Modeling of charge-transport processes for predictivesimulation of OLEDsCitation for published version (APA):Cottaar, J. (2012). Modeling of charge-transport processes for predictive simulation of OLEDs. TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR740068

DOI:10.6100/IR740068

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Modeling of charge-transport processes forpredictive simulation of OLEDs

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voorPromoties in het openbaar te verdedigen

op dinsdag 18 december om 16.00 uur

door

Jeroen Cottaar

geboren te Eindhoven

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr. R. Coehoornenprof.dr. M.A.J. Michels

Copromotor:dr. P.A. Bobbert

A catalogue record is available from the Eindhoven University of Technology Library.

ISBN: 978-90-386-3291-9

Cover design: Paul Verspaget

Printed by: Universiteitsdrukkerij Technische Universiteit Eindhoven

This research forms part of the research programme of the Dutch Polymer Institute (DPI),project #680.

Copyright© 2012 by Jeroen Cottaar

Contents

1 Introduction 41.1 Organic light-emitting diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 White OLED design challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Predictive simulation of OLEDs . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Charge transport in organic semiconductors . . . . . . . . . . . . . . . . . . . . 181.5 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Numerical methods 302.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 General considerations for 3D computations . . . . . . . . . . . . . . . . . . . . 332.3 3D Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 3D master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 1D drift-diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6 Time-dependent calculations with the 3D master equation . . . . . . . . . . . 46

3 A scaling theory for the charge-carrier mobility 523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Scaling formula for the charge-carrier mobility . . . . . . . . . . . . . . . . . . 553.3 Application to different hopping models . . . . . . . . . . . . . . . . . . . . . . . 603.4 Consequences for charge transport . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.A Density of states for dipole-correlated disorder . . . . . . . . . . . . . . . . . . . 71

4 Field-induced detrapping in host-guest systems 744.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 Generalized Hoesterey-Letson detrapping model . . . . . . . . . . . . . . . . . 784.3 Field-dependent occupation function . . . . . . . . . . . . . . . . . . . . . . . . . 824.4 Device applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 Charge transport at non-zero electric field 925.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2 The electric-field dependence of the charge-carrier mobility . . . . . . . . . . . 95

CONTENTS 3

5.3 Device applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4 Variable-range hopping and lattice disorder . . . . . . . . . . . . . . . . . . . . 1025.5 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Charge transport across disordered organic heterojunctions 1086.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2 Three-dimensional and one-dimensional modeling . . . . . . . . . . . . . . . . 1116.3 Improvements to the one-dimensional model . . . . . . . . . . . . . . . . . . . . 1136.4 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Conclusions and outlook 1227.1 Main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.2 Outlook on applications and further research . . . . . . . . . . . . . . . . . . . 1267.3 Coda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A Unified 1D drift-diffusion method 131A.1 Zero electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.2 Host-guest systems at non-zero electric field . . . . . . . . . . . . . . . . . . . . 133A.3 Internal interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Summary 135

List of Publications 139

Curriculum Vitae 141

Acknowledgements / Dankwoord 143

Chapter 1

Introduction

Abstract

Intensive research is taking place into alternative light sources to replace incandescent andfluorescent lamps. Organic light-emitting diodes (OLEDs) show great promise, with theirmain potential advantages being high energy efficiency, cheap roll-to-roll production, excel-lent color rendering and a unique form factor. However, significant challenges must still beovercome, particularly in the areas of luminous efficacy, lifetime and manufacturing. Sev-eral approaches to overcoming these challenges have been proposed, but it is difficult todesign optimized devices around these approaches. This is because at present this designtakes place through trial and error, which makes investigating the full parameter spaceof material choices and layer stack design virtually impossible. To improve this designprocess, a predictive OLED model is needed.

A full predictive OLED model takes as input the layer stack design, deposition methodsand chemical structures of the materials involved, and gives as output the angle-dependentemission spectrum and current-voltage characteristics of the device. This involves molec-ular dynamics, density functional theory, charge-transport modeling, excitonics and pho-tonics. However, charge-transport modeling by itself already yields useful results, suchas current-voltage characteristics and exciton generation locations. In such modeling onethree-dimensionally (3D) simulates the charge transport in organic semiconductors, whichtakes place by hopping of charge carriers between localized sites. Since this 3D simula-tion is computationally expensive, the results must be translated to a fast one-dimensional(1D) drift-diffusion approach to be of use in the OLED design process. In this translation,3D simulation is used in two ways: to determine parameters like the charge-carrier mo-bility, and to validate the results of the 1D approach. Charge-transport modeling and the3D-to-1D translation are the focus of this thesis.

5

Contents

1.1 Organic light-emitting diodes . . . . . . . . . . . . . . . . . . . . . . 61.1.1 Organic electronics . . . . . . . . . . . . . . . . . . . . . . . . 71.1.2 From organic electroluminescence to modern OLEDs . . . 81.1.3 White OLED device structures . . . . . . . . . . . . . . . . . 8

1.2 White OLED design challenges . . . . . . . . . . . . . . . . . . . . . 101.2.1 Luminous efficacy . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Device lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.3 Large-scale low-cost manufacturing . . . . . . . . . . . . . . 13

1.3 Predictive simulation of OLEDs . . . . . . . . . . . . . . . . . . . . 141.3.1 Modeling steps . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 The role of charge-transport modeling . . . . . . . . . . . . 161.3.3 One-dimensional modeling . . . . . . . . . . . . . . . . . . . 161.3.4 Non-predictive modeling . . . . . . . . . . . . . . . . . . . . 18

1.4 Charge transport in organic semiconductors . . . . . . . . . . . . . 181.4.1 The hopping model . . . . . . . . . . . . . . . . . . . . . . . . 191.4.2 History of simulating hopping transport . . . . . . . . . . . 22

1.5 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Introduction

With current energy sources being rapidly depleted and alternative sources emerging onlyslowly, it is crucial to make the most of what energy we have. About 2200 TWh/year, 17.5%of global electricity use, is used worldwide for lighting.1 This has spurred the develop-ment of alternative light sources to replace incandescent and fluorescent lamps. Presently,the main emerging technology is solid-state lighting based on inorganic light-emittingdiodes. A less mature but world-wide intensively studied type of alternative light source isbased on organic light-emitting diode (OLED) technology.2 The main potential advantagesof OLEDs are high energy efficiency,3 cheap roll-to-roll fabrication,4 and excellent colorrendering.5 In addition, OLEDs have a unique form factor: they are a very thin and homo-geneous area light source, which can be made transparent, flexible, and even stretchable.6

This makes new applications possible, for example in smart textiles.

(a) (b) (c)

Figure 1.1: Some examples of current and future OLED applications. (a) Philips Lumiblade, a high-end light source for design and architectural purposes. Source: Philips. (b) A prototypefor a room lit entirely by OLEDs. Source: Universal Display Corporation. (c) SamsungYoum, a flexible display for use in mobile devices. Source: Samsung.

However, OLEDs have not yet been able to break into the general lighting market,despite already being widely used in display applications. In this introduction, we arguethat predictive modeling, which at present is not often included in the design process,will play a crucial role in further developing OLEDs for lighting applications. First, wegive some general background on OLEDs (section 1.1). We then discuss the challengesthat must be overcome for OLEDs to play a role in general lighting, focusing on luminousefficacy, device lifetime, and manufacturing (section 1.2). Next, we show how predictivemodeling, and charge-transport modeling in particular, can help overcome these challenges(section 1.3). We then discuss this model for the charge-transport process, and how tosimulate it (section 1.4). Finally, we give an overview of the contents of this thesis (section1.5).

1.1 Organic light-emitting diodesAn OLED consists of one or more layers of organic materials; these can be small-moleculeor polymer based. These layers are sandwiched between two semiconductor or metal elec-trodes, at least one of which is transparent. When a sufficiently large voltage is applied,

1.1 Organic light-emitting diodes 7

electrons and holes will be injected into this device. These electrons and holes then meetand generate excitons.* Finally, these excitons recombine radiatively, emitting visible light.

In this section, we first give an introduction to conduction in organic materials in gen-eral. Next, we will discuss the timeline of OLEDs: history, current situation and futureoutlook. Finally, we will go over the device structures used in modern white OLEDs.

1.1.1 Organic electronics

Organic materials are typically thought of as insulators. However, π-conjugated materi-als, which contain chains of alternating single and double carbon bonds, have molecularelectron orbitals delocalized along these chains.7 In materials such as graphene these elec-trons can be delocalized enough to form valence and conduction bands comparable to thosein inorganic semiconductors.8 However, OLEDs are typically based on amorphous organicsemiconductors.† In these materials the delocalization only extends to single moleculesor polymer segments, on a scale of nanometers. These materials conduct charge by thetunneling of electrons between the localization sites. The charge transport properties arestrongly determined by the energetic disorder, which results from the amorphous natureof these materials.

Significant conductivity in organic semiconductors was first observed in 1963 in thepolymer polypyrrole.10 In 1974 the first organic device, a voltage-controlled switch basedon the polymer melanin, was built.11 Around this time organic photoconductors also startedseeing their first commercial use in xerographic devices, such as copying machines.12 Themodern era of organic-electronics research truly took off in 1977 with the Nobel prize win-ning work of Shirakawa et al. showing that the conductivity of organic materials can bedramatically increased by doping.13,14 A wide range of devices based on organic semi-conductors is now available or being developed. Apart from OLEDs and the aforemen-tioned xerographic devices, these include photovoltaic cells,15,16 field-effect transistors17

and sensors.18

The most important advantage of organic semiconductors over their inorganic counter-parts is their potential ease of processing. They can in principle be deposited from solutionin a roll-to-roll process under ambient conditions or in a room temperature nitrogen en-vironment, while inorganic materials often have to be treated in vacuum and/or at hightemperatures. This easy processing also makes it much easier to produce flexible devices.Another advantage is that the whole organic chemistry toolbox is available, allowing highchemical tunability of material properties. Finally, organic materials are often cheaper andlighter than their inorganic counterparts. A significant disadvantage is faster degradation,particularly under the influence of oxygen and water.

*In literature, the term ‘recombination’ is often used to describe an electron and a hole generating an exciton. Wewill reserve this term for the (radiative or non-radiative) decay of this exciton.

†We assume amorphous structures throughout this thesis; however, organic semiconductors can in practice showvarying degrees of order based on the deposition method.9

8 Introduction

1.1.2 From organic electroluminescence to modern OLEDsElectroluminescence in small-molecule organic materials was first observed in 1953 byBernanose et al. in acridine orange and quinacrine,19 and in polymers in 1983 by Partridgein polyvinylcarbazole.20 The first thin-film OLED was developed by Tang and Van Slyke in1987 based on small molecules.21 Since 1990, polymer OLEDs (also known as PLEDs) arealso being investigated.22,23 The first white OLED was developed by Kido et al. in 1994.24

In 1997, the first commercial OLED product appeared on the market: a monochromatic cardisplay manufactured by Pioneer.

In the flat-panel display market OLEDs now play a significant role. Many high-endmobile phones use OLED displays, with their main advantages being high image qual-ity, thinness, and high power efficiency compared to LCD displays if a dark backgroundis used.* However, an OLED suitable for display applications is not necessarily a goodcandidate for general lighting. In particular, lighting applications typically need higherbrightness, higher luminous efficacy, larger areas, lower manufacturing costs and improvedcolor stability.25 So far this has limited commercially available OLEDs, such as the PhilipsLumiblade,26 to high-end design purposes.

The first broad use of OLED lighting is expected to be in applications where light qualityrequirements trump other factors such as production cost and power efficiency, for examplein the automotive and aviation sectors.4 To go beyond this to residential and office lighting,further improvements will have to be realized. For commercial deployment an efficacy ofabout 60 lm/W or more is needed; so far, 124 lm/W has been achieved under laboratory con-ditions (although it should be noted that this device did not meet the color point and colorrendering requirements for general lighting applications).3 A roadmap by the U.S. Depart-ment of Energy27 foresees 168 lumen per watt efficacy coupled with a 100000 hour lifetimeby 2020; if this can be achieved together with low-cost roll-to-roll production, OLEDs couldbecome one of the dominant products in the general lighting market.

1.1.3 White OLED device structuresTo create white light, one can either use a combination of blue and yellow emitters, or ofblue, red and green emitters. Using blue and yellow emitters is simpler but leads to anintrinsically lower light quality as measured by the color rendering index (CRI),28 which iswhy most state-of-the-art white OLEDs use blue, red and green emitters. It is important tobalance these three colors to get pure white light at different brightness levels and as thedevice ages. This can be achieved with various device structures,4 as schematically shownin figure 1.2.

The simplest approach is to use only single-emitter OLEDs. One can operate blue, redand green OLEDs in parallel, for example in a striped pattern [figure 1.2(a)].29 A significantadvantage is that the three colors can be separately addressed, allowing easy color tuningfor mood lighting or to compensate for differential aging of the emitters. However, this

*LCD displays work with a backlight, and so their power consumption does not depend on the displayed image.The power consumption of an OLED display, however, is directly related to the brightness of the image. This iswhy OLEDs are relatively efficient when using dark backgrounds.

1.1 Organic light-emitting diodes 9

Charge-generation

layers

Transport layersIndium tin oxide

Emissive layer

Metal electrode

Light-conversion

layers

(a)

(b)

(c)

(d)

(e)

Figure 1.2: White OLED device structures discussed in this section. (a) Parallel. (b) Blue-emittingonly. (c) Mixed emissive layer. (d) Multilayer. (e) Stacked multilayer.

method requires patterning on the microscale, making fabrication of these devices muchmore complicated. Another approach is to use only a blue-emitting OLED, and then convertthe blue light to red and green by using a layer of lumiphors or phosphors [figure 1.2(b)].30

This makes the device structure itself very simple. However, blue is typically the hardestemitter to design, and the least efficient. In addition, these devices typically have shortlifetimes at high brightness.

Most modern OLEDs combine the three emitters in one device structure. These emit-ters can be blended in a single material or separated into distinct emissive layers. Thesingle-emissive-layer approach [figure 1.2(c)] is easier to fabricate, but care must be takento prevent phase separation of the various components. This can be achieved for exampleby copolymerizing all components into a single polymer.31 However, the need to combine somany functionalities in one layer forces compromises on the charge transport and emissionproperties. For this reason, the multilayer approach is generally preferred [figure 1.2(d)].Typically, this includes not only emissive layers but also hole and electron transport andblocking layers. The main challenges in designing such a device are minimizing Ohmiclosses and balancing the emission between the three colors. The latter can be achieved bybalancing the exciton generation itself among the different emissive layers, or by excitondiffusion from a single generation zone to the different layers.32

A variation of the multilayer OLED is the stacked OLED [figure 1.2(e)].33 In this devicestructure the three different emissive layers are separated by charge-generation layers.As a result, one injected hole/electron pair results in three excitons, and so these devicesoperate at a higher voltage and a lower current density. The higher operating voltage canmake the device easier to connect to for example the 12 V grids of cars,4 while the lowercurrent density enhances the device lifetime. However, it is not easy to find a suitabletransparent material for the charge-generation layer.

The multilayer structure, with or without charge-generation layers, appears to be the

10 Introduction

dominant one in the current generation of OLEDs, although this is difficult to confirm sincemany structures are proprietary. It is also used (without charge-generation layers) in themost efficient OLED created to date.3 For these reasons, this structure will be the primaryfocus of our modeling work. However, several of our results can also be applied to otherstructures.

1.2 White OLED design challengesSix main characteristics are used to compare the performances of lighting technologies:4

• Luminous efficacy, i.e. the brightness as perceived by the human eye, expressed inlumen, divided by the input power, expressed in Watt.

• Lifetime, typically expressed as LT70, i.e. the time until the luminance is at 70% ofthe original value.

• Manufacturing costs.

• Light quality, both the emission color point and the color rendering index (CRI).

• Driving, i.e. the difficulty of operating the light source on a given power grid, such asthe standard 110 V/230 V AC grid.

• Environmental impact, including manufacturing, operation and disposal.

Of these six characteristics, the final three are strong points of OLEDs while the first threeare challenge areas.

OLED light quality is typically high, with the emission color well tunable during theproduction process. Current white OLEDs can achieve CRI values up to 90 (out of a possible100), which indicates that the color of illuminated objects is shown properly. These valuescompare favorably to fluorescent and inorganic LED lighting.5 On the other hand, colorstability is an issue; typically, an OLED’s color point will shift as a function of lifetime andbrightness. Pulse width modulation, in which the OLED is switched on and off at highfrequency, can reduce this problem.34

Driving is not difficult, since OLEDs operate at low DC voltage. Although this meansthat unlike incandescent lighting they cannot operate directly on the AC power grid, therequired down-conversion is much easier to implement than the up-conversion required forfluorescent tubes and bulbs.

The environmental impact of OLEDs is low compared to fluorescent tubes and inorganicLEDs. There are (ideally) no high-temperature processes involved in the manufacturing,and they contain few or no toxic components like mercury. They can typically be combusted,making disposal straightforward. Further improvement is being achieved by replacing theenvironmentally taxing indium tin oxide (ITO) typically used as transparent cathode35

and by reducing the amount of solvents needed for synthesis and processing of the organiccomponents.36

1.2 White OLED design challenges 11

The other three performance characteristics (efficacy, lifetime and manufacturing costs)are challenges in the design of white OLEDs. Although they can be optimized quite wellindividually, combining good performance in all three aspects is still very difficult. Nopublicly disseminated OLED has so far combined high efficacy with a high lifetime, andthe best OLEDs are made in laboratories with techniques and materials unsuited to massproduction. In the remainder of this section we will discuss the problems and possiblesolutions associated with each of these characteristics.

1.2.1 Luminous efficacyThe overall luminous efficacy ηoverall of an OLED, i.e. the perceived light brightness dividedby the input power, is the product of several factors:

ηoverall = ηlumhνeV

ηgenηradηout, (1.1)

with ηlum the luminous efficacy of radiation, hν the (average) energy of the emitted pho-tons, e the unit charge, V the applied voltage, ηgen the fraction of electrons and holesthat generate excitons, ηrad the fraction of excitons that recombines radiatively and ηoutthe fraction of photons that are emitted from the device. The product ηgenηradηout is alsoreferred to as the external quantum efficiency, and ηgenηrad as the internal quantum effi-ciency. We will discuss each of the factors in this equation, and some methods to optimizethem.

The luminous efficacy of radiation ηlum, expressed in lm/W, is the ratio of the perceivedbrightness of the emitted light to its power. It is determined purely by the emissive spec-trum. The maximum possible value of ηlum, 683 lm/W, is for green emission at 555 nm,which is why record-efficacy OLEDs typically have a greenish hue. However, for mar-ketable white OLEDs color point and color rendering will always trump a slight efficacygain, so ηlum ≈ 350 lm/W is determined by these concerns and cannot be optimized.

The second factor, hν/eV , is the ratio of the energy of emitted photons to the energy ofthe injected electron/hole pairs. To optimize this factor, the voltage drop must be mostlyover the exciton generation zone(s); otherwise, there are Ohmic losses, for example intransport layers and due to charge injection inefficiencies. Both of these losses can belimited by doping the transport and injection layers37 and by optimizing the layer stackdesign. There is a fundamental limit for this efficiency because eV must be high enoughfor the blue emitter, so that energy is always wasted in the red and green emitters. Astacked configuration with charge injection layers (see section 1.1.3) can help to reducethese losses.

The third factor ηgen is the exciton generation efficiency, the fraction of electrons andholes that meet and generate excitons. This can be brought to unity by the use of chargeblocking layers, which ensure that electrons and holes can never reach the opposite elec-trode and thus must generate excitons. In some cases a hole transport layer can also serveas an electron blocking layer (and vice versa) given the right molecular orbital energy lev-els.

The fourth factor ηrad is the fraction of excitons that recombine radiatively. This is

12 Introduction

determined by the competition between radiative and non-radiative recombination chan-nels. An important point here is the singlet-triplet ratio: in fluorescent emissive layers,about 25% of the excitons formed are in the singlet configuration, while the remaining 75%are in the triplet configuration.* Radiative recombination of the triplet configuration isquantum-mechanically forbidden, so that they eventually recombine non-radiatively. Thiscan be avoided by using phosphorescent emitters containing a heavy metal ion.41 A sig-nificant issue with this approach is that it is difficult to create stable blue phosphorescentemitters.42 One way to solve this problem is to use fluorescent blue emitters to harvest thesinglet excitons, while the long-lived triplet excitons diffuse to red and green phosphores-cent emitters away from the recombination zone.32,43 Another non-radiative recombinationchannel is exciton quenching by the electrodes; this is one of the reasons why transport lay-ers are necessary to separate the emissive layers from the electrodes.

The fifth factor ηout is the outcoupling efficiency, the fraction of generated photons thatare emitted from the device. For state-of-the-art OLEDs it is the dominant loss factorout of the five presented. The organic materials used typically have a refractive index ofabout 1.7, which leads to significant losses due to total internal reflection; without anyoptimizations, about 80% of the light never makes it out of the device.44 The most impor-tant techniques currently in use to improve outcoupling are external scattering foils45 andhigh-index or scattering substrates. Examples of outcoupling techniques in developmentare optical coupling layers,46 resonant cavities,47 excitation of surface plasmons,48 and theuse of microlenses.49 To optimize such techniques, it is crucial to accurately know and con-trol where exciton generation and recombination takes place through careful layer stackdesign.28,50,51 This is particularly true for methods which rely on modifying the internalstructure of the OLED, such as the resonant cavity approach.

1.2.2 Device lifetimeOLEDs for general lighting purposes need to have operating lifetimes of about 50000 hoursand shelf lifetimes of about 125000 hours. LT50 operating lifetimes, i.e. the time for theluminance to degrade by 50%, of up to 130000 hours have been reported at a brightnessof 1000 cd/m2.52 It has been suggested, however, that it might be more relevant to con-sider LT70 at 2000cd/m2, which is expected to be an order of magnitude lower, i.e. about13000 hours and thus not sufficient.2 There are two main causes of degradation in OLEDs:the formation of dark spots, which occurs no matter whether the device is on or off, andphotochemical degradation, which occurs only during operation.

Dark spots are caused by water diffusing into the device through pinholes in the elec-trode or encapsulating layer, leading to delamination of the cathode.53 These areas nolonger conduct current, and grow as more water diffuses through the pinhole. Due toadvances in encapsulation,54 this is no longer the primary cause of luminance degrada-tion. Still, even a few dark spots can significantly reduce visual quality unless a scatteringoutcoupling layer or diffusor is used.

Photochemical degradation takes place because of the chemical instability of the ex-

*The exact singlet-triplet ratio is a matter of debate.38–40

1.2 White OLED design challenges 13

cited states of emitter molecules.55 This has been conclusively shown by the absence ofsuch degradation in single-carrier devices.56 This means that this degradation dependsstrongly, typically superlinearly, on the brightness at which the device is operated. Threesolutions are increasing the device efficacy, designing more stable emitters and improvingthe layer stack design. Increases in device efficacy will boost device lifetimes because asthe external quantum efficiency increases, less excitons have to be generated for the samebrightness and so less chemical degradation occurs. Materials research to develop morestable emitters will obviously improve lifetimes. Improved layer stack design can help bywidening the exciton generation zone, thus reducing the emission load on individual emit-ters. In the same vein, a good understanding of the charge transport may lead to methodsto reduce the occurrence of generation hotspots. These are emitters which see far moreexciton generation events than others and thus are more susceptible to degradation.

1.2.3 Large-scale low-cost manufacturingOne of the main selling points of organic electronics has always been ultra-low-cost produc-tion due to roll-to-roll deposition from solution at ambient conditions.57 Ironically, large-scale low-cost manufacturing has now emerged as one of the most serious challenges forOLEDs. The two main techniques used are gas-phase deposition and processing from solu-tion. Gas-phase deposition techniques, which are typically used for the high performanceOLEDs developed in laboratories, are often unsuitable for large-area production. On theother hand, processing from solution comes with serious limitations to device structureand material choices. Research in this area has intensified since it has become clear thatOLEDs can be commercially viable, and several improvements to both methods have beendeveloped or proposed.

Gas-phase deposition is usually based on vacuum thermal evaporation (VTE). The or-ganic material is evaporated by heating in a high-vacuum chamber and subsequently de-posited on the cooled substrate. The main advantage of this technique is its versatility: anynumber of layers can be formed, with no restrictions on the type of materials that can belayered on top of each other. It is possible to deposit host-guest systems by evaporating twomaterials simultaneously,58 and even to create concentration gradients.59 However, thereare significant disadvantages. The need for high vacuum increases the cost of installationand maintenance. Furthermore, the material yield is typically low. Finally, deposition isslow and often inhomogeneous due to the difficulty of uniformly evaporating the typicallythermally insulating organic materials.

In-line configuration, in which the sample is moved at a constant speed through severalVTE chambers, one for each device layer, can improve the throughput and homogeneityof gas-phase deposition.60 Another promising technique is organic phase vapor deposition(OPVD).61 The difference with VTE is that an inert carrier gas such as nitrogen is used.This reduces the vacuum requirements (from ∼10−9 bar to ∼10−3 bar). In addition, theevaporation process in OVPD saturates the carrier gas and is thus more controlled thanin VTE, in which the evaporation it is a non-equilibrium process. Finally, OVPD allowsgreater control over the film morphology.62

Solution-processing methods are based on applying a uniform layer of the dissolved

14 Introduction

organic material through spin-coating or printing techniques, and then evaporating thesolvent. Printing in particular has the potential to be the cheapest method to produceOLEDs on an industrial scale, because the process is relatively simple and can be carriedout at room temperature and ambient pressure. The most important disadvantage is thatthe solvent used for a layer can redissolve the previous layers, limiting the possibilitiesfor multilayer devices. Several workarounds are available, such as the use of orthogonalsolvents,63 baking between deposition steps,64 and cross-linking polymers after depositionto render them insoluble.65 Still, the possibilities for multi-layer stack design remain lim-ited compared to gas-phase deposition.

1.3 Predictive simulation of OLEDsIn the previous section, several possible approaches to improving white OLEDs were dis-cussed. However, it is difficult in practice to design optimized devices based on these ap-proaches. This also makes it hard to evaluate these approaches, since one cannot be surethat both the reference device and the ‘improved’ device are properly optimized.66 Thereason that optimization is difficult is that there are many parameters to consider in thedesign process, such as material choices, layer thicknesses, and deposition methods. Theparameter space is simply too large to exhaustively inspect through trial and error. Thisis why predictive modeling is a requirement to develop truly optimized OLEDs. Ideally, amodel would take as input the layer stack design, the associated deposition methods, andthe chemical structures of the materials involved. As output, it would give the light outputcharacteristics and power consumption, ideally as a function of device lifetime.

A predictive model as described above does not exist yet, but much work has been doneon each of the steps that together could form such a model. In this section, we will firstdiscuss exactly what these steps are. Next, we zoom in on the role of charge-transportmodeling, and demonstrate how it can yield valuable results on its own. We then discusshow three-dimensional (3D) simulations of this charge transport can be translated to a one-dimensional (1D) approach. Finally, we briefly discuss non-predictive modeling, in whichthe goal is not to quantitatively simulate a device but to analyze scenarios and suggestdesign rules.

1.3.1 Modeling stepsThe steps necessary to develop a full predictive OLED model are visualized schematicallyin figure 1.3. We will discuss each of these steps in turn.

The first modeling step is the determination of the film morphology using moleculardynamics or a Monte Carlo approach, possibly combined with phase-field modeling67 at in-terfaces. In principle this should be done on the basis of the chemical structures and the de-position method. The second step is then to use density functional theory to determine thecharge localization sites, the energies of charge carriers occupying them, and the hoppingrates between them. This calculation must also include polaronic effects, i.e. the deforma-tion of the molecular lattice by the presence of a charge carrier. It is also very important to

1.3 Predictive simulation of OLEDs 15

Chemical structures, layer stack design,

and deposition methods

Morphology

Charge localization sites and hopping rates

Exciton generation sites

Internal photon emission

Light output characteristics Current-voltage characteristics

Model input

Model output

Molecular dynamics

Density functional theory

Hopping simulation

Excitonics

Optical modeling

Figure 1.3: Modeling steps in a 3D predictive OLED device model.

critically examine the exchange and correlation functionals used in the density-functional-theory calculations.68 Work on these first two steps has only recently begun.69–71 Typicallythese systems are too small to describe a full device, which means that multi-scale model-ing is needed to describe the statistics of the site and energy distribution, which can thenbe used to generate a hopping network for a full device.

The third step is electrical or charge-transport modeling. Here, the hopping process onthe site network computed in the first two steps is simulated. This part of the model isthe focus of this thesis. The results are the current-voltage behavior of the device, whichis the first measurable output of the modeling process, and the locations where holes andelectrons generate excitons.

The fourth step is the modeling of the dynamics of these excitons.72,73 The goal hereis to simulate their diffusion through Förster74 or Dexter75 transfer and their eventualradiative or non-radiative recombination. This step is particularly important for lifetimemodeling and for devices based on the diffusion of triplet excitons to phosphorescent emit-

16 Introduction

ters away from the exciton generation zone. After this step we know where, in whichdirection and at which wavelengths light is generated inside the device. The fifth and fi-nal step is then optical modeling to determine how much of this light is actually emittedfrom the device, as a function of wavelength and viewing angle. This is the final outputof the model, together with the current-voltage characteristics obtained from the charge-transport modeling.

In all five of these steps, both the bulk of the used materials and the interfaces betweenthem have to be considered. Except for the final step (optical modeling), this has not yetbeen a research priority in predictive OLED modeling. However, the behavior of morphol-ogy, charge transport, exciton generation and excitonics at interfaces is far from trivial andcan have a large impact on device performance.76,77

1.3.2 The role of charge-transport modelingA full OLED model as described above would revolutionize OLED design, but is not yetfeasible. Especially the first two steps, modeling the morphology and determining thecharge transport sites and hopping rates, are far from being ready for use in a designsetting.

For modeling to play a significant role in the development of the current generationof white OLEDs, a simplified version of the above model must be developed which canprovide results in the short term. For this reason we start in this thesis at the third step:charge-transport modeling. This requires making assumptions about the charge transportsites and hopping rates; we will discuss these assumptions in more detail in section 1.4.1.Using current methods it is feasible to perform all further modeling steps to obtain theemission spectrum; indeed, commercial software is already available for this purpose.78,79

Apart from its status as a vital link in the modeling chain, there are specific reasons whichmake investigating charge transport worthwhile on its own:

• It gives results which are directly measurable and/or useful in the design process:the current-voltage characteristics and the exciton generation efficiency ηgen. Thedistribution of exciton generation events between the different emitters also gives arough idea of the color balance (which is also determined by excitonics and opticaloutcoupling).

• It allows simplified lifetime modeling, for instance by giving every exciton generationevent a low chance to deactivate the emitter on which it takes place. This modelingis simplified because a full analysis should also take exciton diffusion into account.

• It reveals the underlying physics of charge transport in organic electronics, which isstill not fully understood.

1.3.3 One-dimensional modelingEven when we follow the approach discussed in the last section, i.e. focusing only on themodeling of charge transport, a typical three-dimensional (3D) simulation of the charge

1.3 Predictive simulation of OLEDs 17

transport process is still quite time-consuming (on the order of days on a 2.5 GHz processorfor a typical multilayer device). Since OLEDs are laterally homogeneous, we can insteadtry to use a one-dimensional (1D) approach based on computing the densities of holes andelectrons and the electric field strength as a function of distance from the anode. Any suchmethod is based on the balance of electron current, hole current and exciton generation.The two main approaches are the continuum drift-diffusion approach80,81 and the discrete1D master equation approach.82,83 We will focus in this thesis on the 1D drift-diffusion (1D-DD) approach for single-carrier devices, but our results can be straightforwardly appliedto the 1D master equation (1D-ME) approach as well.

The drift-diffusion equation for single-carrier devices is given by:

J = eµ(n,F)n(x)F(x)− eD(n,F)dndx

. (1.2)

where µ is the charge-carrier mobility (the average velocity of a charge carrier divided bythe applied electric field), D is the diffusion coefficient, n is the charge-carrier density, F isthe electric field strength and x is the distance from the anode. The diffusion coefficient Dis related to the mobility by the generalized Einstein expression:84

D(x)= µ(n,F)ne

dEF

dn, (1.3)

where EF is the Fermi energy. The field and carrier density are also related by Gauss’ law:

dFdx

= en(x)ϵ0ϵr

, (1.4)

where e is the elementary charge, ϵ0 is the electrical permittivity of the vacuum and ϵr isthe relative dielectric constant of the organic material. Eqs. (1.2) and (1.4) together form asystem of differential equations which can be solved for n and F.

The development of the 1D-DD approach is closely linked to 3D simulation in two ways.First, 3D simulation is needed to determine parameters, such as the charge-carrier mobilityused above. It may also be necessary to compute the diffusion coefficient for F = 0 inthis way, since the generalized Einstein expression can be derived only at F = 0. Indeed,Nenashev et al. used numerical results to show that Eq. (1.3) does not hold at F = 0.85

For double-carrier devices we also need to determine the exciton generation coefficient.*Typically, 3D simulation to determine parameters is performed on systems with periodicboundary conditions and constant n and F. Second, 3D simulation is used for validation.Here, we use 3D simulation of full devices to verify the accuracy of the 1D-DD approach,and to identify new physical effects which must to be taken into account. We will seeseveral examples of both of these links between 1D and 3D methods in this thesis.

The link between 3D and 1D modeling is similar to the link between experiment andmodeling as a whole. Also in that case experiments serve on the one hand to providethe parameters needed for the modeling, such as the strength of the energetic disorder

*In principle the exciton generation coefficient (also known as the recombination coefficient) is given by Langevin’sexpression,86 but Van der Holst et al. used 3D simulation to show that there are subtleties to take into account.87

18 Introduction

and density of hopping sites. In fact, a specialized parameter extraction method has beendeveloped for this purpose.88 On the other hand, experiments are of course the final arbiterin validating any developed model.

1.3.4 Non-predictive modelingSome problems have come up in practice while developing the 1D drift-diffusion approachdescribed above. In the first place, the 3D to 1D translation described in the previoussection turns out to be quite complicated, with new physical effects that have to be takeninto account regularly popping up. In addition, the 1D approach can only be applied tosystems for which the mobility and its dependence on n and F have been precomputedusing a 3D method, limiting its flexibility. Finally, even the 3D simulation approach isnot fully trusted to be quantitatively accurate due to the need for assumptions about, forexample, the spatial distribution of the hopping sites, the type of hopping, and the presenceor absence of correlation in the energetic disorder. Even comparison with experimentshas not been able to irrefutably determine which assumptions to use.89,90 As a result ofthese issues, the method is still difficult to apply in practice even after several years ofdevelopment.

