Optical properties of highly polarized InGaN light-emitting diodes modified by plasmonic metallic grating

We implement finite-difference time-domain (FDTD) method to simulate the optical properties of highly polarized InGaN light emitting diodes (LEDs) coupled with metallic grating structure. The Purcell factor (Fp), light extraction efficiency (LEE), internal quantum efficiency (IQE), external quantum efficiency (EQE), and modulation frequency are calculated for different polarized emissions. Our results show that light polarization has strong impact on Fp and LEE of LEDs due to their coupling effects with the surface plasmons (SPs) generated by metallic grating. Fp as high as 34 and modulation frequency up to 5.4 GHz are obtained for a simulated LED structure. Furthermore, LEE, IQE and EQE can also be enhanced by tuning the coupling between polarized emission and SPs. These results can serve as guidelines for the design and fabrication of high efficiency and high speed LEDs for the applications of solid-state lighting and visible-light communication. ©2016 Optical Society of America OCIS codes: (230.3670) Light-emitting diodes; (250.5403) Plasmonics; (230.5440) Polarization-selective devices. References and links 1. M. R. Krames, J. Bhat, D. Collins, N. F. Gardner, W. Götz, C. H. Lowery, M. Ludowise, P. S. Martin, G. Mueller, R. Mueller‐Mach, and S. Rudaz, “High‐Power III‐Nitride Emitters for Solid‐State Lighting,” Phys. Status Solidi 192(2), 237–245 (2002). 2. D. Feezell, S. S. James, P. D. Steven, and S. Nakamura, “Semipolar InGaN/GaN light-emitting diodes for highefficiency solid-state lighting,” J. Disp. Technol. 9(4), 190–198 (2013). 3. F. Qian, S. Gradecak, Y. Li, C. Y. Wen, and C. M. Lieber, “Core/multishell nanowire heterostructures as multicolor, high-efficiency light-emitting diodes,” Nano Lett. 5(11), 2287–2291 (2005). 4. J. J. McKendry, R. P. Green, A. E. Kelly, Z. Gong, B. Guilhabert, D. Massoubre, E. Gu, and M. D. Dawson, “High-speed visible light communications using individual pixels in a micro light-emitting diode array,” IEEE Photonics Technol. Lett. 22(18), 1346–1348 (2010). 5. P. Waltereit, O. Brandt, A. Trampert, H. T. Grahn, J. Menniger, M. Ramsteiner, M. Reiche, and K. H. Ploog, “Nitride semiconductors free of electrostatic fields for efficient white light-emitting diodes,” Nature 406(6798), 865–868 (2000). 6. Y. Zhao, S. Tanaka, C. C. Pan, K. Fujito, D. Feezell, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Highpower blue-violet semipolar (2021) InGaN/GaN light-emitting diodes with low efficiency droop at 200 A/cm,” Appl. Phys. Express 4(8), 082104 (2011). 7. H. Fu, Z. Lu, X. Zhao, Y. H. Zhang, S. P. DenBaars, S. Nakamura, and Y. Zhao, “Study of low efficiency droop in semipolar (20-2-1) InGaN light-emitting diodes by time-resolved photoluminescence,” J. Disp. Technol. (to be published). 8. Y. Zhao, S. H. Oh, F. Wu, Y. Kawaguchi, S. Tanaka, K. Fujito, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Green semipolar (202 1 ) InGaN light-emitting diodes with small wavelength shift and narrow spectral linewidth,” Appl. Phys. Express 6(6), 062102 (2013). 9. Y. Zhao, Q. Yan, C. Y. Huang, S. C. Huang, P. S. Hsu, S. Tanaka, C. C. Pan, Y. Kawaguchi, K. Fujito, C. G. Van de Walle, and J. S. Speck, “Indium incorporation and emission properties of nonpolar and semipolar InGaN quantum wells,” Appl. Phys. Lett. 100(20), 201108 (2012). #259477 Received 15 Feb 2016; revised 1 Apr 2016; accepted 9 Apr 2016; published 18 Apr 2016 © 2016 OSA 16 May 2016 | Vol. 24, No. 10 | DOI:10.1364/OE.24.00A856 | OPTICS EXPRESS A856 10. Y. Zhao, R. M. Farrell, Y. R. Wu, and J. S. Speck, “Valence band states and polarized optical emission from nonpolar and semipolar III–nitride quantum well optoelectronic devices,” Jpn. J. Appl. Phys. 53(10), 100206 (2014). 