1. Introduction

Light color conversion through absorption and reemission is an important process for extending a spectral coverage range. It is a useful technique in implementing color micro-light-emitting-diode (LED) display based on a blue-emitting quantum-well (QW) LED overlaid with green- and red-emitting colloidal quantum dots (QDs) [1,2]. The color conversion efficiency depends on three factors, including the emission efficiency of the energy donor, the absorption cross section and emission efficiency of the energy acceptor. Although a color conversion process usually uses the far-field absorption mechanism, a near-field interaction such as resonance energy transfer (RET) can produce a higher conversion efficiency [3–5]. In other words, if the distance between the donor and acceptor is small, RET can effectively convert energy, particularly when the effect of surface plasmon (SP) coupling is applied to the conversion system [6–12]. However, even though SP coupling can extend the spatial range of RET, a structure for making the distance between donor and acceptor shorter than tens nm still requires a careful design. Typically, the required thickness of the top p-type layer in an LED for achieving a high electrical performance is as large as ∼100 nm [13,14]. In this situation, the RET between the QWs of an LED and surface QDs is quite weak. Search for other structures to reduce the distance between the QWs and QDs is demanded for enhancing the color conversion efficiency through RET.

Recently, subsurface GaN porous structures have been fabricated based on electrochemical etching [15–20]. In such a porous structure, tubular voids extend essentially along the current flow direction in a layer of high conductivity. Typically, a highly Si-doped n-type GaN (n⁺-GaN) layer is grown in a sample for electrochemical etching. Such an oriented tubular pore of tens nm in cross-sectional size can accommodate colloidal QDs (∼15 nm in size) if a QD insertion technique can be built. When a subsurface porous structure is fabricated above the QWs in a sample, the inserted QDs are located between the QWs and top surface. Such a structure can lead to the following advantages in color conversion. First, the distance between QWs (donor) and inserted QDs (acceptor) becomes smaller for producing stronger RET. Second, when metal nanoparticles (NPs) are placed on the top surface for inducing SP coupling with the QWs, the QDs inside the porous structure lie between the surface metal NPs and QWs, i.e., in the hot spot (the location of strong near-field distribution) [21,22]. In this situation, QD absorption can be enhanced for increasing the overall color conversion efficiency. Third, the porous structure can generate a certain scattering effect to further enhance the QW-produced field intensity at the location of an inserted QD and its radiated power, particularly under the condition of SP coupling, leading to an even higher color conversion efficiency. Fourth, the oriented tube-like pores in such a structure can result in polarization-dependent emission and RET behaviors, which can enrich device function. Finally, when synthesized metal NPs are also inserted into a porous structure, three-body SP coupling among the metal NPs, inserted QDs, and QWs can further enhance the energy transfer from the QWs into QDs. Based on the discussions above, an oriented subsurface porous structure is useful for enhancing the energy transfer from a QW into a QD. In this regard, it is important to first understand its effects on the emission and RET behaviors of the QW and QD, and the related fundamental mechanisms based on simulation study. The results of this simulation study can provide us with the guidelines for designing an LED device consisting of a GaN porous structure, inserted QDs, and surface Ag NPs to implement effective RET and SP coupling such that the color conversion efficiency can be enhanced. In particular, the polarization dependence is studied for developing advanced display applications.

In this paper, the radiation behaviors of a donor dipole and an acceptor dipole of the same orientation for representing a QW and a QD, respectively, in a porous structure are numerically studied. The field intensity produced by the donor at the position of the acceptor, which is located inside a tubular void, is evaluated for showing the variation of acceptor absorption power among different sample structures. The polarization-dependent behaviors in changing the dipole orientation are of great concern. Based on those results, we can evaluate the polarization-dependent enhancement factors of color conversion caused by either the porous structure or SP coupling. In section 2 of this paper, the sample structures under study and the simulation method are described. The radiation behavior and near-field intensity distribution produced by the donor dipole are discussed in section 3. Then, the radiation behavior of the acceptor and the enhancement of RET from donor into acceptor are reported in section 4. More discussions about the results are given in section 5. Also, the designs of practical LED devices for effective color conversion are illustrated in this section. Finally, conclusions are drawn in section 6.