For this reason non-predictive modeling is now also being considered as a part of theOLED design process. This type of modeling does not attempt to quantitatively model spe-cific devices, but instead uses simulation to analyze scenarios and suggest design rules.One could for example examine the sensitivity of the device characteristics to the thick-nesses of certain layers. This could then give an indication of how to optimize the layerstack design. The optimization must ultimately be done experimentally through trial anderror, but the modeling significantly reduces the size of the parameter space. When thistype of modeling is used in the design process, one typically does not have to simulateas many devices as one does when trying to optimize a device using predictive modeling.Therefore, non-predictive modeling is usually carried out using 3D simulation, and notwith the faster 1D approach. This means that one does not have to worry about the com-plications of the 3D to 1D translation.

1.4 Charge transport in organic semiconductorsWe will now work out the charge-transport model introduced in the previous section.Charge transport in organic semiconductors is typically analyzed in terms of the charge-carrier mobility µ (the average velocity of a charge carrier divided by the electric fieldstrength). The first measurements of this mobility were performed using time-of-flightexperiments,91 and were analyzed using a Poole-Frenkel dependence on the electric field:

µ(F,T)=µ(0,T)exp(γ(T)F1/2), (1.5)

with some empirical temperature-dependent γ(T).Using this empirical formula to describe current-voltage characteristics of devices led

to an unsatisfactory dependence of µ(0,T) on the device thickness. This was typically ex-

1.4 Charge transport in organic semiconductors 19

plained by assuming a dependence of the morphology on film thickness. However, it wasalso observed that there was a mobility difference of up to three orders of magnitude be-tween LED and field effect transistor devices,92 which could hardly be explained by mor-phology differences. Tanase et al. showed that the cause of these discrepancies was theneglect of the dependence of the mobility on carrier concentration.93 Simulations were car-ried out to determine the mobility as a function of both field and carrier concentration,with the results successfully describing the current-voltage characteristics of devices.94,95

Another significant advantage of the simulation approach is that the material parameters,such as the energetic disorder strength σ, are physically interpretable, unlike for exampleγ(T) in Eq. (1.5).

Here, we will focus on this simulation approach to the charge-carrier mobility. We willfirst describe the hopping model for charge transport in organic semiconductors, and thendiscuss some techniques to simulate this model and simulation results found in literature.

1.4.1 The hopping modelAs described in section 1.1.1, charge transport in organic semiconductors takes place byhopping of charge carriers between discrete localization sites, which can be polymer seg-ments or small molecules. The carriers can be holes or electrons; this makes no differenceto the basic physics. We assume that Coulomb interactions prevent two charges from oc-cupying one site. We will focus primarily on molecular organic semiconductors, althoughmany results apply equally well to polymers.

An obvious question now is how the localization sites are distributed in space. As men-tioned, this should be determined based on the chemical structures and deposition method,but for the time being this is computationally too expensive. In some simulations, the amor-phous nature of the organic materials is taken into account by randomly and uniformly dis-tributing these sites; this approach was used for example by Vissenberg and Matters.96 Inothers, the sites are assumed to be ordered on a simple-cubic (SC) lattice, for example in thework of Bässler.97 The random-position approach correctly models the spatial distributionof sites as disordered, but is too simple: sites can be very close together, much closer thanthe molecule size, and there can be large voids. We therefore choose the lattice approach,but will use various methods to analyze the sensitivity of the results to this choice. In thefirst place, we will consider a face-centered-cubic (FCC) lattice in addition to the SC lattice.In the second place, we will consider lattice disorder and variable-range hopping, discussedlater in this section. We will write Nt for the site density, and use a = N−1/3

t as a typicallength scale; note that for the simple-cubic lattice a is identical to the nearest-neighbordistance.

Due to the disordered nature of the materials, the energy of a charge carrier E i dependson which site i it is occupying. The site energy E i is a fixed quantity and does not includethe effect of electrostatic interaction with other carriers. It corresponds to the highest oc-cupied molecular orbital (HOMO) energy for holes and to the lowest unoccupied molecularorbital (LUMO) energy for electrons. The distribution of site energies is described by thedensity of states (DOS), which we write as g(E). Based on the assumption that there areseveral factors determining this energy, the central limit theorem suggests that this dis-

20 Introduction

tribution should be (approximately) Gaussian.97 This is also supported by the absorptionspectra of organic materials.98 The Gaussian DOS is given by:

g(E)= 1p2πσ

exp(− E2

2σ2

), (1.6)

where σ is the standard deviation of the DOS. For typical organic semiconductors, σ≈ 0.1eV. Note that we define the average HOMO or LUMO energy as E = 0.

On its own the DOS g(E) does not fully specify the statistics of the energy landscape,since it does not describe possible correlations between the energies of nearby sites. Suchcorrelations can be caused for example by thermally induced torsions of polymer chains99

or randomly oriented small molecules with electric dipoles.100 In this thesis, we considertwo cases: uncorrelated disorder and dipole-correlated disorder. In the uncorrelated case,we simply take the site energies independently according to the (usually) Gaussian DOS.In the dipole-correlated case, we place a dipole di with fixed magnitude d and randomorientation on every site i. The site energies are then given by:

E i =−∑j =i

ed j · (R j −Ri)

ϵ0ϵr∣∣R j −Ri

∣∣3 , (1.7)

where Ri is the position of site i. An appropriate choice of d then yields an approximatelyGaussian DOS with width σ (see section 3.A). The energies of nearby sites are correlated,with the correlation decreasing asymptotically as 1/R.

We now consider the hopping (tunneling) rates of charge carriers between the localiza-tion sites. To satisfy detailed balance, the rate ωi j for a carrier to hop from an occupied sitei to an unoccupied site j must be of the form

ωi j =ωi j,symm exp(∆E i j/2kBT), (1.8)

where kB is Boltzmann’s constant and T the temperature. The energy difference ∆E i jconsists of two terms: the difference in site energies E i and E j and the difference in theelectrostatic potential due to interactions with other charge carriers and due to the appliedvoltage. The symmetric part of the hopping rate ωi j,symm = ω ji,symm specifies the type ofhopping. We only consider hops between nearest neighbors (see also the discussion at theend of this section).

Two types of hopping are discussed in literature, one based on phonon-assisted hoppingand one based on polaronic effects. The phonon-assisted hopping model was first suggestedby Conwell and Mott101,102 and is used in most charge transport simulations. In this model,the energy needed for a hop upwards in energy is supplied by the absorption of a phonon,while a phonon is emitted for a hop downwards in energy. This means that all hops down-wards in energy share the same rate, while hops upwards in energy are less likely for ahigher energy difference because a phonon with the right energy must be available. Thisleads to the following hopping rate, proposed by Miller and Abrahams:103

ωi j,symm =ω0 exp(−|∆E i j|/2kBT), (1.9)

1.4 Charge transport in organic semiconductors 21

where ω0 is a constant prefactor that scales with the square of the transfer integral J20 .

Note that indeed all hops downwards in energy share the same rate, as can be seen bycombining Eqs. (1.8) and (1.9). We will refer to this type of hopping as Miller-Abrahams(MA) hopping.

The phonon-assisted MA hopping model was developed for hopping between impuri-ties in inorganic semiconductors, and is not considered appropriate to describe hopping inorganic semiconductors.70 Instead, a hopping model proposed by Marcus104 based on pola-ronic effects is used. A polaron is a charge carrier combined with the deformation of thelattice caused by its presence. The energy associated with this deformation is called thereorganization energy Er.* The hopping rate associated with this model is given by:

ωi j,symm =ω0 exp(−∆E2i j/4ErkBT), (1.10)

where ω0 is not constant like for MA hopping, but depends on Er and T:

ω0 ≡J2

0

ħ√

π

ErkBTexp(−Er/4kBT). (1.11)

We will typically express our results for the charge-carrier mobility in terms of ω0, whereit should then be kept in mind that the temperature dependence due to Eq. (1.11) is nottaken into account. We will refer to this type of hopping as Marcus hopping.

As mentioned, we will typically only consider hops between nearest neighbors. This isconsistent with recent first-principles studies of charge transport in tris(8-hydroxyquino-line) aluminum (Alq3).70,105,106 However, that work showed that positional disorder due torandom molecular packing107 can play a large role. When we take this into account, weintroduce a parameter Σ describing the strength of disorder. Because of the exponentiallydecaying wave functions, we vary the transfer integral J0 per bond according to J0,i j =J0 exp(ui j), where ui j = u ji is uniformly distributed between −Σ and Σ. The hopping ratesωi j themselves scale with the square of J0,i j [see Eq. (1.11)]. In other words, this approachleads to new values of ω0 in Eqs. (1.9) and (1.10) given by:

ω0,new,i j =ω0,old exp(2ui j). (1.12)

We will also in some cases consider variable-range hopping instead of restricting ourselvesto nearest-neighbor hopping. We then replace the prefactors by:

ω0,new,i j =ω0,old exp(2α[Ri j −a]), (1.13)

where α is the inverse wave-function decay length and Ri j is the distance between the sites.Note that this form ensures that for nearest neighbors ω0,new,i j = ω0,old. The exponentialdependence is again based on the exponential decay of the wave functions. Eq. (1.13) ap-plies to SC lattices; for an FCC lattice, we must replace a by 21/6a to reflect the differentnearest-neighbor distance.

*In literature, one sometimes encounters the polaron activation energy Ea instead of the reorganization energyEr. They are related by Er = 4Ea.

22 Introduction

A device, such as an OLED, consists of one or more layers of organic materials sand-wiched between two metal electrodes. The charge transport in each individual layer ismodeled as above. The electrodes are handled using a simple approach,108 which is suffi-cient for the low-injection-barrier devices considered in this thesis. At both electrodes anextra layer of sites is added to the lattice, representing the anode and the cathode. Unlikethe organic sites, these sites are not considered to be occupied or unoccupied; carriers canalways hop to and from them, with the same rate as hops inside the organic material. Thesite energy of these electrode sites is fixed to minus the work function of the metal, with noenergy disorder. Since the electrodes are metals they cause image charges for each chargecarrier in the device, which interact both with that carrier and with other carriers.

1.4.2 History of simulating hopping transportThe first 3D simulation of charge transport in organic semiconductors was performed byBässler in 1993.97 In that work, he performed Monte Carlo simulations of the hoppingtransport of a single charge carrier in a simple-cubic lattice with Miller-Abrahams hoppingand uncorrelated Gaussian disorder. This model is referred to as the Gaussian disordermodel (GDM). From these simulations, he obtained the field and temperature dependenceof the charge-carrier mobility in the zero-carrier-concentration limit. A Poole-Frenkel de-pendence [see Eq. (1.5)] was found in a limited range, smaller than the range found ex-perimentally. This led Gartstein and Conwell to introduce the correlated disorder model(CDM), using the same assumptions as the GDM but with dipole-correlated disorder.109 Inthe CDM, the Poole-Frenkel dependence on electric field holds over a much larger rangethan in the GDM.

Both the GDM and the CDM neglect the effect of carrier concentration. In 2003, how-ever, Tanase et al. showed that the dependence of the mobility on carrier concentration is atleast as important as the field dependence.93 The carrier-concentration dependence is dueto state-filling effects.110 The existence of this dependence had already been demonstratednumerically in 2000 by Yu et al. using the 3D master equation (3D-ME) approach.99 Thisapproach is based on computing the probability pi that a site i is occupied by a charge car-rier. The master equation states that the amount of carriers hopping onto each site mustbe balanced by the amount hopping away:∑

j

[ωi j pi(1− p j)−ω ji p j(1− pi)

]= 0, (1.14)

where j runs over all neighboring sites (or sites up to some cutoff radius if variable-rangehopping is taken into account). This equation must be solved for every pi, after which thecurrent density and carrier mobility can be straightforwardly computed.

In the past few years our group has been intensively simulating the hopping modelin systems with periodic boundary conditions and in devices, and we will focus on thiswork in the remainder of this section. In 2005, our group systematically studied thefield, carrier-concentration and temperature dependence of the mobility in the GDM us-ing the 3D-ME approach, and provided a parameterization of the results.94 This model isreferred to as the extended Gaussian disorder model (EGDM), with ‘extended’ referring

1.5 Scope of this thesis 23

to the inclusion of the carrier-concentration dependence. Bouhassoune et al. performed asimilar analysis of the CDM, leading to the extended correlated disorder model (ECDM).95

Both the EGDM and the ECDM were successfully applied to a wide range of single-carrierdevices.89,90,94,95,111,112

All of the approaches described so far in this section used simulation to determinethe charge-carrier mobility in systems with periodic boundary conditions. This mobilitywas then implemented in a 1D drift-diffusion approach (see section 1.3.3) to determinethe current-voltage characteristics of devices. However, it is important to check that thephysics of the hopping model is preserved in the 3D to 1D translation. For this reason, Vander Holst et al. performed 3D-ME simulations directly on single-layer single-carrier de-vices, i.e. including the electrode layers.113 In that work, an excellent agreement betweenthe 1D and the 3D results was found.

An important limitation of the 3D-ME approach is that it cannot take into account cor-relations between the site occupation probabilities pi,114 which is especially troublesomewhen considering Coulomb interactions and exciton generation. Although the influence ofspace charge on the electric field [see Eq. (1.4)] is taken into account in the 3D-ME ap-proach, short-range Coulomb interactions between individual carriers cannot be included.Moreover, double-carrier devices containing both holes and electrons cannot be modeled atall. For these reasons, a multi-particle 3D Monte Carlo (3D-MC) method was developedin our group to accurately model double-carrier devices, including all effects of Coulombinteractions.108 Within the European FP7 project AEVIOM (Advanced Experimentally Val-idated Integrated OLED Model), this method was used to for the first time simulate acomplete multilayer OLED.

1.5 Scope of this thesisThe focus of this thesis is on the development of a predictive charge-transport model. Wewill focus especially on the relationship between 3D simulation and 1D modeling. As de-scribed in section 1.3.3, there are two aspects to this relationship: using 3D simulation tofind parameters required for 1D modeling, and using 3D simulation to validate the accuracyof 1D modeling. We will see examples of both throughout this thesis.

Several numerical techniques were developed or improved in the course of this work.These are discussed in chapter 2. The main focus of that chapter is on using Newton’smethod to solve the 3D master equation (3D-ME). We also describe how to use the 3D-MEto perform time-dependent simulation; this is not used for the work in this thesis but isexpected to play a major role in future research. The 3D Monte Carlo (3D-MC) approachand a fast solver for the 1D drift-diffusion (1D-DD) equation for single-carrier devices arealso covered in this chapter. A reader more interested in the charge-transport results thanin the numerical details may skip this chapter.

The state-of-the-art models to describe the charge-carrier mobility in organic semicon-ductors, the EGDM and ECDM (see section 1.4.2), assume Miller-Abrahams hopping ona simple-cubic lattice. The influence of these assumptions has not yet been numericallyexamined. In chapter 3, we consider different types of hopping (Miller-Abrahams and

24 Introduction

Marcus), lattice (simple cubic and face-centered cubic) and disorder (uncorrelated anddipole-correlated). Instead of parameterizing each case separately, we develop a scalingtheory for the zero-field charge-carrier mobility based on the concept of percolation. Unlikeprevious percolation theories, we take into account multiple critical bonds instead of justone. This scaling theory leads to a general and compact expression for the mobility thataccurately describes the results of 3D-ME simulations. An important result is that thecarrier-concentration dependence is universal, depending only on the shape of the densityof states and not on the type of hopping, lattice or energy correlation. We also discuss howour results may help to experimentally distinguish different hopping models.

Many realistic organic semiconductors intentionally or unintentionally contain guests(charge-carrier traps). As long as the guest concentration is low enough, charge is trans-ported only by carriers that are detrapped from the guest to the host. At zero field, theamount of such carriers is accurately described by Fermi-Dirac (FD) statistics. However,at finite field there is an additional contribution due to field-induced detrapping (FID).In chapter 4, we develop a quantitative approach to describe FID by introducing a field-dependent generalized FD distribution. This distribution depends only on the shape of thehost density of states (DOS), and not on the guest DOS or concentration. We parameterizethis distribution for the case of the Gaussian host DOS, and show that one can then ac-curately predict the charge-carrier mobility found using 3D-ME simulation for host-guestsystems with a Gaussian host. We also use 1D-DD modeling to show that the effect of FIDis relevant under conditions typical of OLEDs.

In chapter 5, we combine results from chapters 3 and 4 to describe charge transport atnon-zero electric field. We show that the charge-carrier mobility factorizes into an ‘intrinsic’factor and a ‘detrapping’ factor. These factors represent separate physical effects, whichinfluence charge transport in devices in different ways. This means that the value of themobility by itself does not fully describe charge transport at non-zero electric field. Weintroduce a new form on the 1D-DD method which does not use the mobility but correctlycontains the intrinsic factor and the detrapping factor separately. Chapters 3 to 5 areexamples of the parameters aspect of the relationship between 3D and 1D modeling.

As discussed in section 1.1.3, a typical OLED consists of multiple layers. To accuratelymodel such devices using the 1D-DD approach, it is important to describe the charge trans-port across the interfaces between these layers in a manner consistent with 3D simulation.In chapter 6, we show that the results of 3D-MC simulation of a bilayer device are not wellreproduced by straightforward 1D-DD modeling. This is because of three physical effects:(1) non-equilibrium charge transport across the interface, (2) the formation of a surfacecharge layer just before the interface, and (3) the reduction of the effect of this surfacecharge due to short-range Coulomb interactions. We describe how to take these effects intoaccount in the 1D-DD solver, which leads to a good agreement with the 3D-MC results.This chapter is an example of the validation aspect of the relationship between 3D and 1Dmodeling.

We finish this thesis with our main conclusions and an outlook on applications andfurther research in chapter 7. We also present a unified 1D-DD method which combinesall charge-transport results of earlier chapters in appendix A.

References 25

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28 Introduction

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Chapter 2

Numerical methods

Abstract

In this thesis, we focus on 3D modeling of charge transport in organic semiconductors andtranslating the results to a 1D method. We use two 3D methods, 3D Monte Carlo (3D-MC) and 3D master equation (3D-ME), and one 1D method, the 1D drift-diffusion (1D-DD)equation.

The 3D-MC method simulates the hopping model for charge transport exactly for single-or double-carrier devices, with the effects of Coulomb interactions between individualcharge carriers fully accounted for. However, the need for time averaging can make themethod slow and makes it hard to obtain detailed statistics at the site level.

The 3D-ME method determines the probability for each localization site to be occupiedby a charge carrier by solving the master equation, which is a balance equation for theseprobabilities. It is suitable for single-carrier devices only, and Coulomb interactions canonly be taken into account in the form of the long-range space-charge effect. On the otherhand, since no time averaging is needed it is very suitable for site-level analysis and fortime-dependent calculations. To solve the master equation we apply Newton’s method fornon-linear equations. This involves linearizing the master equation. This linearization isalso used in numerical techniques for time-dependent simulation such as dark-injectiontransients or small-signal analysis.

The 1D-DD method solves the drift-diffusion equation for the electric field and charge-carrier density in the device, using the charge-carrier mobility obtained from 3D simulationof systems with periodic boundary conditions. We introduce a fast and reliable method tosolve this equation for single-carrier devices.

31

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 General considerations for 3D computations . . . . . . . . . . . . . 332.3 3D Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 3D master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.1 Full device calculations . . . . . . . . . . . . . . . . . . . . . 352.4.2 Periodic boundary conditions calculations . . . . . . . . . . 382.4.3 Solving the master equation using the method of Yu et al. 392.4.4 Solving the master equation using Newton’s method . . . 412.4.5 Obtaining and solving the linearized master equation . . . 43

2.5 1D drift-diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . 452.6 Time-dependent calculations with the 3D master equation . . . . 46

2.6.1 Dark-injection transients . . . . . . . . . . . . . . . . . . . . 462.6.2 Small-signal analysis . . . . . . . . . . . . . . . . . . . . . . 48

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

32 Numerical methods

2.1 IntroductionIn this chapter we will discuss the numerical methods used to obtain the results in thisthesis. These can be subdivided into 3D methods (introduced in section 1.4.2), which nu-merically simulate the model described in section 1.4.1, and the 1D drift-diffusion method(introduced in section 1.3.3), which solves a simplified 1D model using input from the 3Dsimulations. The overall goal of all these methods is to determine the current density ina device as a function of applied voltage. The 3D methods can also be used to obtain thecharge-carrier mobility in a system with periodic boundary conditions.

We will first discuss lattice and energy disorder generation for 3D systems (section 2.2).We will then briefly explain the 3D Monte Carlo method (section 2.3) and thoroughly dis-cuss the method most used in this thesis, the 3D master equation (section 2.4). Next,we will discuss a fast way to solve the 1D drift-diffusion equation for single-carrier de-vices (section 2.5). Finally, we will introduce a new application of the 3D master equationmethod: time/frequency-resolved calculations (section 2.6).

0 1 2 3 4 5 6 7 8 9 1010-2

10-1

100

101

102

curr

ent d

ensi

ty [A

/m2 ]

voltage [V]

3D Monte Carlo 3D Master Equation 1D Drift Diffusion

Figure 2.1: Current density as function of voltage for the example device, computed using threedifferent methods: 3D Monte Carlo (symbols), 3D master equation (black curve) and 1Ddrift diffusion (gray curve).

All methods described in this chapter will be applied to an example device with a simple-cubic lattice, Miller-Abrahams hopping, width of the Gaussian DOS σ= 0.122 eV, site den-sity Nt = 4.28×1026 m−3, hopping attempt rate ω0 = 5.77×109 s−1, and relative dielectricconstant ϵr = 3.2. For the device length we take L = 122 nm, corresponding to 91 organiclayers. At the anode we take no injection barrier, while at the cathode we take a work func-tion 1.8 eV under the highest occupied molecular orbital (HOMO), i.e. Vbi = 1.8 V. Theseparameters correspond to a hole-transporting polyfluorene-triarylamine (PF-TAA) devicestudied by van Mensfoort et al.1 The values used here are slightly different from those re-

2.2 General considerations for 3D computations 33

ported in that work; they were recently determined for the same device after three yearsof aging. Current density-voltage (J-V ) characteristics for this device, obtained using themethods described in this chapter, are shown in figure 2.1.

2.2 General considerations for 3D computations

Two types of 3D calculations are performed in this thesis: full device (FD) calculations,where we model charge transport through one or more organic materials sandwichedbetween two electrodes, and periodic boundary conditions (PBC) calculations, where wemodel charge transport in an infinite medium with fixed electric field and carrier concen-tration. For both types of computations, we start by generating either a simple-cubic orface-centered-cubic lattice of sites and applying periodic boundary conditions in the y andz directions.* For PBC calculations, we also apply periodic boundary conditions in the xdirection, and apply an electric field in this direction. For FD calculations one additionallayer of sites is added at each end in the x direction, representing the electrodes. We willrefer to these as electrode sites, and to all other sites as organic sites.

After generating the lattice, each organic site is assigned a site energy E i. In the caseof uncorrelated disorder, this energy is simply sampled for every site from the energy dis-tribution as given by the density of states (DOS) g(E). It takes very large lattices to getrepresentative statistics; even a 100×100×100 lattice is sometimes not enough. This sug-gests that a few outlying low-energy sites determine the charge transport properties. Toobtain better statistics for smaller lattices, one could imagine first taking energy levelsrepresentative of the DOS non-randomly, as many as there are sites, and then randomlydistributing these values over the sites. However, in practice this approach does not leadto improved accuracy; we will come back to this issue in section 2.4.2.

In the case of correlated disorder, we apply a brute force approach to generate the en-ergy landscape: a randomly-oriented dipole with magnitude d is placed on every site, afterwhich Eq. (1.7) is used to compute E i for every site i. Every site is counted only once in thesummation, always using the closest periodic copy of this site. This calculation takes in theorder of a few days on a 2.5 GHz processor for a 100×100×100 lattice. Since calculationtime scales with the square of the number of sites, using larger lattices is not feasible withthis approach. The performance may be improved by choosing a suitable cut-off radiusfor the summation in Eq. (1.7); however, it is not obvious how one would evaluate whichradius to take, since the outliers in the distribution are crucial in determining the charge-transport properties. Another way to generate larger lattices would be parallelization ofthe code.

*We assume throughout this chapter that the sites are placed on a lattice and take only nearest-neighbor hopsinto account (see also section 1.4.1). However, all methods described can be straightforwardly modified to applyto non-lattice systems or systems with variable-range hopping.

34 Numerical methods

2.3 3D Monte CarloThe 3D Monte Carlo (3D-MC) method developed in our group simulates the hopping modelas described in Section 1.4.1 with no further approximations. We will provide here a briefoverview of the main features of this method; a complete description can be found in Ref. 2.

The method keeps track of the full state of the system, i.e. the locations of all chargecarriers. A simulation step consists of choosing and carrying out one of the possible hopsfrom an occupied site to an empty one (or to/from one of the electrodes), with the probabilityof each hop weighted by its rate as given by Eq. (1.8). This choice of hop is made efficientlyby keeping track of all hopping rates using a binary search tree. Typically, we start withan empty system and run the simulation long enough to achieve a steady state. After that,we continue running the simulation for some time while measuring the desired quantities,such as the current density.

The method takes Coulomb interactions between individual charge carriers (and theirimage charges) into account when computing the energy difference ∆E associated with ahop. However, calculating every pair-wise interaction separately is not feasible. Instead,we split them into short-range direct interactions and long-range space-charge interac-tions. Direct interactions are calculated explicitly within some Coulomb cutoff radius RC.Outside this sphere, we consider each layer of the lattice as a uniform sheet charge, withthe surface charge density determined by the total number of carriers in that layer. To pre-vent double counting of charges, we do not include the surface charge of the circular partsof these layers that are inside the RC-sphere. The interaction of carriers with these sheetcharges is straightforward [see Eq. (1.4)]. For RC →∞, this method is exact; for all resultsin this thesis, RC = 8a was used and found to be sufficient, i.e. increasing RC further doesnot significantly affect the results.

The main advantage of the 3D-MC method is that it can fully simulate our hoppingmodel with no simplifications. Unlike the 3D master equation method discussed below itcan also handle actual OLEDs, where both holes and electrons hop through the device andgenerate excitons. The main disadvantage is that the method can be slow, since one needsto first allow the system to relax and then run long enough to collect sufficient statistics.This problem is especially severe when the current density is low. It also makes it moredifficult to obtain detailed statistics at the site level, such as the occupation probabilities.

2.4 3D master equationIn the 3D master equation (3D-ME) approach, we determine for every site i the steady-state probability pi that it is occupied by a charge carrier. To link these probabilities, weuse the requirement that in the steady state the rate at which carriers hop onto a site mustmatch the rate at which they hop away from it:∑

jωi jP(site i is occupied and j is not)−ω jiP(site j is occupied and i is not)= 0 (2.1)

for all sites i, where the summation runs over all neighboring sites. To translate this tothe probabilities pi, we make the following approximation: the occupation probabilities

2.4 3D master equation 35

are independent of the occupation of other sites. This approximation is central to themaster equation approach. Note that this makes it impossible to take into account short-range Coulomb interactions as discussed in the previous section, and even without theseinteractions the current may be affected.3* Using this approximation in Eq. (2.1) yields thestandard form of the master equation:∑

jωi j pi(1− p j)−ω ji p j(1− pi)= 0. (2.2)

Once the pi are known, the current density J follows straightforwardly from the bondcurrents:

J = eLxL yLz

∑i, j

ωi j pi(1− p j)Ri j,x, (2.3)

where Lx, L y and Lz are the dimensions of the lattice and Ri j,x is the difference in the xcoordinates between sites i and j.

Instead of solving for pi we can also solve for the electrochemical potential energy µi(not to be confused with the charge-carrier mobility µ), which is defined at every site interms of pi:†

pi = 11+exp([E i − µi]/kBT)

. (2.4)

This is mathematically equivalent but will in some cases prove more convenient.The main advantage of the 3D-ME method is that no time averaging is needed. As a re-

sult, it is often faster than the 3D-MC approach and is especially useful for time-dependentmodeling, such as transient or alternating currents. Another advantage is that since pi isknown for every site it is much easier to analyse the behavior of the system at the scaleof single sites. The most important disadvantage of the 3D-ME approach is that it is notpossible to take the Coulomb interactions between individual carriers into account, or tomodel real OLEDs with both holes and electrons. Another issue is that solving Eq. (2.2) re-liably and in a stable way is often quite difficult, while the 3D-MC approach is guaranteedto converge eventually.

In this section, we will first discuss some issues arising when formulating the masterequation for full devices (section 2.4.1) and for periodic boundary conditions (section 2.4.2).We will then discuss two methods for solving the master equation: the method of Yu etal.4 (section 2.4.3) and Newton’s method (section 2.4.4), the latter of which is used for mostresults in this thesis. An important part of Newton’s method is linearizing the masterequation and solving this linearized form (section 2.4.5).

2.4.1 Full device calculationsFor full device (FD) calculations, a distinction must be made between the organic sites andthe electrode sites. As described in section 1.4.1, the electrode sites all have the same en-*Only a small effect on the current was found in Ref. 3 for the case of nearest-neighbor hopping. It is possible thatthe effect may be larger for variable-range hopping. This should be checked by comparison with 3D-MC results.

†We use here the solid-state-physics definition of electrochemical potential energy. This means that it is notnecessarily constant across a device in equilibrium. In electrochemistry, µ would be referred to as the chemicalpotential energy.

36 Numerical methods

ergy (the work function of the electrode material) and are neither occupied nor unoccupied:a carrier can always hop to or from one. This is implemented in the simulation by placingtwo layers of sites at each electrode, one unoccupied (pi = 0) and one occupied (pi = 1).Each of these layers is directly accessible from the adjacent layer of organic sites.

Coulomb interactions are taken into account through the long range space charge effectonly; it is not possible to take into account short range interactions between individualcarriers in the master equation approach. This complicates the method somewhat since thehopping rates ωi j in Eq. (2.2) now depend on the occupation probabilities pi. To computethis dependence explicitly, we determine the electric potential Vi at every site i, which isiteratively defined by

Vi =Vi−1 −Fi−1(xi − xi−1), (2.5)

where Fi is the electric field between sites i and i+1 and xi is the x coordinate of site i,with the indices chosen so that the sites are ordered in the x direction. We set V1 = 0 byconvention. The field Fi is determined by the space charge approximation, i.e. we spreadout the charge on site i over the full lateral layer:

Fi = Fi−1 + epi

ϵ0ϵrNtL yLz, (2.6)

where F0 must be chosen so that the total voltage over the device matches the desiredvoltage:

VN =V −Vbi, (2.7)

where N is the total number of sites and Vbi is the built-in voltage, i.e. the difference inwork function between the electrodes. We efficiently compute Vi by first setting F0 = 0 anditeratively calculating the resulting electric potential, which we call V (0)

i , from Eqs. (2.5)and (2.6). From there we can straightforwardly calculate the correct value of F0,

F0 = (V −Vbi −V (0)N )/L. (2.8)

It is not necessary at this point to redo the calculation for Vi; they are simply given by

Vi =V (0)i −F0xi. (2.9)

These resulting Vi obey Eqs. (2.5) through (2.7), and can be used to compute the energydifference used in the hopping rates. Note that this approach can also be applied when thesites are not ordered in layers.

The exact solution at zero voltage can also be determined using this approach. An extrawrinkle is now that pi, required in Eq. (2.6), is initially unknown. However, by the timewe need it Vi has already been computed. Since thermal equilibrium applies, we haveµi = eVi + eΦleft, with Φleft the work function of the left electrode. We can then obtainpi from Eq. (2.4). When we have done this for all sites, we check whether VN satisfiesEq. (2.7); if not, we adjust our initial guess for F0 and rerun the method until it does.

An important application of 3D-ME simulation is determining field and density profilesas a function of position in the device. A complication in determining the density profiles isthe energy disorder: every layer of the lattice contains a relatively limited number of sites,

2.4 3D master equation 37

which causes a large statistical error in the carrier density. This effect is demonstrated bythe gray curve in figure 2.2, which shows the density profile in the example device for a50×50 lateral lattice size. The large variations, especially when the carrier density is low,complicate the analysis of such a profile.

0 20 40 60 80 100 1201021

1022

1023

1024

1025

carr

ier d

ensi

ty [m

-3]

distance from anode x [nm]

Unsmoothed profile Smoothed profile

V = 3 V

Figure 2.2: Carrier density as a function of position as found from 3D master equation modeling ofthe example device (see section 2.1) with an applied voltage V = 3 V, with and withoutthe smoothing method described in the text. The lateral lattice size is 50×50.

To smooth the profile, we make use of the exact expression for the charge-carrier den-sity:

n = 1Nt

∫dEg(E)p(E), (2.10)

where p(E) is the average occupation probability of a site with energy E. We can approx-imate p(E) by taking the value of pi for the site i with energy closest to E in the layerbeing sampled. This already smoothes the profile considerably. A further improvement ispossible by using the electrochemical potential energy µi, defined by Eq. (2.4), instead ofpi:

n = 1Nt

∫dEg(E)

11+exp([E− µ(E)]/kBT)

. (2.11)

The results of this approach are shown by the black curve in figure 2.2; a significant im-provement is achieved over the unsmoothed profile. This smoothing method is especiallyeffective when close to equilibrium, i.e. for low voltage, since in that case µi = EF and henceEq. (2.11) is exact.

38 Numerical methods

2.4.2 Periodic boundary conditions calculationsIn periodic boundary conditions (PBC) calculations, our goal is to determine the charge-carrier mobility µ (the average velocity of a charge carrier divided by the electric field).This requires uniform conditions, so instead of using electrodes as boundary conditions inthe x direction we use periodic boundary conditions, just like we do for all cases in the yand z directions. In addition, no space charge effects are taken into account; the electricfield F, which we take in the x direction, is uniform throughout the lattice. An additionalequation must be added to the system of equations given by Eq. (2.2) to fix the carrierconcentration c:

1N

∑j

p j = c. (2.12)

Without this additional equation the system is singular, i.e. it allows multiple solutions.This can be verified by summing Eq. (2.2) over all i, which yields 0 = 0. After solving theset of equations (2.2) and (2.12) for pi, the current and carrier mobility µ= J/ecNtF followfrom Eq. (2.3).

In equilibrium, i.e. F = 0, the solution is given by a constant electrochemical potentialµi = EF with EF the Fermi energy, which must be chosen such that Eq. (2.12) is obeyed.Written in terms of pi this solution is the Fermi-Dirac distribution:

pi = 11+exp([E i −EF]/kBT)

. (2.13)

It can be verified straightforwardly that this solution indeed obeys Eq. (2.2) and leads toJ = 0 in Eq. (2.3).

A significant problem in determining µ is that at low carrier concentration the chargetransport is largely determined by the few sites with lowest energy. This is because thesesites trap most of the charge carriers. For large disorder, the number of such sites canvary significantly between realizations of the disorder, even for very large lattice sizes upto 100×100×100.* We can of course average the mobility over multiple realizations, butthis does not solve the issue because this average is not necessarily equal to the actualL → ∞ value of the mobility. This problem is demonstrated by the circles in figure 2.3,which shows the mobility for various lattice sizes L for moderate disorder σ and low car-rier concentration c. For low values of L the mobility is significantly higher than the actualvalue, even after averaging over multiple realizations of the disorder. Lukyanov and An-drienko proposed simulating small systems at high temperature and then extrapolatingthe low temperature behavior,5 but this requires a priori knowledge of the temperaturedependence.