11. H. Masui, H. Yamada, K. Iso, S. Nakamura, and S. P. DenBaars, “Optical polarization characteristics of InGaN/GaN light-emitting diodes fabricated on GaN substrates oriented between (1010) and (1011) planes,” Appl. Phys. Lett. 92(9), 091105 (2008). 12. M. Ueda, M. Funato, K. Kojima, Y. Kawakami, Y. Narukawa, and T. Mukai, “Polarization switching phenomena in semipolar In x Ga 1− x N/GaN quantum well active layers,” Phys. Rev. B 78(23), 233303 (2008). 13. M. Kubota, K. Okamoto, T. Tanaka, and H. Ohta, “Temperature dependence of polarized photoluminescence from nonpolar m-plane InGaN multiple quantum wells for blue laser diodes,” Appl. Phys. Lett. 92(1), 011920 (2008). 14. Y. Zhao, S. Tanaka, Q. Yan, C. Y. Huang, R. B. Chung, C. C. Pan, K. Fujito, D. Feezell, C. G. Van de Walle, J. S. Speck, S. P. DenBaars, and S. Nakamura, “High optical polarization ratio from semipolar (202 1) blue-green InGaN/GaN light-emitting diodes,” Appl. Phys. Lett. 99(5), 051109 (2011). 15. Y. Zhao, Q. Yan, D. Feezell, K. Fujito, C. G. Van de Walle, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Optical polarization characteristics of semipolar ( 3031) and (3031) InGaN/GaN light-emitting diodes,” Opt. Express 21(101), A53–A59 (2013). 16. S. H. Park, D. Ahn, and S. L. Chuang, “Electronic and optical properties of a-and m-plane wurtzite InGaN–GaN quantum wells,” IEEE J. Quantum Electron. 43(12), 1175–1182 (2007). 17. W. G. Scheibenzuber, U. T. Schwarz, R. G. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B 80(11), 115320 (2009). 18. E. Matioli, S. Brinkley, K. M. Kelchner, Y. L. Hu, S. Nakamura, S. DenBaars, J. Speck, and C. Weisbuch, “High-brightness polarized light-emitting diodes,” Light Sci. Appl. 1(8), e22 (2012). 19. J. Vučković, M. Lončar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000). 20. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004). 21. K. Okamoto, I. Niki, A. Scherer, Y. Narukawa, T. Mukai, and Y. Kawakami, “Surface plasmon enhanced spontaneous emission rate of InGaN/ GaN quantum wells probed by time-resolved photoluminescence spectroscopy,” Appl. Phys. Lett. 87(7), 071102 (2005). 22. I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564 (1999). 23. X. Feng, F. Liu, and Y. Huang, “Calculated plasmonic enhancement of spontaneous emission from silicon nanocrystals with metallic gratings,” Opt. Commun. 283(13), 2758–2761 (2010). 24. D. Fattal, M. Fiorentino, M. Tan, D. Houng, S. Y. Wang, and R. G. Beausoleil, “Design of an efficient lightemitting diode with 10 GHz modulation bandwidth,” Appl. Phys. Lett. 93(24), 243501 (2008). 25. N. Gao, K. Huang, J. Li, S. Li, X. Yang, and J. Kang, “Surface-plasmon-enhanced deep-UV light emitting diodes based on AlGaN multi-quantum wells,” Sci. Rep. 2, 816 (2012). 26. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946). 27. Y. Xu, J. S. Vučković, R. K. Lee, O. J. Painter, A. Scherer, and A. Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B 16(3), 465–474 (1999). 28. E. Sakalauskas, Ö. Tuna, A. Kraus, H. Bremers, U. Rossow, C. Giesen, M. Heuken, A. Hangleiter, G. Gobsch, and R. Goldhahn, “Dielectric function and bowing parameters of InGaN alloys,” Phys. Status Solidi, B Basic Res. 249(3), 485–488 (2012). 29. D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. Höpler, R. Dimitrov, O. Ambacher, and M. Stutzmann, “Optical constants of epitaxial AlGaN films and their temperature dependence,” J. Appl. Phys. 82(10), 5090– 5096 (1997). 30. E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express 17(10), 7790–7799 (2009). 31. T. Suhr, N. Gregersen, K. Yvind, and J. Mørk, “Modulation response of nanoLEDs and nanolasers exploiting Purcell enhanced spontaneous emission,” Opt. Express 18(11), 11230–11241 (2010). 32. M. Nami and D. Feezell, “Optical properties of Ag-coated GaN/InGaN axial and core–shell nanowire lightemitting diodes,” J. Opt. 17(2), 025004 (2015).


