in the atomistic data and X n is a (positive) Gamma-distributed random variable, mimicking the freedom in molecular orientation. The procedure was repeated until the desired density of ρsite = 1.59 nm−3 was obtained, in agreement with the atomistic data. The stochastic model for site positions has been validated by comparing the atomistic and stochastic pair correlation functions g(r) (see the supporting information for details). Site energies with spatial correlation were generated using a moving-average procedure [337, 516]. The main idea is to decompose the site energy into two contributions, one that is a site-specific Gaussian distributed random variable and another one that is the sum running over all neighbors, with another set of previously fixed Gaussian distributed random variables. Since any sum of independently Gaussian distributed variables is again Gaussian distributed, we obtain Gaussian distributed random variables with spatial correlation. The necessary parameters for the method are obtained by fitting the resulting correlation function κ(r) against its atomistic counterpart (see Figure 8.4). The neighbor list of the stochastic model was generated by picking a point and connecting to it all neighboring points in a sphere of a radius rmin = 0.633 nm, chosen since in the atomistic data any two points of distance r ≤ rmin are connected. In a second step, points within a sphere of a larger radius rmax = 2.5 nm are connected with a random acceptance criterion chosen such that the average and minimum coordination numbers observed in the atomistic data are reproduced. This algorithm

8.2 Results

| 135

reproduces the probability of two sites being connected (see the supporting information) and retains the same average number of connections per site (coordination number). Transfer integrals J ij for allconnected  sites have been modeled by mimicking the

Gaussian distributions of log10 J 2ij /eV2 for sites at a fixed separation, for which the distance-dependent mean and variance values have been calculated from the atomistic data. With point positions, site energies, and electronic couplings at hand we evaluated hopping rates using Equation (8.1). The use of the stochastic model allowed us to study eight times larger systems and, consequently, a wider range of charge densities, down to ρ = 0.97 × 1017 cm−3 . In order to eliminate finite-size effects at small charge densities, an extrapolation procedure has been used, where mobilities are first simulated at elevated temperatures (for the energetic disorder of 0.253 eV an estimate for a non-dispersive temperature regime is T & 1700 K [261]) and then fitted to the relation      b μ0 a 2 − , (8.2) μ(T) = 3 exp − T T T2

which then gives a non-dispersive mobility value at room temperature. In order to fit the mobility-density dependence of the atomistic and stochastic models, Gaussian disorder models where used. They are based on Monte Carlo simulations in lattices that were used for a parametrization of μ(ρ, T, F) and thus allow the extraction of, e.g., the disorder parameter from experiments. The original version [22] was later extended to include correlated disorder [140], finite charge density in the extended Gaussian disorder model (EGDM) [114] as well as a combination of both in the extended, correlated Gaussian disorder model (ECDM) [141]. Key equations are recapitulated in the supporting information.

8.2 Results The mobility versus field dependencies for the microscopic (atomistic and stochastic) models are shown in Figure 8.2(a). One can see that the mobility increases by four orders of magnitude with the increase of charge density from 1016 cm−3 to 1019 cm−3 . The reason for this increase is that at high densities deep energetic traps are filled, detrapping the rest of carriers. Since detrapping is stronger than the slow-down due to blocked pathways, the overall mobility increases [367]. One can also see that the stochastic and microscopic simulations agree at high charge densities but are off by orders of magnitude at small densities. The stark disagreement is due to finite-size effects: sampling of the density of states is limited to a relatively small number of site energies (since periodic boundary conditions are used), resulting in logarithmically slow convergence of the carrier energy with system size and hence overestimation of the mobility [261]. Since the microscopic system is eight times smaller than the stochastic one, the finite size effects are significantly more pronounced (hotter carriers in a

136 | 8 Parametrization of Lattice Models (a) 10−4

(b) 10−15 stochastic atomistic

−5

−3

×10 cm 62.2

10−6

31.1

−7

10

15.6 7.78

10−8

3.89

V/cm V/cm V/cm V/cm

10−16

10−7

1.94

10−9

9 × 105 7 × 105 5 × 105 3 × 105

10−6 µ (cm2 /Vs)

10

µ (cm2 /Vs)

17

10−5

0.97

10−10 10−11 500

0.00

600

700 800 √ F (V/cm)1/2

900

1000

10−17 0

1

10−8

1018

1019 ρ (cm−3 )

Fig. 8.2. (a) Mobility versus the square root of applied field for charge densities ranging from 9.7 × 1016 (light color) to 6.2 × 1018 cm−3 (dark color). The zero-density limit is given in black. Solid lines correspond to the atomistic system (4096 molecules) while dashed lines represent a stochastic system (ca. 32000 hopping sites). (b) Microscopic mobilities (squares) fitted to the EGDM (solid lines) and ECDM (dashed lines) models.

smaller simulation box), leading to orders of magnitude differences in mobility values. For the atomistic and stochastic models, the energetic disorder, σ, zero-field mobility, μ0 (300 K), and an effective lattice spacing, a, can be obtained directly from the distribution of site energies, diffusion constant (using the Einstein relation), and matep rial density, respectively. Assumingq a cubic lattice, we obtain a = 3 V /N = 0.86 nm, 2 1 P the variance of site energies σ = i ( E i − h E i) = 0.253 eV, and μ 0 (300 K) = N

3.5 × 10−10 cm2 /Vs (for the system of 32000 molecules). These results can be compared by fits of EGDM and ECDM to the obtained mobility dependence. A direct fit of the data as obtained by the microscopic simulations yields parameters which are completely unphysical, e.g., μECDM (300 K) ≈ 1013 cm2 /Vs. The 0 reason for this is that the zero-density mobility of the microscopic model is subject to substantial finite-size effects [261]. Extrapolating the microscopic data to a system of an infinite size, as described in the methodology section, we obtain a non-dispersive zero-density mobility of ∼ 10−17 cm2 /Vs, lowering the finite size biased value by seven (!) orders of magnitude. Including this value in the fit then yield physically reasonable parameters. Fitting results are summarized in Figure 8.2(b) and Table 8.1. Apart from a zerodensity point, densities between 7.76 × 1017 cm−3 and 6.22 × 1018 cm−3 were used, for which the transport is non-dispersive at room temperature. One can see that the values for σ and μ0 (300 K) obtained from the fit to the EGDM model are very similar to the microscopic model. The value of the lattice constant is, however, significantly

