Light emitting diodes (LED) and organic light emitting diodes (OLED) structures often utilize complex materials and nano structures to improve the performance of solid-state lighting. To accurately model these devices, we need to build up the incoherent response, while varying many design parameters, and optimizing several figures of merit simultaneously. The number of required simulations can quickly grow beyond belief. We approach the problem by using analytical methods in 1D to accurately capture the interference and microcavity effects of multilayer stacks for rapid characterization of far field emission and internal field profiles. In 2D we sweep design parameters and dipole positions using FDTD to extract important performance metrics such as light extraction efficiency and radiative decay enhancement. For final parameter verification we discuss symmetry methods to increase throughput of 3D FDTD simulations. An overview on relevant figures of merit and exporting results to raytracing software is also included.
1D Multilayer Stacks
For planar OLEDs/LEDs with no patterning, analytical methods of calculating the optical response are considerably faster than direct simulations of Maxwell’s equations. Lumerical's Stack Solver provides a set of 1D optical stack functions that accurately captures the interference and microcavity effects in multilayer stacks which can be used for rapid analysis of these devices. Typically, the optical response of a specific structure is obtained in less than one second of calculation time.
Layer thickness and dipole position
STACK solver is accessed through script commands or using the stack GUI depending on the user’s preferences. The first step is to define the stack properties via the GUI options, or as a set of n (index), and d (thickness) vectors. Indices can be defined as \( \mathbb{C} \) numbers or passing the material names from the material library.
To improve the spontaneous emission enhancement, maximize the field intensity in the active region. Using stackfield to find the fields of all relevant wavelengths simultaneously. Alternatively, look at suppressing spontaneous emission to change the color. Farfield emissionIn order to find the farfield power density as function of angle call stackpurcell. The dipole position is swept through the active layer to further improve the far field power density. Although the layer dimensions involved with LEDs make them intractable with FDTD one can use STACK to analyze the far field power distribution.
Slab mode analysis
Physical insight into how the radiation will propagate in the substrate can be understood by looking at the mode structure with FDE. This is relevant when considering which modes will be supported and determining how to outcouple these slab modes. FDE makes it possible to model lossy, and dispersive materials for sweeping the bandwidth.
2D Patterned Structure
FDTD is a high performance optical solver that can capture the effects of wavelength-scale patterning and its impact on the efficiency of the device. This is important for OLED and LED simulations because typical designs include many thin layers of highly dispersive materials and complex wavelength-scale patterning is used to increase the quantum efficiency.
Structure
The conventional OLED layers can be constructed using rectangles. OLED patterning should be parameterized using a structure group, which allows one to sweep the design parameters to achieve optimal efficiency.
Simulation region
PML (absorbing) boundaries should be used on all sides except possibly below the metallic cathode layer, where we it is appropriate to use a metal boundary condition. The simulation span, tangent to plane, should be large enough to include multiple periods of the patterning. Normal to the plane of the substrate, the simulation span needs to be large enough that the near fields from the structure do not interact with the PML boundary conditions. Periodic BC's are not applicable, even though the structure is periodic, since this would would imply an infinite array of coherent dipoles. To start we recommend using a low mesh accuracy as it is much easier to debug shorter simulations. Further, simulation efficiency is achieved by placing coarse mesh override regions; on top of metals and high index layers with no patterning. See CMOS image sensor for further discussion.
Sources: simulating incoherent, isotropic emission
OLED\LED active regions are engineered for controlled and enhanced spontaneous emission via the recombination of electrons and holes (or molecular transitions of pi electrons). Each photon produced through this process has a random direction, phase and polarization i.e. it is incoherent. Although a precise treatment of the electroluminescence would require a Quantum Mechanical description; however, in practice it is possible to treat the generated light classically using electromagnetic point dipole sources. The average electromagnetic field intensity of an ensemble of incoherent, isotropic dipole emitters in a small spatial volume can be calculated:
$$ \lt |\mathbf{E}|^2 \gt = \dfrac{{|p_0|}^2}{3}[{|E_a|}^2 + {|E_b|}^2 + {|E_c|}^2] $$
Where \(E_a\), \(E_b\) and \(E_c\) are the electromagnetic fields generated by single dipole along the x, y and z axes. Since FDTD is a coherent method, this means we have to run 3 separate simulations of the same dipole oriented along the x, y, z axes and then sum up the results incoherently. Furthermore, the dipole positions should be swept through the active layer as recombination will occur throughout the active region. One can think of this a statistical sampling process. Although, it would be impossible to sample all possible positions and orientations; with an unbiased sampling technique and appropriate sample size you will converge on the true incoherent response of the system. To reduce the confidence interval simply increase the number of samples.