Introduction
III-nitride InGaN light emitting diodes (LEDs) enable wide range of applications in solid-state lighting [1,2], full-color displays [3], and high-speed visible-light communication (VLC) [4].Conventional InGaN quantum well (QW) LEDs grown on the polar c-plane orientation, however, suffer from quantum confined Stark effect (QCSE) due to the large internal polarization-related fields, which leads to a reduced radiative recombination rate and device efficiency.Furthermore, due to the tilted QW profile induced by QCSE, c-plane LEDs have larger carrier lifetime [5], which significantly limits their performance in achieving fast modulation speed for high speed communication.To circumvent these negative effects, novel nonpolar and semipolar InGaN LEDs have been proposed and demonstrated with reduced QCSE, higher efficiency, and smaller carrier lifetime [6,7].Other advantageous features including high efficiency, improved performance in green spectral region, and polarized emission, were also reported for nonpolar and semipolar devices [8][9][10][11][12][13][14][15][16][17][18].
Recently, surface plasmons (SPs) coupled emissions from LEDs have been proposed and demonstrated with improved efficiency and modulation speed [19][20][21][22][23][24][25].Since the emission frequency of the InGaN/GaN QW is located near the resonance frequency of SPs at the metal/GaN interface, exploiting the plasmonic effect to enhance spontaneous emission is promising to further reduce the carrier lifetime for InGaN LEDs [20][21][22][23].On the other hand, it has been reported that nonpolar and semipolar InGaN/GaN QWs have highly-polarized emission due to the separation of light hole (LH) band, heavy hole (HH) band and crystalfield split-off hole (CH) band [10][11][12][13][14][15][16][17][18].Compared with conventional c-plane devices, optical properties such as F P and carrier lifetime can be further enhanced by the polarized emissions from nonpolar and semipolar InGaN QW LEDs, because the coupling efficiency from the spontaneous emission to the surface plasmon modes is dependent on the transverse magnetic (TM) component of the light [24,25].This SPs-QW coupling effects for polarized nonpolar and semipolar InGaN LEDs has never been explored and is the topic of this study.
In this work, we investigate the optical properties of highly polarized nonpolar and semipolar InGaN/GaN QW LEDs with metallic grating structure using finite difference time domain (FDTD) method.Key parameters such as Purcell factor (F P ), light extraction efficiency (LEE), internal quantum efficiency (IQE), external quantum efficiency (EQE), and modulation bandwidth will be studied systematically.The light source is simulated by classical dipoles and their orientations representing the polarized radiation from the QW.We find that all the optical properties are greatly impacted by the polarization emission coupled with SPs, and the device performance can be optimized by properly engineering the light polarization and metallic grating.The paper is organized as the following: in section 2, we describe the theoretical background and simulation methods; in section 3.1, F P is calculated with various polarization and position of dipoles; in section 3.2, we study the LEE modified by the orientations and positions of the dipoles; in section 3.3 and 3.4, the device performance such as IQE, EQE, and modulation bandwidth are evaluated for the SPs enhanced InGaN LEDs.