Another possible solution to this problem is to use fixed energy levels. In this approach,we first choose the energy levels by solving i/N = Γ(E i), where Γ is the cumulative distri-bution function of the density of states, i.e.

Γ(E)=∫ E

−∞g(E). (2.14)

*In some cases, these finite-size effects may actually be physical. A 100×100×100 lattice typically correspondsto about 100 nm cubed, which is comparable to OLED device thicknesses.

2.4 3D master equation 39

In other words, the energy levels are non-randomly chosen to be representative of the DOS,with the correct number of outliers given the lattice size. To generate the actual disorderwe distribute these energies randomly over all sites.* The results of applying this approachare shown by the triangles in figure 2.3; there is no improvement over using fully randomenergy levels (circles). It may be possible to improve this approach by choosing the energylevels in a different way; but this will not be developed further within this thesis.

Another approach is to fix the Fermi energy instead of the carrier concentration. Wefirst determine the Fermi energy corresponding to the desired carrier concentration usingthe Fermi integral:

c =∫ ∞

−∞g(E)

1+exp([E−EF]/kBT)dE. (2.15)

After solving this equation for EF, we use as zero current solution µi = EF. This leads toa carrier concentration in the system that is not necessarily equal to the desired carrierconcentration c; we simply accept this concentration as the one to use in Eq. (2.12), whichthen becomes: ∑

jp j =

∑j

11+exp([E j −EF]/kBT)

. (2.16)

The method proceeds as usual from there, i.e. we solve the system given by Eqs. (2.2) and(2.16) for the desired field F. The advantage of this approach is that the effect of outlierstrapping carriers is reduced. For example, suppose that a certain realization of the disorderhas more outliers than usual. This simply leads to a higher right hand side in Eq. (2.16);in other words, we are adding additional carriers to fill these trapping sites. The results ofapplying this approach are shown by the squares in figure 2.3. Indeed, the dependence onlattice size is significantly reduced. This method will be used for almost all PBC results inthis thesis. We do note that when considering large fields the effectiveness of this methodis reduced, because the Fermi energy concept no longer applies. The dependence of theresults on lattice size is then much stronger.

2.4.3 Solving the master equation using the method of Yu et al.Yu et al. introduced an explicit iterative method to solve the master equation Eq. (2.2).4 Wewill not directly apply this method in this thesis. However, we do use a modified versionof the iteration step in this method as a part of Newton’s method, discussed in the nextsection. In addition, the method of Yu et al. has been the dominant solution method forseveral years. For these reasons it merits discussion.

We start with the equilibrium solution as described in Sections 2.4.1 and 2.4.2. Theprobabilities pi are then updated one by one by solving Eq. (2.2) for pi, yielding:

pi = 1/[1+

∑j ωi j(1− p j)∑

j ω ji p j

]. (2.17)

Whenever a probability is updated according to this equation, that updated value is used

*Note that this approach works only for uncorrelated energetic disorder.

40 Numerical methods

0 20 40 60

10-11

10-10

carr

ier m

obili

ty [

0ea2 /

]

lattice size L [a]

No adjustments Using fixed energy levels Fixing Fermi level

instead of carrier density

Uncorrelated Gaussian disorderMiller-Abrahams hoppingSimple-cubic lattice

/kBT = 4, c = 10-4, F = 0

Figure 2.3: Charge-carrier mobility as a function of lattice size L. The different symbols representdifferent ways of smoothing the variation in the mobility between realizations of the dis-order (see text). All results have been logarithmically averaged over enough realizationsof the disorder to make the error bars smaller than the symbol sizes.

for all further calculation within the same iteration. These iterations are repeated untilsatisfactory convergence is achieved.

Both FD and PBC calculations require some specific modifications. For FD calculations,updating pi will also change all hopping rates ωi j through the space charge effect. Inpractice we keep the rates fixed while applying the iterative method. Once this method hasconverged, we recompute the rates. These two steps are repeated until overall convergenceis satisfactory.6

For PBC calculations, we must make sure that the requirement of fixed carrier concen-tration Eq. (2.12) is satisfied. The initial equilibrium distribution satisfies this require-ment, but the iterations defined by Eq. (2.17) do not conserve carrier concentration. Thisis solved by first allowing the iterative method to converge and then determining the elec-trochemical potential energy µi from Eq. (2.4). We then shift µi by a constant value for allsites, chosen so that Eq. (2.12) is satisfied. This running of the method of Yu et al. followedby rescaling the potential is repeated until both Eqs. (2.2) and (2.12) are satisfied to withinspecified tolerances.

Although the method of Yu et al. has been successfully applied in several cases,4,6–10

it does not reliably converge for large disorder (σ/kBT & 6). When it does converge, ittypically achieves reasonable accuracy relatively quickly compared to Newton’s method,but is very slow in obtaining further precision. For these reasons it is not used for any ofthe results obtained in this thesis, except as an intermediate step in Newton’s method.

2.4 3D master equation 41

2.4.4 Solving the master equation using Newton’s methodTo apply Newton’s method to solve non-linear equations like the 3D master equation, onefirst linearizes the equation around some initial guess and then solves the linearized equa-tion to obtain a (hopefully) improved guess. This is repeated until satisfactory convergenceis achieved.

To simplify the notation we write our system of equations in the general form:

g(p)= 0, (2.18)

where g and p are N-dimensional vectors; p is the vector of all pi (excluding electrodesites). For full device (FD) calculations, we simply take each component of g to be one ofthe equations defined by Eq. (2.2). For periodic boundary conditions (PBC) calculations,we also have to take into account the requirement of fixed carrier concentration Eq. (2.12).We could of course just replace a random component of g by this equation, but this wouldlead to the different components being mismatched in terms of scale. Instead, we ‘spread’Eq. (2.12) over all others:

g i(p)=[∑

jωi j pi(1− p j)−ω ji p j(1− pi)

]+ tcω0 pi(1− pi)√∑

j p2j (1− p j)2

[c− 1

N

∑j

p j

]= 0. (2.19)

The factor ω0 pi(1−pi) ensures that the scales of the two terms are comparable. The choiceof the concentration tuning factor tc will be discussed below Eq. (2.24).

When applying an iteration of Newton’s method, we start from some guess for the so-lution p(n). We then try to obtain an improved guess p(n+1) by finding ∆p such that itapproximates the difference with the exact solution, ∆p ≈ p(n) −p. Linearizing Eq. (2.18)in ∆p yields

J∆p= g(p(n)), (2.20)

where J is the Jacobian matrix of g at p(n), given by

Ji j = ∂g i

∂p j

∣∣∣∣p(n)

. (2.21)

Methods to determine J and solve the linearized equation (2.20) are discussed in section2.4.5. The next guess for the solution is now given by

p(n+1) =p(n) −∆p. (2.22)

Note that p(n+1) would be the exact solution if g were truly linear. This method convergesrapidly, typically quadratically, if we start close enough to the actual solution.

We repeat these Newton iterations until satisfactory convergence is achieved. The con-vergence criterion is given by

||g(p(n))||2 <βω0N1/2, (2.23)

where ||g(p(n))||2 ≡ [∑j

g j(p(n))2]1/2 is the 2-norm of g(p(n)) and β is an adjustable parame-

ter. For any calculation β must be chosen so that the error in the resulting current density

42 Numerical methods

or carrier mobility is acceptable; we typically allow a relative error of 1%. We choose toscale Eq. (2.23) with N1/2 so that we do not have to change β when repeating a calculationfor a different lattice size. Directly related to the choice of β is the choice of the concen-tration tuning factor tc [see Eq. (2.19)]; allowing a maximum relative error in the carrierconcentration of 10−4 yields

tc = 110−4

βN1/2

c. (2.24)

Typically Newton’s method will only converge when the initial guess is close to theactual solution. This convergence regime can be increased by using damping. To adddamping we replace Eq. (2.22), which defines the next guess in the method in terms of thesolution of the linearized equation, by

p(n+1) =p(n) −α(n)∆p, (2.25)

with 0 < α(n) ≤ 1. By taking smaller steps, the method converges slower but does notdiverge as easily. We pick α(n) at each step by minimizing ||g(p(n+1))||2.

Newton’s method with damping works well when the solution is close to the initialequilibrium guess, i.e. at low applied voltage/electric field. However, it usually does notconverge for typical experimentally relevant voltages. The solution to this is to increase thevoltage in small steps. After each voltage step, we apply Newton’s method to determine thevalues of pi. We then use this as the initial guess for the next voltage step. This ensuresthat we are never far from the solution, so that the method converges. There is an extracomplication for devices with a built-in voltage Vbi, as the equation is very difficult to solvein the blocking regime, V < Vbi. Although it is not very important to get results for thisregime, it also makes it impossible to get J-V characteristics for V >Vbi because we cannotstart from V = 0. We solve this by starting from V = Vbi = 0 and increasing both V and Vbisimultaneously in small steps, until the desired value of Vbi is reached. From there we onlyincrease V , and so get J-V characteristics for V >Vbi.

Another issue is that Eq. (2.25) can lead to unphysical values of pi under 0 or above1. This can be rectified by solving the equations in terms of the electrochemical potentialenergy µi, defined by Eq. (2.4), instead of pi. Note that µi can physically take on anypositive or negative value. The linearized equation remains unchanged, since it does notmatter in which variable we linearize. The equivalent of Eq. (2.25) now becomes

µ(n+1)i = µ(n)

i +α(n) dµi

dpi∆pi = µ(n)

i +α(n) kBT∆pi

p(n)i (1− p(n)

i ). (2.26)

Typically using µi will converge faster and more reliably than using pi. However, whenthe carrier concentration is very low the master equation is linear in pi; in this situationusing pi is generally faster.

A final trick used to stabilize the method is to occasionally follow a step of Newton’smethod by a single iteration of the method of Yu et al., described in the previous section.This can help to eliminate outliers and stabilize the overall solver, while hardly adding tothe computation time.

2.4 3D master equation 43

2.4.5 Obtaining and solving the linearized master equation

In this section we will discuss how to obtain the Jacobian J, defined by Eq. (2.21), andhow to solve the resulting linearized master equation (2.20). The techniques discussedhere also apply to the similar equations (2.42) and (2.53) encountered in time-dependentcalculations.

A complication in determining J is that it is not sparse. For full device (FD) calculations,this is because all sites interact through the space charge effect. For periodic boundaryconditions (PBC) calculations, the requirement of fixed carrier concentration causes allvalues of J to be nonzero [see Eq. (2.19)]. This non-sparsity is not necessarily a problem;after all, we do not need to compute J directly. All we need is an efficient way to computethe product of J with an arbitrary vector dp, i.e. we need to compute the change in g(p)caused by an infinitesimal change in p. By ‘efficient’ we mean that the computation timeshould scale linearly with the number of sites N (a brute force computation of the fullJacobian would scale with N2).

The Jacobian J0 of the ‘basic’ master equation Eq. (2.2), ignoring the requirement offixed carrier concentration and space charge effects, can be determined straightforwardly:

J0,ii = ∑k ωik(1− pk)+ωki pk;

J0,i j = −ωi j pi −ω ji(1− pi) (i = j). (2.27)

These matrix values can be precomputed, i.e. they do not have to be reevaluated whenmultiplying J with a different vector. Note that the matrix J0 is sparse, since only a smallnumber of sites j are accessible from any one site i.

We will now explain how to compute the additional terms for FD and PBC calculations.For FD calculations, we need to include the space charge effect. Specifically, a change inthe occupation probabilities dp will cause a change in the electric potential, dV. This inturn will cause a change in the hopping rates. We can calculate these changes as follows.First, dV is calculated by linearizing the method used to calculate V itself, Eqs. (2.5) to(2.9). The change in ωi j is then given by:

dωi j = e(dVi −dVj

) dωi j

d∆E i j. (2.28)

This is all we need to linearize Eq. (2.2) for FD calculations:

(Jdp)i = (J0dp)i +dωi j pi(1− p j)−dω ji p j(1− pi)= (J0dp)i + e

(dVi −dVj

)[ dωi jd∆E i j

pi(1− p j)+ dω jid∆E ji

p j(1− pi)]

.(2.29)

Note that the term in square brackets can be precomputed.For PBC calculations, the hopping rates do not depend on p. However, we now have to

linearize the equations including the requirement of fixed carrier concentration, Eq. (2.19).

44 Numerical methods

This yields:

(Jdp)i = (J0dp)i + tcω0 pi(1−pi)

N√∑

j p2j (1−p j)2

∑j dp j

+ tcω0(1−2pi)

N√∑

j p2j (1−p j)2

[c− 1

N∑

j p j]

+ tcω0 pi(1−pi)

N[∑

j p2j (1−p j)2

]3/2

[c− 1

N∑

j p j]∑

j([3p2j −2p3

j − p]dp j).

(2.30)

Note that none of the sums in this equation depends on i, so they only have to be calculatedonce and can then be used for all values of i. As a result, the computation time of Eq. (2.30)scales linearly with N, as required.

Since we do not know the actual matrix elements of J, Eq. (2.20) must be solved us-ing an iterative method. Typically, applying such a method only requires us to be able tocompute the vector product of J with an arbitrary vector as described above. Several it-erative methods were tested and evaluated on the basis of their speed and reliability; thebest method turned out to be the NAG library implementation of the biconjugate gradientstabilized (4) method (Bi-CGSTAB(4)).11,12 The convergence criterion for the linear solveris:

||g(p(n))−J∆p||2 <βlinearω0N1/2. (2.31)

If the master equation were truly linear and βlinear = β, this criterion is equivalent tostating that the next approximation p(n+1) satisfies Eq. (2.23), i.e. that we accept it as asolution. In practice, non-linearity will always add a small additional error term, so it isimportant to take βlinear <β. We generally take βlinear =β/1.2.

To speed up the method, a suitable preconditioning method is needed. Preconditioninga linear equation means modifying it to make it easier to solve. We will discuss herehow to use preconditioning for the linear equation as encountered when applying Newton’smethod, Eq. (2.20). Typically left preconditioning is used, which for this equation wouldbe:

A−1J∆p=A−1g(p(n)). (2.32)

for some preconditioner A. Note that this is mathematically identical to Eq. (2.20). Theidea is that A somehow approximates J while being easy to invert. This can then help toeliminate extreme time scales for the problem. However, the most effective preconditioningmethod for our problem was found to be a right preconditioning method, written as follows:

JA−1y= g(p(n)). (2.33)

Note that ∆p = A−1y, so after solving Eq. (2.33) for y we straightforwardly obtain ∆p aswell. For the preconditioning matrix A we choose the matrix J0 with all non-diagonalelements set to zero.

There is a physical basis for this preconditioning method, which can be seen most clearlywhen considering PBC calculations in the low density limit. We are then considering thehopping of a single charge carrier through the lattice, and Eq. (2.2) is reduced to:∑

jωi j pi −ω ji p j = 0. (2.34)

2.5 1D drift-diffusion equation 45

Note that this equation is linear already. It is difficult to solve because of the vastly differ-ent timescales involved. Specifically, the carrier will be trapped in energetically low-lyingsites most of the time. Consider now the preconditioned version of this equation as givenby Eqs. (2.33) and (2.27):

yi −∑

j

(ω ji∑k ω jk

)yj = 0. (2.35)

This can again be viewed as the master equation related to a hopping process, with the ratefrom site i to j now given by ωi j/

∑k ωik. As a result, the sum of all hopping rates away from

any site is equal to one. This means that the average time between hops does not dependon the current location of the carrier, and that the trapping problem is eliminated. Indeed,the Bi-CGSTAB(4) method converges significantly faster for Eq. (2.35) than for Eq. (2.34),and this also holds when comparing Eqs. (2.20) and (2.33) instead of the low density limit.

2.5 1D drift-diffusion equationIn this section, we will discuss a fast and reliable solver for the one-dimensional drift-diffusion equation for single-carrier devices. Another method, based on an approach firstintroduced by Bonham and Jarvis,13 can be found in Ref. 14. A more general approachsuitable for double-carrier devices is discussed in Ref. 15.

The differential equation to be solved is given by Eqs. (1.2) and (1.4):

J =µ(n,F)n(x)[

eF(x)− dEF

dndndx

]; (2.36a)

dFdx

= en(x)ϵ0ϵr

, (2.36b)

where we have made use of Eq. (1.3) to work out the diffusion coefficient D. We also needto specify boundary conditions at the electrodes. In the 3D simulation, the first and lastlayers of the organic material are at a distance a from the electrodes (for a simple-cubiclattice). We emulate this in our 1D solver by only solving Eq. (2.36) between a and L−a.The boundary conditions at these points are given by assuming thermal equilibrium withthe electrodes, with no image charge effects taken into account. These assumptions areaccurate as long as the injection barrier is not too high.6 The resulting equations are:

EF(a)=Φleft + eaF(a);EF(L−a)=Φright − eaF(L−a), (2.37)

where Φleft and Φright are the work functions of the electrodes. The densities n(a) andn(L−a) follow from these values of EF.

In an experiment one typically applies a chosen voltage and measures the current.When solving the above equations, it is easier to fix the current and determine the volt-age, since otherwise J has to be treated as an unknown. At this point the problem is awell-posed boundary value problem. Typically one would now determine the solution on agrid in the x direction using finite element or finite difference techniques. However, for our

46 Numerical methods

specific case of single carrier devices one can convert the problem to an initial value prob-lem, which is easier to solve. To accomplish this, we guess F(L− a) instead of specifyingn(a).* Since we now know both F(L−a) and n(L−a), we can solve for F(x) and n(x) usingMathematica 8.0’s default differential equation solver. We then simply check if the valueof n(a) is consistent with Eq. (2.37); if not, we try a new guess of F(L−a).

Using this approach, it is possible to determine J-V characteristics and field/densityprofiles quickly, reliably and accurately for all devices considered in this thesis.

2.6 Time-dependent calculations with the 3D masterequation

In section 2.4 we considered the steady state using the 3D master equation approach. It isalso possible to use this approach to carry out time-dependent calculations. Although thesecalculations will not be used in this thesis, we expect them to become an important tool inthe further analysis of charge transport in organic semiconductors. We will consider theresponse of the current in a device to a sudden change in voltage (dark-injection transients)and to a small AC voltage applied over a background DC voltage (small-signal analysis).

The basic idea is to determine the time-dependent probabilities pi(t) that site i is occu-pied by a charge carrier at time t. The equation governing the evolution of these probabili-ties is:

dpi

dt=−∑

jωi j pi(1− p j)+ω ji p j(1− pi)=−g i(p), (2.38)

where g(p) is the vector form of the master equation [see Eq. (2.18)]. This equation canbe applied to both PBC and FD calculations. For PBC calculations, it is not necessary totake into account the requirement of fixed carrier concentration given by Eq. (2.19), sinceEq. (2.38) inherently keeps the carrier concentration constant in time. For FD calculations,we must keep in mind that the hopping rates ωi j depend on pi and thus on t through thespace charge effect.

The current due to the movement of charge carriers may now be position dependent,but when the displacement current is taken into account the current density is uniformover the device. It is given by Eq. (2.3) with a term added for the displacement current:

J(t)= eLxL yLz

∑i, j

ωi j pi(1− p j)Ri j,x + ϵ0ϵr

Lx

dVdt

. (2.39)

2.6.1 Dark-injection transientsIn a dark-injection experiment, the voltage applied over a device is instantaneouslychanged and the time-dependent current response is measured. To simulate this, we firstdetermine the steady state solution p(0) at the initial voltage using Newton’s method (seesection 2.4.4). We then change the voltage to the new value at t = 0 and use Eq. (2.38) to

*Alternatively, one could guess F(a), but working from right to left turned out to be faster.

2.6 Time-dependent calculations with the 3D master equation 47

determine p(t), and Eq. (2.39) to determine the time-dependent current density J(t). Themain challenge encountered in this time integration is the huge range of time scales in-volved: typically the fastest hop rate in the system is in the order of nanoseconds, whilethe relaxation time of the system as a whole is in the order of milliseconds or even seconds.

The basic approach is to iteratively determine the probability vectors p[n] at discretetimes t[n], starting from p[0] at t[0] = 0. The most straightforward way to do this is theexplicit Euler method:

p[n] =p[n−1] −∆t[n]g(p[n−1]), (2.40)

where ∆t[n] = t[n] − t[n−1]. Typically, this method is applied with fixed time steps ∆t[n]. Adisadvantage is that if this time step is too large the method becomes unstable. Specifically,∆t must be taken comparable to the fastest time scales in the system. Since the range oftime scales is so large, this makes the required number of steps prohibitively high. Higher-order explicit methods (the so-called Runge-Kutta methods) also share this problem.

The solution to this problem is the use of an implicit method, such as the implicit Eulermethod:

p[n] =p[n−1] −∆t[n]g(p[n]). (2.41)

Note that we now use the value of the derivative g at the new value of p. This eliminatesthe instability encountered in the explicit methods. We must initially still choose a smalltime step, but as the faster timescales in the system relax we can increase it. In practicewe can generally take the time step proportional to the elapsed time, ∆t[n] ∝ t[n], so thatthe number of time steps per order of magnitude in time is constant.

A complication arising in the use of the implicit Euler method is that Eq. (2.41) needsto be solved for p[n]. This is again done using Newton’s method: starting from some guesspguess we determine the difference ∆p with the actual value of p[n] by linearizing Eq. (2.41)around pguess, yielding:

(I+∆t[n]J)∆p=pguess −p[n−1] +∆t[n]g(pguess), (2.42)

where I is the identity matrix and J is the Jacobian matrix defined by Eq. (2.21). Meth-ods to determine J and to solve this equation are discussed in section 2.4.5. Part of thesolution method is the choice of the preconditioner A; we take the diagonal of the matrixI+∆t[n]J0. Once we have obtained ∆p, we use it to improve our guess. This is repeateduntil satisfactory convergence is achieved, specifically:

||pguess −p[n−1] +∆t[n]g(pguess)||2 <β∆t[n]ω0N1/2. (2.43)

To apply Newton’s method, it is important to start with a suitable initial guess, referred toas the predictor. A good choice for this predictor is the result of the explicit Euler method:

pguess =p[n−1] −∆t[n]g(p[n−1]). (2.44)

In practice a second order implicit method turns out to work best. This method is givenby

p[n] = 12ρ+ρ2

[(1+ρ)2p[n−1] −p[n−2] −ρ(1+ρ)∆t[n] g(p[n])

], (2.45)

48 Numerical methods

where ρ =∆t[n−1]/∆t[n]. A suitable initial guess for the Newton solver is

pguess =p[n−1] −∆t[n][(

1+ 12ρ

)g(p[n−1])− 1

2ρg(p[n−2])

]. (2.46)

To show that this method is feasible for realistic devices, we have applied it to theexample device (see section 2.1) in figure 2.4. The similarities and differences with experi-mentally obtained dark-injection curves are currently being studied within our group.

10-9 10-7 10-5 10-3 10-1

100

101

102

103

curr

ent d

ensi

ty [A

/m2 ]

time [s]

Figure 2.4: Simulated current density as a function of time for the example device. At t = 0, thevoltage over the device has been instantaneously increased from 1.5 V to 3 V.

2.6.2 Small-signal analysisIn a small-signal analysis experiment, the response of the current in a device to a smallAC voltage with frequency f is measured. This AC voltage is applied over a backgroundDC voltage V . The result is typically expressed in terms of the complex impedance per unitarea Z, which is the ratio of the applied AC voltage exp(2πi f t)∆V to the current densityresponse exp(2πi f t)∆J (where i =p−1):

Z = ∆V∆J

. (2.47)

The complex impedance is related to the resistance R and capacitance C by:

R =Re(Z) (2.48)

andC = Im(1/Z)

2π f. (2.49)

2.6 Time-dependent calculations with the 3D master equation 49

To simulate such an experiment, we determine the response of the probabilities to theAC voltage, starting from the steady-state solution p0. We assume that the AC voltageis small enough to linearize all responses. This allows us to write the time-dependentprobability vector p as:

p(t)=p0 +exp(2πi f t)∆p. (2.50)

Taking the time derivative yields:

dpdt

= 2πi f exp(2πi f t)∆p. (2.51)

On the other hand, the time derivative of p is given by Eq. (2.38):

dpdt

=−g(p)=−g(p0)−exp(2πi f t)∆V∂g∂V

−exp(2πi f t)J∆p. (2.52)

The second equality is the linearization of g(p) around p0, containing both the response ofg to ∆V and the response to ∆p. Note that g(p0) = 0 since it is the steady-state solution.Combining Eqs. (2.51) and (2.52) then yields:

(2πi f I+J)∆p=−∆V∂g∂V

, (2.53)

where I is the identity matrix. This equation can be solved for ∆p using the methodsdescribed in section 2.4.5, with the diagonal of 2πi f I+J0 as preconditioner. The currentresponse ∆J is then obtained straightforwardly from Eq. (2.39). Note that the dependenceof V and ωi j on time must be taken into account.

As a demonstration of the technique, we have applied it to the example device (see sec-tion 2.1) in figure 2.5. Also shown are experimentally and semi-analytically obtained re-sults for the same capacitance-voltage curve.16 The semi-analytical results were obtainedusing the 1D-ME approach with a multiple-trapping model to describe carrier relaxation.It should, if the assumptions used for that model are accurate, match the numericallyexact results of our 3D-ME approach. From the figure we thus conclude that the multiple-trapping approach is not sufficient, and that 3D effects must be taken into account. How-ever, the results of the 3D-ME method still do not match the experimental results, indicat-ing that there are inaccuracies in our physical model of this device.

50 Numerical methods

0 1 2 3 4

1

1.1

1.2

1.3

1.4

f = 100 Hz

capa

cita

nce

[C0]

voltage [V]

Experimental Multiple-trapping model 3D master equation

Figure 2.5: Capacitance as a function of voltage at frequency f = 100 Hz for the example device, inunits of the geometric capacitance C0 = ϵ0ϵr/L. Black curve: experimental results, asfound in Ref. 16. Gray dashed curve: multiple-trapping model results, also from Ref. 16.Gray solid curve: 3D master-equation results, as described in this section.

References1. S. L. M. van Mensfoort, S. I. E. Vulto, R. A. J. Janssen, and R. Coehoorn, Phys. Rev.

B 78, 085208 (2008).2. J. J. M. van der Holst, F. W. A. van Oost, R. Coehoorn, and P. A. Bobbert, Phys. Rev.

B 83, 085206 (2011).3. J. Cottaar and P. A. Bobbert, Phys. Rev. B 74, 115204 (2006).4. Z. G. Yu, D. L. Smith, A. Saxena, R. L. Martin, and A. R. Bishop, Phys. Rev. B 63,

085202 (2001).5. A. Lukyanov and D. Andrienko, Phys. Rev. B 82, 193202 (2010).6. J. J. M. van der Holst, M. A. Uijttewaal, B. Ramachandhran, R. Coehoorn, P. A.

Bobbert, G. A. de Wijs, and R. A. de Groot, Phys. Rev. B 79, 085203 (2009).7. Z. G. Yu, D. L. Smith, A. Saxena, R. L. Martin, and A. R. Bishop, Phys. Rev. Lett.

84, 721 (2000).8. W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M. Blom, D. M.

de Leeuw, and M. A. J. Michels, Phys. Rev. Lett. 94, 206601 (2005).9. F. Jansson, S. D. Baranovskii, F. Gebhard, and R. Österbacka, Phys. Rev. B 77,

195211 (2008).10. M. Bouhassoune, S. L. M. van Mensfoort, P. A. Bobbert, and R. Coehoorn, Org. Elec.

10, 437 (2009).11. H. A. van der Vorst, S. J. Sci. Statist. Comput. 13, 631 (1992).

References 51

12. G. L. G. Sleijpen and D. R. Fokkema, ETNA 1, 11 (1993).13. J. S. Bonham and D. H. Jarvis, Aust. J. Chem. 31, 2103 (1978).14. S. L. M. van Mensfoort and R. Coehoorn, Phys. Rev. B 78, 085207 (2008).15. E. Knapp, R. Häusermann, H. U. Schwarzenbach, and B. Ruhstaller, J. Appl. Phys.

108, 054504 (2010).16. W. C. Germs, J. J. M. van der Holst, S. L. M. van Mensfoort, P. A. Bobbert, and R.

Coehoorn, Phys. Rev. B 84, 165210 (2011).

Chapter 3

A scaling theory for thecharge-carrier mobility

Abstract

We present a scaling theory for charge transport in disordered molecular semiconductorsthat extends percolation theory by including bonds with conductances close to the percolat-ing one in the random-resistor network representing charge hopping. A compact expressionfor the dependence of the charge-carrier mobility on temperature and carrier concentrationis derived from this theory, with parameters determined from numerically exact results.We determine these parameters for Miller-Abrahams and Marcus hopping on simple-cubicand face-centered-cubic lattices. We do this both for uncorrelated Gaussian energetic dis-order and for correlated energetic disorder as obtained from randomly oriented moleculardipoles. In addition, we show that for Miller-Abrahams hopping the theory can also beapplied to uncorrelated non-Gaussian energetic disorder, without parameter changes. Ourexpression for the charge-carrier mobility predicts a universal charge-concentration de-pendence, which depends only on the shape of the density of states and not on the typeof hopping or lattice. The temperature dependence is model-specific, but can still be quitesimilar for different hopping models. We suggest alternative approaches that could help toexperimentally distinguish hopping models.

53

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Scaling formula for the charge-carrier mobility . . . . . . . . . . . 55

3.2.1 The random-resistor network . . . . . . . . . . . . . . . . . 553.2.2 Percolation theory and the scaling ansatz . . . . . . . . . . 563.2.3 Determining the critical conductance . . . . . . . . . . . . . 58

3.3 Application to different hopping models . . . . . . . . . . . . . . . . 603.3.1 Determining the scaling parameters . . . . . . . . . . . . . 613.3.2 Miller-Abrahams hopping with uncorrelated disorder . . . 643.3.3 Effect of lattice disorder . . . . . . . . . . . . . . . . . . . . . 67

3.4 Consequences for charge transport . . . . . . . . . . . . . . . . . . . 683.4.1 Concentration dependence of the charge-carrier mobility . 683.4.2 Temperature dependence of the charge-carrier mobility . . 69

3.5 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . 703.A Density of states for dipole-correlated disorder . . . . . . . . . . . . 71References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

This chapter was adapted with permission from Refs. 1 and 2:

J. Cottaar, L. J. A. Koster, R. Coehoorn, and P. A. Bobbert, Scaling theory for percolative charge transportin disordered molecular semiconductors, Phys. Rev. Lett. 107, 136601 (2011).Copyright (2011) by the American Physical Society.

J. Cottaar, R. Coehoorn, and P. A. Bobbert, Scaling theory for percolative charge transport in molecularsemiconductors: Correlated versus uncorrelated energetic disorder, Phys. Rev. B 85, 245205 (2012).Copyright (2012) by the American Physical Society.

54 A scaling theory for the charge-carrier mobility

3.1 Introduction

The charge-carrier mobility µ (the average velocity of a charge carrier divided by the ap-plied electric field) is the basic element of our technique of reproducing three-dimensionalOLED simulation results with one-dimensional models (see section 1.3). This mobility canbe determined from 3D simulations of periodic systems. The dependence on factors suchas the carrier concentration c, temperature T and electric field strength F can then be pa-rameterized and used in a one-dimensional drift-diffusion approach. An overview of suchnumerical work is given in section 1.4.2.

Relying only on these numerical results has two important drawbacks. First, they pro-vide no physical insight into the factors determining the value of µ. Second, they areoften limited to one specific hopping model [typically Miller-Abrahams (MA) hopping on asimple-cubic (SC) lattice], without giving any hints as to how sensitive the value of µ is tothis assumption. A more general approach is provided by semi-analytic methods, whichcan be roughly divided into effective-medium3–5 and percolation6–9 theories. The hoppingsystem is then often considered as a random-resistor network. The idea of percolation theo-ries is that at low temperatures, when due to the disorder the spread in resistances is large,the conductivity is determined by a single critical bond in this network. This critical bondhas a conductance Gcrit such that all bonds with conductance G ≥Gcrit just form a percolat-ing cluster.6 However, the results of percolation theories do not agree quantitatively withthe numerically exact results,9 the reason being that also bonds with conductances aroundGcrit influence the conductivity. Dyre and Schrøder introduced the term ‘fat percolation’ forthis, and applied this concept to AC conduction.10

In this chapter, we develop a scaling theory based on the concept of fat percolation thataccurately describes the zero-field mobility function µ(T, c). Our goal is twofold: i) to pro-vide fundamental understanding of charge transport in organic molecular semiconductorsand ii) to provide a general and compact expression for µ(T, c) that can be used in the mod-eling of organic devices. The parameters in this expression are found from numericallyexact results obtained with the master equation (ME) method (see section 2.4). At the con-sidered carrier concentrations of at most a few percent, the ME method properly accountsfor the dominant effect of Coulomb interactions, which is to prevent the presence of twocarriers on one site.11,12 The hopping model itself is described in section 1.4.1. We considernearest-neighbor hopping with MA as well as Marcus rates, and uncorrelated Gaussiandisorder as well as dipole-correlated disorder. For the specific case of MA hopping, we willalso consider non-Gaussian uncorrelated disorder. The influence of the lattice structure isinvestigated by considering next to simple-cubic (SC) also face-centered-cubic (FCC) lat-tices and the effect of lattice disorder.

We will first develop the scaling theory in section 3.2, culminating in a simple expres-sion for the mobility function, Eq. (3.16). Next, we will demonstrate how to apply thisexpression to different hopping models in section 3.3, including an analysis of the effect oflattice disorder. We will then discuss the temperature and concentration dependence of themobility in detail in section 3.4, and finally present the main conclusions and an outlookon applications in section 3.5.

3.2 Scaling formula for the charge-carrier mobility 55

3.2 Scaling formula for the charge-carrier mobilityIn this section we will introduce our scaling theory and derive from it a simple and gen-eral expression for the charge-carrier mobility. First, we map the hopping problem onto arandom-resistor network (section 3.2.1). Next, we use a visualization of the current andpower-dissipation distribution in this network to demonstrate the concepts of percolationand fat percolation. Based on fat percolation we then introduce a scaling ansatz for thecharge-carrier mobility in terms of the critical conductance Gcrit and the number of bondswith such conductance (section 3.2.2). Finally, we will show how to compute this conduc-tance and number of bonds, which together with the scaling ansatz leads to a simple andgeneral expression for the charge-carrier mobility (section 3.2.3).

