Ray Tracing for Light Extraction Efficiency (LEE) Modeling in Nitride LEDs

Abstract

We describe simulations of the light extraction efficiency (LEE) as a function of the major materials parameters and geometries in the three main LED structures used today, namely nitride LEDs on GaN substrates, on patterned sapphire substrates and flip chip nitride LEDs. We use ray tracing simulations of LEDs to show how the various extraction schemes operate. The simulations of this chapter show that both surface roughening and PSS lead to high efficiencies, although based on somewhat different mechanisms. The results appear here for the same device parameters (most of them with conservative values), which allow meaningful comparisons. Some industry results may be higher, either due to better materials quality (lower absorption in particular for ITO and metals) or more aggressive designs (smaller lossy contact areas). For LEDs on GaN substrates, the LEE is determined by residual substrate absorption. A desirable feature is the ability to scale the extraction efficiency for large chips in order to reduce costs. In almost all cases, the use of large surface LEDs is detrimental due to the long ray paths to reach the sidewalls except for those extraction schemes where light is continuously extracted or when substrates have very low absorption.

A chapter for “III-nitride-based light-emitting diodes and applications.”

Prof. T.-Y. Seong (Korea University), Prof. H. Amano (Nagoya University), Prof. J. Han (Yale University), Prof. H. Morkoç (Virginia Commonwealth University) eds.

Appendices

Appendix A

11.1.1 Discussion of the Origins of the Effects of Surface Roughness and of Sapphire Patterning

The Effect of GaN Surface Roughening: Randomization

We discussed above the increased single pass extraction due to surface roughness. A second essential effect is the randomization of light. Even if a ray is not extracted on first try, it will be reflected at a random angle, and may escape on the next attempt. We can quantify this using ray tracing by considering the fate of a random collection of rays and their angular distribution for each incident angle. Figure 11.24 shows the angular distribution of reflected rays as a function of incidence angle and Fig. 11.25 gives some examples of reflected rays distributions for a given incidence angle.

Fig. 11.24

Angular distribution of reflected rays as a function of incidence angle for a roughened surface (hexagonal nonrandom pyramids, with a top angle of 28°, for a GaN/air interface)

Fig. 11.25

Example of reflected light angular distribution pattern on a roughened surface (hexagonal nonrandom pyramids, with a top angle of 28°) for 25°, 50° and 75° incidence angles

It is shown that for a given angle of incidence from an isotropic source, the reflected light pattern is complicated and very different from a specular reflection pattern. The angular distribution of reflected light is obtained by summing over all incidence angles (Fig. 11.26).

Fig. 11.26

a Surface transmission as a function of incident angle (solid) and angular distribution of the reflected rays integrated on all incidence angles (dashed) from a rough surface. b Similar results for a flat GaN/air surface

These reflected rays will hit the header and will be specularly reflected back to the roughened surface for a second attempt. As we know their angular distribution, the second attempt extraction can be computed using the transmission coefficient as a function of the incidence angle of Fig. 11.26a. It is 15.3%, slightly more than the first attempt probability (13.4%). We can iterate this process and compute the nth attempt extraction probability (Fig. 11.27). Due to randomization of the rays, nearly the same fraction of the light is extracted after each attempt (around 12.9% after a few bounces). These values were computed for a GaN/air interface; adding a high index encapsulating epoxy would significantly increase them.

Fig. 11.27

Successive nth attempt extraction probability for a roughened (diamonds) and flat surface (circles)

In contrast, a flat surface with no randomization effect exhibits specular reflection and the second attempt rays are mostly outside of the transmission window of the surface. Thus, the extraction probability drops to zero after two to three successive attempts (Fig. 11.27).

The Effect of Patterned Sapphire Substrate (PSS)

We can also study PSS interface the same way. We will consider a cone shaped pattern in a square packing arrangement. The exact geometry is given in Fig. 11.28. This results in a filling factor of 34.9%.

Fig. 11.28

Cone patterning of the GaN/sapphire interface

The angular reflection pattern of the PSS is given in Fig. 11.29. We can see that, compared to the GaN surface, the pattern is overall simpler because of the cylindrical symmetry of the cone compared to the hexagonal pyramid. We can also see that there is high intensity along the main diagonal; this is specular light that is reflected exactly at the incidence angle. It comes from the fact that contrary to GaN roughening, the filling factor is not 100%; some of the interface is still flat.

Fig. 11.29

Angular distribution of reflected rays as a function of incidence angle (intensity is on a log scale)

We can quantify the intensity of specular reflection, randomized reflection and transmission as a function of incidence angle by averaging over the reflected and transmitted solid angles (Fig. 11.30).