2. Sample structures and simulation methods

Figures 1(a)–1(f) schematically show the structures of the six samples for numerical study. In the four samples shown in Figs. 1(a)–1(d), a tubular void of d in diameter is embedded horizontally in a GaN template at the depth of a (the depth of its upper boundary). The void tube axis is designated to be along the x-axis. A radiating dipole is located at the depth of b below the lower boundary of the void tube to act as the donor in the RET system. Another radiating dipole is located at the cross-sectional center of the void tube to act as the acceptor. The schematic illustrations in Figs. 1(a) and 1(c) [1(b) and 1(d)] for samples V-y and VR-y (V-x and VR-x), respectively, show the structures in the y-z (x-z) plane, in which both donor and acceptor dipoles are oriented along the y- (x-) axis. In sample V-y (V-x), an Ag NP is placed on the top surface for inducing SP coupling. The center of the Ag NP, acceptor dipole, and donor dipole are vertically aligned. No Ag NP exits in sample VR-y (VR-x). The structure of sample C (N), as shown in Fig. 1(e) [1(f)], is the same as that of sample V-y (VR-x) except that the void tube is removed. It is noted that although the orientations of the donor and acceptor dipoles in sample C (N) are designated to be along the y- (x-) axis, the dipole radiation behaviors are independent of dipole orientation since the sample structure is circularly symmetric along the z-axis. Among the six samples, SP coupling effects exist only in samples V-y, V-x, and C. Although we draw the donor and acceptor dipoles in each part of Fig. 1, in the following numerical computations, only either donor or acceptor dipole is considered in a computation case.

 

Fig. 1. (a)-(f): Schematic illustrations of the structures of samples V-y, V-x, VR-y, VR-x, C, and N, respectively.

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The numerical simulation method has been discussed in one of previous publications of ours and is summarized below [21]. To evaluate the radiation behavior of a radiating dipole, we first compute its radiated electromagnetic field when it is situated in a homogeneous spherical background region. Then, the total field is calculated in the real structure of the problem based on the software COMSOL. By subtracting the radiated field of the dipole from the total field, we can obtain the scattered field, which is used for evaluating the feedback effect on the dipole radiation behavior. With the available scattered field, the optical Bloch equations are solved to give the strength and orientation of the modified dipole. From this modified dipole, the final electromagnetic field distribution and the total radiated power can thus be calculated numerically. The computation domain of COMSOL must be large enough to ensure that the electromagnetic field is negligibly small around its edge. In our numerical computation, the experimental data of the dielectric constant for Ag are used [23]. With the feedback process mentioned earlier, the effect of the SP resonance field, which is induced by the radiating dipole, on the radiation behavior of the dipole is included. In practical operation, it is an iteration process until a steady state is reached. With the feedback process in numerical computation, the Purcell effect is taken into consideration [24].

To evaluate the power of RET from donor into acceptor, the absorption coefficient of acceptor must be available. However, it is normally difficult to obtain such information for a QD acceptor because it is not a classical absorbing material. Instead of evaluating the transferred power or transfer efficiency of RET, we will compute its enhancement ratio when a factor for changing RET behavior is considered. If the acceptor absorption efficiency is not significantly influenced by SP coupling, the transferred power of RET is proportional to the field intensity produced by the donor dipole at the position of acceptor. Therefore, the ratio of this field intensity with SP coupling over that without SP coupling can be regarded as the enhancement factor of the absorbed power of acceptor under SP coupling. By combining this factor with the effect of SP coupling on the radiated power of acceptor, we can evaluate the enhancement factor of transferred power in RET. In numerical computations, we set a = 10 nm, d = 30 nm, and b = 20 nm. The surface Ag NP is a truncated ellipsoid with the horizontal semi-axis at 26.411 nm, the vertical semi-axis at 25 nm, and the bottom-truncated height at 10 nm. The refractive index of GaN is fixed at 2.4.

