A tutorial on Drift-Diffusion simulations

Case Study: Understand the Response of OLEDs & Optical Upconversion Devices

Organic upconversion devices (OUCs) are a promising technology for converting infrared light into visible light, with potential applications in process control, imaging, and communication. These devices typically consist of an organic infrared photodetector (IRPD) and an organic light-emitting diode (OLED) connected in series. Here, we describe how to fabricate, characterize and simulate high image-contrast, narrowband OUCs that convert near-infrared light to visible light. Transient photocurrent measurements show that the response speed decreases when lowering the NIR light intensity. This is contrary to conventional organic photodetectors that show the opposite speed-versus-light trend. It is further found that the response speed increases (when using a phosphorescent OLED) or decreases (for a fluorescent OLED) when increasing the driving voltage.

This blog focuses on using our simulation software, Setfos, to explain the behaviour of these devices via a thorough numerical simulation. Our results show that the electron mobility in the OLED primarily determines the response speed behaviour. It is proposed that the low electron mobility in the emitter layer sets a fundamental limit to the response speed of OUCs.

What is a Drift-Diffusion Simulation

The drift-diffusion model is a mathematical framework used to describe the behaviour of charged particles, such as electrons or ions, in semiconductor devices like transistors and diodes. The model is based on the combined effects of two physical processes: drift and diffusion. Drift refers to the directed movement of particles in response to an electric field. The electric field causes charged particles to accelerate and move in a particular direction. Diffusion refers to the random movement of particles due to thermal energy. At any given time, particles are randomly moving in different directions, leading to a leveling of the particle distribution. The drift-diffusion model is mathematically described using the continuity equation, the Poisson equation and the drift-diffusion equation, which can be solved numerically to determine the distribution of free particles at any given point in time and space.

The drift-diffusion equation is for electron:

and for holes:

where q is the elementary charge, µn,p is the electron or hole mobility, respectively; n and p denote the density of electrons or holes, respectively; E is the electric field, Dn,p the diffusion constant of electron or hole, respectively and ∇ is the nabla operator. The first summand of the equation describes the contribution from the drift, while the second summand (subtrahend for the whole equation) describes the diffusion part. The diffusion constant in this model is given by Einstein's law:

with kB the Boltzmann constant and T the temperature. This relation makes the dependence of then diffusion on temperature directly visible. The continuity equation is used to calculate the continuous transport of charge carriers from one site to the neighbouring site also taking into account sources and sinks for charge carriers. In the simulation, charged particles like electrons can be generated by the absorption of photons and splitting of the exciton into a free electron-hole pairs (electron transfer from the HOMO to the LUMO). Also, electrons and holes can be removed by recombination with a charge carrier with opposite charge or be trapped in trap states. These processes are considered by implementing generation, recombination, and trapping terms into the continuity equation for electrons and holes:

RLangevin is the bimolecular recombination of a free electron in the LUMO with a free hole in the HOMO, Rnt and Rpt describe the rate of change in free electrons or holes due to interaction with trap states, gopt is the generation efficiency and Gn,p are the generation rates for electrons or holes, respectively. The bimolecular recombination can be expressed by:

with η the Langevin prefactor, ε the relative permittivity of the semiconductor and ni the intrinsic free charge carrier density. A further extension of the drift-diffusion model is the implementation of trap states – states with an energy within the bandgap. To consider these states, a second set of continuity equations is introduced. These take the form of:

electron trap sites and and hole trap states, respectively. The first two terms denote capturing and release of an electron from the LUMO (corresponding to Rnt) while the last two describe the capturing and release of electrons from the HOMO (Schematic). In the previous equation, the first two terms are capturing and release of a hole from/into HOMO (corresponding to Rpt) and the last two terms are capturing and release of holes from/into the LUMO (Schematic).

If those processes are in equilibrium, a rate for Shockley-Read-Hall recombination can be calculated and gives:

In this formula, cn,p is the capture rate for electrons or holes, respectively, Nt is the total number of electron trap sites, and nt the number of filled trap sites. Correspondingly, Pt is the number of total hole trap sites and pt the number of filled hole trap sites. En,p is the release rate from the trap sites for electrons and holes, which is related to the respective capture rates via

for electron emission and hole emission, respectively. N0,n is the density of states (DOS) for LUMO and N0,p is the DOS for HOMO. The external applied voltage difference and the contribution from charged particles and ions within the device determine the electric field in the drift-diffusion simulation. The internal electric field is expressed by the Poisson’s equation

The drift-diffusion simulation framework can also be used to calculate non-equilibrium cases where one external parameter like temperature or voltage is changed during simulation. This so-called transient simulation takes on the form:

where St is the current simulation step that is updated by a small rate of change α in the external parameter with appropriate time step dt to yield the next simulation step St+1. It is important to note that each simulation step is at steady state while the overall simulation shows how the system changes with time. Parameters dependent on the external parameter that changes during the transient simulation must be adjusted at each time step.

Advantages and Limitations of Drift-Diffusion

Drift-diffusion simulation is a model that predicts the behaviour of charged particles on a macroscopic level. It can be solved using numerical methods and allows for the simulation of large systems with reasonable accuracy while using only small computational resources and time.

Drift-diffusion simulation can predict parameters such as efficiency at the device level. It also enables the local investigation of recombination rate, charge carrier mobility, charge carrier density, electric field, and doping concentration, which are not usually observable in experiments.

Drift-diffusion simulation can also be coupled with optical or thermodynamic models to calculate additional parameters such as absorption and emission of light, energy transfer, and heat dissipation. This coupling provides a more comprehensive simulation of the device and allows for the prediction of local device temperature and internal quantum efficiency. For LEDs, the luminance and color of the emission can be predicted, while for solar cells, a prediction for spectral response, open circuit voltage, and short-circuit current is possible.

Drift-diffusion simulation is a useful tool for testing different materials and optimizing their properties, such as layer thickness, without extensive experimental adjustment. However, there are limitations to its use. The model does not take quantum effects into account and assumes local equilibrium, which might lead to errors, especially at small-length scales. In such situations, simulation models like non-equilibrium Green’s function yield more accurate results. Additionally, it is not possible to depict charge transport on a molecular level, and anisotropies in charge carrier transport due to the molecular structure of the material are better simulated using models like kinetic Monte Carlo. What cannot be described correctly with the drift-diffusion are non-classical transport processes such as tunnelling, non-thermalized populations/hot-carrier effects, and the impact of the local energy landscape in inhomogeneous media.

Finally, the reliability of the simulation results strongly depends on the accuracy of the input parameters. If input parameters such as mobility or recombination rate cannot be determined accurately, only qualitative trends can be simulated, and no quantitative values can be obtained. Thanks to the speed of drift-diffusion simulations, optimization algorithms can be used in order to fit experimental data, obtain reliable input parameters and predict performances of slightly modified stacks.

On the Response Speed of Narrowband Organic Optical Upconversion Devices

Here we show how the simulation can help in understanding unexpected behaviours in an electronic device.

Together with our collaborators, we fabricated and characterized a new type of organic upconversion device (OUC) that is narrowband, meaning it can only convert near-infrared (NIR) light from a narrow range of wavelengths into visible light. [1] This is useful for applications where it is important to distinguish between different types of NIR light. The authors achieved narrowband operation by using a heptamethine J-aggregated polymethine cyanine dye as the absorber material in a photodetector. Polymethine dyes have a themselves a single absorption maximum in the NIR, but they can be self-assembled into J-aggregates to further narrow down the absorption band.

We investigated the response speed of the OUCs and observed a counterintuitive decrease in the response speed of the OUCs as NIR light intensity was reduced. This finding stands in contrast to the expected response speed enhancement typically observed in organic photodetectors. Additionally, we discovered that increasing the driving voltage (VDD) resulted in a rise in response speed for phosphorescent OLEDs but a decline for fluorescent OLEDs. Fig 2 (a - d) shows the characteristics of the investigated OUCs.

Fig 2. ,c) Performance characteristics of OUCs with and without NIR light using a fluorescent (a) or phosphorescent (c) OLED.

The impact of the material composition on the response speed of the OLED is nuanced and significant. We discovered that there are contrasting behaviours in phosphorescent and fluorescent OLEDs under varying voltage conditions. Specifically, the response speed increased with phosphorescent OLEDs, while it decreased with fluorescent OLEDs as the driving voltage was raised​​. This divergence in performance is attributed to the complex interplay between charge mobility in different layers of the devices.

Fig. 4: Speed of Response in OUCs Under Pulsed NIR Illumination.

Drift-diffusion Simulations on OUCs

To understand these surprising results, we carried out a numerical drift-diffusion modelling with the simulation software Setfos. The focus lies on the devices with SY emitting layer. But a similar study could be repeated for the Ir-OUC.

Simulation of transient photocurrent response measurement

To simulate the transient response of our devices, we used the transient models implemented in Setfos. In a transient simulation, the response of a device to a change of a parameter is simulated as a function of time. The transient simulation usually starts from a previously calculated, well-defined steady state. At some point in time a parameter is changed. We simulated the device behaviour in a range of 0.1 μs to 10 s with 50 steps for each decade. The steady state used as a starting point for the transient simulation is calculated in the dark at a constant voltage (6 V unless otherwise specified) and at t = 5 us an illumination source is activated. At the end of the transient simulation, a new steady state is reached. We adjusted the illumination intensity in such a way that the current density in the new steady state at 6 V corresponded to 0.3 mA/cm2. This is the current density at 3dB frequency measured in the transient photocurrent response measurement. To be able to compare the frequency from the transient photocurrent response measurement with the time to reach steady state in the transient simulation, the time constant was calculated as the inverse of the frequency.

Figure 5: The rise of the current density after the device is illuminated. The onset of illumination is at t=5 us. The illumination profile is marked with a green dotted line. b) normalized current during turn-on with zoom to the region around 1 to 7 ms. The blue rectangle marks the region, which is magnified in the inset. To not obscure interesting features, the rectangle is larger than the zoomed in region. The light intensity for all simulations is 0.76 mW/cm2, corresponding to 100% light intensity.

Figure 5 shows the simulation results of the illumination turn-on for different applied voltages. When the illumination is turned on (indicated by the green dotted line), the current starts to increase in two to three distinct steps. The steady-state current, which is the current at the end of the transient simulation, increases with the applied voltage up to a value of 0.31 mA/cm2 at 6 V and then reaches a plateau. Further increasing the voltage does not increase the current any further. This is different from the speed experiment, where the current continues to increase with voltage and reaches 2.08 mA/cm2 at 10 V. However, in the experiment, the current reaches 0.14 mA/cm2 at 4 V while it plateaus at 0.2 mA/cm2 in the simulation. The difference in current density between the experiment and simulation is attributed to the increase in efficiency of charge generation in the Jcy layer with increasing voltage. In the simulation, a charge generation efficiency of 100% is assumed, while in the experiment, it is lower. For this reason, the current density at 6 V at the end of the transient simulation corresponds to the maximum photocurrent density achievable at an illumination of 0.76 mW/cm2. The maximum photocurrent density depends only on the illumination intensity and not on the applied voltage, leading to the plateau at and above 6 V seen in the simulation. At 4 V, the generation efficiency of charge carriers in Jcy is lower than at 6 V, reducing the current density in the experiment further than the current density in the simulation. In the first sharp current increase, the device with the highest applied voltage is the fastest, while the speed decreases for decreasing voltage. To see if this trend is also visible in the second increase, the data was normalized (Figure 5b), and the relevant section is magnified in the inset of Figure 5b. In the second current rise, we can see the opposite trend from the first current rise. For increasing voltage, the time for the current to rise increases. Since this is the last current increase step before the device reaches a steady state, this step determines the overall time needed to reach steady state. Therefore, the device slows down with increasing voltage. This trend is also observed in the transient photocurrent experiment. The initial increase in current is determined by the generation and separation of charge carriers in the Jcy layer. The speed of spatial charge carrier separation is faster at higher voltages, which is reflected by the faster increase in current. The second sharp increase in current is determined by the movement of photogenerated charge carriers through the device and the change in recombination rate in the recombination zone.

Figure 6: Development of the current density at turn on for different light intensities. b) normalized current density during turn on. Inset: zoom to the region denoted by the blue rectangle. The voltage is 6 V for all simulations.

Figure 6 shows a simulation of the current that changes over time for different light intensities at 6 V. To better understand how long it takes for the system to reach a steady state, the data was normalized and presented in Figure 6b. The inset shows a relevant portion of the curve. Like in the case of different voltages, there are two distinct times when the current density sharply increases. The first increase is independent of the chosen illumination intensity, but the current densities are different. This confirms that the initial rise is due to the generation of charge carriers. The number of generated charge carriers differs for the different light intensities, leading to different current densities. However, the separation of charge carriers is equally fast for all cases since the same voltage is applied. The speed is therefore the same for all light intensities. The second increase in current exhibits different speeds depending on the illumination intensity. The higher the intensity, the faster the device reaches a steady state. The experiment shows the same trend. In the transient photocurrent experiment on SY-OUD, only one-step increase in current was observed. Partially, this can be attributed to the low time resolution of the experiment. The experimental time resolution was 0.3 ms while the initial current density rise in the simulation has a duration of 3 μs. The same experiment on an OUD with a different emitter (iridium-based host-guest system) recorded at higher time resolution exhibits two distinct increases in current, as observed in the simulation. The current increase and the time to steady state exhibited by both the SY and the iridium-based OUD follow the trends observed in the simulation.

Figure 7 a) zoom in to one period of the transient photocurrent experiment at 6 V. b) one period of a photocurrent transient experiment on a device with an iridium host-guest emitter at different light intensities with high resolution in time. A fast and a delayed increase in current is observed. c),d) The time needed to reach steady state for experiment (black, variation denoted with bars) and simulation (red line) for different voltage (c) and different light intensity (d).

Summary of the Results from the Simulation

The simulations revealed that the electron mobility (μe) in the OLED, particularly in the emitter layer, significantly impacts the response speed of the OUC. A lower μe results in a slower electron transport process, which in turn slows down the overall response speed of the device. This observation aligns with the counterintuitive decrease in response speed observed as NIR light intensity decreased. With the help of the simulations, we could reveal a decreased and increased electric field in the absorber and SY layer, respectively, with increasing NIR light intensity. The increased electric field compensates for the intrinsically low charge carrier mobility in the SY layer and leads to the observed faster OUC response time.

In contrast, we experimentally found that increasing the driving voltage (VDD) resulted in a different response speed behaviour depending on the type of OLED. For phosphorescent OLEDs, increasing VDD led to an increase in response speed, while for fluorescent OLEDs, it caused a decrease. Solely by changing the electron mobility in the emitting layer from 6e-7 cm2/Vs (for the SY-OUC) to 2e-5 cm2/Vs (for the Ir-OUC), the simulation was able to reproduce the experimental trend qualitatively. [1]

Overall, our drift-diffusion simulations provided valuable insights into the complex interplay of electron mobility, NIR light intensity, and driving voltage in governing the response speed of OUCs.

Conclusions

The simulations performed for this work helped in understanding unexpected experimental results. They revealed that the electron mobility (μe) in the OLED plays a crucial role in determining the response speed of the OUC. Therefore, it has been proposed that the low electron drift velocity in the emitter layer imposes a fundamental limitation on OUC response speed.

Simulations using drift-diffusion revealed that an intricate interplay between the charge mobility values in the different layers governs the response speed of narrowband OUCs. Notably, the low electron drift velocity in the emitter layer dictates a fundamental limit to the response speed of OUCs. To address this limitation, our simulations suggest that employing a very high electron mobility value in the emitter layer could potentially enhance the response speed. However, further studies are required to determine if this is achievable experimentally.

The authors' findings provide valuable insights into the fundamental mechanisms governing OUC response speed and demonstrate the potential for narrowband OUCs in selective NIR-to-visible light conversion applications. Moreover, their identification of electron mobility as a key determinant of response speed offers avenues for optimizing OUC performance.

This work was possible thanks to a fruitful collaboration from FLUXiM, EPFL and EMPA. What reported here is part of the PhD thesis of Dr. Camilla Arietta VAEL-GARN.

We invite researchers, industry experts, and innovators in the field of OLEDs, solar cells, and displays to explore the capabilities of Fluxim’s simulation software. Our tools are not just about analyzing current technologies but about envisioning and creating the future of photonic and electronic devices.

References

[1] Hu, et al. (2022). On the Response Speed of Narrowband Organic Optical Upconversion Devices. Advanced Optical Materials, Adv. Optical Mater. 2022, 10, 2200695. doi.org/10.1002/adom.202200695