8.2 Results

| 137

Table 8.1. Lattice spacing, energetic disorder, and mobility at zero field and density of the atomistic, EGDM, and ECDM models. μ0 is calculated at 300 K.

microscopic EGDM ECDM

a [nm]

σ [eV]

μ0 (300 K) [cm2 /Vs]

0.86 1.79 0.34

0.253 0.232 0.302

2.0 × 10−17 2.1 × 10−17 3.3 × 10−18

larger. The physical reason for this is the absence of spatial correlations in EGDM, which is effectively compensated by an increased hopping range. The ECDM yields a much smaller lattice constant, larger energetic disorder, a smaller value of μ0 (300 K). Overall, the ECDM provides a worse fit of the data (smaller slope at high densities), which is surprising since it includes spatial site energy correlations. Both models, of course, incorporate certain assumptions, e.g., they are parametrized on simulations using Miller-Abrahams rates [205], i.e., without accounting for polaronic effects [217]. The energetic disorder considered in the models ranges from σ = 0.05 eV up to 0.16 eV (for T = 300 K), while an energetic disorder of 0.253 eV is predicted by the microscopic simulations, which is outside the parametrization interval. 0.7 atomistic stochastic ∝ r −1 fit

0.6

energy correlation κ(r )

0.5 0.4 0.3 0.2 0.1 0 -0.1

0

0.5

1

1.5 2 r (nm)

2.5

3

3.5

Fig. 8.3. Site energy correlation function, κ(r). While the agreement between atomistic and stochastic model is good, the proportionality κ(r) ∝ r−1 assumed in the ECDM fails especially at small distances.

The main issue, however, lies in spatial correlations. ECDM site energies, E i , are evaluated as
X qpj rj − ri Ei = − (8.3) 3 , ε r j − r i =j6 i

138 | 8 Parametrization of Lattice Models where pj is a randomly oriented dipole moment of fixed absolute value and ε is the material’s relative permittivity. The resulting energetic disorder has a width of σ = 2.35 q p/ε a2 [140, 253] and the energy correlation function for a simple cubic lattice can be approximated by [140, 141, 254] κ(r) = κ0 a σ2 r−1 with a constant coefficient κ0 . The energetic correlation in this model is fully determined by the energetic disorder, σ, while in microscopic simulations based on realistic morphologies materials with the same energetic disorder may have different strengths of correlation. Furthermore, a comparison of the r−1 proportionality in the ECDM to the observed microscopic correlation (see Figure 8.3) shows that, especially at small distances, the model is unable to capture the spatial correlation. A possible reason is that in DCV4T not only the orientation of dipoles is random, but also their absolute values can differ significantly for different molecules. Indeed, an analysis of the dipole moments in the DCV4T morphology shows that the two most frequent molecular geometry conformations have absolute values of 1.4 Debye and 12.3 Debye Experimental OFET mobility measurements in polycrystalline (or partially amorphous) films report mobilities of 10−4 cm2 /Vs [11, 518] (for a carrier density of about 5 × 1018 cm−3 ). Though the simulated values are reasonable, a direct comparison to experiment is not possible since we study a completely amorphous system: due to increased energetic disorder we obtain two orders of magnitude lower mobility values.

8.3 Conclusions To summarize, we have demonstrated how the dependence of mobility on external field and charge density can be parametrized using atomistic simulations. Two important steps in these procedures are (i) the parametrization of a stochastic network model, which helps to treat large systems and to increase the range of accessible charge densities; (ii) the extrapolation of dispersive mobilities at small charge densities to non-dispersive values, thus removing artifacts related to finite size effects inherently present in small periodic systems. For an organic dye considered here, DCV4T, the EGDM provides reasonable fits of the Poole-Frenkel dependencies, compensating for spatial correlations, which are present in amorphous DCV4T, by a large lattice constant. The ECDM yields worse fitting, in spite of explicit treatment of spatial site energy correlations. This has been shown to be a result of the model correlation differing from the microscopically observed one. The proposed approach relates the chemical structure, morphology, and microscopic transport parameters to an analytical expression for mobility and therefore can be used to optimize organic devices on a macroscopic scale.

8.4 Supporting Information | 139

8.4 Supporting Information 8.4.1 Parametrization of Stochastic Model Center of mass positions are generated using thinning of a Poisson process and validated by comparing the pair correlation function g(r), which is a measure of the density of neighboring points at a distance r. The resulting pair correlation functions for the microscopic and the stochastic model are shown in Figure 8.4. An analysis of the distributions of transfer integrals, J ij , for all pairs of neighboring molecules shows that log10 (J ij ) is Gaussian distributed for molecules whose distance r is within a certain fixed interval, with mean and variance of the Gaussian changing with distance, cf. Figures. 8.5(a) and (b). After determining the distance dependence from the microscopic model for each pair, values are drawn following a Gaussian distribution with the respective parameters. Figure 8.5(c) shows the overall frequency of rates after plugging in site positions, energies and electronic couplings.

8.4.2 Extend Gaussian Disorder Model (EGDM) The EGDM model [114], parametrized on Monte Carlo simulations (cubic lattice, charges interact via an exclusion principle) postulates a factorization into three terms μ(T, ρ, F) = μ0 (T)g(T, ρ)f (T, F),

(8.4)

where T is the temperature, ρ the spatial carrier density (carriers per volume), and F the absolute value of the electric field. The functions μ0 , g and f are defined by h i μ0 (T) = 1.8 × 10−9 μ0 exp −C σˆ 2 , C = 0.42, 1.2

pair correlation g(r )

1

0.8

0.6

0.4

0.2

0

atomistic stochastic 0

0.5

1

1.5 r (nm)

2

2.5

3

Fig. 8.4. Pair correlation function for the microscopic and the stochastic model.

140 | 8 Parametrization of Lattice Models (b)

0.5

1.0

1.5 r (nm)

2.0

(c)

4.5

3.5 3 2.5 2

0.01

1.5 1

2.5

0.1

atomistic stochastic

4

relative frequency

-3 -3.5 -4 -4.5 -5 -5.5 -6 -6.5 -7 -7.5 -8 -8.5

variance of log10 (Jij2 /eV2 )

mean of log10 (Jij2 /eV2 )

(a)

0.5

1.0 1.5 r (nm)

2.0

2.5

0.001 -5

0

5 10 log10 (ωij · s)

15

  Fig. 8.5. Mean (a) and variance (b) of distributions of log10 J ij2 /eV2 and (c) distributions of logarith
mic transfer rates log10 ω ij · s .

