Coherent vertical beaming using Bragg mirrors for high-efficiency GaN light-emitting diodes

We propose a dielectric Bragg mirror that utilizes coherent coupling with multiple quantum wells (MQWs) to significantly enhance light extraction from GaN light-emitting diode (LED). Full vectorial electromagnetic simulation showed that, under constructive interference conditions, the Bragg mirror consisting of two dielectric (SiO2/TiO2) stacks and a silver layer led to >30% enhancement in light extraction, as compared to a single silver mirror. Such significant enhancement by a pre-designed Bragg/metal mirror was ascribed to the vertically oriented radiation pattern and reduced plasmonic metal loss. In addition, the gap distance between the MQWs and a Bragg mirror at which the constructive interference takes place could be controlled by modulating the thickness of the first lowrefractive-index layer. Moreover, a two-dimensional periodic pattern was incorporated into an upper GaN layer with the designed Bragg mirror and it was shown that a lattice constant of ~800 nm was optimal for light extraction. We believe that tailoring the radiation profile of light emitters by coherent coupling with designed high-reflectivity mirrors will be a promising route to overcome the efficiency limit of current semiconductor LED devices. ©2013 Optical Society of America OCIS codes: (260.3160) Interference; (230.3670) Light-emitting diodes; (230.1480) Bragg reflectors. References and links 1. M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Display Tech. 3(2), 160–175 (2007). 2. A. Laubsch, M. Sabathil, J. Baur, M. Peter, and B. Hahn, “High-power and high-efficiency InGaN-based Light Emitters,” IEEE Electron. Lett. 57, 79–87 (2010). 3. J. J. Wierer, A. 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Introduction
High-efficiency GaN light-emitting diodes (LEDs) are environmentally friendly energysaving devices, thus making their development highly desirable [1,2].Over the past decade, most GaN LED chips have been used in mobile backlights units or small-sized light bulbs.Recently, the market trend has shifted to high-output applications such as TV backlight and general illumination, and this has resulted in rapid expansion of the market size.However, to replace conventional lighting with LEDs, the efficiency of LEDs must be improved.The efficiency of GaN LEDs is primarily determined by heteroepitaxial growth techniques, and the extraction efficiency can be improved as long as all photons generated from multiple quantum wells (MQWs) are not extracted into an ambient medium [3].
Thus far, several strategies for enhancing the light extraction efficiency of GaN LED devices have been proposed and demonstrated.The most common method for significantly increasing the light extraction efficiency is the introduction of random surface texturing [4,5] or periodic patterning into a GaN layer [3,6,7], bottom reflector [8,9], or substrate [10].The non-periodic or periodic patterns can even diffract light waves whose wave vectors are outside the light cone (e.g.θ~23.6° for air as an ambient medium) [11,12].Thus, light trapped inside a medium with high refractive index can be extracted until it is completely absorbed [13].Another important strategy is to exploit coherent coupling between MQWs and a highreflectivity mirror [14,15].The distribution of wave vectors of radiation generated from MQWs can be controlled by different interference conditions.When the condition of constructive interference holds for a normal direction, most of the wave vectors steer nearby the normal direction.Vertically oriented radiation by coherent coupling with a mirror does not undergo total internal reflection or does not require diffraction of low-order guided modes [3,11,12].In order to harness this vertical beaming effect, thus far, only a metallic (e.g.silver) mirror has been used in LED structures [14,15].This work differs from previous studies that used coherent coupling between MQWs and a metallic mirror, in that here we introduced a dielectric Bragg mirror as a bottom reflector for GaN LED structures, for which the angular reflectance is substantially different from that of a metallic mirror.Different reflective layers and angular reflectance may influence resonant conditions for the vertical beaming effect which determines the thickness of p-doped GaN layer, and the spatial distribution of radiation generated from MQWs which determines the extraction efficiency.By performing finite-difference time-domain (FDTD) simulation, we calculated the extraction efficiency of GaN LEDs with the Bragg mirrors designed for different interference conditions.In addition, we quantitatively investigated how much light is additionally extracted following the introduction of a two-dimensional periodic pattern into an upper GaN layer.

