Based on the analysis of the evanescent wave from total internal reflection, a light-emitting diode (LED) structure with a plasmonic Ag nanostructure and indium tin oxide (ITO) grating was proposed to enhance the extraction efficiency. The two-dimensional finite-difference time-domain method was used to study the spectral properties of the hybrid structure and the effects of structure parameters on extraction enhancement. The results demonstrate that the plasmonic Ag nanostructure can couple the evanescent wave to a propagation wave around the GaN/ITO interface, and then the photons are scattered out of the LED chips by the ITO grating with high extraction efficiency. Under the optimal parameters, the light extraction efficiency can reach approximately three times the original value at a relatively longer wavelength.
©2012 Optical Society of America
GaN-based light-emitting diodes (LEDs) have attracted much scientific and commercial interest since the realization of a practical LED device with emission frequencies in the visible and ultraviolet regions [1–4]. Studies have focused on the production of economical LEDs with the desired colors and white light sources [5, 6]. However, GaN-based LEDs with inherently low efficiency caused by low internal quantum efficiency (IQE) and difficulty of extracting the generated photons out of the device. The IQE of GaN-based LEDs has been drastically improved with the progress of GaN-based epitaxial growth and device fabrication technologies [7, 8]. Meanwhile, the light extraction efficiency (LEE) remains unsatisfactory because of the total internal reflection (TIR) at the emitter/air interface. Many attempts have been accordingly made to maximize the LEE (external quantum efficiency) of LEDs. These methods include substrate modification  and incorporation of scattering medium [10, 11], micro-lenses , nanogratings , corrugated microstructures , photonic crystals , and so on. However, LEE has still a lot of room for improvement.
In this research, we used a metal nanostructure and medium grating to enhance LEE. Ag nanostructure placed ten nanometers from the emitter/air interface can induce a localized surface plasmon resonance (LSPR) because Ag can resonate with surface plasmons (SPs) on the nitride semiconductors within the visible-light range [16–20]. A localized surface plasmon (LSP) represents the behavior of local collective oscillations of conduction electrons at the interface of metallic nanoparticles and dielectric materials [20–23]. At a resonance condition, this effect allows a metallic nanostructure to capture trapped light in the GaN layer of an LED device and thus enhance the extraction efficiency of light . Compared with the SPR of a metal layer, LSP offers a unique advantage with its tenability. That is, the optical properties resulting from LSP can be easily varied by altering the type, size geometry, and interparticle distance of metallic nanoparticles. Hence, the effects of the structural parameters of Ag nanostructure on extraction enhancement were studied. In our LED model, a transparent and conductive indium tin oxide (ITO) layer is perforated into grating to decrease the TIR in the air/ITO interface . The combined effects of Ag nanostructure and ITO grating for increasing LEE were discussed.
2. Theoretical foundation and calculation model
2.1 Evanescent waves at the GaN/ITO interface
The index contrast between air and a typical semiconductor used in LED chips positions the TIR of most spontaneously emitted waves at the interface of air and the LED medium. Most of the emitted light remains trapped in the LED and is absorbed by metal contacts, basal layer, and active layer, which produce heat and a compound of electrons and cavities without radiation.
The occurrence of TIR at the interface between two materials forms an evanescent wave. The attenuated length of the evanescent wave can be calculated as
where k0 is the wave vector in vacuum; Im(…) represents the imaginary part of the complex; n1 and n2 are the refractive indices of GaN and ITO, respectively; and is the incidence angle of electromagnetic waves. For the GaN/ITO interface, the effect of the incident angle and wavelength on the attenuated length is shown in Fig. 1 .
As shown in Fig. 1, the critical angle at the interface between GaN and ITO is approximately 47°. Within the region of the evanescent mode, the attenuated length la increases with the wavelength but decreases with the incident angle. For the visible region, la is mainly below 230 nm.
2.2 Calculation model
A schematic diagram of the combined Ag nanostructure and ITO grating of the proposed GaN-based LEDs is shown in Fig. 2 . The model consists of a sapphire layer, a GaN layer, an Ag nanostructure, and an ITO grating. The GaN layer, with a refractive index of 2.5, behaves as the epitaxial layer. The sapphire layer possessing a refractive index of 1.76 represents the LED substrate. The ITO electrode layer, with a refractive index of 1.8, can avoid the optical losses of an electrode and provide more uniform current spreading, serving as a tunnel contact to p-GaN in a real LED die [11, 26]. In Fig. 2, W, L, and d represent the period, width, and depth, respectively, of the Ag nanostructure or the ITO grating, and the subscripts A and I represent Ag and ITO, respectively. h denotes the distance between the Ag and the ITO layer. The filling ratio f is defined as f = (W - L)/W. The red line represents the surface that receives the radiation energy escaping from the upper surface of the chip. The parameter t is the distance between the surface of the energy detector and the upper surface of the chips. In the model, tITO, tGaN, and tsapphire represent the thickness of the ITO, GaN, and sapphire layers, respectively. The entire structure is surrounded by the perfectly matched layer (PML), which is used to absorb radiation in the finite-difference time-domain (FDTD) simulation. The side lengths of the two-dimensional simulation region are expressed as Lx and Ly along x- and y-axes, respectively.
