The radiated power enhancement and more congregated radiation of a radiating dipole within a GaN material when it is coupled with the localized surface plasmon (LSP) resonance modes induced on a surface Ag nanoparticle (NP) are numerically demonstrated. The numerical study is based on an algorithm including the induction of LSP resonance on the Ag NP by the source dipole and the feedback effect of the LSP resonance field on the source dipole behavior. The spectral peaks of radiated power enhancement correspond to the substrate LSP resonance modes with mode fields mainly distributed around the bottom of the Ag NP such that the coupling system radiates mainly into the GaN half-space. By moving the radiating dipole laterally away from the bottom of the Ag NP, the spectral peaks of radiated power enhancement red shift and their levels diminish with increasing lateral distance. The radiation patterns in the GaN half-space show more congregated radiation around the vertical direction, indicating that the light extraction efficiency can be enhanced in an LSP-coupled light-emitting device with surface metal NPs.
© 2014 Optical Society of America
Surface plasmon (SP) coupling with a quantum well (QW) in an InGaN/GaN QW light-emitting diode (LED) for enhancing its emission efficiency has been widely studied and demonstrated [1–4]. It is usually believed that such a coupling process not only can enhance the internal quantum efficiency (IQE) of emission, but also can increase the light extraction efficiency (LEE) of the device. From the experimental viewpoint, the overall device emission enhancement can be monitored through electroluminescence measurement. Also, the IQE enhancement can be estimated through temperature-dependent photoluminescence measurement. However, so far it is still difficult to experimentally prove that SP coupling can improve the LEE of an LED. In an SP coupling process, the SP mode coherently couples with a light emitter through near-field interaction to form an emission entity. Therefore, whether such an SP coupling process can improve the LEE depends on the radiation pattern of the coupled emission entity. Generally speaking, if the SP coupling process can increase the emission directly into air or reduce the radiation distribution angle with respect to the out-of-plane direction inside the device, the LEE can be enhanced. Although theoretical and numerical investigations on the emission enhancement of an SP coupling process have been reported [3–8], few studies about the LEE issue in such an SP coupling process could be found in literature.
For implementing strong SP coupling, various metal nanostructures have been fabricated near the QWs, including surface metal nanoparticle (NP) [9–11], thin-film [12, 13], and nano-grating [14, 15] structures on the p-GaN surface, Ag protrusion array into the p-GaN layer [16, 17], and Ag and Au NPs embedded into the p-GaN or n-GaN layers [18–20]. Among those metal nanostructures for inducing SP coupling, the simplest and possibly the most inexpensive method is the fabrication of surface metal NPs on the device surface. Although the implementations of such SP-coupled devices have been reported [9–11], their coupling behaviors have not been well studied yet. In particular, detailed understanding through theoretical/numerical investigations is still lacking. Meanwhile, as mentioned above, the LEE of such an SP-coupled LED structure is still unclear. The LEE is an important issue for understanding the overall emission efficiency improvement of the device with surface metal NPs.
In this paper, we numerically study the localized surface plasmon (LSP) coupling effects of a surface Ag NP with a radiating dipole located in the QW layer of an InGaN/GaN QW LED. We evaluate the distributions of radiated power into the upper-half-space of either air or a coverage layer and the lower-half-space of GaN material. Also, we plot the radiation patterns of the LSP-dipole coupling system and the solid-angle-dependent radiated powers for understanding the variations of LEE. In numerical simulation, we first use the COMSOL software to compute the scattered field of the radiating dipole due to the surface NP. Then, the scattered field is used for solving the optical Bloch equations and evaluating the feedback effect of the induced LSP resonance on the source dipole . The dependencies of radiated power enhancement on the GaN capping layer thickness and the lateral position of the radiating dipole will be evaluated for the study of radiation pattern. In section 2 of this paper, the problem geometry is defined and the used numerical methods are briefly described. Then, the dependence of radiated power enhancement on the GaN capping layer thickness is reported in section 3. Next, the dependence of radiated power enhancement on the lateral position of the radiating dipole is presented in section 4. The radiation pattern and LEE of an SP-coupled LED with surface metal NPs are discussed in section 5. Discussions of certain related issues are given in section 6. Finally, conclusions are drawn in section 7.