Coherent coupling with a single dielectric layer
In our FDTD simulation, MQWs were modeled as an appropriately combined set of electric dipole sources with orthogonal polarization.As an electric dipole is placed close to an interface across which the refractive index changes, an imaginary dipole is created in an opposite plane (Fig. 1(a)).Then, a real dipole interferes with its corresponding imaginary dipole and such interference can modulate the radiation's profile as well as the spontaneous emission rate [16].
To systematically study the interference effect, we first considered a single dielectric substrate.Randomly polarized dipole sources with a center wavelength of 450 nm and a bandwidth of 25 nm were embedded in a GaN (n = 2.5) layer onto the dielectric substrate (Fig. 1(b) right subpanel).Although the thickness of the light element was 5 nm (equivalent to the spatial resolution of our simulation), this model readily supports real InGaN/GaN MQWs wherein most of the radiation's recombination occurs in the first well [17].The total thickness of the GaN layer was set to 3 μm.By carrying out the FDTD simulation, we calculated the extraction efficiency of the GaN structure while varying the positions of the dipole sources from the dielectric layer (Fig. 1(b), left subpanel).Periodic boundary conditions for both x-and y-axes were employed to describe the indefinite propagation of light along in-plane directions.Air (n = 1.0) and sapphire (n = 1.8) were examined as dielectric substrates.In fact, a dielectric interface can be regarded as a weakly reflecting mirror.The calculated result revealed several important design rules related to the interference effect.First, the efficiency of extraction exhibited an oscillatory behavior as a function of the gap distance, d.Second, the difference between maximum and minimum values of extraction efficiency was higher for higher refractive index's contrast of the dielectric layers.Notably, the difference in extraction efficiency between air and sapphire substrates was beyond the difference in reflectivity at each interface.Third, local maxima or minima in extraction efficiency occurred at different gap distances for air and sapphire substrates.These observations illustrate that larger reflectivity at an interface induces more noticeable modulation of extraction efficiency.In addition, the interference condition for which extraction efficiency was maximal relied upon substrate materials, because light reflected at the interface experiences a specific phase change.Therefore, to enhance the extraction efficiency of GaN LEDs by the interference effect, it is important to design a high-reflectivity mirror and to choose an appropriate gap distance between the MQWs and the mirror.