Fig. 11.30

Fate of the incident rays for a PSS surface

This figure is the result of the superposition of two effects: the specular reflection by the flat part of the GaN–sapphire interface and the randomization by the cones. Below the critical angle (42.6°), the cones transmit part of the light and reflect the rest at random angles. The flat surface transmits all of the light that strikes it (modulo the Fresnel transmission coefficient). For higher angles, the flat surface reflects all the rays striking it, and because the filling factor is ~35%, this explains the 65% specular reflection for incidence angle around 43° (Fig. 11.31a). However, for angles larger than 45°, some of the specularly reflected light can immediately hit another cone and be randomized (Fig. 11.31b). This explains the decrease in specular reflection. This also corresponds to the “shading” of the flat surface by the cones (Fig. 11.31c, d).

Fig. 11.31

a 44° specularly reflected ray, b high angle ray undergoing specular and random reflection, c high angle ray that is directly randomized by a pattern that “shades” the flat surface, and d a ray impinging at the critical angle beyond which there is no specular reflection

The purpose of patterning the sapphire differs subtly from that of roughening the surface of GaN. The GaN roughening is used both for its randomization effect and improvement of first pass extraction. However, in the case of GaN/sapphire patterned interface, the situation is different: extracting the rays from GaN to the sapphire substrate is not the final purpose of the texture. Like for the GaN-on-GaN chip with roughened backside, it is the randomization effect that is the most beneficial.

Indeed, getting the light into the sapphire has two opposite effects: due to the sapphire’s lower index and absorption, sidewall extraction is facilitated, but on the other hand, light in the substrate will most likely hit the lossy mirror. So, contrary to flip chip roughening, we cannot simply compute the nth pass extraction in sapphire to determine the efficacy of the structure. We have to study in more detail the extraction process.

As seen in Fig. 11.32, the PSS interface will extract a downwards traveling ray in two ways: (a) it can reflect the ray into the GaN extraction cone or (b) it can refract the ray into the sapphire extraction cone, which leads to the ray being extracted after it bounces back on the LED rear reflector [1]. For (a), rays reflected into GaN must be below 23.6° (GaN/air TIR angle), and for (b), transmitted rays must be in the sapphire/air extraction cone (36.0°).

Fig. 11.32

First attempt extraction rays: a reflected at low angles in GaN and b transmitted at low angles in sapphire

The fraction of incident rays reflected into the GaN light cone will be referred to as the “GaN efficacy.” The smaller number of rays that are transmitted into the sapphire light cone can be extracted, too. For a simplified first pass calculation, we will not take into account the effect of the second randomization of rays going back through the sapphire/GaN interface (Fig. 11.32(2)). The “sapphire efficacy” is then defined as the fraction of rays inside the sapphire light cone (Fig. 11.32b). The total first pass efficacy is then defined as the sum of the GaN efficacy and of the sapphire efficacy. It estimates the fraction of the incident rays that will be extracted on first attempt via scattering by the patterned interface into the air cone directly into GaN, or through the sapphire and back through the GaN into the air cone.

One should note that this model does not take into account the reflectivity of the GaN/air interface inside the light cone (some of the low angle incident rays will still be reflected according to Fresnel laws), the losses on the bottom mirror (taken as zero), and the effect of the sidewall extraction. However, all these effects are mostly independent of the pattern choice, so the total first pass efficacy is a good relative indicator to compare different patterns.

To find the optimal pattern to achieve maximal first pass efficacy, we first find the optimal shape, comparing a flat surface, half sphere, and cone. The cone pattern is still according to Fig. 11.28, and the half sphere has the same filling factor as the cone (34.9%). The total efficacy, for different incidence angles is given in Fig. 11.33.

Fig. 11.33

Comparison of total efficacy (sapphire efficacy + GaN efficacy) for different pattern geometries (cone, half sphere with 34.9% filling factor, and flat surface)

We can compute the average efficacy over all incidence angles for the three surfaces. We find 8.6% for the flat surface, 13.0% for the half sphere and 14.8% for the cone pattern. The higher performance of the cone can be explained by the high efficacy at high angles.

In more detail, we can see that this high angle efficacy comes from GaN efficacy: the cone pattern reflects some of the light incident at high angles into GaN extraction cone (Fig. 11.34a). Qualitatively, we can understand it from simple ray tracing (Fig. 11.34b).