Theoretical background
In 1946, Purcell proposed that the spontaneous emission can be enhanced by modifying its dielectric environment [26], where F P is defined as the ratio between spontaneous emission rate modified by cavity and spontaneous emission rates in bulk.In FDTD method, F P is calculated by the power ratio between the emitted power from a classical dipole surrounded by the cavity, and the power from the same dipole located in bulk material, which is proved to be equal the emission rate ratio [27].In our simulation with grating structure, the Purcell factor F P is given by: where c λ is the wavelength in vacuum, n is the refractive index of bulk material, Q is the quality factor, V is the mode volume, R g is the spontaneous emission rate with grating structure, and R 0 is the spontaneous emission rate in bulk material without grating.P g and P 0 are the corresponding powers from the emitter inside the grating coated cavity and the emitter inside bulk material.FDTD method (from Lumerical FDTD Solutions) are employed in this study to calculate the F P as well as the power emitted from the top surface of the emitter, which will be described in section 3.1 and 3.2.
It is difficult to simulate the LEE of a realistic device because LEE is influenced not only by the metallic grating but also the substrate and the cavity geometry.Since the purpose of our research is to study the coupling between metallic grating and polarized emission, we define the LEE as the ratio between the emitted power measured above the metallic grating to the total power emitted by the dipole source inside the device, which will be normalized in the plots.
Combining the F P and LEE, we further estimate the IQE and EQE in order to study the impact of metallic grating on device performance.Here we define the initial IQE and EQE of the QW emitter without metallic grating as IQE 0 and EQE 0 , the IQE and EQE with the metallic grating structure as IQE g and EQE g , respectively.For the device with grating structure, the IQE g is given by: The injection efficiency and the distribution of the carriers are also important to calculate the total EQE g .The injection efficiency in this study is assumed to be unity.The carrier distribution is estimated using full lateral diffusion approach and no lateral carrier diffusion approach.In full lateral diffusion approach, we assume that the carriers are distributed uniformly across the lateral dimension, while for no lateral diffusion approach, the carriers are only uniformly distributed beneath the metal.
The photon emission inside QW is represented by dipoles with different orientations.For those nonpolar and semipolar InGaN QW LEDs, the emissions are not totally polarized along one direction, which is widely discussed in various literatures [10][11][12][13][14][15] by involving the polarization ratio ( ) / ( ) with different definition of I ⊥ and I  .For InGaN with moderate indium composition, the light polarized vertical to the growth plane (TM polarized) is relatively weak compared with the in-plane polarized light (TE polarized).Thus light polarized vertical to the growth plane is assumed to be zero when we estimate the performance of InGaN LEDs grow on different planes.For simplicity, we only considered the polarization along the optimal direction and the polarization vertical to it.The polarization ratios as well as the main polarization directions of different polar, nonpolar and semipolar InGaN LEDs in our study are listed in Table 1.(1022) semipolar 0.36 (1100) [12] (202 1) semipolar 0.6 [12 10] [13] (303 1) semipolar 0.56 [12 10] [14] m-plane nonpolar 0.9 [12 10] [15] #259477

Set-up of the 3D simulation
The 3D simulation set-up is shown in Fig. 1(a).Perfectly matched layer (PML) boundaries are used to terminate the propagation of the electromagnetic field.The size of simulation region is 2.59 µm × 1 µm × 2 µm for x, y and z directions, respectively.We overlap the PML boundaries on y-z plane with the metal intentionally in order to define the dielectric function on the PML layer accurately.The x-z plane at the top is 200 nm above the grating structure.
Each PML boundary has 8 layers.In order to measure the radiated power, the dipole is surrounded by a 12 nm × 12 nm × 12 nm small power monitor box.A power monitor is located 50 nm above the grating to measure the LEE at the surface.The convergence of all the set-ups mentioned above are rigorously tested in order to obtain accurate results with proper computing speed.
