## 1 Introduction

Common fabrication technologies of organic semiconductor materials, such as high vacuum evaporation deposition of small molecule based materials or solution processing of polymers, often lead to a disordered, amorphous or nearly amorphous, structure. In such systems, with only a weak intermolecular interaction, injected electrons or holes reside on distinct molecular sites or (in the case of polymers) on distinct conjugated segments. The charge transport is due to hopping, i.e., to thermally assisted quantum-mechanical tunneling in between the localized sites. Experimental evidence favoring this picture is the observation of an increase of the charge-carrier mobility with increasing temperature, as measured in unipolar devices in which the transport is due to a small concentration of charges injected from the electrodes. At high carrier concentrations, as obtained, e.g., by strong molecular doping, a cross-over to metallic transport can take place, characterized by a decrease of the conductivity with increasing temperature 1.

In this chapter, we discuss recent advances in our understanding of the charge-carrier transport in disordered organic semiconductors in the hopping regime. We focus on the development of models for the effective mobility and exciton formation rates in organic semiconductors, based on numerically exact three-dimensional (3D) simulations, and on applications to OLEDs. Application to single-layer OLEDs shows that predictive modeling of the current density and light emission efficiency is presently feasible. For an excellent discussion on charge transport in organic semiconductors, providing a more fundamental basis and including transport under conditions which are more relevant in other types of devices, we refer to a recent comprehensive review paper of Köhler and Bässler 2.

In its most ambitious form, building a charge transport model for a disordered organic semiconductor would require the realization of the sequence of the following three steps:

- (1)Prediction of the molecular-scale (disordered) structure using molecular dynamics (MD) or Monte Carlo (MC) methods, appropriately taking the deposition conditions into account.
- (2)Calculation of the electronic structure and interaction parameters which determine the hopping rates between pairs of molecules, using density functional theory (DFT).
- (3)Calculation of the charge carrier mobility as a function of the temperature, electric field, charge carrier density, and other relevant parameters, by solving the hopping problem on a large grid of sites representing the molecules.

Recently, for the first time this approach was indeed followed to develop a mobility model, viz. for hole transport in the archetypical OLED material tris(8-hydroxyquinoline)-aluminum(III) (Alq_{3}) 3. Figure 1 shows the result of a supercomputer simulation 3 of a 10 nm × 10 nm × 10 nm box containing 1137 Alq_{3} molecules, showing their disordered packing and revealing the non-uniformity of the hole transport. A Gaussian shape of the density of states (DOS) was found, with a width *σ* ≈ 0.18 eV. As a result of the disorder, the current density is not uniform but filamentary, so that only a small fraction of the molecules participates in the transport process. This pioneering work has opened the prospect that predicting the mobility from first principles will become a feasible option. However, no complete analysis of the accuracy of the methods used and of comparisons with experiment has yet been made. Making such an analysis is hampered by the CPU-intensiveness of the computational tasks involved and by the well-known sensitivity of the thin-film structure and the resulting mobility of organic electronic materials such as Alq_{3} to the deposition conditions.

Debates about the appropriateness of more simple models have been a recurring theme in the literature. Such simplifications concern the description of the energy level structure (shape of the DOS and spatial correlations between the energy levels of nearby molecules), the molecular positions (positional disorder) and the hopping mechanism. Most fundamentally, the question has been raised whether the energetic disorder is actually the dominant factor determining the mobility. Energetic disorder leads to a wide distribution of the intermolecular hopping rates and the resulting percolative nature of the transport process is an obstacle for obtaining a large mobility. However, it has been argued that often the limiting factor for the mobility is not the disorder, but the Marcus reorganization energy, i.e., the energy involved in the change of the atomic structure of a molecule upon positive or negative charging 4–7 during the hopping process. Both models have been discussed in a review by Coropceanu et al. 8. Experimentally, a distinction between the two points of view can be made from a study of the temperature (*T*) dependence of the low-field mobility of single carriers, which is expected to be proportional to exp[−*C*(*σ*/(*k*_{B}*T*))^{2}] in the case of sufficiently strong Gaussian disorder and to exp[−*E*_{A}/(*k*_{B}*T*)] in the absence of disorder. In the latter case, the intramolecular reorganization energy provides the only contribution to the effective activation energy, *E*_{A}. Here, *σ* is the width (standard deviation) of the DOS, *k*_{B} the Boltzmann constant and *C* is a coefficient of the order 1. A 1/*T*^{2} dependence was first obtained within the Gaussian disorder model (GDM) developed by Bässler 9, who investigated the effects of energetic disorder in the absence of a reorganization energy using MC modeling. Recently, Cottaar et al. showed that a 1/*T*^{2} dependence is also expected for transport in a Gaussian DOS when reorganization plays a role. The *C* coefficient is a function of the ratio between the reorganization energy and *σ*, and its measurement may thus be used to determine the relative role of reorganization 10, 11. Time-of-flight (TOF) mobility measurements 12 and analyses of the current–voltage (*J*(*V*)) characteristics of unipolar sandwich-type single layer devices yield indeed often a 1/*T*^{2} temperature dependence 12, although also a 1/*T* dependence has been obtained from such studies 13.

