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Abstract
To model the carrier transport in organic light-emitting diodes (OLEDs) with random dopant effects in the emitting layer, two-dimensional simulation was used. By including the Gaussian shape density of states and field-dependent mobility in the Poisson and drift-diffusion solver, the carrier transport, trapping in the dopant state, and radiative recombination were accurately modeled. To examine the model, the current-voltage characteristics of organic light-emitting devices were compared. The host material in the emitting layer was 2,2-bis(1-phenyl-1H-benzo[d]imidazol-2-yl)biphenyl (BImBP), which was doped with bis[2-(4,6-difluorophenyl)pyridinato-C2,N](picolinato)iridium(III) (FIrpic) at various concentrations. By including the random doping model, the trend of mobility was altered and the radiative efficiency fitted experimental values well.
© 2017 Optical Society of America
1. Introduction
Organic light-emitting diodes (OLEDs) have received considerable attention since their first demonstrations by Tang et al. [1, 2], and have been gradually more frequently used in display technologies over the past decade. Compared with liquid-crystal display technology, OLEDs have a wider viewing angle, higher brightness, higher power efficiency, and higher contrast ratio, and they can be fabricated on flexible substrates [3]. Because of these advantages, OLEDs are expected to play a major role in display technology and have high potential for use in the next generation of lighting technology.
Recently, several numerical algorithms have been used to model light-emitting diode transport behaviors [4]. The first is the Monte Carlo method, which considers random site energies on a microscopic level. This method is somewhat time consuming and usually applied only in carrier transport studies [5]. The second method is based on hopping transport, also called the master equation. This method solves the probability evolution equation [6, 7]. The last method uses the Poisson and drift-diffusion model [8–10], which is presented in this paper with some additional new algorithms. Traditionally, the Poisson and drift-diffusion model is used to simulate band diagrams, carrier transport, and recombination in semiconductor devices, to which it is well fitted [11–13]. However, unlike traditional semiconductors, the bandgap and density of states (DOS) of organic materials are not well defined. There are tail states above the highest occupied molecular orbital (HOMO) and below the lowest unoccupied molecular orbital (LUMO) in organic materials because the molecules are amorphous and disordered [14, 15]. The process of carrier transport, also called the hopping process, is quite different from the traditional diffusion process of inorganic semiconductors [16, 17]. Therefore, we had to modify the traditional drift-diffusion equations by considering the Gaussian DOS [18–20] and field-dependent mobility [21, 22].
In a phosphorescent emission system, the emitting layer (EML) is doped with another material such as Bis[2-(4,6-difluorophenyl)pyridinato-C2,N](picolinato)iridium(III) (FIrpic) to form a host-guest system to improve the quantum efficiency (QE) [23–26]. The behavior of carrier transport is affected by the doping concentrations of guest materials, and the relationship between mobility and doping concentration is not linear nor easily predictable [27]. Furthermore, studies have shown that under a very low concentration of the guest material, trap-like behaviors are observed in the host material and the hopping ability of carriers is significantly reduced [28]. Predicting the transport behavior by using a one-dimensional (1D) model is very difficult since the model cannot account for random doping effects [29, 30]. The 1D Poisson and drift-diffusion model can be used mostly to fit known experimental results; it is difficult to predict the transport behavior of unknown materials with the 1D model. In the guest-host system, to model the randomly distributed dopants in the host material, at least two-dimensional (2D) simulation model is preferred to describe the random distribution. Therefore, in this paper, we apply a 2D simulation program to deal with this problem. The following sections present the 2D simulation program developed in this study to handle the random dopant effects.
2. Simulation method
This section explains the Poisson and drift-diffusion solver developed in this study and presents the modified models, including the Gaussian DOS, field-dependent mobility, random dopant model, and exciton diffusion models.