The coordinate system in our simulation is defined in Fig. 1(b).θ and φ are defined as the angles from TE 1 and TE 2 to TM oriented dipoles respectively, and ϕ is the in-plane rotation angle from TE 1 to TE 2 .For typical QWs on nonpolar/semipolar planes with high polarization ratios (e.g., QWs on m-plane), θ and φ are equal to 0° approximately and the ϕ is dependent on how we design the grating structure.For QWs on c-plane or other planes with low polarization ratios, the device performance is approximated by considering the emission from two or more dipoles with different ϕ and different weight factors.For QWs with high indium composition or extra strains, the θ and φ are no longer 0° and could be engineered [25].
The simulated structure is an InGaN/GaN QW coupled with a silver grating with period of 400 nm and duty circle of 50%.The parameters are not optimized but can give moderate performance based on our preliminary study.The p-type layer and the width of well are 12 nm and 3 nm, respectively.The indium composition is chosen to emit light at wavelength of 470 nm.The dielectric function of InGaN and GaN can be found in Ref [28].and [29], respectively.For the dielectric function of silver, we choose the built-in Palik model in the Lumerical FDTD solutions.The spontaneous emission is represented by radiation from a classical dipole source located at the center of the QW, which has a central wavelength of 470 nm with finite linewidth.The highly polarized radiation from QW is simulated by dipoles with different orientations.

Purcell factor
In this section, we study the light emission from dipoles with different orientations at different locations.The orientation of TE 1 , TE 2 , and TM dipoles are defined below, where location A and C refer to the dipole located at the center of the gap and the center of sliver stripe, while B refers to the dipole located beneath the edge of the silver stripe.Different rotations are defined in Fig. 1(b).For dipoles located at position A, a large increasing of F P is observed if the θ and φ are increased.Such an increase of F P is also observed in structure with silver thin film (result not shown).With increased θ and φ, the TM radiation component towards the metal/semiconductor interface becomes larger.Since the SPs can be coupled with TM radiation component more efficiently, higher F p could be achieved when θ and φ are increased.Previous theoretical studies only considered the light polarization directions of InGaN LEDs to be random and averaged all the directions when calculating the optical properties [22,23].This could lead to an overestimation of the F P since only one dominant polarization exist in most highly polarized InGaN LEDs.Furthermore, when the dipole is located at A and B, the distance between dipoles and the metal stripe increases, thus the coupling efficiency between the spontaneous emission and the SP modes decrease.For position A, since dipoles are more than 100 nm away from the metal, the plasmonic effect is relatively weak.And the reflected spherical wave from the top surface 12 nm above will have strong "interference" with the dipole source in the simulation, leading to different radiated power through constructive or destructive "interference".Such numerical phenomenon does not happen in reality but is corresponding to the manipulated photon lifetime, or equivalently, the manipulated quality factor (Q) for the radiative mode beneath the metallic grating.For TE 1 and TE 2 oriented dipoles at position A, the F p is higher than that of the TM oriented dipoles since it avoids the destructive "interference" which result in lower quality factor.Moreover, F P smaller than 1 is observed for TM oriented dipoles at position A (F P = 0.49), which is owing to the destructive "interference" between reflected wave from the top layer and dipole source in the FDTD simulation.For TE 1 dipoles at position B, maximum F P is achieved around θ = 60° instead of 90°, which is resulted from the larger TM radiation component towards the metal/semiconductor interface when θ = 60°.In Fig. 2(c), we map the F P with different locations on x direction.As the dipole is moving towards the center of silver strip, the F P increases due to the increased mode coupling efficiency.
We also studied the effect of in-plane rotation on the F P .Figure 2(d) shows that F P is almost constant when the dipole is rotating in-plane at A and C due to the invariant TM component.For dipoles located at B, the F P is rotation-dependent since the silver stripe is no longer right above the dipole geometrically.
