We report a numerical analysis of the lightextraction efficiency (LEE) of lightemitting diodes (LEDs) on patterned sapphire substrates (PSSs). We considered various
n
sided, regular convex pyramids, where
n
is an integer and
n
≥ 3. We then considered four cross sections: extruded, subtracted, truncatedextruded, and truncatedsubtracted. Raytracing simulations were carried out with these polygonal pyramid patterns, and the dimensions of the patterns were systematically varied. Optimized pattern shapes were determined for large LEE. An extruded circular pyramid with a slant angle of 45° was found to be the optimal patterned shape.
I. INTRODUCTION
Lightemitting diodes (LEDs) are solidstate light sources that exploit electroluminescence in semiconductor materials to generate light. LEDs are semiconductor
p

n
junction diodes, from which light is emitted due to the optical recombination of electrons and holes. The wavelength of the emitted light is determined by the band gap of the semiconductor, and is given (in µm) by λ = 1.24/
E_{g}
, where
E_{g}
is the band gap (in eV). LEDs have attracted much attention as light sources for applications such as indicator lights on devices, traffic lights, automobile lighting, and indoor and outdoor lighting, including street lighting. This is due their low energy consumption, long lifetime, robustness, absence of a warmup period, favorable controllability, and good color rendering. To include a wider range of applications, improvements in the efficiencies of LEDs are required.
The lightextraction efficiency (LEE) of LEDs is an important parameter. The LEE of an LED is defined as the ratio of the number of photons emitted into free space from an LED chip to the number of photons emitted from the active region inside the LED chip
[1]
. It is limited by total internal reflection (TIR) inside the LED semiconductor chip, which typically has a larger refractive index than the surrounding material. When the light generated from the active region inside the semiconductor is incident upon the interface between the semiconductor and the surrounding space, if the angle of the emitted light exceeds the critical angle, TIR occurs and light is trapped inside the semiconductor, eventually being dissipated as heat. The LEE of an LED is typically small, because there is a large difference in refractive index between the semiconductor and the surrounding space
[2]
. For example, the refractive index of gallium nitride (GaN) is 2.5, and the LEE of a GaNbased LED chip with a simple rectangular shape into free space is only 4％
[1
,
2]
. Numerous approaches have been used to improve the LEE, including photon recycling
[3]
, flipchip configurations
[4]
, resonant cavities
[5]
, a transparent surface contact layer
[6]
, sidewall treatment
[7]
, surface roughening
[8]
, photonic crystals
[9]
, chip shaping
[10]
, and patterned sapphire substrates (PSSs)
[11

14]
. PSSs employ arrayed patterns on the entire top surface of the sapphire that forms the LED substrate. Scattering and multiple reflection of the light inside the LED chip can be enhanced with these structures, breaking the TIR condition and therefore improving the LEE. The PSS can also serve to reduce the threadingdislocation density during the growth of LED epitaxial layers.
Monte Carlo raytracing simulations are one of the most popular optical simulation methods for the analysis of photonic devices
[15

18]
. Geometrical optical phenomena in photonic devices can be analyzed using this method, including ray scattering, absorption and transmission in the semiconductor layers, Fresnel losses of the media due to different refractive indices, and the randomly emitted light from the active region of an LED
[19]
. Although a number of studies of the numerical design of PSSs have been reported, a systematic study of
polygonal pyramid patterns
in PSSs has not yet been carried out. Polygonal pyramid patterns may be the most important shapes for use in PSSs, because they may be realized on the sapphire using only simple, twodimensional (2D) photolithography patterning and conventional etching processes.
Here we describe numerical simulations for LEDs using the Monte Carlo raytracing method. We examined the LEE of LEDs using PSSs, focusing on PSSs with various polygonal pyramid patterns. By systematically varying the possible shapes of the polygonal pyramid patterns, we determined the pattern shape and size for optimal LEE.
II. RAYTRACING SIMULATION PROCEDURES
We used the software application LightTools to implement raytracing simulations.
Figure 1
shows a schematic diagram of the simulated GaNbased PSSLED structure. The size of the chip was 300 × 300 μm
^{2}
. The parameters for each layer in the PSSLED structure are listed in
Table 1
. The device consisted of a 330μmthick sapphire substrate, a 2μmthick GaN layer, a 0.05μmthick (Al,Ga)N layer, a 2μmthick
n
type GaN layer, a 0.1μmthick multiple quantum well (MQW) layer, and a 0.05μmthick
p
type GaN layer. The refractive indices of these layers were 1.78, 2.4, 2.4, 2.42, 2.54, and 2.45 respectively
[20]
. The absorption coefficient of the MQW layer was assumed to be 10
^{4}
cm
^{−1}
[19
,
21]
, and the reflectivity of bottom surface of the sapphire was 90％
[22]
. To simplify the LED structure, the mesa and electrode structures were neglected in our simulations, and the PSS patterns were modeled on the top surface of the sapphire substrate. We considered only an LED chip structure without a package (PKG) structure, and therefore the material outside is assumed to be simply air in our study. We think, however, our results without PKG can also be applied in general to LEDs with PKG, at the qualitative level.