3.2.1 The random-resistor networkOur goal is to determine the charge-carrier mobility at low electric field. We will showin this section how this can be reformulated in terms of a random-resistor network.6 Themaster equation, Eq. (2.2), can be written in terms of the electrochemical potential energyµi defined by Eq. (2.4):

∑j

eωi j,symm sinh[µi−µ j+eFRi j,x

2kBT

]2cosh

[E i−µi2kBT

]cosh

[E j−µ j2kBT

] = 0, (3.1)

where Ri j,x is the difference in the x coordinates between sites i and j and F is the strengthof the electric field, which is applied in the x direction. Since we are working at low F, wecan linearize this equation in F, µi −EF, and µ j −EF to obtain

∑j

eωi j,symm(µi − µ j + eFRi j,x

)4kBT cosh

[E i−EF2kBT

]cosh

[E j−EF2kBT

] = 0. (3.2)

This can also be read as Kirchhoff ’s law of current conservation, with µi − µ j + eFRi j,x thevoltage difference and the bond conductance G i j given by

G i j =e2ωi j,symm

4kBT cosh[

E i−EF2kBT

]cosh

[E j−EF2kBT

] . (3.3)

The problem of determining the charge-carrier mobility µ is equivalent to determining thenetwork conductance Gnetwork of this random-resistor network. The relationship with µ isstraightforward:

µ= Lx

L yLz ecNtGnetwork, (3.4)

where c is the charge-carrier concentration. Note that the random-resistor network ap-proach is not an approximation; up to this point, both formulations of the problem aremathematically identical.

56 A scaling theory for the charge-carrier mobility

Σ�kBT=1 Σ�kBT=3 Σ�kBT=10

Figure 3.1: Normalized current (line opacity) in bonds of a 15×15 square lattice. The red circlesindicate bonds with a power dissipation of at least 30% of the maximum power dissipa-tion. The results shown are for uncorrelated Gaussian disorder, Marcus hopping withreorganization energy Er→∞ and carrier concentration c = 10−5. A small electric fieldhas been applied from left to right.

3.2.2 Percolation theory and the scaling ansatzTo show the physical basis of our scaling theory, we consider the spatial distribution ofcurrent and power dissipation in the random-resistor network, as shown for a 2D systemin figure 3.1. In the case of low disorder (left panel), the current and power distributions arevery homogeneous. Although there are small local variations, these do not extend to a scaleof more than a few bonds. This regime can be accurately analyzed using effective-mediumtheory,3 in which the average effects of the random resistors are described by an effectivemedium. This theory matches the simulation results for σ/kBT . 2 in a 3D system (seethe dashed curve in figure 3.2). However, it is not accurate in the experimentally relevantregime 3.σ/kBT . 6.

Let us consider the opposite limit, high disorder (right panel in figure 3.1). In this case,the current follows only the path of least resistance. Along this path, the bond with lowestconductance determines the overall conductance (this is the circled bond in the figure). Wewill call this bond the critical bond, and its conductance the critical conductance, Gcrit.According to this reasoning, we should expect Gcrit = Gnetwork, but this would lead to asystem size dependence of the mobility [see Eq. (3.4)]. For this reason, percolation theoriesfor the charge-carrier mobility generally take the following form:6–9

µ= HN2/3

t ecGcrit, (3.5)

for some constant H which does not depend on T or c. This standard percolation approach,however, does not quantitatively match the simulation results (see the dotted curve infigure 3.2).

Let us see what we can learn from the case of intermediate disorder (middle panelin figure 3.1). We clearly see percolation at work here, with the current being funneled

3.2 Scaling formula for the charge-carrier mobility 57

0 1 2 3 4 5 6 7 810-5

10-4

10-3

10-2

10-1

100

Uncorrelated Gaussian disorderMarcus hopping (Er ), SC lattice

Car

rier m

obili

ty

[e0/N

2/3

t]

/kBT

Simulation Effective-medium theory Percolation theory Scaling ansatz

Carrier concentration c = 10-2

Figure 3.2: Dependence of the charge-carrier mobility µ on temperature T for Marcus hopping withreorganization energy Er→∞, a simple-cubic (SC) lattice and uncorrelated Gaussiandisorder. Triangles: master equation (ME) simulation results. Solid curve: scalingansatz, Eq. (3.7), with A = 1.8 and λ = 0.85. Dotted curve: standard percolation the-ory, Eq. (3.5), with H = 0.3. Dashed curve: effective-medium theory [Eq. (5.4) in Ref. 3].

through high conduction pathways. However, there are now multiple bonds with highpower dissipation. This indicates that the mobility is determined not only by the criticalconductance but also by the amount of bonds with such conductance. Dyre et al. introducedthe term ‘fat percolation’ for this phenomenon.10 To quantify this ‘number of bonds’, weuse the partial density function of the bond conductances f . We only use the value of thisfunction at Gcrit, f (Gcrit). This is justified when the disorder is high enough; bonds withconductance well above Gcrit can then be considered as perfectly conducting, and thosewith conductance well below Gcrit as perfectly insulating. Concluding that the mobilityonly depends on Gcrit and f (Gcrit), and using the fact that it must scale linearly with Gcrit,leads to

µ= 1N2/3

t ecGcrith[Gcrit f (Gcrit)] (3.6)

for some dimensionless function h. Since percolation can be viewed as a critical phe-nomenon, with a critical point at f (Gcrit) = 0, we propose as a scaling ansatz for this func-tion h a power-law form:

µ= AN2/3

t ecGcrit[Gcrit f (Gcrit)]λ, (3.7)

where the constants A and λ do not depend on T or c. The theory does not predict thevalues of these constants; they must be fitted to simulation data and may well depend onthe type of lattice, hopping and energy disorder. We assume here that the topology of thepercolating network does not depend on T or c; if it did, we would expect A and λ to depend

58 A scaling theory for the charge-carrier mobility

on them too. We will justify this assumption later. Eq. (3.7) does not apply if the chargetransport is dispersive, i.e. if the mobility does not have a well-defined value for an infinitelattice size.

This scaling ansatz is tested by comparison to simulation results in figure 3.2, withthe values of A and λ fitted to the data. The values of Gcrit and f (Gcrit) were determinedusing the methods described below. We see that for σ/kBT & 1 our scaling theory matchesthe simulation accurately. For σ/kBT . 1, not only f (Gcrit), but the whole distributionf (G) becomes important and the approach fails. Our new scaling approach has two fittingparameters (A and λ), while the standard percolation approach has only one (H) and theeffective-medium approach has none. This means that our scaling approach would be moreaccurate even if it had no physical basis; however, the accuracy is improved so significantlythat we can safely conclude that this is not the case.

3.2.3 Determining the critical conductanceTo derive a simple expression for the charge-carrier mobility from Eq. (3.7) we need tocompute Gcrit and f (Gcrit). To find Gcrit, let us consider the percolation problem in detail.There is a percolation threshold pbond, such that the portion pbond of bonds with highestconductivity just forms an infinitely large connected network, the percolating network.6

The critical conductance Gcrit is the lowest conductance occurring in this network. Gcritand pbond are related through

1−Φ(Gcrit)= pbond, (3.8)

with Φ(G) the cumulative distribution function of the distribution of bond conductances,i.e., Φ(G) is the probability that a randomly chosen bond has a conductance lower than orequal to G. Since G i j depends only on the energies of the bond sites E i and E j, we canwork in the (E i,E j)-space to obtain

1−Φ(Gcrit)= pbond =Ï

G(E i ,E j)>Gcrit

g(E i)g(E j)dE idE j, (3.9)

where g(E) is the density of states.* In words, Gcrit is determined by the requirement thatthe G(E i,E j) = Gcrit-contour encloses a portion pbond of bonds. This concept is illustratedin figure 3.3(a), where the black contour corresponds to G =Gcrit.

To proceed, we will approximate the exact bond conductances given by Eq. (3.3). If theFermi energy EF is well below the site energies E i and E j, the hyperbolic cosine termsbecome exponentials, leading to

G i j =e2ωi j,symm

kBTexp

(EF

kBT− E i +E j

2kBT

). (3.10)

In general, EF is not low enough for this approximation to be accurate for all bonds. How-ever, this only matters in determining Gcrit if bonds are incorrectly determined to be above

*For simplicity, we have assumed the energy disorder to be uncorrelated; in the case of correlated disorder, wewould have to consider the joint density of states g(E i ,E j) in Eq. (3.9), but this does not affect our results.

3.2 Scaling formula for the charge-carrier mobility 59

Ei @ΣD

E j @ΣD

1

1

-6 -5 -4 -3 -2 -1

-6

-5

-4

-3

-2

-1

Increasingconductance

Ei @ΣD

E j @ΣD

1

1

-6 -5 -4 -3 -2 -1

-6

-5

-4

-3

-2

-1

Figure 3.3: (a) Contours of constant bond conductance [using Eq. (3.3)] in the (E i ,E j)-space forMiller-Abrahams (MA) hopping, Fermi energy EF = −3, and σ/kBT = 4 (correspondingto c = 0.0033). The black curve is the contour corresponding to the critical conductanceGcrit for uncorrelated Gaussian disorder and a simple-cubic lattice. E i and E j are theenergies of the sites linked by the bond. The arrow indicates the direction of increas-ing conductance. (b) Same Gcrit contour, using the exact bond conductances as givenby Eq. (3.3) (black) and using the approximation Eq. (3.10) (dark gray). The check-ered area indicates the bonds erroneously considered to have conductance above Gcritby Eq. (3.10).

or below Gcrit. These bonds are indicated by the checkered area in figure 3.3(b). This areais located at low energies and so contains few bonds (about 0.1% of all bonds in this exam-ple). This means that using Eq. (3.10) instead of Eq. (3.3) will not significantly affect thevalue of Gcrit. We will therefore use Eq. (3.10) for the remainder of this chapter. We willsee later that this is accurate for c . 0.01.

The final step in deriving Gcrit is realizing that G i j can now be written as

G i j = e2ω0

kBTexp

(EF −E(E i,E j)kBT

), (3.11)

where E is an energy function of E i and E j that does not depend on T or c. Note that theenergy dependence of ωi j,symm is also included in this function. This allows us to rewriteEq. (3.9) as

1−Φ(Gcrit)= pbond =Ï

E(E i ,E j)<Ecrit

g(E i)g(E j)dE idE j, (3.12)

with the critical energy Ecrit related to the critical conductance Gcrit by

Gcrit =e2ω0

kBTexp

(EF −Ecrit

kBT

). (3.13)

60 A scaling theory for the charge-carrier mobility

Eq. (3.12) defines the percolation problem independently of T and c. Thus, Ecrit is itself in-dependent of T and c, and so Eq. (3.13) gives the dependence of Gcrit on T and c. Eq. (3.12)also shows that the percolating network itself is independent of T and c, a fact which wasuntil now assumed [see the discussion under Eq. (3.7)]. We note that this does not implythat the current filament structure is independent of T and c; indeed, it can be seen infigure 3.1 that it does depend on temperature.

To complete our expression for the mobility we also need to find f (Gcrit). By definition,the partial density function f is the derivative of the cumulative distribution function Φ, soto find f (Gcrit) we can, slightly abusing the notation, take the derivative to Gcrit of Φ(Gcrit)as found above:

f (Gcrit)=dΦ

dGcrit= dEcrit

dGcrit

dΦdEcrit

= kBTGcrit

ddEcrit

[ÏE(E i ,E j)<Ecrit

g(E i)g(E j)dE idE j

]. (3.14)

Here, dEcrit/dGcrit is found from Eq. (3.13) and dΦ/dEcrit from Eq. (3.12). The second factorcan be computed numerically. Since it is independent of T and c anyway, we include it in anew constant B:

B ≡ AWλ

(d

dEcrit

[ÏE(E i ,E j)<Ecrit

g(E i)g(E j)dE idE j

])λ, (3.15)

with W the width of the DOS, which we introduce to make B dimensionless. For Gaussiandisorder we use W =σ. The choice of W is somewhat arbitrary, but does not affect the finalresult.

Combining Eqs. (3.7), (3.13), (3.14) and (3.15) now yields a simple expression for thetemperature and carrier concentration dependence of the zero-field charge-carrier mobility:

µ(T, c)= Beω0

N2/3t Wc

(W

kBT

)1−λexp

[EF(T, c)−Ecrit

kBT

]. (3.16)

This is the central result of this chapter. The parameters B, λ and Ecrit do not depend on Tor c, although they typically do depend on the type of lattice, hopping and energy disorder.

3.3 Application to different hopping modelsIn this section we will show how to apply the scaling expression derived above to differenthopping models, i.e. different types of lattice, hopping rate and energy disorder. We firstshow how to find the parameters in the scaling theory (A, B, λ, pbond and Ecrit), and listtheir values for several hopping models (section 3.3.1). Next, we examine the specific caseof Miller-Abrahams hopping with uncorrelated energetic disorder, leading to an expressionwhich predicts the charge-carrier mobility for different densities of states (section 3.3.2).Finally, we examine the validity of the scaling theory when applied to a system with latticedisorder (section 3.3.3).

3.3 Application to different hopping models 61

3.3.1 Determining the scaling parameters

The simple scaling expression derived in the previous section, Eq. (3.16), applies to a widerange of hopping models, but we need to find the values of the parameters involved for eachmodel. Specifically, we need to find the percolation threshold pbond, prefactor A and scalingexponent λ [see Eq. (3.7)]. From these we can also derive the critical energy Ecrit [throughEq. (3.12)] and prefactor B [through Eq. (3.15)].

2 4 6 810-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Car

rier m

obilit

y [e

0/N2/

3t

]

(a)

Uncorrelated Gaussian disorderc = 10-3

/kBT

Miller-Abrahams SC Miller-Abrahams FCC Marcus (Er ) SC Marcus (Er ) FCC

x10

2 4 6 810-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

(b)

Dipole-correlated disorderc = 10-3

/kBT

x10

Figure 3.4: (a) Dependence of µ on T for different lattices and hopping rates, with uncorrelatedGaussian disorder. Symbols: ME. Curves: scaling theory, Eq. (3.16), with values of B,λ, and Ecrit as given in table 3.1. For clarity the mobilities for Marcus hopping havebeen multiplied by 10. (b) Same for dipole-correlated disorder, with parameter values asgiven in table 3.2.

We first consider the case of uncorrelated disorder. We start by determining pbond froma percolation analysis. Specifically, we generate the 3D lattice and energy disorder, andcalculate the bond conductances using Eq. (3.10). We then find the critical path from left toright, defined as the path for which the minimum bond conductance along this path is thehighest among all paths. To find the critical path we use a modified version of Dijkstra’sshortest path algorithm13 with binary heap sorting.14 The critical bond is the bond withminimum conductance along the critical path. pbond is then simply the portion of bondswith conductance at or above the conductance of the critical bond Gcrit. Note that thisapproach also directly gives the values of Ecrit and f (Gcrit). Next, we use the 3D master-equation (ME) method (described in section 2.4) to numerically determine the temperaturedependence of the charge-carrier mobility. We then fit Eq. (3.7) to these values, with A andλ as fitting parameters. The value of B finally is calculated from Eq. (3.15). The parametervalues thus obtained for uncorrelated Gaussian disorder are listed for different types ofhopping and lattice in table 3.1, and the accuracy of the resulting mobility is shown infigure 3.4(a).

For Marcus hopping, the dependence of the parameter values on the reorganizationenergy Er requires some extra attention. In principle we should consider each value of Eras a separate hopping model, with its own values of the scaling parameters. However, we

62 A scaling theory for the charge-carrier mobility

found that A and λ depend only weakly on Er; the values of A and λ for Er→∞ given intable 3.1 can also safely be used at finite Er. The dependence of the percolation thresholdpbond on Er cannot be neglected, but pbond can be found from the percolation analysisdescribed above, not requiring ME calculations. This also leads to different values of Band Ecrit, as listed in table 3.1. For typical values of T and c, figure 3.5(a) shows that thedependence of µ on Er is well described by this approach. We note that the dependence ofω0 on Er, not included in the figure, leads to a net decrease of µ with Er [see Eq. (1.11)].

100 101 102

10-3

10-2

Car

rier m

obilit

y [e

0/N2/

3t

]

(a)

SC FCC

c = 10-3

Reorganization energy Er [ ]

Uncorrelated Gaussian disorderMarcus hopping

/kBT = 5

100 101 102

10-3

10-2

(b)

c = 10-3

Reorganization energy Er [ ]

Dipole-correlated disorderMarcus hopping

/kBT = 6

Figure 3.5: (a) Dependence of µ on Er for different lattices, with uncorrelated Gaussian disorder.Curves: scaling theory, Eq. (3.16), with values of B, λ, and Ecrit as given in table 3.1.Interpolation was used for values of Er not listed in this table. Note that the prefactorω0 depends on Er [see Eq. (1.11)], which leads to a net decrease of µ with Er. (b) Samefor dipole-correlated disorder, with parameter values as given in table 3.2.

We now consider dipole-correlated energy disorder. Figure 3.6 shows that the topologiesof the percolating networks for uncorrelated and correlated disorder are very different;both the high-current bonds and the critical bonds are much more clustered for correlateddisorder. This clustering makes it very difficult to use the percolation analysis describedabove for correlated disorder; even lattices with 100×100×100 sites are not big enough. Inorder to circumvent this problem, we fitted the parameters B, λ, and Ecrit directly to MEmobility results, using Eq. (3.16). The results for the two different lattices and hoppingtypes are listed in table 3.2. The values of pbond and A are not included in the table, sincethey are not used in this approach. We note that the different topology of the percolatingnetwork for correlated and uncorrelated disorder is reflected in the value of the criticalexponent λ, which is around two for correlated disorder and around unity for uncorrelateddisorder. The accuracy of the resulting mobility is shown in figure 3.4(b).

The reorganization energy needs to be handled slightly differently for correlated dis-order. For uncorrelated disorder, we assumed that A and λ are independent of Er. Thisapproach cannot be used for correlated disorder because we do not know the value of A.Instead, we keep λ constant and fit B and Ecrit to ME calculations, using Eq. (3.16). Theresults are listed in table 3.2. Interestingly, no dependence of Ecrit on Er is found, contraryto the case of uncorrelated disorder (compare with table 3.1). In other words, the depen-

3.3 Application to different hopping models 63

Uncorrelated Gaussian disorder

Lattice Hopping Er[σ] pbond A λ B Ecrit[σ] CSC MA N/A 0.097 2.0 0.97 0.47 −0.491 0.44SC Marcus ∞ 0.139 1.8 0.85 0.66 −0.766SC Marcus 10 0.131 1.8 0.85 0.63 −0.748 0.69SC Marcus 3 0.118 1.8 0.85 0.59 −0.709 0.49SC Marcus 1 0.104 1.8 0.85 0.51 −0.620 0.44FCC MA N/A 0.040 8.0 1.09 0.7 −0.84 0.40FCC Marcus ∞ 0.058 8.0 1.10 1.2 −1.11FCC Marcus 10 0.054 8.0 1.10 1.1 −1.09 0.66FCC Marcus 3 0.048 8.0 1.10 1.0 −1.06 0.45FCC Marcus 1 0.042 8.0 1.10 0.8 −0.98 0.40

Table 3.1: Bond percolation threshold pbond, prefactor A, critical exponent λ in Eq. (3.7), prefactorB, and critical energy Ecrit in Eq. (3.16), for uncorrelated Gaussian disorder. The lastcolumn gives the value C in an optimal fit of the low carrier-concentration mobility µ0(T),as given by Eq. (3.25), to exp(−Cσ2) in the range 2 ≤ σ≤ 6, with σ=σ/kBT. The numberof digits given in each entry is compatible with the accuracy with which the parameterscould be obtained.

Dipole-correlated disorder

Lattice Hopping Er[σ] λ B Ecrit[σ] CSC MA N/A 2.0 0.36 −1.26 0.33SC Marcus ∞ 1.7 0.43 −1.37SC Marcus 10 1.7 0.42 −1.37 0.61SC Marcus 3 1.7 0.38 −1.37 0.38SC Marcus 1 1.7 0.29 −1.37 0.32FCC MA N/A 2.2 0.78 −1.43 0.31FCC Marcus ∞ 2.2 1.1 −1.56FCC Marcus 10 2.2 1.1 −1.56 0.60FCC Marcus 3 2.2 1.0 −1.56 0.38FCC Marcus 1 2.2 0.7 −1.56 0.31

Table 3.2: λ, B, Ecrit for dipole-correlated disorder. The last column gives the value C in an optimalfit of µ0(T), as given by Eq. (3.26), to exp(−Cσ2) in the range 2≤ σ≤ 6.

64 A scaling theory for the charge-carrier mobility

Uncorrelated disorder, c = 10-3

, Σ�kBT = 3 Correlated disorder, c = 10-3

, Σ�kBT = 3

Figure 3.6: Normalized current (line opacity) in bonds of a 30×30 square lattice with uncorrelatedGaussian energetic disorder (left) and dipole-correlated energetic disorder (right). Thered circles indicate bonds with a power dissipation of at least 30% of the maximumpower dissipation. The results shown are for Marcus hopping with reorganization en-ergy Er→∞, c = 10−3, and σ/kBT = 3. A small electric field has been applied from leftto right.

dence of µ on Er occurs only via the prefactor B. This can be understood by considering theeffect of Er on the hopping rates, as given by Eq. (1.10): a large value reduces the hoppingrate when the energy difference between the sites involved is large. This energy differenceis diminished by the correlation of the energy levels, thus reducing the effect of the reor-ganization energy. The validity of these results is demonstrated in figure 3.5(b). Again, wemust keep in mind that there is an additional dependence on Er through the prefactor ω0.

3.3.2 Miller-Abrahams hopping with uncorrelated disorderIn this section we will show that for the specific case of Miller-Abrahams hopping withuncorrelated energetic disorder the scaling parameters A and λ are not just independentof T or c, but also independent of the shape of the density of states (DOS). This allowsus to derive an alternative form of the scaling expression Eq. (3.16) which can predict thecharge-carrier mobility µ and its dependence on T and c for any DOS. This is an importantresult because the charge transport is known to be highly dependent on the exact shape ofthe DOS.15

We start by concluding from Eqs. (1.9) and (3.10) that for MA hopping the energy func-tion E(E i,E j) [see Eq. (3.11)] is given by

E(E i,E j)=max(E i,E j). (3.17)

3.3 Application to different hopping models 65

This allows us to rewrite Eq. (3.12) as

pbond =Ï

max(E i ,E j)<Ecrit

g(E i)g(E j)dE idE j = Ecrit∫−∞

g(E i)dE i

2

. (3.18)

Recall that Eq. (3.12) assumes uncorrelated disorder, so all results in this section are onlyvalid for that case. The bracketed integral is simply the cumulative distribution functionof the energy distribution, which we will call Γ(E). We then have

ppbond =Γ(Ecrit). (3.19)

This means we simply have site percolation: the percolating network consists of exactlythose sites with energy below Ecrit.* In other words, the percolation problem dependsonly on the ordering of the site energies and not on their values, and so is independentof the shape of the DOS. We concluded earlier that the fact that the percolation problemis independent of T and c implies that the scaling parameters A and λ are independentof T and c. By the same token we must then conclude that for MA hopping A and λ areindependent of the shape of the DOS as well.

However, we cannot directly apply Eq. (3.16) to predict the mobility for different shapesof the DOS because the prefactor B does depend on the shape of the DOS. Instead, we willrederive this expression in terms of A by using the specific form of the energy functionEq. (3.17). Working out Eq. (3.14) gives the following expression for f (Gcrit):

f (Gcrit)= 2kBTGcrit

g(Ecrit)p

pbond. (3.20)

Combining Eqs. (3.7), (3.13) and (3.20) then gives the expression for µ for MA hopping:

µ(T, c)= Aeω0

N2/3t kBTc

[2kBT g(Ecrit)

ppbond

]λ exp[

EF(T, c)−Ecrit

kBT

]. (3.21)

When applying this expression to different shaped of the DOS we must keep in mind thatEcrit depends on the shape of the DOS. However, Ecrit can be found straightforwardly frompbond using Eq. (3.19). The parameters A, λ and pbond do not depend on T, c or the shapeof the DOS (as long as the energy disorder is uncorrelated), although they do depend onthe type of lattice. We note that if g(Ecrit) ≈ 0, i.e. if there are very few sites around Ecritavailable, Eq. (3.21) only holds at very low temperatures. This is because it is only at thesetemperatures that the few sites close to Ecrit determine the transport. Such a situationmay occur for example in host-guest systems for guest concentrations at the cross-over inbetween the low guest-concentration regime in which the guest molecules act as traps andthe high guest-concentration regime in which the transport takes place via direct hoppingbetween the guest sites.

*Since site percolation is a standard problem, this also means values of pbond can be found in general percolationliterature.16

66 A scaling theory for the charge-carrier mobility

2 4 6 8 10 12

10-13

10-11

10-9

10-7

10-5

10-3

10-1

SC latticeUncorrelated disorder

Miller-Abrahams hoppingc = 10-3

Car

rier m

obilit

y [e

0/N2/

3t

]

W/kBT

Gaussian DOS Exponential DOS Uniform DOS

Figure 3.7: Dependence of µ on T for different densities of states for MA hopping and uncorrelateddisorder. In all cases W is a measure for the width of the DOS. Symbols: ME results.Curves: scaling theory, Eq. (3.21), with A = 2.0 and λ = 0.97 for all three curves, andEcrit determined from Eq. (3.19) with pbond = 0.097.

In figure 3.7 we show the validity of Eq. (3.21) by plotting the temperature dependenceof the mobility for MA hopping with uncorrelated disorder for three different shaped of theDOS: a Gaussian, an exponential, and a constant DOS. In all three cases, W is a measurefor the width of the DOS. For a Gaussian DOS, W = σ is the standard deviation. For theexponential DOS W = kBT0 is the decay energy: g(E) = exp(E/kBT0)/kBT0 for E ≤ 0 andg(E) = 0 for E > 0. For the constant DOS W is half the size of the energy region in whichthe DOS is non-zero: g(E) = 1/2W for −W ≤ E ≤ W and g(E) = 0 otherwise. The results ofEq. (3.21) (curves) describe the ME results (symbols) very well for W /kBT & 1, using thesame values A = 2.0, λ= 0.97, and pbond = 0.097 for each shape of the DOS.

We will consider the exponential DOS in more detail, because this shape of the DOS isoften used in analyzing the mobility of organic field-effect transistors (OFETs).17 ApplyingEq. (3.16) to this case gives:

µ(T, c)= Beω0

N2/3t kBT0c

(T0

T

)1−λ (exp(Ecrit/kBT0) c

Γ(1−T/T0)Γ(1+T/T0)

)T0/T, (3.22)

where Γ(z) ≡ ∫ ∞0 dyexp(−y)yz−1 and Eq. (2) in Ref. 17 was used to express EF in terms of

c. For MA hopping on an SC lattice, we have from Eq. (3.19) Ecrit = −1.17kBT0 leadingto exp(Ecrit/kBT0) = 0.32. We compare this to the result given by Vissenberg and Matters(VM)17 for MA hopping:

µ(T, c)=σ0,VMe

Ntc

(πNt(T0/T)3c

(2α)3BcΓ(1−T/T0)Γ(1+T/T0)

)T0/T

, (3.23)

3.3 Application to different hopping models 67

where σ0,VM is a conductivity prefactor and the critical number Bc ≈ 2.8. Eq. (3.23) was de-rived within a percolation analysis of a different system from the one we consider: variable-range hopping on a randomly and uniformly distributed collection of sites, with a decaylength α−1 of the wave-functions localized at the sites.* We note that the concentrationdependence in Eqs. (3.22) and (3.23) is the same. The temperature dependence is similarin the sense that the dominant factor is Arrhenius-like, µ(T)∝ exp[−Ea/kBT], with an ac-tivation energy Ea. In our case we have Ea = Ecrit−EF (see Eq. (3.16)), whereas in the VMcase Ea = kBT0 ln

[πNt(T0/T)3/(2α)3Bc

]−EF. We note that the value of Ea in our approachconverges to a finite value for T → 0 while in the VM case it diverges to infinity. It could beworthwhile to reanalyze mobilities in OFETs with the present theory.

3.3.3 Effect of lattice disorder

0 2 4 6 810-4

10-3

10-2

10-1 Uncorrelated Gaussian disorderSC lattice

/kBT = 5, c = 10-3

Car

rier m

obilit

y [e

0/N2/

3t

]

Transfer integral disorder

Miller-Abrahams Marcus (Er )

Figure 3.8: Dependence of µ on transfer-integral-disorder strength Σ. Symbols: ME. Curves: scalingansatz, Eq. (3.7), with values of A and λ as given in table 3.1.

We consider the effect of lattice disorder as described by Eq. (1.12). It is not a prioriclear that Eq. (3.16) can be applied, but we can still determine Gcrit and f (Gcrit) from thepercolation analysis described in section 3.3.1 and apply the basic scaling ansatz Eq. (3.7),assuming no dependence of A and λ on the lattice disorder strength Σ. The results of thisapproach are compared with ME results for typical values of T and c in figure 3.8; we seethat the scaling theory still provides an excellent description of the mobility, even for largedisorder Σ= 6. We also note that for Σ. 3 the mobility is almost independent of Σ, so thatEq. (3.16), valid for Σ= 0, can still be applied in this case.

*A brief comparison of this random-position approach with our lattice approach can be found in section 1.4.1.

68 A scaling theory for the charge-carrier mobility

3.4 Consequences for charge transportIn this section we will analyse the consequences of Eq. (3.16) for the concentration andtemperature dependence of the mobility, and compare these results with the available lit-erature.

3.4.1 Concentration dependence of the charge-carrier mobilityAn important conclusion drawn from Eq. (3.16) is that the dependence of the charge-carriermobility µ on the concentration c is in all cases given by

µ∝ exp(EF(T, c)/kBT) /c, (3.24)

containing no parameters depending on the type of hopping or lattice. For MA hoppingthis dependence was already found in Ref. 9. We now conclude that it also holds for Marcushopping, at variance with a previous claim.5 We note that our conclusion agrees with thenumerically exact mobilities, as shown in figure 3.9. When the carrier concentration is toohigh, the assumption of low Fermi energy used in deriving Eq. (3.10) no longer holds, andso the above dependence also fails. The requirement for uncorrelated Gaussian disorder isc . 0.03, and for dipole-correlated disorder c . 0.01. The higher threshold for uncorrelateddisorder is caused by the higher value of Ecrit.

10-5 10-4 10-3 10-2 10-1

10-5

10-4

10-3

10-2

10-1

100

101

102

Miller-Abrahams SC Miller-Abrahams FCC Marcus (Er ) SC Marcus (Er ) FCC

(a) /kBT = 5Uncorrelated Gaussian disorder

Car

rier m

obilit

y [e

0/N2/

3t

]

Carrier concentration c

x10

10-6 10-5 10-4 10-3 10-2 10-1

10-5

10-4

10-3

10-2

10-1

100

101

102

Assuming Gaussian DOS Using correct DOS, Eq. (3.29)

(b)Dipole-correlated disorder

/kBT = 6

Carrier concentration c

Figure 3.9: (a) Dependence of µ on carrier concentration c for different lattices and hopping rates,with uncorrelated Gaussian disorder. Symbols: ME. Curves: scaling theory, Eq. (3.16),with values of B, λ, and Ecrit as given in table 3.1. For clarity the mobilities forMarcus hopping have been multiplied by 10. (b) Same for dipole-correlated disorder,with parameter values as given in table 3.2. The dashed curve indicates the result ofEq. (3.16) assuming a perfectly Gaussian DOS, while the solid curve uses the correctedform Eq. (3.29).

When applying Eq. (3.24) to the case of dipole-correlated disorder, we must keep in mindthat the DOS is not perfectly Gaussian (see section 3.A). Instead, we describe the DOSusing Eq. (3.29), which is itself an approximation of the exact DOS Eq. (3.27). This leads

3.4 Consequences for charge transport 69

to a slightly weaker concentration dependence than for uncorrelated Gaussian disorder[see figure 3.9(b)], consistent with the extended correlated disorder model (ECDM) resultsfound by Bouhassoune et al.18

3.4.2 Temperature dependence of the charge-carrier mobilityThe temperature dependence of the charge-carrier mobility is typically analyzed in thelimit of low carrier concentration c → 0, i.e. for a single non-interacting carrier. For un-correlated Gaussian disorder the mobility in this limit, µ0(T), is given by [starting fromEq. (3.16)]:

µ0(T) = Beω0

N2/3t σ

σ1−λ exp[−Ecrit/kBT] limc→0

exp(EF(T, c)/kBT)c

= Beω0

N2/3t σ

σ1−λ exp[−σ2/2−Ecrit/kBT

]. (3.25)

This expression does not apply to the dipole-correlated case because the DOS is not exactlyGaussian for that case. Instead, we make use of Eq. (3.31) to obtain for correlated disorder:

µ0(T)≈ Beω0

N2/3t σ

σ1−λ exp[−0.56σ1.9 −Ecrit/kBT

]. (3.26)

This is an approximate expression because Eq. (3.31) is an approximation of the exact limitEq. (3.30). Keep in mind that in the case of Marcus hopping ω0 depends on T via Eq. (1.11),leading to an additional temperature dependence that is not explicitly shown in Eqs. (3.25)and (3.26).

In Ref. 9 the expression µ0(T) ∝ Tγ exp(−bσ2 − aσ) (σ ≡ σ/kBT) was derived witha = 0.566, γ = −1 and b = 1/2 for nearest-neighbor MA hopping with an SC lattice anduncorrelated Gaussian disorder. Our expression for µ0(T) is of the same form, also withb = 1/2. However, the values of a and γ differ: for MA hopping we have a = Ecrit/σ andγ=λ−1, and for Marcus hopping, accounting for the T dependence of ω0, a = (Ecrit+Er/4)/σand γ= λ−3/2. Note that the sign of a found by us for MA hopping (see Ecrit in table 3.1)is opposite to that in Ref. 9, leading to a significantly different T dependence.

The temperature dependence of the mobility is often expressed as µ0(T) ∝ exp(−Cσ2).We find that this provides a quite accurate description of Eqs. (3.25) and (3.26) when consid-ering a limited temperature range 2≤ σ≤ 6. To facilitate the comparison with earlier work,we have included the value of C in such a fit in tables 3.1 and 3.2, taking into account thedependence of ω0 on T for the case of Marcus hopping. For correlated disorder the muchlower value of Ecrit leads to a significantly weaker temperature dependence, i.e. a lowervalue of C. This is consistent with the ECDM results.18. For the case of an SC lattice withuncorrelated Gaussian disorder and MA hopping, the obtained value of C (0.44) is similarto the best-fit value C = 4/9 found from an MC simulation of this system by Bässler.19 Thisresult is often interpreted as if the temperature dependence of the mobility is determinedby the rate of hops from the average carrier energy −σ2/kBT to a ‘transport level’ with anenergy around −(5/9)σ2/kBT. We note that the origin of the similar factor exp(−(1/2)σ2)

70 A scaling theory for the charge-carrier mobility

in Eq. (3.25) is very different: it originates from the limit taken in deriving this equationand results purely from the physics of carriers obeying Boltzmann statistics in a GaussianDOS, not from the transport properties.

