3D Master Equation for Exciton dynamics in oleds

We describe a novel simulation approach for excitonic organic light-emitting diodes (OLEDs) which combines a continuous one-dimensional (1D) drift-diffusion (DD) model with a three-dimensional (3D) master equation (ME) model. This approach describes the exciton dynamics in a multilayer OLED stack with an additional coupling to a thin-film optics solver.

Our approach effectively combines the computational efficiency of a 1D drift-diffusion solver with the physical accuracy of a discrete 3D ME model, where excitonic long-range interactions for energy transfer can be taken into account. We show that such a hybrid approach can efficiently describe the behavior of recently reported TADF OLED.

This brand-new simulation tool is now available in Setfos.

Introduction

The exact mechanisms and interactions between charge carriers, excitons, and photons in organic semiconductors for OLEDs are complex, and no analytical solutions describing the dynamics are available. Numerical methods have to be applied to investigate the processes in organic light-emitting diodes (OLEDs) and to support the development process.

Two widely used frameworks are continuum approaches, such as deterministic drift-diffusion (DD) models, and atomistic approaches like the probabilistic kinetic Monte Carlo (kMC) method. A master equation (ME) model is situated in between these two approaches as it is atomistic like the kMC yet deterministic like the DD. The ME model can be used to describe the evolution of the states' probability mass function over time. The computational costs of solving such a system are also to be found in between those of the DD and the highly demanding kMC approaches.

In the standard Setfos version, the exciton dynamics is analysed by solving multiple excitonic rate equations per semiconducting layer. An ME approach to describe the exciton dynamics provides certain advantages in the physical accuracy over the former as transport can be accurately described using established expressions for Förster and Dexter energy transfer rates where long-range interactions are taken into account. Furthermore, the atomistic nature of ME models renders the modeling of complex host-guest systems as well as multilayer structures with various interfaces straightforward.

Recently, our team demonstrated the potential of combining a 1D charge transport model based on drift-diffusion with a 3D master equation (ME) model for exciton transport [Zeder]. Since Setfos 5.2, this model is available and can be used to simulate exciton dynamics. The ME is fully coupled to the the emission and drift-diffusion module of Setfos, such that a fully coupled OLED simulation can be executed.

Implementation of the Simulation

The ME model allows considering long-range transport of excitons via Förster and Dexter processes also across layer interfaces. Here, we will present a fully coupled OLED simulation that uses the ME model for excitons. The TADF OLED stack and several input parameters (excitonic rates, energy levels, emission spectra) were taken from a publication by the group of Prof. Adachi [Adachi].

The energy level diagram of the structure under analysis is shown in Fig. 1(a). This is how Setfos shows the alignment of the energetic levels of each material in the stack. The EML is a host-guest system, which is modeled as described later. The transport levels are set according to the HOMO and LUMO positions of the host (mCBP). The density of states N0 is reduced to 8·10^26m-3 in order to represent the weight ratio of 0.8. The guest material is mimicked as hole and electron traps using a trap density of 2·10^26m-3 (to represent the 0.2 weight ratio in the EML) and a trap depth that reflects the nominal differences of HOMO and LUMO levels of the two materials. Recombination on the host is considered as Langevin recombination, while trap-trap recombination is enabled for the guest material.

Fig. 1. (a) Energy level diagram as shown in Setfos. The host-guest EML is modelled by considering HOMO/LUMO of the host (mCBP) and the guest as hole and electron traps. (b) Used exciton energies (full black lines) and band gap energies (blue dashed line) of all materials.

The master equation model is enabled for all semiconducting layers, except for the HAT-CN injection layer. All ME transport layers consist of just one material, thus, one only has to define material 1 parameters. Generation of excitons occurs through Langevin recombination, with prefactors of 0.25 and 0.75 for singlets and triplets, respectively. Binding energies are set according to the nominal difference of exciton energies and bandgap energy of the layer (see Fig. 1(b)).

For the EML we define two materials with a ratio of 0.8:0.2. As described in the previous section, Langevin recombination is the source for exciton generation on the host, and trap-trap recombination is used to generate excitons on the guest (material 2). Fig. 2 shows a screenshot of the Graphical User Interface (GUI) of Setfos for ME settings of the guest material. As we are dealing with a TADF emitter, we enable the Triplet –> Singlet transfer to calculate also the reverse intersystem crossing (risc) contribution.

Fig 2: Implementation of a simulation in the Graphical User Interface of Setfos. Master equation settings for the guest (material 2) of the EML.

General Förster and Dexter radii are set to 1.5 and 0.8 nm, respectively, in the “MASTER EQUATION setting” tab.

Note that Setfos currently only considers one Förster and one Dexter radius. Despite this model restriction, host to guest transfer is still more likely due to the lower exciton energy of the guest. The same also holds for transfer to other layers.

