This article will be included, in edited form, as a chapter of the forthcoming book “Physics of Organic Semiconductors,” edited by W. Brütting and C. Adachi (Wiley-VCH, Weinheim, 2012), ISBN 978-3-527-41053-8.
In this review, we discuss recent advances in our understanding of charge transport and exciton generation in disordered organic semiconductors with a Gaussian DOS, with a focus on applications to organic light-emitting diodes (OLEDs). Three-dimensional (3D) modeling shows that the actual current density in OLEDs based on materials with a Gaussian electron and hole DOS is filamentary. However, it is possible to accurately calculate the average current density by solving a one-dimensional (1D) drift-diffusion equation, making use of compact expressions for the temperature, electric field, and carrier density dependent mobility which have been derived from 3D-modeling. For the cases of spatially uncorrelated energetic disorder and spatially correlated disorder due to random dipole fields, these models are called the extended Gaussian disorder model (EGDM) and extended correlated disorder model (ECDM), respectively. We discuss how the effects of trapping on guest molecules can be included, and how exciton generation is described. The application of these models to hole and electron transporting polymer and small molecule materials is discussed, with an emphasis on the modeling of the transport and emission of blue-emitting OLEDs based on a polyfluorene-derivative.
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) (Alq3) 3. Figure 1 shows the result of a supercomputer simulation 3 of a 10 nm × 10 nm × 10 nm box containing 1137 Alq3 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 Alq3 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/(kBT))2] in the case of sufficiently strong Gaussian disorder and to exp[−EA/(kBT)] in the absence of disorder. In the latter case, the intramolecular reorganization energy provides the only contribution to the effective activation energy, EA. Here, σ is the width (standard deviation) of the DOS, kB the Boltzmann constant and C is a coefficient of the order 1. A 1/T2 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/T2 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/T2 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/T2 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 γPF 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 Alq3 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.
2 Mobility models for hopping in a disordered Gaussian DOS
2.1 The extended Gaussian disorder model
The extended Gaussian disorder model (EGDM) 18 has been developed on the basis of 3D ME modeling of the current density due to hopping between the sites on a cubic lattice with a spacing a. The energy level Ei on each site i is taken randomly from a Gaussian distribution with a shape , with Nt ≡ a−3 the total volume density of hopping sites and σ the width. For a fixed carrier concentration, c, and for a uniform electric field, F, the time-averaged occupation probabilities pi on each site are calculated in a selfconsistent way, taking into account that the hopping probability to a neighbor site j is reduced by a factor (1 − pj) due to the on-site Coulomb repulsion, which prevents double occupation of a site. The hopping rate wij from a site i at position to a site j at a position is assumed to be given by the Miller–Abrahams (MA) expression
where ν0 is a hopping attempt frequency, α is the inverse of an effective wavefunction decay length and H is the Heaviside step function (H(x) = 0 for x ≤ 0 and H(x) = 1 for x > 0). The hopping rate varies thus exponentially with hopping distance and with the site energy difference. The MA approach is known to be a good approximation if all hops are accompanied by single-phonon processes. The range of validity of the MA approach and the sensitivity of the calculated transport properties to the hopping model used are subjects of continued debate in the literature 3, 30. In the 3D supercomputer modeling using ME or MC methods, cells containing 106 sites or more are considered, with periodic boundary conditions, in order to appropriately model the filamentary current. The current density is obtained from a summation of the currents through a plane perpendicular to the field direction.
In all published 3D-modeling work on the (E)GDM a fixed value of the product αa, equal to 10, has been used. This choice was motivated by Bässler, who noted that a typical value for the nearest-neighbor hopping distance is 1 nm whereas the effective values of the wavefunction decay length which follows from experimental studies of the site-density dependence of the mobility in dilute host–guest systems is typically 0.1 9. When using a cubic lattice the current density is then under common conditions almost exclusively due to nearest-neighbor hops, so that varying the αa-product has no significant effect on the temperature, field, or carrier density dependence of the mobility 31.
Figure 3a and c (symbols) shows the mobility for various values of the dimensionless disorder parameter as a function of the carrier concentration and the dimensionless electric field (with e the elementary charge), respectively, as obtained in Ref. 18 from ME calculations. The mobility is found to be independent of the carrier concentration for sufficiently small carrier densities, when each carrier may be viewed as independent. The average carrier energy is then independent of the carrier concentration, and equal to the so-called thermal equilibrium energy
In this so-called Boltzmann regime, the zero-field mobility is to an excellent approximation described by the expression
with c1 = 1.8 × 10−9 and c2 = 0.42. These are empirical parameters, determined from a fit to the numerical data. We note, however, that c1 is very close to the value of the factor exp(−2αa) = 2.1 × 10−9 which would be expected for the case of nearest-neighbor hopping in the high-temperature limit. Furthermore, we note that c2 is very close to the value of ∼0.44 obtained by Bässler from MC calculations 9.
A cross-over to a regime in which the mobility is carrier density dependent is found to take place when the Fermi energy EF exceeds E0, which may be shown to correspond to a cross-over concentration given by
For , corresponding at room temperature to σ ≈ 0.1 eV, ccross-over ≈ 10−4. With decreasing temperature or increasing σ, ccross-over decreases rapidly. Percolation theory shows that for sufficiently small carrier concentrations the mobility may be viewed as being determined by the energy difference between the Fermi energy in the tail of the DOS, which increases with increasing carrier concentration, and a fixed so-called “transport level,” more close to the top of the DOS 32–34. It may be shown that the carrier concentration dependence of the mobility is then to a good approximation given by 19
It follows from Eqs. (2), (4), and (5) that at the cross-over concentration the enhancement is precisely equal to a factor of 2. The effect becomes important for EF > E0 when due to filling of the deepest states the average charge-carrier energy starts to increase with increasing energy. Trapping in the already highly occupied deepest states is then less probable, so that the mobility starts to increase. Note that this is purely a density-of-occupied-states effect, and independent of the specific transport model assumed. Figure 3a shows that the carrier concentration dependence of the mobility can give rise to an enhancement of the mobility over many orders of magnitude, as expected from Eq. (5).
Figure 3c shows that an electric field also enhances the mobility, as might have been anticipated from Eq. (1), until for F ≈ 4σ/(ea) the field is so large that most forward hopping attempts are actually “downhill.” The current density is then field and temperature independent and the mobility is proportional to 1/F. Up to high values of the carrier concentration, essentially equal results are obtained using MC modeling 35, within which the actual trajectories between carriers are calculated, taking into account that hops to already occupied sites are forbidden. At concentrations above c ∼ 0.01, depending on the site density, the repulsive Coulomb interaction which was neglected in the ME calculations starts to give rise to a decrease of the mobility 35.
It was shown in Ref. 18 that these 3D modeling results can be described well using an analytical 1D-expression for the mobility of the form
with . The carrier density and the field-dependence occur as separate factors. A motivation for the specific parameterization function used to describe the carrier density dependence and an analysis of its precision have been given in Appendix D of Ref. 19. The parameterization is shown by the full curves in Fig. 3a and b. No attempts have been made so far to develop an accurate parameterization for the large carrier concentration and high field regimes. In actual calculations using the EGDM, often a cut-off is introduced around c = 0.1, and at high fields 36. In the case of OLEDs, the actual value of the cut-off concentration, and the precise assumptions made concerning the very high mobility close to that concentration are seldom of significant impact to calculated device properties. Such very high concentrations would in practice only occur in very thin regions near the interfaces of well-injecting metallic electrodes, whereas the device properties are determined by the much lower mobilities in the bulk of the layers. The dimensionless cut-off-field would typically correspond to F ≈ 0.2 V nm−1, which will only rarely be attained under realistic measurement conditions.