In a short summary, F P is determined by two different mechanisms, for those dipoles located at C, the plasmonic effect is strong.F P is determined by the mode coupling efficiency and is increasing with θ and φ.For dipoles at A, the plasmonic effect is relatively weak, and the TE oriented dipoles give higher quality factor to the radiative mode since it avoid destructive "interference".Thus F P decreases with θ and φ.The simulation on F P also shows that the F P will decrease when the dipole is moved away from the metal stripe due to reduced coupling efficiency to plasmonic modes.In reality, even though the TM emission component of InGaN LEDs is relatively low when indium composition is moderate, it's still possible to add extra strains in the QWs to get more TM components [24] to achieve higher modulation speed.

LEE
In this section, we simulate the LEE of devices with different polarized emissions.The simulation configurations and the definitions of the locations and rotations are the same as we used in the previous discussion.In the simulation, LEE is defined as the ratio of radiated power through metallic grating to the total power emitted from the dipole.In order to clarify how grating structure manipulates the polarized emissions, we normalized the LEE to the highest value which is achieved at position A by TE dipoles.
Figure 3(a) shows LEE as a function of dipole orientation at position A, B and C. At position C, LEE decreases with increasing dipole orientation angle φ or θ.With larger φ or θ, TM polarization component is dominant which leads to a stronger SPs-QW coupling.However, radiation coupled with plasmonic modes do not contribute to the power extraction due to the localized electromagnetic field, which results in the reduction of LEE.For dipoles locate at position A, since the dipole is far away from the metal stripe, the plasmonic effect is relatively weaker compared to the case at position C. Therefore the extraction mechanism is dictated by the reflection of light at the surface.In addition, we also compare the LEE results to the cases without grating structures.The result shows that at position A, the LEE without grating structure is 1.01 for the TE dipole and is 0.946 for the TM dipole under the same normalization condition.This indicate that the grating structure performs as a scattering object which balances the metal loss by enhancing the LEE. Figure 3(b) shows that there's a large difference between the LEE of TE 1 and TE 2 oriented dipole (0.115 for TE 1 and 0.058 for TE 2 ).This is due to the fact that for TE 1 the radiation from the dipole is propagating along the direction with grating periodicity, while for TE 2 the direction is along the metal strip thus has lower chance to be scattered out. Figure 3(c) further verified such deduction by showing the mapped LEE with the distance from A to C. At the edge of the grating (position B) the LEE of TM oriented dipole is higher than TE 2 oriented dipole, meaning the SPs could be scattered into free propagating photons efficiently.By efficiently coupling the spontaneous emission to the SPs then efficiently scattering the SPs out as suggested in Ref [25], the LEE could be enhanced by plasmonic effect.In a short summary, dipoles perpendicular to the grating stripe (TE 1 ) has higher LEE than dipoles parallel to the grating stripe (TE 2 ).These results can help us to improve the LEE of InGaN/GaN QWs based devices where the optical transitions are TE 1 and/or TE 2 oriented.Since the F P of TE 1 and TE 2 oriented dipoles are almost the same, by carefully designing the orientation of the polarized radiation from QW and the metallic grating, we will be able to achieve a high LEE while keeping a moderate F P at the same time.