2.1. Poisson and drift-diffusion model with Gaussian density of state
For the description of electrical characteristic in OLEDs, the basic method is using the finite element method to solve 2D Poisson and drift-diffusion equations at steady state. The set of equations is expressed as follows:
where ε is the permittivity; φ is the electrostatic potential; n and p are the local free electron and hole carrier densities; Efn and Efp are the quasi-Fermi levels of electron and hole; ntrap is the trapped carrier density; Jn and Jp are the electron and hole current densities, respectively; q is the elementary charge; and μn and μp are the electron and hole mobility, respectively. Radiative recombination and the Shockley-Read-Hall (SRH) recombination are also considered. B0 is the intrinsic radiative recombination coefficient; ni and Ei are the intrinsic carrier density and intrinsic energy level, τn and τp are the nonradiative carrier lifetimes; Etrap is the trap energy level; kB is the Boltzmann constant, and T is temperature (300 K in this paper).As previously mention, the carrier transport property is quite different from that of traditional semiconductors and organic materials. Due to the amorphous and disordered molecules, a number of states exist above LUMO and below HOMO; thus, carriers can hop into the lower energy states through these tail states or be trapped by the trap states. Studies [5, 31–34] have shown that the Gaussian distribution can be adopted to illustrate the carrier transport (Fig. 1 (a)) according to the absorption spectrum of organic materials. The Gaussian DOS is expressed as Eq. (6).
In Eq. (6)Ntail is the total number of tail states, σtail is the standard deviation, and Etail is the mean of the distribution. Thus, the total free carrier density is modified as follow: where fe is the Fermi-Dirac distribution function. Similarly, we model the trap states by using the Gaussian distribution, expressed as follows: where Ntrap is the total number of trap states, σtrap is the standard deviation of trap states, and Etrap is the mean of the trap state distribution. Hence, through Gummel’s iteration, these equations can be solved until the results converged.
Fig. 1 (a) Illustration of DOS in an organic material. Ndos is the traditional three-dimensional DOS in semiconductor model. Ndos,tail is the actual DOS in organic materials. (b) Energy level diagram of OLED used in this study. (c) Energy level diagram of the electron-only device (EOD) for BImBP. (b) Chemical structures of NPB, mCP, TAZ, BImBP, and FIrpic.
2.2. Field-dependent mobility model
Carrier mobility in organic materials is influenced by disordered molecules and frequently observed that it is electric field-dependent [21, 22]. This study applied the form, often called Poole-Frenkel-like mobility, to handle this problem.
In Eq. (9), μ is the mobility, μ0 is the zero-field mobility, β is the factor of the field dependence, and F is the localized electric field. The electric field is obtained from Poisson equation, and the updated mobility is fed back into drift-diffusion model. These three equations are solved self-consistently.2.3. Random dopant model
Since modeling random dopant effects in a 1D simulation is difficult, a 2D random dopant model is required to describe this behavior. The fundamental concept is that the host material is doped in the guest material randomly [35]. A similar method was successfully applied to study random alloy effects in quantum wells [36]. To describe a random molecular distribution, we set a basic mesh area of 1.2 by 1.2 nm, which is close to the typical expanded molecular size of FIrpic or the host material [37]. We the had to decide whether the area would be fully occupied by the guest or host material, or partially occupied by the guest material. In practice, if such an area is not fully occupied by the guest material, its molecular orbital properties may be a mix of guest and host materials. Hence, we further divided this basic area into different numbers of sub-domains to determine the guest material’s local average composition. One, two, four, and six sub-domains were used to examine which averaging method provides the optimal fitting results for the transport properties. As shown in Fig. 2(a), each basic area would be FIrpic or the host material if the number of sub-domain was one. If the number of sub-domains in the basic area were larger than one; for example, six (Fig. 2(d)), the total number of sub-domains would be six times higher; consequently, we would need to generate six times as many random numbers to determine the average composition in the basic area. The local average FIrpic composition in the basic area is the average of the six sub-domains. For instance, the illustration in the upper right of Fig. 2(d) shows that three of six sub-domains are FIrpic in the area. As a result, the FIrpic composition of the area is 50% which is calculated from the average result in six sub-domains.
Fig. 2 Distribution of FIrpic composition with (a) one sub-domain, (b) two sub-domains, (c) four sub-domains, and (d) six sub-domains.