Within the EGDM, the mobility is in practice described by three material-specific parameters, viz. σ, Nt, and µ0,EGDM. The consistency can be judged from the measured temperature dependence of µ0,EGDM, which is expected to vary as 1/T2 with a slope parameter c2 in the range 0.38–0.50 (see Eq. (3) and Ref. 19). In general, also the value of the inverse of the wavefunction decay length, α, will affect the physics of the hopping process. The value of α determines the relative contributions of hops over different distances. This is particularly important in multi-component systems (e.g., dye-doped emissive layers in OLEDs) in which the transport may occur predominantly via long-distance hops between molecules which are not each other's direct neighbor. However, for single-component materials (neat films) with a disorder parameter in the range of values for which the EGDM was derived, essentially only nearest-neighbor hops are found to contribute to the hopping transport at small or moderate fields. A possible deviation from the assumed relationship αa = 10 will then lead to a deviation in µ0,EGDM, but it will not affect the accuracy of the mobility functions obtained.
The EGDM may thus be used when the transport is in the nearest-neighbor hopping regime, which extends to values of the disorder parameters around σ/(kBT) ≅ 8. For typical organic semiconductors with a Gaussian DOS, with values op σ up to 0.15 eV, the model is therefore applicable at room temperature. When σ = 0.1 eV, it may be applied down to approximately 150 K.
2.2 The extended correlated disorder model
In TOF experiments, the field-dependence of the mobility is often found to be described well by the empirical Poole–Frenkel (PF) expression . Within the (E)GDM such a field-dependence is indeed found, but only in a narrow field range. Gartstein and Conwell 37 argued that obtaining a PF field-dependence in a wider field range requires that the energy levels are assumed to be spatially correlated. The physical origin of such correlations could be the dipole moments present on some organic molecules 37–39, the quadrupolar moments inherently present for all π-conjugated systems, long-range thermal fluctuations in the molecular geometries (for the case of a polymer 40), or short-range order due to a specific molecular packing. An extended range of validity of the PF law was indeed found from 1D 38 and 3D 39 calculations of the mobility in the Boltzmann regime (correlated disorder model, CDM). Using a 3D-ME approach analogous to that described above, Bouhassoune et al. 27 developed an extension of the CDM including the effect of the carrier density dependence of the mobility (ECDM). The site energies were obtained by assuming correlated disorder resulting from randomly oriented electrical dipole moments, as in Ref. 39. This leads to a Gaussian DOS with a pair correlation function of the site energies that decreases by a factor of ∼2 within approximately 1.5 average intersite distances and which decreases at large distances (r) as 1/r37. Figure 4 shows the distribution of energies on a part of the system containing 31 × 31 × 31 sites.
Figure 3b and d (symbols) show the mobility for various values of the dimensionless disorder parameter as a function of the carrier concentration and the dimensionless electric field. As in the case of the EGDM, the mobility is independent of the carrier concentration for sufficiently small carrier densities, when each carrier may be viewed as independent, whereas it becomes carrier concentration dependent above a value of ccross-over close to that given by Eq. (4). However, the carrier concentration dependence is smaller than for the EGDM, whereas the field-dependence is stronger. The square-root field-axis used in Fig. 3d reveals that indeed over a substantial field range an approximate Poole–Frenkel behavior holds. The cross-over to the down-hill hopping regime takes already place when F ≈ σ/(ea), corresponding for the case of and a = 1 nm to a field equal to 0.1 V nm−1. This relatively low field, which could be realistically obtained in actual devices, made it necessary to develop a parameterization scheme including the high-field range 27. In the low field regime, the mobility is written as
with g and f dimensionless mobility enhancement functions described by analytical expressions given in Ref. 27 and with
where c3 = exp(−20) = 2.1 × 10−9 and with c4 = 0.29. Within the ECDM the temperature dependence of the mobility is thus smaller than within the EGDM 18, reflecting that for a given value of σ the energy level landscape is effectively smoother than in the EGDM due to the spatial correlation: close to each occupied state many other low-lying states are present to which hops can take place which contribute significantly to the total mobility. The full curves in Fig. 3b and d show the results of the parameterization, using for the carrier density dependence a high-concentration cut-off at c = 0.025.
2.3 The mobility in host–guest systems
The EGDM and the ECDM can under certain conditions also be used to accurately describe the mobility of host–guest systems, within which the DOS deviates from a simple Gaussian shape due to the presence of additional low-lying guest-derived states. Such a situation arises often when describing the electron mobility, as a result of a non-intentional small concentration of low-lying trap states, or when describing the mobility in emissive layers containing a relatively large concentration of fluorescent or phosphorescent dye-molecules.
Experimental studies show that, in general, four transport regimes may be distinguished, depending on the relative guest molecule concentration, x41, as illustrated in Fig. 5. We emphasize that the boundaries between the regimes depend on the details of the material system studied, and that a strict separation in four regimes is not always possible. As a concrete example we consider the case of a bimodal Gaussian DOS without spatial correlations, for which the guest concentration dependence of the mobility was obtained from ME modeling or from a basically equivalent percolation model 42. Within this example equal widths of the host and guest DOS, (i.e., σ ≈ 0.1 eV at room temperature) and equal values of their wavefunction decay length α−1 were assumed. Three values of the depth Δ of the trap states are considered, corresponding to 0.3, 0.45, and 0.6 eV at room temperature, and the carrier concentration is 10−4.
Figure 5a shows the four transport regimes. At very small guest concentrations (regime A) the mobility is only due to host–host hopping so that it is not affected by the guest molecules (a). For very high guest concentrations (here for x > 0.3, regime D), the mobility is exclusively due to hopping between the guest sites. The increase of the mobility with increasing guest site density as a result of the decreasing intersite-distance is described well using the EGDM (not shown).
A more interesting situation arises in regime B, when the guest molecules start to act as trap sites. For very small electric fields, the mobility is then to an excellent approximation given by a multiple-trap-and-release “thermal detrapping” model, used in conjunction with the EGDM. Within this approach, the mobility is equal to the fraction of carriers which according to Fermi–Dirac (FD) statistics populate the host states, times the EGDM host mobility at the corresponding (lowered) “free” carrier density 43. The mobility is carrier density dependent for two reasons: (i) the occupation of the host states increases with increasing carrier density and (ii) the mobility of the carriers in the host states increases with increasing carrier density (if the carrier concentration is above the critical value given by Eq. (4)). In earlier theoretical work 17, 44–46, the latter effect was not taken into account. In Fig. 5b, the prediction as obtained from this model (full curves, “thermal detrapping”) is compared with the results as obtained from the benchmark ME model (symbols), showing excellent agreement in the relevant guest concentration range (10−4 < x < 5 × 10−2).
At finite values of the electric field this simple picture is not anymore applicable, as the field creates a “hot” non-equilibrium occupation of the DOS. More electrons than would be obtained from the FD distribution occupy then host states, so that the mobility is larger than expected. The effect is called “field-induced-detrapping” (FID). It is illustrated in Fig. 6 for a system with a set of parameters as given in the figure caption. The figure shows that under realistic conditions, for fields up to approximately 0.15 V nm−1, the thermal detrapping model can underestimate the mobility by a factor up to approximately 3. Recent work by Cottaar et al. 47 has shown that it is not possible to solve this problem by replacing the actual temperature by an effective larger temperature in conjunction with the standard FD expression. Instead, it was shown to be solvable with high accuracy by making use of a field-dependent generalized FD distribution which depends only on the shape of the host DOS and not on that of the guest DOS. An easy-to-use parameterization scheme for this distribution was derived from 3D-ME calculations. Its accuracy is excellent, as may be judged from Fig. 6 by comparing the exact 3D-ME results with the “thermal detrapping + FID + EGDM” prediction. When the mobility in the host can be described within the EGDM, a full extension to include a guest DOS with any arbitrary shape is thus available within regime B. For systems described by the ECDM, the FID effect has not yet been investigated.