   δ  1 2 3 ˆ ˆ g(T, ρ) = exp , σ − σ 2ρa 2 "



f (T, F) = exp 0.44 σˆ

3/2

− 2.2

 q

2 1 + 0.8Fred

−1

#

,


ln σˆ 2 − σˆ − ln(ln 4) δ=2 . σˆ 2 μ0 is a material specific property, Fred = eaF/σ, and σˆ = σ/kB T.

8.4.3 Extended Correlated Gaussian Disorder Model (ECDM) The ECDM [141] accounts for spatial correlations of site energies. It postulates the following mobility dependence μ(T, ρ, F) = μ0 (T)g(T, ρ)f (T, F, ρ), where h i μ0 (T) = 1.0 × 10−9 μ0 exp −C σˆ 2 , C = 0.29,

h 

δ i  ˆ 2 + 0.7σˆ 2ρa3 exp 0.25 σ ,      3 ρa < 0.025 h i g(T, ρ) =
δ 2  ˆ ˆ exp 0.25 σ + 0.7 σ 2 × 0.025  ( )     ρa3 ≥ 0.025,

(8.5)

8.4 Supporting Information |

"





f (T, Fred , ρ) = exp h(Fred ) 1.05 − 1.2 ρa 

× σˆ

3/2

 p

3

 r(σ) ˆ # 

1 + 2Fred − 1

−2

141

,

eaF * ˆ = 0.7σˆ −0.7 , Fred = r(σ) , Fred = 0.16, σ  F*  Fred < 2red  43 FFred * , red    2 h(Fred ) = * Fred *  , −1  1 − 43 FFred * 2 ≤ F red ≤ F red , red

δ = 2.3


ln 0.5σˆ 2 + 1.4σˆ − ln(ln 4) . σˆ 2

8.4.4 Extrapolation to Non-Dispersive Transport The non-dispersive mobilities were extracted by performing simulations at high temperatures (nondispersive regime) and then extrapolating to room temperatures [261]. The minimal temperature at which transport becomes non-dispersive was estimated from an empirical relation lnN = 0.95



σ kB T

2

+ 5.4,

(8.6)

where N is the number of molecules (sites), and σ is energetic disorder. For 4096 molecules in the microscopic system with an energetic disorder of σ = 0.253 eV the equation predicts Tmin = 1686 K. Simulations at six different temperatures above this value (from 1800 K up to 57600 K) were fitted to the relation      μ a 2 b μ(T) = 03 exp − − (8.7) T T T2 which was then used to extract mobility at T = 300 K, see Figure 8.6. This was done for four different (small) field strengths F, yielding slightly different values in the order of 10−17 cm2 /Vs. Dispersive transport simulations at 300 K yield values that are around seven orders of magnitude higher (see circles in Figure8.6).

Acknowledgements This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) under the Priority Program “Elementary Processes of Organic Photovoltaics” (SPP 1355),

142 | 8 Parametrization of Lattice Models 100 10−2 10−4

µ (cm2 /Vs)

10−6

dispersive regime

non-dispersive regime

10−8 10−10 10−12 10−14

×107 9 × 105 7 × 105 5 × 105 3 × 105

10−16 −18

10

100

1000

10000

V/cm V/cm V/cm V/cm 100000

T (K)

Fig. 8.6. Extrapolation of high temperature mobilities to obtain a non-dispersive mobility at 300 K. Squares represent non-dispersive transport simulations at temperatures from 1800 K to 57600 K, which allow extrapolating down to lower temperatures (solid lines). This was done for four different (small) field strenghts, however, curves lie almost on top of each other. Circles are the uncorrected dispersive values from simulations at 300 K.

BMBF grant MESOMERIE (FKZ 13N10723) and MEDOS (FKZ 03EK3503B), and DFG program IRTG 1404. We are grateful to Moritz Philipp Hein for helpful discussions regarding the OFET measurements and to Carl Poelking, Falk May, Christian Lennartz, and Kostas Daoulas for a critical reading of the manuscript.

9 Drift–Diffusion with Microscopic Link Reproduced with permission from the PCCP Owner Societies from: Pascal Kordt, Sven Stodtmann, Alexander Badinski, Mustapha Al Helwi, Christian Lennartz, Denis Andrienko Parameter-free continuous drift–diffusion models of amorphous organic semiconductors Physical Chemistry Chemical Physics (2015). DOI: 10.1039/C5CP03605D

Continuous drift–diffusion models are routinely used to optimize organic semiconducting devices. Material properties are incorporated into these models via dependencies of diffusion constants, mobilities, and injection barriers on temperature, charge density, and external field. The respective expressions are often provided by the generic Gaussian disorder models, parametrized on experimental data. We show that this approach is limited by the fixed range of applicability of analytic expressions as well as approximations inherent to lattice models. To overcome these limitations we propose a scheme which first tabulates simulation results performed on small-scale off-lattice models, corrects for finite size effects, and then uses the tabulated mobility values to solve the drift–diffusion equations. The scheme is tested on DPBIC, a state of the art hole conductor for organic light emitting diodes. We find a good agreement between simulated and experimentally measured current–voltage characteristics for different film thicknesses and temperatures.