Coherent coupling with Bragg mirrors
As a high-reflectivity mirror we have considered a system of distributed Bragg reflectors (DBRs) that were composed of two different dielectrics (Fig. 2(a)).In the DBR, a quarterwave SiO 2 (n = 1.5) and TiO 2 (n = 2.5) were stacked in an alternating way over a specific period [18].The DBR can be used in thin-film flip chip [19] or vertical slab GaN LEDs [3,5,10,12] where the sapphire substrate is removed by laser lift-off technique.First, we calculated the reflectivity of 2-pairs and 4-pairs of DBRs as a function of incident angle, and we also calculated the reflectivity of a silver mirror as a reference (Fig. 2(b)).For this reflectivity calculation, an analytical transfer-matrix method was used [20].The results of reflectivity calculation showed that the reflectivity of the DBRs had a local maximum at normal incidence and reached unity beyond a critical angle which was determined by the two materials, GaN and SiO 2 , a low refractive index layer in the DBRs [21].The overall reflectivity was proportional to the number of the pairs of dielectrics.On the other hand, a silver mirror exhibited a slight gradual increase in reflectivity with increasing incident angle.
To quantitatively investigate the interference effect between the random dipole sources and the DBR or a silver mirror [22], we calculated the extraction efficiency of a GaN structure using FDTD simulation while varying the gap distance, d (Fig. 2(c)).The calculated result showed that the extraction efficiency was greatly enhanced following the introduction of high-reflectivity mirrors, as compared to a single dielectric layer (Fig. 1(b)).For the DBRs, the extraction efficiency increased as the number of the dielectric stacks increased, which was in accord with the results obtained for the reflectivity (Fig. 2(b)).The local maxima in extraction efficiencies of 2-pairs and 4-pairs DBRs were found at gap distances of d = 10, 90 and 180 nm.Because for each dielectric, a DBR is composed of a quarter-wave stack, the phase change that light undergoes at the reflection of each layer is always zero.Therefore, the extraction efficiency in DBR becomes maximized as the distance d approaches zero.Specifically, in-plane dipoles, which are responsible for vertical radiation, satisfy the lowest order requirement of constructive interference condition at d = 0 nm.As the order of constructive interference increases, the extraction efficiency at the local maximum decreases gradually, because additional constructive interference takes place at other off-angles.In the case of a metal mirror, the phase change for reflected light is given by π + 2α, where α = 2πl/λ and l is the skin depth of a metal [14,15].This illustrates that the extraction efficiency in a silver mirror is minimized close to d = 0 nm and is maximized at d = 40 and 120 nm.We note that the maximal extraction efficiency of 2-pairs and 4-pairs DBRs surpassed that of a silver mirror.The interference effect becomes much clearer after the radiation profiles of dipole sources in a GaN structure are calculated.We considered the 2-pairs of SiO 2 /TiO 2 DBR (Fig. 2(d)) and a silver mirror (Fig. 2(e)).Under constructive (destructive) conditions, for both mirrors, the radiation pattern had an intensity anti-node (node) in the vertical direction.However, it was clearly observed that, under constructive conditions, the vertical radiation pattern of the DBR had less angular spreading, compared to that of the silver mirror.We postulate that higher efficiency from the DBRs is achieved due to the narrow beam divergence along with the absence of plasmonic metal loss that becomes severe when dipoles are located nearby a metal surface [23].Lastly, we studied the possibility of tuning a gap distance at which constructive interference occurs.Although extraction efficiency of the DBRs at d = 10 nm was larger than that of a silver mirror at d = 120 nm, a gap distance of 10 nm would not be acceptable in real GaN LED devices because current spreading is very poor at such a small gap distance.To resolve this issue, we reduced the thickness of the first SiO 2 layer in the DBR from a quarter-wave length (75 nm) to 25 nm [21].In this modified DBR, the first maximum of extraction efficiency was observed at d = 30 nm (Fig. 2(c), dashed red), while the extraction efficiency remained almost unchanged.Taken together, a designed dielectric DBR mirror exhibits enhanced vertical beaming effect as compared to a metallic mirror, resulting in the increased extraction efficiency of a DBR.Significantly, an optimal condition that promises the maximal extraction efficiency can be controlled by the rational design of each one of the dielectric layers in the DBR [24].
To further improve the vertical beaming effect and its resultant extraction efficiency, we designed a SiO 2 /TiO 2 DBR combined with an underlying silver mirror (Fig. 3(a), inset).Although Bragg/metal mirrors have been explored in prior works [25,26], the mirrors positioned far away from MQWs were used only for improving the reflectivity.First, we calculated the angular reflectance of the DBR/silver mirror as a function of the number of dielectric stacks in the DBR (Fig. 3(a)).The calculated result showed that the reflectance nearby 0° was enhanced noticeably compared to the system that had only a DBR or a silver mirror (Fig. 2(b)).However, multiple dips emerged at off-axis angles in the angular reflectance, which stems from amplified optical absorption on the silver mirror by optical resonances.The optical resonances were observed more clearly with increasing the number of dielectric stacks.Next, we calculated the extraction efficiency of the DBR/silver mirror at the first constructive interference condition (d = 10 nm) as a function of the number of dielectric stacks in the DBR (Fig. 3(b)).The result showed that the extraction efficiency was maximized when 2-pairs of SiO 2 /TiO 2 dielectric stacks were used for the DBR.More importantly, the extraction efficiency of 18.5% was obtained, which constituted ~30% improvement over the maximal value that could be obtained from a silver mirror (Fig. 2(c)).We infer that the gradual decrease of DBRs in extraction efficiency with increasing number of dielectric stacks is attributed to the multiple dips in angular reflectance (Fig. 3(a)).Consequently, a dielectric DBR combined with a metallic mirror provides excellent reflectivity nearby the normal direction, which leads to significant enhancement in extraction efficiency as a result of the interference effect.The DBR/silver mirror can be further tailored toward higher extraction efficiency by the vertical beaming effect.For example, the SiO 2 layer used as a low index layer can be replaced by air in order to further increase the contrast in refractive index between the two layers [24,27].

Two-dimensional periodic pattern with a designed Bragg mirror
In a semiconductor LED device, introduction of random texturing [4,5] or periodic patterning into a dielectric [3,6,7,10,12] or metallic [8,9] surface is essential to extract light trapped by total internal reflection.To investigate how surface patterning interacts with the vertical beaming effect, we incorporated a two-dimensional square-lattice pattern into a GaN medium with a bottom DBR/silver mirror (Fig. 4(a)).The 2-pairs of SiO 2 /TiO 2 dielectric stacks were used for the DBR onto a silver mirror.Randomly polarized dipole sources were generated at a certain gap distance (d) to induce the vertical beaming effect (d = 10 nm for the DBR/silver mirror and d = 120 nm for a single silver mirror).Then, using the FDTD simulation, we calculated the extraction efficiency of the periodically patterned GaN structures with the DBR/silver mirror (Fig. 4(b), solid red) and a single silver mirror (Fig. 4(b), solid gray) as a function of propagation distance r.Here, propagation distance denotes the distance to which the photons, generated by the MQWs, can travel inside the GaN medium.The lattice constant a and the depth of the periodic pattern were fixed at 800 and 600 nm, respectively.No material absorption except for silver mirror was accounted.The details on the FDTD calculation are described elsewhere [6,8,12,13,21].The calculated result showed that extraction efficiencies for the DBR/silver and single silver bottom mirrors increased steadily with the increasing propagation distance.However, for all propagation distances, the LED structure with the DBR/silver mirror was characterized by a more significant light extraction than the structure with single silver mirror.In addition, we calculated the extraction efficiencies for both LED structures while varying the lattice constant of the periodic pattern (Fig. 4(b), inset).In this calculation, we set a cut-off propagation distance as 300 μm to define the extraction efficiency [11].In a real LED device, the cut-off propagation distance is determined by interior optical absorption such as material absorption, free carrier absorption or inter-band absorption [13,28].The result showed that both structures had their maximal efficiency at a = 800 nm.Such a lattice constant is much larger than λ/n that provides the phase-matching condition to the fundamental guided mode, and is favorable for diffraction of light if the wave-vector is distributed in the vicinity of light cone [3,11,12].Since high-order guided modes with wave vectors nearby light cone are diffracted more efficiently than loworder guided modes propagating horizontally [11,12], the vertical beaming effect can boost the light extraction by a periodic surface pattern.At optimal a, the extraction efficiency of the structure with the DBR/silver mirror was ~15% higher than that of the structure with single silver mirror.The enhancement can be increased if real absorption is taken into account and thus, the propagation distance under consideration is shorter.