3. Radiation behavior and near-field intensity of donor

Figure 2 shows the spectra of the normalized absorbed powers (absorbed by the surface Ag NP) of samples V-y, V-x, and C excited by the donor dipole when the acceptor dipole is ignored. The results are normalized with respect to the total radiated power of the dipole when it is placed in a homogeneous space of GaN. Such a spectral curve shows the wavelengths of LSP resonance features and their relative resonance strengths of the surface Ag NP. Here, each sample shows the strong fundamental LSP resonance mode between 500 and 550 nm in wavelength and two weaker higher-order LSP resonance features around 480 and 365 nm. With the tubular void in samples V-y and V-x, the fundamental LSP resonance features are slightly blue shifted, when compared with that in sample C. When the dipole orientation is perpendicular (parallel) to the void tube axis, each LSP resonance feature becomes stronger (weaker) than that of the reference structure of no tubular void in sample C. Figure 3 shows the spectra of the normalized radiated powers of the donor dipole in the six samples under study when the acceptor dipole is ignored. Here, the results are also normalized with respect to the total radiated power of a dipole in the homogeneous space of GaN. In the three samples with SP coupling (samples V-y, V-x, and C), three peaks of radiated power corresponding to the three LSP resonance features can be observed. As far as the peaks corresponding to the fundamental LSP resonance feature are concerned, the variation trends of radiated power level and spectral position among the three samples are the same as those of absorbed power shown in Fig. 2. Within the concerned spectral range, the radiated power of sample V-y is always significantly higher than that of sample V-x. Among the three samples without SP coupling (samples VR-y, VR-x, and N), the radiated power spectra show slowly varying curves. The radiated power levels of samples VR-x and N are about the same while that of sample VR-y is significantly higher. Here, a vertical dashed blue line is plotted to indicate the wavelength of 450 nm for later discussion.

 

Fig. 2. Normalized absorption spectra of the Ag NP excited by the donor in samples V-y, V-x, and C.

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Fig. 3. Normalized radiated power spectra excited by the donor in the six samples under study. The vertical dashed line indicates the wavelength of 450 nm, which is designated as the emission wavelength of the donor for RET enhancement evaluation.

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Figures 4(a)–4(c) show the charge distributions on the Ag NP viewed from its bottom along the z-axis for the LSP resonance at 450 nm in wavelength excited by the donor dipole (the acceptor dipole ignored). The blue and red colors here represent the opposite charges. On the interface between Ag NP and GaN, we can see the quadrupole structure of charge distribution in each sample. Figures 5(a)–5(f) show the field strength (norm) distributions (in log scale) in the y-z or x-z plane passing through the donor dipole, the acceptor position, and the center of the Ag NP at 450 nm in wavelength for the six samples under study when the acceptor dipole is ignored. Here, one can see that the field strength inside the void in sample V-y (VR-y) is generally stronger than those in samples V-x (VR-x) and C (N). Also, the field strength inside the void in sample V-y (V-x) is stronger than that in sample VR-y (VR-x). Such variation trends can also be seen in Figs. 6–8, in which the spectra of the ratios of radiated powers (continuous curves) and polarized field intensities at the acceptor position (dashed curves) among different samples are shown when the acceptor dipole is ignored. The polarized field intensity means the local intensity with polarization along the acceptor dipole orientation. We consider only the intensity in this polarization because it corresponds to the power to be absorbed by the acceptor. Here, the vertical dashed lines indicate the wavelength of 450 nm. In Fig. 6, we can see that blow 575 nm in wavelength, all the ratios are higher than unity. The over-unity ratios of V-y/C indicate that under the SP coupling condition, the tubular void can further enhance the radiated power and polarized field intensity at the acceptor position when the donor dipole orientation is perpendicular to the void tube. Also, the over-unity ratios of V-y/VR-y indicate that with the tubular void, SP coupling can further enhance the radiated power and polarized field intensity at the acceptor position. In Fig. 7, we can see that the ratios of V-x/VR-x are higher than unity below 560 nm in wavelength indicating that SP coupling can also lead to the enhancements of the radiated power and polarized field intensity at the acceptor position when the donor dipole is oriented parallel to the void tube. However, with this donor dipole orientation, under the condition of SP coupling the spectral range, in which the tubular void can enhance the radiated power and polarized field intensity at the acceptor position, is limited. It is noted that the ratio range in Fig. 6 is significantly larger than that in Fig. 7. In Fig. 8, the curves of VR-y/N show that the void tube can enhance the radiated power and polarized field intensity when the donor dipole orientation is perpendicular to the void tube axis. However, when the donor dipole orientation is parallel to the void tube axis, either radiated power or polarized field intensity is not significantly increased, as shown in the curves of VR-x/N of Fig. 8. From the results in Figs. 6–8, one can see that when the donor dipole orientation is perpendicular to the void tube axis, the total radiated power and polarized field intensity at the acceptor position can be significantly enhanced under either condition with or without SP coupling. The ratios of polarized field intensity at the acceptor position with various combinations at 450 nm are summarized in column 2 of Table 1. These intensity ratios are the same as the absorbed power ratios of the acceptor if the acceptor absorption capability is the same among different sample structures.