For multiple quantum wells (MQWs) the dipole oriented perpendicular to the well is suppressed and thus only horizontal dipoles exist due to the heterostructure. For GaN grown along the c-axis the dominating electron-hole recombination process can be described by horizontal dipole . In our LED model, the horizontal diplole as radiation source simulate electron-hole recombination process in MQWs, and its polarized direction is parallel to the sapphire layer.
2.3 FDTD method
The FDTD method is used to theoretically analyze the electromagnetic problem of SPR [28, 29]. The solution of Maxwell’s equations is accomplished by approximating the partial differential equations using the finite differences, both in time and space. The Maxwell equations in free space for one-dimensional simulation are written as:
The electric and magnetic fields are shown to be separated in both space and time by half a step. In the FDTD algorithm, only two of the Maxwell equations are used to update the electromagnetic field. If the remaining two equations are satisfied at the initial time moment, the separation of the two fields is shown to be conserved in the course of the computations. A metal with the real part of the dielectric constant less than unity requires the solution of the auxiliary differential Eq. (4), which is the time-domain counterpart of the metal dispersion relation. The Drude model in time-domain analysis has the form30] from the study of Johnson and Christy.In the FDTD simulation, the mesh size is assigned as ∆x = 6 nm and ∆y = 2 nm, whereas the time increment is ∆t = 4.0 × 10−18 s. After 150,000 time steps, the Poynting vector distribution of radiation is received on the energy detector surface.
3. Results and discussions
A half Ag nanostructure chip model was simulated to demonstrate the effect of plasmonic Ag on the enhancement of LEE. The values of the parameters of the model are listed in Table 1 . The radiation source in MQWs is a transverse-magnetic (TM)-polarized wave with a wavelength of 635 nm. The Poynting vector distribution of the radiation on the energy detector surface is shown in Fig. 3 . The inset at the bottom of Fig. 3 represents the half Ag nanostructure chip model.
For the structure with half Ag nanostructure separated from the interface of GaN and ITO by 10 nm spaces, the Poynting vectors of the side with Ag nanostructure are increased significantly, whereas those of the side without Ag remain almost unchanged. In the absence of Ag nanostructures near the interface of GaN and ITO, the Poynting vectors on the detector surface are mainly in the direction of the normal incidence of radiation source due to TIR (Fig. 3). The discussion above suggests that the plasmonic Ag nanostructure can improve the extraction efficiency of the LEDs.
The enhancement factor Fλ describes the enhancement effect on light emission and is defined as the total energy flow Pλ across the detection surface with Ag structure or ITO grating divided by the corresponding total energy flow without Ag structure or ITO:
Four LED models designated as Case 1 to Case 4 are designed to analyze the effect of Ag nanostructure and ITO grating on the extraction efficiency. Case 1 corresponds to the LED model without Ag nanostructure and ITO grating. Case 2 is for the LED model with combined Ag nanostructure and ITO grating. Case 3 and Case 4 represent the models that exclusively have either Ag nanostructure or ITO grating, respectively. The unique parameters of each model are listed in Table 2 , and the other parameters are listed in Table 1.
For the TM- and TE-polarized radiation sources in the MQWs, the enhancement factors of the four LED models are graphed in Figs. 4(a) and 4(b), respectively. The enhancement of LEE is located in different wavelength bands, as shown in Fig. 4(a). In the short wavelength band, the improvement of the LEE is mainly attributed to the scattering of the guided modes in the ITO layer caused by the combined Ag nanostructure and ITO grating. Within the 300 nm to 370 nm wavelength band, the LEE improvement of Case 2 is less than that of Case 3 because of the absorption of the Ag nanostructure. The enhancement of LEE in the long wavelength band can be attributed to the extraction of guided modes in the GaN layer. The enhancement factor reaches more than three times the original value at a wavelength of 640 nm. Figure 4(b) shows the enhancement factor of the four chip models when the radiation source in MQWs is TE polarized. The absorption of the electromagnetic wave by metals is relatively small because the TE wave cannot stimulate the SPs in non-magnetic metals. Thus, the TE-polarized wave has higher enhancement factors for the four LED models (Fig. 4(b)) compared with the TM-polarized wave (Fig. 4(a)) in the short wavelength band. Based on Fig. 4(b), the LEE enhancement of Case 2 with combined Ag nanostructure and ITO grating is almost the same as that of Case 3 with only perforated ITO grating. In this case, the only function of the Ag nanostructure is to scatter photons as the real metal grating enhances LEE. The distribution of energy flux at 635 nm wavelength in the steady state is shown in Fig. 5 , wherein Figs. 5(a) and 5(b) correspond to Case 1 and Case 2, respectively. As shown in these figures, the GaN layer is located between 0.5 μm and 1.25 μm of the y-axis. Above the GaN layer, the ITO layer is positioned from 1.25 μm to 2.25 μm along the y-axis. When the combined Ag nanostructure and ITO grating are not applied in the chip model (Case 1), most of the electromagnetic waves are located in the GaN and ITO layers, and few waves are emitted outside the chip (Fig. 5(a)). For the guided modes propagating in the GaN, the energy of electromagnetic wave is reduced gradually because it is absorbed by the MQWs. In Case 2, the Ag nanostructure can transfer the guided models from the GaN layer to the ITO layer, and then the photons are scattered out of the chips by the ITO grating (Fig. 5(b)).