3.5 Conclusions and discussionWe have developed a very general scaling theory for percolative charge transport in dis-ordered molecular semiconductors, describing the dependence of the charge-carrier mo-bility on temperature and carrier concentration. The theory is valid in a scaling regimecorresponding to a wide temperature range that includes most relevant cases. We havedemonstrated how it can be applied to uncorrelated and dipole-correlated energetic disor-der, Miller-Abrahams and Marcus hopping, and simple-cubic and face-centered-cubic lat-tices. The mobility can be obtained for all these cases by applying Eq. (3.16) and simplylooking up the appropriate values of the parameters λ, Ecrit, and B in table 3.1 or 3.2. In thecase of dipole-correlated disorder one should keep in mind that the density of states (DOS)is not perfectly Gaussian, which affects the dependence of the Fermi energy EF on thecarrier concentration, and thereby the carrier-concentration dependence of the mobility. Aparameterization of the exact DOS for this case is given by Eq. (3.29). For uncorrelated dis-order and Miller-Abrahams hopping it is even possible to obtain the mobility for any shapeof the DOS using Eq. (3.21), with values of A, λ and pbond as found in table 3.1. Ecrit canbe easily calculated from Eq. (3.19). No such straightforward formula is known for Marcushopping in a non-Gaussian DOS; Eq. (3.16) applies to this case, but the parameters λ, Ecrit,and B have to be determined specifically for each shape of the DOS.

To determine which model correctly describes the underlying physics, it is importantto develop methods to use experimental data to distinguish Miller-Abrahams and Marcushopping, and between correlated and uncorrelated disorder. The concentration dependenceis nearly identical for all cases and so cannot be used for this purpose. The temperaturedependence is different for every case and so can in principle be used to distinguish models.However, typically experiments take place in the range 2≤σ/kBT ≤ 6, in which for all casesthe temperature dependence is well described by µ0(T) ∝ exp[−C(σ/kBT)2] for some C.The value of C alone is not enough to tell apart all cases (see tables 3.1 and 3.2). We expectthat the field dependence of the mobility, which has been determined for Miller-Abrahamshopping18,20 but has yet to be determined for Marcus hopping, may make it possible tofully distinguish the different models.

Another possibility to distinguish models is to consider the time or frequency dependentac mobility as probed in transient experiments (see also section 2.6). In such experimentscharge-carrier relaxation effects give rise to a time-dependent current density, which re-flects the finite time needed for the carriers to relax to the dynamic thermal equilibriumdistribution. Recently, Germs et al. found that the energy Ecrit as obtained from the presentscaling theory for the case of Miller-Abrahams hopping on a simple-cubic lattice is close tothe effective ‘conduction’ energy level that may be used in a multiple-trapping model de-scribing charge-carrier relaxation in a hole-transporting polyfluorene-based polymer. Thedifferential capacitance of the sandwich-type metal/polymer/metal devices was found to be

3.A Density of states for dipole-correlated disorder 71

very sensitive to the value of Ecrit. This suggests that it will be possible to validate the pre-dicted values of Ecrit as obtained in this paper from analyses of ac current density in, e.g.,admittance, dark-injection, or time-of-flight experiments. The finding that Ecrit is muchlower for systems with correlated disorder than for systems with uncorrelated Gaussiandisorder suggests that the ac response of systems with correlated disorder is less stronglyaffected by relaxation effects. Quantitative analyses of the ac response are thus expected toprovide a method for probing the presence of spatial correlation between the energy levelsin disordered organic semiconductors.

3.A Density of states for dipole-correlated disorderWe consider the density of states g(E) for a lattice site in a dipole-induced correlated energylandscape, as described in section 1.4.1. For convenience we put this site at the origin. InRef. 21, Novikov and Vannikov derived that

g(E) = 12π

∫ ∞

−∞dyexp[i yE+S(y)],

S(y) ≡ ∑j

ln[sin(yz j)

yz j

], z j ≡ ed

ϵR2j, (3.27)

where i =p−1 and the summation runs over all other lattice sites j. Note that this yieldsdifferent densities of states for simple-cubic (SC) and face-centered-cubic (FCC) lattices,though the differences are minor.

Since the main contribution to the integral is from small values of y, it is useful toreplace S(y) by its Taylor expansion around y= 0:21

S(y)=− A6

(edyN2/3

t

ϵ

)2

+O(y4), A ≡∑j

(R j N1/3t )−4. (3.28)

For the SC lattice we have A ≈ 16.532 and for the FCC lattice A ≈ 15.962. Using only thisterm in Eq. (3.27) and setting d =

(p3/A

)σϵ/eN2/3

t yields a Gaussian DOS with standarddeviation σ. It is possible to refine this approximation by continuing the expansion andparameterizing the resulting DOS, yielding

g(E)≈ 0.383σ

exp(−0.44|E/σ|2.11)

, (3.29)

which is accurate to within 10% for −4σ≤ E ≤ 4σ for both SC and FCC lattices.Finally, we determine the limit necessary to describe the carrier mobility at low concen-

tration c → 0 in Eq. (3.26):

limc→0

exp(EF(T, c)/kBT)c

=[∫ ∞

−∞dEg(E)exp(−E/kBT)

]−1= exp(−S(i/kBT)), (3.30)

where g(E) is given by Eq. (3.27) and the resulting double integral is solved using contourintegration. Using only the first term in the Taylor expansion as given by Eq. (3.28) yields

72 A scaling theory for the charge-carrier mobility

the usual exp(−σ2/2) dependence for the Gaussian DOS. Unfortunately, continuing thisexpansion is not fruitful because it diverges for large values of y. For this reason we chooseto parameterize the result of evaluating S explicitly using Eq. (3.27), leading to:

limc→0

exp(EF(T, c)/kBT)c

≈ exp(−0.56σ1.9)

, (3.31)

which is accurate to within 20% for σ≤ 7 for both SC and FCC lattices.

References1. J. Cottaar, L. J. A. Koster, R. Coehoorn, and P. A. Bobbert, Phys. Rev. Lett. 107,

136601 (2011).2. J. Cottaar, R. Coehoorn, and P. A. Bobbert, Phys. Rev. B 85, 245205 (2012).3. S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973).4. B. Movaghar and W. Schirmacher, J. Phys. C 14, 859 (1981).5. I. I. Fishchuk, V. I. Arkhipov, A. Kadashchuk, P. Heremans, and H. Bässler, Phys.

Rev. B 76, 045210 (2007).6. V. Ambegaokar, B. I. Halperin, and J. S. Langer, Phys. Rev. B 4, 2612 (1971).7. B. I. Shklovskii and A. L. Efros, Electronic properties of doped semiconductors

(Springer-Verlag, Berlin, 1984).8. S. D. Baranovskii, O. Rubel, and P. Thomas, Thin Solid Films 487, 2 (2005).9. R. Coehoorn, W. F. Pasveer, P. A. Bobbert, and M. A. J. Michels, Phys. Rev. B 72,

155206 (2005).10. J. C. Dyre and T. B. Schrøder, Rev. Mod. Phys. 72, 873 (2000).11. J. Zhou, Y. C. Zhou, J. M. Zhao, C. Q. Wu, X. M. Ding, and X. Y. Hou, Phys. Rev. B

75, 153201 (2007).12. J. J. M. van der Holst, F. W. A. van Oost, R. Coehoorn, and P. A. Bobbert, Phys. Rev.

B 83, 085206 (2011).13. E. W. Dijkstra, Numerische Mathematik 1, 269 (1959).14. J. Williams, Comm. Assoc. Comput. Mach. 7, 347 (1964).15. W. S. C. Roelofs, S. G. J. Mathijssen, R. A. J. Janssen, D. M. de Leeuw, and M.

Kemerink, Phys. Rev. B 85, 085202 (2012).16. C. D. Lorenz and R. M. Ziff, J. Phys. A 31, 8147 (1998).17. M. C. J. M. Vissenberg and M. Matters, Phys. Rev. B 57, 12964 (1998).18. M. Bouhassoune, S. L. M. van Mensfoort, P. A. Bobbert, and R. Coehoorn, Org. Elec.

10, 437 (2009).19. H. Bässler, Phys. Stat. Sol. B 175, 15 (1993).20. W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M. Blom, D. M.

de Leeuw, and M. A. J. Michels, Phys. Rev. Lett. 94, 206601 (2005).21. S. V. Novikov and A. V. Vannikov, Sov. Phys. JETP 106, 877 (1994).

References 73

Chapter 4

Field-induced detrapping inhost-guest systems

Abstract

In a disordered organic semiconducting host-guest material, containing a relatively smallconcentration of guest molecules acting as traps, the charge transport may be viewed asresulting from carriers that are detrapped from the guest to the host. Commonly usedtheories include only detrapping due to thermal excitation, described by the Fermi-Dirac(FD) distribution function. In this chapter, we develop a theory describing the effect offield-induced detrapping (FID), which provides an additional contribution at finite electricfields. It is found from three-dimensional simulations that the FID effect can be describedby a field-dependent generalized FD distribution that depends only on the shape of thehost density of states (DOS) and not on the guest DOS. For the specific case of a Gaussianhost DOS, we give an accurate and easy-to-use analytical expression for this distribution.The application of our theory is demonstrated for sandwich-type devices under conditionstypical of organic light-emitting diodes.

75

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.1.1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1.2 Chapter structure . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Generalized Hoesterey-Letson detrapping model . . . . . . . . . . 784.2.1 Thermal detrapping . . . . . . . . . . . . . . . . . . . . . . . 784.2.2 Field-induced detrapping . . . . . . . . . . . . . . . . . . . . 81

4.3 Field-dependent occupation function . . . . . . . . . . . . . . . . . . 824.3.1 Computing the occupation function from 3D simulation . 824.3.2 Parameterizing the occupation function . . . . . . . . . . . 83

4.4 Device applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . 89References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

This chapter was adapted with permission from Ref. 1:

J. Cottaar, R. Coehoorn, and P. A. Bobbert, Field-induced detrapping in disordered organic semiconductinghost-guest systems, Phys. Rev. B 82, 205203 (2010).Copyright (2010) by the American Physical Society.

76 Field-induced detrapping in host-guest systems

4.1 Introduction

Host-guest systems play an important role within the field of organic electronics and areencountered in several applications. In organic light-emitting diodes (OLEDs), the emis-sive layers are often doped, and both phosphorescent2 and fluorescent3 guest emitterscan behave as traps of charge carriers. Also, doped organic layers can assist in chargeinjection.4 But also nominally undoped systems can sometimes act as host-guest systems.In particular, many organic semiconductors contain electron traps leading to vastly re-duced electron mobilities as compared to hole mobilities.5–9 Recently, Nicolai et al. foundthat the energy level of these electron traps is about 3.6 eV below the vacuum level in awide range of materials, and proposed hydrated oxygen complexes as the origin of this uni-versal trap level.10 The study of host-guest systems is also relevant for organic photovoltaiccells,11 organic field-effect transistors,12 xerography,13 and organic lasers.14

Charge transport in host-guest systems has been studied since the nineteen-sixties.15

The first semi-analytical result was given by Hoesterey and Letson (HL),16 based on a sys-tem with states at only two energy levels: a transport state (host) and a trap state (guest).Within their approach, the population of host and guest states is obtained assuming localthermal equilibrium, i.e. using the Fermi-Dirac (FD) distribution function, and the chargetransport is determined by the fraction of carriers, the ‘free’ carriers, that occupy the hostdensity of states (DOS). The HL model shows that at extremely low guest concentrationsthe transport is dominated by hopping in between host sites, so that the presence of theguest can be ignored, and that above a certain minimum guest concentration hopping be-tween host and guest sites becomes important, leading to trapping of charge carriers bythe guest and a dramatic reduction of the charge-carrier mobility.17 The model loses itsvalidity when at high guest concentrations direct hopping in between guest sites starts tooccur, leading to an increase of the mobility until eventually the charge transport is fullydominated by guest-to-guest hopping. This concept of thermal detrapping may be readilygeneralized to more complex shapes of the host and guest DOS.18,19 The dependence of theeffective mobility on the charge-carrier density is then not only related to the fraction ofdetrapped charge carriers, but also to the density dependence of the mobility in the hostDOS.20–22 However, the model is only valid in the limit of zero electric field. At finite valuesof the electric field, the fraction of ‘free’ charge carriers exceeds the fraction that is obtainedfrom the FD distribution function. This ‘field-induced detrapping’ (FID) effect gives rise toan additional contribution to the mobility. So far, this effect has only been studied for spe-cific host-guest systems using various semi-analytical approximations,23–30 and a generalmodel for the effect that may be readily used in drift-diffusion device simulations of organicelectronic devices is lacking.

In this chapter, we will develop an accurate and easy-to-use model for the mobility inhost-guest systems with a Gaussian host DOS and a general guest DOS, including theeffect of FID. The model is valid in the regime of low guest concentrations, where guest-to-guest hopping can be neglected. The approach is based on the results of three-dimensional(3D) charge-transport simulation using the master-equation (ME) approach. In an ear-lier study,31 it was already found that the effect of FID on the mobility as obtained fromME simulation is significantly different from the effect as predicted from a semi-analytical

4.1 Introduction 77

effective-medium theory. Using the ME results, we show that the HL model can be ex-tended to finite values of the electric field by using a generalized FD distribution function,which we call the occupation function, that depends only on the shape of the host DOS andthe electric field, and not on the guest DOS. Although one might loosely say that the fieldgives rise to a ‘hot’ out-of-equilibrium distribution, we show that the shape of the occu-pation function is not well enough represented by introducing an effective field-enhancedtemperature. In order to facilitate application of our work in device simulations, we presenta parameterization of the occupation function for a wide range of widths of the GaussianDOS. We furthermore demonstrate the role of FID in sandwich-type metal / organic semi-conductor / metal devices.

4.1.1 SystemWe will use in this chapter the hopping model described in section 1.4.1, with Miller-Abrahams hopping, a simple-cubic lattice and uncorrelated Gaussian energetic disorderin the host. For pure host transport, the resulting model, with a dependence of the mobil-ity on temperature, electric field and carrier concentration as parameterized in Ref. 21, isknown as the extended Gaussian disorder model (EGDM). Our results apply only to thisspecific model, but we will also discuss how the methods we develop could be applied todifferent hopping models. All numerical results in this chapter are obtained using the 3Dmaster-equation (ME) method described in section 2.4 and Ref. 21.

The DOS of the host-guest system ghg is given by

ghg(E)= (1− x)gh(E)+ xgg(E), (4.1)

where gh is the Gaussian DOS of the host [Eq. (1.6)], gg is the DOS of the guest and x isthe guest concentration. We assume that the prefactors ω0 [see Eq. (1.9)] for host-to-host,host-to-guest and guest-to-guest hops are equal; we will show later that this choice doesnot affect the mobility. This means that the only difference between a pure host systemand the host-guest systems we consider is the shape of the DOS.

We will use a specific host-guest system as a framework for the development of ourmethod for determining the mobility. We will later show how to extend the obtained resultsto a general situation. In this specific system the energy levels of all guest sites are locatedat −5σ, i.e.

gg(E)= 1σδ(

E+5σσ

), (4.2)

where δ is the Dirac delta function. The DOS ghg for this system is shown in figure 4.1.

4.1.2 Chapter structureWe first develop our generalized Hoestery-Letson model for the mobility in host-guest sys-tems (section 4.2). This leads to the concept of the generalized Fermi-Dirac distribution,described by the field-dependent occupation function. We show how to numerically com-pute and parameterize this occupation function, with a form based on the observed spatial

78 Field-induced detrapping in host-guest systems

-6 -4 -2 0 2 40.0

0.1

0.2

0.3

0.4

0.5

Guest

Den

sity

of s

tate

s g(

E) [

-1]

Energy E [ ]

Host

Figure 4.1: Density of states for the example host-guest system with guest concentration x = 0.01.The host site energies follow a Gaussian energy distribution with width σ, while theguest sites all have a site energy −5σ.

structure of the field-dependent occupation probabilities (section 4.3). Next, we apply themodel to a single-carrier device, showing that field-induced detrapping can significantly al-ter current-voltage characteristics (section 4.4). We conclude with a summary, conclusionsand an outlook (section 4.5).

4.2 Generalized Hoesterey-Letson detrapping modelIn this section, we will first discuss the standard HL model, which considers only ther-mal detrapping at zero electric field, and then generalize it to also include field-induceddetrapping.

4.2.1 Thermal detrappingWithin the standard HL model, applicable in the zero-field limit, we assume that for a givenvalue of the carrier concentration in the host ch the guests do not influence the chargetransport in the host. The guests then only influence the mobility by trapping carriers,which causes ch to be only a fraction of the total carrier concentration c. The mobility as afunction of c is then given by

µ(c)= ch

cµh(ch), (4.3)

where µh(ch) is the mobility in the pure host material. ch may be found by solving thefollowing set of equations:

c = xcg + (1− x)ch, (4.4)

4.2 Generalized Hoesterey-Letson detrapping model 79

where cg is the carrier concentration in the guest, and

cg =∫ ∞

−∞gg(E)p(E;EF(ch,T))dE, (4.5)

with p(E;EF(ch,T)) the Fermi-Dirac (FD) distribution function

p(E;EF(ch,T))= 1exp[(E−EF(ch,T))/kBT]+1

, (4.6)

which gives the probability of a charge carrier being present at a site with energy E ata specific host-carrier-concentration- and temperature- dependent Fermi energy EF(ch,T).Since the carrier concentration in the host is now determined purely by thermal equilib-rium, we refer to this as thermal detrapping.

Master equation Hoestery-Letson model

10-4 10-3 10-2 10-110-15

10-14

10-13

10-12

10-11

10-10

10-9 /kBT = 3F = 0

c = 0.001

Car

rier m

obili

ty

[e0/N

2/3

t]

Guest concentration x

c = 0.01

Figure 4.2: Dependence of the charge-carrier mobility µ on the guest concentration x for the examplehost-guest system. The symbols represent the results of master-equation (ME) calcula-tions on the host-guest system, while the curves represent the results of the Hoesterey-Letson (HL) model Eq. (4.3), with the host carrier concentration ch obtained by solvingEqs. (4.4) and (4.5) with the Fermi-Dirac (FD) distribution function Eq. (4.6).

We first consider the dependence of the mobility on guest concentration x, picturedin figure 4.2. The results clearly show the cross-over from a low-concentration regime inwhich the HL model is well applicable to the high-concentration regime in which this modelfails due to the contribution of the guest sites to the charge transport. The cross-over con-centration is approximately 5% in the example considered. In general, this concentrationwould depend on the specific values of the attempt frequencies for guest-guest, guest-hostand host-host hopping (taken all equal in this example). It may be possible to describe thisregime using the scaling theory developed in chapter 3; however, the low value of g(Ecrit)limits the effectiveness of this approach [see the discussion under Eq. (3.21)].

80 Field-induced detrapping in host-guest systems

0 2 410-3 10-2 10-110-13

10-12

10-11

10-10

10-5

10-4

10-3

10-2

10-1

Electric field F [ N1/3t /e]

HL model+ EGDM

Exact density +EGDM

Exact (ME) density

/kBT = 3F = 0

(d)

(c)

(b)

(a)

Thermal and field-induceddetrapping + EGDM

Thermal detrapping+ EGDM

Hos

t car

rier c

once

ntra

tion

c hC

arrie

r mob

ility

[e0/N

2/3

t]

Total carrier concentration c

/kBT = 3c = 0.01

HL model

Pure host system

HL model with thermal andfield-induced detrapping

HL model with thermal detrapping only

Exact (ME) mobility

Pure host system + EGDM

Figure 4.3: Dependence of the carrier concentration in the host ch [(a) and (c)] and mobility µ [(b)and (d)] on the total carrier concentration c at F = 0 [(a) and (b)] and on the electric fieldF at c = 0.01 [(c) and (d)], for the example host-guest system with x = 0.01. In (a) thecarrier concentration in the host is shown without taking the guest into account (dottedcurve) and by applying the HL model Eq. (4.3) (dash-dotted curve), obtained by solvingEqs. (4.4) and (4.5) with the FD distribution function Eq. (4.6). The triangles repre-sent the actual carrier concentration in the host as found from ME calculations for thehost-guest system. In (c) the field dependence of the carrier concentration in the host isshown. Two HL model results are shown here, for thermal detrapping only (dash-dottedcurve, based on the FD function Eq. (4.6)) and for thermal and field-induced detrapping(solid curve, based on Eq. (4.9)). In (b) and (d) the mobilities resulting from these ‘free’carrier concentrations in the host by applying the extended Gaussian disorder model(EGDM) are shown. Also included here are the mobilities found from the ME calcula-tions for the host-guest system (squares).

4.2 Generalized Hoesterey-Letson detrapping model 81

We now consider the carrier-concentration and electric-field dependence of the mobil-ity. Figure 4.3 shows the host carrier concentration ch and mobility µ for the case of afixed guest molecule concentration x = 0.01. Figures 4.3(a) and 4.3(b), which give ch andµ as a function of the total carrier concentration c at zero field, respectively, show that thestandard HL model provides an excellent prediction of the carrier-concentration depen-dence of the mobility. We note that already in the pure host system the mobility is slightlycarrier-concentration dependent and that we have used the EGDM to describe that effect.In contrast, figures 4.3(c) and 4.3(d), which give ch and µ as a function of the electric fieldF at a fixed carrier concentration (c = 0.01), respectively, show that the standard HL ther-mal detrapping model fails to provide an accurate description.* However, it may also beseen that an excellent prediction of µ [squares in figure 4.3(d)] is obtained if µ is calculatedusing the enhanced ch that has been calculated using the ME approach [triangles in fig-ure 4.3(c)]. This proves that the general HL picture, within which the mobility is viewedas resulting from the fraction of carriers residing in host states, is still valid, but thatthe field-enhanced host carrier concentration should be used instead of the concentrationobtained assuming thermal detrapping only.

4.2.2 Field-induced detrapping

For practical device-simulation applications, within which the shape of the guest DOS isoften not a priori clear, the ME approach used will in general be computationally too ex-pensive and involved. Repeating it for every host-guest system one may encounter is un-desirable. We argue that, as an alternative, the field-dependent host carrier concentrationmay be obtained by using in Eq. (4.5) a generalized field-dependent occupation functionp(E;F, ch,T) instead of the FD distribution function. p(E;F, ch,T) gives the average prob-ability that a site at energy E is occupied at given values of F, ch and T. It may be un-derstood as follows that such a function, which should apply to both host and guest sites,indeed exists. Firstly, while the energies of host and guest sites are drawn from a differentDOS, a guest site at a certain energy cannot be distinguished from a host site that happensto have the same energy. This means that one single occupation function describes the av-erage occupational probabilities of both host and and guest sites. Secondly, this occupationfunction does not depend on the shape of the guest DOS and on the guest concentration.This follows from the assumption that for a given value of ch the presence of guest sitesdoes not influence the charge-transport properties of the host, which we showed is valid atthe low guest concentrations that we consider in the present work.

Before finding this occupation function, we will argue that a difference in the hoppingattempt frequency ω0 for host-host and host-guest hops does not influence it. Considerthe probability pi that a guest site i is occupied. Within the ME approach described byEq. (2.2), the relationship between the pi and the occupational probabilities p j of the sur-

*We remind the reader that we define the site energy levels E i to be independent of the electric-field strength;instead, this field strength is invoked when considering the energy difference in the hopping rate [see also thediscussion below Eq. (1.8)].

82 Field-induced detrapping in host-guest systems

rounding sites is given by32

pi =∑

j ω ji p j∑j[ωi j(1− p j)+ω ji p j]

. (4.7)

Due to detailed-balance considerations, the attempt frequencies for host-guest and guest-host hops are equal. If site i is a guest site surrounded by host sites, a change in thehost-guest attempt frequency therefore changes ωi j and ω ji by the same factor for all j.Additionally, the p j are not affected by this change because of the assumption that guestsites do not influence the charge transport in the host. As a result, the numerator anddenominator in Eq. (4.7) are changed by the same factor and pi is unaffected. In conclusion,the occupation function p(E;F, ch,T) is not affected by a difference in hopping attemptfrequency between host-host and host-guest hops.

4.3 Field-dependent occupation functionIn this section, we will explicitly determine the occupation function p(E;F, ch,T), i.e. theaverage probability that a site at energy E is occupied by a charge carrier. We will firstdescribe how to compute it numerically, and then parameterize the results. To develop thisparameterization, we will also have a closer look at the physics determining the distribu-tion of charge carriers.

4.3.1 Computing the occupation function from 3D simulationTo determine the occupation function p(E;F, ch,T) numerically, we simply carry out an MEcalculation for a pure host system at the specified values of F, ch and T. We then take theaverage of pi for sites i with energy close to E. An issue with this approach is that wetypically need to have a value for p at very low values of E which are unlikely to occur forany reasonable lattice size (we typically use 106 sites), since a guest DOS may well includethese very low energies. A method to find these values of p was discussed in the paper onwhich this chapter was based (Ref. 1), but this method was later found to apply only to thespecific case of Miller-Abrahams hopping with uncorrelated disorder. We will describe herean alternative method which is generally applicable.

To compute p at values of E which do not naturally occur in the lattice, we simply re-place a small number of site energies by the desired energy E, thus artificially creatingsome guest sites in what is otherwise a pure host system. This does not affect the oc-cupation function because of the central assumption that a few guest sites do not affectthe charge-transport properties of the host. We can therefore find p(E;F, ch,T) from theaverage occupation probability of these artificial guest sites. This calculation must be per-formed for each required value of E, typically for several lattice realizations per value toget sufficient statistics.

As an example, the gray solid curve in figure 4.4 shows the energy dependence of theoccupation function p(E;F, ch,T) for typical values of F, ch and T. The reduction of theoccupation function at low energies, as compared to the FD distribution at zero field (black

4.3 Field-dependent occupation function 83

-8 -7 -6 -5 -4 -3 -2 -1 00.0

0.2

0.4

0.6

0.8

1.0

-8 -6 -4 -2 0 2 410-9

10-7

10-5

10-3

10-1

Fe/N1/3t = 2

Occ

upat

ion

func

tion

p(E

;F,c

,T)

Site energy E [ ]

Fermi-Dirac Master equation Parametrization Effective

temperature

F = 0

/kBT = 3

c = 3x10-5

Figure 4.4: Dependence of the occupation function p on the site energy E, i.e. the average probabilitythat a site with energy E is occupied. The gray solid curve shows the numerical MEresult. The dashed curve shows the parameterization given by Eqs. (4.9)-(4.13). Thedotted curve shows the best fit, under the condition of conservation of the number ofcarriers, to an FD distribution with an effective temperature. The black solid curveshows the FD distribution, which is equal to the occupation function at zero field. Inset:same, in a linear-log plot.

solid curve), is clearly visible. On the other hand, the inset shows that the occupationfunction is larger than the FD distribution in the energy region around E = 0, where themajority of the host sites is located. This accommodates for the conservation of the totalnumber of charges. Values were obtained using the method from Ref. 1; however, themethod discussed above gives similar results for this case.

4.3.2 Parameterizing the occupation functionIn order to develop an accurate parameterization of the occupation function, we have firstinvestigated whether we could employ the concept of an effective temperature.33–35 Indeed,we observe from figure 4.4 that a finite electric field leads to an widening of the energyrange in which the occupation function changes from unity to zero. However, we found thatmodeling of the occupation function by an FD distribution with an effective temperature(dotted curve in figure 4.4) is not accurate enough for our purposes.

A more accurate approach was found after studying the spatial structure of the occu-pational probabilities pi at finite electric fields. We can relate these probabilities to thesite-resolved electrochemical potential energy µi, first introduced in section 2.4, by

pi = 1exp[(E i − µi)/kBT]+1

. (4.8)

84 Field-induced detrapping in host-guest systems

At F = 0, µi is equal for all sites, and is given by the Fermi energy EF. When an elec-tric field is applied, µi is no longer equal for all sites. In order to visualize the result-ing electrochemical potential distribution, we have performed ME calculations for a sys-tem with 30×200×200 lattice sites, with a monomodal Gaussian DOS with a very largedisorder parameter in order to emphasize the effects (σ/kBT = 20), for a large carrierconcentration (c = 0.1), and for a relatively small dimensionless electric-field parameter,F ≡ Fe/N1/3

t σ= 0.01. In the upper part of figure 4.5, the calculated electrochemical poten-tial landscape is shown in a plane parallel to the electric field. It is clearly visible that inmost regions the potential gradually rises in the direction of the field. In between these‘ramps’, we find abrupt ‘cliffs’ where the potential suddenly drops. This behavior is evenmore clearly visible in the lower part of the figure, which shows the electrochemical poten-tial along the one-dimensional path in the direction of the field indicated in the upper partof the figure by an arrow. The observed spatial variation of the electrochemical potentialis typical for the percolative character of the charge transport in disordered systems of thetype studied here.

The electrochemical potential landscape depicted by figure 4.5 suggests that the occu-pation function may be viewed as a result of a distribution of electrochemical potentialvalues, related to the distribution of the sizes of the gradually rising ramps. Very largeand very small values of the electrochemical potential, as compared to the average, wouldthen correspond to those very rare large ramps which result from a major local ‘obstacle’ inthe energy landscape. We now make the approximation that the electrochemical potentialof individual sites follows a statistical distribution f (µ;F, ch,T) that is independent of theenergy of those sites,* so that the occupation function may be expressed as

p(E;F, ch,T)=∫ ∞

−∞1

exp[(E− µ)/kBT]+1f (µ;F, ch,T)dµ. (4.9)

We find that this approach indeed provides an excellent description of the occupation func-tion if a Gaussian distribution of the electrochemical potential is taken:

f (µ;F, ch,T)= 1

θ(F)p

2πexp

(−1

2

[µ−EF(ch,T)−∆(F,T)

θ(F)

]2), (4.10)

with a field-dependent width

θ(F)= [1−exp(−0.67|F|)]σ, (4.11)

and with a shift ∆(F,T) of the center away from the Fermi energy, which is parameterizedby:

∆(F,T) ={ −στ(T)F2 for |F| < 0.15/τ(T),

σ(∆0(F,T)−∆0(0.15/τ(T),T)−0.0225/τ(T)) otherwise, (4.12)

*This is an approximation, since the ‘cliffs’ in the landscape of the electrochemical potential should be correlatedwith sites having energies far from the Fermi energy, leading to ‘difficult hops’. Apparently, this correlation isnot very important.

4.3 Field-dependent occupation function 85

0 50 100 150 200-2x10-4

2x10-4

50

100

150

200

y [a

] [

]

x [a]

Figure 4.5: Electrochemical potential landscape for σ/kBT = 20 and c = 0.1 in a pure Gaussian hostsystem. The upper part of the figure shows the electrochemical potential µ in a planeparallel to the electric field. A two-dimensional slice of the three-dimensional lattice isshown. An electric field F = 0.01σ/ea is applied from left to right. Light areas correspondto a potential above and dark areas to a potential below the Fermi energy. The lowerpart of the figure shows the electrochemical potential relative to the Fermi level alongthe path indicated by the arrow. The resulting Gaussian distribution of µ is sketched inthe bottom-right.

86 Field-induced detrapping in host-guest systems

with

∆0(F,T) = 0.35χ(T)−0.65|F|− log{2cosh[1.05(|F|−χ(T))]}/3, (4.13a)

τ(T) = 0.214×exp[0.57(σ−2)1.428], (4.13b)

χ(T) = 2.07+0.225σ−|0.34−0.085σ|, (4.13c)

with σ=σ/kBT. The dependence of θ and ∆ on the dimensionless electric field F is shownin figure 4.6.

0 1 2 3 4 5

-3

-2

-1

0

1

/kBT = 6/kBT = 4

Wid

th

[]

Shi

ft [

]

Electric field F [ N1/3t /e]

/kBT = 2

Figure 4.6: Dependence on the electric field F of the width θ and shift ∆ in the parameterized occu-pation function Eqs. (4.9)-(4.13), for three dimensionless disorder strengths.

In the zero-field limit the parameterized occupation function reduces to the Fermi-Diracdistribution. At small fields, θ varies linearly with the field. However, it may be verifiedthat the effect on the occupation function and thus on the mobility is of second order in thefield, as should be the case. For high fields, the width of the distribution of electrochemicalpotentials becomes equal to the width of the DOS, as might have been anticipated from thefact that in that limit all states participate equally in the transport process. Interestingly,the parameterizations for θ and ∆ are independent of the carrier concentration. This isin agreement with the observation that the field dependence in a pure host system canbe taken into account in the EGDM by an enhancement factor that does not depend onthe carrier concentration.21 The parameterization is accurate for the range of energies Ewhere p & 0.001. Significant relative deviations of the occupational probabilities of guestsites only occur at high energies where p . 0.001.* For a guest concentration within therange of validity of the present model, a few percent at maximum, the maximum (worst-case) error in the space-charge concentration on the guest sites is then of the order of

*Consequently, the parameterization does not conserve the total number of charges, but this does not affect itspractical usefulness.

4.4 Device applications 87

10−5 electron charges per site. In all practical situations this is sufficiently small to givea negligible contribution to the electric field, so that it is expected that in all practicaldevice-modeling studies our parameterization scheme can be safely applied.

Figure 4.4 shows that this approach indeed yields a very accurate description of theoccupation function when p & 0.001 for the example system studied, more so than theeffective-temperature approach. Also, the parameterization yields accurate predictionsfor the host carrier concentration and hence for the charge-carrier mobility in host-guestsystems. For the example host-guest system discussed in the previous section, this is shownin figures 4.3(c) and (d) (solid curves). We remind the reader that this parameterizationapplies only to the specific case of Miller-Abrahams hopping on a simple-cubic lattice withuncorrelated Gaussian disorder.

4.4 Device applicationsIn order to demonstrate the effect of field-induced detrapping on the current density J indevices, we apply the model to sandwich-type single-layer single-carrier devices based onan organic semiconductor with a bimodal Gaussian DOS with equal widths of the host andguest DOS. The mean energy of the guest molecules is chosen 0.65 eV below that of the hostmolecules, and for the guest concentration x = 0.01 is taken. These values are realisticfor emissive host-guest systems used in OLEDs. The other parameters determining thecharge transport are taken equal to those measured for the hole transport in a polyfluorenederivative: σ= 0.13 eV, a = 1.19 nm, ν0 = 1.29×1010 s−1 and a relative dielectric constant ϵr= 3.2.36 Figure 4.7 shows that also for this more realistic host-guest system, as compared tothe system discussed in the previous sections, the model yields an accurate description ofthe field-dependent mobility. The arrow at the field-axis indicates the electric field F = 0.11V/nm that corresponds to a dimensionless field F = 1. Under realistic conditions, fields upto approximately 0.15 V/nm (i.e. 15 V across a 100 nm device) can occur. Neglecting field-induced detrapping (dash-dotted curve) would thus underestimate the mobility by a factorup to approximately 3. In view of the high sensitivity of the performance of OLEDs to thebalance between the hole and electron mobilities, the effect may thus be regarded as quitesignificant.

Figure 4.8 shows the J-V characteristics at room temperature for device thicknesses of20, 30, and 50 nm, as calculated using the 1D drift-diffusion method described in section2.5. There are no injection barriers at either electrode, so that the current is space-chargelimited and the effect of image charges at the electrodes is negligible. It follows from thisfigure that field-induced detrapping can be very relevant. An increase of the current den-sity of up to half an order of magnitude is obtained at voltages (indicated by the verticalarrows) corresponding approximately to the maximum realistic field mentioned above. Fig-ures 4.7 and 4.8 show that field-induced detrapping should be taken into account when theaverage field in the device V /L &σN1/3

t /e. This is comparable to the value above which themobility in the host shows a significant field-dependence.21 As a rule of thumb we thus con-clude that field-induced detrapping becomes important when the dependence of the mobilityon the electric field in the host becomes important.