The most complex situation arises when at intermediate guest concentrations the hopping takes place via host and guest sites (5 × 10−2 < x < 0.3, regime C). Figure 5b shows that the mobility as obtained from the benchmark ME model is then underestimated by the application of the EGDM in conjunction with the thermal-detrapping model as well as by an application of the EGDM which assumes that all transport is due to guest–guest hopping. The figure shows that even the highly refined semi-analytical Movaghar–Schirmacher percolation model (see also Refs. 43 and 48 for details) still underestimates the mobility in this regime. The ME results indicate that close to the mobility minimum random statistical variations of the local guest concentration, to smaller and to larger concentrations, lead to regions with an enhanced mobility. This intermediate transport regime occurs in many emissive layer materials used in OLEDs, within which typical phosphorescent dye concentrations are 1–10%. It would therefore be important to extend the EGDM (and the ECDM) to include this situation.
3 Modeling of the recombination rate
3.1 Recombination in systems with a Gaussian DOS
The position dependent exciton formation rate in OLEDs is generally described using the Langevin formula,
where ε is the dielectric permittivity, µe and µh are the electron and hole mobility, and ne and nh are the electron and hole carrier density 49. R is often called “recombination rate,” as it is equal to total rate of all radiative and non-radiative electron–hole recombination processes if no exciton dissociation takes place. The derivation is based on a consideration of the interaction between a single electron and a single hole, and assumes that even when the electron and hole are within each other's capture radius, approximately 15 nm at room temperature for typical organic semiconductors with a relative dielectric permittivity εr ∼ 3, diffusive transport characterized by multiple hops precedes the final exciton formation process 50. The mobility can be position dependent as a result of its carrier density and field dependence. In the case of large carrier density or field gradients, Eq. (9) is obviously expected to be a zeroth-order approximation at best. Furthermore, one might wonder whether (i) in the case of a wide Gaussian DOS the capture process is described well using the same carrier density and field dependent mobilities as used for describing the (filamentary) charge transport, (ii) the recombination rate is affected by the Coulomb interactions between carriers at high densities, and (iii) (deep) trap states would affect the applicability of Eq. (9).
The effect of Gaussian disorder on the recombination rate has for the first time been investigated by Albrecht and Bässler 51 using MC calculations of the capture process in systems containing a single electron–hole pair. Their studies showed that Eq. (9) is applicable in the zero-density limit and up to intermediate fields (∼0.05 V nm−1) for systems studied at room temperature with a wide range of disorder parameters σ, up to 0.25 eV. More recently, Groves and Greenham 52a and van der Holst et al. 52b carried out MC calculations for systems with Gaussian disorder as a function of the electron and hole density. Both groups found that the recombination rate can be significantly smaller than as expected from the Langevin formula, up to 70%. However, van der Holst et al. found that the Langevin formula is well obeyed if the bipolar mobilities as obtained for the actual double-carrier devices are used, instead of the unipolar mobilities which are obtained in single-carrier devices (see Fig. 7a). In a bipolar system with a sufficiently high carrier concentration, the electrostatic fields created by the second carrier give rise to additional disorder, so that the bipolar mobilities are smaller than the unipolar mobilities. The difference was found to be larger for the case of perfectly anticorrelated HOMO and LUMO site energies than for the case of perfectly correlated energies (see Fig. 7b). Varying local electrostatic potentials (anticorrelation) could occur as a result of randomly oriented molecular dipole moments and varying energy gaps (correlation) could occur as a result of varying degree of conjugation (e.g., in the case of polymers with varying conjugation lengths). The correlation between the HOMO and LUMO energy levels in actual materials is not well-known.
The applicability of the Langevin formula when bipolar mobilities are used and the effect of correlation are shown in Fig. 7c for a wide range of disorder parameters, for electron and hole concentrations equal to 10−4, and for the zero-field limit. The difference between unipolar and bipolar mobilities increases with increasing disorder parameter. At finite realistic fields (<0.1 V nm−1) only small deviations from the Langevin formula were found. Interestingly, the location of the sites at which the recombination takes place preferentially was found to be independent of the location of the current filaments (see the next section) in the corresponding single-carrier devices (studied when switching off the current due to the other carrier). Instead, these locations coincide with energetically low-lying sites for electrons or holes.
So far, this study of the validity of the Langevin formula and the effect of using bipolar mobilities has not yet been extended to cases with unequal electron and hole densities or unequal disorder parameters, nor to systems with correlated disorder (ECDM). Furthermore, there is not yet an easy-to-use parameterization scheme for the bipolar mobility. Another issue which needs further attention is the electric field dependence of the recombination rate. Albrecht and Bässler 51 found from 3D-MC calculations in the Boltzmann regime that at finite (large) fields the actual rate is slightly larger than as obtained from Eq. (9) (including the field-dependence of the mobility). van Mensfoort and Coehoorn 36 rationalized this result by describing the recombination process on a cubic grid as a result of contributions from two longitudinal hops (parallel and antiparallel to the field), with rates determined by F, and from four transverse hops, with rates independent of F. We have adopted this approach in all our 1D-device simulations presented in this chapter. However, van der Holst et al. 52b obtained from 3D-MC calculations outside the Boltzmann regime a more complex picture, with a (small) initial decrease of the recombination rate with increasing field.
3.2 Recombination in host–guest systems with a Gaussian host DOS
In actual OLEDs, the densities of HOMO and LUMO states are seldomly both single Gaussians. For example, the electron transport in polymer and small-molecule organic semiconductors is often significantly influenced by the presence of electron traps (see Section 4). A natural extension of the Langevin formula within which the effect of electron traps is included is given by 53
For notational simplicity, the position dependence is here not explicitly given. Eq. (10) expresses that the total recombination rate is a sum of contributions due to capture of “free” electrons, i.e., electrons residing in the Gaussian host DOS, by holes and capture of holes (which are all “free”) by electrons (“free” or “trapped”). The electron and hole mobilities used are given by the EGDM expressions, evaluated for the “free” electron density and the total hole density. An application of this expression is discussed in Section 5.2.
i.e., as the sum of a Langevin-type term resulting from bimolecular recombination between “free” carriers and a term resulting from trap-assisted recombination of free holes with trapped electrons. A similar approach was recently presented by Kuik et al. 54, within which the trap-assisted recombination term was calculated using the Schottky–Read–Hall (SRH) formalism 55. Both approaches are equivalent if within the SRH formalism the coefficient describing the rate of electron capture in a trap state is taken essentially infinite, so that the hole capture process becomes the rate-determining step (as we have silently assumed above). The hole capture coefficient used within the SRH formalism, which is treated as a free parameter in Ref. 54, is then equal to Ch = (e/ε)µh.
Trap-assisted recombination is expected to be most important and in some cases even dominant over the free carrier contribution at low voltages, when the fraction of electrons occupying trap states is largest. This was indeed seen in the two studies referred to above 53, 54 for the case of two different polymer OLED systems.
4 OLED device modeling
4.1 Single-layer OLEDs – analytical drift-only theory
In order to be able to analyze the effects of energetic disorder on the current density and recombination in OLEDs, we first give a brief review on the device physics of metal/organic semiconductor/metal sandwich systems based on intrinsic semiconductors with a constant mobility 56, 57 and taking only the drift-contribution Jdrift = eµnF to the current density into account. The contribution due to charge-carrier diffusion is neglected. For unipolar devices with equal and ideally injecting metallic contacts, i.e., with an infinite carrier density at the injecting electrodes, the current density is then given by the Mott–Gurney square-law formula
with L is the thickness of the organic semiconductor layer. The space charge and the field are not uniform. For the case of hole transport with the anode and cathode at x = 0 and x = L, they vary as
with and respectively. Under these assumptions, the device is thus completely empty at V = 0, and is filled upon the application of a voltage difference. Also for the steady-state mobility in systems with traps 58 and for the transient current density as obtained, e.g., in capacitance–voltage 59 and dark-injection 60 experiments useful analytical drift-only expressions are available.
If the electrodes are not ideal, so that there are energy differences Δ0 and Δ1 between the states in the organic semiconductor responsible for the charge-carrier transport and the Fermi level in the electrodes at the injecting and exit contacts, respectively, the device stays empty and no current flows until at the built-in voltage Vbi = (Δ1 − Δ0)/e the so-called “flat-band” condition (F = 0) is reached. This term is actually not very appropriate for disordered organic semiconductor devices within which the transport is due to hopping between localized states instead of transport of delocalized charge carriers in a valence or conduction band. Nevertheless, it is often used, and will be adopted as well in this chapter. For the case of an ideal injecting contact at x = 0, the current is for V > Vbi given Eq. (12) after replacing V by (V − Vbi). In the presence of an injection barrier at the x = 0 contact, the effective carrier density at that contact decreases to a value which depends on the detailed charge-carrier injection model assumed. Often, local thermal equilibrium at the electrode contacts is assumed, resulting from the large rates of hopping expected between the “reservoir-like” electrodes in which the carrier density is very large and the molecules directly adjacent to the electrodes. Using Fermi–Dirac statistics, the charge-carrier density at the interface is then given by
with Nt the total hopping site density. It may be shown that the current density is not significantly affected by the injection barrier as long as n0 >> n1/2. This is called the space-charge-limited current (SCLC) regime, and the contact is called Ohmic. On the other hand, if n0 << n1/2, the carrier density and the field are uniform across the device and equal to n0 and V/L, respectively. In this so-called injection-limited current (ILC) regime the current density is given by
where again V should be replaced by V − Vbi in the case of a built-in voltage. In the ILC regime J(V/L) curves measured for different layer thicknesses thus coincide. Under the assumptions made (no diffusion and no disorder) the boundary between the SCLC and ILC regimes is reached when n0 = n1/2, i.e., at a barrier height . For typical devices, with Nt = 1027 m−3, L = 100 nm, and εr = 3, studied at T = 300 K and V = 5 V, this happens for Δ0 ≈ 0.25 eV.
Properly modeling charge carrier injection is actually more involved than as discussed above, as the injection barrier is effectively lowered due to the interaction of the injected charge with its image charge in the electrode. The situation is most simple if the injection barrier is very high, so that it is sufficient to consider only a single carrier and its image charge. In the presence of a positive applied field F0, the local image-charge induced field gives rise to the position dependence of the potential shown in Fig. 8b (full curve), so that the effective injection barrier is lowered. As argued by Emtage and O'Dwyer, the effective carrier density boundary condition may then be taken equal to the value given by Eq. (13) with a reduced injection barrier
Deep in the ILC regime, F0 is equal to V/L. However, in general F0 is affected by the space charge in the device, from which it must be calculated selfconsistently using the Poisson equation. In the SCLC regime, F0 is negative (Fig. 8a) due to the space charge near the injecting electrode, so that Eq. (16) cannot be applied. In Sections and 4.3 we further discuss the carrier density boundary condition in the SCLC regime.
For the case of double-carrier OLEDs with (i) ideally injecting electron and hole contacts, (ii) equal and constant electron and hole mobilities, and (iii) bimolecular recombination as given by the Langevin formula (Eq. (9)), Parmenter and Ruppel have shown that the drift-only current density shows the same voltage and layer thickness dependence as given by the Mott–Gurney formula, but that the current density is a factor of 256/(9π2) ≈ 2.88 larger than the corresponding single-carrier current density 61. This factor is larger than 2 as the positive and negative space charges cancel partially, so that the applied field is to a smaller extent opposed by the presence of space charge in the device. When holes are injected at x = 0 and electrons are injected at x = L, the space charge and the field are given by
with and respectively. The hole and electron densities at the cathode and anode are zero, which already shows that the recombination efficiency is equal to 1: all injected carriers give rise to recombination. The recombination rate is uniform across the device, as the product of the electron and hole densities is position-independent, and equal to R = (J/e)/L. Parmenter and Ruppel 61 have also given solutions for the case of an arbitrary ratio between the electron and hole mobilities and for the case Langevin recombination as enhanced or decreased by an arbitrary prefactor, γL. A methodology for obtaining exact semi-analytical solutions of the drift-only single-layer double-carrier problem for arbitrary values of the injection barriers has been developed by Martin 62. For γL > 1 or γL < 1 the recombination becomes peaked in the device center or near the two electrodes, respectively. A large µh/µe ratio yields recombination peaked near the cathode. And large injection barriers can give rise to a decreased recombination efficiency, viz. when the carrier density in the device is so low that not all carriers meet an oppositely charged carrier before reaching the other electrode.
4.2 The role of charge-carrier diffusion
In the presence of a gradient in the charge-carrier density, diffusion gives rise to an additional contribution Jdiff = −qD(dn/dx) to the current density, with q the charge and D the diffusion coefficient. In the absence of disorder, the diffusion coefficient is related to the mobility via the Einstein equation D = (kBT/e)µ.
In single-layer OLEDs, the effect of charge-carrier diffusion is most prominent at low voltages. Within the drift-only approximation, the current density in devices with ideal contacts varies quadratically with V as the voltage fulfills two functions: it gives rise to an injected space charge and it provides the driving force for the current. However, charge carrier diffusion gives even at V = 0 already rise to a space charge in the device. At small voltages, the diffusion contribution to the current density is therefore linear in V and larger than the drift-only current density. For the case of a constant mobility, exact semi-analytical expressions are available for the current density including diffusion and for arbitrary charge-carrier density boundary conditions 63. In the case of ideal contacts (n0 = n1 = ∞) the carrier density at V = 0 is given by
with n1/2 ≡ 2π2 = 2π2εkBT/(eL)263. The current density at low voltages may be obtained by viewing the device as a resistor with a position-dependent resistivity ρ(x) = 1/[eµn(x)], which yields
From a comparison with the Mott–Gurney formula, it follows that a cross-over between diffusion and drift-dominated transport is expected around a voltage Vcross-over = (32π2/9) × kBT/e. At room temperature, Vcross-over ≈ 0.9 V. The effect of including diffusion is shown by Fig. 9, in which in both panels curves A and B give the current density without and with diffusion for a symmetric device with ideal contacts and with a material with a constant mobility. At 1 V, the diffusion contribution is still significant, but at 10 V it is almost negligible. This explains why the effect of diffusion is often most important in thin devices, which are investigated at relatively small voltages. Crăciun et al. 64, e.g., studied the hole transport in single-layer PPV-based devices and found for devices with a thickness well below 100 nm an effective mobility (as deduced from an analysis of the J(V) curves using the drift-only Mott–Gurney formula) exceeding the value obtained for devices thicker than 100 nm by more than one order of magnitude.
In Fig. 10, calculated J(V) curves are given for devices for which Vbi = 1 V, due to a 1 V barrier at the exit contact. The dotted and dashed J(V) curves give the drift-only and total current density (including diffusion) for devices with a constant mobility. The figure shows that as a result of diffusion the effective onset voltage, Vonset, is significantly smaller than the built-in voltage. This implies that Vbi cannot be obtained accurately from an analysis of the J(V) curve using the Mott–Gurney formula. To our point of view, the built-in voltage is given by the value of the onset-voltage, extrapolated to zero temperature. A temperature-dependent difference between the onset voltage and Vbi has indeed been observed 65. The square-root of the current density is down to voltages close to Vbi a linear function of the voltage 66, so that Vonset can be defined by extrapolation. Its value is to an excellent approximation given by 67
For the case of a good injecting contact n0 ≈ 1027 m−3 (i.e., of the order of the site density) and for ≈ 1021 m−3 (using L = 100 nm and T = 300 K), Vonset can at room temperature be 0.3 eV smaller than Vbi.
Methods which show in alternative ways the shift of the effective onset voltage due to diffusion include (i) measurements of the low-frequency capacitance, which show a distinct peak close to Vonset68, (ii) electro-absorption experiments 69, and (iii) measurement of the compensation voltage as obtained from the net photocurrent density 70. All these experiments yield characteristic voltages which are close to (although not precisely identical to) Vonset. If the temperature dependence of Vonset is not available, Vbi and the injection barriers Δ0 and Δ1 can be deduced using device modeling. It is shown in the next section that the presence of disorder gives rise to an enhanced onset voltage shift.