The finding of a 1/*T* dependence in the effective mobility does not necessarily imply that energetic disorder is absent or relatively small. Such an effective temperature dependence can also occur in the presence of strong Gaussian disorder, when the experiments used to probe the mobility are carried out at a relatively large charge-carrier density, when due to the filling of low-energy states the mobility is carrier density dependent. This was already known to be a relevant effect in organic-field effect transistors based on amorphous organic semiconductors with an exponential shape of the DOS 14. The application of a gate voltage can then give rise to a large carrier concentration (charge per molecule) in the accumulation channel near the gate dielectric, up to 0.01 or even higher. However, analyses of the *J*(*V*) characteristics of unipolar sandwich-type single-layer devices revealed that the mobility can also be strongly carrier density dependent at much lower concentrations 15. Three-dimensional numerical modeling using a master equation (ME) approach 16 and various semi-analytical studies 17 showed that a carrier density dependence of the mobility is not only expected for the case of an exponential DOS, but also for a Gaussian DOS. Pasveer et al. showed for the case of a spatially uncorrelated Gaussian DOS that from the results of 3D-ME calculations effective compact expressions for the temperature, field, and carrier density dependence of the mobility may be found which provided for conjugated polymer single-carrier (hole-only) sandwich-type devices an excellent explanation of the available experimental current–voltage curves 18. The analysis made it possible to make use of efficient one-dimensional (1D) device modeling, in spite of the fact that the actual current density is filamentary instead of uniform. This extension of the GDM, including the carrier density dependence of the mobility, has been called the extended Gaussian disorder model (EGDM). It is now evident that in many transport experiments the density dependence of the mobility plays an important role, and that in simplified analyses of the *J*(*V*) characteristics of unipolar sandwich-type single layer devices which neglect this effect the *effective* mobility will often show to a good approximation a 1/*T* temperature dependence, whereas the value of the mobility in the low-concentration limit as determined properly from the experiments shows a 1/*T*^{2} dependence 19, 20. It thus follows that the observation of a 1/*T* dependence of the effective mobility is not necessarily inconsistent with the point of view that the effect of disorder on the mobility is significant.

In Fig. 2, our view on the development of OLED modeling is visualized. Conventionally, OLED models 14, 21–26 employed a mobility function with an empirical Poole–Frenkel electric field (*F*) dependence (with *γ*_{PF} an empirical temperature dependent Poole–Frenkel factor) and a 1/*T* temperature dependence as would be expected in the absence of (substantial) disorder. No charge-carrier density dependence was assumed. Within these 1D “first-generation” models the current density is assumed to be spatially uniform. “Second-generation” OLED models are based on 3D supercomputer modeling using ME or MC methods of the transport between mathematical point sites, in practice situated on a regular lattice, taking the energetic disorder into account but neglecting molecular-scale details such as orientational disorder and the Marcus reorganization energy. The resulting current density is filamentary and the recombination takes preferentially place at a small fraction of the molecular sites. The EGDM is based on the spatially non-uniform current density as obtained from such numerically exact 3D modeling, from which after spatially averaging an analytical description of the effective mobility function is obtained. An analogous approach has been used for the case of spatially correlated disorder, resulting in the extended correlated disorder model (ECDM) 27. Recently, OLED simulation software tools based on both models have become commercially available 28, 29. Future “third generation” OLED models will include all molecular-scale details on the charge-carrier transport, as in the pioneering work on Alq_{3} discussed above, and molecular-scale modeling of excitonic processes.

It has been a purpose of our work in the past years to investigate whether 3D second-generation OLED modeling, using the point-site approximation, provide a sufficiently accurate description of the charge transport and exciton generation in OLEDs, and whether it is possible to replace the full 3D calculations by much faster 1D device simulations using expressions for the effective mobility and recombination rates that follow from 3D simulations. This will be the main subject of this review. The EGDM and ECDM are both examples of such an effective mobility model. Within both models, the mobility depends only on a few physically meaningful parameters, although if needed additional parameters describing a trap DOS can be included. We also show how in 1D device simulations the modeling of injection at the electrode interfaces and transport across organic–organic interfaces is included, and to what extent the EGDM and the ECDM may be applied to application-relevant materials and devices.