More information:
- Understanding coherence in FDTD simulations
- Incoherent Unpolarized Dipole
- Spatially Incoherent Dipole
- Spatial Incoherence
- Calculating the power radiated from incoherent isotropic point sources
3D Final Parameter Extraction
Although 2D OLED models offer useful insights into the physics, it is often necessary to run full 3D OLED simulations for final verification with experimental results. The methodology is very similar to 2D but given that 3D simulations can be memory and time intensive, we demonstrate methods of improving throughput. This is especially important for OLEDS as we must sweep the source position and orientation.
Structure and simulation region.
As with 2D simulations one should parametrize the structure, and construct material models that match the optical properties of the layers materials. PML should be used for boundaries.
Sources and symmetry
To model the devices incoherent nature, we must sweep many dipole positions and orientations. This is done by choosing a predefined grid of dipole positions or randomly choosing the orientation and position. Consider the following unit cell which we divide into a 4x4 grid.
For each dipole location, we need 3 simulations for the orientations (x/y/z). In principle, this implies 4x4x3=48 simulations. Use the parameter sweep tool, or script the process to automate running all dipole locations and orientations.If the device has symmetry it is possible to drastically reduce the number of required simulations. We will outline how to perform this process in the following pages.
Figures of Merit
Decay rates
FDTD based simulations of OLED devices often involve measuring enhancements to the radiative decay rate of the emitter. The following definitions will be helpful to understand the results that can be obtained from FDTD based OLED simulations.
Radiative decay rate \( \gamma_{rad} \):
The decay rate of excitations to photons that can be collected and used in the device. For an OLED structure, this would be the decay rate of excitations to photons that propagate from the OLED structure into the air within a useful range of angles. Calculations involving the radiative decay rate are within the scope of an FDTD simulation.
Loss decay rate \( \gamma_{loss} \):
The decay rate of excitations to photons that are absorbed or otherwise lost in the device. Photons absorbed in lossy material, trapped by TIR in high index layers, or radiate outside a desired range of angles are included in this category. Calculations involving the loss decay rate are within the scope of an FDTD simulation.
Total electromagnetic decay rate \( \gamma_{em} = \gamma_{rad} + \gamma_{loss} \):
The total electromangetic decay rate. This is simply the sum of \( \gamma_{rad} \) and \( \gamma_{loss} \). This is within the scope of an FDTD simulation.
Non-radiative decay rate \( \gamma_{nr} \):
The decay rate of excitations to non-radiative processes. Excitations that decay into phonons (heat) are included in this category. Calculations involving the non-radiative decay rate of an emitter are beyond the scope of an FDTD simulation.
Excitation decay rate \( \gamma_{excitation} \):
The excitation rate of the emitter. The emitter is typically excited electrically. Calculations involving the excitation rate are generally beyond the scope of an FDTD simulation. In cases where the system is pumped optically, then enhancements to this quantity can be calculated by FDTD. For simulations involving optical pumping, the 4-level 2-electron laser model may be a helpful starting point.
Light extraction efficiency (LEE)
The light extraction efficiency of an OLED is defined as the fraction of optical power generated in the active layer of the OLED that escapes into the air above the OLED within a desired range of angles.$$LEE = \frac{ \gamma_{rad} }{ \gamma_{rad} + \gamma_{loss} }$$We can also define the light extraction efficiency enhancement as the ratio of the light extraction efficiency for two different designs, such as a patterned vs un-patterned OLED structure as in this example.$$LEE_{enhancement} = \frac{ LEE_{pattern} }{ LEE_{no pattern} }$$When calculating the light extraction efficiency, we measure the fraction of useful power emitted from the OLED device relative to the total power emitted from the active layer. We may consider the fraction of useful power to be the light that escapes to the glass substrate, or we may consider the light escaping to the glass substrate within a particular solid angle (eg. bounded by the TIR critical angle). It is important to remember that the glass substrate is usually quite thick, which means the glass-air interface cannot be directly modeled in the FDTD simulation. Instead, the far field projection functions are used to include any reflection and refraction effects that occur at this interface. The following are the steps for analyzing the extraction efficiency, once the simulations have completed:
- Use a far field projection to calculate the angular distribution of light into the glass (or eventually into air)
- Integrate the far field distribution to calculate the fraction of light that escapes into the air. Note that if we are only interested in the total amount of light that escapes into the glass, then a near field transmission calculation will be sufficient. The more time-consuming far field projections are only required when it is necessary to calculate the power within some range of angles, or when including effects from the glass-air interface.
- Steps 1 & 2 must be repeated for each simulation (dipole orientation and position). Finally, the results must be averaged to obtain the response (light extraction efficiency or enhancement) to replicate the incoherent isotropic source.