In our LED model, the LEE enhancement is mainly attributed to the scattering function of the ITO grating and the LSPR of the Ag nanostructure. As shown in Fig. 5(b), the photons are scattered by the ITO grating near the ITO/air interface. The Ag nanostructure induces an LSPR.
The inset of Fig. 5(b) shows that the magnitude of the electromagnetic wave increases near the Ag nanostructure. In the LED model, the enhancement of LED light emission by LSP could be considered as a two-stage process. The first is the propagation of the guided models in the GaN layer accompanied by the energy transfer to the surface plasmon polaritons (SPPs). The second is the extraction of light from the excited SPPs by scattering them onto the surface of the Ag nanostructure. Using the perforated ITO grating scatters the high-density evanescent wave onto the propagation wave to enhance the efficiency of light extraction.
When the distance h between the Ag nanostructure and the GaN layer has changed, the interaction between metal Ag and the evanescent wave must be affected. As h increases, the enhancement factor greatly changes in the long wavelength band, whereas the factor experiences a smaller variation in the short wavelength band (Fig. 6 ). This phenomenon can be attributed to the increase in the attenuated length of the evanescent wave with increasing wavelength (Fig. 1). As shown in Fig. 6(a), the enhancement factor decreases with increasing h, especially in the long wavelength band. This distance dependence may be ascribed to the coupling of the plasmatic Ag with the evanescent wave that exponentially decays with increasing distance from the GaN/ITO surface. Therefore, the Ag nanostructure should be located within the attenuated length of the wave, so that the evanescent wave can be coupled to the propagation wave and the LEE is increased.
The graph of the enhancement factor versus the depth of the Ag nanostructure at different wavelengths is shown in Fig. 7 . The values of the local peaks decrease with depth, especially in the short wavelength band. When the depth of Ag nanostructure is thicker than the skin depth, a large amount of energy is captured in the metal Ag. This captured energy cannot be transferred to emitted light; thus, it does not contribute to the front-emission light. The positions of all local peaks are nearly unchanged for different h and d values because of the unchanged period and the filling factor of the Ag nanostructure (Figs. 6 and 7). However, the magnitudes of the local peaks vary among the different h or d values. This phenomenon is suggestive of tuning the peak value to obtain enhanced extraction efficiency at the desired wavelength.
For a specific wavelength, if the stimulation of the SPs would be coupled to the propagation wave, the period of the metal nanostructure should be within the proper range. The dependence of the enhancement factors on the period of the Ag nanostructure is demonstrated in Fig. 8 . The local peak positions of the enhancement factor vary with the different periods. In the long wavelength band, the value of the enhancement factors obviously changes. However, the peak values do not show such sensitivity to the period in the short wavelength band. This result can be attributed to the mismatch of the periods and the wave vectors of the SPs that hamper the coupling of the SPs with the evanescent wave. In addition, very small periods (<220 nm) of the Ag nanostructure are found to lose their light extraction advantages in the long wavelength band (Fig. 8).
The dependence of the enhancement factors on the filling ratio of the Ag nanostructure is shown in Fig. 9 . The value and positions of all local peaks are nearly unchanged in the short wavelength band. However, the positions of local peaks vary with the filling ratio in the long wavelength band because the scattering function of the Ag nanostructure depends on the filling ratio. Based on Figs. 8 and 9, the peak position can be altered via different values of w and f to obtain enhanced extraction efficiency at the desired position.
Based on the analysis of evanescent waves in TIR, a LED model has been proposed. In this model, a plasmonic Ag nanostructure is placed around the GaN/ITO interface, and the transparent conductive ITO layer is perforated into grating at the air/ITO interface. The LEE enhancement factors of the model have been studied using the two-dimensional FDTD method. The simulation results indicate that the light-emitting efficiency of the LED model with Ag nanostructure and ITO grating can be well enhanced, especially within the relatively long wavelength regions. Such enhancement is attributed to the coupling of the LSPs with the evanescent waves around the GaN/ITO interface and the photon scattering of the ITO grating into the air/ITO interface. The enhancement factor of the structure can reach up to approximately three times the original value at a wavelength of 640 nm. The local peak values of the enhancement factors vary with the depth and distance between the Ag nanostructure and the GaN/ITO interface. The local peak position of the enhancement factors is clearly affected by the period and filling ratio of the Ag nanostructure, especially in the long wavelength band. When combined, these parameters can be used to fine-tune the peak value and position to obtain the desired enhanced extraction efficiency.
This work was financially supported by the National Natural Science Foundation of China (Grant No 50874079, 51002102 and 21101111), the Natural Science Foundation Project of CQ CSTC (Grant No cstc2012jjA90017), the Natural Science Foundation of Shanxi Province (No. 2011011022-1) and the Graduate innovation projects of Shanxi Province (Grant No 20113039 and 20113045).
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