Schematic diagram of the PSSLED structure used in the raytracing simulations.
Parameters for each layer in the raytracing simulations of the PSSLEDs
Parameters for each layer in the raytracing simulations of the PSSLEDs
We chose to consider only polygonal pyramid patterns, because these patterns can be expected to be fabricated reasonably on the sapphire substrate via simple control of the 2D etching mask pattern shape, in addition to the etching conditions. We did not consider pillars because the slant angle of the pillar pattern is fixed at 90°, and so trapped light cannot efficiently outcouple via multiple reflections.
We aimed to cover many possible shapes of the polygonal pyramid patterns. We considered pyramids with bases of
n
sided regular convex polygons, where
n
= 3, 4, 6, and ∞, (i.e., pyramids with triangular, square, hexagonal, and circular bases). Here, the pyramid pattern with a circular base is also known as a cone. Four different cross sections of the pyramids were considered: extruded, subtracted, truncatedextruded, and truncatedsubtracted. The global arrangement of the patterns on the sapphire substrate was a regular hexagonal array (or honeycomb), which is widely used due to its high degree of integration.
Figure 2
shows the different polygonal pyramid patterns that were investigated. To analyze and compare the LEE of the differently shaped PSSs as a function of the size of patterns, we considered the circumcircle diameter of the pyramid base.
Figure 3
shows a planview of the array patterns of the four different bases. In the examples shown in the figure, the diameter of the circumcircle is the same as the pitch, i.e. the pattern is a closepacked array of the circumcircle. We independently varied the circumcircle diameter and the pitch in addition to the vertical height of the patterns, and then carried out simulations for each pattern. We defined the relative LEE of PSSLED as the ratio of the LEE of PSSLED to that of a reference LED, which did not have the PSS pattern. Using this normalized LEE, we determined the relationships between the LEE and the patterns used on the PSS.
The various polygonal pyramid patterns considered in this work.
The circumcircle of the base of the polygonal pyramids in closepacked PSS patterns. The dashed yellow lines are the circumcircles of the bases, and the red lines show the pitch p of the patterns. (a) Triangular pyramid, (b) square pyramid, (c) hexagonal pyramid, and (d) circular pyramid.
III. RESULTS AND DISCUSSION
Figure 4
shows simulated relative LEEs of the PSSLEDs with various polygonal pyramid patterns. We started with a pitch of
p
= 3.0 µm, which is typical of commercially used PSSs
[23
,
24]
. We then varied the diameter of the circumcircle in each pattern in the range 1.2 ≤
d
≤ 3.0 μm, i.e. with
d
in the range 0.4 ≤
d
/
p
≤ 1. The height of the pattern was
h
=
d
/2, so that the angle between slant edge and base was 45°. The four regular pyramid bases (i.e.,
n
= 3, 4, 6, and ∞) were independently considered, and the extruded type was considered for the cross section of the pyramid.
The LEE for the various PSSLEDs, relative to that of an LED without the PSS, for pyramids with various polygonal bases circumcircle diameters less than the pattern pitch of 3 µm, and pattern heights of onehalf the diameter.
For each base shape, with a fixed pattern pitch, the LEE increased with the circumcircle diameter. In other words, the LEE increased as the fraction of the patterned surface area increased, and reached a maximum when the circumcircle diameter was at its maximum, i.e.
d
=
p
, a closepacked array. This result is to be expected, as one may anticipate that the scattering of the trapped light inside the structure will increase as a function of the fraction of the surface that is covered by the PSS patterns.
With a fixed circumcircle diameter, the circularbase pyramid exhibited a larger LEE than those with
n
= 3, 4, or 6. This is consistent with the above result, as the circular base has the largest proportion of patterned surface area. The hexagonalbase pyramid exhibited a larger LEE than the squarebase pyramids; however, the triangularbase structures exhibited a larger LEE than the square or hexagonalbase pyramids, despite the fact that the triangular structures had the largest proportion of unpatterned planar surface area. The
n
= 3 and
n
= ∞ pyramids are therefore preferred. When the ratio with the circumcircle diameter remained fixed, so that we had a closepacked array of circumcircles (and the angle between the slant edge and the base was 45°, and the proportion of unpatterned planar surface was also unchanged) the LEE did not change as a function of pattern pitch. It follows that we may have freedom of choice over the pattern height, so that we have flexibility in terms of the conditions for sapphire etching and epitaxy for the growth of the LED structure.
