To confirm the enhancements of QD emission and FRET efficiencies when QDs are inserted into a metal NH, simulation studies are undertaken with the sample structures illustrated in Figs. 5(a)-5(d). As illustrated in Fig. 5(a), we design a metal NH simulating the experimental NH structures shown in Figs. 2(a) and 2(b). This metal NH structure is designated as NH-XX/GaN with XX = Ag or Au. The NH is designed as a semi-ellipsoid with its long axis coinciding with the axis of the cylindrical GaN NH. The semi-major axis of the ellipsoid is equal to d, which is the depth of the GaN NH. The semi-minor axis is b/2 - w, where b is width of the GaN NH. The semi-ellipsoid NH is filled with the photoresist of 1.577 in refractive index. Ag or Au is deposited in the space between the GaN wall and the photoresist-filled semi-ellipsoid NH. The thickness of metal deposition at the center of the GaN NH bottom is t. Therefore, it is assumed that on the top surface of a sample, a metal layer of t in thickness is also deposited. Two QDs or radiating dipoles are placed inside the metal NH at its semi-major axis, one serves as the donor and the other serves as the acceptor in an FRET process. The origin of the coordinate system is set at the center of the GaN NH bottom. In Fig. 5(b), the reference sample of the NH-GaN structure is illustrated for comparing the QD emission and FRET efficiencies between the cases with and without metal deposition. Figure 5(c) shows the structure of homogeneous photoresist (structure R), which is used as the normalization base in numerical computations. Figure 5(d) illustrates a cylindrical metal NH structure (NH-Ag) for understanding the cavity resonance effect in a metal NH, as to be further discussed in the next section.
For simulation studies, the dielectric constant, \(\epsilon = {\epsilon }^{{\prime }}+i\epsilon "\), of the used metal needs to be assigned first. Figures 6(a) and 6(b) show the real and imaginary parts (e’ and e”), respectively, of the wavelength-dependent dielectric constants we use for simulation studies. For simulating the experimental conditions, we use the experimental data of dielectric constant for Ag and Au, as labeled by Exp. (Ag) [20] and Exp. (Au) [21] in Figs. 6(a) and 6(b). To understand the SP coupling effect in cavity resonance behavior, we also consider the dielectric constants of Ag based on the Drude model, as to be discussed in section 4. Here, a negative value of the real part of dielectric constant implies the possible excitation of SP resonance and hence SP coupling in the visible range. We can see that the magnitude (negative value) of the real part of the dielectric constant in Au is smaller than that of Ag. Also, the imaginary part of Au is smaller (larger) than that of Ag for a wavelength longer (shorter) than 600 nm in the visible range. The high peak of the imaginary part of Au around 400 nm in wavelength is caused by the electron interband transition between the lower d band and the higher s band. The method for the numerical simulation studies has been described in a few earlier publications of ours [22, 23]. It is based on the exact electromagnetic theory. This method includes the feedback effect from the scattered field on the radiation behavior of the source dipole. In other words, the Purcell effect, either near- or far-field portion, is taken into account [22, 23].
To numerically demonstrate the simulation results, we fix the GaN NH depth, d, at 320 nm and consider the following two sets of GaN NH width and two conditions of metal deposition thickness. In a wider (narrower) GaN NH, which is denoted by the symbol “W” (“N”), the GaN NH width, b, is 280 (200) nm. In the case of a larger (smaller) metal deposition thickness, which is denoted by the symbol “L” (“S”), t = 80 (40) nm, w = 40 (20) nm, the donor coordinates at (0, 0, 200 nm) [(0, 0, 160 nm)], and the acceptor coordinates at (0, 0, 230 nm) [(0, 0, 190 nm)]. Therefore, we have four combinations in total, including the cases of W/L, W/S, N/L, and N/S. In all cases, the distance between the donor and acceptor is kept at 30 nm. For comparison, we also consider the structures of NH-GaN with b = 200 (N) and 280 (W) nm. In either case, the coordinates of the donor and acceptor are (0, 0, 120 nm) and (0, 0, 150 nm), respectively. Figures 7(a) and 7(b) show the normalized radiated power spectra of the acceptor as an x- and z-dipole, respectively, in an Ag NH shown in Fig. 5(a). Besides the aforementioned four cases of Ag NH, the results of the narrow and wide GaN NHs are shown. As shown in Fig. 7(a) for an x-dipole, in the case of either narrower or wider NH, the radiated power shows a sharper peak when metal deposition is thicker. With a thinner metal deposition, the radiated power is weaker and shows a two-peak spectrum. When the GaN NH is narrower and hence the metal NH becomes smaller, the spectral peaks are blue-shifted and become sharper. In the cases of GaN NH, slowly increasing radiated power spectra can be observed. The difference between the narrower and wider GaN NHs is small. In this figure, we can see that in a large spectral range, an Ag NH sample can lead to a stronger radiated power, when compared with that of a GaN NH sample. The vertical dashed line in Fig. 7(a) indicates the wavelength of RQD emission (625 nm). At this wavelength, except case N/L, the radiated powers are all enhanced (normalized radiated power > 