88 Field-induced detrapping in host-guest systems

0.0 0.1 0.2 0.3 0.4 0.510-8

10-7

10-6

10-5

10-4

Exact (ME)mobility

Thermal and field-induceddetrapping + EGDM

Thermal detrapping + EGDM

T = 295 Kc = 0.01

Car

rier m

obili

ty

[cm

2 /Vs]

Electric field F [V/nm]

F = N1/3t e

Pure host system + EGDM

Figure 4.7: Dependence of the room-temperature mobility on the electric field for the host-guestsystem described on page 87, for a total carrier concentration c = 0.01. See figure 4.3 fora description of the curves and symbols.

0 2 4 6 8 10 12

10-4

10-3

10-2

10-1

100

101

V/L = N1/3t e

L = 50 nm

L = 30 nm

L = 20 nm

Thermal and field-induced detrapping

J [A

/m2 ]

V [V]

T = 295 KThermal detrapping

Figure 4.8: Room-temperature current-density vs. voltage characteristics for a single-layer single-carrier device with various thicknesses L of the host-guest system described on page 87.The dash-dotted curves are the model results with thermal detrapping only, while forthe solid curves also field-induced detrapping (FID) is taken into account. The verticalarrows indicate the voltage values in each device where the average field in the deviceV /L is equal to σN1/3

t /e = 0.11 V/nm.

4.5 Conclusions and discussion 89

4.5 Conclusions and discussionWe have shown how the Hoesterey-Letson model for describing the mobility in host-guestsystems in the limit of zero electric field may be generalized to the case of finite electricfields. Furthermore, we have developed an easy-to-use method for calculating the mobilityin an organic semiconducting host-guest system with a Gaussian host DOS and a generalshape of the guest DOS. The model is applicable when the guest concentration is suffi-ciently small, so that no guest-to-guest hopping occurs. We have demonstrated that, as inthe standard Hoesterey-Letson model, also at finite fields the mobility may be viewed asbeing due to the fraction of charge carriers, the ‘free’ carriers, that are detrapped from theguest sites and reside on the host sites. We have shown that by using a practical parame-terization scheme, which provides this field-dependent fraction of free charge carriers, andby using the known field dependence of the mobility in the pure host system, the field de-pendence of the mobility in host-guest systems may be efficiently calculated. Within theparameterization scheme, the free charge-carrier density is calculated using the occupationfunction, a field-dependent generalization of the Fermi-Dirac function.

We have considered only one specific hopping model: a simple-cubic lattice, Miller-Abrahams hopping and uncorrelated Gaussian energetic disorder. Applying the methodto a different lattice should be straightforward; the occupation function will not be thesame, but can be numerically computed and parameterized using the same methods. Thisshould also be possible for Marcus hopping. However, the fact that a different technique isneeded to numerically determine the occupation function (see section 4.3.1) is worrying; itindicates that all other aspects, including the validity of the generalized Hoesterey-Letsonmodel itself, should be carefully analyzed. It is also not certain that the same parameteriza-tion technique, i.e. assuming a site-energy-independent distribution of the electrochemicalpotential energy, will be effective for this case. Correlated disorder finally is expected tohave the largest effect on our method. This is because a guest site is now not identical toa host site at the same energy, since the host site is likely to be surrounded by sites withsimilar energy while the guest site is not. This puts the entire concept of the occupationfunction on shaky ground, and it may be necessary to describe host-guest systems withcorrelated disorder in a different way.

The field dependence of the mobility in host-guest systems is a combination of the in-trinsic field dependence of the mobility of the host material and field-induced detrapping.We have shown that field-induced detrapping becomes quite relevant for fields at whichalso the intrinsic field dependence of the mobility in the host becomes relevant. Applica-tion of the model to typical doped single-layer single-carrier model devices has revealedthat under realistic experimental conditions the effect of field-induced detrapping on thecurrent density can be significant. We infer from the analysis that the effect is also rele-vant in (double-carrier) OLEDs and foresee that our model can be readily combined withexisting software for OLED device simulations, making it possible to study the effects of(emissive) dopants in these devices with enhanced accuracy.

90 Field-induced detrapping in host-guest systems

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Chapter 5

Charge transport at non-zeroelectric field

Abstract

Charge transport in organic semiconductors is often described using the charge-carriermobility. In chapter 3, we analyzed the mobility in the zero-field limit for a wide range ofhopping models. In chapter 4, we analyzed the dependence of the mobility on electric fieldfor host-guest systems. Here, we combine both these results to describe charge transportat non-zero field in a general way, applicable to various hopping models including thosewith variable-range hopping or lattice disorder. We show that the mobility factorizes into acarrier-concentration-independent ‘intrinsic’ factor and a ‘detrapping’ factor. These repre-sent separate physical effects, and they affect charge transport in devices in different ways.This means that the value of the mobility by itself does not fully describe charge transportat finite electric field.

We introduce a new form of the one-dimensional drift-diffusion method that explicitlysplits these two factors instead of using the charge-carrier mobility. Using this method,we perform the first quantitative comparison of one-dimensional and three-dimensionalmodeling for a device with Marcus hopping. We also show charge-transport results fordevices with variable-range hopping and lattice disorder that can only be explained withthis new method.

To apply our results in practice, the intrinsic and detrapping factors must be parameter-ized or described semi-analytically. This is expected to be easiest for the intrinsic factor butis expected to be a major challenge for the detrapping factor. The field-induced-detrappingresults of chapter 4 may be useful here.

93

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2 The electric-field dependence of the charge-carrier mobility . . . . 95

5.2.1 Factorizing the charge-carrier mobility . . . . . . . . . . . . 955.2.2 The role of detrapping . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Device applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.1 The generalized Einstein expression at non-zero electric

field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.2 A new form of the 1D drift-diffusion equation . . . . . . . . 1005.3.3 Device results . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Variable-range hopping and lattice disorder . . . . . . . . . . . . . 1025.4.1 Effect on the charge-carrier mobility . . . . . . . . . . . . . 1025.4.2 Effect on device characteristics . . . . . . . . . . . . . . . . 104

5.5 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . 105References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

94 Charge transport at non-zero electric field

5.1 Introduction

The charge-carrier mobility in the hopping model described in section 1.4.1 and its depen-dence on electric field, carrier concentration, and temperature, can be determined fromthree-dimensional simulation. These results must then be parameterized or describedsemi-analytically to enable their use in the one-dimensional drift-diffusion (1D-DD) ap-proach. In chapter 3, we considered the temperature and concentration dependence of themobility in the limit of zero electric field, using a scaling theory to obtain an analytic ex-pression. However, in typical OLED applications also the field dependence of the mobilityis relevant. The scaling theory cannot be directly applied at finite fields, because it relieson linearization in the electric field.

In literature, the field dependence of the mobility as obtained from three-dimensionalmaster-equation (3D-ME) simulations was analyzed for two hopping models. The firstof these is the Gaussian disorder model (GDM):1,2 Miller-Abrahams (MA) hopping on asimple-cubic (SC) lattice with uncorrelated Gaussian energetic disorder. The second is thecorrelated disorder model (CDM):3,4 MA hopping on an SC lattice with dipole-correlateddisorder. For both of these models, the numerically obtained dependence of the mobility onfield, carrier concentration and temperature was parameterized. These parameterizationsare known as the extended GDM (EGDM) and extended CDM (ECDM) respectively, with’extended’ referring to the inclusion of the carrier-concentration dependence of the mobility.

In this chapter we will use elements of the scaling theory in chapter 3 to describe chargetransport at finite electric field in a more general way. We will also use the field-induced-detrapping theory in chapter 4; after all, there is nothing that distinguishes low-energysites in the Gaussian density of states (DOS) from guest sites in a host-guest system, andso similar physics can be expected to play a role. Our main goal is to provide insightinto the physical nature of charge transport at finite electric field. We will show that thecharge-carrier mobility can be factorized into an ‘intrinsic’ factor and a ‘detrapping’ factor.However, this is not just a convenient notation: we will demonstrate that these factorsrepresent separate physical effects, which affect charge transport in devices in differentways. This indicates that the value of the charge-carrier mobility by itself is not sufficientto describe charge transport at finite field. We should therefore not focus on parameterizingthe mobility, but on parameterizing the intrinsic and detrapping factors. We will show thatparameterizing these two factors may actually be easier than parameterizing the mobilityitself.

Our results apply to several hopping models: different types of lattice, different typesof hopping, different densities of states, presence or absence of correlation in the disor-der, etc. We will use two such hopping models as examples. The first is the well-knownGDM; the second is a model for which the field dependence has not yet been analyzed:Marcus hopping on an SC lattice with dipole-correlated energetic disorder, with the reor-ganization energy Er taken to be 2σ. For convenience we will refer to this model as theMarcus correlated disorder model (MCDM). All numerical results are obtained using the3D-ME method, as described in section 2.4, with a 100×100×100 lattice size for mobilitycalculations and a 100×100 lateral lattice size for device calculations.

We first consider the electric-field dependence of the charge-carrier mobility and show

5.2 The electric-field dependence of the charge-carrier mobility 95

that it can be factorized into an intrinsic and a detrapping factor (section 5.2). We thenshow how the 1D-DD method can be rewritten in terms of these factors instead of themobility itself, and apply this new method to a GDM device and an MCDM one (section5.3). We then analyze the effects of variable-range hopping (VRH) and lattice disorder. Weshow that our theory still applies, and that certain effects of VRH and lattice disorder ondevice current density-voltage characteristics can only be explained using our new 1D-DDmethod (section 5.4). We finish with our main conclusions and a discussion (section 5.5).

5.2 The electric-field dependence of the charge-carriermobility

In this section, we will show how the field dependence of the mobility can be written as theproduct of two factors. We will then argue that one of these factors is determined by carrierdetrapping, and so refer to the two factors as the ‘intrinsic’ factor and the ‘detrapping’factor.

5.2.1 Factorizing the charge-carrier mobility

0 1 2 3

10-13

10-12

10-11

10-10

10-9

10-8

c = 10-1

c = 10-2

c = 10-3

c = 10-4

(a)

GDM (Uncorrelated Gaussian disorder,Miller-Abrahams hopping,SC lattice)

/kBT = 5

Car

rier m

obili

ty

[e0/N

2/3

t]

Electric field F [ N1/3t /e]

Simulation Factorizing in F and c [Eq. (5.1)] Factorizing in F and [Eq. (5.5)]

0 1 2 3(b)

MCDM (Dipole-correlated disorder,Marcus hopping [Er = 2 ], SC lattice)

/kBT = 5

Electric field F [ N1/3t /e]

Figure 5.1: (a) Charge-carrier mobility µ as a function of electric field F for different values of thecharge-carrier concentration c, for the Gaussian disorder model (GDM) with σ/kBT =5. Solid black curves: three-dimensional master equation (3D-ME) simulation results.Gray dashed curves: results of Eq. (5.1), which assumes factorization of the F and cdependence, with c = 10−3 as reference curve. Gray solid curves: results of Eq. (5.5),which assumes factorization of the dependence on F and on the average electrochemicalpotential energy µ, with values of µ and h as found in figure 5.2. (b) Same, for theMarcus correlated disorder model (MCDM).

We will develop a new expression to describe the field, carrier-concentration and tem-perature dependence of the charge-carrier mobility µ. Figure 5.1 shows the field depen-dence for the GDM and the MCDM, as found from 3D-ME simulation (black solid curves).

96 Charge transport at non-zero electric field

In the case of the GDM, we can see that at moderate fields the F dependence is (roughly)independent of the c dependence. Indeed, the EGDM parameterization in Ref. 2 was basedon factorizing the F and c dependence:

µ(T, c,F)≈µ(T, c,0)g2(F,T), (5.1)

where g2 describes the field dependence of the mobility (the notation g2 is taken fromRef. 5). The dashed curves in figure 5.1 show the results of applying this equation. Forboth the GDM and the MCDM, we determine g2 from the simulated c = 10−3 curve, anduse the simulated mobility values for µ(T, c,0). Consequently, the results of Eq. (5.1) matchthe simulation results exactly for F = 0 and for c = 10−3. From the figure we conclude thatthe factorization is indeed approximately correct for the GDM for F . 2σN1/3

t /e. However,it clearly does not apply to the MCDM. In fact, we have checked that the factorization doesnot apply to any system except MA hopping with uncorrelated disorder. This is consistentwith the results of Bouhassoune et al.,4 who showed that the mobility also does not factorizefor MA hopping with dipole-correlated disorder.

0 1 2 3

10-9

10-8

10-7

-4

-3

-2

-1

1 2 3

(c)

Fiel

d-de

pend

ence

func

tion

h [e

0/N2/

3t

]

Electric field F [ N1/3t /e]

GDM

GDM

(a)

Ave

rage

ele

ctro

chem

ical

pote

ntia

l ene

rgy

[]

c = 10-4

c = 10-3

c = 10-2

c = 10-1

(d)

Electric field F [ N1/3t /e]

MCDM

MCDM(b)

Figure 5.2: Average electrochemical potential energy µ and field-dependence function h as a func-tion of F, for the GDM and the MCDM with σ/kBT = 5.

Clearly, we need a more general way to describe the interplay of the F and c dependenceof the mobility µ. Recall that we showed in section 3.4.1 that at F = 0 the c dependence of

5.2 The electric-field dependence of the charge-carrier mobility 97

µ is given byµF=0(c)∝ exp[EF(T, c)/kBT] /c, (5.2)

where EF is the Fermi energy. This dependence applies when c . 0.03 for uncorrelatedGaussian disorder and when c . 0.01 for dipole-correlated disorder. To apply this expres-sion to the non-equilibrium F > 0 case we must first generalize the concept of the Fermienergy to non-zero field. Consider for this purpose the electrochemical potential energy µi(first introduced in section 2.4), which is defined at the site level by

pi = 11+exp([E i − µi]/kBT)

, (5.3)

where pi is the probability that site i is occupied by a charge carrier. Consider now µ, theaverage over all sites of this electrochemical potential energy. At F = 0, µi = EF for all sitesi and so µ= EF. Consequently, µ is a candidate for the generalized field-dependent Fermienergy, and in analogy to Eq. (5.2) we propose as an ansatz that the carrier-concentrationdependence of the mobility at any field is described by

µ(c)∝ exp[µ(T, c,F)/kBT

]/c. (5.4)

The full expression for the mobility, i.e. the analogue of Eq. (5.1), is then

µ(T, c,F)= h(T,F)exp[µ(T, c,F)/kBT

]/c, (5.5)

for some field-dependence function h which does not depend on c.Since Eq. (5.4) is only an ansatz, we must still test the accuracy of Eq. (5.5). First, we

determine the functions µ and h numerically. µ can be calculated directly from the 3D-MEresults for pi. The resulting dependence of µ on c and F is shown in figures 5.2(a) and(b). The function h can now be found by using Eq. (5.5) in combination with the simulatedmobility results for any value of c; we choose to use c = 10−3. The resulting dependence ofh on F is shown in figures 5.2(c) and (d). The results of Eq. (5.5) using these values of h andµ are shown by the gray solid curves in figure 5.1. We see that the simulation results areexcellently reproduced when the carrier concentration is low enough. At c = 10−1, however,Eq. (5.5) is not accurate; this should not surprise us, since Eq. (5.2) also does not apply atthis value of the carrier concentration.

5.2.2 The role of detrappingIt is important to realize that on its own Eq. (5.5) provides no help in describing or pa-rameterizing the mobility; indeed, without knowledge of µ(T, c,F) any mobility functionµ(T, c,F) can be written in this form. Compare this to Eq. (5.1), which makes the EGDMparameterization much easier. Instead, the benefit of Eq. (5.5) lies in revealing the physicsbehind charge transport at finite electric field. It splits the field dependence of the mobilityinto two factors. The first of these is the field-dependence function h; we will call this theintrinsic factor of the field dependence. The second factor depends on the relationship be-tween the carrier concentration c and the average electrochemical potential energy µ; wecall this the detrapping factor.

98 Charge transport at non-zero electric field

To show that the relationship between c and µ is indeed determined by detrapping, wewill follow an approach analogous to the one used in chapter 4 for detrapping in host-guestsystems. Consider first the relationship between c and µ at zero field:

c =∫ ∞

−∞g(E)

1+exp([E− µ]/kBT)dE, (5.6)

where g(E) is the DOS. We can generalize this to finite field by generalizing the Fermi-Dirac distribution function. We will use for this the occupation function p(E;T, µ,F), whichgives the average probability that a site at energy E is occupied at given values of T, µ andF. This occupation function was first introduced in chapter 4; there, it was written in termsof the host carrier concentration ch instead of µ.* Note that p(E;T, µ,0) is the Fermi-Diracdistribution function. Eq. (5.6) now generalizes to:

c =∫ ∞

−∞g(E)p(E;T, µ,F)dE. (5.7)

This makes it clear that the relationship between c and µ is determined by the distributionof charge carriers between sites with different energies, i.e. by detrapping.

Comparing Eqs. (5.1) and (5.5), we see that the factorization of the field and carrier-concentration dependence in the GDM implies that in the GDM µ(T, c,F)− µ(T, c,0) =µ(T, c,F)−EF(T, c) is independent of the carrier concentration c. This can be seen in figure5.2(a): the shape of the µ curves does not depend on c for F . 2σN1/3

t /e, i.e. when Eq. (5.1)applies. Roughly speaking, this means that the detrapping is independent of c. This is con-sistent with the results of section 4.3.2, where we showed that the statistical distributionof the electrochemical potential energy relative to the Fermi energy does not depend on c.

To apply Eq. (5.5) in practice, both the intrinsic and the detrapping factor must beparameterized or described analytically. The intrinsic factor of the field dependence, de-scribed by the field-dependence function h, is expected to be relatively easy to parame-terize. This is because h is independent of several parameters. We have already shownthat it is independent of c. We will also see in section 5.4.1 that it is barely affected whenconsidering moderate variable-range hopping instead of nearest-neighbor hopping as useduntil now. Finally, it does not change when considering a host-guest system with low guestconcentration. This is because the effect on the mobility is entirely due to detrapping (seechapter 4), and so is described completely by the detrapping factor of the field dependence.

The detrapping factor is expected to be much harder to parameterize because it isstrongly affected by any change in the hopping model. Of course, if we could somehow de-rive or at least parameterize the occupation function discussed above we could use Eq. (5.7);we will discuss an outlook on achieving this in section 5.5. However, we must keep in mindthat ultimately we are less interested in the value of the charge-carrier mobility than in itsapplication in the one-dimensional drift-diffusion (1D-DD) method. We will show in section5.3.2 that it is possible to take advantage of Eq. (5.5) to carry out 1D-DD calculations with-out explicitly calculating the mobility itself. We will see that the difficult-to-parameterizedetrapping factor of the field dependence is sometimes not even needed.*The order of the arguments is also different (here T, µ,F; in chapter 4 F, ch,T) for consistency with publishedwork.

5.3 Device applications 99

5.3 Device applicationsIn this section, we will introduce a new form of the one-dimensional drift-diffusion (1D-DD) method that takes advantage of the factorization of the mobility into an intrinsic anda detrapping factor by Eq. (5.5). For a basic discussion of the 1D-DD method, we refer tosection 1.3.3. The differential equations to solve are given by Eqs. (1.2) and (1.4), writtenhere in terms of the carrier concentration c instead of the carrier density n (they are relatedby n = Ntc):

J = eNt

[µ(T, c,F)c(x)F(x)−D(T, c,F)

dcdx

]; (5.8a)

dFdx

= eNtc(x)ϵ0ϵr

, (5.8b)

where D is the diffusion coefficient.We will first introduce an alternative version of the generalized Einstein expression.

We will then use this new expression together with the mobility results of the previoussection to derive a new form of the 1D-DD method, in which the intrinsic factor of the fielddependence appears only in the drift-diffusion equation and the detrapping factor only inthe space-charge equation. Finally, we show how this new form can quantitatively describethe current-voltage characteristics of space-charge-limited devices for both the GDM andthe MCDM.

5.3.1 The generalized Einstein expression at non-zero electric fieldThe generalized Einstein expression relates the diffusion coefficient D to the charge-carriermobility µ. For F = 0, it is given by:6

D = µce

dEF

dc. (5.9)

This expression is also frequently used at finite values of F, but this is physically inaccuratebecause the Fermi energy EF is not properly defined then. We propose a similar approachas the one used for the mobility in the previous section: we replace the Fermi energy EFby its field-dependent generalization, the average electrochemical potential energy µ. Wethen have:

D = µce

dµdc

. (5.10)

To test whether Eq. (5.10) is more accurate, we should compare 1D-DD results usingEq. (5.9) and 1D-DD results using Eq. (5.10) with the results of 3D simulation. How-ever, we have been unable to find a device structure for which the choice between Eq. (5.9)and Eq. (5.10) yields significant differences in the current-voltage characteristics or thefield/concentration profiles. This means that we are free to use either one. In this chapter,we choose to use Eq. (5.10) because it seems physically more accurate and because it leadsto a more elegant final form of the 1D-DD method.

100 Charge transport at non-zero electric field

We apply the same reasoning to the boundary conditions at the electrodes, given byEq. (2.37):

µ(a)=Φleft + eaF(a);µ(L−a)=Φright − eaF(L−a). (5.11)

Again, this is physically more accurate but does not significantly affect the results of the1D-DD method, since we consider only space-charge-limited devices.

5.3.2 A new form of the 1D drift-diffusion equationUsing the results derived earlier in this chapter for the mobility [Eq. (5.5)] and the diffusioncoefficient [Eq. (5.10)] in the drift-diffusion equation [Eq. (5.8a)] yields:

J = eNth(T,F)exp[µ(T, c,F)/kBT

][F(x)− 1

edµdc

dcdx

]. (5.12)

We can simplify this equation by not solving for c and F as a function of x, but for µ and F.The complete set of equations then becomes:

J = eNth(T,F)exp[µ(x)/kBT

][F(x)− 1

edµdx

]. (5.13a)

dFdx

= eNtc(T, µ,F)ϵ0ϵr

. (5.13b)

Note that the carrier concentration c no longer appears in the drift-diffusion equation, butonly appears in the space-charge equation. We now need to know c as a function of µ

instead of the other way around as in Eq. (5.5), but this is actually easier if Eq. (5.7) isused.

With this new form of the 1D-DD method, we have neatly split the two factors of themobility over the two differential equations. The intrinsic factor, described by h, appearsonly in the drift-diffusion equation, Eq. (5.13a), while the detrapping factor, specified by therelationship between c and µ, appears only in the space-charge equation, Eq. (5.13b). Animportant practical advantage is that we do not need to know the relationship between cand µ when the carrier concentration is low enough for space-charge effects to be irrelevant.In particular, we can model low-carrier-density layers and injection-limited devices withouthaving to be concerned with it at all.

Our charge-transport results outside this chapter are written in terms of the charge-carrier mobility and the standard 1D-DD method [Eq. (5.8)]. All these results can howeveralso be rewritten in terms of the intrinsic and detrapping factors and the new 1D-DDmethod [Eq. (5.13)]; these rewritten results can be found in appendix A.

5.3.3 Device resultsWe will now apply the derived 1D-DD method, Eq. (5.13), to an example device and com-pare the results with 3D-ME simulation. We use a similar example device as the one in

5.3 Device applications 101

chapter 2, with material parameters corresponding to a hole-transporting polyfluorene-triarylamine (PF-TAA) device studied by van Mensfoort et al.7 After three years of ag-ing, the material parameters were found to be: DOS width σ = 0.122 eV, site densityNt = 4.28×1026 m−3, hopping attempt rate ω0 = 5.77×109 s−1, and relative dielectric con-stant ϵr = 3.2. These parameters were determined assuming the GDM as the appropriatehopping model, but we will also use them together with the MCDM. For the device lengthwe take L = 122 nm, and we take no injection barrier at either electrode. This means thatVbi = 0, unlike the device used in chapter 2; we make this choice to simplify the calculations.

1 1010-1

100

101

102

103

104

105

T = 295 KGDM

curr

ent d

ensi

ty [A

/m2 ]

voltage [V]

Simulation New 1D-DD method Standard 1D-DD method,

using EGDMparameterization

1 10

T = 295 KMCDM

voltage [V]

Figure 5.3: Current density-voltage characteristics for the polyfluorene-triarylamine (PF-TAA) de-vice structure described in the text, using as hopping model (a) the GDM and (b) theMCDM. Black solid curves: 3D-ME simulation results. Gray solid curves: 1D-DD re-sults using the new method [Eq. (5.13)]. Gray dashed curve in (a): 1D-DD results usingthe standard method [Eq. (5.8)] and the EGDM parameterization of the mobility.

Simulated current density-voltage characteristics for this device are shown in figure5.3(a) for the GDM, and in 5.3(b) for the MCDM. The results of our new 1D-DD method(gray solid curves) match the 3D-ME results (black solid curves) very well. We note thatthis is the first such analysis of a device with Marcus hopping. The required values of hand c were determined by interpolating the results of 3D-ME simulation for several valuesof µ and F. Since there are no injection barriers, we have in both devices c > 0.03 close tothe anode, which means that Eq. (5.5) does not apply there. Apparently this does not havea significant effect on the current density.

The new 1D-DD method provides an elegant way to quantitatively describe these space-charge-limited devices, but there is for this case no real practical advantage compared tothe standard 1D-DD method. Indeed, this standard method, together with the EGDMparameterization of the mobility given by Pasveer et al.2 and the improved field depen-dence given by van Mensfoort et al.,5 also correctly describes the GDM device [gray dashedcurve in figure 5.3(a)]. Our new method does have a practical advantage for injection-limited devices, since we can then ignore the detrapping factor completely. This leavesonly the relatively easy-to-parameterize intrinsic factor (see section 5.2.2). Unfortunately,our boundary condition at the electrodes, Eq. (5.11), is too simple to describe injection-

102 Charge transport at non-zero electric field

limited devices. However, we will see in section 5.4.2 examples of charge-transport effectsin such devices that can be explained only with our new method, and not with the standardmethod.

5.4 Variable-range hopping and lattice disorderAll results discussed so far in this chapter are based on nearest-neighbor hopping on asimple-cubic lattice. In a real disordered organic semiconductor, however, the charge-localization sites are not ordered on a lattice. To investigate the sensitivity of our resultsto the lattice assumption, we will analyze the effects of variable-range hopping (VRH) andlattice disorder. Another reason to consider these effects is that devices with VRH and lat-tice disorder have current density-voltage characteristics which can only be understood byconsidering the intrinsic and detrapping factors of the charge-carrier mobility separately,and not by considering just the value of the mobility. This necessitates the use of our new1D-DD method [Eq. (5.13)], illustrating that this new method is more than just an elegantway of rewriting the standard method.

The implementation of VRH and lattice disorder in our hopping model is described insection 1.4.1. The strength of VRH is described by the inverse wave-function decay lengthα [Eq. (1.13)], and the strength of lattice disorder by the disorder parameter Σ [Eq. (1.12)].Nearest-neighbor hopping with no lattice disorder corresponds to α → ∞ and Σ = 0. Wenote that we have defined the hopping rate such that nearest-neighbor hops are unaffectedby the value of α, which is why charge transport still take place in the limit α→∞.

We will first consider the effects of VRH and lattice disorder on the charge-carrier mo-bility, and show that the factorization into an intrinsic factor and a detrapping factor[Eq. (5.5)] still holds. We will then show that the effects of VRH and lattice disorder ondevice characteristics can only be fully explained by our new 1D-DD method, and not bythe standard one.

5.4.1 Effect on the charge-carrier mobilityWe first consider variable-range hopping. We take α = 10/a, following Refs. 2 and 8. Infigure 5.4(a), we compare the mobility in the nearest-neighbor-hopping case (solid curve)with the mobility in the VRH case (dash-dotted curve). The effect of VRH is small at F = 0,but it plays a much larger role at finite electric field. To determine the nature of this effect,we also show the field dependence of the average electrochemical potential energy µ infigure 5.4(b) and of the field-dependence function h in figure 5.4(c). We see that h is barelyaffected by VRH, while µ differs significantly. In other words, VRH affects the mobilitythrough enhanced detrapping. This reinforces our earlier conclusion that detrapping is akey element in describing charge transport at non-zero field.

In the EGDM work of Pasveer et al.2 it was stated that at α= 10/a varying α predomi-nantly changes the prefactor of the mobility. Our results here show that this is correct onlyat zero field. The EGDM parameterization therefore applies only to α= 10/a and is not asgeneral as previously thought.

5.4 Variable-range hopping and lattice disorder 103

10-12

10-11

10-10

-3

-2

0 1 2 310-9

10-8

(c)

(b)

(a)

GDM/kBT = 5

c=10-3

Car

rier m

obilit

y [e

0/N2/

3t

] = 0 = 10 / a = 0 = 2

Aver

age

elec

troch

emic

alpo

tent

ial e

nerg

y [

]

Fiel

d-de

pend

ence

func

tion

h [e

0/N2/

3t

]

Electric field F [ N1/3t /e]

Figure 5.4: Dependence on F of (a) µ, (b) µ, and (c) h, for the GDM. Solid curves: nearest-neighborhopping without lattice disorder. Dash-dotted curves: variable-range hopping, with in-verse wave-function decay length α= 10/a. Dotted curves: lattice disorder, with disorderstrength Σ= 2.

104 Charge transport at non-zero electric field

Next, we consider lattice disorder. We analyzed the F = 0 case in section 3.3.3 and foundthat for moderate values of Σ (Σ. 3) there is no significant effect on the mobility. However,we see in figure 5.4(a) that even a low value Σ = 2 already affects the field dependence ofthe mobility (dotted curve). This is similar to the VRH case, but the nature of the effect isvery different. This can be seen from the F dependence of µ and h in figures 5.4(b) and (c).This time, both µ and h depend strongly on F, and the two effects cancel each other outto a degree in the final mobility result. It is not immediately obvious why lattice disorderbehaves so differently from VRH. We have checked that Eq. (5.5) does still apply for latticedisorder, i.e. h is still independent of the carrier concentration.

In summary, both VRH and lattice disorder have a stronger effect on charge transportat finite field than at zero field. The nature of these effects is different, however: VRHaffects only the detrapping factor, while lattice disorder affects both the intrinsic and thedetrapping factor.

5.4.2 Effect on device characteristicsIn this section, we will consider the effects of VRH and lattice disorder on charge transportin devices. We will consider both the space-charge-limited GDM device described in section5.3.3, and an injection-limited version of this device with 1 eV injection barriers at bothelectrodes. To prevent injection physics from dominating the device characteristics in theinjection-limited device, we place the electrodes at the same location as the first layer oforganic sites, instead of at a distance a as we have done for all other devices in this thesis.Simulated current density-voltage characteristics for both devices are shown in figure 5.5.

Let us first try to understand these results using the standard 1D-DD method, Eq. (5.8).We start with the space-charge-limited device [figure 5.5(a)]. Here, we see that latticedisorder (dotted curve) has almost no effect on the current (compare with solid curve).However, VRH (dash-dotted curve) significantly increases the current at high voltage. Toexplain these results, consider the field dependence of the charge-carrier mobility as shownin figure 5.4(a) (the maximum voltage shown in figure 5.5 V = 25 V corresponds to anaverage field F ≈ 2.2σN1/3

t /e). We see that lattice disorder has only a moderate effect,explaining why the device current is not strongly affected. VRH significantly increases themobility, but only at high field; this explains why we see a current increase at high voltagein the device. Applying this reasoning to the injection-limited device [figure 5.5(b)], wewould expect similar characteristics there. However, this time lattice disorder increasesthe current at high voltage, and VRH has almost no effect. There is no obvious way toexplain this using the standard 1D-DD method.

Let us see if our new 1D-DD method, Eq. (5.13), can do better. We must now explain thedevice characteristics in terms of the intrinsic factor of the field dependence, described bythe field-dependence function h [figure 5.4(c)], and the detrapping factor, described by therelationship between c and µ [figure 5.4(b)]. Consider first again the space-charge-limiteddevice. Lattice disorder increases h, but also reduces detrapping (i.e. µ as a function ofc decreases) which leads to a higher space charge. These two effects mostly cancel eachother out, so the device current does not change much. VRH does not affect h, but doesincrease detrapping at high field and so leads to a higher current at high voltage. In the

5.5 Conclusions and discussion 105

1 10

100

101

102

103

104

(a) Space-charge limitedT = 295 K

GDM

Cur

rent

den

sity

[A/m

2 ]

voltage [V]

= 0 = 10 / a = 0 = 2

1 1010-9

10-8

10-7

10-6

10-5

(b) Injection limited

voltage [V]

Figure 5.5: (a) Current density-voltage characteristics for the PF-TAA device structure describedin section 5.3.3. See figure 5.4 for a description of the curves. (b) Same, but with 1 eVinjection barriers at both electrodes.

injection-limited-device, detrapping is irrelevant; we therefore only consider h. Latticedisorder increases h at high field and so leads to a higher current at high voltage. VRH,however, does not affect h and so does not affect the current in this device. Both of theseresults are consistent with the device characteristics shown in figure 5.5(b). In conclusion,using our new 1D-DD method we can understand all results shown in figure 5.5.

We have shown that our new 1D-DD method can explain device results which cannot beexplained using the standard 1D-DD method. This shows that the two factors describingthe mobility, intrinsic and detrapping, represent different physical effects and must beconsidered separately. In other words, the charge-carrier mobility by itself does not fullydescribe charge transport, since it only reflects the combined result of the intrinsic anddetrapping factors.

5.5 Conclusions and discussionWe have shown how to factorize the field dependence of the charge-carrier mobility into an‘intrinsic’ factor, which is described by a concentration-independent field-dependence func-tion h, and a ‘detrapping’ factor. This detrapping factor is described by the relationshipbetween the carrier concentration c and the average electrochemical potential energy µ,which is a field-dependent generalization of the Fermi energy. We have shown that thisfactorization is not merely a convenient way to write the mobility; the intrinsic and de-trapping factor are actually separate physical effects, which can be distinguished in devicecharge-transport results. This means that the value of the mobility does not by itself fullydescribe charge transport at finite electric field. We have introduced a new way to writethe one-dimensional drift-diffusion (1D-DD) method which correctly takes both factors intoaccount separately; the intrinsic factor appears in the drift-diffusion equation, and thedetrapping factor in the space-charge equation.

106 Charge transport at non-zero electric field

We have shown that our new 1D-DD method can quantitatively describe device current-voltage characteristics. This was achieved using interpolation of numerical results for hand µ. To complete our method, h and µ have to be determined semi-analytically or param-eterized for different hopping models. This is expected to be easiest for the intrinsic factorof the field dependence (h), since it does not depend on carrier concentration and does notchange when considering moderate variable-range hopping or a host-guest system with lowguest concentration. The detrapping factor (µ) unfortunately does not share these benefi-cial properties and will likely be much harder to parameterize. An advantage, however, isthat it is only needed in the 1D-DD method when considering space-charge effects, and sodoes not need to be known at very low carrier concentration. This is especially beneficialwhen considering low-carrier-density layers or injection-limited devices.