Conclusion
We studied the interference effect of dipole sources as they are placed close to a dielectric DBR at a distance of approximately one wavelength of light.Under the condition of constructive interference, the dipole sources generated radiation patterns with narrow beam divergence, which resulted in the improved extraction efficiency of a GaN LED structure.The vertical beaming effect and its resultant extraction efficiency were further improved when the dielectric DBR was combined with an underlying silver mirror.The maximal extraction efficiency of the GaN structure with a rationally designed DBR/silver mirror was enhanced by ~30% compared to that of the GaN structure that contained only a silver mirror.Considering that the reflectivity of a real silver mirror is degraded due to its surface roughness [29], such an enhancement from the dielectric DBR mirror is likely to be more pronounced in experimental situations.In addition, the vertical beaming effect allowed for a relatively large lattice constant (a ~800 nm) for effective extraction of light trapped in a GaN medium.Our structure can be fully demonstrated in vertical GaN slab or thin-film GaN flipchip structures where n-doped GaN layer is periodically patterned.Manipulating the wave vectors of light emitters by using rationally designed mirrors will provide a feasible and powerful strategy for the development of high-efficiency semiconductor LED devices.

Fig. 1 .
Fig. 1. (A) Schematic of electric dipoles and their image dipoles with polarization normal (left) or parallel (right) to a mirror plane.(B) Extraction efficiency of a GaN structure with a bottom substrate composed of air (solid red) or sapphire (solid black), as a function of distance, d, between random electric dipole sources and the substrate.Right: schematic of the calculated GaN structure.

Fig. 2 .
Fig. 2. (A) Schematic of a GaN structure with a bottom DBR.(B) Reflectance of 2-(solid red) and 4-pairs (solid brown) of SiO 2 /TiO 2 DBRs and a silver mirror (solid gray), as a function of incident angle.(C) Extraction efficiency of a GaN structure with 2-(solid red) and 4-pairs (solid brown) of SiO 2 /TiO 2 DBRs and a silver mirror (solid gray), as a function of distance, d, between random electric dipole sources and the mirrors.The dashed red curve denotes the values from the GaN structure with 2-pairs of modified SiO 2 /TiO 2 DBR.(D, E) Electric field intensity profiles of the GaN structure with 2-pairs of DBR (D) and a silver mirror (E) when d is 10 (upper, D), 100 (bottom, D), 70 (upper, E) and 120 nm (bottom, E), respectively.

Fig. 3 .
Fig. 3. (A) Reflectance of 1-(solid black), 2-(solid red), 3-(solid blue) and 4-pairs (solid green) of SiO 2 /TiO 2 DBRs combined with a silver mirror, which were calculated as a function of incident angle.Inset: schematic of the calculated DBR/silver mirror.(B) Extraction efficiency of a GaN structure with 2-pairs of SiO 2 /TiO 2 DBRs with a silver mirror, calculated as a function of the number of dielectric stacks in the DBR.Inset: schematic of the calculated GaN structure with a bottom DBR/silver mirror.

Fig. 4 .
Fig. 4. (A) Schematic of a GaN patterned structure with a bottom DBR/silver mirror.(B) Extraction efficiency of the GaN structure with the DBR/silver (solid red) and single silver mirror (solid gray), as a function of propagation distance, r.Inset: extraction efficiency of the GaN structure with the DBR/silver (solid red) and a single silver mirror (solid gray), as a function of lattice constant, a.For the DBR, 2-pairs of SiO 2 /TiO 2 stacks were used.