In this example, we use the STACK optical solver and FDTD to characterize a cylindrical micro-LED and extract the emitted power and radiation pattern.

## Overview

Understand the simulation workflow and key results

Similar to OLEDs, micro-LEDs are challenging devices to simulate. They require a large number of simulations and some post-processing is needed to analyze the devices' performance.

Here we will use the STACK optical solver to characterize the emission properties of a planar device, and FDTD to simulate a 3D micro-LED.

### Step 1: Characterize the 1D stack using STACK Optical Solver

The incoherent, unpolarized emission from the active layer of the planar LED is obtained using the STACK optical solver.

This approach can be used to quickly characterize and optimize the layers' thickness and materials before simulating the full 3D device with FDTD.

### Step 2: Simulate the cylindrical micro-LED using 3D FDTD

We use a series of FDTD simulations to extract the incoherent, unpolarized emission of the micro-LED and determine the light extraction efficiency.

## Run and Result

Instructions for running the model and discussion of key results

### Step 1: Characterize the 1D stack using STACK Optical Solver

- Open the simulation file [[micro_LED_cylindrical.fsp]] in FDTD
- Open and run the script file [[micro_LED_STACK.lsf]]

In this script, we use the STACK optical solver to analyze the planar emission of the device. The script constructs the layers and extracts material properties from the material database.

[[Note:]] although we do not run the FDTD simulation in this step, it is best to load the fsp file to ensure all the materials needed for the simulation are loaded. More information about the materials used can be found in the appendix. |

The script uses the stackpurcell command to calculate the Purcell factor as well as the power density per steradian emitted into the sapphire as a function of angle and wavelength. It also integrates the power density to calculate the total upward power (normalized to the power the dipole would radiate in a homogeneous medium). These results will be easy to compare with the 3D FDTD simulation.

Additionally, we use stackdipole to get the luminance and radiance as a function of angle for a given dipole spectrum, as well as the X, Y, Z tristimulus values.

### Step 2: Simulate the cylindrical micro-LED using 3D FDTD

- [[step]]Open and run the simulation file micro_LED_cylindrical.fsp
- Open and run the script file plot_single_dipole_results.lsf to plot the following figures

We start by running a first simulation at low accuracy (mesh accuracy set to 1), with a source position set to 0, oriented along the x-axis. We can then visualize the field intensity in the cross-section from the monitor y-normal (shown here in log scale), or from the monitor farfied::z2 (shown in linear scale).

The “farfield” analysis script will calculate the total power flowing out of the box, as well as the normalized Poynting vector in the far-field (at a distance of 1m). This is normalized so the integral of the Poynting vector is equal to the total power propagating in the entire upper half-space, normalized to the power that would be emitted by the dipole in a homogeneous medium of the same refractive index as the MQW layer. In addition, the analysis group calculates P_vs_theta, the average value of the Poynting vector over all azimuthal angles, and, therefore, represents the Poynting vector as a function of polar angle for a ring of incoherent dipole emitters at the same radial position from the center of the device.

Lastly, it calculates the Purcell factor on the same wavelength grid as the other results (the Purcell factor can be easily calculated on any wavelength/frequency grid from the dipole source).

These results can be then used to perform further calculations. For example, we calculate the total extraction efficiency into a cone of 34.4 degree (extraction from sapphire to air):

- [[step]]Run the parameter sweep "position"
- Open and run the script file dipole_position_analysis.lsf

The nested sweep stores all the results of the “farfield” analysis group for various positions and polarizations of the dipole. To save the memory, we can remove the far-field result which consumes most memory.

The sweep is set to calculate 11 positions for each of the 3 dipole polarizations, for a total of 33 simulations. To reduce the simulation time, run this sweep with a mesh accuracy set to 1, and set the property “project all wavelengths” to 1 (true).

The script dipole_position_analysis.lsf will take the value of P_vs_theta and take the average of the 3 dipole orientations to calculate the result for unpolarized dipole emission (see the *"*Important model settings*"*). Then, it will calculate weights for the contribution of each dipole position, assuming a uniform dipole density per unit area. Each dipole is therefore multiplied by a weight that is proportional to the area of the ring it represents, as shown in the following figure:

These weights can be easily adjusted if more information is known about the current density in each region.