The effect of diffusion on the boundary conditions in the case of a large injection barrier, deep in the ILC regime, has been discussed by Scott and Malliaras 71. In the bulk of the device diffusion plays then no role, as the carrier density is in that regime uniform and equal to n0, but at the interface the field is position dependent due to the image potential effect. The authors find an expression for n0 which is very close to the image-potential modified expression given by Eqs. (15) and (16), modified by a weakly field-dependent prefactor of the order 1. Experimental support for the prediction that in the ILC regime the effective boundary condition n0 does not depend on the mobility, so that the JILC ∝ V (Eq. (15)), has been given by Shen et al. 72.
4.3 The role of Gaussian disorder – 1D device simulations
Methods for performing 1D OLED device simulations including the effects of Gaussian disorder as described by the EGDM or the ECDM have been developed by various groups 36, 73, 74. A first additional element of such calculations is the use of the generalized Einstein equation D = n/[e(dn/dEF)]µ, with EF the Fermi energy, to obtain the diffusion coefficient from the mobility 75. As a result, the diffusion coefficient in a Gaussian DOS is enhanced with respect to the value (kBT/e)µ which would be expected in the absence of disorder by a factor which increases with increasing carrier concentration (see Fig. 1 in Ref. 36). The enhancement is precisely equal to a factor of 2 for the concentration at which EF = E0/2 (see Appendix A in Ref. 36), where E0 is the thermal equilibrium energy given by Eq. (2). This characteristic concentration is relatively large, ranging from ∼10−1 to ∼2 × 10−3 for = 3 and 6, respectively. As in OLEDs the carrier density is generally smaller, except in certain cases at the electrodes and near internal interfaces, the effect on the J(V) curves is in practice quite moderate. However, it affects the low-voltage parts of the curves and the effective onset-voltage.
Secondly, the effect of disorder on the carrier density boundary conditions should be taken into account. As suggested in Refs. 36 and 76, this may be done by assuming (again) thermal equilibrium at the electrode interfaces. The carrier density boundary condition for the case of a Gaussian DOS with width σ and site density Nt is then given by the Gauss–Fermi integral
where Δ0,eff is obtained from the selfconsistently determined electric field using Eq. (16) when F0 > 0. When F0 < 0, a situation which arises as a result of charge-carrier diffusion in the case of sufficiently small barriers, no image charge correction will be included. In another way, an image potential effect is then still taken into account, viz. as the laterally uniform space charge density gives rise to a surface charge density on the electrodes. The effect of disorder on n0 is twofold. Firstly, at a given temperature and injection barrier it gives rise to a larger value of n0, due to the filling of low-energy tail states. A higher value of the injection barrier is then needed to enter the ILC regime. Secondly, it may be readily verified that the decrease of n0 with decreasing temperature is much smaller than in the absence of disorder, which is again consistent with the picture that the effective activation barrier for injection is smaller than in the absence of disorder. These effects were already predicted by Arkhipov et al. 77, although the temperature dependence of the predicted current density is not in good agreement with the results of more recent 3D-device modeling 76. The strong effect of disorder on the temperature dependence of the transport in the ILC regime was demonstrated experimentally by van Woudenbergh et al. 78 in a study of PPV-based devices. We refer to the publications mentioned above for the technical details of the 1D-simulations, and focus in this section on the results of calculations for model systems.
Figure 9 shows the effect of including disorder on the current density of a symmetric single-carrier device (Vbi = 0) with essentially ideal contacts, as obtained from 3D-ME modeling 76. At V = 0, the carrier concentration in the center of the device is approximately 10−5. For the case of weak disorder (, (a)) this concentration is well within the Boltzmann regime, so that the effect of disorder is only visible at high voltages where the carrier density is outside the Boltzmann regime and where the mobility becomes field dependent. However, for the case of strong disorder (, (b)) the carrier density dependence of the mobility gives already at small voltages, in the diffusion regime, rise to a significant increase of the current density as compared to the reference constant-mobility case (curve B). It was shown in Ref. 36 that in the latter case apparent mobilities that vary over more than two orders of magnitude with the layer thickness are obtained if the J(V) curves are (incorrectly) analyzed using the drift-only Mott–Gurney formula. Meaningful analyses of transport in OLED materials should thus take the carrier density and field dependence of the mobility induced by the disorder into account.
Figure 10 (full curves) shows the effect of disorder on the current density in asymmetric single-carrier devices, with an ideal injecting contact at x = 0 and a 1 eV barrier at the exit contact, so that Vbi = 1 V. In this case the linear current density in the diffusion regime is determined by the carrier density at the exit contact 36, which from Eq. (21) can be shown to increase with increasing disorder. The figure furthermore shows that with increasing disorder the effective current density onset voltage shift, caused by the diffusion of charge near the injecting electrode (Section 4.2), increases. This is consistent with experimental analyses. For example, electroabsorption 69 experiments on a polyfluorene-based copolymer with have revealed shifts of approximately 0.6 V.
Gaussian disorder has also a strong effect on the current density and recombination profile in double-carrier devices 73. As an example, Fig. 11 shows the calculated recombination profiles (thin curves) as a function of the disorder parameter for 100 nm symmetric OLEDs with ideal contacts and equal electron and hole mobilities, at 3 and 10 V. The thick curves show for comparison the profiles in the absence of disorder. As a result of charge carrier diffusion and the assumption of thermal equilibrium at the interfaces, the recombination rate goes to zero in thin regions near the electrodes. It may be seen that the effect of diffusion is more important at low voltage. In the center of the device, the recombination rate is uniform, as expected from drift-only Parmenter–Ruppel theory (Section 4.1). The effect of disorder is seen to depend on the detailed conditions. In the case of weak disorder and a large voltage, the field-dependence of the mobility gives rise to a double-peaked profile, as known already for materials with a Poole–Frenkel type mobility 79. The carrier density in the device center (where the field is largest (Eq. (17c)) is then relatively small, due to the mobility enhancement, so that the recombination becomes largest in regions more close to the electrodes. Upon the introduction of strong disorder, the recombination profile becomes more confined to the device center. This striking effect is a result of the carrier density dependence of the mobility. At either side of the device center the carrier densities are strongly unbalanced, as may be already seen from Eq. (17a,b). In the case of disorder this effect is enhanced as the mobility drops then strongly for those carriers which have just passed the device center, entering a low-density region. On the other hand, the mobility of the other carrier then increases. Therefore, most carriers which just passed the device center will recombine quickly. In the same study, disorder was found to have a strong effect on the emission profiles in devices with unequal hole mobilities. Surprisingly, stronger disorder was found to give rise to wider profiles, less strongly confined to a narrow region close to one of the electrodes.
In Ref. 73, the disorder was also found to have a strong effect on the current density. Whereas the Parmenter–Ruppel theory would predict that the current density in double-carrier devices is enhanced by a factor of 2.88 as compared to that in otherwise identical single-carrier devices, the enhancement factor as calculated for the case is almost a factor 8. The effect was shown to be to a large extent due to the enhanced carrier density and field in the device center as a result of the disorder, and the corresponding mobility enhancement.