 

Fig. 4. (a)-(c): Charge distributions on the Ag NP at the wavelength of 450 nm excited by the donor viewed from its bottom along the z-axis in samples V-y, V-x, and C, respectively.

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Fig. 5. (a)-(f): Field strength (norm) distributions at the wavelength of 450 nm excited by the donor on the y-z or x-z plane passing through the donor, acceptor, and Ag NP center for samples V-y, V-x, VR-y, VR-x, C, and N, respectively.

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Fig. 6. Spectra of the ratios of samples V-y/VR-y, V-y/C, and V-y/N for the radiated power and polarized field intensity at the acceptor position produced by the donor dipole.

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Fig. 7. Spectra of the ratios of samples V-x/VR-x, V-x/C, and V-x/N for the radiated power and polarized field intensity at the acceptor position produced by the donor dipole.

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Fig. 8. Spectra of the ratios of samples VR-y/N, VR-x/N, and C/N for the radiated power and polarized field intensity at the acceptor position produced by the donor dipole.

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Table 1. Absorption enhancement at 450 nm and emission enhancement at 528 nm of the acceptor and RET enhancement under various comparison conditions.

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4. Radiation behavior of acceptor and energy transfer enhancement

Now, we turn our attention to the radiation behavior of the acceptor dipole when the donor dipole is ignored. Figure 9 shows the spectra of the normalized absorbed powers (absorbed by the surface Ag NP) of samples V-y, V-x, and C induced by the acceptor dipole when the donor dipole is ignored. Again, the results are normalized with respect to the total radiated power of the dipole when it is placed in a homogeneous space of GaN. The behaviors of those absorption spectra are similar to the corresponding curves in Fig. 2 for the case of donor radiation. However, the absorbed power levels are significantly increased. The higher absorption of the surface Ag NP in the current case of acceptor radiation is caused by the shorter distance between the dipole and Ag NP (25 nm). The shorter distance leads to a stronger coupling effect and hence a stronger field distribution within the Ag NP, resulting in a higher metal absorption. Figure 10 shows the normalized radiated power spectra of the acceptor dipole when the donor dipole is ignored. Here, we can see that the normalized radiated power levels are also generally higher than the corresponding values in Fig. 3 of donor radiation. The relative levels of the radiated power of the acceptor among the sample structures under study are similar to those of donor radiation. The vertical dashed line in Fig. 10 indicates the wavelength of 528 nm, which is the emission wavelength of a green-emitting QD. The ratios of the acceptor radiated powers among different samples are evaluated and listed in column 3 of Table 1.

 

Fig. 9. Normalized absorption spectra of the Ag NP excited by the acceptor in samples V-y, V-x, and C.

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Fig. 10. Normalized radiated power spectra excited by the acceptor in the six samples under study. The vertical dashed line indicates the wavelength of 528 nm, which is designated as the emission wavelength of the acceptor for RET enhancement evaluation.

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In columns 2 and 3 of Table 1, we show the ratios of the polarized field intensity at the acceptor position and the radiated power of the acceptor, which can be regarded as the absorption and emission enhancement factors of the acceptor, respectively, in RET. The product of these two ratios in the same row of Table 1, which is given in column 3 of Table 1, is equivalent to the enhancement factor of the RET efficiency. In rows 2 and 3 of Table 1, the ratios of VR-y/N and VR-x/N show the effects of the tubular void on RET efficiency in the cases of two different dipole orientations. In the case of perpendicular orientation, the RET enhancement factor is significantly larger than unity. However, in the case of parallel orientation, it becomes lower than unity, indicating the strong polarization dependence of RET behavior caused by the oriented tubular void. Similar behaviors can be observed under the condition of SP coupling in rows 4 and 5 of Table 1, which show the ratios of V-y/C and V-x/C. In either dipole orientation case, based on the tubular void structure, the application of SP coupling can result in a significant enhancement, as supported by the ratios of V-y/VR-y and V-x/VR-x in rows 6 and 7, respectively, of Table 1. Compared with the situation of no tubular void, as shown in the ratio of C/N in row 8, the effect of SP coupling on RET enhancement based on the tubular void structure is weaker. The combined effect of tubular void and SP coupling leads to the ratios of V-y/N and V-x/N in the two bottom rows of Table 1, in which we can see the significant difference between the two dipole orientations. For the dipole orientation perpendicular (parallel) to the void tube, the RET enhancement is as larger as 21.28 (only 2.73). Between the two factors of acceptor absorption and emission (columns 2 and 3 in Table 1), generally the absorption enhancement is larger than the emission enhancement in the case of perpendicular dipole orientation. However, no clear variation trend can be seen in the case of parallel dipole orientation.