Introduction Macroscopic and mesoscopic properties of organic semiconductors, such as charge carrier mobility or the width of the density of states, are often extracted by fitting the solution of the drift–diffusion equation [246, 366, 474, 525, 526] to the experimentally measured current–voltage (IV) characteristics. The mobility and diffusion constant of charge carriers, which enter these equations, depend on charge carrier density, ρ, electric field, F, and temperature, T [114, 22]. For one-dimensional transport and specific rate expressions these dependencies can be obtained analytically [363, 255]. In three dimensions, semi-empirical analytic expressions based on fits to lattice models have been obtained [22, 114, 140, 141, 365]. The extended Gaussian disorder model (EGDM) [114], for example, provides a parametrization of the mobility, μ(ρ, F, T), for uncorrelated, Gaussian-distributed site energies, while the extended correlated disorder model (ECDM) [141] additionally accounts for spatial site energy correlations due to long-range charge–dipole interactions. The aforementioned approach has become a standard tool for analyzing experimental data [527]. It has, however, several issues: (i) Gaussian disorder models are

144 | 9 Drift–Diffusion with Microscopic Link parametrized only for materials with moderate energetic disorder, σ ≤ 0.15 eV at room temperature, while many amorphous materials have a higher σ. (ii) The spatial correlation of site energies in the ECDM is material-independent and has an (approximate) 1/r decay, where r is the intermolecular distance, but recent studies show that this decay may be different. [244] (iii) Due to the non-Gaussian shape of the density of states [250], the energetic disorder and the lattice constant are different from those provided by microscopic calculations [244], thus making them merely fitting parameters without a comprehensive link between macroscopic properties and the chemical composition of the material. In this paper we propose an approach which does not have these limitations. In a nutshell, the mobility dependence on charge density, field, and temperature is first tabulated by combining quantum mechanical, classical atomistic and coarse-grained stochastic models for charge transfer and transport. These tables, corrected for finitesize effects, are then used to solve the drift–diffusion equations. To illustrate the advantages of the method, we compare it to the ECDM and the Mott–Gurney model [133] as well as to experimental measurements performed on amorphous layers of Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-phenylene]Ir (DPBIC), a hole-conducting material used in organic light emitting diodes (OLEDs) [528] and organic photovoltaic cells (OPVs) [529]. The paper is organized as follows. In the Methods section we describe the coarsegrained, off-lattice transport model, the procedure used to tabulate the charge carrier mobility, the algorithm used to solve drift–diffusion equations, and the parametrization of the extended correlated Gaussian disorder model. The entire workflow is summarized in Figure 9.1. We also recapitulate the main results of the Mott–Gurney model and provide details of experimental measurements. The IV curves, electrostatic potential, and charge density profiles are then compared in Section 9.2, where we also validate the transferability of the method by studying different layer thicknesses and temperatures. A short summary concludes the paper.

9.1 Methods 9.1.1 Tabulated Mobility To tabulate charge carrier mobility as a function of temperature, field, and charge density, we first simulate amorphous morphologies of N = 4000 molecules using molecular dynamics simulations in the NPT ensemble with a Berendsen barostat and thermostat [330]. The simulation box is equilibrated at 700 K for 1 ns, which is well above the glass transition temperature, and then quenched to 300 K during 1.3 ns. The force-field is tailored for the DPBIC molecule as described elsewhere [247] by performing potential energy scans using density functional theory (DFT) calculations with the B3LYP

9.1 Methods

| 145

Fig. 9.1. Overview of the method. (a) Chemical structure used to parametrize atomistic force field. (b) Amorphous morphology obtained using molecular dynamics simulations. (c) A coarse-grained model for the charge transport network. (d) Kinetic Monte Carlo simulations are used to tabulate the mobilities. (e) Solution of drift–diffusion equations.

146 | 9 Drift–Diffusion with Microscopic Link functional and the 6-311g(d,p) basis set. The Gaussian package [530] was used for all energy calculations. The charge transport network is then generated as follows. A list of links is constructed from all molecules with adjacent conjugated segments closer than 0.7 nm. For each link a charge transfer rate is calculated using Marcus theory, i.e., in the hightemperature limit of the non-adiabatic charge transfer theory [207], "
2 # J 2ij ∆E ij − λ ij 2π p exp − , (9.1) ω ij = ~ 4λ ij kB T 4πλ ij kB T

where ~ is the Planck constant, kB the Boltzmann constant, and T the temperature. Electronic couplings J ij are evaluated for each dimer by using the dimer projection method [393], the PBE functional and the def2-TZVP basis set. These calculations were performed using the TURBOMOLE package [531]. Note that the values of electronic couplings can deviate by up to 50 %, depending on the functional and the basis set size [450, 532]. This deviation is, however, systematic and will result in a constant prefactor for the mobility, i.e., we do not expect any changes in functional dependencies on the external field, charge density, or temperature. The hole reorganization energy [369], λ ij = 0.068 eV, was evaluated in the gas phase using the B3LYP functional and 6-311g(d,p) basis set. Site energy differences, ∆E ij = E i − E j were evaluated using a perturbative scheme [414] with the molecular environment modeled by a polarizable force-field, parametrized specifically for these pol el calculations. In this approach, the site energy E i = Eint + qF · r i is the i + Ei + Ei int sum of the gas phase ionization potential, E i = 5.87 eV, an electrostatic part, Eel i , an induction contribution, Epol , and the contribution due to an external electric field, i qF · r i . The mean value of these energies gives an ionization potential of EIP = 5.28 eV. The electrostatic contribution was evaluated using the Ewald summation technique [242, 388] adapted for charged, semi-periodic systems [441, 533] and distributed multipole expansions [439, 534]. Note that using an interaction cutoff would yield a shifted energetic landscape with an underestimated spatial correlation of energies [390]. The induction contribution, Epol i , was calculated self-consistently using the Thole model [336, 440] with a 3 nm interaction range. Note that the set of Thole polarizabilities were scaled in order to match the volume of the polarizability ellipsoid calculated using the B3LYP functional and 6-311g(d,p) basis set. This step is required to account for larger polarizabilities of conjugated, as compared to biological, molecules. The resulting charge transport network is used to parametrize the coarse-grained model, by matching characteristic morphological and transport properties of the system, such as the radial distribution function of molecular positions, the list of neighboring molecules, the site energy distribution and spatial correlation, and the distance-dependent distribution of transfer integrals [244, 337]. The coarse-grained model allows to study larger systems, here of 4 × 104 and 4 × 105 sites, which are re-

9.1 Methods |

147

quired to perform simulations at low charge carrier densities, in our case from 0.025 down to 10−5 carriers per site. Charge transport is modeled using the kinetic Monte Carlo (KMC) algorithm. Note that charge carriers interact only via the exclusion principle, i.e., a double occupation of a molecule is forbidden. Charge mobility is evaluated by averaging the carrier velocity along the field, μ = hvi · F/F 2 . KMC simulations are repeated for eight different temperature values, from 220 K to 992 K, and twelve field values, in the range of 2.5 − 30 × 107 V/m. To avoid finite size effects, an extrapolation procedure [261, 394] is used for small charge carrier densities. The mobility is simulated at a range of higher temperatures, where mobilities are non-dispersive and hence system-size independent. The extrapolation to lower temperatures is performed by parametrizing the analytic mobility versus temperature dependence available for one-dimensional systems [363] or, alternatively, using the box-size scaling relation [394]. The tabulated mobility is finally interpolated and smoothed by the scattered data interpolation method using radial base functions [535], which can treat manydimensional, unstructured data.