2.4. Exciton diffusion model
When excitons are formed by electron and hole pairs, they diffuse and decay in the materials. Understanding the exciton distribution is necessary to analyze the internal quantum efficiency (IQE). We employed the following exciton diffusion equation [41–45]:
where nex is the exciton density; Dex is the diffusion coefficient of the excitons; τ is the relaxation time of the excitons; Kr and Knr are the radiative and nonradiative recombination rates, respectively; γ is the annihilation rate constant; and G is the exciton generation rate computed by the solver, which is equivalent to the recombination rate B · n · p (Eq. (4)). In a steady state, the exciton distribution nex can be solved at each position. Therefore, the radiative current density is expressed as follows: The IQE was calculated as Jrad/Jtotal. Finally, we could compare the efficiency with the experimental data.3. Result and discussion
In the first stage, we had to calibrate the random dopant model before simulating the whole device with different dopant concentrations. Fig. 1(b) shows the energy level diagram of the OLED device which was composed of a 50-nm-thick N,N′-diphenyl-N,N′-bis(1-naphthyl) (1,1′-biphenyl)-4,4′diamine (NPB) for a hole injection layer (HIL), a 10-nm-thick N,N′-dicarbazolyl-3,5-benzene (mCP) for a hole transporting layer (HTL), a 35-nm-thick 2,2′-bis(1-phenyl-1H-benzo[d]imidazol-2-yl)biphenyl (BImBP) [46] for an emitting layer (EML), and a 30-nm-thick 3-(4-Biphenylyl)-4-phenyl-5-tert-butylphenyl-1,2.4-triazole (TAZ) for an electron transporting layer (ETL). In the EML, FIrpic was utilized as a blue emitting phosphorescent dopant. Notably, we set the mesh size very small to model the random dopant effects. Furthermore, large memories and computation times were required, so we set the simulated structure to have a width of 100 nm to maintain computing efficiency. However, a modeled area of only 100 nm may not have represented the whole large area. We implemented different random seeding maps to observe how the results converged as more maps were averaged. Initially, 100 random maps were calculated at different biases levels to determine current variations with different random maps. In Fig. 3, the blue lines denote the current density of the maps at (a) 3 V and (b) 11 V. The results show an approximately 10% variation in the current densities of different random maps. The average current density of 100 maps is plotted as the dashed black line. It still needs an inefficient and long computation time to simulate 100 maps. Thus, we needed to determine a sufficient number of maps. Subsequently, we calculated the average results of the first map to the Nth maps, which are plotted as the red lines. Compared with the average current densities calculated based on the different numbers of seeding maps, the results show that the errors are less than 2% when more than 10 different random maps were used at different bias levels. To prevent the variations from affecting our conclusion, we adopted 20 different seeding maps to obtain a more convergent result from which to draw conclusions.
Fig. 3 Current densities with different seeding maps used at (a) 3.0 V and (b) 11.0 V. The blue dots represent the current densities in each seeding map and the dashed black lines represent the average current density. The red lines are the errors of the average current densities calculated based on the different numbers of seeding maps used.
As previously discussed, we needed to identify the best fitting method for representing the transport properties through the random dopant model. Accordingly, we determined how many sub-domains can provide the best fit to the mobility behavior at different doping concentrations. Therefore, we demonstrated an EOD because it wold be affected only by the effective mobility. Fig. 1(c) shows the energy level diagram of the EOD for the BImBP doped proportions of FIrpic, and we set both contacts as ohmic so that the contact resistance could be neglected. The Gaussian DOS and field-dependent mobility model were added in each layer, and the parameters are listed in Table 1. The mobilities of FIrpic were referenced from the literature, the others are determined based on the previous fitting results from our laboratory’s experiments [46]. Figs. 4(a) and 4(b) show the LUMO distributions at a FIrpic of 12% with one sub-domain and six sub-domains. The contacts were positioned at both ends (the left and the right ends of the map), where the film thickness was 70 nm. The LUMO of FIrpic was below that of BImBP; hence, the yellow and blue areas represent FIrpic, whereas the other areas represent BImBP. Although the amount of FIrpic in Fig. 4(b) appears to more than that in Fig. 4(a), the FIrpic concentrations were equal because the FIrpic in Fig. 4(b) had more sub-domains (i.e., the composition of each FIrpic area in Fig. 4(b) was lower than that in Fig. 4(a)), and the DOS of FIrpic was smaller in Fig. 4(b). To obtain the effective mobility, an electric field of 3 × 105 V/cm was applied on both sides (at 2.1 V), and the space-charge-limited current (SCLC) method [47] was used to calculate the effective mobilities with different concentrations of FIrpic (Fig. 4(c)). As the number of sub-domains increased, the change from trapping to detrapping occurred at a lower concentration of FIrpic; for example, the lowest effective mobility occurred at approximately 20% FIrpic with one sub-domain. By contrast, the lowest effective mobility occurs at approximately 3% FIrpic with six sub-domains. If more sub-domains are used to determine the average composition in each mesh element, the dopants (FIrpic) are more diffused. Fig. 4(b) shows that carriers which fall in the dopant state find the percolation path more easily. Therefore, mobility starts to increase again at a relatively lower doping concentration. However, if only one sub-domain is used, the doping concentration in each mesh element will be either 0 or 1 (Fig. 4(a)). Hence, a higher number of carriers are localized and the behavior is similar to that of a trapped charge. After comparison with experimental results, we found that six sub-domains provided the optimal fitting result.