4.4 The role of Gaussian disorder – 3D device simulations
The emergence of a filamentary current density as a result of energetic disorder has been revealed from various 3D studies, e.g., for systems with a uniform field and a uniform (average) carrier density 16, 80, in studies of the injection from a metallic electrode 81, of the unipolar 82, and bipolar 83 transport across organic–organic heterojunctions, and in devices with two metallic electrodes with varying injection conditions 76, 84. The filamentarity is strongest at small fields, decreases with increasing field, and vanishes at very high fields when almost all forward hops are “downhill” so that disorder has no effect anymore on the hop rates 16, 80. The filamentarity resulting from injection into a system with correlated disorder is much stronger than for the case of spatially random disorder 81. In case of charge transport across an organic–organic interface, with a step in the average on-site energies, a thin highly charged layer will be present just in front of this step. Because of the steep gradients in the carrier density, the usual continuum approaches for charge transport break down and one has to resort to MC approaches in which the Coulomb interaction between the carriers is explicitly accounted for 82. The Coulomb repulsion in this thin layer pushes away the charges from each other, leading to charge ordering. The reduced charge density around a specific charge leads to an electric field felt by that charge that is much smaller than in the case of a continuum treatment, in which charges are assumed to be smeared out in this layer 82. In condensed-matter theories this would be called a “self-energy” effect. We expect that by accounting for this effect it might be possible to construct continuum theories that do properly describe the current across organic–organic interfaces.
Jurić et al. 83 investigated the effect of 3D-disorder on the recombination rate near an organic–organic heterojunction at which there are equal hole and electron barriers. With increasing disorder energy, a decreased effective recombination cross-section at the barrier (leading to exciton or exciplex charge-pairs) was found, as may be expected from the presence of rare low-energy states which lead to an increased transmission probability through the barrier. A similar effect was found when comparing the transmission through a molecularly sharp interface and through an interface with one or two mixed interface layers.
Important questions to be solved are to what extent the current density, recombination profiles, and other observables predicted from such 3D calculations can be obtained as well from 1D device simulations, and to what extent the filamentarity of the current density gives rise to statistical variations. For the case of single-layer devices, this issue has been studied by van der Holst et al. 76. Figure 12 shows the 3D current density across 22 nm thick symmetric single-carrier devices with a 1 eV injection barrier, studied at room temperature at 2 V using a 3D-ME approach. The calculation was done for a box with 13 × 50 × 50 sites on a cubic grid, with periodic boundary conditions in the lateral direction. The intersite distance is 1.6 nm. The figures show that the current density is already weakly filamentary for σ = 75 meV, and becomes strongly filamentary for σ = 150 meV. In the latter case, almost all the current flows through a relatively small number of sites.
As a consequence of the filamentarity of the current density, the total current through a certain small area varies statistically. Figure 13 shows the current density distributions for the same 80 nm × 80 nm devices as studied in Fig. 12, as obtained using the ME model (light gray) and as obtained using the Burin–Ratner (BR) ILC model (dark) within which the current density is viewed as a sum of strictly 1D filaments 85. For σ = 75 meV, the current density distribution is relatively narrow. It is centered around the average current density (arrow in the figure), with a full width at half maximum equal to approximately 40% of the average current density. In contrast, for σ = 150 meV the distribution is quite asymmetric, with a long high-current tail. In the ensemble of 3200 samples studied, even a case with a current density slightly larger than 40 times the average value was found. In both cases, the distributions as obtained using the BR model were found to be displaced to lower current densities, showing that the actual filaments are not strictly 1D and that side-jumps play a role.
Figure 14 gives an analysis of the current density as calculated as a function of the injection barrier using various methods. The 1D-continuum model (green curve) yields for σ = 75 meV an excellent approximation to the results of the 3D-ME method (red spheres) if the image charge effect is included when F0 > 0 (see Section 4.3). For σ = 150 meV, the overall trend is excellent, but in the SCLC and ILC regimes the continuum model underestimates the current density by a factor ∼4 and ∼8, respectively. The discrepancy observed for the ILC-regime has been attributed to a breakdown of the mobility concept: the assumption, underlying the 1D calculations, of a local mobility that varies continuously breaks down for thin devices 84. For thicker (102 nm) devices, the agreement was found to be excellent. Concerning the SCLC regime, it should be noted that within both methods the image charge effects were treated in an approximate way. In the 3D-ME model a layer-averaged potential was calculated from the laterally averaged charge density, and a site-specific image potential correction was added to the energy at each site. In the 1D-continuum model, no image potential correction to the injection barrier was made in the SCLC regime. Within a more recent MC-study for the same devices, no such approximations were made 84. The open spheres show the results, obtained for four values of the injection barrier. At 2 V, the current density is for Δ = 0 and for σ = 75 meV found to be only ∼25% smaller than the 1D-continuum model result, whereas for σ = 150 meV the discrepancy at 2 V was found to be even smaller. Apparently, the 1D-continuum model provides a rather good approximation, although improvements would still be required when using it for example for the purpose of accurate parameter extraction. In particular, refined modeling to remove the unphysical discontinuity in the slope of the J(Δ) curves at the cross-over from the SCLC regime to the ILC regime (see Fig. 14, arrows) would be needed.
5 Experimental studies
The theoretical discussion on the device physics of OLEDs, presented in the previous section, has made clear that the effects of disorder can be strong, so that they cannot be neglected when developing a predictive model. Second-generation modeling methods, based on the EGDM or the ECDM, take the effects of Gaussian disorder into account. An important consequence is that in practice no simple relationship between the directly measured observables and the parameters describing the mobility function exists. The Mott–Gurney square law, e.g., cannot be used as a simple means for deducing the mobility. It does not include the effect of diffusion and it disregards the fact that the mobility is actually a complex function which in sandwich-type devices is position-dependent and which varies as a function of the voltage due a field and carrier density dependence. Methods used to determine the OLED materials parameters are therefore nowadays based on least-squares-fitting algorithms 28, 29. Such approaches are already used successfully in many other fields in which inverse problems are solved, e.g., seismology and 3D-X-ray medical imaging, but they have only recently emerged in the field of OLED materials parameter extraction. In its generalized form the outcome of a parameter extraction procedure is a set of optimal parameters plus a covariance matrix stating the uncertainties of each parameter and their correlation coefficients 86. A similar inverse approach is also needed for obtaining other information about the functioning of OLEDs. The internal recombination profile, e.g., may be deduced using least-squares-algorithms from the complete wavelength, angular and polarization dependent emission intensity 87, 88. Simulation-assisted methods are also needed for deducing the built-in voltage from the results of C(V) measurements or electro-absorption measurements (see Section 4.2).
Traditionally, the mobility in OLED materials has often been studied using TOF measurements 10. In such experiments the carrier concentration used is relatively small, typically 10−5. One has to judge in each case carefully whether the transport in the Boltzmann regime is probed, yielding correctly µ0,EGDM or µ0,ECDM, or whether an error is made due to the carrier density dependence of the mobility outside that regime. Clear evidence for such an error was found for the case of a TOF study of host–guest systems, providing a solution to a previously puzzling guest-concentration dependence of the mobility 43. It is now evident that the TOF mobility cannot be used in predictive OLED models, as it provides at best a lower limit to the real, concentration dependent, mobility. A further complication, which arises in any transient experiment, is the role of charge-carrier relaxation 33. As a result of a sudden increase of the carrier density in a disordered organic semiconductor, the mobility is initially high, and decreases subsequently due to energetic relaxation to lower-energy states. The effect is directly observable as a time-dependence of the mobility in TOF experiments, and will also affect the analysis of other transient experiments. It has been demonstrated recently for the case of capacitance–voltage experiments how the effect can be included as an extension of the EGDM in device modeling, without the introduction of additional parameters 89.
For all these reasons, we do presently regard the use of steady-state J(V) measurements on sandwich-type devices, carried out for a wide range of voltages, temperatures, and layer thicknesses, as the preferred method for determining the validity of the EGDM or the ECDM (including if needed a trap distribution), and for deducing the EGDM or ECDM parameters. In Table 1, an overview is given of such studies, performed so far only for a small number of polymers and small-molecule systems. Examples of systems studied are the copolymer PF-TAA (see Section 5.2) and the small-molecule materials α-NPD 90 and BAlq 91. The picture which emerges, based on this limited work, is that
(1)The EGDM and ECDM are more appropriate to polymers and small-molecule materials, respectively. However, it is not clear how general this is. A distinction between both models has in all cases studied been made from the hopping site density Nt as obtained from an analysis of a large set of measured J(V) curves for various layer thicknesses and temperatures. For the small-molecule materials α-NPD and BAlq, e.g., an ECDM analysis was found to yield values of Nt within a factor of ∼2 from the known molecular density. For some materials a consistent model could not yet be obtained 92.