9.1.2 Drift–Diffusion Modeling Macroscopic dynamics of electrons/holes (n/p) is modeled using one-dimensional drift diffusion equations Jn/p = ±ρn/p μn/p ∇ψ − Dn/p ∇ρn/p , ∂ρn/p = −∇ · Jn/p , ∂t

(9.2) (9.3)

coupled to the Poisson equation ∆ψ = −

ρn − ρp . ε0 εr

(9.4)

Here ψ denotes the electrostatic potential, D is the diffusion constant, ε0 the vacuum permittivity and εr the relative permittivity. J = I/A is the current density, where I is the current and A the electrode area. In case of DPBIC we are interested in hole transport only, hence the electron current density, J n , and density, ρ n , are set to zero and only the hole equations need to be solved. To simplify the notation we omit the index n/p. Here we are interested only in the steady state, i.e., ∂ρ/∂t = 0 in Equation (9.3). Since charge carriers occupy energetic levels according to Fermi–Dirac statistics, the carrier density is related to the quasi-Fermi level, η, as N ρ (η) = V

Z∞

−∞

−1   E−η g(E) 1 + exp dE, kB T

(9.5)

148 | 9 Drift–Diffusion with Microscopic Link where g(E) is the density of states and V the box volume. The diffusion coefficient and mobility in Equation (9.2) are related via the generalized Einstein relation [468]   ρμ ∂ρ . (9.6) D= e ∂η Equations (9.2) – (9.6) are solved using an iterative scheme, until a self-consistent solution for electrostatic potential, ψ, density, ρ, and current, I, is found [79]. First the equations are rescaled to ensure numerical stability, which is necessary since carrier density and electrostatic potential vary by several orders of magnitude. Then they are discretized according to a scheme proposed by Scharfetter and Gummel [536], linearized [537], and solved by using the Gummel iteration method [538], adapted to organic semiconductors at finite carrier density. This method is less sensitive to the initial value than a Newton algorithm and thus is the method of choice despite its slower convergence [539] in terms of iteration steps. The tabulated mobility values, μ (F, ρ, T ), computed in section 9.1.1, are used while solving Equations (9.2) – (9.6). We use Dirichlet boundary conditions for the electrostatic potential, ψ, by setting the potential difference at the boundaries to ψeff = Vapp − Vint , where Vapp is the applied potential and Vint the built-in potential, defined as the difference of the materials’ work functions. For ITO and Aluminum we use experimental values: The work function of ITO is reported to lie in the range from 4.15 eV to 5.30 eV [540, 541, 542, 543], and for Aluminum from 4.06 eV to 4.26 eV [544]. Here we assume average values of 4.73 eV for ITO and 4.16 eV for Aluminum. In combination with the calculated DPBIC solid-state ionization potential (IP) of 5.28 eV, which is the mean value of the site energies, E i , that are calculated as described before, this yields injection barriers of ∆EITO = 0.55 eV and ∆EAl = 1.12 eV. The charge density at the electrodes is fixed to the density resulting from inserting ∆EITO/Al into Equation (9.5). To model the doped interlayers (see Section 9.1.5) within a five nanometer range from both electrodes, an additional charge concentration of 3 × 10−4 carriers per site, estimated from previous calcuations [545] is added in these regions when solving the Poisson equation (9.4), leading to high hole densities in the doped regions even without space-charge limited effects.

9.1.3 Lattice Model To test the validity of lattice models, we have also parametrized the extended correlated disorder model (ECDM) [141] by fitting the simulation results to the ECDM expression for mobility. The fit was performed for charge densities, ρ, in the range of 8.7 − 140 × 1023 m−3 , including an extrapolated, non-dispersive zero-density mobility [261], and electric fields in the range of 3 − 9 × 107 Vm−1 . For the extrapolation temperatures from 1200 K to 50000 K, giving non-dispersive transport in the small system, were used. All simulations were performed at 300 K.

149

9.1 Methods |

The fit to the ECDM model yields a lattice constant of a = 0.44 nm, an energetic disorder of σ = 0.211 eV, and a zero-field zero-density mobility of μ0 (300 K) = 1.8 × 10−13 m2 V−1 s−1 [247]. These values serve mainly for providing a fitting and extrapolation function as they differ from the values observed in microscopic simulations (a = 1.06 nm, σ = 0.176 eV, μ0 (300 K) = 3.4 × 10−12 m2 V−1 s−1 ).

9.1.4 Mott–Gurney Model The Mott–Gurney, or trap-free insulator model [133], predicts a current density of J(V) =

9 V2 εr μ 3 , 8 d

(9.7)

where d is the thickness of the sample and εr the material’s relative permittivity (here we have chosen εr = 3). This expression is only valid under the assumptions of (i) holeonly (or electron-only) transport, (ii) no doping, (iii) constant mobility and relative permittivity and (iv) no injection barriers. The electrostatic potential and hole density throughout the sample are then given by 3 d−x 2 , d 3 εr V 1 √ ρ(x) = , 4 qd 32 d − x

Vint (x) = V



(9.8) (9.9)

where 0 ≤ x ≤ d. A mobility of μ = 3 × 10−22 m2 /Vs has been chosen to provide the best match of the experimental data.

9.1.5 Experimental Measurements IV curves were measured for three film thicknesses: 203 nm, 257 nm and 314 nm, including two interlayers of DPBIC doped with molybdenum trioxide (MoO3 ) of 5 nm thickness on both sides of the DPBIC film. These serve to enhance the injection efficiency, which has been taken into account in our model by the previously mentioned additional charge in these regions. The hole-conducting DPBIC layer was sandwiched between a 140 nm indium tin oxide (ITO) anode and a 100 nm Aluminum cathode. To control the temperature, the samples are placed into the oil reservoir of a cryostat, which allows for a variation between 220 K and 330 K. The voltage was varied between 0 V and 20 V. All films were fabricated by vacuum thermal evaporation of DPBIC on a glass substrate, patterned with the ITO layer. Thicknesses were determined by optical ellipsometry after a simultaneous deposition of the same amount of DPBIC on a silicon wafer.