The script calculates the power density vs theta at a single wavelength near 625nm or spectrally integrated over a dipole spectrum about 10nm wide near the same wavelength.

It also calculates the total power emitted in a 34.4 degree cone where it can be emitted to air.

## Important model settings

Description of important objects and settings used in this model

### "farfield" analysis group

The *"*farfield*"* analysis group has a setup script to create a box of monitors of the desired size, which can be positioned over the LED structure.

Since it is on a GaAs substrate, we can choose not to include the monitor in the z min boundary (property "include_z_min_boundary" set to 0).

The following analysis properties can be modified by the user:

**theta min, theta max, N theta**: min, max, and number of points for polar angle theta.**N phi per half**: number of values of phi between 0 and 180 degrees. The Poynting vector from 180 to 360 degrees is calculated assuming symmetry.**project all wavelengths**: can be 0 (false) or 1 (true). If true, all wavelengths will be projected which can take 10s of seconds or more. If false, the single wavelength closest to the*"*target wavelength*"*will be chosen for projection.**near field points per wavelength**: the near fields can be down-sampled for far-field projection, which substantially speeds up the projections. Values of 3 or 4 are typically sufficient. Increasing this will result in longer far-field projection times without improving accuracy.

### Mesh accuracy

In the above steps, we use a mesh accuracy of 1 to speed up the simulations. For better accuracy, we can set up the mesh accuracy to 2 or 3 (See the *"*Convergence*"*).

### Dipole orientation

The emission of such micro-LED is incoherent and unpolarized. While FDTD is fundamentally a coherent simulation method where sources have a well-defined polarization, we can extract incoherent, unpolarized results by adding incoherently the fields from simulations with orthogonal polarizations. This means we have to run 3 simulations for each dipole position, 1 for each possible dipole orientation, x, y, and z (see Incoherent, unpolarized dipole).

The dipole orientation is set in the *"model"* object. This group is also parametrized for the dipole position along the x-axis. It will automatically adjust the symmetric and anti-symmetric boundaries of the simulation, assuming the dipole is located on the x-axis, at \(y=0\) to reduce the simulation volume where possible. The properties of *"*model*"* are shown below.

The model setup script will force the x, y position and orientation of the dipole source, and set the x min, y min boundary conditions of the FDTD region. These changes will overwrite any direct changes made to those properties.

[[Note:]] the parameter sweep stores the results for each polarization. We could take the mean of all the results with respect to the dipole orientation, but this would prevent future analysis of any particular orientation. |

## Updating the model

Instructions for updating the model based on your device parameters

The micro-LED structure is parametrized in a structure group that allows us to easily change the layer thicknesses and materials.

The dipole is centered in the middle of the MQW layer. This position can be changed by modifying the value of "z" in the source properties.

## Taking the model further

Information and tips for users that want to further customize the model

### Materials

The materials used in this example are taken from the reference paper. Other materials can be used provided you know their optical properties over the bandwidth of interest. See the appendix for more information about the materials used and how they were added to the FDTD simulation.

### Geometry

The device studied here is a cylindrical micro-LED. Other geometries can be used, but note we use the symmetries extensively to reduce the simulation size as well as the number of dipole positions. This should be taken into consideration when updating the geometry.

### Dipole spectrum

FDTD will normalize the fields recorded by frequency monitors to the source spectrum, so these monitors will return the impulse response of this device. It is then easy to calculate the response to an arbitrary power spectrum. We simply multiply the impulse response of the system by the new power spectrum:

$$ f_{spectrum} = Power_{density}(\lambda) \cdot f_{impulse}(\lambda) $$

Where \(f_{impulse}\) is the impulse response and \(Power_{density}\) the new power spectrum.

This can be performed as a post-processing step. Therefore, you can calculate the response to any spectrum without having to re-run the FDTD simulations.

### Export to ray tracing

Designing OLED/LED devices requires a combination of nanoscale and macroscale optics. The individual pixels often have sub-wavelength features such as thin dispersive layers, scattering structures, that require an electromagnetic field solver, while the emission from the macroscopic device requires a ray-based tool.

For these applications, one can calculate the angular distribution of a pixel with the STACK optical solver or FDTD and then load the result into a ray-tracing tool in the form of ray sets.