5. Discussions

The polarization-dependent behaviors of donor and acceptor emissions and RET discussed above are caused by the near field distribution in the structure of tubular void and the far-field radiation characteristics. It is noted that a distribution of strong (weak) near-field intensity does not necessarily lead to a strong (weak) far-field radiation power. Because the diameter of the void tube is only 30 nm in the current study, the void cannot support a significant cavity resonance feature, either Fabry-Perot or whispering gallery mode. The observed emission and RET behaviors can be caused by a certain polarization-dependent scattering mechanism of the near field in the tubular void structure. However, the detailed mechanism is still unclear to us and deserves further investigation. In this regard, the geometry of the tubular void structure can be an important issue for further study. In practice, such a subsurface porous structure consists of multiple oriented tubular voids forming a layer of a lower effective refractive index. In other words, the factor of a subsurface layer of a lower effective refractive index needs to be considered in numerical simulation. One more issue deserving mentioning is that the geometry of the surface Ag NP can be more carefully designed such that its LSP resonance features match the donor and acceptor emission wavelengths. In this situation, both acceptor absorption and emission enhancements in Table 1 can be further increased under the SP coupling condition.

The obtained simulation results can be used to guide the design of a blue-emitting QW LED with an n⁺-GaN layer for fabricating a GaN porous structure either above or below the QWs. It is noted that a GaN porous structure cannot be fabricated in a p-type GaN layer because its conductivity is not high enough. In Fig. 11(a), we schematically illustrate an LED device of an n-GaN/QW/p-GaN configuration with a fabricated porous structure above the QWs. In this device, we can fabricate surface Ag NPs at the top to induce the SP coupling with the QWs such that the red-emitting QDs (RQDs) or green-emitting QDs (GQDs) inserted into the porous structure lie in the hot spot of the SP coupling process for enhancing color conversion. As shown in Fig. 11(a), we can also insert chemically synthesized Ag NPs into the porous structure to reinforce the SP coupling effect. However, because usually a p-GaN/QW/n-GaN configuration is used for fabricating an LED device, the implementation of the device structure shown in Fig. 11(a) can be difficult. A more practical design is schematically illustrated in Fig. 11(b), in which a p-GaN/QW/n-GaN configuration is used for fabricating a porous structure below the QWs. In this situation, we can use the synthesized Ag NPs inserted into the porous structure to induce the SP coupling with the QWs and inserted QDs. The QDs, which are close to the Ag NPs, are also located in the hot spot of the SP coupling process. In the LED devices of Figs. 11(a) and 11(b), the distances between QWs, porous structures, and surface Ag NPs are well controlled such that effective RET and SP coupling can occur. Also, the injected current does not flow through the region of the porous structure such that current leakage can be minimized. Meanwhile, for color display application, the device structure shown in Fig. 11(a) or 11(b) is suitable for micro-LED fabrication [25].

 

Fig. 11. Schematic illustrations of two LED devices with porous structures and inserted RQDs, GQDs, and synthesized Ag NPs. (a): The LED device with the porous structure above the QWs in an n-GaN/QW/p-GaN configuration. Surface Ag NPs are fabricated to induce SP coupling. (b): The LED device with the porous structure below the QWs in a p-GaN/QW/n-GaN configuration

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6. Conclusions

In summary, the numerical study on the polarization-dependent emission and RET behaviors of a donor dipole and an acceptor dipole in the structure of a subsurface tubular void has been undertaken for simulating the energy transfer from a QW into a QD located within an oriented porous structure above the QW. When the dipole orientation was parallel to the tube axis, the radiated power and the polarized field intensity inside the void produced by the donor were significantly lower, compared with those in the case of perpendicular orientation, particularly when a surface Ag NP was added for inducing SP coupling. By assuming that the absorption efficiency of the acceptor was not affected by SP coupling, the polarized field intensity produced by the donor at the acceptor position inside the void was proportional to the absorbed power of the acceptor. Based on this assumption, the RET enhancement could be evaluated by multiplying the ratio of the polarized field intensity at the acceptor position with that of acceptor radiated power when we intended to understand the effects of tubular void and SP coupling. The tubular void structure could produce strongly polarization-dependent RET enhancements, particularly when the SP coupling effect was included.