1), when compared with that in structure R. However, except case W/S, the radiated powers in samples Ag NH are lower than those of samples GaN NH. We believe that the sidewall metal deposition in experiment is thinner than that designated in simulation such that the curve of N/S is red-shifted. In this situation, the radiated power in the N/S case of an Ag NH sample becomes higher than that of a GaN NH sample. As shown in Fig. 7(b) for a z-dipole, in the cases of narrower NH, the major portions of the radiated power peaks fall into the ultraviolet range. In the cases of wider NH, the radiated power peaks are quite sharp and located far away from the RQD emission wavelength. At this wavelength, the normalized radiated powers in all cases are lower than unity, indicating that it is unlikely for a z-dipole to contribute to QD emission enhancement in either an Ag NH or a GaN NH sample. Such contributions originate mainly from the x- and y-dipole. Figures 8(a) and 8(b) show the normalized field intensity spectra at the position of the acceptor produced by an x- and a z-dipole donors, respectively. Generally speaking, with a narrower NH, the spectral peak of the donor intensity is located at a shorter wavelength. Also, with a thicker Ag deposition, the donor peak intensity is higher. At the GQD emission wavelength, as indicated by the vertical green dashed line, the normalized donor intensities produced by an x-dipole under all the sample conditions, including the NH-GaN structures, are larger than unity, i.e., enhanced. However, those produced by a z-dipole are not significantly enhanced except the case of W/S. The donor intensity enhancements at the emission wavelength of GQD imply the increased energy absorption of the acceptor (RQD) and hence the improvement of the efficiency of the FRET from GQD into RQD.
Figures 9(a) and 9(b) show the spectra of the normalized radiated power of the acceptor and the normalized field intensity at the position of the acceptor produced by the donor, respectively, in the samples of Au NH with the structure shown in Fig. 5(a). Both results of an x- and a z-dipole are shown in either Fig. 9(a) or 9(b). The geometries used for the Au NH samples are the same as those for the Ag NH samples. Here, only the cases of N/S and N/L are considered. Two strong peaks of radiated power produced by an x-dipole acceptor can be seen in Fig. 9(a). In the concerned wavelength range, the radiated power produced by a z-dipole acceptor is quite low. Fano-like oscillations can be observed around 550 nm in wavelength for donor intensity in all cases, as shown in Fig. 9(b). The intensity peak levels produced by an x-dipole donor are higher than those produced by a z-dipole donor. The emission wavelengths of RQD and GQD are also indicated by the vertical dashed lines in Figs. 9(a) and 9(b), respectively.
Table 2 shows the normalized field intensity at the position of the acceptor produced by the donor (abbreviated by “donor intensity”), the normalized acceptor radiated power, and the color conversion factor, which is the product of the last two values, excited by x- and z-oriented dipoles under various metal and GaN NH conditions. The energy absorbed by the acceptor or the transferred energy in an FRET process is proportional to the donor intensity at the position of the acceptor. Hence, the enhancement of donor intensity can be regarded as the increment of FRET efficiency. Therefore, the product of the normalized donor intensity and normalized acceptor radiated power represents the improvement factor of color conversion. In Table 2, we can see that in most cases of z-dipole, either donor intensity or acceptor radiated power is suppressed or weakly enhanced. However, in most cases of x-dipole, either donor intensity or acceptor radiated power is enhanced, leading to the increase of the color conversion factor. These variation trends are true for both metal and GaN NH samples. However, the increments of the color conversion factors in the metal NH samples are generally larger than those in the GaN NH sample, indicating the stronger nanoscale-cavity effect in a metal NH. In the N/S case, the Au NH sample results in a higher color conversion factor, when compared with the corresponding Ag NH sample. However, this variation trend cannot be regarded as a general rule in the comparison between the Au and Ag NH samples. A change of metal deposition or NH geometry condition can reverse the relative color conversion factors between the samples of the two metals.
Table 2
Normalized field intensity at the position of the acceptor produced by the donor (donor intensity), the normalized acceptor radiated power, and the color conversion factor, which is the product of the last two values, excited by x- and z-oriented dipoles under various metal and GaN NH conditions.
| Donor intensity | Acceptor radiated power | Color conversion factor |
| x-dipole | z-dipole | x-dipole | z-dipole | x-dipole | z-dipole |
Ag-N/S | 1.135 | 0.864 | 1.688 | 0.056 | 1.916 | 0.048 |
Au-N/S | 1.719 | 1.122 | 3.543 | 0.087 | 6.090 | 0.098 |
Ag-N/L | 0.025 | 0.878 | 0.114 | 0.013 | 0.003 | 0.011 |
Au-N/L | 2.126 | 1.123 | 0.385 | 0.024 | 0.819 | 0.027 |
Ag-W/S | 1.869 | 1.476 | 5.433 | 0.103 | 10.154 | 0.152 |
Ag-W/L | 4.228 | 0.945 | 1.193 | 0.008 | 5.044 | 0.008 |
GaN, N | 1.582 | 1.032 | 1.720 | 0.708 | 2.721 | 0.731 |
GaN, W | 1.449 | 0.919 | 1.845 | 0.452 | 2.673 | 0.415 |