2. Problem geometry and numerical methods
Figure 1 shows the problem geometry, in which an Ag NP is placed on the surface of a thick GaN layer with an embedded thin QW layer at the depth of d. A radiating dipole, which is represented by a thick (red) arrow, is located in the QW layer at the coordinate of (x, 0, -d). Because the radiating dipoles in an InGaN/GaN QW mainly lie in the QW plane based on theoretical studies [21–24], only the x- and y-oriented dipoles (will be briefly called x- and y-dipole) are considered in this study. The Ag NP is a truncated ellipsoid with the semi-lengths of the minor- and major-axis, which are along the x- (or y-) and z-axis, respectively, being a and b, respectively. A length of s along the major-axis of the ellipsoid is truncated to form the Ag NP situated on the GaN surface. Above this surface, the Ag NP is surrounded by air or a material, which is a dielectric medium with a dielectric constant of ε2 in the visible spectral range (such as a transparent conductor in practical application). The dielectric constants of GaN (including the QW material) and Ag are designated to be ε1 and εm, respectively. It is noted that the geometry of the Ag NP is designed based on the scanning electron microscopy observations of experimental implementations of surface Ag NPs [25, 26]. However, as to be discussed in section 6, the geometry of the Ag NP does not affect the major LSP coupling behaviors to be reported in this paper.
The three-dimensional numerical simulations are carried out with the assistance of the commercial software COMSOL, which is based on the numerical algorithm of the finite-element method. For simulation, the computation domain is chosen to be a sphere region. It contains the radiating dipole embedded in the lower-hemisphere of GaN and the truncated Ag ellipsoid embedded in the upper-hemisphere of air or a dielectric. To simulate the infinite extension of the background environment, a spherical layer, known as the perfectly matched layer, is placed right outside the computation domain.
We first compute the radiated electromagnetic field (the incident field) of a radiating dipole situated in a homogeneous (GaN) spherical background region. Then, the total field is calculated for the real problem geometry including the radiating dipole and the Ag NP embedded in the aforementioned two half-spaces. By subtracting the incident field from the total field, we can obtain the scattered field, which is to be used for computing the feedback effect on the dipole radiation behavior. With the available scattered field, the optical Bloch equations are solved to find the resultant strength and orientation of the modified dipole . Based on this modified dipole, the final total electromagnetic field as well as the radiated power can thus be calculated numerically. A radiation pattern is obtained through a surface integration over an indented surface near the interface of the two half-spaces but circumventing the radiating dipole and Ag NP. For this purpose, the computation domain of COMSOL must be enlarged to ensure that the electromagnetic field is negligibly small around the edge of the aforementioned indented surface. For numerical computations, the geometrical parameters of the Ag NP are chosen to be a = 50 nm, b = 60 nm, and s = 30 nm. Also, ε1 = 5.76, and experimental data are used for the dielectric constant εm in Ag .
3. Numerical results – dependence on capping layer thickness
Figure 2 shows the normalized downward radiated powers of the coupling system as functions of wavelength for d = 30, 50, 70, 90, and 120 nm when the dipole is located exactly below the Ag NP, i.e., at x = 0. The downward radiated power with the surface Ag NP is normalized by the total radiated power of the dipole when the Ag NP is removed from the air/GaN interface. The Ag NP is surrounded by air above GaN. The downward radiated power means the total power entering the lower-half-space of GaN. In Fig. 2, for comparison, the normalized downward radiated power in the case of no Ag NP when d = 70 nm is also plotted as the dashed curve. Here, one can see that in each curve there are a long-wavelength broad hump and one or more short-wavelength narrower peaks (with relatively lower levels in the curves for d = 70, 90, and 120 nm). The long-wavelength broad humps are caused by the couplings with the major LSP resonance modes of the Ag NP. The short-wavelength features originate from the couplings with the higher-order LSP resonance modes. The peak level of the long-wavelength feature increases first and then decreases with increasing d value. Also, the spectral position of this peak blue shifts first and then red shifts with increasing d value. In Fig. 2, the maximum normalized radiated power, i.e., the maximum radiated power enhancement ratio, can reach 3.6 at ~625 nm when d = 70 nm. By comparing the results of different d values in Fig. 2, one can see that the radiated power enhancement due to LSP coupling does not necessarily increase with decreasing distance between the dipole and the metal nanostructure. Also, the enhancement ratio is strongly dependent on the emission wavelength of the radiating dipole. Even when d is as large as 120 nm, a peak enhancement ratio of 1.8 can still be obtained. However, the spectral location of this peak is shifted almost into the infrared range (~680 nm).