(2)For describing hole transport a Gaussian DOS is found most appropriate, whereas for electron transport a DOS formed by the superposition of a Gaussian and an exponential DOS is often used. The nature of the (additional) exponential DOS for electrons (intrinsic or extrinsic trap states?) is yet not known.
(3)The size of injection barriers is often a priori not very clear due to uncertainties related to the interpretation of spectroscopic measurements and due to the possible formation of interface dipole layers, e.g., due to the presence of a net polarization of organic thin film 93. It is therefore advantageous to make use of symmetric sandwich-type devices within which the electrodes are made of equal and highly conductive molecularly n or p doped layers with small injection barriers 94.
(4)So far no direct methods have emerged from which the shape of the (tail of the) DOS can be deduced from experiment with sufficient (∼10 meV) accuracy.
Table 1. Examples of analyses of the charge-carrier mobility and OLED luminance using second-generation OLED modeling.
In Section 5.2, a full model based on experimentally determined mobility functions is discussed for the case of a single-layer blue-emitting OLED based on the co-polymer PF-TAA. Second-generation modeling of full multilayer white OLEDs, also based on separately measured electron and hole mobility functions for each of the layers, is in progress.
5.2 PF-TAA based polymer OLEDs
Intensive studies have been carried out of the effects of disorder on the charge-carrier transport in blue-emitting polyfluorene-(7.5 mol% triarylamine) (PF-TAA) copolymers, from the Lumation™ Blue Series commercially available from Sumation Co., Ltd. The structure of these co-polymers is given as an inset in Fig. 15 and a schematic energy level scheme is given in the top part of Fig. 16 for the hole-only and electron-only devices studied. The hole transport is due to hopping in between the TAA units, whereas the electron transport is due to hopping via the PF-LUMO states. Experimental evidence that for the 7.5% concentration used the hole transport is in the guest–guest hopping regime has been obtained from a study of the current density of otherwise identical devices as a function of the TAA concentration, shown in Fig. 1595. The presence of the TAA units gives rise to an improved efficiency by (i) facilitating the injection of holes (there is a ∼0.6 eV injection barrier from the PEDOT:PSS anode to the polyfluorene HOMO states) and (ii) by reducing the hole mobility so that it is better balanced with the electron mobility. Studies on similar types of OLEDs were reported in Refs. 98 and 99.
Based on separate EGDM hole-only (HO) 20 and electron-only (EO) 96 device studies a combined experimental and modeling study of the current density and luminous efficacy of double-carrier (DC) devices was recently presented by van Mensfoort et al. 53. The devices studied were
EO: glass/AlOx/PF-TAA/LiF/Ca/Al, and
respectively. Figure 16a and b show that for the HO devices the conventional model (including diffusion but neglecting the carrier concentration dependence of the mobility and assuming a PF field dependence) would lead to a rather good fit at room temperature, but that a clear inconsistency is found when studying the layer thickness dependence at low temperatures. In contrast, the J(V) curves were found to be described excellently using the EGDM, as shown in Fig. 16c and d. Similarly, the EGDM was found to provide a good description of the J(V) curves of the EO devices, as shown in Fig. 16e and f for three layer thicknesses and for a range of temperatures, provided that in addition to the Gaussian DOS also an additional exponential density of low-energy (trap) states was assumed. In Fig. 17a an overview is given of the parameters describing the shapes of the HOMO and LUMO densities of states. The analysis revealed hole and electron injection barriers from the PEDOT:PSS and LiF/Ca/Al electrodes equal to ∼0 and 0.3 ± 0.1 eV, respectively.
The width of the hole DOS, σh = 0.13 ± 0.01 eV, is close to the value obtained for two different PPV-derivatives 18. The HOMO site density obtained, Nt,h = (6 ± 1) × 1026 m−3, may be compared to the estimated TAA volume density, ∼ 1.8 × 1026 m−3100. De Vries et al. 95 showed that an analysis within the framework of the ECDM could provide an equally good fit to the data, for all layer thicknesses and temperatures, with σh = 0.085 ± 0.005 eV and Nt,h = (5 ± 2) × 1027 m−3. This showed that a successful analysis of the J(V) curves using the EGDM or the ECDM by itself does not yet convincingly prove that the disorder is completely random or correlated. However, the large value obtained using the ECDM for Nt, more than one order of magnitude larger than the estimated experimental value, was argued to provide a clear indication that for PF-TAA the site energies are uncorrelated. In the same study it was shown that, in general, the EGDM and the ECDM lead to distinctly different predictions concerning the mobility in devices in which large carrier densities can occur, e.g., in OLEDs with internal interfaces at which blocking takes place.
For the case of electron transport, van Mensfoort et al. 96 observed that the width of the Gaussian DOS, 0.07 ± 0.02 eV, is significantly smaller than that of the hole DOS. It coincides with the 0.07–0.10 eV range reported for hole transport in poly-octyl-fluorene (PFO) 101, 102. This is consistent with the point of view that the electron transport is due to hopping in a Gaussian DOS formed by the PF-derived LUMO states, with a similar width of the DOS as that of the PF-derived HOMO states in a polymer which does not contain TAA units. The site density obtained, Nt,e = (1.0 ± 0.5) × 1027 m−3, is consistent with the estimated average intersite distance a ≈ 1.1 nm as obtained from the volume density of fluorene monomer units in PF-TAA. The authors demonstrated that the electron transport could not be described properly without the assumption of the presence of an additional trap DOS. As shown in Fig. 17a, the authors assumed an exponential electron trap DOS of the form g(E) = Ntrap/(kBT0) exp[E/(kBT0)]. Following earlier literature, the 1/e decay length is expressed as kBT0, with T0 the “characteristic trap temperature.” In the figure, the trap DOS is strongly enlarged in order to make it clearly visible. Only the low-energy tail of the trap states is relevant to the transport process. At higher energies, the large Gaussian DOS dominates. The detailed shape of the trap DOS in that energy region is not probed by transport measurements. In the figure, we have depicted the trap DOS as if it continues until the top of the Gaussian DOS. Ntrap is then equal to the total volume density of trap states. With no consequence to the predicted transport properties, it would be possible to assume a different shape of the part of the exponential DOS in the energy region in which it is buried under the high Gaussian DOS, e.g., by choosing a cut-off at a lower energy or by assuming a peak in the trap DOS 103. That was not done, in order to avoid the introduction of additional parameters which could not be validated experimentally. The parameter values describing the trap DOS, Ntrap = (1.0 ± 0.5) × 1024 m−3 and T0 = 2100 ± 300 K, are similar to the values obtained in previous reports on a variety of organic semiconductors.
Figure 17b shows the temperature dependence of µ0, the mobility in the limit of zero carrier density and field. As expected from the EGDM, it varies with temperature as , with a slope parameter C equal to 0.39 ± 0.01 for hole transport 20 and 0.34 ± 0.10 for electron transport 96. Both values are consistent with the range of values of 0.38–0.46 following from semi-analytical percolation theory (see Section 4 and Ref. 19), depending on the wavefunction decay length.
A comparison of the current–voltage curves of the HO and EO devices shows that the latter curves are much steeper, and that for an equal device thickness the curves cross at a certain voltage. The large slope of the EO curves is the result of the presence of traps, as may be understood already semi-quantitatively from the (oversimplified) Mark-Helfrich model 58. For 100 nm single-carrier PF-TAA devices, equal electron and hole current densities are observed at approximately 2.8 V above Vbi. The DC devices, with Vbi ≈ 3 V, would then be expected to be optimally efficient around 6 V. This highly simplified picture was indeed found to provide a useful view on the functioning of these OLEDs, as shown below.