It may be possible to describe detrapping by using the occupation function p(E;T, µ,F),the average probability that a site at energy E is occupied at given values of T, µ andelectric field F [see Eq. (5.7)]. Recall that the occupation function was used successfullyin chapter 4 to describe field-induced detrapping in host-guest systems. However, it is notimmediately clear how we can determine this function. It can be computed using 3D sim-ulation, using techniques described in section 4.3.1, but it would then be more efficient tojust compute the mobility directly. In section 4.3.2 we showed how the occupation functioncan be parameterized for the Gaussian disorder model, using the physics of the distribu-tion of charge carriers as a starting point. This physical basis allows us to be cautiouslyoptimistic that there is some semi-analytical way to describe p(E;T, µ,F), perhaps in theform of a general expression with model-dependent parameters, similar to the mobility ex-pression in chapter 3 [Eq. (3.16)]. On the other hand, the occupation function described insection 4.3.2 applies only to sites with low energy, while in order to use Eq. (5.7) we needto know it over the full energy range. In addition, we have to keep in mind here that wehave already seen several results that apply only to MA hopping with uncorrelated disor-der, such as Eqs. (3.21) and (5.1). The results of section 4.3.2 may also be in this category,which would make them harder to generalize to other hopping models.

References1. H. Bässler, Phys. Stat. Sol. B 175, 15 (1993).2. W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M. Blom, D. M.

de Leeuw, and M. A. J. Michels, Phys. Rev. Lett. 94, 206601 (2005).3. Y. N. Gartstein and E. M. Conwell, Chem. Phys. Lett. 245, 351 (1995).4. M. Bouhassoune, S. L. M. van Mensfoort, P. A. Bobbert, and R. Coehoorn, Org. Elec.

10, 437 (2009).5. S. L. M. van Mensfoort and R. Coehoorn, Phys. Rev. B 78, 085207 (2008).6. Y. Roichman and N. Tessler, Appl. Phys. Lett. 80, 1948 (2002).7. S. L. M. van Mensfoort, S. I. E. Vulto, R. A. J. Janssen, and R. Coehoorn, Phys. Rev.

B 78, 085208 (2008).

References 107

8. C. Tanase, E. J. Meijer, P. W. M. Blom, and D. M. de Leeuw, Phys. Rev. Lett. 91,216601 (2003).

Chapter 6

Charge transport acrossdisordered organicheterojunctions

Abstract

Organic light-emitting diodes often consist of a sandwich structure containing several lay-ers of disordered organic semiconductors. In the modeling of such devices it is essential thatthe charge transport across the organic heterojunctions is properly described. The presenceof energetic disorder and of strong gradients in both the charge density and the electric fieldat the heterojunction complicates the use of continuum drift-diffusion approaches to cal-culate the electrical current, because of the discrete positions of the sites involved in thehopping transport of charges. We use the results of three-dimensional Monte Carlo sim-ulations to construct boundary conditions in a one-dimensional continuum drift-diffusionapproach that accurately describe the charge transport across the junction. These bound-ary conditions take into account three important physical effects: non-equilibrium chargetransport across the interface, the formation of a surface charge layer just before the in-terface, and the reduction of the effect of this surface charge due to short-range Coulombinteractions. The developed approach is expected to have a general validity.

109

Contents

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2 Three-dimensional and one-dimensional modeling . . . . . . . . . 111

6.2.1 Three-dimensional Monte Carlo modeling . . . . . . . . . . 1116.2.2 One-dimensional drift-diffusion modeling . . . . . . . . . . 112

6.3 Improvements to the one-dimensional model . . . . . . . . . . . . . 1136.3.1 Deviation from equilibrium . . . . . . . . . . . . . . . . . . . 1146.3.2 Non-continuous carrier distribution . . . . . . . . . . . . . . 1146.3.3 Short-range Coulomb interactions . . . . . . . . . . . . . . . 116

6.4 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . 118References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

This chapter was adapted with permission from Ref. 1:

J. Cottaar, R. Coehoorn, and P. A. Bobbert, Modeling of charge transport across disordered organic hetero-junctions, Organic Electronics 13, 667 (2012).Copyright (2012) by Elsevier Science.

110 Charge transport across disordered organic heterojunctions

6.1 Introduction

Organic light-emitting diodes (OLEDs) typically consist of several layers, such as emissionlayers and charge injection, transport, and blocking layers (see section 1.1.3).2–5 In suchdevices a proper understanding of the charge transport across the interface between twoorganic semiconductors is crucial for accurate modeling of the optoelectronic characteris-tics. The energetic disorder present in organic semiconductors has a profound influence onthis charge transport.

Our goal in this chapter is to use the one-dimensional drift-diffusion (1D-DD) method(see section 2.5) to reproduce the J-V characteristics of a bilayer device as found from 3DMonte Carlo (3D-MC) simulation (see section 2.3). This requires an appropriate boundarycondition at the interface. We consider here only the case of an energy barrier, but othereffects at the interface, such as different mobilities, can be treated similarly. Staudigel etal. obtained a boundary condition by considering the forward and backward hopping ratesover the interface in conjunction with a reduction of the energy barrier by the local electricfield.6 Arkhipov et al. considered in addition the probability that a carrier that has hoppedover the interface ‘escapes’ further into the organic layer behind the interface.7 Severalmodifications and applications of these semi-analytical approaches have been proposed.8–11

Houili et al. were the first to perform 3D-MC simulations of charge transport across anorganic heterojunction, including Coulomb interactions between the charges, and made aqualitative comparison with the semi-analytical approaches.12 However, a systematic wayto improve the 1D-DD approach using the results of 3D-MC simulations is still lacking.

-2 0 2 4 6 8 10 12 14 16 18 20 22

x (nm)

Anodea

a = 1.7 nm xI

0.2 eV

a

Cathode

Figure 6.1: Energy diagram for the device to which we apply our modeling, with a barrier ∆ betweenthe tops of the Gaussian densities of states - indicated by the shading - of the two disor-dered organic semiconductors. The energies shown are hole energies. The position xI ofthe interface is indicated.

6.2 Three-dimensional and one-dimensional modeling 111

We will apply all modeling in this chapter to an example hole-only device with a simple-cubic lattice, Miller-Abrahams hopping and uncorrelated Gaussian energy disorder (foran explanation of these terms and a description of the charge-transport model itself, seesection 1.4.1). The device structure and the involved energy levels (neglecting the effectof the space charge) are sketched in figure 6.1. Apart from an energy barrier ∆ betweenthe tops of each Gaussian density of states (DOS) we take the two organic materials tobe identical, using typical parameters: σ = 0.09 eV, W0 = 105 s−1 and Nt = 2× 1026 m−3

(corresponding to a ≈ 1.7 nm). At room temperature (295 K), the mobility in the limit ofzero carrier density and zero field is then 1.6×10−14 m2/Vs. The relative dielectric constantϵr is taken to be 3. The total thickness of the device is L = 12a ≈ 20.4 nm. This rather smallthickness ensures that the interface plays a large role in determining the current density-voltage (J-V ) characteristics. The organic-organic interface is located between the fifth andthe sixth layers of organic sites. We define the location of the interface xI = 5 1

2 a ≈ 9.4 nmas the distance of the midpoint of these layers to the left electrode. At the left side of thedevice we take an injection barrier of 0.2 eV, defined as the difference between the Fermienergy of the electrode and the top of the Gaussian DOS of the left organic layer; see figure6.1. Although we consider a specific model device, the conclusions that we will reach havea general validity.

We will first discuss the 3D-MC results, and introduce a basic way to handle the inter-face in the 1D-DD approach by assuming thermal equilibrium (section 6.2). We will thenintroduce three improvements to the 1D-DD method which together allow it to reproducethe 3D-MC results (section 6.3). We finish with our main conclusions and a discussion ofour results, including an analysis of the effect of a rough interface (section 6.4).

6.2 Three-dimensional and one-dimensional modeling

6.2.1 Three-dimensional Monte Carlo modeling

The 3D-MC device simulations proceed as described in section 2.3 and Ref. 13. As described,we take Coulomb interactions into account both through the long-range space charge-effectas well as the direct interaction between holes within a radius Rc = 8a. In the lateraldirection we take 100×100 sites. With this system size and the parameters given aboveit is sufficient to consider only a single disorder configuration. The error in the currentsshown in the plots below is then of the order of or smaller than the symbol sizes. Allsimulations have been performed at room temperature (T = 295 K).

The J-V results obtained by applying the 3D-MC method to our model device are shownby the black squares in figure 6.2, for ∆= 0.6 eV (main figure) and ∆= 0, i.e. no barrier (in-set). For comparison, the current density obtained when neglecting short-range Coulombinteractions is shown by the gray circles. We observe that these interactions strongly affectthe current when an internal interface is present, but not in the absence of the interface(the black squares and gray circles in the inset of figure 6.2 coincide).

112 Charge transport across disordered organic heterojunctions

6.2.2 One-dimensional drift-diffusion modeling

2 3 4

10-4

10-3

10-2

10-1

100

101

1 2 3 410-1

100

101

102

(1) + (2) + (3)(1) + (2)

= 0

J (A

/m2 )

Voltage (V)

= 0.6 eV

(1)

Figure 6.2: Current density-voltage characteristics for the device shown in figure 6.1 with ∆ =0.6 eV. Black squares: 3D Monte Carlo (3D-MC) results. Black triangles: 3D-MC resultsincluding surface roughness. Gray circles: 3D-MC results without short-range Coulombinteractions. Dash-dotted curve: 1D drift-diffusion (1D-DD) results assuming thermalequilibrium across the interface. Dotted curve: 1D-DD results with effect (1) (deviationfrom equilibrium) included. Dashed curve: 1D-DD results with effects (1) and (2) (non-continuous charge distribution) included. Solid curve: 1D-DD results with effects (1),(2) and (3) (short-range Coulomb interactions) included. Inset: same for ∆ = 0, i.e. nointernal interface.

In the continuum 1D-DD approach the carrier density n(x) and electric field F(x) in thedevice are determined as a function of the distance x from the left electrode by solving thedrift-diffusion equation. The method is described in detail in section 2.5 and Ref. 14. Thedrift-diffusion equation itself is given by Eq. (2.36):

J =µ(n,F)n(x)[

eF(x)− dEF

dndndx

]; (6.1a)

dFdx

= en(x)ϵ0ϵr

. (6.1b)

The dependencies of the mobility µ on the carrier density n and electric field F are de-scribed by the extended Gaussian disorder model (EGDM).15 The carrier densities at theboundaries of the device are obtained by assuming thermal equilibrium between theseboundaries and the electrodes, taking into account the potential change due to the field atthese boundaries over a distance a [see also Eq. (2.37)]. It is not necessary to take intoaccount the interaction of individual charges with their own images, because of the fairly

6.3 Improvements to the one-dimensional model 113

low injection barrier.16 Note that we do not use here the new 1D-DD method described inchapter 5; for a description of how the results of this chapter can be implemented in thatmethod, see appendix A. The 1D-DD approach is accurate for the basic case of the devicewith no internal interface (∆ = 0), as may be seen in the inset of figure 6.2 (dash-dottedcurve).

When modeling the J-V characteristics for the system with an organic-organic inter-face, a boundary condition should be imposed to connect the carrier density just to the rightof the interface at x = xI +a/2, nR, to the carrier density just to the left of it at x = xI −a/2,nL. As an initial ansatz we will assume thermal equilibrium at the interface, which isexact only for J = 0:

EF,R(nR)= EF,L(nL)+ eaF(xI), (6.2)

where F(xI) is the electric field at the interface. The dependence of the Fermi energiesEF,R(nR) and EF,L(nL) on the densities nR and nL is obtained through the Gauss-Fermiintegral Eq. (2.15), in which the energy barrier ∆ should be accounted for. The dash-dottedcurve in figure 6.2 shows the J-V curve obtained by using this assumption. Clearly, thecurrent density predicted by this 1D-DD approach is too high compared to the 3D-MCbenchmark results.

6.3 Improvements to the one-dimensional modelIn this section we will discuss three physical effects at the interface, which when taken intoaccount allow the 1D-DD model discussed above to accurately reflect the results of 3D-MCmodeling. These three effects are:

(1) Deviation from equilibrium: the assumption of thermal equilibrium made in Eq. (6.2)is not correct; we will show how we can take the effect of the actual non-equilibriumsituation into account.

(2) Non-continuous charge distribution: in the 1D-DD model the carrier density profilen(x) is continuous, while in the model device all carriers are located in two-dimensionallayers of sites. The effect of this discreteness is especially important in the last layer ofsites before the interface, where a large charge density build-up occurs.

(3) Short-range Coulomb interactions: the effective field experienced by a charge attempt-ing to cross the interface is lower than one would expect based purely on consideringthe space charge, since the charge does not interact with itself and other charges willbe pushed away by the Coulomb repulsion.12

The results of including these effects are shown in figure 6.2. Including effect (1) leads tothe dotted curve, while including effects (1) and (2) leads to the dashed curve. We notethat this result coincides with the 3D-MC simulations without short-range Coulomb in-teractions (circles). Finally, including all three effects (solid curve) leads to an accuratedescription of the 3D-MC results in which all effects of Coulomb interactions are included.We note that the one-dimensional master-equation method17,18 includes effects (1) and (2)

114 Charge transport across disordered organic heterojunctions

in a natural way, since the discreteness of the 3D-MC model is included in this approach.Without any further corrections it provides a fairly good fit (up to about a factor 2) of the3D-MC results without short-range Coulomb interactions (results not shown).

In the remainder of this section, the methods used to include effects (1)-(3) in the 1D-DDapproach will be discussed in detail.

6.3.1 Deviation from equilibriumTo obtain a more realistic boundary condition than Eq. (6.2), we must describe the chargetransport over the interface in a manner similar to the bulk by defining a field, a carrierdensity, and a carrier mobility at the interface. We first note that Eq. (6.1a) can be rewrittenas

J = eµ(n,F)n(x)[F(x)− 1

edEF

dx

]. (6.3)

The bracketed term can be considered as an effective field, consisting of the actual electricfield and the gradient in the electrochemical potential. This leads us to the considerationof an effective field across the interface

Feff = F(xI)+ (EF,L(nL)−EF,R(nR))/ea. (6.4)

Regarding the carrier density and carrier mobility we propose taking the geometric averageof these quantities at both sides of the interface, leading to the following form of the drift-diffusion equation at the interface:

J = e√

µ(nL,Feff)µ(nR,Feff)nLnR Feff. (6.5)

Like in the case of Eq. (6.2), the energy barrier ∆ comes into play through the Gauss-Fermi integral Eq. (2.15), which provides the dependence of the Fermi energies EF,L(nL)and EF,R(nR) on nL and nR. We show the results of applying the 1D-DD approach usingthis boundary condition to our model device in figure 6.2 (dotted curve). We see that thereis some improvement, but that the agreement is still far from satisfactory.

6.3.2 Non-continuous carrier distributionIn the top half of figure 6.3 we plot the carrier density distribution in the model deviceat an applied voltage of 3 V, obtained with 3D-MC modeling (symbols) and the 1D-DDapproach with the improved boundary condition discussed above (dotted curve). An obviousdifference is that in the 1D-DD approach the distribution n(x) is a continuous function ofthe position x, whereas in the 3D-MC model charges can only be present at the discretepositions of the layers of sites of the cubic lattice. This affects the field profile F(x), shown inthe bottom half of figure 6.3: in the 1D-DD approach the field increases continuously, whilein the 3D-MC model it increases discontinuously at the layers of the lattice. This differenceis not important in the bulk of the organic layers, because the gradients in n(x) and F(x) aresmall far from the interface, but it is very significant at the layer just before the interface,where a large build-up of carriers occurs. We note that in a realistic disordered organic

6.3 Improvements to the one-dimensional model 115

1021

1022

1023

1024

1025

1026

0 2 4 6 8 10 12 14 16 18 20

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

Car

rier d

ensi

ty (m

-3)

Ele

ctric

fiel

d (V

/nm

)

x (nm)

xI

Figure 6.3: Charge-carrier density (top) and electric field (bottom) as a function of position in thedevice shown in figure 6.1, with ∆= 0.6 eV and an applied voltage of 3 V. Black squares:3D-MC results. Gray circles: 3D-MC results without short-range Coulomb interac-tions. Dotted curve: 1D-DD results with effect (1) (deviation from equilibrium) included.Dashed curve: 1D-DD results with effects (1) and (2) (non-continuous charge distri-bution) included. Solid curve: 1D-DD results with effects (1), (2) and (3) (short-rangeCoulomb interactions) included. The vertical line indicates the position of the interfacexI.

116 Charge transport across disordered organic heterojunctions

semiconductor the sites are not ordered on a lattice, but for a sharp enough interface non-continuum effects at least qualitatively similar to those described here are nonethelessexpected to occur.

To account for this effect in the 1D-DD approach, we eliminate the unphysical contribu-tion to the space charge of the carrier distribution within a distance a/2 of the last layer ofsites before the interface. Instead, we add a sheet charge with surface charge density eanLat this layer, which causes a jump in the field equal to eanL/ϵ0ϵr. The field profile result-ing from the 1D-DD approach with this improvement is shown by the dashed curve in thebottom half of figure 6.3. As described, the field stops increasing at a distance a/2 beforethe last layer of sites of the left semiconductor, since we do not consider the contributionto the space charge there, and then jumps at this layer of sites. This field profile matchesthe 3D-MC results without short-range Coulomb interactions (circles) quite well, and thesame holds for the corresponding density profile. The description of the corresponding J-Vcharacteristics is also excellent (dashed curve and circles in figure 6.2).

6.3.3 Short-range Coulomb interactionsThe 3D-MC results displayed in figure 6.1 show that short-range Coulomb interactions playan important role. One might surmise that due to the large carrier density at the interfacethe charge-carrier mobility is affected by the short-range interactions, and that this shouldbe accounted for in order to obtain an accurate J-V curve. However, we found that, due tothe large competition between the drift and diffusion components of the current near theinterface, the precise value of the mobility close to the interface is not so relevant for thefinal current density.

Instead, we find that short-range Coulomb interactions are relevant as a result of theeffect of the surface charge just before the interface on the field across the interface. Thiseffect was first noted by Houili et al.12 Up to now we have considered the surface charge atthe layer of sites before the interface as laterally homogeneous, which gives an inaccuratedescription of hops to and from this layer for two reasons: i) the homogeneous surfacecharge incorrectly contains a contribution due to the very charge that is hopping (‘self-interaction’), ii) the homogeneous surface charge incorrectly neglects the fact that chargesaround the one that is hopping will have been pushed away (‘Coulomb hole’). As a result,the electric field felt by a carrier hopping over the interface is strongly reduced.

To quantify this reduction of the field, we performed equilibrium Monte Carlo simula-tions on a collection of holes in a two-dimensional (2D) square lattice, which represents thelayer of sites just before the interface. In such a simulation we place a number of holes inthe lattice and randomly select hops from an occupied site to an unoccupied site. If the hopreduces the total energy of the system (taking into account Coulomb interactions betweenthe particles and the energetic disorder), we always allow it to take place; if it increases thetotal energy by ∆E, we allow it to take place with probability exp(−∆E/kBT) and select anew hop otherwise. After equilibration we can then determine the potential difference dueto the Coulomb interactions for a charge attempting to hop out of this layer and comparethis to the case of a homogeneous surface charge. This procedure accounts for the effects ofboth the self-interaction and the Coulomb hole.

6.3 Improvements to the one-dimensional model 117

By varying the carrier density nL in this layer, the disorder strength σ, and the char-acteristic energy scale of the Coulomb interactions U ≡ e2/4πaϵ0ϵr, we have found that thereduction is well described by multiplying the homogeneous surface charge in this layer bya factor 1−C, with C given by

C = 0.462+0.538exp(−5.5nL/Nt)[1−exp(−0.4{U /kBT}0.65)]. (6.6)

Fractions C/2 of this surface charge are placed in the two adjacent layers of sites to guaran-tee charge conservation. The first term in Eq. (6.6) corrects for the self-interaction and thesecond for the Coulomb hole. We found that this formula applies for U /σ& 2 (for our modeldevice we have U /σ = 3.1). This formula does not depend on the form of the hopping rateas it does not enter the 2D simulation, and it does not depend on the energetic disorder aslong as it is not too strong. However, it does depend on the lattice and would have to berecalculated for non-simple-cubic lattices. In figure 6.4 we show the behavior of Eq. (6.6) asa function of nL and U/kBT (curves) and compare it to the Monte Carlo results (symbols).Note that for two cases no Coulomb hole is formed and so C = 0.462, the self-interactionvalue: for nL = nT, when all sites are occupied, and for U /kBT → 0, when there are noCoulomb interactions.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.4

0.5

0.6

0.7

0.8

0.9

1.0

Sur

face

cha

rge

redu

ctio

n fa

ctor

C

Carrier density left of the interface nL [Nt]

MC simulation results with U = 0.28 eV (U/kBT = 11)

U = 0.08 eV (U/kBT = 3) U = 0.03 eV (U/kBT = 1) Results of Eq. (6.6)

= 0.09 eVT = 295 K

Figure 6.4: Factor C by which the surface charge due to the last layer of charge carriers beforethe interface is reduced for a carrier hopping across the interface, as a function of thecarrier density in this layer nL. Results are shown for different values of the Coulombinteraction energy U ; U = 0.28 eV corresponds to the parameters used for the otherfigures in this section. Symbols: results of the two-dimensional Monte Carlo methoddescribed in the text. Curves: results of Eq. (6.6).

In the 1D-DD approach we apply this procedure only to the last layer of sites beforethe interface, for which the correction is the most important. This means that the sheetcharge with surface charge density eanL obtained in the previous subsection is multiplied

118 Charge transport across disordered organic heterojunctions

by a factor 1−C, while fractions C/2 of this sheet charge are placed in the adjacent layers.The resulting field and density profiles (solid curves in figure 6.3) show excellent agree-ment with the 3D-MC results with all Coulomb interactions included (squares). The sameapplies to the J-V characteristics; see figure 6.2.

6.4 Conclusions and discussionIn this chapter the following drift-diffusion approach was developed and shown to matchMonte-Carlo results. Charge transport within the bulk of the organic layers is describedas usual by the drift-diffusion equation, Eq. (6.1). Three effects must be taken into accountat the interface. (1) The boundary condition at the interface, linking the carrier densitiesnL and nR at the left and right side of the barrier, is given by Eq. (6.5). This boundarycondition accounts for the deviation from equilibrium across the interface due the presenceof a current. (2) The unphysical space charge within a distance a/2 of the last layer ofsites before the barrier, where the carrier density is extremely high, is replaced by a sheetcharge at the location of this layer. This gives rise to a field jump at this layer. (3) Thissheet charge is multiplied by a factor 1−C, where C is given by Eq. (6.6), accounting for theself-interaction of the hopping charge and for the Coulomb hole of reduced charge densityaround this charge, while fractions C/2 of this sheet charge are moved to the adjacentlayers of sites. The combination of (2) and (3) leads to three field jumps: one at the positionof the last layer of sites before the barrier, with a size (1−C)eanL/ϵ0ϵr, and two at distances±a from this layer, with a size CeanL/2ϵ0ϵr.

Although the study in this chapter has been performed for a specific model device withMiller-Abrahams hopping, a simple-cubic lattice and uncorrelated Gaussian energetic dis-order, we expect that the conclusions reached and the approach followed have a generalvalidity for charge transport across organic heterojunctions. Our method should apply toMarcus hopping with no modifications, apart from using the appropriate mobility func-tion.* Eq. (6.6) in particular does not depend on the type of hopping. A different latticeshape also is not expected to affect the method, although a weak dependence of Eq. (6.6) onthe lattice shape is to be expected. Correlation of the energy disorder, however, is expectedto have a large effect. An important distinction here is whether the correlations extendacross the interface; this may affect the boundary condition Eq. (6.5).

With vacuum deposition of organic molecules it is nowadays possible to define layerthicknesses with nanometer precision, i.e. at the scale of the size of a molecule.20 However,interface roughness at this scale will still occur, while the methods described above apply toa sharp interface. In order to investigate the effects of interface roughness at a scale of onelayer of sites we consider a checkered interface, where in the layer before the interface sitescorresponding to both organic semiconductors are arranged in a checkerboard pattern. TheJ-V characteristics resulting from 3D-MC simulations of the modified device are shownby the black triangles in figure 6.2. Although the current is somewhat affected by theinterface roughness, it is clear that the three other effects considered above are much more

*A detailed discussion of the influence of the type of hopping on charge transport across organic heterojunctionscan be found in Ref. 19.

6.4 Conclusions and discussion 119

significant. Hence, it is useful to consider the effects discussed in the present work andto apply the developed techniques to obtain an improved modeling by the drift-diffusionapproach also to cases for which interface roughness is present.

We note that a special role is played by the distance a in the approach. In our study it isthe lattice constant of the simple-cubic lattice we have considered in our Monte-Carlo simu-lations. The finite value of a expresses the positional discreteness of the system of hoppingsites. It is this discreteness and not the specific lattice that is the essential complication incontinuum drift-diffusion approaches. We therefore expect that our approach can also beapplied to realistic organic semiconductors, having a spatially more random distribution ofhopping sites. We propose that in that case a should be treated as an effective parameter,which is expected to be of the order of the typical distance between the hopping sites.

Finally, we remark that when modeling OLEDs it is important to consider not onlycharge transport across organic heterojunctions, but also the exciton generation there. Thisinterface exciton generation may behave very differently from that in the bulk.21 For acomplete description of interfaces in a 1D-DD approach, a similar quantitative analysis asperformed here for charge transport should be performed for exciton generation.

References1. J. Cottaar, R. Coehoorn, and P. A. Bobbert, Organic Electronics 13, 667 (2012).2. C. W. Tang and S. A. Van Slyke, Appl. Phys. Lett. 51, 913 (1987).3. S. A. Van Slyke, C. H. Chen, and C. W. Tang, Appl. Phys. Lett. 69, (1996).4. G. He, M. Pfeiffer, K. Leo, M. Hofmann, J. Birnstock, R. Pudzich, and J. Salbeck,

Appl. Phys. Lett. 85, 3911 (2004).5. C. H. Chang, C. C. Chen, C. C. Wu, S. Y. Chang, J. Y. Hung, and Y. Chi, Org. Electron.

11, 266 (2010).6. J. Staudigel, M. Stossel, F. Steuber, and J. Simmerer, J. Appl. Phys. 86, 3895 (1999).7. V. I. Arkhipov, E. V. Emelianova, and H. Bässler, J. Appl. Phys. 90, 2352 (2001).8. E. Tutiš, M. N. Bussac, B. Masenelli, M. Carrard, and L. Zuppiroli, J. Appl. Phys.

89, 430 (2001).9. T. van Woudenbergh, J. Wildeman, and P. W. M. Blom, Phys. Rev. B 71, 205216

(2005).10. H. Houili, E. Tutis, and L. Zuppiroli, Synth. Met. 156, 1256 (2006).11. S. W. Tsang, Y. Tao, and Z. H. Lu, J. Appl. Phys. 109, 023711 (2011).12. H. Houili, E. Tutiš, I. Batistic, and L. Zuppiroli, J. Appl. Phys. 100, 033702 (2006).13. J. J. M. van der Holst, F. W. A. van Oost, R. Coehoorn, and P. A. Bobbert, Phys. Rev.

B 83, 085206 (2011).14. S. L. M. van Mensfoort and R. Coehoorn, Phys. Rev. B 78, 085207 (2008).15. W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M. Blom, D. M.

de Leeuw, and M. A. J. Michels, Phys. Rev. Lett. 94, 206601 (2005).

120 Charge transport across disordered organic heterojunctions

16. J. J. M. van der Holst, M. A. Uijttewaal, B. Ramachandhran, R. Coehoorn, P. A.Bobbert, G. A. de Wijs, and R. A. de Groot, Phys. Rev. B 79, 085203 (2009).

17. R. Coehoorn and S. L. M. van Mensfoort, Phys. Rev. B 80, 085302 (2009).18. M. Schober, M. Anderson, M. Thomschke, J. Widmer, M. Furno, R. Scholz, B.

Lüssem, and K. Leo, Phys. Rev. B 84, 165326 (2011).19. I. Juric, I. Batistic, and E. Tutiš, Phys. Rev. B 77, 165304 (2008).20. S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo,

Nature 459, 234 (2009).21. N. C. Greenham and P. A. Bobbert, Phys. Rev. B 68, 245301 (2003).

References 121

Chapter 7

Conclusions and outlook

Abstract

In this thesis, we have analyzed charge transport in organic semiconductors. A scaling the-ory for percolative charge transport in organic semiconductors was developed, leading toan expression describing the dependence of the zero-field charge-carrier mobility on tem-perature and carrier concentration. We analyzed field-induced-detrapping in host-guestsystems. This was generalized to show that charge transport at finite field is always de-scribed by an ‘intrinsic’ and a ‘detrapping’ factor. Finally, we analyzed charge transportacross organic-organic interfaces. All results have been translated to a one-dimensionaldrift-diffusion (1D-DD) setting.

In the short term, this 1D-DD method can be used for the characterization of organicsemiconductors. Indeed, this approach (without the results in this thesis) has alreadybeen successfully used in several materials to determine parameters such as the hoppingsite density for a given choice of hopping model. By considering the time or frequencydependence of the mobility it may also be possible to determine which hopping model isappropriate. In the middle to long term, the 1D-DD method will be a valuable tool in OLEDdevice simulation. The three-dimensional Monte Carlo (3D-MC) method has also provento be useful in this kind of simulation. These methods are especially powerful when usedto complement each other. Double-carrier charge transport and excitonics will have to beanalyzed further to complete the 1D-DD method. In the long term, our most valuable resultis most likely the insight we have gained into the physics underlying charge transport inorganic semiconductors. This will allow us to truly understand OLED operation and comeup with new concepts and designs. Some of our results, such as the scaling theory, mayalso be applicable to other fields of study.

123

Contents

7.1 Main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.2 Outlook on applications and further research . . . . . . . . . . . . 126

7.2.1 Characterization of organic semiconductors . . . . . . . . . 1267.2.2 Predictive simulation of OLEDs . . . . . . . . . . . . . . . . 1277.2.3 Understanding charge transport in organic semiconductors128

7.3 Coda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

124 Conclusions and outlook

Our goal in this work was the development of a predictive charge-transport model forOLEDs, and in particular the transfer of this model to a one-dimensional drift-diffusion(1D-DD) setting. These 1D-DD results should match those of three-dimensional (3D) sim-ulation of the hopping model. To achieve this, we have analyzed and derived expressionsfor the charge-carrier mobility (chapters 3, 4 and 5), and examined the charge transportacross internal interfaces (chapter 6). A unified 1D-DD method combining all our charge-transport results can be found in appendix A. An overview of the progress made and chal-lenges remaining in the transfer of 3D results to the 1D-DD setting can be found in table7.1.

In this chapter, we will first summarize the main conclusions of each chapter (section7.1). Next, we give an outlook on possible applications of our results and of the 1D-DDmethod in general, including recommendations for further research (section 7.2). We con-clude with a short coda (section 7.3).

7.1 Main conclusionsIn chapter 3, we developed a general scaling theory for percolative charge transport in or-ganic semiconductors, leading to an expression describing the dependence of the zero-fieldcharge-carrier mobility on temperature and carrier concentration. The theory is valid ina scaling regime corresponding to a wide temperature range that includes most relevantcases. We demonstrated how it can be applied to uncorrelated and dipole-correlated ener-getic disorder, Miller-Abrahams and Marcus hopping, and simple-cubic and face-centered-cubic lattices. For uncorrelated disorder and Miller-Abrahams hopping, we showed howone can even obtain the dependence of the mobility on the shape of the density of states.

In chapter 4, we generalized the Hoesterey-Letson model for describing the mobilityin host-guest systems in the limit of zero electric field to the case of finite electric fields. Inthis model, the mobility may be viewed as being due to the fraction of ‘free’ charge carriers,i.e. carriers that are detrapped from the guest sites and reside on the host sites. The freecharge-carrier density is calculated using the occupation function, a field-dependent gener-alization of the Fermi-Dirac function. We provided a parameterization of this function forMiller-Abrahams hopping on a simple-cubic lattice with uncorrelated Gaussian energeticdisorder. We showed that these effects are relevant for fields at which also the intrinsicfield dependence of the mobility in the host becomes relevant.

In chapter 5, we analyzed charge transport at non-zero electric field. We showed thatthe mobility factorizes into an ‘intrinsic’ factor and a ‘detrapping’ factor. This factorizationis not just for convenience; the intrinsic and detrapping factors represent separate physicaleffects, and affect charge transport in devices in different ways. This means that the valueof the mobility does not by itself fully describe charge transport at non-zero field. Weintroduced a new form of the one-dimensional drift-diffusion method that properly accountsfor both factors, and used this new method to perform the first quantitative comparisonbetween one-dimensional and three-dimensional results for a device with Marcus hopping.

In chapter 6, we showed how charge transport across organic-organic interfaces canbe described in a one-dimensional drift-diffusion approach in a way that is consistent with

7.1 Main conclusions 125

Effect How can we simu-late it in 3D?

Have the 3D results been quantitativelydescribed for use in a 1D method?

Charge transport atzero electric field

3D master equa-tion (3D-ME) or3D Monte Carlo(3D-MC)

Yes, with the scaling theory in chapter3 and the resulting expression for thecharge-carrier mobility Eq. (3.16).

Charge transport atnon-zero electric field

3D-ME or 3D-MC Only for Miller-Abrahams hopping ona simple-cubic lattice with uncorrelatedGaussian or dipole-correlated disorder.1,2

For the case of uncorrelated Gaussian dis-order, host-guest systems can also be de-scribed using the results of chapter 4. Forother hopping models, the results of chap-ter 5 may provide a starting point.

Charge transportacross interfaces

3D-MC Yes, in chapter 6, by taking into accountthe effects of Coulomb interactions and ofnon-equilibrium charge transport acrossthe interface.

Time or frequencydependent chargetransport

3D-ME or 3D-MC The multiple-trapping approach sug-gested by Germs et al.3 may be applicablebut needs further verification.

Exciton generation 3D-MC Yes, Langevin’s expression4 correctly de-scribes the exciton generation rate if theeffect on the mobility of Coulomb interac-tions between individual holes and elec-trons is taken into account.5 This mobil-ity effect itself, however, has not yet beenquantitatively described. Exciton gener-ation at interfaces must also be analyzedseparately.6

Table 7.1: Charge-transport and exciton-generation effects relevant in OLEDs, how they can be sim-ulated in 3D, and whether the results of such simulation have been quantitatively de-scribed for use in 1D modeling.

126 Conclusions and outlook

three-dimensional simulation. We showed that there are three important physical effectsto take into account, and described how to do so quantitatively. The three effects are non-equilibrium charge transport across the interface, the formation of a surface charge layerjust before the interface, and the reduction of the effect of this surface charge due to short-range Coulomb interactions.