Table 1. Simulation parameters of the electron-only device (EOD) and the whole devices.
Fig. 4 LUMO distribution with 12% FIrpic in (a) one sub-domain and (b) six sub-domains. (c) Simulated results for the dependence of the mobility on FIrpic concentration with different number of sub-domains at 3 × 105 V/cm. (d) Illustration of the random method for six sub-domains.
After determining the number of sub-domains for modeling the random dopant effects, we modeled the OLED devices. The energy level diagram is shown in Fig. 1(b). The same model was used for the experiments, and the chemical structures of each material are illustrated in Fig. 1(d). As mentioned previously, to obtain a convergent result, the average of the results from 20 different maps was calculated. Fig. 5(a) shows the J-V curve fitting results with FIrpic doping concentration of 0% and 12%. All the parameters are shown in Table 1. We found that the J-V curve of the doped device decreases significantly. The simulation results show that the carrier transport is limited by the trapped state which originated from the dopants. Because FIrpic has a much lower energy level, the carriers that fall into this lower energy state can only (1) recombine with either electrons or holes, (2) re-escape from the trap states to the host material, or (3) percolate through the lowest energy path created by FIrpic states. This results in the decreases of mobility and current density. As shown in Fig. 5, the J-V curve fits our random dopant model well, confirming that the trapping behavior is the cause of the limited the carrier mobility. Then, we modeled devices doped with other concentrations of FIrpic such as 9% and 15% (Fig. 5(b)) by changing only the seeding concentration. On the logarithm scale, the trend of all the results are very similar to those of the experimental data. The current density increases with the doping concentration because the higher number of dopants provides more percolation paths for carrier transport. If the concentration were more than 20%, the transport behavior would more closely resemble that of a pure FIrpic device because carriers can considerably more easily percolate between FIrpic dopant states. In the BImBP layer, because the zero-field electron mobility is 1000 times higher than the hole mobility, almost all carriers accumulate at the interface between mCP and BImBP despite some FIrpic molecules being added in this region. Consequently, almost all excitons accumulate around the interface between mCP and BImBP (Fig. 6(a) and 6(b)). If FIrpic is added in this region, it would have a higher concentration than other neighboring regions. However, under a higher voltage, part of light-emitting area would not be at the interface (Fig. 6(c)) because holes could be transported into a deeper region by the increasing electric field to overcome the trapping energy.
Fig. 5 (a) J–V curve fitting results with 0% and 12% FIrpic. (b) J–V curve fitting results with 9% and 15% FIrpic.
Fig. 6 Exciton distributions for (a) 12%, (b) 15% FIrpic at 6.0 V, and (c) 12% FIrpic at 10.0 V. In each figure, EML is between the two white dashed lines.
According to previous studies [45, 48, 49], the Dex of the triplet excitons with the FIrpic is approximately 10−8 cm2/s, τ is approximately 10 μs, and γ is approximately 10−13 cm3/s. We fine-tuned these parameters, as shown in Table 2, to obtain similar quantum efficiency results. Fig. 7 shows that the normalized quantum efficiency curves of 12% and 15% doping fit well with experimental results under a low current density. However, because of the thermal effect, the experimental results for 12% and 15% doping are lower than those of simulations.
Table 2. Setting of the exciton diffusion parameters.
4. Conclusion
In this study, we developed a 2D random dopant model to simulate the dopant effects by considering the Gaussian density of state and field-dependent mobility models. The simulated results show that the number of sub-domains affects the behaviors of the dopant effects, determining the concentration of the guest material with the lowest effective mobility. When we select a suitable number of sub-domains, thex results are consistent with those of the experiments. The results adequately explain why the mobility of doped systems decreases at a very low doping concentration and then increases with the doping concentration. The trapping and percolation behaviors are well explained by the random dopant model. The J-V curve fitting of the OLED with different doping concentration fits well with the experimental result, which verifies that this model can be used for device analysis and optimization.
Funding
Ministry of Science and Technology in Taiwan (MOST) (103-2221-E-002-133-MY3, 103-3113-E-155-001, 104-3113-E-155-001, 105-3113-E-155-001, 105-2113-M-002-001, 106-3113-E-155-001-CC2).
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