Four crosssectional shapes were considered: extruded, subtracted, truncatedextruded, and truncatedsubtracted. We fixed the pattern pitch at
p
= 3.0 μm, and the diameter of the circumcircle at
d
=
p
. The vertical height of the pattern was also fixed, at
h
=
d
/2 for the nontruncated types and
h
=
d
/4 for the truncated shapes, so that the slant angles were fixed at 45° in all cases. The LEE was calculated for the four regular, convex pyramid bases, as shown in
Fig. 5
. The effect of the crosssectional shape on LEE differed with base shape. The circular and triangular bases exhibited the largest LEE with the extruded cross section, followed by the truncatedextruded, the subtracted, and then the truncatedsubtracted cross sections. For the square and hexagonal bases, the truncatedextruded cross section had the largest LEE, followed by the truncatedsubtracted, the subtracted and then the extruded cross section. The LEE was largest with the circular base and the extruded cross section.
The LEE for the different PSSLEDs, relative to that of an LED without the PSS, for pyramids with various polygonal bases and crosssectional shapes with a pitch of 3 µm and a circumcircle diameter of 3 µm.
After determining the optimal geometry of the base and the cross section of the pyramid, we varied the vertical height, thereby varying the slant angle.
Figure 6
shows the LEE with the extruded circularbase pyramid patterns as a function of pattern height, for various diameters of circumcircles with a pitch of 3.0 μm. For a given height of the pattern, the LEE increased with the diameter of the circumcircle. For a given circumcircle diameter, when
d
≤ 2.5 μm, the LEE increased with the height of the pattern. In practical epitaxial growth, however, the height of the pattern should be limited to less than the thickness of the
n
type GaN layer. For
d
greater than about 2.5 μm, the pattern has the optimum height for a maximum LEE under the given circumcircle’s diameter. For example, with
d
= 3.0 μm (a closepacked array of circumcircles), the LEE initially increased with increasing
h
, reached a maximum at
h
= 1.5 μm, and then decreased with further increase in
h
.
The LEE for the various PSSLEDs, relative to that of an LED without the PSS, for the extruded circularbase pyramid pattern, as a function of the pattern height for various circumcircle diameters, and with a pattern pitch of 3 µm.
Based on these results, we can conclude that the extruded cross section and circularbase pyramid pattern with
p
=
d
=
h
/2 (i.e. a slant angle of 45°) has the largest LEE. The actual physical values of
p
,
d
and
h
appear to be irrelevant, so long as the constraint
p
=
d
=
h
/2 is observed, which provides some flexibility in terms of the epitaxial growth of the LED.
Additionally, we want to make the following points regarding the basic concern about the validity of the raytracing model for the calculation of LEE in PSSLEDs with micronsized structures. It is known that most researchers have successfully simulated light extraction in LEDs using the raytracing method when they considered LEDs with mixed structure sizes, i.e. some micronsized structures and some submicronsized, like our LEDs
[14
,
16
,
17
,
19
,
20]
, even though they used a strict method such as the finitedifference timedomain (FDTD) method when they considered LEDs with entirely nanoscale structures. The validity for such a use of the raytracing method can be explained: The key point in the phenomenon of light extraction with PSSLEDs is the breaking of TIR, so that the light’s nature as a ray is more conspicuously considered than as a wave. Recently, it has also been reported that simulation results for LEE enhancement of LEDs with entirely nanoscale structures are similar for raytracing and FDTD, the difference not exceeding 8.5％
[18]
, which justifies our reasoning above.
IV. CONCLUSION
We have described the results of Monte Carlo raytracing simulations of the LEE of PSSLEDs with various polygonal pyramid patterns. We systematically varied the geometry of the polygonal pyramid patterns, and found relationships between the LEE of PSSLEDs and the geometry of the patterns. The LEE increased as the fraction of the unpatterned planar surface area decreased. Triangular and circularbase pyramids yielded a maximal LEE for an extruded cross section, whereas square and hexagonalbases had maximal LEEs for a truncatedextruded cross section. The LEE was largest for a circular base, followed by a triangular base. In general, the LEE increased with the height of the pattern when there was a large fraction of unpatterned planar surface. However, when the fraction of the unpatterned planar surface was below a certain critical value, the pattern exhibited an optimum height. The circularbase pyramid pattern with an extruded cross section,
p
=
d
=
h
/2 (i.e. a slant angle of 45°), and a 2D closepacked hexagonal pattern had the largest LEE.
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