Figure 18 shows the experimental and modeling results obtained for the DC devices with a 100 nm PF-TAA layer. Figure 18a shows that for a wide range of temperatures the measured current–voltage curves (symbols) are in excellent agreement with the prediction (full curves) from a 1D-ME double-carrier OLED model using the electron and hole mobility functions obtained from the separate studies discussed above. Langevin recombination was assumed, including recombination of holes with trapped electrons as described in Section 3.2. Figure 18b shows the current efficiency at room temperature, as measured (open spheres) and predicted (closed squares) for the case of perpendicular emission. The emission was obtained from the calculated voltage dependence of the recombination profile, using an optical microcavity light-outcoupling model (LIGHTEX 104) in a manner described in detail in ref. 53. Only the fraction of singlet-excitons formed was regarded as a free parameter. The best fit, shown in the figure, yielded ηS = 22%, very close to the quantum-statistical value of 25% and within the interval obtained from an independent study for the same material yielding 10–25% 97. The figure shows that the model excellently predicts the measured maximum in the current efficiency, and it explains this maximum as the result of a voltage dependence of the light-outcoupling efficiency. As shown in Fig. 18c the emission profile is predicted to shift from a region more close to the cathode at low voltages, to a region more close to the anode at high voltages. At intermediate voltages, the recombination is least quenched due to the presence of nearby electrodes. The light-outcoupling efficiency and hence the current efficiency are then largest. This shows that predictive 1D device modeling of OLEDs is feasible.
6 Conclusions and outlook
On the basis of 3D ME and MC modeling, we have developed a set of compact expressions describing the temperature, electric field and charge carrier density dependence of the mobility in organic semiconductors for the cases of random and spatially correlated Gaussian disorder. The resulting EGDM and ECDM mobility models are the basis of “second-generation” OLED device models, which also include the consequences of the disorder on the recombination rate (bipolar mobility, Section 3), the injection from the electrodes and the transport across internal organic–organic interfaces. Commercial second-generation simulation tools have recently become available 28, 29. In order to theoretically validate these OLED device models, the 1D-modeling results were compared with the results of 3D-modeling of complete devices including the effects of energetic disorder, injection barriers, and short-range-Coulomb interactions on the current density, and showing its filamentarity. Furthermore, experimental validation studies using steady-state J(V) curves of single-carrier devices were carried out for various OLED-relevant materials. Predictive EGDM-based OLED modeling was demonstrated for single-layer OLEDs based on a blue-emitting copolymer. We find that extending the 3D simulations to complete multilayer OLEDs, including the effects of internal interfaces and of fluorescent or phosphorescent emitters, is computationally feasible 105. That work is beyond the scope of the present chapter, and will be reported elsewhere.
Even if a complete theoretical framework is available for describing the mobility in OLED materials, the development of an accurate description of the mobility function of a specific novel material has turned out to be more involved than was thought some years ago. Due to the carrier density dependence of the mobility and the role of diffusion, often-used simple analytical expressions describing the steady-state or transient current density are usually not applicable. Simulation-assisted parameter-extraction methods are required, making use of data sets which are sufficiently “rich” by including experimental results for at least a wide range of voltages, temperatures and layer thicknesses, and if necessary by also including the results of non-steady-state measurements. For host–guest systems within which the transport takes place by direct hopping between the guest molecules or (in polymers) hopping sites (regime D in Fig. 5a), the validity of the EGDM may be investigated by comparing for a series of materials with various guest concentrations the guest site density as obtained from an analysis of the J(V) curves with the known chemical site density. This was recently demonstrated successfully for the case of a series of polyspirofluorene-based polymers with a varying hole-transporting unit (HTU) concentration 106.
There are several unresolved or only partially resolved issues, such as (i) the relationship between the deposition conditions, the molecular-scale structure, the shape of the DOS, and (for a Gaussian DOS) the experimental parameters σ and µ0, (ii) the occurrence and type of spatially correlated disorder, (iii) the microscopic origin of electron trap states, (iv) the description of the mobility in host–guest systems at intermediate guest concentrations (as occurring in many phosphorescent OLEDs), (v) the effect on the transport across interfaces of a net polarization of layers due to non-random molecular dipole orientation 24, and (vi) the effect of disorder on excitonic processes such as exciton diffusion and exciton-exciton or exciton-charge interactions. Concerning the shape of the DOS, it should be mentioned that neither for polymers nor for small-molecule materials it is a priori evident that the shape of the DOS should be Gaussian. We note that recent studies of the shape of the DOS in various types of organic semiconductor materials, studied in lateral (field-effect transitor type) devices, suggest that it can have a more complex structure, in some cases being more close to an exponential shape 107. Concerning the microscopic origin of electron trap states, we remark that recently for a large variety of polymer semiconductors evidence has been obtained that they have a common origin, most likely related to hydrated oxygen complexes 108. The position and volume density of such such states was essentially identical for all polymers investigated. Interestingly, a recent theoretical study of the DOS in a typical two-component (host–guest) small molecule material as used in OLEDs shows that in the case of a change of the molecular polarizability when it becomes charged, the interaction with the randomly oriented dipole moments on the surrounding molecules will give not only give rise to a shift and an (approximately Gaussian) DOS broadening, but also to an exponential tail of the DOS 109. Although these effects could qualitatively already be understood within the framework of a lattice model, with point charges and (polarizable) point dipoles on molecular sites, they were found to be orders of magnitude larger when taking the actual charge distribution within the molecules into account.
In view of the large number of outstanding issues and these recent developments, a continued development of novel or refined experimental and theoretical methods will be needed. Within the “third-generation” OLED modeling methodology that we envision (see Fig. 2), the material-specific mobility parameters will be predicted by microscopic modeling of the morphology of OLED materials by MD or MC methods and the determination of the hopping rates from DFT calculations of the charge-transfer integrals and reorganization energies. After extending such studies to the interfaces, this will allow fully predictive modeling of the charge transport through OLED devices. A further extension, to include the motion and interactions of singlet and triplet excitons, will open the perspective of obtaining a fully predictive electro-optical OLED model.
The complexity of high-efficiency OLEDs presently grows rapidly due the use of many layers of different organic semiconductors with specialized functions. As a result, improving the performance of OLEDs using trial-and-error methods is becoming less feasible. We envisage that the use of “second-generation” OLED modeling as described in this chapter can contribute to solving this problem by providing rational design strategies and by providing a predictive description of their functioning, based on measured materials parameters for each layer. The future development of “third-generation” OLED modeling would further contribute by providing deepened insights in the relationships between the molecular and thin film structure of the organic semiconductors used and the resulting charge transport and excitonic properties.
We gratefully acknowledge the contributions to the work presented in this chapter by J. Billen, M. Bouhassoune, M. Carvelli, J. Cottaar, W.Chr. Germs, H. Greiner, J.J.M. van der Holst, H.-P. Loebl, R.A.J. Janssen, S.L.M. van Mensfoort, F.W.A van Oost, W.F. Pasveer, B. Ramachandhran, V. Shabro, R.J. de Vries, S.I.E. Vulto, and Y.Y. Yimer. The research described has received funding from the European Community's Seventh Framework Program under Grant No. 213708 (AEVIOM, contribution R.C.).
Reinder Coehoorn is Research Fellow of Philips Research Laboratories, Eindhoven, The Netherlands, in the Department of Photonic Material and Devices, and he is part-time professor at the Eindhoven University of Technology, in the group Molecular Materials and Nanosystems. His present field of research is the development, experimental validation, and application of device models for organic light-emitting diodes. Previously he worked on photoelectron spectroscopy, band structure theory, permanent magnets, and giant magnetoresistance materials.
Peter Bobbert is Associate Professor in Theory of Functional Materials in the Department of Applied Physics at Eindhoven University of Technology. He is active in the field of organic electronics, focusing on theory and simulation of charge transport and spin dynamics, with applications in organic light-emitting diodes, organic field-effect transistors and organic spintronics. He has previously worked on Green-function techniques for the calculation of electronic excitations in semiconductors and on the theory of quantum phase transitions.