Figure 3 shows the spectral variations of the normalized upward radiated powers, i.e., the powers entering the upper-half-space of air, for the conditions the same as those in Fig. 2. Here, again, the dashed curve represents the condition of no Ag NP at the air/GaN interface when d is 70 nm. One can see that without the Ag NP at d = 70 nm, the upward radiated power is always smaller than 15% of the total radiated power in the spectral range of 350-800 nm. This result simply confirms the low LEE of an LED unless a light extraction scheme is applied. With the Ag NP, except within certain spectral ranges in the cases of d = 70, 90, and 120 nm, the upward radiated power is generally suppressed. Even though the upward radiated powers are enhanced in those spectral ranges, the overall normalized upward powers are significantly lower than those entering the lower-half-space of GaN. In other words, the LSP coupling mainly enhances the downward radiated power. Generally speaking, the percentage of radiated power entering the air is reduced after the LSP coupling process is applied.
The generally enhanced and suppressed radiated powers in the downward and upward directions, respectively, due to LSP coupling are attributed to the effective coupling of the source dipole only with the substrate LSP modes of the Ag NP. Three categories of LSP resonance modes can be induced on a surface metal NP, including the in-plane air mode, in-plane substrate mode, and out-of-plane mode . The mode field of an in-plane air mode is mainly distributed in the upper-half-space. That of an in-plane substrate mode is distributed around the interface between the metal NP and substrate. The electrical oscillation of an out-of-plane mode is along the vertical direction in Fig. 1. When the radiating dipole is located exactly below the Ag NP, i.e., x = 0, either x- or y-dipole can only induce in-plane LSP modes for coupling. In this situation, the major LSP modes coupling with the source dipole belong to the category of in-plane substrate mode such that the coupling induced radiation mainly propagates into the lower-half-space. To understand the LSP resonance modes coupling with the source dipole, we plot the surface charge distributions of LSP coupling at 620 and 470 nm for the case of d = 30 nm, and at 625 and 490 nm for the case of d = 70 nm in Figs. 4(a)–4(d), respectively. Here, the arrows below the surface charge distributions indicate the source dipole orientation. The long-wavelength maximum feature of downward radiated power in the case of d = 30 nm is located at ~620 nm, at which the surface charge distribution shown in Fig. 4(a) illustrates an image dipole at the bottom face of the Ag NP. Although this image dipole looks like it is oriented in the opposite direction from the source dipole, the phase difference between the image and source dipoles is not necessarily equal to 180 degrees. The radiations of the image and source dipoles interfere essentially constructively to form the long-wavelength peak around 620 nm. The similar argument based on the dipole charge distribution in Fig. 4(c) can be used to explain the maximum radiated power at ~625 nm in the case of d = 70 nm. Such an argument can also be applied to the long-wavelength (or major) maxima of radiated power in other cases of different d values. For the short-wavelength maximum feature of downward radiated power in the case of d = 30 nm (at ~470 nm), as shown in Fig. 4(b), the surface charge distribution indicates that the radiation interference between the source dipole and the induced quadrupole on the bottom face of the Ag NP leads to relatively strong radiated power around this wavelength. In the case of d = 70 nm, at the short-wavelength (or minor) maximum feature of radiated power (~490 nm), the radiation interference between the source dipole and the induced quadrupole on the bottom face of the Ag NP [see the charge distribution in Fig. 4(d)] leads to the minor peak of radiated power. Because of the relatively weaker LSP resonance at this wavelength, the radiated power level is not high. This argument can also be applied to the cases of d = 50, 90, and 120 nm.