150 | 9 Drift–Diffusion with Microscopic Link 102

current density (mA cm−2 )

(a)

101

100

10−1

10−2

experiment tabulated mobilities ECDM Mott–Gurney

0

2

4

6

8

10

12

14

voltage (V)

(b)

tabulated mobilities ECDM Mott–Gurney

electrostatic potential (V)

4

3

2

1

0 0

50

100

150

200

250

300

250

300

position (nm)

(c) 1026

tabulated mobilities ECDM Mott–Gurney

hole density (cm−3 )

1025 1024 1023 1022 1021 1020 0

50

100

150

200

position (nm)

Fig. 9.2. (a) Current–voltage characteristics, (b) electrostatic potential profiles, and (c) hole density profiles. Slab thickness 314 nm, temperature 300 K. (b) and (c) are plotted for an external voltage of 4 V.

9.2 Validation

| 151

9.2 Validation We first compare the current–voltage characteristics, the electrostatic potential and the hole density profiles calculated using tabulated and ECDM mobilities, which are shown in Figure 9.2, together with the experimentally measured current–voltage characteristics. One can see that the experimentally measured current–voltage characteristics are well reproduced using the tabulated mobilities. The ECDM underestimates the current by an order of magnitude and there is a clear mismatch of the slope, as it can be seen in Figure 9.2(a). It also predicts a negative electrostatic force at the beginning of the slab, Figure 9.2(b), and a very steep charge accumulation at the injecting anode, Figure 9.2(c). The disagreement is due to the high energetic disorder obtained from the fit, σ = 0.211 eV, which is outside the range used to parametrize the ECDM expression. In addition, the ECDM does not reproduce the spatial correlation of site energies well. Finally, the Mott–Gurney model does not reproduce experimental results even qualitatively: it neither takes into account doped layers nor field- or density-dependence of the mobility. To illustrate the transferability of the proposed method we also compare current– voltage characteristics for different temperatures and different film thicknesses. Figure 9.3 shows that for high temperatures the agreement between theory and experiment is excellent. At 233 K deviations are significant and can be attributed to the breakdown of the drift–diffusion description, since at low temperature and large energetic disorder charge transport becomes dispersive, showing anomalous diffusion [545]. Its description using equilibrium distributions, mobility and diffusion constant cannot be justified in this situation. Moreover, Marcus theory only applies to sufficiently high temperatures. The crossover temperature below which Miller–Abrahams rates [205] become a more appropriate description has been estimated to be about 250 K [216]. 100

233 K

current density (mA cm−2 )

10 1 100

293 K

10 1 100

313 K

10 203 nm 257 nm 314 nm

1 0.1 0

2

4

6

8

10

12

14

voltage (V)

Fig. 9.3. Current-voltage characteristics for different temperatures and slab thicknesses simulated using tabulated mobilities (lines) and measured (symbols).

152 | 9 Drift–Diffusion with Microscopic Link

9.3 Conclusions To conclude, we have proposed a parametrization scheme for drift–diffusion equations which is based on evaluation of charge transfer rates, simulation of charge transport in a coarse-grained charge transport network, and tabulation of charge carrier mobility as a function of field, charge density and temperature. The method is rather general, in part because it is not limited to functional dependencies build into the ECDM and EGDM models and, therefore, allows to treat systems with large energetic disorder and material-specific spatial site energy correlation functions. Using this scheme, we have simulated IV characteristics of a single-layer device, and found them to be in a good agreement with the experimentally measured IV curves, whereas significant deviations have been observed for the ECDM and Mott– Gurney models.

Acknowledgements This project has received funding from the “NMP-20-2014 - Widening materials models” program (project MOSTOPHOS) under grant agreement No 646259. The work was also partially supported by Deutsche Forschungsgemeinschaft (DFG) under the Priority Program “Elementary Processes of Organic Photovoltaics” (SPP 1355), BMBF grant MESOMERIE (FKZ 13N10723) and MEDOS (FKZ 03EK3503B), and DFG program IRTG 1404. We are grateful to Carl Poelking, Anton Melnyk, Paul Blom, Kurt Kremer, and Aoife Fogarty for a critical reading of the manuscript.

Conclusions and Outlook In this work I develop strategies for the prediction of the mobility and current–voltage characteristics in organic semiconductors, linked to the chemical structure. Applying a stochastic model allows to go to system sizes about two orders of magnitude larger than in microscopic simulations, an important ingredient for linking the different scales. For going from the hopping transport description and the mobility to the continuous space drift–diffusion description I develop two different strategies. The first one is a parametrization of the mobility function given by GDM models, the second one a parameter-free coupling of tabulated mobilities after interpolation and smoothing. If systems are not too large there is another option, namely to perform Kinetic Monte Carlo simulations directly at the device scale, employing the stochastic model. One of the biggest achievements of this work is the almost parameter-free prediction of current–voltage characteristics of amorphous organic small molecule semiconductors, taking only the chemical structure as an input. A validation of my results against experimental data, measured at BASF, shows good agreement. Missing ingredients towards a prediction entirely independent of experimental input and model assumptions are (i) a quantum-mechanical calculation and understanding of charge carrier injection from the electrode materials into the organic layers and (ii) a calculation to determine the doping efficiency and thus the extra charge density in doped interlayers, that are used in OLEDs to increase the doping efficiency. This is clearly not an easy task. However, comparing to what has already been achieved in the field it looks to me that it will be doable and thus not too much is missing for completely parameter-free predictions starting from the chemical structure. The missing building blocks constitute interesting topics for future research. An important figure of merit for understanding the conduction properties and charger carrier mobility is their density in the material. In the course of this work I implemented a Monte Carlo algorithm for hopping transport of multiple charge carriers in a morphology obtained from microscopic simulations. To the best of my knowledge, this has been done for the first time. The dependency of the mobility on the charge carrier density did not show any surprising results compared to what has already been known from lattice simulations of charge transport. However, only with the implementation of the multiple charge Monte Carlo in realistic morphologies it was possible to couple the mobility results to drift–diffusion equations via parametrized GDM functions or interpolated functions. Using this coupling allowed for studies of the charge density distribution in an organic semiconductor film, as it is used in an OLED as a hole-conducting layer. At both electrodes a large charge accumulation is observed (∼ 1025 cm−3 ) and the density throughout the device varies by about three orders of magnitude. The density profile obtained from this approach differs slightly from ECDM lattice calculations. Completely different is the prediction of the heuris-