Fig. 11.34

a Breakdown of the efficacy of the cone pattern between the sapphire efficacy (dashed blue) and GaN efficacy (solid red), b Simple 2D model of the high angle GaN efficacy

Once the cone pattern was identified as the optimal pattern, simulations were performed to determine the cone apex angle that maximizes the efficacy. This was done by varying the height of the cones (while keeping the fill factor and all other parameters the same) from 0.15 to 7 µm. These cone heights correspond to cone apex angles from 85.7° to 17.7°. For each geometry, the average low angle transmission and reflection were computed. The results are given in Fig. 11.36 (Fig. 11.35).

Fig. 11.35

Effect of cone apex angle on PSS single pass efficacy

We can see that the fraction of light transmitted to the sapphire extraction cone does not change much with the cone shape. On the contrary, the reflection in GaN is very sensitive to the cone angle. It reaches a maximum for a cone apex of 66°. We can understand why if we compare the GaN efficacy as a function of incidence angle for four geometries (Fig. 11.36).

Fig. 11.36

Comparing the GaN efficacy for different cone geometries

For very flat cones (high apex angle), the result is close to that of a flat surface, with little reflection at low angles and diminishing to nearly zero for angles beyond the GaN/air critical angle (23.6°). An angle of 67° is approximately optimal, with good reflection at high angles (as explained in Fig. 11.34b). For more peaked cones, we lose this high angle extraction, but this is partly balanced by an increase in reflection for angles around 40°–60°. Then for an even more peaked cone, this reflection diminishes.

Finally, the optimal filling factor for these 66° cones was determined by varying the distances between cones (2–20 µm) in the simulations (Fig. 11.37). As the patterns are in square packing, this results in filling factors from 3.1% to 79%. We can see that the part of light reflected in GaN increases with the filling factor. The light extracted in sapphire is almost constant (8.2%–8.5%).

Fig. 11.37

Effect of varying the filling factor for a given cone shape on the pattern first pass efficacy. To obtain an even better filling factor, a hexagonal packing can be used, resulting in about 91% filling factor. For this filling factor, the resulting efficacy is 18.1% compared to just 8.4% for a flat GaN/sapphire interface

These results can qualitatively predict the LEE, for example [32], showed that increasing the filling factor leads to better extraction. However, both the full chip modeling and the experimental results show that the incremental improvement in LEE with increasing filling factor tends to diminish for high filling factors. This is because only the first pass efficacy was computed. In reality, if the light is not extracted on first attempt, it can reach the patterned interface again, where it would be more likely to be extracted after fewer attempts for a high filling factor.

Appendix B

11.2.1 Comparison of Roughening and Angled Sidewalls for GaN Substrate LEDs

A comparison was done to demonstrate the hierarchy of techniques of roughening, out of plane angled sidewalls (truncated inverted pyramid structure) and high refractive index encapsulation. An angle of 15° was used for the sidewalls (Fig. 11.38). The most effective extraction techniques have high extraction values at low index that saturate earlier with increasing index. The combination of roughening and the sidewall angling gives the highest extraction for low- index encapsulants and the benefit of increasing encapsulant index saturates at ~n = 1.8.

Fig. 11.38

Comparison of roughening and chip shaping techniques for light extraction

The advantage of combining both roughening and sidewall angling over sidewall angling alone, however, is small, differing at most by less than 2%. Roughening without sidewall angling follows after that, indicating that at least for the chip dimensions modeled here, sidewall angling into a truncated pyramid structure is a more effective extraction method. It should be noted, however, that for larger chip sizes, backside roughening may gain an advantage over chip shaping. This is because the improvement from angled sidewalls diminishes with increasing chip area as the rays must travel further through the device to reach a wall where they may be extracted or redirected.

Appendix C

11.3.1 Simulations of Periodic “Rough” Surfaces Versus Random Rough Surfaces

The first pass extraction probability can be quantified by analyzing the intensity of light emission through a roughened surface as a function of incidence angle. In real systems, the surface texturing is not of constant size or regularly distributed on the surface, as for GaN roughening from a chemical etching. Generating random patterns in ray tracing considerably increases the complexity of the simulations, so a simpler model was considered, which uses a periodic pattern rather than a random one (Fig. 11.39). For comparison, the transmission function was computed in the two cases of random and periodic surfaces (Fig. 11.40). Little difference was found between the two, not surprisingly as ray tracing is a geometric optics approach, so the size of the pyramids does not affect the results. Randomizing the position of the pyramids can only modify the trajectory of those already extracted rays that hit a second pyramid, which is very rare. Therefore, the periodic pattern will be further used to simplify the simulations without losing accuracy compared to the actual random patterns.

Fig. 11.39

Random simulated pyramids (left) and evenly spaced pyramids (right)

Fig. 11.40

Transmission function of GaN/air interface with random or periodic rough patterns