4. Numerical results – dependence on the lateral position of source dipole
Radiating dipoles are distributed along the QW plane below the Ag NP. To understand the LSP coupling effects of the radiating dipoles at different lateral locations in the QW, in Figs. 5 and 6, we show the normalized upward [see part (a) of either figure] and downward [see part (b) of either figure] radiated powers as functions of wavelength when the x- and y-oriented source dipoles, respectively, are located at x = 0-6h with the z coordinate being fixed at d = 70 nm. Here, h = 31/2a/4. In the upper-half-space, the Ag NP is still surrounded by air. The curves labeled by x = 0 are the duplicates from those labeled by d = 70 nm in Figs. 2 and 3. From Fig. 5 or 6 for either x- or y-dipole, one can see that as the dipole is moved away from the bottom of the Ag NP, the upward (downward) radiated power level is generally enhanced (reduced) until the NP-dipole distance becomes quite large (x = 5h or 6h) where the upward radiated power is decreased with increasing x. The major radiated power still propagates downward into the lower-half-space of GaN. The increasing and decreasing trends of the upward and downward radiated powers, respectively, with increasing x can be attributed to the increasing (decreasing) coupling strength of the source dipole with the in-plane air mode and out-of-plane mode (the substrate mode) of LSP resonance. In this situation, the radiation into the upper-half-space is increased until the NP-dipole distance becomes large and LSP coupling strength diminishes. The decreasing trend of downward radiated power is partly caused by the increasing distance between the Ag NP and the source dipole. In the case of x-dipole, because of the change of the distance between the Ag NP and the dipole, the interference condition between the induced LSP dipole resonance and the source dipole is varied such that the major peak of the downward radiated power drops dramatically with increasing x. However, in the case of y-dipole, the interference condition does not significantly change such that the peak level decay of the downward radiated power with increasing x is not so dramatic.
5. Radiation patterns and light extraction efficiency
To study the LEE of an SP-coupled LED, in Figs. 7–10, we plot the radiation patterns of downward radiated power at 625 nm in wavelength in the cases of x = 0, 2h, and 4h when the Ag NP is surrounded by air in the upper-half-space and d is fixed at 70 nm. The wavelength of 625 nm corresponds to the spectral peak of normalized radiated power when d = 70 nm and x = 0, as shown in Fig. 2. Figures 7–10 show the results in the x-z and y-z planes for x-dipole and those in the x-z and y-z planes for y-dipole, respectively. For comparison, the radiation patterns under the condition of no Ag NP in the x-z and y-z planes are also plotted in Figs. 7–10. The radiation patterns of the bare source dipole (no Ag NP) are different from the usually observed dipole radiation pattern because of the existence of the air/GaN interface. This interface can reflect radiation to interfere with the directly radiated light for forming the observed patterns. As shown in Fig. 7, with an x-dipole, in the x-z plane, the radiation pattern of the bare source dipole shows two major lobes in the directions of ~154° in polar angle. By adding the Ag NP for inducing LSP coupling, when the source dipole is located at x = 0, the two major lobes in the directions of ~154° can still be seen. However, strong radiation can be observed within the angle range between 154° and 180°. When the source dipole is moved to the position of x = 2h, the radiation pattern becomes asymmetric with the first and second major lobes in the directions of ~153° (the left-hand side) and ~157° (the right-hand side), respectively. Graded side-lobes exist between the two major lobes. Also, a side-lobe is formed at ~120° on the right-hand side. When the source dipole is moved to the position of x = 4h, the radiation pattern is similar to that of the case of x = 2h. However, now the intensities of the two major lobes become comparable. As shown in Fig. 8, with an x-dipole, in the y-z plane, the radiation pattern of the bare source dipole shows two symmetric, broad major lobes in the directions of ~120° in polar angle. By adding the Ag NP for inducing LSP coupling, for any x coordinate of the source dipole position, the symmetric radiation pattern is essentially the same as that of the bare source dipole except that the major lobes become broader and side-lobes are formed between the two major lobes. The formation of the side-lobes around the vertically downward direction can increase the LEE because more radiated power propagates in the directions with the angles smaller than the critical angle between GaN and air.