154 | Conclusions and Outlook tic Mott–Gurney model which also neglects the doped interlayers. This model is still frequently used in the analysis of experimental current–voltage characteristics and, with the mentioned results in mind, it is questionable that this allows for a meaningful quantitative analysis. Another aspect is the comparison of parametrized lattice models to microscopic calculations. The conclusion here is that lattice models can serve as a reasonable parametrization of the mobility, yet they do not capture everything. The results found for the current differ from the microscopic simulation and the experimental findings by more than one order of magnitude. Two primary weaknesses of lattice models were identified. First, they are parametrized up to an energetic order of about 0.15 eV at room temperature and many amorphous organic semiconductors have higher values of energetic disorder. Second, they either neglect spatial correlation of the site energies or they assume a model that fixes the autocorrelation function apart from the lattice constant as the only free parameter. This is a result of assuming a constant absolute value of the dipole moment, while in microscopic simulations I observed wide distributions that can differ significantly for different molecule geometries. Neglecting this fact leads to fitting values for the lattice constant that are usually overestimated in the EGDM (no correlation) and underestimated in the ECDM (fixed correlation function). Overall, the observation is that the simpler, uncorrelated model surprisingly agrees better with microscopic calculations. The reason behind is probably an over-estimation of the correlation in the ECDM. To account for the importance of correlations I developed an algorithm for the reproduction of correlation functions found in microscopic calculation. This algorithm was an important part of the above mentioned stochastic model. It could also be applied to lattice models for transport if the microscopic correlation function is known. In practice, simulations of charge transport can only be conducted at limited system sizes. Every researcher in the field faces the question what size he needs and whether or not the feasible system size is enough to obtain converged values of the mobility, that do not depend on the size of the system, N. In this work I developed a strategy for an analytic a priori estimate of the system size. I found that the necessary system size depends not only on the energetic disorder of the system but also on the charge density. While for high densities smaller system are sufficient, large system are necessary at low densities or in the limit of zero charge carrier density. For the latter case a 1/N dependence of the logarithmic mobility is found. This dependence allows for an extrapolation to obtain the mobility in the thermodynamic limit. As part of this work the amorphous phase of two different molecules was studied, DCV4T and DPBIC. Depending on charge density and electric field, DCV4T showed hole mobilities between 10−11 and 10−5 cm2 V−1 s−1 . For DPBIC mobilities in the range of 10−8 up to 10−2 cm2 V−1 s−1 were observed. The values of the energetic disorder obtained from microscopic simulations were 0.253 eV for DCV4T and 0.176 eV for DPBIC. In both cases the value is higher than the range covered in the derivation of the GDM mobility. DCV4T is mainly used as a material in organic solar cells in the semi-

Conclusions and Outlook | 155

crystalline or crystalline phase, where the energetic disorder is smaller and, consequently, mobilities are higher. DPBIC is used in OLEDs as a hole-conducting material and the values obtained here show that the mobility can go up to comparably high values (for an organic semiconductor) at high charge carrier densities. Kinetic Monte Carlo simulations at the device scale have the potential of extending the model to study other effects on the microscopic scale. Possible interesting topics for future research include electron–hole recombination, exciton transport properties, exciton creation upon illumination in solar cells, exciton loss processes, and radiative exciton decay in OLEDs. This would pave the way towards in silico efficiency studies for organic semiconductor devices. However, there are many variables and, if it will ever be done, it is still a long way to go towards such predictions without experimental input. Another interesting application for future studies is the drift–diffusion description with microscopic input. In principle it has most ingredients for an ab initio study of complete multilayer structures in OLEDs or organic solar cells. The limiting factor at the moment is mainly that the force field of each molecule has to be parametrized by hand. If this procedure and the calculation steps were automated it would be possible to obtain mobilities from chemical structures automatically, provided the computational resources are large enough. This, again, could be coupled to statistical machine learning methods. After identifying important molecule parameters, a training set of molecules could be designed and after training the statistical model it might be possible to sample a wide range of new chemical structures and test them for their charge carrier mobility properties. Once suitable candidates for high mobility are found by the statistical approach, the computational expensive procedure for an exact evaluation of their mobility could be performed to test if the mobility is indeed as high as predicted by the statistical model – if not there would at least be another training point for the statistical model, improving the statistic prediction. Such an approach would allow to search for high mobility materials in the infinite space of possible organic molecules using a strategic approach and not only chemical intuition. Due to the increasing abilities and accuracy of computational methods one may expect simulations to play an increasingly important role in the field of organic semiconductors. The unique properties of these materials allow for interesting applications. With an increased understanding as a result of research effort some of them will probably soon be realized.

A Molecule Abbreviations Table A.1. List of all abbreviations of molecular structures that are mentioned in the text. For each structure an exemplary reference, out of many possibilities, is given. Abbreviation

Name

Reference

Alq3 BCP BEDT-TTF Bphen CoCp2 DCV4T DPBIC F4 -TCNQ ITO MEH-PPV P3HT PCBM PCPDTBT

aluminum-tris(8-hydroxychinolin) bath-ocuproine-4,7-diphenyl-2,9-dimethyl-1,10-phenanthroline bis(ethylenedithio)-tetrathiafulvalene 3-(4-biphenylyl)-4-phenyl-5-tert-butylphenyl-1,2,4-triazole bis(cyclopentadienyl)-cobalt(II) or cobaltocene dicyanovinyl-substituted quaterthiophene Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-phenylene]Ir 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane indium tin oxide, composite of SnO2 (10%–20%) and In2 O3 (90%–80%) poly(2-methoxy,5-(2’-ethylhexyloxy)-p-phenylenevinylene) poly(3-hexylethiophene) [6,6]-phenyl-C61-butyric acid methyl ester poly[(4,4’-bis(2-ethylhexyl)dithieno[3,2-b:2’,3’-d]silole)2,6-diyl-alt-(2,1,3-benzothiadiazole)-4,7-diyl] bis(terpyridine)ruthenium 6,13-bis(triisopropylsilylethinyl)pentacene tetrathianaphthacene zinc phthalocyanine