7.2 Outlook on applications and further researchIn this section, we will discuss current and future applications of our results and the 1D-DD method in general. We will also indicate recommended directions for future research.

In the short term, our results will be especially useful in the characterization of or-ganic semiconductors. Indeed, the 1D-DD method has already been successfully appliedseveral times in this context (section 7.2.1). In the middle to long term, the 1D-DD methodwill be an important tool in predictive OLED simulation. The three-dimensional MonteCarlo (3D-MC) method has also proven its value for such simulation. The combination ofboth methods is especially powerful (section 7.2.2). In the long term, the most enduringresults of this work may well be the deeper understanding we have gained of the physicsunderlying charge transport in organic semiconductors (section 7.2.3).

7.2.1 Characterization of organic semiconductorsThe 1D-DD method has already been applied extensively in the characterization of or-ganic semiconductors. Typically, such a characterization assumes a certain hopping modeland matching parameterization of the mobility, usually the extended Gaussian disordermodel (EGDM)1 for polymers and the extended correlated disorder model (ECDM)2 forsmall molecules. The goal of the characterization procedure is then to determine the pa-rameters in this model, usually the width of the density of states σ, hopping site densityNt and hopping rate prefactor ω0. For electron transport, this list also includes the trapdepth and concentration.

To perform such a characterization, a single-layer space-charge-limited device using thematerial in question is fabricated and the current-voltage characteristics are measured ina range of temperatures. Next, the device is simulated, trying out various values of theparameters and choosing those for which the simulated characteristics best match the ex-perimental ones. Since the parameter space is rather large, an efficient method to convergeto the optimal solution is needed; an example of such a method is discussed in Ref. 7. Still,several simulation runs for different parameter values must be performed. These runscannot be performed in parallel, since the parameter choices depend on the results of theprevious one. Consequently, 3D simulation is too slow and the 1D-DD method must beused. This characterization method has already been applied successfully to a wide rangeof materials,8–10 using the EGDM and ECDM parameterizations of the mobility. The field-induced-detrapping results of chapter 4 may help to more accurately characterize electrontransport, in which traps usually play a role.

In the future, one could use this approach to determine which hopping model (type of

7.2 Outlook on applications and further research 127

hopping, type of lattice, presence or absence of correlated disorder, etc.) applies to a givenmaterial, instead of just determining parameters based on an assumed hopping model. Weshowed in section 3.4 that the temperature and concentration dependence of the mobility isvery similar for different hopping models, which means that these dependences cannot beused to discriminate between hopping models. The field dependence may be more useful,but still has to be determined and parameterized for Marcus hopping and for non-simple-cubic lattices (see also table 7.1); the results of chapter 5 may be a useful starting pointfor this. However, full knowledge of the temperature, concentration and field dependenceof the mobility will not be enough; De Vries et al. already showed that the current-voltagecharacteristics of a polyfluorene-based copolymer device cannot on their own determinewhether the EGDM or the ECDM is appropriate.8 A next step would be to also consider thetime or frequency dependence, using the response of the current in a device to a suddenchange in voltage (dark-injection transients) and to a small AC voltage applied over abackground DC voltage (small-signal analysis). The results of such experiments can bequite sensitive to the charge-transport properties of the organic semiconductor.3 We havedescribed how to simulate these experiments using the 3D master-equation approach insection 2.6.

7.2.2 Predictive simulation of OLEDsThe focus of this work has been translating 3D simulation results into a fast 1D-DD methodfor describing charge transport in organic semiconductors. This 1D-DD method is envi-sioned to form a part of a full predictive OLED model, to be used in the OLED designprocess (see section 1.3). The main reason to use the 1D-DD method instead of full 3Dsimulation is that it is much faster. In fact, when this work started it was still unknownwhether 3D simulation of a full OLED device structure was computationally feasible at all.The two main commercial packages for OLED simulation, SimOLED11 and SETFOS12, arebased on the 1D-DD approach.

In the past few years, it has become clear that 3D simulation of realistic OLED devicestructures using the Monte Carlo (3D-MC) approach is feasible. Although this approachis much slower than the 1D-DD one, it has the advantage that the hopping model canbe simulated with no approximations. However, we must remain aware that the hoppingmodels themselves contains approximations and assumptions, such as Miller-Abrahamshopping, a simple-cubic lattice and a perfectly Gaussian density of states (when usingthe GDM or CDM). Typically these assumptions do not affect bulk single-carrier chargetransport, since even if the hopping model does not reflect the underlying physics it is stillpossible to choose the parameters in such a way that this charge transport is accuratelydescribed. But this is no guarantee that other effects such as charge transport acrossinterfaces and exciton generation and diffusion are also correctly simulated.

In the coming years, the 3D-MC and 1D-DD approaches will become important com-plementary techniques. The 3D-MC approach can be used for initial analysis and as abenchmark for the 1D-DD method. The 1D-DD method on the other hand can quicklyanalyze several different devices, which is important in an iterative design process. An-other advantage of using the techniques side by side is that new physical effects can be

128 Conclusions and outlook

identified. If 1D-DD results do not match 3D-MC ones, a thorough analysis can identifyand describe the underlying physical cause. This can then be used to improve the 1D-DDmethod, ultimately leading to deeper insight into OLED device physics. A clear example ofthis synergy between the two methods was seen in chapter 6 of this thesis, where the dif-ferences between 3D-MC and 1D-DD results for charge transport across interfaces pointedus to several physical effects playing a role there.

Some aspects of the 1D-DD method still need to be developed further for it to giveaccurate results for full OLED device structures. The most important of these are double-carrier physics (including exciton generation; see also table 7.1) and exciton physics. Re-garding double-carrier physics, it was shown by Van der Holst et al. that the hole mobilityis affected when electrons are present due to Coulomb interactions, and vice versa.5 Thiswork also showed that if this effect is taken into account exciton generation in the bulk iswell described by Langevin’s expression.4 However, exciton generation at interfaces mayfollow different rules and must be analyzed separately.6

Exciton physics, which has recently been implemented in the 3D-MC code, must also beconsidered in the 1D-DD method. This involves the diffusion, dissociation and recombina-tion of excitons, and also their interaction with charge carriers by quenching and with eachother by triplet-triplet annihilation. Rates for all these processes should be determinedfrom 3D simulation of systems with periodic boundary conditions, just like was done forthe charge-carrier mobility and exciton generation rate. A challenge here will be that theshort lifetime of excitons means that their energy distribution is strongly out of equilib-rium. An advantage is that they are uncharged, which means that their transport is notaffected by charge carriers or other excitons. On the other hand, this makes it very diffi-cult to experimentally probe exciton behavior, complicating experimental validation of thesimulation results. As in all other cases we also have to carefully analyze what happens atinterfaces.

A 1D-DD method containing both double-carrier and exciton physics solves for fourvariables as a function of position in the device: the electric field, and the hole, electron,and exciton densities. It may also be necessary to consider singlet and triplet excitonsseparately. Alternatively, we could solve for the average electrochemical potential energiesinstead of the densities if the notation of chapter 5 is used. The four differential equationsgoverning these quantities are the space-charge equation and the hole, electron, and ex-citon balance equations. Knapp et al. introduced a solution method for the double-carriercase, which we also expect to be applicable when excitons are included.13 Their approach isbased on Newton’s method for non-linear equations, which was used in this thesis to solvethe 3D master equation.

7.2.3 Understanding charge transport in organic semiconductorsThe choice of hopping model has played an important role in this work. We have consideredmainly the choice of hopping rate, lattice and presence or absence of correlation in theGaussian energy disorder. This is already a simplification: the real hopping system isspatially disordered and the density of states is never exactly Gaussian. To conclusivelydetermine the appropriate hopping model, we would have to determine the morphology

7.3 Coda 129

using molecular dynamics or Monte Carlo techniques, and the hopping rates using densityfunctional theory. However, this approach is fraught with challenges and uncertainties of itown, such as the high computational difficulty and the resulting need for approximations.

Since the appropriate hopping model is so hard to determine, the most important re-sults of this work are those that are independent of the choice of hopping model and aregenerally applicable, such as the scaling theory for the charge-carrier mobility and theresulting universal dependence on carrier concentration in chapter 3, the occupation func-tion concept in chapter 4, the factorization of charge transport at finite electric field intoan intrinsic and a detrapping factor in chapter 5, and the presence of a surface charge atinternal interfaces in chapter 6. We showed how these results help us to quantitatively de-scribe charge transport in organic semiconductors for a given choice of hopping model. Butmore importantly, they give us deeper insight into the physics behind this charge transport,independent of the choice of hopping model.

This deeper insight into the underlying physics provides more than just an academicbenefit. Simulating devices is important, but only with true understanding can we comeup with new concepts and push the envelope on OLED design. This is also the most impor-tant reason for the continuation of this work, for example in the direction of excitonics asdiscussed in the previous section.

Some of our results may also be applicable to other fields of study. The scaling theoryin chapter 4 in particular is envisaged to have many other uses. Many percolation prob-lems are based on networks of bonds with a large spread in bond strength. We use ‘bondstrength’ here as a very general term; it may refer to electrical conductivity, mechanicalstiffness, pore permeability, etc. In any such system, we expect the overall properties todepend on the strength of some critical bond and on the number of bonds with similarstrength. This can then be quantitatively described using our scaling theory.

7.3 CodaThe results of this work make it possible to accurately model single-carrier charge trans-port in OLEDs using a one-dimensional approach. With further research all electronic pro-cesses, including double-carrier transport and excitonics, can be implemented. This willallow for fast one-dimensional simulation of full OLED device structures, with the resultsmatching those of much slower three-dimensional simulation. Such one-dimensional simu-lation will be an invaluable tool in optimizing the design of OLEDs. With this tool in hand,challenges currently facing OLED light sources, such as increasing the luminous efficacyand the device lifetime, and reducing the manufacturing cost, can be tackled effectively,leading to a bright OLED future.

References1. W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M. Blom, D. M.

de Leeuw, and M. A. J. Michels, Phys. Rev. Lett. 94, 206601 (2005).

130 Conclusions and outlook

2. M. Bouhassoune, S. L. M. van Mensfoort, P. A. Bobbert, and R. Coehoorn, Org. Elec.10, 437 (2009).

3. W. C. Germs, J. J. M. van der Holst, S. L. M. van Mensfoort, P. A. Bobbert, and R.Coehoorn, Phys. Rev. B 84, 165210 (2011).

4. P. Langevin, Ann. Chim. Phys. 28, 433 (1903).5. J. J. M. van der Holst, F. W. A. van Oost, R. Coehoorn, and P. A. Bobbert, Phys. Rev.

B 80, 235202 (2009).6. N. C. Greenham and P. A. Bobbert, Phys. Rev. B 68, 245301 (2003).7. R. de Vries, Development of a charge transport model for white OLEDs, Ph.D. Thesis,

Eindhoven University of Technology (2012).8. R. J. de Vries, S. L. M. van Mensfoort, V. Shabro, S. I. E. Vulto, R. A. J. Janssen, and

R. Coehoorn, Appl. Phys. Lett. 94, 163307 (2009).9. S. L. M. van Mensfoort, V. Shabro, R. J. de Vries, R. A. J. Janssen, and R. Coehoorn,

J. Appl. Phys. 107, 113710 (2010).10. S. L. M. van Mensfoort, R. J. de Vries, V. Shabro, H. P. Loebl, R. A. J. Janssen, and

R. Coehoorn, Org. Elec. 11, 1408 (2010).11. sim4tec, SimOLED - OLED Simulation Software , URL: http://www.sim4tec.com/

?Products.12. FLUXiM, SETFOS: SEmiconducting Thin Film Optics Simulation software , URL:

http://www.fluxim.com/Home-OLED-and-Solar.9.0.html.13. E. Knapp, R. Häusermann, H. U. Schwarzenbach, and B. Ruhstaller, J. Appl. Phys.

108, 054504 (2010).

Appendix A

Unified 1D drift-diffusionmethod

In this appendix we will formulate the differential equations and boundary conditions fora one-dimensional drift-diffusion (1D-DD) method unifying all charge-transport effects de-scribed in this work. The purpose of this method is to compute current-voltage charac-teristics of any space-charge-limited single-carrier device, matching those found by three-dimensional simulation based on the hopping model described in section 1.4.1. These de-vices may contain internal interfaces and/or host-guest systems.

In the standard 1D-DD method, one solves for the carrier concentration c and electricfield F as a function of distance from the anode x. However, we found in chapter 5 thatit is advantageous to solve for the average electrochemical potential energy µ, which is afield-dependent generalization of the Fermi energy EF, instead of c. The following form ofthe differential equations was derived [Eq. (5.13)]:

J = eNth(T,F)exp[µ(x)/kBT

][F(x)− 1

edµdx

], (A.1a)

dFdx

= eNtc(T, µ,F)ϵ0ϵr

, (A.1b)

where J is the current density, e the elementary charge, Nt the hopping site density, kBTthe thermal energy, ϵ0 the electrical permittivity of the vacuum and ϵr the relative dielectricconstant of the organic material. h is the field-dependence function, which depends on thechoice of hopping model, on F, and on T but not on µ.

To complete the method, we specify the boundary conditions [Eq. (5.11)]:

µ(a)=Φleft + eaF(a);µ(L−a)=Φright − eaF(L−a), (A.2)

where Φleft and Φright are the work functions of the electrodes and a = N1/3t is the nearest-

132 Unified 1D drift-diffusion method

neighbor distance for the simple-cubic lattice.*Eqs. (A.1) and (A.2) completely specify the 1D-DD method for a single-layer single-

carrier device. They can be solved using one of several methods (see section 2.5 and refer-ences therein). However, we do need to know the dependence of h on T and F, and of c onT, µ, and F. This depends on the specific hopping model: the type of hopping, the type oflattice, the shape of the density of states (DOS), the presence or absence of correlation inthe energy disorder, presence and nature of traps, etc.

In the remainder of this appendix, we first specify the dependences of h and c on µ andT explicitly for the low-field limit (section A.1). We then discuss how to describe chargetransport in host-guest systems at finite field, using only charge-transport properties ofthe host system (section A.2). Finally, we extend the 1D-DD method to multilayer devicesby considering boundary conditions and space-charge effects at internal interfaces (sectionA.3).

A.1 Zero electric fieldWe will now describe the dependence of the carrier concentration c on µ and T and thedependence of the field-dependence function h on T at zero electric field. Since at F = 0the average electrochemical potential energy µ is simply the Fermi energy EF, c can bestraightforwardly computed using the Fermi-Dirac distribution:

c(T, µ,0)=∫ ∞

−∞g(E)

1+exp([E− µ]/kBT)dE, (A.3)

where g(E) is the DOS. The dependence of h on T can be found using the scaling theorydescribed in chapter 3. We rewrite Eq. (3.16) in terms of h instead of the charge-carriermobility µ:

h(T,0)= Beω0

N2/3t W

(W

kBT

)1−λexp(−Ecrit/kBT) , (A.4)

where ω0 is the hopping rate prefactor [see Eq. (1.11)] and W is a measure for the widthof the DOS; for uncorrelated Gaussian or dipole-correlated disorder W = σ, the standarddeviation. The values of the prefactor B, critical exponent λ and critical energy Ecrit canbe found for Marcus or Miller-Abrahams hopping on a simple-cubic or face-centered-cubiclattice in tables 3.1 (uncorrelated Gaussian disorder) and 3.2 (dipole-correlated disorder).For the specific case of Miller-Abrahams hopping with uncorrelated disorder, we can evengive an expression for h which is valid for non-Gaussian densities of states [Eq. (3.21)]:

h(T,0)= Aeω0

N2/3t kBT

[2kBT g(Ecrit)

ppbond

]λ exp(−Ecrit/kBT) , (A.5)

where the prefactor A and bond percolation threshold pbond can be found in table 3.1. Thecritical energy Ecrit depends on the shape of the DOS and so should not be taken from this

*For a face-centered-cubic lattice, Eq. (A.2) must be modified to reflect the different distance between the electrodeand the first layer of organic sites.

A.2 Host-guest systems at non-zero electric field 133

table for non-Gaussian disorder; it can instead be found from Eq. (3.19):

ppbond =

∫ Ecrit

−∞g(E)dE. (A.6)

A.2 Host-guest systems at non-zero electric fieldFor the case of finite electric field, we have not found expressions similar to Eqs. (A.3) and(A.4) to calculate c and h. We have, however, developed in chapter 4 a general method todescribe charge transport in host-guest system with low guest concentration. This methodgives c(T, µ,F) and h(T,F) in the host-guest system, using only charge-transport propertiesof the host system. Specifically, we need to know c and h in the host system, written asch(T, µ,F) and hh(T,F). We also need to know the occupation function p(E;T, ch,F), whichgives the average probability that a site with energy E is occupied at given values of T, chand F. For F = 0 it is simply the Fermi-Dirac distribution function.

We showed that at low guest concentration charge transport takes place only amonghost sites. This means that h is not affected:

h(T,F)= hh(T,F). (A.7)

The carrier concentration is given by the sum of the carriers in the guest and those in thehost [Eq. (4.4)]:

c(T, µ,F)= xcg(T, µ,F)+ (1− x)ch(T, µ,F), (A.8)

where x is the guest concentration. The carrier concentration in the guest cg is given byEq. (4.5):

cg(T, µ,F)=∫ ∞

−∞gg(E)p(E;T, ch,F)dE, (A.9)

where gg is the guest DOS. For the case of the Gaussian disorder model, Eqs. (4.9)-(4.13)give an explicit parameterization of the occupation function p. For other hopping models,p must first be determined for the host system using one of the techniques discussed insection 4.3.1.

A.3 Internal interfacesConsider an internal interface at x = xI (with xI defined as the midpoint between the lay-ers of sites adjacent to the interface), with the HOMO/LUMO energy levels as the onlydifference between the two materials. We now need a boundary condition to describe therelation between µL = µ(xI−a/2) and µR = µ(xI+a/2). This boundary condition was derivedin chapter 6. We have modified Eq. (6.5) to match the notation used in this section:

J = eNth(T,Feff)exp([µL + µR]/2kBT)Feff, (A.10)

with Feff the effective field over the interface. This field is given by Eq. (6.4), which we havealso modified into:

Feff = F(xI)+ (µL − µR)/ea. (A.11)

134 Unified 1D drift-diffusion method

Typically, there will be a large build-up of charges before the interface. These charges forma surface charge, the effect of which is reduced by short-range Coulomb interactions. Thismust be taken into account as follows. The unphysical space charge within a distance a/2 ofthe last layer of sites before the interface is removed. Instead, we apply three field jumps:one at the position of the last layer of sites before the barrier, with a size (1−C)eacLNt/ϵ0ϵr,and two at distances a from this layer, with a size CeanLNt/2ϵ0ϵr. Here, cL is the carrierconcentration in the last layer of sites before the interface and C is (for a cubic lattice)given by Eq. (6.6):

C = 0.462+0.538exp(−5.5cL)[1−exp(−0.4{e2/4πaϵ0ϵrkBT}0.65)

]. (A.12)

With these adjustments to the field profile the 1D-DD results accurately match those ofthree-dimensional simulation of charge transport across an internal interface.

Summary

Modeling of charge-transport processes for predictive simulation of OLEDs

Intensive research is taking place into alternative light sources to replace incandescent andfluorescent lamps. Organic light-emitting diodes (OLEDs) show great promise, with theirmain potential advantages being high energy efficiency, cheap roll-to-roll production, excel-lent color rendering and a unique form factor. However, significant challenges must still beovercome, particularly in the areas of luminous efficacy, lifetime and manufacturing. Sev-eral approaches to overcoming these challenges have been proposed, but it is difficult todesign optimized devices around these approaches. This is because at present this designtakes place through trial and error, which makes investigating the full parameter spaceof material choices and layer stack design virtually impossible. To improve this designprocess, a predictive OLED model is needed.

A full predictive OLED model takes as input the layer stack design, deposition methodsand chemical structures of the materials involved, and gives as output the angle-dependentemission spectrum and current-voltage characteristics of the device. This involves molec-ular dynamics, density functional theory, charge-transport modeling, excitonics and pho-tonics. However, charge-transport modeling by itself already yields useful results, suchas current-voltage characteristics and exciton generation locations. In such modeling onethree-dimensionally (3D) simulates the charge transport in organic semiconductors, whichtakes place by hopping of charge carriers between localized sites. Since this 3D simula-tion is computationally expensive, the results must be translated to a fast one-dimensionaldrift-diffusion (1D-DD) approach to be of use in the OLED design process. In this transla-tion, 3D simulation is used in two ways: to determine parameters like the charge-carriermobility, and to validate the results of the 1D approach. Charge-transport modeling andthe 3D-to-1D translation are the focus of this work.

In this thesis, two 3D simulation methods are used, based on the master equation (3D-ME) and on Monte Carlo simulation (3D-MC). In the 3D-ME method, we determine foreach site the probability that there is a charge carrier on this site. We use this methodto determine the charge-carrier mobility in bulk systems. Advantages of this method areits speed and direct insight into the spatial distribution of charge carriers. An importantdisadvantage is that Coulomb interactions between individual charge carriers cannot betaken into account. In the 3D-MC method, we evaluate the full hopping model throughexplicit Monte Carlo simulation, including all Coulomb interactions. We use it to model

136 Summary

charge transport in multilayer structures relevant for modern OLEDs.A first result is a new scaling theory which describes the dependence of the charge-

carrier mobility on temperature and carrier density at zero electric field (chapter 3). Thisscaling theory is based on percolation theory, in which one critical bond determines thecharge-transport properties of the entire system. We expand on percolation by consideringnot just this single critical bond but also the distribution of almost equally difficult bondsin the system. This leads to an accurate, closed-form expression for the mobility, contain-ing three parameters which depend on the specific system considered (such as the type oflattice, the expression for the charge carrier hopping rate between two sites and the shapeof the energy disorder). This theory makes it possible for the first time to analyze the effectof assumptions like the lattice and hopping rate.

The second result is a description of the electric-field dependence of the mobility inhost-guest systems (chapter 4). These are systems in which a small amount of sites act ascharge-carrier traps, such as dopants, dyes or naturally occurring electron traps. At lowguest concentration charge transport takes place only through the host. The effect of theguest sites is then purely to immobilize a number of carriers. At low electric field, thisnumber can be determined from equilibrium Fermi-Dirac statistics. At finite fields thisno longer applies because of field-induced detrapping: the field assists carriers in escapingthe guest sites. We quantify this effect by generalizing the Fermi-Dirac distribution toa numerically determined occupation function. This allows an accurate prediction of themobility, leading to improved simulation of OLEDs containing host-guest systems.

The third result is a general description of charge transport at non-zero electric field(chapter 5). This result combines elements of the scaling and field-induced-detrappingtheories described above. We show that the mobility factorizes into an ‘intrinsic’ factor anda ‘detrapping’ factor. These are physically separate effects, and we show that they affectthe charge transport in devices in different ways. This means that the value of the mobilityby itself does not fully describe charge transport at finite electric field. We present a newform of the 1D-DD method that explicitly splits these two factors of the field dependenceinstead of using the charge-carrier mobility. This method can be used to more accuratelymodel devices in which high electric fields occur.

The fourth and final result is a description of how to accurately implement internalorganic-organic interfaces in the 1D-DD method (chapter 6). By comparing 3D-MC simu-lations to 1D-DD results, we determine and quantitatively describe three effects that mustbe taken into account. First, the charges at the interface are not in equilibrium, whichmust be taken into account in the boundary condition. Second, there is a discrete surfacecharge caused by charge carriers accumulating before the interface. Third, the Coulombrepulsion of this surface charge is reduced by Coulomb interactions between the carriers.All three effects can significantly influence the current in multilayer OLEDs and must betaken into account in an accurate 1D model.

These charge-transport results have applications in the short, middle and long term(chapter 7). In the short term, the 1D-DD method can be used in characterization of organicsemiconductors. Indeed, this approach (without the results in this thesis) has already beensuccessfully used in several organic semiconductors to determine material parameters fora given choice of hopping model. By considering the time or frequency dependence of the

137

mobility it may also be possible to determine which hopping model is appropriate. In themiddle to long term, the 1D-DD method will be a valuable tool in OLED device simulation.The 3D-MC method has also proven to be useful in this kind of simulation. These meth-ods are especially powerful when used to complement each other. Double-carrier chargetransport and excitonics will have to be analyzed further to complete the 1D-DD method.In the long term, our most valuable result is most likely the insight we have gained intothe physics underlying charge transport in organic semiconductors. This will allow us totruly understand OLED operation and come up with new concepts and designs. Some ofour results, such as the scaling theory, may also be applicable to other fields of study.

138 Summary

List of Publications

1. J. Cottaar, R. Coehoorn, and P. A. Bobbert, Scaling theory for percolative chargetransport in molecular semiconductors: Correlated versus uncorrelated energetic dis-order, Phys. Rev. B 85, 245205 (2012).

2. J. Cottaar, R. Coehoorn, and P. A. Bobbert, Modeling of charge transport across dis-ordered organic heterojunctions, Organic Electronics 13, 667 (2012).

3. J. Cottaar, L. J. A. Koster, R. Coehoorn, and P. A. Bobbert, Scaling theory for percola-tive charge transport in disordered molecular semiconductors, Phys. Rev. Lett. 107,136601 (2011).

4. J. Cottaar, R. Coehoorn, and P. A. Bobbert, Field-induced detrapping in disorderedorganic semiconducting host-guest systems, Phys. Rev. B 82, 205203 (2010).

5. J. Cottaar and P. A. Bobbert, Calculating charge-carrier mobilities in disorderedsemiconducting polymers: Mean field and beyond, Phys. Rev. B 74, 115204 (2006).

6. K. D. Meisel, W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M.Blom, D. M. de Leeuw, and M. A. J. Michels, Charge-carrier mobilities in disorderedsemiconducting polymers: Effects of carrier density and electric field, Phys. Stat. Sol.(c) 3, 267 (2006).

7. W. F. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. A. Bobbert, P. W. M. Blom, D. M.de Leeuw, and M. A. J. Michels, Unified description of charge-carrier mobilities indisordered semiconducting polymers, Phys. Rev. Lett. 94, 206601 (2005).

8. W. F. Pasveer, J. Cottaar, P. A. Bobbert, and M. A. J. Michels, Temperature, chargecarrier density, and electric field dependence of mobilities in disordered conjugatedpolymers: Simulation results, Synth. Met. 152, 157 (2005).

140 List of Publications

Curriculum Vitae

Jeroen Cottaar was born in Eindhoven on April 25, 1984. He grew up in the Netherlands,Belgium and the United States. He attended the ’gymnasium’, the highest level of Dutchsecondary education, at the Bernardinuscollege in Heerlen and obtained his diploma cumlaude in 2001. He then pursued two Bachelor’s degrees, in Applied Physics and in Ap-plied Mathematics, at the Eindhoven University of Technology. He obtained these twodegrees cum laude in 2006, after an internship on the topic of numerically determiningthe charge-carrier mobility in organic semiconductors, supervised by dr. ir. P.A. Bobbert,ir. W.F. Pasveer and dr. F. Redig. Next, he pursued two Master’s degrees, in AppliedPhysics and in Computer Science and Engineering, at the same university. This involvedan internship at the University of Cambridge, supervised by prof. N.C. Greenham anddr. ir. P.A. Bobbert, on the topic of the pressure sensitivity of charge transport in organicsemiconductors. He obtained both degrees cum laude in 2008, after a final project inves-tigating mixing in two-dimensional point-vortex systems, supervised by dr. ir. L.P.J. Kampand prof. dr. R.M.M. Mattheij.

In 2008 Jeroen started his Ph.D. research in the group “Theory of Polymers and SoftMatter” (TPS) at the department of Applied Physics of the Eindhoven University of Tech-nology, with prof. dr. R. Coehoorn en prof. dr. M.A.J. Michels as promotors and dr. P.A. Bob-bert as day-to-day supervisor and copromotor. The subject of this research is charge trans-port in organic semiconductors, particularly in organic light emitting diodes (OLEDs). Hehas presented this work at leading international conferences, such as in Atlanta, Brussels,Veldhoven and Thessaloniki.

Jeroen Cottaar werd geboren op 25 april 1984 in Eindhoven. Tijdens zijn jeugd woondehij in Nederland, België en de Verenigde staten. Hij volgde het gymnasium aan het Ber-nardinuscollege in Heerlen, waar hij in 2001 cum laude zijn VWO-opleiding afrondde.Vervolgens combineerde hij de Bachelorstudies Technische Natuurkunde en ToegepasteWiskunde aan de Technische Universiteit Eindhoven. Hij behaalde zijn diploma in beiderichtingen cum laude in 2006, na een interne stage op het onderwerp van het numeriekbepalen van de mobiliteit in organische halfgeleiders, begeleid door dr. ir. P.A. Bobbert,ir. W.F. Pasveer en dr. F. Redig. Hij vervolgde met twee Masterstudies, Applied Physics enComputer Science and Engineering, aan dezelfde universiteit. In de loop hiervan liep hij

142 Curriculum Vitae

een externe stage aan de University of Cambridge onder begeleiding van prof. N.C. Green-ham en dr. ir. P.A. Bobbert, op het onderwerp van de drukgevoeligheid van ladingstransportin organische halfgeleiders. Uiteindelijke behaalde hij in 2008 beide Masterdiploma’s cumlaude, na een afstudeeronderzoek aan menging in tweedimensionale puntvortexsystemenbegeleid door dr. ir. L.P.J. Kamp en prof. dr. R.M.M. Mattheij.

In 2008 begon Jeroen aan zijn promotieonderzoek in de groep “Theory of Polymers andSoft Matter” (TPS) aan de faculteit Technische Natuurkunde van de Technische Univer-siteit Eindhoven, met als promotoren prof. dr. R. Coehoorn en prof. dr. M.A.J. Michels enals dagelijkse begeleider en copromotor dr. P.A. Bobbert. Het onderwerp van deze promotieis ladingstransport in organische halfgeleiders, met name in organische licht emitterendediodes (OLEDs). Hij heeft dit werk gepresenteerd op toonaangevende internationale con-ferenties, onder andere in Atlanta, Brussel, Veldhoven en Thessaloniki.

Acknowledgements /Dankwoord

I would now like to thank all those who contributed to this thesis through their professionalor personal support.

Als eerste wil ik mijn dagelijkse begeleider en copromotor Peter Bobbert, mijn eerstepromotor Reinder Coehoorn, en mijn tweede promotor Thijs Michels bedanken. Peter, jijhebt ervoor gezorgd dat ik met beide benen op de grond bleef, al was het misschien nietaltijd makkelijk om mijn werk in de goede richting te sturen. Reinder, jouw enthousiasmevoor het onderzoek werkt aanstekelijk. Door jou ben ik altijd gedwongen ook in termenvan toepassingen te denken, en dat heeft mijn werk veel goed gedaan. Peter en Reinder,ik wil jullie verder allebei bedanken voor de vele tijd en moeite die jullie gestoken hebbenin het doornemen van mijn artikelen en proefschrift. Ik heb ook wat dit betreft veel vanjullie geleerd. Thijs, het was een eer in jouw groep te mogen werken, en ik hoop dat je gaatgenieten van je welverdiende pensioen. Dit proefschrift zou niet zijn wat het is zonder jouwscherpe suggesties.

Next, I would like to thank the other members of my reading committee: Wolfgang Wen-zel, Neil Greenham, and Martijn Kemerink. Thanks for taking the time to read my thesisand helping me to improve it. Neil, thanks also for introducing me to the experimental sideof the organic semiconductor story during my time in Cambridge in 2006.

Ik wil mijn werkgever, het Dutch Polymer Institute, bedanken voor het mogelijk makenvan dit onderzoek. Hierbij noem ik in het bijzonder John van Haare en Frans de Schryver.Jullie hadden altijd scherpe vragen bij mijn voortgangspresentaties, die mij vaak op nieuweideeën brachten.

Ook de ondersteuning van mijn werk door de secretaresse van de onderzoeksgroep,Helmi van Lieshout, is van onschatbare waarde geweest. En niets was mogelijk geweestzonder degenen die onze supercomputers (en mijn laptop) aan de praat hebben gehouden:Arieh Tal, Erik Smeets, Tarik Gammoun en Willem-Pieter Sukkel.

I’d like to thank my colleagues in the group Theory of Polymers and Soft Matter overthe years: Abhinav, Sander, Frank, Leon, Jeroen van der Holst, Jeroen de Groot, Mu-rat, Andrea Massé, Andrea Muntean, Adrian, Wouter, Liza, Henry, Chrysostomos, Dmitri,Hamed, Daria, Saber, Ewa, Remy, Charley, Joris, Alexey, Paul, and Cees. The discussionsduring the coffee and lunch breaks were always an attractive distraction, perhaps some-

144 Acknowledgements / Dankwoord

times a bit too attractive. I also want to thank my colleagues at Philips for the inspirationI got from our (somewhat) monthly meetings, and mention in particular Rein, Marco andHarm. Rein, as direct collaborators in one DPI project we always worked together well,even if our research paths diverged somewhat.

Belangrijk was ook de nodige ontspanning. Hiervoor wil ik eerst mijn vrienden bijstudententafeltennisvereniging ESTTV TAVERES, bridgeclub BC70, en studentenscherm-vereniging Hoc Habet bedanken. Van TAVERES wil ik vooral noemen: Dion, Paul, Mark,Lenneke, Martijn, Gilbert, Hans, Maarten en Jolanda, Benjamin en Marieke, en Mauriceen Selma. Bij BC70 wil ik met name Vincent, mijn bridgepartner tijdens deze vier jaar, be-danken. Bij Hoc Habet gaat het vooral om de ‘ouwe lollies’ waar ik vroeger mee schermde,want ik heb tijdens mijn promotie geen wapen aangeraakt: Harm, Alke, Maurice, Mar-tijn, Marloes, Peter en Ana Karla, Jos en Winni, en Joost en Anke. I’d also like to thankour other friends for the happiness they brought to our home, mentioning especially玉锺and Roel, 奕, 靓慧, 天玮, 路和荣和若琳, 颖渊 and John and Daniël, and Michelle andAlessandro and Céria.

Ik wil verder mijn ouders Ward en Alice en mijn broers en zussen (en aanhang) Sanne,Michiel, Kasper en Dorien en Lucas, en Jolijn en Daniel bedanken voor hun onvoorwaar-delijke steun en liefde. Papa en mama, jullie hebben mij altijd gedreven om het besteuit mijzelf te halen en niet met minder tevreden te zijn. Zonder de nieuwsgierigheid enzelfstandigheid die jullie in mij gekoesterd hebben was ik nooit zo ver gekomen.我亲爱的书霞，无法用语言表达，在人群中被你找到的我，是多么的幸运。你的支持

是价值连城的，我相信我们家的未来会越来越好。我不能没有你。En Roosje, mijn kleineRoosje,我的小华兰, ik weet niet meer hoe het leven was voor jij erbij kwam. Jouw lach ishet mooiste wat er is.