IQE and EQE
In this section, we study in detail how the IQE g and EQE g of devices can be modified by the interaction between metallic grating and polarized emission.The definitions of locations and dipole rotations are the same as we used in the previous section.In reality, the lateral carrier diffusion and the non-uniform recombination rate due to the Purcell effect will influence the carrier distribution.For simplicity, the full lateral diffusion approach and no lateral diffusion approach is employed for this work as we discussed in the previous section.Figure 4(a) shows the IQE g at position A, B and C as a function of IQE 0 .IQE g at position C is higher than that at position B due to stronger SPs-QW coupling effects.At the same position, the IQE enhancement for TM polarization is larger than other polarizations, which can be attributed to the larger F p of TM polarization as shown in Fig. 2(c).Although IQE g is larger than IQE 0 , the enhancement of IQE (IQE g /IQE 0 ) decreases with increasing IQE 0 , indicating that Purcell effect will enhance LEDs with relatively low IQE 0 more effectively.Figure 4(b) gives the estimated normalized EQE g at position A, B and C. All of the values are normalized to the EQE g of TE 2 oriented dipoles at position A with IQE 0 = 0.9.When IQE 0 is larger than 0.2 which is almost always the case for InGaN LEDs, TE polarized dipoles has the highest EQE g at all of the locations due to its highest LEE and moderate F P .To give a more accurate estimation, we integrated EQE g over all space and the results are shown in Fig. 4(c).For TE 1 polarized emission, the integrated EQE g is 2.5 times higher than that of the TE 2 polarization without lateral carrier diffusion, even though the F P are almost the same for these two polarization.Such observation is potentially important for the design of high speed LED since a high F P and moderate LEE is desirable in such device.When lateral carrier diffusion is considered, the EQE g of TE 1 oriented dipole is till 1.14 times larger than that of TE 2 dipoles, which is attributed to the large difference in LEE at position B. The performance of the real device is highly dependent on the carrier distributions within the active layer, which is influenced by the lateral diffusion and the non-uniform recombination rate across the active layer.Moreover, since the light radiated from the nonpolar/semipolar InGaN/GaN QW is strongly polarized, the orientation between the dipoles and the metallic gratings might have significant impacts on the EQE g performance of these device.

Device performance
The modulation speed of LEDs and LDs on nanoscale are theoretically analyzed in Ref [30,31].The calculation in [31] shows that the modulation speed is almost Q independent due to the balancing between the large F P and the narrow Lorenzian of cavity mode.For QW LEDs, the highest achievable modulation speed is limited to tens of GHz due to the strong cavity effect.However, the Q factor of our simulated structure is relatively small due to the weak cavity confinement, which eliminates the restraints on the modulation frequency.In our device, the Purcell effect is a result of shrinking the mode volume by the plasmonic effect, which might potentially provide another path to increasing modulation speed.
For the carriers inside the QW, the lifetime modified by Purcell effect can be estimated by [32]: eff r nr 1 1 where r τ is radiative recombination lifetime, nr τ is the non-radiative recombination lifetime, and eff τ is the effective recombination lifetime.Considering the cavity confinement, the 3dB frequency can be written as [31,32]: ω is the lifetime of photons inside the cavity.For device with strong cavity effect, the Q factor is extremely high and the photon lifetime will restrict the modulation speed of device.For device with weak cavity effect (our case), the Q factor is relatively small and the p τ is negligible.Using proposed approach, we estimated the EQE g and the modulation bandwidth with different dipole orientations for nonpolar m-plane InGaN LEDs and the results are presented in Table 2.The radiative recombination lifetime of nonpolar mplane LED is chosen to be r τ = 0.45 ns as measured in Ref [5].using the time resolved photoluminescence (TRPL) method, the non-radiative recombination lifetime is chosen to be nr τ = 0.5 ns as analyzed in Ref [32].For m-plane InGaN/GaN device, modulation speed from 3.5 GHz to 5.4 GHz is achievable if the grating and the growth plane is carefully oriented.As shown in Table 1 in the previous section, the polarization ratios are not unity in nonpolar and semipolar QWs.Therefore the emission cannot be totally polarized at the optimal direction.However, it is quit intuitive that with a large polarization ratio, more emissions could be efficiently polarized at the TE 1 direction where the highest total EQE is observed, leading to better overall device performance.Therefore the m-plane LED performs better compared with devices on other orientations since it has the largest polarization ratio.The estimated EQE g for polar, nonpolar and semipolar InGaN LEDs are summarized in Table 3, where the EQE g are normalized to the m-plane device values for both the cases of full lateral diffusion and no lateral diffusion.For the EQE g with full lateral carrier diffusion, QWs on different planes show similar performance.For the EQE g without lateral carrier diffusion, the nonpolar m-plane device showed much higher performance compared to polar c-plane device.Since the effective F p is 14.5 beneath the metal, the radiative recombination rate is much higher beneath the metal.Therefore in real devices, we can expect that more carriers will be recombined beneath the metal, leading to a higher EQE g at TE 1 direction compared to TE 2 .Thus the devices based on m-plane could provide more output power compared to other devices due to the high polarization ratio and the large radiative recombination rate beneath metal.