FDTD supports the export of the angular distribution to ray sets for Breault Research ASAP, and ZEMAX OpticStudio (requires an additional license).

## Convergence

Tips for ensuring that your model is giving accurate results

The script dipole_position_analysis.lsf makes it possible to use less than 11 dipole positions originally simulated, to understand if fewer positions could be used. In the following figures, we compare 11 dipole positions, with 6 and 3 positions only.

This comparison shows that 6 dipole positions may be sufficient for many analyses, while 3 is not.

We can also run the parameter sweep with a FDTD mesh accuracy of 2 (targeting \(dx=\lambda/10\) instead of \(dx=\lambda/6\)) in all media. Mesh accuracy of 1 is sufficient for rapid simulation and optimization, but errors of 10 to 20% are typical due to the extremely coarse FDTD mesh size. The change from mesh accuracy 2 to 3 (\(dx=\lambda/14\)) is typically only 1-2%.

## Additional resource

Additional documentation, examples, and training material

### Related publications

- R. Horng, H. Chien, K. Chen, W. Tseng, Y. Tsai and F. Tarntair, “Development and Fabrication of AlGaInP-Based Flip-Chip Micro-LEDs,” in IEEE Journal of the Electron Devices Society, vol. 6, pp. 475-479, 2018 (doi: 10.1109/JEDS.2018.2823981).

### See also

### Related Lumerical University courses

## Appendix

Additional background information and theory

### Materials

For materials, we use:

**(AlxGa1-x)0.5In0.5P latticed matched to GaAs**from Kato, Hirokazu & Adachi, Sadao & Nakanishi, Hiroshi & Ohtsuka, Kouji, Optical Properties of (AlxGa1-x)0.5In0.5P Quaternary Alloys. Japanese Journal of Applied Physics. 33. 10.1143/JJAP.33.186 (1994).

We modified the imaginary part of the permittivity with a Heaviside-like function to better model the transition to transparency near 600nm. The script file micro_LED_AlGaInP_mdf.lsf will create the necessary models and add them to the current material database. It will remove any pre-existing materials of the same name. It also reproduces Figure 10 of Kato et al:

**GaAp**from https://refractiveindex.info/?shelf=main&book=GaP&page=Aspnes (yaml file included).**Sapphire**(Sellmeier model) from https://refractiveindex.info/?shelf=3d&book=crystals&page=sapphire. Note that in FDTD simulations this will use the refractive index value at the center frequency of the simulation and will be non-dispersive.**ITO**from https://refractiveindex.info/?shelf=other&book=In2O3-SnO2&page=Konig.

The materials used have been saved in micro_LED_materials.mdf. The default FDTD fit settings on some of the materials have been modified from their defaults to improve fitting over 550-700nm for FDTD simulation:

- ITO and GaP use a fixed fit range of 400-700nm
- MQW material uses a maximum number of coefficients of 1. This forces it to use a simple lossless dielectric model. The lossless nature is useful for calculating the Purcell factor in FDTD.

### Far-field projections

The farfield projection is typically performed from a single surface which is fast because it can use ffts. However, the fields must decay to 0 at the edges of the surface. While filtering of the near field can be used to reduce these effects it still can require larger simulation volumes. If we look at the near field at "farfield::z2", we see that the field is not zero near the edges of the monitor.

A comparison of projection methods below shows that the filter of the near field (farfieldfiler > 0) cannot be used to recover accurate results. The method of projecting from the 5 sides of the box gives the best results, as shown in the bottom right.

- On the left, fft based projection from "farfield::z2" only, with "farfieldfilter"
*=*0; on the right, fft based projection from "farfield::z2" only, with "farfieldfilter"=1

- On the left, far-field exact projection from only "farfield::z2" (gives same results as fft based projection); on the right, current projection from 5 sides of the box.

In principle, the far-field projection needs to be from a closed surface in a homogeneous medium. In this case, due to the substrate, we have excluded the lower z surface. As a result, the power integrated over the open box in the near field and the power integrated over the upper half-space in the far-field are not expected to perfectly agree. Inaccuracies in the radiated power will predominately affect very steep emission angles, far from normal. Here is a comparison of the two integration methods: