This section shows how a 3D multi-layer OLED structure with square PC patterning can be simulated efficiently with FDTD.

## Square symmetry

If we consider a rectangular lattice structure, we might want to use 4x4 dipole locations across the unit cell (more may be required, but we have found this to work quite well). For each dipole location, we need 3 simulations for the 3 dipole orientations (x/y/z). In principle, this means that we need 4x4x3=48 simulations. However, in this case, we can follow the steps below to apply symmetry and reduce the total number of simulations to 7 for the patterned device, and 2 for the unpatterned device.

### Step 1:

Starting with the unit cell, we have the 4x4 dipole locations that may be necessary to fully characterize the OLED structure.

### Step 2:

We recognize that only 1/4 of the unit cell needs to be simulated because the structure has symmetry across the x and y planes, the remaining simulations can be reproduced using symmetry operations. Note that if your unit cell is rectangular, or if it does not have 45 degree rotational symmetry, one will not be able to reduce the 1/4 unit cell any further (ie. this leads to 4 dipoles locations x 3 dipole orientations = 12 simulations).

### Step 3:

Since the unit cell is square and has 45 degree rotational symmetry, we can further reduce the number of simulations to 1/8 of the unit cell so that only 3 dipole locations are required.

### Step 4:

Lastly, we recognize that for the 2 locations along the diagonal, the dipole oriented along the y axis can be determined by the results from the dipole oriented along the x axis. This means that we can avoid simulating both x and y oriented dipoles.

In summary, we now have 3 dipole locations to simulate, with 2 simulations for the two dipole along the diagonal (x/z orientations) and 3 simulations for the third dipole (x/y/z orientations). This means that we have reduced the total number of simulations from 48 to 7 for the patterned device.

## Simulation setup

We will consider a structure similar to the one propose by Chutinan et al. Initially, we let all materials, except the cathode, be loss less dielectrics, but this can be easily updated later. Starting with a simulation span of only 8x8 um^{2} in the x and y plane, the simulation file OLED_3D_square.fsp contains a parametrized OLED structure. Please note that all the children of this structure are fully deleted each time a parameter is changed, so you should not add or remove objects from this group.

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- Open OLED_3D_square.fsp and verify that your computer resources are effectively configure. Take a look at the Setup-Variables tab in the model analysis group. Here, px/py are the locations of each dipole source (as a fraction of the unit cell). Weight is used to keep track of the area contribution from each dipole as we unwrap the results, defined in OLED_params_3D_square.lsf. For example, if there are 16 dipole locations (4x4) in a unit cell, then each dipole location occupies 1/16 of the unit cell area. If the sqaure lattice symmetry applies, an on-diagonal dipole location only occupies 1/32 unit area (0.5*1/16).
- The script file OLED_params_3D_square.lsf can be used to generate the values for the parameters listed under Setup-Variables. This is based on the technique described in the previous section for "PC with square symmetry". This file will generate the values for px/py/orientation/weight for the patterned PC. These values can then be entered directly into the parameter sweep project:

Note that the last two columns (Value_8 and Value_9) correspond to the 2 simulations for the non-patterned case (the px/py/orientation/weight values are irrelevant here, and are set to 0 by default). The default settings correspond to 4x4 dipole locations in a full unit cell, which should work well for most OLEDs with square PC patterning. These values have already been entered into the parameter sweep project in this example (shown above).

Before proceeding to run the parameter sweep project, one can use the Animate feature to "watch" the sweep without running the simulations.

## Results and Discussion

### Radiative decay rate enhancement

The process of calculating the internal quantum efficiency is similar to the approach used in Simple 2D OLED. Here, we use the script *OLED_internal_QE_analysis_3D_square.lsf* to calculate the power emitted by the dipole. We cannot use the Mean Operation in the parameter sweep project to calculate the average power in this case, since using symmetry means that the contribution from each dipole orientation and location may be not the same, and we need to multiply the results for each simulation by different weight factors as we unwrap the results.

The result for the power emitted by the dipole source (normalized to the power emitted in a homogeneous medium) is shown below. Note that there is not much difference between the case with and without the PC structure, which tells us that the excitation lifetimes will not be modified by more than a few percent by the periodic patterning. We calculated the results by 2 different methods and both give similar results to within a few percent..

### Extraction efficiency analysis

Again, we should no rely on only using the parameter sweeping tool to average the far field results over different dipole orientations and locations (unfolding the unit cell requires taking the transpose and applying different weight factors to the results for the simulated dipoles). In this case, we rely on FDTD' scripting environment to carry out this analysis. The script OLED_total_QE_analysis.lsf will load all the individual simulations and calculate the angular distribution of the light in either the substrate (glass) or the air. The projection in the air accounts for the reflection and refraction that will eventually occur at a glass/air interface. This script uses the results of the small number of simulations and, through symmetry operations, reconstructs the angular distribution from the incoherent superposition of all the dipole locations in the unit cell. (Note that this process can take up to several hours depending on the size of the simulation used). The final results are stored in OLED_farfield_results.ldf and this file contains all the information necessary to characterize your OLED. You will likely want to make a backup copy of this file because it can be useful even if the simulation files are deleted.

We calculate the emission enhancement as a function of wavelength. The result is shown below. For this structure, the enhancement is quite modest, but could be improved with some design modifications. Note that these results show the enhancement in emission to air.

### Including the emission spectra

We calculate the actual angular emission pattern that would be observed for different operating wavelengths of an actual OLED. For this, we choose 3 different center wavelengths and a FWHM bandwidth for our OLED. In this case, we have chosen to evaluate operation at 460nm, 510nm and 620nm with a FWHM of 15nm. The emission spectra are plotted below.

We calculate the angular emission by doing a weighted average of the previous results over the above spectrum. The resulting angular distributions are shown below.

Color filtered angular emissions

The color filters tend to smooth some of the features observed with the unfiltered angular distribution. To plot the unfiltered angular distribution for all wavelengths (51 in this case), set the variable plot_all_wavelengths to 1 in *OLED_total_QE_analysis_3D_square.lsf*. An example of three of the figures produced are shown below.

Raw angular emissions

## Next Steps

In order to make these simulation run quickly, a simulation span of only 8x8 \(\mu\)m^{2} was used. This is almost certainly not large enough to obtain accurate angular emission spectra. In general, spans of 12x12 \(\mu\)m^{2} , 15x15 \(\mu\)m^{2} and even sometimes 30x30 \(\mu\)m^{2} may be necessary. The entire set of simulations can take a few hours to, although the entire set of simulations and analysis for a consider distributing the jobs see: Managing resources using the resource configuration utility

It may also be necessary to reduce the size of the mesh, particularly dz, to accurately resolve the effects of thin emission layers.

Testing of higher numbers of dipole locations has shown very little change in results for most structures. In general, we do not recommend using more than the 16 dipole locations described here which requires only 7 simulations.

### Related publications

- H. Greiner and J. Pond, "Simulation of Light Extraction from OLEDS using FDTD", presented at NFO9, Lausanne, Switzerland, September 2006
- A. Chutinan et al., Organic Electronics, 6, 3-9 (2005)
- Y.J. Lee et al., "A high-extraction-efficiency nanopatterned organic light-emitting diode", Applied Physics Letters, Vol.82, no.21, 3779-3781, 2003.

We gratefully acknowledge the collaboration of Horst Greiner of Philips Research and Dave Stegall of 3M in the development of this application example.