Conclusion
We simulate the Purcell factor F P , LEE, IQE, EQE and modulation frequency of InGaN/GaN QWs LEDs coupled with silver grating, which not only supports the plasmonic mode but also enables electrical injection.Results indicate that SPs-QW coupling strongly impacts highlypolarized emission from the InGaN QW as well as the LED device performance.First of all, TM polarization has largest F P and smaller LEE compared with TE polarization.This is because F P is related to plasmonic mode with non-radiative property, while LEE is mainly determined by the coupling efficiency to radiative mode.Second, the distance between SPs and dipole also impacts F P and LEE.By moving the dipole away from grating, the coupling becomes weak (smaller F P ) while LEE increases for TE polarized dipoles.This is due to the competition between plasmonic mode and radiative mode.At the edge of the grating, the LEE of TE 1 dipole is 2 times higher than that of the TE 2 dipole due to their different radiation directions.Third, the in-plane polarization (TE 1 and TE 2 ) shows minimum change in F P while LEE highly depending on the rotation of dipole.TE 1 polarization shows higher LEE than TE 2 polarization, which means TE 1 polarization is more efficient emitter with higher EQE.Such observation might potentially guide the design of highly polarized InGaN QWs for high speed LEDs, which is dominated by TE polarization.In this calculation, the achievable modulation speed is up to 5.4 GHz.The highest EQE is achieved using nonpolar m-plane InGaN QW due to its largest polarization ratio.In addition, carrier distribution also plays an important role in the device performance.Further investigations are required to simulate more realistic device structures (e.g.multi-QW structures), and study the trade-off between the high modulation speed and the output power of the LEDs.

Fig. 1 .
Fig. 1.(a) The schematic view of GaN/InGaN QW with silver grating.The period of the grating is 400 nm with 50% duty circle.The dash box indicates the FDTD simulation region.All of the materials are homogenous along the z direction.(b) The schematic view for different polarizations and rotations studied in this work.θ is polarization angle between TE1 and TM; φ is the polarization angle between TE2 and TM; φ is the polarization between TE1 and TE2.

Fig. 2 .
Fig. 2. (a) The schematic views for the TE1, TE2 oriented dipoles.Position A is at the center of the gap between metal strips; position B is at the edge of the metal strip, and position C is at the center of metal strip.(b) FP of the dipoles located A, B and C with different orientation.0, 90 degrees indicate the TE1/TE2 oriented dipole and TM oriented dipole, respectively.(c) Distribution of FP along x axis.(d) FP of the dipoles rotate in-plane located at A, B and C.

Figure 2 (
Figure 2(b) shows the F P as a function of dipole orientation.For dipoles located at position A, a large increasing of F P is observed if the θ and φ are increased.Such an increase of F P is also observed in structure with silver thin film (result not shown).With increased θ and φ, the TM radiation component towards the metal/semiconductor interface becomes larger.Since the SPs can be coupled with TM radiation component more efficiently, higher F p could be achieved when θ and φ are increased.Previous theoretical studies only considered the light polarization directions of InGaN LEDs to be random and averaged all the directions when calculating the optical properties[22,23].This could lead to an overestimation of the F P since only one dominant polarization exist in most highly polarized InGaN LEDs.

Fig. 3 .
Fig. 3. (a) The normalized LEE of dipoles located at A, B, and C with different orientations.Dipoles are rotated by changing θ and φ . (b) The normalized LEE of the dipoles located at A, B, and C with different orientations.Dipoles are rotated by changing ϕ . (c) Mapped normalized LEE along x axis.

Fig. 4 .
Fig. 4. (a) IQEg with metallic grating as a function of IQE0 at A, B and C. (b) EQEg with grating structure as a function of IQE0 at A, B and C. (c) The integrated EQEg over all space versus IQE0 for cases of full lateral diffusion and non-lateral diffusion.