Increasing the light extraction efficiency of an OLED increases the fraction of emitted power that escapes to the air (or into a desired solid angle). This is essentially a question of redirecting and extracting light to prevent it from being trapped by TIR. In an LED or an OLED, light extraction inefficiencies exist because light generated within a high-index material has difficulty propagating into the surrounding lower-index medium. The constituent layers of the OLED or LED can be textured with micro-scale or nano-scale patterns in order to improve the light extraction efficiency. The texturing leads to:
- Macroscopic optical effects: this mainly involves patterning or roughening the air/glass interface which can be used to reduce TIR. Also, some authors have concluded that patterning near the emission layers can lead to increased efficiency for the same reason: the light makes multiple round trips from the emission layers to the glass/air interface, and scattering from the grating at the emission layer can change the direction of the reflected light and prevent TIR at the air/glass interface. For example, Peter Vandersteegen, in "Modeling of the Optical Behavior of Organic LEDs for illumination" (Doctoral Thesis, 2008), concludes in section 5.4.6 that a substantial efficiency improvement is available for certain designs due to multiple round trip scattering events, but that the efficiency improvement is negligible when these multiple round trip events are ignored. This effect can be studied using FDTD either by calculating the angular emission of the light in the glass and then combining these results with considerations of the multiple round trip effects or by artificially reducing the thickness of the glass layer such that it can be directly modeled by FDTD and yet remain thick enough to obtain accurate results.
- Microscopic optical effects: This is typically done with gratings or patterning in and near the emission layers. Ideally these gratings can assist in extracting light that is trapped in the emission layers and extracting it into the glass at angles where it will not suffer TIR at the glass/air interface. We can study this effect directly with FDTD.
Radiative decay rate enhancement
The electromagnetic decay rate of the emitter will be influenced by the surrounding structures (eg. the metal cathode, PC patterning, etc.). Calculation of the absolute electromagnetic decay rate is beyond the scope of an FDTD simulation, but it is possible to calculate how the surrounding structure enhance this decay rate. This is possible because the electromagnetic decay rate enhancement is equal to the enhancement of the local density of radiative states in the active layer, which in turn is equal to the power radiated by a dipole source in an FDTD simulation normalized to the power radiated in a homogeneous medium. Fortunately, this last quantity is straightforward to calculate with FDTD. Indeed, this decay rate enhancement is also known as the Purcell factor and is a standard result returned by dipole sources. Therefore, by measuring the power radiated by the sources in a particular OLED design, it is possible to calculate the decay rate enhancement relative to a homogeneous region of the emitting region. $$\frac{ \gamma_{em}}{ \gamma_{em}^0 } = \frac{ dipolepower }{ sourcepower } = Purcell factor$$where \( \gamma_{em} \) is the total radiative decay rate with patterning, \( \gamma_{em}^0 \) is the total radiative decay rate without patterning. It then allows us to calculate the decay rate enhancement between those designs.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 electromagnetic field solvers, 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.It is possible to increase the radiative decay rate by modifying the environment of the emitting dipoles with the addition of microscopic patterning. Increasing the radiative decay rate can allow for a higher electrical excitation rate. It will also increase the internal quantum efficiency because radiative decay processes can compete more efficiently with non-radiative processes. By relating the radiative decay rate to the power radiated by a dipole source, we can study this effect using FDTD.Finally, it should be noted that introducing patterning may have effects on the electrical properties of the device. Modeling these effects is beyond the scope of FDTD.
Chromaticity
In display applications the color of a device is very important. The vast majority of human perceptible
colors cannot be described monchromatically, but are mixtures of different parts of the visible light
spectrum. White light for example is a mixture of all the spectral components. Although it may at first seem subjective; human perception of color is remarkably reproducible. The biological basis for this is
trichromacy, where the human eye employs three types of cone cells with different spectral sensitivites.
Trichromacy is the reason that colors are often described as a wheel; although, we know that light
is a continuous spectrum we intuitively understand that colors can be described by mixing
compatible primaries such as red, blue, and green RBG or cyan, magenta, and yellow CMY.
To precisely quantify and compare color a mathematical description is needed and trichromacy allows us to define a 3D vector space for this purpose. This is commonly referred to as a color space with spectra as basis functions. Different choices of these basis functions define different colorspaces, and they need not be orthogonal or correspond to the sensitivity of the cone cells. Two common color spaces come from the society for illumination engineers CIE 1931(CIE XYZ) and CIE 1976(CIE ULV).
More commonly the apparent brightness of the color space is normalized so that the unique color information can be displayed in a 2D plot known as chromaticity diagrams. Below are plots of the CIE 1931 and CIE 1976 are given.
It is helpful to think of these chromaticity diagrams as a quantitative color wheel, where the coordinates describe colors uniquely. The stackdipole commands will directly return the XYZ tristimulus values, or you can use the colormatch functions. Notice that the pure monochromatic spectrum is along the N and NW boundary, and that all other color are mixtures of these. In particular the white point is in the middle.
Interface with Ray Tracing
See the following example for more information on exporting ray sets to geometric optical solvers.