In the case of a y-dipole, as shown in Fig. 9, the radiation pattern of the bare source dipole in the x-z plane is exactly the same as that of an x-dipole in the y-z plane. With the Ag NP right above the y-dipole (x = 0), the radiation pattern in the x-z plane is also exactly the same as that of an x-dipole in the y-z plane. When x is nonzero, although the radiation patterns in the x-z plane are essentially similar to those of an x-dipole in the y-z plane, they become asymmetric. The major lobes on the right-hand side become stronger. As shown in Fig. 10, with a y-dipole, in the y-z plane, the radiation patterns are always symmetric. With LSP coupling, the major radiated power is confined within an angle range between 154° and 180° in polar angle.
Figure 11 shows the percentages of the downward radiated power in a circular cone around the vertically downward direction at 625 nm in wavelength as functions of half-subtended angle when x = 0, 2h, and 4h in the case of an x-dipole. The condition of d = 70 nm is considered for obtaining the results in Fig. 11. For comparison, the corresponding result in the case of no Ag NP is also shown in Fig. 11. It is noted that as shown in Figs. 5 and 6, the downward radiated power is much higher than the upward radiated power under the conditions of Fig. 11. In Fig. 11, one can see that when the Ag NP is added to the radiating system, with x < 4h, the radiated power percentages are increased as the half-subtended angle is smaller than 60°. This increasing trend diminishes when the dipole is further moved away from the location of x = 0. Here, the vertical dashed line indicates the half-subtended angle of 24.6°, beyond which total internal reflection at a GaN/air interface occurs and emitted photons are trapped in the device structure if all the interfaces are flat and the structure extends laterally to infinity. Based on the linear interpolation shown in Fig. 11, the radiated power percentage increases from 4.4% in the case of no Ag NP to 15.6% in the case of Ag NP at x = 0, indicating that the LEE can be significantly enhanced. In a real device, because of the unintentional and intentional rough surface structures and the limited lateral dimension, the LEE can be higher than the given numbers above. The larger radiated power percentages in the range of small half-subtended angle shown in Fig. 11 imply significantly higher LEEs no matter an intentional surface roughing scheme is used or not. In other words, with the same surface roughing scheme, the LEE is expected to be higher in a device with the above-discussed LSP coupling process. Figure 12shows the similar results to those in Fig. 11 in the case of a y-dipole. The variation trends with a y-dipole are the same as those with an x-dipole. However, the diminishing trend of increasing radiated power percentage in the small half-subtended angle range in increasing x is weaker when a y-dipole is considered.
In practical application to LED, the device surface is usually covered by a transparent conductor for current spreading or an epoxy for packaging purpose. With such a thick cover layer, the increase of the refractive index in the upper-half-space may red-shift the LSP resonance wavelengths induced on the Ag NP. Also, this increase of refractive index may enhance the radiation power percentage into the upper-half-space. Figures 13(a) and 13(b) show the normalized upward and downward radiated powers, respectively, of various cases of d values at 30, 50, 70, 90, and 120 nm when the refractive index surrounding the Ag NP in the upper-half-space is increased to 2 and the dipole is located exactly below the Ag NP, i.e., at x = 0. Here, one can see that although the percentage of upward radiated power is significantly increased, the major radiated power propagates into the lower-half-space of GaN, particularly when the LSP coupling process is applied. Also, it is noted that all the LSP resonance features are red-shifted when compared with the results in Fig. 2. Meanwhile, most of radiated power features are broadened. Some of the peak levels become lower. However, the major behaviors of those curves are similar to those in Fig. 2. The similar behaviors are attributed to the weak influence of the upper-half-space refractive index on the behaviors of the substrate LSP resonance modes induced around the Ag NP bottom. The refractive index change in the upper-half-space can strongly influence the behaviors of the in-plane air and out-of-plane LSP resonance modes induced around the Ag NP body. The source dipole can significantly interact with these LSP resonance modes only when it is moved away from the Ag NP bottom. Nevertheless, in this situation, the increased distance between the Ag NP and source dipole will reduce the coupling strength. By comparing the radiated power levels in Figs. 2 and 13, it is found that the increase of the upper-half-space refractive index generally reduces the radiated power enhancement ratio.