Finally, we couple the radiative recombination in the EML, EBL, and HBL to 4 emitters. This is defined in the emitter settings, using the option ME exciton coupled with the respective exciton type as distribution. The calculated exciton densities and selected rates are shown in Fig. 3. While the generation of excitons occurs mostly at the EML/HBL interface (see Fig. 3(b)), the steady-state distribution shows a less steep decrease of excitons in the EML but also in the transport layers (see Fig. 3(a)). This is the effect of the exciton diffusion which we have taken into account by 3D Förster and Dexter transfers. The radiative decay of singlets on material 2 in the EML (i.e. the emitter) is dominating all the other radiative rates in the stack. Therefore, the emission spectrum (not shown here) is almost solely given by the guest spectrum, and surely the cavity, but shows almost no feature of the transport layers or the host.

In the above first example, we can see that the emission is predominantly occurring from the EML/ETL interface. This is mainly due to the high barrier for electrons at this interface (see charge density profile). However, this is not what the authors of reference [Adachi] reported. As the guest concentration in their EML is 20%, it is reasonable that electrons are not injected and transported on the host (mCBP), but through the TADF guest material itself. In order to mimic this situation in Setfos, we, therefore, adapt the simulation as such:

The only layer properties that need to be adapted are the EML parameters. First, we set the LUMO level in Setfos to the value of the guest, i.e. 3.3 eV. Since the hole trap is still there to represent the guest, the recombination on guest molecules would now be considered as Shockley-Read-Hall (SRH) recombination. Therefore, we enable SRH recombination in the drift-diffusion tab and reduce the Langevin recombination efficiency. Lastly, in the master equation tab of the EML, we change the generation type of material 2 excitons to SRH and adjust the binding energies of both materials in the EML to the new "band gap" of 2.8 eV.

In this adapted simulation, the charge accumulation at the HBL/EML interface is strongly reduced, leading to a higher current through and emission from the device. First, we note that the total recombination is dominated by the SRH contribution in the EML (see Fig. 4a). This confirms the suitability of the chosen recombination settings. Most importantly, the exciton densities (Fig. 4b) are now highest at the EML/EBL interface, which agrees with the report by Noda et al. [Adachi]

Figure 4: (a) Recombination profile and (b) exciton densities for the adapted simulation in Setfos.

For further examples of the coupled drift-diffusion - master equation model and comparison with literature data, the reader is referred to ref [Zeder].

Conclusions

A hybrid model for OLEDs consisting of a 1D Drift-Diffusion (DD) model for the electronic part and a 3D Master Equation model for the excitonics has been presented.

The two models are coupled through the charge carrier densities (through polaron quenching), recombination rates, and the electric field (weak electric field dependency of Dexter transfer rates). Furthermore, the 3D ME model can be directly coupled to an optical thin-film simulation providing the necessary information of cavity effects and light outcoupling.

The 1D DD model can be solved very efficiently and fast. At the same time, the 3D ME model allows for physically sound transport modeling using Förster and Dexter transfer rates. This can take into account long-range interactions, which could be additionally introduced to describe non-local quenching processes for even higher physical accuracy. Furthermore, the discrete nature of the model allows for very easy treatment of layer interfaces, as well as interspersed host-guest complexes. Despite the lower dimensionality of the DD model, the subsequent ME simulation for the exciton dynamics yields physically accurate results with the benefit of lower computation times compared with full 3D ME or kinetic Monte Carlo (kMC) approaches for both the charge carrier and the exciton dynamics.

The scope of applicability of the present model includes OLEDs, quantum dots, and other optoelectronic devices featuring materials with strongly discrete structures. It can be used for steady-state as well as transient simulations of such devices. The modeling of host-guest complexes is also possible through trapping dynamics in the 1D DD model, which is then translated into different densities and rates for host and guest molecules in the 3D ME formalism.

The ME model is able to take into account a multitude of physical processes at still moderate computational cost.

This blog is based on the work carried on by Simon Zeder and colleagues: “Coupled 3D master equation and 1D drift-diffusion approach for advanced OLED modeling” as well as the Setfos tutorial example.

References:

• [Zeder] S. ZEDER, C. KIRSCH, U. AEBERHARD, B. BLÜLLE, S. JENATSCH, B. RUHSTALLER “Coupled 3D master equation and 1D drift-diffusion approach for advanced OLED modeling”, J. Soc Inf Displ, 28 (2020) DOI: 10.1002/jsid.903

• [Adachi] H. NODA, H. NAKANOTANI, C. ADACHI “Excited-state engineering for efficient reverse intersystem crossing”, Science Advances, 4 (2018) DOI: 10.1126/sciadv.aao6910

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