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[Ru(terpy)2 ]0 TIPS-pentacene TTN ZnPc

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Index ab initio 70 absorption 32 absorption spectroscopy 29 absorption spectrum 4 adiabatic approximation 35 adiabatic basis 35 AMOLED 3, 13 amorphous 108 Andersen thermostat 76 Arrhenius dependence 69 autocorrelation function 56, 97, 138 band theory 40 Berendsen barostat 144 Berendsen thermostat 76 binary tree 121 Boltzmann inversion 80 bonded interaction 74 Born–Oppenheimer approx. 35, 71 brightness 17 bulk heterojunction 4 canonical ensemble 75 capacitance–voltage analysis 29 CDM 56, 94 CELIV 26, 28 CFL 3 Chapman-Kolmogorov equation 41 charge carrier energy 85 charge density 5, 27, 90, 151 charge mobility 5, 21, 44, 119 charge transfer 35, 109 Child’s law 23, 67 Child–Langmuir law 67 coarse-graining 6, 78, 112, 146 compact fluorescent lamp 3 continuous time random walk 59 convection–diffusion equation 65 coordination number 46 correlation 56, 94, 97, 134, 138 Coulomb interaction 49 CS state 10 CT state 10 CTRW 59 current density 147

current–voltage characteristics 23, 24, 149, 151 degradation 5 density 5, 27, 90, 151 density functional theory 71 detailed balance 43 device simulations 121 DFT 71 diabatic basis 35, 36 diffusion coefficient 148 diffusion constant 44 diode laser 18 dipole 111, 138 dispersive mobility 83 doping 30 DPBIC 107 drain electrode 17 drift–diffusion equation 66, 122, 147 driving voltage 33 ECDM 57, 94, 117, 131, 148, 152 effective medium approximation 60 efficiency 10 EGDM 56, 117, 131 Einstein relation 45, 148 electric pumping 18 electrode 13, 17, 148 electroluminescence 13 electron affinity 110 electronic coupling 112, 133 electrostatic energy 77 electrostatic interaction 146 electrostatic potential 151 emission 12 energetic disorder 55, 133, 149 energy gap 77 energy levels 10 EQE 10 escape rate 46 Ewald summation 78, 110, 146 exchange–correlation energy 72 excited state 14 exciton 10, 14, 105 exciton decay 12 exciton recombination 15

190 | Index

exclusion principle 46 external quantum efficiency 10 extrapolation 84, 135, 141, 147 Fermi–Dirac statistics 46, 85, 147 FET 17 field effect transistor 17, 21 film thickness 149 finite size effect 83, 92 finite size error 92 fluorescence 5, 13 Fokker–Planck equation 65 foldable displays 17 force field 74, 108, 132 force matching 82 four-level laser 19 Frenkel exciton 4, 9 gate electrode 17 Gaussian disorder model 24, 55, 94, 117, 148 GDM 24, 55, 94, 117, 131, 148 generalized Einstein relation 148 Gillespie algorithm 45 Gummel iteration 148 GW method 6 Hartree–Fock method 71 Henderson theorem 79 Hohenberg–Kohn theorem 71 HOMO 4, 10, 16, 30, 112 hopping transport 40 host–guest system 13, 104 hot carriers 120 hybridization 1 IBI 80 impedance spectroscopy 27 induction 76, 110, 146 injection 13 inter-system crossing 15 internal quantum efficiency 10 inverse Monte Carlo 81 ionization potential 110, 146, 148 IQE 10 isobaric-isothermal ensemble 75 iterative Boltzmann inversion 80 ITO 13 Kelvin probe measurement 28

kinetic Monte Carlo 45, 120, 134, 147 KMC 45 Kohn–Sham equations 72 KPFM 28 Kramers–Moyal expansion 65 laser 18 lasing threshold 19 lattice constant 149 lattice model 55, 143, 148 LCD display 2 Lennard–Jones potential 74, 79, 108 lifetime 5 light-emitting OFET 17 lighting 3 liquid crystal display 2 loss mechanisms 16 luminescence 13 luminosity 3, 17 LUMO 4, 10, 16, 30, 112 Marcus rate 36, 85, 109 Markov process 41 master equation 43, 119 MD 70, 73 microcanonical ensemble 75 microscopic model 107 Miller–Abrahams rate 36, 94 minimum image convention 75 mobility 5, 21, 44, 88, 98, 119, 149 molecular dynamics 70, 73, 109 molecular mechanics 70, 73 morphology 108 Mott–Gurney model 23, 68, 149, 152 multiple trapping and release 58 n-type doping 31 nondispersive mobility 83 Nosé–Hoover thermostat 76 NPT ensemble 75, 132, 144 NVE ensemble 75 NVT ensemble 75 OFET 21, 134 off-lattice Monte Carlo 125 OLED 12, 104 OLED display 2, 17 one-dimensional hopping 58 open circuit voltage 4

Index

OPLS 95, 132 optical pumping 18 optical spacers 32 OPV 4 organic light emitting diode 12 p-n junction 9, 32, 68 p-type doping 30 partial charge 74 partial charges 133 Pauli principle 46 percolation theory 61 periodic boundary conditions 75 phosphorescence 5, 13, 104 photo-CELIV 26 photoluminescence 13 π orbital 1 π stacking 2 Poisson equation 147 Poisson process 42 polarization 76 Poole–Frenkel dependence 69, 94, 138 population inversion 18 printing 3 QM/MM 70

source electrode 17 space-charge limited current 23 spatial correlation 56, 94 spectroscopy 108 spin 14 spin–orbit coupling 15, 104 spontaneous emission 18 stimulated emission 18 stochastic model 112, 134, 139 system size 83 tabulated mobility 144 TADF 14, 15 tandem cell 32 TD-DFT 6 TFT 18 thin film transistor 18 Thole model 77, 146 time-of-flight measurement 25 TOF 25 transfer integral 112 transport energy 63 triplet 10, 14 triplet harvesting 15 triplet–polaron quenching 16 triplet–triplet quenching 16

radial distribution function 78, 146 RDF 78 recombination 12, 15 reorganization energy 36, 98, 111, 133, 146

UPS 33

Shockley diode equation 68 singlet 10, 14 site energy 77, 110, 133 solar cell 4

Weiss–Dorsey rate 37 work function 148

vacuum deposition 3 VSSM 120, 134

ZINDO 133

| 191