A similar numerical study on the radiating dipole coupling with LSP resonance on an Ag NP embedded in the p-GaN layer of an LED has been performed . As shown in Fig. 5 of , with an Ag nanosphere of 10 nm in radius embedded in p-GaN, the coupling of an x- or y-oriented (orbital) radiating dipole 30 nm away from the Ag nanosphere with the LSP resonance leads to radiated power enhancement by 35 to 50% in the spectral range of 500-540 nm. Compared with the results of this work with a surface Ag NP, as shown in Fig. 2, the radiated power is enhanced by 50 to 90% in the same spectral range when the dipole is located right below the Ag NP with d = 70 nm. In other words, if an LED of a reasonably good electrical property can be fabricated with the total thickness of the p-GaN, p-AlGaN, and QW barrier layers between 70 and 80 nm, the LSP coupling effect of a surface Ag NP can be stronger than that of an embedded Ag NP. The use of a surface Ag NP array has the advantages of lower cost, higher planar NP density, and higher flexibility in controlling the NP geometry.
The geometry of Ag NP can affect the LSP resonance wavelength. Similar results to those in Fig. 2 are obtained with an Ag hemisphere of 50 nm in radius on the GaN surface. The spectral behaviors of normalized radiated power are similar to those shown in Fig. 2. However, all the LSP resonance features are red-shifted by a few to several tens nm. Also, the peak levels are varied in a complicated manner. Usually, it is supposed that at a given wavelength, the LSP coupling effect diminishes with the distance between the metal structure and radiating dipole. However, as shown in Fig. 2, at the wavelength of 520 nm for example, the radiated power level is enhanced first when d is increased from 30 to 50 nm and then is reduced as d is further increased. In Fig. 2, the major peak level of radiated power first increases, then decreases, and eventually diminishes with increasing d value. However, the spectral location of the major peak keeps red-shifting as d increases except when d is smaller than 50 nm.
In summary, we have numerically demonstrated the radiated power enhancement and more congregated radiation when a radiating dipole within a GaN material coupled with the LSP resonance modes induced on a surface Ag NP. The numerical study was based on an algorithm including the induction of LSP resonance on the Ag NP by the source dipole and the feedback effect of the LSP resonance field on the source dipole behavior. The radiated power enhancement showed the spectral peaks corresponding to the substrate LSP resonance modes with the mode fields mainly distributed around the bottom of the Ag NP such that the enhanced radiated power mainly propagated in the downward direction. By moving the radiating dipole laterally away from the bottom of the Ag NP, the spectral peaks of radiated power enhancement red shifted and their levels diminished with increasing lateral distance. The radiation patterns in the lower-half-space showed more congregated radiation, indicating that the LEE could be enhanced in an LSP-coupled light-emitting device with surface metal NPs.
This research was supported by National Science Council, Taiwan, The Republic of China, under the grants of NSC 101-2221-E-002-153, NSC 99-2221-E-002-123-MY3, NSC 101-2622-E-002-002-CC2, NSC 101-2120-M-002-013, by the Excellent Research Projects of National Taiwan University (101R890952 and 101R890951), and by US Air Force Scientific Research Office under the contract of AOARD-12-4068.
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